Context. Understanding the total flux and polarization signals of Earth-like planets and their spectral and temporal variability is essential for the future characterization of such exoplanets. Aims. We provide computed total (F) and linearly (Q and U) and circularly (V) polarized fluxes, and the degree of polarization P of sunlight that is reflected by a model Earth, to be used for instrument designs, optimizing observational strategies, and/or developing retrieval algorithms. Methods. We modeled a realistic Earth-like planet using one year of daily Earth-observation data: cloud parameters (distribution, optical thickness, top pressure, and particle effective radius), and surface parameters (distribution, surface type, and albedo). The Stokes vector of the disk-averaged reflected sunlight was computed for phase angles α from 0° to 180°, and for wavelengths λ from 350 to 865 nm. Results. The total flux F is one order of magnitude higher than the polarized flux Q, and Q is two and four orders of magnitude higher than U and V, respectively. Without clouds, the peak-to-peak daily variations due to the planetary rotation increase with increasing λ for F, Q, and P, while they decrease for U and V. Clouds modify but do not completely suppress the variations that are due to rotating surface features. With clouds, the variation in F increases with increasing λ, while in Q, it decreases with increasing λ, except at the largest phase angles. In earlier work, it was shown that with oceans, Q changes color from blue through white to red. The α where the color changes increases with increasing cloud coverage. Here, we show that this unique color change in Q also occurs when the oceans are partly replaced by continents, with or without clouds. The degree of polarization P shows a similar color change. Our computed fluxes and degree of polarization will be made publicly available.

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Because clouds scatter and absorb incident radiation, they play crucial roles in the radiative balance, the atmospheric chemistry, climate, and thus the habitability of a planet. Clouds usually affect the planet’s observables. We investigate here the effect of partial and variable cloud coverage on the light that is reflected by an Earth-like exoplanet, exploring not only the total flux but also the linearly and circularly polarized fluxes.@en

PYMIEDAP (the Python Mie Doubling-Adding Programme) is a Python-based tool for computing the total linearly and circularly polarized fluxes of incident unpolarized sunlight or starlight that is reflected by solar system planets or moons, respectively, or by exoplanets at a range of wavelengths. The radiative transfer computations are based on an doubling-adding Fortran algorithm and fully include polarization for all orders of scattering. The model (exo)planets are described by a model atmosphere composed of a stack of homogeneous layers containing gas and/or aerosol and/or cloud particles bounded below by an isotropically depolarizing surface (that is optionally black). The reflected light can be computed spatially resolved and/or disk-integrated. Spatially resolved signals are mostly representative for observations of solar system planets (or moons), while disk-integrated signals are mostly representative for exoplanet observations. PYMIEDAP is modular and flexible, and allows users to adapt and optimize the code according to their needs. PYMIEDAP keeps options open for connections with external programs and for future additions and extensions. In this paper, we describe the radiative transfer algorithm that PYMIEDAP is based on and the principal functionalities of the code. We also provide benchmark results of PYMIEDAP that can be used for testing its installation and for comparison with other codes.

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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 harbor abundant and diverse moons to target in the search for extraterrestrial habitats. Exomoons contribute to exoplanetary signals but are virtually undetectable with current methods. Aims. We identify and analyze traces of exomoons in the temporal variation of total and polarized 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 polarized 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 the 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 the F and P of sunlight reflected by planet-moon systems and could be searched for in exoplanet flux and/or polarization phase functions.@en

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.

Aims. We compute Pc, the degree of circular polarization, of light that is reflected by rocky (exo)planets to provide insight into the viability of circular spectropolarimetry for characterizing (exo)planetary atmospheres.

Methods. We compute the Pc of 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.

Results. 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 Pc. Spatially resolved, Pc varies between ± 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 Pc is typically smaller than ± 0.025%, with maximum for α between 40° and 70°, and 120° to 140°. As expected, the disk-integrated Pc is virtually zero at α = 0° and 180°. The disk-integrated Pc is also very small at α ≈ 100°.

