A Quest for Perspectives: Selected Works of S. Chandrasekhar: With Commentary (Volume 2)

[669]

With this reduction of the last term of equation (38), we can write

(41)

where it may be recalled that (42)

The variational principle requires that
7P

dj^dx dr *- Jdr

dr* dr) r*

ds)\

0 . (43)

The fact that the variational principle leads to a characteristic equation which is quadratic in er2 corresponds to the circumstance (first noted by Cowling 1942) that the characteristic frequencies belonging to a particular spherical harmonic, apparently fall into two distinct spectra with normal modes of widely different attributes. Thus, it appears that one may distinguish between a set of modes which are principally radial—the ^-modes of Cowling—and a set of modes which are principally transversal— the g-modes of Cowling. From the present point of view, the extreme aspects of these two sets of

[670]

modes will be described by the following equations whi -\\ (43) by setting x = 0 and

Jo

™ r* 7o U?\dr)

+

p(/r

iUo

""Gained from equation

drM^')]~

and

The approximation considered by Cowling and others f the gravitational potential during the oscillations is odu'iv i"° g " K t h e variation of term (PW) in equation (38); and when this term'is aunnr. i1 l 0 su PP r essing the last PPrissed, equation (43) becomes X

,lr


dry drj r 2

IV. TWO ILLUSTRATIONS

Detailed applications of equation (43) to the problem of I lnm istic frequencies of non-radial oscillations of polytropes -j| , g t n e charactercons paper. In this paper we shall limit ourselves to illustr'it j n 't 'dered in a later two known results. ' ** 3 applications by deriving a) The Variational Principle for Purely Hmlinl
PW

r* Jo

iyP\drJ+~p^d;(pr)\d^

The last term on the right-hand side of equation (47) ,•-,„ i manner: ' n ,Je r «luced in the following

[671]

= iirG fR ~~(p^)i=iTG ["p^2^ Jo r dr Ja rJo J0

r*dr\

dr J

J0

(48)

dr dr V r*/

p dr dr \ riJ

Inserting the result of this reduction in equation (47) and simplifying, we are left with

It can be readily verified that this last equation is equivalent to the one which was first derived by Ledoux and Pekeris (cf. Ledoux and Walraven 1958, p. 465) for treating radial oscillations. b) The "Kelvin Modes" As a second illustration, consider the following particular case which leads to a simple class of solenoidal velocity fields (cf. n. 1 above): lA = X = rl+l .

(so)

For this choice of the radial functions dp 5p = r1,-1 , -r-. dr

(5,i)

With the substitutions (50) and (51), equation (41) becomes 21+1 I

[Rpr"dr=

JQ Jo

JO Jo

p p

fRr-¥^dr-^GfRp>r-dr Jo

dr dr

(52)

= -G f Jo

2l 2

R 2 u

^M(r)r - dr-4*G dr

f p r d,

J0

On further simplification, we obtain ; _ 1 } 21(1-1)

21+1

fa',r«-M(r)dr R " Ir prnus dr J oi ..us.

2 l ( l

~

2l

_

+

1 ) { 2 l

l

_

fpr^dr

l }

r/

R

(S3)

pr2lc pr«ar

Joi 'o The analogy with the Kelvin modes of non-radial oscillations of an incompressible sphere is manifest.

[672]

Equation (53) is a generalization of the results known for the cases / = 2 and / = 3 (Chandrasekhar and Lebovitz 1962, eq. [3], and 1963, eq. [64]). The research reported in this paper has in part been supported by the Office of Naval Research under contract Nonr-2121(24) with the University of Chicago. APPENDIX THE FORM OF THE VARIATIONAL PRINCIPLE FOR CYLINDRICAL SYSTEMS The variational principle formulated in the paper, in the context of spherical systems, can be adapted readily to cylindrical systems in which, in the equilibrium state, the physical variables are functions only of the distance a from the axis. The appropriately modified form of equation (19) is clearly

(A.l)

rted£djLdx_rx)s,&(x)d

+

where the integrations over V are now to be interpreted as averages per unit length of the cylinder; also, in equation (A.l), £ a is the ffi-component of the Lagrangian displacement i — (£o, £*>, £*) in a cylindrical system of coordinates (n,
ip (us) • cos kz cos m

m £ = — x ( n ) c o s kz sin m
*

(A.2)

n s

and £ . = & x ( ) i n kz cos m

where \f/ and x re two radial functions. For £ of this chosen form

Uv ? =

[orfS+ ( S " + * S ) x ] c o s

kzco m

* *

(A.3)

and 8p(x)

= — div p 3; = — dp ( a ) cos kz cos m

<*••*>

where

5p(a)=i-^-(p^)+(^+FN)px.



For Sp{x) given by equation (A.4), 533, determined as the solution of Poisson's equation, is given by 5 5 5 = -±vG[lm(ka)£

a'8p(us')Km(ka')da' (A.6)

+ Km(ka)

I cc'Sp (s>') /„, ( ka')da'

cos kz cosOT^>,

where 7 m and i
[673]

With the foregoing expressions for the various quantities which occur in equation (A.l), find that the present analogue of equation (41) is

Jo -STG

f Jo

aus

Lo da

daaSp(a)Km(ka)

\ar

/ J

Jo

we

p da da ra

f" da'a'B p (a') I „, Jo

{km').

As in Section III of the paper, we may, by writing A\f/ and Bx in place of \]/ and x in equation (A.7), derive a characteristic equation of order 2 for
[674]

T H E PULSATIONS A N D T H E D Y N A M I C A L S T A B I L I T Y O F GASEOUS MASSES I N U N I F O R M R O T A T I O N S. CHANDRASEKHAR AND N . R. LEBOVITZ

University of Chicago Received August 24, 1967 ABSTRACT A variational principle, applicable to axisymmetric oscillations of uniformly rotating axisymmetric configurations, is established. On the assumption that the Lagrangian displacement (describing the oscillation) at any point is normal to the level surface (of constant total potential) through that point, it is shown how the variational expression, for the frequencies of oscillation, can be reduced to simple quadratures. The reduction is explicitly carried out for certain stratifications of special interest. Some new results on the oscillations of slowly rotating configurations are included; and a number of related observations on their stability are also made. I. INTRODUCTION A slow uniform rotation affects the radial modes of adiabatic oscillation of an initial spherical distribution of mass in two essentially different ways (cf. Cowling and Newing 1949; see also Chandrasekhar and Lebovitz 1962d): first, the term in the centrifugal acceleration in the equation governing equilibrium modifies the initial distribution of density and pressure; and second, the term in the Coriolis acceleration, in the linearized equations governing small departures from equilibrium, further modifies the characteristic frequencies. In general both these effects are of order ft2 (the square of the angular velocity of rotation); and they are found to contribute terms of opposite signs to
i=(3

7

_4)i^i

+

|(S_3

7

)0«

i

(,)

where SB is the gravitational potential energy and

/ = f^p(x)\x\*dx

(2)

(where the integration is effected over the domain 9J occupied b y the fluid) is the moment of inertia about the center of mass. (Note that in eq. [1] both SB and / refer to the rotating configuration and therefore include, implicitly, terms of order O2.) 1 In case y is not a constant, the condition is replaced by y > $, where y is a pressure weighted average fcf. {IVa, however). In this introductory section, we shall restrict our remarks to the case y = constant; toe restriction will not be made in the later analytical developments.

£267]

While Ledoux originally derived his formula from an application of the scalar form of the virial theorem,2 it was soon shown by Cowling and Newing (1949) that the formula equally follows from a variational expression for a2 with the same linear substitution for the Lagrangian displacement j ; (in terms of which a2 is expressed) that was made by Ledoux. And since in the limit 7 —»• f, the correct proper solution for £ is indeed a linear function of the coordinates (cf. Rosseland 1949, p. 20, remarks following eq. [3.14]), it follows that the formula |2Bo 2 2 2

= (37-4)^^+ffi +0(0<)

[ 7 - f = 0(ft )] (3,

(where 3Bo and Io now refer to the non-rotating configuration) is an exact one. The critical value of T for marginal stability is therefore 7 c

(Rot.)=i-^-+0(ft*).

(4)

The result (4) is of importance when effects besides rotation alter the value of yc from its "classical value" f. Thus, it is known that the post-Newtonian effects of general relativity have a destabilizing effect on radial pulsations and that (cf. Chandrasekhar 19646)

7c(G.R.)=f + * ! s + o ( ^ ) ,

«

where if is a certain calculable constant, Rs = 2GM/c2 is the Schwarzschild radius, and R is the radius for the configuration. Now if the effects of general relativity and of rotation are both present, and are both considered as first-order effects, then under their combined influence we must have , . ( R . t . + G.R., . , + , & _ ^ L

+

0

( „ . ^ | , f i ) .

This last result is the essential content of some recent papers by Fowler (1966) and Durney and Roxburgh (1967). While the stabilizing effect of rotation in the limit 7 —* ^ is an unambiguous result, it is not clear how small 7 — § must be to be of "order ft2" in a given practical situation. For the destabilizing effect of the distortion of the spherical distribution is the more dominating effect for centrally condensed configurations when 7 is not too close to the value 5 (see Table 1 in § IVc below). And it is pertinent to observe in this connection that the stabilizing contribution fft2 in equation (3) represents in turn a somewhat delicate balance between the stabilizing contribution (+8ft 2 /3) derived from the requirement of the conservation of angular momentum and the destabilizing contribution (—2ft2) derived from the distortion of the configuration (see § IV7> below). It is natural to wonder whether this balance might not be upset when the distortion ceases to be of order ft2. The question can be stated differently. Consider the variation of yc along a sequence of configurations of increasing ft2. We know that 7c(ft2), along such a sequence, has a negative slope at ft2 = 0. What is the behavior of the curve 7c(ft2) as ft2 increases? Does it always remain below3 7 = §? The present paper is the first of two devoted to a consideration of these questions. 2 For a somewhat more complete treatment, along the same lines but without any restriction on fi2, based on the tensor form of the virial theorem see Chandrasekhar and Lebovitz (1962a, § VII); and for the relation of this treatment to a variational formulation see Clement (1964, § III). 3 It is known that for compressible Maclaurin spheroids this is the case (Chandrasekhar and Lebovitz 1962c, Fig. 1). But these configurations are, of course, atypical since their mass distribution is highly unrealistic.

[268]

The plan of this paper is as follows. In § II, we formulate and discuss the variational principle appropriate for axisymmetric oscillations of a uniformly rotating axisymmetric configuration. In § III we introduce a class of trial functions that appears specially convenient for the present problem. The formulae of § III are applied in § IV to some familiar problems; and it is shown how they lead to results in agreement with those obtained by other methods. In § V we introduce a coordinate system that enables the reduction of the various multiple integrals appearing in the variational expression to simple quadratures; and in §§ VI, VII, and VIII we carry out the necessary reductions for some special choices of equilibrium stratification that we intend to consider in our second paper. II. THE VARIATIONAL PRINCIPLE

In this section we shall show that the characteristic value problem for axisymmetric perturbations of an axisymmetric uniformly rotating configuration can be put in the standard Rayleigh-Ritz form4 (see also Lynden-Bell and Ostriker 1967) just as in the absence of rotation (cf. Chandrasekhar 1964a). The equation governing equilibrium is grad p = p grad U ,

(7)

where p is the pressure, p is the density, and U = S3 + i02(x!2 + *22) •

(8)

In equation (8) 33 denotes the gravitational potential and it has been assumed that the rotation is about the *3-axis. Equation (7) implies that surfaces of constant p, constant p, and constant U all coincide. Accordingly, we may now write p = p(U)

and

p = p(U) .

(»)

The linear equations governing small perturbations of such an equilibrium configuration are (cf. Clement 1965; Lebovitz 1967)

where Z, is the Lagrangian displacement and the operator L is defined by L[Z] = —grad Ap + Ap grad U + p grad AU ,

(u)

where Ap, Ap, and AU are the Lagrangian changes in the respective variables resulting from the displacement £. These changes are given by Ap = —p div ?,

Ap = — ip div ? ,

(12)

and AU = 593+ I- grad U .

(u)

In equation (13), 593 represents the Eulerian change1 in 93 and is given by

4

One of us (N. R. L.) wishes to thank Dr. James Bardeen for a useful conversation on this point. Quite generally the operations A and 5 leading to the Lagrangian and the Eulerian changes, respectively, are related by A = 5 + % • grad . 8

[269]

It can be proved in great generality (cf. Clement 1964; also Lynden-Bell and Ostriker 1967) that the operator L is symmetric, i.e., for any smooth functions J; and n, f^n>L[Z]dx=f^-L[n]dx.

(is>

(It is this symmetry of L that insures the general variational formulation of the underlying characteristic value problem.) We now suppose that the perturbation considered is also axisymmetric. In cylindrical polar coordinates a, z( = x3), and
(16)

where we have integrated with respect to I, and/(o,z) is the "constant" of integration. As one can readily verify, equation (16) expresses the conservation of the 3-component of the angular momentum per unit mass. In what follows we shall set /(&J,z) = 0. For the present, we shall justify this assumption by observing that, for a normal mode for which Z(x,t) = $(x)e*" , (17) where a denotes a characteristic frequency, /(s,z) necessarily vanishes as long as a ^ 0 (for a further justification see below). Considering next the a-component of either side of equation (10), we have

2 a :)=£*m, t

<»>

or eliminating £v with the aid of equation (16) (with/ = 0), we have d2£a r

dt*

= !«[%] - 4 p f i 2 £ 0 .

(!»)

The z-component of either side of equation (10) gives dt*

= £.[?].

(2«)

Since neither LB[%] nor Z,,[£] involves £»,, equations (19) and (20) contain no reference to £v: it has effectively been eliminated. For this reason it is convenient to redefine £ to be the two-component vector K = fclo + U * ,

(21)

and this redefinition underlies the rest of this paper. As a consequence ^ no longer defines the complete three-component Lagrangian displacement. To obtain the latter, we must add to | the vector £„1,,, where £„ is to be determined in terms of Jo with the aid of equation (16) (with/set equal to zero). With J; redefined in the manner (21), we may formally combine equations (19) and (20) into the single equation

[270]

where 2 is now an operator acting on vectors in the (lo,l,)-space and whose components are given by &>[£] = £oR] - 4p$22£D and (23)6

8.M = !.[«• The symmetry of this operator 2 follows trivially from that of L. With a dependence of ? on time of the form (17), equation (22) leads to a characteristic value problem associated with the equation -0*PK = S[fl •

(24)

From the symmetry of the operator 2 it follows that the formula

(23)

f%plK{2dx

f^P(^2+^)dx

provides a variational basis for the determination of the least characteristic value of u2: it is tlte minimum value that the ratio appearing in equation (25) can assume for any smooth function £. This minimum principle for 0,

(26)

the equilibrium configuration is unstable; for the least characteristic value a-2 is then necessarily negative and leads to an exponentially growing mode. That the inequality (26) is a sufficient condition for instability has been deduced directly from equation (22) and independently of the theory of characteristic values by Laval, Mercier, and Pellat (1965). Their result is important, from the mathematical point of view, in our present context for two reasons -.first, the theory in terms of which we have argued has not, to the authors' knowledge, been rigorously established in sufficient generality to apply to the problem being considered; and second, since a solution of equation (22) of unstable type leads to a solution of equation (10), also of unstable type, through the simple expedient of choosing an initial value for d^/dl so as to make /vanish in equation (16), we are justified in setting/equal to zero as long as we are interested only in establishing sufficient conditions for instability. n i . THE FORM OF THE VARIATIONAL EXPRESSION FOR a2 FOR A SPECIAL CHOICE OF ?

In using the variational expression for a2 given in equation (25), we must exercise judgment in the choice of suitable trial functions for £: they must satisfy the requirements which experience and physical considerations suggest as necessary; at the same time they must make the evaluation of the various integrals as simple as possible. Now the principal obstacle to the requirement of simplicity is the presence of the term in 593 in the operator L; for 593 must be determined in terms of the chosen £, either from equation (14) or, equivalently, from a solution of Poisson's equation, V2593 =

-4TTG5P

= 4irG div (p£),

(27)

' Note that under the assumption of axisymmetry L, like 2 , is also an operator in the (10,1«) -space.

[271]

together with appropriate conditions on the boundary dTft of 9?. The choice of £ we shall presently make (eq. [34] below) is motivated, in part, by the desire to avoid this auxiliary calculation. First, we observe that equation (27) implies, quite generally, that (28)7

grad 533 = 4irGpi. + curl A

for some vector A. [Note that since the right-hand side of eq. (27) does not involve £„, and the left-hand side of eq. (28) has no ^-component, eq. (28) is in fact an equation in the (lo,lj)-space.] The contribution to

by the term involving 533 is, therefore, J^pt, -grad 8%dx = 47rGjTp21 ^ | Hx + f^pi, -curlAdx .

(29)

The integral involving A on the right-hand side of equation (29) can be made to vanish if we choose j[ so that p£ = grad and, on 39J, <j> = o = constant;

Uo)

for, then, I grad <j> • curl Adx = fdiv(cur\A)dx

Jut

-/$R

= I <j> curl A • d S

J&Sl

(31)

= 0o / I curl A • dS = 0 (where dS is the vector element of area of d9?). Hence for ^ of the chosen form f^pt

-grad 3 %dx =

4TG^P2

| ? | Hx (32)

=

4TrGf^pHtf+^)dx;

and this contribution to u2 (derived from the term in 533) is formally the same as in the absence of rotation. The substitution (30) corresponds to the correct solution both when rotation is absent and also when allowance is made for it to 0(fl2) (see § IVb below); it, therefore, appears as a reasonable choice for the general case even though it cannot correspond to the exact solution. In this paper, we shall further specialize 4> by requiring it to be a function of U only (which automatically insures the constancy of <j> on d$t); and since p is also a function of U (cf. eq. [9]), the assumption is equivalent to ? = x(^) grad U .

(33)

The present assumption (33) implies that, at every point x, ?(x) is normal to the level surface through x, while the original assumption (30) required this normality only on the boundary d9? of the configuration. (But as we shall see in § IV6 even this more specialized choice for £ is in agreement with the exact solution for the case of slow rotation, and a fortiori for zero rotation.) 7 In the absence of rotation A = 0 (cf. Rosseland 1949).

[272]

a) The Reduction of the Variational Expression for a"1 for the Chosen Form of the Trial Function It is convenient to define the effective gravity g at any point; it is given by g - [grad U\ .

(34)

It follows from this definition that for f, of the form (33) S-grad U = g2x,

OS)

div 3; = g2x + xV2Z7 ,

(36)

and where a prime denotes differentiation with respect to the argument U. By making use of the known relation V2U = - 4 x G p + 2 n 2 , (37) we can rewrite the relation (36) in the form div £ = g2X' - (47rGp - 2fi 2 ) X .

(38)

We turn now to the reduction of the variational expression for
• 2 [ * ] dx = - 4fi2 j f

P

tfdx - f^

• grad Apdx (39)

+ Cf • (Ap grad U + p grad

AU)dx.

With the expression for Ap given in equations (12), the second integral on the righthand side of equation (39) becomes - f^Z-grad

Apdx = f^Ap

div Zdx

J^Jpidiv

l)Hx

,

(40)

where we have integrated by parts and used the condition that Ap vanishes on 39?. Similarly, by making use of equations (12), (13), and (32), the third integral on the right-hand side of equation (39) becomes f t • (Ap grad U + p grad AU ) dx

=X

p t _ ( d i v ? ) U g r a d t / ) + 4 , r G p | ? | 2 + ? g r a d U g r a d U)]d X .

'

'

(41)

'

Now combining equations (39), (40), and (41) and making further use of the relations (34), (35), and (38), we find 'f^S2dx

= f^yp[

g*x' ~ drGp

-2W)xY-dx (

2

2

-j^PX [2(4TGp-0 )g

2

2

2

2

+ gradt/-gradg -40 (^) ](fx.

In this form, the variational expression, besides the known pressure and density distributions in the unperturbed configuration, involves only the single scalar function x-

[273]

There are two alternative forms of equation (42) which we shall find useful. The first of these is obtained with the help of the definition C( x) = div 1 v = div f- grad tf) (43)

2

= —i(gV U — grad [/-grad g). s Apart from a factor — 2, C represents the mean curvature of the level surface passing through x. With the aid of equation (37) and the definition (43), we find (cf. the integrand of the second integral on the right-hand side of eq. [42]) 2g 2 (4irGp-£2 2 ) + grad £/• grad g2 = 2g 2 (0 2 - VU) + g r a d t/-grad g2

- 2 ^- 2|V . OT [(^)" + (f)']_ 2fV .

(44)

Inserting this last relation in equation (42), we obtain

al

JstPX*gtdx

=

Jxyp[g2%'~

(4irGp

~2n2)xVdx

(45)

The second form of equation (42) which we shall need is obtained by noting that if p vanishes on d3J, j^PX 2 grad f/ -grad gHx = - f^[ px2g2VU

+ g 2 grad U -grad( P X 2 ) ]dx (46)

= j ^

2

2

[PX S (4XGP

2

2

- 2tt ) - g* -jjjiPX )

]dx

and hence

v* f^PX2g2dx = f^Yplx'V

~ (4^P

-

2W)x]2dx (47)

IV. APPLICATIONS TO PREVIOUSLY CONSIDERED STRATIFICATIONS

In this section we shall apply the formulae of § III to the non-rotating and the slowly rotating cases, and also to spheroidally stratified configurations. a) The Spherical Case: Q = 0

In this case the following formulae hold: ?==_

d%

77'

dp

1 d ,

Tr=~gp>

2

,

, _

y2j-r(r2g) = ^GP, (48)

1

x =

dx

—g77'

A

and

2

T

c = _

7[274]

Also the radial Lagrangian displacement, £r, is given by %r = -gx •

<•»)

Inserting these relations in equation (45), letting 4* = r2^ ,

(so)

and simplifying, we readily obtain the formula

where R denotes the radius of the configuration. Equation (51) is one of the standard forms in which the variational principle for the radial oscillations of a spherical mass is generally expressed (cf. Chandrasekhar 1964a, eq. [49]; also Ledoux and Walraven 1958). An alternative form of equation (51), which we obtain by integrating by parts the term in dp/dr in equation (51), is

In view of the minimum character of the underlying variational principle (see remarks following eq. [25]), the least proper value of a2 must be less than any value given by equation (52) so long as the chosen | r (i.e., \f/) is smooth. And with the particular choice f = r3 or £r = r, (53) we obtain the inequality 9fR{y~%)prHr /

f%r*dr

pr*dr

/

.

pr*dr

where, in the latter two forms, y denotes the pressure weighted average of y. From the inequality (54) it follows that a sufficient condition for instability is that y < ^. This result is of course well known: the basic formula, in precisely the form given in equation (54), occurs in one of Ledoux's early papers (1946, eq. [6]).8 A slightly sharper result (than the one we have stated) can be deduced from equation (52) in case y is not constant. Writing ^ = r3 + i{r)

(55)

(where e will be assumed small), we find from equation (52) that


+ 6efR(y-±)p^-dr+0(^).

(56)

If, for the sake of simplicity, we suppose that y = ^, we easily see that the right-hand side of equation (56) can be made negative by appropriate choices of (r) and of«, the "appropriateness" depending on the precise nature of the variability of y. In any event, it follows that when y is not a constant and 7 = §, the configuration is already dynamically unstable. * It should also be noted that Ledoux made use of the relation (54) (with the equality sign) to estimate the extent of the regions in which y may take different constant values (7 < J in some regions and y > J in other regions) before instability can set in.

[275]

6) The Case When it2 Is Small and Effects of Order O2, Only, Are Taken into Account In this case, we write f(r,li) = Mr) + ff/ifr.M) + 0(Q«) , (57) where /x is the cosine of the colatitude a n d / stands for any of the variables p, p, U, etc. The variables distinguished with the subscript "0" represent some appropriately chosen undistorted configuration (cf. Chandrasekhar 1933, or Chandrasekhar and Lebovitz \962d, § II). In the present approximation, in which all terms of order Q* and higher are neglected, it will suffice to extend the range of integration, in evaluating an integral, over the volume of the undistorted spherical configuration provided only the integrand vanishes over the boundary (for an explicit proof see, e.g., Clement 1965, Appendix I). For the integrals that appear in equation (45), the required condition is met if we restrict our consideration to equilibrium configurations for which the density p vanishes on the boundary; and this we shall do in the present subsection. A conclusion of some interest may be drawn immediately from equation (45) if we suppose that the trial function x may also be decomposed in the same manner (57). In that case the last term of equation (45) makes a contribution to a1 of the amount LPXH

2W Z* f^PxH

{dU/dwy-

(dU/dz)*]dx .

(dU/da)*

+

(M)

(dU/dz)*]dx

Consistently with neglecting terms of 0(S22), we must suppose that in the foregoing expression p, U, and x a r e functions of r only. Accordingly, we may write dU dU . —— = -j— sin 0t> da dr

, and

dU dU —— = —— cos 0. dz dr

(59)

On evaluating the expression (58) under these conditions, we obtain the value f fi2. This stabilizing contribution, already discussed in § I, is the only contribution of 0(Q2) to
m

fpx2(dU/day-dx

(dU/da)*+

(dU/dz)*]dx

and the remaining contribution — 202 may be traced to the distortion of the equilibrium configuration and the perturbation of the gravitational potential. The particular coefficients § and —2 that we find here are due to the replacement of p, U, and x by functions of r only. If the configuration is strongly distorted, and these replacements become invalid, it may be that this stabilizing influence is strongly modified or even destroyed. Reference to this possibility has already been made in § I. Returning to equation (57), we next observe that/i(r,/i) will normally have the further decomposition Mr,n) = Mr) + f»(r)P*(ji) , W corresponding to the fact that, in the present approximation, the effect of the rotation is a ^-deformation superposed on a uniform expansion (cf. Milne 1923). We shall adopt the same decomposition (61) for the trial function x, as well.

[276]

We have already remarked that, under the assumption that p vanishes on the boundary, it is legitimate to extend all integrations only over the undistorted sphere. Under these circumstances, the coefficients pn, pn, X12, etc., do not contribute to any of the integrals. And since we shall prescribe xo (for example, identify it with the proper solution belonging to
dp

dr

On substituting for x and U in accordance with equations (57) and (61), we obtain \dr

dr/

dfi

\dr

dr/

dp

Neglecting terms of O(O'), as we must, and substituting for xi and Ui in accordance with equation (61), we are left with dxo Tr -j-

_ dUg

U12 = - j —

X12 •

«4)

dr dr Equation (64) uniquely specifies X12 if xo is prescribed; but xio is left entirely arbitrary. We now return to equation (42) and simplify the right-hand side of this equation correctly to 0(fi 2 ) when the functions representing the various variables have the decompositions assumed. To facilitate comparison with the non-rotating case, we shall write (cf. eqs. [49] and [50]) r2dUa/

dr

Since the coefficients of Pi(n) in the expansion of the various quantities do not contribute to any of the integrals, it will suffice to include only the /io-terms in writing out the expansions of the various quantities to 0(0 2 ). Thus, we find C= and

--+. r r4

.. ,

(66)

r4

where dUio/dr dUa .... 7 7 T 7 7 - ^° + r T~ Xio • (68> dUo/dr dr I n equations (66) and (67) the dots stand for those terms of order O2 that occur with the factor P 2 (M) ; and for the reasons we have stated, we have not written out these terms. Turning now to the reduction of equation (45) and considering the first integral on the right-hand side, we find after a straightforward calculation, 1

[277]

=

In considering the second integral on the right-hand side of equation (45), we observe

Consequently, by virtue of equations (66) and (67), we have

(71)

A

f *. d /W\ dx

r , dpa dx

where we have performed an integration by parts. With the foregoing reductions, equation (45) becomes

(72)

We can recover Ledoux's formula from equation (72). (Note that the variables p and p still contain terms of orderft2.)Putting \f/a = r3 we obtain ^ P x V ^ x = 9 j ^ ( 7 - § ) ^ x + f ^ p „ r ^ x + 2 f 2 ^ ( 3 7 - 4 ) / » „ ^ ^ | , (73, after an integration by parts in the third integral on the right-hand side. Since xio is still at our disposal, we may choose it in such a way that q vanishes. By equation (68), we accomplish this with the choice dUio/dr With xio so chosen, equation (73) becomes (cf. also eq. [67]) ^J^prHx

= 9 ^ ( 7 - %)pdx + \Q*J^Por*dx

.

(75)

And equation (75) reduces to Ledoux's formula (1) if we make use of the relation

if^pdx=\m\-im\

(76)

provided by the virial theorem in the case of slow rotation. We now return to equation (72) to complete the reduction. Substituting for p and p their expansions according to equations (57) and (61) and writing a2 —
and

(78)

+J*»Hffl-<''h<$)$-'*J^»'-Z+*J***% [278]

If we now assume that i^o is the proper solution belonging to the characteristic value (To2, then \po satisfies the Euler equation 9 d /ypo
4 dp0 r3 dr

po^o r2

Equation (79) not only implies equation (77) but also implies (as one can readily verify) that the first term on the right-hand side of equation (78) vanishes for arbitrary q.10 We therefore obtain

(80)

dT r' ^ dT Pio^o2 — 7 + 1 / PO^O2 —To r JQ T* We notice the remarkable fact that this formula does not contain xio. Consequently to find the coefficient and p10) and the proper solution t/-0 appropriate to the undistorted configuration. For the special case of polytropes, a formula equivalent to (80) was derived by Clement (1965, eq. [65]). The present formulation of the variational principle with the trial function selected according to equation (33) is now seen to imply no loss of generality, since it leads to equation (80) which, in view of Clement's work, is exact.

J

c^

c) The Evaluation of a? for Distorted Polytropes As we have already stated, the form which equation (80) takes for polytropes was derived by Clement. Using his formula, we have evaluated cr22 for polytropic indices n = 1.5, 2.0, and 3.0 and for various values of 7. The results are listed in Table 1. We see that of becomes negative when 7 is sufficiently different from ^; and also that the value of 7 at which o-22 changes sign decreases with increasing n. We have referred to these facts in § I. However, it should be noted that there is some ambiguity in the comparison implied by the listings in Table 1, namely, that the characteristic frequencies of oscillation of a non-rotating and a rotating polytrope having the same parameters, n, pc (central density), and K (the constant of proportionality in the pressure-density relation), are strictly comparable. It is by no means obvious that this is the case: the two configurations that are "compared" have, for example, different masses and different volumes (cf. Chandrasekhar and Lebovitz 1962^, § LX). d) The Compressible Maclaurin Spheroid The pulsation frequency a2 and the critical value yc of the ratio of specific heats for marginal stability have been worked out exactly for this model (Chandrasekhar and Lebovitz 1962c). This model therefore provides a useful test case for the application of equation (45). The equilibrium configuration is described by the solution p = PU

and

U = K(I-—

2

-—\


where K = irGpaiWAz

(82)

• Eddington's pulsation equation in this context. 10 In fact the integral in question is proportional to the first variation of
[279]

and the notation is that of Chandrasekhar and Lebovitz (19626). With this solution, we readily find that 2 \l/2

gradt/=-2K(^l0 + ^ l , ) , and

c=

-(^+iiy

3/2

, = 2K(g + il) 7

(83)

rf-+-')-+--l

W a-SJ LW a32/<*i4 And as a suitable trial function we shall choose

aL2 a 3 4 J'

X = constant

(84)

since, in the absence of rotation, this choice leads to the exact solution. With the foregoing substitutions, we find from equation (45), after elementary integrations, -lK\(l

i 2 f l * ' + a »'1 I 2 C r 2 f l 3 2 - g l 2

I ' ^ T

(85)

or, in terms of the eccentricity e of the meridional section, we have 2

*2

= 1[ V2I S( 3—- V2/>,* 2I
( 3

=

•KGP

-

fi e 2 ) ( 1 e 2 ) — 2- I 4.-1-7-

U+2-i Gp

3-2e

-jri

J

l-2e2 3-2e2

(86)

For small values of e, equation (86) gives •KGP

= 4 ( 7 - § ) + i | ( 5 - 3 7 ) e 2 + 0(e4);

(87)

this agrees with the exact formula (and with eq. [1]) through terms of 0(ft2). TABLE 1 VALUES OF O02 AND O-22 FOR POLYTROPES* 7

(TO2

ff22

y

0.007755 .019334 .030850 .053710 .098812 .121094 0.150560

+0.30985 +0.27430 +0.23837 +0.16546 +0.01594 0 -0.06051 -0.16394

» = 2.0

1.40 1.45 1.5076 1.55 1.60 1.6667

0.024752 .042805 .077773 .094757 0.116956

U22

» = 3.0

B = 1.5

1.35 1.375 1.40 1.45 1.55 1.5605 1.60 1.6667

(TO2

1.35 1.375 1.40 1.4140 1.45 1.50 1.55 1.60 1.65 1.6667 1.70

0.004010 .009755 .015182 .025110 .033894 .041680 .048642 .054951 .056935 0.060755

+0.27697 +0.18085 +0.06946 0 - 0 20294 —0 54011 —0 92946 -1.35145 -1.78510 -1.93002 -2.21784

+0.21248 +0.11626 0 -0.08913 -0.19761 -0.34747

* The values of
[280]

*•- Go t o H -

O

H - W M t O U On O J ON O O C*J t-» NO tO --J

O

O "*4 -"-Irf*-C*>

OJ <>> O J CKJ OJ

0 \ * 0 w 4-> ^ *

O

O

O

n

A

1

(exact)

(exact)

(approx.)

