Typically, once these stars can no longer fuse hydrogen in their cores, gravity will take over, compress the core, and increasing its temperature. This, in turn, allows the fusion of helium into carbon via the triple-alpha process. As the heat from the core increases, it pushes the outer layers of the star out, making the star a Class III (giant) star. When the helium in the core stops fusing, the outer layers collapse again and recompress the core. The temperature of the core will not get hot enough to fuse the carbon (and possibly nitrogen and oxygen) into anything heavier. Instead, something else will occur.
The outer layers will be pushed out and off the star but the rebound of the core. What is left are the naked core (the white dwarf) and the outer layers of the star which becomes a planetary nebula. The planetary nebula is named such because when they were first discovered, they were thought to be the beginning of a solar system. After initial observations, the first planetary nebulae were discovered to have a white dwarf at the center.
Helix Nebula. Note the white dot at the center. This is the white dwarf
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What makes the white dwarf stable is something called electron degeneracy. Electron degeneracy is the electromagnetic force from electrons in a confined space counteracts the gravity wanting to squash the white dwarf. However, there is a limit to the mass that will be stable with electron degeneracy. This limit is called the Chandrasekhar limit and is around 1.4 solar masses. Any higher, and the white dwarf will collapse farther and lead to a Type I supernova (later topic). Any main sequence star with an initial mass of 10 solar masses or lower will end up as a white dwarf.
.Sirius A and B (with Sun for comparison). Sirius B is the closest known white dwarf to the Sun
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