JWST Transit Spectra I: Exploring Potential Biases and Opportunities in Retrievals of Tidally-locked Hot Jupiters with Clouds and Hazes

Tidally-locked hot Jupiters are the most thoroughly modelled and observed class of exoplanet to date (Heng & Showman 2015). Their large radii, short periods, and high temperatures allow one to obtain infrared phase curves, emission spectra and transit spectra with relatively high $SNR$’s compared to other types of exoplanets (eg: Charbonneau et al. 2000; Henry et al. 2000; Cowan et al. 2007; Cowan et al. 2012; Knutson et al. 2007; Knutson et al. 2009; Knutson et al. 2009, Knutson et al. 2012; Borucki et al. 2009; Demory et al. 2013; Maxted et al. 2013; Stevenson et al. 2014; Zellem et al. 2014; Wong et al. 2015). Due to the extreme irradiation directed at the permanent day-side of these planets, it is theorized that their atmospheres will have large differences in temperature between the day side and the night side, and that extremely fast-flowing equatorial jets will emerge (Showman & Guillot 2002; Cooper & Showman 2005; Fortney et al. 2008; Showman et al. 2008; Lewis et al. 2010; Rauscher 2010; Cowan & Agol 2011; Menou 2012; Perna et al. 2012; Perez-Becker & Showman 2013; Heng & Showman 2015; Kataria et al. 2016; Ginzburg & Sari 2016; Komacek & Showman 2016). These winds should also cause an east-west asymmetry in addition to the day-night differences in the atmospheric properties (Kempton et al. 2017; Powell et al. 2019).

Observations of hot Jupiters have demonstrated a diverse range of behaviors, and theory can explain some, but not all, of what is going on. For example, phase curves have shown that the fractional difference between day-side and night-side emission and the degree of hot-spot offset can vary significantly from planet to planet (Komacek & Showman 2016; Keating et al. 2019). Earlier observations indicated an increasing trend in day-night temperature contrast with level of stellar irradiation (Komacek & Showman 2016). Theoretical modeling with Global Circulation Models (GCM) could explain this trend, at least qualitatively, as the result of the competition between the timescale for advection against the timescale for radiative cooling and heating (Komacek & Showman 2016). A recent reanalysis by Keating et al. 2019 indicates that night-side temperatures of hot Jupiters seem to clump around 1100 K with a slight upward trend for only the most highly irradiated objects. The favored explanation for this is that night-side clouds with similar properties on all these objects re-radiate heat at around 1100 K. Regardless of the precise mechanisms underlying the cause of day-night temperature gradients, they are almost certainly present. Linking the interpretation of transit spectra to phase curve observations by attempting to extract multi-dimensional information, could provide a more complete picture of the physical conditions on these distant bodies (Heng & Showman 2015; Lee et al. 2019).

Considering both the observational evidence, and theoretical models, it seems safe to say that the limbs of hot Jupiters probed by transit spectra will be very heterogeneous. Many researchers have noted this, and investigated how the full 3D properties of hot Jupiters as modelled by GCMs may manifest in transit spectra. These studies generally pull out the atmosphere’s structure in the annulus right around the terminator of a converged GCM, then use some physically motivated assumptions to post-facto fill in chemistry and aerosols, and finally compute transit spectra (Burrows et al. 2010; Fortney et al. 2010; Dobbs-Dixon 2012; Parmentier et al. 2013; Charnay et al. 2015; Charnay et al. 2015; Lee et al. 2015; Helling et al. 2016; Lee et al. 2016; Line & Parmentier 2016; Kempton et al. 2017; Lines et al. 2018; Caldas et al. 2019; Helling et al. 2019; Powell et al. 2019; Lee et al. 2019; Pluriel et al. 2020; MacDonald et al. 2020; Taylor et al. 2020). Comparing these 3D transit spectra to 1D transit spectra has revealed detectable differences. When theoretical 3D spectra are fed into 1D retrieval frameworks and compared to ground truth parameters, it is evident that current state of the art approaches will lead to biases once we have higher quality data. Most studies of inhomogeneous limbs have focused on variation about the azimuthal extent of the terminator, not differences along the transverse path. But the assumption of homogeneity along the transverse path does not always hold, and it is most likely to fail for tidally-locked hot Jupiters (Caldas et al., 2019).

