90

ASTROFtZtt~

T HE EQUILIBRIUM O F A S T A R DURING A PHASE TRANSITION Z. F. Seidov A s t r o f i z i k a , VoL 3, No. 2, pp. 189-201,

1967

UDC 532. 877 The equilibrium of a star whose matter is undergoing a phase transition is considered, and the behavior of the stellar parameters near the transition point is investigated. It is shown that dM/dR is everywhere continuous. A derivation is given of the critical density ratio at phase transition, which leads to loss of stability immediately after the appearance of the new phase. Polytropic stars undergoing a phase transition and having ~ = 0 and-fi = 1 are considered in detail, and analytical expressions are Obtained for the star parameters' as a function of the centrat equilibrium.

i. INTRODUCTION The ultimate fate of stars is an important problem in astrophysics. One of the alternatives is a dense cold star with a mean density of i06-i08 g/cm 3 (white dwarf) or i014-i016 g/cm 3 (neutron star) [1-4]. The dependence of mass on the central density for cold stars, which is of major evolutionary significance, has attracted the attention of many workers. It has been shown that one of the reasons for the existence of a maximum mass (with finite central density) in the case of white dwarfs is "inverse fi-deeay" or "neutronization" [5-8]. (Relativistic effects [19-11] are also important. ) The following point has not, however, been taken into account so far. During neutronization, when nuclei with given atomic number and charge (Ai, Z I ) and a certain electron Fermi energy are transformed into other nuclei (A2, Z2), the electron density and hence the electron pressure P0 (the pressure of nuclei is negligible) remain constant, but the density changes f r o m Pi to P2, w h e r e q = P2/Pl = A2Z1/A1Z2. The function P(p) thus has a r e g i o n Pi -< P -< P2 in wh i ch P = Po = const. * In such c a s e s the d e n s i t y in an e q u i l i b r i u m s t a r is no l o n g e r a continuous function like the p r e s s u r e , and it is t h e r e f o r e m o r e c o r r e c t to r e g a r d al l the s t a r p a r a m e t e r s as functions of the c e n t r a l p r e s s u r e Pc and n o t of the c e n t r a l d e n s i t y Pc. T h e p r e s e n t p a p e r d e a l s with the b e h a v i o r of the m a s s M, r a d i u s R, and t o t a l e n e r g y e as we p a s s f r o m s t a r s with P c < Po to s t a r s with Pc > Po, when a r e g i o n of a new d e n s e r p h a s e a p p e a r s at the c e n t e r . A n a l y t i c a l p r o p e r t i e s of the a s y m p t o t i c b e h a v i o r a r e e x a m i n e d f o r s m a l l a m o u n t s of the new p h a s e . It is shown that f o r any P(p) the d e r i v a t i v e s dM/d@, dR/d@, d e / d e do not have a d i s c o n t i n u i t y at the p h a s e t r a n s i t i o n point (for q ~ 1.5). In t h e s e e x p r e s s i o n s @ = - G M / R (Section 2). We s h a l l d e t e r m i n e the

* A n o t h e r exaxnple of p h a s e t r a n s i t i o n l e a d i ng to this f o r m of P(p) is the c o n v e r s i o n of c o l d h y d r o g e n f r o m a m o l e c u l a r into an a t o m i c c r y s t a l [ 1 2 - 1 4 ] , wh i ch o c c u r s in l a r g e h y d r o g e n p l a n e t s .

critical value of q for which the s~ar becomes unstable when the new phase is formed (Section 3). This critical value is q = i. 5 for any equation of state. Two examples will be considered of stars undergoing a phase transition and having a simple equation of state, i. e., an incompressible fluid (n = 0) mad the polytrope n = I, where n = [d In P/d In o - I] -i. In these simple cases it is possible to obtain a full analytical solution of the problem (Sections 4 and 5L The case of an arbitrary equation of state will be considered in Section 6. It will be shown that the curves constructed by a number of authors using numerical calculations are incorrect near the point of phase transition. 2. A T H E O R E M ON THE C H E M I C A L P O T E N T I A L O F MATTER When T = 0, the e q u i l i b r i u m co n d i t i o n f o r a s p h e r i c a l l y s y m m e t r i c s t a r , which is u s u a l l y w r i t t e n in t h e form

can be expressed

dP

G [, M (r)

dr

r"2

(1)

in the form of a conservation

H(I,)

