Principles of the Gravitational Method, Volume 41 (Methods in Geochemistry and Geophysics) (Methods in Geochemistry and Geophysics)

PRINCIPLES OF THE GRAVITATIONAL METHOD METHODS IN GEOCHEMISTRY AND GEOPHYSICS (Volumes 1–28 are out of print) 29. V.P...

Author: Alex A. Kaufman | Richard O. Hansen

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PRINCIPLES OF THE GRAVITATIONAL METHOD

METHODS IN GEOCHEMISTRY AND GEOPHYSICS (Volumes 1–28 are out of print) 29. V.P. Dimri – Deconvolution and Inverse Theory – Application to Geophysical Problems 30. K.-M Strack – Exploration with Deep Transient Electromagnetics 31. M.S. Zhdanov and G.V. Keller – The Geoelectrical Methods in Geophysical Exploration 32. A.A. Kaufman and A.L. Levshin – Acoustic and Elastic Wave Fields in Geophysics, I 33. A.A. Kaufman and P.A. Eaton – The Theory of Inductive Prospecting 34. A.A. Kaufman and P. Hoekstra – Electromagnetic Soundings 35. M.S. Zhdanov and P.E. Wannamaker – Three-Dimensional Electromagnetics 36. M.S. Zhdanov – Geophysical Inverse Theory and Regularization Problems 37. A.A. Kaufman, A.L. Levshin and K.L. Larner – Acoustic and Elastic Wave Fields in Geophysics, II 38. A.A. Kaufman and Yu. A. Dashevsky – Principles of Induction Logging 39. A.A. Kaufman and A.L. Levshin – Acoustic and Elastic Wave Fields in Geophysics, III 40. V.V. Spichak – Electromagnetic Sounding of the Earth’s Interior 41. A.A. Kaufman and R.O. Hansen – Methods in Geochemistry and Geophysics

Methods in Geochemistry and Geophysics, 41

PRINCIPLES OF THE GRAVITATIONAL METHOD

A.A. Kaufman 158 Del Mesa Carmel, Carmel, CA 93923, USA and

R.O. Hansen EDCON-PRJ, Inc., 171 S. Van Gordon St., Suite E, Lakewood, CO 80228, USA

Amsterdam – Boston – Heidelberg – London – New York – Oxford – Paris San Diego – San Francisco – Singapore – Sydney – Tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2008 Copyright r 2008 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://www.elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-52993-0 ISSN: 0076-6895 For information on all Elsevier publications visit our website at books.elsevier.com Printed and bound in The Netherlands 08 09 10 11 12 10 9 8 7 6 5 4 3 2 1

Contents

Introduction

ix

List of Symbols

xi

Chapter 1 Principles of Theory of Attraction 1.1. Newton’s Law of Attraction 1.1.1. Interaction between a particle and an arbitrary body 1.1.2. The gravitational constant, k 1.2. The Field of Attraction and Solution of the Forward Problem 1.2.1. Example 1.3. Different Types of Masses and their Densities 1.4. Two Fundamental Features of theRAttraction Field 1.4.1. Independence of the integral l gdl on the path of integration 1.4.2. Relation between the ﬂux of the attraction ﬁeld and its sources, (masses) 1.5. System of Equations of the Field of Attraction 1.6. Laplace’s and Poisson’s Equations 1.7. The Potential and its Relation to Masses 1.8. Fundamental Solution of Poisson’s and Laplace’s Equations 1.9. Theorem of Uniqueness and Solution of the Forward Problem 1.9.1. The ﬁrst boundary value problem 1.9.2. The second boundary value problem 1.9.3. The third boundary value problem 1.9.4. The fourth boundary value problem 1.10. Green’s Formula and the Relationship Between Potential and Boundary Conditions 1.11. Analytical Upward Continuation of the Field 1.12. Poisson’s Integral 1.13. Behavior of the Attraction Field 1.13.1. The attraction ﬁeld of a spherical mass 1.13.2. The attraction ﬁeld of a thin spherical shell, Fig. 1.12b 1.13.3. Evaluation of a mass of the arbitrary body 1.13.4. The normal component of the attraction ﬁeld due to planar surface masses 1.13.4.1. Case one: A planar surface of an inﬁnite extent 1.13.4.2. Case two: A plane of ﬁnite extension, (Fig.1.13d) 1.13.4.3. Case three: A plane surface has a form of a disk with radius a 1.13.5. Field caused by a volume distribution of masses in a layer with thickness h and density d 1.13.6. Determination of layer density 1.14. Legendre’s Functions and a Solution of Laplace’s Equation 1.14.1. Expansion of the function 1/Lqp in the power series 1.14.2. Laplace’s equation and Legendre’s functions

1 3 4 6 7 9 10 11 13 14 18 20 21 25 28 29 30 32 33 37 40 42 42 46 47 47 49 50 50 51 53 54 55 57

vi

Contents

Chapter 2 Gravitational Field of the Earth 2.1.

Forces Acting on an Elementary Volume of the Rotating Earth and the Gravitational Field 2.1.1. Equation of motion of an elementary volume 2.1.2. The ﬁeld gs of the surface forces in a ﬂuid 2.1.3. The gravitational ﬁeld g 2.1.4. The centrifugal force 2.1.5. Frame of reference rotating with a constant angular velocity (two-dimensional case) 2.1.6. Frame of reference rotating with the constant angular velocity (three-dimensional case) 2.2. Gravitational Field of the Earth 2.2.1. General features of the ﬁeld g on the earth’s surface 2.3. Potential of the Gravitational Field of the Earth 2.3.1. Level surfaces and plumb lines 2.3.2. Curvature of level surfaces and vector lines 2.3.3. Potential and distribution of density of mass 2.3.4. Poincare theorem 2.4. Potential and the Gravitational Field due to an Ellipsoid of Rotation 2.4.1. Formulation of the boundary value problem for the potential U a 2.4.2. The system of coordinates of oblate spheroid 2.4.3. The system of coordinates of an oblate spheroid 2.4.4. Solution of Equation (2.129) by the method of separation of variables 2.4.5. Expressions for the potential 2.4.6. The gravitational ﬁeld due to the rotating ellipsoid 2.4.7. The gravitational ﬁeld on the surface of the ellipsoid, e ¼ e0 2.4.8. Relation between the reduced and geographical latitudes 2.5. Clairaut’s Theorem 2.5.1. The linear approximation 2.6. Potential of the Gravitational Field in Terms of Spherical Harmonics 2.7. Geoid and Leveling 2.7.1. Geoid and quasi-geoid 2.7.2. A height and an elevation 2.7.3. Leveling 2.8. Stokes’s Formula 2.8.1. Bruns’s formula 2.8.2. Boundary condition on the geoid surface 2.8.3. Boundary value problem 2.8.4. Spherical approximation of the boundary condition 2.9. Molodensky’s Boundary Problem 2.9.1. Molodensky’s problem and Bruns’s formula 2.9.2. Boundary condition for the disturbing potential T 2.9.3. The boundary value problem for the function T 2.9.4. Solution of the boundary value problem

59 59 61 64 65 66 70 72 74 75 77 78 82 82 84 85 85 87 90 91 96 97 98 100 102 106 114 116 118 119 120 121 122 123 124 128 129 132 132 134

Contents

2.10. Attraction Field of the Spheroid 2.10.1. Integration of Equation (2.318) 2.10.2. Potential caused by masses of a homogeneous spheroid 2.11. Spheroid and Equilibrium of a Rotating Fluid 2.11.1. Equation of equilibrium and level surfaces 2.11.2. Relationship between density, angular velocity, and spheroid eccentricity 2.11.3. Solution of Equation (2.344) 2.11.4. Oblate spheroid with very small eccentricity 2.11.5. About a stability of a ﬁgure of equilibrium 2.12. Development of the Theory of the Figure of the Earth (Brief Historic Review) 2.12.1. I. Newton, 1643–1727 2.12.2. Ch. Huygens, 1629–1695 2.12.3. C. MacLaurin, 1698–1746

vii 135 139 142 143 143 145 145 147 148 149 149 153 153

Chapter 3 Principles of Measurements of the Gravitational Field 3.1. History of Measurement of the Gravitational Field 3.2. Principles of Ballistic Gravimeter 3.2.1. Equations of motion 3.2.2. Two methods of ﬁeld measurements of the ﬁeld 3.2.3. Non-symmetrical motion 3.2.4. Symmetrical motion 3.3. Pendulum Devices 3.3.1. Small oscillations 3.3.2. Oscillations with an arbitrary amplitude 3.3.3. Physical pendulum 3.3.4. Equation of a motion 3.3.5. Reversing pendulum 3.4. Inﬂuence of Coriolis Force on Particle Motion 3.4.1. Equation of motion 3.4.2. Motion of a free particle 3.4.3. Foucault’s pendulum 3.5. Vertical Spring–mass System 3.5.1. Vertical spring balance and lever spring balance 3.5.2. Vertical spring balance 3.5.3. Mechanical sensitivity of the system 3.5.4. Equation of free vibrations of a mass 3.5.5. Limiting case, n ¼ 0 3.5.6. Three types of attenuation of free vibrations 3.5.6.1. Weak attenuation, nox0 3.5.6.2. Critical attenuation, n ¼ x0 3.5.6.3. Strong attenuation 3.5.7. Forced vibrations 3.5.8. Mechanical sensitivity and stability of vertical spring–mass system 3.5.9. Stability of the vertical spring–mass equilibrium

161 163 163 165 166 167 169 170 172 175 176 178 180 180 183 184 187 188 188 189 190 192 192 193 193 193 193 196 197

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Contents

3.6. Spring with an Initial Compression and Hooke’s Law 3.7. Torsion Spring–mass System 3.7.1. Three types of points of equilibrium 3.7.2. Equation of mass rotation 3.7.2.1. Unstable equilibrium, @s=@b40 3.7.2.2. Stable equilibrium, @s=@b40 3.7.2.3. Indifferent equilibrium, @s=@b ¼ 0 3.7.3. Mechanical sensitivity of the torsion balance 3.8. Lever Spring–mass System 3.8.1. Zero lever spring system 3.8.2. About measurements in the presence of a high-frequency noise 3.9. Measurement of Second Derivatives of the Potential of Gravitational Field 3.9.1. Resultant force and moment 3.9.2. Equation of equilibrium and motion 3.9.3. Second derivatives of the gravitational potential 3.9.4. Equation of equilibrium in the system of coordinates x, Z, z 3.9.4.1. Case one 3.9.4.2. Case two

197 201 202 203 204 204 205 205 206 207 208 210 210 211 213 214 215 215

Chapter 4 Uniqueness and the Solution of the Inverse Problem in Gravity 4.1. Concept of Uniqueness and the Solution of the Inverse Problem 4.1.1. Uniqueness and its application 4.1.2. Example 4.2. Solution of the Inverse Problem and the Inﬂuence of Noise 4.3. Solution of the Forward Problem (A Calculation of the Field of Attraction) 4.3.1. Two-dimensional model 4.3.1.1. Thin two-dimensional layer 4.3.2. Three-dimensional body 4.3.2.1. The ﬁrst approach 4.3.2.2. The second approach

217 221 223 225 229 230 233 234 235 235

Bibliography

237

Appendix

239

Subject Index

243

Introduction The subject of this monograph is physical and mathematical principles of gravitational method. The ﬁrst chapter is devoted entirely to the ﬁeld of attraction caused by masses. Proceeding from Newton’s law of attraction we introduce the concept of this ﬁeld and describe its fundamental features. Special attention is paid to the system of equations of the attraction ﬁeld at regular points and interfaces where a density is discontinuous function. Then after introduction of the potential we perform transition from the system of equations to Poisson’s and Laplace’s equations and discuss their fundamental solutions. Also the theorem of uniqueness and different boundary value problems are studied in detail. As illustration, we consider two examples, which play an important role in the exploration and global geophysics, namely, analytical upward continuation of the ﬁeld and Poisson’s integral. At the end of this chapter there are several examples, which characterize a ﬁeld behavior inside and outside masses. The second chapter considers the gravitational ﬁeld of the earth. At the beginning, we study forces acting on elementary volume of the rotating Earth in an inertial frame. Then, considering the second Newton’s law in non-inertial system, we introduce the centrifugal force and gravitational ﬁeld, as well as its potential. Special attention is paid to the gravitational ﬁeld and potential, caused by an ellipsoid of rotation. This study allows us to describe Clairaut’s theorem. Also, taking into account results derived in Chapter 1, we represent the potential of the gravitational ﬁeld in terms of spherical harmonics. This chapter also discusses such concepts as geoid, Stokes formula, and Molodensky’s boundary problem which play an important role in studying Earth’s ﬁgure. In conclusion, we brieﬂy consider equilibrium of rotating ﬂuid and history of development of the theory of the earth’s ﬁgure. Principles of measurement of the gravitational ﬁeld are discussed in Chapter 3. We describe the theory of ballistic gravimeter, pendulum, and different types of static gravimeters, including such questions as stable and unstable equilibrium, ‘‘zero-length’’ spring and mechanical sensitivity. Also we consider the principles of measuring of the second derivatives of the potential. Finally, in Chapter 4 we focus on a solution of the inverse problems in the gravitational method and discuss uniqueness and non-uniqueness, ill and well-posed problems, stable and unstable parameters, regularization, as well as different methods of a solution of the forward problem.

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Methods in Geochemistry and Geophysics

ACKNOWLEDGMENTS We express our thanks to Dr. M. Brodsky, Dr. I. Fuks, Dr. T. LaFehr, Dr. A. Levshin, Dr. K. Naugolnykh, Dr. T. Niebauer, Dr. L. Ostrovsky, Dr. W. Torge, Dr. J. Wahr for very useful comments and suggestions. We also want to thank Marianna Borukaeva for her technical assistance. Finally, we thank our wives, Irene and Kathleen for their patience and support.

List of Symbols a a A b C d,d0,d* dF do dl dH E e F1 Fr F f f* G G g ga gc gR, gy, gf gx, gy, gz gi, ge ga gt, gn gU gN gs H Hor h h1, h2, h3 I i, j, k iR , iy , if J20 J j ¼ (1)1/2 k Lqp L

major semi-axis of spheroid acceleration moment of inertia minor semi-axis of spheroid moment of inertia functions elementary force elementary solid angle elementary displacement elementary height Young modulus, harmonic function outside a geoid deformation Coriolis force restoring force force ﬂattening of ellipsoid, frequency parameter characterizing a change of gravitational ﬁeld, (gravity ﬂattening) Green’s function normal ﬁeld gravitational ﬁeld attraction ﬁeld centripetal ﬁeld components of the ﬁeld in spherical system of coordinates components of the ﬁeld in Cartesian system of coordinates ﬁelds caused by masses inside and outside of small sphere secondary ﬁeld tangential and normal components of the ﬁeld useful signal noise ﬁeld of surface forces height, mechanical ellipticity orthometric height elevation metric coefﬁcients moment of inertia, mean moment of inertia unit vectors in Cartesian system of coordinates unit vectors in spherical system of coordinates parameter mean curvature imaginary unit gravitational constant distance between points q and p rod length

xii l l0 M MN m N n P p Pn(m) Pn(e), Q(e) Pn(je), Q(je) R R, R1 r0 S S* S(c) t T T(p) t t0 U(p) Uc U0 Uav v W(p) a a0 g l m j e,Z,j s o

Methods in Geochemistry and Geophysics string length reduced length mass, mass of the earth function describing system of coordinates of oblate spheroid ratio of centrifugal force to gravitational one at the equator function describing system of coordinates of oblate spheroid, distance unit vector pressure observation point, pressure Legendre’s function of ﬁrst kind Legendre’s functions of the ﬁrst and second kind with real argument Legendre’s functions with imaginary argument radius of sphere radius vectors radius vector of the center of mass surface, weight of particle small spherical surface function in Stokes formula time period potential caused by irregular part of masses moment, angle pre-tension moment potential of gravitational ﬁeld, potential of normal ﬁeld potential of centrifugal ﬁeld potential of level surface average value of potential linear velocity potential parameter, angle pre-tension angle magnitude of normal gravitational ﬁeld latitude, linear density elastic parameter angle of twist, coordinate in the spherical system of coordinates coordinates in spheroidal system of coordinates surface density, normal stress solid angle, angular frequency

Chapter 1 Principles of Theory of Attraction 1.1. NEWTON’S LAW OF ATTRACTION The phenomenon of attraction of masses is one of the most amazing features of nature, and it plays a fundamental role in the gravitational method. Everything that we are going to derive is based on the fact that each body attracts other. Clearly this indicates that a body generates a force, and this attraction is observed for extremely small particles, as well as very large ones, like planets. It is a universal phenomenon. At the same time, the Newtonian theory of attraction does not attempt to explain the mechanism of transmission of a force from one body to another. In the 17th century Newton discovered this phenomenon, and, moreover, he was able to describe the role of masses and distance between them that allows us to calculate the force of interaction of two particles. To formulate this law of attraction we suppose that particles occupy elementary volumes DV(q) and DV(p), and their position is characterized by points q and p, respectively, see Fig. 1.1a. It is important to emphasize that dimensions of these volumes are much smaller than the distance Lqp between points q and p. This is the most essential feature of elementary volumes or particles, and it explains why the points q and p can be chosen anywhere inside these bodies. Then, in accordance with Newton’s law of attraction the particle around point q acts on the particle around point p with the force dF(p) equal to dFðpÞ ¼ k

DmðqÞDmðpÞ Lqp L3qp

ð1:1Þ

where k is a coefﬁcient of proportionality, called the gravitational constant. In the International System of Units (SI) its value is k ¼ 6:674 1011 m3 =kg s2 Dm(q) and Dm(p) are masses which may have arbitrary values, and they are measured in kilograms. As follows from Newton’s second law, mass is a quantitative measure of inertia, since with an increase of mass the rate of a change of the particle velocity for a ﬁxed force becomes smaller. Also Lqp is the vector: Lqp ¼ Lqp L0qp

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Methods in Geochemistry and Geophysics

(a)

(b) L qp • dm(p)

dm(q)

dm(p)

M

F

dm(q)

•

(c)

(d)

M

m Lqp q

•p

m M

Fig. 1.1. (a) Newton’s law of attraction, (b) illustration of Equation (1.1), (c) ﬁeld caused by an arbitrary mass, (d) Cavendish experiment.

here L0qp is the unit vector directed from the point q to the point p. As was mentioned above, DV(q) and DV(p) are elementary volumes. It is clear that only in this case the force of interaction of the particles does not depend on the position of the points p and q within the volume, and with an increase of the distance Lqp Equation (1.1) gives a more accurate value of the force. Note that dimensions of elementary volumes can change in different problems from very small to extremely large ones. Thus, in accordance with Equation (1.1) the mass Dm(p) is subjected to the force dF(p) which is directly proportional to the product of both masses and inversely proportional to the square of the distance between them, and it has a direction opposite to Lqp, (the presence of minus at the right hand side of Equation (1.1) illustrates this fact). This extremely simple formula describing the basic physical law of the gravimetry may need some comments. 1. Values of masses can be different. 2. Newton’s law of attraction states that the force of interaction of particles is inversely proportional to the square of the distance between them. However, in a general case of arbitrary bodies the behavior of the force as a function of a distance can be completely different. 3. In the SI system of units the distance is measured in meters, mass in kilograms, and the force in Newtons. 4. Equation (1.1) does not contain the physical parameters of the medium where the masses are located, and this means that the force of interaction between two masses is independent of the presence of other masses. For instance, if we place a mass M between masses Dm(q) and Dm(p), Fig. 1.1b, the force caused

Principles of Theory of Attraction

5.

3

by Dm(q) remains the same. Certainly, it is very interesting that the mass M does not inﬂuence the transmission of the force from the particle q to that near point p. Thus, the medium surrounding particles does not have any inﬂuence on the force of interaction between them. By analogy with Equation (1.1) the force acting on mass Dm(q) caused by the mass Dm(p) is dFðqÞ ¼ k

DmðqÞDmðpÞ Lpq L3qp

Inasmuch as L3qp ¼ L3pq and Lpq ¼ Lqp, we see the validity of the Newton’s third law: dFðqÞ ¼ dFðpÞ Note that the notation dF indicates that we deal with the force caused by elementary masses. Since the gravitational constant is extremely small, it is natural to expect that the force of interaction is also very small too. For illustration, consider two spheres with radius 1 m and mass 31.4 103 kg made from galena, with the distance between their centers 10 m. Then, the force of interaction, Equation (1.1), is F ¼ 6:6 104 N Indeed, it is a very weak force. 1.1.1. Interaction between a particle and an arbitrary body In order to determine the force with which an arbitrary body acts on a particle located around the point p, we mentally divide the volume of the body into many elementary volumes, so their dimensions are much smaller than the corresponding distance from the particle p. It is clear that the magnitude and direction of each force depends on the position of the point q inside a body. Now, applying the principle of superposition, we can ﬁnd the total force acting on the particle p. Summation of elementary forces gives: FðpÞ ¼ k

X DmðqÞ n¼1

L3qp

Lqp DmðpÞ

In the limit when elementary volumes tend to zero we have Z dðqÞdV Lqp FðpÞ ¼ kDmðpÞ L3qp V

ð1:2Þ

ð1:3Þ

Here d(q) is the volume density of mass: dðqÞ ¼ lim

DmðqÞ DV

ð1:4Þ

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Methods in Geochemistry and Geophysics

if DV ! 0 or dm ¼ ddV

ð1:5Þ

3

The dimension of the density is kg/m . It is appropriate to note that to carry out numerical integration the elementary volumes have to satisfy two conditions, namely, a. Their extensions are much smaller than the distance to the observation point p. b. The masses are practically uniformly distributed within each elementary volume. Thus, integration over an arbitrary volume allows us to ﬁnd the force caused by any distribution of masses. It is essential that the particle p can be located either outside or inside of a body and at any distance from its surface. Equation (1.3) describes the total force that is a result of a superposition of the elementary forces, vectors, at the same point. Correspondingly, this force can cause a translation of the particle only. It is also instructive to consider the force F generated by the particle and acting on an arbitrary body. Each elementary volume is subjected to the force dFðqÞ ¼ k

DmðqÞDmðpÞ Lpq L3qp

and, therefore, the total force has the same magnitude as F(p), but the opposite direction. In other words, we again observe the Newton’s third law to be valid. At the same time, it may be appropriate to comment: inasmuch as the elementary forces dF(q) are applied at different points of a body, their action usually causes both a translation and a rotation. Of course, the motion of a particle and a body can drastically differ from each other, and one vivid example is the system consisting the earth and a particle. Applying the same approach it is a simple to ﬁnd an expression for the force of interaction between two arbitrary bodies. It is obvious that the force acting on any elementary volume of a body is the sum of the forces due to other body and the force caused by different elements of the same body. In particular, the resulting force due to body 1 acting on body 2 is Z Z dm2 dm1 F ¼ k L12 L312 V2 V1 Thus, we have derived a generalization of Newton’s law of attraction. 1.1.2. The gravitational constant, k The coefﬁcient k is one of the fundamental constants of physics and astronomy. From Kepler’s laws it is possible to express the masses of all planets in terms of the mass of the sun. However, in order to ﬁnd the mass of the sun and,

Principles of Theory of Attraction

5

correspondingly, masses of other planets we have to know the gravitational constant. In general, its knowledge allows us to calculate the force of interaction between two arbitrary bodies. The numerical value of the gravitational constant depends on the system of units in which force, mass, and distance are deﬁned. Its magnitude is found experimentally by measuring the force of the gravitational attraction between two bodies of known masses, located at a given distance. For bodies of a moderate size this force is extremely small, and for this reason the value of this constant remained unknown for more than two hundred years. Finally, this task was solved by Cavendish in 1792 using a torsion balance. Earlier the same approach was applied by Coulomb to study forces of electrical attraction and repulsion. In principle, the Cavendish balance consists of two small spheres of the same mass m, Fig. 1.1d, mounted at opposite ends of a light horizontal rod, which is suspended at its center by a thin vertical ﬁber (quartz thread). A small mirror mounted on the ﬁber reﬂects a beam of light onto a scale. In order to use the balance two relatively large spheres of mass M, usually made of lead, are brought close to masses m. The forces of gravitational attraction between the large and small spheres form a couple, which twists the ﬁber, as well as mirror. Correspondingly, the light beam moves along a scale. At equilibrium the elastic force of the ﬁber compensates the external one, and we have FL ¼ mj Here F is the force of interaction between masses m and M, L the rod length, m the elastic parameter of the ﬁber, and j the angle of the twist. By using a very ﬁne ﬁber the deﬂection of the light beam may be sufﬁciently large so that the gravitational forces, depending on the unknown constant, k, can be measured quite accurately. Of course, this requires the reduction of different types of noise, such as the inﬂuence of charges on the surfaces of the masses. Note, that the expression for F implies that the spherical masses interact as particles and this assumption will be proved later. To appreciate this experiment we compute the force of the gravitational attraction between the large and small spheres in the Cavendish balance when m ¼ 103 kg and M ¼ 0.5 kg and the distance between their centers is 0.05 m. Then F¼

6:67 1011 103 0:5 ¼ 0:13 1010 N 25 104

Certainly, this is an extremely small value and is the reason why the determination of the gravitational constant with very high accuracy is a rather complicated experiment. During the last two hundred years there were many measurements of this constant, but still only three digits after decimal point are reliable. One can say that due to Cavendish’s measurements it became possible to develop the theory of gravity and evaluate mass of the earth. In fact, determination of this mass was the main goal of this experiment.

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Methods in Geochemistry and Geophysics

1.2. THE FIELD OF ATTRACTION AND SOLUTION OF THE FORWARD PROBLEM In accordance with Equation (1.3) the force acting on the particle around the point p is directly proportional to the mass, Dm(p). Now let us imagine that this mass decreases. Then, the force dF(p) decreases too, but the ratio dFðpÞ dm remains the same and it characterizes a quantity g, which is called the ﬁeld of attraction at the point p: Z dðqÞLqp gðpÞ ¼ k dV ð1:6Þ L3qp V Assuming that the distribution of masses inside the volume V is given, this vector function g(p) depends only on the coordinates of the observation point p, and by deﬁnition it is a ﬁeld. It is appropriate to treat the masses in the volume V as sources of the ﬁeld g(p). In other words, these masses generate the ﬁeld at any point of the space, and this ﬁeld may be supposed to exist whether a mass is present or absent at this point. When we place an elementary mass at some point p, it becomes subject to a force equal to FðpÞ ¼ DmðpÞgðpÞ

ð1:7Þ

that causes motion of the mass. In essence, all measurements in the gravitational method are based on the use of Equation (1.7). One can say that at each point the attraction ﬁeld is ‘‘waiting’’ for a mass to create a force and cause a motion. The function g(p) can be also interpreted in a completely different way. As follows from Newton’s second law, g(p) represents the acceleration of an elementary mass, caused by the force F(p). We will mainly apply a concept of a ﬁeld, but sometimes both approaches will be used. Inasmuch as the interaction of masses usually does not inﬂuence their density, it can be speciﬁed. In other words, the density of each material is a known parameter, and this is a very important fact, because it means that Equation (1.6) allows us to solve the forward problem by integration. However, in general it is not true and, for instance, a rotation of a compressible ﬂuid causes a change of its density. In other geophysical methods generators of the ﬁeld cannot be usually determined before the calculation of the ﬁeld, and for this reason solution of the forward problem requires solution of a boundary value problem. Certainly, the remarkable simplicity of the solution of the forward problem in the ﬁeld of attraction is exceptional. Equation (1.6) can be applied at any point of space, including a volume where masses are located. In such a case the distance Lqp may tend to zero, and this fact naturally causes a suspicion that the use of Equation (1.6) is invalid in these places. However, as was already pointed out and will be proved later, the singularity of the integrand is removable and Equation (1.6) gives the correct value of the attraction ﬁeld inside a mass. It is clear that with an increase of separation between the observation point

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Principles of Theory of Attraction

p and a body, the vector Lqp becomes practically independent of the point q. Correspondingly, in the limit when Lqp tends to inﬁnity, it can be taken out of the integral in Equation (1.6) and we obtain R dðqÞdV m gðpÞ ¼ k V 3 Lqp ¼ k 3 Lqp ð1:8Þ Lqp Lqp where q is an arbitrary point of a body and m its mass. Thus, regardless of the dimensions and shape of a body, it generates at relatively large distances practically the same attraction ﬁeld as that of an elementary particle. 1.2.1. Example To illustrate Equation (1.8), consider a solution of the forward and inverse problems in the simplest possible case, when the ﬁeld is caused by an elementary mass. Suppose that a particle with mass m(q) is situated at the origin of a Cartesian system of coordinates, Fig. 1.2a, and the ﬁeld is observed on the plane z ¼ h. Then, as follows from Equation (1.8), the components of the attraction ﬁeld at the point p(x,y,h) are gx ðx; y; hÞ ¼ k

mx my mh gy ðx; y; hÞ ¼ k 3 gz ðx; y; hÞ ¼ k 3 3 Lqp Lqp Lqp

ð1:9Þ

because Lqp ¼ xi þ yj þ hk (a)

(b)

z gx x h 0 m

gz

x (c) gx

x

Fig. 1.2. (a) Elementary mass beneath an observation plane, (b) behavior of the horizontal and vertical components of the ﬁeld, (c) inﬂuence of mass position.

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Methods in Geochemistry and Geophysics

Here i, j, k are unit vectors along the coordinate axes. Thus, the solution of the forward problem is trivial. In particular, along the line where y ¼ 0 and z ¼ h we have: gx ðx; 0; hÞ ¼ k

mx ðx2 þ h2 Þ3=2

gy ¼ 0 gz ðx; 0; hÞ ¼ k

mh ðx2 þ h2 Þ3=2

ð1:10Þ

The behavior of both components as functions of x is shown in Fig. 1.2b, and these can be easily explained. First of all, the horizontal component vanishes above the body, since the ﬁeld is directed toward the body. Also gx(x) has two extremes, and their position is deﬁned from the condition: @gx ¼0 @x Taking a derivative we ﬁnd: h xe ¼ pﬃﬃﬃ 2

ð1:11Þ

and with decrease of the depth h of the particle an extreme is observed at smaller values of x, but its magnitude increases, Fig. 1.2c. For negative values of x the horizontal component, gx, is positive, and this is understandable because the ﬁeld is directed toward the mass and it forms an angle with the x-axis which is smaller than 901. On the contrary, if x40 this angle exceeds 901 and the component is negative. It is clear that the vertical component jgz j has a maximum at x ¼ 0 and gradually decreases with the distance x. At relatively large distances we have: gx4gz, which follows from the fact that the ﬁeld becomes almost horizontal. Note, that at such distances the vertical component decreases more rapidly than L2 qp . The solution of the inverse problem is very simple too. For instance, if we know only the component gx then its zero value determines the position of the body (x-coordinate). At the same time, the depth h is deﬁned from Equation (1.11). It can be calculated in different ways; taking the ratio of this component at two points, we obtain the equation with respect to h: gx ðx1 Þ ðx22 þ h2 Þ3=2 x1 ¼ gx ðx2 Þ ðx21 þ h2 Þx2

ð1:12Þ

Repeating these calculations with different pairs of gx(x) we may increase the accuracy of the evaluation of h. Next, making use of the value of this component at any point, the mass m is evaluated. In the case when only the vertical component is known, the determination of the position of mass and its value is similar. Here it is appropriate to notice the following. Inasmuch as an arbitrary body, located at a large distance from an observation point p, creates a ﬁeld, known always with some error, often it cannot be practically distinguished from that of an elementary particle, and for this reason we are able to determine only the product of volume and density, mass, but each of them remains unknown. It is the ﬁrst illustration of the fact that the solution of the inverse problem in gravity, as well as in other geophysical methods, is an ill-posed one, because some parameters of a body

Principles of Theory of Attraction

9

cannot be deﬁned. In Chapter 4 we will discuss this subject in some details, including such questions as uniqueness and non-uniqueness, the well- and ill-posed problems and regularization.

1.3. DIFFERENT TYPES OF MASSES AND THEIR DENSITIES As was pointed out earlier, Equation (1.6) allows us to ﬁnd the attraction ﬁeld everywhere, but it requires a volume integration, that in general is a rather cumbersome procedure. Fortunately, in many cases the calculation of the ﬁeld g(p) can be greatly simpliﬁed. First, consider an elementary mass with density d(q), located in the volume DV. Now let us start to increase the density and decrease the volume in such a way that the mass remains the same. By deﬁnition, these changes do not make a noticeable inﬂuence on the ﬁeld because the observation point p is far away. In the limit, when DV ! 0

and

d!1

we arrive at the mathematical concept of the point mass, m(q). Correspondingly, its attraction ﬁeld is deﬁned by Equation (1.8). Of course, it is impossible to place a mass into a point, and a point mass is only an approximate representation of a real body. Suppose that an elementary volume is a cube with sides of length h. Therefore, the transition to the point mass leads to an increase of the volume density as h3. Next, we assume that the volume V(q) has the shape of a rod, Fig. 1.3a, with cross section S ¼ h1h2 and distance to the observation points satisfying conditions: Lqp h1

and

Lqp h2

Then, we can reduce each elementary volume dV ¼ h1h2dl and increase the volume density so that the elementary mass: dm ¼ dh1 h2 dl does not change. Here dl is an elementary displacement along the rod. In the limit, when the cross section S becomes a point of the line l, we obtain a linear (a)

(b)

dS q dl h(q) h2

h1

Fig. 1.3. (a) Linear distribution of masses, (b) surface distribution of masses.

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Methods in Geochemistry and Geophysics

distribution of masses with linear density l: lðqÞ ¼ dðqÞh1 h2 where dðqÞ ! 1

and

h1 ! 0; h2 ! 0 2

It is clear that d(q) tends to inﬁnity as h , and we deal with the second mathematical concept of a mass. Respectively, the expression for the attraction ﬁeld becomes Z lðqÞ gðpÞ ¼ k Lqp dl ð1:13Þ 3 l Lqp and the volume integration is reduced to a linear one that is much simpler. Finally, suppose that masses are situated in a relatively thin layer, so that its thickness h(q) is much smaller than the distance Lqp, Fig. 1.3b. Let us consider an elementary volume dV ¼ h(q)dS. Its mass is dm ¼ dðqÞhðqÞdS. Now, reducing the layer thickness and preserving the mass dm, we obtain in the limit a surface mass with the density s(q): sðqÞ ¼ dðqÞhðqÞ

ð1:14Þ

In this case the volume density d(q) tends to inﬁnity as h1, when h(q) approaches zero. Correspondingly, the expression for the ﬁeld is Z sðqÞ gðpÞ ¼ k Lqp dS 3 S Lqp and the volume integration is replaced by the surface one. Thus, the ﬁeld of attraction caused by volume, surface, linear, and point masses is "Z # Z Z X mðqÞ dðqÞ sðqÞ lðqÞ gðpÞ ¼ k Lqp dV þ Lqp dS þ Lqp dl þ Lqp ð1:15Þ 3 3 3 L3qp V Lqp S Lqp l Lqp We have found that the volume density tends to inﬁnity differently near point, linear, and surface masses, and this fact inﬂuences the ﬁeld behavior in the vicinity of such places. Of course, Equation (1.6) always allows us to calculate the ﬁeld of attraction g. At the same time, in many cases the use of Equation (1.15) greatly simpliﬁes this procedure.

1.4. TWO FUNDAMENTAL FEATURES OF THE ATTRACTION FIELD Inasmuch as Equation (1.6) allows us to solve the forward problem for any distribution of masses, we may say in this sense that the theory of the gravitational method is completely developed. However, in order to understand better the behavior of the ﬁeld of the earth and sometimes to improve the quality of the

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Principles of Theory of Attraction

solution of the inverse problem, it is useful to study general features of the attraction ﬁeld and introduce also the concept of potential. 1.4.1. Independence of the integral

R l

gdl on the path of integration

First, consider the ﬁeld of an elementary mass and evaluate the integral along the straight line shown in Fig. 1.4a. This gives Z p2 Z p2 Lqp dl dLqp 1 1 km ¼ km ¼ km ð1:16Þ 2 Lqp2 Lqp1 L3qp p1 p1 Lqp since Lqp dl ¼ Lqp dl

and

dl ¼ dLqp

It is clear that the integral along an arc of the circle with radius Lqp is equal to zero, because the ﬁeld g(p) is perpendicular to the path of integration. For this reason, as before, the integral along the path shown in Fig. 1.4b is deﬁned by the interval along the straight line. Next consider a more complicated path, shown in Fig. 1.4c. Integration between points p1and p3 gives: Z p3 1 1 1 1 gdl ¼ km þ Lqp3 Lqp2 Lqp2 Lqp1 p1 or Z

p3 p1

1 1 gdl ¼ km Lqp3 Lqp1

ð1:17Þ

Comparison of Equation (1.16) to Equation (1.17) shows that in both cases the value of the integrals is deﬁned by the distance from the particle to terminal points (a)

(b)

Lqp dl p1

q

p2

p3

dl p2

m(q) p1

q

p3

(d)

(c)

(e) p2

p2

d ln

p1 p1

q Fig. 1.4. (a–e) Different paths of integration of ﬁeld g.

d l2n

d l1n

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Methods in Geochemistry and Geophysics

of the path. Now we generalize this result for an arbitrary path, Fig. 1.4d. Each elementary displacement can be represented as a sum of two perpendicular vectors, Fig. 1.4e: dln ¼ dl1n þ dl2n Here dl1n is the displacement along the radius vector, drawn from the point q, but dl2n is the element of the arc of the radius Lqp. Then we have: gðpÞdln ¼ gðpÞdl1n and the integral along the elementary displacement of the path dln can be written as: Z

pnþ1

gdln ¼ km pn

1

Lqpnþ1

1 Lqpn

ð1:18Þ

Taking into account the fact that the interval dl is very small, the integral is equal to gdl1n and the right hand side of Equation (1.18) describes this product. Performing a summation of these integrals along the path with terminal points p1 and p2, we obtain: Z

p2

p1

1 1 gðpÞdl ¼ km Lqp2 Lqp1

ð1:19Þ

As in the ﬁrst case, the integral is deﬁned by the position of the terminal points alone, while the shape and length of the path do not have any inﬂuence on its value. We have demonstrated that in the case of a single elementary mass, the integral in Equation (1.19) is path independent. Now suppose that there is an arbitrary distribution of masses. Applying the principle of superposition we conclude that the same behavior holds in the general case when the ﬁeld is caused by any distribution of masses. This result can be formulated differently, namely, that the work performed by the attraction ﬁeld between two points does not depend on the path of integration. Certainly, this is an amazing fact which would be difﬁcult to predict. Indeed, changing the path connecting two given points we deal with different values of the ﬁeld, different orientation of the displacement, length of the path, and the dot products of vectors g and dl, but the value of the integral remains the same. It seems as if there is communication between the ﬁeld g at different points of the path, so that they control the behavior of each other and produce the same value of the sum of dot products. In order to emphasize this feature of the ﬁeld, let us imagine the following experiment. Suppose that the distance between points a and b is 1 m. The ﬁrst path is a straight line connecting these points. Performing an integration of the dot product gdl, we obtain some value of the integral. The second path is completely different, and it goes through all mountains of the earth, as well as oceans and ﬁnally returns to the point b. During this journey we measure the attraction ﬁeld at each point of such path and form the product gdl, but its summation gives exactly the same result as integration along the short straight line!

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Principles of Theory of Attraction

(a) p

(b) dS1

dS

g(p)

dS2

Lqp

Lqp

dS3

q m(q) z (c)

gθ

(d) gθ gθ

gR

θ p

gθ a gR

0

(e)

R Fig. 1.5. (a) Flux through elementary surface, (b) ﬂux as a sum of elementary ﬂuxes, (c) ﬁeld components due to spherical mass, (d) symmetry of ﬁeld, (e) the ﬁeld inside and outside spherical mass.

1.4.2. Relation between the flux of the attraction field and its sources, (masses) Now we establish the second remarkable feature of the attraction ﬁeld. As before, at the beginning consider the ﬁeld of a point mass, m(q). By deﬁnition, the ﬂux of the ﬁeld through an elementary surface dS, Fig. 1.5a, is gðpÞdS ¼ k

m Lqp dS L3qp

ð1:20Þ

As is known, Lqp dS ¼ do L3qp

ð1:21Þ

is the solid angle, which the surface dS subtends at the point q. Thus, in this case the ﬂux can be represented as: gðpÞdS ¼ kmðqÞdoðqÞ

ð1:22Þ

Note, that the ﬂux of the attraction ﬁeld is a purely mathematical concept; it does not have any physical meaning. Next, we evaluate the ﬂux through the surface S,

14

Methods in Geochemistry and Geophysics

Fig. 1.5b. Performing the integration, we obtain Z gðpÞdS ¼ kmðqÞoðqÞ

ð1:23Þ

where o(q) is the solid angle which the surface S subtends at the point q. It can be positive, negative, or zero. Now consider the most interesting case, when the surface S is closed. Then, as is well known, the solid angle is equal to 4p if the point q is located inside the volume V, enclosed by the surface S, and it is zero, when q is outside the volume. Correspondingly, Equation (1.23) can be written as: I gðpÞdS ¼ 0 ð1:24Þ S

if the mass is located outside the volume V, and I gðpÞdS ¼ 4pkmðqÞ

ð1:25Þ

if it is somewhere inside V, (p is a point of the closed surface S). In accordance with Equation (1.24) the ﬂux caused by the mass located outside V is zero regardless of a position of the mass m(q). Certainly, this is a remarkable fact. Indeed, at each point of the surface this mass creates a ﬁeld and the dot product gdS in general differs from zero. However, the sum of these elementary ﬂuxes over a closed surface is zero. Moreover, a change of the surface leads to a change of elementary ﬂuxes, but their sum, integral, vanishes as long as the closed surface does not intersect the mass. Thus, the ﬂux is insensitive to mass located outside the volume V. At the same time, as follows from Equation (1.25), the ﬂux through a closed surface deﬁnes the amount of mass inside V. It is essential that the ﬂux remains the same, regardless of the location of this elementary mass and a shape and position of the closed surface surrounding it. Applying now again the principle of superposition, we obtain for an arbitrary distribution of masses: I gdS ¼ 4pkM ð1:26Þ S

where M is the total mass situated inside the volume V that is enclosed by the surface S. This equation describes the second fundamental feature of the attraction ﬁeld, and it can be also treated as ‘‘the bridge’’ between sources (masses) and the ﬁeld. It may be appropriate to notice that the similar relationship holds for any ﬁeld and is often called Gauss’s formula.

1.5. SYSTEM OF EQUATIONS OF THE FIELD OF ATTRACTION In the previous section we established two fundamental features of the attraction ﬁeld, and both of them follow from Newton’s law of attraction and the principle of

15

Principles of Theory of Attraction (b)

(a) g b

L1

g

a

dl

g L g

L2

g

(c)

(d) g2

n

g2

n2

n

t

n1

g1

g1

Fig. 1.6. (a, b) Illustration of Equation (1.28), (c) tangential component of the ﬁeld at interface, (d) normal component of the ﬁeld at the interface.

superposition. The ﬁrst one, independence of the work of the attraction ﬁeld on a path between two points, is written as Z Z gdl ¼ gdl ð1:27Þ L1

L2

Here L1 and L2 are two arbitrary paths, connecting the same terminal points, Fig. 1.6a. Changing the direction of one of the paths, for example, L2, we have Z Z Z Z gdl ¼ gdl or gdl þ gdl ¼ 0 ð1:28Þ L1

L2

L1

L2

Since these paths form a closed path L, Equation (1.27) can be rewritten as I g dl ¼ 0 ð1:29Þ that is, the circulation of the attraction ﬁeld is always equal to zero. This means that vector lines of this ﬁeld are always open, and their terminal points are either located inside the masses or at inﬁnity. The second fundamental feature of the ﬁeld g is described by Equation (1.26), and it states that the ﬂux of this ﬁeld through any closed surface characterizes amount of mass in the volume V enclosed by this surface. Both Equations (1.26 and 1.29) represent the system of equations of the attraction ﬁeld in the integral form: I I gdl ¼ 0 and gdS ¼ 4pkm ð1:30Þ l

S

Here m is the total mass inside V. Inasmuch as this system does not contain derivatives of the ﬁeld, it is valid at any point, including boundaries of media with

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Methods in Geochemistry and Geophysics

different densities. These equations can be treated as relationships between values of the ﬁeld g at different points, located at any distance from each other, and, correspondingly, they are integral equations with respect to the ﬁeld g. Next, we obtain the system of ﬁeld equations in differential form, that is, relationships between values of the ﬁeld and the density of masses in the vicinity of the same point. First, assume that this point is regular, and, therefore, the ﬁeld has ﬁrst derivatives in that point. Taking into account the deﬁnition of curl and divergence, we arrive at the system of equations at regular points: curl g ¼ 0 and

div g ¼ 4pkd

ð1:31Þ

Here d is the volume density at a point. For instance, at points where masses are absent div g ¼ 0. Let us discuss the physical and mathematical content of these equations. The ﬁrst one clearly shows that the attraction ﬁeld does not have vortices and, correspondingly, the work done by this ﬁeld is path independent. In other words, the circulation of the ﬁeld is equal to zero. At the same time, the second equation demonstrates that the ﬁeld g is caused by sources (masses) only. As illustration, consider the set of these equations in the Cartesian system of coordinates: i j k @gx @gy @gz @ @ @ þ þ ¼ 4pkd ð1:32Þ @x @y @z ¼ 0; @x @y @z gx gy g z Respectively, in place of the ﬁrst equation we have @gz @gy ¼ ; @y @z

@gx @gz ¼ ; @z @x

@gx @gy ¼ @y @x

ð1:33Þ

Thus, we deal with a system of four differential equations of the ﬁrst order and, in general, there are four unknown functions: gx, gy, gz, and d. If the distribution of masses is known, then we do not need to use the set (1.31) to ﬁnd the ﬁeld. In a fact, this task is solved by integration, using Equation (1.6): Z Lqp gðpÞ ¼ k dðqÞ 3 dV ð1:34Þ Lqp As follows from Equation (1.33) there are relationships between different components of the ﬁeld, and they indicate that each component of the ﬁeld contains the same information about a distribution of masses. Note, that Equation (1.33) directly follows from Equation (1.34). By deﬁnition, we have: Z dðqÞðxp xq Þ gx ðpÞ ¼ k dV L3qp V Z

dðqÞðyp yq Þ

gy ðpÞ ¼ k V

L3qp

dV

Principles of Theory of Attraction

Z gz ðpÞ ¼ k

17

dðqÞðzp zq Þ dV L3qp V

Taking the corresponding derivatives, we again arrive at Equation (1.33). The system (1.31) is written for points where the density d is deﬁned. However, there are exceptions; for instance, an interface between media with different densities, Fig. 1.6c, since in such places the density of masses is a discontinuous function. Now, making use of Equation (1.30), it is easy to derive a surface analogy of Equation (1.31). Let us calculate the circulation along the path shown in Fig. 1.6c. From Equation (1.29) it follows: g2 dl2 þ g1 dl1 ¼ 0

ð1:35Þ

since displacements dl are small, the integrals can be replaced by dot product of the ﬁeld and the displacement, while the integrals along the path h, perpendicular to the surface, vanish when h tends to zero. Taking into account that dl2 ¼ dl1 we obtain g2t dt g1t dt ¼ 0 or g1t ¼ g2t

ð1:36Þ

where gt is the tangential component of the ﬁeld. Equation (1.36) is the surface analogy of the ﬁrst equation of the attraction ﬁeld, and it shows that the tangential component of g is a continuous function at the boundary between media with different densities. Next, imagine an elementary cylinder around some point q of this surface, Fig. 1.6d. Then, applying Equation (1.26), we have Z g2 dS2 þ g1 dS1 þ gdS ¼ 4pkm Sl

where dS2 ¼ dSn

and

dS1 ¼ dSn

but S1 is the lateral surface of the cylinder, n the unit vector directed from the back to front side of the surface. In the limit, when the cylinder height tends to zero, mass m also vanishes, and it gives g2n ¼ g1n

ð1:37Þ

The latter is the surface analogy of the second equation of the ﬁeld. Thus, both components are continuous functions at an interface, Fig. 1.6d. As a summary, it is useful to illustrate a transition from Newton’s law of attraction to the system of equations in the form of diagram. The arrows show that all equations are derived from Newton’s law of attraction, and they do not contain more information than this law. At the same time, the system (1.38) allows us to

18

Methods in Geochemistry and Geophysics

understand better the general features of the ﬁeld and often leads to some simpliﬁcation in solving of the forward and inverse problems of the gravitational method.

1.6. LAPLACE’S AND POISSON’S EQUATIONS As was pointed out, the set (1.31) is a rather complicated system of differential equations, and it is natural to attempt to replace them by much simpler equations. This is the ﬁrst motivation for introducing the potential of the attraction ﬁeld. Taking into account the equality curl grad U ¼ 0 Newton’s law of attraction

g( p) = − k ∫

(q)

V

3

L qp

Lqp dV

∫ gdl = 0

ð1:38Þ ∫ g d S = − 4π km

l

S

g1t = g2t

g1n = g2n

the ﬁrst equation of the set (1.31) gives gðpÞ ¼ grad U

ð1:39Þ

Thus, we have expressed the vector ﬁeld g in terms of a scalar function, U(p), by a relatively simple operation, gðpÞ ¼

1 @U 1 @U 1 @U i1 þ i2 þ i3 h1 @x1 h2 @x2 h3 @x3

ð1:40Þ

where h1, h2, and h3 are metric coefﬁcients; x1, x2, and x3 are coordinates of the observation point; and i1, i2, and i3 are unit vectors of the coordinate system. For instance, in the Cartesian system of coordinates h1 ¼ h2 ¼ h3 ¼ 1, and the unit vectors are i, j, k. Correspondingly, grad U ¼

@U @U @U iþ jþ k @x @y @z

It is clear that Equation (1.39) deﬁnes the potential U up to a constant, since grad C ¼ 0, that is, an inﬁnite number of potentials describe the same ﬁeld g. For this reason, it is appropriate to consider the potential as an auxiliary function introduced with mainly one purpose, namely, to simplify the analysis of the more complicated ﬁeld g. Our next step is obvious: we have to ﬁnd an equation,

Principles of Theory of Attraction

19

describing the behavior of U. In order to introduce the potential we already used the ﬁrst equation, curl g ¼ 0. Now, substituting Equation (1.39) into the second equation of the system (1.31), we obtain: div grad U ¼ 4pkd

or

r2 U ¼ 4pkd

ð1:41Þ

Thus, we have obtained Poisson’s equation, and, as is well known, Equation (1.41) can be represented in the form: 1 @ h2 h3 @U @ h1 h3 @U @ h1 h2 @U þ þ ¼ 4pkd h1 h2 h3 @x1 h1 @x1 @x2 h2 @x2 @x3 h3 @x3 At the same time, in the vicinity of points where masses are absent, (d ¼ 0), in place of Equation (1.41) we have Laplace’s equation: r2 U ¼ 0

ð1:42Þ

Both Poisson’s and Laplace’s equations describe the behavior of the potential at regular points where the ﬁrst derivatives of the ﬁeld exist. To characterize the behavior of the potential at the boundary of media with different densities, let us make use of Equation (1.39) according to which a component of the ﬁeld along some direction l is equal to the derivative of the potential in this direction: @U @l Thus, instead of the surface analogy of the ﬁeld equations we obtain: gl ¼

ð1:43Þ

@U 2 @U 1 @U 2 @U 1 ¼ and ¼ ð1:44Þ @t @t @n @n where U1 and U2 are values of the potential at the back and front sides of the surface, respectively. It is obvious that the continuity of tangential derivatives of the potential follows from the continuity of the potential itself, and therefore the ﬁrst equality of the set (1.44) can be replaced by much simpler one: U1 ¼ U2

ð1:45Þ

Note, the equality of the normal derivatives does not follow from the continuity of the potential, because the normal derivatives also depend on values of the potential above and beneath of the interface. Thus, the behavior of the potential is described by the system given below r2 U ¼ 4pkd and @U 2 @U 1 ¼ on S ð1:46Þ @n @n Certainly, the system of equations for the attraction ﬁeld is much more complicated than that for the potential. Before we continue it may be appropriate to make the following comment. In all geophysical methods the ﬁelds, such as the particle displacement caused by elastic waves, the constant and time-varying electric U2 ¼ U1

20

Methods in Geochemistry and Geophysics

and magnetic ﬁelds can be expressed either in terms of the scalar or vector potentials or both of them. Respectively, all methods may be equally called the potential methods, and in this sense the gravitational method is not an exception.

1.7. THE POTENTIAL AND ITS RELATION TO MASSES Now we establish the relation between the potential U and masses. First, consider an elementary mass dm ¼ ddV. In accordance with Equation (1.8) we have gðpÞ ¼ k

dm 1 Lqp ¼ kdm gradp 3 Lqp Lqp

ð1:47Þ

since Lqp 1 ¼ gradp Lqp L3qp where the index p means that the gradient is considered in the vicinity of the point p. Comparing Equations (1.39 and 1.47) we can conclude that the function U, corresponding to the ﬁeld of the elementary mass located at the point q, is UðpÞ ¼ k

dm þC Lqp

ð1:48Þ

because when the gradients of two functions are equal, then the functions may differ by at most a constant. Taking into account the fact that the ﬁeld g caused by mass the dm tends to zero at inﬁnity, it is natural to assume that its potential also vanishes if Lqp ! 1. Then from Equation (1.48) it follows that C ¼ 0 and we have: UðpÞ ¼ k

dm Lqp

ð1:49Þ

Next, applying the principle of superposition, we arrive at an expression for the potential caused by a volume distribution of masses: Z dðqÞdV UðpÞ ¼ k ð1:50Þ Lqp V Equations (1.6 and 1.50) clearly show that the potential is related to masses in a much simpler way than the ﬁeld g, and this is the second reason for its introduction. Along with volume masses, it is possible to introduce other types of masses, and we have Z Z Z X mðqÞ dðqÞ sðqÞ lðqÞ UðpÞ ¼ k dV þ dS þ dl þ ð1:51Þ Lqp V Lqp S Lqp l Lqp where mi is an elementary mass and s and l surface and linear density, respectively. It is useful to discuss one more application of the potential, and with this purpose in mind we consider the change of this function, dU in the vicinity of some

Principles of Theory of Attraction

21

point. As is well known, dU ¼

@U @U @U dl 1 þ dl 2 þ dl 3 @l 1 @l 2 @l 3

ð1:52Þ

where dl1, dl2, and dl3 are elementary displacements along coordinate lines x1, x2, and x3. It is easy to see that the right hand side of this expression can be written as a dot product of two vectors, namely, dl ¼ dl 1 i1 þ dl 2 i2 þ dl 3 i3 and grad U ¼

@U @U @U i1 þ i2 þ i3 @l 1 @l 2 @l 3

ð1:53Þ

Here dl 1 ¼ h1 dx1 ;

dl 2 ¼ h2 dx2 ;

dl 3 ¼ h3 dx3

Therefore, dU ¼ dlgrad U ¼ g dl

ð1:54Þ

After integration of this equation along an arbitrary path with terminal points a and b, we obtain Z b UðbÞ UðaÞ ¼ gdl ð1:55Þ a

Thus, the integral of the ﬁeld g along a path is expressed by the difference of potentials at terminal points of this path. Certainly, it is much simpler to take a difference of the scalar at these points: U(b)U(a), than to perform an integration, and this fact demonstrates another advantage of using the potential.

1.8. FUNDAMENTAL SOLUTION OF POISSON’S AND LAPLACE’S EQUATIONS Now we return to Poisson’s and Laplace’s equations, which describe the behavior of the potential inside and outside masses, respectively. Earlier we have already derived an expression for the potential: Z dðqÞdV UðpÞ ¼ k Lqp V that is valid at any point. Therefore, this function is a solution of Poisson’s equation inside the masses, and it satisﬁes Laplace’s equation outside them. If an integration is performed over all masses then U(p) represents the fundamental solution of these equations. It is instructive to show in a different way that U(p) obeys these

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Methods in Geochemistry and Geophysics

equations. For instance, in the case when the observation point is located outside the masses, that is, p6¼q, we have: Z 1 r2 UðpÞ ¼ k dðqÞr2p dV L qp V where r2p UðpÞ ¼

@2 1 @2 1 @2 1 þ þ @x2p Lqp @y2p Lqp @z2p Lqp

and 2 2 2 1=2 L1 qp ¼ ½ðxp xq Þ þ ðyp yq Þ þ ðzp zq Þ

Taking the ﬁrst and then the second derivatives, we ﬁnd: @2 1 2 5 ¼ L3 qp þ 3ðxp xq Þ Lqp @x2p Lqp @2 1 2 5 ¼ L3 qp þ 3ðyp yq Þ Lqp @y2p Lqp @2 1 2 3 ¼ L3 qp þ 3ðzp zq Þ Lqp @z2p Lqp that is, r2

1 ¼0 Lqp

and correspondingly the Laplacian of the potential is equal to zero, if p6¼q. In a similar manner, but taking into the account an inﬂuence of masses in the vicinity of the point q, we will demonstrate later that U(p) obeys the Poisson’s equation inside the masses. By deﬁnition, the Laplacian of U represents the divergence of the attraction ﬁeld, and, correspondingly, its value characterizes the density of masses at same point. Now the following question arises. What does the Laplacian tells us about the behavior of the potential? To answer this question we ﬁrst consider the simplest case, when U depends on one argument, x, Fig. 1.7a. Then, we can represent the derivatives as: dUðxÞ 1 Dx Dx ¼ U xþ U x dx Dx 2 2 and dU x Dx d 2 UðxÞ 1 dU x þ Dx 2 2 ¼ dx2 Dx dx dx

ð1:56Þ

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Principles of Theory of Attraction

(a)

z

(b)

y

U(x)

x h = Δx x- Δx x x +Δx

x

Fig. 1.7. (a) Illustration of Equation (1.59), (b) elementary cube around point x, y, z.

