The stellar endpoint leading to its collapse to a black hole was described in Chapter 12. This chapter explores the other possibility, where a star is supported against gravity by a nonthermal source of pressure. This is realized in nature by white dwarf stars and neutron stars. Unlike black holes, which can be understood entirely in the context of general relativity, an understanding of the stars at the endstate of stellar evolution requires almost all of the rest of physics in some way. We cannot hope to review the range of physics necessary for a complete understanding of the equilibrium endstates of stellar evolution, but we can isolate the essential role of gravitational physics and discuss the overall structure of these stars. To an excellent approximation, the properties of the matter relevant for the gross structure of neutron stars and white dwarfs can be summarized by an equation of state relating the pressure of an ideal matter fluid to its energy density.

In this chapter, we concentrate on a particular endpoint of stellar evolution – the state of ongoing gravitational collapse leading to a black hole. This possibility must exist in nature because there is a maximum amount of nonrotating matter that can be supported against gravitational collapse by Fermi pressure or nuclear forces. This mass is in the neighbourhood of two solar masses. (The exact value is uncertain because our knowledge of the properties of matter above nuclear densities is uncertain.) There are many stars more massive than this upper limit, so it is likely that some must wind up in a state of ongoing collapse. This chapter also explores the properties of this state.

The mass–radius relation for polytropes is introduced and analyzed. The Newtonian Lane–Emden equation is derived analytically. Known analytic solutions are discussed. The famous Chandrasekhar mass is obtained as a solution of the Lane–Emden equation. Corrections to the equation of state for white dwarfs are pointed out by estimating the Coulomb corrections for a lattice of nuclei immersed in a sea of electrons. The different layers of a typical white dwarf and the typical sizes are worked out in detail. Thermal effects for the mass relation are examined. Finally, astrophysical observations of white dwarfs are shown and discussed in terms of the overall composition of white dwarfs and the resulting mass–radius relation.

Fermi--Dirac statistics lead to specific thermodynamic consequences at low temperatures.A key quantity is the Fermi energy, which is equal to the chemical potential at zero temperature, and can be used to define a temperature scale, the Fermi temperature.At temperatures that are small compared to the Fermi temperature, thermodynamic quantities may be calculated using the Sommerfeld expansion.The properties of metals and the existence of compact stars such as white dwarfs and neutron stars are a direct consequence of Fermi--Dirac statistics.