aa r X i v : . [ a s t r o - ph . E P ] J u l Asteroids Were Born Big
Alessandro Morbidelli
Observatoire de la Cˆote d’AzurBoulevard de l’ObservatoireB.P. 4229, 06304 Nice Cedex 4, France
William F . Bottke
Southwest Research Institute1050 Walnut St, Suite 300Boulder, CO 80302 USA
David Nesvorn´y
Southwest Research Institute1050 Walnut St, Suite 300Boulder, CO 80302 USA
Harold F . Levison
Southwest Research Institute1050 Walnut St, Suite 300Boulder, CO 80302 USA
Received ; accepted 2 –
ABSTRACT
How big were the first planetesimals? We attempt to answer this question byconducting coagulation simulations in which the planetesimals grow by mutualcollisions and form larger bodies and planetary embryos. The size frequency dis-tribution (SFD) of the initial planetesimals is considered a free parameter in thesesimulations, and we search for the one that produces at the end objects with aSFD that is consistent with asteroid belt constraints. We find that, if the initialplanetesimals were small (e.g. km-sized), the final SFD fails to fulfill these con-straints. In particular, reproducing the bump observed at diameter D ∼
100 kmin the current SFD of the asteroids requires that the minimal size of the ini-tial planetesimals was also ∼
1. Introduction
The classical model for planet formation involves 3 steps. In Step 1, planetesimals form.Dust sediments towards the mid-plane of the proto-planetary disk and starts to collide with 3 –each other at low velocities. The particles eventually stick together through electrostaticforces, forming larger fractal aggregates (Dominik and Tielens, 1997; Kempf et al., 1999;Wurm and Blum, 1998, 2000; Colwell and Taylor, 1999; Blum et al., 2000). Furthercollisions make these aggregates more compact, forming pebbles and larger objects. Abottleneck for this growth mode is the so-called meter-size barrier . The origin of this barrieris two-fold. On the one hand, the radial drift of solid particles towards the Sun due to gasdrag reaches maximum speed for objects roughly one meter in diameter. These meter-sizeboulders should fall onto the Sun from 1 AU in 100 to 1,000 years (Weidenschilling, 1977),i.e. faster than they can grow to significantly larger sizes. On the other hand, becausegas drag is size dependent, bodies of different sizes spiral inwards at different velocities.This leads to mutual collisions, with relative velocities typically several tens of meters persecond for bodies in the centimeter- to meter-size range. Moreover, in turbulent disks,even equal-size bodies collide with non-zero velocities due to turbulent stirring. This effectis again maximized for meter-size boulders (Cuzzi and Weidenschilling, 2006; see alsoDominik et al., 2007). Current theories predict the destruction of the meter-sized objects atthese predicted speeds (Wurm et al., 2005). In the absence of a well understood mechanismto overcome the meter-size barrier, it is usually assumed that Nature somehow managesto produce planetesimals of 1 to 10 km in diameter, objects that are less susceptible togas-drag and all of its hazardous effects.In Step 2, planetary embryos/cores form . Collisional coagulation among the The embryos are objects with Lunar to Martian masses, precursor of the terrestrialplanets, that are expected to form in the terrestrial planets region or in the asteroid belt.In the Jovian planet region, according to the core-accretion model (Pollack et al., 1995), thegiant planet cores are multi-Earth-mass objects that eventually lead to the birth of the giantplanets by gas accretion. 4 –planetesimals allows the latter to agglomerate into massive bodies. In this step gravitystarts to play a fundamental role, bending the trajectories of the colliding objects;this fact effectively increases the collisional cross-section of the bodies by the so-called gravitational focussing factor (Greenzweig and Lissauer, 1992). At the beginning, if thedisk is dynamically very cold (i.e. the orbits have tiny eccentricities and inclinations), thedispersion velocity of the planetesimals v rel may be smaller than the escape velocity of theplanetesimals themselves. In this case, a process of runaway growth begins, in which therelative mass growth of each object is an increasing function of its own mass M , namely:1 M d M d t ∼ M / v rel , (Greenberg et al., 1978; Wetherill and Stewart, 1989). However, as growth proceeds, thedisk is dynamically heated by the scattering action of the largest bodies. When v rel becomesof the order of the escape velocity from the most massive objects, the runaway growthphase ends and the accretion proceeds in an oligarchic growth mode, in which the relativemass growth of the largest objects is proportional to M − / (Ida and Makino, 1993; Kokuboand Ida, 1998). The combination of runaway and oligarchic growth produces in the innerSolar System a population of planetary embryos, with Lunar to Martian masses (Wetherilland Stewart, 1993; Weidenschilling et al., 1997). In the outer Solar System, beyond theso-called snowline (Podolak and Zucker, 2004), it is generally expected that the end resultis the formation of a few super-Earth cores (Thommes et al., 2003; Goldreich et al., 2004;Chambers, 2006) that, by accretion of a massive gaseous atmosphere from the disk, becomegiant planets (Pollack et al., 1996; Ida and Lin, 2004a,b; Alibert et al., 2004, 2005).In Step 3 the terrestrial planets form. The system of embryos in the inner Solar Systembecomes unstable and the embryos start to collide with each other, forming the terrestrialplanets on a timescale of several 10 to ∼ years (Chambers and Wetherill, 1998; Agnoret al., 1999; Chambers, 2001; Raymond et al., 2004, 2005, 2006, 2007; O’Brien et al., 2006; 5 –Kenyon and Bromley, 2006). Of all the steps of planet formation, this is probably theone that is understood the best, whereas Step 1 is the one that, because of the meter-sizebarrier, is understood the least.How can the meter-sized barrier be overcome? Two intriguing possibilities come from arecent conceptual breakthrough; new models (Johansen et al., 2007; Cuzzi et al., 2008) showthat large planetesimals can form directly from the concentration of small solid particlesin the turbulent structures of the gaseous component of the protoplanetary disk. Here webriefly review the models of Johansen et al. (2007) and Cuzzi et al. (2008).Johansen et al. (2007) showed that turbulence in the disk, either generated bythe Kelvin-Helmoltz instability (Weidenschilling, 1980; Johansen et al., 2006) or byMagneto-Rotational Instability (MRI; Stone et al., 2000), may help the solid particlespopulation to develop gravitational instabilities. Recall that turbulence generates densityfluctuations in the gas disk and that gas drag pushes solid particles towards the maximaof the gas density distribution. Like waves in a rough sea, these density maxima comeand go at many different locations. Thus, the concentration of solid particles in theirvicinity cannot continue for very long. The numerical simulations of Johansen et al. (2006,2007), however, show that these density maxima are sufficiently long-lived (thanks also tothe inertia/feedback of the solid particles residing within the gas, the so-called streaminginstability; Youdin and Goodman, 2005) to concentrate a large quantity of meter-sizeobjects (Note that the effect is maximized for ∼
50 cm objects, but we speak of meter-sizeboulders for simplicity). Consequently, the local density of solids can become large enoughto allow the formation of a massive planetesimal by gravitational instability. In fact, thesimulations in Johansen et al. (2007) show the formation of a planetesimal with 3.5 timesthe mass of Ceres is possible within a few local orbital periods.The model by Cuzzi et al. (2008) is built on the earlier result (Cuzzi et al., 2001) 6 –that chondrule size particles are concentrated and size-sorted in the low-vorticity regionsof the disk. In fact, Cuzzi et al. (2008) showed that in some sporadic cases the chondruleconcentrations can become large enough to form self-gravitating clumps. These clumpscannot become gravitationally unstable because chondrule-sized particles are too stronglycoupled to the gas. Consequently, a sudden contraction of a chondrule clump would causethe gas to compress, a process which is inhibited by its internal pressure. In principle,however, these clumps might survive in the gas disk long enough to undergo a gradualcontraction, eventually forming cohesive planetesimals roughly 10-100km in radius,assuming unit density. This scenario has some advantages over the previous one, namelythat it can explain why chondrules are the basic building blocks of chondritic planetesimalsand why chondrules appear to be size-sorted in meteorites. Interestingly, Alexander et al.(2008) found evidence that chondrules must form in very dense regions that would becomeself-gravitating if they persist with low relative velocity dispersion.The models described in Johansen et al. (2007) and Cuzzi et al. (2008) should beconsidered preliminary and semi-quantitative. There are a number of open issues in eachof these scenarios that are the subject of on-going work by both teams. Moreover, thereis no explicit prediction of the size distribution of the planetesimals produced by thesemechanisms or the associated timescales needed to make a size distribution. Both areneeded, if we are to compare the results of these models with constraints. Nevertheless,these scenarios break the paradigm that planetesimals had to be small at the end of Step 1;in fact, they show that large planetesimals might have formed directly from small particleswithout passing through intermediate sizes. If this is true, Step 2 was affected and visibletraces should still exist in the populations of planetesimals that still survive today: theasteroid belt and the Kuiper belt.Thus, the approach that we follow in this paper is to use Step 2 to constrain the 7 –outcome of Step 1. Our logic is as follows. We define the initial
