Let us consider a sphere of hydrogen gas with mass . If the electrons are degenerate then the radius of the sphere is completely determined. The degenerate pressure is given by , where is Planck's constant, is the proton mass and is the electron number density. If the density is roughly constant then where is the proton mass and is the radius of the sphere. In a hydrostatic equilibrium, this pressure balances gravity, so
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
The closest approach is
where is the elementary charge. The de Broglie wavelength is
. We have when
where is the fine structure constant.
Substituting the radius of a degenerate object we get
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.