Radiative Transfer for Exoplanet Atmospheres

Radiative Transfer for Exoplanet Atmospheres


Authors:
Heng et al

Abstract:

Remote sensing of the atmospheres of distant worlds motivates a firm understanding of radiative transfer. In this review, we provide a pedagogical cookbook that describes the principal ingredients needed to perform a radiative transfer calculation and predict the spectrum of an exoplanet atmosphere, including solving the radiative transfer equation, calculating opacities (and chemistry), iterating for radiative equilibrium (or not), and adapting the output of the calculations to the astronomical observations. A review of the state of the art is performed, focusing on selected milestone papers. Outstanding issues, including the need to understand aerosols or clouds and elucidating the assumptions and caveats behind inversion methods, are discussed. A checklist is provided to assist referees/reviewers in their scrutiny of works involving radiative transfer. A table summarizing the methodology employed by past studies is provided.

Reexamination of the Inflated hot Jupiters

On the Radius Anomaly of Hot Jupiters: Reexamination of the Possibility and Impact of Layered Convection

Authors:

Kurokawa et al

Abstract:

Observations have revealed that a significant number of hot Jupiters have anomalously large radii. Layered convection induced by compositional inhomogeneity has been proposed to account for the radius anomaly of hot Jupiters. To reexamine the impact of the compositional inhomogeneity, we perform an evolutionary calculation by determining convection regime at each evolutionary time step according to the criteria from linear analyses. It is shown that the impact is limited in the case of the monotonic gradient of heavy element abundance. The layered convection is absent for the first 1 Gyr from the formation of hot Jupiters and instead overturning convection develops. The super-adiabaticity of the temperature gradient is limited by the neutrally stable state for the Ledoux stability criterion. The effect of the increased mass of heavy elements essentially compensates the effect of the delayed contraction on the planetary radius caused by compositional inhomogeneity. In addition, even in the case where the layered convection is artificially imposed, this mechanism requires extremely thin layers (~ 10^1-10^3 cm) to account for the observed radius anomaly. The long-term stability of such thin layers remains to be studied. Therefore, if the criteria adopted in this paper are adequate, it might be difficult to explain the inflated radii of hot Jupiters by monotonic gradient of heavy element abundance alone.

Do Cooler Protoplanetary Disks NOT Form Gas Giants?

The Role of the Cooling Prescription for Disk Fragmentation: Numerical Convergence & Critical Cooling Parameter in Self-Gravitating Disks

Authors:

Baehr et al

Abstract:

Protoplanetary disks fragment due to gravitational instability when there is enough mass for self-gravitation, described by the Toomre parameter, and when heat can be lost at a rate comparable to the local dynamical timescale, described by t_c=beta Omega^-1. Simulations of self-gravitating disks show that the cooling parameter has a rough critical value at beta_crit=3. When below beta_crit, gas overdensities will contract under their own gravity and fragment into bound objects while otherwise maintaining a steady state of gravitoturbulence. However, previous studies of the critical cooling parameter have found dependence on simulation resolution, indicating that the simulation of self-gravitating protoplanetary disks is not so straightforward. In particular, the simplicity of the cooling timescale t_c prevents fragments from being disrupted by pressure support as temperatures rise. We alter the cooling law so that the cooling timescale is dependent on local surface density fluctuations, a means of incorporating optical depth effects into the local cooling of an object. For lower resolution simulations, this results in a lower critical cooling parameter and a disk more stable to gravitational stresses suggesting the formation of large gas giants planets in large, cool disks is generally suppressed by more realistic cooling. At our highest resolution however, the model becomes unstable to fragmentation for cooling timescales up to beta = 10.

