Statistical analysis of C and S Main Belt Asteroids

In this paper we compare the observable properties of 962 numbered MBAs (Main Belt Asteroids) of Tholen/SMASSII C and S class, with diameter in the range 1-500 km, not belonging to families or binary systems. Above 20 km, the diameters distributions of C and S are similar while under 20 km there is a clear observative bias in favour of small S asteroids which prevents a direct comparison. There is a significant correlation between rotation frequency and diameter both for C and S: if the diameter decreases the rotation frequency tends to increase. There is also a significant correlation between the lightcurve amplitude and the diameter for both samples: if the diameter decreases the lightcurve amplitude tends to increase. For larger diameter the C amplitude tends to be systematically higher than S amplitude of about 0.1 magnitude, but the difference is not very significant. Between 48 and 200 km, the C asteroids have a rotation frequency distribution compatible with a Maxwellian. On the other side, for S asteroids, the compatibility with the Maxwellian concerns diameters greater than 33 km. Considering the rotational properties and the lightcurve amplitude it appears that there are no substantial differences between the samples of C and S asteroids taken into account, and this indicates a good homogeneity in the processes of collisional evolution.     

Asteroid rotation rates

The dependence of rotational frequency on diameter, taxonomic type, and family membership is analyzed for 217 main-belt asteroids with statistically useful periods extracted from the file published by Harris and Young ((1983). Icarus54, 59–109). It is shown that for asteroids with diameters ? 120 km, mean rotational frequency increases with increasing diameter. This trend is equally present in all subsets of M-, S-, and C-type asteroids, for both family and nonfamily members alike, and cannot be accounted for by observational selection. For asteroids with diameters ? 120 km, mean rotational frequency increases with decreasing diameter; however, within this group there is a subset of asteroids with exceptionally long rotational periods. This marked change in the distribution at diameter ～ 120 km could separate primordial asteroids from their collision products. However, it is probable that the sample is biased in favor of small asteroids with short rotational periods and that the apparent increase of mean rotational frequency with decreasing diameter for small asteroids is at least partly the product of observational selection. An observational program that could test this hypothesis is described. If asteroids of any one diameter are considered, then, on average, M asteroids rotate faster than S asteroids which in turn rotate faster than C asteroids. This shows that asteroids which have been classified by their surface properties alone have different bulk properties. There is also some evidence that for all asteroidal types, of all diameters, family members rotate faster than nonfamily members.     

Rotational properties of asteroids: Correlations and selection effects

Rotational data on 321 asteroids observed as of late 1978 are analyzed. Selection effects within the sample are discussed and used to define a data set consisting of 134 main-belt, nonfamily asteroids having reliably determined periods and amplitudes based on photoelectric observations. In contrast to A. W. Harris and J. A. Burns (1979, Icarus40, 115–144) we found no significant correlation between rotational properties and compositional type. Smaller asteroids have a greater range of rotational amplitudes than the largest asteroids but are not, on the average, appreciably more elongated. While no definite relationship between asteroid size and rotation rate was found the distribution is not random. The largest asteroids have rotation periods near 7 hr compared with 10 hr for the smaller. A group of large, rapidly rotating, high-amplitude asteroids is recognized. A pronounced change in rotational properties occurs near this size range (diam = 200 ± 50 km) which also corresponds to the size at which a change of slope occurs in the size frequency distribution. We believe this size range represents a transition region between very large, rapidly rotating, low-amplitude (primordial?) objects and smaller ones having a considerably greater range of periods and amplitudes. Asteroids in this transition size range display an increase in rotational amplitude with increasing spin rate; other than this, however, there is no correlation between period and amplitude. The region of low spatial density in the asteroid belt centered near 2.9 AU and isolated from the inner and outer belt by the 2:5 and 3:7 commensurabilities is shown to be a region in which non-C or -S asteroids are overrepresented and which have marginally higher rotational amplitudes than those located in more dense regions. We attribute disagreements between our results and other studies of this type to the inclusion of non-main-belt asteroids and photographic data in the earlier analyses.     

The rotations of 128 Nemesis and 393 Lampetia: The longest known periods to date

Extensive photoelectric lightcurves of the asteroids 128 Nemesis and 393 Lampetia show for both objects extremely long rotational periods, the longest known to date for minor planets. For 393 Lampetia the combined results suggest with high probability a period of 38.7 hr with a maximal amplitude of 0.14 mag; a double-wave characteristic of the lightcurve must be assumed. For 128 Nemesis the complete double-wave lightcurve was observed and a period of 39.0 hr with a total amplitude of only 0.10 mag was deduced. Observations of 128 Nemesis confirm without doubt the presence of small-scale features of amplitude 0.01 to 0.02 mag, corresponding to small topographic features of about 15 km in height and width on the surface.     