Inasmuch as

dU x þ Dx 1 2 ½Uðx þ DxÞ UðxÞ ¼ Dx dx dU x Dx 1 2 ½UðxÞ Uðx DxÞ ¼ Dx dx

we have d 2 UðxÞ 1 ¼ ½Uðx þ DxÞ þ Uðx DxÞ 2UðxÞ 2 dx ðDxÞ2

ð1:57Þ

d 2 UðxÞ 2 ¼ ½U av ðxÞ UðxÞ 2 dx ðDxÞ2

ð1:58Þ

or

Here Uav(x) is the average value of the potential at the point p: Uðx þ DxÞ þ Uðx DxÞ ð1:59Þ 2 Thus, the ﬁrst derivative deﬁnes the rate of change of the function U(x), while the second derivative shows how its average value differs from the value of the function at the same point. For instance, if U 00 ðxÞo0, we have UðxÞ4U av ðxÞ and there is a maximum. Next, suppose that within some interval of x the average value of the function coincides at each point with the value of this function: U av ðpÞ ¼

U av ðxÞ ¼ UðxÞ

ð1:60Þ

and therefore d 2 UðxÞ ¼0 ð1:61Þ dx2 The last equation represents the simplest class of functions in the one-dimensional case, namely, the linear functions, for which the condition (1.61) is met. Correspondingly, we can say that the second derivative is a measure of how the behavior

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Methods in Geochemistry and Geophysics

of the function in the vicinity of some point differs from that of a linear function. In other words, the second derivative characterizes the curvature of a line describing the function. In order to generalize this study for three-dimensional case, imagine an elementary cube around some point p(x,y,z), so that its sides are directed along the coordinate lines, Fig. 1.7b. The length of each side is 2h. In accordance with Equation (1.58) the second derivatives along the coordinate axes are @2 U 1 ¼ ½Uðx þ Dx; y; zÞ þ Uðx Dx; y; zÞ 2Uðx; y; zÞ @x2 ðDxÞ2 @2 U 1 ¼ ½Uðx; y þ Dy; zÞ þ Uðx; y Dy; zÞ 2Uðx; y; zÞ 2 @y ðDyÞ2

ð1:62Þ

@2 U 1 ¼ ½Uðx; y; z þ DzÞ þ Uðx; y; z DzÞ 2Uðx; y; zÞ 2 @z ðDzÞ2 Taking into account the fact that Dx ¼ Dy ¼ Dz ¼ h and substituting Equation (1.62) into the expression for the Laplacian: @2 U @2 U @2 U þ 2 þ 2 ¼ r2 U @x2 @y @z we obtain

" # 6 1 X U i 6UðpÞ r U¼ 2 h i¼1 2

or

2 r2 U ¼

6 P

3

Ui 7 66 6i¼1 7 UðpÞ 6 7 5 h2 4 6

ð1:63Þ

Here Ui is the value of the potential on ith face of the cube, while U(p) is its value at the center of the cube. It is clear that the term 6 P

Ui

i¼1

6 is the average value of the potential at the point p. Thus, the Laplacian can be written as 6 ð1:64Þ r2 U ¼ 2 ½U av ðpÞ UðpÞ h that is, it is again a measure of the difference between the average value of the function and its value U at the same point. For example, if the average value exceeds the value of the function, the Laplacian is positive. In the case of the gravitational ﬁeld the latter is always negative, that is, UðpÞ4U av ðpÞ, and this

Principles of Theory of Attraction

25

follows from the fact that the right hand side of Equation (1.41) is negative, if d6¼0. Now, making use of Equation (1.64) we obtain the simplest form of Laplace’s equation: U av ðpÞ UðpÞ ¼ 0

ð1:65Þ

Therefore, if the function U satisﬁes the Laplace’s equation, then it possesses a remarkable interesting feature, namely, its average value calculated around some point p is exactly equal to the value of the function at this point. A certain class of functions has this feature only, and such functions are called harmonic. Correspondingly, we conclude that the potential of the attraction ﬁeld is a harmonic function outside the masses. In accordance with Laplace’s equation the sum of the second derivatives along coordinate lines, x, y, and z, equals zero, provided that U(p) is a harmonic function. At the same time we know that in the one-dimensional case there is a class of functions for which the second derivative is equal to zero, that is, d 2U ¼0 dx2 From this comparison of the behavior of the second derivatives, it is natural to consider harmonic functions as an analogy of the linear functions and expect that they have similar features. Let us outline some of them. 1. It is clear that if values of a linear function are known at terminal points of some interval of x, then it can be calculated at every point inside. In the same manner, if a harmonic function is given at each point of the boundary surface surrounding the volume, it can be determined at any of its point. 2. If a linear function has equal values at terminal points of the interval, then it has the same value inside it, that is, the linear function is constant. By analogy, if a harmonic function has same value at all points of the boundary surface, then it has the same value at any point within the volume. Of course, both statements can be proved from the theorem of uniqueness for the attraction ﬁeld. In addition, it is appropriate to comment: a linear function reaches its maximum at terminal points of the interval. The same behavior is observed in the case of harmonic functions, which cannot have their extreme inside the volume. Otherwise, the average value of the function at some point will not be equal to its value at this point, and, correspondingly, the Laplacian would differ from zero. At the same time, saddle points may exist.

1.9. THEOREM OF UNIQUENESS AND SOLUTION OF THE FORWARD PROBLEM Suppose, ﬁrst, that a distribution of masses is known everywhere. Then, as was shown earlier, the potential of the attraction ﬁeld is Z dðqÞ UðpÞ ¼ k dV ð1:66Þ V Lqp

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Methods in Geochemistry and Geophysics

(b)

(a) S

S

V

V •p

U = ϕ1

•

p

(c) S

•

p

U=C

U = ϕ2 n

M

Fig. 1.8. (a) Dirichlet’s problem, (b) Neumann’s problem, (c) the third boundary value problem.

and at regular points this function obeys Poisson’s equation DU ¼ 4pkd

ð1:67Þ

Besides, the potential and its ﬁrst derivatives are continuous at boundaries where a volume density is discontinuous function. It is obvious that in this case the solution of the forward problem is unique. Now consider a completely different situation, when a density d(q) is given only inside some volume V surrounded by a surface S, Fig. 1.8a. Inasmuch as the distribution of masses outside V is unknown, it is natural to expect that Poisson’s equation does not uniquely deﬁne the potential U, and in order to illustrate this fact let us represent its solution as a sum: UðpÞ ¼ U i ðpÞ þ U e ðpÞ

ð1:68Þ

where Ui(p) and Ue(p) are potentials in the volume V caused by masses inside and outside the volume V, respectively, DU i ðpÞ ¼ 4pkd and

DU e ðpÞ ¼ 0

ð1:69Þ

Therefore, we can write DU ¼ DðU i þ U e Þ ¼ DU i þ DU e ¼ 4pkd

ð1:70Þ

This means that Poisson’s equation deﬁnes the potential with an uncertainty of a harmonic function Ue. Regardless of a distribution of masses outside the volume the potential Ue remains harmonic function inside V and, correspondingly, there are an inﬁnite number of potentials U which satisfy Equation (1.70), and they can be represented as: Z ddV UðpÞ ¼ k þ U e ðpÞ ð1:71Þ V Lqp where the last term on the right hand side is unknown. It is clear that along with Poisson’s equation we need additional information, which allows us to ﬁnd the

27

Principles of Theory of Attraction

harmonic function Ue(p). It turns out that a behavior of the potential on the boundary S gives this information, and the latter are called boundary conditions. In essence, they contain the information about the distribution of masses outside the volume V. To ﬁnd these conditions we will proceed from Gauss’s theorem, which is the natural ‘‘bridge’’ between values of the ﬁeld inside the volume and those at the boundary surface: Z I I div XdV ¼ XdS ¼ X n dS ð1:72Þ V

S

S

Here n is the unit vector perpendicular to the surface S and directed outward, and Xn the normal component of an arbitrary vector X, which is a continuous function within volume V. As was pointed out, Equation (1.67) has an inﬁnite number of solutions; let us choose any pair of them, U1(p) and U2(p), and form their difference: U 3 ðpÞ ¼ U 2 ðpÞ U 1 ðpÞ

ð1:73Þ

To derive the boundary conditions, we introduce some vector function X(p): X ¼ U 3 grad U 3 ¼ U 3 rU 3 Substitution of Equation (1.74) into Equation (1.72) gives Z I rðU 3 rU 3 ÞdV ¼ U 3 rn U 3 dS V

ð1:74Þ

ð1:75Þ

S

where rnU3 is the component of the gradient along the normal n, and rn U 3 ¼

@U 3 @n

ð1:76Þ

Since r is a differential operator, we have rðU 3 rU 3 Þ ¼ U 3 r2 U 3 þ rU 3 rU 3 ¼ ðrU 3 Þ2

ð1:77Þ

because r2 U 3 ¼ DU 3 ¼ DðU 2 U 1 Þ ¼ 4pkd þ 4pkd ¼ 0 Taking into account Equations (1.76 and 1.77), we can rewrite Equation (1.75) as Z I @U 3 2 dS ð1:78Þ ðrU 3 Þ dV ¼ U 3 @n V S

This equality is a special form of Gauss’s theorem, and it will allow us to ﬁnd several boundary conditions, which provide uniqueness of a solution of the forward problem. First, we make three comments: a. The volume V can be enclosed by several different surfaces. b. The integrand of the volume integral in Equation (1.78) is non-negative and this fact plays a vital role for deriving boundary conditions.

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Methods in Geochemistry and Geophysics

The equality (1.78) relates the values of the function inside the volume V to its values on the boundary surface S, and the harmonic function U3 is the difference of two arbitrary solutions of Poisson’s equation. Now we are prepared to formulate boundary conditions for the potential of the attraction ﬁeld, which uniquely deﬁne this ﬁeld inside the volume V. With this purpose in mind suppose that the surface integral on the right hand side of Equation (1.78) equals zero. Then Z ðrU 3 Þ2 dV ¼ 0 ð1:79Þ c.

V

and taking into account the fact that its integrand cannot be negative, we have to conclude that at every point of the volume grad U 3 ¼ 0

ð1:80Þ

This means that the derivative of function U3 in any direction l is zero @U 3 ¼0 @l that is, this function is constant, and therefore the derivatives of solutions of Poisson’s equation are equal to each other @U 1 ðpÞ @U 2 ðpÞ ¼ @l @l In other words, if the surface integral in Equation (1.78) vanishes, these solutions can differ by a constant only: U 2 ðpÞ ¼ U 1 ðpÞ þ C

ð1:81Þ

where C is constant, that is, the same for all points of the volume V, including the surface. In particular, this constant can be zero. Next, we will deﬁne conditions under which the surface integral vanishes: I @U 3 dS ¼ 0 ð1:82Þ U3 @n S

and correspondingly Equation (1.81) becomes valid. Several such conditions are described below. 1.9.1. The first boundary value problem Suppose that the potential U(p) is known on the boundary surface; that is, UðpÞ ¼ j1 ðpÞ on S

ð1:83Þ

and we consider a solution of Poisson’s equation that satisﬁes the condition (1.83). Let us assume that there are two different solutions to this equation inside the volume, U1(p) and U2(p), which coincide on the boundary surface: U 1 ðpÞ ¼ U 2 ðpÞ ¼ j1 ðpÞ on S

Principles of Theory of Attraction

29

Then their difference U3 on this surface becomes zero: U 3 ðpÞ ¼ 0 on S and, consequently, the surface integral in Equation (1.82) vanishes. Therefore, in accordance with Equation (1.81) solutions of Poisson’s equation satisfying the condition (1.83) can differ from each other by a constant only. Moreover, this constant is known and it is equal to zero since on the surface S all solutions should coincide. In other words, we have proved that equations: DU ¼ 4pkd inside V and UðpÞ ¼ j1 ð pÞ on S

ð1:84Þ

uniquely deﬁne the potential U as well as the attraction ﬁeld g, since g ¼ grad U Equation (1.84) form Dirichlet’s boundary value problem, which can be either exterior or internal one, Fig. 1.8a, and it has several important applications in the theory of the gravitational ﬁeld of the earth. It is worth to notice that in accordance with Equation (1.83) we can say that along any direction tangential to the boundary surface, the component of the ﬁeld gt is also known, since gt ¼ @U=@t. Consequently, the boundary value problem can be written in terms of the ﬁeld as curl g ¼ 0 div g ¼ 4pkd and gt ¼

@j1 on S @t

ð1:85Þ

This ﬁrst case vividly illustrates the importance of the boundary condition. Indeed, Poisson’s equation or the system of ﬁeld equations have an inﬁnite number of solutions corresponding to different distributions of masses located outside the volume. Certainly, we can mentally picture unlimited variants of mass distribution and expect an inﬁnite number of different ﬁelds within the volume V. In other words, Poisson’s equation, or more precisely, the given density inside the volume V, allows us to ﬁnd the potential due to these masses, while the boundary condition (1.83) is equivalent to knowledge of masses situated outside this volume. It is clear that if masses are absent in the volume V, the potential U is a harmonic function and it is uniquely deﬁned by Dirichlet’s condition. 1.9.2. The second boundary value problem Now let us assume that two arbitrary solutions of Poisson’s equation within the volume V; U 1 ðpÞ and U 2 ðpÞ, have the same normal derivatives on the surface S;

30

Methods in Geochemistry and Geophysics

that is, @U 1 @U 2 ¼ ¼ j2 ðpÞ @n @n

on

S

ð1:86Þ

where j2 ðpÞ is a known function. From this equality it instantly follows that the normal derivative of a difference of these solutions vanishes on the boundary surface: @U 3 ¼0 @n

on

S

Therefore, the surface integral in Equation (1.78), as in the previous case, equals zero and correspondingly inside the volume we have rU 3 ¼ 0 This means that any solutions of Poisson’s equation, for instance U1(p) and U2(p), can differ from each other at every point of the volume V by a constant only, if their normal derivatives coincide on the boundary surface S. Thus, this boundary value problem deﬁnes also uniquely the ﬁeld of attraction, and it can be written as ðinside V Þ

DU ¼ 4pkd and

@U ¼ j2 ðpÞ on S @n

ð1:87Þ

or curl g ¼ 0 div g ¼ 4pkd and gn ¼ j2 ðpÞ on S

ð1:88Þ

and it is called Neumann’s problem. Unlike the previous case, Equation (1.87) deﬁne the potential only to within a constant, but of course the ﬁeld of attraction is determined uniquely. 1.9.3. The third boundary value problem We suppose that the boundary surface S is equipotential, that is, UðpÞ ¼ C

ð1:89Þ

on the boundary surface S, and C is some constant. In addition, it is assumed that the total mass M in the volume V, surrounded by the surface S, is known, Fig. 1.8c. As follows from the Gauss’s theorem: I I I @U dS ¼ 4pkM ð1:90Þ gdS ¼ gn dS ¼ @n S

S

S

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Principles of Theory of Attraction

This means that

I

@U dS ¼ j3 @n

ð1:91Þ

S

where j3 ¼ 4pkM is known. Now we will show that two solutions of Poisson’s equation, U1(p) and U2(p), satisfying Equations (1.89 and 1.90), can differ from each other by a constant only. Consider again the surface integral in Equation (1.78): I @U 3 dS U3 @n S

Inasmuch as the boundary surface is an equipotential surface for both potentials U1 and U2, their difference is also constant on this surface and correspondingly we can write 8 9 I I I
S

S

S

and in accordance with Equation (1.78) inside the volume V rU 3 ¼ 0 or U 2 ðpÞ ¼ U 1 ðpÞ þ C Thus, boundary conditions (1.89) and (1.90) deﬁne the potential within the volume V up to some constant. Correspondingly, the third boundary value problem can be formulated as: 1. Inside the volume the potential obeys Poisson’s equation DU ¼ 4pkd 2.

On the level surface S, surrounding this volume, we have I @U dS ¼ j3 @n

ð1:92Þ

S

where j3 is given and it is directly proportional to the total mass inside the volume. If the surface S includes also some surface at inﬁnity, where the potential tends to zero, then the potential is deﬁned uniquely, too. In terms of the ﬁeld we have curl g ¼ 0 div g ¼ 4pkd and

I gn dS ¼ j3 S

ð1:93Þ

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Methods in Geochemistry and Geophysics

Here gn is the normal component of the ﬁeld and its magnitude coincides with that of the total ﬁeld, since the tangential component vanishes on the equipotential surface. This boundary value problem plays the fundamental role in the theory of the gravity. To illustrate this fact, suppose that the shape of the level surface S, surrounding the earth, and its mass, M, are known. At the same time, masses in space outside S are absent. Then, the third case shows that these two conditions uniquely deﬁne the potential and its ﬁeld. In other words, regardless of a distribution of masses of the earth the ﬁeld outside remains the same. For this reason, we can use the ﬁeld and its potential to study the ﬁgure of the earth. The uniqueness of this boundary value problem was proved by Stokes and, respectively, it is called Stokes’s problem. In this light, it may be appropriate to make two comments: a. The ﬁeld, observed outside this level surface, does not allows us to determine the distribution of masses, if we know only the total mass, M. b. Uniqueness of solution of the Stokes’s problem was proved when masses are absent outside the level surface S. However, as was demonstrated, this condition is not necessary and even in their presence the ﬂux of the ﬁeld through the surface S is still deﬁned by only the masses inside.

1.9.4. The fourth boundary value problem Suppose that in the volume V the potential obeys Poisson’s equation: DU ¼ 4pkd and at points of the surface S we know a linear combination of the potential U and its normal derivative @U=@n: dU þ hU ¼ m ð1:94Þ dn here p and h are either constants or functions of a point and they are positive: p

p40

and

h40

ð1:95Þ

If U1 and U2 are solutions of Poisson’s equation and obey the boundary condition (1.94), their difference U3 ¼ U2U1 satisﬁes Laplace’s equation and the condition on the surface S: dU 3 þ hU 3 ¼ 0 dn Multiplying the latter by U3, we have p

pU 3

dU 3 hU 23 dn

ð1:96Þ

Now consider again the equality (1.78); its left hand side can be either zero or positive, while, as follows from Equation (1.96), it is negative or zero. Therefore, Equation (1.78) takes place if both integrals are equal to zero and, correspondingly,

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Principles of Theory of Attraction

in whole space of V: grad U 3 ¼ 0 that is, two solutions of this boundary problem may differ by a constant only. If the volume V is also surrounded by some surface at an inﬁnity, where the potential tends to zero, this constant becomes equal to zero. Note that this boundary value problem plays an important role in deriving Stokes’s formula, which allows one to obtain a position of the geoid with respect to the reference ellipsoid. In conclusion, let us summarize the main results derived in this section. 1. Four types of boundary conditions have been formulated and each of them uniquely deﬁnes the ﬁeld of attraction within the volume V. 2. As was pointed out above, the volume can be surrounded by several surfaces, and at every point on them one of these conditions has to be given. For instance, the potential is known at one surface, while its normal derivative is speciﬁed on the other. 3. The procedure of determination of these conditions, based on the use of Gauss’s formula, is called the theorem of uniqueness. 4. Boundary value problem can be either the exterior or internal one, and each boundary condition is equally applied to both cases.

1.10. GREEN’S FORMULA AND THE RELATIONSHIP BETWEEN POTENTIAL AND BOUNDARY CONDITIONS In essence, the theorem of uniqueness formulates the steps that have to be undertaken to ﬁnd the attraction ﬁeld and its potential. These steps are 1. the solution of Poisson’s equation and 2. the selection among these solutions of such functions that satisfy boundary conditions. At the same time it is worth to notice that in modern numerical methods of a solution of boundary value problems, based on replacement of differential equations by ﬁnite difference, these steps are performed simultaneously. In accordance with the theorem of uniqueness, the ﬁeld inside the volume V is deﬁned by a distribution of masses inside this volume and boundary conditions, and correspondingly it is natural to derive an equation establishing this link. With this purpose in mind we will again proceed from Gauss’s theorem, Z I rXdV ¼ XdS ð1:97Þ V

S

assuming that the vector function X(q) and its ﬁrst derivatives exist in the volume V. Here, rX is the divergence of the vector X. Let us express the vector X with help of two scalar functions jðqÞ and cðq; pÞ in the following way: X ¼ jðqÞrcðq; pÞ cðq; pÞrjðqÞ

ð1:98Þ

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Methods in Geochemistry and Geophysics

Here, j and c are continuous functions with continuous ﬁrst and second derivatives, p is an observation point where the potential is determined, and q is an arbitrary point. Substituting Equation (1.98) into Equation (1.97) and taking into account the fact that @c @j c XdS ¼ X n dS ¼ j dS @n @n and div X ¼ jrc þ jr2 c rjrc cr2 j ¼ jr2 c cr2 j we obtain Z

ðjr2 c cr2 jÞdV ¼

V

I @c @j j c dS @n @n

ð1:99Þ

S

This relationship is called the second Green’s formula and it represents Gauss’s theorem when the vector X is given by Equation (1.98). In particular, letting c ¼ constant we obtain the ﬁrst Green’s formula: I Z @j dS ð1:100Þ r2 jdV ¼ @n V S

As was mentioned above, our goal is to express the potential of the ﬁeld U(p) within the volume V in terms of both the masses inside the volume and values of the potential and its derivatives on the boundary S, ðS ¼ S 1 þ S2 Þ, Fig. 1.9. To solve n

S*

p q

S1 V

Fig. 1.9. Illustration of Equation (1.104).

S2

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Principles of Theory of Attraction

this problem we will make use of Equation (1.99), assuming that the function j describes the potential U, but the function c ¼ Gðq; pÞ obeys the following conditions: a. It is a solution of Laplace’s equation: r2 G ¼ 0 b.

ð1:101Þ

everywhere except the observation point p. In approaching the point p this function tends to inﬁnity as Gðq; pÞ !

1 as Lqp ! 0 Lqp

ð1:102Þ

Here Lqp is the distance between points q and p. Note that G(q, p) is called a Green’s function. There are an inﬁnite number of such functions and all of them have a singularity at the observation point p. Inasmuch as the second Green’s formula has been derived assuming that singularities of the functions U and G are absent within volume V, we cannot directly use this function G in Equation (1.99). To avoid this obstacle, let us surround the point p by a small spherical surface S* and apply Equation (1.99) to the volume enclosed by surfaces S and S*, as is shown in Fig. 1.9. Further we will be mainly interested by only cases, when masses are absent inside the volume V, that is, DU ¼ 0

ð1:103Þ

Taking into account Equations (1.101 and 1.103), the volume integral in the second Green’s formula vanishes and we obtain I I @G @U @G @U U U G dS þ G dS ¼ 0 ð1:104Þ @n @n @n @n S

S

Our task is to derive an explicit expression for the potential U proceeding from this equation. This means that we have to take the function U out of this integral. With this purpose in mind consider the limiting value of the second integral, when the radius of the spherical surface r tends to zero. Since both the potential and its derivatives are continuous functions inside the volume, we have UðqÞ ! UðpÞ and

@UðqÞ @UðpÞ ! @n @n

That is, functions U(q) and @UðqÞ=@n on the surface S* approach their values at the observation point, respectively. Also from Fig. 1.9 it follows that the normal n and radius vector r on this surface are opposite to each other, and therefore for points on this surface we have G!

1 1 ¼ Lqp r

and

@G @1 1 ! ¼ @n @r r r2

as

r!0

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Methods in Geochemistry and Geophysics

Then, applying the mean value theorem, the surface integral around the observation point p can be represented as I

S

I I @ 1 1 @UðqÞ 1 1 @UðqÞ dS UðqÞ dS ¼ 2 UðqÞdS @n Lqp Lqp @n r r @n S

ð1:105Þ

The terms on the right hand side are I 1 UðpÞ UðqÞdS ¼ 2 4pr2 ¼ 4pUðpÞ r2 r S

and 1 r

I

@UðqÞ 1 @UðpÞ dS ¼ 4pr2 ! 0 @n r @n

S

Therefore, in the limit the integral over the surface S* is I

@G @UðqÞ UðqÞ G dS ¼ 4pUðpÞ @n @n S

ð1:106Þ

This is the most important result in our derivations since it permits us to obtain an expression for the potential at any point of the volume in terms of boundary conditions. Substitution of Equation (1.106) into Equation (1.104) gives I

1 @UðqÞ @Gðq; pÞ UðpÞ ¼ Gðq; pÞ UðqÞ dS ð1:107Þ 4p @n @n S

Here q is the point of the boundary surface and p the observation point. Thus, in order to determine the potential we have to know at the surface S both the potential and its normal derivative. At the same time, as follows from the ﬁrst and second boundary value problems, in order to ﬁnd the potential it is sufﬁcient to know only one of these quantities on S. This apparent contradiction can be easily resolved by the appropriate choice of Green’s function and we will illustrate this fact in the two following sections. At the beginning we assumed that masses are absent inside the volume V and, correspondingly, Equation (1.107) describes the potential of the attraction ﬁeld, caused by masses located outside V. Now suppose that there are also masses in this volume and their distribution is characterized by the density d. Then, applying the principle of superposition, we obtain Z

dðqÞ 1 UðpÞ ¼ k dV þ 4p V Lqp

I

S

@U @G G U dS @n @n

ð1:108Þ

37

Principles of Theory of Attraction

p

Lqp S0 q Lqs s

z Fig. 1.10. Illustration of Equation (1.115).

1.11. ANALYTICAL UPWARD CONTINUATION OF THE FIELD Now we will demonstrate one application of Equation (1.107): I

1 @UðqÞ @Gðq; pÞ UðpÞ ¼ Gðq; pÞ UðqÞ dS 4p @n @n

ð1:109Þ

S

for calculation of the ﬁeld in the upper half space, Fig. 1.10. It is appropriate to note that this procedure is sometimes applied for the reduction of geological noise. Taking into account that in the upper half space (above the earth’s surface S0) masses are absent, the potential satisﬁes Laplace’s equation DU ¼ 0. Suppose that at the plane S0 the vertical component of the attraction ﬁeld, that is, the derivative @U=@n, is known. Inasmuch as we consider a ﬁeld, caused by masses of a conﬁned body, at sufﬁciently large distances from S0 the ﬁeld becomes small and, correspondingly, we can assume that the potential U tends to zero at semi-spherical surface with an inﬁnitely large radius, ðr ! 1Þ. Since gz ¼ @U=@z, Equation (1.109) becomes UðpÞ ¼

@Gðq; pÞ dS @z S0 I

1 @Gðq; pÞ Gðq; pÞgr UðqÞ þ dS 4p @r

1 4p

I

Gðq; pÞgz UðqÞ

Sr

where the z-axis is directed downward and gr is normal component of the ﬁeld on the spherical surface of the radius r. Further we assume that with an increase of radius r of the hemispherical surface the function G decreases, as the potential at least inversely proportional to r, and their ﬁrst derivatives @U=@r and @G=@r tend to zero when r is r2. Correspondingly, the integrand of the second integral in the last

38

Methods in Geochemistry and Geophysics

equation decreases as r3 and making use of the mean value theorem we obtain I I @G C C Ggr U dS ! 3 dS ¼ 3 4pr2 ! 0 @r r r Sr

as r ! 1

Sr

Here C is some constant. Thus, the expression for the potential is UðpÞ ¼

1 4p

Z

Gðq; pÞgz ðqÞ UðqÞ S0

@Gðq; pÞ dS @zq

ð1:110Þ

We have derived a formula for calculation of the potential of the ﬁeld everywhere in the upper half space if both the potential U and the vertical component of the attraction ﬁeld are known at the earth’s surface, S0. Having taken the derivative from both sides of Equation (1.110) we obtain for the vertical component of the attraction ﬁeld at the point p: Z

1 @G @2 G gz ðpÞ ¼ g ðqÞ UðqÞ dS ð1:111Þ 4p S0 z @zp @zp @zq where zp and zq indicate that derivatives of Green’s function are taken with respect to coordinate z near points p and q, respectively. As was pointed out in the previous section, we can imagine an inﬁnite number of Green’s functions. The simplest of these is Gðq; pÞ ¼

1 Lqp

ð1:112Þ

Indeed, it satisﬁes Laplace’s equation everywhere except at the point p, since it describes up to a constant the potential of a point mass located at the point p. Also, it has a singularity at this point and provides a zero value of the surface integral over the hemisphere when its radius r tends to inﬁnity. Correspondingly, we can write Z

1 @ 1 @2 1 gz ðpÞ ¼ gz ðqÞ UðqÞ dS ð1:113Þ 4p S0 @zp Lqp @zp @zq Lqp However, this expression is impractical since the potential U is not measured on the earth’s surface, and therefore we have to choose a Green’s function such that the derivative @G=@z on the earth’s surface would be zero. In this case Equation (1.110) is greatly simpliﬁed: Z 1 UðpÞ ¼ Gðq; pÞgz ðqÞdS ð1:114Þ 4p S0 where gz ðqÞ is the measured vertical component of the ﬁeld on the earth’s surface and G(q,p) is unknown function that satisﬁes the following conditions:

Principles of Theory of Attraction

1.

39

in the upper space, (zo0), Green’s function is a solution of the Laplace’s equation everywhere except at the observation point p: DG ¼ 0,

2.

it has a singularity of type L1 qp , that is, in approaching point p, Gðq; pÞ !

3. 4.

1 , Lqp

At the hemispherical surface it decreases at least as 1/r with an increase of r, and ﬁnally At the earth’s surface the derivative @Gðq; pÞ=@zq vanishes @G ¼ 0. @zq

As we already know a determination of the function G(q, p) satisfying all these conditions represents a solution of the boundary value problem; and in accordance with the theorem of uniqueness these conditions uniquely deﬁne the function G(q, p). In general, a solution of this problem is a complicated task, but there are exceptions, including the important case of the plane surface S0, when it is very simple to ﬁnd the Green’s function. Let us introduce the point s, which is the mirror reﬂection of the point p with respect to the plane of the earth’s surface, Fig. 1.10, and consider the function G1 ðp; q; sÞ equal to G1 ¼

1 1 þ Lqp Lqs

ð1:115Þ

where q is a point at the earth’s surface, and Lqp ¼ fðxq xp Þ2 þ ðyq yp Þ2 þ ðzq zp Þ2 g1=2 Lqs ¼ fðxq xs Þ2 þ ðyq ys Þ2 þ ðzq zs Þ2 g1=2 Taking the derivative @G1 =@zq , we obtain zq zp zq zs @G1 ¼ @zq L3qp L3qs Inasmuch as at every point at the earth’s surface zq ¼ 0 and zp ¼ zs , but Lqp ¼ Lqs the derivative @G 1 =@zq is equal to zero. It is obvious that other conditions are also met. Therefore, in accordance with Equation (1.114), we obtain for the potential and vertical component of the attraction ﬁeld at the point p: Z Z 1 1 @G 1 UðpÞ ¼ G1 gz ðqÞdS and gz ðpÞ ¼ gz ðqÞ dS ð1:116Þ 4p S0 4p S0 @zp

40

Methods in Geochemistry and Geophysics

where zp @G 1 ¼ 3 @zp Lqp Equation (1.116) allows us to calculate the vertical component of the ﬁeld in the upper half space when it is known at the earth’s surface. Correspondingly, this transformation is called upward continuation and it is used to reduce an inﬂuence of geological noise, caused by heterogeneities with relatively small dimensions.

1.12. POISSON’S INTEGRAL Now we will establish a relationship between the potential U(p) at any point p of the volume V and its values on the spherical surface, surrounding all masses, Fig. 1.11. The reason why we consider this problem is very simple; it plays the fundamental role in Stokes’s theorem, which allows one to determine the elevation of the geoid with respect to the reference ellipsoid. As in the previous section it is natural to start from the Green’s formula I

1 @UðqÞ @Gðq; pÞ UðpÞ ¼ Gðq; pÞ UðqÞ dS ð1:117Þ 4p @n @n S

Here the unit vector n and radius vector R have opposite directions. The volume V is surrounded by the surface S as well as a spherical surface with inﬁnitely large radius. In deriving this equation we assume that the potential U(p) is a harmonic function, and the Green’s function is chosen in such a way that allows us to neglect the second integral over the surface when its radius tends to an inﬁnity. The integrand in Equation (1.117) contains both the potential and its derivative on the spherical surface S. In order to carry out our task we have to ﬁnd a Green’s function in the volume V that is equal to zero at each point of the boundary surface: G¼0

on S

ð1:118Þ

r 0

S

r1 s Ψ R

ρ

n

q V

Fig. 1.11. Illustration of Equation (1.117).

p

Principles of Theory of Attraction

Then, we will obtain UðpÞ ¼

1 4p

I UðqÞ

@G dS @n

41

ð1:119Þ

S

In the case of the plane boundary surface we represented the Green’s function as a combination of potentials due to point masses located at points p and s, which are mirror reﬂection of each other with respect to the plane S. By analogy, let us attempt to describe the Green’s function as a difference 1 1 Gðq; p; sÞ ¼ a ð1:120Þ r sq Here s is a point located somewhere inside the sphere along the line 0p, r and sq the distances between points q, p and q, s, respectively, and a a constant. Now we demonstrate that for each observation point p it is possible to ﬁnd such a point s and coefﬁcient a, that the Green’s function is equal to zero at all points of the sphere. In order to prove it, we ﬁrst choose the position of the point s from the condition: rr1 ¼ R2

ð1:121Þ

that is, the product of distances of points p and s from the origin 0 is equal to the square of the radius of the sphere. For instance, with an increase of the distance r the point s approaches the origin. On the contrary, when the point p becomes closer to the boundary, the point s also moves toward this surface. Next, consider two triangles s0q and q0p, shown in Fig. 1.11. They have a common angle c and sides, forming this angle, are proportional. In fact, from Equation (1.121) we have r R ¼ ð1:122Þ R r1 Therefore, these triangles are similar, and we can write sq r1 1 R ¼ ¼ or r sq rr1 R Taking into account Equation (1.122) the latter can be represented as 1 R r r ¼ ¼ sq r R2 rR and Equations (1.118–1.120) give r1 R and we have for points of the volume V and the surface S: 1 r1 1 Gðq; p; sÞ r R sq a¼

Here r ¼ fr2 þ R2 2Rr cos cÞ1=2 ;

sq ¼ ðr21 þ R2 2Rr1 cos cÞ1=2

ð1:123Þ

42

Methods in Geochemistry and Geophysics

Substitution of Equation (1.123) into Equation (1.119) gives I 1 @ 1 r1 1 UðpÞ ¼ UðqÞ dS 4p @n r R qs

ð1:124Þ

S

First, consider the derivatives of the functions r1 and ðsqÞ1 : @ 1 @ 1 R r cos c ¼ ¼ @n r @R r r3

and

@ 1 R r1 cos c ¼ @n sq ðsqÞ3

Whence @G R r cos c r1 R3 ¼ ðR r1 cos cÞ 3 3 3 @n r R r1 r ¼ r13 fðR r cos cÞr21 ðR r1 cos cÞR2 g r12 1

r2 R4 R2 2 cos cÞR ¼ 3 4 ðR r cos cÞ 2 ðR r r r R 1 ¼ 3 2 ½R3 rR2 cos c Rr2 þ R2 r cos c r R ¼

1 ðR2 r2 Þ r3 R

Thus, Equation (1.117) becomes I I 1 r2 R 2 r2 R 2 dS UðpÞ ¼ UðqÞ dS ¼ UðqÞ 3 4pR r r3 4pR S

ð1:125Þ

S

and we have found a formula, which allows us to determine the potential in the volume V, if it is known on the surface S; that is, Dirichlet’s problem is solved.

1.13. BEHAVIOR OF THE ATTRACTION FIELD Now we begin to study the attraction ﬁeld caused by different distributions of masses, and start from a very simple case. 1.13.1. The attraction field of a spherical mass First, consider the ﬁeld of a homogeneous sphere with radius a. Taking into account the spherical symmetry of the mass distribution, it is natural to introduce a spherical system of coordinates with its origin at the center of the sphere, Fig. 1.5c. Then, the vector g(p) is in general characterized by three components: gR ; gy ; and gj

Principles of Theory of Attraction

43

By deﬁnition, any plane y ¼ constant is a plane of symmetry. In other words, there are always two elementary masses, which are equal to each other, and located at opposite sides of this plane but at the same distance. As is seen from Fig. 1.5d, the sum of y-components, caused by both masses is equal to zero. Representing the total mass as a sum of such pairs we conclude that the y-component, gy, due to the spherical mass is absent at every point outside and inside the body. In the same manner we can prove that gf ¼ 0. Of course, volume integration, Equation (1.6), can prove this fact, but this procedure is much more complicated. Thus, the attraction ﬁeld has only a radial component, gR, and the ﬁeld is directed toward the origin, 0. In order to determine this component we will proceed from Equation (1.26) and consider a spherical surface with radius R, as is shown in Fig. 1.5c. Inasmuch as dS ¼ dSiR and the scalar component gR is constant at points of the spherical surface, we have for the ﬂux: Z Z gdS ¼ gR dS ¼ 4pR2 gR ¼ 4pkM S

S

Therefore, the attraction ﬁeld,

geR ,

outside the sphere is equal to

geR ¼ k RM2

if

R a

ð1:126Þ

while inside the sphere: giR ¼ k Rm2

if

R a

ð1:127Þ

Here m is the mass enclosed by the surface of integration, S. As follows from Equation (1.126), the ﬁeld outside coincides with that caused by a point source with the same mass, M, located at the sphere center. This is a well-known result, which is hard to predict. In fact, this behavior occurs regardless of how close the observation point is to the sphere, and it results from the superposition of ﬁelds, caused by elementary masses. This is rather an exception, since in general the ﬁeld differs from that generated by an elementary mass. Next consider the ﬁeld inside the sphere. Since m¼

4p 3 dR 3

we have 4p kdR ð1:128Þ 3 and the ﬁeld magnitude increases linearly from the sphere’s center to its surface. Certainly, it is not inversely proportional to the square of the distance! Due to spherical symmetry, it is clear that the ﬁeld vanishes at the center 0. As follows from Equation (1.127), masses located within the interval 0–R create the same ﬁeld as a point mass situated at the origin, while masses between a and R do not affect the ﬁeld. In other words, the resultant ﬁeld of elementary masses within this range is zero. The behavior of the radial component as a function of R is shown in Fig. 1.5e. giR ¼

44

Methods in Geochemistry and Geophysics

(a)

(b)

a2

p

0 (d)

0

(c)

gR

a1

a1

R

a2

(e) n

dS1

S

p R→∞

M

S0

dS 2 Fig. 1.12. (a) The ﬁeld of attraction and its potential inside mass, (b) spherical shell of ﬁnite thickness, (c) ﬁeld of attraction due to shell masses, (d) ﬁeld inside a shell, (e) illustration of Equation (1.137).

Note that at the sphere’s surface, as well as everywhere else, the ﬁeld is a continuous function. In particular, if R ¼ a, Equations (1.127 and 1.128) give: geR ¼

4p kda ¼ giR 3

Certainly, Equations (1.127 and 1.128) can be derived directly from Equation (1.6), but it requires a rather cumbersome integration. This example also allows us to illustrate the fact that the gravitational ﬁeld has a ﬁnite value inside any mass. With this purpose in mind, imagine that the observation point p is located at the center of a small and homogeneous sphere, Fig. 1.12a. Then, the total ﬁeld can be represented as a sum: gðpÞ ¼ gi ðpÞ þ ge ðpÞ where gi and ge are ﬁelds caused by masses inside and outside the small sphere, respectively. Inasmuch as the ﬁeld gi is equal to zero, but sources of the ﬁeld ge are located at the small but ﬁnite distance from the point p, we conclude that the ﬁeld g(p) has a ﬁnite value inside the masses. It is also useful to ﬁnd an expression for the potential of this ﬁeld. Since the function U vanishes at inﬁnity, we start from the

45

Principles of Theory of Attraction

external medium, R4a. As follows from Equation (1.55) UðRÞ ¼

R1 R

k RM2 dR ¼ k M R ;

R a

and, as we can expect, the potential coincides with that of a point mass. In particular, on the spherical surface we have UðaÞ ¼ k

4p 2 da 3

because the sphere is uniform. Now, knowing the potential at the surface, it is simple to ﬁnd it inside the sphere. Applying again Equation (1.55), we obtain Z

R

4pd k g dR ¼ 3 i

UðRÞ UðaÞ ¼ a

Z

R

RdR ¼ a

2pd kða2 R2 Þ 3

Whence 2 2 UðRÞ ¼ 2pkd 3 ð3a R Þ if

Ra

Thus, the potential reaches a maximum at the sphere’s center and then decreases as a parabolic function. A completely different behavior is observed outside the sphere. This simple problem allows one to demonstrate again that the potential obeys Poisson’s equation. Consider the potential at the point p of an arbitrary body, Fig. 1.12a, assuming that the density may change from point to point. Let us mentally draw a spherical surface around the point p. If its radius is sufﬁciently small, we can suppose that this sphere is homogeneous. The potential at the point p can be written as UðpÞ ¼ U i ðpÞ þ U e ðpÞ where U i ðpÞ and U e ðpÞ are potentials, caused by masses inside and outside the sphere, respectively. Inasmuch as the potential depends on the coordinate R only, we have DU ¼ DU i ðpÞ ¼

i 1 @ 2 @U R ¼ 4pkd @R R2 @R

Substituting the expression for the potential Ui, derived above, we see that it is a solution of Poisson’s equation. In other words, the integral Z UðpÞ ¼ k

dðqÞdV Lqp V

satisﬁes the Poisson’s equation inside masses, and it is true for an arbitrary distribution of their density. Let us also notice that to some extent the ﬁrst example is useful in studying the normal attraction ﬁeld of the earth. In this light consider the next example.

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Methods in Geochemistry and Geophysics

1.13.2. The attraction field of a thin spherical shell, Fig. 1.12b Applying again the Gauss’s formula and taking into account the spherical symmetry, we ﬁnd that inside the shell, Roa1 , the ﬁeld is absent, while outside, R4a2 , it behaves as a point source situated at the origin. Thus, we have giR ¼ 0 if Roa1

geR ¼ k

and

M R2

if

R4a2

ð1:129Þ

We see that a summation of ﬁelds of elementary masses outside of the shell produces the same result as a point source, placed at the center of the shell. This is the second example of such equivalence, and again it is an exception. Consider a spherical surface S with radius R ¼ a1 þ roa2 . Then, the radial component inside the mass is gR ¼ k

m R2

ð1:130Þ

where m is a mass located inside the volume, surrounded by the surface S. Assuming that the thickness of the shell: h ¼ a2 a1 is small with respect to a1, we can represent the mass as: m ¼ 4pR2 rd Correspondingly, in place of Equation (1.130) we have: gR ¼ 4pkdr

ð1:131Þ

The behavior of gR as a function of R is shown in Fig. 1.12c, and, of course, it is a continuous function. Now let us mentally decrease the thickness h and increase the volume density so that the mass remains the same. In such a way we arrive at a distribution of masses with a surface density, and this replacement does not change the ﬁeld outside the shell, but it leads to a discontinuity of the ﬁeld at the surface masses. It is instructive to demonstrate why the ﬁeld inside the shell, Roa1 , is zero. Let us take any point p, Fig. 1.12d, and form an elementary cone with its apex at this point. The lateral surface of the cone intersects the shell and generates inside two elementary surfaces dS1 and dS2. By deﬁnition we have: dS1 ¼ oR21

and

dS 2 ¼ oR22

where o is the solid angle included by the cone. Therefore, the ﬁelds caused by the elementary masses of these surfaces are gð1Þ R ¼ k

m ¼ kso; R21

gð2Þ R ¼k

m ¼ kso R22

and they cancel each other. Since the shell can be represented as a system of such pairs, we see that the total ﬁeld inside the shell is zero. Note, that the potential is constant inside the shell, but outside it behaves as the potential of a point source. It is obvious, that its equipotential surfaces are spherical, with their center

Principles of Theory of Attraction

47

coinciding with the origin 0. Next example is interesting from the practical point of view. 1.13.3. Evaluation of a mass of the arbitrary body Suppose that we know the ﬁeld at points of some plane and that it is caused by an arbitrary body, located beneath this surface of observation. Also, the ﬁeld gz, Fig. 1.12e, vanishes far away from the body. The lower half space is a volume where all the sources are located and it is surrounded by the observation plane S and a hemi-spherical surface S0 with relatively large radius where the ﬁeld can be treated as that of a point source. Correspondingly, the ﬂux through this surface S0 is Z Z M gdS ¼ k 2 dS ¼ 2pkM R S0 S0 For this reason the ﬂux through a closed surface surrounding this volume is expressed in terms of surface integral over the plane of observation. Thus, we have I Z gdS ¼ gz dS 2pkM ¼ 4pkM ð1:132Þ S

since on the plane of observation: gdS ¼ gz dS Here gz is the magnitude of the vertical component of the ﬁeld on surface S. Therefore, we have found the mass, which causes the ﬁeld and it is equal to Z 1 M¼ g dS ð1:133Þ 2pk S z

1.13.4. The normal component of the attraction field due to planar surface masses Suppose that masses are distributed within a plane layer whose thickness is much smaller than the distance from these masses to the observation point, Fig. 1.13a. In other words, the distance between the observation point and any point of the elementary volume is practically the same. Taking into account this fact, we can replace this layer by a plane surface with the same mass, located somewhere at the middle of the layer, Fig. 1.13b. Inasmuch as every elementary volume contains the mass dm ¼ dðqÞhdS;its distribution on the surface can be described by dm ¼ sðqÞdS; where sðqÞ ¼ dðqÞh

ð1:134Þ

and sðqÞ is a surface density of surface masses. Of course, every element of the surface S bears the same mass as a corresponding elementary volume of the layer. Our next step is to ﬁnd the ﬁeld caused by these surface masses. First of all, we will distinguish the back and front sides of the surface S and choose the direction of the normal n(q) at every point of the surface from the back to the front side. The ﬁeld

48

Methods in Geochemistry and Geophysics

(a)

(b)

p dg

p

dg n

n

dS q

S

q

h

(c)

(d)

1

p

2

3

S q

Fig. 1.13. (a) Thin layer and surface mass, (b) normal component of the ﬁeld due to surface masses, (c) illustration of a solid angle near surface, (d) normal component along proﬁles 1, 2, and 3.

caused by an elementary mass on the surface dS is dm sdS dgðpÞ ¼ k 3 Lqp ¼ k 3 Lqp Lqp Lqp

ð1:135Þ

At every point above and beneath the masses, the ﬁeld can be represented as a sum of the tangential and normal components of the ﬁeld: ð1:136Þ gðpÞ ¼ gt t þ gn n where t and n are unit vectors, tangential and normal to the plane, respectively. As is seen from Fig. 1.13b, the normal component of the ﬁeld at the point p caused by an elementary mass is sdS ð1:137Þ dgn ¼ ndg ¼ k 3 Lqp n Lqp Further we assume that the masses are distributed uniformly; that is, s ¼ constant. Applying the principle of superposition we obtain for the normal component of the ﬁeld due to all surface masses: Z dS Lqp gn ðpÞ ¼ ks ð1:138Þ L3qp S since dS ¼ dSn. Taking into account the fact that Lqp ¼ Lpq and L3qp ¼ L3pq , we have for the normal component of the ﬁeld: Z dS Lpq gn ðpÞ ¼ ks ð1:139Þ L3pq S

Principles of Theory of Attraction

49

As is well known, the integrand describes the solid angle under which we see the elementary surface from the point p. Respectively, the integral at the right hand side of this equation characterizes the solid angle for the whole surface: Z dS Lpq oðpÞ ¼ ð1:140Þ L3pq S Thus, the normal component of the attraction ﬁeld caused by masses uniformly distributed on the planar surface can be expressed as ð1:141Þ gn ðpÞ ¼ ksoðpÞ In the general case, when the density varies, Equation (1.137) gives Z gn ðpÞ ¼ k sðqÞdoðq; pÞ

ð1:142Þ

S

Now we will describe the behavior of the normal component gn in several cases. 1.13.4.1. Case one: A planar surface of an inﬁnite extent The solid angle under which the plane is seen from the front and back sides does not depend on the position of the point p and is equal to 82p, respectively. In other words, planar surface masses with inﬁnite extension and constant density create a uniform ﬁeld in each half space: gz ¼ 2pks if

z40

and gz ¼ 2pks

if

z40

Here the masses are located on the plane z ¼ 0 and the z-axis is directed along the normal n. We see that regardless of the distance from the plane the ﬁeld remains the same. Qualitatively, this can be explained in the following way. The ﬁeld gn ðpÞ is a sum of normal components caused by all surface elementary masses. If the point p is situated very close to the surface z ¼ 0, then the nearest element gives the main contribution, since it is seen at the solid angle which is almost equal to 72p, (Fig. 1.13c). At the same time, the angles for other elements are very small, and correspondingly their contribution to this ﬁeld component is negligible. With an increase of distance from the plane the ﬁeld caused by the ﬁrst element decreases, but the inﬂuence of more remote elements becomes stronger. As we demonstrated earlier these two effects compensate each other, and the ﬁeld is independent of z. This result will be used later in studying the ﬁeld inside and outside a layer of ﬁnite thickness. From Equation (1.141) it follows that gþ n gn ¼ 4pks

gþ n and

g n

ð1:143Þ

where are values of the ﬁeld at the front and back sides of the surface, and this discontinuity arises due to replacement of volume masses by surface masses. Inasmuch as any plane perpendicular to the surface masses is the plane of symmetry, the tangential component gt is equal to zero. This means that we can always ﬁnd two equal elementary masses, located at the same distance from the plane of

50

Methods in Geochemistry and Geophysics

symmetry. It is clear that their normal components have the same direction, but the tangential components have the same magnitude but oppose each other and, correspondingly, their sum is zero. 1.13.4.2. Case two: A plane of ﬁnite extension, (Fig. 1.13d) Consider a behavior of the component gz along three proﬁles perpendicular to the plane z ¼ 0, (Fig. 1.13d). The ﬁrst proﬁle intersects masses of this plane, and with an increase of the distance from them the solid angle becomes smaller. Therefore the normal component of the ﬁeld decreases. On other hand, in approaching the surface the solid angle tends to its limiting values: 2p and 2p on the front and back sides of the surface, respectively. It is essential that in the vicinity of the plane surface the component gn is deﬁned by only an elementary mass, located near the observation point. The linear dimensions of this element have to exceed greatly the coordinate z of the point p. The second proﬁle goes through the hole, where masses are absent. If the point of observation is situated inside this hole the normal component vanishes, since the ﬁeld caused by each elementary mass has only a tangential component. The same result follows from the fact that the solid angle under which the plane is seen from any point of the hole is zero. Thus, with an increase of the distance z the magnitude of gz increases and reaches a maximum, and then it gradually becomes smaller. Finally, the behavior of this component along the third proﬁle is similar to the previous case. 1.13.4.3. Case three: A plane surface has a form of a disk with radius a The normal component of the ﬁeld at points of the z-axis is gz ¼ 2pksð1 cos aÞ

ð1:144Þ

since the solid angle under which the disk is seen from points of the z-axis is o ¼ 2pð1 cos aÞ and

cos a ¼

jzj ðz2 þ a2 Þ1=2

With an increase of the distance z, the angle a tends to zero, and therefore the ﬁeld gz decreases. Expanding the radical ðz2 þ a2 Þ1=2 in a series 1=2 a2 a2 3a4 2 2 1=2 1 1 ¼ j zj 1þ 2 jzj 1 2 þ 4 ðz þ a Þ z 2z 8z we obtain gz k

pa2 s 3a2 1 z2 4z2

or m 3a2 gz k 2 1 2 ; z 4z

if a jzj

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Principles of Theory of Attraction

(a)

(b)

z

z y

x

p

h

0

0 (c)

(d)

z

z x

z

gz

gz

Fig. 1.14. (a) A layer of ﬁnite thickness, (b) evaluation of ﬁeld inside a layer, (c) ﬁeld due to plane surface masses, (d) behavior of the ﬁeld due to volume and surface masses.

where m is the total mass of the disk. From this equation it is very simple to evaluate the minimal distance z at which the ﬁeld of masses located on the disk coincides with that of a point mass. In approaching the disk the ﬁeld tends to its limit, equal to 82pks, that corresponds to the ﬁeld due to an inﬁnite plane surface. Applying again the expansion of the same radical, we have j zj gz ðzÞ 2pks 1 if a jzj ð1:145Þ a 1.13.5. Field caused by a volume distribution of masses in a layer with thickness h and density d First, introduce a Cartesian system of coordinates with its origin at the middle of the layer and z-axis directed perpendicular to its surface. Let us note that the layer has inﬁnite extension along the x and y axes, (Fig. 1.14a). At the beginning, suppose that the observation point is located outside the layer, that is, jzj4h=2: Then we mentally divide the layer into many thin layers which in turn are replaced by a system of plane surfaces with the density s ¼ dDh, where Dh is the thickness of the elementary layer. Taking into account the inﬁnite extension of the surfaces, the solid angle under which they are seen does not depend on the position of the observation point and equals either 2p or 2p. Correspondingly, each plane surface creates the same ﬁeld: dgz ¼ 2pkdDh

if z4

h 2

and dgz ¼ 2pkdDh

if z4

h 2

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Methods in Geochemistry and Geophysics

Therefore, after summation the total ﬁeld due to all masses in the layer is h gz ¼ 2pkdh if z 2 h gz ¼ 2pkdh if z 2

ð1:146Þ

It is interesting to notice that these formulas are used to calculate the Bouguer correction. Now we will study the attraction ﬁeld inside a layer when the coordinate of the observation point z satisﬁes the condition, (Fig. 1.14b):zoh=2. First, suppose that z40. Then, the total ﬁeld can be presented as a sum of two ﬁelds: one of them is caused by masses with thickness equal h=2 z;and the second one is due to masses in the rest of the layer, having the thickness z þ h=2, (Fig. 1.14b). In accordance with Equation (1.146) these ﬁelds are h ð1Þ gz ¼ 2pkd z 2 ð1:147Þ h gð2Þ þ z ¼ 2pkd z 2 Correspondingly, for the total ﬁeld we have ð2Þ gz ¼ gð1Þ z þ gz ¼ 4pkdz

if 0 z

h 2

and by analogy gz ¼ 4pkdz

if

h z0 2

ð1:148Þ

The behavior of the ﬁeld gz caused by masses of the layer is shown in Fig. 1.14c. Thus, for negative values of z the ﬁeld component gz inside the layer is positive, since the masses in the upper part of the layer create a ﬁeld along the z-axis, and this attraction prevails over the effect due to masses located below the observation point. At the middle of the layer, where z ¼ 0, the ﬁeld is equal to zero. Of course, every elementary mass of the layer generates a ﬁeld at the plane z ¼ 0, but due to symmetry the total ﬁeld is equal to zero. For positive values of z the ﬁeld has opposite direction, and its magnitude increases linearly with an increase of z. As follows from Equations (1.146–1.148) the ﬁeld changes as a continuous function at the layer boundaries. One more feature of the ﬁeld behavior is worth noting. Inasmuch as the layer has inﬁnite extension in horizontal planes, the distribution of masses possesses axial symmetry with respect to any line parallel to the z-axis that passes through the observation point. For this reason, it is always possible to ﬁnd two elementary masses such that the tangential component of the ﬁeld caused by them is equal to zero. Respectively, the ﬁeld due to all masses of the layer has only a normal component gz. Let us consider now a transition from a layer with a ﬁnite thickness to a plane of surface masses, where z ¼ 0. Suppose that the layer thickness tends to zero, but the volume density increases in such way that their product, s ¼ dh remains constant. Then, the ﬁeld outside the layer does not change and equals its original value, while

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Principles of Theory of Attraction

inside it varies more rapidly and in the limit a discontinuity takes place, (Fig. 1.14d). In reality, regardless of how small the layer thickness is, the ﬁeld is always a continuous function of z, but it appears that there is a singularity when the ﬁeld is not studied inside the layer. Next, we consider the potential of this attraction ﬁeld. As was shown earlier: Z z UðzÞ UðaÞ ¼ gdl ð1:149Þ a

Since the ﬁeld does not vanish at inﬁnity, we assume that the potential is known at some point either outside or inside the layer. For instance, suppose that U ¼ 0 at the plane z ¼ 0. Taking into account that the integral in Equation (1.149) is path independent, we choose dl ¼ dz This gives Rz

Rz

h ð1:150Þ 2 Thus, inside the layer the potential is a parabolic function and, in particular, at the boundary we have: h h pkh2 d U ð1:151Þ ¼U ¼ 2 2 2 UðzÞ ¼

0

gz dz ¼ 4pkd

0

zdz ¼ 2pkdz2

if jzj

This allows us to ﬁnd the potential outside the layer. In fact, Equation (1.149) gives pkdh2 h UðzÞ 2pkdh z 2 2 or UðzÞ ¼ 2pkdhz þ

pkdh2 2

if

z

h 2

ð1:152Þ

By analogy, we have pkdh2 h ð1:153Þ if z 2 2 The behavior of the function U(z) is shown in Fig. 1.15a. In the limiting case of surface masses the potential remains a continuous function, but its derivative, @U=@z; has a discontinuity at the plane z ¼ 0, Fig. 1.15b. In deriving Equations (1.152 and 1.153) we used the fact that the potential is a continuous function at the layer boundaries; otherwise the ﬁeld would be inﬁnitely large. UðzÞ ¼ 2pkdhz þ

1.13.6. Determination of layer density Now let us consider the ﬁeld inside a layer when it is surrounded by other layers, but their density depends on the coordinate z only. Then, as was demonstrated

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Methods in Geochemistry and Geophysics

(a)

(b)

z

U

0

z

U

0

(c)

z

1(z) z1

• 0•

x

z2

Fig. 1.15. (a) Potential caused by a layer of ﬁnite thickness, (b) potential due to surface masses, (c) illustration of Equation (1.159).

above, the ﬁeld inside the layer with the constant density d can be represented as gz ðzÞ ¼ g1z þ g2z 4pkdz if

h h z 2 2

where g1z and g2z are the ﬁeld components caused by the masses of the upper and lower media; neither of these ﬁelds change within the layer, Fig. 1.15c. This remarkable feature permits us to determine the density d. In fact, the ﬁeld at two arbitrary points inside the layer can be written in the form: gz ðz1 Þ ¼ g1z þ g2z 4pkdz1

and

gz ðz2 Þ ¼ g1z þ g2z 4pkdz2

Therefore, the difference of ﬁelds is independent on a surrounding medium and it gives d¼

gz ðz1 Þ gz ðz2 Þ 4pkðz2 z1 Þ

ð1:154Þ

In principle, this equation illustrates an important feature of measuring the rock density in the borehole.

