A burnt-out star is basically a gas of electrons and ions. As the
star collapses, its density increases, and so the mean separation between its
constituent particles decreases. Eventually, the mean separation becomes
of order the de Broglie wavelength of the electrons, and the electron
gas becomes *degenerate*. Note, that the de Broglie wavelength of the
ions is much smaller than that of the electrons, so the ion gas remains
non-degenerate. Now, even at
zero temperature, a degenerate electron gas exerts a substantial pressure,
because the Pauli exclusion principle prevents the mean electron separation
from becoming significantly smaller than the typical
de Broglie wavelength (see
previous section). Thus, it is possible for a burnt-out star to maintain
itself against complete collapse under gravity via the *degeneracy pressure*
of its constituent electrons. Such stars are termed *white-dwarfs*.
Let us investigate the physics of white-dwarfs in more detail.

The total energy of a white-dwarf star can be written

where is the gravitational constant, is the stellar mass, and is the stellar radius.

From the previous subsection, the kinetic energy of a degenerate electron gas is simply

The interior of a white-dwarf star is composed of atoms like
and which contain equal numbers of protons, neutrons, and
electrons. Thus,

Equations (516)-(519) can be combined to give

(521) | |||

(522) |

The equilibrium radius of the star, , is that which

(523) |

(524) |

where is the solar radius, and the solar mass. It follows that the radius of a typical solar mass white-dwarf is about 7000km:

Note from Eqs. (518), (519), and (525)
that
. In other words, the mean energy of the
electrons inside a white dwarf *increases* as the stellar mass increases.
Hence, for a sufficiently massive white dwarf, the electrons can become
*relativistic*. It turns out that the degeneracy pressure for
relativistic electrons only scales as , rather that ,
and thus is unable to balance the gravitational pressure [which also
scales as --see Eq. (520)]. It follows that electron
degeneracy pressure is only able to halt the collapse of a burnt-out star
provided that the stellar mass does not exceed some critical value, known
as the *Chandrasekhar limit*,
which turns out to be about times the mass of the Sun. Stars
whose mass exceeds the Chandrasekhar limit inevitably collapse to
produce extremely compact objects, such as neutron stars (which are
held up by the degeneracy pressure of their constituent neutrons),
or black holes.