Accessibility of Main-belt Asteroids and Trajectory Design for Multi-target Exploration

The mission designed to explore asteroids has nowadays become a hot spot of deep space exploration, and the accessibility of the explored objects is the most important problem to make clear. The number of asteroids is large, and it needs an enormous quantity of calculations to evaluate the accessibility for all asteroids. In this paper, based on the direct transfer strategy, we have calculated the accessibility for the different regions of the solar system and compared it with the distribution of asteroids. It is found that most main-belt asteroids are accessible by the direct transfer orbit with the launch energy of C₃ = 50 km²/s², and that with an additional small velocity correction, the designed trajectory is able to realize the multi-target flyby mission. Such a kind of multi-target flyby can reach the same effect of the orbit manoeuvre in the ΔV-EGA trajectory scheme^[1,2]. Being assisted by the earth's gravity, the accompanying flight with asteroids or the exploration of more distant asteroids can be realized with a lower energy. In the end, as an example, a trajectory scheme is given, in which the probe flies by multiple main-belt asteroids at first, then with the assistance of the earth's gravity, it makes the accompanying flight to a more distant asteroid.     

MIDI observations of 1459 Magnya: First attempt of interferometric observations of asteroids with the VLTI

The Very Large Telescope Interferometer (VLTI) of the European Southern Observatory (ESO) can be used to obtain direct determination of the sizes and the albedos of asteroids. We present results of the first attempt to carry out interferometric observations of asteroids with the Mid Infrared Interferometric Instrument (MIDI) at the VLTI. Our target was 1459 Magnya. This is the only V-type asteroid known to exist in the outer main-belt, and its IRAS-albedo turns out to be rather low for an object of this taxonomic class. Interferometric fringes were not detected, very likely due to the fact that the flux emitted by the asteroid was lower than expected and below the MIDI threshold for fringe detection. However, by fitting the Standard Thermal Model to the N-band infrared flux measured by MIDI in photometric mode and to the visible absolute magnitude, obtained from quasi-simultaneous B- and V-band photometric observations, we have derived a geometric visible albedo of 0.37±0.06 and an effective diameter of 17±1 km. This new estimate of the albedo differs from that previously obtained by IRAS and is more consistent with the V-type taxonomic classification of 1459 Magnya.     

Physical characteristics of Hayabusa target Asteroid 25143 Itokawa

In March 2001, the Hayabusa spacecraft target, Asteroid 25143 Itokawa, made its final close approach to Earth prior to the spacecraft's launch. We carried out an extensive observing campaign from January to September 2001 to better characterize this near-Earth asteroid. Global physical properties of the surface of Itokawa were characterized by analyzing its photometric properties and behavior. Results included here capitalize on analysis of broadband photometric observations taken with a number of telescopes, instruments, and observers. We employed a Hapke model to estimate the surface roughness, single particle scattering albedo, single particle scattering characteristics, phase integral, and geometric and bond albedo. We find that this asteroid has a higher geometric albedo than average main belt S-class asteroids; this is consistent with results from other observers. The broadband colors of Itokawa further support evidence that this is an atypical S-class asteroid. Broadband colors show spectral characteristics more typically found on large-diameter main-belt asteroids believed to be space-weathered, suggesting the surface of this small diameter, near-Earth asteroid could likewise be space-weathered.     

Shape, size and multiplicity of main-belt asteroids: I. Keck Adaptive Optics survey

This paper presents results from a high spatial resolution survey of 33 main-belt asteroids with diameters >40 km using the Keck II Adaptive Optics (AO) facility. Five of these (45 Eugenia, 87 Sylvia, 107 Camilla, 121 Hermione, 130 Elektra) were confirmed to have satellite. Assuming the same albedo as the primary, these moonlets are relatively small (∼5% of the primary size) suggesting that they are fragments captured after a disruptive collision of a parent body or captured ejecta due to an impact. For each asteroid, we have estimated the minimum size of a moonlet that can positively detected within the Hill sphere of the system by estimating and modeling a 2-σ detection profile: in average on the data set, a moonlet located at 2/100×RHill (1/4×RHill) with a diameter larger than 6 km (4 km) would have been unambiguously seen. The apparent size and shape of each asteroid was estimated after deconvolution using a new algorithm called AIDA. The mean diameter for the majority of asteroids is in good agreement with IRAS radiometric measurements, though for asteroids with a D<200 km, it is underestimated on average by 6-8%. Most asteroids had a size ratio that was very close to those determined by lightcurve measurements. One observation of 104 Klymene suggests it has a bifurcated shape. The bi-lobed shape of 121 Hermione described in Marchis et al. [Marchis, F., Hestroffer, D., Descamps, P., Berthier, J., Laver, C., de Pater, I., 2005c. Icarus 178, 450-464] was confirmed after deconvolution. The ratio of contact binaries in our survey, which is limited to asteroids larger than 40 km, is surprisingly high (∼6%), suggesting that a non-single configuration is common in the main-belt. Several asteroids have been analyzed with lightcurve inversions. We compared lightcurve inversion models for plane-of-sky predictions with the observed images (9 Metis, 52 Europa, 87 Sylvia, 130 Elektra, 192 Nausikaa, and 423 Diotima, 511 Davida). The AO images allowed us to determine a unique photometric mirror pole solution, which is normally ambiguous for asteroids moving close to the plane of the ecliptic (e.g., 192 Nausikaa and 52 Europa). The photometric inversion models agree well with the AO images, thus confirming the validity of both the lightcurve inversion method and the AO image reduction technique.     

