What happens to the remaining core? For a star roughly 1 solar mass or less, the core never reaches the ignition temperature of carbon burning. The core cannot contract and heat up to a temperature needed to initiate carbon fusion. In about 75,000 years it forms a white dwarf star, composed mostly of carbon.
As a star contracts into a white dwarf, its surface becomes very hot. But without energy, it cannot sustain itself. Because the core is out of fuel, the white dwarf will eventually cool to a black dwarf. This will take many billions of years to cool. Degenerate gas pressure inside the core continues to support the star so it doesn’t collapse.
The smallest white dwarfs are the most massive, and the largest white dwarfs are the least massive. Why is this? The more massive star, because of its gravitational force, is able to squeeze itself into a smaller, more densely packed object than can a less massive star. This conclusion led to the prediction that smaller and more massive stars must exist. Known as neutron stars, these objects are thought to be remnants of supernova events. In a white dwarf, the electrons are pushed close to the nucleus. In a neutron star, the electrons are force to combine with protons to produce neutrons (hence the name).