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1.
As planetary embryos grow, gravitational stirring of planetesimals by embryos strongly enhances random velocities of planetesimals and makes collisions between planetesimals destructive. The resulting fragments are ground down by successive collisions. Eventually the smallest fragments are removed by the inward drift due to gas drag. Therefore, the collisional disruption depletes the planetesimal disk and inhibits embryo growth. We provide analytical formulae for the final masses of planetary embryos, taking into account planetesimal depletion due to collisional disruption. Furthermore, we perform the statistical simulations for embryo growth (which excellently reproduce results of direct N-body simulations if disruption is neglected). These analytical formulae are consistent with the outcome of our statistical simulations. Our results indicate that the final embryo mass at several AU in the minimum-mass solar nebula can reach about ∼0.1 Earth mass within 107 years. This brings another difficulty in formation of gas giant planets, which requires cores with ∼10 Earth masses for gas accretion. However, if the nebular disk is 10 times more massive than the minimum-mass solar nebula and the initial planetesimal size is larger than 100 km, as suggested by some models of planetesimal formation, the final embryo mass reaches about 10 Earth masses at 3-4 AU. The enhancement of embryos’ collisional cross sections by their atmosphere could further increase their final mass to form gas giant planets at 5-10 AU in the Solar System. 相似文献
2.
S.J. Weidenschilling 《Icarus》1980,44(1):172-189
The behavior of solid particles in a low-mass solar nebula during settling to the central plane and the formation of planetesimals is examined. Gravitational instability in a dust layer and collisional accretion are considered as possible mechanisms of planetesimal formation. Non-Keplerian rotation of the nebula results in shear between the gas and a dust layer. This shear produces turbulence within the layer which inhibits gravitational instability, unless the mean particle size exceeds a critical value, ～1 cm at 1 AU. The size requirement is less stringent at larger heliocentric distances, suggesting a possible difference in planetesimal formation mechanisms between the inner and outer nebula. Coagulation of grains during settling is expected in the solar nebula environment. Van der Waals forces appear adequate to produce centimeter-sized aggregates. Growth is primarily due to sweepup of small particles by larger ones due to size-dependent settling velocities. A numerical model for computing simultaneous coagulation and settling is described. Relative velocities are determined by gas drag and the non-Keplerian rotation of the nebula. The settling is very nonhomologous. Most of the solid matter reaches the central plane as centimeter-sized aggregates in a few times 103 revolutions, but some remains suspended in the form of fine dust. Drag-induced relative velocities result in collisions. The growth of bodies in the central plane is initially rapid. After sizes reach ～103 cm, relative velocities decrease and the growth rate declines. Gas drag rapidly damps the out-of-plane motions of these intermediate-sized bodies. They settle into a thin layer which is subject to gravitational instability. Kilometer-sized planetesimals are formed by this composite process. 相似文献
3.
The influence of gas drag and gravitational perturbations by a planetary embryo on the orbit of a planetesimal in the solar nebula was examined. Non-Keplerian rotation of the gas causes secular decay of the orbit. If the planetesimal's orbit is exterior to the perturber's, resonant perturbations oppose this drag and can cause it to be trapped in a stable orbit at a commensurability of order j/(j + 1), where j is an integer. Numerical and analytical demonstrations show that resonant trapping occurs for wide ranges of perturbing mass, planetesimal size, and j. Induced eccentricities are large, causing overlap of orbits for bodies in different resonances with j > 2. Collisions between planetesimals in different resonances, or between resonant and nonresonant bodies, result in their disruption. Fragments smaller than a critical size can pass through resonances under the influence of drag and be accreted by the embryo. This effect speeds accretion and tends to prevent dynamical isolation of planetary embryos, making gas-rich scenarios for planetary formation more plausible. 相似文献
4.