+ ~ (r) = a) = c o , ~ t ,

law (2)

w h e r e q0(r) is t h e g r a v i t a t i o n a l p o t e n t i a l , and H(p) is the s p e c i f i c en t h al p y , o r h e a t c o n t e n t , d e f i n e d by

dH=

2-dP, p

H=

E f t ) -v- -'~- , '/

(3)

where E(p) is the specific energy. In the absence of an external field, H is also the analogy of the chemical potential; the addition of a unit mass to the matter contained in 1 cm 3 changes its energy by

--•p--p

(p,E(~))

d,~,

'

p

In the presence of a field, @ is a natural generalization of the chemical potential and is equal to de/dN. From the general variational principle it follows that de/dlV[ does not depend on the position of the region of the star to which the unit mass has been added [15], and hence 4~ is constant throughout the star. The properties of the star are determlined by specifying @, in exactly the same way as by specifying M. Since the derivatives dM/d$, dR,/d4~, de/d@ do

ASTROPHYSICS

91

not s u f f e r a d i s c o n t i n u i t y when a new p h a s e i s p r o duced, it follows that the d e r i v a t i v e s d M / d R and d e / d M a r e a l s o c o n t i n u o u s (at the point w h e r e p(P) is d i s c o n t i n u o u s ) . We s h a l l p r o v e now t h e s e r e s u l t s . C o n s i d e r the a p p e a r a n c e at the c e n t e r of the s t a r of a s m a l l r e g i o n of the new p h a s e , with r a d i u s r t << << R. Doth H and ~ a r e e v i d e n t l y c o n t i n u o u s on the b o u n d a r y r 1. We s h a l l c o m p a r e the s o l u t i o n c o r r e sponding to this s i t u a t i o n with the boundai~y s o l u t i o n , o r its a n a l y t i c a l c o n d i n u a t i o n without the new phase. The change in q?(r D due to the f o r m a t i o n of the new p h a s e is 4~

.

0 aM

.

r1

Pl)

-- G - ~ -

.

--

.

8

r,

4-

rl

(5)

N e a r the c e n t e r of any s p h e r i c a l l y s y m m e t r i c body in e q u i l i b r i u m

P(r) --P(O)

,(,-) = ~ ( o ) + 2=o H(r)

dqb

9 ACp = 2 ~ G

~

= Po

(< 1

1

Pe

) .Ml

~

:,

rl,

2~.G

~

f'c r%

(9)

--~

~'[ r~,

"? (r,) =:: qb - - H,,. (10)

T h i s follows f r o m Eq. (9), and f r o m the fact that o n t h e phase b o u n d a r y P = P0, H = H0, w h e r e P 0 and H0 a r e the v a l u e s of P and H at the c e n t e r of the s t a r b e f o r e the new phase b e i n g s to f o r m . In the " , " s o l u t i o n (continued a n a l y t i c a l l y , without a new phase) c o r r e s p o n d i n g to Eq. (8), we have

9- (rl)'•

H(O)

[% r -,,,

The l a s t equation follows f r o m Eq. (2). To find the c o n n e c t i o n b e t w e e n Ar and r t , we p r o ceed as follows. In the "+" s o l u t i o n we have

dP~ C o n s e q u e n t l y , AG ~ r l 2 [ m o r e p r e c i s e l y , A@ ~ (q _ -- 1.5)r12; see below, f o r m u l a (15)]. The c h a n g e s in M, R, and e a r e then i n d e p e n d e n t of the c a u s e of the given change A@ = @ - @0. The a d d i t i o n a l m a s s in the r e g i o n of the new phase is AM ~ (q - 1) r~ ~ (AG) 3/2. The i n c r e m e n t s in r a d i u s , i n t e r n a l p o t e n t i a l , and total e n e r g y due to this a d d i t i o n a l m a s s a r e also p r o p o r t i o n a l to r~:

2r.G .... -~ f,~ r-,

~--qb

0 - -

1

9~G,o

dPr

- -

~

(11)

In the "+" s o l u t i o n (p+(ri) d i f f e r s f r o m ~ _ ( r J by the additional d e n s i t y (P2 - P J at the c e n t e r , which c o r r e s p o n d s to the a d d i t i o n a l m a s s :

The e x p a n s i o n s of M, R, and s a r e t h e r e f o r e of the f o r m a + b(G - G0) + c(G - G0)3/~, w h e r e only the c o e f f i c i e n t c depends on the p r o p e r t i e s of the new p h a s e for q ~ 15. H o w e v e r , c does not c o n t r i b u t e to the d e r i v a t i v e d/d@ ] G0, which should thus be the s a m e b e f o r e and a f t e r the p h a s e t r a n s i t i o n . T h i s concludes the p r o o f for q ~ 1.5. In the d e g e n e r a t e c a s e , when q = 1 . 5 , AG will be p r o p o r t i o n a l to a h i g h e r p o w e r of r 1 (to r~), and the t h e o r e m is then no l o n g e r valid.

