$T$ 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.

ii

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Solution

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The thermal optical depth increases inward, so an atmospheric thermal inversion requires

d T / d τ < 0

. Differentiating the semi-grey irradiated atmosphere profile gives

\frac{d T ^{4}}{d τ} = \frac{3}{4} [T_{int}^{4} - f_{r} T_{irr}^{4} (γ^{2} - 1) e^{- 3 γ τ}] .

(1)

Therefore the inversion criterion with intrinsic planetary flux is

f_{r} T_{irr}^{4} (γ^{2} - 1) e^{- 3 γ τ} > T_{int}^{4} .

(2)

An inverted layer exists at the top precisely when

γ^{2} > 1 + \frac{T _{int}^{4}}{f _{r} T _{irr}^{4}} .

(3)

For

T_{int} > 0

, it extends over

0 \leq τ < \frac{1}{3 γ} ln [\frac{f _{r} T _{irr}^{4} ( γ ^{2} - 1 )}{T _{int}^{4}}],

(4)

provided the logarithm is positive. Otherwise the profile has no inverted interval. For a strongly irradiated hot Jupiter,

T_{int} ≪ T_{irr}

and the threshold is approximately

γ > 1

; the intrinsic flux eventually restores an inward-increasing temperature at greater depth.

Large visible absorption opacity deposits stellar heat high in the exoplanet atmosphere. Gas-phase titanium monoxide and vanadium monoxide are candidate absorbers in sufficiently hot layers. An atmospheric cold trap or other condensate loss can remove them, while sufficiently high temperature and vigorous mixing can help keep them in the gas. The condition concerns absorbing opacity: highly reflective scattering alone does not deposit the required heat.

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Solution

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For uniform composition, the Schwarzschild criterion gives stability against convection when

\nabla_{rad} \equiv \frac{d ln T}{d ln P} < \nabla_{ad} \equiv (\frac{\partial ln T}{\partial ln P})_{s} .

(1)

A parcel displaced upward expands approximately adiabatically. If the ambient logarithmic temperature gradient is shallower than the adiabatic temperature gradient, the parcel cools more than its surroundings, becomes denser, and sinks back. Equality is neutral stability; a steeper gradient drives buoyant growth.

For a diatomic ideal gas with active translational and rotational degrees of freedom,

c_{p} / c_{v} = 7/5

and

\nabla_{ad} = 2/7

. If the infrared opacity is constant,

P \propto τ

, so the profile's radiative temperature gradient is

\nabla_{rad} = \frac{τ}{4 T ^{4}} \frac{d T ^{4}}{d τ} .

(2)

An atmospheric thermal inversion has negative

\nabla_{rad}

and is stable by this criterion. Composition gradients require the Ledoux criterion; molecular dissociation and changing heat capacities can also change the numerical adiabatic temperature gradient.

b

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Solution

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Take

z

to increase inward, so the positive coefficient

A

describes an inward-increasing temperature. Constant gravity and hydrostatic equilibrium give

d P / d z = ρ g

, hence

\frac{d T}{d P} = \frac{A}{g T ^{3}}, \nabla_{rad} = \frac{A P}{g T ^{4}} .

(1)

Matching the radiative temperature gradient to the adiabatic temperature gradient gives the formal local boundary relation

P_{b} = \frac{g}{A} \nabla_{ad} T_{b}^{4} .

(2)

The same result follows from radiative diffusion: for constant upward thermal flux

F

and Rosseland mean opacity

κ

A = 3 κ F / (16 σ_{SB})

There is an important limitation to treating

A

as constant over the entire radiative layer. Integration from an irradiated outer boundary gives

T^{4} = T_{0}^{4} + \frac{4 A}{g} P, \nabla_{rad} = \frac{1}{4} (1 - \frac{T _{0}^{4}}{T ^{4}}) < \frac{1}{4} .

(3)

For a diatomic ideal gas,

\nabla_{ad} = 2/7 > 1/4

, so this profile cannot actually reach a radiative-convective boundary. Formally, imposing a constant

\nabla_{ad}

would give

P_{b} = \frac{g T _{0}^{4}}{A} \frac{\nabla _{ad}}{1 - 4 \nabla _{ad}},

(4)

which is positive only for

0 < \nabla_{ad} < 1/4

. A finite boundary for a normal molecular atmosphere requires additional opacity, flux, or thermodynamic variation. The local matching formula is usable near a real boundary, but constant