Co-Accretion of Chondrules and Dust in the Solar Nebula
aa r X i v : . [ a s t r o - ph ] F e b D raft version N ovember
5, 2018
Preprint typeset using L A TEX style emulateapj v. 03 / / CO-ACCRETION OF CHONDRULES AND DUST IN THE SOLAR NEBULA
C.W. O rmel
Kapteyn Astronomical Institute, University of GroningenP.O. Box 800, 9700 AV Groningen,The Netherlands and
J.N. C uzzi , A.G.G.M. T ielens
NASA Ames Research CenterMail Stop 245-3,Mo ff ett Field, CA 94035, USA Draft version November 5, 2018
ABSTRACTWe present a mechanism for chondrules to stick together by means of compaction of a porous dustrim they sweep up as they move through the dusty nebula gas. It is shown that dust aggregatesformed out of micron-sized grains stick to chondrules, forming a porous dust rim. When chondrulescollide, this dust can be compacted by means of rolling motions within the porous dust layer. Thismechanism dissipates the collisional energy, compacting the rim and allowing chondrules to stick. Thestructure of the obtained chondrule-dust agglomerates (referred to as compounds) then consists ofthree phases: chondrules, porous dust, and dust that has been compacted by collisions. Subsequently,these compounds accrete their own dust and collide with other compounds. The evolution of thecompound size distribution and the relative importance of the phases is calculated by a Monte Carlocode. Growth ends, and a simulation is terminated when all the dust in the compounds has beencompacted. Numerous runs are performed, reflecting the uncertainty in the physical conditions atthe chondrule formation time. It is found that compounds can grow by 1-2 orders of magnitudesin radius, upto dm-sizes when turbulence levels are low. However, relative velocities associated withradial drift form a barrier for further growth. Earlier findings that the dust sweep-up by chondrules isproportional to their sizes are confirmed. We contrast two scenarios regarding how this dust evolvedfurther towards the densely packed rims seen in chondrites.
Subject headings: solar system: formation — planetary systems: formation and protoplanetary disk INTRODUCTION
Protoplanetary nebulae have been studied in increasingdetail from visual to microwave wavelengths (Meyer et al.2007; Watson et al. 2007), and hundreds of extrasolar plan-etary systems have been discovered, but the ‘primary ac-cretion’ stage of the planetary formation process – thatwhich leads from interstellar grains to planetesimals largeenough to decouple from the nebula gas (asteroid- andcomet-nucleus size objects) – remains obscure. In this par-ticle size range, coupled particle-gas dynamics dominatesthe evolution, as reviewed recently by Cuzzi et al. (2005),Cuzzi & Weidenschilling (2006), and Dominik et al. (2007).The main processes which have been hypothesized for pri-mary accretion include i) incremental growth by sticking ofsmall grains to each other and to larger particles, ii) variouskinds of instabilities occurring in a particle-rich midplane re-gion, and iii) formation of planetesimals from dense zones ofparticles that form in turbulence due to vorticity or pressuree ff ects.A critical but unknown nebula property in this stage iswhether turbulence is present, and if so, what its intensity is(Stone et al. 2000; Gammie & Johnson 2005). If the nebula isnonturbulent, particles of all sizes can settle into a dense layernear the midplane where incremental growth is fairly robustfor expected, but still poorly known and therefore somewhatad-hoc, sticking properties (Weidenschilling & Cuzzi 1993; Electronic address: [email protected] address: [email protected], [email protected]
Cuzzi et al. 1993; Weidenschilling 1997, 2000, 2004). Thisis because the dense particle layer drives the entrained gasto corotate, and relative velocities between equal-size parti-cles would largely vanish. The high local mass density en-sures that growth is rapid (Weidenschilling 2000) – perhapstoo rapid (Cuzzi et al. 2005). Various instabilities in sucha layer, mostly gravitational, have been studied for decades(Goldreich & Ward 1973; Sekiya 1998; Youdin & Shu 2002;Youdin & Goodman 2005), but these are precluded if the neb-ula is even weakly turbulent (Cuzzi & Weidenschilling 2006).Astronomical and planetary observations seem tobe most naturally reconciled with turbulent nebu-lae (Dullemond & Dominik 2005; Cuzzi et al. 2005;Brownlee et al. 2006; Zolensky et al. 2006; Ciesla & Cuzzi2007). A number of studies indicate that turbulence excitesmeter-size particles to relative velocities at which theyprobably disrupt each other (Weidenschilling 1988; Benz2000; Sirono 2004; Langkowski et al. 2007), posing a barrierto further growth. However, some recent studies suggest thatturbulence itself can concentrate particles of di ff erent sizes,in di ff erent ways, and trigger rapid planetesimal formation(Cuzzi et al. 2001, 2007; Rice et al. 2006; Johansen et al.2007). Thus, in spite of the ongoing uncertainty in justhow turbulence may be maintained (Stone et al. 2000;Mukhopadhyay 2006), it is sensible to consider its e ff ect inmodel studies. This collisional disruption limit, combinedwith the rapid inward drift of m-size particles by which theyare ‘lost’ from the local region, led to the concept of a ‘m-size C.W. Ormel, J.N. Cuzzi and A.G.G.M. Tielensbarrier’ or bottleneck to growth; once large particles exceedthis barrier, relative velocities become lower, allowing themto grow further and to drift less rapidly out of the accretedregion. Our studies were initially motivated by a desire to seeif growth into loose fractal clusters and subsequent packingcould allow the m-size barrier to be crossed.In the conceptually simplest models, growth occurs bysimple sticking of particles (Weidenschilling 1997, 2000,2004; Blum 2004). While it goes against our earthboundintuition that macroscopic particles can stick to each other,some microgravity and earthbased experiments show that,while bouncing transpires at intermediate velocities, stick-ing prevails for both low (less than a meter / second) and high(for 13 −
25 m s − ) relative velocities (Blum & Wurm 2000;Blum & Schr¨apler 2004; Wurm et al. 2005; Marshall et al.2005; Dominik et al. 2007). Other experiments indicate thatcertain solids (water and methanol ice, organic material) are‘stickier’ than others (silicates) (Bridges et al. 1996). How-ever, no significant amount of these especially sticky materi-als has been found in primitive meteorites. Still, entire chon-drites (and by inference entire parent bodies) are composed ofsmall silicate objects that seem to have been gently assembledand compacted, at least initially (Metzler et al. 1992; Brearley1996; Cuzzi & Weidenschilling 2006); how did this happen?The meteorite record (discussed in more detail below) showsthat many mm-sized solid objects are encased in rims of mi-cron and submicron-sized mineral grains. One obvious possi-bility is that these rims form by nebula accretion of grains ontothe underlying core particles (Nagahara 1984; Metzler et al.1992; Paque & Cuzzi 1997; Hua et al. 2002; Zega & Buseck2003). In § ff ected by aqueous alteration,while Ordinary Chondrites contain very little matrix and havegenerally incurred only limited aqueous alteration. Violentcollisional processes occurred after primary accretion whicha ff ected the contents and appearance of most meteorites, andto best understand the primary accretion process one mustlook back through this stage where possible to the rare, un-brecciated subset of rocks and rock fragments called ‘primarytexture’ (Metzler et al. 1992; Brearley 1993).The dust in chondrites is found to have two physi-cally defined components: rims and inter-chondrule ma-trix (Huss et al. 1981; Scott et al. 1988; Brearley 1996;Brearley & Jones 1998). Fine-grained rims are clearly as-sociated with individual chondrules and other macroscopicparticles in microscopic images, and usually even stay at- tached to the chondrules when they are disaggregated fromthe host rock (Paque & Cuzzi 1997). Some studies report thatthe composition of these fine-grained rims is uniform acrossa wide range of underlying mineral types, including more re-fractory (higher-temperature) oxides which formed much ear-lier (Brearley 1993; Hua et al. 2002) and some find dramaticvariations between the composition of rims on adjacent chon-drules (Taylor et al. 1983; Scott et al. 1984). Generally, thechondrules cooled completely before accreting these fine dustgrains (Brearley 1993). Interchondrule matrix, more gener-ally dispersed between all the macroscopic components of therock, is also made of fine-grained material. The grain sizesin fine-grained rims are noticeably smaller than the ubiqui-tously enveloping matrix, even though the compositions ofthe rims and matrix are very similar or identical (Ashworth1977; Brearley 1993, 1996; Zolensky et al. 1993). It has beenreported that the rim porosity is also smaller than that of thesurrounding matrix (Ashworth 1977; Trigo-Rodriguez et al.2006). The relative abundance of rim and matrix materialto chondrule material varies from one meteorite class to an-other (Scott et al. 1988); however, the rim mass (or thickness)is often found to be proportional to the mass (or radius) ofthe underlying chondrule (Metzler et al. 1992; Paque & Cuzzi1997).Several di ff erent model e ff orts have attempted to explainsome of these properties in the context of nebula sweep-up oraccretion of the fine-grained rims by chondrules and their like.Morfill et al. (1998) hypothesized that if a particle had a speedrelative to the gas which was proportional to its radius, and ifthe chondrules in a region sweep up all the local dust in a one-stage event (no ongoing replenishment of dust), the observedrim-core correlation would be obtained. Cuzzi (2004) showedthat chondrule-size particles in turbulence plausibly exhibitjust the appropriate (near-linear) dependence of relative ve-locity on size, even though most particles obey a square-rootdependence on radius. Cuzzi (2004) relied on collisional out-comes proposed by Dominik & Tielens (1997) for porous ag-gregates of fine grains, and suggested that for particles muchlarger than chondrules, the velocity relative to the gas in-creases to a point where they enter an erosional regime.On the other hand, Sears et al. (1993) andTrigo-Rodriguez et al. (2006) question whether fine-grained rims are nebula accretion products at all.Trigo-Rodriguez et al. (2006) point out in particular thatthe fine-grained rims in CM chondrites, such as discussedby Metzler et al. (1992), have a porosity of 10-20%, muchlower than the high-porosity structures formed by, e.g.,Blum & Wurm (2000) or Blum & Schr¨apler (2004). Less isknown quantitatively about the porosity of fine-grained rimsin other chondrite types, although Ashworth (1977) states thatrim porosities are less than 6-15% in ordinary chondrites. Thealternate that Trigo-Rodriguez et al. (2006) and Sears et al.(1993) prefer, while they di ff er in the details, is that thefine-grained rims seen in CM chondrites, in particular, arecreated on the parent body from a generic enveloping matrix,by some combination of compaction and pervasive aqueousalteration. This suggestion might make it harder to explainwhy the grain size is smaller than in the nearby envelopingmatrix. Nevertheless, the discussion shows that the porosityof fine-grain rims is an important diagnostic of their origin.In this paper, we develop a detailed collision modelto study the rimming and accretion processes of chon-drules simultaneously and, in a statistical study, quantifythe growth that can be obtained under a wide range ofccretion of chondrules in the solar nebula 3(uncertain) nebular conditions. Our model treats multi-ple components: solid ‘chondrules,’ submicron grains andtheir very porous nebula aggregates, porous accretion rimsformed by direct accretion of monomers and aggregatesonto chondrules, compact rims formed by collisional com-pression of pre-existing porous rims, and compound ob-jects formed by sticking of rimmed objects, which them-selves might become rimmed in dust. Our collisional out-comes use physical guidelines set by laboratory and the-oretical models (Dominik & Tielens 1997; Blum & Wurm2000; Blum & Schr¨apler 2004; Langkowski et al. 2007). Weuse quantitatively correct closed form relative velocity ex-pressions for particles in turbulence of varying intensity(Ormel & Cuzzi 2007), which capture the increase in rela-tive velocity as particles grow by accretion of other parti-cles. We use a Monte Carlo approach to calculate the prob-ability of di ff erent outcomes, over a wide range of nebulaparameters (level of turbulence, gas and solid density). Weassess i) the extent to which fine-grained rims can dissipatecollisional energy and allow growth by sticking to proceed,and ii) the extent to which these dissipative collisions com-pact initially porous dust rims into lower porosity states. Weleave for future study the physics of disruptive collisions andthe details of vertically varying particle density and turbu-lent intensity, such as might occur if the global turbulent in-tensity is vanishingly small (Cuzzi et al. 1993; Sekiya 1998;Dobrovolskis et al. 1999; Weidenschilling 1997).This paper is organized as follows. In § ff erent sourcesof relative velocities particles can obtain in the nebula, andcalculate the timescales involved in the various accretion pro-cesses. We end this discussion with a brief summary of theenvisioned collisional scenario. § § § ff ects of a parti-cle dominated environment caused by settling of compoundson the growth of compounds. We also discuss several obser-vational implications, emphasizing in particular the relationbetween the dust in our model to the fine-grained rims seenaround chondrules in meteorites. We summarize our resultsin § MODEL
Outline
The central theme of this paper is to model the process ofdust accretion onto chondrule surfaces and explore whethercompaction of this dust during inter-chondrule collisions actsas a sticking agent, with which significant growth can beachieved. For dust aggregates this compaction mechanism iswell known (Dominik & Tielens 1997; Blum & Wurm 2000;Wada et al. 2007): by restructuring of the constituent grainsthe excess collisional energy is dissipated. For dust-rimmedchondrules we argue the situation is analogous, except thatpart of the aggregate’s interior is now replaced by a chon-drule. The prerequisite for such a scenario is the presenceof a reservoir of dust that is accreted fractally by the chon-drules, preserving its flu ff y structure. Here, we follow theMorfill et al. (1998) ‘closed box’ scenario in which a fixed TABLE 1L ist of frequently used symbols
