On degeneracies in retrievals of exoplanetary transmission spectra

Accurate estimations of atmospheric properties of exoplanets from transmission spectra require understanding of degeneracies between model parameters and observations that can resolve them. We conduct a systematic investigation of such degeneracies using a combination of detailed atmospheric retrievals and a range of model assumptions, focusing on H$_2$-rich atmospheres. As a case study, we consider the well-studied hot Jupiter HD 209458 b. We perform extensive retrievals with models ranging from simple isothermal and isobaric atmospheres to those with full pressure-temperature profiles, inhomogeneous cloud/haze coverage, multiple molecular species, and data in the optical-infrared wavelengths. Our study reveals four key insights. First, we find that a combination of models with minimal assumptions and broadband transmission spectra with current facilities allow precise estimates of chemical abundances. In particular, high-precision optical and infrared spectra along with models including variable cloud coverage and prominent opacity sources, Na and K being important in optical, provide joint constraints on cloud/haze properties and chemical abundances. Second, we show that the degeneracy between planetary radius and its reference pressure is well characterised and has little effect on abundance estimates, contrary to previous claims using semi-analytic models. Third, collision induced absorption due to H$_2$-H$_2$ and H$_2$-He interactions plays a critical role in correctly estimating atmospheric abundances. Finally, our results highlight the inadequacy of simplified semi-analytic models with isobaric assumptions for reliable retrievals of transmission spectra. Transmission spectra obtained with current facilities such as HST and VLT can provide strong constraints on atmospheric abundances of exoplanets.


