Abstract– Here, we show that several geochemical indicators point to number densities during chondrule formation that were far higher than can be accounted for by known nebula processes. The number densities implied by compound chondrules and nonspherical chondrules are shown to be significantly higher than estimated in previous studies. At the implied chondrule number densities, if a chondrule formation region survived a formation event it would have been gravitationally bound and would have collapsed quite rapidly to form an asteroidal-sized body. The diversity of chondrule compositions and textures in a chondrite group could have formed in a single event in subvolumes of a formation region that were chemically isolated from one another because of slow diffusion in the gas. Within these subvolumes, equilibration between chondrules with different compositions would have been fairly rapid, although small isotopic mass fractionations in elements like Fe, Si, Mg, and O may persist. This could explain the existence of the small isotopic mass fractionations in these elements that have been observed in chondrules. However, the evidence for recycling of chondrules requires that there was more than one chondrule formation event prior to formation of a parent asteroid. Finally, we argue that OC and CO chondrule Mg-Al systematics are both consistent with single ages or narrow ranges of ages, and that the CO, and possibly the OC, ages date parent body alteration. This would resolve the conundrum of needing to preserve in a turbulent nebula physically and chemically distinct CO and OC chondrule populations for 1–2 Myr.
Chondrules are either a major or the dominant component of most chondrites. The exceptions are the CI chondrites. Even in the CI chondrites olivine grains that have survived the extensive aqueous alteration all CIs experienced are probably remnants of chondrules (Leshin et al. 1997). Chondrule-like objects, unusual in their small size, have also been found in returned samples of comet Wild 2 (Nakamura et al. 2008). Chondrules or chondrule fragments have not been identified in interplanetary dust particles (IDPs), which could be derived from comets, but this may be the result of biases toward low-density particles imposed by atmospheric entry—the lower the density, the less heating they experience. The apparent ubiquity of chondrules in primitive solar system bodies suggests that they formed by one of the most widespread processes operating in the early solar nebula. As few, if any, primitive meteorites are chondrule-free, it is even possible that chondrule and planetesimal formation are linked. Chondrules formed at high temperatures, suggesting that they are also the products of one of the more energetic processes in the solar nebula. Given the apparent importance of chondrules, it is not surprising that they have been the objects of such fascination to meteoriticists over the years. Yet despite a considerable amount of work, we still know remarkably little about how and under what conditions they formed.
In principle, the physical, mineralogical, and chemical properties of chondrules can be used to place limits on their formation conditions. The survival of relict grains suggests that heating and cooling must have been relatively rapid. Crystal textures suggest that peak temperatures generally approached, but did not necessarily exceed, chondrule liquidi, and that chondrules subsequently cooled at rates of the order of 1–1000 °C h−1 (Hewins et al. 2005; Miyamoto et al. 2009). The absence in chondrules of large, systematic isotopic fractionations associated with free evaporation has been used to infer relatively high chondrule + dust (solid) densities prior to heating—solid/gas ratios of 100s to 1000s times solar at P = 10−4–10−3 bars, and higher at lower total pressures (Alexander 2004; Cuzzi and Alexander 2006). Not surprisingly, these are the conditions needed to stabilize ferromagnesian melts in the nebula, although, as recognized by Ebel and Grossman (2000), the equilibrium liquids even under these conditions differ in composition from the mesostasis compositions of typical chondrules. Based on the frequencies of compound chondrules, the number densities of chondrules during formation have been estimated to be about 0.1–30 m−3 (Ciesla and Hood 2002; Desch and Connolly 2002; Cuzzi and Alexander 2006), which for 1 mm diameter chondrules with densities of 3 × 103 kg m−3 and a total pressure of 10−4 bars is a solid/gas enrichment relative to solar of roughly 10–4000.
