Towards planetesimals- dense chondrule clumps in the protoplanetary nebula

8/14/2019 Towards planetesimals- dense chondrule clumps in the protoplanetary nebula

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Astrophys. J., Accepted, April 11 2008Preprint typeset using L A TEX style emulateapj v. 10/09/06

TOWARDS PLANETESIMALS: DENSE CHONDRULE CLUMPSIN THE PROTOPLANETARY NEBULA

Jeffrey N. Cuzzi 1,4 , Robert C. Hogan 2 , and Karim Shariff 3

Astrophys. J., Accepted, April 11 2008

ABSTRACTWe outline a scenario which traces a direct path from freely-oating nebula particles to the rst

10-100km-sized bodies in the terrestrial planet region, producing planetesimals which have propertiesmatching those of primitive meteorite parent bodies. We call this primary accretion . The scenariodraws on elements of previous work, and introduces a new critical threshold for planetesimal for-mation. We presume the nebula to be weakly turbulent, which leads to dense concentrations of aerodynamically size-sorted particles having properties like those observed in chondrites. The frac-tional volume of the nebula occupied by these dense zones or clumps obeys a probability distributionas a function of their density, and the densest concentrations have particle mass density 100 timesthat of the gas. However, even these densest clumps are prevented by gas pressure from undergo-ing gravitational instability in the traditional sense (on a dynamical timescale). While in this stateof arrested development, they are susceptible to disruption by the ram pressure of the differentiallyorbiting nebula gas. However, self-gravity can preserve sufficiently large and dense clumps from rampressure disruption, allowing their entrained particles to sediment gently but inexorably towards theircenters, producing 10-100 km sandpile planetesimals. Localized radial pressure uctuations in thenebula, and interactions between differentially moving dense clumps, will also play a role that mustbe allowed for in future studies. The scenario is readily extended from meteorite parent bodies toprimary accretion throughout the solar system.Subject headings: solar system:formation; accretion disks; minor planets: asteroids; turbulence; insta-

bilities

1. INTRODUCTION

There is no currently accepted scenario for the forma-tion of the parent bodies of primitive meteorites whichaccounts for the most obvious of their properties. Theseproperties (reviewed by Scott and Krot 2005 and dis-

cussed in more detail in section 2.1) include (a) dom-inance by aerodynamically well-sorted mineral particlesof sub-mm size; (b) class-to-class variation in well-denedphysical, chemical, and isotopic properties; (c) a spreadof 1 Myr or so between the formation times of the old-est and youngest objects found in the same meteorite;(d) a spread of 1-3 Myr in radiometric ages of differentmeteorite types; and (e) a dearth of melted asteroids,with model results for even some melted asteroids whichimply Myr delays in formation relative to ancient miner-als. In recent years, meteoritic evidence has appeared forsome very early-formed planetesimals, which represent aminority of both meteorites and asteroids. This impliesthat primary accretion started early and continued for

several million years - thus, it was fairly inefficient anddid not run quickly to completion.Most current models for this primary accretion stage

(reviewed by Cuzzi and Weidenschilling 2006 and Do-minik et al 2007) can be classied as either (a) incre-mental growth, where large particles sweep up smallerones by inelastic collisions involving porous surfaces, and

1 Space Science Division, Ames Research Center, Moffett Field,CA 94035, USA

2 BAER inc., Sonoma, CA3 NASA Advanced Supercomputing Division, Ames Research

Center4 To whom correspondence should be addressed; E-mail: jef-

[email protected]

growth proceeds hierarchically; or (b) instability, wherephysical sticking is irrelevant and collective effects drivecollapse to km-sizes or larger on very short timescales.Those who favor instability models, most of which relyon gravity and occur in a particle-rich nebula midplane,are concerned by the poorly understood sticking of min-eral particle aggregates and the apparent difficulty of growing beyond meter size due to rapid inward migrationand collisional disruption. Those who favor incrementalgrowth have noted that midplane instability models areprecluded by even very weak global turbulence, and that,in the dense midplane layers that form when turbulenceis absent, incremental accretion is at low relative veloc-ity and the meter-size barrier is not a problem. However,the most sophisticated models of incremental accretion innonturbulent nebulae nd it to be so efficient that largeplanetesimals grow in only 10 4 105 years throughoutthe asteroid belt region (Weidenschilling 2000), a shorttime which is difficult to reconcile with constraints (a-e)above. A third class of scenario suggests that a com-plex interplay between several nonlinear processes - tur-bulence, pressure gradients, and gravity - may concen-trate appropriately sized particles and lead to planetes-imal growth (Cuzzi et al 2001, 2005, 2007; Johansen etal 2006, 2007). Scenarios which involve preferentiallymeter-sized objects encounter concerns about whethermeter-sized objects can survive their own high-velocitymutual collisions in this sort of turbulent environment(cf. Sirono 2000, Langkowski et al 2007, Ormel et al2007). However, in principle, such boulder-concentrationscenarios can proceed independently in the same envi-ronment as discussed in this paper, where we focus onmm-size particles.
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Most of the above scenarios provide no natural ex-planation for observed meteorite properties (a) and (b)above. In particular, the evidence for the H chondriteclass (and perhaps all the ordinary chondrites) suggeststhat entire asteroids of 10-100km diameter formed di-rectly from a physically, chemically, and isotopically ho-mogeneous mix of dust-rimmed particles of similar size(section 2.1). In this paper, we outline a possible path bywhich entire batches of mm-size, aerodynamically sortedparticles might proceed directly in turbulence (even if sporadically) to planetesimals having the properties out-lined above. We nd that sufficiently large and denseclumps of mm-size particles can form by turbulent con-centration such that, even if classical gravitational in-stability cant operate, their self-gravity may still allowthem to survive disruptive forces and slowly sedimentinto a sandpile planetesimal. This simple analyticalmodel is backed up by some numerical simulations thatsupport the basic idea. In sections 2.2 and 2.3, we reviewthe most relevant physics that determines the propertiesof dense, particle-rich zones or clumps in turbulent neb-ulae. In section 3, we address the fate of these denseclumps using analytical and numerical models of theirevolution. We derive the combination of size and massdensity a clump must have to evolve into a sandpilehaving some degree of internal strength. In subsequentstages not modeled here, we imagine that collisions andthermal sintering transform these sandpiles into the co-hesive rocky parent bodies we see today. However, in theouter solar system, lower energy collisions and weakerthermal processing might well allow planetesimals to re-tain their initial low-strength states, as seen for somecometary objects ( eg., Asphaug and Benz 1996).

2. BACKGROUND

2.1. Meteoritics background and evidence for inefficient accretion in a turbulent environment

One can extract a number of clues from primitive (un-melted) meteorites regarding the primary accretion pro-cess by which their parent bodies rst formed (Scottand Krot 2005, Taylor 2005). For the best evidence,one must look back through extensive subsequent evo-lutionary stages. Even unmelted bodies in the 100kmsize range have incurred extensive collisional evolution(Bischoff et al. 2006), producing compaction, fragmen-tation, and physical grinding and mixing on and beneaththeir surfaces, which may obscure the record of primaryaccretion. Model studies ( e.g. , Petit et al 2001, Kenyonand Bromley 2004, 2006; Bottke et al 2005; Chambers2004,2006; Weidenschilling and Cuzzi 2006) suggest that

the collisional stage occurred after dispersal of the nebulagas allowed the orbital eccentricities of primitive bodiesto grow. We are concerned with an earlier stage, whenthe still-abundant nebula gas led to a more benign en-vironment with fewer and gentler collisions. The directproducts of primary accretion might be most clearly visi-ble in the rare primary texture seen in some CM (Met-zler et al 1992) and CO (Brearley 1993) chondrites. Inthese objects, or more specically in unbroken fragmentswithin them, the texture consists of similarly-sized, dust-rimmed particles packed next to each other as if gentlybrought together and compressed, with no evidence forlocal fracturing or grinding.

Even after collisional effects associated with subse-

quent stages of growth have blurred this signal, perhapseven mixing material from different parent bodies, ev-idence remains in the bulk properties of all chondriteclasses. Chondrite classes are dened by their distinctivemineral, chemical, and isotopic properties ( e.g. Gross-man et al 1988, Scott and Krot 2005, Weisberg et al2006). Large samples of material with a quite well de-ned nature were accumulated at one place and/or time,and material of a quite different, but equally well-dened,nature was accumulated at another place and/or timeinto a different parent body. Amongst the most obviousaspects of these class properties is a dominance withinchondrites of sub-mm size mineral particles (genericallychondrules; Grossman 1988, Jones et al 2000, 2005;Connolly et al 2006) which are aerodynamically well-sorted. Aerodynamic sorting of chondrite componentswas rst emphasized by Dodd (1976), Hughes (1978,1980), and Skinner and Leenhouts (1993), and has sincebeen discussed by a variety of authors (see Cuzzi andWeidenschilling 2006 for a summary of the observationsand arguments supporting aerodynamics). Chondruleshave a size which varies from class to class, but is distinc-tive and narrowly dened within a given class or chon-drite.

The evidence suggests that chondrules do not com-prise merely a thin surface layer swept up by a largeobject (Scott 2006). In the case which we suggest as anarchetype, the H-type ordinary chondrite parent bodyis widely believed to be a 100 km radius object (per-haps the asteroid Hebe), initially composed entirely of a physically, chemically, and isotopically homogeneousmix of chondrules and associated material, which wasthermally metamorphosed by accreted 26 Al to a degreewhich varied with depth, into an onion-shell structure,and subsequently broken up in several stages (Taylor etal 1987, Trieloff et al 2003, McSween et al 2002, Grimm

et al 2005). Bottke et al (2005) conclude from modelsof collisional evolution in the primordial (massive) andthe current (depleted) asteroid belt, that the primordialasteroid mass distribution was dominated by objects hav-ing diameter of around 100km, rather than having a pow-erlaw size distribution somewhat like that of the currentpopulation.

