Fragment-collision model for compound chondrule formation: Estimation of collision probability
aa r X i v : . [ a s t r o - ph ] N ov Accepted for publication in Icarus, November 1, 2007
Fragment-Collision Model for Compound ChondruleFormation: Estimation of Collision Probability
Hitoshi Miura , , , Seiji Yasuda , , , and Taishi Nakamoto Theoretical Astrophysics Group, Department of Physics, Kyoto University,Kitashirakawa, Sakyo, Kyoto 606-8502, Japan Research Fellow of the Japan Society for the Promotion of Science Corresponding Author E-mail address: [email protected] Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro, Tokyo152-8551, Japan Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tenno-dai, Tsukuba305-8577, Japan
Pages: 34Tables: 0Figures: 8 1 roposed Running Head:
New Model for Compound Chondrule Forma-tion
Editorial correspondence to:
Dr. Hitoshi MiuraTheoretical Astrophysics Group, Department of Physics, Kyoto UniversityKitashirakawa, Sakyo, Kyoto 606-8502, JapanPhone: +81-75-753-3885Fax: +81-75-753-3886E-mail: [email protected] bstract
We propose a new scenario for compound chondrule formationnamed as “fragment-collision model,” in the framework of the shock-wave heating model. A molten cm-sized dust particle (parent) is dis-rupted in the high-velocity gas flow. The extracted fragments (ejec-tors) are scattered behind the parent and the mutual collisions be-tween them will occur. We modeled the disruption event by analyticconsiderations in order to estimate the probability of the mutual col-lisions assuming that all ejectors have the same radius. In the typicalcase, the molten thin ( ∼ µ m) andthen they are blown away by the gas flow in a short period of time( ∼ .
01 s). The stripped layer is leaving from the parent with the ve-locity of ∼ − depending on the viscosity, and we assumed thatthe extracted ejectors have a random velocity ∆ v of the same order ofmagnitude. Using above values, we can estimate the number densityof ejectors behind the parent as n e ∼
800 cm − . These ejectors occupy ∼
9% of the space behind the parent in volume. Considering that thecollision rate (number of collisions per unit time experienced by anejector) is given by R coll = σ coll n e ∆ v , where σ coll is the cross-sectionof collision (e.g., Gooding & Keil 1981, Meteoritics 16, 17), we obtain R coll ∼
36 collisions / s by substituting above values. Since most colli-sions occur within the short duration ( ∼ .
01 s) before the ejectors areblown away, we obtain the collision probability of P coll ∼ .
36, whichis the probability of collisions experienced by an ejector in one dis-ruption event. The estimated collision probability is about one orderof magnitude larger than the observed fraction of compound chon-drules. In addition, the model predictions are qualitatively consistentwith other observational data (oxygen isotopic composition, texturaltypes, and size ratios of constituents). Based on these results, we con-cluded that this new model can be one of the strongest candidates forthe compound chondrule formation.It should be noted that all collisions do not necessarily lead to thecompound chondrule formation. The formation efficiency and the fu-ture works which should be investigated in the forthcoming paper arealso discussed.
Keywords: meteorites, Solar System origin, Solar Nebula Introduction
Chondrules are millimeter-sized, once-molten, spherical-shaped grainsmainly composed of silicate material. They are abundant in chon-dritic meteorites, which are the majority of meteorites falling onto theEarth. They are considered to have formed from chondrule precursordust particles about 4 . × yr ago in the solar nebula (Amelin et al.2002); they were heated and melted through flash heating events inthe solar nebula and cooled again to solidify in a short period of time(e.g., Jones et al. 2000, and references therein). Typical chondrulesare single spherical objects, while compound chondrules are composedof two or more chondrules fused together. They are rare in all chon-drules ( ∼ f the shell, and formed the secondary chondrule on the surface of theprimary (Sanders & Hill 1994). (4) Relict grain model; fine-graineddust particles accreted on the surface of already-formed primary weremelted in second heating event (Wasson 1993, Wasson et al. 1995).The relict grain model is good for enveloping type if considering thatfine-grains accreted uniformly on the primary surface, but it seems tobe difficult to explain other types. In the eruption model, the sec-ondary was formed from the inner residual melt after the formation ofthe solidified shell on the primary surface, however, the chemical com-positions of some sets of compounds are inconsistent with this scenario(Akaki & Nakamura 2005). The random collision model is considerednot to account for the observed fraction of compound chondrules be-cause of the low density of matter in the nebula (Gooding & Keil 1981,Wasson et al. 1995, Sekiya and Nakamura 1996). Collision betweenindividual chondrules in shock-wave heating can account for the ob-served fraction of compound chondrules if the enhancement of the dustparticle in the pre-shock region is ∼
400 times that expected undercanonical conditions (Ciesla 2006). However, such a highly dust-richregion is unfavorable to explain the scarcity of isotopic fractionationof sulfur if the shock wave has a large spatial extent in solar nebula(Tachibana & Huss 2005, Miura & Nakamoto 2006).In this paper, we propose a new scenario for compound chondruleformation. The shock-wave heating model is one of the most plausiblemodels for chondrule formation (e.g., Connolly & Love 1998). In thismodel, the dust particles are exposed to a high-velocity gas flow andheated by the gas frictional heating. It has been suggested that themaximum sizes of chondrules are regulated by the gas flow becauselarge dust particles should be disrupted by the strong gas ram pres-sure when they melt (Susa & Nakamoto 2002, Kato et al. 2006). Re-cently, we carried out three-dimensional hydrodynamics simulations ofmolten dust particle exposed to the gas flow and showed that moltencm-sized dust particle is disrupted into many small pieces in a typi-cal setting of nebula shocks (Miura & Nakamoto 2007). These pieceshave many chances of mutual collisions to form compound chondrulesbecause the local number density of them behind the disrupted par-ticle is enhanced. We name this scenario “fragment-collision model”and think that it can be a strong candidate for compound chondruleformation model. This model seems very similar to the model of col-lision between individual chondrules in the shock-wave heating at thepoint that compound chondrules are formed in the gas flow (Ciesla r e in this paper. Wedescribe the formulations for estimating the collision probability in § §
