dm = (4πr^{2}dr) ρ(r)

M = ∫_{0}^{R} dm = ∫_{0}^{R} 4πr^{2}ρ(r)dr

M = 4πρ ∫_{0}^{R} r^{2}dr

M = 4πρ ∫_{0}^{R} r^{2}dr = 4πρ |\frac{1}{3}r^{3}|_{0}^{R} = \frac{4}{3}πρR^{3}

\frac{dM_{r}}{dr} = 4πr^{2} ρ

F_{net} = F_{grav} + F_{top} + F_{bot}

F_{net} = −\frac{GM_{r} dm}{r^{2}} − A P_{r+dr} + A P_{r}

F_{net} = −\frac{GM_{r} (ρAdr)}{r^{2}} − A (P_{r+dr} − P_{r})

F_{net} = −\frac{GM_{r} (ρAdr)}{r^{2}} − A (P_{r} + \frac{dP}{dr} dr − P_{r})

F_{net} = −\frac{GM_{r} (ρA)}{r^{2}} dr − A (\frac{dP}{dr} dr)

\frac{F_{net}}{A} = −\frac{GM_{r} ρ}{r^{2}} dr − \frac{dP}{dr} dr

\frac{dP}{dr} = −\frac{GM_{r} ρ}{r^{2}}

\frac{ΔP}{R} = \frac{P_{surface} − P_{center}}{R} ∼ −\frac{GMρ}{R^{2}}

P V = NkT  or  P = nkT

n = \frac{ρ}{m}

n = \frac{ρ}{m̄}

P = \frac{ρ}{m̄} kT

μ = \frac{m̄}{m_{H}}

P = \frac{ρ}{μm_{H}} kT

P_{rad} = \frac{4σ}{3c} T^{4} = \frac{1}{3} aT^{4}