We investigate classical planetesimal accretion in a binary star system of separation ab?50 AU by numerical simulations, with particular focus on the region at a distance of 1 AU from the primary. The planetesimals orbit the primary, are perturbed by the companion and are in addition subjected to a gas drag force. We concentrate on the problem of relative velocities Δv among planetesimals of different sizes. For various stellar mass ratios and binary orbital parameters we determine regions where Δv exceed planetesimal escape velocities vesc (thus preventing runaway accretion) or even the threshold velocity vero for which erosion dominates accretion. Gaseous friction has two crucial effects on the velocity distribution: it damps secular perturbations by forcing periastron alignment of orbits, but at the same time the size-dependence of this orbital alignment induces a significant Δv increase between bodies of different sizes. This differential phasing effect proves very efficient and almost always increases Δv to values preventing runaway accretion, except in a narrow eb?0 domain. The erosion threshold Δv>vero is reached in a wide (ab,eb) space for small <10-km planetesimals, but in a much more limited region for bigger ?50-km objects. In the intermediate vesc<Δv<vero domain, a possible growth mode would be the type II runaway growth identified by Kortenkamp et al. [Kortenkamp, S., Wetherill, G., Inaba, S., 2001. Science 293, 1127-1129]. 相似文献
5.
A. G. W. Cameron 《Meteoritics & planetary science》1995,30(2):133-161
Abstract— The 26Al/27Al ratio in a large number of calcium-aluminum inclusions (CAIs) is a rather uniform 5 × 10?5, whereas in chondrules the ratio is either undetectable or has a much lower value; the simplest interpretation of this is that there was an interval of a few million years between the times that these two meteoritic constituents formed stable solids. The present investigation was undertaken as an exploration of the physics of the processes in the solar nebula during and after the accumulation of the Sun. Understanding the time scales of events in this nebular model, to see if this would cast light on this apparent CAI to chondrule time interval, was the major motivation for the exploration. There were four stages in the history of the solar nebula; in stage 1, a fragment of an interstellar molecular cloud collapsed to form the Sun and solar nebula; in stage 2, the nebula was in approximate steady state balance between infall from the cloud and accretion onto the Sun and was in its FU Orionis accumulation stage; in stage 3, the Sun had been mainly accumulated and there was a slow residual mass flow into the Sun while it was in its classical T Tauri stage; and in stage 4, the nebula had finished accreting material onto the Sun (now a weak-lined T Tauri star) and was in a static condition with no significant dissipation or motions, other than removal at the inner edge due to the T Tauri solar wind and photoevaporation beyond 9 astronomical units (AU). It is found that the energy source keeping the nebula warm during stages 3 and 4 is recombination of ionized H in the ionized bipolar jets and the T Tauri coronal expansion solar wind. The parameters of the heating model were adjusted to locate the ice sublimation line at 5.2 AU. In this work, a nebular model is used with a surface density of 4.25 × 103 gm/cm2 at 1 AU and a variation with radial distance as the inverse first power. Under normal conditions in the nebula, there is a negative pressure gradient that provides partial radial support for the gas, which thus circles the Sun more slowly than large solid objects do. Large objects undergo a slow inward spiral due to the gas drag; very small objects move essentially with the gas but have a slow inward drift; and intermediate objects (e.g., 1 m) have a fairly large inward drift velocity that traverses the full radial extent of the nebula in considerably less than the CAI to chondrule time interval. Such objects are thus lost unless they can grow rapidly to larger sizes. Near the inner edge (bow) of the nebula during stage 4, the pressure gradient becomes positive, creating a narrow zone of zero gas drag toward which solids drift from both directions, facilitating planetesimal formation in the inner solar nebula. Recent theoretical and experimental results on sticking probabilities of solids show that icy surfaces have the best sticking properties, but icy interstellar grains can only stick together when subjected to impact velocities of less than 2000 cm/sec. However, if the solid objects are very underdense, then a collision leads to interpenetration and many points at which the small constituent grains can adhere to one another, and thus coagulation becomes possible for such underdense objects. Simulations were made of such coagulation in the outer solar nebula, and it was found that the central plane of the nebula quickly becomes filled with meter-sized and larger bodies that rapidly accumulated near the top of the nebula and rapidly descended; in a few thousand years this quickly leads to gravitational instabilities that can form planetesimals. These processes led to the rapid formation of Jupiter in the nebula (and the slightly less rapid formation of the other giant planets). The early formation of Jupiter opens an annular gap in the nebula, and thus a second region is created in the nebula with zero gas drag. It is concluded that CAIs were formed at the end of stage 2 of the nebula history and moved out into the nebula for long-term storage, and that most chondrules were formed by magnetic reconnection flares in the bow region of the nebula during stage 4, several million years later. Carbonaceous meteorites should be formed on the far side of the Jovian gap, with the chondrules being heated by flares on the early Jupiter irradiating materials in the nearby zone of zero gas drag, and they should have essentially the same 26Al ages as the CAIs (this will be very hard to confirm owing to scarcity of Al mineral phases in these chondrules). 相似文献
6.
