The Hidden Depths of Planetary Atmospheres

The bending of light due to refraction in a planetary atmosphere is essentially controlled by two quantities: the density scale height (H) and the refractivity (ν) of the atmosphere, both evaluated at the grazing radius of a ray—the deepest region reached by a ray. Although the refractivity of the atmosphere depends on its composition, it depends predominantly on its number density (n), which varies exponentially with altitude. We use the term "scale height" to signify the density scale height of the atmosphere, not the pressure scale height, which we always label explicitly.

A starting point for computing the lensing power of an atmosphere is the tabulated refractivity (see, e.g., Table 1 in Bétrémieux & Kaltenegger 2015) of different molecules (${\nu }_{{\mathrm{STP}}_{j}}$) measured at standard temperature and pressure (STP). Indeed, one first computes the STP refractivity of the atmosphere (${\nu }_{\mathrm{STP}}$) using

$$\begin{eqnarray}&&{\nu }_{\mathrm{STP}}=\displaystyle \sum _{j}{f}_{j}{\nu }_{{\mathrm{STP}}_{j}},\end{eqnarray} \tag{ 1 }$$

which depends on the sum of the STP refractivities of the individual species (${\nu }_{{\mathrm{STP}}_{j}}$) weighed by their abundances, quantified by their mole fraction (f_j). One then obtains the refractivity of the atmosphere at the desired density with

$$\begin{eqnarray}&&\nu =\left(\displaystyle \frac{n}{{L}_{0}}\right){\nu }_{\mathrm{STP}},\end{eqnarray} \tag{ 2 }$$

where L₀ is the number density of the atmosphere at STP, also known as the Loschmidt constant. The unit of atmospheric density that equates L₀ is an amagat. It is clear from Equation (1) that only the most abundant species contribute significantly to the atmospheric refractivity. Below a planet's homopause, the species making up the bulk of the atmosphere have a constant mole fraction with altitude, so that the STP refractivity of the atmosphere is also constant with altitude. Although one might argue that this statement is not true of exoplanets with a high proportion of condensable species, such as "ocean-planets" (Léger et al. 2004), the existence of these exoplanets is at this point speculative.

The path of a refracted ray through an atmosphere is described by an invariant that is equal to the ray's projected distance to the center of the planet with respect to the observer (Phinney & Anderson 1968), also known as the ray's impact parameter (b). The ray's impact parameter is related to its grazing radius (r) by

$$\begin{eqnarray}&&b=r(1+\nu ).\end{eqnarray} \tag{ 3 }$$

Without bending by refraction—ν is equal to zero—the impact parameter and the grazing radius are the same. Equation (3) can be rewritten to emphasize how much deeper a ray probes the atmosphere compared to what the impact parameter suggests when refraction is ignored. The relation

$$\begin{eqnarray}&&b-r=r\nu \end{eqnarray} \tag{ 4 }$$

shows that the difference in altitude between the region perceived to be probed (given by b) and the actual region probed (given by r) increases with the refractivity—and hence density—of the atmosphere at the grazing radius, and thus with decreasing altitudes, as illustrated in Figure 1.

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In an altitude region with a constant density scale height, the corresponding difference between the density probed (n) and the perceived probed density (n*), or apparent density, is then given by

$$\begin{eqnarray}&&\displaystyle \frac{{n}^{* }}{n}={e}^{-(b-r)/H}={e}^{-(r\nu /H)},\end{eqnarray} \tag{ 5 }$$

where the apparent density is the largest density reached if a ray was not bent by refraction. It is interesting to note that the factor $(r\nu /H)$ is a factor that also appears in the improved analytic expressions (Bétrémieux & Kaltenegger 2015) for the column abundance along a ray, as well as that for the ray deflection, and shown again in Section 2.5. Since this factor is a positive quantity, Equation (5) shows that the apparent density probed is always lower than the actual density probed. We want to stress that this factor is only a mathematical construct to map a change of altitude to a virtual change of probed density. In reality, rays do actually bear the signature of the actual density probed through the atmospheric extinction that they experience. Also, Equation (5) does not hold for an atmosphere with a nonconstant scale height. In that case, one must determine the change in density corresponding to the difference in altitude between the impact parameter and the grazing radius from the density altitude profile of the atmosphere.

The lowest region that stellar radiation can pass through an atmosphere without spiraling into the planet is the lower refractive boundary. This lower boundary (represented with the lb subscript throughout the paper) occurs at an altitude that satisfies the condition

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\nu }_{\mathrm{lb}}}{1+{\nu }_{\mathrm{lb}}}\right)\displaystyle \frac{{r}_{\mathrm{lb}}}{{H}_{\mathrm{lb}}}=1,\end{eqnarray} \tag{ 6 }$$

as derived by Bétrémieux & Kaltenegger (2015). If the atmosphere is thin compared to the planetary radius (R_P), then ${r}_{\mathrm{lb}}\approx {R}_{P}$, and one can then solve for the atmospheric refractivity at this lower boundary with

$$\begin{eqnarray}&&{\nu }_{\mathrm{lb}}=\displaystyle \frac{1}{({r}_{\mathrm{lb}}/{H}_{\mathrm{lb}})-1}\approx {H}_{\mathrm{lb}}/{R}_{P},\end{eqnarray} \tag{ 7 }$$

which is then much smaller than 1. In this approximation, we can see that the condition describing the location of the lower boundary (Equation (6)) is simply ${(r\nu /H)}_{\mathrm{lb}}\approx 1$. Knowing ${\nu }_{\mathrm{lb}}$, we can then use Equation (2), with the assumed bulk composition of the atmosphere, to determine the density where the lower boundary is located.

Since the difference in altitude between the apparent and actual probed region increases with density (Equation (4)), the maximum difference occurs at this lower boundary. With the help of Equation (6), we can see that the difference between the apparent and actual altitude of the lower boundary is given by

$$\begin{eqnarray}&&({b}_{\mathrm{lb}}-{r}_{\mathrm{lb}})={r}_{\mathrm{lb}}{\nu }_{\mathrm{lb}}={H}_{\mathrm{lb}}(1+{\nu }_{\mathrm{lb}}).\end{eqnarray} \tag{ 8 }$$

Since ${\nu }_{\mathrm{lb}}$ is typically much smaller than 1, this difference is about one density scale height, as illustrated in Figure 1. This expression is exact even in an atmosphere with a nonconstant scale height.

The fact that the difference between the apparent and actual altitude probed increases with density implies that the density scale height appears to change to an external observer even in an atmosphere where the scale height is constant with altitude. Figure 1 illustrates how rays that are evenly spread in grazing radius—corresponding to intervals of constant ratios of probed densities—map into their respective impact parameters for an atmosphere with a constant scale height. Figure 1 clearly shows that constant grazing radius intervals are mapped to smaller impact parameter intervals the closer a ray gets to the lower refractive boundary. Since the probed densities have not actually changed, the same ratio of density occurs over a smaller altitude interval, and the scale height appears to have decreased. The effective, or apparent, density scale height (H*) is given by

$$\begin{eqnarray}&&\displaystyle \frac{{H}^{* }}{H}=\displaystyle \frac{{db}}{{dr}}=(1+\nu )-\left(\displaystyle \frac{r\nu }{H}\right)\end{eqnarray} \tag{ 9 }$$

(Bétrémieux & Kaltenegger 2015) and shown in Figure 2. Six scale heights above the lower boundary, the effective scale height is essentially the same as the actual scale height. However, 2 scale heights from the lower boundary, the ratio of the effective to the actual scale height drops to less than 0.9, and it decreases dramatically to zero in the bottom scale height. It can be shown that the dominant term describing the curve in Figure 2 is simply

$$\begin{eqnarray}&&\displaystyle \frac{{H}^{* }}{H}\approx 1-{e}^{-(r-{r}_{\mathrm{lb}})/H}\end{eqnarray} \tag{ 10 }$$

for an atmosphere with a constant scale height. The fact that the effective density scale height decreases as the lower boundary is approached implies that the associated spectral features that probe these atmospheric regions also decrease in size (Bétrémieux 2016).

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A thin refractive boundary layer that distorts the apparent scale height of the atmosphere occurs only at relatively high densities, as shown for Jovian atmospheres of different temperatures in Table 3 of Bétrémieux (2016). Whereas this boundary could be observed in planets with thick atmospheres in our solar system in spectral regions of low opacity, the observational geometry of transiting exoplanets can hide these deeper regions (Sidis & Sari 2010) because it creates a refractive boundary located at a higher altitude.

This critical boundary (Bétrémieux & Kaltenegger 2014) occurs at an altitude where the atmospheric density deflects light from the stellar limb opposite the planetary limb toward the observer (see their Figure 2). This critical deflection (${\omega }_{c}$), in turn, depends on the planet–star geometry and is given by

$$\begin{eqnarray}&&\sin {\omega }_{c}=\displaystyle \frac{{r}_{c}+{R}_{\star }}{d},\end{eqnarray} \tag{ 11 }$$

where r_c is the critical grazing radius ($\approx {R}_{P}$), R_⋆ is the stellar radius, and d is the planet–star distance.