On the asteroid belt's orbital and size distribution

For absolute magnitudes greater than the current completeness limit of H-magnitude ∼15 the main asteroid belt's size distribution is imperfectly known. We have acquired good-quality orbital and absolute H-magnitude determinations for a sample of small main-belt asteroids in order to study the orbital and size distribution beyond H=15, down to sub-kilometer sizes (H>18). Based on six observing nights over a 11-night baseline we have detected, measured photometry for, and linked observations of 1087 asteroids which have one-week time baselines or more. The linkages allow the computation of full heliocentric orbits (as opposed to statistical distances determined by some past surveys). Judged by known asteroids in the field the typical uncertainty in the (a/e/i) orbital elements is less than 0.03 AU/0.03/0.5°. The distances to the objects are sufficiently well known that photometric uncertainties (of 0.3 magnitudes or better) dominate the error budget of their derived H-magnitudes. The detected asteroids range from HR=12-22 and provide a set of objects down to sizes below 1 km in diameter. We find an on-sky surface density of 210 asteroids per square degree in the ecliptic with opposition magnitudes brighter than mR=23, with the cumulative number of asteroids increasing by a factor of 10^0.27/mag from mR=18 down to the mR?23.5 limit of our survey. In terms of absolute H magnitudes, we find that beyond H=15 the belt exhibits a constant power-law slope with the number increasing proportional to ¹⁰0.30H from H?15 to 18, after which incompleteness begins in the survey. Examining only the subset of detections inside 2.5 AU, we find weak evidence for a mildly shallower slope for H=15-19.5. We provide the information necessary such that anyone wishing to model the main asteroid belt can compare a detailed model to our detected sample.     

Polarization of the reflected light of asteroid 433 Eros

Linear polarizations measured for asteroid 433 Eros at various wavelengths and at solar phase angles ranging from 9° to 53° are presented. The polarization results are entirely typical of main-belt S asteroids, and indicate a dusty surface with geometric albedo 0.20. The derived effective diameter at photometric maximum is 21 km. Eros is quite uniform polarimetrically; no dependence on aspect is detected, and the polarization is shown to be constant during a single rotation with a precision of one part in forty.     

Thermal inertia of near-Earth asteroids and implications for the magnitude of the Yarkovsky effect

Thermal inertia determines the temperature distribution over the surface of an asteroid and therefore governs the magnitude the Yarkovsky effect. The latter causes gradual drifting of the orbits of km-sized asteroids and plays an important role in the delivery of near-Earth asteroids (NEAs) from the main belt and in the dynamical spreading of asteroid families. At present, very little is known about the thermal inertia of asteroids in the km size range. Here we show that the average thermal inertia of a sample of NEAs in the km-size range is . Furthermore, we identify a trend of increasing thermal inertia with decreasing asteroid diameter, D. This indicates that the dependence of the drift rate of the orbital semimajor axis on the size of asteroids due to the Yarkovsky effect is a more complex function than the generally adopted D⁻¹ dependence, and that the size distribution of objects injected by Yarkovsky-driven orbital mobility into the NEA source regions is less skewed to smaller sizes than generally assumed. We discuss how this fact may help to explain the small difference in the slope of the size distribution of km-sized NEAs and main-belt asteroids.     

Lightcurves and Synodic Periods for Asteroids 1998 ST49, (13154) and (27529)

The first photometric observations of the near-Earth asteroid 1998 ST₄₉ and the main belt asteroids (13154) and (27529) are reported. The synodic rotation periods and amplitudes of the lightcurves are:

? 2.3017 h, 0.11 mag for 1998 ST₄₉;

? 2.98502 h, 0.18 mag for (13154);

? 4.1290 h, 0.51 mag for (27529).

