Small Planetesimals Research Articles

Enhanced collisional growth of a protoplanet that has an atmosphere

Once a protoplanet becomes larger than about lunar size, it accumulates a significant atmosphere that surrounds the solid core. When a planetesimal approaches the protoplanet, it interacts with the atmosphere. If enough energy of the planetesimal is lost by gas drag of the atmosphere, it is captured in the atmosphere even if its original trajectory would not lead to a direct collision with the solid core of the protoplanet. This increases the collision rate, resulting in faster growth of the protoplanet. We have derived the analytical calculations for the collision rate, and calculated the structure of the atmosphere and the trajectories of the planetesimals in the atmosphere. As a result of their large gas drag, small planetesimals are easily captured, resulting in a large rate of collision with the protoplanet. A collision rate of a protoplanet of Earth size with a planetesimal of 100 m radius is, for example, enhanced by a factor of ∼10. These effects play an essential role in the study of formation of solid cores of gas giant planets by the core accretion model.

Dynamical Evolution of Planetesimals in Protoplanetary Disks

The current picture of terrestrial planet formation relies heavily on our understanding of the dynamical evolution of planetesimals—asteroid-like bodies thought to be planetary building blocks. In this study we investigate the growth of eccentricities and inclinations of planetesimals in spatially homogeneous protoplanetary disks using methods of kinetic theory. Emphasis is put on clarifying the effect of gravitational scattering between planetesimals on the evolution of their random velocities. We explore disks with a realistic mass spectrum of planetesimals evolving in time, similar to that obtained in self-consistent simulations of planetesimal coagulation: the distribution scales as a power law of mass for small planetesimals and is supplemented by an extended tail of bodies at large masses representing the ongoing runaway growth in the system. We calculate the behavior of planetesimal random velocities as a function of the planetesimal mass spectrum both analytically and numerically; results obtained by the two approaches agree quite well. Scaling of random velocity with mass can always be represented as a combination of power laws corresponding to different velocity regimes (shear- or dispersion-dominated) of planetesimal gravitational interactions. For different mass spectra we calculate analytically the exponents and time-dependent normalizations of these power laws, as well as the positions of the transition regions between different regimes. It is shown that random energy equipartition between different planetesimals can only be achieved in disks with very steep mass distributions (the differential surface number density of planetesimals falling off steeper than m-4) or in the runaway tails. In systems with shallow mass spectra (shallower than m-3) the random velocities of small planetesimals turn out to be independent of their masses. We also discuss the damping effects of inelastic collisions between planetesimals and of gas drag, and their importance in modifying planetesimal random velocities.

Formation of gas giant planets: core accretion models with fragmentation and planetary envelope

We have calculated formation of gas giant planets based on the standard core accretion model including effects of fragmentation and planetary envelope. The accretion process is found to proceed as follows. As a result of runaway growth of planetesimals with initial radii of ∼10 km, planetary embryos with a mass of ∼10 27 g (∼ Mars mass) are found to form in ∼10 5 years at Jupiter's position (5.2 AU), assuming a large enough value of the surface density of solid material (25 g/cm 2) in the accretion disk at that distance. Strong gravitational perturbations between the runaway planetary embryos and the remaining planetesimals cause the random velocities of the planetesimals to become large enough for collisions between small planetesimals to lead to their catastrophic disruption. This produces a large number of fragments. At the same time, the planetary embryos have envelopes, that reduce energies of fragments by gas drag and capture them. The large radius of the envelope increases the collision rate between them, resulting in rapid growth of the planetary embryos. By the combined effects of fragmentation and planetary envelope, the largest planetary embryo with 21 M ⊕ forms at 5.2 AU in 3.8×10 6 years. The planetary embryo is massive enough to start a rapid gas accretion and forms a gas giant planet.

N-body Simulations of Planet Formation

Accretion from many small planetesimals to planets is reviewed. Solid protoplanets accrete through runaway and oligarchic growth until they become isolated. The isolation mass of protoplanets in terrestrial planet region is about 0.1-0.2 Earth mass, which suggests giant impacts among the protoplanets in the final stage of terrestrial planet formation. On the other hand, the isolation mass in Jupiter's and Saturn's orbits is about a few to 5 Earth masses, which may be massive enough to trigger gas accretion onto the cores. The isolation mass in Uranus and Neptune's orbits is as large as their present cores. Extending the above arguments to extrasolar planetary systems that are formed from disks with various initial masses, we also discuss diversity of extrasolar planetary systems.