Size Frequency Distribution(SFD) as the planetesimal SFD that existed at the end of the planetesimal formation phase(end of Step 1, beginning of Step 2). We then attempt to simulate Step 2, assuming thatthe initial SFD is a free parameter of the model. By tuning the initial SFD, we attempt toreproduce the size distribution of the asteroid belt that existed at the end of Step 2. Thesesimulations should allow us to glean insights into the initial SFD of the planetesimals and,therefore, into the processes that produced them. For instance, if we found that the sizedistribution of the asteroid belt is best reproduced starting from a population of km-sizeplanetesimals, this would mean that the “classical” version of Step 1 is probably correct andthat planetesimals formed progressively by collisional coagulation. If, on the contrary, wefound that the initial SFD had to have been dominated by large bodies, this would providequalitative support for the new scenarios (Johansen et al., Cuzzi et al.), namely that largeplanetesimals formed directly from small objects by collective gravitational effects. In thiscase, the initial SFD required by our model would become a target function to be matchedby these scenarios or by competing ones in the future.A caveat to keep in mind is that there might be an intermediate phase between Steps 1and 2 in which the planetesimals, initially “fluffy” objects with low strength, are compressedinto more compact objects by collisions and/or heat from radioactive decay. This phase,while poorly understood, should not significantly modify the SFD acquired in Step 1; we donot consider it here.The approach that we follow in this paper required us to develop several new tools andconstraints. First, in order to simulate Step 2, we had to develop
Boulder , a statisticalcoagulation/fragmentation code of the collisional accretion process. We built this codealong the lines of previous works (e.g., Wetherill and Stewart, 1993; Weidenschillinget al., 1997; Kenyon and Luu, 1999; Kenyon and Bromley, 2001). The description of 8 –
Boulder
2. Reconstructing the properties of the post-accretion asteroid belt
According to our best models (discussed below), the reconstructed main belt had thefollowing properties:i) The SFD for
D >
100 km bodies was the same as the current main belt SFD.ii) The SFD experienced a significant change in slope to shallower power law values near D ∼
100 km. This left a “bump” that can still be seen in the current main belt SFD(Fig. 1).iii) The number of D = 100-1,000 km objects was much larger than in the currentpopulation, probably by a factor of 100 to 1,000iv) The main belt included 0.01-0.1 Earth mass ( M ⊕ ) planetary embryos.Below, we discuss how we obtained properties (i)–(iv) of the reconstructed post-accretion asteroid belt (the expert reader can skip directly to sect. 3). We are confident thatthe reconstructed belt is a reasonable approximation of reality because it was worked outwithin the confines of a comprehensive model that not only explains the major propertiesof the observed asteroid belt but also those of the terrestrial planets (Petit et al., 2001;O’Brien et al., 2007). Therefore, we argue it is reasonable to use the reconstructed belt totest predictions from planetary accretion simulations. The observed SFD of main belt asteroids is shown in Fig. 1. The asteroid populationthroughout the main belt is thought to be complete down to sizes of at least 15 km indiameter, possibly even 6-10 km (Jedicke et al., 2002; Jedicke, personal comm.). Thus, theSFD above this size threshold is the real asteroid SFD. 10 –
An estimate of the total mass of the main asteroid belt can be obtained from theabove SFD, where the largest asteroids have known masses (Britt et al., 2002), and from ananalysis of the motion of Mars, which constrains the contribution of asteroids too small tobe observed individually (Krasinsky et al., 2002). The result is ∼ × − Earth masses or3 . × g. As we argue below, this mass is tiny compared to the mass of solids that hadto exist in the main belt region at the time of asteroid formation. The primordial mass ofthe main belt region can be estimated by following several different lines of modeling work.First, we consider the concept of the so-called minimum mass solar nebula (MMSN;Weidenschilling, 1977; Hayashi, 1981). The MMSN implies the existence of 1-2.5 Earthmasses of solid material between 2 and 3 AU. Accordingly, this means that the main beltregion is deficient in mass by a factor 1,500-4,000. Using the same procedure, Mars’ regionalso appears deficient in mass, though only by a factor of ∼
10. We stress that thesedepletion factors are actually lower bounds because they are estimated using the concept ofthe minimum mass solar nebula.Second, we can consider estimates of the mass of solids needed in the main belt regionfor large asteroids to accrete within the time constraints provided by meteorite data, e.g.within a few million years (Scott, 2006). Published results from different accretion models,i.e. those using collisional coagulation (e.g. Wetherill, 1989) or gravitational instability(Johansen et al., 2007) consistently find that several Earth masses of material in the mainbelt region were needed to make Ceres-sized bodies within a few My. Thus, it seemsunlikely that the current asteroids were formed in a mass-deficient environment.Third, models of chondrule formation that assume they formed in shock waves(Connolly and Love, 1998; Desch and Connolly, 2002; Ciesla and Hood, 2002) require asurface density of the disk (gas plus solids) at 2.5 AU of ∼ , give or take a 11 –Fig. 1.— The size-frequency distribution (SFD) of main belt asteroids for
D >
15 km, assuming,for simplicity, an albedo of p v = 0 .
092 for all asteroids. According to Jedicke et al. (2002),
D >
15 km is a conservative limit for observational completeness. factor of 3. Assuming a gas/solid mass ratio of ∼
200 in the main belt region, this valuewould correspond to a mass of solids of at least 3 Earth masses between 2 and 3 AU.Therefore, the available evidence is consistent with the idea that the asteroid belt haslost more than 99.9% of its primordial mass. This makes the current mass deficit in themain belt larger than a factor of 1,000, with probable values between 2,000 and 6,000.
If so much mass once existed in the primordial main belt region, collisional evolution,dynamical removal processes, or some combination of the two were needed to get rid of itand ultimately produce the current main belt population. Here we list several argumentsdescribing why the mass depletion was unlikely to have come from collisional evolution ofthe main belt SFD. 12 –
1: Constraints from Vesta
The asteroid (4) Vesta is a D = 529 ±
10 km differentiated body in the inner main beltwith a 25-40 km basaltic crust and one D = 460 km impact basin at its surface (Thomaset al., 1997). Using a collisional evolution model, and assuming various size distributionsconsistent with classical collisional coagulation scenarios, Davis et al. (1985) showed thatthe survival of Vesta’s crust could only have occurred if the asteroid belt population wasonly modestly larger than it is today at the time the mean collision velocities were pumpedup to ∼ D ∼
35 km projectile (Thomas et al., 1997). The singular nature of this crater means thatVesta, and the asteroid belt in general, could not have been repeatedly bombarded by large(i.e. ∼
30 km-sized) impactors; otherwise, Vesta should show signs of additional basins(Bottke et al., 2005a, 2005b; O’Brien and Greenberg, 2005). More specifically, given thecurrent collision probabilities and relative velocities among objects of the asteroid belt, theexistence of one basin is consistent with the presence of ∼ D >
35 km (i.e.the current number; see Fig. 1) over the last ∼
2: Constraints from asteroid satellites
Collisional activity among the largest asteroids in the main belt is also constrainedby the presence of collisionally-generated satellites (called SMATS; Durda et al., 2004).Observations indicate that ∼
2% of
D >
140 km asteroids have SMATS (Merline etal., 2002; Durda et al., 2004). It was shown in Durda et al. (2004) that this fraction isconsistent, within a factor of 2–3, with the collisional activity that the current asteroid beltpopulation has suffered over the last 4 Gy. If much more collisional activity had taken 13 –place, as required by a collisional grinding scenario, one should also explain why so fewSMATS are found among the
D >
140 km asteroids. Like above, this constraint appliessince the time when the asteroid belt acquired the current orbital excitation.
