On physical interpretations of the reference transit radius of gas-giant exoplanets
aa r X i v : . [ a s t r o - ph . E P ] S e p MNRAS , 1–7 (2019) Preprint 30 September 2019 Compiled using MNRAS L A TEX style file v3.0
On physical interpretations of the reference transit radiusof gas-giant exoplanets
Kevin Heng ⋆ University of Bern, Center for Space and Habitability, Gesellschaftsstrasse 6, CH-3012, Bern, Switzerland
Accepted 2019 September 27. Received 2019 September 17; in original form 2019 January 21
ABSTRACT
Two theoretical quandaries involving transmission spectra of gas-giant exoplanets areelucidated. When computing the transit radius as a function of wavelength, one needsto specify a reference transit radius corresponding to a reference pressure. Mathemati-cally, the reference transit radius is a constant of integration that originates from eval-uating an integral for the transit depth. Physically, its interpretation has been debatedin the literature. Jord´an & Espinoza (2018) suggested that the optical depth is discon-tinuous across, and infinite below, the reference transit radius. B´etr´emieux & Swain(2017, 2018) interpreted the spherical surface located at the reference transit radiusto represent the boundary associated with an opaque cloud deck. It is demonstratedthat continuous functions for the optical depth may be found. The optical depth belowand at the reference transit radius need not take on special or divergent values. In thelimit of a spatially uniform grey cloud with constant opacity, the transit chord withoptical depth on the order of unity mimics the presence of a “cloud top”. While thesurface located at the reference pressure may mimic the presence of grey clouds, itis more natural to include the effects of these clouds as part of the opacity functionbecause the cloud opacity may be computed from first principles. It is unclear howthis mimicry extends to non-grey clouds comprising small particles.
Key words: planets and satellites: atmospheres
An exoplanet transiting its star produces an obscuring disccorresponding to some transit radius, which is generally afunction of wavelength. This transit radius corresponds toa sightline from the observer to the star that is a chord (inthe mathematical sense), with an optical depth on the orderof unity, passing through the exoplanetary atmosphere.In the limit of an isothermal transit chord and constantacceleration due to gravity, the transit radius is (Fortney2005; Lecavelier des Etangs et al. 2008; de Wit & Seager2013; Heng et al. 2015; B´etr´emieux & Swain 2017;Heng & Kitzmann 2017; Jord´an & Espinoza 2018) R = R + H [ γ + E ( τ ) + ln τ ] , (1)where R is a reference transit radius, H is the isothermalpressure scale height, γ is the Euler-Mascheroni constant, τ is the optical depth corresponding to the reference transitradius and E ( τ ) is the exponential integral of first order(e.g., Abramowitz & Stegun 1970; Arfken & Weber 1995) E ( τ ) ≡ Z ∞ y − e − yτ dy, (2) ⋆ E-mail: [email protected] (KH) which has the mathematical property thatlim τ →∞ E ( τ ) = 0 . (3)Mathematically, R is a constant of integration that re-sults from evaluating an integral. Its physical interpretationhas been debated in the literature. The goal of the currentstudy is to elucidate two quandaries involving the physicalinterpretation of R and provide possible resolutions to thesequandaries. The first quandary concerns whether the reference transitradius corresponds to a special physical location within theexoplanet, across which the optical depth is discontinuous.Consider only gas-giant exoplanets without rocky surfacessuch that a discontinuity associated with the interface be-tween the atmosphere and rocky surface cannot be claimed.Heng & Kitzmann (2017) reasoned that the referenceoptical depth ( τ ) does not need to take on any par-ticular value, but one may choose a value of R such c (cid:13) Heng that τ ≫