We see the essential challenge as understanding howsuch a large object can be assembled from such a ho-mogeneous mixture of mm-size constituents, while otherobjects (arguably the parents of the L and LL type ordi-nary chondrites as well, and logically then the parents of the chondrites of all classes) are assembled from distinct,yet comparably homogeneous, ensembles of qualitativelysimilar particles. Moreover this assembly, or primary ac-cretion, phase of nebula evolution must persist for a du-ration of 1Myr or more between the formation times of the oldest and youngest meteorites (Wadhwa et al 2007)and even of objects found in the same meteorite. The >1 Myr age spread between ancient refractory inclusions(CAIs) and chondrules is well known (Russell et al 2006),but there may even be a comparably extended age spreadbetween different chondrules in a given meteorite (Kitaet al 2005).

In recent years, radiometric dating of achondrites(melted meteorites) has revealed some early-formed (andearly melted) planetesimals (Kleine et al 2005, 2006).One expects early formation and melting to go together,


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because objects larger than only 10km or so, which ac-creted at the same time that refractory inclusions formed(4.567 Byr ago), would have incorporated their full com-plement of 26 Al and would have melted (LaTourette andWasserburg 1998; Woolum and Cassen 1999, Hevey andSanders 2006). Thus, early accretion implies melting,and lack of melting requires late accretion. Model re-sults imply that, to avoid complete melting, 100km bod-ies must accrete only after a typically 1.5-2 Myr delayrelative to ancient CAI minerals (reviewed by McSweenet al 2002 and Ghosh et al 2006). While unknown se-lection and sampling effects may inuence the limitedmeteorite data record, spectral analysis of the entire as-teroid database also suggests that melted asteroids arerare. That is, while debate continues as to whether smallamounts of melt might be found on the surfaces of manyasteroids, which might be caused by impacts ( e.g. Gaffeyet al 1993, 2002), only Vesta and a handful of other ex-amples of fully melted basaltic objects have been found inspite of extensive searches (Binzel et al 2002; Moskovitzet al 2007).

Put together, this body of information suggests thatprimary accretion started early but continued for severalmillion years - thus, it was inefficient, or at least spo-radic. By comparison, midplane incremental accretionin a nonturbulent nebula accretes numerous Ceres sizebodies and tens of thousands of 10-100km size objects inonly 105 years (Weidenschilling 2000); thus, it is highlyefficient.

Models suggest that the density, temperature, andcomposition of the nebula evolved signicantly over sev-eral million years (Bell et al 1995, DAlessio et al. 2005,Ciesla and Cuzzi 2006). Thus one expects the propertiesof forming planetesimals to change with time even thoughthe chemical, isotopic, and mineralogical properties of the mixture of solids from which they formed might have

been fairly uniform in space at any given time (see Cuzziet al 2003 and Ciesla 2007 for more detailed discussions).Reality might have been even more complicated - there issome evidence that chondrites of very distinct chemicaland isotopic properties might have formed at roughly thesame time (Kita et al 2005), suggesting a combination of spatial and temporal gradients.

In this paper we present the overview of a scenario bywhich accumulation of a well-dened particle mix into alarge planetesimal - making it a snapshot of the localnebula mix - might occur sporadically or inefficiently inweak turbulence, over an extended period of time andthroughout the terrestrial planet region. The challengefor the future is to show that the accretion processesdiscussed here can quantitatively account for the massneeded to build planetesimals of the mass needed to cre-ate planetary embryos and planets during the age of thenebula (Chambers 2004, 2006; Cuzzi et al 2007).

2.2. Turbulence

A variety of astronomical and meteoritical evidencesupports weak, but widespread and sustained, nebulaturbulence. The observed abundance of small particlesat high altitudes above the midplane in many visibleprotoplanetary nebulae for millions of years is most eas-ily explained by ongoing turbulence - both to regeneratethe small grains in collisions and to redistribute them tothe observed altitudes (Dullemond and Dominik 2005).

The survival of ancient, high-temperature mineral inclu-sions (CAIs) in chondrites, in spite of their tendencyto drift inwards towards the sun, can also be explainedby turbulent diffusion (Cuzzi et al 2003, 2005). More-over, the recent discovery by STARDUST of both high-and moderate-temperature crystalline silicates in cometWild2 (Brownlee et al 2007, Zolensky et al 2007) can beeasily understood in this way (Ciesla 2007).

It remains a subject of active theoretical debate as tohow, or even whether, the nebula can maintain itself ina turbulent state. Ongoing dissipation by molecular vis-cosity m on the smallest, or Kolmogorov, lengthscalerequires a production mechanism to continually generateturbulent kinetic energy. Although the very existence of ongoing disk accretion into the star, with the ensuinggravitational energy release, provides an ample energysource, the vehicle for transferring that energy into tur-bulent gas motions is undetermined. Baroclinic instabil-ities require substantial opacity at thermal wavelengths,but might be suitable to generate turbulence during thefairly opaque chondrule-CAI epoch in the inner solarsystem (Klahr and Bodenheimer 2003). The widely ac-cepted magnetorotational instability (MRI) (Stone et al2000) may or may not be precluded at the high gas den-sities of the inner solar system (Turner et al 2007). Therole of pure hydrodynamics operating on radial shearremains unclear, because Keplerian disks are nominallystable to linear (small) perturbations (Balbus et al 1996,Stone et al 2000). However, a number of current theoret-ical studies suggest turbulence might indeed be present(eg. Arldt and Urpin 2004, Busse 2004, Umurhan andRegev 2004, Dubrulle et al 2005, Afshordi et al 2005,Mukhopadhyay et al 2005, Mukhopadhyay et al 2006).Much of this recent work focusses on nonlinear (nite am-plitude) instabilities occurring at the very high Reynoldsnumbers characterizing the nebula, which are well be-

yond the capability of current numerical and even labo-ratory models.In this paper, we simply assume the presence of weak

nebula turbulence. The intensity of turbulence maybe characterized by a nondimensional parameter =(vt /c )2 , where vt is the velocity of the most energeticeddies and c is the sound speed. Values of in therange 10 4 1 are consistent with observed nebula life-times, and can be sustained by only a few percent of thegravitational energy released as the nebula disk ows intothe sun. One then estimates the most energetic (largest)eddy size as L H 1/ 2 , where H is the nebula verti-cal scale height, or disk thickness. A cascade to smallerscales ensues, where the smallest (Kolmogorov) eddy

scale is = LRe 3/ 4

1 km and Re = cH/ m 107

for 10 4 is the nebula Reynolds number (Cuzzi etal 2001).

2.3. Turbulent concentration

It has been found both numerically and experimen-tally that particles of a well dened size and densityare concentrated into very dense zones in realistic 3Dturbulence (Eaton and Fessler 1994). The optimallyconcentrated particle has a gas drag stopping time t swhich is equal to the overturn time of the Kolmogorovscale eddies. In the nebula, ts = r s /c g where r ands are the particle radius and material density, and c


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Fig. 1. Cumulative probability distribution, or volume frac-

tion F p , of particles having local concentration factor larger thansome value of C (from Cuzzi et al 2001; see also section 2.3). Thecurves with symbols represent comparison of the predictive theorydescribed in Cuzzi et al (2001) with 3D numerical simulations atmoderate Reynolds numbers; the curves without symbols are ex-tension of the same theory to nebula conditions denoted by theirvalues of . Concentration factors C 100 of chondrule precur-sors (leading to = p / g 1) are fairly common for a range of nebula turbulent intensities (denoted by ), and help explain lackof chondrule isotopic fractionation (Cuzzi and Alexander 2006). Inthis paper we focus on the much less common, but much denser,regions towards the bottom right side of the plot ( C 104 or 100). More general PDFs which separate out the role of vorticity and incorporate the effect of particle mass loading arediscussed in section 2.3 and Appendix A.

and g are the gas sound speed and density. The Kol-

mogorov eddy frequency scales with the orbital fre-quency as = Re 1/ 2 , and particles are most ef-fectively concentrated when ts 1 (Eaton and Fessler1994). Chondrule-size silicate particles are concentratedfor canonical nebula properties if 10 5 10 3; more-over the shape of the concentrated particle size distribu-tion is parameter-independent, and agrees very well withthat of chondrules (Paque and Cuzzi 1997, Cuzzi et al2001). We dene the concentration factor C p/ p,where we denote the local volume mass density of parti-cles as p and p is its nebula average value, and the par-ticle mass loading p/ g = C p/ g . For instance,gure 1 (Cuzzi et al 2001) shows the probability distri-bution function (PDF) for C determined from numerical

simulations at moderate Reynolds number (curves withsymbols) and predicted for plausible nebula Reynoldsnumbers (curves without symbols). Note that the vol-ume fraction having C > 100 is quite substantial, whichhelps explain some mineralogical and isotopic propertiesof chondrules (Cuzzi and Alexander 2006). 5 Not surpris-ingly, the volume fraction decreases for larger C . In thiswork we focus on the less abundant, but higher mass den-

5 Here, = p / g is different from the denition in Cuzzi andAlexander (2006), which is equivalent to p /A g , where A 10 2is the fractional abundance of solids to hydrogen gas by mass. Thusvalues of in Cuzzi and Alexander (2006) are the equivalent of p / g 1 - a much more common occurrence than the situationwe study here.

sity, concentrations towards the lower-right hand side of the PDF.