Conclusions. 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 such as those associated with life on Earth. Confirmation of the detectability of such signals requires better knowledge of the strength of circular polarization signals of biological sources and in particular of the angular distribution of their scattering.
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Earth-like, potentially habitable exoplanets are prime targets in the search for extraterrestrial life. Information about their atmospheres and surfaces can be derived by analyzing the light of the parent star reflected by the planet. We investigate the influence of the surface albedo A _s, the optical thickness b _cloud, the altitude of water clouds, and the mixing ratio of biosignature O₂ on the strength of the O₂ A-band (around 760 nm) in the flux and polarization spectra of starlight reflected by Earth-like exoplanets. Our computations for horizontally homogeneous planets show that small mixing ratios (η < 0.4) will yield moderately deep bands in flux and moderate-to-small band strengths in polarization, and that clouds will usually decrease the band depth in flux and the band strength in polarization. However, cloud influence will be strongly dependent on properties such as optical thickness, top altitude, particle phase, coverage fraction, and horizontal distribution. Depending on the surface albedo and cloud properties, different O₂ mixing ratios η can give similar absorption-band depths in flux and band strengths in polarization, especially if the clouds have moderate-to-high optical thicknesses. Measuring both the flux and the polarization is essential to reduce the degeneracies, although it will not solve them, especially not for horizontally inhomogeneous planets. Observations at a wide range of phase angles and with a high temporal resolution could help to derive cloud properties and, once those are known, the mixing ratio of O₂ or any other absorbing gas.

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We have developed a method for measuring the linear polarisation of planets which is accessible to experienced amateur astronomers. The method requires a telescope with an aperture of about 20cm or more together with a linear polarising filter and a planetary imaging camera. Many suitable cameras are available and they can record uncompressed video at frame rates of 10 to 60 per second. Typically this rate will depend on the brightness of the source and size of the telescope. An ideal camera will be monochrome and is used with separate colour filters and a polarising filter. The method is to attach the colour and polarising filters to the camera and record a series of video clips. After recording each video clip the camera and filters are rotated by about 20 degrees until the total rotation is over 180 degrees. Each video clip is then stacked to produce a single low noise image. Most stacking software can sort the video frames according to quality, so the stack is based on a selected percentage of the best frames. There are several freeware stacking programs available which are primarily used for planetary imaging in general but are very suitable for polarisation. Original videos are mostly 8 bit but noise allows the combined stack to have a higher effective resolution and it is saved in 16 bit format. The stacked images are currently processed in Matlab, although the algorithms are being incorporated in Winjupos which is freeware. Results so far have been primarily for Jupiter, but we also have some data for Venus. The Matlab code is used to register the stacked frames (removing any camera rotation) and in the case of Jupiter, compensate for rotation of the planet during the video capture process. Accurate image registration is crucial for this method. A disk function is also applied to allow for the changing illumination angle as the planet rotates. A least squares function calculates the best fit cos squared curve for the variation of light at each point in the image as a function of the angle of the polariser. The coefficients of the multivariate fit are used to calculate the polarisation and its direction (the phase angle of the curve). Statistical analysis of the results gives a one standard deviation confidence interval for the polarisation. Initial results for Jupiter show the strong polarisation at the poles and variation in the belts and zones. The confidence interval is typically less than 1 percent. Analysis of the results is at an early stage but has the potential to help our understanding of the structure and composition of planetary atmospheres. In addition the involvement of amateur astronomers allows for plenty of available telescope time and the potential for long term monitoring of possible variations in polarisation patterns.@en

