Spherical Albedo Research Articles

Disk-integrated and disk-resolved photometry of the Moon with GaoFen-4 space observations

The GaoFen-4 (GF-4) satellite obtained high-resolution images of the entire lunar disk in a single exposure without atmospheric disturbance. These data are favorable for the study of both disk-resolved and disk-integrated photometric properties of the Moon. Using the GF-4 observations, we modeled the lunar reflectance phase functions with the Hapke model and analyzed the lunar photometric properties. The results indicate that the lunar surface exhibits pronounced backscattering, and the maria appear to be smoother compared to the highlands. The disk-resolved and disk-integrated models produced different best-fit parameters for the whole Moon, with the disk-integrated model fitting the small phase angle data better, and the disk-resolved model showing a more favorable appearance at large phase angles. By calculating the disk-integrated phase function with a disk-resolved method, the study confirms the equivalence of the two models, albeit with a small discrepancy due to an imperfect approximation of the roughness function. This discrepancy is due to the different forms of the phase function used in the models. The geometric albedo of the Moon derived from the disk-resolved model is 0.137 and the spherical albedo is 0.063 for Band 3 of GF-4 at 560.6 nm. This study observed an arched phase curve of the ratio of reflectance in two wavelengths, indicating a phenomenon of phase reddening at small phase angles and bluing at large phase angles.

Temporal Variation of the Shortwave Spherical Albedo of the Earth

The Earth’s spherical albedo describes the ratio of light reflected from the Earth to that incident from the Sun, an important variable for the Earth’s radiation balance. The spherical albedo has been previously estimated from satellites in low-Earth orbits, and from light reflected from the Moon. We developed a method to derive the Earth’s spherical shortwave albedo using the images from the Earth Polychromatic Imaging Camera (EPIC) on board National Oceanic and Atmospheric Administration’s (NOAA) Deep Space Climate Observatory (DSCOVR). The satellite is located in the Lagrange 1 point between the Earth and the Sun and observes the complete illuminated part of the Earth at once. The method allows us to provide continuously updated spherical albedo time series data starting from 2015. This time series shows a systematic seasonal variation with the mean annual albedo estimated as 0.295±0.008 and an exceptional albedo maximum in 2020, attributed to unusually abundant cloudiness over the Southern Oceans.

Spi-OPS: Spitzer and CHEOPS confirm the near-polar orbit of MASCARA-1 b and reveal a hint of dayside reflection

Context. The light curves of tidally locked hot Jupiters transiting fast-rotating, early-type stars are a rich source of information about both the planet and star, with full-phase coverage enabling a detailed atmospheric characterisation of the planet. Although it is possible to determine the true spin–orbit angle Ψ – a notoriously difficult parameter to measure – from any transit asymmetry resulting from gravity darkening induced by the stellar rotation, the correlations that exist between the transit parameters have led to large disagreements in published values of Ψ for some systems. Aims. We aimed to study these phenomena in the light curves of the ultra-hot Jupiter MASCARA-1 b, which is characteristically similar to well-studied contemporaries such as KELT-9 b and WASP-33 b. Methods. We obtained optical CHaracterising ExOPlanet Satellite (CHEOPS) transit and occultation light curves of MASCARA-1 b, and analysed them jointly with a Spitzer/IRAC 4.5 μm full-phase curve to model the asymmetric transits, occultations, and phase-dependent flux modulation. For the latter, we employed a novel physics-driven approach to jointly fit the phase modulation by generating a single 2D temperature map and integrating it over the two bandpasses as a function of phase to account for the differing planet–star flux contrasts. The reflected light component was modelled using the general ab initio solution for a semi-infinite atmosphere. Results. When fitting the CHEOPS and Spitzer transits together, the degeneracies are greatly diminished and return results consistent with previously published Doppler tomography. Placing priors informed by the tomography achieves even better precision, allowing a determination of Ψ = 72.1−2.4+2.5 deg. From the occultations and phase variations, we derived dayside and nightside temperatures of 3062−68+66 K and 1720 ± 330 K, respectively.Our retrieval suggests that the dayside emission spectrum closely follows that of a blackbody. As the CHEOPS occultation is too deep to be attributed to blackbody flux alone, we could separately derive geometric albedo Ag = 0.171−0.068+0.066 and spherical albedo As = 0.266−0.100+0.097 from the CHEOPS data, and Bond albedoAB = 0.057−0.101+0.083 from the Spitzer phase curve.Although small, the Ag and As indicate that MASCARA-1 b is more reflective than most other ultra-hot Jupiters, where H− absorption is expected to dominate. Conclusions. Where possible, priors informed by Doppler tomography should be used when fitting transits of fast-rotating stars, though multi-colour photometry may also unlock an accurate measurement of Ψ. Our approach to modelling the phase variations at different wavelengths provides a template for how to separate thermal emission from reflected light in spectrally resolved James Webb Space Telescope phase curve data.

