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Theory of planetary photometry
Abstract
Modern definitions of photometric quantities and their units are
reviewed, including the bidirectional reflectance-distribution function.
These ideas are applied to the scattering of light by a spherical planet
under the most general reflection laws, and also more restricted cases
including Lambertian and Lommel-Seeliger scattering laws and phase laws
dependent on phase angle alone. The theory is given in terms of
'physical' concepts such as exitance and irradiance as well as in terms
of 'astronomical' concepts such as magnitude and albedo.
... Taking inspiration from the approach employed in [1], PF i is computed analytically as function of the phase angle α, parametrizing the brdf to the longitude ϕ and latitude θ angles referred to the XYZ frame shown in Figure 1. While in [1] the integral is computed altogether for the full sphere, however, here each discretized surface cap is considered separately to obtain the point cloud. ...
... Taking inspiration from the approach employed in [1], PF i is computed analytically as function of the phase angle α, parametrizing the brdf to the longitude ϕ and latitude θ angles referred to the XYZ frame shown in Figure 1. While in [1] the integral is computed altogether for the full sphere, however, here each discretized surface cap is considered separately to obtain the point cloud. The brdf is scaled by a constant γ which embeds the Bond albedo coefficient p. ...
This paper outlines the development and capabilities of a radiometric-consistent render to image at any observer pose a resolved spherical object and to compute pixel-wise spectral radiometric quantities within any user-defined waveband. Several reflection models are available and albedo, displacement and normal maps can be included to ensure radiometric fidelity. The tool is validated against real images of the Moon taken by the amie camera on-board of smart-1 mission.
... To avoid the inclusion of ORIs' observation-specific peculiarities, surface appearance can be physically modelled based on the surface shape (described solely by the DTM), desired lighting conditions, viewing geometry and a physical model of the reflection/scattering properties of the surface. Panels a2 and b2 of Figure 7.9 show the same view as a1 and b1, and the same datasets 5.1 and 5.4 for a2 and b2 respectively, but surface appearance is calculated according to Lambertian reflection (Lester, et al., 1979). With this physically based (albeit very simplified) model of reflection, the two datasets result in almost identical images. ...
... Here, a description of the radiative transfer model employed in the rendering code will be described (a more general and thorough description of radiative transfer is given by Chandrasekhar (1960)). In an attempt to be consistent with planetary photometry, notation of some quantities will differ slightly from Chandrasekhar's, and will instead follow the conventions used in Lester et al. (1979) (see their Table 1). ...
In this thesis, a novel approach to spaceborne imaging is investigated, building upon the scan imaging technique in which camera motion is used to construct an image. This thesis investigates its use with wide-angle (≥90° field of view) optics mounted on spin stabilised probes for large-coverage imaging of planetary environments, and focusses on two instruments. Firstly, a descent camera concept for a planetary penetrator. The imaging geometry of the instrument is analysed. Image resolution is highest at the penetrator’s nadir and lowest at the horizon, whilst any point on the surface is imaged with highest possible resolution when the camera’s altitude is equal to that point’s radius from nadir. Image simulation is used to demonstrate the camera’s images and investigate analysis techniques. A study of stereophotogrammetric measurement of surface topography using pairs of descent images is conducted. Measurement accuracies and optimum stereo geometries are presented. Secondly, the thesis investigates the EnVisS (Entire Visible Sky) instrument, under development for the Comet Interceptor mission. The camera’s imaging geometry, coverage and exposure times are calculated, and used to model the expected signal and noise in EnVisS observations. It is found that the camera’s images will suffer from low signal, and four methods for mitigating this – binning, coaddition, time-delay integration and repeat sampling – are investigated and described. Use of these methods will be essential if images of sufficient signal are to be acquired, particularly for conducting polarimetry, the performance of which is modelled using Monte Carlo simulation. Methods of simulating planetary cameras’ images are developed to facilitate the study of both cameras. These methods enable the accurate simulation of planetary surfaces and cometary atmospheres, are based on Python libraries commonly used in planetary science, and are intended to be readily modified and expanded for facilitating the study of a variety of planetary cameras.
... In this section, only the essential definitions and equations are given. The detailed mathematical derivations can be found in the literature [7,8,15]. ...
... For such a Lambert sphere the geometric albedo is p = 2r 3 (respectively p = 2 3 for a lossless sphere) and the phase function is given by (all angles given in rad, the detailed mathematical argumentation for this can be found in [15]): ...
