the equilibrium of self-gravitating rings - NASA/ADS

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THE EQUILIBRIUM OF SELF-GRAVITATING RINGS

J. OSTRIKER* Yerkes Observatory

Received May 16, 1964

ABSTRACT In this paper, slender, uniformly rotating, polytropic gaseous rings are considered, with the intention

of determining the equilibrium distributions of pressure, density, and gravitational potential, and the associated rate of rotation. Allowance is made for the possible presence of a spherical body at the center of the ring. Solutions in closed form are obtained for w = 0 (liquid), n = 1, and w —> «> (isothermal perfect gas). Simple expressions are found for various integral properties of rings in equilibrium : the internal, potential, and total energy, and the angular momentum.

I. INTRODUCTION

Of the various forms that a self-gravitating body may take, the ellipsoidal, or more particularly the oblate spheroidal, has received by far the most attention. This emphasis has resulted naturally from observations of planets and normal stars, from the requirement that bodies with small angular momentum be spheroidal, and from the relative ease with which the potential of ellipsoids can be treated. If, however, one were inter- ested in understanding the often complicated forms of galaxies, and of stars in the initial (see Jeans 1929) and perhaps the final (see Limber 1964) stages of their evolution, one might profitably investigate other configurations.

The doubly connected ring-shaped or annular body is probably the next stage, in order of complexity, from the ellipsoids or pseudo-ellipsoids encountered in tidal-rotational problems. Partly for this reason, and partly through the effort to understand Saturn’s rings, the annulus seems to be the only other gravitational system that has been studied in detail. Basic work on the subject includes contributions by Laplace (1789), Maxwell (1885), and Poincaré (1891) and was, in a sense, completed by Dyson’s (1893) compre- hensive treatment of the equilibrium and stability of fluid rings. Modern observations have shown that rings commonly occur in spiral galaxies of both the ordinary and barred types (see Randers [1940] and the classification scheme of de Vaucouleurs [1959]).The same observations led Randers (1942) to combine and extend Laplace’s and Dyson’s studies in the context of galactic dynamics.

These discussions of the equilibrium of rings have generally made the following sim- plifying assumptions: (1) The rings are slender, i.e., minor axis <3C major axis; (2) the rotation is uniform, i.e., like a solid body; (3) the rings are homogeneous, i.e., liquid. The first assumption, which is reasonable in many applications, greatly facilitates treatment of the most difficult aspect of this problem, that of finding the forces due to self-gravitation. It will be retained in the present work. The second, which follows if viscous forces are not strictly absent, is also retained. The last and most restrictive assumption will be replaced here by the requirement that pressure and density satisfy a poly tropic equation of state,

p = Kpwln , (i)

where k and n are given constants. Such a relation can represent a broad spectrum of different possible configurations; for example, n — 0 represents a homogenous liquid, ^ = f a monatomic gas in adiabatic equilibrium, and w —» co an isothermal perfect gas.

The principal mathematical technique to be applied in this analysis derives from the assumption of slender rings. The assumed small ratio between the thickness and diameter

* National Science Foundation graduate fellow 1963, 1964.

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of a ring is utilized to introduce a small quantity ß. Then ß is treated as an expansion parameter, and series solutions are obtained that are accurate to the first and occasional- ly to the second power in ß. Since in the limit ß —» 0 rings degenerate into infinite cylinders, it is clear that all the leading terms of the indicated series expansions will be identical to the analogous expressions derived for cylindrical self-gravita ting polytropes;1 and it is in this context that an earlier discussion of polytropic cylinders (Ostriker 1964a; re- ferred to hereafter as “Paper I”) will be used. We may note here that, as the cylinders were shown to have finite mass and extent, so all rings (having 0 < n < oo ) may be expected to possess the same properties.

In Section II solutions of Laplace’s equation are derived in a coordinate system appropriate for the description of rings. Certain preliminary results are then established, such as the potential of a thin hoop and that exterior to a heterogeneous ring. Section III contains a discussion of solidly rotating ring-shaped polytropes in equilibrium. The effects of a spherically symmetrical mass at the center of the ring is also treated. In Sec- tion IV, isothermal rings with and without central bodies are considered.

II. THE POTENTIAL OE RINGS

a) Coordinate System—Solutions of Laplace's Equation

We define a set of coordinates (r, </>, 0) by

x=(R-\-r cos 0) cos 0 , y = (Æ + r cos </>) sin 0 , z = r ûn <j>, (2)

where the origin of the Cartesian coordinate system (x, y, z) is at the center of the ring. The coordinate r is the distance from a reference circle of radius R (later chosen to be the major radius of the ring) ; 0 and <j> are polar angles. The coordinates are orthogonal, and, in addition, it is also clear from the line element,

fa2 = hr2 + r2b<t>2 + (7? + r cos <t>)2 dd2, 0)

that in the limit R/r-^- &>, (r, <£, Rd) reduce to the cylindrical polar coordinates (r, 0, z). The Laplacian corresponding to the line element (3) is

2 _ d2 , 1 + 2 ( r/R)cos <j> J. d . 1 d2

dr2 1 + ( f/iOcos <£ r dr r2 d<p2

(4) sin 0 1 d 1 1_ d2

1{r/R) cos (p Rr d<t> [ 1 + ( r/TOcos <£ ]2 i£2 d 02 *

Now let aZ = r , (5)

where a is a constant (unspecified for the present) and of the same order as a, the minor radius of the ring. And also let

ß=-. (6) r

Then, since a <£R (by hypothesis), we may be sure that

ß « 1 , (7)

1 An exception must be made of expressions for the potential where different conventions are used for cylinders and rings.