A photoelectric lightcurve survey of small main belt asteroids

A survey to obtain photoelectric lightcurves of small main-belt asteroids was conducted from November 1981 to April 1982 using the 0.91- and 2.1-m telescopes at the University of Texas McDonald Observatory. A total of 18 main-belt asteroids having estimated diameters under 30 km were observed with over half of these being smaller than 15 km. Rotational periods were determined or estimated from multiple nights of observation for nearly all of these yielding a sample of 17 small main-belt asteroids which is believed to be free of observational selection effects. All but two of these objects were investigated for very short periods in the range of 1 min to 2 hr using power spectrum analysis of a continuous set of integrations. No evidence for such short periods was seen in this sample. Rotationally averaged B(1,0) magnitudes were determined for most of the surveyed asteroids, allowing diameter estimates to be made. Imposing the suspected selection effects of photographic photometry on the results of this survey gives excellent agreement with the results from that technique. This shows that the inability of photographic photometry to obtain results for many asteroids is indeed due to the rotational parameters of those asteroids.     

The rotation,color, phase coefficient,and diameter of 1915 quetzalcoatl

Photoelectric observations of 1915 Quetzalcoatl on March 2, 1981 show that this asteroid has a rotational period of 4.9 ± 0.3 hr and a lightcurve amplitude of 0.26 magnitudes. B-V and U-B colors are found to be 0.83 ± 0.04 and 0.43 ± 0.03, respectively, consistent with Quetzalcoatl being an S-type asteroid. Additional observations from March 31, 1981 give a linear phase coefficient of 0.033 mag deg^?1 and a mean B(1,0) magnitude of 20.10. The resulting estimated mean diameter for Quetzalcoatl is only 0.37 km, making it one of the smallest asteroids for which physical observations have yet been made.     

Asteroid rotation: I. Tabulation and analysis of rates,pole positions and shapes

Data are presented for the 182 asteroids whose rotational properties are available in the literature. Plots are provided for the asteroid rotational frequency f and lightcurve amplitude Δm versus asteroid size; the latter is determined using standard methods if data are available but otherwise is estimated from asteroid albedos, selected depending on taxonomic type or orbital position. A linear least-squares fit to all the data shows that f increases with decreasing size, confirming McAdoo and Burns' (1973) result; this is demonstrated to be primarily caused by relatively more small non-C than C asteroids in our sample, coupled with a slower mean rotation rate for C asteroids (P ≈ 11 hr) than non-C asteroids (P ≈ 9 hr). In terms of the collisional theory of Harris (1979a), this means that the C's are less dense than the other minor planets. Any slight tendency for smaller asteroids to spin faster, even within a taxonomic type, could be due to selection effects; our data are not extensive enough to determine whether the very smallest (? 10-km diameter) spin especially fast. The minor planets of our survey become more irregular at smaller sizes, disputing the conclusions of Bowell (1977b), Degewij (1977), and Degewij et al. (1978), based on other, perhaps more complete, data; selection effects may account for this disagreement. Shapes do not appear to depend on taxonomic type. The dispersion of asteroid rotation rates from the mean is found to be in excellent agreement with a three-dimensional Maxwellian distribution, such as would be developed in a collisionally evolved system. The rotation axes, therefore, appear to be randomly oriented in space. Rotation pole positions are also tabulated and calculated to likely be constant in space over the extent of past observation. Observers are encouraged to measure the rotational properties of faint objects and asteroids of unusual taxonomic types, and to carry out long-time studies of asteroids which over short periods do not seem to vary.     

Triaxial equilibrium ellipsoids among the asteroids?

We present a physical model to explain the existence of a class of large-lightcurve-amplitude, rapidly rotating asteroids found most commonly among objects in the size range 100–300 km diameter. A significant correlation between rotation period and lightcurve amplitude exists for asteroids in this size range in the sense that those with larger amplitudes spin more rapidly and hence these objects have high rotational angular momenta. Since this is a property of Jacobi ellipsoids, we have investigated whether these asteriods might be examples of triaxial equilibrium ellipsoids. We find that objects rotating with periods of 6 hr must have densities between 1.1 and 1.4 g cm^?3, while those rotating in 4 hr would have densities between 2.4 and 3.2 g cm^?3. If this model is valid then at least some of these asteroids have rather low mean densities. The reality of this result and its interpretation in terms of collisional evolution of the asteroids is discussed.     

