Experimental study on the velocity of fragments in collisional breakup

The motion of fragments following a catastrophic destruction by either a normal or an oblique impact at 2.5–2.9 km sec^?1 into cubic and spherical basalt targets was studied with a high-speed framing camera. Velocities at the antipodes of the targets vary as (E/M)^0.75 (E = impact energy; M = target mass) and are lower than 200 m sec^?1 at E/M ? 10⁹ ergs g^?1. Excluding fine-grained particles from the impact site, 70 to 80% by mass fraction of the fragments have velocities lower than twice the antipodal velocity. Comminution and ejection energies wasted in this mass fraction were a few percent of the impact energy at E/M ? 5 × 10⁷ ergs g^?1. During a catastrophic impact into asteroids some of the fragmented bodies can be reconcentrated by mutual gravitation.     

A Search for the Collisional Parent Bodies of Large NEAs

A search for the most likely parent bodies of multi-km near-Earth asteroids (NEAs) is attempted, in the framework of a scenario based on a few simple assumptions. (1) Multi-km NEAs are produced by collisional fragmentation of single parent bodies. (2) The fragments are injected into either the 3/1 mean-motion resonance with Jupiter or the ν₆ secular resonance, or they achieve Mars-crossing orbits. (3) The collisional events responsible for the production of multi-km NEAs do not produce observable dynamical families. We show that a limited number of potential parent bodies of multi-km NEAs compatible with the above assumptions do exist in the asteroid Main Belt. It is not clear whether these objects can likely explain the current inventory of known NEAs having sizes around 1-2 km. Our results seem to indicate that the assumed scenario is not completely adequate to justify the number of observed NEAs larger than 2 km. This preliminary analysis must be complemented by a more precise analysis of the rates of occurrence of NEA-feeding events. If present results are confirmed, the conclusion that the origin of multi-km NEAs must be explained by different models, based on long-term dynamical diffusion produced by the interplay of collisional, gravitational, and nongravitational mechanisms in the Main Belt, plus a possible cometary contribution, will be strengthened.     

Orbital evolution of the Gefion and Adeona asteroid families: close encounters with massive asteroids and the Yarkovsky effect

Asteroid families are groupings of minor planets identified by clustering in their proper orbital elements; these objects have spectral signatures consistent with an origin in the break-up of a common parent body. From the current values of proper semimajor axes a of family members one might hope to estimate the ejection velocities with which the fragments left the putative break-up event (assuming that the pieces were ejected isotropically). However, the ejection velocities so inferred are consistently higher than N-body and hydro-code simulations, as well as laboratory experiments, suggest. To explain this discrepancy between today’s orbital distribution of asteroid family members and their supposed launch velocities, we study whether asteroid family members might have been ejected from the collision at low speeds and then slowly drifted to their current positions, via one or more dynamical processes. Studies show that the proper a of asteroid family members can be altered by two mechanisms: (i) close encounters with massive asteroids, and (ii) the Yarkovsky non-gravitational effect. Because the Yarkovsky effect for kilometer-sized bodies decreases with asteroid diameter D, it is unlikely to have appreciably moved large asteroids (say those with D > 15 km) over the typical family age (1-2 Gyr).For this reason, we numerically studied the mobility of family members produced by close encounters with main-belt, non-family asteroids that were thought massive enough to significantly change their orbits over long timescales. Our goal was to learn the degree to which perturbations might modify the proper a values of all family members, including those too large to be influenced by the Yarkovsky effect. Our initial simulations demonstrated immediately that very few asteroids were massive enough to significantly alter relative orbits among family members. Thus, to maximize gravitational perturbations in our 500-Myr integrations, we investigated the effect of close encounters on two families, Gefion and Adeona, that have high encounter probabilities with 1 Ceres, by far the largest asteroid in the main belt. Our results show that members of these families spreads in a of less than 5% since their formation. Thus gravitational interactions cannot account for the large inferred escape velocities.The effect of close encounters with massive asteroids is, however, not entirely negligible. For about 10% of the simulated bodies, close encounters increased the “inferred” ejection velocities from sub-100 m/s to values greater than 100 m/s, beyond what hydro-code and N-body simulations suggest are the maximum possible initial ejection velocity for members of Adeona and Gefion with D > 15 km. Thus this mechanism of mobility may be responsible for the unusually high inferred ejection speeds of a few of the largest members of these two families.To understand the orbital evolution of the entire family, including smaller members, we also performed simulations to account for the drift of smaller asteroids caused by the Yarkovsky effect. Our two sets of simulations suggest that the two families we investigated are relatively young compared to larger families like Koronis and Themis, which have estimated ages of about 2 Byr. The Adeona and Gefion families seems to be no more than 600 and 850 Myr old, respectively.     

