FANDOM


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{G M^2}{R^4} \Rightarrow R \approx \frac{\hbar^2}{G m_e m_p^{5/3} M^{1/3}} $

A sphere whose mass is below the Chadrasekhar mass cannot contract below this radius. This therefore sets an upper limit on the temperature of a non - degenerate sphere

$ k T \approx \frac{G M m_p}{R} $

If thermonuclear reactions require a certain threshold temperature ($ T_i \approx 10^7 \rm K $) then below a certain mass degeneracy pressure will brake the collapse before thermonuclear reactions kick in

[1]$ M_c \approx \frac{\left(k T_i \right)^{3/4} \hbar^{3/2}}{G^{3/2} m_p^{2} m_e^{3/4}} \approx 0.015 M_{\odot} $

This mass is therefore an upper limit on the mass of a brown dwarf.