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and equation (4) may be expanded in terms of ß. Thus, if we write

V2 = Vo2 + ßVi2 + i^Vü2 + . - . ,

we find

v2 = Ji.+AA+l_£i Vo d^2"*" ? ^ £2 d02’

. a i . , a Vi2 = cos <¡> — — j sin —,

d ô V2

2=-I cos2 —-(-sinocos «Í.— + — , ..

(8)

(9)

(10)

(11)

We now seek solutions of Laplace’s equation,

V2 = 0 ,

of the form ^ + ßfii + ßV2 + • . • •

On substituting equations (8)-(ll) and (13) into equation (12) and equating the coeffi cients of ßm, we obtain

dVo , 1 dô , 1 dVo

(12)

(13)

0 d£2 ' ^ d? ' ^2 d2 ’

r> _ j. 1 • , aô , dVi , i aî i i aVi 0-cos<t> dç I sm ^ d^ d^2 ? d| ^2 d4>2 ’

0 = - I cos2 ^||5 + sin t cos 4> + ^ + ~ ^ sin * JJ

. d2\¡/2 . 1 dfa . 1 d2^

(14)

(15)

(16)

' d£2 ' £ ' f2 d</>2 ’

The general axisymmetrical solution to equation (14) which also has a plane of symmetry (as required for an equilibrium self-gravitating body) is

ô(£, <t>) =00 In £ + 00+ S (0>»S_’’,+ i,»>£+m)cos (17) m = l

where the (am, 6W) are arbitrary constants. Now, on substituting this result into equation (IS) we have the solution

iM£, 0) = In £ cos*-i£ cos(m+l) 771 = 1 (18)

+ Ôm£m+1 cos( 7» — 1 )<^] +¿>0 In £ + ?0+ (ÿm£_m+ ?m£+m)C0S 7», m = l

where the (ÿm, gw) are a new set of arbitrary constants. Similarly, if equations (17) and (18) are substituted into equation (16) the latter can be solved for and gives

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= jW2£2ln £ + £2) + (-£ffI£ ln H-^ói¿3)cos^In ^

Xcos 2^» + ^¿ [(^£^)aMr<“-2> + (^p)ó^“+2]cos ^

cos(f» + 2 )<£ + bmÇm+2cos(rn — 2)0] — In ^ cos0 (19)

00

~cos(m+ 1)0 + ?m£m+1 cos( w — 1)0] + r0 In ^ m = l

oo

+ s0+ ^ (i'OT?-”î+ im^+™)cos w0, ?n = l

where the (rm, Sm) are further constants. We recall that the solutions of Laplace’s equation derived in this section are general;

that is, apart from the easily removed restriction to axisymmetric bodies, they do not depend on the shape (ring or otherwise) of the equilibrium body which produces the potential.

b) Potential of a Thin Hoop

In order to apply the results of the last section to the case of rings, it will be useful to consider first the potential of a thin hoop. The potential at a point (r, 0) is

S3 s

(20)

where <r is the mass per unit length along the hoop, R its radius, and s the distance between the point of integration (0, 0, 6') and the point of observation (r, $, 0). In the ring coordinates defined by equation (2),

s2 = 4:R(R + r cos $) sin210' + r2. (21)

If we substitute equation (21) into equation (20), we can write

4G(tR rir/2 da 23

: r*/*

Jo TT where

a = ^ 6'

+ n2 sin2 a ) x/2 r

9 _ 4R2+ 4Rr cos(j>

k'K(k),

»1,

(22)

(23)

k2 = n-

1+ + 1, k'2=l-k2

1 + + «1.

and K(k) is the complete elliptic integral of the first kind. Using the expansion of K valid in the limit of small k', namely,

Z = ln^ + |(ln^-l)r2 + ^(ln^-|)^+... , (24)

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the potential is now

23(r, <*>) = 2G<r jln^-J^ln^- l) cos0

(25)

+TÍP [0 I- 3) + (3 1„ 4) cos 2*] + . . . ¡.

Finally, we transform to the dimensionless variables given by equations (5) and (6); and, noting that the mass of the hoop is M = lirRv, we may write the potential of a thin hoop in the form

$B(£, <#>) =^[[ln|_ ln S+ßf - ! (ln|- i) £ + §£ in cos <*>

+ ß* (2 ln|- pin ? + (3 In 4) P In ^]cos I* | (26)

+ •••]•

It must be kept in mind that this expression is appropriate only in the region r<KR.

c) The External Potential of Rings

Let us consider a slender ring in which the density depends in any arbitrary way on the coordinate r, but is dependent on <£ only to first- and higher-order terms in ß, and is independent of 6. Then, as long as it has a finite thickness a (p[r] = 0 for r > a), a region

a^r<^R

will exist, far enough from the ring so that its potential must approach that of the equivalent hoop, but near enough so that the expansions given earlier are still valid. Then we can match equation (26) with the solutions of Laplace’s equation derived in Section lia, and determine in this manner the values of some of the arbitrary constants (an, bn, pn, gn, . . . Thus fixed, these values are applicable everywhere outside the ring. In this way we find that the external potential of a slender ring must be of the form

S5°(P <t>) £+0 |[ai£_I+§£ In £ - è (ln|_ ^ £>os*

+ cosmtf> j +/32 j s0+ foin £ +Jg- (2 In- — 3) P~-|P In £ m=2 \ H /

(27)

+ ^i£-1 cos^)+[/>2£_2-¿ai+3^ (3 ln^— 4) p —T3^pln £]cos2<¿

CO

III. RING-SHAPED POLYTROPES

d) Introduction

The method to be followed here and in a later section on the equilibrium of isothermal rings is similar to that developed by Milne (1923), Chandrasekhar (1933) and Chan- drasekhar and Lebovitz (1962), in their investigations of the equilibrium of rotationally

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distorted polytropes. In their work, the undistorted equilibrium configuration was spherically symmetrical, the purpose being to find the effect of centrifugal perturbing forces in a slowly rotating body. In the present work, the undistorted body is taken to be one of the cylindrical polytropes, and the “perturbing forces” are due to rotation, the curva- ture of the clyinder, and the possible presence of a body at the center of the ring.