Atmospheric regions below the critical boundary cannot be probed because they are not back-illuminated by the host star from the observer's viewpoint and thus appear dark. Indeed, the atmosphere acts as a lens that severely distorts the imaged stellar disk. Although the imaged stellar disk varies substantially during the course of a planetary transit (Sidis & Sari 2010; García Muñoz & Mills 2012; Misra et al. 2014), one can use the simple geometry when the planet occults the center of its star to determine the average effects of refraction during the transit. Indeed, the actual locations of the imaged stellar limb on opposite sides of the planetary limb straddle the one provided by this simple picture. Furthermore, all radiative transfer codes, used in retrieval algorithms to interpret transit spectra, assume this simple geometry.

Determining the altitude where the critical boundary occurs must be done numerically because there exist no analytic expressions for the ray deflection at high densities. Indeed, the formula for the ray deflection (ω) in a constant scale height atmosphere can be expressed as a Taylor series by

$$\begin{eqnarray}&&\omega =\sqrt{\displaystyle \frac{2\pi r}{H}}\nu \left[1+\displaystyle \sum _{j=0}^{\infty }{D}_{j}{\left(\displaystyle \frac{r\nu }{H}\right)}^{j}\right].\end{eqnarray} \tag{ 12 }$$

A similar type of behavior exists for the column-integrated abundance (N) along the ray, namely,

$$\begin{eqnarray}&&N=\sqrt{2\pi {rH}}n\left[1+\displaystyle \sum _{j=0}^{\infty }{C}_{j}{\left(\displaystyle \frac{r\nu }{H}\right)}^{j}\right].\end{eqnarray} \tag{ 13 }$$

Here C_j and D_j are coefficients for the column abundance and deflection, respectively, of which only the first few are determined (Bétrémieux & Kaltenegger 2015). Both of these expressions are expected to diverge at the lower boundary. The factors outside the parentheses of both expressions are well-known formulae (see, e.g., Baum & Code 1953; Smith & Hunten 1990; Fortney 2005) valid at low densities. Throughout the paper, we use the new numerical ray-tracing method described by Bétrémieux & Kaltenegger (2015) to determine ω and N through the atmosphere, rather than use Equations (12) and (13). Percentage differences of the values of ω and N from the numerical method, as well as from Equations (12) and (13), compared to the well-known simple formulae, have already been shown in Figures 6 and 7 of Bétrémieux & Kaltenegger (2015) for a temperate Jovian and an Earth-like planet.

Bétrémieux & Kaltenegger (2014) showed that for low densities, the critical density (n_c)—atmospheric number density at the critical boundary—is given by

$$\begin{eqnarray}&&\left(\displaystyle \frac{{n}_{c}}{{L}_{0}}\right)=\displaystyle \frac{{\omega }_{c}}{{\nu }_{\mathrm{STP}}}\sqrt{\displaystyle \frac{H}{2\pi r}},\end{eqnarray} \tag{ 14 }$$

where the ratio inside the parentheses on the left-hand side of the equation is a number density expressed in units of amagat. This simple recipe, which ignores the concept of the lower boundary, cannot be applied to hot Jupiters because the critical boundary is located at high densities, where Equation (14) breaks down. Indeed, numerical results show that the critical boundary of a 1200 K isothermal hot Jupiter, orbiting a star cooler than an F0 spectral class, is located within 1 scale height of the lower boundary (see Table 3 of Bétrémieux 2016), well inside the refractive boundary layer. Using Equation (14) when

$$\begin{eqnarray}&&{\omega }_{c}\gtrsim \sqrt{\displaystyle \frac{2\pi H}{r}}\end{eqnarray} \tag{ 15 }$$

yields densities larger than at the lower boundary, which is unphysical.

Since the ray deflection diverges at the lower boundary, the critical boundary, which causes a finite ray deflection, is always located above it and defines the atmospheric region that can be probed in the course of an exoplanet's transit. Limb observations of a planetary atmosphere from an orbiting spacecraft, via solar or stellar occultations, are not similarly restricted because there will always be an atmospheric layer that bends the light sufficiently to reach the spacecraft. The only issue is whether opacity and differential refraction have sufficiently decreased the transmitted flux to the point of nondetection (see review of occultations by Smith & Hunten 1990). Observing the brightness of lunar surfaces during a lunar eclipse is potentially more powerful, as it provides a simultaneous coverage of the transmission of many atmospheric layers. The drawback is that scattered light contribution now comes from the entire limb of the planet. Both methods can in principle probe an atmosphere to its lower boundary.

The difference in altitude (or density) between the lowest atmospheric regions probed by exoplanet transmission spectroscopy (critical boundary) and solar/stellar occultations and lunar eclipses (lower boundary) can be problematic when trying to infer the transmission spectrum of an exoplanet from transmission data of a solar system analog, as was done for Titan (Robinson et al. 2014) and Saturn (Dalba et al. 2015). Indeed, the greater the difference in altitude, the more the spectral features in the exoplanet transit will be decreased compared to the spectrum built up from transmission data of its solar system analog. If the lower boundary and the critical boundary are widely separated, it is entirely possible for refraction to hardly impact the spectrum of solar system planets and to cut down most of the absorption features in an identical exoplanet observed in transit to the point of resulting in a nearly flat spectrum.

To determine whether refraction plays an important role in shaping the transmission spectrum of planetary atmospheres, one must determine the location of the refractive continuum with respect to spectral features of interest. Until recently, the few analytical formalisms (Lecavelier des Etangs et al. 2008; de Wit & Seager 2013) attempting to explain what shapes exoplanet transmission spectra could only do so for clear atmospheres. However, Bétrémieux & Swain (2017) developed an analytical formalism that explains how "surfaces" (surfaces, thick cloud decks, and refractive boundaries) decrease the contrast of absorption features in transmission spectra. At the heart of their formalism is the only published solution (see Equation (27) in Bétrémieux & Swain 2017) to the integral describing the effective radius of a transiting exoplanet whose atmosphere is completely opaque below a "surface." Their formalism is the only one that gives the expected result that as the optical thickness of the atmosphere decreases, the effective radius of the exoplanet approaches asymptotically that of the location of the "surface" without going below it. Indeed, the effective radius of a terrestrial planet without an atmosphere is simply the radius of the planet—the location of its surface.

Bétrémieux & Swain (2017) also showed that, in lieu of a detailed radiative transfer calculation, one can estimate the effective atmospheric thickness (h)—atmospheric contribution to the transmission spectrum—by comparing the mean atmospheric cross section (σ) to a "surface" cross section (${\sigma }_{s}$) with

$$\begin{eqnarray}&&h/H=\mathrm{ln}(\sigma /{\sigma }_{s})\end{eqnarray} \tag{ 16 }$$

and setting negative results to zero. This simple method produces at most half of a scale height error (see Figure 5 in Bétrémieux & Swain 2017) when $\sigma ={\sigma }_{s}$, i.e., when the atmospheric contribution is small. This is equivalent to the ad hoc method used to incorporate the effects of clouds in a few past analyses (Berta et al. 2012; Sing et al. 2015, 2016), except that it is here done in opacity space and that the errors associated with that method are now known. Equation (16) assumes that the atmosphere not only has a constant scale height but also is well mixed so that the mole fraction of species (f_j) are constant with altitude. The mean atmospheric cross section, which is given by

$$\begin{eqnarray}&&\sigma =\displaystyle \sum _{j}{f}_{j}{\sigma }_{j},\end{eqnarray} \tag{ 17 }$$

is thus also constant with altitude and is sufficient to characterize the opacity of an atmosphere. When the extinction in a spectral region is due predominantly to one species, the mean atmospheric cross section is simply given by the product of the abundance and the extinction cross section of that species ($\sigma \approx {f}_{j}{\sigma }_{j}$). The "surface" cross section is computed with

$$\begin{eqnarray}&&{\sigma }_{s}=\displaystyle \frac{{e}^{-{\gamma }_{\mathrm{EM}}}}{\sqrt{2\pi {b}_{s}H}{n}_{s}},\end{eqnarray} \tag{ 18 }$$

where ${\gamma }_{\mathrm{EM}}$ is the Euler–Mascheroni constant (≈0.577), while b_s and n_s are the impact parameter (Equation (3)) and the atmosphere number density, respectively, at the "surface."

Although the well-mixed and isothermal assumptions are not valid over all altitudes in planetary atmospheres, they are usually deemed sufficient when it comes to the retrieval of atmospheric parameters from transmission spectra. Indeed, most exoplanet atmosphere parameter retrieval algorithms (e.g., Madhusudhan & Seager 2009; Benneke & Seager 2012; Waldmann et al. 2015; Barstow et al. 2017; MacDonald & Madhusudhan 2017) assume that atmospheres are well mixed in the range of altitudes probed by transmission spectra. Regarding the effects of temperature on transmission spectra, it is clear that a vertical change in temperature changes the scale height, which in turn should impact the size of spectral features. In Section 3.4, we discuss further the effects of a non-isothermal temperature profile on transmission spectra, but for the present discussion we assume that the isothermal profile assumption holds over the altitude range probed by transmission spectroscopy.

