The Importance of Optical Wavelength Data on Atmospheric Retrievals of Exoplanet Transmission Spectra

Over the past two decades, significant advances have been made in characterizing exoplanet atmospheric properties through transits (e.g., Charbonneau et al. 2002; Kreidberg et al. 2015; Sing et al. 2016; Wakeford et al. 2018; Alderson et al. 2023; Xue et al. 2023), secondary eclipses (e.g Deming et al. 2005; Todorov et al. 2014; Evans et al. 2018; Coulombe et al. 2023), and full phase curves (e.g., Knutson et al. 2012; Stevenson et al. 2017; Mikal-Evans et al. 2022; Kempton et al. 2023).

Space-based transmission spectroscopy provides a powerful window into exoplanet atmospheres (e.g., Seager & Sasselov2000; Brown 2001). Low-resolution spectra can be obtained from transit light curves of telescopes such as the Hubble Space Telescope (HST), Spitzer, and JWST. A majority of early data have been collected from the optical (0.3–1 μm) to near-infrared (>1 μm) using HST's Space Telescope Imaging Spectrograph (STIS) and Wide Field Camera 3 (WFC3) instruments (e.g., Deming et al. 2013; Kreidberg et al. 2015; Sing et al. 2016; Spake et al. 2021). Photometric observations from Spitzer's IRAC instrument provide additional data points at the infrared end of the spectrum near 3.6 and 4.5 μm (e.g., Knutson et al. 2011). JWST now fills the gap between 0.6–5 μm, providing spectroscopic measurements across the whole near-infrared (e.g., Ahrer et al. 2023; Alderson et al. 2023; Feinstein et al. 2023; Fournier-Tondreau et al. 2024; Rustamkulov et al. 2023; Xue et al. 2023), and access to the mid-infrared beyond 5 μm with the Mid-Infrared Instrument (e.g., Dyrek et al. 2023; Grant et al. 2023).

The shape of exoplanet transmission spectra is driven by a planet's chemical composition, cloud composition, and temperature structure, which in turn are influenced by the underlying physical processes in an exoplanet atmosphere. Transmission spectra often show an optical scattering slope. This feature can be produced by different physical processes. Rayleigh scattering of photons by particles smaller than the wavelength of the incident radiation produces a characteristic λ⁻⁴ slope, whereas scattering from aerosol species can produce steeper gradients (Lecavelier des Etangs et al. 2008). The optical slope can also be influenced by stellar activity (Pont et al. 2013; Barstow et al. 2017; Rackham et al. 2018), where the relative number of starspots shifts the transit depth, with this effect being largest toward the shortest wavelengths. High temperatures can also lead to steep optical slopes due to the inflation of the scale height of the atmosphere (Barstow 2020) or absorption of high-temperature UV species (Lothringer et al. 2022).

Absorption due to the presence of aerosols/clouds in planetary atmospheres causes the atmosphere to become opaque at pressures greater than the cloud-top pressure and thus limits the depth to which we can probe an atmosphere. If clouds are present at the pressures accessed by transmission spectra, they will have the effect of removing the baseline of spectral features. This affects the relationship between feature amplitude and species abundance (Barstow 2021).

Disentangling aerosol properties from gas phase species (e.g., Na, K, H₂O) has been the focus of a number of population studies due to the implications of H₂O abundance on the formation and evolution of exoplanets. Exploring this problem, one of the first major comparative studies (Sing et al. 2016) examined the spectra of 10 hot Jupiters in transmission. To make comparisons between the planets, they defined two indices ΔZ_UB−LM and ΔZ_J−LM (measuring the strength of scattering to molecular absorption and the relative strength between the mid-infrared continuum and mid-infrared molecular absorption, respectively) and measured the strength of the characteristic 1.4 μm H₂O absorption feature. A correlation between a strong scattering index and muted water absorption was found leading to the conclusion that clouds and hazes are responsible for weak spectral features, rather than an intrinsic low abundance of H₂O.

A series of atmospheric retrieval studies have been performed on exoplanet transmission spectra to quantify the impact of aerosols on the H₂O abundance (e.g., Barstow et al. 2017; Pinhas et al. 2019; Welbanks et al. 2019) finding that all spectra are likely affected by cloud opacity to some degree but that a large number of planets also have a low intrinsic abundance of H₂O compared to solar values. Studies performed just using HST WFC3 data from 1.1–1.7 μm covering a single H₂O band also showed that gray clouds are favored over nongray scattering for retrievals on near-infrared-only data (e.g., Fisher & Heng 2018; Tsiaras et al. 2018). It is additionally noted in Fisher & Heng (2018) that the 1.1–1.7 μm wavelength range is insufficient to overcome degeneracies between modeling choices.

A number of studies looking at individual planetary data sets have demonstrated that the omission of observational measurements in the optical can result in major differences in the retrieved information (e.g., Wakeford et al. 2018; Pinhas et al. 2019; Alderson et al. 2022). However, there does not exist a detailed systematic study on this effect across a population of planets, and you cannot infer this information by combining multiple study results across different model setups and data ranges due to the differing assumptions used in each separate analysis.

The aim of this study is to explore the dependence of spectral range on the retrieved atmospheric parameters of exoplanet transmission spectra, focusing on the impact of data at optical wavelengths. We explore 14 exoplanets (see Table 1) using transmission spectra that come from previously published reductions of, primarily, HST and Spitzer observations which span from 0.3–4.5 μm. This is especially pertinent as JWST only goes down to a minimum of 0.6 μm and there is not yet a clear assessment of how crucial this short-wavelength information may be on inferred atmospheric properties below this cutoff. We describe our retrieval setup and testing in Section 2. We then present our sample of 14 exoplanet spectra and the results of our retrievals in Section 3, with a table of our data sets and planet parameters in the Appendix. We calculate the information content (IC) obtained with increased wavelength coverage in Section 4. In Section 5 we discuss trends across the population in comparison to the planetary equilibrium temperature. We summarize our results and conclusions in Section 6.

To investigate the wavelength dependence of retrieved parameters from exoplanet transmission spectra, we use the open-source retrieval code, POSEIDON (MacDonald & Madhusudhan 2017b; MacDonald 2023), which couples a forward model and radiative transfer treatment with a Bayesian retrieval framework for parameter estimation and model selection. Underpinning the retrieval framework is the nested sampling package, PyMultiNest (Buchner et al. 2014). We limit the implementation of POSEIDON to free-chemistry retrievals for a one-dimensional atmospheric parameterization appropriate for the precision of the spectral data analyzed. To preserve a consistent retrieval framework across the study, we test a range of model initializations on the 0.3–4.5 μm transmission spectrum of HD 209458b and define a base retrieval setup with an appropriate level of complexity for each of the spectra considered in our study. We then test a range of wavelength cutoffs on the spectra of HD 209458b and WASP-39b to determine a base study for our 14 exoplanets. The retrieval configurations for our population are described below.

2.1. Forward Model and Retrieval Configuration

There are two practical limits to the retrieval complexity that we considered: the number of model parameters must be sufficiently small to perform a statistically valid retrieval, and computational complexity must be minimized such that a population analysis is feasible. We define our base atmospheric model with 12 free parameters (see Table 2 for the parameters and their prior ranges) and describe below the selection of these parameters and the overall retrieval setup. We first test the different parameterizations available in POSEIDON for the pressure–temperature (P-T) structure of the atmosphere. After retrieving a profile consistent with an isothermal atmosphere using a five-parameter P-T profile (Madhusudhan & Seager 2009), we choose to adopt an isothermal profile, where a uniform temperature prior is defined between (0.4 and 1.15) T_eq. This reduces the number of free parameters in the fit for the P-T profile from five to one. The reference radius at which the atmosphere has a pressure of 10 bar (R_P,ref) is also allowed to vary as a free parameter.

