text:
\section{Introduction}

Clouds are present in all solar-system atmospheres and have been shown to be ubiquitous in exoplanet atmospheres~\citep{iyer2016characteristic,sing2016continuum,Wakeford2019}. They often hinder our ability to probe large regions of the atmosphere and are currently the main source of uncertainty when estimating the abundances of chemical species from exoplanet spectra \citep{Deming2013}, a situation that will still remain a challenge with the arrival of the James Webb Space Telescope (JWST)
~\citep{greene2016characterizing}.

Currently, most studies addressing the role of clouds in atmospheric retrievals have tackled the case of transmission spectra \citep{2014Lee,Barstow2016,2017Macdonald,wakeford2017hst,fisher2018retrieval,tsiaras2018population,Benneke2019,2020Carter}. They have shown that different cloud parameterisations lead to similar retrieved abundances~\citep{Mai2019,2020BarstowClouds}, but that neglecting the presence of clouds or neglecting their spatial variability could lead to incorrect retrieved abundances
~\citep{Line2016,2019Welbanks,2019Pinhas}.

Another way to sense exoplanet atmospheres is through their emission spectrum. Phase curves can be used to measure the emission spectrum of the planet at different phase angle, from the dayside to the nightside. Phase curves have been used to build temperature and cloud maps, showing the prevalence of clear and hot daysides \citep{kreidberg2014precise,Parmentier2016,stevenson2017spitzer,2020Irwin} and cloudy and cold nightsides \citep{Showman2009,Kataria2015,Parmentier2016,Helling2019,Beatty2019,Keating2019}. Furthermore, the radiative effect of nightside clouds have been shown to complicate the link between observable quantities, such as the phase curve amplitude and fundamental atmospheric properties such as the heat redistribution efficiency \citep{Parmentier2021}.

Several cloud formation models of varying degrees of complexity have been used to investigate the formation of clouds and the diversity of their compositions in exoplanetary atmospheres, \citep[e.g.][]{2001Ackerman, 2003Woitke,Sharp2007, 2008Helling,2018Ohno, 2018Powell,Samra2020}. Most recently, \citet{Gao2020} use microphysical models to predict the most likely outcome of cloud formation in planetary atmospheres, concluding that for planets with a temperature greater than 800K the clouds would be silicate dominated. The impact of these various cloud compositions have been thoroughly explored in transmission \citep{wakeford2015transmission,pinhas2017signatures,Kitzmann2018,Mai2019}, and it has been found that the variation in cloud optical properties leads to a wide range in spectral shapes.

The importance of clouds, particularly their scattering properties, in exoplanet emission spectra has been highlighted in the context of directly imaged planets and brown dwarfs \citep{2013Lee,2017Lavie,Burningham2017,2020Molliere}.
However, the effects of scattering on the emission spectra of transiting exoplanets has been relatively unexplored; it was thought that the complicated effects of multiple scattering were negligible for the data quality. \citet{2014Barstow} used optimal estimation and a grid model search to study the effects of multiple scattering on the dayside reflectance spectrum of HD189733b, but found little evidence of strong reflection. \citet{deKok2011} explore the effects of multiple scattering using forward models and their findings suggest that the scattering of clouds in thermal emission can cause up to a 10\% difference in spectra compared to their cloud-free counterpart. Our work builds upon these by more deeply exploring the biases and degeneracies involving multiple scattering in the thermal infrared.

Exoplanet observations by JWST and ARIEL \citep{tinetti2018chemical} will improve our understanding of clouds by increasing the wavelength range of observations in the infrared compared to current HST capabilities. The phase curves planned as part of the JWST transiting exoplanets Early Release Science (ERS) program \citep{2018Bean} and the many more to come will allow for the study of night-side clouds, as well as the clouds at different phase angles, and thus reveal the cloud distribution in more detail.  \citet{Parmentier2016} predict that the day-side of hot Jupiters should be relatively cloud free. However, this may not be the case for cooler objects, with longer wavelength spectra accessible by JWST. Hence, when used to observe the night-side of hot Jupiters or to observe cooler planets, which are likely to sustain day-side clouds, the scattering effect will be prominent and might bias atmospheric parameter estimates. With this in mind, it is important that we begin to explore the effects of cloud droplet multiple scattering on thermal emission spectra over the broad wavelengths likely probed by JWST, as well as the corresponding biases and degeneracies that might arise within common parameter estimation frameworks.

An additional consideration is the way in cloud parameterisations are treated within Bayesian retrieval frameworks, usually relying upon \textit{a priori} knowledge of which condensate species are most prominent. For example, the retrievals conducted in \citet{Venot2020} assume\footnote{The emission retrievals with \textsc{TauREx} and \textsc{Pyrat Bay} used in this work did not account for multiple scattering in emission} the species of the cloud \textit{a priori}. Hence, they use fixed properties for the cloud such as its mean molecular weight and SSA. Given that such assumptions cannot generally be made \textit{a priori} when exploring the atmosphere of an unknown planet, it would be preferable to take a more agnostic approach, e.g.\ by using general parameterisations styles that can encompass a range of cloud compositions and configurations. Given that the scattering properties of clouds depend on the real and imaginary refractive index as well as on particle size, these three properties are frequently retrieved simultaneously for the solar system planets (which generally entail in situ high resolution, signal-to-noise spectra at multiple viewing angles) \citep{2015Irwin,2019Toledo,2020Braude}. This technique requires a large number of free parameters, which is problematic for typical exoplanet spectral coverage and precision (even for JWST quality). Instead, we propose a "middle ground" parameterization that fits for a simplified SSA wavelength dependence of the cloud species, as this property encompasses the effects of both the real and imaginary index spectra and the particle size. In this approach, we are agnostic to the chemical and physical properties of the cloud.

This study aims to understand the role of clouds, and particularly thermal scattering, in shaping the emission spectra of exoplanets. This will help to anticipate the challenges that will arise when interpreting JWST quality spectra, and to provide mitigation strategies that are flexible and independent of the numerous cloud parameters that are both hard to predict and hard to constrain. In Section \ref{Section3} we describe our model set up, including how we parameterise the clouds. Section \ref{Section2} discusses why scattering in emission is important and how it effects the spectrum. Section \ref{Section4} explores a simplistic atmospheric set up, where the SSA for the cloud is independent of wavelength. In Section \ref{Section5} we then use more realistic optical properties for the clouds, based on a flexible three-point parameterisation of the SSA, which captures the main behaviour of the different cloud species.

\section{Numerical model setup}
\label{Section3}

In this section we outline the various models and techniques that will be used in this study. We specifically outline two different temperature-pressure (TP) parameterisations and three different cloud assumptions, and introduce a new cloud retrieval approach.

\subsection{NEMESIS}
We use the \textbf{N}on-linear optimal \textbf{E}stimator for \textbf{M}ultivariat\textbf{E} spectral analy\textbf{SIS} code \textsc{NEMESIS} \citep{irwin2008nemesis} to compute our model spectra. \textsc{NEMESIS} uses the correlated-$k$ approach \citep{lacis1991description} to model the spectra, which has been shown to be effective and accurate when compared to using the line-by-line or cross-section approaches \citep{garland2019effectively}. For this study the molecules used are: H$_2$O \citep{polyansky2018exomol} and CO \citep{li2015rovibrational}, which were formatted using the techniques presented in \citet{Chubb2020}. As we are modelling an atmosphere which is H$_2$-dominated we also modelled the H$_2$-H$_2$ and H$_2$-He collisionally induced absorption \citep{richard2012new}. \textsc{NEMESIS} calculates multiple scattering using the matrix operator method \citep{plass1973matrix}.

\subsection{Temperature-Pressure parameterisations}
\label{TP_param}
We consider two different parameterisations for the temperature-pressure (TP) profile in our study. The first is a simplistic three-parameter model, which allows us to fit for the gradient of the temperature profile. It takes the form:
\begin{equation}
\label{tp_1}
    T = T_0 \times \Big(\frac{P}{P_0}\Big)^{\alpha}
\end{equation}
where $T_0$, $P_0$ and $\alpha$ are free parameters. This parameterisation allows us to investigate how the gradient of the TP profile varies for the different model assumptions. We will only be using this TP parameterisation in Section \ref{non_iso}.
The second is the double gray analytic TP profile from  \citet{Parmentier2014}, which has five free parameters: $T_{\text{irr}}$, $\kappa_{\text{IR}}$, $\gamma_1$, $\gamma_2$ and $\alpha$. $T_{\text{irr}}$ is the irradiation temperature, $\kappa_{\text{IR}}$ is the infrared opacity, the parameters $\gamma_{1}$ and $\gamma_{2}$ are the ratios of the mean opacities in the two visible streams to the thermal stream: $\gamma_{1} = \kappa_{v1} / \kappa_{IR}$ and $\gamma_{2} = \kappa_{v2} / \kappa_{IR}$. The parameter $\alpha$ ranges between 0 and 1, and controls the weighting used between the two visible streams, $\kappa_{v1}$ and $\kappa_{v2}$. For each case we fix the internal temperature of the planet to be $T_{\text{int}}$ = 200\,K unless otherwise stated. For details of how this TP profile is implemented in \textsc{NEMESIS}, see \citet{Taylor2020}.

