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# Example temperature profiles for a strongly irradiated (F☉/Fi = 104) gas giant for two different values of k, and taking n = 2 and τ0 = 1. Pressure has been normalized to p0, and temperature, shown as σT4, has been normalized to the net absorbed stellar flux. Thick portions of the curves indicate where the T–p profile is unstable to convection for a dry adiabat with γ = 1.4 (suitable for a world with an atmosphere dominated by H2).

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We present an analytic one-dimensional radiative–convective model of the thermal structure of planetary atmospheres. Our model assumes that thermal radiative transfer is gray and can be represented by the two-stream approximation. Model atmospheres are assumed to be in hydrostatic equilibrium, with a power-law scaling between the atmospheric pressu...

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... These include compositional changes such as those due to condensation (e.g. Robinson & Catling 2012), or the CO ⇔ CH 4 changes at the L-to T-type spectral transition (Tremblin et al. 2015(Tremblin et al. , 2019. The upper atmosphere can be heated by a cloud deck, or by breaking gravity waves (e.g. ...
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Cold, low-mass, field brown dwarfs are important for constraining the terminus of the stellar mass function, and also for optimizing atmospheric studies of exoplanets. In 2020 new model grids for such objects were made available: Sonora-Bobcat and ATMO 2020. Also, new candidate cold brown dwarfs were announced, and new spectroscopic observations at lambda ~4.8 um were published. In this paper we present new infrared photometry for some of the coldest brown dwarfs, and put the new data and models together to explore the properties of these objects. We reconfirm the importance of mixing in these atmospheres, which leads to CO and NH_3 abundances that differ by orders of magnitude from chemical equilibrium values. We also demonstrate that the new models retain the known factor >~3 discrepancy with observations at 2 <~ lambda um <~ 4, for brown dwarfs cooler than 600 K. We show that the entire 1 <~ lambda um <~ 20 energy distribution of six brown dwarfs with 260 <= T_eff K <= 475 can be well reproduced, for the first time, by model atmospheres which include dis-equilibrium chemistry as well as a photospheric temperature gradient which deviates from the standard radiative/convective equilibrium value. This change to the pressure-temperature profile is not unexpected for rotating and turbulent atmospheres which are subject to diabatic processes. A limited grid of modified-adiabat model colors is generated, and used to estimate temperatures and metallicities for the currently known Y dwarfs. A compilation of the photometric data used here is given in the Appendix.
... whereĖ total is the total flux of energy (see "Thermal Budget" subsection), σ sb is the Stefan-Boltzmann constant, ǫ r = 0.9 is the infrared emissivity factor (Henning et al., 2009), and R is the radius of the moon. The lower zone of the atmosphere is in the convective regime if (Sagan, 1969;Weaver and Ramanathan, 1995;Robinson and Catling, 2012) ...
... In principle, the opacity of the atmosphere should be calculated dividing the radiation spectrum into several energy bins to cover the key spectral features of the chemical species involved. However, for our aims, this can be reasonably approximated by using an averaged mean opacity over the whole spectrum (Hansen, 2008;Guillot, 2010;Robinson and Catling, 2012;Parmentier and Guillot, 2014). We decided to employ the Rosseland mean opacity because of the absence of stellar radiation, i.e. ...
... Since the amount of ammonia depends on the moon's formation history (e.g. Mandt et al. 2014), in our case we assume that CO 2 alone controls the thermal evolution of a FFP atmosphere as it happens in our Solar System, and we therefore employ it in our model, assuming γ = 1.3 (Robinson and Catling, 2012). ...
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A free-floating planet is a planetary-mass object that orbits around a non-stellar massive object (e.g. a brown dwarf) or around the Galactic Center. The presence of exomoons orbiting free-floating planets has been theoretically predicted by several models. Under specific conditions, these moons are able to retain an atmosphere capable of ensuring the long-term thermal stability of liquid water on their surface. We model this environment with a one-dimensional radiative-convective code coupled to a gas-phase chemical network including cosmic rays and ion-neutral reactions. We find that, under specific conditions and assuming stable orbital parameters over time, liquid water can be formed on the surface of the exomoon. The final amount of water for an Earth-mass exomonoon is smaller than the amount of water in Earth oceans, but enough to host the potential development of primordial life. The chemical equilibrium time-scale is controlled by cosmic rays, the main ionization driver in our model of the exomoon atmosphere.
... In principle, the opacity of the atmosphere should be calculated dividing the radiation spectrum into several energy bins to cover the key spectral features of the chemical species involved. However, for our aims, this can be reasonably approximated by using an averaged mean opacity over the whole spectrum (Hansen 2008;Guillotm 2010;Robinson and Catling 2012;Parmentier and Guillot 2014). We decided to employ the Rosseland mean opacity because of the absence of stellar radiation, i.e. ...
