Download Central Temperature and Density of Stars in Gravitational Equilibrium

504
Progress of Theoretical Physics, Vol. 36, No. 3, September 1966
Central Temperature and Density of Stars in
Gravitational Equilibrium
Katsuo TAKARADA, Humitaka SATO and Chushiro HAYASHI
·Department of Physics, Kyoto University, Kyoto
(Received April 20, 1966)
The relation between the central temperature and the central density is investigated for
a polytropic star and a polytropic (or isothermal) stellar core, which are in gravitational
equilibrium. The ranges of the temperature and density cover the cases of non-degenerate
and degenerate, non-relativistic as well as relativistic, electrons. It is shown that, when the
mass of the star or of the core is smaller than a certain critical value which depends slightly
on the polytropic index N and is equal to Chandrasekhar's limiting mass for N =3, the
central temperature takes a maximum value depending on the mass at the onset stage of the
electron degeneracy.
§I.
Introduction and summary
Throughout the life of a star, the central temperature and density changes
to a considerable extent. The features of the over-all evolution of the star are
determined mainly by how far the central temperature rises in its whole life
and accordingly how far the synthesis of the chemical elements proceeds 1n
the interior.
For example, it is well known that a star smaller than about one-tenth of
the solar mass is unable to undergo the hydrogen burning phase, since at a
certain stage of its pre-main-sequence contraction electron degeneracy sets in
and afterwards the central temperature has to decrease. 1), 2),B) On the other hand,
if a star is massive enough, it will follow all the phases of nuclear burning,
from hydrogen burning to the formation of iron, developing a shell structure
composed of zones of different chemical compositions. The evolution of such
a massive star has already been studied up to the central carbon-burning phase
for a stellar mass of 16 M 0 . 4 ), 5 ) It has been found that, in the stages before
the onset of carbon burning, electrons in the central region are relativistically
degenerate if the energy loss by neutrino emission is taken into account.
In order to investigate the over-all features of stellar evolution and their
dependence on the mass, it will be necessary first to know the over-all variations in the central temperature and density which are allowed for stellar configurations in gravitational equilibrium. Strictly speaking, these variations depend
on the details c;>f the development of the shell structure. In this paper, however,
an attempt is made to find an approximate relation which holds between the
central temperature and density of a star or of a stellar core, which satisfies
the following simplified conditions.
Central Temperature and Density of Stars in Gravitational Equilibrium 505
First, for the case , of a homogeneous star a polytropic relation is assumed
to hold between the pressure and density throughout the whole interior. Secondly, for a star having a shell structure our consideration is confined to its
innermost homogeneous core. The pressure at the core boundary is assumed
to be so small compared to the central pressure that the existence of the outer
regions can be neglected (the single star approximation) . This assumption is
well satisfied for most of the advanced evolutionary phases which have so far
been studied for stars of different masses. 4 ),B) Furthermore, the polytropic relation or the isothermal condition is assumed to hold in this core.
Under the above approximations, a relation between the central pressure
and density is first obtained from the condition of gravitational equilibrium
alone, as described in § 2. This relation depends on the star or of the core,
but it is rather insensitive to the polytropic index.
A general expression for the equation of state of a relativistic electron gas,
which include the cases of non-degeneracy and degeneracy, is given in § 3.
By means of this expression, a relation between the central temperature Tc and
the central density Pc is obtained, in § 4, as a function of the mass.
The numerical results are illustrated by Fig. 2. As is well known, Tc
increases as p/13 at low densities where the electrons are non-degenerate. When
the mass is smaller than a certain critical value, which is equal to Chandrasekhar's limit when the polytropic index is three, Tc attains a maximum value
because of the onset of electron degeneracy and afterward Tc decreases with
increasing Pc. On the other hand, Tc increases without limit when the mass is
greater than the critical value.
Using the above results, the critical stellar masses for nuclear burning are
estimated in § 5. Finally, a discussion is given of the limitations on the gravitational equilibrium, i. e. of the onset of instability caused by the creation of
electron pmrs at high temperatures or inverse beta-decays at high densities.
§ 2.
Relation between the central pressure and density
We consider spherically symmetric configurations of stars which are in
gravitational equilibrium. If a polytropic relation, Pocpi+l/N (N is a constant),
holds between the pressure and the density throughout a star or throughout a
stellar core, we have7)
M = -4n asP ~2dfJ
r
.
c
d~,
(1)
where Pc and Pc are the central pressure and density, Mr is the mass contained
in a sphere of radius r, and fJ (~) is the Emden function of index N which is
defined by fJN = p/ Pc and r =a~.