The Earth’s albedo, as the fraction of sunlight that is directly reflected back into space from the surface or clouds, is a key factor in modeling Earth’s climate. In tasks such as the simplified estimation of the mean temperature, it is usually given as a constant, without mentioning its determination and the associated difficulties. In fact, the albedo can be determined by a basically simple procedure based on the observation of a well-known phenomenon: Earth’s shine. A comparison of the intensities between the directly illuminated side of the Moon and the side illuminated by Earth’s shine provides the mean albedo for a large part of the Earth’s surface. In this paper, the procedure will be reproduced using simple instruments. Assuming the reflection properties of a Lambert sphere—an ideal diffuse reflecting body—for the Earth, and using measurements of the phase function—as a description for the angular distribution of the scattered light—of the Moon, the Earth’s albedo is determined from self-acquired data. Even with these simple conditions it is possible to come quite close to the value of the Earth’s albedo.
... Unlike the surface albedo, which is defined for a point on a sphere, the geometric and Bond albedos pertain to the entire sphere. The geometric albedo is the ratio of the radiance scattered back to the source by a sphere to that scattered by a normally illuminated ideal Lambertian disc for the same source [14]. Its value coincides with P λ in the backscattering direction, i.e., g λ = P λ (Ω 0 , −Ω 0 ). ...
... Its value coincides with P λ in the backscattering direction, i.e., g λ = P λ (Ω 0 , −Ω 0 ). The Bond albedo is the fraction of the total solar energy intercepted by the sphere reflected in all directions [14]. Obviously, ρ λ can be obtained by integrating π −1 P λ (Ω 0 , Ω) over all directions of reflection Ω. ...
Earth’s reflectivity is among the key parameters of climate research. National Aeronautics and Space Administration (NASA)’s Earth Polychromatic Imaging Camera (EPIC) onboard National Oceanic and Atmospheric Administration (NOAA)’s Deep Space Climate Observatory (DSCOVR) spacecraft provides spectral reflectance of the entire sunlit Earth in the near backscattering direction every 65 to 110 min. Unlike EPIC, sensors onboard the Earth Orbiting Satellites (EOS) sample reflectance over swaths at a specific local solar time (LST) or over a fixed area. Such intrinsic sampling limits result in an apparent Earth’s reflectivity. We generated spectral reflectance over sampling areas using EPIC data. The difference between the EPIC and EOS estimates is an uncertainty in Earth’s reflectivity. We developed an Earth Reflector Type Index (ERTI) to discriminate between major Earth atmosphere components: clouds, cloud-free ocean, bare and vegetated land. Temporal variations in Earth’s reflectivity are mostly determined by clouds. The sampling area of EOS sensors may not be sufficient to represent cloud variability, resulting in biased estimates. Taking EPIC reflectivity as a reference, low-earth-orbiting-measurements at the sensor-specific LST tend to overestimate EPIC values by 0.8%to 8%. Biases in geostationary orbiting approximations due to a limited sampling area are between -0.7% and 12%. Analyses of ERTI-based Earth component reflectivity indicate that the disagreement between EPIC and EOS estimates depends on the sampling area, observation time and vary between -10% and 23%.
... A rigorous definition of the mean reflectance over the sunlit Earth can be found in [13], and can be expressed as: ...
... It characterizes mean TOA reflectance per unit of sunlit Earth area. In the backscattering direction, Equation (3) is the geometric albedo [13]. The difference between Equations (1) and (2) is that in Equation (2), pixels are weighted by cosine of the viewing zenith angle. ...
NASA’s Earth Polychromatic Imaging Camera (EPIC) onboard NOAA’s Deep Space Climate Observatory (DSCOVR) satellite observes the entire sunlit Earth every 65 to 110 min from the Sun–Earth Lagrangian L1 point. This paper presents initial EPIC shortwave spectral observations of the sunlit Earth reflectance and analyses of its diurnal and seasonal variations. The results show that the reflectance depends mostly on (1) the ratio between land and ocean areas exposed to the Sun and (2) cloud spatial and temporal distributions over the sunlit side of Earth. In particular, the paper shows that (a) diurnal variations of the Earth’s reflectance are determined mostly by periodic changes in the land–ocean fraction of its the sunlit side; (b) the daily reflectance displays clear seasonal variations that are significant even without including the contributions from snow and ice in the polar regions (which can enhance daily mean reflectances by up to 2 to 6% in winter and up to 1 to 4% in summer); (c) the seasonal variations of the sunlit Earth reflectance are mostly determined by the latitudinal distribution of oceanic clouds.
... The quantity ℎ is the directional-hemispherical reflectance in the notation of Hapke, 2012, which in general is a function of wavelength . It is called the point-albedo or normal-albedo in the notation of Lester et al., 1979. ...
