Signatures of Clouds in Hot Jupiter Atmospheres: Modeled High-resolution Emission Spectra from 3D General Circulation Models

We compute synthetic emission spectra directly from the GCM simulations previously presented in RR19 and summarized as follows. RR19 used a GCM (Rauscher & Menou 2010, 2012, 2013) that solves the primitive equations of meteorology, coupled to a double-gray, two-stream radiative transfer scheme based on Toon et al. (1989). Atmospheric temperatures, winds, radiative fluxes, and idealized cloud distributions were simulated for a hot Jupiter with a mass, radius, and irradiance temperature based on Kepler-7b (Latham et al. 2010; Demory et al. 2011, 2013, model input parameters listed in Table 1).

The GCM's radiative transfer calculations include the effects of aerosol scattering and absorption with condensate clouds parameterized as Mie scatterers with horizontal spatial distributions solely dependent on the temperature field. In this idealized approach, aerosol scattering was included where the atmospheric temperatures fell below the expected condensation temperature for given condensable species. Four species were included: MnS, Al₂O₃, Fe, and MgSiO₃. The scattering properties (i.e., single-scattering albedo and asymmetry parameter) were precomputed for specific cloud species assuming particles were 0.2 μm in radius at two wavelengths—0.65 and 5.0 μm—representative of the visible and thermal channels of the double-gray model. Where clouds of different species overlapped, the scattering properties were averaged, weighted by the relative optical thickness of each species. Aerosol abundances were taken to be proportional to the compositional abundance of the condensing gas with constant volume mixing ratio, but scaled by an adjustable parameter to control the overall optical thickness of the cloud.

The GCM vertically resolved the atmosphere using a sigma coordinate system, in which the computed temperatures, winds, and clouds were defined at the discrete pressure levels corresponding to these discrete sigma levels, which were ultimately converted into altitude assuming hydrostatic equilibrium. There was no spatial averaging beyond this resolution.

RR19 focused on four cases that explored basic assumptions regarding the cloud thickness and extent. Half the simulations assumed optically thicker clouds equivalent to condensing 1/10 of the gaseous species; the other simulations assumed thinner clouds with only 1/100 of the gas condensed. For each assumed thickness, two different assumptions regarding the vertical extent of the cloud were explored. In one pair of simulations, clouds were truncated ∼1.4 scale heights (five vertical layers) above the cloud base pressure, representing an atmosphere with weak vertical mixing. These two cases were referred to as the compact thick and compact thin cases. For contrast, in the other pair of simulations, clouds were allowed to extend to the 0.1 mbar pressure level (temperature permitting and regardless of the base pressure). These were referred to as the extended thick and extended thin models, and represented atmospheres with more efficient vertical mixing.

For each of these cloud models, RR19 sought to evaluate the importance of cloud radiative effects in shaping the atmospheric temperatures and fluxes. As such, they performed two versions of each simulation. In the first version, clouds were included throughout the duration of the 2000-orbit simulation, continually forcing and responding to the temperature field. They referred to this as active cloud modeling since the clouds actively influenced the thermal and dynamical structure of the atmosphere. For each of these active cloud simulations, RR19 ran a contrasting model in which clouds were absent throughout the simulation (i.e., a clear model), and only post-processed by comparing condensation curves to the final temperature field. The latter post-processing approach has had precedence in the literature in predicting and interpreting hot Jupiter phase curve data (e.g., Kataria et al. 2016; Parmentier et al. 2016). In RR19 the difference between the active and post-processed cloud cases demonstrated the role of aerosol radiative forcing and feedback in the model.

We generated thermal emission spectra at high resolution (R ∼ 10⁶) over the 2.308–2.314 μm wavelength range, corresponding to a strong CO band head that has been the target of exoplanet atmospheric characterization with the CRyogenic high-resolution InfraRed Echelle Spectrograph instrument on the Very Large Telescope (e.g., Brogi et al. 2013, 2014, 2016; de Kok et al. 2013, 2014; Schwarz et al. 2015). From each GCM output from RR19, we calculated spectra in the planet's rest frame, with and without including Doppler effects from atmospheric motion (winds and rotation), at 24 evenly spaced orbital phases, corresponding to different viewing geometries. 3D temperatures, winds, and cloud optical depths (computed at 5.0 μm) throughout the planet's atmosphere were taken from the last time step of the GCM simulations.

