Kumar's Brown Dwarf Limit

Let us consider a sphere of hydrogen gas with mass $M$. If the electrons are degenerate then the radius of the sphere is completely determined. The degenerate pressure is given by $\hbar ^{2}n^{5/3}/m_{e}$, where $\hbar$ is Planck's constant, $m_{e}$ is the proton mass and $n$ is the electron number density. If the density is roughly constant then $n\approx M/m_{p}R^{3}$ where $m_{p}$ is the proton mass and $R$ is the radius of the sphere. In a hydrostatic equilibrium, this pressure balances gravity, so

${\frac {\hbar ^{2}}{m_{e}}}\left({\frac {M}{m_{p}R^{3}}}\right)^{5/3}\approx {\frac {GM^{2}}{R^{4}}}\Rightarrow R\approx {\frac {\hbar ^{2}}{Gm_{e}m_{p}^{5/3}M^{1/3}}}$

A sphere whose mass is below the Chadrasekhar mass cannot contract below this radius.

Let us now consider a star. The nuclear burning at its core is enabled by quantum tunnelling. The condition for tunnelling is that the classical closest approach between two proton should be comparable to the proton's de Broglie wavelength. The average kinetic energy of protons at the centre of the star is

$e\approx {\frac {GMm_{p}}{R}}$

The closest approach is

$d\approx {\frac {q^{2}R}{GMm_{p}}}$

where $q$ is the elementary charge. The de Broglie wavelength is

$\lambda \approx {\frac {\hbar }{\sqrt {m_{p}e}}}$. We have $\lambda \approx d$ when

${\frac {GM}{Rc^{2}}}\approx \alpha ^{2}$

where $\alpha$is the fine structure constant.

Substituting the radius of a degenerate object we get

$M\approx m_{e}^{3/4}m_{p}^{1/4}\left({\frac {r_{e}c^{2}}{Gm_{p}}}\right)^{3/2}$

When blob of gas lighter than this mass gravitationally collapses, electron degeneracy pressure kicks in before nuclear fusion, and the result will be a brown dwarf or a planet. When a blob of a higher mass collapses, nuclear fusion will kick in before gravitational pressure and the result will be a star.