tokens: (), by using equation ( 36 ) for a clear atmosphere The red curve in Fig. 4 depicts our approximation, which follows the solution of Lecavelier des Etangs etal. ( 2008 ) when s , and is zero otherwise. Fig. 5 shows the amount by which our approximation underestimates the effective atmospheric thickness. It is at most E 1 (e ) 0.492 when s 0.561. We can apply the same line of reasoning to quickly determine which CIA features rise above the surface continuum. We use instead of equation ( 43 ), where k is the modified mean collision-induced cross-section with its minimum set at the surface collision-induced cross-section ( k s ), given by We then determine the spectral modulation between any two spectral regions with equation ( 34 ). The surface decreases the spectral modulation only if s or k s k in one of the two spectral regions. Figure 5. Amount by which the effective thickness of the atmosphere (in units of scaleheight) is underestimated for q 1 (see equations 11 and 12 ) with the approximation outlined in Section 3.4 as a function of the surface slant optical depth. Can we use this approximation to quickly determine the location of a surface from observations? If the observed spectral modulation is due to a single species and is less than that expected in a clear atmosphere, we can relate the observed spectral modulation, ( h / H ) obs , with respect to a local maximum in its cross-section ( i ) by Unfortunately, the surface location cannot be unambiguously determined because the mole fraction ( f i ) of the species is unknown. Indeed, equation ( 46 ) shows that the spectral modulation of a species increases with abundance in the presence of a surface, as demonstrated numerically by Btrmieux ( 2016 ) for a refractive boundary. The only way to break this degeneracy are two-fold: either the spectral signature is that of a major species (i.e. f i 1) and we can determine the location of the surface, or we can predict the location of the surface and then determine the abundance of the minor species. Given that refractive boundaries are dependent on the bulk properties of the exoplanetary atmosphere as well as the planetstar geometry, quantities that are retrieved during the analysis of transit data, location of refractive boundaries can be predicted in principle and used to place a lower limit on the abundance of detected minor species (Btrmieux 2016 ). 3.5 Transmission spectrum versus formalism To illustrate the impact of the choice of formalism on a computed transmission spectrum, we apply the following formalisms on simulations by Btrmieux ( 2016 ): the formalism of Lecavelier des Etangs etal. ( 2008 ), the exact solution embodied by equation ( 35 ) with q 1, and the approximation discussed in Section 3.4 . The chosen atmospheric model is that of a Jovian-like planet with an H 2 He atmosphere identical to Jupiters, a 600K isothermal profile, and a constant H 2 O abundance with altitude with a mole fraction of 10 8 . We choose the
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