We also note that the assumption of a constant mean cross section with altitude is not valid in the denser atmospheric regions where collisional broadening becomes important (see the Appendix for a detailed discussion about the impact of collisional broadening on transmission spectra). However, not only do prominent absorption bands in exoplanet transmission spectra typically probe lower pressures where cross sections are constant with altitude, but it is impossible to ascertain the impact of collisional broadening in a hot Jupiter's atmosphere because broadening parameters by H₂ and He are nonexistent except for a few molecules (Wilzewski et al. 2016; Barton et al. 2017). Even for air as a broadening agent, parameters are usually available only around 300 K (Hedges & Madhusudhan 2016). Given this lack of data, one can use our simple method because cross sections can only be computed for zero pressure and are thus independent of altitude in an isothermal atmosphere.

Equations (16)–(18) are valid for Rayleigh scattering, or atomic and molecular absorption. Since the absorption coefficient of CIA is proportional to the square of the density, the relevant equations for CIA are slightly different (Bétrémieux & Swain 2017) and are given instead by

$$\begin{eqnarray}&&{(h/H)}_{\mathrm{CIA}}=\displaystyle \frac{1}{2}\mathrm{ln}(k/{k}_{s}),\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&k=\displaystyle \sum _{i,j}{f}_{i}{f}_{j}{k}_{i,j},\end{eqnarray} \tag{ 20 }$$

and

$$\begin{eqnarray}&&{k}_{s}=\displaystyle \frac{{e}^{-{\gamma }_{\mathrm{EM}}}}{\sqrt{\pi {b}_{s}H}{n}_{s}^{2}},\end{eqnarray} \tag{ 21 }$$

where k and k_s are the mean atmospheric cross section and the "surface" cross section for CIA, respectively, while ${k}_{i,j}$ is the CIA cross section between species i and j. Simply put, the relevant length scale that determines the size of CIA features is in fact half the scale height (de Wit & Seager 2013; Bétrémieux & Swain 2017). Throughout the paper, we use the subscript c and lb instead of s with both types of cross section (σ and k) to identify them as "surface" cross section at the critical boundary and lower boundary, respectively.

What number density should one use in Equations (18) and (21) to determine, to first order, the various "surface" cross sections of the refractive boundary? One should use the apparent density (see Section 2.2) of the refractive boundary appropriate for the type of observations: apparent critical boundary (${n}_{s}={n}_{c}^{* }$) for exoplanet transits, and apparent lower boundary (${n}_{s}={n}_{\mathrm{lb}}^{* }$) for solar/stellar occultations and lunar eclipse observations (see Section 2.5). Indeed, the apparent density essentially translates the refractive case into a nonrefractive framework when it comes to the location of the ray. The concept of the "surface" cross section then determines, inside a nonrefractive framework, the required opacity of a homogeneous clear isothermal atmosphere that causes the effective radius of the exoplanet to be located at the "surface."

To illustrate how refraction impacts in a different way limb observations of solar system planets and exoplanet transits, we determine the location of refractive boundaries for solar system planets with thick atmospheres (Venus, Jupiter, Saturn, Uranus, and Neptune) for both observational geometries. We first use the ideal gas law on published temperature–pressure altitude profiles (see Table 1) for the various planets considered to determine the number density of their atmosphere as a function of altitude. We then interpolate densities on a finer altitude grid assuming that the density follows an exponential law with a constant density scale height between altitudes with known values. Temperatures are interpolated linearly with altitude. The altitude spacing on the fine grid is set at 1/3000 of the density scale height at the bottom of the published atmosphere. The temperature–pressure profiles listed in Table 1 come from the Galileo probe (Jupiter), a combination of American and Russian probes and orbiters (Venus), and occultation of radio signals from the Voyager 2 spacecraft by the planet's atmosphere (Saturn, Uranus, Neptune).

Table 1.  Published Temperature–Pressure Altitude Profiles Used in This Work, and the Corrective Factor Applied to the Temperature and Pressure to Account for an Atmosphere Composition Different from the Referenced Source (See Section 3.1)

Planet	References	($T/{T}_{0}$)	($P/{P}_{0}$)
Venus	Seiff et al. (1985)	⋯	⋯
Jupiter	Seiff et al. (1998)	⋯	⋯
Saturn	Lindal et al. (1985)	1.084	1.123
Uranus	Lindal et al. (1987)	1.141	1.075
Neptune	Lodders & Fegley (1998),	1	1
	and Lindal (1992)

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The detailed temperature–pressure altitude profile from the Voyager 2 radio occultations (Lindal et al. 1985, 1987; Lindal 1992) was retrieved simultaneously with the composition of the atmosphere. If one uses a different atmospheric composition than these published compositions, one must modify the temperature and pressure according to a scaling law (e.g., Lindal et al. 1985; Conrath & Gautier 2000), given by

$$\begin{eqnarray}&&\displaystyle \frac{T}{{T}_{0}}=\displaystyle \frac{\overline{m}}{{\overline{m}}_{0}}\end{eqnarray} \tag{ 22 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{P}{{P}_{0}}=\left(\displaystyle \frac{\overline{m}}{{\overline{m}}_{0}}\right)\left(\displaystyle \frac{{\nu }_{0}}{\nu }\right),\end{eqnarray} \tag{ 23 }$$

where the subscript 0 marks the old values of the temperature (T), pressure (P), mean molecular weight of the atmosphere ($\overline{m}$), and refractivity (ν). We use more recent determinations of the composition of the atmospheres of Saturn and Uranus (see the compositions in Table 2), and Table 1 lists the scaling factors for the temperature and the pressure that we apply to the published profiles prior to computing the densities.

Table 2.  Refractive Boundaries of Solar System Planets with Thick Atmospheres

Parameters	Venus	Jupiter	Saturn	Uranus	Neptune
Inputa
Volumetric radius (km)	6051.8	69911	58232	25362	24622
Mass (1024 kg)	4.8676	1898.3	568.36	86.816	102.42
Semimajor axis (au)	0.7233	5.204	9.752	19.201	30.047
Atmospheric Compositionb
H2	⋯	0.8620	0.8766c	0.825	0.80
He	⋯	0.1356	0.1189d	0.152	0.19
CH4	⋯	1.81 × 10−3	4.5 × 10−3	0.023	0.01
CO2	0.965	⋯	⋯	⋯	⋯
N2	0.035	⋯	⋯	⋯	⋯
Derivede
${P}_{c}^{* }$ (mbar)	154	24.7	15.3	6.5	4.9
Pc (mbar)	168	25.2	15.4	6.5	4.9
(rν/H)c	0.081	0.025	0.0091	0.0038	0.0024
${P}_{\mathrm{lb}}^{* }$ (bar)	2.3	1.9	${2.0}^{\star }\mbox{--}4.6$	${3.0}^{\star }\mbox{--}36$	${3.2}^{\star }\mbox{--}2.4$
Plb (bar)	8.7	12.2	5.6⋆–25	8.1⋆–460	8.8⋆–13
(rν/H)${}_{s}$ f	4.8	1.3	0.26	0.30	0.71
σc (10−27 cm2 molec−1)	2.7	1.5	1.6	5.0	8.3
${\sigma }_{\mathrm{lb}}$ (10−29 cm2 molec−1)	30	2.5	0.62–1.1⋆	0.14–1.1⋆	1.2
kc (10−45 cm5 molec−2)	0.88	1.6	2.1	11	29
klb (10−49 cm5 molec−2)	102	5.3	0.56–1.7⋆	0.025–0.81⋆	1.0
$\mathrm{ln}({\sigma }_{c}/{\sigma }_{\mathrm{lb}})\,$ g	2.2	4.1	5.0⋆–5.5	6.1⋆–8.2	6.5

Notes.

^aSource: planetary fact sheets available on the NASA Space Science Data Coordinated Archive (NSSDCA) website (https://nssdc.gsfc.nasa.gov/). ^bSource: Lodders & Fegley (1998), unless specified otherwise. Quantities are mole fractions. ^cAdjusted so that the sum of all mole fractions is 1. ^dSource: Conrath & Gautier (2000). ^eLower boundaries for Saturn, Uranus, and Neptune are located deeper than the highest pressures in their published temperature–pressure profile (Table 1). Quantities related to the location of the lower boundary usually show two values that bracket the range of possible solutions: one, marked with a ^⋆, that assumes a constant $(r/H)$ ratio below the deepest layer, and another one that assumes an adiabatic profile. For Neptune, both solutions give about the same value. ^fEvaluated at the bottom of the atmosphere model listed in Table 1. The lower refractive boundary is located within the extent of the atmosphere model when this value is greater than 1. ^gDifference in effective atmospheric thickness between the two refractive continuums, expressed in density scale height.

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Although we use the same composition and temperature–pressure profile for Neptune as originally published by Lindal (1992), the model atmosphere of Neptune presents its own set of challenges, namely, that the altitudes at which a given pressure and temperature occur are not specified. Only in Lodders & Fegley (1998), based on Lindal (1992), are the altitudes specified, but only for pressures of 1 bar or lower. We compute the various altitude intervals between adjacent points for pressures greater than 1 bar by assuming that the temperature and pressure follow an adiabatic law with a constant ratio of specific heat (see Section 3.4), which is itself determined empirically from the temperature–pressure power-law index in each altitude interval of the model atmosphere.