Symbol Description ∆ v or ∆ v relative velocity Ω local Keplerian orbital frequency α turbulent strength parameter ( § γ surface energy density δ fractal growth parameter ( § ǫ size ratio ( ǫ ≤ η nebula pressure parameter (eq. [7]) λ mean free path (gas) ( § ν m , ν T molecular / turbulent kinematic viscosity ( § φ, φ PCA , φ pd filling-factor (PCA / porous dust) (Table 3) σ collisional cross section τ f friction time (eqs. [5] and [6]) ρ X gas density over MSN ( § ρ d , ρ g spatial dust / gas density ( § ρ (s)c , ρ (s)d specific material chondrule / dust density (Table 3) ρ , ρ internal particle density C ij collision rate between particles i , j ( § E collisional energy ( § E roll , E br monomer rolling / breakup energy (eq. [12]) H g gas scaleheight ( § R heliocentric radiusRe Reynolds number ( § R gc spatial gas-chondrule mass-ratio ( § R cd spatial chondrule-dust mass ratio ( § § P pressure T temperature V geometrical (total) compound volume (eq. [20]) v g , v pg gas and particle-gas turbulent velocity ( § a geometrical (total) radius ( § a µ reduced radius, a a / ( a + a ) a dust dust aggregate radius ( § a ch chondrule radius a monomer radius ( § c g sound speed (gas) ( § f comp required dust fraction at collision for sticking (eq. [16]) f d compound dust fraction by mass (Table 3) f geo geometry factor ( § f p compound porous dust fraction with respect to total dust mass(Table 3) m , m d , m ch (dust / chondrule) massmw-.. or h .. i m mass-weighted averages (see eq. [30]) t dd , t dc , t cc dust-dust / dust-chondrule and chondrule-chondrule collisiontimes ( § t L / t s large / small eddy turn-over time ( § v K orbital (Kepler) velocity v r particle radial drift velocity (eq. [8])N ote . — List of frequently used symbols that re-occur over various sections inthe paper. amount of dust is injected instantaneously to the chondrulepopulation. The compound objects (or, simply, compounds)thus obtained are modeled to consist of three phases: chon-drules, compact (i.e., restructured) dust, and porous (i.e., frac-tally accreted) dust. The restructuring mechanism also holdsfor collisions between compounds, again at the expense ofthe porous phase. In this way a coagulation process is initi-ated by which chondrules are accreted into large compounds.This coagulation is only stopped when the compounds run outof porous dust, such that the collisional energy can no longerbe dissipated; at this stage, all the free-floating dust has beenaccreted and no more porous dust remains.The environment in which these processes take place is akey ingredient that enters the coagulation model. A violent,low dust density environment leads to high velocity collisionswhich quickly pack down the porous rim, limiting its capabil-ity to allow further sticking, while even higher velocities willlead to break-up of compounds. On the other hand, if (rela-tive) velocities are modest and remain so during the phase in C.W. Ormel, J.N. Cuzzi and A.G.G.M. Tielens TABLE 2G as and dust parameters (1) (2) (3) (4)gas density ⋆ ρ g / . / .
16 10 − g cm − sound speed c g ./ . / . cm s − mean free path (gas) λ . / / T / /
89 Kpressure parameter η . / . / . × − large eddy turn-over time t L = Ω − . / . / . ⋆ α − inner eddy turn-over time ⋆ t s . / /
131 10 sgas-chondrule ratio ⋆ R gc ⋆ R cd a ch µ mmonomer dust size a µ mfractal growth parameter δ . ⋆ γ
19 erg cm − N ote . — Parameters characterizing the gas (upper rows) anddust / chondrules (bottom rows). Gas parameters correspond to a minimummass solar nebula (MSN) model of total gas mass of 2 . × − M ⊙ inside10 AU with power law exponents of, respectively, − . − .
5, for the sur-face density and temperature structure as function of radius (Takeuchi & Lin2002). Columns denote: (1) parameter description; (2) symbol, R gc = ρ g /ρ c and R cd = ρ c /ρ d ; (3) corresponding value with multiple values denoting con-ditions at 1, 3 and 10 AU, respectively (3 AU is the default); (4) unit. Param-eters indicated by a ⋆ are variables (default model values are given). which compounds accrete other compounds and porous dust,this could lead to significant growth. In this work we use vari-ous sources for relative velocities: thermal, turbulent and sys-tematic. The relative velocity is further determined by theinternal structure (density) of the compounds, which a ff ectstheir coupling to the gas. The internal structure of compoundsis reflected in the definition of the ‘geometric size’: the sizethat corresponds to the e ff ective aerodynamic cross section.The evolution of the internal structure during the accretionprocess is therefore of key importance: i.e., do collisions fol-low ‘hit-and-stick’ behaviour in which growth proceeds frac-tally, or do collisions keep the filling factor constant. The turbulent nebula
Disk physical structure and model components
The start of the model is defined as the point at which pop-ulations of chondrules and dust interact. Before this point,the two populations were either spatially isolated or one didnot exist. This paper is not concerned with the history of thetwo populations – specifically, we avoid the nagging chon-drule formation question – but merely define the zero time ofthe model ( t =
0) as the point where the two populations mix.Of course, the history of the two populations determines toa large extent the conditions that prevail at the start of themodel, e.g., the size of the dust particles. We take 1 µ mas the monomer size (i.e., the smallest constituent size of adust grain), which roughly corresponds to the sizes of the finegrains observed in chondrites (Ashworth 1977). These grainsmay have formed by condensation onto seed grains (Chiang2004) and could have aggregated into larger (flu ff y) dust par-ticles before they interact with the chondrules. For instance, ifdust condenses out in a region of (recently formed) chondrules(Wasson & Trigo-Rodriguez 2004; Scott & Krot 2005b), thesize distribution will be dominated by monomers. However,if chondrules encounter a dust cloud only after a certain timesince its formation ( ∼ .. yr), larger aggregates are ex-pected to have formed through monomer collisions. In thiswork, though, the size of the dust aggregates ( a dust ) is simplytreated as a free model parameter ( § R = ρ g = . × − g cm − , a thermal speed of c g = . × cm s − anda mean free path of λ =
82 cm. Studies of chondrule forma-tion indicate, however, that gas densities may be much higher(e.g., Desch & Connolly 2002; Cuzzi & Alexander 2006) andwe will therefore treat ρ g as a free parameter, denoting with ρ X the density enhancement over MSN. We further assumethe gas surface density ( Σ ) and temperature ( T ) profiles arepower-laws of heliocentric radius ( R ) and fix the exponents at − . − .
5, respectively. This choice is consistent with asteady disk in which ˙
Σ = M , inde-pendent of radius. The gas-to-chondrule and the chondrule-to-dust density ratios, R gc = ρ g /ρ c ; and R cd = ρ c /ρ d , (1)are free parameters; for example, R cd = R gc =
100 is thestandard gas to solids ratio. In our model chondrules follow alog-normal size distribution, for which we take parameters of300 µ m (the mean) and a (log-normal) width parameter of 0 . h a i m = µ m. Note that the initial distribution is notnecessarily equivalent to the distribution that ends up in me-teorites, or the distribution extracted from meteorites by thethin section method (Eisenhour 1996). An overview of all pa-rameters characterizing the gas and solids is given in Table 2.We assume that the gas in the disk is in a turbulent state ofmotion. After Shakura & Sunyaev (1973), the turbulent vis-cosity is parameterized as ν T = α c g H g = α c / Ω , (2)where H g is the scaleheight of the gas disk, Ω the local (Keple-rian) rotation velocity, and α a scale parameter that determinesthe strength of the turbulence (Shakura & Sunyaev 1973).Values for α are very uncertain. If the magneto-rotationalinstability is active it may be up to 10 − (Balbus & Hawley1991; Hawley & Balbus 1991); however in regions of lowionization it can be much lower (Gammie 1996; Sano et al.2000). The extent of the turbulence is determined by theReynolds number, Re, defined as Re = ν T /ν m , with ν m themolecular viscosity, ν m = c g λ/ ℓ ), velocity ( v ), and turn-over time ( t ), between an outer (orintegral) scale L and an inner (or Kolmogorov) scale ℓ s . Fol-lowing previous works, t L , the largest eddy turn-over time,is taken equal to the inverse orbital frequency, t L = / Ω ,and v L = α / c g (e.g., Dubrulle et al. 1995; Cuzzi et al. 2001;Schr¨apler & Henning 2004). The eddy properties at the tur-bulence inner scale then follow from the Reynolds number: t s = Re − / t L ; ℓ s = Re − / L ; v s = Re − / v L . (3) Thermal motions
When gas molecules collide with a larger (dust) particlemomentum is transferred, changing the motion of the dust Provided the α -turbulence model (eq. [2]) is assumed and boundary con-ditions are neglected (see, e.g., Pringle 1981). ccretion of chondrules in the solar nebula 5particle. These kicks occur stochastically, resulting in a ve-locity behaviour known as Brownian motion. The ensuingvelocity di ff erence between two particles of mass m and m is highest for low masses and high temperatures, ∆ v BM = r k B T ( m + m ) π m m , (4)where k B is Boltzmann’s constant. For micron-sized parti-cles Brownian velocities are a few mm / s; but since ∆ v BM de-creases with the − / Systematic motions
The key parameter that determines the coupling of solids tothe gas is the friction time, τ f . In the Epstein regime the sizeof the particle, a , is small with respect to the mean-free-pathof gas molecules, λ , and the friction time is given by τ f = τ Epf = c g ρ g m π a . ( a ≤ λ ) (5)For solid 1 µ m-grains the friction time is τ f ∼ /ρ X sfor the default nebula parameters of 3 AU (Table 2), while foran a ∼ µ m chondrule it takes ∼ /ρ X hours before thetraces of its initial motion are ‘erased.’ Note that equation (5)defines a as the geometrical radius of the particle, i.e., theradius corresponding to the angularly-averaged projected sur-face area of the particle. If the particle is a flu ff y aggregate itsfriction time is therefore much less than a compact-equivalentwith the same mass. If significant growth takes place, parti-cles will no longer obey the Epstein drag law; friction timesare then enhanced with respect to τ Epf , i.e., τ Stf = a λ p C τ Epf . ( a ≥ λ ) (6)Here, Re p = a v pg /ν m is the particle Reynolds number, whichdetermines the constant C , and v pg the particle-gas velocity.Within the physical conditions of the simulations in § p =
1, for which C = Re − (Weidenschilling 1977). Friction times are thenindependent of v pg .One of the well-known problems in the planet-formationfield is the strong inward radial drift particles of a specificsize experience, e.g., meter-sized particles at ∼ AU radii orcm-sized at ∼
100 AU radii (see Brauer et al. (2007) for arecent review). This inward radial drift is caused by the exis-tence of gas pressure gradients, resulting in a gas velocity thatis somewhat less than Keplerian by a di ff erence of magnitude ηv K (Weidenschilling 1977; Nakagawa et al. 1986) with η thedimensionless pressure parameter, defined as η ≡ − R Ω ρ g ∂ P ∂ R ≈ c /v . (7)However, particles do not experience this pressure term andinstead attempt to move at Keplerian velocities, faster thanthe gas. The ensuing drag force removes angular mo-mentum from the particle resulting in a radial velocity of Compared to, e.g., Weidenschilling (1977) the definition of C has beenscaled down by a factor of 24. That factor is already present in equation (6). F ig . 1.— Comparison of systematic and turbulent velocities as functionof particle Stokes number. Plotted are radial velocities (eq. [8], solid greycurve) and turbulent velocities with ε = ≃ Re − / . Turbulent velocities dominate oversystematic velocities for St . α/η (provided this is & Re − / ). The Stokesnumber corresponding to a 300 µ m-sized chondrule at the default nebularconditions of 3 AU (see Table 2) is also indicated. All velocities peak atSt = (Weidenschilling 1977) v r = − + St ηv K , (8)where we have defined St = τ f Ω . This systematic radial driftvelocity peaks at St =
1. Chondrule-sized particles, however,are generally su ffi ciently well coupled to the gas (St ≪ α is reallylow). However, when particles grow in size, systematic mo-tions may take over from turbulent velocities (see Fig. 1). Turbulent motions
For a Kolmogorov spectrum, turbulence leads to mean(large scale) velocity fluctuations of v g = (3 / / v L = (3 / / α / c g (Cuzzi & Hogan 2003). Due to their inertia,solids do not instantaneously follow these fluctuations but re-quire a time τ f before their motions align. This leads to a netrelative motion, v pg , between the gas and the solid particle of(Cuzzi & Hogan 2003) v pg = v g s St (Re / − + / + , (9)where the Stokes number, St, is the ratio between the frictiontime and the large eddy turn-over time, i.e., St = τ f / t L = τ f Ω . The limiting expressions of equation (9), v pg = St v g for St ≪ v pg = v g for St ≫
1, respectively, correspond to the In equation (8) we have not accounted for collective e ff ects when theparticle density is comparable to or higher than the gas density. Equation (8)then changes (Nakagawa et al. 1986). Angular momentum exchange betweenthe dust and gas dominated layers (Youdin & Chiang 2004) is another processto be accounted for, but its significance is relatively modest (Brauer et al.2007). St = τ f / t L is the formal definition for the Stokes number. In the α -turbulence model t L = Ω − and the definitions for St in equations (8) and (9)coincide. C.W. Ormel, J.N. Cuzzi and A.G.G.M. Tielenscases of particles that are well coupled (small particles) andpoorly coupled (larger particles) to the gas.The calculation of particle-particle relative velocities doesnot follow directly from the v pg ’s since particle velocities canbecome very incoherent in turbulence (e.g., ∆ v , | v − v | ).Consider, for example, two small particles entrained in thesame eddy. If their motions are aligned, no relative ve-locity is present; it is only within a time τ f after beingcaught in the eddy that these particles have the chance todevelop relative motions, provided their friction times dif-fer (i.e., τ , τ ). The problem of finding suitable (i.e.,closed-form) expressions for ∆ v is important since these arekey to any model of dust coagulation (e.g., Weidenschilling1997; Suttner & Yorke 2001; Dullemond & Dominik 2005;Nomura & Nakagawa 2006) including this work. Follow-ing earlier works of V¨olk et al. (1980), Weidenschilling(1984), Markiewicz et al. (1991) and Cuzzi & Hogan (2003),Ormel & Cuzzi (2007) have presented closed-form analyticalexpressions for ∆ v (with a margin of error of ∼ ∆ v v g ! = Re / (St − St ) for τ < t s h y ∗ a − (1 + ε ) + + ε (cid:16) + y ∗ a + ε y ∗ a + ε (cid:17)i St for 5 t s ≃ τ . t L (cid:16) + St + + St (cid:17) for τ ≥ t L (10)In these expressions τ (or St ) always corresponds to the par-ticle of the largest friction time and ε = τ /τ ≤
1. Nearthe τ = t s turning point the expression for ∆ v is somewhatmore complex (see Ormel & Cuzzi 2007). y ∗ a is a numeri-cal constant of value y ∗ a ≃ .
