= Hydrostatic atmosphere with a constant temperature lapse rate
{title2=$\rho(z)/\rho_0=[1-\alpha z/T_0]^{g/(\mathcal R\alpha)-1}$}
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 $p'=-(g/\mathcal R T)p$ 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 $\alpha\to0$ is $\rho/\rho_0=\exp[-gz/(\mathcal R T_0)]$.
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