arXiv subjects

Daphne M. Stam

Publications and source records attributed to Daphne M. Stam.

At least 19 recordsLinked to original sources

arXiv2026

Planet-wide, Concentric Density Waves in Venus's Upper Atmosphere Revealed through Polarimetry?

We report observations of faint ($10^{-6}$), concentric, planet-wide rings in the polarized flux of sunlight that is reflected by Venus, obtained during a serendipitous, 36-minute run in 2010, with the highly sensitive Extreme Polarimeter (ExPo) on the William Herschel Telescope. The rings appear to be centered slightly downwind of the sub-solar point, are visible in different filters across the visible, and are not obvious in the simultaneous total flux observations. ExPo's dual-beam exchange and double-differencing design strongly suppresses first-order instrumental polarization, and we could not identify an instrumental cause of the observed pattern. Because ExPo was dismantled before the rings were identified in the data, this is the only set of observations of these rings. We are therefore careful in claiming the detection of a new atmospheric phenomenon on Venus. However, numerical radiative transfer simulations show that planet-wide rings in polarization can arise due to density variations of 5 to 10% in the gas above the clouds, consistent with a gravity wave. Our simulations also show that such density variations would not show up in total flux observations. By presenting our observations and numerical simulations, we hope to motivate new polarimetric observations of Venus that could confirm or refute the presence of such planet-wide waves.

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arXiv2026

The power of polarimetry for characterising exoplanet atmospheres, clouds, and surfaces with NASA's Habitable Worlds Observatory

The Habitable Worlds Observatory (HWO), planned for launch in the 2040s, represents the next major step in exoplanet characterisation. HWO will, for the first time, enable detailed studies of the atmospheres and surfaces of Earth-like exoplanets through high-contrast reflection spectroscopy across the UV, optical, and near-infrared. These wavelength ranges provide access to key molecular absorption features, including O2, O3, H2O, CO2, and CH4, as well as potential surface biosignatures such as the vegetation red edge or ocean glint, making HWO a cornerstone mission for assessing planetary habitability. Clouds are a dominant factor in determining planetary climate and observability, yet their properties remain highly degenerate when constrained using reflected flux alone. Spectropolarimetry, a measure of the polarisation state of reflected light as a function of wavelength and orbital phase, provides a powerful complementary diagnostic. Polarisation is highly sensitive to cloud particle size, composition, shape, vertical distribution, and surface type, enabling degeneracies between atmospheric and surface models to be broken. Numerous studies have demonstrated the value of polarimetry for characterising a wide range of exoplanets, from hot Jupiters to cooler potentially habitable worlds. HWO's proposed instrument suite includes a coronagraph, a high-resolution imager, and a candidate high-resolution spectropolarimeter, offering multiple pathways to exploit polarimetry across diverse planetary regimes. This white paper argues that incorporating polarimetric capability into HWO instruments would significantly enhance the mission's scientific return. We highlight the unique opportunity for UK leadership in both instrument development and theoretical modelling, and advocate for a strong UK role in shaping HWO's polarimetric capabilities to maximise its impact on exoplanet science.

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arXiv2024

A general polarimetric model for transiting and non-transiting ringed exoplanets

We explore the potential of polarimetry as a tool for detecting and characterizing exorings. For that purpose, we have improved the publicly available photometric code Pryngles by adding the results of radiative transfer calculations that fully include polarization and scattering by irregularly shaped particles. With this improved code, we compute the total and polarized fluxes and the degree of polarization of a ringed gas giant along its orbit. We vary key model parameters such as the orbit inclination, ring size and orientation, particle albedo and optical thickness, and demonstrate the versatility of our code by predicting the total and polarized fluxes of the "puffed-up" planet HIP41378f assuming this planet has an opaque dusty ring. We find that spatially unresolved dusty rings can significantly modify the flux and polarization signals of the light that is reflected. Rings are expected to have a low polarization signal and will generally decrease the degree of polarization as the ring casts a shadow on the planet and/or blocks part of the light the planet reflects. During ring-plane crossings, when the thin ring is illuminated edge-on, a ringed exoplanet's flux and degree of polarization are close to those of a ring-less planet and generally appear as sharp changes in the flux and polarization curves. Ringed planets in edge-on orbits tend to be difficult to distinguish from ring-less planets in reflected flux and degree of polarization. We show that if HIP41378f is surrounded by a ring, its reflected flux (compared to the star) will be of the order of $10^{-9}$, and the ring would decrease the degree of polarization in a detectable way. The improved version of the photometric code Pryngles that we present here shows that dusty rings may produce distinct polarimetric features in light curves across a wide range of orbital configurations, orientations and ring optical properties.

