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

Charlotte Fairman HH Wills Physics Laboratory, University of Bristol, Tyndall
Avenue, Bristol, BS8 1TL, UK  Hannah R. Wakeford HH Wills Physics Laboratory,
University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK  Ryan J. MacDonald
NASA Sagan Fellow  Department of Astronomy, University of Michigan, 1085 S.
University Ave., Ann Arbor, MI 48109, USA

###### Abstract

Exoplanet transmission spectra provide rich information about the chemical
composition, clouds and temperature structure of exoplanet atmospheres. Most
exoplanet transmission spectra only span infrared wavelengths ($\gtrsim$ 1 m),
which can preclude crucial atmospheric information from shorter wavelengths.
Here, we explore how retrieved atmospheric parameters from exoplanet
transmission spectra change with the addition of optical data. From a sample
of 14 giant planets with transit spectra from 0.3–4.5 m, primarily from the
Hubble and Spitzer space telescopes, we apply a free chemistry retrieval to
planetary spectra for wavelength ranges of 0.3–4.5 m, 0.6–4.5 m, and 1.1–4.5
m. We analyze the posterior distributions of these retrievals and perform an
information content analysis, finding wavelengths below 0.6 m are necessary to
constrain cloud scattering slope parameters ($\log{a}$ and $\gamma$) and
alkali species Na and K. There is limited improvement in the constraints on
the remaining atmospheric parameters. Across the population, we find limb
temperatures are retrieved colder than planetary equilibrium temperatures but
have an overall good agreement with Global Circulation Models. As JWST extends
to a minimum wavelength of 0.6 m, we demonstrate that exploration into
complementing JWST observations with optical HST data is important to further
our understanding of aerosol properties and alkali abundances in exoplanet
atmospheres.

††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, 2017;
MacDonald, 2023), pymultinest (Buchner et al., 2014).

##  1 Introduction

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 & Sasselov, 2000; Brown, 2001). Low
resolution spectra can be obtained from transit light curves of telescopes
such as Hubble (HST), Spitzer and, JWST. A majority of early data have been
collected from the optical (0.3–1 m) to near-infrared ($>$1 m) 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 m (e.g., Knutson et al., 2011). JWST now fills the gap
between 0.6–5 m, providing spectroscopic measurements across the whole near-
infrared (e.g., Alderson et al., 2023; Ahrer et al., 2023; Feinstein et al.,
2023; Rustamkulov et al., 2023; Xue et al., 2023; Fournier-Tondreau et al.,
2023), and access to the mid-infrared beyond 5 m with the MIRI instrument
(e.g., Grant et al., 2023; Dyrek 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 $\lambda^{-4}$ 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 towards 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 $\Delta Z_{UB-LM}$ and $\Delta
Z_{J-LM}$ (measuring the strength of scattering to molecular absorption and
the relative strength between the mid-infrared continuum and mid-infrared
molecular absorption respectively) and measure the strength of the
characteristic $1.4\mu m$ 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
1.1–1.7m covering a single H2O band also showed that gray clouds are favored
over non-gray 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 1.1–1.7m wavelength range is insufficient to overcome
degeneracies between modelling choices.

A number of studies looking at individual planetary datasets have demonstrated
that the omission of observational measurements in the optical can result in
major differences on 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 set-ups 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
m. This is especially pertinent as JWST only goes down to a minimum of 0.6 m
and there is not yet a clear assessment of how crucial this short wavelength
information may be on inferred atmospheric properties below this cut-off. We
describe our retrieval set-up 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 datasets and planet parameters in Appendix References. We
calculate the information content 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.

Table 1: Transmission spectra sources for our 14 planet optical data
investigation.

Planet | Instruments | Reference
---|---|---
HAT-P-1 b | STIS G430/G750, WFC3 G141, Spitzer IRAC 3.6/4.5 | Sing et al. (2016)
HAT-P-11 b | STIS G430/G750, WFC3 G102/G141, Spitzer IRAC 3.6/4.5 | Chachan et al. (2019)
HAT-P-12 b | STIS G430/G750, WFC3 G141, Spitzer IRAC 3.6/4.5 | Sing et al. (2016)
HAT-P-32 b | STIS G430/G750, WFC3 G141, Spitzer IRAC 3.6/4.5 | Alam et al. (2020)
HD189733 b | STIS G430/G750, WFC3 G141, NICMOS photometry, Spitzer IRAC 3.6/4.5 | Sing et al. (2016)
HD209458 b | STIS G430/G750, WFC3 G141, Spitzer IRAC 3.6/4.5 | Sing et al. (2016)
WASP-6 b | VLT FORS2, STIS G430/G750, WFC3 G141, Spitzer IRAC 3.6/4.5 | Carter et al. (2020)
WASP-12 b | STIS G430/G750, Spitzer IRAC 3.6/4.5 | Sing et al. (2016)
| WFC3 G141 | Kreidberg et al. (2015)
WASP-17 b | STIS G430/G750, WFC3 G141, Spitzer IRAC 3.6/4.5 | Alderson et al. (2022)
WASP-19 b | STIS G430/G750, WFC3 G141, Spitzer IRAC 3.6/4.5 | Sing et al. (2016)
WASP-31 b | STIS G430/G750, WFC3 G141, Spitzer IRAC 3.6/4.5 | Sing et al. (2016)
WASP-39 b | VLT FORS2, STIS G430/G750, WFC3 G102/G141, Spitzer IRAC 3.6/4.5 | Wakeford et al. (2018)
WASP-121 b | STIS G430/G750 | Evans et al. (2018)
| WFC3 G141 | Evans et al. (2016)
WASP-127 b | STIS G430/G750, WFC3 G141, Spitzer IRAC 3.6/4.5 | Spake et al. (2021)

Note. — The planets considered in this investigation are chosen based on the
availability of at least one transit observation in either of the optical STIS
bandpasses (G430/G750) and one observation in either of the WFC3 bandpasses
(G102/G141).

##  2 Designing a robust and consistent retrieval set-up

To investigate the wavelength dependence of retrieved parameters from
exoplanet transmission spectra, we use the open-source retrieval code,
POSEIDON (MacDonald & Madhusudhan, 2017; 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 1-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 $0.3$ to $4.5$ m 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 set-up. We first test the different parameterisations available in
POSEIDON for the pressure-temperature (P–T) structure of the atmosphere. After
retrieving a profile consistent with an isothermal atmosphere using a
5-parameter P-T profile (Madhusudhan & Seager, 2009), we choose to adopt an
isothermal profile, where a uniform temperature prior is defined between
$(0.4-1.15)\ T_{\mathrm{eq}}$. This reduces the number of free parameters in
the fit for the P-T profile from 5 to 1. The reference radius at which the
atmosphere has a pressure of 10 bar ($R_{\mathrm{P,ref}}$) is also allowed to
vary as a free parameter.

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. HCN, NH3) are excluded from the base set
up 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 (2017). A cloud deck with
infinite opacity across all wavelengths is defined at a pressure level
$\log{P_{cloud}}$ and coupled with a wavelength dependent scattering haze. Two
parameters describe the behavior of the haze: the scattering slope $\gamma$,
and $\log{a}$, that acts as a scaling factor to the H2 Rayleigh scattering
cross Section defined at 350 nm. To model inhomogeneous cloud cover,
$\bar{\phi}_{clouds}$ parameterizes the terminator cloud fraction. We tested
models with and without $\bar{\phi}_{clouds}$ 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 parameterisation 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 $R$ =
5,000 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 $N$ = 4,000 live points to
effectively explore the prior phase space.

Table 2: Priors for our POSEIDON retrievals.

