© The Authors 2025

1 Introduction

Correctly constraining the chemical structure of exoplanetary atmospheres is a crucial element in the inversion of spectro-scopic data, since it provides important information about their physical and climatic characteristics, and above all about their evolutionary history (Öberg et al. 2011; Madhusudhan et al. 2016; Mordasini et al. 2016; Brewer et al. 2017; Eistrup et al. 2018; Turrini et al. 2021; Lothringer et al. 2021). The chemical structure of an exo-atmosphere was originally recovered using either a constant-with-altitude profile (a single free parameter representing each molecule) or by assuming thermochemical equilibrium (White et al. 1958; Madhusudhan & Seager 2009; Lee et al. 2012; Line et al. 2013; Waldmann et al. 2015; Al-Refaie et al. 2021), which requires computing the chemistry state by minimising the Gibbs free energy of the system. The latter is advantageous in terms of degrees of freedom, as it requires only two free parameters – metallicity and the carbon-to-oxygen (C/O) ratio – chosen for their natural links to planetary formation and processes. However, assuming thermochemical equilibrium is a strong assumption, especially given the orbital geometry of exoplanets. While proximity to their host stars implies very high atmospheric temperatures – promoting rapid chemical kinetics and thus chemical equilibrium – other processes that can drive the atmosphere out of equilibrium are also enhanced. For instance, intense irradiation in the UV range triggers significant photodissociation processes, driving the chemical composition away from chemical equilibrium (and thus missing from equilibrium calculations). Examples of planetary atmospheres in our solar system and recent JWST results (Tsai et al. 2023; Dyrek et al. 2024) suggest that exo-atmospheres are incompatible with this thermochemical equilibrium assumption. Some attempts have been made to explore other and more complex chemical assumptions in exo-atmosphere retrievals, including a two-layer profile chemistry (Changeat et al. 2019), a chemical relaxation disequilibrium (Kawashima & Min 2021), and an equilibrium offset (Madhusudhan & Seager 2009). However, all these chemical assumptions consist of approximate solutions to chemical disequilibrium and do not directly handle kinetic reactions.

The need to account for disequilibrium processes (vertical mixing, including quenching and photodissociation) in spectral data inversion calculations is all the more obvious following the latest observations of SO₂, which is produced by processes initiated by photochemistry (Tsai et al. 2023; Dyrek et al. 2024). The lack of photochemistry processes in data retrieval calculations remains one of the main sources of error. To reinforce this argument, simulations using kinetic calculations (i.e. accounting for disequilibrium processes such as mixing and photochemistry) have proven that chemical equilibrium is inadequate in many scenarios (e.g. Moses et al. 2011, 2013, 2016; Venot et al. 2012, 2014, 2020a,b; Morley et al. 2017; Molaverdikhani et al. 2019; Mollière et al. 2020; Kawashima & Min 2021; Tsai et al. 2021). To ensure unbiased interpretations of exoplanet observations, it is crucial to properly represent the chemical processes within exo-atmospheres, using in particular modern and future telescopes, whose resolution enables a more precise and complete physical and chemical characterisation. See the first analyses of James Webb Space Telescope data (JWST Transiting Exoplanet Community Early Release Science Team 2023; Tsai et al. 2023; Dyrek et al. 2024).

For this purpose, the retrieval model TauREx 3.1 (Al-Refaie et al. 2021) is coupled with FRECKLL, the disequilibrium chemical kinetic Python model, thanks to its plugin functionality (Al-Refaie et al. 2024). In this study, the authors present this new coupled model, together with retrieval tests on simulated HD 189733 b observations at JWST resolution, to demonstrate the viability of total disequilibrium chemistry retrievals and JWST’s ability to detect disequilibrium processes.

In this paper, we aim to address some of the exoplanet and observational analysis questions raised by population studies (Sing et al. 2016; Tsiaras et al. 2018; Min et al. 2020; Estrela et al. 2022; Changeat et al. 2022; Edwards et al. 2023; Saba et al. 2025), using the TauREx-FRECKLL model for the first time on real observations. We consider ten hot-Jupiter spectra, from the HST-WFC3 G141 grism, observed in both eclipse and transit. To that end, we first consider whether the addition of a kinetic chemical model to the retrieval algorithm improves constraining metallicity and C/O, as this would make them viable observables for advancing our understanding of exoplanet formation. While Changeat et al. (2022) note that, for retrieval calculations, C/O and metallicity are difficult to constrain under either a free or a thermochemical equilibrium hypothesis in HST observations, both hypothesis remain crucial for precisely determining planetary formation. If hot Jupiters form via a three-step process (Mizuno 1980; Bodenheimer & Pollack 1986; Ikoma et al. 2000) – solid core accretion, runaway gas accretion, and migration – their present-day composition should largely remain the same from the early stages of planetary formation; that is, sub-stellar in heavy elements, such as C, O, and refractory elements, since most of the heavy elements would be sequestered in the cores. Hence, we attempt to better constrain C/O and metallicity by accounting for kinetic chemistry calculations in HST retrievals.

Second, we consider how important the chemical species, TiO and VO, are in the exoplanet atmospheric spectra. Because HST instruments are not sensitive to N- and S-bearing molecules, the detection and subsequent analysis of elemental ratios such as N/O and S/O (Turrini et al. 2021) have remained unexplored with HST data and thus could not help determine planet formation scenarios. Another way to infer planetary formation is through the analysis of refractory elements (Lothringer et al. 2021). Consequently, previous observational studies have focus on refractory species as TiO, VO, and FeH, which have been detected in HST eclipse spectra. For instance, Changeat et al. (2022) show that, in a population study of 25 hot gaseous planets observed in eclipse, the hottest planets in their sample (T > 2000 K) have inverted thermal profiles with signatures from thermal dissociation (H⁻) and refractory species (TiO, VO, or FeH). Moreover, the spectra in the HST wavelength range are inconsistent with simple blackbody emission. However, the authors note that equilibrium chemistry retrievals are not the preferred solution for many planets in their sample population and strongly suggest that disequilibrium mechanisms may be important. Therefore, by considering disequilibrium hypotheses in our retrieval calculations, we explore the impact of including TiO and VO on improving spectra inversion.

Third, we consider whether using the kinetic chemistry model together with the retrieval model allows for consistent eclipse and transit retrieval for individual exoplanets in terms of their temperature and chemistry structures. Transit spectra are primarily sensitive to the atmospheric scale height of the planet, and this sensitivity depends on the planet’s radius, mass, and mean temperature. In transit view, the mean temperature does not account for possible inhomogeneities between the morning and evening terminator compositions, implying large degeneracies for retrieval calculations (for both temperature and chemical structures), due to the geometry of the observed spectra. Hence, the temperatures of exoplanets inverted from transit data and reported in the literature are biased by several hundred degrees in the best cases, to more than a thousand degrees in the worst cases. This systematic bias arises from 1D retrieval codes that describe an approximate chemical composition – either following a free or a thermochemical equilibrium assumption – introducing additional significant biases in the retrieved chemical abundances and, consequently, in the retrieved temperature structure. To compensate for this bias in 1D models, it is essential to turn to 2D or 3D inversion calculations to account for the inhomogeneities between morning and evening limbs (MacDonald et al. 2020). Bi- or tri-dimensional retrieval calculations, with disequilibrium chemistry assumptions, are beyond the scope of the present paper. However, coupling TauREx 3.1 with FRECKLL permits verification of the consistency between eclipse and transit inverted spectra within a 1D retrieval model. This point aims to assess how much information about the temperature structure can be extracted from HST observations in transit by considering a kinetic chemical scheme in retrieval calculations.