1.3052 1.2863 1.2533 1.1848 1.0009

rf

-J

1

H W

§ >

X

O

to

^

rfl

a

>

o H > o H

« w M

o •z. o

C/3

>

o

o

> w1 tw

0 0 S M M 0 \ 0 > O I

- — O O O O O O O O Q O

O O i U ' - O O O O ^ ' ^ O v O O v 00t*>GJ-*j4*t#Jtn00C*»*O-*-lO\ vO»—GtOirf*.00>—GJNiC/i^O

0 0 0 ^ 0 * ,* - » O i 4 * ' U ) C * i O J { > J O o U i H-»N3 O0\O»tnOtn^0J*k»-='

1

C n K i O O O J - ^ l O ,- 4 H - . t — 4 ^ . 0 ) 0 0 O ^ O r > s - l - ^ ^ i - ' O i - 4 00

O C ^ H I M S ) [ O * » . 0 0 C * J O V I 0 \

O j O J C M U Ul M t O, f M tOtOtOig 004^Nii-*O O00 c-l"^O\0vOv

tr-h-wOvUUi^OOOjQ •OONCnO\ONOGit--»0\*©

O O O O V ^ O J I O H O O Q O

S OS OOI pI &OW OW OI - O' OO OQQ OO

vlSQl>lK]tO)U00UO O M O O J O a i t O C n O O 00O4^0NCn--jO0sO00

I I I I I o

>-^ •—*»—»H*H— t-» fcON00*OO»--(OtOOJ0J

I I I1 o3 O O O

i ^ O^ OO O O H - N J t o OJ OJ O t>) r - 00 O ' O -v| 00 VJ (O

I I I I I I I I I I O O OOOO 1 "**"**"* 1 - * H-» t-*

OOUiOtnSlnOlnoln

0N0N00*-00CnOtOC*iO4-«O

O O O O O O O O O O O O 00~JsON>-a4^*~*'000'^OvO*

^JH»0\,OK)Ui{ktflC>0\t>0\ tosOj^^JOOOOONH-CnCCO* h-^l^JWiE'OtOH-ON^i^OO

O O O O O O O O O O O O

»f*OOi-»if».OOKi~aC*l000^1t> C*J SO 4*. " >t» H- tO

K 3 O O C 0 0

•»^ -*J »^j O s O * 0 * 0 > 0 0 * 0 \ C > > 0 \

>OiOCnOC/iOCnOCn-

693

V. THE REDUCTION TO QUADRATURES OF THE INTEGRALS IN THE VARIATIONAL EXPRESSIONS FOR a2

In order to facilitate the evaluation of the integrals appearing in equation (42), and in its alternative forms (45) and (47), we shall introduce a system of coordinates s, t>, and

{x:S(x,s)=0},

With

(88) $R(*) = {x:S(x,s) <0] .

For definiteness we shall suppose that 35 — < as

0

in$R,

(89)

where SR is denned by equation (88) with s set equal to its maximum value si (say), i.e., 9? = $R(5i) .

(90)

The condition (89) implies in a natural way that ®(/i) is interior to ©(£>) if h < k. It further implies that there is a unique surface © passing through a given point x. Finally, we shall also require that there is a minimum value of s, so (say), such that S(x,s<>) is positive unless x = 0; this insures that So corresponds to the origin. Now let the position vector of a point lying on the surface ®(s) be given by * = X{s,&,
(9i)

where the surface parameters d- and

dX , _ .. -—-grad5 = 0 , ai>

, and

dX . „ _ -—grad 5 = 0 . dip

(94)

The last two of these equations imply that there exists a scalar function a(x) such that dX , dX . . , ., -r^rX — = a ( x ) g r a d . S . ov- a
(95)

[282]

The expression (93) for the Jacobian now becomes J = -—agradS=— a—. (98) ds ds Eliminating a in equation (96) with the aid, once again, of equation (95), we finally obtain •T, \(.dX/d»)X(.dX/dy)\ dS m

=

m

17'

\^dl]

where the minus sign has been inserted to accord with our convention that dS/ds is negative. Returning to our principal problem, we are primarily interested in evaluating integrals whose integrands have the form G(s)H(x), where H, in general, depends on # and
f'lG(s)W(s)ds,

pi)

where W(s)= ff\J\H{X)dMv,

(99)

where, as the notation implies, the dependence of H on x must be replaced, by means of the transformation equation (91), by its dependence on s, #, and
SPHEROIDAL STRATIFICATIONS

It is possible to show that, in a rigorous theory, a uniformly rotating configuration cannot be stratified on spheroids unless the density is everywhere the same (cf. Dive 1952). Since, however, on an approximate theory, slowly rotating configurations are stratified on spheroids, it is reasonable, when seeking approximate equilibrium configurations, to assume spheroidal stratification and try to find that set of spheroidal surfaces which is, in some sense, the best. This is the approach taken by Roberts (1962, 1963) and by Hurley and Roberts (1964a, b), who have constructed rapidly rotating polytropes selecting the "best" stratification on the basis of a variational principle. We shall adapt equation (47), appropriately, for a study of the oscillations of these models of Hurley and Roberts. The function 5 of equations (88) may, in the present case, be written as 5(fflAl)=i(-J» + B , + J ^ i ) ,

(100)

where e is, in general, a function of s. The coordinate s is the equatorial radius of a given level surface and e(s) is the eccentricity of its meridional section. From equation (100) it follows that dS , ee' , / , de\ —

ds and

r--Z2

= - 5 + - ^ -

(1 — e 2 ) 2

^rad5i=[a2+u^l *2

[283]

[e'=

- _ ) ,

\

ds/

(101)

-)l/2 (102)

It is convenient to introduce the coordinates s, &, and

and

x3 = (1 — e*)Uis cos t? ; (103)

t? is then the "mean anomaly" and tp is the usual azimuthal angle. We now calculate \J\ as given by equation (97). Since e'(s) > 0 in all cases of interest (cf. Hurley and Roberts 19646; in the slowly rotating case, this inequality follows from Clairaut's equation) we may define, without ambiguity,

Pis ) = ( r ^ 7 i ) 1 / 2

<«'£<»• (io<>

With this definition, equation (101) takes the alternative form — = - 5 ( 1 ~ / 3 2 COS 2 #).

(10S)

as Our earlier requirement (89) that dS/ds < 0 now implies that /3 < 1; we shall assume that this is the case. [It can be verified that the difficulty encountered by Hurley and Roberts (1964a) with the crossing of the level surfaces is related to a violation of this inequality.] A straightforward calculation now gives / = (1 - ^ " ^ ( l - /32 cos2 d) sin & ;

doc)

and equation (99), for the stratification we are presently considering, becomes W{s)

=2ir(l-e2)u*s*f+\l-Pn*)H(s,lx)dii,

d07)

where we have used n = cos i> as the variable of integration. Inspection of equation (47) now shows that to reduce the various integrals occurring in this equation to quadratures over s, it is necessary to evaluate four functions W(s) with the choices

*-,. *-(£)". *-,-(£)•+(£)'.

-w.«-

We shall denote the corresponding functions by Wi(s) (i = 0, 1, 2, and 3). It will appear that all these functions can be expressed in terms of elementary functions. First, we note that since the potential U is a function of s only, we may write (to avoid ambiguity) U = M(J) ;

(109)

and we have

/auy

/duy/dsy

„

/duy

and

Cw-CirKis)

/duy /asy

br)=UKw-

On the other hand, the derivatives of S(a,z) may be obtained by implicit differentiation of equation (100); we find dS _ ( 1 - M 2 ) 1 / 2

da

1-^2M2

.

a5

dz

=

M

(1 - e2)1/2(l - ^ 2 M 2 ) "

U1

[284]

Writing out the functions Wi(s), and simultaneously defining the new functions Vi(s) (for later convenience), we have *J o

(112)

= 47r(l-e2)l/2(l-i/32)52,

^G9*".">-^">-'(»->'»-GI) , / , T^'" and

w(#"",*<'>-™.)-«w1-..)«,.(Eru The integrals over /* appearing in the definitions of the functions V2(s) and V3(s) are also readily evaluated, and we find

and

fn5)

+

(5 3 i

i +

(i-'y)-[ ~ ' '" '

i^

(1+
(5 8

3)

' ' (i-' t -)' ' '~ 1i'

where

With the foregoing reductions equation (47) finally takes the form

H/>.(g)V.„./;g;)^(px-)^, -4^,,px2(^J[(37rGP-n2)72-fi2K1]tf5. This equation represents a generalization, to the case when rotation is present, of equation (51) valid in the absence of rotation. The Euler equation based on equation (117) will provide a generalization of Eddington's pulsation equation analogous to Roberts' generalization of the Lane-Emden equation (Roberts 1962, eq. [2.31]). However, we shall have no reason to write down this generalization.

[285]

VH. THE ROCHE STRATIFICATION

The spheroidal stratification considered in § VI is an extrapolation to larger rotations of the stratification that is known to be correct for slow rotation. In contrast, a stratification that will apply to all rotations, with sufficient approximation under most circumstances of practical interest, is that of Roche which one obtains when all the mass (M) is concentrated at the center. The level surfaces in this limit are given by (cf. Jeans 1929 p. 252) f~*7(JF

U Roche

-Kn2r2sin2#.

(lis)

While these level surfaces are strictly applicable only in the limit of infinite central condensation, they do provide a very good approximation to the true level surfaces of rotating masses of even moderate central condensations such as a polytrope of index n = 3. For example, the ellipticity e of a polytrope n = 3 for slow rotation is given by (cf. Chandrasekhar 1933) € = 0.7717^5-^—

(no)

(where pm denotes the mean density), while for the Roche surfaces £ = 0.75^-^—

(P'» = r~™)-

(120)

The discrepancy, for slow rotation, is thus less than 3 per cent even for the outermost boundary. The assumption that even for rapid rotation the Roche surfaces provide a very good approximation to the true level surfaces of rotating masses, with central condensations comparable to the polytrope n = 3, has been confirmed by Monaghan and Roxburgh (1965a, b) and by Ostriker and Mark.11 Indeed Roxburgh, Griffith, and Sweet (1965) have made the approximation provided by the Roche surfaces the first step in a rapidly converging iteration scheme for obtaining the exact solutions under a wide variety of circumstances. It would therefore appear that, for a discussion of the oscillations and the stability of rapidly rotating masses, the reduction of equation (45) for the Roche stratification will provide a good starting point. It is important to observe that the foregoing remarks on the good agreement of the level surfaces in rotating configurations with the Roche surfaces are not meant to suggest that the march of U in these configurations agrees well with the march of U in the Roche model. They are meant rather to point out that the level surfaces in rotating configurations (of even moderate central condensations) can be geometrically well approximated by trie Roche surfaces so that in the same approximation we may consider U as a function of Z/Roche, regarding the latter only as a labeling parameter. In other words, the assumption that we propose to make is U = /(^Koche) ,

(12D

where / is some function (to be determined empirically from the integrations such as those of Ostriker and Mark). Returning to Roche's equation (118), we first observe that the condition, that for equilibrium the effective gravity should be directed inward, requires that the equatorial radius of the outermost level surface does not exceed the critical value Oc = (GM/ff) 1 ' 3 .

(122)

11 We are greatly indebted to Dr. J. Ostriker and Mr. J. Mark for providing us with the results of their exact numerical integrations for rapidly rotating polytropes.

[286]

By choosing ac as the unit of length and 0 as the unit of frequency, we can rewrite equation (118) in the form v = - + f r 2 sin2 &, (123) r if i/Roche is measured in the unit (GMQ)213; and we suppose that U (also measured in the same unit) is some determinate function only of v. Accordingly, the corresponding form of the function S of equations (88) is S = S(r,»,s)=j+%rHl-S)-v[U(s)].

(m)

We now write the variational expression (42) for
© , jt« , ' ,rf *-^[(^) , « 4 - 4(2 e- 1) ^« ,+4(2 G- 1 )v>

X

(12S)

where g is measured in the unit (GAfQ4)"3, and Q = 7rGp/fi2 ;

(i2«)

also the unit in which p is measured is (GMQ,)213 times the unit of p. Since we have supposed that U is some function of v only, g2 = ( - J - ) I grad ? | 2 ,

grad U = —^ grad v, and gradJ7-grad g* = (-j~J

[ grad x • grad ( |gradv| 2 ) ] +2 (— J —— |gradi/| 4 ; (127)

whereas from equation (123) we find

and

(us) 2

grad.'-grad(|gradH )=-

4

?-2(-^

2

2

+ ^-r )(l-M ).

Further, taking s = — v (the minus sign is needed because v decreases outward, whereas s, by assumption, must increase outward), we find that the required Jacobian is '""

i_ri(„>M)(i-M«)

3 - 2 ^ ^ ) '

where the latter form is obtained by making use of equation (123). With the definitions (cf. eq. [107]) a Eq. (129) follows directly from a formula given in § VIII (eq. [133]) which is applicable to entirely general axisymmetric level surfaces: we have only to identify S in that formula with the surface S defined by equation (123) and let s — —v.

[287]

VO(P) =v* f \J\dn, -'o 1 r1

Vx{v) = f \J\ |grad*| *
VAv)=-iJg\J\\g™dv\*dfi,

Vi(v)=j^ \J\{j^)d»,

1

/-i

Vs(v) = - j J

v- J0

< 1 3 «> 1 3

\J\ grade- grad( [ grad x 12)dlj.,

equation (24) may now be written in the form

-M"«-n(f)'K, + ,(f7, + 2 ^K.)(f) , -^a.„ J , M(®'(f)>'-«"«-"^^"-+*' c-'' «'5>+ where the integrations are over the appropriate range of v. Table 4 gives V0, Vi, V2, V3, and Vi as functions of 1/v. In applying equation (129), we may substitute for p(v) and p{v) the distributions derived for configurations of at least moderate central condensations: as we have stated, the Roche stratification may be expected to provide a sufficiently good approximation for them. VIII. GENERAL AXISYMMETRIC CONFIGURATIONS

A general axisymmetric configuration may be specified by giving the distance R(s,n) from the center to a point on the level surface at colatitude d- = cos -1 p. The associated coordinate transformation (91) has the form .ti = R(s\n)(l — M2)"2 cos
x3 = i2(5,/i)jt ; (U2)

and a straightforward calculation gives (cf. eq. [97]) T

_„ .

.

dS/ds

We may take S ( V , M ) = r - R(s,fj.) ,

(134)

as the equation defining the level surfaces; but to be in agreement with our convention (89) regarding s we must assume dR/ds > 0. With this definition of 5 equation (131) gives

where, as in the preceding section (see n. 12), we have taken s = — U. It also follows from equation (132) (with s =— U) that

/dU\*_(R+pdR/dy)*

2

R*+(l~S)(dR/d^

13 The factor vs in the definition of Vt,(v) and the factor v~* in the definitions of ViM and VJ(K) have been inserted in order that these functions may not vary over too wide a range of values.

[288]

700 TABLE 4 T H E WEIGHT FUNCTIONS FOR THE ROCHE STRATIFICATION V,

0.0. .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 .23 .24 .25 .26 .27 .28 .29 .30 .31 .32

.33 .34 .35 .36 .37 .38 .39 .40 .41 .42 .43 .44 .45 .46 .47 .48 .49 .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 .61 .62 .63 .64 .65 0.66

[289]

.000000 1.000000 .000000 0.999988 .000001 0.999990 .000048 0.999980 .000125 0.999956 .000248 0.999916 .000430 0.999855 .000685 0.999771 .001024 0.999658 .001460 0.999513 002004 0.999332 002670 0.999111 003470 0.998846 004417 0.998532 005524 0.998165 006805 0.997742 .008273 0.997258 009943 0.996708 .011829 0.996089 013948 0.995396 .016313 0.994623 018944 0.993768 021856 0.992824 025068 0.991788 .028600 0.990654 032472 0.989416 036706 0.988071 .041327 0.986611 046358 0.985032 051828 0.983328 057766 0.981492 064204 0.979518 071176 0.977399 078721 0.975128 086879 0.972697 095698 0.970099 105227 0.967325 115522 0.964367 126646 0.961214 138669 0.957856 .151669 0.954282 165733 0.950481 180962 0.946439 197470 0.942143 215386 0.937577 234860 0.932725 256065 0.927569 279202 0.922087 304508 0.916258 332263 0.910056 362800 0.903452 396521 0.896416 433917 0.888910 475589 0.880892 522291 0.872313 574979 0.863115 634889 0.853231 703651 0.842577 783476 0.831052 877447 0.818531 990038 0.804849 128060 0.789791 302565 0.773057 533135 0.754208 859343 0.732543 380898 0.706770 506312 0.673723

1.000000 1.000008 0.999978 0.999912 0.999788 0.999584 0.999281 0.998857 0.998294 0.997572 0.996669 0.995568 0.994247 0.992688 0.990872 0.988778 0.986387 0.983682 0.980642 0.977250 0.973488 0.969337 0.964780 0.959801 0.954382 0.948507 0.942161 0.935329 0.927996 0.920148 0.911773 0.902858 0.893392 0.883365 0.872767 0.861590 0.849827 0.837472 0.824521 0.810971 0.796820 0.782069 0.766721 0.750779 0.734250 0.717144 0.699471 0.681245 0.662484 0.643209 0 623444 0.603216 0.582559 0.561510 0.540112 0.518414 0.496473 0.474353 0.452127 0.429880 0.407711 0.385734 0.364084 0.342927 0.322466 0.302972 0.284843

0 0.040000 0.079999 0.119996 0.159986 0.199967 0.239931 0.279872 0.319781 0.359650 0.399466 0.439218 0.478893 0.518474 0.557947 0.597294 0.636495 0.675531 0.714380 0.753018 0.791422 0.829564 0.867419 0.904955 0.942143 0.978951 1.015343 1.051284 1.086737 1.121661 1.156015 1.189754 1.222832 1.255202 1.286810 1.317605 1.347529 1.376522 1.404521 1.431459 1.457267 1.481868 1.505185 1.527132 1.547620 1.566553 1.583827 1.599333 1.612951 1.624552 1.633996 1.641129 1.645782 1.647770 1.646884 1.642890 1.635520 1.624470 1.609379 1.589821 1.565273 1.535080 1.498384 1.453995 1.400126 1.333695 1.247630

0.666667 .666667 .666662 .666646 .666616 .666567 .666494 .666392 .666257 .666083 .665866 .665600 .665282 .664905 .664465 .663957 .663375 .662715 .661972 .661139 .660211 .659183 .658049 .656804 .655440 .653952 .652333 .65057/ .648676 .646624 .644412 .642033 .639478 .636739 .633806 .630668 .627316 .623739 .619924 .615858 .611527 .606916 .602010 .596789 .591234 .585324 .579036 .572342 .565213 .557617 .549517 .540869 .531625 .521730 .511118 .499712 .487419 .474128 .459702 .443965 .426694 .407587 .386227 .361994 .333888 .300056 0.255881

It is now clear from equations (98) and (99) that to reduce equation (47) to simple quadratures over s, we need the following functions: V0(U)=f_+il\J\dli,

Vl{U)=f*\*\J\dli,

V2(U)=f_*\*\J\d»,

and

(U7)

In terms of these functions equation (47) becomes

*f

• u l'

„„ ,.r

PX'v1dU=

JUt

ru>

Jul Vl

V

+ f yp[(^J ^

d ,

ru*

« , ,„

w f '-^(px'Widu-if

'PXMOTGP-n^-nv.jiiy

du

JUt 2

~ 2(4xGp ~ 20 )x jffVi+

(138) G

2

(** P ~ 2&) )cVo]dU

.

Equation (136) should be especially suitable for the numerical evaluation of a2 for the rotating polytropes of James (1964, 1967). IX. CONCLUDING REMARKS

That slow rotation has a stabilizing effect, in lowering the critical values of y (for marginal stability) below the value ^, is an unambiguous theoretical result. When one seeks the origin of the stabilizing effect in the analytical treatment, one finds that it is directly traceable to the Coriolis term in the equations of motion which leads to the relation £,, = (2jft/o-)£o (cf. eq. [16]) between the
[290]

REFERENCES Chandrasekhar, S. 1933, M.N.R.A.S., 93, 390. . 1964a, Ap. J., 139, 664. . 19644, ibid., 140, 417. Chandrasekhar, S., and Lebovitz, N. R. 1962a, A p. J., 135, 248. . 19626, ibid., 136, 1037. . 1962c, ibid., p. 1069. . 1962a", ibid., p. 1082. Clement, M. 1964, Ap. J., 140, 1045. _ . 1965, ibid., 141, 210. Cowling, T. G., and Newing, R. A. 1949, Ap. J., 109, 149. Dive, P. 1952, Bull. Sci. Math., 76, 38. Dumey, B. R., and Roxburgh, I. W. 1967, Proc. Roy. Soc. London, A, 296, 189. Fowler, W. 1966, Ap. J., 144, 180. Hurley, M., and Roberts, P. H. 1964a, Ap. J., 140, 583. . 19646, Ap. J. Suppl., 11, 95. James, R. A. 1964, Ap. J., 140, 552. • . 1967 (private communication). Jeans, J. H. 1929, Astronomy and Cosmogony (2d ed.; Cambridge: Cambridge University Press; reprinted 1964, Dover Publications, New York). Laval, G., Mercier, C , and Pellat, R. 1965, Nucl. Fusion, 5, 156. Lebovitz, N. R. 1967, Ap. J., 150, 203. Ledoux, P. 1945, Ap. J., 102, 143. . 1946, ibid., 104, 333. Ledoux, P., and Walraven, Th. 1958, Hdb. d. Phys., ed. S. Flugge (Berlin: Springer-Verlag). Lvnden-Bell, D., and Ostriker, J. 1967 ^I.N.R.A.S., 136, 293. Milne, E. A. 1923, M.N.R.A.S., 83, 118. Monaghan, J. J., and Roxburgh, I. W. 1965a, M.N.R.A.S., 129, 5. . 19654, ibid., 131, 13. Roberts, P. H. 1962, Ap. J., 137, 1129. . 1963, ibid., 138, 809. Rosseland, S. 1949, The Pulsation Theory oj Variable Stars (Oxford: Clarendon Press; reprinted 1964 Dover Publications, New York). Roxburgh, I. W., Griffith, J. S., and Sweet, P. A. 1965, Zs. f. Ap., 61, 203. Copyright 1968. The University of Chicago. Printed in U.S.A.

[291]

SOME ELEMENTARY APPLICATIONS OF T H E VIRIAL THEOREM TO STELLAR DYNAMICS S. Chandrasekhar and Donna D. Elbert (Received 1971 September 1) SUMMARY The dynamical evolution of spherical and spheroidal systems of mass points is examined with the aid of the scalar and the tensor forms of the virial theorem. Spherical systems with positive total energy tend to disperse to infinity while those with negative total energy execute periodic oscillations of finite amplitude. Spheroidal systems with positive total energy exhibit expansion like the spherical systems; but they also become less oblate (and sometimes actually become prolate) if they are initially oblate, and less prolate if they are initially prolate. Spheroidal systems with negative total energy collapse to smaller volumes while enhancing their initial oblateness or prolateness.

I . INTRODUCTION T h e most common (if not the sole) application of the virial theorem to stellar dynamics has been for the purposes of estimating the kinetic energy, %, of a group ('cluster') of stars (or galaxies) from their dimensions. T h e basis of this estimation is, of course, the relation 25C + 1JI = o

(1)

provided by the scalar form of the viral theorem under stationary conditions. T h e gravitational potential-energy Wi that occurs in equation (1) is, in practice, estimated from a dimensional relation of the form

M = -

2

-

r

,

(2)

where R is some average measure of the linear dimension of the system and M is its mass. In this paper we shall outline some other elementary applications of the nonstationary form of the virial theorem (in both its scalar and tensor forms) to suggest the scope of its applications that remains to be exploited.

2 . THE VIRIAL THEOREM

For a system consisting of a number (AO of discrete mass-points (or particles) under their mutual gravitational attractions, the tensor form of the virial theorem takes the standard form (cf. Chandrasekhar 1964) --^l=2®ij+mv,

(3)

Figures 3 and 5 have been replaced to correct oversights. I am grateful to Dr. R. Miller for pointing out the errors and providing corrected figures—S. Chandrasekhar. [435]

where 7y, Giy, and M y are, respectively, the moment of inertia, the kinetic-energy, and the potential-energy tensors given by Iij = E »!<«>*«<•>*><•>,

l y = J £ jn<«)tt(Wlly(«),

a

(4)

a

and

w„ = - 1 G ? _ s r , « » ^ = ^ ! I s ^ l .

(5)

In equations (4) and (5) the Greek superscripts distinguish the different mass points while the Latin subscripts distinguish the different Cartesian components of the position x and the velocity u of the different particles. The contracted version of equation (3) provides the usual scalar form of the virial equation, namely, 1 dH 2 ^ 2 = a S + W = a
(6)

where IB =% + W

(7)

is the total energy of the system that remains constant. In applying the virial theorem to a system of particles described statistically in terms of a density distribution p(x), the common procedure is to use the same equations (3) and (6) altering only the definitions of Zy, {Jy, and M y as sums to integrals; thus, hi =

pxiXj dx,

£y = £

puiUj dx,

(8)

and M y = - \G j J p(x)p(x>) (*'

* l ) ( g r 3 * ' - 3 dx dx'.

(9)

But this procedure is not strictly justified. For, if one wishes to describe a stellar system in terms of distribution functions, then one must start with an ensemble of systems described by a density function /W(X<1>, X<2>

X(iV)

; u d ) , U<2>

U<*>; t)

(lO)

defined in the six iV-dimensional phase space of the system and consider the hierarchy of equations that one obtains by integrating the Liouville equation g o v e r n i n g / ^ over the coordinates and velocities of all particles except a particular one, of all particles except a particular pair, and so on. And whatever integral relation one uses must be a consequence of this hierarchy of equations. Equation (3) with the definitions (8) and (9) is not such a consequence. However, the correct generalization of equation (3) to stellar dynamics has been derived in an earlier paper (Chandrasekhar & Lee 1968). Thus, by separating the ' mean ' (U*1*) from the ' peculiar ' (v*1') velocities in the manner «,(« = w (i)+ <«,(»> = vtW+Utm, and defining a pressure tensor Py

(1)

(11)

by the relation

7i = n(«<w<1)^<«> + n<«L^<1)I/y<1;)

[436]

(where «<1>(x<1); t) is the one-particle distribution and the angular brackets denote ensemble averages), the following integral relation was derived: j 9 j

I

~j-:g =atr«+n t f +w y ,

(13)

2 at*

where

« y = 1« f n(i>^(DC//i) <&<«,

(14)

n « = f P « ( " <**<«, ® y = - ^G7VW2 J J nd. »(x(W, x<2); t) - [ — ' ^ ' f f ^ . ! ^ ^ <*x<1) ^ (2) > and n*1- 2>(x(1), x<2>; <) is the two-particle distribution defined by „(1, 2) = f

/J

(IS)

f /(iV) JX(3) , U<2); *) dllW

rfu<2).

(16)

In deriving the foregoing equations the further simplifying assumption has been made that all the particles have the same mass m. From equation (15) it is clear that it is only when we ignore the two-particle correlations and assume that ad. 2) = _L „<1>(XU>; /) «(D(x<2); *),

(17)

does this new definition of Wiy coincide with the common definition (9). When there are no mass motions and U*1* = o, equation (13) takes the more familiar form 1 d2Ii] 2 d&

= ny+w«,

(18)

where

Ut, = f Pw(xw, t)
(19)

and

Mi] = - \G J J pd. 2)(X(1), X(2); t) [ * < W T^ 8 > l [ *ff|7^ ( 8 > ] dxW &<».

(20)

And these last equations are equivalent to equations (3), (8) and (9) only if we ignore two-point correlations and assume that ptt. 2>(x; f) = pd)(x(«, t) p(1>(x(2), t).

(21)

In Sections 3 and 4 which follow, the two-point correlations will be ignored and the analysis will be based on equations (3) and (6) together with the definitions (8) and (9). In Section 5 a first attempt will be made to include the two-point correlations. [437]

3 . THE DYNAMICAL EVOLUTION OF A SPHERICAL SYSTEM

Suppose that at some initial instant, a cluster is characterized by a certain moment of inertia and kinetic and potential energies. The evolution of such a cluster, while it must be consistent with equation (6), it cannot, of course, be uniquely determined by equation (6). On the other hand, if we choose to describe the cluster ' grossly ' by a certain linear size, then it would be proper to inquire how this size changes with time. Such a description can, at best, give one only the crudest indication of what might happen. The following analysis based on equation (6) is undertaken with the hope that even crude inferences concerning the evolution of isolated clusters (of stars or galaxies) are of some interest. In the spirit of the foregoing remarks, it should not also matter (within limits) how we choose to describe the internal distribution of density in the cluster. Accordingly, to be specific, we shall choose a spherically symmetric cluster of iV equal mass-points with a gaussian distribution of density P(') = ^ 2 e X P ( - r 2 / * 2 ) '

(")

where a = a(t) is a measure of the radius of the cluster. For clusters in which the distribution (22) continues to be maintained (by assumption!) equation (6) leads to the following differential equation for a: 3 *7 d2<*2 «, GN2m2 1 3iVOT_=2(B+I_yi7I-.

, , (23)

It should be noted here that if instead of a gaussian distribution, we had assumed a homogeneous distribution inside a sphere of radius a, we should have obtained an exactly similar equation with only somewhat different numerical coefficients; and the equation will be identically the same in the non-dimensional form given below (equation (25)). The particular assumption (22) is therefore not an essential one so long as we assume that the evolution proceeds homologously. Letting a = aoz

(24)

where ao is the value of a at some time to be specified later and measuring time in the unit _r/VV/2

ao3]1'2,

(25)

we find that equation (23) becomes 5-2=2(sgn
(26)

where sgn
o-liSir

(27) [438]

and TSBo is the potential energy when a — OQ. It is seen that equation (26) admits the first integral I -j- I = (sgn (g) Qz2 + z + constant.

(28)

At this point it is convenient to distinguish systems with positive or negative total energies. (a) Systems with positive total energies In this case, suppose that a = ao, z = 1 and

dz — = o at time dt

t = o.

(29)

Then equation (28) can be written alternatively in the form

Z2(J)2=(*-I)(!?*+2+I)-

(3°)

The required solution of this equation is ' =g[(*-I)(2*+2+I)]

1 / 2

-^

2

tanh-i

0(*-i) [Qz + Q+i

1/2

(3i)

and for the particular case Q = o, the solution is t = | ( * + 2 X « - l) 1 / 2 -

(32)

The variations of z with t that follow from equations (31) and (32) are illustrated in Fig. 1 for various initially assigned values of Q. As might be expected, the solutions show that systems with positive total energy are eventually dispersed.

10

1-5

20

25

30

3-5

40

4-5

5 0

FIG. I . The evolution of a spherical cluster with a gaussian distribution of density; the case of positive total energy. The abscissa (z) measures the mean radius (a) of the cluster in terms of its initial value (ao). The ordinate measures t in the units specified in equation (25). The curves are labelled with the values of Q ( = |(E|/|tSE|)to which they belong. [439]

(b) Systems with negative total energies In this case, it is convenient to rewrite equation (28) in the form

„(*)'_A.- B (,-£)•,

,33,

where A 2 is now the constant of integration. The solution of equation (33) is

f

= ^7i[A'-(*-^)T/,+^0™"1 &(*"^)]'

(34

>

if we ignore an inessential arbitrary constant defining the origin of t. The solution (34) makes it manifest that when the total energy of the system is negative, it executes periodic oscillations of finite amplitude. The period and the amplitude of the oscillation are given by period: TT/03/2

( 3S )

and amplitude of oscillation: + A + —— ~Z z ~z — A + —-. 2Q 2Q

(-26) u

;

The requirement that z > o (by definition) implies that A < ^

(37)

It can be verified that the critical value A = 1J2Q corresponds to the system having a vanishing kinetic energy when t = o and z = i/Q; that under these circumstances the system should collapse to a point—as it does after a time t = TTJZQ3^—is to be expected. The variation of z ( = a/ao) during a half-period of oscillation is illustrated in Fig. 2 for a number of selected values of Q and A.

4 . THE DYNAMICAL E V O L U T I O N OF A S P H E R O I D A L SYSTEM

The analysis of Section 3 applies to the case when a single linear dimension provides an adequate description of it. We now turn to the slightly more general case when the cluster departs so much from even a gross spherical symmetry that at least two linear dimensions are required to describe it. Under these circumstances it might suffice to idealize the system as being homogeneous and spheroidal (either oblate or prolate). We shall suppose then that we are given a system that is initially homogeneous and spheroidal with semi-axes a\, a% ( = a{), and a$ (which is not necessarily the least axis) and a given total energy (g. We shall further suppose that the distribution of velocities is (locally) isotropic so that I n = f 22 = &33 = i®-

(3 8 )

<£ = ® + 2l!tii + lE33 = constant

(39)

On these assumptions

[440]

7-

0=06 I

I 6

c

5-

05 1 25 o 1-0 075 050

l

-•\

10 0

1-5

20

2-5

30

625 0 = 0-8

0-425 025~~~"

-

0-25

2 5 2 0 1-5 10

0=0-4

____|_25

i~o

I

i

i

i l l

i

0-75 O50 0 25

-

0 5 1

2 0

oT? •

1

1

1-00

1

1

l\

0=0-5

0-50~15 p-25*"*~ z 10 0-5

0

1

1

l

!