3. We consider appropriatesituations in which our model can be applied in §
4. We compare ourmodel with observational data of compound chondrules in §
5. Finally,we make conclusions in § The number of collisions per unit time experienced by each ejector(collision rate) is given by R coll = σ coll n e ∆ v , where σ coll is the colli-sional cross-section ( σ coll = 4 πr ), n e is the number density, and ∆ v is the velocity dispersion of ejectors (Gooding & Keil 1981, Sekiya &Nakamura 1996). The original point of our model is to estimate n e resulting from disruption of the parent. Considering the total numberof ejectors torn away from the parent in an infinitesimal duration δt , δN , and the volume of the region in which these ejectors are scat-tered, δV , we obtain n e = δN/δV . Assuming that all ejectors justafter ejection are parting from the parent with a velocity of ∼ ∆ v , weobtain the volume in this phase as δV ∼ πr ∆ vδt , where r p is theradius of parent (see Fig. 1a). It should be noted that ejectors arejumping out of rear side of the parent surface, not of front one. Afterejection, the motions of ejectors are affected by the ambient gas flow.We simply assume that ejectors are accelerated with a constant accel-eration a in the direction of the gas flow ( z -axis), on the other hand,in the direction perpendicular to the gas flow ( r -axis) they move witha constant velocity of ∼ ∆ v (see Fig. 1b). The acceleration is given by a = 3 p fm / r e ρ mat , where p fm is the gas ram pressure and ρ mat is thematerial density inside of the molten dust particle. In this later phase,the region in which ejectors are scattered is getting wider steeply withtime t and its volume is given by δV t ∼ π (∆ vt ) atδt . Approximating V ≃ δV + δV t , we obtain the number density of ejectors as n e ∼ (cid:16) R eject / πr ∆ v (cid:17) h t/t ∗ ) i − (1)and the collision rate R coll ∼ (cid:16) r R eject /r (cid:17) h t/t ∗ ) i − , (2)where R eject is the total number of ejectors extracted from the parentper unit time (ejection rate) defined by R eject ≡ δN/δt . The time t ∗ is defined as t ∗ ≡ (cid:16) ρ mat r r e / vp fm (cid:17) / and it gives the timescalewithin which most collisions will occur (see § [Figure 1] The probability of collisions that each ejector experiences during thetime from t to t is given by integrating the collision rate R coll overthe time t from t to t as (see Eq. 2) P coll ( t , t ) ≡ Z t t R coll dt. (3)In order to simplify the integration, we approximate R coll as R coll = ( r R eject /r for t ≤ t ∗ , r R eject /r ( t/t ∗ ) for t > t ∗ . (4)Using above approximation, we obtain P coll ( t , t ) as P coll ( t , t ) = r R eject /r ( t − t ) when t ≤ t ≤ t ∗ ,r R eject /r h t ∗ − t − ( t /t ∗ ) − t i when t < t ∗ < t ,r R eject t ∗ /r (cid:16) t − − t − (cid:17) when t ∗ ≤ t ≤ t . (5)In this paper, we count all collisions that expected to occur behindthe parent. Finally, we obtain the probability of collisions experiencedby each ejector in one disruption event (collision probability) as P coll ≡ P coll (0 , ∞ ) = 3 r R eject t ∗ r = ρ mat r R r ∆ vp fm ! / . (6) n addition, we find that P coll (0 , t ∗ ) /P coll = 2 /
3. It suggests thatmost (67% of) collisions occur within t ∗ , therefore, the time t ∗ can beconsidered as the typical timescale of the mutual collision. In order to estimate ∆ v and R eject in Eq. (6), we consider the hy-drodynamics of molten parent particle exposed to gas flow. Beforemelting, since the cm-sized parent is too large to homogenize internaltemperature due to the thermal conduction, the temperature is higherat the parent surface facing to the gas flow than at the center (Yasuda& Nakamoto 2005, 2006). It causes to form liquid layer at the parentsurface. Kato et al. (2006) obtained the internal velocity of the liquidlayer by analytically solving the steady hydrodynamics equations for acore-mantle structure with a linear approximation. According to theirresults, we can approximate a maximum tangential velocity of the liq-uid layer as v max ≃ . p fm h/µ , where h is the width of the liquidlayer and µ is the viscosity of molten dust particle (see Appendix A).Considering that ∆ v is about the same order of magnitude of v max ,we obtain ∆ v ∼ v max ∼ p fm h/ µ. (7)The total volume of liquid layer can be roughly estimated as ∼ hr .When the whole part of the liquid layer fragments into ejectors, thetotal number of ejectors is N e ∼ hr /r . Since it is considered thatthe fragmentation proceeds in about a fluid crossing time t cross ∼ r p /v max ∼ µr p /p fm h, (8)we obtain the ejection rate as R eject ∼ N e /t cross ∼ r p p fm h / µr . (9) The liquid layer should be thick enough to cause disruption. Kadono& Arakawa (2005) carried out aerodynamic experiments in which aliquid layer was attached to solid cores, and the breakup of this layer There is an exact solution of the integration. The exact solution of P coll (0 , ∞ ) issmaller than that with the approximation of Eq. (4) by about 19%. In addition, the ratio P coll (0 , t ∗ ) /P coll is about 69%, instead of 2 / ccurred by means of the interaction with a high-velocity gas flow.They discussed that the breakup did not occur at the Weber numberdefined by W ′ e ≡ p fm h/γ s , where γ s is the surface tension, less than10 −
20. This result is similar to the finding that the threshold ofbreakup of liquid droplets without solid cores is W e ∼
10 at O h < . W e ≡ p fm r p /γ s is the Weber number for a completely-moltenparticle and O h ≡ µ/ √ ρ mat r p γ s is the Ohnesorge number (e.g., Fig. 1of Hsiang and Faeth 1995). Based on their results, we consider thatejection will occur at the time when W ′ e = 10. Therefore, the widthof liquid layer is given by h ∼ γ s /p fm . (10)It should be noted that h does not depend on r p . Strictly speaking, weshould write h as h ∼ min [10 γ s /p fm , r p ] because h cannot exceed r p .However, we do not consider the case in which h > r p in this paper,so we simply use Eq. (10). The parent particle is assumed to be mainly composed of forsterite.The physical parameters adopted in this paper are ρ mat = 3 g cm − and γ s = 400 dyne cm − (Murase & McBirney 1973). For other physi-cal parameters, we adopt r p = 1 cm, r e = 300 µ m, p fm = 3 × dyne cm − ,and µ = 10 g cm − s − as a standard set of parameters. We also showresults for other sets of parameters in § Substituting Eqs. (7), (9), and (10) to Eq. (6), we obtain the expres-sion of the collision probability as P coll ∼ . × ρ mat γ r p r p µ ! / . (11)Substituting a standard set of parameters (see § h ∼ . v ∼ . − , and R eject ∼ . × s − . From Eq. (1),we obtain the number density of ejectors behind the parent as n e ∼
796 cm − for t ≪ t ∗ , where t ∗ ∼ .