If the polytropic relation holds throughout a star of mass M, we have
from Eq. (1) a relation between Pc and Pc as a function of M, i.e.
506
K. Takarada, H. Sato and C. Hayashi
(2)
with
(3)
where the_ subscript 1 denotes the stellar surface and f is a constant which
depends only on the polytropic index N, as shown in Table I. The values in
Table I have been calculated using the surface values, ( -ed ()/~) 1 , of the
Emden functions tabulated by Chandrasekhar.7) It is to be noticed that the stellar radius is infinite for N = 5, but f takes a finite valve. In realistic stars
of homogeneous composition, th~ polytropic index is not generally constant,
but its local value lies mostly in the range between 1.5 and 3.0 and thus the
value off i~ limited within ·a relatively narrow range, as is seen from Table I.
Table I. The value off for a polytropic star of index N.
N
f
0
0.8060
0.5
0.6379
1.0
0.5419
1.5
0.4780
2.0
0.4310
N
f
3.0
0.3640
3.5.
0.3379
4.0
0.3146
4.5
0.2925
5.0
0.2687
2.5
0.3942
For a polytropic index greater than 5, there is no surface boundary of a
star. In this case, the U- V curve of an Emden solution starting from the
center (U = 3, V = 0) attains a maximum value of V, as shown in
·Fig. 1, and afterwards approaches
the point
N-3
N+1'
U=~-
V=2N+1
N-1'
(4)
by. spiraling around it. For an
evolved star in which a shell structure has been developed by nuclear
burning, the innermost homogeneous core will, in some cases, be
described approximately by a polytrope of index greater than 5.
For a polytropic core as mentioned above, the same expressions
as Eq s. (2) and (3) are obtained
from Eq. (1), if the core mass
is taken as M and the core boudary is denoted by the subscript 1.
The value off depends generally
u
Fig. 1. The U- V curves of the Emden functions of
polytropic index N which start from the center,
U=3 and V=O. The dots represent the points
where V take maximum values.
Central Temperature and Density of Stars in Gravitational Equilibrium 507
on the position of the core boundary on the U- V curve in Fig. 1, but
its change along the U- V curve is very small in so far as the core
boundary lies in a region of small U (where the single star approximation as
mentioned in § 1 is valid), i.e. in a region near or exterior to the point where
V takes a maximum value. The Emden functions and the values of f have
been computed on the Kyoto University KDC II for several values of N. For
N = ?O, the table of Chandrasekhar and Wares 8 ) has been used. The values of
f, P/Pe, U and V at the points of maximum V are tabulated in Table II.
Again the value off is insensitive to the polytropic index.
Table II.
The boundary values of a polytropic core of index N.
N
5.0
6.0
8.0
20.0
f
0.2777
1.182X 10-3
u1
0.2687
0
0
V1
6
0.2809
5.9126 X 10-3
0.5845
3.744
0.2863
1.851X 10-2
0.8623
2.907
Pt!Pe
0.3563
4.509
I
I
00
0.2904
3.113X10-2
1.000
2.5176
i
I
Finally, we consider a partially-degenerate isothermal core in which relativistic effects are taken into account. By using the equation of state for an
electron gas which will be described in § 3, the solutions for the jsothermal
core, which are specified by the two parameters a=kT/mc 2 and ¢e (the degeneracy at the center), have been tabulated by Hayashi and Cameron9) for the case
Table III.
a
.10
c/Je
The boundary values of a partially-degenerate isothermal core.