The angular momenta of solar system bodies: Implications for asteroid strengths

The angular momentum H is plotted versus mass M for the planets and for all asteroids with known rotation rates and shapes, primarily taken from D. C. McAdoo and J. A. Burns [Icarus18, 285–293 (1973)]. An asteroid's angular momentum is derived from its rotation rate as determined by the period of its lightcurve, its shape as indicated by the lightcurve amplitude, and where possible its size as given by polarimetry or radiometry. The asteroid is assumed to be rotating about its axis of maximum moment of inertia. As previously found by F. F. Fish [Icarus7, 251–256 (1967]) and W. K. Hartmann and S. M. Larson [Icarus7, 257–260 (1967)], H is approximately proportional to $\text{M}{}^{\mathrm{<mtext>5</mtext><mtext>3</mtext>}}$, which shows that the asteroids and most planets spin with nearly the same rate. The very smallest asteroids on the plot deviate from the above reaction, usually containing excess angular momentum. This suggests that collisions have transferred substantial angular momentum to the smallest asteroids, perhaps causing their internal stress states to be substantially modified by centrifugal effects.The forces produced by gravitation are then compared to centrifugal effects for a rotating, triaxial ellipsoid of density 3 g cm^?3. For all asteroids with known properties the gravitational attraction is shown to be larger than the centrifugal acceleration of a particle on the surface: thus the observed asteroid regoliths are gravitationally bound. Poisson's equation for the gravitational potential is investigated and it is shown by mathematical and physical arguments that any arbitrarily shaped ellipsoid with the attractive surface force boundary condition found above will have only attractive internal forces. Thus the internal stress states in asteroids are always compressive so that asteroids could be internally fractured without losing their integrity.     

The asteroids as outcomes of catastrophic collisions

The role of catastrophic collisions in the evolution of the asteroids is discussed in detail, employing extrapolations of experimental results on the outcomrs of high-velocity impacts. We determine the range of the probable largest collision for target asteroids of different sizes during the solar system's lifetime, and we conclude that all the asteroids have undergone collisional events capable of overcoming the material's solid-state cohesion. Such events do not lead inescapably to complete disruption of the targets, because (i) for a previously unfractured target, experiments show that fragments of significant size can survive breakup, depending on the energy and geometry of the collision; (ii) self-gravitation can easily cause a reaccumulation of fragments for targets exceeding a critical size, which seems to be of the order of 100 km. In the intermediate diameter range 100?D ?300 km, where formation of gravitationally bound “rubble piles” is frequent, the transfer of angular momentum can be large enough to produce objects with triaxial equilibrium shapes (Jacobi ellipsoids) or to cause fission into binary systems. In the same size range, low-velocity escape of collisional fragments can also occur, leading to the formation of dynamical families. Asteroids smaller than ～100 km are mostly multigeneration fragments, while for $\text{D?300}\text{km}$ the collisional process produces nearly spheroidal objects covered by megaregoliths; whether their rotation is “primordial” or collisionally generated depends critically on the past flux of colliders. The complex and size-dependent phenomenology predicted by the theory compares satisfactorily with the observational evidence, as derived both by a classification of asteroids in terms of their size, spin rate, and lightcurve amplitude, and by a comparison between the rotational properties of family and nonfamily asteroids. The fundamental result of this investigation is that almost all asteroids are outcomes of catastrophic collisions, and that these events cause either complete fragmentation of the target bodies or, at least, drastic readjustments of their internal structure, shape, and spin rate.     

Asteroidal binary systems: Detection and formation

Recent occultation data and an analysis of some photometric lightcurves have shown the possible existence of asteroidal binary systems.A simple geometrical model taking into account mutual shadowing effects shows some peculiar features of the lightcurve which can be recovered in several previously observed objects; therefore the hypothesis of a relatively high frequency of binary asteroids should be seriously considered.On the other hand, while the rotational period distribution of large asteroids (D>200 km) is sharply peaked at about 5–8 hours, the surprisingly higher dispersion towards longer periods for intermediate size objects (50<D<150 km) could be connected with a larger probability of binary nature within this class.From a theoretical point of view, the collisional fragmentation of asteroids could originate gravitationally bound fragments, with a tidal transfer of rotational into orbital angular momentum, causing a rapid synchronization of the system. This kind of processes could more easily occur for intermediate objects since: (a) for large ones, very massive colliding bodies are needed for fragmentation, that means a very rare event; (b) for smaller asteroids, solid state interactions are stronger than the gravitational ones, so that a breakage probably causes a complete disruption of the gravitational binding. Further collisional events could disintegrate some systems, so that the present frequency of binary asteroids could be lower than that of the objects whose rotational period was increased by such processes.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

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.     

