 Solution

Assume a well-mixed ideal gas with constant $g,c_p$, no latent heat, and sufficiently strong surface heating to maintain dry convection near the surface. An adiabatically displaced unit mass has specific volume $V=1/\rho$, so the stated relation and hydrostatic equilibrium giveThis is the dry neutral lapse rate; infrared absorption alone does not guarantee that a radiative profile is steep enough to trigger convection. Combining the same equations with the ideal gas law yields $T\propto p^{R_a/c_p}$ along that adiabat.

High above the main infrared-emitting region, optical depth to space becomes small. A transparent-to-sunlight atmosphere has no substantial direct solar heating there; a thin grey layer absorbs the upwelling planetary infrared flux $Q=(1-\alpha )S_0/4$ and emits from both faces. In the optically thin radiative-equilibrium limit,The factor four in $Q$ is projected-area versus whole-sphere averaging; the additional factor two counts emission upward and downward by the tenuous layer. For Earth-like $S_0\simeq 1360\mathrm{W}{\mathrm{m}}^{-2}$ and $\alpha \simeq 0.30$, this is about $214\mathrm{K}$. Thus temperature approaches a height-independent skin value as $\tau \to 0$, rather than continuing the dry adiabat to zero. This is an asymptotic grey-atmosphere approximation, not a proof that an entire finite stratosphere is exactly isothermal.