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Tyler D. Robinson

Publications and source records attributed to Tyler D. Robinson.

At least 37 records · Page 2Linked to original sources

HWO Yield Sensitivities in the NIR and NUV

Habitable Worlds Observatory (HWO) will search for biosignatures from Earth-size exoplanets in the habitable zones of nearby stars. The wavelength range for biosignatures used by the HabEx and LUVOIR mission concept studies was 200 nm to 2 microns and, as such, this is a candidate wavelength range for HWO. The visible wavelength range (500-1000 nm) provides for detection of water, oxygen, and Raleigh scattering; the near-ultraviolet is valuable for detection of ozone; and the near-infrared enables detection of carbon dioxide and methane for Earth-like atmospheres. Damiano et al. 2023 showed the significant improvement in spectral retrieval reliability when the NUV and NIR are both used with the visible. However, the challenge of the NUV, in addition to the technological and engineering challenges of starlight suppression in the NUV, is the drop in flux of host stars. In the NIR, the challenge is the geometric access to the habitable zone due to the wavelength dependency of the inner working angle limit of coronagraphs. For these reasons, exoplanet yields are lower in the NUV and NIR than in the visible (Morgan et al. 2023, Morgan et al. 2024) and some instrument parameters are more critical for improving NUV and NIR yields than others. In this paper we present a new capability for performing a large number of end-to-end yield modeling simulations to enable large, multivariate parameter sweeps. We utilize this capability to calculate the Visible, NIR, and NUV yield sensitivities to the instrument parameters: aperture diameter, coronagraph core throughput, contrast, and inner working angle (IWA). We find that parameter interactions are important in determining yield, the most important of which is the interaction between contrast and IWA, but that the strength of that interaction is different in each of the three wavebands.

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Microphysical Prescriptions for Parameterized Water Cloud Formation on Ultra-cool Substellar Objects

Water must condense into ice clouds in the coldest brown dwarfs and exoplanets. When they form, these icy clouds change the emergent spectra, temperature structure, and albedo of the substellar atmosphere. The properties of clouds are governed by complex microphysics but these complexities are often not captured by the simpler parameterized cloud models used in climate models or retrieval models. Here, we combine microphysical cloud modeling and 1D climate modeling to incorporate insights from microphysical models into a self-consistent, parameterized cloud model. Using the 1D Community Aerosol and Radiation Model for Atmospheres (CARMA), we generate microphysical water clouds and compare their properties with those from the widely-used EddySed cloud model (Ackerman & Marley 2001) for a grid of Y dwarfs. We find that the mass of water condensate in our CARMA water clouds is significantly limited by available condensation nuclei; in models without additional seed particles for clouds added, the atmosphere becomes supersaturated. We incorporate water latent heat release in the convective and radiative parts of the atmosphere and find no significant impact on water-ice cloud formation for typical gas giant compositions. Our analysis reveals the CARMA cloud profiles have a gradual decrease in opacity of approximately 4% per bar below the cloud base. Incorporating this gradual cloud base falloff and a variable $f_{sed}$ parameter allows spectra generated from the parameterized Eddysed model to better match those of the microphysical CARMA model. This work provides recommendations for efficiently generating microphysically-informed water clouds for future models of cold substellar objects with H/He atmospheres.

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Influence of Orbit and Mass Constraints on Reflected Light Characterization of Directly Imaged Rocky Exoplanets

Survey strategies for upcoming exoplanet direct imaging missions have considered varying assumptions of prior knowledge. Precursor radial velocity surveys could have detected nearby exo-Earths and provided prior orbit and mass constraints. Alternatively, a direct imaging mission performing astrometry could yield constraints on orbit and phase angle of target planets. Understanding the impact of prior mass and orbit information on planetary characterization is crucial for efficiently recognizing habitable exoplanets. To address this question, we use a reflected-light retrieval tool to infer the atmospheric and bulk properties of directly imaged Earth-analogs while considering varying levels of prior information and signal-to-noise ratio (SNR). Because of the strong correlation between the orbit-related parameters and the planetary radius, prior information on the orbital distance and planetary phase yield tight constraints on the planetary radius: from $R_{\rm{p}}=2.95^{+2.69}_{-1.95}~R_{\oplus}$ without prior knowledge, to $R_{\rm{p}}=1.01^{+0.33}_{-0.19}~R_{\oplus}$ with prior determination of the orbit for $\rm{SNR}=20$ in the visible/near-infrared spectral range, thus allowing size determination from reflected light observations. However, additional knowledge of planet mass does not notably enhance radius ($R_{\rm{p}}=0.98^{+0.17}_{-0.14}~R_{\oplus}$) or atmospheric characterization. Also, prior knowledge of the mass alone does not yield a tight radius constraint ($R_{\rm{p}}=1.64^{+1.29}_{-0.80}~R_{\oplus}$) nor improves atmospheric composition inference. By contrast, because of its sensitivity to gas column abundance, detecting a Rayleigh scattering slope or bounding Rayleigh opacity helps to refine gas mixing ratio inferences without requiring prior mass knowledge. Overall, apart from radius determination, increasing the SNR is more beneficial than additional prior observations.