... This substitution occurs through a separate approximation, over and above the point wise radiative equilibrium assumption, and is done to relate the models to observational evidence. Rather than observing the albedo (reflectance) ℎ at a point on the surface, we instead observe the geometric albedo which is the albedo for a sphere (Hapke, 2012; Lester et al., 1979). In addition, we know that asteroids are not perfect Lambertian spheres, and have complex phase laws (3-5). ...
Multiple terrestrial and space-based telescopes have been proposed for
detecting and tracking near-Earth objects (NEOs). Detailed simulations of the
search performance of these systems have used complex computer codes that are
not widely available, which hinders accurate cross- comparison of the proposals
and obscures whether they have consistent assumptions. Moreover, some proposed
instruments would survey infrared (IR) bands, whereas others would operate in
the visible band, and differences among asteroid thermal and visible light
models used in the simulations further complicate like-to-like comparisons. I
use simple physical principles to estimate basic performance metrics for the
ground-based Large Synoptic Survey Telescope and three space-based instruments
- Sentinel, NEOCam, and a Cubesat constellation. The performance is measured
against two different NEO distributions, the Bottke et al. distribution of
general NEOs, and the Veres et al. distribution of earth impacting NEO. The
results of the comparison show simplified relative performance metrics,
including the expected number of NEOs visible in the search volumes and the
initial discovery rates expected for each system. Although these simplified
comparisons do not capture all of the details, they give considerable insight
into the physical factors limiting performance. Multiple asteroid thermal
models are considered, including FRM, NEATM, and a new generalized for of FRM
(GFRM). I describe issues with how IR albedo and emissivity have been estimated
in previous studies, which may render them inaccurate. A thermal model for
tumbling asteroids is also developed and suggests that tumbling asteroids may
be surprisingly difficult for IR telescopes to observe.
... The surface brightness of the sunlight reflected by this dust distribution is proportional to σ(r, β) multiplied by a light-scattering function and integrated along an observer's line-of-sight. The flux density of sunlight that is reflected by dust in a small volume element dV is dF = σ(r, β)Φ(ϕ)(L ⊙ /4πr 2 )dV /∆ 2 (Lester et al. 1979) where dV = Ω∆ 2 d∆ and Ω is the solid angle of the volume element as seen by an observer a distance ∆ away; see Fig. 6 for the definition of all the geometric quantities used here. The scattering phase function Φ(ϕ) is related to the phase law ψ(ϕ) via Φ(ϕ) = (a/π sr)ψ(ϕ) where a is the dust geometric albedo and ϕ is the scattering angle. ...
Using the Moon to occult the Sun, the Clementine spacecraft used its navigation cameras to map the inner zodiacal light at optical wavelengths over elongations of 3-30 degrees from the Sun. This surface brightness map is then used to infer the spatial distribution of interplanetary dust over heliocentric distances of about 10 solar radii to the orbit of Venus. We also apply a simple model that attributes the zodiacal light as being due to three dust populations having distinct inclination distributions, namely, dust from asteroids and Jupiter-family comets (JFCs), dust from Halley-type comets, and an isotropic cloud of dust from Oort Cloud comets. The best-fitting scenario indicates that asteroids + JFCs are the source of about 45% of the optical dust cross-section seen in the ecliptic at 1 AU, but that at least 89% of the dust cross-section enclosed by a 1 AU radius sphere is of a cometary origin. When these results are extrapolated out to the asteroid belt, we find an upper limit on the mass of the light-reflecting asteroidal dust that is equivalent to a 12 km asteroid, and a similar extrapolation of the isotropic dust cloud out to Oort Cloud distances yields a mass equivalent to a 30 km comet, although the latter mass is uncertain by orders of magnitude.
... There are different definitions of the albedo. In this work, I treat in particular the geometric albedo, given by the brightness of a body at zero degree phase angle relative to a theoretical, perfectly reflecting disk of equal crosssection (Lester et al. 1979). This theoretical disk is a diffuse scatterer, hence, the geometric albedo may be larger than 1 if the physical body has a preferred direction of reflectance. ...
