= 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 $p(z)=p_0e^{-z/H}$ and $\rho(z)=\rho_0e^{-z/H}$, where the <scale height> is $H=k_BT/(mg)$.
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>.
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