In our radiative transfer calculations, we modeled clouds as a source of pure thermal absorption/emission. We note, however, that aerosols can potentially scatter light back into the outgoing beam and partly mitigate the overall observed extinction, particularly if single-scattering albedos are high and scattering phase functions are strongly forward-scattering. ⁸ Here, because the total cloud distributions are optically thin in the infrared (IR; see Figure 1), and because MgSiO₃, Al₂O₃, and MnS are near-isotropic scatterers in the IR (RR19), we assume that the effects of non-isotropic scattering are relatively small. Hence, we neglect the contribution of scattering to the observed planet flux and leave a more complete scattering treatment to future work. ⁹

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Following Zhang et al. (2017), we calculated the emission spectrum from the GCM output by summing the intensity of all emergent lines-of-sight from the planet's visible side. The emergent intensity of each ray was calculated using the standard radiative transfer equation in the thermal emission approximation

$$\begin{eqnarray}&&{I}_{\lambda }={B}_{\lambda }({\tau }_{\max }){e}^{-{\tau }_{\max }}+{\int }_{{\tau }_{\max }}^{0}{B}_{\lambda }(\tau ){e}^{-({\tau }_{\max }-\tau )}\,d\tau ,\end{eqnarray} \tag{ 1 }$$

where τ is the slant optical depth along the sight line (implicitly wavelength dependent; defined below), B_λ (τ) is the (Planckian) source function evaluated at the local temperature, and ${B}_{\lambda }({\tau }_{\max })$ is the source function evaluated at the base layer of the atmosphere, defined here as ∼100 bar in pressure (RR19). Here, the first term accounts for radiation emitted from the bottom of the atmosphere and attenuated by gas and clouds along the line of sight (LOS). The second term accounts for radiation emitted and absorbed by each parcel of gas and clouds in the path of the initial light ray.

The slant optical depth is defined as

$$\begin{eqnarray}&&{\tau }_{\lambda }=\int ({\kappa }_{\mathrm{gas}}+{\kappa }_{\mathrm{cloud}}){dl},\end{eqnarray} \tag{ 2 }$$

where κ_gas is the wavelength-dependent gas opacity evaluated at local atmospheric temperature and pressure, κ_cloud is the local cloud opacity, and dl is the LOS path length through each grid cell encountered by the ray. We assume solar composition gas in local thermochemical equilibrium and that cloud opacities are gray (i.e., wavelength independent) over the narrow wavelength range of the calculations pursued in this work.

For these LOS calculations, we used altitude as the vertical coordinate after interpolating the GCM output to a grid of constant Δ-altitude. We note that because different parts of the planet had different scale heights, not all of the T–P profiles extended upward to the same altitude—the local opacities were set to zero at altitudes where the pressure dropped below the upper boundary of the atmosphere in the GCM simulations (∼57 μbar; RR19).

For Equation (2), we converted the cloud optical depth d τ_cloud of each condensate species in a grid cell of the GCM (expressed at 5.0 μm by RR19) into an effective extinction opacity at 2.3 μm with the following transformation:

$$\begin{eqnarray}&&{\kappa }_{\mathrm{cloud}}=\frac{d{\tau }_{\mathrm{cloud}}}{{ds}}\frac{{Q}_{2.3\mathrm{\mu m}}}{{Q}_{5.0\mathrm{\mu m}}},\end{eqnarray} \tag{ 3 }$$

where the ratio of the extinction efficiencies (Q_{2.3 μm}/Q_{5.0 μm}) is used to convert extinction at 5.0 μm to the extinction at 2.3 μm, and ds is the differential path length defined in the radial direction. The calculation described in Equation (3) was performed individually for each of the four cloud species from the GCM (given in Table 2), and then the total cloud opacity κ_cloud was taken as the sum of the single-species opacities.

