Lunar gravimetry

Lunar Gravimetry

This is Volume 35 in INTERNATIONAL GEOPHYSICS SERIES A series of monographs and textbooks Edited by WILLIAM L. D O N N A complete list of the books in this series appears at the end of this volume.

Lunar G ravi metry

M. U. SAGITOV

B. BODRl

Department of Gravimetry Sternberg State Astronomical Institute University of Moscow Moscow, USSR

Department of Geophysics Eotvos University Budapest, Hungary

V. S. NAZARENKO

Kh. G. TADZHIDINOV

Department of Gravimetry Sternberg State Astronomical Institute University of Moscow Moscow. USSR

Department of Gravimetry Sternberg State Astronomical Institute University of Moscow Moscow, USSR

1986

ACADEMIC PRESS

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Copyright @ 1986 by ACADEMIC PRESS INC. (LONDON) LTD All rights reserved. No part of this book may be reproduced in any form by photostat, microfilm, or any other means, without written permission from the publishers

B r i t i s h Library Cataloguing in Publication D a t a Lunar gravimetry. 1. Moon-Gravity I. Sagitov, M. U. 523.3’1 QB581 ISBN 0-12-61 4660-8

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Preface

Gravimetry is the science dealing with the gravitational fields of the Earth and other planets and permitting studies, based on this knowledge, into their figures and internal structure. Investigating the structural characteristics and the features of variations of the planetary gravity fields, gravimetry is also instrumental in computing the trajectories of rockets and artificial satellites, as well as in inertial navigation. The branch of gravimetry specifically concerned with the Moon is known as lunar gravimetry. Like other natural sciences, gravimetry is based on measurement, and its development is closely interrelated with the enhancement of measurement accuracy. Under terrestrial conditions, gravity is now measured with a maximum accuracy of about lop6cm s - ~ ,while its gradients are measured to within lo-" s - ~ .The advent of artificial satellites has been marked by the development of new methods for studying the gravitational fields of the Earth and planets. The anomalous part of the gravitational field was first defined from observations of perturbations in the motion of satellites. The gravitational field of the Moon has become better understood with the introduction of methods designed solely for investigation of the lunar field. For example, the line-of-sight components of the force of gravitational attraction, which are often used as basic quantities for characterizing the gravity field of the Moon, are obtained through Doppler tracking of artificial lunar satellites (ALS). Some of the lower harmonics of the gravitational field have been derived from observations of the Moon's physical librations. Our knowledge of the gravitational field on the Moon's far side is less certain. Only a few single gravity measurements have been carried out directly on the Moon's surface. Moreover, the lunar gravitational field differs from the terrestrial one not only in that the gravity on the Moon is six

vi

Preface

times lower, in absolute terms, as that on the Earth, but also in the spectrum of gravity anomalies. Some series expansions achieved for the Earth’s gravitational field are not applicable to the Moon because of the lack of clearly defined flattening of the latter and the corresponding predominant gravitational field harmonic. Yet, in spite of all the differences, it should be emphasized that the physical and mathematical principles underlying the theory and methods of terrestrial gravimetry remain valid also in lunar gravimetry. Gravimetric studies of the Moon expand our knowledge of the terrestrial gravitational field. Results have been obtained on the Moon with the aid of gravimetric methods, which have later been found to be applicable to the Earth. The active gravimetric studies of the Moon carried out over the last 15 years have yielded a wealth of data. The need has arisen to summarize and generalize this material to some extent, and this is what we have attempted to do. Not all chapters are equal in scope, mainly because some problems of lunar gravimetry are better investigated than others, but also because we have chosen to concentrate on some aspects more than others. We have included many tables, some of which are summaries compiled for the book from other sources, which may help the reader to better comprehend the subject. In some instances, we have derived mean values by averaging a great number of previously found ones. Following the old Chinese saying that “one picture replaces ten thousand words”, we have included numerous illustrations. As part of his contribution to the book, M. U. Sagitov has drawn on his lunar gravimetry lectures delivered over a number of years at the astronomical division of the physics department of Moscow University. The present monograph differs from that published by M.U.S. in 1979 under the same title in that it includes a new chapter dealing with time variations of gravity and their use in studies into the Moon’s structure. Many findings concerning the Moon’s gravitational field have been revised and updated. Some of the previously published text has been excluded (theory of sample functions, orbital motion of the Moon, Moon’s origin and evolution). The first three chapters have been written by M. U. Sagitov, V. S.Nazarenko and Kh. G. Tadzhidinov, the author of the fourth chapter is B. Bodri. We are grateful to Dr Y. Nakamura (University of Texas, Galveston) for providing numerical listings of the parameters of the four radially heterogeneous lunar models investigated in Chapter 4. Moscow and Budapest June 1986

M. U. Sagitov B. Bodri V. S. Nazarenko Kh. G. Tadzhidinov

Contents

Preface

V

List of Symbols

ix

Chapter One The Gravitational Field of the Moon: Methods for Its Determination The Law of Universal Gravitation and Various Gravitational Constants Problems Solvable from the Gravitational Field Lunar Gravitational and TidalLCentrifugal Potentials: Gravity Quantities Characterizing the Lunar Gravitational Field Comparison of Methods Used for Studying the Gravitational Fields of the Moon and the Earth 1.6 Elements of ALS Orbits and the Concept of Perturbations in Motion as a Result of Irregularities of the Lunar Gravitational Field 1.7 Methods for Determining the Harmonic Coefficients of the Gravitational Field from ALS Tracking Data 1.8 Determination of Line-of-Sight Accelerations due to the Earth’s Rotation and Orbital Motions of the Moon and ALS 1.9 Determination of the Gravitational Field from Variations in the Line-of-Sight Velocity of the Circumlunar Satellite 1.10 Gravity Measurement Concepts and Requirements of Lunar Gravimeters 1.11 Direct Gravity Measurements on the Moon’s Surface 1.12 Studying Second Derivatives of the Lunar Gravity Potential 1.13 Generalized Model of the Lunar Gravitational Field

1.1 1.2 1.3 1.4 1.5

Chapter Two 2.1 2.2

1

5 7 14 17 19

25 31 41 49 54 59 73

Normal and Anomalous Gravitational Fields of the Moon

Structure of the Gravitational Field and Its Role in the Evolution of the Moon Expansion of Lunar Gravity Potential Derivatives in Spherical Functions

99 101

viii

Conrents

Expansion for Gravity Selenoid Figure of the Normal Moon Distribution of Normal Gravity Surfaces of Equal Gravity and Equal Radial Gravity Gradient Anomalies of the Lunar Gravitational Field Relation between the Coefficients of Expansion of Different Parameters of the Moon’s Gravitational Field and Figure 2.10 Moon’s Relief and Gravitational Field

2.3 2.4 2.5 2.6 2.7 2.8 2.9

Chapter Three

4.1 4.2 4.3 4.4 4.5 4.6

131 134

Spatial Variations in the Lunar Gravitational Field and Their Use in Studying the Figure and Internal Structure of the Moon

3.1 Covariance Analysis of the Moon’s Gravitational Field 3.2 Degree Variances and Covariance Functions of Various Characteristics of the Gravitational Field and Figure of the Moon 3.3 Degree Variances for Horizontal Lunar Attraction Components 3.4 Comparative Analysis of the Lunar and Terrestrial Gravitational Fields 3.5 Comparison of the Gravitational Fields of Terrestrial Planets 3.6 The Selenocentric Constant of Gravitation: The Mass and Mean Density of the Moon 3.7 Centres of the Moon’s Figure and Mass 3.8 General Geometric Figure of the Moon 3.9 Dynamic Figure of the Moon 3.10 Hydrostatic Equilibrium Figure of the Moon 3.1 1 Gravimetric Figure and Distribution of Plumb-Line Deflections on the Moon 3.12 General Comments on the Moon’s Internal Structure 3.13 Density Irregularity Parameter 3.14 Density Model of the Moon 3.15 Variations in Gravity, Its Radial Gradient, and Pressure with Depth 3.16 Mascons

Chapter Four

105 108 111 118 122 125

147 153 156 158 171 173 180 186 190 200 206 201 211 218 224 228

Inconstant Lunar Gravity

Periodic and Secular Variations of Lunar Gravity Tidal Potential on the Moon Brief Theory of Solid Tides Tidal Deformation of the Moon Tides of the Anelastic Moon Tidal Friction and Secular Variations of Lunar Gravity

239 242 25 1 259 272 276

References

28 1

index

289

List of Symbols

G a

r R

Newton’s constant of gravitation planetary radius radius vector distance between the Moon’s centre and that of the tide raising body planetary mass symbols referring respectively to the Moon, Earth and Sun zenithal distance and its mean value, respectively order of spherical harmonic function Legendre polynomials surface spherical harmonic function selenographic coordinates colatitude = 90” - cp selenocentric equatorial coordinates Moon’s mean longitude mean longitude of lunar perigee mean longitude of the node of the Moon’s orbit Sun’s mean longitude mean longitude of solar perigee gravity deviations of the vertical density undisturbed density time shear modulus, Lam6 constant bulk modulus

List of Symbols

X

displacement vector components of the stress tensor components of the strain tensor Kronecker delta hydrostatic pressure tidal potential hydrostatic gravitational potential gravitational potential due to tidal distortion gravitational potential of the deformed planet = Vo Love numbers frequency of the perturbing force imaginary unit, (- 1)1/2 phase lag of planetary tides quality factor

+ Vt

Chapter One

The Gravitational Field of the Moon: Methods for Its Determination

1.1 The Law of Universal Gravitation and Various Gravitational Constants

The basic postulate of gravimetry is the law of universal gravitation, stating that two particles of matter attract each other with a force F having a magnitude proportional to the product of their masses m and . m land inversely proportional to the square of the distance r between them: F=G-

mm 1 r2

(1.1.1)

where G is the constant of gravitation, or gravitational constant. In mechanics, force is defined from Newton's second law as the product of the mass m l by acceleration a: F

= mla

(1.1.2)

Force has the following dimensions: [F] = [M][L][7'--2, M being mass, L, length and T,time. For the force F in (1.1.1) to have the same dimensions as in (1.1.2), the dimensions of G must be

G

= [M]-1[L]3[T]-2

2

Lunar Gravimetry

The mass rnl in (1.1.1) and (1.1.2) exhibits different properties. In the former formula, it possesses the property of gravitational interaction, while in the latter, it is a measure of inertia. It is assumed that the gravitational mass (in the law of gravitation) is identical with the inertial mass (in Newton’s second law). This is consistent with the equivalence principle experimentally verified by L. Eotvos, J. Renner, V. Dicke, and V. B. Braginsky. As was shown by Braginsky, this principle holds to within 10- 1 2 . The question as to further enhancement of this accuracy remains open. It would be appropriate at this juncture to trace the history of the discovery of the law of gravitation (Sagitov, 1969), which is intimately linked with that of exploration of the Moon. The law of universal gravitation, as it is still used today, was first formulated by Isaac Newton (1643-1727) in his fundamental work “Philosophiae Naturalis Principia Mathematica” (Mathematical Principles of Natural Philosophy) published in 1687. Other scientists before Newton also tried to establish the rules governing the attraction between bodies, but their concepts were purely speculative and incomplete and definitely cannot be regarded as laws. Proceeding from the hypotheses of his precursors, Newton had arrived at a precise statement of the law, followed by its analytical description. Newton claimed priority, firstly for having derived an analytical expression representing changes in gravity, and secondly, for having proved the identity of the gravity exerted by the Earth with the attraction of the planets, just as that of all other bodies, to one another. According to his own statements in his letters to Edmond Halley (1 656-1 742), Newton established that gravitation varies with distance as far back as 1666. The lack of reliable data as regards the Earth’s radius and the distance to the Moon, which were necessary along with knowledge of the Earth’s gravity and the Moon’s period of revolution around the Earth to check numerically the law of universal gravitation, was the main reason why publication of Newton’s law was delayed. After Jean Picard (162s1682) had computed the radius of the Earth from his measurements of degree, Newton recalculated the Moon’s orbit and achieved an excellent fit of the precomputed positions of the Moon to observation results. This had provided practical proof of the new law. Newton then used it to explain the motion of planets, their satellites, comets, as well as tidal motions in the sea. The constant G of gravitation of (1.1.1) is used in different forms depending on the application of the law of gravitation. Since different systems of units are employed to measure mass, length, and time, the constant of gravitation may have different values. Each of these constants is known under a different name: Cavendish’s, Gaussian, Einstein, geocentric, selenocentric, heliocentric, and so on. Table 1.1 lists some gravitational constants, their numerical values, dimensions, and units of mass, length, and time measurement. All constants are defined in terms of Newton’s law of gravitation. In so far as

3

The Gravitational Field of the Moon

TABLE 1.1 Different constants of gravitation, their values and units of measurements

Constant Cavendish’s constant of gravitation Einstein constant of gravitation Gaussian constant of gravitation Geocentric constant of gravitation

Numerical value

Units of measurement Length Mass Time

(6.6742 f 0.0008) m x lo-” m3 kg-’ s - ~ (1.865 & 0.001) m x m kg-’ 0.017,202,098,95 a.u.

(398,603 1) 109 r n 3 s - 2 Selenocentric constant of gravitation (4902.7 f 0.1) x i09m3s-*

m

m

mass of the Sun mass of the Earth mass of the Moon

day S

S

certain astronomical problems are concerned, it is convenient to use, as the basic equation, the formula of the third law stated by Johannes Kepler (15711630), which, in the final analysis, is also derived from Newton’s law of gravitation. Kepler’s third law is used, in particular, to derive the selenocentric gravitational constant G M , from the motion of a satellite of the Moon. If this motion is due only to the gravity field of the Moon, which is assumed to be uniform and spherical, then the following relation exists between the period T of the satellite’s revolution around the Moon, the semimajor axis a . of the satellite’s orbit, and GM,: (1.1.3) This constant has significantly gained in importance after the launching of space vehicles toward and onto the Moon. It is used to calculate the trajectories of artificial lunar satellites (ALS) and spacecraft sent to other planets of the solar system, in studying the overall density distribution inside the Moon, in determining its dynamic figure, and for other purposes. Determined in a similar manner is the geocentric gravitational constant G M e which plays an important role in studying the motion of space vehicles and satellites in the gravitational field of the Earth. The Gaussian constant K of gravitation was initially derived assuming that the Sun’s mass is Ma = 1 and the semimajor axis of the Earth’s orbit around the Sun is Aa = l,,while the period T and relation ( M @+ Mu)/Ma were determined from observation. However, since the latter two quantities have been constantly refined during observations, the currently accepted value of K is that given in Table 1.1, whereas variations in T and ( M e + M t ) / M o are compensated by slight

4

Lunar Gravimetry

changes in A. The unit of distance becomes a derivative quantity corresponding to a definite, predetermined value of the Gaussian constant K of gravitation and is referred to as astronomical unit (AU). The tabulated numerical value of K was for the first time obtained by Karl Gauss (17771855) in 1809. However, Newton had derived it within a lower order of magnitude (K = 0.017,202,12) as early as 120 years before. The term “Gaussian constant” has taken root out of respect to Gauss for his having introduced the law of gravitation into celestial mechanics, rather than for priority in deriving this constant. The Einstein constant of gravitation enjoys currency in theoretical physics. Paradoxical as it may seem, the most “Earthbound” constant of gravitation derived on the assumption that mass, length, and time are expressed in the universally adopted metric system is known only in the roughest approximation. It is known as Cavendish’s constant G of gravitation. It is determined experimentally,by measuring the force of mutual attraction of test bodies spaced an exactly known distance apart (the masses of these bodies are also known exactly) and calculating it with due account for the shape of these bodies and the distance between them. The first experiment in which the contant G of gravitation was determined was carried out by Henry Cavendish (1731-1810). The significance of Cavendish’s experiment is not restricted to numerical definition of this constant. More important by far was experimental corroboration of applicability of Newton’s law of gravitation, not only to celestial bodies, but also small terrestrial ones. The importance of this finding cannot be overestimated in view of the assumption made after the discovery of the law of universal gravitation to the effect that the force of mutual attraction reaches sizeable proportions only between celestial bodies. As regards terrestrial bodies, the consensus at that time was that their small size made it impossible to observe the attraction between them. This conclusion had been drawn after Newton’s miscalculation. Table 1.2 summarizes the results of recent measurements of Cavendish’s constant G of gravitation. In all experiments, a torsion balance TABLE 1.2 Determinations of Cavendish’s constant of gravitation

Workers, country, year of publication Hey1 and Chrzanowski (USA), 1942 Rouse, Parker, Beams et al. (USA), 1969 Renner (Hungary), 1970 Facie, Pontikis and Lucas (France), 1972 Sagitov, Milyukov, Monakhov et al. (USSR), 1978 Laser and Towler (USA), 1982

Value of constant G ( l o - ” m3kg-ls-’) 6.673 f 0.005 6.674 f 0.004 6.670 f 0.008 6.6714 f 0.0006 6.6745 & 0.0008 6.6726 f 0.0005

The Gravitational Field of the Moon

5

was used. In experiments based on the dynamic method, the force attracting test bodies to each other was determined from the torsional oscillation period, whereas in rotational experiments the torsional system was made to rotate about the axis of the torsion wire for more accurate measurements of attraction. Earlier definitions of the constant of gravitation are reviewed in the monograph by Stegena and Sagitov (1979). Using Cavendish’s (G) and the selenocentric ( G M u ) constants of gravitation, one can determine the mass M u and mean density O, of the Moon. The determination of G M , , M , , and O, will be treated at greater length in Chapter 3.

1.2 Problems Solvable from the Gravitational Field

The complexity of the lunar gravitational field stems from the uneven density distribution within the Moon and non-uniformity of its figure. The gravitational field merely provides intermediate data on the Moon’s internal structure and figure. The final picture is much more complex and diverse. As far as selenophysical problems involving the internal structure and solved with the aid of the gravitational field are concerned, they can be listed in the following order of decreasing importance: (1) determination of the radially oriented planetary changes in the density of the lunar rocks; (2) investigation of density changes in a tangential direction; (3) determination of the Moon’s mass and mean density; (4) finding the difference in internal structure between the near and far sides of the Moon; ( 5 ) studying the departure of the Moon from hydrostatic equilibrium as well as estimation of the stresses inside the Moon; (6) estimation of the isostatic state of large lunar regions and craters; (7) definition of the boundary between the mantle and crust in the highlands and maria; (8) determination of the crustal structure in zones of transition from highlands to maria; and (9) examination of small-scale features of crater boundaries, rille regions, faults, and so on. Associated with the selenodetic aspects of gravitational field applications are such problems as: (a) determination of the selenocentric constant of gravitation; (b) more accurate location of the lunar centre of mass with respect to the geometric centre of the Moon; (c) determination of the distances between individual points on the physical surface and the lunar centre of mass; (d) construction, from satellite data, of the gravitational field model optimal from the standpoint of elimination of the systematic errors which are as high as 100 to 200 mGal in various models; (e) determination of plumb-line deflections on the Moon; (f) more accurate definition of the Moon’s figure and physical libration parameters; and (g) gravity distribution at different points on the physical surface or standard Moon.

6

Lunar Gravimetrv

We have already mentioned studies into spatial changes in the gravitational field. However, as is known, the gravitational field at every point on the Moon changes in the course of time. Secular changes occur on the Moon just as surely as on the Earth, although they seem to be extremely small. Even on the Earth there are no reliable ways of measuring them. As regards periodic gravity variations on the Moon, they are greater in magnitude than on the Earth, but since their period is longer, they are more difficult to measure. Periodic variations in gravity are used to determine the elastic properties of the lunar rocks. Moreover, knowledge of the lunar gravitational field is instrumental in calculating the trajectories of space vehicles in the neighbourhood of the Moon. So far, only problems solvable with the aid of the gravitational field have been mentioned. Many of these problems will be discussed in more detail in the following chapters. Even the above listing attests to their great diversity. Obviously, solution of different problems calls for different levels of knowledge about the gravitational field, different sets of its parameters (gravitational potential, gravity, horizontal gravity components, gradients of gravity, etc.), and different degrees of accuracy. Some of the above problems are solved using gravity data supplied by satellite observations. To solve all problems necessitates determination of gravity and its gradients directly on the lunar surface. To this end, gravimetric instruments mounted on modules performing “soft” landing on the Moon’s surface can be used. Of particular importance for gravimetric survey of the Moon are all kinds of lunar roving vehicles. The gravimetric surveying may involve area and profile measurements, and in some cases single measurements of absolute gravity must be taken. The selenodetic problems (b)-(d), namely, more accurate location of the lunar centre of mass and construction of a gravitational field model from satellite data, require absolute gravity determinations. It is necessary that the absolute measurement stations be spaced as widely apart as possible. The problem (c), or determination of the distances between individual points and the lunar centre, also calls for absolute gravity determinations. An accuracy of * 5 mGal in such measurements is quite adequate for solution of these problems at present and in the immediate future. Even measurements with an accuracy of f20mGal permit the position of the centre of mass to be determined to within f 1 0 0 m and the systematic errors in the lunar gravitational field models to be minimized. To solve the overwhelming majority of selenophysical and selenodetic problems, it is recommended to resort to relative gravity determinations, using for reference abs.olute gravity determinations at a few stations. For example, the terrestrial survey experience tells us that gravity determinations to within f 1-5 mGal are sufficient for solution of the selenophysical problems (1)-(6). Solution of

The Gravitational Field of the Moon

7

problem (9wefinition of small-scale details such as crater boundaries, rilles, and so on+alls for an accuracy within fractions of a mGal. The relative gravity determination range may span 500 mGal. In view of the ambiguity of solution of the inverse problem of gravimetry-determination of the anomalous mass distribution pattern from a known gravitational field-gravimetric methods should preferably be combined with other geophysical techniques (seismic, electrical sounding, etc.). If selenophysical problems do not go beyond local gravimetric surveys, most of selenodetic ones are difficult to solve without knowing the gravitational field of the entire Moon. It is precisely in the case of selenodetic problems that elimination of systematic errors in the gravitational field is highly desirable. It is well known that measurement of lunisolar tides is a difficult task even on the Earth, although the tidal period here is of a diurnal duration as opposed to the monthly period on the Moon. Studies into lunisolar tidal variations in gravity necessitate stationary gravimeters with a measurement range of about 3 mGal and an extremely high accuracy of fO.OO1 mGal (Chapter 4). The basic requirement of tidal gravimeters is stability of their zero-point. Measurements must be taken continuously, at least over a period of several months. The above figures are, of course, very approximate. In practice, different accuracies may be needed, although definitely no different than by a factor of two or three. Prediction is a risky affair. The validity of forecasts is affected by the leaps and bounds in the evolution of science and technology. This is especially true in space exploration, where scientific and technical advances are most spectacular.

1.3 Lunar Gravitational and Tidal-Centrifugal Potentials. Gravity

Let S be the surface of the Moon and R its volume (Fig. 1.1). We shall use a fixed-in-the-Moon system of rectangular coordinates X , Y, and Z . The origin 0 of the coordinates. corresponds to the lunar centre of mass. The axis Z is directed along the rotational axis of the Moon, and the axes X and Y are oriented in its equatorial plane so that the former will pass through the zeromeridian plane. The point P at which the gravitational field is examined has coordinates (x, y, z), while (5, q, c) are those of the mass element drn at the current point M of the lunar body. We assume that the point P accommodates unit mass; then, the force acting upon it is numerically equal to acceleration. This mass is acted upon by several accelerations of different origins: acceleration gl of lunar mass attraction, acceleration g2 of attraction of the masses of the Earth and other celestial bodies, and centrifugal acceleration g3.By gravity or, to be more precise, gravity acceleration g, is meant the

8

Lunar Gravimetry

Fig. 1.1. Coordinates, angles and distances used in deriving the lunar attraction potential.

resultant of all acting acceleration. The potential of lunar mass attraction is expressed as a triple integral over the Moon’s volume R:

(1.3.1) where drn = Q dR, dR((, q, [) is a volume element, and o((, u, [) is the volume element density. Let us now introduce a spherical coordinate system with the same origin (Fig. 1.1), where p, cp and 1stand, respectively, for the polar distance, latitude, and longitude of the point P at which the potential of lunar mass attraction is considered; p l , cpl, ill are the coordinates of the current point M of the lunar body. The distance between P and M is r

where

= (p2

+ pt - 2pp1 cos $)1’2

\I/ is the angle between the radius vectors

p and p l . Expand the

9

The Gravitational Field o f the Moon

quantity l/r in the integrand of (1.3.1) into a series of Legendre polynomials (1.3.2) where cos $ = sin cp sin cp, + cos cp cos cp, cos (1 - Al), and P,(cos $) is a Legendre polynomial of the nth order. Using the theorem of Legendre polynomial addition, one can express the polynomial with argument cos I(/ in terms of functions of ql,1 1 , cp, A: Pn(cos $) = f'no(sin cP)Pno(sin PI) +2 m=l

( n - m ) ! (cos m l cos mil, (n m ) !

+

+ sin mil sin m1,)

x P,,(sin

cp)P,,(sin cp,)

(1.3.3)

where P,,(sin cp) and P,,(sin cp,) are associated Legendre functions. Substituting the expansion for l/r (1.3.2) into (1.3.1) with due account for (1.3.3), we can write the potential V of attraction as VP, cp,

4=

~

P

2

n=O

y:(

(C.,cos

mil+ S,, sin ml)P,,(sin cp)

(1.3.4)

m=O

where R is the mean lunar radius. C,, and S,, are harmonic coefficients, sometimes referred to as Stokes coefficients. They have nothing to do with the coordinates of the observation stations but depend only on the density distribution a(pl, cpl, 2 , ) within the moon and its figure R:

c nm

- (n

2(n - m)! ~~[op;P..(sin + m)!MuR"

cp,) cos m l , dR

(1.3.5)

R

Snm

+

= (n 2(nm)! -m MuR" ) ! jj!op;P,,,,,(sin

cpl) sin mAl dR

n

where dR = pf cos cpl dcpl d l l dp, is a volume element in spherical coordinates. Consider now the potentials corresponding to the accelerations g2 and g3. The contribution of the Sun to tidal lunar gravity variations can be ignored because it is two orders of magnitude less than that of the Earth. We shall use

10

Lunar Gravimetry

t’

Earth

Fig. 1.2. Derivation of the tidal potential on the Moon.

the selenocentric equatorial spherical coordinate system (Fig. 1.2). Let P(p, cp, i) be the point at which the variations in the tidal potential, due to the Earth’s attraction, and variations in the centrifugal potential, due to the Moon’s rotation about its own axis and around the Earth, are examined. The Earth’s coordinates will be denoted (A, b,, l,). Having assumed the Earth to be a point body with mass M,, we can write the following expression for the potential V Q of terrestrial attraction at the point P:

where A = (A2 + p z - 2 p A cos z ~ ) ’ ’ is~ the distance between the Earth’s centre of mass and the point P , A is the distance between the terrestrial and lunar centres of mass, and z, is the angle with vertex at the origin of coordinates between the directions toward the point P and the Earth. This angle equals the zenith distance z, of the Earth at the point P. Let us expand

11

The Gravitational Field of the Moon

P e ( P ) into a series of Legendre polynomials: (1.3.6)

The potential of attraction at the lunar centre of mass is (1.3.7)

The difference of potentials (1.3.6) and (1.3.7) gives rise to a tide-generating potential on the Moon due to the effect of the Earth. Lacking the properties of an absolute solid, the Moon undergoes partial deformation which manifests itself in tides within the solid Moon. The term with spherical function P,(cos zQ) in (1.3.6) is excluded because the origin of the selected coordinate system coincides with the lunar centre of mass. Restricting ourselves to the second harmonic in expansion (1.3.6) in view of the smallness of the subsequent terms as a result of the rapidly decreasing relation @/A)”, we obtain V,(P)

=

P,(P)

-

Pe(0)= ___ G’@p2 P2(cos z e ) A3

(1.3.8)

Let the angle zQ be expressed in terms of angle $ which is the angle between the directions toward the point P and the mean position of the Earth, coinciding with the direction of the coordinate axis X . It is known that the angular coordinates of the Earth ( b e , l Q ) vary, from a lunar observer’s point of view, within -6”. Then, it is easy to find that COS’ ZQ = COS’

$ + F($,

~ p ,I ,

b e , I@)

where F($, cp, I , b e , l e )

%

2 cos cp(cos cp sin I sin le

+ cos I sin cp sin b Q )

is a time-dependent function because of the varying terrestrial coordinates (A, b e , lQ). Using the latter expression, we can rewrite (1.3.8) as

Using the addition theorem for Pzo(cos $) and bearing in mind that cos II/ = cos cp cos I , we shall write the final expression for the terrestrial tidal potential on the Moon:. VQ(P ) = GMQp2[ -2P2,(sin cp) 4A3 ~

+ Pzz(sin cp) cos 21 + 6F($, cp, I , be, re)] (1.3.10)

12

Lunar Gravimetry

Thus, the tidal potential on the Moon comprises two parts. The first, invariable part is independent of time, while the second part is timedependent. Both parts are dependent on the coordinates (cp, A) of the tidal phenomena observation station. The centrifugal acceleration potential due to the Moon's rotation is much smaller than that due to the Earth's rotation because of the low angular velocity of the Moon's rotation: w = 0.26616995 x lop5rad s-'. The centrifugal potential Q at the point P is

The value of w corresponds to a sidereal month which is determined according to Kepler's third law, from 0 2 =

~

GMQ A3

Use of this expression for w z gives the following formula for the centrifugal potential: (1.3.1 1) Addition of (1.3.10), in which the time-dependent part of the potential is omitted, and (1.3.11) gives the following expression for the tidal-centrifucal potential on the Moon: U ( P ) = ___ G M Q p [2 - 5Pzo(sin cp) 6A3

+

Pzz(sincp) cos 2 4 (1.3.12)

Thus, the tidalxentrifugal potential U is essentially the sum of the centrifugal (Q) and tidal ( W) potentials. It depends on the geocentric constant G M e of gravitation, the lunar point coordinates ( p , cp, A), and the distance A between the Earth and the Moon. As can be inferred from (1.3.12), the potential U is independent of the internal structure of the Moon, and therefore its definition does not seem to present any of the difficulties normally associated with poor knowledge of the lunar density distribution and figure, as was the case with the lunar potential of attraction (1.3.4).In fact, this is not so. Under the effect of the tidal-centrifugal potential, the Moon undergoes deformation: its mass is redistributed, and the coordinates of the observation station change. The result is uncertainty in the knowledge of the rigidity or susceptibility of the Moon to deformations (see Chapter 4). We have already mentioned at the beginning of this section that the gravity acceleration g is the vector sum of all accelerations acting at a given point.

13

The Gravitational Field of the Moon

The potential W of the gravity acceleration also equals the sum of potential V of acceleration due to lunar mass attraction and potential U of tidalcentrifugal acceleration: W ( P )=

V P , cp, A) + V(P,cp, A)

=-

P

(C,,,, cos mA n=O

+ S,,

sin mA)P,,(sin cp)

m=O

(1.3.13)

The full value of the gravity acceleration g(p, cp, A) will be found if the latter expression is differentiated with respect to the three orthogonal axes (p, cp, A), their squares are added, and the square root is taken: 1

(3"'"

(1.3.14)

Obviously, g is determined primarily by a W p p . Assuming the potential W(x,y, z) as a function of the coordinates x, y, and z and equating it to a constant, we obtain the following implicit equation of a surface: W(x,y , z) = const.

(1.3.15)

It exhibits some gravimetrically important properties. This is an equation for equipotential, or level, surface. By changing the value of the constant, we can obtain a family of level surfaces. Since the gravity acceleration component along a direction I is

the increment in the potential over an elementary distance dl can be determined using the formula d W = g dl cos (g?). If the direction of g coincides with 1, then cos (g?) = 1 and the increment d W is maximal. The equality of cos (g?) to zero indicates that no increment in the potential occurs in moving along the level surface. It should be emphasized that the equality of potentials on the level surface is far from implying that gravity is constant on the same surface. Also, if we constructed a surface of equal gravity, the potentials at different points on such a surface would not be equal.

14

Lunar Gravimetry

1.4 Quantities Characterizing the Lunar Gravitational Field

In addition to the such purely terrestrial characteristics of gravitational field as gravity potential, acceleration components along different coordinate axes, and gradients of these acceleration components, the Moon is also characterized by line-of-sight acceleration which is an acceleration component in the direction from the artificial lunar satellite to a terrestrial Doppler tracking station. It can be identified, to a sufficient degree of approximation, with the acceleration along the Earth-Moon line. Only in the central parts of the lunar disk do line-of-sight accelerations coincide with those oriented radially to the Moon. The acceleration component orthogonal with respect to the line-of-sight acceleration a W/ap, that is tangential to the spherical surface of the Moon, can be expressed as

where grp= (1/R)(aW/acp) is the latitudinal acceleration component, and g A = 1/(R cos cp)(aW/aA)is the longitudinal one. In view of the smallness of g1 in comparison with W/dp, in most problems the radial gradient a g / a p of gravity may be considered equal to a2 W/ap2.The line-of-sight acceleration in a rectangular coordinate system, selected such that the axis X with its origin in the Moon’s centre is directed earthward, is

a

An important characteristic of the gravitational field is the plumb-line deflection. The deflection component along the meridian is ( = 206265 (l/R)(aW/acp)

9

(in seconds of arc)

and that along the parallel is ( = 206265 (l’R)(a

w’aA)

9 cos cp

(in seconds of arc)

The full plumb-line deflection is 1 = ((2

+

t+)1/2

Not all of the above characteristics can be measured directly, yet all of them can be calculated, provided the distribution of one of them over a closed surface enveloping the Moon is given. Such calculations usually involve

15

The Gravitational Field of the Moon

methods employed to solve boundary-value problems in potential theory. The problem is stated as follows: all over a known closed surface S there exists a gravitational potential Ws(r,cp, A ) = f ( r , cp, A) generated by the masses randomly distributed within S. Find the potential or its derivatives at points in the space external with respect to S. A simple analytical solution of this problem exists only for an ellipsoid, a sphere and a plane. If the same potential function given on a sphere is determined at different points in the surrounding space (Dirichlet problem), the solution takes the form of Poisson's integral (Fig. 1.3): (1.4.1) S

In the case of the Moon, which is taken to be a sphere S of radius R, the following notation is used p, 0 and A are the sphkrical coordinates of the point P at which the potential W(p,(D, A) is determined, and W(R,cp, A) is the potential given at the current points M ( R , cp, A ) on the surface of the sphere, and r =' ( p 2 R 2 - 2Rp cos is the distance between the points P and

+

1'

Fig. 1.3. Angular coordinates and distances determining the Poisson integral equation for a sphere.

16

Lunar Gravimetry

M . The derivative of a potential of any order along the axes X , Y and 2 can A) if both parts of (1.4.1) are differentiated be obtained at the point P ( p , 0, along these directions: axm ayn

aZI =

11

W ( R , cp, A)K(R, cp,

A, p, a, A) dS

(1.4.2)

S

where =

[

r

am+n+l

axm ayn

azt

2 -RZ)] 4 a ~ r 3 xy==PpC Ocosasin S@COS~ A z=psin@

is the kernel of the transformation p = (XZ

+ y2 +

= C(X = (p2

z2)1/2

0’ + (Y - v)2 + ( z - 02 1 1 / 2

+ R 2 - 2Rp cos $)’/’

+ sin cp sin @ cos (A - A) And if we want to determine the gravity component a W/ap along the radial cos $ = cos cp cos @

direction p, the kernel is

a

K p = - a( p

p2 - R 2

4nRr3

)=

- p 2 ( R cos $ + p ) + R2(5p - 3R cos $) 4aRr

When components tangential to the spherical surface S (latitudinal g@and longitudinal gA) are determined, the kernels take the respective forms: K @ = - (a

pz - ~2

~ a @4nRr3 a K A = - - - (1

p2

RCOS@

aA

)=

3 ( p z - R Z )a~cos $

47cRr5

)=

- RZ

4nRr3

a@

3(pz - ~ z ) p a c o s +

~ ~ R ~ ’ c o saA@

Finally, here is another expression of the transformation kernel for calculating the second radial derivative a2 W/ap2of the potential: KPP

=

R2(5R2- 23p2 + 28Rp cos $ - 15R2 C O S ~$ - p2 COS’ $) 4aRr’

+ 2R cos $) + 2p3(p 4nRr’ Thus, calculation of a particular derivative of the lunar gravity potential boils down to integration of a given potential W ( R , cp, A) over a spherical surface with a respective kernel.

17

The GravitationalField of the Moon

In conclusion, we should like to point out an interesting possibility. If all derivatives of the gravity potential could be determined at a single point in space, this would be sufficient, in principle, to define the gravitational field within the entire space, the only prerequisite being that this space should not contain any masses that have generated the field. However, measurement of what amounts to third derivatives of the potential involves insuperable technical difficulties from the standpoint of the precision necessary to produce components of an appropriate instrument and ensuring stable conditions for the measurement. Table 1.3 presents some of the above-mentioned characteristics with their dimensions, full values, and the possible regional anomalies on the lunar surface. It also gives the corresponding terrestrial parameters for comparison. Note the much more pronounced anomaly of the Moon's gravitational field, as compared to that of the Earth. 1.5 Comparison of, Methods Used for Studying the Gravitational Fields of the Moon and the Earth

Investigations of the lunar gravitational field began with the launching of artificial lunar satellites (ALS). Interestingly, the chronological sequence in which the methods for determining the gravitational fields of the Earth and the Moon were used has been different. In the case of the Earth, satellite methods started being used when its gravitational field had already been studied in general with the aid of gravimeters and pendulum instruments. TABLE 1.3

Some characteristicsof the lunar and terrestrial gravitational fields Magnitude on the surface (possible anomaly) Characteristics; dimensions

Unit of measurement

Moon

Earth

2824 ( f12)

626,37 (k70)

Gravity acceleration potential,

K CLY, C T Z

10' cmz s - ~

Gravity acceleration, g; [ L ] ,

crl-2

Radial gravity acceleration gradient, dg/dp; [ f l - 2 Attraction components tangential to the surface, W,,W,;

cLIcrl-2 Plumb-line deflection components, t, rl

ems-' = 1 mGal 162,700 (k500) 980,600 (f300) s2 = 1 eotvos(E)

1870 (k500)

3086 ( f 100)

cm s - ~= 1 mGal

(k600)

( k 300)

* 500

+ 60

seconds of arc (")

18

Lunar Gravirnetry

Each method, satellite and gravimetric, has its advantages and drawbacks. The former provides more reliable means for determining the gravitational field harmonics of lower orders, whereas the latter is more suitable for higher harmonics. A practical approach has been to combine both methods with recourse to astronomical and geodetic data. As regards the Moon, satellite methods were the first to be used in investigations of its gravitational field, if we do not count some evaluations of changes in gravity with the selenocentric latitude and longitude, based on the lunar moments of inertia established by astronomical observations (Grushinsky and Sagitov, 1962). The first detailed definition of the lunar gravitational field, in the form of expansion into a series of spherical functions, was carried out by Akim (1966) using data supplied by the artificial lunar satellite Luna 10. Data from some American spacecraft, including Lunar Orbiters 1-5, Explorers 35 and 49, Apollos 14-17, and Apollo 15 and 16 subsatellites, as well as the Soviet spacecraft Luna 10 and 24, have widened our knowledge of the Moon's gravitational field. Direct gravity measurements on the lunar surface have just begun. So far, only four measurements with the aid of gravimetric instruments installed on lunar probes that have performed soft landing on the Moon (Apollos 11, 12, 14 and 17) are known to have been carried out. Gravity profile measurements on the Moon's surface were conducted with a special gravimeter mounted on the American lunar roving vehicle delivered by Apollo 17. Although ALS will continue to play a predominant role in investigations of the overall gravitational field of the Moon, we must not underestimate direct gravity measurements using soft-landing lunar vehicles and probes. The advantages of satellite methods for investigating the gravitational field of the Moon, as compared to that of the Earth, are as follows: (1) The flight of ALS can be watched from the same terrestrial tracking

station all the time it is not behind the Moon. Two tracking stations located at longitudes 180" apart will permit uninterrupted tracking. (2) Since the Moon rotates more slowly than the Earth, its gravitational effect on the ALS orbit is averaged to a lesser extent. (3) The dense and time-dependent atmosphere of the Earth produces a strong perturbing effect on Earth-orbiting satellites. The virtual absence of atmosphere on the Moon means that the motion of ALS is unperturbed by this factor. (4) Owing to the lack of atmosphere on the Moon, its satellites may orbit very close to the lunar surface. The pericentres of some Apollo spacecraft used to study the gravitational field of the Moon were at distances of only 15-20 km from the lunar surface. The shortest distance at which satellites may orbit the Earth is about 200 km. Low-

The Gravitational Field of the Moon

19

orbiting spacecraft are more prone to the effect of gravitational field anomalies. ( 5 ) The ratio of gravitational field anomalies to the overall field is greater for the Moon than for the Earth, which is why ALS are more susceptible to the perturbing effect than those orbiting the Earth. Of course, there are some factors that make investigation of the lunar gravitational field more difficult. These are, primarily, the highly complex procedure for launching ALS and their prohibitive cost. From the technical standpoint, tracking also imposes some limitations. When a spacecraft swings around the far side of the Moon, it escapes tracking and the efficiency of deriving the lunar gravitational field suffers. This drawback is inherent in the currently used Moon-orbiting satellite tracking techniques. Placing two spacecraft in the lunar orbit at a time will permit retransmission of signals from the satellite which becomes invisible from the Earth, whereby more reliable data on the far-side gravitational field can be obtained. Errors in determination of the lunar gravitational field arise because the exact position of the ALS relative to the Moon is not known and as a result of inexact subsequent reductions of the measured field parameters. Serious errors occur during mathematical processing of Doppler tracking data, which includes filtering operations and conversion of the Doppler frequency variations into the gravitational field characteristics. In addition, various instrumental errors are inevitable as well as errors due to refraction of radio waves in the Earth’s ionosphere and uncertainty as regards the position of the Moon-orbiting spacecraft with respect to the terrestrial tracking station. This uncertainty stems from inexact knowledge of the lunar ephemerides and the position of the tracking station on the Earth. Some of the above errors are systematic. They can be eliminated by closely studying their sources through observations of other celestial objects rather than the Moon-orbiting satellite. Particularly promising for studies into at least the high-frequency portion of the gravitational field are ALS-mounted gradiometers. Apart from high resolution, gradiometers offer a number of advantages for investigating the gravitational field on the far side of the Moon. Since they are more or less selfcontained and remain functional beyond the zone of visibility from the Earth, they can be used to measure the gradient of accelerations on the Moon’s far side. 1.6 Elements of ALS Orbits and the Concept of Perturbations in Motion as a Result of Irregularities of the Lunar Gravitational Field

Let point 0 stand for the Lunar centre of mass and S be an ALS (Fig. 1.4). The origins of the rectangular (X, Y, Z) and spherical (p, CD, A) coordinate

20

Lunar Gravimetry

Point of the vernal equinox Fig. 1.4. ALS orbit elements.

systems coincide with the lunar centre of mass, and Y is the vernal equinox. The equatorial plane of the Moon coincides with the plane XO Y. The orbital plane of the ALS intersects the equatorial plane along the nodal line K K 1 .It is known that the spatial position of a satellite at any instant is characterized by its radius vector (X,Y, 2 ) and velocity vector (X, Y, Z). Instead of these six parameters, one can use six elements of the Keplerian ellipse having one of its foci at the origin of coordinates. The elements representative of the ALS position in its orbital motion are (Fig. 1.4): R, longitude of the ascending node; i, inclination of the orbital plane of the ALS to the equatorial plane of the Moon; a and b, semimajor and -minor axes of the orbit; e = [(a’ - b’)/~’]’’~,its eccentricity; o,angular distance of the pericentre from the node; and M, mean anomaly M

271

2a T

= - ( t - to) = -t - Mo

T

where T is the satellite’s orbital period, 2 4 T is its mean motion (mean angular velocity), and to is the instant of transit of the ALS through the

21

The Gravitational Field of the Moon

pericentre. In addition, true ( u ) and eccentric (E) anomalies are used. The latter is related to the mean anomaly by the Kepler equation M = E - e sin E

while the true anomaly u is expressed in terms of the eccentric one as follows: tan

2

=

(g)"' ):( tan

The true (u) and eccentric (E) anomalies are necessary to determine the distance p between the satellite S and the origin of coordinates, coinciding with one of the foci of the elliptical orbit: a(l -- e2) p = ( X 2 + y2 + Z2)1/2 = = a(1 - ecos E) (1.6.1) 1 + ecosu Sometimes, use is made of the pericentre's longitude p=R+o

It comprises angles R and o measured in different planes. The differential equations of the satellite's motion in the irregular gravitational field of the Moon, in the rectangular system of coordinates ( X , Y, Z ) , are d2X G M , =+p3X=d2Y -+dt2

aT

ax

aT y=p3 ay

GM,

d2Z G M , =+p3Z=-

aT

az

where G M , is the selenocentric constant of gravitation, and T is the perturbing potential of the Moon, due to the difference of the true lunar potential from that of the point mass. The same equations written in the selenocentric spherical coordinate system take the following forms: GM, (p2:)

+ pz&y

aT

sin @ cos @ = aT

am

.(1.6.2)

22

Lunar Gravimetry

in this case, P = p(a, e, i, 0, Q, M )

CD = @(a,e, i, o,R, M)

(1.6.3)

A = A(a, e, i, o,R, M) It is convenient to represent the perturbing potential Tof the Moon expanded into a series of spherical functions:

2 [(ty

P n=2 T = G“,

(C,, cos mA

m=O

+ S,,

sin mA)P,,(sinCD)

1

(1.6.4)

Here, R is the mean radius of a spherical Moon. The expansion begins with n = 2 because the perturbing potential does not contain the zeroth harmonic corresponding to the point Moon potential; first-order harmonics are absent because the origin of the selected system of coordinates is assumed to coincide with the lunar centre of mass. If T = 0, which means that no perturbations affect the ALS motion, then the satellite is propelled by the gravitational field only of the point Moon. In that case, the motion of the ALS would have followed an elliptical orbit and obeyed Kepler’s laws. The presence of perturbations T complicates the spatial motion of the ALS. The elements of its orbit change in the course of time or, in other words, undergo perturbations. The magnitude and character of these perturbations depend on the coefficients C,, and S,, which represent the irregularity of the lunar gravitational field. Observational data are used to compute the perturbations affecting ALS orbit elements, from which the numerical values of C,, and S,, are determined. Let us provide a general scheme for derivation of the formulae correlating the perturbations of the elements with C., and S,,. To do this, we shall turn from the differential equations system (1.6.2) to a system of six first-order differential equations known as Lagrange equations (see, for instance, Caputo, 1967). They can be used to represent the perturbations for any element (daldt, deldt, di/dt, do/dt, dR/dt, dMu/dt) in terms of the derivatives of the perturbing function T with respect to the elements (dT/da, dT/de, dT/di, dT/do, dT/dR, dT/dMu) and the elements themselves (a, e, i, o,R, Mu). Consider in greater detail only the perturbations in the ascending node longitude R and angular pericentre distance o. Lagrange equations give dR -

dt

cosec i dT (CMu)’”[a(l - e2)]”2 di

(1.6.5)

23

The Gravitational Field of the Moon

and do

dt

=-

dT cot i (GMu)’/’[u(l - e2)]’/2 di

dT (GMu)’/2(ae)’/2de 1-e’

-+

(1.6.6)

Turn now from the time argument t to the true anomaly argument u, using the equation du 1 - = - (GMU)’/’[a(l - e2)]’/’ dt p’ Equations (1.6.5) and (1.6.6) can be rewritten as

dsz _ du

p’ cosec i

dT

GMaa(l - e’)

di

(1.6.7)

and do_ _ du

-

p’coti dT p2 dT -+-GMua(l - e’) di GM(ae de

(1.6.8)

Sequential transformations may yield the final expressions for dR/du and do/du. The result is cumbersome expressions whose arguments will be the desired harmonic coefficients, orbital elements, and the mean lunar radius R: dR - dR (C,,, S,,, R, orbital elements) du du d o -d o du du

(C,,, S,,, R, orbital elements)

Thus, in determining the zonal harmonics, assuming that rn = 0 for dR/du, we obtain

(:Yc2.

-

i + -32 (1 cot -

e2)3

x (1

):(

+

sin2 (o u)(l+ e cos u) 3

+

+

c30[5 sin3 (a u) sin’ i - sin (o u)]

+ 2e cos u + e’

+ -5 (1cosi - e2)4) :( ~ x (I + 3e cos u + 3e’ 4

cos’ u)

+

~ sin4~ (o[ u ) 7sin2 i - 3 sin2 (o+ u)]

cos’ u

+ e3 C O S ~u) + . . . (1.6.9)

24

Lunar Gravimetry

The perturbations in the orbital elements are secular and periodic. The periodic perturbations are, in turn, divided into long- and short-period perturbations. In particular, in the case of dR/dv, short-period perturbations are due to changes in v-that is, with a period close to the orbital period of the ALS. To eliminate them, it is sufficient to carry out averaging over v :

-=-I-& 2r

-

dR dv

dR dv

1 271

0

Then,

In order to eliminate the long-period perturbations as well, the averaging should also be performed over o: 15

('p40(

1

1 - sin2 i)

m=O

x(l

+iez)+E 6 (1

- e2)6 ('Y ac60

1+5e2+-e4 l5

8

) ] + . a .

(1.6.11)

In conclusion, here is a brief summary of secular variations in other elements, as a function of the harmonic coefficient CzO:

(1.6.12)

The odd zonal harmonics of the gravitational field,just as the tesseral ones,

25

The Gravitational Field of the Moon

do not give rise to secular perturbations. The odd zonal harmonics are responsible for long-period variations in orbital elements. For example, these variations are related to C30 in the following manner:

-

):(

3ecoso

("y ( y

cot i -sin2

= 2(1 - e2)3

1 + 4e2

3 sin o 3esino

(p

.

)

i -1

(1.6.1 3)

sin i - e cot i)(y sin2 i -

)

cot i -sin2 i - 1 (:)3

c 3 0

C30

Obviously, the best way to determine the harmonic coefficients of the lunar gravitational field would be from the secular perturbations in the orbital elements of a Moon-orbiting satellite. The maximum secular variations occur at the ascending node longitude Iz and at the distance w of the pericentre from the node. Long-period perturbations are predominant in the inclination i and eccentricity e. For instance, over 46 revolutions of Luna 10, the variations in Iz amounted to -7.7", and those in o,to + 11.8".

1.7 Methods for Determining the Harmonic Coefficients of the Gravitational Field from ALS Tracking Data

The problem of defining the lunar gravitational field is inverse to that of studying the ALS motion in this field. Tracking data are used to determine the perturbations in the ALS motion, due to anomalies in the gravitational field. It is assumed that the left-hand sides of (1.6.9)-(1.6.13) are known from tracking data, and the approximate values of the ALS orbital elements, used in the calculations of the factors of the coefficients C,, and S,, in the righthand sides of these equations, have been defined. Such a method for determining the coefficients C,, and S,, may be conventionally referred to as analytical because it is based on analytical solution of the differential equations of the ALS motion. The analytical method has provided several modifications of the solution, based on different initial tracking data, as well as several procedures for their processing. The processing procedures differ in

26

Lunar Gravimetry

the arrangement of the ALS tracking data obtained over the entire period when the ALS is under observation. Distinction is made between the method of long orbital arcs covering tens and hundreds of ALS revolutions and that of short arcs covering just a few revolutions. Experience has shown that, due to insufficient diversity of ALS orbit parameters, in the early determinations the harmonic coefficients C,, and S,, were cross-correlated. For example, as was shown by Lorell and Sjogren (1968), derivations of harmonic coefficients up to the fourth order yielded coefficients of correlation between Cz0 and c40, Goand c60, CZland c61,c 3 0 and GO, s 3 2 and C42r c 3 2 and s42, S33 and C43, C40 and c60, Cs0 and CT0 exceeding 0.8. Therefore, in later investigations, use was made of tracking data processing procedures that ensured a smaller cross-correlation between the harmonic coefficients to be determined. Let us dwell on the tracking data processing procedure whereby the harmonic coefficients are determined through analytical solution of equations of ALS motion, with reference to Ferrari (1973, 1977) and Ferrari and Ananda (1977). The Doppler tracking data are used first to derive the Keplerian elements (a, e, i, o,Q) for each orbit. Each element is approximated, with the aid of splines, by a time function. Differentiation of the latter permits calculating the long-period variations in the orbital elements (6, i, h, 0).The semimajor axis a is not perturbed by the gravitational field, therefore a = 0. Furthermore, Lagrange equations are used to calculate the harmonic coefficients C,, and S,, from long-period perturbations. The equations of orbital element perturbations can be written in the following vector form which is more compact: E = F(E)P, (1.7.1) where E is the vector of mean elements (a, i, e, Q, o,M ) , and F ( E ) is a matrix composed of partial derivatives of element rates, with respect to harmonic coefficients:

[; __

.. .

ac20

ah ai

ai

acnm

as21

...

]

ai

as,,

(1.7.2)

F(E) =

ah G

o

...

-

ac,,

ah ~

... an as,, ~

The matrix dimensions are [(n + 1)2 - 41 x 4. Let P be the vector of the desired harmonic coefficients C,, and S,, of expansion. of the lunar gravitational field into a series of spherical functions. Its matrix has the 1)' - 41 x 1 and takes the form dimensions [(n

+

P = C C Z O , C Z I , C Z ZCnrn,S21,S~~,..*,Snrnl ,...,

27

The Gravitational Field of the Moon

In the mean-square approximation, the solution algorithm for (1.7.1) is written as follows (Bryson, 1969): p =(FTw-~F

+ K - 1 ) - 1 ( ~ -1p* + ~ T w -E1)

(1.7.3)

where W is a weighting matrix for long-period element rates, having dimensions 4 x 4. It is assumed to be diagonal-that is, the covariance between the derivatives of different elements is neglected: 2

Oi 2

W=

Oi.

(1.7.4)

4 2 u6J

The variances ui can be calculated using their approximate relationship with the element errors 6 E . The above-mentioned quantity K is a covariance matrix for the desired harmonic coefficients. In the case of lower harmonics, its elements are determined from an a priori estimated vector P*, whereas in the case of higher harmonics, they are determined from an empirically established law of monotonically decreasing variances numbered n. The matrix K is also assumed to be diagonal

I 0

","

C

Using this technique, Ferrari (1977) derived harmonics up to order (16, 16) from Lunar Orbiter 5 and Apollo 15 and 16 subsatellite tracking data. Apart from determining the harmonic coefficients through analytical solution of equations of ALS motion in the irregular lunar gravitational field, a technique based on numerical integration of these equations has been used. The vector form of the differential equations of ALS motion can be written as

A=GM,V

(:-+- AJ

where A is the radius vector from the lunar centre of mass to the ALS, V is a Hamiltonian operator, and Tis a perturbation function which takes the form of (1.6.4) in the case of expansion in terms of spherical functions. However, it can also be expressed differently, particularly, in the form of the gravitational

28

Lunar Gravimetry

potential of a plurality of mass points: -

L

d

i=l

ri

where mi are mass anomalies within the Moon, and ri is the distances between these anomalies and the ALS. In practice, this boils down to differential determination of harmonic coefficients or mass points. To this end, a lunar gravitational field model characterized by coefficientsC,, and S,, (or masspoint parameters) in a zeroth-order approximation is derived. This field model is used to calculate the intermediate ALS orbit, then the observed orbit is compared with it. It is assumed that the difference between the two arises from the errors inherent in the adopted model of the lunar gravitational field. Differential corrections are introduced to handle the gravitational field coefficients of the initial model. Such an approach is more direct in certain respects than the analytical method described above. The latter calls for, first, determination of the orbital element rates from which the gravitational field coefficients are then derived. The tracking data are used to find the coordinates and velocity of the ALS, which is why the coefficients are related to them only through element rates. The discrepancy between the intermediate and observed ALS orbits stems from the disparity between the initial gravitational field model and the actual field. The model must be adequately representative of the gravitational field on the near and far sides of the Moon. Therefore, the lunar gravitational field coefficients determined by the above selection method must correspond, in principle, to both sides of the Moon. Errors may occur, of course, due to the fact that all gravitational field coefficients are determined from the tracking data covering only the near side-that is, the observations are not equally valid. The method of numerical selection of the coefficients C,, and S,, of the lunar gravitational field model has been used in some works, particularly those of Michael et al. (1970) and Michael and Blackshear (1972). It should be emphasized that the analytical methods are indispensable for qualitative analysis of the behaviour of solutions of the equations of ALS motion in the irregular gravitational field of the Moon. They are necessary for finding optimal solutions, optimizing ALS tracking programmes, and the like. It is quite obvious that reliable determination of the gravitational field coefficients requires sufficiently long tracking periods (weeks, months). Besides, a greater number of ALS with various orbital elements (inclination, eccentricity, semimajor axis) is desirable. Tables 1.4-1.6 list 'the ALS orbit elements used in determining the coefficients C,, and S,, of the lunar gravitational field. Theoretical analysis has shown that the optimal ALS orbits for studying the gravitational field are those with minimal semimajor

29

The Gravitational Field.of the Moon

TABLE 1.4 Basic parameters of some ALS used for determination of the lunar gravitational field: a, semimajor axis; e. eccentricity; i, inclination of the orbit to the lunar equator; T. period

ALS

a(km)

e

i

T(min)

Lunar Orbiter 1 Lunar Orbiter 2 Lunar Orbiter 3 Lunar Orbiter 4 Lunar Orbiter 3 Lunar Orbiter 4 Lunar Orbiter 5 Explorer 35 Explorer 49

2670 2702 2688 3751 1968 6150 2832 5980 2803

0.327 0.371 0.332 0.516 0.062 0.654 0.317 0.576 0.002

12 18 21 84 21 85 85 169 62

206 210 208 344 130 72 1 225 690 220

Lengths of arcs used for analysis (days)

Number of observations

9.7 8.5 15.9 8.6 17.1

2076 2662 2502 3739 3322

19.9 2138 831

5847 133 64

axes a and maximal inclinations i. In this respect, the orbital elements of Lunar Orbiters 1-3 were not optimal because of the small inclination of their orbits, whereas in the case of Lunar Orbiter 4 the value of a was too high. The latter becomes an advantage when only lower harmonics are to be derived. In this case, the perturbing effect of the high-frequency gravitational field components is minimized. For the gravitational field of the Moon to be defined, the ALS must not be subject to any accelerations other than the gravitational ones due to irregularities of the internal structure of the Moon and its figure. The satellite must not execute propulsive manoeuvres. However, since ALS are normally launched to accomplish a broad range of tasks in addition to gravity studies (photography, investigation of radiation, meteorites, and other phenomena in the circumlunar space), they have to perform manoeuvres, which is why the time alotted for gravity studies is limited. For example, with regard to all five Lunar Orbiters, throughout their active lifetime (45 satellite-months), only 24 individual orbits between 3 and 20 days long were available for gravity determinations. The most tangible perturbing effect on the ALS motion is exerted by sunlight pressure. Its magnitude can be calculated using the formula F=

CSo(1 C

+ K ) cos u

where So = 1.39 x erg cm-2 s-' is the solar constant representing the solar radiation intensity per unit area per unit time, C is the cross-sectional area of the satellite, c is the velocity of light, K is reflectance (0 < K < l), and a is the angle of incidence of sunlight on the satellite's surface. To minimize this effect use should be made of satellites with a small cross-sectional areato-mass ratio. The pressure of sunlight on, for example, Lunar Orbiter l led

30

Lunar Gravimetry

TABLE 1.5 ALS parameters used by Akim and Vlasova (1983) to derive harmonic coefficients of the Moon

T (min)

ASL Luna 10 Luna 11 Luna 12 Luna 14 Luna 15(1) Luna 15(2) Luna 15(3) Luna 16(1) Luna 16(2) Luna 16(3) Luna 17(1) Luna 17(2) Luna 18(1) Luna 18(2) Luna 18(3) Luna 19(1) Luna 19(2) Luna 19(3) Luna 19(4) Luna 19(5) Luna 20(1) Luna 20(2) Luna 21(1) Luna 21(2) Luna 21(3) Luna 22(1) Luna 22(2) Luna 22(3) Luna 22(4) Luna 22(5) Luna 23(1) Luna 23(2) Luna 23(3) Luna 24(1) Luna 24(2) Luna 24(3)

242 1 2417 2662 2250 1866 1896 1801 1848 1808 1798 1824 1796 1837 1875 1805 1869 1969 1969 1969 1970 1835 1800 1838 1806 1812 1978 1978 2543 2543 2543 1837 1197 1799 1853 1805 1804

178.1 177.7 205.4 159.6 120.6 123.5 114.3 118.9 115.0 114.0 116.5 113.8 117.8 121.4 114.7 120.9’ 130.7 130.7 130.7 130.7 117.6 114.2 117.8 114.8 115.3 131.5 131.6 191.8 191.8 191.8 117.7 113.9 114.1 119.3 114.7 114.6

e 349 129 101 163 44 97 16 99 21 15 83 21 73 90 15 126 12 83 134 161 89 25 89 7 15 194 234 171 172 193 95 17 17 108 14 10

0.138 0.228 0.309 0.155 0.045 0.032 0.026 0.006 0.028 0.025 0.001 0.020 0.014 0.025 0.029 0.003 0.08 1 0.075 0.050 0.036 0.005 0.021 0.006 0.034 0.033 0.023 0.003 0.249 0.249 0.240 0.002 0.023 0.024 0.004

0.029 0.03 1

71.9 9.7 17.8 41.8 125.8 125.9 126.7 70.2 70.5 70.8 141.5 141.1 35.2 35.7 35.4 40.6 40.6 40.6 40.2 40.9 64.6 64.1 62.1 62.8 62.7 19.7 19.6 19.3 19.3 19.9 137.7 137.4 137.5 119.4 120.5 120.3

132.3 80.3 78.7 311.6 65.2 40.9 159.8 222.0 320.8 343.7 137.3 98.2 14.4 89.4 4.7 103.9 261.6 294.9 329.5 202.7 223.8 347.8 82.3 131.7 137.3 144.6 267.9 77.7 1 19.4 220.4 52.7 159.4 157.1 140.8 159.5 155.4

’

113.9 225.1 237.9 293.7 285.6 286.9 287.7 61.6 60.9 60.0 40.8 43.2 61.3 57.7 57.1 25.5 339.7 288.2 163.5 73.2 54.6 53.8 226.0 225.1 224.6 7.0 302.4 93.5 354.6 298.3 226.4 228.6 229.0 235.3 237.1 237.1

to its acceleration equal to 1.2 x 10-5cms-2. Accurate account of the perturbations due to light pressure is rendered difficult by the fact that its perturbing effect on the orbiting circumlunar satellite changes abruptly each time the latter disappears in the Moon’s shadow and re-emerges from it. Moreover additional perturbations are caused by the tidal action of the Earth, the Sun and planets of the solar system. The perturbing effect due to the Earth can be expressed, to a sufficient degree of approximation, as T e = G M , ,P2 P,,(cos $)

A

31

The Gravitational Field of the Moon

TABLE 1.6 Some ASL parameters used by Ferrari (1977) to derive the harmonic coefficients of the Moon

ASL Apollo 15 subsatellite Apollo 16 subsatellite Lunar Orbiter 5

Length of arcs used Inclination (days) (deg)

222 34 8

150 170 85

Coordinate variations Period, T(min)

120 120 191

Eccentricity e cP(deg)

0.02 0.02 0.28

+27 +10

+40

I(deg) 0 - 360 0+3m +50

where G M Q is the geocentric constant of gravitation, A is the Earth-Moon distance, $ is the angle between the radius vectors A and p from the lunar centre to the satellite. The perturbing potential TQ comprises invariable and time-dependent portions. A similar formula is applicable to the perturbing effect produced 'by the Sun, which is much less pronounced. The perturbations due to the Earth, the Sun and planets are of secular and long-period nature. For example, the perturbing effect exerted by the Earth at an altitude of 2000 km is roughly equal to that due to lunar oblateness. The effect of the Sun amounts to 0.5 x low2of that of the Earth, while the effect of the planets is even less. Relativistic effects are responsible for secular motion of the perigee: d o - 3(GMg)3/2 dt c2a5/2(l- eZ) Other possible perturbations are of electromagnetic origin, if it is borne in mind that the ALS travels through an ionized medium and in a field of solar radiation. All these effects contribute to the satellite's drag. Also, the attractive forces of the Earth and the Sun bring about the redistribution of mass within the Moon, which must also affect the satellite's motion. However, with the accuracies currently achieved in ALS, the effects of electromagnetic origin and due to mass redistribution inside the Moon are ignored. As regards the perturbations due to sunlight as well as the tidal action of the Earth and the Sun, one must find the derivatives, along the respective coordinates, of the perturbing potential of these effects and introduce appropriate terms into the right-hand sides of equations (1.6.2) 1.8 Determination of Line-of-Sight Accelerations due t o the Earth's Rotation and Orbital Motions of the Moon and ALS

A new satellite method started being used in the late nineteen sixties for studying the lunar gravitational field (Muller and Sjogren, 1968), which had

32

Lunar Gravimetry

never been applied before to the gravitational field of the Earth. According to this new method, the ALS is regarded as a test body which changes its direction and velocity, during its orbital motion, under the effect of the irregular gravitational field of the Moon. The satellite velocity changes are measured from terrestrial tracking stations, using the Doppler technique. This permits the measurement of only one component of the velocity, namely, the projection on the ALS-terrestrial tracking station direction, known as the line-of-sight velocity. To determine the line-of-sight velocity of the ALS with respect to a terrestrial tracking station by the Doppler method, a highly stable generator of standard frequency radio signals is mounted on board the satellite. These signals are received by some tracking stations on the Earth. It is known that, if the distance between the source of radio signals and tracking station changes, the Doppler effect will manifest itself in an observed frequency different from that of the radio signals transmitted from the satellite. The difference between the transmitted signal (fo) and observed (f) frequencies is

where V, is the component of the relative velocity along the line from the ALS to the terrestrial tracking station, and V, is the radio signal propagation velocity. The "+" sign corresponds to shortening of the distance between the satellite and station, and " -"indicates that this distance increases. Doppler measurements yield what amounts to line-of-sight velocity averaged over time T:

0

0

where K = V,J0 is the factor of conversion of the averaged frequency difference Af into velocity. It can be easily seen that the line-of-sight velocity V, may vary for a number of reasons, namely: the tracking station moves about the axis of the Earth's diurnal rotation; the lunar centre of mass revolves, over a complex orbit, around that of the Earth-Moon system; the satellite moves along a Keplerian orbit around the point Moon; and the irregularity of the lunar gravitational field distorts the Keplerian orbit. The satellite isalso affected by perturbations due to the Sun, planets, light pressure, and other factors. All these factors render the time variations in the observed line-of-sight velocity highly complex. Let us now examine the line-of-sight velocity components one by one and assess the contribution of each component to the observed

The Gravitational Field of the Moon

33

line-of-sight velocity V,. Determination of the anomalous gravitational field of the Moon calls for elimination of the velocity components which are due to all of the above factors, except for the component arising from the irregularity of the lunar gravitational field. We shall attempt to describe analytically variations in the line-of-sight velocity and line-of-sight acceleration, due to (1) diurnal rotation of the Earth, (2) orbital motion of the Moon, and (3) orbital motion of the ALS in the central gravitational field of the Moon. First, let us introduce a geocentric ecliptic rectangular system of coordinates (X, Y, 2) (Fig. 1.5) and take this system as the main one as opposed to some other coordinate systems which will be auxiliary. Thus, we shall denote the coordinates of the terrestrial Doppler tracking station P, in the main system, by X p , Y p ,and Z p , while the coordinates of the lunar centre of mass (0,)will be ( X ( , Y,, Z,) and those of the satellite’s centre (0,) will be (Xs,Y,, 2,). The geocentric equatorial coordinate system will have the following notation: ( x , y , z ) . Its origin coincides with that of the main coordinate system (X, Y, 2).

