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

\frac{dP_{rad}}{dr} = \frac{1}{3c} σT^{3} \frac{dT}{dr} = \frac{4}{3} aT^{3} \frac{dT}{dr}

\frac{dP_{rad}}{dr} = −\frac{κ̄ρ}{c} F_{rad}

−\frac{κ̄ρ}{c} F_{rad} = \frac{4}{3} aT^{3} \frac{dT}{dr}

F_{rad} = \frac{L_{r}}{4πr^{2}}

−\frac{κ̄ρ}{c} \frac{L_{r}}{4πr^{2}} = \frac{4}{3} aT^{3} \frac{dT}{dr}

\frac{dT}{dr} = −\frac{3κ̄ρL_{r}}{16πa c r^{2} T^{3}}

P V = NkT  ⇒  P = nkT = \frac{ρ}{m̄} kT

P = \frac{k}{m̄} ρT , T = \frac{m̄}{k} \frac{P}{ρ} , ρ = \frac{m̄}{k} \frac{P}{T}

\frac{dP}{dr} = \frac{k}{m̄} \frac{dρ}{dr} T + \frac{k}{m̄} ρ \frac{dT}{dr}

\frac{dP}{dr} = \frac{P}{ρ} \frac{dρ}{dr} + \frac{P}{T} \frac{dT}{dr}

P = C ρ^{γ}

\frac{dP}{dr} = C γρ^{γ−1} \frac{dρ}{dr} = γC \frac{ρ^{γ}}{ρ} \frac{dρ}{dr} = γ \frac{P}{ρ} \frac{dρ}{dr}

\frac{P}{ρ} \frac{dρ}{dr} + \frac{P}{T} \frac{dT}{dr} = γ \frac{P}{ρ} \frac{dρ}{dr}

\frac{dT}{dr} = \frac{T}{P} [(γ − 1) \frac{P}{ρ} \frac{dρ}{dr}]

\frac{dT}{dr} = \frac{T}{P} [γ \frac{P}{ρ} \frac{dρ}{dr} − \frac{P}{ρ} \frac{dρ}{dr}]

\frac{dT}{dr} = \frac{T}{P} [\frac{dP}{dr} − \frac{1}{γ} \frac{dP}{dr} ]

\frac{dT}{dr} = \frac{T}{P} (1 − \frac{1}{γ} ) \frac{dP}{dr}

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

\frac{dT}{dr} = − (\frac{m̄}{k}) (1 − \frac{1}{γ} ) \frac{GM_{r}}{r^{2}}

|\frac{dT}{dr} |_{actual} >= |\frac{dT}{dr} |_{adiabat} = (\frac{m̄}{k}) (1 − \frac{1}{γ} ) \frac{GM_{r}}{r^{2}}

P ∝ T^{α}  ⇒  ln P = α ln T

\frac{d ln P}{d ln T} = α

\frac{d ln P}{d ln T} = \frac{γ}{γ − 1}

\frac{d ln P}{d ln T} < \frac{γ}{γ − 1}