It has long been recognized that volatile elements (e.g., alkalis) can place some of the most stringent constraints on chondrule formation conditions (e.g., Tsuchiyama et al. 1981; Hewins 1991), particularly if their abundances at various temperatures during formation can be determined. The Ca-rich pyroxene (CPX) to glass distribution coefficients (Kds) show that during CPX crystallization, approximately 1000–1200 °C (Ebel and Grossman 2000; Alexander and Grossman 2005), Na was present in most chondrules at similar levels to the present abundances (Jones 1994; Libourel et al. 2003; Alexander and Grossman 2005). Sodium zoning profiles in olivine phenocrysts suggest that this was the case from the onset of crystallization, that is, at near-liquidus temperatures of approximately 1600 °C (Alexander et al. 2007, 2008; Borisov et al. 2008; Kropf and Pack 2008; Kropf et al. 2009). This implies that chondrules behaved as essentially closed systems for Na. The chondrule and dust densities that are required for essentially closed-system behavior of Na during chondrule formation at near-liquidus temperatures are very high (≥106 times solar solid/gas ratios) (Alexander et al. 2008). Such high solid/gas enrichments are very hard to achieve in the nebula by known or hypothesized mechanisms, such as turbulent concentration (Cuzzi et al. 2001). These very large enrichments, equivalent to number densities of 1 mm diameter chondrules of >103–104 m−3 depending on the PH2, are far higher than estimated from compound chondrule abundances. There are also no known nebular mechanisms for producing the large energy densities necessary to melt chondrule precursors when their concentrations are so high.
The ages of CO carbonaceous chondrite and ordinary chondrite (OC) chondrules, based on 26Al measurements, are very similar and each have total spreads of about 1.5–2 Myr (Kurahashi et al. 2008). On the other hand, physically and chemically the CO and OC chondrules are quite distinct. How were these distinct chondrule populations, and the chemical/physical conditions that produced them, maintained for up to 1.5–2 Myr when mixing time scales for the asteroid belt could have been ≤0.1 Myr and were no longer than approximately 1 Myr (Alexander 2005; Cuzzi et al. 2010)? Equally perplexing is the difference in the abundances, sizes, and types of refractory inclusions (CAIs and ameboid olivine aggregates or AOAs) in COs and OCs—refractory inclusions are relatively abundant in COs, for instance, but are rare in OCs (Hezel et al. 2008; Rubin 2010). Thus, we are faced with two contradictory sets of observations: (1) the requirement of rapid transport or storage in the disk to preserve inclusions, and to place chondrule-like objects and inclusions in comet-forming regions, and (2) slow transport, at least in the asteroid belt, to preserve discrete chondrule populations (if the chondrule ages are real) and CAI abundances/populations for long periods of time.
Given the difficulty of (1) explaining the chondrule formation conditions, inferred from chondrule elemental and isotopic compositions, with known nebular processes, and (2) reconciling the fact that chondrule ages and CAI abundances in COs and OCs require low levels of turbulence while disk dynamics, disk transport, and planetesimal formation require fairly vigorous turbulence, here we re-examine many of the chondrule observations to see if they can reasonably be reconciled with current astrophysical expectations for the solar nebula.
Chondrule Concentrations Re-Examined
Is it possible that the solid/gas enrichments estimated from the chondrule elemental, particularly Na, and isotopic data have been overestimated?
Fundamentally, the chemical constraints depend on the estimates of the equilibrium vapor pressures of the chondrules. For ease of calculation, the equilibrium vapor pressures are generally calculated for constant PH2. At high temperatures and dust enrichments, H2 is not the only, or even the major, H-bearing species in the equilibrium vapor—H2O, H, and OH are also important. Figure 1 shows the total vapor pressures of the type IA and IIA chondrule compositions as a function of PH2 and temperature. The chondrule compositions are given in Table 1. At near-liquidus temperatures, the equilibrium vapors at high PH2 are dominated, respectively, by Na, H2O, and H2, while at low PH2 they are dominated by Na and O2. To calculate the solid/gas enrichments (dgsolar), relative to a system of solar composition (solid mass fraction ∼0.005), as would have existed prior to heating, these other H-bearing gases must also be taken into account, i.e.,
where S (g m−3) is the solid density, R the gas constant (J mole−1 K−1), T the temperature (K) and Px are the partial pressures of the H-bearing species (N m−2).
Table 1. The bulk chondrule compositions (wt%) and type I mesostasis composition (wt%) used in calculating chondrule vapor pressures and viscosity. The chondrule compositions are from Alexander et al. (2008).