1. Such a choice implies that the E ( τ )term in equation (1) must vanish. The obscuring disc hasthe area (de Wit & Seager 2013; B´etr´emieux & Swain 2017;Heng & Kitzmann 2017), πR = πR + A ( R , ∞ ) , (4)where πR is the area of the secondary obscuring disc cor-responding to the reference transit radius, A ( R , ∞ ) is thearea of a thin annulus defined by (Brown 2001) A ( r , r ) = Z r r (cid:0) − e − τ (cid:1) πr dr, (5) τ ( r ) is the optical depth and r is the radial coordinate.Jord´an & Espinoza (2018) offered an alternative expla-nation motivated by equation (3) of de Wit & Seager (2013)as a starting point. They reasoned that, since πR = A (0 , ∞ ) = A (0 , R ) + A ( R , ∞ ) , (6)and one necessarily needs to have A (0 , R ) = Z R (cid:0) − e − τ (cid:1) πr dr = πR , (7)this implies that the atmosphere immediately below thereference transit radius must possess large optical depths.Specifically, they remarked that, “ R satisfies the conditionof being a radius below which the planet is fully opaque.”This quoted statement contains a footnote that states, “If R is not chosen to be at an optically thick region (i.e.,a region where τ → ∞ ), then it is not possible to write A (0 , R ) = πR .”At face value, it seems challenging to reconcile these twoviewpoints. Equation (7) indeed trivially integrates to yield πR if one allows τ → ∞ within the integrand. However, ifone asserts that the optical depth needs to be a continuousfunction, then this implies that τ ≫ E ( τ ) termin equation (1) is permanently absent. In order to assertthat τ → ∞ and retain the E ( τ ) term in equation (1),one has to assume that the optical depth is a discontinuous,piecewise function, τ = ( τ e ( R − r ) /H r ≥ R ∞ ≤ r < R , (8)which one may argue lacks generality. In the absence of arocky surface, the origin of this discontinuity is unclear.The first goal of the present study is to reconcile theseviewpoints and demonstrate that the optical depth need notbe discontinuous across R . In the limit of constant opacity,one may demonstrate that A (0 , R ) ≃ πR for any value of τ . The spherical surface associated with R has previouslybeen interpreted by B´etr´emieux & Swain (2017, 2018) torepresent the boundary associated with an opaque (opti-cally thick) cloud deck. Furthermore, B´etr´emieux & Swain(2017, 2018) claim that variations of equation (1), as derived The final equation in Jord´an & Espinoza (2018) is an expres-sion for πR that contains this E ( τ ) term. by Lecavelier des Etangs et al. (2008) and de Wit & Seager(2013), are valid only for describing cloudfree atmospheres.For example, the abstract of B´etr´emieux & Swain (2017)states, “Although the formalism of Lecavelier des Etangset al. is extremely useful to understand what shapes trans-mission spectra of exoplanets, it does not include theeffects of a sharp change in flux with altitude gener-ally associated with surfaces and optically thick clouds.”As another example, Section 2.6 of B´etr´emieux & Swain(2018) states, “Until recently, the few analytical formalisms(Lecavelier des Etangs et al. 2008; de Wit & Seager 2013)attempting to explain what shapes exoplanet transmissionspectra could only do so for clear atmospheres.”When computing the transmission spectrum, one needsto specify the cross section or opacity (cross section per unitmass) as a function of wavelength, temperature and pres-sure. Physically, the opacity function includes contributionsfrom the extinction (absorption and scattering) of radia-tion by atoms, ions, molecules and aerosols/hazes/clouds,whether in the form of spectral lines or a continuum. Thesecontributions are weighted by their relative abundances (i.e.,mass or volume mixing ratios). Sources of spectral continuainclude collision-induced absorption and Rayleigh scatter-ing. The shape of the continuum due to extinction by cloudsdepends on the size of the constituent particles. A cloud par-ticle is small or large only in comparison to the wavelengthof