Table 2. Priors for Our POSEIDON Retrievals

Parameter	Prior Distribution	Prior range
Rp,ref	uniform	(0.85–1.15) Rp
Tp	uniform	(0.4–1.15) Teq
log H2O	uniform	−12 to −1
log CO2	uniform	−12 to −1
log CO	uniform	−12 to −1
log CH4	uniform	−12 to −1
log Na	uniform	−12 to −1
log K	uniform	−12 to −1
$\mathrm{log}a$	uniform	−4 to 8
γ	uniform	−20 to 2
$\mathrm{log}{P}_{\mathrm{cloud}}$	uniform	−6 to 3 dex
ϕclouds	uniform	0–1
High-temperature Species
log TiO	uniform	−12 to −1
log VO	uniform	−12 to −1
Stellar Parameters
fhet	uniform	0–0.5
Thet	uniform	(0.6–1.2) Teff
Tphot	Gaussian	μ = Teff, σ = 100 K

Note. The first 12 parameters define the base retrieval configuration. The reference radius is defined at 10 bar. R_p is the planetary white-light radius. The high-temperature species (TiO and VO) are added for planets with T_eq > 2000 K, while the stellar heterogeneity parameters are included for known active stars.

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We assume a bulk atmosphere of H₂ and He, with a fixed He/H₂ ratio of 0.17, and include six chemical opacity sources Na and K (Barklem & Collet 2016), H₂O (Polyansky et al. 2018), CO₂ (Tashkun & Perevalov 2011), CO (Li et al. 2015), and CH₄ (Yurchenko et al. 2018). For each we apply uniform priors on the log₁₀ mixing ratios of −12 to −1 dex. The presence of alkali species Na and K in the optical and H₂O in the near-infrared can be inferred from Hubble STIS and WFC3 observations, respectively. Additional opacity from carbon species covers the spectral range probed by the Spitzer photometric points. Opacity from nitrogen species (e.g., $\mathrm{HCN}$, NH₃) is excluded from the base setup to reduce model complexity. Additional continuum opacity arising from H₂–H₂ and H₂–He collision-induced absorption (Karman et al. 2019) and Rayleigh scattering are included within the model setup. The final four parameters encode cloud and haze opacity into the atmospheric model, using a deck-haze prescription as defined in MacDonald & Madhusudhan (2017b). A cloud deck with infinite opacity across all wavelengths is defined at a pressure level $\mathrm{log}{P}_{\mathrm{cloud}}$ and coupled with a wavelength-dependent scattering haze. Two parameters describe the behavior of the haze: the scattering slope γ and log a, which act as a scaling factor to the H2 Rayleigh scattering cross Section defined at 350 nm. To model inhomogeneous cloud cover, ${\bar{\phi }}_{\mathrm{clouds}}$ parameterizes the terminator cloud fraction. We tested models with and without ${\bar{\phi }}_{\mathrm{clouds}}$ and found that it was necessary to improve the statistical fit to the data in a range of cases and therefore include it in our base retrieval.

We initialize the atmospheric model on a pressure grid of 100 layers, uniform in log pressure, ranging from 100–10⁻⁹ bar. The pressure minimum is selected to allow for the haze parameterization to fit spectra with strong optical scattering slopes, where the shortest wavelengths probe the lowest atmospheric pressures in transmission. To decrease computation time, opacities are sampled from a high-resolution line-by-line database onto a user-defined resolution grid. A range of opacity grid resolutions were tested and R = 5000 was selected, as this provided the same level of consistency across the wavelengths covered in this study as an R = 10,000 model while reducing the computation time. Each retrieval was computed with N = 4000 live points to effectively explore the prior phase space.

2.2. Optical Data Cutoffs: Case Studies for HD 209458 b and WASP-39b

We first consider the impact of optical data wavelength cutoffs on retrievals of HD 209458b and WASP-39b. We do not consider measurements from JWST in this study to enable a consistent retrieval and analysis across a larger set of planets. We sequentially cut off short-wavelength data points that fall below a threshold ${\lambda }_{\min }$, in intervals of 0.1 μm, from 0.3 to 1.1 μm. Due to the absence of WFC3 G102 data for HD 209458b, no retrieval is run for ${\lambda }_{\min }$ = 1.0 μm for this planet as there is no data change from the surrounding cutoffs.

Figure 1 shows the median retrieved spectra for each cutoff range combined with the observed data for HD 209458b (left) and WASP-39b (right). Increasing wavelength coverage does not lead to a convergence between retrieved spectra until data are included below 0.5 μm. Without data below this point the retrieved median spectra are not well fit at optical wavelengths, where models are unconstrained by observations. For WASP-39b, the decrease in optical transit depths with increasing wavelength coverage is driven by the downturn in data points preceding the sodium line. Wavelengths below 0.5 μm are necessary to fit the observed scattering slope.

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Cloud opacity and alkali species are the main parameters driving the differences between retrieved spectra. A comparison of the median retrieved parameters and their 1σ errors for each wavelength range of HD 209458b and WASP-39b can be seen in Figure 2. For both planets, K abundances are constrained for data below 0.7 μm corresponding to spectra where the wavelength range encompasses the line peak at ∼ 0.770 μm. The characteristic Na doublet is found at ∼ 0.589 μm. WASP-39b shows constraints for sodium abundance below 0.5 μm; however for HD 209458b, a minimum wavelength of 0.6 μm is sufficient to improve constraints on sodium due to the evidence of strong line broadening around the NaI feature seen in the spectrum.

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Of the four cloud parameters, for HD 209458b the retrievals across all wavelength ranges can converge on a solution for the cloud-top pressure, $\mathrm{log}{P}_{\mathrm{cloud}}$, and patchy cloud, ϕ_clouds, parameters. The variation in the median patchy cloud parameter is what drives the variation in the base transit depths between the H₂O and CH₄ features in the 2–3 μm wavelength range. In contrast, the scattering slope parameters $\mathrm{log}a$ and γ are only well constrained in the full 0.3–4.5 μm retrieval. For WASP-39b, constraints for $\mathrm{log}{P}_{\mathrm{cloud}}$ and ${\bar{\phi }}_{\mathrm{clouds}}$ consistently improve with increased short-wavelength coverage. $\mathrm{log}a$ and γ require wavelengths shortward of 0.5 μm for retrievals to constrain the parameter space.

While the H₂O constraints become tighter when shorter-wavelength information is included in retrievals, the median H₂O abundance of HD 209458b for the ${\lambda }_{\min }$ = 0.3 μm retrieval ($-{4.46}_{-0.35}^{+0.46}$ dex) only marginally falls outside of the 1σ errors of the ${\lambda }_{\min }$ = 1.1 μm retrieval. The WFC3 G141 data points alone can constrain the water abundance to $-{3.59}_{-0.73}^{+0.85}$ dex despite the large range of potential cloud solutions. Water is retrieved at the upper boundary of the prior for WASP-39b across all wavelength ranges. This is consistent with the H₂O abundance found by Wakeford et al. (2018). Those species with opacities only accessed by the two Spitzer photometric points (CO, CO₂, CH₄) remain unconstrained with consistent distributions across all retrieved wavelengths, due to the sparse data.

We now extend our investigation on the wavelength dependence of retrieved atmospheric parameters to a total sample of 14 planets: HAT-P-1b, HAT-P-11b, HAT-P-12b, HAT-P-32b, HD 189733b, WASP-6b, WASP-12b, WASP-17b, WASP-19b, WASP-31b, WASP-121b, and WASP-127b, additional to HD 209458b and WASP-39b. The transmission spectra used come primarily from previously published reductions of HST and Spitzer observations, with data originating from five instrument modes: HST STIS/G430L, HST STIS/G750L, HST WFC3/G102, HST WFC3/G141, and the Spitzer IRAC photometric bandpasses centered at 3.6 and 4.5 μm. We also consider additional ground-based observations from the Very Large Telescope (VLT) FORS2 for WASP-39b and WASP-6b, as reduced by Wakeford et al. (2018) and Carter et al. (2020), and include the HST NICMOS photometric points for HD 189733b that were also used by Sing et al. (2016) and Barstow et al. (2017) in their analysis. Our full sample and data sources are listed in Table 1.