The detection of moons around extrasolar planets is one of the main focuses of current and future observatories. These silent companions contribute to the planets' observed signals but are barely detectable with current methods. Numerous gaseous exoplanets are known to orbit in the habitable zones of stars, and the expected abundance of natural satellites and their diversity in composition make them ideal targets when looking for habitable celestial bodies. And moons are suspected to play key roles in stabilizing a planet's rotational axis and hence its climate. We show that an exomoon orbiting an Earth-like exoplanet could be identified by measuring the flux and polarization of starlight reflected by the planet-moon system, allowing the characterization of their orbital motions and physical properties. @en

Context. Clouds have already been detected in exoplanetary atmospheres. They play crucial roles in a planet’s atmosphere and climate and can also create ambiguities in the determination of atmospheric parameters such as trace gas mixing ratios. Knowledge of cloud properties is required when assessing the habitability of a planet.

Aims. We aim to show that various types of cloud cover such as polar cusps, subsolar clouds, and patchy clouds on Earth-like exoplanets can be distinguished from each other using the polarization and flux of light that is reflected by the planet.

Methods. We have computed the flux and polarization of reflected starlight for different types of (liquid water) cloud covers on Earth-like model planets using the adding-doubling method, that fully includes multiple scattering and polarization. Variations in cloud-top altitudes and planet-wide cloud cover percentages were taken into account.
Results. We find that the different types of cloud cover (polar cusps, subsolar clouds, and patchy clouds) can be distinguished from each other and that the percentage of cloud cover can be estimated within 10%.
Conclusions. Using our proposed observational strategy, one should be able to determine basic orbital parameters of a planet such as orbital inclination and estimate cloud coverage with reduced ambiguities from the planet’s polarization signals along its orbit.
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In this work, we analyse linear polarization data of the planet at a distance, obtained with the Extreme Polarimeter (ExPo) on the William Herschel Telescope on La Palma. These spatially resolved, high-accuracy polarization observations of Venus show faint circular patterns centered on the sub-solar point that are absent in the flux observations. So far, careful analyses have ruled out instrumental effects which leaves us to wonder about atmospheric properties on Venus as the cause of the circular patterns. Using numerical simulations of the flux and polarization of sunlight that is reflected by Venus, we have investigated the relation between the observed patterns and several atmospheric properties, such as variations in particle sizes, composition, density and altitude. We discuss the plausibility of the possible causes in the view of the current knowledge of the composition and dynamical processes in Venus’s atmosphere.@en

In this work, we analyse linear polarization data of the planet at a distance, obtained with the Extreme Polarimeter (ExPo) on the William Herschel Telescope on La Palma. These spatially resolved, high-accuracy polarization observations of Venus show faint circular patterns centered on the sub-solar point that are absent in the flux observations. So far, careful analyses have ruled out instrumental effects which leaves us to wonder about atmospheric properties on Venus as the cause of the circular patterns. Using numerical simulations of the flux and polarization of sunlight that is reflected by Venus, we have investigated the relation between the observed patterns and several atmospheric properties, such as variations in particle sizes, composition, density and altitude. We discuss the plausibility of the possible causes in the view of the current knowledge of the composition and dynamical processes in Venus’s atmosphere.@en

ESA’s Venus Express mission has revealed our neighbouring planet to be a highly dynamic world, with everchanging cloud properties and structures, wind speeds that increase in time, and variable concentrations of atmospheric trace gases such as SO2. The SPICAV-IR instrument on Venus Express has provided us with close-up linear polarization data of sunlight reflected by Venus’s clouds and hazes, that allows a characterisation of their composition and particle sizes. Here, we analyse linear polarization data of the planet at a distance, obtained with the Extreme Polarimeter (ExPo) on the William Herschel Telescope on La Palma. These spatially resolved, high-accuracy polarization observations of Venus show faint circular patterns centered on the sub-solar point that are absent in the flux observations. So far, careful analyses have ruled out instrumental effects which leaves us to wonder about atmospheric properties as the cause of the circular patterns. Using numerical simulations of the flux and polarization of sunlight that is reflected by Venus, we have investigated the relation between the observed patterns and several atmospheric properties, such as variations in particle sizes, composition, density and altitude. We discuss the plausibility of the possible causes in the view of the current knowledge of the composition and dynamical processes in Venus’s atmosphere. @en