For each planet, we compute the deflection for 80 rays evenly spread in grazing altitude from the refractive boundary upward across the vertical extent of its atmosphere with MAKEXOSHELL, which uses the new ray-tracing algorithms described by Bétrémieux & Kaltenegger (2015) and is accurate over a large dynamical range of densities. This Fortran code numerically integrates along the path of rays on the fine altitude grid, starting from the density profile of the atmosphere and its composition. The required refractivity is determined from the density and composition of the atmosphere with Equation (2), while the density scale height is derived directly from the altitude dependence of the density profile. Early versions of MAKEXOSHELL (Bétrémieux & Kaltenegger 2015; Bétrémieux 2016) required human intervention at a few intermediate steps. Given the aim of this paper, which requires computing the location of the refractive boundary over some part of parameter space, we have substantially improved MAKEXOSHELL such that all of the steps described in Sections 3.1 and 3.2 (except for Equations (22) and (23), which modify the input temperature–pressure profile) are now fully automated.

Most of the improvement pertains to the automatic determination of the location of the refractive boundary itself and the subsequent setup of the rays, which is done in the following fashion. We first determine the location of the lower boundary by evaluating the left-hand side of Equation (6) on the fine altitude grid and finding the lowest altitude where that value is just barely less than 1. We set up the rays using this boundary as the location of the deepest ray and compute the deflection of each ray across the limb. We then interpolate the deflection values onto the fine altitude grid by assuming that the ray deflection follows an exponential with a constant scale height between the computed values. We also compute on the fine altitude grid the critical deflection of the planet–star system (Equation (11)) using the inputs from Table 2 and the radius of the Sun (6.96 × 10⁵ km). We choose for our critical boundary the highest altitude where the ray deflection is still larger than the critical deflection. We then determine, from their grazing altitudes, the impact parameter of both boundaries (Equation (3)). From a planet's temperature–pressure altitude profile, we can translate grazing altitudes and impact parameters to actual and apparent densities, as well as pressures, by interpolation rather than from Equation (5), which is valid only for an isothermal atmosphere. From the densities, we finally compute the various "surface" cross sections (Equations (18) and (21)).

The results from our calculations are displayed in Table 2. As previously demonstrated by Bétrémieux & Kaltenegger (2014) and Bétrémieux (2016), the critical pressure is smaller for a planet farther away from its star. We can also see that at the low pressures of the critical boundary the ${(r\nu /H)}_{c}$ factor is much smaller than 1 and the actual and apparent critical pressures are similar. For our Jovian planets, the critical pressure is located at low pressures, and refraction hides their tropospheric clouds. For Venus, the situation is reversed, and the critical boundary is located below the optically thick clouds of Venus, whose top is situated at an average pressure of 37 mbar (or an average altitude of 70 km), as previously pointed out by García Muñoz & Mills (2012). Thus, for a Venus analog, dominated by hazes and clouds, refraction plays very little role on its transmission spectrum.

For solar system planets, lower boundaries are located much deeper in the atmosphere than critical boundaries. For Venus and Jupiter, where in situ measurements are available, we determine that the lower boundary occurs at pressures on the order of 10 bar, deep in their troposphere, which translates to apparent pressures of about 2 bar. For Saturn, Uranus, and Neptune, the model atmospheres retrieved from Voyager 2 radio occultations do not go deep enough to probe the lower boundary. Indeed, the ${(r\nu /H)}_{s}$ factor at the bottom of the model atmosphere is for these three planets less than 1, and we cannot use the method described earlier to determine the location of the lower boundary. Instead, we consider two possible solutions that bracket the possible range of temperature gradients in these deeper regions. One solution assumes that the temperature–pressure profile follows an adiabatic law. In this case, we extrapolate the temperature profile to larger pressures using the same ratio of specific heat as in the deepest region of the model atmosphere and then proceed as before. The other solution, almost equivalent to an isothermal profile, assumes that $(r/H)$ is constant downward from the bottom of the model atmosphere. In this approximation, the density at the lower boundary is simply given by ${\nu }_{\mathrm{lb}}\approx {H}_{s}/{r}_{s}$ (see also Equation (7)), where quantities are evaluated at the bottom of the measured atmosphere. It is interesting to note that in the case of Neptune, even though the two solutions of the pressure of the lower boundary differ, the "surface" cross section is identical to two significant digits. This clearly shows that knowing the deepest pressure probed by refraction is not enough to understand what happens to absorption features because the density scale height of the atmosphere also matters.

The large difference between the critical and the lower boundary, shown in Table 2, can potentially be a problem when trying to infer the transmission spectrum of an exoplanet analog to a solar system planet, as we already pointed out in Section 2.5. For example, Dalba et al. (2015) recently used solar occultations viewed from the Cassini spacecraft to build up a transmission spectrum of Saturn's atmosphere. They claimed that refraction was responsible for the observed continuum, which they estimated was located at a pressure of 1.0 ± 0.5 bar. They further showed that the continuum had been shifted upward by refraction from its actual location of about 2 bar, resulting in about a 10 part-per-million (ppm) increase in the transit depth of the continuum and a corresponding decrease of the size of spectral features to a maximum size of about 90 ppm. They concluded that since cold planets such as Saturn could show as much as 90 ppm of spectral modulation, it demonstrated that transmission spectroscopy was a viable technique to study cold long-period exoplanets analog to Saturn.

Our results demonstrate that this is not the case. For a solar occultation, the relevant refractive boundary is the lower boundary, which their observations support. Indeed, their continuum is located at a pressure of about 1 bar, a slightly lower pressure than the possible range of pressures of 2–4.6 bar of the apparent lower boundary of Saturn. We can only deduce that the continuum in their spectrum must be due not only to refraction but also to some opacity from clouds and gas extinction. However, we determine that the critical boundary of Saturn is located at about 15 mbar, a much lower pressure than the claimed location of their continuum. We can estimate the change in transit depth due to the change in the location of the continuum. We compute that Saturn's critical boundary is located about 150 km (Δz) above its 1 bar pressure level, which results in a change of transit depth ($2{R}_{P}{\rm{\Delta }}z/{R}_{\star }^{2}$) of about 36 ppm. Figure 7 in Dalba et al. (2015) shows five prominent spectral features from 1 to 5 μm, three of which have peaks about 45 ppm above the continuum, and the other two stand at 55 and 90 ppm above the continuum. We can see that a 36 ppm upward shift of the continuum severely decreases the size of all spectral features except for the largest one, which is still reduced by about 40% in size. Hence, the transmission spectrum of a cold long-period exoplanet is probably nearly featureless except for the strongest features, contrary to the conclusions reached by Dalba et al. (2015).

A quick way to determine how much spectral features are reduced by refraction, which does not require a detailed radiative transfer computation, is through the "surface" cross sections, which we provide for both refractive boundaries in Table 2. We can compare the mean cross section of the atmosphere to the "surface" cross section, which essentially defines the minimum mean cross section of the atmosphere that produces spectral features. Any spectral features below that "surface" cross section will be severely decreased as demonstrated in Figure 6 in Bétrémieux & Swain (2017). This comparison is trivial in the case in which one species is predominantly responsible for the opacity in a given spectral region, as only the abundance and the cross section of that species matter (Equation (17)). For CIA, we can hold the exact same type of reasoning, except that the mean cross section depends on the product of the abundances of two responsible species (Equation (20)) and the comparison must be done to the "surface" CIA cross section.

As an example, let us suppose that the extinction cross section of a species of interest ranges from 10⁻²⁵ to 10⁻²¹ cm² molec⁻¹ inside an observed spectral range. The maximum spectral modulation—change of effective thickness, expressed in scale height or transit depth—from a single species is simply given by the natural logarithm of the ratio of the maximum and minimum values of the cross section (see the detailed discussion in Sections 3.3 and 3.4 in Bétrémieux & Swain 2017), which is here 9.2 scale heights. For CIA, the same ratio of cross sections would lead to half the spectral modulation (Equation (19)). The "surface" cross section of Uranus is 5 × 10⁻²⁷ cm² molec⁻¹. If our species of interest makes up the bulk of the atmosphere and f ≈ 1, then no part of its cross section is hidden by refraction. If the abundance of our species is 10⁻⁴, then the mean cross section ranges from 10⁻²⁹ to 10⁻²⁵ cm² molec⁻¹, and the maximum spectral modulation that can be observed is given by $\mathrm{ln}({10}^{-25}/5\times {10}^{-27})$, or about 3 scale heights. In this particular example, the maximum size of the spectral feature of our species has been reduced by more than a factor of three from 9.2 to 3 scale heights. If the abundance of our species is only 10⁻⁶, no part of the mean cross section has values above the "surface" cross section, and that species is to first order well hidden by refraction. Thus, the abundance of a species controls the size of its spectral features above the refractive continuum (Bétrémieux 2016) and above the continuum created by clouds and surfaces (Bétrémieux & Swain 2017).

We can evaluate how many more scale heights of atmosphere the refractive continuum in an exoplanet transmission spectrum hides compared to a spectrum constructed from solar system observations. This is done by computing $\mathrm{ln}({\sigma }_{c}/{\sigma }_{\mathrm{lb}})$, whose values for the different planets are listed at the bottom of Table 2. This difference is at least 4 scale heights for the giant planets and generally increases as the orbital distance increases. Uranus displays a much larger value for its solution of the adiabatic extrapolation because the lower boundary is located at a high pressure of 460 bar, a direct consequence of the steep temperature gradient at the bottom of its model atmosphere. It is clear from Table 2 that the shift in the refractive continuum is large for exoplanets that are analogs to the cold long-period gas giants in our solar system, and that transmission spectra inferred from solar system observations overestimate the size of spectral features compared to exoplanet transits.