The turbulent concentration PDFs of gure 1 did notallow for the feedback of the concentrated particles onthe gas. This mass loading would be expected to dampgas turbulent motions once the particle mass density sig-nicantly exceeds that of the gas, and lead to a satura-tion of concentration at some value. Recent work whichincludes the feedback effects, through drag, of particledensity on the damping of turbulence, has shown thatvalues of 100 can indeed be achieved - although lesscommonly than in the unloaded models such as shown ingure 1. This cascade model of turbulent concentration(Hogan and Cuzzi 2007) is summarized in Appendix A.Based on these results, we adopt = 100 as an upperbound in all stability and evolution modeling.

3. PRIMARY ACCRETION OF PLANETESIMALS

The ability of gravity to overwhelm opposing forces isa well-known theme in astrophysics, dating back 80 yearsto early studies of star formation by Jeans. Over 30 years

ago, gravitational instability (GI) of solids in a denselayer near the nebula midplane was proposed to lead di-rectly to solid planetesimals on a dynamical timescale(the orbit time) (Safronov 1969, Goldreich and Ward1973). Traditional GI thresholds require the dynami-cal (collapse) time of a dense region tG = (G p) 1/ 2 ,where G is the gravitational constant, to be shorter thanboth the transit time due to random velocity and thelocal shear (vorticity) timescale (Toomre 1964). Thesearguments imply that GI occurs when the local par-ticle density exceeds a few times the Roche densityR = 3 M / 4a 3 , where M is the solar mass and ais the distance from the sun (Safronov 1991, Cuzzi etal 2001). However, this scenario is frustrated by self-

generated midplane turbulence even if the nebula is notglobally turbulent ( e.g. Weidenschilling 1980,1995; Cuzziet al 1993,1994; Johansen et al 2007). That is, a denselayer of cm-size and larger particles orbits at a differ-ent velocity than the pressure-supported gas, leading toa vertical velocity shear which becomes turbulent andstirs the particle layer. A more recent avor of GI re-quires the particles to be small enough that the particle-rich midplane acts as a single uid, like fog, so that itsvertical density stratication stabilizes it against self-generated turbulence (Sekiya 1998, Sekiya and Ishitsu2001, Youdin and Shu 2002, Sekiya and Takeda 2003,Youdin and Chiang 2004). Because the particles must bevery small to satisfy this one-phase requirement (mm-size

and smaller), their settling towards the midplane is frus-trated by even the faintest breath of nebula turbulence( 10 9). If the particles are prevented from settlinginto a sufficiently dense layer that their mass density canstratify the total density (eg, Dubrulle et al 1995, Cuzziand Weidenschilling 2006), the conditions needed to trig-ger the instability are never achieved.

In the following subsections, we study the evolution of a dense clump such as might form towards the lower rightdomain of gure 1, albeit in a simplied way. We treata clump as an isolated object when in reality it exists ina context of surrounding material of varying density. Insection 4, we discuss the implications of more realisticconditions.


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0 0.2 0.4 0.6 0.8 1t / t

G

0.3

0.35

0.4

0.45

0.5

0.55

r a

d i a l c e n

t r o

i d

o f p a r

t i c

l e d e n s

i t y

incomp.Ma = .00125Ma = .0025Ma = .005Ma = .01 0 = 100

Fig. 2. Results of a 1D compressible numerical model, show-ing how gas pressure precludes dynamical timescale gravitationalinstability (Sekiya 1983) as described in more detail in section 3.1.Here, Ma = ( l/c )( G g )1 / 2 = 1 / is like a Mach number. Neb-

ula values of c and g lie in the top set of nearly linear curves,allowing only slow inward sedimentation on the timescale t sed (for0 = 100). That is, under nebula conditions of gas density andtemperature, and particle loading, the gas behaves as if it wereincompressible and prevents dynamical timescale GI for particleswith stopping time much less than the dynamical collapse time.

3.1. Gas pressure precludes particle gravitational instability

The role of the gas pressure in GI for nebula parti-cles has been almost universally ignored, even thoughits importance was pointed out decades ago by Sekiya(1983). We encountered it ourselves independently. Totest our GI thresholds, we initially ran numerical modelsof static, but very dense, clumps ( = 1000), expectingthem to collapse with gravitational free-fall or dynami-cal collapse times tG = (G p) 1/ 2 = (Gg ) 1/ 2 andvelocities V G l(G p)1/ 2 / 2, where l is the clump di-ameter. We assumed a spherically symmetrical denseparticle clump with density p, embedded in gas withdensity g , where = p/ g . Instead, only very slowshrinkage ensued. We suspected that the collapse wasbeing articially blocked by our incompressible code.

We then developed a fully compressible 1D modelwhich conrmed our incompressible calculations andclearly demonstrated a much higher, gas-pressure-dependent threshold for GI in the limit when t s tG .Results from the model are shown in gure 2.

Figure 2 shows results from this fully compressiblemodel, illustrating how > c/ (l(G

g)1/ 2) is required

for traditional (dynamical timescale) instability to occur.The initial condition was a Gaussian blob having den-sity distribution p(r ) = 0g exp( r 2 /l 2). Slow shrink-age of the particles through the gas is seen in the sta-ble, incompressible regime where nebula parameterslie, well within the at curves at the top of the plot withMa 10 4 for l 104km, c 105cm/s, and g 10 9gcm 3 . The high value of Ma required for true dynamicaltimescale collapse (given the chosen value of = 100)would require smaller gas sound speed or larger clumpsize l and gas density g , by orders of magnitude.

The slow shrinkage in the stable regime results fromparticles sedimenting inwards towards their mutual cen-

ter under their own self-gravity at their terminal velocityvT = gt s , where g = 2 Glg is the local gravitational ac-celeration due to the clumps mass, and t s = r s /c g isthe particle stopping time. Thus vT = 2 Glr s /c V Gfor ts tG , and the sedimentation timescale is

tsed =l

2vT =

14G(r s /c )

=c

4Gr s=

14Gg t s

,

(1)roughly 30-300 orbit periods at 2.5 AU for 300 radiuschondrules and = 1000 100; that is, much longer thantG . It seems inappropriate to describe this slow ongoingevolution as an instability - certainly in midplane scenar-ios where it was slow vertical sedimentation by individualparticles toward the midplane that produced the un-stable situation in the rst place, and where only fur-ther slow sedimentation transpires. The situation is moreakin to star formation mediated by ambipolar diffusionthan by traditional gravitational instability. Gas pres-sure even precludes the Safronov-Goldreich-Ward GI inthe form it was originally proposed, where cm-sized par-ticles were envisioned (although tsed would be faster forliterally cm-sized particles, and more closely approachestG ).

The results of Sekiya (1983) have apparently been over-looked by all subsequent workers, who have invariably as-sumed a Toomre-Safronov-Goldreich-Ward type particleensemble which is decoupled from the gas, and stabilized(on small scales) only by particle random velocities. Thisgas pressure constraint is, however, crucial for particlesin the chondrule size regime where t s an hour andtG a year, and must be applied to all GI models in-volving chondrule-sized particles. The good news is thatfor the interesting range of nebula gas density and par-ticle concentration values, the gas does actually behaveas if it were incompressible (validating the use of an in-compressible code in numerical models such as shown insection 3.4).

The physics is easily understood in the one-phaseregime ( ts tG ). The particles feel no direct pres-sure force, but are rmly trapped to the gas which does.The (inward) gravitational force is dominated by parti-cles for 1. The particles drag and compress thegas with them as they begin to collapse under their self-gravity, producing a radial gas density gradient dg /dl ;this translates into a large outward pressure gradientc2dg /dl which acts on the gas, and thereby also on thestrongly coupled particles.

The gravitational force per unit volume is f G =4GM p/l 2 2G22g l, where M is the clump mass andR its radius. After some incipient shrinkage and com-pression occurs, the outward pressure gradient force perunit volume becomes roughly f P = c2 dg /dl c2g /l ,where c is the gas sound speed. Then for GI to be possi-ble, f G /f p = G2 l2g /c 2 > 1, giving the criterion for GIas > , where = c/l (Gg )1/ 2 . is like an inverseMach number Ma 1 , where Ma = ( l/c )(Gg )1/ 2 . An al-ternate (perturbation) approach compares the incremen-tal changes f p and f G associated with a small gravi-tationally induced shrinkage characterized by l l. Itis straightforward to show (for t s tG ) that f p/f G c2g /GM 105 under nebula conditions. Thus theclump is stiffened by pressure and even a small incre-mental shrinkage is self-limiting. Inspection of gure


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2 shows small oscillations which suggest that a formallinear stability analysis might lead to an improved sta-bility criterion; this is, however, beyond the scope of thepresent paper.

For l a few 103 km and g 10 9 g cm 3 (De-sch et al 2005), 104 . This criterion for GI canbe compared with the traditional criterion for (marginal,or maximal) instability which is widely used as the crite-rion for gravitational instability, roughly twice the Rochedensity R = 3 M / 4a 3 (Goldreich and Ward 1973,Safronov 1991, Cuzzi et al 1993). The ratio of the twothresholds is:

g2R

=g

2R Ma

2c(Gg )1/ 2

2 l

2 6 103

(l/ 103km),

for g = 10 10 to 10 9. For the midplane GIs which havebeen widely discussed in the past (reviewed by Cuzziand Weidenschilling 2006), l 102km, so the traditionalGI criterion falls short by a factor of more than 10 4(cf. Sekiya 1983)! This degree of mass concentrationis unlikely under any plausible circumstances, especially

for particles which are already well trapped to the gas.For the ubiquitous dense clumps we discuss here, whichform at all elevations, l 104 km. Thus the pressure-supported gas phase prevents the tightly coupled parti-cles from undergoing GI until the particle mass loading ishundreds of times larger than the traditional GI criterion(which is on the order of the Roche density R ).