A bootstrap approach to Pluto's origin

The solar nebula theory hypothesizes that planets are formed from an accretion disk of material that, over time, condenses into dust, small planetesimals, and that the planets should have, on average, coplanar, nearly circular orbits. If the orbit of Pluto has a different origin to the other planets in the solar system, then there will be tremendous repercussions on modelling the spacecrafts for a mission to Pluto. We test here the nebula theory for Pluto, using both parametric and non-parametric methods. We first develop asymptotic distributions of extrinsic means on a manifold, and then derive bootstrap and large sample distributions of the sample mean direction. Our parametric and non-parametric analyses provide very strong evidence that the solar nebula theory does not hold for Pluto.

Noble gas systematics of the Réunion mantle plume source and the origin of primordial noble gases in Earth’s mantle

New noble gas data of ultramafic xenoliths from Réunion Island, Indian Ocean, further constrain the characteristics of primordial and radiogenic noble gases in Earth’s mantle plume reservoirs. The mantle source excess of nucleogenic 21Ne is significantly higher than for the Hawaiian and Icelandic plume reservoirs, similar to excess of radiogenic 4He. 40Ar/ 36Ar of the Réunion mantle source can be constrained to range between 8000 and 12 000, significant 129Xe and fission Xe excess are present. Regarding the relative contribution of primordial and radiogenic rare gas nuclides, the Réunion mantle source is intermediate between Loihi- and MORB-type reservoirs. This confirms the compositional diversity of plume sources recognized in other radioisotope systematics. Another major result of this study is the identification of the same basic primordial component previously found for the Hawaiian and Icelandic mantle plumes and the MORB reservoir. It is a hybrid of solar-type He and Ne, and ‘atmosphere-like’ or ‘planetary’ Ar, Kr, Xe (Science 288 (2000) 1036). 20Ne/ 22Ne ratios extend to maximum values close to 12.5 (Ne-B), which is the typical signature of solar neon implanted as solar corpuscular radiation. This suggests that Earth’s solar-type noble gas inventory was acquired by small (less than km-sized) precursor planetesimals that were irradiated by an active early sun in the accretion disk after nebular gas dissipation, or, alternatively, that planetesimals incorporated constituents irradiated in transparent regions of the solar nebula. Previously, such an early irradiation scenario was suggested for carbonaceous chondrites which follow common volatile depletion trends in the sequence CI–CM–CV–Earth. In turn, CV chondrites closely match Earth’s mantle composition in 20Ne/ 22Ne, 36Ar/ 22Ne and 36Ar/ 38Ar. This indicates that mantle Ar could well be a planetary component inherited from precursor planetesimals. However, a corresponding conclusion for mantle Kr and Xe is less convincing yet, but this may be just due to the lack of appropriate ‘meteoritic’ building blocks matching terrestrial composition. Alternatively, heavy noble gases in Earth’s mantle could be due to admixing of severely fractionated air, but this effect must have affected all mantle sources to a very similar extent, e.g. by global subduction before the last homogenization of the mantle reservoirs.

Formation and early evolution of the atmosphere

Abstract The tectonic activity of the Earth allowed exchange of volatile elements (H, C, N, rare gases) between the surface of the Earth (atmosphere, crust, sediments, oceans) and the mantle. However, some of these elements still present elemental and isotopic heterogeneities that allow us to reconstruct the volatile composition of the terrestrial mantle. The protosolar nebula supplied a significant fraction of helium and neon, which were presumably trapped during the major phase of the Earth’s accretion and were possibly hosted by accreting dust and/or small porous planetesimals. Surprisingly, volatile elements are in chondritic proportion despite their drastic (10 −3 ) depletion in the mantle relative to chondrites, in a way that recalls the case of highly siderophile elements. From stable isotope systematics, we find that the contribution of comets to the volatile inventory of the Earth was very limited. The integrated flux of chondritic-like material necessary to provide water, carbon and nitrogen is consistent with that required for the formation of the lunar craters as well as that necessary to account for the inventory of siderophile elements in the mantle. A consequence of this scenario is that the Earth’s surface was oxidized very early. Alternatively, volatile and siderophile elements of the mantle could be the remnant of small patches of chondritic material that did not equilibrate with the core nor drastically degas.