0.00
I
20
l
100
log (Pel Jl.E)
f
0.4594
5.5x10-3
2.1x10-2
-0.5
0.3450
7.6765
Pt!Pe
PtfPe
¢1'
U1
V1
a
c/Je
log (Pel Jl.E)
f
Pt!Pe
PtfPe
¢1
ul
V1
0.4690
0.4772
7.7X 10-4 2.7x 10-6
5.8x 10-3 1.1 X 10-4
-0.8
-2.5
0.1104
0.0021
14.0714 71.1787
I
10
I
10
6.305
0.4264
5.4x10-3
1.9X 10-2
-0.3
0.3657
6.9980
I
20
6.883
0.4125
6.6X 10-4
4.2x w-3
-0.5
0.1204
11.6236
20
I
100
I
10
I
I
100
I
100
0.10
0.08
I
20
8.073
6.531
7.357
5.475
5.978
5.986
0.3839
0.4253
0.4031
0.4345
0.4448
0.4429
5.1x10-3 7.8x10-4 1.8x 10-6 5.2x10-3 7.4>< J0-4 1.2X 10-6
I1.9X 10-2 5.4x10- 3 5.7x 10-5 1.9X 10-2 4.9x1o-3 3.6x10-5
-2.0
-0.5
-0.6
-2.1
-0.4
-0.5
0.0021
0.1245
0.3526
0.3393
0.1212
0.0023
I
7.1877 12.2379 45.2222
7.4120 13.0217 52.6272
0.06
I
0.04
0.02
)1
10
I
20
8.524
6.542
7.148
0.3759
0.4214
0.4034
1.1X 10-6 6.1xl0-3 6.4X 10-4
2.9x w-5 2.0x10-2 3.9X lQ-3
-1.8
-0.4
-0.1
. 0.0021
0.3971
0.1242
41.1895
11.1298
6.8668
I
100
10
I
20
7.362
8.857
6.734
0.3965
0.3718
0.4161
8.8x10-7 6.3x10-3 6.4X 10-4
2.4X lQ-5 2.1x10- 2 3.8x w- 3
-0.3
-1.7
0.0
0.4095
0.1283
0.0019
38.6477
10.7278
6.6984
100
I- - - - - - -
I
I
9.121
0.3694
7.8xl0- 7
2.1X10- 5
-1.6
0.0019
36.9007
508
K. Takarada, H. Sato and C. Hayashi
of negligible ion pressure. In terms of these solutions, the expressiOn for
which replaces Eq. (3) is given by
f= (4rr)lf3 G(cfJe, a)/cp1 2;3{F(¢e, a)} 4; 3 ,
where cp
IS
(5)
a variable defined by Eq. ( 4C. 26) in reference 3),
Mr =
J1E
2
I.
e.
(2m kT) 3f4
(2m G) 3; 2 cp '
h3f2
4n
f
I-J2
the subscript 1 denotes the core boundary as before, and G and F are functions
defined by Eqs. (11) and (12) in § 3. The core boundary is chosen as a point
of maximum V on the U- V curve, as mentioned previously. The results,
which are given in Table III, show that the value off lies intermediately between
the cases, N = 1.5 and 3.0, of a polytropic star.
§ 3. Pressure and density of an electron gas
When relativistic effects are taken into account, the number density n and
the pressure P of a gas composed of free electrons are given by7)
(6)
(7)
with
(8)
where JlE and cpkT are the mean molecular weight and the thermodynamic
potential of an electron, respectively.
The above expressions can be re-written as
n=A F(cp, a), P=AkTG(cjJ, a),
4n (2mkT) 3
A=~
ha
;
2
'
a = _kT_ = _T_____:_(°K_)
mc 2 5.93 X 109
(9)
(10)
'
where
(11)
(12)
Central Tenzperature and Denszty of Stars zn Gravztatwnal Equtlzbrzum 509
are the generalized Fermi-Duac functiOns which reduce to F1; 2 ( cjJ) and (2 /3)
X F 3; 2 (¢), respectively, In the non-relativistic limit, a =0
We have a general
relatiOn
F(¢, a)
=
aG (¢, a) ja¢
(13)
In the case of non-degeneracy (¢~ -1), the functiOns F(¢, a) and G (¢, a)
can be expressed In power senes
00
F= (2a) -1!2~ ( -1) n+1n-1en<t+1fa) K2 (n/ a) '
(14)
n=l
00
G = (2a) -1!2~ ( -1) n+1n-2en<t+1fa) K2 (n/ a) ,
(15)
n=1
where K 2 IS the modified Bessel functiOn In the case of extreme degeneracy
(¢~ 1), the asymptotic expressiOns for F and G are giVen by
(16)
where
x = (2a¢ + a2 ¢2) 1f2 '
(18)
r/
f(x) =3ln{x+ (1 +x2 2} +x(2x2-3) (1 +x2) 112
-
8x5/5- 4x7/7 + x 9/3 + 0 (x11 )
2x4 -2x 2+3 ln 2x-7 /4+ 0 cx- 2)
1
for x~1
(19)
for x~1
For the Intermediate case of Incomplete degeneracy, the functiOns F and G
have been tabulated by Hayashi and Cameron 9) by direct IntegratiOn with an
IBM 7090 In the ranges and Intervals, cjJ = -4 0 (0 1) 20 0 and a= 0 00 (0 02) 0 38
§ 4.