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arXiv2023

Modelling reflected polarised light from close-in giant exoplanet WASP-96b using PolHEx (Polarisation of Hot Exoplanets)

We present the Polarisation of Hot Exoplanets (PolHEx) code for modelling the total flux (F) and degree of linear polarisation (P) of light spectra reflected by close-in, tidally locked exoplanets. We use the output from a global climate model (GCM) combined with a kinetic cloud model of hot Jupiter WASP-96b as a base to investigate effects of atmospheric longitudinal-latitudinal inhomogeneities on these spectra. We model F and P-spectra as functions of wavelength and planet orbital phase for various model atmospheres. We find different materials and sizes of cloud particles to impact the reflected flux F, and particularly the linear polarisation state P. A range of materials are used to form inhomogeneous mixed-material cloud particles (Al2O3, Fe2O3, Fe2SiO4, FeO, Fe, Mg2SiO4, MgO, MgSiO3, SiO2, SiO, TiO2), with Fe2O3, Fe, and FeO the most strongly absorbing species. The cloud particles near the relatively cool morning terminator are expected to have smaller average sizes and a narrower size distribution than those near the warmer evening terminator, which leads to different reflected spectra at the respective orbital phases .We also find differences in the spectra of F and P as functions of orbital phase for irregularly or spherically shaped cloud particles. This work highlights the importance of including polarisation in models and future observations of the reflection spectra of exoplanets.

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arXiv2023

Chasing rainbows and ocean glints: Inner working angle constraints for the Habitable Worlds Observatory

NASA is engaged in planning for a Habitable Worlds Observatory (HabWorlds), a coronagraphic space mission to detect rocky planets in habitable zones and establish their habitability. Surface liquid water is central to the definition of planetary habitability. Photometric and polarimetric phase curves of starlight reflected by an exoplanet can reveal ocean glint, rainbows and other phenomena caused by scattering by clouds or atmospheric gas. Direct imaging missions are optimised for planets near quadrature, but HabWorlds' coronagraph may obscure the phase angles where such optical features are strongest. The range of accessible phase angles for a given exoplanet will depend on the planet's orbital inclination and/or the coronagraph's inner working angle (IWA). We use a recently-created catalog relevant to HabWorlds of 164 stars to estimate the number of exo-Earths that could be searched for ocean glint, rainbows, and polarization effects due to Rayleigh scattering. We find that the polarimetric Rayleigh scattering peak is accessible in most of the exo-Earth planetary systems. The rainbow due to water clouds at phase angles of ${\sim}20-60^\circ$ would be accessible with HabWorlds for a planet with an Earth equivalent instellation in ${\sim}{46}$ systems, while the ocean glint signature at phase angles of ${\sim}130-170^\circ$ would be accessible in ${\sim}{16}$ systems, assuming an IWA${=}62$ mas ($3λ/D$). Improving the IWA${=}41$ mas ($2λ/D$) increases accessibility to rainbows and glints by factors of approximately 2 and 3, respectively. By observing these scattering features, HabWorlds could detect a surface ocean and water cycle, key indicators of habitability.

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arXiv2023

From exo-Earths to exo-Venuses -- Flux and Polarization Signatures of Reflected Light

Terrestrial exoplanets in habitable zones are ubiquitous. It is, however, unknown which have Earth-like or Venus-like climates. Distinguishing different planet-types is crucial for determining whether a planet could be habitable. We investigate the potential of polarimetry for distinguishing exo-Earths from exo-Venuses. We present computed fluxes and polarisation of starlight that is reflected by exoplanets with atmospheres in evolutionary states from current Earth to current Venus, with cloud compositions ranging from pure water to 0.75 sulfuric acid solution, for wavelengths between 0.3 and 2.5 microns. The polarisation of the reflected light shows larger variations with the planetary phase angle than the total flux. Across the visible, the largest polarisation is reached for an Earth-like atmosphere with water clouds, due to Rayleigh scattering above the clouds and the rainbow near 40 deg phase angle. In the near-infrared, the planet with a Venus-like CO2 atmosphere and thin water clouds shows the most prominent polarisation features due to scattering by the small cloud droplets. A planet around Alpha Centauri A would leave temporal variations on the order of 10E-13 W/m3 in the reflected flux and 10E-11 in the degree of polarisation along the planet's orbit for a spatially unresolved star-planet system. Star-planet contrasts are on the order of 10E-10. Current polarimeters cannot distinguish between the possible evolutionary phases of spatially unresolved terrestrial exoplanets, as a sensitivity near 10E-10 is required to discern the planet signal on the background of unpolarised starlight. Telescopes capable of reaching planet-star contrasts lower than 10E-9 should be able to observe the variation of the planet's resolved degree of polarisation as a function of its phase angle and thus to discern an exo-Earth from an exo-Venus based on its clouds' unique polarisation signatures.

