though the methods are easily be extended to other kinds of observations (transmission spectra) and exoplanets (hot-Neptunes, mini-Neptunes, super-Earths etc.) with relatively minor modifications to the forward model. For future instruments, with broader spectral coverage and higher spectral resolution, the forward models can increase in sophistication.

Lacking sufficient data (these are low signal-to-noise, low resolution spectra), we simplify our atmosphere to 8 parameters that characterize the temperature structure and gas concentrations. For sake of simplicity, we use an analytic temperature profile formulated by Guillot (2010), and since then modified by Parmentier & Guillot, (in preparation) to include three channels. The profile, derived using a 3 channel approximation, is given by $$T^4(\tau)=\frac{3T^4_{int}}{4}(\frac{2}{3}+\tau)+\frac{3T^4_{irr}}{4}(1-\alpha)\xi_{\gamma_{1}}(\tau)+\frac{3T^4_{irr}}{4}\alpha \xi_{\gamma_{2}}(\tau)$$ where $$\xi_{\gamma_{i}}=\frac{2}{3}+\frac{2}{3\gamma_{i}}[1+(\frac{\gamma_{i}\tau}{2}-1)e^{-\gamma_{i}\tau}]+\frac{2\gamma_{i}}{3}(1-\frac{\tau^{2}}{2}){\rm E_{2}}(\gamma_{i}\tau)$$ with $\gamma_{1}=\kappa_{v_{1}}/\kappa_{IR}$ and $\gamma_{2}=\kappa_{v_{2}}/\kappa_{IR}$, where $\kappa_{v_{1}}$, $\kappa_{v_{2}}$, and $\kappa_{IR}$ are the visible and infrared (thermal) opacities, respectively. The parameter $\alpha$ (range 1 to 0) partitions the flux between the two visible streams, and $E_2(\gamma \tau)$ is the second order exponential integral function. The internal heat flux (from the net cooling history) is represented by the temperature T$_{int}$, while the solar flux at the top of the atmosphere is represented by T$_{irr}$; these two temperatures are fixed. Assuming zero albedo and unit emissivity, $T_{irr}$ is $$T_{irr}=(\frac{R_{*}}{2a})^{1/2}T_{*}$$ 