Our planet selection was based on several factors. First, the planet must have transit observation in the optical STIS bandpasses (G430/G750) and one observation in either of the WFC3 bandpasses (G102/G141). While a number of planets in our study are being observed with JWST for consistency in analysis methods we choose not to include those data sets here. Second, we select planets with a range of T_eq spanning a range of predicted cloud properties (e.g., Gao et al. 2020; Ohno & Kawashima 2020). Finally, we limit our sample to the 14 shown due to limitations in computation time for this study. The planetary and stellar properties for each system are summarized in the Appendix, Table 5.

We explore multiple optical wavelength cutoffs throughout our planet sample. From our initial findings for HD 209458b and WASP-39b (Section 2.2), we select three cutoffs: ${\lambda }_{\min }$ = 0.3, 0.6 and 1.1 μm. The 1.1 μm cutoff marks the transition between HST WFC3 G141 and G102 observations, and hence when using this cutoff, the main opacity sources are the 1.15 μm and 1.4 μm water features. We select 0.6 μm as the next cutoff. Not only does this mark the minimum wavelength probed by a majority of JWST instrument modes, but with this cutoff there can be a partial indication of the pressure-broadened Na resonance doublet for relatively clear atmospheres. Our final cutoff, ${\lambda }_{\min }$ = 0.3 μm, considers the full spectral range of the HST STIS data.

3.1. Population Retrieval Configuration

3.2. Results: Population-level Analysis

Figures 3 and 4 present the retrieved transmission spectra across the population for our three optical data cutoffs. The planets are ordered from the lowest retrieved optical slope Rayleigh enhancement factor (WASP-17b), $\mathrm{log}a$, to the highest retrieved value (HD 189733b). The corresponding retrieved atmospheric properties (posterior median and 1σ confidence intervals) for each planet and spectral range are shown in Figure 5. Full posterior distributions for all planets are provided in the supplementary online material ⁴ . Below, we discuss the effect of changing the spectral range on model parameters.

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3.2.1. Aerosol Properties

Of the four cloud parameters, $\mathrm{log}a$ is the best constrained when using the full wavelength range, ${\lambda }_{\min }$ = 0.3 μm. We find the average 1σ error range for $\mathrm{log}a$ decreases from 6.54 to 5.66 to 3.05 for the 1.1, 0.6, and 0.3 μm cutoffs, respectively, corresponding to a precision improvement of 46% when adding data from 0.6 to 0.3 μm. The scattering slope, γ, also sees improved constraints with increased wavelength coverage, but in several cases, it remains largely unconstrained even with the addition of spectral information down to 0.3 μm, likely due to large uncertainties at these wavelengths. For γ, the error range between the 1.1 and 0.6 μm cutoffs shows limited improvement, decreasing from 11.7 to 10.5. However, with the addition of 0.3–0.6 μm data, the 1σ error range decreases by 30% to 7.2. A significant proportional decrease in the error range still leaves large regions of degenerate parameter space between $\mathrm{log}a$ and γ. Consequently, it is this shortest wavelength range that is crucial for constraining the scattering slope parameters, where the primary gain in IC comes from the Rayleigh enhancement factor (see Section 4).

In contrast to the scattering slope parameters, constraints on the cloud-top pressure, $\mathrm{log}{P}_{\mathrm{cloud}}$, do not improve significantly with additional optical wavelength coverage. Even for the full wavelength range down to ${\lambda }_{\min }$ = 0.3 μm, for many planets, $\mathrm{log}{P}_{\mathrm{cloud}}$ is largely unconstrained. Between 1.1 and 0.3 μm, the average 1σ error is only reduced by 0.57 dex from 4.31 dex because a majority of spectra result in only an upper limit corresponding to a nondetection of an optically thick cloud deck. Of the four planets with cloud pressure constraints, HD 209458b, HD 189733b, WASP-31b, and WASP-39b, HD 209458b is the only one constrained at ${\lambda }_{\min }$ = 1.1 μm. Because the cloud pressure contributes opacity across the entire spectrum particularly affecting the amplitude of spectral features, in the case of HD 209458b there is sufficient information from the infrared spectrum to constrain the cloud deck. WASP-31b, WASP-39b, and HD 189733b are important counter-examples to this trend, where the addition of optical wavelengths can improve cloud deck constraints if there is evidence of high-altitude, gray clouds.

The cloud fraction, ${\bar{\phi }}_{\mathrm{clouds}}$, does show some improvement with shorter-wavelength coverage. The addition of wavelengths between 1.1 and 0.3 μm decreases the average 1σ error range from 0.45 to 0.32. Those planets with well-constrained cloud fractions (HD 189733b, HAT-P-32b, HAT-P-11b) follow a general trend of tighter cloud fractions with increasing $\mathrm{log}a$, as can be seen in Figure 5. Therefore, optical wavelengths can improve our knowledge of the terminator cloud fraction, which must be well constrained to also precisely measure the temperature and chemical composition of exoplanet atmospheres (e.g., Line & Parmentier 2016; MacDonald & Madhusudhan 2017b).

Overall, our population analysis shows that increasing the wavelength coverage into the optical can significantly improve constraints on aerosol scattering parameters. With only near-infrared observations, these scattering parameters often add unnecessary complexity to the retrieval model. A similar conclusion was drawn by Fisher & Heng (2018) when retrieving only WFC3 1.1–1.7 μm data, where they showed that only gray clouds were required to fit the data. Wavelengths shortward of 0.6 μm are necessary to obtain constraints on $\mathrm{log}a$ and γ, with additional marginal improvement on $\mathrm{log}{P}_{\mathrm{cloud}}$ and ${\bar{\phi }}_{\mathrm{clouds}}$.

3.2.2. Atmospheric Temperature

Across the population, we do not see an improvement in retrieved temperatures from adding optical data. This average 1σ temperature range between the ${\lambda }_{\min }$ = 1.1 and 0.3 μm spectra decreases from 363 to 344 K, with the error range increasing marginally for the ${\lambda }_{\min }$ = 0.63 μm to 373 K. As the H₂O feature in the near-infrared is often the most prominent absorption signature at these wavelengths, this feature influences the convergence of the temperature parameter through the atmospheric scale height. In principle, Rayleigh scattering can infer temperature from the optical spectrum (for a clear atmosphere), but since we are fitting for a free optical slope this method of determining the temperature is degenerate with the scattering properties (e.g., Barstow 2020; Barstow & Heng 2020). The lack of additional temperature information with optical wavelengths is also a consequence of implementing free-chemistry retrievals. In equilibrium chemistry retrievals of WASP-39b, Wakeford et al. (2018) find improved temperature constraints with the inclusion of optical data, driven by the opacity of optical species influencing the best fit model temperature. Additionally, degeneracies between the reference pressure and temperature that occur across the population are generally not broken by the addition of observations at optical wavelengths.