radiation it is absorbing or scattering. Let the radius ofa spherical cloud particle be r cloud and the wavelength be λ . When 2 πr cloud /λ ≪ πr cloud /λ ≫ τ ∼ τ is the chord optical depth). Assuming thata radial pressure gradient exists within the atmosphere, thetransit chord corresponds to a “cloud top” pressure of (Heng2016) P = 0 . gκ r H πR , (9)where g is the acceleration due to gravity and κ is the opac-ity. Even though the opacity is constant, the cloud is opti-cally thin at lower pressures or higher altitudes and exerts anegligible influence on the spectrum. It is the same radiativetransfer principle for why one observes an edge to the Sun,even though no sharp boundary exists. This thought exper-iment suggests that as long as an opacity function may bespecified in the formula for the transit radius, the formulamay be used to model cloudy atmospheres, contrary to theclaim of B´etr´emieux & Swain (2017, 2018).The second goal of the present study is to demonstratethat it is not necessary to impose a boundary associatedwith an opaque cloud via the reference transit radius, eventhough it is possible for the surface associated with R to MNRAS , 1–7 (2019) n physical interpretations of the reference transit radius of gas-giant exoplanets mimic the effects of a grey cloud deck. Such mimicry doesnot straightforwardly extend to non-grey clouds consistingof small particles. It is useful to visualize the gas-giant exoplanet as consistingof two regions. The region corresponding to 0 ≤ r ≤ R is referred to as the “interior” of the exoplanet and it en-compasses the vast majority of its mass. The region corre-sponding to r ≥ R is referred to as the “atmosphere” of theexoplanet and the mass enclosed is negligible. The demar-cation between the two regions is not meant to be sharp.A different set of approximations is applied to each region.The ideal gas law is expected to be a good approximationwithin the atmosphere, but it breaks down deeper into theinterior as the pressure increases.Within the interior of the exoplanet, the solutions to theLane-Emden equation are used to describe the mass densityprofile, ρ ( r ) (Chapter 4 of Chandrasekhar 1967). Analyticalsolutions to the Lane-Emden equation exist for polytropeswith indices of 0, 1 and 5. The current study examines onlythe first two cases, which correspond to the simplest as-sumption (constant ρ ) and a reasonable approximation forhydrogen-helium mixtures at high pressures (e.g., Figure 2of Stevenson 1982). Upon specifying ρ ( r ), one may then eval-uate the optical depth, τ = Z ρκ dr. (10)Since the opacity function for 0 ≤ r ≤ R cannot be easilyspecified because of poorly known physics (e.g., Stevenson1982; Guillot 2005; Valencia et al. 2013), κ is assumed to beconstant in this study. A polytrope of index 0 corresponds to a constant mass den-sity, ρ (Chapter 4 of Chandrasekhar 1967). While this ap-proximation lacks physical realism, it serves as a mathemati-cal prelude to the more realistic case of a polytrope of index1. Furthermore, one may argue that assuming a constant ρ is no worse than assuming a constant optical depth for0 ≤ r ≤ R , i.e., equation (8).By demanding that τ = τ at r = R , one obtains τ = τ c (cid:18) − rR (cid:19) + τ , (11)where τ c ≡ ρκR . At r = 0, one obtains an optical depth of τ c + τ . It is important to emphasise that τ is the“zero point”for the optical depth, whereas τ c is the difference in opticaldepth between the center of the exoplanet and r = R . Itis analogous to the distinction between displacement anddistance. Thus, we expect τ c to be large, but no assumptionneeds to be made on τ . Demanding that τ c ≫ l mfp ≪ R , (12)where l mfp = 1 /ρκ is the photon mean free path. The opticaldepth is neither discontinuous nor constant, as it goes froma value of τ c at the center of the exoplanet to τ at r = R by construction (as a boundary condition). τ c -9 -8 -7 -6 -5 -4 -3 -2 -1 F e − τ τ =0τ =1τ =10 Figure 1.