Fig. 1.5. Systems of coordinates of the terrestrial Doppler tracking station ( P ) , the Moon’s centre of mass (00, the ALS centre (OJ, and the distances between them.

34

Lunar Gravimetry

1

The coordinates X p , Y p ,and Z p vary with time t in the following manner:

[ [A =

0

[

0 COSE

si:&]

-sin&

COSE

pp

cos ( p p cos A p

ppcos(ppsinAp p ~ s i n( p p

(1.8.1)

where E is the inclination of the Earth's equatorial plane to the ecliptic; p p , ( p p , and AP are the geocentric coordinates of the station P , A p = (27c/n/T)(t- to), T being the diurnal period of the Earth's rotation and t o being the initial instant at which the point P passes through the plane Oxz. Now, let us find the expression for time changes in the position of the lunar centre of mass. We have already mentioned the complexity of the true orbital motion of the Moon even in the context of the main lunar motion problemthat is, with due account for the perturbing effect of the Sun as the only factor affecting this motion and assuming that all three bodies (Moon, Earth, Sun) are points and that the centre of mass of the Earth-Moon system revolves around the Sun along a Keplerian orbit. We shall resort to the formulae of lunar motion theory developed by Brown (1 919), which represent the ecliptic geocentric spherical coordinates of the lunar centre of mass (A, &, L,) as the following sums:

B,

=

A=

1i bj sin

$j

(1 3.2)

a@

c cj cos

*j

j

Here, Xis the mean longitude of the Moon, u @is the equatorial radius of the Earth ll/i = k l l kzl' k3D k4F are combinations of angles 1, l', D and F. They are dependent on the lunar orbital elements and time t. The coefficients k l , k 2 , k 3 and k4 assume, in various combinations, the values 0, f 1, f 2 , . . .. The subscript j stands for one of the combinations of coefficients k l , k 2 , k3 and k4. The coefficients a j , bj and c j of the trigonometric functions in (1.8.2) are complex series in terms of the manner in which the mean motions of the Sun and the Moon, the semimajor axes of their orbits, and the inclination of the Moon's orbit plane to the ecliptic are correlated. The coefficients uj, bj and c j are expressed in angular units. If we restrict ourselves only to the linear relation of I, I', D and F with time t, we have

+

+

+

1 = 296O6'25.31"

+ 17,179,167.085,94t"

35

The Gravitational Field of the Moon

L‘ = 358’28’33.60” + 1,295,977.415,16tr’

+ 16,029,616.645,69t” F = 11’15’11.92“ + 17,395,266.093,19t”

D

=

350’4423.67”

where t is expressed in Julian years reckoned from the epoch 1900. The harmonics in (1.8.2) representing periodic and secular variations in L,, B, and A arise from perturbing motions and are referred to as inequalities. Using the equations (1.8.2), we can express the rectangular coordinates of the lunar centre of mass as a function of time t in the following manner:

Xu = A cos B, cos L,

=

Y, = A cos B, sin L,

=

Z, = A sin B,

=

a& cos

[cjbj sin $j(t) cos I + Cj

sin $j(t)I

Qj

Cj cj cos $j(O

+ Cj aj sin +hj(t)]

cos [Cj bj sin t,bj(t) sin 1 Cj cj cos $j(t)

acBsin [Ij bj sin IClj(t>] Cj cj cos $j(t)

(1.8.3)

Finally, let us examine the changes, in time t , in the coordinates of the satellite executing an unperturbed motion. The rectangular coordinates of the ALS in the main coordinate system (X, Y, Z ) will be represented as the sum of the corresponding coordinates of the lunar centre of mass (Xu, Y,, Z , ) and the relative coordinates of the ALS (X,,j,,Z,). The system of relative coordinates X,, j , and 2, has its origin 0,at the lunar centre of mass, which is to say that it is selenocentric,its axes being parallel to the corresponding axes of the main system (X, Y, Z). The satellite’s coordinates in the main system are therefore

x,= x,+ x,,

Y, = Y, + y,,

z,

=

z,

+ 5,

(1.8.4)

The coordinates X,, j , and 5, will be expressed in terms of the ecliptic selenocentric spherical coordinates p,, b, and I,:

X, = p , cos b, cos I, J,

=p,

cos b, sin 1,

(1.8.5)

Z, = p , sin b,

Let R be the longitude of the satellite’s ascending node, reckoned from the vernal equinox to the line of intersection of the ALS orbit plane with the ecliptical plane, w be the angular distance from the node to the pericentre, i be the inclination of the ALS orbit plane to that of the ecliptic, a be the semimajor axis, e be the eccentricity of the ALS orbit, and u be the true anomaly. The ecliptic latitude b, and longitude I, are related to R, i, w and u

36

Lunar Gravimetry

by the formulae 1, = R + tan-' [cos i tan (o+ u)] 6, = sin - [sin i sin (w + u)]

'

(1.8.6)

It is known that the radius vector p, is expressed in terms of orbital elements as a(1 - e') ps = 1

(1.8.7)

+ e cos u

Substitution of (1.8.6) and (1.8.7) into equations (1.8.5) gives

1 + 1 +

a(1 - eZ) a 2, = 1 +ecOSu[2sin i sin (o u)

+

x cos [Q

+ tan-'

- - sin i sin (o

Ys =

x -

z, =

cos i tan (o+ u)]

sin [Q

+ tan-'

a(1 - e')

1 + e cos u

u)

cos i tan (o u)]

(1.8.8)

sin i sin (o+ u )

Thus, the selenocentric rectangular coordinates (X,, Y,,27), are expressed in terms of ALS orbital elements. If it is assumed that the satellite moves in the central gravitational field without any perturbations, then it will follow a Keplerian orbit. Only the true anomaly u will vary in time t , the rest of the orbital elements remaining constant. The distance between the Doppler tracking station on the Earth and the monitored ALS will be denoted by

6

=

(l' + q'

+

[')1'2

(1.8.9)

where <,q and [ are projections of this distance on the axes of the main system (X, Y, 2). The expressions for 5, q and [ will be derived from (1.8.1) and (1.8.3) through (1.8.5): &t) =

-pp

2a cos ~p cos - (t - To) T

+ A cos Bg(t) cos Lg(t)

+ p, cos b,(t) cos I&) 2x

q(t)= -ppcoscpsin-(t T

-To)cos~-ppsincpsin~

+ A cos Bu(t)sin Lu(t)+ p, cos b,(t) sin l,(t)

37

The Gravitational Field of the Moon

27T T

l(t) = p p cos cp sin -(t - t,)sin E - p p sin cp cos E

+ A sin B,(t) + p, sin b,(t) Substituting these expressions into the formula (1.8.9), we obtain

d2 = A2 + p:

+ p; + ~AP,(COS B, cos L, cos b, cos 1, + cos B, sin L, cos b, sin I, + sin B, sin b,) - 2App sin cp(cos B, sin L, sin E + sin B, cos - 2pSppsin cp(cos b, sin 1, sin + sin b, cos 2n - 2pp cos cos - (t - Fo)(A cos B, cos L, + ps cos b, cos 1,) T E)

E)

E

~p

- 2 p p cos cp

n sin - (t T

-

Fo)[A(cos B, sin L, cos E

+ sin B, sin + p,(cos b, sin I, cos + sin b, sin E)

E

E)]

Consequently, the distance 6 is a function of time t through the medium of trigonometric functions with arguments (2n/T)(t - to), Bu(t),L&t), b,(t), l,(t), as well as distances A(t) and p,(t). The direction cosines for 6 are

'

v cos (Y,) = -, 6

5 cos ( X , ) = -, 6

cos ( Z , ) = 6

(1.8.10)

What we are interested in is the component of acceleration of the Doppler tracking station with respect to the ALS, projected on the direction 6 or, in other words, the line-of-sight acceleration ga. The Doppler tracking data are used to determine the line-of-sight velocity V,, whereas the acceleration ga is calculated by differenting V,. Apparently, the observed line-of-sight velocity V, and, consequently, line-of-sight acceleration ga comprise components that are due to the above three motions and those due to the perturbation caused by the irregularity of the Moon's figure and internal structure. Consider separately the components of accelerations of the diurnal rotation of the Doppler tracking station along each axis of the main coordinate system ( X , y, Z ) : (gp)x

ry;

2n T

= - - pp cos cp cos -(t - To)

(g) 2

(gP),, =

27T p p cos cp cos E sin -(t - to) T

(1.8.1 1)

38

Lunar Gravimetry

(F) 2

(gP)* =

2a T

p p cos cp sin e sin - (t - to)

For the line-of-sight acceleration, equations (1.8.9H1.8.11) give

cos cp

= ($)lpp

f {p p cos cp - (A cos B, cos L, + p, cos b, cos l,) cos &(Acos B, sin L,

T - sin &(Asin B,

+ p, cos b, sin 1,)

11

+ p, sin b,) sin 2T71 ( t - 6 ) -

(1.8.12)

1.8.1 Orbital Motion of the Moon

Using equations (1.8.3), we shall find the components of acceleration of the lunar centre of mass along the axes of the main coordinate system ( X , Y, 2): ( g & = & cos B, cos L, -&A - BfA

sin B, cos L, - LuA cos B, sin L,

cos B, cos L, - LfA cos B, cos L,

- 2AB,

sin B, cos L,

- 2ALucos B, sin L,

+ 2BuL,A sin B, sin L, (g,), =

(1.8.13)

cos B, sin L, - BuA sin B, sin L,

+ LuA cos B, cos L, - BfA cos B, sin L, -

LfA cos B, sin L, - 2AB, sin B, sin L,

+ 2ALucos B, cos L, - 2BaLaAsin B, cos L, ( g o Z = A sin B, + &A cos Ba + 2ABucos B, - BtA sin B, The line-of-sight acceleration Fa due to orbital motion of the lunar centre of mass is, respectively,

ra = ( g d x j5 + ( g d Y sv + ( g d Z55

( 1.8.14)

Equations (1.8.8) and (1.8.9), hence (1.8.13), include the first and second derivatives of the coordinates A, B, and L,. Expressions for these can be

39

The Gravitational Field of the Moon

found with the aid of (1.8.2):

A = -A2 1

cjt+Jjsin $ j

aQ

j

+ C ajt+Jjcos

Lu = I

( 1.8.15)

$j

j

B,

=

c B $ j cos

$j

j

cj$f cos $ j

+C~

)

~sin3$ j ; ~

j

E,

=

C b j q j cos

$j

-

j

L,

=

C bj$f

(1.8.16)

sin $ j

j

X + C ajqj cos t,hj - 1aj$? sin

$j

i

j

In (1.8.15) and (1.8.16), the coefficients a j , bj and cj are known constants, and qj stands for known functions of time t . They can be found in tables of lunar motion.

1.8.2 Motion of the ALS in an Elliptical Orbit

For the ALS acceleration components we have, similarly to (1.8.13), (g,), = ps cos b, cos 1, - 6,p, sin b, cos 1,

- lips cos b, sin 1,

- bfp,

cos b, cos 1, - ifp, cos b, cos 1, - 2p,& sin b, cos 1,

- 2p,i,

cos b, sin I,

+ 2b,i,p,

(gJY = p, cos b, sin 1, - 6,p,

sin b, sin I,

sin b, sin 1,

+ lip, cos b, cos 1,

(1.8.17)

- b % p , cos b, sin 1, - ifp, cos b, sin 1, - 2psb, sin b, sin 1,

(g,),

+ 2p,i, cos b, cos I, - 2b,i,p, sin b, cos I , = p, sin b, + 6,pS cos b, + 2psb, cos b, - hfp, sin b,

Using (1.8.6) and (1.8.7), which correlate the selenocentric ecliptic coordinates (p,, b,, Is) with the ALS orbital elements, we obtain the desired

40

Lunar Gravirnetry

derivatives from the following equations: Ps

=

a(1 (1

- e’)e

sin u

+ e cos 0)’

d

i is = 1 - sin icos d sin’ (w + u)

b, =

+ +

sin i cos (o u ) d [1 - sin’ i sin’ (o u)] ‘1’

a(1 = (1

s’

1;

=

( 1.8.18)

- e’)

+ e cos u)’

vsin u

+ cos u 1++2ee -cose ucos’ u d’]

cos i sin2 i sin 2(cu u) d’] 1 - sin’ i sin’ (o u ) u + 1 - sin i sin’ (w u )

++

+

( 1.8.19)

b- [I

- sin’

cos’ i sin (o+ u) sin i vcos (0+ u)1 - sin’ i sin’ (w + u ) i sin2 (o+ u)]”Z

The line-of-sight acceleration Tsdue to the unperturbed motion of the ALS in the central gravitational field of the Moon is rs =

(SS),

3t + ( S J y s rl + (SSL s 5

(L8.20)

Time variations in the above line-of-sight acceleration components are shown in Fig. 1.6. The accelerations due to the Earth’s rotation (r,) and orbital motion of the ALS in the central gravitational field of the Moon (r,) are illustrated in Fig. 1.6a. Their amplitudes are, respectively, 3 and 150 Gal. It can be seen that the variation pattern of the line-of-sight accelerations T, is strongly dependent on the eccentricity e and inclination i of the ALS orbit. As regards the acceleration Ta due to the orbital motion of the Moon, its variations are slow and its amplitude amounts to several tens of mGal (Fig. 1.6b). 1.8.3 Perturbed ALS Motion

This motion is caused by the gravitational effect of density irregularities within the Moon and the irregularity of its figure. It is also affected by light pressure, the gravitational perturbation due to the Sun and large planets, the oblateness of the Earth, tidal phenomena on the Moon itself, and other factors. The latter influences are insignificant; they can be calculated

The Gravitational Field of the Moon

r,

41

(Gal)

(b) FigA.6. (a) Time variations in: (1) line-of-sight acceleration due to the Earth‘s rotation ( t ) ; (2.3) line-of-sight acceleration due to the movement of the ALS in the central gravitational field of the Moon with the following orbit parameters: (2) a = 2700 km, i = 0.209 rad, e = 0; ( 3 ) a = 2700 km. i = 0.209 rad, e = 0.327. (b) Time variations in line-of-sight acceleration r( due to orbital motion of the Moon.

theoretically and taken into account. Elimination of the above-described effects of the Earth’s rotation, ALS motion in the central gravitational field, and the orbital motion of the Moon leaves the desired line-of-sight accelerations due to anomalies in the Moon’s internal structure and figure. It is precisely this anomalous field that is of particular interest in gravimetric studies of the Moon. 1.9 Determination of the Gravitational Field from Variations in the Line-ofSight Velocity of the Circumlunar Satellite

Making the ALS perform a new function, namely, that of a test body in the anomalous gravitational field of the Moon, has made it possible to get a

42

Lunar Gravimetry

much better insight into the near-side gravitational field. Muller and Sjogren (1968) were the first to map the near side of the Moon, from the Doppler tracking data involving Lunar Orbiter 5, with line-of-sight acceleration r isolines being shown on the map. It is precisely this map that revealed for the first time the extremely curious lunar formations known as mascons. The line-of-sight velocity of the ALS was determined from departures of the radio signals transmitted from the satellite from a standard frequency (2300 MHz). A 1 Hz change in frequency corresponded to a line-of-sight velocity variation by 65 mm s- l . The frequency variations were determined to within 0.02 Hz, or 1.3 mm s-'. This is where one cannot help admiring the perfection of modern technology. At a distance of 380,000 km, changes in the velocity of an object are measured to within fractions of cm s - l and those in acceleration, to within thousandths of cm s - '. Plotting of the above-mentioned map of line-of-sight accelerations involved averaging over minute-long time intervals. The satellite orbited at an altitude exceeding 100 km. The map is confined to +60° latitude and & 60" longitude. To derive gravitational anomalies or, in other words, accelerations radial to the Moon, one must know, apart from the line-of-sight component, two more spatial acceleration components which are not measured. In the central parts of the Moon, they are negligibly small but become sizeable toward the periphery. This is why line-of-sight accelerations may be identified with gravitational anomalies only in the central region. The above method for direct determination of anomalous line-of-sight accelerations of satellites orbiting at low altitudes above the lunar surface has been instrumental in detailed gravimetric mapping of individual areas on the near side of the Moon. The maps are based on the data provided by Doppler tracking of the command modules of Apollos 14-17 (Muller et al., 1974; Sjogren et al., 1972a,b, 1974a,b) and Apollo 15 and 16 subsatellites (Sjogren et al., 1974~).The pericentres of their orbits were a few tens of km above the lunar surface, that of the Apollo 15 command module being 12 km above the centre of Mare Serenitatis. About ten revolutions of each Apollo command module were available. Small changes in the satellites' orbits provided line-of-sight accelerations within a longitudinally extending strip about 10 to 70 km wide. The sufficiently high measurement accuracy permitted isolines to be drawn on the map at 10 mGal intervals. The maps show that lunar formations of volcanic type occur at line-of-sight acceleration isolines. The pronounced gravitational anomalies coincide with large unfilled craters. Standing prominently out among other maria on the near side is Mare Orientale. Its central portion exhibits a positive anomaly surrounded by a ring of negative anomalies. The peculiarity of Mare Orientale can be explained by the fact that this feature is located in the highland region far from other maria on the

43

The Gravitational Field of the Moon

TABLE 1.7 Line-of-sightaccelerationsrdeterminedat thesamepointsabove the Moon’ssurfacefrom observationsof different Apollos

r (mGal), determined from observation of I (deg)

Apollo 14

Apollo 16

Apollo 17

- 60

0

0

- 43 - 42

30

30 53

+7 12 16 20 24 60

+ + + +

+

90 50 -40 60 80

66 40 - 34 66

107 66

57

Apollo 16 subsatellite

Difference (mGal) 0 0

55

2 24 - 10 -6 -6 -21 9

near side of the Moon. Another possible reason for its uniqueness is the thinner crust in this part of the Moon. Detailed surveying has made it possible to interpret more carefully the gravitational field over individual mascons. Radically new facts have been obtained about these interesting lunar formations. They do not occur at depths of about 50 km in the form of compact masses, as was believed earlier, but are near-surface accumulations of dense material in the shape of disks. Being isostatically uncompensated, excess masses in the order of 800 kg cmP2 (Muller et al., 1974) give rise to stresses in the upper layers of the lunar crust. There was a possibility of controlling the accuracy of the determination of the line-of-sight acceleration anomalies during intersection of the paths of the different Apollos. Table 1.7 summarizes the differences between the accelerations of different Apollo command modules, the first two and the last lines corresponding to Apollos 16 and 17, lines four through eight corresponding to Apollos 14 and 16, and line three, to the command module of Apollo 17 and the subsatellite of Apollo 16. Similar differences are observed on the Earth, during gravimetric measurements at sea. It is, of course, preferable to have maps showing gravitational anomalies rather than line-of-sight accelerations, as they are easier to interpret. As regards the obtained line-of-sight acceleration anomalies, we should like to make the following comments. Firstly, they are merely projections of gravity accelerations at respective points on the line of sight from a terrestrial observer. Secondly, the line-of-sight acceleration anomalies are associated with points at different altitudes corresponding to that of the satellite’s orbit. Thirdly, the actual extreme values of line-of-sight accelerations become 2& 30% lower as a result of the approproximation and weighted averaging operations performed on the observed Doppler velocities (Gottlieb, 1970).

44

Lunar Gravimetry

We have already mentioned the methods for conversion of the observed line-of-sight accelerations r = a V / a x (x being in the Moon-Earth direction) into the simple layer mass distribution on a surface approximating that of the Moon. Given the simple layer distribution, one can determine the gravitational potential Vat any point in an external space, and hence any one of its derivatives (Sagitov, 1979). The r field transformation with the aid of a simple layer offers certain advantages, including the fact that the nonsphericity of the Moon and the random positions of the points at which the derivatives of r are measured are taken into account. This is especially important in view of the fact that, for example, the ALS whose observations have allowed Muller and Sjogren (1968) to produce the above-discussed map of line-of-sight accelerations on the near side of the Moon was about 100 km distant from the lunar surface in the equatorial zone, this distance increasing to 300 km at +60” latitude. Given a set of line-of-sight accelerations r for all points of a closed surface, one can, in principle, determine the gravitational potential and any one of its derivatives by solving the boundary-value problem of potential theory. This has already been discussed in the general case. Consider now two analytical solutions of the boundary-value problem, based on the assumption that the boundary surface is a sphere and that in (1.4.2) rn = 1, and I = n = 0. The derivations are given in accordance with Moritz (1969). The same problem was solved independently by Brovar (1970). Write the Poisson integral providing a solution to the Dirichlet problem involving a sphere, applicable to the line-of-sight acceleration: (1.9.1) where x,y and z are the coordinates of the point at which the values of r have been measured, and 5, q and [ are the current coordinates on the surface S which is assumed to be spherical,

+ y2 + z y R = (t2+ 1’ + [2)”2 p

= (x2

r

=

[ ( x - O2

+ ( y - q)2 + ( z -

Assume that the function

[)2]1’2

= (R2

+ p 2 - 2Rp cos +)I/*

r satisfies the condition (1.9.2)

45

The Gravitational Field of the Moon

which means that this function comprises only alternating components. Having multiplied both sides of (1.9.2) by (4nRp)-', use the products to derive the following expression from (1.9.1): (1.9.3) Multiply both sides of the latter equation by dx and integrate them for x going from co to x. The result will be an expression for the potential V at the point (x, y, z):

s

o

0

By virtue of the potential's regularity,

u r n , Y , z)

=0

Given

where K is the kernel of transformation, write as follows the equation in which the gravitational potential V is expressed in terms of predetermined values of r on the surface S:

S

It now remains only to determine the analytical form of the function K . Integration of (1.9.4) gives, for points outside and on the sphere S of radius R

for x 2 0 and x2 + yz + z 2 2 R 2 ;

46

Lunar Gravirnetry

-=

for x 0 and x2 + y 2 + z2 2 R 2 . The validity of (1.9.6) and (1.9.7) can be easily checked by simple differentiation. Thus, integral (1.9.5) permits the potential V ( x , y, z ) at points outside the sphere S to be calculated from a given distribution of line-of-sight accelerations T(S) on the surface of the sphere S. Let us now turn to an alternative method for determining the lunar gravitational potential. It is based on formulae by which the coefficients of line-of-sight acceleration field and lunar gravitational potential expansion in spherical functions are related. With due respect to the elegance of the formulae derived by Moritz (1969), they involve a coordinate system in which the x and z axes are interchanged. Since no one has ever defined harmonic coefficients in this coordinate system, these formulae are impractical. On the other hand, the formulae developed by Brovar (1970) are in the usual coordinate system, where the z axis is directed along the Moon's rotation axis. However, they are intended for non-normalized spherical functions. Normalization will yield equations for normalized harmonic coefficients. Let us use expansions of the lunar gravitational potential and line-of-sight acceleration in the form:

x (Cnmcos mA

+ Sn, sin mA)Pnm(sincp)

(1.9.8)

+

x (Qn+l,mcosml Ln+l,,sinm~)Pn+l,,(sincp)

Given the coefficients Cnmand Snm,one can find formulae - [(Zn + l)n(n + Q n + l , O = -cn1 2(2n 3)

Qnm

+

Qn+l,l

En, from the

I")

= {Go- Cn2[2(n n(n - l) 2 ) +

+

+ l)(n + 2)(n + 3) x [ 2(2n + 3) n(n 1)(2n + 1 ) = [ 4(2n + 3) (n

-

Ln+l,l

and

Sn2

1

1

(1.9.9)

47

The GravitationalField of the Moon

If the line-of-sight acceleration field expansion coefficients are known, it is possible to derive the selenopotential expansion coefficients from the formulae: -

2

c,, = ___ 1 + bmO

n-m-Zj>O

C

j=o

Qn+l,m+l+~j

(1.9.10)

(2n

+ 3)(n + rn)! ( n - rn)!

where bmO= 1 at m = 0 and bmO= 0 at rn # 0. The practical realization of the above two methods for transforming the observed field of line-of-sight accelerations r into the gravitational potential V encounters difficulties due to the absence of r values over the entire closed surface S , to say nothing of the fact that the surface S is far from spherical because of the ellipticity of the ALS orbits. The impossibility of monitoring the line-of-sight accelerations when the satellite is above the far side of the Moon adversely affects the derivation of its general gravitational field. With this in view, Brovar and Ganifaeva (1974, 1975) considered the following problem. A closed surface S enveloping the entire Moon and satisfying Liapunov’s condition is given. Also given are line-of-sight accelerations T(x, y, z ) at points on that part of the surface S which faces the Earth and will be denoted by Sl-that is, the following boundary

48

Lunar Gravimetry

condition is set:

w,Y , 4 Is,

=f1k

(1.9.11)

Y , 2)

It is assumed that the rest of the surface ( S , ) is characterized by some other given parameter of the gravitational field. These may be potential increments along the surface Sz or gravitational potential derivatives tangential to it. In view of the future ALS projects involving auxiliary relay satellites placed in high orbits in addition to the main satellite, it may be expected that lunar gravitational potential derivatives along two orthogonal directions p and q tangential to the surface S2 will be determined. The boundary conditions associated with them can be written as

Proceeding from these boundary conditions, one can determine the potential V(x, y, z) at points on the surface Sz to within constant C

V(x, y, 4 Is, =f4(x, y, z)

+c

The above-described exterior mixed boundary-value problem has a unique solution at boundary conditions (1.9.11) and (1.9.12), provided the Moon's mass is known, the direction of x nowhere coincides with that of the tangent to S , , and the boundary conditions at the line joining the surfaces S1 and Sz coincide. The line-of-sight accelerations T(x, y, z) at points of .a selected sphere So are determined, from given line-of-sight accelerations Ti(x, y, z) at N points (xi,yi,zi)of the external space above the surface S1 and from the tangent derivatives d V/dp and d V/dq on the surface S , , by minimizing the functional

FCW, Y , 41 = [[[fl Sl

I'

Y , z)K(x1 - x,Yl - Y , 21 - z)dSo

SO

where K(xl - x, yl - y, z1 - z) is the kernel of Poisson's integral (1.9.1), while K 1 and K 2 are the kernels in the formulae of Moritz and Brovar, which

The Gravitational Field of the Moon

49

take the form of (1.9.6). Minimization (1.9.13) of the functional is performed on a set where the desired function of r belongs to Hilbert’s space L2 and is bounded in the norm. In addition, r satisfies the bounding condition of the type of orthogonality with respect to the known sets of spherical harmonic of Y, on S o . Mathematically, these conditions for the desired function can be written as

The constant C depends on known lunar gravitational potential expansion coefficients. Narrowing the class of functions to be found enhances the stability of the sought-after solution. The mixed boundary-value problem under consideration belongs to ill-posed ones. Minimization (1.9.13) of the functional F(T) is performed using the penalty function method in combination with the method of conditional gradients.

1.10 Gravity Measurement Concepts and Requirements of Lunar Gravimeters

Lunar gravity, just like terrestrial gravity, manifests itself in many phenomena. The free fall acceleration, pendular oscillation frequency, deformation of springs and torsion wires with test bodies attached to them, and the like depend on the magnitude of gravity. Quantitative measurements of these manifestations of gravity may yield its actual value. The instruments for gravity measurements can be divided into two major groups: (1) static and (2) dynamic (Veselov and Sagitov, 1959). The instruments based on the dynamic principle are used to measure time-a quantity in one or another way associated with gravity (time of free fall of a test body from a particular height, period of oscillations, etc.). Let us begin our description with the static methods of gravity measurement.

1.1 0.1 Static Methods

These methods are used to determine the magnitude of a balancing force equivalent to gravity. The function of such a reference force may be performed by the elastic force of a deformed spring, a torsion wire, or a compressed gas, the force of a magnetic or electrostatic field, the centrifugal force, and so on. This reference force must be invariant with time and unaffected by temperature, pressure, and other factors. In studying the

50

Lunar Gravirnetry

gravitational field of the Earth, the most commonly used instruments are gravimeters in which the reference force is produced by elastic elements in the form of various springs and torsion wires. In terms of design, such gravimeters can be divided into two groups differing in the motion of the test body under the effect of gravity, which may be (a) linear or (b) rotary. They are shown schematically in Figs 1.7a and b, respectively. The balance equations for these gravimeters can be written as mg

+ W(x) = 0,

mglk(cp)

+ %R(cp)

=0

(1.10.1)

Here, g is gravity, 1 is the distance between the centre of gravity of the test body of mass m and the axis of rotation, x is the displacement, cp is the angle of rotation of the test body, W(x) is the reference force, W(cp) is the reference force moment, and mglk(cp) is the gravity moment. k(cp), W(cp), and %R(x) are nonlinear functions of cp and x, whose concrete analytical form depends on the gravimeter design. Equations (1.10.1) can be used to derive expressions for the sensitivity of gravimeters; that is, a relationship between the linear or angular motion and gravity acceleration:

Let us now carry out some estimations of the gravimeter systems in which the balancing force W(x) and force moment W(cp) are linear with respect to x and cp,

w-4= t1(x - xo),

W(cp) = %(cp

- cpo)

( 1.10.3)

where t1 and z2 are constants of the elastic elements (“rigidity”), xo and cpo are the initial deviations corresponding to the deformation of the elastic elements at g = 0. In the case of the design shown in Fig. 1.7b, k(q) = cos cp. Substitution of (1.10.3) for (1.10.2) gives the following simple expressions for the gravimeter sensitivity: dx_- -_ m _ dg

tl’

dcpml cos cp dg mglsincp - z2

--

(1.10.4)

The first equation indicates that the sensitivity depends on rigidity t and mass m. As can be inferred from the second equation, the sensitivity depends not merely on the mass m but on its product by arm 1. The sensitivity of rotary gravimeters (see Fig. 7b) can be enhanced, in accordance with (1.10.4), by ensuring that the gravity moment mgl is balanced out by the elastic force of the torsion wire twisted once or several times so that cp x 2pn. To estimate the sensitivity of gravimeters, it is useful to resort to relationships between sensitivity and the period of the oscillation of the test

51

The Gravitational Field of the Moon

(a)

(b)

Fig.l.7. Gravimeterswith (a) linearand (b) angularmotionofthetest bodyundertheeffectofgravity.S. spring; 0, axis of the horizontal torsion wire.

body in the gravimeter:

T 2 rnlk(0) -

1 T2

14n2 In other words, the sensitivity of static gravimeters is proportional to the square of the period T of free oscillation of the test body.

1.1 0.2 Dynamic Method

The classical dynamic instrument for measuring gravity g is the pendulum instrument. Its basic component is the pendulum undergoing free oscillation. It is known that g=-

471’1

MlT

where I is the moment of inertia of the pendulum with respect to the axis of oscillation, while M , I and T stand for the mass, reduced length, and free oscillation period, respectively, of the pendulum. Practical uses of pendulum instruments include relative measurements of gravity-that is, determination

52

Lunar Gravimetry

of the gravity difference between the measurement points and the datum point at which gravity go is assumed to be known. Then, the problem of measuring gravity at a point is reduced to determining only one quantity, namely, the change in the period of the pendulum’s free oscillation with respect to the datum point. It is assumed that the mass M , moment of inertia I , and reduced length 1 of the pendulum remain invariant. If the free oscillation periods at the measurement and datum points are T and To, respectively, then gravity g at the measurement point is 2

9 = go(;)

Pendulum measurements of gravity involve various correction factors introduced into the measured periods and taking into account the damping of free oscillation, differences in oscillation amplitudes, time variations of the pendulum temperature and its gradient, air density, sympathetic oscillation of the tripod, and so on. Should the need arise to measure lunar gravity to within f 1 mGal, the period of a quarter-second pendulum would have to be determined to within + 8 x l o w ’ s (Korzhev, 1979). In recent years, so-called vibrating-string gravimeters began to be used. Their operating principle is very simple. The upper end of an elastic string is secured to a frame, while suspended from the lower end is a test body. The gravity-dependent quantity is the frequencyf of natural transverse vibrations of the string, by virtue of its being related to tension under the effect of gravity by the relation ( 1.10.5)

where M and m are the masses of the test body and string, respectively, and L is the length of the string. The idea of using a vibrating string loaded with a test body for gravity measurements was proposed as far back as 1935 (Melikyan, 1938), yet the first gravimeter built on this principle was described only in 1949 (Gilbert, 1949). Subsequently, various designs of such gravimeters were proposed in the Soviet Union and abroad. They have been used to measure gravity from sea-going vessels, including submarines, on the bottom of shallow water basins, and in drilling wells. Vibrating-string gravimeters then began to be employed as accelerometers to measure acceleration in inertial navigation systems. Dynamic gravimeters also include instruments based on a balanced gyro. The amount of precession of such a gyro, which depends on the moment of gravity, is also a measure of gravity. Measurement of the absolute gravity on the Earth is one of the most difficult tasks of modern science: its determination on the Moon is even more

The Gravitational Field of the Moon

53

complicated. The absolute gravity determination provides a vivid example of the difficulties involved in absolute measurements of even a relatively simple physical quantity having length and time as its only dimensions. The most accurate and promising technique of determining absolute gravity is currently considered to be the free fall method. Gravity g is determined from measurement of the time it takes a test body to fall freely within an exactly known range of heights. On the Earth, an accuracy of several pGal has been achieved. No measurements of this kind have been carried out on the Moon, but lunar conditions are highly favourable for absolute gravity determinations, primarily because of the natural high vacuum. Also, the lower magnitude of lunar gravity, as compared to terrestrial gravity, results in a slower fall of a test body, making measurements easier. Of course, metrological measurements in outer space without direct participation of man are an extremely difficult task. The gravimeters intended for lunar gravity measurements must meet a number of special requirements: (1) First, the wide difference between the terrestrial and lunar gravitiesas great as 820Gal-must be compensated. The compensation may, in principle, be attained in a number of ways, depending on the gravimeter design (by tensioning a special compensating spring, by varying the voltage applied to a compensating capacitor, by removing some of the mass from the test body, by changing the tilt of the instrument, etc.). (2) The measuring system of the automatic lunar gravimeter must provide for digital data output as the most convenient and noise-immune form for transmission to Earth. (3) The lunar gravimeter must have an adequate thermostatic temperature control to maintain the temperature inside the instrument stable within 0.0014.01"C.Under lunar conditions, such temperature control is extremely difficult in view of variations in the external temperature from -100°C to + 120°C. (4) The instruments designed for unattended operation must have a sufficiently sophisticated automatic remote control system. Provision must be made for automatic levelling of the gravimeter, de-arresting of its sensitive system, activation of the measuring system, remote measurement of gravity, processing of the measurement results in a code suitable for transmission to the Earth, and transmission of the code. Then, the instrument must be arrested and ready for transportation to a new site. The accuracy with which gravimeters must be levelled is confined to a few minutes of arc. This can be done by positive shifting of the instrument into the necessary position with the aid of the sensor of the vertical designed on the free pendulum or level principle.

54

Lunar Gravimetry

The data recording and transmission procedures can be simplified if the measurement is carried out by an astronaut. Lunar gravimeters must withstand impact loads and considerable accelerations due to vibration of the running booster engine. ( 5 ) Another difficult task is calibration of lunar gravimeters or, in other words, matching the gravity values exactly with the gravimeter readings. Under the natural conditions of the Earth, gravimeters can be calibrated only within 5.3 Gal (difference between the gravity on the pole and the equator) near the mean value of 980Gal. Use of a gravimeter calibrated in this fashion on the Moon calls for reliable extrapolation into the range of g x 162 Gal. Some gravimeters can be calibrated additionally by the reversal method which will be considered in what follows in the context of a particular vibrating-string gravimeter design. (6) Lunar gravimeters are limited in mass, size, and power consumption. Overall analysis of various gravimeter designs does not permit one to select the optimal design for lunar gravity measurements. The selection of a gravimeter depends on, among other things, the tasks to be accomplished by gravimetric surveying-that is, on whether absolute or relative gravity values are the target, whether the scope of measurement is global or confined to detailing of a gravitational field within a limited area, and whether we are interested in the spatial distribution of the gravitational field or in its time variations at a certain point.

1.ll Direct Gravity Measurements on the Moon’s Surface

No matter how accurate, satellite methods cannot replace direct gravity measurements on the Moon with the aid of pendulum instruments and gravimeters. Satellite observations permit defining the anomalous portion of the gravitational field, averaged for certain areas. The linear dimensions of these areas amount to hundreds or, in the best case, tens of kilometres. It should be remembered that an ALS flying over the Moon’s surface covers about two kilometres per second. The acceleration acting upon a test body (in this case, the ALS) can be measured more accurately if it persists for a sufficiently long period of time, which is why the slower movements of the ALS and, of course, at low altitudes are preferable. The finer details of the gravitational field, associated with the density heterogeneities of the upper layers of the Moon and its surface features, can be detected using gravimetric instrumentation positioned directly on the lunar surface. So far, only four direct gravity measurements have been performed on the Moon’s surface, involving the Apollo 11, 12, 14 and 17 spacecraft (Nance, 1969, 1971; Talwani et al., 1973; Talwani and Kahle, 1976). According to

55

The Gravitational Field of the Moon

Nance (1969), the accuracy of gravity determination from Apollo 11 was & 13 mGal, which is about the same as for Apollos 12 and 14 where similar instruments were used. The instruments were in fact updated threecomponent accelerometers known as PIPA-pulsed integrating pendulous accelerometers, previously employed for gravity determinations at sea and in air from moving craft (Bowin et al., 1969). The accelerometers were mounted on board the lunar module which performed soft landing on the Moon's surface. If x, j ; and i' are the accelerometer readings in three orthogonal directions, then gravity g = (9+ j j z + zz)l'z. The absolute gravity measurements during landing of Apollo 17 were carried out by the astronauts using the traverse gravimeter TG developed especially for lunar gravity measurements at the Massachusetts Institute of Technology. Talwani and Kahle (1976) estimated the accuracy of absolute gravity determination to be k 5 mGal. This undertaking may be considered as the first sufficiently accurate measurement of the difference between the terrestrial and lunar gravities. Table 1.8 summarizes the results of direct lunar gravity measurements with the aid of the above instruments. In addition to the observed gravity values gobs, the table also gives the corresponding normal gravity values y. The latter were calculated assuming that the Moon is a uniform sphere. The Moon's rotation was ignored, and the selenocentric constant of gravitation was assumed equal to GM,

= 4902.71

km3 sec-2

The first gravimetric profile about 20 km long was obtained on the Moon with the aid of the above-mentioned TG gravimeter mounted on the lunar roving vehicle. Designed similarly to the TG is the vibrating-string gravimeter in which gravity is determined by measuring the changes in natural frequency of transverse vibrations of the string under the effect of gravity. The basic sensing element of the gravimeter is the vibrating-string accelerometer (VSA). It comprises (Fig. 1.8) two test bodies suspended from elastic strings TABLE 1.8 The results of direct gravity measurements on the Moon

Spacecraft

cp

I

Elevations of landing sites under sphere with R = 1736 km (km)

11

0'40 N 3"12'S 3"W s 20"13' N

23"29' E 23"24 W 17"28 W 30"42' E

-0.53 0 0.39 1.19

Landing site coordinates

Apollo Apollo Apollo Apollo

12 14 17

Observed gravity value, G (mGal) 162852 13 162674 162653 162695 & 5

56

Lunar Gravimetry

Fig. 1.8. Sensitive element of a vibrating-string accelerometer. 1, test bodies; 2, strings; 3, spring; 4 solenoids.

made of beryllium bronze. The test bodies are interconnected by a soft spring and linked with the housing via braces eliminating transverse displacements of the test bodies. The strings are arranged between the poles of permanent magnets, and, when an a.c. voltage is applied to the strings, they start vibrating transversely. As has already been mentioned, frequency is dependent on the gravity acceleration g acting upon the test bodies with mass M and on the string tension F o due to the intermediate spring. In the case of a vibrating-string accelerometer with a tensioning spring, the frequency f of transverse vibration is given by the formulaf= $[(Fo + M g ) / L m ] ” 2 , rather than (1.10.5).Because of technical difficulties, it is impossible to make L, M and rn exactly equal in the upper and lower accelerometers, which is why the difference between their frequencies (fi -fi) is also dependent on the even power exponents of g : F

=fi

-fz

= ko

+ k l g + kzg2 +

1 . .

(1.11.1)

In general, F is a nonlinear function of g . If L, M and m had been equal, all the terms with even power exponents of g would have been equal too. The feequency F was measured at each measurement point, and the gravity difference at these points was determined from the difference between the corresponding values of F. It is assumed that the coefficients ko, k l and k z in

57

The Gravitational Field of the Moon

(1.1 1.1) are known. The terms nonlinear with respect to g are necessary only in determining wide gravity differences, such as in the case of terrestrial and lunar gravities. The gravimeters were designed to allow for periodic checking of the coefficient ko on the Moon, To this end, the sensitive system of the gravimeter may be turned through 180". Equation (1.11.1) holds for the reversed system with substitution of -g for g.

F,

=

ko - klg

+ kzg2

(1.11.2)

Measurement of F and F , at the same point permits the coefficient ko to be determined using (1.1 1.1) and (1.1 1.2). Such measurements were performed at every point along the path of the lunar roving vehicle. This served as an additional check of the zero point drift. The frequency was measured in the gravimeter automatically by way of comparison with the reference frequency of 100 kHz of the quartz oscillator mounted inside the gravimeter. The gravity measurement results were digitized and transmitted to the astronauts in the form of a pulsed code. The vibrating-string accelerometer was placed in a gimbal mount which allowed the accelerometer to rotate within 30". The rotation was imparted by a step motor associated with the sensor of the vertical. The sensor was essentially a pendulum with two degrees of freedom. It produced a continuous signal indicative of position with respect to the vertical. The faster levelling was performed up to gravimeter tilts equal to 32', whereas the slower levelling was carried out within k 32'. A step of the motor corresponded to a + 1 ' tilt of the instrument. The overall levelling accuracy is estimated to be f7'. The instrument was provided with a twostep thermostat rated at 50°C. The outer thermostat module operated on an on/off principle, while the inner one worked continuously. The temperature stability inside the instrument was maintained to within +0.005"C. Additional thermal insulation was provided (multilayered coatings, gold-plated surface, etc.). The total capacity of the battery supplying power to the entire gravimeter was 300 Wh over a period of fifteen days. Special shock absorbers were used to minimize the effect of vibration, impacts and g-loads both during flight toward the Moon and at landing. The gravity measurement range was 170 Gal. The overall dimensions of the gravimeter were 48 cm x 26 cm x 23 cm. It weighed a total of about 15 kg. Figure 1.9 shows the profile of "free-air" gravity anomalies in the Taurus-Littrow Valley near the south-eastern edge of Mare Imbrium, obtained by with the aid of the T G gravimeter (Talwani and Kahle, 1976). The measurement accuracy is k 2 mGal. Of particular interest is the possibility of using self-contained capsules with gravimetric instrumentation, hopping (flying) from one lunar site to another. Relative gravity measurements will be taken at each new site with the horizontal and vertical coordinates of the capsule being determined with a

+

58

Lunar Gravimetry

LM

Lo ,~ I

-6

I

, I

-4

1. Y

, I

LM Y

0

-2 0 Distance (km)

2

Fig. 1.9. Profile of gravity anomalies across the Taurus-Littrow Valley, derived with the aid of the TG gravimeter. 1, gravities measured with respect to the landing site of the LRV; 2, free-air anomalies.

high degree of accuracy if accelerations are measured in transit with the aid of a three-component accelerometer. This approach to investigating the gravitational field seems to be energetically more efficient than the use of lunar roving vehicles. Let us now estimate the possible accuracy of determining the coordinates (x, y, z) of the observation point with respect to the initial point. If use is made of accelerometers whose acceleration , accuracy of determinmeasuring accuracy is E . ~= ci; = E! = 0.01 cm s - ~ the ing the increments Ax, Ay and Az of the initial coordinates is

Let the flight over a distance of 5 km take 10 s. This means that, while measuring acceleration with the above accuracy, the acelerometer will permit the coordinates of the new point to be determined, relative to.the initial ones, with an accuracy of E, = cy = E, = f0.5 cm. This chapter has been concerned with gravity measurements. Of greater interest are measurements of gravity gradients on the lunar surface; however, none have been carried out so far.

59

The Gravitational Field of the Moon

1.12 Studying Second Derivatives of the Lunar Gravity Potential

The main advantage of the method proposed by Muller and Sjogren is that it provides information about the gravitational field “free of charge” (as Forward put it) because this method does not require any special instrumentation using, instead, measurements of the carrier frequency of the telemetering transmitter. However, the method has a drawback-measurements are possible only on the near side of the Moon. Information from the far side can be transmitted only via a special relay satellite placed in a high-altitude circumlunar orbit. Yet, even when the experiment is thus made much more complicated and expensive, about 40% of the far-side trajectory will be beyond the tracing zone. Therefore, global mapping of line-of-sight accelerations will take several months with the terrestrial stations operating full time. Hence, it would be of particular interest to carry out independent measurements of the gravitational field with the aid of gravimetric instruments on board ALS. The results of such measurements, as opposed to Doppler tracking data, can be stored and transmitted to the Earth at any time convenient for communication. By virtue of the equivalence principle, a freely flying spacecraft cannot be used to measure the gravitational intensity-that is, the vector g = V V. The physically measurable quantity in this case may only be the difference between the accelerations of test bodies separated in space, which is determined, by the second and higher spatial derivatives of the gravitational potential V(r). This is why independent measurements of the gravitational fields of planets, their satellites, and asteroides from spacecraft are possible only using gradiometers. The second derivatives of the gravitational potential V make up a symmetrical tensor of rank 2, which takes the following form in the local rectangular basis:

Vik

=

Vxx

Vxy

Vxz

Vyx

Vyy

Vyz

Vzx

VZY

Vzz

The diagonal components are known as in-line gradients, and the nondiagonal ones are referred to as cross-gradients. The spur of the tensor Vik equals zero, according to the Laplace equation

therefore, this tensor is unambiguously determined by five independent

60

Lunar Gravimetry

tY

Fig. 1.10. Force moment acting upon a dumbbell in a nonuniform gravitational field. x, y, z, local rectangular base.

quantities, namely, three cross-gradients V,,, V,,, V,,, and any two in-line gradients. All existing gradiometers and those currently under development can be classed as being in one of two groups. The instruments belonging to the first group measure the force moment M acting upon an extended body in a nonuniform gravitational field. For example, if two point bodies with mass m are interlinked by a weightless rod whose length is 2R (Fig. l.lO), such a dumbbell is acted upon by,a force moment rotating it about the axis z : M,(cp)

= 2mR2( VAsin

2cp

+ V,, cos 2cp)

(1.12.1)

In Eotvos’ gradiometer, the dumbbell is suspended from a thin torsion wire, and the gravitational moment is measured by the angle of its twisting. Such a gradient measurement method may be called static. Another modification of the static method is used in the gradiometer developed at the Charles Stark Draper Laboratory (CSDL) (Fig. 1.1 1). In this gradiometer, the dumbbell is enclosed in a spherical envelope immersed into a special liquid where it has zero buoyancy. Such a suspension provides the dumbbell with all six degrees of freedom. Highly sensitive transducers of linear and angular displacements control electromagnetic servomotors maintaining the orientation of the dumbbell along the x axis. The outputs of the gradiometer is compensating force moments with respect to the y and z axes: M , = VxyA J ,

M y = - V,, AJ

where AJ = J,, - J,, N J,, is the difference between the principal moments of inertia of the sensor. Under laboratory conditions, the sensitivity of the gradiometer has been

61

The Gravitational Field of the Moon

',

I

a- , '

'\

m

.I

Fig. 1.11. CSDL gradiometer

brought to tenths of an eotvos at an averaging time of 10 s (Ames et al., 1977). Development of a low-noise liquid suspension has turned out to be a technically difficult task; in particular, thermostatic control to within 10-6"F was necessary. In the existing form, the three-axis version of the CSDL gradiometer is rather cumbersome (100 dm3, 100 kg) and consumes about 20 W. It should be pointed out, however, that the instrument is intended for terrestrial and marine measurements. There is every possibility that in the satellite modification some of the technical problems will be obviated. According to Heller (1979), the most promising application would be the use of the sensing element of the CSDL gradiometer as a test body in a drift-free satellite. Under conditions of complete weightlessness, the liquid suspension would no longer be needed and, theoretically, the intrinsic noise could be reduced by several orders of magnitude. There is no other information concerning the development of a satellite version of the CSDL gradiometer. The static method is not the only way to measure gravity gradients. Indeed, let us consider the equation describing the motion of a dumbbell suspended from a torsion wire: J$

+ Hqj + D(cp - cpo) = J ( V Asin 249 + V,, cos 249)

'

(1.12.2)

Here, cp is the angular coordinate of the dumbbell in the xy plane, J = 2mRZ is the moment of inertia, H is the coefficient of viscous friction, D is the torsional rigidity of the wire, cpo is the angular coordinate of the dumbbell at which the moment of elastic forces is nil. Since the second derivatives of the

62

Lunar Gravimetry

potential of real gravitational fields are small, the deflections of cp from the balanced cpo, caused by them, may also be considered small:

*=

cp - cpo << 1

Expand sin 243 and cos 243 on the right-hand side of (1.12.1) into a series in terms of cp near cpo; retaining the first two terms of this series gives: sin 243

2:

sin 243,

+ 21,b cos 2cp0

cos 2cp N _ cos 2cp0 - 21) sin 2q0 Then, (1.12.1) takes the form:

II;+ 26$

+ (a;- 2r2)$ = rl

(1.12.3)

where the following notation is introduced: 6

H/2J;

O;

= D/J

rl = VA sin 2cp0 + V,, cos 2q0 r2= V , cos 2q0 - V,, sin 243,

(1.12.4)

As can be seen from (1.12.3), a nonuniform gravitational field not only shifts the balanced position of the torsion balance (term rl) but also introduces an additional gravitational “rigidity” so altering the frequency o of free torsional oscillations:

This effect can also be used in gradient measurements in view of the fact that modern instruments permit frequency to be measured with an extremely high degree of accuracy. Such a gradient measurement technique (which may be called dynamic) has been used with a great deal of success in some laboratory experiments aimed at determining the absolute value of the constant of gravitation (Sagitov et al., 1979). In the mid nineteen sixties, yet another method was proposed to measure gradient with the aid of a torsion balance, which may be termed the “modulation method”. Its basic idea is that a torsion balance, together with a transducer of angular displacements, are rotated about the axis of the torsion wire (axis z in Fig. 1.10) at a constant angular speed p . In this case, cpo = p t , and the gravity moment is modulated, according to (1.12.1), in amplitude with frequency 2p: M,(t) = J ( VA sin 2pt

+ Vxycos 2pt)

Thus, in a modulation gradiometer, gravity gradients excite forced torsional

63

The Gravitational Field of the Moon

Fig. 1.12. RGG gradiometer.

oscillations of the dumbbell near the balanced position. The amplitude of these oscillations will be maximal if the rotational speed is selected such that the .following resonance condition is met: 2p = 0 0

The modulation method was implemented for the first time in the rotating gravity gradiometer (RGG) developed by the Hughes Research Laboratbries (Bell et al., 1970). The sensing element of the RGG consists of two identical dumbbells crossed at a right angle and interlinked by a rigid torsion spring (Fig. 1.12). The natural frequency of the relative torsional oscillations of the dumbbells is 35 Hz. In the course of measurement, the sensing element rotates about its axis at 17.5 rps. Each dumbbell is acted upon by a periodic force moment of the same magnitude but opposite sign:

The result is forced vibrations of the dumbbells in phase opposition at the resonant frequency. These vibrations are sensed by a piezoelectric transducer converting deformations of the torsion spring into an electric signal. In addition, a special optical transducer senses the instantaneous angular position of the rotating sensing element with respect to a fixed frame of reference and produces reference signals proportional to sin 2pt and cos 2pt. These signals enable phase detection of the dumbbell oscillations and determination of the values of Vxyand Va in the selected fixed coordinate system.

64

Lunar Gravimetry

The performance of the instrument under laboratory conditions was demonstrated in measurements of the second derivatives of the field of test bodies moving near the RGG (Bell, 1970). The sensitivity of the RGG is 1 E at an averaging time of 10 s, its drift being 10 E per day (Forward, 1979). Instruments of the second group are used to measure the difference between the readings of spatially separated linear accelerometers. Figure 1.13 shows how two accelerometers oriented differently can be used to measure, within a given local basis x, y and z: (a) in-line gradients Vxx;(b) crossgradients V,..; (c) the difference between in-line gradients VA = %V,, - Vxx); and (d) linear combinations of the following type: Va sin 2cp + Vxycos 2cp. Obviously (b) and (c) are particular cases of (d) at cp = 0 and cp = n/4, respectively. Practical designs of gradiometers of this type call for an extremely high sensitivity of the accelerometers, especially when the instrument is compact. For example, if the distance between the accelerometers is about 10 cm, for gradients to be measured with an accuracy of 1 E their sensitivity must be better than 0.01 pGal or lo-” g. Bell Aerospace is currently working on a satellite gradiometer based on miniature electrostatic accelerometers (MESA) having a sensitivity of lo-’’ g. Four such accelerometers are evenly spaced over the perimeter of a disk with a radius of 10 cm. Just like Forward’s rotating gravity gradiometer, this one is intended for measurements by the modulation method. To this end, the disk rotates, during measurement, about its z axis at a constant speed p so that the output of the gradiometer is a periodic signal with a frequency of 2p: S(t)

-

VA sin 2pt

+ V,,

cos 2pt

Synchronous detection of the output signal permits the determination Vxy and VA. The laboratory version of the Bell Aerospace gradiometer has a volume of 100 dm3, weighs 100 kg, and its intrinsic noise level is in the neighbourhood of 3 E Hz-’I2. In the satellite version, the modulation is going to be achieved by spinning the satellite itself. Elimination of a noisy mechanical drive, the absence of microseismic accelerations on board the satellite, increasing the base from R = 10 cm to R = 30 cm, and thermostatic control to within 0.01”C will permit, in the designers’ opinion, a sensitivity of about 0.03-0.1 E Hz-”’ to be attained. The French national aerospace agency (Office national d’ktudes et de recherches aerospatiales) plans to develop a similar satellite gradiometer. It will be based on SuperCACTUS accelerometers with a sensitivity of lo-” g (Bernard et al., 1980), which are an improved version of the CACTUS accelerometer (CACTUS is the French acronym standing for “capteur &accelerations capacitif a trois axes ultrasensitif”-ultrasensitive triaxial ’

ly (a Fig. 1.13. Measurement of various gravity gradients and their combinations with the aid of uniaxial linear accelerometers. Vectors ei indicate the orientation of their Sensitivity axes.