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Impact of Planetary Parameters on Water Clouds Microphysics

Potentially habitable exoplanets are targets of great interest for the James Webb Space Telescope and upcoming mission concepts such as the Habitable Worlds Observatory. Clouds strongly affect climate and habitability, but predicting their properties is difficult. In Global Climate Models (GCMs), especially those aiming at simulating Earth, cloud microphysics is often crudely approximated by assuming that all cloud particles have a single, constant size or a prescribed size distribution and that all clouds in a grid cell are identical. For exoplanets that range over a large phase space of planetary properties, this method could result in large errors. In this work, our goal is to determine how cloud microphysics on terrestrial exoplanets, whose condensable is mainly water vapor, depend on aerosol properties and planetary parameters such as surface pressure, surface gravity, and incident stellar radiation. We use the Community Aerosol and Radiation Model for Atmospheres as a 1D microphysical model to simulate the formation and evolution of clouds including the processes of nucleation, condensation, evaporation, coagulation, and vertical transfer. In these 1D idealized experiments, we find that the parameters that determine the macrophysical thermal structure of the atmospheres, including surface pressure and stellar flux, impact cloud radiative effect (CRE) most significantly. Parameters such as gravity and number density of aerosols working as cloud condensation nuclei affect the microphysical processes of cloud formation, including activation and vertical transfer. They also have a significant, though weaker effect on CRE. This work motivates the development of more accurate GCM cloud schemes and should aid in the interpretation of future observations.

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Exomoons & Exorings with the Habitable Worlds Observatory I: On the Detection of Earth-Moon Analog Shadows & Eclipses

The highest priority recommendation of the Astro2020 Decadal Survey for space-based astronomy was the construction of an observatory capable of characterizing habitable worlds. In this paper series we explore the detectability of and interference from exomoons and exorings serendipitously observed with the proposed Habitable Worlds Observatory (HWO) as it seeks to characterize exoplanets, starting in this manuscript with Earth-Moon analog mutual events. Unlike transits, which only occur in systems viewed near edge-on, shadow (i.e., solar eclipse) and lunar eclipse mutual events occur in almost every star-planet-moon system. The cadence of these events can vary widely from ~yearly to multiple events per day, as was the case in our younger Earth-Moon system. Leveraging previous space-based (EPOXI) lightcurves of a Moon transit and performance predictions from the LUVOIR-B concept, we derive the detectability of Moon analogs with HWO. We determine that Earth-Moon analogs are detectable with observation of ~2-20 mutual events for systems within 10pc, and larger moons should remain detectable out to 20pc. We explore the extent to which exomoon mutual events can mimic planet features and weather. We find that HWO wavelength coverage in the near-IR, specifically in the 1.4 micron water band where large moons can outshine their host planet, will aid in differentiating exomoon signals from exoplanet variability. Finally, we predict that exomoons formed through collision processes akin to our Moon are more likely to be detected in younger systems, where shorter orbital periods and favorable geometry enhance the probability and frequency of mutual events.

