Nuclear Physics A364 (1981) 527-532 @ North-Holland Publishing Company

MASS LIMITS FOR NON-DEGENERATE WHITE DWARFS*

GERD SCHMIDT, HANS-THOMAS ELZE and JOHANN RAFELSKI

Institut fiir Theoretische Physik, Johann Wolfgang Goethe Universitiit, 6000 Frankfurt am Main, Germany

Received 15 December 1980

Abstract: We obtain the dependence of the upper mass limit of white dwarf stars on the specific entropy per electron. For non-degenerate stars significantly larger masses are found. Stellar collapse as an entropy-producing process can therefore have a self-stabilizing influence on the stellar evolution.

After a star has burned its thermonuclear fuel, it can no longer withstand the gravitational pressure, and collapses. With increasing density the Fermi pressure of the electrons may balance the gravitational pressure and stop the collapse. Such stars with high density and small radius, stabilized by degenerate electron gas pressure, are called white dwarfs ‘). These are one of the stable points in stellar evolution. The largest allowed mass of a white dwarf is about 1.4 solar masses (Chandrasekhar mass). The fact that most known stars have masses larger than this limit suggests that they should evolve into a neutron star, explode by forming a supernova, or lose mass during the burning process.

Our aim in this paper is to calculate the limiting mass for a non-degenerate electron gas. The question of what happens if we take a non-degenerate electron gas is not only of academic interest, since in nature white dwarfs need not be- cold. Furthermore their cooling time is very long because of their small surface - of the order of the age of the universe ‘). Therefore over long times their temperature and their entropy can be treated as constant.

1. Fundamentals of stellar structure

The mass to radius ratio of white dwarfs is of the order of lop3 to lop4 and therefore general relativistic effects are small. For static, uncharged, non-rotating and spherically symmetric objects the newtonian equation for the pressure is:

dp(r)=_ M(r)&) dr r2 ’

where

M(r) = lo’47rp(r’)r’” dr’

* Supported by the Deutsche Forschungsgemeinschaft (DFG).

527

(1)

528 G. Schmidt et al. / Mass limits

is the mass within the radius r and p(r) is the mass density of the star. Given an equation of state, p = p(p), eq. (1) may be solved. For constant entropy per particle the equation of state relates p and p only and can be written as

P =KP”, (2)

where in principle y may also depend on p. From textbooks 3, we obtain the radius R and the mass h4 for newtonian stars having such an equation of state withconstant y

R = ( 4mz l))1’2p(‘2)i2 (O)%,

M = _4,#-4)/*

(0)(47r(; 1) >

3’2W2dU(z)

-X-’

(3)

(4)

where 2 is the zero of the Lame-Emden function U(X) to the index l/(y - 1).

2. Relativistic ideal electron gas at finite temperature

We are interested in the equation of state for a relativistic electron gas in the case where the chemical potential p is much larger than the restmass m, of the electrons. We derive the necessary quantities from the grand canonical potential R defined by:

fi=-$v I 1 1 (e-_p)/T+ 1 +e(E+rvT+ 1 1 5 (5)

where g (= 2 for electrons) counts the spin degeneracy of the single-particle energy levels and V is the volume. The effect of the electromagnetic interaction can also

be included, but it is of little relevance in our present discussion 4), Analytically? (with p B m, but arbitrary T) the electron pressure becomes 5,

gv p.V=-R=24 s[S(rrr)4+2~2(RT)Z+CL4],

the electronic entropy is

s = -g = ~i+rT)3+r*(7rT)1,

and the total charge Q of the electrons

(Q>= -$=$(aT)*+p3]=Z, which is equal to the number of protons 2.

(8)

’ The neglect of the electron mass in the second part of eq. (S), although not quite justified, does not introduce a significant error into the integral of electrons and positrons taken together.

G. Schmidt et al. / Mass limits

The mass density of the star is given by (neglecting the small mass defect)

529

p = m,$!= m,+= mN+[p(aT)z+p3], (9)

where n is the ratio A/Z and mN is the mass of the constituting nucleons. Further we have neglected the mass of the electrons.