1.14. LEGENDRE’S FUNCTIONS AND A SOLUTION OF LAPLACE’S EQUATION In studying the attraction and gravitational ﬁelds of the earth it is very useful to represent the potential in terms of Legendre’s functions; and this is mainly related to the fact that the earth is almost spherical and the position of observation points is

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Principles of Theory of Attraction

characterized by spherical coordinates. Besides, these functions are very important in solving many boundary value problems, which include a solution of Laplace’s equation as well as other partial differential equations of the second order. 1.14.1. Expansion of the function 1=Lqp in the power series First, we introduce Legendre’s functions, considering the power series of the function j ¼ L1 qp

ð1:155Þ

where Lqp ¼ ðR2 þ R21 2RR1 cos yÞ1=2 is the distance between two points. Here R and R1 are the radius vectors with the origin at the point 0 and y is the angle between them, Fig. 1.16. We can write 1 1 ¼ ð1 þ r2 2rmÞ1=2 Lqp R1

if RoR1

1 1 ¼ ð1 þ r2 2rmÞ1=2 Lqp R

if R4R1

and ð1:156Þ

Here m ¼ cos y, ð1 m þ1Þ and r is either equal to R=R1 o1 or R=R1 41. It is essential that in both cases this parameter is less than unity. Now we expand the function jðr; mÞ ¼ ð1 þ r2 2rmÞ1=2

ð1:157Þ

in a power series with respect to r, provided that 0oro1 and 1 m þ1. By deﬁnition, jðr; mÞ ¼

X jðnÞ ð0; mÞ n¼0

n!

rn ¼

X

Pn ðmÞrn

ð1:158Þ

n¼0

and Pn ðmÞ are called Legendre’s functions or Legendre’s polynomials; they depend on only one parameter, m ¼ cos y, which changes from 1 to +1. Lqp q R1 θ

0 Fig. 1.16. Illustration of Equation (1.155).

p R

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Methods in Geochemistry and Geophysics

Differentiating the function j with respect to r and substituting r ¼ 0, we obtain jð0Þ ¼ 1 " j ð0Þ ¼ " ð2Þ

j ð0Þ ¼

#

rm

ð1Þ

ð1 þ r2 2rmÞ3=2

3ðr mÞ2 ð1 þ r2 2rmÞ5=2

¼ m ¼ cos y

ð1:159Þ

r¼0

#

1 ð1 þ r2 2rmÞ3=2

¼ 3cos2 y 1 r¼0

and so on. Correspondingly, Equation (1.158) is written in the form: jðr; mÞ ¼

1 ½1 þ r2 2rm1=2

¼ 1 þ r cos W þ r2

3cos2 y 1 þ 2

ð1:160Þ

and therefore P0 ðmÞ ¼ 1;

P1 ðmÞ ¼ cos y;

3 1 2 cos y ; P2 ðmÞ ¼ 2 3

5 3 P3 ðmÞ ¼ ðcos3 y cos yÞ 2 5 ð1:161Þ

In principle, in the same manner we can obtain the Legendre’s functions of any order n. Now, substituting Equation (1.158) into Equation (1.156) we arrive at the following equations: 1 1 P R n ¼ Pn ðcos yÞ if RoR1 Lqp R1 n¼0 R1 and 1 1 P R1 n ¼ Pn ðcos yÞ Lqp R n¼0 R

if R4R1

ð1:162Þ

These formulas are of a great importance, because they allow one to describe the potential of the attraction ﬁeld, caused by an arbitrary distribution of masses, in terms of Legendre’s functions and the distance from the origin, R. From the physical point of view, it is clear that Equation (1.162) characterizes the potential due to a unit mass. Then, applying the principle of superposition, it is easy to generalize them and obtain formulas for the potential inside and outside of any masses, and this approach will be applied in the next chapter. Legendre’s polynomials have been studied in detail and they posses many remarkable features. For instance, these functions, like the trigonometric and Bessel functions, are orthogonal; more precisely, Legendre’s polynomials of different order are orthogonal over the interval (–1, 1): R1 1 Pn ðmÞPm ðmÞdm ¼ 0 if man

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Principles of Theory of Attraction

and R1 1

2 P2n ðmÞdm ¼ 2nþ1

if m ¼ n

This is one of the main reasons why these functions play a very important role in solving boundary value problems. Also, between Legendre’s polynomials of different order, there is a simple recursive relationship: ðn þ 1ÞPnþ1 ðmÞ mð2n þ 1ÞPn ðmÞ þ nPn1 ðmÞ ¼ 0 and this can be used for calculating Pn ðmÞ42, since P0 ðmÞ ¼ 1 and P1 ðmÞ ¼ m. Let us notice that due to orthogonality of Legendre’s polynomials many functions can be represented as a series, which is similar to Equation (1.162), and this fact is widely used in mathematical physics. Now, we will derive the differential equation, one of the solutions of which are Legendre’s functions. 1.14.2. Laplace’s equation and Legendre’s functions Let us demonstrate that with help of Legendre’s functions we can ﬁnd a solution of Laplace’s equation. As is well known, Laplace’s equation has the following form in the spherical system of coordinates: @ @U @ @U 2 R sin y sin y þ ¼0 ð1:163Þ @R @R @y @y provided that the function U is independent of the coordinate j. This is a partial differential equation of second order, and U depends on two spherical coordinates: R and y. To solve Equation (1.163) we suppose that its solution can be represented as the product of two functions, so that each function depends on one argument only; and consequently we have U ¼ TðRÞPðyÞ

ð1:164Þ

Substituting Equation (1.164) into Equation (1.163) we obtain @ @ @P 2 @T R sin y PðyÞ sin y þ TðRÞ ¼0 @R @R @y @y Dividing both parts by PðyÞTðRÞ sin y, we have 1 @ @T 1 @ @P R2 sin y þ ¼0 TðRÞ @R @R PðyÞ sin y @y @y

ð1:165Þ

On the left hand side of Equation (1.165) it is natural to distinguish two terms: 1 @ @T R2 Term 1 ¼ TðRÞ@R @R and 1 1 @ @P sin y Term 2 ¼ PðyÞsin y@y @y

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Methods in Geochemistry and Geophysics

At ﬁrst glance it seems that they depend on the arguments R and y, respectively, and Equation (1.165) can be represented as Term 1ðRÞ þ Term 2ðyÞ ¼ 0 However, such an equality is impossible, since by changing one of arguments, for example, R, the ﬁrst term varies while the second one remains the same, and correspondingly the sum of these terms cannot be equal to zero for arbitrary values of R and y. Therefore, we have to conclude that neither term depends on the coordinates and each is constant. This fact constitutes the key point of the method of separation of variables, allowing us to describe the function U as a product of two functions, each of them depending on one coordinate only. For convenience, let us represent this constant in the form 7m2, where m is called a constant of separation. Thus, instead of Laplace’s equation we have two ordinary differential equations of second order: 1 d 2 dT R ¼ m2 TðRÞdR dR and 1 d dP sin y ¼ m2 PðyÞ sin ydy dy

ð1:166Þ

Letting m2 ¼ nðn þ 1Þ and choosing the minus sign at the right hand side of the second equality we obtain Legendre’s equation; one of its solutions is the function Pn ðcos yÞ: d dP sin y þ nðn þ 1Þ sin yPðyÞ ¼ 0 ð1:167Þ dy dy Correspondingly, the equation for the function T(R) is d dT R2 nðn þ 1ÞTðRÞ ¼ 0 dR dR

ð1:168Þ

and it is simple to see that functions Rn and Rn1 obey this equation. Note that Legendre’s equation has a second solution, but it is not considered here. As follows from Equation (1.164) for each n we have a partial solution, which can be represented in the form: U n ðR; yÞ ¼ ðAn Rn þ Bn Rn1 ÞPn ðcos yÞ

ð1:169Þ

Performing a summation with respect to n, we obtain the general solution, which is independent of n: X ðAn Rn þ Bn Rn1 ÞPn ðcos yÞ ð1:170Þ UðR; yÞ ¼ n¼1

In essence, Equation (1.162) is an example, when the function L1 qp is presented as a sum of partial solutions of Laplace’s equation. Further, we will use Equation (1.170) to describe the gravitational ﬁeld of the earth.

Chapter 2 Gravitational Field of the Earth 2.1. FORCES ACTING ON AN ELEMENTARY VOLUME OF THE ROTATING EARTH AND THE GRAVITATIONAL FIELD As is well known, the earth is mainly a ﬂuid; the upper crust is an exception, but it is extremely thin layer with respect to the earth’s radius. For this reason it is natural to expect that rotation around its axis makes the shape of the earth practically the same as if it was a ﬂuid, and we will follow this conventional point of view. Suppose that during this motion the mutual position of all elementary volumes of the earth remains the same, and correspondingly each of them is involved only in rotation with angular velocity o. This means that the effect of different types of currents inside the earth is neglected and we deal with hydrostatic equilibrium. 2.1.1. Equation of motion of an elementary volume Consider a rotation of the earth around the z-axis in which every particle, elementary volume, of the earth moves along the horizontal circle with the radius r. Our ﬁrst goal is to ﬁnd the distribution of forces inside the earth; and with this purpose in mind we will derive an equation of motion for an elementary volume of the ﬂuid. Let us introduce a Cartesian system of coordinates with its origin 0, located on the z-axis of rotation. Since this frame of reference is an inertial one, it does not move with the earth, we can write Newton’s second law as ma ¼ F

ð2:1Þ

where m is the mass of the elementary volume, a the centripetal acceleration, and F the real force applied to mass. Suppose that the elementary volume is a cube and its central point q has coordinates x, y, and z. There are two types of forces acting on this volume element dV ¼ dxdydz, Fig. 2.1a. One of them is the force of attraction Fa; it obeys Newton’s law of attraction and is caused by all other masses. The second type, Fs, is the surface force, and it is described in terms of the pressure, applied to different parts of the surface surrounding the volume. At the beginning, consider a motion of the particle along the x-axis. By deﬁnition, the scalar component of the force of attraction is F ax ¼ mgax ¼ gax ddV

ð2:2Þ

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Methods in Geochemistry and Geophysics

(a)

(b) z gs θ

z y dz

r

q dy

ga

dx

λ

r p

x Fig. 2.1. (a) Elementary volume of a ﬂuid, (b) attraction and gravitational ﬁelds on the earth’s surface.

Here d is the volume density and gax the scalar component of the attraction ﬁeld, that is, the force per unit mass. To describe the effect of surface forces, consider front- and the backsides of the volume, dS(x+dx/2,y,z) and dS(xdx/2,y,z), which are perpendicular to the x-axis. Due to the action of the neighboring part of the medium the surface force acting on backside is x dx x dx F ax ; y; z ¼ P ; y; z dS ð2:3Þ 2 2 Here P is the pressure, which is uniformly distributed over the face of the cube. This force causes a deformation of the elementary volume and, as a result, it gives rise to a force on the medium in front of the cube. In accordance with the Newton’s third law this side is subjected to the force: x þ dx x þ dx F ax ; y; z ¼ P ; y; z dS ð2:4Þ 2 2 Therefore, the resultant force, applied to opposite sides of the cube is x þ dx x dx @P F ax ¼ P ; y; z P ; y; z dS ¼ dV 2 2 @x where the derivative is taken at the middle point q. Correspondingly, the equation of motion along the x-axis is

@P dV þ gax ddV ¼ ax ddV @x

or

@P þ gax d ¼ ax d @x

ð2:5Þ

In the same manner a motion along the two other coordinate lines is described by the equations:

@P þ gay d ¼ ay d @y

and

@P þ gaz d ¼ az d @z

ð2:6Þ

Multiplying Equations (2.5 and 2.6) by the corresponding unit vectors i, j, and k and adding them, we obtain the equation of motion of an elementary volume inside

Gravitational Field of the Earth

61

the earth: grad P þ ga d ¼ ad

ð2:7Þ

provided that the hydrostatic equilibrium holds. Inasmuch as the earth rotates around its axis, there is only a centripetal acceleration and it is equal to a ¼ o2 r ¼ o2 rr0

ð2:8Þ

Here r0 is the unit vector perpendicular to the z-axis, and r is its distance to the point q. From this equation it follows that the sum of the two true forces represents the centripetal force. In other words, the distribution of the attraction and surface forces per unit volume: F1 ¼ ga d

and

Fs ¼ rP

ð2:9Þ

is such that at every point inside the earth the resultant force is located in the horizontal plane, directed toward the rotation axis and its magnitude provides the given angular velocity of each elementary volume. Note also that the force Fs shows a direction along which there is a maximal decrease in the pressure. Taking into account Equation (2.8), in place of Equation (2.7) we have rP þ ga d ¼ do2 rr0

ð2:10Þ

and its left hand side characterizes the centripetal force per unit volume. It is appropriate to emphasize again that each element is subjected to the action of only two forces, which differ essentially from each other. The attraction force is caused by all elements of the earth and directed inside it, while the surface force is caused by a change of the pressure on the surface surrounding the elementary volume and directed away from the central part of the earth. Both the attraction and surface forces vary with a change of the angular velocity, because the distribution of the density of a ﬂuid depends on o. Of course, Equation (2.10) cannot be applied for elementary volume of the solid crust; however, the rotation is still caused by only the attraction and surface forces, and the latter are described by a stress tensor. 2.1.2. The field gs of the surface forces in a fluid Inasmuch as formulas for the attraction and centripetal forces are known, it is natural to express the surface forces in terms of these quantities. With this purpose in mind it is convenient to introduce the ﬁeld of the surface forces as Fs ¼ rP ¼ dgs

ð2:11Þ

gs þ ga ¼ o2 r

ð2:12Þ

Then, Equation (2.10) becomes

where r is the radius vector, perpendicular to the axis of rotation. Thus, gs ¼ ðga þ o2 rÞ

ð2:13Þ

Later we will demonstrate that the gravitational ﬁeld g has the same magnitude as the ﬁeld of the surface forces, gs, but the opposite direction. Until now we have

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Methods in Geochemistry and Geophysics

studied the motion of an elementary located inside the ﬂuid Earth. Next, suppose that some object is located on the earth’s surface and it is at rest. Due to the attraction force this body produces in its vicinity a deformation of the earth’s surface and, in accordance with Newton’s third law, a force of reaction arises and acts on this mass. It is clear that this force is directed away from the earth and is described by the ﬁeld gs, which is located in the same plane as the attraction ﬁeld and the centripetal acceleration. Otherwise, it would be impossible to achieve equilibrium. Thus, in the same manner as inside, these forces have to provide on the earth’s surface a rotation with the given angular velocity. Now we demonstrate that the orientation of the ﬁeld gs has to satisfy the following conditions, Fig. 2.1b: gs i40

and

gs k40

ð2:14Þ

Then, this ﬁeld shows, Equation (2.11), that the pressure inside the ﬂuid becomes lesser while approaching the earth’s surface and the resultant ﬁeld is much smaller than the attraction ﬁeld. To illustrate the necessity of such an orientation of the reaction (surface) force, suppose that this force is absent and the point q is located at the earth’s surface in the vicinity of the equator. In accordance with Equation (2.12) g 1=2 o¼ a a where a is a major semi-axis of the earth. Approximately, a ¼ 6.4 106 m and ga ¼ 9.8 m/s2. This gives o ¼ 1:2 103 s1 At the same time the angular velocity of the earth is equal to 2p 7:3 105 s1 24 36 102 that is, it is two orders of magnitude smaller than the angular velocity caused by the attraction force only. Next, assume that the surface force is present and it is oriented along the same line as the attraction force, but in the opposite direction. However, this is also impossible because in such a case the z-component of the resultant force is not equal to zero, which contradicts the condition of equilibrium, Equation (2.12). We may expect this result. In fact, when motion is absent the sum of the attraction and surface forces is zero, but in the presence of a rotation with a very small angular velocity the magnitude and orientation of the surface force change only slightly. The orientation of both ﬁelds is shown in Fig. 2.1b, and in order to determine components of the ﬁeld gs, we will proceed from Equation (2.13). First, let us use the cylindrical system of coordinates where o¼

gs ¼ gsr ir þ gsz iz Then, Equation (2.13) gives gsr ¼ ðgar þ o2 rÞ

and

gsz ¼ gaz

ð2:15Þ

Gravitational Field of the Earth

63

As is seen from Fig. 2.1b gs cos l ga cosðl yÞ þ o2 r ¼ 0

or

gs cos l ¼ ga cosðl yÞ o2 r

and gs sin l ¼ ga sinðl yÞ

ð2:16Þ

Here l is the angle between the ﬁeld gs and unit vector r0, and y the angle between ﬁelds gs and gs. Note, that the last inequality of the set (2.14) is necessary to provide equilibrium along the z-axis. From the last two equations we have for the magnitude of the surface force gs ¼ ðg2sr þ g2sz Þ1=2 ¼ ½g2a cos2 ðl yÞ 2o2 rga cosðl yÞ þ o4 r2 þ g2a sin2 ðl yÞ1=2 or

o2 r cosðl yÞ gs ga 1 ga

ð2:17Þ

It is also instructive to consider components of the ﬁeld gs along and perpendicular to the direction of the ﬁeld of attraction. From Equation (2.13) we have gs cos y ¼ ½ga o2 r cosðl yÞ and gs sin y ¼ o2 r sinðl yÞ Since the angle y is small we again obtain Equation (2.17), which can be written as gs ¼ ga o2 r cos l ð2:18Þ At the same time the inclination of the reaction ﬁeld gs to the direction of the attraction ﬁeld is o2 r sin l ð2:19Þ y sin y ¼ ga It is equal to zero at poles and at the equator and reaches a maximum at the latitude l ¼ p/4: o2 r ð2:20Þ ymax 0:7 ga By deﬁnition, the reaction ﬁeld gs at any point p is normal to its level surface at this point and for this reason the direction of gs is called the vertical that corresponds to the direction of a plumb line. Thus, we have found the components of the surface force and they are 1. along the attraction force ga þ o2 r cos l 2.

perpendicular to the attraction force o2 r sinðl yÞ

ð2:21Þ

This last component is deﬁned by the angular velocity and coordinates of the point and its dependence on the attraction force is through o.

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Methods in Geochemistry and Geophysics

Now we demonstrate one interesting feature of the magnitude of the surface force as a function of the latitude. First, assuming that the ﬂuid Earth is almost a sphere with a radius a, we can represent the distance r as r ¼ a cos l. Then, Equation (2.18) has the form gs ¼ ðga o2 acos2 lÞ

or

gs ¼ ½ðga o2 aÞ þ o2 asin2 l

ð2:22Þ

At points of the equator we have ges ¼ ga o2 a

ð2:23Þ

gs ðlÞ ¼ ges ð1 þ qsin2 lÞ

ð2:24Þ

and, therefore, we obtain

Here the parameter q¼

o2 a ges

ð2:25Þ

is very small. In fact, assuming that gs ¼ 9:8 m=s2 ;

a ¼ 6:4 106 m;

and

o ¼ 7:3 105 s1

we have 1 ð2:26Þ 289 This parameter can be interpreted in the following way. At the pole Equation (2.24) gives q¼

gps ¼ ges ð1 þ qÞ

or

q¼

gps ges ges

ð2:27Þ

Equations (2.24 and 2.27) look like as Clairaut’s formulas which will be derived later. However, this similarity is superﬁcial, since the former do not contain the ﬂattening of the earth. A variation of the ﬁeld magnitude, gs with latitude, Equation (2.24), is caused by only a change of the centripetal acceleration on the spherical surface. 2.1.3. The gravitational field g In order to study the attraction of masses of the earth which moves around the axis of rotation, it seems appropriate to use the ﬁeld gs, which depends on the distribution of masses and the angular velocity, as well as coordinates of the point. Besides, it has a physical meaning of the reaction force per unit mass. However, it has one very serious shortcoming, namely, unlike the attraction force it is directed outward. In other words, it differs strongly from the attraction ﬁeld, in spite of the fact that the contribution of rotation is extremely small. To overcome this problem we introduce the gravitational ﬁeld g which differs from the reaction ﬁeld in direction only: gðpÞ ¼ gs ðpÞ

ð2:28Þ

Gravitational Field of the Earth

65

and in accordance with Equation (2.13) gðpÞ ¼ ga ðpÞ þ o2 rðpÞ

ð2:29Þ

Inasmuch as the second term at the right hand side of this equation is relatively small, the vector ﬁelds of the gravity and attraction have almost the same magnitude and direction. It is important to emphasize again that the magnitudes of the reaction and gravitational ﬁelds coincide and the former is measured. Notice that all derivations in this chapter are based on Equation (2.29) and in most cases we assume that the particle is at rest for an observer rotating together with the earth. 2.1.4. The centrifugal force As a rule, geophysical literature describes the rotation of a particle on the earth surface with the help of the attraction force and the centrifugal force. It turns out that the latter appears because we use a system of coordinates that rotates together with Earth. As we know Newton’s second law, ma ¼ F, is valid only in an inertial frame of reference, that is, the product of mass and acceleration is equal to the real force acting on the particle. However, it is not true when we study a motion in a system of coordinates that has some acceleration with respect to the inertial frame. For instance, it may happen that there is a force but the particle does not move. On the contrary, there are cases when the resultant force is zero but a particle moves. Correspondingly, replacement of the acceleration in the inertial frame by that in a non-inertial one gives a new relation between the acceleration, mass, particle, and an applied force: m

d 2 r1 ¼ F ma0 dt2

ð2:30Þ

Here r1 is the radius vector of the particle in a non-inertial system of coordinates and a0 is the acceleration of the moving frame. Certainly, this is not Newton’s second law since the right hand side of this equation contains not only the real force but also an additional term. However, even in such an accelerated frame of reference we may say that Newton’s second law is valid, if this second term is treated as force. Unlike real forces (gravitational, electric, and magnetic) this force is not exerted by any agent and for this reason it is usually called a ﬁctitious or pseudo force. For an observer in a non-inertial frame of reference, the pseudo force appears as a real one because of our strong belief in Newton’s second law in any frame. Such a classiﬁcation into real and ﬁctitious forces is justiﬁed within the framework of classical mechanics where the appearance of pseudo forces is merely a result of a transformation of the acceleration from a ﬁxed to a moving system of coordinates. There is another approach, which assumes that Newton’s second law is correct in any frame of reference and, correspondingly, all forces are real. We will follow the ﬁrst approach, when the equation of a motion along with the real forces contains additional terms. The most familiar pseudo force is the centrifugal force. Before we derive its expression would beappropriate to make one comment. There are cases when the centrifugal force is a real force. For instance, imagine a small body of

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Methods in Geochemistry and Geophysics

(a)

(b)

y y1

p x1

0

r

r 0

θ

x

Fig. 2.2. (a) Centripetal and centrifugal forces, (b) rotation of frame of reference (two-dimensional case).

mass m attached to a pin by a cord of length r and set revolving about it with the angular velocity o, Fig. 2.2a. It has a constant tangential velocity v and radial acceleration ar ¼ o2r. As follows from Newton’s second law in the inertial frame, the force acts on a body and produces a radial acceleration, and the direction of both the force and acceleration has to be the same, that is, toward the center of the circle. For this reason, it is called the centripetal force. The word ‘‘centripetal’’ means ‘‘seeking the center’’. Inasmuch as ma ¼ F

and

a¼

v2 ¼ o2 r r

the magnitude of the centripetal force is F ¼m

v2 ¼ mo2 r r

ð2:31Þ

This inward force has a real agent, namely, a tension of the cord, which is applied to the body. The same cord acts on the pin with a force, which has the same magnitude as that of the centripetal one but opposite direction. This force has the same physical origin, (deformation of cord) and it is called the centrifugal force. The term ‘‘centrifugal’’ means literally ‘‘ﬂeeing a center’’. Centripetal and centrifugal forces always form a pair of action and reaction and they are applied to different parts of the system. In our case these parts are a small body and a pin. Now we focus on pseudo forces and, ﬁrst, consider a two-dimensional case. 2.1.5. Frame of reference rotating with a constant angular velocity (two-dimensional case) Assume that origins of two Cartesian systems of coordinates are located at the same point and the frame of reference P0 rotates about a point 0 of the frame P with constant angular velocity o. Let us imagine two planes, one above another, so that the upper plane P0 rotates and, correspondingly, unit vectors i1and j1 change their direction, Fig. 2.2b. Consider an arbitrary point p, which has coordinates x, y on the plane P and x1, y1 on P0 , and establish relationships between these pairs of coordinates. For the radius vector of the point p in both frames we have r ¼ xi þ yj ¼ x1 i1 þ y1 j1 ¼ r1

ð2:32Þ

Gravitational Field of the Earth

67

and this is valid at any instant of time. Because of the rotation and translation of the point p in the frame P0 the coordinates x, y and x1, y1, as well as unit vectors of the rotating frame, are in general functions of the angle y between corresponding axes, Fig. 2.2b, where y ¼ ot. At the same time the unit vectors i and j are constants. Thus, for the velocity in the inertial frame we have v¼

dr dx dy dx1 dy di1 dj ¼ iþ j¼ i 1 þ 1 j 1 þ x1 þ y1 1 dt dt dt dt dt dt dt

ð2:33Þ

If a change of the angle y is very small, then the unit vector and its change are perpendicular to each other. For instance, for the unit vector i1 we have i1 ðy þ DyÞ i1 ðyÞ ¼ j1 Dy and di1 ¼ j1 dy

or

di1 ¼ oj1 ¼ xxi1 dt

ð2:34Þ

In the same manner dj1 ¼ oi1 ¼ xxj1 dt

ð2:35Þ

Substitution of Equations (2.34 and 2.35) into Equation (2.33) gives v¼

dr ¼ x_ 1 i1 þ ox1 j1 þ y_ 1 j1 oy1 i1 dt ¼ ðx_ 1 oy1 Þi1 þ ðy_ 1 þ ox1 Þj1

ð2:36Þ

This equation establishes the relationship between the velocities in both frames of reference. Performing one more differentiation we obtain for the acceleration in the inertial system P: a ¼ ðx€ 1 oy_ 1 Þi1 þ ðx_ 1 oy1 Þoj1 þ ðy€ 1 þ ox_ 1 Þj1 ðy_ 1 þ ox1 Þoi1 or a ¼ ðx€ 1 2oy_ 1 o2 x1 Þi1 þ ðy€ 1 þ 2ox_ 1 o2 y1 Þj1

ð2:37Þ

We see that the acceleration in the inertial frame P can be represented in terms of the acceleration, components of the velocity and coordinates of the point p in the rotating frame, as well as the angular velocity. This equation is one more example of transformation of the kinematical parameters of a motion, and this procedure does not have any relationship to Newton’s laws. Let us rewrite Equation (2.37) in the form a ¼ a1 þ a2 þ a 3

ð2:38Þ

Here a1 ¼

d 2 r1 dt2

a2 ¼ 2ðv1 xxÞ

a3 ¼ o2 r1

ð2:39Þ

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Methods in Geochemistry and Geophysics

and r1 ¼ x1 i1 þ y1 j1 v1 ¼ x_ 1 i1 þ y_ 1 j1 , where a1 is the acceleration of the particle, when an observer is located in noninertial frame of reference, x ¼ ok the vector of the angular velocity, directed along the axis of rotation and i1 j1 k1 ð2:40Þ a2 ¼ 2 x_ 1 y_ 1 0 0 0 o Substitution of Equation (2.38) into the Newton’s second law in the inertial frame of reference: ma ¼ F gives ma1 ¼ F þ F1 þ F2 ð2:41Þ where F is the real force, while F1 ¼ 2mðv1 xxÞ

and

F2 ¼ mo2 r1

ð2:42Þ

are simply two additional terms in the equation which relates the acceleration a1 with the real force. Thus, in a non-inertial frame of reference rotating with constant angular velocity the product of mass and acceleration is not equal to the real force and this means that Newton’s second law is invalid. As was pointed out earlier, we can still use this law if additional terms on the right hand side of Equation (2.41) are treated as forces, even if their agents are unknown. In accordance with Equation (2.42) components of these forces are F 1x1 ¼ 2moy_ 1 and F 1y1 ¼ 2mox_ 1 ð2:43Þ Also, F 2x1 ¼ mo2 x1

and

F 2y1 ¼ mo2 y1

ð2:44Þ

Correspondingly, in terms of components Equation (2.41) becomes ma1x1 ¼ F x1 þ 2moy_ 1 þ mo2 x1 and ma1y1 ¼ F y1 2mox_ 1 þ mo2 y1

ð2:45Þ

where Fx1 and Fy1 are components of the real force in a non-inertial frame of reference. The ﬁrst pseudo force, F1, is called the Coriolis force, and its magnitude is directly proportional to the angular velocity of the rotating frame of reference and the linear velocity of the particle in this frame. By deﬁnition, this force is perpendicular to the plane where vectors v1 and x are located, Fig. 2.3a, and depends on the mutual position of these vectors. The second ﬁctitious force, F2, is called the centrifugal force. Its magnitude is directly proportional to the square of the angular velocity and the distance from the particle to the center of rotation. It is directed outward from the center and this explains the name of the force. It is obvious that with an increase of the angular velocity the relative contribution of this force

69

Gravitational Field of the Earth

(a)

(b) z

v1

Fr

m 2r1

ω

2m( v 1 xω)

p

Fa

0 (c)

(d)

(e)

z

z1 r

0

z

p y

ω

r1

r0 01

R

p

x1

z

i1

r

r1

θ

x

0

r0

01

Fig. 2.3. (a) Direction of pseudo forces, (b) orientation of real forces, (c) position of point p in the inertial and non-inertial frames of reference, (d) illustration of Equation (1.53), (e) illustration of Equation (2.57).

becomes stronger. In particular, when the particle does not move, v1 ¼ 0, the inﬂuence of Coriolis force disappears. Before we continue, let us again point out that a. In principle, we can study a motion in any frame of reference and, correspondingly, proceed either from Newton’s second law ma ¼ F or Equation (2.41) ma1 ¼ F þ 2mðv1 xxÞ þ mo2 r1

ð2:46Þ

This relationship can be also treated as Newton’s second law in a non-inertial frame of reference and, in particular, it shows that the acceleration a1 and the true force F may have different directions. c. Usually the last two terms on the right hand side of Equation (2.46) are called pseudo forces because they do not have an agent causing them. The other approach implies that there is no difference between the real and pseudo forces and the latter are called the inertial forces. Finally, if we would like it is possible to avoid any names for these terms and simply use Equation (2.46) in order to study a motion in a non-inertial frame of reference. As an illustration, consider again the case of the earth, rotating with constant angular velocity, Fig. 2.3b, and suppose that the particle p on its surface remains at rest in the system of coordinates moving together with the earth. Since in this case the particle rotates with Earth in the horizontal plane perpendicular to the z-axis, we can use Equation (2.41), (two-dimensional case), and this gives b.

0 ¼ F þ F2 ¼ F þ mo2 r1

ð2:47Þ

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Methods in Geochemistry and Geophysics

where F and F2 are the real and centrifugal forces, respectively. In accordance with Equation (2.47) the sum of these forces is zero, and this is natural because for the observer in this frame the point p is at rest. As we know, the real force is a superposition of the force of attraction, Fa, and the reaction of the earth, Fr, and Equation (2.47) can be written as 0 ¼ ðFa þ mo2 r1 Þ þ Fr

ð2:48Þ

After a division by a mass m we again obtain for the gravitational ﬁeld (Equation (2.29)): g ¼ ga þ o 2 r 1 It is clear that in the case of a particle, moving in the frame P0 , the term corresponding to Coriolis force appears and the equation of motion becomes ma1 ¼ fFa þ 2mðv1 xxÞ þ mo2 r1 g þ Fr Then, the gravitational ﬁeld for a moving particle has the form: g ¼ ga þ 2ðv1 xxÞ þ o2 r1

ð2:49Þ

2.1.6. Frame of reference rotating with the constant angular velocity (threedimensional case) In order to study an arbitrary motion of a particle on the earth’s surface we consider a more general case, when corresponding coordinate planes in both frames are not parallel to each other and they have different origins, Fig. 2.3c. The moving frame P0 rotates about the z-axis of the frame P and the distance between origins remains the same. We introduce three radius vectors r, r0, and r1, characterizing a position of a point p in both frames, as well as the origin 01. It is obvious that r ¼ r0 þ r1 or xi þ yj þ zk ¼ x0 i þ y0 j þ z0 k þ x1 i1 þ y1 j1 þ z1 k1

ð2:50Þ

Here i, j, k and i1, j1, k1 are unit vectors in the inertial and rotating frames of reference, respectively. Performing a differentiation with respect to time we obtain for the velocity of the point p v¼

dr dx1 dy dz1 di1 dj dk1 ¼ v0 þ i1 þ 1 j1 þ k 1 þ x1 þ y1 1 þ z 1 dt dt dt dt dt dt dt

ð2:51Þ

Here v0 is the velocity of the origin 01. It is convenient to represent the unit vector i1 as a sum of two components: one of them is perpendicular to the z-axis, while the

Gravitational Field of the Earth

71

other is parallel and the latter does not change during a rotation. Then, as is seen from Fig. 2.3d, and by analogy with two-dimensional case di1 ¼ xxi1 dt In the same manner, dj1 ¼ xxj1 dt

and

dk1 ¼ xxk1 dt

Bearing in mind that the distance r0 between origins remains constant, we also have v0 ¼ xxr0 ð2:52Þ Thus, the velocity of the particle in the frame P can be represented in the form dx1 dy dz1 v ¼ v0 þ i1 þ 1 j1 þ k1 þ xxr1 ð2:53Þ dt dt dt Taking one more derivative, we obtain a ¼ a0 þ a1 þ xxv1 þ xxv1 þ xxðxxr1 Þ Here a0 is the acceleration of the origin 01 in the frame P. In accordance with Equation (2.52), it is equal to a0 ¼ xxðxxr0 Þ Thus, a ¼ a1 þ xxðxxrÞ þ 2ðxxv1 Þ

ð2:54Þ

Since xxðxxrÞ ¼ ðx rÞx o2 r we have a ¼ a1 2ðv1 xxÞ þ ðx rÞx o2 r

ð2:55Þ

Here v1 and a1 are the velocity and acceleration of the point p in the rotating frame of reference, respectively. Substitution of Equation (2.55) into Newton’s second law gives an equation of motion in the non-inertial frame: ma1 ¼ F þ 2mðv1 xxÞ þ mo2 r mðx rÞx

ð2:56Þ

Taking into account the fact that the vector of angular velocity is constant, the right hand side of this equation can be simpliﬁed and with this purpose in mind we make use of Fig. 2.3e, where the vector R is perpendicular to the axis of rotation. By deﬁnition, x r ¼ or cos y where y is the angle between the axis of rotation and the vector r. Thus, we have mo2 r mro2 k cos y ¼ mo2 ðr r2 Þ ¼ mo2 R

ð2:57Þ

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Methods in Geochemistry and Geophysics

This gives ma1 ¼ F þ 2mðv1 xxÞ þ mo2 R

ð2:58Þ

and we have arrived at the same equation as that in the two-dimensional case.

2.2. GRAVITATIONAL FIELD OF THE EARTH As was shown in the previous section the gravitational ﬁeld of the earth consists of two parts. One of these describes the ﬁeld of attraction but the other is caused by rotation, and we have Z dðqÞ gðpÞ ¼ k Lqp dV þ o2 r ð2:59Þ 3 V Lqp where p(xp, yp, zp) and q(xq, yq, zq) are an observation point and an arbitrary point of the earth, respectively. Lqp is the distance between these points and Lqp is the vector directed from the point q to point p. Also, in the Cartesian system of coordinates, where the z-axis coincides with the axis of rotation: Lqp ¼ ðxp xq Þi þ ðyp yq Þj þ ðzp zq Þk and r ¼ xp i þ y p j Correspondingly, the components of the gravitational ﬁeld are Z ðxp xq Þ gx ðpÞ ¼ k dm þ o2 xp L3qp V Z

ðyp yq Þ

gy ðpÞ ¼ k

L3qp

V

dm þ o2 yp

ð2:60Þ

Z gz ðpÞ ¼ k

ðzp zq Þ dm L3qp V

By deﬁnition, at each point p the gravitational ﬁeld is tangent to the vector line and its equation is dx dy dz ¼ ¼ gx gy gz

ð2:61Þ

Note that vector lines of ﬁelds ga and g differ from each other only slightly. Both terms on the right hand side of Equation (2.60) have the same dimension, and the gravitational and attraction ﬁelds are measured in Galls. 1 m=s2 ¼ 100 Gal ¼ 105 mGal ¼ 108 mGal

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Gravitational Field of the Earth

The gravitational ﬁeld is a sum of two ﬁelds: g ¼ ga þ gc

gc ¼ o2 r ¼ o2 ðxi þ yjÞ

and

Earlier we demonstrated that the system of equations of the attraction ﬁeld is curl ga ¼ 0

and

div ga ¼ 4pkd

ð2:62Þ

Performing differentiations we ﬁnd that the ﬁeld gc is also the source ﬁeld and curl gc ¼ 0

and

div gc ¼ 2o2

ð2:63Þ

This ﬁeld of the centrifugal force, unlike the attraction ﬁeld, is ﬁctitious, and correspondingly, we observe a volume distribution of ﬁctitious sources with a density proportional to o2. A summation of the ﬁrst and second Equations (2.62 and 2.63) gives the system of equations of the gravitational ﬁeld at regular points curl g ¼ 0 and div g ¼ ð4pkd 2o2 Þ

ð2:64Þ

As in the case of the attraction ﬁeld, applying the integral form of this set, we ﬁnd that the tangential and normal components of the ﬁeld g are continuous functions at the interface between media with different densities of mass: g1t ¼ g2t

and

g1n ¼ g2n

ð2:65Þ

In a rotating frame of reference, the gravitational ﬁeld is a sum of the attraction and centrifugal ﬁelds, and its orientation has one very important feature, namely, this ﬁeld is normal to the surface of the geoid at each of its point. Such behavior is understandable, because a geoid is the ﬁgure of equilibrium of the ocean. In fact, in order to prevent water from a motion the component of the gravitational ﬁeld tangential to the geoid has to be zero. In other words, the ﬁeld is directed along the normal to this surface, as is shown in Fig. 2.4. As we see the ﬁeld of attraction is not strictly directed to the center of the earth, since it differs from a sphere. At the beginning, with an increase of the distance from the earth the ﬁeld magnitude z

geoid

g

ga

0 Fig. 2.4. Direction of the gravitational and attraction ﬁelds.

r

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Methods in Geochemistry and Geophysics

decreases, since the attraction ﬁeld becomes smaller and the centrifugal force, acting in the opposite direction, grows. Then, there is a distance where both components almost compensate each other and with further increase of distance the gravitational ﬁeld is practically directed from the earth and its magnitude increases. However, we are not interested in the behavior of the ﬁeld at such large distances from the earth’s surface. As concerns this ﬁeld inside the earth, its dependence on the distance essentially depends on the distribution of density. Certainly, in the vicinity of the center of mass there is a point where the ﬁeld is equal to zero, and we call this point the center of the earth; in approaching this point ﬁeld becomes smaller.

2.2.1. General features of the field g on the earth’s surface There are regular and irregular changes of the gravitational ﬁeld on the earth’s surface. Two factors deﬁne the regular change of this ﬁeld. One of them is due to the rotation effect, which is equal to zero at the poles and reaches a maximum at the equator. The second is the earth’s ﬂattening, which causes an increase of the ﬁeld in approaching the poles. A change of the gravitational ﬁeld from the equator to poles is approximately equal to 5 Gal, so that gp ge 1 ¼ 189 ge

ð2:66Þ

At the same time, earlier we demonstrated that the ratio of the centrifugal force to the attraction one at the equator is q¼

o2 a 1 ¼ ge 289

Correspondingly, the difference gp ge 1 1 1 q¼ 189 289 550 ge and this characterizes the change of the ﬁeld, caused by ﬂattening. We see that an inﬂuence of rotation is dominant. At the same time, irregular changes of the ﬁeld are relatively small and they are caused by anomalous changes of density of masses beneath the earth’s surface; their magnitude does not exceed several hundreds of milliGalls. Besides these sources of the ﬁeld, related to the earth, there is also an inﬂuence of sun and moon and other planets. However, their effect is relatively small and has a periodical character. Correspondingly, they are often treated as a periodical perturbation of the ﬁeld. But these have to be accounted for in data processing. The maximal contribution caused by the moon is about 0.165 mGal, while the ﬁeld due to sun is 0.076 mGal. General characteristics of the ﬁeld on the earth’s surface are given in Table 2.1.

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Gravitational Field of the Earth Table 2.1. General characteristics of the ﬁeld on the earth surface.

Gravitational ﬁeld

Values in mGal

Total change of g from pole to equator Change of g due to rotation Change of g from pole to equator due to ﬂattening Maximal anomalies of the ﬁeld Change of g due to a change of elevation by 1 m Maximal magnitude of the Moon–Sun perturbations

979,700 5200 3400 600 0.30 0.24

2.3. POTENTIAL OF THE GRAVITATIONAL FIELD OF THE EARTH As was shown earlier the gravitational ﬁeld is a superposition of the attraction and centrifugal ﬁelds: Z dðqÞLqp dV þ o2 r gðpÞ ¼ k L3qp V In order to simplify the study of this ﬁeld we attempt to describe it with the help of potential, as it was done in the case of only the attraction: ga ¼ grad U a

ð2:67Þ

where Z U a ðpÞ ¼ k

dðqÞ dv V Lqp

ð2:68Þ

In the same manner we introduce the potential of the ﬁeld, gc, describing a rotation gc ¼ o2 r ¼ grad U c

ð2:69Þ

This equality directly follows from the ﬁrst equation of this ﬁeld: curl gc ¼ 0. To determine the function Uc we make use of a cylindrical system of coordinates when the z-axis coincides with the axis of rotation of the earth and take into account the fact that this ﬁeld has only a radial component. Then, in place of Equation (2.69) we have @U c ¼ o2 r @r Its integration gives 1 U c ðpÞ ¼ o2 r2 þ C ð2:70Þ 2 Here C is some constant whose value does not inﬂuence the ﬁeld gc. Assuming that this potential is equal to zero at the axis of rotation, we obtain 1 U c ðpÞ ¼ o2 r2 2

ð2:71Þ

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Methods in Geochemistry and Geophysics

Thus, the potential of the gravitational ﬁeld can be represented as Z dðqÞ o2 r2 UðpÞ ¼ k dv þ 2 V Lqp

ð2:72Þ

and g ¼ grad U

ð2:73Þ

Certainly, the expression for the potential is much simpler than that for the ﬁeld, and this is a very important reason why we pay special attention to the behavior of this function U(p). As follows from the behavior of the gravitational ﬁeld, the potential U has a maximum at the earth’s center and with an increase of the distance from this point it becomes smaller, since the ﬁrst derivative in the radial direction, that is, the component of the gravitational ﬁeld, is negative. At very large distances from the earth the function U has a minimum and then it starts to increase, but this range is beyond our interest. In the ﬁrst chapter we demonstrated that the potential of the attraction ﬁeld obeys Poisson’s and Laplace’s equations inside and outside the earth, respectively: DU a ¼ 4pkd and DU a ¼ 0 ð2:74Þ At the same time, the potential of the ﬁeld gc satisﬁes Poisson’s equation everywhere. In fact, taking the second derivatives of the potential U c ðpÞ ¼

o2 2 ðx þ y2 Þ 2

ð2:75Þ

we ﬁnd @2 U c @2 U c ¼ ¼ o2 @x2 @y2 and therefore DU c ¼ 2o2

ð2:76Þ

As follows from these formulas, at a ﬁnite distance from the earth the potential of the centrifugal force, as well as its ﬁrst derivatives, have ﬁnite values and are continuous. The fact that Uc and grad Uc increase unlimitedly with an increase of x and y should not confuse us because our study takes place either on the earth’s surface or relatively close to it. Besides, as we know, the value of o2 is very small. In this light it is proper to notice that the earth’s ﬁgure is mainly deﬁned by the force of attraction because the inﬂuence of rotation is very small. As concerns the Laplacian, that is, a sum of second derivatives, it is constant outside the earth. Thus, the potential of gravitational ﬁeld obeys Poisson’s equation at regular points inside the earth DU ¼ ð4pkd 2o2 Þ

ð2:77Þ

DU ¼ 2o2

ð2:78Þ

as well as outside the earth:

Gravitational Field of the Earth

77

This suggests that the second derivatives have a discontinuity at places where the density changes abruptly. In particular, a discontinuity takes place at the earth’s surface. 2.3.1. Level surfaces and plumb lines As was pointed out in the Chapter 1, it is very useful to study a scalar ﬁeld with the help of equipotential or level surfaces. At each point of such a surface the potential is constant, and correspondingly its equation is Uðx; y; zÞ ¼ C

ð2:79Þ

where C is some constant. These surfaces are always closed and never intersect each other. Otherwise two different level surfaces would have the same value of the potential. We can imagine an inﬁnite number of equipotential surfaces, which literally ﬁll whole space, and every point is located on one such surface. In accordance with Equation (2.72) in place of Equation (2.79) we have Z dðqÞ 1 k dv þ o2 r2 ¼ U 0 ð2:80Þ 2 V Lqp Here U0 is the value of the potential on the surface. It is proper to notice that potentials of the attraction ﬁeld and the centrifugal force usually vary on the level surface of the gravitational ﬁeld. Changing the value of the constant, U0, we obtain different level surfaces, including one which coincides in the ocean with the free undisturbed surface of the water and, as was pointed out earlier, this is called the geoid. As follows from Equation (2.73) the projection of the ﬁeld g on any direction l is related to the potential U by gl ¼

@U @l

ð2:81Þ

that is the component gl characterizes the rate of the change of the ﬁeld in the ldirection. Inasmuch as the potential does not change on the level surface, the projection of the ﬁeld on the plane, tangential to this surface, is equal to zero. In other words, the gravitational ﬁeld is always perpendicular to the level surface and, therefore, the vector lines of the ﬁeld are normal to these surfaces. These lines are slightly curved and in physical geodesy and gravimetry they are also called plumb lines. Usually the position of the level surface is associated with a horizontal surface and for this reason the vector line is often named the vertical. Thus, the plumb line, the vertical, and the vector line describe the same thing, namely, the line which is tangential to the gravitational ﬁeld and perpendicular to the level surface in a given point. This is the reason why the equipotential surfaces of the gravitational ﬁeld play so important role. For instance, in the 19th century C. Gauss suggested treating the mean surface of the oceans as a mathematical model of the earth, and later this equipotential surface was called the geoid, Fig. 2.5. As an important illustration, let us introduce the concept of the height H above sea level, which is also called the orthometric height. It is measured along the curved plumb line from the geoid.

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Methods in Geochemistry and Geophysics

Plumb line

P

H

U=const g

Fig. 2.5. Level surface and plumb line.

If we take an elementary displacement dl along the vector line, then jdlj ¼ dH This means @U ð2:82Þ @H This relationship between this geometrical concept and the gravitational ﬁeld plays a very important role for determination of heights. dU ¼ gdl ¼ gdH

g¼

or

2.3.2. Curvature of level surfaces and vector lines To study geometric features of a potential ﬁeld in detail, consider the curvature of the level surfaces. As is well known, the curvature of a curve y ¼ f(x) is deﬁned as w¼

y00 ð1 þ

y02 Þ3=2

¼

1 r

where r is the radius of curvature and dy d2y and y00 ¼ 2 dx dx Thus, the curvature depends on the argument x and the ﬁrst and second derivatives. At the point of a maximum or minimum, Fig. 2.6a, y0 (x) ¼ 0 and the curvature is equal to the second derivative: y0 ¼

w¼

1 ¼ y00 ðxÞ r

ð2:83Þ

that is, the inﬂuence of the ﬁrst derivative vanishes. Now we make use of Equation (2.83) and ﬁnd an expression for the curvature of the level surface. Let us choose the Cartesian system of coordinates with the origin at the point p, located on the level

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Gravitational Field of the Earth

(b)

(a) z

Plumb line p

z p

y=y(x) Level surface

x g Fig. 2.6. (a) Illustration of Equation (2.83), (b) plumb line and level surface.

surface, Fig. 2.6b. First, consider the intersection of the level surface Uðx; y; zÞ ¼ U 0 with the xz-plane, where y ¼ 0. It is essential that the xy-plane is tangential to the level surface at the point p. Then, the curvature of the level surface in the xz-plane is d 2 zðxÞ ð2:84Þ dx2 Now we express this curvature in terms of the gravitational ﬁeld and potential. Since the coordinate z depends on x, differentiation of U with respect to x gives at the point p w1 ¼

dUðx; zÞ dz ¼ Ux þ Uz ¼0 dx dx Differentiating again we obtain dz dz dz d 2z U xx þ U xz þ U zx þ U zz þ Uz 2 ¼ 0 dx dx dx dx

ð2:85Þ

or 2 dz dz d 2z U xx þ 2U xz þ U zz þ Uz 2 ¼ 0 dx dx dx

ð2:86Þ

Here @U @2 U ; U xz ¼ ; and so on @x @x@z Taking into account the fact that at the point p, Fig. 2.6b, the x-axis is tangent to the curve z ¼ z(x), Ux ¼

dz ¼0 dx

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Equation (2.86) is greatly simpliﬁed and we have d2z U xx ¼ 2 dx Uz At the point p, the z-axis coincides with the vertical (plumb line) and, therefore, Uz ¼

@U @U ¼ ¼ g @z @H

Then Equation (2.84) becomes w1 ¼

U xx g

ð2:87Þ

In the same manner we obtain an expression for the curvature of the function z ¼ z(y), which is the intersection of the level surface at the point p with the zyplane: w2 ¼

U yy g

ð2:88Þ

It is proper to point out again that the simplicity of Equations (2.87 and 2.88) is related to the fact that the plane x0y is tangent to the level surface at the point p and, correspondingly, the plumb line coincides with the z-axis. Next, we introduce the concept of the mean curvature of the level surface as the arithmetic mean of the curvatures of the curves z ¼ z(x) and z ¼ z(y): J¼

U xx þ U yy w1 þ w2 ¼ 2 2g

ð2:89Þ

Since the right hand side contains the sum of the second derivatives, we will make use of the Laplacian inside the earth and express the second derivative of the potential along the vertical through the mean curvature, density, and angular velocity. Then, we have DU ¼ U xx þ U yy þ U zz ¼ 4pkd þ 2o2

or

2Jg þ U zz ¼ 4pkd þ 2o2

Bearing in mind that @U ¼ g; @z

@2 U @g @g ¼ ¼ U zz ¼ 2 @z @z @H

we ﬁnally obtain @g ¼ 2gJ þ 4pkd 2o2 @H At the same time outside the earth:

ð2:90Þ

@g ¼ 2gJ 2o2 @H This equation was derived by Bruns, and it establishes a relation between the derivative of the ﬁeld along the vertical and the mean curvature of the level surface.

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Now we express the curvature of the plumb line in terms of the gravitational ﬁeld, and with this purpose in mind consider its elementary displacement dl: dl ¼ dxi þ dyj þ dzk which is parallel to the ﬁeld g. Therefore, (Chapter 1): dx dy dz ¼ ¼ Ux Uy Uz

ð2:91Þ

As is seen from Fig. 2.6b, the ﬁrst derivative dx/dz at the point p is equal to zero and correspondingly the curvature of the projection of the plumb line on the xz-plane is @2 x @z2 The similarity with the case of the level surface is obvious. From Equation (2.91) we obtain for points in the vicinity of p k1 ¼

dx U x ¼ dz U z Differentiation with respect to z gives d 2x 1 dx dx ¼ U z U xz þ U xx U x U zz þ U zx dz2 U 2z dz dz

ð2:92Þ

Since the ﬁeld at the point p is directed along the z-axis, Fig. 2.6b, the components gx and gy are absent at this point, that is, Ux ¼ Uy ¼ 0 Thus, Equation (2.92) becomes d 2 x U xz U zx ¼ ¼ dz2 Uz Uz

or

d 2x 1 @ @U ¼ dz2 U z @x @z

Bearing in mind that @U ¼ g @z we ﬁnally have k1 ¼

1 @g g @x

ð2:93Þ

and the curvature of projection of the vector line on the xz-plane is expressed in terms of the ﬁeld and its derivative. Similarly, we have for the projection on the y, zplane k2 ¼

1 @g g @y

ð2:94Þ

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From analytical geometry follows that the total curvature of the plumb line is k ¼ ðk21 þ k22 Þ1=2

2.3.3. Potential and distribution of density of mass From the physical point of view it is obvious that there is a relationship between the distribution of density of a ﬂuid and the geometric properties of the scalar ﬁeld U. To illustrate this we will proceed from the equation of motion rP þ dðqÞrU a ¼ dðqÞrU c

or

dðqÞrU ¼ rP

ð2:95Þ

In the simplest case when a medium is homogeneous we have a constant PðqÞ ¼ dUðqÞ

ð2:96Þ

Thus, the potential of the gravitational ﬁeld and pressure differ by a constant multiplier, and the pressure plays the role of the potential even though its meaning is completely different. Next, we discuss the general case, when the density is an arbitrary but continuous function of the point. Taking the curl of both sides of the second equation of the set (2.95), we obtain rxðdrUÞ ¼ 0

ð2:97Þ

since curl grad U ¼ 0. The left hand side of Equation (2.97) can be written in the form rxðdrUÞ ¼ dðqÞðrxrUÞ þ rdðqÞxrU and this equation becomes grad dðqÞx grad U ¼ 0

ð2:98Þ

We see that the gradient of the density and that of the gravitational ﬁeld are parallel to each other. This means that at each point the ﬁeld g has a direction along which the maximal rate of a change of density occurs. The same result can be formulated differently. Inasmuch as the gradient of the density is normal to the surfaces where d is constant, we conclude that the level surfaces U ¼ constant and d ¼ constant have the same shape. For instance, if the density remains constant on the spheroidal surfaces, then the level surfaces of the potential of the gravitational ﬁeld are also spheroidal. It is obvious that the surface of the ﬂuid Earth is equipotential; otherwise there will be tangential component of the ﬁeld g, which has to cause a motion of the ﬂuid. But this contradicts the condition of the hydrostatic equilibrium. 2.3.4. Poincare theorem Fortunately, the earth rotates with a relatively small angular velocity, when the force of attraction plays the dominant role. It is interesting to raise the following

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question. What happens with the earth if this velocity starts to increase? In principle, this problem can be solved in both an inertial and non-inertial frames of reference. It is more preferable to consider the non-inertial frame, where we deal with the pseudo force, which helps us to see more vividly the effect of rotation. In a fact, with an increase of o the inﬂuence of the centrifugal force grows and there is a frequency, where the gravitational ﬁeld ceases be directed into the earth. Then, the earth is broken. Certainly, this explanation is rather superﬁcial since it does not describe what really happens inside the earth. In order to establish the maximal possible frequency we will proceed from Gauss’ theorem, (Chapter 1) Z I I div gdV ¼ g dS ¼ gn dS V

S

S

or in terms of potential and its normal derivative Z I @U ds DUdV ¼ @n V

ð2:99Þ

S

Here S is a surface surrounding the earth and the normal n directed outward. To preserve the earth, the component of the gravitational ﬁeld along the normal has to be negative and this means that the surface integral satisﬁes an inequality I @U dSo0 ð2:100Þ @n s

Therefore, making use of Poisson’s equation, we have Z Z ð4pkd þ 2o2 ÞdV o0 or 4pk dðqÞdV þ 2o2 V o0 V

ð2:101Þ

V

Here V is the volume of the earth and the integral represents its total mass. Introducing the average density, dm, of the earth, Equation (2.101) becomes 4pkdm V 2o2 V 40 Since V as only positive values, we obtain o2 o2pkdm

ð2:102Þ

This relationship was derived by Poincare and deﬁnes the range of frequencies, where the earth or any planet is not broken. The remarkable feature of this inequality is the fact that it is independent of the dimensions of the planet, and only the density deﬁnes the maximal permissible frequency. Introducing the period T, we represent Equation (2.102) as 2p 1=2 T4 ð2:103Þ kdm

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Since k ¼ 6:67 1011 m3 =kg s2 we have T43:07 105

1 dm

1=2 s

Assuming that the average value of density is 5500 kg/m3, we ﬁnd for the minimal period: T min 1:15 h Let us look at inequality (2.103) differently and with this purpose in mind consider the case of a ﬂuid. In accordance with Equation (2.10) we have rP ¼ dga þ o2 rr0 For instance, in the direction of the radius vector we obtain: @P @U ¼ dgaR þ do2 R sin y ¼ gR ¼ @R @R

ð2:104Þ

From Equation (2.100) it follows that the ﬂuid Earth is held together when the pressure decreases in approaching the outer surface, and the maximal frequency corresponds to the case when the pressure is independent of the distance R.

2.4. POTENTIAL AND THE GRAVITATIONAL FIELD DUE TO AN ELLIPSOID OF ROTATION Now we will start to apply the theory of the potential U(p) and its ﬁeld g(p) to study a gravitational ﬁeld caused by masses of the earth. Earlier, it was pointed out that the behavior of the gravitational ﬁeld on the earth’s surface has mainly a regular character, while the irregular part is very small, less than 0.1%. Correspondingly, it is natural to divide the mass of the earth into two parts: 1. The regular part, which is symmetrical with respect to the axis of rotation and the equatorial plane of the earth. Also we assume that this mass is equal to the total mass M of the earth, and that the center of this mass and that of the earth coincide. 2. The irregular part, which is relatively small, and is obtained by subtracting the regular part from the whole Earth. The density of this part has to be negative in some places, because the mass of the regular part is M. Up to a certain moment we will ignore the inﬂuence of these masses and focus on the regular part. Suppose that the regular part forms an ellipsoid of rotation, which is ﬂattened at the equator and that its outer surface is equipotential. These assumptions are based on geodesic measurements on the earth’s surface, as well as theoretical studies, started by Newton in 17th century and continued by great scientists during more than two centuries. In order to determine the gravitational ﬁeld

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85

caused by this rotating body, we can apply different methods; one of them is described here. It is based on a solution of the boundary value problem.

2.4.1. Formulation of the boundary value problem for the potential Ua First, we formulate the boundary problem for the potential of the attraction ﬁeld, which has to satisfy the following conditions: a. Outside the masses it obeys Laplace’s equation: DU a ¼ 0 b.

c.