66

Lunar Gravimetry

capacitive acceleration transducer). The latter was successfully tested on g) is not sufficient to measure gravity board spacecraft but its sensitivity ( gradients with the desired accuracy. Particularly promising is the use of cryogenic accelerometers in gradiometry, in which the displacements of test bodies are measured by extrasensitive SQUID-based magnetometers. Such accelerometers were created, in particular, at Stanford University to detect ultralow-amplitude vibrations (10- l 6 cm) of a solid-state gravitational wave antenna (Paik, 1976). Currently under development at Maryland University is a triaxial gradiometer with a E Hz-”’, intended for measurement of in-line sensitivity of about 2 x gradients as shown in Fig. 1.13a and cross-gradients as shown in Fig. 1.13b (Paik, 1980). Paik sees the following advantages in a cryogenic gradiometer. Cooling test bodies to the temperature of liquid helium (4.2 K) minimizes the level of Brownian noise. At helium temperatures, the thermal coefficients of linear expansion and elastic properties of solids are reduced considerably; besides, liquid helium at the phase-transition temperature is a highly effective passive thermostat. SQUID-based magnetometers are characterized by an extremely low drift level, their intrinsic noise having a uniform spectral density up to frequencies in the neighbourhood of lo-’ Hz. All this gives us every reason to believe that the cryogenic approach will yield a gradiometer with an unprecedentedly low intrinsic noise level and a highly stable scaling factor without any need to modulate the gravitational effect. The absence of rotating parts substantially simplifies the design of the gradiometer, particularly the trivial one. The first phase of the Maryland University programme envisages development of a triaxial in-line gradiometer with a sensitivity of 0.01 E HzP1I2.According to the published reports, laboratory models of uniaxial in-line gradiometers have been tested so far. Their sensitivity was found to be 0.2 E Hz- ‘1’ at a base of 10 cm and, according to the authors, it was determined by the noise inherent in the commercial SQUIDS. The resolution of any gradiometer is determined, on the one hand, by that of the test body displacement transducer and, on the other, by the intensity of the external noise or, in other words, random forces acting upon the test bodies. Such forces may have a wide variety of sources. Some of them can be eliminated in some or another way, such as thorough shielding, seismic filters, and the like. However, there are factors that cannot be eliminated in principle and, therefore, determine the theoretical limit of the attainable gradiometer sensitivity. These include, primarily, the Brownian motion of test bodies (thermal fluctuations). Let a gradiometer be in the form of the above-discussed torsional oscillator (Fig. 1.10) which is in a thermal equilibrium with the environment having absolute temperature T.Then, the oscillator is acted upon by a random force

67

The Gravitational Field of the Moon

moment m ( t ) caused, for example, by collisions of the test bodies with the molecules of the surrounding gas. In the classical approximation, m ( t ) is a white noise with the spectral density N o determined by the Nyqyist formula N o = 4 x T H , where x = 1.38 x erg deg-' is the Boltzmann constant, and H is the coefficient of the friction between the dumbbell and the environment. The response of the oscillator to the noise m(t) is in the form of Brownian fluctuations. The limiting accuracy with which gravity gradients can be measured against the background of Brownian fluctuations depends on the measurement technique and can be established in the following fashion (Nazarenko, 1982). The low-amplitude oscillations of the torsional balance in a nonuniform gravitational field with due account for thermal fluctuations are given by the equations

3; + 264 + w;+

=

rl + 2r2$ + p ( t )

(1.12.5)

or

3; + 26$

+

=

V,, cos 2pt

+ VA sin 2pt + p ( t )

(1.12.6)

where the first equation relates to the static and dynamic methods and the second relates to the modulation method. The term p ( t ) is used to denote the random function m(t)/J which is white noise with a uniform spectral density N1

= NoJ-'

(1.12.7)

=~xTHJ-'

If the transducer of the angular displacement of the dumbbell is ideal, its output is a signal which is an exact analogoue of oscillations $(t). Let the gravity gradients V,, and VA (as well as rl and r2)be constant within the measurement interval 0 < t < z. Then, the + ( t ) signal processing problem boils down to establishing the exact values these parameters had within the observation interval. To this end, the following quantity must be formed from the $(t) signal: y ( t ) = 3;

+ 26lj + o$$

This operation can be performed by an analogue device comprising differentiating, scaling and summing amplifiers. According to (1.12.5) and (1.12.6), the y ( t ) signal is the sum of the white noise p ( t ) and useful signals representative of the parameters rl,r2 or V,,, VA: y ( t ) = l-1

+ 2r'$(t) + d t )

y(t) = V,, cos 2pt

+ VA sin 2pt + p(t)

.

(1.12.8) (1.12.9)

As can be seen from (1.12.8) and (1.12.9), the parameters rl,T2,Vxyand Va may be regarded as unknown amplitudes a of signals of a known waveform

68

Lunar Gravimetry

aS(t), where Sl(t) = 1, S , ( t ) = 2$(t), S , ( t ) = cos 2pt and S,(t) = sin 2pt. The problem of estimating the amplitude of a determined useful signal received against the background of white noise has been closely studied in statistical communication theory. It is known, in particular, that in the case of lowamplitude noise, the estimation with the smallest possible variance involves the maximum likelihood method which leads to the following algorithm (Tikhonov, 1966): r

(1.12.10) 0

where y(t) is the sum of the useful signal and noise, S ( t ) is the waveform of the useful signal, and Eo =

s

S’(t) dt

0

is the energy of a useful signal with unit amplitude. For instance, in the case of a resonance gradiometer, sin 2pt

s(t)

=

(,,, Zpt)

then, Eo = 2/2 (taking into account the natural condition 2p2 = o o z >> l), and the optimal algorithm for finding VxYand V,, will be sin 2pt cos 2pt 0

which corresponds to synchronous detection of the y(t) signal. Since noise p(t) is present in the y(t) signal, (1.12.10) yields a random value. It can easily be shown that its variance is D ( 4 = N1IEo Substitution of N 1 = ~ x T H J into - ~ this formula and calculation of Eo for each measurement method gives: (1) For the static method: S(t) = 1;

Eo

= 2;

4x TH

D(f1) = Nl/z =J 22

(1.12.11)

69

The Gravitational Field of the Moon

(2) For the dynamic method: S ( t ) = 211/(t) = 2t,b0 sin (mot + Do);

(1.12.12) 2xTH D(f,) = Eo = 211/;z; 21+@ ll/$Jzz (ll/o is the dumbbell oscillation amplitude determined by the initial conditions). (3) For the modulation method: 8x TH D(P,,) = D(P,) = (1.12.13) J 2z N1

~

~

Formulae (1.12.11)-(1.12.13) permit the different methods for gravity gradient measurement to be compared in terms of the limit of accuracy, determined by thermal fluctuations. As can be inferred from (1.12.12), the thermal limit of accuracy of a dynamic gradiometer depends on the initial dumbbell oscillations amplitude. Since the initial equation (1.12.5) holds only for lowamplitude oscillations when ll/o << 1, equations (1.12.1lk(1.12.13) suggest that, all things being equal, the dynamic gradiometer has the highest thermal theshold. For example, if the torsional balance has the following parameters: coo = 2a rad s - l , J = lo4 g cm2, Q = J o o / M = 1000, and T = 300 K, at a measurement time z = 10 s and an initial oscillation amplitude +o = 0.1 rad in the dynamic method, the thermal threshold trT = D1’2 equals 0.1 E in the static case, 0.15 E in the modulation case, and 0.75 E in the dynamic case. Based on the assumptions just made, the thermal limit of the static and modulation gradiometers is almost the same. However, the modulation method offers a number of important advantages. Modulation shifts the spectrum of the gravitational signal from the neighbourhood of zero frequency to the higher frequency region near the modulation frequency 2p. This may minimize the effect of the flicker noise (drift) inherent in both mechanical and electrical components of the gradiometer. Encoding of a gravitational signal with exactly known frequency and phase facilitates its segregation from interference (particularly, mechanical accelerations). Decoding of the gradiometer output with the aid of orthogonal reference sin 2pt and cos 2pt signals permits simultaneous and independent determination of Vxyand VA, whereas static and dynamic gradiometers measure only linear combinations of these parameters-F1 or TZ. Regardless of thermal fluctuations, the resolution of a gradiometer may be confined to the intrinsic noise of the displacement transducer. Assume that external random factors (including thermal fluctuations) do not affect the gradiometer or are negligible. Then, the gradiometer output will take the following form:

w = 4w +

70

Lunar Gravimetry

where n(t) is the equivalent noise of the displacement transducer and $(t) stands for unperturbed oscillations. The function $(t) is known a priori for each measurement method because it is found by solution of the motion equations (1.12.5) or (1.12.6) at m(t) = 0 and is explicitly dependent on gravity gradients. This brings us to the problem of establishing the parameters of the determined useful $(t) signal received against the background of the displacement transducer noise. If this noise is assumed to be white with a spectral density N,, solution of this problem by the maximum likelihood method gives the following expressions for variances of estimated gravity gradients (Nazarenko, 1982): D(f1) =o:N,/T

D(f2) = 60gN,/$ir3

(1.12.14)

D( PXy) = D( Va) = 240gN,/z3

These formulae permit estimating, for each measurement method, the resolution of the angular displacement transducer, which is necessary for measuring gradients with a predetermined accuracy rmin. The resolution of = (N,/z)”’, which is the transducer may be characterized by quantity numerically equal to the r.m.s. value of the intrinsic noise of the transducer in a frequency band l/z wide. Such a change in notation allows formulae (1.12.14) to be rewritten as: for the static case

rmin/d

for the dynamic case z

( 1.12.15)

for the modulation case

For example, at oo= 2.n rad s - l , z = 10 s, and $o = 0.1 rad, we obtain the following resolution estimates necessary for measuring gravity gradients with an accuracy of 0.1 E: 2.5 x lo-’’ rad in the static case, 6.5 x lo-’’ rad in the dynamic case, and 3 x lo-” rad in the modulation case. Consequently, even from the standpoint of the instrument sensitivity, the modulation method is preferable. There is nothing surprising in this finding in view of the fact that the modulation method is based on resonance enhancement of the gradiometer’s response. Note that formulae (1.12.11)-(1.12.15) hold also when the resolution of gradiometers belonging to the second group is determined. In this case, quantity mi’ must be substituted for the dumbbell inertia moment J and xmin/lmust be substituted for in these formulae, m being the mass of the

71

The Gravitational Field of the Moon

test body of the accelerometer, 1 being the base, and xminbeing the resolution of the linear displacement transducer. The possibility of carrying out autonomous measurements is not yet a decisive argument in favour of satellite gradiometry. In view of their relatively high complexity and cost, such measurements are justified only if they are capable of yielding more detailed and exact information on the gravitational field, as compared to Doppler measurements. The effectiveness of both methods, as applied to investigation of the lunar gravitational field, was compared by Forward (1976), who resorted to mathematical modelling to find out what information can be obtained if the second vertical derivative V,, is measured with an error of 1 E and the line-of-sight velocity of the ALS is determined with an error of 1 mm sec-'. It was assumed that the errors were not correlated and their only source was the intrinsic noise of the instruments, while the magnitudes of these errors were essentially the standard deviations u of this noise at an averaging time of about ten seconds. The selected values of u = 1 E and 0 = 1 mm s T 1corresponded to the highest accuracies attained by that time. The method of lunar gravitational field anomalies was constructed on the basis of the data provided by Doppler tracking of the Apollo 16 subsatellite in the central region on the near side of the Moon, confined by -22" to +27" longitude and +2" to +9" latitude (Sjogren et al., 1974). An iteration method was used to define a system of point masses (28 large masses inside the Moon and a grid of small masses on its surface with a 15 km spacing) whose field approximated the initial data with a standard deviation of 0.6 mGal. The map of the model field of V, at an altitude of 15 km above' the lunar surface is shown in Fig. 1.14 (from Forward, 1980). It can be seen that the derived model of lunar gravitational field anomalies is close to reality and is marked by a rather complex structure. This model was used to calculate, for altitudes of 15, 30, 60 and 100 km above the lunar surface: (a) the second vertical derivative of the potential V,,; (b) the line-of-sight velocities u, of the ALS; (c) variations in the line-of-sight velocity over the measurement time interval At, corresponding to an orbit stretch 15 km long covered by the satellite; this quantity is related with the anomalies of V, simply as Av, = V, At where At = 15 km/1.7 km s-' N 8.8 s. The calculated values of V,, and Au, were then represented on contours maps. The contours were drawn with a spacing proportional to the measurement accuracy, which allows the.maps representing the dissimilar quantities Av, and V,, to be compared in terms of their detailing. Figure 1.14 represents contour maps of Av, and V,, isolines at an altitude of 15 km, drawn with a spacing of 5 0 or, in other words, 5 mm s-' and 5 E, respectively. Comparison of these maps vividly demonstrates the

2

8

m U

5

VI

a

a

W

0

0

0

A

I

0

a

I

0

A

I

0

A

I

0

0

9

I

0

0

a

0

a

I

0

N 0

8 45

0

a

0

0

I

8

B

a 0

0

4

o_ ? i

9

P

i

a

5E contours

10 m GAL contours

5

m m s-' change contours

The Gravitational Field of the Moon

73

higher resolution of gradiometry, which is particularly evident in the central region. Similar mapping at altitudes of 30 km with a spacing of 50, 60 km with a spacing of 2a, and 100 km with a spacing of indicates that, although the gradient field becomes less detailed with altitude at a faster rate, in comparison with the line-of-sight acceleration field, gradiometry nevertheless yields more reliable results. The Doppler method becomes preferable only at altitudes exceeding 200 km, but even then it reveals only large-scale features of the gravitational field structure. Also estimated in the above-cited work were the distortions due to random measurement errors. To this end, at each node of the grid with 15 km x 15 km squares, a random uncorrelated error with zero average and a deviation of 1 mm s-l and 1 E was added to the calculated values of Avz and V,,. Then, after isolated noise peaks of an extremely large magnitude had been smoothed out, contours of the “noisy” field were drawn. This procedure was repeated four times, and all four sets of random contours were represented on the same map. Figure 1.15 (from Forward, 1976) shows the maps of Av, and Vzzproduced in this manner for the altitude H = 30 km. The contours spacing was 50. These maps indicate that the effective “noise width” of the Doppler velocity contours is substantially greater than 15 km-that is, the size of an elementary cell-reaching in some places 45 km. On the other hand, the contours of the gradient field have a much narrower “noise width. Thus, measurements of the vertical gradient of the lunar gravitational field at altitudes of up to 100 km with an accuracy of 1 E over a ten-second averaging interval are better than Doppler measurements of line-of-sight accelerations with an accuracy of 1 mm s - l in terms of both spatial resolution and the signal-to-noise ratio.

1.13 Generalized Model of the Lunar Gravitational Field

About twenty models of the lunar gravitational field have been published so far. They are based on the observations of numerous ALS with various orbital elements, carried out with different durations and various raw data processing methods. Some of these models have been derived from a relatively small amount of observation data and, therefore, do not adequately represent the gravitational field of the Moon as a whole. A few models have used the same observation data, the only difference between them being in the data processing procedure. It is, therefore, natural to attempt a generalized model of the lunar gravitational field based on the totality of available data. The first lunar gravitational field model used by us was that produced by Ferrari (1977) on the basis of the data supplied by Doppler tracking of the Apollo 15 and 16 subsatellites. Also used were some Lunar Orbiter 5

Avz, vertical Doppler velocity change in 15 km (mm s-'). 4 superimposed additive noise plots.

30 km altitude

Fig. 1.15. Effect of random measurement errors on the isolines Of line-of-sight accelerations and vertical gravity gradients (according to Forward, 1976).

75

The Gravitational Field of the Moon

observation data. This model (we shall call it F77 according to the year of its publication) included estimates of the harmonic coefficients C,,,,, and S,,, to n = rn = 16 and their errors. At the same time, Ferrari mapped the errors of the radial selenopotential derivative au/ar on a sphere with radius I = 1838 km. These errors varied from 20 mGal at the equator to 60 mGal at the poles. The second model of the Moon’s gravitational field, to be used in deriving a generalized one, was that proposed by Akim and Vlasova (1983). This model (let us call it AV) is based on a wealth of Doppler data provided by tracking of the Soviet ALS Luna 10 and 24. A distinctively positive feature of these data is their even distribution over the lunar surface, stemming from the different inclinations of the lunar orbits with respect to the Moon’s equator. Table 1.9 lists the coefficients C,,,,, and S,,, of the AV models and their errors. It should be pointed out that the authors of this model presented not the formal errors involved in the method of least-squares but their possible maximum values determined by the errors and the distribution of the initial data of this model. These errors represent the actual accuracy of determining the coefficients. The third model of this kind must be credited to Ferrari et al. (1980). It is based on the observations involving laser detection of the Moon’s physical libration and on the results of Doppler tracking of Lunar Orbiter 4 at its high orbit stage (perilune of 2700 km and apolune of 6200 km). The model (we shall refer to it as F80 according to the year of its publication) has provided a wealth of geophysical and geodetic data for both the Earth and TABLE 1.9 Normalized harmonic coefficients in the lunar gravitational field model, calculated using the data from Akim and Vlasova (1 983)

n

m

cx

2 2 2 3 3 3 3 4 4 4 4 4 5 6 7

0

-0.926 -0.002 0.376 -0.019 0.262 0.100 -0.029 -0.003 -0.061

1 2 0 1 2 3 0 1 2 3 4 0 0 0

104

0.004

-0.017 0 0.011 0.003 0.074

(Tc x

104

0.029 0.016 0.111 0.027 0.012 0.053 0.122 0.029 0.015 0.054 0.033 0.095 0.036 0.016 0.022

s x 104

0.043 0.170 -

0.069 0.097 - 0.258 -

0.026 0.013 -0.017 0.142 -

u s x 104 -

0.012 0.115

0.029 0.053 0.122 0.013 0.049 0.033 0.095

-

76

Lunar Gravimetry

the Moon. These parameters include the selenocentric constant of gravitation, GM, = 4902.799 f 0.003 km3 s-', and 35 harmonic coefficients Cnm and S,,,,,.All harmonic coefficients were determined up to n = m = 5 plus four coefficients with n = m = 6 and n = m = 7. In this model, the coefficients of the second and third degrees were determined with high accuracy, while the accuracy with which the coefficients of the fourth and fifth degrees were determined was the same as in other works. The analysis carried out by the authors has shown that the harmonic coefficients were determined primarily from Doppler tracking data. The next, fourth source of data for constructing our model of the Moon's gravitational field was a set of 83 point mass bodies representing the field of line-of-sight accelerations on the Moon's near side. This model derived by Brovar and Yuzefovich (1975) as well as by Ogorodova et al. (1975) has made it possible to attain a better approximation, owing to a proper selection of the point mass body coordinates, of the field of line-of-sight accelerations on the near side of the Moon, as compared to the model proposed by Wong et al. (1971) with 580 point mass bodies spread over the surface of a sphere with r = 1738 km and not associated with the gravitational field features. The determination of 83 point masses involved approximation of 11,000 Doppler observations of Lunar Orbiters 1 through 5 as well as Apollos 8 and 12 within a region confined latitudinally by f60" and longitudinally by f 100". All the four parameters (mi, cpi, ,Ii, hi) were determined for each point mass body from a respective local anomaly of line-of-sight acceleration. Their depths, reckoned from a sphere with r = 1738 km, varied from several to a hundred kilometres, and their masses varied within (0.05-2.5) x 10- Mu.Twelve out of the 83 point mass bodies were arranged in such a way as to approximate the influence of mascons. It is precisely these bodies that have the largest masses out of all 83 bodies. We have borrowed Table 1.10 from the work by Brovar and Yuzefovich (1979, listing the rectangular selenocentric equatorial coordinates (x, y, z), the depths h, and the relative masses m/Ma of these 12 main bodies. Ogorodova et al. (1975) present a map of residual accelerations Ar. The quantities 6r are essentially the differences between the observed accelerations and those created by the 83 point mass bodies. They do not exceed 10 mGal in the equatorial zone between +40" latitude and 5 mGal northward and southward. The only exception is the region of Mare Orientale where the residual accelerations are as high as 30 to 40 mGal. The fifth source of data for the generalized model of the lunar gravitational field was a set of four profiles of line-of-sight accelerations over the central portion of the Moon's near side, derived as a result of low-altitude flights of Apollos 14, 15, 16 and 17 modules. These data had never been used before for constructing global lunar field models. We have used these profiles to do two different things: firstly, they have been applied directly to derive conditional

77

The Gravitational Field of the Moon

TABLE 1.10 Parameters of the 12 point mass bodies approximating the influence of mascons

Mascon 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Mare Frigoris Mare Imbrium Mare Serenitatis Sinus Aestuum Mare Crisium Mare Marginis Sinus Medii Crater Julius Caesar Mare Nubium Mare Humorum Mare Hectaris Mare Smythii

x (km)

Y (km)

(km)

h(km)

933 1200 1381 1437 827 29 1457

- 247 -366 459 - 156 1395 1488 - 183

1334 926 664 5 70 487 ,548 127

91 178 133 185 45 152 264

- 1.48 2.47 2.25 -0.99 0.73 -0.59 1.70

1427 1520 1240 1389 83

475 - 320 - 977 892 1611

316

20 1 174 25 39 123

-2.18 -0.92 0.21 0.45 0.88

- 180

-666 -401 86

m/Ma x

equations for constructing a generalized model by the least-squares method; secondly, the profiles have provided the values of a radial potential derivative, calculated from given line-of-sight accelerations. In this case the line-of-sight accelerations were smoothed by one-degree (c. 30 km x 30 km) squares, then, they were represented in terms of line-of-sight accelerations of the point mass bodies. Then, these point mass bodies were used to calculate the values of the radial potential derivative. However, if point mass bodies are used to represent only the profiles, a substantial error may arise in the determination of the vertical gravity gradient, hence in the sought values of gravity. In order to avoid such errors, we have decided to use the point mass bodies to approximate not only the line-of-sight acceleration profiles, but also the values of the radial derivative aV/ar, calculated from the model derived by Bills and Ferrari (1980) in a broader region (lql < 40" and 121 < 80"). These values of aV/ar were calculated at 10" intervals at the altitude H = 100 km, which has resulted in 153 conditional equations of the tYPe

(1.13.1) where E~ are the sought relative masses m i / M , , R i , ( p i and Ai are their selenographic coordinates, N is the number of point mass bodies, and r j , Qj and Lj are the coordinates of the points at which the values of the radial derivative of the anomalous potential had been calcdated using the formula:

(1.13.2)

78

Lunar Gravimetry

Here, C,,,and S,, are the normalized harmonic coefficients of Bills and Ferrari, and P,,, stands for normalized associated Legendre polynomials. Then, 141 conditional equations of the following type were derived for each profile: (1.13.3)

where Iij = [(xj - xi)’ + ( y j - yi)’ + ( z j - zi)’I1/’, x i , yi and zi being the rectangular selenocentric coordinates of the sought point mass bodies, and xj, y j and z j being the coordinates of the line-of-sight acceleration profile points in the same system. The number of point mass bodies and their depths were selected to achieve a satisfactory approximation. The result of the solution was that the number of point mass bodies was about 60, namely, N = 65 for Apollo 14, N = 57 for Apollo 15, and N = 61 for Apollos 16 and 17. In the case of the Apollo 15 profile, the point mass bodies were at a depth of 138 km (R = 1600 km from the Moon’s centre) and, in the rest of the cases, at a depth of 238 km (R = 1500 km from the Moon’s centre). The slightly greater number of point mass bodies (65) and their lower depth (138 km) in the case of Apollo 14 stem from the higher frequency nature of the corresponding profile. Figure 1.16 represents the Apollo 17 profile showing the values of the initial smoothed line-of-sight accelerations (solid line), the residual difference between the initial field and the approximation result, and the radial potential derivative at an altitude of 100 km above a sphere with r = 1738 km. The magnitude of the residual field is much smaller than the observation errors. The behaviour of the radial potential derivative calculated from the found point mass bodies is strikingly similar to that of the line-of-sight accelerations because they are concentrated in the equatorial zone of the Moon’s near side, but is smoother in nature due to the recalculation for the altitude of 100 km. The residual variance at the points where the radial potential derivative aV/ar was defined from the model derived by Bills and Ferrari (1980) was (10 mGa1)’. The generalized model of the Moon’s gravitational field was derived by us from the data provided by the five sources using two different methods. The first method is based on the orthogonality of spherical functions. This property of the latter permits estimating the values of the harmonic coefficients C,,,,, and f,,, by integrating the values of some potential characteristic over a sphere. We have selected the radial potential derivative because this parameter is close to the line-of-sight accelerations determined from Doppler tracking data. Moreover, this function exhibits no peculiar behaviour in different

79

The Gravitational Field of the Moon 240c

200

1

A

Fig. 1.l0. Point-mass approximations of the line-of-sight acceleration profile of Apollo 17. The dotted-and-dashed line indicates the initial line-of-sight accelerations, and the dashed line indicates the radial derivative of the selenopotential at an altitude of 100 km.

regions. In the F77 model (Ferrari, 1977), it is precisely for this quantity that an error map had been provided. Each of the five initial data arrays was used to define the radial derivatives aV/ar at different points of a sphere with r = 1838 km (i.e., at an altitude of 100 km above the lunar surface). These points were evenly distributed over the surface of the sphere. The number of points, N = 1225,corresponds to the distance of 5.2", at the equator, or 167 km in linear measure. The linear distance between adjacent points is constant over the entire sphere, and the number of points at parallels is proportional to the cosine of the latitude. Tests have shown that this number of points is quite adequate for determining harmonics up to the 16th order. Derived next was the weighted mean value of aV/ar from five sets of initial data. It is known that, given several independent values of xi of dissimilar degrees of accuracy, the values of x, the best estimate of 2, and that of its

80

Lunar Gravimetry

variance S2 are defined as weighted means:

where I is the number of values of x i , Wi = 1/0? stands for weights, and O: stands for variance estimates. More often than not, the standard deviation S is greater than bi.Joint processing shifts dissimilar systematic errors into the category of random ones, whereby the standard deviation is increased. This means that, in the case of joint processing of dissimilar measurement results, the systematic error in X is usually smaller, while the random error is greater. However, the random error can be reduced by enhancing the accuracy and increasing the number of measurements. In our case, in formulae (1.13.4), 1 = 5 for the central region of the near side (191 < 30", 111 < 70"), 1 = 4 elsewhere on the Moon's near side (IcpI < 60") because of the loss of the low-orbiting Apollos from sight, and 1 = 3 for the rest of the Moon because the model with 83 point mass bodies is lost and three models of expansion in spherical functions remain. Each value of x i = dV/& can be written in the following more detailed form:

av

-(r, ar

cp,

A) = Aizi

(1.13.5)

where i = 1,2, 3,4, 5; zi is the vector of initial data; and Ai is the operator of transition from initial data to the radial derivative values. Used as the initial data in three models out of five for deriving a generalized model were the following sets of harmonic coefficients: zi = {Zfm}

=

{Cf,,

C,}

( 1.13.6)

In these three cases, the operator Ai appears to be similar and represents addition of the initial coefficients Cf, and Pi,,,multiplied by the spherical functions-that is, a linear combination of initial data:

( 1.13.7)

In the other two cases, the initial data are sets of point mass bodies. In these cases, the operator A represents addition over the point sources, the

81

The Gravitational Field of the Moon

coordinates of the point mass bodies being involved in a nonlinear fashion: (1.13.8) where i = 4,5; ki is the number of point mass bodies; E~ = mj/M,; pi, cpj and l j are the coordinates of the point mass bodies; and A, = ( r 2 + pi’ - 2rpj cos $j)? In all five initial models, the harmonic coefficients and point mass bodies are the results of processing of other initial data, namely, Doppler ALS tracking data. Mathematically, this can be written as ( 1.13.9) ~i = BiDi where Di is the set of Doppler tracking data, underlying the ith model, and Bi is the operator of transition from these data to harmonic coefficients or point mass bodies. The operator Bi is usually rather complex and represents a combination of such operations as differentiation, smoothing, integration, and adjustment by the least squares method. Equation (1.13.5) can now be written as

a vi

-(z, ~ p ,A) = AiBiDi

dr

(1.13.10)

If we could determine the radial potential derivative or another parameter, of the lunar gravitational field from (1.13.10) with the processing involving directly the tracking data, their more complete use would have provided the most reliable model of the Moon’s gravitational field. However, we do not know which initial tracking data were used in the models of expansion in spherical functions (F77, AV, F80). The lack of initial data in three out of five models has led us to processing already completed models-that is, beginning with solution of equations (1.13.5), while disregarding equation (1.13.10). The second approach to constructing a generalized model of the selenopotential was based on the least-squares method used to determine the harmonic coefficients of potential expansion and approximating point mass bodies. This approach permits joint use of various gravitational field characteristics. Therefore, we have resorted to the observed values of line-of-sight accelerations from the low-altitude profiles of Apollos 14,15,16 and 17 rather than the gravity values recalculated in terms of point mass bodies at an altitude of 100 km above the Moon’s surface. Another positive feature of the leastsquares method is the possibility of using data not in the form of some mean quantities derived from all models at each point but directly. This means that for each value of the radial potential derivative or line-of-sight accelerations one can produce its own conditional equation with its own weight.

82

Lunar Gravimetry

A third advantage of the least-squares method is the possibility of using data directly at the points where they were determined without recalculation for other surfaces (in our case, a sphere with r = 1838 km). This processing technique, however, suffers from certain drawbacks. Firstly, the results are dependent on the number of parameters to be determined. This leads to redistribution of value estimates and the errors involved in these estimates. Secondly, we are dealing with a smoothing effect-that is, with a spread of parameters determined by this method which is narrower than the actual one. Thirdly, when many values are to be defined, the method becomes rather cumbersome. Two types of difficulties arise: firstly, correlation between individual parameters of interest and inadequate matrix of normal equations and, secondly, too many (tens and even hundreds of thousands) matrix elements to be stored in the computer and, consequently, a long computation time. The system of conditional equations can be written as: AiZi

+ ai = x i ,

(1.13.1 1)

where i = 1, 2, 3, 4, 5; Ai = {Aij}; ai= (aij};x i = { x i j } .Just as with the first method, taken as the “observed” values of xlj, x Z j and x 3 j for three expansions in spherical functions are those of the radial gravity derivative on the surface of a sphere with r = 1838 km. This means that the operators A l , A 2 and A 3 are the same as before, and the elements A i j are

(1.1 3.1 2)

The unknown values of zi are the same as in (1.13.6)-that is, arrays of harmonic coefficients C,, and S,,,,,. The fourth model with 83 point mass bodies was also used in the form of values of a V/dr in the region confined by k 60” latitude and k 90” longitude. In this case, we also derive the same expressions for A , as in equations (1.13.8). The fifth set of initial data, namely, profiles of line-of-sight accelerations of Apollo 14, 15, 16 and 17 modules, was used directly in this version-that is, no approximation in terms of point mass bodies was carried out but the values of line-of-sight accelerations themselves at the points of their measurement were used. To this end, we resorted to expansion of line-of-sight accelerations in spherical functions:

83

The Gravitational Field of the Moon

x

(on+

l,m

cos mA +

sin mi) (1.13.13)

where r, cp and iare the selenocentric equatorial coordinates of the line-ofsight acceleration observation points. Thus, for the fifth model we have

where F ( n , m) stands for formulae (1.9.9) of transition from C,, and S,, to &+I,, and L + l , m . Finally, we have the system of equations Aizi

+

~i

(1.13.15 )

= xi

where i = 1,2,3,4, 5. Solution of this system is found in the form of ( 1.13.16)

where Wi = ( O ; O ~ ) - ~stands for weight matrices. In the solution of (1.13.16), the expression K =

C ATW,A~ [irl

I

-1

( 1.13.1 7)

is a covariance matrix, and its diagonal elements ~ ( z j= ) (Kjj)1'2

(1.1 3.18)

give the formal errors inherent in the coefficients Cnmand S,,, while the nondiagonal ones give the coefficients of correlation between various C,,, and FYnm: (1.13.19)

The model of the selenopotential, obtained by integration of the radial derivative au/ar will be referred to in what follows as INT. Construction of this model using formulae (1.13.4) has involved mapping of the weighted mean values of the radial derivative dV/ar (Fig. 1.17) and its weighted mean

Fig.'1.17. M a p of the moon showing weighted mean values of the radial derivative of the selenopotential at an altitude of 100 km. Scale 1 :20,000,000.

85

The Gravitational Field of the Moon

errors. The formula for deriving the values of the harmonics is obtained from

( 1.13.20)

Multiplication of the left- and right-hand sides of this formula by spherical functions and its integration over the surface of a sphere with radius r gives:

0

-n/Z

(1.13.21) The derivation of (1.13.21) is based on the orthogonality of spherical functions. To define the errors involved in formula (1.13.21) for estimating the harmonics, we have resorted to multiple superposition of the random noise produced by the generator of random normally distributed numbers upon the weighted mean values of the radial derivative of the selenopotential. The amplitude of this noise at each point was proportional to the weighted mean error at that point. A total of 30 noise-affected field realizations have been obtained by such superposition of the noise. In this way, an ensemble of realizations have been created, which had already been used to find the where i = 1,2, . . ., 30. Such a method of creating coefficients Cnmiand Snmi, “pseudo-ensembles’’of observations was used by different workers in cases where estimation of errors with the aid of integral transformations is difficult. Then, we find the mean

where K = 30, n = 0, 1, ..., 16, and m = 0, ..., n. Knowing the mean, we find the errors:

Table 1.11 lists the harmonic coefficients Cnm,Sn, and their errors oc,,, of,,, found in the INT model. For a more complete description of the statistical properties of the derived harmonics, the correlation between them has been calculated. Let us introSnmi},the set of all sets of harduce the following notation: Znmi= {Cnmi,

86

Lunar Gravirnetry

TABLE 1.11 Harmonic coefficients in the INT model

n

m

c

OC

s

n

1 1 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9

0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6 .7 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5

155 147 72 131 69 -9138 -268 148 19 3530 53 -265 2268 106 87 1201 31 1562 47 794 - 533 114 -818 86 - 129 40 - 343 25 - 185 46 -902 104 21 250 - 124 91 31 38 -649 39 84 350 140 92 - 320 41 -246 82 143 49 246 19 727 10 596 79 223 56 -3 128 20 - 59 - 34 65 161 35 - 161 32 13 -965 332 63 21 119 64 69 16 83 47 346 - 309 45 - 347 33 -261 17 21 1 22 - 384 149 72 35 90 212 -243 134 -15 52 63 - 165

-

- 37 525 -231 -

505

-

156 198 84 -

73

-

-

209 -761 -1338 -178 - 12 - 56 415 - 599 1065

119 147 57 14 165 96 51 44 23 92 71 56 24 30 22 30 106 58 55 38 45 19 162 100 72 62 81 49 16 13 60 43 28 89 41

-

- 130 -214 - 289 - 19 - 763 - 832 166 122 -5 25 102 612 820 269 -114 213 - 47 444 -241 -47 - 120 161 -62 188 84 - 276

rn

c

6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4

- 125 -228 -229 -89 59 62 -7 55 - 287 33 - 42 -286 - 160 - 190 - 53 -235 -112 208 161 27 - 14 98 -29 - 273 -256 -251 - 55 - 129 - 68 - 17 61 120 82 69 178 - 38 52 - 190 - 146 25 - 30 15 -43 13 148 - 105 37 -71 - 23 -31

QC

s

~

~

9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11

11 5 11 6 11 7 11 8 11 9 11 10 11 11 12 0 1 12 2 12 3 12 4 12 12 5 6 12 7 12 12 8 9 12 12 10 12 11 12 12 13 0 1 13 2 13 13 13 13 13 13 13 13

32 31 10 13 113 44 50 81 36 69 10 40 31 22 19 21 75 65 46 70 47 43 40 28 16 40 32 63 64 74 48 48 55 45 44 33 20 14 10 17 99 18 57 68 41 38 38 38 28 47

-20 35 -351 85 -

30 33 24 20 112 85 54 20 78 34 38 48 17 12

-326 - 164 187 60 137 - 193 - 349 114 175 -48 84 -265 52 86 79 82 279 83 105 64 - 253 70 - 94 45 35 36 - 52 40 141 160 - 56 224 259 75 - 15 50 -225 62 174 32 113 97 114 56 41 58 59 -237 27 96 -90 32 - 54 20 10 - 35 17 83 - 16 55 -111 70 - 98 121 108 52 71 48 - 83, 35 - 135 38 91 37

87

The Gravitational Field of the Moon

13 10 13 11 13 12 13 13 14 0 1 14 2 14 3 14 14 4 5 14 6 14 7 14 8 14 9 14 14 10 14 11 14 12 14 13 14 14 0 15 1 15 2 15 3 15 4 15 15 5 15 6

69

- 135 -56 64 -21 18 63 101 53 35 42 - 135 23 -22 -66 - 109 - 130 39 78 72 0

100

-60

- 130 -40 -6

22 30 27 10 113 35 76 12 36 77 109 66 40 46 28 27 22 23 13 47 36 28 69 69 32 49

160 -9 - 116 24

32 14 9 9

-

-

- 144 70 134 - 106 -80 153 101 53 38 42 66 21 -47 -8 66 138 45 -235 35 166

56 83 67 51 84 58 30 39 65 29 35 17 59 17 25

35 75 20 56 38

15 7 15 8 15 9 15 10 15 11 15 12 15 13 15 14 15 15 16 0 16 1 16 2 16 3 16 4 16 5 16 6 7 16 16 8 16 9 16 10 16 11 16 12 16 13 16 14 16 15 16 16

175 124 -15 -25 -99 -172 -34 30 1 55 -38 5 110 5 -9 98 7 70 9 114 1 -56 1 47 48 -28

41 25 22 42 12 26 10 11 10 62 87 21 87 69 65 75 14 46 -3 47 24 25 22 12 16 18

87 105 -7 68 - 16 13 -43 18 2 - 24 9 68 49 29 -115 -61 27 -40 47 101 - 33 38 - 86 - 38 71

54 62 47 56 25

90 20 14 17 66 32 56 66 ,69 33

60 69

40 35 44 16 17 25 20 13

monics; Z,, = {en,,S,,}, mean values of the harmonics; nzmm= nc,,, ns,,, standard errors inherent in the mean values. Now, a unified formula can be written as an expression for the coefficient of correlation between the coefficients Z,, and 2,:

In Table 1.12only part of the correlation matrix is given, calculated using formula (1.13.24),because the entire matrix is too large with its dimensions of 289 x 289 (at n = 16). This is the most important part for low values of n. Note that the correlation coefficients do not attain high values. Consider now the results of adjustment of the harmonic coefficients by the least-squares method (model LSM). Usk of formula (1.13.16)has provided the solution LSM-1 with a residual field variance 0: = 120 mGa12-that is, the error of approximation of the initial field is about 11 mGal. Inversion of the matrix of normal equations results in a pseudo-inverse matrix which differs from the exact one due to the effect of the computation errors.

88

Lunar Gravirnetry

‘20 c20 c 21 c22

1

‘21

c22

0.54

-0.04

1

-0.14 1

s21

s22

0.03 -0.15 -0.09 -0.06 0.00 -0.32

c30 c31 c32 c33

c30

c31

-0.11 -0.14

-0.16

-0.39 1

0.29 -0.36 1

0.17

c32

0.01

c33

0.05

-0.27 -0.05 -0.07 -0.28 0.12 -0.10 -0.08 -0.28 1 0.06 1

$31

s32

0.15 -0.42 0.34 -0.24 -0.20 -0.31 0.29 0.15 0.15 -0.16 0.17 -0.02 0.25 0.16

c40 c4 1 c42 c43 c44

Therefore, to refine the solution use was made of a special solution refinement algorithm based on the method of conjugate gradients. The initial approximation in this algorithm was the solution LSM-1. Search for the optimum has yielded an LSM solution with ci = 69 mGa12-that is, the approximation error is in the neighbourhood of 8mGal. The results are summarized in Table 1.13. Approximation of the gravitational field of a planet with the aid of a set of point mass bodies involves the’following simple relation: (1.13.25) where Uy is the total gravitational potential at the point M, Ii is the distance between the ith point mass body and the point M, and E~ is the relative mass of the ith point mass body. The correlation matrix calculated using formula (1.13.17) has been found to be close to that in the INT model. The point masses may be distributed either under the extreme points of the initial field or evenly. We have examined the second case where the point masses were to be found distributed evenly over a sphere with a radius less than the mean radius of the planet. The system of even distribution of points over the surface of a sphere was described by Giocaglia and Lundquist (1972). The spherical coordinates of such points are determined from the following formulae: cos q k = (Akj=&f((

+

cos [ k Z / ( h I)] n + ’ l - k )-2(n+:-k))n

(1.13.26)

where n is the number of the parallels at which the point mass bodies are

89

The Gravitational Field of the Moon

TABLE 1.12 Matrix of correlation between the coefficients in the generalized model

~

-0.11 -0.10 -0.17 0.46 -0.14 0.12 -0.08

-0.22 0.13 -0.51 0.03 -0.32 -0.40 0.28 1

0.26 -0.23 0.11 -0.04 -0.17 0.57 0.39 -0.23 1

0.04 -0.10 -0.21 0.24 -0.21 0.01 0.04 0.01 -0.20 1

0.10 0.33 -0.05 -0.03 0.05 -0.27 0.24 0.15 -0.02 0.17 1

0.16 0.19 -0.15 0.47 0.06 0.02 -0.10 0.06 0.00 0.10 -0.04 1

0.00 -0.06 -0.03 0.38 -0.26 -0.36 0.04 0.00 -0.20 0.13 -0.07 0.22

0.05 -0.40 -0.01 0.23 -0.05 -0.16 -0.34 -0.11 0.05 0.14 -0.21 -0.06

0.05 -0.14 0.01 -0.20 0.13 0.23 -0.05 -0.19 -0.16 0.29 -0.10 -0.08

0.17 0.00 -0.02 -0.10 -0.07 -0.04 0.15 -0.22 0.08 -0.01 0.08 -0.28

,

0.06 0.13 0.38 -0.47 -0.16 - 0.08 -0.22 -0.03 0.00 -0.25 0.40 -0.33

located, N = (n + 1)2 is the number of point mass bodies, ic varies from zero to N,j varies from zero to 2k, and ( ) mark the integer part. As can be seen from formulae (1.13.26), the coordinates (Pk and &j are easily determined, and to know the position of a point it is sufficient to know its number. Just as in determination of the harmonic coefficients Cnmand S,, determination of the point masses by the least-squares method may directly involve, the line-of-sight accelerations r, taken from the results of tracking Apollos 14, 15, 16 and 17 without preliminary approximation of the tracking data in terms of supplementary point masses. To this end, we determined the rectangular selenocentric equatorial coordinates of the point mass bodies of interest and those of the points of line-of-sight acceleration profiles, using the formulae: x=rcoscpcos1 y = rcos cp sin 1

(1.13.27)

z = r sin cp

where r, cp and 1 are known polar coordinates of these points. Thus, 141 conditional equations of the following type have been derived for each profile:

where ei stands for the relative masses mi/Muto be found; Xi, Yi and Z i are the coordinates of the point mass bodies of interest; xi, y j and z j are the coordinates of the line-of-sight acceleration profile points; and dij = [ ( x j - Xi)’ + ( y j - ~ i ) ’ (zj - Z J 2 11 / 2

+

TABLE 1.13 Harmonic coefficients in the LSM model

1

0

1

1

2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5

0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4

5

5

6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9

0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5

58 197 -9003 70 3547 - 350 2329 1267 1592 367 -467 - 750 272 - 325 -88 -752 293 - 16 6 -601 384 91 - 340 - 205 128 289 719 643 163 -133 - 39 87 189 - 162 -968 303 11

205 - 26 360 - 300 - 327 -242 206 - 181 127 68 - 107 -77 - 107

171 95 103 136 53 164 123 79 46 70 96 82 40 29 99 95 74 67 32 29 84 87 78 75 42 31 20 65 68 110 87 52 37 27 23 88 9 62 71 67 52 31 24 20 100

66 67 87 51 52

-

131 126 - 197 -

616 653 - 844

0 173 72 107 95 39

-

-

237 - 586 - 1267 - 196

107 92 50 27 112 98 71 40 22

-

-117 -116 34 1 - 522 1041 -20 -244 - 322 - 60 - 728 - 845 188 -31 30 43 131 619 802 215 -44 100 - 67 386 -296 -48 - 100 42 - 146 183 120 - 301

9 9 9 9 10 10 10

140

10 10 10

10 10 10 10 10 11 11 11

11 11

-

79 86 57 46 38 22 71 82 85 34 31 31 22 -

94 73 64 70 83 35 23 18 65 56 68

60 51

11 .

I1 11 11

11 11

11 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13

6 7

8 9 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9

- 125 -212 - 223 -79 92 0 27 49 - 283 -1

-44 -294 -203 - 209 -71 -39 9 20 1 88 -48 -73 71 2 -246 -264 - 248 - 48 -60 -62 -26 -22 120 50 33 85 - 22 - 14 - 188 - 150 -3 - 19 11

- 172 -9 134 -92 - 12 -113 - 30 22

25 29 24 20 97 73 36 74 75 75 33 39 26 21 22 55 69 57 57 65 49 52 43 33 26 18 15

70 68 64 51

77 54 43 50 41 24 17 15

13 77 47 62 57 63 53 53 33 42 35

-40 50 -321 76 -

- 154 - 90 166 50 36 - 159 - 382 168 153 - 28 20 41 96 353 113 - 265 - 149 54 - 55 148 -45 225 43 - 167 131 66 30 - 20 - 237 130 -46 - 34 - 27 - 32 15 -62 - 149 8 102 80 - 162 33

36 28 24 19 62 79 56 56 66 54 38 37 18 14 77 60 77 60 73 73

44 33 55

148 226 73 57 61 70 65 50 57 43 32 22 14 14 68 85 59 73 51

50 40 45 32

91

The Gravitational Field of the Moon

n

m

13 13 13 13 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 15 15 15 15 15 15 15

10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 4 5 6

C 38

- 102 - 58 48 41 46 31 55 0 - 58 -118 -47 68 56 - 74 - 154 - 106 14 73 96 - 66 111 - 74 -88 -11 -51

uc

S

us

n

m

18 22 19 14 87 52 62 54 37 67 60 39 38 35 26 26 23 18 16 71 58 51 67 53 54 73

168 22 -127 21

29 21 13 19

15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

-

-

-57 -3 94 -96 0 94 56 22 52 31 81 23 -45 -12

49 73 77 56 80 53 33 60 48 28 27 23 14 17

-

-

14 107 14 -219 -6 140

48 54 82 46 73 62

C

uc

21 44 113 39 -27 46 -12 29 -128 22 -157 25 -32 11 27 17 17 11 38 61 -36 72 -11 26 -23 87 -25 44 -84 63 44 62 -37 35 2 4 0 -39 41 20 44 22 33 -71 32 3 21 28 19 40 13 -145 15

S

us

203 150 -71 13 14 23 - 15 24 - 13

60

-

-

-43

79 56 72 58 60 70 55 54 51 31 33 18 21 18 20 13

-5 29 -3 - 34 - 87 -48 80 - 102 30 88 - 22 -4 -31 - 14 66

55 46 35 31 19 21 15 15

Calculated for the other models were the values of the radial potential derivative aV/ar on the surface of a sphere with r = 1838 km-that is, the same procedure was used as in the case of the LSM model described earlier. However, the conditional equations take a different form:

GMU

c

ei(rj - Ri cos $ij) - aV -arj

(1.13.29)

where Ri,cpi and liare the coordinates of the bodies with point masses ei to be found; rj, Q j and Aj are the coordinates of the points at which the values of the radial potential derivative were calculated; and K is the number of the point mass bodies of interest. One thousand two hundred and twenty-five conditional equations have been derived for each of the three models F77,AV and F80 describing the lunar gravitational field as a whole. Obtained for the model with 83 point mass bodies, describing the gravitational field of the equatorial zone on the near side (IcpI < 60°, IAl goo). have been 540 conditional equations. Thus, a total of

-=

K = 564 + 3675 + 540 = 4779

92

Lunar Gravimetry

equations of the (1.13.28) and (1.13.29)types have been derived. Just as in the case of the LSM model for determining the harmonic coefficients, the solution of the system of conditional equations is found from the formula 6=

[

i= 1

AT W i A i ] -

[

i= 1

AT Wix i]

(1.13.30)

where Ai represents matrices of the coefficients at ci in formulae (1.13.28)and (1.13.29); Wi = (aToi)-' represents weight matrices; and x i represents the right-hand sides of (1.13.28) and (1.13.29). Formula (1.13.30) gives an estimate of the point masses ci at which the minimum of the functional

is achieved. This minimum at a given number n of point mass bodies is attained when they are distributed over a sphere with a definite radius p, where p = f ( n ) . While solving the problem of representation of the generalized model of the lunar gravitational field by a set of point mass bodies, we become involved in a search for the number n of these bodies and the depth of their occurrence inside the Moon. This search was aimed at the appropriate values of p for a prescribed number n of point mass bodies. The result was that 225 point mass bodies at p = 1500 km (depth H = 1738 - 1500 = 238 km) approximate the initial data with an error smaller than that inherent in the initial data. At n = 225, in view of the many unknowns, we resorted to a procedure of iterative refinement of the solution. The resulting estimate was 6.2 mGal, as opposed to 7.0 mGal before the refinement. The calculated masses are listed in Table 1.14.The found point masses were used to calculate the values of the harmonic coefficients C,,,and S,,, corresponding to the field created by these masses. The coefficients were calculated using the formula

where ci = m i / M , represents the found masses. The coefficients calculated using formula (1.13.31) turned out to be very close to those determined in the LSM model, which is why they are not presented here. The resulting model of point masses permits the characteristics of the lunar gravitational field to be calculated within shorter computer time, as compared to the models based on expansion of spherical functions. Let us now compare the derived generalized model of the lunar gravitational field with those proposed by other workers. Out of the many such

93

The Gravitational Field of the Moon

TABLE 1.14 Calculated point masses in the generalized model

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45

- 392 5 24 1 297 154 79 89 425 875 550 268 384 1086 829 843 -99 676 763 1051 249 148 905 1227 515 19 56 1 652 321 526 316 - 387 1717 725 -710 1059 447 1161 -428 1014 618 197 922 1019 441 805

46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

1335 193 131 1577 - 162 -279 289 - 147 220 7 95 1086 - 53 76 746 -512 -403 - 240 - 346 237 1223 - 625 415 - 2018 543 613 63 1 -1181 140 195 -904 -457 1026 -447 1447 -478 - 322 -262 202 -694 1560 - 647 -976 125 176

91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135

- 349 2458 1229 -878 1606 -448 - 833 592 1249 577 1558 -4813 808 83 - 24 36 312 -493 -371 - 139 28 1 - 269 -755 138 838 52 1 166 362 35 - 390 - 1021 -411 -638 -1109 618 -247 1348 - 145 - 57 1244 966 -1125 - 1768 -230 396

136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180

- 706 - 1693 47 1 1213 1294 97 - 1446 -1104 - 1383 359 309 - 120 67 1 - 1475 -487 294 988 34 - 1277 - 93 48 1 -793 - 164 1339 -917 -2616 100 45 1 353 316 - 187 33 -489 - 548 56 -626 -472 - 793 579 - 175 - 505 - 705 - 1484 - 1244 - 1430

181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 20 1 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 22 1 222 223 224 225

-831 - 33 - 1345 - 1609 1220 398 -2152 - 1407 257 1491 - 1324 -306 522 - 152 -226 919 - 1428 336 - 243 969 -272 1119 212 - 70 874 -600 - 1054 1351 1702 - 1155 -2087 - 101 -263 - 1356 -409 131 - 288 -999 - 330 764 364 - 327 -415 -451 -733

models we have selected for comparison three models: F77 (Ferrari, 1977), BF (Bills and Ferrari, 1980) and F80 (Ferrari et al., 1980). Up to n = 4, all models are similar in terms of both the values of the coefficients and their

94

Lunar Gravimetry

errors. However, even at n = 4 and 5, the F77 and F80 models differ widely from the rest, the LSM model being more closely similar to the BF model than the INT one. It is difficult to compare the different models in terms of the harmonic coefficients because there are too many of them and sizeable errors are involved. This is why the subsequent comparison will be based on degree variances of the radial potential derivative, which are determined from the formula Zn+4

(.g)" (%y(n

+ 1(:)'

=

m=O

(c&,+ s:,,)

(1.13.32)

Figure 1.18 shows curves showing the trend of the quantity ( D aV'ar),"' or, in other words, the root of the degree variances of the radial gravity potential

012 1

I

3

1

I

5

I

I

7

l

l

9

I

I

11

1

I

13

I

1

15

I

I

17 n

Fig. 1.18. Degree variance in different models of the Moon's gravitational field. x , F77 model; 0, LSM-1 model; 0, BF model; soline line, LSM model.

The Gravitational Field of the Moon

95

derivative from three models: LSM, F77 and BF. Also shown is the trend of the root of the degree variance of the derivative from the LSM-1 model, which is essentially an early version of the LSM model, derived without recourse to the solution refinement algorithm. The degree variances were calculated for r = 1838 km because most of the initial data underlying the models under consideration had been determined at an altitude of about 100 km above the Moon’s surface. As can be seen from Fig. 1.18, the LSM model approaches the BF one but is somewhat below the latter because the mean value of the LSM model coefficientsis lower than in the case of the BF model. The maxima at n = 11 and n = 15, highly prominent in the F77 model and visible in the BF model, are almost imperceptible in the LSM model. Interestingly, the earlier version LSM-1 appears to be intermediate between the F77 model, on the one hand, and the BF and LSM models, on the other. The LSM-1 model is similar to F77 with the difference that, in the latter, the initial data are concentrated in the equatorial zone, while the LSM-1 model is based on both equatorial and circumpolar data from the AV and F80 models. We believe that it is precisely the broader presentation of the initial data that has lowered the extrema at n = 7,9,10,11,14 and 15. A closer scrutiny reveals that the presence of these extrema is due to the following factors; the lack of data for high latitudes has led to inadequate conditioning of the system of normal equations, which has given rise to sizeable errors in the coefficients and that, in turn, has led to a wide spread of the latter and, consequently, to high values of the degree variances. Thus, the prominent extrema in the behaviour of the degree variance in the F77 model are due to the inadequacy of the initial data rather than the actual properties of the lunar gravitational field. It should also be pointed out that the major extrema in the gravitational field of the Moon, associated with mascons and relief features, lie in the equatorial zone. Since the F77 model was constructed from equatorial data (cp < 30”), the above features have crept over into the high-latitude regions of the Moon. Consequently, the F77 model ascribes the properties inherent onIy in the equatorial zone to the entire Moon. The BF model is based on more data which, and that is rather important, are better distributed latitudinally. Furthermore, the authors have based their model on the requirement that the degree variance decrease perceptibly with increasing frequency, which is another factor responsible for a narrower spread of the coefficients. The models may also be compared through calculation of the correlation between the harmonics of every degree and within the entire set of harmonics. Estimation of the correlation for each degree is given by the correlation coefficient

TABLE 1.15 Coefficients of rank and overall correlations of the generalized model with the F77 and BF models

n

4"

F77 BF

2

3

4

5

6

7

8

0.99 0.99

0.97 0.98

0.92

0.32 0.95

0.83 0.87

0.41 0.94

0.68

0.32

0.94

10

11

12

13

14

15

16

q

0.23

0.76

0.21 0.88

0.50

0.75

0.60 0.71

0.41

0.85

0.83

0.77

0.75 0.85

0.85 0.35

0.85 0.94

9

The Gravitational Field of the Moon

97

The coefficient of correlation within the entire set-up to the degree N is given by:

When the correlation coefficient is calculated in this manner, its value is not an exact statistical measure of similarity of the two solutions because the correlation between individual coefficients is not taken into account. However, because the correlation between coefficients is insignificant in our case, the quantities q. and q are useful for examining the similarity between the solutions as a whole. The values of qn and q are listed in Table 1.15. Given in the first line is the degree n, the second line gives the coefficients q. of correlation between the degrees of the LSM and F77 models, and the third line does the same for the LSM and BF models. The last column summarizes the coefficients of correlation within the set as a whole. Analysis of the tabulated results attests to the close similarity of the LSM and BF models. The F77 model correlates well with the LSM model only up to n < 5. The high value of the coefficient q for both cases stems from the marked correlation between the low-degree harmonics whose contribution is significant.

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Chapter Two

Normal and Anomalous Gravitational Fields of the M o o n

2.1 Structure of the Gravitational Field and Its Role in the Evolution of the Moon

The manner in which the Moon’s gravitational field is distributed is rather complex. It reflects the irregularities of the internal structure of the Moon and its figure. However, if the lunar gravitational field is examined in its entirety, one of its striking features is the central symmetry with respect to the Moon’s centre of mass. This symmetry is almost perfect. Indeed, variations in the gravity potential on the lunar surface do not exceed 0.02% of its mean value W = 282 x 1O’O cm2 s-’, while those in the acceleration of gravity do not exceed 0.3% of its mean value equal to 162 cm s - ~ .How could Nature create an entirety so perfect in the midst of the cataclysmic processing that has given rise to the Solar system, including the Moon? Such an ideal gravitational field must have been the result of eons of the Moon’s evolution. Doubtless, the lower absolute value of lunar gravity, as compared to that of the Earth, has made this parameter secondary to other forces determined by temperature, pressure and other factors. This may be the reason why the Moon is farther than the Earth from the state of hydrostatic equilibrium and isostatic compensation: its body retains higher stresses, and the anomaly of its gravitational field is more pronounced. The fact that lunar gravity is lower than terrestrial gravity is also responsible for many other processes and phenomena. Gravity is known to be the major factor determining the presence of an atmosphere on planets because it prevents gas molecules from being scattered. For these molecules to be permanently retained near the

100

Lunar Gravimetry

planet surface requires that the parabolic velocity

v = (2gR)”2 where g is gravity and R is the planet radius, should be far in excess of the root-mean-surface velocity of the gas molecules. Since the lunar gravity g is low, the parabolic velocity is also low: V = 2.38 km s-’. This has resulted in the Moon having lost its atmosphere. The low relative value of g has also been partially responsible for the jagged contours of the Moon’s relief. The lunar gravity cannot overcome the molecular cohesion of the lunar rock matter, whereas on the Earth, where gravity is six times as high as on the Moon, such rock formations would tend to collapse. The Same applies to the steeper slopes of aggregations of loose materials on the Moon, as opposed to the Earth. Another result of the low lunar gravity, albeit indirect, is the peculiar Moon’s landscape. The already mentioned loss of the atmosphere has brought erosion processes to a halt with the consequence that the surface features have remained intact. On the other hand, the lack of atmosphere has allowed meteoric bodies to impinge upon the lunar surface unhindered, and their impacts have pitted the Moon. Examination of the gravitational field over the spherical surface of the Moon makes it evident that its irregularities (anomalies) are superimposed on a permanent gravitational field of a sufficiently high level. It can be readily seen that the anomalies vary in extent. The Moon has long been approximated by a triaxial ellipsoid, which is borne out by observations. The semiaxes of the ellipsoidal Moon differ in length by no more than a kilometre. This difference is responsible for the most extended anomalies in its gravitational field. The next largest anomalies are due to maria and highlands, and these are followed by anomalies due to mascons and other tectonic formations of comparable size. No matter how irregular the variations in the anomalous portion of the gravitational field are, certain statistical characteristics and the pattern of these variations can be established. To solve the various problems associated with these aspects, it is convenient to represent the lunar gravitational field as consisting of two parts-normal and anomalous. The former corresponds to a simple lunar model closely approximating the real Moon. It represents the figure, internal structure, and gravitational field of the Moon without any detail, in a most general form. The changes in the normal field follow a simple pattern which depends on the observation point coordinates and a few constants. As can be inferred from the foregoing, the anomalous gravitational field is irregular in structure. The anomalous field can be regarded as the difference between the actual gravitational field and what has been adopted as the normal one. The division of the gravitational field into normal and anomalous parts is a

101

Normal and Anomalous GravitationalFields

convention and depends on the principle underlying the definition of the normal field. The ratio between the anomalous and normal parts of the gravitational field is to a great degree dependent on the field characteristics under consideration (gravity potential, gravity itself, gravity gradients, third potential derivatives,etc.). The normal part of the gravity potential is several thousands of times greater than the corresponding anomalous part, while the normal vertical gradient is only several tens of times that of the anomalous gradient. As regards the third derivatives of the lunar gravity potential, their normal part is substantially smaller than their anomalies. Also different is the spectral composition of the various derivatives of the lunar gravity potential. In this chapter, we shall analyse the expressions for these derivatives, which are essentially expansions in terms of spherical functions. At present, the gravitational field of the Moon is known in detail given by spherical functions to within the 16th order, except for the narrow areas defined through observations via Apollos 14, 15, 16,17, and the Apollo 15 and 16 subsatellites from altitudes of several tens of kilometres above the Moon’s surface.

2.2 Expansion of Lunar Gravity Potential Derivatives in Spherical Functions

As the initial expansion, let us use an expression for the lunar gravity potential, including the potential of lunar mass attraction and the constant portion of the centrifugal-tidal potential due to the Moon’s rotation and the gravitational tidal effect of the Earth

x (Cnm cos m l

+ S,,

sin ml)P,,(sin cp)

+ GAM @p 2[-1 - 2Pzo(sincp) +

I

cos 21P22(sin4711

(2.2.1)

The centrifugal-tidal potential is the term in square brackets. Let us consider the derivatives of the lunar gravity potential at points ( p , c p , l ) of its surrounding space. To determine the radial derivative, differentiate (2.2.1) with respect to p:

- - - -m=2

x

(Cnm

cos m l

+ S,,

m=O

sin ml)P,,(sin cp)

1

102

Lunar Gravimetry

+GMce p [- - 3Pzo(sinq) + +Pzz(sincp) cos 2 4 A3

(2.2.2)

This derivative has the dimensions of acceleration. It is the major component of gravity acceleration. Now, let us find the other two acceleration components normal to the p axis. One of the components is tangential to the parallel:

(2.2.3) where

The other coefficients of this series are:

K!,: = mSnm,

MLZ = -mC,,

The remaining component is tangential to the meridian:

-- -pdcp

pz

m=O

n=z

+ Snmsin mL)[ -m tan cpPnm(sincp) + 6Pn,,+ I(sin cp)] GMce P (2.2.4) [+PZ1(sincp) + tan qPzz(sin cp) cos 2L] A3

x (Cnmcos rnn

Equation (2.2.4) has been derived using the well-known formula

where 1 0

atm
The complete horizontal acceleration component a W/al will be determined if we find

ai

103

Normal and Anomalous Gravitational Fields

This quantity also has the dimensions of acceleration. In the case of a uniformly spherical Moon, a W p l = 0. Next, consider the higher derivatives of the potential W.The second radial derivative is given by a2W

2GM,

dp2=p3

{ +- (”)’ (;’’ - + 1 1

p-’

O0

A

3

-

n=2

It

(KfL’cosmA

m=O

+ Mf$” sin mA)P,,(sin

cp)

(2.2.5)

where

and the other coefficients are

Find the mean value of the second radial derivative on a sphere with the radius p : -

-a2w ap2

j’ja2w(s)

1 4np2

ap2

S

-

The obtained value of a2 W/ap2may be regarded as the normal second radial gradient of the gravity potential, corresponding to a spherical Moon with an even or concentric density distribution. The second term in square brackets is due to the tidal effect and equals 2.5 x The quantity dZW/ap2 also varies with the distance from the Moon’s centre. Table 2.1 lists the values of a* W/ap2,calculated without taking into account the centrifugal-tidal effect. The selenocentric constant of gravitation has been assumed to be G M , = 4902.7 km3 s - ~ . Finally, consider the third radial derivative of the lunar gravity potential

(;y

-a3w - y{1 f + aP3

n=2

m=O

x (KIP,PP) cos m l

+ MkCP)sin mA)P,,(sin

cp)

104

Lunar Gravimetry

TABLE 2.1 Second and third radial gradients of the lunar attraction potential at different distances p from the Moon’s centre

- 1874.2 - 1870.9 - 1861.1 - 1225.6 -78.3 - 9.8

1736 1737 1738 2000 5000 10,Ooo

323.9 323.1 322.4 183.8 4.7 0.29

where KfW) =

M?,PP) =

(n

+ l)(n + 2)(n + 3 ) C., 6

(n

+ l)(n + 2)(n + 3) S”ln .