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Detectability of Surface Biosignatures for Directly-Imaged Rocky Exoplanets

Modeling the detection of life has never been more opportune. With next generation space telescopes, like the currently developing Habitable Worlds Observatory (HWO) concept, we will begin to characterize rocky exoplanets potentially similar to Earth. However, currently, few realistic planetary spectra containing surface biosignatures have been paired with direct imaging telescope instrument models. Therefore, we use a HWO instrument noise model to assess the detection of surface biosignatures affiliated with oxygenic, anoxygenic, and nonphotosynthetic extremophiles. We pair the HWO telescope model to a 1-D radiative transfer model to estimate the required exposure times necessary for detecting each biosignature on planets with global microbial coverage and varying atmospheric water vapor concentrations. For modeled planets with 0% - 50% cloud coverage, we determine pigments and the red edge could be detected within 1,000 hours (100 hours) at distances within 15 pc (11 pc). However, tighter telescope inner working angles (2.5 lambda/D) would allow surface biosignature detection at further distances. Anoxygenic photosynthetic biosignatures could also be more easily detectable than nonphotosynthetic pigments and the photosynthetic red edge when compared against a false positive iron oxide slope. Future life detection missions should evaluate the influence of false positives on the detection of multiple surface biosignatures.

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Exoplanet Analog Observations of Earth from Galileo Disk-integrated Photometry

The Galileo spacecraft had distant encounters with Earth in 1990 and 1992. Limited Solid State Imager (SSI) data acquired during these encounters has been previously presented, but the majority of the data from these Earth flybys have not been presented in the literature. Observations of Earth taken from afar are both rare and directly relevant to the development of any future exo-Earth direct imaging mission. Here we present a pipeline that vets, calibrates, and measures the disk-integrated brightness of the Earth, in multiple filters, from the complete SSI data sets from both the 1990 and 1992 Galileo flybys. The result is over 1500 usable photometric measurements for Earth as an analog for an exoplanet. The 1990 data set includes full rotational lightcurves in six bandpasses spanning the optical range. The 1992 data set is more limited, with lightcurves only spanning 14 hr. Time-averaged photometry for both encounters is presented while variability and color are discussed relative to findings from NASA's EPOXI mission (which also provided photometric lighturves for Earth). The new Galileo/SSI data are used to further validate the Virtual Planetary Laboratory 3D spectral Earth model, which often serves as a stand-in for true disk-integrated observations of our planet. The revived Galileo/SSI data for Earth is a testament to the ability of NASA's Planetary Data System to maintain data over decades-long timescales. The disk-integrated products derived from these data add to a very short list of calibrated and published whole-disk observations of the Pale Blue Dot.

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Inner Edge Habitable Zone Limits Around Main Sequence Stars: Cloudy Estimates

Understanding the limits of rocky planet habitability is one of the key goals of current and future exoplanet characterization efforts. An intrinsic concept of rocky planet habitability is the Habitable Zone. To date, the most widely used estimates of the Habitable Zone are based on cloud-free, one-dimensional (vertical) radiative-convective climate model calculations. However, recent three-dimensional global climate modeling efforts have revealed that rocky planet habitability is strongly impacted by radiative cloud feedbacks, where computational expense and model limitations can prevent these tools from exploring the limits of habitability across the full range of parameter space. We leverage a patchy cloud one-dimensional radiative-convective climate model with parameterized cloud microphysics to investigate Inner Edge limits to the Habitable Zone for main sequence stars ($T_{\rm eff}$ = 2600 -7200K). We find that Inner Edge limits to the Habitable Zone can be 3.3 and 4.7 times closer than previous cloud-free estimates for Earth- and super-Earth-sized worlds, respectively, depending on bulk cloud parameters (e.g., fractional cloudiness and sedimentation efficiency). These warm, moist Inner Edge climates are expected to have extensive cloud decks that could mute deep atmosphere spectral features. To aid in rocky planet characterization studies, we identify the potential of using $\rm{CO_{\rm 2}}$ absorption features in transmission spectroscopy as a means of quantifying cloud deck height and cloud sedimentation efficiency. Moist greenhouse climates may represent key yet poorly understood states of habitable planets for which continued study will uncover new insights into the search and characterization of habitable worlds.

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Characterizing Exoplanet Habitability

A habitable exoplanet is a world that can maintain stable liquid water on its surface. Techniques and approaches to characterizing such worlds are essential, as performing a census of Earth-like planets that may or may not have life will inform our understanding of how frequently life originates and is sustained on worlds other than our own. Observational techniques like high contrast imaging and transit spectroscopy can reveal key indicators of habitability for exoplanets. Both polarization measurements and specular reflectance from oceans (also known as "glint") can provide direct evidence for surface liquid water, while constraining surface pressure and temperature (from moderate resolution spectra) can indicate liquid water stability. Observations of variability (that indicates weather) from, as well as mapping of, exoplanets can provide indirect evidence of habitability, and measurements of water vapor or cloud profiles that indicate condensation near a surface could also provide evidence for habitability. Approaches to making the types of measurements that indicate habitability are diverse, and have different considerations for the required wavelength range, spectral resolution, maximum noise levels, stellar host temperature, and observing geometry.

