2K + U = 0

K = \frac{3}{2} NkT

U = − \frac{3}{5} \frac{GM^{2}}{R}

N = \frac{M}{μm_{H}} = \frac{\frac{4}{3}πR^{3}ρ}{μm_{H}}

K = \frac{3}{2} kT (\frac{4πR^{3}ρ}{3μm_{H}}) = 2kT (\frac{πR^{3}ρ}{μm_{H}})

U = − \frac{3}{5} \frac{G(\frac{4}{3}πR^{3}ρ)^{2}}{R}

U = − \frac{16}{15} Gπ^{2}R^{5}ρ^{2}

2[2kT \frac{πR^{3}ρ}{μm_{H}}] = \frac{16}{15} Gπ^{2}R^{5}ρ^{2}

\frac{4kT}{μm_{H}} = \frac{16}{15} πGR^{2}ρ

R = \sqrt{\frac{15kT}{4πGρ μm_{H}}}

K = \frac{3}{2} kT (\frac{M}{μm_{H}} )

3kT \frac{M}{μm_{H}} = \frac{3}{5} \frac{GM^{2}}{R}

M = \frac{kT}{μm_{H}} \frac{5R}{G}

M = \frac{4}{3}πR^{3}ρ  ⟹  R = (\frac{3M}{4πρ})^{1/3}

M = (\frac{5kT}{Gμm_{H}} )^{3/2} (\frac{3}{4πρ})^{1/2}

σ_{λ} ∝ \frac{1}{λ}

σ_{λ} ≃ constant

σ_{λ} ∝ \frac{1}{λ^{4}}