6

The values of the derivative a3 W/ap3= -6GM,/p4, corresponding to a uniformly spherical Moon at Merent distances from its centre are given in Table 2.1. The units of this derivative are cm-’ s-’. No instruments have been developed as yet for its direct measurement, but it can be calculated given a potential or gravitational field. As can be seen from the expressions for the coefficients K;:) and Mi:), in the case of the second radial potential derivative a’ Wlap’, they contain the factor ( n l)(n 2)/2, while the coefficients KfLP) and M f g P )in the case of a3 W/ap3contain the factor ( n + l)(n + 2)(n + 3)/6. In other words, one may expect high values of the short-wavelength components of the second, and more especially third, derivatives of the lunar gravity potential. Detailed observations of the Moon’s gravitational field distribution lend support to this assumption. The gravity anomalies shown in Fig. 1.9 correspond to horizontal gravity gradients as large as hundreds of eotvos units (E). The anomalies of the second radial derivative of the gravity potential may be of the same order of magnitude. The above expressions for the derivatives permit their values to be determined at random points in the space around the Moon. To determine them at points on a particular surface (physical surface of the Moon, selenoid, ellipsoid, etc.), one must solve these equations together with that for the radius vector p ( q , A) describing the corresponding surface. Their solution implies exclusion of the radius vector p. Then, the formulae for the derivatives will be functions of only two angular variables rp and I and will give the distribution of these derivatives on respective surfaces. ~

+

+

Normal and Anomalous Gravitational Fields

105

2.3 Expansion for Gravity

It is appropriate now to consider at greater length the derivation of a formula to define gravity g(p, cp, A). Apparently, at any point it can be determined as the square root of the sum of squares of the components along the three orthogonal coordinate axes p, cp and k,

(2.3.1) Since (a W/ap)’ is substantially greater than the other two addends in the right-hand side of (2.3.1), the equation can be rewritten as

This expression represents the first two terms of the expansion of the righthand side of (2.3.1) in terms of the degrees of the relation of the above addends to the square of the radial derivative aW/ap. Using (2.2.3) and (2.2.4), we shall write the expressions for the squares of the components (l/p)(a W/acp),(l/p cos cp)(a WpA). Only those terms will be retained in the namely, the terms with expansions whose relative value exceeds 1 x coefficients CiO, c:2,c20c22 and CZOc31

(2.3.3)

The expansion of gravity into a series will be performed as Bursa (1975) did for terrestrial gravity. Our expansion is different in that we retain the terms (2.3.3) essential for the Moon. Thus, confining ourselves in the expansions for the squares of (l/p)(d W/ap)and (l/p cos cp)(a W@A)only to the terms (2.3.3),

1

106

Lunar Gravimetry

Here and in what follows, we omit the argument sin cp of the Legendre functions P,,(sin cp) for brevity. Substitution of (2.3.4) into (2.3.2) and simple transformation give

In order to eliminate the squares and products of the associated Legendre functions, we shall use the following auxiliary relations:

3 72 6 6 tan2 cpp:, = - - P~~+ 3P22 = -- p40+ - pzo+ 35 35 7 5

4 7 tan cpPZ1 P31 = - PS1 -P31 7 5

+

1 cos2 21 = 2

+

36 -P l l 35

+ cos4n 2

1 cos4n sin2 2A = - - 2 2

These relations enable (2.3.5) to be transformed into g(p, cp,

n) =

~

1

aaPw {

+

(X -

- (c:,

+12c9

(2.3.6)

107

Normal and Anomalous Gravitational Fields

6 12 c20c22~22 cos 2a - - c20c22~42 COS 2a 7 35

--

Substitute expression (2.2.2) for a W/ap into (2.3.7). The transformations give the final expression for the distribution of gravity g at random points (p, cp, A) on the Moon's surface:

The other harmonic coefficients take the form gnrn

= (n

+ l)Cnrn,

hnrn

= (n

+ 1)S.m

(2.3.10)

Thus, the coefficients(2.3.9) and (2.3.10) of expansion of the lunar gravity g are expressed in terms of harmonic coefficients C., and S.,.

108

Lunar Gravimetry

2.4 Selenoid

First of all, a definition must be given to the figure of the Moon. Distinction is made between several figures, each representing particular properties of the Moon, including the geometrical, dynamic, hydrostatic, equal gravity potential (equipotential), equal gravity, and other figures. By the geometrical figure the physical surface of the Moon is understood. All other figures are defined by some imagined surfaces. The dynamic figure is an ellipsoid exhibiting certain dynamic properties displayed by the moving Moon. The hydrostatic figure of the Moon is characterized by an equal-pressure surface. Also used is a figure with equal values of the gravity potential at all points on its surface, known as the level surface. Of great importance in lunar studies may be surfaces of equal gravity, equal radial gravity gradient, and so on. Depending on the problem to be solved, each figure is examined in every detail or a smoothed surface is considered, only generally representing the true figure but simple in shape (sphere, ellipsoid, spheroid) and, therefore, convenient to use. The dynamic figure is always an ellipsoid, namely, that of the inertia of the Moon. All of the above-mentioned figures of the Moon will be discussed below. Here, we should like to dwell on the level surface whose equation is derived by equating the expression (2.2.1) for gravity potential to a constant Wo and the resulting expression is regarded as an equation for a closed surface. This will be an equation with three variables p, @ and R. Varying Wogives other level surfaces. In the case of the Earth, one of the level surfaces coinciding with the averaged surface of the seas and oceans is known as the geoid. By analogy, the Moon is represented by the selenoid. The absence of a water surface on the Moon does not complicate the solution of the selenoid definition problem. In so far as the selenoid is an arbitrary surface, it may be assumed to represent one of the level surfaces sufficiently close to the averaged physical surface of the Moon. In order to fix it, a level surface passing through a point on the Moon is considered. For example, the selenoid may be tied to a point where gravity is known from direct measurements. If G M , is a known value and go is the measured gravity on the Moon’s surface, the constant Wo may be found from

w, =

~

RO

=

(GM,go)’/’

Thus defined, Ro may be named a dimension factor (Bursa, 1969). The level surface equation takes the form

+ S,,

sin m,l)P,,(sin cp)

109

Normal and Anomalous Gravitational Fields

Pzo(sincp)

+ 21 cos 2;1Pz2(sincp)

= GM/Ro

-

(2.4.1)

To describe a surface in terms of (2.4.1) is not convenient, which is why it has to be transformed so that the radius of the surface is a function of the angular coordinates cp and 1,that is, P =

4

It is difficult to find an exact solution of (2.4.1) with respect to p because we are dealing with l/p to various powers. Here, the method of successive approximations is used. Retaining the terms with the squares and products of (2.3.3) in the expansion gives p(cp, A) = R O { 1 +

z2(g)’

(c,, cos rnA

m=O

Pzo(sin cp)

+ s,, sin rnl)P,,(sin

+’-21 cos 21Pz2(sincp)

+ 2C20C22cos 21PZo(sincp)PZ2(sincp) + C:,

cp)

1

cos’ 2IP:,(sin cp)]

I

(2.4.2)

C2oC31 cos lPzo(sin~ ) P ~ ~ (cp)s i n Using the auxiliary relations (2.3.6) as well as the relations 18 Pl0 = jjP40

+ -27P z o + -51

3 72 48 Pg2 = 35P44= - P40 - -P20 35 7 3 2 PZOPZZ = -p42 - - p22 35 7

+ 245

-

(2.4.3)

and grouping the terms with the same orders of spherical functions, we can write with the following expression for the radius vector p of the level surface:

110

Lunar Gravimetry

where the coefficients A,, and B,, are related to the harmonic coefficients C,,, and S,,, as follows: 24

All

= --(-)CZOc31 18 R

7 Ro

A44

=

(&-[c4.

A 5 1 =(&-[c51

-7cZOC3l lo

1

For the rest of the coefficients included in expansion (2.4.4), the following expressions hold: (2.4.6) Thus, expression (2.4.4) respresents the radius vectors of points on the selenoid surface as an expansion in spherical functions. The coefficients of this expansion depend on the harmonic coefficients C,, and S,, of the Moon's gravitational potential, on the value of p - l which is the ratio of the Earth's mass to that of the Moon, and on the adopted constant Wo of the gravity potential. The following features of expansion (2.4.4) should be noted. The mean radius of the selenoid becomes equal, as can be seen from expansion (2.4.4), to

Normal and Anomalous Gravitational Fields

111

R* = RoAoo instead of Ro. Assuming that R/Ro = 1, p-’ = 81.30, Ro = 1738 km, and A = 384,400 km and using formula (2.4.5), we shall find the mean radius increment AR = R* - Ro = 435 cm. The term with the coefficient A l l , which appears in expansion (2.4.4), is indicative of the selenoid volume centre shifting along the x axis with respect to the initial origin of coordinates, coinciding with the Moon’s centre of mass. The amount of this shift has been found to be about 1 cm-negligibly small. The nonlinear corrections to the harmonic coefficients C,, and S,, in the coefficients A,, and B,, (2.4.5) are small, too. The above formulae have been derived for estimations and practical use in the future when more accurate measurement data become available. Since the selenoid is a three-dimensional figure, it is represented as a map of elevations of its surface above the reference surface of a sphere, ellipsoid, or spheroid. Figure 2.1 shows the isolines of selenoid elevations above the sphere with R = 1738 km for the near and far sides, derived from the results presented in $1.13 of Chapter 1. It can be seen that the elevation of the selenoid surface varies within f500m. It should be remembered for comparison that the variation in the elevations of the geoid over a triaxial ellipsoid (Fig. 2.2) does not exceed & 100 m. The geoid is a more regular figure than the selenoid. This becomes more apparent if the variations in the elevations of the geoid and selenoid are related to their radii. In terrestrial measurements, the geoid is a necessity because the elevations of the Earth’s surface are obtained through levelling with respect to the geoid (sea level). The geoid is required as an intermediate surface used as a reference in measuring elevation of the Earth’s surface. Those of the Moon’s surface are determined differently. They are measured by an externally stationed observer virtually as geometric distances between points on the physical surface of the Moon and a geometric centre. Therefore, there is no need to use the selenoid in studying the geometrical figure of the Moon.

2.5 Figure of the Normal Moon

The solution of most problems can be simplified if a simpler level surface is involved instead of the rather complicated selenoid. Corresponding to this surface will be a simpler external gravitational field representing the real field minus small-scale details. This simple surface may be a sphere, a spheroid, an ellipsoid of rotation, a triaxial ellipsoid or whatever. In this case, the angular velocity of the model of the Moon is assumed to coincide with that of the real Moon. It is also assumed that their masses and the parameters representing the invariable part of the tides due to the Earth’s gravitational effects are equal as well. The model of the Moon satisfying all these conditions is

Fig. 2.1. Map of the moon showing selenoid elevations relative to the surface of a sphere with radius R = 1738 km (isolines taken at 100 m intervals). Scale 1 : 20,000,000.

Fig. 2.2. Geoid elevations relative to an ellipsoid of rotation with a polar flattening of 1/298,256 (isolines taken at 10 m intervals)

114

Lunar Gravimetry

referred to as the normal Moon, and its gravitational field is known as the normal gravitational field. That part of the gravitational field which is governed by the irregularities in the Moon’s internal structure and figure is called the anomalous gravitational field. It is smaller in magnitude and has an uneven, irregular distribution. Two approaches are used in constructing the figure of the normal Moon and its normal field. The first approach involves expansion of the real lunar gravity potential in spherical functions (2.3.l), in which several principal terms are retained. The resulting gravitational field is regarded as normal. If the truncated expression for the gravity potential is equated to a constant, the result will be an equation for the surface of the normal Moon’s figure. We have to decide which terms of expansion (2.3.1) can be considered principal. This is a problem because it is difficult to identify one or two dominating terms in expansion (2.3.1) of the lunar gravity potential. If too many terms are assumed to be principal, the normal Moon’s figure will become too complicated. As is known from observations, the most pronounced gravity anomalies on the Moon are given by harmonics with Pzo(sin cp), cos 2RPz2(sincp), and cos RJ’31(sin’cp). Therefore, taken as the surface of the normal Moon may be-a level surface given by a shortened potential expansion:

C31 cosAP,,(sincp) x

[--2 3

5 Pzo(sin cp) 3

+ -21 cos 2APz2(sincp)

(2.5.1)

Apart from the terms associated with the lunar mass attraction, retained in this expression are those representing the invariable part of the centrifugaltidal potential. To find the normal Moon figure, equate the right-hand side of (2.5.1) to a constant GM,/R,. The resulting expression for the radius of the normal Moon’s surface is a particular case of formula (2.4.2) at n < 6. The coefficients of expansion in spherical functions are given in formulae (2.4.5). Since formula (2.5.1) giving the gravity potential of the normal Moon coefficients S,,, only coefficients A,, are left in the equation of the normal Moon’s gravity potential. The expression for the radius of this surface will are taken into have the following form if terms with magnitudes up to account:

115

Normal and Anomalous Gravitational Fields

+ [A1lPll(sin cp) + A31P31(sincp) + A5,PSl(sin cp)] + CA22P22(sin cp) + A42P42(sincp)] cos 2A + A44P44(sincp) cos 4A}

cos A

(2.5.2)

The equation contains zonal, tesseral and sectorial spherical functions; that is, more than the simple “triaxiality” of the normal Moon is taken into account. The polar flattening is given by the terms with A 2 , and A40. Of paramount importance is the harmonic with A2,, which is greater by almost two orders of magnitude than that with ,440. The purely longitudinal variations in the radius of the surface are given by the terms with A 2 2 and A44. In addition to these two harmonics, the axial asymmetry of the surface is provided by those with A 3 1 , A41 and AS1; that is, harmonics whose second subscripts are nonzero. The term with A J 1 is predominant. The other approach to constructing the normal Moon’s figure resides in the following: a figure, such as an ellipsoid of rotation or a triaxial ellipsoid, is chosen first. The ellipsoid is assumed to be a level (equipotential) one, which means that the gravity potential on its surface is constant. It is also assumed that the mass, angular velocity of rotation and tidal effect of the Earth are equal for the real Moon and the model. The parameters of the ellipsoid representing the normal Moon are determined from the following condition:

(2.5.3) where p and pe are the radii of, respectively, the selenoid and normal ellipsoid surfaces. The ellipsoid of rotation is characterized by two parameters: semimajor axis a and polar flattening c1 = (a - c)/a; four parameters determine the figure of the triaxial ellipsoid semimajor axis a, polar flattening a, equatorial flattening a1 = (a - b)/a, and longitude A, of the semimajor axis. The equation of the triaxial ellipsoid in a rectangular system of coordinates is

-+ u2 X2

Y2

Z2

aZ(1 - a1)2 + a2(1 - a)2 =

(2.5.4)

The coordinates x, y and z can be expressed in terms of the polar coordinates p, cp and A: x = pe cos cp cos (A - A,) y = pe cos cp sin (A - A,) z = pe sin cp

116

Lunar Gravimetry

These coordinates are substituted into (2.5.4), which is expanded into a power series in terms of a and al. Retaining the terms of the second order of smallness with respect to a’, a: and aal gives pe(cp, A) = a(1 - a ) ( l - a1)[ 1

+ +E(2 - a ) cos2 cp

+ +a1@ - a l ) sin’ cp + f a l ( 2 - al)sin2 cp

+ )a1(2 - a1)(1 - a)2 cos2 cp cos2 (A - 10) + $a2 C O S ~cp + $a: sin4 cp + $a; C O S ~cp C O S ~(A - A,) + 3aal sin’ cp cos2 cp + 3aa1 C O S ~cp cos’ (1 - no) + 3 4 sin2 cp cos’ cp cos2 (A do) -

(2.5.5)

The following relations

+ cos 2(A - A,)]/2 sin’ cp = (1 + 2PzO)/3 (A - no) = f + 3cos (A - no) + + C O S ~(A - A,)

cos2(A - 2,) = [l

COS~

(2.5.6) c0s4

cp = 8 15 - ‘2 16 p20

+AP4O

= 3p22 - &p42

sin’ cp cos’ cp = &

+ &pz0

=h

p44

- & ~ 4 0=

+ &P42

hpZ2

are used to reduce (2.5.5), giving the radius of the triaxial ellipsoid surface, to the following form:

+ azoPzo + (az2cos 21 + bZ2sin 2 4 P Z 2 + a40P40 + cos 2A + bzz sin 21)PZ2 + cos 22 + b42 sin 2A)P42 + (a44cos 4A + b44 sin 4A)P44

pe(cp, 1) = a00

(a42

(a42

(2.5.7)

where aoo = a[1 a20 a40

-

3.

+ = a[%a2 + = a[-ja

-

ial - $2

3.1

- 3.2

- l5m 21 - 5ma11 4

,

+ &a: + 3aaJ

- +&la]

(2.5.8)

Normal and Anomalous

117

Gravitational Fields

The equation representing the difference between the radii of the selenoid (2.4.5) and the normal Moon’s ellipsoid (2.5.7) is written as follows:

dcp,4- P ~ ( P4 , = Ro* - aoo + R,*All cos AP21(sinrp) N

n

1 C(Ro*Anm - a n m ) cos mA + n1 =2 m=O

+ (R,*Bnm- brim) sin mA]P,,(sin

(2.5.9)

cp)

The radius pe(cp, A) (2.5.7) of the triaxial ellipsoid is related, through coefficients anmand b,,, to several parameters, namely; semimajor axis a, polar flattening a, equator flattening a, and longitude loof the semimajor axis. By varying these parameters one can obtain a triaxial ellipsoid approximating the selenoid as closely as possible. As is known from the previous section, the selenoid is derived from the observed gravitational field of the Moon. Therefore, the coefficients R;Anm and R,*B,, in (2.5.9), coresponding to the selenoid, are considered to be known. The unknowns of interest are the four parameters of the triaxial ellipsoid: a, a, a, and Lo. When the parameters of an ellipsoid of revolution are determined, it is assumed that a, = l o= 0. When a spherical surface is selected as that of the normal Moon, only one parameter needs to be found radius R = a. In his work, Buzuk TABLE 2.2 Maximum values of some characteristicsof the lunar gravitational field, calculated relative to different

reference surfaces Plumb-line deflection components Gravity anomalies,

Perturbing potential,

Ag (mGal)

T (lo6 cm2 s - ~ )

Relative to a sphere -240-+135 - 10,796-+ 8665 Relative to an ellipsoid of rotation -24O-+ 163 -9755-+6438 Relative to a triaxial ellipsoid - 10,734-+ 5383 -246-+ 137

t (“1

v(“)

+221

- 1 8 6 + 263

-213-+ 188

- 186-+ 263

-259-

-201-

+200

-235-+251

118

Lunar Gravimetry

(1975) selected such surfaces which satisfy the minimum condition (2.5.3). The selenopotential model proposed by Gapcynski et al. (1969) was used in constructing the normal Moon’s figure. As a result, the parameters of the optimal sphere (R = 1736.41 km), optimal ellipsoid of revolution (semimajor axis a = 1736.68 km and polar flattening a = 1/3147), as well as optimal triaxial ellipsoid (the same semimajor axis a and polar flattening a, al = 1/5028, and lo = 3” W) have been determined. Table 2.2 lists the maximum values of the perturbing potential T, gravity anomalies Ag, and plumb-line deflection components 5 and q, corresponding to these three surfaces. It can be seen it does not actually matter which surface has been chosen to represent that of the normal Moon. 2.6 Distribution of Normal Gravity

In accordance with the two drastically different approaches to constructing the surface of the normal Moon, the field of normal gravity y can be formulated in two ways. In the case of the first approach, gravity is determined from formulae similar to (2.3.1) and (2.3.8). The only difference is that a “shortened” potential W is used instead of the initial gravity potential (2.2.1). This will be a gravity potential including, in addition to the principal zeroth harmonic of G M , / p , those harmonics with Pzo(sin cp), Pz2(sincp) cos 22, Pz2(sincp) sin 2A, PJl(sin cp) cos A and PJl(sin cp) sin A. The acceleration due to normal gravity y will be determined similarly to gravity g(p, cp, A) in $2.3, with the difference that only the harmonics mentioned above will be retained:

+ hz2 sin 2A)PZ2+

1

+ hJ1 sin A)PJ1

(3

(931

cos A (2.6.1)

The coefficients go, g20, g22, h z 2 , 9 3 1 and hJ1 are expressed in terms of equations (2.3.9). Let the normal Moon’s surface be that of a spheroid:

Acp, 4= ROC1 + CzoPzo(sin cp)l

(2.6.2)

This equation can be used to eliminate p from (2.6.1), bearing in mind that coefficients (2.3.9) are also dependent on p. Performing the ncessary transformations and assuming that R = Ro, we obtain the following

119

Normal and Anomalous Gravitational Fields

expression for normal gravity at points on the normal Moon's surface (2.6.2):

where

(2.6.4)

y g = -35 162 c:, yk:' =

+ 76 P - 1 c 2 0 ( 3

3

-3s 36 c 2 0 c 2 2 - 70 3 p- 1 c 2 , ( 33 40

yyi = --7 c2 0 c 3 1 Normal gravity y at points on the normal Moon's surface (2.6.2) becomes dependent only on latitude cp and longitude 1.The predominant role in the distribution of gravity is played by the harmonics included in the initial equation (2.5.1). The additional harmonics P l l cos 1, P42cos 21 and PS1cos 1,as well as the small corrections to the coefficients of the principal harmonics, have emerged as a result of retention of the terms (2.3.3) in the expansions. The resulting variation in gravity does not exceed 0.01 mGal. The dimension factor Ro calculated from the measured values of lunar gravity (see Table 1.8) is not representative of the Moon in its entirety. The

120

Lunar Gravimelry TABLE 2.3 Values of the coefficients in the normal gravity formula

Values of the codlicients in

Y E:,:Y

The generalized model 0.515 x -0.2044 x 0.683 x -0.447 x 0.980 x 0.218 x 0.193 x 0.479 x 0.286 x

Ferrari's model (1977) 0.615 x -0.1923 x 0.621 x 0.756 x 0.1168 x 0.259 x 0.192 x 0.457 x 0.341 x

10-3 loW4 10-5 10-4

lo-'

1O-j

lo-' 1O-j 10-4

value of Ro is underestimated because gravity was measured only on the Moon's near-side surface, which is nearer to its centre of mass than the farside surface. In our calculations of the normal gravitational field, the dimension factor Ro in formulae (2.6.4)was assumed equal to 1738.0 km, the mean radius of physical surface of the Moon. Using the harmonic coefficients C,, and S,, listed in Table 1.13, taking the specified value of R o , and assuming that Ro/A = 4.57 x lop3and p-' = 81.3, we have calculated the coefficients yb',' and:;7 (2.6.4), and the results are given in Table 2.3. Table 2.4 lists the values of normal gravity y ( q , A) for some points on the surface of a symmetrical normal Moon having the shape of a spheroid. Long before the lunar gravitational field began to be explored from satellites, Grushinsky and Sagitov (1962) derived the formula for normal TABLE 2.4 Values of the normal gravity y ( q , A) at various points on the normal Moon's surface, in mGal

rp (deg)

0

60

120

180

240

300

-90 -60 -30 0 30 60 90

162,285 162,330 162,327 162,310 162,327 162,330 162,284

162,302 162,308 162,300 162,308 162,302

162,271 162,304 162,325 162,304 162,271

162,263 162,316 162,358 162,316 162,263

162,257 162,301 162,333 162,301 162,257

162,291 162,307 162,311 162,307 162,291

The coefficients from the generalized model (Table 1.13) have been used

121

Normal and Anomalous Gravitational Fields

gravity distribution: y(cp,

1) = y.(1

- 0.000,37 sin2 cp

+ O.O00,08 cos2 cp cos 2 4

(2.6.5)

They used the differences in lunar inertia moments, known from astronomical observations of the Moon. The latter formula has led to the interesting finding that lunar gravity diminishes from the equator poleward in spite of the Moon’s polar flattening, while in the case of the Earth gravity increases from the equator toward the poles. The decrease in the normal lunar gravity is about 60 mGal, the range of its longitudinal variation along the equator being about 25 mGal. If only terms with yi‘ and yil.J are left in the normal gravity formula (2.6.3) and the numerical values of harmonic coefficients C,, and S,, from Table 1.13 are used, then y(cp,

1) = 162,306(1 - 0.000,92 sin2 cp

+ 0.000,077 cos2 cp cos 21) mGal (2.6.6)

In deriving expression (2.6.6), it was assumed that in formulae (2.6.4) Ro = 1738.0 km, A = 384,400 km, G M , = 4902.71 x lo9 m3 s - and ~ p - ’ = 81.30. So the poleward decrease in gravity, which looked paradoxical in 1962, has proved to be a fact. The other approach to constructing a normal gravitational field is based on the derivation of formulae descriptive of its distribution over a body of a preselected shape, which is usually an ellipsoid of revolution or a triaxial ellipsoid. It is assumed that we are dealing with a level ellipsoid, which is to say that the gravity potential is constant all over its surface. The formula for gravity distributions over the surface of a triaxial level ellipsoid was derived for the first time by Mineo: ~ ( c p ’1 ‘

+ by, cos2 cp sin2 1 + cy, sin2 cp cos2 cp cos2 1 + b2 cos2 cp sin2 1 + c2 sin2 cp)1/2

ay, cos2 cp cos2 i (a2

(2.6.7)

where a, b and c are the semiaxes of the ellipsoid, y,, Y b and yc are the normal gravities at the ends of the semiaxes. The known Somigliana’s formula for a level ellipsoid of revolution is a particular case of Mineo’s formula. Using the smallness of the following quantities: u = (a - c)/a (polar flattening of the ellipsoid), u1 = (a - b)/u (equatorial flattening), B = (y, - y,)/ya and p1 = (y, - Yb)/Ya (ratios of the gravity differences at the ends of the semiaxes to gravity y,), one can express (2.6.7) as a power series of these small values. In the expansion for the Moon, the terms with a l and PI must be retained to the same order of smallness as tl and p. Restricting ourselves to the second order of smallness of u, /I, u1 and B1, we obtain the following formula for the

122

Lunar Gravimetry

distribution of normal gravity y over the surface of a triaxial level ellipsoid: ~ ( c p , 1.) = 70Cl + ~20P20(sin cp) + y40(sin cp) + 722 P d s i n cp) cos A

+

742 P42(sin cp)

cos 21 + y44P44(sincp) cos 4A]

(2.6.8)

where 70 =

7eCl

+ 9 + +Pl + &-2@

- 2UlP1

+ aal + a l p + pla)] = [@ - fPl + &(a: - a2 2aB - alpl + Mal + a l p + p l a ) ] 3 a2 3 1 alp1 + + - ci: - - (ala + a l p + p l a )] 35 2 16 2 - a2 - a:

720

-

-

g-sl + h a : + $(%PI aal - .1P 1 1 2 742 = i K d T a 1 + alp1 aal - up1 - alp) 1 1 2 744 = m ( r a + N l P d Y22 =

-

(2.6.9)

-P141

-

The similar relations derived by Zhongolovich (1952) for the Earth are a particular case of (2.6.9). To distinguish between the two approaches just described, the former can be defined as “Helmert’s” and the other as “Clauraut’s”. It is by resorting to the first approach that the famous German geodesist Helmert (1 843-1917) derived, from the scant terrestrial gravity measurements at that time, the first formula of normal Earth gravity distribution which still enjoys wide currency. The theory of construction of the external gravitational field of a planet with a preselected level surface was set forth by the remarkable French scientist Alexis Clairaut (1 7 13-1 765) in his book “Theorie de la figure de la terre tir6e des Principes de l’hydrostatique” (Theory of the Earth’s Figure, Based on Fundamentals of Hydrostatics) published in 1743, which was a landmark in the theory of planets’ figures. This theory is based on the second approach to constructing the normal gravity field of a planet.

2.7 Surfaces of Equal Gravity and Equal Radial Gravity Gradient

The same approach used in deriving the equipotential (level) surface can be used to determine the surface of equal gravity and that of equal radial gravity derivative or, in other words, vertical gravity gradient. Sagitov and Tadzhidinov (1983) developed equations for radii of surfaces of equal gravity and equal radial gravity derivatives and obtained the isolines of elevations of these surfaces. In doing so, they equated the right-hand sides of (2.3.8) and

123

Normal and Anomalous Gravitational Fields

(2.2.5) to constants. This procedure yields implicit equations for these surfaces. The equation of equal gravity surface takes the form

x

[+ 1

y:(

$l

io

(g., cos m 1

+ hnm sin mA)P,,(sin

cp)

1

= go

(2.7.1)

For the surface of equal vertical gravity gradient we have

(2.7.2) In deriving the radii of these surfaces, we shall restrict ourselves to the linear terms with respect to g,,, h,,, K,, and M,,, while ignoring the terms representing the centrifugal-tidal . effect. Equation (2.7.1) will give us an explicit equation of the surface, in which the radius pe of the equal gravity surface is expressed in terms of the angular coordinate cp and 1:

+ h,,

sin mA)P,,(sin cp)

1

(2.7.3)

1

(2.7.4)

where po = (GM/go)1/2. The coefficients grim and h,, are expressed in terms of (2.3.9) and (2.3.10). Equation (2.7.2) was used to find the radius pe, of the surface of constant radial gravity gradient:

+ M,,

sin ml)P,,(sin cp)

for ghl) = 162,718 mGal the equal gravity surface is represented in Fig. 2.3a in the form of elevation isolines with respect to a sphere having radius pb1) = 1738.0 km. The spread of elevations of the equal gravity surface is more pronounced than that of elevations of the selenoid. In this case, the elevations are as high as 1500 m. Yet more pronounced must be the spread of elevations of the surface of equal radial gravity gradient (Fig. 2.4a). Let us now see what form a surface on which gravity equals half that

124

Lunar Gravimetry

-120"

-60"

0"

60"

120"

180"

(a) Fig. 2.3. lsolines of equal gravity surface elevations: (a) g = 162,718 mGal, the elevations are taken relative to a sphere with pi" = 1738 km (isolines taken at 500 m intervals); (b) g = 81,356 mGal, the elevations are taken relative to a sphere with p t ) = 2454.82 km (isolines taken at 100 m intervals); (c) @ = 20,340 mGal,theelevationsaretakenrelativetoaspherewithpi5) = 4909.7 km (isolinestakenatloo m intervals).

labelled gh'), that is gh') = 81,359 mGal, will take. The elevation isolines of this surface with respect to a sphere with radius p f ) = 2454.82 km are shown in Fig. 2.3b. Another set of isolines was constructed in order to represent a surface with $2) = i$$ = 20,340 mGal with respect to a sphere having radius po = 4909.7 km (Fig. 2.3~).In a similar manner, surfaces of equal radial gravity gradient (d2g/dp2), were derived for the same distances from the Moon's centre of mass: ph2) = 2454.82 km and pb3) = 4909.7 km. The elevations of the equal gradient surface ( m / d p z = (ag/dp)!,l) = 662.8 x s-l) vary within 3000 m with respect to a sphere with pb2) (Fig. 2.4b),this variation being already within 300 m with respect to a sphere with pb3)(d2W/47'= 82.8 x lo-' s - ~ )(Fig. 2.4~).If the range of the elevations of the equal radial gradient surface was greater by one order of magnitude than that of the elevations of the equal gravity surface near the lunar surface, at ph,) = 4909.7 km they would become the same. This is

Normal and Anomalous Gravitational Fields

125

further proof that the anomalies of the radial gravity gradient diminish rapidly as the distance from the Moon increases.

2.8 Anomalies of the Lunar Gravitational Field

The anomalous part of the gravitational field results from subtraction of the normal component from the observed gravitational field of the Moon. It is due to the uneven density inside the Moon and irregularities of its future. It is precisely the anomalous part that arouses the greatest interest at the current stage of gravitational studies of the Moon. The natural evolution of science implies establishment, at the initial stage, of the general pattern of a phenomenon. Advances in measurement techniques and instrumentation open up new possibilities for investigating the “fine” structure of an object or phenomenon. Anomalies of the lunar gravitational field give a new insight into the internal structure and figure of the Moon. Definition of the gravitational anomalies (gravity, gravity potential, gravity gradient, etc.) is a

126

Lunar Gravimetry

rather conditional operation in the sense that the magnitude of the anomalies depends on the manner in which the normal field has been defined. The normal part may include some or other harmonics of gravitational field expansion, or else the normal Moon is regarded as an ellipsoid of rotation, a triaxial ellipsoid, or a sphere. All these factors are taken into account in using and interpreting particular gravitational anomalies. We have already mentioned, in different contexts, the perturbing potential T(p,cp, A) which is defined as the difference between the real gravity potential (2.2.1) and the normal gravity potential (2.5.1): In the general form, all gravity anomalies can be written as: where y is normal gravity, g is the observed (measured) gravity, and 8, is a reduction correction whose structure determines the nature of the anomaly,

127

Normal and Anomalous Gravitational Fields

90.1 -180"

I

-120"

-60"

I

0"

60"

I

120"

I

180"

Fig. 2.4. lsolinesofequal radial gravitygradientG/dp2surfaceelevations (a) m / d p z = 1874.8 E; the elevations are taken relative to a sphere with pi') = 1738 km (isolines taken at 5000 m intervals). (b) m a p z = 662.8 E;theelevationsaretakenrelativetoaspherewith p,") = 2458 km (isolinestakenat1 00 in intervals). (c) d2W/dpl = 82.8 E;theelevations?wtakenrelativetoaspherewithpp' = 4909.7 km (isolines taken at 100 m).

Ag, of interest. In some cases, 6g takes care of the effects of the relief, while in others it enables reduction of the measured gravity to another surface. The normal gravity y(p, cp, A) at random points in space can be calculated using formula (2.6.1). But if the normal gravity y(cp, A) (2.6.3) corresponding to the normal Moon's surface is involved, some additional reductions are required. Let us represent the quantity y ( p , rp, A) as a power series expansion in terms of elevations H over the normal Moon's surface and restrict ourselves to the first three terms of the expansion

where -y(A, cp) is the normal gravity on the normal Moon's surface, (ay/i3p)o = a* W / a p 2 and ( f i / a p 2 ) o = m / a p 3 are the first and second radial

128

Lunar Gravirnetry

gradients of normal gravity (or first and second normal radial gradients of gravity). Their values have already been given in Table 2.1. In the case of a sphere whose radius is Ro = 1738 km, ($)o

=

-0.1874 mGal m-l

(3) 8p2 = 324 x

(2.8.3) mGal m-2

Remember, for comparison, that the mean vertical gradient of terrestrial gravity is -0.3086 mGal m- ’. In spite of the Moon’s mass being 81.3 times smaller than that of the Earth, the vertical gradient of lunar gravity is only one and a half times smaller than that of terrestrial gravity because of the Moon’s shorter radius. The so-called free-air gravity anomalies can find the widest aplications. They are derived as the difference between the measured and normal gravities

129

Normal and Anomalous Gravitational Fields

60". 200

90" -180"

---

-120"

(C)

at the same point M P , CPY

4= g(P9 CP, 4 - Y(P, CP, 4

= g(P, CP,1) - CY(CP, A) - 0.18748 162 x 1 0 - 9 ~ 2 1

+

where H is given in metres, while Ag, g and y are given in milligals. In studying the distribution of anomalous masses inside the Moon, one must exclude the gravitational effect of all masses except for the anomalous masses of interest. First of all, it is necessary to take into account the effect of the irregularities of the Moon's physical surface. Toward this end, an imaginary horizontal plane is drawn through the observation point. Since the most tangible effect is produced by the relief in the immediate proximity to the observation point, the gravitational effect due to the excess of masses above and their shortage below this plane is calculated. In either case, the masses tend to lower the observed gravity and, therefore, the correction for the gravitational effect of the relief is always positive. Thus, introduction of the correction for relief apparently eliminates the shortage of masses between

130

Lunar Gravimetry

the physical surface and the plane and “neutralizes” their excess above the plane. Now, at each observation point, the anomaly appears to be created by an infinite plane layer of masses, whose thickness depends on the elevation of the relief above normal Moon’s surface. The attraction of such a layer having thickness H and density a at a point whose elevation above the upper surface of the plane layer is h can be derived through evaluation of the following integral: (H [r2

6gn(0,0, h) = G o 0

0

+ h - z)r d+ dr dz + (H + h - z)2]3’2

= 2nGaH

(2.8.4)

0

Here, a cylindrical system of coordinates r, t,b and z is used, with its origin on the lower surface of the plane layer and the z axis passing through the observation point. The lower surface of the layer is taken at a level coinciding with the normal Moon’s surface. Expression (2.8.4) reveals an interesting property of the plane layer’s attraction. This attraction turns out to be independent of the distance h between the observation point and the layer, but depends only on the layer thickness H and density a. Substitution of the numerical values of n and G into the right-hand side of (2.8.4) gives 6g,

= 0.0418aH

(2.8.5)

where a is expressed in g ~ m - H ~ is, in m and 6g is in mGal. The gravity anomaly which includes, apart from the free-air correction, a correction factor excluding the attraction of the plane layer of masses between the physical and normal surfaces of the Moon, is known as Bouguer’s anomaly: AgB(p, ~ p ,A) = g(p, ~ p ,A) - y(v, A) - 0.1871H

+ 0.0418aH

(2.8.6)

It is named after Pierre Bouguer, the French scientist who used this anomaly in gravimetric studies during the famous expedition to South America in the middle of the 18th century. In principle, Bouguer’s anomalies must represent the distribution of only anomalous masses inside the Moon, because they take into account the elevations of the observation points and the effects of the masses, lying intermediate between the observation points and the normal Moon’s surface. Widely used in gravimetric studies of the Earth‘s figure are so-called mixed gravity anomalies. Although they are not so important for the Moon, mentioning them would be appropriate here. They are different from free-air anomalies in that the measured and normal gravities are associated with different surfaces, the measured gravity being related to the selenoid surface and the normal gravity to the normal Moon’s surface, as

131

Normal and Anomalous Gravitational Fields

follows: A.Sin(P, cp,

4= S(P, cp, 4- Y(V, 4

(2.8.7)

As regards the anomalies of the second gravity potential derivatives, the most interesting ones are those of the radial gravity gradient, which will be defined as the difference between the observed and normal radial gravity gradients: ’

a9 T -- (1874 - 0.0022H) x lo-’ s - ~ pp

- ap

(2.8.8)

(The second term in brackets takes into account variations in the normal gradient with height H reckoned from a sphere with R = 1738 km and given in kilometres.) The anomalies of the second radial gravity gradient may be determined from the formula TPPP

=

aFg apz - ($)o

(2.8.9)

In view of the smallness of (a2y/ap2)o(2.8.3), the anomalies Tpppare virtually equal in value to a2g/ap2. When the second and third potential derivatives are involved, the effect of the relief becomes substantially more important. The plane-parallel uniform layer exerts no influence on them. On the Moon, as opposed to the Earth, the anomalies of the second gravity potential derivatives are quite pronounced. Judging from the detailed measurements of line-of-sight accelerations from Apollo and gravity profile measurements from the lunar roving vehicle of Apollo 17, the anomalies of the second derivatives are as high as hundreds of eotvos (lo-’ s-~).

2.9 Relation between the Coefficients of Expansion of Different Parameters of the Moon’s Gravitational Field and Figure

Let us now see how the generalized spherical functions of the same degree for various parameters of the Moon’s figure and gravitational field are interrelated. First, we shall determine the amount 6 of difference between the radial gravity potential derivative a W p p and gravity g (2.3.2) at the same point:

6 = aW(P, cp, aP

4 - 9(P, c p , 4

=

-

i/p2(a w/acp)Z

+ i/p2 C O S ~cp(a w/an)2 WPP)

The numerator of the right-hand side includes squares of the horizontal attraction components tangential to meridians and parallels. Assuming, for

132

Lunar Gravimetry

example, that each of them has the highest possible value of 500 mGal, it can be easily calculated that 6 = 0.25 mGal. Thus, virtually every conclusion drawn as regards the value of a W/ap fully applies to gravity g. Using the expansions of the lunar gravity potential (2.2.1) and radial gravity potential derivatives (2.2.2), (2.2.3) and (2.2.Q and ignoring the nonlinear and centrifugal-tidal terms in the latter, we shall compose a table of factors (Table 2.5). Given the coefficients of expansion of a radial derivative of the lunar attraction potential in this table, one can easily determine those of another derivative. To do that suffices to multiply the harmonic coefficients of the first expansion by the respective tabulated factors. We shall henceforth refer to these as transformation factors. For instance, the harmonic coefficients g,, and h,, of gravity expansion are given. In order to determine the harmonic coefficients K,, and M,, of the expansion of the radial gravity gradient d2 W/ap2,it is necessary to multiply, respectively, g,, and h,, by the transformation factor - (n + 2)/p according to the table. Conversely, given the values of K,, and M,,, to determine g,, and h,, the former must be multiplied by transformation factors - p/(n + 2). But since the transformation factor for any radial potential derivative is independent of m, virtually the entire sum over m is multiplied by the transformation factor-that is, a generalized spherical function representing the derivative under consideration. Consider now the relations between some characteristics of the Moon’s figure and gravitational field anomalies. The perturbing potential T is related TABLE 2.5 Transformation factors for the spherical harmonics of different radial attraction potential derivatives

av -

V

U

aP V

I

av

n+l --

-

aP a2v -

(n

aP2 a2 v -

aP3

I

P

+ l)(n + 2)

n+2 --

P2 - (n

1

P

+ l)(n + 2)(n + 3)

(n

P3

+ 2)(n + 3) PZ

0,

n+3

--

I

P

= Ku,

where u is the initial field characteristic, 0 is the transformationresult and K is the transformation factor.

133

Normal and Anomalous Gravitational Fields

to the elevations [ of the selenoid above the normal Moon’s surface in a rather simple fashion. This relation stems from the property of level surfaces, mentioned in 51.3. In gravimetry, the term “Bruns formula” is applied to the relation T

[=-

(2.9.1)

Y

where, in the Moon’s case 7 is the mean normal gravity between the surfaces of the normal Moon and selenoid. If gravity anomalies Ag are determined from observations, rather than the perturbing potential T, equations through which these characteristics of the gravitational field are related are necessary. Let us write a formula expressing mixed gravity anomalies Ag,(p, cp, A) in terms of the perturbing potential T and its derivative aT/ap, both related to the normal Moon’s surface: (2.9.2) Now we can define the transformation factors between the principal spherical functions representing the figure and gravitational field of the Moon, namely, the elevation (2.9.1) of the selenoid above the normal Moon’s surface, mixed gravity anomalies Ag, (2.9.2),the perturbing potential T (2.8. l), and the radial derivative aT/ap (2.8.4) of the perturbing potential. To this end, the characteristic should be expressed as expansion in generalized spherical functions m

(2.9.3) OD

T=

Tn

(2.9.4)

n=2

(2.9.5) m

C=

1

(2.9.6)

Cn

n=2

Introducing expressions (2.9.3)-(2.9.6) into equation (2.9.2), we find that n = 2 , 3 , ...

(2.9.7)

It is easier to determine the transformation factors between other characteristics, which can be done using expressions (2.8.9) and (2.9.3)-(2.9.6). Table

134

Lunar Gravimetry

TABLE 2.6 Transformation factors for the spherical harmonics of some anomalies of the Moon's gravitational field and figure U

n+l -

($)"

I

P

-1

r

P

I

(n + 1)Y

Y ,

u,,, = Ku,,,,,

where u is the initial field characteristic, 0 is the transformation result and K is the transformation factor.

2.6 summarizes the transformation factors for the generalized spherical functions of the anomalies of the Moon's gravitational field and figure.

2.10 Moon's Relief and Gravitational Field

The Moon's relief has a complex structure. It features vast highlands and plains. It is standard practice to represent the relief in the form of isolines of equal elevation of a smoothed physical surface of the Moon above a reference surface. The easiest way is to take a sphere as the reference surface, whose radius is assumed to be equal to the mean radius of the Moon. For numerical analysis, the relief can most conveniently be represented as an expansion in spherical or sample functions. The radius vector p(cp,A) of the smoothed physical surface of the Moon, written as an expansion in spherical functions, is N

1

+n=l

n m=O

(a,,,,, cos mA + 6,,,sin mA)P,,,,,(sin cp)

1

(2.10.1)

where N is the order of the expansion, dependent on the validity and number of the initial absolute elevations of the lunar surface, and Ro is the radius of the reference sphere. The coefficients a,,,,, and b,,,,, are determined from the available coordinates of points on the Moon's physical surface. If we go to

135

Normal and Anomalous Gravitational Fields

elevations h of the relief above the reference sphere, we have N

n

C n= 1 m=O

RO 1

(ii,m

cos m l

+ 6,, sin ml)Pnm(sincp)

(2.10.2)

where in, and Fnmare normalized coefficients, and Pnm(sincp) stands for normalized Legendre functions. Approximation of the relief by an expansion in spherical functions of the type shown in (2.10.2) inevitably involves two kinds of errors. One of these is due to the finiteness of the series, while the other stems from the expansion coefficients. Let N be the highest order of the retained terms in the expansion. Then, the first of the two errors is m

el = Ro

n

C C n=N+1

(anm cos m l

+ b,,

sin ml)P,,(sin cp)

m=O

If the errors in the expansion coefficients are labelled Aanmand Ab,,, the second error is N

e2

= RO

n

1 1( fAanmcos mA -L Ab,,

sin ml)P,,(sin cp)

n=O m=O

The overall’varianceof

8’

equals the sum of the variances of N

m

E’

=

1 n=N+l

Dn

E:

and

8;:

N

+ nC= O d n = D - C (Dn-dn) n=O

where d, stands for the degree variances of the errors involved in the determination of the coefficients an, and b,,, D , stands for the degree variances of the coefficients anmand b,, themselves, and D is the variance of the Moon’s relief elevations. The latter equation shows that it makes sense to increase the order N of the expansion by one each time Dn+ - d, + > 0;that is, if Dn+ > d,+ 1. Thus, the optimal order of the expansion depends on the ratio between the degree variances of the errors involved in the determination of the coefficients and those of the coefficients themselves. The first expansion of the Moon’s relief in spherical functions up to the eighth order was carried out by Goudas (1968) who used absolute elevations of the near-side relief. The total lack of data on the Moon’s far side has given rise to certain assumptions. Goudas proceeded from symmetry of the relief on the near and far sides. This is why his expansion could adequately describe the near-side relief and have nothing to do with the far-side relief. Since the first expansion of the Moon’s relief, more data on the latter have become available. The early selenodetic catalogues of absolute elevations on the near side were revised and compiled catalogues have been prepared (Gavrilov et al., 1977; Lipsky et al., 1973; and others). The elevations were

136

Lunar Gravimetry

reduced to the Moon’s centre of mass. The basic methods for compiling unified catalogues have been outlined (Gavrilov, 1969; Lipsky et al., 1973). The principles of establishing base networks and their relative deformations, techniques of lunar polygonometry and triangulation have also been described (Gavrilov, 1969; Gurshtein and Slovokhotova, 1971; Habibullin et al., 1972; Light, 1972; Helmering, 1973). The most serious drawback of the available elevation data is the limited knowledge about absolute elevations on the far side. This drawback has begun to be remedied by satellite observations, which have been instrumental in the determination of the selenodetic longitudes, latitudes, and absolute elevations of several fundamental base points. In particular, Wollenhaupt et al. (1972) used the results of numerous optical measurements from Apollos 8, 9, 10, 11, 12, 14 and 15 to determine 31 base points along the equatorial belt of the Moon. Their latitudes range from - 1 1 ” to + 26”;21 out of these points are on the near side and 10 on the far side. The coordinate determination errors are 0.7 ( q ) and 0.6 (A) km, while those of absolute altitude determination do not exceed 0.4 km. The coordinate errors were determined by the accuracy of location of the Apollo spacecraft. Extremely valuable data on absolute elevations were obtained by laser altimetry from Apollos 15, 16 and 17 (Wollenhaupt and Sjogen, 1972; Kaula et al., 1973, 1974; Sjogren, 1977). Elevations of the physical surface were measured along profiles (Fig. 2.5) extending across the entire Moon in its equatorial zone. They are given on the profiles relative to a spherical Moon with a radius of 1738.0 km. The centre of this sphere coincides with the Moon’s centre of mass. Laser measurements were taken from the orbiting Apollo spacecraft at 20 s intervals, which corresponds to 30 km of covered distance. Although the altimeter was sensitive to within about 2 m and the spot of the laser beams on the lunar surface was only about 30 m in diameter, there was no need to probe at shorter intervals because the uncertainty in the Apollo’s coordinates remained considerable. The position of the spacecraft was determined from the Earth by Doppler tracking at 10 s intervals. For intermediate points in time, the position of the spacecraft on its orbit was computed using a model of the lunar gravitational field. The uncertainty in the Apollo’s position in the direction of the laser beams did not exceed an absolute value of 0.4 km, while the error in the relative position between measurements was 0.1 km. Laser altimetry has made it possible, in addition to studying the Moon’s relief, to solve a number of other problems. In particular, the position of the centre of the Moon’s figure with respect to its centre of mass was determined more accurately and the geometrical figure of the Moon was defined more exactly. Some common features of the Moon’s relief were established. Table 2.7 lists the mean elevations of some features on the Moon’s near and far sides

-E *

1

re

4

o

U

-4 -81

I

90"

I

I

120"

I

I

150"

I

I

180"

I

210"

I

1

240"

I

I

270"

I

1

300"

1

I

330"

I

I

360"

I

I

30"

I

I

60"

I

I

90"

I

8-

-

-E @ Y U

1

:i-flkg+q% --4

-

-

-----

------

#

------

Fig. 2.5. Intervals in elevations of the physical surface on the near and far sides of the Moon, relative to a spherical Moon with radius R = 1738 km, measured by laser altimetry from Apollo 15 (top) and Apollo 16 (bottom) (Kaula eta/., 1973, 1974; Siogren and Wollenhaupt. 1972, 1973).

138

Lunar Gravimetry

TABLE 2.7 Mean elevations of some formations on the Moon, relative to a sphere with radius R = 1738 km (Kaula ef a/., 1973)

Mean elevation according to (km) Formation type Far-side highlands Near-side highlands Circular maria Other maria

% of

Apollo

surface

15

57 23 6 14

+ 1.9

Apollo 16

Apollo

f2.1

+0.9 - 1.3 -3.7

- 1.7

- 1.2

-4.1 -2.0

-4.1 -2.5

17

-2.1

Weighted mean elevations (km)

+ 1.8

- 1.4 -4.0 -2.3

above a sphere with R = 1738 km, determined from these measurements (Kaula et al., 1973). The tabulated elevations belong to the equatorial zone and are not adequately representative of the Moon as a whole. The elevation profiles constructed from laser altimetry data (see Fig. 2.5) attest to the qualitative difference in relief between the near and far sides of the Moon. The maria have smooth floors inclined from west to east at about 1 : 500 to 1 :2000 (Kaula et al., 1973). An inverse correlation is observed between the depths of circular maria (Mare Serenitatis, Mare Crisium, Mare Smythii) and their ’diameters. The interferometric measurements with the aid of Earth-based radars, which preceded laser altimetry, allowed determining the elevations of only large features of the lunar surface, their accuracy being in the neighbourhood of 200 m (Zisk, 1972). The first data on the far-side relief elevations in the western hemisphere were provided by the photographs taken from the automatic probe Zond 6. They revealed a wide depression on the southern far side (Rodionov et al., 1971, 1976; Ziman et al., 1975). The level of the physical surface over a sizeable area (Fig. 2.6) was about 4.7 km below the mean level of the Moon’s surface. The upland region in the south, having a relative elevation of about 2.6 km, extends into Mare Australe on the near side. Satellite data on the geometrical figure of the Moon cannot substitute for the knowledge accumulated over more than fifty years of astronomical investigation of lunar surface elevations. Yet they provide the only insight into the Moon’s far side. It was extremely important to obtain information on the overall geometrical figure of the Moon with due account for both the near and far sides as well as its relationship with the centre of mass, which has been partially accomplished by way of laser altimetry. The results of the photogrammetric processing of the pictures taken from Apollos 15,16 and 17 are going to be used in establishing a highly accurate base network covering

139

Normal and Anomalous Gravitational Fields

0"

r

+

Fig. 2.6. The Moon's physical surface elevations, in a section from a plane near to 2. = 180", from the photographs taken from Zond 6 with a 20-fold magnification (Rodionov et al., 1971).

20% of the lunar surface (including its far side) and having a density of one base point per 900 km2 (Light, 1972; Helmering, 1973). The accumulated elevation data have made it possible to achieve more or less reliable expansions of the lunar relief in spherical functions. Bills and Ferrari (1977) handled in their work the results of 5800 laser altimetric measurements, 1400 photographs taken from Apollo spacecraft with an accuracy of f0.3 km, and 3300 elevation measurements based on photographs taken from the Earth. The latter were said to be accurate to within f 1.0 km. The terrestrial measurements were corrected for displacement of the Moon's centre of mass by 1.77(f0.16) km toward a point with coordinates 25"s and 191"E. The elevations were determined with respect to a sphere with R = 1737.46 km. The map of elevations of the smoothed Moon's relief, based on the work by Bills and Ferrari (1977), covers the zone confined within zk 45" latitude. The elevations vary from +5.5 to -2.5 km; that is, the total variation range is 8 km. Proceeding from the up-to-date lunar relief elevation data, Chuikova (1975a, 1975b, 1978) expanded the relief. She used a hyposometric map of the

140

Lunar Gravimetry

Moon’s near side confined within f70” latitudinally and logitudinally, based on Mills’ catalogue, the coordinates of the above-mentioned 3 1 base points (Wollenhaupt et al., 1972), the results of laser altimetry from Apollos 15 and 16 (Sjogren and Wollenhaupt, 1973), the absolute elevations of 68 points of the liberation zone on the far side in the Moon’s western hemisphere, determined from space probes Zond 6 and Zond 8 (Ziman et al., 1973)’ the elevations determined with the aid of Zond 6 along the far-side meridional profile (Rodionov et al., 1976), as well as the catalogue of elevations in the peripheral region, compiled at the Main Astronomical Observatory of the Ukrainian Academy of Sciences of particular interest were the data provided by Zond 6 and Zond 8; they have not been used in other works. This is precisely why on the averaged relief map drawn in this work (Fig. 2.7) the regions in the southern part of the western hemisphere, polar regions, and the peripheral region better represent the real Moon. Consider now the degree variances, not of the relief elevations themselves, but of their horizontal gradients along tangents to meridians and parallels. The expressions for these degree variances as applied to the relief will take the forms n (Dh,)n =

+ Br%)

m=O

(2.10.3) The coefficients A,,,, Brim, R,, and fin,,,can be computed according to the formulae (3.3.3) where the coefficients a,,, and 6,,,of expansion of the relief Figure 2.8 presents graphs elevations must be substituted for Cnmand Snm. showing the degree variations in r.m.s. values ahq = [(Dhq)n/(2n and a h l = [(Dh,),J(2n 1)]’/2 calculated using the relief expansion coefficients (Chuikova, 1978). Autocorrelation functions of the Moon’s and the Earth‘s reliefs are shown in Fig. 2.9. Let us examine the anomalous gravitational field due to the Moon’s relief. The masses constituting the visible relief will be represented condensed on a sphere’s surface as a simple layer with surface density

+

+

On(%

4= aowcp, 4= N

aoR

n

1C n=l

(anm

cos mA

m=O

+ bn, sin ml)P,,(sin

cp)

(2.10.4)

where a. is the mean density of the relief-forming (2.10.4) rock. The gravity potential of the masses in the simple layer is V(P, cp,

4=G

jj S

CP,1)dS

r

Fig. 2.7. Elevations of the lunar relief relative to a sphere with radius R = 1738 km (isolines taken at 0.5 km intervals)

r;

40'

c

I I

'E Y E 30 E

I

I

IIII

20

n Fig. 2.8. R.m.s. values of the spherical harmonics of the horizontal relief elevation gradients along meridians and parallels plotted against position in degrees. (1 ) (uhJn along meridians; (2) ( u , , ~along )~ parallels.

143

Normal and Anomalous Gravitational Fields

where I is the distance between a current point on the surface and the point (p, cp, A) at which the potential is considered. Substitution of the expansion for l/r (1.3.2) and that for 6, (2.10.4) into this equation gives V(p, cp,

A) = GaoR

ss

c N

n

1(anrncos rnl + 6,, sin rnA)Pnm(sincp) n = 1 m=O

S

In expansion (1.3.2),it was assumed that p1 = R. By resorting to the theorem of restoration of spherical functions (Idelson, 1936), we can simplify the righthand side of the latter equation:

x

(anrncos rnl + li,

sin rnA)Pnm(sincp)

(2.10.5)

Comparison of the coefficients of the resulting expansion with the corresponding ones of expansion (2.3.4)gives (2.10.6) Introducing the Moon’s mean density au instead of its mass Mu, we can of the gravitational potential expansion to relate the coefficients Cnmand Snm

Fig. 2.9. Normalized autocorrelation functions KOn($) of the (1) Moon’s and (2) Earths reliefs.

144

Lunar Gravimetry

the corresponding coefficients an, and 6,, of the relief expansion in the following manner: (2.10.7) Assuming that the Moon’s density varies with depth as a(p) = 0 0

+ upp

where u and p are constants governing the density variation pattern, instead of formula (2.10.7), we have (Goudas, 1968, 1973): (2.10.8)

Assuming that u = 0, we have (2.10.6) instead of (2.10.8), which corresponds to the relief density being constant and equal to the mean density of the Moon. Given in Table 2.8 are the values of normalized harmonic coefficients C,,, and S,,, of the gravitational potential, calculated using formula (2.10.6) based on the coefficients of the Moon’s relief expansion executed by Chuikova (1975a,b) as well as by Bills and Ferrari (1977). This table also lists for comparison the harmonic coefficients derived by Ferrari (1 977) and Akim and Vlasova (1977) from the perturbed motion of the ALS. The values of en, and S,,,, corresponding to the relief, are much higher. Also compare the degree variances of the gravity potential calculated with reference to the Moon’s relief and determined from tracking of the perturbed motion of the ALS (Fig. 2.10). The first degree variances are much greater than the corresponding variances determined from the perturbed ALS motion. The TABLE 2.8 Comparison of the normalizedharmonic coefficientsof the gravitationalpotential, calculatedfrom the observed Moon’s relief and determined from ALS tracking data ( x

Go

Gl

From relief (Chuikova, 1975)

-607.6

-603.8

From relief (Bills and Ferrari, 1977)

-212.3 f25.8

-605.8 f17.5

Determination

Go

Ctl

G 2

-21.9

-119.3

-78.98

87.23

-147.5 f13.6

-135.9 f22.1

-149.8 f26.5

11.5 f5.7

-91.52 51.7

2.04 f1.8

f 1.8

-90.03 f 1.0

0.7 f1.4

35.47 f0.5

&I

From ALS tracking data (Ferrari, 1977) From ALS tracking data (generalized model)

0.58 f 1.7

1.97 f9

1.3 k1.4

33.66

145

Normal and Anomalous Gravitational Fields

n Fig. 2.10. R.m.s. values of the spherical harmonics ( u T ) , for the lunar gravitational potential plotted against degree, n. (1 ) Calculated from the Moon's relief; (2) determined from ALS tracking data.

difference diminishes as the order n of the harmonics increases. This pattern can be explained by the different degree of isostatic compensation of masses differing in extent. The relief masses over small areas cannot overcome the strength of the lunar crust and remain isostatically uncompensated, which is to say that they are responsible for a stronger gravitational field. As regards regions characterized by low harmonics, the isotatic compensation is virtually complete; that is, corresponding to excess of relief masses is a

c30

€31

c32

€33

$3 1

67.6

-9.i~

-16.5

63.6

102.4

47.1

23.5 k19.3

167.0 k14.6

62.2 k20.5

102.1 k28.0

47.2 k18.1

250.1 k11.1

53.1 - 18.2

+

51.9 k22.7,

28.9 k11.4

-0.23 k2.3

0.10 k2.2

-4.90 k4.5

27.04 k1.5

11.52 k3.8

24.98 k7.1

6.00 f1.7

1.18 k4.7

-9.02 & 6.8

1.26 k1.7

-1.97 k0.7

3.5 k1.6

23.29 k1.2

12.67 k0.8

15.82

k0.5

* 6.16 1.1

6.53 f0.9

-8.44 k0.4

$21

$22

54.0

1.73

-7.35

146

Lunar Gravirnetry

deficiency of mass below and vice versa. As a result, the anomalous gravitational field approaches zero, as can be seen on the grounds of Fig. 2.10. It can be inferred from Table 2.8 that the geometrical figure of the Moon cannot be derived from the coefficients of expansion of the lunar gravitational field determined from perturbations in the ALS motion. This is precisely what Volkov and Shober (1969) did following Goudas (1968, 1973). Disregarding the internal mass distribution will not produce anything close to the actual relief of the Moon.