T.A. Heppenheimer 《Icarus》1974,22(4):436-447
A theory is presented for determining regions where planets may form in binary star systems. Planet formation by accretion is assumed possible if mean planetesimal collision velocities do not exceed a critical value. Collision velocities are increased by perturbations due to the companion star, treated by secular perturbation theory. Collision velocities are damped by aerodynamic drag within the solar nebula, taken as the linear case of Cameron and Pine.A general feature of planetary systems in binary stars is the existence of two zones. The inner zone has enough damping to permit unimpeded growth by accretion; in the outer zone, growth proceeds to a limited diameter, beyond which damping is insufficient. It is shown that the asteroids could not have failed to coalesce due to Jupiter perturbations in the primitive solar nebula. Binary star systems with semimajor axis < 30AU are not likely to have planets; these include Alpha Centauri and 70 Ophiuchi. Systems possibly possessing planets include Eta Cassiopeiae, 40 Eridani, and Σ 2398. Epsilon Eridani is a marginal case. 相似文献
7.
Runaway growth ends when the largest protoplanets dominate the dynamics of the planetesimal disk; the subsequent self-limiting accretion mode is referred to as “oligarchic growth.” Here, we begin by expanding on the existing analytic model of the oligarchic growth regime. From this, we derive global estimates of the planet formation rate throughout a protoplanetary disk. We find that a relatively high-mass protoplanetary disk (∼10 × minimum-mass) is required to produce giant planet core-sized bodies (∼10 M⊕) within the lifetime of the nebular gas (?10 million years). However, an implausibly massive disk is needed to produce even an Earth mass at the orbit of Uranus by 10 Myrs. Subsequent accretion without the dissipational effect of gas is even slower and less efficient. In the limit of noninteracting planetesimals, a reasonable-mass disk is unable to produce bodies the size of the Solar System’s two outer giant planets at their current locations on any timescale; if collisional damping of planetesimal random velocities is sufficiently effective, though, it may be possible for a Uranus/Neptune to form in situ in less than the age of the Solar System. We perform numerical simulations of oligarchic growth with gas and find that protoplanet growth rates agree reasonably well with the analytic model as long as protoplanet masses are well below their estimated final masses. However, accretion stalls earlier than predicted, so that the largest final protoplanet masses are smaller than those given by the model. Thus the oligarchic growth model, in the form developed here, appears to provide an upper limit for the efficiency of giant planet formation. 相似文献
8.
We present results from 44 simulations of late stage planetary accretion, focusing on the delivery of volatiles (primarily water) to the terrestrial planets. Our simulations include both planetary “embryos” (defined as Moon to Mars sized protoplanets) and planetesimals, assuming that the embryos formed via oligarchic growth. We investigate volatile delivery as a function of Jupiter's mass, position and eccentricity, the position of the snow line, and the density (in solids) of the solar nebula. In all simulations, we form 1-4 terrestrial planets inside 2 AU, which vary in mass and volatile content. In 44 simulations we have formed 43 planets between 0.8 and 1.5 AU, including 11 “habitable” planets between 0.9 and 1.1 AU. These planets range from dry worlds to “water worlds” with 100+oceans of water (1 ocean=1.5×1024 g), and vary in mass between 0.23M⊕ and 3.85M⊕. There is a good deal of stochastic noise in these simulations, but the most important parameter is the planetesimal mass we choose, which reflects the surface density in solids past the snow line. A high density in this region results in the formation of a smaller number of terrestrial planets with larger masses and higher water content, as compared with planets which form in systems with lower densities. We find that an eccentric Jupiter produces drier terrestrial planets with higher eccentricities than a circular one. In cases with Jupiter at 7 AU, we form what we call “super embryos,” 1-2M⊕ protoplanets which can serve as the accretion seeds for 2+M⊕ planets with large water contents. 相似文献
9.