3: Constraints from meteorite shock ages
We also consider meteorite shock degassing ages recorded using the Ar- Ar system.Many stony meteorite classes (e.g. L and H-chondrites; HEDs, mesosiderites; ureilites)show evidence that the surfaces of their parent bodies were shocked, heated, and partiallydegassed by large and/or highly energetic impact events between ∼ . ∼
10 times more populous than it is today (Gomes et al.,2005; Strom et al., 2005). It is possible we are looking at a biased record. For example,because shock degassing ages only record the last resetting event that occurred on themeteorite’s immediate precursor, impacts produced by projectiles over the last 4.0 Gy mayhave erased radiometric age evidence for asteroid-asteroid impacts that occurred morethan 4.0 Gy. On the other hand, many meteorite classes show clear evidence for eventsthat occurred 4.5 Gy. Why did the putative erasure events fail to eliminate these ancientAr-Ar signatures? While uncertainties remain, the simplest explanation is that the mainbelt population experienced a minimal amount of collisional evolution between 4.1-4.5 Gy.Thus, meteorites constrain the main belt regions overall collisional activity from the timewhen impacts among planetesimals became energetic enough to produce shock degassing.Taken together, the above arguments imply that the asteroid belt after its orbital 14 –excitation experienced only a moderate amount of net collisional activity over its lifetime.Using numerical simulations, Bottke et al. (2005a,b) found it to be roughly the equivalentof ∼
10 Gy of collisional activity in the current main belt. It is unlikely that this limiteddegree of collisional activity could cause significant mass loss. In fact, the current dustproduction rate of the asteroid belt is at most 10 g yr − (i.e. assuming all IDPs come fromthe asteroids; Mann et al., 1996). Thus, over a 10 Gy-equivalent of its present collisionalactivity, the asteroid belt would have lost 10 g only 1/3 of the current asteroid belts massand a negligible amount with respect to its inferred primordial mass of ∼ g.If collisions since the orbital excitation time cannot explain the mass deficit of theasteroid belt, then the mass either had to be lost early on, when collisions occurred atlow velocities, or by some kind of dynamical depletion mechanism. In the first case, onlysmall bodies could be collisionally eroded, given the low velocities. Even in the “classical”scenario, where the initial planetesimal population was dominated by km-size bodies, it isunlikely that more than 90% of the initial mass could be lost in this manner, particularlybecause these bodies had to accrete each other to produce the larger asteroids observedtoday. We will check this assertion in sect. 3. Accordingly, and remembering that the totalmass deficit exceeds 1,000, 99% (or more!) of the remaining main belt’s mass had to belost by dynamical depletion, defined here as a a process that excited the eccentricities of asubstantial fraction of the main belt population up to planet-crossing values. These excitedbodies would then have been rapidly eliminated by collisions with the planets, with theSun, or ejection from the Solar System via a close encounter with Jupiter. We describe insect. 2.5 the most likely process that produced this depletion and its implications on thetotal number of objects and size distribution of the “post-accretion asteroid belt”. 15 – The limited amount of collisional grinding that has taken place among
D >
100 kmbodies in the asteroid belt has two additional and profound implications. The sizedistribution of objects larger than 100 km could not have significantly changed since the endof accretion (Davis et al., 1985; Durda et al., 1998; Bottke et al., 2005, 2005b, O’Brien andGreenberg 2005). This means the observed SFD for
D >
100 km is a primordial signatureor “fossil” of the accretional process.
This characterizes property (i) of the post-accretionmain belt population.
Moreover, it was shown that the “bump” in the observed SFD at D = 100 km (seeFig. 1) is unlikely to be a by-product of collisional evolution. Bottke et al. (2005, 2005b)tested this idea by tracking what would happen to an initial main belt SFD whose powerlaw slope for D >
100 km bodies was the same for
D <
100 km bodies. Using a range ofdisruption scaling laws, they found they could not grind away large numbers of D = 50-100km bodies without producing noticeable damage to the main belt SFD at larger sizes( D = 100-400 km) that would be readily observable today. Other consequences includethe following. First, they found that large numbers of D ∼
35 km objects would producemultiple mega-basin-forming events on Vesta. This is not observed. Second, the disruptionscaling laws needed to eliminate numerous 50-100 km asteroids would produce, over thelast 3.5 Gy, far more asteroid families from 100-200 km objects than the 20 or so currentfamilies that are observed. Also, the ratio between the numbers of families with progenitorslarger than 100 and 200 km respectively would be a factor of ∼ D <
100 km objects, their model results did suggest that it couldnot be steeper than what is currently observed.
This sets property (ii) of the post-accretionmain belt population . The power-law slope for
D <
100 km could have been exceedinglyshallow, with the observed SFD derived from generations of collisional debris whoseprecursors were fragments derived from break-up events among
D >
100 km asteroids.For this reason, in Fig. 3 and all subsequent figures, we bracket the possible slopes of thepost-accretion SFD for
D <
100 km by two gray lines: the slanted one representing thecurrent slope and the horizontal one representing an extreme case where no bodies existedimmediately below 100 km.
We now further explore property (iii) of the post-accretion main belt, namely theputative dynamical depletion event that should have removed most (i.e. more than 99% inmass) of the large asteroids as required by (a) the current total mass deficit of the asteroidbelt (a factor of at least 1,000) and (b) the relatively small mass depletion factor that couldhave occurred via collisional grinding of small bodies before the dynamical excitation event(at most a factor of 10). So far, the best model that explains the properties of the asteroidbelt is the “indigenous embryos model” (Wetherill, 1992; Petit et al., 2001; O’Brien et al.,2007). Other models have been proposed (see Petit et al. (2002), for a review) but all haveproblems in reproducing at least some of the constraints, so we ignore them here and detailbriefly only the indigenous embryos model below.According to this model (Wetherill, 1992), planetary embryos formed not only in theterrestrial planet region but also in the asteroid belt. The combination of their mutualperturbations and of the dynamical action of resonances with Jupiter eventually removed 17 –them from the asteroid belt (Chambers and Wetherill, 2001; O’Brien et al., 2006). Beforeleaving the belt, however, the embryos scattered the asteroids around them. This excitedthe asteroids’ eccentricities and inclinations but also forced the asteroids to random-walkin semi-major axis. As a consequence of their mobility in semi-major axis, many asteroidsfell, at least temporarily, into resonance with Jupiter, where their orbital eccentricities andinclinations increased further. By this process, 99% of the asteroids acquired an eccentricitythat exceeded the value characterizing the stability boundary of the current asteroid belt(O’Brien et al., 2007). Thus, their fate was sealed and these objects were removed duringor after the formation of the terrestrial planets.In addition, about 90–95% of the asteroids that survived this first stage should havebeen removed by sweeping secular resonances due to a sudden burst of radial migrationof the giant planets that likely triggered the so-called “Late Heavy Bombardment” of theMoon and the terrestrial planets (Gomes et al., 2005; Strom et al., 2005; Minton andMalhotra, 2009). This brings the dynamical depletion factor of the asteroid belt to a totalof ∼ ∼
100 cannot be excluded. It is unlikely that thedynamical depletion factor could be smaller than this.Large-scale dynamical depletion mechanisms are size independent. Thus, a massdepletion factor of ∼ ∼ This is used to set property (iii) of the post-accretion main belt.
At large asteroid sizes, we are affected somewhat by small number statistics. Forinstance, assuming a dynamical depletion factor of 1,000, the existence of one Ceres-sizebody might imply the existence of 1,000 bodies of this size, but is also consistent, at the 18 –10% level, with the existence of only 100 of these bodies.More precisely, given a population of N bodies, each of which has a probability p tosurvive, the probability to have M specimen in the surviving population is P = (1 − p ) N − M p M N ! / [ M !( N − M )!] . (1)From this, assuming that p = 10 − , one can rule out at the 2- σ level that the population ofCeres-size bodies contained less than 21 objects, because otherwise the odds of having onesurviving object today would be less than 2.1%. Similarly, we can rule out the existence ofmore than 3876 Ceres-size objects in the original population, otherwise the odds of havingonly one Ceres today would be smaller than 2.1%. In an analog way, for the population ofbodies with D >
450 km (3 objects today), the 2- σ lower and upper bounds on the initialpopulation are 527 and 7441 objects, respectively. Fig. 2 shows the cumulative SFD of thereconstructed asteroid belt in the 100-1,000 km range (the current SFD scaled up by afactor 1,000; solid curve) and the 2- σ lower and upper bounds computed for each size asexplained above (dashed lines). The meaning of this plot is the following: consider all theSFDs that could generate the current SFD via a random selection of 1 object every 1,000;then 95.8% of them fall within the envelope bounded by the dashed curves in Fig. 2. Wehave checked this result by generating these SFDs with a simple Monte-Carlo code.We have also introduced a functional norm for these SFDs, defined as D = X D i | log( N ′ ( > D i ) /N ( > D i )) | , (2)where D i are the size bins between 100 and 1,000 km over which the cumulative SFD iscomputed (8 values), N ( > D i ) is the current cumulative SFD scaled up by a factor 1,000and N ′ ( > D i ) is a cumulative SFD generated in the MonteCarlo code. We have found that95.8% of the MonteCarlo-generated SFDs have D < .