013 s (see § / πr n e ∼ .
09, so they occupy about % of the space behind the parent in volume. Finally, the collisionprobability is P coll ∼ .
67. Surprisingly, above estimation is largerthan the observed fraction of compound chondrules by one order ofmagnitude or more. If we assume that all collisions lead to compoundchondrule formation, this result suggests that the fragment-collisionmodel can account for the observed fraction when only ∼
10% or less ofall chondrules formed via the fragmentation events of cm-sized parentdust particles (also see § The expression of collision probability, Eq. (11), is complex to under-stand how each parameter affects the result. In order to see in details,we explicitly write the dependences of physical parameters as follows: h ∝ p − , (12) N e ∝ r r − p − , (13) t cross ∝ r p µ, (14) R eject ∝ r p r − p − µ − , (15) δV ∝ r µ − , (16) n e ∝ r − r − p − , (17) σ coll ∝ r , (18)∆ v ∝ µ − , (19) R coll ∝ r − r − p − µ − , (20) t ∗ ∝ r / r / p − / µ / , (21) P coll ∝ r − / r − / p − / µ − / . (22)Let us see the dependence on the ejector radius r e at first. The totalnumber of ejectors contained in the liquid layer, N e , decreases withincrease of r e as ∝ r − . The ejection timescale t cross does not dependon r e . Therefore, the ejection rate R coll ∼ N e /t cross decreases as r e increases. We consider only the collision at the region close to theparent particle because most (67% of) collisions occur at the earlyphase of t ≤ t ∗ (see § δV does notdepend on r e , the number density of ejectors n e ∝ R eject /δV decreases ith increase of r e as ∝ r − . The collisional cross section σ coll isproportional to r and the velocity dispersion ∆ v does not depend on r e , so the collision rate R coll ∼ σ coll n e ∆ v is inversely proportional to r e . We can roughly estimate the collision probability by P coll ∼ R coll t ∗ ,which is smaller than Eq. (6) by a factor of 2/3, and t ∗ is proportionalto r / , so it is found that P coll decreases with increase of r e .We will show other dependences of P coll on the parent radius r p ,the gas ram pressure p fm , and viscosity of molten parent particle µ asbelow. Fig. 2 shows the value of P coll as a function of r e for various valuesof r p = 0 . . . P coll is larger for smaller parent than for largerone as estimated by Eq. (22). The disruption timescale is longer forlarger r p , on the other hand, N e increases with increase of r p moresteeply, so R eject increases as r p increases. However, for larger r p , ejec-tors are disrupted into more wider region. As a result, n e and R coll are inversely proportional to r p . The time t ∗ increases with increase of r p , but it cannot completely cancel the dependence of R coll . Finally,we obtain P coll ∝ r − / as seen in Eq. (22). For the parameter rangewe adopted in Fig. 2, P coll ranges about from 0 . [Figure 2] Fig. 3 shows the value of P coll as a function of r e for various valuesof p fm = 10 (dashed), 3 × (solid), and 10 dyne cm − (dotted-dashed), respectively. It is found that P coll is larger for weaker gas rampressure than for stronger one as estimated by Eq. (22). The strongergas flow can disrupt thiner liquid layer (Eq. 12), so N e and R eject decreases as p fm increases. This results in that n e and R coll are alsoinversely proportional to p fm . In addition, the strong gas flow scattersdisrupted ejectors rapidly, so t ∗ decreases as p fm increases. Finally, weobtain P coll ∝ p − / as seen in Eq. (22). For the parameter range weadopted in Fig. 3, P coll ranges about from 0 .
06 to 6. [Figure 3] .1.3 Viscosity Fig. 4 shows the value of P coll as a function of r e for various valuesof µ = 10 (dashed), 10 (solid), and 10 g cm − s − (dotted-dashed),respectively. It is found that P coll increases as µ decreases as estimatedby Eq. (22). In a highly viscous case, R eject becomes small becauseof a long ejection timescale (Eq. 15). However, in this case, since thevelocity dispersion ∆ v also decreases and ejectors are disrupted intonarrow region, n e does not depend on µ . Therefore, R coll is inverselyproportional to µ , which is due to the dependence of ∆ v . Consideringthe dependence of t ∗ , we obtain P coll ∝ µ − / as seen in Eq. (22). Forthe parameter range we adopted in Fig. 4, P coll ranges about from0 .
