Asteroid photometric and polarimetric phase curves: Joint linear‐exponential modeling

Abstract— We present Markov‐Chain Monte‐Carlo methods (MCMC) for the derivation of empirical model parameters for photometric and polarimetric phase curves of asteroids. Here we model the two phase curves jointly at phase angles ≤25° using a linear‐exponential model, accounting for the opposition effect in disk‐integrated brightness and the negative branch in the degree of linear polarization. We apply the MCMC methods to V‐band phase curves of asteroids 419 Aurelia (taxonomic class F), 24 Themis (C), 1 Ceres (G), 20 Massalia (S), 55 Pandora (M), and 64 Angelina (E). We show that the photometric and polarimetric phase curves can be described using a common nonlinear parameter for the angular widths of the opposition effect and negative‐polarization branch, thus supporting the hypothesis of common physical mechanisms being responsible for the phenomena. Furthermore, incorporating polarimetric observations removes the indeterminacy of the opposition effect for 1 Ceres. We unveil a trend in the interrelation between the enhancement factor of the opposition effect and the angular width: the enhancement factor decreases with decreasing angular width. The minimum polarization and the polarimetric slope at the inversion angle show systematic trends when plotted against the angular width and the normalized photometric slope parameter. Our new approach allows improved analyses of possible similarities and differences among asteroidal surfaces.     

Opposition polarimetry and photometry of S- and E-type asteroids

The first results of the observational program devoted to simultaneous investigation of asteroid polarimetric and photometric opposition phenomena are presented. UBVRI polarimetric and V-band photometric observations of the S-type Asteroid 20 Massalia and the E-type Asteroids 214 Aschera and 620 Drakonia were carried out in 1996-1999 down to phase angles of 0.08°, 0.7°, and 1.2°, correspondingly. The S-type Asteroid 20 Massalia is characterized by the pronounced brightness opposition surge with an amplitude larger than that observed for the E-type asteroids. A sharp peak of negative polarization at small phase angles was not observed for this asteroid. The value of polarization degree at phase angle α<1° is less than 0.5% for both S and E types. The negative polarization branches of S and especially E-asteroids have an asymmetrical shape. The phase angle at which the polarization minimum occurs is close to the angle at which non-linear increase begins in the asteroid magnitude phase curves. A relation of the observed effects to the mechanism of coherent backscattering is discussed.     

Detection of λ = 2 cm emission from minor planet 704 Interamnia

λ = 2 cm emission from 704 Interamnia was detected on June 28, 1984, using the VLA. At this time, 704 Interamnia was 2.0 AU distant from Earth. A flux density of 953±53 μJy was measured at 14.9 GHz (50-MHz bandwidth). If 338 km is adopted for the diameter, a disk temperature of 190±15°K results. This temperature is consistent with a rapidly rotating black sphere with 704 Interamnia's diameter. Using published 20-μm infrared data, we infer that a dust-like layer (depth ≥3 cm) covers 704 Interamnia. Most probably, the bulk composition of this layer is identical to that of the outermost shell of the asteroid. Comparison with observations of 15 Eunomia reported earlier suggest that smaller main belt asteroids are covered by a few centimeters of impact gardened material. The physical properties of this layer (especially its depth) make it impossible to infer the dielectric properties of the underlying surface with radio measurements at frequencies above 8 GHz.     

Polarization and brightness opposition effects for the E-type Asteroid 44 Nysa

We present new polarimetric and photometric observations of high-albedo E-type Asteroid 44 Nysa in the BVRI wavebands at phase angles ranging from 0.41° to 7.49° during the 2005 opposition. A bimodal phase-angle dependence of polarization was found for Nysa in the V band. The polarization opposition effect was revealed in the form of a secondary minimum of negative polarization with amplitude ∼0.3% centered at a phase angle ∼0.8°. It is superimposed on the regular negative polarization branch with minimal polarization −0.30% at a phase angle 5.8°. We analyzed all available polarimetric data for E-type Asteroids 44 Nysa, 64 Angelina, and 214 Ashera and confirmed the presence of the polarization opposition effect for high-albedo asteroids at phase angle ∼1° with an amplitude ∼0.35%. The magnitude-phase curves reveal the presence of spike-like opposition effect of brightness for 44 Nysa in the BVRI spectral bands. 44 Nysa is the second high-albedo asteroid after 64 Angelina for which both the polarization opposition effect and the brightness opposition effect are detected. The differences between the parameters of the opposition effects for silicate surfaces (44 Nysa, 64 Angelina, Io) and icy surfaces (Europa, Ganymede, Iapetus, Saturn's rings) are discussed. The specific morphological parameters of opposition effects, in particular the angular width of the polarization opposition effect is comparable to that of the brightness opposition effect, provide almost unequivocal evidence that they are caused by coherent backscattering. One of unexpected results of our investigation is that 44 Nysa becomes bluer with increasing phase angle, while 64 Angelina shows phase reddening.     