The Leonard Award Address: Presented 2000 August 30, Chicago, Illinois, USA: Early solar system events and timescales

Abstract— Some recent information on the Mn‐Cr and Al‐Mg systems is reviewed. This information is used to derive constraints on the timing of processes and events, which took place in the early solar system. Using reasonable assumptions, a timeline is constructed where the estimated age of the solar system is ∼4571 Ma. This age is taken to mark the time when most calcium‐aluminum‐rich inclusions (CAIs) were starting to form, a process that may have lasted for several 105 years. Almost contemporaneously small planetesimals have accreted that served to store these CAIs for later dispersal among larger planetesimals. By the time large numbers of planetesimals of several tens of kilometers in size had formed, the interior of these objects started to melt through the decay of 26Al. Collisional disruption of these planetesimals allowed gases, dust, and melt to escape into the surrounding space. The fine droplets of melt reacted with gas and dust to form chondrules, which, after rapid cooling, were partially re‐accreted onto the residual rubble pile. This process of primary chondrule formation, in most cases involving several generations of planetesimals, most plausibly lasted only for ∼2 Ma. Towards the end of this period and during the following 3 to 4 Ma planetary objects of several hundred kilometers in size were formed. They still stored enough energy to continue melting from the inside to finally differentiate into chemically stratified layers, with basaltic volcanism occurring within a few million years.

A New Mechanism Relevant to the Formation of Planetesimals in the Solar Nebula

Current models of the formation of km-sized planetary building blocks, or planetesimals, by collisional accretion require unrealistically low collision velocities or ad hoc assumptions about sticking in order for growth to occur. Collision velocities in the protoplanetary nebula increase with increasing particle size, leading to bouncing and fragmentation of colliding bodies rather than growth. We describe a new and efficient mechanism in planetesimal accretion which leads to net growth in spite of fragmentation, at least for small planetesimals. Our experimental results show that nebular gas flow returns protoplanetary dust grains to a growing planetesimal, resulting in accretion for impact speeds up to at least 10–15 ms −1. This new threshold velocity is three orders of magnitude higher than the threshold velocity for accretion between macroscopic ice spheres and ten times higher than the sticking velocity for micron-sized dust particles.

Chondrule formation by the ablation of small planetesimals

— Numerous models have been proposed to explain the formation of chondrules, but none can be reconciled with the highly diverse properties of these objects. Here the formation of chondrules by the surface melting and ablation of small planetesimals in nebula shock waves is investigated using a numerical model. It is shown that bodies between ∼1 mm and 500 m in diameter would have produced molten droplets by ablation during gas drag in nebula shocks stronger than ∼2.0 Mach. The properties of chondrules produced by ablation are estimated by comparison with meteorite fusion crusts and through consideration of the environment within the bow shock envelope of ablating planetesimals. It is suggested that most ablation chondrules will have broadly chondritic compositions with depletions in siderophile and chalcophile elements and relatively high volatile contents and textures that are mainly porphyritic. The formation of chondrules by ablation of planetesimals in shock waves was probably most important at a late stage in nebula history and occurred at the same time as chondrules formed by the melting of dust particles. The high abundance of dust particles relative to larger bodies at all stages of accretion implies that only a proportion of chondrules may have been formed by ablation and that genetic groups of chondrules with very different origins may coexist in meteorites.

Planetesimal Accretion in Binary Star Systems

Planetesimal accretion in close binary systems is a complex process for the gravitational perturbations of the companion star on the planetesimal orbits. These perturbations excite high eccentricities that can halt the accumulation process of planetesimals into planets also in those regions around the star where stable planetary orbits would eventually be possible. However, the evolution of a planetesimal swarm is also affected by collisions and gas drag. In particular, gas drag combined with the secular perturbations of the secondary star forces a strong alignment of all the planetesimal periastra. Since periastra are also coupled to eccentricities via the secular perturbations of the companion, the orbits of the planetesimals, besides all being aligned, also have very close values of eccentricity. This orbital strongly reduces the contribution of the eccentricity to the relative velocities between planetesimals, and the impact speeds are dominated by the Keplerian shear: accretion becomes possible. This behavior is not limited to small planetesimals but also affects bodies as large as 100 km in diameter. The effects of gas drag are in fact enhanced by the presence of the constant forced component in the orbital eccentricity of the planetesimals. We describe analytically the periastron alignment by using the secular equations developed by Heppenheimer, and we test the prediction of the theory with a numerical code that integrates the orbits of a swarm of planetesimals perturbed by gas drag and collisions. The gas density is assumed to decrease outward, and the collisions are modeled as inelastic. Our computations are focused on the α Centauri system, which is a good candidate for terrestrial planets as we will show. The impact velocities between planetesimals of different sizes are computed at progressively increasing distances from the primary star and are compared with estimates for the maximum velocity for accretion. According to our simulations in the α Centauri system, the formation of a planet within 2 AU of the primary star is possible because of the orbital phasing forced by gas drag.