Detecting Biospheres Through Thermodynamic Disequilibrium in Exoplanetary Atmospheres

On detecting biospheres from thermodynamic disequilibrium in planetary atmospheres

Authors:

Krissansen-Totton et al

Abstract:

Atmospheric chemical disequilibrium has been proposed as a method for detecting extraterrestrial biospheres from exoplanet observations. Chemical disequilibrium is potentially a generalized biosignature since it makes no assumptions about particular biogenic gases or metabolisms. Here, we present the first rigorous calculations of the thermodynamic chemical disequilibrium in the atmospheres of Solar System planets, in which we quantify the difference in Gibbs free energy of an observed atmosphere compared to that of all the atmospheric gases reacted to equilibrium. The purely gas phase disequilibrium in Earth's atmosphere, as measured by this available Gibbs free energy, is not unusual by Solar System standards and smaller than that of Mars. However, Earth's atmosphere is in contact with a surface ocean, which means that gases can react with water, and so a multiphase calculation that includes aqueous species is required. We find that the disequilibrium in Earth's atmosphere-ocean system (in joules per mole of atmosphere) ranges from ~20 to 2E6 times larger than the disequilibria of other atmospheres in the Solar System depending on the celestial body being compared. Disequilibrium in other Solar System atmospheres is driven by abiotic processes, and we identify the key disequilibria in each atmosphere. Earth's thermodynamic disequilibrium is biogenic in origin, and the main contribution is the coexistence of N2, O2 and liquid water instead of more stable nitrate. In comparison, the disequilibrium between O2 and methane constitutes a negligible contribution to Earth's disequilibrium with this metric. Our metric requires minimal assumptions and could potentially be calculated using observations of exoplanet atmospheres. Our Matlab source code and associated databases for these calculations are available as open source software.

Magnetic Turbulence and Thermodynamics in the Inner Region of Protoplanetary Disks

Magnetic Turbulence and Thermodynamics in the Inner Region of Protoplanetary Discs

Author:

Hirose

Abstract:

Using radiation magnetohydrodynamics simulations with realistic opacities and equation of state, and zero net magnetic flux, we have explored thermodynamics in the inner part of protoplanetary discs where magnetic turbulence is expected. The thermal equilibrium curve consists of the upper, lower, and middle branches. The upper (lower) branch corresponds to hot (cool) and optically very (moderately) thick discs, respectively, while the middle branch is characterized by convective energy transport near the midplane. Convection is also the major energy transport process near the low surface density end of the upper branch. There, convective motion is fast with Mach numbers reaching ≳0.01, and enhances both magnetic turbulence and cooling, raising the ratio of vertically-integrated shear stress to vertically-integrated pressure by a factor of several. This convectively enhanced ratio seems a robust feature in accretion discs having an ionization transition. We have also examined causes of the S-shaped thermal equilibrium curve, as well as the thermal stability of the equilibrium solutions. Finally, we compared our results with the disc instability models used to explain FU Ori outbursts. Although the thermal equilibrium curve in our results also exhibits bistability, the surface density contrast across the bistability is an order of magnitude smaller, and the stress-to-pressure ratios in both upper and lower branches are two orders of magnitude greater, than those favored in the disc instability models. It therefore appears likely that FU Ori outbursts are not due solely to a thermal-viscous limit cycle resulting from accretion driven by local magnetic turbulence.

Modeling Convection in Exoatmospheres

THEORY AND SIMULATIONS OF ROTATING CONVECTION

Authors:

Barker et al

Abstract:

We study thermal convection in a rotating fluid in order to better understand the properties of convection zones in rotating stars and planets. We first derive a mixing-length theory for rapidly rotating convection, arriving at the results of Stevenson via simple physical arguments. The theory predicts the properties of convection as a function of the imposed heat flux and rotation rate, independent of microscopic diffusivities. In particular, it predicts the mean temperature gradient, the rms velocity and temperature fluctuations, and the size of the eddies that dominate heat transport. We test all of these predictions with high resolution three-dimensional hydrodynamical simulations of Boussinesq convection in a Cartesian box. The results agree remarkably well with the theory across more than two orders of magnitude in rotation rate. For example, the temperature gradient is predicted to scale as the rotation rate to the four-fifths power at fixed flux, and the simulations yield 0.75 ± 0.06. We conclude that the mixing-length theory is a solid foundation for understanding the properties of convection zones in rotating stars and planets.