Correction factor to the projected area of the sphericalexoplanet at r = R for a polytrope of index 1 (see text for defi-nition of F ) as a function of the optical depth difference betweenthe exoplanet center and reference transit radius. The assumptionof R /R = 1 has been made; other choices (e.g., R /R = 0 . τ = 0 curve isolates the effect of F . It follows that A (0 , R ) = πR + 2 πR τ c e − τ (cid:0) − τ c − e − τ c (cid:1) ≃ πR − πR τ c e − τ . (13)A more illuminating way to write the preceding equation is A (0 , R ) πR ≃ − l mfp R e − τ . (14)The correction terms are small if τ c ≫ l mfp ≪ R .Thus, A (0 , R ) ≃ πR for any value of τ . The mass density profile is (Chapter 4 of Chandrasekhar1967) ρ = ρ c sin xx , (15)where ρ c is the mass density at r = 0, x ≡ πr/R and R isthe radius of the exoplanet. By construction, ρ = 0 when r = R . The corresponding pressure profile is (Chapter 4 ofChandrasekhar 1967) P = W GM R (cid:18) ρρ c (cid:19) , (16)where G is the gravitational constant and M is the mass ofthe exoplanet. The constant W = 0 . r = R and x = x ≡ πR /R , the reference massdensity and pressure are ρ = ρ c sin x x , P = W GM R (cid:18) sin x x (cid:19) . (17)Since the profiles of mass density and pressure need to joinsmoothly to the ideal gas law at r = R , one may solve forthe temperature at the reference transit radius, T = W GM R ρ c R (cid:18) sin x x (cid:19) , (18) MNRAS , 1–7 (2019)
Heng where R is the specific gas constant. This exercise demon-strates that if the interior structure of an exoplanet is apriori known, then the conditions at the reference transitradius are completely specified.By again imposing the boundary condition that τ = τ at r = R , one obtains τ = τ c (cid:18) − SS (cid:19) + τ , (19)where the trigonometric integral is S ≡ Z x sin x ′ x ′ dx ′ . (20)The optical depth between the center of the exoplanet andthe reference transit radius is τ c ≡ ρ c κRS /π . The quantity, S ≡ S ( x ) , (21)depends on the chosen value of R /R . Similar to a polytropeof index 0, demanding that τ c ≫ l mfp ≪ R S π , (22)where l mfp = 1 /ρ c κ is the photon mean free path at the center of the exoplanet and is thus expected to be very small.No assumption is made on τ . At r = 0, the optical depth isagain τ c + τ and τ serves as its “zero point” as before.It follows that A (0 , R ) πR = 1 − F e − τ , (23)where the correction factor involves the integral, F ≡ π (cid:18) R R (cid:19) Z πR /R xe τ c ( S/S − dx. (24)Since we expect R /R ∼
1, it suffices to numerically eval-uate F for R /R = 1 as a function of τ c (Figure 1). When R /R = 1, one obtains S ≈ .
852 but in order to evaluatethe integral accurately S needs to be numerically computedto machine precision. It is important to note that the cor-rection to A (0 , R ) /πR = 1 is F e − τ . With τ = 1, thecorrection is about 0 . F ; choices of τ ∼ A (0 , R ) /πR =1 become .
1% for τ = 1 when τ c & . It is worthestimating conservative values for τ c , τ c ∼ (cid:18) ρ c − κ .
05 cm g − P R R J (cid:19) , (25)where R J = 7 . × cm is the radius of Jupiter. Theorder-of-magnitude estimate of the opacity, as well as itslinear dependence on pressure, is taken from Guillot (2005)and is broadly consistent with the more detailed calculationsof Valencia et al. (2013) and Freedman et al. (2014) for P =1 bar and T ∼ ρ c and theopacity are likely to be higher.Figure 2 shows four examples of hot Jovian transmis-sion spectra, where the absolute transit depths are typically ∼ − ; also shown are two choices for τ , which is generallya function of wavelength. The spectral features are typically ∼ − variations in the relative transit depth correspondingto ∼
10% variations in πR . A desired property is for the cor-rection to A (0 , R ) /πR = 1 to be much smaller than thesevariations in πR . Figure 1 shows that with τ c = 10 , one λ (µm) ( R / R ⋆ ) × − gray clouds (P =8 bar)cloudfree (P =8 bar)cloudfree (P =0.9 mbar)non-gray clouds (r cloud =0.01 µm) λ (µm) -3 -2 -1 τ R=1.709R J , P =8 barR=1.968R J , P =0.9 mbar Figure 2.