The periods for the first two asteroids are unambiguous, while the period of 4.1570 h is not ruled out for the third asteroid. Despite fast rotations, none of the presented asteroids exhibited any sign of binarity. Lightcurves were obtained at Modra Observatory.     

Mass distribution in the asteroid belt

The dependence of the cumulative number of numbered asteroids (up to 3720) on their absolute magnitude is investigated. The differential mass index k is derived from these relations for fainter asteroids. A steeper slope (2.2 < k < 2.4) is found in the four most populous asteroid familes (Flora, Koronis, Eos and Themis) and a flatter slope (1.3 < k < 1.6) for non-family asteroids. This indicates that there are two different asteroid populations in the asteorid belt. Total masses of the asteroid families may be greater than it is commonly accepted.     

Hidden Mass in the Asteroid Belt

The total mass of the asteroid belt is estimated from an analysis of the motions of the major planets by processing high precision measurements of ranging to the landers Viking-1, Viking-2, and Pathfinder (1976-1997). Modeling of the perturbing accelerations of the major planets accounts for individual contributions of 300 minor planets; the total contribution of all remaining small asteroids is modeled as an acceleration caused by a solid ring in the ecliptic plane. Mass M_ring of the ring and its radius R are considered as solve-for parameters. Masses of the 300 perturbing asteroids have been derived from their published radii based mainly on measured fluxes of radiation, making use of the corresponding densities. This set of asteroids is grouped into three classes in accordance with physical properties and then corrections to the mean density for each class are estimated in the process of treating the observations. In this way an improved system of masses of the perturbing asteroids has been derived.The estimate M_ring≈(5±1)×10⁻¹⁰M_⊙ is obtained (M_⊙ is the solar mass) whose value is about one mass of Ceres. For the mean radius of the ring we have R≈2.80 AU with 3% uncertainty. Then the total mass M_belt of the main asteroid belt (including the 300 asteroids mentioned above) may be derived: M_belt≈(18±2)×10⁻¹⁰M_⊙. The value M_belt includes masses of the asteroids which are already discovered, and the total mass of a large number of small asteroids—most of which cannot be observed from the Earth. The second component M_ring is the hidden mass in the asteroid belt as evaluated from its dynamical impact onto the motion of the major planets.Two parameters of a theoretical distribution of the number of asteroids over their masses are evaluated by fitting to the improved set of masses of the 300 asteroids (assuming that there is no observational selection effect in this set). This distribution is extrapolated to the whole interval of asteroid masses and as a result the independent estimate M_belt≈18×10⁻¹⁰M_⊙ is obtained which is in excellent agreement with the dynamical finding given above.These results make it possible to predict the total number of minor planets in any unit interval of absolute magnitude H. Such predictions are compared with the observed distribution; the comparison shows that at present only about 10% of the asteroids with absolute magnitude H<14 have been discovered (according to the derived distribution, about 130,000 such asteroids are expected to exist).     

Space weathering of small Koronis family members

The space weathering process and its implications for the relationships between S- and Q-type asteroids and ordinary chondrite meteorites is an often debated topic in asteroid science. Q-type asteroids have been shown to display the best spectral match to ordinary chondrites (McFadden, L.A., Gaffey, M.J., McCord, T.B. [1985]. Science 229, 160–163). While the Q-types and ordinary chondrites share some spectral features with S-type asteroids, the S-types have significantly redder spectral slopes than the Q-types in visible and near-infrared wavelengths. This reddening of spectral slope is attributed to the effects of space weathering on the observed surface composition. The analysis by Binzel et al. (Binzel, R.P., Rivkin, A.S., Stuart, J.S., Harris, A.W., Bus, S.J., Burbine, T.H. [2004]. Icarus 170, 259–294) provided a missing link between the Q- and S-type bodies in near-Earth space by showing a reddening of spectral slope in objects from 0.1 to 5 km that corresponded to a transition from Q-type to S-type asteroid spectra, implying that size, and therefore surface age, is related to the relationship between S- and Q-types. The existence of Q-type asteroids in the main-belt was not confirmed until Mothé-Diniz and Nesvorny (Mothé-Diniz, T., Nesvorny, D. [2008]. Astron. Astrophys. 486, L9–L12) found them in young S-type clusters. The young age of these families suggest that the unweathered surface could date to the formation of the family. This leads to the question of whether older S-type main-belt families can contain Q-type objects and display evidence of a transition from Q- to S-type. To answer this question we have carried out a photometric survey of the Koronis family using the Kitt Peak 2.1 m telescope. This provides a unique opportunity to compare the effects of the space weathering process on potentially ordinary chondrite-like bodies within a population of identical initial conditions. We find a trend in spectral slope for objects 1–5 km that shows the transition from Q- to S-type in the main-belt. This data set will prove crucial to our understanding of the space weathering process and its relevant timescales.     