06 to 6. [Figure 4]
In Eq. (10), we estimated the layer width h from the disruption con-dition of partially-molten parent particle. However, it should be notedthat this estimation is not valid for much larger parent particle be-cause the disruption timescale becomes larger than the thermal con-duction timescale. In this case, the layer width h will increase bythe rapid thermal conduction before disruption as explained below.The thermal conduction timescale in the liquid layer t cond is given by t cond ∼ ρ mat Ch /κ ∼ ρ mat Cγ /κp , in the second sign of similar-ity we substitute Eq. (10), where C is the specific heat per unit massand κ is the heat conductivity. On the other hand, we obtain the dis-ruption timescale t cross ∼ r p µ/γ s by substituting Eq. (10) to Eq. (8).The layer width h would not change significantly during disruption ifthe condition of t cross < t cond is satisfied. We rewrite this condition as r p < ∼ ρ mat Cγ µκp . (23)Adopting C = 10 erg g − K − and κ = 4 × erg cm − s − K − (Murase & McBirney 1973), we find r p < ∼ . § r p is much larger than this value, the layerwidth h changes during disruption because the thermal conductioncan rapidly transfer energy inside the parent and melt deeply. In this tudy, we do not treat such situation because we consider cm-sizedparent particle, which satisfies the condition of Eq. (23). The gas ram pressure p fm should be sufficient to disrupt the moltenparent particle. For simplicity, we consider a completely-molten par-ent particle here ( h = r p ). The disruption will occur when the Webernumber W e ≡ p fm r p /γ s exceeds about 10 (Hsiang & Faeth 1995).Therefore, the condition that p fm should satisfy is given by p fm > ∼ γ s /r p . (24)We find p fm > ∼ × dyne cm − for a standard set of parameters (see § p fm expected to affect the moltenparent particle just behind the shock front (solid lines) as a function ofthe shock velocity v s and the pre-shock gas number density n . Thegray region is the chondrule-forming shock condition in which thegas frictional heating is sufficient to melt the precursor dust particlebut not so strong as it evaporates the dust completely (Iida et al.2001). The gas ram pressure is p fm ∼ − dyne cm − when n ∼ − cm − and v s ∼ −
10 km s − . The shocks associated withgravitational instability (e.g., Boss & Durisen 2005) or planetesimalbow shocks (e.g., Hood 1998, Weidenschilling et al. 1998) might bepreferable for such situation. [Figure 5] The viscosity of molten parent particle should be small enough tocause disruption. The timescale of disruption is given by t cross ∼ r p µ/γ s (see Eqs. 8 and 10). On the other hand, the heating eventceases within the timescale that the parent dust particle stops againstthe ambient gas (Iida et al. 2001). The timescale is the stoppingtime, which is given by t stop ∼ ρ mat r p v g /p fm , where v g is the relativevelocity between ambient gas and dust particle. If t cross > t stop , thegas drag heating ceases before the disruption occurs. Therefore, the Substituting p fm = ρ g v , where ρ g is the gas density, we obtain a well-known expressionof the stopping time t stop ∼ ρ mat r p /ρ g v g . iscosity at the disruption satisfies the condition of t cross < t stop . Werewrite this condition as µ < ∼ ρ mat γ s v g p fm . (25)Substituting v g = v s = 8 km s − (see Fig. 5), we find µ < ∼ . × g cm − s − for a standard set of parameters (see § t cross ∼ r p µ/γ s (see Eqs.8 and 10), and it decreases as µ decreases. For example, we obtain t cross ∼ . × − s for r p = 1 cm and µ = 0 . − s − . However, theliquid layer cannot be accelerated to the velocity given by Eq. (7) insuch a short duration. Considering that the momentum of the gas flowconverts to the motion of the liquid layer effectively, the timescale inwhich the liquid layer is accelerated to ∆ v is given by t acc ∼ m ∆ v/F ,where the mass of the liquid layer is m ∼ πρ mat r h , the force due tothe gas flow is F = πr p fm , and the velocity dispersion is ∆ v ∼ γ s /µ (see Eqs. 7 and 10). Then we obtain t acc ∼ ρ mat γ /p µ. (26)When t acc > t cross , the liquid layer does not reach the steady solutionsbefore ejection. Therefore, the condition of t acc < t cross should besatisfied in our model. This condition can be rewritten as µ > ∼ ρ mat γ r p p ! / . (27)We find µ > ∼ − s − for a standard set of parameters (see § We found that the collision probability estimated by our model, P coll ,is close to unity, which is larger than the observed fraction of com-pound chondrules by one order of magnitude or more (see § ia the fragmentation events. The fraction of chondrules which haveundergone the fragmentation event depends on a size distribution ofprecursor dust particles in chondrule-forming region. Nomura et al.(2007) solved coagulation equations for various sizes of settling dustparticles in the solar nebula. They obtained the dust size distributionsfor various positions in the nebula and ages of the nebula. Accordingto their results, for example, the dust particles with the radii from1 mm to 1 cm have the size distribution similar to dn ∝ r − d (log r ),where r is the dust radius, in regions near midplane at the distanceof 1 AU or 10 AU and at the age of 10 yr (it corresponds to the lineof the slope − of mm-sized dust particles per a cm-sized one. Assuming that thecm-sized particle is disrupted into mm-sized ejectors in a shock-waveheating, total number of ejectors is estimated as ∼ . If we alsoassume that all of them become compound chondrules with two con-stituents, about 500 sets of compounds are expected to form. In thiscase, the fraction of compound chondrules is ∼