Parameter | Prior Distribution | Prior range
---|---|---
R${}_{p,\,\mathrm{ref}}$ | uniform | $(0.85-1.15)\ R_{p}$
Tp | uniform | $(0.4-1.15)\ T_{\mathrm{eq}}$
log H2O | uniform |  $-12$ – $-1$
log CO2 | uniform |  $-12$ – $-1$
log CO | uniform |  $-12$ – $-1$
log CH4 | uniform |  $-12$ – $-1$
log Na | uniform |  $-12$ – $-1$
log K | uniform |  $-12$ – $-1$
$\log{a}$ | uniform |  $-4$ – $8$
$\gamma$ | uniform |  $-20$ – $2$
$\log{P_{\mathrm{cloud}}}$ | uniform |  $-6$ – $3$ dex
$\phi_{\mathrm{clouds}}$ | uniform | $0-1$
High Temperature Species
log TiO | uniform |  $-12$ – $-1$
log VO | uniform |  $-12$ – $-1$
Stellar Parameters
$f_{\mathrm{het}}$ | uniform |  $0$ – $0.5$
$T_{\mathrm{het}}$ | uniform | $(0.6-1.2)\ T_{\mathrm{eff}}$
$T_{\mathrm{phot}}$ | Gaussian | $\mu=T_{\mathrm{eff}},\ \sigma=100\ K$

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
Teq$>$ 2000 K, while the stellar heterogeneity parameters are included for
known active stars.

###  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 cutoff short wavelength data points that fall below a
threshold $\lambda_{\mathrm{min}}$, in intervals of 0.1 m, from 0.3 to 1.1 m.
Due to the absence of WFC3 G102 data for HD 209458b, no retrieval is run for
$\lambda_{\mathrm{min}}$ = 1.0 m for this planet as there is no data change
from the surrounding cut-offs.

Figure 1 shows the median retrieved spectra for each cut-off 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 is included below 0.5 m. 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 datapoints preceding the sodium line. Wavelengths below 0.5 m are necessary
to fit the observed scattering slope.

Figure 1: Median retrieved spectra for the HST and Spitzer observations of HD
209458b (left) and WASP-39b (right) a lower wavelength data cutoff of
$\lambda_{\mathrm{min}}$. Shown are $\lambda_{\mathrm{min}}$ values from 0.3 m
(dark blue) to 1.1 m (dark red) in intervals of 0.1 m. The observed, multi-
instrument transmission spectrum is overlaid (black diamonds). The retrieved
spectrum is plotted at R = 100 for clarity.

Cloud opacity and alkali species are the main parameters driving the
differences between retrieved spectra. A comparison of the median retrieved
parameters and their 1$\sigma$ 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 m corresponding to spectra where the wavelength
range encompasses the line peak at $\sim$ 0.770 m. The characteristic Na
doublet is found at at $\sim$ 0.589 m. WASP-39b shows constraints for sodium
abundance below 0.5 m, however for HD 209458b, a minimum wavelength of 0.6 m
is sufficient to improve constraints on sodium due to the evidence of strong
line broadening around the Na I feature seen in the spectrum.

Figure 2: Retrieved median model parameters and $1\sigma$ errors for HD
209458b (top) and WASP-39b (bottom) from observations spanning a wavelength
range $\lambda_{\mathrm{min}}$ to 4.5 m. $\lambda_{\mathrm{min}}$ takes values
from 0.3 (dark blue) to 1.1 m (dark red) in intervals of 0.1 m.

Of the four cloud parameters, for HD 209458b the retrievals across all
wavelength ranges can converge on a solution for the cloud top pressure,
$\log{P_{cloud}}$, and patchy cloud, $\phi_{clouds}$, 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 m
wavelength range. In contrast, the scattering slope parameters $\log{a}$ and
$\gamma$ are only well constrained in the full 0.3 to 4.5 m retrieval. For
WASP-39b, constraints for $\log{P_{cloud}}$ and $\bar{\phi}_{clouds}$
consistently improve with increased short wavelength coverage. $\log{a}$ and
$\gamma$ require wavelengths shortward of 0.5 m for retrievals to constrain
the parameter space.

Whilst the H2O constraints become tighter when shorter wavelength information
is included in retrievals, the median H2O abundance of HD 209458b for the
$\lambda_{\mathrm{min}}$ = 0.3 m retrieval ($-4.46^{+0.46}_{-0.35}$ dex) only
marginally falls outside of the $1\sigma$ errors of the
$\lambda_{\mathrm{min}}$ = 1.1 m retrieval. The WFC3 G141 datapoints alone can
constrain the water abundance to $-3.59^{+0.85}_{-0.73}$ 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.

##  3 Population Analysis

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 m. We also
consider additional ground-based observations from 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 datasets here. Second, we select
planets with a range of Teqspanning 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 Appendix A,
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: $\lambda_{\mathrm{min}}$ = 0.3, 0.6 and 1.1 m. The 1.1 m 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 m and 1.4 m water
features. We select 0.6 m 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,
$\lambda_{\mathrm{min}}$ = 0.3 m, considers the full spectral range of the HST
STIS data.

###  3.1 Population Retrieval Configuration

We consider three different retrieval configurations for our population
analysis. First, we apply the same base retrieval configuration as in Section
2.1 to all planets with $T_{\mathrm{eq}}<$ 2,000 K. For those planets
exceeding 2,000 K (WASP-12b, WASP-19b, and WASP-121b), we additionally
considered a ‘high-temperature’ configuration including TiO and VO opacity.
Finally, for planets with _a priori_ known active host stars (HAT-P-11b, HD
189733b, WASP-6b, and WASP-19b) we additionally consider a retrieval model
accounting for a single population of unocculted stellar heterogeneities to
asses any evidence of stellar contamination on the transmission spectrum
(e.g., Rackham et al., 2017; Pinhas et al., 2018). For the two planets where
some datasets were not consistently reduced (WASP-121b and WASP-12b), we
initially fitted for a relative offset between these data for a preliminary
retrieval on the full $\lambda_{\mathrm{min}}$ = 0.3 m wavelength range, with
offsets applied to the combined HST WFC3 and Spitzer data. Our final retrieval
results adopt a fixed offset equal to the median value of the retrieved
offset. We note that due to low spectral resolution, resulting in a small
number of measured datapoints, WASP-12b is only analyzed for the 0.6 and 0.3 m
cutoffs and WASP-19b is limited to the full 0.3 m cut-off analysis, such that
the degrees of freedom in the model does not exceed the number of datapoints.

Table 3 summarizes which retrieval configurations were conducted for each
planetary dataset, with the final selected set-up highlighted. All but two of
our spectra could be fit by our base retrieval model, with HAT-P-11b requiring
additional stellar contamination parameters and WASP-121b favoring the
inclusion of high-temperature species. With our optimal retrieval
configurations established for each planet, we proceed to discuss the results
of our population analysis.

Table 3: Retrieval configuration applied to each planet.

Planet | Base | High Temperature | Stellar Contamination
---|---|---|---
HAT-P-1b | ✓$\star$ |  |
HAT-P-11b | ✓ |  | ✓$\star$
HAT-P-12b | ✓$\star$ |  |
HAT-P-32b | ✓$\star$ |  |
HD 189733b | ✓$\star$ |  | ✓
HD 209458b | ✓$\star$ |  |
WASP-6b | ✓$\star$ |  | ✓
WASP-12b | ✓ | ✓$\star$ |
WASP-17b | ✓$\star$ |  |
WASP-19b | ✓$\star$ | ✓$\dagger$ | ✓$\dagger$
WASP-31b | ✓$\star$ |  |
WASP-39b | ✓$\star$ |  |
WASP-121b | ✓ | ✓$\star$ |
WASP-127b | ✓$\star$ |  |

Note. — Ticks (✓) indicate which setups were tested for each planet, while the
stars ($\star$) mark the final adopt retrieval setup for the population
analysis (Section 3.2). For WASP-19b, daggers ($\dagger$) mark reduced
parameter setups, where the carbon species CH4, CO and CO2 are removed from
the retrieval. All planets are tested with the base setup.