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arXiv2023

Polarized Signatures of a Habitable World: Comparing Models of an Exoplanet Earth with Visible and Near-infrared Earthshine Spectra

In the JWST, Extremely Large Telescopes, and LUVOIR era, we expect to characterize a number of potentially habitable Earth-like exoplanets. However, the characterization of these worlds depends crucially on the accuracy of theoretical models. Validating these models against observations of planets with known properties will be key for the future characterization of terrestrial exoplanets. Due to its sensitivity to the micro- and macro-physical properties of an atmosphere, polarimetry will be an important tool that, in tandem with traditional flux-only observations, will enhance the capabilities of characterizing Earth-like planets. In this paper we benchmark two different polarization-enabled radiative-transfer codes against each other and against unique linear spectropolarimetric observations of the earthshine that cover wavelengths from $\sim$0.4 to $\sim$2.3 $μ$m. We find that while the results from the two codes generally agree with each other, there is a phase dependency between the compared models. Additionally, with our current assumptions, the models from both codes underestimate the level of polarization of the earthshine. We also report an interesting discrepancy between our models and the observed 1.27 $μ$m $O_2$ feature in the earthshine, and provide an analysis of potential methods for matching this feature. Our results suggest that only having access to the 1.27 $μ$m $O_2$ feature coupled with a lack of observations of the $O_2$ A and B bands could result in a mischaracterization of an Earth-like atmosphere. Providing these assessments is vital to aid the community in the search for life beyond the solar system.

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arXiv2022

Spectropolarimetry of life: airborne measurements from a hot air balloon

Does life exist outside our Solar System? A first step towards searching for life outside our Solar System is detecting life on Earth by using remote sensing applications. One powerful and unambiguous biosignature is the circular polarization resulting from the homochirality of biotic molecules and systems. We aim to investigate the possibility of identifying and characterizing life on Earth by using airborne spectropolarimetric observations from a hot air balloon during our field campaign in Switzerland, May 2022. In this work we present the optical-setup and the data obtained from aerial circular spectropolarimetric measurements of farmland, forests, lakes and urban sites. We make use of the well-calibrated FlyPol instrument that measures the fractionally induced circular polarization ($V/I$) of (reflected) light with a sensitivity of $<10^{-4}$. The instrument operates in the visible spectrum, ranging from 400 to 900 nm. We demonstrate the possibility to distinguish biotic from abiotic features using circular polarization spectra and additional broadband linear polarization information. We review the performance of our optical-setup and discuss potential improvements. This sets the requirements on how to perform future airborne spectropolarimetric measurements of the Earth's surface features from several elevations.

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arXiv2022

Ocean signatures in the total flux and polarization spectra of Earth-like exoplanets

Numerical simulations of starlight that is reflected by Earth-like exoplanets predict habitability signatures that can be searched for with future telescopes. We explore signatures of water oceans in the flux and polarization spectra of this reflected light. With an adding-doubling algorithm, we compute the total flux F, polarized flux Q and degree of polarization P of starlight reflected by dry and ocean model planets with Earth-like atmospheres and patchy clouds. The oceans consist of Fresnel reflecting surfaces with wind-ruffled waves, foam and wave shadows, above natural blue seawater. Our results are presented as functions of wavelength (from 300 to 2500 nm with 1 nm resolution) and as functions of the planetary phase angle from 90 to 170 degrees. The ocean glint increases F, |Q| and P with increasing phase angle at non-absorbing wavelengths, and causes the spectra of F and |Q| for the various phase angles to intersect. In the near-infrared, Q is negative, i.e. the direction of polarization is perpendicular to the plane through the star, planet, and observer. In the P-spectra, the glint leaves dips (instead of peaks) in gaseous absorption bands. All those signatures are missing in the spectra of dry planets. The dips in P, and the negative Q in the near-infrared, can be searched for at a phase angle of 90 degrees, where the planet-star separation is largest. Those ocean signatures in polarized light do not suffer from false positive glint signals that could be due to clouds or reflecting dry surfaces. For heavily cloudy planets, ocean detection is possible when the glint is (partially) cloud-free. When modelling signals of planets with oceans, using horizontally inhomogeneous cloud covers is thus crucial. Observations spread over time would increase the probability of catching a cloud-free glint and detecting an ocean.

