Stability of a White Dwarf Against Gravitational Collapse.
It is energetically favorable for a body held together by gravita-
tional forces to be as compact as possible. We take the star to be made
up of an approximately equal number ##N## of electrons and protons, since
otherwise the Coulomb repulsion would overcome the gravitational in-
teraction. Somewhat arbitrarily we also assume that there is an equal
number of neutrons and protons. On Earth the gravitational pressure
is not large enough to overcome the repulsive forces between atoms and
molecules at short distance. Inside the sun, matter does not exist in the
form of atoms and molecules, but since it is still burning there is radi-
ation pressure which keeps it from collapsing. Let us consider a burnt
out star such as a white dwarf. Assume that the temperature of the
star is low enough compared to the electron Fermi temperature that the
electrons can be approximated by a ##T = 0## Fermi gas. Because of their
large mass the kinetic energy of the protons and neutrons will be small
compared to that of the electrons.
(a) Show that, if the electron gas is non-relativistic, the electron mass
is ##m_e## , and the radius of the star is ##R##, the electron kinetic energy of
the star can be written:
$$E_{kin} = \frac{3\hbar^2}{10m_e}\bigg( \frac{9\pi}{4} \bigg)^{2/3} \frac{N^{5/3}}{R^2}$$
(b)
The gravitational potential energy is dominated by the neutrons and
protons. Let ##m_N## be the nucleon mass. Assume the mass density
is approximately constant inside the star. Show that, if there is an
equal number of protons and neutrons, the potential energy will be
given by:
$$E_{pot}=-\frac{12}{5}m_N^2G N^2/R$$
where ##G## is the gravitational constant.
(c)
Find the radius for which the potential energy plus kinetic energy
is a minimum for a white dwarf with the same mass as the sun
(##1.99 x 10^{30} kg##), in units of the radius of the sun (##6.96 x 10^8 m##).
(d)
If the density is too large the Fermi velocity of the electrons becomes
comparable to the velocity of light and we should use the relativistic
formula,
$$(2.107)\ \ \ \ \ \epsilon(p) = \sqrt{m_e^2c^4+p^2c^2}-m_e c^2$$
for the relationship between energy and momentum.
It's easy to see, that in the ultra relativistic limit (##\epsilon \approx cp##), the electron kinetic energy will be proportional to ##N^{4/3}/R##, i.e. the ##R## dependence is the same as for the potential energy. Since for large ##N## we have ##N^2\gg N^{4/3}##
we find that if the mass of the star is large enough the
potential energy will dominate. The star will then collapse. Show that the critical value of ##N## for this to happen is:
$$ N_{crit} = \bigg(\frac{5\hbar c}{36\pi m_N^2 G}\bigg)^{3/2} \bigg( 9\pi /4 \bigg)^2$$