\subsection{Cloud parameterisation}
\label{cloud param}

Here we will briefly describe the various cloud parameterisations explored in this work. They all share the following similarities: The cloud is assumed to be a uniformly mixed gray opacity throughout the whole atmospheric column and the scattering phase function is assumed to be isotropic. The variations are as follows:
\begin{description}
    \item {\bf Purely absorbing case:} We set the SSA to 0.0.
    This is analogous to a grey opacity cloud in other retrieval studies \citep[e.g.][]{line2016no}.
    \item {\bf Constant SSA case:} We fit simultaneously for the cloud opacity and SSA, but take both to be constant with wavelength.
    \item {\bf Three-parameters case:} We model the SSA as a step function with one break point. The parameters are the SSA value before the break, the wavelength of the break, and the SSA value after the break. We call this retrieval style the "three-parameter" case as three free parameters are used to constrain the SSA. More break points could be added in future, but we adopted this prescription because it keeps the number of free parameters low, while adequately capturing the shape of the SSA spectrum of many cloud condensing species (Figure~\ref{fig:real_ssa}).
\end{description}

\subsection{Parameter Estimation}
For the retrievals conducted in this study we have wrapped the forward-modelling component of \textsc{NEMESIS} in a Bayesian framework, using a nested sampling approach \citep{feroz2008multimodal,feroz2013importance}, which we implemented using \textsc{pymultinest} \citep{Buchner2014}. We use nested sampling due to its ability to compute both posterior distributions for the parameters, and the Bayesian evidence (or marginal likelihood), which can be used in model comparison and selection.

\section{How thermal scattering shapes the emission spectrum}
\label{Section2}
\subsection{Single Scattering Albedo and Emissivity}
\label{ssa_em}
Cloud particles can interact with photons by either absorption or scattering. The Single-Scattering Albedo (SSA) is the probability that a photon colliding with a particle is scattered rather than absorbed, and can be defined as:
\begin{equation}
    \omega_{\lambda}=\frac{Q_{\rm scat}(\lambda)}{Q_{\rm ext}(\lambda)}
\end{equation}
where the total extinction cross-section, $Q_{\rm ext}$ is the sum of the scattering cross-section $Q_{\rm scat}$ and the absorption cross-section $Q_{\rm abs}$. Additionally, the ability of particles to re-emit light is determined by their emissivity $\epsilon_\lambda$. In general, the source function in an atmosphere can be written as the combination of the scattering and the thermal emission terms:
\begin{equation}
    S_{\lambda}=\omega_\lambda J_\lambda+\epsilon_\lambda B_{\lambda}.
\end{equation}
where $B_\lambda$ is the Planck function and $J_\lambda$ is the first moment of the intensity. Kirchhoff's law states that at any particular wavelength, a particle is as good an emitter as it is an absorber. This means that the emissivity and the SSA must be linked by:
\begin{equation}
    \epsilon_\lambda+\omega_\lambda=1.
\end{equation}
The source function therefore only depends on the atmospheric emissivity (or only on the SSA) and can be used to solve the radiative transfer equations for a plane-parallel atmosphere at local thermodynamic equilibrium. The full solution for non-isothermal atmospheres can be found in~\citet{Rutten2003} and is reproduced in Appendix~\ref{general_rutten} for reference. In the following sections we discuss some simplified cases in more detail.

\subsection{Solution in the isothermal case}
\label{soln_iso_case}
An isothermal atmosphere is often thought to emit like a blackbody. However, when thermal scattering is present the source function is no longer a Planck function and the planetary flux becomes:
\begin{equation}
    F_p = \frac{2\pi}{\sqrt{3}} \frac{2\sqrt{\epsilon_{\lambda}}}{1 + \sqrt{\epsilon_{\lambda}}} B_{\lambda},
    \label{eq::FLux}
\end{equation}
where the numerical prefactor depends on the choices of boundary conditions~\citep[see][Section 2.3.2 for a discussion]{Parmentier2014}. For unit emissivity, $\epsilon_\lambda = 1$, we recover the standard result, namely the planetary flux is equal to the Planck function. For smaller emissivity, however, the outgoing planetary flux is smaller than the blackbody cases, and vanishes for a purely scattering atmosphere.

When the emissivity is constant with wavelength, the resulting planetary spectrum is a diluted blackbody. This is demonstrated in Figure~\ref{fig:diff_opac}, where the purple curve corresponds to a model entirely dominated by a cloud opacity, with SSA $\omega=0.9$, and hence emissivity $\epsilon=0.1$.

In the general case, emissivity should depend on wavelength for two reasons. First, the cloud emissivity itself depends on wavelength (see Section~\ref{ssa_em} and Fig.~\ref{fig:compare_ssa}). Second, the local emissivity is the weighted sum of the cloud and gas emissivities. As we take the latter to be unity in this work, the local emissivity amounts to:
\begin{equation}
    \epsilon_{\lambda}=\frac{\epsilon_{\rm cld} \, \kappa_{\rm cld}+\kappa_{\rm gas}}{\kappa_{\rm cld}+\kappa_{\rm gas}},
\end{equation}
where $\kappa_{\rm cld}$ and $\kappa_{\rm gas}$ are the cloud and gas opacities, respectively.

This equation shows that, for a constant cloud opacity and emissivity, the local emissivity is a monotonically decreasing function of the molecular opacity. The dilution factor before the Planck function in Equation \ref{eq::FLux} is then a monotonically-decreasing function of the molecular opacities. As a consequence, inside molecular absorption bands the emissivity will be high, the dilution factor will be small and the planetary flux will be large, whereas it is the opposite outside molecular bands. In summary, \emph{an isothermal atmosphere with a cloud that has a constant opacity and constant emissivity will mimic molecular emission features}.

Figure~\ref{fig:diff_opac} demonstrates this behaviour for an isothermal atmosphere with H$_2$O and CO in solar-composition proportions and a constant opacity, constant emissivity cloud. If the cloud opacity is much smaller than the gas opacity, the spectrum follows a Planck function as expected. As the cloud opacity increases, it becomes comparable to the gas opacities, and emission bands of H$_2$O and CO start to appear. For very large cloud opacity, the emissivity is entirely dominated by the cloud contribution and the spectrum reverts to that of a blackbody, albeit diluted by a constant factor.

\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{diff_opac_r.pdf}
    \caption{Model atmospheres varying from cloud free ($\text{SSA} = 0$, smooth red line) to completely dominated by the opacity of the cloud ($\text{SSA}=0.9$, smooth purple line). The thinner lines containing features, which are due to H$_2$O in the atmosphere, correspond to isothermal atmospheres with a temperature of 1400\,K and a cloud with a SSA of 0.9, whose opacity increases by an order of magnitude between each line, from red to purple. We transition from a molecule opacity dominated regime to a cloud opacity dominated regime. It can be seen that the opacity of the cloud results in different temperature layers being probed. }
    \label{fig:diff_opac}
\end{figure}

\begin{figure*}
    \centering
    \includegraphics[width=\textwidth]{compare3.pdf}
    \caption{Impact of the SSA on the emission spectra for different TP profiles. The TP profiles corresponding to cases 1, 2 and 3 (top left, top right and bottom left plot, respectively) are shown in the bottom right plot. All models were generated using $\log(\kappa_{\rm cld}) = 3$ and with roughly solar H$_2$O and CO abundances. The non-isothermal TP profiles were generated using the 3-parameter model discussed in Section~\protect{\ref{TP_param}}
    (see Eq.~\protect{\ref{tp_1}}), with $\alpha = 0.1$ and $-0.1$ for cases 1 and 3, respectively.}
    \label{fig:compare_ssa}
\end{figure*}

\subsection{Non-isothermal case}
\label{non iso case}
In the non-isothermal case (see derivation in Appendix \ref{general_rutten}) the situation is more complex and the planetary flux takes the form:
\begin{equation}
        F_p=\frac{2\pi}{\sqrt{3}}\left(\frac{2\sqrt{\epsilon_{\lambda}}}{1+\sqrt{\epsilon_\lambda}}\right)B_{\lambda}\left(\tau_\lambda=\frac{1}{\sqrt{3\epsilon_\lambda}}\right).
\end{equation}
Scattering therefore affects the planetary flux in two ways. First, as for the isothermal case, it multiplies the planetary flux by a dilution factor, which is a monotonically decreasing function of the gaseous opacity. Second, it increases the effective depth of the photosphere by a factor $1/\sqrt{\epsilon_\lambda}$.

For an atmosphere with a temperature that increases with increasing altitude, the flux is generally smaller outside molecular bands. However, that is where the emissivity is low, hence the photosphere probes deeper, colder layers and the flux is reduced. Similarly, the dilution factor is small when the emissivity is low. Thus, for an atmosphere with a temperature inversion and emission features, both effects act jointly to reduce the planetary flux more strongly outside molecular bands than within them, hence increasing the relative contrast of the emission features.

For an atmosphere with a temperature that decreases with increasing altitude, the flux is generally larger outside molecular bands. However, that is a region where the emissivity is low, hence the photosphere probes deeper, hotter layers. We see an opposite effect with the dilution factor, as it is small when the emissivity is low. Therefore, for an atmosphere without a temperature inversion, the two effects of the scattering act in opposition. Whether the amplitude of molecular features is increased or decreased by scattering depends on the strength of the temperature gradient.