1

2

3

4

\

1

I

5

6

FIG. 2. The finite amplitude oscillations of a spherical cluster with a gaussian distribution of density; the case of negative total energy. Each panel illustrates the oscillations for a particular value of Q ( = |<6|/|3H|); and in each panel the curves are labelled by the values of A to which they belong. In each panel the ordinate measures the amplitude (2 = a/ao) while the abscissa measures the time in the unit specified in equation (25). The period and the amplitude of these oscillations are given by TTQ~312 and 2 A , respectively. [441]

and the tensor virial-theorem provides the pair of equations £ ^ T "*«-*«»-*«*»

(4o)

and Yl^

- § « - t W n + iW88.

(41)

The assumption that the cluster is homogeneous and spheroidal provides the further relations Wu--a*GMi/u--£^L*

(42)

and IR33 = -zvGpAzliz

= - i . 9M1 aA A3, 10

a\ ai

Ui) y

W

where Aj and A 3 are the 'index symbols' defined in the theory of the gravitational potential of the ellipsoids (cf. Chandrasekhar 1969, Section 21). T h e values of A] and A3 are expUcitly known in the case of spheroids (cf. Chandrasekhar 1969, p. 43, equations (36) and (38)). W i t h t h e values o f ^ B n andtK!^ given by equations (42) and (43), equations (40) and (41) reduce to d2ax2 20® , GM lax . , a3 (44) + y£(«A1 + *«A,) dt2 2>M a\ \<*3 «i / and d2az2 20(8 GM ( m + ^ai( 4 ^ . - 5 4 (45) dt2 3M a\ \ a3 ax / Making the substitutions ai 2 = a02z, a32 = ai2(i-y) and measuring time in the unit

= a02z(i-y),

(46)

\20|«|/

we find that equations (44) and (45) can be reduced to the forms

T!TS-'+>" and

where

, _<E, . zJ-^y + z *?& = -y + -Q-My), |«| dt2+ dt dt ' v * •

e =^^ and

*

(49)

(50

20 «o<S

/i(j)=J(i->')1/2 + ^

S(y),

*O0 = g - 3 ) ( i - y ) V - « ^ ± 2 S(y),

(5i)

<S* [442]

and (y> o) _ sinh~V — y / „

(y < o).

(S3)

V-y Equations (48) and (49) have been integrated for various values of Q with the initial conditions = ii y = yo (assigned), and — = j = o at / = o. (54)

0 - I O / » --Q 2 jrO'i-Q.r-.Q

5

" = -0-5,JJ=-0-2

-0-5

FIG. 3. The evolution of homogeneous spheroidal clusters for various initially assigned values of Q and yo. The curves are labelled by the values of Q and yo to which they belong. The ordinate y measures the square of the eccentricity (e 2 ) positive for oblate spheroids and negative for prolate spheroids. The abscissa measures the time in the unit specified in equation (47). The full-line curves refer to clusters of positive total energy while the dashed curves refer to clusters of negative total energy. [443]

20

18

16

14

12

"« 10

8

6

4

2

0

10

20

30 40 t FIG. 4. The evolution of homogeneous oblate spheroidal clusters. The ordinate z ( =
The derived solution curves are exhibited in Figs 3, 4 and 5 where it should be noted that y > o corresponds to oblate spheroids while y < o corresponds to prolate spheroids. An examination of these solution curves reveals the following features. Consider first the evolution of systems with positive total energy. The series of integrations for Q = 1 -o and yo = 0-9, 0-5, and o-2 shows that these objects which are initially oblate eventually become prolate (see Fig. 3); and while the eccentricity in all these cases appears to tend to a finite limit the object itself grows considerably in size (see Fig. 4). It will be observed that in all these three cases, the variations of z ( = ai 2 /ao 2 ) with time are so nearly the same that they cannot be distinguished in the scale of Fig. 4. For the same value of Q ( = 1 • o), if the object is initially prolate (yo = - 0 - 2 ) it evolves towards a less prolate and a more spherical shape as it grows in size (see Figs 3 and 5). For the case Q = 0-5, yo = 0-2, the behaviour is essentially the same as for Q — i-o, yo = 0 - 2 except that the object continues to remain oblate though of considerably reduced eccentricity (see Figs 3 and 5). Considering next the integrations for systems with negative total energy (Q = - 0 - 5 , yo = ±0-2 and Q = - i - o , yo = +0-2) we observe (see Fig. 3) that an object which is initially oblate becomes increasingly more oblate while [444]

F I G . 5. Same as Fig. 4 but for other initially assigned values of Q and ya. The curves are labelled appropriately. The full-line curves refer to clusters of positive total energy while the dashed curves refer to clusters of negative total energy.

an object which is initially prolate becomes increasingly more prolate. This last result is in agreement with an earlier investigation of Lin, Mestel & Shu (1965) in which the collapse of an object initially at rest with zero kinetic-energy is considered. For the case Q = —o*5 the increase in the oblateness or prolateness is not accompanied by any substantial changes in the length of the a\ ( = «2)-axis« But in the case Q = — 1 • o, ^0 = +0-2, the increase in the oblateness is accompanied by a substantial decrease in a\ ( = ao). 5. A SEMI-EMPIRICAL ALLOWANCE FOR TWO-POINT CORRELATIONS As we have noted in Section 2, the allowance for two-point correlations in the present context modifies only the expression for the potential energy. And we shall try to allow for it in a semi-empirical fashion by assuming that pQ- 2>(xW, x<2>) = pW(xW) pa>(x<2>)[i+Kexp(-A|xU>-x<2>|)],

( S5 )

where K is a constant and k is a measure of the correlation length. On this assumption, we may define a correlation potential (of the Yukawa form)

^(x)=.Gjp(xO^-^-X'l)^

( 5 6)

and a correlation energy WLCOR. = - h \ f4 d*

(57)

that must be added to the Newtonian potential energy WIN to obtain the true potential energy. [445]

For spherically symmetric systems it can be readily shown that ~—n.. - i rr exp ( — 2) pz dz

*>-^(«p(-oj; — exp ( + r)

exp ( — z) pz dz+2 sinh r

exp ( — z) pz dz\, (58)

where r is measured in the unit ijk. For the two cases (i) p = po = constant and (ii) p = p0 exp ( —r2/a2)

(59)

we find that « ' ) = ^ p ^ [' -

^

<*+ 0 « P ("*>]

( c a s e i)

(60)

(case ii),

(61)

and *') = ^

(

^

«• e x p

(a2/4))

I [* ( r _ ? ) exp ( - r )

+ 0 (- + - ) exp ( + r) —2 sinh r

where r, a, and /? (the radius of the sphere in case i) are all measured in the unit ijk and «>(*) = - ? - ) *exp (-z2)

dz

(62)

is the error function. The corresponding expressions for the correlation energy are ^

O R

WLN

- = K -2- [\R* - (R cosh R - sinh R) exp ( - R) (R + 1)] R*

(case i) (63)

FIG. 6. TAe correlation energy for clusters with homogeneous and gaussian distributions of density for a Yukawa-type potential. The abscissa measures, in units of the correlation length, the radius R (in the case of a homogeneous cluster) and the mean radius a (in the case of a cluster of gaussian distribution). The ordinate gives the ratio of the correlation energy to the Newtonian potential energy. [446]

and «^- J c(f)i/.„p(a./a)J o -[i

+

*(,-4a)] x [ i —
(case ii),

(64)

where WLN denotes the Newtonian potential energy (without correlations) in the two cases. Fig. 6 exhibits the functions WLCORJWLN defined by the foregoing equations. The effect of the correlation energy on the finite amplitude oscillations of spherical systems considered in 3(b) has been examined on the assumptions that K in equation (55) is o-i and that a and x\k are equal and remain equal during the oscillation. The results of these new calculations are contrasted with those of Section 3(b) in Fig. 7. It will be observed that the qualitative features of the solution are not affected.

20 -

FIG. 7. The effect of two-point correlations on the finite amplitude oscillations of clusters with negative total energy. The full-line curves 1, 2, 3, 4 and 5 are the cases illustrated in Fig. 3forQ = 0-5, A = 0-9; Q = 0-5, A = 0-5; Q = 0-5, A = 0-25; Q = o-8, A = 0-425; and Q = o-8, A = 0-225, respectively. The dashed curves refer to these same values of Q and A after allowing for two-point correlations. ACKNOWLEDGMENT T h e research reported in this paper has in part been supported by the National Science Foundation under contract GP-28342 with the University of Chicago. University of Chicago, Chicago, Illinois, U.S.A. REFERENCES Chandrasekhar, S., 1964. Lectures in Theoretical Physics, ed. W. E. Brittin, Vol. 6, p. 1. University of Colorado Press, Boulder. Chandrasekhar, S., 1969. Ellipsoidal Figures of Equilibrium, Yale University Press. Chandrasekhar, S. & Lee, E. P., 1968. Man. Not. R. astr. Soc, 139, 135. Lin, C. C , Mestel, L. & Shu, F. H., 1965. Astrophys. J., 142, 1431. [447]

The stability of a rotating liquid drop BY

S.

CHANDRASEKHAR,

F.R.S.

University of Chicago {Received 18 September 1964) I n this paper, the stability of a rotating drop held together by surface tension is investigated by an appropriate extension of the method of the tensor virial. Consideration is restricted to axisymmetric figures of equilibrium which enclose the origin. These figures form a oneparameter sequence; and a convenient parameter for distinguishing the members of the sequence is 2 = pi22a?/8T, where Q is the angular velocity of rotation, a is the equatorial radius of the drop, p is its density, and T is the interfacial surface tension. I t is shown that 2 ^ 2-32911 (not 1+.J2 as is sometimes supposed) if the drop is to enclose the origin. It is further shown t h a t with respect to stability, the axisymmetric sequence of rotating drops bears a remarkable similarity to the Maclaurin sequence of rotating liquid masses held together by their own gravitation. Thus, at a point along the sequence (where Z — 0-4587) a neutral mode of oscillation occurs without instability setting in at t h a t point (i.e. provided no dissipative mechanism is present); and the instability actually sets in at a subsequent point (where S = 0-8440) by overstable oscillations with a frequency Q. The dependence on Z of the six characteristic frequencies, belonging to the second harmonics, is determined (tables 3 and 4) and exhibited (figures 3 and 4).

1.

INTRODUCTION

The interpretation of phenomena related to nuclear fission has revived interest in the classical problem of the equilibrium and the stability of a rotating liquid drop held together by surface tension. This problem was the subject of a remarkable series of experiments by Plateau (1843-69); and certain aspects of the associated theoretical problem of the allowed figures of equilibrium were investigated by Rayleigh (1914; see also Poincare 1895 and Appell 1932). A brief account of the equilibrium theory, which carries the analytical treatment somewhat further than Rayleigh's, is given in the Appendix. If one restricts oneself to axisymmetric forms and uniform density, the figure of equilibrium depends on the value of the non-dimensional parameter E = pQ2a3j8T,

(1)

where Q denotes the angular velocity of rotation, a the equatorial radius of the drop, p its density, and T the interfacial surface tension. A sequence of equilibrium figures derived on the basis of the formulae given in the Appendix is illustrated in figure 1. When E-+0, the figure tends to an oblate spheroid with an ellipticity e given by (see equation (48) below) e=E

(E^O).

(2)

However, as E increases, the figure rapidly departs from the spheroidal form: at E = 1, the polar regions are flat; and as E increases beyond 1, the drop develops a dimple that deepens quickly and reduces the polar axis to evanescence when E — 2-3291; and for E > 2-3291, the equilibrium figures no longer enclose the origin and are tori. The stability of these axisymmetric figures has never been [11

investigated properly. Indeed, as we shall see, the supposed onset of instability at Z — 1 (cf. Sperber 1962) is incorrect. I n this paper, the stability of the rotating liquid drop will be investigated. I t will appear that the results are remarkably similar to those along the Maclaurin sequence of rotating incompressible masses in gravitational equilibrium. Thus, at a point along the sequence (where Z = 0-4587) a neutral mode of oscillation occurs without instability setting in at that point (i.e. provided no dissipative mechanism is present); and the instability actually sets in at a subsequent point (where Z = 0-8440) by overstable oscillations with a frequency equal to the angular

w FIGURE 1. The figures of equilibrium of a rotating liquid, drop for various values of the parameter 2. The curves labelled 1,2 7 are for E = 0, 0-210478, 0-510790, 1-0, 1-72598, 2-09589, and 2-32911 ( = 2,IMI.), respectively. The radial co-ordinates have been normalized in such a way that the volumes of all the figures are the same. The curve labelled 1 is, of course, a quadrant of a circle; the curve labelled 4 (for 2 = 1) is flat at the poles; and the curve labelled 7 for 2mmx, is the limiting figure enclosing the origin.

velocity of rotation. (Along the Maclaurin sequence, which is a sequence of oblate spheroids, the corresponding points occur at eccentricities of the spheroid equal to e = 0-8127 and 0-9529, respectively.) At its first neutral point (at Z = 0-4587) the rotating drop must be 'secularly' unstable (i.e. unstable if some dissipative mechanism is present), and one may anticipate that at this point a stable sequence of triaxial forms branches off in a manner analogous to the way the Jacobian sequence of ellipsoids branches off from the Maclaurin sequence of spheroids. I n many ways it is remarkable that even though Plateau's original investigations were explicitly motivated by ' cosmogonical' considerations, the almost complete similarity of Plateau's problem to the one, then, under active theoretical consideration by Jacobi, Riemann, Poincare, Bryan, and Darwin (cf. Jeans 1919) seems to have been overlooked. Apparently, the emergence of a sequence of toroidal figures of equilibrium (for Z > 2-3291) and its striking similarity (in [2]

appearance!) with systems such as Saturn s rings succeeded in drawing attention away from the real significance of Plateau's problem to the astronomical problem of the time. In concluding this introductory section, it may be stated that, on the analytical side, the method that will be developed in this paper is an extension, to allow for capillary forces, of the method of the virial (Chandrasekhar i960, 1961) which has found applications to a wide variety of problems in the theory of homogeneous masses in gravitational equilibrium (see the various papers by Chandrasekhar and by Chandrasekhar & Lebovitz in the Astrophysical Journal of the past four years; and for a summary of the principal ideas see Chandrasekhar 1964.) 2.

T H E VTRIAL THEOREM; THE SURFACE-ENERGY TENSOR

Consider a fluid confined to a volume V bounded by a closed surface S on which surface tension (of magnitude T) is operative. To keep the present development at its simplest level, we shall suppose that the external pressure on S is zero. I n virtue of surface tension, the pressure in the interior, immediately adjacent to S, is given by ^ = T d i v n (on S_), (3) where n is the unit outward normal on S. Interior to S and in the volume V the equation of motion governing the fluid is

where ut denotes the velocity and p is the pressure. To obtain the form of the virial theorem appropriate to the problem on hand, we multiply equation (4) by x^ and integrate over the volume V. By familiar transformations, we obtain

^ ^ i , , d x = 22 i ,-J^,gdx, where

2i3- = ~

putUjdx

(5) (6)

is the kinetic-energy tensor. The second term on the right-hand side of equation (5), after an integration by parts, gives

where

dSi = nidS,

(8)

dS being an element of area of S. By making use of the condition (3) obtaining on S, we have r f) r C xj^-dx

= T\

^.divnd^-^.J

pdx.

(9)

Equation (5) can now be written in the form ~ f pu^dx

= 2 $ , , + ©„ + *„ 77,

(10) [3]

where

n = j pdx

and

®,y = - 2 '

(11) (12)

a^divndS;.

We shall call ©^ the surface-energy tensor. From equation (10) it is apparent that the conservation of the angular momentum of the fluid requires the surface-energy tensor (3^ to be symmetric. While (5^-, as defined by equation (12), is not manifestly symmetric in its indices, its symmetry, nevertheless, can be demonstrated. I am indebted to Professor G. Wentzel for the following demonstration. LEMMA.

Let S be a surface bounded by a simple closed contour G. Then L=d> x x ( d x x n ) = - f (xxn)divndS.

(13)*

Js

Jc

Proof. Let (i,j,k) represent the indices (1,2,3) in some cyclic order. Then, suspending the summation convention throughout this proof, we have for the ith component of L Li = j> [dXiin^

+ n^^

+ dx^-niX^

+ dx^-niX/,)].

(14)

Transforming this line integral around C to a surface integral over S by Stokes's theorem, we obtain

Li =

\\n{k{~niXk)~^}~niXi) + iu — (ntx} + nkxk) -T-(-

ntxk)

dx,

j + n,, dx,-(-nixj)-~(njxdx,

+ nkxk)

dS.

(15)

On carrying out the requisite differentiations, we find after some rearrangement,

-/.

- K 4r K + n\) + \x, A (»? + nj) dxt dx,. (8n,-

dn,\

(on,-

en,

(16)

dS, dS:

or, since |n| 2 = 1,

Li = - \

(xj nk - xk nj) div n dS;

(17)

and this establishes the lemma * Contrary to the definition of S that has been used up to the present (and that will be used subsequently) we are, in the statement of this lemma and during the course of its proof, letting S be an open surface bounded by a finite closed contour C. [4]

Now let the surface S in the statement of the lemma tend to become closed. Then the contour G will tend to evanescence and Lit defined as the line integral in equation (14), will tend to zero; and from its equivalent definition as a surface integral in equation (17) it follows that Xjdivn dSk =

xkdivndS^

(S a closed surface);

(18)

and this establishes the symmetry (in fact) of the surface-energy tensor. Returning to equation (10), we observe that under conditions of a steady state 2%ij + <5ij + Sijn=0;

(19)

and if, in addition, no macroscopic motions are present ®tj = -dtjn.

(20)

The contracted versions of equations (19) and (20) are 22 + © + 377=0

and

© = -377,

(21)

where © = © w is the surface energy and % = Zi{ is the kinetic energy. 3.

T H E VIRIAL THEOREM FOR A ROTATING DROP

In a frame of reference rotating uniformly with an angular velocity Q about the x3 axis, the equations of motion take the form

By multiplying equation (22) by x^ and integrating over the volume V occupied by the fluid, we obtain, by the same sequence of transformations that led to equation (10), jj^ where

puiXj dx = 2 2 y + ©„ + £2(7i;- - 8i3I3j) + 8i}n+ 7y =

2Qell3j^ pulXj dx,

px^dx.

(23) (24)

is the moment of inertia tensor. When no relative motions are present in the chosen frame of reference and a state of hydrostatic equilibrium prevails, equation (23) gives

®li+a>(iti-siaiv)

= -8i}n.

(25)

Writing out equation (25) explicitly for the different components, we have ©n + flVii = @82 + G»/ M = ©33 = - n, 2

©12 + £ 7 12 = © 21 + QU21 = 0, © 13 + .Q2713 = © 31 = 0,

and

@23 + i22723 = ©32 = 0.

(26) (27) (28) [5]

From the symmetry of the tensors <Sit and Iip it follows from equations (28) that ©i3 = @23 = 0 and

713 = 723 = 0.

(29)

But it is not required that ©12 and 712 also vanish; all that is required is © „ = -£>«/„.

(30)

I t is often convenient to eliminate 77 from equations (26) and obtain two integral relations t h a t are in no way dependent on any constitutive relations that may exist. Two such integral relations are

and

(S u -(S 2 ! 5 + i 3 2 ( / n - ^ ) = 0

(31)

@ u + @22 - 2© 33 + fl»(/u + J22) = 0.

(32)

I t is particularly to be noted that equations (30) to (32) do not require In = 722 and © u = <S22 a s necessary conditions of equilibrium; and neither is it required that I12 = 0 and
and

7iy=

/hi h* 721 722 \0

0

o \ 0 .

(33)

i j

I t is important to observe that the virial equations provide no substance to the common expectation that symmetry about the rotational axis should be associated, necessarily, with any form produced by an axisymmetric rotational field. Indeed, we may expect that along a sequence of axisymmetric forms a point of bifurcation occurs where a sequence of genuine triaxial forms branches off. 4.

RELATIONS OBTAINING FOB AXISYMMETEIC FIGURES OF EQUILIBRIUM

An axisymmetric figure of equilibrium can be uniquely specified by giving the equation of its meridional section. Let this equation, in the positive hemisphere (x3 > 0) be ., , ,nj. 3

where

z= *,=/»,

(34)

w = J(x\ + x\)

(35)

is the perpendicular distance from the axis of rotation. The equatorial plane of a rotating configuration may in all cases be expected to be a plane of symmetry.* Accordingly, to be valid over the whole sphere, we should write , ,„«\ x,, x 3 = ±/M(36) However, to avoid ambiguities in sign which will arise if equation (36) is explicitly used, we shall (when it is necessary to use this equation) restrict our attention to the positive hemisphere and use equation (34). And to avoid misunderstanding, * For configurations in equilibrium under their own gravitation, a proof of this symmetry about the equatorial plane has been given by Lichtenstein (1933); an analogous proof, valid for configurations in equilibrium under capillary forces, may be constructed. [6]

we shall distinguish integrals extended (under the imposed restriction) over the positive hemisphere b y specifying the domains of integrations as V+ or S+ as the case may be. Letting

* = df/dm,

(37)

we can write (on S + )

Wi = _

va+^j^' n* = ~7v~^¥)i'

and Ws =

v(iT^j;

(38)

and correspondingly (also on S + ) dSx = -fa-L&TB&d, dS2 = -
and dS3 = mdmdd,

(39)

where dd is an element of the cylindrical polar angle 6. From the expressions for the components of n given in (38), we readily verify i. divn =

4> ' I d m*)i wdTn(l+
,,„, (40) '

Fin fa

<S11 = -2T\ z 1 d i v n d S 1 = 2T <j>x\ div n dm dd Js+ Jo Jo - 2TTT I m*divndm;

(41)

and ©22 n a s *hi s same value. On the other hand, @33 = -2T\

Js+

x3 div ndS3 = -2T

\ \ f(m) div nm dm dd Jo Jo

= - 4 T T T Yf(w)wdivndw.

Hence

(42)

S 3 3 - @ u = - 2nT | ° (m* -J- + 2mf\ div n dm = - 2nT (a (m2f)' div n d m . Jo

And since

[•

fa

ff{m)

i"n = 2p \ x\ dx = 27T/3 dtzrw3 Jr+ Jo Jo

(43) dz 3

= 27T/9 | w3/(w) dw = - np \ " (m2f)'m2 dm, Jo Jo

(44)

we have by the virial theorem £>2 = ®M z C i i

=

_ r Jo

(45)

3

Jo

m fdv [7]

If w is measured in units of the equatorial radius a, the foregoing relation takes the form f (s7 2 /)'divnds7 J (n7 2 /)'divndc7 J 0 827 = - - - =^ , (46) w3fdru (ru2f)'rn2 dm where 27 is the non-dimensional parameter defined in equation (1). As an illustration of equation (46) consider a slightly oblate spheroid with an ellipticity e. Then, in the units in which equation (46) is expressed, divn = 2 + 4ettJ2;

(47)

and the second equality in (46) gives 27 = e,

(48)

a known result. 5.

T H E VIRIAL EQUATIONS FOR SMALL OSCILLATIONS ABOUT EQUILIBRIUM

We shall suppose that a uniformly rotating drop, initially in a state of equilibrium, is slightly perturbed; and further that the ensuing motions are described by a Langrangian displacement of the form %(x)eM,

(49)

where A is a parameter whose characteristic values are to be determined. To the first-order in % the general time-dependent equation (23) gives \Wtij - 2XQeiaV;.j = S<5tJ + Q°-(SIU - 8i3SI3j) + 8tjm, where

Vi;j =

p^a^dx

(50) (51)

is the second-order unsymmetrized virial, and #8,,-, SIip and SFI are the first variations in the respective quantities due to the deformations caused by the displacement %. (Note that the index after the semi-colon indicates that a moment with respect to the associated space co-ordinate is involved.) The first variation of Z(;- is readily written down. We have SIu^SJ

pxixJdx=j

pfaxj +

x^dx

= Vi.i + Vi,i. We shall write

(52)

SIif = Vip

(53)

where Vit without the semi-colon is the symmetrized virial.* Turning next to SQ^, we have, by definition, S
j ^divndSi;

* Note that Vn = 2Vl:1, etc. [8]

(54)

and since the operations of displacement and integration are permutable, we can write

*s w = - r f £]divndSi Js

+ f xjS(divn)dSi+ Js

f a;,, div n <J(d& Js

(55)

and we require to express #(divn) and ^(d»S1) in terms of %. We proceed as follows. First, we observe that if F is any explicitly known function of the position, then (cf. Chandrasekhar & Lebovitz 1964, equation (A3)) [dxj

~ 8xL

dxkdxt'

Accordingly,

£(div n) = div (Sn) - ^ ^ , (57) cxk ox 1 where Sn is the Lagrangian change in n as the surface S gets displaced by the application of %. I t is known that

»*2rK-**Sr^'^£x ' CX

(58)

(Note that, as required, Sn is orthogonal to n.) I t remains to determine S(dSt). Let F be any arbitrary continuous function defined in V and on S. Then s{ F&St = [ SFdSi+t F8(dSi). Js Js Ja

(59)

On the other hand, we can also write * f FdSt = s[ f ? d x = f # ( | ^ \ d x + f

^-«J(dx).

But it is known that <J(dx) = div $• dx. Making use of this result and of equation (56), we have

(60) (61)

The first of the three integrals on the right-hand side of equation (62) is clearly expressible as a surface integral over S; and integrating by parts the other two, we have sj FdSt =j SVdSt-j

F^dSk+j

F~(div%)d-K

+ f F div \ dSt - [ F-J-(div I) dx, Js J v cxi or

S f F dSt = f SF dSt + f F /div \ dSt - 1 ^ ' dSk\ .

(63) (64)

Comparing equations (59) and (64), and remembering that F is arbitrary, we conclude that „e J(dS,)=div$
I n particular, if % is required to be divergence-free, as in the case we are presently considering, then *(dS<) = -^dSf i f c

(div? = 0).

(66)

Inserting this last expression for 8(684) in equation (55), we finally have 8
dSfe

(67)

where, it may be recalled that S(div n) = div (8n) -1^' ^

(68)

and

(69)

*».•=

(«traj-^j»f- » , | .

Returning to equation (50), and replacing 87tj by Vy in accordance with equation (53), we have \*Vtii - 2 A £ e ^ ; , = 8
(70)

The nine equations which equation (70) represents fall into two non-combining groups of five and four equations distinguished respectively by their parity (i.e. evenness or oddness) with respect to the index 3. I t is convenient to have these equations written out explicitly. The equations of even parity are *.2V3;3 = 8®33 + 8n, A2F1:! - 2A£P2. x = 8<Sn + QWn + 817, 2

(71) (72)

A F2;2 + 2A£F 1:2 = 3<Sn + &Vn + 8II,

(73)

A 2 F 1 . 2 -2Ai2F a . 2 = £<S12 + .Q2F12,

(74)

\%.1

(75)

+ 2\QV1;1 = 8
and the equations of odd parity are A2F3:1 = 2

A F3.2 = <*<332, AW1.3-2Ai2V2.3

= 8(S13 + QW13,

AW2;3 + 2AQV1.3 = 8<523 + Q%3.

(76) (77) (78) (79)

I t will be observed t h a t while the even equations (71) to (75) involve 817, the odd equations (76) to (79) do not. In general, the specification of 877 in terms of \ will require some supplementary assumption concerning the physical nature of the oscillations, such as for example that they are taking place adiabatically. However, for an incompressible fluid, the Lagrangian displacement is required to be solenoidal, and we can dispense with the evaluation of 877 by eliminating it from [10]

(71) to (75) and considering the eliminant system. I t will appear subsequently that it is appropriate to associate the equation, obtained by subtracting (73) from (72), with the equation obtained by adding (74) and (75); and similarly to associate the equation, obtained by subtracting twice equation (71) from the sum of (72) and (73), with the equation obtained by differencing (74) and (75). Thus, the eliminant system of equations which we shall consider are the two pairs of equations *A«(7U - 722) - 2XQVn = * © u - <J(322 + fl»(7u - 722), A2712 + A £ ( 7 u - 7 2 2 ) = 28<512 + 2QW12, and

(80) (81)

JA»(7U + 722) + 2Afl(7 1; , - 7 2; x) - A«7„ = 8<5J1 + *®n-28®at

+ a*{Vn + Vat),

2

A (7 1 ; 2 -7 2 ; 1 )-A£(7 1 1 + 722) = 0.

(82) (83)

And as already stated, these equations must be supplemented by the solenoidal condition on \. We shall see in detail in §6 below how equations (76) to (83) enable us to determine all the characteristic frequencies of oscillation belonging to the second harmonics. 6.

T H E CHARACTERISTIC EQUATIONS GOVERNING THE OSCILLATIONS BELONGING TO THE SECOND HARMONICS

I t can be shown quite generally that the proper solutions belonging to the different characteristic frequencies of oscillation of a rotating incompressible mass in gravitational equilibrium are polynomials of different degrees in the co-ordinates (cf. Cartan 1922; also Lyttleton 1953). But this is not necessarily the case in the problem we are presently considering. However, we may expect that an adequate approximation for the lowest modes will be provided for the problem on hand (particularly for Z < 1 when the equilibrium figures do not depart appreciably from spheroidal forms) by a form for % which is linear in the co-ordinates (cf Chandrasekhar & Lebovitz 1962 for the analogous treatment of the oscillations of rotating compressible masses). On this approximation the eight virial equations (76) to (83) will suffice to determine all the characteristic frequencies belonging to the second harmonics. For, the nine constants L{j (i,j = 1,2, 3) in the most general linear form for \, namely U^Lti'i

(84)

are reduced to eight by the solenoidal condition on \ which requires the trace of Lit to vanish: ,. _ r ,QK> n l > div5 = L« = 0. (85) I n actual practice it is not necessary to insert the general form (84) for \ in the virial equations and carry out the reductions on that basis, since the equations separate into three non-combining groups characterized by the requirements t h a t only certain of the virials, or certain specific linear combinations of them, do not vanish. The three non-combining groups of equations are, in fact, the equations [ll]

(76) to (79), the equations (80) and (81), and the equations (82) and (83). The requirements appropriate to each of these three groups are as follows: (A) equations (76) to (79): only the ' o d d ' virials V1;3, V3.v V2;3, and V3;2 are non-vanishing; (B) equations (80) and (81): all the odd virials vanish, V1;2 = V2.v Vn = — V2i, and V33 = 0; (C) equations (82) and (83): all the odd virials vanish, V1;2 — — V2;1, V±1 = V22, and 1^3 4= 0. This same separation of the eight virial equations into three non-combining groups happens when one is treating the oscillations of a Maclaurin spheroid (cf. Lebovitz 1961); and as in the case of the Maclaurin spheroids, we shall call the different modes of oscillations that are obtained as proper solutions of the three groups of equations, which we have distinguished by (A), (B), and (C), as the transverse-shear, the toroidal, and the pulsation modes, respectively. Of these three types of oscillations, that we have called toroidal is the most important, since it is by one of these mbdes that the liquid drop manifests both 'secular' and 'ordinary' instability. On this account, we shall treat the toroidal modes of oscillation first. (a) The toroidal modes The form of the Lagrangian displacement appropriate for these modes is £,l = ax1 + ^xz,

g% = /?xx - ax2,

and

£3 = 0,

(86)

where a. and /? are constants unspecified for the present; for, consistent with the requirements for these modes, the only non-vanishing virials that the displacement (86) gives are ,a 2 Vi:2 = \; 1 = ^ / ? J Q ***f(™) dw (87) V22 = 4npa vj3f(m) dm. •/. From equations (68) and (69), we find that for the chosen displacement

and

and

Ki =

(88)

8nx — \a{n\ — n\) + 2{lnxn2\n1 — {anl+pn2),

(89)

Sn2 = [a(wf — w|) + 2/?7i1n2]w2 — (/?% — an 2 ),

(90)

dn3 = [ a ( n f - n l ) + 2/?w1w2]w3,

(91)

(J(divn) = a (n\ - n\) div n + n . grad {n\ - n\) - 2

I8nx \BX-L

8n2\ 8xJ

»i + 2J3 h n2 div n + n . grad•.,{nx n2)> - I/ d—* + dw ^ 2-YJ

(92)

Inserting in equation (92) the explicit forms for nt given in equation (38), we find
QM

+ 2Pxxx£Q{vj),

W2 + 30') w2 vj(l+2)i

6'(P-2y (1 + 0 2 )*.

(93) (94)

Next, we verify t h a t for the displacement we are presently considering 8
and

x28(divn)dS1

8<&11-8<£22 = -2T\

= -2T(

z1S(divn)dS2

(a^d^-^dS^divn).

(95) (96)

Now making use of equations (39) and (93), we readily find that 8
r^Qim)dw

8
(97) (98)

Also, from (87) and (88) we have Va = inp/!J°vfif{m)dvj and

Vn-VZ2 = 8npoc\aTD3f{TD)dw.

