= Solution
This is a <semi-grey irradiated atmosphere> in <radiative equilibrium>. The quantities have these meanings:
* $T$ is the local atmospheric <temperature>.
* $T_{\rm int}$ is the <internal effective temperature of a planet>, defined by the upward intrinsic flux $F_{\rm int}=\sigma_{\rm SB}T_{\rm int}^4$.
* $T_{\rm irr}$ is the <irradiation temperature>, conventionally $T_\star\sqrt{R_\star/a}$, so $\sigma_{\rm SB}T_{\rm irr}^4$ is the incident substellar bolometric stellar flux.
* $f_r$ is the <stellar flux redistribution factor>. With albedo treated separately, uniform global averaging gives $f_r=1/4$ and dayside averaging gives $1/2$. If reflected power is included, the absorbed factor is reduced accordingly; these choices must not be double-counted in $T_{\rm irr}$.
* $\tau$ is the downward thermal <optical depth>, measured from the top with mean infrared <opacity> $\kappa_{\rm IR}$.
* $\gamma=\kappa_{\rm vis}/\kappa_{\rm IR}$ is the ratio of the mean stellar-band absorption <opacity> to the mean thermal <opacity>, not the gas's adiabatic index.
The model assumes a static <plane-parallel atmosphere>, separate grey stellar and thermal bands with constant <opacities>, <local thermodynamic equilibrium> for thermal emission, negligible scattering, and transport by <radiative transfer> rather than <convection>. The <Eddington closure approximation> sets the thermal angular-moment ratio to $1/3$, and the upper boundary supplies the usual $2/3$ term. Treating the incoming radiation with representative direction cosine $1/\sqrt3$ gives its attenuation $e^{-\sqrt3\gamma\tau}$. Intrinsic flux enters from below, while the imposed external stellar flux is absorbed from above.
With constant gravity and <hydrostatic equilibrium>, $\tau\simeq\kappa_{\rm IR}P/g$, allowing conversion to an <atmospheric pressure-temperature profile>. This last relation additionally assumes constant thermal <opacity>. The grey treatment describes the energy balance approximately; it does not resolve individual molecular absorption lines.
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