Nevertheless, Sekiya (1983) also noted that when pis in the range usually cited for GI (a few times R , or a few hundred), a 3D incompressible mode ariseswhich, while not explicitly stated, we interpret as retain-ing an identiable cohort of particles. An incompress-ible mode might resemble a blob of water oscillatingin zero-gravity. Dense zones which result from turbu-lent concentration are candidates for such incompressiblemodes, even at mass loadings too low to induce actual GI.However, the fate of such slowly evolving entities in thepresence of likely perturbations has never been explored.Below we discuss the least avoidable, most pervasive dis-ruptive perturbation which such clumps would encounteronce they form, which we believe to be ram pressure dis-ruption due to systematic velocity differences betweenthe dense zones and the gas, and then derive conditionsunder which these clumps can survive to become plan-etesimals.

3.2. The fate of incompressible clumps

Several kinds of perturbation by the enveloping nebulagas might disrupt a dense clump before the slow sedimen-tation of its constituent particles towards their mutualcenter (on the timescale tsed ) can produce a moderatelycompact object.

Perhaps the simplest to discuss and dismiss are tur-bulent pressure, velocity, and/or vorticity uctuationsencountered by the clump. Turbulent pressure uctua-tions on the scale l have typical intensity gvt (l)2 , wherevt (l) is certainly smaller than the uctuating velocityvt (L) = c1/ 2 of the largest eddy (the dominant energy-containing eddy). The ratio of even the largest eddypressure uctuations to the steady ram pressure fromthe headwind a = vK (a is the semimajor axis)is (vt /v K )2 = c2 / 2v2K . Since c/v K = H/a and

= ( H/a )2 , (vt /v K )2 < / < 1 unless > 10 3.Nebula evolutionary timescales, and parameters leadingto turbulent concentration of chondrule-sized particlesin the asteroid belt region, imply 10 6 < < 10 3 de-pending on gas density and other properties (Cuzzi et al2001). This suggests that turbulent velocity uctuationson clump lengthscales play a small role. We have runsimulations of clumps settling in gravity with and with-out enveloping turbulence, and the typical evolutions arequalitatively similar.

We next address local average vorticities on the length-scale of a clump. It has long been known that localvorticity is a factor in gravitational instability ( Toomre1964, Goldreich and Ward 1973). Because eddies muchsmaller than the largest scale of turbulence typically havelarger vorticity than the Keplerian shear commonly ex-plored ( eg. Toomre 1964), this is a concern in princi-ple. For example, in the inertial range, (l) (L/l )2/ 3(Tennekes and Lumley 1972). Then for l 104km,L H 1/ 2 and 10 4 , (l) 104. However, turbu-lent concentration has the property that dense particlezones preferentially lie in zones of locally low vorticitycompared to the average at their lengthscale (Eaton andFessler 1994). Statistical studies (Hogan and Cuzzi 2007)show that dense zones in high Re environments can formin regions with local vorticity 1-2 orders of magnitudesmaller than the average value expected for that length-scale (see also gure 5). Below we sketch how such aconstraint is derived and applied.

A simplied requirement for gravitational binding of aclump of lengthscale l and mass density p, in the pres-ence of local rotation at angular velocity (l), is 4G p >2(l). Recalling that the local vorticity (l) can varyby orders of magnitude from its average value (l) , wenormalize both sides of the above relationship between p and by the average inertial range enstrophy 2 (l) ,and dene the normalized enstrophy S = ( 2 (l)/ 2(l) ).This quantity provides the horizontal axis on gure 5.Then the simple expression above can be rewritten as > 2(l) S/ 4Gg . Inertial range relationships can beused to express 2(l) = 2(L/l )4/ 3 = 2(L/B )4/ 3where is the Kolmogorov scale and B is some scal-ing factor. Making use of Re = cH/ m = ( L/ )4/ 3as discussed earlier, and substituting nebula parametersat 2.5 AU, we nd that gravitational binding requires > 3 106S . This constraint yields diagonal lines onplots of the sort shown in gure 5 (for an example seeCuzzi et al 2007). Interesting regions of parameter spaceremain accessible to a degree that depends on quantita-tive modeling of the PDF P (, S ).

Obviously, more detailed numerical models of clumpsin realistic turbulence are needed to assess these pertur-bations. We will assume, for the purposes of this study,that turbulent pressure (and associated vorticity) uctu-ations are a minor inuence and that regions which arestable to their own local vorticity exist (see Cuzzi et al2007). This leaves the dominant disruptive process, un-avoidably shared by turbulent and laminar nebulae, asram pressure disruption due to systematic velocity dif-ferences between the clump and the gas, as describedbelow. Incidentally, survival against ram pressure givesthe horizontal stability boundaries in gure 3 of Cuzziet al (2007). Emplacing these constraints on PDFs ob-


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tained from cascade models is somewhat involved, and afull quantitative description of the situation is deferredto a future paper.

3.3. Ram pressure disruption and the Weber number

Here, for simplicity, we envision the evolution of a sin-gle, isolated dense clump with 1, merely as an ex-ample of one of the many dense clumps formed near thelower right hand corner of gure 1. The clump exertsa strong collective inuence on the entrained gas, andthe mixture experiences solar gravity as a unit. It at-tempts to move as an individual large object relative tothe nebula gas, dragging the entrained gas along. For in-stance, a clump forming at some distance above the neb-ula midplane settles downward under the accelerationof the vertical component of solar gravity, whereas thesurrounding gas remains under vertical hydrostatic bal-ance. Also, nebula gas generally orbits at a slower veloc-ity than the Keplerian orbit velocity vK obeyed by mas-sive particles. This is because of its generally outwardradial pressure gradient force (1/ g)dP/da = 2 a2 ,where a is the distance from the sun, and 10 3).Thus, particles experience a headwind with magnitudewg a = vK a few 103 cm/sec. These head-winds result in a ram pressure gw2g / 2 which can disrupta strengthless clump.

For example, a dense drop of ink settling in a glass of water is quickly disrupted by the Rayleigh-Taylor insta-bility and mixes with its surroundings (Thomson andNewhall 1885). However, a dense drop of uid with surface tension can avoid disruption and settle indef-initely at terminal velocity (as do raindrops, or waterin oil). In this situation, the criterion for stability of auid droplet, which determines its terminal velocity andthence its size, is given by the Weber number We, whereWe = 2 r g v2 / , and is the surface tension coefficient.Drops are disrupted when We exceeds some critical valueWe 1 10 (Pruppacher and Klett 1997).

Dense particle clumps in the nebula have no surfacetension, but they do have self-gravity. By direct analogywith the droplet surface tension criterion, we dene agravitational Weber number WeG as the ratio of the rampressure force per unit area to the self-gravitational forceper unit area for a attened disk, which is the initialstage of a strengthless spherical clump of diameter l andparticle density p = g upon encountering a headwindwg (Thomson and Newhall 1885). For a disk of surfacemass density p = pl, the gravitational force per unitarea is G 2 p . Then

WeG =C D g w2

g2G 2 p =C D w2

g2G2g l2 , (2)

where C D is an effective drag coefficient for the clump,on the order of unity (see Appendix B). WeG can bewritten in other ways as well. The orbit frequency at aprovides the useful relationship ( a) = ( GM /a 3)1/ 2 (G)1/ 2 , or G = 2 / , where = M /a 3 3.8 10 8 g cm 3 at 2.5 AU 6 ; then

WeG =C D w2g

22g 2 l2=

C D 2

22

ga2

l2. (3)

6 Note is different from Safronovs (1991) Roche density R =3M / 4a 3 discussed earlier

By analogy with the more familiar surface tension case,there will be some critical gravitational Weber numberfor stability We G, that is probably on the order of unity(Pruppacher and Klett 1997); its value must be con-strained by numerical experiments as described below.Then we require for stability against headwind disrup-tion that We G < WeG or

l > w g / (2GWeG g )1/ 2 =a

(2Gg WeG /C D )1/ 2 . (4)

These expressions determine the combination of clumpdiameter l and particle loading that will stabilize itagainst a headwind of magnitude vK . The headwinddue to nebula radial pressure gradient will occur even if a clump is at the midplane and has zero settling veloc-ity; this gives the lower limit on ram pressure that theclump must be stable against. Neglecting vertical set-tling restricts potentially stable clumps to lying withinabout 0.01 gas scale heights of the midplane, where theheadwind a is comparable to the vertical settling ve-locity. This restriction must be factored into statisticalestimates of planetesimal production (see, e.g. Cuzzi et

al 2007).For typical nebula parameters, we assume gas den-sity to be in the range g = 10 10 g cm 3 (a nom-inal minimum mass value) to 10 9 g cm 3 (a morerecent value supported by nebula evolution and chon-drule formation; cf. Desch et al 2005), semimajor axisa = 3 .8 1013 cm (2.5 AU), pressure gradient/headwindparameter 10 3 (see, eg., Nakagawa et al 1986, Cuzziet al 1993), and solar mass M = 2 1033 g. In thisregime, l > 1.5 5 106(/ 10 3) km. Recall fromsection 2.3 and Appendix A that mass loading limits themaximum achievable mass loading to 100. A clumpsatisfying the above constraint, with l = 1 5 104 km,would have the mass of a 10-100 km radius body of unit

density - thus, these precursors can lead to quite size-able objects. This characteristic size range is intriguinglyclose to the roughly 50 km radius primordial buildingblock size of Bottke et al (2005). The very existence of any preferred size for the primordial population, if true,is an intriguing result.

