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How is Chandrashekhar Limit calculated? What happens to stars below the mass of the Chandrashekhar limit after they collapse?

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The Chandrasekhar mass is a theoretical maximum stellar mass that can be supported by ideal electron degeneracy pressure.

It was originally calculated by Chandrasekhar using polytropic equations of state for white dwarf stars of various compositions.

It is found that if the equation of state is $P \propto \rho^{4/3}$, which is appropriate for ultra relativistic electrons, then equilibrium is only obtained for $M_{CH} = 5.8 \mu_e^{2} M_{\odot}$, where the density becomes infinite and $\mu_e$ is the number of atomic mass units per electron in the gas.

Typical white dwarfs are carbon and oxygen with $\mu_e= 2$, and $M_{CH}= 1.45M_{\odot}$, but in the core of a supernova progenitor made of iron-56 it would be lower - around $M_{CH}=1.25M_{\odot}$.

In more recent years, the Chandrasekhar limit has evolved to colloquially mean the maximum possible mass for a white dwarf. The main corrections to the ideal case considered by Chandrasekhar are Coulomb corrections, the possible onset of inverse beta decay (electron capture) and the instability at a finite density predicted by using GR rather than Newtonian gravity. The latter probably sets the "Chandrasekhar mass" for carbon white dwarfs to be $1.38M_{\odot}$.

A carbon white dwarf that had a mass less than this could be supported by electron degeneracy pressure and would not collapse.

If your question is asking for a derivation of the Chandrasekhar mass, then I suggest you look at the stability analyses that are presented in a standard text on the subject like "Black holes, white dwarfs and neutron stars" by Shapiro and Teukolsky. An approximate result can be obtained using the virial theorem assuming a uniform density star.

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