ð2:105Þ

On the surface of the ellipsoid S0 the potential Ua has to satisfy the equality 1 ð2:106Þ U a ðS 0 Þ ¼ U 0 o2 r2 2 Here U0 is the potential of the gravitational ﬁeld at the surface of the ellipsoid of rotation and r the distance between the axis of rotation and points of this surface. With an increase of the distance from the mass the potential tends to zero and far away it decreases inversely proportional to this distance U a ðpÞ ! 0

ð2:107Þ

As follows from Chapter 1, we have formulated an external Dirichlet’s boundary value problem, which uniquely deﬁnes the attraction ﬁeld. In this light it is proper to notice the following. In accordance with the theorem of uniqueness its conditions do not require any assumptions about the distribution of density inside of the earth or the mechanism of surface forces between the elementary volumes. In particular, these forces may not satisfy the condition of hydrostatic equilibrium. Taking into account the shape of the outer surface of the ellipsoid of rotation, S0, it is convenient to introduce the system of coordinates, where this surface coincides with one of coordinate surfaces. 2.4.2. The system of coordinates of oblate spheroid Inasmuch as this system is not used so often as Cartesian, cylindrical, or spherical coordinates, let us describe it in some detail. First of all, we ﬁnd a condition when a family of non-intersecting surfaces can be a family of equipotential surfaces. Suppose that the equation of the surfaces is F ðx; y; zÞ ¼ C

ð2:108Þ

and that each value of C characterizes one of these surfaces. If these surfaces are level surfaces, then for every value of C we have a certain value of the potential: U a ¼ U ¼ UðCÞ

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and this function obeys the Laplace’s equation. Performing a differentiation with respect to x, we obtain @U @U @C @2 U @2 U @C 2 @U @2 C ¼ and ¼ þ @x @C @x @x2 @C @x2 @C 2 @x In the same manner @2 U @2 U @C 2 @U @2 C ¼ þ ; @y2 @C @y2 @C 2 @y

@2 U @2 U @C 2 @U @2 C ¼ þ @z2 @C @z2 @C 2 @z

Summation of the second derivatives gives Laplace’s equation: DU ¼ U 00 ðCÞðrCÞ2 þ U 0 ðCÞDC ¼ 0 or DC U 00 ðCÞ ¼ FðCÞ ¼ U 0 ðCÞ ðrCÞ2

ð2:109Þ

Thus, the surface F(x, y, z) ¼ C can be equipotential only if the ratio DC ðrCÞ2 is a function of C only. As illustration, consider families of surfaces of the second order deﬁned by the equation x2 y2 z2 þ 2 þ 2 ¼1 þy b þy c þy

a2

ð2:110Þ

where c4b4a and c2oyoN. In order to visualize these surfaces we will change y. Within the interval a2oyoN every term in Equation (2.110) is positive, and the corresponding surfaces are ellipsoids. They represent the ﬁrst family of surfaces. If y ¼ N, we obtain a sphere with an inﬁnitely large radius, while the surface becomes a disk in the xz-plane when y ¼ a2. In the next interval:b2oyoa2, Equation (2.110) describes hyperboloids of one sheet. Finally, within the interval: c2oyob2 this equation characterizes hyperboloids of two sheets. Thus, one surface of each family passes through any point of space. It is a simple matter to ﬁnd equations for the normal to these surfaces and demonstrate that they are orthogonal to each other. Now we show that these surfaces can be eqiupotential surfaces. Let us introduce the functions Mn ¼

x2 y2 z2 þ þ ða2 þ yÞn ðb2 þ yÞn ðc2 þ yÞn

and N¼

1 1 1 þ þ a2 þ y b2 þ y c 2 þ y

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It is clear that M1 represents the left hand side of Equation (2.110). Its differentiation with respect to x gives 2x @y ¼0 M2 ða2 þ yÞ @x

or

@y 2x ¼ @x M 2 ða2 þ yÞ

ð2:111Þ

In the same manner, @y 2y ¼ @y M 2 ðb2 þ yÞ

and

@y 2z ¼ @z M 2 ðc2 þ yÞ

Thus, ðryÞ2 ¼

2

@y @x

þ

2 2 @y @y 4M 2 4 þ ¼ ¼ 2 @y @z M2 M2

ð2:112Þ

Differentiation of Equation (2.111) yields @2 y 2 2x @y 2x 1 2x @y ¼ 2M 3 @x2 M 2 ða2 þ yÞ M 22 ða2 þ yÞ2 @x a2 þ y M 22 ða2 þ yÞ2 @x ¼

2 4x2 4x2 8x2 M 3 2 2 þ 3 3 2 M 2 ða þ yÞ M 2 ða2 þ yÞ M 2 ða2 þ yÞ ða2 þ yÞ2 M 32

Adding similar expressions for the y and z derivatives, we obtain Dy ¼

2N 8M 3 8M 2 M 3 2N þ ¼ M2 M2 M 22 M 32

ð2:113Þ

Substitution of Equations (2.112 and 2.113) into Equation (2.109) gives Dy 2N M 2 N ¼ ¼ 2 M2 4 2 ðryÞ Thus, FðyÞ ¼

1 1 1 1 þ þ 2 a2 þ y b2 þ y c 2 þ y

and this means that the surfaces, described by Equation (2.110), can be level surfaces of the potential. 2.4.3. The system of coordinates of an oblate spheroid Now we demonstrate the system of coordinates, where the ellipsoids of rotation and hyperboloids of one sheet form two mutually orthogonal coordinate families of surfaces. First, we introduce the Cartesian system at the center of the mass and suppose that semi-axes of the ellipsoid of rotation obey the condition boa where the smallest axis is directed along the axis of rotation. As can be seen from Fig. 2.7a, the relation between coordinates of the Cartesian and cylindrical

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Methods in Geochemistry and Geophysics (a)

(b) y

z

n p’

η

ε

ϕ +π

ϕ y

r

0

2

p

0

x

(c) y p’ p y

0

β x

x

b

Fig. 2.7. (a) Spheroidal system of coordinates, (b, c) reduced and geographical latitudes.

systems is x ¼ r sin j

and

y ¼ r cos j

ð2:114Þ

Here r is the distance from the point to the z-axis and j the azimuth angle. Substitution of Equation (2.114) into Equation (2.110) gives z2 r2 þ 2 ¼1 b þy a þy

ð2:115Þ

b2 þ y ¼ ða2 b2 Þy21 ¼ c2 y21

ð2:116Þ

c ¼ ða2 b2 Þ1=2

ð2:117Þ

2

Let Here Then, in place of Equation (2.115) we have z2 r2 ¼1 þ 2 2 2 c y1 c ð1 þ y21 Þ

ð2:118Þ

If b2oyoN, then 0oe2oN, where y21 ¼ 2 , and Equation (2.115) describes a family of the oblate spheroids, and each of these is characterized by a constant value of e. This is the ﬁrst coordinate of a point. If a2oyob2, we write Equation (2.116) as b2 þ y ¼ c2 Z2 Z varies within the range 0oZ2o1 and we obtain a family of confocal hyperboloids of one sheet. This is the second family of coordinate surfaces and Z is the second

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coordinate of the point. The third coordinate is the azimuth angle j. In accordance with Equation (2.115) we have for spheroids z2 r2 ¼1 ð2:119Þ þ c2 2 c2 ð1 þ 2 Þ and the equation of hyperboloids is z2 r2 ¼1 þ 2 2 2 c Z c ð1 Z2 Þ

ð2:120Þ

Eliminating r and z from these equations we obtain z ¼ cZ and r ¼ c½ð1 þ 2 Þð1 Z2 Þ1=2

ð2:121Þ

The coordinate r is always positive but x changes within the range Noxo+N. Taking into account this fact we assume that, Fig. 2.7a, 0oo1 and 1oZ þ 1 ð2:122Þ In order to solve Laplace’s equation we have to ﬁnd also expressions for the metric coefﬁcients. By deﬁnition, the elementary displacements along coordinate lines are ds1 ¼ h1 d; ds2 ¼ h2 dZ; and ds3 ¼ h3 dj ð2:123Þ Here de, dZ, and dj characterize the change of coordinates along corresponding coordinate lines 1 @ @y1 1 @Z @y1 ¼ and ð2:124Þ ¼ ¼j ¼j h1 @n h2 @n @n @n Here derivatives are taken in the direction perpendicular to the corresponding coordinate surface and qn is an elementary displacement of the coordinate line. As follows from Equations (2.112 and 2.116) jryj @y1 1 @y 1 ¼ ¼ 2 ¼ ð2:125Þ 2c y1 @n 2c2 y1 c2 y1 M 1=2 @n 2

Substituting, ﬁrst, e into Equations (2.124 and 2.125) and then Z and the use of Equation (2.125) gives for h1 and h2: 2 1=2 2 1=2 z r2 Z þ 2 2 ¼c ð2:126Þ h1 ¼ c 4 4 þ c 1 þ 2 c4 ð1 þ 2 Þ2 h2 ¼ cZ

z2 r2 þ 2 2 2 4 ðc Z Þ c ð1 Z2 Þ2

1=2 ¼c

2 1=2 Z þ 2 1 Z2

ð2:127Þ

By deﬁnition, we have h3 ¼

rdj ¼ r ¼ c½ð1 þ 2 Þð1 Z2 Þ1=2 dj

ð2:128Þ

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Methods in Geochemistry and Geophysics

As is well known, Laplace’s equation in an orthogonal system of coordinates has the form: 1 @ h2 h3 @U @ h1 h3 @U @ h1 h2 @U DU ¼ þ þ h1 h2 h3 @x1 h1 @x1 @x2 h2 @x2 @x3 h3 @x3 In our case x1 ¼ ;

x2 ¼ Z;

and

x3 ¼ j

Then, taking into account that the distribution of masses inside the spheroid is independent of the azimuth coordinate j, we have for Laplace’s equation @ @U a @ @U a ð1 þ 2 Þ ð1 Z2 Þ þ ¼0 ð2:129Þ @ @Z @ @Z

2.4.4. Solution of Equation (2.129) by the method of separation of variables As in the case of a spherical system of coordinates, Chapter 1, the potential Ua(e, Z) is a solution of the partial differential equation of the second order; and in order to express Ua in terms of known functions we represent the potential in the form of the product U a ð; ZÞ ¼ LðÞTðZÞ

ð2:130Þ

Substituting Equation (2.130) into Equation (2.129) and dividing by U, we obtain 1 @ 1 @ 2 @L 2 @T ð1 þ Þ ð1 Z Þ þ ¼0 LðÞ @ @ TðZÞ @Z @Z The ﬁrst and second terms contain functions of two different arguments and therefore each of them is constant. This allows us to replace the Laplace’s equation by two ordinary differential equations, and they are 1 @ 1 @ 2 @L 2 2 @T ð1 þ Þ ð1 Z Þ and ¼m ¼ m2 L @ @ T @Z @Z or

@ @L ð1 þ 2 Þ m2 L ¼ 0 @ @

and

@ @T ð1 Z2 Þ þ m2 T ¼ 0 @Z @Z

ð2:131Þ

if m ¼ n(n+1). The second equation of this set is a known equation (Chapter 1) and its solutions are Legendre’s functions Pn(Z) and Qn(Z). Now we demonstrate that the ﬁrst equation is also Legendre’s equation and with this purpose in mind introduce a new variable 0 ¼ j

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where j is an imaginary unit. Since @ @ @0 @ ¼ 0 ¼j 0 @ @ @ @ we have

and

ð1 þ 2 Þ ¼ ð1 02 Þ

@ 02 @L ð1 Þ 0 þ nðn þ 1ÞL ¼ 0 @0 @

and its solutions are Legendre’s functions and Pn ðjÞ

Qn ðjÞ

Thus, partial solutions for each equation can be represented as Ln ðÞ ¼ An Pn ðjÞ þ Bn Qn ðjÞ and T n ðZÞ ¼ C n Pn ðZÞ þ Dn Qn ðZÞ Here coefﬁcients An, Bn, Cn, and Dn are independent of the coordinates of a point e and Z. 2.4.5. Expressions for the potential Correspondingly, a general solution for the potential is U a ð; ZÞ ¼

1 X ½An Pn ðjÞ þ Bn Qn ðjÞ½C n Pn ðZÞ þ Dn Qn ðZÞ

ð2:132Þ

n¼0

This function is a solution of Laplace’s equation regardless of the values of constants, and our goal is to ﬁnd such of them that the potential satisﬁes the boundary condition on the surface of the given ellipsoid of rotation and at inﬁnity. In order to solve this problem we have to discuss some features of Legendre’s functions. First of all, as was shown in Chapter 1, the Legendre’s function of the ﬁrst kind Pn(Z) has everywhere ﬁnite values and varies within the range 1oPn ðZÞo þ 1 Examples of this function are given below, (Chapter 1) 1 P2 ðZÞ ¼ ð3Z2 1Þ ð2:133Þ 2 The Legendre’s function of the second kind Qn(Z) has completely different behavior; in particular, it tends to inﬁnity when|Z| ¼ 1. In accordance with Equation (2.121), this happens at points of the z-axis. Since the potential has everywhere a ﬁnite value the function Qn(Z) cannot describe the attraction ﬁeld and has to be removed from Equation (2.132). This ﬁrst simpliﬁcation gives: P0 ðZÞ ¼ 1;

U a ð; ZÞ ¼

P1 ðZÞ ¼ Z;

1 X ½Bn Pn ðjÞ þ An ðÞQn ðjÞPn ðZÞ n¼1

The latter can be also simpliﬁed, and with this purpose in mind consider the behavior of Legendre’s functions with an imaginary argument. As follows from

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Equation (2.133) 1 P2 ðjÞ ¼ ð32 þ 1Þ; and so on 2 Making use of recursive relations between them it is easy to see that all the abovementioned functions, except P0, increase unlimitedly with the distance from masses, e-N which contradicts the condition at inﬁnity. Therefore, terms with Pn(je) have to be discarded too, and this gives one more simpliﬁcation of Equation (2.132) P0 ðjÞ ¼ 1;

P1 ðjÞ ¼ j;

U a ð; ZÞ ¼

1 X

An Qn ðjÞPn ðZÞ

ð2:134Þ

n¼0

The Legendre’s function of the second kind has everywhere ﬁnite values, except e ¼ 0. Examples of this function are given below Q0 ðjÞ ¼ jcot1 ;

Q1 ðjÞ ¼ cot1 1

j Q2 ðjÞ ¼ ½ð32 þ 1Þcot1 3; 2

and so on

or Q0 ðjÞ ¼ jtan

1

1

and

j 2 1 1 Q2 ðjÞ ¼ ð1 þ 3 Þtan 3 2

ð2:135Þ

There is also simple recursive expression which allows one to ﬁnd this function of any order n, if we know the Legendre’s functions of the ﬁrst kind. Now we will demonstrate that the functions Qn(je) tend to zero when the distance from the ellipsoid, (e ¼ e0), increases. Making use of the power series 1 1 1 1 tan1 ¼ 3 þ 5 3 5 it is clear that 1 if ! 1 As follows from the theory of these functions Q0 ðjÞ ! j

1 ð2:136Þ nþ1 On the other hand, in accordance with Equation (2.121) the distance from the origin R is Qn ðjÞ /

R ¼ ½r2 þ z2 1=2 ¼ d½ð1 þ 2 Þð1 Z2 Þ þ 2 Z2 ¼ dð1 þ 2 Z2 Þ and with an increase of the coordinate e the distance R also increases unlimitedly. Thus, the function Ua(e, Z), given by Equation (2.134), obeys the condition at inﬁnity. In particular, the ﬁrst term of the sum, n ¼ 0, decreases inversely proportional to the distance. To simplify slightly derivations and deal only with real

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93

numbers we rewrite Equation (2.134) in the form U a ð; ZÞ ¼

1 X n¼0

An

Qn ðjÞ Pn ðZÞ Qn ðj0 Þ

ð2:137Þ

In order to solve the boundary value problem we have to satisfy the condition for the potential of the attraction ﬁeld at the surface of the earth ellipsoid of rotation, which can be written as 1 X

1 An Pn ðZÞ þ o2 ðx2 þ y2 Þ ¼ U 0 2 n¼0

ð2:138Þ

where x, y are coordinates of the points where e ¼ e0, and U0 the constant potential of the gravitational ﬁeld on this surface. Taking into account Equation (2.121), the latter becomes 1 X 0

1 An Pn ðZÞ þ o2 c2 ð1 þ 20 Þð1 Z2 Þ ¼ U 0 2

or 1 X

1 An Pn ðZÞ þ a2 o2 ð1 Z2 Þ ¼ U 0 2 n¼0

ð2:139Þ

Now we make use of the orthogonality of Legendre’s functions. This means that from the equality 1 X

An Pn ðZÞ ¼ 0

ð2:140Þ

n¼0

it follows the remarkable fact, namely, all coefﬁcients An are equal to zero: An ¼ 0. For this reason we will represent Equation (2.139) in the same form as Equation (2.140). Inasmuch as 1 P2 ðZÞ ¼ ð3Z2 1Þ 2 we have 2 1 Z2 ¼ ½1 P2 ðZÞ 3 and Equation (2.139) becomes 1 X

1 1 An Pn ðZÞ þ o2 a2 o2 a2 P2 ðZÞ U 0 ¼ 0 3 3 n¼0

or 1 X 1 2 2 1 2 2 An Pn ðZÞ ¼ 0 A0 þ o a U 0 P0 ðZÞ þ A1 P1 ðZÞ þ A2 o a P2 ðZÞ þ 3 3 n¼3

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since P0(Z) ¼ 1. From the orthogonality of Legendre’s functions we obtain 1 A0 ¼ U 0 o2 a2 ; 3

A1 ¼ 0;

An ¼ 0

if n 3

and 1 A2 ¼ o2 a2 3

ð2:141Þ

Thus, the expression for the potential of the attraction ﬁeld is greatly simpliﬁed and in place of an inﬁnite sum, Equation (2.134), we have 1 Q0 ðjÞ 1 2 2 Q2 ðjÞ U a ð; ZÞ ¼ U 0 o2 a2 þ o a P2 ðZÞ ð2:142Þ 3 Q0 ðj0 Þ 3 Q2 ðj0 Þ where functions Q0 and Q2 are given by Equation (2.135). Now it is clear that the presence of constant denominators in both terms allowed us to remove the inﬂuence of the imaginary unit. On the surface of the spheroid both the attraction and rotation potentials are 1 1 U a ¼ U 0 o2 a2 þ o2 a2 P2 ðZÞ 3 3 and 1 1 U r ¼ o2 d 2 ð1 þ 20 Þð1 Z2 Þ ¼ o2 a2 ð1 Z2 Þ 2 2

ð2:143Þ

Thus, the potential of the gravitational ﬁeld on the ellipsoid of rotation is 1 1 1 2 2 2 Uð0 ; ZÞ ¼ U 0 þ o a þ P2 ðZÞ þ ð1 Z Þ 3 3 2 Since 3 1 P2 ðZÞ ¼ Z2 2 2 we have

1 1 2 1 1 1 2 Uð0 ; ZÞ ¼ U 0 þ o a þ Z þ Z ¼ U 0 3 2 6 2 2 2 2

and, therefore, the boundary condition for the potential of the attraction ﬁeld 1 U a ð0 ; ZÞ ¼ U 0 o2 a2 ð1 Z2 Þ 2

ð2:144Þ

is satisﬁed. Taking into account Equation (2.136), we see that with an increase of the distance from the spheroid the ﬁrst term in Equation (2.142) becomes dominant and further we have 1 tan ð1=0 Þ U a ð; ZÞ ! U 0 o2 a2 ð2:145Þ 3

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since Q0 ð0 Þ ¼ jcot1 0

and

Q0 ðÞ ! j=

if

!1

Bearing in mind that R2 ¼ r2 þ z2 ¼ c2 ½ð1 þ 2 Þð1 Z2 Þ þ 2 Z2 ¼ c2 ½1 þ 2 Z2 ! c2 2 if e-N, we obtain and Equation (2.145) becomes

U a ð; ZÞ !

R ! c

ð2:146Þ

1 c 1 tan U 0 o2 a2 3 R 0

ð2:147Þ

Thus, we have demonstrated that the potential Ua is a solution of Laplace’s equation and satisﬁes the boundary condition at the surface of the ellipsoid of rotation and at inﬁnity. In other words, we have solved the Dirichlet’s boundary value problem and, in accordance with the theorem of uniqueness, there is only one function satisfying all these conditions. Let us represent the potential U0 in terms of the angular frequency, geometrical parameters of a body, and its mass M. As we know, at large distances from the ellipsoid the potential of the attraction ﬁeld is described as U a ð; ZÞ ! k

M R

Then, comparison with Equation (2.147) gives 1 1 kM 1 1 tan1 þ o2 a2 k M ¼ U 0 o2 a2 c tan or U 0 ¼ 3 0 c 0 3

ð2:148Þ

The latter allows us to determine the potential of the gravitational ﬁeld outside the ellipsoid of rotation, including points of its surface, provided that the mass, geometry, and angular frequency are given. Making use of Equation (2.148), we can represent the potential of the gravitational ﬁeld in the form kM 1 1 Q ðjÞ 1 Uð; ZÞ ¼ tan1 þ o2 a2 2 P2 ðZÞ þ o2 c2 ð1 þ 2 Þð1 Z2 Þ ð2:149Þ c 3 Q2 ðj0 Þ 2 It is interesting to notice that the ﬁrst term on the right hand side is independent of the coordinate Z and the frequency, while the last two terms are functions of the angular frequency and both coordinates. As we see, the potential U(e,Z) contains four constants and they are a; b; o; and M ð2:150Þ Before we continue, let us relate the coordinate 0 with the semi-axes of the given ellipsoid. As follows from Equation (2.121) we have b ¼ c0

and

a ¼ cð1 þ 20 Þ1=2

ð2:151Þ

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With a decrease of e0 the ellipsoid approaches a disk; on the contrary, when e0-N, it tends to a sphere. In order to ﬁnd the gravitational ﬁeld of the rotating ellipsoid, it is convenient to represent Equation (2.149) in a slightly different form. For this purpose introduce notations Z ¼ sin b and, therefore, 3 1 P2 ðZÞ ¼ sin2 b 2 2 Also,

1 2 1 1 3 d ¼ ð1 þ 3 Þtan 2

and

1 2 1 1 d 0 ¼ ð1 þ 30 Þtan 30 2 0

Thus, in place of Equation (2.149) we have kM 1 1 d 1 1 tan1 þ o2 a2 sin2 b Uð; bÞ ¼ þ o2 c2 ð1 þ 2 Þcos2 b c 2 d0 3 2

ð2:152Þ

ð2:153Þ

2.4.6. The gravitational field due to the rotating ellipsoid Proceeding from Equation (2.153) we will derive the expressions for the normal gravitational ﬁeld and the main attention is paid to the ﬁeld on the surface of the ellipsoid. To emphasize the fact that we deal with the normal ﬁeld it is conventional to use the letter g instead of g. Then, at each point outside the mass we have g ¼

1 @U h1 @

and

gZ ¼

1 @U h2 @Z

where expressions for metrical coefﬁcients were given above. Performing a differentiation, we have 2 2 d 2 2 2 h2 g Z ¼ o a o c ð1 þ Þ sin b cos b d0

ð2:154Þ

ð2:155Þ

Taking into account the fact that on the surface of the ellipsoid e ¼ e0 and d ¼ d0 we arrive at the obvious result, namely, at the level surface the tangential component of the gravitational ﬁeld vanishes, gZ ¼ 0. Next, consider the component ge. Again differentiating Equation (2.153) we obtain kM 1 2 1 2 2d sin b h1 g ¼ þo a ð2:156Þ þ o2 c2 cos2 b cð1 þ 2 Þ 6 d0 2 Here d ¼

@d @

From Equation (2.152) it follows that 1 d1 2 1 1 d ¼ 1Þ þ 1 ¼ 3ð1 þ Þðtan 2 ð1 þ Þ ð1 þ 2 Þ

ð2:157Þ

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2.4.7. The gravitational field on the surface of the ellipsoid, e ¼ e0 By deﬁnition, we have for the ﬁeld magnitude on this level surface: kM o 2 a2 c d 1 1 2 1 o 2 a2 b 2 sin cos g ¼ g ¼ 1 b b kM d 0 2 6 kM c2 ð1 þ 20 Þ1=2 ð20 þ sin2 bÞ1=2 ð2:158Þ Here d 1 ¼ 3ð1 þ

20 Þ

1 1 0 tan 1 þ1 0

and

h1 ¼ c

ð20 þ sin2 bÞ1=2 ð1 þ 20 Þ1=2

Inasmuch as a ¼ cð1 þ 20 Þ1=2

0 ¼

and

b ða2

b2 Þ1=2

we have g ¼ g ¼

kM aða2 sin2 b þ b2 cos2 bÞ1=2

o 2 a2 c d 1 1 2 1 o 2 a2 b 2 sin b cos b 1 kM d 0 2 6 kM ð2:159Þ

2

2

Making use of the equality: sin b+cos b ¼ 1 and introducing the parameter m¼

o2 a2 b kM

ð2:160Þ

we obtain me1 d 1 me1 d 1 2 2 g¼ 1 sin b þ 1 m þ cos b 3 d0 6 d0 aða2 sin2 b þ b2 cos2 bÞ1=2 kM

and e1 ¼

c b

ð2:161Þ

Now we express the ﬁeld magnitude g in terms of its values at the pole and the equator. As follows from Equation (2.109) at the equator, (b ¼ 0) we ﬁnd kM me1 d 1 ga ¼ 1mþ ð2:162Þ ab 6 d0 and at the poles, (b ¼ 7p/2), we obtain kM me1 d 1 gb ¼ 2 1 a 3 d0

ð2:163Þ

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From Equations (2.160, 2.162, and 2.163) we have g¼

agb sin2 b þ bga cos2 b ða2 sin2 b þ b2 cos2 bÞ1=2

ð2:164Þ

and this shows the change of the gravitational ﬁeld on the level surface of the ellipsoid as a function of the argument b. Let us introduce two quantities, namely, the ﬂattening of the ellipsoid: f ¼

ab a

and the relative change of the ﬁeld on its surface: g ga f ¼ b ga

ð2:165Þ

ð2:166Þ

It is a simple matter to establish a relationship between these parameters. With this purpose in mind consider the ratio of the ﬁelds at the pole and the equator. In accordance with Equations (2.162 and 2.163) we have: gb b 1 ðme1 =3Þðd 1 =d 0 Þ ¼ ga a 1 m þ ðme1 =6Þðd 1 =d 0 Þ Hence gb b b mð1 1=2ðe1 d 1 =d 0 ÞÞ o2 b e1 d 1 ¼ ¼ 1 ga ga a a 1 m þ ðme1 =6Þðd 1 =d 0 Þ 2 d0 The latter gives a b gb ga o 2 b e1 d 1 þ ¼ 1 a ga ga 2 d0

ð2:167Þ

In the next section, assuming that the ﬂattening is very small, fo o1, we will obtain from Equation (4.63) the approximate formula, derived by Clairaut in 1738. 2.4.8. Relation between the reduced and geographical latitudes We have derived Equation (2.164), which shows how the ﬁeld varies with the reduced latitude b on the surface of the spheroid. The reduced latitude is the angle between the radius vector and the equatorial plane, Fig. 2.7c. Also, it is useful to study the function g ¼ g(j), where j is the geographical latitude. This angle is formed by the normal to the ellipsoid at the given point p and the equatorial plane, Fig. 2.7b. First, we ﬁnd expressions for coordinates x, y of the meridian ellipse. Its equation is x2 y2 þ ¼1 a2 b2

ð2:168Þ

Gravitational Field of the Earth

99

Differentiation with respect to x gives x y dy ¼0 þ a2 b2 dx

ð2:169Þ

The value of the derivative is equal to the tangent of the angle, formed by the tangent with the x-axis: dy p ¼ tan j þ ¼ cot j ð2:170Þ dx 2 Substitution of Equation (2.170) into Equation (2.169) gives x y þ 2 ð cot jÞ ¼ 0 2 a b Thus, b2 x tan j a2 since the eccentricity e is equal to y¼

y ¼ xð1 e2 Þ tan j

or

c e¼ ¼ a

a2 b2 a2

ð2:171Þ

1=2

Next consider Fig. 2.7c. It is obvious, that x ¼ a cos b

ð2:172Þ

x2 þ bp02 ¼ a2

ð2:173Þ

As is seen from the triangle 0bp0

while from the equation of the ellipse a2 2 y ¼ a2 b2 Comparison of two last equations yields x2 þ

ð2:174Þ

a bp0 ¼ y b 0 From Fig. 2.7c we see that bp ¼ a sin b and therefore y ¼ b sin b

ð2:175Þ

Making use of Equations (2.171, 2.172, and 2.175), we obtain y b ¼ tan b ¼ ð1 e2 Þ tan j x a or a b tan b ¼ ð1 e2 Þ tan j0 ¼ ð1 e2 Þ1=2 tan j ¼ tan j b a

ð2:176Þ

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Methods in Geochemistry and Geophysics

Correspondingly, Equation (2.164) can be represented as g¼

aga cos2 j þ bgb sin2 j ða2 cos2 j þ b2 sin2 jÞ1=2

ð2:177Þ

This formula was introduced by the Italian geodesist Somigliana in 1929 and deﬁnes the behavior of the gravitational ﬁeld on the level surface of the spheroid for any distribution of density inside, as long as the outer surface remains equipotential.

2.5. CLAIRAUT’S THEOREM We have derived formulas for the gravitational ﬁeld outside and at the surface of the rotating spheroid with an arbitrary value of ﬂattening f, provided that this surface is equipotential. Such a distribution of the potential U(p) takes place only for a certain behavior of the density of masses. For instance, as follows from the condition of the hydrostatic equilibrium this may happen if the spheroid is represented as a system of confocal ellipsoidal shells with a constant density inside each of them. Now our main attention will be paid to the model of the earth spheroid with very small ﬂattening: f 1

ð2:178Þ

By analogy, the ratio f ¼

gb ga ga

ð2:179Þ

may be called the gravity ﬂattening and it characterizes the relative change of the ﬁeld on the earth spheroid. In accordance with Equation (2.167) the relation between these two parameters is o2 b e1 d 1 f þ f ¼ 1 ð2:180Þ ga 2 d0 Here 1 2 1 1 d 0 ¼ ð1 þ 30 Þtan 30 2 0

and

2 1 1 1 þ1 d 1 ¼ 3ð1 þ 0 Þ 0 tan 0

and e1 ¼

c

1 b

and

a ¼ cð1 þ 20 Þ1=2

ð2:181Þ

Inasmuch as b ¼ c0

the coordinate e0 tends to an inﬁnity, when the spheroid becomes a sphere, (a-b).

Gravitational Field of the Earth

101

Correspondingly, the parameter 1/e0 tends to zero and we can expand the inverse trigonometric function tan1 1/e0 in the power series tan1

1 1 1 1 1 ¼ þ þ 0 0 330 550 770

ð2:182Þ

Its substitution into Equation (2.181) gives d0 ¼

2 4 6 þ 3 530 5 750 7 970

and d 1 ¼ 6

1 1 1 1 1 þ 3 520 5 7 40 7 9 60

Therefore, the right hand side of Equation (2.180) becomes 5 o2 b 9 2 f þf 1 þ e1 2 ga 35

ð2:183Þ

ð2:184Þ

Next, we modify the Equation (2.177): g¼

aga cos2 j þ bgb sin2 j ða2 cos2 j þ b2 sin2 jÞ1=2

as g ¼ ga ½cos2 j þ ð1 þ f Þð1 f Þsin2 j½cos2 j þ ð1 f Þ2 sin2 j1=2 ¼ ga ½1 þ ðf f ff Þsin2 j½1 ð2f f 2 Þsin2 j1=2 Expanding the denominator in the series, we obtain 1 3 2 2 2 2 2 4 g ¼ ga ½1 þ ðf f ff Þsin j 1 þ ð2f f Þsin j þ ð2f f Þ sin j þ 2 8

1 1 2 2 2 2 2 4 ¼ ga 1 þ f sin j þ f ff þ ð2f f Þ sin j þ ð2ff 2f þ 3f Þsin j 2 2 or 1 1 2 2 2 2 4 g ¼ ga 1 þ f sin j ð2ff þ f Þsin j þ ð2ff þ f Þsin j 2 2 Finally, we have g ¼ ga ½1 þ f sin2 j f 1 sin2 2j þ

ð2:185Þ

where 1 1 ð2:186Þ f 1 ¼ ff þ f 2 4 8 In deriving these equations we preserved terms proportional to the ﬁrst and second power of ﬂattening.

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2.5.1. The linear approximation In the ﬁrst approximation, neglecting the term proportional to the square of the ﬂattening, we obtain from Equations (2.184 and 2.185) g ¼ ga ð1 þ f sin2 jÞ

ð2:187Þ

where f ¼ f þ

5 o2 a 2 ga

ð2:188Þ

because aEb. The last two equations were derived by Clairaut, and they demonstrate that the ﬂattening can be obtained from measurements of the gravitational ﬁeld. Note that in this approximation, the parameter m: m¼

o2 a ga

ð2:189Þ

characterizes the ratio of the centrifugal force to the gravitational ﬁeld at the equator. Formulas (2.187 and 2.188) constitute the Clairaut’s theorem. The last one deﬁnes the ﬂattening of the spheroid in terms of the parameter m, known with a sufﬁcient accuracy, and the coefﬁcient f*. The ﬁrst one, Equation (2.187), gives the law of a distribution of the normal ﬁeld on the surface of the spheroid. Now we evaluate a contribution of terms in Equation (2.188). The gravitational ﬁeld varies from the equator to pole by approximately 5 Gal. Then, gb ga 1 189 ga Since a ¼ 6378:24 km; o ¼

2p 1 1 s ; f ¼ ; g ¼ 978:049 Gals; ga gb ¼ 5 Gals 86; 164 297 a

we have f ¼

1 ; 189

f ¼

1 ; 289

and

5 1 m¼ 2 114

and we see that the change of the normal gravitational ﬁeld on the earth’s surface is deﬁned by two factors: ﬂattening and rotation, and the latter plays the dominant role. Taking into account the relation between angles b and j, (Equation 2.176): b tan b ¼ tan j a and assuming the linear approximation, in place of Equation (2.187) we can write g ¼ ga ð1 þ f sin2 bÞ

ð2:190Þ

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It is instructive to derive this equation in a different way. By deﬁnition, the equation of the earth’s surface is x2 þ y2 z2 þ ¼1 a2 a2 ð1 f Þ2 Neglecting the square of the parameter f, this equation becomes x2 þ y2 þ z2 ð1 þ 2f Þ ¼ a2

ð2:191Þ

since 1 1 þ 2f 1 2f As is well known, the relations between the Cartesian and spherical coordinates are x ¼ R sin y cos l;

y ¼ R sin y sin l;

and

z ¼ R cos y

Substituting the latter into Equation (2.191), we obtain R2 sin2 y þ R2 ð1 þ 2f Þcos2 y ¼ a2

or

R2 ð1 þ 2f cos2 yÞ ¼ a2

This gives R ¼ að1 þ 2f cos2 yÞ1=2

or

R ¼ að1 f cos2 yÞ

ð2:192Þ

where f is very small, E1/300. Next, we will again arrive at the equation for the equipotential surface, applying the following argument. Since we consider points, (particles), which are at rest with respect to the non-inertial frame of the earth rotating with a small angular velocity, they are subjected to only one pseudo force, namely centrifugal force. Correspondingly, the components of the gravitational ﬁeld in the Cartesian system of coordinates are gx ¼

@U a þ o2 x; @x

gy ¼

@U a þ o2 y; @y

gz ¼

@U a @z

ð2:193Þ

Here Ua is the potential of the ﬁeld of attraction. Inasmuch as we assume that the earth’s surface is equipotential, the vector lines of the normal gravitational ﬁeld are perpendicular to this surface. This condition can be represented as cds ¼ gx dx þ gy dy þ gz dz ¼ 0

ð2:194Þ

where ds is an elementary vector tangential to the equipotential surface. Substitution of Equation (2.193) into Equation (2.194) gives @U a @U a @U a dx þ dy þ dz þ o2 ðxdx þ ydyÞ ¼ 0 @x @y @z or dU a þ o2 ðxdx þ ydyÞ ¼ 0

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Its integration yields 1 U a þ o2 ðx2 þ y2 Þ ¼ C or U ¼C ð2:195Þ 2 This equation describes any level surface of the potential U of the gravitational ﬁeld g, where x, y are coordinates of a point on the surface, while C is the value of the potential. At the same time, the potential of the attraction ﬁeld varies on this surface. Our next step is to represent the left hand side of Equation (2.195) in the spherical system of coordinates and then, using Equation (2.192), obtain the equation of the equipotential surface, which coincides with the outer surface of the earth spheroid. As was shown earlier, the potential related to a rotation is 1 1 U r ¼ o2 R2 sin2 y ¼ o2 R2 ð1 cos2 yÞ 2 2 Since 3 1 P2 ðcos yÞ ¼ cos2 y 2 2 we can say that this potential is expressed in terms of the second spherical harmonic, P2. For this reason it is natural to expect that the potential of the attraction ﬁeld also contains the term with this Legendre’s function. Applying the results, derived in the last section of the ﬁrst chapter and based on the method of separation of variables, we may write Ua ¼ k

M AP2 ðcos yÞ þ R R3

ð2:196Þ

The presence of the ﬁrst term is explained by the fact that at large distances from the earth its mass M generates the ﬁeld of a point source with the same mass. Thus, Equation (2.195) can be written as k

M AP2 1 2 2 þ 3 þ o R ð1 cos2 yÞ ¼ C R 2 R

ð2:197Þ

Here A is unknown and in order to ﬁnd it we substitute Equation (2.192) into Equation (2.197). This gives M A 3cos2 y 1 1 2 2 ð1 þ f cos2 yÞ þ 3 þ o a ð1 cos2 yÞ ¼ C ð2:198Þ a a 2 2 Note that in the second and third terms we have assumed that R ¼ a. This replacement is justiﬁed because A and o2 are of the same order as the parameter f. Taking into account the fact that the left hand side of Equation (2.198) has to be constant, the coefﬁcient in front of cos2y vanishes, that is, A 1 2fM 2 2 o a k ¼ ð2:199Þ a3 3 a k

As soon as A is found, the terms which are independent of the angle y, allow us to determine the constant C. Thus, the expression for the potential of the attraction

Gravitational Field of the Earth

ﬁeld is Ua ¼ k

M a3 o2 a2 fkM 1 þ 3 cos2 y R R a 3 2

Therefore, the potential of the gravitational ﬁeld has the form: M a3 o 2 a2 fM 1 1 k cos2 y þ o2 R2 sin2 y UðpÞ ¼ k þ 3 R R a 3 2 2

105

ð2:200Þ

ð2:201Þ

By deﬁnition, the angle between the z-axis and the radius vector R is equal to y. At the same time, the gravitational ﬁeld can be represented as a sum of the attraction force and centrifugal one, and it is almost opposite to the radius vector. Therefore, the radial component of the ﬁeld is gR ¼ g cos t Since the angle t exceeds p/2 and the difference: pt is very small, we can neglect the tangential component and assume gR ¼ g

ð2:202Þ

Here g is the magnitude of the gravitational ﬁeld. Differentiation of Equation (2.201) with respect to R yields @U M 3a3 1 2 2 kfM 1 ¼k 2þ 4 o a g¼ ð2:203Þ cos2 y o2 R sin2 y @R a 3 R R 2 where R is given by Equation (2.192). Taking into account the fact that Rn að1 þ nf cos2 yÞ we have g¼k

M 3 2 5 2 kfM o o ð1 þ f Þ a þ a cos2 y a2 2 2 a2

ð2:204Þ

In particular, at points of the equator: ga ¼ k

M 3 ð1 þ f Þ o2 a a2 2

Introducing again the parameter m m¼

o2 a ga

we can rewrite Equation (2.204) as 3 M 1 þ m ga ¼ k 2 ð1 þ f Þ 2 a

ð2:205Þ

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Thus, kMð1 þ f Þ ¼

3 1 þ m a2 g a 2

ð2:206Þ

Correspondingly, Equation (2.203) becomes

1 þ 3=2m f 3 3 5 2 m g ¼ ga 1 þ m m þ cos y 1þf 2 2 2 or 5 2 m f cos y g ¼ ga 1 þ 2 which describes Clairaut’s theorem by Equations (2.187 and 2.188).

2.6. POTENTIAL OF THE GRAVITATIONAL FIELD IN TERMS OF SPHERICAL HARMONICS Earlier we solved the boundary value problem for the spheroid of rotation and found the potential of the gravitational ﬁeld outside the masses provided that the outer surface is an equipotential surface. Bearing in mind that, we study the distribution of the normal part of the ﬁeld on the earth’s surface, where the position of points is often characterized by spherical coordinates, it is natural also to represent the potential of this ﬁeld in terms of Legendre’s functions. This task can be accomplished in two ways. The ﬁrst one is based on a solution of the boundary value problem and its expansion into a series of Legendre’s functions. We will use the second approach and proceed from the known formula, (Chapter 1) which in fact originated from Legendre’s functions 1 1 X R1 n ¼ Pn ðcos cÞ; R R1 ð2:207Þ Lqp R n¼0 R Here R is the distance between an observation point p and the center of mass, where the origin of the spherical system of coordinates is located. R1 is the distance of an elementary mass from the origin; Lqp is the distance between this mass and the point p: Lqp ¼ ðR2 þ R21 2RR1 cos cÞ1=2 and c is the angle formed by the radius vectors R1 and R. In the Cartesian and spherical systems points q and p have coordinates: qðx1 ; y1 ; z1 Þ and qðR1 ; j1 ; l1 Þ;

pðx; y; zÞ and pðR; j; lÞ

ð2:208Þ

The quantities j, l deﬁne the latitude and longitude of some point, respectively. It is also important to express the angle between the vector radii R and R1 in terms of the latitude and longitude. With this purpose in mind we make use of the deﬁnition

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Gravitational Field of the Earth

of the dot product: R R1 ¼ RR1 cos c

or

cos c ¼

R R1 x1 x þ y1 y þ z1 z ¼ RR1 RR1

ð2:209Þ

As is well known, x ¼ R cos j cos l;

y ¼ R cos j sin l;

and

z ¼ R sin j

and x1 ¼ R1 cos j1 cos l1 ;

y1 ¼ R1 cos j1 sin l1 ;

and

z1 ¼ R1 sin j1

ð2:210Þ

Substitution of Equation (2.210) into Equation (2.209) gives cos c ¼ cos j cos j1 cos l cos l1 þ cos j cos j1 sin l sin l1 þ sin j sin j1

ð2:211Þ

Now we are ready to perform the transformation of the expression for the potential of the gravitational ﬁeld Z dm 1 UðpÞ ¼ k dV þ o2 r2 ð2:212Þ L 2 qp V Here r ¼ ðx2 þ y2 Þ1=2 Replacing the inverse distance L1 qp by the series, Equation (2.207), and changing the order of summation and integration, we obtain: 1 Z M kX R1 n 1 UðpÞ ¼ Pn dm þ o2 R2 cos2 j ð2:213Þ R n¼0 0 2 R since r ¼ R cos j This expression for the potential is valid for any distribution of masses, provided that RZR1. Now we focus our attention on the potential of the normal ﬁeld caused by regular part of masses. As was assumed before, their density is independent of the longitude, and the equator is a plane of symmetry. For this reason this part of the potential of the earth has a similar behavior and, correspondingly, Equation (2.213) is greatly simpliﬁed. Let us rewrite it in the form: U ðpÞ ¼

Z Z M Z M k 1 M 1 o2 R3 1 Mþ cos2 j þ 3 R1 P1 dm þ 2 R21 P2 dm þ R31 P3 dm R R 0 2k R 0 R 0 ! Z M Z M 1 X 1 1 4 n þ 4 R1 P4 dm þ R1 Pn dm ð2:214Þ Rn 0 R 0 n¼5

In deriving Equation (2.214) we took into account the fact that P0 ¼ 1 and the integral corresponding to zero term of the series is equal to mass of the earth, M. It may be proper to recall that the z-axis coincides with the axis of rotation and the center of mass is located at the origin of coordinates; j is the latitude. It is clear that

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the ﬁrst term in brackets is the leading term and represents the attraction of the earth. This ﬁeld possesses axial symmetry and it coincides with that of a point mass M located at the origin 0. The integrand of the second term Z 1 M R1 P1 dm ð2:215Þ R 0 contains the ﬁrst order of the spherical harmonic which is equal to P1 ¼ cos c

ð2:216Þ

It is a simple matter to show that for a regular distribution of masses this term vanishes. Substituting Equation (2.216) into Equation (2.215) and making use of the equality (2.211), we have Z M Z M R1 P1 dm ¼ ðsin j sin j1 þ cos j cos j1 cos l cos l1 0

0

þ cos j cos j1 sin l sin l1 ÞR1 dm

ð2:217Þ

First of all, an integration of terms containing R1cos j1cos l1 and R1cos j1sin l1 does not give any contribution, otherwise the potential would depend on cos l and sin l, which contradicts our assumption of symmetry around the axis of rotation. The integral from the term with R1sin j1 also vanishes, because in the opposite case the potential would contain a multiplier proportional to the angle j and, therefore, becomes non-symmetrical with respect to the equator, (where the coordinate z changes sign). Thus, the second term in the brackets disappears and this is the ﬁrst important simpliﬁcation. It turns out that this conclusion is valid for any distribution of masses, that is, for the real Earth. Taking into account the fact that the center of mass coincides with the origin of coordinates, 0, we choose the orthogonal systemx, Z, z, so that z is directed from the origin to the observation point p. The integral then becomes Z M Z M Z M R1 P1 dm ¼ R1 cos cdm ¼ zdm ð2:218Þ 0

0

0

and, by deﬁnition of the center of mass, it is equal to zero. Before we discuss the next two terms in brackets let us consider spherical harmonic terms of odd order, and ﬁrst of all the integral Z M R31 P3 dm ð2:219Þ 0

As we know 5 3 2 cos c cos c P3 ðcos cÞ ¼ 2 5

ð2:220Þ

Substituting Equation (2.211) into the last equality and performing some simple algebra, we can see that the integrand in Equation (2.219) contains either terms

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proportional to l or to the ﬁrst and third power of R sin j. But this contradicts conditions of symmetry of mass distribution, and therefore the integral is equal to zero. The same arguments apply to all terms of the series with odd orders of spherical harmonics. Next consider the term containing the Legendre’s function of the second order: Z M R21 P2 dm ð2:221Þ 0

By deﬁnition, we have 3 1 2 cos c P2 ðcos cÞ ¼ 2 3 Making use again of Equation (2.211) and discarding terms which are obviously functions of l, we have cos2 c ! sin2 jsin2 j1 þ cos2 jcos2 j1 ðcos2 lcos2 l1 þ sin2 lsin2 l1 Þ The expression in brackets can be written as 1 þ cos 2l 1 cos 2l 2 1 cos2 l1 þ sin l1 ! 2 2 2 Here we have also dropped terms that depend on longitude. Thus, cos2 c ! sin2 jsin2 j1 þ

1 3 ð1 sin2 jÞð1 sin2 j1 Þ ¼ sin2 jsin2 j1 2 2 1 2 1 2 1 sin j sin j1 þ 2 2 2

Correspondingly, 9 3 3 1 P2 ! sin2 jsin2 j1 sin2 j sin2 j1 þ ! 4 4 4 4 The sign ‘‘-’’ means that terms with l are not taken into account. Substituting the latter into Equation (2.221) we obtain Z M Z M 3 2 1 3 2 1 2 2 sin j sin j1 R1 dm R1 P2 dm ¼ ð2:222Þ 2 2 0 2 2 0 Let us represent the integrand in the form 3 2 1 3 R2 x 2 y2 sin j1 R21 ¼ z21 1 ¼ z21 1 1 2 2 2 2 2 2 Now we introduce two moments of inertia; one of them, A, around an arbitrary axis in the equatorial plane and the other, C, around the rotation axis, and taking into account the axial symmetry: Z M Z M 2 2 2 x1 þ z1 dm ¼ y1 þ z21 dm A¼ 0

0

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and Z

M

C¼ 0

x21 þ y21 dm

ð2:223Þ

Since z21

x21 y21 z21 þ x21 z21 þ y21 ¼ þ x21 y21 2 2 2 2

we have Z 0

M

R21 P2 ðcos cÞdm

3 2 1 ¼ sin j ðC AÞ 2 2

ð2:224Þ

Introducing the quantity, which is usually called ‘‘the mechanical ellipticity’’ H¼

CA C

in place of Equation (2.224) we obtain Z M 3 1 R21 P2 dm ¼ sin2 j HC 2 2 0

ð2:225Þ

ð2:226Þ

The parameter H is expressed through moments of inertia and characterizes the dynamic ﬂattening of the earth. Applying the same approach and omitting terms containing l, we have Z M Z M 6 2 3 1225 4 6 2 3 4 4 4 R sin j1 sin j1 þ R1 P4 dm ¼ sin j sin j þ dm 7 35 0 64 1 7 35 0 ð2:227Þ Thus, the expression for the potential of gravitational ﬁeld is M CH 3 2 1 o2 R3 1 sin cos2 j j UðpÞ ¼ k þ R 2 2kM MR2 2 D 6 2 3 4 þ 4 sin j sin j þ þ 7 35 R

ð2:228Þ

Here 1 D¼ M

Z 0

M

1225 4 6 2 3 4 R sin j1 sin j1 þ dm 64 1 7 35

ð2:229Þ

Let us recall that this equation was derived assuming that the distribution of masses is characterized by a symmetry with respect to the z-axis and the equator. In addition, the center of mass is located at the origin and the total mass M is equal to that of the earth. Earlier we found the equation for the outer surface of the ideal Earth, proceeding from a solution of the boundary value problem. It is useful also to make use of Equation (2.228) and to derive the same result. To simplify derivations we

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neglect the last term in this equation and introduce the parameter J 20 ¼

CA Ma2

ð2:230Þ

Here a is the earth’s radius at the equator and J20 as well as H characterizes the dynamic ﬂattening of the earth. Then, in place of Equation (2.228) we have kM a a3 m R2 2 2 þ UðpÞ ¼ J 20 ð1 3sin jÞ þ cos j ð2:231Þ a R 2R3 2 a2 The parameter m was introduced earlier m¼

o2 a ga

m

1 289

ð2:232Þ

As we know, it is very small

and gives the ratio of the centrifugal and attraction forces at the equator. The Equation (2.231) describes the potential of the gravitational ﬁeld at points where RZa. In order to obtain the equation of the level surface of the potential which coincides with the surface of our model of the earth, we have to let U ¼ C in Equation (2.231) and specify the potential at one point of this surface. With this purpose in mind assume that R ¼ a and j ¼ 0. Substitution of these values in Equation (2.231) gives kM 1 m Uða; j ¼ 0Þ ¼ C ¼ 1 þ J 20 þ ð2:233Þ a 2 2 Correspondingly, the equation of the level surface of the potential which closely describes the earth’s ﬁgure is 1 1 a2 m R3 1 1 m 2 2 1 þ J 20 2 ð1 3sin jÞ þ 1 þ J 20 þ cos j ¼ ð2:234Þ R 2 2 a3 a 2 2 R Now we arrive at an important result, namely, with an accuracy of the square of the ﬂattening this equation characterizes the ellipsoid of rotation. From the last equation we have R 1 þ 1=2J 20 ða2 =R2 Þð1 3sin2 jÞ þ m=2ðR3 =a3 Þcos2 j ¼ a 1 þ 1=2J 20 þ m=2

ð2:235Þ

Here R and j are coordinates of the equipotential surface and they vary from point to point, but the potential remains the same. Next, we will attempt to ﬁnd the relationship between these coordinates in the explicit form. Taking into account the fact that parameters J20 and m are very small, we will use the following

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approximation 1 1 m 1 J 20 1 þ 1=2J 20 þ m=2 2 2 Then, multiplying the latter by the numerator of Equation (2.235) and keeping only terms of the ﬁrst order in the ﬂattening and m, we obtain R 1 m 1 m 3 m ¼ 1 þ J 20 ð1 3sin2 jÞ þ cos2 j J 20 ¼ 1 J 20 þ sin2 j ð2:236Þ a 2 2 2 2 2 2 Because of small ﬂattening, we let R ¼ a at the right hand side of Equation (2.235). Thus, we again arrive at the known equation of the spheroid in the spherical system of coordinates R ¼ að1 asin2 jÞ

ð2:237Þ

Here 3 m ð2:238Þ a ¼ J 20 þ 2 2 characterizes the ﬂattening of the spheroid. It is a simple matter, performing a differentiation of the potential U(p) and preserving terms proportional to the ﬁrst and second order of ﬂattening, to derive an expression for the normal gravitational ﬁeld. As was shown in the previous section, Equation (2.185), it has the form g ¼ ga ð1 þ f sin2 j f 1 sin 2jÞ

ð2:239Þ

where f ¼

gb ga ga

and

1 1 f 1 ¼ ff þ f 2 ; 4 8

f ¼

ab a

A similar formula describes the dependence of the distance from the origin to any point of the outer surface of the spheroid. In both expressions terms proportional to the third and higher order of ﬂattening are discarded. This reference ellipsoid and its ﬁeld are deﬁned by four constants. The best-known and widely used values are a ¼ 6378388:000 m;

f ¼

1 ; 297:000

ga ¼ 978:049000 Gals,

o ¼ 0:72921151 104 s1 The corresponding international formula for the normal gravitational ﬁeld is g ¼ 978:0490ð1 þ 0:0052884sin2 j 0:0000059sin2 2jÞ Gals The potential on the surface of this ellipsoid is U 0 ¼ 6263978:7 103 Gals m

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The product of the earth’s mass and the gravitational constant is kM ¼ 3:9863290 1014 m3 =s2 and the mass of the earth is equal to M ¼ 5:98 1024 kg Also, J 20 ¼ 0:0010920 The numerical values for coefﬁcients in Equation (2.239) can be obtained differently. The right hand side of this equation contains three unknowns: ga, f*, f*1. Performing measurements at three points of the earth’s surface at different latitudes we obtain three equations to determine these parameters. This approach is valid for the surface of the spheroid and, therefore, it gives only a ﬁrst approximation. In reality on the physical surface of the earth the gravitational ﬁeld differs from the normal ﬁeld, and for this reason the determination of the constants is carried out in the following way. The gravitational ﬁeld is measured at as many as possible points, distributed sufﬁciently uniformly on the earth’s surface. Then, we obtain a system, where a number of equations is much more than the three unknowns. Applying the least-squares method we deﬁne two unknowns ga and f*. Since the last parameter f 1 is very small, it is calculated with a very large error. Because of this f 1 is usually determined analytically. Now let us brieﬂy discuss the importance of the parameter J20 in studying the internal parts of the earth. In this light consider the mean moment of inertia: C þ 3A 3 For instance, in the case of a homogeneous sphere I¼

ð2:240Þ

I s ¼ 0:4Ma2

ð2:241Þ

Here M is the total mass of the spherical Earth and a its radius. Comparison of values of I and Is is useful for understanding general features of the density distribution inside the earth’s surface. In order to ﬁnd I we have to know A and C. The parameter J20 establishes one relationship between them; the other is obtained from astronomy and it gives CA ¼ 0:0032732 C Correspondingly, for the earth we have H¼

I ¼ 0:33076 Ma2 It is convenient to deal with the normalized value of the mean moment of inertia I ¼

I Ma2

ð2:242Þ

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Thus, for the earth and a homogeneous sphere we obtain the following values: I ¼ 0:33

and

I s ¼ 0:40

ð2:243Þ

As calculations show, when the density increases with a distance from the earth’s surface the parameter I* is smaller than 0.4. On the contrary, with a decrease of the density toward the earth’s center we have I*40.4. Inasmuch as in reality I*o0.4, we conclude that there is essential concentration of mass in the central part of the earth. In other words, the density increases with depth and this happens mainly due to compression caused by layers situated above, as well as a concentration of heavy components. In conclusion, it may be appropriate to notice the following: a. In the last three sections, we demonstrated that the normal gravitational ﬁeld of the earth is caused by masses of the ellipsoid of rotation and its ﬂattening can be determined from measurements of the gravitational ﬁeld. b. This ﬁeld is the dominant part of the total ﬁeld, since the irregular part of the masses generates not more than several hundreds of milliGalls, which constitutes less than 0.1% of the resultant ﬁeld. c. The positions of points of the spheroid surface are known with respect to the center of mass, that is, the origin of the coordinate system, but the mutual position of the real surface of the earth and this spheroid remains unknown.

2.7. GEOID AND LEVELING In the previous sections we studied the distribution of the normal ﬁeld caused by masses of an ellipsoid of rotation. Also we demonstrated how to calculate the ﬂattening from measurements of the gravitational ﬁeld. Our next goal is to improve our knowledge of the ﬁgure of the real Earth, and this problem can be solved in different ways. One of them requires knowledge of the distances from the ellipsoid of rotation to points of the physical surface of the earth; these distances are called the geodesic heights. Certainly, the accuracy of determination of the earth’s ﬁgure becomes higher with an increase of number of points where these heights are known. Usually their evaluation consists of two steps. The ﬁrst one deﬁnes the distance from a geoid to points of the physical surface of the earth, while the second is based on the Stokes formula and allows one to determine the position of the geoid with respect to the ellipsoid of rotation. In this section we focus on the concept of the geoid and its position with respect to the physical surface of the earth, and with this purpose in mind consider the potential W(p) of the total gravitational ﬁeld W ðpÞ ¼ UðpÞ þ TðpÞ

ð2:244Þ

where U(p) is the potential of the normal ﬁeld and T(p) is the potential caused by the irregular part of the masses. Taking into account the fact that the normal ﬁeld g(p) is much greater than the secondary one, we can expect that TðpÞ UðpÞ

ð2:245Þ

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As we know, the behavior of the potential W(p) of the gravitational ﬁeld can be illustrated with the help of the equipotential or level surfaces W ðx; y; zÞ ¼ W 0

ð2:246Þ

where W0 is constant at points of such a surface. Therefore, if dl is an elementary vector tangential to the level surface, then grad W dl ¼ 0

ð2:247Þ

that is, the gravitational ﬁeld is normal to the equipotential surface passing through the same point. In this sense, as was pointed out earlier, the level surface has the meaning of the horizontal, but the plumb line of instruments, showing the direction of the ﬁeld, is often called the vertical or normal. It seems that Gauss was the ﬁrst who suggested treating one of the equipotential surfaces of the gravitational ﬁeld of the earth as the mathematical ﬁgure of the earth; later this surface was called the geoid by the German geodesist Listing in 1873. By deﬁnition, the level surface, which coincides with the mean surface of oceans and extends through continents and islands, is called the geoid. It has one important advantage with respect to other models of the earth, namely, it coincides with the physical surface of the earth over the oceans, which constitute almost 70% of the earth’s surface. In the remaining part, inside the continents, it is located mainly at several hundreds of meters beneath the physical surface of the earth, and only in areas of mountains the distance between these surfaces may reach several kilometers. In this light, it may be proper to notice the following. The transition from the sphere to the ellipsoid of rotation was an important improvement of our knowledge of the ﬁgure of the earth, which has order of ﬂattening, that is, about 21 km at the equator. At the same time, the geoid can be hardly treated as the next approximation. The distance between the ellipsoid and geoid does not exceed 100 m, while on the continents the separation between the real earth surface and the geoid can be much greater and reach 8000 m. However, on the oceans the surface of the geoid coincides with the mean surface without taking into account the inﬂuence of waves, and this is already a great achievement. As concerns the geoid inside continents, the separation between its surface and the physical surface of the earth can be determined with a relatively high accuracy using the well-known procedure of geodesy, namely, leveling. Carrying out these measurements, it is possible to determine with a rather small error the elevation of points located at the earth’s surface with respect to the geoid and, therefore, to translate results of gravimetric and geodetic surveys from the complex physical surface of the earth to the smoother surface of the geoid. This procedure is of a great importance for understanding the global behavior of the gravitational ﬁeld of the earth and its shape. Before we continue and describe the concept of height, as well as the main features of leveling, let us make two comments: a. Bearing in mind that the total ﬁeld only slightly differs from the normal one, the shape of geoid is very close to that of an ellipsoid of rotation. At the same time there is a difference. The curvature of the ellipsoid is a continuous function of the points on its surface, but this is not true in the case of the geoid at

116

b.

Methods in Geochemistry and Geophysics

points where a density is a discontinuous function. In fact, as follows from Poisson’s equation the behavior of the sum of the second derivatives of the potential, W(x, y, z), is related to the curvature of the level surface and depends on the density at the same point. Perhaps, this is the main reason why it is preferable to refer results of measurements on the real surface of the earth to the ellipsoid of rotation rather than to points of a geoid. As was pointed out earlier, the determination of the elevation of points of the physical surface of the earth with respect to the geoid is the only the ﬁrst step in studying the shape of the earth. The next step is the calculation of the distance between corresponding points of the ellipsoid of rotation and the geoid, and this subject will be described in the following section.

