The total electron energy (including the rest mass energy) can be written

(8.207) |

by analogy with Equation (8.198). Thus,

(8.208) |

giving

(8.209) |

It follows, from the previous analysis, that the total energy of an ultra-relativistic white-dwarf star can be written in the form

(8.210) |

where

(8.211) | ||

(8.212) | ||

(8.213) |

As before, the equilibrium radius is that which minimizes the total energy. . However, in the ultra-relativistic case, a non-zero value of only exists for . When , the energy decreases monotonically with decreasing stellar radius. In other words, the degeneracy pressure of the electrons is incapable of halting the collapse of the star under gravity. The criterion that must be satisfied for a relativistic white-dwarf star to be maintained against gravity is that

(8.214) |

This criterion can be re-written

(8.215) |

where

(8.216) |

is known as the

(8.217) |

Thus, if the stellar mass exceeds the Chandrasekhar limit then the star in question cannot become a white-dwarf when its nuclear fuel is exhausted, but, instead, must continue to collapse. What is the ultimate fate of such a star?