2KE + GP E = 0

2KE = −GP E

KE = \frac{3}{2}NkT

GP E = −\frac{3}{5} \frac{GM^{2}}{R}

N = \frac{M}{μ m_{H}}

M = N μ m_{H}

2 [\frac{3}{2} \frac{M}{μ m_{H}} kT ] = \frac{3}{5} \frac{GM^{2}}{R}

M = \frac{4π}{3} R^{3} ρ

R = [\frac{3M}{4πρ}]^{1/3}

\frac{kT}{μ m_{H}} < \frac{1}{5}GM^{2/3} (\frac{4πρ}{3})^{1/3}

M_{J} = [\frac{5kT}{μ m_{H}G}]^{3/2} (\frac{3}{4πρ})^{1/2}