Combined effect of YORP and collisions on the rotation rate of small Main Belt asteroids

The rotation rate distribution of small Main Belt asteroids is dominated by YORP and collisions. These mechanism act differently depending on the size of the bodies and give rise to non-linear effects when they both operate. Using a Monte Carlo method we model the formation of a steady state population of small asteroids under the influence of both mechanisms and the rotation rate distribution is compared to the observed one as derived from Pravec et al. (Pravec, P. et al. [2008]. Icarus 197, 497-504). A better match to observations is obtained with respect to the case in which only YORP is considered. In particular, an excess of slow rotators is produced in the model with both collisions and YORP because bodies driven to slow rotation by YORP have a random walk-like evolution of the spin induced by repeated collisions with small projectiles. This is a dynamical evolution different from tumbling and it lasts until a large impact takes the body to a faster rotation rate. According to our model, the rotational fission of small asteroids is a very frequent event and might explain objects like P/2010 A2 and its associated tail of millimeter-sized dust particles. The mass loss during fission of small asteroids might significantly influence the overall collisional evolution of the belt. Fission can in fact be considered as an additional erosion mechanism, besides cratering and fragmentation, acting only at small diameters.     

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.     

Zero secular torque on asteroids from impinging solar photons in the YORP effect: A simple proof

YORP torques, where “YORP” stands for “Yarokovsky-O’Keefe-Radzievskii-Paddack,” arise mainly from sunlight reflected off a Solar System object and the infrared radiation emitted by it. We show here, through the most elementary demonstration that we can devise, that secular torques from impinging solar photons are generally negligible and thus cause little secular evolution of an asteroid’s obliquity or spin rate.     

The effect of YORP on Itokawa

The effect of solar radiation on the near-term rotation rate of Asteroid Itokawa via the YORP effect is predicted using the detailed shape model, rotation pole, mass estimate, and optical properties derived from the Hayabusa mission to Itokawa. Based on these estimates Itokawa is decelerating at a rate which will halve its rotation rate in only 50-90 thousand years, a large deceleration that should be detectable in a future appartion. The implications of such a large deceleration for Itokawa's past history are discussed and related to possible seismic shaking.     

Generalized YORP evolution: Onset of tumbling and new asymptotic states

Asteroids have a wide range of rotation states. While the majority spin a few times to several times each day in principal axis rotation, a small number spin so slowly that they have somehow managed to enter into a tumbling rotation state. Here we investigate whether the Yarkovsky-Radzievskii-O'Keefe-Paddack (YORP) thermal radiation effect could have produced these unusual spin states. To do this, we developed a Lie-Poisson integrator of the orbital and rotational motion of a model asteroid. Solar torques, YORP, and internal energy dissipation were included in our model. Using this code, we found that YORP can no longer drive the spin rates of bodies toward values infinitely close to zero. Instead, bodies losing too much rotation angular momentum fall into chaotic tumbling rotation states where the spin axis wanders randomly for some interval of time. Eventually, our model asteroids reach rotation states that approach regular motion of the spin axis in the body frame. An analytical model designed to describe this behavior does a good job of predicting how and when the onset of tumbling motion should take place. The question of whether a given asteroid will fall into a tumbling rotation state depends on the efficiency of its internal energy dissipation and on the precise way YORP modifies the spin rates of small bodies.     

The dynamical evolution of uniformly rotating asteroids subject to YORP

A detailed derivation is given of the effect of solar radiation on the rotational dynamics of asteroids, commonly called the YORP effect. The current derivation goes beyond previous discussions published in the literature and provides a comprehensive secular dynamical analysis of the effect of solar radiation torques acting on a uniformly rotating body, and the evolution of its rotation state over time. Our predicted model has the global radiation properties of the asteroid as explicit parameters, and hence can be specified independent of these parameters. The resulting secular equations for the rotation rate and rotation pole are characterized by three parameters of the body's shape and explicitly includes the effect of thermal inertia on the evolution of these rotation state parameters. With this detailed model, in conjunction with estimated asteroid shapes and poles, we compute the expected YORP torques and dynamic response of several asteroids and the change in rotation rate for specific shapes as a function of obliquity. Finally, we define a convenient dimensionless parameter that is only a function of the body geometry and that can be used to characterize the effects of YORP.     

