kT = \frac{3}{2}mv^{2}

P = \frac{ρkT}{μm_{p}}

\frac{dP}{dR} = − \frac{GM_{int}ρ}{R^{2}}

M_{int} = − \frac{k}{Gμm_{p}} T R (\frac{R}{T} \frac{dT}{dR} + \frac{R}{ρ} \frac{dρ}{dR})

= − \frac{k}{Gμm_{p}} T R (\frac{d ln T}{d ln R} + \frac{d ln ρ}{d ln R})

\frac{dy/y}{dx/x} = \frac{x}{y} \frac{dy}{dx} = \frac{d ln y}{d ln x}

y = x^{p}

\frac{d ln y}{d ln x} = p

ρ_{gas} = K ρ_{mass}^{β}

β = \frac{μm_{p}σ^{2}}{kT}

ρ_{mass}(r) = K(1 + \frac{r}{R_{c}} )^{−3/2}

\begin{aligned}ρ_{gas}(r) & = K ρ_{mass}^{β} \\ & = C(1 + \frac{r}{R_{c}} )^{−3β/2} \end{aligned}

I(r) ∝ r^{(−3β+0.5)}

M ∼ \frac{Rv^{2}}{G}

t_{relax} ∼ t_{cross} \frac{N}{ln N}