As a first approximation we treat the entropy per electronic charge S/(Q) = constant = S, as constant throughout the white dwarf. Taking the ratio of eqs. (7) and (8) we find for the specific electron entropy

and we define the inverse of eq. (lOa) as function d

d(S,) = rT/,u . (lob)

For non-negative S, the third-order equation (10a) has only one real solution and can be solved by using the cardanian formulae:

Approximately we get

S ~(+rsj(s,)/d-2) ) se>> 7T

d&J = s”cl +&,/7T)2) , se<< 7r. ?T

We can now eliminate T in eqs. (6) and (9) and obtain:

p = mN+(l + d2(se)>P3.

Combining eqs. (11) and (12) we obtain the required equation of state:

(11)

(12)

(13)

= I-z (s,)p4’3 *

Ford = 0 (that is T = 0) we recover’with g = 2 the conventional equationof state for a

530 G. Schmidt et al. / Mass limits

degenerate electron gas in the ultra-relativistic limit 3,

4/3

P4/3 = KP4j3 . (14)

The only effect of finite entropy (non-degeneracy) is the entropy-dependent factor that relates g to K:

a = Ic(S,) = 1+2&s,) +&d”(s,) K

(1 + d2(s,))4’3 * (15)

All results obtained for a degenerate white dwarf in the ultra-relativistic limit can now be extended to non-degenerate white dwarfs if we replace K by k For the limiting mass with non-zero entropy we get from eq. (4) with y = 4/3.

M s = [I+ 2d2(Se) +&sd4Lw13’* M

D + cowl* 01

which behaves as

and where MO is the degenerate Chandrasekhar mass

Mo=5.87 [MO]. rl

(16)

(17)

In fig. 1 we show the dependence of M, on the specific electron entropy S, for small values of S,.

Fig. 1. The ratio of the mass limit for non-degenerate white dwarfs (IL&) to a degenerate white dwarf (MO) as a function of the specific electron entropy S,.

3. Discussion

For increasing entropy per electron the Chandrasekhar mass, given by eq. (16), increases. In fig. 2 we show the mass to radius relations taking the specific electron entropy as a parameter.

G. Schmidt et al. / Mass limits

lo2 MIM,] J

Se,,3 __-----

10 7 ,,j __-___-

5 _______

3 _______ c~lmnt 0 _______

l-

\y

white ear's I.

lo",

xl-*, ' j ,I&,.,' "111,1' "111,1' "C 10 II* VI3 10' RIkml

531

Fig. 2. Mass to radius ratio for white dwarfs. The different branches belong to different values of the specific electron entropy S, and the arrows indicate the direction of increasing central density.

Bethe et al. 6, have derived from nuclear physics a value S, = 1.1 for the specific electron entropy in a white dwarf consisting of 56Fe. Inspecting fig. 1 we find:

M(S,= 1.1) = 1.13 MO,

which means that the Chandrasekhar mass limit is increased by about 13%. Similarly, we think that the thermal pressure will also increase the maximal masses of neutron stars, which are the next step in the stellar evolution. All this has remarkable consequences in astrophysics. Up to now, usually a quite small upper mass limit for stable white dwarfs and neutron stars based on the degenerate Fermi gas has been assumed. As we have shown, this limit can be increased by taking into account the entropy per particle, i.e. the non-degenerate Fermi gas. In stellar evolution there are entropy producing phases. Consequently there may exist white dwarfs and neutron stars with masses greater than the Chandrasekhar mass MO but nevertheless stable against further collapse. So the conventional mass to radius diagram 3, has to be extended by several branches belonging to different specific entropy. The inter- mediate region between the branches of white dwarfs and the branches of neutron stars is not yet analyzed in detail.

In summary, we have derived an analytic expression for the mass limit of non-degenerate electron stars (white dwarfs). For expected values of the entropy per electronic charge a 13% correction to the Chandrasekhar limit is found for con- ventional white dwarfs.

We thank W. Greiner for continuing interest and support during the course of this work and G. Baym for helpful comments.

532 G. Schmidt et al. / Mass limits

References

1) H. van Horn, Physics Today 32 (1979) 23; and references therein 2) L. Mestel, Mon. Not. R. Astron. Sot. 112 (1952) 583 3) S. Weinberg, Gravitation and cosmology (Wiley, NY, 1972) 4) L. Landau and E. Lifshitz, Statistical mechanics, (Pergamon, NY, 1970) 5) H.-T. Elze, W. Greiner and J. Rafelski, J. Phys. 66 (1980) 149 6) H.A. Bethe, G.E. Brown, J. Applegate and J.M. Lattimer, Nucl. Phys. A324 (1979) 487