Relation between the central temperature and density
Now, a relatiOn between the central temperature and density will be obtamed
from Eqs (2) and (9)
To take mto account the arb1trarmess of llE and the
small vanatwn off, the mass In Eq (2) IS measured In units of
_ ( -3)
Mon
1 2
1
(
1 \
~-
/lbH)
2
(
-he-- )
8Gf
3 2
1
_
1 26 M0
11E2 f3!2
(20)
It IS to be noticed that for N = 3, M 0 IS exactly equal to Chandrasekhar's limitIng mass The radiatiOn pressure, which IS Important for a mass greater than
M 0 , IS mcluded, but the IOn pressure, which Is not Important for a gas of heavy
elements, Is neglected for simplicity Then, using
510
K. Takarada, H. Sato and C. Hayashi
P=AkTG(cjJ, a) +a T 4/3, a=8n 5 k4/15c 3 h 3
Eq. (2)
IS
(21)
,
expressed in the form
M
Mo
=
2 1114 ~c 314 {G(cjJ, ac) + (n /~5v~)ac~ }~~
3 112
{F(cfJc, ac)} 2
4
2
(22)
8
I
I
I
I
20
I
I
I
I
7
-0.5
3
4
5
-02 -0.1 -0.05
6
7
8
9
10
.Qog(fc/JJE) ( g/cm3)
Fig. 2. The curves of constant M/Mo in the central temperature-density diag~am. The dashed
curves represent the states of given cPc· The stars become unstable when they enter the shaded
region where r is smaller than 4/3 because of the creation of electron-positron pairs and the
decomposition. of. iron nuclei.
Central Temperature and Density of Stars in Gravitatzonal Equilibriwn 511
Using the asymptotic expressions and the table for the functions F and G,
the values of ac and </Jc (and hence the values of Tc and Pc) which satisfy Eq.
(22) have been calculated for a variety of values of M/M0 • The upper limit
of M/ Mo has been taken to be 10 so that the presence of electron-positron pairs
can be neglected. The results are shown by the curves of constant M/Mo in
the Tc- Pc diagram in Fig. 2. If M 0 is a constant, each of these curves represents the variation 'of Tc with Pc in a star or a stellar core of given mass.
In this case, Tc increases as p/13 as long as the electrons are non-degenerate.
The behavior of Tc in the degenerate region is defferent depending upon
whether the mass is greater or smaller than Mo
For a ~ass greater than M 0 , Tc increases continuously with Pc though the
gradient is relatively small at the stage of incipient degeneracy. This effect of
the degeneracy is conspicuous when the mass is close to M 0 , as shown by the
curve of log (M/ Mo) = 0.01 in Fig. 2. On the other hand, for a mass smaller
than Mo, Tc attains a maximum value which depends on the mass, as shown
m Table IV, and decreases with further increase of PcTable IV. The values of Tc,
• The maximum values of
Pc and ¢c at
Pc ar~ added
the tops of the
for comparison.
Tc-Pc
curves for constant JI.;JjM0 •
II
log(M/M0 )
I!
-------
logTc
The stage of
maximum
¢c
----
----------
max1mum
density
I;
9.096
6805
4.87
log(pc/.Up;)
temperature
---------
-o.o5
-0.10
-0.20
-0.30
-0.50
-1.00
8.968
6.388
3.90
8.768
5.882
3.17
8.601
5.525
2.85
8.291
4.974
2.58
7.593
3.873
2.30
--~------
-- - - - - - - - -
7.562
log (Pel .UE)
-
--------
7.029
6.442
5.459
6.050
4.254
When the mass is sufficiently smaller than M 0 , we can use the non-relativistic approximation and at the same time the radiation pressure can be neglected. In this case, we have from Eqs. (9) and (22) a relation between Tc
and Pc of the form
Tc = fJ (</Jc) (
M)4;s o
K,
Mo
(23)
where the functions fJ C</Jc) and rJ C</Jc) are tabulated in Table V. On each of
the curves of constant M) M 0 in the Tc- Pc diagram, Tc attains a maximum
value at </Jc = 2.276. Relativistic effects become important for M/ M 0 >0.3 as
shown in Fig. 3.