2.7.1. Geoid and quasi-geoid We have introduced the concept of the geoid but do not know yet how to determine its position on continents. First of all, it may be proper to notice that if we knew the gravitational ﬁeld inside the earth the construction of a geoid would be a very simple task, since every vector line of the ﬁeld g is normal to the equipotential surface. Therefore, an extension of the geoid from the ocean into the continents is, indeed, an elementary procedure. In reality, however, the ﬁeld is usually known only on the earth’s surface, and because of this it is appropriate to outline an approximate method, which deﬁnes the position of the geoid, or more precisely, a quasi-geoid with a sufﬁciently high accuracy. At the beginning let us discuss this task without the use of the concept of elevation, that is, a height. By deﬁnition, we have g ¼ grad W

ð2:248Þ

and the linear integral between two points Z

b

g dl ¼ W ðbÞ W ðaÞ ¼ DW

ð2:249Þ

a

is path independent, because DW is a difference of potentials. First, consider a path ab, located on the earth’s surface, Fig. 2.8a. Its initial point a is situated on the ocean’s surface, that is, it belongs to the geoid. We divide the path in to many small displacements dl; each of them is characterized by a magnitude and a direction. Then, measuring its value and an orientation, as well as the ﬁeld, the dot product: g(p) dl(p) is calculated. Performing their summation along the path we obtain the value of the integral, DW, on the right hand side of Equation (2.249). Next, we choose a different path, which is located beneath the earth’s surface but it has the same terminal paths. Its ﬁrst part ac is located on the surface of the geoid, while the second cb coincides with the vector line of the gravitational ﬁeld, but it has the opposite direction. Within the ﬁrst interval ac the

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Gravitational Field of the Earth

(a)

(b) b Hn

H or quasi-geoid

a c d

geoid ellipsoid

(c)

(d)

b

S2

n

b’

a’

m

a S1

Fig. 2.8. (a) Illustration of Equation (2.250), (b) positions of geoid and quasi-geoid, (c) orthometric height, (d) reading at the benchmarks.

dot product g dl is equal to zero and therefore Z b Z b Z b g dl ¼ gdh ¼ gdh ¼ DW a

c

ð2:250Þ

c

since the integral is path independent. Here dh is the element of the vector line, directed upwards and g is the magnitude of the ﬁeld. As was pointed out earlier, if we knew the gravitational ﬁeld at every point inside the earth, the determination of the geoid would be trivial; we only have to plot from a to point b a value DW along the vector line passing through this point that deﬁnes one point of the geoid. Repeating this procedure for different points of the earth’s surface we could obtain the geoid. In this approach it is essential that all points of the path cb belong to the vector line. Taking into account the fact that in general we do not know the ﬁeld beneath the physical surface of the earth, this problem cannot be solved exactly. However, it is possible to make use of an approximate solution, which deﬁnes the position of the geoid with a sufﬁciently high accuracy. Bearing in mind that the total potential W differs only slightly from that of the normal ﬁeld, we make use of two assumptions: 1. The vector line of the ﬁeld g between the earth’s surface and the geoid can be replaced by a straight line which has the same direction as the plumb line, (vertical), at the point b of the earth’s surface. 2. At the points of this line the magnitudes of the normal and total ﬁelds are equal to each other: g ¼ g. Then in place of Equation (2.250) we have Z d DW gds ð2:251Þ b

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where ds is the element of the straight line and g the magnitude of the normal ﬁeld, which can be calculated. Plotting along the straight line the values of gds and performing a summation we determine a position of the point d, where the integral on the right hand side of Equation (2.251) is equal to DW. Speaking strictly this point does not belong the geoid, but it is located on a surface which may be called the quasi-geoid. As we already mentioned, these surfaces are very close to each other, and the word ‘‘geoid’’ really means the quasi-geoid. Of course, such closeness is not occasional and it is explained by the fact that the secondary ﬁeld is several orders smaller than the normal one and distances between the geoid and the physical surface of the earth are relatively small too. 2.7.2. A height and an elevation Until now, in essence, we used a geometrical approach based on the behavior of the gravitational ﬁeld and, in particular, the level surfaces of its potential. This was done in order to emphasize the relationship between the basic concepts of the ﬁeld theory and the fundamental geodetic characteristic, such as a height or elevation. Next, we deﬁne a position of the geoid as well as quasi-geoid in a slightly different but more conventional way. First of all, let us notice again the following. The plumb line of any instrument has the direction of the gravitational ﬁeld at the same point and, therefore, it is perpendicular to the level surface passing through an observation point. Correspondingly, such deﬁnitions as a direction of the ﬁeld, the direction of its vector line, the vertical as well as the direction of the plumb line are synonymous. First, consider the integral in Equation (2.250), where integration is performed along the vector line. Applying the mean-value theorem of calculus we can write Z b gdh ¼ gðpÞH ð2:252Þ c

where p is some point of the vector line, but its position is unknown; g(p) the mean value of the gravitational ﬁeld, and H the height; a and b points on the earth’s surface. It is measured along the curved plumb line and called the orthometric height or elevation of the point b with respect to the geoid. Of course, H is positive since the integral on the left hand side of this equation is negative. By deﬁnition, Rb g dl DW ¼ ð2:253Þ H or ¼ a gðpÞ gðpÞ Note that a change of the potential, DW is sometimes called a geo-potential, and values of orthometric heights deﬁne the position of points on the physical surface of the earth with respect to the geoid. However, in order to ﬁnd such heights we have to know the ﬁeld along the survey line and along the vector line beneath the earth’s surface. The ﬁrst requirement can be satisﬁed by performing measurements of the ﬁeld, but the second condition cannot usually be met, because the distribution of masses is not known exactly. Of course, the direction of the plumb line is also unknown, and for these reasons, the orthometric heights or the position of the geoid

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119

are not determined exactly. This forces us to make the ﬁrst approximation and replace the mean value of the gravitational ﬁeld in Equation (2.253) by that of the normal ﬁeld. This replacement gives the value of normal height H n ðbÞ ¼

DW gav

ð2:254Þ

Unlike Hor(b), the normal height is measured along the straight line, which coincides with the vertical at the point b. Determination of these heights gives a system of normal heights beneath the earth. Plotting the normal heights along the plumb lines, we obtain some surface which is very close to the geoid, and as was pointed out it is called the quasi-geoid. The difference between these surfaces is the same as the difference between the orthometric and normal heights. On the oceans they coincide with each other and the mean surface of oceans. At valleys the difference is not more than several centimeters, and at mountains it does not exceeds 1 m. Certainly, the quasi-geoid is not a level surface, but the change of the potential on this surface is very small. Let us evaluate the difference between the geoid and quasigeoid positions, Fig. 2.8b. By deﬁnition, we have Z c 1 1 gav Dz ¼ H or H n ¼ av av gdh ¼ 1 av H n g g g b The value of the difference gavgav may reach several hundreds of milliGalls. Assuming that it is equal to 100 mGal, gav ¼ 9.8 Gal, and HnE500 m, we obtain DzE5 cm. 2.7.3. Leveling To illustrate Equation (2.253), consider two points a and b located at different equipotential surfaces and assume that they are located so close to each other that the ﬁeld between them changes linearly along the vector line. At the same time the separation of these surfaces may vary. First, choose the path ab0 b, where points b0 and b are at the same plumb line, perpendicular to both surfaces, Fig. 2.8c. Then, Z b Z b DW g dl ¼ gdh or Dh ¼ av ð2:255Þ g a b0 and, by deﬁnition, gðb0 Þ þ gðbÞ 2 As we know, the quantity Dh is called the elevation or an orthometric height of the point b with respect to point b0 . In particular, if both points are located at the same level surface, this elevation is zero. If we take a different path, for instance, aa0 b, Equation (2.255) gives a height of the point a0 with respect to a, which can differ from the ﬁrst one. Suppose that within the interval ab the equipotential surfaces are parallel to each other, that is, the ﬁeld does not change along these surfaces. In such a case the gav ¼

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height, calculated from Equation (2.255), is independent of the path, but this is rather exception, since in the general case the surfaces are not parallel, and in those places where the ﬁeld is stronger we obtain smaller elevation, but with a decrease of the ﬁeld the elevation becomes bigger. With an increase of the separation between points a and b we can imagine a system of many level surfaces between them, and a sum of orthometric heights becomes a function of the path. This happens because in calculating each elevation, Dh, different reference points are used. In this light, it may be proper to brieﬂy describe the leveling survey which is used in geodesy to calculate the elevation of points on the earth’s surface. This method does not require knowledge of the ﬁeld magnitude but takes into account its direction at each observation point. Consider a relatively small interval of the survey path on the earth’s surface; at its middle we place an instrument in such a way that its plate is horizontal. In other words, it is perpendicular to the plumb line, and, correspondingly, this surface is tangential to the level surface, passing through this point, Fig. 2.8d. This condition is usually provided with the help of an air bubble of the level. The benchmarks are placed vertically at the terminal points of each interval, that is, parallel to the plumb line of the instrument. Inasmuch as the distance between these points is sufﬁciently small, we may assume that the equipotential surfaces passing through them and the middle point are parallel. This means that benchmarks are tangential to vector lines. Then, the difference of reading at the benchmarks: Dh ¼ m n gives the elevation of the front point with respect to the back one. Of course, it can be either positive or zero or negative. In principle, the same result can be obtained applying trigonometric leveling or measuring the pressure of air. Performing a summation of these heights along the survey line we ﬁnd an elevation of the last point b over the point a. If the latter belongs to the mean ocean surface, geoid, we obtain the height: X H¼ Dhi ð2:256Þ of the point b of the physical surface of the earth with respect to the geoid. In other words, Equation (2.256) deﬁnes the position of the geoid beneath the point b. However, this procedure, called geometric leveling, gives a sufﬁciently accurate result, if the distance between the terminal points is not too large. As we know, this is related to the fact that the sum in Equation (2.256) in general depends on the path of the survey. This is the reason why the orthometric and normal heights were introduced and the gravitational ﬁeld is measured.

2.8. STOKES’S FORMULA We assume that with the help of leveling we solved our ﬁrst problem and found the separation between the geoid and the points of the physical surface of the earth. Our next step is to determine the position of the geoid with respect to the reference ellipsoid. The solution of this fundamental problem was given by Stokes. To begin,

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we introduce some concepts of the physical geodesy and then derive the Stokes formula. 2.8.1. Bruns’s formula Suppose that the inﬂuence of masses above the geoid is removed and we know the gravitational ﬁeld g(p) on the surface of the geoid. The procedure which allows one to solve this is called reduction of the gravitational ﬁeld, and it is necessary in order to apply Stokes theory. Let us form the difference: Dg ¼ gðpÞ gðqÞ

ð2:257Þ

Here the points p and q are terminal points of the perpendicular drawn from the reference ellipsoid to the geoid; g(p) the total ﬁeld on the surface of the geoid at the point p, but g(q) is the normal ﬁeld at the point q of the ellipsoid, Fig. 2.9a. It turns out that comparison of these ﬁelds allows us to ﬁnd the separation between the spheroid and geoid. For instance, if the difference Dg is zero, then we may say that the geoid coincides with a spheroid. In reality there is no complete coincidence, even though the value of Dg is very small. This indicates that the shape of the geoid is very close to that of the spheroid. Stokes and later independently Poincare realized that the function Dg, given on the surface of the geoid, permits one to ﬁnd its position with respect to the spheroid. There are three reasons why the gravity anomaly differs from zero, namely, (a) the total and normal ﬁelds are considered at different level surfaces, (b) these ﬁelds are caused by different distributions of masses, (c) the difference gg is formed in such a way that the points, where the ﬁelds are calculated, are located on the same normal to either the geoid or the spheroid. It is possible to use any of them, because the angle between them is very small and does not exceeds 10 . Correspondingly, a change of the normal leads to an error equal to @W @W @W a2 @W cos a ¼ ð1 cos aÞ ¼ @n @n @n 2 @n where a is the angle between two normals. Thus, considering magnitudes, the relative error is around 107, that is extremely small. (b)

(a)

S

p geoid N q γ

p r

n 0

g

ρ

Ψ R

Reference ellipsoid

Fig. 2.9. (a) Illustration of Equation (2.257), (b) Poisson’s integral.

q

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Earlier we represented the total mass of the earth as a sum of regular and irregular parts and, correspondingly, the potential of the gravitational ﬁeld at points q and p are W ðpÞ ¼ UðpÞ þ TðpÞ

and

W ðqÞ ¼ UðqÞ þ TðqÞ

ð2:258Þ

The function T(p) is called the disturbing potential and it is very small, (T U). Now we focus our attention on the space between the geoid and the ellipsoid of rotation and assume the potential of the gravitational ﬁeld of the geoid, W(p), and the normal potential of the spheroid, U(q), are equal : UðqÞ ¼ W ðpÞ ¼ C

ð2:259Þ

In other words, we obtain the reference ellipsoid from the condition that its normal potential U has the same value as the resultant potential of the geoid. At the same time, it is assumed that the geoid and the ellipsoid enclose equal masses and have the same angular velocity. Taking into account the fact that the distance between points p and q is very small, the normal potential varies almost linearly and this gives: @U N ¼ UðqÞ gN ð2:260Þ @n Here N is the distance between points p and q measured along the perpendicular from the point q to the geoid, Fig. 2.9a. The linear behavior of the normal potential implies that the ﬁeld g is constant between the geoid and the reference ellipsoid. The change of sign in Equation (2.260) is related to the fact that the ﬁeld has a direction, which is opposite to the direction of differentiation. As follows from the ﬁrst equation of the set (2.258 and 2.260) we have UðpÞ ¼ UðqÞ þ

W ðpÞ ¼ UðpÞ þ TðpÞ ¼ UðqÞ gN þ T

ð2:261Þ

where T(p) ¼ T is the value of the disturbing potential at the surface of the geoid. At the surface of the geoid Equation (2.261) is greatly simpliﬁed and, in accordance with Equation (2.259), it gives T ¼ gN

ð2:262Þ

This is one of the fundamental formulas of the physical geodesy, named Bruns’ theorem and it states that at each point of the geoid the value of the disturbing potential is equal to the product of the normal ﬁeld magnitude and the height with respect to the ellipsoid. Thus, if we knew the function T on the surface of the geoid it would be a simple matter to determine its position with respect to the reference ellipsoid. It may be proper to notice that in deriving this relation we did not make any assumptions about the distribution of masses between the ellipsoid and geoid. 2.8.2. Boundary condition on the geoid surface In order to determine the function T we focus on the space, bounded by the geoid and inﬁnity and formulate the boundary value problem with respect to

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123

unknown function T. In reality, there are masses between the geoid and earth’s surface, but we imagine that their inﬂuence is taken into account, that is they are removed. Then, we can treat the disturbing potential as a harmonic function in this volume, which obeys the Laplace’s equation. One boundary condition is known. In fact, at inﬁnity the ﬁeld of irregular masses, as well as the potential T, tends to zero. Now we have to formulate the second condition at the surface of the geoid. Taking the normal derivative on both sides of Equation (2.261) and bearing in mind that g¼

@W @n

and

g¼

@U @n

we obtain @g @T @T @g or Dg ¼ þN @n @n @n @n Now, making use of Bruns’ theorem we have at points of the geoid gðpÞ ¼ gðqÞ þ N

@T 1 @g þ T ¼ Dg @n g @n

ð2:263Þ

ð2:264Þ

The latter represents the boundary condition for the function T at the geoid since it is assumed that the anomaly of the gravitational ﬁeld, Dg is known at each point. 2.8.3. Boundary value problem In order to determine the function T, and, correspondingly, the height N, we have to solve the boundary value problem, that is, to ﬁnd the function T, which satisﬁes the following conditions: Outside the geoid T is a harmonic function: DT ¼ 0 At an inﬁnite distance the disturbing potential tends to zero T !0 At the surface of the geoid we have

@T 1 @g þ T ¼ Dg @n g @n

where the anomaly of gravitational ﬁeld and the normal ﬁeld as well as its derivative, are known. As was shown in Chapter 1, these conditions uniquely deﬁne the function T. For determination of the disturbing potential we will make use of Poisson’s integral, described in the Chapter 1, which allows one to ﬁnd the harmonic function E outside the spherical surface of the radius R, Fig. 2.9b, if this function, E(p), is known at points of this surface: Z r2 R2 EðpÞ E¼ dA ð2:265Þ 4pR A r3

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where r ¼ ðR2 þ r2 2Rr cos cÞ1=2

ð2:266Þ

2.8.4. Spherical approximation of the boundary condition Before we make use of Equation (2.265), let us transform the boundary condition (2.264) in the following way. With an accuracy of small quantities, which have the same order as the square of the geoid heights, a differentiation along the normal can be replaced by differentiation along the radius vector, and correspondingly the condition (2.264) becomes

@T T @g þ ¼ Dg @r g @r

if r ! R

ð2:267Þ

In essence, in place of the surface of the geoid we use a spherical surface. Now let us make one more approximation, which has the same order as the ﬂattening, and, therefore, causes a very small error in determining the height N. We assume that the normal ﬁeld is inversely proportional to the square of r. This gives @g 2g ¼ @r r and Equation (2.267) is simpliﬁed @T 2T þ ¼ Dg @r r r!R

ð2:268Þ

To facilitate derivations and express the ﬁeld T in terms of the anomaly of gravity let us multiply both sides of this equality by r. Then, in place of Equation (2.268) we have @T 1@ 2 rDg ¼ 2T r ¼ ðr TÞ ð2:269Þ @r r @r Letting E ¼ rDg Equation (2.265) becomes

1@ 2 r2 R 2 ðr TÞ ¼ r @r 4pR

Z

EðpÞ dA 3 A r

ð2:270Þ

It is proper to notice that the function E is harmonic outside the geoid. Now, multiplying both sides of this equation by rdr and integrating within the range: R ro1 we obtain

Z

1

R

@ 2 ðr TÞdr ¼ ½r2 T1 R ¼ @r

Z

Z A

r R2 r dr r3

1 3

EðpÞdA R

ð2:271Þ

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125

First, we perform some transformations of the integrand in the integral by r. By deﬁnition: @ 1 1 @r ð2:272Þ ¼ 2 @r r r @r As follows from Equation (2.266) @r r R cos c ¼ @r r Then

@ 1 r R cos c ¼ @r r r3

The latter allows us to express the integrand in Equation (2.271) in terms of derivatives with respect to r. In fact, multiplying the left and right hand side of the equality (2.272) by 2r2 and subtracting r/r, we obtain: @ 1 r r3 r2 R r 1 2r2 ¼ 3 2r2 ðr R cos cÞ rr2 ¼2 3 @r r r r r r 1 3 ¼ 3 ðr R2 rÞ ð2:273Þ r Thus, we have expressed the integrand in Equation (2.272) as a difference of two terms, allowing us to perform the integration in a relatively simple way. Substitution of Equation (2.273) into Equation (2.271) yields for the integral with respect to r Z 3 Z Z r R2 r 1 r 2 @ dr ð2:274Þ dr ¼ 2 r dr r3 @r r r Both integrals on the right hand side can be easily expressed in terms of elementary functions. First, integration by parts gives Z Z @ 1 r2 r r2 dr dr ¼ 2 @r r r r Correspondingly, Equation (2.274) becomes Z 3 Z r R2 r r2 r dr þ 3 dr ¼ r3 r r

ð2:275Þ

The integral on the right hand side can be found in tables, and it can be written in the form Z rdr r b ¼ 3=2 lnð2c1=2 r þ 2cr þ bÞ ð2:276Þ 1=2 2 c 2c ða þ br þ cr Þ Here a ¼ R2 ;

c ¼ 1;

and

b ¼ 2R cos c

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hence Z

rdr ¼ r þ R cos c ln 2ðr þ r R cos cÞ r ¼ r þ R cos c lnðr þ r R cos cÞ þ R cos c ln 2

The last term is independent of r and after substitution of limits it vanishes, and for this reason it will be discarded. Then in place of Equation (2.275) we have Z

r3 R2 r r2 dr ¼ 2 þ 3r þ 3R cos c lnðr þ r R cos cÞ þ constant r r

Its substitution into Equation (2.271) gives ½r2 T1 R ¼

1 4pR

1 r2 EðpÞ 2 þ 3r þ 3R cos c lnðr þ r R cos cÞ dA ð2:277Þ r A R

Z

Here the point p belongs to the spherical surface A of radius R. In order to ﬁnd the upper limit on the left hand side of this equality, let us recall that T is the disturbing potential. In other words, it is caused by the irregular distribution of masses whose sum is equal to zero. This means that its expansion in power series with Legendre’s functions does not contain a zero term. The next term is also equal to zero, because the origin coincides with the center of mass. Therefore, the series describing the function T starts from the term, which decreases as r3. This means that the product r2T-0 if r-N and 2 ½r2 T1 R ¼ R TðrÞ

We substitute the latter into Equation (2.277) and obtain Z 1 T¼ EðpÞSðr; cÞdA 4pR A

ð2:278Þ

Here EðpÞ ¼ RDg

Sðr; cÞ ¼

1 1 r2 þ 3r þ 3R cos c lnðr þ r R cos cÞ 2 r2 r R

To ﬁnd the expression for the upper limit, we represent r as 1 R2 R r¼r 1þ 2 cos c þ ﬃ r R cos c; 2 r2 r

if r ! 1

ð2:279Þ

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and 1 1 R ﬃ þ 2 cos c r r r Then Sðr; cÞ ¼

1 2r 2R cos c þ 3r 3R cos c þ 3R cos c ln 2r r2 r2 þ2 3r 3R cos c lnðr þ r R cos cÞ r

or Sðr; cÞ ¼

2 3r 1 R cos c r þ r R cos c 2 þ 5 3R cos c ln 2 r r r r 2r

ð2:280Þ

Making use of Equations (2.278 and 2.280) we can calculate the disturbing potential on the spherical surface and outside. In particular, at points of this surface we have r¼R

and

r ¼ 2R sin

c 2

Correspondingly, 1 c c c 2c cos ec 6 sin þ 1 5 cos c 3 cos c ln sin þ sin SðR; cÞ ¼ R 2 2 2 2 1 ð2:281Þ ¼ SðcÞ R and in place of Equation (2.278) we obtain Z 1 TðR; cÞ ¼ DgSðcÞdA 4pR A where T is the value of the disturbing potential on the spherical surface of the radius R. It is obvious that the surface of the geoid differs only slightly from the spherical surface of some radius R and the error of such replacement has the same order as the ﬂattening. For this reason we can calculate the height of the geoid over the reference ellipsoid, assuming that the former has the spherical surface. Applying Bruns’ theorem we have for the height N: Z 1 N¼ DgSðcÞdA ð2:282Þ 4pRg A This is the Stokes formula, which permits us to ﬁnd the elevation of the geoid at point p. Imagine that the z-axis of the spherical system of coordinates goes through the point p. Then for this point the angle c plays the role of the azimuth y and dA ¼ R2 sin cdcdj

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Thus, N¼

R 4pg

Z

2p

Z

p

Dgðj; cÞF ðcÞdjdc 0

ð2:283Þ

0

Here F ðcÞ ¼ SðcÞ sin c

ð2:284Þ

Unlike the function S(c), which has a singularity if c ¼ pn, the function F(c) is continuous and limited, but it is not monotonic. In particular, F(c) does not decrease with an increase of the angle c and has a rather complex character. Because of this, in calculating the geoid heights it is not enough to know the anomalies near the point of investigation. Anomalies, located at 901 and 150–1801 from the point of a study also make a rather noticeable contribution to the height value. Note, that in the practice of modern astronomical geodesy the geoid heights are usually obtained from satellite measurements.

2.9. MOLODENSKY’S BOUNDARY PROBLEM In the previous section we described the Stokes method, which allows us to ﬁnd the distance between the reference ellipsoid and the physical surface of the earth. The ellipsoid, given by its semi-major axis a, ﬂattening a, and elements of orientation inside of the earth can be considered as the ﬁrst approximation to a ﬁgure of the earth. In order to perform the transition to the real earth we have to know the distance along the normal from each point of the spheroid to the physical surface of the earth. Earlier we demonstrated that this problem includes two steps, namely, 1. evaluation of the distance between the reference ellipsoid and the geoid and 2. calculation of the distance between the geoid and the physical surface of the earth. In this formulation of the problem the geoid, or more precisely, quasi-geoid plays role of an auxiliary surface, which facilitates the transition from the ﬁrst approximation to the real surface of the earth. As we know, the ﬁrst task (evaluation of geoid’s heights with respect to the ellipsoid) is solved with the help of Stokes formula. The second task (determination of heights of the physical surface over a geoid) is carried out by leveling. This approach implies that the true ﬁgure of the earth is deﬁned by the following factors: (a) the parameters of the reference ellipsoid; (b) the system of heights N of the geoid over the reference ellipsoid; and (c) system of heights h of the physical surface over the geoid. Inasmuch as the use of Stokes’ formula is based on the assumption that masses are absent between the geoid and a true surface of the earth, the accuracy of evaluation of N is not often sufﬁcient. Also, there are sometimes noticeable errors related to the application of leveling, for instance, the transition from the orthometric to normal heights. The conventional approach requires knowledge of the potential of the gravitational ﬁeld inside masses, and this is its main shortcoming. The method

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129

suggested by Molodensky does not have this problem and using this method, measurements of the gravitational ﬁeld on the earth’s surface give information for the solution of all geodetic problems. In this case the reduction of all measuring elements on the reference ellipsoid is performed more accurately. At the same time, the geoid still remains a fundamental concept of geodesy, and it is difﬁcult to overestimate its importance. It allows one to distinguish from the earth, surrounded by a complicated irregular surface, the main part of the masses. It is enclosed by a geoidal surface and separates the irregular part between the geoid and the physical surface. Perhaps, it is possible to say that new approach contains some essential features that are similar to the conventional one. For instance, there is a surface, located near the physical surface of the earth, which is called the telluroid. Unlike the geoid it is not level surface. However, at each point it is deﬁned as a level surface and can be characterized as a piece-wise level surface. The telluroid divides the geodetic heights, counted from the reference ellipsoid, in two parts. One of them changes smoothly, and it is deﬁned from the solution of the boundary value problem of potential theory. The other is solved by geometrical leveling through the geo-potential. The telluroid and geoid coincide on the ocean surface, and the difference between them increases on continents with an increase of the anomaly of gravity, (gg), and the complexity of the relief at the given area. 2.9.1. Molodensky’s problem and Bruns’s formula Molodensky’s problem can be formulated in the following way. When the earth rotates with constant angular velocity o around some axis, then the surface S of the earth, the external potential, and the ﬁeld g are deﬁned by: (1) a change of the potential with respect to some initial point 0: WSW0; (2) a change of the gravitational ﬁeld with respect to that at the initial point: gSg0; (3) astronomical coordinates. The solution of this problem is unique, if in addition two constants are known: the mass of the earth M and the potential W0 at the initial point 0. These constants can be replaced by measuring an absolute value of the gravitational ﬁeld and the distance between two remote points on the earth’s surface. Before we describe the boundary value problem, let us recall the concept of a height. First, carrying out leveling, we obtain an excess in the heights, Dhi, between two points located close to each other. Here Dhi is the distance along the vector line of the ﬁeld, which is the vertical or plumb line at the front point. Then, a summation of elementary heights gives the height of some point A of the earth over the geoid: X HA ¼ Dhi i

As we know, the sum depends on the path of leveling. For this reason the following quantity was introduced Z A 1 H or ¼ gdh gm O

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z A

W=CA

N U=CA Ocean level Ocean level

A1

B

0

N U = C0

quasi-geoid ellipsoid

B1

Physical surface of the Earth Fig. 2.10. Telluroid, quasi-geoid, and reference ellipsoid (after Grushinsky).

Here gm is the mean value of the ﬁeld magnitude along the vector line between points O and A. The value of the integral is called the geo-potential and it is path independent. In principle, the use of this integral gives a set of the orthometric heights at points of the earth’s surface over the geoid. Unfortunately, the value of the ﬁeld gm is unknown and for this reason in place of the orthometric heights we apply the normal heights, deﬁned as Z 1 A Hn ¼ gdh gm O and, correspondingly, we obtain a system of normal heights. In valleys they may differ from the orthometric heights by several centimeters, but on mountains this difference may reach 1 m. As in the case of Stokes problem we represent the total potential W at each point of the earth’s surface as a sum of the normal and disturbing potentials: W ðf ; l ; HÞ ¼ Uðf ; l ; HÞ þ Tðf ; l ; HÞ

ð2:285Þ

Here f*, l*, H are geodetic coordinates of the point on the physical surface of the earth and H is geodetic height of the point A, Fig 2.10. Correspondingly, the disturbing potential T, characterizing the deviation of the level surface of the total potential from the level surface of the normal potential, is Tðf ; l ; HÞ ¼ W ðf ; l ; HÞ Uðf ; HÞ

ð2:286Þ

since the normal potential is independent of longitude. Consider a point A1 with coordinates f, l, h, located slightly below the point A, and a difference of coordinates f f ¼ Df;

l l ¼ Dl;

Hh¼N

ð2:287Þ

is so small that we can neglect differences of higher order. Here N is equal to the distance between the geoid and reference ellipsoid along the line AB1, that is AA1 ¼ BB1.

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Now let us expend the normal potential near point A1 in a series and represent the potential at the point A of the earth’s surface as Uðf ; HÞ ¼ Uðf; hÞ þ

@U @U Nþ Df þ @z @f

ð2:288Þ

Here the normal potential and its ﬁrst derivatives are taken at the point A1 beneath the earth’s surface. Since within the interval Hh we assume that the potential varies linearly, these derivatives are equal to those on the earth’s surface. At the same time, the potential of the total ﬁeld at the point A we represent as Z W ðf ; l; HÞ ¼ W 0 gdhw ð2:289Þ 0A

Here W0 is the potential of the geoid (quasi-geoid) and dhw an elementary height measured by leveling, g the ﬁeld magnitude, and the presence of the minus in front of the integral is understandable because the earth’s surface is located above the geoid, that is, the integral is positive. Substitution of Equations (2.288 and 2.289) into Equation (2.286) gives for the disturbing potential at the point A of the earth: Z @U @U N Df þ ð2:290Þ Tðf ; l; HÞ ¼ W 0 gdhw Uðf; hÞ @z @f 0A As follows from Equation (2.288), the last term in Equation (2.290) is a small quantity of the second order and it can be neglected. Inasmuch as a geoid is a level surface (W0 ¼ constant), we have: Z Z gdhw þ gdz ¼ 0 0A

AB

or Z

Z

Z

gdhw ¼

0A

gdz Uðf; hÞ Uðf; 0Þ ¼ AB

Z gdz ¼

AB

gdz

ð2:291Þ

A1 B1

This means that we have made two approximations. One of them is replacement of the total ﬁeld by a normal ﬁeld, and the second is a shift of the interval of integration. Since the secondary ﬁeld is very small, as are the intervals BB1 and AA1, we may expect that the errors caused by these approximations are relatively small. Besides, the normal ﬁeld varies relatively slowly within the interval B1A. Correspondingly, U(f,h) and U(f,0) are values of the normal potential at points A1 and B1. From Equations (2.290 and 2.291) we have Tðf ; l; HÞ ¼ W 0 U 0

@U N @z

ð2:292Þ

Here W0 and U0 are the total and normal potentials on the surface of the geoid and on the surface of the reference ellipsoid, respectively. By deﬁnition, g ¼ qU/qz is the magnitude of the normal gravitational ﬁeld. Thus, Equation (2.292) becomes Tðf ; l; HÞ ¼ W 0 U 0 þ gðf; hÞN

ð2:293Þ

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Later we will assume that the difference W0U0 is equal to zero. In this equation N is the height of the quasi-geoid, B1B. It also deﬁnes an excess of the level surface of the potential W, passing through the point A of the physical surface of the earth over corresponding level surface of the normal potential passing through the point A1. Let us represent Equation (2.293) in the form N¼

T U0 W0 þ g g

ð2:294Þ

that is an analogy of Bruns’ formula but it is written for the physical surface of the earth. The importance of Equation (2.294) is that it establishes a relationship between the height anomaly, N, and the disturbing potential T at the same point of the earth. By deﬁnition, we have for the normal height h of the point A over a quasi-geoid Z 1 h¼ gdh gm 0A Taking into account (2.293) we have Z Z 1 1 gdh ¼ gdhw h¼ gm B 1 A 1 gm 0A

ð2:295Þ

which is deﬁned from leveling and the knowledge of the normal ﬁeld. Thus, Equations (2.294 and 2.295) allow us to ﬁnd the value of the height of a physical surface of the earth over the reference ellipsoid: Z 1 T U0 W0 H¼ ð2:296Þ gdh þ þ gm AB g g and this quantity is called the geodetic height. 2.9.2. Boundary condition for the disturbing potential T First of all, we choose the parameters of the ellipsoid in such a way that the normal potential on its surface, U0, is equal to the potential of the total ﬁeld at points of the geoid, W0. Then, Equation (2.294) is greatly simpliﬁed and we obtain N¼

T g

ð2:297Þ

Thus, the determination of heights of the quasi-geoid N requires knowledge of the disturbing potential T on the physical surface of the earth. As in the case of the Stokes problem, in order to calculate N we have to determine the disturbing potential, which obeys some boundary condition on the physical surface of the earth instead of the surface of a geoid. This is the main advantage of a new approach. 2.9.3. The boundary value problem for the function T Taking into account the fact that T ¼ WU, the disturbing potential obeys Laplace’s equation outside the earth surface: DT ¼ 0 and it is a regular function at

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133

inﬁnity lim rT ¼ constant;

if r ! 1

Now we formulate the boundary condition for T on the physical surface of the earth. It is clear that @T @W @U ¼ @n @n @n

ð2:298Þ

Here v is an elementary displacement along a line normal to the ellipsoid. Then, we have qU/qn ¼ g and, neglecting quantities of smaller order, we have @W ¼ g @n Correspondingly, Equation (2.298) becomes @T ¼ ðg gÞ @n Since all quantities are taken on the earth’s surface we can write @T g ¼ g @n hþN

ð2:299Þ

Performing a transition from the physical surface of the earth to the surface S located at a small distance N, we have with an accuracy N2 @T @T @g ¼ and ghþN ¼ gh þ N @n hþN @n h @n Their substitution into Equation (2.299) gives @T @g gN ¼ g @n @n h

ð2:300Þ

or

@T T @g @n g @n

¼ ðg gÞ

ð2:301Þ

h

Here g is the gravitational ﬁeld on the physical surface of the earth, g the normal ﬁeld on the surface S. At the same time, qT/qn and qg/qn have the same values along line n at both surfaces. This is the boundary condition for the disturbing potential and therefore we have to ﬁnd the harmonic function regular at inﬁnity and satisfying Equation (2.301) on the surface S. In this case, the physical surface of the earth is represented by S formed by normal heights, plotted from the reference ellipsoid. In other words, by leveling the position of the surface S becomes known.

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2.9.4. Solution of the boundary value problem First, we imagine that there are ﬁctitious masses at points of the surface S and they are distributed with surface density s in such way that the potential of their ﬁeld is equal T. Then, in accordance with the Newton’s law of attraction this potential is Z sðqÞ TðpÞ ¼ k dS ð2:302Þ S r One can say that we have expressed the disturbing potential in terms of an unknown density. Now we demonstrate that this transition is justiﬁed because it is possible to obtain the integral equation with respect to s. In Chapter 1, it was shown that the discontinuity of the normal components of the ﬁeld at both sides of the surface masses is equal to 2pks. Correspondingly, we have Z @T @ s ¼k dS 2pks cos a ð2:303Þ @n @n S r Here the ﬁrst and second terms describe the ﬁeld on different sides of the surface and a is the angle between the direction of the n line and the normal to the surface S. It is obvious that the function T, given by Equation (2.302), is harmonic and regular at inﬁnity. Next, we have to choose among an inﬁnite number of such functions T one which also obeys the boundary condition on S. The last equation allows us to rewrite Equation (2.301) in the form Z Z @ s 1 @g s k dS 2pks cos a k dS ¼ ðg gÞ ð2:304Þ @n S r g @n S r Thus, we have derived the integral equation with respect to the function s. As in the case of Stokes’ problem it is possible to apply the spherical approximation, that is, the magnitude of the normal ﬁeld at points of the surface S is M @g @g kM 1 @g 2 g¼k 2; ¼ ¼ 2 3 ; ¼ @n @R g @n R R R Then

@ 1 @ 1 R r cos c 1 r2 þ R2 r2 ¼ 3 Rr ¼ ¼ @n r @n R r3 r 2rR ¼

r2 R 2 1 2Rr 2Rr3

ð2:305Þ

Correspondingly, Equation (2.304) becomes Z 2 Z Z r R2 s 2 s ðg gÞ dS 2ps cos a þ dS ¼ sdS 3 R Sr k S 2Rr S 2Rr Letting Dg ¼ gg, we ﬁnally obtain Z Z 2 Dg 3 s r R2 2ps cos a ¼ þ dS þ sdS 3 k 2R S r S 2Rr

ð2:306Þ

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Gravitational Field of the Earth

z y p θ

r

q

0

a

x

b

Fig. 2.11. Illustration of Equation (2.307).

We have derived a linear integral equation of the second kind called Molodensky’s equation. This equation allows us to determine the function s. Its substitution into Equation (2.302) and an integration gives us the disturbing potential. Finally, calculating N from the Bruns’ formula we can evaluate the geodetic heights of the points of the real earth. As is well known, there are different algorithms for the solution of the integral equation; one of them is the method of subsequent approximations. It may be appropriate to notice that we assumed that the function T is harmonic everywhere above the surface S. In fact, we have neglected the inﬂuence of a very thin layer between S and the physical surface of the earth. Inasmuch as N is very small, this assumption is well justiﬁed.

2.10. ATTRACTION FIELD OF THE SPHEROID In Section 2.4 we have studied the behavior of the gravitational ﬁeld of the spheroid outside of masses. Now let us focus our attention on the ﬁeld of attraction inside masses. It may be proper to notice that the determination of the ﬁeld caused by masses in the spheroid and, in general, by an ellipsoid, was a subject of classical works performed by Maclaurin, Lagrange, Laplace, Poisson, and others. As is well known, the equation of the ellipsoid, when the major axes are directed along coordinate lines is x21 Z21 z21 þ þ ¼1 a2 a2 b2

ð2:307Þ

where x1, Z1, and z1 are coordinates of some point q, located on the surface of the spheroid. Let us take a point p inside the spheroid with coordinates x, y, and z, Fig. 2.11. In order to determine the attraction ﬁeld we calculate the ﬁeld caused by each

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elementary mass and then perform their summation: Z dðqÞrdV ga ¼ k r3 Then, the x-component of the attraction ﬁeld at the point p(x, y, z) is Z xx gax ðpÞ ¼ kd dV r3

ð3:308Þ

Here 1=2 r ¼ ðx xÞ2 þ ðy ZÞ2 þ ðz zÞ2 d is the density of mass and dV an elementary volume. Its central point has coordinates x, Z, z. Let us consider the point p as the origin of a spherical system of coordinates: r, y, and l. The polar axis is directed along the x-axis and, correspondingly, the relationship between the coordinates in the systems is x ¼ x þ r cos y;

Z ¼ y þ r sin y cos l;

z ¼ z þ r sin y sin l

For an elementary volume we have dV ¼ r2 sin ydrdydl Substitution of these expressions into Equation (3.308) leads to a great simpliﬁcation and we obtain Z r1 Z p Z 2p gax ðpÞ ¼ kd dr sin y cos ydy dl ð2:309Þ 0

0

0

where r1 is the value of r for the point (x1, Z1, z1), located on the surface of the spheroid. Integration over r gives Z p Z 2p gax ðpÞ ¼ kd r1 sin y cos ydydl ð2:310Þ 0

0

where r1 is a function of both angles y and l. To perform the integration we ﬁrst ﬁnd the equation, which describes the behavior of this function. Inasmuch as the point q(x1, Z1, z1) is located on the spheroid surface, the distance r1 obeys the Equation (3.307): ðx þ r1 cos yÞ2 ðy þ r1 sin y cos lÞ2 ðz þ r1 sin y sin lÞ2 þ þ ¼1 a2 a2 b2

ð2:311Þ

This is an algebraic equation of second order with respect to r1 and has the form: Ar21 þ 2Br1 þ C ¼ 0 where A¼

cos2 y sin2 ycos2 l sin2 ysin2 l þ þ a2 a2 b2

ð2:312Þ

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B¼

x cos y y sin y cos l z sin y sin l þ þ a2 a2 b2 C¼

ð2:313Þ

x2 y2 z2 þ þ 1 a2 a2 b2

Here A is a positive number and C equal to zero for a point located on the spheroid surface, but in accordance with Equation (2.307), inside it is negative. For this reason, the difference: B2AC is positive and greater than B2. Thus, this equation has two real roots, one of which is positive. The latter represents the distance r1 from the point p to any point q(x1, Z1, z1) on the spheroid surface, but the negative root has to be discarded. Respectively, we have r1 ¼

B þ ðB2 ACÞ1=2 A

Introducing this expression into Equation (2.310) we obtain Z p Z 2p 2 Z p Z 2p B ðB ACÞ1=2 sin y cos ydydl þ kd gax ¼ kd sin y cos ydydl A A 0 0 0 0 ð2:314Þ Now we demonstrate that the last integral vanishes and with this purpose in mind consider an interval of integration as a sum of two sub-intervals, shown in Fig. 2.12: 0 yop=2 and 0lp p=2 y p and p l 2p Then, the second integral can be written as Z p=2 Z p 2 Z p Z 2p 2 ðB ACÞ1=2 ðB ACÞ1=2 I¼ sin y cos ydydl þ sin y cos ydydl A A 0 0 p=2 p

λ

2

2 π 1

0

π

π 2

Fig. 2.12. Integration of Equation (2.314).

θ

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In the last integral we make a replacement of variables in the following way: a¼py

and

b ¼ 2p l

Correspondingly, the function B becomes B¼

x cos a y sin a cos b z sin a sin b a2 a2 b2

ð2:315Þ

Therefore, B2, as well as A and C, have the same expressions in both intervals. This means that ðB2 ACÞ1=2 A has the same magnitude and sign in both intervals of integration, but the product sin y cos y changes sign. Since da ¼ dy

and

db ¼ dl

the integrals in the two intervals differ by sign only. This allows us to conclude that the second integral in Equation (2.314) is equal to zero and we obtain Z p Z 2p B gax ðpÞ ¼ kd sin y cos ydydl ð2:316Þ A 0 0 Certainly, it is a strong simpliﬁcation of the formula for the ﬁeld component. Substituting in the expression for B, Equation (2.313), gives Z Z Z Z x p 2p 1 y p 2p 1 2 gax ðpÞ ¼ kd 2 cos2 y sin ydydl kd 2 sin y cos y cos ldydl a 0 0 A a 0 0 A Z Z z p 2p 1 2 sin y cos y sin ldydl kd 2 ð2:317Þ b 0 0 A Applying the same approach, as before it is simple matter to show that the second and third integrals are also equal to zero. In fact, replacing l by p+l, A remains the same but the products sin2 y cos y cos l and sin2 y cos y sin l preserve their absolute value but change sign. For this reason integration of each of two last integrals over l from 0 to p and from p to 2p gives values equal in magnitude but opposite in sign. Thus, we have Z Z x p 2p 1 sin ycos2 ydydl gax ðpÞ ¼ kd 2 ð2:318Þ a 0 0 A Note, that Equation (2.318) does not contain y and z. This means that the component gax is the same in any plane perpendicular to the x-axis, (x ¼ constant) and changes linearly with x. As we will see later, the components gay and gaz are constant on the planes y ¼ constant and z ¼ constant, respectively. Now we will discover the second remarkable feature of the ﬁeld behavior, which is based on the fact that the right hand side of Equation (2.318) depends on the ratio of the semi-axes, b/a. Correspondingly, if we imagine another spheroid with the same density of

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masses with semi-axes, equal na and nb, where n is a positive number, (n41), then the component gax remains the same as that for the original spheroid. This occurs only if the layer conﬁned by surfaces of both spheroids, does not create this ﬁeld component. Of course, the same it is true for the two other components. Thus, we conclude that a homogeneous layer, conﬁned by surfaces of two similar spheroids with the same focus does not create a ﬁeld inside. This means that the ﬁeld caused by elementary masses of such a spheroid layer compensates each other. Certainly, it would be difﬁcult to predict this result, which is an extension of the theorem proved by Newton for the ﬁeld inside a uniform spherical layer. Inasmuch as we did not use the assumption that two axes are equal, we arrive at the conclusion that inside a layer, bounded by similar ellipsoidal surfaces the ﬁeld is also equal to zero. 2.10.1. Integration of Equation (2.318) Applying the same approach, we can show that the whole interval of integration can be represented as four sub-intervals in l and two sub-intervals in y and over each of them the integrand is described by the same function. Therefore, in place of Equation (2.318) we have Z Z x p=2 p=2 1 gax ðpÞ ¼ 8kd 2 sin ycos2 ydydl ð2:319Þ a 0 A 0 First, it is convenient to represent the function A as cos2 y þ sin2 ycos2 l sin2 ysin2 l 1 1 1 A¼ þ ¼ 2 þ 2 2 sin2 ysin2 l a2 a a b2 b 1 ¼ 2 ð1 þ e21 sin2 ysin2 lÞ a Here ða2 b2 Þ1=2 ð2:320Þ b is the second eccentricity of the meridian ellipse which was used earlier. Then, Equation (2.319) becomes Z p=2 Z p=2 dl 2 gax ðpÞ ¼ 8kdx sin ycos ydy ð2:321Þ 2 1 þ e1 sin2 ysin2 l 0 0 e1 ¼

The last integral is performed using the substitution cot l ¼ t. This yields sin2 l ¼

1 1 ¼ 1 þ cot2 l 1 þ t2

and

or dl ¼ sin2 ldt ¼

dt 1 þ t2

dl ¼ dt sin2 l

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Correspondingly, the integral with respect to l becomes Z 1 Z p=2 dl dt tan1 ðt=ð1 þ e21 sin2 yÞÞ 1 ¼ ¼ j 1 þ e21 sin2 ysin2 l 1 þ e21 sin2 y þ t2 0 ð1 þ e21 sin2 yÞ1=2 0 0 p ¼ 2 2ð1 þ e1 sin2 yÞ1=2 Thus,

Z

p=2

gax ðpÞ ¼ 4pkdx

sin ycos2 ydy

ð2:322Þ

ð1 þ e21 sin2 yÞ1=2

0

To perform the integration, we introduce new variable c: cos y ¼

ð1 þ e21 Þ1=2 sin c e1

Then, sin ydy ¼

ð1 þ e21 Þ1=2 cos cdc e1

e21 sin2 y ¼ e21 cos2 c sin2 c

and

Substitution of these expressions greatly simpliﬁes the integrand in Equation (2.322) and it gives Z 1 þ e2 a 2 sin cdc gax ðpÞ ¼ 4pkd 3 1 e1 0 Here a ¼ sin1

e1 ð1 þ

e21 Þ1=2

or

sin a ¼

e1 ð1 þ e21 Þ1=2

or

tan a ¼ e1

For this reason, sin1

e1 ð1 þ e21 Þ1=2

Whence gax ðpÞ 4pdkx It is clear that Z tan1 e1 0

1 þ e21 e31

¼ tan1 e1

Z

tan1 e1

0

tan1 e1 1 sin cdc ¼ ðc sin c cos cÞ j ¼ 2 0 1 e 1 1 tan e1 ¼ 2 1 þ e21 2

Finally, we have gax ðpÞ ¼ 2pkdx

sin2 cdc

1 tan e1 tan c c j 1 þ tan2 c 0

1 þ e21 e1 1 tan e 1 1 þ e21 e31

ð2:323Þ

Gravitational Field of the Earth

Since the directions along x and y axes are equivalent, we also have 1 þ e2 e1 gay ðpÞ ¼ 2pkdy 3 1 tan1 e1 1 þ e21 e1

141

ð2:324Þ

In order to ﬁnd the vertical component of the ﬁeld we can apply the same approach as before, namely, the integration over the volume of the spheroid, only in this case the polar axis of the spherical system should be directed along the z-axis. However, we solve this problem differently and will proceed from the second equation of the gravitational ﬁeld. In Cartesian system of coordinates we have @gax @gay @gaz þ þ ¼ 4pkd @x @y @z Taking into account Equations (2.323 and 2.324) we obtain @gaz 1 þ e21 e1 1 ¼ 4pkd þ 4pkd 3 tan e1 @z 1 þ e21 e1 Then, integration gives 1 þ e21 e1 1 tan e gaz ðpÞ ¼ 4pkdz 1 þ C1 1 1 þ e21 e31 Inasmuch as due to symmetry this component of the ﬁeld is equal to zero at the plane z ¼ 0, we obtain 1 þ e21 e1 1 tan e gaz ðpÞ ¼ 4pkdz 1 1 1 þ e21 e31 or gaz ðpÞ ¼ 4pkd

1 þ e21 ðe1 tan1 e1 Þ e31

ð2:325Þ

Thus, we have for all components of the ﬁeld gax ðpÞ ¼ Px;

gay ðpÞ ¼ Py;

and

gaz ðpÞ ¼ Qz

ð2:326Þ

where 1 þ e21 e1 1 tan e1 P ¼ 2pkd 3 1 þ e21 e1 and Q ¼ 4pkd

1 þ e21 ðe1 tan1 e1 Þ e31

ð2:327Þ

are constants, which are independent of the coordinates of the observation point, p(x,y,z).

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2.10.2. Potential caused by masses of a homogeneous spheroid By deﬁnition, we have g ¼ grad U a or @U a ; @x

gax ¼

gay ¼

@U a ; @y

gaz ¼

@U a @z

Then, from Equation (2.326) it follows 1 1 1 U a ðpÞ ¼ Px2 Py2 Qz2 þ C 2 2 2 2

ð2:328Þ

where C2 is some constant, and this deﬁnes the potential at the center of the spheroid. To determine it, we choose a spherical system of coordinates so that the polar axis is directed along the z-axis and, correspondingly, A¼

sin2 ysin2 l sin2 ycos2 l cos2 y sin2 y cos2 y þ 2 ¼ 2 þ 2 þ a2 a a b b

and B¼

x sin y sin l y sin y cos l z cos y þ þ a2 a2 b2

ð2:329Þ

By deﬁnition, the distance r1 still obeys Equation (2.312) and, in accordance with Equations (2.313 and 2.329) 1 A ¼ 2 ð1 þ e21 cos2 fÞ; B ¼ 0; and C ¼ 1 a Therefore, r1 ¼

1 A

and the potential at the origin of Cartesian coordinates is deﬁned as C ¼ U a ð0; 0; 0Þ Z r1 Z p Z 2p 2 Z dm r sin ydrdydl ¼ kd ¼k r r 0 0 0 Z p Z 2p 1 ¼ kd r21 sin ydydl 2 0 0 or 1 C ¼ kda2 2

Z

p 0

Z 0

2p

sin ydydl ¼ pkda2 1 þ e21 cos2 y

Z

p 0

sin ydy 1 þ e21 cos2 y

The last integral is calculated using a new variable e1 cos y ¼ t, and it gives Z a2 e1 dt a2 C ¼ pkd ¼ 2pkd tan1 e1 ð2:330Þ 2 e1 e1 1 þ t e1

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Thus, the potential of the attraction ﬁeld at any point inside the spheroid is 1 1 1 a2 U a ðpÞ ¼ Px2 Py2 Qz2 þ 2pkd tan1 e1 ð2:331Þ 2 2 2 e1 We have demonstrated that as in the case of the spherical mass the attraction ﬁeld increases linearly approaching the spheroid surface and it is equal to zero at the center. At the same time, the potential has a maximum at this point and then decreases gradually as a parabolic function and reaches a minimum on the surface of the spheroid.

2.11. SPHEROID AND EQUILIBRIUM OF A ROTATING FLUID Now we focus our attention on the conditions of equilibrium for a ﬂuid spheroid rotating about a constant axis. In this case the mutual position of ﬂuid particles does not change and all of them move with the same angular velocity, o. As is well known, there is a certain relationship between the density, angular velocity, and eccentricity of an oblate spheroid in equilibrium. In studying this question we will proceed from the equation of equilibrium of a ﬂuid, described in the ﬁrst section. 2.11.1. Equation of equilibrium and level surfaces In a frame of reference rotating together with the ﬂuid, a particle is at rest and thus the sum of three forces is equal to zero: grad p þ dga þ do2 r ¼ 0

ð2:332Þ

Here p is the pressure, ga the ﬁeld of attraction, d the density of the ﬂuid, and r the vector directed away from the axis of rotation and it is equal in magnitude to the distance between a particle and this axis. The ﬁrst two terms of Equation (2.332) characterize the real forces acting on the particle, namely the surface and attraction ones. At the same time the last term is a centrifugal force, and it is introduced because we consider a non-inertial frame of reference. It is convenient to represent Equation (2.332) as f ¼ grad p ¼ dðga þ o2 rÞ ¼ dg

ð2:333Þ

where g is the gravitational ﬁeld, which is a superposition of the attraction ﬁeld and the centrifugal force. By deﬁnition, f is the gravitational force per unit volume. Let us multiply both sides of Equation (2.332) by an elementary displacement dl: dl ¼ dx i þ dy j þ dz k This gives dl rp ¼ dp ¼ dl f

or

dp ¼ f x dx þ f y dy þ f z dz

ð2:334Þ

This elementary transformation of Equation (2.332) allows us to visualize better geometry of the force f and level surface of its potential. Assume that the surface of

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the rotating ﬂuid is described by an equation Sðx; y; zÞ ¼ 0 Differentiation gives @S @S @S dx þ dy þ dz ¼ 0 @x @y @z

or

rS dl ¼ 0

ð2:335Þ

where grad S is the vector perpendicular to the surface S. If the pressure is constant on the ﬂuid surface S, then, as follows from Equation (2.334), for any two points on this surface located very close to each other we have f x dx þ f y dy þ f z dz ¼ 0

ð2:336Þ

where dl is the distance between these points and dx, dy, and dz are scalar components of the displacement dl. Comparison with Equation (2.235) shows that the force is perpendicular to the surface S. Of course, the force f usually varies on the level surface. Earlier, in Section 2.1, we demonstrated that in the general case when the density changes the level surfaces of the pressure and the density coincide. In what follows it is assumed that the density is constant, and introducing the potential U of the gravitational ﬁeld, we can represent Equation (2.336) in the form @U @U @U dx þ dy þ dz ¼ 0 @x @y @z

ð2:337Þ

Thus, in equilibrium the level surfaces of the potential serve as surfaces of equal pressure; this is not surprising because the potential of the gravitational ﬁeld and pressure may differ by a constant only. Inasmuch as the force f and grad S are parallel to each other then we have fy fx fz ¼ ¼ @S=@x @S=@y @S=@z

ð2:338Þ

The latter shows that in the direction tangential to the level surfaces, including the exterior one, the force components are equal to zero, but the normal component provides a rotation of any particle with the same angular velocity and for this reason equilibrium takes place. In accordance with Equation (2.338) the determination of the ﬁgure of ﬂuid equilibrium is reduced to the following problem: we have to ﬁnd such a surface of the ﬂuid, S(x,y,z), that its partial derivatives should be proportional to the corresponding components of the acting force. As we pointed out, when a ﬂuid rotates uniformly around the same axis the total force can be represented as a sum of the attraction and centrifugal forces, and the former depends on the shape of the ﬂuid mass in a rather complicated way. Besides, in the case of an inhomogeneous ﬂuid the potential of the attraction ﬁeld depends on the distribution of a density of a ﬂuid and for this reason this problem becomes even more complicated.

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Gravitational Field of the Earth

2.11.2. Relationship between density, angular velocity, and spheroid eccentricity We will narrow our task and attempt to choose a function S such that its partial derivatives obey Equation (2.338). Of course, this does not allow us to ﬁnd all possible ﬁgures of equilibrium, but for our purposes it is not important, because we are only interested to show that a rotating homogeneous spheroid, describing Earth, can be under certain conditions a ﬁgure of equilibrium. This fundamental fact was established by MacLauren in the 18th century. In this case we have S¼

x2 þ y2 z2 þ 21¼0 a2 b

ð2:339Þ

Taking partial derivatives, we have @S 2x ¼ ; @x a2

@S 2y ¼ ; @y a2

@S 2z ¼ @z b2

ð2:340Þ

In the previous section we found components of the attraction ﬁeld due to masses of a homogeneous spheroid, Equations (2.326 and 2.327). Correspondingly, the components of the gravitational ﬁeld which include the inﬂuence of the centrifugal force are f x ¼ o2 x Px;

f y ¼ o2 y Py;

and

f z ¼ Qz

ð2:341Þ

since a rotation occurs about the z-axis. Substitution of Equations (2.340 and 2.341) into Equation (2.338) yields a2 ðo2 PÞ ¼ a2 ðo2 PÞ ¼ b2 Q or a2 ðo2 PÞ ¼ b2 Q

ð2:342Þ

If this equality is valid, then the oblate spheroid is indeed a ﬁgure of equilibrium. As follows from Equation (2.342) the frequency is deﬁned as b2 Q ð2:343Þ a2 Taking into account Equation (2.327) and the equality b2 =a2 ¼ 1=1 þ e21 , we have o2 ¼ P

W¼

o2 3 þ e21 3 ¼ tan1 e1 2 3 2pkd e1 e1

ð2:344Þ

2.11.3. Solution of Equation (2.344) Since the left hand side of Equation (2.344) is positive, the following inequality has to hold ð3 þ e21 Þtan1 e1 43e1

or

tan1 e1 4

3e1 3 þ e21

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0.224 0.18 e1 2.529 Fig. 2.13. Function y(e1).

provided that we deal with an oblate spheroid, a4b, rotating about the z-axis. Suppose that the density and an angular frequency are given. Then, the transcendental Equation (2.344) allows us to ﬁnd the ratio of axes when the rotating ﬂuid is in equilibrium. This equation is solved numerically with respect to e1 for every value of W and the function W(e1), shown in Fig. 2.13. One can say that for each frequency we deﬁne the ratio of the semi-axes when the rotating spheroid is in a state of equilibrium. From the solution of Equation (2.344) follows that if e1 ¼ 0, then W ¼ 0, as well as qy/qe1 ¼ 0. With an increase of e1 the function W increases too, and it reaches a maximum, equal to 0.224 when e1 ¼ 2.529. With further increase of the argument the value of W begins to decrease and in the limit it tends to zero. As is seen from Fig. 2.13 if o2/2pkdo0.22467 then Equation (2.344) has two solutions. One of them corresponds to a slightly oblate spheroid, but the other to a strongly oblate one which looks like a plane disk. In the limit, when o2/2pkd ¼ 0 the ﬁrst solution gives e1 ¼ 0, that is, a spheroid is transformed into a sphere, but the other, e1 ¼ N, characterizes the plane. It is understandable, that the latter solution has hardly any interest. The ﬁrst limiting case is obvious, because the surface of a rotating sphere is not an equipotential one, but it becomes so when o ¼ 0 and the inﬂuence of rotation vanishes. At the maximum of the function W(e1) both solutions coincide and the relation between semi-axis of the spheroid is a/ b ¼ 2.72. If o2/2pkd40.22467, the spheroid of a homogeneous ﬂuid cannot be a ﬁgure of equilibrium regardless of a value of the eccentricity. Note that if we know the ﬂuid mass M and from Equation (2.344) an eccentricity e1, it is a simple matter to ﬁnd the semi-axes. In fact, we have 4 M ¼ pda2 b 3

and

a2 ¼ b2 ð1 þ e21 Þ

and this gives b3 ¼

3M 4pdð1 þ e21 Þ1=2

Gravitational Field of the Earth

147

2.11.4. Oblate spheroid with very small eccentricity Suppose that we deal with a slightly oblate spheroid, e1o1. Then, making use of an expansion e3 e5 e7 e9 tan1 e1 ¼ e1 1 þ 1 1 þ 1 3 5 7 9 Equation (2.344) gives o2 4 8 4 ¼ e21 e41 þ e61 15 21 2pkd 15 Solving the latter with respect to e1, we obtain 6 37 e21 ¼ þ 2 þ 4 þ 7 49

ð2:345Þ

Here ¼

15 o2 8 pkd

ð2:346Þ

It is instructive to express e in terms of the ratio of the centrifugal force and the gravitational ﬁeld at points at the equator o2 a q¼ ge and also represent the gravitational ﬁeld at these points as ge ¼ ga o2 a

ð2:347Þ

Here ga is the magnitude of the attraction ﬁeld. Correspondingly, the latter give o2 a qga q¼ or o2 ¼ 2 ga o a að1 þ qÞ Making use of Equation (2.346) we have 15 qga ¼ 8 pkdað1 þ qÞ

ð2:348Þ

From an expression for gax, derived in the previous section, we have 4pkd 1 1 e21 ga ¼ if e1 1 3 5 Its substitution into Equation (2.348) yields 5 q ð1 e21 Þ ¼ 21 þ q or, taking into account Equation (2.346) and discarding all terms except the ﬁrst one we obtain 5 q 1 1 ¼ 21 þ q 5

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Whence 5 15 ¼ q q2 þ 2 4 and we have found a relationship between the parameters q and e. Substitution into Equation (2.345) gives 5 45 e21 ¼ q þ q2 þ 2 28 By deﬁnition, the ﬂattening of a spheroid is a¼

ð1 þ e21 Þ1=2 1 ð1 þ

e21 Þ1=2

1 3 e21 e41 þ 2 8

and making use of the previous series we obtain the linkage between the ﬂattening and parameter q: 5 345 2 q a q 4 224 For small angular velocities we can neglect the second term on the right hand side of this equality, and this gives the well-known relationship 5 a q 4 obtained by Newton. In the case of the earth the second root of Equation (2.344) gives the ratio of semi-axes a/b ¼ 681. 2.11.5. About a stability of a figure of equilibrium Investigations performed by Poincare and Lyapunov have shown that only slightly oblate spheroids are stable and for them o2 0:1872 2pkd The ratio of semi-axes at this upper limit is a ¼ 1:716 b For a uniform ﬂuid having a density equal to an average density of the earth, the minimal period is 2 h and 39 min. With an increase of the angular velocity the ﬁgure becomes unstable. Note that a stable ﬁgure differs from unstable one by the following: under the action of an inﬁnitely small force a stable ﬁgure has an inﬁnitely small deformation and when this force ceases it returns to the original shape. The descending branch of the curve in Fig 2.13 corresponds to spheroids with relatively large ﬂattening, and they are unstable. For this reason such ﬁgures are not met in celestial systems. The ﬂattening of our ﬁgure of equilibrium is deﬁned by the ratio o2/2pdk and depends only on the angular velocity and density of a ﬂuid, but it is independent of a size. Because of this the dimensions of a planet do not affect its ﬂattening.

Gravitational Field of the Earth

149

A planet of smaller density has to have a larger ﬂattening, provided that the angular velocity is the same.