Type I bulk
Type II bulk
Type I mesostasis
As outlined by Alexander et al. (2008), the solid densities needed to approximate closed-system behavior of a volatile element, i, can be estimated by calculating the density of solids required for a fraction fi of the element to remain condensed in chondrules. In this case, the solid density is given by
where Pi is the equilibrium partial pressure of the element above the silicate chondrule (N m−2), Mi the molar weight (g mole−1), and Xi is the mass fraction in the chondrules. For silicate chondrules, the equilibrium vapor pressures were calculated using MELTS to estimate the activities of components in the melt (Ghiorso and Sack 1995). Equation 2 ignores the total volume of the chondrules, which at the densities considered here is justified. Also, Equation 2 is only strictly applicable when fi is close to one because it does not take into account the decrease in Pi and Xi as fi decreases, although provided that Pi is roughly linearly proportional to Xi it is a reasonable approximation. Hence, for the purposes of this paper it will be used in all estimates. It is straightforward to calculate the chondrule number density, assuming a chondrule size and that there is no dust, from Equation 2.
Given the high equilibrium vapor pressures of chondrules (Fig. 1) that are dominated by Na, it is not surprising that to achieve essentially closed system behavior requires very high solid/gas ratios. The necessary solid/gas enrichments are higher the lower the total pressure of the system. Astrophysical models typically have midplane pressures in the asteroid belt of 10−5–10−7 bars (Wood and Morfill 1988; Boss 1996). Shock models suggest that pressures and chondrule densities could increase by approximately one to two orders of magnitude in the post-shock region (Connolly and Love 1998; Iida et al. 2001; Ciesla and Hood 2002; Desch and Connolly 2002; Miura et al. 2002). This would reduce the necessary solid/gas enrichments somewhat. However, to date no shock model has been able to show that there would be sufficient chondrule heating under the extreme chondrule mass loading required by the near closed system behavior of Na. The highest solid enrichments in any published shock models are 300–600 times solar, and under these relatively modest solid enrichments there would be considerable evaporation, not only of the alkalis, but also of Fe, Si, and Mg (Alexander and Desch 2006; Fedkin and Grossman 2007; Fedkin et al. 2008).
Hewins et al. (2010) have inferred that up to 50% of the Na was lost from type II chondrules at peak temperatures. A 50% loss of Na, rather than 10% as assumed by Alexander et al. (2008), may be allowable by the chondrule data, but 90% loss is almost certainly not. Sodium losses of 50 and 90% would imply, respectively, roughly one and two orders of magnitude lower chondrule number densities and solid enrichments than those shown in Fig. 2. It makes relatively little difference to the estimated solid/gas ratios if it is assumed that chondrule phenocrysts formed at several hundred degrees of undercooling, either because crystallization began during the heating-up stage or there was rapid cooling and few crystal nuclei present. The solid enrichments are still uncomfortably high for astrophysical models. Even at PH2 = 10−4 bars and 90% Na loss, chondrule number densities at near-liquidus temperatures would be more than an order of magnitude higher than the upper limit based on compound chondrule frequencies. In the shock models, chondrule peak temperatures both increase with increasing chondrule density. In the model of Desch and Connolly (2002), the cooling rate (K h−1) is almost linearly proportional to the solar-normalized dust enrichment factor (solid/gas ratio) with a slope of close to unity. Extrapolation to the solid enrichments shown in Fig. 2 (≥106) would imply cooling rates of a similar order, which is orders of magnitude faster than current cooling rate estimates for chondrules. However, at present it is not clear that current shock models are applicable to such high chondrule densities.
There are other chemical indicators, in addition to Na, that can be used to place some constraints on the solid enrichments during chondrule formation. For instance, at what might now be considered relatively modest solid enrichments, 1000 times solar at a total pressure of 10−3 bars, type II chondrule phenocrysts would all have very forsteritic cores (Alexander et al. 2008), but this is not the case for actual chondrules. Major element zoning in chondrule phenocrysts is consistent with essentially closed-system behavior of moderately volatile elements like Fe, Mn, and Cr, again requiring high solid densities. Estimates of the conditions necessary to achieve closed system behavior of FeO in type IA and IIA silicate chondrules are shown in Fig. 3.