Caldas et al. 2019 point out that transit spectra of hot puffy atmospheres probe a relatively broad swath of atmosphere about the terminator, and, for tidally-locked exoplanets, this region will likely vary widely in temperature and chemical composition between the day and night side (Pluriel et al. 2020). They demonstrated that day-night differences will affect transit spectra at detectable levels and conducted a series of retrieval experiments to determine how this might bias 1D retrievals. To generate input data for the retrieval experiments, they constructed idealized 3D atmosphere structures which varied continuously from a high temperature on the day side to a cooler temperature on the night side. They simulated a wide range of temperatures and transition widths, then attempted to fit them with single T-P profile transit spectra as is standard practice for retrievals. They found that they could produce satisfactory fits to the simulated data based on goodness-of-fit metrics, but got results which systematically biased the terminator temperature up towards the day-side temperature. This warmer temperature then biased retrieved water abundances. They thus concluded that ignoring the presence of day-night temperature gradients will confuse our understanding of some of the strongest observational targets and potentially hinder attempts to understand overarching trends in large samples of exoplanet atmospheres.

In this paper we extend Caldas et al. 2019’s line of investigation further by modeling day-night temperature gradients with local thermal-equilibrium chemistry. This approach varies the abundances of all major opacity sources continuously across the day-night transition, whereas their study kept the chemistry constant through out the whole atmosphere, and a follow up study (Pluriel et al. 2020) varied only the H^- abundance between the day and night sides of the planet. We also explore the effects of including opacity from a variety of clouds and hazes. Including aerosols is an important step because existing transit spectra directly show that they are present at the limbs of some tidally-locked exoplanets. There are also hints that reflective aerosols may be present on the day-side in some hot Jupiter atmospheres arose when Kepler provided the first observations of optical phase curves (Parmentier et al., 2016). Other observations have indicated that different species may be condensing on the cooler night-side then evaporating on the warmer day side (Keating et al. 2019; Ehrenreich et al. 2020). Gao et al. 2020 expect silicate aerosols to form in atmospheres with equilibrium temperatures between 1000 and 2000 K, and hyrdrocarbon hazes to form for equilibrium temperatures below 1000 K, based upon the results of detailed microphysical models.

In §5 we show the results of similar retrieval experiments to those done by Caldas et al. 2019, where data is simulated for an atmosphere with a day-night temperature gradient but then fit using a single T-P profile. We also show results for retrievals done with the full temperature gradient model, in order to explore the issue from a more positive perspective. If ignored and brushed under the rug, day-night temperature gradients will most likely bias temperature and abundance measurements, but, if acknowledged and included in models, day-night temperature gradients may provide additional information about the extreme and dynamic atmospheres of highly-irradiated planets. Before we show the retrieval results, we illustrate the nature of our temperature gradient model for clear atmospheres in §3 and for hazy or cloudy atmospheres in §4.

METIS produces transit spectra given an atmospheric metallicity and an arbitrary longitude-latitude-altitude grid of temperatures, pressures, and densities (the type of output expected from a General Circulation Model, GCM). METIS is built around our own implementation of the geometrical framework and algorithms of Pytmosph3R Caldas et al. (2019). We used the detailed descriptions in Caldas et al. (2019) to replicate the basic framework of their code, but then paired the geometry and integration approaches with a suite of pre-tabulated equilibrium chemistry and corresponding gas opacities rather than using their on-the-fly approach to chemistry and opacity calculations.

The transit spectrum for a 3D atmosphere can be computed as:

where $\frac{\delta F}{F}\big{(}\lambda\big{)}$ refers to the fractional change in flux during mid-transit or the so-called transit depth, $b$ is the impact parameter of a ray of light passing through the planet’s atmosphere relative to the center of the planet, $\theta$ is the azimuthal angle about a line drawn from the star through the center of the planet to the observer, $\tau(\theta,b,\lambda)$ is the slant optical depth of the atmosphere at the specified $\theta$ and $b$, and $R_{*}$ is the radius of the host star. We construct $\tau(\theta,b,\lambda)$ as:

where $\rho$ is the mass density of the atmosphere, $\kappa$ is the opacity expressed as a cross section over a mass, and $x$ is path length along the light ray described by cylindrical coordinates $b$, $\theta$. Note that $x$ and $b$ can be related to an altitude in the atmosphere $z$ by the Pythagorean theorem: $z=\sqrt{b^{2}+\mid x\mid^{2}}-R_{P}$, where we have set $x$=0 at the terminator and the altitude $z$ is measured relative to a planet radius $R_{P}$.