Introduction
Transmission spectroscopy of transiting exoplanets offers a powerful probe to study their atmospheres. Recent observational advancements have enabled high-precision transmission spectra of exoplanets over a broad spectral range. Such observations have been obtained in low resolution from space using Hubble Space Telescope (HST) spectrographs-the Space Telescope Imaging Spectrograph (STIS) in the NUV/ Optical and the Wide Field Camera 3 (WFC3) in the nearinfrared (e.g., Deming et al. 2013;Kreidberg et al. 2015;Sing et al. 2016). On the other hand, spectra of comparable quality have also been obtained recently, particularly in the visible range, from large ground-based facilities such as the Very Large Telescope (VLT) and the Gran Telescopio Canarias (e.g., Sedaghati et al. 2017;Chen et al. 2018;Nikolov et al. 2018).
The spectral range accessible to current facilities has the capability to constrain a wide range of atmospheric properties. While the near-infrared spectral range (1.1-1.7 μm) of the WFC3 contains strong spectral features, due to H 2 O (Deming et al. 2013), the visible range probes features of several other species expected in hot Jupiters such as Na, K, TiO, VO, etc. (e.g., Nikolov et al. 2016;Sing et al. 2016;Sedaghati et al. 2017). In addition, optical spectra can also provide important constraints on the possibility and properties of clouds and hazes (e.g., Brown 2001;Line & Parmentier 2016;Barstow et al. 2017;MacDonald & Madhusudhan 2017a). Statistical constraints on these various properties have been reported from such data sets using rigorous atmospheric retrieval methods for various planets (e.g., Madhusudhan et al. 2014;Kreidberg et al. 2015; Barstow et al. 2017;MacDonald & Madhusudhan 2017a). It is clear from these studies that reliable estimates of atmospheric properties using retrievals of transmission spectra rely heavily on a thorough understanding of the model degeneracies and the capability of the data to resolve the same.
The role of degeneracies in interpreting transmission spectra has been investigated in some detail since the beginning of the field. Several early studies highlighted the importance of various atmospheric properties (e.g., clouds, temperature, composition) on observable spectral features (e.g., Seager & Sasselov 2000;Brown 2001;Fortney 2005). For example, Brown (2001) alluded to possible degeneracies between chemical abundances, temperature structure, and the presence of clouds.
Later, Lecavelier Des Etangs et al. (2008a) noted the degeneracy between chemical abundance and the reference pressure in the atmosphere. Using transit spectroscopy to measure the effective radius, it was possible to derive the pressure assuming an abundance or assuming a pressure to derive the abundance.
While the above early works sought to explore the degeneracies using semi-analytic or equilibrium forward models, the advent of retrieval techniques in the last decade (Madhusudhan & Seager 2009) allowed this problem to be investigated with a rigorous statistical approach. Benneke & Seager (2012) studied the degeneracies involved in retrieving transmission spectra of super-Earths and mini-Neptunes using synthetic spectra. They explored the interplay among chemical composition, cloud-top pressure, planetary radius, and/or surface pressure in determining the spectral features, and suggested combinations of observables that could resolve the degeneracies in different cases. Benneke & Seager (2013) commented on the degeneracy between the mean molecular mass and cloud-top pressure, which is present in transmission spectra especially for low-mass planets.
De Wit & Seager (2013) showed that the slant-path optical depth at the reference radius depends on the scale height, reference pressure, temperature, and number densities of the absorbers present in the atmosphere in unique ways, making their retrieval possible with high-quality data. Such constraints, in principle, also allow the planetary mass to be determined from the transmission spectrum using the retrieved gravity through the scale height (de Wit & Seager 2013), but can be challenging for low-mass planets (Batalha et al. 2017). Griffith (2014) suggested that there can be a broad range of degenerate solutions to fit infrared data, which make constraining molecular abundances challenging. Nonetheless, they suggest ways in which the degeneracy can be resolved. For example, they suggest measuring the radius of the planet at a wavelength where the atmosphere's opacity is known, e.g., Rayleigh scattering in the optical. Line & Parmentier (2016) explored the influence of nonuniform cloud coverage in transmission spectra. They quantitatively explored the degeneracy between clouds and mean molecular weight within an atmospheric retrieval framework. They found that partial and fully cloudy atmospheres are distinguishable, and that the visible wavelengths offer an opportunity to break degeneracies between mean molecular weight and cloud coverage.
The effects of clouds and other surfaces have been studied by Bétrémieux (2016) and Bétrémieux & Swain (2017, 2018. Among their findings are the conclusions that spectral signatures in the optical encode information useful to break degeneracies between retrieved abundances and the planet's radius, and that collision-induced absorption (CIA) potentially determines the highest pressures that can be probed in exoplanetary atmospheres in the infrared. An alternative to breaking the innate degeneracy between clouds and chemistry was offered by MacDonald & Madhusudhan (2017a) by introducing a two-dimensional inhomogeneous cloud coverage.
Lastly, Heng & Kitzmann (2017) highlighted a potential three-way degeneracy among H 2 O abundance, reference pressure (P ref ), and planet radius (R p ) using semi-analytic models. Their conclusions about this degeneracy were based on assumptions of isobaric and isothermal atmospheres with H 2 O as the only molecular opacity source. Our present work investigates this further.
In the present paper, we conduct a detailed analysis of the effect of model parameterization and spectral coverage of data on atmospheric retrievals of transmission spectra. Such an analysis also helps us explore some of the key degeneracies previously discussed in the literature using semi-analytic models. Employing retrieval techniques, we test a series of atmospheric models with varying levels of complexity. In Section 2, we start by reproducing the results of previous analytic studies. We discuss the validity of their interpretations and use their assumptions as a starting point for our study. In Section 3, we perform a step-by-step analysis of model dependencies with retrievals using the canonical hot Jupiter HD209458b as our case study.
We start with retrievals assuming a simplistic clear, isothermal, and isobaric planetary atmosphere and using infrared data alone. We sequentially improve the model considerations culminating in a realistic atmospheric model with a full pressure-temperature (P-T) profile, inhomogeneous clouds, collision-induced opacities, and multiple chemical species. We also study the impact of including data in the optical wavelengths instead of using only data in the nearinfrared. For each of these cases, we investigate the constraints on the retrieved parameters and our ability to determine the chemical abundances, especially that of H 2 O. In Section 4, we assess the ability of our retrievals to constrain atmospheres with high cloud fractions. Lastly, in Section 5, we revisit the notion of a three-way degeneracy between X H O 2 , R p , and P ref . We show that the degeneracy between R p and P ref is real and well characterized, but has no effect on the abundance estimates, contrary to previous assertions. We also show that the choice of an R p versus P ref as a free parameter is inconsequential to constraining molecular abundances when a full retrieval study is performed. We summarize our findings in Section 6.
2. The R p -P ref -H 2 O "Degeneracy" In this section we illustrate some of the key degeneracies inherent to transmission spectra. We begin with a qualitative illustration using model spectra. We generate four forward models showing different combinations of R p , P ref , and X H O 2 , spanning optical and infrared wavelengths. The forward models are generated using parameters for HD209458b with log 10 (g)= 2.963 in cgs and a stellar radius of 1.155R e  (Torres et al. 2008). The models shown here were chosen by inspection and use a parametric P-T profile with the parameters described by Madhusudhan & Seager (2009) with values of log 10 (P 1 )= −1.65, log 10 (P 2 )=−4.02, log 10 (P 3 )=0.48, α 1 =0.67, α 2 = 0.58, and temperature of T 0 =1435 K. The choice of P-T profile parameters is within 2σ of the best-fit values reported by MacDonald & Madhusudhan (2017a).
The models have 100 pressure layers equally spaced in logpressure between 10 −6 and 10 2 bar. Our prescription considers the effects of H 2 Rayleigh scattering and CIA due to H 2 -H 2 and H 2 -He interactions and is adapted from the recent work of Pinhas et al. (2018). The only other source of opacity considered in these illustrative models is H 2 O. The model setup is discussed in more detail in Section 3.
The models are shown in Figure 1 and depict the degeneracies in cloud-free atmospheres. These models show that some spectral features in the infrared can be mimicked by different combinations of H 2 O abundance, radius, and reference pressure. While the degeneracy among radius, pressure, and molecular mixing ratio allows multiple models to show similar spectral features in the infrared, there are significant differences at shorter wavelengths (i.e., below 1 μm). These differences at shorter wavelengths are the result of setting the baseline of the spectrum to different levels by changing R p and/or P ref .
As alluded to in Section 1, several works in the past have discussed possible degeneracies in transmission spectra (e.g., Lecavelier Des Etangs et al. 2008a;Benneke & Seager 2012;de Wit & Seager 2013;Griffith 2014;Bétrémieux & Swain 2017). One of the often discussed degeneracies is that between chemical abundance and reference pressure in the atmosphere for the observed radius. Such a degeneracy was formally investigated using semi-analytic models by Lecavelier Des Etangs et al. (2008a). Their work presents the effective altitude z of the atmosphere at a wavelength λ as where H is the scale height, and σ abs and ξ abs are the cross section and abundance (volume mixing ratio) of the dominant absorbing species, respectively. τ eq , also known as equivalent optical depth, is the slant optical depth at an altitude z eq such that the contribution of an equivalent planet completely opaque below this altitude produces the same absorption as the planet with its translucent atmosphere. P z=0 is the reference pressure at an altitude z=0 corresponding to R p , the measured radius of the planet. Additionally, g is the gravity of the planet, k the Boltzmann constant, T the temperature of the atmosphere, and μ the mean molecular mass of the atmosphere. This expression is one of the first indications of a degeneracy between the reference pressure and the chemical abundance. Lecavelier Des Etangs et al. (2008a) concluded that to derive an abundance, a reference pressure needs to be assumed or vice versa. Variants of this expression have also been derived from first principles in other studies (de Wit & Seager 2013;Bétrémieux & Swain 2017;Sing 2018). The expression was later used by Heng & Kitzmann (2017, hereafter HK17) in the following form: where R 0 is the radius of the planet at the reference pressure (R p in this work), P 0 is the reference pressure (P ref in this work), H is the scale height, g is the gravity of the planet, and κ is the cross section per unit mass.  (2), which was used to fit an observed transmission spectrum of the hot Jupiter WASP-12b in the near-infrared (∼1.15-1.65 μm) obtained using the HST WFC3 spectrograph (Kreidberg et al. 2015). The model assumed an isothermal atmosphere with isobaric opacities, with H 2 O as the only molecular opacity source. The model was fit to the near-infrared spectrum using a nonlinear least-squares fitting routine to obtain best-fit values of different combinations of parameters for an assumed value of R p . By repeating the fits for a range of R p values, they investigated the degeneracy To investigate the potential three-way degeneracy reported by HK17, we follow two approaches. We first reproduce the results of HK17 using their approach, i.e., their semi-analytic model and least-squares fit to the WFC3 transmission spectrum of WASP-12b. We then reproduce the same results using their semi-analytic model in a Bayesian retrieval approach. We later include additional opacity due to H 2 -H 2 and H 2 -He CIA in the HK17 model to investigate the validity of their assumptions. With CIA included, we follow the same two approaches, i.e., first employing a nonlinear least-squares fit and then a Bayesian retrieval.
We begin by following the approach of HK17 and performing a fit to the WASP-12b WFC3 data using Equation (2) with a least-squares minimization routine (curve_fit in Python). Our model considerations are identical to those in HK17, e.g., isothermal atmosphere and isobaric H 2 O opacity. The top-left panel of Figure 2 shows our results reproducing Figure 7 of HK17. At the outset, we notice two discrepancies. First, we are able to reproduce the fit in HK17 using the log of X H O 2 (P ref /10 bar) versus R p . However, HK17 presented their y-axis as X H O 2 (P ref /10 bar) −1 . We interpret this as a typographical error in HK17. This is especially the case considering that Equation (2) implies the product of X H O 2 and P ref , and also considering Figures 3 to 8 of Fisher & Heng (2018), who used the same model and notation. Second, HK17 claimed from this figure that X H O 2 is strongly degenerate with R p , i.e., that the H 2 O abundance varies by many orders of magnitude with slight changes in R p . However, it is not possible to deduce information about the H 2 O abundance from this figure alone given that only the product X H O 2 (P ref /10 bar) is shown to be degenerate with R p and not X H O 2 or P ref individually.
Next, we study this problem using a Bayesian retrieval approach. Our retrieval code is adapted from the works of Pinhas et al. (2018) to consider the semi-analytic model and assumptions discussed above. We replace the numerical model of Pinhas et al. (2018) with the semi-analytic model of HK17 while retaining the module for Bayesian parameter estimation using the nested sampling algorithm (Feroz et al. 2009;Buchner et al. 2014). The model parameters remain the same as in HK17, namely X H O 2 , P ref , κ cloud , R p ,and T, the isothermal temperature. The prior range for the radius is R p =1.79R J to properties. Spectra in red, blue, and green include variations of two or three parameters that are still capable of generating features similar to the reference spectrum shown in black. There is a clear difference in the spectra at shorter wavelengths.
1.87R J  to match the range shown in Figure 7 of HK17 and the prior range of log 10 (P ref ) is from −6 to 2 in bar. Both the X log 10 H O 2 ( ) (volume mixing ratio) and log 10 (κ cloud ) (m 2 kg −1 ) priors are from −1 to −10. The temperature prior is from 500 to 2000 K. Similarly, we consider an isothermal atmosphere, isobaric H 2 O opacity (at 1 mbar), a fixed mean molecular weight of 2.4 amu, and a fixed gravity of log 10 (g)=2.99 in cgs (Hebb et al. 2009).
The results from the retrieval are shown in green in the topleft panel of Figure 2. We show the posteriors from the retrieval as a two-dimensional histogram of X H O 2 (P ref /10 bar) against the retrieved R p . The bottom four panels of Figure 2 show the posterior distribution of the H 2 O abundance, P ref , κ cloud , and R p in the green histograms. . We find that the posterior distributions from the retrieval closely follow the results from the linear fit (i.e., orange line) as shown in the topmost left panel.
We now investigate the validity of the assumptions of the semi-analytic model above by including CIA absorption as an additional source of opacity. The importance of CIA as a continuum source of opacity is highlighted in several previous studies (e.g., de Wit & Seager 2013; Bétrémieux & Swain 2017, 2018, which makes its inclusion imperative in model spectra of giant planets. We amended the total opacity in the formulation of HK17, shown in Equation (3) where the first two terms remain as explained above. The third term is the opacity due to H 2 -H 2 and H 2 -He CIA. This and other opacity sources are discussed in Section 3. We follow the approach described above by performing a least-squares fit of the amended model to the near-infrared data. The additional opacity source (i.e., CIA) is computed following the same method used to compute H 2 O opacity in HK17, and we preserve the assumption of an isobaric atmosphere by evaluating the opacities at 1 mbar. We find that the inclusion of CIA absorption changes the slope of the resulting linear relationship between X H O 2 (P 0 /10 bar) and R p . Our resulting fit is shown as an orange solid line in the top-right panel of Figure 2, where we also show the fit of HK17 using a dashed black line. This analytic fit shows that the slope of the relation between X H O 2 (P 0 /10 bar) and R p has changed. Again, it is not possible to infer from this result if R p is degenerate with X H O 2 or P ref or both. In comparison, a retrieval approach would provide the necessary insight as pursued above.
We, therefore, now perform a retrieval using the modified model including CIA opacity. Our retrieval approach keeps the previous description although in this case we add the pressuredependent effects of CIA. This retrieval study finds a better constrained H 2 O abundance with a median of . Similarly to the previous retrieval, we present in the background of the top-right panel of Figure 2 the twodimensional histogram of X H O 2 (P ref /10 bar) against R p . We also show the posterior distributions of the retrieved parameters including the H 2 O abundance for this case in the bottom orange histograms. We find that the inclusion of CIA opacity results in a better constraint on the H 2 O abundance even within the framework of this simplistic model.
Our results above demonstrate two main points. First, the retrieved molecular abundance changes with the inclusion of CIA absorption. The inclusion of CIA opacity provides a continuum to the spectrum that sets the maximum pressure probed in the atmosphere, i.e., the line-of-sight photosphere (de Wit & Seager 2013;Line & Parmentier 2016;Bétrémieux & Swain 2018). As a result, the thickness of the atmospheric column probed by the transmission spectrum decreases compared to the non-CIA scenario, thereby requiring a different H 2 O abundance to explain the data. Second, the log-linear behavior seen in both panels of Figure 2 is likely strongly influenced by a relation between P ref and R p , irrespective of the H 2 O abundance. We further discuss this relation in detail in Section 5. The constraint on the H 2 O abundance improves with the inclusion of CIA, irrespective of any degeneracy between P ref and R p . Nevertheless, the H 2 O abundance is still weakly constrained even in the CIA case. However, this is not due to a three-way degeneracy but rather a result of incomplete model assumptions and limited data. We demonstrate this in more detail in the next section.
In summary, these results show that the conclusions of HK17 are likely due to the restricted model assumptions. The lack of consideration of CIA opacity, among other factors, is likely responsible for their conclusions. We discuss this further in Section 5.3. The three-way degeneracy noted in HK17 manifests itself fully under idealized conditions encapsulated in the analytic formalism of Equation (2), namely an isothermal, isobaric, constant mean molecular weight, constant gravity, a single absorber, and a cloud-free atmosphere. In a more realistic atmosphere, this degeneracy is broken in various ways. For example, for high chemical abundances, the mean molecular weight becomes significant enough to affect the scale height and hence the amplitude of the spectral feature (e.g., Benneke & Seager 2012;Line & Parmentier 2016). On the other hand, at low abundances, the CIA provides the continuum level for the spectrum (e.g., de Wit & Seager 2013;Line & Parmentier 2016). Other effects influencing the spectrum include considerations of clouds, non-isothermal atmospheres, multiple-molecular absorbers, etc. Furthermore, constraining the contributions from these various effects require observed spectra in the visible in addition to the infrared spectra. The importance and effects of such considerations are studied in the rest of this work. In what follows, we perform an in-depth study of the effects of model assumptions and data coverage on atmospheric retrievals using transmission spectra.

