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 Incoming links: Atmospheric pressure-temperature profile

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 / 3 / a / Solution Created 2026-10-03 Updated 2026-10-05

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Four mechanisms that can produce disequilibrium chemistry in an exoplanet atmosphere are:

Atmospheric photochemistry. Stellar ultraviolet photons initiate reactions whose products need not follow local thermochemical equilibrium. The ozone layer on Earth is a solar-system example. Calculations for HD 189733 b predict enhanced hydrogen cyanide and acetylene from the processing of methane and ammonia; these are model examples rather than assertions of an unambiguous detection.
Vertical transport and chemical quenching. When the eddy mixing time is shorter than the chemical relaxation time, gas retains a deeper abundance above its chemical quench level. The excess carbon monoxide in Jupiter's cool atmosphere exemplifies carbon monoxide–methane quenching. Models of HD 189733 b predict quenched methane and ammonia abundances differing from their local chemical equilibrium values. The enhancement or depletion depends on the underlying atmospheric pressure-temperature profile.
Horizontal chemical quenching. If the atmospheric advection time is short, winds move chemically processed gas into regions with different irradiation or temperature faster than it can re-equilibrate. Transport of gas within Earth's ozone layer moves material away from its local photochemical production regions. Models of HD 209458 b show that dayside carbon monoxide-rich composition can persist into the cooler nightside instead of forming the local chemical equilibrium amount of methane.
Condensation with sedimentation or rainout. Finite-rate cloud formation can depart from phase equilibrium, while atmospheric condensate rainout removes elements from a layer and changes its gas composition. Earth's water atmospheric cold trap limits the supply of water to the stratosphere. In HD 209458 b models, titanium-bearing condensates can settle and suppress upper-atmospheric titanium monoxide. The remaining gas can still be in local chemical equilibrium with its depleted inventory: rainout is an open-column effect, not necessarily a failure of equilibrium among all gas reactions.

The exoplanet transport and photochemical examples follow kinetic atmosphere calculations and models including horizontal transport; the condensate example is examined in cold-trap calculations.

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 315 / 3 / b / iii / Solution Created 2026-10-03 Updated 2026-10-05

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Four possible molecular signatures of disequilibrium chemistry in an exoplanet atmosphere, measured through an exoplanet transmission spectrum or exoplanet emission spectrum, are:

Excess carbon monoxide in the cool upper layers, where local chemical equilibrium would place most carbon in methane. This can indicate carbon monoxide–methane quenching from deeper hot gas.
Suppressed methane bands relative to the same cool equilibrium model, consistent with transport preventing complete carbon monoxide conversion or with photochemical loss.
An ammonia abundance inconsistent with the local equilibrium nitrogen partition, potentially recording nitrogen–ammonia quenching. Its direction must be evaluated for the actual atmospheric pressure-temperature profile; a depleted value is possible when hotter, molecular nitrogen-rich gas is transported upward.
Enhanced hydrogen cyanide or acetylene bands, consistent with atmospheric photochemistry acting on the transported carbon and nitrogen reservoirs.

The first two signatures can be consequences of the same process and are not independent evidence for two mechanisms. Changes in atmospheric carbon-to-oxygen ratio, atmospheric metallicity of a giant planet, exoplanet cloud coverage, and the atmospheric pressure-temperature profile can mimic abundance changes. A convincing inference compares several molecular bands with a chemically consistent equilibrium model rather than identifying one unusual band alone.

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 315 / 3 / b / i / Solution Created 2026-10-03 Updated 2026-10-05

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Take the logarithm to base ten and write pressure in units of

1 bar

, so its argument is dimensionless. Continuity of the atmospheric pressure-temperature profile gives

C = \frac{1400 - 500}{l o g _{10} ( 1 ) - l o g _{10} ( 1 0 ^{- 2} )}, C = 450 K per pressure decade .

(1)

Thus

T (P) = ⎩ ⎨ ⎧ 500 K, 1400 K + 450 K lo g_{10} (P /1 bar), 1400 K, P \leq 1 0^{- 2} bar, 1 0^{- 2} bar < P < 1 bar, P \geq 1 bar .

(2)

If the logarithm means

ln

, the equivalent constant is

C = 900/ ln 100 ≃ 195.4 K

. A plot of temperature against logarithmic pressure is vertical in each isothermal region and straight between the two endpoints.

The processes can be organized by the supplied pressure ranges, although the exact boundaries require reaction rates, irradiation and mixing information:

At $P > 1 bar$ , the dense $1400 K$ gas can approach thermochemical equilibrium because collisions and reactions are relatively rapid. Deep carbon monoxide and molecular nitrogen can provide reservoirs for transported material.
At $1 0^{- 2} < P < 1 bar$ , the falling temperature slows chemical conversion. Vertical transport can produce a chemical quench level when the chemical relaxation time crosses the eddy mixing time. Horizontal chemical quenching is also possible if dayside and nightside conditions differ. Suitable species can condense and undergo atmospheric condensate rainout where a saturation curve is crossed.
At $P < 1 0^{- 2} bar$ , slow thermal chemistry permits a quenched atmospheric mixing ratio to survive. Atmospheric photochemistry can dominate where stellar ultraviolet photons penetrate, often at still lower pressures; atmospheric haze may form from its products. Extremely high layers can also experience atmospheric escape.

The profile identifies plausible chemical regimes, but does not fix their transition pressures by itself. In particular, cloud formation depends on the species-specific condensation curve, and ultraviolet processing depends on shielding.

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