These different behaviours are shown in Figure~\ref{fig:compare_ssa}. In the first panel, we show the case of a non-inverted atmosphere with a temperature gradient that is weak enough that the effect of the dilution factor dominates, and the amplitude of the molecular features decreases with increasing SSA. In the second panel, we show the case of an isothermal atmosphere, where emission features appear as the SSA increases. Finally, the third panel show that case of an inverted atmosphere, where the strength of the emission features increases with increasing SSA.

\subsection{Expected degeneracies}
\label{degen}
We have shown that thermal scattering can change the shape of the spectral features in a planet atmosphere and even create emission features in the isothermal case. This will naturally lead to degeneracies between parameters: the spectra produced by atmospheres with scattering can look very much like those of atmospheres without scattering, but with a different set of physical properties.

The most obvious degeneracy arises when comparing the brightness temperature of a planet to the actual atmospheric temperature. Indeed, for an isothermal atmosphere with a constant, non-zero emissivity, Eq.~(\ref{eq::FLux}) shows that the spectrum is a diluted blackbody. If we observe the atmosphere at long wavelengths, where the Planck function is linear in temperature, the brightness temperature measured in a given bandpass would become:
\begin{equation}
\label{t_bright}
    T_{\rm bright}=\frac{2\sqrt{\epsilon}}{1+\sqrt{\epsilon}}~T_{\rm atm}.
\end{equation}
For example, measuring the brightness temperature from the spectrum produced by an atmosphere with an emissivity of $\epsilon=0.5$ would lead to a retrieved brightness temperature roughly $20\%$ smaller than the actual atmospheric temperature. With an emissivity of $\epsilon=0.1$, the brightness temperature is roughly $50\%$ smaller than the true atmospheric temperature.  This has far reaching implications for the night-side of hot Jupiters, which have been shown to have cooler than expected temperatures \citep{Beatty2019,Keating2019,Parmentier2021}. These results could be explained by the presence of low emissivity, high single-scattering clouds, such as pure silicate clouds, which would lead to an apparent brightness temperature much smaller than the actual atmospheric temperature.

The second degeneracy arise from the dilution factor itself. \cite{Taylor2020} showed, in the context of a planet with an inhomogeneous thermal structure, that such a dilution factor can be mimicked by the spurious addition of molecules such as methane or ammonia. We therefore expect that the lack of scattering in retrieval models could lead to the spurious detection of molecules in cloudy atmospheres.

The third degeneracy we can expect is linked to the spectral variation of the atmospheric emissivity, which could affect either the inferred strength and sign of the temperature gradient, or the inferred molecular abundances. As highlighted in Fig.~\ref{fig:compare_ssa}, the wavelength dependence of the emissivity generally reduces absorption features and increases emission features. This can be compensated in at least three different ways by a non-scattering retrieval. First, the thermal structure could be biased towards more positive temperature gradients. Second, the molecular abundances could be biased towards lower values for non-inverted atmospheres, so that the increased pressure broadening of the lines reduces the spectral features. Conversely, the abundances could be biased towards larger values for inverted atmospheres, so that the reduction of the pressure broadening of the lines enhances the size of the features. Finally, the change in the strength of molecular features could be compensated by the addition of a gradient in the molecular abundances. As shown by
~\citet{Parmentier2018}, spectral features are increased if abundances increase with height and decreased if abundances decrease with height. A more thorough discussion of these degeneracies, based on an analytical derivation can be found in Appendix ~\ref{Appendix B}

\section{Results from constant cloud single-scattering albedo case}
\label{Section4}

In this section we start with a simplistic model to understand the biases and degeneracies that thermal scattering can induce when extracting information from the emission spectra of exoplanets. For this we focus on the addition of a cloud with a constant opacity and constant emissivity. We first explore the simplest case of an isothermal atmosphere and then move to the non-isothermal case.

\subsection{Isothermal atmosphere}
\label{iso const}

As seen in Section \ref{soln_iso_case} we expect a cloud with a constant SSA in an isothermal solar-composition atmosphere to produce features that mimic emission that could be interpreted as the presence of an inverted thermal structure. We now want to quantify this in a retrieval framework.

We base our physical parameters of the system around the canonical exoplanet WASP-43b \citep{Hellier2011} which have been refined by \citet{gillon2012trappist}. We present the physical parameters and atmospheric parameters used in Table \ref{tab:model_params}.

\begin{table}
   \centering
\begin{tabular}{l|l}
    \hline
    Parameter      & Value   \\
    \hline
    log(H$_2$O)     & -3 \\
    log(CO)     & -3 \\
    Temp [K]     & 1400 \\
    SSA & 0.9 \\
    log(Opacity) & 4

\end{tabular}
\caption{Atmospheric and cloud properties used to create the isothermal model atmosphere.}
    \label{tab:model_params}
\end{table}
We used \textsc{NEMESIS} to generate a forward-modelled spectrum given the atmospheric parameters in Table \ref{tab:model_params} and then we used \textsc{PandExo} to generate the wavelength grid and the error bars for a single eclipse observation of WASP-43b. We consider the observing modes by NIRSpec Prism and MIRI LRS. We generate these observations considering a resolution of R = 50. We do not employ a random noise instance, but rather assume the data points at the exact model value in order to mitigate biases due to random noise instances \citep{Feng2018,Mai2019,Changeat2019}

In order to estimate which wavelengths are most sensitive to the effects of thermal scattering we focus on three different possible observing strategies:
\begin{enumerate}[1.]
    \item[1.] A single eclipse observation using the NIRSpec Prism observing mode. Wavelength coverage: 0.6 - 5.3 $\mu$m
    \item[2.] A single eclipse observation using the MIRI LRS observation mode. Wavelength coverage: 5.0 - 12.0 $\mu$m
    \item[3.] Combining both NIRSpec Prism and MIRI LRS observations to investigate the complete wavelength coverage of JWST (0.6 - 12.0 $\mu$m).
\end{enumerate}

We then perform retrievals of temperature and composition assuming the \citet{Parmentier2014} pressure-temperature profile from Section \ref{TP_param} and an atmosphere with H$_2$O and CO gaseous absorption. Additionally, each of our three retrievals treats the clouds with an increasing complexity:
\begin{enumerate}[1.]
    \item A cloud-free model, shown in blue in upcoming figures.
    \item A pure absorbing cloud with constant opacity as described in Section \ref{cloud param}  This model is shown in red in upcoming figures.
    \item A constant single-scattering albedo (SSA) cloud, with a constant emissivity and a constant opacity as described in Section \ref{cloud param}. This model is shown in purple in upcoming figures.
\end{enumerate}

In order to quantify the fit each model, we calculate the Bayesian evidence and convert the result to a sigma significance following the same method as \citet{Taylor2020}. We compare the cloud free and the pure absorbing cloud model to the scattering cloud model, as the scattering cloud model is representative of the "true" model, and is hence the reference model for the model comparison analysis. In this comparison, the larger the value of the sigma significance, the more rejected the model is.

\subsubsection{Observing scenario 1: NIRSpec Prism (0.6 - 5.3 $\mu$m)}

\label{iso nirspec}

The wavelengths observed with NIRSpec Prism cover a broad array of molecular species encompassing the dominant nitrogen, oxygen, and carbon reservoirs. Hence, NIRSpec is one of the instruments which will be highly used by the exoplanet community \citep{stevenson2016transiting,2018Bean}. The NIRSpec Prism is the lowest resolution instrument at around R $\sim$ 100, but with the broadest simultaneous wavelength coverage.

Fig \ref{fig:NIRSpec_Iso_Pandexo} summarizes the retrieval results (fits, TP profile, and abundance constraints) under the 3 different retrieval model parameterisations. When calculating the sigma significance of excluding the cloud free and the pure absorbing cloud model we find that they are excluded at 8.3-$\sigma$ and 4.8-$\sigma$ respectively. Both the pure absorbing cloud and the cloud-free models require a temperature inversion to be able to fit to the data. However, when considering the scattering and retrieving for the SSA using the constant SSA cloud model, we are able to retrieve the correct profile, as we should given this retrieval forward model is identical to the input forward model.

Interestingly, the non-scattering cloud forward model is able to retrieve the CO and H$_2$O abundances with a similar precision and accuracy as the "true" scattering forward model. Despite this success, however, we would incorrectly retrieve an inverted thermal profile. This suggests, in this scenario, that the scattering-TP profile degeneracy is largely decoupled from the abundances. The cloud-free model is not able to retrieve the correct abundances. In fact the retrieved abundances for H$_2$O and CO are so low that it appears the model is using the H$_2$-H$_2$ collision induced absorption to fit to the synthetic data.

\begin{figure*}
    \centering
    \includegraphics[width=\textwidth]{NIRSpec_Iso_Pandexo_v3.pdf}
    \caption{\textbf{Left}: Best-fitting spectra, temperature profiles and retrieved chemistry for three different retrieval models applied to synthetic NIRSpec Prism observations generated from planet with an isothermal TP profile and cloud opacity of $10^4$. For each model we retrieve the TP structure with the analytical TP profile described in Section \ref{TP_param}. In purple we retrieve on the synthetic data with the constant SSA cloud model. In red we fit using the pure absorbing cloud model. In blue we present the fit using a cloud-free model. We present the retrieved temperature-profiles with the true profile represented by a dashed green line. We present the residuals of the fit with grey shading indicating the error in ppm on the measurement. \textbf{Right}: we show the retrieved chemistry for each case with the true value in black.}
    \label{fig:NIRSpec_Iso_Pandexo}
\end{figure*}

\subsubsection{Observing scenario 2: MIRI LRS (5.0 - 12.0 $\mu$m)}
\label{iso miri}
The 5.0 - 12.0 $\mu$m spectral range contains the strongest H$_2$O features and is crucial in studying the clouds in exoplanet atmospheres as it covers such cloud features as a silicate `bump' \citep{Cushing2006,wakeford2015transmission}. Therefore, we have decided to explore its ability to recover the atmospheric properties from model observations generated using a simple scattering cloud. Fig \ref{fig:MIRI_Iso_Pandexo} summarizes the retrieval constraints (similar to Fig \ref{fig:NIRSpec_Iso_Pandexo}) achieved under the MIRI observational setup.