5% (= 500 / ). Thisis very close to the observed fraction of compound chondrules.However, all collisions do not necessarily lead to compound chon-drule formation if we consider the situation of mutual collision in moredetail. We discuss the efficiency of compound chondrule formation in § In the fragment-collision model we proposed in this paper, the con-stituent chondrules of compounds are likely to have similar compo-sitions because ejectors originate from the same parent. Akaki &Nakamura (2005) measured the oxygen isotopic compositions for 3sets of blurred-type compounds, 6 sets of adhering- or consorting-type In the shock-wave heating model, all of the equations governing the evolution of theprecursor dust particle are scaled with the initial precursor radius r as long as the post-shock gas properties (temperature, density, and so forth) are spatially uniform (see § ompounds, and 2 sets of enveloping-type compounds. It was foundthat in a three-isotopic diagram, all sets of blurred-, adhering-, andconsorting-types fall in the typical range obtained for single chon-drules from the same CV3 chondrites. These results suggest that thetwo constituent chondrules of these compounds originated from thesame dust reservoirs as those single chondrules. These observationsare consistent with our new model for compound chondrule formation.In contrast, in one set of the enveloping-types, the oxygen isotopiccompositions differ between two constituent chondrules. This resultmight suggest that this enveloping-type compound has not formedby our model but by the relict grain model, in which fine-graineddust particles accreted on the surface of already-formed primary weremelted in a second heating event (Wasson 1993, Wasson et al. 1995). There is a dependence of observed fraction of compound chondruleson textural types of component chondrules. Gooding & Keil (1981)found that compound chondrules with non-porphyritic pairs are morefrequent than that with porphyritic pairs. From their thin sectionresults, they estimated that 13% of non-porphyritic and 2% of por-phyritic chondrules are compound or cratered, which are interpretedas products of collisions between plastic chondrules. Akaki & Naka-mura (2005) reported the same tendency. It is considered that por-phyritic and non-porphyritic textures have been formed from partially-and completely-molten dust particles, respectively (Lofgren & Rus-sell 1986). It is also naturally considered that a partially-molten dustparticle is highly-viscous, in contrast, completely-molten one has lowerviscosity. Based on above assumption, the dependence of the observedfraction of compound chondrules on the textural types might reflectthe dependence of the collision probability on the viscosity of moltenparent dust particle. Actually, our model predicts that lower viscosityresults into larger collision probability as shown in Fig. 4. This resultis consistent with the observations.In addition, various types of textures are seen in compound chon-drules, e.g., P-P, P-NP, and NP-NP pairs, where P and NP stand forporphyritic and non-porphyritic textures, respectively (Wasson et al.1995). In contrast, our model seems not to account for compoundchondrules with porphyritic textures because the following proper-ties are implied: (a) completely-molten ejectors are extracted from ompletely-molten parent particle, and (b) temperatures of all ejectorsare the same at the extraction. However, regarding the point (a), itcan be considered that the partially-molten parent particle, which in-cludes tiny unmelted cores inside, behaves as a fluid if the molten partoccupies the most volume of the parent. In this case, the partially-molten ejectors could be extracted from the partially-molten parentand the porphyritic textures might be formed. Regarding the point(b), we can consider the case that the temperatures of ejectors are notuniform at the extraction. If the initial temperatures of ejectors aredifferent, various types of textures can be formed in compound chon-drules. In order to verify above hypotheses, we must quantitativelyinvestigate the thermal evolutions of the parent dust particle and eachejector at disruption. It requires to carry out the three-dimensional(thermo-)hydrodynamics simulation and/or aerodynamical disruptionexperiment.In addition, Connolly & Hewins (1995) reported that porphyritictextures can be reproduced from totally-molten droplets by dust seed-ing. Based on their results, even if ejectors extracted from the parentare totally-molten, they can obtain porphyritic textures inside themwith the help of the dust seeding. Considering that such seeded ejec-tors collide with others before solidification, compound chondrules in-cluding porphyritic textures might be formed. Therefore, the effect ofthe dust seeding in compound chondrule formation is also an impor-tant issue that should be investigated in the future. In § ng rate is given by R cool = 3 σ SB (cid:0) ǫ emit T − ǫ abs T (cid:1) r e ρ mat C , (28)where ǫ emit and ǫ abs are the emission and absorption coefficient, re-spectively, σ SB is the Stefan-Boltzmann constant, T e is the tempera-ture of the ejectors, T rad is the effective temperature of the ambientradiation field (Desch & Connolly 2002, Miura & Nakamoto 2006),and C is the specific heat. The cooling timescale in which the tem-perature decreases by ∆ T is given by ∼ ∆ T /R cool . Here, we assumethat ∆ T = 100 K and 300 K are required for moderate cooling andcomplete solidification, respectively. The effective radiation tempera-ture T rad should be lower than 1273 K in order to prevent the isotopicfractionation of sulfur in chondrules (Miura & Nakamoto 2006). Thecooling rate is reduced by the term of T rad , however, it is a minoreffect in the cooling phase (if we assume T rad = 1000 K, we obtain T /T = 0 . ǫ emit = 1 and T e = 1600 K, compound chondrules can be formedonly in the phase from t cool = 0 .
08 s to t solid = 0 .
24 s (moderate cool-ing phase). Since the timescale of the mutual collision ( t ∗ ∼ .