We have made numerical experiments of the collisional and gravitational interaction of a planetesimal swarm in the early Solar System. In particular we study the dynamical evolution of an initial population of kilometer-size planetesimals subject to collisions (accretion, rebound, cratering, and catastrophic fragmentation). This study is based on a Monte-Carlo statistical method and provides the mass and velocity distributions of the planetesimal swarm as a function of time as well as their distribution in heliocentric distance. Several experiments have been performed and three of them are presented here. They simulate the accretional growth of numerous planetesimals in the absence (or presence) of gaseous drag, with (or without) one larger embryo among them, and with (or without) a size gradient. The results show that (i) for a population of planetesimals submitted to a negative gradient in size as the heliocentric distance increases, the outer planetesimals spiral toward the Sun faster than inner ones, leading after some time to an accumulation of bodies inside the cloud which allows the formation of an embryo; (ii) the growth of one embryo among a population of planetesimals is accelerated by the presence of gas and is warranted as long as its feeding zone is fed by the inward flow of planetesimals due to gas drag. These results offer some complementary new insights in the understanding of the accretional formation of 4–5 terrestrial planets instead of the numerous Moon-size planets generally found in numerical experiments. 相似文献
10.
In this paper we present a new semianalytical model of oligarchic growth of planets considering a distribution of planetesimal sizes, fragmentation of planetesimals in mutual collisions, sublimation of ices through the snow line, random velocities out of equilibrium and merging of planetary embryos. We show that the presence of several planetary embryos growing simultaneously at different locations in the protoplanetary disk affects the whole accretion history, specially for the innermost planets. The results presented here clearly indicate the relevance of considering a distribution of planetesimal sizes. Fragmentation occurring during planetesimal-planetesimal collisions represent only a marginal effect in shaping the surface density of solid material in the protoplanetary disc. 相似文献
11.
Origin of the atmospheres of the terrestrial planets 总被引：1，自引：0，他引：1
A.G.W. Cameron 《Icarus》1983,56(2):195-201
The monotonic decrease in the atmospheric abundance of 36Ar per gram of planet in the sequence, Venus, Earth, and Mars has been assumed to reflect some conditions in the primitive solar nebula at the time of formation of the planetary atmospheres, having to do either with the composition of the nebula itself or the composition of the trapped gases in small solid bodies in the nebula. Behind such hypotheses lies the assumption that planetary atmospheres steadily gain components. However, not only can gases enter atmospheres; they may also be lost from atmospheres both by adsorption into the planetary interior and by loss into space as a result of collisions with minor and major planetesimals. In this paper a necessarily qualitative discussion is given of the problem of collisions with minor planetesimals, a process called atmospheric cratering or atmospheric erosion, and a discussion is given of atmospheric loss accompanying collision of a planet with a major planetesimal, such as may have produced the Earth's Moon. 相似文献
12.
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 ∼1027 g (∼ Mars mass) are found to form in ∼105 years at Jupiter's position (5.2 AU), assuming a large enough value of the surface density of solid material (25 g/cm2) 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 21M⊕ forms at 5.2 AU in 3.8×106 years. The planetary embryo is massive enough to start a rapid gas accretion and forms a gas giant planet. 相似文献
13.
G.P. Horedt 《Icarus》1985,64(3):448-470
We derive first-order differential equations for the late stages of planetary accretion (planetesimal mass >1013 g). The effect of gravitational encounters, energy exchange, collisions, and gas drag has been included. Two simple models are discussed, namely, (i) when all planetesimals have the same mass and (ii) when there is one large planetesimal and numerous small planetesmals. Gravitational two-body encounters are modeled according to Chandrasekhar's classical theory from stellar dynamics. It is shown that the velocity increase due to mutual encounters can be modeled according to the simple theory of random flights. We find analytical equations for the average velocity decrease due to collisions. Gas drag, if present, is modeled in averaged form up to the first order in the eccentricities and inclinations of the planetesimals. Characteristic time scales for the formation of the terrestrial planets are found for the most favorable models to be of order 108 year. The calculated mass of rock and ice of the giant planets is too low as compared to the observed one. This difficulty of our model could be overcome by assuming a several times larger surface density, an enlarged accretion cross section, and gas accretion during the final stages of accretion of the solid cores of the giant planets. Analytical and numerical results are presebted, the evolutionary tracks showing satisfactory agreement with observations for some models. 相似文献
14.
