Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physica 285

temperature is lo8 K and assuming that the ‘ion gas” can be regarded as
a Maxwell-Boltzmann gas, and hence convince yourself that E, >> Ei.
(c) If M is the mass of the star, and if R is its radius, then the gravi-

3GM2

uc = -

In the case in which the internal energy is dominated by extremely rel-

ativistic electrons (as in part (t) above), the virial theorem implies that

the total internal energy is approximately equal to the gravitational po-

tential energy. Assuming equality, and assuming that the electrons do not

contribute significantly to the mass of the star, show that the stellar mass

can be expressed in terms of fundamental physical constants alone. Eval-

uate your answer numerically and compare it with the mass of the sun,

2 x 1030 kg. (It can be shown that this is approximately the maximum

possible mass of a white dwarf.)

(UC, Berkeley)

Solution:

(a) Use the approximation of strong degenerate electron gas and E = pc.

From the quantum state density of electrons, it follows

tational potential energy is given by

5R ’

2 8T

-dp = -E2ds ,

h3 h3c3

then

n, = /,” %E2ds

h3c3

Therefore P sr

(b) When p = lo9 g/cm3,

n, = 14ni = 3 x 1032~m-3 = 3 x

E, = 5 x

Ei = -kT = 2 x

rn-’ ,

• J = 3 MeV ,
  3
  2


• 
  

  J = 1.3~10-~ MeV.

  
  

Obviously, xi << s,.