14. By repeating the MonteCarloexperiment with different (large) decimation factors p we have also checked that the value 19 –Fig. 2.— The size-frequency distribution (SFD) in the 100-1,000 km range, expected for the mainbelt at the end of the accretion process (e.g. for the reconstructed belt), assuming a dynamicaldepletion factor of 1,000. The solid line is obtained scaling the current SFD (Fig. 1) by a factor1,000; the dashed lines show the 2- σ lower and upper bounds, given the current number of objects.The shape of these curves does not change (basically) with the dynamical depletion factor 1 /p ; thecurves simply shift along the y -axis by a quantity log((1 /p ) / of D is basically independent on p , while all the curves in Fig. 2 shift along the y -axisproportionally to p (so that the solid line coincides with the current SFD, scaled up by afactor 1 /p ). These results will be used when testing some of our model results in section 4and 5.The embryos-in-the-asteroid-belt-model in Wetherill (1992) does not only explain thedepletion of the asteroid belt but also the final orbital excitation of eccentricities andinclinations of the surviving asteroids and the radial mixing of bodies of different taxonomictypes (Petit et al., 2001). In addition, it provides a formidable mechanism to explain thedelivery of water to the Earth (Morbidelli et al., 2000; Raymond et al., 2004, 2007).In summary, the “indigenous embryos” model does a good job at explaining the orbital 20 –and physical properties of the asteroid belt within the larger framework of terrestrial planetaccretion. To date, it is the only model capable of doing so. Thus, if we trust this model,embryos of at least one lunar mass had to exist in the primordial asteroid belt. Thischaracterizes property (iv) of the post-accretion main belt population.
A successful accretionsimulation should not only be able to form asteroid-size bodies in the main belt, but also asignificant number of these embryos.
3. The classical scenario: accretion from kilometer-size planetesimals
We start our investigation by simulating the classical version of Step 2 of the accretionprocess. In other words, we assume that kilometer-size planetesimals managed to form inStep 1, despite the meter-size barrier; the accretion of larger bodies occurrs in Step 2, bypair-wise collisional coagulation. We simulate this second step using our code
Boulder .The simulations account for eccentricity e and inclination i excitation due to mutualplanetesimal perturbations as well as ( e, i ) damping due to dynamical friction, gas dragand mutual collisions. Collisions are either accretional or disruptive depending on the sizesof projectiles/targets and their collision velocities.The disruption scaling law used in oursimulations, defined by the specific dispersion energy function Q ∗ D , is the one provided bythe numerical hydro-code simulations of Benz and Asphaug (1999) for undamaged sphericalbasaltic targets at impact speeds of 5km/s. See the electronic supplement for the detailsof the algorithm. However, in section 3.4, we will examine what happens if we use a Q ∗ D function that allows D <
100 km disruption events to occur much more easily thansuggested by Benz and Asphaug (1999), as argued in Leinhardt and Stewart (2009) andStewart and Leinhardt (2009).Here, and in all other simulations (unless otherwise specified), we start with a totalof 1.6 M ⊕ in planetesimals within an annulus between 2-3 AU. By assuming a nominal 21 –gas/solid mass ratio of 200, this corresponds to the Minimum Mass Solar Nebula as definedin Hayashi (1981). The bulk mass density of the planetesimals is set to 2 g/cm , the averagevalue between those measured for S-type and C-type asteroids (Britt et al., 2002). Thesimulations cover a time-span of 3 My, consistent with the mean lifetime of proto-planetarydisks (Haisch et al., 2001) and hence the probable formation timescale of Jupiter. Theinitial velocity dispersion of the planetesimals is assumed to be equal to their Hill speed(i.e. v orb [ M obj / (3 M ⊙ )] / , where v orb is the orbital speed of the object, M obj is its mass, and M ⊙ is the solar mass). The lower size limit of planetesimals tracked in our simulation isdiameter D = 0 . D = 2 km. In this simulation,the total mass lost into dust by collisional grinding is 7 . × g, i.e. more than oneEarth mass but only 76% of the original mass. This is consistent with our claim in section 2that, even starting with km-size planetesimals, low-velocity collisions cannot deplete morethan 90% of the initial mass.The final SFD of the objects produced in the simulation is illustrated by the blackcurve in Fig. 3. We find this SFD does not reproduce the turnover to a shallower slopethat the post accretion asteroid belt had to have at D ∼
100 km (i.e. property (ii) ofthe reconstructed belt). Moreover, in the final SFD shown in Fig. 3, there are about1.25 million bodies with
D >
35 km. Even if we were to magically reduce this populationinstantaneously by a factor 200, in order to reduce the number of
D > ∼ , D >
35 km objects remaining in the system.Recall (section 2) that D ∼
35 km projectiles can form mega-basins on Vesta and thatthe formation of a single basin is consistent with the existence of 1,000 of these objectsin the main belt over 4 Gy. Thus, 6,200 objects would statistically produce 6 basins; the 22 –Fig. 3.—
The gray lines show the reconstructed (i.e., post-accretion) main belt SFD. The solidgray curve shows the observed main belt SFD for 100 < D < ,
000 km asteroids scaled up 200times, so that the number of bodies with
D > km matches that produced in the simulation. Thedashed lines show the upper and lower bound of the main belt power law slope in the 20-100 kmrange (Bottke et al., 2005). The upper bound corresponds to the current SFD slope. The verticaldotted lines show the sizes of Lunar/Martian-sized objects for bulk density 2 g cm − . These sizeembryos are assumed to have formed across the inner Solar System (Wetherill, 1992; O’Brien et al.,2007). The black curve shows the final SFD, starting from 1 . × planetesimals with D = 2 km,at the end of the 3 My coagulation/grinding process. probability that only one mega-basin would form, according to formula (1), is only 1.2%.For all these reasons, we think that this simulation produces a result that is inconsistentwith the properties of the asteroid belt. To test whether these results are robust, weperformed additional simulations as detailed below. 23 – One poorly understood issue is how long the accretion phase should last, i.e. therequired length of our simulations. Thus, we continued the simulation presented above upto 10 My. The result is illustrated in Fig. 4a. We find that the total amount of mass lostinto dust increases only moderately, reaching at 10 My 78% of the mass at t = 0. Also,SFD does not significantly change between 3 and 10 My. The size of the largest embryosdoes grow from slightly less than 6,000 km to about 7,000 km, mostly by agglomeratingobjects smaller than a few tens of kilometers. Accretion and collisional erosion reduce thecumulative number of D > . × to 9 . × . For the size range80 < D < D ∼ D >
Another poorly-constrained parameter is the initial total mass of the planetesimalpopulation. For this reason, we tested a range of options. Here we discuss a simulationstarting with a system of planetesimals carrying cumulatively 5 M ⊕ instead of 1.6 M ⊕ as in Fig. 3. This total mass is of the order but slightly larger than that computed inWeidenschilling (1977) and is the same as assumed in Wetherill (1989). The result is shownin Fig. 4b. As a result of the factor of ∼ ∼ < D < As in Fig. 3, but for additional simulations. (a) Continuation of the simulation of Fig. 3 up to10 My. (b) Starting with 5 M ⊕ of D = 2 km planetesimals (here the solid gray line reproduces the currentSFD scaled up by a factor 600, instead of 200 as in all other panels). (c) Starting with 1.6 M ⊕ of D = 600 mplanetesimals. (d) Starting with 1.6 M ⊕ of D = 6 km planetesimals. (e) Assuming that Q ∗ D is 1/8 of thevalue given in Benz and Asphaug for ice. (f) Introducing turbulent scattering with γ = 2 × − (a run with γ = 2 × − resulted in basically no accretion). See text for comments on these results. Thus, the turn-over of the SFD at D ∼
100 km is still not reproduced. As aconsequence, there are about 2.5 million bodies with
D >
35 km, the putative size of the 25 –basin-forming projectile on Vesta. Invoking an instantaneous dynamical depletion eventcapable of removing a factor of 600 from the population, a value needed to reduce the thenumber of
D >
D >
35 km objects wouldbe left in the system. Thus, about 4 basins should have formed on Vesta; the probabilitythat only 1 would have formed, according to (1) is 6%.For all these reasons, we think that it would be very difficult to claim that thesimulation of Fig. 4b is successful. Notice that also in this case the total amount of masslost in collisional grinding does not exceed 86% of the initial mass.
