Consider a white dwarf of mass \(M\) and radius \(R\). The dwarf
consists of ionized hydrogen, thus a bunch of free electrons and
protons, each of which are fermions. Let the electrons be degenerate
but nonrelativistic; the protons are nondegenerate.

Show that the order of magnitude of the gravitational self-energy is
\(-\frac{GM^2}{R}\), where \(G\) is the gravitational constant. (If
the mass density is constant within the sphere of radius \(R\), the
exact potential energy is \(-\frac53\frac{GM^2}{R}\)).

Show that the order of magnitude of the kinetic energy of the
electrons in the ground state is \begin{align}
\frac{\hbar^2N^{\frac53}}{mR^2}
\approx \frac{\hbar^2M^{\frac53}}{mM_H^{\frac53}R^2}
\end{align} where \(m\) is the mass of an electron and \(M_H\) is
the mas of a proton.

Show that if the gravitational and kinetic energies are of the same
order of magnitude (as required by the virial theorem of mechanics),
\(M^{\frac13}R \approx 10^{20} \text{g}^{\frac13}\text{cm}\).

If the mass is equal to that of the Sun (\(2\times 10^{33}g\)), what
is the density of the white dwarf?

It is believed that pulsars are stars composed of a cold degenerate
gas of neutrons (i.e. neutron stars). Show that for a neutron star
\(M^{\frac13}R \approx 10^{17}\text{g}^{\frac13}\text{cm}\). What is
the value of the radius for a neutron star with a mass equal to that
of the Sun? Express the result in \(\text{km}\).