Z. M. Leinhardt D. C. Richardson G. Lufkin J. Haseltine 《Monthly notices of the Royal Astronomical Society》2009,396(2):718-728
In this paper, we extend our numerical method for simulating terrestrial planet formation to include dynamical friction from the unresolved debris component. In the previous work, we implemented a rubble pile planetesimal collision model into direct N -body simulations of terrestrial planet formation. The new collision model treated both accretion and erosion of planetesimals but did not include dynamical friction from debris particles smaller than the resolution limit for the simulation. By extending our numerical model to include dynamical friction from the unresolved debris, we can simulate the dynamical effect of debris produced during collisions and can also investigate the effect of initial debris mass on terrestrial planet formation. We find that significant initial debris mass, 10 per cent or more of the total disc mass, changes the mode of planetesimal growth. Specifically, planetesimals in this situation do not go through a runaway growth phase. Instead, they grow concurrently, similar to oligarchic growth. The dynamical friction from the unresolved debris damps the eccentricities of the planetesimals, reducing the mean impact speeds and causing all collisions to result in merging with no mass loss. As a result, there is no debris production. The mass in debris slowly decreases with time. In addition to including the dynamical friction from the unresolved debris, we have implemented particle tracking as a proxy for monitoring compositional mixing. Although there is much less mixing due to collisions and gravitational scattering when dynamical friction of the background debris is included, there is significant inward migration of the largest protoplanets in the most extreme initial conditions (for which the initial mass in unresolved debris is at least equal to the mass in resolved planetesimals). 相似文献
15.
16.
Radial drift of planetesimals due to density wave interaction with the solar nebula is considered. The mechanism is most effective for large masses and provides mobility over a size range where aerodynamic drag is unimportant. The process could shorten accretion time scales to ≈O(105–106 years) throughout the solar system. Accumulation stalls down when growing objects are massive enough to open gaps in the gas disk. Implications of this process for current cosmogonic models are discussed. 相似文献
17.
Abstract— We describe results of 32 N‐body planetary accretion simulations that investigate the dependence of terrestrial‐planet formation on nebula surface density profile σ and evolution of the eccentricities of Jupiter and Saturn ej,s. Two surface density profiles are examined: a decaying profile with σ ∝ 1/a, where a is orbital semi‐major axis, and a peaked profile in which σ increases for a < 2 AU and decreases for a > 2 AU. The peaked profiles are generated by models of coagulation in an initially hot nebula. Models with initial ej,s = 0.05 (the current value) and 0.1 are considered. Simulations using the decaying profile with ej,s = 0.1 produce systems most like the observed planets in terms of mass‐weighted mean a and the absence of a planet in the asteroid belt. Simulations with doubled σ produce planets roughly twice as massive as the nominal case. Most initial embryos are removed in each simulation via ejection from the solar system or collision with the Sun. The asteroid belt is almost entirely cleared on a timescale of 10–100 Ma that depends sensitively on ej,s. Most initial mass with a < 2 AU survives, with the degree of mass loss increasing with a. Mass loss from the terrestrial region occurs on a timescale that is long compared to the mass loss time for the asteroid belt. Substantial radial mixing of material occurs in all simulations, but is greater in simulations with initital ej,s = 0.05. The degree of mixing is equivalent to a feeding zone of half width 1.5 and 0.9 AU for an Earth mass planet at 1 AU for the cases ej,s = 0.05 and 0.1, respectively. In simulations with ej,s = 0.05, roughly one‐third and 5–10% of the mass contained in final terrestrial planets originated in the region a > 2.5 AU for the decaying and peaked profiles, respectively. In the case ej,s = 0.1, the median mass accreted from a > 2.5 AU is zero for both profiles. 相似文献
18.