The paper is structured as follows: Section 2 presents the data processing, retrieval procedures, and opacity materials. Results of our retrieval calculations are presented in Section 3. Lastly, discussions and concluding remarks are presented in Section 4.

2 Methodology

2.1 Data and processing procedure

Our study encompasses data for the following ten hot Jupiters observed in eclipse with the HST-WFC3 G141 grism: HAT-P-2b (Salz et al. 2015), HD 189733 b (Addison et al. 2019), HD 209458 b (Barstow et al. 2017), Kepler-13 A b (Batalha et al. 2013), TrES-3 b (Southworth 2011), WASP-4 b (Huitson et al. 2017), WASP-19 b (Cortés-Zuleta et al. 2020), WASP-43 b (Esposito et al. 2017), WASP-74 b (Mancini et al. 2019), and WASP-77 A b (Cortés-Zuleta et al. 2020). It also includes data in transit with the HST-WFC3 G141 grism for four of these planets: HD 189733 b, HD 209458 b, WASP-43 b, and WASP-74 b. For our retrieval calculations, we used the consistent set of reduced data from Iraclis (Tsiaras et al. 2016), obtained in eclipse from Changeat et al. (2022) and in transit from Tsiaras et al. (2016) and Edwards et al. (2023).

All of the HST-WFC3 data used in these three studies were processed following the same procedure to ensure consistency between planets (detailed in Tsiaras et al. 2016; Changeat et al. 2022; Edwards et al. 2023). We summarise this procedure in what follows. Tsiaras et al. (2016); Changeat et al. (2022); Edwards et al. (2023) used the Iraclis pipeline, a highly specialised software for processing WFC3 spatially scanned spectroscopic images (Tsiaras et al. 2016). All data were reduced following an eight-step process, namely: zero-read subtraction, reference-pixel correction, non-linearity correction, dark current subtraction, gain conversion, sky background subtraction, calibration, flat-field correction, and bad-pixel/cosmic ray correction. The white (1.088–1.68 μm) and spectral light curves were extracted from the resulting reduced images, taking into account the geometric distortions caused by the tilted detector of the WFC3 IR channel.

The white light curves for each planet (whether from eclipse or transit data) were fitted using the transit model package PyLightcurve (Tsiaras et al. 2016), allowing only the planet-to-star radius ratio and the mid-transit (mid-eclipse) time as free parameters for the transit (eclipse) data. Time-dependent sys-tematics, the long-term and short-term ‘ramps’, affect each HST visit and each HST orbit, respectively. The first ramp was corrected using a linear behaviour formula, and the second was corrected using an exponential behaviour. The white light curves fitting included this ‘ramps’ correction, as well as the uncertainties per pixel, as propagated through the data reduction process with Iraclis. Additional scatter unexplained by the ramp model was also corrected by scaling the uncertainties in the individual data points so that their median matched the standard deviation of the residuals. The fitting was then repeated, with the planet-to-star radius ratio and the mid-transit (mid-eclipse) time as free parameters for the transit (eclipse) data (Tsiaras et al. 2018).

The spectral light curves were fitted to a transit model (with the planet-to-star radius ratio being the only free parameter), considering a model for the systematics that includes the white light curve (divide-white method, Kreidberg et al. 2014a) and a wavelength-dependent, visit-long slope (Tsiaras et al. 2016). In the same way as for the white light curve, an initial fit was performed using the pipeline uncertainties and then refitted while scaling these uncertainties for their median to match the standard deviation of the residuals.

2.2 Retrieval procedure

Our systematic retrieval calculations were computed using the TauREx 3.1 atmospheric retrieval code (Al-Refaie et al. 2021), and the retrieval parameters used for this study are summarised in Table 1. The modelled planetary atmospheres assume the plane-parallel assumption. Hence, atmospheric column was uniformly discretised in log-pressure with 20 layers per decade, for 80 levels from 10¹ to 10^–3 bar. The temperature profile was determined using a heuristic N-point profile: discretised by five temperature points, one for each decade of pressure. Each of these five temperature points was assigned a prior of 500–3900 K, but their pressure level was fixed (Changeat et al. 2021, a sensitivity study on the number of temperature points to retrieve the temperature profile is available in Appendix A). C/O had a uniform a priori of 0.01–1.1, and the metallicity Z had a logarithmic a priori of 10^–2–10³ (to solar). Here, both metallicity and C/O vary: metallicity controls all the elements heavier than He, while C/O varies from the fiducial solar elemental abundance with the abundance of carbon. Hence, the abundance of oxygen was fixed at the value recommended by Lodders (2010), (i.e. log(He) = 10.925, log(C) = 8.39, log(O) = 8.73, and log(N) = 7.86). For transmission spectra, the planetary radius R_planet, at the reference pressure of 10 bar, has a uniform a priori of 0.5–5.0 R_J. The parameter space was explored using the Multi-Nest nested sampling algorithm (Feroz & Hobson 2008; Feroz et al. 2009; Feroz et al. 2019), implemented by the PyMultiNest python wrapper (Buchner et al. 2014), with 400 live points and a log-likelihood tolerance of 0.5. Because we used the nested sampling algorithm MultiNest, the computation of the Bayesian evidence for each model, here denoted E, was automatic. As a result, the difference in ln(E) between two models M1 and M2 can be used for model selection and to compare the ability of the two models to explain the observed spectra (Kass & Raftery 1995; Tsiaras et al. 2016). To follow, we used Bayesian evidence to compare the results with the ‘free runs’ (assuming constant abundances as a function of altitude of the considered molecular species) from Changeat et al. (2022) (for eclipse data) and Edwards et al. (2023) (for transit data).

The disequilibrium chemical calculations were performed using the FRECKLL model, with the Python plugin developed by Al-Refaie et al. (2024). Contrary to thermochemical equilibrium models, which predict the chemical state of a planet’s atmosphere by minimising the Gibbs free energy of the system, chemical kinetic models integrate a system of differential equations (formed by continuity equations of each species considered, describing the temporal evolution of their abundance) until a steady state is reached. In FRECKLL, the continuity equations of each species evolve using the stiff ordinary differential equation (ODE) solver VODE package (Brown et al. 1989), with the initial atmospheric state initialised at thermochemical equilibrium composition. In addition to the metallicity and C/O parameters, FRECKLL also requires the definition of the vertical eddy diffusion parameter K_zz, given as a constant value in the present study. For the validation study, we employed the full Venot et al. (2020a) network, including 108 species, made of H, He, C, O, and N (with up to two carbon atoms, i.e. C₀–C₂ chemical kinetic network), 1906 reactions, and 55 photodissociations (all the photolysis data are contained in the scheme, and the description of the UV spectrum used for the host star of each planet is detailed in Appendix B).

For each eclipse spectrum, we carried out several retrieval calculations to address the questions formulated in the Introduction: (i) a retrieval test with FRECKLL only (hereafter named FRECKLL-only), a test with FRECKLL plus TiO following the ‘free’ chemical structure (i.e. a constant-with-altitude profile, hereafter named FRECKLL-TiO), (ii) a test with FRECKLL plus VO (‘free’ chemical structure, hereafter named FRECKLL-VO), and (iii) FRECKLL plus both TiO and VO as well, named FRECKLL-TiO-VO. In addition, planets for which the FRECKLL-only retrieval result described a bimodal metallicity distribution (HAT-P-2 b, HD 189733 b, WASP-19 b, and WASP-74 b; see Appendix C), we carried out additional tests: we separated the range of metallicity priors into two sub-ranges ([10^–2:10¹] and [10¹:10³] (to solar)) and repeated the TiO and VO addition tests for each of these two sub-ranges. These additional tests were motivated by the very nature of Multinest sampling: when two solutions were found, the number of live points (representing at first order the resolution of the sampling, here set at 400) was halved to explore the two solutions as best as possible. Artificially separating the metallicity regimes with two distinct retrieval calculations enabled us to maintain the number of live points needed to explore a single solution and to maintain resolution consistency throughout the set of spectra. The Bayesian evidence was then compared in order to determine the most probable solution between the two modes forced through the separation of metallicity regimes. For transit spectra, only two retrieval tests were conducted, both using FRECKLL, with either an isothermal profile for the retrieved temperature calculations or a five-point thermal profile (see Appendix D).

