For a geometrically thin isothermal ideal-gas atmosphere, set $g=GM_p/R_p^2$ and
$$
H=\frac{k_BT_0}{\mu m_Hg}.
$$
Hydrostatic balance gives $P(z)=P_0e^{-z/H}$. The observed transit photosphere at $P_{\rm phot}$ is consequently at the <isothermal pressure-level transit radius>
$$
\boxed{R_{\rm obs}=R_p+H\ln\frac{P_0}{P_{\rm phot}}}.
$$
This assumes $P_{\rm phot}<P_0$, constant $T_0$, composition, and gravity, and an opacity that selects the stated pressure.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 2 / d / Solution

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
$$
R(P)=R_p+H\log\frac{P_0}{P},
\qquad
H=\frac{k_BT}{\mu m_Hg}.
$$


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 Incoming links: Isothermal pressure-level transit radius