The solid enrichments in Figs. 2 and 3 are minimum estimates because metal was not included in the calculations. The presence of metal, which is abundant in FeO-poor type I chondrules but rare in FeO-rich type IIs, would reduce the fO2 of the vapor and increase the vapor pressures of Na, Fe, and other elements whose oxides in the melt mostly dissociate on evaporation (Alexander et al. 2008).
Type I chondrules dominate in carbonaceous chondrites and tend to have lower Na contents than type II chondrules. Could they have formed at lower chondrule densities than type IIs? All but the most FeO-poor chondrules seem to have behaved as roughly closed systems for Na (Alexander et al. 2008). For the most FeO-poor chondrules, low Na abundances and the possible presence of relict cores in most phenocrysts make it difficult to determine whether Na was lost from them. However, Fe metal is abundant in these chondrules and its presence can also be used to place constraints on chondrule densities. The texture of the metal suggests that it was fully molten, which sets a minimum temperature of formation for these chondrules of ≥1500 °C (Connolly et al. 2001). Figure 4 shows the dust enrichments needed to evaporate only 10% of the metal, assuming a CI dust composition, but with all Fe and Ni as metal and all S in the vapor. The equilibrium vapor pressure of the metal, calculated using the model of Chuang et al. (1986), is independent of PH2, or other gaseous components. Consequently, the solid/gas enrichment is inversely proportional to the total pressure, but not to PH2 because at low pressure PH becomes increasingly important.
Evidence for evaporation of metal during chondrule formation has not been reported for most chondrites. There is evidence for evaporation and recondensation of metal in the CR chondrites (Humayun et al. 2010; Connolly et al. 2001), although the fraction of metal that was evaporated is not clear. For 50% metal evaporation at near chondrule liquidus temperatures and a PH2 of 10−4 bars, the dust enrichments that are needed are very high and are only slightly more than an order of magnitude lower than those required to keep type IA and IIA chondrules essentially closed to Na loss at near their liquidus temperatures.
The highly volatile element S may not have evaporated from all chondrules (Hewins et al. 1997; Rubin et al. 1999; Tachibana and Huss 2005), although it is possible that some or all of the S in chondrules entered after cooling. If the S was not completely lost from chondrules, this may ultimately place even more stringent constraints than Na on solid/gas enrichments during chondrule formation. However, to use S to constrain the solid/gas ratios, it is necessary to know the S contents in the Fe-Ni-S melts, and this is hard to measure because of the heterogeneous exsolutions of sulfides during crystallization.
Compound and Nonspherical Chondrules
Given the high inferred chondrule densities, one might expect high abundances of compound chondrules in chondrites. The probability of forming a compound chondrule has been discussed in a number of papers (Gooding and Keil 1981; Ciesla and Hood 2002; Desch and Connolly 2002; Desch et al. 2005). If all chondrules had similar sizes and random relative velocities, the number of collisions that a typical chondrule will experience in a certain time interval is given by
where Dc (m) is the chondrule diameter, nc (m−3) is the number density of the chondrules, νc (m s−1) is their relative velocity, and t (s) is the length of time during which the chondrules can form compound chondrules.
Compound chondrules seem to make up only a few percent of chondrules (Wasson et al. 1995; Rubin 2010). Several things may have mitigated against the formation of easily recognized compound chondrules. To form compound chondrules, the chondrules must have had significant relative velocities, but not so large that a collision resulted in fragmentation. Turbulence and gas drag are two potential sources of relative velocities. Gas drag associated with shocks, or in otherwise rapidly heated and expanding regions, would have produced differential velocities between chondrules of different sizes and/or densities for brief periods of time until the chondrules and gas matched their velocities (minutes). In shocks or some other impulsive heating event, the large differences in velocity would occur when temperatures were near their peak but would quickly be damped. Largely molten chondrules that did collide would probably merge rather than form compound chondrules. Thus, if there were significant numbers of collisions when temperatures were near their peak, chondrule size distributions may not faithfully record the size distributions of their precursors.