The non-scattering cloud and cloud free retrievals produce nearly identical fits to the simulated data. This is due to the pure absorbing cloud model fitting for a near zero opacity which is consistent with a cloud-free atmosphere. When compared to the scattering cloud model, they can be excluded with a sigma significance of 4.3-$\sigma$ and 4.4-$\sigma$ respectively. Both these cases produce a slightly inverted TP profile. The residuals show that the fits are good given the error on the observations, and by not considering the effects of scattering, we could consider this atmosphere to be cloud-free with a temperature inversion. We also see that the retrieved TP profiles for these models are different from the TP profiles retrieved when studying the short wavelength case. The constant SSA cloud model fits best to the observations, with the TP profile converging to the true isothermal input.

The constant SSA model is able to retrieve the abundance of H$_2$O, but not CO. This makes sense as there are no CO features in this wavelength region. The cloud-free and pure absorbing cloud model slightly overestimates the abundance of H$_2$O and find an upper limit on the CO which is higher than the input.

\begin{figure*}
    \centering
    \includegraphics[width=\textwidth]{MIRI_Iso_Pandexo_v3.pdf}
    \caption{Same simulations as Figure \ref{fig:NIRSpec_Iso_Pandexo}, but for the MIRI LRS case.}
    \label{fig:MIRI_Iso_Pandexo}
\end{figure*}

\subsubsection{Observing scenario 3: Full Wavelength Range}
\label{iso all}
Figure \ref{fig:ALL_Iso_Pandexo} summarizes the retrieved constraints when combining the NIRSpec and MIRI modes for full 0.6 - 12 $\mu$m coverage.

By combining the observations the fit to the data is now worse for the cloud-free model, which now can be excluded with a sigma significance of 14.8-$\sigma$, with the most extreme deviations over the MIRI LRS wavelength coverage. It can be seen to have a slight temperature inversion and a retrieved H$_2$O abundance that is 2 orders of magnitude larger than the input value. We also find that we are unable to detect CO, despite it being present in the forward model.

We find that for the cloud-free models, the retrieved TP profiles for the short (0.5 - 5.3 $\mu$m) and long (5.0 - 12.0 $\mu$m) wavelength ranges are extremely different and mutually exclusive. As a consequence, when fitting the full wavelength range the cloud-free model is unable to provide a good fit to the synthetically generated observed data.

Interestingly, similar to the NIRSpec only case, the non-scattering cloud model produces abundance constraints reasonably consistent with the truth (and the scattering cloud constraints). The non-scattering cloud model can be excluded with a sigma significance of 7.1-$\sigma$. Hence, if we did not compare with a scattering cloud model, we could be tricked into believing that this model is the best fitting model and also conclude that this planet has a temperature inversion, which is not, in fact, the case and not what we would conclude using the constant SSA model.

\begin{figure*}
    \centering
    \includegraphics[width=\textwidth]{ALL_Iso_Pandexo_v3.pdf}
    \caption{Same simulations as Figure \ref{fig:NIRSpec_Iso_Pandexo}, but for the combined NIRSpec Prism and MIRI LRS case. We present the full posterior distribution for the scattering cloud retrieval in Fig. \ref{fig:iso_cloud_posterior}.}
    \label{fig:ALL_Iso_Pandexo}
\end{figure*}

\subsection{Non-isothermal atmosphere}
\label{non_iso}

We now explore the retrieval biases expected in a more realistic case of a planet with a temperature that decreases with decreasing pressure. The set up is the same as in Section \ref{iso const}, but this time the TP profile used is that described by three-parameter model in Section \ref{TP_param}, with $T_0$ = 1400 K, $P_0$ = 0.1 bars and $\alpha$ = 0.1. We also reduced the log$\kappa_{(\text{cld})}$ to be 3. Similar to Section \ref{iso const} we consider the same three observing cases.

We then re-retrieve on each data set with the three models described in Section \ref{iso const}. We use the three-parameter temperature profile for the non-isothermal case as we wanted to investigate the retrieved temperature-pressure slope to see if it is consistent with the analytical approach in Section \ref{non iso case}.

\subsubsection{Observing scenario 1: NIRSpec Prism}

We first investigate the three retrieval forward models on the NIRSpec Prism data, just as in Section \ref{iso nirspec}. Fig.  \ref{fig:NIRSpec_Non_Iso_Pandexo} summarises these results.

The residuals show that both the constant SSA model and pure absorbing cloud model have similar fits to the data and that the cloud-free model deviates the most, with significant deviation at around 4 $\mu$m, the region where CO should be present. The cloud free and pure absorbing cloud models have an excluded sigma significance of 9.3-$\sigma$ and 2.4-$\sigma$ respectively, suggesting that the cloud free model is firmly rejected by the data but the non-scattering cloud model cannot be rejected. For the cloud-free model, we retrieve a steeper TP profile compared to the input, as anticipated from the analytic arguments. Only the constant SSA model can retrieve the input TP profile, with the pure absorbing cloud model retrieving a larger temperature at higher pressures.

All models are able to retrieve the correct composition to within 2-$\sigma$, with nearly consistent posteriors. The pure absorbing cloud model is the least accurate and has the widest posterior distribution.

\begin{figure*}
    \centering
    \includegraphics[width=\textwidth]{NIRSpec_Non_Iso_Pandexo_v3.pdf}
    \caption{\textbf{Left:} Best-fitting spectra, temperature profiles and retrieved chemistry for three different retrieval models applied to synthetic NIRSpec Prism observations generated from a planet with three-parameter TP profile and cloud opacity of $10^3$.
    In purple we retrieve on the synthetic data with the 3-parameter TP profile described in Section \ref{TP_param} and model which allows for the single-scattering albedo to be fit as a constant value at all wavelengths. In red we retrieve using a similar model to the one shown in purple. However, we fix the single-scattering albedo to be zero, effectively removing the scattering component of the cloud and making the cloud grey. In blue we present a model that has no cloud. We present the retrieved temperature-profiles with the true profile represented by a dashed green line. We present the residuals of the fit with grey shading indicating the error in ppm on the measurement. \textbf{Right:} The posterior distributions of the retrieved chemistry with the colours corresponding to the models in the left plot. The black line represents the true value.}
    \label{fig:NIRSpec_Non_Iso_Pandexo}
\end{figure*}

\subsubsection{Observing scenario 2: MIRI LRS}

Fig \ref{fig:MIRI_Non_Iso_Pandexo} summarises the retrieval results under the MIRI LRS observational setup.

The residuals show that the cloud-free fit is not as good as either the pure absorbing cloud or SSA cloud models. The cloud free and pure absorbing cloud models are both considered consistent with the data (e.g., neither is rejected with cloud free at 2.3-$\sigma$, and purely absorbing at < 1-$\sigma$).  The same trends that can be seen in the NIRSpec Prism retrieved TP profiles are also seen with MIRI LRS. However, the profiles are less constrained.

The abundance posteriors are much wider than those obtained using NIRSpec Prism, with the cloud-free model not being able to retrieved the correct chemistry. The pure absorbing cloud and the constant SSA cloud model are able to retrieve the H$_2$O abundance with a posterior distribution which spans 5 orders of magnitude. This is a result of the large errors on the data, the data quality is not good enough to distinguish from the different models or say anything meaningful about the abundances.

\begin{figure*}
    \centering
    \includegraphics[width=\textwidth]{MIRI_Non_Iso_Pandexo_v3.pdf}
    \caption{Same as Fig \ref{fig:NIRSpec_Non_Iso_Pandexo} but for the wavelength grid and errorbars of MIRI LRS.}
    \label{fig:MIRI_Non_Iso_Pandexo}
\end{figure*}

\subsubsection{Observing scenario 3: Full Wavelength Range}

Fig \ref{fig:ALL_Non_Iso_Pandexo} shows the constraints under the three retrieval forward model assumptions given the NIRSpec Prism + MIRI LRS "full" wavelength coverage observational scenario. The cloud free and pure absorbing cloud models have an excluded sigma significance of 11.2-$\sigma$ and 3.1-$\sigma$ respectively, this suggests that the cloud free model is firmly rejected, however the pure absorbing cloud is only marginally rejected, with the rejection limit being at 3-$\sigma$.

Interestingly, the cloud-free and the constant SSA cloud cases retrieve relatively similar H$_2$O abundances. However, the cloud-free model constrains a lower value of the CO abundance by around 2-$\sigma$ compared to the input value. The pure absorbing cloud model constrains higher values of both the H$_2$O and CO abundances by around 3-$\sigma$ compared to the input model value. This suggests that the broader wavelength coverage introduces more bias into the retrieved abundances when using the incorrect cloud model.