The first-order method described at the end of Section 3.2 to evaluate the impact of refraction on the size of spectral features assumes that the atmosphere is well mixed and that its scale height is constant with altitude. In this section, we evaluate the impact of a departure from the second assumption by looking in detail at the atmosphere of Jupiter, for which we have an in situ measurement of the temperature–pressure altitude profile from the Galileo probe (Seiff et al. 1998). We ignore on purpose the impact of clouds, as the effects on transmission spectra of an optically thick cloud deck (Bétrémieux & Swain 2017) and their patchy distribution around the planetary limb (Line & Parmentier 2016) are now well understood and are not the focus of this paper.

Since we assume that the atmosphere is well mixed, the opacity of the atmosphere can be described by a single quantity, the mean extinction cross section of the atmosphere (see Equation (17)), which is constant with altitude. We then obtain the altitude-dependent limb optical depth (τ_σ) with

$$\begin{eqnarray}&&{\tau }_{\sigma }=N\sigma ,\end{eqnarray} \tag{ 24 }$$

where N is the altitude-dependent column-integrated abundance (N) along the limb, computed numerically by MAKEXOSHELL (Bétrémieux & Kaltenegger 2015) from the Jovian density altitude profile. We have also upgraded MAKEXOSHELL to compute numerically the column-integrated square of the density ($\langle {N}^{2}\rangle$) along the limb, which we use to compute the altitude-dependent limb CIA optical depth (τ_k) with

$$\begin{eqnarray}&&{\tau }_{k}=\langle {N}^{2}\rangle k.\end{eqnarray} \tag{ 25 }$$

MAKEXOSHELL does these column integrations by first computing the refractivity of the atmosphere from its composition (see Section 2.1) and tracing the ray through the atmosphere outward from a specified grazing altitude (Bétrémieux & Kaltenegger 2015). We treat the nonrefractive case by setting the atmospheric refractivity to zero, irrespective of the atmospheric composition.

Figure 3 shows the limb optical depth altitude profile for atomic and molecular extinctions (triple-dotted-dashed line), as well as CIA (light-blue line), for a range of mean atmospheric cross sections in the nonrefractive case. The sum of both optical depths (black line) is also shown, but only for a mean atmospheric CIA cross-section value of 3 × 10⁻⁴⁷ cm⁵ molec⁻². Under the well-mixed nonrefractive atmosphere assumption, the optical depth traces the density profile of the atmosphere. As expected, we can see from their differing slopes that the relevant scale height for CIA optical depth is about half that of the atomic and molecular extinction optical depth (de Wit & Seager 2013; Bétrémieux & Swain 2017). Figure 3 also shows that the atmosphere can be separated into two regions: a lower-altitude region where CIA is the dominant source of opacity, and a higher-altitude region where atomic and molecular extinctions are. The span of altitudes where the two different opacity sources have about the same optical depth is fairly small. The transition occurs at an altitude where the density $n\approx \sqrt{2}(\sigma /k)$. As the (σ/k) ratio increases, the transition occurs at a higher density, or lower altitude.

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The slope of the limb optical depth with altitude is dependent on the scale height, and hence the temperature, of the atmosphere. Shallower slopes occur in the higher-temperature regions, or the troposphere (low altitude) and the thermosphere (high altitude). The changing temperature with altitude of Jupiter's atmosphere is responsible for the changing slope with altitude of the optical depth. If Jupiter's temperature was isothermal, the slope would be nearly constant with altitude, as gravity only modifies slightly the scale height of the atmosphere with altitude. Refraction modifies the limb optical depth only in the deeper regions of Jupiter's atmosphere (red line) and causes the limb optical depth to increase dramatically with decreasing altitude, essentially mimicking a surface (Bétrémieux & Kaltenegger 2015). Even in the absence of clouds, the effect of Jupiter's refractive boundary layer (also discussed in Sections 2.2 through 2.5) hides the temperature signature of its troposphere in an occultation observation geometry, as the slope of the limb optical depth with altitude never becomes shallower at lower altitudes and instead becomes infinite.

From the altitude-dependent limb optical depth, we can compute the effective atmospheric thickness of Jupiter's atmosphere, viewed in an exoplanet transit geometry, using the generalized vertical integration scheme presented by Bétrémieux & Swain (2017). Figure 4 shows how the atmospheric thickness of Jupiter's atmosphere, referenced to the unrefracted 1 bar pressure level, changes with the mean atomic and molecular extinction cross section of the atmosphere. This is shown for several values of the mean CIA cross section in the nonrefractive case (see figure caption), as well as for refractive cases without any CIA contribution to the total optical depth. Figure 5 focuses on the lower regions of the atmosphere with an additional curve (orange line) that has no contribution from either CIA or refraction. The range of values chosen for the mean CIA cross section spans values of the H₂–H₂ CIA cross section measured from 150 to 1000 K (Borysow 2002).

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The signature of a non-isothermal temperature structure is manifest by the nonconstant slopes of the various lines in Figures 4 and 5 representing the nonrefractive cases (see Section 3.4 for a detailed discussion), except at very low opacities, where the effective atmospheric thickness tends to a minimum and the slope tends to zero. The latter is a signature of the underlying constant mean CIA cross section used in the calculation, which becomes the dominant opacity source at these low mean cross-section values. As the mean CIA cross section increases, this minimum effective thickness also increases. In the case of the orange curve in Figure 5, which does not include any CIA opacity, the minimum effective thickness simply coincides with the bottom of the atmosphere, defined by the lowest altitude probed by the Galileo probe, which we treated as a "surface" below which the atmosphere is completely opaque.

Figures 4 and 5 also show the minimum effective thickness of Jupiter's atmosphere from the refractive continuum for both solar system observations (lower boundary) and exoplanet transits (critical boundary). As with CIA opacity, the effects of refraction are only apparent in spectral regions of low opacity from atomic and molecular sources of extinction. Comparison with published CIA cross sections (e.g., Borysow et al. 2001; Borysow 2002) reveals that Jupiter's lower boundary (dot-dashed line) hides its deep troposphere and fills in partly the minima between H₂–H₂ CIA bands. In contrast, the critical boundary is located almost 90 km above the lower boundary and hides all but the strongest CIA features.

To test how well the "surface" cross-section concept works for refraction in the case of a non-isothermal atmosphere, we need to see whether the effective thickness of a clear nonrefractive atmosphere matches that of the refractive continuum when the mean cross section of the clear atmosphere equals the "surface" cross section. We thus compute the critical boundaries, and their corresponding "surface" cross sections, for a Jupiter-twin orbiting various stars (G2, K5, M2, and M5 spectral class) to see how well these different "surface" cross sections track the effective thickness of the refractive continuum. To determine the orbital distance of a Jupiter-twin around stars with a spectral class different from our Sun, we used the method described in Bétrémieux & Kaltenegger (2014), with the stellar effective temperatures and radii listed in their Table 1. The method finds the orbital distance that holds the radiation flux at the top of the atmosphere identical to that received by Jupiter in our solar system, assuming that the luminosity of the star is given by that of a perfect blackbody at the specified effective temperature.

The top four plus signs in Figure 5 show the surface cross section associated with the continuum of the critical boundary of a Jupiter-twin orbiting a G2, K5, M2, and M5 star (from top to bottom), while the bottom plus sign shows the surface cross section associated with the continuum of the lower boundary. The fact that all the plus signs fall very closely onto the orange curve shows that the "surface" cross section does work to first order, even in a non-isothermal atmosphere, and that it can be used to determine the mean cross-section value below which spectral features may be hidden by the refractive boundary. However, since the scale height is not constant, it is not straightforward to then translate an effective thickness expressed in units of scale height, obtained with Equations (16) and (19), into a transit depth. Instead, this must be computed numerically, as we have done in this section.

Figures 4 and 5 are particularly interesting because they illustrate the potential signatures of a non-isothermal atmosphere in an exoplanet's transmission spectrum, as we now demonstrate. Indeed, Lecavelier des Etangs et al. (2008) have shown that for a clear well-mixed atmosphere the spectral change in the effective atmospheric thickness (h) of the planet is connected to the spectral change of the atmosphere's opacity by

$$\begin{eqnarray}&&\displaystyle \frac{{dh}}{d\lambda }={H}_{P}\displaystyle \frac{d(\mathrm{ln}\sigma )}{d\lambda },\end{eqnarray} \tag{ 26 }$$

where λ is the observed wavelength of the radiation. Bétrémieux & Swain (2017) have shown analytically that the pressure scale height (H_P) should be replaced with the density scale height (H) for sources of opacity besides CIA (see also Equation (16)). For CIA, the relevant length scale is H/2.