Triaxial ellipsoid dimensions and poles of asteroids from AO observations at the Keck-II telescope

Five main belt asteroids, 2 Pallas, 129 Antigone, 409 Aspasia, 532 Herculina, and 704 Interamnia were imaged with the adaptive optics system on the 10 m Keck-II telescope in the near infrared on one night, August 16, 2006. The three axis dimensions and rotational poles were determined for Pallas, Antigone, Aspasia, and Interamnia, from their changing apparent sizes and shapes as measured with parametric blind deconvolution. The rotational pole found for Interamnia is much different from all previous work, including our own at Lick Observatory the previous month. Although images of Herculina were obtained at only two rotational phases, its rotation appears to be opposite to that predicted from the lightcurve inversion model of M. Kaasalainen, J. Torppa, and J. Piironen [2002. Icarus 159, 369-395]. A search for satellites was made in all of the asteroid images, with negative results, but three trailing stars around Herculina (200 km diameter), down to 8.9 magnitudes fainter and between 1 and 115 asteroid radii (100 to 11,500 km) from the asteroid, establishes an upper limit of 3.3 km for any object with the same albedo near Herculina.     

Polarimetry of main belt asteroids: Wavelength dependence

New UBVRI polarimetric observations of ten asteroids, including space mission targets 1 Ceres and 21 Lutetia, are presented. These observations were obtained with the 1.25-m telescope of the Crimean Astrophysical Observatory and have been used to study the wavelength dependence of polarization for a sample of asteroids belonging to the M and low albedo classes. A more general analysis including also a larger data set of UBVRI polarimetric observations available in the literature for more than 50 main belt asteroids belonging to different taxonomic classes shows that the variation of the polarization degree Pr as a function of wavelength is generally well described by a linear trend. It typically does not exceed 0.2% in the studied spectral range 0.37-0.83 microns and tends to increase for increasing phase angle. Asteroids belonging to the S and M classes are found to exhibit a deeper negative branch and smaller positive polarization for increasing wavelength (negative sign of the slope of ΔPr/Δλ). Since the objects belonging to these classes are known to exhibit reddish reflectance spectra, the observed wavelength behavior of negative polarization contradicts the well-known inverse correlation of Pmin and albedo. Low albedo asteroids show larger dispersion of spectral slopes, but the overall trend is characterized by a shallower negative branch and a larger positive polarization for increasing wavelength (positive sign of the slope of ΔPr/Δλ). A few exceptions from this general trend are discussed. The observed variety in the wavelength dependence of asteroid polarization seems to be mainly attributed to surface composition.     

On spectral dependence of polarization of asteroids

From the analysis of all of the data available on the spectral dependence of polarization of light reflected by asteroids, it has been shown that the slope of the spectral dependence of polarization of asteroids changes its sign, when moving from the negative branch of the phase curve of polarization to the positive one. This effect also manifests itself in the spectral behavior of polarization of the Moon and, probably, in the polarization of the other atmosphereless bodies. From the analysis of a population of asteroids of different types, a weak correlation between the spectral slopes of the polarization degree and the albedo has been found.     