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A free-floating planet (FFP) is a planetary-mass object that orbits around a non-stellar massive object (e.g. a brown dwarf) or around the Galactic Centre. The presence of exomoons orbiting FFPs has been theoretically predicted by several models. Under specific conditions, these moons are able to retain an atmosphere capable of ensuring the long-term thermal stability of liquid water on their surface. We model this environment with a one-dimensional radiative-convective code coupled to a gas-phase chemical network including cosmic rays and ion-neutral reactions. We find that, under specific conditions and assuming stable orbital parameters over time, liquid water can be formed on the surface of the exomoon. The final amount of water for an Earth-mass exomoon is smaller than the amount of water in Earth oceans, but enough to host the potential development of primordial life. The chemical equilibrium time-scale is controlled by cosmic rays, the main ionization driver in our model of the exomoon atmosphere.
... similar to equation 1.12 but for a single atmospheric layer). The observed flux is then Guillot, 2010;Heng et al., 2012;Robinson and Catling, 2012;Parmentier and Guillot, 2014). ...
... A number of temperature profile parameterisations have been used for atmospheric retrievals in the literature (e.g. Madhusudhan and Seager, 2009;Guillot, 2010;Heng et al., 2012;Robinson and Catling, 2012;Parmentier and Guillot, 2014;Waldmann et al., 2015). ...
Thesis
Observations of exoplanet atmospheres have flourished in recent years, revealing a remarkable diversity of thermal, chemical and dynamical conditions. In particular, thermal emission observations of such atmospheres provide unique insights into their temperature profiles, chemistry and energy transport mechanisms. In this thesis, I explore the radiative and thermal conditions in exoplanets across a wide range of masses and irradiation conditions, from isolated brown dwarfs to rocky exoplanets. I begin by investigating important considerations for the atmospheric retrieval of isolated brown dwarfs. These objects provide remarkable laboratories for understanding atmospheric physics in the low-irradiation regime, and can be observed more precisely than exoplanets. As such, they provide a glimpse into the future of high-SNR observations of exoplanets. I introduce novel retrieval methods for isolated brown dwarfs, including a new temperature profile parameterisation and a method for including model uncertainty. I demonstrate this retrieval framework on both simulated and real data, showing that excellent precisions can be achieved on the inferred chemical abundances. I further investigate the temperature profiles and thermal emission spectra of hot Jupiters, including thermal inversions in their dayside atmospheres. In particular, TiO has long been proposed to cause thermal inversions in hot Jupiters and its spectral features in the optical and near-infrared have been detected. I investigate how TiO detections can depend on the molecular line list used to interpret the data, and how this sensitivity varies across different types of atmospheric observations. I also explore the occurrence of thermal inversions due to TiO and assess the performance of photometric metrics used to quantify them over a range in chemical composition, irradiation, gravity and stellar type. Recent optical observations of hot Jupiters in secondary eclipse are providing new constraints on their albedos and cloud/haze scattering. I investigate this for three hot Jupiters spanning a range of temperatures and gravities: KELT-1 b, WASP-18 b and WASP- 43 b. Using self-consistent atmospheric models, I interpret the optical to infrared spectra of these three targets and find that they can be explained by pure thermal emission, without the need for optical scattering by clouds or hazes. Furthermore, I find that inefficient day-night energy redistribution is needed to explain the high infrared fluxes from the daysides of these planets, consistent with previous works. I then explore the atmospheric conditions in mini-Neptunes, whose observations are beginning to provide constraints on their chemical and thermal properties, while also providing clues about their interiors and potential surfaces. With their relatively large scale heights and large planet-star contrasts, mini-Neptunes are currently ideal targets towards the goal of characterising temperate low-mass exoplanets. I explore the effects of irradiation, internal flux, metallicity, clouds and hazes on the atmospheric temperature profiles and thermal emission spectra of temperate mini-Neptunes. Building on recent suggestions of habitability of the mini-Neptune K2-18 b, I find a range of physically-motivated atmospheric conditions that allow for liquid water under the H2-rich atmospheres of such planets. I find that observations of thermal emission with the James Webb Space Telescope (JWST) can place useful constraints on the habitability of temperate mini-Neptunes such as K2-18 b. These results underpin the potential of temperate mini-Neptunes such as K2-18 b as promising candidates in the search for habitable exoplanets. Finally, I explore the characterisation of rocky exoplanet atmospheres across a wide temperature range. JWST will allow unprecedented characterisation of the atmospheres of small, rocky exoplanets. In particular, emission spectroscopy is an ideal technique to observe the secondary atmospheres of rocky exoplanets as such observations are not limited by the small scale height of high-mean-molecular-weight atmospheres. I develop a new retrieval framework tailored for rocky exoplanet atmospheres with unknown atmospheric constituents, and use this to assess the observability of promising, known rocky exoplanet targets. I find that the atmospheres of several known rocky exoplanets across a range of temperatures can be characterised using JWST.