60 if τ ≪ t L ; however, when τ ≃ t L it becomes a function of τ and drops to unity at τ = t L . In that case we approximate y ∗ a by an interpolationfunction. In Fig. 1 ∆ v is plotted for two values of α in thelimit of St = St ≫ St (dashed curves). The three regimesof Equation (10) are clearly distinguishable: the linear regimefor τ f . t s (or St . Re − / ; small particles); the square-rootregime, t s . τ f . t L ; and the high Stokes regime, St ≥ ≪
1; whether they fallinto the linear or square-root velocity regime depends on theirsizes in relation to the gas parameters (e.g., α, ρ g ) of the disk.Figure 1 also shows the systematic drift velocity (eq. [8],solid curve). If one assumes that St ≪ St the radial driftcurve also gives the relative velocity a particle with St = St has with a much smaller particle. Actually, for St ≤ (the lower Stokes number); thecurves in Fig. 1 can therefore be interpreted as the typical rel-ative velocity a particle of a given Stokes number has withparticles of similar or lower Stokes numbers. Figure 1 showsthat for very small particles ( τ f ≪ t s or St ≪ Re − / ) turbu-lent velocities only dominate when α & − . Then, whenturbulent velocities flatten out in the square-root regime, theradial drift motion may catch up with the point of intersectionlying at St ≃ α/η , provided α is not either too low or too high.For any model with α . η radial drift motions will eventuallydominate: a regime of high relative velocities ( ∼
10 m s − ) istherefore unavoidable. Collisions between dust-rimmed chondrules
Theoretical studies and laboratory experiments have shownthat the outcome of grain-grain or aggregate-aggregate col-lisions depends on its ratio of the kinetic energy to a criti-cal energy (Dominik & Tielens 1997; Blum & Wurm 2000).
TABLE 3M aterial properties of compounds phases
Phase Specific density Mass fraction a Filling factorchondrule ρ (s)c = − (1 − f d ) φ ch = ρ (s)d = − f d (1 − f p ) φ cd = . ρ (s)d = − f d f p φ pd . . a with respect to entire compound (sum equals 1); f d = mass fractionin dust; f p = porous mass fraction of the dust. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) A CB chondrule porous dustcompact dust F ig . 2.— ( A ) Schematic representation of the three-phase model of dust-rimmed chondrules: chondrule (dark), compact dust (grey) and porous dust(pattern). The inset ( B ) shows the substructure of the dust that consists ofmicron-sized monomers. ( C ) If chondrules collide, the collision energy isdissipated by transferring dust from the porous to the compact dust phase.Figure not to scale. Specifically, porous dust is accreted when chondrules sweepup dust grains, or aggregates of dust grains, at collisional en-ergies ( E ) that stay below the energy threshold for restruc-turing, 5 E roll , where E roll is the energy required to roll onecontact area over the surface of the grain. This can lead toa very open structure of filling factors ( φ pd ) that are lowerthan the filling factors obtained in particle-cluster aggrega-tion, φ PCA ≃ .
15 (see below, § § E > E roll and the dust within the compound will restructure, dissipat-ing a unit of ∼ E roll for each dust grain that is involved in therolling motion. The porous dust that is involved in restructur-ing compacts to a higher filling factor, φ cd ( § f d and the porous dust fraction f p , quan-tify the relative importance of each phase within a compound(see Table 3). The internal structure of each phase is furthercharacterized by its filling factor, φ . A schematic picture ofthe structure is given in Fig. 2. Collisional compaction
The accreted dust mantles surrounding chondrules can havea very porous and fractal structure. Typically, grains in theserims will be bonded to two other grains in large string-likestructures. When two rimmed chondrules collide, contact willbe established between two (or a limited number of pairs of)grains and these grains will bear the full brunt of the collision.Once the force on these grains exceeds the critical rollingforce, they start to roll (restructuring). The rolling of theseccretion of chondrules in the solar nebula 7 F ig . 3.— (solid curve) Compression of PCA aggregates, obtained bystatic compression of ‘dust cakes’ created by random ballistic deposition of a = . µ m SiO spheres (Blum & Schr¨apler 2004; Blum et al. 2006). Theuncertainty in the measurements is denoted by the grey area. Values for φ forseveral amounts of packing configurations are shown. (Data on the compres-sion curve experiments kindly provided by J¨urgen Blum.) grains may enable contact formation between more pairs ofgrains, thereby promoting compaction and at the same timereducing the force per contact. Compaction will stop whenthe force on newly made contacts drops below the rollingforce. Compaction may also stop because the resulting struc-ture is too rigid to allow for further rolling, i.e., the rollinggrain made contact with too many grains. Since forces arepropagated through such compacted structures, this meansthat none of the grains involved experiences a force exceed-ing the rolling force. The compression of the contact area ina collision between two monomers will give rise to an elas-tic repulsion force slowing down and eventually reversing thecollision. The absolute value of the repulsive force will beset by the kinetic energy of the collision; in the Hertzian limitthe sum of the forces on the individual contacts scales withthe remaining kinetic energy to the 3 / . µ m diameter monomers at low velocities where rollingis not a factor and growth occurs through a hit-and-stick pro-cess. The volume filling factor of the resulting aggregates wasmeasured to be 0 .
15, in good agreement with numerical simu-lations of this process (Ossenkopf 1993; Watson et al. 1997).At this volume filling factor, the typical coordination number,i.e., the number of neighbours with which the monomer isin contact, is calculated to be 2 (Ridgway & Tarbuck 1967).These dust cakes were exposed to a unidirectional pressurein a static experiment. Figure 3 shows the resulting volumefilling factor as a function of the applied unidirectional pres-sure (Blum & Schr¨apler 2004). The results show that com- paction is initiated at an applied pressure of ∼ dyn cm − .If we assume that the number of monomers per unit area beingpressed on is given by, N / A = (cid:16) π a φ / (cid:17) − ∼ × cm − (11)with a the radius of the monomer, the force on an individ-ual monomer becomes ∼ × − dyn. This is very close tothe rolling force of 7 × − dyn Blum & Schr¨apler (2004) ex-trapolated from the measurements by Heim et al. (1999). Asthe dust cake compacts and the average coordination numberincreases, the structure becomes more rigid and resistant tofurther compression (under these conditions, see below). Fi-nally, at a pressure of ∼ dyn cm − the structure is denseenough for rolling motions to be inhibited. This correspondsto an average coordination number of 3.9 and a filling factorof 0.33 (Fig. 3).The conditions in the protoplanetary disk under which dustrims are formed by grain accretion and under which theyevolve through collisions with other rimmed grains di ff erfrom those in these laboratory studies. First, the initial struc-ture of the dust rims accreted on the chondrule surface maydi ff er from PCA. Although low velocity collisions are ex-pected, the monomers may collide preferentially among eachother before colliding with a chondrule. In that case the re-sulting structure is referred to as Cluster-Cluster Aggregation(CCA), a process that leads to much lower filling factors thanPCA. Whether PCA or CCA is preferred depends on the rel-ative abundance of dust and chondrules and their relative ve-locities. So, the filling factor of the porous dust componentmay start lower than the experimental one in Fig. 3. How-ever, since this process is directly tied to the rolling force ex-perienced by the monomers that make contact, we expect thatthis di ff erence in initial structure will have no influence on thecritical pressure required for the onset of compaction. We ex-pect, likewise, that uniaxial compression of dust rims grownby CCA will stall at 0.33 filling factor since this is again aproperty of the resulting structure; e.g., at these kinds of vol-ume filling factors, monomers in the dust rims will have beenorganized in ‘stabilizing’ structures.However, under nebular conditions continuous impacts willarise from random directions; it is therefore likely that colli-sional compaction under these conditions will result in highervolume filling factors than the unidirectional compression ex-periments would indicate, possibly even as high as 0.5 (thevalue of φ = .
64 characterizes for Random Close Packing,RCP). We note that Blum & Schr¨apler (2004) and Blum et al.(2006) in their compression studies did approach RCP whenapplying an omnidirectional pressure of ∼ dyn cm − .Note, however, that omnidirectional pressure is not achievedin collisions between two bodies in an open environment; i.e.,the dust has the chance to spread perpendicular to the di-rection of compression, and the obtained high volume fill-ing factor may not be generally attainable. Indeed, for φ > .