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arXiv2021

Biosignatures of the Earth I. Airborne spectropolarimetric detection of photosynthetic life

Context. Homochirality is a generic and unique property of life on Earth and is considered a universal and agnostic biosignature. Homochirality induces fractional circular polarization in the incident light that it reflects. Because this circularly polarized light can be sensed remotely, it can be one of the most compelling candidate biosignatures in life detection missions. While there are also other sources of circular polarization, these result in spectrally flat signals with lower magnitude. Additionally, circular polarization can be a valuable tool in Earth remote sensing because the circular polarization signal directly relates to vegetation physiology. Aims. While high-quality circular polarization measurements can be obtained in the laboratory and under semi-static conditions in the field, there has been a significant gap to more realistic remote sensing conditions. Methods. In this study, we present sensitive circular spectropolarimetric measurements of various landscape elements taken from a fast-moving helicopter. Results. We demonstrate that during flight, within mere seconds of measurements, we can differentiate (S/N>5) between grass fields, forests, and abiotic urban areas. Importantly, we show that with only nonzero circular polarization as a discriminant, photosynthetic organisms can even be measured in lakes. Conclusions. Circular spectropolarimetry can be a powerful technique to detect life beyond Earth, and we emphasize the potential of utilizing circular spectropolarimetry as a remote sensing tool to characterize and monitor in detail the vegetation physiology and terrain features of Earth itself.

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arXiv2020

Design of the Life Signature Detection Polarimeter LSDpol

Many biologically produced chiral molecules such as amino acids and sugars show a preference for left or right handedness (homochirality). Light reflected by biological materials such as algae and leaves therefore exhibits a small amount of circular polarization that strongly depends on wavelength. Our Life Signature Detection polarimeter (LSDpol) is optimized to measure these signatures of life. LSDpol is a compact spectropolarimeter concept with no moving parts that instantaneously measures linear and circular polarization averaged over the field of view with a sensitivity of better than 1e-4. We expect to launch the instrument into orbit after validating its performance on the ground and from aircraft. LSDpol is based on a spatially varying quarter-wave retarder that is implemented with a patterned liquid-crystal. It is the first optical element to maximize the polarimetric sensitivity. Since this pattern as well as the entrance slit of the spectrograph have to be imaged onto the detector, the slit serves as the aperture, and an internal field stop limits the field of view. The retarder's fast axis angle varies linearly along one spatial dimension. A fixed quarter-wave retarder combined with a polarization grating act as the disperser and the polarizing beam-splitter. Circular and linear polarization are thereby encoded at incompatible modulation frequencies across the spectrum, which minimizes the potential cross-talk from linear into circular polarization.

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arXiv2020

LOUPE: Observing Earth from the Moon to prepare for detecting life on Earth-like exoplanets

LOUPE, the Lunar Observatory for Unresolved Polarimetry of the Earth, is a small, robust spectro-polarimeter with a mission to observe the Earth as an exoplanet. Detecting Earth-like planets in stellar habitable zones is one of the key challenges of modern exoplanetary science. Characterising such planets and searching for traces of life requires the direct detection of their signals. LOUPE provides unique spectral flux and polarisation data of sunlight reflected by the Earth, the only planet known to harbor life. This data will be used to test numerical codes to predict signals of Earth-like exoplanets, to test algorithms that retrieve planet properties, and to fine-tune the design and observational strategies of future space observatories. From the Moon, LOUPE will continuously see the entire Earth, enabling it to monitor the signal changes due to the planet's daily rotation, weather patterns, and seasons, across all phase angles. Here, we present both the science case and the technology behind LOUPE's instrumental and mission design.