HD209458b: A Case Study
We now conduct a systematic exploration of the degeneracies in interpreting transmission spectra using fully numerical models within a rigorous retrieval framework. For this study, we choose the canonical hot Jupiter HD209458b, which has the most data available (Deming et al. 2013;Sing et al. 2016) and has been a subject of several recent retrieval studies (e.g., Madhusudhan et al. 2014;Barstow et al. 2017;MacDonald & Madhusudhan 2017a).
We used an atmospheric retrieval code for transmission spectra adapted from the recent work of Pinhas et al. (2018). The code was modified to include the radius of the planet (R p ) as one of the retrieval parameters and, unlike Pinhas et al. (2018), we do not infer any stellar properties. The code computes line by line the radiative transfer in a transmission geometry, assuming hydrostatic equilibrium. We consider a parametric P-T profile using the prescription of Madhusudhan & Seager (2009). We consider a one-dimensional model atmosphere consisting of 100 layers in pressure ranging from 10 −6 -10 3 bar, uniformly spaced in log 10 (P). We use the cloud/ haze parametrization of MacDonald & Madhusudhan (2017a), which allows for cloud-free to fully cloudy models, including nonhomogeneous cloud cover. The haze is included as where γ is the scattering slope, a is the Rayleigh-enhancement factor, and σ 0 is the H 2 Rayleigh scattering cross section (5.31×10 −31 m 2 ) at the reference wavelength λ 0 =350 nm. Cloudy regions of the atmosphere are included as an opaque cloud deck with cloud-top pressure P cloud . The fraction of cloud cover at the terminator is given by f .
The absorption cross sections of the molecular and atomic species are obtained from Rothman et al.  Gandhi & Madhusudhan (2017). Our model assumes that the atmosphere has uniform mixing ratio for each species considered and treats these mixing ratios as free parameters. Unlike the retrievals in Section 2, these retrievals do not fix the mean molecular weight to a specific value and instead calculate it based on the retrieved molecular abundances and assumption of a H 2 -He-dominated atmosphere with a fixed He/H 2 ratio of 0.17 (MacDonald & Madhusudhan 2017a). Lastly, the reference pressure (P ref ) is a free parameter that establishes the pressure in the atmosphere at which the reference radius of the planet (R p ) is located. In summary, our full model has 19 free parameters: R p , P ref , seven chemical species (H 2 O, CO, Each column indicates a model assumption or a free parameter. The WFC3 column indicates the inclusion of data in the near-infrared in the retrieval. On the other hand, optical signifies that data in the optical wavelengths were used in the retrieval. P-T means that we consider a parametric P-T profile in the retrieval. Clouds are implemented in two ways: F stands for a retrieval with full cloud cover in which the cloud fraction is fixed to f =100%, and N represents cases with nonuniform clouds in which the cloud fraction f is a free parameter in the retrieval. CO 2 , HCN, NH 3 , Na, and K), six parameters for the P-T profile, and four parameters for clouds/hazes including the cloud deck pressure P cloud and cloud fraction f . Our goal is to investigate the effect of each model parameter and/or assumption on the retrieved parameters and their degeneracies. We start with the simplest setup and gradually increase the physical plausibility of the model and extent of the data. We start by considering an isothermal and isobaric atmosphere with only one molecule present, H 2 O, to carry on from our reproduction of previous results in Section 2. We later increase the number of considerations until we use a full model with a parametric P-T profile, with multiple molecules (H 2 O, Na, K, NH 3 , CO, HCN, and CO 2 ) and the presence of clouds/ hazes. For our retrievals, we use the spectrum of HD209458b reported in Sing et al. (2016). The spectrum has two wavelength ranges observed with HST: near-infrared (1.1-1.7 μm) obtained using WFC3 and full optical range (0.3-1.01 μm) obtained using the STIS instrument. We compare the retrieved radius values to the value reported by Torres et al. (2008) of R p =1.359 0.019 0.016 -+ R J ,which is consistent with the reported radius by Sing et al. (2016).
The 12 cases in our study are summarized in Table 1. The parameters, priors, and results for all cases are summarized in Tables 2 and 3 in the Appendix. The retrieved median spectra for all the cases are shown in Figure 3. The constraints on the retrieved H 2 O abundances for the different cases are illustrated in Figure 4. The posterior distributions for X H O 2 , P ref ,and R p for all cases are included in the Appendix.