The stars in the Main Sequence and their dynamical parameters

The stars in the Main Sequence are seen as a hierarchy of objects with different massesM and effective dynamical radiiR _eff=R/α given by the stellar radii and the coefficients for the inner structure of the stars. As seen in a previous work (Paper I), during the lifetime in the Main SequenceR _eff(t) remains a near invariant when compared to the variation in the time ofR(t) and α(t). With such an effectiveR _eff one obtains the amounts of actionA _c(M), the effective densities ρ_eff(M)=ρ(M)α³(M), the densities of action and of energy (or mean presures in the stellar interior)a _c(M),e _c(M), and the potential energiesE _p(M). The amounts of action areA _c∝M ^k withk≈1.87 for the M stars,k≈5/3 for the KGF stars, andk≈1.83 for the A and earlier stars, representing very simples conditions for the other dynamical parameters. For instancek≈5/3 means a near invariant effective density α_eff for the KGF stars, while for such stars the mean densities and coefficients α present the strongest variations with masses ρ(M)∝M ^?1.81, α(M)∝M^0.6. The cases for the M stars (e _c(M)∝M ^?1) and for the A and earlier stars (betweena _c(M)=constant and α_eff(M)∝M ^?1) and also discussed. These conditions for the earlier stars also represent reasonable mean values for the whole stellar hierarchy in the range of masses 0.2M _⊙≤M≤25M _⊙. With all this, one can build ‘dynamical’ HR diagrams withA _c(M), E_p(M), α_eff M ^?p , etc., whose characteristics are analogous to these in the photometrical HR diagram. A comparison is made betweenA _c(M) from the models here and the HR diagram with the best known stars of luminosity classes IV, V, and white dwarfs. The comparison of the potential energiesE _p(M)∝M ^?p according to the stellar models used here and the observed frequency function ψ(M∝M _?q (number of stars in a given interval of masses) from different authors suggests the possibility that the productE _p(M)ψ(M) is a constant, but this must be confirmed with further studies of the function ψ(M) and its fine structure. There are analogies between the formulation used here for the stellar hierarchy and other physical processes, for instance, in modified forms of the Kolmogorov law of turbulence and in the formulation used for the hierarchy of molecular clouds in gravitational equilibrium. Besides, the function of actionA _c(M) for the stars has analogous properties to the relations of angular momenta and massesJ(M) for different types of objects. The cosmological implications of all this are discussed.     

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).     

Experimental study on the collisional disruption of porous gypsum spheres

Abstract— In order to study the catastrophic disruption of porous bodies such as asteroids and planetesimals, we conducted several impact experiments using porous gypsum spheres (porosity: 50%). We investigated the fragment mass and velocity of disrupted gypsum spheres over a wide range of specific energies from 3 times 10³ J/kg to 5 times 10⁴ J/kg. We compared the largest fragment mass (m₁/M_t) and the antipodal velocity (V_a) of gypsum with those of non‐porous materials such as basalt and ice. The results showed that the impact strength of gypsum was notably higher than that of the non‐porous bodies; however, the fragment velocity of gypsum was slower than that of the non‐porous bodies. This was because the micro‐pores dispersed in the gypsum spheres caused a rapid attenuation of shock pressure in them. From these results, we expect that the collisional disruption of porous bodies could be significantly different from that of non‐porous bodies.     

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.     