Table 2. Cloud Scattering Parameters

Parameter	MgSiO3	Fe	Al2O3	MnS
Molecular weight, μg [g mol−1]	100.4	55.8	102.0	87.0
Mole fraction, χg	3.26 × 10−5	2.94 × 10−5	2.77 × 10−6	3.11 × 10−7
Particle density, ρ [g cm−3]	3.2	7.9	4.0	4.0
Refractive index, $\tilde{n}$	1.5 + i(4 × 10−4)	4.1 + i8.3	1.6 + i(2 × 10−2)	2.6 + i(1 × 10−9)
Extinction efficiency (λ = 2.3 μm), Q2.3 μm	0.07	1.25	0.12	0.56
Extinction efficiency (λ = 5 μm), Q5.0 μm	0.01	0.16	0.02	0.02
Optical depth per bar (thin clouds), τ/ΔP	29.0	105.1	3.4	1.5
Optical depth per bar (thick clouds), τ/ΔP	290.0	1051.5	34.3	15.3

Note. All scattering parameters are shown for 2.3 μm, unless otherwise specified. Optical depths per bar (τ/ΔP) were calculated from Equations (1) and (2) in RR19, based on the computed extinction efficiencies (${Q}_{e}^{(\lambda )}$) and assumed mole fractions (χ_g ), molecular weights (μ_g ), particle radii (r), particle densities (ρ), planet surface gravity (g), and vapor condensation fractions (f = 0.01 for thin clouds and f = 0.1 for thick clouds) in an atmosphere of Jovian mean molecular weight. The complex refractive indices ($\tilde{n}$) used to calculate ${Q}_{e}^{(\lambda )}$ were taken from Kitzmann & Heng (2018), and χ_g and ρ were taken from RR19, with relevant references therein.

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Values for the extinction efficiencies were calculated using spherical particles and the indices of refraction cited in Table 2. Following RR19, we computed these parameters using the Mie scattering code developed by M.I. Mishchenko (de Rooij & van der Stap 1984; Mishchenko et al. 1999), assuming a log-normal particle size distribution with an effective mean radius of 0.2 μm and variance of 0.1 μm. Further details on the assumed aerosol scattering properties in the GCM can be found in RR19.

We assumed that the planet's orbital radial velocity can be readily isolated from observational data, and we therefore ignored its contribution to the total motion along the LOS in our calculations. Following Zhang et al. (2017), we accounted for the remaining local LOS velocity as ¹⁰

$$\begin{eqnarray}\begin{array}{rcl}{v}_{\mathrm{LOS}} & = & -u\sin (\phi +\varphi )\\ & & -v\cos (\phi +\varphi )\sin (\theta )\\ & & +w\cos (\phi +\varphi )\cos (\theta )\\ & & -{\rm{\Omega }}({R}_{{\rm{p}}}+z)\sin (\phi +\varphi )\cos (\theta ),\end{array}\end{eqnarray} \tag{ 4 }$$

where the first term represents the contribution from the zonal (i.e., east–west) wind speed u, the second term accounts for the meridional (i.e., north–south) wind speed v, the third term accounts for the vertical wind speed w, and the fourth term accounts for the planet's rotation. In the planet's reference frame ϕ is longitude, θ is latitude (both fixed to the planet's rotation), and φ is the phase of the orbit. In the final term, Ω is the angular rotation speed of the planet (in radians per second; assumed to have been tidally locked into synchronous rotation), R_p is the radius of the planet at the base of the atmosphere, and z is the vertical height in the atmosphere above R_p. Assuming an edge-on orbit, we defined the substellar point as ϕ = θ = 0, with transit geometry corresponding to an orbital phase of φ = 0.