2.3 Opacity sources

For our re-analysis study, we selected molecular line lists from the Exomol project (Tennyson et al. 2016; Chubb et al. 2021), HITEMP (Rothman & Gordon 2014), and HITRAN (Gordon et al. 2016). We note that, as the FRECKLL C₀–C₂ chemical network only includes chemical species made of H, He, C, O, and N, chemical species such as TiO and VO are not included in FRECKLL. Hence, our set of exoplanets was selected to be valid for FRECKLL regime i.e. set of exoplanets without inversion in their thermal profile (from previous studies Changeat et al. 2022; Edwards et al. 2023), without TiO and/or VO spectral signature, and with an equilibrium temperature under 2500 K. But in what follows, we show a case for which we can extend the retrieval assumptions and improve the results by including TiO and/or VO following the ‘free’ chemistry assumption (WASP-77 A b). We therefore selected the same molecular cross sections at resolution R = 15 000 as Changeat et al. (2022), namely: H₂O (Barton et al. 2017; Polyansky et al. 2018), CH₄ (Hill et al. 2013; Yurchenko & Tennyson 2014), CO (Li et al. 2015), CO₂ (Yurchenko et al. 2020), as well as HCN (Barber et al. 2014), H₂CO (Al-Refaie et al. 2015), CN (Syme & McKemmish 2021), C₂H₂ (Chubb et al. 2020), C₂H₄ (Mant et al. 2018), NH₃ (Al Derzi et al. 2015; Coles et al. 2019), TiO (McKemmish et al. 2019), VO (McKemmish et al. 2016). The opacity sources for these latter (TiO and VO) are only used by the radiative transfer calculation in TauREx. Atomic and ionic species were not included as they do not absorb in the wavelength range considered here. We also included collision-induced absorption (CIA) of the H₂–H₂ (Abel et al. 2011; Fletcher et al. 2018) and H₂–He (Abel et al. 2012) pairs as well as opacities induced by Rayleigh scattering (Cox 2015).

3 Results

The HST eclipse and transit observations retrieved with the coupled TauREx-FRECKLL model are shown in Figure 1 and Figure 2, respectively. We adopt a conservative approach and show the FRECKLL-only fits for eclipse (except for WASP-77 A b, which is FRECKLL-TiO), and FRECKLL-isothermal fits for transit, unless another retrieval model is significantly preferred – i.e. the difference in ln(E) between two retrieval runs Δln(E) > 5 (see Tables C.1 and D.1 for detailed retrieval results). Hence, for WASP-77 A b, the addition of TiO as a ‘free’ chemical species improved the retrieval fit, resulting in a ln(E) > 199 (for the ‘FRECKLL-only’ case, ln(E) = 195). Therefore, for the population diagnostics figures (Figs. 1, 3), we consider the FRECKLL-TiO retrieval results in eclipse for this planet.

To infer metallicity and C/O of exoplanets, population studies rely on the detection of water vapour, C-bearing species, or from chemical equilibrium calculations. Changeat et al. (2022) (hereinafter C22) argued that, if detected, water vapour can be used as a proxy for metallicity, allowing for the estimation of the O/H ratio. While directly inferring C/O from C-bearing species remains difficult with HST data (as this telescope lacks sensitivity for this family of molecules), our set of planets for this population study depicts a clear H₂O spectral signature, which is known to also affect C/O (Rocchetto et al. 2016; Drummond et al. 2019), and an equilibrium temperature inferior to 2500 K (‘cold’ hot Jupiters) preventing any degeneration of metallicity and C/O from dissociated water vapour (only Kepler-13 A b has an equilibrium temperature warmer than 2500 K to test the limit of our model).

The use of a disequilibrium model to analyse HST data may have a fundamental impact on previous conclusions drawn using other chemistry assumptions. In particular, it could affect the retrieved metallicity and C/O of exoplanets and thus the reservoir for planet formation to which they belong. This is shown in Figures 3 and 4 where we present the metallicity and C/O inferred from our disequilibrium chemical model, respectively, for eclipse and transit data. As shown in previous population studies (Tsiaras et al. 2018; Changeat et al. 2022; Edwards et al. 2023), our results suggest two distinct groups of planets, which is particularly evident for the transit data. This overall picture is not consistent with C22, displaying all planets considered here in the negative range of metallicity. Therefore, for individual planets, our results can differ significantly from results assuming free or chemical equilibrium, with a moderate impact on the Bayesian evidence for disequilibrium chemistry retrievals (results from Table D1 of C22 HST free chemistry retrieval taking into account free chemistry profiles for H₂O, CH₄, CO, CO₂, for all planets and in addition free profiles for TiO, VO, FeH, e^– in the case of WASP-77 A b have been plotted on Figure 3, right panel for easy comparison). For example, for the disequilibrium chemistry case, HD 189733 b has a subsolar metallicity (log(Z) ~ –1.5 in eclipse and ~ –0.5 in transit), with C/O between 0.7 (eclipse) and 0.8 (transit); consistent with the free chemistry retrieval carried by C22, which placed HD 189733 b as a sub-solar metallicity planet (log(Z) ~ –1.5) as well, with C/O equal to 0.8. Similar results are obtained for HD 209458 b, except that in our case C/O is bulked up to 1.0 compared to its 0.66 value in C22. For both planets, Bayesian evidence is impacted: for HD 189733 b the Bayesian evidence decreases by ≈ 2.5 points (from 147.5 for C22 to 145.28 here) and increases by ten points for HD 209458 b (from 198.76 for C22 to 218 here). WASP-4 b, WASP-74 b, HAT-P-2 b and WASP-19 b were clearly subsolar under free chemistry assumptions in the retrieval calculation, but using here the disequilibrium FRECKLL model moved their metallicity towards solar (for WASP-4 b) to strongly supersolar (WASP-74 b, HAT-P-2 b and WASP-19 b), impacting as well their C/O. For the most extreme example, WASP-74 b displayed a subsolar metallicity of log(Z) = −2.1 and C/O of 0.7 in free chemistry; here we obtain a supersolar metallicity of log(Z) = 1.74 and C/O of 0.47. Both chemistry assumptions result in very close Bayesian evidence values of 202.8 (free) and 203.29 (FRECKLL-only). In the same vein, planets with an around solar metallicity in free chemistry (Kepler-13 A b, TrES-3 b, WASP-43 b and WASP-77 A b) are, with FRECKLL, moved to supersolar (Kepler-13 A b and TrES-3 b) and subsolar (WASP-43 b and WASP-77 A b) metallicity conditions. While the C/O of WASP-77 A b, for instance, is roughly the same between the two methods (0.4 for free and 0.5 for disequilibrium), its metallicity has been changed from solar (in free) to subsolar (log(Z) ~ –1) using the disequilibrium model (Figure 3). On the contrary, for TrES-3 b, free chemistry retrieval depicts this planet with a slightly subsolar metallicity (log(Z) ~ –0.4) and C/O of 0.2, but using FRECKLL as chemical model, this planet is here described with a supersolar metallicity (~0.8) and a higher C/O of 0.5. As before, accounting for disequilibrium chemistry in these four cases does not significantly impact the Bayesian evidence. Disequilibrium retrievals carried out here imply an increase of one point in the Bayesian evidence for Kepler-13 A b, an increase of two points for WASP-77 A b, and a decrease of five points for TrES-3 b, and one point for WASP-43 b compared to the results from the C22 free chemistry retrievals. Nonetheless, we note that the results of retrievals using FRECKLL for eclipse and transit data (Figures 3 and 4, respectively) are consistent with each other, describing each planet in the same log(Z)-C/O region in both geometries, except for HD 209458 b where C/O varies from ~1 in eclipse to 0.2 in transit. Metallicity and C/O are therefore model-dependent, and this makes results interpretation in the context of planet formation difficult.