3.4. Numerical model of clump evolution

To obtain a sanity check on the concepts of section3.3, and to constrain the (unknown) value of We G forthis problem, we developed a very simplied numericalmodel of a clump experiencing a steady nebula headwindfrom a more slowly orbiting, pressure-supported gas, us-ing Hills approximation which transforms a cylindrical

system into a cartesian system rotating at some meanrate 0 - useful if the domain covers a sufficiently narrowradial and angular region that the curvature is negligible.The overall orbital motion of particles (the Keplerian ve-locity) is thus subtracted out, and the gas exhibits asmall differential velocity because of its deviation fromKeplerian. A particle clump under inuence of the gaswill slowly lag behind in the y (orbital) direction, andslowly drift to smaller x (radial) values, due to the gasheadwind. The model is not intended to be a high-delityrepresentation of a realistic nebula situation, but merelyto check the basic Weber number model of section 3.3.

The code evolves the velocity eld of a two-phase sys-tem. The gas is perturbed by a clump of particles, which


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are mutually attracted by gravitational forces and arethemselves dragged by, and exchange momentum with,the gas. The Hill frame, with cartesian coordinates(x,y,z ), represents a radially narrow region correspond-ing to some range of radius a, orbital angle , and ver-tical distance Z , respectively. Periodic boundary condi-tions are used in all three dimensions. The instantaneousNavier-Stokes (Eulerian) equations describing the con-servation of mass and momentum for the incompressiblegas are expressed in the Hill frame as,

U = 0 , (5)

and

Ut

+ ( U )U =P + P glob

g+ 2 U

2 0 U + f drag + f sp (6)

where U is uid velocity, g is gas mass density, isgas viscosity, P is local pressure (which uctuates along

with gas velocity and vorticity variations), P glob

=(2g wg 0 , 0, 0) represents the (constant) global nebularadial pressure gradient, and 0 = (0 , 0, 0) is the Ke-pler frequency vector. This setup causes the gas toow towards the clump in the y direction at a uniformspeed wg . We are currently neglecting the Keplerianradial shear term 2 q2xe x , where (a) = 0(a/a o) qand 0 = ( a0), in the gas and particle equationsto get a better comparison with our analytical stabil-ity model. The shear rate associated with this term(dw/dx )Kep = 0 / 2, whereas a typical (uctuating)local shear or strain due only to turbulent motions is(dw/dx )turb (l) > 0 . Because the turbulent shearis so widely varying, and likely to be larger than the

Keplerian shear, in this study we neglect both of thesecomplications, though they are both suitable avenues forfuture work.

In our code, particles are followed in the Lagrangiansense. The term f drag represents the force per unit massimparted to the gas by the particles. It has the generalform

f drag = p

g t s(V U ) (7)

where V is the mean weighted particle velocity at a gridpoint, t s is the particle gas drag stopping time, and pis the particle mass density. To obtain V , a weightedsum is carried out over particles in the eight cell volumesadjacent to each uid grid point. Weighting functions

are used which vary inversely with the distance of theparticle from the grid point (Squires 1990, his section5.1). Because of the periodic boundary conditions, thewake of a clump can impinge articially on the clumpfrom the upwind direction. To avoid this, we include theterm f sp in equation (6) as a sponge force per unit massto restore the gas velocities downstream of the clump atthe outow boundary plane to their initial values U 0 ; thisensures constant inow values at the upstream boundaryplane.

The sponge force has the form

f sp =1

sp(U U 0), (8)

where sp is some time scale. The function 1 / sp is mod-eled as a sigmoid in the y direction very near the outowboundary. The parameters of the sponge were deter-mined through test runs with the goals of minimizingthe sponges spatial extent and maximizing its effective-ness without introducing numerical instabilities. Oncedetermined, the same values were used for all produc-tion runs.

The (Lagrangian) equation describing the motion of particle i subject to the forces of gas drag and mutualgravity is (again neglecting the Keplerian shear term)

dV idt

= 2 0 V i + f dragi + f gravi (9)

The term f dragi describing the drag force per unit masson particle i by the gas takes the form

f dragi =1ts

(U (X i (t)) V i (t)) , (10)

where U (X i (t)) is the gas velocity interpolated to theparticles position X i (t) at time t.

Finally, the mutual gravity force per unit mass on each

particle is

f gravi = GN p

j = i

m jX ij (t)

(|(X ij (t)| + )2(11)

where G is the gravitational constant, and N p is the num-ber of particles. X ij is the unit vector associated withthe distance X ij (t) X i (t) X j (t) between particles jand i, and is a constant that softens the force at smallseparations to prevent numerical singularities, chosen tobe comparable to a grid cell in extent.

Eqs. (6) and (9) are solved using psuedo-spectral meth-ods commonly used to solve Navier-Stokes equations fora turbulent uid (Canuto et al 1987). Periodic boundaryconditions are assumed and the number of nodes used ineach direction is generally 128, 256, 128 with a spacingof 2/ 64. A Fast Fourier Transform (FFT) algorithm isused to evaluate the dynamical variables U at the com-putational nodes. A second-order Runge-Kutta schemeis used to time-advance the gas and particle velocities.A third-order Taylor series interpolation scheme is usedto determine gas velocities at the particle positions fromvalues at the eight nearest neighbor nodes. The mutualgravity calculation is done in a brute force fashion witha N 2 p algorithm to evaluate the force contributions fromall particles. The code is written in Fortran 77 and isparallelized using OpenMP directives. The FFT calcula-tions are done simultaneously on the planes perpendicu-lar to the y and z axes, and all loops involving particleindices are parallelized. Typically, 3000 superparticlesare used, but some runs used more than 20000. Each su-perparticle represents the dynamical effects, and typicalresponse, of a large number of actual chondrules.

Initially the particles are arranged in a uniform spher-ical clump. The initial velocities for the gas and the par-ticles inside and outside the clump are determined fromthe expressions in Nakagawa et al. (1986):

U 0 = wg p

g + p2D

D 2 + 2,

D 2

D 2 + 2

g + p p

, 0(12)

V 0 = wgg

g + p2D

D 2 + 2,

D 2

D 2 + 2, 0 (13)


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Fig. 3. Snapshots from evolutionary models of dense clumpsexperiencing a nebula headwind without (left column) and with

(right column) self-gravity. Top row: projected into a vertical plane(z ); bottom row: projected onto the a z plane ( a projectionsare similar), with the gas inow into the page. In the top leftpanel the open circle shows the initial size and position of theclump. After less than an orbit, ram pressure disrupts the gravity-free clump (left) but the gravitating clump is stable (right) andcontinuing to shrink inexorably, even while being eroded. Moviesshowing this evolution are available online (see Appendix C).

where D = ( g + p)/ g t s . The initial velocity of the gasoutside the clump is found by setting p 0 in equation(12): U 0 = (0 , wg , 0); inside the clump the specic casevalue of is used in equations (12) and (13) to initializethe gas and particle velocities.

Runs with single particles of different t s were madeto verify that the Nakagawa et al. (1986) initial condi-

tions are steady state solutions for the particle equations.The wallclock time to evolve one integration step is 0.7sec using 256 Intel Itanium 2 processors running at 1.5gighertz. The wallclock time to evolve 1 orbital periodis 121 hours.

3.5. Results of the numerical model Figure 3 shows how, as predicted by the simple theory

of section 3.3, certain combinations of clump mass load-ing and dimension l remain stable against ram pressuredisruption for nebula headwinds produced by a pressuregradient characterized by . We is dimensionless, andthe second expression in eqn. (2) can be used to obtainthe critical value of We G which separates stable from un-

stable congurations of the clump, We G = We

G , directlyfrom measured values in code units (see Appendix B).We expect that We G will be in the range 1-10. The com-bination of parameters in the stable case shown (rightpanels) gives We G 1; if self gravity is turned off, theclump is disrupted in tdis 1/ 2 orbits, whereas it willsediment into itself on a timescale tsed 8 times as long(see Appendix B). In the stable cases, a dense core is seento continually shrink and become denser throughout therun, even as material is shed from the periphery of theclump, such that the value of l for the core continuesto increase (see Appendix C, gure 7 ).

The numerical clump models exhibit considerable ero-sion over the duration of the runs from a viscously stirred

surface layer, and this gradual but inexorable mass lossmight lead to their disruption if arbitrarily long runs werepractical at this time. However, as discussed in Appen-dices B and C, this large amount of erosion is a gross over-estimate of what would happen in the nebula, becausenumerical viscosity (and consequently shear around theperiphery of the clump) plays a far larger role in the cur-rent numerical models that it ever would in the actualnebula. That is, the viscously stirred and eroded surfacelayer in the numerical clumps contains orders of mag-nitude more mass than would be the case for a nebulaclump of the size and density required to survive rampressure disruption (see Appendix B). In Appendix C,we describe the movies from which gure 3 is taken andalso show how the protected inner regions of the clumpsare behaving exactly as predicted, both in stable and un-stable regimes. For more realistic, larger clump Re c inthe nebula, a far larger fraction of the clump mass willshow this behavior rather than being articially eroded.