The direct detection of reflected starlight from exoplanets marks the beginning of a new era in the characterization of extrasolar planetary atmospheres. The flux and in particular the linear polarization signals from such planets are sensitive to atmospheric structure and composition, but other effects may also contribute to observed signals. We investigate the influence of an exoplanet’s shape and orbit orientation on its flux and polarization signature, and compare it against the influence of a variable cloud cover.@en

The circular polarisation of light is a property of electromagnetic radiation from which extensive information can be extracted. It is oft-neglected due to its small signal relative to linear polarisation and the need for advanced instrumentation in measuring it. Additionally, numerical modelling is complex as the full Stokes vector must always be computed. Circular polarisation is commonly induced through the multiple scattering of light by aerosols [?] and multiple reflections of light by rough surfaces [?]. Most interestingly, distinctive spectral circular polarimetric behaviour is exhibited by light reflected by organisms due to the homochiral molecular structure of all known organisms [?]. Especially fascinating is the unique circular polarimetric behaviour of light reflected by photosynthesising organisms at the absorption wavelength of the chlorophyll pigment [?]. This presents the previously unexplored possibility of circular polarimetry as a method for identifying and characterising the presence of organisms, a method which could be applied in the hunt for extraterrestrial life. To date, few telescopes exist that measure circular polarisation and none that have been deployed in space. Observations of the circular polarisation reflected by other planets in the solar system have been made with ground-based telescopes, with significant results [?]. However, none of these observations have been made at the phase angles at which exoplanets will be observed. Also, none have been made of the Earth, which is the logical starting point for the study of biologically induced circular polarisation signals. This introduces the need for numerical modelling to determine the extent to which circular polarisation is present in light reflected by exoplanets or the Earth. In this study, we model the multiple scattering and reflection of light using the doubling-adding method [?]. We will present circular polarisation signals for both spatially resolved and spatially unresolved planets, using various atmospheric and surface properties and across a range of phase angles. As a test, the calculated degree of circular polarisation resulting from the multiple scattering of light in an atmosphere with varying properties was compared with results presented by Kawata  and was found to be in agreement. Initial modelling of the atmospheric scattering of light by a planetary disk has shown a presence of degree of circular polarisation in the order of 104. This represents a static case with one cloudy hemisphere, one cloudless hemisphere and a Lambertian surface. Results containing varied patchy cloud patterns shall also be presented in a bid to reflect the random nature of planetary cloud cover. We will also present the calculated degree of circular polarisation of planets with various cloud coverage and a circularly polarising surface in order to discover the influence of organisms on the numerical results. @en

Polarimetry promises to be an important method to detect exoplanets: the light of a star is usually unpolarized[1] while scattering by gas and clouds in an atmosphere can generate high levels of polarization. Furthermore, the polarization of scattered light contains information about the properties of the atmosphere and surface of a planet, allowing a possible characterization [2], a method already validated in the solar system with Venus [3, 4]. We present here Pymiedap (Python Mie Doubling-Adding Program): a set of Python objects interfaced with Fortran radiative transfer codes that allows to define a planetary atmosphere and compute the flux and polarization of the light that is scattered. Several different properties of the planet can be set interactively by the user through the Python interface such as gravity, distance to the star, surface properties, atmospheric layers, gaseous and aerosol composition. The radiative transfer calculations are then computed following the doubling-adding method [5]. We present some results of the code and show its possible use for different planetary atmospheres for both resolved and disk-integrated measurements. We investigate the effect of gas, clouds and aerosols composition and surface properties for horizontally homogeneous and inhomogenous planets, in the case of Earth-like planets. We also study the effect of gaseous absorption on the flux and polarization as a marker for gaseous abundance and cloud top altitude. @en