01 s, see § t cool and t solid , we obtain the collision probabilitybetween t cool and t solid as (see Eq. 5) P ′ coll ≡ P coll ( t cool , t solid ) ≃ r R eject t ∗ r (cid:18) t ∗ t cool (cid:19) , (29)where we neglect the term of t solid because t − ≪ t − . Comparingwith Eq. (6), we find that P ′ coll is smaller than P coll by a factor of Q ≡ P ′ coll /P coll = ( t ∗ /t cool ) / ∼ × − . The physical meaningof Q is the efficiency of compound chondrule formation per a mutualcollision. It means that almost all (99.5% of) collisions do not leadto compound chondrule formation. Taking into account the low effi-ciency, the probability of compound chondrule formation is about oneorder of magnitude smaller than the observed fraction of compoundchondrules, although this estimation is larger than that estimated fromthe random collision model (Gooding & Keil 1981, Sekiya & Nakamura1996).However, we can consider other possibilities which enhance thecollision probability. One possibility is that the gas flow is blockedby the parent particle and ejectors. In this case, the shadowed re-gion in which there is no effect of the gas flow appears behind them. ince the ejectors in the shadowed region are not accelerated by thegas flow, the concentration of ejectors will be higher than the caseneglecting this effect. The second possibility is to consider the colli-sions between different-sized ejectors. The relative velocity betweenthe different-sized ejectors is given by ∼ ∆ v + | v rel | , where | v rel | isthe bulk relative velocity between different-sized ejectors due to thedifference of the acceleration (see § § t cool can be shorter and it leads to theincrease of P ′ coll (see Eq. 29). These possibilities are very importantto investigate in detail, however, they are beyond the scope of thispaper. We are planning to investigate these issues and the results willappear in the forthcoming papers. Wasson et al. (1995) measured median diameters of primaries andsecondaries in compound chondrules and the ratio of the “diameter”of the secondary divided by that of the primary. They found that themean and median of the ratio are about 0.3 and 0.25, respectively.These results suggest that compound chondrules with different-sizedpairs appear more frequently than that with same-sized pairs. On theother hand, in our model, if there is a size difference between collidingtwo ejectors moving in the gas flow, it leads to a large relative velocitybetween them because the acceleration a depends on the ejector radius(see § § a = 3 p fm / r e ρ mat (see § z -direction, v z , and the position from the parent, z , aregiven by v z = at = 3 p fm t r e ρ mat , (30) = 12 at = 3 p fm t r e ρ mat , (31)where t is the time after extraction. It is considered that the compoundchondrule formation occurs after the ejector cools moderately (see § z > z cool ≡ p fm t r e ρ mat . (32)Next, eliminating t from Eqs. (30) and (31), we obtain v z = (cid:18) p fm z r e ρ mat (cid:19) / . (33)Since the velocity v z depends on the ejector radius r e , there is therelative velocity v rel between the large ejector with radius r l and thesmall ejector with radius r s at the same position. It is given by v rel = (cid:18) p fm z r l ρ mat (cid:19) / "(cid:18) r l r s (cid:19) / − . (34)It is found that v rel increases as z increases. The compound chondruleformation can occur only if the condition of v rel < v dest is satisfied,where v dest is the critical destruction velocity, otherwise ejectors willbe disrupted upon impact. This condition is rewritten as z < z dest ≡ r l ρ mat v p fm "(cid:18) r l r s (cid:19) / − − . (35)From Eqs. (32) and (35), the region in which compound chondrulescan be formed is z cool < z < z dest . In other words, the necessarycondition for compound chondrule formation is given by z cool < z dest ,which is rewritten as r s r l > (cid:18) r l ρ mat v dest p fm t cool + 1 (cid:19) − . (36)Substituting r l = 300 µ m, p fm = 3 × dyne cm − , and t cool = 0 .
08 sto Eq. (36), we obtain the appropriate size ratio of the small chondruleto large one as r s /r l > .
99 for v dest = 10 cm s − , r s /r l > .
91 for v dest = 10 cm s − , r s /r l > .
44 for v dest = 10 cm s − , and r s /r l > We substitute r e = r l in Eq. (32). .
28 for v dest = 10 cm s − , respectively (see Fig. 6). Ciesla (2006)mentioned that as chondrules cool they can survive collisions withone another at velocities up to 10 cm s − due to viscous dissipationin the melt. If so, compound chondrule with the size ratio of about > ∼ . r s /r l from observations ( ∼ .
3, Wasson et al. 1995). In order toexplain the observation, it is required that v dest ∼ − or more,however, it seems unrealistic.In order to overcome this difficulty, we can consider the same possi-bilities as discussed in § v rel does not increase with z because ejectorsare not accelerated. If v rel just after entering the shadowed region isless than v dest , we obtain z dest → ∞ because v rel does not increase fur-ther. In this case, the compound chondrule formation is possible forarbitrary r s /r l . It implies that the compound chondrules with smallervalue of r s /r l can be formed in the shadowed region. In addition,we discussed the possibility that the disruption from partially-moltenparent. In this case, t cool can be shorter than that we assumed in thissubsection. The shorter t cool allows the compound chondrule forma-tion for the wider range of the size ratio r s /r l (see Fig. 6). Theseissues will be discussed in detail in the forthcoming paper. [Figure 6] We proposed a new scenario for compound chondrule formation namedas “fragment-collision model,” in the framework of the shock-waveheating model. We modeled the disruption of molten cm-sized par-ent dust particle exposed to a high-velocity gas flow in order to esti-mate the efficiency of mutual collisions between small fragments as-suming that all of them have the same radius. We obtained colli-sion probability P coll for a wide range of parameters (parent radius r p = 0 . − r e = 100 − µ m, ram pressure ofthe gas flow p fm = 10 − dyne cm − , and viscosity of molten par-ent µ = 10 − g cm − s − ). The estimated collision probability was ∼ . − hondrules when about 10% or less of all chondrules formed via thefragmentation events. Since the fraction of chondrules which haveundergone the fragmentation event depends on a size distribution ofprecursor dust particles in chondrule-forming region, it would be diffi-cult to make a conclusion about the fraction of compound chondrules.However, numerical results of dust coagulation equations in the solarnebula seem to match well with our estimation (Nomura et al. 2007).In addition, our model does not require the dust enhancement in thepre-shock region because compound chondrules are formed from a sin-gle large dust particle. This is advantageous to explain the scarcityof isotopic fractionation of sulfur (Tachibana & Huss 2005, Miura &Nakamoto 2006).We also compared our model with other observational data. Akaki& Nakamura (2005) measured the oxygen isotopic compositions of con-stituents of compound chondrules and found that in a three-isotopicdiagram, all compound chondrules except for enveloping-types fall inthe typical range obtained for single chondrules from the same chon-drites. These observations are consistent with our model because twoconstituent chondrules are expected to originate from the same dustreservoirs. Gooding & Keil (1981) and Akaki & Nakamura (2005) re-ported that compound chondrules with non-porphyritic pairs are morefrequent than that with porphyritic pairs. It is thought that these ob-servations can be also explained by our model because the collisionprobability depends on the viscosity of molten parent dust particle.The dependence on the viscosity is consistent with the experimen-tal results that porphyritic and non-porphyritic textures have beenformed from partially- and completely-molten dust particles, respec-tively (Lofgren & Russell 1986). Finally, the size ratios of secondaryto primary in each set of compound chondrule have the mean valueabout 0.3 (Wasson et al. 1995). This result might be explained by ourmodel because two fragments with different sizes are accelerated bythe gas flow with different accelerations. As a result, these two frag-ments obtain large relative velocity and it would enhance the collisionprobability. Therefore, the compound chondrule of different-sized pairtends to be formed more frequently than that of same-sized one.However, it should be noted that all collisions do not necessar-ily lead to compound chondrule formation. For example, undesirablyfast collisions cause disruption of ejectors upon impact. Assuming thatthe upper limit of the collisional velocity for coalescence is 10 cm s − (Ciesla 2006), the appropriate size ratio of secondary (small) chon- rule to primary (large) one is r s /r l > ∼ .