33 rolling motion become impeded and we do expectthat in order to ‘crush’ the dust (rims) to RCP values muchhigher pressures are required. Studies indicate pressures of ∼ dyn cm − in order to reach RCP (Martin et al. 2003;Tanwongwan et al. 2005). This second stage of compactionwould correspond to a very di ff erent collision regime charac-terized by much higher energies. Adopting the Hertzian limit,we expect that this higher pressure, c.q., force in the contactarea corresponds to a impact energy which is higher by a fac-tor (10 / ) / or a collision velocity higher by a factor 10 / over the velocity / energy required to initiate compaction. In C.W. Ormel, J.N. Cuzzi and A.G.G.M. Tielens TABLE 4C ritical energies
Expression Breaking RollingTheoretical a E br = A br γ / a / µ / E / E roll = A roll ξ crit a µ γ DT97 Prefactors a A br ≃ . A roll ≃ b E br = . × − erg E roll = . × − ergEmpirical prefactors c A br = . × A roll = . × N ote . — Comparison between predicted and measured critical energiesfor breakup and rolling. a Theoretically derived expressions for E br , E roll from Chokshi et al. (1993)(for the breakup energy) and Dominik & Tielens (1997) (for rolling) andcorresponding pre-factors, A br , A roll . We define ξ crit = − cm. b Values adopted from Blum & Wurm (2000) for parameters of a = a µ = . × − cm, γ =
19 erg cm − and E = . × dyn cm − . The origi-nal measurements were performed by Poppe et al. (1999) (for the breakupenergy) and Heim et al. (1999) (for the rolling energy). c Empirically derived prefactors from the theoretical expressions with themeasured values for E br and E roll . this study, while acknowledging that higher filling factors areplausible, we have for simplicity assumed that φ = .
33 is thelimiting value.
Acquisition of a porous dust layer
Two critical energies – the breakup and rolling energy – reg-ulate the behaviour of the dust (porous accretion / compaction)upon collision: (Chokshi et al. 1993; Dominik & Tielens1997; Blum & Wurm 2000) E br = A br γ / a / µ E / ; (12) E roll = π ξ crit a µ γ = A roll ξ crit a µ γ, (13)where a µ = a a / ( a + a ) is the reduced radius of the col-lision partners, γ the surface energy density of the materialand E Young’s elastic modulus (assuming the same materialscollide). ξ crit in the E roll expression is some critical distanceused to initiate rolling which Dominik & Tielens (1997) as-sumed to be on the order of the atomic size, ξ crit = − cm.Using these definitions the constants A br and A roll are dimen-sionless. Blum & Wurm (2000) have experimentally deter-mined the breakup and rolling energies (see Table 4) andfound these to be higher than the Dominik & Tielens (1997)theoretical predictions. However, apart from a scale fac-tor, the Blum & Wurm (2000) experiments agreed well withthe Dominik & Tielens (1997) model; that is, collisions canbe separated into the regimes of perfect sticking, restructur-ing and fragmentation. We therefore apply the mechanismput forward by Dominik & Tielens (1997) but use pre-factors( A br , A roll ) from the experimental results (last row of Table 4).Note that for micron-sized particles the rolling and breakupenergy are similar.When two particles meet, direct sticking occurs if the col-lision energy, E , is dissipated at the first point of contact;i.e., E ≤ E stick , where E stick is related to the breakup en-ergy as E stick = . E br (Dominik & Tielens 1997). Writing E = m µ ( ∆ v ) with m µ = m m / ( m + m ) the reduced massand ∆ v the relative velocity, the criterion E ≤ E stick translatesinto a threshold velocity of v st = s E stick m µ = p . A br γ / a / µ E / m / µ = . A / (cid:16) ρ (s)d (cid:17) − / γ / N − / µ a / µ a − / E − / =
35 cm s − N − / µ a µ a ! / a µ m ! − / ρ (s)d − − / × γ
19 erg cm − ! / E . × dyn cm − ! − / , (14)where we have assumed that like materials meet (i.e., same γ, E ) , and where the reduced mass has been parameterized as m µ = N µ m with m = πρ (s)d a / N µ = / N µ = ff erent size particles. Equation (14) shows that micron-sized silicate particles have no problem to stick to each otherat velocities of ∼
10 cm s − . This also holds for collisions be-tween µ m-sized grains and chondrules since it is the reducedsize a µ that enters the equation. However, at higher velocitiesthe grains will bounce o ff .In collisions between chondrules ( a ∼ µ m) the stick-ing velocity falls below ∼ cm s − , lower than the velocities be-tween chondrules for most values of α (see Fig. 1). Also, forchondrules, the assumption of a smooth, spherical surface onwhich the physics behind equation (13) is based breaks down.Although surface roughness increase the sticking capabilitiesfor µ m-sized grains (Poppe et al. 2000), the asperities in chon-drules are probably too large to favour sticking. However, wenow expect the previously accreted porous dust layer to act asthe sticking mechanism through a dynamic restructuring andcompaction of the constituent grains ( § without restructuring is E ≤ E roll (Dominik & Tielens 1997). This translates into a velocity of(using the same substitutions as above) v st , aggr = s E roll m µ = . × cm s − N − / µ a µ a ! / a µ m ! − × ρ (s)d − − / γ
19 erg g − ! − / , (15)in which now N µ > α is high (e.g., α ≥ − and ρ X =
1; Fig. 1) somecompaction is likely to occur. However, in this study we haveignored this e ff ect (for reasons of computational e ffi ciency)and simply assumed that all dust accretion occurs fractally.Although invalid in a violent collisional environment, the con-sequences of this assumption are marginal as the porous duston the chondrule surface is quickly compacted by collidingchondrules in any case ( § Collisions between dust-chondrules compounds
Hatzes et al. (1991) studied collisions between cm-sizedparticles and found that sticking forces increased significantlywhen a frosty layer was present. While they attribute this en-hanced sticking to interlocking of jagged surface structures,this e ff ect probably reflects energy dissipation due to restruc-turing. In the case of chondrules, rimmed by a layer of flu ff ydust, the situation is analogous: the flu ff y structure allows thecollisional energy to be dissipated away. Assuming that eachmonomer (of mass m , size a and internal density ρ (s)d ) inccretion of chondrules in the solar nebula 9 D v , velocity [ cm s − ] − − e , s i z e r a t i o catastrophicdisruption f comp > v i s i b l ec o m p ac ti on negligiblecompaction, f comp ≪ on s e t o ff r a g m e n t a ti on , < f c o m p < − − − F ig . 4.— Contours of f comp (dashed curves) – the fraction of the (com-bined) compound mass that must be involved in restructuring to dissipateaway the collisional energy to stick the compounds (eq. [16]) – as functionof collision velocity ( x -axis) and size ratio ( y -axis). Equal internal densitiesare assumed, ρ = ρ = − and a = µ m The criterion for stickingis f comp ≤ h f d f p i m (see text). For low velocities or size-ratio’s compaction isinsignificant. Collisions with f comp & h f d f p i m compact all their dust. When f comp & § the porous dust layer is capable of absorbing an energy E roll , E / E roll monomers are needed to dissipate the total collisionenergy. Expressed in terms of mass, a porous mass fractionof at least f comp must be available, with f comp the ratio of therequired mass in porous dust to the total mass of the collisionpartners, f comp = m E / E roll m + m = π A roll ξ crit ρ (s)d a γ − m µ m m ( ∆ v ) = . × − m µ m m a µ m ! ∆ v
10 cm s − ! × ρ (s)d − γ
19 erg cm − ! − . (16)This equation reveals a few important results. First, f comp de-creases with smaller dust grains (smaller a ); although moremonomers are required to dissipate the same collision energy,the total mass of the monomers that restructures is less. Also,the dependence on velocity is rather steep; at very low ve-locities the amount of compacted material is negligibly low( f comp ≪ f comp does not dependon the absolute masses of the particles involved, but, throughthe m µ / m m factor, rather on the mass-ratio of the collisionpartners. Thus, a collision between particles of very unequalsize has a lower f comp than equally-sized particles colliding atthe same velocity and, therefore, a higher probability to stick(see Fig. 4). This is of course due to the reduced mass thatenters the collision energy. Collisional recipe
We will now quantify how the collisions a ff ect the struc-tural parameters of the compound, i.e., the f d and f p quan-tities. Equation (16) gives the mass fraction of the colli-sion partners that must be compacted, which, for sticking, aa b b b (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) AO B OC F ig . 5.— Projection of two compounds colliding at an impact parameter b . The ratio of the shaded region ( A ) relative to the cross section of eachcompound ( π a i ) determines the fraction of the porous dust that can be usedduring the collision, i.e., f geo , i = A /π a i . A is obtained by subtracting the kiteO BO C from the two circular sectors spanned up by β and β (eq. [17]). must be less than the mass fraction available in porous dust( f d f p ), averaged over the collision partners, i.e., h f d f p i m = ( m f d , f p , + m f d , f p , ) / ( m + m ). If f comp ≤ h f d f p i m enoughporous dust is present to absorb the collisional energy and thetwo compounds stick. A fraction f comp is then transferredfrom the porous to the compact phase. If f comp > h f p f d i m ,however, restructuring cannot dissipate all the collisional en-ergy. As mentioned before, we do not include other en-ergy dissipation channels but simply consider all collisionsin which f comp > h f d f p i m to result in bouncing; f p is then setto 0 for both particles. This means fragmentation of com-pounds or erosion of the porous rim are neglected (but seebelow, § f geo ) of the compound (and ofits dust) is involved in being compacted and dissipating en-ergy. We estimate f geo from the intersection between the par-ticles’ trajectories. This intersection actually is one betweena cylinder and a sphere, but here we approximate it as a 2dintersection between two circles which meet at an impact pa-rameter b . The area of the intersection, A , can be calculatedby elementary geometry as (see Fig. 5) A ( a , a , b ) = a arccos b + a − a ba + a arccos b + a − a ba − p ( − b + a + a )( b + a − a )( b − a + a )( b + a + a ) , (17)from which f geo , i for each particle is calculated as f geo , i = A /π a i . Equation (17) is valid for impact parameters of | a − a | < b < a + a . For b ≤ | a − a | the intersection equalsthe projected area of the smaller particle, while A = b ≥ a + a . The total mass-fraction of the particles thatparticipates is ( m f d1 f p1 f geo , + m f d2 f p2 f geo , ) / ( m + m ) andthis has to be greater than f comp for sticking. Note, how-ever, that inclusion of the f geo factor might be too restric-tive: since the sound speed inside aggregates ( ∼
30 m s − ;Paszun & Dominik 2008) is usually much higher than ∆ v , theenergy will be quickly transferred along movable structures.For completeness we therefore consider both extremes: f geo determined from equation (17) and f geo = f d = f d m + f d m m + m ; (18) f p = f d f p m + f d f p m − f comp ( m + m ) f d ( m + m ) . (19)For example, in equation (19) the three terms in the numer-ator denote, respectively, the mass in porous dust of particle1, the porous dust mass of particle 2, and the porous masstransferred to the compact dust phase. In order to get the rel-ative amount of porous dust this is divided by the new totaldust mass (calculated in equation (18)) of the compound. Ifthe collision results in a bounce, f d stays the same for bothparticles and f p is reduced by a factor 1 − f geo . Role of fragmentation
As the above formulas for f d and f geo suggest, fragmenta-tion is not included in the collision model. The large numberof particles produced by a fragmenting collision is especiallyproblematic in the context of a Monte Carlo program, wherethe number of particles is limited (see § E ≃ . N c E br withcatastrophic disruption at energies of ≃ N c E br , where N c is the total number of contacts (roughly equal to the numberof monomers, N ) in an aggregate. Recalling from § E br ∼ E roll the catastrophic fragmentation limit corresponds to f comp ∼ cm s − (for equal sized particles;as the mass disparity increases the fragmentation velocity in-creases). The m / s transition for the onset of fragmentationis in agreement with previous studies (Blum & M¨unch 1993;Blum & Wurm 2000) but compact structures at high fillingfactor may require more e ff ort to fragment than their flu ff ycounterparts (D. Paszun, priv. comm.). From Fig. 1 the crit-ical velocity can be translated into a Stokes number. We will a posteriori check in which models fragmentation is expectedto play a dominant role. Evolution of the internal structure
The aerodynamic properties of the compounds, which de-termine their coupling to the gas, alter with accretion of dustand mutual collisions between compounds. These propertiesare quantified in the friction time, τ f (see eq. [5]), of the par-ticles – essentially a measure of its mass-to-surface area ratio, m /π a . It is an important parameter since lower friction timesmean lower relative velocity between the particles ( § § V geo , defined by a , canbe reduced to its three components: i) chondrule, ii) compactdust, and iii) porous dust, i.e.,4 π a ≡ V = V ch + V cd + V pd . (20)In § V i for these phases linearly corresponds to the mass inside thesephases. However, for the porous dust phase, this does nothave to be the case: the dust flu ff -balls that are accreted can be of di ff erent size and porosity. Besides, if the compoundaccretion process itself proceeds fractally, the porous phasebecomes a mixture of porous dust and voids created by thehit-and-stick packing of the compounds. This could lead to amuch reduced filling factor of the porous phase (see § i) dust-dust, ii) chondrule-dust, and iii) compound-compound accretion. Dust-dust aggregation
If the dust distribution initially consists of monomers of size a , the number density of dust particles is likely to be muchlarger than that of chondrules. Therefore, monomers proba-bly form aggregates before they are themselves accreted bychondrules or compounds. Provided the collisional energiesinvolved stay below 5 E roll (see § π a ∝ m δ , or, V ∝ m δ/ with δ = .
