δGPE = −\frac{GM_{r}dm}{r}

dm = (4πr^{2}dr) ρ

M_{r} = \frac{4}{3}πr^{3} ρ

GPE = ∫_{0}^{R} −\frac{16π^{2}}{3} Gρ^{2}r^{4}dr

GPE = −\frac{16}{15} π^{2}Gρ^{2}R^{5}

GPE = −\frac{3}{5} \frac{GM^{2}}{R}

T_{crit} ∼ \frac{2}{3} \frac{Z_{1}Z_{2}e^{2}}{kr}

T_{crit} ∼ \frac{4}{3} \frac{μ_{m}Z_{1}^{2}Z_{2}^{2}e^{4}}{kh^{2}}

μ_{m} ≡ \frac{1}{\frac{1}{m_{1}} + \frac{1}{m_{2}}} = \frac{m_{1} m_{2}}{m_{1} + m_{2}}

ϵ_{pp} ∼ 1 ρX^{2} (\frac{T}{10^{6} K})^{4} \frac{J}{s}/kg/\frac{kg}{m^{3}}

ϵ_{CNO} ∼ 8 × 10^{−19} ρXX_{CNO} (\frac{T}{10^{6} K})^{19.9} \frac{J}{s}/kg/\frac{kg}{m^{3}}

ϵ_{3α} ∼ 5 × 10^{7} ρ^{2}Y^{2} (\frac{T}{10^{8} K})^{41} \frac{J}{s}/kg/ (\frac{kg}{m^{3}} )^{2}

dL = ϵ dm

dL = ϵ (4πr^{2}dr) ρ

\frac{dL}{dr} = 4πr^{2}ρ ϵ

dm = 4πr^{2}dr ρ