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 Incoming links: Stellar flux redistribution factor

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 315 / 2 / a / i / Solution Created 2026-10-03 Updated 2026-10-05

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This is a semi-grey irradiated atmosphere in radiative equilibrium. The quantities have these meanings:

$T$ 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

gives its attenuation

e^{- 3 γ τ}

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

With constant gravity and hydrostatic equilibrium,

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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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 315 / 4 / b / Solution Created 2026-10-03 Updated 2026-10-05

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The geometric albedo

A_{g}

compares the planet's reflected brightness at full phase with that of a perfectly reflecting Lambertian surface in the form of a flat disc of the same projected area, illuminated by the same incident flux. The Bond albedo

A_{B}

is the fraction of total incident stellar power reflected in all directions and over all wavelengths. They need not be equal: the planetary phase integral and the incident spectral weighting connect them.

For a blackbody star, the incident flux at orbital distance

a

S = \frac{L _{⋆}}{4 π a ^{2}} = σ_{SB} T_{⋆}^{4} (\frac{R _{⋆}}{a})^{2} .

(1)

Assume negligible intrinsic luminosity, thermal emissivity one, and efficient day-night heat redistribution giving uniform emission over the whole planet. The absorbed and emitted powers are

π R_{p}^{2} S (1 - A_{B}) = 4 π R_{p}^{2} σ_{SB} T_{e}^{4} .

(2)

Using the Stefan–Boltzmann law gives the planetary equilibrium temperature

T_{e} = T_{⋆} \frac{R _{⋆}}{2 a} (1 - A_{B})^{1/4} .

(3)

For a representative emitting area encoded by a stellar flux redistribution factor

f

, the equivalent expression is

T_{e} = T_{⋆} R_{⋆} / a [f (1 - A_{B})]^{1/4},

(4)

with

f = 1/4

for global uniform emission and

1/2

for uniform dayside-only emission. The Bond albedo, rather than the geometric albedo, belongs in the energy budget. Internal heat, wavelength-dependent emissivity and imperfect redistribution require corresponding modifications.

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