The Roche limit based on vibrational stability

Chin.Astron.Astrophys. 9 (1985) 106-113 Act.AStron.Sin.25 (1987) 365-375

Pergamon Press.

THE ROWE LIMIT BASED ON VIB~TIONAL

SONG Guo-xuan

Printed in Great Britain 0275-1062/85$10.00+.00

STABILITY

Shanghai Observatory, Academia Sinica

Received 1983 September 8

ABSTRACT Starting from the concept of vibrational stability, the Roche limit for a uniform incompressible liquid sphere is derived. It is shown how this method can be generalized to the case where there is a non-uniform matter distribution in the radial direction.

1.

IN~ODU~TION

The problem of the Roche limit is an old and It spells out the important problem. condition where a body can hold itself together under the tidal force of another. lhe earliest application of its theory was the explanation of the formation of the rings of Saturn. In 1963, Chandrasekhar [l] used the Virial theorem and its generalizations and derived the Roche limit for a uniform incompressible liquid sphere. The method has certain limitations, for it is for the case of uniform density that we can use the Virial theorem to find the equilibrium configuration under the combined effect of rotation and tide; it cannot be generlaized to the case of non-uniform density distribution. The Roche limit for the case considered by Chandrasekhar was alternatively found by Kopal and SONGGuo-xuan [2] by means of the In this paper, I shall Roche coordinates. use the method of vibrational stability, developed by Kopal [3,4] to make a renewed discussion on this problem with the hope that a generalization to the case of non-uniform density can be effected. Consider a spherical body of mass m, uniform density pe and radius al. rotating with angular velocity w. It is subject to the tide-raising force of another body of mass M, at a distance R. We suppose the rotation and the orbital revolution are synchronous, that is 2.

(11 For a uniform incompressible liquid sphere, its equilibrium configuration, under the combined effect of rotation and tide, can be expressed as [S]

where r, 8, $ are the spherical a is the equi-potential surface

coordinates, parameter,

(3)

I shall use Kopal’s method of vibrational stability and consider a small perturbation and find under what conditions this distorted equilibrium configuration is stable, hence derive its Roche limit.

BASIC EQUATIONS

configuration For a uniform incompressible medium, I take as its base state, the distorted The linearized (2) under rotation and tide and consider an additional perturbation. equations of the perturbation are

Roche Limit

,+

[ +‘u+wf--&&v t

1

107

-2wr.W&8-~----1-,

BV e$ li3P -22corWcosi3-=-----'-is68 PO68

cb@a

LdLLep, I#( OS

poea>

(4) (5)

where P is the pressureperturbation, [I,v, W are the perturbingvelocitiesin the a, 8, $ directionsand + is the perturbingself-potential.The conditionof in~o~ressibilityis

(7)

?he equationsatisfiedby the JIin the case of uniformdensitydistribution is V’J, - 0.

(81 We expandall the perturbations in power series,

Substituting(4) in (5) and (6) and keepingonly small quantitiesof the first order,we have

(10)

(11)

Substitutin the expressions(9) into (4). (10) and (ll),having firstmultiplied(4) and (10)by sin!! 8 and sine respectively, we have

108

Roche Limit

109

110

(15) where (16)

With the help of

(9),

the perturbing

potential

equation

(8) becomes

(17) where

Eqns. (12) - (15) and (17) are the basic equations that we have to solve. The variables are uNn, vNn, wnm, pm and Jlom. They are all functions of a only. All the quantities E, A, C, K, L, N with various superscripts and subscripts are constant The condition that the coefficients. equations (12) - (15), (17) should hold is that the sum of coefficients of all like ?his gives the terms in them be zero. must equations that u , w JI Obvious“f y &$ %an”Tnfinite set satisfy. of linear differential equations in an

infinite number of unknowns. Let us consider some of the properties. 1. For n fm the unknown functions ummr wm, pm, qm are coupled to one another. &%, it is easily seen that all those with m odd are coupled, and all those with m even are coupled, but the two sets are independent Below, only the case of odd of each other. m will be discussed; similar argument will apply to the case of even m. force makes 2. ‘Ihe appearance of Coriolis Eqns. (5) and (6) no longer identically true. This difficulty is removed by the introduction

Roche Limit

of wm#O, [31. Also, the base state is not asymmetrical, it is easily seen that the waves propagating in opposite directions of the circumference are also coupled. In the particular case m=O, the perturbations must assume the following forms,

111

Eqn. (23) showing we have to know both the perturbing potential inside and the potential outside. Obviously, at the surface, these two are equal, (25)

(19) P-

Eqn. (23) can be expressed

2 P.Xo)PX-ehm,

as

That is, in (12) - (18)) for m- 0, we must replace m and “w,,,, by 0 and wno, respectively. This is different from the case of vibrational stability of rotating stars [3], but is analgous to the case where both rotation and tide are considered, [4]. case, 3. In the uniform incompressible f2' and f2 are constants, hence all the equations are Euler equations and their solutions can be expressed in the form &, Q being the characteristic value. 3.