Our radiative transfer accurately accounted for the 3D geometry of the atmosphere, following the methods outlined initially in Miller-Ricci Kempton & Rauscher (2012) and Zhang et al. (2017). We self-consistently incorporated the effects of LOS motion by evaluating the local opacities (in Equation (2)) at their Doppler-shifted wavelengths according to

$$\begin{eqnarray}&&\lambda ={\lambda }_{0}\left(1-\displaystyle \frac{{v}_{\mathrm{LOS}}}{c}\right),\end{eqnarray} \tag{ 5 }$$

where λ₀ is the rest-frame (i.e., unshifted) wavelength and c is the speed of light.

We calculated a spectrum for a particular orbital phase by dividing the visible hemisphere of the planet into 2304 individual cells, determined by the latitude–longitude grid of the GCM output, and propagating a light ray along the LOS through the atmosphere at each cell according to Equation (1). The disk-integrated flux from the planet was the sum of all emergent light ray intensities weighted by the solid angle subtended by each grid cell.

Longitude–latitude maps of temperatures, winds, and cloud coverage are shown in Figure 1 for the clear atmosphere and post-processed and active cloud models with extended thick cloud properties. The corresponding vertical thermal structures are shown in Figure 2. Temperature and cloud maps and vertical temperature profiles for the other cloud models are also shown in Figures 7 and 8 in the Appendix.

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The clear model reproduced the standard pattern we have come to expect for the hot Jupiter atmospheric circulation regime. The wind structure within the observable atmospheric layers is characterized by eastward flow, dominated by a strong equatorial jet. This jet advects the hottest gas away from the substellar point before it has a chance to efficiently cool, resulting in an eastward offset of the brightest emitting region (see Figure 1).

From the global temperature profiles of the clear model (Figure 2), we can see more typical hot Jupiter behavior. The largest (longitudinal) temperature differences are at the lowest pressure levels (the highest regions of the atmosphere), while temperatures are more homogenized at depth. Large day-to-night temperature differences exist high in the atmosphere, but this transitions to the main temperature difference being between the equator and poles deeper in the atmospheres. In the clear atmosphere the incoming optical photons are absorbed at deeper atmospheric levels than where the thermal IR radiation from the planet is emitted to space, meaning that temperatures decrease with pressure. (Deviations from this at ∼10 bar are the result of advection dominating the temperature structure at those depths.)

When clouds were included in the GCM, the thickest clouds form near the poles and western terminator, where cooler temperatures allow for MgSiO₃, MnS, and Fe to condense. On the dayside, this leads to enhanced cooling at depth as clouds scatter and reflect incoming visible radiation. In contrast, Al₂O₃, with its relatively higher condensation temperature, also forms clouds over the much of the hot dayside and warms the visible atmosphere through absorption of stellar and thermal radiation. The relative importance of these effects depends on the thickness and vertical extent of the clouds, as discussed in RR19. Clouds forming higher in the atmosphere are more effective at altering the albedo and emission, and so the extended clouds have a more significant effect on the atmospheric temperatures. In general, the increased planetary albedo on the dayside and the increased extinction on the nightside lead to a cooler planet. In all cases, thermal emission is reduced on the nightside relative to a clear or post-processed cloud model, although dayside emission is increased in the thickest cloud models as heat escapes through relatively clearer skies on the warmer dayside.

An important characteristic of the active cloud models not previously noted in RR19 is the prevalence of thermal inversions along the western terminator (longitude = −90°) and dayside polar regions of the planet, as can be seen in Figure 2 (see also Figure 8 in the Appendix). We defined the thermal inversion at a particular location in the atmosphere as the maximum continuous increase in temperature from the bottom to the top of the atmosphere. Stronger thermal inversions tended to form as we transition from models with thinner and more compact clouds to models with thicker and more extended clouds.

