Check 1 integrates the slant optical depth along a chord numerically and compares the radius with the closed form. Check 2 compares the radius change between 1.25 and 1.4 um with H ln(sigma2/sigma1).

An atmosphere with scale height H = kT / (mu m_H g) is opaque at the radius where the slant optical depth reaches about 0.56. For an isothermal exponential atmosphere this gives an effective altitude z(lambda) = H ln( n0 sigma(lambda) sqrt(2 pi R0 H) / 0.56 ).

So the transit radius changes with wavelength by dz = H ln(sigma2/sigma1): each e-fold of cross-section adds one scale height. The depth is (R/Rs)^2; the spectral amplitude is only a few hundred ppm, which is why many transits are stacked and the noise falls as 1/sqrt(N).

Simplifications: schematic Gaussian absorption bands, isothermal limbs, equilibrium-like chemistry, a grey cloud deck with a partial cover fraction, photon noise plus an optional systematic floor. Ring thickness in the cross-section is exaggerated.