Closed-form ab initio solutions of geometric albedos and reflected light phase curves of exoplanets

Studying the albedos of the planets and moons of the Solar System dates back at least a century. Of particular interest is the relationship between the albedo measured at superior conjunction, known as the ``geometric albedo", and the albedo considered over all orbital phase angles, known as the ``spherical albedo". Determining the relationship between the geometric and spherical albedos usually involves complex numerical calculations and closed-form solutions are restricted to simple reflection laws. Here we report the discovery of closed-form solutions for the geometric albedo and integral phase function, which apply to any law of reflection that only depends on the scattering angle. The shape of a reflected light phase curve, quantified by the integral phase function, and the secondary eclipse depth, quantified by the geometric albedo, may now be self-consistently inverted to retrieve globally averaged physical parameters. Fully Bayesian phase curve inversions for reflectance maps and simultaneous light curve detrending may now be performed due to the efficiency of computation. Demonstrating these innovations for the hot Jupiter Kepler-7b, we infer a geometric albedo of $0.25^{+0.01}_{-0.02}$, a phase integral of $1.77 \pm0.07$, a spherical albedo of $0.44^{+0.02}_{-0.03}$ and a scattering asymmetry factor of $0.07^{+0.12}_{-0.11}$.

Exploring super-Earth surfaces: Albedo of near-airless magma ocean planets and topography

In this paper we propose an analytic function for the spherical albedo values of airless and near-airless magma ocean planets (AMOPs). We generated 2-D fractal surfaces with varying compositions onto which we individually threw 10,000 light rays. Using an approximate form of the Fresnel equations we measured how much of the incident light was reflected. Having repeated this algorithm on varying surface roughnesses we find the spherical albedo as a function of the Hurst exponent, the geochemical composition of the magma, and the wavelength. As a proof of concept, we used our model on Kepler-10b to demonstrate the applicability of our approach. We present the spherical albedo values produced from different lava compositions and multiple tests that can be applied to observational data in order to determine their characteristics. Currently, there is a strong degeneracy in the surface composition of AMOPs due to the large uncertainties in their measured spherical albedos. In spite of this, when applied to Kepler-10b we show that its high albedo could be caused by a moderately wavy ocean that is rich in oxidised metallic species such as FeO, Fe2O3, Fe3O4. This would imply that Kepler-10b is a coreless or near-coreless body.

Different methods of correcting the readings of the DVGN-01 albedo neutron dosimeter placed behind the IBR-2M protection

The paper presents the results of correction of readings of DVBN-01 albedo dosimeters behind IBR-2M protection, the Neutron Physics Laboratory of the Joint Institute for Nuclear Research (JINR), Dubna, byvarious methods. The neutron spectra were measured at two points behind the IBR-2M protection in the experimental halls, and measurements were carried out with the spherical albedo system at these points. The correction coefficients for DVGN-01 were calculated from the measured spectra based on the data of the spherical albedo system. A good agreement of the coefficients calculated by different methods is shown, which indicates the reliability of the obtained values for the correction coefficients. Based on the results of this work and the data obtained in the other investigations, the values of correction factors recommended for use in individual radiation control (IRC) in the FLNP were obtained.