###  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),
$\log{a}$, to the highest retrieved value (HD 189733b). The corresponding
retrieved atmospheric properties (posterior median and $1\sigma$ 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:
https://doi.org/10.5281/zenodo.10407463 (catalog 10.5281/zenodo.10407463).
Below, we discuss the effect of changing the spectral range on model
parameters.

Figure 3: Retrieved transmission spectra for the population. Retrieved median
spectra for each wavelength range, $\lambda_{\mathrm{min}}$ = 0.3 m,
$\lambda_{\mathrm{min}}$ = 0.6 m and $\lambda_{\mathrm{min}}$ = 1.1 m, are
displayed in purple, blue and orange respectively. The light and dark shaded
regions correspond to the $1$ and $2\sigma$ confidence regions. The datapoints
show the observed spectrum, where the marker colors correspond to data above
the minimum wavelength of the three retrievals, following the same color
scheme as the retrieved spectra. Planets are displayed in order of their
retrieved Rayleigh enhancement factor, $\log{a}$, with the planet with the
lowest value of $\log{a}$ at the top of the figure. The y-axis scale is
expressed in atmospheric scale heights, $H=k_{B}T/\mu g$, where $T$ is taken
as the retrieved temperature from the $\lambda_{\mathrm{min}}$ = 0.3 m
retrieval (+ an arbitrary offset for clarity). Figure 4: Retrieved median
transmission spectra for the population (continued). See Figure 3 for the
caption. Figure 5: Median retrieved parameters and $1\sigma$ errors across the
population. Only parameters from the base retrieval are displayed (HAT-P-11b,
WASP-12b and WASP-121b contain additional parameters in their retrievals). The
retrieval setup of each planet can be found in Table 3. Results for each
wavelength range, $\lambda_{\mathrm{min}}$ = 0.3 m, $\lambda_{\mathrm{min}}$ =
0.6 m and $\lambda_{\mathrm{min}}$ = 1.1 m, are displayed in purple, blue and
orange respectively. Planets are displayed in order of their retrieved
Rayleigh enhancement factor, $\log{a}$, with the planet with the lowest value
of $\log{a}$ at the top.

####  3.2.1 Aerosol Properties

Of the four cloud parameters, $\log{a}$ is the best constrained when using the
full wavelength range, $\lambda_{\mathrm{min}}$ = 0.3 m. We find the average
1$\sigma$ error range for $\log{a}$ decreases from 6.54 to 5.66 to 3.05 for
the 1.1, 0.6 and 0.3 m cutoffs, respectively, corresponding to a precision
improvement of $46\%$ when adding data from 0.6 to 0.3 m. The scattering
slope, $\gamma$, 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 m likely due to large
uncertainties at these wavelengths. For $\gamma$, the error range between the
1.1 and 0.6 m cutoffs show limited improvement, decreasing from 11.7 to 10.5.
However, with the addition of 0.3–0.6 m data, the 1$\sigma$ error range
decreases by 30$\%$ to 7.2. Whilst a significant proportional decrease in the
error range, this still leaves large regions of degenerate parameter space
between $\log{a}$ and $\gamma$. Consequently, it is this shortest wavelength
range that is crucial for constraining the scattering slope parameters, where
the primary gain in information content comes from the Rayleigh enhancement
factor (see Section 4).

In contrast to the scattering slope parameters, constraints on the cloud-top
pressure, $\log{P_{\mathrm{cloud}}}$, do not improve significantly with
additional optical wavelength coverage. Even for the full wavelength range
down to $\lambda_{\mathrm{min}}$ = 0.3 m, for many planets
$\log{P_{\mathrm{cloud}}}$ is largely unconstrained. Between 1.1 and 0.3 m,
the average 1$\sigma$ 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 non-
detection 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 $\lambda_{\mathrm{min}}$ = 1.1 m.
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 m decreases the average 1$\sigma$ 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
$\log{a}$, 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, 2017).

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 m data,
where they showed that only gray clouds were required to fit the data.
Wavelengths shortward of 0.6 m are necessary to obtain constraints on
$\log{a}$ and $\gamma$, with additional marginal improvement on
$\log{P_{\mathrm{cloud}}}$ 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 1$\sigma$ temperature range between the
$\lambda_{\mathrm{min}}$ = 1.1 and 0.3 m spectra decreases from 363 K to 344
K, with the error range increasing marginally for the $\lambda_{\mathrm{min}}$
= 0.63 m 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 to within an order of magnitude. Some improvement in
constraints is realized as the wavelength range of the observed spectrum
increases, where the average 1$\sigma$ error range in the H2O abundance
decreases from 2.00, to 1.36 and 1.15 dex, for the $\lambda_{\mathrm{min}}$ =
1.1, 0.6 and 0.3 m 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 m, whereas for WASP-127b, it is the 0.6
to 1.1 m 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 m 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 m 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
$\lambda_{\mathrm{min}}$ = 1.1 m spectra are unable to constrain the
abundances of Na and K. A minimum wavelength of $0.6$ m is sufficient to
constrain the K abundance (should its atomic line feature be present). For
planets with detections of K ($>$ 3$\sigma$), the 1$\sigma$ constraints span a
consistent range of 2.4 dex for both the $\lambda_{\mathrm{min}}$ = 0.6 and
0.3 m spectra. However, to reliably constrain Na, the full optical wavelength
range is required. For planets with Na detection significances $>$ 3$\sigma$,
the abundance constraint ranges improve from 4.66 dex to 1.94 dex between the
$\lambda_{\mathrm{min}}$ = 0.6 and 0.3 m spectra. We find that, with the
exception of HD 209458b, detecting only the red wing of the Na resonance
feature with HST (i.e. $\lambda_{\mathrm{min}}$ = 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 from
the average $1\sigma$ 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 micron
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 micron point
is at higher transit depths than the 3.6 micron point. This can be fit by a
sharp CO2 peak near 4.5 m. Increasing the wavelength coverage into the optical
decreases constraints by $\sim$ 30$\%$ between the $\lambda_{\mathrm{min}}$ =
1.1 and 0.3 m spectra. Therefore, if infrared data can provide evidence of
carbon species, the inclusion of optical wavelengths can strengthen
constraints on the retrieved mixing ratios.

Table 4: Detection significances for Na, K and H2O.

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

Note. — The equivalent sigma values are computed from Bayesian evidence
ratios. ‘—’ indicates a non-detection (Bayes factor $<$ 1).

###  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 $>$
1$\sigma$ presented in Table 4.

The posterior distributions for individual planets can be fond in the
supplementary online material: https://doi.org/10.5281/zenodo.10407463
(catalog 10.5281/zenodo.10407463), to which we direct the reader. We note a
few general insights to 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 m 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 $\gamma$ and
$\bar{\phi}_{clouds}$ 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 on cloud scattering properties,
$\log{a}$ and $\gamma$, with increasing wavelength coverage. However we see
little improvement in other fit parameters. H2O is well constrained with a
detection significance of 3.7$\sigma$ but Na remains tentative in our analysis
despite a significant “by-eye” deviation at the Na I wavelength.