The primary purpose of the numerical models is merelyto provide an independent sanity check on the physicsof our Weber number model and obtain some idea of We

G , which gives us an order-of-magniture estimate forthe product l which nebula clumps require to surviveheadwinds of a particular g . Future coding improve-ments are needed to allow larger (higher Re c) clumpswhich would more closely approach nebula conditions interms of their balance between pressure forces and vis-cous forces (see Appendices B and C); these improve-ments will probably include an implicit time advance forthe drag terms and perhaps also a tree or particle-in-mesh code for the particles.

4. DISCUSSION AND SUMMARY:

We have shown how self-gravity can stabilize denseclumps of mm-sized particles, which form naturally in

3D turbulence, against disruptive gas ram pressure ontimescales which are sufficiently long for their constituentmineral grains to sediment towards their mutual cen-ters and form physically cohesive sandpiles of order10-100km in size. The essence of the result is a criti-cal gravitational Weber number on the order of unity,in which self-gravity plays the role of surface tension inmore familiar situations such as raindrops. Characteris-tic mass densities and lengthscales are determined whichmeet this requirement. We show numerical results whichare in general agreement with the predictions of the sim-ple theory, for isolated, spherical initial clumps.

The scenario we have sketched out leads from aerody-namically size-sorted nebula particles, having the prop-

erties of chondrules, to sizeable planetesimals formed en-tirely from such particles which contain a snapshot orgrab-sample of the local particle mixture. Turbulent con-centration rst produces the dense zones of size-sortedparticles. The more common, less dense of these re-gions may provide typical chondrule melting environ-ments (Cuzzi and Alexander 2006). Some of the lesscommon, very dense zones, which we have shown else-where can achieve mass densities 100 times larger thanthat of the gas, have the potential to become planetes-imals depending on their lengthscales, nebula location,and local vorticity. It is intriguing that our character-istic stabilized clump masses are not far from the massinferred by Bottke et al (2005) to represent a typical pri-


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mordial object in the pre-dispersal, pre-erosional asteroidbelt.

Ultimately, the sandpiles resulting from completed sed-imentation will become compacted further by inevitablecollisions with other sandpiles, leading to todays fairlydense asteroids, while retaining a physical and chemicalmemory of their parent particle clumps. The mechanismis easily extended to the unmelted aggregate particles of the outer solar system, where, however, if chondrules areabsent, the size-sorting ngerprints of the process (Cuzziet al 2001) might be less evident.

The conclusions of this paper differ from the sugges-tions in Cuzzi et al (2001), who focussed on the possiblerole of ultra-dense clumps with size comparable to a Kol-mogorov scale (0.1 -1 km). Since that time, Hogan andCuzzi (2007) found that particle mass loading saturatesthe value of = p/ g at a value of about 100, redirect-ing our attention to larger clumps in the 10 3 104 km sizerange. Clumps and uid structures of these large sizesare much more accessible with numerical codes, both of the standard direct simulation type and the cascade type(Appendix A).

In this scenario, primitive bodies may not primarilyrepresent spatial, but perhaps temporal, samples of theparticulate contents of the nebula as its chemical, physi-cal, and isotopic properties evolve over several Myr. Thetemporal, as well as the spatial, variation in the physi-cal, chemical and isotopic properties of the concentratedparticles, which are being continually altered by thermalevents and mineralogical alteration in the nebula gas, canthen help account for class-to-class variations betweenthe chondrite groups. An implication of this drawn-out,inefficient process is that younger chondrite types shouldcontain evidence of leftovers or refugees from earliertimes, which escaped primary accretion. Several aspectsof chondrite makeup are compatible with this implica-

tion. Of course, it is well known that ancient CAI min-erals and less-ancient, less refractory Amoeboid OlivineAggregates are found alongside much younger objects inthe same chondrites ( eg. Scott and Krot 2005). Also, theoldest chondrites (CVs) contain primarily type I chon-drules (which have an age observationally indistinguish-able to that of CAIs) and no (more oxidized, heavierO-isotope) type II chondrules, while the younger ordi-nary chondrites contain a mixture of type I and type IIchondrules, some with hints of age variation even withina given chondrite (Kita et al 2005). The oldest andrst-formed objects are all likely to have melted fromtheir abundant 26 Al (Kleine et al 2005, 2006; Hevey andSanders 2006), producing differentiated achondrites and

metallic objects, so it is no longer possible to dissect theirprimordial components.Of course, the real world is more complicated. Some

complications are meteoritic: as only one example, COchondrites and ordinary chondrites appear to be aboutthe same age (younger than CV chondrites), but havevery different chemical and isotopic properties (Kita etal 2005), which would, in the context of this scenario,argue for some degree of spatial (radial?) variation inthe makeup of nebula particulates at that time anyway.

On the theoretical side, of course, self-consistent nu-merical models which follow clump evolution in realis-

tic turbulence with headwinds and vertical settling needto be pursued, to check these preliminary assessmentswhich make simplistic assumptions about the local head-wind and assume isolated clumps. Several recent stud-ies have found that transient pressure ridges arise inthe gas in association with spiral density waves or vor-tical structures (Haghighipour and Boss 2003, Rice etal 2004, Johansen et al 2007); in such regions, the localradial pressure gradient and headwind diminish (otherregions will have unusually large headwinds). Moreover,clumps do not exist in isolation, but more realistically asa dense core within a larger, less dense envelope whichweakens the headwind felt by the dense clump core rel-ative to the nebula average we characterize by . Bothof these conditions would allow some clumps to survivewith smaller l. Other complications include interac-tions between different strengthless objects, leading tomergers or disruptions.

Our stability criteria are most likely to be satised ina region fairly close to the nebula midplane ( 0.01H ,see section 3.3), because at higher altitudes the verti-cal component of solar gravity leads to vertical settlingof dense clumps at higher terminal velocities than areeasily stabilized against. However, this vertical settling,even if it leads quickly to disruption of individual clumps,enhances the downward transport rate of small parti-cles and increases the solid/gas ratio closer to the mid-plane from that generally predicted by simple 1D diffu-sion models ( eg., Bosse et al 2006).

All these effects have implications for the occurrencefrequency of clumps of the requisite l for stability.Quantitative statistical determinations of the volumefraction of stable clumps which can become planetesi-mals at any given time will use, for instance, approachessuch as the cascade models described in Appendix A.Simplied, preliminary work exploring these factors in-

dicates that, while the volume fraction of stable clumpsis low at any given time (so the process is clearly notan efficient one), accretion rates are roughly an Earthmass per Myr in the asteroid belt region (Cuzzi et al2007). However, considerable work remains to establishthe statistical formation rate of appropriate clumps ca-pable of following this evolutionary path and to study theevolution of clumps in realistic turbulence, with variableheadwind velocity and simultaneous vertical settling.

We thank J. Eaton, M. Gaffey, P. Goldreich, W. Hart-mann, G. Laughlin, K. Sreenivasan, K. Squires, and A.Wray for very helpful conversations on various aspects of

this research, and A. Wray, A. Dobrovolskis, P. Garaud,and S. Weidenschilling for thorough reviews of the origi-nal manuscript and a number of useful comments whichhave been included. N. Turner and A. Carballido alsoprovided useful comments on an earlier version. Thiswork was supported by a grant to JNC from NASAsPlanetary Geology and Geophysics program. Generousgrants of cpu time from NASAs HEC program were es-sential to the progress of this research; we thank E. Tu,K. Schulbach, C. Niggley and R. Pesta in particular fortheir help along these lines. We also thank J. Chang forcoding optimization assistance.


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Fig. 4. Conditioning curves for multipliers p(m ); the parameter b depends on the local mass loading ; larger b indicates a narrowerPDF p(m ). Open symbols: for particle concentration C ; lled symbols: for enstrophy ( S = 2 ). The small insets show p(m ) at severalvalues of . From Hogan and Cuzzi (2007).

Fig. 5. Joint (binned) probability 2 P (, 2 ) from the conditioned cascade model, for a 24-level case with initial p / g = 10 2 .The curves of gure 1 are the S averaged equivalent of this plot (and show C instead of ), but do not include the effects of mass loading.Note the attening or saturation at 100, reecting the near-vertical asymptote of the conditioned- b curve of gure 4 at 100. A24-level cascade is in the plausible range for clumps of size 104 km in a nebula with 10 3 10 4 . In this gure, the enstrophy S isnormalized to its mean value for the binning lengthscale (from Hogan and Cuzzi 2007).

APPENDIX

APPENDIX A: MASS LOADING AND THE CASCADE MODEL

This aspect of our work incorporates two related and important concepts: that of intermittency , and that of astatistical cascade process. For instance, it is widely known that dissipation of turbulent kinetic energy occurs atthe Kolmogorov scale; it is less widely known that the spatial distribution of dissipation, like that of the particleconcentration factor C , is highly intermittent . Intermittent quantities are spatially and temporally unpredictable,and uctuate increasingly with increasing Reynolds number Re. However, the statistical properties of intermittentquantities like C (their probability distribution functions or PDFs) are well determined on any lengthscale. In whatfollows, it will be necessary to distinguish between two closely related quantities: the concentration C and the massloading = p/ g = C p/ g . This is because the emergence of particle feedback on the gas due to mass loading (),depends both on C and on the initial value of p. It is ultimately C that is evolved in the cascade, but how it evolvesdepends on . So both quantities will be alluded to in parallel.