3.2.3. Chemical Composition

All three wavelength ranges can broadly constrain the H₂O abundance, with consistent median values within an order of magnitude. Some improvement in constraints is realized as the wavelength range of the observed spectrum increases, where the average 1σ error range in the H₂O abundance decreases from 2.00 to 1.36 and 1.15 dex, for the ${\lambda }_{\min }$ = 1.1, 0.6, and 0.3 μm spectra, respectively. In the case of WASP-17b and WASP-127b constrains improve by > 50%. For WASP-17b, this improvement is primarily from observations between 0.3 and 0.6 μm, whereas for WASP-127b, it is the 0.6 to 1.1 μm wavelength range that contributes to the improved constraints. With the exception of HAT-P-11b (see Section 3.4.4), the retrieved H₂O abundances are consistent between the three wavelength ranges. Primarily, the H₂O mixing ratio is well characterized by infrared observations shortward of 1.1 μm where H₂O molecular bands dominate the opacity. However, optical data can provide improvements on the H₂O constraints. We may expect the water abundance and constraints to change with increased wavelength coverage through a gain of information on cloud parameters. Studies have discussed the degeneracy between cloud pressure level and water abundance (e.g., Welbanks & Madhusudhan 2019) where the 1.4 μm water feature can be fit by a lower-H₂O-abundance, clear atmosphere or a higher-H₂O-abundance atmosphere where the feature is muted by cloud opacity. However, in general, constraints on the cloud pressure level do not significantly improve with the addition of optical wavelengths.

Precise alkali metal abundances are crucially enabled by optical data. With little prominent atomic opacity at long wavelengths, the ${\lambda }_{\min }$ = 1.1 μm spectra are unable to constrain the abundances of Na and K. A minimum wavelength of 0.6 μm is sufficient to constrain the K abundance (should its atomic line feature be present). For planets with detections of K (>3σ), the 1σ constraints span a consistent range of 2.4 dex for both the ${\lambda }_{\min }$ = 0.6 and 0.3 μm spectra. However, to reliably constrain Na, the full optical wavelength range is required. For planets with Na detection significances > 3σ, the abundance constraint ranges improve from 4.66 to 1.94 dex between the ${\lambda }_{\min }$ = 0.6 and 0.3 μm spectra. We find that, with the exception of HD 209458b, detecting only the red wing of the Na resonance feature with HST (i.e., ${\lambda }_{\min }$ = 0.6) is not sufficient to constrain the Na abundance.

The HST and Spitzer data examined here cannot generally constrain the abundances of the carbon-bearing species, nor do the constraints improve with additional wavelength information in the optical. This is demonstrated by the average 1σ abundance constraints between each cutoff. For CO, the range changes negligibly from 6.16 to 5.99 dex between the 1.1 and 0.3 μm cutoffs, and for CO₂ and CH₄, the average constraints change from 4.12 to 4.35 dex and 3.72 to 4.12 dex, respectively. However, WASP-17b and WASP-127b are outliers in this trend. Across all wavelengths, we are able to constrain the CO₂ mixing ratio to within < 2 dex. These constraints are driven by the strong relative offset between the Spitzer points where the 4.5 μm point is at higher transit depths than the 3.6 μm point. This can be fit by a sharp CO2 peak near 4.5 μm. Increasing the wavelength coverage into the optical decreases constraints by ∼30% between the ${\lambda }_{\min }$ = 1.1 and 0.3 μm spectra. Therefore, if infrared data can provide evidence of carbon species, the inclusion of optical wavelengths can strengthen constraints on the retrieved mixing ratios.

3.3. Assessment of Individual Planet Spectra

We next discuss retrieval results for the individual planets that are not captured within the population plots. Detection significances for Na, K, and H₂O are computed from Bayesian evidence ratios, with significances > 1σ presented in Table 4.

Table 4. Detection Significances for Na, K and H₂O

Planet	DSNa	DSK	${{DS}}_{{{\rm{H}}}_{2}{\rm{O}}}$
HAT-P-1b	1.7σ	1.3σ	3.7σ
HAT-P-11b	—	—	4.7σ
HAT-P-12b	—	1.3σ	4.6σ
HAT-P-32b	—	1.1σ	8.5σ
HD 189733b	5.2σ	2.0σ	5.7σ
HD 209458b	7.3σ	5.6σ	8.9σ
WASP-6b	4.5σ	3.5σ	4.8σ
WASP-12b	2.2σ	—	6.1σ
WASP-17b	—	—	8.4σ
WASP-19b	—	1.5σ	3.2σ
WASP-31b	—	3.2σ	2.6σ
WASP-39b	4.5σ	3.2σ	8.9σ
WASP-121b	2.4σ	2.3σ	6.1σ
WASP-127b	2.9σ	1.1σ	14.4σ

Note. The equivalent sigma values are computed from Bayesian evidence ratios. "—" indicates a nondetection (Bayes factor < 1).

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The posterior distributions for individual planets can be found in the supplementary online material ⁵ , to which we direct the reader. We note a few general insights into the relationships between retrieved parameters across the planetary population. Widely present across the population and wavelength cutoffs is a correlation between the reference pressure and temperature. With the exception of WASP-17b, which is mentioned below, this does not improve with the inclusion of optical data.

Considering just the ${\lambda }_{\min }$ = 1.1 μm retrievals, there are many cases where the parameter space is largely degenerate. With the inclusion of optical data, correlations between parameters can emerge as regions of the parameter space are ruled out. Cases where correlations arise with the inclusion of optical data are often between cloud parameters, optical species mixing ratios, or across these two parameter categories. For example, we see for HAT-P-1b that the inclusion of optical data reveals a correlation between γ and ${\bar{\phi }}_{\mathrm{clouds}}$ while still representing a degeneracy between those two parameters. Further emergent correlations for individual planets are presented in their respective sections.

3.3.1. HAT-P-1b

HAT-P-1b shows increasing improvement in cloud scattering properties, $\mathrm{log}a$ and γ, with increasing wavelength coverage. However, we see little improvement in other fit parameters. H₂O is well constrained with a detection significance of 3.7σ but Na remains tentative in our analysis despite a significant "by-eye" deviation at the NaI wavelength.

3.3.2. HAT-P-12b

For the ${\lambda }_{\min }$ = 1.1 μm retrieval HAT-P-12b favors high Na and K abundance modes. This condition is resolved with the addition of observations below 1.1 μm, where Na and K features can be measured, which rule out high-abundance solutions. Large transit depth uncertainties on the STIS data prevent any strong constraints on the cloud parameters, where broad regions of the parameter space remain degenerate. However, the ${\lambda }_{\min }$ = 0.3 μm retrieval finds, although unconstrained, a γ distribution skewed toward values consistent with Rayleigh scattering. This supports the submicron particle size ranges found by Wong et al. (2020) from retrievals implementing Mie particle scattering.

3.3.3. HAT-P-32b

HAT-P-32b is the only planet in our sample that is consistent between all three wavelength cutoffs. Small differences in the H₂O abundance and temperature are seen between ${\lambda }_{\min }$ = 1.1 μm and 0.6 μm, resulting from the nondetection of K and Na. However, the scattering slope measured in the G141 bandpass is maintained throughout the optical wavelengths with no significant change in gradient, unlike that seen in WASP-31b or WASP-39b where there is a clear turning point in the gradient of the slope in the optical wavelengths. This means the water feature alone can constrain $\mathrm{log}a$ and γ to within 1.1 dex.

3.3.4. HD 189733b

HD 189733b shows the highest retrieved value of the Rayleigh enhancement factor, $\mathrm{log}a$, with this parameter constrained even by the ${\lambda }_{\min }$ = 0.6 μm retrieval. This is due to the turning point of the prominent scattering slope extending longwards of 0.6 μm. This scattering has been previously attributed to stellar activity (McCullough et al. 2014); however, our retrievals fitting for stellar inhomogeneities were not statistically favored over the base retrieval performed.

For most of the planets, our retrievals are unable to constrain aerosol parameters with ${\lambda }_{\min }$ = 0.6 μm, since the data from 0.6 and 1.1 μm typically contain a large scatter with a wide range of possible aerosol parameters that could fit the spectrum. For example, HD 209458b and WASP-6b also contain a moderate scattering slope that is only constrained with the ${\lambda }_{\min }$ = 0.3 μm spectrum; however, our retrievals cannot disentangle the cloud parameters at ${\lambda }_{\min }$ = 0.6 μm due to the presence of strong Na absorption.