The composition of M-type asteroids: Synthesis of spectroscopic and radar observations

We have conducted a radar-driven observational campaign of 22 main-belt asteroids (MBAs) focused on Bus–DeMeo Xc- and Xk-type objects (Tholen X and M class asteroids) using the Arecibo radar and NASA Infrared Telescope Facilities (IRTF). Sixteen of our targets were near-simultaneously observed with radar and those observations are described in a companion paper (Shepard, M.K., and 19 colleagues [2010]. Icarus, in press). We find that most of the highest metal-content asteroids, as suggested by radar, tend to exhibit silicate absorption features at both 0.9 and 1.9 μm, and the lowest metal-content asteroids tend to exhibit either no bands or only the 0.9 μm band. Eleven of the asteroids were observed at several rotational longitudes in the near-infrared and significant variations in continuum slope were found for nine in the spectral regions 1.1–1.45 μm and 1.6–2.3 μm. We utilized visible wavelength data (Bus, S.J., Binzel, R.P. [2002b]. Icarus 158, 146–177; Fornasier, S., Clark, B.E., Dotto, E., Migliorini, A., Ockert-Bell, M., Barucci, M.A. [2010]. Icarus 210, 655–673.) for a more complete compositional analysis of our targets. Compositional evidence is derived from our target asteroid spectra using two different methods: (1) a χ² search for spectral matches in the RELAB database, and (2) parametric comparisons with meteorites. This paper synthesizes the results of the RELAB search and the parametric comparisons with compositional suggestions based on radar observations. We find that for six of the seven asteroids with the highest iron abundances, our spectral results are consistent with the radar evidence (16 Psyche, 216 Kleopatra, 347 Pariana, 758 Mancunia, 779 Nina, and 785 Zwetana). Three of the seven asteroids with the lowest metal abundances, our spectral results are consistent with the radar evidence (21 Lutetia, 135 Hertha, 497 Iva). The remaining seven asteroids (22 Kalliope, 97 Klotho, 110 Lydia, 129 Antigone, 224 Oceana, 678 Fredegundis, and 771 Libera) have ambiguous compositional interpretations when comparing the spectral analogs to the radar analogs. The number of objects with ambiguous results from this multi-wavelength survey using visible, near-infrared, and radar wavelengths indicates that perhaps a third diagnostic wavelength region (such as the mid-infrared around 2–4 μm, the mid-infrared around 10–15 μm, and/or the ultraviolet around 0.2–0.4 μm) should be explored to resolve the discrepancies.     

Absolute magnitudes of asteroids and a revision of asteroid albedo estimates from WISE thermal observations

