When a star has completely run out of hydrogen fuel in its core and it lacks the mass to force higher elements into fusion reaction, it becomes a white dwarf star. The outward light pressure from the fusion reaction stops and the star collapses inward under its own gravity. The outcome of a star's struggle between gravity and pressure depends entirely on its birth mass. Stars with masses below about 5 solar masses swell into red giants near the ends of their lives, after which the envelope is ejected as a planetary nebula, while the core becomes a white dwarf. Examples of planetary nebulae are the Ring Nebula, Eskimo Nebula, Helix Nebula and the Cat's Eye Nebula.
A white dwarf shines because it was a hot star once, but there's no fusion reactions happening any more. A white dwarf will just cool down until it becomes the background temperature of the Universe. This process will take hundreds of billions of years, so no white dwarfs have actually cooled down that far yet.
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