Before proceeding with the perturbation treatment, however, let us see what infor- mation can be obtained from a consideration of the virial theorem and the simplest (zero- order) calculations. In this approximation the mass Mr, internal energy U, and potential energy W, are simply 2ttR times the value per unit length of the corresponding quantities in the case of infinite cylinders (see Paper I, eqs. [7], [19], [23], [26]):

and

1 GMr2

7 — 1 47TjR ’

TJ7 GMr2\, SR , i i \ 1

H%--2TRLlnT+i<”+1)J'

(28)

(29)

where R and a are, respectively, the major and minor axes of the ring, and n is the polytropic index. In order to derive equation (29) from equation (26) of Paper I, it is necessary to fix the potential SSoi along the central circle of the ring (£ = 0) which was left undetermined2 in Paper I. This is done by requiring that the potential far from the line ? = 0 given by equation (28) of Paper I agree with the potential far from a hoop of mass Mr given by equation (26).

If there is also a central mass Mc about which the ring rotates, then there will be an additional term of —GMcMr/R in the potential energy. Thus, with the definition

the total potential energy is

»

w= GM* IttR

[in + 2ij + J(»+l)].

(30)

(31)

The kinetic energy is clearly

T = %WR?Mr, (32)

where O is the angular velocity of rotation. The virial theorem for an equilibrium body gives

2r + IT + 3(7- l)£/= 0. (33)

Now, if we substitute for T, W) and U from equations (32), (31), and (28) and solve for Q2, we find

£22

2irGpo ÖT77[1”T+2’’ + l(»-5>]’ (34)

where po and are the central and mean densities of the ring. We will return to this equation later. At this point it suffices to note that, if we define a parameter

_ Ü2

2'irGpo

2 There, the potential was normalized to be 2V at £ = 0.

(35)

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as a dimensionless measure of the angular velocity, then, since a « a, equations (6), (34) and (35) indicate that

v = Oiß2 ln ß) + 0(ß?r)) . (36)

This result is essential for the perturbation analysis; two small parameters (v and ß) appear in the theory, and without some prior knowledge of the relation between them, it would be impossible to construct expansions in terms of only one of them. Although rj need not be small, it is clear that ß2rj must be; this restriction will be tightened sub- sequently.

The equation of equilibrium for a uniformly rotating gaseous body is

Vÿ +V|02i32= 0, (37) P

where æ2 = #2 + y2 and 33" and 33e are the gravitational potentials arising from the body itself and from external matter (here assumed to be a spherical mass at the center of the ring). After substitution for p from the poly tropic equation of state (eq. [1]), equation (37) becomes

V[33i + 33e - K(n + l)p1/w + |î22gj2] = 0 . 08)

Equation (38) will be operated upon in two ways. We may take its divergence and use Poisson’s equation. We obtain

K(n + l)vV/n = -47rGp + 2ß2. 09)

Alternatively, equation (38) may be integrated from the central circle to the point of observation:

33¿ - SSo* + 35e - 330e - K(n + l)(P

1/w - p01/w) + éfí2(a2 - R2) = 0 , Ot»

where the subscript “0” designates values along the central circle. Transforming to the ring coordinates and expanding (33e — 33oe) and (gj2 — R2) in powers of r/Ry we may rewrite equation (40) as

33* = 3V + .£(»+ l)(p1/n— po1/") +^(SSo' —i22i?2)cos^ 2

R (41)

— i 33o'( 1 + 3 cos 20) +ß2JR2( 1 +cos 2<£)] + . .. .

In order to recast equations (39) and (41) in a more transparent form it is convenient to define the following variables :

p = X9” ; 33 = AV»*(K+1)F. («>

For complete polytropes it is expedient to take X = po. When we substitute equations (6), (35), and (42) into equations (39) and (41) we have

V20 = — 0" + i» (43)

and

V{ = FV + 0 — 1 + /3£[F0e - K^2)] cos <t> - + 3 cos 2<t>)

(44) +§(î'Æ-î)(1+cos 2<f>)].

The term in cos </> is of order ß or ßrj, whichever is greater; hence ßrj must be small.

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b) Pure Rings

1. Perturbation analysis.—The poly tropic variable 0(£, 0; ß) may also be expanded as a series in ß. In the present treatment, terms of order ß2 and higher will be neglected. Furthermore, we know that the zeroth-order term must be the cylindrical Lane-Emden function, since, when ß tends toward zero, the ring degenerates into an infinite cylinder. Thus we may assume

00

ß) =:0(£)+ßr^4o/o(£)+/i(£)cos<¿>+ Amfm( £)c°s w$+0(ß2) 1, (45)

having chosen to write the first-order term in the above form for subsequent convenience. The functions 0W(£), which are discussed in Paper I, are defined by the equations:

en"+jen' + dn»=0; 0„(O) = 1; 0„'(O)=O. (46)

The partial differential equation for 0 is

d20

d£2

i ae

£

i a2e ?2 a2

ae ^ i de . ^ ä|cos<i,-Iä^sm^

*— 0W+ z; (47)

from equations (8)-(10) and (43). Now we obtain ordinary differential equations for /m(£) by substituting equation (45) into equation (47) and equating the coefficients of ßl and cos m<l> to zero. From the coefficient of ß° we simply recover the first of equations (46). From the coefficient of ß1 we obtain

/i" + |/1, + (ra0“-i-|i)/1= -0'

and

fm"+j:fm' + (ne*-'-?ÿym = 0, m = 0, 2, 3, 4, 5, ... .