We obtained estimates of the Johnson V absolute magnitudes (H) and slope parameters (G) for 583 main-belt and near-Earth asteroids observed at Ond?ejov and Table Mountain Observatory from 1978 to 2011. Uncertainties of the absolute magnitudes in our sample are <0.21 mag, with a median value of 0.10 mag. We compared the H data with absolute magnitude values given in the MPCORB, Pisa AstDyS and JPL Horizons orbit catalogs. We found that while the catalog absolute magnitudes for large asteroids are relatively good on average, showing only little biases smaller than 0.1 mag, there is a systematic offset of the catalog values for smaller asteroids that becomes prominent in a range of H greater than ～10 and is particularly big above H ～ 12. The mean (H_catalog ? H) value is negative, i.e., the catalog H values are systematically too bright. This systematic negative offset of the catalog values reaches a maximum around H = 14 where the mean (H_catalog ? H) is ?0.4 to ?0.5. We found also smaller correlations of the offset of the catalog H values with taxonomic types and with lightcurve amplitude, up to ～0.1 mag or less. We discuss a few possible observational causes for the observed correlations, but the reason for the large bias of the catalog absolute magnitudes peaking around H = 14 is unknown; we suspect that the problem lies in the magnitude estimates reported by asteroid surveys. With our photometric H and G data, we revised the preliminary WISE albedo estimates made by Masiero et al. (Masired, J.R. et al. [2011]. Astrophys. J. 741, 68–89) and Mainzer et al. (Mainzer, A. et al. [2011b]. Astrophys. J. 743, 156–172) for asteroids in our sample. We found that the mean geometric albedo of Tholen/Bus/DeMeo C/G/B/F/P/D types with sizes of 25–300 km is p_V = 0.057 with the standard deviation (dispersion) of the sample of 0.013 and the mean albedo of S/A/L types with sizes 0.6–200 km is 0.197 with the standard deviation of the sample of 0.051. The standard errors of the mean albedos are 0.002 and 0.006, respectively; systematic observational or modeling errors can predominate over the quoted formal errors. There is apparent only a small, marginally significant difference of 0.031 ± 0.011 between the mean albedos of sub-samples of large and small (divided at diameter 25 km) S/A/L asteroids, with the smaller ones having a higher albedo. The difference will have to be confirmed and explained; we speculate that it may be either a real size dependence of surface properties of S type asteroids or a small size-dependent bias in the data (e.g., a bias towards higher albedos in the optically-selected sample of asteroids). A trend of the mean of the preliminary WISE albedo estimates increasing with asteroid size decreasing from D ～ 30 down to ～5 km (for S types) showed in Mainzer et al. (Mainzer, A. et al. [2011a]. Astrophys. J. 741, 90–114) appears to be mainly due to the systematic bias in the MPCORB absolute magnitudes that progressively increases with H in the corresponding range H = 10–14.     

Possible effect of spatial separation of bright and dark asteroids

The observed albedo distributions in asteroid families as well as numberical calculations suggest that the spatial separation of bright and dark asteroids can be caused by some nongravitational mechanism acting in the solar system. For main-belt asteroids of size 10–50 km, the separation rate can be roughly estimated at 1 AU per 10⁸ yr. The physical mechanism of this effect requires further investigation.     

Critical crater diameter and asteroid impact seismology

Abstract— If impact stress reverberation is the primary gradational process on an asteroid at global scales, then the largest undegraded crater records an asteroid's seismological response. The critical crater diameter D_crit is defined as the smallest crater whose formation disrupts all previous craters globally up to its size; it is solved for by combining relationships for crater growth and for stress attenuation. The computation for D_crit gives a simple explanation for the curious observation that small asteroids have only modest undegraded craters, in comparison to their size, whereas large asteroids have giant undegraded craters. D_crit can even exceed the asteroid diameter, in which case all craters are “local” and the asteroid becomes crowded with giant craters. D_crit is the most recent crater to have formed on a blank slate; when it is equated to the measured diameter of the largest undegraded crater on known asteroids, peak particle velocities are found to attenuate with the 1.2–1.3 power of distance—less attenuative than strong shocks, and more characteristic of powerful seismic disturbances. This is to be expected, since global degradation can result from seismic (cm s^?1) particle velocities on small asteroids. Attenuation, as modeled, appears to be higher on asteroids known to be porous, although these are also bodies for which different crater scaling rules might apply.     

Astrometric masses of 21 asteroids, and an integrated asteroid ephemeris

We apply the technique of astrometric mass determination to measure the masses of 21 main-belt asteroids; the masses of 9 Metis (1.03 ± 0.24 × 10^-11 M_⊙), 17 Thetis (6.17 ± 0.64 × 10^-13 M_⊙), 19 Fortuna (5.41 ± 0.76 × 10^-12 M_⊙), and 189 Phthia (1.87 ± 0.64 × 10^-14 M_⊙) appear to be new. The resulting bulk porosities of 11 Parthenope (12±4%) and 16 Psyche (46±16%) are smaller than previously-reported values. Empirical expressions modeling bulk density as a function of mean radius are presented for the C and S taxonomic classes. To accurately model the forces on these asteroids during the mass determination process, we created an integrated ephemeris of the 300 large asteroids used in preparing the DE-405 planetary ephemeris; this new BC-405 integrated asteroid ephemeris also appears useful in other high-accuracy applications.     