2.12. DEVELOPMENT OF THE THEORY OF THE FIGURE OF THE EARTH (BRIEF HISTORIC REVIEW) 2.12.1. I. Newton, 1643– 1727 The history of the subject of this chapter starts from the time of Newton. In 1672, the French astronomer J. Richer traveled to South America and performed astronomical measurements near the equator. In particular, it was found that a pendulum watch, calibrated in Paris, lost 2.5 min every 24 h. It was necessary to decrease the pendulum length by almost 3 mm in order to observe the same period as in Paris. Similar behavior was observed by other travelers, but nobody tried to explain this behavior until Newton. In the third book of ‘‘Mathematical Principles of Natural Philosophy’’ he wrote that these experiments indicate that ‘‘Earth is slightly higher at an equator’’. He also made an attempt to determine theoretically the ﬁgure of the earth, considering the force of attraction and centrifugal force. It is clear from symmetry that if rotation is absent there is only the force of interaction between particles and the earth has a spherical shape. However, in the presence of rotation centrifugal force arises, acting perpendicular to the axis of rotation, and this should tend to extend the earth in the direction of the equator, where it has a maximal value. This conclusion was supported by the observation of Jupiter, which clearly shows ﬂattening in the direction of its poles. Correspondingly, the following question arose. What is the shape of a rotating planet, if all its particles attract each other in accordance with Newton’s law? Of course, at that time it was impossible to solve exactly this problem and Newton made several assumptions. For instance, he assumed that for a relatively small angular velocity an exterior surface of a planet has the shape of a spheroid with a very small value of ﬂattening, a. Newton found the ﬁeld of attraction due to masses of a spherical layer as well as an ellipsoid with a very small ﬂattening, and he also suggested an approximate method of evaluation of this parameter for the ﬂuid earth. Let us imagine two slender channels with a ﬂuid: A and B, shown in Fig. 2.14. They are connected with each other at the center of the earth. At this point the pressure in both channels has to be the same, otherwise the ﬂuid would not be in equilibrium and we would observe a motion of the ﬂuid from one channel to the other. If rotation were absent then the channels would have the same length, but due to the rotation the centrifugal force arises and it decreases the weight of a ﬂuid in the channel A, but in a channel B this effect of rotation is absent. Correspondingly, one can expect that the length of A is greater than that of B. First of all, Newton demonstrated that at points on the equator the ratio of the centrifugal and attraction forces is equal to q ¼ 1/288, and this relation remains valid for any point of the channel A inside of the earth, because both forces are directly proportional to the distance from the earth’s center. For the centrifugal force this dependence is correct, while it is true for the attraction force only if the density of the earth is constant. Introduction of parameter q and its evaluation was

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b

ga B

− p Δ

A

a ga

r

Fig.2.14. Model for evaluation of ﬂattening.

already an important achievement of Newton. By deﬁnition, the earth’s ﬂattening is equal to ab a¼ a where a and b are the length of the channels. In order to determine these lengths it is necessary to know the difference of the gravitational ﬁelds at the poles and the equator, but at the time of Newton, mathematics did not allow one to solve this problem. As was pointed out, Newton solved this problem approximately assuming that the parameter a is very small. He numerically found three relationships, namely, 1. For a spheroid with semi-axes a and b gap 1 ð2:349Þ ¼ ga ðbÞ 1 4=5a

2.

3.

Here gap is the force of attraction of a spheroid at a pole, but ga(b) is the attraction ﬁeld of the enclosed sphere with radius b. The ratio of the ﬁelds ga(b) and ga(a) on the surface of spheres with radii a and b: ga ðbÞ b 1 ð2:350Þ ¼ ¼ ga ðaÞ a 1 þ a For a spheroid with semi-axes a and b and a sphere of radius a: ga ðaÞ 1 ¼ gae 1 2=5a

ð2:351Þ

Here ga(a) is the ﬁeld of attraction on the sphere surface with radius a, and gae the ﬁeld at points of the spheroid equator. After multiplication of Equations (2.349–2.351), we obtain gap 1 1 ¼ 1 1=5a gae 1 4=5a ð1 þ aÞ 1 2=5a

ð2:352Þ

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The latter shows that the attraction ﬁeld at the equator (10.2a) is smaller than that at the pole. Besides, at points of the equator there is a centrifugal force, which decreases the effect of attraction by the factor q. Therefore, the ratio of gravitational ﬁelds or weights at the pole and equator is gp 1 ð2:353Þ ge 1 1=5a q In order to provide equilibrium of the ﬂuid in both channels Newton used the following equality 1 a ¼ ð2:354Þ 1 1=5a q b Since b ¼ a(1a), Equation (2.354) gives 1 1 aq¼1a 5 Whence 5 a¼ q 4

ð2:355Þ

Assuming that q ¼ 1/288, Newton was able theoretically to determine a value of the parameter of ﬂattening of the earth (a ¼ 1/230) which is somewhat smaller than the real one. Making use of this value, Newton calculated the gravitational ﬁeld and, correspondingly, the length of a pendulum with the same period at different points of the earth’s surface and found a good agreement with experimental data. It convinced him that the earth is a spheroid ﬂattened along poles and it has very small eccentricity. Note that in deriving Equation (2.355) Newton made several assumptions; one of them is that the exterior surface of the earth is a level surface. It is interesting to notice that Newton thought the average density of the earth is approximately in ﬁve to six times greater than that of water, which is in agreement with modern data. It may be useful to assume that we know the equation of equilibrium for a ﬂuid and demonstrate how a model of two channels allows us to evaluate the ﬂattening of earth. First, consider the channel B, where we have grad p þ dga ¼ 0

ð2:356Þ

since particles of a very slender channel are not involved in rotation. Here ga is the ﬁeld of attraction, which is caused by all masses and has only a component directed toward the center. Assuming that a ﬁeld of attraction inside of the earth and along a polar axis coincides with the ﬁeld of a spherical mass we have 4p krd ð2:357Þ 3 where r is the distance from the origin to some point of a channel. Thus, in place of Equation (2.356) we can write ga ðrÞ ¼

dp 4p krd2 ¼ 0 dr 3

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Integration of this equation gives pðrÞ ¼

2pkr2 d2 þC 3

In order to ﬁnd the unknown constant, suppose that on the earth’s surface the pressure is zero. Then, we have C¼

2pkb2 d2 3

and

pðrÞ ¼

2pkd2 2 ðb r2 Þ 3

ð2:358Þ

Thus, the pressure has a maximum at the center and the decreases as a parabolic function and it is equal to zero at the pole. Next, consider the distribution of pressure in the channel A, where both the attraction and centrifugal forces act on any particle. Inasmuch as a difference of a pressure at terminal points of both channels is the same and a4b , it is natural to assume that the attraction ﬁeld in the channel A is smaller and suppose that the correction factor is equal to the ratio of axes, b/a. Correspondingly, a condition of equilibrium is

dp 4p b krd2 þ do2 r ¼ 0 dr 3 a

ð2:359Þ

As we see, the ratio of both forces is independent of the distance from the center. Integration of this equation gives 2pkd2 b 1 2 2 þ do r þ C pðrÞ ¼ 3 a 2 Since at points of an equator the pressure has to be equal to zero, (the earth’s surface is a level one), we obtain: 2pkd2 b 1 2 2 C¼ do a 3 a 2 and the pressure at points of the channel A is 2pkd2 b 1 2 do ða2 r2 Þ pðrÞ ¼ 3 a 2

ð2:360Þ

Inasmuch as the pressure at the bottom of both channels is the same, Equations (2.358 and 2.360) give b 3o2 b 3o2 a a b2 ¼ ¼ 1 or 1q a2 a 4pkd a 4pkda b Since b/aE1a we obtain a¼q¼

1 288

ð2:361Þ

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2.12.2. Ch. Huygens, 1629– 1695 A contemporary of Newton, a Dutch physicist Huygens published a monograph where he described a solution of the same problem with a different result. He found that a ¼ 1/2q and correspondingly the ﬂattening a ¼ 1/576. Besides, Huygens formulated the condition of equilibrium when the ﬂuid covering a planet does not move. He showed that the resultant force has to be directed along the normal to the surface: otherwise we can imagine two components of the force. One of them is perpendicular to the ﬂuid and causes its compression, while the other is a cause of a motion of the ﬂuid. Later, in 1738 the French mathematician A.C. Clairaut made an attempt to explain why Newton and Huygens obtained different values of the parameter a. In calculating the attraction ﬁeld Newton assumed that the spheroid is homogeneous, while Huygens’ evaluations implied that all the mass of the earth is concentrated at its center, but other parts have zero density. In fact, the earth is not uniform and the density of deeper layers is larger. Clairaut showed that the values of ﬂattening derived by Newton and Huygens are limiting values, and the real value for a spheroid is somewhere between them. As we know, Newton gave a much more accurate estimate value of the parameter a. Theoretical studies performed by Newton and Huygens showed that the earth has to be ﬂattened in the direction of the poles. However, geodesic measurements, performed at that time by Cassini, were treated as if the earth were extended along its poles. As result, there were very heated discussions, until French academicians including Clairaut performed angular measurements in the north of Europe and in South America during 1735–1741 and demonstrated that Newton was correct.

2.12.3. C. MacLaurin, 1698– 1746 In studying the ﬁgure of the earth, both Newton and Huygens calculated the attraction ﬁeld on the surface of a spheroid. Newton gave a solution correct only for a small ﬂattening. For the general case this problem was solved by the Scottish mathematician C. MacLaurin. He found the exact expression for the attraction force inside a homogeneous spheroid and proved that an oblate spheroid can be a ﬁgure of equilibrium of a ﬂuid rotating with a constant angular velocity about its minor axis. Thus, Newton’s assumption about the ﬁgure of equilibrium of a ﬂuid was conﬁrmed. Also MacLaurin was able to give a more accurate expression for the ﬂattening of the spheroid. For instance, it turned out that the meridian eccentricity: e¼

ða2 b2 Þ1=2 a

ð2:362Þ

is deﬁned in terms of the parameter q in the following way: 5 15 e 2 ¼ q q2 þ 2 7

ð2:363Þ

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From Equation (2.362) we have a¼

ab 1 1 ¼ 1 ð1 e2 Þ1=2 ¼ e2 þ e4 þ a 2 8

and with accuracy of the ﬁrst term in Equation (2.363) 5 a¼ q 4 which corresponds to Newton’s solution. Shortly after MacLaurin’s studies it became clear that the equilibrium of a ﬂuid spheroid does not hold for all angular velocities. An elementary analysis, performed at that time, showed that at a sufﬁciently large angular velocity, the centrifugal force tears a mass by parts. It is simple to evaluate a maximal velocity o at which such a catastrophe occurs. At points of the equator, where the centrifugal force is maximal, its value is around 1/289 of the attraction ﬁeld. If we increase the angular velocity 17 times, the weight of particles at these points would be zero and with further increase of o any body will be thrown away into space. In this case the period of rotation is equal to 24/17 ¼ 1 h 25 m. However, this result is correct if the radius of an equator remains the same. In reality the ﬂattening of the earth increases with an increase of the angular velocity and, correspondingly, it leads to increase of the centrifugal force. For this reason, the limiting value of the period is larger. The subject of equilibrium of a spheroidal ﬁgure was studied by J. D’Alambert (1717–1783) and P.S. Laplace (1749–1827). It turned out that for existence of such a ﬁgure of equilibrium the parameter q has to satisfy the condition qo0.337, that is, the period T has to exceed 2 h 25 m. For this limiting case the ﬂattening becomes equal 0.6323 and this means that a ¼ 2.7b. The English mathematician T. Simpson, (1710–1761), found that for an angular velocity which is smaller than the limiting one it is possible to have one more ﬁgure of equilibrium which has the shape of a strongly oblate spheroid. For small angular velocities it is transformed into a thin disk. For instance, when q ¼ 1/288, the ﬂattening of this spheroid is a ¼ 0.99853 or a ¼ 681b. This family of spheroids is not met among known celestial bodies. It turned out that this shape cannot exist for a sufﬁciently long time. In other words, it is unstable ﬁgure of equilibrium. MacLaurin proved also that an oblate spheroid can be a ﬁgure of equilibrium, but a ﬂuid prolate spheroid can never be at equilibrium. It is natural that mathematician raised the following question. Are there other ﬁgures, different from oblate spheroids, which can be in equilibrium? The German mathematician K. Jacobi, 1804–1851, found that an ellipsoid with three different semi-axes, rotating about its smallest axis, can be a ﬁgure of equilibrium, if qo0.2807. For the angular velocity of the earth the relation between the axes has to satisfy the condition: a : b : c ¼ 52:4 : 1:0023 : 1 provided that rotation takes place around the axis c. Certainly, such an ellipsoid looks like a cigar, rather than our earth. Finally, H. Poincare (1854–1912), G. Darwin (1845–1912), and A. Lyapunov (1857–1918), investigated ﬁgures of

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equilibrium of pear-shaped type, close to ellipsoids, which are important for modern cosmology theories. At that time the importance of stability of these ﬁgures of equilibrium was already realized. If a ﬁgure of equilibrium is stable, then a ﬂuid mass, rotating with a constant angular velocity, preserves its shape. Any external disturbance, which does not exceed a certain limit, changes a stable ﬁgure only temporarily and when this force ceases to act a disturbed mass tends to return to its original shape. In the case of an unstable ﬁgure we observe a completely different behavior. For instance, even an inﬁnitely small disturbance causes a ﬁnite change of a shape and after the action of an external force a ﬂuid mass does not take its original shape. For this reason an unstable ﬁgure cannot exist but it may appear as a transition from one stable ﬁgure to the other, or before a body is ruptured. Later, it was established that spheroids of a small ﬂattening and ellipsoids of Jacobi are stable if the angular velocity does not exceed a certain limit, but the family of spheroids of large ﬂattening belongs to unstable ﬁgures and this explains why we do not observe celestial bodies of this shape. Until now we discussed the study of ﬁgures of a homogeneous bodies rotating with a constant angular velocity. Since the earth and other planets are inhomogeneous and their shape depend on the distribution of density it is clear that an understanding of equilibrium of such bodies is of a great importance. The main work in this direction was performed by Clairaut, 1713–1765. The theory of Clairaut is based on the following assumption. The earth consists of spheroidal layers of small ﬂattening which have a common center and the same axis of rotation. Every layer is homogeneous but the density may change arbitrary from one layer to another. He did not make any assumption about a substance of each layer: they can be either solid or ﬂuid. Correspondingly, it is possible that the internal layers are not in equilibrium. The condition of equilibrium has to hold for the external surface, where the resultant force has to be directed along a normal. In order to meet this condition it is sufﬁcient to assume that the external layer is a ﬂuid. The ﬂattening of internal layers is small and changes continuously. Proceeding from these assumptions Clairaut calculated components of attraction due to an inﬁnitely thin spheroid shell and then integrating he found the force of attraction of the whole spheroid. Taking into account the fact that the density, as well as the ﬂattening of internal layers, is unknown, the force of attraction is expressed in terms of several integrals. Substituting these integrals into an equation of equilibrium, Clairaut found a relationship between the integrals and, after some algebra it was transformed into a linear differential equation of the second order, which is a linkage between the density and the ﬂattening of layers. This famous differential equation became the foundation for all studies of the density inside the earth. Analysis of this equation allowed Clairaut to come to conclusions of a general character, for instance, if the density increases from the earth’s surface to its center then the ﬂattening of the spheroidal shells should decrease in approaching this center. Also, Clairaut derived an expression for the gravitational ﬁeld on the earth’s surface, which contains three unknown integrals. Using the condition of equilibrium he eliminated them and obtained two equations which are usually known as

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the theorem of Clairaut: g ¼ ge ð1 þ bsin2 jÞ where 5 ð2:364Þ b¼ qa 2 Here q is the ratio of the centrifugal and gravitational ﬁeld at points of the equator. As we already know, the ﬁrst equation gave the law of change of the ﬁeld on the earth’s surface as a function of geographical longitude, where gp ge b¼ ge From the second equation of the set (12.16) follows that 5 ð2:365Þ a¼ qb 2 Thus, Clairaut’s theorem allows one to ﬁnd the ﬂattening of the earth spheroid from observations of the gravitational ﬁeld regardless of the distribution of density inside the earth. At that time this theory was ahead of practice of measurements of the gravitational ﬁeld. For instance, Clairaut himself thought that observations of the gravitational ﬁeld were insufﬁcient to make a conclusion about the ﬂattening of the earth, even though he believed that angle measurements, performed in Paris and in North Europe, showed that the earth is stretched in the direction of the equator. In 1785, A. Legendre (1752–1833), in a monograph, describing the attraction ﬁeld of ellipsoids, used the concept of the potential function. In accordance with Legendre, this function was ﬁrst suggested by Laplace, who later discovered its remarkable properties. Let Fx, Fy, and Fz be components of the acting force along coordinate axes. The potential function or potential is a function U(x, y, z), that has the following feature: @U ¼ F x; @x

@U ¼ F y; @y

@U ¼ Fz @z

The word ‘‘potential’’ was ﬁrst introduced in 1828 by G. Green (1793–1841), but before that it was called the force function. The main advantage of introduction of the potential function is the fact that instead of three components of the force, which usually have different analytical expressions, it is possible to ﬁnd and then investigate only one function U. Besides, in order to obtain the level surface it is sufﬁcient to make a potential equal to some constant: U ¼ constant Because of this a study of mathematical properties of function U led to understanding geometrical and mechanical features of level surfaces. Also, with a help of potential it was proved that external surface of earth with an accuracy of ﬂattening of the ﬁrst order has to be spheroid. The next step in developing the theory of the

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earth it is not difﬁcult to predict. It was attempted to remove any assumptions about a distribution of density of the earth. This great achievement belongs to G. Stokes, 1819–1903. He proved in 1840 that if the level surface of a body is given and it surrounds all its masses and we know the angular velocity, then the gravitational ﬁeld is uniquely deﬁned on this level surface and in all external space, and it is completely independent of the distribution of density inside the earth. This remarkable theorem allows one to ﬁnd the gravitational ﬁeld on a level surface, satisfying Stokes condition, if we know a shape of this surface. For practical applications it is an important inverse problem, namely, when the results of observation of the gravitational ﬁeld are known and we have to ﬁnd the ﬁgure of the geoid; and this problem was also solved by Stokes. If the earth’s surface is a level surface of a spheroid we obtain the ﬁrst formula of Clairaut, Equation (2.364), regardless of the distribution of density inside. We could say that the level surface of the earth would indeed be a spheroid, if this formula exactly describes the observed gravitational ﬁeld. However, the calculated and observed values of the ﬁeld usually differ from each other. This difference between the observations and theoretical data is called the anomaly of the gravitational ﬁeld and its presence indicates a deviation of a geoid from a spheroid. In developing Clairaut’s theory Laplace took into account only the ﬁrst power of a and q. The French mathematician M. Hamy proved in 1887 that in general the surface of an ellipsoid cannot be a ﬁgure of an equilibrium for a rotating inhomogeneous mass. A spheroid, as a particular case of an ellipsoid, is also impossible. However, for small angular velocities these deviations from a ﬁgure of equilibrium are insigniﬁcant and can be ignored. The term of ﬁrst order in Clairaut’s theorem is independent of the distribution of density inside the earth, and the value of the parameter b is obtained from measurements of the gravitational ﬁeld. Terms of the second order were ﬁrst derived by Legendre and then by others, applying either experimental or theoretical arguments, including astronomy and geophysics. For instance, this was done by E. Wiechert and G. Darwin. In spite of differences of assumptions about the distribution of density their results practically coincide, that is, the presence of heterogeneities inside the earth causes very small deviations of a geoid from an ellipsoidal form. In 1749, D’Alembert pointed out that there is a connection between the theory of precession and a ﬁgure of the earth. As is well known, precession is caused by the fact that the resultant force of attraction due to celestial bodies, such as Sun and Moon, does not pass through the center of the earth. Correspondingly, there are couple of forces, which tend to turn Earth in such way that the plane of an equator would go through an attracting body and produce a precession. If the earth had a spherical form, then due to spherical symmetry the resultant force passes through the center. However, the spheroidal form does not have such symmetry. Points of the equator or polar axes are exceptions, since the resultant force passes through the earth’s center. For all other points this condition is not met. Besides, the position of the resultant force depends also on the distribution of a density inside the earth. Let

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us denote by A and C the moments of inertia with respect to the axis, lying in the plane of an equator and the polar axis, respectively. The ratio CA C is called the constant of precession and its value is equal to 1/304.9. For a homogeneous Earth we have CA ¼a C Thus, the difference between the precession constant and ﬂattening of Earth is a measure of heterogeneity of the earth. The Dutch astronomer W. deSitter (1872–1934) found from the value of the precession constant that the ﬂattening of earth is equal to 1/296.92. As was pointed out, the ﬂattening depends on the distribution of density of earth, as well as the moments of inertia, which deﬁne the precession constant. Let us imagine a distribution, which satisﬁes the following conditions: (a) Average density of surface layers is d1 ¼ 2.7 103 kg/m3. (b) Average density of the earth dm ¼ 5.52 103 kg/m3. (c) Density at the center of the earth d0 has to be ﬁnite and does not exceed that of heavy metals. (d) Ratio CA/C ¼ 1/305. (e) Flattening of the external surface a ¼ 1/297. (f) Value of the parameter b in the theorem of Clairaut equal to the observed one, b ¼ 0.00529. All these conditions do not deﬁne uniquely a distribution of a density and it is possible to ﬁnd an inﬁnite number of laws satisfying these conditions, even if the density depends on the distance r only, d ¼ f(r), where r is the distance from the earth’s center, normalized by, for example, the semi-major axis a. It is obvious, that this formula implies that the earth consists of concentric spherical shells. As concerns the function f(r), this function has to increase when r decreases from 1 to 0, that is, from the earth’s surface to its center. Second, it has to contain a sufﬁcient number of arbitrary constants to satisfy all conditions. For instance, Legendre assumed that sin gr r 1 where G and m are constants. If g ¼ 144 m and G ¼ 4.46, we obtain d¼G

d1 ¼ 2:63 103 kg=m3 ;

d0 ¼ 11:2 103 kg=m3 ;

CA 1 ¼ C 304:8

E. Roche, 1820–1883, supposed that dðrÞ ¼ d0 ð1 nr2 Þ and if n ¼ 0.764 and d1 ¼ 10.1 103 kg/m3 we obtain d1 ¼ 2.64 103 kg/m3, while the other conditions are satisﬁed. E. Wiechert took into account the fact that

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seismology indicates the presence of a core with different mechanical properties than the other part of the earth. For this reason, he assumes that the core and a surrounding medium have constant but different densities. Letting the core radius be 0.78a, he obtained d0 ¼ 8:2 103 kg=m3

and

d1 ¼ 3:1 103 kg=m3

In accordance with geophysical data of his time, H. Haalck divided the earth into four parts. Inside each, the density increases slightly and at each boundary it changes abruptly, as is shown below: 1. 2. 3. 4. 5. 6.

Surface layers Depth, 60 km Depth, 1200 km Depth, 1200 km Depth, 2900 km Core

Density Density Density Density Density Density

2700 kg/m3 3100 kg/m3 3400 kg/m3 4800 kg/m3 5200 kg/m3 11,200 kg/m3

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Chapter 3 Principles of Measurements of the Gravitational Field 3.1. HISTORY OF MEASUREMENT OF THE GRAVITATIONAL FIELD Galileo was the ﬁrst to establish a law of fall of a free body and performed experiments, which allowed him to evaluate the gravitational ﬁeld. This happened in 1590. At that time the accuracy of determination of the acceleration, gravitational ﬁeld, of a rapidly falling body was very low. The English physicist D. Atwood invented a device in 1784, which artiﬁcially decreased the acceleration of a body and, correspondingly, the accuracy requirement for measurement of time became lower. Atwood’s machine consists of two masses m1 and m2 connected by a light string passing over a pulley, which has a very small friction, Fig. 3.1. If the masses are equal, then they remain at rest, one balancing the other, and the sum of forces acting on each body is zero: m2 g m1 g ¼ 0

m1 m2 F1

F2 Fig. 3.1. Atwood’s device.

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If m14m2, then mass m2 begins to move downward and the resultant force acting at this system is F ¼ m2 g m1 g Therefore, the equation of motion of these two masses is ðm2 þ m1 Þa ¼ ðm2 m1 Þg where a is the acceleration of these bodies and a¼

m2 m1 g m2 þ m1

It is clear that with a decrease of the difference between the masses the motion becomes slower and we can measure acceleration with a higher accuracy, provided that an inﬂuence of a friction is negligible. Then, last equality permits us to calculate the gravitational ﬁeld g. This method was very useful because the acceleration g is too great to be measured directly with devices available two hundred years ago. During this long period of time there were many improvements and yet direct measurements of g using the free fall had to wait the development of a laser technique and precise clocks capable of registration of distances and times with a very high accuracy until it became the main method for measuring the total gravitational ﬁeld. The ﬁrst application of a pendulum for the determination of g was done by La Condamine in 1735 in Haiti. He measured the length of the pendulum that is related to the ﬁeld g by a simple relation: g 4p2 L, provided that the period T is 1 s. La Condamine obtained L ¼ 990.85 mm and g ¼ 977.9 Gal. The purpose of these measurements was also to determine the unit of length. Later in 1792, J.C. Borda and J.D. Cassini performed measurements of the pendulum length in Paris and found out that g ¼ 980.9 Gal and L ¼ 993.827 mm. At the beginning of 19th century, French physicist J.B. Biot repeated these experiments using more advanced measurements. Also in Germany the famous astronomer F. Bessel determined the gravitational ﬁeld using a combination of the string pendulums. In 1818, the English geodesist H. Kater built a device with a reverse pendulum for measuring the absolute value of the ﬁeld g and this approach has been widely used. Austrian geodesist R. Sterneck developed a method of relative measurement of the ﬁeld and constructed a special type of pendulum, which was later improved by Stukart. The concept of the static gravimeter was introduced by Herschel (1833), who suggested the use of a spring mounted in a € os € invented a metal frame and supporting a mass. The Hungarian physicist R.Eotv gravitational variometer at the end of the 19th century, which allowed one to measure so-called gradients of the gravitational ﬁeld. At the beginning of the last century, gravimetric measurements were introduced in applied geophysics and this initiated the development of new devices for gravitational measurements. One of the ﬁrst static gravimeters was a gas–pressure gravimeter suggested by German geodesist Hecker and was developed around 1920 by Haalck. Its physical principle is similar to that of the barometer. Taking into account some shortcomings of this device, the main efforts were undertaken to develop gravimeters with higher accuracy. All of them are based on the use of

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Hooke’s law, (spring-balance principle). Due to a great progress in measuring of length and time, as well as a quality of materials used in devices, different types of gravimeters have been introduced and they are able to measure a change of the gravitational ﬁeld with an extremely high accuracy. It is conventional to distinguish two groups of gravimeters. The ﬁrst one, dynamic gravimeters, measures either a motion of a falling body or the period of a swinging body under an action of the gravitational ﬁeld, and they allow us to calculate an absolute value of the ﬁeld. The second group consists of instruments based on the spring-balance principle, such as static gravimeters and gradiometers; they provide information about a change of the ﬁeld from point to point, (relative measurements). To begin with we describe the ﬁrst group of these devices, assuming that an inﬂuence of the Coriolis force is negligible; later we will take into account this effect of the earth’s rotation.

3.2. PRINCIPLES OF BALLISTIC GRAVIMETER 3.2.1. Equations of motion In order to outline the main features of measuring the gravitational ﬁeld with the help of ballistic gravimeter imagine that a small body falls inside a vacuum cylinder under the action of the gravitational ﬁeld only. In accordance with Newton’s second law in the inertial frame we have d 2 sðtÞ ¼ F ðtÞ ¼ mgðtÞ ð3:1Þ dt2 Here m is the mass of a small body, s(t) the distance between the mass location at the initial moment and a position of the body at the instant t, and g the gravitational ﬁeld magnitude. Assuming that the amount of air inside a cylinder is very small, the inﬂuence of a friction force can be neglected. During its motion the body approaches the earth’s surface and, correspondingly, the ﬁeld g increases. Inasmuch as the change of distance with respect to the earth’s radius is very small, we can assume that the ﬁeld changes linearly within the interval of distance where we study the motion, that is, m

@g s ¼ g0 þ as ð3:2Þ @s In essence, we have expanded the ﬁeld magnitude in a power series and discarded all terms except the ﬁrst and second ones. It may be proper to point out that we assume that within the interval of measurements the rate of change of the ﬁeld: gðtÞ ¼ g0 þ

a¼

@g @s

ð3:3Þ

is constant. From Equations (3.1 and 3.2) we have d 2s ¼ g0 þ as dt2

ð3:4Þ

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and our goal is to determine the ﬁeld g0 at the initial point. First, let us study the simplest case in which we can neglect the change of the gravitational ﬁeld. Then Equation (3.4) becomes d2s ¼ g0 dt2

ð3:5Þ

To ﬁnd a solution of Equation (3.5) we perform its integration that gives ds ¼ g0 t þ C 1 dt

and

2

sðtÞ ¼ g0 t2 þ C 1 t þ C 2

ð3:6Þ

Both constants are determined from two initial conditions: sð0Þ ¼ s0

and

ds dt ð0Þ

¼ v0

if t ¼ 0

Therefore, the second equation of the set (3.6) becomes sðtÞ ¼ s0 þ v0 t þ

g0 t 2 2

ð3:7Þ

In particular, if s0 ¼ 0 and v0 ¼ 0 we have sðtÞ ¼ g0

t2 2

ð3:8Þ

Next, we suppose that the ﬁeld varies linearly along the path of movement. Then, the fall of a body is described by Equation (3.4): d 2s asðtÞ ¼ g0 dt2

ð3:9Þ

This is an inhomogeneous linear differential equation of second order with constant coefﬁcient a, where g0 is its right hand side. The parameter a is very small, and it is approximately a ¼ 2k

M 2g 103 2 ¼ Gal=m 0:3 mGal=m R R3 6:4 106

Correspondingly, Equation (3.9) can be solved, expanding the solution in a series with respect to this parameter: sðt; aÞ ¼ s1 ðtÞ þ as2 ðtÞ þ ð3:10Þ Its substitution into Equation (3.9) yields d 2 s1 d 2 s2 þ a aðs1 þ as2 Þ g0 ¼ 0 dt2 dt2 Equating terms with the same power of a, we obtain d 2 s1 ðtÞ ¼ g0 dt2

and

d 2 s2 ðtÞ ¼ s1 ðtÞ dt2

ð3:11Þ

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As follows from Equation (3.7) s1 ðtÞ ¼ s0 þ v0 t þ

g0 t 2 2

and this represents the ﬁrst approximation. Substituting the latter into the second equation of the set (3.11) we get d 2 s2 t2 ¼ s þ v t þ g 0 0 0 dt2 2 Integration of this equation gives 1 1 1 s2 ðtÞ ¼ s0 t2 þ v0 t3 þ g0 t4 þ C 1 t þ C 2 2 6 24

ð3:12Þ

From the initial conditions, we have s0 þ C 2 ¼ s0

and

v0 þ C 1 ¼ v0

C1 ¼ 0

and

C2 ¼ 0

or

Then, in accordance with Equation (3.10) we obtain t2 a 1 1 2 s0 þ v0 t þ g0 t t2 sðt; aÞ ¼ s0 þ v0 t þ g0 þ 3 12 2 2

ð3:13Þ

ð3:14Þ

This is a relationship between unknown ﬁeld g0 and two measured quantities, namely, the distance s and time t, provided that we neglect terms proportional to the square of the coefﬁcient a and those of higher order. Besides, this equation contains three unknown parameters, namely, the position of the mass s0 at the moment when we start to measure time, the initial velocity, v0 , at this moment and, ﬁnally, the rate of change of the gravitational ﬁeld, a, along the vertical. Thus, in order to solve our problem and ﬁnd the ﬁeld we have to perform measurements of the distance s at four instants. If s0 is known, the number of these measurements is reduced by one. In modern devices the coefﬁcient of the last term on the right hand side of Equation (3.14) has a value around 100 mGal and it is deﬁned by calculations as a correction factor ðv0 ; g0 ; t; aÞ. In the case when we can let s0 equal to zero, it is sufﬁcient to make measurements at two instances only. 3.2.2. Two methods of field measurements of the field There are two ballistic methods of measurements of the ﬁeld; one of them is nonsymmetrical, in which we observe the free fall of a body. The second is the so-called symmetrical one. In this case we study the motion of the body up and down.

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3.2.3. Non-symmetrical motion In principle, a determination of the ﬁeld can be done in the following way. During its fall a body passes through three stations, positions, s0 ; sðt1 Þ, and sðt2 Þ; and this happens at instances t0 ¼ 0, t1 and t2, respectively. The distances are measured from the station s0 Fig. 3.2a. In accordance with Equation (3.14) we have for moments t1 and t2 g sðt1 Þ ¼ s0 þ v0 t1 þ 0 t21 þ 1 ðv0 ; g0 ; t1 ; aÞ 2 and sðt2 Þ ¼ s0 þ v0 t2 þ

g0 2 t þ 2 ðv0 ; g0 ; t2 ; aÞ 2 2

ð3:15Þ

In particular, if s0 ð0Þ ¼ 0 we have sðt1 Þ ¼ v0 t1 þ

g0 2 t þ 1 2 1

sðt2 Þ ¼ v0 t2 þ

g0 2 t þ 2 2 2

and ð3:16Þ

Multiplying the ﬁrst and second equations of the set (3.16) by t2 and then t1 respectively, we eliminate the term with the initial velocity and this gives 2 sðt2 Þ sðt1 Þ g0 ¼ þ ð3:17Þ t2 t1 t2 t1 where e is the correction factor, depending on a, v0 , g0 , t1 , and t2 , and as it was pointed out this correction is not measured but calculated. We have obtained the equation for evaluation of the gravitational ﬁeld g0 , by making use of observations of a falling body at two intervals. Devices based on this principle allow us to measure the absolute value of the ﬁeld with an error 108 109 with respect to the total ﬁeld on the earth. (a)

(b) 0

.

t0 = 0

s0 s

t1

s1

t2

s2

Fig. 3.2. (a) Non-symmetrical motion, (b) symmetrical motion.

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3.2.4. Symmetrical motion In this approach, a body is thrown up and then it falls along the same path. Observations are made in passing two ﬁxed positions during the movement up and down, Fig. 3.2b. Of course, inside the cylinder where the movement takes place, there is always some air, but this approach allows us to reduce its inﬂuence better than in the previous method. This is related to the fact that the resistance of air is the same during motion in the different directions. However, in one case the friction force and ﬁeld g have the same direction, while in the second case they are opposite to each other. In the symmetrical approach we measure moments when a mass passes the upper and lower stations. The point 0 is origin that corresponds to the highest position of the body where s0 ¼ 0

and

v0 ¼ 0

Then, Equation (3.14) becomes t g t 2 t 4 1 2 2 2 s þ ag0 ¼ 0 24 2 2 2 2 and s

t 1

2

¼

t 4 g0 t1 2 1 1 þ ag0 24 2 2 2

ð3:18Þ

The distance between stations is H ¼ sðt2 Þ sðt1 Þ ¼

g0 2 g t4 t41 ðt2 t21 Þ þ a 0 2 8 24 16

ð3:19Þ

Whence g0 ¼

8H a þ ðt22 þ t21 Þ 2 t1 48

t22

ð3:20Þ

The simple equations, described above, were mainly given for an illustration. At the same time, determination of the ﬁeld can be done continuously during motion of the small body. Suppose that a generator creates a laser wave with a given frequency and there are two reﬂectors: one is attached at the falling body but the other is ﬁxed and, for example, located on the cylinder bottom. Waves, caused by a generator, reach reﬂectors and correspondingly two reﬂection waves arise and their superposition is measured by the sensor, located for instance at the top of the cylinder. These two waves have the same frequency and for simplicity assume that they also have equal amplitudes. Then, the amplitude of their sum is a function of a phase shift between them. During the fall of a body the distance, s(t), traveled by one reﬂected wave changes and it leads to a phase change, too. It is clear that the maximal amplitude is observed when the constructive interference takes place, that is, the phase difference is proportional to 2pn (n is integer) while the amplitude is

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zero in the presence of the distractive interference. Certainly, this process is described by a sinusoidal function: 2ps þ j0 C þ A sin 2 l Here C and j0 are some constants, but l is the wavelength of laser wave and it is a extremely small number. For instance, if a motion is given by Equation (3.8), we have 2pg0 t2 C þ A sin þ j0 l Here multiplier 2 in front of the ﬁrst term of the argument is due to the fact that 2 s is the total length of the incident and reﬂected rays. It is obvious, that the frequency of the measured sinusoid is f ¼

g0 t l

and it linearly increases with time. Measurements of this quantity allow us to determine the ﬁeld at the initial point of the path. The ballistic device has always a vacuum cylinder inside which the body is moved, and the latter has a form of a triangular reﬂector. At the bottom of the cylinder there is a device (elastic element) that throws the body upward. The maximal height of the body position is approximately equal to 0.5 m and the cylinder length is around 1 m. In conclusion of this section, it is appropriate to make the following comments: 1. To achieve the astonishing accuracy of around 5 10–7 m/s2, some remarkable measures to reduce noise are required. Even though the dropping chamber is evacuated to about 10–4 Pa, molecular drag is still a signiﬁcant source of error. This error is virtually eliminated by enclosing the proof mass in a mechanical carriage, which drops with the mass, using servos driven by the optical system to ensure that the carriage maintains its relative position to the mass. The carriage pushes the residual gas molecules ahead of the proof mass, thereby achieving an essentially drug-free environment. As a side beneﬁt, the carriage serves as an elevator to catch the proof mass at the bottom of its fall and return it to its rest position at the top of the chamber. 2. The other major source of noise is vibration from the environment. The entire measuring system is, of course, mounted on appropriate rubber isolation pads. However, these are not adequate for the level of accuracy required. The residual vibration is eliminated from the signal by mounting the lower part of the optical reﬂection system on a spring which is driven by servo electronics that detects its movement. An effective free period for the spring of up to 60 s can be achieved in this way, and virtually all environmental vibration is eliminated from the measured signal. 3. After these measures have been taken, the remaining dominant sources of error are in the optical system and electronics, and the effects of Coriolis force, each

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in the vicinity of 5 10–8 m/s2. However, there are numerous other error sources with effects in the range of 10–8 m/s2.

3.3. PENDULUM DEVICES As was pointed out earlier, the ﬁrst practical method of evaluation of the gravitational ﬁeld was based on measurements of the period of vibrations. To begin with, we will describe the theory of a ﬁctitious so-called mathematical pendulum, which consists of the point mass m, particle, and a weightless thread with length l, Fig. 3.3. This is a ﬁrst approximation of a real pendulum. The heavy particle m is attached to one end of the string while the other end is ﬁxed at the point 0. We assume that the mass is moved from the point x ¼ 0 and z ¼ l and then begins to move in the vertical plane. The gravitational ﬁeld directed along the vertical causes force on the string S oriented toward the origin 0, and this is compensated by the component of the weight, Fn . Thus, a motion of the body occurs along a circular path with radius l under an action of the force tangential to the path: F t ¼ mg sin y

ð3:21Þ

It is simple to visualize this motion. Suppose that the initial position of mass is characterized by the angle y0. At the instant t ¼ 0 it begins to move. Since the force is directed along the motion, the velocity increases and at the lowest point, (x ¼ 0 and z ¼ l), it reaches the maximum value. As soon as a mass passes this point, the velocity begins to decrease because the force component and velocity have opposite directions. Finally, the particle stops and then the motion begins again but in opposite direction. Assuming that friction is absent, we may expect a periodic movement of the mass around the middle point. In fact, equations of a motion of this particle in the Cartesian system of coordinates are mx€ ¼ S sin y and

m€z ¼ S cos y þ mg

0

x θ

S

m

Fn

Ft mg z Fig. 3.3. Illustration of Equation (3.22).

ð3:22Þ

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As is seen from Fig. 3.3 sin y ¼

x l

and cos y ¼

z l

ð3:23Þ

3.3.1. Small oscillations Now we will study the vibrations of the mass about the point of equilibrium, (x ¼ 0, z ¼ l), provided that the amplitude of the displacement is very small. This means that sin y y

and

cos y 1

In this approximation the right hand side of the second equation of the set (3.22) is independent of the angle y, and we can conclude that at any instant the string tension is the weight of the particle: S ¼ mg

ð3:24Þ

Correspondingly, the second equation of the set (3.22) becomes z€ ¼ 0 that is, the motion along the z-axis is not subjected to a force. Taking into account Equation (3.24), the ﬁrst equation of this set becomes x mx€ ¼ mg or x€ þ o20 x ¼ 0 ð3:25Þ l Here g 1=2 o0 ¼ ð3:26Þ l Thus, we have demonstrated that a small motion of a mass is described by the equation of a harmonic oscillator, Equation (3.25), and, as is well known, its solution is xðtÞ ¼ A sin o0 t þ B cos o0 t

ð3:27Þ

The parameter o0 is called the frequency of free vibrations and it is related to the period of a motion of a particle by 1=2 l ð3:28Þ T ¼ 2p g We see that the time interval during which the mass moves from the original position and back to the same place, T, is proportional to the string length and becomes smaller with an increase of the gravitational ﬁeld. The constants A and B are deﬁned only by the initial conditions. For example, if at the instant t ¼ 0, we have xð0Þ ¼ x0

and

_ xð0Þ ¼0

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then Equation (3.27) becomes xð0Þ ¼ x0 cos o0 t

ð3:29Þ

Thus, we have demonstrated that measuring the period of oscillations, T, the pendulum allows us to determine the ﬁeld g. From Equation (3.28) we have g¼

4p2 l T2

ð3:30Þ

Correspondingly, the string length is gT 2 4p2 For instance, if the pendulum period is 1 s and g ¼ 9.8 m/s2, we obtain l¼

ð3:31Þ

l 0:25 m Assuming that an error of measuring the string length is zero, it is simple to obtain a relation between the accuracy dT of a period and dg of the gravitational ﬁeld. As follows from Equation (3.30) T2 ¼

4p2 l g

and its differentiation gives 2TdT ¼ 4p2

ldg dg T 2 ¼ 4pl g2 g 4p2 l

Thus, 2

dT dg ¼ T g

ð3:32Þ

For instance, if we would like to measure a ﬁeld with an accuracy of 1 mgal, that is, dg=g ¼ 106 , it is necessary to know the period 1 s with an accuracy dT ¼ 0:5 106 s. In this light, it is useful to obtain similar relation for the ballistic gravimeter. In accordance with the solution of the equation of motion we have g¼

2s t2

Taking the logarithm of both sides ln g ¼ ln 2s 2ln t and performing a differentiation we obtain dg ds dt ¼ 2 g s t Therefore, if the ﬁeld has to be determined with a relative error 108 , we need to measure the distance and time with the same accuracy.

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3.3.2. Oscillations with an arbitrary amplitude We have studied small oscillations of the mathematical pendulum. Next, we solve the same problem for larger values of displacements, and with this purpose in mind consider both equations of the set (3.22). Multiplying them by unit vectors i and k, respectively, and adding we obtain one equation of motion € þ z€k ¼ gk iS sin y kS cos y xi

or

r€ ¼ gk S

ð3:33Þ

and it can be used to study a motion along any direction. Inasmuch as the tension vector S is directed along the radius vector, the projection of this equation on the direction tangential to the path of a motion is dv d2y ¼ g sin y ¼ l 2 dt dt

ð3:34Þ

Of course, this equation directly follows from Fig. 3.3. Multiplying Equation (3.34) by ds=dt ¼ v, we have dv ds dv dz v ¼ ðg sin yÞ or v ¼ g dt dt dt dt Since dz ¼ ds sin y we may write vdv ¼ gdz Its integration gives 1 2 v ¼ gz þ C 2

ð3:35Þ

where C is a constant of integration. At the initial point p0 we have z ¼ z0

and

v¼0

Hence C ¼ gz0

and

1 2 2v

¼ gðz z0 Þ

or v2 ¼ 2gðcos y cos y0 Þ

ð3:36Þ

since z ¼ l cos y and

z0 ¼ l cos y0

Here it is appropriate to make three comments: a. As in the case of small oscillations we assume that the gravitational ﬁeld is independent of the position of the pendulum; that is, it is a constant equal to the ﬁeld when the angle y ¼ 0. b. The gravitational ﬁeld is a vector sum of the ﬁeld of attraction and centrifugal force per unit mass, that is, we partly take into account the rotation of the earth. The same is valid for the case of a free fall of a mass.

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In the ballistic gravimeter and the pendulum, we study the motion of a mass in a non-inertial frame of reference, and, speaking strictly, have to consider the Coriolis force too. However, the inﬂuence of this factor will be discussed later. Our goal is to ﬁnd a relationship between the period of oscillations and the gravitational ﬁeld and with this purpose in mind we write Equation (3.36) as 2 ds 2 y0 2y 2 sin ¼ 4gl sin v ¼ dt 2 2 c.

Taking into account the fact that ds ¼ ldy, we have 2 dy 4g 2 y0 2y sin sin ¼ dt l 2 2

ð3:37Þ

Note that we used the following equality sin2

j 1 cos j ¼ 2 2

Thus, Equation (3.37) yields 1 l 1=2 dy dt ¼ 1=2 2 g sin2 ðy0 =2Þ sin2 ðy=2Þ

ð3:38Þ

If T is the oscillation time of pendulum or the time the pendulum needs to swing from the angle y0 to y0 , (half period ), then integration of Equation (3.38) gives Z 1 l 1=2 y0 dy T¼ 1=2 2 g y0 sin2 ðy0 =2Þ sin2 ðy=2Þ or T¼

1=2 Z y0 l dy 2 1=2 g 0 sin ðy0 =2Þ sin2 ðy=2Þ

ð3:39Þ

We see that unlike the case of small oscillations the half period T depends on three parameters, namely, the string length l, the initial angle y0 , and the gravitational ﬁeld g, but as before it is independent of the moving mass of the pendulum. Also, in accordance with Equation (3.34), the function yðtÞ is not sinusoidal, but of course it has a periodic character, since the inﬂuence of friction is neglected. To simplify the integral on the right hand side of Equation (3.39), we introduce a new variable y y0 sin ¼ sin sin c 2 2

ð3:40Þ

1 y y0 cos dy ¼ sin cos cdc 2 2 2

ð3:41Þ

Then

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and we have

dy

2 sinðy0 =2Þ cos cdc 2dc 1=2 ¼ cosðy=2Þ sinðy =2Þ cos c ¼ cosðy=2Þ 0 sin ðy0 =2Þ sin ðy=2Þ 2

2

ð3:42Þ

As follows from Equation (3.40), the angle c changes from 0 to p=2, when y varies from 0 to y0 , and, therefore 1=2 Z p=2 1=2 Z p=2 l dc l dc ¼2 T ¼2 1=2 2 g cosðy=2Þ g 0 0 1 sin ðy0 =2Þsin2 c or 1=2 Z p=2 l dc T ¼2 1=2 g 0 1 e2 sin2 c

ð3:43Þ

Here e ¼ sin

y0 2

ð3:44Þ

is the parameter, characterizing the maximal displacement of the mass and this is usually small. Thus, we have expressed the time T in terms of the elliptical integral: Z p=2 dc 1=2 0 1 e2 sin2 c and it depends on the parameter e only. Expanding the integrand: 1 1=2 1 e2 sin2 c in a power series with respect to e, (eo1), and performing an integration of each term we obtain 1=2 l 1 2 y0 9 4 y0 þ sin þ ð3:45Þ T ¼p 1 þ sin g 4 2 64 2 As y0 is very small, of the order of minutes, we can write sin2

y0 y20 y40 ¼ þ 2 4 48

and

sin4

y0 y40 ¼ 2 16

This gives 1=2 l 1 2 11 4 y þ 1 þ y0 þ T ¼p g 16 3072 0

ð3:46Þ

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or

1 11 4 y0 þ T ¼ T 0 1 þ y20 þ 16 3072

Here T 0 is the half period when the amplitude of displacement is very small. If the angle y0 is sufﬁciently small so that we can neglect the term proportional to y40 , then T0 ¼

T y20 ¼ T T 16 1 þ ðy20 =16Þ

ð3:47Þ

The second term is usually called the arc reduction and it is small for amplitudes used in practice. Performing measurements at one point with two pendulums of different lengths we have l 1 : l 2 ¼ T 20 ðl 1 Þ : T 20 ðl 2 Þ that is, the lengths of pendulums are proportional to the square of periods. Suppose that measurements of the oscillation times, (T 1 and T 2 ), are performed at two different points with the same pendulum and the arc reductions are introduced. Then, the ratio of the gravitational ﬁelds is g1 : g2 ¼ T 202 : T 201

ð3:48Þ

and this is the main formula for relative gravity measurements with the pendulum. 3.3.3. Physical pendulum Until now we have considered the theory of the mathematical model of the real pendulum. Next, suppose that a solid body of ﬁnite dimensions swings in the plane XZ around a horizontal axis, and that the motion takes place with the angular velocity oðtÞ, Fig. 3.4. (a)

(b) 01

x

0

r

W1

θ

dm a

l C

• l-a

W2 o S

z Fig. 3.4. (a) Physical pendulum, (b) reversing pendulum.

02

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3.3.4. Equation of a motion Let us mentally represent this body as an inﬁnite system of elementary masses dm, located at different distances r from the axis and, ﬁrst, we obtain an equation for the rotation of some mass dm. By deﬁnition, the linear velocity, v, of each mass is related to the angular velocity by ds ¼ ro dt Here ds is an elementary displacement. As is seen from Fig. 3.4, components of the velocity along coordinate axes can be written in the form: v¼

dx dz ¼ or cos y ¼ zo and ¼ or sin y ¼ xo dt dt Multiplication of these equations by z and –x, respectively, and then their subtraction gives dx dz x ¼ oðz2 þ x2 Þ ¼ or2 dt dt To derive an equation of rotation, we take a derivative of this equality with respect to time and the latter yields z

d 2 x dz dx dx dz d2z do x þ ¼ r2 2 2 dt dt dt dt dt dt dt Taking into account the fact that the second derivatives are components of the gravitational ﬁeld, that is, an acceleration, which has only the vertical component, € ¼ 0 and z€ ¼ g, the last equation is greatly simpliﬁed and we have: g, that is, xðtÞ z

do ¼ gx ð3:49Þ dt This is an equation of rotation of an elementary mass around the y-axis. Here r2 can be treated as the moment of inertia of the unit mass and do=dt is the angular acceleration. The product gx characterizes the torque with respect to the point 0. Multiplying Equation (3.49) by dm and performing integration over the pendulum mass, we obtain Z Z do r2 dm ¼ g xdm ð3:50Þ dt r2

By deﬁnition, the center of mass of the pendulum is characterized by the radius vector Z 1 rC ¼ rdm M and, correspondingly, 1 xC ¼ M

Z xdm

ð3:51Þ

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Here M is the mass of the pendulum. The angular velocity is related to the angle y by o¼

dy dt

and, therefore, do d 2 y ¼ 2 dt dt The integral at the left hand side of Equation (3.50) represents the moment of inertia of the pendulum: Z ð3:52Þ I ¼ r2 dm Thus, Equation (3.50) is written as d2y I ¼ gxC M dt2 Denoting the distance from the origin to the center of mass by a, we obtain d 2 y Ma g sin y ¼ 0 þ dt2 I

ð3:53Þ

since xC ¼ a sin y Thus, have we derived an equation of a motion of the physical pendulum and found parameters, which describe the swinging around the ﬁxed axis. Introducing the ratio: l¼

I Ma

ð3:54Þ

Equation (3.53) is written as d 2y g þ sin y ¼ 0 dt2 l which coincides with Equation (3.34) derived for the mathematical pendulum. This means that the motion of the physical pendulum is described by a periodic function and it swings as fast as the mathematical pendulum with length l. The quantity l is called the ‘‘reduced length’’ of the physical pendulum. In principle, measuring the oscillation time T it is possible to evaluate the gravitational ﬁeld if we know the pendulum mass, the moment of inertia, and the distance between the axis and the center of mass.

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3.3.5. Reversing pendulum In order to avoid the determination of these parameters and errors related with this procedure, Henry Kater suggested at the beginning 19th century so-called the reversing pendulum. To explain the principle of this device let us recall one basic feature of the moment of inertia. First, we make use of the identity r2 ¼ x2 þ z2 ¼ ½xC þ ðx xC Þ2 þ ½zC þ ðz zC Þ2 or r2 ¼ x2C þ z2C þ ðx xC Þ2 þ ðz zC Þ2 þ 2xC ðx xC Þ þ zC ðz zC Þ Letting r2 ¼ ðx xC Þ2 þ ðz zC Þ2 we have r2 ¼ a2 þ r2 þ 2xC ðx xC Þ þ 2zC ðz zC Þ Therefore, the moment of inertia with respect to the origin can be represented as Z Z Z Z I ¼ r2 dm ¼ a2 M þ r2 dm þ 2xC ðx xC Þ dm þ 2zC ðz zC Þ dm By deﬁnition of the center of mass the last two integrals are zero and we obtain Z 2 I ¼ a M þ r2 dm ð3:55Þ The integral on the right hand side represents the moment of inertia of the pendulum with respect to the axis passing through the center of gravity, and Equation (3.55) describes the well-known theorem of mechanics. Bearing in mind that, we already introduced the reduced length l, (Equation (3.54)), let us assume that Z r2 dm ¼ n2 M Then, in place of Equation (3.55) we have I ¼ ða2 þ n2 ÞM

ð3:56Þ

and Equation (3.54) gives a2 þ n2 n2 ¼aþ ð3:57Þ a a that is, the reduced length and the distance a differ by the term n2 =a. Note that the point S, (l ¼ OS), located on the continuation of the line OC at the distance l from the origin, is called the ‘‘swinging center’’ of the pendulum. It is clear that l¼

CS ¼

n2 ¼la a

ð3:58Þ

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Suppose that the same pendulum moves about the axis passing through the point S. Then it is characterized by a new reduced length l0 and, by analogy with Equation (3.57), we have l0 ¼ l a þ

n2 la

ð3:59Þ

Here (la) is the distance between the new axis and the same center of mass. From Equation (3.58) it follows that l0 ¼ l

ð3:60Þ

This means that the oscillation times in both cases are equal to each other. Certainly, this is hardly an expectable equality and it is used in the ‘‘reversing pendulum’’, Fig. 3.4b. In principle, the device works in the following way. First, it swings about the axis 01 , then it is turned upside down and oscillates about the axis 02 , located on the other side of the center of mass. By adjusting the position of weights W 1 and W 2 , we make the periods of swinging about both axes are equal. In this case the distance 01 02 equals the reduced length l of the pendulum. Therefore, making use of Equation (3.28) we can determine the gravitational ﬁeld g, provided that the period and the distance between axes are known. As in all other methods, several factors inﬂuence measurements of the gravitational ﬁeld and their role becomes extremely important because we wish to measure the time and the reduced length with very high accuracy. Let us brieﬂy discuss an effect of some of them. 1. Barometric factor First of all, the presence of air decreases the weight of the pendulum, (Archimedes’ law). On the other hand, a moving pendulum carries some amount of air, and it leads to an increase of the pendulum weight. Also, the friction caused by air reduces the oscillating time. 2. Bending of the stand Experience shows that even a stand with a height 0.2–0.3 m will bend slightly with the pendulum. This means that the motion of the pendulum is not free and is subjected to the action of the force, caused by the deformation of the stand. Correspondingly, the equation of motion of the pendulum is

3.

d 2y g þ y þ f ðtÞ ¼ 0 dt2 l where f(t) is the horizontal component of acceleration of the suspension point of the pendulum. Inasmuch as the force of bending is directed opposite to that of the motion, the time of oscillation T becomes smaller. This is also true for the reduced length l. It turns out that the inﬂuence of this factor, even when measurements are performed on strong ground, may reach 30 107 s. In order to reduce this effect two synchronized pendulums are used, swinging on the same stand and in the same plane. They have to have the same swing amplitudes but opposite phases. A change of temperature leads to a change of the pendulum length and the correction factor is deﬁned as kðt t0 Þ

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where t is an observation time, t0 the reference time, and k constant determined experimentally for each pendulum. In addition, a correction is introduced in order to take into account the fact that temperature makes a different effect on the pendulum and thermometer. It is obvious that a change of clock rate directly affects the value of the swinging time T. For instance, even if we observe a very small change during 1 h, for instance, 0.0007 s, then it causes a change of 107 s for the period of swinging of the pendulum. The use of crystal clocks greatly increased the accuracy of determination of the period, but in the past it was necessary to perform measurements during over 24 h to reduce the effect of the change of the clock rate.

3.4. INFLUENCE OF CORIOLIS FORCE ON PARTICLE MOTION 3.4.1. Equation of motion As was pointed out earlier, when we have considered the physical principles of the ballistic gravimeter and the pendulum an inﬂuence of the Coriolis force was ignored. Now we will try to take into account this factor and consider the motion of a particle near the earth’s surface. With this purpose in mind let us choose a noninertial frame of reference, shown in Fig. 3.5a; its origin 0 is located near the earth’s surface and it rotates together with the earth with angular velocity o. The unit vectors i, j, and k of this system are ﬁxed relative to the earth and directed as follows: i is horizontal, that is, tangential to the earth’s surface and points south, j is also horizontal and points east, k is vertical and points upward. As is shown in Fig. 3.5a SN is the earth’s axis, drawn from south to north, I is the unit vector along 01 0, and K is a unit vector parallel to SN. First, we derive again but in a slightly different way than in Chapter 2 the equation of a motion in a non-inertial frame of reference. As before, r is the position of the moving particle with respect to 0 and 01 0 ¼ r0 . The position of the particle with respect to the origin 01 of the inertial frame is r1 ¼ r0 þ r

z

p λ

01 S

r

r1 K r0

(c)

(b)

(a)

N

ð3:61Þ

b d

l k j

0

I

i

i

k 0

j

0 .

a

y

x Fig. 3.5. (a) Rotating system of coordinates, (b) Foucault’s pendulum, (c) illustration of pendulum path.