(99) (100)

Inserting the foregoing expressions for <£<S12, <J(Sn — 8<S22, V12, and Vn — V22 in (80) and (81), we obtain the pair of equations

and

where

A2a-2A£ytf = -23<x + 2.£2a

(101)

A 2 /?+2A£a = - 2 3 / ? + 2£ 2 /?,

(102)

\a$m*Q(m)dm 23 = - - ^ . P s " m f(w)dTn

(103)

J

Equations (101) and (102) can be rewritten in the forms

and

[A2 + 2(3-i2 2 )]a-2Ai2/? = 0

(104)

[A2 + 2 ( 3 - £ 2 ) ] £ + 2 A . G a = 0;

(105)

and these equations lead to the characteristic equation [A2 + 2(3-.Q 2 )] 2 + 4A 2 £ 2 = 0.

(106)

Setting A2 = — (J2 (so t h a t a real c is a necessary and a sufficient condition for the stability of these modes), we can factorize (106) to give o- 2 -2o-J2-2(3-.Q 2 ) = 0

(107)

and a similar equation with — Q in place of Q. The roots of (107) are


(108) [13]

Equation (108) for the frequencies of the toroidal modes predicts the occurrence of a neutral mode of oscillation at the point where J

a

i <prn Q(m)dvj

^2 = 3 = 7 3 ~ ^

:

(109)

tZ73/(7Z7) (

but since er is real in the vicinity of this point, on either side, instability does not occur here.* However, instability does occur at the point where fi« = 23;

(110)

for Q2 > 23 the rotating drop is unstable by overstable oscillations with a frequency Q. If we measure w in the unit a, and cr and Q in the unit (8T/pa 3 )i, the expression (108) for a takes the form
Ti 32

w3J'(w)dro

( 112 >

•

(6) The pulsation mode The form of the Lagrangian displacement appropriate for this mode—we are using the singular since, as we shall see presently, there is only one non-trivial mode of this kind—is £1 = ax1 + fix2, £2 = -Px-L + aXz, and

£3 = -2ax3,

(113)

where a and /? are constants unspecified for the present. As defined, the displacement is clearly solenoidal; and, moreover, consistent with the requirements for these modes, the only non-vanishing virials which the displacement gives are Vn = F22 = 2 a J n ,

F33 = - 4 a / 3 3 ,

and

7 1;a = - F 2 ; 1 = A i >

In = 2np\ vysf(w)dw and I3a = §7779 wfz{w)dw Jo Jo are the two distinct components of the moment of inertia tensor. From (68) and (69) we now find

where

S(div n) = a Un* + n\ - 2w§) div n + n . grad (n\ + n\ - 2«§) - 2 U^ + -~\

(H*) (115)

(116)

And on inserting the explicit forms for nt given in (38), we obtain S(divn) = ocR(w), where

*(«,) - ^

+

* \

+

^ f f .

(117) (118)

* It should be qualified that the non-occurrence of instability at this point depends on the assumed absence of any dissipative mechanism. [14]

We next verify that for the displacement we are considering at present <J@U = £<S22 = %nTa and

fa

<j>w2R{m) dm

(119)

8<£33 = - InTa I xnfR{w)dm.

(120)

<J@1X + £@22 - 2<S<333 = 4nTa [" (w2f)'R(m)dm.

Hence

(121)

I n view of equations (114) and (121, equations (82) and (83) become 2aA 2 / u + 4/?A£/ u + 4aA2/33 = tnTa. | (tu 2 /)' R(m) dm + 4 a £ 2 J u

(122)

2^A 2 / n = iocAQIn.

(123)

and

If we ignore the possibility that A may be zero—we return to this possibility l a t e r - ( 1 2 3 ) gives

y?A = 2 a £ .

(124)

From equation (122) it now follows, after some rearrangements, that 2

a

1 + 2^3 V

iu/

\a(w>f)'R(Tn)dv7 = 2£ --*^— , P \'w>fdm 2

(125)

where we have written a2 = — A2. If we measure m in the unit a, and er and Q in the unit (8T//0O3)*, the expression (125) for a2 becomes f1{m2f)'R{m)dm = 2I- ——— 8 j m3fdm J

.

(126)

Returning to the possibility that A may be zero, we observe that (122) and (123) do allow this possibility (without requiring a and fi to vanish) provided that Q*Ill = -TrT[a{wif)'R{m)dw,

(127)

or in the units used in expressing <x2 as in equation (126), f 1 {w 2 f)'R(w)dm Z=-J-° 16 m*fdm

.

(128)

The occurrence of a neutral mode, should condition (127) be satisfied, corresponds to the possible existence of an equilibrium configuration, along the axisymmetric sequence, whose figure is invariant to a suitable infinitesimal deformation. The simplest form of such a deformation is represented by the displacement, | x = ctx-y, £2 = az2,

and

§ 3 = — 2ax 3 ,

(129) [15]

since it manifestly preserves the axisymmetry and conserves the volume. Of the three integral relations (30), (31), and (32), which must always obtain under conditions of equilibrium, the first two are clearly invariant to the displacement (129), while the invariance of the last requires 8<S11 + S<522-28<333 + Q2(V11 + V22) = 0;

(130)

it is exactly this last requirement that leads to the condition (127) (cf. (114) and (121)). Along the sequence of the axisymmetric figures of equilibrium of a rotating drop, the condition (128) is not met; therefore, a shape-preserving deformation along the sequence does not exist. However, it may be recalled that, along the Maclaurin sequence, the condition analogous to (130) is met where Q2 attains its maximum value. (c) The transverse-shear modes Finally, turning to the odd equations (76) to (79), we have for the form of the Lagrangian displacement, appropriate to a proper solution of these equations, £x = acx3, £2 = fix3, and

£3 = yx^Sx^

(131)

where a, /?, y, and S are constants unspecified for the present. For this displacement, the only non-vanishing virials (as required) are V1;3, V3;1, V2.3, and V3.2; and these are given by V1;3 = ctl33,

Va:3 = /3Im,

Vail = ylu,

and

F 3;2 = SIllt

(132)

where Ixl and I33 have the same meanings as in equations (115). From equations (68) and (69) we find that for the displacement (131), <J(divn) = (a. + y)(n1n3divn

+ n.grsidn1n3

— cn3l8x1)

+ (fi + 6) (n2n3 div n + n . gradn 2 n 3 — dn3jdx2);

(133)

and when the known expressions for ni are inserted we find *(divn) = [{cc + y)x1 + {fi + 8)x2]P(w), where

P(w) = -

1

(t + 3 JL

727(1+0 2 )* \VJ 1 G7 2 (l+0 2 )4

(134)

"

1+07 1 d VJ T27 2 dT77(l+^ 2 )f'

(135)

From (67) we find that for the displacement we are now considering <5®13 = 2nTy (" (m2f)'divndm + 2nT(a + y) P<j>fm2P(w)dw; Jo Jo or in view of the identity (45), we can write 8<S13 = -yn*In-(a where [16]

L =-2nT

+ y)L,

T^fw2P{m)dru.

(136)

(137) (138)

By the same procedure we also find 8<Sai = +aQ»Iil-(a

+ y)M,

(139)

M = 2nT \ w3P(w)dru.

where

(140)

The expressions (137) and (139) for 8
-2TTT

f a (w 2 /)'divndTEr =

M-L

= 27TTia(vj + ^f)m2P(xn)drn,

(141)

an integral relation which can be established.* Analogous to (137) and (139) we have 8
6>@32 = +/?£ 2 / 1 1 -( / tf + <J)itf.

(142) (143)

Returning to (76) to (79) and inserting for the various first variations the expressions we have found for them, we obtain, after some minor simplifications, the set of equations a(M-Q2Iu)

+ y(M + X2Iu) = 0,

a

2

fl(M - fl /u) + 8(M + A / u ) = 0,

(144) (145)

a(\%3 + L- Q%3) + yL = + 2fi\QIn,

(146)

£(A"JM + L- QH33) + #L = - 2oc\QI33.

(147)

From equations (144) and (145) it is apparent that p = qa

and

8 = qy,

(148)

where q is some constant; from equations (146) and (147) it then follows that g = ±i.

(149)

* The required equality follows from the relations ra ra 2 + d>* vfiPim) dro = —^rodro Jo J o ( l + 02)* ra ra M and I 0/ro2P(ro)dro — — —-—^mdw 1 Jo Jo(l + ^ 2 )* which can be derived, after two or more integrations by parts, from the second form of Pro)) given in (135), and use is made of the further relation f« ra 2-(4 2 2 (ro /)' divndro = — ^wdm Jo Jo(l + 02)* which can be similarly derived when use is made of the expression for divn given in (40). [17]

733 Choosing the lower sign in equation (149) for the sake of definiteness and writing iA = o", we obtain the pair of equations a(M - QH-u) + y(M - a-2In) = 0 2

and

a(cr%3-2
(150)

= 0.

(151)

Equations (150) and (151) lead to the following characteristic equation for cr: (cr2In - M) (a2I33 - 2aQI33 + Q%3 -L)+

L(Q2In -M)

= 0.

(152)

On simplification (after expansion and division by InI33) we find that equation (152) can be brought to the form (
K1 = MjI11

and

+ Q)] = 0,

(153)

K3 = L\I33.

(154)

Thus cr = Q is an allowed root; and the three remaining roots ere determined by the cubic equation ^-cr^Q-cr^ + K^ + Q^-K^ (155) = o. We observe the remarkable similarity of (155) with the equation one obtains for the corresponding modes of a rotating compressible mass on an approximative treatment based on the virial equations (Chandrasekhar & Lebovitz 1962, equations (54) and (55)). In the analogous exact treatment of the Maclaurin spheroid, one initially obtains equations similar to equations (153) and (155); but the coefficients which replace Kx and K3 in that treatment happen to be equal; consequently, a root cr = 0 is obtained and the remaining non-zero roots are given by a quadratic equation. If we measure w in the unit a, and cr and Q in the unit (8T/pa 3 )i, the equation for cr takes the form o - 3 - ( r V r - ( r ( / c 1 + /c3) + (AC1-/C3)Vi:= 0, (156) J w3P(w)dv>

where

J

Kt = - ^ 7 — 8

f C4/TI72P(TO) dm

and

w3fdw

J

K3 = - - ° — -^

.

(157)

vjf3dm

Also, it may be recalled that the relation (141), in the present units requires f1 VJ3P(W) dw+ f1 <j)fw2P{w) dw = 81 P w3fdvj. Jo Jo Jo 7.

(158)

NUMERICAL RESULTS

The characteristic equations (111), (126), and (156), governing the modes of oscillation of a rotating drop belonging to the second harmonics, depend on certain integral properties of the equilibrium figure. The values of the requisite integrals (besides jw3fdm, already included in table A 1 of the Appendix) are listed in table 1 for a few values of S; with their aid, the characteristic frequencies can be evaluated. Considering first the discriminant 2 J — E of the toroidal modes, we find by interpolation among the values listed in table 1, and by subsequent verification and refinement, that along the sequence E = J (the condition for the occurrence of [18]

T A B L E 1.

sin - 1 A; 78° 77° 76° 75° 76° 77°-7509

V A L U E S O F T H E I N T E G R A L S W H I C H OCCUR I N T H E CHARACTERISTIC E Q U A

2

'o

0-275621 0-366754 0-510790 1-00000 1-72598 2-32911

T A B L E 2.

2J-Z

$w*Q(w)dw 1-61355 1-48750 1-31766 0-91779 0-62650 0-57456

+ 0-67736 + 0-56871 + 0-39636 -018180 -0-95377 -1-40324

2 fl 3jo 0064978 0052915 0039003 0015234 00040096 —

-/:

(•mif)'R{w)d 303696 2-98386 2-90118 2-65144 2-45867

S O M E P R O P E R T I E S O F T H E E Q U I L I B R I U M F I G U R E S AT N E U T R A L S T A B I L I

(z„ is the polar radius in units of the equatorial radius; al is the equatorial radius in units of volume; the integrals

Jo

wfdxn and I vfifdw are simply related to the volume and the

Jo

and A = £ - y ( l + 2Z').)

at point of neutral stability X1 A mfdm

at point overstabil

0-458706 30187

0-84404 1-9426

0-46259

0-3646

••$:

Txfifdw «i

/: 2J-

$m*Q(m)dTtj

0-093704

007569

0-67139 1-1295

0-4924 1-2228

1-375444

102215

0-458706

0

a neutral mode) and Z = 2J (the condition for the occurrence of two coincident roots and the onset of overstability) at the points 27=0-458706 and 27=0-84404, (159) respectively. The equilibrium figures for these two values of E are illustrated in figure 2; and some of their properties are listed in table 2. In table 3 the characteristic frequencies of the toroidal modes, derived in accordance with equation (111),

o

FIGURE 2. The equilibrium figures for 27 = 0-458706 (the one on the left) and 27 = 0-84404 (the one on the right). The figure on the left allows a neutral mode of oscillation; and the one on the right is on the verge of overstability.

0

0-4

0-8

1-2

1-6

Z FIGURE 3. The dependence on 27 of the characteristic frequencies of the two toroidal modes of osculation of a rotating liquid drop. At B (27 = 0-45871) the drop allows a neutral mode but does not pass through instability in the absence of any dissipative mechanism; at S (27 = 0-84404) the drop becomes overstable; and the curves labelled 8&(cr) and^{cr) are the j-eal and the imaginary parts of the frequency for larger values of 27. [20]

TABLE 3. T H E FREQUENCIES OF THE TOROIDAL MODES OF OSCILLATION EXPRESSED IN THE UNIT (8T/pa3)t (The values in parentheses were obtained by interpolation applied to the values of 2J — E given in tables 1 and 2.)

0 0-05 0-10 0-20 0-275621 0-366754 0-458706 0-510790 0-75 0-84404

1 (1-1944) (1-2566) (1-3230) 1-3480 1-3597 1-3546 1-3443 (1-1993) 0-9187

1 (0-7472) (0-6242) (0-4286) 0-2980 01485 0 0-0851 (0-5328) 0-9187

10 1-2 1-4 1-6 1-725982 2-329113

1-0000 ±0-4264i (1-0954 ±0-6394i) (1-1832 ±0-7915i) (1-2649 ±0-9121i) 1-3138 ±0-9766i 1-5261+ 11846i


$tf>

0-8

0-4

0-4

0-8

1-2

1-6

FIGUBE 4. The dependence on E of the characteristic frequencies of the pulsation mode (curve labelled 1) and the three transverse-shear modes (curves labelled 2, 3 ,and 4). are listed; and their dependence on 27 is further exhibited in figure 3. I t is of interest to compare the behaviour exhibited in figure 3 with the corresponding behaviour along the Maclaurin sequence (cf. Lebovitz 1961, figure 2) and notice their remarkable similarity. Finally, in table 4 and figure 4 the results for the pulsation and the transverse shear modes are presented.

[21]

T A B L E 4.

T H E FREQUENCIES OF THE PULSATION AND T H E TRANSVERSE-

SHEAR MODES EXPRESSED IN THE UNIT pulsation

transverse-shear modes A

mode

E

^3

Vi


0-275621 0-366754 0-510790 1-000000 1-725982

1-3629 1-4740 1-6426 2-1655 2-8660

1-4959 1-6094 1-7795 2-3491 3-4152

0-9690 0-9990 1-0512 1-2429 1-5573

8.

(8T/pa3)$ •> 0-00193 0-00482 0-01364 010613 0-54413

CONCLUDING REMARKS

I t is clear that the present investigation requires to be extended along several directions. On the side of hydrodynamics, the extension to the case where the medium surrounding the drop is not a vacuum is particularly important: since only then can the results of the theory be compared with experiments (such as Plateau's) in which the drop is freed of the effects of gravity by its suspension in a liquid of the same density. And on the side of nuclear physics (for applications to phenomena related to nuclear fission), the extension to include the effects of volunie charges would appear to be most important. Extensions of the present analysis along both these directions are now being considered. In conclusion, I wish to record my indebtedness to Professor G. Wentzel for helpful discussions and especially for his proof of the symmetry of the surface-energy tensor @i3-; to Mr M. J. Clement for checking the analysis and providing tables of the elliptic integrals for the three critical values of Z (namely, 2-32911, 0-8440, and 0-45871); and to Miss Donna D.Elbert for carrying out the necessary numerical calculations. The research reported in this paper has in part been supported by the Office of Naval Research under contract Nonr-2l21(24) with the University of Chicago. APPENDIX.

T H E AXISYMMETRIC FIGURES OF EQUILIBRIUM OF A ROTATING DROP

The axisymmetric figures of equilibrium of a rotating drop have been considered by Poincare (1895), Rayleigh (1914), Appell (1932), and Sperber (1962). But the extant analytic treatments are not as complete as it is needed for the purposes of this paper. We therefore give a brief account of the theory in the context of the present paper and assemble the basic formulae of the subject. The pressure in the interior of the drop is clearly given by P =Po+hp®2™2

(Al)

where p0 is a constant. Applying equation (A 1) to the boundary z = f(w), we have, by equations (3) and (40), p0 + hpQ2rn* = Tdivn [22]

= - - d mdmj(l+2)'

(A 2)

where 0 = djjdw.

Equation (A 2) directly integrates to give

fc>„w» + \PQW = -T J0P)

+ O,

(A 3)

where C is a constant. We shall suppose that the drop encloses the origin. Then w = 0 is attained; and at this point <j> must vanish by symmetry. From equation (A 3) it follows that under these circumstances the constant C must be set equal to zero; and we have

At the equator (where w = a) cfi->-—co (cf. the remarks following equation (36) with respect to the convention of restricting ourselves to the positive hemisphere when ambiguities in sign arise) and equation (A 4) gives

Hence with E defined as in equation (1)

If we now measure VJ and z in the unit a, we can rewrite equation (A 4) in the form

4>

•TD(1-2+ZW2),

V(i+^2) 0 = # = Y dm

(A 7)

mjl-Z+Z^) [l-w2(l-Z + £w*)2]i

(A8)

From this last equation it follows that w(l-E+Zw2)

-**-F.\ By the transformations where

dw.

w2 = 1 — A tan 2 \6, A = ^(1

+ 21),

(A 9) (A 10) (All)

the integral defining z can be reduced to the form dd ( l+AI)--^-ft 1 + cos 8 V(l-jfc2sin20)' 22V^Jol where

k2 = -

G+r).

1 /l l+

A

1

(A 12) (A 13)

and tp is related to the value of w at the boundary (where z has the value (A 12)) by m = V(l-4tan2£(p).

(A 14) [23]

The integrals appearing in equation (A 12) are expressible in terms of the standard elliptic integrals

^ ( * , T H foV-*Wfl) ^

^.
(A 15)

of the two kinds. We have 1

IZ4A

'(l-AE)F(k,
+ 2AE E(k, )-S^^(1~k2sin2<9)' v 9r> '

I + C0S9

(A 16)

Equations (A 14) and (A 16) define the equation of the boundary of the drop, parametrically, for a given 27. In particular w = 0 when

9 =
(A 17)

and the corresponding value of 2 ( = z0, say) at the poles follows from (A 16). Since the maximum allowed value of 9 is \TT, it follows from equation (A 17) that a necessary (but, as we shall presently see, not a sufficient) condition for the equilibrium figure to enclose the origin is A>\.

(A 18)

From equation (A 11) we find that this lower bound on A requires that E^l+^2.

(A 19)

But it should not be supposed, as Sperber (1962) does, that if 27 < 1 + ^ 2 the equilibrium figure necessarily encloses the origin. Clearly, the maximum value of 27, compatible with the requirement that the origin be enclosed, is determined by the condition that z as given by equation (A 16) vanishes simultaneously with w, i.e. when 9 = 9 m a x . = 2 t a n - 1 [27$/(l + 227)1]; and this condition is more stringent than (A 19): it gives 27max. = 2-32911. (A 20)* Most of the integrals encountered in § 6, in the treatment of the stability, can be reduced to elliptic integrals. We shall state only the results: Zo=(l-27)J0

+ 27J1,

(A 21)

9 mix.

/ .0

J

9max.

wfdm = i[(l -E)J1 + EJ2], m3fdw = |[(1 - E) J2 + EJ3],

(A

22

)

(A 23)

* According to equations (All) and (A 13) V(l+2i7) = 2 ( 2 £ 2 - l ) ± [ 4 ( 2 £ 2 - l ) 2 - 3 ] i (i) The reality of the right-hand side requires that £ 2 > £ ( 2 + V3) or & = sin-1*: 3= 75°. (ii) For & = 75°, the two values of E given by (i) coincide and are equal to 1. For 75° ^ # *£ 90°, the smaller of the two values of £ given by (i) is less than 1 and is an 'allowed' root; but the present requirement that S ^ 2-32911 implies that both the roots of (i) are allowed only for & in the range 75° < d- < 77°-7509 (see table A1). [24]

' fatQim) dvj = 4( 1 - 27) [(1 - Sf

- 27] J2

/ : + [2227(1 - 2 7 ) 3 - 4272] J 3 + 42272(1 -27) 2 J 4 + 34273(1 -27) J 5 + 10274J6, iax,

_

m3P(w)dru = 2 J 0 - 3 ( 1 -EfJx

/ :

(A 24)

[(1 -27) 3 -627] J2

+ (1 -E)

+ [427(1 - 27)3 - 3272] J 3 + 6272( 1 - Ef J 4 + 4273( 1 - 27) J 5 + 27V8>

(A 25)

(w 2 /)' £(nj) dw = 2( 1 - 27) [3( 1 - 27)3 - 427] J& - / :

+ [4227( 1 - 27)3 - 8272] J3 + 90272( 1 - 27)2 J 4 + 78273( 1 - 27) J 5 + 24274J6, J

where

71

1

ii

f

(A 26)

1

ro "+ dro

(A 27)

2 2 J o [ l - t i r 2 ( l - 2 7 +-27w . ) ]i'

By the same transformations t h a t were used in the reduction of the integral (A 12) it can be shown t h a t J„ =

n

1

(A 28)

(l+4)»-™(-24)»J J B (

2 -,

2E^Am=Qm\(n-m}l

dd

where

+

(A 29)

cosd)m<J{l-k2sm2d)'

The integrals I0 and 71 are given by (cf. (A 15) and (A 16)) I0 = F(k, 9)

and

sin 9^/(1 — k2 sin 2 9) ^ = i^/fc, 9) - E(k, 9) + 1 + C0S9

(A 30)

and the integrals of the higher orders can be reduced to I0 and It by means of the recurrence relation

Im+1

2m+1

(4F+l)m/m-2(2m-l)fc2/m_1

+

(TO-l)F4_2

sin 9^(1 — P s i n 2 9 ) +

TABLE A l .

(l+cos9)m+1

(m ^ 1).

(A 31)

CONSTANTS DESCRIBING THE AXISYMMETRIC FIGURES OF EQUILIBRIUM OF A ROTATING DROP

(z0 is the polar radius in units of the equatorial radius; a1 is the equatorial radius in units of the radius of a sphere of the same volume; and the integrals simply related to the volume and the moment of inertia J u sin- 1 *: 80° 79° 78° 77° 76° 75° 76° 77° 77°-75085

F

E

A

0-969846 0-963592 0-956773 0-949397 0-941474 0-933013 0-941474 0-949397 0-954987

0-161182 0-210478 0-275621 0-366754 0-510790 1-000000 1-72598 2-09589 2-32911

7-13442 5-66348 4-51885 3-58995 2-78358 1-73205 1-22247 1-08715 1-02129


41°-0505 45°-5845 50°-3865 55°-6487 61°-8748 74°-4577 84°-2570 87°-6069 89°-3965

! f wfdm Jo 0-57605 0-55351 0-52641 0-49268 0-44699 0-33333 0-21935 0-17173 014312

wfdm and Jo of the drop.) I w'fdw Jo 0-11542 0-11105 0-10582 0099384 0-090784 0-070109 0-050706 0-043139 0-038785

Tjfifdw are Jo

0-86008 0-82391 0-77961 0-72314 0-64400 0-43118 0-18698 0-07281 0

1-0499 10640 1-0819 11061 1-1425 1-2599 1-4485 1-5716 1-6701 [25]

I n the particular case Z = 1, the integrals Jn can be reduced to T-functions;

thus

>r(H^) In table A 1 we collect together a number of constants describing the equilibrium figures for nine values of Z (including ZmSiX = 2-32911); they were evaluated with the aid of the foregoing formulae. REFERENCES

Appell, P. 1932 Tr&ite de Mecanique Rationelle, vol. 4, p. 295. Paris: Gauthier Villars. Cartan, E. 1922 Bull. Sci. Math. 46, 317, 356. Chandrasekhar, S. i960 J. Math. Anal. Appl. 1, 240. Chandrasekhar, S. 1961 Hydrodynamic and hydromagnetic stability, §§117, 118. Oxford: Clarendon Press. Chandrasekhar, S. 1964 The higher order virial equations and their applications to the equilibrium and the stability of rotating configurations in Lectures in theoretical physics, vol. 6. Boulder: University of Colorado Press. Chandrasekhar, S. & Lebovitz, N. R. 1962 Astrophys. J. 135, 248. Chandrasekhar, S. & Lebovitz, N . R. 1964 Astrophys. J. 140, 1517. Jeans, J. H. 1919 Problems of cosmogony and stellar dynamics, p. 40. Cambridge University Lebovitz, N. R. 1961 Astrophys. J. 134, 500. Lichtenstein, I. 1933 Gleichgewichtsfiguren Rotierender Fliissigkeiten. Berlin: Springer Verlag. Lyttleton, R. A. 1953 The stability of rotating liquid masses. Cambridge University Press. Plateau, J. 1843-69 Mem. Acad. Roy. Sci. Lett. Belg., vol. 16-37; various papers with the common title ' Recherches Experimentales et Theorique sur les Figures d'Equilibre d'une Masse Liquide sans Tesanteur'; these papers are available in English translation in The Annual Report'of the Board of Regents of the Smithsonian Institution, 1863 (pp. 207285), 1864 (pp. 286-369), 1865 (pp. 411-435), and 1866 (pp. 255-289) under the common title ' Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity'. Washington: Government Printing Office. Poincar^, H. 1895 Capillarite, p. 118. Paris: George Garre, Editeur. Rayleigh, Lord 1914 Phil. Mag. 28, 161; also Scientific papers, vol. vi, p. 257. Sperber, D. 1962 Phys. Rev. 130, 468.

[26]

743

VII. Relativistic Instabilities and Post-Newtonian Approximations During and after the near completion of his work on hydrodynamic and hydromagnetic stability, Chandra began to think of turning to general relativity, a subject he was first introduced to in his first year as a graduate student in Cambridge. Charmed though he was by Eddington's exposition of relativity, full of fun and humor, he had shied away from a serious study of relativity for over thirty years. This was partly because, at the time, relativity did not seem to be relevant for problems of stellar structure, internal constitution of stars, and other down-toearth problems in astronomy. The situation had changed by the sixties. Rapid discoveries were taking place in astronomy. Quasars, pulsars, radio galaxies, cosmic x-ray sources, cosmic microwave background, and the proclaimed detection of gravity waves created a new arena of research for practical-minded relativists. On the theoretical side, a new discipline, relativistic astrophysics, was shaping up, dominated by youthful personalities that included Kip Thorne, Roger Penrose, James Hartle, James Bardeen, Stephen Hawking, Brandon Carter, and others. "Chandrasekhar (or Chandra, as he encouraged us to call him) was our young-at-heart co-worker," says Kip Thorne, "as new to relativity as we. We had the flexibility of youth, freedom from preconceived notions that is a modest compensation for lack of experience. Chandra had the wisdom of decades of research in fundamental, Newtonian physics and astrophysics — a wisdom that gave him guidance on what problems were worth studying and how to approach them." (In Selected Papers, S. Chandrasekhar, Vol. 5, p. xii.) Once he decided to turn to general relativity, Chandra, in his typical fashion, began the summer of 1960 with an intense study of the subject, followed by teaching an advanced course on general relativity during the fall semester. In 1962, he attended the Warsaw international conference on general relativity, as an observer, mainly as he would say, to get a feeling of "what the experts were thinking." But it was not long before he brought general relativity to its "natural home" — astronomy. In 1964, applying the theory to the study of the stability of spherical gaseous masses in equilibrium under their own gravitational forces, he made a major discovery concerning the instability against gravitational collapse specific to the theory of relativity. Whereas in Newtonian theory, a star modeled after a self-gravitating spherical mass was stable against small radial perturbations, independent of the mass and radius of the star, Chandra showed that this was not the case within the framework of general relativity. This is the subject matter of the first two papers in this section. After that a steady stream of papers followed with the development of a systematic post-Newtonian approximation scheme. It led to the correct

744

description of radiation reaction and the discovery of a radiation-reaction-driven instability (Papers 3 and 4). Paper 5 presents a discussion on conservations laws in general relativity and how they are incorporated in successive post-Newtonian approximations. Paper 6 is an excellent review of the stability of relativistic systems as compared with the corresponding situation in Newtonian theory. Finally, Paper 7 marks the beginning of the investigation of stability and equilibrium of rotating systems in general relativity.

^^^^^ „ * ^ ^

:

^

746

VII. Relativistic Instabilities and Post-Newtonian Approximations

1.

2.

The Dynamical Instability of Gaseous Masses Approaching the Schwarzschild Limit in General Relativity The Astrophysical Journal 140, no. 2 (1964): 417-33

747

The Dynamical Instability of the White-Dwarf Configurations Approaching the Limiting Mass With R.F. Trooper; The Astrophysical Journal 139, no. 4 (1964): 1396-98

764

3.

The Second Post-Newtonian Equations of Hydrodynamics in General Relativity With Yavuz Nutku; The Astrophysical Journal 58, no. 1 (1969): 55-79

4.

The 2/4-Post-Newtonian Equations of Hydrodynamics and Radiation Reaction in General Relativity With F. Paul Esposito; The Astrophysical Journal 160, no. 1 (1970): 153-79

767

792

5.

Post-Newtonian Methods and Conservation Laws Relativity, ed. M. Carmeli, S.I. Fickler, and L. Witten (Plenum, 1970), pp. 81-108

825

6.

The Stability of Relativistic Systems Gravitational Radiation and Gravitational Collapse, ed. C. Dewitt-Morette (D. Reidel, 1974), pp. 63-81

853

7.

Criterion for the Instability of a Uniformly Rotating Configuration in General Relativity With John L. Friedman; Physical Review Letters 26, no. 17 (1971): 1047-50

872

THE DYNAMICAL INSTABILITY OF GASEOUS MASSES APPROACHING THE SCHWARZSCHILD LIMIT IN GENERAL RELATIVITY S. CHANDRASEKHAR University of Chicago Received May 11, 1964 ABSTRACT In this paper the theory of the infinitesimal, baryon-number conserving, adiabatic, radial oscillations of a gas sphere is developed in the framework of general relativity. A variational base for determining the characteristic frequencies of oscillation is established. It provides a convenient method for obtaining sufficient conditions for the occurrence of dynamical instability. The principal result of the analysis is the demonstration that the Newtonian lower limit J, for the ratio of the specific heats y, for insuring dynamical stability is increased by effects arising from general relativity; indeed, is increased to an extent that, so long as y is finite, dynamical instability will intervene before a mass contracts to the limiting radius (>2.25 GM/c*) compatible with hydrostatic equilibrium. Moreover, if y should exceed $ only by a small amount, then dynamical instability will occur if the mass should contract to the radius

I

3

where K is a constant depending, principally, on the density distribution in the configuration. The value of the constant K is explicitly evaluated for the homogeneous sphere of constant energy density and the polytropes of indices n = 1, 2, and 3. I. INTRODUCTION

It is well known that the gravitational field external to a spherical distribution of matter is described by Schwarzschild's metric

~ds>= -(l-^(dx°)>+rHd6*

+ sin>dd4>*) + l_2dGr^/rc2,

»>

where M is the inertial mass of the source of the gravitation. The metric (1) obtains, whether or not the matter which gives rise to the field is endowed with radial motions, so long as the spherical symmetry is preserved. From the form of the metric (1), it is apparent that the spacelike and the timelike characters of the coordinates x° and r are retained only for .2GM , D , r >—r-=Rs(sa.y).