(48)

(49)

Boundary conditions follow from the requirement that the functions /m(0 be finite at the origin and that their values at that point be consistent with the normalization of both 0 and 6 to unity at the origin (cf. eq. [45]). It is found that

MO) = 0 and /w(0) = /m'(0) = 0 , m = 0, 2, 3, 4, 5, . . . . (so)

One more boundary condition on/i(£) remains to be specified. As will be shown later, the Ao and Am of equation (45) are all zero. Consequently, the choice of a second boundary condition on /i(£) is equivalent to a choice of the origin for the ring coordinates. If we wish to place the reference circle (£ = 0) at the center (most condensed part) of the ring, then we must choose

/i'(0) = 0, n>0. (si)

This is a natural choice for all polytropes having n> 0. However, in the case n = 0, when the ring is homogeneous (has no í<center,,), it is most natural to adopt a boundary condition such that the origin is in the center of the circle describing the ring’s cross- section. Formally, we require

MUl) = 0 , ** = 0 , (52)

where £i is the first zero of 0o(£).

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Closed solutions for /i(£) have been found in the same two cases (w = 0, 1) as was possible for 0(£) :

/l(0 = + ¿1=2, /* = 0 , (53)

and

/i(¿) = , ¿i = 2.4048, n=l. (54)

For other values of n, equation (48) requires numerical integration. Computed solutions for a range of values are tabulated in another paper (Ostriker 19646).

It is still necessary that continuity conditions on the gravitational potential and its derivatives be satisfied at the boundary surface of the ring. Let the surface be defined by the equation

00

HO) = £i + |3^/m cosm4>, (55) m = 0

where the C’s are constants to be determined. Now, with equation (55) and TayloFs theorem, we can specify the value of any well-behaved function G(£, (j>) on the surface:

G(H) =G(%i, (j>) +ß VzmcosOT¿ + 0(/32). (se)

Applying equation (56) to 0(£, <£), which is by definition zero everywhere on the surface, we find a set of equations relating the constants lm to previously defined quantities. Thus,

O = 0(E) = 0(£i)+ß I [^4o/o(¿i)+^o0/(fi)] + [/i(¿i)+^i0/(¿i) ]cos<£

oo (57)

+ [Amfm( £i) +/m0/( £i)] cos m = 2

gives

7 */"l(¿l) j 7 A- m j" m ( ^l') nO'J/lC /so\ l\ 6f {^ )* ß/ ^ ^ ^ ^ Oj 2, 3, 4, 5, ... . (58)

The gravitational potentials interior and exterior to the ring are determined by equations (27), (42), (44), and (45).

Vi(^<t>) = Vio o-l + 0U)+/? j[F0 S+AjoU)]

oo (59)

+ [/i(£) — è( vß-2) ^]cos<t>+ ^ [Amïm{0]c-osm<t>\ m — 2

and

F°(£,*) =e[ln!-ln £+,3 | [«i^1 - i (in l)^ + | Í In

oo (60)

+ ^ (am£~m)cos w<dT] , m = 2

where Vq1 has been separated into its zeroth- and first-order components

V0l = Vl0, 0 ßV'i0J 1 (61)

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and

J. OSTRIKER

GMr

Vol. 140

(62) TrR(n+l)K\'/n '

The potentials and their derivatives can now be evaluated along the boundary. We find

Vi(3) = V\ 0-1+/3 ji^o i + AoM^)+loe'(^)] + lfi(^)+he'(^)

CO

— è( fl|8-2)£i]cos <£+ ^ [^4m/m(?i) +lm0'(£i)]COS m<l>l ; m=2 ’

dFi

(S) = e'iM+ß \[Aofo,Ui)-l'>l;i-1e'(i;1)] + [U(!;i)-i(vß-*)

00

— ]cos^+ —ImÇr'O' (£l)] cos ?w<^ | ; m = 2

F«(S) =ein|-ein ii + ^<3 j (ln|- l)

00

+ §£i In ¿j — /i?i-1Jcos<l> + — ZOT|x_1 ]cos m| ;

and

dF° E) = -Ö^x^ + dO jZoíi-2+[-ai|x-2-i(ln|-2-lníi)

00

+ Zi¿i-2]cos<Z>+ [ — niam%i-m-'l + lmi;i-i]cosm<t>\> .

(63)

(64)

(65)

(66)

On equating the right-hand sides of equations (63) and (65) and of equations (64) and (66), we find, with the aid of equation (58),

e=-mil) = Ui,d'ai)i, (67)

F^ o = 1 + I £i0'(£i) I In37- and Fi0i = O. (68)

P Ç1 Also

== 0 , m = 0, 2, 3, 4, 5, . . . . (69)

Now, since two boundary conditions have already been imposed on these functions,

^4m = 0 , w = 0, 2, 3, 4, 5, . . . . (70)

Consequently (cf. eq. [58]),

lm = 0 , m = 0, 2, 3, 4, 5, ... ; am = 0, m = 2, 3, 4, 5, .... (7i)

And finally

<ii = i£i2+ 21 j £i2[]r[ —f1 (íi)] ; (72)

« = ^2[Ui0,Ui)l(ln^-|) + ¿/iUi)+/i,(|i)]. (73)

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Comparing equations (73) and (36) we see that our treatment of the relation between the two small parameters, v and ß, has been self-consistent.