We find that the broad wavelength coverage results in higher precision constraints on the TP profile when using the correct forward model.
\begin{figure*}
    \centering
    \includegraphics[width=\textwidth]{ALL_Non_Iso_Pandexo_v3.pdf}
    \caption{Same as Fig \ref{fig:NIRSpec_Non_Iso_Pandexo} but for the wavelength grid and errorbars of NIRSpec Prism and MIRI LRS.}
    \label{fig:ALL_Non_Iso_Pandexo}
\end{figure*}

\subsection{Inverted Thermal Profile}
We now explore whether the biases exist when the atmosphere has an inverted thermal profile. We use the same setup from Section \ref{non_iso}, this time setting the $\alpha$ parameter to be $\alpha$ = -0.1 when generating our synthetic spectra, this results in an atmosphere with a thermal structure where the temperature increases with height. We then re-retrieve on this spectrum using the three models described in Section \ref{iso const}.

\begin{figure*}
    \centering
    \includegraphics[width=\textwidth]{Inv_NIRSpec.pdf}
    \caption{Same as Fig \ref{fig:NIRSpec_Non_Iso_Pandexo} but for the data generated using an inverted thermal profile.}
    \label{fig:NIRSpec_Inv}
\end{figure*}

For this case, we only consider the observing scenario of NIRSpec Prism. Fig. \ref{fig:NIRSpec_Inv} summaries these results. It can be seen the fit of the scattering cloud and the pure grey cloud models are consistent within the errorbars, however, the cloud free model shows the largest deviation. The cloud free and pure absorbing cloud models have an excluded sigma significance of 15.4-$\sigma$ and 8.0-$\sigma$ respectively, this suggests that both the cloud free and pure absorbing cloud models are firmly rejected. Only the scattering model is able to retrieve the input thermal structure and chemistry.

\subsection{Summary}

In summary, we demonstrated that failure to account for scattering can result in biased results in both the abundance inferences and in the derived thermal structures, though a greater bias appears in the temperature profiles. For the simple isothermal, scattering atmosphere,  the interaction of the cloud opacity with the gas opacity can mimic emission features in a non-scattering atmosphere. A non-scattering retrieval forward model will interpret it as such, resulting in an inverted thermal profile.  Despite the incorrectly retrieved TP profile, the non scattering cloud can accurately retrieve the abundances with a largely negligible bias.

We find for the more realistic case of an atmosphere with a decreasing with height TP profile and a scattering cloud, a non-scattering cloud retrieval model cannot be distinguished between the scattering model. We find that for each wavelength range that retrieved temperature gradient for the non-scattering cloud model is smaller than the truth, with the opposite being true for the cloud free model. The retrieved abundances for the non-scattering cloud model are consistent with the scattering model for the individual instruments, however when combined for the "full" JWST range the abundances become biased by roughly 1 dex for both CO and H$_2$O in the non-scattering cloud case, and by roughly 1 dex for CO in the cloud free case.

\section{Non-constant cloud single-scattering albedo}
\label{Section5}

In the above proof of concept we have used a relatively simple setup, where we consider the single-scattering albedo and the opacity to be constant with wavelength. In this section we present some atmospheric simulations where we have used Mie theory to calculate the single-scattering albedo for some common cloud condensing species.  We present the single-scattering albedo as a function of particle size and wavelength for these common cloud species in Figure \ref{fig:real_ssa}, with  their properties taken from \citet{Kitzmann2018} apart from S$_8$, which was taken from Fuller, K. A. et al. in \citet{PalikEdwardD1998Hooc}.

We can see from Figure \ref{fig:real_ssa} that the single-scattering albedo of the majority of the molecules either follow a straight line or a step like shape. This motivated our three-point cloud parameterisation scheme, as described in Section \ref{cloud param}.

\subsection{Realistic cloud single-scattering albedo and constant opacity}
\label{real_const}

We explore whether our three-point cloud parameterisation scheme is able to account for the scattering properties of potential clouds without introducing bias in the retrieved abundances or thermal structure.  This three-point cloud approach is independent of the choice of specific condensate, making it a favorable approach for a generalized cloudy retrieval. We consider four cloud condensing species: Al$_2$O$_3$, MnS, MgSiO$_3$ and Fe, chosen as these condensates span the range of relevant optical morphologies. For each of the forward models we consider the particle radius to be of 10 $\mu$m and to have a variance of 0.1 following a Gamma distribution as shown in \citet{hansen1974light}. We consider the condensates to be vertically mixed, with a fractional scale height of 1 and to have abundance of log($\kappa_{\text{cld}}$) = 4. As in Section \ref{iso const}, we assume an isothermal atmosphere, resulting in "emission-like" features due to the scattering itself. Simulated observations are produced using the same procedure described in Section \ref{Section4} for the "full" wavelength range.

We then retrieve on the four different condensate set ups with three different models. We use the analytical pressure-temperature profile from Section \ref{TP_param} and an atmosphere with H$_2$O and CO gaseous absorption with abundances the same as those used in Section \ref{Section4}. Additionally, each of our three retrievals treats the clouds with an increasing complexity:
\begin{enumerate}[1.]
    \item A cloud-free model, shown in blue in upcoming figures.
    \item A pure absorbing cloud model, shown in red in upcoming figures.
    \item A three-point cloud model. This is our novel cloud parameterisation, which is discussed in Section \ref{cloud param}. This model is shown in purple in upcoming figures.
\end{enumerate}

Our results for the realistic SSA are similar to the ones with the constant SSA described in the previous section. All retrievals that do not take thermal scattering into account estimate biased abundances and converge towards the wrong thermal structure (see Fig. \ref{fig:al2o3_spec}, \ref{fig:mns_spec}, \ref{fig:mgsio3_spec} and \ref{fig:fe_spec}.)

Importantly, as shown in Fig. \ref{fig:ssa_grid}, we are able to retrieve the overall wavelength dependent shape of the the single-scattering albedo for all species considered. This is a very powerful result, given the approach was completely agnostic to the chemical and physical properties of the cloud, hence verifying that it works with a range of different shaped SSA spectra means that it is possible to learn about the clouds physical properties, this can be done by comparing the retrieved SSA spectrum with various different cloud condensing species.

\begin{figure*}
    \centering
    \includegraphics[width=0.99\textwidth]{SSA_grid.pdf}
    \caption{The best-fitting retrieved single-scattering albedo (SSA) spectrum using the three-point cloud parameterisation. In purple we present the retrieved SSA spectra with 1- and 2-sigma envelopes. In green we present the SSA spectrum for each of the cloud condensing species, which are labelled in each of the plots; each species has been modelled with a particle size of 10 $\mu$m.}
    \label{fig:ssa_grid}
\end{figure*}

\subsection{Realistic cloud single-scattering albedo spectrum and varying opacity}
We now investigate how the three-point cloud model would perform when retrieving on a spectrum that has a more physically motivated cloud. For this we use the \textsc{CHIMERA} \citep{line2013systematic} retrieval tool (\url{https://github.com/mrline/CHIMERA}) which employs the \citet{2001Ackerman} cloud model to produce realistic vertical droplet profiles over a broad range of droplet sizes (for a specific condensate), given a sedimentation efficiency (f$_{\text{sed}}$), eddy diffusivity (K$_{zz}$, simplistically assumed to be $10^{8}$ cm$^2$/s), cloud base pressure, and the condensate mixing ratio at the cloud base (see \citet{Mai2019} for implementation details). Here we assume enstatite (MgSiO$_3$) physical/optical properties, given a condensate mixing ratio of $10^{-5.5}$ at the cloud base (200 mbar).  Along with these cloud properties, we include gaseous opacity due to H$_2$O (4$\times$10$^{-4}$), CO (2$\times$10$^{-4}$), and H$_2$-H$_2$/He (0.86/0.137) CIA, and employ the analytical TP profile as discussed in Section \ref{TP_param} (T$_{\text{irr}}$ = 1400, log$\kappa_{\text{ir}}$ = -1.5, log$\gamma_1$ = -0.7, T$_{\text{int}}$ = 300K), under WASP-43b system properties.  Molecular (and continuum/condensate) opacities are treated within the correlated-K resort-rebin framework (e.g., \citet{Molliere2015,Amundsen2017}).  The \citet{Toon1989} two stream source function technique is used to solve for the fluxes (both up, down, stellar, and planetary) in these non-homogeneous multiple scattering atmospheres (a different approach to the multiple scattering problem than \textsc{NEMESIS}).  While stellar reflected flux is a natural consequence of multiple-scattering atmospheres, here, we explore only the retrieval biases arising from the thermal component.  We assume an extended cloud scenario for these tests (f$_{\text{sed}}$ = 0.1, a nearly constant with altitude condensate profile).

We then used this forward model in \textsc{PandExo} to generate the observations for one eclipse observation of WASP-43b using NIRSpec Prism and MIRI LRS configurations at a resolution of R = 50. These simulations are similar to the ones calculated in Section \ref{Section4}.  However, for this scenario we have kept the random noise instance to emulate a "real observation". We then retrieve on these data with the same three models described in Section \ref{real_const}.

To validate \textsc{CHIMERA} and \textsc{NEMESIS} for this work we performed a benchmark test. We used \textsc{NEMESIS} to retrieve on a cloud free model generated by \textsc{CHIMERA} with an error envelope of 60ppm. We find consistent results as shown in Fig. \ref{fig:chivnem_spec} and \ref{fig:chivnem_triangle}. Therefore the differences that are found due to the 3 parameter SSA modelling approach and the Ackerman $\&$ Marley cloud parameterisation are not due to the choice of radiative transfer assumptions in the respective codes.