5, which does not accountfor the observations (mean value of r s /r l ∼ .
3, Wasson et al. 1995).In addition, ejectors should cool moderately before collision to makecompound chondrule not to fuse into a single droplet. The collisionprobability after the moderate cooling, however, is much smaller thanthat of total collisions. In order to overcome these difficulties, we con-sider other physics that we did not take into account in this paper(e.g., the gas flow is blocked by the parent and/or numerous num-bers of ejectors). These issues will be discussed in the forthcomingpaper. In addition, our model should be tested by other methods,e.g., three-dimensional (thermo-)hydrodynamics simulation or aero-dynamic disruption experiment in the future.
Acknowledgment
We are grateful to Drs. Tomoki Nakamura and Fred J. Ciesla, andan anonymous referee for useful comments in this study. H.M. andS.Y. were supported by the Research Fellowship of Japan Society forthe Promotion of Science for Young Scientists. T.N. was partiallysupported by the Ministry of Education, Science, Sports, and Culture,Grant-in-Aid for Scientific Research (C), 1754021.
A Hydrodynamics in Liquid Layer
Kato et al. (2006) examined the hydrodynamics of the liquid layer byanalytically solving the hydrodynamics equations for a core-mantlestructure with a linear approximation. Fig. 7 shows a schematic pic-ture of the set-up in their analysis. According to their solutions, wecan obtain the radial and tangential velocities, v r and v θ , at arbitraryposition in the liquid layer. The tangential velocity has the maximumvalue v max at θ = 0 . π on the surface of liquid layer. Fig. 8 shows v max as a function of the layer width h . The horizontal axis is a nor-malized layer width h/r p and the vertical one is a normalized velocity v max / ( p fm r p /µ ). The solution obtained by Kato et al. (2006) is a com-plex function (dashed), however, it can be approximated by a linearinterpolation between v max = 0 for h/r p = 0 and v max = 0 . p fm r p /µ for h/r p = 1. The value of v max for h/r p = 1 corresponds to the so-lution obtained by Sekiya et al. (2003), in which they analyzed thehydrodynamics of a completely-molten dust particle. In our model, e adopt the linear interpolation for v max (solid line) as the velocitydispersion of ejectors disrupted from a molten parent particle, whichis given by v max = 0 . p fm h/µ. (37) [Figure 7][Figure 8] References [1] Akaki, T., Nakamura, T., 2005. Formation processes of com-pound chondrules in CV3 carbonaceous chondrites: Constraintsfrom oxygen isotope ratios and major element concentrations.Geochim. Cosmochim. Acta 69, 2907-2929.[2] Amelin, Y., Krot, A. N., Hutcheon, I. D., Ulyanov, A. A., 2002.Lead isotopic ages of chondrules and calcium-aluminum-rich in-clusions. Science 297, 1678-1683.[3] Ciesla, F. J., Lauretta, D. S., Hood, L. L., 2004. The frequency ofcompound chondrules and implications for chondrule formation.Meteorit. Planet. Sci. 39, 531-544.[4] Ciesla, F. J., 2006. Chondrule collisions in shock waves. Meteorit.Planet. Sci. 41, 1347-1359.[5] Connolly Jr., H. C., Hewins, R. H., 1995. Chondrules as productsof dust collisions with totally molten droplets within a dust-richnebular environment: An experimental investigation. Geochim.Cosmochim. Acta 59, 3231-3246.[6] Connolly Jr., H. C., Love, S. G., 1998. The formation of chon-drules: Petrologic Tests of the Shock Wave Model. Science 280,62-67.[7] Desch, S. J., Connolly Jr., H. C., 2002. A model of the thermalprocessing of particles in solar nebula shocks: Application to thecooling rates of chondrules. Meteorit. Planet. Sci. 37, 183-207.[8] Gooding, J. L., Keil, K., 1981. Relative abundances of chondruleprimary textural types in ordinary chondrites and their bearingon conditions of chondrule formation. Meteoritics 16, 17-43.[9] Hsiang, L. -P., Faeth, G. M., 1995. Drop deformation and breakupdue to shock wave and steady disturbances. Int. J. MultiphaseFlow 21, 545-560.