In its inertial range, which is extensive at high Re, turbulence is a scale-free process that is often referred to as acascade and inspired the approach of cascade modeling (Meneveau and Sreenivasan 1991). The physics of transportof kinetic energy, vorticity, and dissipation from their sources at large scales is independent of scale until viscousprocesses enter at the smallest (Kolmogorov) scale. Scale-independence and Re independence are connected becauseRe determines the depth of the inertial range: Re 3/ 4 = L/ (Tennekes and Lumley 1972). Cascade models simplifythis complex, nonlinear, 3D scale-free process into a set of partition rules which are independent of scale, or of level


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in the cascade. Larger Re means a larger number of eddy spatial bifurcations (or levels) between L and , andstronger uctuations in intermittent properties (Meneveau and Sreenivasan 1991). Cascade models dont preserveall the information of full 3D models (such as the tubelike spatial structures characterizing vorticity) but they doreproduce the statistical properties (the PDFs), which are of primary importance for our problem. The successful useof such models to describe the intermittency of dissipation, and other quantities, in turbulence (Juneja et al 1994,Sreenivasan and Stolovitsky 1995) led us to pursue a cascade model for turbulent concentration of particles.

A cascade model consists of a set of multipliers - fractions which dene how a quantity of interest is unequallypartitioned from a uid parcel at some level, into its equal-sized subdivisions at the next level. The PDF of particleconcentration factor C is determined over all the numerous end-level outcomes of applying a multiplier m (chosenfrom its own PDF p(m)) to each volume element as it bifurcates at each level of the cascade. It has been observedthat p(m) is independent of level in the turbulent inertial range (Sreenivasan and Stolovitsky 1995). Hence, multiplierPDFs determined from a direct numerical simulation (DNS) at low Re (with a relatively small number of levels),merely applied repeatedly over additional levels, can predict the properties at higher Re (if nothing changes in thephysics). Our cascade model (Hogan and Cuzzi 2007) uses a common form for p(m): p(m) mb 1(1 m)b 1 , wherethe parameter b determines the width of p(m) (Sreenivasan and Stolovitsky 1995). Both the concentration C and thevorticity are described by their own b-distributions. Multiplier PDFs with small b( 1) are broad, and in themmultipliers much different from m = 0 .5 occur with higher probability, producing a grainier, more intermittent spatialdistribution. PDFs with large b( 1) are narrow and centered on m = 0 .5; because each eddy bifurcation theninvolves a nearly 50-50 partitioning, the resulting spatial distribution is nearly uniform and subsequent growth of becomes more difficult.

The cascade model can then be used to address the issue of how particle mass loading can affect the cascade - thatis, change the physics as the cascade progresses. This dependence is known as conditioning of a cascade. We foundthat the process can be represented quite well as two separate one-phase conditioned cascades for C and in whichmultiplier distributions p(m) for both C and depend only on the local particle mass loading . We used our full 3Dnumerical simulations to establish how b for both these properties depends on local particle or uid properties.

Figure 4 shows conditioning curves for both C and (the latter expressed as enstrophy S = 2), as extracted frommultiple 3D DNS simulations at various Re as large as 2000 (Hogan and Cuzzi 2007). Note how the multiplier PDFsextracted from the DNS results (shown using insets) get narrower (have larger b) as mass loading increases, chokingoff intermittency. In these runs, p/ g = 1, so C = . These results establish the upper limit 100 which canbe obtained by turbulent concentration, regardless of the initial value of p/ g . In the nebula, with a smaller initial p/ g , higher Re or deeper cascades (and larger ensuing C values) are needed to reach this saturation point.

To determine the joint PDF of concentration C and vorticity , we require not only the conditioned multipliers for2 and C (Figure 4 ), but also their spatial correlation. Hogan and Cuzzi (2007) found, on average, a 70-30 preferencefor anticorrelation at each partitioning, consistent with previous observations that particle concentration zones avoidzones of high uid vorticity (Squires and Eaton 1990, 1991; Eaton and Fessler 1994; Ahmed and Elghobashi 2000,2001). Including this partitioning asymmetry factor as a weighted coin gives us a two-phase (particle-gas), conditioned

cascade model which shows very good agreement with the joint PDF P (, 2) directly determined from our full 3Dmass loaded simulations (see Hogan and Cuzzi 2007). However, to match the full 3D DNS simulations at Re = 2000,

the cascade model needs only about 15 levels, taking about 10 cpu-hours (for 1024 realizations) compared to over90000 cpu hours to converge our full 3D simulation.

Using these conditioned multipliers and asymmetry factor, entire PDFs of particle concentration (mass loading) andvorticity can be generated for arbitrary numbers of levels (arbitrary Re). For example, the 24 level model of Figure 5(from Hogan and Cuzzi 2007) is of direct relevance for 10 3 104 km scale dense zones of interest for the nebula. Figure5 illustrates the saturation of the particle concentration at = 100 and the fact that high occurs preferentially atlow enstrophy or vorticity. The upper left quadrant is of most interest for determining the numbers of clumps whichcan survive to become planetesimals (see Cuzzi et al 2007 for more details).

APPENDIX B: SCALING BETWEEN NUMERICAL CODE AND NEBULA

While the primary result of this paper - the existence of some stability regime determined by a critical gravitationalWeber number We G - can be expressed in a nondimensional way, some aspects of the results (specically the numerical

results and the inferred value of l for the nebula) require us to pursue the relationship between nebula parametersand code units more deeply. Values in code units are denoted by primes below. The fundamental quantities in theproblem are (1) the gas and particle densities g and p = g ; (2) the clump diameter l; (3) the local orbit frequency(a) where a is the distance from the sun; (4) the local radial pressure gradient dP/da = dP/dx , where x is in the Hillframe; (5) the particle stopping time t s ; (6) the velocity of material in the Hill frame W , that is, relative to Keplerian,where W represents either the particle or gas velocity; and (7) the gravitational constant G. The principal forces arethe coriolis force due to the rotating frame 20 W , the pressure force ( 1/ g)dP/dx , and the gas drag forces thatcouple the gas and particles (section 3.3). The code unit of length is radians (the domain is 2 radians on the shortdimension ( x or z) and the grid cell size is therefore = 2 /N where N is the number of grid points along that axis).The code time unit (ctu) is 1 / where = 1 is the code value of rotation frequency; thus the orbit period is 2 incode time units or ctu. We treat as equivalent a = R = x.

As noted in section 3.3,

(a) = ( GM /a 3)1/ 2 (G)1/ 2 , or G = 2 /


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where = M /a 3 3.8 10 8 g cm 3 at 2.5 AU (note this is different from Safronovs (1991) Roche densityR = 3 M / 4a 3). To normalize mass and mass density (and ultimately establish the code value of the gravitationalconstant G ) we assume a nebula gas mass density o 10 10 g cm 3 (minimum mass nebula; Cuzzi et al 1993) at2.5 AU; then code quantities g = g / o = 1 and p = p/ o = g = , and = / o . Also, in code units = / 0 1 where o = ( ao) is the actual orbit frequency at the region of interest. Then the gravitationalconstant in code units, G , can simply be written as G = 2 / = 1 / . Finally, the pressure force is written as

1g

dP dx 2a

2

= 2 aG

= 2 wg ,

where 10 3 is typical (Nakagawa et al 1986; Cuzzi et al 1993), G = o/ = 3 10 3. Then the dimensionlessparameters and l are set to satisfy various constraints of the code (resolvable clump, adequate particle statistics,reasonable timescale, etc) and is allowed to vary, to determine empirically the critical value of We G as the largestvalue of We G that remains stable. That is, we want l to be large enough such that the clump is well resolved, but notso large that it lls the entire cross section of the computational box. Typically l is only 10-20 grid cells so far, andas noted below, this exaggerates the viscous stresses and surcial erosion of material. We want to be large enoughthat the clump is much denser than the gas, and we want a large enough number of particles to act like a continuum.The value of ts should ensure that the clump sedimentation time t sed is larger than its ram pressure disruption timetdis (see below), to provide a real test that self-gravity is preserving the clump rather than simple collapse.

Important timescales in the numerical model

The dynamical collapse time of the clump under its own self-gravity, and in the absence of gas pressure, is tG =(G p) 1/ 2 , or in code units t G = (G ) 1/ 2 . The mass loading which produces a clump dynamical time comparableto the orbit time is then obtained using (G ) 1/ 2 = 2 / = 2 or = 1 / 4G 100, again in fair agreementwith prior expectations. However, gas drag prevents clump collapse on this timescale, as described in section 3.1, andactual shrinkage takes a time t sed (eqn. 1). The number of orbits over which we need to follow the clump to ensure itreally survives is roughly

tsed 2

=c

8G r s=

H 2

8G r s=

2

GsH r

18

=

sH r

18

,

where we have used c = H . This expression can be assessed using real quantities, and is 30-300 orbits for particleradius 300 and mass loading = 1000 100. This behavior ( t sed 2/ ) can be controlled in the code, oncethe other parameters are established as above, by adjusting the particle stopping time ts . Because of long code runtimes, at present we are limited to stipulating only that the sedimentation time signicantly exceeds the nominal(self-gravity-free) disruption time; that is, tsed tdis , where

tdis (l/w ) 2/C D .This expression for the disruption time tdis is derived from the ram pressure force (sect 2.3) by setting the distancetraveled in time t by the windward half of the clump, under acceleration by the ram pressure force, equal to the clumpdiameter (assuming the leeward half of the clump doesnt get accelerated), and solving for t. The value of C D for ourclumps is on the order of unity, as we verify below.

