= Isothermal pressure-level transit radius
Let $R_p$ be a reference <radius> at <pressure> $P_0$ in an <isothermal atmosphere>. It can be a reference isobar on a <gas giant>, rather than a solid surface. Assume constant <temperature> $T$, constant <mean molecular weight> $\mu$ in units of hydrogen mass $m_H$, and approximately constant <gravitational acceleration> $g$. The <ideal gas law> and <hydrostatic equilibrium> give
$$
P=\frac{\rho k_BT}{\mu m_H},\qquad
\frac{dP}{dr}=-\rho g,\qquad
\frac{d\log P}{dr}=-\frac1H,\qquad H=\frac{k_BT}{\mu m_Hg}.
$$
Integrating from the reference isobar yields
$$
P(r)=P_0e^{-(r-R_p)/H},\qquad
R(P)=R_p+H\log\frac{P_0}{P}.
$$
Reducing the pressure by a factor $e$ therefore increases the pressure-level <radius> by one <atmospheric scale height>. Constant gravity is a thin-atmosphere approximation over the height range being modeled.
A pressure-level <radius> is not automatically the observed <transit radius>. Transmission depends on the line-of-sight <optical depth>, which also requires composition and <opacity>. Different reference pressures or cloud assumptions can therefore assign different pressure levels to the same observed <exoplanet transmission spectrum>.<Theoretical Transmission Spectra During Extrasolar Giant Planet Transits by Sara Seager and Dimitar Sasselov>[Atmospheric-annulus transmission method and sensitivity to clouds and thermodynamic conditions]{ref}
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