Chapter Three

Spatial Variations in the Lunar Gravitational Field and Their Use in Studying the Figure and Internal Structure of the Moon

3.1 Covariance Analysis of the Moon's Gravitational Field

The currently existing methods for measuring the Moon's gravitational field yield information only within a discrete set of points with inevitable random errors. Handling of the gravitational field in most selenodetic and selenophysical problems reduces to its integration with a particular kernel in a certain region which, as a rule, comprises the entire surface of the Moon. There arise problems of error filtration and interpolation (prediction) of gravity anomalies in areas inadequately covered by measurements, if at all. Kaula (1959, 1967) and Heiskanen and Moritz (1967) demonstrated that these problems can best be resolved by applying the Kolmogorov-Wiener linear filtration method well known in the theory of stochastic processes. The Kolmogorov-Wiener method is based on examination of the statistical relationship involving the parameters of a random process (field) at different points. Such a relationship is characterized by a covariance function determined as the expected value of the product of the values of a random

148

Lunar Gravirnetry

function f(x) at two different points: K = E(f(Xdf(X2))

(3.1.1)

The argument x is essentially time t , if f ( t ) is a random process, or the coordinates of points in space, if f(x) is a random field. The way (3.1.1) is written implies that the random function f(x) is centred; that is, its mean value is zero:

3=~ { f ( x )=l 0

(3.1.2)

Otherwise, deviations f ( x ) -7must be substituted for f ( x ) in expression (3.1.1). E is an operator averaging over all possible realizations of the random process (field f ( x ) . In reality, we are always dealing with a single realization of the gravitational field of any object which is invariable in time, including the Moon. Then, the question arises: can we regard the Moon's gravitational field as a realization of a random field, and if so, what is the exact definition of the averaging operator E? This question and its implications were covered at length by Moritz (1980). Random processes f(t) are known to exist whose covariance can be found unambiguously from any single realization. Substituting the following operator of averaging over the function domain for the expected value operator E , we obtain: E = {f(t)f(t

li

+ At)} = T

f(t)f(t

0

+ A t ) dt

(3.1.3)

The random processes exhibiting this property are referred to as ergodic. As can be inferred from (3.1.3), the covariance function of an ergodic process does not depend on the values of the variables t l and t2 themselves, but is determined by the interval between them or, in other words, it is a function of the variable At. In his work, Moritz (1980) demonstrated that it is possible to find random fields f(0,A) on a sphere which also exhibit the property of ergodicity. The covariance function of such fields K = M{f(P)f(Q))

(3.1.4)

must be dependent only on the angular distance Ic/ between the points P(0, A) and Q(W,A'). Therefore, the averaging operator M is derived a s follows. First, one of the points (say, P ) is fixed, then, the product f(P)f(Q) is averaged over all points at a given angular distance from P (Fig. 3.1). This is done through integration with respect to the azimuth a of the point Q between 0 and 2.n and provides for isotropism of the operator M. The resulting mean value with

+

149

Spatial Variations in the Lunar Field

Fig. 3.1. Averaging of the product over all points at a given angular distance

respect to a is then averaged over all possible positions of the point P on the sphere, which ensures uniformity of the operator M. To simplify the subsequent expressions, we shall denote the averaging over the sphere by operator M p : ’

.

2n r

n r

(3.1.5) .

.

e=o A = O

where 8 and l are the coordinates of the point P. In this case, the complete operator for finding the covariance (3.1.4)can be written as 2n

1

M{.} = 2A

Mp{-}dcr

(3.1.6)

a=O

Assume that the values of the anomalous lunar gravitational potential T (8, A) on a sphere with radius R can be regarded as a realization of an ergodic random field. The validity of such an assumption was treated at length by Moritz (1980). Therefore, we shall not dwell on this problem but proceed directly to computation of the covariance function of the anomalous potential. The term “anomalous lunar gravitational field will be used here and in what follows to denote the field of lunar mass attraction ( V )minus the central term

150

Lunar Gravimatry

For the anomalous potential T defined in this manner, the expansion in spherical functions begins with n = 1: P

n=l

(Cnmcos mA + S,, sin rnA)Fn,(cos e) (3.1.7)

To simplify the formulae, let us introduce the following designation of normalized spherical functions and harmonic coefficients: v

in

With such designations, the series (3.1.7) takes the following form: (3.1.9) where P is the point with coordinates p, 8, and A. The relations of orthonormality of the spherical functions (3.1.8) take the following simple form: M p { ynrn(P)xk(P)} = dnidmk

(3.1.10)

The mean values of the spherical functions are zero: MP{ Ynm(P)} = 0

therefore, the anomalous potential T is centred: MP{T(P)} = 0

and, according to (3.1.4) and (3.1.6), we have the following expression for the covariance function K: 2%

(3.1.11) Let us find K ( $ ) in the form of expansion in Legendre polynomials Pn (cos JI): m

K($) =

C D~P~(cos JI) = i=O 2n

M p { ( T ( P ) T ( Q ) }da 0

(3.1.12)

151

Spatial Variations in the Lunar Field

Multiply both sides of this equation by Pn(cos $) and integrate with respect to $ between 0 and 7c with weight sin$. In view of the orthogonality of Legendre polynomials, we have

11 n

2 2n+ 1D

n=g

Zn

yl=O

P,(cos $)Mp{T(P)T(Q)}sin $ d$ da (3.1.13)

a=O

The right-hand side of (3.1.13) can be rearranged in the following manner:

{

n

Dn = (2n + l)Mp T(P) x

d, j

2n

-

v=O a=O

The double integral in the right-hand side of (3.1.14) is nothing but an operator of averaging over the coordinates of the point Q on the surface of a sphere (see Fig. 3.1): n

Zn

jj

47c -

yl=O

Pn(cos $)T(Q) sin $ d$ dcr = MQ{P,(COS$)T(Q)} (3.1.15)

-

a=O

We shall use the known formula of addition of spherical functions for its computation. In the above-adopted notation, this formula takes the following simple form:

Multiply both sides of (3.1.16) by T(Q) and apply the operator MQto them: in= - n

With the orthonormality relation (3.1.10), at p = R, we have GM MQ{Ynm(Q)T(Q)} = 7 Ynm

(3.1.18)

Substitution of this expression into the right-hand side of (3.1.17) gives

(2n + 1)MQ {Pn(COS $)T(Q)} =

GM R 1

m= -n

Then, according to (3.1.14) and (3.1.15), we have

YnmYnm(P)

(3.1.19)

152

Lunar Gravimetry

Rearranging the right-hand side of this equation and using formula (3.1.18) again, we obtain the coefficients Dn of interest: (3.1.20)

or, if we turn back to the usual harmonic coefficients, (3.1.21)

Thus, the covariance function of the anomalous potential on a sphere with radius R is

A similar approach can be used to find the covariance for points located on different spheres with radii p1 and p2 when P = P (pl, 8, A), Q = Q ( p 2 , 8’,X), and the angular distance between P and Q is still In this case, the same formulae are used, with the difference that a general potential expansion of the (3.1.7) and (3.1.9) type is substituted for T As a result

+.

K(4h P1, P 2 ) =

c (Ky+’ DnPn(cos$) PlP2

(3.1.23)

n=l

where the coefficients Dn are determined using the same formula (3.1.21). Let us assume that P = Q in formula (3.1.4); that is, )I = 0 and p1 = p2 = p. Then, from (3.1.23) we derive

K(P, P ) = M ( T 2 ( P ) } that is, the mean value of the square of the potential anomalies over the entire sphere. If we use statistical terminology, this quantity should be called “anomalous potential variance”. Denoting it by D, we have m

D=K(O) =

1 Dn n=l

As regards D,, they have the meaning of variances of the nth harmonic of the anomalous potential and are known as degree variances: Dn

=

MP { T m }

The quantities Dn considered as functions of the harmonic number n form a spatial spectrum of the gravitational field potential, which makes studies into degree variances a matter of special interest.

153

Spatial Variations in the Lunar Field

3.2 Degree Variances and Covariance Functions of Various Characteristics of the Gravitational Field and Figure of the Moon

The above-derived formula (3.1.21)for degree variances applies not only to the anomalous potential, but also to any other characteristic of the gravitational field if the coefficients of its expansion in spherical functions (3.1.7)are known. Hence, we can immediately write, the expressions for the degree variances of the quantities for which transformation factors had been obtained (Tables 2.5 and 2.6),namely, for the level surface elevations [,mixed gravity anomalies Aqm, gravity anomalies T p , anomalies of the vertical gravity gradient T,, and anomalies of the second vertical gravity gradient Tppp. Here are the formulae for the degree variances of these quantities on a sphere with radius p > R:

(3.2.1)

The covariance functions of the above characteristics of the gravitational field on a sphere with radius p are determined from formula (3.1.12): K { .1 =

2 Dn{ .}Pn(COS $1

(3.2.2)

n=2

where Dn{ . } stands for the corresponding degree variances determined from formulae (3.2.1).For example, m

KT,,(+,P ) =

1 ~n(T,)Pn(COS+)

n=1

=

(Fy2

n=l

(n + 1)2('yn+2pn(cos

+) m = O

(ctm+ s:.) (3.2.3)

The general rule in the computation of covariance functions of various characteristics of the gravitational potential is as follows (Moritz, 1980). Let

154

Lunar Gravimetw

j and g be arbitrary characteristics of an anomalous gravitational field, which can be represented as linear functions of the anomalous potential T: f=FT

(3.2.4)

g=GT where F and G are linear transformation operators. Consider the quantities f ( P ) and g(Q) at two different points P = P ( p , cp, 1)and Q = Q(p', cp', A'), the angular distance between them being II/: cos $

= sin cp

sin cp'

+ cos cp cos cp' cos (1

-

A')

(3.2.5)

According to (3.2.4), these quantities can be written as (3.2.6) where the subscripts in FP and GQ indicate that F operates upon the coordinates of the point P, while G operates upon the coordinates of the point Q. Let us derive the covariance .off and g: K,g = M{FPT(P)GQVQ)) = M{FPGQT(P)T(Q)}= FpGpM(T(P)T(Q)} = FPGQK(P,Q)

where K ( P , Q) is the covariance function of the anomalous potential. Formula (3.2.7) provides the desired law of transformation of covariances. Its derivation involves the commutativity of the averaging operator M [see (3.1.6)] as well as the operators F and G. A proof of this generally obscure fact can be found in Morirz's work (1980). Let us illustrate the application of formula (3.2.7) by the following example. Suppose we want to find the covariance between the selenoid elevations i= T/y on a sphere with radius p = p1 and the mixed gravity anomalies bg, = -(aT/ap) - 2T/p on a sphere with radius p = p z . Then, the operator F takes the form l/y while the operator G = (a/ap) - 2/p. According to formula (3.2.7), we have 1 aK

KC, &,

hence

=

2~

-i [Q-k p']'

155

Spatial Variations in the Lunar Field

In particular, if l and Agmare given on the same sphere with radius R, 1

m

Ks,A~,(+> = - C (n - 1)DnPn(cos yR n = l Here is another example. We want to find the covariance between the horizontal components of the plumb-line deflection:

1 aT ( ( P ) = ~- and YP1 acp

1 v(Q) = yp2 cos cp'

aT

. anl

According to (3.2.7), K<.v=

1 a2K(cos 11/) acp aA' YP1, P2 cos cp'

we have

Formula (3.2.5) for cos 11/ gives

a cos + - cos cp cos cp' sin (A - A') ax

--

Then,

$) + cos cp cos cp' sin (i- A') d2P,(cos a(cos 11/12

Hence, it can be inferred, in particular, that the correlation coefficient between the plumb-line deflection components taken at one point (cp = cp', A = 2)is zero. In a third example, we shall find the covariance function for the longitudinal component of the plumb-line deflection on a sphere with radius R . We have 1

K(v) = 7

2 ~ cos 2

cp cos cpl

a2K

an anl

156

Lunar Gravimetry

Using the derivations from the previous example gives

1 y2R2 $1

’.{

--

x sin (A

Hence, at cp = cp’ and

-

dPn(cos +) x cos(A - A’) d(cos +)

II/) + d2Pn(cos d(cos 1+9)~

1’)cos cp cos cp’ sin (A - A’)

A = 1’ for the plumb-line deflection variance we have

Similarly, it can be shown that 1

D { r >= yZR2

2 D n n(n 2+ 1) ~

Thus, the variances of both plumb-line deflection components are equal.

3.3 Degree Variances for Horizontal Lunar Attraction Components

Consider now the degree variances for the horizontal attraction components in directions tangential to meridians and parallels. The expressions for the horizontal components of gravity acceleration were derived in an earlier chapter (52.2). The horizontal attraction components, which we shall denote by V, and V, can be derived if we ignore the centrifugal-tidal terms in formulae (2.2.2) and (2.2.3). Let us write the expansion of horizontal lunar attraction components Vv and V, on the sphere R as a series of spherical functions:

av -GM v, = C 1 (Aijcos j A + Bij sin jA)Pij(sin cp) Racp R2 w

i

i=oj=o

v,=---- 1 av - GM C RcoscpaA R2

i

i=oj=o

(3.3.1)

(Rijcos j A + M i j sin jA)qj(sin cp)

The coefficients of these expansions may be determined from the following formulae based on the orthogonality of spherical functions:

157

Spatial Variations in the Lunar Field

(n

+ 1) sin cpFnm(sincp) -

{

[(n

+ 1)'

- rn2](2n + I)}'/'

2n+3 cos j n x Fn+l,m(sin cp) Pij(sin cp). {sin j i } d' d i

1-

- {c(n

+

2n

+3

+

1

1)]''2Pn+ j(sin cp) Fij(sin cp) dcp

(3.3.2)

x cos dcp dcp d l

x Pnm(sincp)Ej(sin cp)

12

{?;}

dcp d l

+(W) =

-_

4

n= 2

j{

2'

}Pnj(sin cp)Pij(sincp) dcp

-C n j

(3.3.3)

-(n/2)

There Cij and Sijare harmonic coefficients of gravity potential 6 , = 1 for aj0 = 0 for j # 0. We have been unable to derive more simple analytical expressions for coefficients Aij, Bij, K k j ,M i j , so the degree variances

j = 0 and

" '

(3.3.4)

for both the Moon and the Earth were calculated numerically. The results of these calculations are presented in the next section.

I 5a

Lunar Gravimetry

3.4 Comparative Analysis of the Lunar and Terrestrial Gravitational Fields

In so far as the general characteristics of the lunar gravitational field are concerned, refer to Table 3.1 which gives the normalized values of the harmonic coefficients of gravity potential expansion, corresponding to both the Moon and the Earth. Here, the following three distinctive features should be noted: (1) The values of the dimensionless harmonic coefficients for the Moon, are greater by almost one order of magnitude than those of the corresponding coefficients for the Earth. The exception is C20,which is indicative of the comparative irregularity of the Moon's gravitational field. (2) In the case of the Moon, not a single harmonic coefficient differs drastically from the rest in terms of their absolute value. If, for example, the coefficient C,, for the Earth is greater by two and a half orders of magnitude than the other coefficients, that for the Moon is only several times greater than the others. (3) The values of the harmonic coefficients for the Earth diminish perceptibly with the number n, whereas in the case of the Moon no such trend is discernible. Using the normalized harmonic coefficients C,,, and $ ., of the lunar and terrestrial gravity potentials, let us examine the degree variances and covariance functions for the different characteristics of the selenopotential. of Figure 3.2 presents curves showing the dimensional r.m.s. values (&-)i/2 the spherical harmonics of the gravitational potential plotted against the degree, n, for the Moon and the Earth. It can be seen that (&)i/2 decreases monotonically for the Earth, while for the Moon the decrease is not monotonic. The anomalies (&).'/' stem from the presence of mascon-type masses inside the Moon. They have characteristic dimensions corresponding roughly to the orders at which the variances are anomalous. It is interesting to examine the variation pattern of the degree variances of radial gravity gradient anomalies Tppwith the degree n in the case of both the Moon and the Earth. The curves of Fig. 3.3 represent dimensional r.m.s. values of spherical harmonics, (DTp,>,"', corresponding to the radial gravity gradient, plotted against n. One of the curves corresponds to the Earth, and the other to the Moon. Comparison of the corresponding curves in Figs 3.2 and 3.3 suggests that the higher degree harmonics of the perturbing potential (T), have higher values on the Moon, as opposed to the Earth, which is even more so in the case of harmonics of the radial gravity gradient (Tpp),. Consider the pattern of variation of the degree variances of the perturbing potential T and the anomalous radial gravity gradient Tppof the Moon with the distance from its

TABLE 3.1 Normalized harmonic coefficients of expansion of the lunar and terrestrial gravitational Dotentials ( x lo-*) c20

Earth Moon

-484.16 -91.38 3 33

Earth Moon

1.42 -8.18 €5 1

Earth Moon

-0.05 -9.02

(72 1

0.001 2.68 c 40

0.54 3.94 s 51

-0.10 -0.12

321

-1.40 5.25

(72 2

2.43 3.53

s22

c3 0

c 31

0 -2.31

0.96 -2.65

2.03 22.68

c441

341

c42

342

-0.53 -5.33

-0.47 2.09

0.35 -8.18

0.67 -7.61

G 2

g 52

G3

$5 3

-0.33 -0.56

-0.46 -1.24

-0.22 4.15

0.65 2.50

The coefficients are taken from Table 1.13 and the GEM-1OB model.

€43

0.99 -1.29 G

4

-0.30 0.38

33 1

0.25 5.05

c 3 2

0.89 12.01

332

-0.62 6.26

c33

0.71 15.62

c44

s44

c50

-0.19 -3.43

0.31 -17.80

0.07 -1.85

s54

c55

s 55

0.05 -5.99

0.16 -6.49

-0.66 10.65

343

-0.21 -13.38

160

Lunar Gravimetry

n Fig. 3.2. R.m.s. values of the spherical harmonics of the lunar and terrestrial gravitational potentials. (1) Moon, harmonic coefficients from the generalized model; (2) Earth, harmonic coefficients from SE111.

surface. To this end, we have calculated the r.m.s. values of the spherical functions of the nth degrees for the perturbing potential, (DT)."', and for the second radial gravity gradient, (DTpp);/', at different altitudes H above the lunar surface. Use was made of (3.2.1) and (3.3.3) at p = R + H. The calculation results are presented in Figs 3.4 and 3.5. In Fig. 3.4, the variation in (DT)."' as a function of n is given for H = 0, 100 and 500 km, while in Fig.

161

Spatial Variations in the Lunar Field

0.52L

0.48

I 0.441

I 0.40

-'I I 0.36 -I I - 0.32- I I % -

-P

la

0.24-

0.20-

A

\\

V

/

1'

c/I

PI\

L/P - - ?\

'1

0.16 0.12 -

0.080.04 -2 2

I

I

I

I

I

I

I

4

6

8

10

12

14

16

Fig. 3.3. R.m.s. values of the spherical harmonics of the radial gradients of lunar and terrestrial gravities. (1 ) Moon, harmonic coefficientsfrom the generalized model; (2) Earth, harmonic coefficients from SE-Ill.

3.5, the variation in (BT,,)A/zwith n is given for the same altitudes. The curves indicate,just as was expected, that the higher degree harmonics decrease with increasing altitude at a faster rate. If (BT)ii2 for the perturbing potential Tat H = 100 km decreases by about 2%, as compared to the lunar. surface, the equivalent decrease in for the radial gradient is 15%. Obviously, the radial gravity gradient (DT,,)."z will decrease with altitude and degree n at an even faster rate, which is clearly illustrated in Fig. 3.5. Next, we analyse the variation pattern of the degree variances (BY,)nand

(cT,),!/2

Lunar Gravimetw

162

n Fig. 3.4. R.m.s. values of the spherical harmonics of the lunar gravitational potential at different altitudes above the Moon's surface: (1) 0 km; (2) 100 km; (3) 500 km.

(3.3.3), corresponding to the latitudinal and longitudinal components Vv and V , of lunar and terrestrial mass attraction, respectively. Note the distinctive feature of the Moon's gravitational field (Fig. 3.6). The second-degree variances (DYP)"and (DVA).are small in spite of the fact that the spherical harmonic Pzo(sin cp) dominates in both 'the potential of lunar gravity and in the gravity itself. In computing the degree variances of V , , V, for the Earth (Fig. 3.7) we excluded this harmonic from the anomalous

163

SDatial Variations in the Lunar Field

0.52

0.48

0.44 0.40 -

2

4

6

8

10

12

14

16

n Fig. 3.5. R m s . values of the spherical harmonics of the radial gravity gradient at different altitudes above the Moon's surface: ( 1 ) 0 km; (2) 100 km; (3) 500 km.

potential because it in no way can be treated as stochastic-it is determined by the well-expressed polar flattening of the Earth. Figures 3.6 and 3.7 demonstrate the common feature that is worth mentioning-there are firstdegree and zero-degree harmonics in the spectra of horizontal components of gravity which are absent in the potentials of both lunar and terrestrial gravity fields. When analysing the general gravitational field, it is necessary in some cases to know the variation pattern of the degree variances at high values of n, which have never been obtained from observations. This need arises when

164

Lunar Gravimetry

n Fig. 3.6. R.m.s. values of the spherical harmonics of the attraction components tangential to the Moon’s (1) meridians and (2) parallels.

one estimates the magnitude of the high-frequency portion of the gravitational field and its effect on the covariance function, on the determination of solenoid elevations, on the calculation of various gravity potential derivatives, and so on. Some extrapolation formulae may have to be derived. Kaula (1969) speculated that rocks on the Moon are capable of withstanding almost the same stresses as on the Earth. This has led Kaula to a conclusion that the degree variances of the gravitational potential of the Moon (DTOnand the Earth (DTe)nare related by (3.4.1)

Kaula has suggested that the degree variances for the lunar potential should

165

Spatial Variations in the Lunar Field

1

2

1

1

4

1

1

6

,

1

,

10

8

1

,

12

1

1

14

1

,

16

1

~

~

18

n Fig. 3.7. R.m.s. values of the spherical harmonics of the attraction components tangential to the Earth's (1) meridians and (2) parallels.

be calculated from the corresponding power variances for the Earth, determined from gravimetric and satellite observations. The curves of Fig. 3.8 represent dimensional r.m.s. values of the spherical harmonics of the radial lunar potential derivative (DTp),!'2plotted against n. One of the curves corresponds to a generalized model of the Moon's gravitational field; the other has been derived using Kaula's theoretical method. The degree of the perturbing terrestrial gravitational potential (Standard variances (DT)" Earth-111) were used to calculate the degree variances for the lunar potential on the basis of (3.4.1). Then, the degree variances for the radial derivative of

166

Lunar Gravimetry

I

1

I

I

I

I

I

4

6

8

10

12

14

16

n Fig. 3.8. R.m.s. values of the spherical harmonics of the radial lunar attraction potential derivative: (1 ) from harmonic coefficients in the generalized model; (2) from Kaula's theoretical formula.

the lunar potential, (DTp),,,were calculated according to (3.2.1) to derive the second curve. Comparison of the curves indicates that the observed curve fits reasonably well to the theoretical one, which lends support to Kaula's hypothesis (3.4.1). It is also useful sometimes to have a numerical-analytical expression for variations in degree variances. In particular, Kaula proposed to represent the r.m.s. values of the spherical harmonics of the radial lunar potential gradient

167

Spatial Variations in the Lunar Field

-0.6L Fig. 3.9. Normalized autocorrelationfunctions of the lunar and terrestrial gravitational potentials. (1 ) Moon; (2) Earth.

as

(DT,),!''= 175.43n-'.''

mGal

(3.4.2)

Let us now turn to the correlation functions for various derivatives of the lunar gravitational potential in order to examine the changes they undergo, and compare them, in some cases, with similar functions for the Earth. Figure 3.9 shows normalized autocorrelation functions K:($) for the perturbing potentials of the Moon and the Earth. What we see is a common trend followed by the varying functions. The region of negative values of the functions K$($) lies within 45" < $ < 120". The minima of the functions do not differ significantly either. Yet, in view of the different radii of the Moon and the Earth, we can speak of a less pronounced autocorrelation in terms of distances as regards the Earth's gravitational field, as opposed to that of the Moon. The farther from anomolous masses, the smoother the gravitational field, and its autocorrelation function becomes smoother, too, because, as the distance from the Moon increases, the role of the higher frequency harmonics constituting its gravitational field becomes less important. The latter becomes especially true when we are dealing with higher derivatives of the gravitational potential. To give quantitative substance to this statement, we have calculated the autocorrelation functions for various derivatives of the

168

Lunar Gravimetry

1.o

0.8

0.6

-

3

nL

0.4

0.2 0 -0.2 -0.4

-0.6L Fig. 3.10. Normalized autocorrelation functions on a sphere with radius R = 1738 km of (1) the anomalous potential T ; (2) the radial attractioncomponent T,; (3) thesecond radial potentialgradient Tpo, Normalized autocorrelation functions on a sphere with radius R = 2738 km of (4)the anomalous potential T ; (5) the radial attraction component T,; (6) the second radial potential gradient TPv

perturbing lunar gravitational potential (perturbing potential T radial attraction component Tp, radial gravity gradient Tpp)for two spherical surfaces with radii R = 1738 km and R = 2738 km. The results of these calculations are presented in Fig. 3.10. It should be pointed out that, for most applications, one must know the covariance function K ( $ ) to a rough approximation. It is used in calculations whose results are not so critical as regards the accuracy with which K ( $ ) is known. The covariance function, just as the degree variances, is to be determined more reliably than the initial harmonic coefficients C,, and S,, in various models of the lunar gravitational field. Give in Fig. 3.11 are the autocorrelation functions of the gravitational potential calculated for models of the Moon’s gravitational field, derived by different workers using different sets of initial data. One can see the same common trend in the autocorrelation functions. Both the autocorrelation functions and the degree variances, determined from available coefficients of the gravitational field expansion in spherical functions, are over-smoothed. They do not take into account the highfrequency portion of the gravitational field, which, of course, has features not

169

Spatial Variations in the Lunar Field

-0.2 -

-0.4 -

Fig. 3.11. Normalized autocorrelation functions of the gravitational potential in the Moon models constructed by different workers: (1) Akim and Vlasova (1983); (2) Ferrari (1977): (3) generalized model.

seen on the autocorrelation function curves. An attempt has been made to derive an autocorrelation function for the Moon from the results of detailed Doppler measurements of the line-of-sight accelerations of Apollo 14, with reference to small altitudes (tens of kilometres) and an accuracy of 10 mGal. The autocorrelation function Kr(r) was calculated using the formula L

K,(z) x

L

j

r(x)r(x

+ ).

dx

(3.4.3)

0

The values of T(x) in (3.4.3) were centred in advance. The length L of the profile with the values of T(x) was more than 1800 km. The calculated normalized autocorrelation function KF(t) is shown in Fig. 3.12. As in the case of degree variances, the empirically derived covariance function is approximated by a simple numerical-analytical formula. For instance, Krarup (1969) proposed to use the following expression for K T ( $ , p , p l )

170

Lunar Gravimew

800

600 I

t

I

I

-0.5L Fig. 3.12. Normalized autocorrelation function of line-of-sight acceleration: (1 ) calculated from the acceleration (r)profile taken from Apollo 14; (2) approximated by a numericalanalytical formula.

in which A and po are parameters determined from approximation of the empirical curve by (3.4.4). A more complex approximating function was offered by Lauretzen (1 973). To approximate the empirical curve of the autocorrelation function K;(T) shown in Fig. 3.12, the following simple expression was used: K:(T) = e-Krcos oz

(3.4.5)

where K and o are parameters to be determined. Having determined them by the least-squares method, we derived the following empirical numericalanalytical expression for the autocorrelation function: K:(r)

= e-0.002*13r cos 0.005,12~

(3.4.6)

where T is given in km. This relation suggests that the correlation radius for the lunar anomalies T(x) under consideration is r0 = 150 km. Of course, the autocorrelation function calculated using separate profiles is not representative of the Moon as a whole. In the case of the Earth's gravitational field, we know how widely different are the statistical characteristics for regions with dissimilar tectonics (platforms, geosynclines, depressions, mountain ranges, etc.) (Pellinen, 1970).

171

Spatial Variations in the Lunar Field

3.5 Comparison of the Gravitational Fields of Terrestrial Planets

Figure 3.1 3 presents the degree variances for the gravitational potentials of the Earth, the Moon, Mars and Venus (Phillips et al., 1979). Note the coincident trends of the degree variances, as a function of n, in the pairs comprising the Earth and Venus, the Moon and Mars. Such coincidences seem to be associated with the cosmogonic backgrounds of these planets. Given in Table 3.2 are the mean values of the gravitational potential, its second and third radial derivatives on the surface of the terrestrial planets, and their mean densities. Included for comparison are the corresponding characteristics of Phobos. In addition, the masses and mean radii of the planets are presented. Note that, in spite of the enormous difference between the masses and sizes of the celestial bodies, their second radial potential derivatives W,, differ but insignificantly. For example, the ratio between the Earth's mass and that of Phobos is 6 x lo'', while the ratio between the corresponding values of W,, is only about three. The reason is that W,, on the surface of a spherical body with a uniform or spherical symmetric density distribution is independent of the linear dimensions of the body and has dimension [7'l- *. In this case, W,, is proportional to the mean density omean

1

-4

Fig. 3.13. Degree variances of the gravitational potentials of (1) the Earth; (2) the Moon; (3) Mars and (4) Venus.

172

Lunar Gravirnetry

TABLE 3.2 Planetocentric constants of gravitation G M , mean radii R, mean densities om, mean gravities g on the surface, second and third radial potential derivatives W,, and W,,,, respectively, for terrestrial planets

Planet Earth Venus Mars Mercury Moon Phobos

GM x 104 (km3 s-')

R (km)

39.86 32.49 4.28 2.20 0.490 66.

6311 6050 3389 2439 1737 11.03

9

urn

(g ~

m - ~ ) (cm s C 2 )

5.51 5.25 3.93 5.43 3.34 1.8

982 888 373 370 162 0.55

W,,(E)

-3080 -2930 -2199 -3033 -1870 -990

W,,, (cm-' s - ~ )

1.45 x 1.46 x 1.95 x 3.73 x 3.23 x 2.7 x lo-"

of the planet: 8 3

Wpp(R,CP, 2) = - nGdmean

where G is Cavendish's constant. of gravitation. Consequently, the gravitational field of small bodies must be studied with the aid of gradiometers. On the surface of the normal Earth and an arbitrarily small sphere with a density equal to the mean density of the Earth, W,, will be always equal to 30868. This is why the anomalies of the vertical as well as other gradients at different points on the physical surfaces of the Earth and planets must attain high values (tens of thousands of eotvos). Since the sensing element of gravimetric valiometers is about one metre above the surface, observations yield gravity gradient anomalies in the order of only tens or hundreds of eotvos. Still more indeterminate is the real field of third gravitational potential derivatives. For the normal Earth, the third radial derivative of the potential is w,,,(R,CP,n) = 1.5 x 10-14 cm-1

s-2.

At the same time, on the surface of a sphere with R = 64 cm and mean density of the Earth, W,,, = 1.5 x lo-' cm-' s-'; that is, the anomalous part of the W,,, field is by many orders of magnitude greater than the normal one. Conversions of the field of gravity anomalies W, into the field of W,,, give rise to sizeable errors, which makes the W,,, field unsuitable for quantitative interpretation in gravimetric surveying. Therefore, the W,,, field is used only for qualitative estimations of the anomalous mass distribution. The above statement concerning the second and third derivatives of the terrestrial gravitational potential appears with some assumptions, to be generally true as regards other terrestrial planets as well.

173

Spatial Variations in the Lunar Field

3.6 The Selenocentric Constant of Gravitation: The Mass and Mean Density of the Moon

The selenocentric constant G M , of gravitation plays a major role in computations of the motion of circumlunar spacecraft and artificial satellites. It is one of the fundamental constants of astronomy and astrodynamics. The value of GM, is used to determine the mass Md and mean density (ru of the Moon, which, apart from being interesting in their own right, are necessary in order to formulate the auxiliary conditions for determining the density distribution inside the Moon. In spite of the great interest shown in the value of the selenocentric constant of gravitation by specialists in celestial mechanics, the relative accuracy of its determination had remained low Significant before the advent of the space exploration age (c. 2 x advances in its measurement were made after launching of ALS and spacecraft. It would be appropriate at this juncture to look back on the basic ideas behind the astronomical methods for determining G M , . Some of them are still used today in an updated form applicable to spacecraft. Most of GMu measurement methods are based on the assumption that the Moon produces a gravitational effect on the Earth’s rotation or orbital motion. In the first case, this effect manifests itself in the precession and nutation of the rotation axis of the Earth due to the fact that the Earth lacks central symmetry in mass distribution. The theory of the Earth’s precession and nutation posits that their amount is to a great degree dependent on the ratio p = M ( / M , , where Mu is the Moon’s mass and M e is that of the Earth. Since the geocentric constant of gravitation G M , is currently measured with a relative accuracy of about determination of ,u from observations of the precession and nutation may help calculate the selenocentric constant G M , of gravitation. In what follows, we shall treat the determination of p or GMu on equal terms because either of these quantities can be calculated given the other. The constant of annual lunisolar precession of the Earth is p

=H

cos E ( P +

P

Q)

(3.6.1)

while that of the annual lunisolar nutation is N = HCOSE-’ R 1+,u

(3.6.2)

where H = (2C - A - B)/2C is the dynamic flattening of the Earth, and E is the obliquity of the Earth’s equator to the ecliptic, which varies primarily in a secular manner. The functions P,Q and R are:

174

Lunar Gravimetry

3 n2 (1 - e2)-3/2 p=-T 2 0 l+ml

R = - N I R l sin 1”

N2

where n and n‘ are the mean diurnal sidereal motions of the Sun and the Moon, w is the angular rate of the Earth’s diurnal rotation, T is the tropical year in mean days, zZ1 is the mean motion of linear node per tropical year, e is the orbital eccentricity of the Moon, ml is the mass of the Earth-Moon system expressed in solar masses Ma, that is, ml = (Mu+ Me/Ma, N 1 and N 2 are functions dependent on lunar orbit elements, derived analytically in the lunar theory and considered to be known. Taken into account in (3.6.1) must be a small relativistic correction leading to an effect similar to precession. From (3.6.1) and (3.6.2) one can easily derive a formula for determining the quantity of interest, namely,

P N

p-1= A -

+B

(3.6.3)

The coefficients A and B in this equation are derived from the values of P, Q and R, calculated for a particular epoch on the basis of the respective values of their arguments, given above. The precession constant p is determined from astronomical observations of the changes in stars’ positions, more particularly from meridian observations. The nutation constant N is determined from observations of latitude variations with the aid of zenith telescopes or photographic polar tubes. Nutation does not lend itself to measurement as readily as precession, and, therefore, the main source of error in determining p- is that of nutation. Another astronomical method for determining GMu is based on observations of the lunar inequalities in the Earth’s longitude as the Earth executes its orbital motion. These inequalities stem from the fact that revolving around the Sun is the centre of mass of the Earth-Moon system, whereas the Earth, as does the Moon, revolves at the same time around their common centre of mass. The distance between the centre of mass of the Earth and that of the Earth-Moon system is d = - PA 1+P

where A is the mean distance between the Earth and the Moon, and so d

175

Spatial Variations in the Lunar Field

equals 4670 km. As a result of rotation of the Earth-Moon system around the centre of mass, the terrestrial observer is under the impression that the changes in the Sun’s longitude are irregular or, to be more precise, he will observe variations in the visible Sun’s longitude with a period equal to a sidereal month. If the distance to the Sun were measured, it would vary with the same period. The amplitude of the variation depends on the ratio between the distance to the Sun and the distance d. By lunar inequality is meant the coefficient of periodic displacement of an object an astronomical unit away from the Earth in the lunar orbital plane. The displacement is caused by the Earth’s motion relative to the centre of mass of the Earth-Moon system. Therewith, it is assumed that the Earth’s orbit is circular with a radius equal to the mean distance between the Earth and the Sun. The coefficient of lunar inequality in the Sun’s longitude is (3.6.4) where Aa is the distance between the Earth and the Sun, no and nh are the solar and lunar parallaxes, respectively. We can transform (3.6.4) to

l A 1 (3.6.5) P-l = E G To find G M , , the Sun’s positions relative to the stars are determined from astronomical observations and then used to derive the coefficient L of lunar inequality with an assumption that the distances A. and A are known. Since the amounts of lunar inequality in the motion of the minor planets are greater, because of their closer proximity to the Earth, than in the case of the Sun, they have been used to determine GM,. The minor planets Victoria and, more often, Eros, were observed to this end. One can do without the distances A. and A, if the Moon’s parallactic inequalities determined from astronomical observations are used. By parallactic inequalities are meant the terms in the expansion of the true Moon’s longitude in trigonometric series, whose argument is the difference between the Earth’s and Sun’s longitudes (S - h), or the same difference with odd factors. Derived from lunar theory is the coefficient of lunar parallactic inequality with argument (S - h) whose period is 29.5306 days:

Solution of this equation together with (3.6.4) gives

P

p-’ = 4.137,44-

L

+1

(3.6.6)

176

Lunar Gravimetry

’

The accuracy of determination of p- depends only on that of measurement of jj and L. The values of L and jj are about 6 . 4 and 125”, respectively. Let us write an exact expression for Kepler’s third law as applied to the Moon’s motion around the Earth: (3.6.7) where 4n2/T2 = 2.661,6995 x s-’ and F, is a small correction for the perturbations due to flattening of the Earth, influence of the Sun, planets, position of the centre of mass, and other factors. This correction is calculated theoretically. The geocentric constant of gravitation GMe is assumed to be known. To determine p-’ from (3.6.7), one must know, in addition to T, G M e and F,, the semimajor axis a, of the lunar orbit and the Moon’s radius. Laser detection and ranging permits distances to be determined to within centimetres. The accuracy, of determining the selenocentric constant of gravitation GM, was enhanced considerably with the advent of artificial lunar satellites, lunar spacecraft, and automatic interplanetary stations. To derive GM,, the coordinates and velocities of ALS and various spacecraft were tracked electronically with a high degree of accuracy. The value of GM, is refined on the basis of the difference between the observed coordinates and velocities, on the one hand, and those calculated using an approximate value of G M , on the other. The determination of GM, with the aid of ALS was based on Kepler’s third law, written as follows for the motion of ALS in the lunar gravitation field:

where T is the satellite’s period, a is the semimajor axis of its orbit, and F stands for the corrections taking into account various perturbations, including those due to the noncentrality of the Moon’s gravitational field. The latter is known to be highly anomalous. The influence of the anomalous gravitational field diminishes with increasing distance, and the role of the perturbations caused by the Earth, the Sun, and planets becomes increasingly important. When GM, is determined using ALS tracking data, it is assumed that the satellite moves without any forces acting upon it and with no correction being applied. The value of GM, was derived many times by different workers using individual Lunar Orbiter tracking data as well as in combination. The values of GM( were determined simultaneously with those of the coefficients C,,, and S,,, of expansion of the lunar gravitational field in spherical functions. Attempts were made to determine G M , by retaining different numbers of expansion terms. A summary of major determinations of

177

Spatial Variations in the Lunar Field

the selenocentric constant G M , of gravitation from Lunar Orbiter tracking data is given in Table 3.3. Also used in such determinations were spacecraft with highly elongated orbits in the sphere of lunar attraction. These spacecraft included Rangers and Surveyors. The coordinates and velocities of the spacecraft were tracked electronically from terrestrial stations and compared with the computed values. The computations were based assuming certain approximate parameters of the orbits and gravitational field, including GM,. The differences between the observed and computed coordinates and velocities were used to refine the initial parameters. The determinations of p and GMt based on Ranger and Surveyor tracking data are also summarized in Table 3.3. Yet another class of spacecraft was used to refine the value of GM,, namely, Mariners, Pioneers and Veneras (Table 3.3). In this case, the concept of lunar inequality observations, underlying one of the astronomical methods described above, was employed, with far spacecraft being used as the object of observations instead of the Sun or a minor planet. Furthermore, variations in the distance to the spacecraft and the radial component of its velocity were measured rather than the angular coordinates. Separated from the tracking data was that part of the variation which was due to the motion of the Earth around the centre of mass of the Earth-Moon system. For variations in distance p and radial velocity p we have PA

sp = --

1+P

sp =

cos /3 sin (A - A,)

2npA

T(1 + PI

cos /3 sin (A - A,)

where /3 and A are the geocentric coordinates of the spacecraft I , is the geocentric longitude of the Moon, and T is the sidereal month. The radial velocity p is determined from Doppler tracking data to within f3.6 mm swith respect to the centre of mass of the Earth-Moon system. The value of p-' is deduced from long-term observations (several months) so as to cover several periods T of measurement of Sp and Sp. For example, when p - ' was determined from the Venera tracking data, nine parameters of the trajectory, the coefficient of light pressure, and the geocentric constant GM, of gravitation were also considered to be unknown apart from p-'. At the same time, the perturbations of the spacecraft motion due to the M,oon, the Sun, planets, noncentrality of the terrestrial gravitational field, and light pressure were taken into account. The astronomical unit was assumed equal to 149,597,900km. A total of about 7000 observations of Veneras 5, 6, and 7 were used, taken over a period of three to four months for each spacecraft (Akim et al., 1971).

TABLE 3.3 Determination of the selenocentric constant of gravitation GM, and parameter p - ’ . In cases where the authors of the cited works have not mentioned the values of GM, or p - ‘ , they were computed by us at GM, = 398 601.2 km’s-’. Computed values are given in parentheses

Spacecraft

p - ’ = M,/M,

G M , (km’s-’)

81.3030 (81.3007) (8 1.3034) (8 1.3019)

4902.66 f 0.19 4902.796 4902.64 0.1 1 4902.73 f 0.14

Ranger 6 Ranger 7 Ranger 8 Ranger 9 Rangers 6-9

(81.3030) (81.3050) (81.3035) (8 1.3022) (81.3035 f 0.0012)

4902.66 i 0.19 4902.54 f 0. I7 4902.63 f 0.12 4902.71 f 0.30 4902.63 f 0.07

Surveyor 1 Surveyor 3 Surveyor 4 Surveyor 5 Surveyor 6 Surveyor 7 Pioneer 6 Pioneer 7 Pioneer 8 Pioneer 9 Pioneers 6 and 7 Mariner 2 (to Venus) Manner 4 (to Mars) Manner 5 (to Venus) Mariner 6 (to Mars) Manner 7 (to Mars) Venera 4 Venera 5 Venera 6 Venera 7 Veneras 4-7 Mariners 4, 5 and Pioneer 7 Radar and optical observations of the inner planets

(81.3032) (81.3034) (81.3035) (81.3035) (81.3034) (8 1.3034) 81.3005 _+ 0.0007 81.3021 _+ O.OOO4 81.3004 f 0.0001 81.3008 f O.OOO1 81.3016 f 0.0020 81.3001 k 0.0013 81.3015 f 0.0017 81.3006 k 0.0008 81.3011 f 0.0015 81.2997 0.0015 81.3006 8 I ,022 81.033 8 1.002 81.3005 (81.3016 f 0.002)

4902.65 f 0.24 4902.64 f 0.25 . 4902.63 f 0.25 4902.63 f 0.24 4902.64 f 0.24 4902.64 f 0.24 (4902.81 5 0.04) 4902.72 k 0.02 (4902.82 f 0.01) (4902.79 f 0.01) 4902.75 f 0. I2 (4902.84 f 0.08) (4902.76 f 0.10) (4902.77 f 0.05) (4902.78 k 0.09) (4902.86 f 0.09) 4902.806 4902.702 4902.638 4902.808 4902.716 f 0.10 4909.75 0.12

81.3024 f 0.005

(4902.70 f 0.30)

Lunar Lunar Lunar Lunar

Orbiter 2 Orbiter 4 Orbiters I , 3. 4 Orbiters 1-5

*

G M , (km’s-*)

References Melbourne (1970) Sjogren (1971) Tolson and Gapcynski (1968) Michael and Blackshear (1972) Michael et al. ( I 970) Melbourne (1970) Sjogren et a/. (1966) Tolson and Gapcynski (1968) Tolson and Gapcynski (1968) Vegos and Trask (1967) Melbourne (1970) Vegos and Trask (1967)

398601.86 f 0.01 398601.41 f 0.4 398601.74 _+ 1.4 398601.49 f 0.39 398600.89 398600.26 398600.72 398600.45 398600.37

Anderson et a/. (1970) Anderson and Helt (1969) Anderson et a/. (1970) Anderson ef al. (1970) Anderson et a/. (1970) Anderson et a/. (1970) Tolson and Gapcynski (1968) Akim e t a / . (1971) Akim e t a / . (1971) Akim et al. (1971) Akim et al. (1971) Akim e t a / . (1971) Michael et a/. (1970) Ruskol (1975)

179

Spatial Variations in the Lunar Field

Assuming that the mean values of GMu, determined with the aid of each type of spacecraft (Lunar Orbiter, Ranger, Surveyor, Pioneer, Mariner, Venera) are equally accurate, let us derive the overall mean value of GMu. Taken as the mean value determined from the Lunar Orbiter tracking data was that obtained by Michael et al. (1970) and Michael and Blackshear (1972), who used the results of observation of all five ALS. On the other hand, the mean value based on the Ranger tracking data was obtained simply using the G M , , values for these spacecraft, given in Table 3.3. The same procedure was used in determining the mean values of GM, on the basis of the Surveyor (1, 3 and 7) and Mariner (2,4 and 7) tracking data. The mean value of GMu corresponding to the Venera tracking data was obtained by Akim et al. (1971), who resorted to the results of observations of Veneras 2 , 4 and 7. The same applies to the G M , values based on the Ranger tracking data (Vegos and Trask, 1967; Tolson and Gapcynski, 1968). The mean values of GMu are summarized in Table 3.4. The table also includes the mean value of GM, determined from many years of radar and optical observations of the inner planets of the solar system, which is regarded here as being of equal accuracy. At present, the accuracy of determination of GMu is limited by that of our knowledge of the geocentric constant of gravitation GM,, the astronomical unit, the distance to the Moon and the velocity of electromagnetic waves in the interplanetary medium. Let us now determine the mass Mu and mean density ouof the Moon. M u will be essentially a ratio between the selenocentric constant of gravitation G M , and Cavendish’s constant of gravitation G. Knowledge of the Moon’s mass M uwill be only as accurate as determination of G because the accuracy with which GMu is known by almost two orders of magnitudes greater than that of G. Assuming that G M , = 4902.7 x lo9 m3 sC2 (see Table 3.4) and TABLE 3.4 Mean values of GM,, determined from various spacecraft tracking data and averaged GM,

Tracking data from: Lunar Orbiter Ranger Surveyor Pioneer Mariner Venera Radar and Optical observations Laser probing and Lunar Orbiter 4 Averaged value and r.m.s. error

4902.73 4902.63 4902.64 4902.78 4902.80 4902.72 4902.70 4902.80 4902.72

,

0.06

180

G

= (6.6745

Lunar Gravimetry

k 0.0008)

(see Table 1.2), we obtain

x lo-" m3 kg-'s-'

M~ = 7.3454 x

1025

g

(3.6.8)

The mean density of the Moon is =

MU n

where SZ is the Moon's volume, which can be calculated with reasonable accuracy as that of a triaxial ellipsoid

SZ

= 4zabc

a, b and c being the semiaxes of the ellipsoid, with lengths of, respectively, 1738.04, 1737.68 and 1736.67 km (see 82.10). The above values of the semiaxes a, b, c and the mass Mugive the following mean density of the Moon: au = 3.3433

0.0004 g cm-3

(3.6.9)

Note that enhancement of the accuracy of determination of the Moon's mass and mean density, just as those of some other planets in the solar system, has so far been dependent only on the accuracy of determination of the fundamental constant G in terrestrial laboratory experiments.

3.7 Centres of the Moon's Figure and Mass

In studying the orbital motion of the Moon, its physical libration, the motion of spacecraft in the lunar gravitational field, as well as in determining the ephemeris time and so on, one must know the position of the centre of mass (gravity) of the Moon. It is known not to coincide with the centre of the Moon's figure, the difference between the two being between one and two kilometres. The physical surface of the Moon is sometimes approximated by a simple geometric figure (sphere, ellipsoid of rotation, triaxial ellipsoid, surface given by the sum of several spherical functions). However, identification of the centre of a real figure confined by a physical surface with that of an approximating simple geometric figure is not adequate. The question arises as to whether a sphere, an ellipsoid, or another geometric figure should be chosen for approximation. The best approximation is provided by the condition that the sum of the squares of the differences between the radius vectors of points on the surface of interest and the corresponding points on the physical surface is minimal, or that the sum of moduli of these differences is minimal, or other more complex conditions. The identification will be adequate if the centre of the figure is defined as that of mass of a Moon with

181

Spatial Variations in the Lunar Field

uniform density. Then, the coordinates of the centre 0, of the figure with respect to the origin 0 can be calculated using the following simple formulae:

x, =

sssx n

sss R

;

Yo =

YdR ;

sssz R

zo=

dR

sssdR sssdR sssdQ R

R

R

where R is the Moon’s volume and dR is a volume element. The beauty of this definition of the figure’s centre is that it is based on the same concepts as that of the Moon’s centre of mass, with the difference that the latter depends on the actual nonuniform density distribution. The geometric parameters of the Moon alone are not sufficient to determine the position of its centre of mass. This position manifests itself in the external gravitational field, with the moments of the Moon’s inertia being dependent on it. Therefore, the position of the centre of mass can be determined through tracking of an ALS in the lunar gravitational field, observations of the free fall of bodies on the Moon’s surface, and examination of some peculiarities of the Moon’s motion and rotation. The problem can be stated as follows. The coordinates of points on the Moon’s surface with respect to the centres of figure and mass are known. It is the mutual position of these centres that is to be found. The absolute distances R between the points on the lunar surface and the centre of mass have been determined by various measurements, including those of the time of free fall of the spacecraft Rangers 6-9 onto the Moon’s surface (Sjogren, 1967), the velocity-to-altitude ratio of Lunar Orbiters, and the lunar gravity during landing of Apollos 11, 12, 14 and 17. These distances were derived from the Doppler tracking data involving Surveyors 1 and 6 as they landed on the Moon, laser probing data involving corner reflectors left on the Moon’s surface, and the results of VLBI measurements with the aid of ALSEP transmitters. The above techniques for determining the distance R have so far been used on the near side of the Moon. Only a few photogrammetric measurements and the results of laser altimetry from Apollo spacecraft have been applied to obtain the absolute radii of the farside equatorial zone. Reports about some determinations of the position of the Moon’s centre of mass relative to the geometric figure’s centre can be found in the works by Lipsky et al. (1973) and Sjogren and Wollenhaupt (1973). Lipsky and coworkers used the absolute radii to the Moon’s near side. Sjogren and Wollenhaupt based their report on the results of 15 different procedures for

182

Lunar Gravimetry

TABLE 3.5 Some determinationsof the Moon‘s geometric figure and the position of the centre of mass relativeto the figure’s centre

Semiaxis determination method

Figure Sphere Sphere Ellipsoid Ellipsoid Ellipsoid

LA-15, 16 LA215, 16, RFF, SL, PDBP LA-15, 16 LA-15, 16; RFF, SL, VIA, PDBP LA-15, -16, RFF, SL, VIA, PDBP

Offset of the centre of mass (km)

Semiaxis lengths (km)

Ax

Ay

Az

a

b

C

2.20 2.05

1.10 0.91

1.0* 1.0*

1737.7 1737.8

1737.7 1737.8

1737.7 1737.8

2.07 1.98

1.11 0.75

1.0* 1.0*

1737.8 1738.2

1737.5 1737.4

1738.0* 1738.0’

2.15

0.77

0.07

1738.6

1737.5

1730.8

* Given a priori. Determinationmethod: RFF, free fall of Rangers 6-9; SL, landing of the Surveyors; PDBP, photographic determinationof the base pointson the moon;VIH, velocity-to-altitude ratio of Lunar Orbiter 1;and LA-15, 16, laser altimetry from Apollos 15 and 16.

processing the data on the position of the centre of mass, in some cases with due account for the measurements made on both sides of the Moon. Also used were the results of observations involving spacecraft. The position of the Moon’s centre of mass was deduced with respect to the centres of a sphere and an ellipsoid approximating the Moon. Table 3.5 includes only five such procedures. The observation methods are given in the second column of the table. The offsets of the centre of mass with respect to that of figure are denoted by Ax, Ay and Az. A positive value of Ax means that the centre of figure is behind that of mass, as viewed from the Earth; Ay is positive when the centre of figure is to the left of the centre of mass; and Az is positive when the centre of figure is below that of mass. Some workers tried to define the centre of mass by a method based on the best fit of the solenoid’s surface to the geometric figure of the Moon. As a result of this fit, the centre of the solenoid, assumed to be the centre of mass, is determined with respect to the centre of a geometric figure. This approach is not quite correct because the geometric surface of the Moon and the equipotential surface (seleniod) cannot be regarded as equivalent. Yet we admit that the method may give an insight into an approximate mutual position of these centres. Let us have a closer look at the attempts to determine the position of the centre of mass relative to points on the lunar surface, at which gravity had been measured. As is known, there were four such points (01.1 1). The gravity

183

Spatial Variations in the Lunar Field

distribution is given by

where

6g represents the anomalous portion of the gravity field, which determines

the difference between the real field and that of a uniformly spherical Moon. If it is assumed that g (p, cp, A) has been derived from direct gravity measurements on the lunar surface, one can find the square of the distance between the observation point and the Moon’s centre of mass:

(3.7.1) A single gravity measurement suffices to determine the locus of the centre of mass. This locus is a sphere whose radius equals the distance between the observation point and the centre of mass. To define the centre of mass, gravity must be measured at no less than three’different points on the Moon’s surface. The point of intersection of three spheres will give the centre of mass. Obviously, for a more accurate determination of all three coordinates of the centre of mass, the measurement points on the Moon should preferably be spaced as widely apart as possible. Let S be the physical surface of the Moon (Fig. 3.14); at the points P. of

03 Fig. 3.14. The lunar centre of mass, the centre of the Moon’s figure, and the parameters characterizing their mutual positions.

184

Lunar Gravimetry

which absolute gravity g. measurements are made. A system of coordinates is selected with its origin Ofcoinciding with the centre of figure of the Moon. The rectangular coordinates of the points P,, will be xn,y,, and z,,, whereas the ,, and A,. Let the X axis be directed towards the Earth, polar ones will be I,cp,, the 2 axis coincide with the axis of rotation and the Y axis extend eastward. The Moon’s centre of mass 0, is offset with respect of Ofand has coordinates Ax, Ay, and Az. If the distance between the centre of mass and a point P,, is denoted by p,,, then the following equation can be written: p,” = r.’

+ 1’

- 2rn1cos $,,

(3.7.2)

in which $,, is the angle between the directions toward the point P,, and the centre 0, of mass, with its vertex at the point Of,1 = [(Ax)’ + (Ay)’ (Az)’I1/’ is the distance between Of and Om,

+

cos $,, = sin cp,, sin cp,

+ cos cp,,

cos cpo cos (A,, - A,)

and (cp,, A,) are the spherical coordinates of the direction from the origin Of to the centre 0, of mass. Since Ax sin cpo = Az/l, lo= arctan cos cpo = [(Ax)’ (Ay)’]’/’/l, AY (3.7.3) then Az [(Ax)’ (Ay)’]’’’ cos $,, = -sin cp,, + COS (Pn 1 1

+

+

x

[

3+

(

cos A, cos arctan -

31

(

sin A sin arctan -

After slight transformation, we obtain Ax AY cos $,, = cos cp,, sin A,, -cos cp,, cos A, 1 1

+

Az

+1 sin cp,,

(3.7.4)

Find cos $,, from (3.7.2) and equate it to the right-hand side of the equation (3.7.4). Instead of p i , use its expression from (3.7.1). The result will be GM,ri’Cgn(pn,

cp.9

An) - agnI-’

(Ax)’ =1+

+ (Ay)’ + ( A z ) ~ 2 rn

Ax cos cp,, sin A,,

+ Ay cos cp,,

cos A,,

+ Az sin cp,,

rn

n = 1,2, ...

(3.7.5)

185

Spatial Variations in the Lunar Field

In this equation, the unknowns of interest are Ax, Ay and Az. The known quantities include r,,, (P,, A,,which are the polar coordinates of an observation point P,,g, (p,,, cp,, A,) which represents the gravity values measured directly at the points P,,and 69, which represents the gravity anomalies that can be determined from the known spherical coordinates (P. and A,, of the observation points. In view of the smallness of the ratios Ax/r,, Ay/r, and Az/r,,, one can solve, to a first approximation, a system of linear rather than quadratic equations (3.7.5). Introducing new designations for brevity, we shall rewrite equations (3.7.5) as follows:

d, = a, Ax

+ b, Ay + c, Az,

n

1,2, . . .

=

(3.7.6)

where a,, = -

2 cos (P, sin A,, ,

b,=-

2 cos (Pn cos A, 9

rn

rfl

c,=--

2 sin (P, In

Since the expression for d,, includes the term AX)^ + (Ay)’ + ( A ~ ) ~ ] / r i , the system of equations can be solved by the successive approximation method. The expected accuracy of determination of the coordinates will be estimated proceeding from the following assumptidns. Let the random error of taking into account the gravity anomalies 69, or that of absolute gravity measurement be equal to f 16 mGal. Then, the distance between the adopted origin of coordinates and the centre of mass will be determined with a random error of f 100 m. Increasing the number of independent gravity measurements will reduce the random error of determination of the coordinates of the centre of mass. The effectiveness of the method for determining the position of the centre of mass from gravity measurements on the Moon will be illustrated as follows. To this end, the measured values of gravity (see Table 1.8) and coordinates of the landing sites of Apollos 11, 12, 14 and 17 (see Table 3.9) will be used. The position of the centre 0, of mass will be determined relative to the origin of the coordinate system in which the Apollo landing sites are given. This requires preliminary reduction of the measured values of gravity g, to the level for which a map of gravity anomalies (see Fig. 1.17) is available; that is, 100km above the lunar surface. The reduced gravity values will be designated g h = 100

= gn

+ 61g

where is the correction for changing the altitude to 100 km. Next, using the gravimetric map (Fig. 1.17), we shall determine the gravity anomalies 6g

186

L m a r Gravimetry TABLE 3.6 Gravity at Apollo landing sites

Apollo Apollo Apollo Apollo

11 12 14 17

162,852 162,674 162,653 162,695

145,599 145,436 145,426 145,422

1835.505 1836.014 1836.369 1834.814

-28 50 70 40

+ + +

145,571 145,486 145,496 145,462

9,. measured value; gh=lOO and reduced values; R, = ( p , IOO),distance to the reduction level.

+

from the coordinates of the Apollo landing sites. The gravities taking into account these anomalies will be gh0=100 =

gn

+ 6,s + &I

The values of ghO,loo for four points on the Moon (Table 3.6) were used to define the coordinates Ax, Ay and Az. The calculations have yielded the following values. Ax

=

-300 m,

Ay = -90 m,

Az = 0.

(3.7.7)

Since the initial landing site coordinates were already related to the centre of mass, the values of Ax, Ay and Az must be zero. However, the values obtained for Ax, Ay and Az (3.3.7) reveal the errors inherent in the method for determining the position of the centre of mass from gravity measurements on the Moon. The unfavourable positioning of the gn measurement points on the Moon should be pointed out. The landing sites differed by no more than 24" in latitude. It has already been mentioned that for the method to be effective measurements should be taken at points spread all over the Moon. Yet, even under the existing conditions, the accuracy of determining the centre of mass from gravity measurements was quite high.