Inference of an Optimal Ice Particle Model through Latitudinal Analysis of MISR and MODIS Data

The inference of ice cloud properties from remote sensing data depends on the assumed forward ice particle model, as they are used in the radiative transfer simulations that are part of the retrieval process. The Moderate Resolution Imaging Spectroradiometer (MODIS) Collection 6 (MC6) ice cloud property retrievals are produced in conjunction with a single-habit ice particle model with a fixed degree of ice particle surface roughness (the MC6 model). In this study, we examine the MC6 model and five other ice models with either smoother or rougher surface textures to determine an optimal model to reproduce the angular variation of the radiation field sampled by the Multi-angle Imaging Spectroradiometer (MISR) as a function of latitude. The spherical albedo difference (SAD) method is used to infer an optimal ice particle model. The method is applied to collocated MISR and MODIS data over ocean for clouds with temperatures ≤233 K during December solstice from 2012–2015. The range of solar zenith angles covered by the MISR cameras is broader at the solstices than at other times of the year, with fewer scattering angles associated with sun glint during the December solstice than the June solstice. The results suggest a latitudinal dependence in an optimal ice particle model, and an additional dependence on the solar zenith angle (SZA) at the time of the observations. The MC6 model is one of the most optimal models on the global scale. In further analysis, the results are filtered by a cloud heterogeneity index to investigate cloudy scenarios that are less susceptible to potential 3D effects. Compared to results for global data, the consistency between measurements and a given model can be distinguished in both the tropics and extra-tropics. The SAD analysis suggests that the optimal model for thick homogeneous clouds corresponds to more roughened ice particles in the tropics than in the extra-tropics. While the MC6 model is one of the models most consistent with the global data, it may not be the most optimal model for the tropics.

Dynamic spatial fusion of cloud top phase from PARASOL, CALIPSO, cloudsat satellite data

Cloud phase is one of the important parameters of weather and climate research and core elements of atmospheric cloud parametric inversion. The accuracy of its recognition directly influences the inversion precision of cloud optical thickness, spherical albedo, effective particle radius, ice/liquid water content and other parameters. In this paper, we combine with cloud phase products of spaceborne multi-angle polarimetric radiometer, polarized laser radar, and millimeter wave radar. Next, we put forward a Dynamic Spatial Optimal Fusion (DSOF) algorithm. In this algorithm we construct the spatial optimal fusion rules of Cloud Top Phase (CTP) for multi-source data. Finally, we realize the cloud phase spatial fusion using these rules. We used the typhoon “Lupit” as the research object and verified this method. The standard deviations are 4.67%, 7.18%, 40.14% and 36.51%, respectively, compared the results of fused CTP to that of CALIPSO, CloudSat, POLDER3 and MODIS. The fusion results are most close to the CTP of CALIPSO. The results show that this method can effectively achieve multi-source cloud phase inversion, and provide new technology for the development and data synergy of spaceborne multi-sensor satellite.

Discrete spherical harmonics method for radiative transfer in scalar planar inhomogeneous atmosphere.

The radiative transfer problems in a participating inhomogeneous scalar planar atmosphere, subjected to diffuse or collimated incidence, are investigated using the discrete spherical harmonics method. In developing the method, the radiative intensity is expanded in a finite series of Legendre polynomials and the resulting first-order coupled differential equations of radiance moments are expressed in a set of discrete polar directions. The method is applied to homogeneous/inhomogeneous atmospheres of various anisotropic scattering degrees and thicknesses, and reflective boundary conditions. The discrete spherical harmonics method albedo, transmittance, and radiative intensity predictions agree well with benchmark literature results. Additionally, numerical predictions show that the discrete spherical harmonics method using Mark boundary conditions are more efficient than using Marshak boundary conditions.

Analytic reflected light curves for exoplanets

The disk-integrated reflected brightness of an exoplanet changes as a function of time due to orbital and rotational motion coupled with an inhomogeneous albedo map. We have previously derived analytic reflected lightcurves for spherical harmonic albedo maps in the special case of a synchronously-rotating planet on an edge-on orbit (Cowan, Fuentes & Haggard 2013). In this paper, we present analytic reflected lightcurves for the general case of a planet on an inclined orbit, with arbitrary spin period and non-zero obliquity. We do so for two different albedo basis maps: bright points ($\delta$-maps), and spherical harmonics ($Y_l^m$-maps). In particular, we use Wigner $D$-matrices to express an harmonic lightcurve for an arbitrary viewing geometry as a non-linear combination of harmonic lightcurves for the simpler edge-on, synchronously rotating geometry. These solutions will enable future exploration of the degeneracies and information content of reflected lightcurves, as well as fast calculation of lightcurves for mapping exoplanets based on time-resolved photometry. To these ends we make available Exoplanet Analytic Reflected Lightcurves (EARL), a simple open-source code that allows rapid computation of reflected lightcurves.