3.3.5. HD 209458b

HD 209458b is discussed in detail in Section 2.2. Emergent correlations between cloud parameters are seen between $\mathrm{log}a$ and $\mathrm{log}{P}_{\mathrm{cloud}}$ for ${\lambda }_{\min }$ = 0.3 μm retrievals, and the mixing ratios of H₂O, Na, and K show correlations in both the ${\lambda }_{\min }$ = 0.3 and 0.6 μm retrievals.

3.3.6. WASP-6b

WASP-6b has significant detections of Na, K, and H₂O. The Na line wing is detected in ${\lambda }_{\min }$ = 0.6 μm but is not constrained until ${\lambda }_{\min }$ = 0.3 μm. The significant scattering slope, $\mathrm{log}a\,\sim \,6$ and γ = −10 is well defined but unconstrained until the full data set is used. The inclusion of HST/WFC3 data also allows us to place a near-solar constraint on the water abundance, as also shown by Carter et al. (2020). Correlations between cloud parameters arise at wavelengths below 0.6 μm, most notably $\mathrm{log}a$–γ and T–γ. The H₂O–Na–K correlation present for HD 209458b can also be seen in the ${\lambda }_{\min }$ = 0.3 μm posterior distribution for WASP-6b.

3.3.7. WASP-12b

The HST data for WASP-12b from Sing et al. (2016) only permits loose constraints on cloud properties. With only six data points shortward of 0.6 μm, no inferences can be made about the cloud fraction across the terminator, and only the lowest cloud deck pressures can be ruled out.

3.3.8. WASP-17b

WASP-17b has poorly constrained aerosol parameters due to the low-precision STIS data (caused by a partial transit, as detailed by Alderson et al. 2022). For all wavelength ranges cloud constraints do not improve, contrary to the general trend across the population. However, the inclusion of wavelengths below 0.6 μm improves constraints on the R_p,ref–T degeneracy. WASP-17b is the only planet where this degeneracy is largely broken, as seen in the reference pressure posterior distribution.

However, the short-wavelength data do improve the chemical abundance constraints for WASP-17b. Most notably, correlations between the H₂O, K, and CO₂ abundances collapse with the addition of < 0.6 μm data. Our infrared-only retrieval (${\lambda }_{\min }$ =1.1 μm) finds a high-abundance mode for Na and K (see Figure 1), causing the median retrieved spectrum to show significantly greater transit depths around the Na and K features than for the ${\lambda }_{\min }\,=$ 0.6 and 0.3 μm retrievals. This high-abundance mode corresponds to a set of solutions that fit the data with a high mean molecular weight and high-temperature atmosphere. However, this unphysical mode is lost for WASP-17b with the additional information from the alkali wings at shorter wavelengths. This suggests that the near-infrared HST WASP-17b data can inaccurately and overconfidently predict alkali abundances.

3.3.9. WASP-19b

Our WASP-19b retrieval is limited due to the small number of data points in this planet's spectrum; however, our results broadly agree with previous studies with retrieved parameters consistent with those in Pinhas et al. (2019), despite their study using a more complex 19-parameter retrieval. We note that some retrieval studies have used different data for WASP-19b. In particular, Welbanks et al. (2019) use VLT/FORS2 data from Sedaghati et al. (2017) for their analysis, leading them to find constraints on Na and TiO. We choose not to include the Sedaghati et al. (2017) data in our analysis, as they are reduced with a significantly different analysis method to that of the Sing et al. (2016) data set. Including the VLT/FORS2 data would require additional free offset parameters in our retrievals, as well as the consideration of wavelength-correlated data, which would run counter to our uniform analysis approach. However, we did test retrievals with and without stellar activity for WASP-19b (see, e.g., Espinoza et al. 2019; Sedaghati et al. 2021) and found that stellar heterogeneity is not statistically favored by the HST data.

3.4. WASP-31b

WASP-31b provides a clear demonstration of optical data resolving ambiguous aerosol properties. With only near-infrared data (${\lambda }_{\min }$ = 1.1 μm), there is a bimodal solution in WASP-31b's optical slope between Rayleigh and super-Rayleigh scattering solutions, with the most likely mode being a Rayleigh scattering slope. The addition of data down to 0.6 μm reduces the likelihood of this mode and leads to a flatter, more unconstrained posterior distribution. However, once the data down to 0.3 μm are included, the super-Rayleigh region of parameter space is favored.

We note that other chemical species, beyond those considered here, may also be present in WASP-31b's atmosphere. MacDonald & Madhusudhan (2017a) reported tentative evidence of NH₃ in WASP-31b's WFC3 data, while Braam et al. (2021) proposed the presence of CrH given the STIS observations. Recently, Flagg et al. (2023) confirmed CrH via high-resolution Doppler spectroscopy. Further, our 3.2σ detection of K in WASP-31b is driven by a single data point at 0.77 μm, which may be due to instrument systematics. The validity of this detection has been debated, since ground-based observations have been unable to reproduce the detection (e.g., Gibson et al. 2017, 2019; McGruder et al. 2020)

3.4.1. WASP-39b

WASP-39b is discussed in detail in Section 2.2.

3.4.2. WASP-121b

WASP-121b is one of our high-temperature case studies, for which TiO and VO opacity must be included in the retrieval analysis. The multiple strong peaks between 0.4 and 1.0 μm for the ${\lambda }_{\min }$ = 0.6 and 1.1 μm retrievals indicate prominent contributions from TiO and VO opacity. Without optical data below 0.6 μm, TiO and VO, alongside the aerosol parameters, are unconstrained. Additionally, WASP-121b shows emergent correlations between mixing ratios of VO and H₂O with $\mathrm{log}a$ when observations below 0.6 μm are included. For all wavelength ranges, a strong correlation between VO and H₂O is present, and constraints do not improve with the addition of shorter wavelengths.

Our retrievals find a relatively steep slope is needed to fit the optical spectrum; however, this may result from our model setup. Such an optical slope may be mimicking opacity from other high-temperature species beyond TiO and VO that we do not include in our model. We note that both Evans et al. (2016) and Evans et al. (2018), from which we obtain our spectra, attribute the large transit depth shortwards of 0.4 μm to other opacity sources, without the need for a scattering slope. Therefore, absorbers such as HS and F_e H may provide an alternative explanation for WASP-121b's high transit depth at short optical wavelengths.

3.4.3. WASP-127b

WASP-127b is one of a handful of planets where the temperature constraints worsen with increased optical wavelength coverage (from spanning 184 K for the ${\lambda }_{\min }$ =0.3 μm spectrum to 333 K for the ${\lambda }_{\min }$ = 0.3 μm spectrum). For the ${\lambda }_{\min }$ = 0.3 μm retrieval, a region of higher-temperature parameter space, which is disfavored by the ${\lambda }_{\min }$ = 0.6 and 1.1 μm retrievals, occupies a portion of the probability distribution. This high-temperature region only arises for sub-Rayleigh scattering values of $\mathrm{log}a$ and γ, which are most prominent for the 0.3 μm retrieval and completely absent from the 1.1 μm retrieval. This example demonstrated that shallow slopes in the optical data can alter other retrieval atmospheric properties from the values inferred from the infrared alone.

3.4.4. HAT-P-11b

HAT-P-11b is an outlier within our planet sample, occupying the Neptune regime of the mass–radius space. It orbits a known active K star with a measured spot covering a fraction of ${3}_{-1}^{+6} \%$ (Morris et al. 2017). Despite correcting for spot coverage in the data reduction (Chachan et al. 2019), HAT-P-11b is the only planet in our sample where retrievals support the inclusion of stellar contamination. Therefore, we can use HAT-P-11b to provide a test on the wavelength dependence of retrieved parameters for a stellar contaminated transmission spectrum.

