Solution (source code)

= Solution

Assume an <isothermal atmosphere> in <hydrostatic equilibrium>, with constant gravity $g$, <mean molecular weight> $\mu$, and an <ideal gas> equation of state. Use $h$ for the <atmospheric scale height>, to distinguish it from the total observable height $H$. Then
$$
\frac{dP}{dz}=-\rho g=-\frac{\mu m_Hg}{k_BT}P,\qquad h=\frac{k_BT}{\mu m_Hg},
$$
so the <cloud-top pressure in an isothermal atmosphere> is
$$
\boxed{P_c=P_s\exp(-H_c/h)}.
$$
An opaque, approximately grey <exoplanet cloud deck> can flatten the <exoplanet transmission spectrum> if the remaining gas above it has little wavelength-dependent extinction. If its top nearly reaches the observable atmospheric extent, $H_c\simeq H$, this gives $P_c\simeq P_s e^{-H/h}$. More generally, a top at $N$ <atmospheric scale heights> has $P_c/P_s=e^{-N}$; illustrative values $N=5$ and $10$ give $6.7\times10^{-3}$ and $4.5\times10^{-5}$.

\b[A flat <exoplanet transmission spectrum> and $P_s$ alone do not determine a unique cloud-top <pressure>.] The <temperature>, gravity, <mean molecular weight> and cloud altitude, or equivalent opacity information, are also needed. A large <mean molecular weight> can itself suppress spectral features, so the <exoplanet cloud deck> interpretation is conditional.