The density scale height is given by

$$\begin{eqnarray}&&\displaystyle \frac{1}{H}=\displaystyle \frac{\overline{m}g}{{k}_{B}T}+\displaystyle \frac{{{\rm{\nabla }}}_{z}T}{T}=\displaystyle \frac{1}{{H}_{P}}+\displaystyle \frac{{{\rm{\nabla }}}_{z}T}{T}\end{eqnarray} \tag{ 27 }$$

(Chapter 1 in Chamberlain & Hunten 1987), where $\overline{m}$ is the mean molecular weight of the atmosphere, g is the gravitational acceleration, k_B is the Boltzmann constant, ${{\rm{\nabla }}}_{z}T$ is the vertical temperature gradient, and H_P is the pressure scale height. The pressure and density scale height are identical only in isothermal atmospheric regions where ${{\rm{\nabla }}}_{z}T$ is zero. Atmospheric regions with similar temperatures can have different scale heights because the vertical temperature gradient is not necessarily the same. Regions with negative gradients, such as the troposphere, have a larger scale height than the pressure scale height, while regions with positive gradients, such as the lower thermosphere, have a smaller scale height than the pressure scale height.

How much larger than the pressure scale height can the density scale height be? The largest negative vertical temperature gradient sustained by an atmosphere, before the atmosphere becomes convectively unstable and restores that gradient, is given by

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{z}T=-\displaystyle \frac{g}{{C}_{P}}=-\left(\displaystyle \frac{\gamma -1}{\gamma }\right)\displaystyle \frac{\overline{m}g}{{k}_{B}}\end{eqnarray} \tag{ 28 }$$

for a dry adiabat. Here $\gamma ={C}_{P}/{C}_{V}$ is the ratio of specific heats, where C_P and C_V are the gas specific heat for constant pressure and constant volume, respectively. This means that for a dry adiabat,

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\nabla }}}_{z}T}{T}=-\left(\displaystyle \frac{\gamma -1}{\gamma }\right)\displaystyle \frac{1}{{H}_{P}},\end{eqnarray} \tag{ 29 }$$

which yields, after substitution in Equation (27) and a few algebraic manipulations, the simple result that

$$\begin{eqnarray}&&H=\gamma {H}_{P}.\end{eqnarray} \tag{ 30 }$$

To first order, γ = 5/3 for monatomic gases, and γ = 7/5 for diatomic gases. Hence, the density scale height is significantly larger (about 40%) than the pressure scale height in precipitation-free convective regions of H₂ and N₂ atmospheres.

Since the density scale height is independent of wavelength, Equation (26) (with H_P replaced by H) can be recast as

$$\begin{eqnarray}&&\displaystyle \frac{{dh}}{d(\mathrm{ln}\sigma )}=H,\end{eqnarray} \tag{ 31 }$$

which is simply given by the slopes of the various curves in Figures 4 and 5. Thus, Figures 4 and 5 show how the temperature profile in the Jovian atmosphere modulates the size of spectral features in its transmission spectrum. The same change in $\mathrm{ln}\sigma$ maps to a different change in the effective atmospheric thickness depending on the probed atmospheric region. This is readily seen also by inspecting in Figure 5 how the minimum effective thickness due to CIA changes with CIA opacity. Indeed, a one-order-of-magnitude change from 3 × 10⁻⁵⁰ cm⁵ molec⁻² to 3 × 10⁻⁴⁹ cm⁵ molec⁻² changes the minimum effective thickness of the atmosphere by about 60 km, whereas a change by two orders of magnitude from 3 × 10⁻⁴⁷ cm⁵ molec⁻² to 3 × 10⁻⁴⁵ cm⁵ molec⁻² changes it by only 40 km. Except in the troposphere and the lower thermosphere, where temperature changes are large, the effects of a changing temperature with altitude are otherwise very subtle, as pointed out by Barstow et al. (2013) in their study of the effects of the assumption of an isothermal temperature profile on the retrieved abundances of species from exoplanet transmission spectra.

This modulation of the size of spectral features by an atmosphere's temperature profile occurs in all transmission spectra of exoplanets, and the temperature change from the stratosphere to the thermosphere has been detected in a few exoplanets (HD 189733b, HD 209458b) from high spectral resolution transmission spectra centered on the core of atomic lines (Vidal-Madjar et al. 2011, 2013; Huitson et al. 2012). The transmission spectrum of a non-isothermal atmosphere is distorted compared to that of an isothermal atmosphere in a similar way to how our reflections are distorted by fun house mirrors. Regions of larger scale height stretch in the vertical direction spectral features that probe these regions, while regions of smaller scale height compress them. It is interesting to note that in the case of Jupiter's atmosphere, the lower refractive boundary modifies the slope in the troposphere and masks the signature of its decreasing temperature with altitude in transmission spectra built up from occultation measurements. Jupiter's critical boundary hides it even further. In hotter atmospheres, refractive boundaries are located deeper (see Section 4.2 and Bétrémieux 2016) and may not be as efficient as Jupiter's atmosphere at hiding the temperature signature of the troposphere, but that could be accomplished by clouds instead.

Having demonstrated the use of the "surface" cross-section concept as a metric to quantify the effects of refraction on the giant planets in our solar system, we now extend that concept to exoplanets and explore how the "surface" cross section associated with the critical boundary changes with various parameters. We consider parameters that affect the lensing power of the atmosphere, such as the scale height and refractivity of the planet's atmosphere, as well as the size of the planet, but also the spectral class of the host star, which drives the temperature of the atmosphere. We use the same method as discussed in Section 3.2 to determine the location of the critical boundary. The only difference is the input density altitude profile. The radius of the planet is defined at a reference pressure of 1 bar, while the bottom of the atmosphere is set at a pressure of 500 atm. We spread 80 rays in constant altitude intervals over 30 pressure scale heights, of which the latter is evaluated at the bottom of the atmosphere.

As in Bétrémieux (2016), we assume that exoplanets are perfectly spherical and that their atmospheres follow a density altitude profile corresponding to an isothermal temperature–pressure profile at the mean planetary emission temperature (T_e), with a gravitational acceleration that varies with altitude. We use a Bond albedo of 0.30, a value similar to Earth and the gas giants in our solar system. The mean planetary emission temperature is identical to the equilibrium temperature (T_eq), often discussed in the exoplanet literature (Seager 2008), with an even heat redistribution. This simplifying assumption allows us to relate the planetary temperature to the orbital distance of the planet given the star's spectral type, a necessary step to determine trends in the location of the critical boundary with planetary and stellar temperatures.

As discussed in Section 3, we have upgraded MAKEXOSHELL (Bétrémieux & Kaltenegger 2015) with the capability to compute the column-integrated square of the density along the path of a refracted ray, necessary to compute the CIA optical depth with grazing altitude. We have also automated the computation of the location of the various refractive boundaries from the planetary bulk properties (radius, mass, and atmospheric composition), the planetary atmosphere's temperature, and the spectral class of the host star. The locations of the refractive boundaries are computed by MAKEXOSHELL in terms of their altitude and the density and pressure of the atmosphere, for both the actual and the apparent boundary, which are then converted into their respective "surface" cross sections as well.

We first consider an exoplanet with the size, mass, and bulk atmospheric composition of Jupiter as listed in Table 2. We expand on the parameter space explored in Bétrémieux (2016). Not only do we add M5 stars to the list of stellar spectral type (F0, G2, K5, M2), but more importantly, we consider a much larger number of planetary temperatures than Bétrémieux (2016), which only considered planetary temperatures of 300, 600, and 1200 K. We compute the actual and apparent location of the critical boundary for planetary temperatures ranging from 100 to 2000 K, in increments of 50 K, for a planet orbiting each of the stellar spectral types considered. We show the results in Figure 6.

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We find that the location of the critical boundary goes to regions of lower pressure—higher altitudes—for planets orbiting hotter host stars, as well as lower atmospheric temperatures, in agreement with Bétrémieux (2016). We also find that the differences in the apparent pressure location of the critical boundary, across the range of considered stellar spectral classes, decrease with increasing planetary temperature. This occurs because as the planetary temperature increases, the critical boundary approaches the lower boundary, where not only does a change in the ray deflection require a smaller altitude change in the grazing radius of the ray, but the effective scale height of the atmosphere also decreases (see Section 2.4). This fact had also been demonstrated by Bétrémieux (2016), who showed that for a planetary temperature of 1200 K the location of the refractive continuum was close enough to the lower boundary that its location was almost independent of stellar spectral type.

What had not been demonstrated by Bétrémieux (2016) is how the apparent location of the critical boundary of an isothermal atmosphere with a Jupiter-like composition compares with the pressures probed by CIA, both of which are determined by the bulk composition of the atmosphere. The pressure probed by CIA (P_CIA) is given by

$$\begin{eqnarray}&&{P}_{\mathrm{CIA}}={\left(\displaystyle \frac{{{GM}}_{P}\overline{m}}{\pi }\right)}^{1/4}{\left(\displaystyle \frac{{k}_{B}T}{{R}_{P}}\right)}^{3/4}{\left(\displaystyle \frac{{e}^{-{\gamma }_{\mathrm{EM}}}}{k}\right)}^{1/2}\end{eqnarray} \tag{ 32 }$$

for an isothermal atmosphere. Here the universal constant of gravitation (G) and the planetary mass (M_P) are the only quantities that we have not yet defined. We derive Equation (32) by asking at what pressure does the CIA optical depth along a nonrefracted ray equal ${e}^{-{\gamma }_{\mathrm{EM}}}$ (≈0.56, Lecavelier des Etangs et al. 2008; de Wit & Seager 2013) given the mean CIA cross section (k; see Equation (20)) of a Jovian atmosphere. The pressure probed by CIA as a function of wavelength is shown in Figure 7 for the nonrefractive case for four different temperatures (400, 700, 1000, and 2000 K). We consider only the most important source of CIA absorption in Jupiter's atmosphere, which is from H₂–H₂, and use the H₂–H₂ CIA cross-section values published by Borysow et al. (2001) and Borysow (2002).