Volume and mass distribution in selected asteroid families

The main focus of this paper is calculation of the diameters of asteroids belonging to five families (Vesta, Eos, Eunomia, Koronis, and Themis). To do that, we used the HCM algorithm applied for a data set containing 292,003 numbered asteroids, and a numerical procedure for choosing the crucial parameter of the HCM, called “the cutting velocity” v_cut. It was established with a precision as high as 1 m s^?1. Thereafter, we used the WISE (Wide‐field Infrared Survey Explorer) catalog to set a range of albedo for the largest members of each family considered. The albedo data were supported by the data concerning color classification (SDSS MOC4). The asteroids with albedo out of this range were classified as interlopers and were therefore disqualified as family members. Sizes were calculated for the asteroids with albedo within the acceptable range. For the other asteroids (those chosen by means of the HCM, but with albedo not listed in the WISE), the value of albedo of the largest member of the family was adopted. Results are given in a set of figures showing the families on the planes (a, e), (a, i), (e, i). Diameters and volumes of the asteroids that are the individual members of a family were calculated on the basis of their known or assumed albedo and on their absolute magnitude. Volumes of the parent bodies of the families were found on the basis of the cumulative volume distribution of these families. We also studied the secular resonances of the family members. We have shown that the locations of members of the considered asteroid families are related to the lines of secular resonances z₁, z₂, and z₃ with Saturn.     

Grooves on asteroids: A prediction

The characteristics of the grooves on Phobos indicate that these features were formed by a large impact—an observation which implies that grooves may occur on other small bodies. Published data on fragmentation experiments suggest that the energy of the impact which produced the grooves on Phobos was in a critical range, strong enough to produce fracturing at large distances from the crater, but too weak to fragment the object. Other small bodies, such as asteroids, which have experienced an impact flux sufficient to include one such critical cratering event should show grooves, unless they have been fragmented subsequently by a more severe impact. We estimate that between $\text{1}\text{12}\text{th}\text{and}\text{1}\text{4}\text{th}$ of asteroids smaller than 100 km in diameter may have surface features resembling grooves associated with large impact craters. Several morphologically distinct types of grooves can be expected, depending on the thickness and composition of the regolith of the parent body. The full development of Phobos-like grooves seems to require a thick, and possibly volatile-rich, regolith—conditions which might obtain on certain C-asteroids.     

Energy partition into translational and rotational motion of fragments in catastrophic disruption by impact: An experiment and asteroid cases

Translational kinetic energy E_t and rotational kinetic energy E_r of the fragments produced in the laboratory catastrophic disruption of a basalt sphere by the impact of a high-speed projectile were determined. It was found that the maximum value of E_t/E_t is of the orders of 10⁻², which is not so different in the order of magnitude from the value estimated for family asteroids in spite of the great difference of the scale. However, the laboratory value is significantly higher than the value for the family.     

Formation of Asteroid Families by Catastrophic Disruption: Simulations with Fragmentation and Gravitational Reaccumulation

This paper builds on preliminary work in which numerical simulations of the collisional disruption of large asteroids (represented by the Eunomia and Koronis family parent bodies) were performed and which accounted not only for the fragmentation of the solid body through crack propagation, but also for the mutual gravitational interaction of the resulting fragments. It was found that the parent body is first completely shattered at the end of the fragmentation phase, and then subsequent gravitational reaccumulations lead to the formation of an entire family of large and small objects with dynamical properties similar to those of the parent body. In this work, we present new and improved numerical simulations in detail. As before, we use the same numerical procedure, i.e., a 3D SPH hydrocode to compute the fragmentation phase and the parallel N-body code pkdgrav to compute the subsequent gravitational reaccumulation phase. However, this reaccumulation phase is now treated more realistically by using a merging criterion based on energy and angular momentum and by allowing dissipation to occur during fragment collisions. We also extend our previous studies to the as yet unexplored intermediate impact energy regime (represented by the Flora family formation) for which the largest fragment's mass is about half that of the parent body. Finally, we examine the robustness of the results by changing various assumptions, the numerical resolution, and different numerical parameters. We find that in the lowest impact energy regime the more realistic physical approach of reaccumulation leads to results that are statistically identical to those obtained with our previous simplistic approach. Some quantitative changes arise only as the impact energy increases such that higher relative velocities are reached during fragment collisions, but they do not modify the global outcome qualitatively. As a consequence, these new simulations confirm previous main results and still lead to the conclusion that: (1) all large family members must be made of gravitationally reaccumulated fragments; (2) the original fragment size distribution and their orbital dispersion are respectively steeper and smaller than currently observed for the real families, supporting recent studies on subsequent evolution and diffusion of family members; and (3) the formation of satellites around family members is a frequent and natural outcome of collisional processes.     