Computing the effects of YORP on the spin rate distribution of the NEO population

The overall change of NEO spin rate due to planetary encounters and YORP is evaluated by using a Monte Carlo model. A large sample of test objects mimicking a source population is evolved over a timescale comparable with the Solar System age until they reach a steady state spin distribution that should reproduce the current NEO distribution. The spin change due to YORP is computed for each body according to a simplified model based on Scheeres [Scheeres, D.J., 2007a. Icarus 188, 430-450].The steady state cumulative distribution of NEO spin rates obtained from our simulation nicely reproduces the observed one, once our results are biased to match the diameter distribution of the sample of objects included in the observational database. The excellent agreement strongly suggests that YORP is responsible for the concentration of spin at low rotation rates. In fact, in the absence of YORP the steady state population significantly deviates from the observed one. The spin evolution due to YORP is also so rapid for NEOs that the initial rotation rate distribution of any source population is quickly relaxed to that of the observed population. This has profound consequences for the study of NEO origin since we cannot trace the sources of NEOs from their rotation rate only.     

Extreme sensitivity of the YORP effect to small-scale topography

Radiation recoil (YORP) torques are shown to be extremely sensitive to small-scale surface topography, using numerical simulations. Starting from a set of “base objects” representative of the near-Earth object population, random realizations of three types of small-scale topography are added: Gaussian surface fluctuations, craters, and boulders. For each, the expected relative errors in the spin and obliquity components of the YORP torque caused by the observationally unresolved small-scale topography are computed. Gaussian power, at angular scales below an observational limit, produces expected errors of order 100% if observations constrain the surface to a spherical harmonic order l?10. For errors under 10%, the surface must be constrained to at least l=20. A single crater with diameter roughly half the object's mean radius, placed at random locations, results in expected errors of several tens of percent. The errors scale with crater diameter D as D² for D>0.3 and as D³ for D<0.3 mean radii. Objects that are identical except for the location of a single large crater can differ by factors of several in YORP torque, while being photometrically indistinguishable at the level of hundredths of a magnitude. Boulders placed randomly on identical base objects create torque errors roughly 3 times larger than do craters of the same diameter, with errors scaling as the square of the boulder diameter. A single boulder comparable to Yoshinodai on 25143 Itokawa, moved by as little as twice its own diameter, can alter the magnitude of the torque by factors of several, and change the sign of its spin component at all obliquities. Most of the total torque error produced by multiple unresolved craters is contributed by the handful of largest craters; but both large and small boulders contribute comparably to the total boulder-induced error. A YORP torque prediction derived from groundbased data can be expected to be in error by of order 100% due to unresolved topography. Small surface changes caused by slow spin-up or spin-down may have significant stochastic effects on the spin evolution of small bodies. For rotation periods between roughly 2 and 10 h, these unpredictable changes may reverse the sign of the YORP torque. Objects in this spin regime may random-walk up and down in spin rate before the rubble-pile limit is exceeded and fissioning or loss of surface objects occurs. Similar behavior may be expected at rotation rates approaching the limiting values for tensile-strength dominated objects.     

Orbital YORP and asteroid orbit evolution, with application to Apophis

Photon thrust from shape alone can produce quasi-secular changes in an asteroid's orbital elements. An asteroid in an elliptical orbit with a north–south shape asymmetry can steadily alter its elements over timescales longer than one orbital trip about the Sun. This thrust, called here orbital YORP (YORP = Yarkovsky–O'Keefe–Radzievskii–Paddack), operates even in the absence of thermal inertia, which the Yarkovsky effects require. However, unlike the Yarkovsky effects, which produce secular orbital changes over millions or billions of years, the change in an asteroid's orbital elements from orbital YORP operates only over the precession timescale of the orbit or of the asteroid's spin axis; this is generally only thousands or tens of thousands of years. Thus while the orbital YORP timescale is too short for an asteroid to secularly journey very far, it is long enough to warrant investigation with respect to 99942 Apophis, which might conceivably impact the Earth in 2036. A near-maximal orbital YORP effect is found by assuming Apophis is without thermal inertia and is shaped like a hemisphere, with its spin axis lying in the orbital plane. With these assumptions orbital YORP can change its along-track position by up to ±245 km, which is comparable to Yarkovsky effects. Though Apophis' shape, thermal properties, and spin axis orientation are currently unknown, the practical upper and lower limits are liable to be much less than the ±245 km extremes. Even so, the uncertainty in position is still likely to be much larger than the 0.5 km “keyhole” Apophis must pass through during its close approach in 2029 in order to strike the Earth in 2036.     