\subsubsection{Retrieval results}
Figure \ref{fig:mike_pandexo} shows the best fitting models, their temperature profiles and abundances. The purely absorbing cloud and cloud free retrievals produce nearly identical fits to the simulated data. This is due to the pure absorbing cloud model fitting for a near zero opacity which is consistent with a cloud-free atmosphere. Both models fit for an inverted TP profile and retrieve abundances that are lower than the input model.

The three point cloud model is able to retrieve the correct H$_2$O abundance, however the CO abundance is nearly 2 order of magnitudes larger than the input. This model is able to retrieve a temperature profile that is decreasing with height. Despite the CO abundance, the retrieved results are more consistent with the input, therefore, the scattering cloud model is an improvement compared to the non scattering and cloud free models. We find that the retrieve SSA spectrum is representative of the SSA spectra of the photosphere of the forward modelled atmosphere which is around 10$^{-2}$ to 10$^{-3}$ (Fig. \ref{fig:ssa_chimera}). However, the change of SSA with altitude, related to a change of particle size with altitude cannot be taken into account in our modelling framework and is probably leading to the bias in the retrieved abundances.

We compute the Bayesian evidence to conclude which model is most supported by the data. We find that the data supports the 3-point scattering model over the purely absorbing and cloud free models by 8.25-$\sigma$ and 8.00-$\sigma$ respectfully.

We note that the model used here to test our framework is probably a bit pessimistic. The nightside of the real WASP-43b planet is likely to have a larger thermal gradient than predicted by the semi-grey thermal profile, which should lead to larger spectral features than the ones predicted here.

We note that our 3 parameter SSA method offers an improvement over cloud free or purely absorbing clouds, but does not offer a perfect solution.  This is likely due to the additional simplifications in our cloud model, such as the constant-with-altitude mixing and different choice of particle size distribution.  Future work will focus on implementing more realistic vertical cloud droplet profiles. A hybrid approach between an Ackerman $\&$ Marley type vertical droplet parameterisation and an "N-parameter" SSA retrieval may provide a better representation of "realistic" clouds.  We also do not consider the influence of inhomogeneities (e.g., "holes" or "patchy clouds") which could further complicate thermal emission inferences.

\begin{figure*}
    \centering
    \includegraphics[width=\textwidth]{mike_pandexo_final.pdf}
    \caption{The retrieved results from the data generated using \textsc{CHIMERA}, the error bars and noise were generated using \textsc{PandExo}. Left: The best fitting spectra and retrieved temperature profiles. In purple we present the 3-point cloud model. In red we present the purely absorbing cloud model. In blue we present the cloud free model. In green we present the input temperature profile. Right: The retrieved chemical abundances, with the true value represented by a black line.}
    \label{fig:mike_pandexo}
\end{figure*}
\section{Conclusion}

Clouds have been found to be common in exoplanet atmospheres and with upcoming advanced space based facilities such as JWST we need to understand how best to account for them in our retrieval models. In this study we have explored the impact that a scattering cloud has on the interpretation of an emission spectrum as observed with JWST. We summarise our conclusions as follows:
\begin{enumerate}[1.]
    \item The presence of a scattering cloud reduces the emissivity of the atmosphere and leads to brightness temperatures that are smaller than real temperatures, this can been seen from Eq \ref{t_bright}. We further show that even when the cloud has a constant single-scattering albedo, the emissivity of the atmosphere varies with wavelength. This results in spectral features being present due to the cloud even if the atmosphere is isothermal. Finally we show that the variation of SSA with wavelength can introduce additional cloud features in the emission spectrum.
    \item We show that using a pure absorbing cloud, which is commonly used when interpreting clouds in exoplanet atmospheres, would lead to the inference of a systematically smaller than truth temperature gradient in an atmosphere with a non-inverted temperature profile. In the specific case of an isothermal atmosphere, the emission features produced by the thermal scattering would lead to the spurious detection of a thermal inversion. Similarly we find that when the scattering is not taken into account, we retrieve the incorrect chemical abundance for the molecules in the atmosphere. This is important, as the fits from the pure absorbing cloud and the clouds that account for the scattering are of similar quality. Hence, if the scattering was not considered, we would be tricked into believing that the exoplanet atmosphere has a thermal inversion and the accompanying incorrect chemical abundances.
    \item We develop a new three-point cloud parameterisation that can be used as an agnostic approach to fitting for clouds in emission. We show that the chemical abundance for a range of atmospheres with different cloud condensing species can be retrieved without assuming anything about the cloud. This is an improvement on what current models do, where an assumption about the cloud species is made before performing the retrieval \citep{Venot2020}, this assumption could bias the results.
    \item We show that the three-point cloud parameterisation can also retrieve the general shape of the single-scattering albedo spectrum of the cloud. Despite the approach being agnostic to what sort of cloud is present, the retrieved single-scattering albedo can be compared to the single-scattering albedo of real species to infer the range of possible chemical composition and particle size distribution of the cloud.
    \item We find that our simple 3-point model cannot completely capture the complexities of the Ackerman $\&$ Marley model. This is due to our constant-with-altitude particle size and mixing. Future improvements on this work will include a hybrid approach between the "N-parameter" SSA retrieval and the vertical droplet parameterisation of the Ackerman $\&$ Marley model. Additionally, the spectrum used to test our framework is possibly pessimistic, as the real planet will probably have a larger thermal gradient and thus show larger spectral features.
\end{enumerate}

\section*{Acknowledgements}

Jake Taylor is a Penrose Graduate Scholar and would like to thank the Oxford Physics Endowment for Graduates (OXPEG) for funding this research. We thank Yakari for entertaining Vivien Parmentier's children while working on this manuscript from home. The majority of this manuscript was written while in lockdown due to COVID-19 and it would not have been possible to complete this study without the encouragement and support that Jake received from his supervisors Vivien Parmentier, Patrick Irwin and Suzanne Aigrain. Jake would also like to thank everyone at the University of Oxford's sub department of Atmospheric, Oceanic and Planetary Physics for providing loads of online support and being there for each other during this difficult time. We would like to thank the anonymous reviewer, their comments greatly improved our manuscript.

\section*{Data Availability}
The models generated in this paper are available on request.

