A star that can no longer burn its nuclear fuel becomes one of three types of object: a degenerate dwarf, a neutron star, or a black hole. Small stars such as the Sun become degenerate dwarfs, which are also called white dwarfs. These stars contain up to one and a half times the mass of the Sun in a volume similar to Earth. While the degenerate dwarf is quite dense by earthly standards, it is tenuous by neutron star standards. Neutron stars can contains up to about 3 solar masses in a sphere of 10 to 15 km. These stars are at the density of an atomic nucleus, and their gravitational fields are strong enough to deflect light by a dozen degrees. Stars that are a couple times the Sun's mass collapse down to neutron stars when they exhaust their nuclear fuel, an event that causes a supernova explosion. But the largest stars that collapse after consuming their nuclear fuel do not stop collapsing when the reach neutron star densities; they continue to collapse to a black hole. We have three possible outcomes when a star ceases thermonuclear fusion, because two of the outcomes produce objects that are vulnerable to gravitational collapse.
Degenerate dwarfs and neutron stars, as well as giant gaseous planets like Saturn, are held up by a pressure that is a unique consequence of quantum mechanics. One principle of quantum mechanics is that the electrons, protons, and neutrons in a gas can have only very specific energies, starting at zero energy and increase by finite increments. The spacing between energy levels is larger for low-mass particles than for high-mass particles. A second principle is that only two electrons, protons, or neutrons can simultaneously have the same energy. When a material is cold, the particles occupy all of the lowest energy levels. In this state, the particles are said to be degenerate. The pressure exerted by the particles in this cold material is called degeneracy pressure.
If the density of a degenerate material is low, the most energetic particles have velocities far below the speed of light. This means that a star composed of this material, which has an adiabatic index of 5/3, is stable against gravitational collapse. At a higher pressure, however, the most energetic particles are moving at close to the speed of light, and the adiabatic index of the material drops to 4/3. This is true even when the temperature of the gas is near zero. A star composed of this material is unstable to gravitational collapse. Because the density of a star supported by degeneracy pressure rises with mass, a maximum mass exists for such stars. Stars below this maximum mass are stable forever, but stars above this mass are gravitationally unstable, and rapidly collapse to become a much smaller object.
Within the degenerate dwarf, the degeneracy pressure is provided by electrons, but in the neutron star, the degeneracy pressure is provided by neutrons and the protons. This difference is the reason neutron stars are so much more dense than degenerate dwarfs—electrons, because of their smaller mass, become degenerate at a much lower density and temperature than either protons or neutrons.
Degenerate dwarfs come into existence after a small star has exhausted its supply of hydrogen and helium. A small star that has exhausted its core helium will shrinks until either the electrons at its core are degenerate, in which case it has become a degenerate dwarf composed of carbon, or carbon fusion commences. A star that has exhausted its carbon will shrink until either the electrons are degenerate, in which case it has become a degenerate dwarf composed principally of oxygen, or oxygen fusion commences. This branching progression does not continue with every stage of thermonuclear fusion, however, because at some point the electrons become relativistic, and the star becomes unstable to collapse. Once in this state, the star continues shrinking until thermonuclear fusion commences for heavier elements. So stars can become low-mass degenerate stars composed of carbon, and they can become intermediate-mass degenerate stars of oxygen, but they cannot become degenerate stars with masses above about 1.4 solar masses. This mass limit is called the Chandrasekhar limit. Stars with cores greater than the Chandrasekhar limit become either neutron stars or black holes.
The existence of a maximum mass for the degenerate dwarf is the basis for the hypothesis that type 1a supernovae, which are brilliant explosions that consistently release the same amount of energy, are caused when a degenerate dwarf is pushed above Chandrasekhar limit. The idea is that a degenerate dwarf could acquire mass from a companion star. If it acquires enough mass over time to push it above Chandrasekhar limit, it would collapse. As the star collapses, the light elements that compose it undergo rapid thermonuclear fusion, destroying the star and creating the supernova we see. Degenerate dwarfs at the Chandrasekhar limit are therefore stellar thermonuclear bombs waiting to be triggered.
Stars with stellar cores somewhat more massive than 1.4 solar masses collapse to neutron stars. As with the degenerate dwarf, the neutron star is supported by degeneracy pressure. This time, the degeneracy pressure is supplied by neutrons and protons. But as with the degenerate dwarf, the neutron star is unstable if it is too massive. The upper mass is very uncertain, but it should be around 3 solar masses, and no more than about 5 solar masses. Anything larger must collapse to a black hole.
As long as gravity is consistent with one of our central assumptions in our stability analysis—that gravity increases at least as fast as the inverse square of an object's radius—stars supported by degeneracy pressure are unstable when they are large. General relativity is consistent with this assumption. This natural tendency to instability by degenerate dwarfs and neutron stars is the reason we believe that the core collapse of a very large star produces a black hole rather than a neutron star. It is also the reason we believe the compact objects at the centers of galaxies are black holes: with estimated masses between a million and a billion times that of the Sun, they could be nothing else.