Atmospheric Correction of Multispectral Satellite Images Based on the Solar Radiation Transfer Approximation Model

A model of the solar radiation transfer in the Earth’s atmosphere is developed for explicit calculations of the transfer characteristics of the atmosphere (brightness coefficients, transmission, and spherical albedo) in the visible and near-IR atmospheric windows depending on the surface air pressure (or terrain elevation), aerosol optical parameters, and angles indicating the position of the Sun and the satellite system with respect to the Earth’s surface. The model is based on spectrally dependent approximations obtained using the atmospheric radiative transfer code DISORT. The effect of altitudinal stratification of the atmospheric optical parameters on the accuracy of underlying surface spectral albedo retrieval from the brightness coefficients at the top of the atmosphere is investigated. An algorithm is developed for radiometric correction of multispectral satellite images based on the model suggested. Approbation of the algorithm is performed using MERIS (Medium Resolution Imaging Spectrometer) data. It is shown that accounting for optical properties of the atmosphere and adjacency effects in processing satellite images substantially improves the visibility of the underlying surface and renders their spectral reflectance parameters.

Ground-based characterization of Hayabusa2 mission target asteroid 162173 Ryugu: constraining mineralogical composition in preparation for spacecraft operations

Asteroids that are targets of spacecraft missions are interesting because they present us with an opportunity to validate ground-based spectral observations. One such object is near-Earth asteroid (NEA) (162173) Ryugu, which is the target of the Japanese Space Agency's (JAXA) Hayabusa2 sample return mission. We observed Ryugu using the 3-m NASA Infrared Telescope Facility (IRTF) on Mauna Kea, Hawaii, on July 13, 2016 to constrain the object's surface composition, meteorite analogs, and link to other asteroids in the main belt and NEA populations. We also modeled its photometric properties using archival data. Using the Lommel-Seeliger model we computed the predicted flux for Ryugu at a wide range of viewing geometries as well as albedo quantities such as geometric albedo, phase integral, and spherical Bond albedo. Our computed albedo quantities are consistent with results from Ishiguro et al. (2014). Our spectral analysis has found a near-perfect match between our spectrum of Ryugu and those of NEA (85275) 1994 LY and Mars-crossing asteroid (316720) 1998 BE7, suggesting that their surface regoliths have similar composition. We compared Ryugu's spectrum with that of main belt asteroid (302) Clarissa, the largest asteroid in the Clarissa asteroid family, suggested as a possible source of Ryugu by Campins et al. (2013). We found that the spectrum of Clarissa shows significant differences with our spectrum of Ryugu, but it is similar to the spectrum obtained by Moskovitz et al. (2013). The best possible meteorite analogs for our spectrum of Ryugu are two CM2 carbonaceous chondrites, Mighei and ALH83100.

15-digit accuracy calculations of Ambartsumian-Chandrasekhar’s H $H$ -functions for four-term phase functions with the double-exponential formula

We have established an iterative scheme to calculate with 15-digit accuracy the numerical values of Ambartsumian-Chandrasekhar's H-functions for anisotropic scattering characterized by the four-term phase function: the method incorporates some advantageous features of the iterative procedure of Kawabata (2015) and the double-exponential integration formula~(DE-formula) of Takahashi and Mori (1974), which proved highly effective in Kawabata (2016). Actual calculations of the H-functions have been carried out employing 27 selected cases of the phase function, 56 values of the single scattering albedo $\varpi_0$, and 36 values of an angular variable $\mu(=\cos \theta)$, with $\theta$ being the zenith angle specifying the direction of incidence and/or emergence of radiation. Partial results obtained for conservative isotropic scattering, Rayleigh scattering, and anisotropic scattering due to a full four-term phase function are presented. As a sample application of the isotropic scattering H-function, an attempt is made in Appendix to simulate by iteratively solving the Ambartsumian equation the values of the plane and spherical albedos of a semi-infinite, homogeneous atmosphere calculated by Rogovtsov and Borovik (2016), who employed their analytical representations for these quantities and the single-term and two-term Henyey-Greenstein phase functions of appreciably high degrees of anisotropy, to find that our results are in satisfactory agreement with theirs.