The minor bodies of the Solar System are witnesses of the formation history of our Solar System. Their compositional diversity represents the material present in the protoplanetary disk altered through hundreds of millions of years of collisional evolution, thermal metamorphism,aqueous alteration, and irradiation from the space environment. The history of the Solar System is therefore fossilised in the minor bodie sand the study of their ensemble properties provides insights into large-scale processes that shaped our planetary system. Asteroids, the minor bodies between the Sun and Jupiter, are the remnants of planetesimals which either formed in the inner Solar System or were implanted into it from the outer region via dynamical processes.The asteroid taxonomy is a language used to describe the compositional groups and trends observed among the asteroid population. In this work, I present a new iteration of the asteroid taxonomy. By advancing the established method with an unsupervised machine learning approach, Iimplement three key improvements over the previous method. First, the classification can be applied to observations of visible-, near-infrared, or visible-near-infrared reflectance spectra, providing consistent taxonomic classes in all three cases. Second, the class assignments are probabilistic rather than absolute, enabling fine selections of populations based on their probabilities to belong to anyclass. Third, the visual albedo is reintroduced as classification observable, resolving the spectral degeneracy and mineralogical ambiguity of the featureless reflectance spectra. The method is applied in the largest compositional study of asteroids based on reflectance spectra and visual albedo to date with 5906 observations, 2983 of which are used to derive the class scheme of the taxonomy. The new taxonomy combines the previously established frameworks of Tholen (1984), Bus and Binzel (2002) and DeMeo et al. (2009) with the observational and mineralogical insights into the asteroid population over the past decade. The core of the taxonomy are three complexes, the well-established carbonaceous C and silicaceous S complexes as well as the new M complex, which hosts a variety of metallic-silicaceous compositions and replaces the X complex. A new class Z is introduced for extremely-red asteroids which are found throughout the Main Belt. The method is applied to the spectral matching of asteroids and meteorites while accounting for the unknown properties of planetary surfaces. The connection between CV/CO chondrites to K- and L-type asteroids is studied as well as the link between equilibrated and unequilibrated ordinary chondrites and S-types. I further present two tools for analysis of asteroid data I developed. Rocks simplifies the exploration and acquisition of asteroid parameters, while classy enables the taxonomic classification of asteroid observations in the presented taxonomic scheme.
... First, except for the very small monolith asteroids, most solid planetary bodies are covered by regolith grains produced by impacts and space weathering, as evidenced by recent close-up images of small bodies (e.g., Itokawa (~300 m), Bennu (~500 m), etc.). For most celestial bodies, the Earth-based telescopic observations before the space age were only able to measure the disk-integrated reflectance spectra and thus the measured reflectance was dominated by small particle scattering (e.g., Lester et al., 1979;Helfenstein et al., 1996;Spjuth et al., 2012). Second, although surfaces of natural rocks are also granular materials at a microscopic scale, their microstructures are very different from that of packed layers of individual discrete grains. ...
Most solid planetary bodies in the solar system are covered by a layer of fine particles and the topic of light scattering by small particles has been thoroughly studied in the past decades. In contrast, light reflection from intact rocks has received much less attention, though the spectral features of fresh rocks are more diagnostic than that of highly space-weathered regolith grains. As high spatial-resolution spectral images obtained by modern space-borne and in-situ sensors have become available, it is important to understand the spectral feature links between rocks and powders made by crushing the rocks. In this work, we selected 13 terrestrial igneous rocks with a 1 μm absorption feature and measured the visible and near-infrared reflectance spectra of their slabs and powders in three size fractions, 0-45 μm, 90-125 μm, and 450-900 μm. We have found that the spectral characteristics of these samples can be divided into two groups. For slabs with reflectance lower than 0.1 at 0.5 μm, they have less pronounced 1 μm absorption feature. For slabs with reflectance higher than 0.1, they have pronounced 1 μm feature, consistent with that of their powdered counterparts. By using the equivalent-slab and the Hapke model, we obtained the optical constants and single scattering albedo values of the samples. The dependence of single scattering albedo on effective absorption thickness indicates that the differences between the spectral characteristics of rock slabs and powdered samples are likely controlled by the degree of weak surface scattering contributions. We reconstructed the spectrum of a powdered lunar meteorite which best matches the Chang'E-4 rock and found that the reconstructed rock spectra are very close to the rock spectrum observed in suit by Chang'E-4.
... The angles of incidence and reflection are related to the selenographic coordinates by (Lester, McCall and Tatum, 1979) cos cos cos ; cos cos cos , i ...
We expose and analyze the proposed models of the visual magnitude of the Moon for large phase angles (>150º). We devised a method to determine the luminance and illuminance per unit angular length of the lunar crescent as a function of position and phase angles.
... ¿Y cómo sabemos la relación que hay entre el flujo de luz reflejado por el exoplaneta y el flujo de la estrella? Según Cabrera & Schneider (2007), que utilizan la teoría fotométrica planetaria desarrollada por Lester, McCall & Tatum (1979), tenemos que: 43) siendo A el albedo del planeta, R p su radio, a p su semieje y Ψ(α) lo que se llama ley de fase, que tiene en cuenta elángulo de incidencia de los rayos de luz provenientes de la estrella. ...
