 Grey-atmosphere convection onset with thirteenth-power opacity

In a thin ideal-gas grey atmosphere with $T^4=T_e^4(1/2+3\tau /4)$, constant enclosed mass $M$ and luminosity $L$, and $\kappa ={\kappa }_0\rho T^{13}$, divide the stellar hydrostatic equation by radiative diffusion in a star. This gives $PdP/dT=16\pi acGM\mathcal{R}{T}^{-9}/(3{\kappa }_{0}L\mu )$. Integrating from $P=0$, $T^4=T_e^4/2$, yieldsLogarithmic derivative gives $\nabla =T^8/T_e^8-1/4$. For a monatomic gas gas, the Schwarzschild criterion reaches equality at $\nabla =2/5$, hence $T/T_e=(13/20{)}^{1/8}$. The purely radiative profile cannot be continued past this point as a convectively stable model.