From Magnitudes to Diameters: The Albedo Distribution of Near Earth Objects and the Earth Collision Hazard

A recently published model of the Near Earth Object (NEO) orbital-magnitude distribution (Bottke et al., 2002, Icarus156, 399-433.) relies on five intermediate sources for the NEO population: the ν₆ resonance, the 3:1 resonance, the outer portion of the main belt (i.e., 2.8-3.5 AU), the Mars-crossing population adjacent to the main belt, and the Jupiter family comet population. The model establishes the relative contribution of these sources to the NEO population. By computing the albedo distribution of the bodies in and/or near each of the five sources, we can deduce the albedo distribution of the NEO population as a function of semimajor axis, eccentricity, and inclination. A problem with this strategy, however, is that we do not know a priori the albedo distribution of main belt asteroids over the same size range as observed NEOs (diameter D<10 km). To overcome this problem, we determined the albedo distribution of large asteroids in and/or near each NEO source region and used these results to deduce the albedo distribution of smaller asteroids in the same regions. This method requires that we make some assumptions about the absolute magnitude distributions of both asteroid families and background asteroids. Our solution was to extrapolate the observed absolute magnitude distributions of the families up to some threshold value H_thr, beyond which we assumed that the families' absolute magnitude distributions were background-like.We found that H_thr=14.5 provides the best match to the color vs heliocentric distance distribution observed by the Sloan Digital Sky Survey. With this value of H_thr our model predicts that the debiased ratio between dark and bright (albedo smaller or larger than 0.089) objects in any absolute-magnitude-limited sample of the NEO population is 0.25±0.02. Once the observational biases are properly taken into account, this agrees very well with the observed C/S ratio (0.165 for H<20). The dark/bright ratio of NEOs increases to 0.87±0.05 if a size-limited sample is considered. We estimate that the total number of NEOs larger than a kilometer is 855±110, which, compared to the total number of NEOs with H<18 (963±120), shows that the usually assumed conversion H=18?D=1 km slightly overestimates the number of kilometer-size objects.Combining our orbital distribution model with the new albedo distribution model, and assuming that the density of bright and dark bodies is 2.7 and 1.3 g/cm³, respectively, we estimate that the Earth should undergo a 1000 megaton collision every 63,000±8000 years. On average, the bodies capable of producing 1000 megaton of impact energy are those with H<20.6. The NEOs discovered so far carry only 18±2% of this collision probability.     

Early aqueous alteration,explosive disruption,and reprocessing of asteroids

Abstract— Several classes of chondritic meteorites experienced heterogeneous aqueous alteration and subsequent brecciation before agglomeration into the final parent bodies. These processes resulted in intimate mixing of materials altered to various degrees. In the past, this mixing has been attributed solely to impact processes. However, our investigation of the physical consequences of aqueous alteration processes in bodies accreting from a mixture of silicate and ice grains shows that aqueous alteration was commonly accompanied by substantial gas production. The asteroids may have become so internally pressurized by these gases that the sudden onset of gas release led to partial or complete disaggregation of the body. In some cases, fragments may have escaped completely from the parent asteroid and could potentially have been incorporated into other accreting asteroids. In other cases, fragments originating from different parts of the initial body, having various sizes and temperatures and exhibiting varying degrees of alteration, may have re-accreted into a second-generation object and undergone further stages of alteration. Such events could have been repeated two or three times during the lifetime of ²⁶Al, the likely heat source for these processes.     

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.     

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.     