Thermal inversions along the western terminator originate from excess cloud opacity as cool gas is advected eastward from the planet's nightside to the dayside. The cooler nightside temperatures permit cloud formation in the upper atmosphere along the terminator; as these cooler cloudy regions advect past the terminator to the dayside, stellar radiation is scattered and absorbed at the top of the cloud deck. This simultaneously increases heating in these cloudy layers of the atmosphere and cools the lower layers because radiation is impeded from penetrating below the clouds, leading to a thermal inversion. As gas on the planet's western limb continues to move eastward beyond the terminator and absorbs more radiation, the local temperature eventually increases beyond the condensation limit of the clouds. This leads to evaporation of the clouds and an overall decrease in the optical thickness of the atmosphere. As a result, stellar radiation is allowed to penetrate deeper into the atmosphere, returning the temperature profile to a normal, non-inverted one.

Thermal inversions at the planet's higher dayside latitudes can be explained by a similar process. Advection of cooler gas from the nightside of the planet allows clouds to form in the upper atmosphere along the western terminator, which scatter and absorb radiation and prevent heating of the lower layers of atmosphere. However, because stellar radiation is less intense at higher latitudes than near the equator, heating rates in the upper atmosphere fail to rise sufficiently to evaporate clouds and reverse the temperature inversions as the cooler region advects over the dayside.

We note that the presence and strength of thermal inversions ultimately depend on the composition of the atmospheric gas and aerosols. In their modeling, RR19 include Al₂O₃ and Fe, which tend to strongly absorb solar radiation and lead to thermal inversions (Roman et al. 2021). However, the microphysical models of Gao et al. (2020) suggest strongly scattering silicate clouds should dominate over the absorbing Fe clouds based on arguments of nucleation rates, thus potentially reducing the likelihood of inversions due to clouds. Although a full discussion of the impact of each individual cloud species and assumptions regarding composition are beyond the scope of the this work, the results discussed here nonetheless reveal how clouds with assumed properties can alter the observed spectra.

While these temperature inversions were not discussed explicitly in RR19, we recognize their importance here because of their potential to impact thermal emission spectra via their effect on line shapes (i.e., emission versus absorption). In our radiative transfer calculations, some regions of the planet can be sources of emission lines and other regions can be sources the same lines in absorption, both in view at the same time. The implications of this are discussed in the following section.

From the GCM outputs, we calculated two emission spectra at each orbital phase: one with Doppler effects from LOS wind and rotation velocities turned on, and one with Doppler effects turned off. Each rest-frame spectrum was computed as a template for assessing the net Doppler shift produced in the corresponding Doppler-shifted spectrum and to visually examine the effects of Doppler shifts on our models. Figure 3 shows our computed rest-frame planet emission spectra and a subset of our Doppler-shifted spectra from the clear, post-processed cloud, and active cloud GCMs. Additional Doppler-shifted spectra for all orbital phases and GCM runs are shown with arbitrary continuum flux in Figures 9 and 10 in the Appendix.

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Emission spectra from the post-processed cloudy GCM are qualitatively similar in shape and depth to the spectra from the clear GCM. Because the thermal and dynamical structure of the post-processed atmosphere was identical to that of the clear atmosphere, differences in spectral features are only from gray opacity differences where the cloud deck was optically thick. This is most noticeable in the continuum flux. When the thickest clouds were covering the visible side of the planet, there was a greater deficit in the continuum flux. Since cooler nightside temperatures permitted more cloud formation, the nightside spectra from our post-processed model are the farthest offset from the nightside spectra from the clear model, while the dayside spectra have similar continuum levels in both models. This matches the results presented in RR19 for the broadband thermal phase curves.

The spectra were more drastically affected when we included active clouds in the GCM. Given similar cloud thicknesses above the photosphere in the active and post-processed cases, the additional differences seen in the active cloud case can be primarily attributed to the significant effects clouds have on the temperature field only captured in active modeling. The emission is more strongly altered due to changes to the photosphere temperature and high-altitude thermal inversions as opposed to direct attenuation alone.

First, we note that the continuum fluxes from the active cloud atmosphere are generally lower than the cloud-free and post-processed cloudy atmospheres, which is again consistent with the broadband thermal emission phase curves from RR19. The variations in continuum flux from the planet's brightest side to dimmest side are also smallest for the active cloud model. This is the result of more widespread and optically thick clouds forming in the active cloud model (see Figure 1), which act as an additional gray opacity source that attenuates the continuum flux. The enhanced cloud cover also increases the planetary albedo, which reduces the planet's equilibrium temperature.

