 Solution

The incident stellar power is intercepted over a disk and distributed over the spherical surface. With a black shortwave surface, no clouds and no shortwave atmospheric absorption, the global mean absorbed flux is $F_{\ast }=L_{⊛}/4$. Define $T_e=(F_{\ast }/\sigma {)}^{1/4}$.

To relate a radiating-level temperature to the ground, assume the intervening well-mixed gas follows a dry adiabat with constant $R_g,c_{pg}$:Emission from $p_{\mathrm{rad}}$ balances absorbed sunlight when $\sigma T(p_{\mathrm{rad}}{)}^4=F_{\ast }$. HenceThe final form uses the optically thick result in part (a). Since $p_{\mathrm{rad}}<p_s$, the surface is warmer than the effective emission temperature. Efficient horizontal redistribution alone does not mathematically fix the vertical lapse rate: the adiabatic/convective assumption is needed to obtain the requested dependence on $R_g/c_{pg}$. A different imposed vertical profile would give a different relation.