A burnt-out star is basically a gas of electrons and ions. As the
star collapses, its density increases, 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 (because of the ions' much larger mass), 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 Section 8.15 and Exercise 18.) 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 total kinetic energy of the degenerate electrons (the kinetic energy of the ions is negligible), and is the gravitational potential energy. Let us assume, for the sake of simplicity, that the density of the star is uniform. In this case, the gravitational potential energy takes the form

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

Let us assume that the electron gas is highly degenerate, which is equivalent to taking the limit . In this case, we know, from Section 8.15, that the Fermi momentum can be written

(8.194) |

where

Here,

is the stellar volume, and the total number of electrons contained in the star. Furthermore, the number of electron states contained in an annular radius of -space lying between radii and is

(See Exercise 20.) Hence, the total kinetic energy of the electron gas can be written

where is the electron mass. It follows that

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

where is the proton mass.

Equations (8.192), (8.193), (8.195), (8.196), (8.199), and (8.200) can be combined to give

(8.201) |

where

(8.202) | ||

(8.203) |

The equilibrium radius of the star, , is that which minimizes the total energy, . In fact, it is easily demonstrated that

(8.204) |

which yields

(8.205) |

The previous formula can also be written

where is the solar radius, and the solar mass. It follows that the radius of a typical solar-mass white-dwarf is about 7000km: that is, about the same as the radius of the Earth. The first white-dwarf to be discovered (in 1862) was the companion of Sirius. Nowadays, thousands of white-dwarfs have been observed, all with properties similar to those described previously.