When only considering near-infrared data, we find spurious evidence of CH₄ and CO₂ in HAT-P-11b's atmosphere. As seen in Figure 5, the mixing ratios of CH₄ and CO₂ are bounded and constrained to high values for ${\lambda }_{\min }=1.1$ μm. With an equilibrium temperature of 840 K, HAT-P-11b lies on the boundary between a CO- and CH₄-dominated atmosphere, depending on the metallicity of the atmosphere (Moses et al. 2013), so the presence of CH₄ and CO₂ within the atmosphere is physically plausible. However, once the short-wavelength data are included these detections disappear.

Similarly, the retrieved H₂O abundance is significantly anomalous for the ${\lambda }_{\min }=1.1$ μm retrieval. Without data between 0.3 μm and 1.1 μm, the stellar heterogeneity temperature is retrieved approximately 1200 K below the effective temperature of the star (T_eff = 4780 K), corresponding to unocculted starspots. The resulting low H₂O abundance can therefore be attributed to starspots mimicking the atmospheric H₂O absorption features, which can be seen by the increase in transit depth at optical wavelengths in the ${\lambda }_{\min }\,=1.1$ μm retrieval (see Figure 6).

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However, once shorter-wavelength optical data are included (${\lambda }_{\min }$ = 0.6 and 0.3 μm), the retrieved spectra for HAT-P-11b support solutions with stellar heterogeneities hotter than the surrounding photosphere (i.e., unocculted faculae). This produces a strong negative slope in the transmission spectrum, which then no longer biases the H₂O abundance. Similarly to retrievals without stellar contamination, wavelengths below 0.6 μm are necessary for cloud parameter constraints.

Although the inclusion of stellar contamination improves the model fit for HAT-P-11b, the retrieved model still provides a poor fit to the data (${\chi }_{\nu }^{2}=2.42$) and may not accurately encompass the underlying physics of the planet. A potential drawback of the setup is the limitations of the stellar contamination model used here, which only considers a single stellar heterogeneity population with a single temperature. For active stars, this model may struggle to account for the realistic distribution of heterogeneities due to the inability to model both starspots and faculae together.

While we can show the importance of the inclusion of optical wavelengths in constraining atmospheric parameters through their retrieved median values and errors, we can go further to quantitatively estimate the gain/loss in information with increased spectral range at short wavelengths. As this study focuses on observational data, there is a complex relationship between the underlying composition of the planet's atmosphere, observational uncertainty (through the signal strength, transit depth errors, and binning), model parameterization, and the information gained through retrievals. This makes quantitative conclusions across the population difficult to disentangle. However, on a planet-by-planet basis, we can quantify the information gained through entropy estimation.

IC analysis has been applied to inverse problems within exoplanet atmospheric characterization in several previous studies (e.g., Benneke & Seager 2012; Batalha & Line 2017; Howe et al. 2017). IC analyses can quantify how the knowledge of an atmospheric state changes relative to the prior, following an observation. This knowledge change can be computed from the difference in entropy of the prior and posterior distributions of a retrieval.

The mutual information, or IC of a retrieval is defined as the change in entropy between the prior and the posterior distribution. In this case, the entropy of a distribution is the average information required to encode a parameter value from its prior. Therefore, IC describes the improvement in the confidence of the retrieved atmospheric parameters, given a set of observations, from the initial prior knowledge (Line et al. 2012). We define the IC (I(θ, X)) of a retrieval, where Θ is the set of all parameters θ and X is the set of observations x as

$$\begin{eqnarray}&&I({\rm{\Theta }},X)=H({\rm{\Theta }})-H({\rm{\Theta }}| X),\end{eqnarray} \tag{ 1 }$$

where H(Θ) is the entropy of the prior distribution p(Θ) and H(Θ∣X) is the entropy of the posterior distribution p(Θ∣X).

For a discrete random variable X, entropy is defined as

$$\begin{eqnarray}&&H(X)=-\displaystyle \sum _{x\in X}p(x)\mathrm{log}p(x),\end{eqnarray} \tag{ 2 }$$

which is the expectation of the self-information $h(x)=-\mathrm{log}p(x)$. Extending the entropy to a continuous distribution function F of a random variable X, with an associated probability density function f(x), gives the following definition

$$\begin{eqnarray}&&H(f)=-{\int }_{-\infty }^{\infty }f(x)\mathrm{log}f(x){dx}.\end{eqnarray} \tag{ 3 }$$

The output of our retrievals provides posterior samples of some unknown theoretical distribution function. As such, we use the package scipy.stats.differential _entropy (Alizadeh Noughabi 2015) to implement the Vasicek method as an entropy estimator of our posterior samples. Vasicek (1976) expresses Equation (3) as

$$\begin{eqnarray}&&H(f)={\int }_{0}^{1}\mathrm{log}\left\{\displaystyle \frac{d}{{dp}}{F}^{-1}(p)\right\}{dp},\end{eqnarray} \tag{ 4 }$$

which is then reformulated by replacing F with the empirical distribution function F_n . The derivative of F_n can be estimated from the ordered samples from the probability distribution x_n , where the window size, m, must be a positive integer and m < n/2. This reformulation, HV_mn , is Vasikeck's entropy estimator, defined as

$$\begin{eqnarray}&&{{HV}}_{{mn}}=\displaystyle \frac{1}{n}\displaystyle \sum _{i=1}^{n}\mathrm{log}\left\{\displaystyle \frac{n}{2m}({X}_{i+m}-{X}_{i-m})\right\}.\end{eqnarray} \tag{ 5 }$$

The estimator is consistent, such that HV_mn → H(f) as n → ∞ , m → ∞ , and m/n → 0.

We estimate the IC between the prior and posterior distributions for each model parameter by drawing random samples from the prior distributions and using the output samples from the marginalized posterior distribution. To these samples, we apply Vasicek's entropy estimator, where we measure IC in nats. For an event with probability 1/e, the IC in nats is one. We then sum over the mutual entropy for all parameters to find the total IC gained from the retrieval for a given set of observations. We also assess the IC between the posteriors of retrievals with different spectral ranges.

4.1. Information Content Analysis Results

Figure 7 displays the change in information between the prior and posterior distribution from our retrievals. We show the IC change as a function of the number of data points (left panel) and with the different optical wavelength cutoffs (right panel) for each planet. As a trend, the IC increases with greater wavelength and data point coverage. Most planets have a greater increase in IC per wavelength and per data point between ${\lambda }_{\min }$ = 0.6 μm and 0.3 μm than between the ${\lambda }_{\min }$ = 1.1 μm to 0.6 μm retrievals, with some showing a marked increase in IC with only a few additional data points in this range. Exceptions to this trend are WASP-31b and WASP-127b (and also HAT-P-32b, but only for the data point difference, not the spectral range). The average increase in information between 1.1 and 0.6 μm is 1.39 nats and the average increase in information between 0.6 and 0.3 μm is 2.11 nats for the sample of exoplanets.

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We find that the greatest average information gain across the planet population is found between 0.6 μm and 0.3 μm. Therefore, we explore the contribution of each parameter to the IC in this wavelength range by calculating the change in estimated entropy between the posterior distributions of the ${\lambda }_{\min }$ = 0.6 μm and 0.3 μm retrievals. In this case, it is possible for the change in entropy to be negative, as it is calculated between two final measurements and not a prior and posterior distribution. Figure 8 shows the breakdown of the change in IC for each portion of the model between the ${\lambda }_{\min }$ = 0.6 μm and 0.3 μm retrievals. We first show the contribution from the clouds across all four parameters and then show the breakdown in IC for each cloud parameterization used in the model.