In practice we discretize the integral in equation (2) as a summation over all the atmosphere grid cells passed through by each ray defined by $\theta$ and $b$, varying the path length $dx$ appropriately. We also discretize the integral in equation (1), selecting a lower bound for the impact parameter $b$ that lies below where the planet appears opaque and an upper bound that is sufficiently high for atmospheric extinction to become negligible at all wavelengths. The linear spacings in b are chosen such that resultant transit spectra no longer changed beyond machine precision. This typically takes of order 100 impact parameter bins.

Throughout our study, rather than run tens of thousands of full GCMs, we construct 3D atmospheres with parameterized day-night temperature gradients assuming hydrostatic equilibrium in the vertical direction, thermo-chemical equilibrium, and the ideal gas law. These assumptions are again inspired by the work of Caldas et al. (2019). This structure model uses eight parameters to determine a latitude-longitude-altitude grid of temperatures, pressures and densities: the day-side temperature, T_day, the night-side temperature, T_night, the angle over which the transition from day-side to night-side temperature occurs, $\beta$, the mass of the planet, M_p, the metallicity of the planet’s atmosphere, $Z$, the radius of the planet, R₀, and finally the reference pressure, P₀, to which R₀ corresponds. To be physically motivated in using a single value of R₀ and P₀, they ought to be chosen such that they are deep enough in the atmosphere that the day-night temperature difference imposed by stellar irradiation has faded and the internal state of the planet is dominating (below 10 bars, Heng & Showman 2015).

The temperatures are mapped onto latitude assuming a linear change from night-side temperature to day-side temperature across an angular width $\beta$:

Note that here a latitude of 90^∘ or $\pi/2$ radians corresponds to the terminator, a latitude of 180^∘ or $\pi$ radians corresponds to the substellar point, and a latitude of 0^∘ or 0 radians corresponds to the anti-stellar point. This is flipped from the usual orientations of latitude and longitude because it allows us to easily introduce azimuthal symmetry along the axis defined by star-planet-observer. This azimuthal symmetry allows us to speed up the computation of the transit spectrum by removing the integration over $\theta$ in eq. (1). The day-night variation comes in as one integrates along $dx$ in eq. (2), which is now independent of $\theta$. This speed-up allows us to carry out retrievals.

For each temperature we then assign appropriate pressures to a grid of altitudes according to an isothermal atmosphere with varying gravity:

where z is 0 at $R_{P}$, and increases with decreasing pressure. We begin at $P_{0}$ and z=0 assuming g(0)=$\frac{GM_{P}}{R_{P}^{2}}$, and then integrate upwards from there. Each new pressure level gives a new density following the ideal gas law, and adding on the corresponding mass for that shell and accounting for the new radius gives a new surface gravity. We also allow the mean molecular weight to vary with altitude as the composition of the atmosphere changes, using our pre-tabulated equilibrium chemistry tables to determine the appropriate mean molecular weight.

In reality, transit spectra probe enough of a range in altitude that an isothermal assumption is not fully adequate (Blecic et al., 2017), but this model will still enable us to quantify the effects of having a day-night temperature gradient by representing the limit where all temperature variations are across longitude rather than altitude. An exaggerated example with a 3000 K day-side and 900 K night-side is shown in Figure 1 for three different values of $\beta$. In this figure, the host star would lie towards the top of the Figure and the observer towards the bottom. The impact parameter $b$ would vary along the horizontal axis, and the path along the line-of-sight would run parallel to the vertical axis.

Once the initial atmospheric structure is set up, either from a GCM output, or from our toy model, we can use the temperatures, pressures, and densities in each cell to determine the chemistry and opacities as described in the following §2.3 and §2.4, then carry out the integrals in eqs. (2) and (1).