The solution is completed by substituting equations (67), (68), and (70)-(73) into equations (45), (55), and (59). We can obtain explicit expressions for 0 and the internal and external potentials. In particular,

0(£, <*>; ß) = 0(0 + ßM£) cos 0 (74)

2. Equidensity surfaces.—The equidensity surfaces are found by setting 0 = constant; they are a family of circular tori. When viewed in a plane through the axis of rotation, the equidensity surfaces are circles. It can be shown that the circle of radius £ has its center displaced by the amount

ß MAI

Fig. 1.—The equidensity surfaces—p/p0 = 1.0, 0.8, 0.6, 0 4, 0 2, 0 0—seen in cross-section on a plane through the axis of symmetry (or rotation), for the case n — \ and ß = 0 1.

TABLE 1

The Distortion of Polytropic Rings*

3/2 8 10 12

€» .. 0 41583 0 54316 0 70748 1 1733 1 8757 2 8980 4 3515 6 3854 9 1987 18 312 35 037

* See eq (75)

outward from the center of the ring. Figure 1 shows the equidensity surfaces for the case n = _ 3

2> ß = 0.1. Now let

€n = ^ »(ll)

ÉlÔn'Ul) I (75)

€n then is a measure of the displacement of the boundary circle in units of its own radius. In Table 1, en is given as a function of the poly tropic index. As might have been expected, the distortion increases with increasing poly tropic index.

3. The liquid ring.—As an example and a check of the foregoing analysis of poly tropic rings, we shall consider the polytrope « = 0. The boundary surface is given by (cf. eqs. [55], [58], [71], and [53])

H(*) =2+/3/1cos<*>=2+/3ph£|^-y=2. (76)

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As mentioned earlier, this simple result arises from our choice for the second boundary condition on /i(£) (cf. eq. [52]). The expansion parameter is in this case

p R HiR = ii 2 R'

(77)

Now, from equations (73), (53), (77), and equation (13) of Paper I, the equilibrium angular velocity can be found. We have

Û2

2irGp (78)

which re-establishes the results of Poincare, Dyson, and others. The constant ai appearing in the external potential is given by equations (72), (53), and equation (13) of Paper I:

(79) 01 J

Finally, the internal and external potentials are

r, <t>) GM

^ 1in ?+K1 ~ i) ~4 GXO" 0 í ~15lc°s*

+o(ê)l and (80)

-ri„r]cos*+o(|¡)¡.

These expressions agree with those of Dyson.

c) Rings Rotating about a Central Mass

An expression was derived in Section III# (eq. [44]) for the gravitational potential within a poly tropic ring due to its own mass, when the ring is in the field of a (spherically symmetrical) central body. If we substitute into this expression the expansion for 0(£, ß) given in Section IIIZ> (eq. [45]) and note that

Tr . GMC r GMr V = n Fo RKV-/n{n+\) \-TrRK\l/n{n-\- 1)J\ lfr/ ^ ’

we find

Vi(^<P) = Vio 0-l + m)+i3 x + 40/o(0]

oo (82)

+ [/i( n + Mô4?-§ *0-2)] cos <¿.+ £um/mU) ]cos nut>+0(ß*)\. m = 2

Here we have again separated Fo* into its zeroth- and first-order parts. The form of the ring’s external potential is not changed by a central body, and consequently it is already given by equation (60). Now, as in Section 116, we will impose continuity conditions on the gravitational potential of the ring at its surface (defined by eq. [55]). To avoid needless complications, we consider just that part of the potential due to the ring itself rather than the total potential; the part of the potential due to the central body is cer-

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SELF-GRAVITATING RINGS 1079 No. 3, 1964

tainly continuous. Thus the problem is to match the potentials, given by equations (60) and (82), at the surface. Detailed calculations are, however, not necessary, since equation (82) differs from equation (59) only with respect to the constant coefficient of ß£ cos </>, and the matching of equations (60) and (59) has already been done in Section Hb. Hence we may simply adopt the results derived there (eqs. [67]-[73]), if we first identify the constant coefficients appearing in equations (59) and (82). Then we find

/ _ /i(£i) 1 lm = 0; (83)

w = 0, 2, 3, 4, 5, ... ;

01 — 4 1

2Ui0'(£i) £i2[^/it £i) ““//i £i) J i am — 0; (84)

<2= -M'(£i) = l£i0'Ui)l;

8 FS o= 1+ I £i0'Ui) I In^ï and F*0 i = 0. Psi

Identifying the constants, we find

Qrj - iv(rj)ß-2 = = O)0“2,

or with equation (85)

v(r}) = ß21 £i0'(£i) \ 2rj + v(rj = 0) .

Now, equations (73) and (87) give

V(v) = Uifl'Ui) l(ln-A+21?-f) + ^/1(£1)+/i'(£1) .

(85)

(86)

(87)

(88)

The results of this section are at first somewhat surprising, since it appears that in calculations correct to the first order in ß, introducing a central body does not alter the equilibrium configuration of a ring, but merely requires the ring to rotate more rapidly. The explanation can be found in the short cut of the analysis just taken. Both the centrifugal force and the gravitational force of the central body can be derived from potentials of the form

potential = constant r cos 0 ,

the constants being of opposite sign in the two cases. Consequently, the effect of a central body can be exactly compensated for by an increase in the rate of rotation.