Yarkovsky/YORP chronology of asteroid families

Asteroid families are the byproducts of catastrophic collisions whose fragments form clusters in proper semimajor axis, eccentricity, and inclination space. Although many families have been observed in the main asteroid belt, only two very young families, Karin and Veritas, have well-determined ages. The ages of other families are needed, however, if we hope to infer information about their ejection velocity fields, space weathering processes, etc. In this paper, we developed a method that allows us to estimate the ages of moderately young asteroid families (approximately in between 0.1 and 1 Gyr). We apply it to four suitable cases—Erigone, Massalia, Merxia, and Astrid—and derive their likely ages and approximate ejection velocity fields. We find that Erigone and Merxia were produced by large catastrophic disruption events (i.e., parent body ?100 km) that occurred approximately 280 and 330 Myr ago, respectively. The Massalia family was likely produced by a cratering event on Asteroid (20) Massalia less than 200 Myr ago. Finally, the Astrid family, which was produced by the disruption of a 60-70 km asteroid, is 100-200 Myr old, though there is considerable uncertainty in this result. We estimate that the initial ejection velocities for these families were only a few tens of meters per second, consistent with numerical hydrocode models of asteroid impacts. Our results help to verify that asteroid families are constantly undergoing dynamical orbital evolution from thermal (Yarkovsky) forces and spin vector evolution from thermal (YORP) torques.     

Evolution of NEO rotation rates due to close encounters with Earth and Venus

In this paper we study the statistical effect of planetary flybys on the rotation rates and states of Near Earth Objects (NEOs). Our approach combines numerical and analytical methods within a Monte Carlo model that simulates the evolution of the NEO spin rates. We take as input for the simulation a source distribution of spin states and evolve it to find their steady state distribution. In performing this evolution we track the changes in the spin rate and state distribution for the different components of the NEO population. We show that the cumulative effect of planetary encounters is to spin up the overall population of NEOs. This spin up effect holds on average only, and particular members of the population may experience an overall decrease in rotation rate. This effect is clearly seen across all components of the NEO population and is significant both statistically and physically. For initially slow rotators the spin up effect is strong, lowering the mean rotation period by 32%. For faster rotating populations the effect is less, lowering the spin period by 15% for the intermediate case, 6% for fast rotating rubble piles, and 8% for fast rotating monoliths. Physically, the spin up effect pushes 1% of the fast rotating rubble-pile NEOs over the disruption limit, while 6% of these bodies experience a sub-disruption event that could modify their physical structure. For monolithic NEOs, the spin up effect is self-limiting, reaching a minimum spin period of 1.1 hr, with a strong cut-off between 2-3 hr. This has two implications. First, it may not be necessary to invoke the rubble-pile hypothesis to recover a cut-off in spin period. Second, it shows that planetary flybys cannot account for the extremely rapid rotation rates of some small NEOs. We also tested a different balance between the effects of Earth and Venus by treating the Aten sub-class of asteroids separately. Due to increased interactions with the planets, the spin up effect is more pronounced (10%) and disruptions increase by a factor of three. The slow rotation tails of the spin distributions are increased to longer periods, in general, with rotation periods of over 100 hr occurring for a few tenths of a percent for some component populations. Thus, this mechanism may account for some of the noted excess in slow rotators among the NEOs. Planetary flybys also cause NEOs to enter a tumbling state, with approximately 0.5% of the population being placed into a long-axis rotation mode. Finally, based on the evolution of spin states of different components of the NEO population, we compared the evolved states with the measured distribution of NEOs to estimate the relative populations of these components that comprise the NEOs.     

The YORP effect with finite thermal conductivity

The Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect has been recently suggested to significantly change, on a long-term, rotation state of small asteroids and meteoroids. Though YORP is closely related to the Yarkovsky (orbital) effect, it differs from the latter in two aspects: (i) YORP needs bodies of irregular shape to be effective, and (ii) YORP acts on bodies of zero surface thermal conductivity. To simplify computations, YORP has been so far investigated in the zero surface thermal conductivity limit only. Here we analyze the role of the surface conductivity and we find it substantially changes previous conclusions. Most importantly, unlike in the zero-conductivity limit, (i) YORP preferentially tilts obliquity toward two asymptotic states perpendicular to the orbital plane, and (ii) YORP asymptotically decelerates and accelerates rotation rate in about equal number of cases. Our work also indicates that direct detection of the YORP effect for a small asteroid may significantly constrain its mass.     