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Performing a differentiation with respect to time twice, we obtain the acceleration in the inertial system r€1 ¼ r€ 0 þ r€ ð3:62Þ Here r€ 0 is the acceleration of the origin 0, and since 0 moves in a circle with constant angular velocity x ¼ oK, we have r€0 ¼ r0 o2 I The relationship between unit vectors: I, i, and k, (Fig. 3.5a) gives r€ 0 ¼ r0 o2 ði sin l þ k cos lÞ

ð3:63Þ

and we have expressed the centripetal acceleration of the point 0 in the rotating system of coordinates. In accordance with Newton’s second law, the equation of motion of a particle with an elementary mass m in the inertial frame of reference is m€r1 ¼ mga þ F

ð3:64Þ

where ga is the ﬁeld of attraction caused by masses of the earth and F the force of any other origin. For instance, a. if an elementary mass is located inside a ﬂuid, then F characterizes the surface forces. b. if this mass is situated on the earth’s surface, F describes the force caused by a deformation of the earth and is directed along the vertical. c. if the mass falls on the earth’s surface under only the ﬁeld of attraction, this force is zero. d. in the case of the mathematical pendulum, F is the elastic force of the string or a spring when we deal with static gravimeters. Now we ﬁnd the relationship between forces and the function r€ and with this purpose in mind we make use of Equations (3.62–3.64). They give m€r ¼ mga ðpÞ þ F þ mo2 r0 ði sin l þ k cos lÞ

ð3:65Þ

Let us make the ﬁrst assumption, namely, that a particle moves near the origin 0 and we can assume that variations of the attraction ﬁeld are negligible, that is, ga ðpÞ ¼ ga ð0Þ

ð3:66Þ

By deﬁnition, for the gravitational ﬁeld we have ga þ o2 r0 ði sin l þ k cos lÞ ¼ g ¼ gk

ð3:67Þ

because we assumed that at the point 0 the unit vector k is directed opposite the gravitational ﬁeld at this point. Thus, Equation (3.65) becomes m€r ¼ F mgk

ð3:68Þ

Several factors cause changes of the function r€ and they include a change of coordinates of the point, as well as a change of an orientation of unit vectors: i, j, and k due to rotation. For this reason, r€ does not characterize the acceleration of

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the point p in the non-inertial frame of reference with the origin 0. As was shown in Chapter 2 r€ ¼

d2 r dr þ 2xx þ xxðxxrÞ 2 dt dt

ð3:69Þ

Here d2 r=dt2 and dr=dt are the acceleration and velocity in the non-inertial frame of reference, respectively. In other words, these quantities describe the motion of the point p at given moment, assuming that coordinate axes do not change a direction. Substitution of Equation (3.69) into Equation (3.68) yields m

d2 r ¼ F mgk þ 2mðvxxÞ þ xxðxxrÞ dt2

ð3:70Þ

where v ¼ dr=dt. Thus, as we may expect, the relationship between the acceleration and forces in the rotating system of coordinates differs essentially from Newton’s second law. In fact, three terms on the right hand side of Equation (3.70), except the ﬁrst one, are subjected to the inﬂuence of rotation. The second term includes effect of both the attraction and centrifugal forces, but last two terms vanish together with the rotation. This shows that we can still treat Equation (3.70) as Newton’s second law if we assume that the elementary mass is subjected to the action of real and ﬁctitious forces. Next we introduce the second assumption based on the fact that the angular velocity of the earth is small and it is possible to neglect the last term in Equation (3.70), which is proportional to square of o, where r is the distance from the point to the origin 0, but not the axis of rotation of the earth. Then, in place of Equation (3.70) we obtain d2 r ¼ F mgk þ 2mðvxxÞ ð3:71Þ dt2 and, as we know, the last term characterizes the Coriolis force. Consider components of this equation along the coordinate axes. From Fig. 3.5a it follows that m

x ¼ oK ¼ io cos l þ ko sin l By deﬁnition, we have _ þ yj _ þ z_k v ¼ xi and i j vxx ¼ x_ y_ o cos l 0

o sin l k z_

Therefore, the equation of motion along the coordinate lines becomes mx€ ¼ F x þ 2moy_ sin l my€ ¼ F y 2moðx_ sin l þ z_ cos lÞ m€z ¼ F z mg þ 2moy_ cos l

ð3:72Þ

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and we will proceed from this set to study the inﬂuence of the Coriolis force in two cases where it will be assumed that the gravitational ﬁeld is constant: g(r) ¼ g(0). 3.4.2. Motion of a free particle Consider a fall of a particle under the action of a gravitational ﬁeld only. Then, Equations of set (3.72) are greatly simpliﬁed and we have x€ ¼ 2oy_ sin l y€ ¼ 2oðx_ sin l þ z_ cos lÞ z€ ¼ g þ 2oy_ cos l

ð3:73Þ

Suppose that at the instant t ¼ 0 the particle is at origin, and u1, u2, u3 are components of the velocity. Then, performing integration with respect to time we obtain for the particle velocity: x_ ¼ 2oy sin l þ u1 y_ ¼ 2oðx sin l þ z cos lÞ þ u2 z_ ¼ gt þ 2oy cos l þ u3

ð3:74Þ

Substitution of the ﬁrst and second equations of this set into the second equation of set (3.72) gives y€ ¼ 2oðu1 sin l þ u3 cos l gt cos lÞ provided that terms in o2 are discarded. It is essential that the right hand side of this equation does not contain the coordinates of the particle. The ﬁrst integration yields 1 y_ ¼ 2o u1 sin l þ u3 cos l gt cos l t þ u2 2 and after one more integration we obtain 1 y ¼ u2 t ot2 ðu1 sin l þ u3 cos lÞ þ ogt3 cos l ð3:75Þ 3 Substituting the latter into the ﬁrst and last equations of the set (3.74) and neglecting terms proportional to o2 , we obtain after integration x ¼ u1 t þ ou2 t2 sin l and 1 ð3:76Þ z ¼ u3 t gt2 þ ou2 t2 cos l 2 Equations (3.75 and 3.76) describe the motion of a free particle in a rotating frame of reference, when the angular velocity is relatively small and the z-axis is directed along the plumb line. Certainly, a presence of terms with o is related to the rotation of the earth. Besides, the gravitational ﬁeld also contains a term with this frequency. It may be proper to emphasize again that in deriving these formulas we assumed

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that the gravitational ﬁeld does not depend on r. Suppose that at the beginning, the particle is located at the origin and it is at rest. Thus, the initial conditions are u1 ¼ u2 ¼ u3 ¼ 0 and

u_ 1 ¼ u_ 2 ¼ u_ 3 ¼ 0

ð3:77Þ

z ¼ 12 gt2

ð3:78Þ

From Equations (3.75 and 3.76), we obtain x ¼ 0;

y ¼ 13 ogt3 cos l;

and

and this allows us to make corrections, caused by Coriolis force in measuring the distance inside the ballistic gravimeter. Eliminating the time t from the last two equations, we ﬁrst of all see that the path of a motion is located in the east–west plane, (Y0Z), since x ¼ 0, and the equation of the particle’s path is y2 ¼

8 o2 cos2 l 3 z 9 g

ð3:79Þ

As we see that the deviation from the vertical is toward the east and in accordance with Equation (3.79) is 1=2 1 2h y ¼ o cos l 2h ð3:80Þ 3 g where h ¼ jzj. For instance, if h ¼ 0.5 m, we have 1=2 1 y 7:3 105 cos l 104 m cos l 9:8 Correspondingly, if l 0 the deviation a from the vertical is approximately y ¼ tan a a ¼ 2 104 h It is clear this deviation is absent at poles, ðl ¼ p=2Þ.

3.4.3. Foucault’s pendulum Next we consider the inﬂuence of the earth rotation on the motion of the mathematical pendulum. We will proceed from Equation (3.72). As is seen from Fig. 3.5b, the components of the tension S of the string are x y lz ð3:81Þ X ¼ S; Y ¼ S; and Z ¼ S l l l In deriving the system of Equation (3.72), we assumed that the frequency of rotation is small and a term proportional to o2 was neglected. At the same time the centrifugal force was taken into account. Besides, we imply that the gravitational ﬁeld does not change with the pendulum swing and that its amplitude is small. The last assumption allows us to greatly simplify equations. As we can see from Fig. 3.5b, the displacement along the z-axis is much smaller than that in horizontal direction: zoox and zooy. The same inequalities hold for the corresponding

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derivatives with respect to time. This allows us let Z ¼ S and neglect the second derivative z€. Respectively, the last equation of the set (3.72) is written as _ cos l S ¼ mg 2myo

ð3:82Þ

Substitution of this equality into the ﬁrst equation of the set (3.82) gives x

g x€ ¼ x þ 2oy_ cos l þ sin l l l Since x/loo1 we ﬁnally have x€ 2oy_ sin l þ p2 x ¼ 0

ð3:83Þ

y€ þ 2ox_ sin l þ p2 y ¼ 0

ð3:84Þ

Similarly

where p¼

g 1=2

l and this characterizes the angular frequency of vibration. The system of Equations (3.83 and 3.84) describes the motion of the pendulum in the plane z ¼ 0 around the point 0, and in order to obtain its solution we introduce the complex plane where the position of a point is deﬁned by a coordinate x ¼ x þ iy Then, multiplying the second equation of this set by an imaginary unit i and adding the ﬁrst equation we arrive at one equation with respect to the complex variable x: x€ þ 2iox_ sin l þ p2 x ¼ 0

ð3:85Þ

This is a homogeneous linear differential equation of second order and its characteristic equation is n2 þ i2no sin l þ p2 ¼ 0 The roots of this equation are n ¼ io sin l iðo2 sin2 l þ p2 Þ1=2 Neglecting terms in o2 we have n1 ¼ io sin l þ ip

and

n2 ¼ io sin l ip

ð3:86Þ

It is essential that the roots are pure imaginary and this indicates that the solution is expressed in terms of sinusoidal functions. The general solution of Equation (3.85) is x ¼ A expðn1 tÞ þ B expðn2 tÞ

ð3:87Þ

and after substitution of Equation (3.86), we obtain xðtÞ ¼ CðtÞ expðio sin ltÞ

ð3:88Þ

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where CðtÞ ¼ A expðiptÞ þ B expðiptÞ

ð3:89Þ

A and B are constants and they are independent of time. Bearing in mind that x ¼ x þ iy, it is simple to ﬁnd functions x(t) and y(t), which describe a motion of the pendulum on the earth’s surface. In accordance with Equation (3.88) a solution is represented as a product of two functions. The ﬁrst one characterizes a swinging of the pendulum with the angular velocity p, which depends only on the gravitational ﬁeld and the length l, while the second is also a sinusoidal function and its period is deﬁned by the frequency of the earth’s rotation and the latitude of the point, (Foucault’s pendulum). In order to understand the behavior of the pendulum at the beginning consider the simplest case when a rotation is absent, o ¼ 0: Then, we have xðtÞ ¼ CðtÞ or xðtÞ ¼ ðA þ BÞ cos pt and

yðtÞ ¼ ðA BÞ sin pt

ð3:90Þ

Eliminating the variable pt, we obtain x2 ðtÞ y2 ðtÞ þ 2 ¼1 a2 b Here AþB¼a

and

AB¼b

ð3:91Þ

Thus, in general the pendulum moves along an ellipse on the spherical surface with radius equal to the string length, l. This motion has a periodic character and the period is that of the swinging, (T ¼ 2p/o). In particular, if the initial conditions are xð0Þ ¼ x0

and

_ ¼0 yð0Þ

from Equation (3.90), we have a ¼ x0 and b ¼ 0, that is, xðtÞ ¼ x0 cos pt and

yðtÞ ¼ 0

ð3:92Þ

Therefore, the pendulum moves periodically along the arc, located in the plane XOZ, and this case was studied in detail in the previous section. Comparison of Equations (3.91 and 3.92) shows x0 A þ B and A ¼ 2 Therefore, Equation (3.89) gives CðtÞ ¼ x0 cos pt

ð3:93Þ

In the presence of rotation, from Equation (3.88) it follows that xðtÞ ¼ x0 cos pt cosðot sin lÞ and yðtÞ ¼ x0 cos pt sinðot sin lÞ

ð3:94Þ

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This means that the path of swinging rotates together with the earth and each point of the path moves along a circle. The period of these oscillations depends on the latitude. For instance, at the poles it is the period of the earth and then increases in approaching the equator, where this motion is absent and the path remains in the same plane. In the case when the pendulum path is an ellipse the effect of the earth’s rotation is to cause the ellipse to rotate with an angular velocity equal to ðo sin lÞ. As we know, at the poles it is approximately 7:3 105 s1 . In accordance with Equation (3.94), for an observer located at the z-axis above the earth the rotation is clockwise in the northern hemisphere, ðl40Þ, and counterclockwise in the southern hemisphere, ðlo0Þ. For example, at the North Pole the direction of the pendulum rotation and the earth are opposite to each other. During swinging the pendulum moves from the point a to the opposite point of the path b, which is shifted at some small distance because of the earth’s rotation, Fig. 3.5c. Suppose that the radius of the circle is s0 , then the displacement bd during a half period of swinging, T/2, is bd s0 o

T 2

Correspondingly, the distance ad is ðadÞ2 ¼ ð2s0 Þ2 ðbdÞ2 or "

bd ad ¼ 2s0 1 2s0

2 #1=2

"

# 1 bd 2 2s0 1 2 2s0

Thus ad o2 T 2 1 2s0 32 and the second term on the right hand side characterizes a correction factor due to the rotation of the earth, which is rather small since o2 0:5 108 s2 .

3.5. VERTICAL SPRING– MASS SYSTEM In previous two sections we described principles of measuring devices in which an elementary mass moves under the action of the gravitational ﬁeld. The pendulum and the free fall mass are examples of such methods. Next, we focus our attention on the gravimeters, which are called the static gravimeters, because measurements are performed when the elementary mass does not move. Correspondingly, there is a balance between the gravitational and counteracting forces and usually the latter is caused by elastic springs. It is obvious that before a state of equilibrium takes place the mass is involved in a motion, which can be either a translation or rotation. From this point of view as well as a historic one, it is conventional to distinguish

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two groups of static gravimeters, which are called vertical spring balance and lever spring balance. 3.5.1. Vertical spring balance and lever spring balance In this section we will describe the physical principles of measurements for gravimeters of the ﬁrst group. It is our pleasure to note that Prof. W. Torge gave very extensive and detailed description of both types of gravimeters that greatly helped us to prepare this and the next sections. 3.5.2. Vertical spring balance Consider a vertically suspended spring of length l 0 , shown in Fig. 3.6a, and suppose that at some instant a mass is attached to its lower end. Then, under the action of the gravitational force the mass starts to move and this causes a change of the spring length and the appearance of an elastic force. The motion of this mass will be studied a little later, but now notice that relatively quickly the mass stops. This happens when the weight and elastic force have equal magnitudes and opposite directions. In accordance with Hooke’s law for the spring the magnitude of an elastic force is F ¼ kðl l 0 Þ

ð3:95Þ

Here l is the spring length, l l 0 the extension of the spring, caused by the weight: P ¼ mg

ð3:96Þ

and k is the elastic parameter of the spring, which depends on the spring material, its construction and geometry. It is essential that k is inversely proportional to the spring length: k 1=l. From Hooke’s law it follows that when there is only a normal stress s and a deformation e is considered in the same direction as that of the force then: s ¼ Ee

ð3:97Þ

Here s is the normal stress equal to the ratio of the force F to the elementary surface dS, provided that this force has only the component normal to dS: s¼ (a)

l

(b)

l0

m

F dS

0

s

z

Fig. 3.6. (a) System: spring and mass, (b) illustration of Equation (3.103).

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The strain e is usually very small number and it can be represented as e¼

l l0 oo1 l0

ð3:98Þ

and in place of Equation (3.97), we have F¼

EdS ðl l 0 Þ l0

ð3:99Þ

Here E is Young modulus. Comparison with Equation (3.95) clearly shows that the parameter k, usually called spring stiffness, is inversely proportional to its length. Sometimes k is also called the elastic constant but it may easily cause confusion because of its dependence on length. By deﬁnition, Hooke’s law is valid when there is a linear relationship between the stress and the strain, Equation (3.97). For instance, if l 0 ¼ 0:1 m then an extension (l l 0 ) cannot usually exceed 1 mm. After this introduction let us write down the condition when all elements of the system: mass–spring are at the rest (equilibrium): mg ¼ kðl l 0 Þ

ð3:100Þ

k ðl l 0 Þ m

ð3:101Þ

or g¼

3.5.3. Mechanical sensitivity of the system Thus, measuring the spring extension we can evaluate the gravitational ﬁeld. However, the main goal of the static gravimeter is to determine the change of the ﬁeld as a function of the observation point. Suppose that g2 and g1 are ﬁeld magnitudes at two points. Then, in accordance with Equation (3.101), we have g2 ¼ mk ðl 2 l 0 Þ

g1 ¼ mk ðl 1 l 0 Þ

and, therefore, Dg ¼ g2 g1 ¼

k k ðl 2 l 1 Þ ¼ Dl m m

or m Dg ð3:102Þ k This relationship characterizes the mechanical sensitivity of the vertical spring balance, because it shows the change in displacement due to a change in the ﬁeld; and the ratio m/k is the parameter of this sensitivity. It is clear that with an increase of this ratio in principle, we are able to observe smaller changes of the ﬁeld because the difference Dl becomes larger. This dependence of m and k is almost obvious. For instance, with an increase of mass the gravitational force increases and a greater Dl ¼

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expansion of the spring is necessary in order to provide equilibrium. The same is true with a decrease of the stiffness, since in this case we need a greater extension to compensate the given weight. Later we will consider methods allowing one to increase the mechanical sensitivity, and one of them is based on a decrease of the parameter k of the elastic system. In order to see importance of an increase of sensitivity consider one numerical example. Suppose that the gravitational ﬁeld of the earth (g ¼ 9.8 m/s2) causes an extension of the spring (l l 0 ) equal to 101 m. As follows from Equation (3.101), the parameter of sensitivity is m 102 s2 k Correspondingly, a change of the ﬁeld of around 1 mGal ¼ 108 m=s2 produces an extremely small change of the spring length: Dl ¼ 1010 m ¼ 0:1 nm ¼ 1 A, that is, comparable with the diameter of a hydrogen atom. One can say that the history of development of the static gravimeters is a history of an amazing improvement in the sensitivity of these devices, applying different approaches, and the ability to measure displacements of a mass, which are comparable with atomic dimensions. In this section we will discuss several times the concept of mechanical sensitivity but now let us consider a slightly different subject but which characterizes dynamic features of this device. 3.5.4. Equation of free vibrations of a mass As was pointed out before we observe the equilibrium of the system, consisting of a mass connected with a vertical spring, it is in motion. In order to describe this movement we choose a system of coordinates with its origin at the point of an equilibrium, z ¼ 0, and its z-axis is directed downward, Fig. 3.6b. In accordance with Newton’s second law the displacement of the mass, s, obeys an equation d 2s ¼F ð3:103Þ dt2 Here s(t) is the distance of the mass from the origin and F the total force acting on the elementary mass. Note that at the point of an equilibrium, z ¼ 0, the elastic force of the spring is not equal to zero, and it compensates the weight. If the mass m is taken away from this point an additional elastic force arises and the resultant force F is a superposition of the following forces: 1. Weight P ¼ mg 2. The elastic force F eq ¼ kðl l 0 Þ which together with weight provides equilibrium. The sign ‘‘–’’ means that this force is directed upward, that is, opposite the z-axis. 3. The elastic force F e is due to an additional deformation caused by a motion of the spring, and in accordance with Hooke’s law: F e ¼ ks. Here s is the displacement of the mass from an origin, and the sign ‘‘–’’ indicates that the direction of this force and that of the movement are opposite to each other. In fact, when the mass moves down the spring is expanded and therefore an elastic force tends to move this mass upward. We observe the same in the case of spring compression. m

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During its motion the mass experiences a resistance, caused by different factors, in particular, by air. This force of air resistance opposes the motion and it is directly proportional to the velocity: ds F r ¼ a dt where a is the coefﬁcient of resistance. Thus, Equation (3.103) becomes

4.

d 2s ds þ a þ ks þ mg kðl l 0 Þ ¼ 0 dt2 dt Taking into account Equation (3.100), we obtain m

d 2s ds þ 2n þ o20 s ¼ 0 dt2 dt

ð3:104Þ

Here a k ð3:105Þ ; o20 ¼ m m and these are parameters of the spring–mass system. It is proper to notice that the mass of the spring is neglected with respect to m. We have obtained a homogeneous ordinary linear differential equation of the second order. To determine a solution we represent a function s(t) as 2n ¼

sðtÞ ¼ C expðr tÞ Substitution into Equation (3.104) gives the characteristic equation r2 þ 2nr þ o20 ¼ 0 which has two roots r1;2 ¼ n ðn2 o20 Þ1=2

ð3:106Þ

Correspondingly, a solution of Equation (3.104) is sðtÞ ¼ C 1 expðr1 tÞ þ C 2 expðr2 tÞ

ð3:107Þ

Here C 1 and C 2 are constants which are determined only from the initial conditions, since Equation (3.104) does not contain any information about them. Before we describe the behavior of the function s(t), let us notice the following: at the beginning the system is in equilibrium, since the external force, weight, is compensated by an elastic force caused by an extension of the spring. In other words, the resultant force is zero. Then, at the instant t ¼ 0, the mass is moved from its initial position, z ¼ 0. This may happen by different ways, for instance, by applying an impulsive force, acting during an inﬁnitely short time. Because of the mass displacement, the balance between the weight and an elastic force is distorted and we observe a motion under the action of internal forces only, namely the elastic and friction forces. Correspondingly, this motion is called free vibration and the word ‘‘free’’ emphasizes the fact that during the motion external forces are absent. We have considered the case when a mass is moved from equilibrium at some instant

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t ¼ 0 and then it vibrates due to internal forces. Certainly, the static gravimeter does not operate under such condition, but, as will be shown, free vibrations are very important feature of the motion for real conditions when there is an external force. 3.5.5. Limiting case, n ¼ 0 Now we study the behavior of the function s(t) for different parameters of the system and, ﬁrst of all, consider the limiting case when attenuation is absent, n ¼ 0. As follows from Equation (3.106) its roots are pure imaginary and are r1 ¼ io0

and

r2 ¼ io0

ð3:108Þ

Making use of Euler’s formula expð io0 tÞ ¼ cos o0 t i sin o0 t and substituting it into Equation (3.107), we obtain sðtÞ ¼ ðC 1 þ C 2 Þ cos o0 t þ iðC 1 C 2 Þ sin o0 t After introduction of the notations: C 1 þ C 2 ¼ A sin j

and

iðC 1 C 2 Þ ¼ A cos j

we obtain sðtÞ ¼ A sinðo0 t þ jÞ

ð3:109Þ

Here A is the amplitude, j the initial phase, and o0 the frequency of free vibrations. Thus, in the absence of attenuation free vibrations are sinusoidal functions and this result can be easily predicted since mass is subjected to the action of the elastic force only. In other words, the sum of the kinetic and potential energy of the system remains the same at all times and the mass performs a periodic motion with respect to the origin that is accompanied by periodic expansion and compression of the spring. As follows from Equation (3.105) the period of free vibrations is m 1=2 2p T0 ¼ ¼ 2p ð3:110Þ o0 k Thus, m T 20 ¼ ð3:111Þ k 4p2 and we have expressed the mechanical sensitivity of the vertical spring–mass system in terms of the period of free vibrations. Correspondingly, one can say that the sensitivity increases as the square of the period T 0 . 3.5.6. Three types of attenuation of free vibrations In reality, however, there is always attenuation ðna0Þ and free vibrations may have a completely different behavior. In this light it is appropriate to distinguish three cases.

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3.5.6.1. Weak attenuation, nox0 In accordance with Equation (3.106) roots are complex and ðn2 o20 Þ1=2 ¼ io0 As follows from Equations (3.106 and 3.107) the function s(t) can be represented in the form sðtÞ ¼ Aent sinðpt þ jÞ

ð3:112Þ

p ¼ ðo20 n2 Þ1=2

ð3:113Þ

Here

Thus, the function s(t) is a product of two functions; one of them is a decaying exponential, but the other is a sinusoidal function with a frequency p. For instance, if nooo0 free vibrations are close to a function described by a sinusoid slightly decaying with time, and their frequency is approximately o0 . 3.5.6.2. Critical attenuation, n ¼ x0 With an increase of the parameter n, the exponential decay becomes stronger but the frequency of free vibrations decreases. If n ¼ o0 vibrations are described by a function sðtÞ ¼ ðC 1 þ C 2 tÞ expðntÞ

ð3:114Þ

and periodical behavior disappears. Instead we may observe an increase of the displacement until it reaches a maximum and then gradually returns to equilibrium. 3.5.6.3. Strong attenuation In this case, as follows from Equation (3.106) both roots are negative and the function s(t) is a sum of two exponents: sðtÞ ¼ C 1 expðr1 tÞ þ C 2 expðr2 tÞ

ð3:115Þ

and its behavior is almost the same as in the previous case. 3.5.7. Forced vibrations Now we consider a more general and practical case where free vibrations still play a certain role. Suppose that the vertical spring with mass is in a state of equilibrium until the moment t ¼ 0, and then we apply some external force F e ðtÞ, which acts during the time interval: 0 t t . Correspondingly, Equation (3.103) can be written as m

d 2s ds ¼ F e a ks 2 dt dt

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or d 2s ds þ 2n þ o20 s ¼ f e ðtÞ dt2 dt

ð3:116Þ

where f e ðtÞ ¼

Fe m

We have obtained an inhomogeneous differential equation with constant coefﬁcients. As follows from the theory of linear equations, its solution is a sum sðtÞ ¼ s0 ðtÞ þ s1 ðtÞ

ð3:117Þ

Here s0 ðtÞ is a free vibration, that is, this function is a solution of a homogeneous equation, ðf e ¼ 0Þ, and s1 ðtÞ a partial solution of an inhomogeneous equation, which strongly depends on the external force, F e ðtÞ, and which exists only during the time when an external force is applied. Thus, within the ﬁrst interval, 0 t t , the displacement of a mass is a superposition of free and forced vibrations, but then we observe only free vibrations, t4t . Inasmuch as only the parameters of the system, that is, the roots of the characteristic equation, deﬁne the time dependence of free vibrations, they usually hinder the study of an external force. In order to reduce their effect, attenuation is used so that within the ﬁrst interval there is a time when we start to see only forced vibrations. In other words, unlike free vibrations, attenuation does not inﬂuence the decrease of forced vibrations with time. Now we consider one example that has a direct relation to measurements of the gravitational ﬁeld. Suppose that at some point, where the gravitational ﬁeld is g1 , the mass–spring system is at equilibrium, that is, the weight is compensated by the elastic force of spring. Then the device is moved to the point with a ﬁeld g2 , where g2 ¼ g1 þ Dg, and we begin to measure a displacement at the instant t ¼ 0. Correspondingly, Equation (3.116) becomes d2s ds þ 2n þ o20 s ¼ Dg dt2 dt

ð3:118Þ

Here F e ¼ mDg and s(t) is displacement of mass with respect to an equilibrium at the ﬁrst point. If we assume that this device starts to measure instantly, the function Dg is described by a step function, Fig. 3.7a. In order to ﬁnd the motion of the mass due to the step-function force, forced vibrations, one can apply different approaches, for instance, the Fourier transform. We will solve this problem differently and suppose that the right hand side of Equation (3.118) behaves as Dg ¼ 0

if to0

and DgðtÞ ¼ Dg½1 expðbtÞ

if t40

ð3:119Þ

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(a)

195

(b) eyepiece

micro

source

fe

D

Δg t

0

beam mass

Fig. 3.7. (a) Step-function force, (b) Hartley gravimeter.

where b is a positive number. In the limit when b tends to inﬁnity, we obtain a step function. Applying the trial and error method let us represent a partial solution in the form sðtÞ ¼ A þ B expðbtÞ Its substitution into Equation (3.118) gives ðb2 2nb þ o20 ÞB expðbtÞ þ Ao20 ¼ Dg½1 expðbtÞ Whence A¼

Dg and o20

B¼

Dg b 2nb þ o20 2

In the case of the step function we have B ¼ 0 but the coefﬁcient A remains the same and it is A¼

Dg DP ¼ k o20

ð3:120Þ

and this deﬁnes a static displacement of the mass, caused by a change of its weight between two points. Thus, the total displacement sðtÞ ¼

Dg þ C 1 expðr1 tÞ þ C 2 expðr2 tÞ o20

ð3:121Þ

and constants characterizing free vibrations can be found from the initial conditions. Since our goal, using damping, is to remove the inﬂuence of this part of the motion and provide equilibrium we will not calculate these constants. It is obvious, that one can represent the constant displacement as a difference m s ¼ s2 s1 ¼ Ds ¼ Dg, k

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where s1 and s2 are displacements of a mass at two points under an action of the ﬁeld g1 and g2 , respectively. It may be proper to notice that in an absence of free vibrations the step function force instantly moves the mass to the point of equilibrium. 3.5.8. Mechanical sensitivity and stability of vertical spring– mass system As was shown earlier, the coefﬁcient of proportionality between the change of the displacement and that of the ﬁeld is the mechanical sensitivity m T 20 ¼ k 4p2 From Equation (3.116), it follows also that at equilibrium o20 s ¼ g Performing a differentiation with respect to g we obtain @s m T 20 ¼ ¼ @g k 4p2

ð3:122Þ

Thus, the mechanical sensitivity is the rate of a change of the displacement, caused by a change of the gravitational ﬁeld, and in order to evaluate it in general we have to take a derivative. It may be proper to notice that in accordance with Equation (3.101) @s Dl l l 0 ¼ ¼ @g g g

ð3:123Þ

where Dl is a spring expansion due to the gravitational ﬁeld, and within the elastic range it has to be much smaller than the original length. Assuming that the parameters of the system are constants, (Equation (3.118) is linear one), we deal with a linear system. Besides, there is a linear relationship between the displacement and the ﬁeld change. In order to illustrate the vertical spring–mass system, consider some features of one of the simplest gravimeters, which was designed many years ago by Hartley, Fig. 3.7b. As we see, it consists of a beam, hinge, an elastic element (spring) with a weight, an adjusting spring with micrometer screw, light source, lenses, mirror, eyepiece, and scale. A change of the gravitational ﬁeld causes a change of the position of the mass, and with help of an adjusting spring the mass returns to its original position, corresponding to the ﬁeld at the reference point. The dial of the micrometer allows us to read this shift, which is proportional to the change of the gravitational ﬁeld between the observation and reference points. An arrangement of the optical path permits one to amplify a tiny change of the spring length, and this provides accuracy around 1 mGal. A mechanical device (micrometer) brings mass back after it reaches equilibrium. Later, electrical methods of compensation were introduced which held the mass at all times in the initial position. In both cases it is usually assumed that the elastic force is directly proportional to a displacement, that is, Hooke’s law is valid. Note that modern devices permit one to take into

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account the effect of non-linearity between stress and strain. At the same time during many years the main goal was to increase the mechanical sensitivity of the gravimeter in order to measure relatively small changes of the ﬁeld and only recently, due to outstanding improvement of methods for measuring small signals, this task has become less important. Now let us return to the system: vertical spring with mass. In accordance with Equation (3.122), with an increase of mass or decrease of stiffness the mechanical sensitivity becomes larger. However, in practice the variation of these two parameters is relatively small. For instance, an increase of mass rapidly causes a relatively large displacement, which does not obey Hooke’s law, if the original spring length is sufﬁciently small. The same effect is observed with a decrease of stiffness. Similar problem of an increase of mechanical sensitivity existed for the vertical seismographs and in order to improve this device L. La Coste developed, more than seventy years ago, the so-called zero-length spring. Before we describe the main features of this very important invention, it may be useful to discuss one more question. 3.5.9. Stability of the vertical spring– mass equilibrium The vertical spring and mass is an example of a stable system and by deﬁnition this means that an arbitrary small external force does not cause the mass to depart far from the position of equilibrium. Correspondingly, the mass vibrates at small distances from the position of equilibrium. Stability of this system directly follows from Equation (3.102) as long as the mechanical sensitivity has a ﬁnite value, and it holds for any position of the mass. First, suppose that at the initial moment a small impulse of force is applied, delta function, then small vibrations arise and the mass returns to its original position due to attenuation. If the external force is small and constant then the mass after small oscillations occupies a new position of equilibrium, which only differs slightly from the original one. In both cases the elastic force of the spring is directed toward the equilibrium and this provides stability. Later we will discuss this subject in some detail.

3.6. SPRING WITH AN INITIAL COMPRESSION AND HOOKE’S LAW Next we describe La Coste’s invention and demonstrate that it allows one to make a spring with much smaller stiffness. From the condition of equilibrium it follows that l l 0 1=2 T 0 ¼ 2p ð3:124Þ g and this clearly shows that with an increase of the difference Dl ¼ l l 0 the mechanical sensitivity becomes greater. Since for conventional springs this quantity is extremely small La Coste raised the following question. Is it possible to make such spring that in place of a difference we would have the total length of the spring l ?

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In such a case the period would drastically increase, as well as the mechanical sensitivity, and we obtain 1=2 l T 0 ¼ 2p ð3:125Þ g In accordance with Hooke’s law, l is the length of the spring when it is deformed. Comparing Equation (3.124) with (3.125), we may say that the new spring, when it is not stretched, should have zero length. Certainly, for an ordinary spring this is impossible, but formally everything happens as if l 0 ¼ 0. This is the reason why this spring is often called a zero-length spring, which easily and naturally causes confusion. There are several methods of making this type of the spring and for illustration consider some of them. The ﬁrst one simply requires bending of the spring wire tightly placed on a mandrel. The second approach includes a twist of wire being wound. Finally each turn of the spring at the beginning is placed from one side to the other, and if there is a space between them a reversal of turns takes place and they are tightly placed close to each other. As a result the new spring has a remarkable feature, namely, there is an elastic force F r , even before a weight is applied to the low end of the spring, Fig. 3.8a. When the spring is removed from the mandrel there is the same restoring force F r at its every cross section. It is proper to notice that this force provides a homogeneous deformation of the spring and it has a direction opposite to the weight. Now we mentally perform some experiments and derive Hooke’s law for such spring. First, suppose that a mass is attached to the lower end of the spring and a force F applied to the mass satisﬁes the inequality F oF r

ð3:126Þ

and it is directed downward. Therefore, the total force acting on each element of the spring becomes F Fr

(b)

(a)

F l0 Fr 1

l

2 0

l0

m Fig. 3.8. (a) Spring with force F r , (b) Total force as function of length.

l

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and the interaction between wire turns decreases. Of course, we do not see this change and think that the spring length remains the same, but in reality a very tiny expansion takes place, which makes the compression smaller. With an increase of the force F the deformation decreases and at the instant when F ¼ Fr it disappears. With further increase of the force we may start to see an expansion of the spring, since F 4F r , and the magnitude of the elastic force due to an expansion of the spring is k0 ðl l 0 Þ Here l 0 is the initial length of the spring with restoring force F r . Correspondingly, the total force applied to mass and directed downward is a sum of the force equal to the restoring one and a force caused by a spring expansion. Both of them have the same direction F ¼ F r þ k0 ðl l 0 Þ

ð3:127Þ

Here k0 is the stiffness of the spring with the restoring force and Equation (3.127) can be treated as Hooke’s law for such spring. Suppose that the mass performs vibrations near a point of equilibrium. Then the equation of motion changes slightly and we have d 2s ds 1 þ 2n þ ½F r þ k0 ðs s0 Þ ¼ g dt2 dt m and the condition of equilibrium when F ¼ mg has the form F r þ k0 ðl l 0 Þ ¼ mg

ð3:128Þ

Now assume that the spring is made in such a ‘‘clever’’ way that the restoring force is directly proportional to the original length l 0 and the coefﬁcient of proportionality is exactly equal to the elastic parameter k0 : F r ¼ k0 l 0

ð3:129Þ

Then, in place of Equation (3.127), we obtain F ¼ k0 l

ð3:130Þ

and the equation of motion becomes d 2s ds þ 2n þ o20 s ¼ g 2 dt dt where 1=2 k0 o0 ¼ m At the same time the condition of equilibrium gives k0 l ¼ mg

ð3:131Þ

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Thus, we demonstrated that if the restoring force obeys Equation (3.129) instead of the period

l l0 T 0 ¼ 2p g

1=2

we obtain T 0 ¼ 2p

1=2 l g

ð3:132Þ

and this means that the square of the period increases l=l l 0 times. Certainly, this is a strong increase of a mechanical sensitivity of the system and that was the purpose of L. La Coste’s invention. For illustration the behavior of the total elastic force applied to mass is shown in Fig. 3.8b. For comparison the elastic force (line 2) shows the dependence of the elastic force for an ordinary spring. In this case there is a minimal length l 0 , ðl 0 a0Þ, where the elastic force vanishes and this means that the spring is not deformed. In the presence of a restoring force a different behavior is observed. If the initial length of the spring is given then at all its cross sections the force has the same value. However, when we start to decrease the length l 0 the restoring force also becomes smaller and this linear dependence is shown as a dashed line. In this case the restoring force disappears if the length becomes zero, and it is hardly a strong argument to call a real spring with a ﬁnite length a spring with zero length. Let us note that there is a simple way to verify that a given spring has a ‘‘zero’’ length. Suppose that at equilibrium the spring length is l and a free motion of the mass around this point takes place with the period T 0 . If this value obeys Equation (3.132) it means that the spring has ‘‘zero’’ initial length. As was shown earlier, this type of spring increases the mechanical sensitivity l=l l 0 times and, taking into account the size of real gravimeters, this ratio may reach 20–30. Certainly, this was a great improvement of sensitivity and this spring found a broad application, specially, in gravimeters where a mass is involved in a rotation. It is useful to make one comment and with this purpose in mind return to Equation (3.127). Assume that the restoring force satisﬁes the inequality F r 4k0 l 0 Then, it can be written as F r ¼ k0 l 0 þ k0 l 0 Its substitution into Equation (3.127) gives F ¼ k0 ½l ðl 0 Þ

ð3:133Þ

and this illustrates why sometimes the very strange name ‘‘a spring with a negative length’’ is used. Next we consider systems in which a mass rotates, and start from the simplest case.

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3.7. TORSION SPRING– MASS SYSTEM The main features of this device are shown in Fig. 3.9a. The z-axis is directed along the ﬁeld (the vertical), and the lever with a small mass m rotates around a horizontal axes, passing through the point 0. The length of the lever is a. The torsion spring is installed on the axis, and its torsion constant is equal m. This is a coefﬁcient of proportionality between the moment t and the rotation angle a: t ¼ ma Also suppose that the spring has pre-tension moment: t0 ¼ ma0

ð3:134Þ

where a0 can be called the pre-tension angle, and it characterizes the deformation of the spring before the action of the gravitational force. As is seen from Fig. 3.9a, a rotation of the mass takes place due to two moments oriented along the horizontal axis 0 which have opposite directions. The ﬁrst one is due to the torsion of the spring and moves the mass counterclockwise: ts ¼ mða þ a0 Þ

ð3:135Þ

Here a is the angle between the vertical and lever. The second moment is caused by weight and its magnitude is equal to product of the lever length and the projection

(b)

(a)

(c)

τ

τ

τs

a α

α mg

0

τs

τg

τg

z

α

0

α

0

(d)

(e) 1

τ

α1

α2

0 τ τg

0

τs α

-1

-2 0

40

80 α°

120

160

Fig. 3.9. (a) Torsion mass–spring system, (b–d) function t(a), (e) resultant moment as a function of the angle.

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of the gravitational force on the direction perpendicular to the lever. Thus, we have ð3:136Þ tg ¼ mga sin a and it rotates the mass clockwise. Equilibrium takes place when the magnitudes of both moments are equal: mða þ a0 Þ ¼ mga sin a

ð3:137Þ

where m, a, m, a0 are parameters of the system. It is clear that equilibrium may take place for any angle a and its value depends on the gravitational ﬁeld g. Let us notice that the torsion moment depends linearly on the angle a, but there is a non-linear relation between this angle and the second moment. Examples of behavior of both moments as functions of the angle are shown in Figs. 3.9b–d. At angles where these curves intersect each other, the total moment t: t ¼ mga sin a mða þ a0 Þ ð3:138Þ is zero, and in such places we may observe equilibrium. As an illustration, consider the dependence of the resultant moment on the angle for functions tg and ts given in Fig. 3.9b. If we assume that mgaoma0 , then at small angles the torsion moment prevails and to0. Correspondingly, within the range 0oaoa1 , Fig. 3.9e, the mass moves counterclockwise. At the angle a ¼ a1 , the total moment is zero, and if mass is placed at this position it remains at rest. At greater values of the angle, the moment becomes positive and, therefore, the mass rotates clockwise. With an increase of a, the moment also increases until a maximum and then it becomes smaller. There is one more angle where the moment is zero, a ¼ a2 , and we observe equilibrium, but at greater angles the mass again tends to move counterclockwise. Now we focus our attention on angles where the total moment is zero. 3.7.1. Three types of points of equilibrium First, suppose that the position of the mass m is characterized by an angle a1 and it is at rest, but then is moved toward larger values of the angle. Consider the motion of the mass due to both moments, tg and ts . Inasmuch as the resultant moment is positive, the mass starts to move clockwise, that is, away from the point of equilibrium. The same happens if the mass is moved toward smaller values of angles. In fact, the resultant moment is negative and mass tends to move counterclockwise. Correspondingly, the distance from the initial point again increases. In other words, an arbitrary small displacement leads to motion of mass away from this point. By deﬁnition, the point with a ¼ a1 is called unstable. Notice that at this point the derivative of the moment with respect to angle is positive. Therefore, this point is deﬁned by two conditions: @t ð3:139Þ 40 t¼0 @a From Equation (3.138), it follows @t ¼ mga cos a m @a

ð3:140Þ

Principles of Measurements of the Gravitational Field

This allows us to write conditions (3.139) as t ¼ 0 mga cos a4m

203

ð3:141Þ

Next, we assume that the position of the mass is characterized by the angle a2 . This is the second point of equilibrium, since the resultant moment is zero. As before, let us slightly displace the mass from equilibrium and study its motion due to weight and elastic force of a spring. If its initial position is characterized by larger values of the angle, the resultant moment is negative and the mass moves counterclockwise, that is, closer to the point of equilibrium. The same tendency is observed when the mass is moved in the opposite direction, because in such case the moment is positive. This means that small initial displacements do not take mass far away from the original position and this point is called a stable one. In other words, we have stable equilibrium. As seen from Fig. 3.9e, the conditions of stable equilibrium are @t ð3:142Þ t¼0 o0 @a or t¼0

mga cos aom

ð3:143Þ

We have considered two types of points of equilibrium where the derivative @t=@a is either positive or negative. There is one more case and which occurs if @t ð3:144Þ t¼0 ¼0 @a In order to study this case as well two others, let us consider free vibrations of the mass. 3.7.2. Equation of mass rotation As before, suppose that at the beginning the spring extension compensates a weight and the lever spring system is in equilibrium. Then at some instant the mass is moved away from this position and an external force is removed. After this moment we start to observe a rotation of mass due to the resultant moment. As follows from Newton’s second law, a motion of elementary mass is described by the equation @2 a ð3:145Þ ¼ t or I a€ t ¼ 0 @t2 Here I ¼ ma2 is moment of inertia, a€ angular acceleration, and t the resultant moment. Note that we have neglected attenuation but in reality, of course, it is always present. This equation characterizes a motion for any angle a, but we consider only the vicinity of points of equilibrium. For this reason, the resultant moment in the linear approximation can be represented as I

tðaÞ ¼ tða Þ þ

@t ða a Þ þ @a

ð3:146Þ

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In essence, we have expanded the moment in a power series in the vicinity of the point of equilibrium, a ¼ a and assumed that within this range the derivative @t=@a is constant. Taking into account the fact that at points of equilibrium tða Þ ¼ 0 and introducing the notation b ¼ a a Equation (3.145) can be written as @t b ¼ 0 or b€ n2 b ¼ 0 ð3:147Þ @b Here b is angular displacement of mass with respect to the point of equilibrium, I b€

@2 b @2 a b€ ¼ 2 ¼ 2 @t @t

;

n2 ¼

1 @t 1 @t ¼ : I @b I @a

Finally, @t @t ¼ @b @a is the rate of change of the moment as a function of the angle at the point of equilibrium. Now we demonstrate that Equation (3.147) allows us to see again the difference between three types of points of equilibrium. 3.7.2.1. Unstable equilibrium, @s=@b40 In this case the coefﬁcient n2 is positive and a solution of Equation (3.147) is bðtÞ ¼ A expðntÞ þ B expðntÞ Thus, with an increase of time, the mass m moves away from the point of equilibrium. 3.7.2.2. Stable equilibrium, @s=@b40 This means that n2 o0 and Equation (3.147) can be written as b€ þ o20 b ¼ 0 Here 1=2 1 @t I @b and the angular displacement is described by a sinusoidal function: o0 ¼

ð3:148Þ

bðtÞ ¼ A expðiotÞ þ B expðiotÞ ð3:149Þ Therefore, the mass vibrates around the point of equilibrium and in the presence of attenuation it returns to its original position b ¼ 0. It is clear that the period of free vibrations is ( )1=2 I T 0 ¼ 2p ð3:150Þ t_ b

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3.7.2.3. Indifferent equilibrium, @s=@b ¼ 0 As follows from Equation (3.140) this means that mga cos a ¼ m Correspondingly, in place of Equation (3.147) we have b€ ¼ 0 and an integration gives bðtÞ ¼ At þ B

ð3:151Þ

Assuming that at the initial instant the angular velocity @b=@t ¼ 0, we conclude that the mass m, placed at any point around the point of equilibrium, remains at rest. Of course, it is only an approximation, because we preserved in the power series, (Equation (3.146)), only the linear term and discarded terms of higher orders. Formally, this case is characterized by inﬁnitely large period of free vibrations T0 ! 1

ð3:152Þ

but from a physical point of view this means that in the vicinity of the point where t_ a ¼ 0 any point is the point of equilibrium. However, in reality, there is always some initial velocity and the mass does not remain at rest but moves away from the point of equilibrium.

3.7.3. Mechanical sensitivity of the torsion balance Now we determine the mechanical sensitivity of this system at the point of equilibrium, that is, the relationship between a change of the ﬁeld and that of the angle. Performing a differentiation of Equation (3.137) with respect to g, we obtain m

da da ¼ ma sin a þ mga cos a dg dg

Thus, da ma sin a ¼ dg m mga cos a

ð3:153Þ

and the dimension of sensitivity is rad/m/s2. In the case of the vertical spring–mass system the sensitivity depends on only two parameters, k and m, but here there is also an inﬂuence of the angle a. Making use of the condition of equilibrium, Equation (3.137) in place of Equation (3.153), we obtain da a þ a0 ða þ a0 Þ tan a ¼ ¼ dg g½1 ða þ a0 Þ cot a g½tan a ða þ a0 Þ

ð3:154Þ

Suppose that the parameters of the system are chosen in such a way that the torsion moment compensates the moment due to the gravitational ﬁeld g when a ¼ p=2,

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and correspondingly measurements of small variations of this ﬁeld take place around this position. For mechanical sensitivity we have da p=2 þ a0 ¼ dg g

ð3:155Þ

As follows from Equation (3.154) a change of the angle a may strongly increase a mechanical sensitivity of the system and this is an important advantage of the lever spring system. In this light, suppose that tan ac ¼ ac þ a0

ð3:156Þ

This means that the mechanical sensitivity tends to inﬁnity, that is, small changes of the ﬁeld result in unlimited change of the angle, and a system becomes unstable. It is clear that within some interval of angles close to the critical one, ac , the mechanical sensitivity can be very large but the system approaches to a state of unstable equilibrium. A choice of parameters of the system which satisﬁes this condition is called astatization and this procedure allows one to greatly increase the mechanical sensitivity of the elastic system of a gravimeter. At the same time, this system has to be stable, that is, the mass has to return to its original position after the action of a small force, and we see that the mechanical sensitivity and stability of a system are closely related to each other near equilibrium.

3.8. LEVER SPRING– MASS SYSTEM Next consider the condition of equilibrium of the lever spring–mass, shown in Fig. 3.10, which has common features with La Coste–Romberg gravimeter. We use following notations: 0A ¼ d is the distance between the horizontal axis of rotation 0 and the upper end of the spring where it is suspended, AC ¼ l is the spring length, 0B ¼ h is perpendicular from point 0 to the spring. 0C ¼ b and 0D ¼ a, d is the angle between the z-axis and line 0A, a is the angle formed by line 0A and the lever

A τs B d D

h δ

α

b

a C mg

0

z Fig. 3.10. Principle La Coste–Romberg gravimeter.

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207

0D. As before, there are two moments, caused by the weight and the spring, and they are tg ¼ mga sinða þ dÞ and ts ¼ kðl l 0 Þh

ð3:157Þ

It is convenient to express this moment in terms of the angle a and with this purpose in mind consider the triangle A0C. In accordance with the sine theorem of trigonometry sin a sin A ¼ l b At the same time from the triangle A0B we have sin A ¼

h d

Thus, sin a h ¼ l bd and bd sin a l Therefore, the total moment of the system is ts ¼ kðl l 0 Þ

ð3:158Þ

bd sin a ð3:159Þ l Here m, g, k, d, a, b, d, l0 are parameters of the system. The condition of equilibrium is obvious: tðaÞ ¼ mga sinða þ dÞ kðl l 0 Þ

bd sin a ð3:160Þ l and for each value of the gravitational ﬁeld we may expect two positions of the mass where either stable or unstable equilibrium is observed. Considered in some detail in the following case. mga sinða þ dÞ ¼ kðl l 0 Þ

3.8.1. Zero lever spring system If l 0 ¼ 0, the inﬂuence of length l disappears and Equation (3.160) is simpliﬁed to mga sinða þ dÞ ¼ kbd sin a

ð3:161Þ

Taking the derivative of t, (e.g., Equation (3.159)), with respect a and letting l 0 ¼ 0, we obtain @t ¼ mga cosða þ dÞ kbd cos a @a

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Then, substitution of Equation (3.161) gives at points of equilibrium @t sin d ¼ kbd @a sinða þ dÞ

ð3:162Þ

If ða þ dÞop, the equilibrium is stable, but in the opposite case, ða þ dÞ4p, it is unstable. In the case when the point of suspension A is located on the z-axis, d ¼ 0, indifferent equilibrium is observed. Next we ﬁnd an expression for the mechanical sensitivity. Differentiation of Equation (3.161) with respect to g gives ma sinða þ dÞ þ mag cosða þ dÞ

da da ¼ kbd cos a dg dg

Thus, da ma sinða þ dÞ ¼ dg kbd cos a mga cosða þ dÞ Taking into account the condition of equilibrium, Equation (3.161), the last equality is simpliﬁed da ma sinða þ dÞ ¼ sin a dg mga sinða þ dÞ cos a mga cosða þ dÞ sin a or da sinða þ dÞ sin a ¼ dg g sin d

ð3:163Þ

It is clear that by decreasing the angle d it is possible to perform astatization of the system and increase the mechanical sensitivity. The same approach is applied in the general case when we deal with a non-zero-length spring or a system of them, and Hooke’s law has a conventional form: F ¼ kðl l 0 Þ. 3.8.2. About measurements in the presence of a high-frequency noise Finally, consider one more important feature of the vertical or lever spring–mass systems. This is related to application of gravimeter in marine geophysics, where there are at least three additional sources of noise: changes of orientation of the gravimeter, the other is Coriolis force and the inﬂuence of forces caused by waves. During a motion of a boat the gravitational ﬁeld and wave forces vary with time, and the magnitude of the latter can be many orders higher than that of the function g(t). Fortunately, they have completely different spectra and this fact allows us to greatly reduce this noise. For illustration suppose the vertical spring–mass system is subjected to an action of sinusoidal force with frequency o and, correspondingly, the equation of motion has the form: d2s ds þ 2n þ o20 s ¼ f cos ot 2 dt dt

ð3:164Þ

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209

In order to ﬁnd the displacement of the mass caused by this force, we introduce the function z(t) as zðtÞ ¼ sðtÞ þ iuðtÞ

or

sðtÞ ¼ Re zðtÞ

ð3:165Þ

and in place of Equation (3.164) consider the following equation: d 2z dz þ 2n þ o20 z ¼ f expðiotÞ 2 dt dt

ð3:166Þ

We will look for its partial solution in the form zðtÞ ¼ A expðiotÞ

ð3:167Þ

where A is unknown. Substitution of the function z(t) into Equation (3.166) gives ðo2 2ion þ o20 ÞA ¼ f or A¼

f o20

o2

2ino

ð3:168Þ

Therefore, zðtÞ ¼

f expðiotÞ o20 o2 2ino

and, in accordance with Equation (3.165), forced vibrations of the mass caused by waves are also sinusoidal oscillations: expðiotÞ o20 o2 2ino

sðtÞ ¼ f Re

Taking the real part of this expression, we obtain sðtÞ ¼

f ½ðo20

o 2 Þ2

þ 4n2 o2 1=2

cosðot jÞ

ð3:169Þ

where jðoÞ ¼ tan1

2no o2

o20

As follows from Equation (3.169), if the frequency of free vibrations obeys the condition: o0 ooo, then the displacement s(t) becomes very small: sðtÞ / 1=o2 . Taking into account this fact, it is possible to greatly reduce the inﬂuence of waves. Besides, a relatively strong attenuation allows one to decrease an effect of free vibrations caused by this force.

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3.9. MEASUREMENT OF SECOND DERIVATIVES OF THE POTENTIAL OF GRAVITATIONAL FIELD Until now we described methods of measuring of the gravitational ﬁeld, which is the ﬁrst derivative of the potential along the vertical (plumb line). Next, we consider devices that allow one to measure the change of the ﬁeld, that is, second derivatives of the potential. The ﬁrst such instrument (gravitational variometer) was invented € os € around 1880 in and developed by the Hungarian geophysicist Roland von Eotv order to evaluate the departure of the earth’s surface from spherical one. This is related to the fact that the second derivatives of the potential characterize a curvature of equipotential surfaces. Later several types of these devices were developed € os € and were widely used in exploration geophysics. The essential element of the Eotv torsion balance is a bar with a pair of masses at its ends, so that its center of mass is suspended from a thin elastic thread. This bar with masses can rotate around the thread. If such a bar is moved from a position of equilibrium, it starts to vibrate with a relatively large period; this motion occurs due to the gravitational forces acting on masses and the elastic force of the thread. Because of attenuation, the motion amplitude slowly decreases and ﬁnally the bar stops. This position of equilibrium essentially depends on the change of the ﬁeld g between masses. For instance, if the device is placed into a homogeneous gravitational ﬁeld, where vector lines of the ﬁeld are parallel to each other, this ﬁeld does not produce a rotation and € os € torsion balance measures the a torsion force is zero. This means that the Eotv deviation of the ﬁeld from a homogeneous one. 3.9.1. Resultant force and moment In order to derive the equation of rotation as well as the condition of equilibrium let us represent the sum of two gravitational forces as a superposition of the resultant force and the moment. Suppose that one gravitational force is F1 , Fig. 3.11a. Now we imagine that forces F1 and F1 are also applied to the center of mass. It is obvious that the action of all three forces is the same as one force acting on the mass m1 , and they can be represented as a combination of the force F1 applied at the center of mass and the moment s1 , formed by the pair of forces: F1 and F1 acting at different points. In the same manner, we replace the gravitational force F2 moving mass m2 by the force F2 at the center of mass and the torque s2 . Inasmuch as both forces are applied at the same point (the center of mass) we can say that the action of two gravitational forces is equivalent to the resultant force F and the torque s: F ¼ F1 þ F2

and

s ¼ s1 þ s2

ð3:170Þ

The resultant force F is compensated by the reaction of the thread and, correspondingly, it does not inﬂuence the motion of the mass. In general, the total torque s has arbitrary direction, but we will be interested in only its vertical component, that causes a rotation in the horizontal plane. At the same time special measures are taken to remove an inﬂuence of motion in the vertical plane.

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Principles of Measurements of the Gravitational Field (b)

(a)

z

z

F2

F2 F1

F1 m

y y 0

−F1

−F2

x

m

ξ η (d)

(c)

l 0

m

m

m

0

h

ζ

m

Fig. 3.11. (a) Superposition of forces applied at different points, (b) system of coordinates with origin at the center of mass, (c) the ﬁrst type of gradiometer, (d) the type L of the gradiometer.

3.9.2. Equation of equilibrium and motion First, we introduce a Cartesian system of coordinates with an origin 0 at the center of mass, Fig. 3.11b. The x and y axes are directed to north and east, respectively, and the z-axis is along the vertical. By deﬁnition, the moment of the force F(x, y, z) with components Fx, Fy, and Fz with respect to point 0 is i j k y z and r ¼ xi þ yj þ zk s ¼ rxF ¼ x Fx Fy Fz is the radius vector of the point where force F is applied. Therefore, the vertical component of the moment of the force, acting on elementary mass dm is dtg ¼ xF y yF x where F y ¼ gy dm

and

F x ¼ gx dm

are components of the force along the y and x axes. The total component of the moment acting on all masses and caused by gravitational ﬁeld is obtained by integration over all masses Z tg ¼ ðxgy ygx Þdm ð3:171Þ

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In the case when masses can be treated as elementary and it is possible to neglect the bar’s mass, the volume integral disappears and we have tg ¼ m

2 X

½xn gy ðxn ; yn ; zn Þ yn gx ðxn ; yn ; zn Þ

n¼1

At equilibrium the moment tg is balanced by the moment of the elastic force of the thread ts ¼ mða a0 Þ

ð3:172Þ

where m is the elastic constant of the thread and a0 the initial torsion angle. Correspondingly, the equation of equilibrium is Z ðxgy ygx Þdm ¼ mða a0 Þ ð3:173Þ and the measured angle a gives us an information about a distribution of the gravitational ﬁeld. As in the case of a gravimeter, in the vicinity of equilibrium the motion of the bar with masses is deﬁned by the total moment: Z t ¼ ðxgy ygx Þdm mða a0 Þ ð3:174Þ and in accordance with Newton’s second law the equation of motion is I a€ þ ka_ þ mða a0 Þ ¼ tg

ð3:175Þ

Here k is the coefﬁcient of resistance and I the moment of inertia of the system. Comparison with the gravimeter clearly shows that the expression for the moment tg in the gradiometer is much more complicated and, in particular, it contains values of the gravitational ﬁeld at all points of the bar. Now we perform a great simpliﬁcation of this expression and demonstrate that it is deﬁned by the behavior of the ﬁeld at the center of the bar and the parameters of this linear system. Inasmuch as the bar length is relatively small it is natural to assume that the horizontal components of the ﬁeld g change linearly within the bar. Then, carrying out an expansion of the functions gx and gy in a power series and discarding all terms except the ﬁrst two terms, we obtain for both components @gx @gx @gx gx ðx; y; zÞ ¼ gx ð0Þ þ xþ yþ z þ @x 0 @y 0 @z 0 and gy ðx; y; zÞ ¼ gy ð0Þ þ

@gy @gy @gy xþ yþ z þ @x 0 @y 0 @z 0

ð3:176Þ

The index ‘‘0’’ means that derivatives are taken at the origin where x ¼ y ¼ z ¼ 0. Inasmuch as the thread is oriented along the vertical (plumb line), the gravitational ﬁeld at the origin has only a vertical component, that is, gx ð0Þ ¼ gy ð0Þ ¼ 0

ð3:177Þ

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213

Introducing notations for the second derivatives of the potential U at the origin: @gy @gy @gy @gx @gx @gx ¼ U xx ; ¼ U xy ; ¼ U xz ; ¼ U yx ¼ U xy ; ¼ U yy ; ¼ U yz @x @y @z @x @y @z in place of Equation (3.176) we have gx ðx; y; zÞ ¼ U xx x þ U xy y þ U xz z and gy ðx; y; zÞ ¼ U yx x þ U yy y þ U yz z

ð3:178Þ

Substitution of the latter into Equation (3.173) gives the equation of equilibrium in terms of the second derivatives of the potential at the origin of coordinates: Z Z Z Z U D xydm þ U xy ðx2 y2 Þdm þ U yz xzdm U xz yzdm ¼ mða a0 Þ ð3:179Þ where U D ¼ U yy U xx

ð3:180Þ

We have obtained a relationship between four second derivatives of the potential and the measured angle a, where the integrals on the right hand side of Equation (3.169), as well as m and a0 , are parameters of the device. 3.9.3. Second derivatives of the gravitational potential It is clear that there are six derivatives of the second order and they are @2 U ; @x2

U yy ¼

@2 U ; @y2

U zz ¼

@2 U @z2

@2 U ; @x@y

U xz ¼

@2 U ; @x@z

U yz ¼

@2 U @y@z

U xx ¼ and U xy ¼

There are different types of the gravitational variometers, for instance, one of them measures four derivatives: U xz ; U yz ; U xy and the difference U D ¼ U yy U xx while the other measures only U xz and U yz : Inasmuch as the potential satisﬁes Poisson’s equation DU ¼ U xx þ U yy þ U zz ¼ 2o2 the derivative U zz can be calculated if two others are known. The second derivatives of the potential are measured in s2 and the unit, 1 109 s2 is called Eo¨tvo¨s.