####  3.3.2 HAT-P-12b

For the $\lambda_{\mathrm{min}}$ = 1.1 m retrieval HAT-P-12b favors high Na
and K abundance modes. This condition is resolved with the addition of
observations below 1.1 m, 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
$\lambda_{\mathrm{min}}$ = 0.3 m retrieval finds, although unconstrained, a
$\gamma$ distribution skewed towards values consistent with Rayleigh
scattering. This supports the sub-micron 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 $\lambda_{\mathrm{min}}$ = 1.1 m and 0.6 m,
resulting from the non-detection 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
$\log{a}$ and $\gamma$ to within 1.1 dex.

####  3.3.4 HD 189733b

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

For most of the planets, our retrievals are unable to constrain aerosol
parameters with $\lambda_{\mathrm{min}}$ =0.6 m, since the data from 0.6 and
1.1 m typically contains 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
$\lambda_{\mathrm{min}}$ = 0.3 m spectrum, however, our retrievals cannot
disentangle the cloud parameters at $\lambda_{\mathrm{min}}$ = 0.6 m 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 $\log{a}$ and $\log{P_{cloud}}$ for
$\lambda_{min}$ = 0.3 m retrievals and the mixing ratios of H2O, Na and K show
correlations in both the $\lambda_{min}$ = 0.3 and 0.6 m retrievals.

####  3.3.6 WASP-6b

WASP-6b has significant detections of Na, K, and H2O. The Na line wing is
detected in $\lambda_{\mathrm{min}}$ = 0.6 m but is not constrained until
$\lambda_{\mathrm{min}}$ = 0.3 m. The significant scattering slope,
$\log{a}\,\sim\,$6 and $\gamma$ = -10 is well defined but unconstrained until
the full dataset is used. The inclusion of HST/WFC3 data also allows us to
place a near solar constraint on the water abundance, also shown by Carter et
al. (2020). Correlations between cloud parameters arise at wavelengths below
0.6 m, most notably $\log{a}$ – $\gamma$ and T – $\gamma$. The H2O–Na–K
correlation present for HD 209458b can also be seen in the $\lambda_{min}$ =
0.3 m 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 datapoints shortward of 0.6 m,
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, inclusion of wavelengths below
0.6 m improves constraints on the $R_{p,\,\mathrm{ref}}$–$T$ 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 does 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 m data. Our infrared-only
retrieval ($\lambda_{\mathrm{min}}$ = 1.1 m) 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
$\lambda_{\mathrm{min}}=$ 0.6 and 0.3 µmretrievals. 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 over-confidently predict alkali abundances.

####  3.3.9 WASP-19b

Our WASP-19b retrieval is limited due to the small number of datapoints 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 it
is reduced with a significantly different analysis method to that of the Sing
et al. (2016) dataset. 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 ($\lambda_{\mathrm{min}}$ =
1.1 m), 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 m reduces
the likelihood of this mode and leads to a flatter, more unconstrained
posterior distribution. However, once the data down to 0.3 m are included the
super-Rayleigh region of parameter space favored.

We note that other chemical species, beyond those considered here, may also be
present in WASP-31b’s atmosphere. MacDonald & Madhusudhan (2017) 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 3.2$\sigma$ detection of K in WASP-31b is driven by a single data point at
0.77 m, 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., McGruder et al., 2020; Gibson et al., 2019,
2017)

####  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 m and 1.0 m for the $\lambda_{\mathrm{min}}$ = 0.6 and 1.1 m
retrievals indicate prominent contributions from TiO and VO opacity. Without
optical data below 0.6 m, TiO and VO, alongside the aerosol parameters, are
unconstrained. Additionally, WASP-121b shows emergent correlations between
mixing ratios of VO and H2O with $\log{a}$ when observations below 0.6 m 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 m to other opacity sources, without the need
for a scattering slope. Therefore, absorbers such as HS and FeH 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
$\lambda_{\mathrm{min}}$ = 0.3 m spectrum to 333 K for the
$\lambda_{\mathrm{min}}$ = 0.3 m spectrum). For the $\lambda_{\mathrm{min}}$ =
0.3 m retrieval, a region of higher temperature parameter space, which is
disfavored by the $\lambda_{\mathrm{min}}$ = 0.6 and 1.1 m retrievals,
occupies a portion of the probability distribution. This high-temperature
region only arises for sub-Rayleigh scattering values of $\log{a}$ and
$\gamma$, which are most prominent for the 0.3 m retrieval and completely
absent from the 1.1 m 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 mass-radius space. It orbits a known active K star with a measured spot
covering fraction of $3^{+6}_{-1}\%$ (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
$\lambda_{\mathrm{min}}=1.1$ m. 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 is included these detections
disappear.

Similarly, the retrieved H2O abundance is significantly anomalous for the
$\lambda_{\mathrm{min}}=1.1$ m retrieval. Without data between 0.3 m and 1.1
m, the stellar heterogeneity temperature is retrieved approximately 1200 K
below the effective temperature of the star ($T_{\mathrm{eff}}$ = 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 $\lambda_{\mathrm{min}}=1.1$ m retrieval (see Figure 6).

Figure 6: Retrieved transmission spectrum for HAT-P-11b. Retrieved median
spectra for each wavelength range, $\lambda_{\mathrm{min}}$ = 0.3 m,
$\lambda_{\mathrm{min}}$ = 0.6 m and $\lambda_{\mathrm{min}}$ = 1.1 m, are
displayed in purple, blue and orange respectively. Light and dark shaded
regions correspond to the $1$ and $2\sigma$ bounds. The datapoints show the
observed spectrum, where the marker colors correspond to data above the
minimum wavelength of the three retrievals, following the same color scheme as
the retrieved spectra. The y-axis scale is expressed in atmospheric scale
heights, $H=k_{B}T/\mu g$, where $T$ is taken as the retrieved temperature
from the $\lambda_{\mathrm{min}}$ = 0.3 m retrieval.

However, once shorter wavelength optical data is included
($\lambda_{\mathrm{min}}$ = 0.6 and 0.3 m), 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 m 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
($\chi^{2}_{\nu}=2.42$) 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.

Figure 7: The information content (measured in nats) between the posterior and
prior distribution for $\lambda_{min}$ = 0.3, 0.6 and 1.1m retrievals. Left
panel: the change in information content with the number of datapoints. Right
panel: the change in information content with data wavelength cutoff. Planets
are ordered in terms of their retrieved Rayleigh enhancement factor,
$\log{a}$, with the planet with the highest value of $\log{a}$ (HD 189733b) in
yellow and the lowest value of $\log{a}$ (WASP-17b) in purple.

##  4 Information content analysis

Whilst 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 gain
through entropy estimation.

Information content (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). Information
content 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 information content 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, information content
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 information content ($I(\theta,X)$) of a
retrieval, where $\Theta$ is the set of all parameters $\theta$ and $X$ is the
set of observations $x$ as

$I(\Theta,X)=H(\Theta)-H(\Theta|X)$ (1)

where $H(\Theta)$ is the entropy of the prior distribution $p(\Theta)$ and
$H(\Theta|X)$ is the entropy of the posterior distribution $p(\Theta|X)$.

For a discrete random variable $X$, entropy is defined as

$H(X)=-\sum_{x\in X}p(x)\log{p(x)}$ (2)

which is the expectation of the self information $h(x)=-\log{p(x)}$. 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

$H(f)=-\int^{\infty}_{-\infty}f(x)\log{f(x)}dx$ (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

$H(f)=\int^{1}_{0}\log{\Bigg{\\{}\frac{d}{dp}F^{-1}(p)\Bigg{\\}}dp}$ (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

$HV_{mn}=\frac{1}{n}\sum^{n}_{i=1}\log{\Big{\\{}\frac{n}{2m}(X_{i+m}-X_{i-m})\Big{\\}}}$
(5)

The estimator is consistent, such that $HV_{mn}\to H(f)$ as $n\to\infty$,
$m\to\infty$, and $m/n\to 0$.