Rough surfaces: Is the dark stuff just shadow?: “Who knows what evil lurks in the hearts of men? The shadow knows!”

Remote observations of the surfaces of airless planetary objects are fundamental to inferring the physical structure and compositional makeup of the surface material. A number of forward models have been developed to reproduce the photometric behavior of these surfaces, based on specific, assumed structural properties such as macroscopic roughness and associated shadowing. Most work of this type is applied to geometric albedos, which are affected by complicated effects near zero phase angle that represent only a tiny fraction of the net energy reflected by the object. Other applications include parameter fits to resolved portions of some planetary surface as viewed over a range of geometries. The spherical albedo of the entire object (when it can be determined) captures the net energy balance of the particle more robustly than the geometric albedo. In most treatments involving spherical albedos, spherical albedos and particle phase functions are often treated as if they are independent, neglecting the effects of roughness. In this paper we take a different approach. We note that whatever function captures the phase angle dependence of the brightness of a realistic rough, shadowed, flat surface element relative to that of a smooth granular surface of the same material, it is manifested directly in both the integral phase function and the spherical albedo of the object. We suggest that, where broad phase angle coverage is possible, spherical albedos may be easily corrected for the effects of shadowing using observed (or assumed) phase functions, and then modeled more robustly using smooth-surface regolith radiative transfer models without further imposed (forward-modeled) shadowing corrections. Our approach attributes observed “powerlaw” phase functions of various slope (and “linear” ranges of magnitude-vs.-phase angle) to shadowing, as have others, and goes in to suggest that regolith-model-based inferences of composition based on shadow-uncorrected spherical albedos overestimate the amount of absorbing material contained in the regolith.

Investigation of ice particle habits to be used for ice cloud remote sensing for the GCOM-C satellite mission

Abstract. In this study, various ice particle habits are investigated in conjunction with inferring the optical properties of ice clouds for use in the Global Change Observation Mission-Climate (GCOM-C) satellite programme. We develop a database of the single-scattering properties of five ice habit models: plates, columns, droxtals, bullet rosettes, and Voronoi. The database is based on the specification of the Second Generation Global Imager (SGLI) sensor on board the GCOM-C satellite, which is scheduled to be launched in 2017 by the Japan Aerospace Exploration Agency. A combination of the finite-difference time-domain method, the geometric optics integral equation technique, and the geometric optics method is applied to compute the single-scattering properties of the selected ice particle habits at 36 wavelengths, from the visible to the infrared spectral regions. This covers the SGLI channels for the size parameter, which is defined as a single-particle radius of an equivalent volume sphere, ranging between 6 and 9000 µm. The database includes the extinction efficiency, absorption efficiency, average geometrical cross section, single-scattering albedo, asymmetry factor, size parameter of a volume-equivalent sphere, maximum distance from the centre of mass, particle volume, and six nonzero elements of the scattering phase matrix. The characteristics of calculated extinction efficiency, single-scattering albedo, and asymmetry factor of the five ice particle habits are compared. Furthermore, size-integrated bulk scattering properties for the five ice particle habit models are calculated from the single-scattering database and microphysical data. Using the five ice particle habit models, the optical thickness and spherical albedo of ice clouds are retrieved from the Polarization and Directionality of the Earth's Reflectances-3 (POLDER-3) measurements, recorded on board the Polarization and Anisotropy of Reflectances for Atmospheric Sciences coupled with Observations from a Lidar (PARASOL) satellite. The optimal ice particle habit for retrieving the SGLI ice cloud properties is investigated by adopting the spherical albedo difference (SAD) method. It is found that the SAD is distributed stably due to the scattering angle increases for bullet rosettes with an effective diameter (Deff) of 10 µm and Voronoi particles with Deff values of 10, 60, and 100 µm. It is confirmed that the SAD of small bullet-rosette particles and all sizes of Voronoi particles has a low angular dependence, indicating that a combination of the bullet-rosette and Voronoi models is sufficient for retrieval of the ice cloud's spherical albedo and optical thickness as effective habit models for the SGLI sensor. Finally, SAD analysis based on the Voronoi habit model with moderate particle size (Deff = 60 µm) is compared with the conventional general habit mixture model, inhomogeneous hexagonal monocrystal model, five-plate aggregate model, and ensemble ice particle model. The Voronoi habit model is found to have an effect similar to that found in some conventional models for the retrieval of ice cloud properties from space-borne radiometric observations.

