Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 315 / 2 / a / i

Solution

$T$ is the local atmospheric temperature.
$T_{int}$ is the internal effective temperature of a planet, defined by the upward intrinsic flux $F_{int} = σ_{SB} T_{int}^{4}$ .
$T_{irr}$ is the irradiation temperature, conventionally $T_{⋆} R_{⋆} / a$ , so $σ_{SB} T_{irr}^{4}$ 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 $T_{irr}$ .
$τ$ is the downward thermal optical depth, measured from the top with mean infrared opacity $κ_{IR}$ .
$γ = κ_{vis} / κ_{IR}$ 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/ 3

e^{- 3 γ τ}

. Intrinsic flux enters from below, while the imposed external stellar flux is absorbed from above.

τ ≃ κ_{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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