... The positive correlation between solar absorption and entropy production rates is also corroborated in the simple analytic radiative-convective model (ARCM) of Robinson and Catling (2012) and Tolento and Robinson (2019) (Fig. 4, red lines). The model approximates the atmosphere as a gray gas in the longwave with two shortwave channels for stratospheric and tropospheric/surface absorption. ...
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A recent paper by Kato and Rose reports a negative correlation between the annual mean entropy production rate of the climate and the absorption of solar radiation in the CERES SYN1deg dataset, using the simplifying assumption that the system is steady in time. It is shown here, however, that when the nonsteady interannual storage of entropy is accounted for, the dataset instead implies a positive correlation; that is, global entropy production rates increase with solar absorption. Furthermore, this increase is consistent with the response demonstrated by an energy balance model and a radiative–convective model. To motivate this updated analysis, a detailed discussion of the conceptual relationship between entropy production, entropy storage, and entropy flows is provided. The storage-corrected estimate for the mean global rate of entropy production in the CERES dataset from all irreversible transfer processes is 81.9 mW m ⁻² K ⁻¹ and from only nonradiative processes is 55.2 mW m ⁻² K ⁻¹ (observations from March 2000 to February 2018).
... where for a Venus-like CO 2 -dominated atmosphere, the ratio of specific heats γ = 1.3, ζ = 0.78 (Robinson and Catling, 2012), the pressure at the radiative-convective boundary P rc ∼ 0.1bar (Robinson and Catling, 2014), and the equilibrium heat flux is 230(0.72/a) 2 W m − 2 . A Venus twin orbiting interior to 0.23 au, or out to about 0.3 au if the atmosphere was thicker or the planet more massive, or had a lower albedo, would have T s > 1300K. ...
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The magma ocean concept was first conceived to explain the geology of the Moon, but hemispherical or global oceans of silicate melt could be a widespread lava world” phase of rocky planet accretion, and could persist on planets on short-period orbits around other stars. The formation and crystallization of magma oceans could be a defining stage in the assembly of a core, origin of a crust, initiation of tectonics, and formation of an atmosphere. The last decade has seen significant advances in our understanding of this phenomenon through analysis of terrestrial and extraterrestrial samples, planetary missions, and astronomical observations of exoplanets. This review describes the energetic basis of magma oceans and lava worlds and the lava lake analogs available for study on Earth and Io. It provides an overview of evidence for magma oceans throughout the Solar System and considers the factors that control the rocks these magma oceans leave behind. It describes research on theoretical and observed exoplanets that could host extant magma oceans and summarizes efforts to detect and characterize them. It reviews modeling of the evolution of magma oceans as a result of crystallization and evaporation, the interaction with the underlying solid mantle, and the effects of planetary rotation. The review also considers theoretical investigations on the formation of an atmosphere in concert with the magma ocean and in response to irradiation from the host star, and possible end-states. Finally, it describes needs and gaps in our knowledge and points to future opportunities with new planetary missions and space telescopes to identify and better characterize lava worlds around nearby stars.
... where D = 1.66 is the diffusivity factor and F + and F − are the upwelling and downwelling longwave radiative fluxes. Integrating these two equations and plugging in the solar flux and the convective temperature profile provides, upon further manipulation, expressions for the upwelling thermal flux and the temperature in both the convective and non-convective region, in terms of incomplete gamma functions that can be handled numerically (for derivation, see Robinson and Catling (2012); Tolento and Robinson (2019)). The two constraints -that the upwelling radiative flux and temperature be continuous across the radiativeconvective boundary -then allows the model to be solved for two free parameters (for example, τ rc and T 0 ) by standard root-finding methods. ...
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There is ongoing interest in the global entropy production rate as a climate diagnostic and predictor, but progress has been limited by ambiguities in its definition; different conceptual boundaries of the climate system give rise to different internal production rates. Three viable options are described, estimated and investigated here, two of which -- the material and the total radiative (here 'planetary') entropy production rates -- are well-established and a third which has only recently been considered but appears very promising. This new option is labelled the 'transfer' entropy production rate and includes all irreversible processes that transfer heat within the climate, radiative and material, but not those involved in the exchange of radiation with space. Estimates in three model climates put the material rate in the range $27$-$48$ mW/m$^2$K, the transfer rate $67$-$76$ mW/m$^2$K, and the planetary rate $1279$-$1312$ mW/m$^2$K. The climate-relevance of each rate is probed by calculating their responses to climate changes in a simple radiative-convective model. An increased greenhouse effect causes a significant increase in the material and transfer entropy production rates but has no direct impact on the planetary rate. When the same surface temperature increase is forced by changing the albedo instead, the material and transfer entropy production rates increase less dramatically and the planetary rate also registers an increase. This is pertinent to solar radiation management as it demonstrates the difficulty of reversing greenhouse gas-mediated climate changes by albedo alterations. It is argued that the transfer perspective has particular significance in the climate system and warrants increased prominence.