(2)

The radius Rs defined by this inequality is the Schwarzschild limit appropriate to the mass M. If we should contemplate a situation in which the matter giving rise to the field is confined within a radius R < Rs, then the character of the coordinates x° and r must change as we cross the Schwarzschild limit; and the nature of space-time under these circumstances has been discussed widely in the literature (cf. Landau and Lifshitz [1962], p. 329). But these questions do not arise if we suppose that the matter, which gives rise to the external metric (1), is in a state of hydrostatic equilibrium and is further described by a pressure p and an energy density e (= — To0); for it can then be shown that the radius R of the configuration is necessarily restricted by the inequality (Buchdahl 1959),

[417]

In other words, we are not called upon to consider what might happen if r < Rs: it simply does not happen! The question which now occurs is whether the existence of the limit (3) has any bearing on the dynamical stability of the mass. It is known that, in the framework of the Newtonian theory of gravitation, the condition for dynamical stability is that the "ratio of the specific heats" y (in case it should be a constant) must exceed ^. The question is now essentially this: is the Newtonian lower limit J on y affected by general relativity? We shall show that general relativity does affect the condition for dynamical stability in a very striking way: for any finite y, dynamical instability always intervenes before the limit on the radius set by the inequality (3) is reached; and, more particularly, if y should exceed J only by a small amount, then for dynamical stability D

^

K

2GM

where K is a constant which depends, principally, on the density distribution. According to the asymptotic requirement (4), dynamical instability can intervene long before the Schwarzschild limit is reached. The fact that instability may arise under certain circumstances, by virtue of the effects of general relativity, has been considered by Iben (1963) and Fowler (1964). These authors introduce the concept of a "binding energy" and argue in terms of it. While such arguments are physically plausible (and sometimes give the correct results), it is known that they are not always conclusive. In any event, it is clear that the question of the dynamical stability of a gaseous mass can be answered without any ambiguity by an analysis of its normal modes of radial oscillation. In this paper such an analysis will be provided; and on its basis the condition for dynamical stability will be established in a number of special cases. The principal results of this paper have been announced in preliminary Letters (Chandrasekhar 1964a, b; see also Chandrasekhar and Tooper 1964). After the present paper was mostly completed, Dr. A. H. Taub drew my attention to an earlier paper of his (1962) in which the problem considered in this paper has been formulated by him in one special case. But the variational base for discriminating dynamical stability and the fact that a relation of the form (4) emerges in general relativity do not appear to have been considered before. n . THE FIELD EQUATIONS

We consider a spherically symmetric system with motions, if any, only in the radial directions. There is then no loss of generality in supposing that the metric is of the form - ds- = - e'(dxa)2 + r\dd2 + sin2 8d
(5)

where v and X are functions of r and x° only. The Einstein field equations appropriate to the metric (5) are well known (see, e.g., Landau and Lifshitz [1962], p. 325); they can be written in the forms c4 p i

[418]

\r dr &TGT1

_x/l

rv

r2

dv , 1 \

r 2 dr 1

r-

(6)

(7)

8irG ^ „

Rf -\R=Ri-\R= t 3 " - i"~

SirG

j- 7V = c*

j - T3> c* (8)

e

~

and

^

6

L 2 dr> L

i

i

dr dr +

dx» dx"+

pi-_M

2

r

t

KdrJ

dx° \dx°J ^

+

2 r dr

4

(V

M

J

\dx°J J '

i__!^ii

(9)

where 2?,-' denotes the contracted Riemann-Christoffel tensor and TV the energy-momentum tensor. We shall assume that Tf is of the form Tj* = (p + e)u% + pSf ,

do)

where

is the contravariant four-velocity, p is the pressure, and « is the energy density (arising from all causes). In the case under consideration only the components w° and ul are different from zero; therefore, TV = T33 = p, (iz) whether or not motions are present. A useful relation which follows from equations (6) and (7) is

— f(x+,)=^(^-r0»).

»3)

r dr c* Equations (6)-(9) are not all independent since the covariant divergence of U>* — R5j{ vanishes identically; and this last fact leads to the relation (TV);; = 0

(U)

as a necessary identity. In the case under consideration equation (14) leads to the two relations dTJ> dTp1 •JOT -2Y) ~ + Tal f \ -f (\+v) +-] = 0 (is) dx° dr ' o r L or rJ and

where it may be noted that TV = - e^Ti0

.

(17)

HI. THE EQUATIONS GOVERNING EQUILIBRIUM

In a state of hydrostatic equilibrium none of the quantities depends on x°; and ul is moreover zero. Distinguishing the quantities describing this equilibrium state by a subscript zero, we have TV = TV = T33 = p0,

and

To0 = - e0 ;

da)

[419]

and as the independent held equations we shall take (cf. eqs. [6], [7], and [16]) d (, r e - x .o ). = l. dr

8TG

— r2e0,

(19)

c4

e~ x ° dva 1 . . . 8irG ,_ — - } - = - ! ( 1 - e-x»)H r / » 0 ,

r

and

dr dpo

r2

c* , , . .

<20)

r

, <Mi

We also have the identity (cf. eq. [13]) — — ^ - ( X o + V o ) = — " J - (pO + to).

«2)

r ar c4 Eliminating vo from equations (20) and (21), we obtain the following pair of equations which are usually taken to describe hydrostatic equilibrium in general relativity: /,

2GMT\dp0

1 .

,

JGMr

,

4TTG

\

and Mr = ~Y C

f iQr2dr Jo

.

(24)

IV. EQUATIONS GOVERNING INFINITESIMAL RADIAL OSCILLATIONS

We shall now suppose that an equilibrium configuration governed by the equations of Section III is perturbed in a manner that its spherical symmetry is not violated. As a consequence of such a perturbation, motions in the radial directions will ensue. In obtaining the equations governing the perturbed state, we shall neglect all quantities which are of the second and higher orders in the motions and retain only the terms which are linear in them. Then Ml = e~"«liV ,

U\ — fixo-»o'2j) , (25)

u° = e~"°n ,

and

«o = — e"«'-,

where dr is the velocity in the radial direction in units of the velocity of light. To the same order as equations (25), TV = TV = T33 = p , T0° = - « ; (27) TQ1 = (pO + 6O)U1UQ = — (p0 + «l>)*> ,

and

(28) 2Y = (pa + «o)«°«i = e^-'ipa + «o> .

(Notice the absence of the subscripts zero for p and t in eq. [27].) Now let X = X0 + SX ,

[420]

v = v0 + Sv ,

P = po+ &p,

and

« = «o + Be (29)

be the values of the various quantities in the perturbed state. In view of equations (27), equations (19) and (20) are valid as they stand if Xo and v0 are replaced by X and v. The corresponding linearized equations governing the perturbations are, therefore, = ^ r28e c*

4- (re~^S\) dr

(30)

and

— (4- 3,-£«x)-^ax+M a ,.

r \dr dr J r2 c4 As the remaining two field equations, we may use the appropriately linearized forms of equations (9) and (16). We have (cf. eqs. [28]) r

e4

dx°

and e*°-'°(p0 + e0) -^i + ^-Sp dx° dr

+ iipo + to) 4-S"+^(Sp dr

+ 8i) ^ = 0 . dr

(33)

It is convenient to introduce a "Lagrangian displacement" £ with respect to the worldtime x° defined by v =

(34)

zv dx°

„.

8irG . i x , . (p0 + e0)Z,

S X = to - —give j Equation (32) then directly integrates

5X = -

$ -r- ( X o + V o ) .

or in view of equation (22), Equations (30) and (35) now give

(36)

dr

11 ^=-ZiTzlr*(po r2 dr

or alternatively,

(35)

+ ^)i],

(37)

* e = - * ; j S ! - « ^ - ( * > + «o)Ji7r(r»f). < 38 > 2 dr dr r dr Substituting for dp0/dr in this last equation from equation (21), we can also write **=-Sj?-iPo

+ «)*:£j;(r*e->.<*S).

(3.)

Considering next equation (31) and substituting for 8\ in accordance with equation (35), we obtain e~xo d or in view of equation (22), (#o + «o) £

« * = [ « * - (jo + «o)(f^ + ^ ) $ ] £

(X 0 + ,o).

(«>

[421]

We shall now suppose that all the perturbations have a dependence on x° of the form eiax\

(42)

where ca is a characteristic frequency to be determined. Letting S\, 8v, 8p, and 5e now stand for the amplitudes of the respective quantities with the time-dependence (42), we can rewrite equation (33) in the form <72^.-"o(p0 + «0)£ = -f5p+8p-^dr

dr

(iXo+O+ISef5 dr (43)

WJ._I_

-^P>

\fdvo , 1\ /d\0

+ <*\Tr+-r)\Tr-

. dv0\

+

d-r)^

where we have substituted for (po + eo)dSv/dr in accordance with equation (41). Also, it should be recalled that 5e is expressed in terms of £ and the unperturbed variables by equations (38) or (39). V. THE CONSERVATION OE THE BARYON NUMBER

The field equations have enabled us to express the perturbations in all the variables, except the pressure, in terms of the Lagrangian displacement £ and the unperturbed variables. To express Sp similarly in terms of £, we need an additional assumption which will make the physical content of the present theory equivalent to the physical content of the corresponding non-relativistic theory of the adiabatic radial oscillations of a gaseous mass. Since the energy-momentum tensor (10) which we have assumed includes no dissipative terms, the first law of thermodynamics is, in effect, already included in the theory (cf. Landau and Lifshitz [1962], p. 382). What requires to be added to the field equations is, therefore, some statement which will be equivalent to the separation of «, in the classical theory, into a part which is related to the rest mass of the constituent particles of the system and a part which is related to what is conventionally the internal energy of the system. While the additional assumption by which we must supplement the field equations can be formulated in a variety of ways, it is perhaps simplest (particularly for the contexts in which the present theory may find application) to state the supplementary condition as that required by the conservation of the baryon number.1 li N is the number of baryons (anti-baryons counted negatively) per unit volume, its conservation in general relativity will require that (Nuk)-k = 0

(44)

^ J ( # « * ) + tf**^ log V - g = 0 .

(45)

or

In the framework of the present linearized theory in which the non-vanishing velocity components are given by equations (25), equation (45) becomes ox"

dr

dx° (46)

+ Ne-,a/2V

A |^i ( x +*)+!] = 0.

Now writing N = N0(r) + SN(r, x")

(47)

* The procedure we follow here is essentially the same as that of Landau and Lifshitz (1959, p. SOI) in their formulation of the equations of relativisticfluiddynamics.

[422]

in an obvious notation, we obtain from equation (46) (again neglecting all terms which are quadratic and higher in v) dx°

r1 dr

dx° (48)

L (A 0 +„o) = 0 + |iVoe-"«/2j, — dr With v expressed in terms of the Lagrangian displacement £, equation (48) integrates to give 5iV + - ^ £ : ( i V „ r 2 ^ - ' ' . / 2 ) + i i V 0 [ 5 X + a ^ : ( X 0 + J ; 0 ) ] = 0.

(49

The last term on the right-hand side of equation (49) vanishes on account of equation (36). We are thus left with e"o/2 d 5,V= (tfr.{e-r,/*) (SO) r2 dr or alternatively, dr r 2 dr Now if N = N((, p) (52)

is the equation of state, then, it follows from equations (39) and (51) that

where y is the "ratio of the specific heats" defined in the manner V I . THE PULSATION EQUATION AND THE VARIATIONAL PRINCIPLE

With Se and 8p given by equations (37) and (53), equation (43) becomes

-i'^)(^)(^+S)i-»S!^i<^+«'^+7<^'^!'"> — e-(*.+2"o)/2 — r e (x,+2»„)/J^2 e*a/t— ( r s e — , / 2 t ) ] . ar L r2 iir J Substituting for dpo/dr its value given by equation (21) in the first two terms on the right-hand side of equation (55) and combining them with the two following terms, we find , \{d2v0 ! d\0 dv0 1 d\0 3 dv0\ .. w . \dr2 dr dr r dr r dr/ * For an inclosure containing matter and radiation, y denned by this equation is identical with the coefficient r , as customarily denned in stellar structure (cf. Chandrasekhar [1939], pp. 55-59).

[423]

On the other hand, according to equation (8), under conditions of equilibrium, 167r<

d2v0 dr2

^ J. x c * 4

i, d\0 dva dr dr

1

/dv0\2 \dr/

.Id r dr

Using this relation, we find that expression (56) becomes

c

dr \r c4

r rt r

r

r

po + e0 dr / ^0 + e 0 \ r f r /

where use has again been made of equation (21). With this reduction of the four terms, equation (55) takes the form o + e o K = - 4 ^ £ - e-<xo+2"»>/2-^--[ e<\+u,)fl 2h2 *. ( r 2 e - o / 2 £ ) l r dr dr I r dr J (59)

*>*G x „ / A i > f 1 /^M*, — ^o(/>o + . o ) $ - ^ ^ ( / - J | . This is the required "pulsation equation." Solutions of this equation must be sought which satisfy the boundary conditions £ = 0 at r = 0

and

Sp = 0 at r = R .

(eo)

Equation (59) together with the boundary conditions (60) constitute a characteristic vulue problem for a2. It is apparent that this problem is self-adjoint and that a variational base for determining a2 is provided by multiplying equation (59) by r2£ exp [j(Xo + "o)\ and integrating over the range of r. We thus obtain «7* f Jo +

e<-^-")/2(p + e)r2t2dr

=4f Jo

e<>+')/n-r ^ dr

£2dr

|fte(x+,)/2l|[^(r2e-,/2n]2^-^He(x+,/2(^)2l!il,r H—V / C

W1)

e( 3N +')/ 2 />(/>-r-e)r 2 | 2 (fr,

JQ

where we have now suppressed the distinguishing subscript zero as no longer necessary. Associated with the variational equation (61) is the orthogonality relation f

e(3X-,)/2(^_|_e) r2 £(i)|(;') dr=0

(i^j),

(62)

Jo

where £') and $>~> are the proper solutions belonging to different characteristic values of
[424]

VII. THE HOMOGENEOUS MODEL

We shall first apply the condition for dynamical instability derived in Section V I to the homogeneous sphere of constant energy-density e and constant ratio of specific heats y . As is well known, equations (23) and (24) governing hydrostatic equilibrium allow of immediate integration in this case. Thus, writing y

2

r2 = l - —2 a

and

yf = 1

R2 r, a'

(63)

where a2 = ^ " ,

(64)

we find that the solutions for the relevant physical quantities can be expressed in the forms y - yi * e' = \{3yi-y)*. <«s> ex and 3y!-y y2 We may parenthetically note here that according to the present solution for p, a necessary condition for the pressure to be positive everywhere is 3yi > 1

R2 < f a 2 .

or

(66)

Expressing this last inequality in terms of the inertial mass we have

M = |7ri? 3 e/c 2 ,

(67)

2GM R2 . . Re = — < s

(68)

or (cf. eq. [3])

As we have already stated in Section I, Buchdal (1959) has shown that the lower limit to R set by the inequality (69) is absolute in the sense that it obtains independently of any constitutive relations between p and e that may exist. Now inserting the solutions (65) in equation (61), we obtain A 2 2 , f* vH2dy rn 2 y 2 - l - 9 y i 2 2 4a2
2

+ i7/""(y-y 1 )(3y 1 -y) r-f („»«-/»$ jl'-^L where TJ

= r/a

and

771 = R/a,

(71)

and 7 (as stated) has been assumed to be a constant throughout the mass. For the "trial function" £ = r,e"12 = | , ( 3 y i - y) , equation (70) becomes ( a o - ) 2 y i / 3 —.dv = \yi J^y Ja

(2y2-l-9yl2)

(72)

—,dv y3 (73)

rii

+ I7/

n2

( y - y i ) ( 3 y i - y ) 2 —dTj.

[425]

With the substitutions (cf. eqs. [63] and [71]) y = cos 0

and

t] = sin 9 ,

(74)

equation (73) reduces to 4 r'i sin r"i sin40 sin400 2 d9 [ -}LT^d9 = \cos9l ( 2 c o s 2 0 - l - 9 c o s 2 0 i ) cos 0 2 o cos 9 Jo + l7j (cos9-cosei)(3cos9 1 -cos9) ! ! sin 2 9(f0,

J

(75)

where 0i = sin"1 (JR./a) .

(76)

The various integrals in equation (75) are elementary and can be evaluated. And we find from the resulting equation (with a2 set equal to zero) that for any assigned value of 0i ( < sec -1 3 = 70?529) dynamical instability must occur if y is less than a certain yc (which depends on 0i). The values of yc determined in this manner are listed in Table 1. Table 1 also includes the results of similar calculations based on the alternative TABLE 1 THE LOWER LIMITS FOR y FOR DYNAMICAL STABILITY OF A HOMOGENEOUS SPHERE OF CONSTANT ENERGY DENSITY « FOR VARIOUS VALUES OF Rc'/IGM (= R/Rs) yc EVALUATED WITH THE TRIAL FUNCTION »1

0° 20 30 40 45

ye EVALUATED WITH THE TRIAL FUNCTION

R/Rs

00

8.549 4.000 2.420 2.000

R/Rs

9l

{ = ,W<

«-«« 1'

1.3333 1.3940 1.4890 1.6769 1.8375

1.3333 1.3938 1.4879 1.6704 1.8257

1.704 1.490 1.333 1.217 1.125

50° 55 . . . 60 65 70.529

{ = ,«»/<

{ = ,«»/"

2.0899 2.5204 3.3703 5.5802

2.0627 2.4520 3.1652 4.7146

OO

CD

trial function $ = i\e'u; it is seen that this trial function is somewhat "better" than f = ye'/2, particularly for the larger values of 0i. It is evident that in the limit 0i —> 0, the results obtained are exact: for, in this limit, the chosen trial functions tend to the true proper solutions. This is the reason for the close agreement of the results derived with the aid of the two trial functions as 0i —> 0. When 0i —• 0, we find from equation (75) that (aff)2 = M(3T - 4) - ^0i 2 (54 7 - 53)]

(01 - > 0) . (77)

In this limit, the condition for dynamical instability is, therefore, J? 2 -7 , _ 4 < 1 9 t 10 , 2 = 1 9 _ ^^42 ' 4 T a2-

(78)

Expressed in terms of the mass M, inequality (78) gives 7

[426]

3

*2

RC2

(79)

19

2GM

42(7-J)

c2

'

The meaning of this last inequality is that if 7 should be in excess of ^ by only a small amount, then dynamical instability will intervene when the mass has contracted to the radius given by the right-hand side of the inequality. The inequality (80) is the expression, in the present context, of the relation (4) mentioned in the Introduction. VIII. RELATIVISTIC POLYTROPES

Tooper (1964) has recently constructed, in the framework of general relativity, models which are generalizations of the classical models of Lane and Emden. In these relativistic models, the pressure is assumed to be proportional to some power of the energy density (rather than the density as in the classical models). Expressing the pressure and the energy density in terms of a single function 9 in the manner P = pc&n+l

e = ec9n

and

(si)

(where n is the "polytropic index" and the subscript c distinguishes the values at the center), and letting r = at)

where

a = -—. L

'*—

iir(jtc

and J

q= —,

<82>3

ec

we may reduce equations (23) and (24) appropriately to obtain differential equations which will determine 9 as a function of i\. Tooper has carried out the necessary reductions; and I am grateful to him for providing me with his tables of the function 0 in advance of publication. In terms of the variables 9 and r;, equation (60) becomes f ^ ? /""eC3X-,)/2e»( i + q Jo

qQ)v^2dv

= y

/"" e( x+3,)/2 en +i [ A ( ^ g - ^ ) ] 2 ^ Jo Lari J 17•" (83)

+ 2(n+l)q

fl

e^+')'iQi"+1 ( 1 + qB)

n^2dr,,

J{\ '0

whsre it may be noted that, in the present context, equation (21) takes the form dv _ drj

2(n+\)q I + 59

Also, y has been assumed to be a constant. With the trial functions £ = Tje"74 and

dQ drj '

(84)

Z = V,

<8S>

the values of 7 at which marginal stability occurs, for some assigned values q, were determined with the aid of equation (83) for the case n = 3. The results of the calculation are given in Table 2. We observe that while the trial function £ = 77 is slightly "better" than J = rje"'*, the differences are insignificant in all cases of physical interest

(7 < f).

* The notations used in this paper are slightly different from Tooper's: what he denotes by 9, £, and a are here denoted by 9, TJ, and q, respectively.

[427]

a) The Post-Newtonian Approximation In a post-Newtonian approximation, in which the effects of general relativity are treated as first-order corrections, we may expect relations of the forms •±

=C

TT —

^

pc n — = Cq

(86)

PcC

and Rc =

K 2GM y— ±3 '

(87)

where C and K are constants which depend, principally, on the density distribution in the configuration. The results given in Tables 1 and 2 are in agreement with these expectations; and it appears, moreover, that the relations (86) and (87), which are expected to be valid only asymptotically, actually predict the onset of dynamical instability (for the polytrope n = 3) with unexpected precision for y < §. It would, therefore, appear TABLE 2 THE LOWER LIMITS FOR y FOR DYNAMICAL INSTABILITY OF RELATIVISTIC POLYTROPES OF INDEX 3

FOR DIFFERENT VALUES OF q = pc/ec 7c EVALUATED WITH THE TRIAL FUNCTION q

{-,«»/«

0,015 .040 ,100 0.200

1.3732 1.4411 1.6088 1.8894

f= i

1.9144

that most cases of physical interest can be treated adequately in a post-Newtonian approximative treatment. Such a post-Newtonian treatment will now be considered for the polytropes. It is shown in the Appendix that in the post-Newtonian approximation, 9 = 9 + q

«- x = l +

dd_ 2(n+l)qv dv'

and

* " = l - 2 ( n + l ) g ( 0 + in|0i'|),

(88)

where is a function which is tabulated in the Appendix for n = 1, 2, and 3; B(ri) is the classical Lane-Emden function; 771 its first zero; and |0i'| = — (dd/dri)v=Vi . Using the relations in the theory of the polytropes which express pc and pc in terms of M and R (or more immediately from the fact that at the boundary e _x = 1 — 2GM/Rc2), we find that 1 2GM (89) 1 = 2 ( n + l ) i | i | 0 i ' | Re2 The constants C and K in equations (86) and (87) are, therefore, related (in the case of the polytropes) by K =

(90)

2(»+l)i71|9i'|

[428]

Inserting now the relations (88) in equation (83), expanding the various quantities in a power series in q, and neglecting all terms which are of order higher than the first in q, we obtain after some reductions

—

[fVl6nv2Z2dv

+ q / V - l F ( v ) vH2dv]

_ Tj fV»[£ (rt) ]-£ +4( . +1)J rvg,w, +( . +1)TfJ rv[,_,(, + ,« +ftlvl )][^ ( ,. {) ]^ + 2(n+l)yq

/ " V + 1 7ai\^ f T at] -(i?,{)rfij+(n+l)?

•>ts

(91)

f1en-lS(v)v^dv,

JQ

where F(n) = - 3 ( « + l ) > ; 0 ^ + a??

+ 2)02+(M+l)7h|0i'|0

W4>+U

<«>

and art \

drj

/

at\ (93)

It is known that in the Newtonian limit dynamical stability requires that y > ^. Therefore, in the post-Newtonian approximation, dynamical stability will require that 7 > 7c = § + e, where e is a small quantity which will depend linearly on q (in a firstorder theory such as the present). It is also known that in the Newtonian limit the proper solution at marginal stability is proportional to »/. Consequently, with the trial function £ = t\ we should obtain from equation (91) the exact condition for marginal stability in the post-Newtonian approximation. And equation (91) with £ = J; and cr2 = 0 gives

9(7-#)

r'O'+Wdr, (94)

1

+ ( n + l ) j [f"o~- S(Lv)r,*dn

+ yf'"o«R(r,)r,*dv]

= 0,

where R(n)=9[-d(9

+ V^-+rn\81'\^]

+ 6r,6^-.

(95,

Equation (94) leads to the following criterion for the onset of dynamical instability:

7 . - | = - q 9ii5i~9ni,l2[fQ'hd''~^(v)v'dv

+ ^foVle"R(v)v^v],

(96)

where we have made use of the relation Jo

o—n

well known in the theory of the polytropes.

[429]

We observe that equation (96) is in agreement with the expectation expressed in equation (86). Moreover, for the constant C we now have the value C=

5—n

[f^^-'Si

v) V3dV +%f'h6»R(r,)v*dv]

;

(98)

and the value of the constant K in the associated relation (87) follows from equation (90). The values of the constants C and K were determined in accordance with equations (90) and (98) for the cases n = 1, 2, and 3; these values are listed in Table 3. The values of the same constants for the homogeneous sphere derived in Section VII are also included in Table 3 under n = 0. I X . CONCLUDING REMARKS

The principal result established in this paper is the demonstration that, in the framework of general relativity, dynamical instability by a mode of radial oscillation will intervene before a gaseous mass contracts to the limiting radius compatible with hydrostatic equilibrium. Under most natural circumstances, when a mass approaches a state where effects arising from general relativity may be expected to play a role, the ratio of the specific heats y also approaches the value ^ so that dynamical instability (under these circumstances) will, in fact, intervene long before the Schwarzschild limit is reached. TABLE 3 THE VALUES OF THE CONSTANTS C AND K IN THE ASYMPTOTIC RELATIONS (86) AND (87) FOR DIFFERENT POLYTROPES n 0 1 2 3

C

K

1.8095 2.2615 2.4968 2.6325

0.4524 0.5654 0.7513 1.1245

As an illustrative example, consider a gaseous configuration with a mass currently assigned to the quasi-stellar radio sources, namely, 108 solar masses. For a mass as large as this, the gas pressure can only be a minute fraction ( = 4.3 X 10-4) of the total pressure; and this fact entails the consequence that the effective ratio of the specific heats differs from | by only 7.2 X 10 -5 . If we assume that the configuration is a polytrope of index 3, then dynamical instability will occur when the mass has contracted to the radius (see Table 3) 1.12 45 2GM (99) = 4.7X 1 0 " cm = 0 . 5 light-year, R=

7-3 and it is suggestive that this is of the same order as the radii presently estimated for the quasi-stellar objects. A second example which illustrates the theory is the occurrence of dynamical instability along the sequence of the degenerate white-dwarf configurations before the limiting mass is reached and when the mean density is still only 6 X 108 gm/cm 3 (Chandrasekhar and Tooper 1964). Finally, it should be emphasized that the instability considered in this paper is entirely relativist'ic in its origin. An unambiguous demonstration that the instability is manifested in nature when the conditions for its occurrence, as required by equation (61), are fulfilled will provide a unique confirmation for the general theory of relativity.

[430]

I am greatly indebted to Miss Donna D. Elbert for her assistance with all the numerical work. I am also grateful to Professor G. Wentzel for helpful discussions. The research reported in this paper has in part been supported by the Office of Naval Research under contract Nonr-2121(24) with the University of Chicago. APPENDIX THE POST-NEWTONIAN APPROXIMATION FOR POLYTROPIC EQUILIBRIUM When the pressure p and the energy-density € in a configuration are related in the manner (cf. eq. [81] in the text) p = pcQn+l and t = ecQn, (AD the equations governing hydrostatic equilibrium (namely, eqs. [23] and [24]) can be reduced to the pair of equations (cf. Tooper 1964) 1 + 50

ay

*

dr]

and £^=,,20"

(A3)

dr]

where q — pc/ec and r\ is the dimensionless radial coordinate defined in equation (82). In the post-Newtonian approximation, we treat q as a small constant and seek solutions of equations (A2) and (A3) of the forms e = 0 + g0+O(?2)

and

V = v + qw+0(

?2) ,

(A4)

where 8 is the classical Lane-Emden function satisfying the equations

J. * ( „ , * £ ) _ _ , , . T d-q \

dr]/

and V =

~

7J2 —

(A6)

.

dt\ Inserting for 9 and V the forms assumed and neglecting all quantities of orders higher than the first in q, we obtain from equations (A2) and (A3)

v2^-\e-2(n+i)v^-] d-q

L

3 n l

v^+v

dr]S

e + +w

=o

aw

dr]

and

dw -r- = «n 2 0 n - 1 . dr) Eliminating w between equations (A7) and (A8), we find after some reductions that

(A8)

^(''^)--"»-'*-[""'-3(»+1"-S-(2"+ 3 , (0]- And the boundary conditions with respect to which equation (A9) must be solved are dd> = -p-=0 dr]

at

77 = 0 .

(Aio)

Solutions of equation (A9) satisfying the boundary conditions (AIO) have been obtained by numerical integration for the cases n = 1, 2, and 3; these solutions together with their first derivatives are tabulated in Appendix Table 1.

[431]

O

-> tsj Cn 00 OO > : t o 4 ^ O* --4 f- 1

O O O O O O O O O

..._

.—

O O O

O

O

O

O

O

O

O

O

O

O

IIM

O

O

O

O

O

O

O

O

O

O

O

IIIII1IIM

IIIM

IM

IiII I

JftMOiOoci-'xOifrK'CvOJ

tnC/iCnC/»C/»Cyitn(jiCyi4J.4*^th»0^lsJ>—'00>--'NiOJtf^

O

+ ++++ ++ ++ +++ ++ +++ M I I M

N)^OJC*J4*>4^t/iC/iO\0>^^00(XOOOO^OOOOCOOO \Oc^ONUccw<3^iNU)Cotn^Oi-4i-'Owocoi-'ai N300CnO»00O'OO>-t4^'^'f*^'-tOiN)00>-'^OC«(>Jit*

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

-+++++++++++++++++++++++++++++++

OOOOOOOOOOOOOOOOOOOOOOO^-^-^-o^^^-*'—^>——'^'-'H-H-^-H-oo.-i

t1ItM

IIIIIII I

H- (— ^,-*h-*->-*-*->-*»-*t-»l— l—l— h - H - H - H - h - H - l -

I IIIIIIIIIIIII

ojw»-o i ooo^a.oi*»utot-o , oc»Maoi^u

0vOl*'v00000i-tOMOO\O^(Xtsj0J\0v0t/iLnO'f"-'Hai0J00

+++++111111111111111111111

OjOJ4^^CnCn^4^^C^tNjH-Ooo-^Q\4^C^H-OCXi'^p\Oi4^C^ p>i{*ooooo^ooC)OJvioi>ooK»tfl1uouooo\OH^o^o

i i i i i i i i i i i i i i I i i i i I I I i i i I

i - O O O o v j O s t n ^ U W ^ - ' O ' O O O M O ^ t / i ^ W W K O * O O O vj

ONOvOvONONONOsOvOvCnCyiC/iLnt^t^C/iUiC/iC^**^*-^^^^****^^)^

I IIIItII M

<3

1 1111 1 I I 11 11111 1 o oooooooooooooooo OOOOOvOOvOvO'OOOOOvjOtjiUKJt--

O^JiO»0^0»-'OLn(»i-'i&

H - O O O O O O O O O O O O O

1 1111 1 1111I I 1 1 I I

n-i-*t-*i-*i-n-*t-*0O©OO©00O0 OvOi^OHOt-OOCO-NtOstn^uWi-.

.. . ^

o o . —

I I I I I I I I I I I I

* - J O J ^ > O I - K> h j G J Cyi VO C\

I IIIIII!II II ooooooooooooo

i— h - i — O O O O O O O O O O

00666 02653 05931

The expressions for the metric coefficients ex and e" are, in general, given by (cf. Tooper 1964) . - * - l - 2 ( « + l ) « - ^ -

and

*' =

( 1 +

qQ]^X)

(AH)

where iji is the first zero of 0. In the post-Newtonian approximation, the foregoing expressions become j a

e-*=l

+ 2(n+l)qr,-r

and

e* = 1 - 2 ( » + 1) q ( 6 + vi I 0i' I ) ,



where T/I is now the first zero of d and |fli'| = — (<W/d?;),_,, . REFERENCES Buchdahl, H. A. 1959, Phys. Rev., 116, 1027. Chandrasekhar, S. 1939, An Introduction to the Study of Stellar Structure (Chicago: University of Chicago Press; Dover reprint, 1957). . 1964a, Phys. Rev. Letters, 12, 114. 19646, ibid., p. 437. Chandrasekhar, S., and Tooper, R. F. 1964, Ap. J., 139, 1396. Fowler, W. F. 1964, Rev. Mod. Phys., 36, 545. Iben, I. 1963, Ap. /., 138, 1090. Landau, L. D., and Lifshitz, E. M. 1959, Fluid Mechanics (London: Pergamon Press). . 1962, The Classical Theory of Fields (London: Pergamon Press). Taub, A. H. 1962, Collogues internationaux du Centre Nationale de la Recherche Scientifique, 91, 173. Tooper, R. F. 1964, Ap. J., 140, 434.

[433]

THE DYNAMICAL INSTABILITY OF THE WHITE-DWARF CONFIGURATIONS APPROACHING THE LIMITING MASS* It is known that completely degenerate configurations built on the relativistic FermiDirac equation of state for the electrons have a mass-radius relation which predicts that the radius tends to zero as the mass tends to the finite limiting value (Chandrasekhar 1931, 1935; and for a general account Chandrasekhar 1939) Mi = 5.75 M r 2 O ,

(i)

where y.e is the mean molecular weight per electron. It is also known that the effective "ratio of the specific heats" (7) tends to the value f over most of the configuration as the limiting mass is approached. We should therefore expect, on the basis of a result which has recently been established (Chandrasekhar 1964a, b), that, if the effects of general relativity are allowed for, the white-dwarf configurations should become unstable long before the Schwarzschild limit, Rs = ^ ~ =

16.94

Me~

2

km,

W

is reached. It is of particular interest to verify this prediction in detail, since it will provide an exact result of general relativity based on a self-consistent physical model. II The equations governing completely degenerate matter are (Chandrasekhar 1939, chap, x, eqs. [19]-[23]) p=Af(x),

pc* =

^x*,

and

(3) 2

t = pc + Ekin =

8/4

3

{x +

lqg(x)\,

where .

7rrae4c5

^=^F-'

q=

me

^H<

(4)

f{x) = x(2x2 - 3)(x2 + 1) I/2 + 3 sinrr 1 x ,

(5)

g{x) = 8x3[(*2 + l) l / 2 - 1] - fix) .