(12)

~ (r0 = ~ _ ( r , ) - ~ t ~ . ~ - ~ ) ~ f .

F r o m Eqs. (10), (11), and (12) we obtain the r e q u i r e d relationship 1 h

dPc dq)

_"

5r

2 =G ~ ( 3 h - - 2h) r f

(13)

o r , u s i n g Eqs. (10) and (13) 3. CRITICAL DENSITY I~ATIO We s h a l l c o n s i d e r that the f u n c t i o n s P(p), H(p), P - Pl and M(Pc), R(Pc), e(Pc), Pc -< P0 a r e known. To c a l c u l a t e G, we a s s u m e that on the s u r f a c e of the s t a r p = 0, P = 0, H = 0, and ~ = - G M / R , so that

cp = - - G M ( P r

(Pc) : q) (Pc).

3 - 2q .___dP" + ' q2 dCo

,-~., .

q=

(14)

rq

F i n a l l y , u s i n g the c o n t i n u i t y of d/d@, we a r r i v e at the i m p o r t a n t r e l a t i o n s h i p

(7)

C o n s i d e r a s t a r with a s m a l l r a d i u s rt of the new p h a s e , m a s s M, r a d i u s R, t o t a l e n e r g y e, and G --= @ + AG. A c c o r d i n g to Section 2, both i n the "+" a n d in the " - " solution

*o ._Xc~, R = R o 4 - ~ - ~dR . [ ~ . Sq), M = Mo T. - d~M

~ = % + ~--~ d~ a,o - "0.

dP~ _ = dOp

(8)

dM ] dPr I -

3-- 2q q"-

dM dP~

_ 9

(15)

The s a m e is t r u e f o r R and e. Thus the d e r i v a t i v e s of M, R, and a e x p r e s s e d as f u n c t i o n s of Pc have a d i s c o n t i n u i t y at the point c o r r e sponding to the f o r m a t i o n of the new phase. F u r t h e r m o r e , M, R, and e a r e monotonic functions of Pc and G only- when q < 1.5. When q > 1 . 5 , the d e r i v a t i v e s with r e s p e c t to Pc change sign at the c r i t i c a l point,

/4STROFIZiKA

92

and the function of I, have a turning point. In p a r t i c u l a r , f o r q > 1.5, the function hi(Pc) has a sharp m a x i m u m at the c r i t i c a l point, which leads to a }oss of the s t a r ' s stability. The relationships given by Eq. (15) w e r e obtained in [15] for the case where p(P) was continuous for P < Po, but a s s u m e d the value qPl = const, where Pi = = P(P0) for P > P0, which c o r r e s p o n d s to the c a r e when the new phase is an i n c o m p r e s s i b l e fluid. Howe v e r , in point of fact f o r m u l a (15) is valid f o r any equation of state before and after the phase transition. It is thus e a s y to see that Eq. (9), on which the d e r i vation of Eq. (15) is b a s e d , is valid f o r any equation of state. 4. I N C O M P R E S S I B L E

FLUID, n = 0

where r I and NIi are the radius and mass of the new phase. The pressure distribution in the core is then of the form

(is) and i n the outer shell (the region of the old phase)

P=4~Qp~(C

M Mo

I

I

(R):'

I

"~o

( -

1--

=/_R? ta/

q

).,

x

2q:

I

0.8

I

n-O I

__^q---I/ --.~ \, z

I

M _I Mo .

