x}\mbox{,}\label{eqn_F_def}\\
I & = & \frac{(1-A)G}{r^2}\cos(\theta^*)\cos\left(\frac{2\pi t}{\tau}\right)\mbox{,}\label{eqn_W_def}\\
R & = & \gamma\:\frac{(1-A)G}{r^2}\cos(\theta^*)\cos\left(\frac{2\pi t}{\tau}\right) = \gamma\:I_\mathrm{max}\mbox{, 
and}\label{eqn_I_def}\\
W & = & \kappa\:F\mbox{.}\label{eqn_R_def}\end{aligned}$$

Here, $\epsilon$ is the surface emissivity of the planet, $\sigma$ is the Stefan-Boltzmann constant, $T_\mathrm{planet}$ is the planet’s temperature, $k$ the coefficient of heat conduction, $A$ the planet’s bolometric albedo, $G$ the Solar constant (i.e. the energy flux density of Solar irradiation at the earth’s mean distance), $r$ the distance of the planet to the sun in astronomical units, $\tau$ the planet’s rotation period and $\theta^*$ the geographical latitude of the radiation absorbing surface element. The temperature distribution in the interior of the planet at radial position $x$ is controlled by the heat conduction equation: $$c\cdot\frac{\upartial T_\mathrm{planet}}{\upartial t} = k\cdot\frac{\upartial^2 T_\mathrm{planet}}{\upartial x^2}\mbox{,}$$ with the specific heat per unit volume $c$.

In our model, the heat loss $R$ of the planet’s surface due to conduction to the planet’s atmosphere is taken to be a constant fraction of the heat flux $F$ outward from the interior of the planet, the constant of proportionality being $\kappa$, for which we assumed $\kappa=0.1$. Similarly, the heat gain by back-scattering radiation by the atmosphere $R$ was assumed to be a constant fraction $\gamma$ of the 