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I was reading on how Type II Supernovae release as much as 99% of their energy as neutrinos. So I was thinking: suppose there was a large star and black hole binary system, but the two were not close enough for the BH to draw mass out of the star, so the star aged naturally instead.

If one were close enough to observe when the star's core collapses - but before the supernova bursts through the star's surface - would it be possible to see the black hole immediately grow in size for no apparent reason as it eats up the neutrinos from the companion star's collapse?

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Yes. Neutrinos which hit the BH will add to its mass.

But you may be thinking that the neutrinos pass through the nascent supernova without interaction. In a dense enough environment, neutrinos do interact, and absorption of neutrinos by the layers beyond the core is a major source of the energy that disrupts the star.

In a bit more detail, the core collapse happens through inverse beta decay, where a proton and an electron react to form a neutron and a neutrino. This basically turns the star's iron core to neutrons. The energy is carried away by the neutrinos and the absorption of this enormous flux of neutrinos by the outer layers drives the SN. The Wikipedia article has a good (but IMO somewhat incoherent) longer discussion.

Still the many neutrinos which do escape do so before the shock wave blows the star apart and the energy they carry will bulk up a nearby BH.

I doubt that it would easily be observable, though. Stellar-mass BHs are small in cross-section and would only absorb neutrinos from a patch of the sky proportional to their area. If a 10km BH was orbiting a million km star right at its surface, it would only absorb one part in 1010 of the neutrinos, producing roughly a 10-9 fractional increase in size. Very difficult to measure!

If you're using this to provide an early warning of the core-collapse (perhaps so you can start you FTL drive and get outta there) it's fairly useless. First, the neutrino flux is probably high enough to fry people who are close enough to need the early warning. And second, if you're far enough away not to be fried by the neutrinos, you have plenty of time to see the star rapidly swelling and get moving before the comparatively slow shock wave gets near you.

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