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Retrievals Applied To A Decision Tree Framework Can Characterize Earth-like Exoplanet Analogs

Exoplanet characterization missions planned for the future will soon enable searches for life beyond our solar system. Critical to the search will be the development of life detection strategies that can search for biosignatures while maintaining observational efficiency. In this work, we adopted a newly developed biosignature decision tree strategy for remote characterization of Earth-like exoplanets. The decision tree offers a step-by-step roadmap for detecting exoplanet biosignatures and excluding false positives based on Earth's biosphere and its evolution over time. We followed the pathways for characterizing a modern Earth-like planet and an Archean Earth-like planet and evaluated the observational trades associated with coronagraph bandpass combinations of designs consistent with The Habitable Worlds Observatory (HWO) precursor studies. With retrieval analyses of each bandpass (or combination), we demonstrate the utility of the decision tree and evaluated the uncertainty on a suite of biosignature chemical species and habitability indicators (i.e., the gas abundances of H$_2$O, O$_2$, O$_3$, CH$_4$, and CO$_2$). Notably for modern Earth, less than an order of magnitude spread in the 1-$σ$ uncertainties were achieved for the abundances of H$_2$O and O$_2$, planetary surface pressure, and atmospheric temperature with three strategically placed bandpasses (two in the visible and one in the near-infrared). For the Archean, CH$_4$ and H$_2$O were detectable in the visible with a single bandpass.

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Inferring Chemical Disequilibrium Biosignatures for Proterozoic Earth-Like Exoplanets

Chemical disequilibrium quantified via available free energy has previously been proposed as a potential biosignature. However, exoplanet biosignature remote sensing work has not yet investigated how observational uncertainties impact the ability to infer a life-generated available free energy. We pair an atmospheric retrieval tool to a thermodynamics model to assess the detectability of chemical disequilibrium signatures of Earth-like exoplanets, emphasizing the Proterozoic Eon where atmospheric abundances of oxygen-methane disequilibrium pairs may have been relatively high. Retrieval model studies applied across a range of gas abundances revealed that order-of-magnitude constraints on disequilibrium energy are achieved with simulated reflected-light observations at the high abundance scenario and signal-to-noise ratios (50) while weak constraints are found at moderate SNRs (20\,--\,30) for med\,--\,low abundance cases. Furthermore, the disequilibrium energy constraints are improved by modest thermal information encoded in water vapor opacities at optical and near-infrared wavelengths. These results highlight how remotely detecting chemical disequilibrium biosignatures can be a useful and metabolism-agnostic approach to biosignature detection.

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Earth as a Transiting Exoplanet: A Validation of Transmission Spectroscopy and Atmospheric Retrieval Methodologies for Terrestrial Exoplanets

The James Webb Space Telescope (JWST) will enable the search for and characterization of terrestrial exoplanet atmospheres in the habitable zone via transmission spectroscopy. However, relatively little work has been done to use solar system data, where ground truth is known, to validate spectroscopic retrieval codes intended for exoplanet studies, particularly in the limit of high resolution and high signal-to-noise (S/N). In this work, we perform such a validation by analyzing a high S/N empirical transmission spectrum of Earth using a new terrestrial exoplanet atmospheric retrieval model with heritage in Solar System remote sensing and gaseous exoplanet retrievals. We fit the Earth's 2-14 um transmission spectrum in low resolution (R=250 at 5 um) and high resolution (R=100,000 at 5 um) under a variety of assumptions about the 1D vertical atmospheric structure. In the limit of noiseless transmission spectra, we find excellent agreement between model and data (deviations < 10%) that enable the robust detection of H2O, CO2, O3, CH4, N2, N2O, NO2, HNO3, CFC-11, and CFC-12 thereby providing compelling support for the detection of habitability, biosignature, and technosignature gases in the atmosphere of the planet using an exoplanet-analog transmission spectrum. Our retrievals at high spectral resolution show a marked sensitivity to the thermal structure of the atmosphere, trace gas abundances, density-dependent effects, such as collision-induced absorption and refraction, and even hint at 3D spatial effects. However, we used synthetic observations of TRAPPIST-1e to verify that the use of simple 1D vertically homogeneous atmospheric models will likely suffice for JWST observations of terrestrial exoplanets transiting M dwarfs.