The nature of pristine noble gases in mantle plumes

High-precision noble gas data show that the Hawaiian and Icelandic mantle plume sources contain uniquely primitive neon that is composed of moderately nucleogenic neon-21 and a primordial component indistinguishable from the meteoritic occurrence of solar neon. This suggests that Earth's solar-type rare gas inventory was acquired during accretion from small planetesimals previously irradiated by solar wind from the early sun. However, nonradiogenic argon, krypton, and xenon isotopes derived from the mantle display nonsolar compositions and indicate an atmosphere-like fingerprint that is not due to recent subduction.

Accretion in the Early Outer Solar System

We describe calculations of the evolution of an ensemble of small planetesimals in the outer solar system. In a solar nebula with a mass of several times the minimum mass solar nebula, objects with radii of 100-1000 km can form on timescales of 10-100 Myr. Model luminosity functions derived from these calculations agree with current observations of bodies beyond the orbit of Neptune (Kuiper Belt objects). New surveys with current and planned instruments can place better constraints on the mass and dynamics of the solar nebula by measuring the luminosity function at red magnitudes, mR ≥ 28.

Capture of Trojans by a Growing Proto-Jupiter

We have studied the capture of planetesimals in Trojan orbits by a growing proto-Jupiter by integrating numerically the equations of motion of planetesimals in a four-body problem Sun–Jupiter–Saturn–planetesimal. The masses of Jupiter and Saturn increase exponentially with time. Nebular gas drag on small planetesimals is taken into account. We show that a fraction of the planetesimals near Jupiter's orbit and almost all planetesimals originally in horseshoe orbits are trapped into stable Trojan orbits. This mechanism was probably an important factor together with collisional diffusion, for delivering a large amount of planetesimals into Trojan orbits during the final growth of Jupiter by gas in-fall. We find also that the libration amplitude of trapped Trojans is damped by the increase of Jupiter's mass reinforcing the stability of the resonance lock. For planetesimals small enough to be perturbed by nebular gas drag, we find an asymmetry between the trapping rate in L5 compared to L4. Peale (1993,Icarus106,308–322) noted also that the stability of the L4 resonant region is reduced in presence of an intense frictional drag. We present here a simple semi-analytical explanation for the trapping asymmetry based on the evolution of the angular difference between the planetesimal's and proto-Jupiter's longitude of perihelion. The asymmetry between L4 and L5 could have led to the capture of a larger number of small planetesimals in L5. This may have favored a higher degree of collisional activity in L5 and might explain an alleged asymmetry in the size distribution of the two swarms.

Distribution of Planetesimals around a Protoplanet in the Nebula Gas

We investigated the long-term evolution of the spatial and velocity distribution of a planetesimal swarm around a protoplanet, calculating gas drag force and gravitational force of the protoplanet. Mutual scattering between planetesimals is taken into account, which removes the effect of mean motion resonances with the protoplanet. We simulated orbital evolution of 2000 planetesimals around a protoplanet for ∼105years and found that scattering by a protoplanet, coupled with the gas drag, cleans up planetesimals in the feeding zone of the protoplanet. Our results show that planetesimals are trapped outside the feeding zone without the help of resonances. Sufficiently small planetesimals, on which the gas drag effectively acts, can remain in the feeding zone. We obtained the trapping condition in this nonresonance case. It is almost the same as the condition obtained by Karyet al. (1993,Icarus106, 288–307), who considered “resonance trapping.” Our results mean that the trapping of planetesimals does not depend on the strength of resonances against mutual scattering between planetesimals. Furthermore, the cleanup of the planetesimals occurs rapidly, so that the protoplanet cannot feed most of the planetesimals before the cleanup and growth of the protoplanet will slow down. That is, the effect of gas drag halts rather than prompts protoplanet growth by supply of material due to radial migration. We also examined the multiple-protoplanet case and considered the effect of gas on the formation of protoplanets.