En esta Tesis Doctoral, realizada por Carlos Vázquez Monzón y propuesta y dirigida por el prof. José Ángel Docobo Durántez, se hace un profundo estudio de los exoplanetas y también de los satélites que se mueven en torno a ellos, los exosatélites. En el Capítulo 1, se describen los principales métodos de detección y, en el caso de exosatélites, se proponen algunos métodos para intentar su descubrimiento. En el Capítulo 2 se trata el proceso por el cual se determina si un exoplaneta o exosatélite ha sido detectado, aplicando la teoría desarrollada a varios ejemplos. El Capítulo 3 versa sobre la dinámica y estabilidad de las órbitas de estos cuerpos, en distintos escenarios, estando el Capítulo 4 dedicado a la posible habitabilidad de los mismos.
... Bidirectional reflectance-distribution function (BRDF) [15,16,17] f r is defined as the ratio between the reflected radiance L r and total irradiance E of the surface element as ...
Light scattering from self-affine homogeneous isotropic random rough surfaces is studied using the ray-optics approximation. Numerical methods are developed to accelerate the first-order scattering simulations from surfaces represented as single-connected single-valued random fields, and to store the results of the simulations into a numerical reflectance model. Horizon mapping and marching methods are developed to accelerate the simulation. Emphasis is given to the geometric shadowing and masking effects as a function of surface roughness, especially, to the azimuthal rough-surface shadowing effect.
... The scalar products |Ω 0 · Ω n | and (Ω · Ω n ) give cosines of the SZA and satellite view zenith angle, respectively. In backscattering direction, P λ (Ω 0 , −Ω 0 ) is the geometric albedo (Lester et al., 1979), a major variable in exoplanet studies (Jiang et al., 2018). The EPIC observations, therefore, allow us to estimate geometric albedo. ...
Abstract We performed a detailed analysis of Earth Polychromatic Imaging Camera (EPIC) spectral data. We found that the vector composed of blue and near‐infrared (NIR) reflectance follows a counterclockwise closed‐loop trajectory from 0 to 24 UTC as Earth rotates. This nonlinear relationship was not observed by any other satellites due to limited spatial or temporal coverage of either low Earth orbit or geostationary satellites. We found that clouds play an important role in determining the nonlinear relationship in addition to the well‐known cloud‐free land‐ocean reflectance contrast in the two bands. The nonlinear relationship is the result of three factors: (1) a much larger cloud‐free land‐ocean contrast in the NIR band compared to the blue band, (2) significantly larger difference between cloudy land and cloudy ocean reflectance in the NIR band compared to the blue band, and (3) the periodic variation of fractions of clear land, clear ocean, cloudy land, and cloudy ocean in the sunlit hemisphere as Earth rotates. We found that the green vegetation contributes significantly to the NIR global average reflectance when the South and North Americas appear and disappear in the EPIC's field of view. The blue and NIR relationship can be useful for exoplanet research. Clouds impose a strong impact on global spectral reflectance, and the reflectance response to a change in cloud cover depends on whether the change is over land or over the ocean. On average, an increase of 0.1 in cloud coverage will lead to a 7% increase in spectrally integrated global average reflectance.
... These were taken from the NASA-Goddard Mars Factsheet (Williams 2017). The phase function P (χ) is often approximated as the diffuse reflection from a Lambertian sphere (Lester et al. 1979) ...
This paper examines the effectiveness of small star trackers for orbital estimation. Autonomous optical navigation has been used for some time to provide local estimates of orbital parameters during close approach to celestial bodies. These techniques have been used extensively on spacecraft dating back to the Voyager missions, but often rely on long exposures and large instrument apertures. Using a hyperbolic Mars approach as a reference mission, we present an EKF-based navigation filter suitable for nanosatellite missions. Observations of Mars and its moons allow the estimator to correct initial errors in both position and velocity. Our results show that nanosatellite-class star trackers can produce good quality navigation solutions with low position () and velocity () errors as the spacecraft approaches periapse.
... Table 3 shows the models for Minimum, Nominal, and Maximum predicted brightness of Bennu at 550 nm. Figure 11 show the data for Bennu as compared with our models. We note that the BRDF value at i=0 o , e=0 o , and α=0 o , calculated using the BRDF models is almost 1/π times the Geometric Albedo (0.045±0.015) as expected for Lommel-Seeliger spheres (Lester et al. 1979) and published in the Design Reference Asteroid Document (this document). ...