Figure 8: Information content (measured in nats) broken down by parameter
group for $\lambda_{\mathrm{min}}$ = 0.6 to 0.3m. Top: breakdown of parameters
by $R_{p,\mathrm{ref}}$, T, alkali species, H2O, carbon species, cloud
parameters, high temperature species (TiO/VO) and stellar parameters. Bottom:
breakdown of the information content change per cloud parameter. Planets are
ordered from highest to lowest (left to right) retrieved Rayleigh enhancement
factor, $\log{a}$.

We estimate the information content 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 information content in nats. For an event with
probability $1/e$, the information content in nats is one. We then sum over
the mutual entropy for all parameters to find the total information content
gained from the retrieval for a given set of observations. We also assess the
information content between the posteriors of retrievals with different
spectral ranges.

Figure 9: Information content (measured in nats) broken down by parameter
group for $\lambda_{\mathrm{min}}$ = 1.1 to 0.3m. Top: breakdown of parameters
by $R_{P,\mathrm{ref}}$, T, alkali species, H2O, carbon species, cloud
parameters, high temperature species (TiO/VO and stellar parameters. Bottom:
breakdown of the information content change per cloud parameter. Planets are
ordered from highest to lowest (left to right) retrieved Rayleigh enhancement
factor, $\log{a}$.

###  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 information content change as a
function of the number of datapoints (left panel) and with the different
optical wavelength cutoffs (right panel) for each planet. As a trend, the IC
increases with greater wavelength and datapoint coverage. Most planets have a
greater increase in IC per wavelength and per datapoint between
$\lambda_{\mathrm{min}}$ = 0.6 m and 0.3 m than between the
$\lambda_{\mathrm{min}}$ = 1.1 m to 0.6 m retrievals, with some showing a
marked increase in IC with only a few additional datapoints in this range.
Exceptions to this trend are WASP-31b and WASP-127b (and also HAT-P-32b, but
only for the datapoint difference, not the spectral range). The average
increase in information between 1.1 m and 0.6 m is 1.39 nats and the average
increase in information between 0.6 m and 0.3 m is 2.11 nats for the sample of
exoplanets.

We find that the greatest average information gain across the planet
population is found between 0.6 m and 0.3 m. 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 $\lambda_{\mathrm{min}}$ = 0.6 m and 0.3 m 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 $\lambda_{\mathrm{min}}$ = 0.6 m and 0.3 m
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.

We see that the change in information content does not show the same ordering
as the median retrieved Rayleigh enhancement factor ($\log{a}$). It can be
seen that cloud parameters dominate the IC gain between 0.6 m and 0.3 m. If we
further break down the cloud parameters, the greatest contribution to
information gain does come from $\log{a}$, followed by $\gamma$. 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 m and 0.3 m 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 the information
content 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 $R_{\mathrm{P,ref}}$, 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 m.
Whilst the IC between 0.6 m and 0.3 m is negligible, between 1.1 m and 0.3 m
(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 m. 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.

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

##  5 Cloud and Temperature Trends Across the Population

Cloud scattering slopes are generally attributed to small aerosol particles in
the upper atmosphere causing Rayleigh-like ($\gamma$ = $-4$) to super-Rayleigh
profiles in optical spectra. On average we retrieve scattering slopes
2.2$\times$ Rayleigh for $\lambda_{\mathrm{min}}$ = 0.3 m 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 $\gamma$ values
against Teq. The relationship between scattering slope and Teq assumes a drag
free atmosphere at a pressure of 1 mbar for particle tracers of 0.01 m 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.

Figure 10: Retrieved scattering slope, $\gamma$, against Teq. The purple and
grey lines show the slopes for soot and tholin aerosols, respectively, from
Ohno & Kawashima (2020) for two different aerosol mass fluxes ($F$). The
models do not extend beyond 2,250 K and we find no clear trend to any single
aerosol profile across the full population of our study. Figure 11: Retrieved
cloud fraction, $\bar{\phi}_{\mathrm{clouds}}$, trends with retrieved water
abundance (left) and planetary equilibrium temperature (right). Dashed and
dotted lines represent the solar abundance values for temperatures $<$ 1200 K
and $>$ 1200 K calculated by Madhusudhan (2012) respectively. The planets are
colored by equilibrium temperature from the hottest (WASP-12b) in yellow to
the coldest (HAT-P-11b) in purple.

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 towards 50:50 cloud
coverage suggesting the potential for morning/evening asymmetries.

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 which
account for a wider range of parameters and micro-physics may be necessary to
account for any non-linear 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. Whilst 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 GCM predictions
which simulate the radiative convective structure of a planetary atmosphere.
We evaluate the retrieved temperatures with respect to a population of GCM
models presented by Kataria et al. (2016). Figure 12 presents our retrieved
limb temperatures against Teqalongside 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.

Across the retrieved limb temperatures the general trend follows that
$T_{\mathrm{limb}}<$ Teq. Although we include patchy clouds that have been
shown to reduce biases towards 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 Teqfor the ultra-hot Jupiters WASP-12b and WASP-121b. Our
retrieved temperatures reproduce the extreme difference in Teq– Tlimb seen in
literature (Kreidberg et al., 2015; Evans et al., 2018; Pinhas et al., 2019;
Welbanks et al., 2019) for ultrahot Jupiters where Teq\-
$T_{\mathrm{limb}}\sim$ 1,000 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, seven of the 11 planets agree with the GCM temperatures to
within $1\sigma$. Fitting a linear relation to
$T_{\mathrm{limb}}-T_{\mathrm{eq}}$ and $T_{\mathrm{GCM}}-T_{\mathrm{eq}}$
gives gradients of 0.36 $\pm$ 0.13 and 0.57 $\pm$ 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 Teqis 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 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.

Figure 12: Retrieved limb temperatures plotted against equilibrium
temperatures (purple). Plotted in green are GCM limb average temperatures for
11 planets from Kataria et al. (2016). The gray dashed line marks where limb
temperature is equal to planetary equilibrium temperature. Linear fits and
$1\sigma$ errors are found for the retrieved limb temperatures and GCM limb
temperatures, shown by the purple and green lines and shading respectively.

##  6 Conclusions

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 (§ 2.2), to select three spectra ranges to apply to a larger
population of planets. We implement retrievals for minimum wavelength ranges,
$\lambda_{\mathrm{min}}$ = 0.3, 0.6, and, 1.1 m for a sample of 14 exoplanets
with transit spectra from 0.3–4.5 m. We evaluate the retrieved parameters
across the population considering their median values and 1-$\sigma$
uncertainty for each $\lambda_{\mathrm{min}}$ 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 information content analysis on the posterior distributions of
our retrievals and break down the information content per parameter (§ 4). As
this analysis covers a population of exoplanets, in § 3 we search for trends
between atmospheric and planetary parameters. We compare our retrieved limb
temperatures to temperatures derived from GCM models and investigate the
relationship between cloud parameters, water abundance and temperature. More
data is required to draw conclusions about population trends.

From the spectral range investigation, we find that wavelengths below 0.6 m
are necessary to constrain alkali species and cloud scattering parameters
$\log{a}$ and $\gamma$, although the scattering slope $\gamma$ is not
consistently constrained across the population. This is supported by the
information content analysis, where the largest information gains between the
$\lambda_{\mathrm{min}}$ = 1.1, 0.6 and 0.3 m cutoffs are from the cloud
parameters. We find 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 m provides comparable constraints and retrieved median values to the full
0.3 to 4.5 m 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
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 Teq, finding a
tentative suggestion that cloud fraction increases with Teq up to 1,600 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 that predicted by GCMs related to Teq.