Fig. 4c and 4d illustrate how the results depend on the size of the initial planetesimals.The simulation in Fig. 4c starts from 1.6 M ⊕ of material in D = 600 m planetesimalsinstead of D = 2 km as in the nominal simulation. The final SFD is indistinguishable fromthat of the nominal simulation up to D ∼ D = 6 kmplanetesimals. The final SFD has an excess of 10-200 km objects relative to the SFD in thenominal simulation, but the SFDs are similar in the D = 40-4,000 km range.Thus, these cases can be rejected according to the same criteria applied in Sec. 3.2 . In the previous simulations we assumed that the planetesimals have size-dependentspecific disruption energy ( Q ∗ D ) characteristic of undamaged basalt targets being hit at 26 –several km/s (see Benz and Asphaug, 1999). Leinhardt and Stewart (2009) have arguedthat the original planetesimals might have been weak aggregates with little strength.Moreover, Stewart and Leinhardt (2009) showed that early planetesimals should have low Q ∗ D also because impact energy couples to the target object better at low velocities. Inthese conditions, Q ∗ D might be more than an order of magnitude weaker than the one thatwe adopted at all sizes. To test how the results change for extremely weak material, wehave re-run the coagulation simulation starting with 1.6 M ⊕ in D = 2km planetesimals(that of Fig. 3), this time assuming Q ∗ D is one eighth of that reported by Benz and Asphaug(1999) for competent ice struck at impact velocities of 1km/s. This is fairly close to thevalue found by Leinhardt and Stewart (2009) for strenghtless planetesimals.The resulting SFD is shown in Fig. 4e. Overall, the outcome is not very different fromthat of the nominal simulation. Despite of the weakness of the objects, the total mass lostin collisional grinding (8 . × g) does not exceed 90% of the initial mass, as we argued insection 2. Interestingly, though, this simulation fails to form objects more massive than ourMoon. Thus, in conclusion, the change to a new disruption scaling law produces a worse fitto the constraints than before, particularly because constraint (iv) of the reconstructed belt(e.g. the existence of Lunar-to-Martian mass embryos) is not fulfilled. In all previous simulations we have implicitly assumed that the gas disk in which theplanetesimals evolve is laminar. Thus, the gas can only damp the velocity dispersion ofthe planetesimals. In this case, the sole mechanism enhancing the planetesimal velocitydispersion is provided by mutual close encounters, also named viscous stirring (Wetherilland Stewart, 1989; see section. 1.4.1 of the electronic supplement). In reality, the diskshould be turbulent at some level. As discussed in Cuzzi and Weidenschilling (2006), local 27 –turbulence contributes by stirring the particles and increasing their velocity dispersion.This effect is maximized for meter-sized objects. In addition, however, turbulent disksshow large-scale fluctuations in gas density (Papaloizou and Nelson, 2003). The fluctuatingdensity maxima act as gravitational scatterers on the planetesimals, providing an additionalmechanism of excitation for the velocity dispersion that is independent of the planetesimalmasses (Nelson et al., 2005). To distinguish this mechanism from that discussed by Cuzziand Weidenschilling, we call it turbulent scattering hereafter. Ida et al. (2008) showedwith simple semi-analytical considerations that turbulent scattering can be a bottleneck forcollisional coagulation because it can move collisions from the accretional regime to thedisruptive regime. Here, we check this result with our code.
Boulder accounts for turbulent scattering using the recipe described in Ida et al.(2008) and detailed in sect. 1.4.7 of the electronic supplement of this paper. In short, inthe equations for the evolution of the velocity dispersion, there is a parameter γ governing“turbulence strength”. The effective value of γ in disks that are turbulent due to themagneto-rotational instability is uncertain by at least an order of magnitude. Simulationby Laughlin et al. (2004) suggest that γ ∼ − –10 − , but values as low as 10 − cannot beexcluded (Ida et al., 2008). The relationship between γ and the more popular parameter α that governs the viscosity in the disk in the Shakura and Sunyaev (1973) description hasbeen recently investigated in details by Baruteau (2009). He found that α ∝ γ / ( h/a ) where h/a is the scale-height of the gas disk; for h/a = 3%, γ = 10 − corresponds to α ∼ × − .We have re-run the coagulation simulation of Fig. 3, assuming γ = 2 × − (whichcorresponds to α ∼ × − according to Baruteau’s scaling).. In this run we adopt aninitial velocity dispersion of the planetesimals that is larger than that assumed in thenon-turbulent simulations illustrated above. Recall that in all previous simulations the 28 –initial velocity dispersion of the objects was set equal to their Hill velocity. These velocitiesare too small for a turbulent disk. If we adopted them, we would get a spurious initialphase of fast accretion, before the velocities were fully stirred up by the turbulent disk.Thus, we need to start with velocity dispersions that represent the typical values achievedin the disk. More precisely, for D = 2 km objects, we assume initial eccentricities andinclinations that are the equilibrium values obtained by balancing the stirring effect of theturbulent disk with the damping effects due to gas drag and mutual collisions (Ida et al.,2008). We used Boulder to estimate what these values should be by suppressing collisionalcoagulation/fragmentation and letting the velocity dispersion evolve from initially circularand co-planar orbits. We found that at equilibrium we get e ∼ i ∼ . × − . This valueis attained in about 50,000 years, whereas e ∼ i ∼ . × − is attained in 5,500 years.In the simulation performed with this set-up, growth is fully aborted. The largestplanetesimals produced in 3 My are just 2.5 kilometer in diameter, whereas 9 . × g arelost in dust due to collisional grinding. This result is due to the fact that collisions becomerare (because the gravitational focussing factor is reduced to unity by the enhanced velocitydispersion) and barely accretional even for “strong” Q ∗ D disruption functions that is used inthis simulation (for basaltic targets hit at 5km/s; Benz and Asphaug, 1999). We also rana simulation where we did not modify the initial velocity dispersion of the planetesimals,although we consider this unrealistic for the reasons explained above. In this case there isa short initial phase of growth, as expected, which rapidly shuts off; the largest objectsproduced have D = 40km. These results confirm the analysis of Ida et al. (2008); accretionis impossible in turbulent disks if all planetesimals are small.To investigate how weak “turbulence strength” should be to allow accretion from D = 1 km planetesimals, we also ran a simulation assuming γ = 2 × − . In this case,we set as initial values e = 2 i = 2 . × − . Using the Baruteau’s scaling, this value of γ
29 –corresponds to α ∼ × − , that is well below a minimum reasonable value in a turbulentdisk; however, it might be acceptable for a dead zone , e.g. a region of the disk where themagneto-rotational instability is not at work. The solid curve in Fig. 4f shows the finalSFD in this simulation. It now looks similar to that obtained in the nominal simulation ofFig. 3, which had no turbulent scattering. Thus, this very low level of turbulence does notinhibit growth, but like the nominal simulation in Fig. 3, the resulting SFD is inconsistentwith that of the reconstructed main belt. From the simulations illustrated in this section, we conclude that the SFD of the initialplanetesimals were not dominated by objects with sizes the order of one kilometer. In fact,in a turbulent disk, 1 km planetesimals would not have coagulated to form larger bodies. Ina dead zone, collisional coagulation would have produced a final SFD that is inconsistentwith the current SFD in the main asteroid belt because the bump at D ∼
100 km is notreproduced; also we find it unlikely (at the few percent level) that only one big basinformed on Vesta with such a SFD, even in the case of an instantaneous dynamical depletionevent of the appropriate magnitude. While we were writing the final revisions of this paper,we became aware that Weidenschilling (2009) reached the same conclusions with similarnon-turbulent simulations performed with a different code.Obviously, there is an enormous parameter space left to explore, and -strictlyspeaking- an infinite number of simulations would be necessary to prove that the SFD ofthe reconstructed post-accretion main belt is incompatible with the classical collisionalaccretion model starting from km-size planetesimals. Nevertheless, we believe that the 9simulations presented above are sufficient enough to argue that our result is reasonablyrobust. 30 –Given this conclusion, in the next sections we try to constrain which initial planetesimalSFD would lead, at the end of Step 2, to the SFD of the reconstructed main belt.
