Ideal-gas atmosphere in uniform gravity

An isothermal noninteracting gas in potential $mgz$ has an exponential vertical density and scale height $k_{B}T/(mg)$.

An isothermal atmosphere has constant temperature. Under constant gravity and ideal-gas hydrostatic equilibrium, its pressure and density decay exponentially with altitude.

For constant gravitational acceleration and temperature,

The pressure and density decay length in an isothermal atmosphere is $H=k_{B}T/(mg)$.

A static fluid in a uniform downward gravitational field obeys $dp/dz=-\rho g$.

A scale height is a characteristic distance over which a vertically stratified physical quantity varies. For a decreasing exponential $q(z)=q_0{e}^{-z/H}$, it is $H=-(d\log q/dz{)}^{-1}$. The local pressure scale height applies this logarithmic measure to pressure; density and pressure scale heights need not coincide when temperature varies.

A pressure scale height is the local distance over which pressure would change by a factor $e$ if its logarithmic gradient were constant. In hydrostatic equilibrium, $H_P=P/(\rho g)$; for an ideal gas it is $k_{B}T/(\mu m_{u}g)$. A small logarithmic pressure interval has depth approximately $H_P\Delta \log P$.