The lightcurve and rotation period of asteroid 139 Juewa

Results of broad-band photoelectric photometry of 139 Juewa during 5 consecutive nights in March 1974 are presented. The synodic period found is 20.9 hr. A linear phase coefficient, β = 0.080 ± 0.004, is determined between phase angles of 0.9° to 1.5°. This value is similar to that for the lunar highlands and for three other asteroids (4 Vesta, 20 Massalia, 110 Lydia) at similar phase angles, indicating that these surfaces have comparable porosities. The composite lightcurve presented covers 80% of the rotational period; short timescale features in the lightcurve are seen which correspond to topography a few kilometers in size.     

A possible YORP effect on C and S Main Belt Asteroids

A rotating frequency analysis in a previous paper, showed that two samples of C and S-type asteroids belonging to the Main Belt, but not to any families, present two different values for the transition diameter to a Maxwellian distribution of the rotation frequency, respectively 48 and 33 km. In this paper, after a more detailed statistical analysis, aiming to verify that the result is physically relevant, we found a better estimate for the transition diameter, respectively D_C = 44 ± 2 km and D_S = 30 ± 1 km. The ratio between these estimated transition diameters, D_C/D_S = 1.5 ± 0.1, can be supported with the help of the YORP (Yarkovsky-O’Keefe-Radzievskii-Paddack) effect, although other physical causes cannot be completely ruled out.In this paper we have derived a simple scaling law for YORP which, taking into account the different average heliocentric distance, the bulk density, the albedo and the asteroid “asymmetry surface factor”, has enabled us to reasonably justify the ratio between the diameters transition of C-type and S-type asteroids. The same scaling law can be used to estimate a new ratio between the bulk densities of S and C asteroids samples (giving ρ_S/ρ_C ≈ 2.9 ± 0.3), and can explain why the asteroids near the transition diameter have about the same absolute magnitude. For C-type asteroids, using the found density ratio and other estimates of S-type density, it is also possible to estimate an average bulk density equal to 0.9 ± 0.1 g cm⁻³, a value compatible with icy composition. The suggested explanation for the difference of the transition diameters is a plausible hypothesis, consistent with the data, but it needs to be studied more in depth with further observations.     

Angular momentum drain: A mechanism for despinning asteroids

It is proposed that a new mechanism—angular momentum drain—helps account for the relatively slow rotation rates of intermediate-sized asteroids. Impact ejecta on a spinning body preferentially escape in the direction of rotation. This material systematically drains away spin angular momentum, leading to the counterintuitive result that collisions can reduce the spin of midsized objects. For an asteroid of mass M spinning at frequency ω, a mass loss δM correspond to an average decrease in rotation rate $\text{δω ≈}\text{ωδM}\text{M}$. A. W. Harris' (1979), Icarus40, 145–153) theory for the collisional evolution of asteroidal spins is significantly altered by inlusion of this effect. While the modified theory is still somewhat artificial, comparison of its predictions with the data of S. F. Dermott, A. W. Harris, and C. D. Murray (1984, Icarus57, 14–34) suggests that angular momentum drain is essential for understanding the statistics of asteroidal rotations.     

Coordinated observations of asteroids 1219 Britta and 1972 Yi Xing

Identified as possible flyby targets for the Galileo spacecraft, Asteroids 1219 Britta and 1972 Yi Xing became the focus of a coordinated observing program. Although a subsequent change in the launch date removed these asteroids from consideration for the Galileo mission, the ground-based observing program yielded a substantial amount of information on these previously unobserved asteroids. Britta's sideral rotation period is found to be 5.57497 ± 0.00013 hr and its rotation is retrograde. The lightcurve amplitude ranged from 0.60 to 0.70 mag, depending on phase angle. Britta can be classified as an S-type asteroids based on its measured spectra and albedo. The absolute magnitude and slope parameter derived from the lightcurve maxima are H₀ = 11.67 ± 0.03 and G₀ = 0.03 ± 0.04. A 0.002 mag deg⁻¹ phase reddening in B·V was also measured. 1972 Yi Xing was less well observed but a unique synodic period of 14.183 ± 0.003 hr was determined. The observed lightcurve amplitude was 0.18 mag. Five-color measurements are consistent with an S-type classification. For an assumed slope parameter G = 0.25, Yi Xing's (lightcurve maximum) absolute magnitude H₀ = 13.32 ± 0.01.     