More interestingly, the combined effects from additional gray cloud opacities and different thermal structure and dynamics give the spectra from these models features that are distinct from the clear and post-processed models in terms of continuum flux, absorption line depth, and Doppler effects (for the latter, see Section 3.3). A particularly notable result is the flattening of absorption features in the dayside emission spectra from the active cloud model. Recall that the active cloud atmosphere has strong, inhomogeneous temperature inversions (Section 3.1). Because emission features arise from regions of the visible side of the planet that exhibit temperature inversions along the LOS, these emission features overlap with and subsequently cancel out absorption features (of the same line) from regions with non-inverted T–P profiles. This effectively flattens out the features we see in the emission spectra at certain orbital phases.

Moreover, because the absorption/emission lines originate from different regions of the planet, they are created by light rays that traveled along different paths through the atmosphere, and hence encountered grid cells with different temperatures and velocities. Therefore, absorption lines can have different Doppler shifts compared to the same lines in emission. When the light rays from all regions of the planet's visible side are combined, the central wavelengths of the absorption and emission features do not necessarily coincide, leading to complex line shapes set by a superposition of both emission and absorption features.

For example, consider the complex shape of the Doppler-shifted ∼2.30928 μm H₂O feature in the active cloud spectrum at φ = 225° (Figure 3). This feature appears to have both a blueshifted absorption component and a redshifted emission component. At this orbital phase, much of the temperature-inverted region (at higher dayside latitudes) has crossed the sub-observer longitude and is receding away from the observer. Additionally, on the western limb of the planet, a thermally noninverted region is approaching the observer. The combined effect of this geometry is a spectrum that contains a redshifted emission line from the receding portion of the atmosphere and a blueshifted absorption line from the approaching side of the planet.

This Doppler splitting of spectral features is a unique characteristic of the active cloud GCM because of the active clouds' ability to create inhomogeneous thermal inversions in the upper atmosphere. Post-processed cloud models lack these spectral features because the clouds do not directly influence the temperature structure in the GCM, and hence cannot create inhomogeneous thermal inversions. We continue our discussion of Doppler splitting in the following section.

From Figure 3 (and Figures 9 and 10 in the Appendix) we can see that Doppler effects strongly influence the emission spectrum line profiles, and that these effects are phase dependent. The shape, depth, and position of each line are dependent on a complex combination of wind and rotation velocities along the LOS, inhomogeneous clouds that act as an additional source of opacity, and atmospheric temperatures that vary in latitude, longitude, and altitude. As discussed previously, spatially nonuniform temperature inversions (caused by clouds) can also cause spectral lines to appear simultaneously as emission and absorption features, but with different Doppler signatures.

To quantify how the sum of these factors affects the overall Doppler shift of the disk-integrated spectra as a function of orbital phase, we performed the following analysis. We determined the net Doppler shift at each orbital phase from our computed emission spectra by cross-correlating the Doppler-shifted spectrum with the corresponding unshifted template spectrum (calculated with Doppler effects turned off in the radiative transfer routine). We took the net Doppler shift at each phase to be the velocity shift corresponding to the peak of the cross-correlation function (CCF). The normalized CCFs from the clear and active cloud GCM spectra are shown for 24 orbital phases in Figure 4. The net Doppler shift as a function of orbital phase is indicated by the line connecting the CCF peaks in Figure 4, and this is shown for all the active cloud cases, along with the clear case and the post-processed extended/thick cloud case in Figure 5.

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For the clear atmosphere, the cross-correlation functions are well behaved, and we recovered Doppler shifts similar to those found by Zhang et al. (2017) for cloudless 3D GCMs representative of HD 209458b, WASP-69b, and HD 189733b. Broadly speaking, the net Doppler shifts are dominated by the relative motion of the planet's brightest region (i.e., the hotspot to the east of the substellar longitude). Net Doppler blueshifts were produced when the brightest region of the planet was approaching the observer and net Doppler redshifts were produced when the brightest region was receding.