3.8 General Geometric Figure of the Moon

In spite of the long history of studies involving the geometric figure of the Moon, our knowledge is still sparse. By the geometric figure of the Moon is implied the figure of its physical surface. Depending on the problem being solved, the general geometric figure of the Moon may be a. triaxial ellipsoid, an ellipsoid of rotation, or a sphere. The concepts as regards the geometrical figure of the Moon, shaped by terrestrial astronomical observations of its near side, were refined after the Moon started being explored using spacecraft. On 4 September, 1959, the automatic interplanetary station Luna

187

Spatial Variations in the Lunar Field

3 circumnavigated the Moon and took pictures of its far side. Subsequently, Soviet automatic stations of the Luna and Zond series as well as American Ranger, Lunar Orbiter and Apollo spacecraft gained more information about the Moon's topography (52.10). The basic task of selenodesy is to establish a selenodetic network of points on the lunar surface. Such networks were already mentioned in 52.10. We should only like to add and that a comprehensive coverage of selenodetic networks can be found in the work by Slovokhotova (1973), where reference is made to 16 selenodetic catalogues. Consider the general geometric figure of the Moon, approximated by a simple mathematical surface. Approximation by a sphere calls for determination of only four parameters (three coordinates of the sphere's centre relative to the centre of mass and its radius). Obviously, the coordinates of the figure's centre with respect to the centre of mass are equal in absolute value to those of the centre of mass relative to the figure's centre, but are opposite in sign (53.7). The radius to be determined must be mean for the entire Moon and not only for the part of the sphere representing the near side. This brings to mind a fallacy sometimes encountered in the literature to the effect that the centre of the figure, determined fro'm the major portion of the near side, is close to that of the Moon as a whole. Listed in Table 3.7 are the mean radii deduced from the results of laser altimetry of the Moon's equatorial zone (Kaula et al., 1974). The table also includes the positions of the centre of mass relative to that of sphere, determined from the same data. A further approximation of the general geometric figure of the Moon is an ellipsoid. As many as nine parameters must be known for its definition, namely, three coordinates of the figure's centre, three direction cosines determining the orientation of the ellipsoid, and three of its semiaxes. In the case of an ellipsoid of rotation, the number of unknown parameters is reduced to seven. Table 3.5 already listed the parameters of the ellipsoids approximating the geometric figure of the Moon. As was assumed a priori in these approximations, the axes of the ellipsoids are parallel to those of the selenocentric system of coordinates, in which the X axis is directed toward

TABLE 3.7 Mean lunar radius determined by laser altimetry of the Moon's equatorial zone (Kaula et a/., 1974)

Spacecraft

Mean radius (km)

Offset of the centre of mass

Direction of offset

Apollo 15 Apollo 16 Apollo 17

1731.3 1738.1 1737.4

2.1 2.9 2.3

25"E 25"E 23"E

Weighted mean

1737.7

2.55

24"E

188

Lunar Gravimetry

the Earth, while the Z axis extends along that of rotation. This makes three out of the above-mentioned nine parameters known (direction cosines). Bills and Ferrari (1977) also approximated the real geometric figure of the Moon by an ellipsoid, with the difference that its orientation was assumed to be random. The initial data had been provided by 5800 laser altimetric measurements from Apollo spacecraft. Their accuracy was crl = f0.3 km. In addition, use was made of 1400 determinations of the absolute radius vectors with their origins at the centre of mass, which were made from satellite photographs with an accuracy oz = k0.3 km. Added to these must be 3300 determinations of the radius vectors of points on the Moon's surface from terrestrial photographs, the accuracy of these being o3 = k 1.0 km. As a result, the optimal ellipsoid was selected (Table 3.8). Listed in the fourth and fifth columns of Table 3.8 are the directions of the ellipsoid semiaxes. The mean radius of the spherical Moon, corresponding to the same initial data, is 1737.46 L- 0.04 km. Approximation in terms of an ellipsoid with a random orientation was made even earlier, but the initial data had not been as accurate. Thus, the geometric figure of the Moon is approximated by a triaxial ellipsoid with the orientation of its axes being widely different from that of the axes of the selenodetic system of coordinates. The minor axis of the ellipsoid does not coincide with that of rotation, and- the entire geometric figure is substantially asymmetrical with respect to the equatorial plane. As will be shown below, such differences in orientation do not occur in the case of the dynamic figure of the Moon. This means that the masses constituting the Moon's relief are distributed in depth with a compensation which, in the final analysis, brings the axes of the dynamic figure close to those of the selenodetic coordinates. Of course, the geometric figure characterized by the parameters listed in Fig. 3.3 is not final in spite of the fact that the accuracy with which the semiaxes of the ellipsoid have been defined is essentially high. However, this is merely an estimate as regards the internal convergence, what with the scantiness of the data on elevations in the circumpolar regions and on the far side of the Moon. TABLE 3.8 Parameters of the ellipsoid approximating the surface of the Moon (Bills and Ferrari. 1977)

Ellipsoid axes Major Intermediate Minor

Semiaxis lengths R.m.s. error (km) (km) 1738.04 1737.68 1736.67

kO.09 k0.06 k0.13

Semiaxis directions Offset of the centre cp 1 of mass (km) 23" 33 48

20"E 94"E 317"E

1.57 0.30 0.75

R.m.s. error (km)

k0.08 f0.04 +0.13

189

Spatial Variations in the Lunar Field

TABLE 3.9 Exact rectangular and polar coordinates of the corner reflectors and ALSEP transmitters placed on the Moon’s surface (King et al., 1976)

Corner reflectors Apollo 11 1592.506 Apollo 14 1652.333

689.537 -522.205

20.973 - 109.762

1735.505 1736.362

23.41205 - 17.53870

0.69244 -3.62427

ALSEP Apollo 12 Ap0110 14 Apollo 15 Apollo 16 Apollo 17

-690.866 -522.178 96.992 458.816 831.357

-90.558 - 109.748 765.007 -270.452 599.313

1736.014 1736.369 1735.509 1737.453 1734.814

-23.48471 - 17.53768 3.56965 15.43634 30.70781

-2.99013 -3.62379 26.15478 -8.95506 20.21006

1590.046 1652.350 1554.782 1654.363 1399.730

Finally, here are some recent data that can be used in refining the general geometric figure of the Moon. What we have in mind are the very exact spatial coordinates of some sites of Apollo and Lunokhod landings on the Moon. Five of these probes (Apollos’ll, 14 and 15; Lunokhod 1 and 2) left corner reflectors on the Moon’s surface. The reflectors carried by the Lunokhod had been developed by French specialists. Laser detection and ranging of corner reflectors from the Earth permits atcurate measurement of the distance between a reflector and the terrestrial station. In addition, astronauts from Apollos 12, 14 and 17 left radio transmitters operating at a frequency of 2.3GHz on the Moon. They were intended for a series of experiments with instrumentation known as the Apollo Lunar Surface Experiments Package (ALSEP). The intention was to receive signals from these transmitters with the terrestrially based very long base interferometer (VLBI). Such instrumentation permits measurement of the relative positions of points on the Moon to within 1-3 m and the rotary motions about the centre of mass to within 1” or so. Laser detection and ranging along with VLBI observations will enable a broad range of experiments to be carried out to gain more information about the Moon and its gravitational field. The experimental programme included checking whether the observed position of the Moon coincides with that calculated in accordance with the existing lunar theories, refining the selenodetic coordinates of the corner reflectors and ALSEP transmitters, studying the physical libration, a more careful determination of the dynamic flattening of the Moon, determination of the 3rd- and 4th-order harmonic coefficients of expansion of the lunar gravitational field, a more accurate definition of the ratios between the lunar and terrestrial masses p - ’ and between the mass of the Sun and that of the Earth and Moon combined, and determination of the flattening of the Sun. Other items were verification of the equivalence principle as applied to large

190

Lunar Gravimetry

masses, verification of the Dicke-Brans theory of gravitation, examination of the effects produced by gravitational waves, measurement of time variations in the constant G of gravitation, and so forth (Bender et al., 1973). As a result of 97 individual observations involving the ALSEP transmitters, made over a period of 16 months, and 1184 laser probings of the corner reflectors, made over a period of three and a half years, some rather interesting conclusions have been drawn concerning the motion, rotation and librations of the Moon, as well as certain aspects of the theory of gravitation. In particular, the rectangular and spherical coordinates of the positions of the corner reflectors and ALSEP transmitters relative to the lunar centre of mass have been determined with a high degree of accuracy (King et al., 1976) (Table 3.9). The error of determination of the x coordinate did not exceed 30 m, while in the case of the other two coordinates-y and z-the error was less than 10 m. 3.9 Dynamic Figure of the Moon

It is known that the Moon’s motion in space comprises the translational motion of its centre of mass and rotation about it. Knowing the mass of the Moon and the position of its centre of mass is not sufficient to described lunar rotary motions. This also calls for determination of its other dynamic characteristics, namely, second-order moments of inertia, totalling nine, only six of which are independent. They can be written in the following dimensionless form: I

f

o(y2 + 2’) dR

1

B=-JMtR2

U(X’

+ z2)dR

n

1 c=-j M ~

Ro(x2~ + y2) dR

(3.9.1)

n

D = - j1

MuR2

oyz dR

n 1

r

E=-JMUR2 oxz dR n

(3.9.2)

191

Spatial Variations in the Lunar Field

F=-

1

MCR2

J axy dR R

where a is the density of the Moon, R is the volume of the Moon, Mc is its mass, R is its radius, A, B and C are the axial moments of inertia ( A with respect to the X axis, B with respect to Y , and C with respect to Z ) , and D, E and F are the products of inertia, each being dependent on the distances from two planes. Note that the axial moments are always positive, whereas the products of inertia may have any sign. As is known from theoretical mechanics, the directions of the three orthogonal coordinate axes X , Y and Z may be selected such that the products of inertia (3.9.2) will be zero. The axial moments A, B and C will characterize an ellipsoid of inertia whose axes are called principal axes of inertia. If the origin of the coordinates coincides with the centre of mass, the ellipsoid of inertia is referred to as the central ellipsoid of inertia, and its axes are known as the central axes of inertia. The central ellipsoid of inertia of the Moon is taken as its dynamic figure. The dynamic figures of planets differ only in the parameters of their central ellipsoids of inertia. In general, when it comes to the dynamic figure of a planet, nobody is interested in the actual shape it takes. Usually, a fictitious body is implied, exhibiting predetermined dynamic characteristics, namely, definite axial moments of inertia. It may so happen that bodies with different shapes and internal structures have the same dynamic figure. Thus, the dynamic figure in a system of coordinates whose origin coincides with the centre of mass and having its axis coincident with the central axes of inertia is characterized by principal moments of inertia A, B and C . Equations of motion involve so-called dynamic flattenings given by a=-

C-B A

p=-

1

C-A B

7

y=-

B-A C

(3.9.3)

We shall see in what follows how A, B and C are expressed in terms of the harmonic coefficients C,,, and S,,, of expansion of the lunar gravitational field in spherical functions and how they are determined from observational data. But first let us establish a relationship between C,,, and S,,,, on the one hand, and degree moments of inertia Jpqr, on the other. The general expression for dimensionless degree moments of inertia in the rectangular system of coordinates will take the form Jpqr

axPyQzrdQ,

=

p , 4, r = 0, 1,2 . . .

R

where p

+ q + r = n is the order of the degree moment.

(3.9.4)

192

Lunar Gravimetry

For a given n, there will be (n + l)(n + 2)/2 different combinations of subscripts p, q and I, which is to say, precisely this number of different moments Jpqr are possible. On the other hand, there will be only 2n + 1 different harmonic coefficients C,,, and S,, of the nth order. Consequently, the degree moments of the nth degree will exceed in number the harmonic coefficients of the same order by n(n - 1)/2. Let us write the expressions for the harmonic coefficients C,,, and S,,, in the rectangular system of coordinates at n = 0, 1, 2, 3 (these expressions will be used later): 1 Coo= -So

MU

dR;

Clo =

n

~

‘S

MUR

oz dR;

R

R

Czo=-

oydR,

2MeR2

R

j o [ 2 z 2 - (x’

+ y’)]

dR

R

‘s

c21=-

1

C l l = -S o x dR MUR

o x z dR;

M , R ~R

S2,

=

1

MUR

j o y z dR; R

(3.9.5)

1 c30

=

S 0 z [ 2 z 2 - 3(x2

+ y’)] dR

R

c31=-

‘s

OX[Z’

McR3

R

- $(x’

+ y’)] dR;

193

Spatial Variations in the Lunar Field

1 az(x2 - yz) dR; c 3 Z = - 4MaR3 S R

OX(X’

1

s3Z

=7 [axyz dR 2M(R n

- 3 y z ) dR;

n s33

= - ay(3x2 - yz) dR

24MaR3 l S n

Already these few expressions give an idea of the structural relationship between the harmonic coefficients and the degree moments of inertia. However, in the general case, the degree moments cannot be expressed in terms of the harmonic coefficients. Only the zeroth-order degree moment is equal to the harmonic coefficient of the same order: Jooo = c o o = 1 There are three degree moments of the first order (n = l), which corresponds to the number of first-order harmonic coefficients, and these are: ClO = J100;

c 1 1

-

= J010;

s 1 1 = JOOl

The first-order harmonic coefficints represent the position of the centre of mass. The coordinates of the centre of mass (Ax,Ay, Az) relative to the origin of the system of coordinates in which C,, and S,, have been determined are

AX = RClo;

Ay = RC11;

AZ = RS11

(3.9.6)

There are a total of five second-order harmonic coefficients and six degree moments of the same order. Three degree moments are expressed directly in terms of harmonic coefficients. These are the products of inertia JllO

= 2Sn,

JlOl

= Cz1;

JOll

= SZl

The other two second-order harmonic coefficients CZOand Cz2are expressed only in terms of a combination of three second-order degree moments Jzo0, Joz0 and JOo2.Equations (3.9.4) and (3.9.5) give c z o = Jooz - &Jzoo

+ Jozo);

czz = W Z O O - 5020)

(3.9.7)

The degree moments Jzoo, Joz0,and JOo2 can be expressed in terms of C20 and Cz2 if the dynamic flattening 8 is used (8 and y are determined from observational data more reliably than a). Equations (3.9.l), (3.9.3) and (3.9.4) give Jzoo - 3002 (3.9.8) 8 = Jzoo + Jooz

194

Lunar Gravimetry

The following relations can be derived from Eqs. (3.9.7) and (3.9.8):

(3.9.9)

Seven third-order harmonic coefficients may be expressed in terms of combinations of ten degree moments of inertia of the same order: c 3 0 = -$(5201 s31

= -kc5210

c 3 3 = 2%5300

+ 5021 - 25003); + 5030 + 45012); - 351~0);

1

c31 = - d 5 3 0 0 c32

= &JZOl

s 3 3 = 2%35210

f 5120 - 45102)

- 5021);

s32

= 45111

- 5030)

On the other hand, to express degree moments of inertia in terms of harmonic coefficients requires three more equations in which the degree moments would be related to some three additional observable parameters representative of the Moon’s dynamic properties. Still more such equations are needed for degree moments of higher orders. Using expressions (3.9.1) qnd (3.9.5), we shall state relations between the Moon’s principal moments of inertia A, B, C and the harmonic coefficients C20 and C22: (3.9.10)

Attempts to express each of the moments A, B and C separately through two harmonic coefficients have not been successful. This is possible only for a combination principal moments of inertia. From (3.9.10) one can easily derive B - c = c20 C-A

+ 2c22

= 2C22 - C20

(3.9.11)

A - B = -4c22

The Moon’s moments of inertia were initially obtained exclusively through observations of the motions of the Moon itself, including the physical libration, motions of the perigee and node of the lunar orbit. A new possibility for refining A, B and C has arisen in connection with the determination of the harmonic coefficients CzOand CZ2through observations involving ALS. To find the three unknowns (A, B and C) there are two

195

Spatial Variations in the Lunar Field

equations (3.9.lo), in principle, relating them to the harmonic coefficients C z o and C Z 2 as , well as three equations (3.9.3) relating A, B and C to the dynamic flattenings M, B and y. However, the equation involving M (3.9.3) is virtually useless for finding A, B and C because of the unreliable determination of M from observation. A, B and C can be determined from the remaining four equations by the least-squares method. The equations can be written as follows: czo

=

A+B-2C ; 2

B-A CZZ = *4 ’

B=-

C-A B

B-A y=C

(3.9.12)

Let us calculate anew the Moon’s principal moments of inertia A, B and C . At the same time, equations (3.9.12) will be used to equate the observed values of B, y, C z o and C z z .To do this, the mean values of the latter quantities will have to be calculated first from. their observations. The values of the harmonic coefficients C z o and CZ2were determined fifteen times by different workers who used different ALS tracking data. Table 3.10 lists the main results of such determinations. They were obtained using various methods and different sets of initial tracking data. Obviously, the determinations differ in accuracy, most of them being interdependent because the same data had been used, primarily those involving Lunar Orbiters. Some of the determinations have been completely independent of one another, including those involving only Lunar Orbiters or only Lunas. To assess the true accuracy of C z o and C z z is extremely difficult. We have decided to deduce the weighted mean. Taken as the weight was a quantity inversely proportional to the variance in the determination of C z o and Cz2.Ten workers have specifieid the r.m.s. errors of their determinations of these pairs of harmonic coefficients. Therefore, we first computed the weighted mean values of C z o and C z z from these ten determinations and their r.m.s. errors. The latter were for C z z . These errors found to be k3.3 x for C z o and k1.7 x were then ascribed to the other five pairs of C z o and CZ2values for which no errors had been specified by their authors. Next, the weighted means were again calculated for all 15 pairs of harmonic coefficients: Czo

=

-(201.41 & 0.24) x

Cz2 = +(22.10 k 0.16) x

(3.9.13)

The next step was to find the values of the dynamic flattenings B and y for the Moon. The results of some determinations of these parameters are summarized in Table 3.11, including those by Koziel (1967), who used the

196

Lunar Gravirnetry

TABLE 3.10 Harmonic coefficients C , and C,,determined from ALS tracking data

Value and r.m.s. error ( x 10-6) Reference

C20

c22

+

Akim (1966) Tolson and Gapcynski (1968) Lorell and Sjogren (1968) Michael et al. (1970) Lorell (1970) Sjogren (1971) Liu and Laing (1971) Michael and Blackshear (1972) Ferrari (1972) Bryant et al. (1974) Sinclair et al. (1976) Blackshear and Gapcynski (1977) Ananda (1977) Ferrari (1977) Akim and Vlasova (1977) Bills and Ferrari (1980) Ferrari et al. (1980) Akim and Vlasova (1983)

206 f 2.2 205.96 f 14 202.63 k 1.43 207.1 f 3.3* 195.64 f 3.3* 204.8 k 3.0 199.6 f 2.0 203.8 3.3* 205.6 f 3.3* 201.24 f 0.19 202.91 f 1.33 202.19 f 0.91 210.9 f 3.97 204.6 f 3.8 200.1 & 1.3* 202.4 f 1.4 202.1 5 1.2 207.0 & 6.6

14 1.2 20.42 f 2.9 21.91 f 2.5 22.42 f 1.7 15.87 f 1.7* 22.1 f 0.5 23.59 5.3 24.85 f 1.7* 22.58 f 1.7* 20.2 f 1.9 22.4 f 0.2 22.21 f 1.23 22.1 f 0.08 21.5 f 1.2 25.0 f 1.7* 22.3 f 1.2 22.3 f 1.2 24.3 7.2

Weighted mean

201.44 f 0.22

22.1 f 0.2

* The r.m.s. errors have been taken a priori,

data produced by almost forty years of astronomical observations of the Moon, the results of laser detection and ranging of corner reflectors (Bender et al., 1974; Williams et al., 1974), and the most exact values of 8 and y derived by King et al. (1976), who combined the results of many years of laser observations with the data supplied by the ALSEP transmitters. Using the values of 8 and y given in Table 3.11, we computed the corresponding weighted means. Taken as the weights were quantities inversely proportional to the squares of r.m.s. errors. The weighted mean values of the dynamic flattenings were found to be as follows: 8 = (631.26 k 0.05) x (3.9.14) = (327.45 k 0.32) x The high accuracy attained in the determination of the dynamic flattenings 8 and y from the results of laser detection and ranging of corner reflectors and observations involving VLBI interferometers permits the vahe of CZ2to be calculated, assuming that C20 is known: (3.9.15)

197

Spatial Variations in the Lunar Field

TABLE 3.11 Some determinations of dynamic flattenings of the Moon

Reference

( x 10-9) P

Method

Y ( x 10-9)

Heliometric observations 629.4 & 0.6 231.0 f 3.2 (1877-191 5) Bender et al. (1974) Laser observations of corner 631.1 & 0.4 226.8 & 1.0 reflectors Williams et al. As above 631.26 & 0.3 227.37 & 0.6 (1974) 631.27 & 0.3 227.7 & 0.7 King et a!. (1976) As above, the VLBI observations Ferrari et a/. As above, and Doppler 631.687 f0.132 228.022 f0.100 (1980) tracking of Lunar Orbiter 4 Koziel (1967)

a* (X

398.4 404.3 403.89 403.6 403.665

' Determined by calculations involving p and y from a = b - y

Substitution of the values given in (3.9.13) and (3.9.14) into (3.9.15) gives Czz = 22.15 x lop6 Since Cz0,Czz,p and y are related to one another by equations (3.9.12),it would be more appropriate to find their values by mutually equating them with due regard for the accuracy of their' determination from observations. Let us now determine the principal moments of inertia A, B and C of the Moon and find the equated values of CzO,Czz,p and y. To solve this problem with the aid of the system of equations (3.9.12),we shall resort to the found weighted means of Czo, Cz2 (3.9.13) and p, y (3.9.14). The results of these calculations are presented in Table 3.12. Table 3.12 also gives the differences 6Cz0, 6Czz,Sp and 6 y between the initial and equated values of CzO,Cz2,p and y. We consider the tabulated values to be the best available and shall use them in our subsequent calculations. The mutually compatible system of parameters had been derived earlier by Williams et al. (1973), but the data he had at his disposal were not as complete as in our case. TABLE 3.12 Results of calculating the principal moments of inertia of the Moon (A, B, C) and the equated values of C,, C , b and y

Parameter c20

6C20 c22

sc22

P SP

Value

Parameter

Value

-201.40 x -0.01 x lo-; 22.10 x 100.00 631.26 x 0.00

Y 6Y

227.13 x 10-6 0.320 x 0.388,550,44 0.388,462,08 0.388,707,36

A B C

198

Lunar Gravirnetrv

The Moon’s principal moments of inertia A, B and C are included in the equations for the inequalities in the motions of the perigee dn and node dR of the lunar orbit: dn = 20.138(6.27B - 4.27A - 2C) dR = -0.334(C - A )

(3.9.16)

The values of dn and dR are determined from observations. Meshcheryakov and co-workers (Meshcheryakov and Zazulyak, 1975) used these equations to calculate the moments of inertia A, B and C, adding equations (3.9.16) to the system of equations (3.9.12). The initial harmonic coefficients were taken by them from other sources (Akim, 1966; Mottinger and Sjogren, 1967; Gapcynski et al., 1969; Lorell, 1970), as well as the dynamic flattenings (Bender et al., 1973) and the results of observation of the perigee dn and node dR (Eckert, 1965). Later, Meshcheryakov et al. (1976) refined further the values of Cz0, CZ2,P, y, dR and dn by mutually equating them, for which purpose they used different initial harmonic coefficients Czo and Czz. The dynamic figure of the Mo.on determines the character of its physical libration. We shall now delve into the determination of the physical libration parameter f from parameters characterizing the dynamic figure of the Moon and its gravitational field. It is known that the actual rotation of the Moon is more complex than the rotation described by Cassini’s laws. The difference between the real rotation and the simplified rotation postulated by Cassini’s laws corresponds to the Moon’s rotary motion known as physical libration. It is small in magnitude but extremely complex in its variation pattern. The physical libration is usually considered in terms of three components of rotation. Each of the components, an angle, is given by a sum or periodic terms. The associated expansion include an important physical libration parameter f which depends on the Moon’s principal moments of inertia: (3.9.17) Although the history of determination of the parameter f spans more than a century, the accuracy with which it was known left much to be desired. Kulikov and Gurevich (1972) presented a summary of about 50 determinations off in this century alone. The summary shows that the values off vary from 0.5 to 0.9. It has already been proposed to discontinue further terrestrial observations aimed at refining the value off: The advent of space vehicles performing soft landings on the Moon and circumlunar satellites has opened up three possibilities for refining the physical libration parameter f: One of these involves astrotracking from the Moon with the aid of automatic

199

Spatial Variations in the Lunar Field

astrometric lunar stations. The second possibility resides in investigation of the fine details of the Moon’s rotation through laser detection and ranging of corner reflectors placed on its surface. Finally, the third approach is based on studies of the lunar gravitational field using ALS tracking data. According to (3.9.3), (3.9.1 1) and (3.9.17), the physical libration parameter f can be expressed in terms of the harmonic coefficients CzOand C 2 2in the following manner: c20

+ 2c22

(3.9.18)

= c20 - 2c22

Of the above-mentioned three approaches to refining f, two are currently in use owing to recent advances in space techniques. Because the harmonic coefficients C20 and C Z 2characterizing the lunar gravitational field have now been determined more accurately, we can use the equated values of C20 and C Z 2to find the parameter f from (3.9.18). Also, the equated values of the dynamic flattenings /3 and y can be used to calculatef with the aid of (3.9.17). Values of the parameter f calculated.in both these ways are listed in Table 3.13. The values off, determined from two radically different sets of tracking data (involving the gravitational field and rotary motions of the Moon), agree well. Comparison of these values of f with those deduced from astronomical observations (Table 3.13 gives only three such values) attests to the much greater reliability of the first of them (Eckert, 1965). The second basic parameter of physical libration is the obliquity of the lunar equator to the ecliptic

I=

3@(1 + rn + irn2)1/2 (1 rn)(2rn rn2 - 38)

+

(3.9.19)

+

where i is the inclination of the lunar orbit plane to the ecliptic, and in is the ratio of the period of the Moor’s rotation to the period of precession of the TABLE 3.13

Some determinations of the physical libration parameter, f Reference Eckert (1965) Jeffreys (1961) Koziel(l967)

Method

f

From motions of the lunar perigee and orbital node From physical libration From heliometric observations From harmonic coefficients CZOand CZ2 found from ALS tracking data From dynamic flattenings B and y, found from laser observations of corner reflectors and measurements involving ALSEP transmitters

0.638 0.639 & 0.014 0.633 & 0.006 0.6401 0.0019 0.6402

0.0010

200

Lunar Gravimetry

lunar orbit node (0.004,019). The determination of I is restricted, primarily, by parameters that have nothing to do with the Moon’s gravitational field and, therefore, will not be considered here in greater detail.

3.10 Hydrostatic Equilibrium Figure of the Moon

The model of the hydrostatic equilibrium figure of the Moon can be constructed theoretically. Its construction calls for an assumption that the Moon behaves as a liquid body with respect to constant forces applied to it over a long period of time. Ideal liquids are characterized by a lack of cohesion between their particles, which is why they are free of shear stresses but governed by hydrostatic pressure. The latter is the result of every unit mass of the Moon being acted upon by volume forces, such as the attraction of the lunar mass itself, the tidal attractions of the Earth and the Sun, and the centrifugal forces generated by the Moon’s rotation. In principle, pressure is also affected by such surface forces as light pressure and meteoritic impacts, but these are negligibly small. ‘The hydrostatic pressure inside a liquid depends on the coordinates of the unit volume under consideration, yet it is equal in any direction within this volume. If the potential of lunar mass attraction is denoted by V, the potential of centrifugal acceleration due to the Moon’s rotation is denoted by U , the potential of the tidal acceleration from the Earth is denoted by W,, and the potentials of accelerations due to other factors are ignored by virtue of their insignificance, the condition of equilibrium of the liquid Moon is determined by equality of the sums of projections of all accelerations along the three orthogonal coordinate axes:

av +-+au 0 ax ax ax au _l a_ p _ -av +-+aY aY 0 aY au _l a_p- _ -av +-+aZ aZ 0 aZ lap

--

aw, ax aw, aY

(3.10.1)

aw,

az

where P is the hydrostatic pressure potential. Multiplication of these dquations by dx, dy and dz, respectively, with subsequent addition gives the. complete differential dP = o d W

(3.10.2)

in which W = V + U + W , is the total potential of all acting forces. If W = const. is a level surface, then, on this surface, the increment d W = 0,

201

SDatial Variations in the Lunar Field

hence, according to (3.10.2), P = const. This means that the level surfaces W = const. are at the same time surfaces of equal pressure. It can also be inferred from (3.10.2) that the hydrostatic pressure on the outer surface is nil. Differentiation of both sides of (3.10.1) with respect to x, y and z may give the following equations:

aa a w - aa. a w -- -aa. a w ax‘ ax a y eay a i aZ ~

from which it follows that the surfaces of equal pressure and level surfaces are identical. If the density a had varied as a function of coordinates in a random manner, it would have been impossible to equalize all forces, and flows would have occurred in the liquid. It should be pointed out that, strictly speaking, the level surface coincides with those of equal pressure and equal density, provided the body of the Moon exhibits isothermal conditions and adiabatic equilibrium. In what follows, the surface of the Moon’s hydrostatic equilibrium will be represented by a sphere deformed by’a second-order spherical function. The radius of this surface is

r(cp, 4= RC1 + &S2(cp, 41

(3.10.3)

where R is the mean radius of the Moon, S2 (cp,1) is a surface spherical function of the second order; and E is a small factor. The problem of deriving the surface of the Moon’s hydrostatic equilibrium reduces to the determination of this factor. In the case of a liquid Moon with uniform density and an outer surface of the (3.10.3) type, the potential of attraction at points P in the surrounding space is

[ +;

V(P)= E5 1

r

)(2;

&S2((P,A)]

(3.10.4)

In order to find the potential V at points P on the surface (3.10.3), we must exclude the raidus r from (3.10.4) with the aid of (3.10.3): GMU [l - &(cp, V(P)= R

A)]

Then, we shall write the potential of centrifugal forces: U ( P ) = $02r2[1 - P2,(sin cp)]

In accordance with Kepler’s third law, the latter equation can be rewritten as (3.10.5)

202

Lunar Gravimetry

where o is the angular velocity of the Moon's rotation, and the mean distance C1 between the Earth and the Moon is used instead of the semimajor axis of the lunar orbit. As regards the potential of tidal acceleration from the Earth (1.3.12),we shall take only the portion that is invariable in time: 1 GMQrz W,(P) = -~ [ -2Pzo(sin cp) 4 c:

+ Pzz(sin cp) cos 2A)]

(3.10.6)

The total potential of the Moon will be found by addition of (3.10.4),(3.10.5) and (3.10.6) after having excluded r from these equations with the aid of (3.10.3):

+ GCMl ,

R 2 [ i - gPzo(sin cp)

+ $Pzz(sin cp) cos 2A] (3.10.7)

Since the surface of hydrostatic, equilibrium coincides with the level surface, the potential Won it must be constant. Therefore, there must be no spherical functions in the expression for W (3.10.7).Hence, in accordance with (3.10.7), we obtain

("Y[i

MQ 2 Mu C1

eSz = -~

5

- ;(sin cp)

1

+ -41 Pz2(sincp) cos 22

(3.10.8)

Substitution of the expression found for &SZinto (3.10.3) gives an equation describing the figure of the Moon which is in a state of hydrostatic equilibrium: r(cp, I.) = R[1

+ A,Pzo(sin cp) + AzPzz(sincp) cos 2 4

(3.10.9)

in which

5

The surface given by (3.10.9) approaches that of a triaxial ellipsoid. If we substitute the numerical values of R, p-' and C1 (1738 km, 81.3 and 384400 km, respectively) into the latter formulae, we obtain the following

203

Spatial Variations in the Lunar Field

expression for the radius of this surface: r(cp, A) = 1738[1 - 1.56 x 10-5P20(sincp)

+ 0.47 x

10-’Pz2(sin cp) cos 211 km

This expression will be used to establish the differences between the three semiaxes of the resulting figure and the mean radius R. For the equatorial semiaxis directed toward the Earth we have

-

r(0,O) - R = Aa = -35 p-1 12

(fly -

I?= 38.1 m

For the other equatorial semiaxis we have

and for the polar semiaxis we have

The Moon’s rotation is responsible only for the polar flattening, while the triaxiality is the result of the Earth’s tidal action. We shall now calculate the axial moments of inertia for the constructed hydrostatic equilibrium figure of the Moon: A =

j-. .j‘

i 0

=

0

=

[

-7112

r(rA)

arf(cos2 cpl sin2 A1

+ sin2 cpl) cos cpl

arf(cos2 cpl cost il

+ sin2 cpl) cos cpl dill dcpl drl

d l l dcp, drl

0 r/[A)

-nl2

(3.10.10)

0

r(rA)

a r t c0s3 cpl d& dcp, drl 0

-n/2

0

The density a is assumed to be constant. Substituted after the first integration for the variable rl was the upper limit of r(cp, A), which is equal to expression (3.10.9) to the fifth power. Expand it in degrees of the small term Alpzo + A2PZ2cos 21 and retain only the first-degree terms in the expansion. As a result of simple but cumbersome transformations, we obtain the following

204

Lunar Gravimetry

expressions for the principal moments of inertia:

15

[4 +

27ca~5 B =15

10

p-I($)']

(3.10.11)

25 15 As can be seen from these equations, the moments of inertia A , B and C depend not only on the density a and radius R, but also on the ratio between the masses of the Earth ( M , ) and the Moon ( M a )as well as on the ratio (R/C1)3.In the case of a uniformly spherical Moon, all moments of inertia are equal to one another and to $MuR2. We can now calculate the dynamic flattening that the Moon would have had, if it had been in a state of hydrostatic equilibrium. Using equations (3.10.11) and the above numerical values of p - l and (R/C1)3 gives the following expressions with a sufficient degree of approximation:

(3.10.12)

We can also find, for the hydrostatic equilibrium Moon, the values of the dimensionless harmonic coefficients CzOand CZ2:

Integration of these equations, assuming that the density is constant, gives:

205

Spatial Variations in the Lunar Field

c 2 0

(&)

3

5 4

= --p-

(3.10.13) 3

3 4

c 2 2 =-"-I(&)

Expressions for these harmonic coefficients can be obtained using the formulae derived earlier for the moments of inertia A, B and C (3.10.11) and formulae (3.9.12). Table 3.14 lists the numerical values of the dynamic flattenings a, p and y, the harmonic coefficients Cz0and Cz2,the physical libration parameterf, and the density irregularity parameter g = (3/2)(G/McR2) (§3.13), calculated assuming that the Moon is in a state of hydrostatic equilibrium. The derivation of these values has involved formulae (3.10.12) and (3.10.13).For comparison with these theoretical values, the table also gives the corresponding values obtained directly or indirectly from observations. The last column includes ratios of theoretical to actual values. Analysis of these ratios indicates that the departure from hydrostatic equilibrium is pronounced in the case of the Moon. To explain the fact that the Moon is far from being in a state of hydrostatic equilibrium, a number of mechanisms have been proposed. Way back in 1915, Jeffreys hypothesized that the equatorial bulge of the Moon is merely a solidified tidal formation that appeared when the Moon was less distant from the Earth than it is today by a factor of 2.7. However, in his later monograph TABLE 3.14 Parameters of the Moon in hydrostatic equilibrium and the real moon

Parameter Dynamic flattenings

Symbol

Ratio of observed to theoretical value

403.81 x 631.26 x 227.45 x

c22

9.39 x 5.64 x lo-6

201.40 x 10-j 22.10 x 1 0 - 6

f

0.25

0.6402

2.5

0.584

0.97

Y

Physical libration parameter Density irregularity parameter

Actual observed value

94 x 1 0 - 7 376 x lo-' 282 x

a

B Harmonic coefficients

Theoretical value for hydrostatic equilibrium

c20

9

0.6

* Calculated from the known ratio between dynamic flattenings a - @ from observations.

10-6'

43 17 8 22 37

+ y = aBy with Band y determined

206

Lunar Gravimetry

(1959), Jeffreys repudiated his own hypothesis as erroneous. Indeed, how can the asymmetry in the Earth-Moon direction, characterized by the third tesseral harmonic of the gravitational potential, be explained in terms of this hypothesis? The same applies to the equatorial asymmetry given by the third zonal harmonic. As regards the dynamic flattening B, it corresponds to the Moon being 140,000 km away from the Earth or to its free axial rotation with a period of 3.5 days. Runcorn and Shrubsall (1968) ascribed the absence of hydrostatic equilibrium to convective flows inside the Moon, which sustain the irregularity. This hypothesis has met with objections from the standpoint of the high viscosity inside the Moon and the high damping rate. Notably, if such flows do exist, they do not seem to affect the surface layers in view of the absence of folded structures on the Moon. Levin (1964,1967) came up with a thermal mechanism which suggests that there is a relationship between the internal temperature of the Moon and latitude, which results in a thinner lunar crust in the equatorial zone, as compared to the polar regions. Hence the isostatic disequilibrium manifest in the different moments of inertia A , B and C . The calculations made by Safronov (1967) indicate that it suffices to assume that the density of the shell is only 3% above that of the crust and that the lunar crust on the polar is 20 to 30 km thicker than on the equator. This is consistent with the 140°C difference in temperature between the equator and the poles. Zharkov and co-workers (1969) have shown that a minor density irregularity, primarily dependent on latitude, is quite sufficient to explain the observed dynamic flattening B and physical libration constant5 For example, if the density varies in the surface layer 0.1 W thick, the variation in a radial direction must amount to 0.015 g cm-3, while the latitudinal variation must be 0.001 g cm-3 less. Urey (1956) contended that the density of the Moon had been irregular ever since it was formed from meteoric matter whose density was variable. The difference in moments of inertia is the result of this accidental density irregularity. According to MacDonald (1961, 1962) and Urey (1952, 1962), the associated elastic stresses are as high as 20 bar; that is, these stresses maintain the hydrostatic disequilibrium of the Moon. The elastic stresses are distributed in the outer hard layers (Urey et al., 1959). The presence of stresses, in turn, is indicative of the absence of large volumes in the upper layers of the Moon’s interior, which are in a molten or half-molten state. 3.11 Gravimetric Figure and Distribution of Plumb-Line Deflections on the Moon

In the previous sections we have examined the geometric, dynamic and hydrostatic equilibrium figures of the Moon. The next, gravimetric figure,

207

Spatial Variations in the Lunar Field

implies a surface of equal gravity potentials (equipotential surface, level surface, or selenoid) ($2.4). Generally speaking, the gravimetric figure may be both a surface of equal gravity or that of equal radial gravity gradient ($2.7). These surfaces were presented in the form of equal elevation lines in Figs 2.3 and 2.4. Let us now turn to the plumb-line deflection on the Moon. By plumb-line deflection is understood the angle, at a given point, between the normal to the selenoid and that to the reference surface (sphere, ellipsoid). For convenience, the overall plumb-line deflection is resolved into two orthogonal components: one extending along the meridian, and the other along the parallel. The meridional (t) and parallel ( q ) components are defined as ratios of the corresponding horizontal attraction components, tangential to the meridian (l/p)(a W/acp) and parallel (l/p cos cp)(dW/aA), to the mean lunar gravity g:

t=--206,265 a w (angular seconds) PS

q=

acp

206,265 aw. (angular seconds) pgcoscp an ~

The horizontal attraction components are given as expansions in spherical functions (2.2.3) and (2.2.4). Bursa (1971) defines the plumb-line deflection as the ratio of the difference in selenoid elevations to the distance between the points at which this difference is determined. The distribution of plumb-line deflections on the Moon was also derived several times from the available harmonic coefficients C,, and S,, in earlier works (Bursa, 1971,1975; Chuikova, 1971). We have decided to plot a map of plumb-line deflections, using the most recently determined harmonic coefficients (see Table 1.13). The computed plumb-line deflection components 5 and q are represented on two maps (Figs 3.15 and 3.16) with isolines showing equal values of these components. In some instances, the full plumb-line deflection is represented in a vectorial manner. An example of such a representation (Bursa, 1971) is given in Fig. 3.17.

3.12 General Comments on the Moon's Internal Structure

Investigation of the internal structure of the Moon is one of the fundamental tasks currently being undertaken. Its accomplishment calls for a combination of various methods. These methods are essentially those used in terrestrial -studiesbut, of course, modified specifically for lunar applications. Seismology remains the predominant method for determining the general structure of the Moon's interior. Gravimetry, on the other hand, may perform the function of

Fig. 3.15. Map of the Moon showing plumb-line deflections t along meridians (isolines taken at 5 0 intervals). Scale 1 :20,000,000.

Fig. 3.16. Map of the Moon showing plumb-line deflections VJ along parallels (isolines taken at 5 0 intervals). Scale 1 :20,000,000.

210

Lunar Gravimetry

Fig. 3.17. Vector representation of the distribution of the full plumb-line deflections for the Moon’s near side (BurSa, 1971).

an interpolating technique through which the results of seismic sounding at separate points may find applications in other areas. Seismic measurements are used to define the boundaries of stepwise changes in the velocity of seismic wave propagation in the Moon’s interior having a layered structure. The layers consist of rocks of different petrographic and chemical compositions and, consequently, different elasticities and densities. The seismic waves are initiated by meteoritic impacts, drops of lunar modules and rocket stages, and explosions of charges. Seismic waves penetrate the Moon all the way down only in very few cases, when triggered by the impact of very large meteorites. Since the deep layers traversed by seismic waves at different velocities have different densities, variations in the boundaries between the layers manifest themselves in changes affecting the gravitational field. Given a few determinations of the layer boundaries from seismic data, one can spread them, essentially with the aid of the gravitational

Spatial Variations in the Lunar Field

211

field, to other regions of the Moon. In practice, however, all this is not as simple as it sounds. The observed gravitational field is affected by the Moon’s relief, and, apart from varying from one layer to another, the rock density may also vary within a single layer. Gravimetry permits the study of individual structures, determining their depth as well as their position and, in some cases, even the shape of the geological bodies responsible for anomalies. A good example of the effectiveness of the gravimetric method as applied to studies into individual structures is the investigation of mascons, which plays a major role in the elucidation of a most important stage in the Moon’s evolution. A prerequisite for effectiveness of the gravimetric method as applied to studies into the distribution of anomalous masses is differentiation of rocks in terms of density. The density of rocks is determined by their mineralogical composition and porosity. Accurate measurement of rock density is as important as that of the gravitational field parameters. Later, we shall treat at greater length the general structure of the Moon’s interior as well as the structure of the lunar crust, highlands, maria and mascons. Knowledge of the Moon’s gravitational field is instrumental in these studies, which, in turn, calls for investigation of the density distribution from the surface centreward. The density distribution determines the variations in pressure, gravity, and its radial gradient with depth. The tidal variations in lunar gravity depend on the Moon’s internal structure, which makes all these relationships an object of selenophysics. Finally, the hypotheses concerning the origin of the Moon and its evolution will be reviewed briefly.

3.13 Density Irregularity Parameter

From the standpoint of its internal structure, the Moon is more primitive than similar terrestrial planets. Yet even the Moon has a complex structure. Its interior is characterized by pronounced density irregularities. Apart from the usual planetary density variations from the surface toward the centre, density also varies horizontally, which manifests itself in anomalies of the lunar gravitational field. The most significant of these anomalies are due to mascons, discovered for the first time on the Moon. There is a possible explanation of why the Moon contains comparatively larger density irregularities than the Earth. This may be a result of the lunar gravity being six times lower than terrestrial gravity, resulting in less pronounced density differentiation of rocks inside the Moon. Another possible reason is the fact that most the Moon’s interior has remained half-molten or solid in the course of its thermal history.

212

Lmar

Gravimetry

Studying the overall distribution of density inside the Moon from gravimetric data alone is impossible. The main difficulty is that the inverse problem of gravimetry-determination of the mass distribution from a given external gravitational field-is multiple-valued. One and the same gravitational field may be represented by an arbitrary number of mass distributions. However, there are certain values which are determined from observations unambiguously and, at the same time, characterize the overall density distribution in the Moon. These are the harmonic coefficients of expansion of the gravitational field in spherical functions, which are a combination of degree moments of inertia (53.9). In particular, it has turned out to be convenient to represent the overall density distribution from the centre toward the surface using the ratio between the second-order axial moment of inertia and the total mass Mu of the Moon. Taking only the highest-value C’ out of the three axial moments of inertia A’, B’ and C for example, one can deduce the following dimensionless parameter characterizing the density irregularity of the Moon: 3 C‘ 2 MuR2

g=--

(3.13.1)

Here and in what follows, in contrast with 43.9, dimensional moments of inertia A’

=

AMUR’;

B‘ = BMUR’;

C’ = CMuR2

are involved, Mu being the total mass of the Moon and R, its mean radius. Assuming spherical density distribution o(r), the total mass is:

M,

=

1 0

j o ( p l ) p ; sin cpl dAl dcp, dp,

(3.13.2)

-n/2 0

where p l , cpl and Il are the current coordinates of the Moon’s volume. The moment of inertia with respect to the axis of diurnal rotation is

Substitution of integration with respect to a sphere’s volume for that with respect to the Moon’s volume produces a negligibly small error in the mass M uand moment of inertia C’. Of course, other combination of moments of inertia could be used to characterize the irregularity, but Muand C‘ have been chosen because they are more reliably determined from the observed

213

Spatial Variations in the Lunar Field

gravitational field and astronomical observations. The harmonic coefficients C20and CZ2are expressed in terms of the moments of inertia A, B and C (3.9.10): C20 = [C’

-

$(A’

+ B’)]/M~R2

(3.13.4)

Observations of singularities in the Moon’s rotation allow us to obtain the dynamic flattenings: C’ - B

@

= -.

A‘

’

p=-* C’ - A’ B ’

y=-

A’

-

C’

B

(3.13.5)

Solvingjointly equations (3.13.4) and (3.13.5) and taking into account that the flattening P is most reliably determined for the Moon, we have

(3.13.6)

Substitution of the expression for C‘ into (3.13.1) gives the following formula for the irregularity parameter:

The second term in the right-hand side of the latter equation, as opposed to the first term, has a value of about 6 x This is why, at the currently attained accuracy of determination of CZ0and C22,it is ignored. Equations (3.13.6) indicate that the differences between A, B and C are of the order of the value of b, and therefore the irregularity parameter g’ could be found using any one of the three moments of inertia A’, B’ and C‘. Consider a two-layer model of the Moon, where density varies stepwise in the radial direction. Assume that the Moon consists of an outer spherical layer and a central spherical portion. The spherical layer may be the lunar crust or the entire shell, depending on how the problem is Stated. Let the density of the outer spherical layer be cl, that of the central spherical portion be 0 2 ,the radius of the latter be p, and that of the Moon as a whole be R.

214

Lunar Gravimetry

Then, it can be easily found that (3.13.8)

and

471

Mu = 7[ a l R 3

+

(02

- al)p3]

(3.13.9)

In this case, the irregularity parameter can be written as

where

It can be readily seen that at.5 < 1 and q < 1; that is, when a2 > al then 1 - q > 0. The positive number 1 - q in the numerator is multiplied by ts, while in the denominator the same number is multiplied by t3 which is greater than t5.Hence, at a2 > al,g < 0.6. Conversely, if a2 c al,g > 0.6. All these cases, including those involving the two-layer model, are represented in Fig. 3.18. Corresponding to the concentration of mass on a spherical surface-that is, on a completely void model of the Moon-is g = 1. Next, let us see how the moment of inertia C' varies with the radius p of the sphere dividing the Moon into two parts with different densities (two-layer model). We shall vary p as well as a1 and a2 so that the total mass M u of the Moon remains constant. This condition can be written as 471

471

MU= auR3= - [alR3 3 3

+ (a2- 01)p3]

(3.1 3.10)

where au is the mean density of the Moon. In the case of a uniform Moon with density a<,the moment of inertia Cb is 871 Cb = - auRs 15

(3.13.1 1)

From (3.13.8), (3.13.10) and (3.13.11) we have the following variation in the moment of inertia for the two-layer Moon: (3.13.12)

Spatial Variations in the Lunar Field

215

Type of density distribution

Magnitude of g

I . The density decreasing.

g < 0.6

2. Two layers with constant densities. U l , u2; u1

> u2

9 < 0.6

3. The density is constant in whole sphere.

g = 0.6

4. The density increasing from centre to the surface.

g > 0.6

5. Two layers with constant densities. U l , u 2 ; Ul

g

< u2

6. The whole mass distributed on the surface of the sphere.

> 0.6

g = 1.0

density irregularity parameter.

in which

AD = 02

- 01

Figure 3.19 shows graphically the variation in AC'/MuRZas a function of the normalized distance p = p / R from the Moon's centre. In the calculations, it is assumed that Co = 0.400 McR2 and (rU = 3.34 g cm-3. The differential density A c is essentially a parameter. The graph indicates that the variation in AD near the centre has little effect on AC'. Let us find the extremum of the

216

Lunar Gravimetry

-0.10

t

P Fig. 3.19. Increment AC'/M,R'at a given difference between the layers in the two-layer model of the Moon plotted against normalized radius p .

function AC'/M,R2 of the variable p . Obviously, it occurs at the radius of the boundary sphere p = (0.6)'12.The extremum of the function AC'/M,R2 indicates that the increment of the moment of inertia AC'/M,R2 at a given density difference varies to the greatest Ao extent when the radius of the sphere dividing the two parts with different densities is (0.6R)'/2. In view of AC'/MuR2being a double-valued function of p , one and the same value of AC'/MuR2at a given Ao can be attained at boundary spheres with two radii: greater and smaller than p = (0.6)f.However, it should be borne in mind that, although the density difference Ao is the same in both cases, the density o1is different in each case. It is determined from condition (3.3.10)and equals 3

o1=ou-(&)

Ao

that is, it is dependent on the radius p of the boundary sphere. Particular interest in the problem of radial distribution of the Moon's density arose after publication of Eckert's work (1965).He had analysed the theory of secular motion of the node and perigee of the lunar orbit and. recalculated them. According to Eckert, the motion dR, of the node is related to the irregularity parameter g by the expression

3 C' - -0.0346 dR, 2M , R ~

9=---

while the physical libration constant f is related to motions of the node (dQ,)

217

Spatial Variations in the Lunar Field

and perigee (da,) as C - B - 0.681 - 0.381 3 dx j=-C-A dRU where dx, and dR, are expressed in angular seconds per century. In new calculations, Eckert found that dR, = + 3.1” and dx, = -27.9“, hence g = 0.965 a n d f = 0.638. The calculated value of g suggests that the density of the Moon must diminish considerably toward the centre; that is, most of the Moon’s mass is concentrated in the surface layers. This has led to the paradox of the “void Moon”. While analysing this work, Jeffreys (1967) found the results to be erroneous and pointed out, among the possible errors, the systematic errors in the node motion, arising either in observations or in reductions, the errors inherent in the physical libration constant used in the calculations, and others. Kaula (1969) believed that the error stemmed from the observations of motion dR, of the node. The latter was determined from the formula

dR, - L1910 - L1750 -dt 1910 - 1750

- F1910 - F1840 1910 - 1840

(3.13.13)

in which L is the Moon’s longitude, F is the argument of the latitude, equal to the angle along the orbit from the ascending node to the Moon, and 1750, 1840 and 1910 are the epochs of observations. In these calculations, time was determined from the Earth’s rotation. However, if the irregularity of the Earth’s diurnal rotation is taken into consideration in determining time according to (3.13.13), the resulting value of g is 0.615, which is much lower than the value derived by Eckert. Although we can now say that the paradox of the “void Moon” in Eckert’s sense is nonexistent, some uncertainty in the radial density distribution pattern remains. Initially, it was due to insufficient knowledge of the dynamic flattening p in (3.13.7). The value of B was deduced from astronomical observations of the Moon. As was demonstrated by subsequent measurements with the aid of radio signals from the ALSEP transmitters and laser probing of corner reflectors, the previously used value of B was too low (King et al., 1976), hence, the parameter g had a value that was too high. The latter was determined several times (Kaula, 1969; Blackshear and Gapcynski, 1977; Caputo and Panja, 1979) using different values of the harmonic coefficients CzO and Cz2. After the accuracy of measurement of /? and y had been enhanced, it occurred to investigators to use the quantity y instead of the unreliably determined coefficient Cz2and to determine the parameter g from the formula (3.13.14)

218

L m a r Gra vimetry

rather than from (3.13.7). In our opinion, the best way to determine g from (3.13.14) is by using the values of Czo,p and y resulting from adjustment of all observed values of C 2 0 , CZ2,p and y, taking due account for their weights (3.9). The values of C Z 0 p, and y, adjusted in this manner, give g = 0.5838

0.0007

(3.13.15)

It should be remembered, for comparison, that the irregularity parameter g is 0.49 for the Earth, 0.58 for Mars and 0.37 for Jupiter. Of the above planets, Mars has the structure closest to being regular, while the density distribution in Jupiter is the most irregular.

3.14 Density Model of the Moon

Determination of density inside a planet is a complex task in the sense that it involves gravimetric, astronomical and seismic observations. One must know the heat flows in the planet’s interior, its chemical and mineralogical composition, and the pressure .at different depths. Various approaches have been used to determine the density of lunar rocks at different depths. The density of the uppermost layer, or lunar soil, was determined directly on the Moon as well as in terrestrial laboratories on lunar rock samples. In addition, some indirect methods were used. The basement rocks also lent themselves to direct measurements. Pieces of them are scattered all over the lunar surface, and in some places they are exposed. As regards rocks at great depths, their density can only be determined indirectly, using a variety of techniques. Of these, the gravimetric techniques predominate, but are not sufficient in view of the fact that the inverse problem of gravimetry is multiple-valued in principle. However, with some simplifying assumptions, the problem can be solved on the strength of gravimetric data alone. Of course, this can only be done at the expense of some details, primarily those that might characterize the actual horizontal density variations. Yet, something must be sacrificed if we want to find a solution, even if it will approximate the reality less closely. Two constraints have been imposed on the density distribution in the simplest solution, the first being that the distribution of interest should be such that the total mass of the Moon equals a known quantity M , . This constraint has already been written as (3.13.2). The second constraint (3.13.3) calls for the axial moment of inertia C in the density distribution model to be consistent with the value determined from observations. The accuracy of determining the density distribution can be improved if seismic data are also involved. According to Williamson and Adams (1973), an additional constraint can be

219

Spatial Variations in the Lunar Field

imposed on the density distribution by introducing an equation relating the density 0 of the matter in the Moon's interior and its elasticity to the velocities of longitudinal ( Vp) and transverse ( V , ) seismic waves:

(3.14.1) where K , is the adiabatic modulus of bulk compression. The above relation has been derived theoretically and corroborated experimentally. What is more important, seismic observations permit the establishment of the boundaries between media with different elastic properties. Such boundaries also mark the interfaces between media of different chemical and mineralogical compositions or different phase states of matter having the same composition. They also divide media with different densities. In the early nineteen sixties, observations of oscillations in the Earth caused by strong earthquakes prompted a new approach to determine more accurately the mass distribution inside planets. The free frequency of spheroidal and torsional oscillations of a planet depend on density distribution and elastic characteristics (Bolt, 1960; Carr and Kovach, 1962; Zharkov et al., 1966; Derr, 1969; Zharkov, 1978). The important point is that the frequencies, of which there is a whole spectrum, are selective with respect to different internal spherical layers of mass. This permits depth probing of density and elastic properties of planets. Thus, the equations of free frequencies are essentially new constraints on density distribution, which are similar in a way to (3.13.2) and (3.13.3). The number of such constraints equals that of observed frequencies of spheroidal and torsional oscillations. This method has not yet been applied to the Moon because of the lack of data on its oscillations. We shall now make some estimates concerning the density distribution inside the Moon, without taking into account the variations in pressure and temperature in its interior. The lunar density distribution will be determined assuming that it exhibits a central symmetry; that is, the density is a function of only the radius p and does not depend on spherical coordinates. First, consider a simple model of density distribution inside the Moon, assuming that its variations are given by the second-degree equation

The coefficients c0 and K are determined from equations (3.13.1)-(3.13.3):

K=-

35Ma 167cR3 (59 - 3);

15Ma

00

= --

('9 - 5,

220

Lunar Gravimetry

Using the values of M , from (3.6.8) and g from (3.13.15) and assuming that R = 1738 km, we obtain

CY

~ ( p=) 3.81 - 0.78 -

(3.14.2)

Hence, the density in the centre of the Moon is oc = 3.81 g ~ m - whereas ~ , near the surface it is 6,= 3.03 g cm-3. The latter density is in agreement with that observed in the surface layers. It is known (Wood et al., 1970, 1971), that the highlands are made up of anorthosites and basalts whose densities are 2.8-2.9 and 3 . 0 g ~ m - ~respectively. , The maria, on the other hand, are composed of mare basalts having a density of 3.3-3.4 g cm-3. According to Turkevich (1973), the average density of maria is 3.19 g cm-3 and that of highlands is 2.97 g cmP3. The highlands may be 3 4 km above the mean lunar surface level, yet their density is lower than that of the maria. Therefore, if the ratio between their areas is taken into account then it appears that the real upper crust may be represented by an equivalent model 25 km thick and having density o1 = 3.0 g cm-3 (Turkevich, 1971; Solomon, 1974). To estimate more accurately the density of the Moon’s interior, one must take into account the effects of pressure and temperature on density. As the temperature rises with depth, the density decreases because the subsurface matter expands, and vice versa the latter contracts as the pressure increases with depth. Density also varies with depth as a result of changes in the chemical composition or phase state of the subsurface matter. It can be stated with a sufficient degree of approximation, taking into

account the effect of pressure on density, that the Moon is in a state of

hydrostatic equilibrium. In this case, the variation in pressure with depth can be given by the differential equation (3.14.3) in which gravity is (3.14.4) and mass is

The variation in mass with depth can be written in the form of the differential

221

Spatial Variations in the Lunar Field

equation dm

- = 47ca(p)p2

dP

(3.14.5)

Let us now turn to the temperature regime inside the Moon. Apart from affecting the density of matter, temperature also influences the velocity of seismic waves which, as has already been mentioned, can also be used in studying the density distribution. The calculation of the temperature inside the Moon is based on a heat conductivity equation for a body with internal sources of heat and, in some instances, with due account for the solar heat. To perform the calculations, one must know the distribution of the heat sources, the mechanism of heat transfer, the composition of the subsurface matter, its heat capacity, heat conductivity, melting point at high pressures, and initial temperature. In addition, one must take into account the latent heat of fusion, the redistribution of the heat sources, the changes affecting the mechanism of heat transfer, temperature-dependent variations in heat conductivity, and so on. The mere listing of the factors determining the Moon's temperature testifies to the complexity of the problem. The temperatures inside the Moon are also determined from observations of electrical conduction. A large body of literature deals with the thermal history of the Moon, which is related directly to the current temperature reginie. Figure 3.20 shows several models of temperature variations with depth (Solomon and Toksoz, 1973; Toksoz and Johnston, 1974). The solidus corresponds to mare basalts. As can be seen from the curves, the temperature rises rapidly to depths of 500 to 600 km. Then, the rise slows down all the way to the Moon's centre. The temperature gradient in the range &550 km is in the neighbourhood of 2.3"Ckm-', and from 550 km down to the centre the gradient is about 0.1"Ckm- In the uppermost 20-m thick layer, the temperature gradient is about 1.6"Cm-', which is indicative of the low heat conductivity of this layer, its pronounced porosity and low density. To take into account the relationship between the density of the subsurface matter and the chemical compositior. as well as phase state, characterized by the atomic weight W, use is made of the empirical Birch diagram in which three parameters are related: pressure p, seismic wave velocity V , and mean atomic weight W In the case of lunar rocks, W is quite stable and is equal to 21-22. The differential equation of density variations with depth can be written as. a function of three quantities: gradient of pressure p , gradient of temperature T, and gradient of composition W of the subsurface matter: (3.14.6)

222

Lunar Gravimetrv

c1500 -

-v 1000 I-

!?

3

c

Fa

a

E

I-

0

500

1000

1500

Depth (km) Fig. 3.20. Models of the Moon's temperature variations with depth: (1) model by Solomon and Toksoz (1973). consistent with the heat flux on the lunar surface, observed during landing of Apollo 15; (2,3) models by Toksoz (1972): model 3 is based on a convective heat transfer mechanism in which the lower lithosphere is predominant; (4) model derived from observations of electrical conductivity (Sonnet et a/., 1971).

where

is the isothermal coefficient of bulk compression, the reciprocal of PT being known as the isothermal modulus of bulk compression KT = l/PT; a(p, T, W ) = In o/aT)T,wis the coefficient of cubical expansion, and u( p , 7; W) = In o/a W) is the coefficient of density variation as a function of composition of the subsurface matter. This equation can be reduced to the following form:

-(a (a

do dp

-

Gm

dW

o

p 2 Vf-'VZ 3

- NOT - A 0 s

(3.14.7)

dP

in which z is the difference between the true and adiabatic temperature

223

Spatial Variations in the Lunar Field

gradients, ACT= rs2 - o1,o1 and a2 being the densities of matter of two compositions such that l/a = (W/o,) (1 - W)/a2. This is an alternative interpretation of W. Equation (3.14.7) is solved together with differential equation (3.14.5) for the mass m(p). The boundary conditions are

+

m(p) I p = ~ = M U ; m(p) I p = 0 = 0 In the case of a planet with a homogeneous composition, the term Aa(dW/dp) in the right-hand side of (3.14.7) is zero. The advantage of using (3.14.5) and (3.14.7) for determining (T is because they take into account the temperature, density and composition variations with depth. Moreover, when used to determine the distribution of density (T, in addition to gravimetric data they are also based on the results of observations of the seismic wave velocities associated with the Moon rock density. Figure 3.21 shows two alternative density distribution inside the Moon with due regard for the pressure and temperature in its interior, as a function of the density irregularity parameter. Corresponding to g = 0.592 is the density (T, = 3.64 g ~ m in,the - ~ Moon’s centre. Accordingly, for g = 0.585 (see 53.13) we have oC = 3.79 g cm-3. These curves are merely an illustration of the method in which the variations in pressure, temperature, and composition of the lunar rocks are taken into account. Further refinement of the method for determining the density distribution is possible through observations of the Moon’s free frequency. Models of natural spheroidal and torsional oscillations of the Moon already exist in theory (Carr and Kovach, 1962; Derr, 1969). Of the available modifications, the model that fits the observational data the best should be selected.

-

3.8 n -

m

-

0

100

300 500

700

900 1100 1300 1500 1700

z (km) Fig. 3.21. Two possible distributions of the Moon’s density taking due account for the pressure and temperature inside the Moon, as a function of the density irregularity parameter.

224

Lunar Gravimetry

We have already mentioned the planetary distribution of density with depth, density being regarded as a function solely of the radius p . To describe horizontal density variations, these are approximated with the aid of polynomials whose variables are the polar coordinates p , cp and A (Meshcheryakov, 1973; Meshcheryakov and Dayneka, 1975; Shcherbakov, 1978). The horizontal density variations in the surface layers of the Moon can be investigated by gravity surveying (Fedynski, 1964; Veselov and Sagitov, 1968) where a whole array of techniques is available. There is not a single common method of studying the distribution of anomalous masses. The interface between media of different densities can be clearly defined when it is a single boundary or several conforming ones. Application of the gravimetric method is most effective in defining the interface between different media when it is used for interpolation between points at which the depth of the boundary is determined by seismic sounding. A case in point is marking the lower boundary of the lunar crust which has a density contrasting with that of the underlying mantle. A good example of how gravimetric measurements were used to determine the density distribution in the upper layers of the Moon can be found in the work by Talwani and Kahle (1976). They used the profile of gravity anomalies, taken with the aid of a TG gravimeter during a trip of the lunar rover driven by the Apollo 17 astronauts. The profile ran across the TaurusLittrow Valley. The latter extends along the southern rim of Mare Serenitatis, between two mountain masses as high as 2 km above its level. The samples brought to the Earth indicate that the mountains are composed of brecciated rocks. The Taurus-Littrow Valley is considered to be a graben formed at the same time as the basin of Mare Serenitatis itself. For geological interpretation using the observed gravity values, the Bouguer anomalies were calculated (Fig. 3.22). The problem was approached as a two-dimensional one. The excess density of the basaltic rocks occurring under the day surface and responsible for the Bouguer anomaly, with respect to the density of the surrounding brecciated rocks, was assumed equal to 0.8 g cm-3. The profile of Bouguer anomalies has shown that the valley is underlain by a basaltic block 800 m thick and about 10 km wide.