Formation of the Giant Planets by Concurrent Accretion of Solids and Gas

New numerical simulations of the formation of the giant planets are presented, in which for the first time both the gas and planetesimal accretion rates are calculated in a self-consistent, interactive fashion. The simulations combine three elements: (1) three-body accretion cross sections of solids onto an isolated planetary embryo, (2) a stellar evolution code for the planet's gaseous envelope, and (3) a planetesimal dissolution code within the envelope, used to evaluate the planet's effective capture radius and the energy deposition profile of accreted material. Major assumptions include: The planet is embedded in a disk of gas and small planetesimals with locally uniform initial surface mass density, and planetesimals are not allowed to migrate into or out of the planet's feeding zone.All simulations are characterized by three major phases. During the first phase, the planet's mass consists primarily of solid material. The planetesimal accretion rate, which dominates that of gas, rapidly increases owing to runaway accretion, then decreases as the planet's feeding zone is depleted. During the second phase, both solid and gas accretion rates are small and nearly independent of time. The third phase, marked by runaway gas accretion, starts when the solid and gas masses are about equal. It is engendered by a strong positive feedback on the gas accretion rates, driven by the rapid contraction of the gaseous envelope and the rapid expansion of the outer boundary, which depends on the planet's total mass. The overall evolutionary time scale is generally determined by the length of the second phase.The actual rates at which the giant planets accreted small planetesimals is probably intermediate between the constant rates assumed in most previous studies and the highly variable rates used here. Within the context of the adopted model of planetesimal accretion, the joint constraints of the time scale for dissipation of the solar nebula and the current high-Zmasses of the giant planets lead to estimates of the initial surface density (σinit) of planetesimals in the outer region of the solar nebula. The results show that σinit≈ 10 g cm−2near Jupiter's orbit and that σinit∝a−2, whereais the distance from the Sun. These values are a factor of 3 to 4 times as high as that of the “minimum-mass” solar nebula at Jupiter's distance and a factor of 2 to 3 times as high at Saturn's distance. The estimates for the formation time of Jupiter and Saturn are 1 to 10 million years, whereas those for Uranus fall in the range 2 to 16 million years. These estimates follow from the properties of our Solar System and do not necessarily apply to giant planets in other planetary systems.

Rocky relationship

The core of the Earth is a little younger than previously thought, according to two University of Michigan geochemists. Studying the composition of rocks and meteorites, the pair believe they have found direct evidence that Earth's metallic core and its moon formed no less than 62 million years after the creation of the solar system. Previous estimates had placed core formation anywhere from 15–50 million years after the solar system got started.Der‐Chuen Lee and Alexander Halliday used a process known as MC‐ICPMS (multiple collector inductively coupled plasma magnetic sector mass spectrometry) to compare Hafnium and tungsten isotopes in chondrites (primitive meteorites), silicate Earth rocks, lunar basalt from the Apollo 14 mission, and two iron meteorites, which are considered to be remnants of the original cores of small planetesimals.