\bibliographystyle{mnras}

redpajama_set_name: RedPajamaArXiv


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Arizona is well-known for its beautiful landscapes and scenery. These beautiful, must-experience places are bucket-list worthy; some are well-known while others are hidden gems you might not have known about. From national landmarks to historical towns and breathtaking outdoor landscapes, here are 16 places to visit on your next Arizona road trip.
Grand Canyon © Rex Vogel, all rights reserved
The most obvious landmark and Arizona road trip (and the most breathtaking of them all) is the Grand Canyon. If you have never experienced the sight of this outstanding view, you absolutely have to add this to your bucket list. The hiking trails will leave you speechless. Plus many photo opportunities! Check out the El Tovar Hotel, a historic property that opened its doors in 1905 and has entertained celebrities and presidents for over 100 years. Just steps away from the Grand Canyon's edge, the dining room is as close to the canyon as you can get as well.
Bisbee © Rex Vogel, all rights reserved
One of Arizona's best-kept secrets is the historic town of Bisbee. The former mining town is a small, unique community that sits high in the mountains in the far southeast corner of Arizona. With plenty of things to do, activities, events and festivals, shops, and galleries plus hiking, birding, gallery-gazing, or dining, Bisbee offers a plethora of choices to keep you entertained.
Glen Canyon National Recreation Area © Rex Vogel, all rights reserved
Glen Canyon National Recreation Area
Home to Lake Powell, The Glen Canyon National Recreation Area is a stunning region of blue water with desert landscape and dramatic stone walls. One of the largest manmade lakes in the United States, this area is known for both land-based and water-based recreational activities. You can enjoy a summer's day with perfect weather, cool water, amazing scenery, and endless sunshine. This is the perfect place to escape to and rent a houseboat, stay at a campground, or enjoy lodging.
Montezuma Castle National Monument © Rex Vogel, all rights reserved
Montezuma Castle National Monument
Montezuma Castle, near Camp Verde, has nothing to do with Montezuma, nor is it a castle. The Sinagua built the five-story, 20-room pueblo about 1150 but abandoned it in the early 1400s, almost a century before Montezuma was born. Montezuma Castle is built into a deep alcove with masonry rooms added in phases. A thick, substantial roof of sycamore beams, reeds, grasses, and clay served as the floor of the room built on top.
Hoover Dan © Rex Vogel, all rights reserved
Linking Arizona and Nevada, Hoover Dam is one of America's great engineering marvels and a fantastic Arizona road trip. Completed in 1935, this massive and hard-to-miss structure crosses the Colorado River and sits at a total of 726 feet high and 1,244 feet long. You are able to walk across the dam or take a tour. The visitor center provides information on the tours and has a café where you can stop for some basic grub.
Jerome © Rex Vogel, all rights reserved
An old mining town-turned ghost town-turned tourist attraction, Jerome sits on a mountainside just above the desert floor. Jerome is unique and quirky, to say the least, with the Sliding Jail in Jerome that was originally built around 1928. While you're there, you can visit the town's most appreciated historical landmarks including the Gold King Mine Museum and the Jerome State Historic Park.
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Monument Valley © Rex Vogel, all rights reserved
Along a 17-mile one-way gravel road, you will find the heart of the valley, Monument Valley Navajo Tribal Park. While visiting this area, which straddles the border between Arizona and Utah, you'll experience the true Arizona desert feel with miles and miles of beautiful landscape and scenery of mesas and buttes, shrubs and trees, and windblown sand, creating all the wonderful and majestic colors of the Valley.
Prescott © Rex Vogel, all rights reserved
With its small-city feel and defined seasons, Prescott has tall Ponderosa pine trees, lakes, and the occasional sprinkle of snow. This charming town has many things to offer, including the old courthouse, Whiskey Row, Elks Theatre, and numerous other tourist attractions. You can grab a bite to eat at one of the downtown restaurants.
Saguaro National Park © Rex Vogel, all rights reserved
One of Tucson's most popular attractions is Saguaro National Park which is a great place to experience the desert landscape around this well-known town and see the famous saguaro cacti up close. With an east and west portion, the park has two sections, approximately 30 minutes apart. Both sections of the park offer great opportunities to experience the desert and enjoy hiking trails.
Picacho Peak State Park © Rex Vogel, all rights reserved
Picacho Peak State Park
Jutting out of the Sonoran Desert some 1,500 feet, you can't help but see Picacho Peak for miles as you drive along Interstate 10 between Phoenix and Tucson. Travelers have used the peak for centuries as a landmark and continue to enjoy the state park's 3,747 acres for hiking, rock climbing, spring wildflowers, and camping
Tombstone © Rex Vogel, all rights reserved
After getting its start as a silver mining claim in the late-1870s, Tombstone grew along with its Tough Nut Mine becoming a bustling boomtown of the Wild West. From opera and theater to dance halls and brothels, Tombstone offered much-needed entertainment to the miners after a long shift underground. The spirits of Wyatt Earp, Doc Holliday, and the Clanton Brothers live on in the authentic old west town of Tombstone, home of Boothill Graveyard, the Birdcage Theatre, and the O.K. Corral.
Papago Park © Rex Vogel, all rights reserved
Just minutes from downtown Phoenix, Papago Park offers great hiking and a wide array of recreational facilities. Comprised primarily of sandstone, the range is known for its massive buttes that rise and fall throughout the park. Papago is home to two of the region's most visited attractions, the Phoenix Zoo and Desert Botanical Garden.
Sedona © Rex Vogel, all rights reserved
Sedona is a well-known hotbed of energy—one that's conducive to both meditation and healing—and this is one of the reasons 4.5 million travelers flock here annually. That and the region's red rocks: stunning sandstone formations that jut upward thousands of feet and change colors from orange to rust to crimson as the sun passes through the sky.
Canyon de Chelly National Monument © Rex Vogel, all rights reserved
A comparatively little-known canyon, Canyon de Chelly has sandstone walls rising up to 1,000 feet, scenic overlooks, well-preserved Anasazi ruins, and an insight into the present-day life of the Navajo, who still inhabit and cultivate the valley floor. From the mesa east of Chinle in the Navajo Nation, Canyon de Chelly is invisible. Then as one approaches suddenly the world falls away—1,000 feet down a series of vertical red walls.
Tucson © Rex Vogel, all rights reserved
Surrounded by mountains, Tucson is a beautiful city set in the Sonoran Desert and is the second-largest city in Arizona. With many historic sites and cultural attractions, Tucson is a place to unwind and explore. Highlights include the Arizona-Sonora Desert Museum, Saguaro National Park, Sabino Canyon, El Presidio Historic District, and Old Tucson Studios. You will also discover hiking trails, and afterward, you can find a bite to eat at one of the many wonderful restaurants Tucson has to offer.
Organ Pipe National Monument © Rex Vogel, all rights reserved
Organ Pipe Cactus National Monument
The remote Organ Pipe Cactus National Monument is a gem tucked away in southern Arizona's vast the Sonoran Desert. Thanks to its unique crossroads locale, the park is home to a wide range of specialized plants and animals, including its namesake. The organ pipe cactus can live to over 150 years in age, have up to 100 arms, reach 25 feet in height, and will only produce its first flower near the age of 35.
Worth Pondering…
The trip across Arizona is just one oasis after another. You can just throw anything out and it will grow there.
Author vogelontheroadPosted on November 25, 2021 November 23, 2021 Categories National Park, Road Trips, Snowbirds, State ParkTags Arizona, Arizona road trips, Bisbee, Canyon de Chelly National Monument, Glen Canyon National Recreation Area, Grand Canyon National Park, Hoover Dam, Jerome, Montezuma Castle National Monument, Monument Valley, Organ Pipe National Monument, Papago Park, Picacho Peak State Park, Prescott, Sedona, Tombstone, Tucson
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Danceline Clinic will be Friday, March 22, 2019. (PLEASE NOTE THE CHANGE IN DATE). The cost is $20. Deadline to turn in the registration form to the elementary or high school office is Wednesday, March 13, 2019.


Karim is from Colombia, she is Occupational Therapist and holds a PhD in Social Sciences, her work, among others, has been oriented to reflection about how the people learn about disability, especially with higher education students and social communities. She designed a subject called Disability and Society since 2001, for undergraduate students at School of Medicine and Health Sciences of Universidad del Rosario in Bogotá, she is associate professor there.
During her fellow in iHuman, she wants to deepen and systematize her pedagogical practice (thoughts, reasoning and attitudes) in Disability Teaching and explore opportunities of joint research around this thematic.


Sharks drown when they stop swimming. Inactivity kills. Keep moving, be the shark that is always swimming, always on the move. Be the shark… Be consistent.
Another year is upon us and by now you're sick and tired of all the campaigns for health and wellness. The ad campaigns like "It's the best time to start working towards a new you!", are on repeat 24/7. The truth is that there is no better time than the present to get started regardless of the date. It's about consistency, getting fit is not an impulse because of the season.
Set goals. Start small and work your way up to bigger goals. Every small victory is another notch in the belt that will help you eventually reach your ultimate goal. That goal might be to get faster, get stronger, lose weight, lower cholesterol or just look good in a swimsuit for the upcoming vacation on the beach. Whatever the goal, consistency with training will help get you there.
Measure your success. Would you go shopping for groceries on an empty stomach without a list? Sure, maybe if you are a frat boy, but those of us who have a little life experience know that going in to something unprepared and without a game plan is foolish. You need to see the fruits of your labor by keeping track of your results and measure the success by keeping times, PR's, weight loss/gain, etc. This will keep you motivated. Consistency is key.
Push yourself past your limits and know when you plateau. Anyone that has experienced the "get fit quick" bug, where they spend 2-4 weeks killing themselves trying to get in shape will know what this means. Athletes will know what this feels like when they are peaking too soon or coming into another season without enough rest. Pushing yourself past the normal comfort zone is hard but important for advancing towards reaching your fitness goal. It's when you stop pushing yourself or you don't change the workout up enough that you begin to see a stagnant plateau. The weight stops dropping & weight gain (if trying to lose) or decrease in strength seem to be among the first side effects. Avoid this by consistently changing your workout up and varying reps and sets frequently.
So… If sharks don't ever stop swimming because they know they'll drown… Do the fish that they stalk as prey ever stop swimming too? Keep moving. Inactivity kills. Be the fish that is consistently being chased by the shark who never rests… Be consistent.
Matt Lindland is a Greco-Roman Wrestling Olympics medalist and UFC mixed martial arts world champion. He currently coaches wrestling, kickboxing, MMA and professional and amateur fights through Team Quest, first launched with fellow athlete Randy Couture. Lindland founded North West Kids and has coached two of the top wrestling clubs in the USA, won a NJCAA National title, and is one of the most respected wrestlers in the world. His insight on staying fit offers a wealth of health information.
You can connect with us on Facebook and Google Plus.