Case 0: Reproducing the Semi-analytic Model
Before conducting our case-by-case study, we first consider case zero, which presents a numerical analog of semi-analytic models. Case zero has the simplest model considerations, i.e., of an isothermal and isobaric atmosphere with H 2 O absorption as the only source of opacity. In addition, the mean molecular weight and gravity are fixed quantities. The isobaric assumption means evaluating the molecular cross section at only one pressure, in this case 1 mbar. Following the models in Section 2, the mean molecular weight is fixed to a value of 2.4 amu, that of an H 2 -rich atmosphere with solar elemental abundances. The fixed value for gravity is log 10 (g)=2.963 in cgs for HD209458b (Torres et al. 2008).
While generally our numerical model spans a pressure range of 10 −6 to 10 3 bar as discussed above, in the present case, the retrieval is strongly sensitive to the edges of the pressure range, due to the limited opacity sources. The deepest pressure level in the model atmosphere effectively acts as an opaque surface. In order to circumvent this edge effect, we consider a model atmosphere with an unrealistically extreme range in pressure, from 10 −14 to 10 14 bar, uniformly spaced in log 10 (P) using 400 layers.
We use this model for a retrieval using a near-infrared WFC3 spectrum of HD209458b, similarly to our retrievals in Section 2. The model parameters are  (Heng & Kitzmann 2017). However, it is important to note that the degeneracy is a result of unrealistic model assumptions. In addition to the factors discussed in Section 2 and later in this section, several other factors make this case unphysical. First, it is unrealistic to have an atmosphere expanding to such high pressures (e.g., 10 14 bar) while maintaining the isobaric assumption for the cross sections, especially evaluating them at 1 mbar. Second, such a deep atmosphere would become opaque at much lower pressures, due to the effects of CIA (e.g., de Wit & Seager 2013; Bétrémieux & Swain 2018); this is further explored in Section 3.3. Third, assuming a fixed mean molecular weight is unrealistic at high H 2 O abundances explored in the retrieval such as X 10 H O 2 2  -(e.g., Benneke & Seager 2012;Line & Parmentier 2016). Fourth, maintaining a fixed gravity over the entire atmosphere spanning many orders of magnitude in pressure is also unrealistic.  Table 1. Infrared and optical data from Sing et al. (2016) are shown using green markers. While all models produce some degree of fit to the data in the infrared, only cases 8-12 produce a good fit to all of the data.
Nevertheless, the present case clearly demonstrates the threeway degeneracy among R p , P ref , and X H O 2 obtained for such a simplistic model while fitting near-infrared data alone.
We now perform a case-by-case retrieval study using more realistic model atmospheres as explained at the beginning of Section 3. All of the cases henceforth consider models with a height-dependent g, a variable mean molecular weight, and a pressure range of 10 −6 -10 3 bar.

Case 1: Isobar, Isotherm, H 2 O Only, and WFC3 Data
The initial model we now consider is that of an atmosphere that is best described by an isotherm at a temperature T and an isobar with only one molecule present, H 2 O. For clarity, we specify that the isobaric assumption means evaluating the molecular cross section at only one pressure, while density, pressure, and gravity are still changing with height. Making only these assumptions in our model means ignoring CIA opacity due to H 2 -H 2 and H 2 -He. Furthermore, we apply this model on WFC3 data only in order to test the retrievals with a limited wavelength range.
The molecular cross sections are evaluated at 1 mbar following HK17.  (Torres et al. 2008). However, the reference pressure is not tightly constrained, and the retrieved H 2 O abundance is ∼4 orders of magnitude smaller than that in other studies (Madhusudhan et al. 2014;Barstow et al. 2017;MacDonald & Madhusudhan 2017a). The retrieved H 2 O abundance in this case is also sensitive to the bottom of the model atmosphere for the same reason as in Section 3.1. In this case, the bottom of the atmosphere is at P=10 3 bar, which limits the amplitude of the H 2 O feature in the model spectrum, similar to the effects of an opaque surface. Changing the bottom pressure of the atmosphere can result in different H 2 O abundance constraints. Nevertheless, for the present demonstration, we have assumed a physically realistic pressure range of 10 3 -10 −6 bar. Regardless of the pressure range, the present case is inevitably unrealistic, due to the lack of various other model considerations, which are incorporated in subsequent cases below. More importantly, this edge effect is not relevant once CIA opacities are considered.
3.3. Case 2: Case 1+H 2 /He CIA We now consider a slightly more realistic model that includes CIA opacities due to H 2 -H 2 and H 2 -He given that the test case of HD209458b is a gas-giant planet with a H 2 -dominated atmosphere. All other assumptions about the isothermal and isobaric characterization of the atmosphere in the model stay the same as in the previous retrieval. However, while we still evaluate the molecular cross sections at 1 mbar, we consider the CIA to be pressure dependent. We also consider gaseous Rayleigh scattering due to H 2 in this and all subsequent cases.
The inclusion of CIA has resulted in a value for the H 2 O abundance that is consistent with other studies while keeping R p consistent with the white light radius within 2σ. This highlights the importance of considering CIA for constraints on the molecular abundances, as also discussed in Section 2. We find that ignoring CIA leads to erroneous results. CIA opacity determines the highest pressures that can be probed, and as a result provides the continuum to the spectrum (Line & Parmentier 2016;Bétrémieux & Swain 2018). The inclusion of CIA raises the slant photosphere of the planet to a higher altitude compared to the previous case. By decreasing the thickness of the observed slant column of the atmosphere along the line of sight, a higher abundance is required to explain the same features. In comparison, case 1, where we did not have CIA opacity, the effective column of the atmosphere is larger and hence requires less H 2 O abundance to explain the same features. Our results show that the molecular abundance is much less biased upon the inclusion of CIA.  Table 1 and Section 3. The abundance (i.e., mixing ratio) for case 1 has been increased by 10 4 to be in the same range as the abundances of other cases.
While the isobaric assumption makes for a simplified problem construction in analytic models, it is not necessary when numerical methods are available. It is computationally inexpensive to evaluate the molecular opacities at the corresponding pressure in the atmosphere instead of assuming a constant pressure of 1 mbar.

Case 3: Case 2 without an Isobar
We now remove the assumption of an isobar for the calculation of H 2 O opacities. Instead, we calculate the molecular opacities at the corresponding pressure in the atmosphere rather than at a fixed pressure of 1 mbar. We maintain the remaining assumption of an isotherm for the temperature profile of the atmosphere. Our retrievals obtain an isothermal profile with  Tables 2 and 3.
While the consideration of pressure-dependent CIA is essential, assuming molecular line cross sections to be isobaric does not make a significant difference compared to the present case given current data quality. This is because the atmosphere is mostly probed at low pressures as discussed in Section 5. However, the isobaric assumption cannot be maintained when considering the effects of CIA as the CIA opacity has a stronger dependence on pressure, being proportional to the pressure squared (de Wit & Seager 2013).

Case 4: Case 3+P-T Profile
We now remove the assumption of an isothermal atmosphere and consider a full P-T profile in our retrieval. We implement the parametrization used in Madhusudhan & Seager (2009), which involves six parameters that capture a typical P-T profile. Along with this, we retrieve X H O 2 , P ref , and R p . This allows the atmosphere to have any P-T profile the data requires.
With the inclusion of the parametric P-T profile, we retrieve nine parameters in total. This retrieval results in R p =1. , and temperature of T 0 =870.11 49.12 82.12 -+ K. The retrieved values did not change significantly compared to the assumption of an isothermal atmosphere as in case 3. These numerical results agree with analytic studies that predict that while non-isothermal atmospheres distort the spectrum of an isothermal one, the effects are subtle considering present data quality with HST (Bétrémieux & Swain 2018). The retrieved mixing fraction of H 2 O is consistent with that of other studies (Madhusudhan et al. 2014;Barstow et al. 2017;MacDonald & Madhusudhan 2017a).

Case 5: Case 4 + Full Cloud Cover
We continue to remove assumptions from our model and now consider the possibility of clouds being present in the atmosphere of the planet. There is no a priori information to assume that the atmosphere of HD209458b is cloud free. We consider the cloud prescription of MacDonald & Madhusudhan (2017a) as explained at the beginning of this section. We include four parameters for clouds and hazes. For hazes, we use a, the Rayleigh-enhancement factor, and γ, the scattering slope. For clouds, P cloud and f characterize the pressure level of the optically thick cloud deck and cloud coverage fraction, respectively. In this particular case, instead of considering a clear atmosphere like we did in case 4, we consider the presence of a fully cloudy planet atmosphere by fixing f =100%.
The inclusion of a fully cloudy deck increases the number of retrieved parameters from nine to 12.
. Although the value of the retrieved planetary radius is still consistent with the observed radius, we see that the 1σ limits have increased. Similar effects are seen with the retrieved H 2 O abundance.
The interesting effect of the inclusion of a fully covering cloud deck is that the H 2 O abundance is now hardly constrained. Because the pressure at which this cloud deck could be located spans several orders of magnitude, so does the H 2 O abundance. In this case, the cloud deck mimics a surface (Bétrémieux & Swain 2017), and the pressure at which the cloud is located is fully degenerate with the retrieved H 2 O abundance. A cloud deck at a higher altitude requires higher H 2 O abundance to account for the same features, while a lower cloud deck can explain the same features with a lower molecular abundance (e.g., Deming et al. 2013;Barstow et al. 2017). An alternate way to explain this is similar to what happened with the inclusion of CIA in case 2. By lowering the cloud deck altitude (i.e., increasing the cloud-top pressure), we are increasing the effective column of the observable atmosphere, which requires lower abundance than a smaller observable atmosphere corresponding to a cloud deck at a higher altitude (i.e., decreasing the pressure). We also notice that the lowest H 2 O abundance is consistent with the lowest abundance found in case 2, due to CIA providing the continuum opacity. . The inclusion of nonhomogeneous clouds does not significantly change the retrieved P-T profile parameters. It, however, allows a constraint on the cloud fraction to be placed at ∼68%. Furthermore, R p and P ref are consistent with those of cases 4 and 5.
While the median value of the retrieved H 2 O abundance is consistent with that of case 5, the uncertainty is smaller when nonhomogeneous clouds are considered. Considering a nonhomogeneous cloud cover allows for a better H 2 O constraint compared to the assumption of a fully cloudy atmosphere. It is true that the constraints in the case of a clear atmosphere are even tighter (e.g., case 4), but the validity of this assumption is not evident. Furthermore, previous studies suggest that failure to consider nonhomogeneous cloud cover can bias molecular abundance findings (Line & Parmentier 2016).
We now look into other factors that could help further constrain molecular abundances. Until now, we have only considered HST WFC3 data in the near-infrared for retrievals with different model assumptions. Given that the main differences in spectra with clouds manifest in the optical wavelengths, we now incorporate data in the optical.