The Diagnosis of Complex Rotation in the Lightcurve of 4179 Toutatis and Potential Applications to Other Asteroids and Bare Cometary Nuclei

The analysis of radar observations of the asteroid 4179 Toutatis by Hudson and Ostro (1995, Science270, 84-86) yielded a complex spin state. We revisit the visible lightcurve data on Toutatis (Spencer et al. 1995, Icarus117, 71-89) to explore the feasibility of using a rotational lightcurve to recover the signature of an excited spin state. For this, we apply Fourier transform and CLEAN algorithm (WindowCLEAN). WindowCLEAN yields clear and precise frequency signatures associated with the precession of the long axis about the total angular momentum vector and a combination of this precession and rotation about the long axis. For a long-axis mode state, our periodicities for Toutatis yield a mean long-axis precession period, P_φ, of 7.38 days and a rotation period around the long axis, P_ψ, of 5.38 days, which compare well with the respective periods of 7.42 and 5.37 days derived by Ostro et al. (1999, Icarus137, 122-139) and represent an independent confirmation of these values. We explain why the dramatic change in the Earth-Toutatis-Sun geometry during the time that the lightcurve was obtained has little effect on the final results obtained. Using the Toutatis example as a guide, we discuss the capabilities as well as the limitations on deriving information about complex spin states from asteroidal lightcurves.     

1580 Betulia: An unusual asteroid with an extraordinary lightcurve

Results of UBV photometry and polarimetry of 1580 Betulia during its 1976 apparition are presented. The synodic period of rotation is found to be 6.130 hr. The linear phase coefficient and absolute magnitude of the primary maximum in V are 0.032 mag/deg and 14.88, respectively. No color variations with rotation or solar phase angle detected, the mean colors being B?V = 0.66 and U?B = 0.24. Betulia's lightcurve is unique among asteroids studied to date in that it displays three maxima and three minima within one rotational cycle, indicative of a region of greater roughness and/or a dark spot on one of its broad faces. Polarization results indicate a low albedo and a mean diameter of about 7 km, establishing Betulia as the first C type asteroid to be found among the Mars crossers. A model accounting for most features of Betulia's lightcurve is given by a prolate spheroid rotating about one of its shorter axes having an axis ratio of 1:1.21 with a major topographic feature on one of its broad faces.     

Asteroid rotation: IV. 1979 observations

Results of photoelectric lightcurve observations made during 1979 are reported. Of a total of 53 asteroids observed, reliable rotation periods are reported for 22 asteroids for which no previous values are known, 7 periods are reported which are revisions of previously reported values, and for 12 other asteroids periods are suggested which are admittedly of low reliability and those objects should be reobserved. In addition, phase relations are presented for many of the asteroids, fitted to the theoretical phase function of Lumme and Bowell (Astron. J., 86, 1705, 1981). Adopting their formalism, mean absolute magnitudes at zero phase angle, $\text{V}\text{(0°)}$, for 52 asteroids, and values of the multiple scattering parameter, Q, for 22 asteroids are reported. For comparison purposes, the absolute magnitude, V(1,0) and the linear phase coefficient, βv, in the traditional system are computed. In the appendixes (1) the methods of observation and data reduction are discussed, which are recommended to other lightcurve observers in the hope of standardizing reporting practices as much as possible; and (2) a cumulative index of all asteroid rotation data of which the authors are aware is presented.     

Photometric properties of outer planetary satellites

We present optical broadband photometry for the satellites J6, J7, J8, S7, S9, U3, U4, N1, and polarimetry for J6, obtained between 1970 and 1979. The outer Jovian satellites resemble C-type asteroids; J6 has a rotational lightcurve with period ～9.5 hr. The satellites beyond Jupiter also show C-like colors with the exception of S7 Hyperion. S9 Phoebe has a rotational lightcurve with period near either 11.25 or 21.1 hr. For U4 and N1 there is evidence for a lightcurve synchronous with the orbital revolution. The seven brighter Saturnian satellites show a regular relation between the ultraviolet dropoff and distance to the planet, probably related with differences in the rock component on their surfaces.