__=(R 1 +

r = 1x ~ , % ' Po

6q ~

=0'

(20)

,~). ~, Po :

I

Let M0, 1%0, e0, and ~0 be the p a r a m e t e r s of a s t a r consisting of an i n c o m p r e s s i b l e fluid of density" Pi, the c e n t r a l p r e s s u r e of which is equal to the threshold p r e s s u r e P0 f o r phase transition. Then for a s t a r consisting of this phase, with a c e n t r a l p r e s s u r e Pc = P0 + a [18], we have

-~o]

a x~':~ (X/ +

5(q-1

P-o0/

tPo)

I

'to

Fig. i

(I +

q'

F r o m these general e x p r e s s i o n s we can readily obtain the asymptotic behavior for sma}l e/t)~:

*/*o

M Mo

Gg}

:'o/

__--2q--3 a ) R q~ ~ Ro

1.0

z~2V.A %X;"

.

.

;/R\ 5 , . 5 q--1/R\~/ 'o = t R 0 ) + 2 e (')R.

0.8

i -6-

The constants C and D and thus ali p a r a m e t e r s of the star can readily be found f r o m Eqs. (17}-(19). * We shall give only the final result:

In this case, first examined by R a m s e y [17], the density does not depend on the external pressure.

l

D

-;- -

Ro

~

r qb~

3(2q--3} 2q ~ 1

2q

3 q2

9 Po

- - : - ~ RI Ro

a Po

a ~o

!.

2q ..... 3 2r

,~ Po (21)

5(2q .... 3) ~. 2q ~ P~I

Expressions analogous to Eq. (15) c a a be obtained f r o m (16) and (21). When q = 1. 5, the t e r m p r o p o r tional to a / P 0 in Eq. (21) b e c o m e s equal to zero. Using Eq. (20) we then find that M M~

~'",

~-~]

~ _ = ( 1 - b - ~ao /'~':, 9 (16) ~o

3 / R \ :i 2 \ Ro /

4

-

! Y 2 P~

Ro

.

Suppose now that a sphere of the new phase (incomp r e s s i b l e fluid of density Pz) is p r e s e n t at the center of the star. On the b o u n d a r y between the p h a s e s

F r o m F_Ats. (16) and (22) we deduce that the gunctiona N(R), e(M), etc. have a b r e a k at the transition point for q = 1.5 (the s a m e is true of M(~), e(r etc.).

= [ 3~, 'l'"', "=" ,,, = Po,

t i . ap~;J

*P(r} and M(r) are continuous on the interphase

4'~

M = M, = ~-

~,-,~,

(17)

boundary

M(r)

p dr

dr

; see Section 2~

ASTROPHYSIC

S

93

5. THE P O L Y T R O P E n = 1

..... 2 ( q - - l )

.M Mo

,1 \

~

,

-Ro

Po /

"q

1 =--"hq

1,

(sin.q_ .qcos.ra) '

sin "ta

~q:~

F o r a s t a r with the e q u a t i o n of s t a t e P = Kip 2 [18]

"%,'q

arctg 1 ....

(l - - q tg'q)'

q

The a s y m p t o t i c b e h a v i o r f o r s m a l l c~/P 0 is a s follows:

7-~o) v:' )L__.%1 "-~'Po

~-~-=(lqb,

(23) R Ro

S u p p o s e now t h a t at the c e n t e r of the s t a r t h e r e is a r e g i o n of a new p h a s e with the equation of s t a t e P = K~p ~. We then h a v e 9_L = ( K~ "~'".

(24)

1

q---1 ( 3.%~!:~ 3 r.q:' \ Po ]

M Mo

1

3.-2q 2q ~

:t Po

(30) dO = 1 ~ 3 .... 2q Oo 2q:

....

~ Po

~ %

j

1

3-2q q~"

a Po

E q u a t i o n s (26) and (30) l e a d to r e l a t i o n s h i p s analogous

to Eq. (15). The s o l u t i o n f o r the c o r e i s of the f o r m [

f~ == ?r sin

i

i

i

E,

t.2 ;

(,

?

(25)

o1.0

On the c o r e b o u n d a r y p = P2, r = r t, ~ = ~t, and therefore

~E 0.8

1

=

sin ~

1

_

sin (r,/'~.A

~

,

(26)

o*

r~/i~

i i 0.6 I

f r o m Which we o b t a i n the c o n n e c t i o n b e t w e e n t h e r a d i u s r~ of t h e new p h a s e and a . The m a s s of the new p h a s e is

n=O

io. I

0.8

110

_. I

RIRo

r,

Fig. 2

1141= 4 ~ ~ ~ r'dr = ~J 0

9.- 4

=~.3~ .,. [1-~--~--")'", ~ (sin % \

"%:co~ "q).

to/

(27)

When q = 1 . 5 , the t e r m p r o p o r t i o n a l to a / P 0 in Eq. (30) b e c o m e s equal to z e r o . We t h e r e f o r e o b t a i n t h e following e x p a n s i o n t e r m s (it is m o r e c o n v e n i e n t to expand in t e r m s of 71 and not in t e r m s of ~ / P 0 ) :

The s o l u t i o n f o r the s h e l l o u t s i d e the c o r e is of the form . sin(~:, = ~.

