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If two brown dwarfs got close enough to exchange gas with each other, could it trigger nuclear fusion within their cores? Is it possible for two brown dwarfs to orbit so closely and become a rocheworld in the first place?

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If two brown dwarfs got close enough to exchange gas with each other, could it trigger nuclear fusion within their cores?

Very roughly, a star can undergo fusion once it is approximately 7.5-8.3% the mass of the Sun, depending on the metallicity of the brown dwarfs. If one of the brown dwarfs is on the brink of fusion, the gases that it would accrete from its less massive counterpart would make it theoretically possible for the dwarf to be pushed over its threshold and become a star, similar to type Ia supernovae. Of course, this would only happen to one brown dwarf.

Is it possible for two brown dwarfs to orbit so closely and become a rocheworld in the first place?

Orbiting brown dwarf binaries are unable to exchange gas between one another. The only way that binary star systems are able to exchange gas is if one of the stars exceeds its Roche lobe, which is the region around a star in a binary system where orbiting material is bound to the star, and the other star attracts the first star's material. As brown dwarfs do not grow in radius, it is impossible for a brown dwarf's matter to exist outside of its Roche lobe.

Sources:
Science (vol 313, p 936)
"Mass transfer and accretion in close binaries – A review" by C.B. Boyle in Vistas in Astronomy

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It is possible for two brown dwarfs to form a contact binary. This would not, of itself, cause core temperatures in either to rise.

If the two brown dwarfs were to merge, the mass of the resulting body could be enough to generate sufficient heat and pressure in the core for nuclear reactions to start. If both brown dwarfs had a mass of about 60 Jupiters, the combined size of 120 Jupiters would certainly be big enough to form a red dwarf.

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