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,
$\log{a}$ and $\gamma$, are most affected by the inclusion of sub 0.6 m data
with improvements in their constraint of over 30%.

The authors thank the anonymous referee for their 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
number 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.

Data availability: Supplementary corner plots, spectra and results are
available on Zenodo: https://doi.org/10.5281/zenodo.10407463 (catalog
10.5281/zenodo.10407463)

## References

  * Addison et al. (2019) Addison, B., Wright, D. J., Wittenmyer, R. A., et al. 2019, PASP, 131, 115003, doi: 10.1088/1538-3873/ab03aa
  * Ahrer et al. (2023) Ahrer, E.-M., Stevenson, K. B., Mansfield, M., et al. 2023, Nature, 614, 653
  * Alam et al. (2020) Alam, M. K., López-Morales, M., Nikolov, N., et al. 2020, AJ, 160, 51, doi: 10.3847/1538-3881/ab96cb
  * Alderson et al. (2022) Alderson, L., Wakeford, H. R., MacDonald, R. J., et al. 2022, Monthly Notices of the Royal Astronomical Society, 512, 4185, doi: 10.1093/mnras/stac661
  * Alderson et al. (2023) Alderson, L., Wakeford, H. R., Alam, M. K., et al. 2023, Nature, 614, 664, doi: 10.1038/s41586-022-05591-3
  * Alizadeh Noughabi (2015) Alizadeh Noughabi, H. 2015, Annals of Data Science, 2, 231
  * Anderson et al. (2011a) Anderson, D. R., Smith, A. M. S., Lanotte, A. A., et al. 2011a, MNRAS, 416, 2108, doi: 10.1111/j.1365-2966.2011.19182.x
  * Anderson et al. (2011b) Anderson, D. R., Collier Cameron, A., Hellier, C., et al. 2011b, A&A, 531, A60, doi: 10.1051/0004-6361/201016208
  * Barklem & Collet (2016) Barklem, P. S., & Collet, R. 2016, A&A, 588, A96, doi: 10.1051/0004-6361/201526961
  * Barstow (2020) Barstow, J. K. 2020, Monthly Notices of the Royal Astronomical Society, 497, 4183, doi: 10.1093/mnras/staa2219
  * Barstow (2021) —. 2021, Astronomy & Geophysics, 62, 1.36, doi: 10.1093/astrogeo/atab044
  * Barstow et al. (2017) Barstow, J. K., Aigrain, S., Irwin, P. G. J., & Sing, D. K. 2017, The Astrophysical Journal, 834, 50, doi: 10.3847/1538-4357/834/1/50
  * Barstow & Heng (2020) Barstow, J. K., & Heng, K. 2020, Space Science Reviews, 216, 82, doi: 10.1007/s11214-020-00666-x
  * Batalha & Line (2017) Batalha, N. E., & Line, M. R. 2017, The Astronomical Journal, 153, 151
  * Benneke & Seager (2012) Benneke, B., & Seager, S. 2012, The Astrophysical Journal, 753, 100
  * Bonomo et al. (2017) Bonomo, A. S., Desidera, S., Benatti, S., et al. 2017, A&A, 602, A107, doi: 10.1051/0004-6361/201629882
  * Bourrier et al. (2020) Bourrier, V., Ehrenreich, D., Lendl, M., et al. 2020, A&A, 635, A205, doi: 10.1051/0004-6361/201936640
  * Braam et al. (2021) Braam, M., van der Tak, F. F. S., Chubb, K. L., & Min, M. 2021, A&A, 646, A17, doi: 10.1051/0004-6361/202039509
  * Brown (2001) Brown, T. M. 2001, ApJ, 553, 1006, doi: 10.1086/320950
  * Buchner et al. (2014) Buchner, J., Georgakakis, A., Nandra, K., et al. 2014, A&A, 564, A125, doi: 10.1051/0004-6361/201322971
  * Carter et al. (2020) Carter, A. L., Nikolov, N., Sing, D. K., et al. 2020, Monthly Notices of the Royal Astronomical Society, 494, 5449, doi: 10.1093/mnras/staa1078
  * Chachan et al. (2019) Chachan, Y., Knutson, H. A., Gao, P., et al. 2019, The Astronomical Journal, 158, 244
  * Chakrabarty & Sengupta (2019) Chakrabarty, A., & Sengupta, S. 2019, AJ, 158, 39, doi: 10.3847/1538-3881/ab24dd
  * Charbonneau et al. (2002) Charbonneau, D., Brown, T. M., Noyes, R. W., & Gilliland, R. L. 2002, The Astrophysical Journal, 568, 377, doi: 10.1086/338770
  * Collins et al. (2017) Collins, K. A., Kielkopf, J. F., & Stassun, K. G. 2017, AJ, 153, 78, doi: 10.3847/1538-3881/153/2/78
  * Cortés-Zuleta et al. (2020) Cortés-Zuleta, P., Rojo, P., Wang, S., et al. 2020, A&A, 636, A98, doi: 10.1051/0004-6361/201936279
  * Coulombe et al. (2023) Coulombe, L.-P., Benneke, B., Challener, R., et al. 2023, Nature, 1
  * Delrez et al. (2016) Delrez, L., Santerne, A., Almenara, J. M., et al. 2016, MNRAS, 458, 4025, doi: 10.1093/mnras/stw522
  * Deming et al. (2005) Deming, D., Seager, S., Richardson, L. J., & Harrington, J. 2005, Nature, 434, 740, doi: 10.1038/nature03507
  * Deming et al. (2013) Deming, D., Wilkins, A., McCullough, P., et al. 2013, ApJ, 774, 95, doi: 10.1088/0004-637X/774/2/95
  * Dyrek et al. (2023) Dyrek, A., Min, M., Decin, L., et al. 2023, arXiv e-prints, arXiv:2311.12515, doi: 10.48550/arXiv.2311.12515
  * Espinoza et al. (2019) Espinoza, N., Rackham, B. V., Jordán, A., et al. 2019, MNRAS, 482, 2065, doi: 10.1093/mnras/sty2691
  * Evans et al. (2016) Evans, T. M., Sing, D. K., Wakeford, H. R., et al. 2016, The Astrophysical Journal Letters, 822, L4
  * Evans et al. (2018) Evans, T. M., Sing, D. K., Goyal, J. M., et al. 2018, The Astronomical Journal, 156, 283
  * Feinstein et al. (2023) Feinstein, A. D., Radica, M., Welbanks, L., et al. 2023, Nature, 614, 670
  * Fisher & Heng (2018) Fisher, C., & Heng, K. 2018, Monthly Notices of the Royal Astronomical Society, 481, 4698, doi: 10.1093/mnras/sty2550
  * Flagg et al. (2023) Flagg, L., Turner, J. D., Deibert, E., et al. 2023, ApJ, 953, L19, doi: 10.3847/2041-8213/ace529
  * Fortney et al. (2008) Fortney, J. J., Lodders, K., Marley, M. S., & Freedman, R. S. 2008, ApJ, 678, 1419, doi: 10.1086/528370
  * Fournier-Tondreau et al. (2023) Fournier-Tondreau, M., MacDonald, R. J., Radica, M., et al. 2023, MNRAS, doi: 10.1093/mnras/stad3813
  * Gao et al. (2020) Gao, P., Thorngren, D. P., Lee, E. K. H., et al. 2020, Nature Astronomy, 4, 951, doi: 10.1038/s41550-020-1114-3
  * Gibson et al. (2019) Gibson, N. P., de Mooij, E. J., Evans, T. M., et al. 2019, Monthly Notices of the Royal Astronomical Society, 482, 606
  * Gibson et al. (2017) Gibson, N. P., Nikolov, N., Sing, D. K., et al. 2017, Monthly Notices of the Royal Astronomical Society, 467, 4591
  * Grant et al. (2023) Grant, D., Lewis, N. K., Wakeford, H. R., et al. 2023, ApJ, 956, L29, doi: 10.3847/2041-8213/acfc3b
  * Harris et al. (2020) Harris, C. R., Millman, K. J., Van Der Walt, S. J., et al. 2020, Nature, 585, 357
  * Howe et al. (2017) Howe, A. R., Burrows, A., & Deming, D. 2017, The Astrophysical Journal, 835, 96
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Karman et al. (2019) Karman, T., Gordon, I. E., van Der Avoird, A., et al. 2019, Icarus, 328, 160
  * Kataria et al. (2016) Kataria, T., Sing, D. K., Lewis, N. K., et al. 2016, The Astrophysical Journal, 821, 9, doi: 10.3847/0004-637X/821/1/9
  * Kempton et al. (2023) Kempton, E. M. R., Zhang, M., Bean, J. L., et al. 2023, Nature, 620, 67, doi: 10.1038/s41586-023-06159-5
  * Knutson et al. (2011) Knutson, H. A., Madhusudhan, N., Cowan, N. B., et al. 2011, The Astrophysical Journal, 735, 27
  * Knutson et al. (2012) Knutson, H. A., Lewis, N., Fortney, J. J., et al. 2012, The Astrophysical Journal, 754, 22, doi: 10.1088/0004-637X/754/1/22
  * Knutson et al. (2014) Knutson, H. A., Fulton, B. J., Montet, B. T., et al. 2014, ApJ, 785, 126, doi: 10.1088/0004-637X/785/2/126
  * Komacek et al. (2019) Komacek, T. D., Showman, A. P., & Parmentier, V. 2019, ApJ, 881, 152, doi: 10.3847/1538-4357/ab338b
  * Komacek et al. (2017) Komacek, T. D., Showman, A. P., & Tan, X. 2017, ApJ, 835, 198, doi: 10.3847/1538-4357/835/2/198