(«)

and Also, in equations (4), the various atomic constants have their standard meanings. For matter governed by equations (3)-(6), the equations of hydrostatic equilibrium in general relativity can be reduced to the pair of equations

and

p-=r,tlx'

+ tqg(x)],

(8)

* The research reported in this paper has in part been supported by the Office of Naval Research «wder contract Nonr-2121(24) with the University of Chicago.

11396]

where y = V(x2 + 1) ,

(«

and rj measures the coordinate radius r in the unit °2l (10,! (32xG,4)V2And the boundary conditions with respect to which equations (7) and (8) must be solved are y = yo and v = 0(ij3) for rj = 0 , (u) where yo(>l) is some initially assigned constant. For each assigned value of yo, equations (7) and (8), together with the boundary conditions (11), determine a unique configuration whose mass and radius are determined by the value ijj of 17 where y = 1. Ill The equation which provides the variational base for determining the frequency (c
f Jo

+/

e^+'l/Hp

*

+ ^rWdr I2 dr

Vd e(x+3,)/27£

I " ( r 2^,)

fBe^+i')^r^-4>idr dr

= 4 Jo '0 rR

8ir<7

fLL + £ ^

/

eH\+>)/tp(p

+ e)r*4>idr

(12)

-/"•*~"?E(£)'<where we have used, instead of the Lagrangian displacement £, the variable \p = e-'n$ .

(13)

x

In equation (12), the metric coefficients e and e' have, in the context of equations (7) and (8), the values 1 1 - 2GM/R c2 2 J -x \-2qv/n *and e" = -r-r-, U + g ;C—y - l. )^]-,«• d*) Also, for matter governed by equations (3)-(6) •v = 7

»

3

(15)

/(*) V(*2+i)'

Substituting from equations (3), (14), and (15) in equation (12) and changing to the variable i), we obtain ? ( 1 - 2 G M / J ? c 2 ) y0

(1-2W?)3/2

'

•ri^js + *7[^^)]'i(i-2w^p[i+ } ( y -i)-p < i 6 ) /•'. f{x)x3+2r,2dv r\ xhl,*r,Hdy/dri)*dv U (l-2?e/ij)»/»[l + g ( y - l ) ] » V„ ( l - 2 9 » / i i ) 1 / « [ l + j ( y - l ) ] 4 " 1 This unit differs by a factor 1/yo from the one customarily used in the theory of the white dwarfs (cf. Chandrasekhar 1939, chap, xi, eq. [27]). ,

x 4

[1397]

IV We have determined a2 with the aid of equation (16) for a number of configurations governed by equations (7) and (8) (with //« = 2 and q = 2.743 X 10~4) and a chosen trial function of the form yp = AT\ + Bi)3,

(17)

where A and B are two variational constants. The calculations show that along the sequence of the degenerate configurations dynamical instability sets in at a point where y0 = 22.581,

M = 1.4176 Q ,

R= 1.0267 X 103 km , (18)

Pa = 2.328 X 1010 gm/cm3, and p mean = 6.220 X 108 gm/cm 3 . Thus, in agreement with the general predictions of the theory, dynamical instability sets in along this particular sequence when the radius is 246 times the corresponding Schwarzschild limit. S. CHANDRASEKHAR ROBERT F. TOOPER Received May 3, 1964 UNIVERSITY OF CHICAGO AND I.I.T. RESEARCH INSTITUTE, CHICAGO

REFERENCES Chandrasekhar, S. 1931, M.N., 91, 456. . 1935, ibid., 95, 207. . 1939, An Introduction to the Study of Stellar Structure (Chicago: University of Chicago Press). . 1964a, Phys. Rev. Letters, 12, 114. . 19646, ibid., p. 437.

[1398]

T H E ASTROPHYSICAL JOURNAL, Vol. 158, October 1969 © 1969. The University of Chicago. All rights reserved. Printed in U.S.A.

THE SECOND POST-NEWTONIAN EQUATIONS OF HYDRODYNAMICS IN GENERAL RELATIVITY S. CHANDRASEKHAR AND YAVUZ NUTKU University of Chicago Received 1969 February 26 ABSTRACT In this paper the equations of hydrodynamics governing a perfect fluid in the second post-Newtonian approximation to general relativity are derived; in this approximation all terms inclusive of 0(c~*) are retained consistently with Einstein's field equations. And the equation 2 s 'y = 0 governing the energymomentum tensor is also derived to 0(0"*), The various conserved quantities to 0(c~*) are isolated with the aid of the Landau-Lifshitz complex 6 ' ' . In particular, to obtain the conserved energy to 0(c~4) it is necessary to evaluate the metric coefficient gajS to 0(c~6), i.e., to an order appropriate to the third post-Newtonian approximation. I . INTRODUCTION

In an earlier paper (Chandrasekhar 1965; this paper will be referred to hereinafter as Paper I) the equations of hydrodynamics governing a perfect fluid, in the first postNewtonian approximation to general relativity, were derived; in this approximation all terms of order c~~2 are retained consistently with Einstein's field equations. In the present paper, the corresponding equations in the second post-Newtonian approximation are derived; in this approximation all terms inclusive of order c~4 are retained. It will appear that in this second approximation the conserved quantities (except the total baryon number) can no longer be obtained by a simple inspection of the equation of motion T% = 0 ,

(1)

expressing the vanishing of the covariant divergence of the energy-momentum tensor Tij as it was possible in the first approximation. However, the equation 6« y = 0 ,

(2)

expressing the vanishing of the ordinary divergence of the Landau-Lifshitz complex 0 " , enables us to determine the conserved quantities by a simple and direct algorism (cf. Chandrasekhar 1969; this paper will be referred to hereinafter as Paper II). In this paper, besides obtaining the explicit form of equation (1) in the second postNewtonian approximation, the various conserved quantities are also explicitly determined. I I . T H E INFORMATION PROVIDED BY T H E EQUATIONS OF THE FIRST POST-NEWTONIAN APPROXIMATION

The method by which we shall obtain the equations in the second post-Newtonian approximation is similar to the one by which the equations of the first post-Newtonian were obtained from the equations and relations valid in the Newtonian limit. Thus, with the improved knowledge of the metric coefficients provided by the first post-Newtonian approximation in Paper I, we can evaluate the components of the energy-momentum tensor to one higher order than was known before; this improved knowledge of the energy-momentum tensor will enable us to solve the field equations for the metric coefficients to one higher order; and this improved knowledge of the metric coefficients, 55

56

S. CHANDRASEKHAR AND YAVUZ NUTKU

Vol. 158

in turn, will enable us to obtain the equations governing the fluid in the second post-Newtonian approximation. The coefficients of the metric tensor as determined in Paper I are

g°o = 1 + **L + - i (£/2 + 2$) ,

goo = 1 - -J- + j ; (U> - 2*) , 1 / go* = g0a = -j Pa, g«fi =° -{1

2U\ + ~j)

8afl ,

(3) f 2U\ g* = - [\ - — J Sa0 ,

and

where U denotes the Newtonian gravitational potential due to the prevailing distribution of p, and and P a are further potentials determined as solutions of the equations V^=_47rGp0, and

<*> = »»+ tf + j n + 2f*, P

(4)

v2p

" " 4v2f/» " * a £ ^ = " 16^"° + S •

(5)

Also, it should be noted that the solution for the metric coefficients given in equations (3) is obtained in the gauge dP, . ,dU

,{dU„

, dU\

„/l\

...

The expressions for the components of the four velocity that follow from equations (3) are «P = 1 + ^ {W + V) + ± (|»« + §»»£/• + §f72 + 2$ - vf,) , (Jt* + £/) + - i (|»« + **U + §t72 + 2$ - ^ P , ) ] ,

W = -cva[l+-^

(7)

uo = 1 + -i- (^ 2 - 17) + -Jj (ftf1 + | *2t7 + §i/2 - 2$) , and u«= -~va c

+ ^[Pac

&* + 3U)va] .

And from the definition Ta = p(c2 + n + p/p)uiUj - pgij, we deduce

(8)

Taf> - \g*T = §Pc25a? + p [».»„ + ( V + §n - J £) 5a3] ,

(9)

Toa - hgOaT = -PCVa + ^ p [iP a - (V + 2 [7 + U + 2) *.] ,

(10)

and

r M - hooT = | P c 2 + P (*2 - u + in + f£) (il) 4

2

1

2

s

+ 4r p T» + 2^ Z7 + U - 2$ + n(j< - U) + -p (w - 3*7)1J • cz L With the foregoing information provided by the first post-Newtonian approximation, we can proceed to obtain the equations appropriate to the next higher approximation.

No. 1, 1969

SECOND POST-NEWTONIAN EQUATIONS

57

III. THE SOLUTION OF THE FIELD EQUATIONS APPROPRIATE TO THE SECOND POST-NEWTONIAN APPROXIMATION

We now suppose that the metric coefficients in the second post-Newtonian approximation have the forms / . . 2U\ t , 1 _ g*0 = — yl -h —fj $aP + ~i <Jaf) , gOa = ^Pa+^Qoa,

(12)

and

l~™+ji{Ui-2*)+±Qa>,

goo =

where the Qi/s are certain coefficients to be determined. a) The Equation Determining Qap With the assumed form for the metric coefficients, the (a^-component of the Ricci tensor can be evaluated to ()(£**). We find RaP ~ ~, V USaP ~ ^

|_V QaP + ^ ^

+ 1 T_ _JL_ m* + 2$) + 2^™

-*(%

" ^

[ ^

+ ~^~) j

- i l ^ +^

(13)

+ *"•)]•

And since (cf. eqs. [4] and [9])

~(Tap-h*pT)=^V>U8aP (14) 2

+ - i [2V $5„? - 8TG P ( ^

- V^SaP ~ 2 £ «„,)]

the (a,/3)-component of Einstein's field equation gives

v

^~^U^ _ 5 ^7~^A^r~*~^r/ ~Saff'

where 5„^ = - 2 (v*8ap + -Q^fc) d (dPa dt \ dx$

dP„\ dXa/

(15)

(U2 + 2$) + 16TTGP («.»„ - v^p - 2 £ S^) 2K

d*U ° dt2

idU

dU dxa dxp '

There is an integrability condition for the solvability of equation (15). Thus, by contracting equation (15), we obtain

w°> - Sfe = **-; while by differentiating it with respect to x&, we obtain _d_ dx, V

dXpdXy/

dxp

^

770 58

S. CHANDRASEKHAR AND YAVUZ NUTKU

Vol. 158

Accordingly, we must have —• (Sap ~ i8*S„)

= 0.

(19)

Now it can be verified that Sn/s defined in equation (16) does indeed satisfy the condition (19). Thus, by contracting equation (16) we find

S„ = -8v 2 (t/ 2 + 2$) - 167rG(2Pi>2 + 6p) + 4 ( J ^ J (20) = 32TTGP(T? + iU

+ n) -

12 ( ~ y

,

where we have made use of equation (4) defining $. A direct evaluation now leads to the result

•k {S« ~ ^M = 1M> [ft <**•> + -k

M

~ "I

+

^] *

(21)

Since we are here concerned with the terms of the highest order that we are presently retaining, we are justified in evaluating these terms in the Newtonian limit; and in this limit the right-hand side of equation (21) does indeed vanish as required. With the integrability condition (19) satisfied, the general solution of equation (15), as we shall now show, involves an arbitrary vector function. Thus, letting dx$

dxa

we can rewrite equation (15) in the form

^••-^!f+S;

™

V2Qap = S,

and the contraction of this equation yields V2<2„ = S„ + 2 ^

.

(24)

With the aid of equations (23) and (24) we readily verify that, by virtue of the integrability condition being satisfied, jZ

v2

(Q* ~ **«/#«) =

v2w

( 25 )

«;

and we recover the defining equation (22). Therefore, we may consider wa in equation (23) an arbitrary vector function in space-time. b) The Equation Determining Qoa With the assumed form for the metric coefficients, the (0,a)-component of the Ricci tensor can be evaluated to 0(e~6). In evaluating it, we shall let (cf. eq. [6])

cified function in spacewhere w is, for the present, an unspecified space-time. We find -^°° ~

—

2 ~J V2Ua

+ L(iUV2TJ c6 \

1 /_,_ . .

dw .... dU

+ P

"

' " dx„dxa

dwa -. 9V

dP

" dx« dxa

5dU6U). dt dxj '

•

771 No. 1,1969

SECOND POST-NEWTONIAN EQUATIONS

59

With the aid of equation (10), we now find that the (0,a)-component of the field equation gives V2Q0« = -

16TTGP (V1

+ 4*7 + n + 2)va + 8irGpPa

V

+ 2p 11

PJ

d2U

dXpdXa

(28)

2 ^ — - 10 — — + — + —". dxa dx„

dt dxa

3za

dt

It remains to verify that equation (26) is consistent with equations (24) and (28). By straightforward reductions we find

*K£+^)+K£-» i Sr)]->-

(29)

where ° =

P

+ ^P(v*

+ 2U + n + £y

(30)

In view of the equation of continuity in the first post-Newtonian approximation (Paper I, eq. [64]), the right-hand side of equation (29) reduces to V2w/c2; and we recover the defining equation (26). Therefore, we may consider w in equation (28) an arbitrary function in space-lime. c) The Equation Determining Qoo In evaluating the (0,0)-component of the Ricci tensor, we find that we need the contravariant component g°p of the metric tensor. We readily verify that g* = - ( l - ^ ) *„ - ji (Q„ + W84,) .

(31)

With the assumed form of the metric coefficients, Ra
-

_ _ L (V20

j i 8TGP (V* -

-2w

^

U + JII + f

£) d U

- 2— - 4^

*

}

+ 1 [(*„ + W%v) ££• + WW* - 2*) + g ^ dx,, dx„ \ dt) dx, \dx„ dx„)\ ' where V2WV = a>„ (see eq. [40] below) and V22,». = S„r. The (0,0)-component of the field equation now gives mn V2Qoo = o2w„

d U

w .h 12 d—+1

+

16TTGP [»2 (i>2 + iU

+

2

dW

4 -"T - ^

A

d

'

U

+ n + ^

-

(t/ 2 +

2$)1

Z , - ^ - - 12£/f—Y+12 — — + 2 ^ ^ "' dx.dx, \dxj dxu dx. dx„ dt _ 6 f d i A 2 _ <>P± (dP? _ dPA \ dt / dx, \ Sat, #:<:„/

^

S. CHANDRASEKHAR AND YAVUZ NUTKU

60

Vol. 158

d) The Choice of the Gauge wa = w = 0 We have seen that the solutions for the coefficients Qa involve four arbitrary functions: w and the vector function w a . This is consistent with the requirements of covariance that we have the freedom to effect arbitrary coordinate transformations that do not affect the choice already made in the first post-Newtonian approximation. I t is convenient to work in a particular gauge; and unless we explicitly state otherwise, the gauge we shall adopt will be based on the choice wa = w = 0 .

(34)

In this gauge

» ;>

A and

dxg

(35)

- °

dxa

dx, and we may further identify £,,„ with Q„, in equation (33). (Strictly speaking, we should write on the right-hand sides of eqs. [35] and [36] 0(c~~2), and Oic-4), respectively.) In the chosen gauge, the equation governing Q„„ is (cf. eqs. [20] and [24])

V 2 C = 327rGp(J<2 + iU + n) - 12 (—J

.

(37)

An alternative form of this equation which we shall find useful is V 2 «?„„ +

6U*)

= 16TGP(2V2

+ m

+ SU).

(38)

If the occasion arises when we wish to determine the effect of the chosen gauge on the results which are obtained, we shall distinguish the quantities derived on the gauge (34) by asterisks, e.g., Q*,,- instead of simply Qy. Then, the solutions in a general gauge (the unasterisked quantities) are related to the solutions in the chosen gauge by the transformations

n - n* 4- dW° 4- dWl3

a

- cf

4- ?

dW

' (39)

Qoa = Q\a + | ^ + ^

,

and

QM = Q\ + 2 ^

+

2W0,

where

V W . = wa,

VW = w ,

and

V ¥ , = w,—

+ 2d-^-^~--

(40)

IV. CHRISTOFFEL SYMBOLS AND RELATED QUANTITIES IN THE SECOND POST-NEWTONIAN APPROXIMATION

With the metric coefficients given in equations (12) we readily find that

g= ~ [ l + ^ + - ~ j ( 2 £ / 2 - 4 $ - £ , „ ) ] ,

(41)

- 1 (£/2 + 2$ + JQ„) ,

(42)

V-g = 1 + ^

1 We are grateful to Drs. F. B. Estabrook, B. K. Harrison, and H. D. Wahlquist for a full clarification of the relationship of the transformations (39) to the general covariance requirements of general relativity. It should, however, be emphasized that in this paper we mean by a "gauge transformation" the transformation (39) which enables us to pass from solutions for the metric coefficients with a particular choice of the functions w and wa to a different choice of the same functions.

No. 1, 1969

SECOND POST-NEWTONIAN EQUATIONS

61

and

log v-g = ^ - 7 (3U2 + 2 $ + *e«) •

(43)

From equation (43) it follows that y

_dlogV-g dxa

,a

*

_ 2 dU c2 dxa

1 df c4 dxa '

and

(44) _ y ° ~

rt.

_ _ a i o g V - g _ 2 dU ~ dx0 c3 dt

i0

where

I df c6 dt '

/ = 3£/2 + 2$ + J C -

(45)

We shall further let (see eqs. [47] below) F = |U 3 - 4*7$ + §<2oo •

(46)

Now evaluating the various Christoffel symbols with the known metric coefficients in the second post-Newtonian approximation (and in the chosen gauge) and including only those terms that we are entitled to retain, we find 1

°°

1 0a

_

C* dt

~

c2 dxa

+

l

J> V" dX„

c* dxa

+

dt )

<* Ldx. {*

+

^ '

2r

" dx,

3/

i ^ 1 /6P.

~<# -

i

n

at/

+0^1•

dPe

~ 2 oTa I 3T7 _ dx,. ITT ~ c3 Vdxp

26

dU\

«0 ~^"^ dt.

r

" dt J '

62

S. CHANDRASEKHAR AND YAVUZ NUTKU

Vol. 158

and r

1 /dU .

, dU .

W

\

t

'^ = 7 Ur y *«* + a^ *•* ~ a^ 5" V -* l2U \dxy ** + a*,5"* ~ a^ 8"V + * fe

+

"a^" ~ 1£VJ •

Also, in the chosen gauge

c6 * dz„ .

w

c2 a» a

c4 a x a

v

It should be noted that for the purposes of deriving the various equations in the second post-Newtonian approximation, we do not need the terms of 0(c~7) in r°oo and the terms of 0(c~6) in r°oa; i.e., for these particular Christoffel symbols the expressions derived for them with the knowledge provided by the first post-Newtonian approximation are sufficient. (However, we shall need the additional terms in § X below.) V. THE EQUATIONS GOVERNING THE FLUID IN THE SECOND POST-NEWTONIAN APPROXIMATION

First we observe that with our present knowledge of the metric coefficients, the expressions for the contravariant components of the energy-momentum tensor are T00 = pc2 + p(i>2 + 2 * 7 + 11) l + i-

P

r / *vi [D 4 + 6v2U + 2U2 + 4$ - 2vllPll + 2UIL + v* IU + ^ ) J ,

(49)

T°a = f>cva + — pvjv* + W + n + £) C

V

pJ

(50)

+ ^ p \Va ( / + 6 v 2 U + 2 U * + 4 $ - 2v»p» + ^ +2f/) ( n + - ) ] - -p>\ > and T*e = pvj>f + PS*? + ^ [p».», (v2 + 2U + n + £ ) - 2pU5ap] + ^ jp».5(, [»« + 6zW + 2Z72 + 4$ - 2 ^ + (v* + 2£/) ( n + ^ ) ]

(51)

+ M* #**./>+ &<»)[ . The relevant equations governing the motions in the fluid in the second post-Newtonian approximation can be obtained by simply writing out the equation r«;y = 0

(52)

with the aid of equations (47) and (49)-(51). We need not, however, write down the 0-component of equation (52) since this equation is no more than the expression of the constancy of the volume integral of pu°V — g. Evaluating this quantity with the aid of equations (7) and (42), we find

puW-g = P [l + 7 W + 3C/) + 7 (f*4 + VU + W ~ *&- - VP,)] • (53)

775

No. 1, 1969

SECOND POST-NEWTONIAN EQUATIONS

63

The reduction of the a-component of equation (52) is long but essentially straightforward. We obtain (j{ + div * ) [pB. + -^ p [v. (v* + 6U + n + ^ ) - Pa]

+

7 ip"a (V + 1(W + i3t/2 + 2 $ - 2 VP, - K?«. + (^2 + 6 c / ) ( n + - ) ] - P (&o + »/&*) - PP*(f + 4J7 + n + £ ) | ]

-

P[

W - 3[/2 + 2$ + 2J7 (n + 2) - 2vP„ - i<2,J | ^

+ '(' + " + • + { ) ( * £ - < - ' £ ) It is worth noticing here that equation (54) is gauge invariant, i.e., it is valid as it is written independently of the choice of the functions w and wa in the solutions for Q^. VI. 9 0 0 AND THE CONSERVED ENERGY IN THE FIRST POST-NEWTONIAN APPROXIMATION

It has been shown in Paper I I in the context of the first post-Newtonian approximation that an algorism for determining the conserved quantities is to evaluate the LandauLifshitz complex 6 « = -g(T*i+t*f), (55) where the symmetric "pseudo tensor" /*' is defined in Paper II, equations (32) and (33). It will appear that in approximations higher than the first post-Newtonian the LandauLifshitz complex provides the sole means of determining the conserved quantities. In this section we shall consider the implications of the conservation law feMdx = S(-g)(T»

+ f°)dx .

(56)

Evaluating tao with the aid of the Christoffel symbols listed in equations (47), we find

+

2? l~d^ \te~e

+

~tej ~ 6 \~dt) J \ •

64

S. CHANDRASEKHAR AND YAVUZ NUTKU

Vol. 158

Now making use of equations (41), (49), and (57), we find

e<x> = - g ( r » + JP[V'

;oo) = pC 2 +

+

p(l)2 + 6U + n)

_ _L_ 0 0

+ 6i?U + 4£/2 - Q„ - 2^PM + 2f/n + v* (u + £)]

^ AirGc* Uxa dxa {IU

^ ^"")

+

8

Ldxf, dxp + dxa dx0

(58)

° V at) J J '

We have already noted in Paper II (eq. [49]) that

Also,

SG^--fe^ = ^

(m ddiv)

°

(60)

-

Accordingly, we may write eoo

=

pc* +

p

(B2 +

SJJ + n )

+ )j p [»* + l(to2£/ + 12£/2 + 6C/II + J;2 (ll + ^) - 2 ^ p J 327rGc2 L 8x3 dxp

dxp dxa

(61)

\ dt J J

Again, we have, modulo divergence, = — PaV2Pa = 16irGpVaPa — Pa

axa a/ = 16rGpvaPa — 3 ( - r r ) (mod div) and 8P. 3P, _ BK 8P, = dxp dxa dxa dx$

9

(3U_y

(mQd d.v)

_

(63)

Hence we may write

e°° = Pc* + P(v2 + \u + n) (64) + -~2p[vi+

10^2f/ + 12t/2 + 6UU + v* (n + £\ -

f^pj

No. 1, 1969

SECOND POST-NEWTONIAN EQUATIONS

65

From equations (53) and (64) we obtain

® = e°° - c>pu°V-g = p(iv* -hu + n) + ^ P[¥* + ^

u

U2

+ *r

+

6Un

v

(65)

n

v p

+ * ( + " ) + ten* ~ ? " "]

In this expression for @ we may replace the term \pQm by (cf. eq. [38]) §P&M

= - ! J ! (Q«. + 6C/2) - 3PC/2 S

V2(C

" 8^

"" +

6C/2)

~

3plP

= -p(13C/ 2 + 4 J W + 4Z/H)

(66)

(mod div) .

Hence, we may define the energy in the first post-Newtonian approximation by

<s = p ( ^ - § c / + n)

+ ^ p [V + ¥u - W +2Un + * (n + ~) - KJ'J ; and the integral of this quantity over the three-dimensional space gives the conserved total energy, I t is of interest to verify that the expression (67) for (§ is, modulo divergence, the same as the one that was obtained in Paper I from a direct inspection of the equation of motion and assuming the condition for isentropic flow. The expression for (g given in Paper I, equation (166), is

e = (<7-ip> 2 + p*n-i P *c/* 4

+ -^ P(-I*< + ^u

(68) 2

+ iu* - H P - i* n - A
where a has the same meaning as in equation (30) and where

P* = p + i p ( i " 2 + 3£/) and

(69) V

= U + -%0 , c

where

V2© = -4irGp(iu 2 + 3CO .

Accordingly,

p*zr = Pu + ^ Pu(iv* + 3co + -^ PO (70) s

pu

+ ^pU(^

+ 3U)

(mod d i v ) .

Replacing a, p*, and p*U* in equation (68) by their equivalents, we recover the expression (67) for (§. Thus, in agreement with what was stated in Paper II, the conserved energy in the first post-Newtonian approximation is all that can be derived at this stage.

66

S. CHANDRASEKHAR AND YAVUZ NUTKU

Vol. 158

VII. THE CONSERVED LINEAR AND ANGULAR MOMENTA IN T H E SECOND POST-NEWTONIAN APPROXIMATION

Evaluating the (O,a)-component of the pseudo tensor in accordance with equations (32) and (33) of Paper II, we find i \l2dUdU i 16irGc t dt dxa

^

+

c2 L dt dxa ^

+

dU dp

dP

( f> dxp\dxa

i

M

L

">

a

3x,s dxa 3x 7 \ dxa

A dxpj +

dx* dt

\dxpj

44(7

dt dxa

dxp \ dxa

dxp)

dxy / J )

With the aid of equations (41), (50), and (71), we now obtain A 00a = _ A g ^ a -L. JO.) = po. + -^ Pva (v2 + 6U + n + ^ ) + -^ P jua [n4 + 10»2£/ + 12Z72 + (v2 + 6U) (u + £\

1

ri23^3£/

+

16TTGC2 L

+

16^Gc4 L 3* 3x a ^

6/

ldU/dPi,

3xa

3x0 \ 3xa 2V

";

(72)

gP.N-j Sx^/ J

3x a d<

+

3/ 3* a

+H4(f/2+2*+^+S4(c/2+2$ - ^ + 4 P 3x^ ^ - 6 P a\3x0/ r ^ Y + 4 3x0 ^ ( \^ 3x f i - 3x^ ^ A/ f l ^3x a

a

3P0 /3Q0Y _ 3Q„0\1 dxy \ 3x„ 3xY / J After a lengthy reduction in which the equations governing P„ and Q^ as well as the particular gauge conditions (35) and (36) are used, we find (cf. Paper II, eqs. [58] and [59]) -&* c

= *. + %*, 3x0

(73)

No. 1, 1969

67

SECOND POST-NEWTONIAN EQUATIONS

where Ta = Pv* + ^

P

[v* (v* + 6U + H + £ ) - Pa]

+ ^ \pv* [v* + lOzW + 13C/2 + 2$ + (i>2 + 6U) (ll + ^j-

1%P, - 40„,] (74)

- P«2a0 + ^ ) - pPa (f + 4tf+ n + £) I

+

rGc4 \

' dxfi dx* ~r

2V

"

dtdxj

and U

" 4^

4

l^ W

+

W

+

at/ 2

a3x

,,

**" 3*

"a/3*.**,

- 4 ( c g + e» g ) - £ (i-,e„ - ii>.e..>] • Since Q0a/c and TT« are equal modulo divergence and 0„/3 is moreover equal, modulo divergence, to a symmetric (pseudo) tensor, it is clear that they will yield the same total linear, as well as angular, momentum when integrated over the whole of the threedimensional space. We notice that the expression (74) for T„ contains two terms which are not confined to the volume occupied by the fluid. In § VIII we show how in a suitable gauge these terms can be transformed, modulo divergence, into quantities which are nonvanishing only in the volume occupied by the fluid. Also, it may be noted here that the quantity in equation (54) on which (d/dt + div v) operates is precisely 7r„ except for the two terms which do not vanish outside the volume occupied by the fluid. Vin. THE EFFECT OF THE CHOICE OF GAUGE ON THE CONSERVED QUANTITIES

Since we have evaluated the conserved quantities in §§ VI and VII in a particular gauge, it is of interest to see how they are affected by this choice. We have already noticed in § I I K how the solutions for Qag, Q„a, and Qoa in a general gauge are related to these same coefficients in the particular gauge w = wa = 0. If in accordance with equations (39) and (40) we now distinguish the quantities derived in the chosen gauge by asterisks (hitherto not so distinguished), then by equations (41)-(44) .

*

y =

°

2 dWM dx„ '

l aw, y ~ JdldZ '

, v

, and

. 6

v

. 6

1 dW„ c4 3z„

i aw„ * = * - saxjxZ 6

(76)

.

Also, while r°oo, r°oa, and T'oo are to the required orders—0{
68

S. CHANDRASEKHAR AND YAVUZ NUTKU

ra^r

a,

+_ ^ - ,

r ^ - r „--

Vol. 158

— ,

and

(77)

Now re-evaluating <°°, we find that in addition to the terms given in equation (57) (in which-ft,, is now replaced by Q*m), we have 1 (/ i^L.dU I. °„dUdU d*W, PWtS w 1 dx* ° dx* dxadXu/ \6vGc \ (78)

In 0 00 , in addition to the term (78) derived from i°° we have —IpdWp/cPdx,, derived from the factor — g; thus, altogether, the additional term in 0 00 is c*

H

dx,c

The additional term in cVw°V — g, derived from the factor V — g, is

~^p-axf-

W

Hence G00 —
dW*\ dxp )

dPp d /dW„ dx„ dxp \ dx*

dW*\ dx? )

. .

eu-i

It can be readily verified that with these additional terms Q**^ and Q*o«, which now occur in equation (71), can be replaced by Q*p and ( V in accordance with the transformation (39) and that the only terms not so absorbed are 1

(

dP*

dU\

....

The further reduction of 9°'/c> modulo divergence, using the general gauge conditions (22) and (26) leaves in the definition (74) for T* (in which the Q,/s are now defined in the general gauge) the two additional terms

No. 1, 1969

69

SECOND POST-NEWTONIAN EQUATIONS 1 _ \l*Uw 16irGc4 L a*a

dPjL_d dz,, dxB

"I < J

(g3)

In the expression (83) we have retained the divergence of —wpPa since this is not symmetric in a and /3. With the choice

• --P,f£

and ^ - - * ^ ,

(84)

the terms in ira, not confined to the volume occupied by the fluid, become 1 16TTGC4

r , _ 8U6U . lf. L dZfl dz„

96irGc4 L dxt, dxa 96TTGC4 LdXff \y" =

d*U . . dU , S/Sla d*a v

°° dtdxa

dxj

~ 9cVG7 ~dx~t, lQ" \~dxj

dPff 3xa

dxf, \ dxp

d . 0*0

I J

a

Jl

dXf, \ dXg V J +

i t j ~ Jx~f, ^'Pa)J

'

where in passing from the second to the third line we have made use of equation (6). Thus with the choice of the gauge (84), we can reduce T* modulo divergence to the volume occupied by thefluid.And the conserved angular momentum can also be defined in terms of the same reduced linear momentum since the final expression in (84') is, modulo divergence, symmetric in o and /3. IX. THE EQUATION 6"',,- = 0

To write the equation governing 9", namely,

h it K-«>

(85>

in the different approximations, we need f* to different orders: to 0{c~2) in the first postNewtonian approximation and to 0{c~*) in the second post-Newtonian approximation. Certain peculiarities that are noteworthy appear in the evaluation P" to these required orders; we shall clarify their nature in the framework of the first post-Newtonian approximation. a) The Equation 0ai,< = 0 in the First Post-Newtonian Approximation As we have noted, in the context of the first post-Newtonian approximation, V* is needed to 0(c~2). Since t*' is quadratic in the Christoffel symbols and appears with a factor c*, it is clear that to evaluate t* consistently to 0(c~2), we should want to know the necessary Christoffel symbols to <Xc~4). While some of the Christoffel symbols (e.g., r°oo and TV) are known to the required orders, some others (e.g., rapy) are not. However, since the Christoffel symbols listed in equations (47) (with our present knowledge of the second post-Newtonian approximation) provide the needed information, we can evaluate P* consistently to 0(e~2). And when the evaluation is

70

S. CHANDRASEKHAR AND YAVUZ NUTKU

Vol. 158

4

carried out, a remarkable thing happens: all the terms ofOicr ) in t°& which derive from an explicit knowledge of the second post-Newtonian approximation cancel separately. In other words, we should formally have obtained the correct final result had we simply ignored (blindly!) all the terms in t# of 0(c -2 ) which depend on terms in the Christoffel symbols known only in the" second post-Newtonian approximation. By a careful inspection of why this cancellation occurs in the first post-Newtonian approximation, one can convince oneself that the same cancellation must also occur in the higher approximations; e.g., the terms of Oicr*) in f# which depend on the knowledge of the Christoffel symbols in the third post-Newtonian approximation will cancel. Returning to i°P, we have to 0(c~2) 1 16TG

/,

4U\r.dUdU dx* dxg

16irGc2 L \dxa dxp

afi

\dXp) J

dXff dxaJ

/dUdPf,

dxu"•"ndxu

"f

dUdPa

dUdP,} -8"B^-W)

+ 4KdxZ^T + d^-df dP* /dPj, _ dPts\ dx„ \ dxp dxj (dP„ dP„ x "*" 2 "Vd*, dx,

/du\*

-6S" Vat)

(86)

dPj. /dPf, _ dPA dxa \ dXf. dx$J

dP„ dPr\-\ dxr dxj] '

and the corresponding expression for 9°^ (see eq. [90] below) can be written down with the aid of the known expressions for T°P and g. The further reduction of the equation 1 d9 0a

i

3Qae

^ - + ^- = °.