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Scattering Transparency of Clouds in Exoplanet Transit Spectra

The presence of aerosols in an exoplanet atmosphere can veil the underlying material and can lead to a flat transmission spectrum during primary transit observations. In this work, we explore forward scattering effects from super-micron sized aerosol particles present in the atmosphere of a transiting exoplanet. We find that the impacts of forward scattering from larger aerosols can significantly impact exoplanet transits and the strength of these effects can be dependent on wavelength. In certain cloud configurations, the forward-scattered light can effectively pass through the clouds unhindered, thus rendering the clouds transparent. The dependence of the aerosol scattering properties on wavelength can then lead to a positive slope in the transit spectrum. These slopes are characteristically different from both Rayleigh and aerosol absorption slopes. As examples, we demonstrate scattering effects for both a rocky world and a hot Jupiter. In these models, the predicted spectral slopes due to forward scattering effects can manifest in the transit spectrum at the level of $\sim$10s to $\sim$100s of parts per million and, hence, could be observable with NASA's James Webb Space Telescope.

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Exploring and Validating Exoplanet Atmospheric Retrievals with Solar System Analog Observations

Solar System observations that serve as analogs for exoplanet remote sensing data can provide important opportunities to validate ideas and models related to exoplanet environments. Critically, and unlike true exoplanet observations, Solar System analog data benefit from available high-quality ground- or orbiter-derived "truth" constraints that enable strong validations of exoplanet data interpretation tools. In this work, we first present a versatile atmospheric retrieval suite, capable of application to reflected light, thermal emission, and transmission observations spanning a broad range of wavelengths and thermochemical conditions. The tool -- dubbed rfast -- is designed, in part, to enable exoplanet mission concept feasibility studies. Following model validation, the retrieval tool is applied to a range of Solar System analog observations for exoplanet environments. Retrieval studies using Earth reflected light observations from NASA's EPOXI mission provide a key proof-of-concept for under-development exo-Earth direct imaging concept missions. Inverse modeling applied to an infrared spectrum of Earth from the Mars Global Surveyor Thermal Emission Spectrometer achieves good constraints on atmospheric gases, including many biosignature gases. Finally, retrieval analysis applied to a transit spectrum of Titan derived from the Cassini Visual and Infrared Mapping Spectrometer provides a proof-of-concept for interpreting more feature-rich transiting exoplanet observations from NASA's James Webb Space Telescope (JWST). In the future, Solar System analog observations for exoplanets could be used to verify exoplanet models and parameterizations, and future exoplanet analog observations of any Solar System worlds from planetary science missions should be encouraged.

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A Radiative-Convective Model for Terrestrial Planets with Self-Consistent Patchy Clouds

Clouds are ubiquitous\, -- \,they arise for every solar system planet that possesses an atmosphere and have also been suggested as a leading mechanism for obscuring spectral features in exoplanet observations. As exoplanet observations continue to improve, there is a need for efficient and general planetary climate models that appropriately handle the possible cloudy atmospheric environments that arise on these worlds. We generate a new 1D radiative-convective terrestrial planet climate model that self-consistently handles patchy clouds through a parameterized microphysical treatment of condensation and sedimentation processes. Our model is general enough to recreate Earth's atmospheric radiative environment without over-parameterization, while also maintaining a simple implementation that is applicable to a wide range of atmospheric compositions and physical planetary properties. We first validate this new 1D patchy cloud radiative-convective climate model by comparing it to Earth thermal structure data and to existing climate and radiative transfer tools. We produce partially-clouded Earth-like climates with cloud structures that are representative of deep tropospheric convection and are adequate 1D representations of clouds within rocky planet atmospheres. After validation against Earth, we then use our partially clouded climate model and explore the potential climates of super-Earth exoplanets with secondary nitrogen-dominated atmospheres which we assume are abiotic. We also couple the partially clouded climate model to a full-physics, line-by-line radiative transfer model and generate high-resolution spectra of simulated climates. These self-consistent climate-to-spectral models bridge the gap between climate modeling efforts and observational studies of rocky worlds.