Case 7: Case 6 + Optical Data
Our seventh case considers the inclusion of an optical spectrum of HD209458b. We included data from 0.30 to 0.95 μm from Sing et al. (2016). The addition of optical data helps constrain the Rayleigh slope and cloud properties (Benneke & Seager 2012;Griffith 2014;Line & Parmentier 2016). This also allows us to evaluate the effects of more data considered in our retrieval. We keep the number of parameters the same as in case 6, for a total of 13. We report a retrieved . As expected, the parameters most affected, compared to case 6, are those responsible for clouds and hazes. The inclusion of data in the optical allows us to place tighter constraints on a and γ, which characterize the slope in the optical. The cloud parameters are consistent with those of case 6, with f mostly unchanged. However, the uncertainty in log 10 (P cloud ) is smaller by almost a factor of 6 compared to the values in case 6. Naturally, given that we now have information in the wavelength range where the scattering slope manifests itself, our cloud and haze prescription can fit for it, in contrast to previous cases where we fit for the slope without adequate data in the optical range.
Furthermore, by constraining the baseline of the spectrum, we are now able to place better constraints on the H 2 O abundance. The uncertainties on the H 2 O abundance are half as small as the ones from case 6. Thus, it is evident that the inclusion of optical data allows for better estimates of chemical abundances. Our numerical results show the importance of short wavelengths in breaking key degeneracies and in better constraining molecular abundances in agreement with previous analytic studies (e.g., Benneke & Seager 2012;Griffith 2014;Line & Parmentier 2016).
The last step in increasing the physical reality of our model is to allow for the presence of more molecules in our atmosphere. This would prevent our models from trying to explain every spectroscopic feature with only one molecule. Furthermore, a possible way to break the degeneracy between R p and mixing ratios is to consider the absorption features of different absorbers (Benneke & Seager 2012). In the next cases, we incorporate several species that can be prominent in hot Jupiter atmospheres, e.g., Na, K, NH 3 , CO, HCN, and CO 2 (Madhusudhan et al. 2016).
3.9. Case 8: Case 7 + Na and K The first species we incorporate are the alkali atomic species Na and K. Given that their spectroscopic features are present in the range covered by the additional optical data, we investigate the impact these species have on the retrieved H 2 O abundances.  Figure 4, and all retrieved parameters are summarized in Tables 2 and 3.

Our retrieved H 2 O abundance for this and other cases is shown in
The inclusion of Na and K has further decreased the 1σ spread of the retrieved H 2 O abundance almost by a factor of 2. While the retrieved H 2 O abundance is consistent with that of case 7, it is important to note that the posterior distribution has shifted toward lower H 2 O abundance by ∼0.5 dex. This shift in the median value is as much as the shift between case 6 and case 7 due to the inclusion of optical data. This suggests that the Na and K, which themselves are constrained by the optical data, also strongly affect the retrieved H 2 O abundance. This is due to better fitting the features in the optical. An additional effect is the change in the retrieved cloud fraction from ∼70% in case 7 to ∼50%. Evidently, these results will be sensitive to the absorption cross sections being used. Nonetheless, it is clear that including molecules that have signatures in the optical allow us to fit the data in those wavelengths better and further constrain the H 2 O abundance. This has little effect on the retrieved R p and P ref, which continue to be well constrained.
3.10. Case 9: Case 8 + NH 3 Next, we include NH 3 as a source of opacity. The retrieval gives the following results for molecular abundances: log 10 (X  , which are also consistent. Similar to NH 3 , our constraint on HCN is also weaker given current data. Our constraint, however, is consistent with the mixing ratio of 10 6 -, which was required to detect HCN on the dayside of the planet (Hawker et al. 2018).

Case 12: Case 11 + CO 2
We add one last molecule, CO 2 , in order to have what we refer to as a full retrieval. This is the equivalent to a state-ofthe-art retrieval in which several molecules and atomic species, a parametric P-T profile, and nonhomogeneous clouds are considered, totaling 19 free parameters. This retrieval gives us an atmosphere with the following molecular abundances: log 10 (X Overall, with the inclusion of all the effects discussed, we find that the combination of near-infrared and optical data allows strong constraints on several important parameters and in resolving key degeneracies. The H 2 O abundance is tightly constrained, and it is consistent with values of previous studies (e.g., Barstow et al. 2017;MacDonald & Madhusudhan 2017a). Other chemical species are less well constrained owing to their weaker opacities in the observed range. Nonetheless, retrieved abundance estimates are consistent with studies that investigate their presence in the planet's atmosphere, e.g., detection of HCN (Hawker et al. 2018). While the abundance of H 2 O is retrieved, P ref and R p are also retrieved, with the latter being consistent with the observed photometric radius of R p =1.359 0.019 0.016 -+ R J  (Torres et al. 2008). The full retrieval has resolved the degeneracy between X H O 2 , R p ,and P ref . Simultaneously, the cloud fraction is retrieved with tight constraints on its value, indicating that the planet is not cloud free. The inclusion of multiple absorbers in our retrievals helps break key degeneracies in our results. One of the advantages of the retrieval technique is that robust models (i.e., those that consider parametric P-T profiles, with many molecules, and partial clouds) can be implemented efficiently. The full posterior distributions for all retrieved parameters are included in the Appendix.

Key Lessons
Here we summarize the results from our case study of HD209458b based on retrievals with various model assumptions. Overall, with the inclusion of all the effects, we find that the combination of near-infrared and optical data are responsible for strong constraints on several important parameters resolving key degeneracies. The combination of data and accurate models allows for high-precision retrievals that impose tight constraints on the H 2 O abundance, R p , P ref , and the cloud fraction. The retrieved H 2 O abundances under different model assumptions are shown in Figure 4. The full retrieval is able to also estimate the abundance of other chemical species like HCN.
The retrieval's ability to constrain the H 2 O abundance is not affected by R p and P ref . We find that is is possible to simultaneously retrieve both R p and P ref and find values for R p in agreement with the observed photometric radius. We also analyze the impact of the cloud fraction and the potential degeneracy between this parameter, and the planetary radius and the H 2 O abundance. We first find that there are strong differences in the retrieved H 2 O abundances between a cloudfree and fully cloudy atmosphere. Assuming a fully cloudy atmosphere introduces a degeneracy between the H 2 O abundance and the pressure at which the cloud deck is located, because the cloud deck has the same effect as the optically thick photosphere on the transmission spectrum. An alternative to this is to consider nonhomogeneous cloud coverage in the atmosphere of the planet as there is no a priori information that favors a cloud-free atmosphere or a 100% cloudy atmosphere. On the other hand, theoretical models suggest the presence of partial clouds at the day-night terminators, i.e., the limbs, of planets (e.g., Kataria et al. 2016;Parmentier et al. 2016). We also find that in order to better constrain the clouds and hazes, it is important to consider data points in the optical wavelength range where clouds and hazes manifest themselves. We find that there is no degeneracy between cloud fraction and radius of the planet. Furthermore, it can be seen that it is not necessary to assume a fixed cloud fraction, and instead it is better to allow for the cloud fraction to be a free parameter in the retrieval.
A crucial lesson of our study is that CIA opacity is key in constraining molecular abundances in both clear and cloudy atmospheres. The lack of CIA due to H 2 -H 2 and H 2 -He in the model skews the retrieved H 2 O abundance by several orders of magnitude. Once CIA contribution is considered, the retrieved abundances are consistent within one order of magnitude. The CIA opacity strictly limits the location of the planetary photosphere and, hence, the column of the atmosphere above the photosphere that is probed by the observed spectrum. Without CIA, the photosphere will lie deeper in the atmosphere, increasing the observable column. As such, the molecular abundances will be higher when considering CIA in comparison to models without CIA.
The inclusion of optical data in retrievals is paramount to provide highly constrained H 2 O abundances while helping constrain the range of possible planetary radii and their associated reference pressures. In addition, we find that Na and K absorption lines in the optical significantly affect the constraints on H 2 O abundances. The availability of a broad spectral range between optical and near-infrared helps provide joint constraints on the H 2 O abundance and the reference pressure or the planetary radius. On the other hand, strong degeneracy still persists between the R p and P ref without affecting the H 2 O abundance. This relationship is further discussed in Section 5. Optical data also allow for tight constraints on the cloud fraction of the planet, making it possible to assess whether a planet is cloud free or not. In the next section, we investigate the effectiveness of the cloud parametrization and its ability to constrain the cloud fraction in the utmost case of a fully cloudy atmosphere.