In cases where we do not wish to consider a day-night temperature gradient, we can simply assign the same temperature regardless of longitude, and proceed with the same calculations. We refer to such atmospheres as “uniform” from here on out.

We assign mixing ratios for important species using pre-tabulated Tables of equilibrium chemistry computed by Burrows & Sharp 1999 and Sharp & Burrows 2007. The calculations used to construct these tables start with the solar composition of 27 chemical elements scaled depending on the metallicity of the table (solar abundances taken from Anders & Grevesse 1989). Then a Newton-Raphson solver is used to minimize the total free energy of the system tracking 330 gas-phase species, including the monatomic forms of the elements, as well as about 120 condensates. The calculations also include electrons, H^-, and H⁺. Refractory elements (Ti, V, H₂O, and Fe) are withdrawn from the equilibrium amounts in stoichiometric ratios via an ad hoc algorithm to approximate condensate rainout. For metallicities other than those for which we have Tables, we interpolate linearly in log(Z/Z_⊙).

The tables record the mixing ratios of 30 species for a grid of 805 temperatures evenly spaced in log between 50 K and 5000 K, and 108 pressures evenly spaced in log between 400 atm and 7.982$\times 10^{-9}$ atm.

When aerosols are included, the material tied up in the aerosols is not taken into account in the calculations of gas phase chemistry, so we are essentially assuming that the time scales for photochemical processes and condensation are long compared to gas-phase interactions. We also assume that replenishment of new material from deeper in the atmosphere keeps the gas phase unchanged, or that the amount of material tied up in aerosols is negligible compared to the gas-phase abundances of relevant atomic species.

It can be useful to determine which pressure levels and longitudes impact the transit spectrum as the structure of the atmosphere and the properties of the aerosols in it vary. Our code lends itself well to determining which portions of the atmosphere are probed by transit spectra both by altitude and by longitude. If one computes the transit spectrum when the opacity in a single pressure level or in a single longitude slice is set to zero and finds a difference compared to the transit spectra of the full atmosphere, this indicates that the portion of the atmosphere in question is probed by the transit spectrum.

Using this method, we can compute the width of the transit limb by determining the maximum and minimum longitudes slices which cause a non-zero change in the transit spectrum. This can be split into a day-side width and a night-side width, since, as we will show in the following section, they are often not the same.

We begin with clear atmospheres to evaluate how previous results hold up when the abundances of all the significant opacity sources are assumed to vary, and to provide a point of comparison for the subsequent studies which incorporate clouds and hazes. This is also the first study to directly try to fit a model with a day-night temperature gradient. This allows us to evaluate whether spectra could contain sufficient information to actually constrain separate day- and night-side temperatures and constrain how quickly or slowly the atmosphere transitions from one to the other. Such information could inform our understanding of heat-redistribution and circulation on tidally-locked exoplanets, providing a useful link between phase curve observations and transit observations.

Figure 10 shows the simulated transit spectra (colored dots with shaded error envelope) along with the model corresponding to the median values of the posterior distributions mapped out by the MCMC retrievals (black dashed lines). The left panel shows the control case: a uniform temperature atmosphere fit by a single T-P profile. The center panel shows the spectra for atmospheres with day-night temperature gradients fit by a model which includes a day-night temperature gradient. The right panel shows the test-case: data simulated for an atmosphere with a day-night temperature gradient, but fit using a single isothermal T-P profile. One can readily see that the best-fit spectra mostly fit the data very well. This agrees with the finding of Caldas et al. 2019 that, for clear atmospheres, there is not an indication from typical goodness-of-fit estimators that a single T-P model is inadequate, even if it is causing biased results. The only apparent discrepancy is actually when HAT-P-7b is fit by a model which accounts for the day-night temperature gradient. The best-fit spectra underestimates the strength of the CO feature around 4.5-5.5 $\mu$m.

In the clear uniform temperature control case, all the retrieved values are very accurate and the posteriors are approximately Gaussian (see Figure 11). The varying width of the posteriors track with the varying $SNR$ of the targets. These same variations in $SNR$ will affect the rest of the fits as well.