Polarization and brightness opposition effects for the E-type Asteroid 64 Angelina

We present new polarimetric and photometric observations of the high-albedo Asteroid 64 Angelina in the UBVRI wavebands at phase angles ranging from 0.43° to 13.02° during oppositions in 1995, 1999, and 2000/2001. The polarization opposition effect has been observed in the form of a sharp peak of negative polarization with amplitude of about −0.4% centered at αmin≈1.8°, which is superimposed on the regular negative polarization branch. The amplitude of the polarization opposition effect appears to be apparition-dependent. Our photometric data confirm the early detected by Harris et al. [1989. Phase relations of high-albedo asteroids: The unusual opposition brightening of 44 Nysa and 64 Angelina. Icarus 81, 365-374] of a very strong and unusually narrow opposition spike, i.e., brightness opposition effect, for Angelina. Thus, 64 Angelina is the first asteroid for which both the polarization opposition effect and the brightness opposition effect have been detected. We observed that the polarization opposition effect as well as the regular negative polarization branch depends on the wavelength of scattered light, but in different manners. In addition, the colors B-V and V-R show little phase-angle dependence, while the color U-B increases with increasing phase angle, thus indicating that the amplitude of the brightness opposition effect is larger in the U band and almost the same in the B, V, and R bands. It appears that all colors indices begin to increase with decreasing phase angle to zero. The composite lightcurve computed with a period of 8.752 h has amplitude of 0.13 magnitude.     

Material composition assessment and discovering sublimation activity on asteroids 145 Adeona, 704 Interamnia, 779 Nina,and 1474 Beira

Spectrophotometric observations of 145 Adeona, 704 Interamnia, 779 Nina, and 1474 Beira—asteroids of close primitive types—allowed us to detect similar mineralogical absorption bands in their reflectance spectra centered in the range 0.35 to 0.92 μm; the bands are at 0.38, 0.44, and 0.67–0.71 μm. On the same asteroids, the spectral signs of simultaneous sublimation activity were found for the first time. Namely, there are maxima at ～0.35–0.60 μm in the reflectance spectra of Adeona, Interamnia, and Nina and at ～0.55–075 μm in the spectra of Beira. We connect this activity with small heliocentric distances of the asteroids and, consequently, with a high insolation at their surfaces. Examination of the samples of probable analogues allowed us to identify Fe³⁺ and Fe²⁺ in the material of these asteroids through the mentioned absorption bands. For analogues, we took powdered samples of carbonaceous chondrites Orgueil (CI), Mighei (CM2), Murchison (CM2), and Boriskino (CM2), as well as hydrosilicates of the serpentine group. Laboratory spectral reflectance study of the samples of low-iron Mg serpentines (<2 wt % FeO) showed that the equivalent width of the absorption band centered at 0.44–0.46 μm strongly correlates with the content of Fe³⁺ in octahedral and tetrahedral coordinations. Our conclusion is that this absorption band can be used as a qualitative indicator of Fe³⁺ in the surface matter of asteroids and other solid celestial bodies. The comparison of the listed analog samples and the asteroids by parameters of the spectral features suggests that the silicate component of the asteroids' surface material is a mixture of hydrated and oxidized compounds, including oxides and hydroxides of bivalent and trivalent iron and carbonaceous-chondritic material. At the same time, the sublimation activity of Adeona, Interamnia, Nina, and Beira at high surface temperatures points to a substantial content of water ice in their material. This contradicts the previously existing notions on the C-type and similar asteroids as bodies containing water only in the bound state. Moreover, since the sublimation process simultaneously occurs in four primitive-type bodies at small heliocentric distances, we may suppose that this phenomenon is common for the main-belt asteroids.     

Combining asteroid models derived by lightcurve inversion with asteroidal occultation silhouettes

Asteroid sizes can be directly measured by observing occultations of stars by asteroids. When there are enough observations across the path of the shadow, the asteroid’s projected silhouette can be reconstructed. Asteroid shape models derived from photometry by the lightcurve inversion method enable us to predict the orientation of an asteroid for the time of occultation. By scaling the shape model to fit the occultation chords, we can determine the asteroid size with a relative accuracy of typically ∼10%. We combine shape and spin state models of 44 asteroids (14 of them are new or updated models) with the available occultation data to derive asteroid effective diameters. In many cases, occultations allow us to reject one of two possible pole solutions that were derived from photometry. We show that by combining results obtained from lightcurve inversion with occultation timings, we can obtain unique physical models of asteroids.     