Solutions to Homogeneous Cloud Cover
Here we investigate the robustness with which clouds can be constrained. In particular, we focus on the ability to retrieve the cloud fraction of the atmosphere f in the worst case scenario of a fully cloudy atmosphere (i.e., f =100%). It can be argued that a 100% cloud deck leads to an entirely degenerate set of solutions for the H 2 O abundances, as seen in Section 3.6. This leads to the question of whether an inhomogeneous cloud prescription can resolve this problem. In order to answer this question, we investigate the potential of retrievals to estimate the cloud fraction covering a planet's atmosphere. For this, we consider the median values for the full retrieval of HD209458b, performed in Section 3.13, which includes data in the near-infrared and optical ranges, multiple molecules, a parametric P-T profile, and clouds. We use these values to generate three synthetic data sets with three different cloud fractions. The simulated data has the same resolution, error, and wavelength range as the data in Sing et al. (2016). In our simulated data, we add random error to the binned transit depth drawn from a normal distribution. The simulated models have cloud fractions of 100%, 90%, and 80%. Figures 5-7 show the results of our retrievals along with the values of the parameters used in the simulated data. For all cloud fractions (f ), our retrieved molecular abundances are consistent with the input value within 2σ. H 2 O can be reliably estimated for f 80%. For higher f , only upper limits are found but the f is accurately retrieved. f is always retrieved within ∼1σ. Furthermore, the retrieved f , R p , and P ref are consistent with the input values in all cases. These results demonstrate that the retrieval technique can discern the cloud fraction covering the planet's atmosphere without compromising the ability to retrieve other properties.
The worst case scenario would be an atmosphere with 100% cloud coverage at a very high altitude, as in the present case. Such a high-altitude cloud deck mutes almost all spectral features, resulting in a flat spectrum. Although no molecular  abundances are reliably constrained for this case, the cloud fraction is still correctly retrieved to be consistent with 100% as shown in Figure 7. In principle, a 100% cloud deck at a lower altitude, e.g., at 10 mbar pressure level, would still have some spectral features. Stronger spectral features result in better constraints on the model parameters even for 100% cloud coverage given adequate data in the optical and infrared. Lower cloud fractions are naturally retrievable in all these cases. These results are consistent with the studies of MacDonald & Madhusudhan (2017a) and agree that nonuniform cloud coverage in models allows for a more precise determination of chemical abundances in transmission spectra in comparison to models that assume a fixed cloud fraction, effectively breaking the cloud-composition degeneracies. These results also show that nonhomogeneous and homogeneous cloud scenarios are distinguishable, in agreement with Line & Parmentier (2016).

The R p -P ref Degeneracy
As discussed in Section 1, several recent studies have highlighted possible degeneracies between chemical abundances, clouds/hazes, and reference radius in interpreting transmission spectra (e.g., Lecavelier Des Etangs et al. Recently, HK17 inferred a three-way degeneracy among R p , P ref ,and X H O 2 as a fundamental hindrance for deriving chemical abundances. They argue that one way to break the three-way degeneracy is to find a functional relationship between R p and P ref . In this section, we interpret our results from Sections 2 and 3 and present an empirical relation between R p and P ref . We show that previous suggestions of a three-way degeneracy are the result of model simplifications and inadequate data, and that the primary degeneracy is between R p and P ref . The relationship between R p and P ref is explored when a fit or retrieval is performed. We briefly revisit our reproduction of previous semi-analytic results from Section 2 and our retrievals from Section 3. We begin by revisiting Figure 2, which shows a linear relationship between R p and log 10 (X H O 2 P ref ) obtained from fitting a near-infrared WFC3 spectrum of the hot Jupiter WASP-12b. We find that the slope can be described as m=−1/(H ln 10), where H is the atmospheric scale height. The slope of the linear fit obtained by HK17 and reproduced by us is m=−85.77. Using the above relationship, this slope is consistent with a scale height of 362 km, matching the estimated value for this planet reported in HK17.
We now investigate this empirical finding using the retrievals from Section 3, under different model assumptions. The correlations between log 10 (P ref ) and R p for each of our retrievals of Section 3 are shown in Figure 8, along with a linear fit and the corresponding slope. The fit is obtained using polyfit included in NumPy (Oliphant 2015). Accompanying this figure we have Figure 9, where we show log 10 (X H O 2 ) as a function of R p for the same cases. Figure 9 shows the posterior distributions of X H O 2 , which become more localized as different assumptions are removed from the retrievals. It is clear from Figures 8 and 9 that while log 10 (P ref ) and R p are strongly degenerate, there is almost no degeneracy between log 10 (X H O 2 ) and R p in most of the cases. The only exception is case 5, with an assumed cloud fraction of 100%. This assumption introduces a degeneracy between the cloud level (i.e., P cloud ) and R p . The different combinations of R p and P cloud that  explain the spectrum for an assumed f =100% result in the wide spread of H 2 O abundances observed in Figure 9, case 5.
From Figure 8, it can be observed that there is a log-linear relation between P ref and R p . The superimposed linear fit gives us an idea of what the scale height for each model is, i.e., m=−1/(H ln 10), as we did above with our analysis of Figure 2. While the slopes vary between cases (m=−28 to −81), they converge to a value of m=−58 as the model and data in our retrieval become more robust (i.e., case 7 and above). Figure 8 also shows a temperature estimate for the photosphere of the planet, which we obtain using the slope of the linear fit and assuming a mean molecular weight of 2.4 amu and a planet gravity of log 10 (g)=2.963 in cgs. We find that these temperature estimates range between ∼1012 K and ∼1910 K for all cases except case 5; we discussed the exception of case 5 previously. The temperature values converge in case 12 to 1430 K, which is consistent with the equilibrium temperature of the planet as well as the photospheric temperature estimated in previous studies (e.g., MacDonald & Madhusudhan 2017a). These findings suggest that the relationship between R p and P ref is indeed governed by the atmospheric scale height.
As our retrieval cases build toward full model considerations and adequate data, the estimated slope and the scale height converge. This is to be expected as data at short wavelengths help constrain the continuum and, hence, the molecular abundances, the mean molecular mass, and the scale height (Benneke & Seager 2012;de Wit & Seager 2013). As such, the spread in H 2 O abundances seen in Figure 9 is not a result of the R p -P ref degeneracy, but a result of data quality and model assumptions. The better the data and model, the better the constraints we can impose on the molecular abundances.
Here, we investigate a possible justification for the log-linear relationship we empirically observe between P ref and R p . Generally, the pressure and distance in a planetary atmosphere are related by the consideration of hydrostatic equilibrium. We explore whether the same can explain the observed P ref -R p relation.
An observed transmission spectrum consists of transit depths, i.e., (r/R s ) 2 , as a function of wavelength. By knowing the radius of the star, we know the observed radius (or effective radius) of the planet as a function of wavelength. An observed effective radius should correspond to an effective height in the atmosphere and the corresponding pressure level, where the atmosphere has a slant optical depth of τ λ ∼τ eq (Lecavelier Des Etangs et al. 2008a). The equivalent slant optical depth (τ eq ) corresponding to the observed spectral features is discussed in more detail in Section 5.1.
The pressure (P) and distance (r) in the atmosphere are related by hydrostatic equilibrium as Here, R p and P ref are a reference planet radius and the corresponding pressure, respectively. This equation can be solved if the temperature profile with distance is known.
Assuming an isotherm, P and r are related by where H=k B T(μg) −1 is the scale height. This relation is linear in ln(P) and r with a slope of −1/H and an intercept of ln(P ref )+R p /H. We rewrite Equation (6) as The observed radii also provide another constraint. Given a set of observations r λ,i , the corresponding P λ,i are those for which the slant optical depths satisfy τ λ,i ∼τ eq . From a procedural point of view, a retrieval tries to find the best-fitting model parameters for which the distances in the model atmosphere at r=r λ,i satisfy τ λ ∼τ eq . The atmospheric model consists of a fixed pressure grid, as discussed in Section 3. For a given R p and P ref , among other parameters drawn in a model fit, the pressure grid is related to a distance grid using hydrostatic equilibrium as shown in Equation (6). These properties in turn are used to create a grid of slant optical depths corresponding to the altitude, or pressure, as a function of wavelength. The differential optical depth along the line of sight is given by dτ λ =nσ λ ds, where σ λ is the absorption cross section, n is the number density, and s is the distance along the line of sight. Thus, the model has a distance grid on a one-to-one correspondence with the pressure grid and an associated τ λ map. In a retrieval, the acceptable fit parameters are those for which the locations of the observed r λ,i in the model distance grid have τ λ ∼τ eq . Thus, from Equation (7), given a set of observations r λ,i the values of a and b can be constrained. a independently constraints the scale height of the planet and the slope of hydrostatic equilibrium because a=1/H. Similarly, b helps determine a unique relationship through b=R p /H+ln(P ref ). Rearranging for ln(P ref ), we obtain ln(P ref )=−R p /H+b, where it is evident that R p and P ref , by construction, will also need to satisfy hydrostatic equilibrium with the same slope determined by a. Figure 9. Correlation between the retrieved planetary radius (R p in units of R J ) and H 2 O mixing ratio (X H O 2 ). The 12 panels correspond to the cases explained in Section 3. The spread in the retrieved values changes under different assumptions, with case 12 being the most general case. The H 2 O abundance (i.e., mixing ratio) in case 1 has been multiplied by 10 4 to be in the same range as the H 2 O abundance of other cases.
We can thus conclude that there is indeed a degeneracy between P ref and R p but it is well defined and it does not affect the retrieved molecular abundance. It is the functional form of b in Equation (7) that seems to explain the −1/H behavior seen in Figures 2 and 8, and what defines the relationship between P ref and R p . Now, we inspect more closely the requirement imposed for the observed radii to correspond to a constant slant optical depth τ λ ∼τ eq .