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A comparison of Figures 6 and 7 is instructive. The critical boundary of Jovian exoplanets hotter than 1000 K is located at an apparent pressure greater than 10 bar, and CIA absorption features are not impacted by refraction because all CIA features probe lower pressures than this at these temperatures. At 400 K, the critical boundary can mask various parts of the near-infrared CIA features, depending on the spectral class of the host star. Below 200 K, even the size of the strongest CIA features is significantly reduced by refraction. Robinson et al. (2017) reached similar conclusions from simulations done at three temperatures: 150, 300, and 500 K.

A different but equivalent approach consists in comparing values of the "surface" cross section to the mean atmospheric cross section (see example in Section 3.2). Figures 8 and 9 give the corresponding "surface" cross section to the apparent location of the critical boundary shown in Figure 6 for atomic and molecular extinction and for CIA, respectively. The advantage of these two figures over Figure 6 is that observers can get a quick answer as to whether or not their favorite spectral features are reduced or hidden by refraction by simply looking up extinction cross sections, factor in the expected abundances of the responsible species, and determine whether any part of the computed mean cross section is lower than the "surface" cross section.

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We caution the readers that if refractive boundaries are located deep enough that they should lie inside the planet's troposphere, where the temperature decreases with altitude, then the refractive boundaries will in fact be located deeper than we predict owing to the atmosphere's larger scale height. In such cases, refraction has even less of an effect on transmission spectra than our simple method suggests. However, it is also very likely that refraction did not have much of an impact to begin with, such that our point might be moot.

Table 3 lists the location of the critical boundary for some of the first exoplanets that will be targeted by JWST through the GTO and the ERS (Early Release Science) program, which we rank in order of decreasing temperatures. We chose the properties of the exoplanets by first perusing through the NASA Exoplanet Archive (Akeson et al. 2013) and selecting sources with a complete coherent set of parameters sufficient to compute the location of the critical boundary. We assume that the bulk composition of their atmosphere is the same as Jupiter's. All of these planets are about the size of Jupiter, except for HAT-P-26b, which is Neptune-sized. Although Kepler-167e (Kipping et al. 2016), a true Jupiter analog, is not on the GTO or ERS target list of JWST, we have included it to emphasize the dramatic effect that the planetary temperature has on the location of the critical boundary. For all the hot exoplanets, the critical boundary is located close to the lower boundary, as the factor $(r\nu /H)$ evaluated at the critical boundary is fairly close to 1. Indeed, assuming a constant scale height, ${(r\nu /H)}_{c}\approx ({n}_{c}/{n}_{\mathrm{lb}})$. Among the hot exoplanets, this factor is smallest for HAT-P-26b essentially because of its smaller size. The low values of the CIA "surface" cross section (k_c) imply that spectral features from CIA are not impacted by refraction, and that CIA—not refraction—potentially determines the highest pressure that can be probed in the infrared in these hot exoplanets. For the cold Kepler-167e, exactly the opposite occurs.

Table 3.  Refractive Boundaries of Some Giant Exoplanets Assuming a Jovian Atmospheric Composition

Planet	Tea	${P}_{c}^{* }$ (bar)	σc (10−28 cm2 molec−1)	kc (10−48 cm5 molec−2)	(rν/H)c	Sourceb
WASP-17b	1615	172	0.0020	0.00037	0.45	Bento et al. (2014)
HD 209458b	1305	62.7	0.0098	0.0040	0.51	Knutson et al. (2007)
WASP-43b	1262	18.0	0.092	0.13	0.89	Hellier et al. (2011)
HD 189733b	1090	24.4	0.037	0.0330	0.70	Southworth (2010)
WASP-39b	1025	69.9	0.0053	0.0015	0.35	Maciejewski et al. (2016)
HAT-P-26b	909	80.3	0.0075	0.0017	0.23	Hartman et al. (2011)
Kepler-167e	134	0.0465	9.0	500	0.051	Kipping et al. (2016)

Notes.

^aAssumes a Bond albedo of 0.30. ^bReference for planetary and stellar parameters.

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The importance of refraction in transmission spectra can be a different story for terrestrial exoplanets than for Jovian exoplanets. Not only is the lensing power of an atmosphere lower for smaller planets, but the composition of their atmospheres can be quite different, and the amount of gas making up the atmosphere is finite compared to gas giants. Thus, the actual pressure at the surface may define the largest pressure that can be probed if the atmosphere is thin. However, we focus here on thick atmospheres with a surface pressure of 500 atm and explore how the location of the critical boundary depends on the bulk composition of the atmosphere. We assume that the radius of the planet is located at a reference pressure of 1 bar, as we did for Jovian planets.

Figures 10 and 11 show the "surface" cross section of the critical boundary for exoplanets with the size and mass of Earth, orbiting M5, M2, K5, G2, and F0 stars, for atomic and molecular extinction and for CIA, respectively. As for Jovian exoplanets, we vary the atmospheric temperature from 100 to 2000 K in 50 K intervals, but we now compute the critical boundary for two different types of atmosphere: a pure H₂ and a pure CO₂ atmosphere. Both figures show that the "surface" cross sections of the critical boundary increase by several orders of magnitude from an H₂ to a CO₂ atmosphere. This occurs because both the refractivity and the mean molecular weight of CO₂ are larger than H₂. For an H₂ atmosphere, the 500 atm surface can be probed if the atmospheric temperature is above 900 K when the planet orbits an M5 star, as indicated by the part of the curve with the positive slope, which tracks the apparent location of the surface. The atmospheric temperature at which the critical boundary reaches the surface increases as the temperature of the host star increases.

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Comparison between Earth-sized and Jovian-sized planets of the two types of "surface" cross section reveal that the "surface" cross sections of H₂ atmospheres are lower for Earth-sized planets. The "surface" cross sections of a pure CO₂ atmosphere of an Earth-sized planet are, however, higher than that of a Jovian planet because the higher lensing power of a CO₂ atmosphere more than compensates for the lower lensing power of a smaller planet. Just as in Jovian planets, the pure CO₂ atmosphere of an Earth-sized planet also exhibits a decrease in spread of "surface" cross-section values between the different host stars with increasing temperatures. This is a telltale sign that the critical boundary approaches the lower boundary well inside the refractive boundary layer at these hotter temperatures.

With the discovery of the TRAPPIST-1 system (de Wit et al. 2016; Gillon et al. 2016, 2017)—a system of at least seven terrestrial planets orbiting an M8 star—characterization of the composition of the atmosphere of terrestrial exoplanets is a reality that is taking shape now. Some of the members of that system will be targeted by JWST during its GTO program. Given the importance of the atmospheric bulk composition on the location of the critical boundary and the fact that terrestrial planets can accommodate a large diversity of atmospheres, one might wonder how the bulk composition of the atmosphere might affect the integration time required to detect various chemical species. We provide a method to estimate this in the presence of refractive effects for the TRAPPIST-1 planets by exploring how the critical boundary changes with the bulk composition of their atmosphere and expressing it in terms of its "surface" cross section. We use the planetary and stellar parameters that appear in Gillon et al. (2017). We do not include TRAPPIST-1h in our calculations because there are no estimates of its mass. We compute the atmospheric temperature in the same manner as the Jovian planets (see Section 4.2) and list them in Table 4.

Table 4.  Refractive Boundaries of Some Terrestrial Exoplanets for Different Pure Atmospheric Compositions

			P${}_{c}^{* }$ (bar)				${\sigma }_{c}$ (10−28 cm2 molec−1)				kc (10−48 cm5 molec−2)
Planet	Tea	(rν/H)cb	He	H2	N2	CO2	He	H2	N2	CO2	He	H2	N2	CO2
GJ 1214b	550	0.22–0.70	102	40.6	2.68	1.19	0.011	0.020	1.0	2.9	0.0011	0.0053	4.2	27
TRAPPIST-1b	366	0.12–0.45	79.2	29.7	2.63	1.24	0.018	0.036	1.3	3.5	0.0016	0.0088	3.6	20
TRAPPIST-1c	313	0.12–0.45	35.9	13.6	1.16	0.55	0.047	0.089	3.7	9.7	0.0079	0.040	19	110
TRAPPIST-1d	264	0.061–0.25	33.6	12.3	1.29	0.64	0.042	0.083	2.6	6.7	0.0064	0.035	11	54
TRAPPIST-1e	230	0.055–0.23	19.1	7.0	0.73	0.36	0.063	0.12	4.1	10	0.015	0.079	25	130
TRAPPIST-1f	200	0.044–0.19	12.4	4.5	0.49	0.25	0.077	0.15	4.9	12	0.024	0.13	40	190
TRAPPIST-1g	181	0.051–0.22	6.5	2.4	0.25	0.12	0.17	0.33	12	29	0.092	0.48	170	830

Notes.

^aAssumes a Bond albedo of 0.30. ^bLowest value is that of a pure H₂ atmosphere, while the highest is that of a pure CO₂ atmosphere.