Opposition Effects in Brightness,Color, and Polarization of Comet 1P/Halley: Comparison with Atmosphereless Solar System Bodies

Systematic and uniform sets of photometric and polarimetric observations of comet 1P/Halley have been analyzed. The phase dependence of brightness for comet Halley was obtained at phase angles α ranging from 1.4° ≤ α ≤ 65°. The following parameters were determined: the amplitude of the opposition effect Δm = 0.75^m ± 0.06^m; the half-width at a half-maximum of intensity HWHM = 6.4° ± 1.6°; the linear phase coefficient β = 0.0045 ± 0.0001 mag/deg for α from 30° ≤ α ≤ 65°; and the phase angle at which a nonlinear increase in brightness starts, α _opp ≈ 31°. For the first time, the phase-angle dependence was obtained for the color of the dust of comet Halley: the color index BC-RC systematically decreases with increasing phase angle. Such a phase behavior of the dust color can be caused by the decrease in the mean size of dust particles that occurs when the comet approaches the Sun. For comet Halley, the negative polarization branch is almost symmetric; the minimal value of polarization is P _min = −1.54% at a phase angle α_min = 10.5°, and the inversion angle is α_inv = 21.7°. A comparative analysis of the phase functions of brightness and polarization has been performed for the cometary dust and atmosphereless bodies. Among the latter are low-albedo asteroids of the P and C types (102 Miriam and 47 Aglaja, respectively), as well as Deimos; high-albedo objects, such as the E-type asteroid 64 Angelina and the icy satellite of Jupiter Europa; and the Moon with its intermediate albedo. The possibility of a weak depression in the negative polarization branch of comets Halley and 47P/Ashbrook-Jackson at phase angles smaller than 2° is discussed.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 4, 2005, pp. 353–363.Original Russian Text Copyright © 2005 by Rosenbush.     

Polarization of Asteroid (387) Aquitania: The newest member of a class of large inversion angle asteroids

We present new imaging polarimetric observations of two Main Belt asteroids, (234) Barbara and (387) Aquitania, taken in the first half of 2008 using the Dual-Beam Imaging Polarimeter on the University of Hawaii 2.2 m telescope, located on Mauna Kea, Hawaii. Barbara had been previously shown to exhibit a very unusual polarization-phase curve by [Cellino, A., Belskaya, I.N., Bendjoya, Ph., di Martino, M., Gil Hutton, R., Muinonen, K., Tedesco, E.F., 2006. Icarus 180, 565-567]. Our observations confirm this result and add Aquitania to the growing class of large inversion angle objects. Interestingly, these asteroids show spinel features in their IR spectra suggesting a mineralogical origin to the phase angle-dependent polarimetric features. As spinel is associated with calcium-aluminum-rich inclusions and carbonaceous chondrites, these large inversion angle asteroids may represent some of the oldest surfaces in the Solar System. Circular as well as linear polarization measurements were obtained but circular polarization was not detected.     

Asteroid observations at low phase angles. I. 50 Virginia, 91 Aegina and 102 Miriam

We present observations of magnitude-phase dependences of three low-albedo asteroids down to phase angles of 0.1–0.2°. Data were obtained during 40 nights from 1994 to 1995 within the joint observational program at ESO and Kharkiv Astronomical Observatories with the aim to reach as low phase angles as possible. All three low-albedo asteroids may display a small nonlinear increase in magnitude-phase dependence at subdegree phase angles. The phase curves of 50 Virginia and 102 Miriam are poorly approximated by the HG function. Rotation periods of the asteroids were also determined: 14.310±0.010 hours for 50 Virginia, 6.030±0.001 h for 91 Aegina and 15.789± 0.003 h for 102 Miriam.     

Observations of asteroids in the infrared range (JHK) at the AZT-24 telescope

In 2007–2010, ten asteroids were observed in the JHK bands. The color indexes J-H and H-K were determined from the observations. Their positions in the two-color diagram generally correspond to the areas occupied by the asteroids of the corresponding taxonomic types. For two asteroids, (624) Hektor and (762) Pulcova, the light curves in the J and H bands and the curves of the color index J-H were constructed. The changes in the color index J-H from 0.05 ^m to 0.1 ^m were revealed; it may be supposed that they are caused by the heterogeneity in structure or mineral composition of the surface. For all of the asteroids, the astrometric positions were obtained. It was found that the best reference catalogue for the astrometric processing of observations of asteroids in the JHK bands under a small field of view is the 2MASS survey.     

Polarimetric observations of 9 Metis, 15 Eunomia, 89 Julia,and other asteroids

White-light polarization curves for a number of asteroids are presented. All have well-developed negative branches, suggestive of relatively dark, texturally complex surfaces—possibly regoliths. Geometric albedos, estimated from the positive branch of the polarization curves, show that a significant dispersion in surface albedo exists within the asteroid belt.     

