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We know the mass a white dwarf needs to be. That's well defined by the Chandrasekhar limit, but before a main sequence star turns into a white dwarf it tends to lose a fair bit of its matter in a stellar nebula.

According to this site, the white dwarf that remains is about half the mass of the main sequence star, with larger stars losing a bit more.

So, the question: Is it correct to say that a star with a mass of about three solar masses will eventually go supernova, similar to a type 1 supernova, even when it's not part of a binary system? Has that kind of supernova ever been observed?

Or does something else happen like in the final stages of that star? Does it keeps going though collapse and expand cycles, losing enough mass that when it finally becomes a white dwarf it's below the Chandrasekhar limit in mass?

Mostly, what I've read on supernovae says that type 1 supernovae happen when a white dwarf accretes extra matter and reaches the limit and type 2 supernovae are much larger and require about 8-11 solar masses to generate the iron core which triggers the supernova. What happens with the death of the star between three solar masses and eight solar masses?

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This is ground that is probably duplicated in a variety of questions here and on Physics SE, so I'll keep it brief. You have also mixed in several different questions.

The Chandrasekhar mass has very little to do with determining what initial mass of object will end up as what particular type of stellar remnant (or black hole).

Whether a star will end up exploding as a supernova depends primarily on its initial mass, but also whether it has a binary companions. There are (basically) two routes to supernovahood.

  1. If the star is more massive than about $8M_{\odot}$ it will progress through several nuclear burning stages. The core of the star does not become degenerate and continues to get more dense and hot through each burning stage. It ends up as iron. Once the core mass of iron exceeds about $1.2M_{\odot}$ (which is the Chandrasekhar mass for an iron composition), then it collapses and we get a type II (core-collapse) supernova.

In this route a $3 M_{\odot}$ star gets nowhere near being able to go supernova. It will burn hydrogen and helium, produce a degenerate core of carbon and oxygen. This degenerate core can cool whilst maintaining the same pressure. The outer layers are shed through thermal pulsations and a dense stellar wind in the asymptotic giant branch phase, leaving behind a white dwarf. The relationship between the initial mass of the progenitor and the final mass of the white dwarf is not a straightforward fraction. It probably is about 50% for a star like the Sun, but the fraction is more like 15% for a $7M_{\odot}$ initial mass. The maximum mass of a white dwarf formed in this way is probably about $1.1-1.2M_{\odot}$ and some way below the Chandrasekhar limit for a C/O white dwarf ($\simeq 1.39M_{\odot}$).

The preceding paragraph is more-or-less what should happen for all stars between about $0.6 M_{\odot}$ (except they haven't had time to do so yet) and $8M_{\odot}$, except that there is a small "grey area" at the upper mass end ($7-9M_{\odot}$) where you might produce slightly more massive O/Ne white dwarfs.

  1. Once a white dwarf has formed and if it is in a binary system, then the white dwarf could merge or accrete more mass. At some point close to the Chandrasekhar limit, it ignites. This causes a type Ia (detonation or deflagration) supernova explosion (or at least this is the leading model for how this works). This is really the only route whereby a star with initial mass $<8M_{\odot}$ could end up going supernova.
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