10] Jones, R. H., Lee, T., Connolly Jr., H. C., Love, S. G., Shang, H.,2000. Formation of chondrules and CAIs: Theory vs. observation.In: Boss, A. P., Russell, S. S. (Eds.), Protostars and Planets IV.Univ. of Arizona Press, Tucson, pp. 927-962.[11] Kadono, T., Arakawa, M., 2005. Breakup of liquids by high veloc-ity flow and size distribution of chondrules. Icarus 173, 295-299.[12] Kato, T., Nakamoto, T., Miura, H., 2006. Maximal size of chon-drules in shock wave heating model: Stripping of liquid surfacein a hypersonic rarefied gas flow. Meteorit. Planet. Sci. 41, 49-65.[13] Lofgren, G., Russell, W. J., 1986. Dynamic crystallization ofchondrule melts of porphyritic and radial pyroxene composition.Geochim. Cosmochim. Acta 50, 1715-1726.[14] Miura, H., Nakamoto, T., 2005. A shock-wave heating model forchondrule formation: effects of evaporation and gas flows on sili-cate particles. Icarus 160, 258-270.[15] Miura, H., Nakamoto, T., 2006. A shock-wave heating model forchondrule formation: Prevention of isotopic fractionation. Astro-phys. J. 651, 1272-1295.[16] Miura, H., Nakamoto, T., 2007. Shock-wave heating modelfor chondrule formation: Hydrodynamic simulation of moltendroplets exposed to gas flows. Icarus 188, 246-265.[17] Murase, T., McBirney, A. R., 1973. Properties of some commonigneous rocks and their melts at high temperatures. Geol. Soc.Am. Bull. 84, 3563-3592.[18] Nomura, H., Aikawa, Y., Tsujimoto, M., Nakagawa, Y., Millar, T.J., 2007. Molecular hydrogen emission from protoplanetary disks.II. Effects of X-ray irradiation and dust evolution. Astrophys. J.661, 334-353.[19] Sanders, I. S., Hill, H. G. M., 1994. Multistage compound chon-drules and molded chondrules in the Bovedy (L3) meteorite. Me-teoritics 29, 527-528.[20] Sekiya, M., Nakamura, T., 1996. Condition for the formation ofthe compound chondrules in the solar nebula. Proc. NIPR Symp.Antarct. Meteorites 9, 208-217.[21] Sekiya, M., Uesugi, M., Nakamoto, T., 2003. Flow in a liquidsphere moving with a hypersonic velocity in a rarefied gas—An nalytic solution of linearized equations. Prog. Theor. Phys. 109,717-728.[22] Susa, H., Nakamoto, T., 2002. On the maximal size of chondrulesin shock wave heating model. Astrophys. J. 564, L57-L60.[23] Tachibana, S., Huss, G. R., 2005. Sulfur isotope composition ofputative primary troilite in chondrules from Bishunpur and Se-markona. Geochim. Cosmochim. Acta 69, 3075-3097.[24] Wasson, J. T., 1993. Constraints on chondrule origins. Meteoritics28, 14-28.[25] Wasson, J. T., Alexander, N. K., Lee, M. S., Rubin, A. E., 1995.Compound chondrules. Geochim. Cosmochim. Acta 59, 1847-1869.[26] Yasuda, S., Nakamoto, T., 2005. Inhomogeneous temperaturedistribution in chondrules in shock-wave heating model. LunarPlanet. Sci. 36, 1252-1253.[27] Yasuda, S., Nakamoto, T., 2006. Possible size of porphyritic chon-drules in shock-wave heating model. Lunar Planet. Sci. 37, 1674-1675. z -axis),and a constant-velocity motion perpendicular to the gas flow ( r -axis).27igure 2: Collision probability P coll are plotted as a function of ejector radius r e for various parent radii of r p = 0 .
2, 1 .
0, and 5 . p fm = 3 × dyne cm − and µ = 100 g cm − s − (see § p fm = 10 ,3 × , and 10 dyne cm − . Other parameters are described in § µ = 10, 10 , and10 g cm − s − . Other parameters are described in §
10 100 p r e - s h o c k g a s n u m b e r d e n s i t y n [ c m - ] shock velocity v s [km s -1 ]10
10 100 p r e - s h o c k g a s n u m b e r d e n s i t y n [ c m - ] shock velocity v s [km s -1 ] p fm [dyne cm -2 ] c h o nd r u l e - f o r m i n g c o nd iti o n evaporatenot melt Figure 5: Gas ram pressure affecting the molten parent particle behind theshock front. The horizontal axis is the shock velocity v s and the vertical axisis the pre-shock gas number density n . The gray region is the chondrule-forming shock condition in which the gas frictional heating is sufficient tomelt the precursor dust particle but not so strong as it evaporates the dustcompletely (Iida et al. 2001). 31 s i z e r a t i o r s / r l v dest [cm s -1 ] 0.80.080.008 t cool [sec]mean & median values of observations Figure 6: Appropriate ranges of size ratio of the small chondrule (radius of r s ) to large one (radius of r l ) for compound chondrule formation are displayedas a function of assumed destruction velocity v dest in the case of r l = 300 µ mand p fm = 3 × dyne cm − . According to Ciesla (2006), v dest ∼ cm s − .The solid, dashed, and dotted curves are criteria of the destructive collisionfor the timescales of the moderate cooling t cool of 0 .
008 sec, 0 .
08 sec, and0 . r c is surrounded by a liquid layer (white) of width h , where the dustradius r p = r c + h . The radial and tangential fluid velocities inside the liquidlayer, v r and v θ , were obtained as a function of r and θ , where r is a distancefrom the center. The ambient gas flows from up to bottom in this figure.33igure 8: Maximum tangential velocity at the surface of liquid layer v max asa function of a width of liquid layer h . The solution obtained by Sekiya et al.(2003) was for a completely-molten particle, so it corresponds to h/r p = 1 aspointed by a filled circle. The solutions for Kato et al. (2006, dashed curve)were for arbitrary values of h/r p from 0 to 1 and v max → h/r p → v max = 0 at h/r p = 0 to v max = 0 . p fm r p /µ at h/r pp