  * Kreidberg et al. (2015) Kreidberg, L., Line, M. R., Bean, J. L., et al. 2015, ApJ, 814, 66, doi: 10.1088/0004-637X/814/1/66
  * Lam et al. (2017) Lam, K. W. F., Faedi, F., Brown, D. J. A., et al. 2017, A&A, 599, A3, doi: 10.1051/0004-6361/201629403
  * Lecavelier des Etangs et al. (2008) Lecavelier des Etangs, A., Pont, F., Vidal-Madjar, A., & Sing, D. 2008, Astronomy & Astrophysics, 481, L83, doi: 10.1051/0004-6361:200809388
  * Li et al. (2015) Li, G., Gordon, I. E., Rothman, L. S., et al. 2015, The Astrophysical Journal Supplement Series, 216, 15, doi: 10.1088/0067-0049/216/1/15
  * Line & Parmentier (2016) Line, M. R., & Parmentier, V. 2016, The Astrophysical Journal, 820, 78, doi: 10.3847/0004-637X/820/1/78
  * Line et al. (2012) Line, M. R., Zhang, X., Vasisht, G., et al. 2012, The Astrophysical Journal, 749, 93
  * Lothringer et al. (2022) Lothringer, J. D., Sing, D. K., Rustamkulov, Z., et al. 2022, Nature, 604, 49, doi: 10.1038/s41586-022-04453-2
  * MacDonald (2023) MacDonald, R. J. 2023, Journal of Open Source Software, 8, 4873
  * MacDonald et al. (2020) MacDonald, R. J., Goyal, J. M., & Lewis, N. K. 2020, The Astrophysical Journal, 893, L43, doi: 10.3847/2041-8213/ab8238
  * MacDonald & Madhusudhan (2017) MacDonald, R. J., & Madhusudhan, N. 2017, Monthly Notices of the Royal Astronomical Society, 469, 1979, doi: 10.1093/mnras/stx804
  * MacDonald & Madhusudhan (2017) MacDonald, R. J., & Madhusudhan, N. 2017, ApJ, 850, L15, doi: 10.3847/2041-8213/aa97d4
  * Madhusudhan (2012) Madhusudhan, N. 2012, ApJ, 758, 36, doi: 10.1088/0004-637X/758/1/36
  * Madhusudhan & Seager (2009) Madhusudhan, N., & Seager, S. 2009, The Astrophysical Journal, 707, 24, doi: 10.1088/0004-637X/707/1/24
  * Mancini et al. (2018) Mancini, L., Esposito, M., Covino, E., et al. 2018, A&A, 613, A41, doi: 10.1051/0004-6361/201732234
  * McCullough et al. (2014) McCullough, P. R., Crouzet, N., Deming, D., & Madhusudhan, N. 2014, ApJ, 791, 55, doi: 10.1088/0004-637X/791/1/55
  * McGruder et al. (2020) McGruder, C. D., López-Morales, M., Espinoza, N., et al. 2020, The Astronomical Journal, 160, 230
  * Mikal-Evans et al. (2022) Mikal-Evans, T., Sing, D. K., Barstow, J. K., et al. 2022, Nature Astronomy, 6, 471
  * Moran et al. (2023) Moran, S. E., Stevenson, K. B., Sing, D. K., et al. 2023, ApJ, 948, L11, doi: 10.3847/2041-8213/accb9c
  * Morris et al. (2017) Morris, B. M., Hebb, L., Davenport, J. R. A., Rohn, G., & Hawley, S. L. 2017, ApJ, 846, 99, doi: 10.3847/1538-4357/aa8555
  * Mortier et al. (2013) Mortier, A., Santos, N. C., Sousa, S. G., et al. 2013, A&A, 558, A106, doi: 10.1051/0004-6361/201322240
  * Moses et al. (2013) Moses, J. I., Madhusudhan, N., Visscher, C., & Freedman, R. S. 2013, ApJ, 763, 25, doi: 10.1088/0004-637X/763/1/25
  * Nikolov et al. (2014) Nikolov, N., Sing, D. K., Pont, F., et al. 2014, Monthly Notices of the Royal Astronomical Society, 437, 46, doi: 10.1093/mnras/stt1859
  * Ohno & Kawashima (2020) Ohno, K., & Kawashima, Y. 2020, ApJ, 895, L47, doi: 10.3847/2041-8213/ab93d7
  * Pinhas et al. (2019) Pinhas, A., Madhusudhan, N., Gandhi, S., & MacDonald, R. 2019, Monthly Notices of the Royal Astronomical Society, 482, 1485, doi: 10.1093/mnras/sty2544
  * Pinhas et al. (2018) Pinhas, A., Rackham, B. V., Madhusudhan, N., & Apai, D. 2018, Monthly Notices of the Royal Astronomical Society, 480, 5314
  * Polanski et al. (2022) Polanski, A. S., Crossfield, I. J. M., Howard, A. W., Isaacson, H., & Rice, M. 2022, Research Notes of the American Astronomical Society, 6, 155, doi: 10.3847/2515-5172/ac8676
  * Polyansky et al. (2018) Polyansky, O. L., Kyuberis, A. A., Zobov, N. F., et al. 2018, MNRAS, 480, 2597, doi: 10.1093/mnras/sty1877
  * Pont et al. (2013) Pont, F., Sing, D. K., Gibson, N. P., et al. 2013, Monthly Notices of the Royal Astronomical Society, 432, 2917, doi: 10.1093/mnras/stt651
  * Rackham et al. (2017) Rackham, B., Espinoza, N., Apai, D., et al. 2017, ApJ, 834, 151, doi: 10.3847/1538-4357/aa4f6c
  * Rackham et al. (2018) Rackham, B. V., Apai, D., & Giampapa, M. S. 2018, ApJ, 853, 122, doi: 10.3847/1538-4357/aaa08c
  * Rustamkulov et al. (2023) Rustamkulov, Z., Sing, D., Mukherjee, S., et al. 2023, Nature, 614, 659
  * Seager & Sasselov (2000) Seager, S., & Sasselov, D. D. 2000, ApJ, 537, 916, doi: 10.1086/309088
  * Sedaghati et al. (2017) Sedaghati, E., Boffin, H. M., MacDonald, R. J., et al. 2017, Nature, 549, 238
  * Sedaghati et al. (2021) Sedaghati, E., MacDonald, R. J., Casasayas-Barris, N., et al. 2021, MNRAS, 505, 435, doi: 10.1093/mnras/stab1164
  * Seidel et al. (2020) Seidel, J. V., Lendl, M., Bourrier, V., et al. 2020, A&A, 643, A45, doi: 10.1051/0004-6361/202039058
  * Showman (2021) Showman, A. P. 2021, in ExoFrontiers; Big Questions in Exoplanetary Science, ed. N. Madhusudhan, 19–1, doi: 10.1088/2514-3433/abfa8fch19
  * Sing et al. (2016) Sing, D. K., Fortney, J. J., Nikolov, N., et al. 2016, Nature, 529, 59, doi: 10.1038/nature16068
  * Southworth (2010) Southworth, J. 2010, MNRAS, 408, 1689, doi: 10.1111/j.1365-2966.2010.17231.x
  * Southworth (2011) —. 2011, MNRAS, 417, 2166, doi: 10.1111/j.1365-2966.2011.19399.x
  * Spake et al. (2021) Spake, J. J., Sing, D. K., Wakeford, H. R., et al. 2021, Monthly Notices of the Royal Astronomical Society, 500, 4042, doi: 10.1093/mnras/staa3116
  * Stevenson et al. (2017) Stevenson, K. B., Line, M. R., Bean, J. L., et al. 2017, The Astronomical Journal, 153, 68
  * Tashkun & Perevalov (2011) Tashkun, S., & Perevalov, V. 2011, Journal of Quantitative Spectroscopy and Radiative Transfer, 112, 1403, doi: https://doi.org/10.1016/j.jqsrt.2011.03.005
  * Todorov et al. (2014) Todorov, K. O., Deming, D., Burrows, A., & Grillmair, C. J. 2014, The Astrophysical Journal, 796, 100, doi: 10.1088/0004-637X/796/2/100
  * Tregloan-Reed et al. (2015) Tregloan-Reed, J., Southworth, J., Burgdorf, M., et al. 2015, MNRAS, 450, 1760, doi: 10.1093/mnras/stv730
  * Tsiaras et al. (2018) Tsiaras, A., Waldmann, I. P., Zingales, T., et al. 2018, The Astronomical Journal, 155, 156, doi: 10.3847/1538-3881/aaaf75
  * Vasicek (1976) Vasicek, O. 1976, Journal of the Royal Statistical Society Series B: Statistical Methodology, 38, 54
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Wakeford et al. (2018) Wakeford, H. R., Sing, D. K., Deming, D., et al. 2018, The Astronomical Journal, 155, 29, doi: 10.3847/1538-3881/aa9e4e
  * Wang et al. (2019) Wang, Y.-H., Wang, S., Hinse, T. C., et al. 2019, AJ, 157, 82, doi: 10.3847/1538-3881/aaf6b6
  * Welbanks & Madhusudhan (2019) Welbanks, L., & Madhusudhan, N. 2019, AJ, 157, 206, doi: 10.3847/1538-3881/ab14de
  * Welbanks et al. (2019) Welbanks, L., Madhusudhan, N., Allard, N. F., et al. 2019, The Astrophysical Journal, 887, L20, doi: 10.3847/2041-8213/ab5a89
  * Wong et al. (2020) Wong, I., Benneke, B., Gao, P., et al. 2020, The Astronomical Journal, 159, 234
  * Xue et al. (2023) Xue, Q., Bean, J. L., Zhang, M., et al. 2023, arXiv e-prints, arXiv:2310.03245, doi: 10.48550/arXiv.2310.03245
  * Yurchenko et al. (2018) Yurchenko, S. N., Al-Refaie, A. F., & Tennyson, J. 2018, A&A, 614, A131, doi: 10.1051/0004-6361/201732531