8)

,

r =

/&)',

,~i,

,~ = ~

..... 1 RO

q- -__~1r~, 3 v,q ~ - I

w h e r e the X and 6 a r e c o n s t a n t s d e t e r m i n e d f r o m the m a t c h i n g c o n d i t i o n s . F r o m Eq. (27)-(30) we f i n a l l y obtain M _ ~,/q ( 1 + ~'~, M0 sin ( ~ ' l / q -- ':) -~ff,]

qbo __ 1

2=

R

= 1 q.-'--~

RO

sin ('~l/q -- ~) ('@q)e

['2 (= -i ~ - -

"q.'q) +

sin'~ (~l/q - - ';)

Q-sin('2Q/q- - 2 g ) ] + 1 2 r.

I - F a / P o [2 ~a __ sin (2.r,~)] . q~

1

2q - 3__ "r,~. q - 1 6q ~ 3 ~,q:~ .% '

, (28)

71

%

M -~o

q:)o

2q

3

,,f.

(3t)

6q ~

From Eqs. (23) and (31) we find again (as for n = 0) that when q = I. 5, the derivatives d/dP c and d/de have a discontinuity at the transition point. 6. DISCUSSION

OF

RESULTS

Figures 1-4 show plots of M(~) and ]M(R) for various n and q. The asymptotic behavior near the phase transition point for q = I. 5 and q ~ i. 5 have a break at the point (i, i). The stable and unstable branches are indicated (crossed once and twice, respectively), the transition from stable to unstable branches and vice versa is determined by the extremum of M (but not of R or ~!).

94

ASTRO:F~ZIKA

w,

I

I

"T

1.6

n=l 1.4

1.5

1.2

s Z

1.0

1.0

k

0.8 0.5 0.5

0.6 I

l

I

I

" [

I

I

I

1.0

I

I

I

1,5

*/r

I

I

0.4

0.6

,

~

..

0.8

RIRo

Fig. 3

Fig. 4

~<1

R

~ R i

t 84

I I

P,

I

i

I

t% Fig. 5. B e h a v i o r n e a r the p h a s e t r a n s i t i o n point for v a r i o u s n (q > 1.5),

|

1.0

I

1.2

ASTROPHYSIC S The p o l y t r o p e n = 1 exhibits a v e r y i m p o r t a n t d i f f e r e n c e when c o m p a r e d with an i n c o m p r e s s i b l e fluid. A s t a r c o n s i s t i n g of two i n c o m p r e s s i b l e fluid p h a s e s does not r e a c h a m a s s m a x i m u m for q _< 1 . 5 , and t h e r e f o r e does not lose s t a b i l i t y . A s t a r with the equation of state P = Kp 2 a f t e r the f o r m a t i o n of a new p h a s e with q < 1.5 l o s e s s t a b i l i t y only a f t e r the t r a n s i t i o n point for a finite a m o u n t of the new p h a s e , and it does so f a s t e r the c l o s e r q i s to 1o 5. F o r s m a l l a m o u n t s of the new p h a s e , the a s y m p t o t i c b e h a v i o r [ f o r m u l a s (23) and (30)] does not depend on the p r o p e r t i e s of this p h a s e , i . e . , on the f o r m of PiP) at p > P2, but only on the p r o p e r t i e s of the old p h a s e (and of c o u r s e on q), :This m e a n s that :the r e s u l t s o b t a i n e d f o r n = 0 a n d n = 1 on the a s y m p t o t i c d e p e n d e n c e of the s t e l l a r p a r a m e t e r s in the c a s e of two p h a s e s n e a r the t r a n s i t i o n point a r e v a l i d for any new phase. The r e l a t i o n s h i p s b e t w e e n the d e r i v a t i v e s of the s t e l l a r p a r a m e t e r s on the r i g h t a n d o n the left of the p h a s e t r a n s i t i o n point do not even depend on the p r o p e r t i e s of the old p h a s e , but only on q. To s u m -

