 Temperature from a transmission-band amplitude

For a thin hydrostatic isothermal atmosphere, $\Delta D\simeq 2R_p{N}_{H}H/R_{\star }^2$ and $H=k_{B}T/(\mu m_{u}g)$. Hence $T\simeq \mu m_{u}g\Delta D{R}_{\star }^2/(2k_B{R}_p{N}_H)$. A measured exoplanet transmission spectrum amplitude constrains $N_{H}H$, so gravity, mean molecular mass and the number $N_H$ of sampled atmospheric scale heights are additional inputs. In a clear fixed-composition model, $N_H\simeq \log ({\kappa }_{\mathrm{band}}/{\kappa }_{\mathrm{cont}})$; clouds or abundance gradients can change it.