Top panel: synthetic transmission spectra adoptingparameter values from WASP-17b (see text). Two of these spectraassume R = 1 . R J and P = 8 bar; one of them is cloudfree,while the other assumes a grey cloud with a constant opacityof 0.01 cm g − corresponding to a transit chord located at 0.9mbar. The third spectrum uses R = 1 . R J and P = 0 . τ involving only the water opacity. already has F < − . Therefore, F e − τ ≪ τ and the correction to A (0 , R ) /πR = 1 isnegligible in the sense that it is much smaller than the vari-ations in the relative transit depth associated with spectralfeatures.The optical depth increases smoothly from a value of τ c at the center of the exoplanet to its boundary-conditionvalue of τ at the reference transit radius. It is neither con-stant within 0 ≤ r ≤ R nor discontinuous at r = R . Equation (1) is derived by solving for h = A ( R , ∞ ) / πR and inserting it into R = R + h . Evaluating A ( R , ∞ ) = Z ∞ R (cid:0) − e − τ (cid:1) πr dr (26)requires that one elucidates the relationship between τ and r . Within the atmosphere, assuming the ideal gas law and MNRAS , 1–7 (2019) n physical interpretations of the reference transit radius of gas-giant exoplanets hydrostatic balance yieldsln (cid:18) ττ (cid:19) = − Z rR mgk B T dr, (27)where m is the mean molecular mass, k B is the Boltzmannconstant and T is the temperature. Evaluating the integralrequires that one specifies g ( r ).There are three different ways of expressing the accel-eration due to gravity: constant g , constant exoplanet mass( g ∝ /r ) or constant bulk density ( g ∝ r ). Consider agas-giant exoplanet where R ∼ R J . If P ∼
10 bar and theinfrared photosphere is located at ∼ ∼
10 pressure scale heights thick. Since
H/R ∼ . H = k B T /mg being the pressure scale height), thismeans that the atmosphere is ∼ . R J thick. In the con-stant exoplanet mass or bulk density approximations, thisimplies that the acceleration due to gravity is changing by ∼
10% within the atmosphere, which provides the motiva-tion for investigating these three ways of deriving τ ( r ).In the standard derivation where g is assumed to beconstant, one obtains the usual expression for hydrostaticequilibrium,ln (cid:18) ττ (cid:19) = R − rH . (28)One recovers equation (S.4) of de Wit & Seager (2013),equation (20) of B´etr´emieux & Swain (2017) or equation (8)of Heng & Kitzmann (2017), h = H Z τ − e − τ τ (cid:20) HR ln (cid:16) τ τ (cid:17)(cid:21) dτ ≃ H [ γ + E ( τ ) + ln τ ] , (29)where the second, approximate equality holds if one assumesthe term involving the logarithm in the integrand to besmaller by a factor of H/R and is hence dropped, whichallows the integral to be evaluated analytically.One could instead assume that g = GM/r with a con-stant M , which yieldsln (cid:18) ττ (cid:19) = R H (cid:18) R r − (cid:19) , (30)where H ≡ k B T /mg , g ≡ GM/R and G is the gravita-tional constant. It follows that h = H Z τ − e − τ τ (cid:20) H R ln (cid:18) ττ (cid:19)(cid:21) − dτ ≃ H [ γ + E ( τ ) + ln τ ] . (31)Again, the integral may only be evaluated analytically if the ∼ H /R term within the integral is dropped.Alternatively, one may assume the mass of the exo-planet to be given by M = 4 π ¯ ρr /
3, where ¯ ρ is an averagebulk mass density that is assumed to be constant for r ≥ R .This assumption yields g = 4 πG ¯ ρr/
