A lot is now in flux on the theoretical side. As we have added increasingly sophisticated physics to our supernovae models, we have experienced phases where either everything explodes or nothing explodes. It’s been harder to develop models that give us a good picture of why some stars explode and others might not.
Still, our best current models agree that neutrinos are essential to the process. Neutrinos are produced in massive quantities both by the complex fusion reactions that occur during a supernova and by the formation of neutron star material in the core of the collapse (which occurs even if the collapse continues to result in the formation of a black hole). And these numbers matter for the fate of matter outside the dying star’s core.
As fewer photons emerge from the star’s core, this matter lacks the energy to resist gravity’s pull and begins to flow toward the core. On its way, it encounters the shock wave of neutron star/black hole formation traveling in the opposite direction. When left alone, these forces roughly balance each other out, slowing the shock wave and allowing gravity to take over.
Neutrinos change the equation. While they don’t frequently interact with matter, the sheer number of particles they pour out ensures that enough shocks hit the material around the blocked shock wave. This transfers energy and heats it to such an extent that gravity is overcome and the shock wave can escape, destroying the star. On the other hand, if this process fails, almost the entire contents of the star would collapse into a black hole, killing the star without causing an explosion.
Tastefully
A potential problem with the models that show neutrino heating is that they treat neutrinos as a single factor. Neutrinos don’t want to be held like that. There are three types or flavors of neutrinos (electron, muon and tau). But each particle is in a superposition of all three flavors and can switch between them in a process called taste oscillation. So even if the events within the supernova produced only electron neutrinos, they would likely oscillate between the other two identities several times before reaching the star’s surface.