##  Appendix A Planetary and stellar parameters

Planet | $R_{*}\ (R_{\odot})$ | $T_{\mathrm{eff}}\ (K)$ | [Fe/H] | $\log{g}\ (cgs)$ | $R_{p}\ (R_{J})$ | $M_{p}\ (M_{J})$ | $T_{\mathrm{eq}}\ (K)$
---|---|---|---|---|---|---|---
HAT-P-1 b a | 1.17 | 5980 | 0.13 | 4.36 | 1.32 | 0.53 | 1320
HAT-P-11 b b | 0.70 | 4780 | 0.31 | 4.66 | 0.40 | 0.08 | 840
HAT-P-12 b c | 0.68 | 4670 | -0.20 | 4.61 | 0.92 | 0.20 | 960
HAT-P-32 b d | 1.37 | 6000 | -0.16 | 4.22 | 1.98 | 0.68 | 1840
HD 189733 b | 0.77 e | 5010 f | 0.01 f | 4.49 f | 1.12 e | 1.16 e | 1210 e
HD 209458 b | 1.16 g | 6040 f | 0.05 f | 4.30 f | 1.38 g | 0.17 g | 1460 g
WASP-6 b h | 0.86 | 5380 | -0.15 | 4.49 | 1.23 | 0.49 | 1180
WASP-12 b | 1.66 i | 6360 i | 0.33 i | 4.16 i | 1.94 j | 1.47 j | 2590 j
WASP-17 b | 1.57 k | 6490 f | 0.00 f | 4.08 f | 1.99 k | 0.49 k | 1770 k
WASP-19 b | 1.01 l | 5620 l | 0.15 m | 4.42 l | 1.42 l | 1.15 l | 2110 l
WASP-31 b | 1.25 n | 6300 n | -0.20 o | 4.76 p | 1.55 n | 0.48 n | 1580 q
WASP-39 b | 0.94 c | 5460 f | -0.01 f | 4.41 f | 1.28 c | 0.28 c | 1170 c
WASP-121 b | 1.46 r | 6340 f | 0.24 f | 4.17 f | 1.75 s | 1.16 s | 2360 r
WASP-127 b | 1.33 t | 5850 f | -0.16 f | 4.22 f | 1.31 t | 0.16 t | 1400 u
Table 5: Stellar and planetary parameters for the sample of 14 exoplanets used
in this investigation, where values have been obtained from the following
studies: Nikolov et al. (2014)a; Southworth (2011)b; Mancini et al. (2018)c;
Wang et al. (2019)d; Addison et al. (2019)e; Polanski et al. (2022)f;
Southworth (2010) g; Tregloan-Reed et al. (2015) h; Collins et al. (2017) i;
Chakrabarty & Sengupta (2019) j; Anderson et al. (2011a) k; Cortés-Zuleta et
al. (2020) l; Knutson et al. (2014) m; Anderson et al. (2011b) n; Bonomo et
al. (2017) o; Mortier et al. (2013) p; Sing et al. (2016)q; Delrez et al.
(2016) r; Bourrier et al. (2020) s; Seidel et al. (2020) t; Lam et al. (2017)
u.