The Slant Photosphere
Following the previous section, we investigate how the observed radius at a given wavelength corresponds to a pressure through τ λ . The one-to-one correspondence between a set of observations r λ,i and their associated pressures is determined by the slant optical depth τ of the photosphere. In this section, we explore further this notion of the equivalent slant optical depth and how this helps constrain R p and P ref . To illustrate, we use the retrieved values of HD209458b for case 12 in Section 3 and generate a model spectrum for a cloud-free and isothermal atmosphere with temperature set to the retrieved T 0 . For each wavelength in our model, we obtain the slant optical depth as a function of the pressure in the atmosphere corresponding to the impact parameter. We show a contour of τ λ in P−λ space in Figure 10. Figure 10 shows both a pressure axis and a transit depth axis that are related by our selection of R p and P ref and hydrostatic equilibrium. The color map shows that the equivalent slant photosphere appears at pressures between 0.1 and 0.01 bar for most wavelengths. Furthermore, it is clear from the model spectrum that the slant optical depth at the apparent radius is ∼0.5. This τ λ surface is close to τ λ =0.56, a value first encountered numerically by Lecavelier Des Etangs et al. (2008a) and later discussed extensively by de Wit & Seager (2013). The cumulative contribution of the atmosphere to the spectrum is consistent with an opaque planet below the τ eq surface. This factor provides an additional constraint when fitting Equation (7). Following Figure 10 we, find that τ0.5 generally determines the equivalent radius and motivates the condition τ λ ∼τ eq discussed in the previous section. This condition is true for hot Jupiters and for most planetary atmospheres as long as R p /H is between ∼300 and ∼3000 (Lecavelier Des Etangs et al. 2008a).

Retrieving R p versus P ref
So far, the retrievals presented here have both R p and P ref as parameters in the retrieval. We have shown above that the degeneracy between these variables can be characterized through an empirical relationship. Several retrieval analyses use only one of R p or P ref as a free parameter and assume a fixed value for the other (e.g., Benneke & Seager 2012;Kreidberg et al. 2015;Line & Parmentier 2016;Sedaghati et al. 2017;Chen et al. 2018;Wakeford et al. 2018;von Essen et al. 2019). Here, we conduct retrievals that assume P ref and retrieve R p and vice versa, in addition to case 12 in Section 3.13, where both were considered to be free parameters. We compare the results and discuss whether the retrievals are sensitive to these assumptions.
We start by assuming a reference pressure and retrieving a planetary radius. Our retrieval is set up in the same way as in Section 3.13 for case 12: the model includes volatiles, a parametric P-T profile, inhomogeneous cloud cover, and uses data in the near-infrared and optical.  Then, we perform the retrieval in which we assume a planetary radius and retrieve the reference pressure. Here we assume a radius of R p =1.359 R J using the value reported by Torres et al. (2008)  . Again, the retrieved values are consistent with those of Section 3.13 to within 1σ.
We show the retrieved H 2 O abundances and their error bars in Figure 11 for the three cases. First, we show the retrieval in Section 3.13 where we retrieved both R p and P ref . Second, we show the retrieval assuming an R p and retrieving P ref . The third panel shows the remaining permutation where we assume a P ref and retrieve R p . All three retrieved H 2 O mixing fractions are consistent with each other. These results confirm that it is not necessary to retrieve both R p and P ref , assuming one will retrieve the other and both will be used to determine the atmospheric structure as discussed in Section 5. While our paper was under review, a follow-up paper to HK17 was published (Fisher & Heng 2018), which retracts some of the claims of HK17. They suggest breaking the three-way degeneracy of HK17 for cloud-free atmospheres by deriving a P ref for an R p assuming it is associated with a part of the atmosphere opaque to optical and infrared radiation, a similar procedure to that suggested in previous studies (e.g., Griffith 2014). They come to a similar conclusion that it is not necessary to retrieve both R p and P ref , and that it is possible to assume the value of one or the other, as is common practice in the literature (e.g., Benneke & Seager 2012;Kreidberg et al. 2015;Line & Parmentier 2016;Sedaghati et al. 2017;Chen et al. 2018;Wakeford et al. 2018;von Essen et al. 2019). However, here we show that no such assumption of wavelength is necessary. The R p can be fixed to any measured value and the retrieval will automatically derive the corresponding P ref .

