Collisional growth of planetesimals

Safronov's (1972) demonstration that relative velocities of planetesimals would be comparable to the dominant size bodies' escape velocities, combined with a plausible size distribution that has most mass in the largest bodies, yielded his evolution model with limited growth of the largest planetesimal with respect to its next largest neighbors. A numerical simulation of planetesimal accretion (Greenberget al., 1978) suggests that at least over one stage of collisional accretion, velocities were much lower than the escape velocity of the largest bodies, because the bulk of the mass still resided in km-scale bodies. The low velocities at this early stage may conceivably have permitted early runaway growth, which, in turn, would have kept the velocities low and permitted continued runaway growth of the largest bodies.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.     

A Monte Carlo estimate of the fraction of comets developing into sizeable asteroidal bodies

We present results of Monte Carlo simulations of orbital evolution showing that assuming a steady state there are roughly 50 extinct comets per active one in the Mars-crossing Jupiter family. The large number of extinct comets thus expected compared with the absence of observed apollo or Amor asteroids with aphelion distances greater than 4.2 AU indicates that less than five percent of the extinct comets survive as sizeable asteroidal bodies.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.     

Numerical simulation of the final stages of terrestrial planet formation

Three representative numerical simulations of the growth of the terrestrial planets by accretion of large protoplanets are presented. The mass and relative-velocity distributions of the bodies in these simulations are free to evolve simultaneously in response to close gravitational encounters and occasional collisions between bodies. The collisions between bodies, therefore, arise in a natural way and the assumption of expressions for the relative velocity distribution and the gravitational collision cross section is unnecessary. These simulations indicate that the growth of bodies with final masses approaching those of Venus and the Earth is possible, at least for the case of a two-dimensional system. Simulations assuming an initial uniform distribution of orbital eccentricities on the interval from 0 to e_max are found to produce final states containing too many bodies with masses which are too small when e_max < 0.10, while simulations with e_max > 0.20 result in too many catastrophic collisions between bodies thus preventing rapid accretion of planetary-size bodies. The e_max = 0.15 simulation ends with a state surprisingly similar to that of the present terrestrial planets and, therefore, provides a rough estimate of the range of radial sampling to be expected for the terrestrial planets.     

Dynamics,interrelations and evolution of the systems of asteroids and comets

The dynamics of larger interplanetary bodies is reviewed, with emphasis on evolutionary problems, interrelations, and open questions. Observational biases distinguishing the sample of known objects from the whole population are briefly discussed. A schematic division of the solar system into zones with different regimes of motion, and a rough taxonomy of orbit types are attempted. The role of individual major planets in controlling the dynamical evolution of interplanetary objects, in particular by stabilizing resonances and destabilizing close encounters, is compared. There are significant evolution asymmetries due to boundary conditions and preferential evolutionary paths; individual major displacements in the phase space of orbital elements conserve the Tisserand invariant with respect to the planet responsible, thus favouring certain evolutionary sequences against others. Very limited lifetimes of some orbit types imply a long-term balance between source and sink, and require a continuous supply of objects from other types of orbits. In this respect, the ultimate fate of extinct comets is of particular interest. Under very specific conditions, nongravitational effects of mass loss can result in stabilization of a formerly unstable orbit. Since the dividing line between the two basic interplanetary populations distinguished by origin and composition — the asteroids and the comets — is essentially that between stable and unstable motion, orbital data can be used to specify which of the known asteroid-like objects may be devolatilized cometary nuclei.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.     

Ion emission from micrometeorite impacts on atmosphereless planets

Conclusions have been drawn on the generation of intensive high-velocity ions and atoms fluxes on the surfaces of the Moon, Mercury and the atmosphereless satellites of Jupiter by micrometeorite impacts. About 30% of each flux is ejected at small angles and detained by planetary reliefs, forming surface-active layer of alkalines. The combined erosional effect of that layer, the thermocycling and the radiational tractks have been studied. The ion fluxes leaving Jupiter's atmosphereless satellites may well be one of the main sources of the short UV-emission in their vicinities.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.From September 1979.     

Shape of small solar system bodies

The morphometric parameters are examined for the shape of fragments of ordinary chondrites, iron meteorites, S- and C-class stony asteroids, metallic asteroids, and icy small bodies of the Solar System. All small Solar System bodies are shown to have, depending on their composition and, hence, physical and mechanical properties, a specific shape that is unique to a given composition. C-class asteroids, the strength of which is almost three times less than that of S asteroids, differ from the latter in their less elongated shape. No systematic change is observed in the morphometric parameters (increased roundness or sphericity) of small bodies of differing compositions depending on their mass, which suggests that the hypothesis of creep in small Solar System bodies is unlikely to be true. The absence of creep confirms that, regardless of their composition, all small Solar System bodies are solid elastic bodies having an ultimate strength (tensile strength and compressive strength) and a yield strength.     