The Design Reference Asteroid (DRA) is a compilation of all that is known
about the OSIRIS-REx mission target, asteroid (101955) Bennu. It contains our
best knowledge of the properties of Bennu based on an extensive observational
campaign that began shortly after its discovery, and has been used to inform
mission plan development and flight system design. The DRA will also be
compared with post-encounter science results to determine the accuracy of our
Earth-based characterization efforts. The extensive observations of Bennu in
1999 has made it one of the best-characterized near-Earth asteroids. Many
physical parameters are well determined, and span a number of categories:
Orbital, Bulk, Rotational, Radar, Photometric, Spectroscopic, Thermal, Surface
Analog, and Environment Properties. Some results described in the DRA have been
published in peer-reviewed journals while others have been reviewed by
OSIRIS-REx Science Team members and/or external reviewers. Some data, such as
Surface Analog Properties, are based on our best knowledge of asteroid
surfaces, in particular those of asteroids Eros and Itokawa. This public
release of the OSIRIS-REx Design Reference Asteroid is a annotated version of
the internal OSIRIS-REx document OREX-DOCS-04.00-00002, Rev 9 (accepted by the
OSIRIS-REx project on 2014-April-14). The supplemental data products that
accompany the official OSIRIS-REx version of the DRA are not included in this
release. We are making this document available as a service to future mission
planners in the hope that it will inform their efforts.
... The surface brightness of the sunlight reflected by this dust distribution is proportional to σ(r, β) multiplied by a light-scattering function and integrated along an observer's line-of-sight. The flux density of sunlight that is reflected by dust in a small volume element dV is dF = σ(r, β)Φ(ϕ)(L ⊙ /4πr 2 )dV /∆ 2 (Lester et al. 1979) where dV = Ω∆ 2 d∆ and Ω is the solid angle of the volume element as seen by an observer a distance ∆ away; see Fig. 6 for the definition of all the geometric quantities used here. The scattering phase function Φ(ϕ) is related to the phase law ψ(ϕ) via Φ(ϕ) = (a/π sr)ψ(ϕ) where a is the dust geometric albedo and ϕ is the scattering angle. ...
Using the Moon to occult the Sun, the Clementine spacecraft used its navigation cameras to map the inner zodiacal light at optical wavelengths over elongations of 3≲ϵ≲30° from the Sun. This surface brightness map is then used to infer the spatial distribution of interplanetary dust over heliocentric distances of about 10 solar radii to the orbit of Venus. The averaged ecliptic surface brightness of the zodiacal light falls off as Z(ϵ)∝ϵ−2.45±0.05, which suggests that the dust cross-sectional density nominally falls off as σ(r)∝r−1.45±0.05. The interplanetary dust also has an albedo of a≃0.1 that is uncertain by a factor of ∼2. Asymmetries of ∼10% are seen in directions east–west and north–south of the Sun, and these may be due the giant planets' secular gravitational perturbations.We apply a simple model that attributes the zodiacal light as due to three dust populations having distinct inclination distributions, namely, dust from asteroids and Jupiter-family comets (JFCs) having characteristic inclinations of i∼7°, dust from Halley-type comets having i∼33°, and an isotropic cloud of dust from Oort Cloud comets. The best-fitting scenario indicates that asteroids + JFCs are the source of about 45% of the optical dust cross section seen in the ecliptic at 1 AU but that at least 89% of the dust cross section enclosed by a 1-AU-radius sphere is of a cometary origin. Each population's radial density variations can also deviate somewhat from the nominal σ(r)∝r−1.45. When these results are extrapolated out to the asteroid belt, we find an upper limit on the mass of the light-reflecting asteroidal dust that is equivalent to a 12-km asteroid, and a similar extrapolation of the isotropic dust cloud out to Oort Cloud distances yields a mass equivalent to a 30-km comet, although the latter mass is uncertain by orders of magnitude.
... The flux density of sunlight (here, power per area per wavelength interval) reflected by a – 7 – spherical grain of radius R toward an observer a distance ∆(t) away is ((Lester et al. 1979 ...
Pre-impact observations of Comet Shoemaker–Levy 9 (S-L9) obtained with the Hubble Space Telescope are examined, and a model of an active, dust-producing comet is fitted to images of fragments G, H, K, and L. The model assumes steady isotropic dust emission from each fragment's sunlit hemisphere. Best-fit results indicate that the dominant light-scatterers in these fragments' comae were relatively large dust grains of radii 10 μm ≲ R ≲ 3 mm. The fragments' dust size distributions were rather flat in comparison to other comets, d N (R) ∝ R−2.3±0.1, and the dust ejection speeds were ∼0.5–1.5 m/s. The S-L9 fragments themselves were not detected directly, and upper limits on their radii are 1.0–1.5 km assuming an albedo a=0.04. However, these fragments' vigorous production of dust, which ranges from 6 to 22 kg/s, places a lower limit of ∼100 m on their radii at the moment of tidal breakup. Any fragments smaller than this limit, yet experiencing similar mass loss rates, would have dissipated prior to impact. Such bodies would fail to leave an impact scar at Jupiter's atmosphere, as was realized by fragments F, J, P1, P2, T, and U.
