When a star has nearly bumed up its intemal fuel, it may become a white dwarf.
It is crushed under its own enonnous gravitational forces to the point at
which the exclusion principle for the electrons becomes a factor. A smaller
size would decrease the gravitational potential energy, but assuming the
electrons to be packed into the lowest energy states consistent with the
exclusion principle. "squeezing" the potential well necessarily increases the
ener gies of all the electrons (by shortening their wavelengths). If
gravitation and the electron exclusion principle are the only factors, there
is a mini. mum total energy and corresponding equilibrium radius.
(a) Treat the electrons in a white dwarf as a quantum gas. The minimum energy
allowed by the exclusion principle (see Exercise 67 ) is
$$
U_{\text {elecimns }}=\frac{3}{10}\left(\frac{3 \pi^{2} \hbar^{3}}{m_{e}^{3 /
2} V}\right)^{2 / 3} N^{5 / 3}
$$
Note that as the volume \(V\) is decreased, the energy does increase. For a
neutral star. the number of electrons, \(N\), equals the number of protons.
Assuming that protons account for half of the white dwarf's mass \(M\) (neutrons
accounting for the other half). show that the minimum electron energy may be
written
$$
U_{\text {electrons }}=\frac{9 \hbar^{2}}{80 m_{e}}\left(\frac{3 \pi^{2}
M^{5}}{m_{\mathrm{p}}^{5}}\right)^{1 / 3} \frac{1}{R^{2}}
$$
where \(R\) is the star's radius.
(b) The gravitational potential energy of a sphere of mass \(M\) and radius \(R\)
is given by
$$
U_{\operatorname{mav}}=-\frac{3}{5} \frac{G M^{2}}{R}
$$
Taking both factors into account, show that the minimum total energy occurs
when
$$
R=\frac{3 h^{2}}{8 G}\left(\frac{3 \pi^{2}}{m^{3} m_{p}^{5} M}\right)^{1 / 3}
$$
(c) Evaluate this radius for a star whose mass is equal to that of our Sun,
\(\sim 2 \times 10^{30} \mathrm{~kg}\).
(d) White dwarfs are comparable to the size of Eath. Does the value in part
(c) agree?