Oscillating peak model of basin and crater formation

A model of crater and basin formation is presented in which the interior morphology is most strongly influenced by the amount of central rebound occurring rapidly after the initial crater excavation. In large craters the rebound is so great that it has started to collapse again under its own weight, and in small basins this collapse is so rapid that a second interior depression is formed. In large basins such as Orientale, the central region is considered to have undergone a more extensive damped vertical oscillation.Field evidence, particularly stratigraphical relations in Orientale and the morphometry of central peaks and basin inner rings, strongly support this theory.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.     

Cratering rates in the outer Solar System

This paper is a compilation by table, graph, and equation of impact cratering rates from Jupiter to Pluto. We use several independent constraints on the number of ecliptic comets. Together they imply that the impact rate on Jupiter by 1.5-km-diameter comets is currently ?(d > 1.5 km) = 0.005−0.003+0.006 per annum. Other kinds of impactors are currently unimportant on most worlds at most sizes. The size-number distribution of impactors smaller than 20 km is inferred from size-number distributions of impact craters on Europa, Ganymede, and Triton; while the size-number distribution of impacting bodies larger than 50 km is equated to the size-number distribution of Kuiper Belt objects. The gap is bridged by interpolation. It is notable that small craters on Jupiter’s moons indicate a pronounced paucity of small impactors, while small craters on Triton imply a collisional population rich in small bodies. However it is unclear whether the craters on Triton are of heliocentric or planetocentric origin. We therefore consider two cases for Saturn and beyond: a Case A in which the size-number distribution is like that inferred at Jupiter, and a Case B in which small objects obey a more nearly collisional distribution. Known craters on saturnian and uranian satellites are consistent with either case, although surface ages are much younger in Case B, especially at Saturn and Uranus. At Neptune and especially at Saturn our cratering rates are much higher than rates estimated by Shoemaker and colleagues, presumably because Shoemaker’s estimates mostly predate discovery of the Kuiper Belt. We also estimate collisional disruption rates of moons and compare these to estimates in the literature.     

Ice-silicate fractionation among icy bodies due to the difference of impact strength between ice and ice-silicate mixture

Laboratory experiments on the impact disruption of ice-silicate mixtures were conducted to clarify the accretion process of small icy bodies. Since the icy bodies are composed of ice and silicates with various porosities, we investigated the effect of porosity on the impact disruption of mixtures. We tested the mixture target with the mass ratio of ice to silicate, 0.5 and with 5 different porosities (0, 12.5, 25, 32, 37%) at the impact velocities of 150 to 670 m/s. The silicate mass ratio was changed from 0 to 0.5 in steps of 0.1 at a porosity of 12.5% and a constant impact velocity of about 300 m/s. The impact strength of the mixture was found to decrease with increasing porosity and the silicate mass ratio between 0.1 and 0.5 could enhance the strength of the icy target. The observed dependence of the impact strength on the porosity is opposite to that observed for pure ice. This difference could play an important role in ice-silicate fractionation during the accretion process. Because, ice rich bodies are easily broken as the porosity decreases in their evolution, the collisional growth could be prohibited. On the other hand, among the silicate rich bodies the collisional growth could be enhanced.     

Numerical modelling of heating in porous planetesimal collisions

Collisions between planetesimals at speeds of several kilometres per second were common during the early evolution of our Solar System. However, the collateral effects of these collisions are not well understood. In this paper, we quantify the efficiency of heating during high-velocity collisions between planetesimals using hydrocode modelling. We conducted a series of simulations to test the effect on shock heating of the initial porosity and temperature of the planetesimals, the relative velocity of the collision and the relative size of the two colliding bodies. Our results show that while heating is minor in collisions between non-porous planetesimals at impact velocities below 10 km s⁻¹, in agreement with previous work, much higher temperatures are reached in collisions between porous planetesimals. For example, collisions between nearly equal-sized, porous planetesimals can melt all, or nearly all, of the mass of the bodies at collision velocities below 7 km s⁻¹. For collisions of small bodies into larger ones, such as those with an impactor-to-target mass ratio below 0.1, significant localised heating occurs in the target body. At impact velocities as low as 5 km s⁻¹, the mass of melt will be nearly double the mass of the impactor, and the mass of material shock heated by 100 K will be nearly 10 times the mass of the impactor. We present a first-order estimate of the cumulative effects of impact heating on a porous planetesimal parent body by simulating the impact of a population of small bodies until a disruptive event occurs. Before disruption, impact heating is volumetrically minor and highly localised; in no case was more than about 3% of the parent body heated by more than 100 K. However, heating during the final disruptive collision can be significant; in about 10% of cases, almost all of the parent body is heated to 700 K (from an initial temperature of ∼300 K) and more than a tenth of the parent body mass is melted. Hence, energetic collisions between planetesimals could have had important effects on the thermal evolution of primitive materials in the early Solar System.     

