Hydrostatic atmosphere with a constant temperature lapse rate ($\rho (z)/{\rho }_0=[1-\alpha z/T_0{]}^{g/(\mathcal{R}\alpha )-1}$)

For an ideal gas with constant composition, set $\mathcal{R}=k_B/\bar{m}$, where $\bar{m}$ is the mean particle mass. Constant downward gravitational acceleration $g$ and $T=T_0-\alpha z>0$ give $p^{\prime }=-(g/\mathcal{R}T)p$ by hydrostatic equilibrium. Integration gives $p/p_0=(T/T_0{)}^{g/(\mathcal{R}\alpha )}$, and the ideal gas law gives the displayed mass density ratio. The isothermal limit $\alpha \to 0$ is $\rho /{\rho }_0=\exp [-gz/(\mathcal{R}{T}_0)]$.