PX282 - D2b - radiation

P_{r a d} = \frac{1}{3} a T^{4}

\begin{aligned} \frac{d P_{r a d}}{d r} & = - ρ κ \frac{F}{c} \\ = \frac{d}{d r} (\frac{1}{3} a T^{4}) \\ - ρ κ \frac{F}{c} & = \frac{4}{3} a T^{3} \frac{d T}{d r} \\ ∴ F & = - \frac{4 a c}{3 κ ρ} T^{3} \frac{d T}{d r} \end{aligned}

L_{r} = 4 π r^{2} F (r) = - \frac{16 π r^{2} a c}{3 κ ρ} T^{3} \frac{d T}{d r}

L_{r} \propto \frac{d T}{d r}