Only in the temperature range where chondrule mesostases were very viscous but not solid/brittle, presumably as temperatures approached the mesostasis glass transition temperatures, could compound chondrules have formed. Even then, compound chondrules would not have stuck together after every collision—perhaps only in essentially head-on collisions would compound chondrules have formed. Equation 3 does not include an efficiency factor that probably would depend on temperature, mesostasis composition and modal abundance, velocity, relative chondrule masses, and collision angle. It seems likely that often collisions would have resulted in deformation of the colliding chondrules rather than formation of compounds. Indeed, most chondrules are not the perfect spheres that one might have expected surface tension to produce in a largely molten object that formed in isolation in the nebula. Rubin and Wasson (2005) showed that nonspherical chondrules make up roughly 60% of chondrules in COs and OCs, and the figure may be higher if all chondrules could be viewed in 3-D (Ebel and Rivers 2007). Rubin and Wasson (2005) argued that the nonspherical chondrules formed by very low degrees of partial melting in the final chondrule-forming event that they experienced. However, it seems possible that these irregular chondrules formed in typical events and are the products of collisions that occurred when the cooling system approached the glass transition temperatures of the chondrules’ mesostases, and the high viscosities of the mesostases prevented them from relaxing back to spherical shapes. Also, the textures that have previously been interpreted as evidence for hot accretion of chondrites (Hutchison et al. 1979, 1980) might instead be the products of chondrule-chondrule collisions in densely populated formation regions.
Ultimately, it will be important to make realistic estimates of the numbers of compound and deformed chondrules that would be expected as a function of chondrule density, but this will depend on a number of still poorly understood parameters that go into the efficiency factors for forming compound and nonspherical chondrules. Nevertheless, if we assume an efficiency factor of 1, p =0.6, νc = 0.1 m s−1, Dc = 1 × 10−3 m, and t = 3600 s (100 K interval at a cooling rate of 100 K h−1), we obtain a first order chondrule number density estimate of approximately 375 m−3. The relative velocity is that expected for a moderately turbulent disk, but this would be damped out quite rapidly by gas drag and chondrule-chondrule collisions unless there was a source of turbulence (or other causes of relative velocities between chondrules) in the chondrule forming regions. An approximate 100 K temperature interval during which deformed chondrules produced by collisions would be preserved was determined by: (1) estimating a glass transition temperature (Tglass) of 1060 K (Tangeman and Lange 2001) for a type IA chondrule mesostasis composition (Table 1), and (2) estimating the temperature at which the relaxation time for a deformed chondrule (Rubin and Wasson 2005) exceeds that of the cooling time to the glass transition. The latter is given by
where η (N s m−2) is the dynamic viscosity of the mesostasis and γ is its surface tension (∼0.36 N m−1; Walker and Mullins 1981). The assumption is that below the glass transition temperature the chondrules do not deform after a collision.
In calculating the relaxation times, the viscosity of the mesostasis was estimated using two models (Shaw 1972; Giordano et al. 2008) that gave very different viscosities at low temperatures, implying temperature intervals of 30–125 K for a cooling rate of 100 K h−1. This analysis does not take into account the large mass fraction of crystals that is likely to be present at these low temperatures, which will probably inhibit the relaxation of a distorted chondrule and thereby increase the temperature/time interval over which nonspherical chondrules can be preserved. However, taking into account the crystals is beyond the scope of this paper. The type IA mesostasis composition was chosen because it had high Ca and Al contents and a roughly chondritic Ca/Al ratio (i.e., it had experienced little CPX crystallization). Once the CPX starts to crystallize, the silica content of the mesostasis and its viscosity tends to rise rapidly. In the future, more detailed estimates of chondrule densities using the frequencies of compound and nonspherical chondrules will have to take into account the evolution of the mesostasis composition during cooling that can be particularly marked for type II chondrules.
The enrichment in the solid/gas ratio, relative to solar, as a function of chondrule number density is roughly given by
where P is the PH2 (bars) and the assumed chondrule density is 3 × 103 kg m−3. A chondrule number density of 375 m−3 represents a solid enrichment of about 5 × 104 relative to solar at a PH2 of 10−4 bars and T = 1000 K, which while high is still much less than required to explain the behavior of Na in chondrules at similar PH2 (Fig. 2). However, this estimate is almost certainly a lower limit for these assumed conditions because: (1) the efficiency factor is unlikely to be one for making compound chondrules or sufficiently deformed chondrules in an impact, (2) the frequency of nonspherical chondrules in chondrites may have been underestimated because of the absence of the third dimension in thin section studies, and (3) many nonspherical chondrules may have experienced more than one collision. On the other hand, chondrule sizes in most chondrites are typically about half those assumed above (Rubin 2010), which would reduce the dust enrichments by a third if only the chondrule number density was also allowed to change.