Cometary dust properties retrieved from polarization observations: Application to C/1995 O1 Hale-Bopp and 1P/Halley

The analysis of the polarized light scattered by cometary dust particles provides information on the physical properties of the solid component of cometary comae for C/1995 O1 Hale-Bopp and 1P/Halley. A model of light scattering by a size distribution of aggregates of up to 256 submicron-sized grains (spherical or spheroidal) mixed with single spheroidal particles has been developed, with its parameters adjusted to fit the phase angle and wavelength dependence of the polarization observations. The particles are built of two materials: a non-absorbing silicates-type material and a more absorbing organic-type material. The model reproduces accurately the inversion angle and the positive branch of the polarization phase curves from the visible to the near-infrared spectral domains. A negative branch of the polarization phase curves appears in our model, although the negative branch is not deep enough to reproduce accurately the observations. Significant differences are shown between the two comets, with dominance of small grains in the coma of Comet C/1995 O1 Hale-Bopp, well fitted by a distribution of the volume-equivalent diameter, a, following a^−3.0 with a lower cutoff around 0.20 μm and an upper cutoff of at least 40 μm. For 1P/Halley, the size distribution follows a^−2.8 with a lower cutoff around 0.26 μm and an upper cutoff of about 38 μm. The relative amount of organic-type particles is larger for 1P/Halley while the amount of aggregates, significant for both comets, is larger for C/1995 O1 Hale-Bopp.     

The negative polarization parameters of the light scattered by the lunar surface: Mapping

The images of the southwestern part of the lunar disk showing the distributions of the negative polarization parameters of the light scattered by the lunar surface are presented. The distributions of the negative polarization minimum P _min, the inversion angle α_inv, and the polarization slope at the inversion point h significantly differ from the albedo image. This testifies to the fact that polarimetry yields independent information on the structure and optical properties of the lunar regolith.     

Spectropolarimetry of comets Austin and Churyumov-Gerasimenko

Spectropolarimetric observations from 5000 to 8000 Å have been obtained for comets P/Austin (1982g) and P/Churyumov-Gerasimenko (1982f). The observations were spaced over phase angles of 50–125° for comet Austin and 10–40° for comet Churyumov-Gerasimenko. The use of spectropolarimetry allowed an evaluation of continuum polarization without molecular line contamination. Especially for comet Churyumov-Gerasimenko, the curve of polarization versus phase angle resembles curves for asteroids, where the polarization is negative (electric vector maximum parallel to the scattering plane) for phase angles less than 20° and the most negative polarization is from ?1 to ?2%. The negative polarization at backscattering angles may be due to multiple scattering in agglomerated grains, as assumed for asteroids, or to Mie scattering by small dielectric particles. If multiple scattering is important in comet dust, polarization measurements may imply a low albedo, less than 0.08. The polarization of comet Austin remained steady during a large change in the dust production rate. Both comets increased continuum flux by a factor of 2 near perihelion. The continuum of comet Churyumov-Gerasimenko had the shape of the solar spectrum with derivations less than 5%. The equivalent width of spectral features of C₂, NH₂, and O varied as r^?2.     

Asteroid photometric and polarimetric phase curves: empirical interpretation

A method for interpretation of asteroid phase curves, based on empirical modeling and laboratory measurements, is outlined and preliminary results are presented. A linear-exponential function is used to describe the opposition peaks and negative polarization surges of various asteroids and laboratory samples and a statistical algorithm is used in parameter estimation. The linear-exponential function describes well the phase curves, but dense phase angle coverage, particularly at small phase angles must be obtained to improve the results. Major emphasis should also be put on laboratory study: with an extensive library of laboratory measurements, a stronger connection between the phase curve properties and surface characteristics is possible.     

Discovering New V-Type Asteroids in the Vicinity of 4 Vesta

We present results on the identification of two new V-type asteroids, 809 Lundia and 956 Elisa. These asteroids are located in the neighborhood of Asteroid 4 Vesta, but they do not belong to Vesta's dynamical family. Their spectra in the visible wavelength are consistent with the spectra of the Vesta family members (Vestoids) and of other V-type objects in the nearby region too. The possible existence of two spectroscopic groups of V-type asteroids in the Vesta region is discussed.