In the clear case fit with a temperature gradient (Figure 12), we see that some objects fall in an area of parameter space with large degeneracies between $\beta$, T_day and T_night (HD189733b, HD209458b, HD149026b, and HAT-P-7b), while other spectra are able to constrain the more complex model (WASP-103b and WASP-12b). The correct day-side temperature is always within the 16-84 percentile range of the posterior, but there is a long tail towards higher temperatures. The night-side temperature posteriors always contain the true value within the 16-84 percentile range, but it also has a tail, this time tending towards lower temperatures. These skews towards high day-side temperatures and low night-side temperatures correspond to the very long tail in the posterior of $\beta$, extending up towards large angles. This degeneracy simply reflects the geometry of our toy-model and the finite angular extent of the atmosphere probed by transit spectroscopy. If you lower T_night and raise T_day, but broaden $\beta$ you can get the same range of temperatures with longitude to fall within the region about the terminator probed by the transit spectrum. For WASP-12b and WASP-103b, there is an actual constraint placed on $\beta$, so it seems the CO feature which is formed at higher altitudes in hot atmospheres is key to cutting off the long tail of $\beta$ towards high values. The metallicities retrieved for the day-night temperature gradient control case are slightly biased towards lower metallicities in the objects which don’t have a constraint on $\beta$, but still agree with the true metallicity within their error bars. WASP-12-b actually has a tighter metallicity constraint in the temperature gradient case than in the uniform temperature case. WASP-103b has a multi-modal metallicity posterior with one peak centered on the correct metallicity.

Figure 13 shows all the posteriors for the clear test case, where we have fit a temperature gradient with a single T-P profile. The retrieved temperatures fall between the day- and night-side temperature, favoring the day-side very strongly for HD149026b, HAT-P-7b, WASP-12b and WASP-103b. This is consistent with results reported in Caldas et al. 2019, and with what we expect from our calculations demonstrating that the day-side contributes more to the transit spectrum than the night-side (Figure 3). For HD189733b and HD209458b the retrieved temperature is much closer to the average of the day-side and night-side temperatures. The retrieved values of $P_{0}$ and $Z$ deviate from the ground truth in an attempt to mimic the effects of the day-night temperature gradient, while forced to use a single temperature. Metallicity is important because it is tied to planet formation theories, but there are other parameters we may be misunderstanding. None of the metallicity posteriors peak at the true value. For HD209458b and WASP-12b the 16-84 percentile range excludes the true metallicity entirely. For HAT-P-7b and HD209458b the true value falls right at the edge of the 16-84 percentile range. For HD149026b and HAT-P-7b, with the lowest $SNR$, the 16-84 percentile range includes the true metallicity. For HD189733b, $P_{0}$ is biased slightly high and $Z$ is biased slightly low, but the values just barely agree with the true values within the error bars. This object has a relatively small day-night temperature gradient but a high $SNR$, so it is right at the boundary of having sufficient $SNR$ that accounting for the effects of its small day-night gradient matters. It is definitely not necessary to account for the day-night temperature gradient in HD149026b. Other objects retrieve the wrong value for one or both of $P_{0}$ and $Z$, so it will be necessary to account for day-night temperature gradients to get unbiased results.

The clear MCMC experiment results indicated that the presence of a day-night temperature gradient on a tidally-locked hot Jupiter has a significant enough effect on its transit spectrum to bias our retrieval results when it is unaccounted for, provided the $SNR$ is not too low, and the difference in temperature between the day-side and night-side is larger than around 400 K or so. These findings are consistent with previous studies. We have also found that, for some planets, there is sufficient information encoded in the transit spectrum to constrain a separate day-side and night-side temperature.

Now we show results for the same MCMC experiment using transit spectra of planets with a thick enstatite slab aerosol in their atmosphere. This haze is present across the day- and night- side extending up to pressures of 10^-4.5 bars. The thick, high altitude haze introduces some new degeneracies between $Z$ and aerosol properties like $F$ and a_m. It also strengthens the degeneracy between $Z$ and P₀ since key gaseous features in the optical wavelength range are obscured. Adding in aerosols always complicates matters regardless of the presence of a day-night temperature gradient. This is well-established by previous retrieval studies. In this sub-section and the following sub-section 5.3, we build up from there to explore interplay between aerosols and day-night temperature gradients. Are degeneracies exacerbated or broken? If day-night temperature gradients are not accounted for in hazy atmospheres, do we see similar biases in the retrieved metallicities as has been noted for clear atmospheres?