Abrupt alteration of Asteroid 2004 MN4's spin state during its 2029 Earth flyby

We predict that when Asteroid 2004 MN4 passes 5.6±1.4 Earth radii from Earth's center on April 13, 2029, terrestrial torques during the flyby will alter its spin state in a dramatic manner that will be observable using groundbased telescopes. Although the asteroid will most likely not undergo catastrophic disruption, it may be subject to localized failure across its surface and interior, providing a unique opportunity to measure otherwise inaccessible mechanical properties of an asteroid.     

On the shapes and spins of “rubble pile” asteroids

We examine the shape of a “rubble pile” asteroid as it slowly gains angular momentum by YORP torque, to the point where “landsliding” occurs. We find that it evolves to a “top” shape with constant angle of repose from the equator up to mid-latitude, closely resembling the shapes of several nearly critically spinning asteroids imaged by radar, most notably (66391) 1999 KW4 [Ostro, S.J., Margot, J.-L., Benner, L.A.M., Giorgini, J.D., Scheeres, D.J., Fahnestock, E.G., Broschart, S.B., Bellerose, J., Nolan, M.C., Magri, C., Pravec, P., Scheirich, P., Rose, R., Jurgens, R.F., De Jong, E.M., Suzuki, S., 2006. Science 314, 1276-1280]. Similar calculations for non-spinning extremely prolate or oblate “rubble piles” show that even loose rubble can sustain shapes far from fluid equilibrium, thus inferences based on fluid equilibrium are generally useless for inferring bulk properties such as density of small bodies. We also investigate the tidal effects of a binary system with a “top shape” primary spinning at near the critical limit for stability. We find that very close to the stability limit, the tide from the secondary can actually levitate loose debris from the surface and re-deposit it, in a process we call “tidal saltation.” In the process, angular momentum is transferred from the primary spin to the satellite orbit, thus maintaining the equilibrium of near-critical spin as YORP continues to add angular momentum to the system. We note that this process is in fact dynamically related to the process of “shepherding” of narrow rings by neighboring satellites.     

Secular spin dynamics of inner main-belt asteroids

Understanding the evolution of asteroid spin states is challenging work, in part because asteroids have a variety of orbits, shapes, spin states, and collisional histories but also because they are strongly influenced by gravitational and non-gravitational (YORP) torques. Using efficient numerical models designed to investigate asteroid orbit and spin dynamics, we study here how several individual asteroids have had their spin states modified over time in response to these torques (i.e., 951 Gaspra, 60 Echo, 32 Pomona, 230 Athamantis, 105 Artemis). These test cases which sample semimajor axis and inclination space in the inner main belt, were chosen as probes into the large parameter space described above. The ultimate goal is to use these data to statistically characterize how all asteroids in the main belt population have reached their present-day spin states. We found that the spin dynamics of prograde-rotating asteroids in the inner main belt is generally less regular than that of the retrograde-rotating ones because of numerous overlapping secular spin-orbit resonances. These resonances strongly affect the spin histories of all bodies, while those of small asteroids (?40 km) are additionally influenced by YORP torques. In most cases, gravitational and non-gravitational torques cause asteroid spin axis orientations to vary widely over short (?1 My) timescales. Our results show that (951) Gaspra has a highly chaotic rotation state induced by an overlap of the s and s₆ spin-orbit resonances. This hinders our ability to investigate its past evolution and infer whether thermal torques have acted on Gaspra's spin axis since its origin.     

The spin state of 433 Eros and its possible implications

In this paper, we show that Asteroid (433) Eros is currently residing in a spin-orbit resonance, with its spin axis undergoing a small-amplitude libration about the Cassini state 2 of the proper mode in the nonsingular orbital element sinI/2exp(?Ω), where I the orbital inclination and Ω the longitude of the node. The period of this libration is ?53.4 kyr. By excluding these libration wiggles, we find that Eros' pole precesses with the proper orbital plane in inertial space with a period of ?61.4 kyr. Eros' resonant state forces its obliquity to oscillate with a period of ?53.4 kyr between ?76° and ?89.5°. The observed value of ?89° places it near the latter extreme of this cycle. We have used these results to probe Eros' past orbit and spin evolution. Our computations suggest that Eros is unlikely to have achieved its current spin state by solar and planetary gravitational perturbations alone. We hypothesize that some dissipative process such as thermal torques (e.g., the so-called YORP effect) may be needed in our model to obtain a more satisfactory match with data. A detailed study of this problem is left for future work.     