4. Accretion from 100 km planetesimals
We start our search for the optimal initial planetesimal SFD by assuming that allplanetesimals originally had D = 100 km. Note that no formation model predicts that theinitial planetesimals had to have the same size. We make this assumption as a test caseto probe the signature left behind in the final SFD by the initial size of the objects. Morespecifically, we attempt to satisfy property (ii), the turnover of the size distribution at D ∼
100 km, assuming that this might be the signature of the minimal size of the initialplanetesimals.As before, our input planetesimal population carries cumulatively 1.6 M ⊕ . This impliesthat there are initially 9 . × planetesimals. The coagulation simulation covers a 3 Mytime-span. No turbulent scattering is applied.The final SFD is shown in Fig. 5a (solid curve). This SFD has the same properties ofthat obtained by Weidenschilling (2009) starting from D = 50km planetesimals. A sharpturnover of the SFD is observed at the initial planetesimal size. This is in agreementwith the observed “bump” (i.e. property (ii) of the reconstructed belt). However, the finalSFD is much steeper than the SFD of the current asteroid belt. Nevertheless, it wouldbe premature to consider this simulation unsuccessful because we showed in Fig. 2 thatthe slope of the SFD of the reconstructed asteroid belt has a large uncertainty. Thus, inFig. 5b, we replot the final SFD against the 2- σ bounds of the reconstructed main beltSFD. These bounds have been taken from Fig. 2 and are “scaled up” by a factor of 10 sothat they match the total number of D >
100 km objects found in the simulation. As one 31 –Fig. 5.— (a): Non-turbulent simulation starting with 1.6 M ⊕ in D = 100 km objects. The bulletshows the initial size and total number of planetesimals. The black curve reports the final SFDobtained after 3 My of collisional coagulation. The grey lines sketch the reconstructed asteroid beltSFD as in Figs. 3 and 4. (b): like panel (a), but in this case the grey solid lines report the 2- σ bounds of the SFD of the reconstructed asteroid belt, from Fig. 2. Moreover, all grey lines havebeen moved upwards by a factor of 10, to match the number of D > can see, the final SFD falls slightly out of the lower bound of the reconstructed SFD. Thismeans that the result is inconsistent, at 2- σ , with the data (i.e. with the current SFD).Another way to check the statistical match between the simulation SFD and thereconstructed main belt SFD is through the parameter D defined in (2). The SFD resultingfrom this simulation has D = 8 .
20; only 0.5% of the SFDs generated from the current SFDin a Monte-Carlo code have D larger than this number. Thus, we can actually reject theresult of this simulation as inconsistent with the reconstructed main belt at nearly the 3- σ level.Rejecting this simulation, however, is not enough to exclude the possibility that theinitial planetesimals were ∼
100 km in size. Before accepting this conclusion, we need tomore extensively explore parameter space. The simulation reported in Fig. 5 is indeedsimplistic because it did not account for the effects of turbulence in the disk. Recall, 32 –however, that the works that motivated us to start with large planetesimals (Johansenet al., 2007; Cuzzi et al., 2008) assumed (and required) a turbulent disk, so we need tocope with turbulence effects. Turbulence should affect our simulation in two respects:(I) theoretical considerations (Cuzzi et al., 2008) indicate that planetesimals should formsporadically over the lifetime of the gas disk, in qualitative agreement with meteorite data(Scott, 2006), whereas in the previous simulation we introduced all the planetesimals at t = 0; (II) turbulent scattering should enhance the velocity dispersion of the planetesimals,as we have seen in sect. 3.5.. With a new suite of more sophisticated simulations, we nowattempt to circumvent our model simplifications. We do this in steps, first addressing issue(I), still in the framework of a laminar disk, and then (II).To account for (I), we randomly introduce planetesimals in Boulder over a 2 Mytime-span in two different ways. In case-A, we assume all the mass was initially in smallbodies. Every time a 100-km planetesimal is injected in the simulation, we remove an equalamount of mass from the small bodies. In case-B, we inject equal mass proportions ofsmall bodies and planetesimals. This second case mimics the possibility that planetesimalformation is regulated by the availability of ‘building blocks’. Note that chondrulesmay be such building blocks; they are an essential component of many meteorites andthey appear to have formed progressively over time (Scott, 2006). In both cases, wemodel the small body population with D = 2 m particles, which might be considered astracers, representing a population of smaller bodies of the same total mass (for instancechondrule-size particles in the model of Cuzzi et al., or meter-size boulders in the model ofJohansen et al.). In the previous section, bodies of any size accreted or disrupted dependingon the impact energy relative to their specific disruption energy, consistent with the classicalscenario of planetesimal accretion. Here, we change our prescription. We assume that oursmall-bodies/tracers do not disrupt or accrete upon mutual collisions. The rationale for thiscomes from the models of Johansen et al. and Cuzzi et al. and is twofold. First, bodies so 33 –small have difficulty sticking to one other, so that they can not grow by binary collisions;when they form large planetesimals, they do so thanks to their collective gravity. Second,a large number of small bodies have to be in the disk at all times, in order to be able togenerate planetesimals over the spread of timescales shown by meteorite data (Scott, 2006).However, the typical relative velocities of the small bodies are not very small, because ofthe effects of turbulence. Thus, either the small bodies are very strong or, if they break,they must be rapidly regenerated by whatever process formed them from dust grains in firstplace.The solid black line in Fig. 6a shows the final SFD obtained in case-A. The availabilityof small bodies promotes runaway growth among the 100-km planetesimals introduced atearly times into the simulation. This leads to very distinctive signatures in the resultingSFD: a steep fall-off above the input size of the planetesimals; the presence of very largeplanetary embryos; a very shallow slope at moderate sizes (in this case, from slightly morethan 100 to several 1,000km) and an overall deficit of objects in this size-range. As a result,the SFD that does not match that of the reconstructed main belt even within the 2- σ boundaries.For completeness, we present in Fig. 6b two additional variants of this nominalsimulation. In one, inspired by the Cuzzi et al. work, we assume that our 2 m-particles aretracers for chondrule-size objects. Chondrules would be strongly coupled with the gas, so weassume, for simplicity sake, that the particles are perfectly coupled with the gas. In practice,instead of letting our particles evolve in velocity space according to the damping/stirringequations of Boulder (as in the nominal simulation), we force them to have the samevelocity of the gas (i.e. 60 m/s) relative to Keplerian orbits. The result is illustrated by theblack solid curve. In the second variant, inspired by the Johansen et al. work, we assumethat our 2 m-particles are tracers for meter-size boulders. These objects should migrate 34 –Fig. 6.—
Additional coagulation simulations with 100 km initial planetesimals. The black curves showthe final SFDs; the grey lines sketch the reconstructed asteroid belt SFD and its 2- σ bounds as in Figs. 5b.(a) Non-turbulent case-A simulations, where initially all the mass is in D = 2 m particles and the 100 kmobjects are introduced progressively over 2 My. Here, the velocity dispersion of the 2 m-particles evolvesaccording to the damping (collisional & gas drag) and viscous stirring equations. (b) The same as (a) butwith different assumptions on the dynamical evolution of the particles. The dashed curve refers to the casewhere the radial drift speed of the 2 m particles due to gas drag is also taken into account. The solid curverefers to the case where the particles are assumed to be tracers of much smaller bodies, fully coupled withthe gas. (c) Non-turbulent case-B simulation, where equal masses of 2 m-particles and 100km-planetesimalsare introduced progressively over 2 My, up to a total of 3.2 M ⊕ . (d) Case-A simulations where the velocitydispersion of particles and planetesimals is stirred by turbulent scattering. The solid curve is for γ = 2 × − ,the dashed and dotted curves for γ = 2 × − ; in the case shown by the dotted curve we impose that theminimal eccentricity of the D = 2 m particles cannot decrease below 0.001 and the inclination below 5 × − . very quickly towards the Sun by gas drag. We neglect radial migration in Boulder becausethe annulus that we consider (2–3 AU) is too narrow. This is equivalent to assuming that 35 –the bodies that leave the annulus through its inner boundary are substituted by new bodiesdrifting into the annulus through its outer boundary. The drift speed, however, shouldbe included in our calculation of the relative velocities of particles and planetesimals.Accordingly, we add a 100m/s radial component to the velocities of all our tracers. Theresult is illustrated by the black dashed curve. We find the dashed and solid curves are verysimilar to the solid curve of panel a. Thus, none of the considered effects appear to havemuch effect in changing the final SFD. Based on this, we believe it will be reasonable toneglect these corrections to the velocity of our particles in the remaining simulations. Thisreduces the number of cases to be investigated and simplifies our discussion.The solid black line in Fig. 6c shows the final SFD obtained in case-B. Again, thesignature of runaway growth, triggered by the availability of a large amount of mass insmall particles, is highly visible. Consequently, the SFD does not match at all that of thereconstructed main belt. In particular, it shows a strong deficit of 100-1,000 km bodies.In order to get a better match with the SFD of the main belt, we would need tosuppress/reduce the signature of runaway growth. One potential way to do this is toenhance the dispersion velocities of small bodies via turbulent scattering (see sect. 3.5).Thus, we proceed to the inclusion of this effect, which addresses issue (II) mentioned abovein this section.We start by assuming that the parameter γ = 2 × − (relatively small compared toexpectations). For the 2 m-particles we assume initial eccentricities and inclinations thatare the equilibrium values obtained by balancing the stirring effect of the turbulent diskwith the damping effects due to gas drag and mutual collisions ( e ∼ i ∼ × − ). Forthe 100km-planetesimals that are injected in the simulation, we assume that eccentricityand inclination are 1/2 of their equilibrium values (accounting also for tidal damping; Idaet al., 2008). This means e = 0 .
