Disequilibrium chemistry in an exoplanet atmosphere

Disequilibrium chemistry in an exoplanet atmosphere occurs when transport or externally driven reactions prevent the composition from reaching local thermochemical equilibrium. Deep gas brought upward can retain its earlier composition if chemical reactions become too slow in cooler layers. At higher altitudes, atmospheric photochemistry can instead destroy or create species using stellar photons.

Models coupling chemical kinetics with vertical diffusion for HD 189733 b and HD 209458 b demonstrate that chemical quench levels and photochemical destruction can affect the same molecule at different heights.^[ref] An atmospheric retrieval that assumes equilibrium everywhere can consequently infer a different composition or atmospheric pressure-temperature profile from one that allows transport.

Atmospheric photochemistry is the network of chemical reactions initiated when absorbed photons dissociate or ionize atmospheric species. The initial fragments can react with other molecules, so a small abundance of short-lived products can control the chemistry of much more abundant gases. The reaction rates depend on the stellar spectrum, absorption and the amount of shielding above a layer.

The Chapman ozone equilibrium illustrates coupling between photodissociation, molecular formation and destruction. In an exoplanet atmosphere, such reactions can change spectral absorbers or supply material for atmospheric haze. A photochemical steady state balances production and loss while reactions continue; it need not be thermochemical equilibrium, because stellar radiation continuously drives the system.

A photochemical steady state has approximately constant local abundances because production and loss balance for each species. If a molecule is produced at a rate $P$ per unit volume and removed with a first-order rate coefficient $k$, then $dn/dt=P-kn$ and the stationary number density is $n=P/k$. The relaxation time is $1/k$ when the coefficients stay fixed.

This state is useful for short-lived intermediates in atmospheric photochemistry. It does not mean that molecular reactions stop or that thermodynamic equilibrium has been reached. Transport adds another term to the abundance balance; if it is important, the local chemical production and loss rates need not balance each other by themselves.

Chapman ozone equilibrium balances production and destruction of atomic oxygen and ozone in a minimal oxygen-only atmospheric photochemistry model. Write square brackets for number densities. Define photolysis rate coefficients $J_1,J_3$ and collisional rate constants $K_2,K_4$ byThe collision partner $\mathrm{M}$ removes excess formation energy without being consumed. The third-body requirement reflects conservation of energy and momentum conservation during ozone formation.^[ref]

Adding the reaction rate equations for atomic oxygen and ozone cancels their rapid interconversion. A local steady state givesIf ozone loss is predominantly photolytic, $J_3\gg K_4[\mathrm{O}]$, its separate balance isCombining the two balances yieldsThe two factors of two in oxygen-atom production and destruction cancel in the first balance. This is a sunlit local steady state, with specified temperature, radiation and background number densities; transport and additional catalytic chemistry need not obey this four-reaction estimate.

An ozone layer is an atmospheric altitude range with enhanced ozone abundance. Oxygen-dissociating ultraviolet radiation is strongest aloft while three-body formation is fastest at greater density, so their competition can concentrate ozone at intermediate altitude.

The eddy mixing time over one atmospheric scale height is ${\tau }_{\mathrm{mix}}\simeq H^2/K_{zz}$, where $K_{zz}$ is the vertical eddy diffusion coefficient.

The vertical eddy diffusion coefficient parametrizes turbulent transport in an atmosphere by a diffusive flux proportional to an abundance gradient. Over distance $L$, the corresponding eddy mixing time is $L^2/K_{zz}$. Its units are area per time, so $1{\mathrm{m}}^2{\mathrm{s}}^{-1}=10^4\mathrm{c}{\mathrm{m}}^2{\mathrm{s}}^{-1}$.

A chemical quench level is an atmospheric region where the chemical relaxation time becomes comparable to a transport time. Below it, sufficiently fast chemical reactions maintain near-local thermochemical equilibrium. Above it, slower reactions cannot keep up with transported gas, allowing the composition to retain information about deeper conditions.

For vertical mixing over length $L$, a useful estimate is ${\tau }_{\mathrm{mix}}\sim L^2/K_{zz}$ with vertical eddy diffusion coefficient $K_{zz}$. Comparing this with a reaction-network relaxation time locates an approximate quench region. The quenched atmospheric mixing ratio can remain nearly constant even though number density falls upward. Different species can quench at different depths, and atmospheric photochemistry or atmospheric condensate rainout can subsequently alter their abundances.

For a vertical eddy diffusivity $K_{zz}$, mixing over a length $\ell$ takes ${\tau }_{\mathrm{mix}}\sim {\ell }^2/K_{zz}$. The condition that transport outrun a chemical relaxation time therefore gives $K_{zz}\gtrsim {\ell }^2/{\tau }_{\mathrm{chem}}$. Taking $\ell =0.1H$ instead of one atmospheric scale height $H$ changes this estimate by a factor of one hundred. The atmospheric scale height itself depends on both temperature and gravity.

Atmospheric advection time shorter than the chemical relaxation time allows winds to carry a composition into regions where it differs from local thermochemical equilibrium. In a hot Jupiter, this can transport dayside carbon chemistry into the cooler nightside.

Above a chemical quench level, rapid eddy mixing can keep a species' number mixing ratio $f_A$ approximately fixed at its deeper value, provided condensation, photochemical loss, and molecular diffusion remain negligible. The species' number density is then $n_A=f_{A}p/(k_{B}T)$; constant mixing ratio does not imply constant number density. In an isothermal atmosphere, $n_A$ decreases exponentially with altitude on the atmospheric scale height.

Carbon monoxide–methane quenching preserves a transported carbon partition after local reactions become too slow to restore thermochemical equilibrium. A useful net reaction isThis is a net stoichiometric balance, rather than a single elementary reaction whose rate can simply be read from the equation. Temperature, pressure and the kinetic pathway determine the preferred composition and conversion time.

In models of hot Jupiters, vertical transport can maintain methane away from its local equilibrium abundance, while atmospheric photochemistry removes it at higher levels.^[ref] A methane band therefore probes both chemistry and circulation, rather than acting as a unique local temperature measurement.

Nitrogen–ammonia quenching occurs when transport preserves a nonequilibrium partition between molecular nitrogen and Ammonia. Their net conversion may be writtenbut its actual relaxation time depends on a multistep reaction network. Near a chemical quench level, that time exceeds the eddy mixing time, so the abundance can retain a deeper value instead of following local thermochemical equilibrium.

Coupled diffusion and chemistry models illustrate ammonia enhancement by transport and destruction at high altitudes by atmospheric photochemistry.^[ref] Interpreting an ammonia feature therefore requires an atmospheric pressure-temperature profile and a transport model, not only a molecular identification.