... Bidirectional reflectance-distribution function (BRDF) [15][16][17] f r is defined as the ratio between the reflected radiance L r and total irradiance E of the surface element as ...
Light scattering from self-affine homogeneous isotropic random rough surfaces is studied using the ray-optics approximation. Numerical methods are developed to accelerate the first-order scattering simulations from surfaces represented as single-connected single-valued random fields, and to store the results of the simulations into a numerical reflectance model. Horizon mapping and marching methods are developed to accelerate the simulation. Emphasis is given to the geometric shadowing and masking effects as a function of surface roughness, especially, to the azimuthal rough-surface shadowing effect.
This paper explores a space mission concept in the geostationary orbit (GEO) region to build a catalogue of space objects down to 10 cm and provide satellite inspection services by using a mother satellite (EaglEye) and two smallsats (Harriers). The mothercraft carries the daughter smallsats, is launched in geostationary transfer orbit (GTO), and reaches the operational sub-GEO circular equatorial orbit at 3000 km from the GEO ring. The Harriers are deployed on the same operational orbit and phased using their propulsion system to be positioned at +/- 800 km from the mothercraft. The RSO measurement phase can then start with the objective to build a complete catalogue of all RSOs in less than one year, down to 10 cm size and up to 15 deg inclination. For that purpose, each spacecraft carries the same wide-field imager, providing a field of view (FOV) of about 700 square degrees with a pixel angular resolution of about 13 arcsec. The three satellites continuously and simultaneously observe a common FOV sliding on the GEO ring. We show that the RSO position and orbit are derived from the multispacecraft measurements, and the catalogue construction can be completed in about 7.5 months. The Harriers are designed to operate in either inspection or RSO detection mode. To limit the carried mass and save delta V, each Harrier is sent back to the mothercraft for refuelling as needed. The mission profile is flexible and can alternate inspections and RSO catalogue updates or perform both in parallel. Also, the use of both Harriers for inspections allows 3D stereo imaging. The mission lifetime is eight years with chemical propulsion and exceeds 15 years with electric propulsion while complying with a medium-size launcher.
The JUICE (JUpiter ICy moons Explorer) satellite of the European Space Agency (ESA) is dedicated to the detailed study of Jupiter and its moons. Among the whole instrument suite, JANUS (Jovis, Amorum ac Natorum Undique Scrutator) is the camera system of JUICE designed for imaging at visible wavelengths. It will conduct an in-depth study of Ganymede, Callisto and Europa, and explore most of the Jovian system and Jupiter itself, performing, in the case of Ganymede, a global mapping of the satellite with a resolution of 400 m/px. The optical design chosen to meet the scientific goals of JANUS is a three mirror anastigmatic system in an off-axis configuration. To ensure that the achieved contrast is high enough to observe the features on the surface of the satellites, we also performed a preliminary stray light analysis of the telescope. We provide here a short description of the optical design and we present the procedure adopted to evaluate the stray-light expected during the mapping phase of the surface of Ganymede. We also use the results obtained from the first run of simulations to optimize the baffle design.
We used ground-based photometric phase curve data of the OSIRIS-REx target asteroid (101955) Bennu and low phase angle data from asteroid (253) Mathilde as a proxy to fit Bennu data with Minnaert, Lommel-Seeliger, (RObotic Lunar Orbiter) ROLO, Hapke, and McEwen photometric models, which capture the global light scattering properties of the surface and subsequently allow us to calculate the geometric albedo, phase integral, spherical Bond albedo, and the average surface normal albedo for Bennu. We find that Bennu has low reflectance and geometric albedo values, such that multiple scattering is expected to be insignificant. Our photometric models relate the reflectance from Bennu’s surface to viewing geometry as functions of the incidence, emission, and phase angles. Radiance Factor functions (RADFs) are used to model the disk-resolved brightness of Bennu. The Minnaert, Lommel-Seeliger, ROLO, and Hapke photometric models work equally well in fitting the best ground-based photometric phase curve data of Bennu. The McEwen model works reasonably well at phase angles from 20o to 70o. Our calculated geometric albedo values of , and for the Minnaert, the Lommel-Seeliger, and the ROLO models respectively are consistent with the geometric albedo of 0.045±0.015 computed by Emery et al. (2014) and Hergenrother et al. (2014). Also, our spherical Bond albedo values of , and for the Minnaert model, Lommel-Seeliger, and ROLO models respectively are consistent with the value of 0.017±0.002 presented by Emery et al. (2014). On the other hand, the semi-physical models such as the Hapke model, where several assumptions and approximations were necessary, and the McEwen model are not supported by the global disk-integrated data, indicating that disk-resolved measurements will be necessary to constrain these models, as expected.