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We see that the change in IC does not show the same ordering as the median retrieved Rayleigh enhancement factor ($\mathrm{log}a$). It can be seen that cloud parameters dominate the IC gain between 0.6 and 0.3 μm. If we further break down the cloud parameters, the greatest contribution to information gain does come from $\mathrm{log}a$, followed by γ. It is unsurprising that the scattering parameters provide the greatest information gain, as this supports the finding in Section 3.2 that optical wavelengths provide the strongest constraints for scattering parameters.

Significant contributions to information between 0.6 and 0.3 μm also come from the alkali species. In particular, we find the greatest gains in information for those planets with the highest detection significances of Na (see Table 4). For planets where there is no detection of alkali species (e.g., WASP-17b and HAT-P-32b), there is still a contribution to IC due to retrievals ruling out high abundances of these species. Interestingly, as the wavelength range of retrievals increases, for some planets, we lose information on R_P,ref, temperature, and carbon species. These changes are likely related to the increased wavelength probing a wider pressure range.

WASP-17b does not follow the trend of cloud parameters dominating the increase in information with data over an extended optical wavelength range. Instead, the increase in information is driven by improved constraints on H₂O, alkali species, and the reference pressure. The lack of cloud information reiterates the findings of Section 3.2, where limited constraints on the cloud parameters are found, due to the large uncertainties of the STIS data.

HAT-P-11b also acts as an outlier. It is the only planet where the information on cloud parameters decreases with the addition of wavelengths below 0.6 μm. While the IC between 0.6 and 0.3 μm is negligible, between 1.1 and 0.3 μm (Figure 9) a large decrease in information is seen where the stellar parameters no longer converge on an unocculted cold spot solution. The stellar contamination information is driven by optical data below 1.1 μm. However, due to the ability of stellar contamination to mimic atmospheric molecular features, stellar contamination will directly impact the retrieved abundances of species such as H₂O, which itself impacts the temperature of the atmosphere. As such, HAT-P-11b shows a large change in IC across many parameters.

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Finally, Figure 9 displays the same IC breakdown by parameter as Figure 8, but between the ${\lambda }_{\min }$ = 1.1 μm and 0.3 μm retrievals. This yields the same result in IC as between 0.6 to 0.3 μm but increases the impact of alkali species as the contribution of K opacity between 0.6 and 1.1 μm is added. Of the planet's population, HD 189733b has the greatest increase in IC between 1.1 and 0.3 μm but not 0.6 to 0.3 μm, which can be explained by the scattering slope extending to wavelengths greater than 0.6 μm. The scattering parameters $\mathrm{log}a$ and γ are well constrained for the ${\lambda }_{\min }$ = 0.6 μm retrieval such that less additional information is gained with the inclusion of wavelengths below 0.6 μm.

Cloud scattering slopes are generally attributed to small aerosol particles in the upper atmosphere causing Rayleigh-like (γ = −4) to super-Rayleigh profiles in optical spectra. On average we retrieve scattering slopes 2.2× Rayleigh for ${\lambda }_{\min }$ = 0.3 μm across all 14 planets in this study. Enhanced Rayleigh scattering can be attributed to high eddy diffusion coefficients and moderate aerosol mass fractions (Ohno & Kawashima 2020). In Figure 10, we show the profiles for soot and tholin as a proxy for photochemical hazes, along with our retrieved γ values against T_eq. The relationship between the scattering slope and T_eq assumes a drag-free atmosphere at a pressure of 1 mbar for particle tracers of 0.01 μm based on simulations using different eddy diffusion coefficients (Komacek et al. 2019). We find that the measured scattering cannot be explained by a single aerosol model, suggesting that multiple mechanisms (e.g., mineral clouds, Grant et al. 2023) may also cause enhanced Rayleigh scattering in exoplanet atmospheres. Further measurements, including extended UV or infrared data, may help understand the cause of enhanced scattering in giant-exoplanet atmospheres.

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For many planets, clouds are not expected to be present across the entire surface due to changing temperature structures from day to night. To account for this, patchy clouds can be invoked to describe the fractional coverage of clouds around the transmitting limb. Our retrievals and IC analysis demonstrate that patchy clouds should be considered alongside other cloud parameters when assessing exoplanet transmission spectra. We investigate the relationship between patchy clouds, temperature, and water abundance shown in Figure 11. For planets with water abundances below solar values, a trend of increasing cloud coverage with water abundance could be inferred. However, there is only weak evidence to support this from the uncertainties associated with both parameters. The planets in our sample with the highest water abundance values (WASP-39b, HAT-P-1b, and HAT-P-32b) tend toward 50:50 cloud coverage, suggesting the potential for morning/evening asymmetries.

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We find no apparent trend in changing cloud fraction with equilibrium temperature across the whole range sampled by our 14 planets. We do however note a slight increase in cloud fraction with increasing temperate up to 1600 K, beyond this uncertainties are too large to determine if there is a turnover or the trend continues to higher temperatures. More complex relations that account for a wider range of parameters and microphysics may be necessary to account for any nonlinear trends; for example, considering the effect of temperature on the condensation of different species (Gao et al. 2020). To achieve this, tighter cloud constraints are needed. While observationally, higher precision measurements may improve constraints, extending observations to combine transmission and emission spectra (or phase curves) can provide vital spatial information on temperature. We also note that extending population investigations to both larger sample sizes and a wider parameter space can help disentangle trends.

Through their interaction with internal and external radiation, clouds impact the overall energy budget of a planet and therefore the temperature structure of the atmosphere. The formation of clouds along the limbs can be both an indicator and driver of the 3D temperature structure of exoplanet atmospheres. However, we often compare atmospheric thermal profiles to general circulation model (GCM) predictions, which simulate the radiative–convective structure of a planetary atmosphere. We evaluate the retrieved temperatures with respect to a population of GCMs presented by Kataria et al. (2016). Figure 12 presents our retrieved limb temperatures against T_eq alongside average GCM limb temperatures, evaluated across between 10⁻² and 10⁻⁴ bar (representing an estimated range of the observable photosphere) for 11 of the 14 planets.

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Across the retrieved limb temperatures the general trend follows that T_limb < T_eq. Although we include patchy clouds that have been shown to reduce biases toward cool retrieved temperatures that arise from modeling atmospheres with differing compositions between morning and evening terminators MacDonald et al. (2020), we still find low limb temperatures compared to T_eq for the ultrahot Jupiters WASP-12b and WASP-121b. Our retrieved temperatures reproduce the extreme difference in T_eq − T_limb seen in literature (Kreidberg et al. 2015; Evans et al. 2018; Pinhas et al. 2019; Welbanks et al. 2019) for ultrahot Jupiters where T_eq - T_limb ∼ 1000 K. This is likely demonstrating where for extreme temperatures, the advective timescale is too large to efficiently transport the vast amount of heat incident on the dayside to the limbs with a majority re-radiated more efficiently from the atmosphere (e.g., Fortney et al. 2008; Komacek et al. 2017; Showman 2021).

However, when comparing the retrieved limb temperatures of the population to GCM temperatures, 7 of the 11 planets agree with the GCM temperatures to within 1σ. Fitting a linear relation to T_limb–T_eq and T_GCM–T_eq gives gradients of 0.36 ± 0.13 and 0.57 ± 0.03, respectively. The differences in the retrieved and GCM limb temperatures reflect the different aspects of physics they capture. The GCM temperatures are derived from a solar-metallicity, cloud-free atmosphere, where the divergence from T_eq is due to the effects of longitudinal heat transport. In contrast, encoded within our retrieved limb temperatures is the impact of cloud opacity. These two distinct mechanisms have produced similar results within our margin of uncertainty, potentially demonstrating the role of clouds in the transport of heat around a planet. Future comparison between modeled and observed limb temperatures may shed more light onto the underlying physical mechanisms driving heat transport in hot-Jupiter planets.