Clearly there are large potential uncertainties in the estimates of chondrule number densities. Until these estimates can be made more realistically, and we have a better understanding of the frequency of compound chondrules and the origins of the irregular shapes of most chondrules (i.e., collisions versus low degree partial melting), these textural properties of chondrules cannot be used to constrain chondrule densities during formation with any great certainty.
One Formation Event or Many Per Chondrite Group?
Alexander et al. (2008) pointed out that if chondrule number densities in their formation regions were as high as required by essentially closed-system behavior of Na and if the sizes of the formation regions were like those estimated by Cuzzi and Alexander (2006), i.e., 150–6000 km across, the chondrule formation regions would be gravitationally bound if they survived the formation event. If not subject to any other destructive processes, then on cooling these regions would collapse relatively rapidly to form asteroidal-sized objects. Very dense regions can form spontaneously in turbulent disks (Cuzzi et al. 2010), but the abundance and ubiquity of chondrules in chondrites hint at a link between chondrule and chondrite formation. If there was no link, chemically primitive, chondrule-free meteorites (but perhaps rich in chondrule precursors) should be far more common. Could chondrule formation have promoted rapid collapse of a region, while chondrule-free regions collapsed more slowly and were generally destroyed by other disk processes? Answering this question is beyond the scope of this paper. However, the relatively rapid collapse of gravitationally bound regions after chondrule formation at first sight seems to be in conflict with several petrologic and chemical features of chondrules. For instance, how was the diversity of chondrule compositions and textures that are found within a chondrite produced? Why is this diversity so similar in all meteorites from a single chondrite group when their metamorphic grades suggest that they formed at very different depths (and therefore times) within their parent body? Matrix and CAIs would not have survived chondrule formation. Can they be introduced into the collapsing regions on short enough time scales and uniformly enough to maintain the uniform bulk compositions and petrology of meteorites from the same group?
The Diversity of Chondrule Compositions and Textures
It is still not clear how much diversity a single formation event could generate. Chondrule chemistries and textures will have depended on: the composition and grain size of their precursors, the formation conditions, and the size of the formation region. Is it possible that the range of chondrule textures and compositions found in individual chondrites were produced in a single event if chondrule precursors and conditions (e.g., solid/gas ratio) were very heterogeneous?
Cuzzi and Alexander (2006) showed that in large formation regions, gas diffusion rates would have limited the sizes of the volumes within which chondrules could have communicated chemically with one another. Consequently, within a large formation region there could have been multiple sub-regions that experienced different conditions and produced chondrules with different compositions and textures. Although many of the parameters that determined chondrule textures are qualitatively understood, it is still not possible to predict chondrule textures given a set of formation conditions and precursor properties. However, it seems likely that a range of textures would have been generated in a single event since peak temperatures, cooling rates and solid densities will not have been uniform across a formation region. A range of textures could even be produced in a sub-region if, for instance, the precursors had different liquidus temperatures, mineralogies, and/or initial grain sizes. It is argued below that re-equilibration between chondrules of different compositions is fairly rapid at near-liquidus temperatures. The fact that in about 40% of compound chondrules the two chondrules have distinct compositions and textures from one another (Wasson et al. 1995) is evidence that there were sub-regions with different conditions within chondrule forming regions, and that these regions began to mix once temperatures (and re-equilibration rates) had fallen to the point that chondrules had largely solidified but were still plastic.
