Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 317 / 2 / i

In the thin radiative envelope, take enclosed mass $m(r)\simeq M$, luminosity $L_r\simeq L$, and constant mean molecular weight $\mu$. Combining radiative diffusion in a star with the hydrostatic pressure support equation givesThe ideal gas relation is $\rho =AP/T$, where $A=\mu m_u/k_B$, so the opacity law impliesHere $a_{\mathrm{rad}}$ is the radiation constant, distinct from the exponent $a$. Since $\nabla =(P/T)dT/dP$, separation gives $P^{a-1}dP=C{T}^{b-1}dT$. With photospheric $T=T_{\mathrm{eff}}$, the power-law opacity radiative envelope therefore hasThis expression assumes $a,b\ne 0$ and a positive bracket on the physical branch. If $b=0$, replace $(T^b-T_{\mathrm{eff}}^b)/b$ by $\log (T/T_{\mathrm{eff}})$; if $a=0$, the integrated relation is $\log (P/P_{\mathrm{phot}})=C(T^b-T_{\mathrm{eff}}^b)/b$, with the same logarithmic limit when $b=0$.