Emulation of Leaf, Canopy and Atmosphere Radiative Transfer Models for Fast Global Sensitivity Analysis

Physically-based radiative transfer models (RTMs) help understand the interactions of radiation with vegetation and atmosphere. However, advanced RTMs can be computationally burdensome, which makes them impractical in many real applications, especially when many state conditions and model couplings need to be studied. To overcome this problem, it is proposed to substitute RTMs through surrogate meta-models also named emulators. Emulators approximate the functioning of RTMs through statistical learning regression methods, and can open many new applications because of their computational efficiency and outstanding accuracy. Emulators allow fast global sensitivity analysis (GSA) studies on advanced, computationally expensive RTMs. As a proof-of-concept, three machine learning regression algorithms (MLRAs) were tested to function as emulators for the leaf RTM PROSPECT-4, the canopy RTM PROSAIL, and the computationally expensive atmospheric RTM MODTRAN5. Selected MLRAs were: kernel ridge regression (KRR), neural networks (NN) and Gaussian processes regression (GPR). For each RTM, 500 simulations were generated for training and validation. The majority of MLRAs were excellently validated to function as emulators with relative errors well below 0.2%. The emulators were then put into a GSA scheme and compared against GSA results as generated by original PROSPECT-4 and PROSAIL runs. NN and GPR emulators delivered identical GSA results, while processing speed compared to the original RTMs doubled for PROSPECT-4 and tripled for PROSAIL. Having the emulator-GSA concept successfully tested, for six MODTRAN5 atmospheric transfer functions (outputs), i.e., direct and diffuse at-surface solar irradiance ( E d i f , E d i r ), direct and diffuse upward transmittance ( T d i r , T d i f ), spherical albedo (S) and path radiance ( L 0 ), the most accurate MLRA’s were subsequently applied as emulator into the GSA scheme. The sensitivity analysis along the 400–2500 nm spectral range took no more than a few minutes on a contemporary computer—in comparison, the same analysis in the original MODTRAN5 would have taken over a month. Key atmospheric drivers were identified, which are on the one hand aerosol optical properties, i.e., aerosol optical thickness (AOT), Angstrom coefficient (AMS) and scattering asymmetry variable (G), mostly driving diffuse atmospheric components, E d i f and T d i f ; and those affected by atmospheric scattering, L 0 and S. On the other hand, as expected, AOT, AMS and columnar water vapor (CWV) in the absorption regions mostly drive E d i r and T d i r atmospheric functions. The presented emulation schemes showed very promising results in replacing costly RTMs, and we think they can contribute to the adoption of machine learning techniques in remote sensing and environmental applications.

Application of general invariance relations reduction method to solution of radiation transfer problems

A brief analysis of different properties and principles of invariance to solve a number of classical problems of the radiation transport theory is presented. The main ideas, constructions, and assertions used in the general invariance relations reduction method are described in outline. The most important distinctive features of this general method of solving a wide enough range of problems of the radiation transport theory and mathematical physics are listed. To illustrate the potential of this method, a number of problems of the scalar radiative transfer theory have been solved rigorously in the article. The main stages of rigorous derivations of asymptotical formulas for the smallest in modulo elements of the discrete spectrum and the eigenfunctions, corresponding to them, of the characteristic equation for the case of an arbitrary phase function and almost conservative scattering are described. Formulas of the same type for the azimuthal averaged reflection function, the plane and spherical albedos have been obtained rigorously. New analytical representations for the reflection function, the plane and spherical albedos have been obtained, and effective algorithms for calculating these values have been offered for the case of a practically arbitrary phase function satisfying the Hölder condition. New analytical representation of the «surface» Green function of the scalar radiative transfer equation for a semi-infinite plane-parallel conservatively scattering medium has been found. The deep regime asymptotics of the “volume” Green function has been obtained for the case of a turbid medium of cylindrical form.