Because the brightest region of the planet is advected to a longitude approximately 45° east of the substellar point, it rotates into view on the approaching limb of the planet when the orbital phase is about 45° past transit (φ ≈ 45°). Emission from this region dominates over dimmer emission from the receding limb, creating the maximum net blueshift of ∼0.5 km s⁻¹ we see at φ ≈ 45° as shown in Figures 4 and 5. Around the time when the bright spot reaches the sub-observer longitude (φ ≈ 135°), the Doppler shift transitions from a net blueshift to a net redshift as the bright spot begins moving away from the observer. The brightest region then reaches the receding limb of the planet at an orbital phase of φ ≈ 225°, lagging secondary eclipse by approximately 45°, creating the maximum redshift of ∼1 km s⁻¹ at φ ≈ 225°. Smaller-scale variations in the Doppler shift pattern are more difficult to explain qualitatively because they result from the combined effects of various properties of the complex 3D atmosphere, as mentioned above.

Spectra from our model with post-processed clouds had the most similar Doppler shifts to those from the clear atmosphere. This result is reasonable because the temperatures and wind structure of these two models were identical. Differences in the Doppler signatures therefore arose purely from the excess gray opacity from clouds. Since the cloud coverage was nonuniform, the deviations of the Doppler shifts from the clear model depend on the position of optically thin regions in the cloud distribution and hence are phase dependent. Consider, for example, the Doppler shifts around orbital phase φ ≈ 345° in Figure 5 (indicated by the dashed pink line for the post-processed cloud model). Here we see an enhanced blueshift relative to the clear model, which can be explained by fact that clouds on the planet's dayside are generally thinner than the clouds on the nightside, especially at longitudes east of the substellar point (see Figure 1). At φ = 345°, the visible longitudes range from +105° to −75°; thus, the approaching limb of the planet is relatively clear of clouds, while the receding limb is almost fully concealed by clouds. This attenuates the emission from the redshifted side of the planet, resulting in a stronger net blueshift. The inverse of this effect can be seen at an orbital phase of φ ≈ 255°, when the approaching limb of the planet is covered by clouds and the receding limb is relatively cloud-free.

It is obvious by looking at Figures 4 and 5 that the Doppler shifts from the active cloud model are much more complex and variable than those from the clear (or post-processed) model. The active cloud cross-correlation functions do not have the same quasi-Gaussian shape as the CCFs for the clear model and they vary in a more unexpected manner as a function of phase. An exact diagnosis of the shapes of the CCFs is challenging because the Doppler signals come from a mixture of spectral lines that originate from different spatial locations on the planet (and different heights in the atmosphere), and are each affected by a complex combination of rotation speed and inhomogeneous winds, temperatures, and cloud opacities.

We can attempt to understand the general behavior of the net Doppler shifts by tracking the relative motion of the regions of the atmosphere with temperature inversions (located around the higher dayside latitudes and along the meridian at a longitude of about −90°; see Figure 2) and the planet's brightest region. To aid with our interpretation of the Doppler shifts, we utilized an animation showing the visible temperature structure of the planet throughout the planet's orbit to evaluate when the planet's brightest region and thermally inverted regions were either approaching or receding away from the observer. A representative snapshot from the animation of the active cloud atmosphere with extended thick cloud properties is shown in Figure 6. Note that the brightest region of the active cloud atmosphere is shifted east of the substellar longitude by about 30°, less than it was in the clear atmosphere (as noted by RR19 ).

At φ ≈ 0° (i.e., transit geometry), both the bright spot and the thermally inverted regions are hidden from view. The visible temperature pattern is relatively uniform, so this alignment produces a spectrum similar to one from the clear atmosphere, as well as a simple CCF and almost negligible net Doppler shift. As the bright spot starts to become visible on the approaching limb, the spectra become slightly blueshifted. However, this blueshift is weaker than the clear model because emission features from the thermally inverted higher latitudes effectively decrease the strength of absorption features on the approaching limb relative to those on the receding limb.