As previously mentioned, this treatment has assumed that rj is of the order of or smaller than ß-1. Gunnar Banders’ (1942) discussion of rings with central bodies, based on Laplace’s work on Saturn’s rings, implicitly assumes3 that the mass of the ring is many orders of magnitude smaller than the mass of the central body, and thus his con- clusions cannot be compared with the results derived here.

d) Integral Properties of Polytropic Rings

The most important integral properties associated with a ring are the following :

Mass, Mr= f pdr ; (89) Jv

3 He ignores all terms in the self-gravitation but the zeroth and keeps up to second-order terms in the external field.

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Moment of inertia, 1=1 P®2d r Î Jy

Potential energy, W = f p^dr — f p%$edr ; J V J y

Internal energy, u^-^r f p£dr=-ï-rf ppl/ndr . y — \ Jy p y— 1 jy

Vol. 140

(90)

(91)

(92)

Other important quantities, such as the angular momentum and the kinetic energy, can be derived from the above four integrals.

To begin, it will be shown that the first-order contributions to these integrals are all zero. It is convenient to use polytropic variables and ring coordinates. We note that to the first order in ß

dr = r(R + r cos </>) drd<t>dd = jR2an2(l + cos 0) ^d^dc^dd ; (93)

= (R + r cos </>)2 = J?2(l + 2ß% cos 0) ; (94)

p = \0n = \(0 + ßfi cos <£)n = \(0n + ßndn~lf\ cos <¡>) ; (95)

and

Sß^Cw+^ZXVnF^Cw+DZXi/^ö+l ?10,(|1)|ln-g|-+^(/1-!;/3-^)cos«?>]. (96)

If equations (93)-(96) are substituted into equations (89)-(92), we see that all the integrals to be evaluated are of the general form

Q= f d<t> f cos 4>]d%. (97) ''O •'O

Here A(£) and B(£) are of order unity, ^4(0 being the product of all the zeroth-order terms. Substituting in equation (97) for the boundary E(0) from equations (55) and (83) gives

Q= rr^+ßB^costWdi; 'S 0 '/0

/27r /»fi+jíU, COS0 / [A(t;)+ßB(!;)cOS<t>]dct>d!;+Om

- -'f* (98)

/f, /*2t /• fi+zSZ. COS0 AU)di:+fo d<t>J(i

X{A(^)+ßB(^)cos<t>+[A'(^)+ßB'(^)cos<l>](^-^)+. . . + }di-+Om,

where, in the second integration, we have expanded the integrand in a Taylor series about £ = £i. By performing successively the £- and ^-integrations, we have

Q=2tt f(l A ( £ )<f £ +/3 f^dthAi^) cos<*> + 0(d2) '/Q ‘/0

= 2ir /'il^4 ( £)d£+0(/32),

(99)

as was to be proved.

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Now the required integrals are simply written down:

Mr = 47T2i?aw2X f*l£dnd£ = 47r2£an

2\ | £i0'( ) | ;

I = 4ir2R*an*\ t dH £ =RWr = 4T2R*an2\ | £i0'( £i) | ;

(100)

(101)

W= -2w2Ran2(n+l)k\1+1/n

(102)

= -2xy?aB2(W + l)¿\1+V"|£10'(£x)p[ln^ + 217+p^^^V+1íá£]

= -lirtRanHn+DKV+^l 1:0'( ^i) 12[ln ^|-+2,? + | (« + 1 ) ] ;

and

u-~r[ i îiô'(îi) i* (Ui,,;ii)i=/y+'iit) (103)

= (^|) x^^X'+V" 1^0'( ^ ) I 2 ;

to the first order in ß. Here we have used the values of the first moments of 0n and 0n+1 given in Paper I,

equations (19) and (22). In evaluating W, equations (82)_and (86) were also used. The kinetic energy of the ring is

T = J/ß2 = IwWRWMïie'tii) 1 + 0(ß2) . (104)

Since the first-order contributions to £/, T, and W are zero, the expression for the angular velocity, which was derived in Section lia (eq. [34]) with the aid of the virial theorem by naively neglecting the distortion from cylindrical symmetry, is correct to the first order in ß. At first glance, however, equation (34) appears to differ from the results found in the perturbation analysis by matching potentials at the surface. The expression found there for ß2 (cf. eq. [73]),

ß2

2'TrGpo In -^+ 2ij —§-+

a 1

M'Ui) ^/l'( ^l) +T~ /l( ?l) J I , (105)

explicitly contains the first-order perturbation function /i. It can be shown, however, that the two results are identical. If we multiply the defin-

ing equation for/i(£) (eq. [48]) by ^'(Ö, we find

£0'/i" + 6'fi + SM”-1 0' - r20')/i = —£0'2. do*)

Differentiating the defining equation for 0(£) (eq. [46]), we obtain an expression for ndn~lQf, which, when substituted into equation (106), gives

- £= £d’Jx" + 07/ - ?0"7i - 0,7i 07/ + 07i + £ rA) • <“>*>

By integrating now from £ = 0 to £ = £i we obtain

[//(¿i) Ul/Ul)| ■ (io8)

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Equation (108) can be simplified with the aid of an identity derived in Paper I (eq. [22]) :

i(w+l) Uiö'Ui) |2= /'%U0n)á¿= - /'%0'2d£. (109) *^o */o •/o

The second and third equalities follow from equation (46) and an integration by parts. Equations (108) and (109) give

[/i,(Si)+^/iUi)] = !u+im10'u1)i. (110)

On substituting equation (110) into equation (105), we recover equation (34), as was to be proved.

The most important integral properties will be rewritten here in convenient physical form. First we have

Mr=2ir2Ra2(p), I = MrR\ and Û2 = |^[ln 217 - 1 (5 - w)] . (m)

From these we derive

4:TrR

GMrzRY. 8Æ T9 T0^ GMr6R[, XR t / r ,1 /2 = 72fí2 = ln 2t] — 1(5 — n) ,

2 ?r l -I

(112)

(113)

8R

and

U

2tR

1 GMr2

7 — 1 47r2?