3, which yieldsln (cid:18) ττ (cid:19) = R H (cid:18) − r R (cid:19) . (32)It follows that h = H Z τ − e − τ τ dτ = H [ γ + E ( τ ) + ln τ ] , (33) where we again have H ≡ k B T /mg , but g ≡ GM /R and M is the mass of the exoplanet enclosed by r = R .There is no ∼ H/R or ∼ H /R correction term to dropand the integral is evaluated exactly.The constant g , g = GM/r and g = 4 πG ¯ ρr/ h and hence R , despite thedifferent functional forms for τ ( r ). This is a mathematicalcoincidence and arises only because small correction termsin the integrand for A ( R , ∞ ) were dropped in order to eval-uate the integral analytically.The optical depth may be constructed using equation(11) or (19) for 0 ≤ r ≤ R and equation (28), (30) or(32) for r ≥ R . For all 6 combinations, the optical depthis continuous across the reference transit radius and finiteeverywhere. From equation (12) of Heng & Kitzmann (2017), the refer-ence optical depth within equation (1) is τ = κP g r πR H . (34)Note that equation (9) of de Wit & Seager (2013) expresses τ (denoted by them as A λ ) in terms of a reference num-ber density and cross section. B´etr´emieux & Swain (2017)write τ as τ s in their equation (26), but do not explicitlyprovide an expression for it beyond their equation (42). Thenormalisation degeneracy (Benneke & Seager 2012; Griffith2014; Heng & Kitzmann 2017; Fisher & Heng 2018), whichis the three-way degeneracy between R , P and κ (whichcontains the relative abundances of atoms and molecules) isnot explored in detail by either de Wit & Seager (2013) orB´etr´emieux & Swain (2017).The wavelength-, temperature- and pressure-dependentopacity function is κ = X cloud σ cloud m + X i κ i X i m i m . (35)The sum is over all of the atoms, ions and molecules in theatmosphere. The opacity of each species is denoted by κ i .The volume mixing ratio of each species is X i ; it is worthnoting that the mass mixing ratio is X i m i /m , where m i isthe mass of each species. The cloud volume mixing ratio andcross section are denoted by X cloud and σ cloud , respectively.In the current study, the only molecule considered is wateras this suffices to construct the necessary arguments.Assuming a monodisperse cloud (i.e., particles of a sin-gle radius), the cloud cross section is σ cloud = Qπr , (36)where Q is the extinction efficiency. It may be com-puted using Mie theory (e.g., Kitzmann & Heng 2018).Kitzmann & Heng (2018) provide a convenient fitting func-tion, Q = Q Q X − a + X . , (37) MNRAS , 1–7 (2019)
Heng which is calibrated to full numerical calculations. The di-mensionless size parameter is given by X = 2 πr cloud /λ . Thisfitting function for Q smoothly connects the regimes of small( X ≪
1; Rayleigh) and large ( X ≫
1) particles. As an il-lustration, I adopt the calibration for forsterite (Mg SiO ): Q = 11 . Q = 4 .
16 and a = 4 .
05 (see Table 2 ofKitzmann & Heng 2018).It is worth noting that Mie theory specifies the wave-length dependence of the cloud cross section, but not itsspatial dependence. The latter is driven by poorly known de-tails of the formation, evolution and interaction of the cloudwith radiation hydrodynamics and disequilibrium chemistry(Marley et al. 2013; Helling 2018). It is possible to prescribethe spatial boundaries of the cloud deck in a phenomeno-logical manner, as has been implemented in the study ofthe atmospheres of brown dwarfs (e.g., Burrows et al. 2006,2011). Over a limited wavelength range and at low spec-tral resolution, such as by the Hubble Space Telescope WideField Camera 3 (HST-WFC3), it has been shown that thetransmission spectrum probes a limited range of pressuresand the transit chord may be approximated as being isobaric(Heng & Kitzmann 2017), rendering the spatial dependenceof the cloud cross section a non-issue.