As more regions with temperature inversions became visible, the emission from these regions effectively dilutes the absorption features on the approaching limb enough that the absorption features on the receding limb began to dominate the net signal. This leads to a gradual transition from a net blueshift to a net redshift, which occurs at an orbital phase of φ ≈ 60°, followed by a steady increase in the amplitude of the net redshift up to ∼2 km s⁻¹ at φ ≈ 105°.

The planet's brightest region then approaches and crosses over the sub-observer longitude, at an orbital phase of φ ≈ 120°. At this point, the distribution of visible high-latitude temperature inversions becomes approximately symmetric (the inversions along the western terminator are still hidden from view). Hence, the net Doppler shift of the spectra is approximately zero.

Note that the high-altitude thermal inversions can sometimes produce emission reversals near to the core of certain strong spectral lines (e.g., the ∼2.30892 μm feature in the active cloud Doppler-shifted spectra in Figure 9 (Appendix) has a small perturbation in the absorption line that moves from shorter wavelengths to longer wavelengths as phase increases from φ ≈ 90°–φ ≈ 135°). This effect is akin to the more familiar emission reversals in the cores of the Ca ii H and K lines in the spectra of main-sequence stars, although in our case the line shapes are more complex due to the asymmetry in the temperature field caused by the planet being irradiated from one side only. In our spectra, this effect can lead to the complex (sometimes bimodal) appearance of the CCFs shown in Figure 4.

Because the net Doppler shift is determined only by the peak of the CCF, which corresponds the to wavelength associated with the deepest part of the lines, it is sensitive to the relative positions of both the brightest regions and the temperature inversions. This explains why there is a transition from redshift to blueshift at φ ≈ 120°, as shown in Figure 5. The sensitivity of the net Doppler shift signal demonstrates a limitation of interpreting observed spectra using 1D models, which cannot capture the complexity of the spectra that arises from inherently 3D effects.

Next, there is a net blueshift as absorption features originating from the approaching limb dominate over the signal from the receding limb. This gradually transitions back to a net redshift, reaching a net zero Doppler shift at a phase of φ ≈ 150° and a maximum redshift of ∼2.5 km s⁻¹ at φ ≈ 210°. At φ ≈ 180°, the temperature inversions along the western terminator start to become visible on the approaching side of the planet, creating strong blueshifted emission lines. As before, these emission lines create perturbations in the absorption features, which gradually transition from shorter wavelengths to longer wavelengths as phase increases (Figure 9). Again, this leads to spectral lines with two local minima, giving rise to complex CCF shapes. This explains the bimodal CCF and sharp transition from the maximum redshift to the maximum blueshift (∼3 km s⁻¹) at φ ≈ 225° (Figures 4 and 5).

The Doppler shift functions for the other active cloud models shown in Figure 5 lie somewhere between the behavior for the atmosphere with extended thick clouds and the clear atmosphere. These can be explained in a similar way to the extended thick cloud case, since all of the active cloud GCMs exhibit stronger temperature inversions than the clear atmosphere (see Figure 8 in the Appendix). As expected, there are increasingly strong temperature inversions as we transition from GCMs with thinner and more compact clouds to GCMs with thicker and more extended clouds. Likewise, the Doppler shift functions become more distinct from the clear model (with greater shifts and sharper discontinuities) as the temperature inversions in the GCMs become more significant.

We note that while Doppler broadening is another interesting feature of the spectra, extracting information from it is not straightforward. Zhang et al. (2017) showed that the broadening has significant contributions from both rotation and winds (which are largely in the same direction as the rotation), with the relative contributions depending on the particular planet. An important consequence of this is that we cannot strongly constrain the rotation rate because it does not uniquely define the broadening (Beltz et al. 2021). Moverover, the broadening in the cloudy versions of the spectra is complex because the shapes of the CCFs are superpositions of multiple components. Therefore, we do not include a detailed interpretation of Doppler broadening here because we are unable to provide an unambiguous interpretation.