= Slant optical depth of an isothermal atmosphere
For an isothermal thin atmosphere with <number density> $n(z)=n_0e^{-z/H}$ 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 depth>
$$
\tau_\lambda(z)\simeq\sigma_\lambda n(z)\sqrt{2\pi R_pH}.
$$
The effective transit height is near $\tau_\lambda\sim1$, so $z(\lambda)=H\log\sigma_\lambda+\mathrm{constant}$ if composition is constant. This explains the <scattering slope of a transmission spectrum>.
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