(87)

c dt dxp with 6°° as given in equation (57) of Paper II, yields the equation

(l+^0l^+7'K'+6l7+n+?)-p-]}-'S (88)

We observe that this equation is identical with the equation T"-,i = 0 in the first postNewtonian approximation as reduced in Paper I (eq. [68]). b) The Equation Qa\i = 0 in the Second Post-Newtonian Approximation In the second post-Newtonian approximation, <"" is needed to Oic*). To evaluate t# to this order, one should strictly want to know the Christoffel symbols to 0(G~~*). The symbols listed in equations (47) do not provide this information for all of them. But as we have explained in § a above, the terms in l°0 of 0(e~*) which depend on terms in the Christoffel symbols known only in the third post-Newtonian approximation will cancel. Therefore, ignoring these contributions to P8, we find that the terms of 0(c~4) in 16TTGI°* are

No. 1, 1969 8C/2 +

SECOND POST-NEWTONIAN EQUATIONS

71

32$)^^-8|i|i-40^(^|i + ^ f ) dxa axa oxa axa \dxa axa oxa axj

_ *H_L rp.rs _ ?E_L fp.2) dxa dxB ^ X ; ax„ ax0 ^ X ;

_2~—(PPa) ^ axM dx, V*"

- ( £ + © > + » + «-> + * £ ( < + < ) \5xa 3 ^ ~1~

3x„ \

a

dx„ ax„/

\3x M 3x,s

3x„ 3x a /

d x j " \ 3 x a 3xM " r 3x0 3x„/

5x^

- f ^ ' + ^ - t ^ + iM-f^

aQ _ 20 » _ *&• + d<M + ^ f * & ° _ £>2e° + a < M ya/5 f ~ Y + t ^ f V3x„y ~~ axa V ax„ ax^ "*" a* / "^ ax^ V 3x„ 3xa ^ dt J

_dPJidQ^_dPa

(dQao _ dQ„0\ __ afV /aQ a 0 _ 3Q M 0 \

a/ ax„ 3x„ \ 3x„ dxa) ax„ \ 3x„ _ ! aQJT dQw , aQ„„ aQ^ t2BQ^ SQ^ * ax0 ax^ dx, dx, ax„ ax^ ~t~

,dU/dQao 3 x A a/

1 dQm\ 2

(dQ^dQ,,

dQ0lidQ,,\

\ ax, 3x3

,dUfdQa0 dxB\ dt

dxa)

1

(89)

ax„ /

agooN 2 dxJ

ax, ax0 /

«*[-» (ID"+<0+<
dx„ dx,,

" dx„ dx,

dx„ \ dx, dxj

+ f £ <8f" + 2"> + 2 1 ^ «* + 2*> " 2 « - f f +

au;/aQoo_

aQ»0\

a i , l 3x„

ac/ag,,

dt J ^

dt

t/a(?,.y

dt ^

i

\dxltJ

_ dP, /dQuo _ ag^ + a&A _ i a&„ a&, dx„ \ dx,

dx,,

dt J

* dx, dx,

t aQ„

2

dQ„i

dx, dx,, J

72

S. CHANDRASEKHAR AND YAVUZ NUTKU With the aid of equations (41), (51), and (89) we now find

e*f> = p».», + pKf> + ^ [pvw (v* + 6U + n + 2 ) + 2pUdae] + -^ \Pvav» [v* + \WU + 12C/2 + (s* + 6tf)(n + - ) - 2»MP, - Q „ ] + # [ & * - (2C/2 + 4$ + C ) M | +

+

16TG

IL4 dxa 6x0

^

\6xJ

c2 I \dz« da^ + 6x0 dxj \dxa

dxg

L

+ i j(_48C/2 c4 (

+

Vdxa dt

dx„ dx„J

dx„ dx„

dxp dt )

KdXp dx$

\ dt J

dx„ dx*J

dx„ dt

dx„ \dx„

dx,/\)

+ 16$ _4^)^^-8|i|i-8^r^^

dxa dX0

a*.a*/

u

dx,,dxaK

'

zap* , 9PA a ,

5*,, \

dxp

dxa 0X0

dx„ dx„K .

6xJ

6x0 dxa/

a w

at/ /

* \6xa 6x„

+ ^|^

\dxa 6x0

dt/ az7\

6x0 dx^/

+<£«+*£)+(£+&)£< D, +»+«-> zap, _ aPA /a&o _ dQjA , /ap; _ d£i\ /a&o _ aQM0\ \aa; 0

dx„J \ dx,,

6x0 J

\6x0

6xJ \dx„

3P, agft, , 6P, 6Qa> _ 6P, dQrf _ x 6Q^ dQ^ 4 6xa dt 6x0 dt dt dx„ dxa 6x0 dxa \ dt

6x0 /

, dQaj, 60^ _ /dQ.» 3 & , dx, dx, \ dxv dX0

dx0 \ dt dQfr dQg,\ dx, 6xa)

dxa /

2

a*a/ tdQrdQv 6xa 6x0

+ 5tf

[(2.0. - 8* + *>..) (H) + S ( £ ) '

16, | | | |

+

(90,

_ 2 ^ A m. + 2 « - 4P dJLsJL + SJL W _ 2 p aft£^ dt dt " dx„ 3/ Sx,, dx„ ,2dQI£_d_(„)dQI1aP!.dQszd£_ ~r dx„ dx„ y ^ ' """ dx^ dt "*" dt dt 2

317 /3QOQ _

dXp \ dx,, i fdQ„\*

V 3*„ /

0a

2

dftjoN _ dP, /flQ„ _ dQ*

dt )

dXy. \ dx,

" dx„ dx, dUdU Vl " dx» dx, dQ^\

dx^

dt )

_ i4 a f t , agg, ,1_ i2 dQ^ agjj-i> TI

ax, ax, ~

dx, dx„ J > JJ *

3

With 8 and G" given by equations (72) and (90), respectively, we find that equation (87), after considerable simplifications, reduces to the same equation (54). Thus in this second approximation also the equations satisfied by 0*' and T*> are the same (see Notes added in proof). X. THE CONSERVED ENERGY IN THE SECOND POST-NEWTONIAN APPROXIMATION

To obtain the conserved energy in the second post-Newtonian approximation, we must evaluate 8 00 and tfpu0 V — g to 0(0**) (i.e., in the third post-Newtonian approximation) in order that their difference may be known to 0(e~4). As we shall see presently, the evaluation of 6 00 and c^pu0 V — g to 0(c~4) requires a knowledge of gap to 0(e~6). Letting, then,

^ = - ( i +^ ) « ^ +^ g ^ +^G^».

(9i)

we find

- * =

1 +

f

+ ±(.2U>-4*-QJ

(92)

+ ^ (-4E/* - 24C/$ - 2VQ„ - &„<» + P* + Qm) ,

V-s = i + ^r - 7 (^ + 2 $ + i G J (93)

+ 1 ( - 8C/$ + ify + i&o - i&,„<3)), and M

° = 1 + 4 (i^2 + tO + 4 (f«4 + \*U + §t/2 + 2$ - vP,) + 7 [^ 6 + ¥"4!7 + ^ 2 f / 2 _ i*7* + ^ + 2 C / ) $

(94)

With the aid of these equations we find

p w V - g = P + ^ Pi¥2 + 3V) + ± p&* + pu+*lP+ - 1 p j ^ S + 33.„4tf + ^ 2 £ / 2 _ 1U3

+

v.P,- IQJ 2 z ) 2$

17)^1. ++ i*V W -- K V + v)Q„ - (|o»+ (|o» + S5t/)vP„

(95)

74

S. CHANDRASEKHAR AND YAVUZ NUTKU

Vol. 158

Next, we note that to O^c^) J00 = Pc* + p(i>2 + 2U + K) + -^p[vi

+

6viU

2U

+

* + 4$ + 2UU + v* (u + £ ) - 2D^PJ

+ -^ P[» 6 + 10»4Z7 + 16JW 2 + 80 2 + 2CO* + 2(C/2 + 2$)n

(96)

+ (z>4 + 6iW - 2^PM) ( n + - ) - 4(D2 + 2£f) V M — 2vllQll0 — v ^ p — Qooj . And the terms of Ofa-*) in /00 are found to be

~

2jP

" a ^ ax7 ~ 16U d£~d£~ 6Ql" La^Jx~, + dPv aft., j / a p , ap,\ afl,0 t aQ„ aQ w Sx,, 5<

\dx,

i

dxj dx,

2hv

\fa) J (97)

dx, dx,

\j*n

~H

i

2 dx, dx« dx, dxugiven indxequation u dxu )(57), we find that the terms Combined with the terms of lower orders 4 00 of 0(
4" P\V* + 14iW + 42zW2 + 8U3 + 4w2$ + 12J/2n C

L

+ (»4 + 1WU - 2^PM) (n + £\ - 4(v* + 4t/)vP„ + P„2 - (»2 + 4C/ + n)G„ - v»G« - 2»„&o] + _ J _ r _ 4 P ^ ^ - ^ ^ - 2 P ^ ^ - 12tf(^Y ^ 167TGC4 L " ax„ a/ dx„ dx„ " ax„ ax„ V a// dP,/dP, dP,\ (WV_ d£dR + 2U + 2Q 60 dx, \dx,

dx^J

• aft.o (dP* • aPA ax, \dx,

_ ! ag„ aft„ * ax, dx,

dxj

\dXpJ

aP> aft, , ! aft, aft,, 4

dx„ dt

t aQ„ 2

dXp dx,

ag„ ax, ax„

2

dx, dx,

at/ /aft,„ dt \ ax„

t 2

aQ.Ai a^ / J

(98) *• ;

No. 1, 1969

SECOND POST-NEWTONIAN EQUATIONS

75

These terms can be reduced, modulo divergence, to A P [» 6 + 14iW + 42iW 2 + SU3 + 4»2$ + 12U2U + \P* + (n4 + I O I W - 2ifePM) ( n + ^ ) - 4(D2 + 4f/)^P M

- J(»2 + iu + u)Q„ - iv&o - v&»] (99) lOTrGtr L

ax„ dt

2 y " \dxj

\et) xdU

+ 2

(dQ,a

d/ V a*„

dx„ dx,

+

oxv \ dx, y

"" a*M a*, +

dx^J 4

a*, ax,

idQ..X\

* a* / ] '

Now making use of the earlier result (67) and including the contribution of 0(c~4) that follows from equations (95) and (99), we find G00 - c*puW-g

= P ( ^ 2 - \U + n)

+ A p [§»« + **u - ft/2 + 2c/n + tf (n + £) - fvP„] + A jp[i|z> 6 + ™v*U + l|iz; 2 /7 2 + -x,f£/3 + 2o2$ + 12C/2n

+ (^4 + low - 2vP„) (n + £\ - (|»2 + n c o w + \p? (100) - i(2Z7 + TL)Q„ - i v ^ i - + i C < 3 ) ] I

1

I" 16irGc4 L

_ 127/ f?EY \dtj +

auseg* dt \ dx„

IT» iEi£-2p " ax,, a/ - sn &" +

(WV \dxj

^ \ dt J

^w + " Ox,, dx,

2U9P>(W

+

dx„ \ax M


- fsD dUdU I 4 idQ»dQ» °^' dx, dx, "•" dx, dx,

_ w *(*S* _ , ^ 1 1 . dt\ dx„

dt J A)

The integral of this last expression over the whole of the three-dimensional space gives the required conserved energy in the second post-Newtonian approximation. But we observe that it includes the third post-Newtonian term \pQ„m; and we must eliminate its appearance. For this last purpose we need the equation which governs Q„m; i.e., we must obtain the third post-Newtonian approximation to gap. a) The Equation Determining Qa0<3> With the metric coefficient g^ given in equation (91), we find that the terms of 0(c~6) in Rtf are

76

S. CHANDRASEKHAR AND YAVUZ NUTKU dxp J -t- ^ t / -I- a ^

+ ,

^ + U», dXiidXy ** i" ^ ^

2

Vol. 158 2

dxp \ dXp

dX)i

i - U«. ^

dx*

^

3a;a \ w " 3 ^ / da* V^"" d ^ 7 ~r 2 W o dxp ~r dx0 dxa J a2Q,g dQm\ , /dP t t 3Pff 3P„ a P A i^Grf 1 3 /3ft, 2 2 3/2 3/ \ 3a* 3* a / \ dXp dXn dxa dxpj P

a2^

dxadxp

l2 pM

3a;,, \3a*

d# a /

dt dXp

dt \dxf, ^ dxj ^ dt\ " dxp ~r * dxj ^ d2U3 /dU\2 d2U

(101)

32t/

a / a p . , aPA 3P„3t/

* dtdXp 2

dt \dxp

d2U dU 3 — 4[/ 4$ ° u _ _ £ _ (ip _|_ 2$)5as dxadxp dxadxp 3xM dx^ ' while the terms of 0(c~2) in T^ — \ga&T are (z* - ig^r)® = P».»/I (»* + 6t/ + n + 2 ) -

dxj

d2$

o v

p ^ , + ^p a ) (102)

+ 5„0P£/(n-£)-§pQa0. Making use of these results, we find that the equation governing Qap(3) can be brought to the form

vQ

s * dxz\rw~*^r)~^\~^r~^~^r)« where S«»

>

(103)

d2U* /dUV = 2S«U + f V W , + 4 _ - ,U ( - ) ** /3U\2 d2U d2* 32f/ 2 \ dt J dt dxadxp dxadxp d2P 4 3x * £ *dXp * « . + 2P"dXadxp " | dPadPp | dXp dx^ M

dP dP

" "

dxa dx$

+

dt Kdxf, + dxj AU dt\dx9 "*" axJ *^M a/a*/" 3 dP.dU d / dU du\ a /ap a aPA 2 + 2 + p - ^ ^ ^ ^ V ° a ^ ^a^;~ ''ax;W + ^ a^ax^

+

dx„dxr "

dUdQ„ . dU dQ„ + dxa dxp dxp dx*

"^ 3/ \ dt

dxp 2

2

ax„ a^ dxa V & a*J

( 0 )

dxadxp 2

ax3 \Qa" dxj

dxa J

+ 16TrGP[vaVf, (v + 6U + n + £) - vjp - vt,Pa +u(ll-£)

Sa0 - J Q ^ ]

No. 1, 1969

77

SECOND POST-NEWTONIAN EQUATIONS 3)

Equation (103) governing Q 0 / must be considered together with equation (15) governing Qap; and when they are considered together, their solvability simultaneously requires that the equation §S« _ i £ ^ 2 dxp dxa

+

1 (dS*em _ i dS°°(3)\ 2 c2 \ dxp dxa )

= 0

(105)

be satisfied as an integrability condition. It can be verified (though the verification is long and tedious) that this integrability condition is indeed satisfied by virtue of the equation of motion (88) in the first post-Newtonian approximation. In view of the integrability condition (105) being satisfied, we can extend the gauge condition (35), used in the second post-Newtonian approximation, to dQat dxp

i dQ„ dxa

1 MV» _ 2 \ dxp

dQal

K^-»^)-

+c

<'°«

in the third post-Newtonian approximation. With this choice of the gauge condition, the equation determining (?a/s<3) takes the form

VW 3 ) = *W3) •

(107)

Now contracting equations (104) and (107), we obtain

• 40 d*U I 4 dU dQ°° ^ VliV dx^dx, ""*" dx„ dx„

2

d

fdQ"° dt \ dx„

1 2

dQ "\ dt )

(108) 2

2

2

+ le-rGphv* + i6v u + nu + sun + 2v (n + £) 4vP„ - Q„] • 6) The Conserved Energy in the Second Post-Newtonian Approximation Returning to equation (100), we may replace (modulo divergence) the term %pQc<,w which occurs in it by hQa/3)

= hp(Q,*m - I2t/ 3 ) + 6Pu*

= - £ g ( Q ^ - i 2 ^ ) + 6pC/3

(iQ9)

= - J ^ . V2(<2„„(3> - 12U3) + 6PU3 (mod div) , where we may further substitute for V2(<2(r/3) ~~ 12£/3) the expression on the right-hand side of equation (108). With this replacement and after some further reductions, modulo divergence, we find that the conserved energy in the second post-Newtonian approximation is the integral of the quantity,

78

S. CHANDRASEKHAR AND YAVUZ NUTKU

Vol. 158

+ Jr P[V + ¥*u ~ W + 2UU + v* ( n + ~) ~ M j + 1 . p ^ B « + 47^17 + |t,2^2 - ^jT/3 + 2»2$ _ 5J/2TT

+ (v* + 6v*U - 2vfd(TL + |(7Z7 — 2n)Q„ +

1 r 16irSc4 L ,

,p 1

(HO)

iv»&»]

dUdU " dx„ dt

dQrdQ,.

+ - ) - (f*2 + 3C0-PA + \Pf

dPvdU " dx, dx, ^ . 1dU/dQ<¥

2

2

"*"* a*, ax, ^ dtKdx,^

BU_dV_ V>" dx„ dx, 1dQ„\]

et Jr

over the whole of the three-dimensional space. We observe that this expression for S, like the one for 7ra (eq. [74]), includes terms that do not vanish outside the volume occupied by the fluid; but these terms in @, unlike the analogous terms in x a , depend explicitly on the metric coefficients in this approximation. TABLE 1 INFORMATION ON THE METRIC COEFFICIENTS THAT IS NEEDED IN THE VARIOUS APPROXIMATIONS ORDERS* OF THE METRIC COEFFICIENTS NEEDED EQUATIONS OF MOTION

Newtonian First post-Newtonian Second post-Newtonian First radiative corrections

gag

goa

0 2(4) f 4(6) f 5

1 3 5 6

fto

2 4 6 7

* Defined by the inverse power of c in the relevant expansions. t The order in the parentheses is needed to determine the conserved energy if the condition for isentropic flow is not supplemented to the equations. X I . CONCLUDING REMARKS

The present paper completes the solution of Einstein's field equations to the extent that is necessary to solve any problem in the hydrodynamics of self-gravitating perfect fluids with proper allowance for the effects of general relativity to O^c^. Also, the various integrals which the equations admit have been isolated to the same order. The information on the metric coefficients which is needed to obtain the relevant equations of motion in the different approximations is summarized in Table 1. As is indicated in this table, it is expected that the effects of gravitational radiation on the behavior of the system will first manifest itself when the next half-step is successfully taken. But it appears that entirely new considerations will be needed before we can properly take this next half-step. We hope to be able to return to these considerations in the near future.

No. 1, 1969

SECOND POST-NEWTONIAN EQUATIONS

79

One of us (S. C.) is grateful to Drs. Kip S. Thome and James W. Bardeen for many fruitful discussions on topics related to this paper. The research reported in this paper has in part been supported by the Office of Naval Research under contract Nonr-2121(24) with the University of Chicago. Notes added in proof: 1. By an accidental omission of a term in the evaluation of la& (eq. [89]), the equation 9 i j j = 0 at first did not agree with equation (54). That this discrepancy might be due to an error in the evaluation of t^ was suggested to us by Drs. F. B. Estabrook, B. K. Harrison, and H. D. Wahlquist. We are immensely grateful to them for their interest and for their critical examination of a preprint version of this paper. 2. Professor J. Stachel has kindly informed us that the gauge transformation 1,84) was independently discovered by Dr. Elliott Krefetz while reading an earlier preprint version of this paper. We are also grateful to Professor Stachel for some clarifying remarks. 3. Since writing this paper it has been possible to extend the analysis along lines initiated by Drs. W. L. Burke and Kip Thorne (but much more in the spirit of an early paper by A. Trautman [Bulletin de I'Academie Polonaise des Sciences, 6, 627-633, 1958]) to obtain the equations of the 2|-post-Newtonian approximation in which the terms representing radiation reaction appear explicitly. The results of this investigation will be published separately by one of us (S. C ) . REFERENCES Chandrasekhar, S. 1965, Ap.J., 142, 1488 (Paper I). . 1969, ibid., 158, 45 (Paper I I ) .

T H E ASTROPHYSICAL JOURNAL, Vol. 160, April 1970

© 1970. The University of Chicago. All rights reserved. Printed in U.S.A.

THE 2^-POST-NEWTONIAN EQUATIONS OF HYDRODYNAMICS AND RADIATION REACTION IN GENERAL RELATIVITY S. CHANDRASEKHAR AND F. PAUL ESPOSITO University of Chicago Received 1969 September 17 ABSTRACT In this paper the equations of hydrodynamics in the 2§-post-Newtonian approximation to general relativity are derived. In this approximation all terms of 0(c~h) are retained consistently with Einstein's field equations; it is also the approximation in which terms representing the reaction of the fluid to the emission of gravitational radiation by the system first make their appearance. The paper is in four parts. In Part I (by S. C.) the lowest-order terms in the metric coefficients are derived which are consequences of the imposition of the Sommerfeld radiation-condition at infinity. It is shown (following an early investigation of Trautman) that these terms are of 0(c~s) in goo, of 0(c~*) in goa, and of 0(e - 6 ) in gap. Unique expressions are obtained for these terms. They are found to be purely of Newtonian origin. In Part II (by S. C. and F. P. E.) the equations of motion governing the fluid in the 2j-post-Newtonian approximation are derived. In addition to the coefficients already determined, these equations depend on a knowledge of the term of 0(c~7) in goo- This term is determined by an explicit appeal to the field equation. It is further shown that this approximation brings no change to the density (c2pu°ij — g) and the linear momentum (ira) that are conserved in the second post-Newtonian approximation. In Part III (by S. C.) it is shown that the terms of 0(e - 6 ) in the equations of motion contribute principally to the dissipation of the energy and the angular momentum that are conserved in the second post-Newtonian approximation. The rates of dissipation of energy and of angular momentum that are predicted are in exact agreement with the expectations of the linearized theory of gravitational radiation. Finally, in Part IV (by S. C. and F. P. E.) the energy, 6 00 — e!/)«0 V —g, to be associated with the 2J-post-Newtonian approximation is derived by evaluating the (0, 0)-component of the Landau-Lifshitz complex and the conserved density in the 3j-post-Newtonian approximation.

PART I THE LOWEST-ORDER TERMS IN THE METRIC COEFFICIENTS THAT DERIVE FROM THE OUTGOING-RADIATION CONDITION AT INFINITY S. CHANDRASEKHAR I. INTRODUCTION

In two earlier papers (Chandrasekhar 1965 and Chandrasekhar and Nutku 1969; these two papers will be referred to hereafter as Papers I and II, respectively) the equations of hydrodynamics governing a perfect fluid were derived in the first and the second postNewtonian approximations to the equations of general relativity. In these two approximations the equations of motion include all terms of orders c -2 and c -4 , respectively, that arise in a systematic solution of Einstein's field equations in an expansion1 in inverse powers of c. The orders to which the different metric coefficients have to be known in order to obtain the equations of motion, as well as the corresponding conserved quantities, are set out in Table 1 in § II below (see also Table 1 in Paper II). It is known that no term representing the reaction of the fluid to the emission of 1 Precisely, the expansion is in powers of v/c, V U/c, V (p/p)/c, and V U/c, all of which are considered to be of the first order of smallness and comparable.

153

154

S. CHANDRASEKHAR AND F. PAUL ESPOSITO

Vol. 160

gravitational radiant energy appears in the first or the second post-Newtonian approximation; also, that on the standard linearized theory of gravitational radiation, the reaction of the fluid to the emission of gravitational radiation by the system must first manifest itself in the next "^-approximation" by the explicit appearance in the equations of motion of certain "damping terms" of 0(c~6). The question, how the succession of terms of higher even powers of 1/c, that follow one another on the standard postNewtonian schemes, gets broken at precisely the order c~6, has concerned many authors including Infeld (1938; see also Infeld and Plebanski 1960), Hu (1947), Trautman (1958a, b), and Peres (I960), all in the framework of the original theory of Einstein, Infeld, and Hoffmann of n mass points. But the results of these authors were inconsistent with one another, inconclusive, or incomplete. More recently, in the framework of the hydrodynamics of a perfect fluid, Thorne (19696) has, in the context of his exact treatment of the nonradial oscillations of highly relativistic neutron stars (Thorne 1969a), shown (developing certain ideas of Burke; see below) that in the limit of weak fields his exact results on the rate of emission of gravitational radiation are in accord with the expectations of the linearized theory. More generally, Thorne has concluded that the secular effects of the emission of quadrupole gravitational radiation, in the lowest order, can be formally included by modifying the Euler equation of Newtonian hydrodynamics, dva , dp

P

+

dU p

Yt -d^- ^

. . =

°'

(1)

in the manner dva,dp__ it dxa

dU_ _ 2G_ (^ #!„ _ ^ &I, ( ^ - 3 * , ^ f ) = 0, dxa 15 c1

(2)

where Iafi = J"pXaX0dx (3) v denotes the moment-of-inertia tensor. However, as Thorne has himself emphasized, the inclusion of the damping terms of 0(c"6) in the Newtonian equation of motion overlooks terms of the same order and, indeed, lower-order terms which describe nonsecular effects (of greater importance over shorter intervals of time). A somewhat more general framework for Thome's ideas (but less explicit in important details) has been described by Burke (1969). In this paper, we shall develop an alternative approach to this same problem of the reaction of the fluid to the emission of gravitational radiation that is in harmony with the post-Newtonian scheme as set out in Papers I and II and in a related paper (Chandrasekhar 19696; this paper will be referred to hereafter as Paper III) on the conservation laws in general relativity. It should, however, be stated that the present paper derives its basic ideas from Trautman's (1958a, b) discussion of this same problem in the framework of the original theory of Einstein, Infeld, and Hoffmann. In that discussion Trautman failed to get agreement with the predictions of the linearized theory of gravitation; but this disagreement arose, as we shall see, from a simple oversight. When this is corrected, Trautman's procedure (as applied and extended in this paper in the framework of hydrodynamics) becomes consistent with the predictions of the linearized theory. And to this writer it appears that Trautman's approach to this problem is the simplest and the most direct that has been devised so far. I I . PRELIMINARY CONSIDERATIONS

First it is useful to recall why the post-Newtonian schemes as normally developed for obtaining the equations of motion as a series in inverse powers of c automatically generate an even series.

No. 1, 1970

2i-POST-NEWTONIAN

EQUATIONS

155

The starting point of all post-Newtonian schemes is provided by initial values goo = 1 — W/&

+ ...,

goa = 0 + . . . ,

and

gafi = — 8^ + . . . ,

(4)

2

where the term —2U/c in goo is demanded by the principle of equivalence. The first iteration of Einstein's field equations with the initial values (4) leads to an improvement in the metric coefficients by determining further terms of 0(0"*) in goo, of 0(c~3) in g0a, and of 0(e~ 2 ) in ga$—it is these "improvements" that lead to the equations of motion in the first post-Newtonian approximation. Had we supposed, in developing the expansion for the metric coefficients, that terms of 0(c~3) in goo, of 0(c~2) in goa, and of 0(c~l) in gap occur, then we should have found that these terms satisfy homogeneous equations (unlike the terms of one higher order which satisfy inhomogeneous equations); they can, consequently, be set equal to zero b y a suitable choice of gauge. 2 The same phenomenon will be repeated when we proceed to the second post-Newtonian approximation: the terms of 0(c~5) in goo, of 0 ( c - 4 ) in g0a, and of 0(c~d) in gap will again satisfy homogeneous equations; and again, they can be set equal to zero by a suitable choice of gauge. By induction it follows that we shall continue to skip the " o d d " steps indefinitely if we continue the scheme of iterations without any modifications. The question arises: How are we to break this chain by providing a nonzero source that will provide "starting values" for a first nontrivial odd step (even as the principle of equivalence originally provided the "source" —2U/c2 in gm for starting the even series)? Clearly, the reason why the standard post-Newtonian scheme fails to provide a source for a nontrivial odd step is that nowhere in the scheme do we impose on the solutions the Sommerfeld radiation-condition, namely, that at "infinity" there is only outgoing radiation. This condition cannot, however, be applied in any straightforward manner to the solutions obtained in a "slow-motion" approximation as the post-Newtonian approximations are, for these approximations, based as they are on the assumption that v/c < 1, require that the operation of d/dx0 ( = d/cdt) on any quantity lowers its order by one. This last fact implies that the solutions obtained on the basic assumptions of the post-Newtonian scheme can be valid only in the near zone where r < ct. What is required, then, is a "matching" (in the sense of Thorne and Burke) of the solutions appropriate to the near zone with those appropriate to the far zone (where the Sommerfeld condition is to be imposed). A way in which this matching can be "painlessly" accomplished is the principal burden of Trautman's 1958 papers. As we have already stated, an oversight in Trautman's paper will have to be corrected. But his (uncorrected) considerations are still useful (and, in the author's opinion, necessary) for answering the two main questions: (1) Why is there no nontrivial lf-postNewtonian approximation?; and (2) What are the lowest orders (in powers of c _1 ) in which the outgoing-radiation condition at infinity modifies the metric coefficients in the near zone? We start with the equation which provides the basis for the standard linearized theory of gravitational radiation. Letting yik _ ^ik _ gifc _ yik _ gik^/

— g

f

(5)

where i)ik is the diagonal Minkowskian metric, ( + 1 , — 1, — 1, —1), and imposing on yik the de Donder condition 7ik,k = 0 ,

(6)

we have the equation

• ,.-««,»

(n=v=-i|i),

(7)

2 By a choice of gauge we mean here the same as in Paper II (n. 1 on p. 60), namely, the choice of the four arbitrary functions that naturally occur in the solutions of the field equations in each successive approximation.

156

S. CHANDRASEKHAR AND F. PAUL ESPOSITO

Vol. 160

ik

where T is ascribed (in the first instance) its Newtonian value: T00 = pc2,

r 0 " = pcva,

and

T«f> = pvavs + pd^.

(8)3

In writing equation (7), we do not foreclose the possibility that its use, together with the imposed gauge on yik and the choice of Tik, may be inconsistent with the solutions already derived (in a different gauge) in the first and the second post-Newtonian approximations. Indeed, as we shall see in § III below, for consistency Tik must be replaced by the Landau-Lifshitz complex 9". On this account, it should be emphasized that our present objective is only to obtain some preliminary answers to the two questions which we have raised. The solution of equation (7) which satisfies the outgoing-radiation condition is 7 tt (x, t) = - ^ f • / * ' • rk(x', t-\x-x'\/c). (9) c v |* — * | By this choice of the particular solution of equation (7), expressed in terms of the "retarded potentials," we have automatically excluded the possibility of any incoming radiation at infinity. Also, it should be noted that equation (9) represents an exact solution of equation (7) which satisfies the Sommerfeld radiation-condition at infinity (albeit eq. [7] is itself not an exact consequence of the field equations). We now expand Tik(x', t — | x — x' \ /c) as a series in inverse powers of c in a manner that is appropriate for the near zone. Thus 7**(x, /) = - — y Cr v

. dx' \X — JC |

Afi . 2 . (— 1V+1 /)»

(10)

Considering first the (0, 0)-component of equation (10) and writing out the first few terms explicitly, we have

v---f-f|/p(*',oi«-*'i^ (11)

We recognize that the terms of 0(c~2) and 0(c~*) in the solution (11) are the same as those that occur in the solutions for the metric coefficients in the Newtonian and in the first post-Newtonian approximations.4 We also observe that the law of the conservation of mass, in its Newtonian form

£ fP(x', t)dx'- = 0 ,

(12)

at v 3 The convention regarding the indices is the same as in Papers I and I I : Latin indices take the values 0 , 1 , 2, and 3, and Greek indices take only the values 1, 2, and 3 referring to the spatial coordinates; and the summation convention will be restricted to their respective ranges. Also, Xo will be replaced by ct when the notation of ordinary Cartesian tensors is used; and when the notation of Cartesian tensors is used, the Greek indices will always be written as subscripts; and the summation over repeated Greek (Cartesian) indices will also be assumed. 4 However, in making the "identifications," it must be borne in mind that the solutions in the postNewtonian approximations are expressed in a different gauge.

No. 1, 1970

2f POST-NEWTONIAN EQUATIONS

157

700 = 0 .

(13)

already ensures that 3

(By the numeral 3 below 700 we indicate that we are referring to the term of 0(c~3) in the series expansion of 700 in the inverse powers of c. This notation for referring to the terms of different orders in the expansion of a quantity will be adopted in the rest of this paper.) More precisely than equation (12), we know from the results of the post-Newtonian approximations that ftSp(*',t)dx'

= 0(c->)

(14)

and is a function of time only. Again from the laws of the conservation of mass and of linear momentum (in their Newtonian forms) it follows that the term of 0(c~5), already present in the solution (11), namely, ^fp{x>,t)\x-x'\Hx'

= ^fP{x',t){\x\*

+ WV-lx^Jdx'

= ^f,

(15)

is also a function of time only. From the results (14) and (15) we may conclude that the lowest-order nonvanishing odd term in 7 00 is of 0{c~h): 700 7^ 0

and is a function of time only .