The use of a disequilibrium chemical model during retrieval calculations highlights the fact that the conclusions drawn from HST observations are first dependent on the retrieval parameters, priors and more broadly on the model used, then are dependent on the reduction method. This is shown through the example of the planet WASP-43 b (Figures 5–8), using the reduced data from C22 and Kreidberg et al. (2014b) (hereinafter K14). Retrieved parameters, prior setup and stellar and planetary configurations (see Table 1) were kept the same across both datasets to ensure consistency during the comparison. In addition, we considered the Bayesian evidence for each retrieval test carried out to address this point in Table 2.

Overall, the thermal profiles for the two reduced datasets are consistent, especially for ‘FRECKLL-only’, ‘FRECKLL-TiO’ for both datasets, as well as ‘FRECKLL-VO’ for K14 (Figures 5 and 6). The above-mentioned retrieval tests result in a thermal profile describing a decrease of temperature with altitude up to ~ 10⁵Pa for C22, and up to ~ 10⁴Pa for K14, in addition to a second thermal inversion at ~ 10³Pa for K14. For lower pressures, the temperature increases with altitude, suggesting two distinct atmospheric layers. This increase is within larger error bars, which suggests that greater care should be taken when reading these temperature and pressure profiles. However, ‘FRECKLL-VO’ and ‘FRECKLL-TiO-VO’ retrievals for C22 dataset depict a three-layer-atmosphere in the thermal profile: (bottom-up) a first layer of temperature decreasing up to ~ 10⁴Pa, a second layer of temperature increasing from ~ 10⁴Pa to ~ 10³Pa, and a third layer of temperature decreasing from ~ 10³Pa to the top of the modelled atmosphere (10²Pa). Moreover, the retrieval test ‘FRECKLL-TiO-VO’ for K14 dataset, presents a thermal profile entirely different (Figure 6). The thermal profile in the case of ‘FRECKLL-TiO-VO’ for K14 depicts an inversion of temperature (from decrease to increase of temperature with the decreasing pressure) at 10⁵ Pa and a second one at 10⁴ Pa from which the temperature slightly decreases with altitude. For both, the temperatures at the bottom of the atmosphere are constrained, within a small error envelope and roughly equivalent. Despite these above-named consistencies, the chemical structure for the reference ‘FRECKLL-only’ run is largely impacted. In particular for the C₂H₂, C₂H₄, and CN profiles, for which the quenching level has increased by up to one decade of pressure. As expected, C/O remains very challenging to be retrieved because HST observations with the WEC-3 instrument (whatever the reduction methods) lack sensitivity to C-bearing species due to a limited spectral range, this is particularly the case for the C22 dataset (Figure 5), for which each retrieved tests depict a broad posterior distribution for C/O. In the case of K14 dataset, this posterior distribution for C/O tends to a value of about 0 (i.e. a carbon-poor atmosphere), for most of the retrieval tests, except the ‘FRECKLL-TiO-VO’ run presenting a C/O tending to 1 (Figure 6). Reduced data from K14 and C22 present a large difference in resolution (data binning), as well as larger errobars for C22 than K14. Difference in data binning might be one of the sources for the difference in retrieval results between C22 and K14 datasets. However, population studies, such as Tsiaras et al. (2018) and Changeat et al. (2022) found that, with HST data, difference in data binning does not impact much their retrieval results (when the low/high resolution reductions come from the same pipeline), so other effects could also contribute to the difference in retrieval results between C22 and K14 datasets.

Retrieved results from both reduced datasets agree on the metallicity (log(Z) ~ −2), considering WASP-43 b as a subsolar-metallicity planet. However, those retrieval results, whether with C22 or K14 reduced datasets, are in strong opposition to the results obtained by those previously mentioned studies, (e.g. 0.8×solar in log for C22 and 0.18×solar in log for K14, converted from the linear value of 1.5×solar in the K14 paper). To consistently compare the effects of the FRECKLL model during retrieval calculations, we carried out additional retrieval tests on both reduced datasets for WASP-43 b with the same set of retrieval runs; this consisted namely of a ‘FRECKLL-only’, a ‘FRECKLL-TiO’, a ‘FRECKLL-VO’, and a ‘FRECKLL-TiO-VO’ run, but constraining the metallicity prior to the supersolar region from 1 to 10³ (to solar, in log). Results from these additional retrievals are shown in Figure 7 and Figure 8.

Overall, for both reduced datasets, restricting the metallicity priors largely impacts the Bayesian evidence of the retrievals (see Table 2), with a Δln(E) up to ten points lower than when the prior is restricted to the positive region. With a metallicity prior constraint to the supersolar region for the C22 dataset, we obtain a bimodal distribution of metallicity, with a most probable value of 2.5 to solar (in log) and a second probable value of 0 to solar (in log, i.e. equal to solar metallicity) for all retrieval tests, except for the ‘FRECKLL-TiO-VO’ retrieval (for which the metallicity posterior distribution peaks at zero, suggesting an overly constraining priority in this case). Despite consistency with the results obtained by C22 for WASP-43 b metallicity, this close proximity to one of the prior boundaries suggests that the model did not have enough freedom to explore suitable(s) value(s) to properly constrain the metallicity. Concerning the K14 dataset, metallicity distribution is constrained only in the case of ‘FRECKLL-only’. For the rest of the retrieval tests, the retrieved metallicity displays a sparse distribution around 2.5 to solar (in log). In addition, ‘FRECKLL-only’ retrieval results in a thermal profile approaching the vertical variation of the results obtained by the full-prior of metallicity retrieval for either C22 or K14 (Figures 5 and 6). Constraining the metallicity prior breaks the consistency in the thermal structure between the two data reduction methods, leads to thermal profiles with a lot of temperature inversions, one temperature inversion per decade of pressure (in particular for the C22 dataset), and this is even reinforced when considering TiO or/and VO in the chemical calculations. Those thermal profiles also impact the chemical structure of the planet for both reduced datasets, particularly in the case of C22.

Therefore, with WASP-43 b datasets produced by two different reduction methods, we highlight that the retrieval calculations are lightly impacted by the reduction method, but above all sensitive to the retrieval prior. With too many constraints on the parameter space, the model is not able to properly investigate and ‘pulls’ on the prior by determining retrieved values on the prior boundary. Uncertainties remain concerning the actual metallicity of this planet as our results are opposed to previous studies. Both the K14 and C22 retrieval calculations of WASP-43 b retrieved the abundances of chemical species following a free model (e.g. constant-with-altitude profiles of abundance). Since these two retrieval calculations differ in the method used to reduce the observations, as well as the retrieval model used (although following the same calculation assumptions), obtaining supersolar metallicity for WASP-43 b in both the K14 and C22 cases reinforces the conclusion that this planetary atmosphere probably possesses such metallicity, even if the value obtained varies from one method to another (0.18×solar for K14 and 0.8×solar for C22). However, using the disequilibrium chemical model FRECKLL, the planetary atmosphere of WASP-43 b appears as a subsolar metallicity atmosphere, with a consistent-enough thermal and chemical structures across both C22 and K14 reduced datasets.