Table V.
-4.0
I
log 0
log cr
7.7896
1.8619
1
1
-2o
•
I
I
The values of {) (¢) and cr (¢) in Eq. (23).
o.o
-------- ---------
8.3406
1
3.5398 .1
8.7661
4.9503
2.o __ I
__!-~~~-- J -~~_]__2?___
8.9152 . 8.9168
5.7411
5.8032
I
I
8.8771
6.0467
I
I
10.0
15.0
1
8.74781
6.1942
- - -
8.6283
6.2411
8.4726
6.2692
512
K. Takarada, H. Sato and C. Hayashi
llog·fc(''K)
75 3~------~------~5--~---L~s---L--~7
flog UUJ.lE) (g/cm3 )
Fig. 3. The curves of constant M/M0 in the Tc-Pc
diagram. The solid and dashed curves represent
the relativistic and non-relativistic cases, respectively.
§ 5.
Application and discussion
For a star to experience a certain kind of nuclear burning in its life, its
mass has to be greater than a certain critical value. The results in § 4 will be
used in the following to estimate the minimum masses for burning of hydrogen,
helium, carbon and so on.
The effect of the ion pressure, which has so far been neglected f<?r simplicity, is important for the accurate determination of the critical masses, especially for burning of hydrogen and helium. This effect can be included by
replacing G (</Jc, ac) in Eq. (22) by G C</Jc, ac) + (!1E/ !lr) FC</Jc, ac)' where /lr is the
mean molecular weight of the ions. The result is to lower the curves of constant M/Mo in Fig. 2 and to shift their maximum points to slightly higher
densities.
For the case of hydrogen burning, we abopt !1E = 1.24 and flr= 1.40, corresponding to the chemical composition X=0.61 and Y=0.37. Further, we take
f = 0.478 which corresponds to a wholly convective structure (N = 1.5) in the
pre-main-sequence stage. For helium burning, we take /1E = 2, /lr = 4 and f = 0.453
which corresponds to a point-source model of constant opacity and negligible
radiation pressure. 10 ) The temperatures of hydrogen burning and- helium burning are simply taken to be 5 X 10 6 °K and 1.0 X 10 8 °K, respectively. The final
results for the corresponding minimum masses and the central densities are given
in Table VI. These values are 1n good agreement with the results of other
authors. 1 ). 2 ).s).u)
For carbon burning, we adopt flg=2, !1r=12 and f=0.453 for the above
Central Temperature and Density of Stars in Gra·vitational Equilibrium 513
point-source model. The burning temperature Tc has been determined as follows.
The minimum mass and the central density are calculated as a function of Tc
for the case of non-relativistic degeneracy. Then, the luminosity is obtained
using the mass-luminosity relation of the above stellar model, which is given by
__L =21.2/14 ( M)
Le
!Co
(24)
s'
Me
where the opacity /Co is taken to be 0.2 crn 2/ g and the effect of degeneracy Is
adequately included by taking 1/,u=1/,uEA+l/J1r with A=0.7.
On the other hand, the nuclear luminosity is obtained using Eq. (4D. 4)
in reference 3) and the nuclear energy production rate given by Reeves 12 ) where
the fractional abundance by mass of carbon is taken to be 0.5. The above two
luminosities agree with each other for Tc = 5.5 x'10 8 °K. The minimum mass
for this temperature and the corresponding central density are given in Table
VI. It is to be noticed that in the case of relativistic degeneracy the critical
mass will be somewhat larger than the tabulated value since the polytropic
index in the convective core is slightly larger than 1.5 and correspondingly the
value of f is smaller than the adopted value.
Table VI.
The minimum masses for nuclear burning.
Relativistic degeneracy
Helium
Carbon
1Xl08
5.5Xl08
0.30
0.72
3.66
3.88
Non-relativistic degeneracy
5.004
0.32
4.16
3.38
6.265
0.89
2.66
2.902
4.939
5.925
Recently, Deinzer and Salpetey1 3 l have obtained 0.697 Me for the carbonburning critical mass in the case of non-relativistic degeneracy and no neutrino
loss. This low mass value is probably due to their incorrect normalization of
the physical quantities in their stellar model. For example, the values of Tc
and IIc in Table I of their paper gives for Pc only a half of the tabulated value.