The extent to which an initial chemical diversity in the precursors could be preserved in a sub-region will have depended on the rates of re-equilibration between chondrules. Chondrules do show small variation in isotopic mass fractionation in Mg, Fe, Si, and O (Davis et al. 2005), which may be the result of incomplete re-equilibration. Figure 5 shows the results of a simple model in which a type IA chondrule equilibrates in a region that is dominated by type IIA chondrules. The calculations used the model of Alexander (2004) and assumed isothermal conditions of 1600 °C, a total pressure of 10−4 bars, a solid/gas ratio of 104 times solar, and that the chondrules were fully molten. Thus, the calculations should be regarded as illustrative rather than quantitative. Nevertheless, the calculations suggest that chondrules can approach chemical equilibrium on fairly short time scales. In the calculations, significant isotopic variations persist for some time, and both the sign and the size of the variations differ from element to element. Of course, in nature the rates of equilibration will have depended on many parameters (e.g., thermal history, composition, crystallization rates, etc.), but the calculations do provide an explanation for why chondrules preserve some isotopic variations and why, in general, those variations do not exhibit any systematic behavior among the different elements. Re-equilibration between chondrules of different composition (e.g., SiO2-rich and SiO2-poor) might even explain features of many type I chondrules that have been attributed to SiO2 recondensation (Tissandier et al. 2002; Libourel et al. 2006).
Relict Grains, Igneous Rims, and Matrix
Perhaps the most problematic petrologic observations for models in which chondrule formation in very dense regions induced rapid formation of a planetesimal is the evidence for recycling of chondrule material in the form of relict grains that were mostly inherited from a previous generation of chondrules (Nagahara 1981; Kracher et al. 1983; Steele 1986; Jones 1996; Ruzicka et al. 2008) and igneous rims that appear to have formed by melting of a previously formed accretionary rim (Rubin 1984; Rubin and Wasson 1987; Krot and Wasson 1995; Rubin and Krot 1995). Recycling requires that either: (1) the recurrence time of chondrule forming events was shorter than the collapse times, or (2) chondrule formation often failed to induce collapse and that chondrule forming regions dispersed and reformed multiple times, but did so on time scales that were short compared to mixing time scales in the asteroid belt.
The presence in chondrites of pristine CAIs, matrix, and chondrules with accretionary rims is also problematic. Primitive rim and matrix material, including organics and presolar grains, could not have been even briefly heated to temperatures of more than a few hundred degrees centigrade. The water responsible for aqueous alteration in OCs and CCs was probably accreted as ice along with the rims and matrix, in which case the rim/matrix material may never have seen temperatures above approximately 170 K. Thus, the rim and matrix material must somehow have largely avoided being heated by chondrule formation. CAIs often have lower liquidus temperatures than chondrules and would have equilibrated as rapidly as chondrules during chondrule formation. Like the matrix, the CAIs in chondrites must have avoided chondrule formation. Alexander et al. (2008) suggested that even relatively low levels of turbulence would have allowed dust and CAIs to diffuse into a chondrule formation region after it cooled but before it collapsed. Whether accretionary rims of sufficient thickness could form on these time scales is less clear. This turbulent mixing would also explain why the distribution of chondrule types in meteorites from a given group are so similar despite the fact that they may have come from quite different depths within their parent body.
The ages of CO and OC chondrules, based on 26Al measurements, are very similar and each have total spreads of about 1.5–2 Myr (see the discussion of Kurahashi et al.  and the references therein). On the other hand, physically and chemically the CO and OC chondrules are quite distinct. Equally problematic are the differences in the distributions of refractory inclusions (CAIs and AOAs) between COs and OCs—refractory inclusions are relatively abundant in COs but are rare in OCs (Hezel et al. 2008). Is it possible to maintain these distinct chondrule and inclusion populations for so long when mixing time scales for the asteroid belt could have been ≤0.1 Myr and certainly <1 Myr (Alexander 2005; Cuzzi et al. 2010)? Cuzzi et al. (2010) estimated an upper limit of approximately 1 Myr based on the mixing time scale between chondrule populations that formed at 2 and 4 AU. Were the OC and CO formation regions so well separated? The OCs show evidence for aqueous activity (Hutchison et al. 1987; Alexander et al. 1989; Grossman et al. 2000, 2002), so they presumably accreted some ice and therefore formed beyond the snowline. Whether there was aqueous activity in the COs is less clear (Brearley 2006). Assuming that the abundance of water/ice in a chondrite is a crude measure of its formation distance relative to the snowline, it seems doubtful that the COs formed very far beyond it. In this case, the distance between CO and OC formation locations may have been a small fraction of the width of the asteroid belt, and mixing time scales would have been much shorter than 1 Myr.