While using a kinetic chemical model to calculate the chemical structure during retrieval calculations, contributions of TiO and VO (not implemented in FRECKLL, they are described as constant profiles) do not improve the retrieved results for almost all planets considered in this study. This statement is consistent with the design of our exoplanet sample (i.e. no inversion in the thermal profile, no TiO and/or VO spectral signature, and an equilibrium temperature under 2500 K, as described in Section 2). Table C.1 encompasses the retrieval results for each test and each planet studied in the present paper, showing that only for two planets, TrES-3 b and WASP-77 A b, retrieval results benefit from the addition of TiO and VO contributions (by obtaining a Δln(E)>5) compared with the ‘FRECKLL-only’ retrieval results.

In both cases, the ‘FRECKLL-TiO’ retrieval is associated with the higher Evidence, with ln(E) = 117.12 for TrES-3 b (Δln(E) = 6 compared to ‘FRECKLL-only’ case) and ln(E) = 199.5 for WASP-77 A b (Δln(E) ≈ 5 compared to ‘FRECKLL-only’ case). However, regarding retrieval results in the case of TrES-3 b (Figure C.5), ‘FRECKLL-TiO’ configuration does not clearly improve the convergence of the retrieved parameters compared to other retrieval configurations. All temperature points, C/O, and metallicity posterior distributions present equivalent variations whatever the retrieval configurations considered, without any clear convergence towards a precise value. Metallicity posterior describes a slightly bimodal distribution, with a preferential value of log(Z)=1 implying a supersolar atmosphere. C/O could be any value between 0 and 1, confirming the challenge to constrain such planetary parameters with HST observations. Thermal structures are not constrained as well for all cases, with a large 1000-K-wide envelope of probability on both sides of the profiles. The resulting atmosphere consists of a bottom layer in which temperature decreases with altitude up to 10⁵Pa, followed by a layer of a slow increase of temperature with altitude (except the ‘FRECKLL-TiO-VO’ case, which shows a slow decrease with a smaller gradient than the bottom layer), and a top layer with a large positive gradient of temperature. Moreover, we would like to highlight that two teams produced widely different datasets for TrES-3 b (Changeat et al. 2022; Ranjan et al. 2014), likely due to the staring mode used for this observation. Therefore, despite the improvement in Evidence, we preferred to focus on the conservative model for this planet, i.e. the ‘FRECKLL-only’ results.

On the contrary, for WASP-77 A b – which is a more recent observation in scanning mode that enables us to discuss the presence of TiO more robustly – the inclusion of TiO as a chemical contributor in the retrieval improves constrains on (i) the thermal structure, with narrow error bars on the temperature points, (ii) the metallicity, and (iii) the TiO mean abundance. However, C/O remains a challenge for these particular retrieval results (Figure C.9). Among all retrieval configurations, ‘FRECKLL-TiO’ is the only one resulting in a subsolar metallicity atmosphere with log(Z) ≈ –1, whereas the three other configurations describe WASP-77 A b as a supersolar metallicity atmosphere, all agreeing on log(Z) ≈ 2.5. This disagreement between ‘FRECKLL-TiO’ and other configurations of retrieval is also reflected in the C/O posterior distribution: all retrieval tests result in an equivalent posterior of C/O, which would seem to converge towards 0.2. Adding only TiO as a chemical contributor implies a large and imprecise distribution of C/O value, with a maximum value of 0.48 (that could not really be identified as a ‘most probable’ value regarding the shape of the posterior). As in the TrES-3 b case, adding a degree of complexity by adding the chemical contribution of TiO weakens the constraints on metallicity and C/O, preventing any firm conclusion about the plausibility of one value over another. However, adding TiO improves the constraints on the lower part of the retrieved thermal profile for the modelled atmosphere. Indeed, the three lowest temperature points (T_surface to T₂) precisely agree towards a temperature value, with only a 250-K-wide envelope of uncertainties, when other configurations to retrieve this planet predict a 700-K error on either side of the thermal profile. Overall, all retrieval configurations agree about the shape, gradient, and temperature inflexion points for the top layer of the atmosphere (from 10³ to 10²Pa), but in the ‘FRECKLL-TiO’ case, the topmost temperature point T_top posterior follows a wide distribution and is less constrained.

While using the FRECKLL kinetic chemical model during the retrieval calculations, retrieved temperature profiles from transit and eclipse observations are almost consistent with each other for the lowest layers of the modelled atmosphere. To verify the consistency between eclipse and transit spectra retrieval while considering the disequilibrium chemistry assumptions, we carried out two types of tests for a reduced set of planets (HD 189733 b, HD 209458 b, WASP-43 b, and WASP-74 b). Both retrieval tests only use FRECKLL, with either an isothermal profile for the retrieved temperature calculations or a five-point thermal profile (see Figures D.1 to D.4). The use of an isothermal or a five-point thermal profile largely impacts the uncertainty in several parameters such as planet radius, C/O, and metallicity, for which the width of the posterior distribution is largely impacted. Planet radii are, for all cases, consistent between isothermal and 5-point thermal profile retrieval, with the smallest error envelope in the case of the isothermal profile retrieval. Transit observations are firstly sensitive to scale height, which is at first order controlled by the radius, the planet mass, and afterwards controlled by the mean temperature. Using a 5-point profile provides more flexibility, but it is also more complex (i.e. it leads to larger uncertainties on the T-p profile). As a consequence, the planet radius is better constrained by the isothermal assumption. Concerning C/O and metallicity, the use of an isothermal profile or a 5-point profile leads to diametrically opposite results: if C/O converges towards ≈1 for one type of temperature profile, it converges towards ≈0 for the other, and the same result is obtained with metallicity (except for WASP-74 b) that varies from largely subsolar (log(Z) < –1) to largely supersolar (log(Z) > 1). In the case of WASP-74 b (Figure D.4), metallicity describes a bimodal posterior distribution for the isothermal retrieval, with a maximum value at log(Z) = 2.04 (see Table D.1), consistent with the fairly constrained value obtained using a 5-point thermal profile (log(Z) = 2.21).

Concerning the thermal profile, the use of five inflection points to retrieve the temperature profile improves consistency between eclipse and transit observation inversion in the lower part of the modelled atmosphere – though not systematically across all the planets considered in our sample – whereas the upper part of the atmosphere depicts an isothermal variation with altitude, consistent in value with the transit isothermal retrieval. In the case of HD 189733 b (Figure D.1), the overall shape as well as the value of the temperature profile of the transit 5-point retrieval is consistent with the eclipse retrieval results in the case of log(Z)>0 (purple curves on Figure C.2). Within the error bars, we obtain a bottom temperature between 2200 K and 2500 K considering both eclipse and transit observations, with a top layer atmosphere around 1000 K, but the inversion level is deeper in the atmosphere for the transit retrieval: at 10⁴Pa, whereas the inversion occurs at 10³Pa for eclipse observation. In addition to a consistency between eclipse and transit observation inversion for the thermal profile in the case of HD 189733 b, the eclipse log(Z) > 0 retrievals and the transit 5-point retrieval are consistent with each other for the metallicity (ensured by the restrictive prior for this parameter), and for C/O (also shown by Figures 3 and 4). Similar conclusions are drawn in the case of WASP-74 b (Figures C.8 and D.4), except for C/O: transit 5-point retrieval predicts a C/O that tends towards 1, while eclipse log(Z) > 0 retrievals agree for a C/O that tends towards 0.

For the two other planets, HD 209458 b and WASP-43 b, increasing the complexity in the temperature inversion with the five-point retrieval yields nearly identical values for the lowest temperature point of the modelled atmosphere (2000–2500 K). Higher in the modelled atmosphere, the five-point retrieval thermal profile converges towards the variation and temperature values of the temperature profile obtained with the isothermal retrieval, diverging further from the results predicted by the eclipse retrieval configurations. Contrary to HD 189733 b and WASP-74 b, transit isothermal retrieval implies consistent results with the eclipse retrieval results for metallicity (log(Z) ≈ −2), while the transit 5-point retrieval induces supersolar metallicity atmosphere and breaks consistency analysis between eclipse and transit for a particular planet.