marize: (I) q ~ 1.5. The curves of RIM), M(~), etc. do not in this case have breaks or discontinuities, but only turning points. The R(M) curves given by some authors [8, 12, 14] are therefore incorrect. The RiPc), M(Pc) , and ~(Pc) curves have a break at the transition point, and the R(pc), M(Pc) , and ~(Pe) curves are in any case not physical, since the density as the center of the star is not a continuous variable when the threshold reaction of phase transition is taken into account. (2) q = I. 5. Here the considerations of Section 3 do not apply, and a. s p e c i a l a n a l y s i s is r e q u i r e d for e v e r y i n d i v i d u a l c a s e (see Section 4 and 5). (3) q > 1.5. A c c o r d i n g to Section 3, the m a s s has a s h a r p m a x i m u m and a d i s c o n t i n u i t y in d M / d P c at the t r a n s i t i o n point. Since dR/dM s h o u l d be c o n t i n u o u s e v e r y w h e r e , the d e r i v a t i v e d R / d P c should change s i g n when d M / d P c c h a n g e s sign [the s a m e is t r u e of e(Pc) ]. T h i s m e a n s that i m m e d i a t e l y a f t e r the t r a n s i t i o n point the r a d i u s of a s t a r with effective p o l y t r o p e e x p o n e n t - n < 1 should d e c r e a s e , while for > 1 it should i n c r e a s e . C u r v e s for a r b i t r a r y - ~ a r c g i v e n in Fig. 5. (4) When q < 1.5 t h e r e is no m a s s m a x i m u m at the t r a n s i t i o n point. A f t e r the f o r m a t i o n of the new phase, the M(Pe) curve rises to a maximum, as a result of which the s t a r loses its stability, and the rate at which this occurs increases with increasing

95

q and n" Salpeter's curves [8] for cold white dwarfs, which take neutronization into account (q < i. 5 for all the reactions considered) and ha~e a mass maximum at the transition point, are therefore incorrect. We note in conclusion that the statement about the continuity of the derivatives of M, R, and e with respect to ~, and about the critical value q = i. 5, is valid not only in Newton's theory of gravitation but also in general relativity. In the latter case M is of course M 0, i, e., the sum of the rest masses of the particles making up the star (nucleons), and e is the total energy of the star, including internal, gravitational, and M0c ~. RE FERENC

ES

i. Ya. B. Zel'dovich and L D. Novikov, UFN, 84, 377, 1964. 2. Ya. B. Zel'dovich and I. D. Novikov, UFN, 86, 449, 1965. 3. Gravitiation and Relativity [Russian translationl, IL, Moscow, 1965. 4. B. K. Harrison, K. S. Thorne, M. Wakano, and J. A. Wheeler, Gravitational Theory and Gravitational Collapse, Chicago, 1965. 5. E. Shatsman, Astron. zh., 33, 800, 1956. 6. G. S. Saakyan and E. V. Chubaryan, Soobshch. Byur. obs., 34, 99, 1963. 7. E. E. Salpeter, Ap. J., 134, 669, 1961. 8. T. Hamada and E. E. Salpeter, Ap. J., 134, 683, 1961. 9. S. A. Kaplan, Ueh. zap. LGU, seriya fiz.mat., 15, 109, 1949. i0. S. A. Kaplan and I. A. Klimishin, Tsirkulyar astron, obs. L'vovskogo Un-ta, no. 27, 17, 1953. ii. S. Chandrasekhar, Phys. Rev. Left., 12, 114, 437, 1964; 14, 241, 1965;Apo J., 140, 417, 1964. 12. W. H. Ramsey, M. N., 113, 427, 1951. 13. A. A. Abrikosov, Astrom zh., 31, 112, 1954. 14. A. A. Abrikosov, Voprosy kosmogonii, 3, ii, 1954. 15. Ya. B. Zel'dovich, ZhETF, 42, 1667, 1962. 16. M. J. Lighthill, M. N., ii0, 339, 1950. 17. W. H. Ramsey, M. N. 110, 325, 1950. 18. S. Chandrasekhar, Introduction to the Study of Stellar Structure [Russian translation], IL, Moscow, 1950. 6 December

1966

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