text:
Leaked emails just exposed embarrassing internal civil war at Fox news over white supremacy
by Natalie Dickinson
Fox News has been a blight on this country ever since it first launched, a conduit pumping unadulterated paranoia and right-wing hatred into the homes of millions of Americans, slowly warping their worldviews and dividing them along partisan lines, but under Donald Trump, it's somehow managed to morph into something even more sinister.
Fox is now little more than a state propaganda machine, completely untethered from reality, constantly spewing pro-Trump rubbish which has, in turn, caused it to increasingly embrace unvarnished white nationalism. Tucker Carlson's show alone is one of the vilest, racist shows to ever air in this country.
Now, The Daily Beast has obtained Fox emails in which one of the organization's reporters told two of his coworkers that they sounded like a "white supremacist chat room" for trying to defend Trump's "very fine people on both sides" comments after the Charlottesville debacle. The emails were originally published by FTV Live.
Yesterday, during his presidential bid election announcement, former Vice President Joe Biden zeroed in on Trump's Charlottesville rhetoric as proof that he lacks the moral backbone to serve as president, decrying the fact that Trump "assigned a moral equivalence between those spreading hate and those with the courage to stand against it."
The Biden comments prompted Fox reporter Doug McKelway to rush to Trump's defense, and he penned an email to numerous Fox employees which stated that he was "Putting this Biden statement out there, next to Trump's original presser, and a live interview I did in C-ville with 'good people on both sides'" The email included a defense of Trump's false rhetoric that he never really said what people believe he did about the Charlottesville Neo-Nazis.
McKelway brought up a quote from one of his own past interviews with Brian Lambert, one of the men who attended the Unite the Right Rally:
"They're denying people their right to assemble. They're denying their right to speak freely, however hateful their views may be," he said.
Some of those who received McKelway's email, like Fox News Digital Senior Editor Cody Derespina, voiced their agreement with his sentiments. Derespina also sent out an email containing an interview with Jarrod Kuhn, a man who marched at Charlottesville and who insisted he is not a white supremacist but that he didn't want a statue of Confederate general Robert E. Lee removed.
Not everyone on the email list agreed.
"I really don't understand the point you are making. Jarrod Kuhn was one of those individuals in Charlottesville holding a tiki torch while the mob chanted 'Jews will not replace us," responded Fox News Radio White House Correspondent Jon Decker.
In response, McKelway backed down a bit, and linked to an article from C-Ville that revealed Brian Lambert, the man whose interview he quoted earlier in the email chain, had in fact been exposed as a white nationalist who put up rebel flag stickers around his town and used white power hand signals at his court sentencing.
"[I]t appears Lambert revealed himself to be not the squeaky clean 1st amendment supporter he claimed to be on live TV," McKelway conceded.
Decker reprimanded him further and added:
"Based upon the slew of emails that I've received today, both of you should send an apology to your Fox News colleagues—many of whom are hurt and infuriated by your respective posts. Your posts read like something you'd read on a White Supremacist chat room."
Decker should be commended for standing up to his colleagues. Now, more than ever it's imperative to confront this ideology head-on. Trump and his racist beliefs are a tumor on this nation that can only be truly excised at the ballot box, but hatred can still be battled each day in our interactions with others.
Natalie Dickinson
Natalie is a staff writer for the Washington Press. She graduated from Oberlin College in 2010 and has been freelance blogging and writing for progressive outlets ever since.
Russia's backchannel to the NRA just got sentenced to hard time
Trump just sparked cries of racism with a controversial tweet about last night's NFL draft
Natalie Dickinson April 26, 2019


The name Aquis probably doesn't mean much to you if you live outside Far North Queensland. If you do, it's a name that's been on everyone's lips for the last year and has dominated local media.
Aquis Great Barrier Reef Resort is a mega casino proposed for the sleepy Cairns beach suburb of Yorkeys Knob – a beautiful, but environmentally sensitive and flood prone area on the edge of the Great Barrier Reef.
At an estimated $8.15 billion to build, it's a development of unprecedented scale in a city of 150,000 people, and its bold promises to reinvigorate the region have won many followers. None more so than than the Queensland Government, who have granted it one of two new regional casino licences up for grabs, provided conditions are met.
The Newman Government has been salivating over Queensland's potential for gambling revenue for some time, obsessed with turning the state into Australia's own Nevada. Queensland's economy has been suffering over the last few years, largely due to the downturn in the mining sector. Though still hell bent on selling off Queensland's environment to the highest bidder, Campbell Newman now sees casino revenue as a much needed quick buck, to line the state coffers during these economically challenging times.
Since Newman opened up bidding for three new casino licences last year, one in Brisbane and two in regional Queensland, developers have been chomping at the bit to get a piece of the action.
Not content with Cairns' unique, natural wonders – the Great Barrier Reef, the Daintree Rainforest, the endemic wildlife – his vision is one of bright lights, glitz and 24 hour gambling. And for Fung, the bigger, the better – everything in his 'resort' is super-sized.
His target market? Chinese high rollers, who can gamble their days away while their families enjoy the other facilities – theatres, horse riding, a mega Aquarium, artificial lagoons, shopping, sports stadiums, restaurants, golf courses – visitors won't ever need to leave the resort. They won't even need to use local taxis – on arrival into Cairns airport, they will be chauffeured or transferred by courtesy bus directly to the resort.
At the heart of Fung's 'resort' is a gigantic gambling hub, consisting of two casinos. Aquis is requesting the same number of pokies as Sydney's Star City Casino, and more gaming tables than Australia's two largest casinos (The Crown & Star City) combined. Fung is also in the midst of taking over Cairns' existing casino, the Reef Hotel.
Fung submitted his initial Aquis application to the Queensland government last July, who declared it a 'Coordinated Project' a mere six days later. For a multi-billion dollar development next to the world heritage listed waters of the Great Barrier Reef, this is alarmingly fast. The normal, stringent process of environmental and social assessments can take months, even years to complete.
Aquis has bypassed the proper environmental approval process from the beginning. In a submission to the Department of Environment in April, Fung maintained that Aquis does not require a commonwealth environmental assessment process, as any impacts on the surrounding environment are not significant enough to warrant it.
Queensland is going through some economically challenging times. With high levels of unemployment, particularly amongst the youth, I understand the need to boost the Cairns economy and to stimulate much needed growth and development in the region. But using a mega casino as a bandage to fix a flawed economic model will not accomplish this.
Gambling is not a stable source of state revenue. The success of Aquis is completely dependent on the economies of the Asian tourists that Fung is trying to lure. Most people realise that the Cairns economy is too reliant on tourism and needs to diversify in order grow sustainably. Diversification ensures that the local economy survives through the tourist low season, as well as any global economic downturns which may reduce overseas visitors dramatically.
Even the Cairns Chamber of Commerce listed diversification as one of their top 5 federal election priorities in 2013. Hedging all our bets on Asian tourists to solve economic problems is doomed for failure and is not aligned with the Chamber's own top priorities.
Perceived benefits to local businesses from the increase in tourist numbers is one of the main reasons that people support the new casino. But the reality is that local business suffer when you centralise shopping, restaurants, hotels, entertainment into one facility. With everything in one resort, guests will have significantly reduced need to leave the complex, meaning small business may have to downsize or close altogether.
Tourism and business aside, do we really want Cairns to become the new Macau of Fung's dreams? Macau has been plagued by a host of problems since opening up its economy to foreign casinos in 2002 including organised crime, prostitution and environmental degradation. It's certainly not something to aspire to, and I question anyone who thinks otherwise.
This means that the front of house staff will need to be Chinese. In recent months, the Federal Government has been trying to negotiate a free trade deal with China. In order to clinch the deal, it is willing to consider visa options for skilled workers to come to Australia to work on major Chinese projects. The free trade deal will certainly benefit Tony Fung if he wishes to use Chinese workers.
But the biggest bone of contention for many is the casino itself. Does Cairns really need two more casinos?
The social impacts of these mega casino in a city which already has high levels of problem gambling have not been adequately explored. Per capita, Cairns is Queensland's highest spending pokie city, with the average resident spending $45.41 per month on pokies in 2012. Those behind Aquis, including the Newman government, insist that the casino will be frequented mostly by wealthy Asian tourists and that negative community impacts will be minimal.
But the Productivity Commission found that high rollers only count for 11% of revenue in casinos, with the rest coming from locals playing cards and pokies. In addition, studies have proven that big pokies venues are the most dangerous to local communities.
The debate is growing outside of the Far North, with recent national and even international coverage with amusing headlines like, "Hippie Town Seen as New Macau With World's Biggest Hotel".
South Australian Independent Senator Nick Xenophon, has also lent his vocal support to the growing campaign against Aquis. On June 20th, he will speak at a community forum hosted by Aquis Aware, warning locals of the "fools gold" that is Aquis and the dangers of a gambling resort.
The dream of Aquis has an undeniable lure, especially for those who have been hit hard over the past few years. It has the potential to change the face of Cairns forever, but I fear it won't be for the better.


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The architect of the Capital (AOC), the attending physician's office is recruiting for a nurse. Employees provide nursing services to buildings of Congress, staff and visitors in the jurisdiction of AOC.
The mission of the architect of the Capital (AOC) is to serve the Congress and the Supreme Court, protects America's capital and inspires memorable experiences. Stephen T. Under the leadership of Ayers, FAIA, Lead AP, the roles and responsibilities of the architect of Capitol cover a huge and diversified portfolio. AOC is responsible for the maintenance, operation, development and conservation of 553 acres of total 174 million square feet of buildings in Capitol Hill. AOC U.S Through the Capital Visitor Center and US Botanical Gardens provides a welcome and educational environment for millions of visitors. AOC offers professional expertise on conservation of architectural and artistic elements assigned to its care.
You-must be a U.S. Citizen.
You-should be able to pass a drug test.
Your-resume and answers to the questions should show KSA related to the job.
You-have to fulfill the definition of special experience.
Registration: Applicants should be present in the state as a professional nurse, District of Columbia, Commonwealth of Puerto Rico or an area of the United States, current registration.


A bare-bones user avatar display with support for falling back to using the user's initials on a colored background as the avatar.
React component for rendering a gravatar profile image. Adjusts automatically to HiDPI displays.


Refined ladylike style dresses and outfits to dress up your little ones on special occasions. For Sarabanda girls floral patterns or checks, lace details and natural tones, stylish micro polka dots patterns or checks for boys. They will be the most chic of the party!


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Rebekah Jean & Anthony Papaleo like you've never heard them before!
Playing their favorite jazz numbers at this fantastic community event in Cuyahoga Falls, Papaleo's exquisite guitar playing paired with Rebekah's voice on songs of Billie Holiday, Julie London, Chet Baker, and more.
Bebop on the Block is the kickoff event to "Falls Downtown Fridays."