Fig. 1 FRECKLL-only fit spectra for all planets in this study for HST eclipse observations. HD 189733 b is excluded: restricting the metallicity prior to the subsolar region (FRECKLL-log(Z)<0) improved ln(E) by 30 compared to the reference ‘FRECKLL-only’ retrieval. WASP-77 A b is also excluded: a FRECKLL-TiO retrieval, with TiO added, really improved the fitting spectrum. Individual analyses and additional retrievals can be found in Appendix C.

Fig. 2 FRECKLL-isothermal fit spectra of the four planets considered in this study for the HST observations in transit. Individual analyses and additional retrievals can be found in Appendix D.

Fig. 3 Eclipse retrievals. (Left panel) Metallicity (O/H) and C/O retrieved from the FRECKLL-only retrievals. HD 189733 b is excluded: restricting the metallicity prior to the subsolar region (FRECKLL-log(Z) improved ln(E) by 30 compared to the reference ‘FRECKLL-only’ retrieval. WASP-77 A b is also excluded: a FRECKLL-TiO retrieval, with TiO added, really improved the fitted spectrum. C/O remains very difficult to retrieve because HST observations lack sensitivity to carbon-bearing species. (Right panel) From Changeat et al. (2022) (Table D.1): metallicity and C/O retrieved from HST free-chemistry retrievals, taking into account free chemistry profiles for H2O, CH4, CO, CO2 for all planets. WASP-77 A b is excluded, for which additional free profiles – TiO, VO, FeH and e– are included.

Fig. 4 Transit retrievals: Metallicity (O/H) and C/O retrieved from the FRECKLL-isothermal retrievals. C/O remains very difficult to retrieved because HST observations lack sensitivity to carbon-bearing species.

Fig. 5 Detailed retrieval results for WASP-43 b. Upper row (from left to right): fitting spectra for each retrieval configuration tested through this population study, the retrieved temperature profile, as well as the chemical structure of the planet (for the ‘FRECKLL-only’ retrieval). Bottom row: posterior for each test, as normalised distribution. Equivalent figures for the rest of the population sample is available in Appendix C for eclipse retrieval results and Appendix D for transit retrieval results.

Fig. 6 Detailed retrieval results for WASP-43 b using data reduced by Kreidberg et al. (2014b). Same display layout as Figure 5. Here, retrieved parameters, prior setup, and stellar and planetary configurations are kept the same as before. We only changed the data to that reduced by Kreidberg et al. (2014b).

Fig. 7 Detailed retrieval results for WASP-43 b with a restrictive metallicity (log(Z)) prior. Upper row (from left to right): Fitting spectra for each retrieval configuration tested through this population study, the retrieved temperature profile, as well as the chemical structure of the planet (for the ‘FRECKLL-only’ retrieval). Bottom row: Posterior for each test, as normalised distribution.

Fig. 8 Detailed retrieval results for WASP-43 b using data reduced by Kreidberg et al. (2014b) with a restrictive metallicity (log(Z)) prior. Same display layout as Figure 7. Here, retrieved parameters, prior setup, and stellar and planetary configurations are kept the same as before. We only changed the data to that reduced by Kreidberg et al. (2014b).

4 Discussion and conclusions

In this paper, we tested the TauREx-FRECKLL model on Hubble Space Telescope WFC3 observations of ten hot-Jupiter atmospheres, to better constrain the metallicity, C/O, thermal structure, and, above all, the chemistry structure of those planets by applying a more realistic chemistry hypothesis for spectral observations inversion. The TauREx-FRECKLL model allows us to account for disequilibrium kinetic chemistry calculations – thanks to the plugin to FRECKLL – simultaneously of the state-of-the-art retrieval calculations made by TauREx-3.1 for exoplanets observations.

For this purpose, we conducted a series of retrieval experiments, considering both the entire population and specific planets to study: (i) the impact of the disequilibrium chemistry hypothesis on the metallicity and C/O inference; (ii) the impact of the reduction method of HST/WFC3 observations; (iii) the impact of including refractory elements TiO and VO under the disequilibrium chemistry hypothesis; and (iv) whether the disequilibrium chemistry hypothesis can reconcile transit and eclipse retrieval results for a specific planet. The following are our conclusions for each of the above-mentioned subjects.

(i) Considering the whole exoplanet population selected for this study, the disequilibrium chemistry hypothesis can strongly impact the conclusion drawn from the analysis of HST data under other and simpler chemistry hypotheses (as free or equilibrium, for example). Metallicity and C/O values retrieved from transit and eclipse, while consistent with each other, are in some cases in opposition with previous studies using the same data. Thus, the characteristics retrieved with the disequilibrium hypothesis can have consequences for planet formation studies, by changing the group to which the considered exoplanet belongs.

(ii) We explored the impact of the reduction method on the retrieval of HST/WFC3 observations of WASP-43 b with our TauREx-FRECKLL coupled model. We used reduced data from Kreidberg et al. (2014b) (K14) and Changeat et al. (2022) (C22). Although both datasets agreed in characterising the metallicity of the atmosphere of WASP-43 b as supersolar, our disequilibrium chemistry retrievals result in a subsolar metallicity for both datasets. Disequilibrium retrievals conducted in the present paper are consistent with each other, for both datasets, in metallicity value, thermal structure, and slightly in chemistry structure considering the simpler retrieval configurations (i.e. FRECKLL-only assumption). However, constraining the retrieval prior of metallicity to match previous conclusions from both K14 and C22 reveals an inconsistency in the metallicity distribution between these two datasets. Additionally, the retrieved metallicity distribution is too close to the limit values of the parameter space. Hence, using a disequilibrium chemical model during retrieval calculations highlights the fact that the conclusions drawn from HST observations depend primarily on the retrieval parameters, priors, and – more broadly – the model used, and to a lesser extent on the reduction method.

(iii) We find that the refractory elements, TiO and VO, do not improve the fit for most of the exoplanets analysed in this study when using a kinetic chemical model to calculate the chemical structure during retrieval calculation. This is expected since most planets belong to the hot Jupiter category (i.e. below the metaldriven thermal inversion regime, C22). Although TiO and VO have been previously detected in eclipse spectra using HST data and shown in the literature to help constrain the recovered metallicity and C/O – thus aiding in the inference of planet formation scenarios – we noticed a sensitive improvement in the fit for only two exoplanets, TrES-3 b and WASP-77 A b. In these cases, the addition of TiO alone increased ln(E) by at least five points. While accounting for TiO does not impact the convergence of retrieved parameters for TrEs-3 b, the addition of TiO actually impacts the convergence of both WASP-77 A b thermal and the chemical structures. However, in this case, C/O and metallicity are in opposition compared to the results of other retrieval experimentation.

(iv) In transit view, spectral observations are primarily sensitive to the atmospheric scale height, depending on the planet radius, planet mass, and the mean temperature. In the present study, by comparing eclipse and transit data retrievals, we obtain retrieved temperature profiles consistent between both views (i.e. eclipse and transit) for the lowest layers of the modelled atmosphere. This consistency is achieved when the thermal profile is discretized into five temperature-pressure points, allowing up to three inflection points for the thermal structure of the planet (i.e. tests using an isothermal profile for the retrieval still depict a bias of about a thousand kelvin compared to retrievals using eclipse data). Hence, up to 10⁴Pa, the temperatures retrieved from transit data are within the error bars of those obtained from eclipse data. Adding complexity to the thermal profile while retrieving transit observations is also necessary for JWST observations. Schleich et al. (2024) found that a two-point profile is sufficient to retrieve the known atmospheric parameters, whereas in the presence of an atmospheric temperature inversion, it is necessary to use an even more complex temperature profile. For the uppermost part of the modelled atmospheres, temperature profiles exhibit an isothermal variation with altitude, closely matching the results obtained with the isothermal profiles and thus maintaining an offset of nearly 1000 K compared to the results of analysis of eclipse observations. This latter result is consistent with the fact that both the eclipse and transit views do not probe the same region of the atmosphere.

