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In terms of astronomy, a nova is when a star suddenly becomes a lot brighter, then gradually dims back to normal over several months. This can happen multiple times with the same star.
Novae happen in binary star systems where one of the two stars is a white dwarf. A white dwarf is a type of dead star. They have cores made of carbon and oxygen atoms that are packed as closely together as possible without violating the Pauli exclusion principle. This quantum mechanics law states that electrons can't share the same quantum state with other electrons - or, put another way, two electrons can't occupy the same space at the same time. The close packing of the electrons creates a pressure, called degenerate-electron pressure, that holds the star up against the strength of its gravity.
In a normal star, gravity is balanced by nuclear fusion reactions occurring in the star's core - the same ones that create a star's light. But a white dwarf, being dead, no longer has these reactions going on. Left on its own, a white dwarf slowly cools down and dims over time until its temperature reaches absolute zero.
When a white dwarf is in a binary system with a normal star or red giant star, however, hydrogen from the companion star can fall on the white dwarf's surface if they orbit each other closely enough. Temperatures rise steadily as the hydrogen accumulates, until it explosively ignites, fusing hydrogen into helium and creating the bright flash of light that we see back on Earth as a nova.
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Nova vs. X-Ray Burster
A nova is similar to an X-ray burster. These also happen in binary star systems, but instead of a white dwarf, the dead star is a neutron star. Neutron stars are smaller and denser than white dwarfs, with cores made of neutrons packed as closely together as possible. Neutron stars hold up against their own gravity by degenerate-neutron pressure. (They've passed the maximum mass limit for the Pauli exclusion principle, causing all the electrons to fuse with the protons to create neutrons. The limit is called the Chandrasekhar limit and is about 1.4 times the mass of our Sun.)
Hydrogen falling on the surface of a neutron star instantly fuses into helium due to the higher temperatures and pressures. When enough helium accumulates, it explosively ignites to fuse helium into carbon and oxygen, producing a burst of X-rays.
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Nova vs. Type II Supernova
A nova is not the same thing as a supernova. They look similar when viewed from Earth, in that both involve a sudden burst of light, then gradual dimming over several months. However, the reasons behind the bursts of light are completely different. Novae occur after a low-mass star has already died and become a white dwarf. Supernovae are the death throes of a high-mass star on its way to becoming a neutron star. Novae can happen multiple times with the same binary pair as new hydrogen falls continuously onto the white dwarf. Supernovae are one-time ultimate destruction events.
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Type I Supernova
It's also possible for a white dwarf that periodically novas to someday supernova. As hydrogen continues to accumulate from the normal companion star, the mass of the white dwarf can eventually surpass the Chandrasekhar limit. When that happens, temperatures rise high enough for carbon burning to begin in the white dwarf's core - but without the usual balance of rising pressure and expanding surface that a normal (not dead) star has. This causes it to explode.
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Reference and Image Credit
Kaufman, W.J. III. Universe 2nd ed. W.H. Freeman and Company, 1987.
Nova image created by NASA.