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.     

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.     

Photoelectric photometry of asteroids 9 Metis, 18 Melpomene, 60 Echo, 116 Sirona, 230 Athamantis, 694 Ekard,and 1984 KD

Photoelectric observations of seven asteroids were made from Gila Observatory between October 14, 1983, and June 21, 1984. The following synodic rotational periods and amplitudes are reported: 9 Metis, P = 5.04 hr, ΔM = 0.05; 18 Melpomene, P = 11.570 hr, ΔM = 0.22; 60 Echo, P = 25.208 hr, ΔM = 0.22; 116 Sirona, P = 12.028 hr, ΔM = 0.42; 230 Athamantis, P = 23.99 hr, ΔM > 0.20; 694 Ekard, P = 5.925 hr, ΔM = 0.50; 1984 KD, P = 1.97 hr, ΔM = 0.26. The rotational periods reported for asteroid 60 Echo, 116 Sirona, 694 Ekard, and 1984 KD represent completely new results. The synodic rotational period reported for asteroid 1984 KD is tentative and is based largely upon the observations of a single night. The reported synodic periods of the remaining six asteroids are based upon a minimum of 3 nights of photometric observations.     

Photoelectric lightcurves of asteroids 42 Isis, 45 Eugenia, 56 Melete, 103 Hera, 532 Herculina,and 558 Carmen

Photoelectric observations of six asteroids are presented. The following synodic periods of rotation and amplitudes of variation are reported: 42 Isis, P = 13^h.59, Δm = 0.32; 45 Eugenia, P = 5^h.70, Δm = 0.30; 56 Melete, P = 13^h.7 or 19^h.0, Δm = 0.06; 532 Herculina, P = 9^h.408, Δm = 0.15; 558 Carmen, P ≈ 10^h, Δm ≈ 0.25. The asteroid 103 Hera exhibited no periodic variation in excess of about 0.03 magnitude. The period found for 532 Herculina is one half that previously reported by other observers.     

Asteroid rotation rates: II. A theory for the collisional evolution of rotation rates

A model for the evolution of the mean rotation rate of asteroids arising from mutual collisions yields reasonable agreement with observed rotation rates. The mean rotation rate of large asteroids for which gravitational binding energy exceeds material strength should be constant with respect to size. Since collisional erosion of small asteroids is more rapid than collisional spin-up, the onset of increased mean rotation rate occurs at a considerably smaller radius than the size at which material strength begins to dominate gravitational binding energy. For strong igneous rock, increased rotation rates are not expected among bodies larger than a few kilometers. If there is a real trend toward more rapid rotation among asteroids of ≈1?km radius (Degewij and Gehrels, (1976). Bull. Amer. Astron. Soc.8, 459), then a substantial population of strong asteroids in that size range is implied by this model. The slower mean rotation rate of C-type asteroids than other types (paper I) implies a ratio of densities of ≈2:3 between those types, in the context of this model.     

Lightcurves and phase relations of the asteroids 82 alkmene and 444 Gyptis

The asteroids 82 Alkmene and 444 Gyptis were observed photoelectrically at Table Mountain Observatory and at Torino Observatory during their 1979 oppositions. The rotation periods and amplitudes of variation observed were, for 82: P_syn = 12.^h999, Δm = 0.55; and for 444: P_syn = 6.^h214, Δm = 0.15. The phase relation of 82 Alkmene can be well fit to the theory of K. Lumme and E. Bowell (Astron. J. (1981), 86, 1705). It showed a probable decrease in brightness of ～0.04 mag from 1 month before opposition to 2 months after opposition, which can be attributed to the changing viewing aspect coupled with polar flattering of the asteroid. The phase relation of 444 Gyptis is poorly fit by the Lumme and Bowell theory when only Q and V(0η) are treated as variables. A good fit can be obtained by adjusting some of the other parameters of their theory, but the physical interpretation is ambiguous.     

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.     

A possible rotation period for the minor planet 164 Eva

The minor planet 164 Eva passed through opposition on December 1, 1975 with a magnitude B_opp = 11.3 mag. Photoelectric observations at the Observatory of Torino, Italy, were carried out in two nights on Oct. 27/28 and Nov. 11, each with a run of about 3 hr. Two further successful photoelectric observations were carried out at the OHP, France, each with a run of about 6 hr. From all observed parts of the lightcurve a resulting synodic period of rotation of about 27.3 hr can be deduced, with a range of the total amplitude of at least Δm = 0.07 mag. With this period of 27.3 hr the minor planet 164 Eva is one more long period object, falling now between 654 Zelinda (H. J. Schober, 1975, Astron. Astrophys.44, 85–89) and 139 Juewa (J. Goguen et al., 1976, Icarus29, 137–142), at the high end in the histogram of the distribution of minor planet rotation periods.     

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.     

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.     

Zur Methodik der Bestimmung von Entdeckungswahrscheinlichkeiten veränderlicher Sterne

The Bernoullian statistics give for rare types of variable stars only inaccurate results concerning the probability of discovery p. In these cases p may be reasonably calculated by means of the „Quality Function”︁ Q(Δm, m), defined by Borgman. Q(Δm, m) is the probability of the discovery of a magnitude difference Δm near the apparent magnitude m. Three methods are discussed which allow to find Q(Δm, m). An application to the Sonneberg material shows, as expected, that Q(Δm, m) will become smaller with Δm decreasing and m increasing. Moreover the psychologically interesting fact, already supposed by Borgman, is confirmed that even well-trained observers will overlook about 10% of all large magnitude differences < 1 mag in using the blink-, stereo- or electronic scanning method for the discovery of variable stars.     