A comprehensive study of Nova V1500 Cygni 1975 is presented. All
previous work is carefully reviewed to arrive at a self-consistent set
of fundamental physical properties for the system. These data are then
combined with old and new observations to derive models for the ejecta
and the remnant and to arrive at a possible explanation for the extreme
nature of the outburst.
We study the variability of major atmospheric absorption features in the
disk-integrated spectra of the Earth with future application to Earth-analogs
in mind, concentrating on the diurnal timescale. We first analyze observations
of the Earth provided by the EPOXI mission, and find 5-20% fractional variation
of the absorption depths of H2O and O2 bands, two molecules that have major
signatures in the observed range. From a correlation analysis with the cloud
map data from the Earth Observing Satellite (EOS), we find that their variation
pattern is primarily due to the uneven cloud cover distribution. In order to
account for the observed variation quantitatively, we consider a simple opaque
cloud model, which assumes that the clouds totally block the spectral influence
of the atmosphere below the cloud layer, equivalent to assuming that the
incident light is completely scattered at the cloud top level. The model is
reasonably successful, and reproduces the EPOXI data from the pixel-level EOS
cloud/water vapor data. A difference in the diurnal variability patterns of H2O
and O2 bands is ascribed to the differing vertical and horizontal distribution
of those molecular species in the atmosphere. On the Earth, the inhomogeneous
distribution of atmospheric water vapor is due to the existence of its exchange
with liquid and solid phases of H2O on the planet's surface on a timescale
short compared to atmospheric mixing times. If such differences in variability
patterns were detected in spectra of Earth-analogs, it would provide the
information on the inhomogeneous composition of their atmospheres.
By September 2006, more than 200 planets have been discovered (see the Extrasolar Planets Encyclopedia at http://exoplanet.eu for a permanent update). In our solar system, 6 of the 8 planets have from 1 to several tens of satellites and numerical simulations show that they should be common (Ida, Canup & Stewart 1997). The search for satellites of extrasolar planets is relevant for the understanding of planetary system's evolution and for the perspective of their habitability (Williams, Kasting & Wade 1997, Nature, 385, 234). We study the possible interac-tions between satellites and their host planets in extrasolar systems and how these phenomena will affect their detection.
A numerical model of a circumstellar debris disk is developed and applied to observations of the circumstellar dust orbiting β Pictoris. The model accounts for the rates at which dust is produced by collisions among unseen planetesimals, and the rate at which dust grains are destroyed due to collisions. The model also accounts for the effects of radiation pressure, which is the dominant perturbation on the disk's smaller but abundant dust grains. Solving the resulting system of rate equations then provides the dust abundances versus grain size and dust abundances over time. Those solutions also provide the dust grains' collisional lifetime versus grain size, and the debris disk's optical depth and surface brightness versus distance from the star. Comparison to observations then yields estimates of the unseen planetesimal disk's radius, and the rate at which the disk sheds mass due to planetesimal grinding. The model can also be used to measure or else constrain the dust grain's physical and optical properties, such as the dust grains' strength, their light-scattering asymmetry parameter, and the grains' efficiency of light scattering Qs. The model is then applied to optical observations of the edge-on dust disk orbiting β Pictoris, and good agreement is achieved when the unseen planetesimal disk is broad, with 75 r 150 AU. If it is assumed that the dust grains are bright like Saturn's icy rings (Qs = 0.7), then the cross section of dust in the disk is Ad 2 × 1020 km2 and its mass is Md 11 lunar masses. In this case, the planetesimal disk's dust-production rate is quite heavy, M
⊕ Myr–1, implying that there is or was a substantial amount of planetesimal mass there, at least 110 Earth masses. If the dust grains are darker than assumed, then the planetesimal disk's mass-loss rate and its total mass are heavier. In fact, the apparent dearth of any major planets in this region, plus the planetesimal disk's heavy mass-loss rate, suggests that the 75 r < 150 AU zone at β Pic might be a region of planetesimal destruction, rather than a site of ongoing planet formation.
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