Through free-chemistry retrievals with POSEIDON, we set out to explore how retrieved atmospheric parameters inferred from exoplanet transmission spectra improve with the addition of optical data. Initially, the wavelength dependence of retrieved parameters is explored in detail for WASP-39b and HD 209458b (Section 2.2), to select three spectra ranges to apply to a larger population of planets. We implement retrievals for minimum wavelength ranges, ${\lambda }_{\min }$ = 0.3, 0.6, and, 1.1 μm for a sample of 14 exoplanets with transit spectra from 0.3–4.5 μm. We evaluate the retrieved parameters across the population considering their median values and 1σ uncertainty for each ${\lambda }_{\min }$ data range to determine the impact of expanding wavelength range on atmospheric properties. To quantify how our knowledge of the atmosphere changes with wavelength range, we implement an IC analysis on the posterior distributions of our retrievals and break down the IC per parameter (Section 4). As this analysis covers a population of exoplanets, in Section 3 we search for trends between atmospheric and planetary parameters. We compare our retrieved limb temperatures to temperatures derived from GCMs and investigate the relationship between cloud parameters, water abundance, and temperature. More data are required to draw conclusions about population trends.

From the spectral range investigation, we find that wavelengths below 0.6 μm are necessary to constrain alkali species and cloud scattering parameters $\mathrm{log}a$ and γ, although the scattering slope γ is not consistently constrained across the population. This is supported by the IC analysis, where the largest information gains between the ${\lambda }_{\min }$ = 1.1, 0.6 and 0.3 μm cutoffs are from the cloud parameters. We find a limited impact of the wider optical wavelength coverage on the remaining parameters and abundances. In particular, we note that in general, constraints and median values improve, but not significantly, for cloud pressure level where in most cases, the WFC3/G141 and Spitzer data below 1.1 μm provide comparable constraints and retrieved median values to the full 0.3–4.5 μm spectrum.

The impact of stellar activity is of particular interest when looking at M-star planets (e.g., Moran et al. 2023). We show that JWST may not be able to resolve this issue on its own. HAT-P-11b was an outlier in our planet's population due to its location in mass–radius space and that it was the only planet to favor retrievals that account for stellar contamination. For this reason, it provides an important test case for the impact of stellar contamination on retrieved parameters. Without the optical wavelengths, our retrievals do not converge on the same solutions for the stellar parameters as when optical wavelengths are included. With a different stellar spectrum, this leads to vastly different inferences being drawn about the cloud properties and species abundances (particularly water) in the atmosphere.

Across our population we find a wide range of values for the measured water abundance, in contrast to previous studies which favor low, subsolar abundances, but find no trends with retrieved cloud coverage fractions. We additionally investigate trends of cloud coverage with T_eq, finding a tentative suggestion that cloud fraction increases with T_eq up to 1600 K, above which cloud fraction uncertainties become too large to discern significant trends. We show that our retrievals that consider patchy clouds get comparable temperatures to those predicted by GCMs related to T_eq.

Overall this study demonstrates that optical wavelengths below the reach of JWST are vital to evaluate the cloud properties of exoplanet atmospheres. Studies such as Grant et al. (2023) have already demonstrated the utility of optical wavelengths on constraining cloud location and particle size when constraining the composition of clouds via infrared absorption signatures. This supports our analysis, which shows that the scattering properties, $\mathrm{log}a$ and γ, are most affected by the inclusion of sub 0.6 μm data with improvements in their constraint of over 30%.

The authors thank the anonymous referee for the suggestions and comments. We thank L. Alderson for help with defining the idea behind the investigation while writing HST proposals and wishing something like this existed, N.E. Batalha for helpful comments and discussion around the IC content analysis, and T. Kataria for providing the GCM pressure–temperature profiles presented in Kataria et al. (2016). We also thank A. Young and J. Barstow for evaluating the MSc by research thesis in which this work was first presented.

C.F. is funded by the University of Bristol School of Physics PhD Scholarship Fund. H.R.W. was funded by UK Research and Innovation (UKRI) under the UK government's Horizon Europe funding guarantee for an ERC Starter Grant [grant No. EP/Y006313/1]. R.J.M. is supported by NASA through the NASA Hubble Fellowship grant HST-HF2-51513.001, awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.

Facilities: HST (STIS) - , HST (WFC3) - , Spitzer (IRAC) - , VLT FORS2 - , HST (NICMOS) - .

Software: numpy (Harris et al. 2020), scipy (Virtanen et al. 2020), matplotlib (Hunter 2007), POSEIDON (MacDonald & Madhusudhan 2017b; MacDonald 2023), pymultinest (Buchner et al. 2014).

Supplementary corner plots, spectra, and retrieved parameters are available on Zenodo: 10.5281/zenodo.10407463.

We include stellar and planetary parameters used within the analysis in Table 5.

Table 5. 

Planet	R* (R⊙)	Teff (K)	[Fe/H]	$\mathrm{log}g\ (\mathrm{cgs})$	Rp (RJ )	Mp (MJ )	Teq (K)
HAT-P-1b (a)	1.17	5980	0.13	4.36	1.32	0.53	1320
HAT-P-11b (b)	0.70	4780	0.31	4.66	0.40	0.08	840
HAT-P-12b (c)	0.68	4670	−0.20	4.61	0.92	0.20	960
HAT-P-32b (d)	1.37	6000	−0.16	4.22	1.98	0.68	1840
HD 189733b	0.77 (e)	5010 (f)	0.01 (f)	4.49 (f)	1.12 (e)	1.16 (e)	1210 (e)
HD 209458b	1.16 (g)	6040 (f)	0.05 (f)	4.30 (f)	1.38 (g)	0.17 (g)	1460 (g)
WASP-6b (h)	0.86	5380	−0.15	4.49	1.23	0.49	1180
WASP-12b	1.66 (i)	6360 (i)	0.33 (i)	4.16 (i)	1.94 (j)	1.47 (j)	2590 (j)
WASP-17b	1.57 (k)	6490 (f)	0.00 (f)	4.08 (f)	1.99 (k)	0.49 (k)	1770 (k)
WASP-19b	1.01 (l)	5620 (l)	0.15 (m)	4.42 (l)	1.42 (l)	1.15 (l)	2110 (l)
WASP-31b	1.25 (n)	6300 (n)	−0.20 (o)	4.76 (p)	1.55 (n)	0.48 (n)	1580 (q)
WASP-39b	0.94 (c)	5460 (f)	−0.01 (f)	4.41 (f)	1.28 (c)	0.28 (c)	1170 (c)
WASP-121b	1.46 (r)	6340 (f)	0.24 (f)	4.17 (f)	1.75 (s)	1.16 (s)	2360 (r)
WASP-127b	1.33 (t)	5850 (f)	−0.16 (f)	4.22 (f)	1.31 (t)	0.16 (t)	1400 (u)

Note. Stellar and Planetary Parameters for the Sample of 14 Exoplanets used in this Investigation, where values have been obtained from the following studies: (a) Nikolov et al. (2014); (b) Southworth (2011); (c) Mancini et al. (2018); (d) Wang et al. (2019); (e) Addison et al. (2019); (f) Polanski et al. (2022); (g) Southworth (2010); (h) Tregloan-Reed et al. (2015); (i) Collins et al. (2017); (j) Chakrabarty & Sengupta (2019); (k) Anderson et al. (2011a); (l) Cortés-Zuleta et al. (2020); (m) Knutson et al. (2014); (n) Anderson et al. (2011b); (o) Bonomo et al. (2017); (p) Mortier et al. (2013); (q) Sing et al. (2016); (r) Delrez et al. (2016); (s) Bourrier et al. (2020); (t) Seidel et al. (2020); (u) Lam et al. (2017).

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