95 (Ossenkopf 1993; Paszun & Dominik 2006) as com-pared to V ∝ m / (or δ = /
3) for compact growth. Using therelation V = V N δ/ , where V denotes the volume of a sin-gle monomer and N the number of monomers the aggregatecontains, the filling factor evolves as φ dust = NV V = VV ! / δ − = a dust a ! /δ − = N − δ/ . (21)We consider two mechanisms through which dust aggre-gates can form: i) Brownian motion and ii) di ff erential veloc-ities due to turbulence. For simplicity, equal particle sizes areassumed at all times. The timescales involved are determinedby the particle number density ( n d ), size ( a dust ) and relativevelocities ( ∆ v ) between the particles, i.e., t dd = ( n d σ ∆ v ) − = ρ (s)d ρ d a dust φ ∆ v , (22)in which we have used n d = ρ d / m , σ = π ( a + a ) = π a for the collisional cross section, and m = π a φρ (s)d / ρ (s)d the specific material densityof the dust. The relative velocities in the case of Brownianmotion and turbulence read, respectively, ∆ v BM = s k B T π m µ = π s k B T ρ (s)d a − / φ − / ; (23) ∆ v T = v s t s ρ (s)d c g ρ g a dust φ, (24)If τ f < t s , turbulent relative velocities are in fact determinedby the spread within their friction times (see § v s / t s = Re / α / c g Ω .For Brownian motion, the timescale t dd then becomes t BMdd = π ( ρ ( s )d ) / R gd ρ g √ k B T φ / a / ≈ . × yr ρ − X φ / a dust µ m ! / , (25)ccretion of chondrules in the solar nebula 11and for turbulence, t Tdd ≈ × R gd Re − / α − / Ω − ≈ × yr ρ − / X (cid:18) α − (cid:19) − / , (26)where the expressions are evaluated for the default parametersof the R = R gd = R gc R cd = τ f so that thetimescales for turbulence become independent of size and φ and growth progresses exponentially; for Brownian motionthe growth of aggregates (in mass) is proportional to t (Blum2004).Equations (25) and (26) show that aggregate formation isinitiated by Brownian motion. Turbulence can take over athigh α but the dust is then also quickly swept up by chon-drules. At high gas densities aggregates can grow large. Dust-chondrule / compound accretion The size of the dust aggregates at which they are accreted bychondrules / compounds ( a dust ) depends on the dust-chondruleaccretion time t dc in relation to t dd . The timescale, t dc , for adust particle to encounter a chondrule of size a ch , is t dc = n ch π a ∆ v = ρ ( s )c a ch ρ c ∆ v = R gc / α / Ω ≈
240 yr ρ − / X (cid:18) α − (cid:19) − / , (27)where a monodisperse distribution of chondrules that domi-nates the cross section ( a ch ≫ a dust ) and the velocity field isassumed, and R gc = ∆ v ∼ v s τ ch / t s , although chondrules might also fall inthe square-root part of the velocity regime ( § R cd ∼
1, chondruleswill sweep up the dust before significant aggregation takesplace and a dust ∼ a ; contrarily, if Brownian motion domi-nates (or when τ ch ≫ t s ), the monomers will first collide witheach other before being accreted by chondrules.The question that remains is what this means for the poros-ity of the rim. Assuming a hit-and-stick process, in which theaccreting dust particles are all of equal size and much smallerthan the chondrule / compound, the structure of the rim will re-semble that of particle-cluster agglomeration (PCA). Thus, ifthe dust particles are solid monomers the filling factor of therim indeed equals φ PCA . On the other hand, if the accretingdust particles are flu ff y aggregates, but still smaller than thechondrule, the resulting filling factor of the rim will be lessthan φ PCA . The precise filling factor will be determined by theamount of interlocking between the aggregates but as a crudeupper limit the aggregated may be approximated as a homo-geneous porous sphere such that the packing process of thedust aggregates is PCA. Then, the filling factor of the porousdust, φ pd , is just the product of φ PCA (caused by hit-and-stickpacking) and the porosity the dust aggregates already contain( φ dust ), i.e., φ pd = φ PCA φ dust .In yet another collisional growth scenario we envision thatchondrules are mixed into a dust region after this dust hasaggregated into dust balls but before the dust balls are com-pacted (Ormel et al. 2007). In any case we assume here that the dust consists of porous aggregates and parameter-ize its filling factor by the φ pd = φ PCA φ dust relation. Usingequation (21), a dust is the parameter that regulates the flu ffi -ness of the dust accretion process and we run models at dif-ferent values of a dust to test its importance and sensitivity. Compound-compound accretion
The timescale for chondrule-chondrule accretion, t cc , issimilar to the dust-chondrule timescale, equation (27). Dur-ing the collision, a fraction of the porous dust is lost to thecompact dust phase. Since the porous phase filling factor isalways less than that of the compact phase there is always anet loss in geometrical volume when two compounds collides,i.e., V < V + V . This, we call the ‘conservative approach’(no fractal accretion of compounds). Alternatively, collisionsof compounds (consisting of one or more dust-rimmed chon-drules) may be in the hit-and-stick regime. This would occurif the impact energy is absorbed locally and is not communi-cated to other parts of the compound. In that case the com-pound packing proceeds fractally. Ormel et al. (2007) providean expression for the growth of V in the hit-and-stick case forparticles of di ff erent size, derived by an interpolation from thePCA and CCA limiting cases, i.e., V = V + V V ! δ , (28)where V is the volume of the largest of the two particles thatmeet and δ ≃ .
95. The growth of the porous phase V pd then results from the gain in V through equation (20). Theporous phase is then a mixture of porous dust and voids andthe geometrical volume becomes a balance between hit-and-stick packing of chondrules (increasing V ) and compaction ofporous dust (decreasing V ). This, contrary to the conserva-tive approach in which the total volume always decreases atcollision.For example, if V = V , equation (28) shows a volume0 . V is added to the porous phase, decreasing its fillingfactor. It then depends on f comp how the net growth of theporous phase turns out. In the initial stages of coagulation f comp is often very low and, therefore, fractal accretion ofcompounds can become very important in enhancing the geo-metrical volume of the compounds.Although fractal accretion of compounds increases the vol-ume of the porous phase, no mass is transferred to it. It isonly the filling factor that is a ff ected, in its turn a ff ecting theaerodynamic properties of the compound. Eventually, due tocompaction, all models run out of porous dust and the dust in-side the final objects – whether fractal accretion is involved ornot – has the same filling factor φ cd = .
33. The lower fillingfactor of the porous phase during the collision process merelyreflects the voids between the chondrules that are created inthe models with fractal accretion. As velocities increase, how-ever, the fractal structure must collapse. We suspect, further-more, that structures of very low filling factor are too weak tosurvive the more violent collisions (see, e.g., Paraskov et al.2007).
Collisional scenario
We have proposed a model where chondrules – in the pres-ence of dust particles – acquire rims of fine-grained dust,which help them stick together, and discussed various colli-sional scenarios for this growth process. Here we briefly sum-marize the envisioned scenario from a chronological view-2 C.W. Ormel, J.N. Cuzzi and A.G.G.M. Tielenspoint, and emphasize the (free) parameters the model con-tains.At the start of our simulation ( t =
0) a population of chon-drules encounters a reservoir of dust particles of characteris-tic size a dust (see § ρ X , turbulent strength α ; see § / compounds.Another important parameter is the spatial density of dust ( ρ d )or, rather, the dust to chondrule density ratio, R cd , since thisdetermines the thickness of the rims. These, and other phys-ical conditions at the start of the simulation determine thesubsequent accretion process. First, chondrules start to ac-crete the dust aggregates (see § φ pd . When rimmed chondrules collide,part of this porous structure collapses to φ cd = .
33 fillingfactor through the initiation of rolling motions. This dis-sipates the collisional energy and, if enough porous dust ispresent by the criterion of equation (16), the two chondrulesstick and a compound is created. In this way compounds arecreated and many chondrules can be stuck together until theaccretion process is terminated when both the amount of free-floating dust and the porous dust inside the compounds havebecome insignificant. The end product is an inert populationof compact-dust rimmed chondrules and compounds that onlybounces. Collisional fragmentation is not explicitly includedin the model, but we can a posteriori compare the velocitieswith a critical threshold ( ∼ m s − ) to verify its importance. MONTE CARLO COAGULATION
The physical model of chondrule accretion contains manyfree (i.e., unknown) parameters. In a statistical study of com-pound coagulation, we will sample these free parameters atdiscrete intervals such that a grid of models is created (see § m , f d and f p to determine the mass inside each phase, and φ pd for the filling factor of the porous phase. Therefore, a MonteCarlo code, rather than the multi-variate Smoluchowski equa-tion, is the obvious way to solve the collisional evolution. Inour code we do not keep track of the individual positions ofeach constituent unit (the monomers) within a compound as ine.g., Kempf et al. (1999) but identify each compound by thesefour numbers. In this way we have a good characterization ofthe internal structure of the compound, although the preciseinternal structure cannot be retrieved.The code we use is called event driven ; i.e., the timestep ∆ t defines the time between two consecutive events (Gillespie1975; Spouge 1985; Li ff man 1991). Here, events are colli-sions between two compounds (see below for the dust). Thecollision rate C i j gives the probability of a collision – C i j ∆ t isthe probability of collision in the next (infinitesimal) timestep ∆ t involving compounds i and j , i.e., C i j = σ i j ∆ v i j V ; and C tot = N X i N X j > i C i j , (29)in which σ i j is the collision cross section between particles i and j , ∆ v i j the relative velocity between the two compounds, In general one must distinguish between the collisional cross section(which gives the reaction rate of the two species) and the geometrical crosssection (which is the average projected area of the particle that determines thecoupling to the gas). Here we will simply equate them as in σ ij = π ( a i + a j ) and ignore the small discrepancy (see Krause & Blum 2004). and N the total number of particles in the simulation. Thevolume of the simulation, V , is determined from the spatialdensity in chondrules, the constant ρ c , and the total mass inchondrules, i.e., V = P Ni = m i (1 − f d , i ) /ρ c . From these quanti-ties the timestep is defined by ∆ t = − C − ln r , with r a randomdeviate. The particles that are involved in the collisions arealso determined randomly, weighted by their collision rates C i j . Then, using the recipes outlined in §
2, the outcome ofthe collision – sticking or bouncing – is determined. In eithercase, the parameters of the new or modified compounds arere-computed. (In the case of bouncing the change is reflectedin a smaller size, a , due to the compaction.) Subsequently, thenew collision rates of the particles (i.e., the { C ik } and, if thesecond particle due to bouncing is still present, the { C jk } for k = . . . N and k , i , j ) are re-computed. These updates ofthe collision rates are the most CPU-intensive part of the code.With it one cycle is completed, after which a new stepsize ∆ t is determined and the steps repeat themselves.Equation (29) involves the total relative velocity betweenthe particles. To calculate ∆ v i j we use thermal, turbulent andsystematic velocities (eqs. [4], [8] and [10]), adding them upin quadrature. Strictly speaking, the zero-dimensional natureof the MC-model is inconsistent with dispersal of particles(particles do not have a positions); however, when the driftis modest the change in the physical environment is negli-gible and we can still use the MC-approach. For the radialdrift this assumption applies only when the total drift is smallcompared to the initial location of the particle, i.e., ∆ R ≪ R ,such that the same physical conditions apply throughout thesimulation. We will a posteriori check whether radial drift issignificant.Apart from collisions between compounds, we also keeptrack of dust accretion. This is, however, not implemented ina Monte Carlo fashion: it would have made the code very slowsince the tiny dust particles far outnumber the chondrules. In-stead, we catalogue the cumulative dust mass that is accretedby the compounds over the timesteps; i.e., for compound i weincrease the amount by π a i ∆ v i d ρ d ( t ) ∆ t . Only when this massexceeds a certain fraction (say f upd = − ) of the total massof the compound, this quantity is added as porous dust to thecompound and the f d , f p parameters as well as the collisionrates are updated (eq. [29]). Although this procedure makesthe program still a bit slow at the initial stage of the simula-tion, it is definitely much faster than updating all N ( N − every timestep. The (decreasing) amount offree-floating dust, ρ d ( t ), is computed in this way. We haveexamined the sensitivity of this mechanism on f upd and foundthat f upd = − is accurate, while much more e ffi cient (faster)than, e.g., 10 − .The strong point of the MC code is that it can deal withmany structural parameters and that it is transparent andstraightforward; the weak point, however, is its low numer-ical resolution. Given the complexity of the model and thelarge number of models we intend to run, the number of par-ticles ( N ) we use in the simulations is a few thousands atmost. To prevent the resolution from deteriorating (a colli-sion resulting in sticking decreases N by one) we artificiallystabilize the total particle number by a procedure called dupli-cation . In this process, one particle is randomly chosen andduplicated from the existing population (Smith & Matsoukas1998). Subsequently, V is increased proportionately suchthat the total density in chondrules, ρ c , stays constant. Thisprocedure is called the constant- N algorithm – an algorithmccretion of chondrules in the solar nebula 13far superior in terms of accuracy to the constant V algo-rithm (Smith & Matsoukas 1998), and we have previouslyshown that it is able to calculate large orders of growth, espe-cially when the size distribution is narrow (Ormel et al. 2007).Through the duplication mechanism, furthermore, a distinc-tion can be made between ‘duplicates’ and ‘distinct species,’and it is actually the latter that we keep constant, such that thetotal number of compounds involved can be much larger thana few thousands, also improving the e ffi ciency of the model. RESULTS