Results from the use of a disequilibrium chemistry model coupled with the retrieval algorithm are highly dependent on the spectral resolution of the data. With this first application to real data, we demonstrate the feasibility of such complex retrievals, although the low spectral coverage of HST does not offer the possibility of obtaining stronger constraints than in previous studies assuming simpler chemistry assumptions (free or thermodynamic equilibrium). In a subsequent study, it will be interesting to conduct an equivalent population analysis using James Webb Space Telescope observations, which provide broader spectral coverage and a higher spectral resolution. This will enable the acquisition of additional and different molecular signatures to constrain the thermal and chemical structures as well as the metallicity and C/O of the atmosphere.

Acknowledgements

Bardet and Venot acknowledge funding from Agence Nationale de la Recherche (ANR), project ‘EXACT’ (ANR-21-CE49-0008-01). In addition, Venot acknowledges funding from the Centre National d’Études Spatiales (CNES). The authors acknowledge the exceptional computing support from Grand Equipement National de Calcul Intensif (GENCI) and Centre Informatique National de l’Enseignement Supérieur (CINES). All the simulations presented here were carried out on the ADASTRA cluster hosted at CINES. This work was granted access to the High-Performance Computing (HPC) resources of CINES under the allocations A0140110391, and A0160110391 made by GENCI. The total number of computing hours used to carry out this study amounts to 874207 hCPU. All retrieval results and parameter files of each retrieval runs are available on the EXACT project website at https://www.anr-exact.cnrs.fr/fr/retrieval/. All code used for generation of figures is available on GitHub at https://github.com/debbardet/EXACT_plotter.

Appendix A Sensitivity exploration on the dependence of the retrieved thermal profiles on the number of temperature points

Setting the number of parameters necessary to retrieve and analyse spectral observations of exoplanets can be really difficult, in particular for the temperature vertical profile. Changeat et al. (2021) have studied in detail the impact of retrieval parameters on the resulting temperature profile of WASP-43 b using HST eclipse data, especially the number of N-points to describe the temperature profile (see their appendix D). In their study, they have found it very difficult to determine the ideal number of N-points. In Schleich et al. (2024) study, the authors have shown that an isothermal temperature profile for analysing HST data is sufficient, but for JWST data, it is necessary to use at least two points. In the latter case, they have shown that adding more points does not necessarily bias the result.

In the present population study of ten hot-Jupiter exoatmospheres, we adopted a consistent retrieval methodology including, among others, a five-point thermal profile to retrieve the temperature vertical structure. To explore the dependence of the resulting temperature profile on the number of N-points set as retrieval parameters, we conducted additional retrieval calculations considering only three and four temperature points, with and without the addition of TiO as a constant abundance profile in the case of WASP-77 A b, and all the other retrieval parameters remain the same as described in Section 2. For the 3-point method, we have a unique inflection point at the fixed pressure of 10⁴ Pa. For the 4-point method, we set an additional inflection temperature point at the fixed pressure of 10⁵ Pa. Results of those additional retrieval calculations are shown in Figure A.1 and Table A.1.

Regardless of the number of points chosen for the temperature profile, including the TiO compound among the retrieval parameters significantly improves the representation of the spectrum (Figure A.1), yields consistent temperature profiles across all cases (except for the ’FRECKLL, 4-point’ run), and results in all retrieval tests with TiO converging on similar abundance values (≈10^–7 in volume mixing ratio). In general, determination of C/O, as well as the metallicity, remains difficult, but some of those retrieval calculations depict consistent values. For instance, the C/O of WASP-77 A b is estimated between 0.22 and 0.48, with this latter being determined with three-point and five-point temperature profiles, both including TiO. In addition, both ’4-point’ retrieval runs (i.e. with and without TiO), as well as the ’3-point’ retrieval run result in close C/O values, evaluating it between 0.34 and 0.39. For the metallicity, it is more complex to draw any trend; depending on the case, WASP-77 A b is determined as solar to super solar in metallicity. Finally, for runs without TiO, the addition of points in the temperature profile impacts the Bayesian evidence by a reduction of only 1.3, while the reduction is 2.5 when TiO is taken into account.

As Changeat et al. (2021) and Schleich et al. (2024), we find it complex to determine the optimum number of points to describe the retrieved temperature profile using the N-point methods, and, in our case, with the example of WASP-77 A b, the addition of points does not bias the results. Rather, it provides greater freedom to the Multinest sampling and offers a more detailed description of the vertical thermal structure. This sensibility study will be interesting to conduct on JWST observations.

Fig. A.1 Detailed retrieval results for the sensitivity analysis of the dependence of the retrieval thermal profiles on the number of temperature points, using WASP-77 A b as an example, with and without the addition of TiO. The upper row presents (from left to right) the fitting spectra for each retrieval configuration tested through this population study and the retrieved temperature profile. The lower row presents the posterior for each test, as normalised distribution.

Appendix B Stellar spectral data

To obtain stellar spectra for each of the stars considered here, we use the referenced data from type F, G, K stars – from the Virtual Planetary Laboratory, Spectral Database & Tools database http://depts.washington.edu/naivpl/content/spectral-databases-and-tools – closest to our sample stars. When available, we also use data for the 1–120 nm part of the UV stellar spectra from the X-Exoplanets database (http://sdc.cab.inta-csic.es/xexoplanets/jsp/homepage.jsp) and the MUSCLES Treasury Survey database (https://cos.colorado.edu/~kevinf/muscles.html). For instance, as HD 209458 is a G0 star (Salz et al. 2015), TrES-3 is a G4 star (O’Donovan et al. 2007), and WASP-4, WASP-19 and WASP-77 A are G8 stars, (Triaud et al. 2010; Hebb et al. 2010; Salz et al. 2015, respectively), we use for those 5 stars the UV spectral irradiance of the Sun (Thuillier et al. 2004) scaled to correspond to the radius and effective temperature of each of them.

For missing spectral regions in the recorded data, we use a theoretical Kurucz spectrum model to model our star ensemble (http://kurucz.harvard.edu/grids.html). By selecting the grid file corresponding to a star with the closest metallicity [Fe/H] of each of our individual stars, we model a theoretical stellar flux as a function of the effective temperature and gravity (in log scale) of the desired star.

Then, we carry out a step of normalisation of the theoretical and observed spectra to obtain the stellar data for each star used in the retrieval calculations, taking into account photodissociation.

Appendix C Individual planet analysis in eclipse

Fig. C.1 Detailed retrieval results for HAT-P-2 b. This planet is part of the group of planets for which the ’FRECKLL-only’ retrieval test depicted a bimodal distribution for the metallicity. Therefore, we carried additional tests: we tested the effect of FRECKLL and the effect of FRECKLL plus TiO, VO, and both TiO and VO on two priors for the metallicity (one on the subsolar side log(Z) < 0, and another one on the supersolar side log(Z) > 0). The upper row presents (from left to right) the fitting spectra for each retrieval configuration tested through this population study, the retrieved temperature profile, and the chemical structure of the planet for the ’FRECKLL-only’ retrieval. The lower row presents the posterior for each test, as normalised distribution.