Submillimeter photometry and lightcurves of Ceres and other large asteroids

Photometry and thermal lightcurves of six large asteroids (1-Ceres, 2-Pallas, 3-Juno, 12-Victoria, 85-Io and 511-Davida) have been observed at 870 μm (345 GHz) using the MPIfR 19-Channel Bolometer of the Heinrich-Hertz Submillimeter Telescope. Only Ceres displayed a lightcurve with an amplitude (∼50%, peak to peak) that was significantly greater than the uncertainty in the observations. When thermal fluxes and brightness temperatures are corrected for heliocentric distance and albedo, there is a significant relation with the sub-solar latitude of the asteroid, or the local season of the asteroid. No such trend can be found between observations with solar phase angle. These results are evidence that most of the submillimeter thermal radiation is emitted from below the diurnal thermal wave. Comparing the observed trend with model output suggests that the submillimeter radiation from all the asteroids we observed is best modeled by surface material with low thermal inertia (<15 J m⁻² s^−0.5 K⁻¹, consistent with mid-infrared observations of large main-belt asteroids) and a refractive index closer to unity relative to densities inferred from radar experiments, implying a veneer of material over the asteroid surface with a density less than 1000 kg m⁻³. More data with better signal-to-noise and aspect coverage could improve these models and constrain physical properties of asteroid surface materials. This would also allow asteroids to be used as calibration sources with accurately known and stable, broadband fluxes at long wavelengths.     

The collisional and dynamical evolution of the main-belt and NEA size distributions

The size distribution of main belt of asteroids is determined primarily by collisional processes. Large asteroids break up and form smaller asteroids in a collisional cascade, with the outcome controlled by the strength-size relationship for asteroids. In addition to collisional processes, the non-collisional removal of asteroids from the main belt (and their insertion into the near-Earth asteroid (NEA) population) is critical, and involves several effects: strong resonances increase the orbital eccentricity of asteroids and cause them to enter the inner planet region; chaotic diffusion by numerous weak resonances causes a slow leak of asteroids into the Mars- and Earth-crossing populations; and the Yarkovsky effect, a radiation force on asteroids, is the primary process that drives asteroids into these resonant escape routes. Yarkovsky drift is size-dependent and can modify the main-belt size distribution. The NEA size distribution is primarily determined by its source, the main-belt population, and by the size-dependent processes that deliver bodies from the main belt. All of these effects are simulated in a numerical collisional evolution model that incorporates removal by non-collisional processes. We test our model against a wide range of observational constraints, such as the observed main-belt and NEA size distributions, the number of asteroid families, the preserved basaltic crust of Vesta and its large south-pole impact basin, the cosmic ray exposure ages of meteorites, and the cratering records on asteroids. We find a strength-size relationship for main-belt asteroids and non-collisional removal rates from the main belt such that our model fits these constraints as best as possible within the parameter space we explore. Our results are consistent with other independent estimates of strength and removal rates.     

The Rotational Periods of Three C-Type Asteroids 431, 521 and 524

The determination for asteroids’ spin parameters is very important for the physical study of asteroids and their evolution. Sometimes, the low amplitude of light curves and kinds of systematic errors in photometric data prevent the determination of the asteroids’ spin period. To solve such a problem, we introduced the de-correlation methods developed in searching for exoplanetary transit signal into the asteroid’s data reduction in this paper. By applying the principle of Collier Cameron (MNRAS 373:799–810, 2006) and Tamuz et al. (MNRAS 356:1466–1470, 2005)’s, we simulated the systematic effects in photometric data of asteroid, and removed those simulated errors from photometric data. Therefore the S/N of intrinsic signals of three selected asteroids are enhanced significantly. As results, we derived the new spin periods of 18.821 ± 0.011 h, 28.202 ± +0.071 h for (431) and (521) respectively, and refined the spin period of (524) as 14.172 ± 0.016 h.     

The large crater on the small Asteroid (2867) Steins

The maximum size of impact craters on finite bodies marks the largest impact that can occur short of impact induced disruption of the body. Recently attention has started to focus on large craters on small bodies such as asteroids and rocky and icy satellites. Here the large crater on the recently imaged Asteroid (2867) Steins (with crater diameter to mean asteroid radius ratio of 0.79) is shown to follow a limit set by other similar sized bodies with moderate macroporosity (i.e. fractured asteroids). Thus whilst large, the crater size is not novel, nor does it require Steins to possess an extremely large porosity. In one of the components of the binary Asteroid (90) Antiope there is the recently reported presence of an extremely large depression, possibly a crater, with depression diameter to mean asteroid radius ratio of ∼(1.4–1.62). This is consistent with the maximum size of a crater expected from previous observations of very porous rocky bodies (i.e. rubble-pile asteroids). Finally, a relationship between crater diameter (normalised to body radius) is proposed as a function of body porosity which suggests that the doubling of porosity between fractured asteroids and rubble-pile asteroids, nearly doubles the size (D/R value) of the largest crater sustainable on a rocky body.