REFLECTED LIGHT CURVES, SPHERICAL AND BOND ALBEDOS OF JUPITER- AND SATURN-LIKE EXOPLANETS*

ABSTRACT Reflected light curves observed for exoplanets indicate that a few of them host bright clouds. We estimate how the light curve and total stellar heating of a planet depends on forward and backward scattering in the clouds based on Pioneer and Cassini spacecraft images of Jupiter and Saturn. We fit analytical functions to the local reflected brightnesses of Jupiter and Saturn depending on the planet’s phase. These observations cover broadbands at 0.59–0.72 and 0.39–0.5 μm, and narrowbands at 0.938 (atmospheric window), 0.889 (CH4 absorption band), and 0.24–0.28 μm. We simulate the images of the planets with a ray-tracing model, and disk-integrate them to produce the full-orbit light curves. For Jupiter, we also fit the modeled light curves to the observed full-disk brightness. We derive spherical albedos for Jupiter and Saturn, and for planets with Lambertian and Rayleigh-scattering atmospheres. Jupiter-like atmospheres can produce light curves that are a factor of two fainter at half-phase than the Lambertian planet, given the same geometric albedo at transit. The spherical albedo is typically lower than for a Lambertian planet by up to a factor of ∼1.5. The Lambertian assumption will underestimate the absorption of the stellar light and the equilibrium temperature of the planetary atmosphere. We also compare our light curves with the light curves of solid bodies: the moons Enceladus and Callisto. Their strong backscattering peak within a few degrees of opposition (secondary eclipse) can lead to an even stronger underestimate of the stellar heating.

Probing exoplanet clouds with optical phase curves

Kepler-7b is to date the only exoplanet for which clouds have been inferred from the optical phase curve--from visible-wavelength whole-disk brightness measurements as a function of orbital phase. Added to this, the fact that the phase curve appears dominated by reflected starlight makes this close-in giant planet a unique study case. Here we investigate the information on coverage and optical properties of the planet clouds contained in the measured phase curve. We generate cloud maps of Kepler-7b and use a multiple-scattering approach to create synthetic phase curves, thus connecting postulated clouds with measurements. We show that optical phase curves can help constrain the composition and size of the cloud particles. Indeed, model fitting for Kepler-7b requires poorly absorbing particles that scatter with low-to-moderate anisotropic efficiency, conclusions consistent with condensates of silicates, perovskite, and silica of submicron radii. We also show that we are limited in our ability to pin down the extent and location of the clouds. These considerations are relevant to the interpretation of optical phase curves with general circulation models. Finally, we estimate that the spherical albedo of Kepler-7b over the Kepler passband is in the range 0.4-0.5.

Spherical albedo of a Lommel-Seeliger scattering ellipsoidal asteroid

AbstractWe compute the spherical albedo for a Lommel-Seeliger scattering ellipsoidal asteroid with a realistic disk-integrated phase function. The spherical (or Bond) albedo gives the ratio of the fluxes incident on and scattered by an asteroid. Thus, it plays a key role in the determination of the flux absorbed and afterwards thermally emitted by the asteroid at longer wavelengths. We provide extensive computations for the spherical albedo of low-albedo and moderate-albedo asteroids by utilizing the analytical disk-integrated brightness of a Lommel-Seeliger ellipsoid. In doing so, we utilize realistic triaxial models of known asteroids as well as idealistic prolate or oblate models of substantial elongation or flatness, respectively. We show that the spherical albedos can vary significantly as a function of the rotational pole orientation, rotational phase, and the triaxial ellipsoidal shape: variations of the order of 5-10% are realistic, with a tendency to grow with increasing elongation or flatness of the shape.