In our models we generally recognize three stages in thegrowth process: hit-and-stick dust accretion (increasing theporosity), compound accretion (i.e., growth), and compactionwith accompanied stalling of the growth. The balance be-tween these phases controls the size of the resulting com-pounds, while their relative importance and ‘timing’ are deter-mined by the adopted model parameters. In § § Individual model runs
Figures 6 and 7 show detailed results for two individualruns of the simulation with (default) parameters of gas den-sity ρ X = a dust = µ m, R gc =
100 and R cd = γ =
19 erg cm − at a distance of 3 AU (see Table 2). In thesefigures panels A correspond to a model with α = − , while α = − in panels B. In Fig. 6 several (mass-averaged) quan-tities are shown as function of time, while in Fig. 7 the sizedistributions of compounds are shown at three points duringtheir evolution: i) t = ii) the time at which 50% of the dust is accreted; and iii) the time at which a negligible amount of porous dust re-mains (the final distribution). The negligible criterion is metwhen both the porous dust mass within all compounds as wellas the density of free-floating dust are less than 0.1% of theinitial dust mass.In Fig. 6 we make use of mass-weighted averages. For ex-ample, the mass-weighted average size of the population isdefined as h a i m = P i m i a i P i m i , (30)where the summation is over all particles of the simulation. Itgives the mean size in which most of the mass of the popu-lation resides, and is more appropriate to describe the popu-lation than the average size, h a i . In particular, adding a largenumber of small particles with negligible mass (density) tothe population, decreases h a i but leaves h a i m una ff ected. Inthe following the prefix ‘mw-’ is used as an abbreviation for‘mass-weighted average of the distribution.’Figure 6A shows that dust is accreted on timescales of a few10 yr, which agrees well with previous studies (Cuzzi 2004).This causes the size distribution (Fig. 7) to shift towards largersizes: the accretion of porous dust particles at low filling fac-tor significantly increases the geometrical size of the com-pounds. The dotted curve in Fig. 6A shows the mw-fillingfactor of the accreted dust. At the start of the simulationthis equals φ pd = φ dust φ PCA ; however, collisions are energeticenough to compact the porous dust on a global scale. The de-crease in friction time (solid grey curve), caused by the accre-tion of porous dust, therefore is only modest. (Note that evenaccretion of φ = φ cd =
33% dust on chondrules would cause the friction time to decrease). Consequently, the sticking ratenever increases much beyond ∼ ∼ yr the ac-cretion of porous dust cannot keep pace with the compactionand sticking fails, resulting in a decrease of the compoundsgeometrical size (dashed curve). This results in a ‘retrogrademotion’ of the final size distribution curve in Fig. 7A. In the α = − model (Fig. 6B) the collision velocities are muchlower and, di ff erent from the α = − model, the porous dustdoes not experience compaction for a long time. The stickingrate then increases to almost 100%. However, depletion ofdust triggers the end of the growth phase; growth is quicklyterminated by the mutually enforcing processes of rim com-paction and increasing velocities. From these panels it is clearthat much growth can be achieved when relative velocities arekept low during the dust and chondrule accretion.Although the Monte Carlo code does not keep track of theposition or size distribution of chondrules within compounds,we can still extract useful statistical information from themodel runs. One such statistic is the distribution of the dustover compounds of di ff erent size: is the dust primarily ac-creted by individual chondrules or by large compounds con-taining many chondrules? The results are presented in thehistogram of Fig. 8. The x − axis denotes the number of chon-drules a compound contains ( N ) and is divided into logarith-mic bins of base 2, i.e., the first bin corresponds to single-chondrule compounds, the second to compounds that contain2 or 3 chondrules, the third to 4 − y − axis gives the relative fraction of the dust that first ac-cretes onto a chondrule or compound with size in each bin;that is, Fig. 8 only reflects the dust accretion history and doesnot include the subsequent re-distribution of dust due to co-alescence of compounds (which would shift the dust-rimmedchondrules to a larger compound bin). The relatively highlevel of the first bin (single-chondrule compounds) reflectsdust accretion during the early phase of the simulation whereindividual chondrules provide a high surface area and the den-sity of free-floating dust is highest. In simulations with strongturbulence this fraction becomes very high: the dust is thenaccreted by single chondrules only. But even in the case oflow α single chondrules are responsible for a significant shareof the dust sweep-up, as the α = − results show. Besides,larger compounds also have a larger surface to spread this dustover; rims created by dust accretion are therefore thickest onchondrules. Parameter study
Figure 9 presents the results of the parameter study. Thefree parameters (Table 5) are distributed over a grid such thateach grid point corresponds to a unique model. In total a fewthousand distinct models are run. Each model is run a fewtimes to account for stochastic e ff ects in their results (typi-cally ∼
10% or so). For each free parameter the models areordered by the grid-values of the parameter, corresponding tothe panels in Fig. 9. Two output values are shown: the ra-tio of final mw-radius to initial mw-radius, or growth factor(crosses) and the time at which the dust is depleted and thesimulation terminated, or simulation time (diamonds) (theyshare the same y -axis; for clarity the timescale error-bars areslightly o ff set in the x -direction). The symbols denote the log-arithmic averages of all models at the grid-values and the errorbars indicate the range in which 50% of the models fall. Thisspread can be huge since it is primarily determined by thespread in the other parameters (and therefore nowhere closeto Gaussian). The same holds for the averages: these can be4 C.W. Ormel, J.N. Cuzzi and A.G.G.M. Tielens time [yr] − − − initial h a i m max. comp. A r d ( t ) h a i m [cm] h f i m h St i m GF time [yr] − − − initial h a i m max. comp. B r d ( t ) h a i m [cm] h f i m h St i m GF F ig . 6.— ( A ) A run of the compound accretion model with α = − , ρ X = a dust = µ m, γ =
19 erg cm − , R = R gc =
100 and R cd = ρ dust ( t = = . × − g cm − );the mass-averaged size of the population (dashed-line); the mass-averaged filling factor of the dust within the compounds (dotted curve); and the mass-averagedStokes number of the population (solid grey curve). Shown is also the definition of the growth factor, GF . All quantities share the same y -axis. ( B ) Like ( A ) butwith α = − . − − compound size, a [ cm ] − − − − − m a ss d e n s i t y , m · a · f ( a ) [ g c m − ] A initial50% dustfinal − − compound size, a [ cm ] − − − − − m a ss d e n s i t y , m · a · f ( a ) [ g c m − ] B initial50% dustfinal F ig . 7.— Size distributions of compounds corresponding to the runs in Fig. 6 for the α = − model ( A ) and the α = − model ( B ). Shown are the initialdistribution (crosses), the distribution at the time where 50% of the dust has been accreted (plus-signs) and the final distribution (diamonds). Note that compactionhas the e ff ect of shifting the distribution to the left. arbitrarily scaled up or down by giving more weight to ex-treme models in the parameter study.However, the value of Fig. 9 lies not in its absolute num-bers but in the trends that emerge from the parameter varia-tion. The lines indicate this trend and their slopes are givenin each panel. From these, it is seen that timescales are pri-marily determined by turbulent α (velocities), nebula location R (densities) and the chondrule density (panel E). Parame-ters that favor large growth of compounds are low α (panelA), high gas densities (panel C), low chondrule-to-dust ra-tios (panel F), and high surface energy densities (panel G).Growth is favored in these models due to the moderate rel-ative velocities (panels A and C) or better sticking capabili-ties (panels F and G). Other parameters are sometimes sur- prisingly irrelevant. For example, the dependence on the sizeof the dust flu ff -balls, a dust , defining their porosity (eq. [21]),is only modest (panel B), and also the latter two panels donot show clear trends. Panels H and I directly give the ratiobetween the two data points, instead of the exponent of thepower-law fit. Panel H shows the e ff ects of taking the geom-etry of the collision into account. f geo = X where 0 < X ≤ f geo is determined by the geometry of the collisionas discussed in § f geo = f geo also reduces the maximum amount of dust that is com-pacted. Apparently, these two e ff ects largely compensate. Asimilar insensitiveness is shown in panel I: whether we allowfor fractal accretion of compounds (‘1’) or not (‘0’) does not,ccretion of chondrules in the solar nebula 15 compound size [number of chondrules] . . . . . . . . . r e l a t i v e f r a c t i o n a = − a = − F ig . 8.— Relative fraction of dust accreted by compounds of di ff erent sizefor the α = − and α = − simulations. The compounds are placed inbins according to the number of chondrules they contain. The bins are expo-nentially distributed by factors of two. The histogram shows the distributionof the dust over the compound sizes (in terms of number of chondrules insidethe compound) at the time of the dust accretion. Single chondrules (first bin)accrete a significant fraction of the dust.TABLE 5 free model parameters (1) (2) (3) (4) (5)turbulent strength α − —10 − ]size of dust aggregates a dust cm 3 [10 − —10 − ]gas density a ρ X g cm − R AU 3 [1, 3, 10]gas-chondrule ratio R gc R cd b γ ergs cm − c f geo d ote . — List of free model parameters. Columns denote: (1) pa-rameter description; (2) symbol; (3) unit; (4) number of grid points perparameter; (5) parameter range, with a grid point at every factor of 10,unless otherwise indicated. See also Table 2 for other (fixed) parame-ters. a ρ X : gas density over MSN model at 3 AU. b γ : energy surface density. The high γ model corresponds to ice as thesticking agent (10 AU models only). c f geo = ′ X ′ : f geo is computed after the procedure outlined in § f geo =
1: use f geo = d Whether a hit-and-stick packing model for compounds (leading to afractal structure, § on average, make a di ff erence. Note, however, that the valuesshown in the panels are averages; in some individual simula-tions we do see a notable increase when fractal accretion isturned on.Panel A shows that the positive correlation between growthfactor and decreasing α breaks down for values below α < − : the growth flattens out and reaches a constant level. Thereason is that for low values of α , and, subsequently, largecompounds, radial drift motions quickly take over from tur-bulent motions such that the evolution becomes insensitive to α . The high radial drift obtained when particles approach theSt = α com-bined with high gas densities delay this transition since com-pounds are now much better coupled to the gas (lower Stokes numbers) meaning much growth early on. Yet, Stokes num-bers inevitably grow to values near unity, and in most casesthe resulting ηv K drift velocities (eq. [8]) stall growth below 1meter.Panel D shows that growth depends only modestly on neb-ula radius, R . Here, the positive correlation with growth factoris a bias resulting from the high γ ‘ice models’ (panel G) – icepromotes sticking – that are only present at R =
10 AU. Thus,despite the fact that almost all nebula parameters scale with R , their combined e ff ect does not result in a clear trend thatfavors growth. For example, larger nebula radii mean lowerdensities and higher Stokes number, increasing the velocity inthe initial stages, but this is o ff set by a (slightly) lower soundspeed, and the better sticking agents that are available.In Fig. 10 all models are combined in a scatter plot of to-tal (mass-weighted) radial drift against the final mw-size ob-tained in the simulation. The few models that cluster aroundthe meter size are all ice models ( γ =
370 and R =
10 AU,indicated by triangles). Some of them do manage to crossthe h St i m = ρ X = ∆ R ∼ R the local assumption is justified; for modelsthat drift over several AU-distances, however, the approxima-tion we used in the calculation of the collisional evolution,i.e., that the physical conditions stay the same, breaks down.Note, however, that the drift distances in Fig. 10 are upperlimits: radial drift slows down with decreasing R due to abetter coupling to the gas, or may diminish when collectivee ff ects become important (see § Importance of fragmentation
The models neglect the possibility that high velocity colli-sions will fragment, rather than merely compact or bounce,compounds. In § ∼ m s − velocities, fragmentation becomes likely, starting with ero-sion, followed by catastrophic disruption of the compound.This threshold can now be compared to the maximum veloc-ities attained at a particular Stokes number (see Fig. 1), i.e., ∼ η / c g St and ∼ α / c g St / for the systematic drift and tur-bulent velocities, respectively. Inserting the final mw-Stokesnumber into these expressions, we obtain a criterion whetherfragmentation is of importance. As the critical velocities wetake 2 m s − for turbulence and 6 m s − for radial drift (seeFig. 4). It can be shown that for systematic velocities scalingproportional to size the collisional energy peaks at size ratiosof ǫ ∼ .