Mass dispersal and angular momentum transfer during collisions between rubble-pile asteroids

We perform N-body simulations of impacts between initially non-rotating rubble-pile asteroids, and investigate mass dispersal and angular momentum transfer during such collisions. We find that the fraction of the dispersed mass (Mdisp) is approximately proportional to , where Qimp is the impact kinetic energy; the power index α is about unity when the impactor is much smaller than the target, and 0.5?α<1 for impacts with a larger impactor. Mdisp is found to be smaller for more dissipative impacts with small values of the restitution coefficient of the constituent particles. We also find that the efficiency of transfer of orbital angular momentum to the rotation of the largest remnant depends on the degree of disruption. In the case of disruptive oblique impacts where the mass of the largest remnant is about half of the target mass, most of the orbital angular momentum is carried away by the escaping fragments and the efficiency becomes very low (<0.05), while the largest remnant acquires a significant amount of spin angular momentum in moderately disruptive impacts. These results suggest that collisions likely played an important role in rotational evolution of small asteroids, in addition to the recoil force of thermal re-radiation.     

Variation with interplanetary sector of the total magnetic field measured at the OGO 2, 4 and 6 satellites

Variations in the scalar magnetic field (ΔB) from the polar orbiting OGO 2, 4 and 6 spacecraft are examined as a function of altitude for times when the interplanetary magnetic field is toward the Sun and for times when the interplanetary magnetic field is away from the Sun. In both cases, at all altitudes, the total field variations form a region of positive ΔB between about 22 hr and 10 hr MLT and a region of negative ΔB between about 10 hr and 22 hr MLT. This morphology is basically the same as that found when all data, irrespective of inter-planetary magnetic sector, are averaged together (Langel, 1974a, b). Differences in ΔB occur, both between sectors and between seasons, which are similar in nature to variations in the surface ΔZ found by Langel (1973). The altitude variation of ΔB at sunlit local times, together with variations in the vertical component ΔZ at the Earth's surface, demonstrates that the ΔZ and ΔB which varies with sector has an ionospheric source. Langel (1974b) showed that the positive ΔB region in the dark portion of the hemisphere is due to at least two sources, the westward electrojet and an unidentified non-ionospheric source (s). Comparison of magnetic variations between season/sector at the surface and at the satellite, in the dark portion of the hemisphere, indicates that these variations are caused by variations in the latitudinally narrow electrojet currents and not by variations in the non-ionospheric source of ΔB.     

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.     

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.     

Photometric survey of binary near-Earth asteroids

Photometric data on 17 binary near-Earth asteroids (15 of them are certain detections, two are probables) were analysed and characteristic properties of the near-Earth asteroid (NEA) binary population were inferred. We have found that binary systems with a secondary-to-primary mean diameter ratio Ds/Dp?0.18 concentrate among NEAs smaller than 2 km in diameter; the abundance of such binaries decreases significantly among larger NEAs. Secondaries show an upper size limit of Ds=0.5-1 km. Systems with Ds/Dp?0.5 are abundant but larger satellites are significantly less common. Primaries have spheroidal shapes and they rotate rapidly, with periods concentrating between 2.2 to 2.8 h and with a tail of the distribution up to ∼4 h. The fast rotators are close to the critical spin for rubble piles with bulk densities about 2 g/cm³. Orbital periods show an apparent cut-off at Porb∼11 h; closer systems with shorter orbital periods have not been discovered, which is consistent with the Roche limit for strengthless bodies. Secondaries are more elongated on average than primaries. Most, but not all, of their rotations appear to be synchronized with the orbital motion; nonsynchronous secondary rotations may occur especially among wider systems with Porb>20 h. The specific total angular momentum of most of the binary systems is similar to within ±20% and close to the angular momentum of a sphere with the same total mass and density, rotating at the disruption limit; this suggests that the binaries were created by mechanism(s) related to rotation near the critical limit and that they neither gained nor lost significant amounts of angular momentum during or since formation. A comparison with six small asynchronous binaries detected in the main belt of asteroids suggests that the population extends beyond the region of terrestrial planets, but with characteristics shifted to larger sizes and longer periods. The estimated mean proportion of binaries with Ds/Dp?0.18 among NEAs larger than 0.3 km is 15±4%. Among fastest rotating NEAs larger than 0.3 km with periods between 2.2 and 2.8 h, the mean proportion of such binaries is (66⁺¹⁰₋₁₂)%.     

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.     

A lightcurve survey of asteroids with Schmidt telescopes: Observations of nine asteroids during oppositions in 1977

The results of photographic photometry of nine asteroids with the Schmidt telescope at the Uppsala Southern Station are presented. The results tend to confirm earlier results [Lagerkvist, (1978) Astron. Astrophys. Suppl.31, 361] that the smaller asteroids are collisional products. Rotation periods were determined for the asteroids 700, 1267, 1523, and 1562. Absolute magnitudes were determined for the asteroids 525, 700, 1207, 1267, and 1717.     

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.     

Asteroid rotation: III. 1978 Observations

Photoelectric observations of 32 asteroids observed from Table Mountain Observatory during the second half of 1978 are reported. Rotation periods were obtained for most objects. Absolute magnitudes and phase functions were not determined for any of these asteroids. The geometric mean rotation period of the 32 asteroids observed is 14.2 ± 1.6 hr, as compared to 9.38 ± 0.35 hr for 182 asteroids analyzed in Paper I (A. W. Harris and J. A. Burns, 1979, Icarus 40, 115–144). We attribute this difference to an observational selection effect which favors detection of fast rotation, as discussed in Paper I. If this is true, then the present sample contains the reverse bias, since it is complete in that a period (in some cases very approximate) was obtained for each object observed, but fast rotators are underrepresented due to prior discovery of their rotation properties.