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arXiv2019

A snapshot full-Stokes spectropolarimeter for detecting life on Earth

We present the design of a point-and-shoot non-imaging full-Stokes spectropolarimeter dedicated to detecting life on Earth from an orbiting platform like the ISS. We specifically aim to map circular polarization in the spectral features of chlorophyll and other biopigments for our planet as a whole. These non-zero circular polarization signatures are caused by homochirality of the molecular and supramolecular configurations of organic matter, and are considered the most unambiguous biomarker. To achieve a fully solid-state snapshot design, we implement a novel spatial modulation that completely separates the circular and linear polarization channels. The polarization modulator consists of a patterned liquid-crystal quarter-wave plate inside the spectrograph slit, which also constitutes the first optical element of the instrument. This configuration eliminates cross-talk between linear and circular polarization, which is crucial because linear polarization signals are generally much stronger than the circular polarization signals. This leads to a quite unorthodox optical concept for the spectrograph, in which the object and the pupil are switched. We discuss the general design requirements and trade-offs of LSDpol (Life Signature Detection polarimeter), a prototype instrument that is currently under development.

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arXiv2019

Blue, white, and red ocean planets - Simulations of orbital variations in flux and polarization colors

An exoplanet's habitability will depend strongly on the presence of liquid water. Flux and/or polarization measurements of starlight that is reflected by exoplanets could help to identify exo-oceans. We investigate which broadband spectral features in flux and polarization phase functions of reflected starlight uniquely identify exo-oceans. We compute total fluxes F and polarized fluxes Q of starlight reflected by cloud-free and (partly) cloudy exoplanets, for wavelengths from 350 to 865 nm. The ocean surface has waves composed of Fresnel reflecting wave facets and whitecaps, and scattering within the water body is included. Total flux F, polarized flux Q, and degree of polarization P of ocean planets change color from blue, through white, to red at phase angles alpha ranging from 134-108 deg for F, and from 123-157 deg for Q, with cloud coverage fraction fc increasing from 0.0 to 1.0 for F, and to 0.98 for Q. The color change in P only occurs for fc ranging from 0.03-0.98, with the color crossing angle alpha ranging from 88-161 deg. The total flux F of a cloudy, zero surface albedo planet can also change color, and for fc=0.0, an ocean planet's F will not change color for surface pressures ps > 8 bars. Polarized flux Q of a zero surface albedo planet does not change color for any fc. The color change of P of starlight reflected by an exoplanet, from blue, through white, to red with increasing alpha above 88 deg, appears to identify a (partly) cloudy exo-ocean. The color change of polarized flux Q with increasing alpha above 123 deg appears to uniquely identify an exo-ocean, independent of surface pressure or cloud fraction. At the color changing phase angle, the angular distance between a star and its planet is much larger than at the phase angle where the glint appears in reflected light. The color change in polarization thus offers better prospects for detecting exo-oceans.

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arXiv2018

Spectral and Temporal Variability of Earth Observed in Polarization

We present a comprehensive set of spectropolarimetric observations of Earthshine as obtained by FORS2 at the VLT for phase angles from 50degree to 135degree (Sun-Earth-Moon angle), covering a spectral range from 430nm to 920nm. The degree of polarization in BVRI passbands, the differential polarization vegetation index, and the equivalent width of the O2A polarization band around 760nm are determined with absolute errors around 0.1 percent in the degree of polarization. Earthshine polarization spectra are corrected for the effect of depolarization introduced by backscattering on the lunar surface, introducing systematic errors of the order of 1 percent in the degree of polarization. Distinct viewing sceneries such as observing the Atlantic or Pacific side in Earthshine yield statistically different phase curves. The equivalent width defined for the O2A band polarization is found to vary from -5nm to +2nm. A differential polarized vegetation index is introduced and reveals a larger vegetation signal for those viewing sceneries that contain larger fractions of vegetated surface areas. We corroborate the observed correlations with theoretical models from the literature, and conclude that the Vegetation Red Edge(VRE) is a robust and sensitive signature in polarization spectra of planet Earth. The overall behaviour of polarization of planet Earth in the continuum and in the O2A band can be explained by existing models. Biosignatures such as the O2A band and the VRE are detectable in Earthshine polarization with a high degree of significance and sensitivity. An in-depth understanding of Earthshines temporal and spectral variability requires improved models of Earths biosphere, as a prerequisite to interpret possible detections of polarised biosignatures in earthlike exoplanets in the future.