The total energy of the ring is

E=T+W+U;

GM 2 „ GMr^, &R , ^ , 1 , , 1 1

(114)

(115)

(116)

(117)

If we eliminate the minor radius of the ring a with the aid of the equation for the angular momentum (eq. [113]), we find

F= J2 GMr

2 /3y — 4\ 2R2Mr 47rR V 7 - 1 /

(118)

We notice that neither the poly tropic index nor the central mass appears in this equation,4 and that, as is often the case, the value 7 = § occupies a special place. For 7 > J the energy is always negative, and cooling causes a decrease in R. But for 1 < 7 < I the energy is positive for sufficiently large R, and cooling will produce a general expansion of the ring.

4 Eq. (118) can be proved under quite general conditions. Essentially it depends on the requirement that 1.

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IV. ISOTHERMAL RINGS WITH CENTRAL MASSES

a) Perturbation Analysis

Given the isothermal equation of state,

„ kT -r p = —— p = Kip , pm0

the equations analogous to (39) and (41) are

KiV* In p = -4xGp + 202

and

5ß 4= Sßo' + In -^- + ^ ( «o6 - 02i22 ) cos ^ P0 Iv

- ( 1 + 3 cos 2 ^ ) + 02i?2 ( 1 + cos 2 ) ] +

(119)

(120)

(121)

Now let 1/2

\47rG!po/

Equations (120) and (121) may now be written

y2^r = — V ;

p = Xe^r=p06'ir; and ^S — KiV. (122)

and

= IV _ * + ^[F0e - 1(^2)] cos 0 + 3 cos 2$)

(123)

(124) + |(^2)(1 + COS )] + . . . .

We expand A? in powers of ß; to the first order it is 00

^(£>$jß) =:^(£)+ß£AUgo(£)+gl(£) COS 0 + Am grad) COS m<j>^ , (125) m = 2

where yf/^) is the cylindrical isothermal function defined by the equations

t(0) =ip'(0) = 0.

The solution of equations (40) is (cf. Paper I, Sec. Ill)

,K£) =21n(l+f!2).

(126)

(127)

From equations (122), (125), and (127), we see that, at a great distance from the ring, the density falls off at approximately the rate

~ 64 ¡y> i p~pojr> l»i- (128)

Thus the mass of an isothermal ring is finite, but its extent infinite. The equation for ’P, analogous to equation (47), is

a2^ , i a* , i a2* , ^ /a* , i a* . A d£2+£ a£ + £2a4>2+/3 (a£cos |dAinv e v'

(129)

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If we substitute equation (125) into equation (129), we find

gi"+j + = —t'

and

gm"+j gJ + Çe^- gm = 0 , w = 0, 2, 3, 4 . . . . (i3D

The boundary conditions are

gm(0) = gj(0) = 0 , w = 0, 1, 2, 3, . . . , (132)

by the same reasoning as that given in Section Illft. The potential within the ring is derived by substituting equation (125) into equation (124) and separating Va‘ into its zeroth- and first-order components:

FKS, <*>) = FVo-^U)+d CO (133)

X j[ FVi—.4o£o(£)] + [ —giiiO+ifCPi?-3»ß-2)]cos0— ^ ,4mgm(£)cos m<t>\.

Vol. 140

(130)

Here we have defined

so that

p-GMr ttRK i ’

(134)

(135)

At a great enough distance from the central circle of the ring, (£ )>> 1 but still £ <3C ß~l) the potential must tend to that of an equivalent hoop since the mass of the ring is finite. That is (cf. eq. [26]),

FHí, «í>)=Tjln|-ln í+/3[-é (ln|-l)+Uln?]cos0+O(/32)|, £» 1 • <136>

Now, from the boundary conditions and the requirement that the potential ultimately be given by equation (136), all the except g\ can be eliminated. If we substitute for ^(£) in equation (131), the differential equation for g0 becomes

go"+| go' + jyq7|ygy¿ g0 = °- <I37>

Equation (137) has no solution satisfying the boundary conditions (cf. eq. [132]) except the trivial one,

go(£) = 0, (all£), (138)

which we must therefore adopt. For w > 2, equation (131) becomes

1 r 1 gm"+J [( J _|_ £2/gy2 —■£?]£»> = 0> (139)

It can be shown from analysis of the asymptotic (£ « 1 and £ » 1) properties of gm that

«»(0 = ^”, m>2, £—>o°. a«)

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Substituting equation (140) into equation (133), we observe that the potential will not have the correct asymptotic form (given by eq. [136]) unless

Am = 0 , tn>0, (141)

which we must therefore adopt. Thus it is necessary to consider only one of the perturbation functions—gi(£)—which satisfies equation (130). This equation can be integrated to give

x. (142)

Now if we let z^i + ie,

we find, after some transformations, that equation (142) becomes

gl( 2 ) = - 8V2 (^)1/2[3 -2z + z (J^l) In 0 + 2L( z) ] .

Here L(z) is the dilogarithmic function (for tables see Fletcher [1962]) :

r / \ — f‘ Inz'dz'

Series solutions for gi(£) are easily obtained:

gx(£)= + £<V8;

and

gi(S) = -£lnU2+2£-! (ln|^)+|

£ > V 8

(143)

(144)

(145)

(146)

(147)

Now, having obtained solutions for the functions yK£) and gm(£) we may substitute in equation (133) and find the resultant potential. In the limit £ » 1, we have

V'iÇ, <t>) = 4 [[(JF’o.o+Jln 8) — In £ (148)

(|ln 8+l + £»;8~2—5-P»?)£ + l£ln£]cos0}]] .