For clarity of discussion, the specific case study of WASP-17b is used. Fisher & Heng (2018) have previously esti-mated that R = 1 . R J at P = 8 bar based on theinference made by Heng (2016) that the HST-STIS transitchord of WASP-17b is cloud-free (see also Fisher & Heng2019). The surface gravity of WASP-17b is g = 316 cms − , while the stellar radius of WASP-17 is R ⋆ = 1 . R ⊙ (Southworth et al. 2012). A temperature of T = 1700 K isadopted, which is roughly the retrieved transit chord tem-perature reported by Fisher & Heng (2018) for WASP-17b(see their Table 2). For illustration, I adopt X H O = 10 − ,which is not an uncommon value for the retrieved volumemixing ratio of water (see Figure 29 of Fisher & Heng 2018).With these numbers, H ≈ H/R ≈ . R = 1 . R J and P = 8 bar. One of these spectra adds aconstant term of κ cloud = 0 .
01 cm g − to the opacity func-tion in equation (35), which represents a grey cloud compris-ing large particles. For a grey cloud, the cloud cross sectionand mixing ratio may be subsumed into a single number.Even though this grey cloud is assumed to be spatially uni-form, it corresponds to a pressure of 0.9 mbar for the transitchord using equation (9). One may mimic the effect of thisgrey cloud by setting P = 0 . R that corresponds tothis pressure ( R = 1 . R J ), as shown in Figure 2. It isimportant to note that it is the E ( τ ) term in equation (1)that allows for this mimicry.There are two concerns with this mimicry. First, thevalue of κ cloud may be specified from first principles by speci-fying the cloud particle radius and computing the extinctionefficiency and cross section using Mie theory, whereas it isless clear how intrinsic cloud properties may be related to R . Second, the mimicry does not extend to clouds compris-ing small particles. In Figure 2, an example is shown with r cloud = 0 . µ m and X cloud = 10 − . This cloud produces a non-flat spectral continuum between 0.8 and 1.3 µ m. Itis difficult to see how such wavelength-dependent behaviourmay be specified from first principles via R .The reference transit radius and reference pressureare not independent quantities. Rather, they specify awavelength-independent reference point within the exo-planet, analogous to the radiative-convective boundary ingas giants. From a phenomenological point of view, the roleof a wavelength-independent R ( P ) is well-established inatmospheric retrievals (Fisher & Heng 2018). If one needs tofit for a different value of R at each wavelength, then thenumber of fitting parameters will always exceed the numberof data points. There are several implications of the current study.(i) The second footnote of Jord´an & Espinoza (2018) maybe disregarded.(ii) Equation (1) may be used without assuming τ ≫ E ( τ ) term may be retained.(iii) The E ( τ ) term in equation (1) should not be usedas a proxy for a cloud deck.(iv) Atmospheric retrievals should continue to fix thevalue of R or P and include the other quantity as a fittingparameter (Fisher & Heng 2018), unless the interior struc-ture of the exoplanet is a priori known. The anonymous referee is credited with providing three thor-ough reviews that led to an improved manuscript. Besidesthe intellectual stimulation provided by Jord´an & Espinoza(2018), I am grateful to Andres Jord´an and N´estor Espinozafor useful and constructive conversations. I benefited from acolloquium given on 28th February 2019 at the Jet Propul-sion Laboratory, where Yan B´etr´emieux was part of the au-dience. I thank Yann Alibert for useful email exchanges onpolytropes and the Lane-Emden equation. I acknowledge par-tial financial support from the Center for Space and Habit-ability (CSH), the PlanetS National Center of Competencein Research (NCCR), the Swiss National Science Founda-tion, the MERAC Foundation and an European ResearchCouncil (ERC) Consolidator Grant (number 771620).
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