005 and i = e/
2. A simulation of the evolution of the 36 –eccentricity/inclination of a 100 km-planetesimal in a turbulent disk shows that these valuesare achieved in ∼ ,
000 y starting from a circular orbit within the disk’s mid-plane. Thesolid black curve in Fig. 6d shows the result for the case-A simulation with this settings.Even with this small amount of turbulent scattering, the accretion is strongly inhibited andthe largest objects do not exceed D = 200 km. In a second simulation, we decreased γ bya factor of 10, as well as the initial eccentricities and inclinations. This makes turbulentscattering so weak that runaway growth turns back on, making the final SFD similar tothose shown in panel b. We remark, though, that the initial eccentricity and inclinationof the particles are much smaller than what one might expect, due to simple diffusion dueto local turbulence (Cuzzi and Weidenschilling, 2006). This might have favored runawaygrowth. In reality, turbulent diffusion should prevent the eccentricities and inclinations ofsmall bodies to become smaller than e ∼ i ∼ − (Cuzzi, private communication). Thus,we did a third simulation, still adopting γ = 2 × − , but imposing that e and i of ourparticles/tracers do never decrease below these minimal values. The result is shown by thedotted curve. Runaway growth is now less extreme than in the previous case (the slope ofthe SFD just above D = 100 km is shallower and the final embryos are smaller), but it isstill effective. Again, the final SFD is inconsistent with the asteroid belt constraints. Thus,we conclude that the accretion process in the presence of a large mass of small particlesis very sensitive to the effects of turbulent scattering: if turbulent fluctuations are tooviolent, accretion is shut off; if they are too weak, runaway growth occurs. In both cases,no match can be found for the reconstructed main belt.We conclude from these simulations that the initial planetesimal SFD had to span asignificant range of sizes; our best guess would be upwards from 100 km. In the next twosections we will try to constrain the size ( D max ) of the largest initial planetesimals and theSFD in the 100 km– D max range that are necessary to achieve a final SFD consistent withthat of the reconstructed belt. 37 –
5. Accretion from 100–500 km planetesimals
Here we start with planetesimals in the D = 100-500 km diameter range, with aninitial SFD whose slope is the same as the one observed in the reconstructed (and current)SFD of the asteroid belt.In the first simulation, all the planetesimals are input at t = 0, as in the simulationof Fig. 5a. In order to place 1.6 M ⊕ in these bodies, we have to assume they were ∼ ,
000 times more numerous than current asteroids in the same size range. As in theprevious sections, no turbulent scattering is taken into account in this first simulation. Thefinal SFD is shown in Fig. 7a. An important result is that the slope of the input SFD ispreserved to the end of the simulation. The turn-over of the final SFD at D ∼
100 km isrecovered and a few Lunar-mass embryos are produced. Notice, though, that the final SFDshows a sharp break at the initial planetesimals’ maximum size (
D >
500 km); for sizeslarger than this threshold, the slope is steeper than the initial slope in the 100-500km range.The observed SFD of the asteroid (middle grey solid line in the figure) does not show thisbehavior. As discussed in section 2, however, the observed SFD is determined by a singleobject (i.e. Ceres) and therefore the determination of the SFD of the reconstructed belt isaffected by small number statistics. With a 95.8% probability, the post-accretion SFD ofthe asteroid belt should be confined between the upper and lower solid gray curves shownin the figure. We find the final SFD in our simulation does fulfill this requirement very well.One might be tempted to claim success on the basis of this simulation, but we cautionthat this run is overly simplistic for the reasons that we enumerated in the previous section:we assumed that (I) all planetesimals are introduced at time = 0 My and (II) turbulentscattering was not taken into account. We lift these approximations below.A simulation conducted with the case-A set-up discussed in the previous section(Fig. 7b) exacerbates the break of the SFD at D ∼ Like Fig. 6, but starting from initial planetesimals in the 100-500 km range and a mainbelt-like SFD in this size interval, as shown by the filled dots in panel (a). The three solid graycurves reproduce the reconstructed SFD and its 2- σ bounds (see Fig. 2), rescaled so that the totalnumber of D > planetesimals introduced early in the simulation efficiently gobble up the small bodies andform embryos more massive than Mars via runaway growth. The final SFD is in fact typicalof this growth mode (see sect. 4): it shows a steep slope just above the initial size of theplanetesimals and a deficit of ∼ σ boundaries of the reconstructed belt’s SFD, with only 13 bodies with diameters between 500and 2,000 km (5 bodies if D is restricted to be larger than 530 km). Given the dynamicaldepletion factor of 1,000 required to bring the number of D > D ∼ γ = 2 × − . Unlike the run in Fig. 6d, this valueof turbulence strength does not inhibit accretion in this case, such that the signature ofrunaway growth is evident in the final SFD (this simulation does not produce a good matchto the main belt SFD, as in the cases of panels b and c). We defer to section 6.1 a discussionon which values of γ allow accretion as a function of planetesimal sizes. Conversely, if γ isincreased to γ = 2 × − , accretion is inhibited and the final SFD above the D = 500 kmdrops vertically. As in the previous section, we conclude that the accretion process is veryunstable with respect to turbulent scattering: if turbulence is too violent, accretion is shutoff; if it is too weak, runaway growth occurs.Thus, from all our runs, we conclude that it is unlikely that the asteroid belt SFDcan be reproduced if we start with planetesimals solely in the 100–500 km size range.Our insights from these runs also suggest that reducing the size of the largest initialplanetesimals is only going to make the match more problematic. Thus, we argue that theinitial planetesimals had to span the full 100–1,000km range, with a power law slope similarto that of the main belt SFD. In the next section we check whether this initial planetesimaldistribution does indeed lead to a final distribution matching all asteroid belt constraints. 40 –
6. Accretion from 100–1,000 km planetesimals
In this section, we redo all the runs presented in the previous section but extend thesize distribution of the initial planetesimals up to Ceres-size bodies ( D ∼ M ⊕ , instead of the 1.6 M ⊕ used in all other previous simulations, we placethe remaining mass (0.7 M ⊕ ) in D = 2 m bodies. These bodies are treated like normalplanetesimals in this run: they can accrete or break in mutual collisions. We find that ∼
10% of the meter-size bodies coagulate with the large planetesimals, while the rest areeliminated by collisional grinding. The final SFD, shown by the black solid curve in Fig. 8a,is now consistent with properties (i)-(iv) of the reconstructed post-accretion asteroid belt.Our results, once properly scaled, are consistent with those found by direct N -bodysimulations also starting with large planetesimals (e.g. Kokubo and Ida, 2000; KI00). Thesimulation in KI00 lasts 500,000 y in an annulus centered on 1 AU. This is equivalent toour simulations where we examine what happens over 2 My to an annulus centered around2.5 AU. After 2 My of coagulation, our biggest object has a mass of 1 . × g. In KI00, itsmass is 2 × g. In KI00 there are 7 bodies more massive than 10 g in their ± .