When preplanetary bodies reach proportions of ∼1 km or larger in size, their accretion rate is enhanced due to gravitational focusing (GF). We have developed a new numerical model to calculate the collisional evolution of the gravitationally-enhanced growth stage. The numerical model is novel as it attempts to preserve the individual particle nature of the bodies (like N-body codes); yet it is statistical in nature since it must incorporate the very large number of planetesimals. We validate our approach against existing N-body and statistical codes. Using the numerical model, we explore the characteristics of the runaway growth and the oligarchic growth accretion phases starting from an initial population of single planetesimal radius R0. In models where the initial random velocity dispersion (as derived from their eccentricity) starts out below the escape speed of the planetesimal bodies, the system experiences runaway growth. We associate the initial runaway growth phase with increasing GF-factors for the largest body. We find that during the runaway growth phase the size distribution remains continuous but evolves into a power-law at the high-mass end, consistent with previous studies. Furthermore, we find that the largest body accretes from all mass bins; a simple two-component approximation is inapplicable during this stage. However, with growth the runaway body stirs up the random motions of the planetesimal population from which it is accreting. Ultimately, this feedback stops the fast growth and the system passes into oligarchy, where competitor bodies from neighboring zones catch up in terms of mass. We identify the peak of GF with the transition between the runaway growth and oligarchy accretion stages. Compared to previous estimates, we find that the system leaves the runaway growth phase at a somewhat larger radius, especially at the outer disk. Furthermore, we assess the relevance of small, single-size fragments on the growth process. In classical models, where the initial velocity dispersion of bodies is small, these do not play a critical role during the runaway growth; however, in models that are characterized by large initial relative velocities due to external stirring of their random motions, a situation can emerge where fragments dominate the accretion, which could lead to a very fast growth. 相似文献
19.
We have performed N-body simulations on the stage of protoplanet formation from planetesimals, taking into account so-called “type-I migration,” and damping of orbital eccentricities and inclinations, as a result of tidal interaction with a gas disk without gap formation. One of the most serious problems in formation of terrestrial planets and jovian planet cores is that the migration time scale predicted by the linear theory is shorter than the disk lifetime (106-107 years). In this paper, we investigate retardation of type-I migration of a protoplanet due to a torque from a planetesimal disk in which a gap is opened up by the protoplanet, and torques from other protoplanets which are formed in inner and outer regions. In the first series of runs, we carried out N-body simulations of the planetesimal disk, which ranges from 0.9 to 1.1 AU, with a protoplanet seed in order to clarify how much retardation can be induced by the planetesimal disk and how long such retardation can last. We simulated six cases with different migration speeds. We found that in all of our simulations, a clear gap is not maintained for more than 105 years in the planetesimal disk. For very fast migration, a gap cannot be created in the planetesimal disk. For migration slower than some critical speed, a gap does form. However, because of the growth of the surrounding planetesimals, gravitational perturbation of the planetesimals eventually becomes so strong that the planetesimals diffuse into the vicinity of the protoplanets, resulting in destruction of the gap. After the gap is destroyed, close encounters with the planetesimals rather accelerate the protoplanet migration. In this way, the migration cannot be retarded by the torque from the planetesimal disk, regardless of the migration speed. In the second series of runs, we simulated accretion of planetesimals in wide range of semimajor axis, 0.5 to 2-5 AU, starting with equal mass planetesimals without a protoplanet seed. Since formation of comparable-mass multiple protoplanets (“oligarchic growth”) is expected, the interactions with other protoplanets have a potential to alter the migration speed. However, inner protoplanets migrate before outer ones are formed, so that the migration and the accretion process of a runaway protoplanet are not affected by the other protoplanets placed inner and outer regions of its orbit. From the results of these two series of simulations, we conclude that the existence of planetesimals and multiple protoplanets do not affect type-I migration and therefore the migration shall proceed as the linear theory has suggested. 相似文献
20.
Zoë M. LeinhardtDerek C. Richardson 《Icarus》2002,159(2):306-313
We present results from direct N-body simulations of collisions between gravitational aggregates of varying size as part of a study to parameterize planetesimal growth in the Solar System. We find that as the ratio of projectile to target mass departs from unity, the impact angle has less effect on the outcome. At the same time, the probability of planetesimal growth increases. Conversely, for a fixed impact energy, collisions between impactors with mass ratio near unity are more dispersive than those with impactor mass ratio far from unity. We derive an expression for the accretion probability as a function of mass ratio. For an average mass ratio of 1:5, we find an accretion probability of ∼60% over all impact parameters. We also compute the critical specific dispersal energy Q*D as a function of projectile size. Extrapolating to a projectile size of 1 m with a 1-km target, we find Q*D=103−104 J kg−1, in agreement with several other collision models that use fundamentally different techniques. Our model assumes that the components of each gravitational aggregate are identical and indestructible over the range of sampled impact speeds. In future work we hope to incorporate a simple fracture model to extend the range of applicable speeds and we plan to implement our results in a large-scale planetesimal evolution code. 相似文献