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arXiv2018

Polarized scattered light from self-luminous exoplanets. Three-dimensional scattering radiative transfer with ARTES

Direct imaging has paved the way for atmospheric characterization of young and self-luminous gas giants. Scattering in a horizontally-inhomogeneous atmosphere causes the disk-integrated polarization of the thermal radiation to be linearly polarized, possibly detectable with the newest generation of high-contrast imaging instruments. We aim to investigate the effect of latitudinal and longitudinal cloud variations, circumplanetary disks, atmospheric oblateness, and cloud particle properties on the integrated degree and direction of polarization in the near-infrared. We have developed a three-dimensional Monte Carlo radiative transfer code (ARTES) for scattered light simulations in (exo)planetary atmospheres. The code is applicable to calculations of reflected light and thermal radiation in a spherical grid with a parameterized distribution of gas, clouds, hazes, and circumplanetary material. The disk-integrated degree of polarization of a horizontally-inhomogeneous atmosphere is maximal when the planet is flattened, the optical thickness of the equatorial clouds is large compared to the polar clouds, and the clouds are located at high altitude. For a flattened planet, the integrated polarization can both increase or decrease with respect to a spherical planet which depends on the horizontal distribution and optical thickness of the clouds. The direction of polarization can be either parallel or perpendicular to the projected direction of the rotation axis when clouds are zonally distributed. Rayleigh scattering by submicron-sized cloud particles will maximize the polarimetric signal whereas the integrated degree of polarization is significantly reduced with micron-sized cloud particles as a result of forward scattering. The presence of a cold or hot circumplanetary disk may also produce a detectable degree of polarization ($\lesssim$1%) even with a uniform cloud layer in the atmosphere.

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arXiv2018

Traces of exomoons in computed flux and polarization phase curves of starlight reflected by exoplanets

Context: Detecting moons around exoplanets is a major goal of current and future observatories. Moons are suspected to influence rocky exoplanet habitability, and gaseous exoplanets in stellar habitable zones could harbour abundant and diverse moons to target in the search for extraterrestrial habitats. Exomoons will contribute to exoplanetary signals but are virtually undetectable with current methods. Aims: We identify and analyse traces of exomoons in the temporal variation of total and polarised fluxes of starlight reflected by an Earth-like exoplanet and its spatially unresolved moon across all phase angles, with both orbits viewed in an edge-on geometry. Methods: We compute the total and linearly polarised fluxes, and the degree of linear polarization P of starlight that is reflected by the exoplanet with its moon along their orbits, accounting for the temporal variation of the visibility of the planetary and lunar disks, and including effects of mutual transits and mutual eclipses. Our computations pertain to a wavelength of 450 nm. Results: Total flux F shows regular dips due to planetary and lunar transits and eclipses. Polarization P shows regular peaks due to planetary transits and lunar eclipses, and P can increase and/or slightly decrease during lunar transits and planetary eclipses. Changes in F and P will depend on the radii of the planet and moon, on their reflective properties, and their orbits, and are about one magnitude smaller than the smooth background signals. The typical duration of a transit or an eclipse is a few hours. Conclusions: Traces of an exomoon due to planetary and lunar transits and eclipses show up in F and P of sunlight reflected by planet-moon systems and could be searched for in exoplanet flux and/or polarisation phase functions.

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arXiv2018

Circular polarization signals of cloudy (exo)planets

The circular polarization of light that planets reflect is often neglected because it is very small compared to the linear polarization. It could, however, provide information on a planet's atmosphere and surface, and on the presence of life, because homochiral molecules that are the building blocks of life on Earth are known to reflect circularly polarized light. We compute $P_c$, the degree of circular polarization, for light that is reflected by rocky (exo)planets with liquid water or sulfuric acid solution clouds, both spatially resolved across the planetary disk and, for planets with patchy clouds, integrated across the planetary disk, for various planetary phase angles $α$. The optical thickness and vertical distribution of the atmospheric gas and clouds, the size parameter and refractive index of the cloud particles, and $α$ all influence $P_c$. Spatially resolved, $P_c$ varies between $\pm 0.20\%$ (the sign indicates the polarization direction). Only for small gas optical thicknesses above the clouds do significant sign changes (related to cloud particle properties) across the planets' hemispheres occur. For patchy clouds, the disk--integrated $P_c$ is typically smaller than $\pm 0.025\%$, with maximums for $α$ between $40^\circ$ and $70^\circ$, and $120^\circ$ to $140^\circ$. As expected, the disk--integrated $P_c$ is virtually zero at $α=0^\circ$ and 180$^\circ$. The disk--integrated $P_c$ is also very small at $α\approx 100^\circ$. Measuring circular polarization signals appears to be challenging with current technology. The small atmospheric circular polarization signal could, however, allow the detection of circular polarization due to homochiral molecules. Confirmation of the detectability of such signals requires better knowledge of the strength of circular polarization signals of biological sources.

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