If the universal Fermi interaction between electrons and neutrinos is assumed,
the energy loss by neutrino production becomes important at temperatures above
3 X 108 °K. In the the phase of gravitational contraction, the existence of this
energy sink has the effect of lowering the temperature gradient in the central
region. Therefore, the polytropic index in this region increases, at least, when
the electrons are non-degenerate. Thus, the value off is lowered correspondingly
and the value of M 0 increases according to Eq. (20). This means that the
curves of constant mass in the Tc- Pc diagram are lowered in the high ternperature region where neutrino loss is effective. Thus, the minimum masses
for the onset of carbon burning and burning of the heavier elements are increased.
Now, we consider the phase of carbon burning with neutrino loss. Since
514
K. Takarada, H. Sato and C. Hayashi
the energy liberated by carbon burning is mostly carried away by neutrinos,
more energy is required to maintain a stellar luminosity. Therefore, the central
temperature becomes slightly higher than in the case of no neutrino loss. In
the ranges of temperature and density relevant to carbon burning, the photoneutrino process is predominant. The minimum mass and the corresponding
central temperature are estimated by the same method as in the case of no
neutrino loss, except that the neutrino luminosity is taken into account. We adopt
the same value off as in the case of no neutrino loss, since f is determined essentially by the structure or the polytropic in,dex of a relatively inner region of a
star. 14 ) In this manner, we obtain Tc = 7.0 X 108°K, and Mmm = 1.06 M 0 (nonrelativistic case) or 0.82 M 0 (relativistic case). The minimum mass given by
Deinzer and Salpeter13 ) is not in agreement with the above value 1.06 M 0 ,
probably for the same reason as stated previously.
So far, our consideration has been confined to the stable configuration of
gravitational equilibrium. It is known that instability sets in when the adiabatic
exponent, r = (d ln P 1d ln p) ad, of a gas becomes smaller than 4/3 or when the
effects of general relativity become important. At high temperatures, r becomes
smaller than 4/3 owing to the creation of electron pairs and the decomposition
of Fe56 into He 4 and neutrons, as shown by the shaded region in Fig. 2, which
is taken from the results of Sato. 15 ) Another region of instability is known to
exist at high densities and at relatively low temperatures. At densities above
1010 g/cm3 , the electron Fermi energy is higher than about 10 lVIeV and most of
the stable nuclei are subject to inverse beta-decays. Further, Chandrasekhar
and Tooper 16 ).l 7 ) have recently shown that an instability due to general relativistic
effects occurs at a density of about 1011 g/cm3 •
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
S. S. Kumar, Astrophys. J. 137 (1963), 1121.
C. Hayashi and T. Nakano, Prog. Theor. Phys. 30 (1963), 460.
Y. Osaki, Publ. Astron. Soc. Japan 15 (1963), 336.
C. Hayashi, R. Hoshi and D. Sugimoto, Prog. Theor. Phys. Suppl. No. 22 (1962).
C. Hayashi, Stellar Evolution (edited by R. F. Stein and A. G. W. Cameron, Plenum
Press, New York, 1965), p. 253.
C. Hayashi, R. Hoshi and D. Sugimoto, Prog. Theor. Phys. 34 (1965), 885.
S. Chandrasekhar, An Introduction to the Study of Stellar Structure (The University of
Chicago Press, Chicago, 1939 and Dover Publications, New York, 1957).
S. Chandrasekhar and G. W. Wares, Astrophys. J. 109 (1949), 551.
C. Hayashi and R. C. Cameron, unpublished.
J. P. Cox and R. T. Giuli, Astrophys. ]. 133 (1961), 755.
]. P. Cox and E. E. Salpeter, Astrophys. J. 140 (1964), 485.
H. Reeves, Stellar Structure (Vol. 8 of Stars and Stellar Systems, edited py L. H. Aller
and D. B. McLaughlin, The University of Chicago Press, Chicago, 1965), p. 113.
W. Deinzer and E. E. Salpeter, Astrophys. J. 142 (1965), 813.
D. Sugimoto, Prog. Theor. Phys. 32 (1964), 703.
H. Sa to, Prog. Theor. Phys. 35 (1966), 241.
S. Chandrasekhar and R. F. Tooper, Astrophys. J. 139 (1964), 1396.
S. Chandrasekhar, Astrophys. J. 140 (1964), 417.