5, which, according to Fig. 4, corresponds to a frag-mentation velocity of ∼ − , or a ∼ − radial driftvelocity for the largest particle. For ice models (triangles)the fragmentation threshold is increased by another factor offour, reflecting their higher γ . In Fig. 10 the models that havecrossed the threshold velocity are indicated by an open square,whereas black squares indicate velocities that stay below thethreshold. The grey squares are models in which collectivee ff ects could have had a significant reduction in relative ve-locities and drift rates, due to concentration of compoundsnear the midplane (see below, § not taken into account in the simulationand it remains unclear whether fragmentation is an importantphenomenon in models for which settling is important.For most models in Fig. 10 fragmentation is not a seri-ous concern. This is a natural result as compaction precedesfragmentation and growth stalls before reaching the fragmen-tation threshold. However, for large, flu ff y compounds the6 C.W. Ormel, J.N. Cuzzi and A.G.G.M. Tielens Accretion mode, FA g r o w t h f a c t o r / t i m e [ y r ] m . m . I growth factortime chondrule dust ratio , R cd = r c / r d g r o w t h f a c t o r / t i m e [ y r ] − . . F growth factortime gas density enhancement , r X g r o w t h f a c t o r / t i m e [ y r ] . − . C . − . C . − . C . − . C growth factortime − − − fluff-ball size , a dust g r o w t h f a c t o r / t i m e [ y r ] . . B . . B . . B . . B growth factortime X 1 collision efficiency factor , f geo g r o w t h f a c t o r / t i m e [ y r ] m . m . H growth factortime gas chondrule ratio , R gc = r g / r c g r o w t h f a c t o r / t i m e [ y r ] − . . E growth factortime − − − − − − − turbulence strength a g r o w t h f a c t o r / t i m e [ y r ] − . − . A − . − . A growth factortime dust composition, g g r o w t h f a c t o r / t i m e [ y r ] . . G growth factortime heliocentric radius , R [ AU ] g r o w t h f a c t o r / t i m e [ y r ] . . D growth factortime F ig . 9.— Results of the parameter study. Each panel sorts the data according to a free parameter ( x -axis), from which the logarithmic mean and variance arecalculated. Two output values are shown on the same y -axis: growth factor (crosses) and simulation time (diamonds). The lines show the trend in variationof the parameter and the ‘best-fit’ power-law exponent is given. ( G ) Data from R =
10 AU models only, comparing silicate dust ( γ =
19 erg cm − ) with ice( γ =
370 erg cm − ). ( H ) f geo = X indicates f geo is a free parameter calculated after equation (17), while f geo = I ) 0 and 1 denote,respectively, that fractal accretion of compounds is turned o ff or on. In these latter two panels the numerical factor next to the m gives the ratio in growth factorand timescale between the two modes (not the power-law exponent). ccretion of chondrules in the solar nebula 17 − − − radial drift, D R / R − fi n a l m w - s i z e [ c m ] , h a i m F ig . 10.— Scatter plot of the fractional inward radial drift covered during the aggregation process ( ∆ R / R , x -axis) against the final mass-averaged size of thecompounds ( h a i m , y -axis). The results of all model runs are shown. Models are separated into the low-velocity regime ( ∆ v . cm s − , black squares) and thehigh velocity regime ( ∆ v & cm s − , open squares). Triangles denote ‘ice models’ ( γ =
370 at R =
10 AU). In the models shown by grey squares (or triangles)the collective e ff ect could have prevented high drift velocities but this is not incorporated in the present models (see text). The vertical dashed line corresponds toa drift of 1 AU. The dashed horizontal lines indicate compound sizes of 1 cm and 1 m, respectively. compaction is more pronounced, resulting in a significant de-crease in surface area-to-mass ratio, increasing the Stokes,and thereby possibly breaching the threshold for fragmenta-tion. Also note the fragmentation models (open symbols) atthe bottom of Fig. 10: in these the fragmentation thresholdwas already exceeded at the start of the simulation.Figure 11 shows a small subset of models from Fig. 10 thattakes away the redundancy (caused by less influential param-eters) and focuses on the more plausible scenarios. Morespecifically, Fig. 11 shows models limited to the followingparameters: R gc = R cd = . a dust = − cm, ρ X = R = γ =
19 erg cm − ; furthermore, we assumeonly local compaction ( f geo = X) and assume collisions be-tween compounds are not in the hit-and-stick regime ( § α (all 6 distinct values) and ρ X (2 values). Table6 shows various output values corresponding to the ‘top ten’models of Fig. 11, ordered after final mw-size; for example, the maximum velocities due to systematic and turbulent mo-tions. This shows that for these low- α models systematic driftvelocities (Col. (7)) quickly become dominant over turbulentmotions (Col. (8)). DISCUSSION
Collective e ff ects in a settled layer Despite the ability of the chondrule-sticking model to tweakmany parameters to optimize the growth, compounds neverachieve planetesimal sizes. Ultimately, m / s or higher veloci-ties are unavoidable in any model due to the radial drift; thatis, compounds inevitably reach (and have to cross) the St = ff y dust – is simply too weak to grow past theSt = − − − radial drift, D R / R − fi n a l m w - s i z e [ c m ] , h a i m r X high X d ecre a s i n g a r low F ig . 11.— A selection of 12 models from the scatter plot of Fig. 10,indicating systematic trends. Compared to Fig. 10 results are limited to: a dust = − , ρ X = R gd = , R cd = , R = f geo = X(local compaction), and no fractal accretion of compounds. See Table 6 forquantitative results. TABLE 6D etailed results α ρ X h a i m h St i m ρ p /ρ mid g ∆ v turb ∆ v sys ∆ v sys , CE [cm] [cm s − ] [cm s − ] [cm s − ](1) (2) (3) (4) (5) (6) (7) (8)10 −
10 8 . . × − . . . . −
10 8 . . × − . . . . −
10 4 . . × − . . . . − . . × − . . . . − . . × − . . . . −
10 0 . . × − . . . . − . . × − . . . . −
10 0 . . × − . . . . − . . × − . . . . − . . × − . . . . − . . × − . . . . −
10 0 . . × − . . . . ote . — Detailed results from 12 selected models (see text), ordered afterfinal mw-size, Col. (3). The columns denote: (1) turbulent- α ; (2) gas-densityenhancement (restricted to 1 or 10); (3) final mw-size; (4) final mw-Stokesnumber; (5) final midplane dust-gas density ratio would settling have been in-cluded; (6) turbulent velocity contribution, α / St / c g ; (7) systematic drift afterequation (8); (8) systematic drift due to collective e ff ects after equation (31) withCol. (4) for St. turbulence is weak enough, in addition to moving radially,compounds can also settle into a dense layer at the midplaneas their Stokes numbers increase. When the density of solidsat the midplane exceeds the gas density, the gas is draggedwith the particles (instead of the other way around), result-ing in gas velocities that tend to become closer to Keplerian,which subsequently diminishes the radial drift and relative ve-locities of particles. Nakagawa et al. (1986) have solved theequations of motion in such a two-fluid medium analyticallyfor a single particle size (or Stokes number); the radial driftvelocity now becomes (instead of eq. [8]) v r = + (1 + ρ p ( z ) /ρ g ) ηv K , (31)where ρ p ( z ) is the total density of particles at a height z above the midplane. For a generalized solution over a particlesize distribution see Weidenschilling (1997) or Tanaka et al.(2005). Thus, in a dust-dominated layer the radial drift of in-dividual particles depends through ρ p ( z ) on the density of allother particles: a collective e ff ect. The particle concentrationcan be found by balancing the gravitationally induced settlingrate with the di ff usion rate of a particle, assuming a steady-state distribution. The scaleheight of the resulting particledistribution, h p , can be calculated as (Dubrulle et al. 1995) h p = H g √ + S , (32)where S = St /α . Under conditions of initial cosmic abun-dances, in order to reach ρ p ∼ ρ g the particles must settleinto a layer of thickness one-hundredth of the gas scaleheight,requiring S > (Cuzzi et al. 2005). This may occur forchondrules in very low- α environments, or, at moderate α ,only for large compounds during their growth and settlingstage. In Table 6 we have calculated the density enhancement( ρ p /ρ g , Col. (5)) and the corresponding velocities ( ∆ v sys , CE ,Col. (8)) for a few selected models at the end of their simu-lation, where we fixed most parameters at their default 3 AUvalues, except for α and ρ X . Note that in the context of ourcurrent model setup collective e ff ects are purely hypothetical(we treat ρ g /ρ c = R gc as a constant); the columns of Table 6therefore merely provide an indication of what could be ex-pected had settling-e ff ects been included. In these calcula-tions we have used the mass-averaged Stokes number of thepopulation (Col. (4) of Table 6) as the Stokes number that en-ters equations (31) and (32). The last two columns of Table 6show that collective e ff ects ( ρ p /ρ g >
1) quickly reduce the ra-dial drift. In a future study, we intend to investigate the e ff ectsof the particle concentration on the compounds’ growth.There is yet another subtlety involved when collective ef-fects (i.e., a dust-dominated midplane) become important.This is the Kelvin-Helmholtz instability (Weidenschilling1980), caused by the shear between the two fluids now mov-ing at a relative velocity of ∆ V , the azimuthal velocity di ff er-ence between the gas in the particle-dominated and the gas-dominated layer. For shear turbulence the turbulent viscos-ity is ν T ∼ ( ∆ V ) / Ω Re ∗ (Cuzzi et al. 1993), where Re ∗ is acritical Reynolds number at which the flow starts to becometurbulent, which Dobrovolskis et al. (1999) found to be Re ∗ ∼ −
30. Also, the large eddy turnover frequency in shearturbulence ( Ω e ) can become much larger than Ω , dependingon the thickness of the shear layer (see Weidenschilling 2006for how Ω e depends on the particle density structure, ρ p ( z )).Equating ν T ∼ ( ∆ V ) / Ω Re ∗ with ( v shearL ) / Ω e then providesthe expression for the shear turbulent (large eddy) velocity, v shearL , v shearL ∼ Ω e Ω ! / ∆ V Re ∗ ≈ . Ω e Ω ! / ∆ V . . η / c g , (33)where the upper limit assumes ( Ω e / Ω ) ∼ Re ∗ =
30 and ∆ V = ηv k = η / c g . This corresponds to the situation wherethe shear layer is thin (meter-size or larger particles; theshear layer cannot become thinner than the Eckman layer,see Cuzzi et al. 1993). In that case, setting v shearL = α / c g the equivalent α value for shear turbulence becomes α shear ∼ × − . This is an upper limit; for smaller particles, or asize-distribution of particles, both Ω e and ∆ V are lower and α shear decreases as well. Shear turbulence may therefore bemuch more conducive to compound growth.ccretion of chondrules in the solar nebula 19Future studies must show whether these e ff ects enablegrowth to planetesimal sizes. Recently, Johansen et al. (2007)have suggested that concentration of meter-size particles (St ∼
1) in certain azimuthally-oriented near-midplane high pres-sure zones, which form between large turbulent eddies, mightlead to gravitationally bound clumps with the mass of plan-etesimal size objects. The results from Johansen et al. (2007)were most pronounced when the turbulent intensities weremoderately high (this leads to the largest radial pressure con-trast), suggesting values of α ∼ − . However, our resultssuggest that it is di ffi cult to grow a population of meter-sizeboulders in the first place under such conditions. The maxi-mum growth (in terms of Stokes number) our models achievefor α = − is St ∼ × − at 3 AU (essentially no growthat all: just dust-rimmed chondrules). 10 AU ice models dosomewhat better: St ∼ . × − . Even if they can form, apopulation of meter-sized boulders may be di ffi cult to main-tain if these originated from dust-coated, solid chondrulesas modeled in this paper. In § ∼ cm s − , 30times smaller than the expected value of St = α = − . This translates into a specific kinetic energy fordisruption of Q ∗ = erg g − , much lower than the criti-cal Q ∗ Johansen et al. (2007) adopt (for aggregates of solidbasalt objects, as taken from Benz 2000). Thus, our resultsindicate that it may be di ffi cult for the instability described byJohansen et al. (2007) to become viable in the turbulent inner(ice-free) nebula.In the outer solar system, however, conditions may be morefavorable to growth in a turbulent environment. First, ifice acts as the sticking agent Q ∗ may be over an order ofmagnitude larger, reflecting the scaling with the surface en-ergy density parameter, γ . Second, if chondrule formation isnot common the particles grow directly from aggregates oftiny grains to larger aggregates and therefore contain roughlytwice as much mass in small grains as dust-rimmed solidchondrules. Moreover, Stokes numbers for the same parti-cles increase with larger heliocentric radii ( R ) due to the lowergas densities. Therefore, at large R the St ∼ ffi cult to disrupt (i.e., higher Q ∗ ) than m-sizebodies (Housen & Holsapple 1990; Benz 2000). (In our sim-ple estimate of Q ∗ we do not have a size dependence, though.)Still, it is hard to see that &
10 m s − velocity collisions be-tween equally-sized particles, even under these most favor-able conditions, will not result in disruption; but this shouldof course really be tested by experiments.On the other hand, it also seems sensible to pursue incre-mental growth scenarios which take place in quiescent (orlow- α ) nebulae. Due to the relatively low e ff ective α -valuesfor shear turbulence derived above, it is the radial drift mo-tions that will provide the limits to growth. However, even amodest reduction of radial drift motion by a few factors dueto collective e ff ects may already be su ffi cient to prevent catas-trophic collisions as particles reach St = ff ects will vary with height,however, it is di ffi cult to predict how these e ff ects will unfold, f =33% compact dust F ig . 12.— A sketch of a cross-cut through a compound at the final state ofour model. The compound contains two phases, present in equal proportionby mass: chondrules (black) and φ = .