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We consider four different gases (H₂, He, N₂, and CO₂) and characterize the different atmospheric compositions in terms of the resulting mean molecular weight of the mixture of these gases. We only consider binary mixtures of gases with neighboring mean molecular weight (H₂–He, He–N₂, and N₂–CO₂) to avoid possible compositional degeneracies for a given mean molecular weight. Since lighter gases escape planetary atmospheres more readily than heavier gases, this potentially covers different end products of atmosphere-stripping scenarios where a terrestrial planet that has undergone little atmospheric loss is composed of primordial H₂ and He, whereas it is predominantly composed of He (Hu et al. 2015), N₂ (Earth-like), or CO₂ (Venus-like) with increasing loss of the primordial atmosphere.

The "surface" cross sections for atomic and molecular extinction and for CIA, are shown in Figures 12 and 13, respectively, as a function of the mean molecular weight of the atmosphere for the TRAPPIST-1 planets. As expected, the "surface" cross section generally increases as the orbital distance of the planet increases, because the refractive continuum is located at a higher altitude. The only exception to this trend is between TRAPPIST-1c and TRAPPIST-1d, where the much smaller radius of TRAPPIST-1d decreases the lensing power of the atmosphere sufficiently that it overcomes the distance dependence. The smallest "surface" cross section occurs for a pure He atmosphere, because helium's low refractivity decreases its lensing power compared with an H₂ atmosphere, in spite of its higher mean molecular weight. A pure CO₂ atmosphere has the highest "surface" cross section. The difference in "surface" cross sections between the He and CO₂ atmospheres amounts to a difference in location of the critical boundary of about $\mathrm{ln}({\sigma }_{c}({\mathrm{CO}}_{2})/{\sigma }_{c}(\mathrm{He}))\approx 5$ scale height—a substantial difference. This implies that as the mean molecular weight of the atmosphere increases, not only do the sizes of spectral features decrease because of the decreasing scale height, but the refractive continuum also shifts to higher altitudes, except in the H₂-to-He transition, reducing the size of spectral features further.

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The apparent locations of the critical boundary, as well as their conversion into "surface" cross sections, for the various pure atmospheres are also summarized in Table 4, not only for the TRAPPIST-1 planets but also for GJ 1214b—an exoplanet that has received a lot of attention—for which we used the maximum probability value of Anglada-Escudé et al. (2013) for the planetary and stellar parameters. The hotter temperature of GJ 1214b generally causes the critical boundary to be located at substantially larger pressures than any of the TRAPPIST-1 planets except for the N₂ and CO₂ atmospheres. For these atmospheres, the critical boundary is located close to the lower boundary, as shown with an ($r\nu /H$) factor ($\approx {n}_{c}/{n}_{\mathrm{lb}}$ for a constant scale height) close to 1. The much larger radius of GJ 1214b causes that factor to be significantly larger, and the resulting upward shift of the apparent boundary with respect to the actual one causes the apparent pressure of the critical boundary in a CO₂ atmosphere to be lower for GJ 1214b than for TRAPPIST-1b. Irrespective of the location of the critical boundary, changing the atmosphere of GJ 1214b from H₂ to CO₂ changes the transit depth per scale height, evaluated at the critical boundary, from about 350 to 21 ppm, smaller than the spectral modulation of about 100 ppm observed by Kreidberg et al. (2014). However, the lack of correlation between the transmission spectrum and that of a pure CO₂ atmosphere favors clouds to explain the fairly flat spectrum.

To illustrate how to use Figures 12 and 13, let us consider an example with TRAPPIST-1e (green line) for atomic and molecular extinction (Figure 12). Let us suppose that a molecule of interest has a cross section that ranges from 10⁻²⁵ to 10⁻²² cm² molec⁻¹. If atmospheric models predict an abundance of 10⁻² (mole fraction), then the mean cross section ranges from 10⁻²⁷ to 10⁻²⁴ cm² molec⁻¹, assuming that the contribution from other sources of opacity is negligible. Since the "surface" cross section never goes above these values, except barely for a pure CO₂ atmosphere, refraction does not reduce the size of that spectral feature, and its spectral modulation is $\mathrm{ln}({10}^{-24}/{10}^{-27})\approx 7$ scale height. If atmospheric models predict an abundance of 10⁻⁵, then the mean cross section ranges from 10⁻³⁰ to 10⁻²⁷ cm² molec⁻¹, and the effects of refraction depend on the bulk composition of the atmosphere. For a pure H₂ atmosphere, the "surface" cross section is above 10⁻²⁹ cm² molec⁻¹, so that its spectral modulation is less than $\mathrm{ln}({10}^{-27}/{10}^{-29})\approx 4.6$ scale height. In this example, refraction decreases the size of the spectral feature by at least a third from 7 to 4.6 scale heights. For a pure CO₂ atmosphere, the "surface" cross section is larger than the maximum mean cross section, and the spectral feature is completely hidden by refraction. One can use the same method with CIA on Figure 13, except that one must multiply the CIA cross section by the product of the abundance of the two species responsible for the CIA feature to obtain the mean CIA cross section. To obtain the spectral modulation expressed in scale height, one must further divide by 2 the natural logarithm of the ratio of relevant CIA cross sections (Equation (19)).

This comparison is potentially complicated by the fact that the extinction cross sections of gases vary with temperature and pressure, which can lead to non-negligible changes in the extrema of opacities and in the atmospheric pressures probed. Indeed, cross sections computed at higher pressures and temperatures generally have a lower contrast between the minimum and maximum values of the bands. However, some sources of opacity, such as Rayleigh scattering, hardly change with pressure and temperature, and this comparison can be done easily. One should ideally use cross sections at the highest possible spectral resolution and at the relevant conditions of temperature and pressure and determine in the spectral regions of interest how the computed mean cross section compares to the "surface" cross section. Since we only consider isothermal atmospheres, it is sufficient for the comparison to use a cross section close to the equilibrium temperature of the atmosphere for which the "surface" cross section was computed. We provide these equilibrium temperatures in Tables 3 and 4 and show them in Figures 8–11. Since cross sections have no significant pressure dependence below a transition pressure (see Equation (34) and the detailed discussion on collisional broadening in the Appendix), cross sections at low pressures can be approximated by a cross section computed at zero pressure. Hence, if one wants to know how many scale heights the peak of an absorption features rises above the refractive continuum, one can compare the "surface" cross section to the mean cross section computed at zero pressure because most prominent absorption bands probe these low pressures. If, instead, one wants to know whether the refractive continuum hides the continuum created by molecular extinction, one should use the mean cross section at the apparent critical pressure. Differences between these two scenarios can yield the spectral modulation present in the transmission spectrum. Indeed, the transmission spectrum of an exoplanet transitions from one computed using zero-pressure cross sections near the peak of absorption bands to one using cross sections computed at the apparent critical pressure in the deeper regions.

Choosing a cross section at the appropriate temperature and pressure is a luxury that is not always available depending on the spectral region or the molecular species of interest. In the ultraviolet, for instance, extinction cross sections are predominantly measured, rather than computed from first principles, and this only for a limited range of pressures and temperatures. Having at one's disposal a cross section at a lower spectral resolution, and hence lower contrast, may present some ambiguity in interpretation. Indeed, if the "surface" cross section is higher than part of said cross section, it is clear that refraction plays a role in shaping the transmission spectrum. If the "surface" cross section is always lower, one might wonder if that still would have been the case compared to a higher spectral resolution cross section that has a higher contrast. In that particular scenario, it is not so clear that refraction has no impact on the spectrum. Even in the infrared, where line lists are available, the parameters for collisional broadening are typically known only for self-broadening or air, and only around 300 K. They are poorly known at the high temperatures and for the broadening agents (H₂ and He) relevant to hot Jupiters, and HITRAN and EXOMOL—two line list databases—have only recently compiled broadening parameters for H₂ and He for a few molecules (Wilzewski et al. 2016; Barton et al. 2017). When broadening parameters are not available, one can use our method in an isothermal atmosphere because computed cross sections cannot change with altitude or pressure.

Whether the atmosphere is probed deep enough that collisional broadening becomes important or not is of secondary concern because our method is not a substitute for a radiative transfer computation, but only meant to help ascertain to first order whether refraction plays a likely role in modifying the transmission spectrum of the exoplanet, given a set of extinction cross sections and relative abundances of several species of interest. One may simply take a quick look at a published cross section and factor in various molecular abundances to get a feel for the range of abundances that can create spectral features above the refractive continuum, or for which spectral features are hidden by refraction.

With this in mind, once the spectral modulation, expressed in scale height (${\rm{\Delta }}h/H$), of a specific spectral feature is determined, one can estimate the corresponding transit depth (${\rm{\Delta }}F/{F}_{\star }$) associated with this feature with

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\rm{\Delta }}F}{{F}_{\star }}\right)=\left(\displaystyle \frac{{\rm{\Delta }}h}{H}\right)\left(\displaystyle \frac{2{R}_{P}H}{{R}_{\star }^{2}}\right).\end{eqnarray} \tag{ 33 }$$

Here ΔF is the stellar flux drop caused by the spectral feature, ${F}_{\star }$ is the stellar flux, and ${R}_{\star }$ is the stellar radius. Figures 12 and 13 and Table 4 are thus valuable to get first-order effects of refraction for the TRAPPIST-1 planets in order to help determine the exposure time required to detect spectral features with various JWST instruments, from an assumed composition of the atmosphere.