Exoplanet Rotation has Big Impacts on Oceanic Heat Transport & Habitability

The Importance of Planetary Rotation Period for Ocean Heat Transport

Authors:

Cullum et al

Abstract:

The climate and, hence, potential habitability of a planet crucially depends on how its atmospheric and ocean circulation transports heat from warmer to cooler regions. However, previous studies of planetary climate have concentrated on modeling the dynamics of atmospheres, while dramatically simplifying the treatment of oceans, which neglects or misrepresents the effect of the ocean in the total heat transport. Even the majority of studies with a dynamic ocean have used a simple so-called aquaplanet that has no continental barriers, which is a configuration that dramatically changes the ocean dynamics. Here, the significance of the response of poleward ocean heat transport to planetary rotation period is shown with a simple meridional barrier—the simplest representation of any continental configuration. The poleward ocean heat transport increases significantly as the planetary rotation period is increased. The peak heat transport more than doubles when the rotation period is increased by a factor of ten. There are also significant changes to ocean temperature at depth, with implications for the carbon cycle. There is strong agreement between the model results and a scale analysis of the governing equations. This result highlights the importance of both planetary rotation period and the ocean circulation when considering planetary habitability.

How Hot Jupiter WASP-3b Cooks

The thermal emission of the exoplanet WASP-3b

Authors:

Rostron et al

Abstract:

We report the detection of thermal emission from the transiting hot Jupiter WASP-3b at 3.6, 4.5 and 8.0 μm using the Spitzer Space Telescope. We obtain planet-to-star flux ratios of 0.209+0.040−0.028, 0.282 ± 0.012 and 0.328+0.086−0.055 per cent at these wavelengths, respectively, implying infrared brightness temperatures of T3.6μm=2280+210−150K, T4.5 μm = 2400 ± 80 K and T8.0μm=2210+390−250K. We find that WASP-3b falls into an emerging class of highly irradiated planets whose measured temperatures suggest that the planets are dark and redistribute heat around the planet inefficiently. The latter is similarly concluded from 1D atmospheric model comparisons, which also favour the presence of an atmospheric temperature inversion. We compare the WASP-3 system to the proposed inversion–activity relation, finding that it hints at a more complex relation than a simple cut-off in activity implied by previous data. Using eclipse timings we also constrain e cos ω to be −0.0006+0.0010−0.0006, suggesting that the eccentricity of WASP-3b can only be large for a narrow range of ω.

Modeling Exoplanetary Atmospheres' Radiative Transfer

Analytical Models of Exoplanetary Atmospheres. II. Radiative Transfer via the Two-Stream Approximation

Authors:

Heng et al

Abstract:

We present a comprehensive analytical study of radiative transfer using the method of moments and include the effects of non-isotropic scattering in the coherent limit. Within this unified formalism, we derive the governing equations and solutions describing two-stream radiative transfer (which approximates the passage of radiation as a pair of outgoing and incoming fluxes), flux-limited diffusion (which describes radiative transfer in the deep interior) and solutions for the temperature-pressure profiles. Generally, the problem is mathematically under-determined unless a set of closures (Eddington coefficients) is specified. We demonstrate that the hemispheric (or hemi-isotropic) closure naturally derives from the radiative transfer equation if energy conservation is obeyed, while the Eddington closure produces spurious enhancements of both reflected light and thermal emission. We further demonstrate that traditional non-isothermal treatments of each atmospheric layer lead to unphysical contributions to the fluxes. We concoct recipes for implementing two-stream radiative transfer in stand-alone calculations and general circulation models. We use our two-stream solutions to construct toy models of the runaway greenhouse effect. We present a new solution for temperature-pressure profiles with a non-constant optical opacity and elucidate the effects of non-isotropic scattering in the optical and infrared. We derive generalized expressions for the spherical and Bond albedos and the photon deposition depth. We demonstrate that the value of the optical depth corresponding to the photosphere is not always 2/3 (Milne's solution) and depends on a combination of stellar irradiation, internal heat and the properties of scattering both in optical and infrared. Finally, we derive generalized expressions for the total, net, outgoing and incoming fluxes in the convective regime.