Isothermal atmosphere

An isothermal atmosphere has the same temperature at every height in the region modeled. With an ideal gas, uniform downward gravity and constant mean particle mass $m$, hydrostatic equilibrium gives $p(z)=p_0{e}^{-z/H}$ and $\rho (z)={\rho }_0{e}^{-z/H}$, where the scale height is $H=k_{B}T/(mg)$.

The local gradient is therefore set by a balance: larger thermal energy spreads particles over a greater height, whereas larger particle mass or gravity concentrates them lower down. Measuring the ratio of pressures at two heights yields $H=-\Delta z/\log (p_2/p_1)$ within this model. A real atmosphere can depart from the exponential law because temperature, composition or gravity changes with height. Being isothermal specifies a thermal profile; it does not independently establish that the fluid is at rest or in hydrostatic equilibrium.