Limitations of Semi-analytic Analysis
Based on our above results, here we summarize some key factors that may have limited some previous studies using semi-analytic models for constraining chemical abundances (e.g., Heng & Kitzmann 2017;Fisher & Heng 2018). These key factors include ignoring the effects of CIA opacity, incorrect inferences from least-squares fits, and generalized conclusions drawn from inadequate data.
As we show in Section 3, ignoring the effects of CIA leads to an incorrect estimate of molecular abundances by several orders of magnitude. In the work of HK17, CIA effects were not considered, thereby rendering their analytic solution incomplete. While the assumption of isobaric opacities for molecular line absorption does not affect the retrieved molecular abundances substantially, the inclusion of CIA is incompatible with an isobaric assumption. Molecular features in current data are less strongly affected by the pressure dependence, because the spectrum probes lower pressures (P0.1 bar) for these features. On the other hand, CIA has strong dependence on the pressure. In other words, CIA absorption is proportional to P 2 (de Wit & Seager 2013), and its impact on the spectrum is underestimated if evaluated at only one pressure (e.g., Fisher & Heng 2018) or completely ignored (e.g., Heng & Kitzmann 2017).
The more problematic assumption in the work of HK17 comes in their inferred H 2 O abundances obtained from best fits to the observed WFC3 spectrum of WASP-12b as shown in their Figure 7. In that figure, they claim to be showing values of X H O 2 (P 0 /10 bar) −1 as a function of the assumed planetary radius R 0 . A close inspection of this graph suggests that they are instead showing X H O 2 (P 0 /10 bar) as a function of the assumed planetary radius. The linear trend they obtain in their figure is likely a manifestation of the relationship between P ref and R p , and not X H O 2 . They claim that a small change in the assumed planetary radius leads to a large change in the ordinate and hence the H 2 O abundance. While this claim may be partly true, their inference regarding the H 2 O abundance is manifestly incorrect. A large change in the product of X H O 2 (P 0 /10 bar) is not because of a change in H 2 O abundance but a change in the reference pressure. Changing the assumed planetary radius will change the associated reference pressure. As shown in Sections 3 and 5, the inferred H 2 O abundance is largely unaffected by the P ref -R p degeneracy.
Lastly, their work considers only the interpretation of WFC3 data and ignores the effects of optical data. On the other hand, as we have shown here, it is the inclusion of optical data that helps constrain molecular abundances the most. The inclusion of optical data helps constrain the effects of clouds, the reference pressure, and the scale height. These in turn improve the constraint on the H 2 O abundance. Thus, retrievals that do not take these factors into account are inherently biased toward incorrect chemical abundances (e.g., Fisher & Heng 2018). As such, their abundance estimates (e.g., of HD 209458b) do not agree with retrievals that use cloudy models and optical data Figure 11. Retrieved H 2 O abundances for three model considerations, median value, and 1σ error bars. In blue, we show the case were both R p and P ref are retrieved. In yellow, only P ref is retrieved for an assumed R p , and in green, the opposite is shown. All three retrievals provide a consistent H 2 O abundance, showing that the assumption of a radius or reference pressure is inconsequential in the retrieval of H 2 O abundance.
In this work, we have shown an empirical relationship between R p and P ref that seems to be related to hydrostatic equilibrium. Furthermore, we show that this relationship is independent of X H O 2 , effectively breaking the three-way degeneracy. Complementary to this is the importance of choosing the right models and assumptions in the model atmospheres. Models ignoring CIA, considering only H 2 O opacity, along with constant gravity and mean molecular weight, lead to poor constraints in the retrieved H 2 O abundance. Equally important are the consideration of inhomogeneous cloud coverage and inclusion of optical data in constraining molecular abundances.

Summary and Discussion
We conduct a detailed analysis of degeneracies in transmission spectra of transiting exoplanets. We investigate the effect of various model assumptions and spectral coverage of data on our ability to constrain molecular abundances. Utilizing atmospheric retrievals, we test simple isobaric and isothermal atmospheric models for their ability to constrain molecular abundances using infrared spectra alone. We later remove these assumptions one by one until a realistic atmospheric model composed of H 2 /He CIA, multiple-molecular species, a full P-T profile, inhomogeneous cloud coverage, and the inclusion of broadband data spanning infrared to optical is obtained. We conduct this investigation using the canonical example of HD209458b, a hot Jupiter that has the best data currently available.
We identify several key properties that need to be accounted for in models for reliable estimates of chemical abundances, in particular H 2 /He CIA opacities, a full P-T profile, and possible inhomogeneities in cloud cover. The inclusion of CIA has the most effect in accurately constraining molecular abundances as it provides a natural continuum in the model spectrum.
We find that the degeneracies between molecular abundances and cloud properties can be alleviated by the inclusion of optical data. Optical data provide constraints on the scattering slope in the optical as well as a continuum for the full spectrum. As such, optical data are key to constrain molecular abundances using transmission spectra. When optical data are included, considering Na and K absorption significantly influences the retrieved H 2 O abundances, making them more precise. We also find that assuming a cloud fraction a priori (e.g., 100% cloud cover) leads to erroneous estimates on molecular abundances. Leaving the cloud fraction as a free parameter allows for molecular estimates more accurate than those obtained when assuming a fixed cloud cover fraction.
We show, using simulated data, that even in the case of an atmosphere with 100% cloud cover, the retrievals are able to closely retrieve abundances and other properties. In principle, R p is degenerate with the pressure level of the cloud top. However, the range of altitudes, and hence pressures, of the cloud top where it affects the spectrum is limited. A 100% cloud deck must be at an altitude higher than the line-of-sight photosphere and lower than the level where the atmosphere is optically thin. In the former case, the cloud deck does not contribute significantly, and in the latter case, the cloud deck causes a featureless spectrum, contrary to observed features. On the other hand, we show that an inhomogeneous cloud model accurately retrieves the cloud fraction even for a 100% cloudy case. Among the considerations that we leave unexplored in this set of retrievals are the effect of stellar activity on the transmission spectra and its consequence in resolving degeneracies, and the presence of shifts or offsets between data taken from different instruments. Other aspects to consider in the future include inhomogeneities across the limb due to the 3D structure of an atmosphere (e.g., Caldas et al. 2019), refraction in the atmosphere (e.g., Bétrémieux 2016), height-varying chemical abundances (e.g., Parmentier et al. 2018), and various cloud properties (e.g., Vahidinia et al. 2014;Barstow et al. 2016). Future work and retrieval frameworks like that of Pinhas et al. (2018) could help elucidate these aspects. Overall, the quality of the data and the wavelengths they span are fundamental in breaking degeneracies and retrieving molecular abundances.
We also discuss the limitations of semi-analytic studies in fully assessing degeneracies in transmission spectra. One important finding is that the degeneracy between P ref and R p does not lead to an inability to determine molecular abundances in transit spectroscopy, contrary to previous suggestion. We show an empirical relationship between the planetary radius and the reference pressure that characterizes their degeneracy. We find that ln(P ref ) and R p have a linear relationship with a slope of −1/H and suggest that this behavior is rooted in hydrostatic equilibrium. For each R p ,there is an associated P ref and vice versa. As such, it is redundant to perform retrievals that consider both quantities as free parameters. Instead, we demonstrate that it is justified to assume a value for one quantity and to retrieve the other. Some studies assume an R p and retrieve a P ref ; here we demonstrate that the inverse is also consistent, i.e., that it is possible to assume a P ref and retrieve the radius of the planet corresponding to that P ref .
We investigate the origins of spectral features in transmission spectra by following the line-of-sight opacity of the planet as a function of the vertical pressure level and wavelength. This allows us to calculate the height and pressure levels in the atmosphere at which the observed features are generated and compare it to the white light radius. We show that the effective radius corresponding to the observed transit depth at a given wavelength corresponds to a level in the atmosphere with a slant optical depth of τ0.5, as also suggested by previous studies.
Overall, our study demonstrates the effectiveness of highprecision spectra and realistic models to retrieve atmospheric abundances. Data with current facilities such as HST and VLT over the visible and near-infrared can already provide valuable constraints on abundances of key species such as H 2 O, Na, K, etc. The upcoming James Webb Space Telescope and groundbased facilities therefore hold great promise for characterizing exoplanetary atmospheres using transmission spectra.
L.W. is grateful for research funding from the Gates Cambridge Trust. We thank the anonymous reviewer for their thoughtful comments on the manuscript. L.W. thanks Siddharth Gandhi for helpful discussions and Anjali Piette for inputs on Figure 10. distributions for X H 2 O, P ref , and R P and the correlations between them for cases 0 to 12. In Figure 13 we show the full posterior distributions for all parameters for the full retrieval case, case 12, discussed in Section 3.13. Tables 2 and 3 show the parameters retrieved for all the 12 cases along with the prior ranges and retrieved values. Figure 13. Full posterior distribution for case 12 as explained in Sections 3 and 3.13. This is a full retrieval of HD209458b using data in the near-infrared and optical wavelengths from Sing et al. (2016). The model includes the effects of H 2 -H 2 and H 2 -He CIA opacity; absorption from H 2 O, Na, K, NH 3 , CO, HCN, and CO 2 ; a parametric P-T profile; and the presence of clouds/hazes. Both R p and P ref are simultaneously retrieved.  Note. The uniform priors and the parameter estimates are shown for Cases 1-7; the remaining cases are shown in Table 3. * The log 10 (H 2 O) prior for Cases 4 and 5 is −12, −2.   Note. The uniform priors and the parameter estimates are shown for Cases 8-12; the remaining cases are shown in Table 2.