Radar observations of E-class Asteroids 44 Nysa and 434 Hungaria

We observed the E-class main-belt Asteroids (MBAs) 44 Nysa and 434 Hungaria with Arecibo Observatory's S-band (12.6 cm) radar. Both asteroids exhibit polarization ratios higher than those measured for any other MBA: Nysa, μc=0.50±0.02 and Hungaria, μc=0.8±0.1. This is consistent with the high polarization ratios measured for every E-class near-Earth asteroid (NEA) observed by Benner et al. [Benner, L.A.M., and 10 collegues, 2008. Icarus, submitted for publication] and suggests a common cause. Our estimates of radar albedo are 0.19±0.06 for Nysa and 0.22±0.06 for Hungaria. These values are higher than those of most MBAs and, when combined with their high polarization ratios, suggest that the surface bulk density of both asteroids is high. We model Nysa as an ellipsoid of dimension 113×67×65 km (±15%) giving an effective diameter Deff=79±10 km, consistent with previous estimates. The echo waveforms are not consistent with a contact binary as suggested by Kaasalainen et al. [Kaasalainen, M., Torppa, J., Piironen, J., 2002. Astron. Astrophys. 383, L19-L22]. We place a constraint on Hungaria's maximum diameter, Dmax?11 km consistent with previous size estimates.     

A spectroscopic comparison of HED meteorites and V-type asteroids in the inner Main Belt

V-type asteroids in the inner Main Belt (a < 2.5 AU) and the HED meteorites are thought to be genetically related to one another as collisional fragments from the surface of the large basaltic Asteroid 4 Vesta. We investigate this relationship by comparing the near-infrared (0.7-2.5 μm) spectra of 39 V-type asteroids to laboratory spectra of HED meteorites. The central wavelengths and areas spanned by the 1 and 2 μm pyroxene-olivine absorption bands that are characteristic of planetary basalts are measured for both the asteroidal and meteoritic data. The band centers are shown to be well correlated, however the ratio of areas spanned by the 1 and 2 μm absorption bands are much larger for the asteroids than for the meteorites. We argue that this offset in band area ratio is consistent with our currently limited understanding of the effects of space weathering, however we cannot rule out the possibility that this offset is due to compositional differences. Several other possible causes of this offset are discussed.Amongst these inner Main Belt asteroids we do not find evidence for non-Vestoid mineralogies. Instead, these asteroids seem to represent a continuum of compositions, consistent with an origin from a single differentiated parent body. In addition, our analysis shows that V-type asteroids with low inclinations (i < 6°) tend to have band centers slightly shifted towards long wavelengths. This may imply that more than one collision on Vesta’s surface was responsible for producing the observed population of inner belt V-type asteroids. Finally, we offer several predictions that can be tested when the Dawn spacecraft enters into orbit around Vesta in the summer of 2011.     

The distribution of basaltic asteroids in the Main Belt

We present the observational results of a survey designed to target and detect asteroids whose photometric colors are similar to those of Vesta family members and thus may be considered as candidates for having a basaltic composition. Fifty basaltic candidates were selected with orbital elements that lie outside of the Vesta dynamical family. Optical and near-infrared spectra were used to assign a taxonomic type to 11 of the 50 candidates. Ten of these were spectroscopically confirmed as V-type asteroids, suggesting that most of the candidates are basaltic and can be used to constrain the distribution of basaltic material in the Main Belt. Using our catalog of V-type candidates and the success rate of the survey, we calculate unbiased size-frequency and semi-major axis distributions of V-type asteroids. These distributions, in addition to an estimate for the total mass of basaltic material, suggest that Vesta was the predominant contributor to the basaltic asteroid inventory of the Main Belt, however scattered planetesimals from the inner Solar System (a<2.0 AU) and other partially/fully differentiated bodies likely contributed to this inventory. In particular, we infer the presence of basaltic fragments in the vicinity of Asteroid 15 Eunomia, which may be derived from a differentiated parent body in the middle Main Belt (2.5<a<2.8). We find no asteroidal evidence for a large number of previously undiscovered basaltic asteroids, which agrees with previous theories suggesting that basaltic fragments from the ∼100 differentiated parent bodies represented in meteorite collections have been “battered to bits” [Burbine, T.H., Meibom, A., Binzel, R.P., 1996. Meteorit. Planet. Sci. 31, 607-620].