N-Body Simulations of Late Stage Planetary Formation with a Simple Fragmentation Model

We present results of two-dimensional gravitationalN-body simulations of the late stage of planetary formation. This stage is characterized by the direct accretion of hundreds of lunar-sized planetesimals into planetary bodies. Our simulation code is based on the Hermite Individual Timestep integration algorithm, and gravitational interactions among all bodies are included throughout the simulations. We compare our simulation with earlier works that do not include all interactions, and we find very good agreement. A previously published collisional fragmentation model is included in our simulation to study the effects of the production of fragments on the subsequent evolution of the larger planetary bodies. It is found that for realistic two-body collisions that, according to this model, both bodies will suffer fragmentation, and that the outcome of the collision will be a relatively large core containing most of the mass and a few small fragments. We present the results of simulations that include this simple fragmentation model. They indicate that the presence of small fragments have only a small effect on the growth or orbital evolution of the large planet-sized bodies.     

On the existence of periodic solutions of Poincare's third sort in the general problem of three bodies in three dimensions

A proof is offered of the existence of periodic solutions of the general problem of three bodies, of the third sort envisaged by Poincaré, that is, arising by analytic continuation from unperturbed keplerian motion of each of two bodies about a primary, in which the two orbits are of commensurable periods, of zero eccentricity, but lying in different planes, provided that the inclination of the two planes is sufficiently small (but not zero).Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.     

Steady-state size distributions for collisional populations:: analytical solution with size-dependent strength

The steady-state population of bodies resulting from a collisional cascade depends on how material strength varies with size. We find a simple expression for the power-law index of the population, given a power law that describes how material strength varies with size. This result is extended to the case relevant for the asteroid belt and Kuiper belt, in which the material strength is described by 2 separate power laws—one for small bodies and one for larger bodies. We find that the power-law index of the small body population is unaffected by the strength law for the large bodies, and vice versa. Simple analytical expressions describe a wave that is superimposed on the large body population because of the transition between the two power laws describing the strength. These analytical results yield excellent agreement with a numerical simulation of collisional evolution. These results will help to interpret observations of the asteroids and KBOs, and constrain the strength properties of those objects.     

Accretion of the asteroids: Implications for their thermal evolution

Thermal models of asteroids generally assume that they accreted either instantaneously or over an extended interval with a prescribed growth rate. It is conventionally assumed that the onset of accretion of chondrite parent bodies was delayed until a substantial fraction of the initial ²⁶Al had decayed. However, this interval is not consistent with the early melting, and differentiation of parent bodies of iron meteorites. Formation time scales are tested by dynamical simulations of accretion from small primary planetesimals. Gravitational accretion yields rapid runaway growth of large planetary embryos until most smaller bodies are depleted. In a given simulation, all asteroid‐sized bodies have comparable growth times, regardless of size. For plausible parameters, growth times are shorter than the lifetime of ²⁶Al, consistent with thermal models that assume instantaneous accretion. Rapid growth after planetesimal formation is consistent with differentiation of parent bodies of iron meteorites, but not with the assumed delay in formation of chondritic bodies. After the initial growth stage, there is an interval of slower evolution until the belt is stirred and the embryos are dynamically removed. During this interval, a fraction of asteroid‐sized bodies experience large accretional impacts, allowing bodies of the same final size to have very different histories of radius versus time. Accretion from small primary planetesimals leaves some fraction of material in bodies small enough to preserve CAIs while avoiding heating by ²⁶Al. Unheated material can be a significant fraction of the mass that remains after large embryos are removed from the Main Belt.     