But this must be the same as the potential of the equivalent hoop given by equation (136). Comparison then shows that

4=P = GMr

F*o 0 = 2 In

tRKí

8

or Mr = ÍttRK, = 4ttK

G fxwiçfj RT

ß2' and

F ^ i = 0;

8 ,= 8d2(iln^-l + ^)

With the aid of equations (122) and (149) we find that

Mr

( 47t )2Rpi and ß2 =

Mr R2 (4x)2J?Vo'

(149)

(ISO)

(151)

(152)

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It is useful to consider, as a characteristic minor radius of the ring, ay 2, the radius within which one-half the mass is contained :

#1/2 =: :=:: OL% \/8 , (153)

from which we obtain 1 #1/2

V 8 R

Here we have taken £i/2 from Paper I, equation in an isothermal ring is, to the first order in ß,

P = Po£ * = Po (l + £2/8)2 [1

(154)

(61). Finally, the density distribution

— ßgi(£)cos 0]. (15S)

b) Integral Properties of Isothermal Rings

One can show by the same type of argument as given in Section Hid that the first- order terms in ß vanish, and that consequently zeroth-order calculations of the integral properties will suffice.

Thus the moment of inertia about the rotation axis is

I = MrR2 4tR^T

IJlMqG (156)

The equilibrium angular velocity of rotation is found by transforming equation (151) into physical variables with the aid of equations (35), (152), and (154):

ti2 GMr' A SR 2tRz Vn #1/2

2^-2 )•

(157)

One can show that this is in fact the limiting form taken by equation (111) (for polytropic rings) as w > oo. This result, which is not apparent from the form of equation (111), can be proved with the aid of formulae derived in the Appendix to Paper I. The kinetic energy of the ring is

GMr2 A SR

47t£ Vn #1/2 2 77-2

)• (158)

The angular momentum J is given by

J2 = J2Q2 GM/R A SR 2ir \ ai/2

217-2 )■

(159)

The potential energy may be found by substituting for ív and p in equation (91):

W = GMr2f, SR

2TrRVn «1/2 (160)

The internal energy is simply

Ki r KiMr_ 1 GMr2

7_ ! JPdr -T_ 1 7_ ! 47rR- (161)


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One can verify that these expressions for T, W, and U satisfy the virial theorem (eq. [33]). The total energy of the ring (cf. eq. [116]) is

E = GM,

4îri? ( In

8i? 2»? + 1

It)- (162)

«1/2 T

If we eliminate from equation (162) with equation (159), we find the simple result

J2 GMS /Sy - 4' E =

2R2Mr 4:ttR my (163)

As expected, the total energy is not explicitly dependent on the temperature, the half- radius, or the presence of a central mass, and is given by the same expression as the total energy of poly tropic rings.

V. CONCLUDING REMARKS

Although the most obvious application of this work is to stellar systems (cf. Sec. I), it may also be useful in giving a quantitative foundation to discussions of rings in other astronomical contexts. For example, the observations of certain planetary nebulae (cf., among others, Minkowski and Osterbrock 1960), the extended atmospheres of some early-type stars (cf. Underhill 1960), and perhaps extragalactic radio sources (cf. Maltby and Moffet 1962), have caused rings to be mentioned as possible models. The integral relations given in equations (111)-(118) and (156)-(163) are of a simple and perhaps readily applicable form.

But for all these applications it will be necessary to investigate the dynamical stability of rings as well as their equilibrium; and as in the equilibrium studies, it will be useful first to consider cylinders, the normal modes and oscillations of which will be treated in succeeding papers.

I should like to thank Professor S. Chandrasekhar for suggesting this problem, and for his continuous guidance during the course of the work.

The computations were supported by National Science Foundation grant GP-9-75.

REFERENCES

Chandrasekhar, S. 1933, M.N., 93, 390. Chandrasekhar, S., and Lebovitz, N. 1962, Ap. /., 136, 1082. Dyson, F. W. 1893, Phil. Trans. Roy. Soc., London, A, 184, 43-95, 1041-1107. Fletcher, A. 1962, An Index of Mathematical Tables (2d ed.; Oxford: Blackwell Scientific Publishers Ltd.),

1, 551. Jeans, J. 1929, Astronomy and Cosmology (2d ed.; Cambridge: Cambridge University Press), Sec. 300. Laplace, P. S. 1789, “Mémoire sur la théorie de Fanneau de Saturne,” Mém. Acad. Sei. Limber, D. N. 1964, Ap. /., 139, 1251-1266. Maltby, P., and Moffet, T. 1962, Ap. J. Suppl., 7, No. 67, 141-163. Maxwell, J. C. 1859, “On the Stability of the Motions in Saturn’s Rings,” Adams Prize Essay reprinted

in Scientific Papers of James Clerk Maxwell (Cambridge: Cambridge University Press), 1, 286-377. Milne, E. A. 1923, M.N., 83, 118. Minkowski, R., and Osterbrock, D. 1960, Ap. J., 131, 537-540. Ostriker, J. 1964a, Ap. J. (in press) . 19646, Ap. J. Suppl, (in press). Poincaré, H. 1891, summarized in F. Tisserand, Mécanique Celeste (Paris: Gauthier-Villars), 2, chap. xi. Randers, G. 1940, Ap. J., 92, 235-246. . 1942, Ap. J., 95, 88-111. Underhill, A. B. 1960, Stellar Atmospheres, ed. J. L. Greenstein (Chicago: University of Chicago Press),

pp. 411-433 Vaucouleurs, G. de. 1959, Handbuch der Physik (Berlin: Springer-Verlag), 53, 275-311.


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