Atmospheric scale height ($H$)

This is the atmospheric instance of a scale height.

For an isothermal ideal-gas atmosphere, $H=k_{B}T/(\mu m_{H}g)$. A strong transmission feature spanning $N_H$ scale heights has approximate transit-depth amplitude $2R_p{N}_{H}H/R_{\ast }^2$.

For an isothermal thin atmosphere with number density $n(z)=n_0{e}^{-z/H}$ and extinction cross-section ${\sigma }_{\lambda }$, a tangent ray near radius $R_p+z$ has height $z+{\ell }^2/(2R_p)$ at distance $\ell$ along the ray. The resulting Gaussian integral gives slant optical depthThe effective transit height is near ${\tau }_{\lambda }\sim 1$, so $z(\lambda )=H\log {\sigma }_{\lambda }+\mathrm{constant}$ if composition is constant. This explains the scattering slope of a transmission spectrum.

If an isothermal thin atmosphere has surface radius $R_p$, surface pressure $P_0$, and nearly constant gravity, the radius of a pressure level $P$ is

The column mass between pressures $P_0$ and $P_{\mathrm{top}}$ is $(P_0-P_{\mathrm{top}})/g$. A thin atmosphere over a spherical planet therefore haswhen its top pressure is negligible.