Figure 14 shows the simulated data (colored dots with shaded error envelope) and spectra corresponding to the median values of the posteriors (black dashed lines). We again show the uniform temperature and temperature gradient control cases and the mixed fit test case. One can see that the haze has filled in a lot of the troughs in gaseous absorption. For the hottest planets with a temperature gradient, the resonance feature for MgSiO₃ is very prominent around 10 $\mu$m. All the best-fit spectra seem to agree well with the simulated data except for the mixed fits for WASP-103b and WASP-12b, the two objects with the hottest day-side temperatures and the largest difference between their day-side and their night-side temperature. The single temperature model has trouble fitting both the aerosol and the top of the CO absorption at 4.5 $\mu$m. The posteriors for all the parameters in the hazy uniform-temperature control case and the hazy temperature gradient control case are shown in Figures 16 and 15 respectively. The posteriors for all the parameters in the hazy test case are shown in Figure 17.

For the hazy uniform-temperature control case (Figure 15), the temperatures are still accurate, though some precision is lost relative to the clear atmosphere. The reference pressure, P₀, has a looser constraint than the clear atmospheres as well. It even becomes a lower limit for some objects, but it is still consistent with the true value. The cloud-top pressure is precise and accurate for all the objects. $F$ is always just a lower limit. The metallicity has a flat posterior (HD189733b, HAT-P-7b, WASP-103b, and WASP-12b) or a lower bound within our priors (HD209458b and HD149026b). The poor constraint on metallicity is due to a strong degeneracy between the fraction of available material incorporated into the haze ($F$), the modal particle size (a_m), and the metallicity ($Z$). The reference pressure (P₀) is also degenerate with metallicity when aerosols obscure gas features in the optical wavelength range. The modal particle size and size-dispersion are retrieved for HD189733b, but their posteriors only provide upper limits for HD209458b, HD149026b, HAT-P-7b, WASP-103b, and WASP-12b.

The results for the hazy temperature gradient control case (Figure 16) show that, rather than heightening degeneracies, the presence of both a day-night temperature gradient and a thick haze provided tighter constraints on P₀ and $Z$, than a thick haze in a uniform atmosphere, and provided tighter constraints on temperatures and $\beta$ than a clear atmosphere with a temperature gradient. The presence of the haze tightens the constraints on both day- and night-side temperatures significantly because it enables a much better constraint on $\beta$. Recall that, in §4, we saw that the presence of a slab aerosol at altitude tends to steepen the dependence of the transit spectrum on $\beta$. Similar to the hazy uniform-temperature control case, the posteriors for P₀ are broader than when clear, and they have a tail towards higher values. However the effect is not so severe as for the uniform-temperature hazy spectra. The posteriors for $F$ still only provide a lower limit, but the other aerosol properties are all accurately and relatively precisely retrieved. For the day-night temperature gradient control case, we get a flat posterior or a lower bound within the posteriors for all the objects except WASP-12b. The presence of a day-night temperature broke degeneracies between $Z$ and P₀ for hazy WASP-12b, allowing us to retrieve a relatively precise and accurate metallicity measurement, even in the presence of a thick haze. However, it is not as precise as the measurement for a clear atmosphere. WASP-12b has both the highest day-side temperature and also a higher SNR than the next hottest object WASP-103b, so it has the tightest constraint on $\beta$ and P₀ which allows a tighter constraint on $Z$. For the other objects, there is only a weak lower limit on metallicity, but the presence of a thick haze can weaken or erase constraints the metallicity in many planets, regardless of whether they exhibit day-night temperature gradients.