BOUNDARY CONDITIONS

(26)

From the above, we see that the appropriate boundary conditions are the following. At the centre, 4. PIazO= finite

We have ended with an infinite set of coupled differential equations in our infinite number of unknowns. For practical calculation, we limit the index to n=2. We then have a total of 19 unknowns, u2, _2,

w

G I s=o = finite At the surface, %mla=al = D

(22)

uo ,O’ “z,ar

rqo,or JIO’K

!!E 8n

I4

w

--&-

I

6

- - 4&Aj(.,)

PROCESSOF SOLUTION ANDRESULT

(23)

The last equation arises as follows. Because of the additional perturbation, there is a further distortion of the original, distorted equilibrium surface, so we must consider a displacement of the matter of the surface and this displacement can be regarded as a surface density, hence Eqn. (23), where n is the unit outward normal of the equilibrium surface and u is the radial displacement. From the expression for the radial velocity, we have

w2,01

U2,2r w2,2r

+2io’.$2.2’ e so ution

V2,_2’ P2,-2’ $,-2’

consists

V2,O’

PO,O’

V2,2* P2,O’

W2,_2’ P2,2r

of the following steps. 1. The four equations (12) - (15) are ordinary linear equations in u, v, w, and p. The non-linear coefficients involve $I. the conditions 2. Using respectively, that the solution is finite at the centre and tends towards 0 at infinity, we solve Eqn. (17) for the inside and outside potentials. We also use Eqn. (25) to fix in advance certain parameters in these two potentials, solution 3. Substitute the interior obtained into the right hand of the equations of step 1. Thus, using the ’ boundary conditions (20) and (22)) we find

SOW&

112

for

the

expression

5.

CASE OF RADIALVARIATIONOF DENSITY

u,

vr

w,

p,

$.

4. Substitute the obt;rined expressions for u, $+, $- in the boundary condition (23) gives the set of homogeneous equations for the four adjustable constants;, For a nantrivial salution, the determinant of the coefficients must be zero, hence the equation for the characteristic frequency o. This can be real or complex, depending on the values of s2 a and 92. If cr is real, then the original equilibrium state is stable; if a is complex, then the stability depends on the sign af its imaginary part. The eritlcai state corresponds to the imaginary part of o being 0, and this gives the Roche limit. As is known, in the solution of the above differential equations, we first solve the characteristic equation, and precisely the latter gives the Roche limit. Therefore, it is very difficult to solve the characteristic

equation by direct methods. We propose an indirect method as follows. First, pick an arbitrary value of B I &/&pO, and make a solution as described above, Because of the arbitrary nature of this value, we expect a to come out complex. By varying B, we get a series of values for 0, We keep varying B until we get a u with a zero imaginary point. For the case m<<# (&handrasekhar*s _@I case), the present method gave a Roche limit of t3=5.58B4. The value obtained in I21 is 0.0875 and Cbandrasehkar’s value in f23 is 5.555568. Although there is still some discrepancy, our result can be improved by taking more terms in Eqns. (12) I (17). But the important point is that the present method can be generalised to the case orF non-uniform density distribution.

Roche

Limit

113

At the surface, P=O, the inside and the gradient of the perturbing potential is finite. and the outside potentials are equal (cf. Eqn.(25)), and there is a jump in the perturbing potential across the surface (cf. Eqn.(23)). The expressions for the perturbations are the Substituting these in (27) - (33) then gives a set of same as before, i.e., Eqns.(9). The Roche limit is then ordinary differential equations with a as the independent variable. found as before, only here we must resort to numerical methods when solving the set of differential equations. ACKNOWLEDGEMENT This work was completed during the author’s The author thanks Professor Kopal for the guidance and help

REFERENCES [ 11

Qundnsekhlr,S., Ap. I., 138(1%3), 1182.

[ 2 1 Kopd, Z., Song,G. X., Arhophyr. and Space Science, % (1983), 381. [ 3 1 Kopnl,Z.. Astrophys. and Space S&me, 76(1981), 187. [ 4 1 Kc@, Z., Sons,G. X., Astrophyr. swd Spocc Science, 92 (1983), 3. [ 5 1 Kopll, Z., Dvnuoicaof Uoac BinarySyrtenw. D. ReidalPublishingCo. (1978). [ 6 1 Love, A. E. H., A Treatiseon the hinthematicalThcorv of Pm

(1959).

I7 1 Kopnl,Z.. ~rtmplryr.md .PJMC~ Science,70 (1980), 407.

EL&city, GmbrLlge

stay given.

at Manchester

University,