ORBITAL EVOLUTION OF IMPACT EJECTA FROM MARS

The orbital evolution of material ejected from Mars into heliocentric orbits is investigated, with particular emphasis on the origin of the shergottite, nakhlite, and chassignite achondrites. Two models are considered. In the first, meteorite-size bodies are ejected directly from Mars. In the second, the ejecta are ∼ 15 m diameter bodies, that are subsequently fragmented by collisions in space. In both cases a large fraction (∼ 35%) of the objects that will ever reach Earth do so within 10 m.y. For the “small body” model, it is found that about 0.03% of the Mars crater ejecta must be accelerated to the Mars escape velocity; the “large body” model requires an efficiency of 0.4%. Assuming that the acceleration of large bodies to be less probable, the results indicate that meteorites originating as small bodies should dominate the terrestrial flux of Mars ejecta. This result is in general agreement with data from SNC meteorites, but the reported short exposure age of EETA 79001 is hard to understand in a pure small body model. The yield of meteorites from Mercury is found to be at least a factor of 100 lower than from Mars.     

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.     

The effects of impacts on the cooling rates of iron meteorites

Iron meteorites provide a record of the thermal evolution of their parent bodies, with cooling rates inferred from the structures observed in the Widmanstätten pattern. Traditional planetesimal thermal models suggest that meteorite samples derived from the same iron core would have identical cooling rates, possibly providing constraints on the sizes and structures of their parent bodies. However, some meteorite groups exhibit a range of cooling rates or point to uncomfortably small parent bodies whose survival is difficult to reconcile with dynamical models. Together, these suggest that some meteorites are indicating a more complicated origin. To date, thermal models have largely ignored the effects that impacts would have on the thermal evolution of the iron meteorite parent bodies. Here we report numerical simulations investigating the effects that impacts at different times have on cooling rates of cores of differentiated planetesimals. We find that impacts that occur when the core is near or above its solidus, but the mantle has largely crystallized can expose iron near the surface of the body, leading to rapid and nonuniform cooling. The time period when a planetesimal can be affected in this way can range between 20 and 70 Myr after formation for a typical 100 km radius planetesimal. Collisions during this time would have been common, and thus played an important role in shaping the properties of iron meteorites.     

Particles in the nebular midplane: Collective effects and relative velocities

Abstract– In the absence of global turbulence, solid particles in the solar nebula tend to settle toward the midplane, forming a layer with enhanced solids/gas ratio. Shear relative to the surrounding pressure‐supported gas generates turbulence within the layer, inhibiting further settling and preventing gravitational instability. Turbulence and size‐dependent drift velocities cause collisions between particles. Relative velocities between small grains and meter‐sized bodies are typically about 50 m s^?1 for isolated particles; however, in a dense particle layer, collective effects alter the motion of the gas near the midplane. Here, we develop a numerical model for the coupled motions of gas and particles of arbitrary size, based on the assumption that turbulent viscosity transfers momentum on the scale of the Ekman length. The vertical distribution of particles is determined by a balance between settling and turbulent diffusion. Self‐consistent distributions of density, turbulent velocities, and radial fluxes of gas and particles of different sizes are determined. Collective effects generate turbulence that increases relative velocities between small particles, but reduce velocities between small grains and bodies of decimeter size or larger by bringing the layer’s motion closer to Keplerian. This effect may alleviate the “meter‐size barrier” to collisional growth of planetesimals.     

Are the cometary outbursts caused by impacts with interplanetary vagabonds probable?

The possibility of impacts and their results in relation to the cometary outbursts between comets and other small bodies in the solar system has been investigated. Taking into consideration certain physical features of cometary nuclei and impacting bodies, the probability of impacts of small bodies moving in the main asteroid belt with hypothetical comets which represent three types: Jupiter family comets, Halley family comets and long period comets has been computed. The probability of impacts between comets and meteoroids at large heliocentric distances has also been estimated. Potential consequences of these events in relation to outbursts of the cometary brightness have been discussed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Method of determining the orbits of the small bodies in the solar system based on an exhaustive search of orbital planes

A universal method of determining the orbits of newly discovered small bodies in the Solar System using their positional observations has been developed. The proposed method suggests determining geocentric distances of a small body by means of an exhaustive search for heliocentric orbital planes and subsequent determination of the distance between the observer and the points at which the chosen plane intersects with the vectors pointing to the object. Further, the remaining orbital elements are determined using the classical Gauss method after eliminating those heliocentric distances that have a fortiori low probabilities. The obtained sets of elements are used to determine the rms between the observed and calculated positions. The sets of elements with the least rms are considered to be most probable for newly discovered small bodies. Afterwards, these elements are improved using the differential method.