33 compact dust (grey). The cross-cut introduces a selection e ff ect and shifts the chondrule size distribution tobigger chondrules. The chondrules are placed at random but a certain distancebetween the chondrules is preserved (see text). and which parameters are key. Clearly, additional modelingis needed, where we may even combine these two di ff erentmodes of turbulence since it is quite natural to expect that dif-ferent physical processes operate at di ff erent heights (Ciesla2007). However, incremental growth in the dense, particle-dominated midplanes of nonturbulent models then proceedsextremely rapidly (Cuzzi et al. 1993; Weidenschilling 2000)which is contrary to the evidence from meteorites and aster-oids (see Cuzzi et al. 2005 or Cuzzi & Weidenschilling 2006for a discussion). Dust rim and matrix
Figure 12 provides an illustration of the internal structureof the objects obtained at the end of our simulation. In Fig. 12it is assumed that each chondrule (black circles) is surroundedby a dust rim at least ∼
40% of the chondrule’s mass, corre-sponding to the amount of dust accreted by individual chon-drules (see Fig. 8). This translates into an outer rim radiusthat is a factor of 1 . i) a periodof nebula dust sweep-up and compaction; or ii) shock wavesin the parent body.The first scenario concerns a moderately intense turbulentenvironment (i.e., high α ) in which chondrules are largely un-able to stick, so that most of the dust is accreted by individ-ual chondrules. These bouncing chondrules quickly compacteach other’s rims, while grazing collisions may also result indust being partially stripped away or eroded from the rims.Presumably, a steady-state between rim accretion and erosionis established, where some of the dust is firmly attached toeach chondrule and compacted, while another, more flu ff y,component is continuously eroded o ff and reaccreted to thechondrule surfaces. Any of this latter, loosely bound phasewhich remains attached to chondrules at the point they are ac-creted to their parent planetesimal would be easily strippedaway in the abrasive environment of the accreting planetesi-mal to become ‘matrix.’Alternatively, a much more gentle collisional environ-ment may be considered in which big compounds form veryquickly, and then continue to grow to planetesimal sizes. Thedust is then primarily accreted by compounds, though, as wehave argued in § ff on the parent plan-etesimal. In this scenario, the fine-grained dust rims mightresult from later processes on the parent body. Specifically, ithas been suggested that shock waves through these planetes-imals (caused, for example, by violent collisions with otherplanetesimals) will compact the dust (Trigo-Rodriguez et al.2006). In the Trigo-Rodriguez et al. (2006) model the highestcompaction of the dust takes place near the solid chondrulesurface. Thus, it is only during the planetesimal stage thatrims become distinguishable from the matrix.There are a number of constraints the rim formation mecha-nisms must satisfy. For instance, collisions must be energeticenough to compact the rims significantly to explain the highfilling factors observed in chondrites. In the nebula forma-tion scenario, therefore, more energetic collisions are requiredthan provided by the model we present here. From the argu-ments given at the end of § / higher than thesticking velocity (eq. [14]), or v ∼ . − . Clearly, othercompaction processes are needed than can be provided by col-lisions in turbulence, even between compounds approachingSt = − − − m ch − − − − − m r i m A accr. time [yr] − − m r i m / m c h B a = − a = − F ig . 13.— Model predictions for the thickness of the chondrule dust rim.The amount of dust accreted before the chondrule’s incorporation into a com-pound determines the dust rim mass ( m rim ). ( A ) Scatter plot of chondrulemass m ch against m rim for 200 chondrules, chosen randomly at the conclu-sion of the run. Two models are shown: α = − and α = − with thelatter population being shifted by a factor of 10 for clarity. The least-squarespower-law fits are given by the dashed lines, which have exponents of 0.93and 0.91, respectively. ( B ) Dust-chondrule mass ratio ( y -axis) at the time ofits accretion into a compound ( x -axis). quantitatively by, e.g., sophisticated numerical simulation.Yet a third observational constraint is the linear correlationbetween chondrule radius and rim thickness (Metzler et al.1992; Paque & Cuzzi 1997). Figure 13 illustrates this point inthe context of our accretion model. In Fig. 13A the chondrule-mass ( m ch , x -axis) is plotted against the rim mass ( m rim , y -axis) for the two models discussed in detail in § α = − model and circles for the α = − model.(In Fig. 13A the α = − points are arbitrarily o ff set verti-cally by a factor of 10 for reasons of clarity.) m rim is definedas dust that is accreted by individual chondrules, before theybecome incorporated into a compound. The m ch - m rim relationis shown for 200 chondrules, randomly selected from the ini-tial distribution. The dashed lines show the best fit havingslopes of 0 .
93 and 0 .
91, respectively. The near-linear trend of m rim with chondrule mass is obvious but the spread is large,as seen in actual chondrites (Metzler et al. 1992). Figure 13Bshows the accretion history of these compounds: the massratio, m rim / m ch , is plotted ( y -axis) against the time at whichthe chondrule is swept up by a compound. The ‘ α = − chondrules’ lie to the left of the ‘ α = − chondrules,’ re-flecting their shorter collision times. The initially linear trendbreaks down at later times as the density of free-floating dustdecreases. Although many processes contribute to the spreadin the data points of Fig. 6A – for example, di ff erences in ve-locity field (linear / square-root regime) during the simulationand the bouncing history of chondrules – the stochasticity inthe chondrule-compound accretion time is the main contribu-tor. Note also the pile-up of particles in the α = − modelnear t ∼ yr, the final time of the simulation: these are thechondrules that remained single during the entire simulation.The relation of rim thickness with chondrule size can beccretion of chondrules in the solar nebula 21naturally understood as the outcome of a nebula accretion pro-cess (Morfill et al. 1998; Cuzzi 2004; Fig. 13). The observedlinear relationship in chondrites therefore suggests this rela-tionship should somehow have survived further processing.As dust rim accretion in the violent collisional environmentdi ff ers from the non-fragmentation environment in which oursimulations are performed, it still remains to be shown that thelinear relationship is maintained after fragmentation / erosionsets in. Alternatively, if the imprints of nebula dust-accretionare destroyed during parent body accretion, a di ff erent mech-anism must explain the observed relationship.Future work – e.g., experimental work on rim-chondrulesize ratios and more advanced theoretical models – must de-termine which of the two scenarios described above is morelikely. Dust fragmentation and additional compaction mech-anisms may be included into the present model. Increas-ingly energetic collisions (when compounds grow towards theSt = ff of individual chondrules, and indoing so may compact the surviving fine grained rims fur-ther than the φ cd = .
33 limit we have adopted in this study.Also, size distributions in the fine-grained component mightalso allow a greater degree of packing than in our models andBlum & Schr¨apler (2004) expect, in which the grains are allmonodisperse. SUMMARY
We have investigated a chondrule-dust aggregation mecha-nism in which the fine-grained dust acts as the glue that allowschondrules to stick. We argue that the energy in collisions issu ffi cient to compress directly accreted material, which ini-tially has a porous ‘fairy-castle’ structure, into a more com-pact state having a porosity that is roughly 67% (based oncompaction measurements by Blum & Schr¨apler 2004 andtheoretical arguments). We have applied this model to a vari-ety of questions regarding the meteoritic record: the relationof individual chondrules to their fine-grained dust rims, theinternal structure of the chondrites, and the ability of growthby sticking to surpass the meter-size barrier. This study onlystarts to address these questions; more sophisticated modelsare needed to answer detailed questions on the structure ofthe meteorites.We find that porous accretion rims do indeed cushion col-lisions and facilitates growth to compound objects contain-ing many rimmed chondrules, but this growth is limited to30 −
100 cm radius objects under the most favorable condi-tions. This is because the chondrule component sweeps upall the local dust in a short time (10 − yr, dependingon nebular location) and these compounds experience higherrelative velocities during their growth stage. Subsequent col-lisions merely pack the existing rims down further, so that thesystem ultimately reaches a dead-end steady state where col-lisions only result in bouncing, or possibly disruption. Otherconclusions from this study are: • Compound growth works best in a quiescent environ-ment (high gas density, low α values). In a more violentcollisional environment ( α ∼ − .. − ) it is di ffi cult toaccrete dust fractally on chondrules surfaces and the en-ergetic collisions between compounds quickly compactthe remainder such that collisional growth is quicklyterminated. • The importance of the other parameters on the accre-tion process is mostly minor. The radial location doesnot a ff ect the final growth of the compounds, althoughtimescales are longer at larger R . Ice, rather than sil-icate, as the sticking agent will lead to bigger com-pounds (but we note icy grains do not dominate the me-teoritic record). • In no single model do compounds grow to planetes-imal sizes. Either turbulent or systematic velocitiesare too high for the porous dissipation mechanism, theSt = ff ects in low turbu-lence nebulae. • We anticipate that the dust accreted by individual chon-drules – before chondrules coagulate into compounds –will finally end up as the chondrule rim. In strong tur-bulent models this fraction is very high, but it remainssignificant (tens of percents) even in the collisionallygentle models. • However, at the current state of the art of this model,fine-grained accretion rims have a porosity significantlylarger than seen in actual rims. Other compaction pro-cesses are not hard to envision, such as higher velocitycollisions by larger mass compound objects, or nebulashock waves (peripheral to those energetic enough tomelt chondrules). These remain to be modeled. • When we define the rim as the dust swept up by indi-vidual chondrules, we find very good agreement withthe nearly linear (average) correlation between rimthickness and underlying chondrule radius seen in CMand CV chondrites (Metzler et al. 1992; Paque & Cuzzi1997).Future work will focus on two aspects of our coagulationmodel: • An improvement of the collisional physics, i.e., includ-ing fragmentation as a collisional outcome for veloci-ties above ∼ m s − ; and a refinement of the characteri-zation of the compound structure, e.g., to allow dust tocompact to higher filling factors. • Inclusion of a proper description of the vertical struc-ture of the nebula, i.e., taking account of phenomenasuch as settling, collective e ff ects, and shear turbu-lence. 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