(16)

Considering next the (0, a)-component of equation (10), we have y0„ =

_ 4G f

P(x'tK(x>

c v

0 dx, _2G»

X — X \

c

j

.

^ ,K(x,

t), x

or v

_ x,, d%, x'\dx' (17)

+ ~ j ; fP(x',

t)va(x', t)dx' +£6-~

fP(x',

t)va(x', t)\x-x'\Hx'

.

The terms of 0(c -3 ) and 0(c -5 ) on the right-hand side of equation (17) occur in the solutions for the metric coefficients in the first and in the second post-Newtonian approximations. On the other hand, since we know from the post-Newtonian results that

t)va(x', t)dx' = 0(C"2) ,

4 SP(X',

(18)

at v we may conclude without any ambiguity that 7°« = 0

while

4

7 0a 5^ 0 .

(19)

6

Turning finally to the (a, /3)-component of equation (10), we have

7"(x, t) = - % f £ ^ T dx' + ^ l fT*V, t)dx' .

(20)

c v \x — x \ c 01 v Therefore, the lowest-order nonvanishing odd term in 7"^ is 5 6

^

We are not writing here the explicit form for faff(l) suggested by the solution (20) since we shall find in § III below that the suggested form is in fact incorrect.

158

S. CHANDRASEKHAR AND F. PAUL ESPOSITO

Vol. 160

The Newtonian forms of the laws of the conservation of mass and of linear momentum have thus allowed us to infer that ,y00 _

yO
,yO|8 _

3

2

3

Q

and

(22) 7°° = -6 F(t),

7"" = f6 U(t) ,

and

0a T

= 0,

though it should be noted that the present considerations do not enable us to specify F(l) and fap(t) in a unique manner. TABLE 1 INFORMATION ON THE METRIC COEFFICIENTS THAT IS NEEDED IN THE VARIOUS APPROXIMATIONS

ORDERS OF THE METRIC COEFFICIENTS NEEDED EQUATIONS OF MOTION

Newtonian i-post-Newtonian 1-post-Newtonian U-post-Newtonian 2-post-Newtonian 2i-post-Newtonian§ 3-post-Newtonian 3 i-post-Newtonian

«O0

go*

«oo

-Saff Of -2USa0C^ 0t (. . .)c~* Q^C't (. . .")c 6|j QajS<7)c~'||

0 Of Pac~3 Of (. . .)c~6 Q<mi')(T't

l-2U/c2* Of 2(£/ 2 -2*)
*The term —2U/c2 in goo is demanded by the principle of equivalence. t These terms vanish by virtue of the laws of the conservation of mass and of linear momentum. t These are the lowest-order terms in the metric coefficients that derive from the imposition of the Sommerfeld radiation-condition at infinity. (All the other terms in the table are obtained by solving the field equations.) § This is the approximation in which radiative-reaction terms first appear. || These terms are needed to determine the energy in the one lower approximation.

The foregoing results for y*' when expressed in terms of the metric coefficients g,y (with the aid of eqs. [41] below) imply gOO = gOa = ga/3 = 0 , 3

2

(23)

1

and goo = -6
C

and

g0a = g«e = 0 . 4

(24)

3

The vanishing of the terms listed in (23) establishes why there is no nontrivial |-postNewtonian approximation. Also, even though we do not as yet know precisely what Qoo(6) is, the fact that it is a function of time only is sufficient to ensure that by a gauge transformation (involving only the function W in the transformation equations given in Paper II, eq. [39]) among the coefficients (24), we can reduce Qoo<5) to zero. In other words, there is also no nontrivial 1^-post-Newtonian approximation. These results are in agreement with the known expectation that the terms representing radiation reaction must first manifest themselves in the 25-post-Newtonian approximation. In Table 1 the conclusions to which we have presently arrived are summarized. I I I . T H E UNIQUE SPECIFICATION OF QooW,

Qoa(6),

AND

Qa0(i)

The preliminary considerations of § II have indicated that the lowest-order terms in the metric coefficients that result from the imposition of the Sommerfeld radiation-condi-

No. 1, 1970

2f POST-NEWTONIAN EQUATIONS 6

6

159

5

tion at infinity are 0(c~ ) in goo, 0(c~ ) in g0a, and 0(e~ ) in gap. We shall denote these terms by Soo = 76 Coo'6' ,

goa = £ Coa(6) ,

and

gafl = £ Q^

.

(25)

Our problem now is to obtain unique expressions for these coefficients. For this purpose, we start with the field equation written in terms of the Landau-Lifshitz complex 9*'*. We have (cf. Paper III, eqs. [31] and [32]) (0*Vm - $%km),i.m = ^r-

Qik

•

(26)

If in equation (26) we let 0 * have the value 0**(« determined in the first post-Newtonian approximation and the metric coefficients have the values g**(2) appropriate to the second post-Newtonian approximation, then the two sides of the equation will be balanced with respect to the order of the terms that are retained on each side; and, moreover, the equation will be satisfied to that order. Let yik denote the lowest-order term in g** that derives from the imposition of the boundary condition at infinity. From our considerations of § II, we expect them to be 700,

70a,

5

6

and

y*?>.

(27)

5

Now make the substitution (cf. eq. [5]) B a = fltt9> - yik (28) on the left-hand side of equation (26) and linearize with respect to yik: the terms that will thus be ignored will be of orders higher than any that are retained. We shall then obtain ($ik(v
Q'k.

So long as the y's are of the orders specified in (27)—and we shall presently verify that they are—we may consistently replace the g-(2)'s, that occur as factors of the 7"'s, by the corresponding Minkowskian coefficients. Also, we may impose on the 7-'s a gauge independently of the one chosen in the solution for the g'(2)'s: there will be no conflict since the g-(2)'s do not include any terms of the orders of the 7-'s. And we shall find it convenient to impose on the 7-'s the same de Donder condition (6). On these assumptions and restrictions, equation (29) becomes (BttwB'-
.

(30)

The terms in the g^o's on the left-hand side of equation (30) do not include any of the orders specified in (27); and, moreover, as we have already noted, these terms (derived from the g-(2)'s), when consistently expanded, will just make up 167rG9aa)/c4. Accordingly, &
(31)

This equation differs from equation (7) only by 9,A:(i) having replaced Tik; and this difference, as it will appear, is crucial for the unique specification of yik. Notice also that by virtue of the property, (32) Qik . = e « = o

160

S. CHANDRASEKHAR AND F. PAUL ESPOSITO

Vol. 160

of the Landau-Lifshitz complex, equation (31) is consistent with the de Donder condition imposed on the y's (to the order required). We may now treat equation (31) in the same way as we treated equation (7) in § II. Thus, by considering the (0, 0)-component of equation (31), we shall obtain (cf. eq. [11])

7

°° = ¥ Tt fe°°m

{x t)dx +

' f ? 5 fe°°a) (*'>') I * ~ *' IHx'» <33)

''

where we have not written the terms of even order. (It may be recalled here that the dominant term in Q00 is pc2.) For 900(i) determined in the first post-Newtonian approximation | / e ° % ( x ' , 0 < 2 x ' = O(c- 2 ).

(34)

The first term on the right-hand side of equation (33) is, therefore, of 0(c~7) in contras' to the second term which is of 0(c~6). We thus obtain without any ambiguity that

r-BBfM'M'-'w-

(35)

Considering next the (0, a)-component of equation (31), we similarly obtain (cf. eq. [17], but recall that now the dominant term in 9 0a is pcva) 7 a

° = 1* ft Se0°m(*',tW

+ | f fs /e»» (I) (x',t)\x- x'Iw.

(36)

For 60a(D determined in the first post-Newtonian approximation, j t S&*mtf,t)dxr

= 0(
(37)

The first term on the right-hand side of equation (36) is, therefore, of 0(c~i) in contrast to the second which is of Oicr6). We thus obtain, again, without any ambiguity that

r = 3 ? 5 /p(x>' t)v°(x'> ° > x " x> > w •

(38)

Finally, considering the (o, |8)-component of equation (31), we obtain

y^ = ^ftfB^x',t)dx',

(39)

where it will suffice for our present purposes to substitute for 91"3 its Newtonian expression (cf. Paper III, eqs. [11] and [12]) Q"e = e a / J = pvav& + p8*o + Lp ,

where

1 °3

(^E)2]

r dU dU

16TTG L dxa dxp

a

(40)

KdXp) J

It is important to observe that the expression (39) for ya& differs from that which follows from equation (20) by the fact that G""3 now includes the gravitational contribution to the total stress; and this difference, it will appear, is decisive. The "oversight" in Trautman's treatment to which we have referred earlier occurs precisely here: he has T°-& where he should have had 6a/3.

No. 1, 1970

2|-POST-NEWTONIAN EQUATIONS

161

The expressions (35), (38), and (39) that we have found for the y's can be transformed into expressions for the corresponding g..'s with the aid of the formulae goo = !(7°° + 7 a a ) ,

go. = - 7 0 a ,

n

n

and

n

n

n

(41)

g* = y'e + H*(7°° - 7""), n

n

n

n

which follow from the general relation (Trautman 19586, eq. [27])

gn = vikvn(ykl - hklvr*y") • n

We thus obtain Qnm = \G ~

n

fp{x',

&a(6) = - fG ^

(42)

n

t) I x - x' | Hx' + 1G |

fe„dx

,

/ > ( * ' , <)».(*', 01 x - *' | Hx',

(43) (44)

and Q«m = 4 G ^ ye a3 rfx + S ^ G ^ / P ( x ' , t)\x-x'\Hx'

- 2G^ ye^rfx] . (45)

It is satisfactory in many ways that the expressions which we have found for the lowest-order terms in the metric coefficients in the near zone, which reflect the outgoingradiation condition at infinity, are purely of Newtonian origin in the sense that they do not involve explicitly any quantity defined in the higher approximations. a) Alternative Expressions for £?ooC5\ (?c>a(6), and Qafl(5) in Terms of the Moment-o]'-Inertia Tensor We have already seen in § II (eq. [15]) how by expanding | x — x'\2 and making use of the Newtonian laws of the conservation of mass and of linear momentum, we can write

T* / p ( ' . O I * - r W = ^ .

(46)

Similarly, we can transform the integral expression Qo*w to give d3

•»

-Qp Jp(x',t)va(x',t)\x

- x'\2dx' =

d3

-^fpvAxVdx (47) d?

~2x"~df SP' ^V^X''

0*V*' •

The further simplification of the expressions (43)-(45) depends on the following lemmas which derive from the tensor virial theorem and the associated definitions and relations (for a brief account of this theory, see Chandrasekhar 1969a, chap. 2). LEMMA 1: The tensor he = - i p S W (mod div) , (48)6 where «*(*) = Gfp(x') ( « • - « ' • ) ( « » - * ' ' » > dx> (49) v \x — x I denotes the tensor potential. 6 As defined in Paper I I I (§ IV), two functions are said to be equal modulo divergence if they differ by the divergence of a vector which vanishes sufficiently rapidly at infinity that their integrals over the whole space (assuming that they exist) are equal.

162

S. CHANDRASEKHAR AND F. PAUL ESPOSITO

Vol. 160

PROOF: Clearly, 4 — —- = - 2 —- —- V2x = 2V2U 2 (mod dxa dxp dxa dXff dxadxg

dlv

) (50)

--8rfp^<m«idiv). Also (cf. Paper III, eq. [49]), -25 a / 3 (IJjpY s -8irGpUdal) (mod div) .

(51)

From equations (50) and (51) and the known relation

^=s3x^+us">

(52)

the result stated follows. LEMMA 2: For 9 a/3 defined as in equation (40),

fQa,dx = \d^f.

(53)

PROOF: From Lemma 1 and the definition of 9ag it follows fQtfdx = f(pvaV() + p8ag + ta&)dx = f[p(wfi - J 2 U + p6*]dx = 2£„0 + 9Ba/J + ^

where

(54)

,

% = fpdx (55) v and £ a £ and 2BOJ3 denote the kinetic-energy and the potential-energy tensors, respectively. The result stated now follows from the standard form of the tensor virial theorem. LEMMA 3: jj/>.**** = 4^ T -

(56)

PROOF: The conservation of angular momentum ensures the symmetry of the integral on the left-hand side with respect to a and /?. Therefore, we may write instead h jf fpfaxe

+ vpxa)dx = | -j f p -j( (xaxp)dx = | ——•,

(57)

and this is the result stated. By making use of the relations (46) and (47) and the foregoing lemmas, the expressions (43)-(45) for the metric coefficients can be brought to the forms Q^ d*T (

Qoa « =• IGx,^f

,d%, = tG^>dl?

(58)

d3

- f G | i fpva\x\*dx,

(59)

No. 1, 1970

2i-POST-NEWTONIAN EQUATIONS

163

and Qa^ = 2G^f-iGSa^.

(60)

An important consequence of the formula (60) for Qapi5) is = 0 ,

(61)

(6)

(i)

i.e., QO0 is traceless. We shall see that this traceless character ofQafi consistency of the whole development. Further consequences of the foregoing formulae are

dQ,^=2eQ^ at

and

dxa

is essential for the

ae^=w«_ dxp

dxa

The first of these relations is equivalent to the de Donder condition that was imposed on the 7"'s in obtaining the solutions, and the second is an expression of the conservation of the Newtonian angular momentum. PART I I THE EQUATIONS OF MOTION IN THE 2i-POSTNEWTONIAN APPROXIMATION S. CHANDRASEKHAR AND F. PAUL ESPOSITO IV. THE SOLUTION FOR Qf

(7)

In Part I, we have seen how the metric coefficients go„ and gap, to orders required for the 2^-post-Newtonian approximation, can be deduced by supplementing the solutions obtained in the second post-Newtonian approximation by terms of 0(c - 6 ) in g0a and of Oic"6) in g„0—terms which are required by the outgoing-radiation condition at infinity. But the equations of motion in the 25-post-Newtonian approximation cannot be written down without a knowledge of the term of 0{c~'1) in goo. Our earlier considerations have determined (?oo<6); this is not sufficient for our purposes. We need to know "Qoo<7)"; and to determine it an appeal must be made to the field equation Rw =

r (T°° ~ hiooT) .

(63)

We shall presently see how our knowledge of Qoo(s), (?oa<6), and Qap(b) just suffices to determine Qoo(7). It is perhaps significant that an explicit appeal to the field equations is necessary before the radiation-reaction terms in the equations of motion can be made determinate: it emphasizes the essential nonlinear character of the theory. Considering equation (63), we readily verify that no contributions of 0(c~7) to Rm are made by terms in the Ricci tensor which are quadratic in the first derivatives of g„— terms which are usually written as products of the Christoffel symbols. Precisely, the contributions of 0(c~7) to i?oo arise from the terms j? - 1 * d2goc, 7

dx.dXy 7

d A dg„ dx0 \

dXo

dg„o\.

(Ms

OX, J

7

and we find *oo = - * W

- ,fl„«> ^ — - goo -Tf(i

-d-t

^ - ) •

(65)

164

S. CHANDRASEKHAR AND F. PAUL ESPOSITO

Vol. 160

Remembering that goo = - 2 * 7 ,

(66)

2

and making use, also, of the relations (61) and (62), we can rewrite equation (65) in the form d>U ,13^0 ,«) ^ = - i v W " +Q , - ^ +! ^ - . (67) Considering next the right-hand side of equation (63), we readily find that = §p<2oo<6) •

Tm-kmT

(68)

The equation governing Qoom is, therefore, VW» =8 . < W

+ 2&,«> £ L

+ ^

f ,

(69)

where it may be recalled that(Poo(6) andQ,,/ 5 ' are, according to equations (43) and (45), functions of time only. a) The Solution for Qoo<7) The part of the integral of equation (69) which derives from the first two terms on the right-hand side of equation (69) is readily written down. We clearly have

-2^)u-^<Sk-

(7o)

It is less straightforward to write down the part of the integral which derives from the last term. The principal question here is one of uniqueness. It appears that the question can be resolved as follows. First, we state the following readily verifiable lemma. LEMMA 4: If ^ « ( x ) = y>(;t') | x - x' | ndx', (71) where \f/(x) is a good function,7 then W w ( i ) = n{n + l)fiP(x')\x

- x'y-Hx'

= n(n + l)*<"-2)(x) .

(72)

In writing the integrals corresponding to terms such as (cf. eq. [43]) d20nn(6)

d5

a?~ = *

G

|i

d3

, SPW>

2

') I * ~ *' I d*' + 2G^ fQM,

t)dx'

(73)

in equation (69), we shall make use of the converse of Lemma 4 in the operational form (V 2 ) _ 1 ^ ( n ) (x) = — (74) K W) * W („ + 2)(n + 3) " ' The justification for this method of inversion is that in this way the solutions are unambiguously and uniquely expressed in terms of the sources. But more importantly, the only terms (such as d2Qoo<6)/di2 in eq. [69]) which require this procedure, when so inverted, lead precisely to those terms in the expansion of the solution, - ^

f~~r{Qi"{x',t-\x-x'\/c),

(75)

7 In the following sense defined by Lighthill (1958, p. 15): "a good function is one which is everywhere differentiable any number of times and such that it and all its derivatives are 0 ( ] x \ ~") as | x \ —> <*> for all N."

No. 1, 1970

2i-POST-NEWTONIAN EQUATIONS

165 k

of equation (31) which appear in the requisite order as (cf. eq. [10] with T' replaced by 0*)

^H^^r ( *' /)|x -*' |n ~ w -

(76)

where 0** is the Newtonian expression for the complex. Thus when we are considering (?oo<7) the requisite order is c - 7 ; and, in accordance with expression (75) and equation (42), the term in question is

fp(x

'' »\x-x'\W

*[lf 5 2

+ H Jji fQ^x'> 01 x - x' | W] ; (77)

whereas, inverting the expression on the right-hand side of equation (73) in accordance with equation (74), we have ioG £> SPV,

0 I * - *' | W + \G ^

fG^x',

t) | x - *' | Hx' •

(78)

and we observe that the two results agree. Combining the results (70) and (78), we can now write the complete integral of equation (69); we have Q^

= -2<2M<«U - ,

t)\x-x'\W (79)

+ \G^

fQM,t)\x-x'\Hx'

.

Since Q,,/ 6 ' is traceless, we can in view of the relation (52) rewrite the solution (79) in the form Q^

- &„<«2V + ioGJp

= -2Qm«W

SP(X''

01* ~ x ' | W (80)

+ i G ^ ye wl (x',0|x-x'|w. V. CHRISTOFFEL SYMBOLS AND RELATED QUANTITIES

With the metric coefficients Qoo°\ (?oa(6), (?«0(5), and Qm(6) determined, we readily find that f> = - 4 Coo(6) ,

«* = - i < V 6 ) ,

5

5

C

and

C

V-|, = ^ v 5


(81)

iO

With this knowledge, we find on evaluating the various Christoffel symbols that the terms in them which follow those listed in Paper II, equations (47), are

e

2c6 \ dxp

dxa

dt J

5

166

S. CHANDRASEKHAR AND F. PAUL ESPOSITO

Vol. 160

Also, 1 dQ (5) y0 = d log V - g/dxo = — - ^ -t— *

6

s/

2 c*

'

and

dt

ya = 0 .

(83)

5

6

VI.

THE EQUATIONS GOVERNING T H E FLUID IN THE 2J-POST-NEWTONIAN APPROXIMATION

First we observe that our present increased knowledge of the metric coefficients enables us to supplement the expressions for the components of the energy-momentum tensor given in Paper II, equations (49)-(51), by the additional terms 6

£C

3

4

C

"

and

(84) C"

6

The relevant equations governing the motion of the fluid in the 2J-post-Newtonian approximation can now be obtained by simply writing out the equation T% = 0

(85)

to the required order with the aid of equations (82)-(84). Considering first the 0-component of equation (85), we find that there are no additional terms beyond those included in the second post-Newtonian approximation. The reason for this circumstance is that the term of 0(c~5) in the conserved density, pw°V—g, vanishes (cf. eqs. [81] and [84]): pu°V - g = 0 .

(86)

In other words, no alteration in the baryon number, conserved in the second post-Newtonian approximation, is introduced in this higher approximation. This fact is consistent with the requirement that the present "intermediate" approximation does not alter the expression for the density conserved in the third post-Newtonian approximation given in Paper II, equation (95) (which in turn is required in the determination of the energy conserved in the second post-Newtonian approximation). Considering next the o-component of equation (85), we find (6)

\ T- - i [~4 «**»*. + Q*»* + (V»> + m dXa W5) ,

, i

+ ^

~dT +

n (6) du pQw

^

, p aQoo<7>i

(87)

+ * ~l3xV\ •

By making use of the relations (62), we find that the foregoing equation can be brought to the somewhat simpler form i^a,-

=

! T

n m^«

pQm

7 r"" * i ~

I

dQm{i)

Tt - ^

d ,

., .

-^r-pjtiQ"^ (88)

p

- -8T

+ pQo

°

^

+

p

-* ~dxV\-

Finally, inserting in equation (88) the solution (80) for Qma) and simplifying, we are left with

No. 1, 1970

2^-POST-NEWTONIAN EQUATIONS

l f , , - l [ - PO... & _ t 30

_, <( o ^ -

^

(6)

- P ^ g - + ^PG ^ a3

+ i p G ^ SQM,

167 ipS

f p(x', 0 | * - x' 12(*a - x'Mx'

(89)

<)(*. ~ xa>)dx'~\

VII. THE COMPONENTS OF THE LANDAU-LIFSHITZ COMPLEX IN THE 25-POST-NEWTONIAN APPROXIMATION

Expressions for the components of the Landau-Lifshitz complex in the second postNewtonian approximation have been given in Paper II, equations (58), (72), and (90). The terms of one higher order that must be added to these expressions in the 25-postNewtonian approximation can now be evaluated with the aid of the Christoffel symbols listed in Paper II, equations (47), supplemented by equations (82) of the present paper. We find that e 00 = 0 and 0°" = 0 , (90) 3

4

while G"" = -^ [-pQm»h.vf 5

+ pQ*™ +

QOO(6)(PWS

+ fa + 4*)]

C

+

16TTGC6

I

^a"

\dxj Vox,,/

8y

°°

3*. dxs d#a dxp «>

2

\d£,3

d£a/

^SdUdQyM \dxa dt

<&

dt

ap.dn..m

dP„ d&,<»

dXa

dXfi

dUdQ0aM\ dxp dt )

^UdQ^m dx M dt

dt dt

2/dUdQm™

\dxa

dP,dQ„<*> dx, dt

dxp

0

dUdQmw\ dxp dxa )

2dUdQaP-\\ dXp dx? J )

Accordingly, the (0, 0)- and the (0, a)-components of the Landau-Lifshitz complex retain, in the 2\-post-N ewtonian approximation, the same values as in the second post-Newtonian approximation. These facts are consistent with what we shall find in Part III, namely, that the terms (89) of 0(c~b) in the equation of motion governing the fluid principally contribute only to the dissipation of the energy and the angular momentum that are conserved in the second post-Newtonian approximation.8 Next by evaluating the divergence of the terms (91) (with respect to 0), we find that we simply recover the terms (87). Therefore, in view of equations (90), we may write - 0"'.y = e'P.0 = - T^j . C 4

5

(92)

C 4

8 The statement is strictly correct only with respect to the angular momentum; a slight amplification is necessary with respect to the energy (cf. Part IV).

168

S. CHANDRASEKHAR AND F. PAUL ESPOSITO

In other words, the identity of the equations, 9a',y = 0 and

Vol. 160

T*>.j = 0 ,

(93)

verified in thefirstand the second post-Newtonian approximations, continues to be maintained in the 2\-post-Newtonian approximation. PART III THE DISSIPATION OF ENERGY AND ANGULAR MOMENTUM COMPARISON WITH THE LINEARIZED THEORY OF GRAVITATIONAL RADIATION S. CHANDRASEKHAR VIII. THE RATE OF DISSIPATION OF ENERGY

The equations governing the fluid motions in the 2j-post-Newtonian approximation can be written as the sum of the second post-Newtonian terms, included in Paper II, equation (54), and the terms of 0(c~5), included in equation (89). Symbolically, we may write -

T";j + - T";j = 0 .

C 2d post-N

(94)

C 4

Contracting this equation with va and integrating over the volume occupied by the fluid, we obtain - fva C V

T<j dx + - fvaT'Kjdx 2d post-N

C V

= 0.

(95)

4

The first term on the left-hand side of this equation represents d(tf(&(2)dx)/dt, where (S(2> (given by Paper II, eq. [110]) is the energy conserved in the second post-Newtonian approximation (cf. Chandrasekhar 1970). We can, therefore, write 4: f^dx + - SvaT"\idx = 0 . (96) at v cv 4 We shall presently show that this equation predicts a rate of dissipation of energy by the system that is in accord with the predictions of the linearized theory of gravitational radiation. First, we shall simplify the last two terms on the right-hand side of equation (89). By expanding |x — x'\2 that occurs under the integral sign in the first of them and ignoring those terms which vanish by virtue of the conservation of mass and of linear momentum, we obtain ^fP(x',t)\x-x'\\xa-x'a)dx' or v = x*£5 SPW, - ^ fp(x', at v

t) (I x I * + I x' 12 -

Ix.x'Mx'

0*'a(|*| 2 + l*'l 2 -

2xlix'li)dx' (97)

x^fP{x',t)\x'\Hx' dt° y

- £b Sp(x',tW«(\x'\* u- J. uu

° dfi

Qf J- tiua,

dfi

I

+

s\

CL J. UQ

* dt>

2x,x',)dx'

No. 1, 1970

2J-POST-NEWTONIAN EQUATIONS

169

where we have let hey = fpXtXpXydx

(98)

v

denote the third-order moment of inertia. Next, by making use of Lemma 2, we can reduce the last term on the right-hand side of equation (89) in the manner d3 ^5 fO^ix',

(p *)(*.. - x'a)dx' = xa-^ fQ^dx

(p - -^

fQ^Xadx (99)

~ 2 * a dfi

dP

J ii

>"'x'ax

•

Also, b y equation (59),

" P^f = »*£/>.W**< - «p^f^-

(wo)

Substituting the results of these reductions in equation (89), we obtain

~0) - hpQ^

^ f

(ii) + \pxJS ^

(HI)

- lPxfi

g k

(101)

(IV)

5

3

<2 /

<24


^

„ (V)

.

1

.

We shall now write down the results of contracting the different groups of terms in equation (101) b y va and integrating over the volume occupied by the fluid. T h e first three groups of terms give (I): - e » < » . / > . ~ , ^ - \ ^ y at at (II): - Qjnfpv^'dx v at

- % ^ at

fp\v\Hx=-±v

(Q„™ZJ ,

(1021

at

y>.v** v

(103)

= —O «) — and

(III): - hQ^fpVa^dx

=

hQ„Mf~(pva)^dx (104)

—

iVl"

J

- . -OpydX — 2yM"

j .

These three groups of terms combine to give

(i) + (ii) + (in): - jt(ov5>3;„ + 2ov«£j + hQ«m(2 ^ f + ^ t ) - (105)

170

S. CHANDRASEKHAR AND F. PAUL ESPOSITO

Vol. 160

or in view of the traceless character of Qa/1w and the tensor virial theorem, we can write (I) + (II) + (III) : - ft (&,«>£„, + 2Qa^ZJ

+ i<2„„(6> ^f

.

(106)

Finally, substituting forQ a „ (6) its expression given in equation (60), we obtain (I) + (II) + (III) : - j t (Qoo^

+

+

2Q^L,) (107)


Returning to equation (101), we find that the contribution by the group of terms IV is (IV) : \G - ^ fpXtvJx _ __ j

r T5

j_

r

10

i

(? diIlia dl„a _ dt'Iw dlca\ V dP it dP dt ) / , d*ha dH^ \ dP dP

, irA(*

~TO

- §G —^ f pxav„dx

dt \

(108)

ffil„
d

lL^ §!hi _ Urn *LL - -i #La. dl^ + #I» 4l»\ dP dP

dP dP

dP dt

+

dP dt ) '

The contribution by the last group of terms V can be directly written down and calls for no special comment. Combining then the results of the foregoing reductions, we obtain

I

j y-i &_ l o ^ *aii

& * CM

^ ™ 7t\ ~W~IP G x

& * MM ^ -^ffg

-j z_~J*a

® Ma _|_ ^ •* MM z_

a
\

(\ r\C\\

{

W~W~HF~dr~*"~dP~!u~J

'

+ 0 1 />- i i •** - A ^r - * I •/* w*) /w*] •

In terms of the tensor, Dap = 3/as — Sa/siw >

(HO)

defining the quadrupole moment of the system, equation (109) can be written more compactly in the form CPDgf, dWgf, dP

~c fT
= 1 [ - j t (0»«%M. + 2QaMTJ + &G dp

, , „ d fcPDa& (PD* ~i"^rJdt\ dP dP +

G

\*dP

ff>v*\x\Hx

d'D*,, dDaf)\ dP dt ) ~ Tk~dP~ ~ *df* J%e»»x*dx)

{111)

ff>v*dx\

•

No. 1, 1970

2|-POST-NEWTONIAN EQUATIONS

169

where we have let Itfy = J* pXaXpXydx (98) v denote the third-order moment of inertia. Next, b y making use of Lemma 2, we can reduce the last term on the right-hand side of equation (89) in the manner d3 ^ fQ,»(x',

d3 - -^

d? t)(xa - x'a)dx' = xa-^ fQ^dx 2

dbT " dfi

d* dP

fQmxadx (99)

J

x ax

^"" ' -

Also, by equation (59), " "^T

- i*£Sn\*\U* - ifGx/^.

(100)

Substituting the results of these reductions in equation (89), we obtain

0)

(5)

'

'

-*pe^^ +^ g ^ - f ^ ^ (HI) _1pG

^ ~diT

(ioi)

(IV)

~ ^pGltl ^Q^x"-dx + * p G ^ 7>a|*| 2 <^ J • (V)

We shall now write down the results of contracting the different groups of terms in equation (101) by va and integrating over the volume occupied by the fluid. T h e first three groups of terms give (I): -Qoo^fpva^dx v at (II): - Qa™ fpva v

- \ ^ at

% dx - % ^ at at

fp\v\*dx v v

= - I (Qoo^XJ , at

(102\

fpv^dx (103)

and (III): - iQ^fPva^dx

=

hQia<*>S-£-(?v.)%iadx (104)

= _ i O ^ C — ft dx = hO

(5)

^ ^

These three groups of terms combine to give (I) + (II) + (III) : - |

(Qoo^

+ 2Q^ZJ

+ !&„<» ( 2 ^

+ ^ )

,

(105)

170

S. CHANDRASEKHAR AND F. PAUL ESPOSITO

Vol. 160

( :

or in view of the traceless character of Qa„ - and the tensor virial theorem, we can write rl (I) + (II) + (III) : - j t ( Q o o ^ + 2Qa^XJ

d3T + !&„«> ^ f .

(106)

Finally, substituting for(3 a/ / 6) its expression given in equation (60), we obtain (I) + (II) + (III) : - j t ( ( V 8 ^ , +

IQ^ZJ (107)

\G{Z

J_ XC* I i

^

dP

^

dP

S<J

-

dP dt3 J"

Returning to equation (101), we find that the contribution by the group of terms IV is

(IV) : \G ^ f fpx^M -T1OG(

- |G ^f

_ d J. fia dl pa

dtb

^r/^cPh«d?ha 10 3 V dt

fpx^dx

d J. fifi Cllffa'

dt6 dt J

dt

(108)

3

dt3

_ d*I„ d h„\ dP dt3 )

d3!^ d2!^ _ d3!^ d2I,a _ , d?%a dim dP dP dt3 dP dp dt +

, ! r d^ / To dt \

+

d%,^ dh£\ dP dt ) "

The contribution by the last group of terms V can be directly written down and calls for no special comment. Combining then the results of the foregoing reductions, we obtain d3!^ d3!^ dt3 dP I

1 s-* d_ ( 2 ® *ap & * ap

"•"^ + G

~di\

Z_B±

\.-l

2 d 1 fig Cllfia

x

^

d x

~^dl$r

1 & -*MM __aa

\

(\ (\r\\

~d¥~~dt~"t~"Wlit)

dP dP ~~~dP~ dp

(f § / ^

d3!^ d3!^ dP dP j

~\^f%^dx)

K

fpvjx]

'

.

In terms of the tensor, •Da/9 = 3/a/9 — S^I^ ,

(110)

defining the quadrupole moment of the system, equation (109) can be written more compactly in the form

I /?••*** - M - £ «****» + 2QM + & ^ 3

d (d Dall d?Dat "+" 3 0 dt\ dP dP + G i

( $fpv*Wx

^

dWrt dDal)\ dP dt J

~ ™^W ~hfp SQ»x«dx) fp*«dx] •

U

;

No. 1, 1970

21-POST-NEWTONIAN EQUATIONS

169

where we have let I any = f pxaXffXydx (98) v denote the third-order moment of inertia. Next, by making use of Lemma 2, we can reduce the last term on the right-hand side of equation (89) in the manner d3 d? ^ / © / * ( * ' , t)(xa - x'*)dx' = xa-^ fe„„dx j