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3.9.4. Equation of equilibrium in the system of coordinates x; Z; z It is convenient to introduce a system of coordinates which moves together with the lever, and this is shown in Fig. 3.11b. The formulas which relate the old and new coordinates are x ¼ x cos y Z sin y;

y ¼ x sin y þ Z cos y;

and

z¼z

Substitution of Equation (3.181) into Equation (2.179) gives R 2 R R 2 1 2 2 2 U D ½sin 2y ðx Z Þdm þ 2 cos 2y xZdm þ U xy ½cos 2y ðx Z Þdm R R R R 2 sin 2y xZdm þ U yz ½cos y xzdm sin y Zzdm U xz ½sin y xzdm R þ cos y Zzdm ¼ mða a0 Þ

ð3:181Þ

ð3:182Þ

In order to simplify this equation of equilibrium we take into account the following. First of all, the lever is usually made from a thin and light tube, with masses at its ends, and they can be treated as elementary masses. This means that the distance along axis x is much larger than the dimensions of masses: x44Z. This allows us to write: Z Z ðx2 Z2 Þdm ðx2 þ Z2 Þdm ¼ I ð3:183Þ where I is the moment of inertia of the lever with respect to the vertical axis. As in the case of a gravimeter the moment of inertia can be determined by measuring the period of free vibrations, Equation (3.175). Since the plane x0z is a plane of symmetry, there are always two equal masses located at distances Z and Z with respect to this plane. Thus, the integrals in Equation (3.182), containing the ﬁrst power in Z, are equal to zero and we have: Z 1 IU D sin 2y þ IU xy cos 2y þ U yz cos y xzdm 2 Z U xz sin y

xzdm ¼ mða a0 Þ

ð3:184Þ

As we know, there is always a level surface which passes through any point of the space, including the origin of coordinates. Correspondingly, U D and U xy characterize the curvature of this surface at the point 0. At the same time, U xz and U yz describe the rate of change of the ﬁeld gð0Þ ¼ gz ð0Þ along the x and y axes. I and t are known parameters, since their values are found in the laboratory, as well as the integral at the left hand side of Equation (3.184). The angle a is measured, but a0 is unknown. Finally, the azimuth of the lever y is known with a sufﬁcient accuracy at each observation point. In general, Equation (3.184) contains ﬁve unknowns and in order to ﬁnd them we have to perform measurements for ﬁve values of azimuth: y1 ; y2 ; y3 ; y4 , and y5 . This gives a linear system of ﬁve equations with ﬁve unknowns. For illustration consider two systems.

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215

3.9.4.1. Case one Suppose the lever is suspended horizontally and, correspondingly, the masses are located symmetrically with respect to the plane Z0z, (Fig. 3.11c). This means that the integrals in Equation (3.184) vanish and we obtain 1 IU D sin 2y þ IU xy cos 2y ¼ mða a0 Þ ð3:185Þ 2 Taking into account the fact that the masses are situated in the same horizontal plane, it is natural that derivatives of the potential in the vertical direction are not measured. 3.9.4.2. Case two In this example we deal with gradiometer of type L, shown in Fig. 3.11d. Let us assume that the masses at the lever ends are elementary but the mass of the bar connecting them is negligible. Then, it is simple to calculate the integral in Equation (3.184); it is Z xzdm ¼ lhm and equation of equilibrium becomes 1 IU D sin 2y þ IU xy cos 2y þ U yz lhm cos y U xz lhm sin y ¼ mða a0 Þ ð3:186Þ 2 Here l is the distance between masses and h the z-coordinate of one of masses. Note that as in the case of static gravimeters the mechanical sensitivity is equal to the rate of change of the measured angle as a function of a parameter (the second derivative), which is evaluated.

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Chapter 4 Uniqueness and the Solution of the Inverse Problem in Gravity 4.1. CONCEPT OF UNIQUENESS AND THE SOLUTION OF THE INVERSE PROBLEM In the previous chapters our attention was paid mainly to the study of the ﬁgure of the earth and with this purpose in mind we represented the total ﬁeld as a sum of the normal and secondary ﬁelds. In this chapter, we will discuss a completely different application of the gravity method, related to exploration geophysics, in which the gravitational ﬁeld is measured in order to study lateral changes of the density near the earth’s surface. By analogy, we also represent the gravitational ﬁeld as a superposition of two ﬁelds: G þ ga Here G can be also called the normal ﬁeld, but in general it differs from the normal ﬁeld of the earth and is more often called the ‘‘regional ﬁeld’’. It changes from place to place and allows one to emphasize the secondary ﬁeld ga and its relation to the distribution of density in given area of investigation. Notice that the gravitational ﬁeld G includes the effect of the earth’s rotation, centrifugal force, but ga is a ﬁeld of attraction and is caused by masses only. Correspondingly, the latter depends on several factors, such as the elevation, topography, position of the observation point, lateral changes of the earth’s density, and so forth. Suppose that we measure this ﬁeld, for instance, on the earth’s surface and our goal is to determine a distribution of masses, which causes this ﬁeld. In other words, we have to solve the inverse problem or perform an interpretation of gravitational data, measured by a gravimeter. Certainly, this is the most important element of any geophysical method. To outline this subject we begin with the simplest model and gradually approach more complicated ones. First, suppose that measurements of the gravitational ﬁeld are performed on the surface of a horizontally layered medium as is shown in Fig. 4.1a. As we know, the ﬁeld has only a vertical component and depends on the density and thickness of each layer. Since at each point of the horizontal plane of observation we obtain the same value of the ﬁeld, we conclude that gravimetry is not able to perform soundings, that is to determine the parameters of each layer. In fact, at

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(b)

(a)

δ1

h1

δ2

h2

δ3

h3

gz x

0 δ0

δ

δ4

z (d)

(c)

h1 h2 δ0 = 0

δ

Δδ

Fig. 4.1. (a) Horizontally layered medium, (b) ﬁeld in the presence of a conﬁned body, (c) secondary ﬁeld, (d) prism with density d.

each point the ﬁeld is deﬁned by the same set of parameters in exactly the same manner. In other words, at each observation point we obtain an identical linear equation with respect to the unknowns. Thus, in order to obtain information about the density beneath the earth’s surface we have to have lateral changes. The next example is shown in Fig. 4.1b. Here d0 and d are the densities of the surrounding medium and some body, respectively. We assume that the density d0 is in general a function of the coordinate z, and in particular, the medium surrounding the body can be a horizontally layered one. As was already demonstrated in the absence of the body, the gravitational ﬁeld does not change along a proﬁle and, correspondingly, the anomaly (the secondary ﬁeld) vanishes. However, in the presence of a body a secondary ﬁeld arises and its magnitude becomes greater when the density difference rd ¼ dd0 increases. In fact, we have Z ga ðpÞ ¼ k

dðqÞ Lqp dV ¼ k 3 V Lqp

Z

d0 ðqÞ Lqp dV k 3 V 0 Lqp

Z

d Lgp dV 3 L V b qp

or Z

Z d0 ðqÞ d0 ðqÞ L dV k Lqp dV qp 3 3 L V0 V b Lqp qp Z Z d0 ðqÞ dðqÞ þk Lqp dV k Lqp dV 3 3 V b Lqp V b Lqp

ga ðpÞ ¼ k

Uniqueness and the Solution of the Inverse Problem in Gravity

Z

d0 ðqÞ ga ðpÞ ¼ k Lqp dV k 3 V Lqp

Z

DdðqÞ Lqp dV 3 V b Lqp

219 ð4:1Þ

Here V ¼ V0+Vb is the volume of the whole space, but V0 and Vb are volumes of the surrounding medium and a body, respectively. Also DdðqÞ ¼ dðqÞ d0 ðqÞ is the difference of densities. The ﬁrst integral on the right hand side of Equation (4.1) is independent of the point of observation and it plays the role of the background for the secondary ﬁeld. This simple analysis shows that in this case we can present the original model of the medium as a combination of much simpler models, namely, 1. A half space with density d0(z) that creates a background for a secondary ﬁeld. 2. A body with density Dd ¼ dd0(z) surrounded by free space, which generates the secondary ﬁeld. Undoubtedly, this model is simpler than the original one, and in calculating the secondary ﬁeld by Newton’s law, it allows us to perform integration only within the body volume. Two obvious conclusions follow: a. As was pointed out above the ﬁeld does not change on the plane surface of a horizontally layered medium; that is, the information about the medium due to measurements at any point along the proﬁle is the same. However, the ﬁeld depends on many parameters of this model, such as the density and thickness of layers, and therefore their determination from the gravity ﬁeld is impossible. In other words, as was pointed out, this geophysical method does not permit carrying out soundings. b. To apply the gravitational method there must be lateral changes of rock density. Now we are ready to discuss some aspects of interpretation of gravitational data and at the beginning suppose that the ﬁeld of attraction is caused by only some conﬁned body, Fig. 4.1c, and from measurements along a proﬁle or a system of proﬁles one component of the attraction ﬁeld, for example, the vertical one, is known. Then, the main purpose of interpretation is to determine the location, shape, dimensions, and density of the subsurface body. This task is often called the inverse problem of gravitational ﬁeld theory, since it is necessary to ﬁnd the distribution of masses when the ﬁeld caused by them is known along some proﬁle or in some area. It is essential that the ﬁeld is not known in the volume occupied by masses, since measurements are almost always performed at some distance from them, and this is the main reason why interpretation becomes a rather complicated problem. First let us analyze the measured ﬁeld. In accordance with Newton’s law the ﬁeld can be represented at every observation point as a sum of ﬁelds caused by elementary masses of the body, and their contributions depend on the size and location of these volumes with respect to the observation point. In particular, those masses within a body located relatively far away from the observation point only slightly affect the ﬁeld magnitude. On the contrary, masses situated closer to the point of observation have a stronger inﬂuence. Strictly speaking, at every

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observation point the ﬁeld is subjected to the inﬂuence of all parameters of the body, although to different extent, but each element of a body, regardless of its position, makes a contribution to the measured ﬁeld. Their relative effect varies from point to point since they have different positions with respect to body. In other words, the inﬂuence of different parts of a body changes with the position of the point of observation. Thus, in principle, all information about the density and geometry of a body is contained in the measured ﬁeld. Taking into account this simple but fundamental fact let us formulate the main steps of interpretation. 1. First, we will make some assumptions about the distribution of masses, and correspondingly ascribe values to parameters of the body that characterize its geometry and density contrast. Such a step is usually called the ﬁrst guess or the ﬁrst approximation. It is mainly based on speciﬁc geological information and data obtained from other geophysical methods, and they approximately deﬁne the number of parameters and their numerical values. For example, if gravimetry is carried out for detecting salt domes, there is usually some information about the density of the surrounding rocks and salt domes, as well as their shape and location. Of course, the difference between the ﬁrst approximation and the factual values of the body parameters can vary signiﬁcantly depending on our knowledge of the geology. 2. The second step of interpretation consists of calculating synthetic values for the ﬁeld along a proﬁle, using the ﬁrst approximation and comparing the measured and calculated ﬁelds. A reasonable coincidence of these ﬁelds may suggest that the chosen parameters of the model are close to the real ones. 3. If there is a difference between the measured and calculated ﬁelds, all parameters of the ﬁrst approximation or some of them are changed in such a way that a better ﬁt to these ﬁelds is achieved. Thus, we obtain a second approximation of mass distribution. Of course, in those cases when even the new set of parameters does not provide a satisfactory match of these ﬁelds, this process of calculation has to be continued. As we see from this process, every step of the interpretation requires application of Newton’s law. Let us recall that this procedure, based on the use of Newton’s law, is often called the solution of the forward problem of the ﬁeld of attraction, and by deﬁnition we have Z dðqÞ ga ¼ k Lqp dV ð4:2Þ 3 V Lqp In summary we can say that the process of interpretation for the simplest case, shown in Fig. 4.1c, includes three main elements, namely, a. Formulation of the ﬁrst approximation for parameters of the body, the ‘‘ﬁrst guess’’. b. A solution of the forward problem making use of eq. 4.2. c. A change of parameters of the model to provide a better ﬁt between the measured and calculated ﬁelds. Later, we will add one more element, caused by the fact that the ﬁeld containing information about the parameters of a body (useful signal), is never known exactly. Inasmuch as in the process of gravity interpretation every step is reasonably well deﬁned, we may arrive at the impression that the solution of the inverse problem is

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221

straightforward and does not contain any complications. Unfortunately, in reality this is not true and if there are exceptions, then they have only a purely theoretical interest. In order to realize some of the difﬁculties of solution of the inverse problem it is useful to start from an unrealistic situation. 4.1.1. Uniqueness and its application Suppose that both the calculated and measured ﬁelds of attraction are known with inﬁnitely high accuracy. Of course, it is impossible to imagine that we can know the value of the ﬁeld without any error. This means that any digit after the decimal point, describing the ﬁeld, is known, regardless of how small its contribution. At the same time, any computer or gravimeter provides a value of the ﬁeld, for which we know only its ﬁrst digits. In spite of the fact that we are going to consider this unrealistic case, it is very useful to discuss this subject for understanding of principles of interpretation in the gravitational method. Thus, assume that we know the ﬁelds exactly and perform all steps of the interpretation described above. Suppose that sequentially repeating the solution of the forward problem at each step and comparing calculated and measured values of the ﬁeld, we obtain a set of parameters such that difference between these ﬁelds is inﬁnitely small. Then the following question arises. Does this mean that by providing an unrealistic ideal ﬁt between the measured and calculated ﬁelds, it is always possible to determine with an inﬁnitely small error the shape, dimensions, and density as well as the location of the masses that create the given ﬁeld? In general, the answer is negative and the solution of the inverse problem sometimes is not unique; that is, different distributions of masses may create exactly the same ﬁeld along a proﬁle or a system of proﬁles. In other words, in general, but not always, different bodies can generate a ﬁeld that provides an exact match to the measured ﬁeld. The simplest example of such non-uniqueness is the very well-known case in which the ﬁeld is caused by different spheres with the same mass and common center but different densities and radii. At the same time, if we look more carefully at this subject, then it becomes clear that the phenomenon of non-uniqueness is hardly obvious. In fact, Newton’s law of attraction tells us that a change of mass distribution, Equation (4.2), should result in a change of the ﬁeld. However, from a non-uniqueness it follows that different mass distributions can create outside them exactly the same ﬁeld, even if we know the ﬁelds without error. In other words, it is impossible to detect the difference between ﬁelds generated by such masses. It is difﬁcult to get rid of the impression that non-uniqueness is an amazing, unexpected fact which is more natural to treat as a paradox rather than as an obvious consequence of the behavior of the gravitational ﬁeld. In this light let us imagine for a moment that nonuniqueness is always present. Then it is clear that in such a case the interpretation of gravitational data would always be impossible. In fact, having determined parameters of a body that generates a given ﬁeld, we have to also assume that due to non-uniqueness there are always other distributions of masses which create exactly the same ﬁeld. Certainly, we can say that such an ambiguity would be a

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disaster for the application of the gravitational method. Fortunately, whole this subject of uniqueness is not relevant in practice, because we never know the ﬁelds exactly and for this reason it would be natural to avoid the discussion of this topic. At the same time it is worth clarifying some aspects of uniqueness in solving the inverse problem. First of all, as the theory of the potential shows, our treatment of non-uniqueness as a paradox is often correct. There are at least two classes of bodies for which a solution of the inverse problem is unique. One of them is a prism, Figs. 4.1d and 4.2a, M. Brodsky proved that if we know the ﬁeld of attraction caused by masses of some prism exactly then there is only one prism which generates this ﬁeld. Other prisms cause different ﬁelds. In other words, the solution of the inverse problem is unique. For illustration, consider a single prism, characterized by a density, dimensions of a cross-section, and distances from the top and bottom to the plane of observation. Regardless how small the prism size is with respect to the distances to the observation points a solution of inverse problem will give exact values of the parameters of the prism. To emphasize this fact we can imagine that the distance between the prism and the observation points is comparable with that from the earth to the moon. If we assume that a mass is located in a different prism the ﬁeld will be of course different too, and a solution of the inverse problem will give new parameters for the body. This means that the solution of the inverse problem for bodies which belong to the class of prisms is unique. The same result holds if instead of a single prism we have a system of prisms of different sizes and densities, Fig. 4.2a. It is interesting to notice that an arbitrary body can be usually represented as a system of different prisms. Another class of bodies for which the solution of the inverse problem is unique is so-called starshaped bodies, which are characterized by a more general shape. By deﬁnition, every ray drawn from any point within the body volume intersects the body surface only once. Unlike prisms, in this class of bodies uniqueness requires knowledge of the density. This theorem was proved by P. Novikov. The simplest example of starshaped bodies is a spherical mass. Of course, prisms are also star-shaped bodies but due to their special form, that causes ﬁeld singularities at corners, the inverse problem is unique even without knowledge of the density. It is obvious that these two classes of bodies include a wide range of density distributions; besides it is very possible that there are other classes of bodies for which the solution of the inverse problem is unique. It seems that this information is already sufﬁcient to think that non-uniqueness is not obvious but rather a paradox. Assuming that the measured and calculated ﬁelds, caused only by masses of a body, are known exactly it is simple to outline the main steps of interpretation and, as was pointed out earlier, it is a straightforward task. Suppose that we deal with a class of bodies for which uniqueness holds. Then, the main steps of interpretation were formulated above and they are 1. Proceeding from an observed ﬁeld and making use of additional information we approximately deﬁne the parameters of the body (ﬁrst guess). 2. Substituting values of these parameters into Equation (4.2) we solve the forward problem and compare the measured and calculated ﬁelds.

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(b)

(a)

q 1

2

3

p Fig. 4.2. (a) System of prisms, (b) star-shaped body.

This process of comparison allows us to determine how the parameters of the ﬁrst guess have to be changed in order to decrease the difference between the measured and calculated ﬁelds. 4. Performing a solution of the forward problem with the new parameters we again compare ﬁelds, and this process can continue until the accuracy of a determination of the parameters satisﬁes our requirements. It is essential that in performing the solution of the inverse problem we can in principle reduce the error in evaluating the parameters of a body to zero. Note that if the ﬁrst guess contains parameters which do not characterize a body and they are introduced by error, this procedure of solution of the inverse problem allows us to detect and eliminate them. Of course, there are classes of bodies for which the solution of inverse problem in gravimetry is not unique. The same may be true if bodies of different classes are considered. This fact is not surprising, and is also observed in other geophysical methods. For this reason it is very difﬁcult to understand why the literature about the gravitational method so often emphasizes the fact that ‘‘the solution of inverse problem in this method is not unique’’ without any reference to other methods. At the same time, theorems of uniqueness are proven for certain well-speciﬁed class of bodies, such as prisms or star-shaped bodies and etc. For instance, a solution of the inverse problem for prisms does not mean that we cannot ﬁnd some body different from a prism, which creates exactly the same ﬁeld as that of the prism. In this light it may be proper to recall one example, which often helps to create confusion about uniqueness. 3.

4.1.2. Example Suppose that masses are located inside a volume of an arbitrary shape and imagine that the potential of the ﬁeld due to these masses is constant on some surface S, surrounding the masses, Fig. 4.3. Also assume that the potential tends to zero at inﬁnity. Then, as was shown in Chapter 1, the potential at any point p outside the surface S is equal to I I 1 @UðqÞ @Gðq; pÞ UðpÞ ¼ dS U dS ð4:3Þ Gðq; pÞ 4p @n @n S

S

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p R q δ

V S

Fig. 4.3. Illustration of Equation (4.3)

since the surface S is equipotential and the potential U can be taken out of the integral. Here G(q,p) is a Green’s function and among of them we choose the simplest one: 1 ð4:4Þ Gðq; pÞ ¼ R In this case the last integral on the right hand side of Equation (4.3) becomes I I @ 1 dS R dS ¼ ¼ oðpÞ @n R R3 S

S

Here o(p) is the solid angle under which the closed surface S is seen from the point p, and as is well known it is equal to zero. Correspondingly, Equation (4.3) is greatly simpliﬁed and we obtain I 1 1 @U dS ð4:5Þ UðpÞ ¼ 4p R @n S

Letting 1 @UðqÞ ¼ ksðqÞ 4p @n we obtain

I UðpÞ ¼ k

sðqÞ dS R

ð4:6Þ

ð4:7Þ

S

This equation describes the potential due to masses distributed with the density s(q) on the level surface S. On the other hand, in accordance with Newton’s law the same potential is equal to Z dðqÞ UðpÞ ¼ k dV ð4:8Þ V Rðp; qÞ Here q is a point inside the volume V. Hence, it is impossible to distinguish between the ﬁeld caused by a volume distribution of masses and the ﬁeld generated by masses on the equipotential surface S, provided that the condition (4.6) is met and the observation point is located outside S. As a rule, a three-dimensional body and

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the surface S belong to different classes of bodies and it is not surprising that in this case we can observe non-uniqueness. Moreover, using the same Green’s formula it is simple to show the existence of equivalent surface layers in other geophysical methods. Thus, this example hardly illuminates the concept of uniqueness or nonuniqueness. As was pointed out earlier, uniqueness of a solution of an inverse problem has purely mathematical interest because it implies that the calculated and measured ﬁelds are known exactly. Since in reality this does not hold true so this subject will be put aside.

4.2. SOLUTION OF THE INVERSE PROBLEM AND THE INFLUENCE OF NOISE Now we are ready to make one step forward and discuss some aspects of interpretation for real conditions when the gravitational ﬁeld is measured with some error; that is, the numbers that describe the ﬁeld are accurate to only some decimal places. This is a fundamental difference from the previous case where we assumed that the ﬁeld generated by masses of some body is known exactly. The presence of error is caused by the two following factors, and these are 1. Measurements are always accompanied by errors, which depend on the design of the instrument as well as external factors, such as change of air pressure, variation of temperature, etc. 2. The measured ﬁeld consists of two parts: one of them gU, caused by masses of a body, which has to be found (anomaly); this is usually called the useful signal. The other part, gN, is due to masses surrounding the body as well as due to changes of the position of the observation point with respect to the earth’s center and it represents a noise. Thus, the measured ﬁeld ga is a sum: ga ¼ gU þ gN

ð4:9Þ

Here ga is any component of the ﬁeld. It can be measured with a relatively high accuracy, but the error of determination of the useful signal: gU ¼ ga gN

ð4:10Þ

is also dependent on the contribution of noise. Note that reduction of this noise is one of the most important elements of interpretation of gravitational data, since the determination of parameters of a body is based on a comparison of a calculated ﬁeld, solution of forward problem, with not the measured ﬁeld but the useful signal, which only contains information about the body. Bearing in mind that at any observation point we never know the value of the noise signal but rather an interval of its change, it is appropriate to speak also about an interval of variation of the useful signal. To emphasize this fact consider curves in Fig. 4.4a,b. On the left side of Fig. 4.4 we show a graph of the measured signal, while in Fig. 4.4b the interval of a change of the useful signal, as well as the graph of gU, (dotted line). Along the horizontal axis we plot the coordinate x of the observation point.

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ga

(a)

(b)

gU

gU

0

x

0

x

Fig. 4.4. (a) The measured ﬁeld, (b) interval of a change of useful signal.

Because of the inﬂuence of noise, the value of gU is located somewhere inside of the interval (bar). Its boundary is obtained from Equation (4.10), assuming a certain level of noise. If at an observation point the measured ﬁeld and that due to noise have different signs, then we have: gU 4ga . In contrary, when ga and gN have the same sign, the value of the useful signal is smaller than the measured ﬁeld. Notice that a boundary of a bar, where the useful signal is located, is deﬁned approximately and based on some additional information like knowledge of the inﬂuence of topography and the medium surrounding a body. From this consideration it is clear that the accuracy of the ﬁeld calculation, forward problem, can be practically the same as that of the measured ﬁeld, and because of this there is always a difference between these ﬁelds. For this reason any attempt to achieve ﬁtting of the calculated ﬁeld and useful signal with an accuracy exceeding that of their determination has no meaning. Taking into account the fact that the useful signal is known with some error which sometimes reaches several percent, let us consider the inﬂuence of this factor on the interpretation. First, as was pointed out, any component of the ﬁeld can be represented at every point as a sum of ﬁelds caused by different masses within a body, and their contribution depends essentially on the location and distance of these masses from the observation point. In particular, masses located closer to the observation point give larger contributions, while remote parts of the body produce smaller effects. It is obvious that there are always masses within the body such that their contribution to the useful signal is so small that within a given accuracy of its measurement it cannot be detected. For instance, we can imagine such changes of shape, dimensions, and location of a body, as well as density, that the useful signal would remain somewhere inside an interval, (bar). In other words, due to the presence of noise there can be an unlimited number of different distributions of masses that generate practically the same useful signal. For instance, this may happen when the bodies belong to the class of prisms, the same prisms for which a solution of the inverse problem is unique when the ﬁeld is known exactly. Inasmuch as the secondary ﬁeld is caused by all masses within the body––that is, an integrated effect is measured––some changes of masses in relatively remote parts of the body can be signiﬁcant; but their contribution to the ﬁeld would still remain small. At the same time similar changes in those parts of the body closer to observation

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points will result in much large changes of the ﬁeld. For this reason, in performing an interpretation it is natural to distinguish at least two groups of parameters describing the distribution of masses, namely: 1. Parameters that have a sufﬁciently strong effect on the ﬁeld that is relatively small changes of their values produce a change of the useful signal that can be detected. 2. Parameters that have a noticeable inﬂuence on the ﬁeld only if their values are signiﬁcantly changed. This simply means that they cannot be deﬁned from a useful signal measured with some error. Therefore, we can say that an interpretation or a solution of the inverse problem consists of determining the ﬁrst group of parameters of the body even though they incompletely characterize the distribution of masses. It is clear that this so-called stable group of parameters describes a model of the body that differs to some extent from the actual one, but both of them also have common parameters. For instance, these can be the depth to the top of the body, or the product of its thickness and its density, or others. Certainly, the most important factor which in essence deﬁnes all features of the interpretation, is the fact that the useful signal or the ﬁeld caused by only masses of a body is known with some error and because of this the error of evaluation of some parameters, unstable ones, can be unlimitedly large. In other words, these parameters cannot be practically determined. Such inverse problems are called ill-posed problems. In general, an inverse problem in gravity, as well as in other geophysical methods, is ill-posed. To illustrate this fact, let us write the following relation between the change of an useful signal DgU and that of a body parameter, Dpi: Dpi ¼ ki DgU

ð4:11Þ

Here ki is the coefﬁcient of proportionality for the ith parameter of the body, and DgU the change of the useful ﬁeld, an interval width, at an observation point; and this varies from point to point. The coefﬁcient ki is different for different parameters; if it is small, then for a given interval width the change of the parameter pi is also relatively small. This means that by performing a solution of the inverse problem we can determine the value of this parameter with a sufﬁciently high accuracy. In contrast, when ki is large and DgU has a ﬁnite value the range of change of the parameter pi can be great, and we cannot evaluate this parameter. In principle, an upper limit cannot be established for some coefﬁcients ki and this is the most important feature of the ill-posed problem. In other words, even an unlimited change of some parameters of the body does not produce a noticeable variation of the ﬁeld, exceeding an interval, and as a result of this, it is impossible to deﬁne these parameters. Thus, there are always two groups of parameters: they are called stable and unstable parameters of the body and our goal is to separate them and determine the stable ones. The latter characterize a new model of a real body and the coefﬁcients ki for its parameters are relatively small. In this case we arrive at a wellposed problem, but are not able to determine some parameters, and this fact reﬂects a reality of the solution of the inverse problem in geophysics, in particular, in gravity. The transition from an ill-posed problem to a well-posed one is called the

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regularization of the inverse problem, and it is of a great practical interest. It is obvious that the interpretation of gravitational data is useful if the parameters of a model, approximating a real distribution of masses, are deﬁned within such narrow range of values that is sufﬁcient from a practical point of view. Usually a choice of this group of parameters is automatically selected making use of the corresponding algorithm of the inverse problem. Of course, with an increase in the number of model parameters the approximation of a real distribution of masses can in principle be better. However, the error with which some of these parameters are determined also increases. As in the case when the ﬁeld is known exactly we can expect that the interpretation of gravitational data is greatly facilitated by the presence of additional information about a distribution of masses usually derived from geology and other geophysical methods. Now let us formulate the main steps of a solution of the inverse problem, taking into account the fact that the useful signal is subjected to the inﬂuence of error. These steps are: 1. Making use of preliminary information about parameters of a body we formulate a ﬁrst guess and completely repeat what was done earlier in the case of uniqueness. 2. The second step is a solution of the forward problem, applying Newton’s law of attraction Z dðqÞ gU ðpÞ ¼ k Lqp dV 3 V Lqp 3.

4.

As result of this calculation we obtain a set of values of the ﬁeld component which can be graphically represented as a curve gU(x). Suppose that this curve is situated beyond the interval, Fig. 4.4b. Then, changing parameters of the body we again use Newton’s law and obtain a curve of the ﬁeld which is closer to the interval of the useful signal. This process continues until a curve of the calculated ﬁeld is located inside the interval. So far the steps of interpretation are identical to those when we considered the case of uniqueness. Now we will observe a fundamental difference. As soon as the values of a calculated ﬁeld are located inside the observation interval further improvement of matching between the measured and calculated ﬁelds does not have any meaning, because we do not know where inside the interval the useful signal is located. Therefore, we stop the process of ﬁtting of ﬁelds and start a new procedure which also requires a solution of the forward problem. In the last stage of matching, we obtained the set of parameters p1 ; p2 ; p3 ; :::; pn that places the calculated useful signal inside the interval. Our goal is to determine the range of change of each parameter so that the calculated useful signal remains inside the interval. This procedure is usually repeated several times and it is accompanied by a solution of the forward problem. Of course, every step causes a movement of the curve of the useful signal, and as long as

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its position is inside the interval it is equivalent to the previous one. As result of these steps we obtain for each parameter its range: max min max pmin 1 op1 op1 ; :::::::::::::::::: pn opn opn

As was pointed out earlier, within these ranges the degree of ‘‘matching’’ with the measured useful signal is the same for any set of parameters. Certainly, knowledge of variation of these parameters is the most important step in solving the inverse problem, because this table allows us to separate the stable from unstable parameters, and correspondingly, perform a transformation from the ill-posed to a well-posed problem. Thus, the interpretation gives us a set of stable parameters of a body which represents a distribution of masses causing the useful signal. It is natural to raise the following question. Is it possible that there are other distributions of masses that produce the same ﬁeld gU? Without any doubts the answer is positive, but due to additional information about geology and previous geophysical surveys in the same area or similar ones, this ambiguity is often reduced to a minimum. In this light, it may be appropriate to notice that uniqueness and this ambiguity are not the same. For instance, if the useful signal is caused by a prism or a system of them, the solution of the inverse problem is unique, however we will still observe ambiguity as soon as the useful signal is deﬁned with some ﬁnite error. From a review of the solution of the inverse problem it is clear that with a decrease of the interval width the range of each possible parameter of a body decreases too and the number of unstable parameters may become smaller. At the same time stable parameters can be determined with higher conﬁdence. For this reason reduction of different types of noise such as corrections for topography, elevation, and large-scale sources, is a very important subject. We outlined the main features of interpretation for classes of models where uniqueness takes place. It turns out that if models can be described by ﬁnite number of parameters, the solution of inversion will be stable and one can get inequalities for potential errors. As concerns an interpretation within class of models with no uniqueness situation is completely different. We can say that a solution of inverse problem in such case is hardly possible, (speaking strictly it is senseless). In fact, even in an absence of a noise we always have inﬁnite number of models which create the same measured ﬁeld. At the same time it may be possible to determine some generalized characteristics of a body such as its total mass. Thus, uniqueness theorems that appear to be of mainly academic interest are very important in solving practical inverse problems.

4.3. SOLUTION OF THE FORWARD PROBLEM (A CALCULATION OF THE FIELD OF ATTRACTION) Inasmuch as determination of the ﬁeld of attraction is an important element of interpretation of gravitational data, let us derive some equations allowing us to simplify the calculation of the useful signal. As follows from the Newton’s law of

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attraction the gravitational ﬁeld is expressed in terms of the volume integral: Z dðqÞLqp ga ðpÞ ¼ k dV ð4:12Þ L3qp V Proceeding from Equation (4.12) we consider several methods of calculation of the vertical component gz, which allows us to simplify the integration. Let us start with a two-dimensional case. 4.3.1. Two-dimensional model Suppose that a body is strongly elongated in some direction and with sufﬁcient accuracy it can be treated as the two-dimensional. In other words, an increase of the dimension of the body in this direction does not practically change the ﬁeld at the observation points. We will consider a two-dimensional body with an arbitrary cross section and introduce a Cartesian system of coordinates x, y, and z, as is shown in Fig. 4.5a, so that the body is elongated along the y-axis. It is clear that if at any plane y ¼ constant the behavior of the ﬁeld is the same. To carry out calculations we will preliminarily perform two procedures, namely, 1. Mentally divide the cross section of a body into many elementary areas. Correspondingly we can treat the model as a system of many elementary prisms. The dimensions of every elementary cross section are much smaller than the distance between an observation point and any point in this area.

(b)

(a) 0

gy

x

p gz

r

0*

y

p

h

gr

gy

z q y

(c) x

0 p Lqp

z

q

z Fig. 4.5. (a) Field due to two-dimensional body, (b) ﬁeld caused by two elementary masses, (c) the ﬁeld due to thin two-dimensional layer.

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2.

231

Each elementary prism is replaced by an inﬁnitely thin line directed along the y-axis with the same mass per unit length as that of the prism. These two steps allow us to replace the two-dimensional body by a system of inﬁnitely thin lines which are parallel to each other, and the distribution of mass on them is deﬁned from the equality dm ¼ dðqÞdSðqÞdy ¼ lðqÞdy since lðqÞ ¼ dðqÞdS

ð4:13Þ

where l(q) is the linear density on the line passing through point q, and dS the area of the elementary cross section of the prism. Let us note that the density is a function of coordinates x and z, but it does not depends on y. Derivation of the formula for the attraction ﬁeld caused by an inﬁnitely thin line with the density l is very simple, and is illustrated in Fig. 4.5b. We will consider the ﬁeld at the plane y ¼ 0. Due to the symmetry of the mass distribution, we can always ﬁnd a pair of elementary masses ldy and ldy, which when summed do not create the ﬁeld component gy directed along the y-axis, and respectively the total ﬁeld generated by all elements of the line has only the component gr, located in the plane y ¼ 0. Here r is the coordinate of the cylindrical system with its origin at the point 0*, and the line with masses is directed along its axis. As is seen from Fig. 4.5b the component dgr at the point located at the distance r from the origin 0* is dgr ¼ k

ldy r dy ¼ klr 3 2 R R R

or

dgr ¼ klr

dy ðy2

þ r2 Þ3=2

Here R ¼ ðr2 þ y2 Þ1=2 At the beginning, we assume that the line length is 2l. Then, summation of ﬁelds caused by all elementary masses of the line gives Z l dy gr ðpÞ ¼ klr 3=2 l ðy2 þ r2 Þ Introducing a new variable j : y ¼ r tan j, we have dy ¼ r sec2 jdj and Z kl j0 2kl sin j0 gr ðpÞ ¼ cos jdj ¼ r j0 r where tan j0 ¼ l/r. Since sin j ¼

tan j ð1 þ tan2 jÞ1=2

we obtain gr ¼ 2k

l l r ðl 2 þ r2 Þ1=2

ð4:14Þ

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Because it is assumed that l44r, let us represent the latter in the form of a series. l 1 2kl 1 r2 1 r4 1 2 þ 4 ð4:15Þ gr ¼ 2k r 1 þ ðr2 =l 2 Þ1=2 r 2l 8l The ﬁrst term of this series describes the ﬁeld caused by the inﬁnitely long line, and in this case 2kl ð4:16Þ r Comparison of two last equations allows us to determine the error that occurs when we replace a line of ﬁnite length with an inﬁnitely long one, and then apply this result to a real elongated body. For instance, if the length of the line is 10 times greater than the distance from the observation point, this error is less than one-half percent. As follows from Fig. 4.5a for the vertical component of the ﬁeld we have gr ðrÞ ¼

zp zq 2kl ¼ 2 ðzq zp Þ r r In particular, on the earth’s surface, where zp ¼ 0, gz ðpÞ ¼ gr

gz ðpÞ ¼

2klzq 2kdðqÞdSðqÞzq ¼ r2 r2

and r2 ¼ ðxp xq Þ2 þ z2q

ð4:17Þ

Here pairs xq, zq and xp, zp are Cartesian coordinates of any point of a body and observation points, respectively. Equation (4.17) describes the vertical component of the ﬁeld caused by masses within an elementary prism. Correspondingly, for the ﬁeld gz, due to a two-dimensional body, we have Z dðqÞdSðqÞ gz ðpÞ ¼ 2k zq if zp ¼ 0 ð4:18Þ r2 S Thus, instead of a volume integral, the ﬁeld is represented as a surface integral, which, of course, greatly simpliﬁes calculations. If the function d(q) is constant, we have Z zq gz ðpÞ ¼ 2kd dS 2 S r or Z gz ðpÞ ¼ 2kd

zq dxdz ; ðx xq Þ2 þ z2q S p

if zp ¼ 0

ð4:19Þ

For those cases when the cross section of the body has a relatively simple shape, the integral on the right hand side of this equation can be expressed through elementary functions. However, in more complicated cases, determination of the ﬁeld is performed by numerical integration of Equation (4.19).

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4.3.1.1. Thin two-dimensional layer To further simplify calculations consider the special case when a two-dimensional body is oriented along the y-axis, and its thickness is much smaller than the distance between the body and observation points; that is, hoorqp, where h is the body thickness. Then, it is obvious that if the body were replaced by a two-dimensional strip, (Fig. 4.5c), bearing the same mass and placed somewhere inside the body, the ﬁeld would not change at the observation point. Now Equation (4.18) is greatly simpliﬁed and we have Z x2q sðxq Þdxq gz ðpÞ ¼ 2kzq ð4:20Þ 2 2 x1q ðxp xq Þ þ zq Here zq is a coordinate of a strip point along the z-axis, while sðxq Þ ¼ dðqÞh is the surface density of masses within the strip, and x1q, x2q are coordinates of terminal points of the strip. Note that such a model may be useful in calculating, for example, the gravitational ﬁeld caused by two-dimensional structures (anticlines, depressions) on the surface of the basement, wherein their amplitudes are small with respect to the sediment thickness. If the surface density s is constant, in place of Equation (4.20) we have Z x2q dxq gz ðpÞ ¼ 2kdhzq 2 2 x1q ðxp xq Þ þ zq Applying again substitution ðxp xq Þ ¼ zq tan j the ﬁeld gz due to masses of the strip is expressed as gz ðpÞ ¼ 2khdðj1 j2 Þ

ð4:21Þ

where j1 ¼ tan1

xp x1q xp x2q ; j2 ¼ tan1 zq zp if x1q o0 and x2q 40

In the limiting case, when a strip becomes the plane of an inﬁnite extension, we arrive at the well-known formula for the Bouguer correction. In fact, if j1 ¼ p=2 and j2 ¼ p=2, we have gz ðxp Þ ¼ 2pkd Suppose that the observation point is located in the plane xp ¼ 0 and x1q ¼ x2q. Then, we can write Equation (4.21) in the form gz ðxÞ ¼ 4khd tan1

jxj zq

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Assuming thatjxj4zq we can expand tan1 jxj=zq in a series tan1

jxj p zq 1 z3 þ zq 2 jxj 3 x3

and this gives zq 2 z3q gz ðpÞ 2kphd 1 þ jxj 3p x3

!

It is clear that this equation allows us to evaluate the difference of the ﬁelds caused by the plane and a strip with a ﬁnite width. Of course, this evaluation can be done directly proceeding from Equation (4.21). Now we will show that, making use of this equation, it is possible to calculate the gravitational ﬁeld caused by masses in a two-dimensional body with an arbitrary cross section. With this purpose in mind, let us mentally divide the body cross section into a sufﬁcient number of relatively thin layers with the thickness hi. Then, applying the principle of superposition and Equation (4.21) for elementary layer, we have gz ðpÞ ¼ 2k

N X

hi di ðj1i j2i Þ

ð4:22Þ

i¼1

Here gz(p) is the ﬁeld caused by all masses of the body, hi ¼ ziþ1 zi ; ziþ1 , and zi are vertical coordinates of the bottom and top of the i-layer, N is the number of elementary layers, and j2i ¼ tan1

xp x2i ; z0i

j1i ¼ tan1

xp x1i z0i

where ziþ1 þ zi 2 x2i, x1i are horizontal coordinates of terminal points of the i-layer, and di is its density. In particular, if di is constant within the body, we have z0i ¼

gz ðpÞ ¼ 2kd

N X

ðziþ1 zi Þðj2i j1i Þ

ð4:23Þ

i

Note, that with an increase in the number of elementary layers the accuracy of ﬁeld determination increases too. 4.3.2. Three-dimensional body Next, suppose that the gravitational ﬁeld is caused by masses in a three-dimensional body. In principle, the ﬁeld can be calculated from Equation (4.12) directly using known algorithms of integration, but even with fast computers a numerical integration over the volume for many observation points requires a lot of time. For

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this reason, it is natural to apply methods that allow us to simplify this procedure. Two of them are described in the next sections. 4.3.2.1. The ﬁrst approach By analogy with the previous case, let us represent the three-dimensional body as a system of elementary layers located in horizontal planes whose thickness is much smaller than the distances from them to the observation points. In such a case, every layer can be replaced by a horizontal plane of ﬁnite dimensions with the surface density sðqÞ ¼ dðqÞhðqÞ Correspondingly, the vertical component of the ﬁeld caused by all masses of the body can be represented as a sum of ﬁelds giz due to elementary thin layers: gz ¼

N X

giz

ð4:24Þ

i¼1

As was shown in Chapter 1, the ﬁeld generated by surface masses is expressed through the solid angle oi (p) under which the surface is seen from the observation point p. Assuming that the density, si (q), is constant at the ith surface, we have giz ðpÞ ¼ ksi ðzq Þoi ðpÞ

ð4:25Þ

Thus, for the ﬁeld gz, we obtain gz ðpÞ ¼ k

N X

dðzq Þhðzq Þoi ðpÞ

i¼1

or gz ðpÞ ¼ k

N X

dðzq Þðziþ1 zi Þoi ðpÞ

ð4:26Þ

i¼1

where hi ¼ ziþ1 zi is the thickness of the elementary layer. In particular, if the density of a body is constant we have gz ðpÞ ¼ kd

N X

ðziþ1 zi Þoi ðpÞ

ð4:27Þ

i¼1

Correspondingly, determination of the vertical component of the ﬁeld due to masses in a three-dimensional body may consist of calculating a set of solid angles. 4.3.2.2. The second approach Now we will proceed from the known equality of the vector analysis Z I q grad TdV ¼ TdS V

S

ð4:28Þ

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Methods in Geochemistry and Geophysics

where S is the surface surrounding the volume of the body, dS ¼ dSn, and n the unit vector normal to the surface element and directed outward, T a continuous function in the volume V. Index ‘‘q’’ means that the gradient is taken with respect to the point q. Assuming that the density is constant and taking into account the equality Lqp 1 ¼ gradq 3 L Lqp qp we can rewrite Equation (4.12) as Z gðpÞ ¼ kd

Lqp dV ¼ kd 3 V Lqp

Z

gradq V

1 dV Lqp

Now making use of Equation (4.28) we obtain I

dS Lqp

gðpÞ ¼ kd S

Respectively, for the vertical component of the ﬁeld we have I gz ðpÞ ¼ kdi3

dS Lqp

S

where i3 is the unit vector directed along the z-axis. Inasmuch as dS i3 ¼ dS cos b where b is the angle between n and i3, which depends on point q of the surface, we have I gz ðpÞ ¼ kd

dSðqÞ cos bðqÞ Lqp

ð4:29Þ

S

Similar expressions can be written for horizontal components of the ﬁeld of attraction. Thus, instead of the volume integral, we have derived an expression for the ﬁeld that requires integration only over the surface. The formulas described in this section allow us, in many cases, to simplify the solution of the forward problem.

Bibliography Blakely, R., 1996. Potential Theory in Gravity and Magnetic Applications. Cambridge University Press. Garland, G., 1971. Introduction to Geophysics. W.B. Saunders Company. Grant, F. and West, G., 1965. Interpretation Theory in Applied Geophysics. McGraw-Hill Book Company. Grushinsky, N., 1976. Foundation of Gravimetry. Publisher House ‘‘Nauka’’ Moscow. Grushinsky, N., 1976. Theory of Earth’s Figure. Publisher House ‘‘Nauka’’ Moscow. Heiskanen, W.A. and Moritz, H., 1967. Physical Geodesy. W.H. Freeman and Co. Hofmann-Wellenhof, B., Lichtenegger, H. and Collins, J., 1992. Global Positioning System: Theory and Practice, 2nd edn. Springer-Verlag. Kaufman, A., 1992. Geophysical Field Theory and Method, Part A. Academic Press, Inc. Michaelov, A., 1939. Course of gravimetry and ﬁgure of Earth. Moscow. Nabigian, M.N., Ander, M.E., Grauch, V.J.S., Hansen, R.O., LaFehr, T.R., Li, Y., Pearson, W.C., Peirce, J.W., Philips, J.D. and Ruder, M.E., 2005. Historical development of the gravity method in exploration: Geophysics, 70. Nettleton, L., 1940. Geophysical Prospecting for Oil. McGraw-Hill Book Company. Nettleton, L., LaCoste, L. and Harrison, J.C., 1960. Tests of an airborne gravity meter: Geophysics, 25: 181–202. Niebauer, T.M., Sakagawa, G.S., Faller, J.E., Hilt, R. and Klopping, F., 1995. A new generation of absolute meters: Metrologia, 32: 159–180. Smythe, W., 1950. Static and Dynamic electricity. McGraw-Hill, New-York. Synge, J.L. and Grifﬁth, B.A., 1959. Principles of Mechanics. McGraw-Hill Book Company. Torge, W., 1989. Gravimetry. de Gruener, Berlin. Torge, W., 2001. Geodesy. de Gruener, Berlin. Yakosky, J., 1949. Exploration Geophysics. Trija Publishing Company. Wahr, J., 1997. Geodesy and Gravity: Class Notes. Samizdat Press.

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Appendix

SATELLITE METHODS FOR A STUDY OF THE GRAVITATIONAL FIELD OF THE EARTH The global-positioning system (GPS) is just one of family of satellite-positioning systems which have become available since about 1995. However, since all have the same operating principles, and the GPS is the most widely used, we will refer to these systems generically as GPS. The reason for our interest in these systems is that they have produced radical changes in both geophysical surveying and study of the gravitational ﬁeld. It is obvious that precise determination of position coordinates is very important element in all geophysical methods, including the gravitational method too. Besides, GPS allows one to investigate lateral variations of the gravitational ﬁeld of the earth. The GPS system consists of a set of satellites, called a ‘‘constellation’’, which are essentially at the same elevation, orbiting the earth at about 12-h period. The orbits are distributed so as to make approximately the same number of satellites visible at any point and time in the sub-polar regions. Radio signals from the known stations on the earth’s surface allow us to know orbits of satellites with great precision and this is vitally important for the use of GPS in geodesy and gravity. In addition to position information, the system requires that the satellites carry accurate times, which are obtained from onboard atomic clocks. As was pointed out, each satellite transmits a radio signal that carries satellite location information and timestamp for that information. Receivers use the data from several satellites to triangulate their locations. A simple receiver available at a consumer electronic store can achieve an accuracy of around 10 m and more elaborate receiver systems employing different corrections can yield accuracies of better than 0.01 m. Thus, traditional geodetic measurements have almost entirely superseded by GPS.

GPS AND A STUDY OF GRAVITATIONAL FIELD OF THE EARTH It turns out that the GPS allows one to obtain information about the gravitational ﬁeld of the earth and we will brieﬂy outline here two different approaches. The ﬁrst approach is based on the study of the satellite orbit while the other requires measurements of the satellite height.

240

Methods in Geochemistry and Geophysics

The first approach As the ﬁrst approximation, we assume that a motion of satellite on its orbit is described by Newton’s second law and only the gravitational ﬁeld of the earth is present. In other words, we neglect by a force, caused by other bodies, such as Moon and planets, as well as by the inﬂuence of air and corrections, which follows from the general theory of relativity. If we also suppose that an earth is uniform or concentrically uniform, then the satellite will move along an elliptical or circular orbit, which is deﬁned by constant parameters pi, such as large semi-axis of the orbit, its eccentricity, inclination of the orbit with respect to the plane of equator, and others. It is essential that these parameters remain the same at any point of the orbit and correspondingly their derivatives with respect to time are equal to zero. Completely different picture occurs when there are lateral changes of the density or irregularities in the earth shape. Then, parameters of the orbit are not constant and they begin to change, that is, we observe perturbation of the orbit caused by a deviation of the gravitational ﬁeld from the normal ﬁeld. In this case for each instant there is a system of parameters, which characterizes the orbit at this given moment, and moreover it deﬁnes a motion of the satellite at following times if the gravitational ﬁeld stops to act. Thus, in principle, we can obtain a system of such sets of parameters for each instant and in our approximation they completely characterize a position of the satellite with respect to the earth. Such system is called the system of osculating elements and their perturbations with time are relatively slow. Celestial mechanics describes equations that relate the rate of a change of the orbit parameters with these parameters and components of the secondary gravitational ﬁeld, caused by lateral changes of the density. These equations written for each parameter form the system of Lagrange’s equations. Applying an expansion by spherical harmonics, it is possible to solve this system with respect to unknown components of the secondary ﬁeld for each harmonic and represent results in the form of maps for coefﬁcients of Legendre’s polynomial, which are related to the dynamic parameters of the earth. At the beginning, we made several assumptions and did not take into account an inﬂuence of other factors, which change the parameters of the satellite orbit. In reality, these corrections are always introduced and ﬁnally we again use Lagrange’s system of equations. The second approach Since early 1990s, a series of satellites have been deployed which measure their distance from the surface of the earth from the return time of radar signals beamed to the earth’s surface and reﬂected back to a receiver onboard the satellite. The vertical accuracy of these measurements is a few times 0.01 m averaged over a horizontal distance of around 10 km. Although the primary purpose of these satellites is oceanographic, when derived elevations are averaged over sufﬁcient time to eliminate the effects of tides and storms at ocean. Besides, these measurements allow us to determine an elevation x of the geoid, quite ocean surface, over the reference ellipsoid. The idea of this method is very

241

Appendix

S a S

b

Earth surface

h

h

hn

geoid

geoid ellipsoid

re

ellipsoid 0

re 0

Fig. 1. (a, b) Determination of geoid elevation over the reference ellipsoid.

simple and as is seen from Fig. 1a we have x ¼ rs re h Here rs and re are distances from the earth center to the satellite and the point of the ellipsoid beneath of the satellite, respectively, and they are known. At the same time h is the height of the satellite, measured by radio-altimeter. For the points of the land, Fig. 1b we have a similar expression x ¼ r s r e hn h where hn is the normal height of the earth’s surface over the geoid. Knowledge of the elevation x allows us to study an anomaly of the gravitational ﬁeld. In fact, in Chapter 2 we demonstrated that the Stokes’ theory allows us to ﬁnd the function x when the secondary ﬁeld Dg is given. Now using the satellite-derived geoid, that is, the elevation x, Stokes’ formula can be applied in reverse to obtain the function Dg over the oceans. These values are again by far the best available except where localized shipborne gravity surveys have been performed. Current global gravity models are composite of satellite-derived gravity over the open oceans, shipborne gravity near shore, and traditional measurements on land.

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Subject Index

acceleration, 6, 59, 61–62, 64–69, 71, 161–162, 176, 179, 181–182, 203 angular acceleration, 176, 203 angular velocity, 59, 61–64, 66–71, 80, 82, 103, 122, 129, 143–145, 148–149, 153–155, 157, 175–177, 180–183, 186–187, 205 attenuation, 192–194, 197, 203–204, 209–210 attraction, 1–7, 9–19, 21–23, 25, 27–31, 33, 35–39, 41–47, 49, 51–57, 59–65, 70, 72–77, 82, 85, 91, 93–95, 103–105, 108, 111, 134–136, 143–145, 147, 149–157, 172, 181–182, 217, 219–222, 228–231, 236 auxiliary function, 18 ballistic gravimeter, 163, 171, 173, 180, 184 boundary conditions, 27–28, 31, 33, 36 boundary value problem, 6, 26, 28–33, 36, 39, 55, 57, 85, 93, 95, 106, 110, 122–123, 129, 132, 134 Brun’s equation, 80 Brun’s formula, 121–122, 129–132, 135

earth rotation, 184 eastic force, 5, 181, 188, 190–192, 194, 196–200, 203, 210, 212 elastic parameters, 5, 188, 199 elementary surface, 13, 46, 49, 188 elementary volume, 1–4, 9–10, 47, 59–61, 85, 136 ellipsoid of rotation, 84–85, 87, 91, 93–95, 111, 114–116, 122 equilibrium, 5, 59, 61–63, 73, 82, 85, 100, 143–146, 148–149, 151–155, 157, 170, 187, 189–191, 193–197, 199–200, 202–208, 210–215 equipotential surface, 31–32, 46, 77, 85, 103–104, 106, 111, 115–116, 119–120, 210, 224

centrifugal force, 65–66, 68, 70, 73–74, 76–77, 83, 102–103, 143–145, 147, 149, 151–152, 154, 172, 182, 184, 217 centripetal force, 61, 66 Clairaut’s formulas, 64 Clairaut’s theorem, 100–106, 156–157 Coriolis force, 68–70, 163, 168, 173, 180, 182–184, 208

ﬁeld of attraction, 6, 10, 14, 30, 33, 44, 63, 72–73, 103, 135, 143, 149–151, 172, 181, 217, 219–220, 222, 229, 236 ﬂattering of ellipsoid, 98 force, 1–6, 59–66, 68–70, 73–74, 76–77, 82–83, 102–103, 105, 143–145, 147–151, 153–157, 162–163, 167–170, 172–173, 179–184, 188–203, 206, 208–212, 217, 240 forced vibrations, 193–194, 209 forward problem, 6, 8, 10, 25–27, 220–223, 225–226, 228–229, 236 Foucault’s pendulum, 180, 184, 186 free vibrations, 170, 190, 192–196, 203–205, 209, 214 friction, 161–163, 167, 169, 173, 179, 191

damping, 195 density, 3–4, 6, 8–10, 16–17, 20, 22, 26, 29, 36, 45–47, 49, 51–54, 60–61, 73–74, 77, 80, 82–85, 100, 107, 113–114, 116, 134, 136, 138, 143–146, 148–149, 151, 153, 155–159, 217–222, 224, 226–227, 231, 233–236, 240 Dirichlet’s problem, 26, 42

Gauss’s theorem, 27, 30, 33–34 geo-potential, 118, 129–130 geographical latitude, 88, 98 Geoid, 33, 40, 73, 77, 114–124, 127–132, 157, 240–241 gravitational constant, 1, 3–5, 113 gravitational ﬁeld, 24, 29, 44, 54, 58–61, 63–65, 67, 69–79, 81–85, 87, 89, 91, 93–107,

244

Subject Index

109–123, 125, 127–129, 131, 133, 135, 137, 139, 141, 143–145, 147, 149–151, 153, 155–157, 159, 161–167, 169–173, 175–177, 179, 181, 183–187, 189–191, 193–197, 199, 201–203, 205, 207–213, 215, 217–219, 221, 225, 230, 233–234, 239–241 Green’s formula, 33–35, 40, 225 Green’s function, 35–36, 38–41, 224 harmonic function, 25–29, 40, 123, 133 ill-posed problem, 9, 227 inertia, 1, 109–110, 113, 158, 176–178, 203, 212, 214 inertial frame of reference, 65, 68–69, 143, 173, 180–182 initial compression, 197 interaction, 1–6, 149, 199 inverse problem, 7–8, 11, 18, 157, 217, 219–223, 225–229, 231, 233, 235, 237 Laplace’s equation, 19, 21, 25, 32, 35, 37–39, 54–55, 57–58, 76, 85–86, 89–91, 95, 123, 132 Legendre’s functions, 54–57, 90–94, 106, 126 level surface, 31–32, 63, 77–82, 85, 87, 96–98, 100, 104, 111, 115–116, 118–121, 129–132, 143–144, 151, 156–157, 214, 224 leveling, 114–115, 119–120, 128–129, 131–133 lever spring balance, 188 linear functions, 23, 25 mass, 1–15, 17, 20, 29–32, 38, 42–48, 50–52, 56, 59–60, 62, 64–66, 68, 70, 73–74, 82–85, 87, 95–96, 104, 106–110, 113–114, 122, 126, 129, 136, 143–144, 146, 151, 153–155, 157, 162–163, 165, 167–170, 172–174, 176–179, 181–182, 187–212, 215, 220–222, 229, 231, 233 mechanical sensitivity, 189–190, 192, 196–198, 200, 205–206, 208, 215 metric coefﬁcients, 18, 89 moment of inertia, 113, 176–178, 203, 212, 214 Neumann’s problem, 26, 30 Newton’s law of attraction, 1–2, 4, 14, 17–18, 59, 134, 221, 228

non-inertial frame of reference, 65, 68–69, 143, 173, 180, 182 non-symmetrical motion, 166 non-uniqueness, 9, 221–222, 225 normal component, 15, 27, 32, 37, 47–50, 52, 73, 134, 144 normal gravitational ﬁeld, 96, 102–103, 112, 114, 131 normal height, 119–120, 128, 130, 132–133, 241 oblate spheroid, 85, 87–88, 143, 145–148, 153–154 observation point, 4, 6, 8–9, 18, 22, 34–36, 39, 41, 43–44, 47, 50–52, 54, 72, 106, 108, 118, 120, 141, 189, 214, 217–220, 222, 224–227, 230, 232–235 orthometric height, 77, 117–120, 130 physical pendulum, 175, 177 plumb line, 63, 77–82, 115, 117–120, 129, 183, 210, 212 Poisson’s equation, 18–19, 21–22, 26, 28–33, 45, 76, 83, 116, 213 Poisson’s integral, 40, 121, 123 potential, 11, 18–40, 42, 44–46, 53–54, 56, 75–80, 82–85, 87, 90–91, 93–95, 100, 103–108, 110–112, 114–119, 122–123, 126–135, 142–144, 156, 192, 210, 213, 215, 222–224, 229, 236 pressure, 59–62, 82, 84, 120, 143–144, 149, 152, 162, 225 quasi-geoid, 116–119, 128, 130–132 radius of curvature, 78 reduced latitude, 98 reference ellipsoid, 33, 40, 112, 120–122, 127–133, 240–241 regular points, 16, 19, 26, 73, 76 reversing pendulum, 175, 178–179 rotating earth, 59 secondary gravitational ﬁeld, 240 singularities, 35, 222 solid angle, 13–14, 46, 48–51, 224, 235 spherical harmonics, 106, 109, 240 spheroid eccentricity, 145 stability, 148, 155, 196–197, 206

Subject Index

stable parameters, 227, 229 Stokes formula, 114, 121, 127–128 superposition, 3–4, 12, 14–15, 20, 36, 43, 48, 56, 70, 75, 143, 167, 190, 194, 210–211, 217, 234 theorem of uniqueness, 25, 33, 39, 85, 95 torque, 176, 210 Torsion balance, 5, 205, 210 total mass, 14–15, 30–32, 43, 51, 83–84, 110, 113, 122, 229

unit vector, 2, 8, 17–18, 27, 40, 48, 60–61, 63, 66–67, 70, 172, 180–181, 236 unstable parameters, 227, 229 well-posed problem, 229 work, 12, 15–16, 155 zero-length spring, 197–198, 208 zero lever spring system, 207

245

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