In the test case with a mixed fit (Figure 17), we see that, even when a haze is present and spans the full limb, one will obtain biased results if they ignore day-night temperature gradients. The MCMC fits find much broader posteriors for the temperature than for the clear results. The fits for WASP-103b and WASP-12b are actually multi-modal with one peak at the true terminator temperature and one peak skewed towards the day-side temperature. The reference pressure formally agrees with the true value for all the objects, but one of the modes for WASP-103b and WASP-12b pushes up against the deepest allowed reference pressure. None of the posteriors formally disagree with the true metallicity, but we do see different results than the control case with uniform temperatures and the control case with a temperature gradient. A wider prior on metallicity would be better to fully tease out the implications, but we can see a few hints from the posteriors. WASP-12b and HD189733b take on the shape of a lower-bound that is tending towards excluding the true metallicity. HD209458b has shifted to an almost uniform posterior across the allowed metallicities, whereas for the fit to a truly uniform temperature atmosphere it was possible to rule out the lowest metallicities. HAT-P-7b and HD149026b are largely unaffected. The recovery of the aerosol size distribution performs roughly as well as or better than the uniform-temperature hazy control case. This speaks to the nature of the aerosol spectral signatures imprinted in our simulated data. When the data is generated using a day-night temperature gradient for the hotter objects the wavelength-dependent cross section of MgSiO₃ is clearly imprinted on the spectra, but when the data was generated using a single temperature the haze imparts a predominantly gray opacity on the spectrum. The posteriors for $F$ still show a loose lower bound for all the objects. However, the posterior for HD189733b formally excludes the true value of $F$. This was the object that seemed to be adequately fit by a single T-P profile when it was clear, due to it’s small difference between day-side and night-side temperature. Adding in a haze has made it possible and necessary to use a model with a temperature gradient instead.

Some evidence hints that clouds may be condensing on the night sides of hot Jupiters and evaporating on the day sides, so we decided to include an experiment that approximates this behavior. We simulated transit spectra which include a cloud of MgSiO₃ according to the phase equilibrium cloud model described in §2.4.2. Figure 18 shows the simulated data and best-fit models. For the data simulated with a uniform temperature atmosphere, it should be noted that only HD149026b had a temperature that would allow MgSiO₃ to condense and contribute significantly to the transit spectrum (see left panel). When the planets had a day-night temperature gradient, MgSiO₃ could condense somewhere within the limb for all the objects except HD189733b (see center and right panels). The mixed fits with an equilibrium cloud for the cooler objects (HD149026b, HD209458b, and HD189733b) would pass a goodness-of-fit criterion, but the hotter objects (HAT-P-7b, WASP-103b, and WASP-12b) would certainly fail. We examine the posteriors of these retrievals in more detail below.

Figure 19 shows all the posteriors for the control case with a uniform temperature. Since only HD149026b can form a significant enstatite cloud, that is the only posterior which places a constraint on the particle size distribution (maroon lines and dots). The retrieved values for modal particle size and size dispersion are both accurate and precise. The posterior for $\alpha$ only provides a very week upper bound, implying that the majority of the cloud opacity is contributed near the cloud base pressure, so varying $\alpha$ has little effect on the shape of the transit spectrum. The reference pressure is still accurate but less precise than for a clear atmosphere, due to the degeneracy with $Z$. The retrieval for HD149026b can only place a lower limit on metallicity (within our priors) due to a degeneracy with $\alpha$ and P₀. HD209458b shows a slight hint of a cloud (lighter purple lines and dots). Posteriors for all the other objects have essentially flat posteriors for a_m, $\sigma_{a}$, and $\alpha$, then similar posteriors to the clear atmospheres for P₀, $Z$, and $T$. Such a result from your fit would indicate that you should run it again leaving the cloud parameters out.

Figure 20 shows all the posteriors for the cloudy control case simulated and fit with a temperature gradient. It seems that, if condensed clouds are present and well-described by equilibrium models, then a lot can be learned about both the cloud particle size distribution and the day-night temperature structure. Similar to the results for the slab aerosol, the presence of a cloud improves the constraint on $\beta$ and thus on the day- and night-side temperatures relative to the clear case. This comes at the cost of broadening the constraint on metallicity relative to the clear case, but, for four of the objects at least, the posterior still shows a distinct peak centered on the true metallicity. For HD189733b, HD209458b, and HD149026b, the constraint on metallicity is actually more accurate than the clear case with a temperature gradient, though it is also much less precise. The modal particle size and the dispersion of the log-normal size distribution are well constrained for all the equilibrium cloud fits, but $\alpha$ is just an upper bound in every case. Rather than confusing the picture, it seems that the combined presence of condensing clouds and temperature gradients can provide deeper insight into the atmospheres of tidally-locked planets.