3.15 Variations in Gravity. Its Radial Gradient and Pressure with Depth

The amount of gravity deep down depends on density distribution. Let us ignore the density variations in the horizontal direction as well as the centrifugal acceleration due to the Moon’s rotation and the effect of the tidal accelerations from the Earth and the Sun. The radial density distribution

225

Spatial Variations in the Lunar Field

Observed gravity value (mGal)

-.-g c

2000 1500 1000 500[ South masses

North masses

P O U -500 -1000

go_

500

- 1000

Distance (km) Fig. 3.22. Variations in gravity along a profile passing across the Taurus-Littrow Valley: (1) measured gravity values; (2) Bouguer anomalies. The hatched portion shows the cross-section of an anomalous body found as a result of interpretation of Bouguer anomalies (Talwani and Kahle, 1976).

inside the Moon will be assumed to be the same as in 93.14. It is known that, in the case of a spherically symmetrical density distribution o(p), the spherical layers external to the point at which the radial acceleration of attraction is examined does not exert any attractive action on it. Attraction is due only to the internal layers. The radial acceleration of attraction can be considered equal to gravity g(p) with a high degree of accuracy:

0 rr/2 0 P

(3.15.1) 0

Figure 3.23 shows g(p) plotted against the dimensionless Moon’s radius ij

= p/R, where R is the mean radius of the Moon. Also shown for comparison

is a g(p) curve for a Moon having uniform density and a mass equal to the actual Moon’s mass M , . It can be seen that the difference is negligible, which is not so in the case of the Earth. We shall now see how the radial gravity gradient aglap varies as a function

226

Lunar Gravirnebv

P Fig. 3.23. Gravity g ( p ) inside the Moon plotted against normalized distance p = p / R . The dashed line shows g ( p ) for a uniform Moon with its mean density.

of p. To this end, differentiate both sides of (3.15.1) with respect to p:

At a given density distribution o(p1), one can easily calculate the values of the radial gravity gradient ag(p)/ap for different depth. The variation in ag/dp as a function of the normalized radius p = p/R is shown in Fig. 3.24. aglap increases with depth, reaching the maximum value in the Moon’s centre. In the case of a regular Moon with a constant density o 0 , the radial gradient agjap in it is constant regardless of the distance to the centre, namely:

1200

t

0 0.2 0.4 0,6 0.8 1.0 P Fig. 3.24. Radial gravity gradient inside the Moon plotted against normalized distance dashed line shows ag/ap for a uniform Moon with its mean density.

p

= p / R . The

227

Spatial Variations in the Lunar Field

As one passes through the Moon's surface, when p > R , the radial gradient changes abruptly. Here, it is given by another analytical expression

(3.15.3) This expression is derived from (3.15.2). The first term in the right-hand side of (3.15.2) equals zero because the density o (p > R ) = 0. Integration of the addend between 0 and p with Ma = 47cooR3 gives (3.15.3). Finally, consider pressure variations with depth. For an increment of pressure p we have dP(P) = - 4 P M P ) dP The total pressure at a point located at a distance p from the Moon's centre is P(P) = -

f

4 P M P l ) dP1

0

The curve representing the pressure variation with depth is given in Fig. 3.25.

-

P

Fig. 3.25. Pressure P inside the Moon plotted against normalized radius p = p/R. The dashed line shows P for a uniform Moon with its mean density.

228

Lunar Gravimetry

The pressure inside the Moon reaches its maximum at the centre where it approaches 50 kbar. A similar pressure is attained inside the Earth at a depth of only 150 km. Figure 3.25 also illustrates pressure variations, assuming that the density of the Moon is uniform.

3.16 Mascons

Among the new discoveries made on the Moon at the dawn of the space era, that of mascons is one of the most astonishing. Mascons (mass concentrations) coincide with lunar maria, the largest ones being found in Mare Imbrium and Mare Serenitatis as well as in Maria Crisium, Nectaris, Humorum and others. Mascons are responsible for pronounced local anomalies of the gravitational field. They occur at relatively shallow depths and have a high density, as compared to the enclosing rocks (do = 0.3 g ~ m - ~For ) . example, the gravity anomaly over the mascon of Mare Imbrium at an altitude of 100km above the Moon’s surface reaches 250 mGal. All anomalies (except for that in Sinus Iridium) above mascons are positive. A model of the lunar gravitational field, in which the anomalies due to mascons were shown for the first time, was published by Muller and Sjogren (1968). The discoverers of mascons were awarded the gold Magellan Medal of the American Philosophical Society. In subsequent years, the parameters associated with mascons were refined through more accurate measurements of the lunar gravitational field (Table 3.15). The table lists the coordinates -(q, A ) of the mascons’ centres, the maximum gravity anomaly over the mascons at an altitude of 100 km above the Moon’s surface, the diameters D of mascons, and their excess masses m. The diameters D are essentially those of the craters enclosing mare lava outpourings. The masses m are determined by selecting such a mass distribution as would correspond to the observed gravitational anomaly. Another way to determine m is through integration of gravity anomalies Ag(S) due to the mascon:

where S is the lunar sphere (part of the sphere is quite adequate for the purpose). The main difficulty associated with this approach is delineation of the local gravitational field corresponding to the mascon under consideration, excluding the effect of all other lunar masses. It should be pointed out that the table lists excess masses m related to the actual masses ma as

TABLE 3.15 Mascon parameters

Coordinates Mascon

cp

(ded Mare Imbrium Mare Serenitatis Mare Orisium Mare Nectaris Sinus Aestuum Mare Humorum Mare Humboldtianum Mare Orientale Mare Smythii Unnamed Unnamed Crater Grimaldi Sinus Iridium Mare Moscoviense Unnamed Unnamed Unnamed Total

38 28 16 - 16 10 - 25 57 -20 -4 -7 - 17 -6 45 26 - 37 1

-3

- 18

18 58 34 -8 -40 82 -95 86 27 70 -68 -31 147 151 - 128 - 157

Maximum values of gravity anomalies g,,, 100 km above the mascon (mGal) 220 220 130 120 80 65 52 52 52 52 40 26 -90

’

Inertia moment difference MaR2)

Diameter D and r.m.s. error (km)

Mass and r.m.s. error (10-6 M,)

B-A

625 f 35 650 f 35 450 f 25 420 f 20

10.26 14.04 - 4.95 3.63

300f35 300 315 f 15 370 20

19.40 f 1.65 22.44 f 1.73 11.52 f 1.50 9.67 f 1.41 7.19 f 1.41 4.31 f 0.87 4.68 5.25 f 0.5 7.25 f 0.15

0.63 - 1.31 -4.64 - 7.20

1.41 -3.31 -0.53 0.01

200 250 250 190

2.05 3.23 3.23 1.84

0.67 1.08 -0.73 1.29

0.76 0.36 1.25 1.55

12.77

24.44

C-A

4.29 1 1.30

1.81 5.55

230

Lunar Gravimetry

where a. is the mean density of the mascon-enclosing rocks, and Aa is the excess density of the mascon with respect to the enclosing rocks. By approximate estimates, a. = 3.0g cm-3 and Aa = 0.3 g cm-'. The errors in the determination of the mascon masses m may be as high as 20%. It can be easily calculated that the actual masses of individual mascons constitute between 1 x and 3 x l o w 4of the total Moon's mass; in other words, they are large anomalous inclusions in the Moon's body. Finally, the table gives the differences between Moon's moments of inertia A(B - A ) and A(C - A), due to mascons, expressed in M , R 2 . Calculations show that the contribution of all mascons to the differences B - A and C - A amounts to about lo%, which is quite significant. We shall use the mascon of Mare Nectaris as an example illustrating how mascons have been investigated more extensively. Figure 3.26 is a gravimetric profile resulting from Doppler tracking of Apollo 12 at an altitude of 116 km. An acceleration profile was us,ed to construct several versions of the mascon model (Chuikova, 1971). One of the models was in the form of a sphere with a M a ,located at a depth of 100 km. Two other models were mass of 9.1 x material disks near the surface, with radii of 150 and 200 km and masses of

100-

-

$ E

..

4

80 -

/ - - _

60-

z

40-

.-$

20-

c

Q

8

a

0-20-

-40 -60 1

l

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Fig. 3.26. Profiles of accelerations r for the mascon in Mare Nectaris at an altitude of 116 km: (1) from Apollo 1 2 tracking data; (2) calculated for a disk having a radius of 200 km and a mass of 9.1 x 10-'Ma; (3) calculated for a disk having a radius of 150 km and mass of 5.1 x Ma;(4) calculated for a sphere (Muller and Sjogren, 1975).

231

Spatial Variations in the Lunar Field

I

I

28"

l

l

30"

I

I

32"

I

I

1

34"

1

36"

1

1

38"

I

I

40"

I

I

42"

Longitude Fig. 3.27. Profiles of accelerations r for the mascon in Mare Nectaris at an altitude of 25 krn: (1 ) from Apollo 14 tracking data; (2) calculated for a sphere; (3) calculated for a disk 150 krn in radius; (4) calculated for a disk 200 krn in radius.

5.1 x Mu and 9.1 x lop6M,, respectively. The same agreement with the observed gravitational field is displayed by both the field due to the sphere at the depth of 100 km and that due to the material disks near the surface. Accelerations were then measured using the Apollo 14 tracking data over the same mascon but at lower altitudes-28 km (Fig. 3.27). There is a wide difference between the observed field and that computed for the first spherical model of the mascon. The difference is less pronounced in the case of the disk with the radius of 150 km; the gravitational field due to the disk with radius of 200 km has been found to agree quite well with the observed field. This example shows how important it is, in estimating the depth of mascons, to take into account their shape and to have a gravitational field in which the shape of the anomalous body is manifest. The gravitational fields over Maria Serenitatis and Crisium were interpolated (Muller et al., 1974) as fields due to material disks. Two modifications are shown for Mare Serenitatis (Fig. 3.28). One of them is a material disk with M ( , located at a depth of 17 km. a radius of 230 km and a mass of 18 x In the other modification, two disks located at depths of 17 and 1 km and Mu and radii of 230 and Mu and 0.15 x having masses of 15.8 x

232

-120 O*/0"

Lunar Gravimetry

10"

1 5"

20"

25"

Longitude Fig. 3.28. Profiles of accelerations r for the mascon in Mare Serenitatis: (1 ) from Apollo 15 tracking data; (2) calculated for a disk 230 km in radius, located at a depth of 17 km; (3) calculated for two disks 30 and 230 km in radius, respectively, located at depths of 1 and 17 km. The disk centres have the following coordinates: cp = 26". 1' =9.5", cp =26",A =18.5" (Muller eta/., 1974).

30 km, respectively, were selected. The second, smaller disk, whose centre is offset by nine degrees relative to that of the main disk, accounts for the small singularity of the gravitational field due to the mascon in Mare Serenitatis. The gravitational field of Mare Crisium is represented by a mascon model (Fig. 3.29) in the form of a material disk at a depth of 14 km, having a radius of 200 km, and a mass of 12.8 x MC.The gravitational fields shown in the last two figures have been deduced from Apollo 15 Doppler tracking data. The field above Mare Serenitatis is associated with altitudes ranging from 15-20 km, and that above Mare Crisium, with altitudes of 2&25 km. Analysis of finer details of the gravitational field above the mascons indicates that only negative gravitational anomalies occur around a mascon. This is particularly so in the case of the mascon associated with Mare Orientale. In Chuikova's work (1979, the gravitational fields due to the mascons of Maria Imbrium and Serenitatis are approximated by disks and material annuli coaxial therewith, having a negative density. The mascon parameters are derived using a combination of values of the anomalous gravitational potential W, the vertical attraction component W,, and the vertical gradient W,,, calculated at a certain altitude above the lunar surface on the axis of the disk under consideration. Safronov (1971) constructed his mascon model in the form of a material disk with a density which was a function of its radius. Thus, interpretations of the gravitational fields due to mascons have

233

Spatial Variations in the Lunar Field

'"[

P

200

d

-E

Y.

160

-

120

-

-160

-200 45"

I

50"

55"

,

60"

\\ 65"

Longitude Fig. 3.29. Profiles of accelerations for the mascons in Mare Crisium: (1 ) from Apollo 15 tracking data; (2) calculated for a disk 200 km in radius and 12.8 x Mcin mass, located at a depth of 1 4 km. The disk centre has the following coordinates: cp = 17". I = 58" (Muller et al., 1974).

established that mascons are distributed anomalous rather than compact masses resembling circular disks in shape. Furthermore, the mass of a disk diminishes toward the edge; that is, the structure of mascons takes the shape of segments of a sphere. The horizontal boundaries of mascons can be defined reasonably well, and they are essentially near-surface features. As far as mascons go, mention should be made of the works (O'Leary et al., 1969a,b) in which a combination of mascons occurring at a depth of 300 km and approximated by circular disks with the same surface density of lo5 g cm-2 was determined from the gravitational field. The investigators predicted a mascon on the far side of the Moon, five times bigger than the largest near-side mascon associated with Mare Imbrium. It even received a name-Occultum (hidden). Subsequent works have disproved the existence of such a mascon. The above-cited work was nevertheless important because a large anomalous formation was postulated on the far side. It is now known that this formation is the central plateau responsible for a pronounced positive anomaly over a vast area. Ever since mascons were discovered, various hypotheses have been set forth to explain their origin. The abundance of such hypotheses is comparable to the number of possible explanations of pulsars-the phenomenal objects recently discovered by astrophysicists. Although the former are less extravagant than the pulsar hypotheses, quite a few original ideas do crop up among them. At present there is enough known about mascons which can be regarded as firmly established facts. Therefore, whatever hypotheses are still in circulation must somehow account for these facts. We are speaking

234

Lunar Gravimetry

primarily about the pronounced positive gravitational anomalies in regions with depressed relief. The lithosphere must be able to bear a rather large excess mass (c. 800 kg cm-’) over a long period of time (about 3 x lo9 years) without isostatic compensation. The mare basins with which mascons coincide were not filled immediately after they had been formed but after some 0.5 x lo9 years. There is no doubt that some constraints have been imposed on the hypotheses concerning the origin of mascons by studies into the lunar gravitational field. These constraints, however, are indirect in the sense that the gravitational field has been used to establish a number of facts relating to the structure of mascons and their distribution. The hypotheses must explain why mascons are masses distributed near the lunar surface rather than compact masses deep inside the Moon. Why does isostatic compensation not take place? Why do mascons exist on the near side? It has been extremely important to find out whether a mascon is a dome or a funnelshaped body. Knowing this may provide the answer to the question of whether mascons owe their origin to intrusive or impact processes. To know exactly the shape of mascons in all details, one must have a more intimate knowledge of their gravitational .fields. We shall restrict ourselves to a brief review of the hypotheses, proposed so far. The discoverers of mascons, Muller and Sjogren (1968), believed at first that they were merely nickel-iron meteorites of a density exceeding that of the lunar crust rocks, which had fallen on the Moon and remained embedded in the lunar crust. Then, the authors revised their hypothesis as follows (Muller et al., 1974; Muller and Sjogren, 1975). First, a crust about 50 km thick is formed on the Moon. It is less dense than the underlying mantle rocks. A body of asteroidal dimensions falls on the Moon and creates a mare depression which gives rise to a small negative gravitational anomaly. Next, the reasoning goes, the depression is immediately filled with lava outpourings of the mantle material from the Moon’s interior, which has a density higher than that of the crustal rocks. The filling continues till the state of isostatic equilibrium is reached. After that, the lunar crust becomes so strong that it can bear the surface load created by the additional mass without any perceptible deformation. The basin continues being filled with material, and an excess anomalous mass is formed. Since no isostatic compensation takes place, significant positive gravitational anomalies occur. The hypothesis of lava outpourings was supported by Cone1 and Holstrom (1968). They speculated that basaltic lava of normal density fills the crater in the lowdensity lunar crust. Baldwin (1968) and O’Keefe (1970) suggested that abnormally dense lava fills the depression in the crust which has normal density. However, currently available data attest that the lava efflux and filling of depressions did not follow immediately but after about 0.5 x lo9 years. It is therefore assumed that during that period the lunar crust and the

Spatial Variations in the Lunar Field

235

depression formed in it eventually reach a state of isostatic equilibrium. The initial negative gravitational anomaly disappears. The lithosphere gradually becomes thicker, and the crust becomes strong enough to support additional heavy loads. When outpourings of mare basalts occur, the crust withstands the resulting load without being isostatically compensated for a period of about 3 x lo9 years. Thus, the effluents filling the mare depressions and having a density in excess of that of the crustal rocks, form mascons responsible for sizeable positive gravity anomalies. The idea of mascons being iron meteorites was also pursued by Stipe (1968), who contended that the meteorites forming mascons after their impact against the Moon are rather large. Stipe has calculated the sizes of the meteorites that had given rise to six major maria and found them to be in the range of 1&60 km. Upon impact, the meteorites penetrate to a depth between seven and eleven times their diameter. In the case of Mare Serenitatis, the mass of the iron meteorite must be 930 x 1015kg, while that of the meteorite associated with Mare Nectaris must be 17 x 1015kg. This hypothesis is at variance with the fact that mascons occur near the surface. Urey and MacDonald (1968, 1973), who expressed their doubts about the possibility of the mantle material melting inside the Moon, contended that the currently existing mascons used to be large celestial bodies, known as planetesimals, that had fallen onto the Moon. These planetesimals are not from the asteroid belt of the solar system but celestial bodies from the neighbourhood of the Earth-Moon system and are characterized by low velocities. Being in disagreement with Stipe (1968), Urey and MacDonald did not believe that the falling bodies penetrated so deep inside the Moon. They suggested that upon impact against the lunar surface, the planetesimals are flattened and melt partially, as do the lunar rocks on the impact site from which matter is scattered over the Moon’s surface. Instead of the ejecta, the crater contains the flattened high-density planetesimal. The hypothesis put forward by Urey and MacDonald, however, does not explain why mascons are observed on the Moon’s near side. Many of the hypotheses differed in details as well. For example, some hypotheses (Arkani Hamed, 1974) stated that, apart from forming a crater, the meteoritic impact also reduced the pressure beneath it and caused partial melting of the mantle material. As a result of the increasing pressure in the melting zone, basalts separated out of the mantle material, and a lowerdensity body emerged in the melting area. These basalts then flowed through the cracks resulting from the impact down onto the basin’s bottom. In some areas, the basins overflowed, and the mantle material spread beyond the basin’s edge, forming irregularly shaped maria. If isostatic equilibrium was reached, the gravitational anomalies disappeared and no mascons were formed. In other areas where no isostatic compensation had taken place,

236

Lunar Gravimetry

mascons have been found. The weak point of the hypotheses based on lava outpourings is that an assumption must be made to the effect that the crust must, on the one hand, support additional mass loads without isostatic compensation and, on the other, let through the hot lava emerging from below. Kaula (1971) obviated this difficulty by assuming that the Moon undergoes compression. As the spherical lunar crust is compressed, it squeezes out basaltic lavas which fill mare basins through cracks. Putting the formation of mascons in the context of the thermal evolution of the Moon, Arkani Hamed (1974) speculated that the impact of a celestial body incident upon the Moon exposed a layer of high heat conductivity. This must have resulted in a rapid growth of the lithosphere under the basin. At the same time, because the highland area around the impact crater disintegrated and was covered by ejected matter, its heat conductivity dropped. The subsurface rocks started warming up more intensely as a result of accumulation of heat due to radioactive decay. The rocks melted, and only some 0.5 x lo9 years after the impact they started flowing into the depression, filling it, and forming mascons. Gilvary’s hypothesis (1 969) concerning the formation of mascons states that depressions were formed by meteoritic impacts, yet the major role in mascon formation was played by the atmosphere and hydrosphere that had existed before. This hypothesis is not consistent with the established fact (Safronov, 1971) that the hydrosphere disappeared over a period of time which was by three orders of magnitude shorter than that required for mascons to form according to this hypothesis. Wise and Yates (1970) hypothesized that the positive gravitational anomalies above mascons are merely excess masses in the form of plugs in the lunar crust. The material of the plug is essentially mantle rock that had risen from below and filled the hole made in the crust by the falling meteorite. Consequently, mascons are batholiths composed of high-density rocks emerging from the mantle underlying the lunar crust (Kane et al., 1969). According to other hypotheses, mascons are the result of chemical differentiation of matter and have nothing to do with the impacts of meteorites or planetesimals or the efflux of magnetic material. In conclusion, we should like to mention the original hypothesis stating that mascons have formed as a result of cooling of the surfaces of maria during lunar eclipses (Buhl, 1968). This proposition is hardly plausible. A critical analysis of some hypotheses concerning the origin of mascons can be found in Kaula’s work (1968). We prefer the hypothesis according to which in the course of the Moon’s evolution, its rotation became synchronous with the revolution around the Earth. This resulted in an asymmetry of the thermal regime on the near and far sides. The far-side lithosphere became thicker than on the near side and more “impenetrable”. The thinner crust on

Spatial Variations in the Lunar Field

237

the near side eventually lost its integrity, giving rise to new effluxes of matter. Craters were formed, the material scattered, and the heat capacity changed causing local heating, partial melting, and outpouring of magmas on the surface, into the craters of future mascons. This is why mascons exist on the near side of the Moon. The hypothesis also explains the pronounced positive anomalies above mascons, their depth and structure. Over the fifteen years that have elapsed since mascons were discovered, more than a hundred works have been published dealing with different aspects of these formations which are not found anywhere on the Earth. It has even been proposed to use mascons for detecting gravitational waves emanating from pulsars (De Sabbata, 1970), thus putting three phenomenal objects-mascons, pulsars, and gravitational waves-into one basket! Mascons are regarded in this case as test bodies elastically coupled with the Moon. The gravitational wave arriving at a frequency resonant with the natural frequency of a mascon (c. 1 Hz) induces its oscillations. A seismograph placed on the Moon near the mascon will sense and record, among other seismic events, the response of the mascon to the gravitation wave.

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Chapter Four

Inconstant Lunar Gravity

4.1 Periodic and Secular Variations of Lunar Gravity

The gravity field of the Moon exhibits complicated variations with time. Compared to the magnitude of lunar gravity, these variations are small and can be classified as periodic and secular variations. The more important constituent, the periodic variations, are of tidal origin, i.e. these are due to the gravitational attractions of the Earth ,and the Sun, which are continually changing as the positions of these bodies change in the sky. Having determined the position of any tide-generating celestial body with respect to the Moon, and considering the latter as a perfectly rigid planet, the tidal potential (and also the tidal force in any given direction) exerted on the Moon by the perturbing mass can be expressed analytically. This is the primary tidal gravity field. It is different from point to point on the lunar surface, and inside the Moon, and varies periodically with time. The solid body of the Moon yields elastically under the changing value of gravity. Therefore the coordinates of its points and also the density in the lunar interior change slightly with respect to the corresponding initial values. This elastic deformation gives rise to a secondary tidal gravity field; which, in general, is small compared to the primary one. The elastic distortion of the Moon under the tidal forces evidently depends on the distributions of density and elastic moduli in its interior. A further, and even smaller effect results from the influence of anelasticity

240

Lunar Gravimetry

of the Moon on its gravity field. Due to the imperfect elasticity of lunar material there is a phase lag of the tides with respect to the acting potential. This means that the tide on the Moon is high and low not when the tideraising potential has its maximum and minimum values there, but at some later times. Thus, similarly to the case of tides on the Earth, a comparison of the measured tidal amplitudes with those calculated from astronomical theory for a rigid moon should yield information on the elastic properties and the distribution of density in the lunar interior, while the determination of phase differences would add to our knowledge of the imperfections of elasticity of the material of the Moon. The most characteristic feature of planetary tides is that these are the planets’ only deformation phenomenon for which the forces at work are known exactly, and one is thus able to calculate these with high accuracy. The study of Earth tides has recently become a major field of geophysical research. Besides the traditional use of the tidal records to obtain basic information on the internal constitution of our planet (particularly in respect of the liquid core), it has been recognized that the measurements of many phenomena on the Earth are disturbed by tidal effects. These measurements obviously must be corrected for the tides before interpretation. In order to be able to calculate the necessary corrections, the tidal phenomenon must be carefully investigated and clearly understood. The tidal deviations of the vertical, for example, are sufficient to affect some precise fundamental astronomical observations (variations of latitude). Since tides produce systematic effects on high precision levelling and gravity measurements, the study of earth tides is also of basic interest for geodesy. The maximum tidal variation of gravity on the Earth is of the same order as the gravity anomalies being sought in gravimetric prospecting. As a result of a continual increase of precision of the gravity measurements, gravimetric prospecting has become generally used in the search for new riches on the Earth, and this in turn has contributed to the rapid development of the earth tide research. Several aspects of tidal effects are also of interest in oceanography, hydrology and volcanology. Tidal variations of the Earth’s potential have a detectable effect on the orbits of artificial satellites, and these effects must be taken into account in the corresponding problems of space dynamics. The accuracy of laser distance measurements to the Moon and satellites recently has increased so much that exact tidal correctiqns (obtained by experimental determination of the tides at the sites concerned) are also needed in such. measurements. Due to significant differences between earth tides and lun.ar tides, and also the different internal structures of the Earth and the Moon, some of the tidal effects above are not relevant on the Moon. However, present development of lunar physics and possibly the future exploration of the Moon will probably

Inconstant Lunar Gravity

241

reveal such links between lunar tides and some other phenomena which are not known, or do not exist, on the Earth. It has already been established that a close correlation exists between lunar seismic activity and moon tides. Investigation of the lunar tides in certain respects can prove to have more perspectives than the earth tides have. This is because tides on the Moon are relatively larger than those on the Earth. Thus, the tide-raising potential on the Moon is about 6 times larger, and the relative variations of gravity are approximately 75 times larger, than those produced by the Moon on the Earth. The periodic variations of the lunar gravity field are surely large enough to be checked by measurements on the Moon’s surface. It will be shown in the next section that the amplitude of the variations in lunar gravity and in the vertical is about 104-105 times the noise levels of the best corresponding instruments available at present. The problem of the secular variations of the lunar gravity field is much less defined than that of the periodic variations. Such variations can result from astronomical and selenophysical processes. Astronomical factors causing the Moon’s gravity field to vary arise from tidal evolution of the Earth-Moon system. This evolutionary process ‘means that the parameters specifying the Earth’s rotation, and also the orbital motion and the rotation of the Moon, change over time, as the result of tidal friction. As known for the present configuration of the Earth-Moon system, tidal energy dissipation in the Earth leads to secular retardation of the Earth’s rotation, and also causes the semimajor axis of the lunar orbit and the period of the orbital revolution of the Moon to increase with time at slow rates. The first observational evidence of this effect was found by Halley in 1695, who revealed an apparent acceleration of the Moon in longitude with respect to universal time (determined and counted by the Earth’s rotation). On the basis of present trends in the development of laser location of the Moon, one can expect that it will soon be possible to check the rate of increase of its semimajor axis by direct measurement. Thus, due to the increasing Earth-Moon distance, the primary tidal gravity field exerted on the Moon at present is slowly decreasing. Tidal friction also gives rise to secular changes in the eccentricity and inclination of the Moon’s orbit and the obliquity of the Earth’s equator to the ecliptic. These variations, however, are too small to become significant over historical time intervals. Some insignificant changes in the lunar gravity field can occur due to variations of the angular speed of its rotation. Variations of the rotation rate may be caused by tidal friction in the Moon itself and changes of the moment of inertia about the axis of rotation. The moment of inertia can vary owing to natural redistribution of material inside the Moon and also in a forced way, due to tidal deformation. At present, the effects of these secular variations on the lunar gravity are minute, but, if they are operative over a long period of time, the accumulated effects can become

242

Lunar Gravimetry

quite noticeable. The astronomical factors of secular variations of the lunar gravity field will be treated in more detail later in this chapter. Selenophysical processes leading to secular variations of gravity include gravitational differentiation of material within the Moon, isostatic adjustment, phase transitions that can occur in the lunar interior, temperature changes, i.e., all processes causing the density and elastic properties of the lunar material to change. Thermal convection, possibly existing in the deep interior of the Moon, can, in principle, also exercise influence on its gravity field. And finally, meteoritic impacts which increase the total mass of the Moon, could also contribute to the secular variations of the lunar gravity. Particularly, this mechanism could have an especially significant role during the early evolution of the Moon. On the basis of our present understanding of the evolution of the Moon itself and the Earth-Moon system, it is hardly possible to carry out any definite quantitative analysis of secular changes of the lunar gravity field due to the selenophysical processes mentioned above. Therefore, in the following we will deal only with the periodic variations of the lunar gravity, and those resulting from its orbital evolution. 4:2 Tidal Potential on the M o o n

In this section we give an analytical description (suitable for numerical calculations) of the tidal potential field (called earlier the primary tidal gravity field) that would arise on an absolutely rigid moon. As a matter of fact, the tidal potential at a given point P (Fig. 4.1) of the Moon is determined as the difference of the gravitational potentials exerted by an external tide-raising body at P and the centre of mass of the Moon. Let M be the mass of the perturbing body fixed at a point E at a distance of R from the centre of mass 0 of the Moon. We let P E = R’, O P = a, and define z as the selenocentric zenithal distance of the external body at the considered point. Then the tidal potential W may be written as the following development into series of zonal spherical harmonics (see e.g., Bartels, 1957). R

(4.2.1) n=2

where P,(cos z ) is a zonal spherical harmonic function of order n and G is the constant of gravitation. The ratio (a/R)”diminishes rapidly with increase in one n. Since for the Earth and the Sun au/Re = 5 x and au/Ro = can immediately realize that each subsequent term in (4.2.1) is less than the respectively. Therefore preceding one by factors of lo-’ or worse, and if when calculating the tidal potential from the Earth an accuracy of 1% is

243

Inconstant Lunar Gravity

Fig. 4.1. Geometrical relationship used for determination of the tidal potential.

sufficient, one can neglect the third-order term in (4.2.1), but it should be included if a higher precision is required: As to the solar tidal potential, it is usually only necessary to retain the leading term. The solar tide-raising potential is smaller than that induced by the Earth in the ratio

that is, the maximum solar tide on the Moon amounts only to a small fraction (less than 1 part in lo-’) of the tide of terrestrial origin. Therefore in calculations of the lunar tides the solar effects are usually neglected. To understand the variation of the tidal gravity field from point to point in the Moon and with the varying position of the tide-generating body, the zenithal distance z must be written in terms of the selenographic coordinates (cp, A) of the point considered and the selenocentric equatorial coordinates (b, I ) of the perturbing body. The fundamental formula of the position triangle E’PN of spherical astronomy (Fig. 4.1, where N is the North pole of the Moon) yields cos z = sin cp sin b

+ cos cp cos b cosl(l- A)

(4.2.2)

Substituting the latter expression into (4.2.1) and using the theorem of addition ofspherical harmonics, the leading term in W can be rewritten as

1 GMa2 W , ( P ) = -~ [12Pzo(sin cp)Pzo(sinb) 12 R 3

+ 4Pzl(sin cp)PZ1(sinb) cos (1 - A)

+ Pzz(sin cp)Pzz(sinb) cos 2(1 - A)]

(4.2.3)

244

Lunar Gravimetry

This expression represents the main harmonic of the tidal potential as being separated into three terms corresponding to the three types (zonal, tesseral and sectorial) of surface spherical harmonic functions of the second order. Such a representation of the potential is due to Laplace, who first pointed out its remarkable meaning and geometrical characteristics. A principal zonal tidal wave on the Moon results from the eccentricity of the lunar orbit. The period of this tide is 1 anomalistic month with an average length of 27.555 days. The zonal tides vary over the lunar surface only with the latitude: they are symmetrical relative to the lunar equator and the function corresponding to these tides has as nodal lines (lines where the function is zero) the parallels cp = k35'16'. Since solar perturbations of the lunar orbit give rise to significant variations (from 0.0432 to 0.0666) in lunar orbital eccentricity, the maximum and minimum Earth-Moon distances also vary with a period of about 206 days (7.5 anomalistic months). This longer period variation is superimposed upon the monthly period of the principal zonal tide. A principal sectorial tidal wave is produced by the Moon's libration in longitude with a period also of 27.555 days. This libration results from the fact that the axial rotation velocity of the Moon is constant (period of 1 sideric month, 27.322 days) but it has varying velocity of revolution in an elliptic orbit (period of 1 anomalistic month). The difference between the velocities *ofrotation and orbital revolution results in an oscillation of the lunar limb in longitude with maximum amplitudes of 8". Variations of the orbital eccentricity of the Moon with a period of 206 days obviously produce similar variations in its period of orbital revolution. This, in turn, causes the amplitude of the longitudinal libration to vary between about k 5" and L- 8" in a period of 7.5 anomalistic months. This kind of tides has zero amplitude (nodal lines) along the meridians located at 45" on either side of the meridian of the perturbing body. These meridians divide the sphere into 4 sectors where the tidal function is alternately positive (areas of the high tides) and negative (regions of the low tides). The tidal pattern is symmetrical about the meridian of the tide-raising body. Maximum sectorial tides arise at the lunar equator when the latitude of the perturbing body is zero. The libration of the Moon in latitude gives rise to a principal tesseral wave with a period of 27.212 days (1 draconitic month). The latitudinal libration arises because the rotation axis of the Moon is not strictly perpendicular to its orbital plane but it is inclined to this plane at a constant angle. Therefore, as the Moon is travelling along its orbit, during a part of the anomalistic month it is inclined towards the Earth with its North pole and in other part of the month with its South pole. The result of this is an alternating periodic displacement of positions on the lunar disk in latitude by the amount of inclination of the Moon's equator to its orbital plane of 6.68" with a period of

245

Inconstant Lunar Gravity

1 draconitic month. The function representing the tesseral tides has as nodal lines the lunar equator and a meridian situated 90" to the meridian of the perturbing body. Tesseral tides are symmetrical about the meridional nodal line. The nodal lines divide the sphere into areas where the tides change sign with the latitude of the tide-raising body. The amplitudes of this type of tides are maximum at cp = +45' when the latitude of the perturbing body is also maximum. From the considerations given above it is clear that the lunar tides are significantly different from those arising on the Earth. The periods of the principal tidal waves on the Moon are equal or differ slightly, so the zonal, sectorial and tesseral terms all relate to identical waves of long period which produce tides of comparable magnitude. However, the relative importance, or the contribution of each term into the global tidal effect, varies greatly with location on the Moon. The general distribution of the resultant is very complicated, contrary to the situation on the Earth where all principal waves have strongly different frequencies. On the moon the terms cannot be separated. Therefore, in the case of lunar tides it is preferable to employ direct calculations for each point by the formulae (4.2.1) and (4.2.2) where only 3 quantities: R, b and 1 are variable. These quantities can be calculated by expansions into trigonometric series of the next five arguments: s Moon's mean longitude measured from mean equinox of date (MED); p mean longitude of lunar perigee measured from MED; N mean longitude of ascending node of Moon's orbit measured from MED; h Sun's mean longitude measured from MED; p s mean longitude of solar perigee measured from MED.

The latter variables are explicit functions of time determined as

+ 481267.89057'T + 0.001980~2+ 0 . 0 0 0 0 0 2 ~ ~ 3 p = 334.32956" + 4069.03403"T s = 270.43659'

- 0.01032'T2 - 0.00001"T3

N = 259.18328'

-

1934.14201'T

+ 0.00208'T2 + O.OOOOO2"T3 h = 279.69668' + 36000.76892'T + 0.00030"T2 + 1.71902"T + 0.00045'T2 + 0.000003'T3

ps = 281.22083'

(4.2.4)

246

Lunar Gravimetry

where T is the time for which the calculation is needed counted in Julian centuries of 36,525 mean solar days from noon Greenwich mean time, 31 December, 1899. At the beginning of this century Brown (1919) set forth his famous theory of the Moon's motion and issued tables for calculating the lunar ephemerides. Later on, with the appearance of high-speed computers, these tables have been refined, thus allowing us to obtain the Moon's ephemerides to almost any desired accuracy. A detailed discussion of the formulae written below, used for calculations of the coordinates of the Moon, will not be given here; such an analysis has been carried out, e.g., by Moutsoulas (1971). The inverse ratio of the actual Earth-Moon separation to its mean value Ro as given by Brown's theory, is

+ 10-6[54501 cos p ) + 8249 cos 2(s - h) + 2970 cos 2(s - p ) + 1025 cos + p - 2h) + 902 cos (3s - 2h - p ) + 560 cos (2s - 3h + ps) + 422 COS'(S- 3h + p + p,) + 337 cos ( S - p - h + ps) - 286 cos (S - h) - 277 cos ( S + h p - ps) - 208 cos (S + p ) + 182 cos 3(s - p ) + 176 cos (3s - 4h + p ) - 117 cos (h - ps) + 109 cos 2(s + p - 2h) - 89 cos 2(h - p ) - 88 cos (2s - ps - h) + 83 cos (4s - 2p - 2h) + 76 cos 4(s - h) + 67 cos (3s - 3h p + ps) (4.2.5) - 66 cos + p - h - p,) + ..*I

Ro/R = 1

(S -

(S

-

-

(S

In equation (4.2.5) the expansion of R o / R into harmonic components is From this limited to the terms the amplitudes of which exceed 5 x equation one can obtain (R0/R)3and (Ro/R)",figuring in the first terms of the second and third order of spherical harmonic development of the tidal potential, and which are of practical interest as far as the calculational procedure is considered. (R0/R)3 should be determined with the same accuracy as the first power of this ratio, whereas in the expansion for (R0/R)4 the terms with coefficients less than 1 x may be neglected. The necessary equations are

+ 0.0046 + 0.1461 cos (s - p) + 0.0249 cos 2(s - h) + 0.0134 cos 2(s - p )

(R0/R)3= 1

247

Inconstant Lunar Gravity

+ 0.0044 cos + p - 2h) + 0.0041 cos (3s - 2h - p ) + 0.0017 cos (2s - 3h + p,) + 0.0013 cos - 3h + p + ps) + 0.0010 cos (s - p - h + p,) + 0.0010 cos 3(s - p ) - 0.0009 cos - h) - 0.0008 cos - p + h - ps) - 0.0006 cos + p ) + 0.0005 cos (3s - 4h + p ) (S

(S

(S

(S

(S

- 0.0004 cos (h

- ps)

+ 0.0004 cos 2 ( 2 ~- p - h)

+ 0.0003 cos (2s - 2p - 4h) - 0.0003 cos (2s - ps - h) + 0.0003 cos 4 ( -~ h) + 0.0002 cos (3s - 3h - p - p,) - 0.0002 cos (s + p - h - p,) + ... and

+ 0.0009 + 0.219 cos - p ) + 0.033 cos 2 ( ~ h) + 0.013 cos 2 ( ~ p) + 0.004 cos (s. + p - 2h) +

(R,3/R)4 = 1

(4.2.6)

(S

(4.2.7)

The corresponding expansions of the selenocentric latitude and longitude of the perturbing body into harmonic terms are

+ 49 sin (2s - p - N ) - 48 sin (p - N ) - 30 sin (s - 2h + N) + 10 sin (2s - 2h + p - N) + 8 sin (-2s + 3p - 2p, + N) + 6 sin (3s - 2h - N ) + 3 sin (3s - 2p - N) + 2 sin (2s - p - 2h + N ) - 2 sin (-s + 2p - N ) + sin(s - 3h + ps + N) + sin(-s + 2p - 2h + N) + sin (4s - p - 2h - N) - sin (s - h - ps + N ) ] (4.2.8) I = 10-4[1098 sin (s - p) + 222 sin (s - 2h + p) + 115 sin 2(s - h) + 37 sin 2(s - p) + 32 sin (-h + p,)

-b =

10-4[895 sin (s - N)

-20sin2(s-N)+

10sin2(-h+p)

+ 10 sin (s + p - 3h + ps) + 9 sin (3s - p - 2h)

+ 8 sin (2s - 3h + p,) + 7 sin (s - h - p + ps) - 6sin(s - h) + 5 sin(-s + p - h + ps) + 2 sin 3(s - p) + 3 sin 2(N h) - 2 sin (3s p - 2N) -

+ 2 sin (3s + p - 4h) - 2 sin (s + p - 2N)

-

248

Lunar Gravimetry

+ sin (2s + 2p - 4h) - sin (s - h + p - ps) - sin (2s + h - 3p,) - sin (-h + p) + sin (s - ps) + sin (3s - p - 3h + ps) + sin (4s - 2 p - 2h) + sin 4(s - h) + sin (-s - 2h + 3p)l

(4.2.9)

where only the terms with coefficients not less than 1 x lop4 radians are included. At any point on the Moon the zenithal distance of the Earth varies,only within a few degrees relative to some mean angular position corresponding to the mean zenithal distance Z of the Earth at the point considered. The mean zenithal distance varies over the lunar surface as cos z = cos cp cos 1 Deviations of the Earth’s position from this direction can be characterized by the angles of optical libration b and 1 which are small quantities. Therefore it is preferable to represent the tidal potential as separated into two parts. One of the constituents does not vary with time, depending only upon the mean zenithal distance of the perturbing body. The second one, however, is time dependent and can be expressed by power series expansions with respect to b and 1. We note that the presence of a term in the tidal potential on the Moon representing its fixed deformation towards the Earth is another peculiar feature of the lunar tides having no equivalent with the earth tides. P2(cos z) and P~(COS z) expressed in terms of Z,b and I are P2(c0s z ) = P2(c0s .T)

+ b($ sin 2cp cos 1)

+ I($

cos’ cp sin 2A) + $b2(1 - cos’ Z - cos’ cp)

- $1’

cos’ cp cos 22 + $bl sin 2cp sin A

- b3 sin 2cp cos 1 - l 3 cos’ cp sin 22 - +lb2 cos’ cp sin 21 - $b12 sin 241 cos A

(4.2.10)

P3(cos z) = P~(COS 2) - $b sin cp cos I(cos2 z - +)

-9 1 sin 1 cos cp(cosz z - +)

5 COS’

1 - 15)

z 3 COS’

cp - &)

- y b 2 cos Z(COS’Z -

- ?’I

cos qcos’

-

+ q b l sin cp sin 1(cos22 - &)

(4.2.11)

The expansion above for P2(c0s z) is limited to terms with up to the third

249

Inconstant Lunar Gravity

powers of the arguments, while the power series representation of P3(cos z ) includes only terms with up to the second powers of b and 1. Thus, the variation of the Earth-induced tidal potential in time and with position on the lunar surface is completely determined. As has already been mentioned earlier, in the spherical harmonic development of the tidal potential due to the Sun, only the leading term need be considered. The equations necessary to evaluate the time-dependent Moon-Sun distance and the selenocentric latitude and longitude of the Sun are simpler than (4.2.5), (4.2.8)and (4.2.9). However, due to the relatively insignificant contribution of solar tides to the total tidal effect, and for the sake of brevity, we will not write those here. Formulae for the tides due to the Sun can be found, e.g., in Harrison (1 963). Having determined the tide-raising potential, one can easily calculate the tidal variations of gravity and deviations of the vertical along a meridian and a parallel of latitude as

eA =

1

aw

-

1 aw ga cos cp 3T

(4.2.12)

respectively, where g is gravity. The series expansions above allow us to calculate the variations of gravity and the deviations of the vertical in both horizontal directions with an accuracy of 0.5 pGal and 0.01 ” respectively. Earlier expansions, such as that of Harrison (1963), yield an accuracy for the theoretical gravity tide of 10 pGal. The need to calculate the theoretical tides on the Moon with very high accuracy will be shown later. Figure 4.2 shows the variation of the geosolar tidal potential over the year 1983 at different points on the lunar surface, as calculated by the algorithm above. Due to the more complete series expansions of the Earth-raised tides, this potential differs on average by 5% from that calculated by Harrison’s development. More significant differences (reaching in some cases 2&30%) occur between the corresponding values of the derivatives of the potential with respect to the angular coordinates. This may become important when the tangential displacements and stresses in the Moon are investigated. It can be seen in Fig. 4.2 that the geosolar tidal potential at the centre of the lunar disk oscillates around a constant value of 21 m2 s - with ~ an amplitude of about 3 m2 s - ~ ,that is the total range of variation of the potential reaches here 6-1 m2 s - ~ .

250

Lunar Gravimetry

206 days

I J l F l M l A l M l J l J I A I S I O I N I D ] Total year of 1983 Fig. 4.2. Variation of the tidal potential at different points of the Moon's surface over the year 1983. The selenographic latitudes and longitudes of the points from (a) to (d) respectively are: (V, V), (30". 0"),(0",30") and (30". 30").

Since the Moon's radius is 0.2725 of the Earth's and the Earth represents a tide-raising mass 81 times larger, formula (4.2.1) shows that the tide on the Moon is about 6 times larger than that raised by the Moon on the Earth. However, comparing lunar tides and earth tides, it is more convenient to consider variations in the gravity and the deviations of the. vertical (usually measured in tidal practice). Using formulae (4.2.1 2) and also the algorithm for calculation of the tidal potential given above, one can show that the total variation of gravity on the lunar surface is approximately 2.5 mGal at maximum, and the amplitude of the variation of the vertical in both

Inconstant Lunar Gravity

251

directions may be up to about 2.3". It is interesting to note that the total variations in both the gravity and the vertical consist of corresponding constant terms (varying only with position on the lunar surface) and variable, or time-dependent, parts. The contribution of the latter constituent to the total effect reaches about 1 mGal and 1.0 respectively. For comparison, it can be remembered that the maximum lunisolar tidal variation of gravity on the Earth is about 0.24 mGal and the deviation of the vertical is not more than 0.05". The solar tides on the Moon are extremely small; in a similar way as indicated above, one can ascertain that the solar tide in gravity is only about 6 pGal at most, and the deviation of the vertical due to the Sun's tidal effect does not exceed 0.01'' or so. Since the noise levels of the best tidal gravity meters and horizontal pendulums are 0.015 pGal and 0.010 milliseconds of arc (Melchior, 1978), one can see that the amplitude of tidal variations on the Moon even in the case of the clinometric tide (detectable with less accuracy than the gravimetric one) is about 104-105 times the internal precision of the measuring instruments.

4.3 Brief Theory of Solid Tides

In the previous section we described analytically the primary tidal gravity field that would arise on the Moon if it were an absolute rigid body. As it has already been mentioned earlier, this field is completed by an additional gravity field which is due to the tidal distortion of the Moon. On the Earth this effect is quite considerable, being about 0.30 of the total tidal potential and 0.16 of the gravimetric tide that is affected by the elastic yielding of the Earth. The problem of calculating the earth tides becomes highly complicated if the effects of rotation of the Earth are also considered. Rotation must be taken into account when the tides in the Earth's fluid outer core are investigated. Tides in the liquid core can be calculated on the basis of the dynamic tidal theory, when inertial terms are also included in the equations of motion. This means that the different harmonic components of the tides in this case are no longer separable, and instead of a finite set of equations corresponding to static tides one obtains an infinite set of second-order differential equations, where terms of the second order are accompanied by those of the fourth, sixth, etc. orders. Such as a set of equations can obviously be solved with certain difficulties. Nevertheless, the dynamic effects of the liquid core must be taken into account in earth tides since these effects can reach as much as 20%in the amplitudes of certain diurnal tidal waves having their frequencies close to that of a nearly diurnal-free wobble of the Earth (Molodensky, 1961; Bodri and Bodri, 1977). Considering lunar tides, the situation is simpler. According to the

252

Lunar Gravimetrv

calculations of Bolt (1960, 1961) and Takeuchi et al. (1961), the longest free oscillation period of the Moon does not exceed 15 min. Compared to this time interval the periods of the principal tidal waves on the Moon (some 27 days) can be considered as very long, and the tidal phenomenon can be treated using static theory. Mathematically this means that in the basic equations of motion the frequencies of both the perturbing force and the rotation of the Moon can be set to zero. The theoretical study of the tides of a nonrotating spherical moon, even in the case where it is taken as heterogeneous and compressible, presents no major problem. To describe tidal deformation of the Moon, we start with the equations of motion, the continuity of deformations, and, since the effects of self-gravitation in present problem are essential, also Poisson's equation defining the total gravitational potential of the body. Supposing that the deformations are small, these equations in a spherical system of coordinates (r, 8, cp) respectively are a2u.

p 2= p grad ( W at2

+ U ) + (div nr,div be, div nV)

and

where u is the displacement vector, p is the mass density, t is time, aijare the components of the stress tensor, ni is the stress acting on a surface normal to the i axis, W is the external perturbing potential described in the previous section, U is the total gravitational potential of the deformed Moon, and G is Newton's constant of gravitation. When considering tides, we are interested in tidal displacements and stresses and also the changes in the gravitational potential due to tidal deformation of the Moon. In order to determine these quantities (which do not figure explicitly in above equations), we make the assumptions: (1) Following Love (1911) we assume that the Moon is initially in hydrostatic equilibrium with an unperturbed density of po. This assumption enables us to define the displacement vector as that related to the initial state considered as an undeformed state. The stress tensor can be represented as consisting of the initial hydrostatic stress o$') and the additional elastic tidal stress a::). The initial constraints reduce themselves to an initial hydrostatic

253

Inconstant Lunar Gravity

pressure Po governed by Euler's equation

(4.3.2)

grad P o = p o grad U o

where U o is the initial gravitational potential of the planet corresponding to hydrostatic equilibrium and determined as

(4.3.3)

v2Uo = -4nGpo

(2) The dynamic stress-strain relationship characteristic of the Moon is that for an isotropic and perfectly elastic Hooke body, i.e., g!!) LJ =

K6..A V + $pe.. lJ

(4.3.4)

where A = div u is the dilatation, p is the shear modulus, K is the bulk modulus (expressed by p and the compressibility 1 as K = A 2p/3), dij is the Kronecker delta, and eij is the strain tensor. (3) In the initial state, the equipotential surfaces in the planet coincide with surfaces of equal density, elastic parameters, etc. With this assumption we can write

+

J P O - Pb a u o _

axi an axi

ug axi

auo ub axi A!

(4.3.5)

where x i is any of the spherical coordinates and the primes over the symbols indicate the total derivatives of the corresponding variables. Making use of the latter assumption and also the continuity equation, the perturbations of volume density due to tidal deformation can be expressed as p - po

=

Pb -div (pou) = -poA - r]

vb

(4.3.6)

where q = u grad U o is the work done by the deformation. At this stage only the tensor @r of the nontidal quantities is still undetermined. Since the initial hydrostatic pressure at a point r in the deformed state is the hydrostatic pressure at the point originally at (r - u), one can write cr$" = - P o ( r - u)hij= - [Po(r)- u grad P 0 ] 6 , = -(PO

- POV) 6ij

(4.3.7)

254

Lunar Gravimetry

Now the fundamental equations (4.3.1)can be reduced to such a form that they included as unknowns only quantities characterizing the tidal deformation of the Moon. These are the tidal displacements and the perturbation of the gravitational potential induced by tides. Substituting expression (4.3.6) into Poisson's equation, the total gravitational potential of the planet figuring in (4.3.1)can be represented as the sum of the hydrostatic potential U Oand an additional potential U , due to tidal deformation, where

With the help of (4.3.3)-(4.3.7), the components of the vector equation of motion can be written as a set of three partial differential equations of the second order with respect to the three displacement components

a Po

Po

(4.3.9) -A

( u e r2 sin2 8

a2u, at2

-

r sin8 acp Po v2u,

"".acp)I

+ 2 cos 8-

+ r sin 8 aq

7 -

255

Inconstant Lunar Gravity

where

2 I=K--p 3

1 au, 1 aue r sin 8 acp’ r sin 8 acp’ r sin 8

--*-_*--

acp

Equations (4.3.8) and (4.3.9) are a spherical analogue of those obtained by Molodensky (1953, 1961) in a Cartesian reference frame. The equations of motion (4.3.9) can be simplified. With the help of these equations one can show that the radial component of the curl of the displacement vanishes, while the remaining components are different from Zero. Assuming that the frequency o of the perturbing body force in a common time factor exp (iot) becomes zero (static approximation), the displacement can be represented by m

(4.3.10) Y3 W

‘v=,,Z2=

asfl(8, cp) acp

where Sn(8, cp) is a surface spherical harmonic function of degree n. The additional gravitational potential U,is assumed to take the form

ut = Y S W f l ( 8 ,cp)

(4.3.1 1)

In static approximation, the equations governing tidal deformation of the

256

Lunar Gravimetry

planet can be written separately for each S,. Substituting formulae (4.3.10) and (4.3.11) into (4.3.8) and (4.3.9), one obtains for each n three second-order ordinary differential equations with respect to y l , y3 and y,. Introducing some additional variables determined as 21 An(n + 1) Y Z = (A + 2P)Y; + I Y l Y3 r

+ F)

y4 = p(y; y6

(4.3.12)

= y; - 471GpOyl

the set of these second-order equations can be reduced to six simultaneous differential equations of the first order, suitable for numerical integration. It can be shown that functions yz, y4 and y6 are the radial factors in the radial and tangential stresses and the gradient of the potential Ut, respectively. The fundamental equations governing the static tides of the Moon, and written in terms of the functions yi (i = 1,2, ..., 6), are Y; =

-2AY 1 (A 2p)r

+

+-A +Yz2p A

y;=

Yl Y3 --+-+-

r

r

Y4

p

+ l)Y3

+

(A

+ 2p)r

+ 2p

(4.3.13)

257

Inconstant Lunar Gravity

With appropriate boundary conditions equations (4.3.13) can be solved easily for any distribution of density and elastic moduli in the planet, by any of the standard integration techniques used for the solution of linear ordinary differential equations. The above equations are to be solved under the conditions of: (1) Regularity at the origin. ( 2 ) Vanishing of the stresses on the deformed surface of the Moon. (3) Equality of the values of the internal and the external potentials and their respective derivatives at the surface of the Moon. At an internal surface of discontinuity all of the quantities sought for (except the tangential displacement) must be continuous. At a solid-fluid interface we must also have vanishing of the tangential stress. Boundary conditions at the lunar surface can be formulated in terms of the functions y i as Yz

y6

= Y4 = 0 = (2n

(4.3.14)

+ l ) g - (n + 1) r Ys

at r

=a

The problem of boundary conditions at the origin (taken to be the centre of mass of the Moon) needs some special treatment, since equations (4.3.13) have a regular singular point there. To avoid the singularity problem, it has been customary in most of the related studies to start numerical integration from the surface of a small sphere located at the centre of the Moon. As boundary values on this surface, the classical Kelvin’s solution for the tides of a homogeneous incompressible planet has been introduced. This sphere may be made small enough so that the properties inside it do not affect significantly the result. Nevertheless, to eliminate even the uncertainties involved in this technique, we present here an algorithm providing a possibility to start integration directly from the centre of the planet. A similar approach has been applied by Pekeris (1966) and Crossley (1975) in studies of free oscillations of the Earth in connection with the current emphasis on the use of computed eigenfrequencies to refine earth models. Derivation of the formulae suitable for direct numerical integration from the origin is based on the fact that the solution of equations (4.3.13) may be represented by the following power-series expansion (4.3.15)

where none of the coefficients aiois equal to zero. In this expression we have as unknowns the coefficients aik and the indices a i . Substituting (4.3.15) into

258

Lunar Gravimetry

equations (4.3.13), and equalizing those of the coefficients a i k which appear at the same powers of r, we obtain recurrence relationships suitable for subsequent determination of aik, and also obtain the set of characteristic algebraic equations allowing us to obtain the parameters cli. The set of the latter equations, under the regularity condition at the origin (ai 2 0), has three independent solutions. A linear combination of these solutions yields the general solution at the origin we are interested in. In a particular solution, corresponding to some actual set of the parameters ai, at least one of the coefficients aik cannot be determined from the recurrence relationships. Therefore in the general solution of equations (4.3.13) three coefficients remain undetermined. These can be found from the boundary conditions at the surface of the Moon. In principle, a solution of type (4.3.15) can be obtained for any model of the internal constitution of the Moon. The final expressions related to the general case, however, are rather complicated and lengthy, so for brevity we present here only such a solution of the equations (4.3.13) which is obtained under the assumption that pb = p' =.A' = 0 at r = 0. The general solution of the tidal equations in this case takes the form

(4.3.16)

where

(4.3.17)

259

Inconstant Lunar Gravity

a60

= n(a50 - 3ya30)

a62

= (a

+ 2 b 5 2 - 3ya12

where y = (4nG/3)pO. The quantities p o , p and I are assumed in above expressions to have their values at r = 0. Coefficients ~ 3 0a32 , and a50 are arbitrary and (as we already have mentioned earlier) these must be obtained from boundary conditions at the free surface. Coefficients appearing at higher powers of r in (4.3.15) are also determined by expressions of type (4.3.17). Formulae (4.3.16) and (4.3.17) enable us to initiate the numerical integration directly from r = 0.