... We use a semianalytic vertical column Energy Balance Model with a gray atmosphere (hereafter the "RC model") where the net upward and downward propagating energy fluxes balance throughout the atmosphere (Robinson & Catling, 2012). The upper atmosphere is assumed to be in radiative-only equilibrium, while convection is assumed to ensue in the lower atmosphere when the radiative lapse rate exceeds the moist adiabatic lapse rate (Manabe & Strickler, 1964). ...
... The highly idealized nature of underlying model naturally makes our results subject to several approximations and limitations. The approximations inherent to the RC model are discussed in Ramanathan and Coakley (1978) and Robinson and Catling (2012). Hence, we focus the discussion here to the two additional approximations made to derive the surface-anomaly relationships: ...
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Changes in the atmospheric composition alter the magnitude and partitioning between the downward propagating solar and atmospheric longwave radiative fluxes heating the Earth's surface. These changes are computed by radiative transfer codes in Global Climate Models and measured with high precision at surface observation networks. Changes in radiative heating signify changes in the global surface temperature and hydrologic cycle. Here, we develop a conceptual framework using an Energy Balance Model to show that first‐order changes in the hydrologic cycle are mainly associated with changes in solar radiation, while those in surface temperature are mainly associated with changes in atmospheric longwave radiation. These insights are used to explain a range of phenomena including observed historical trends, biases in climate model output, and the intermodel spread in climate change projections. These results may help identify biases in future generations of climate models.
... The thermal structure of even a strongly irradiated gas planet with an internal flux does eventually transition to a convectiondominated region in the deep atmosphere. There, the temperature profiles are expected to closely follow the convective adiabat, as discussed in many previous papers, e.g., Hubbard & Smoluchowski (1973), Marley & Robinson (2015), and Robinson & Catling (2012). According to the radiativeconvective equilibrium temperature profile calculated in Miller-Ricci & Fortney (2010), for an atmosphere of solar composition on GJ 1214b, the radiative-convective boundary Figure 13. ...
... As another example of inaccuracies of the gray approximation, the gray model in Fig. 1b, which was tuned to exhibit the same 170 W m 22 columnintegrated cooling as RFM, also yields an OLR of 170 W m 22 , which is a serious underestimate 17 of RFM's OLR value of 325 W m 22 . Despite such errors, however, gray models are still in use in both astronomy (e.g., Parmentier and Guillot 2014;Rauscher and Menou 2012;Robinson and Catling 2012;Heng et al. 2011) as well as terrestrial atmospheric sciences, both for understanding (Hu and Vallis 2019;Goessling and Bathiany 2016;Vallis et al. 2015) and also as radiation schemes for idealized aquaplanet models (e.g., Frierson et al. 2006;see Maher et al. 2019) and Jeevanjee et al. (2017) for extensive further references]. The SSMs could prove useful as alternative, cheap, clear-sky radiation schemes that still only depend on a few parameters (cf. ...
Article
Atmospheric radiative cooling is a fundamental aspect of the Earth’s greenhouse effect, and is intrinsically connected to atmospheric motions. At the same time, basic aspects of longwave radiative cooling, such as its characteristic value of 2 K/day, its sharp decline (or ‘kink’) in the upper troposphere, and the large values of CO 2 cooling in the stratosphere, are difficult to understand intuitively or estimate with pencil-and-paper. Here we pursue such understanding by building simple spectral (rather than gray) models for clear-sky radiative cooling. We construct these models by combining the cooling-to-space approximation with simplified greenhouse gas spectroscopy and analytical expressions for optical depth, and we validate these simple models with line-by-line calculations. We find that cooling rates can be expressed as a product of the Planck function, a vertical emissivity gradient, and a characteristic spectral width derived from our simplified spectroscopy. This expression allows for a pencil-and-paper estimate of the 2 K/day tropospheric cooling rate, as well as an explanation of enhanced CO 2 cooling rates in the stratosphere. We also link the upper tropospheric kink in radiative cooling to the distribution of H 2 O absorption coefficients, and from this derive an analytical expression for the kink temperature T kink ≈ 220 K. A further, ancillary result is that gray models fail to reproduce basic features of atmospheric radiative cooling.