Figure 21 shows the results for the test case, using a single temperature to fit data simulated with a temperature gradient and an equilibrium cloud. Most notably, in the three planets with the hottest day sides, a single temperature cannot account for both the tall CO feature around 5 $\mu$m and the formation of an enstatite cloud within our equilibrium cloud parameterization. The MCMC model ends up favoring a higher temperature, so there is no constraint on particle size, width of distribution, or the relative scale height of gas and aerosol for HAT-P-7b, WASP-103b, and WASP-12b. This can be seen in the right-hand panel of Figure 18. The black dashed lines overlying the yellow, orange, and pink data don’t have any clouds in them. WASP-12b, WASP-103b, and HAT-P-7b almost exactly retrieve their day-side temperature alone. In order to bring the gaseous absorption in the optical up to near the depths of the transit spectra with a cloud, the reference pressures are biased to higher values. Meanwhile, to bring the absorption in the mid infrared down and flatten things out, the metallicities are biased to lower values. The mixed fits for HD209458b and HD149026b are able to replicate the data very well. To form the cloud base at the right pressure level, HD209458b retrieves a temperature even higher than its day-side temperature, while HD149026b retrieves a temperature just below its day-side temperature. The correct reference pressure, modal particle size, and dispersion of the particle size distribution are constrained for these objects, but the metallicities skew towards values that are too low to get approximately the right balance between gaseous opacity and cloud opacity in the transit spectrum. For HD189733b, there is not much cloud at all, so the posteriors for P₀, $Z$, and $T$ are similar to the clear mixed fits. Some very small amount of cloud must be present though, since posteriors for a_m, $\sigma_{a}$, and $\alpha$ are not simply flat lines both here and in the temperature gradient control case. The posterior for a_m is mostly flat, but it has a peak that is much smaller than the true a_m, whereas the posterior for a_m peaks at the correct value in the cloudy temperature gradient control case. Both the structural parameters and the aerosol-related parameters end up biased in this case.

These results show that if the aerosols in tidally-locked hot Jupiters are condensing species, then ignoring day-night temperature gradients in models can lead to severe and complicated biases. For planets with extremely hot day-sides, the inadequacy of the single temperature model will be apparent, but for others it won’t be. When the day-night temperature gradient is accounted for, one obtains accurate constraints on structural and aerosol-related parameters in most cases.

Taken all together, the results of these experiments show that the importance of day-night temperature gradients is highly dependent on the properties of the object in question, the $SNR$ of the data, and the presence or absence of aerosols. When HD189733b contains a haze, it is important to account for the day-night temperature gradient, but, when it is clear, the day-night temperature gradient can almost be ignored. This planet has the smallest difference between its day-side temperature and its night-side temperature. HD209458b has a similar difference between its day-side temperature and night-side temperature, but a larger $SNR$, especially when the atmosphere is clear. If there is a haze, then fitting HD209458b with a single T-P profile does not retrieve significantly biased results for any parameter, but, if it is clear, then a single T-P profile biases metallicity to the point that it formally disagrees with the true value. If it has a cloudy atmosphere, then the correct aerosol properties are retrieved, but the wrong metallicity and temperature. It is always important to account for the day-night temperature gradient in WASP-103b and WASP-12b whether clear, hazy, or cloudy. These two planets have the largest difference between their day-side temperature and their night-side temperature. If condensed clouds are present in the atmospheres of WASP-103b and WASP-12b, it may be apparent that a single T-P profile model is inadequate. HAT-P-7b and HD149026b have large differences between their day- and night-sides, but also have lower $SNR$, and thus, larger uncertainties in their retrieved parameters. For these two planets, the biases induced by ignoring temperature gradients didn’t shift retrieved metallicities to the extent that they formally disagreed with the true values. The exception is if HAT-P-7b contains a condensed cloud. In that case, the best fit using a single T-P profile retrieved an upper bound on the metallicity that excluded the true value, but the corresponding spectra would likely fail to satisfy goodness-of-fit criteria. In the cases investigated here, the effects of aerosols and the effects of day-night temperature gradients are not degenerate. In fact, if both are present in the atmosphere and accounted for in the retrieval model, their combined effects can actually tighten constraints on relevant parameters.