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Detecting Oceans on Exoplanets with Phase-Dependent Spectral Principal Component Analysis

Stable surface liquid water is a key indicator of exoplanet habitability. However, few approaches exist for directly detecting oceans on potentially Earth-like exoplanets. In most cases, specular reflection of host starlight from surface bodies of water -- referred to as ocean glint -- proves to be an important aspect of liquids that can enable detection of habitable conditions. Here, we propose that spectral principal component analysis (PCA) applied to orbital phase-dependent observations of Earth-like exoplanets can provide a straightforward means of detecting ocean glint and, thus, habitability. Using high-fidelity, orbit-resolved spectral models of Earth, and for instrument capabilities applicable to proposed exo-Earth direct imaging concept missions, the extreme reddening effect of crescent-phase ocean glint is demonstrated as the primary spectral component that explains phase-dependent variability for orbital inclinations spanning 60--90 degrees. At smaller orbital inclinations where more-extreme crescent phases cannot be accessed, glint can still significantly increase planetary brightness but reddening effects are less pronounced and, as a result, glint is not plainly indicated by phase-dependent spectral PCA. Using instrument models for future exoplanet direct imaging mission concepts, we show that brightness enhancements due to glint could be detected across a wide range of orbital inclinations with typical exposure times measured in hours to weeks, depending on system distance and mission architecture. Thus, brightness increases due to glint are potentially detectable for Earth-like exoplanets for most system inclinations and phase-dependent spectral PCA could indicate reddening due to glint for a subset of these inclinations.

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Impacts of Water Latent Heat on the Thermal Structure of Ultra-Cool Objects: Brown Dwarfs and Free-Floating Planets

Brown dwarfs are essential targets for understanding planetary and sub-stellar atmospheres across a wide range of thermal and chemical conditions. As surveys continue to probe ever deeper, and as observing capabilities continue to improve, the number of known Y dwarfs -- the coldest class of sub-stellar objects, with effective temperatures below about 600 K -- is rapidly growing. Critically, this class of ultra-cool objects has atmospheric conditions that overlap with Solar System worlds and, as a result, tools and ideas developed from studying Earth, Jupiter, Saturn and other nearby worlds are well-suited for application to sub-stellar atmospheres. To that end, we developed a one-dimensional (vertical) atmospheric structure model for ultra-cool objects that includes moist adiabatic convection, as this is an important process for many Solar System planets. Application of this model across a range of effective temperatures (350, 300, 250, 200 K), metallicities ([M/H] of 0.0, 0.5, 0.7, 1.5), and gravities (log $g$ of 4.0, 4.5, 4.7, 5.0) demonstrates strong impacts of water latent heat release on simulated temperature-pressure profiles. At the highest metallicities, water vapor mixing ratios reach an Earth-like 3%, with associated major alterations to the thermal structure in the atmospheric regions where water condenses. Spectroscopic and photometric signatures of metallicity and moist convection should be readily detectable at near- and mid-infrared wavelengths, especially with James Webb Space Telescope observations, and can help indicate the formation history of an object.

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Linearized Flux Evolution (LiFE): A Technique for Rapidly Adapting Fluxes from Full-Physics Radiative Transfer Models

Solar and thermal radiation are critical aspects of planetary climate, with gradients in radiative energy fluxes driving heating and cooling. Climate models require that radiative transfer tools be versatile, computationally efficient, and accurate. Here, we describe a technique that uses an accurate full-physics radiative transfer model to generate a set of atmospheric radiative quantities which can be used to linearly adapt radiative flux profiles to changes in the atmospheric and surface state -- the Linearized Flux Evolution (LiFE) approach. These radiative quantities describe how each model layer in a plane-parallel atmosphere reflects and transmits light, as well as how the layer generates diffuse radiation by thermal emission and by scattering light from the direct solar beam. By computing derivatives of these layer radiative properties with respect to dynamic elements of the atmospheric state, we can then efficiently adapt the flux profiles computed by the full-physics model to new atmospheric states. We validate the LiFE approach, and then apply this approach to Mars, Earth, and Venus, demonstrating the information contained in the layer radiative properties and their derivatives, as well as how the LiFE approach can be used to determine the thermal structure of radiative and radiative-convective equilibrium states in one-dimensional atmospheric models.

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