Dead stars are considered to fall into four categories, brown dwarfs, white dwarfs, neutron stars and black holes. White dwarfs and neutron stars are both composed of degenerate matter, so they will be considered here.
As it approaches the end of its life a star has burnt up the hydrogen in its core, converting it to helium. The nuclear reactions are no longer able to be sustained, and the star begins to collapse. This collapse increases the density of the hydrogen envelope just outside the core which gains enough energy to itself ignite in a thermonuclear reaction. This causes an expansion of the remaining envelope and is called the red giant phase.
The mass of the star determines what occurs next. In stars of mass less than 0.8 solar masses, the hydrogen shell burning continues until the shell is close enough to the surface that radiation pressure blows the remaining unburnt hydrogen into space. This forms a "planetary nebula", with the core of the star remaining in the centre. The temperature of the remaining core's surface is over 25,000 °K.
The gravitational collapse of the core causes a corresponding increase in its temperature. For stars of between 0.8 and 3 solar masses the core temperature can increase to 108 °K. This is enough to start a new nuclear reaction: 3He4->C12+7.5 eV. However, the degenerate helium lattice that makes up the core prevents expansion of the core. This increases the temperature of the core, and further increases the rate of C12 production, until finally the core explodes. This effectively stops the nuclear reactions in the star, and the star begins to collapse again. Eventually the helium-carbon reaction can begin again, but now the core is no longer degenerate, so expansion is possible.
Eventually the core's supply of helium is used up, and a degenerate carbon core is produced. A similar process to the low mass stars occurs, only this time helium, not hydrogen, is burnt in a shell around the core, forming a "supergiant". Again a planetary nebula is formed, with a degenerate core in the centre of surface temperature of twice that for the low mass stars.
For stars of mass greater than 3 solar masses, the carbon itself can fuse to form other elements, and for stars of mass greater than 10 solar masses further fusion reactions can occur up to the production of Fe56. In the latter, the densities are low enough that the cores are not degenerate until the production of Fe56. These higher mass stars probably end their lives in supernova explosions resulting again in a degenerate neutron star, or in extreme cases, perhaps a black hole.
The degenerate cores that result from the deaths of stars less than 3 solar masses is known as a white dwarf. Their radius is small, approximately that of the Earth's, which means that, despite their high temperature, they have low luminosity, due to the relation that
This gives a luminosity of about 10-3 times that of our own Sun.
Formation of Degenerate Stars