## FANDOM

148 Pages

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.