4.4 Tidal Deformation of the Moon

It has been shown in the previous section that the tides of the Moon can be described by six functions yi, characterizing the tidal displacements, stresses and the perturbation of the gravitational potential with its respective gradient. One can also see that these tidal functions are completely determined by distributions of the density and the elastic parameters within the lunar interior. The role of the elasticity of the considered planet in planetary tides can be demonstrated by the following simple example. If the Earth were an absolute rigid body, it would not undergo any deformation under the lunisolar tidal forces. If it were a perfect liquid, the radial displacement of its surface (in other words the height of the static tides) determined as u, = W/g, would show a total variation of 78 cm. As the physical properties of the Earth are somewhere between these two extremes, it undergoes some intermediate deformation. Formula (4.3.10)shows that the actual radial displacement can be represented as the product of the function yl and of the height of the static tides. Since the value of yl at the Earth's surface is about 0.6 (Table 4.1), one finds that the total range of the effective radial deformation of the Earth is about 50 cm. Now let us consider how the Moon may react to the tidal forces. For the Moon, where on account of the small total mass the effects of selfcompression on the &nsity even at the central part of the satellite are not larger than a' few per cent, until recently a homogeneous structure model has been considered as a reasonable approximation to the real planetary structure. The parameters defining this homogeneous model 'are: p = 3340 kg mW3,up = 8.1 km s - l and us = 4.67 km s - l , where up and us represent the compressional and shear seismic wave velocities, respectively. In the following we shall consider this simplest variant of the lunar structure and the parameters characteristic of its tidal deformation as standard values,

260

Lmar

Gravimetry

TABLE 4.1 Calculated values of the Love numbers, gravimetric and clinometric amplitude factors for the considered planetary models

Planetary models Homogeneous moon LM 101 LM 201 LM 301 LM 761 Earth, 1066A

h

1

k

s

Y

0.0349 0.0470 0.0439 0.0497 0.0503 0.6092

0.0097 0.0128 0.0123 0.0130 0.0133 0.0849

0.0202 0.0274 0.0254 0.0287 0.0281 0.3013

1 .W46 1.0059 1.0058 1.0068 1.0082 1.1573

0.9853 0.9804 0.9815 0.9789 0.9778 0.6921

suitable for comparison with those related to certain more refined structural models. To investigate the effects arising from .the elastic yielding of the Moon, we calculated the values of the parameters characterizing its tidal response and also the actual tidal displacement and stress fields for four further different models of the lunar interior. Three of these (models LM 101, LM 201 and LM 301 of Nakamura and Latham, 1969) had been elaborated before lunar seismic data became available. In the procedure developed for calculation of these models, temperature and compositional effects, as well as pressure effects on density, have been taken into account. The range of permissible variations of the density in such models is well controlled by the total mass and the observed moment-of-inertia values of the Moon. The values of seismic velocities, however, depend greatly on the assumption of its composition. Calculating seismic velocities, Nakamura and Latham used a density-seismic velocity-mean atomic weight diagram. The position of the lunar material in this diagram was fixed by a least-squares fit to a large number of observed pu values of terrestrial rocks. Model LM 101 (a somewhat refined version of the standard homogeneous one) is chemically homogeneous, its parameters being influenced only by temperature and pressure. The relatively high value of the surface density and probably too large a moment of inertia obtained in this model imply, however, that a conqentration of lighter material must occur near the surface, i.e., that the Moon carhot be homogeneous throughout. Model LM 201 has a 20-km thick basaltic surfaie layer and a continuously heterogeneous interior. A three-layer model, LM 301, includes a basaltic outer layer with thickness of 20 km and a liquid iron core of radius 348 km; that is, the change of composition along the radius is discontinuous. The presence of a core is not a necessity to satisfy the moment-of-inertia criteria-the composition may change with depth also gradually. And finally, the most sophisticated model of the Moon’s structure that we considered

Inconstant Lunar Gravitv

261

(model LM 761 of Nakamura et al.; 1976a,b) is based on lunar seismic data and consists of five zones: crust, upper, middle and lower mantles, and a core (tentative). The most specific feature of the outermost zone of the Moon is that it is almost twice as thick as the Earth’s crust. The crust has a thickness of 60 km in LM 761. Due to transition of the nonconsolidated lunar material into the consolidated state under increasing pressure and temperature, the seismic velocities in this zone increase rapidly with depth as the material becomes more rigid. At the bottom of the crust the density and both Lam6 constants reach values of 3200 kg m-3 and 7.0 x lo4 MPa respectively. The crust, the upper mantle and the middle mantle of the Moon constitute its cold rigid lithosphere which extends to a depth of 900 km and is able to bear significant non-hydrostatic stresses. The density of the lithosphere has values close to the mean density of the Moon, the shear modulus p in this region decreases from about 7.0 x lo4 MPa at the top to about 4.6 x lo4 MPa near the bottom. Parameter 1, ranging from some 7.0 x lo4 MPa in the crust to about 1.2 x lo5 MPa at the base of the middle mantle, shows a different variation with depth in the lithosphere. The Poisson’s ratio increases with depth in this section from 0.25 to 0.36. The large thickness, the different variation of Lamt’s parameters along the radius, the increase in Poisson’s ratio with depth, and low attenuation of the seismic waves; all these features are very specific of the lunar lithosphere and are very different from those of the Earth’s lithosphere. In the lower part of the middle mantle (at 800 to lo00 km depths) the physical properties of the lunar material seem to change rapidly. It has been firmly established that deep moonquakes are concentrated in this relatively narrow transition zone between the lithosphere and the lower mantle. The region below the level of deep moonquakes, starting at a depth of about lo00 km, in certain respects shows some similarity to the Earth’s asthenosphere. This region, known as the lower mantle of the Moon, is characterized by high attenuation of the shear waves. Seismic P waves, however, do not appear to show any variations from that in the overlaying middle mantle. Such a propagation of the seismic waves implies that this zone is likely to be in a state of partial melting. Available seismic data allow the existence of a small molten core (with a radius of 250 km in the model LM 761) presumably consisting of an Fe-FeS mix. The characteristic parameters of the central core in LM 761 are: p = 7500 kg m-3, p = 75 MPa and 1 = 1.5 x lo5 MPa. For comparison of the tidal deformations arising in cases of different models of the internal structures of the Earth and the planets, it is convenient to consider the values of the functions yi which these models give at the surfaces of the planets in question. In tidal research and geodynamics these characteristics are generally known as the Love numbers. The above

262

Lunar Gravimetry

approach has been introduced because the measurements of the earth tides are carried out on the surface of the Earth, with instruments attached to the crust. The observed amplitudes and phases for each of the tidal waves are compared with those calculated for a model Earth. Such a comparison can obviously yield information on the validity only of the surface values of functions yi. It has become customary to use the following notation for the Love numbers: Yl

=h

Y3

=1

ys=l+k,

atr=a

The most common method in tidal practice is to measure the gravimetric 6 = 1 + h - 3k/2 and the clinometric y = 1 + k - h Love numbers. Determination of the Love numbers is the most important information that can be obtained from observations of the tides, and the values of these numbers may be used as basic criteria for the comparison of different models of the planetary structures. Table 4.1 shows the values of the Love numbers calculated by ourselves for the structural models of the Moon briefly described above. For comparison we calculated and also included in this table the Love numbers related to the earth model 1066A of Gilbert and Dziewonski (1975). Looking at these numbers, one can see that the effects of elasticity of the Moon on the lunar tides are at least by an order of magnitude less than those on the Earth. In other words, the data presented in Table 4.1 indicate that the Moon appears to be close to the case of a perfectly rigid planet. The differences between the corresponding Love numbers calculated for slightly different earth models (such as, e.g., 1066A and B1 or B2 of Bullen and Haddon, 1967) are as small as some units in To differentiate between the various earth models, one should measure the earth tides to a relative accuracy of the order of 1%. With present instruments and methods of analysis this accuracy can certainly be achieved. On the Moon, however, only an accuracy of about 0.1%will allow, for example, the homogeneous model to be distinguished from LM 761. Since such a high accuracy has not yet been achieved even at the best earth tide stations in long series of observations, one must admit that the measurements of lunar tides does not appear to be a promising method of exploring the Moon’s interior. Even the problem of a relatively rough estimation of the Love numbers from observations of the lunar tides would meet certain difficulties, for the periods of the principal tidal waves on the Moon are so long that the instrumental drift would probably put strong limitations on the accuracy of the measurements. We show, however, that even if the tides are not too informative in respect

263

Inconstant Lunar Gravity

of the Moon’s structure, nevertheless one should calculate the theoretical tides on a rigid moon to a very high accuracy. This is necessary in order to be able to introduce certain tidal corrections with the desirable accuracy. One must remember that despite the fact that actual tides on the Moon are much less than the static tides (this is not the case on the Earth), they exceed the tides actually arising on the Earth. Figure 4.3 shows the radial and tangential displacement profiles in the Moon that we calculated for both a homogeneous model and LM 761. One can see in this figure that the actual displacements of the Moon’s surface at the considered moment in radial and lateral directions have values of almost 1 m and about one-third of that, respectively. We have already mentioned at the beginning of this section that the maximum effective radial deformation of the Earth’s surface is not more

Fraction of lunar radius Fig. 4.3. Tidal displacement profiles for t w o lunar models at 0.00 h Greenwich mean time, 1 January 1983. The radial displacement profiles are calculated along a radius at the centre of the lunar disk and the tangential displacements along a radius 45” from the centre of the disk. Solid lines represent displacements for model LM 761 and the dashed lines correspond to a homogeneous model.

264

Lunar Gravimetry

than some 30 cm, and this varies in the total range of about 50 cm. At the centre of the Moon both the radial and the tangential displacements are zero, the former having its maximum value in the upper mantle at depths of 100150 km, and the latter attaining its maximum amplitude at a depth of 450650 km in the region of the middle mantle. As is known, laser measurements of the Earth-Moon distance at present are accurate to a few cm, and for measuring direction, the VLBI method yields an accuracy of +0.005”. A further increase in accuracy of both these techniques is expected in the near future. Since tidal displacements of the lunar surface, and particularly the variations of the vertical (details of tidal variations of the vertical were given in §4.2), are much larger than uncertainties of the laser distance measurements and VLBI-determinations of directions, no doubt tidal corrections should be introduced into such measurements if it is intended to determine from those the parameters characterizing the relative position of the Earth and the Moon. In order not to reduce the accuracy that laser measurements can yield, the displacements of the Moon’s surface should be calculated to an accuracy not worse than -0.1 cm. This, in turn, necessitates the calculation of the tidal potential to a relative accuracy of OS%, i.e. to the accuracy that series expansions in $4.2 yield. Thus, as in the case of terrestrial geodesy, the problem of the exact tidal corrections should not be left out of consideration in extraterrestrial geodesy. The interest in the tidal stresses in the Moon has increased recently because there is strong evidence that moonquakes are controlled by tides. It has been found in the course of lunar seismic experiments that the seismicity of the Moon differs greatly from that of the Earth. A characteristic feature of the lunar seismicity is that the great majority of the moonquakes (except the high-frequency teleseismic events believed to be shallow tectonic moonquakes) occur deep in the lunar interior in the depth range from 800 to 1100 km, near the lithosphere-asthenosphere boundary. Their occurrence displays well-defined tidal periodicities. The moonquake activity peaks occur near times of maximum eccentricity of the lunar orbit and at extreme latitudinal and longitudinal librations. It can be shown that the tidal potential presented in Fig. 4.2, and consequently also the tidal distortion of the Moon, attain their extreme values just at such instants of time. Tidal origin of the deep moonquakes seems to be so obvious that their occurrence probably can be predicted by calculating the tidal stresses in the Moon. Having obtained the functions y i for a given structural model of the Moon, the components of the tidal stress tensor can be calculated as. urr

= Y2S”

265

Inconstant Lunar Gravity

Crq

= Y4

1

asn -

sin9 acp

~

(4.4.1)

+2p-

;;;

(sin12 ~g +cot 9-

The above expressions show that tidal stresses depend on time, depth and position within the Moon. Once the stress tensor is determined, using the secular equation one can calculate the three principal stresses. The magnitude of the total shear stress is calculated as half the difference between the largest and the smallest principal stresses. Figure 4.4 presents the maximum principal stress and the total shear stress profiles at 0.00 h Greenwich mean time, 1 January 1983, calculated for a homogeneous moon and for the model LM 761. As seen in Fig. 4.4, the tidal stresses in the Moon are relatively small, and at any particular depth these do not exceed 0.1 MPa. Such small stresses are insufficient to generate moonquakes in unfractured rock but they can activate faults that are near failure at certain pre-existing zones of weakness. Besides, one can suppose that at times of maximum seismic activity the tidal stresses not only attain their extreme values but their orientation, varying with time, is also favourable to generate thrust faulting. Differences between the various structural models of the Moon have an especially marked influence on the distributions of tidal stresses within the planet. In the case of a homogeneous model the stresses decrease monotonically from maximum values at the centre of the Moon to zero at the surface. In model LM 761 maximum radial stresses occur in the lower mantle, at depths of 1 2 W 1400 km. The lateral stresses attain their largest values near the base of the middle mantle at depths from 80CL1000 km, where the zone of partial melting is believed to start. It is remarkable that the depth range of the maximum total shear stress coincides well with the observed range of moonquake focal depths. A maximum of J m-3 in the strain energy density also occurs at these depths. This means that even in the case where it is supposed that faulting proceeds in blocks with a characteristic size as small as 1 km, and not more than 1% of the tidal strain energy is stored in the seismic waves, the tides are sufficient to activate individual seismic events.

-

266

Lunar Gravimetry

7 0 ?\ 4I

I

I

I

I

I

I

\"

60 -

\

\ - 50

-m

-40

& I-

ln

I

IOL 10

E

-30 tj L

m 0, c ln

-20

3

z

-10

Fraction of lunar radius Fig. 4.4. Radial section of the Moon showing tidal stress profilesfor lunar model L M 761 (solid lines) and a homogeneous moon (dashed lines) at 0.00 h Greenwich mean time, 1 January 1983. The radial stress profiles are calculated along a radius at the centre of the lunar disk and the total shear stress along a radius 45" from the centre of the disk.

The tidal stresses vary along the lunar disk in a complicated fashion. Figure 4.5 shows the calculated spatial distribution of all six components of the tidal stress tensor at 900-km depth (the region of maximum lateral stresses) at 0.00 h Greenwich mean time, 1 January 1983. The instantaneous subearth point at the considered moment is at the middle of the lunar disk, but due to librations it is not exactly at the centre of the projection. The diagonal components of the stress tensor, being proportional to S 2 , are symmetrical about the lunar equator and the 0" and 90" 'meridians. They attain their maxima, and are tensile, at the middle of the disk. They have zero amplitudes at points satisfying the condition cos cp cos I x cos 55"; then they become compressive and reach their maximum negative values on the limb. Compared to the normal stresses, the deviatoric stress components vary

Fig. 4.5. Components of the tidal stress tensor calculated for lunar model LM 761 at 900-km depth, at 0.00 h Greenwich mean time, 1 January, 1983. The contours are in kPa, with 10-kPa contour intervals (except for uovwhich is given in Pa, with 5-Pa contour intervals).

268

Lunar Gravirnetry

along the lunar disk in a more complicated way since these depend on S2 and also on certain derivatives of this surface spherical harmonic. All these stresses vanish at the lunar limb. Component ore also becomes zero at the equator and has its extreme values on the zero meridian, at cp = +45". Component or,,, is the same as ore rotated by 90", and these are antisymmetrical about the zero meridian and the lunar equator, respectively. Component oe,,,, being the smallest among all stresses, is antisymmetrical with respect to both the equator and the zero meridian, and attains its extreme values at points about 45" away from the instantaneous subearth point. Having considered the variation of the tidal stresses with depth, it appears that the lateral stresses are of importance in the generation of moonquakes. It was shown that the observed range of moonquake focal depths corresponds to the zone of maximum total shear stress within the Moon. Comparing the distributions of the stress components along the lunar disk with locations of the marked seismic belts of the Moon, one finds that the tangential stresses seem to be of interest also in this respect. The zones of maximum normal stresses along the disk do not coincide with the moonquake belts, while the most significant western seismic belt, for example, trending nearly N-S along the 2W-40" meridians, corresponds well to a zone of maximum of the deviatoric stress or,,,.However, selenophysical factors connected with the possible existence of lateral heterogeneities in the deep interior of the Moon can probably exercise a more significant influence on spatial distribution of the moonquake epicentres. According to a general view, the region near the base of the lunar lithosphere may consist of meteorite material which had been compressed initially, and could never be melted completely. This implies that certain cracks, faults, zones of weakness and other types of heterogeneities causing stress concentrations may exist in the lunar interior. Therefore the distribution of the moonquake foci within the satellite is likely to be controlled simultaneously by tidal stresses and stress concentrations due to heterogeneities above. Considering the problem of prediction of the occurrence times of future moonquakes, it is interesting to examine the temporal variation in the tidal stresses. We calculated, and present in Figs. 4.6 and 4.7, the components of the tidal stress tensor at moonquake foci A1(10.8"S, 31.3"W, 900-km depth) and A2,(20.8"N, 27.0°W, 915-km depth) as a function of time. Focus A l is an extremely interesting one, since it is situated in the most active section of the western seismic belt. For the registration period indicated in Fig. 4.6, almost 30% of the total number of the moonquakes were generated in this section and half of those occurred just in A l . We indicate in Fig. 4.6 only four of the six stress components because has values and phases practically equal to and no,,, is much smaller than the remaining ones. Also plotted those of o,,,,,,,

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Inconstant Lunar Gravity

15 10 5

m

40 30

!2m

20

1

a ml

g v)

-20 -10

0 30

20 10

12

5 10 2

.-c

8

? i 6 4 2

aE

0

A, Moonquakes

I

Fig. 4.6. Temporal variation of tidal stresses at the A, moonquake focus, calculated for lunar model LM 761, The A, moonquakeoccurrence times are indicatedas dots on the stress curves.Also shown below the stress curves are the recorded amplitudes of the moonquakes. uvq z urn,and uopz 0.

below the stress curves are the occurrence times and the amplitudes of the A l moonquakes as recorded on the seismograms. The latter characteristics and also the foci coordinates are taken from Lammlein (1977). The Earth-Moon distance varies between about 357,000 and 407,000 km around its mean value of 384,000 km in each anomalistic month. Since both the tidal potential and its derivatives with respect to the angular coordinates vary during a tidal cycle as l/R3, the amplitudes of the tidal stresses at perigee are about 25% larger, and at apogee some 20% smaller, than the correspond-

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Lunar Gravimetry

15 . a++ 10

5 35 30

iii 25 a Y 20 ln

a l ln

!! 30

iz

20 10

25 20 15

a

A, Moonquakes

$ 6 . s 4

$

3

2

s o

E a

II

I

I

I

Fig. 4.7. Same as Fig. 4.6, but corresponding to the A, moonquake focus

ing mean values. Due to variations in lunar orbital eccentricity, the maximum and minimum Earth-Moon distances also vary with a period of about 206 days. This longer period variation evidently also gives rise to corresponding pulsations in the stresses. The above effects obviously are independent on positions of the moonquake foci within the Moon. One can easily identify both the 27.5-day and 206-day periodicities on the calculated stress curves. Comparing the temporal variation of the stress components with the occurrence times of the moonquakes, one finds that the moonquakes at A l occur near perigee and when the zenithal distance of the Earth, being equal to about 30°, is less than its mean value. Moreover, the strongest moonquakes arise at minimum perigee and near maximum northern latitudinal and maximum western longitudinal librations. The moonquake occurrence times

.

Inconstant Lunar Gravity

271

show a clear correlation with the extreme values of the ore stress tensor component. At the A z o moonquake focus, located in the northern region of the western seismic belt, the moonquakes were detected near perigee at times of near average zenithal distances of the Earth. Unlike the A l moonquakes, the deep seismic events here usually occur during the maxima of orcp and these times correspond to the maximum negative gradient of ore. A comparison of the temporal variation of the tidal stresses with the occurrence times of the moonquakes at several foci indicates that at each focus the time dependence of the moonquakes may be controlled by one particular component of the tidal stress tensor. This specific stress component is usually the same for a number of near-seated groups of the moonquake foci. In addition, the particular tidal stress orientation existing at the times of moonquake occurrences, remains more or less fixed over a period of several years at each focus. The above facts, taken with a number of other characteristic features of moonquakes (which we do not intend to analyse in the present work), imply that the moonquakes may be induced in the zone of the lithosphereasthenosphere contact by fault displacement of blocks of rock appearing along single faults or series of near-seated faults. The orientation of such a displacement, and the specific tidal phase that holds at times of moonquake occurrences at a certain group of foci, are most probably determined by the complicated relief of the lithosphere-asthenosphere transition zone and/or the features of the convective motion possibly existing in the lunar asthenosphere. An argument in favour of the latter mechanism is that the tide-induced cubic expansion on the Moon has an amplitude ten times that found on the Earth. The cubic dilatation of the Earth is reflected in alternate rises and falls of water level in wells, and this also applies to oil, volcanic lava and other liquids contained in the Earth’s crust. On the Moon, at times of high tide there is certain expansion in the lithosphere. The relatively hot, lower viscosity material of the asthenosphere in such a tidal situation can ascend along the faults between the separate blocks of rock, thus serving as a kind of “lubricant” between these blocks. In principle, it is even possible that the location of the moonquake epicentre zones in some way can reflect the geometry of the convection flow within the Moon. Finally we note that due to recent efforts made in investigating the Moon, it is now recognized that the lunar tides have a close relation to certain selenophysical processes, and that tidal deformations of the Moon must be taken into account to a very high accuracy in different space geodetic measurements. Since lunar tides have amplitudes greater than those on the Earth, one can expect that further development of lunar physics will reveal newer phenomena in some way also related to moon tides.

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Lunar Gravimetry

4.5 Tides of the Anelastic Moon

In the previous sections we have assumed that the Moon was a perfectly elastic body. Under this assumption we were able to use Hooke’s law for derivation of the equations governing tidal deformation of the Moon. The real phenomenon, however, is more complicated. The Moon travels with synchronous rotation along an eccentric orbit about the Earth. If the Moon were perfectly elastic, the tides there would have maximum amplitudes at times of minimum Earth-Moon separation. Owing to friction, however, the tides are high not at perigee but at some later time. A schematic representation of tidal friction in the Moon is given in Fig. 4.8b. It is convenient to characterize the frictional retardation of the tides by an angle 6,. Besides retarding the tides in phase, the frictional processes in the Moon can, in principle, also reduce the amplitudes of the tidal deformations. Both of these effects can be calculated in a relatively simple way. It can be shown on a firm theoretical basis (for brevity we omit the detailed analysis) that under the following conditions the basic tidal equations derived in 94.3, can be extended also for the case of an anelastic tidal response, so that the shear modulus and functions yi in (4.3.13) are modified in the anelastic case in a certain fashion. It was mentioned earlier that an elastic, isotropic medium is characterized by two elastic parameters: the shear and the bulk moduli. Now we assume that energy dissipation in bulk compression is much less than in shear deformation. This assumption is reasonable, as it is consistent with experimental data that any substance behaves almost perfectly elastically under hydrostatic stress alone, while the situation is quite different in the case of shear stress. This feature can be formulated mathematically by taking the shear modulus as a complex quantity while the incompressibility remains a real function. Thus, one can write

P*

= POU

+ j$),

$ << 1

(4.5.1)

where j = (- l ) l i 2 , and po$ and p o are the imaginary and real parts of the shear modulus respectively. Applying the theory of small oscillations with damping to the present problem, it can be shown that in case of internal friction the time dependence of the displacements can be characterized by the multiplicative factor expCjot), where o is the frequency of the oscillation considered and t is time. Therefore the displacements and stresses, and consequently all the unknown functions in equations (4.3. L3), should be considered complex, i.e. yi = yoi

+ jyli,

i = 1, . . ., 6

(4.5.2)

where y l i are the imaginary parts of the functions characterizing the tides of

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273

Fig. 4.8. Diagrammatic representation of tidal friction in the Earth-Moon system. (a) Frictionally retarded tidal bulge on the Earth; (b) delayed radial tides on the Moon. The height of the tide is indicated schematically by the length of the arrows.

an anelastic planet, and yoi are the corresponding real parts. Substituting (4.5.1) and (4.5.2) into (4.3.13), a set of complex equations with respect to yi is obtained. The amplitude and phase shift of a certain tidal deformation will be equal to the magnitude and the argument of the corresponding complex number (4.5.2). Carrying out an analysis of the anelastic tidal equations, one finds that for any particular tidal deformation the phase shift will be of the order of $ and the amplitude will be reduced with respect to the perfectly elastic case by a value of the order of $ 2 . Corresponding to (4.5.2), the boundary conditions in the anelastic case also become complex. Thus, mathematical consideration of the anelastic tidal

274

Lunar Gravimetry

response is equivalent to the change from real to complex arithmetic with the corresponding form of the shear modulus. Since present computers are capable of carrying out operations with complex numbers, transformation of system (4.3.13) into a set of complex equations, i.e. inclusion of the anelasticity effects, does not give rise to any mathematical problems. The only problematical operation is the determination of the complex shear modulus. For this purpose one has to introduce a concrete rheological model of the considered planet. Investigations of the damping of free oscillations, surface waves and bodily seismic waves have produced several data sets on the attenuation of vibrations within the Earth. Analyses of these data show that the dissipation function Q; l, determined as the fraction of the elastic strain energy dissipated in each oscillation cycle, for the rocks of the Earth’s crust and mantle over a wide range of periods depends only weakly on the frequency of the oscillation considered, or is even almost completely independent of it (e.g., Anderson and Hart, 1978). Therefore the favoured rheological model of the anelastic earth mantle, also in the tidal frequency range, is that for a Knopoff-Lomnitz body, where

The subscript in this expression indicates that dissipation is attributed to shear. The problem of applicability of the Knopoff-Lomnitz rheological model to periodical phenomena having greatly different frequencies is studied in two ways. On the one hand, laboratory and geophysical measurements of attenuation of the elastic vibrations in rocks yield data on the functional dependence QJo). On the other hand, introducing such frequency dependence into certain test curves of Q,, one can carry out numerical experiments for the modelling of different phenomena, in order to investigate attenuation at different frequencies. Both these approaches seem to justify the nondependence of Q, on frequency, thus providing a possibility to apply Q,-profiles for the modelling of anelastic deformations of a planet at much longer characteristic periods than those used for the determination of Q,(o). Bodri and Pedersen (1980a,b) have carried out numerical experiments for the determination of the phase shifts of tidal deformations of an anelastic earth. These model calculations were made for the earth model 1066A of Gilbert and Dziewonski (1975) supposed to display Knopoff-Lomnitz rheology, with a Q,-distribution obtained by Anderson and Hart (1978). In spite of the significant complications that arise when making experimental determinations of the earth tide phase shifts, Bodri and Pedersen have found an extremely good agreement between their theoretical results and those obtained by tidal measurements.

Inconstant Lunar Gravity

275

Applying this approach to the problem of tidal deformations of the anelastic moon, we suppose that the anelastic properties of the lunar material in the tidal frequency range can be characterized by a Knopoff-Lomnitz rheological model, with a distribution of the quality factor Q, obtained from lunar seismic data by Nakamura et al. (1976a). Independent of the significant differences in the internal structures of the Earth and the Moon, the radial variation of the dissipation function does not differ greatly within the two bodies. The most remarkable difference is that the lunar lithosphere displays a much higher quality factor than does the Earth's lithosphere. The estimated values of the shear-wave Q in the Moon's lithosphere, ranging from about 6000 in the crust to about 4000 in the upper mantle, are extremely high. These Q-values are about 1&100 times those found for the lithosphere of the Earth. In the transition zone (of about 150 km thickness) between the upper and middle mantles Q, decreases rapidly. In the upper part of the middle mantle it is estimated to be about 1500, and decreases further with increasing depth. Near the bottom of the middle mantle, in the zone of the deep moonquakes, it does not exceed a value of 300, becoming comparable with the Q,-values occurring in the lower lithosphere of the Earth. As depth increases Q, in the lower mantle decreases further, until at the core-mantle boundary it is as small as 100 or less, vanishing almost completely in the core. Due to similar anelastic feqtures of the Moon and the Earth's mantle, the influence of anelasticity of the Moon on the lunar tides is about the same as that in the case of solid tides on the Earth. Exact calculations show that the phase lags of the tidal displacements and stresses on the lunar surface with respect to the acting potential field amount only to some minutes of arc and none of them exceeds 10 minutes of arc. As to the tidal amplitudes, these are almost unaffected by the anelasticity of the Moon. There is, however, a significant difference in phase retardations of the earth tides and lunar tides, and therefore also in the nature of the tidal energy sink on the Earth and the Moon. Based on astronomical evidence (deceleration of the Earth), the rate of tidal energy dissipation in the ocean-solid earth system is exactly determined and at present it is equal to 2.7 x 10l2J s- '. However, astronomical observations do not identify the sink of energy. If the dissipation were attributed to solid friction, then the bodily tides would have a lag (about 2") at least 2&30 times higher than that actually occurring in the solid earth. Therefore it seems to be quite probable that almost the whole dissipation is within the Earth's hydrosphere. On the Moon, however, the tidal phase lags, amounting only to some minutes of arc, reflect the total dissipation in the satellite, since only the mechanism of solid friction operates there. Due to radial variation of the dissipation function within the Moon, the phase shifts of the tidal deformations increase with depth and reach their maxima in the region of the mantle-core boundary. The maximum values,

276

Lunar Gravimetrv

however, whatever type of deformation is considered, do not exceed some 15 minutes of arc. In the light of the above considerations one can conclude that the quality factor characterizing the anelasticity of the lunar material is so high that anelasticity effects appear only in secular changes of the gravity field of the Moon. In case of the periodic tidal deformations the influence of anelasticity on lunar tides is practically negligible, and the Moon can be considered as a perfectly elastic body.

4.6 Tidal Friction and Secular Variations of Lunar Gravity

For the analysis of periodic variations of the lunar gravity field we applied a method involving different degrees of approximation to the real case. First we derived the primary tidal gravity field that would arise on an absolute rigid moon. With the introduction of the elastic yielding, and then, anelasticity of the Moon, the problem has got more and more complicated. It was shown that even if the Moon displays mechanical features rather close to those of a rigid body, nevertheless, analysing different selenophysical phenomena, one has to take into account its elasticity. It has also been pointed out that the effects of anelasticity of the Moon on the periodic variations of its gravity field are negligible. A similar approach can also be used for investigation of the dynamic evolution of the Earth-Moon system, appearing as one of the most significant and most exactly determined mechanisms resulting in secular variations of the lunar gravity field. In the first stage one can treat the Earth and the Moon as point masses. With such an approximation one can determine the actual motion of the Moon (excepting the phenomena of precession and nutations) to an accuracy sufficient for practical astronomy. Introducing each next approximation, the characteristics of motion analysed on the previous stage not only get more complicated, but a number of certain new phenomena should also be taken into consideration. Thus, supposing in the second approximation the Earth and the Moon to be absolute rigid bodies of finite extensions, the features of their rotation with respect to their centres of mass can be described. Taking into account the elasticity of the Earth and the Moon, one finds that their motion is influenced to a certain degree by their own elastic deformations. In these stages we are concerned actually not with’ secular, but with long-period variations in the Earth-Moon system. Some of these, however, over relatively short intervals of time, may also be considered as secular variations. Nevertheless, truly secular changes appear in the EarthMoon system only in the case where both these bodies are no ,longer assumed to have perfect elasticity and the process of tidal energy dissipation

Inconstant Lunar Gravitv

277

in their interiors is taken into consideration. Tidal friction gives rise to significant changes in the orbital parameters of the Moon only over time intervals of some 108-109 a, that is even for the longest period of variations of the lunar orbit of 26,000 a (the period of variation of the mean motion of the Moon, resulting from changes of the eccentricity of the Earth’s orbit) the effects due to frictional processes are minute. The phenomenon of tidal friction in the Moon has been demonstrated in the previous section. The general nature of tidal friction in the Earth is illustrated by Fig. 4.8a. If the Earth were supposed to have perfect elasticity, then the maximum tide raised by the Moon would be symmetrical to the line of centres OE. Tidal friction, however, causes the peak of the tide to occur on the Earth with a certain delay in time. Since the period of orbital revolution of the Moon is longer than that of the Earth’s rotation, the rotating Earth carries the tidal bulge forward; the tide is high at a longitude ~3~ that has already rotated past the line OE. The gravitational attraction on the bulge is asymmetrical to the Earth-Moon line and this gives rise to a torque in the planet-satellite system. Since the Moon exerts a greater attraction for the front than the rear tidal bulges, a component of the torque, tending to retard the rotation of the Earth and to accelerate the Moon, transfers energy and angular momentum from the Earth’s rotation to the Moon’s orbit. Another component of the torque tends to tip both the axis of rotation of the Earth and the Moon’s orbital plane. The transfer of angular momentum is accompanied also by a loss of mechanical energy from the planet-satellite system. Thus, as a result of tidal friction, the Earth rotates at a decreasing rate and the obliquity of its equator to the ecliptic changes with time. As to the Moon, its semimajor axis increases and the inclination of its orbital plane and its eccentricity also vary with time. Tidal interaction is obviously greatest when the Moon is at perigee. Therefore the transfer of energy and angular momentum into the Moon’s orbital motion near perigee is greater than near apogee. Consequently, the apogee moves out more rapidly than the perigee; that is, the orbit as a whole expands and becomes more eccentric. Tidal friction in the Moon supplements the frictional effects of the earth tides. Since the tidal bulge on the Moon lies always along the line OE, no angular momentum is transferred but there is a loss of mechanical energy. Dissipation by radial tides in the Moon causes the eccentricity of the lunar orbit to decrease with time. In the past when the rotation of the Moon was not synchronous, tidal friction in the Moon, besides affecting the orbital elements, resulted also in its own deceleration. There have been many theoretical studies and numerical experiments concerning tidal evolution of the Earth-Moon system, such as those by MacDonald (1964), Kaula (1964, 1971) and Goldreich (1966). Extrapolating

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Lunar Gravimetry

the Moon's orbit back in time, certain difficulties occur owing to uncertainties about its initial state (including the problem of origin of the Moon), probable variations in the anelastic properties of the planetary materials, etc. Apart from these complications, however, the major features of tidal evolution of the Earth-Moon system are understood well enough. The mean orbit of the Moon at present has its semimajor axis equal to 60.3 earth radii with an eccentricity of 0.0549.The obliquity of the Earth's equator to the ecliptic is 23'27' and the Moon's orbital plane is inclined to the ecliptic by 5'9. The sidereal period of the Earth's rotation is 23.93 h. The sidereal period of revolution of the Moon at present is equal to that of its rotation and amounts to 27.32 mean solar days. As it has already been mentioned earlier, the present rate of tidal energy dissipation in the Earth is well determined, and it corresponds to a phase lag of the tidal bulge of 2.16".Estimations of the energy dissipated in the ocean tides, supplemented with high-precision observations of the phase shifts of the solid earth tides and also the results of model calculations of the tides in anelastic planets, seem to indicate that the phase lags of the solid tides in the Earth at present do not exceed some 1&20 minutes of arc. This implies that at present frictional losses in the Earth's hydrosphere are the major sink of energy. Such a conclusion is consistent with the results of several authors concerning the characteristic time of tidal evolution of the Earth-Moon system. These results suggest that if a constant phase lag of about 2" were extrapolated back in time, then the Moon would have been close to the Earth at a time about 1.5 x lo9 a ago. Such a relatively late event is hardly consistent with both certain terrestrial evidence (the existence of tidal stomatolites of age about 2.5 x lo9 a) and lunar data (there are no signs of volcanic activity on the Moon during the last 3.3 x lo9 a). Thus, the characteristic time of tidal evolution of the Earth-Moon system has a length comparable with the age of this system, only if the phase lag of the Earth's tidal bulge is supposed to be smaller in the past than it is at present value. Considerations of thermal history of the Earth imply that in the course of its structural evolution the temperature within the planet has increased from initially lower values, and this in turn resulted in formation of the oceans on the surface at some later stage of the Earth's history. Therefore, tracing back the orbital evolution of the Moon, it seems to be a reasonable approximation decreases as we go back in time. to suppose that the phase lag Concerning tidal friction in the Moon, Goldreich (1965) has shown that the Moon's rotation could become synchronous only in the case where the eccentricity of its orbit in the past was at least 25%less than the present value. It has been noted above that the tides raised on the Earth by the Moon tend to increase the eccentricity of the lunar orbit, while the radial tides on the Moon cause the eccentricity to decrease. To reduce eccentricity in the past,

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279

the lunar tides should have phase lags ranging from about 10 to some 3". Despite the fact that the lower limit of this interval corresponds well to the phase lag values calculated by ourselves and referred to in the previous section, one may not exclude the possibility of the dependence of the dissipation function on the amplitude or the frequency of the considered deformational process over intervals of time characteristic of secular changes of the lunar orbit. It can be shown that if the sign of the phase shifts of the induced tides remains unchanged during the tidal evolution of the Earth-Moon system, then one always obtains only monotonic variation of the semimajor axis of the Moon's orbit with time. This means that in this case the Earth-Moon distance increases monotonically with time. The possible minimum EarthMoon separation, determined by the Roche limit, could not have been smaller than about 3 earth radii. If the coagulation theory of lunar origin is considered to be the most reasonable, then the minimum distance that would have occurred initially would significantly exceed the Roche limit. This results from the analysis of Goldreich (1966), indicating that the Moon can have its present orbit only if it had formed from a cloud, or swarm, of smaller bodies around the Earth in a zone at 10-30 earth radii distant. In this case the primary tidal gravity field on the Moon could have had initially a magnitude of about 100 times the present value at an Earth-Moon separation of 10 earth radii (or about 10 times the present amplitude at an initial distance of 30 earth radii). We note, however, that in the course of evolution the rate of increase of the Earth-Moon distance depends largely on the actual value of the semimajor axis of the lunar orbit. Thus, the process of tidal evolution of the semimajor axis of the Moon's orbit can be considered as consisting of two stages. The first stage that had started from an unknown initial orbit and completed at some 30 earth radii, is the period of relatively rapid changes. This is followed by the stage of slow variations lasting over almost the whole life of the Moon. Therefore one can conclude that the primary tidal gravity field on the Moon and its tidal deformations over the greater part of the evolution process have been monotonically decreasing from values by about an order of magnitude higher than those at present. Going back in time, the eccentricity of the Moon's orbit probably decreases, or, in other words, the lunar orbit possibly was less eccentric in the past than it now is. Consequently, the range of variation of the tidal deformations of the Moon in one tidal cycle could be significantly less than at present. The inclination of the lunar orbit to the ecliptic remained practically unchanged while the Moon was travelling around the Earth at distances of 30-60 earth radii. The rotational angular velocity of the Moon at present is equal to that of its orbital revolution and this state is stable. The rotational history of the Moon can be traced back in time with large uncertainties, since

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Lunar Gravimetry

the energy and angular momentum of the Moon’s rotation constitute negligibly small parts in those of the combined Earth-Moon system. However, it seems to be probable that in case of any initial rate of rotation (less than that leading to rotational instability) the tidal deceleration of the Moon had been some hundred times more effective than the Earth‘s retardation, provided that the quality factor in the Moon had a magnitude to yield such phase lags of the tides as presented above. Therefore one finds that over practically all geological epochs the rotation of the Moon has been synchronous with its orbital revolution and the lunar tides had features similar to those described in the previous sections. Simultaneously with the process of orbital evolution of the Moon, tidal friction gives rise to changes in both the direction of the Earth’s axis of rotation and its rotational angular velocity. These effects, however, are of less importance as far as the secular changes of the lunar gravity field are considered. We note only that the obliquity of the Earth’s equator to the ecliptic at present has a greater value than it had in the past, but the Earth’s axis of rotation and the axis of the ecliptic have never been collinear. At the minimum probable Earth-Moon distance the length of the day on the Earth could be about 5-6 current hours, but it has never approached the value of 2.6 hours corresponding to rotational instability. Generalizing the above arguments, we conclude that during the period of outward motion of the Moon from a distance of about 30 earth radii to the present Earth-Moon separation, lasting over almost the whole history of the Earth-Moon system, the primary tidal gravity field and the tidal deformations of the Moon have decreased by about an order of magnitude. Since the eccentricity of the Moon’s orbit has not increased significantly up to the present and the rotation of the Moon had already become synchronous at an early stage of its life, the main features of the lunar tides have not undergone major changes during its past history. As to the future evolution of the Earth-Moon system, we note t h d starting from the present configuration the semimajor axis of the Moon’s orbit will increase monotonically in the future. According to Goldreich (1966), the maximum Earth-Moon distance that will have been achieved is about 75 earth radii. The primary tidal gravity field corresponding to this situation will be less than at present by a factor of 2. As the maximum Earth-Moon separation will be approached, the period of the Earth’s rotation becomes 1 month and this synchronous rotation will remain stable for all future times. This dynamical situation will change completely the tidal pattern on the Earth. The features of the lunar tides, however, will remain unchanged.

References

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Index

Acceleration centrifugal, 7 gravity, vector sum, 7, 12-13, 16-17 lunar mass attraction, 7 see also Artificial Lunar Satellites, line-ofsight accelerations Accelerometer, vibrating string, 52, 55-57 see also Gradiometers: Gravimeters Artificial Lunar Satellites ALSEP transmitters, 189-190 VLBI measurements, 181 Apollos, 11 to 14 direct gravity measurements, 54-55 Appollo 12, tracking data, M. Nectaris, 230 Appollo 14, 18, 43 data, lunar gravitational field, 76 tracking data, M. Nectaris, 231 Apollos 14 to 17, gravimetric mapping, 4243 Apollos 15 and 16, Doppler tracking, 71 laser altimetry, 136-140, 182, 187 subsatellite orbits, 31 Apollo 17, direct gravity measurements, 18, 54-58, 225 laser altimetry, 136-140, 187 CACTUS accelerometers, 64-66 Doppler tracking, comparison, satellite gradiometry, 71-73 distance, 36-38 errors, mathematical processing, 19 15 km above Moon, 71-72 satellite velocity changes, 32 systems of co-ordinates, 33 terrestrial tracking station, 33, 36-38 see also Line-of-sight accelerations Explorers 35 and 49, 29 gravitational field, measurements (gradiometry), 59-73

see also Gradiometers laser probing, selenocentric constant, 179 line-of-sight accelerations, determination, 31-40 anomalies, 43-49 Doppler method, 32-38 effect of random measurement errors, 74 global mapping, 59 gravitational potential, 44, 46-49 mascons, 42 normalized autocorrelation function, Apollo 14, 169-170 orbital motion, Moon, 3 8 4 0 perturbed ALS motion, W1 point-mass approximation, Apollo 17, 79 radial potential derivative, 77 time variations, 41 unperturbed ALS motion, 35-38 variations, reasons, 32-34 Lunas 10 to 24 parameters of orbits, 30 Luna 10, 18 long period perturbations, 25 Lunar Orbiter 1, basic parameters, 29 velocity-to-altitude ratio, 182 Lunar Orbiters 1 to 5, 18 gravity determinations, 29 Lunar Orbiters 2 to 5, 176-179 basic parameters, 29 velocity-to-altitude ratio, 181 Lunar Orbiter 5, first map, line-of-sight isolines, 42 parameters of orbits, 31 see also Mascons Lunar roving vehicles, 6 Apollo 17, 131 direct gravity measurements, 18, 55-58

290 Artificial Lunar Satellites-contd, Lunokhod 1 and 2 corner reflectors, 189-190 Mariners 2 to 7, 177-179 motion. in elliptical orbit, 3 9 4 0 perturbed, 4 W 1 orbits, elements of, 19-25 perturbations in motion, orbits, 19-25 derivation of Moon’s geometrical figure, I46 Earth, tidal actions, 30, 31 effect of Sun and planets, 31, 34 Lagrange equations, 22-23, 26 periodic, equations, 24-25 secular, equations, 24-25 sunlight pressure, 29-30 unperturbed motion, 35-36 radar and optical observations, selenocentric constant, 179 Rangers, 177-1 79 Rangers 6 to 9, 177-179 free fall times, 181, 182 retransmission of signals, two satellites, 19 selenocentric constant, determination, 176179 Surveyors I to 7, 177-1 79 see also Moon, centre of mass Astronomical unit (AU), 4

Batholiths, see Mascons Bell Aerospace, MESA gradiometer, 64 Bouguer’s anomaly, 130 Taurus-Littrow valley, 224-225 Brownian noise (thermal fluctuations), minimization, 66-67 Brown’s theory, lunar ephemerides, 246 Bruns formula, 133

Cassini’s laws, rotation of Moon, 198 Cavendish, Henry constant of gravitation, 3, 5, 172 various determinations, 4 Charles Stark Draper Laboratory, gradiometry, 6 0 6 1 Clairant, Alexis: Theory of the Earth’s figure, 122 Corner reflectors laser detection, 181, 189-190

Index

physical libration parameter, 199, 205 Crust, Moon, see Moon, structure

Dicke-Brans, theory of gravitation, 190 Dirichlet problem, 15, 44 Doppler technique, see Artificial Lunar Satellites, Doppler tracking Draconitic month, definition, 244

Earth cubic expansion, effects, 271 equator, obliquity to ecliptic, 278 gravitational field, characteristics, 17 gravitational potential, 164, 171, 172 gravity, mean vertical gradient, 128 irregularity parameter g, 218 rotation, gravitational effect of Moon, 173 precession and nutation, 173-174 sidereal rotation period, 278 solid tides, phase lags, 278 stresses, Moon and, compared, 164-166 tidal deformation, differences, Moon, 24% 24 I effect on artificial satellites, 240 maximum effective radial deformation, 263 tidal friction, 273, 277-280 Earth-Moon distance, 269 direct measurement, 264 increase with time, 279-280 laser measurements, accuracy, 264 moonquakes and, 270 tidal stress amplitudes, 269 variation, 270, 279-280 Earth-Moon system, tidal evolution, 277-280 Eigen frequencies, computed, 257 Einstein, Albert, constant of gravitation, 3 Elasticity parameters, shear and bulk moduli, 272-274 Eotvos, L. equivalence principle, 2 gradiometer, 60, 61 Ephemerides, Moon, 246 Equivalence principle, 2 Eros, minor planet, 175

French national aerospace agency, 6 4 6 6

Index

Gapcynski, selenopotential model, gravitational field, 118 Gauss, constant of gravitation, 3, 4 Geocentric constant of gravitation, 3 Geodesy, tidal corrections, 264 Gradient measurements, see Gradiometers Gradiometers Charles Stark Draper, 60-62 comparison, Doppler tracking, 71, 72 cost-effectiveness, 7 I cryogenic accelerometers, 66 Eotvos, 60 independent measurements, planets, 59, 172 methods of measurement, 68-73 miniature electrostatic accelerometers, 64 modulation, 6 2 4 4 , 69 output, 69-70 resolution, 66 resonance, 68 rotating gravity, 63-64 satellite (Bell Aerospace), 54 SQUID-based magnetometers, 66-67 super CACTUS, 64-66 torsional oscillators, 66-69 triaxial, 66 Gravimeters, 50-51 calibration, 54 gyro, 52 linear, 50, 51 PIPA, 55 rotary, 50, 51 SQUID-based magnetometers, 66 super CACTUS, 65-66 tidal, stable zeropoint, 7 traverse (TG), 55-58 vibrating-string, 52, 55-57 Gravitation, constants of, 3 early definitions, 5 Gravitational field, Moon anomalies, 125-1 31 Bouguer’s anomaly, 130 normal and anomalous fields, 1o(rlOI, 126-127 second gravity potential derivatives, 131 coefficients of expansion, 131-134 Bruns formula, 133 radial gravity potential derivative, 131132 covariance analysis, 147-1 52 anomalous potential, 149-1 52

291

ergodic (random) processes, 148 linear filtration method (KolmogorovWiener), 147-148 degree variances and covariance functions, 153-156 horizontal lunar attraction components, 156-1 57 law of transformation, I54 density irregularity parameters, 21 1-218 density model, 218-224 dynamic figure, 190-200 dynamic flattenings, determinations, 197, 213 harmonic coefficients, 191-196 physical libration parameter, 198-200 equal gravity, derivatives, 122-125 constant radial gravity gradient, 123-124 far side, data, 135, 136, 138 see also Moon, far side figure, normal Moon, 11 1-1 18 maximum values, different reference surfaces, 117 normal gravitational field, 114 triaxial ellipsoid, 1 1 5-1 18 general geometric figure, 186-190 generalized model, five sets of data, AV, 75 BF, 93-97 comparison, F77 and BF, 96 other models, 93-97 correlation matrix, 88-90 degree variance, 94 F77, 73-75, 93-97 F80, 75,93-97 integration, radial derivative, 83-87 least-squares method, expansion, 81-92 orthogonality, spherical functions, 78-81 gravimetric figure and plumb line deflections, 206-209 gravity, expansion to define, 105-108 distribution, normal gravity, 118-122 Mineo’s formula, 121 normal, 119-122, 127 normal gravity formula, 119 Somigliana’s formula, 121 gravity potential derivatives, centrifugaltidal potential, 101 second and third radial derivatives, 103104 spherical functions, 101-105, 109-1 10, 114-115

292 Gravitational field, Moon-contd. gravity variations, and depth, 224-228 harmonic coefficients, 9, 75, 86-87, 90-91, 144-145, 191-196 see also Moon, harmonic coefficients horizontal lunar attraction components, 156-157 degree variances, 156, 161-166 hydrostatic equilibrium figure, 200-206 inequalities, periodic and secular variations, 35 internal structure, 207-21 I mascons, see Mascons models, see generalized models, Gravitational field moments of inertia, principal, 191-198 variations, 214-215 periodic variations, 239-241 long-period variations, 276 perturbing potential, normalized autocorrelation functions, 167-170 physical libration parameter, 198-200, 216 planets, comparison, 171-172 primary tidal gravity field, 239, 279-280 evolution, 241 relief, structure, 134-146 anomalous gravitational field, 140 expansion, first, Goudas, 135 mean density, 143-144 semiaxes, difference, 100 Zond 6, elevations from, 139-140 secondary tidal gravity field, 239 secular variations, 241-242 and tidal function, 276-277 selenocentric constant, determination, 173 180 artificial lunar satellites, 176180 mass and mean density, 172-174 mean values, 179 selenoid, definition problem, 108-1 11 level surface, equation, 101 map, 112 mean radius, 11&111 parameters of triaxial ellipsoid, 117 radius, 117 selenopotential model, Gapcynski, 118 tidal force, 239 tidal potential, 239 phase lag, 240

Index

triaxial level ellipsoid, Mineo’s formula, 121 see also Moon, gravity potential Gravitational fields, Earth/Moon attraction components, 17 characteristics, 17 comparative analysis, 158-170 harmonic coefficients, compared, 158166 normalized autocorrelation functions, 166-170 gravitational potential, degree variances, 164, 171-172 gravity acceleration potential, 17 plumbline deflection components, 17, 117 radial gravity acceleration gradient, 17 Gravity measurements, 49 dynamic methods, 51-54 Moon’s surface, direct, absolute gravity determinations, 53-54 Apollos 11 to 17 spacecraft, 54-55 lunar roving vehicle, 57 self-contained capsules, 57-58 traverse gravimeter, 54-58 vibrating-string accelerometer, 52, 55-56 static methods, 49-51 Gravity potential expansion, harmonic coefficients, 9-13

Halley, Edmund, 2 increase, orbital revolution period, Moon, 24 1 letters from Newton, 2 Hamiltonian operator, 27 Harmonic coefficients, see Gravitational field, Moon, harmonic coefficients: Spherical harmonics Helmert’s approach, normal gravity distribution, 122 Hilbert’s space, 49 Hooke’s modules, stress-strain relationship, 253

Inertia, equivalence principle, 2 INT model, selenopotential, 83-87 see also Gravitational field, Moon, generalized models

Index

Jupiter, irregularity parameter g, 218 Julian century, 246 Kelvin, tides of incompressible planet, solution, 257 Kepler, third law, 176 Keplerian elements, 20-21 Knopoff-Lomnitz body, anelastic modelling, 274-275 Lagrange equations, 22-26 Lame constants, Moon’s crust, 261 Laplace equation, 59 Laser altimetry, 136140 Laser location, Moon, semi major axis, increase rate, 241 Laser probing data centre of figure, Moon, 181 selenocentric constant, 179 Least-squares method, potential expansion, 81-82 selenopotential, model, 87-97 Legendre polynomials, 9, 11, 78, 151 Liapunov’s condition, closed surface, 47 Libration angles of optical libration, 248 latitudinal, 244-245 see also Physical libration parameters Light pressure, see Sunlight pressure Line-of-sight accelerations, see Artificial Lunar Satellites, line-of-sight accelerations Longitude measurements Moon, mean equinox of date, 245 Sun, mean longitude, 245 solar perigee, 245 Love numbers calculated values, 260-261 notation, 262 Lunar tides, see Moon, tidal deformation Lunar vehicles, see Moon, lunar roving vehicles Magnetometers, SQUID-based, 66 Maps equal lunar gravity surfaces, 124-126 equal lunar gravity gradient surfaces, 1 27129

293 geoid elevations, 113 lunar relief, 141 plumbline deflections, along meridians, 208 along parallels, 209 selenoid elevations, 112 weighted mean values, selenopotential radial derivative, 84 Mars gravitational potential, 17 1-1 72 irregularity parameter g, 218 Mascons hypotheses, 21 1, 233-237 line-of-sight accelerations, anomalies, 4243 meteoric impacts, 234-236 parameters, 228-230 point mass bodies, coordinates, 76-77 principal Maria, 229 thermal evolution, 236 see also Moon, Mare Nectaris Meteorites, and mascons, 234-236 Mineo’s formula, normal gravity, triaxial level ellipsoid, 121 Miniature electrostatic accelerometers, 64 Mixed gravity anomalies, 130-131 Models, considered planetary, 260-267 Lunar Model, 261 crust thickness, 261 moonquake occurrence, 269 shear stress profiles, 264-266 tidal displacement profiles, 262-263 tidal stress tensor, component distribution, 267 Modulation gradiometers, see Gradiometers Moon asthenosphere, convection flows and faulting, 271 -lithosphere boundary, 264 central core, parameters, 261 centre of mass, determination, 180-186 direct gravity measurements, 55, 186 Crater Grimaldi, 229 Crater Julius Caeser, 77 cubic expansion, tide-induced, 271 density irregularity, parameter, 201-218 mean, 5, 173, 180 model, 218-224, 260 alternative distributions, 223 radial, distribution, 215, 224-227 dynamic flattenings, 197, 205

294

Moon-contd. elasticity, 239-240, 242 equator-polar differences, 206 equatorial bulge, 205 far side, 135-138 asymmetry, thermal regime, 236-237 lack of mascons, 233, 237 laser altimetry, Apollo, 181 Luna 3, 187 “occultum” mascon, disproof, 233 photogrammetric measurements, 181 general geometric figure, approximations, 187 data from comer reflectors, 189 data from transmitters, 189 determinations, 182, 186-190 mean radii, 187 parameters, ellipsoid, 188 geometric centre, 5 gravitational constants, 3 gravitational fields, characteristics, 17 see also Gravitational field, Moon gravity potential, 13-17 degree variances, 171-1 72 distribution of plumbline deflections, 207-209 in hydrostatic equilibrium, 205 INT model, 86-87 LSM model, 9G91 see also Gravitational field, Moon history and future, 279-280 hydrostatic equilibrium figure, 2W206 longitude measurements, 245 lunar inequality, coefficient, 175 Lunar Model, 261, see Models, considered planetary lunar roving vehicles, 6 Apollo 17, 131 direct gravity measurements, 18, 55, 58 Mare Crisium, 77, 229 profiles of acelerations, 232-233 Mare Frigoris, 77 Mare Humboldtianum, 229 Mare Humorum, 77, 229 Mare Imbrium, 77, 229, 233 Taurus-Littrow valley, 57-58, 224-225 Mare Marginis, 77 Mare Moscoviense, 229 Mare Nectaris, 77, 229, 230-23 1 associated meteorite, 235

Index

Mare Nubium, 77 Mare Orientale, 229 negative gravitational anomalies, 232 residual accelerations, 76 uniqueness, mascons, 43 Mare Serenitatis, 77, 224, 229, 231-232 associated meteorite, 235 Mare Smythii, 77, 229 Maria, crustal structure, 5 table of principal mascons, 77, 229 mascons, see Mascons mass, determination, 3, 5, 179-180 mean motion, period of variation, 277 moments of inertia, 194-197 due to mascons, 229-230 variations, 214-21 5 orbital revolution, 38-39 eccentricity, variations, 270 effect of tidal energy dissipation, 241 semi major axis, 241, 278 sidereal period, 278 origins, 279-280 parallactic inequalities, 175 physical libration parameter, 198-200, 205 pressure variation with depth, 227-228 relief and gravitational field, 134-146 rotation, centrifugal acceleration potential, 12-13 Cassini‘s laws, 198 polar flattening, 203 synchronization with Earth, 236-237 satellites, see Artificial Lunar Satellites seismic measurements, 207, 210 see also Moonquakes selenopotential, models, 73-97 see also Gravitational field Moon, generalized models semi major axis, rate of increase, 241 shear waves, anelastic model, 272-275 estimated values, 275 solid tides, 251-259 equations of motion, 254-255 hydrostatic equilibrium assumption, 252 theory, 251-252 see also Moon, tidal deformations stresses, Earth and, compared, 164-166 structure, 99-101 atmosphere, lack, 99-100 lithospheres, 234-235 asthenosphere boundary, 264, 271

295

Index

cubic expansion, tide-induced, 271 density, 261 fault displacement, 265 Lam6 parameters, 261 Poisson ratio, 261 Shear modules, 261 stresses, 164-166 temperature variations, 221-223 thermal convection, 242 tidal-centrifugal potentials, 7-13 tidal deformation, 252, 259-271 actual radial displacement, 255, 263 description, six functions, 257-259 effective radial deformation, earth, 260 elastic yielding, effects, 260 equations, 255-256 homogeneous structure model, 259 Love numbers, calculated values, 260 notation, 262 perturbations of volume density, 253 theoretical tides, calculation, 263 tidal bulge, 273, 277 tidal strain energy, seismic waves, 265 tidal stress tensor, calculation, 264-265 components, spatial distribution, 267 profiles, LM-261, 266 specific stress components, moonquakes, 27 1 temporal variation, and moonquakes, 269, 270 see also Moon, solid tides: Moon, tidal potential field tidal effects, 241 tidal equations, boundary conditions, at origin, 257-258 deformation, 255-256 general solution, 258-259 of motion, 254-255 tidal friction, 272-273, 278-279 tidal potential field, 10, 12-13, 242-251 analytical description, 242-25 1 Brown’s theory, 246 compared with that of Earth, 245, 250 geosolar tidal potential, variation, 249250 measuring instruments, precision, 251 optical libration, angles, 248 sectorial tides, 244, 245 solid tides, theory, 251-259

surface spherical harmonic functions, 244,246248 terrestrial, on Moon, 11-12 tesseral tides, 244-245 theoretical gravity tide, accuracy, 249 tide-raising potential, 248 variations, contribution of Sun, 9 zenithal distance, mean, 243, 248 zonal tides, 244, 245 tides, of an anelastic Moon, 272-276 Earth and Moon, dissipation of energy, 275 phase retardations, 275 secular changes, 276 velocity, rotational angular, 279 volume, 212 see also Gravitational field, Moon Moon-Earth distance, see Earth-Moon distance Moonquakes Al, A20, occurrence and amplitudes, 269, 270 epicentres, spatial distribution, 268 lower mantle, 261, 264, 271 P waves, 261 prediction, 268 seismic velocity calculations, 21G211, 260267 shallow tectonic, 264 shear stress, maximum zone, 268 tidal origins, 264-265

‘

Newton, Isaac law of universal gravitation, 2 Mathematical Principles of Natural Philosophy, 2 second law, 1, 2 tidal motion, 2 Nutation, Earth’s rotation, 173-174 Nyqyist formula, 67

Occultum mascon, disproof of existence, 233 Oscillations, free, in Earth, 257

Pendulum, principle, 51-52 Perturbations in motion, see Artificial Lunar Satellites, perturbations in motion

296 Phobos, gravitation, planetocentric constants, 171-172 Physical libration parameters, 5, 198-200, 205 Picard, Jean, 2 radius of the Earth, 2 Pioneers 6 to 9, 177-179 Planetary models, considered, tidal deformations, 259-260 Planetisimals, origin, 235 Plumb-line deflections components, 17, 117 determination, 5, 14 variances, 1 5 6 156 vector representation, 210 Point mass bodies Apollos 14 to 17, 78-81 estimation, 92 generalized model, 93 see also Gravitational field, Moon, generalized model Poisson’s integral, 15, 48 gravitational potential, total, 254 solution to Dirichlet problem, 44 sphere, 15 Poisson’s ratio, Moon’s crust, 261 Precession, Earth’s rotation, 173-174 Primary tidal gravity field, see Moon, tidal potential field Q values, shearwaves, 275 Roche limit, Earth-Moon separation, 279 Rotating gravity gradiometer, modulation method. 63-64 Satellites, artificial lunar, see Artificial Lunar Satellites (ALS) Seismic data, see Moonquakes Selenocentric constant of gravitation, 3, 5 determination, 173-175 mean values, 179 using satellites, 175-180 Selenocentric equatorial spherical coordinate system, 10

Index

Somigliana’s formula, level ellipsoid of rotation, 121 Sinus Aestium, 77, 229 Sinus Iridium, 229 Sinus Medii, 77 Spherical harmonics surface harmonics, 255 theorem of addition, 243 tidal stresses, spatial distribution, 266-268 see also Moon, tidal potential field Stokes coefficients, 9 see also Harmonic coefficients Stomatolites, tidal, 278 Stress tensor, see Moon, tidal deformation, tidal stress tensor Sun calculation of mass, 3 longitude measurements, 245 luni-solar tides, 7 mean distance to Earth, 174-175 tidal lunar gravity, contribution to, 9 Sunlight pressure, perturbing effect on ALS motion, 29-30, 40

Taurus-Littrow Valley “free-air” gravity anomalies, 57-58 gravity anomalies, 224-225 Tidal, tides, see Earth; Moon Traverse gravimeter, see Gravimeters Triaxial level ellipsoid, gravity distributions, 121

Vehicles, see Moon, lunar roving vehicles Veneras 4 to 7, 177-179 Venus, gravitational potential, 171-172 Victoria, minor planet, 175 VLBI determinations, accuracy, 264

White noise, 67

Zonal tides, Moon, 244, 245 Zond 6, physical surface elevations, 139-140

International Geophysics Series EDITED BY J. VAN MIEGHEM (July 1959-July 1976) ANTON L. HALES (January 1972-December 1979) WILLIAM L. DONN Lamont-Doherty Geological Observatory Columbia University Palisades, New York Volume 1 BENOGUTENBERG. Physics of the Earth’s Interior. 1959 Volume 2 JOSEPHW. CHAMBERLAIN. Physics of the Aurora and Airglow. 1961 Volume 3 S. K. RUNCORN (ed.). Continental Drift. 1962 Volume 4 C. E. JUNGE.Air Chemistry and Radioactivity. 1963 Volume 5 ROBERTG. FLEAGLE AND JOOST.A. BUSINGER. An Introduction to Atmospheric Physics. 1963 Volume 6 L. DUFOURAND R. DEFAY.Thermodynamics of Clouds. 1963 Volume 7 H. U. ROLL.Physics of the Marine Atmosphere. 1965 Volume 8 RICHARDA. CRAIG. The Upper Atmosphere: Meteorology and Physics. 1965 Volume 9 WILLISL. WEBB.Structure of the Stratosphere and Mesosphere. 1966 Volume 10 MICHELECAPUTO.The Gravity Field of the Earth from Classical and Modern Methods. 1967 Volume 11 S. MATSUSHITAAND WALLACEH. CAMPBELL(eds.). Physics of Geomagnetic Phenomena. (In two volumes.) 1967 Volume 12 K. YA. KONDRATYEV. Radiation in the Atmosphere. 1969 Volume 13 E. PALMENAND C. W. NEWTON.Atmospheric Circulation Systems: Their Structure and Physical Interpretation. 1969 Volume 14 HENRYRISHBETHAND OWEN K. GARRIOTT. Introduction to Ionospheric Physics. 1969 C. S. RAMAGE. Monsoon Meteorology. 1971 JAMESR. HOLTON.An Introduction to Dynamic Meteorology. 1972 K. C. YEHAND C. H.LIU. Theory of Ionospheric Waves. 1972 M. I. BUDYKO. Climate and Life. 1974 MELVIN E. STERN.Ocean Circulation Physics. 1975 J. A. JACOBS. The Earth’s Core. 1975 DAVIDH. MILLER.Water at the Surface of the Earth: An Introduction to Ecosystem Hydrodynamics. 1977 Volume 22 JOSEPHW. CHAMBERLAIN. Theory of Planetary Atmospheres: An Introduction to Their Physics and Chemistry. 1978

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Volume 23 JAMESR. HOLTON.Introduction to Dynamic Meteorology. Second Edition. 1979 Volume 24 ARNETTS. DENNIS.Weather Modification by Cloud Seeding. 1980 Volume 25 ROBERTG. FLEAGLE AND JWST A. BUSINGER. An Introduction to Atmospheric Physics. Second Edition. 1980 Volume 26 KOU-NANLIOU.An Introduction to Atmospheric Radiation. 1980 Volume 27 DAVIDH. MILLER.Energy at the Surface of the Earth. 1981 Volume 28 HELMUTE. LANDSBERG. The Urban Climate. 1981

M. I. BUDYKO. The Earth’s Climate: Past and Future. 1982 ADRIANE. GILL.Atmosphere-Ocean Dynamics. 1982 PAOLO LANZANO. Deformations of an Elastic Earth. 1982 RONALDT. MERRILLAND MICHAELW. MCELHINNY.The Earth’s Magnetic Field: Its History, Origin and Planetary Perspective. 1983 Volume 33 JOHN S. LEWISAND RONALDG . PRINN.Planets and Their Atmospheres: Origin and Evolution. 1983 Volume 34 ROLFMEISSNER. The Continental Crust: A Geophysical Approach. 1986 Volume 35 M. U. SAGITOV, B. BODRI,V. S. NAZARENKO AND Kh. G. TADZHIDINOV. Lunar Gravimetry. 1986

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In preparation Volume 36 J. A. JACOBS. The Earth’s Core. Second Edition. 1986