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Is it possible that a binary star doesn't exceed its Roche lobe and so when the second star becomes a red giant, instead of accreting some of its plasma onto the white dwarf that it becomes a white dwarf itself which causes there to be no supernova?

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Of course this is possible. Whether it will fill its Roche lobe depends on the masses and radii of the stars and the distance between the stars. If the distance is big enough then the second (less massive) star will not fill its Roche lobe and continue evolving as a single star would. The (large) majority of binaries will fall in this category, only close binaries will evolve into cataclysmic variables.

For instance the Alpha Centauri/Proxima Centauri system will not evolve into a system where mass transfer takes place via the Roche lobes as the distance between Alpha Cen A/B and Proxima Centauri is much too big.

Even when the stars are very close, a supernova might not occur if the masses of the stars are low. The supernova occurs when the mass of the existing white dwarf increases (because of the accretion) to a mass surpassing the Chandrasekhar limit (1.4 solar masses). If the transferred mass is too low then no supernova will happen.

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