where $R_{*}$ and $T_{*}$ are the stellar radius and temperature, $a$, the star planet separation and $\tau$ is the infrared (thermal) optical depth $$\tau=\frac{\kappa_{IR}P}{g}$$ with $P$ the pressure and $g$ the surface gravity (at 1 bar). In total there are 4 free parameters governing the temperature structure, $\kappa_{IR}$, $\kappa_{v_{1}}$, $\kappa_{v_{2}}$ and $\alpha$. We choose this parameterization with two visible streams as opposed to the traditional one visible stream (Hansen 2005; Guillot 2010) because the extra stream allows more freedom for a temperature inversion, though in some cases (as we shall see below) the second visible stream does not matter.

The remaining 4 parameters are the uniform mixing ratios for H$_2$O, CH$_4$, CO, CO$_2$, expected to be the major molecular opacity sources (Tinetti et al., 2007; Swain et al., 2009a). We choose vertically uniform mixing ratios for two reasons. First, the data lack sufficient information content to actually help resolve vertical structure in abundances, and second, chemical kinetics models (Moses et al. 2011; Line et al. 2010, 2011), show that vertical mixing leads to constant vertical mixing ratios for these species within the IR photosphere, so even if we could resolve detailed vertical information, we would most likely find that the 