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Neon burning is a nuclear reaction which occurs in the core of massive stars (8 solar masses or greater) near the end of their life. It converts neon to oxygen and magnesium atoms, releasing light and heat in the process. Neon burning is so rapid that it only takes place over the course of a few years, a blink of an eye in astrophysics, where timescales are usually measured in millions or billions of years. The neon burning process occurs after carbon burning and before oxygen burning.
For most of a star's lifespan, it will slowly burn hydrogen in its core, fusing the hydrogen nuclei into helium nuclei, slowly raising the percentage of helium in its core. If the star is massive enough, it will begin fusing helium through the triple-alpha process, leaving the main sequence and becoming a giant star. If the star has even more mass, it will start fusing helium into carbon, a process that only takes about 1000 years.
What happens next separates the truly massive stars from the smaller ones. If a star has less than about 8 solar masses, it ejects most of its envelope through solar wind and leaves behind a oxygen/neon/magnesium white dwarf. If it has more, the core condenses in size, heats up, and begins the neon burning. Neon burning requires temperatures in the range of 1.2×109 K and pressures around 4×109 kg/m3. This is about four million metric tons per square meter.
Above the neon burning core, carbon burning, helium burning, and hydrogen burning continue in shells located at progressively greater distance from the core. Neon burning fundamentally relies on photodisintegration -- the process whereby gamma rays of extreme energy are created, and impact atomic nuclei so forcefully that they knock off protons and neutrons, or even break the nucleus in half. The the core of a dying star, photodisintegration knocks alpha particles (helium nuclei) off neon nuclei, producing oxygen and alpha particles as byproducts. The energetic alpha particles then fuse with neon nuclei to create magnesium.
Over time, the star uses up its neon and the core condenses again, at which point oxygen burning begins. If the star keeps burning heavier and heavier nuclei, it eventually reaches iron, which cannot be ignited in a sustainable fashion, and core collapse takes place, followed by a supernova.