04 AUwide annulus. In an annulus that is 7.5 times larger we have 51 bodies more massive thanthis threshold. More importantly, KI00 also finds that the SFD of their initial planetesimalremains essentially unchanged during the simulation (see their Fig. 8).Using the case-A and B set-ups (Figs. 8b and 8c), the final SFDs show the distinctsignatures of runaway growth, though it is more pronounced in case-A than in case-B. Thefinal planetary embryos are also more numerous and massive than in the nominal simulation 41 –Fig. 8.—
Like Fig. 7, but starting from initial planetesimals in the 100-1,000 km range and anSFD slightly shallower than that of the main belt in this size interval, as shown by the bullets inpanel (a). The solid gray line reproduces the current SFD of the main belt, rescaled so that thetotal number of
D > of panel (a). Notice that, in the case-A simulation, the final number of objects in theinput size-range is smaller than in the other cases. The reason is that every time a largeplanetesimal is introduced in the simulation, a number of small bodies of equivalent totalmass is removed. If at some point small bodies are no longer available because they havebeen accreted by the growing embryos, the introduction of new planetesimals is terminated.Fig. 8d shows results of simulations accounting for the excitation of the relativevelocities due to turbulent scattering. We assume γ = 2 × − , which was inhibitingaccretion in the cases with D max ≤ D max =1,000km) we find 42 –that this level of turbulence strength has little effect on the accretion process. The reasonsfor this are discussed in sect.6.1.In summary, all the simulations shown in Fig. 8 give results that are consistentwith the reconstructed SFD of the main belt. The apparent robustness of our results,as opposed to the systematic failures or improbable matches obtained in the previoussections, gives us increased confidence that the initial planetesimal SFD had to span sizesranging from 100 km up to (at least) Ceres-size objects. We argue the initial slope of theinitial planetesimals also had to be similar to the one currently observed in the main beltpopulation. We have shown in Figs. 4f and 6d that if the input planetesimals are not larger than100 km, the introduction of turbulent scattering with γ = 2 × − causes an effectivenegation of the accretion process. However, if the input planetesimals have sizes rangingfrom 100 to 500 km (Fig. 7d) or 1,000 km (Fig. 8d), the same turbulent strength does notchange the outcome of the simulation with respect to the case where no turbulent scatteringis included. Similarly, γ = 2 × − aborts accretion if the initial planetesimals are notlarger than 500 km (Fig. 7d) but not in the case where they are Ceres-size (Fig. 8d).Turbulent scattering provides an ‘external’ excitation of the velocity dispersion of theplanetesimals. By ‘external’, we do not mean generated by the interaction among theplanetesimals themselves. In the simulations, and in our discussion below, what matters isthe magnitude of this external excitation and not the process that causes it. Thus, otherforms of excitation, such as, for example, gravitational stirring from Jupiter’s forming corecan be considered as well. 43 –In collisional coagulation, the key factor is the ratio between the escape velocity fromthe largest planetesimals and the dispersion velocity of the bodies carrying the bulk of thetotal mass, relative to those planetesimals. In absence of external excitation mechanisms,the former is always larger than, or of the same order of, the latter. The first case leads torunaway growth; the second to oligarchic growth. If an external excitation is present, thevelocity dispersion can become much larger than the escape velocities. This slows down thecoagulation process considerably and effectively ends it.For instance, for γ = 2 × − , turbulent excitation pushes 100 km bodies toeccentricities of e ∼ .
01. At 2.5 AU, this corresponds to a velocity of ∼
200 m/s relativeto a local circular orbit. This value is larger than the escape velocity from a 100 km object( ∼ ) but is smaller than the escape velocity froma 500 km or 1,000 km object ( ∼ γ in first approximation. Thus, by theargument described above, one would predict that, in the case with initial planetesimalsup to 1,000 km in size, collisional coagulation is severely inhibited if γ > × − becausethis value would give a velocity dispersion on the order of ∼
500 m/s, comparable to theescape velocity from a 1,000 km object. In reality, we have seen in Fig. 8d that collisionalcoagulation is still effective in case-A even for γ = 2 × − . This is due to the fact that,in case-A, all the mass is initially in small bodies whose velocity excitation is reduced dueto gas drag and mutual collisions (Ida et al., 2008); in turn these small bodies damp thevelocity dispersion of the large planetesimals by dynamical friction. We have checked that,if turbulent excitation is introduced into a simulation with our nominal set-up (like that ofFig. 8a), collisional coagulation is indeed negated for γ > × − . 44 –
7. Conclusions
The first and most basic step of the accretion of planets is the creation of planetesimals.Unfortunately, planetesimal formation is still poorly understood. In the traditional view,planetesimals grow progressively from coagulations of dust and pebbles to kilometer-sizedobjects. Consequently, the simulations of the second step of the accretion process, that inwhich collisional coagulation among the planetesimals leads to the formation of planetaryembryos and giant planet cores, usually starts from a population of kilometer-sizedplanetesimals (e.g. Weidenschilling et al., 1997; Kenyon and Bromley, 2006).However, recent paradigm-breaking work (Johansen et al., 2007; Cuzzi et al., 2008)showed that planetesimals might form big (100km or larger) thanks to the self-gravity ofsmall bodies highly concentrated in the turbulent structures of the proto-planetary disk. Ifthis is true, then there are no km-sized initial planetesimals and the second step of accretionhas to be somehow affected by this change in ‘initial conditions’.In this work, we have assumed the SFD of the ‘initial planetesimals’ is unknown andwe have attempted to constrain it by matching the final SFD produced by the second stepof the accretion process with the SFD of a reconstructed asteroid belt. More specifically,the ‘target SFD’ that we try to reproduce with collisional coagulation simulations is theone that the asteroid belt should have had just prior to it being dynamical excited anddepleted of material. The large body of work on the past history of the asteroid belt, whichwas reviewed in section 2, allows us to define the shape and size of this target SFD.While it is impossible to prove a negative result, we believe we have run enoughsimulations to understand the response of the coagulation process to various initial andenvironmental conditions. Based on these results, we find it likely that the SFD of theasteroid belt cannot be reproduced from an initial population of km-sized planetesimals.It also cannot be reproduced by assuming that the initial planetesimals had sizes up to 45 –some value D max < ∼
100 to several 100 km, probably up to 1,000km, and that their initial SFD had a slopesimilar to that of the current SFD of asteroids in the same size-range. Curiously, this resultis reminiscent of the original intuition by Kuiper (1958) that the original asteroid sizedistribution had to have a Gaussian shape centered around 100 km.Our result provides support for the idea that planetesimals formed big (Johansen etal., 2007; Cuzzi et al., 2008). The precise process that formed these big planetesimals is stillan open issue. Our findings (size range and SFD slope of the initial planetesimals) shouldhelp constrain the planetesimal formation models.We have also shown that, if the initial planetesimals can be as big as 1,000 km, thesubsequent collisional coagulation process leading to the formation of planetary embryosis not seriously affected by the excitation of eccentricities and inclinations due to theturbulence in the disk. This may provide a possible solution for the problem of planetformation in turbulent disks (Nelson, 2005; Ida et al., 2008).Our results also help us explain several interesting mysteries about small body evolutionacross the solar system. For example, if we assume the asteroid belt was initially deficientin
D <
100 km asteroids, its early collisional activity may have been much lower thanpreviously thought. Thus, the constraint provided by the uniqueness of Vesta’s large basin(i.e. that the asteroid belt hosted cumulatively over its history a population of
D >
D > D ∼
100 km (Bernstein etal., 2004) is also a signature of accretion and not one of collisional grinding, unlike what itis usually assumed (e.g. Kenyon and Bromley, 2004; Pan and Sari, 2005).Finally, we have shown that the sudden appearance of large planetesimals in a massivedisk of small bodies boosts runaway accretion of large objects (see the case-A/B simulationsin sections 4 and 5). This result might help in solving the problem of the formation of theJovian planet cores, one of the major open issues in planetary science.
Acknowledgments
This work was done while the first author was on sabbatical at SWRI. A.M. is thereforegrateful to SWRI and CNRS for providing the opportunity of this long term visit and fortheir financial support. This paper had three reviewers: J. Chambers, J. Cuzzi and S.Weidenschilling. The challenges that they set made this manuscript evolve over more thanone year from a Nature-size letter (although with a long SI) to a thesis-size monograph(blame them if you thought that this paper is too long!). However, these challenges alsomade this work more extensive, robust and -hopefully- convincing, and we thank thereviewers for this. We also thank Scott Kenyon for a friendly review of an early version ofthe electronic supplement of this paper, describing our code and its tests.
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