Fig. C.2 Detailed retrieval results for HD 189733 b; as Figure C.1, but showing the chemical structure for the retrieval ’FRECKLL, log(Z) < 0’. Constraining the metallicity prior to the subsolar region improved ln(E) by 30.

Fig. C.3 Detailed retrieval results for HD 209458 b. The upper row presents (from left to right) the fitting spectra for each retrieval configuration tested through this population study, the retrieved temperature profile, and the chemical structure of the planet for the ’FRECKLL-only’ retrieval. The lower row presents the posterior for each test, as normalised distribution.

Fig. C.4 Same as Figure C.3 for Kepler-13 A b.

Fig. C.5 Same as Figure C.3 for TrES-3 b.

Fig. C.6 Same as Figure C.1 for WASP-19 b.

Fig. C.7 Same as Figure C.3 for WASP-4 b.

Fig. C.8 Same as Figure C.1 for WASP-74 b.

Fig. C.9 Same as Figure C.3 for WASP-77 A b. Here, the chemical profiles are from the FRECKLL-TiO retrieval, which is the best fit obtained for this planet (see Table C.1).

Appendix D Individual planet analysis in transit

Fig. D.1 Detailed retrieval results for HD 189733 b. The upper row presents (from left to right) the fitting spectra for each retrieval configuration tested through this population study, the retrieved temperature profile, and the chemical structure of the planet for the ’FRECKLL-5-point’ retrieval. The lower row presents the posterior for each test, as normalised distribution.

Fig. D.2 Same as Figure D.1 for HD 209458 b.

Fig. D.3 Same as Figure D.1 for WASP-43 b.

Fig. D.4 Same as Figure D.1 for WASP-74 b.

References

All Tables

All Figures

Fig. 1 FRECKLL-only fit spectra for all planets in this study for HST eclipse observations. HD 189733 b is excluded: restricting the metallicity prior to the subsolar region (FRECKLL-log(Z)<0) improved ln(E) by 30 compared to the reference ‘FRECKLL-only’ retrieval. WASP-77 A b is also excluded: a FRECKLL-TiO retrieval, with TiO added, really improved the fitting spectrum. Individual analyses and additional retrievals can be found in Appendix C.

In the text

	Fig. 2 FRECKLL-isothermal fit spectra of the four planets considered in this study for the HST observations in transit. Individual analyses and additional retrievals can be found in Appendix D.
In the text

Fig. 3 Eclipse retrievals. (Left panel) Metallicity (O/H) and C/O retrieved from the FRECKLL-only retrievals. HD 189733 b is excluded: restricting the metallicity prior to the subsolar region (FRECKLL-log(Z) improved ln(E) by 30 compared to the reference ‘FRECKLL-only’ retrieval. WASP-77 A b is also excluded: a FRECKLL-TiO retrieval, with TiO added, really improved the fitted spectrum. C/O remains very difficult to retrieve because HST observations lack sensitivity to carbon-bearing species. (Right panel) From Changeat et al. (2022) (Table D.1): metallicity and C/O retrieved from HST free-chemistry retrievals, taking into account free chemistry profiles for H2O, CH4, CO, CO2 for all planets. WASP-77 A b is excluded, for which additional free profiles – TiO, VO, FeH and e– are included.

In the text

	Fig. 4 Transit retrievals: Metallicity (O/H) and C/O retrieved from the FRECKLL-isothermal retrievals. C/O remains very difficult to retrieved because HST observations lack sensitivity to carbon-bearing species.
In the text

Fig. 5 Detailed retrieval results for WASP-43 b. Upper row (from left to right): fitting spectra for each retrieval configuration tested through this population study, the retrieved temperature profile, as well as the chemical structure of the planet (for the ‘FRECKLL-only’ retrieval). Bottom row: posterior for each test, as normalised distribution. Equivalent figures for the rest of the population sample is available in Appendix C for eclipse retrieval results and Appendix D for transit retrieval results.

In the text

	Fig. 6 Detailed retrieval results for WASP-43 b using data reduced by Kreidberg et al. (2014b). Same display layout as Figure 5. Here, retrieved parameters, prior setup, and stellar and planetary configurations are kept the same as before. We only changed the data to that reduced by Kreidberg et al. (2014b).
In the text

	Fig. 7 Detailed retrieval results for WASP-43 b with a restrictive metallicity (log(Z)) prior. Upper row (from left to right): Fitting spectra for each retrieval configuration tested through this population study, the retrieved temperature profile, as well as the chemical structure of the planet (for the ‘FRECKLL-only’ retrieval). Bottom row: Posterior for each test, as normalised distribution.
In the text

	Fig. 8 Detailed retrieval results for WASP-43 b using data reduced by Kreidberg et al. (2014b) with a restrictive metallicity (log(Z)) prior. Same display layout as Figure 7. Here, retrieved parameters, prior setup, and stellar and planetary configurations are kept the same as before. We only changed the data to that reduced by Kreidberg et al. (2014b).
In the text

Fig. A.1 Detailed retrieval results for the sensitivity analysis of the dependence of the retrieval thermal profiles on the number of temperature points, using WASP-77 A b as an example, with and without the addition of TiO. The upper row presents (from left to right) the fitting spectra for each retrieval configuration tested through this population study and the retrieved temperature profile. The lower row presents the posterior for each test, as normalised distribution.

In the text

Fig. C.1 Detailed retrieval results for HAT-P-2 b. This planet is part of the group of planets for which the ’FRECKLL-only’ retrieval test depicted a bimodal distribution for the metallicity. Therefore, we carried additional tests: we tested the effect of FRECKLL and the effect of FRECKLL plus TiO, VO, and both TiO and VO on two priors for the metallicity (one on the subsolar side log(Z) < 0, and another one on the supersolar side log(Z) > 0). The upper row presents (from left to right) the fitting spectra for each retrieval configuration tested through this population study, the retrieved temperature profile, and the chemical structure of the planet for the ’FRECKLL-only’ retrieval. The lower row presents the posterior for each test, as normalised distribution.

In the text

	Fig. C.2 Detailed retrieval results for HD 189733 b; as Figure C.1, but showing the chemical structure for the retrieval ’FRECKLL, log(Z) < 0’. Constraining the metallicity prior to the subsolar region improved ln(E) by 30.
In the text

	Fig. C.3 Detailed retrieval results for HD 209458 b. The upper row presents (from left to right) the fitting spectra for each retrieval configuration tested through this population study, the retrieved temperature profile, and the chemical structure of the planet for the ’FRECKLL-only’ retrieval. The lower row presents the posterior for each test, as normalised distribution.
In the text

	Fig. C.4 Same as Figure C.3 for Kepler-13 A b.
In the text

	Fig. C.5 Same as Figure C.3 for TrES-3 b.
In the text

	Fig. C.6 Same as Figure C.1 for WASP-19 b.
In the text

	Fig. C.7 Same as Figure C.3 for WASP-4 b.
In the text

	Fig. C.8 Same as Figure C.1 for WASP-74 b.
In the text

	Fig. C.9 Same as Figure C.3 for WASP-77 A b. Here, the chemical profiles are from the FRECKLL-TiO retrieval, which is the best fit obtained for this planet (see Table C.1).
In the text

	Fig. D.1 Detailed retrieval results for HD 189733 b. The upper row presents (from left to right) the fitting spectra for each retrieval configuration tested through this population study, the retrieved temperature profile, and the chemical structure of the planet for the ’FRECKLL-5-point’ retrieval. The lower row presents the posterior for each test, as normalised distribution.
In the text

	Fig. D.2 Same as Figure D.1 for HD 209458 b.
In the text

	Fig. D.3 Same as Figure D.1 for WASP-43 b.
In the text

	Fig. D.4 Same as Figure D.1 for WASP-74 b.
In the text