Mutual gravitational potential,force, and torque of a homogeneous polyhedron and an extended body: an application to binary asteroids

Binary systems are quite common within the populations of near-Earth asteroids, main-belt asteroids, and Kuiper belt asteroids. The dynamics of binary systems, which can be modeled as the full two-body problem, is a fundamental problem for their evolution and the design of relevant space missions. This paper proposes a new shape-based model for the mutual gravitational potential of binary asteroids, differing from prior approaches such as inertia integrals, spherical harmonics, or symmetric trace-free tensors. One asteroid is modeled as a homogeneous polyhedron, while the other is modeled as an extended rigid body with arbitrary mass distribution. Since the potential of the polyhedron is precisely described in a closed form, the mutual gravitational potential can be formulated as a volume integral over the extended body. By using Taylor expansion, the mutual potential is then derived in terms of inertia integrals of the extended body, derivatives of the polyhedron’s potential, and the relative location and orientation between the two bodies. The gravitational forces and torques acting on the two bodies described in the body-fixed frame of the polyhedron are derived in the form of a second-order expansion. The gravitational model is then used to simulate the evolution of the binary asteroid (66391) 1999 KW4, and compared with previous results in the literature.     

Relativistic effects and solar oblateness from radar observations of planets and spacecraft

We used more than 250 000 high-precision American and Russian radar observations of the inner planets and spacecraft obtained in the period 1961–2003 to test the relativistic parameters and to estimate the solar oblateness. Our analysis of the observations was based on the EPM ephemerides of the Institute of Applied Astronomy, Russian Academy of Sciences, constructed by the simultaneous numerical integration of the equations of motion for the nine major planets, the Sun, and the Moon in the post-Newtonian approximation. The gravitational noise introduced by asteroids into the orbits of the inner planets was reduced significantly by including 301 large asteroids and the perturbations from the massive ring of small asteroids in the simultaneous integration of the equations of motion. Since the post-Newtonian parameters and the solar oblateness produce various secular and periodic effects in the orbital elements of all planets, these were estimated from the simultaneous solution: the post-Newtonian parameters are β = 1.0000 ± 0.0001 and γ = 0.9999 ± 0.0002, the gravitational quadrupole moment of the Sun is J₂ = (1.9 ± 0.3) × 10^?7, and the variation of the gravitational constant is ?/G = (?2 ± 5) × 10^?14 yr^?1. The results obtained show a remarkable correspondence of the planetary motions and the propagation of light to General Relativity and narrow significantly the range of possible values for alternative theories of gravitation.     

On the mechanism of catastrophic destruction of minor planets by high-velocity impact

For the study of the collisional breakup of minor planets, semiquantitative interpretations of the catastrophic destruction of cubic rock targets by high-velocity impact reported by A. Fujiwara, G. Kamimoto, and A. Tsukamoto (1977, Icarus31, 277–288) are attempted. The conditions for transition and core-type destruction are derived from consideration of the side surface spallation, or central spallation, and the back surface spallation caused by the rarefaction pulses. In the derivation, the strengths of the original P-pulse at the crater rim and the bottom are given by application of the failure criterion. As the condition for complete destruction, an empirical rule is proposed. It is found that the critical sizes for these destructions vary as E^0.40 and the previously reported $\text{E}\text{M}\text{-}\text{scaling}$ is only an approximate rule (E, impact energy; M, target mass). The possibilities of extending of these results to larger bodies of any other shape are 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.     

Orbital evolution of asteroids with the effects of mutual gravitational attraction

The effects of the mutual gravitational attraction between asteroids were analyzed by two N-body calculations, in which N=4,516 (the Sun, the nine planets, and 4,506 asteroids). In one calculation the gravity of the asteroids was taken into account, and in the other it was ignored. These calculations were carried out for a time period of about 100 years. The largest difference in the positions of the asteroids between these two calculations is about 10^?3 AU. For the orbital elements of the semimajor axis, the eccentricity, and the inclination, the largest differences were 9 × 10^?6 AU, 4 × 10^?6, and 5 × 10^?4 degrees, respectively. It was found that the distribution of the differences of the semimajor axis between the two calculations is quite similar to the Cauchy distribution.