Numerical viscosity and the clump Reynolds number:

The code calls for some input viscosity ; with it, the Reynolds number of the clump having diameter l radians,in a headwind of speed wg (in code units of radians per ctu), is Re

c = l w / 2 where = 0 .1 is the dened code

viscosity (radians 2/ctu). We assume an inertial range expression within the wake of the clump to determine the wakesKolmogorov scale: /l = Re 3/ 4c , and thus = (2 l 1/ 3/w g )3/ 4 radians. Our highest resolution runs to date hada gas relative velocity wg = 38 radians/ctu and used a clump size l = 2 radians. The nominal Kolmogorov scale

associated with this ow is = (2(0 .1)(2) 1/ 3 / 38)3/ 4 0.02 radians (compared to the grid cell size of 0.05-0.1 radians);the wake turbulence is thus under-resolved and the codes true viscosity is numerical: n wg (at the boundary of the clump).

While we are not concerned with the ne-scale details of the wake, this marginal resolution introduces some caveats.The ratio of ram pressure force to viscous force (both per unit area) is, in general, ( g w2g / 2)/ (gdwg /dr ) wg l/ 4 =Rec / 2 where Re c is the Reynolds number of a clump, and we have approximated dwg /dr wg / (l/ 2). Because we aredominated by numerical viscosity, n wg , thus in code units Re

c wg l / 2wg = l / 2 5 in many cases so far.

For comparison, the Reynolds number for nebula clumps of interesting sizes is > 106 . This means that viscous stressesaround the periphery of the clump, and the fractional depth of the viscous boundary layer, are grossly exaggeratedin the numerical model relative to the actual nebula regime of interest. The anomalously large role of (numerical)viscosity leads to anomalously large erosion from the surface of our numerical clumps, and degrades our results inthe sense that potentially stable clumps might appear less stable, due to the erosive mass loss experienced over theirsedimentation time. Going to higher resolution cases (Re c 10) signicantly ameliorated this effect but the problem


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Fig. 6. Determination of the effective drag coefficient for our clumps, from plotting estimated disruption times tdis vs. the quantity(l/w g )(2) 1 / 2 ; the slope is then C

1 / 2D . This determination is slightly qualitative as to when we judge the clump to be disrupted; we

decided to use the time at which vortex-driven voids rst appear in a a z planar cross-section through the nominal center of the clump.

has certainly not been entirely removed. Another way to characterize the degree of articial erosion is to estimate thephysical Kolmogorov scale for the actual nebula/clump situation: for = 10 6 cm sec 2, l = 10 4 km, and wg =3800cm/s, we nd 100m, or 10 5 of the size of a 104 km clump, compared to a fractional size of perhaps 10 1 or soin the code, as determined by numerical viscosity. Clearly, surface erosion is a much smaller effect in the nebula casethan in our crude models. While even at this resolution apparently stable congurations can be found, it would bedesirable for future studies to improve on this situation.

Clump drag coefficient C D : The drag coefficient of the clump is Reynolds-number-dependent and must also be scaledto nebula values to obtain an estimate of l. Weidenschilling (1977) gave expressions for the drag force per unit area(giving the pressure force per unit area), experienced by solid spheres of different Re c in the Stokes drag regime of interest here, as C D g w2g / 2. For 1 < Rec < 800, C D = 24 / Re

0.6c , or C D 8 for Rec 5, as in our code. The large

drag coefficient and ram pressure is partly due to low-pressure zones set up immediately behind particles in this rangeof Rec . By comparison, for a solid sphere with Re c > 800 as under nebula conditions, C D = 0 .44. However, neither of these values might be entirely applicable to our clumps, which are not rigid spheres. In practice, we have determinedC D empirically from observed values of tdis with self gravity turned off. Recalling that tdis = ( l/w g )(2/C D )1/ 2 , wesimply plot our estimate of tdis , the point at which sizeable vortex-driven internal voids appear in the clump, againstthe combined parameter ( l/w g )(2) 1/ 2 , giving C

1/ 2D as the slope of the best t straight line ( gure 6 ). It seems that

C D 1 is a good assumption, but this calculation should be redone with higher resolution (lower numerical viscosity)codes. At nebula Re c , C D could reach its theoretical high- Re value of 0.4 and this is what we assume in deriving l.

Finally then, the several stable cases we have run ( eg. gure 3) can be characterized by wg = 38 rad/ctu, = 380(which corresponds to a minimum mass nebula local density g = 10 10 at distance 2.5 AU), = 10 3 , and l 1radian, and while we have not as yet accurately determined the transition value of We G , a value We G 1 is clearlystable. Solving equation (4) of section 3.3 for l under nebula conditions, assuming WeG = 1, taking C D = 0 .44 for ahigh-Re c situation and assuming a nominal = 10 3 , we nd l > 1.5 5 106, independent of g and neglecting aquestionable coefficient of order unity (because of the contribution of viscous stresses in our low-Re c code). Foreseeablerenements (specically, lower numerical viscosity) may increase We G ( the surface tension analogue gives We

G 10)and thus decrease the value of l.

APPENDIX C: TIME EVOLUTIONS AND MOVIES

Quicktime movies showing our time evolutions are available online athttp://spacescience.arc.nasa.gov/users/cuzzi/,with lenames as given below. After publication, the movie les will also be available on the ApJ web site as

mpgs. Each shows a dense clump in the Hill frame, rotating at local Keplerian velocity, with more slowly-orbiting gasimpinging on the clump from its leading side. The two left-hand panes show the view from along the radial axis (inthe Z plane; top) and along the vertical axis (in the a plane; bottom). The large right-hand pane shows theview from the orbitally leading direction (in the a Z plane). The wake formed by the clump is easily visible in thetwo left-hand panes. Cases hill26 .gif and hill29a .gif are sampled at a similar time near their end-point in gure 3. Filehill26.gif shows how a clump with no self-gravity is disrupted in about 1.5 code time units for the initial conditionschosen (the orbit period is 2 code time units), in agreement with our simple estimates of tdis . Files hill29a.gif andhill31a.gif are intended to illustrate the same (stable) Weber number case (We G 1) but we vary two constituentparameters of We G (we decrease both the gas velocity and mass loading by a factor of three) to illustrate similarity;
http://spacescience.arc.nasa.gov/users/cuzzi/http://spacescience.arc.nasa.gov/users/cuzzi/


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Fig. 7. Properties of the dense central regions in three simulations, as a function of time. Cases are shown in each panel as: solidline (stable, self-gravitating case 31a); dashed line (stable, self-gravitating case 29a); and dotted line (unstable, non-gravitating case 26).Not all cases were run to the same stopping point due to resource constraints. Top: cumulative total mass retained by the clump; middlepanel: average mass density of the central mass fraction f ; bottom panel: effective radius containing mass fraction f . These resultsshow that, away from the articially perturbed perimeter regions, W eG -stable clumps are behaving as expected by the simple model, andW eG -unstable clumps do not show this behavior.

because of the specic parameters chosen, hill31a .gif runs for a longer time, but is suffering considerable erosiontowards the end of the run. Nevertheless, as discussed below, it retains a dense core that continues to contract. Asdiscussed in Appendix B, more realistic cases, with higher numerical resolution, would have smaller numerical viscosityand incur less erosion. Case hill29a .gif, while not run as long as hill31a .gif, appears solidly stable with a dense corethat is continuing to shrink at the end of the run. Note some interesting oscillatory behavior, such as hinted at in our1D compressible runs (gure 2).

Behavior of central regions of dense clumps

The simulations which created the movies referred to above can be used to assess the behavior of the central, denseregions of each clump in a quantitative manner. Figure 7 has three panels, showing the time variation of someproperties of the particles which lie within the central regions of the clump (its densest fraction, as sampled out tosome cumulative mass threshold). In the top panel, we show how much mass is being eroded from the clump overall(particles moving faster than some comoving velocity threshold are assumed to have left the clump). All three clumpsare losing mass, naturally, but the non-gravitating clump 26 is losing it much more rapidly than the gravitating clumps29a and 31a. In the central panel we show the mean mass density as measured over some fraction f of the clumpmass lying at the highest densities; in clumps 26 and 29a, we chose f = 0 .5, and in clump 31a, we chose f = 0 .25. Inthe nongravitating clump, the core density never increases and it quickly blows up, while the central regions of thegravitating clumps (removed from the articial mass erosion at their peripheries) inexorably get denser at a steadyand perhaps even slightly increasing rate (an increasing rate is predicted by the simple model). The lower panel showsthe effective average radius of the region being sampled. The non-gravitating clump never shrinks at all, but bothgravitating clumps do. Moreover, as the gravitating clumps shrink, their density increases faster than their lineardimension decreases, so the product l increases - causing them to become more stable as time goes on. This is anargument for stability even in the presence of erosion. As expected, the density growth timescale (the sedimentationtime t sed of equation 1) is three times smaller for the denser clump of case 29a (middle panel). Meanwhile, the apparentagreement of the variation of the fractional radius containing half of the mass seen between cases 29a and 31a in thelower panel is fortuitous, because the total masses of the clumps are changing (and eroding) at different rates and fromdifferent depths in the two cases. Nevertheless, the cores are both shrinking and becoming denser in the two stablecases. The amount of erosion that is incurred from the margins of each clump is related to (perhaps proportional to)the thickness of the viscous boundary layer relative to the radius of the clump, which can be expressed in terms of theReynolds number. As we have argued above in this Appendix, actual nebula clumps capable of remaining stable wouldhave Reynolds numbers ve orders of magnitude larger - impossible to study with numerical models. The relevance of all viscous boundary layer effects, including the erosion that inicts our current numerical runs, will decrease by thatfactor in the real situation.


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