The Effect of the Nebula on the Trojan Precursors

Numerical and analytical calculations demonstrate that small planetesimals exhibit long-term stability of libration about the L4 and L5 Lagrange equilibrium points at a proto-Jupiter's distance from the Sun despite continuous dissipation of energy due to gas drag in the primordial solar nebula. This stability may mean that most of the mass of the Trojan asteroids was collisionally captured into libration and accreted into large bodies while the solar nebula was still there. If this is so, the Trojans represent samples of the material that composed Jupiter's core before the addition of the large mass of volatiles leading to the current largely gaseous planet.For a 13-M⊕. Jupiter core that is assumed not to disturb the mass distribution in the solar nebula, dissipation of orbital energy while the planetesimal is further from the Sun than the stationary point in the frame rotating with Jupiter (or the elliptic fixed point in a surface of section if Jupiter's orbit is eccentric) decreases the amplitude of libration. However, dissipation while the planetesimal is closer to the Sun than this point increases the amplitude of libration. The effects of dissipation outside the stationary point nearly cancel the effects of the dissipation inside, resulting in either a very slow increase or decrease in the amplitude of libration depending on the details of the nebular model. Those models of the nebula leading to a larger ratio of drag forces equidistant outside and inside of Jupiter's orbital distance give the librating planetesimals a higher stability. By higher stability we mean that the planetesimal is more likely to remain permanently in the region with decreasing amplitude of libration, or if the amplitude is increasing, that the rate of increase is lower. The L4 stationary points of Trojan precursors experiencing nebular drag are located more than 60° in front of Jupiter, whereas the L5 stationary points are located less than 60° behind Jupiter. The separation of these stationary points from ±60° increases as the particle diameter d decreases or the drag acceleration is otherwise increased. For a 13-M⊕ Jupiter and a nominal nebular model the stationary points at the L4 equivalent reach a maximum angular separation of about 108° in front of Jupiter for d ≈ 30 m, where smaller planetesimals do not librate in front of Jupiter. In contrast, the stationary points may be located only a few degrees behind Jupiter for d ≈ 1 m. If Jupiter has a nonzero orbital eccentricity, the stability of the trailing, L5 planetesimals is increased, but the stability of the leading, L4 planetesimals is decreased relative to the zero eccentricity case. Even if libration amplitudes are increasing slowly, the planetesimals librate around their particular stationary points in the frame rotating with Jupiter for times which are significant fractions of the nebular lifetime. This means that as Jupiter's core grows, the gentle collisions among the accumulating, librating planetesimals will induce the growth of larger Trojans.When Jupiter's mass approaches its current value, an annular gap in the distribution of nebular material is believed to form. The reversal of the radial pressure gradient on the outside of the gap means that the planetesimals are accelerated from behind when on the outside of their libration trajectories. This reverses the trend toward smaller libration amplitudes on this part of the trajectory, and the libration amplitude is increased on all parts of the libration trajectory. Small planetesimals are quickly lost, although for d > 3 km lifetimes are probably longer than the remaining life of the nebula.The striking asymmetry in the response of the planetesimals to the nebular drag between the L4 and L5 points implies that there should be an asymmetry in the numbers or total mass of Trojans remaining trapped near each stability point after the nebula is dissipated. However, these results favor more material being trapped at the trailing Trojan point, which is apparently contrary to observation. Still, the collisional evolution of the Trojans over 4.6 × 109 years may have skewed the mass distributions in such a way as to mask what was there initially, and current observational uncertainties indicate that the current apparent dominance of the L4 Trojans is not definitive.A possible history of the Trojan swarms of asteroids would have most of the mass being accumulated in libration before the nebula was dispersed, with the elimination of large libration amplitudes and dispersion in orbital inclinations occurring afterward. The dominance of high-velocity collisions after the nebula was dispersed has probably resulted in a drastic change in the size distribution and a net loss in librating mass.

Formation of Planetary Embryos: Effects of Fragmentation, Low Relative Velocity, and Independent Variation of Eccentricity and Inclination

An earlier investigation of the formation of ∼1026 g planetary embryos from much smaller planetesimals (G. W. Wetherill and G. R. Stewart 1989, Icarus 77, 350-357) has been extended to include the effects of collisional fragmentation, the low relative velocity regime in which the effects due to solar gravity are important, and independent perturbations of eccentricity and inclination. In agreement with this earlier work, it is found that at 1 AU runaway growth occurs on a ∼105-year time scale as a consequence of equipartition of energy between large and small planetesimals. It is now seen that the runaway is initiated after ∼104 years, when the relative velocities of the larger bodies temporarily fall into the low-velocity regime, lowering their inclinations and increasing their gravitational capture rates. After ∼2 × 104 years, relative velocities between most bodies emerge from the low-velocity regime, and these higher velocities tend to inhibit further runaway growth. This rapid runaway growth is self-regulated, however, by these same higher velocities, causing fragmentation of the smaller bodies. The velocities of the collision fragments are reduced by gas drag, facilitating their capture by the growing runaway embryos. Variations in which different fragmentation models are used, or long-range forces between nonrunaway bodies are absent, give similar results. When fragmentation is not included, the time scale for growth increases to ∼3 × 105 years as a result of loss of the self-regulating process described above.

On the Statistical Distribution of Massive Impactors

If there is no preferred mass scale in the distribution of small bodies or planetesimals in the Solar System, the mass distribution of impactors striking a target will be a power law, N(>m) ∝ m-γ . For γ ≤ 1 most of the mass is accreted in a few giant impacts. We apply this simple model to cratering and derive probability distributions for quantities such as the largest impactor mass relative to the total mass of impactors and the nth largest impactor relative to the largest. We also use our model to study the distribution of giant impacts that is likely to occur in planet formation.