Given that the conundrum posed by the OC and CO chondrules is largely a consequence of their Al-Mg ages, it is important to take a critical look at these ages. In particular, (1) do the measured ages really require that chondrules in COs and OCs formed over 1.5–2 Ma, and (2) could some or all of them have been disturbed by parent body processes?
To address this first question, in Fig. 6 histograms of the inferred 26Al/27Al ratios for individual chondrules are shown. For both the COs and OCs, the means and standard deviations of the populations were calculated and used to plot normal distributions for comparison. Both the CO and OC chondrule histograms can be approximated reasonably well by normal distributions (Fig. 6). In the case of the OCs, the standard deviation of the reported initial 26Al/27Al ratios (0.19 × 10−5) is almost exactly the average 1σ error of the individual isochrons. Thus, the OC measurements are consistent with all the chondrules having a single age, or at least a narrow range of ages, and with the observed spread reflecting the uncertainties in the measurements and the isochron fits. However, the OC data in Fig. 6a does not include three outlying chondrules with 26Al/27Al ratios of 1.4–2.3 × 10−5, albeit with large 1σ errors of approximately 0.35 × 10−5, nor does it include chondrules for which there were only upper limits. For the CO chondrules, the standard deviation (0.17 × 10−5) is very similar to that of the OCs, although the average of the quoted 1σ measurement uncertainties is somewhat less at approximately 0.11 × 10−5. Whether the CO measurement/isochron uncertainties have been underestimated or there was a significant range of CO chondrule formation ages is explored further next.
The dating of rocks is typically accomplished by using measurements of a number of phases with distinct parent-daughter ratios to construct isochrons. However, in unaltered ferromagnesian chondrules from primitive meteorites, the glass/plagioclase is the only high Al/Mg phase. Thus, the difficulty of using the Al-Mg system to date typical chondrules, particularly if as is often the case the glasses are fairly uniform in composition, is that one normally has what are essentially two-point isochrons. This means that useful isochron ages can only be obtained from typical ferromagnesian chondrules by either including olivine/pyroxene analyses with the glass or plagioclase analyses, or by forcing the fitted Al-Mg isochron line through the origin (normal 26Mg/24Mg). When there are few glass/plagioclase analyses and/or little spread in their Al/Mg ratios, the effect of including the much more precise olivine or pyroxene analyses with their very low Al/Mg ratios is essentially the same as forcing the fit through the origin. In those cases where ion probe analyses of chondrules do exhibit a significant range in Al/Mg ratios, it seems likely that the inclusion of variable amounts of CPX crystallites in the analyses has at least contributed to this range. CPX crystallites are almost ubiquitous in chondrule mesostases and it can be difficult to avoid them even in electron probe studies (even apparently crystallite-free areas may have them just below the surface), let alone in the 25–30 μm diameter and several micrometers deep analysis pits that are typical for ion microprobe analyses. Certainly, few studies have used electron probe images and analyses to demonstrate that the Al/Mg variations recorded in the ion probe measurements are indigenous to the glass. Indeed, to date the causes of any variations in chondrule glass compositions in primitive meteorites (i.e., whether they are primary or secondary) have received very little attention and need to be better understood in order to interpret the isochron data with any certainty.
Including olivine/pyroxene analyses, using mixed glass/CPX analyses or forcing the fit through the origin when calculating an isochron are only valid approaches if the glass/plagioclase data are undisturbed. Often, the glass/plagioclase data alone are almost equally consistent with a horizontal line as they are with the isochron. A horizontal line would be indicative of a late disturbance (after all 26Al has decayed away) that has modified the Al/Mg ratios and/or has re-equilibrated the Mg isotopes in the glass/plagioclase and possibly exchanged them with the matrix. The errors for the glass/plagioclase analyses are generally too large and the range of Al/Mg ratios too small for them to give useful isochrons for individual chondrules on their own. However, if all chondrules from a meteorite group had similar initial 26Al/27Al ratios, as seems possible from Fig. 6 for OC and for CO chondrules, the glass/plagioclase data for all chondrules can be combined to give a more precise bulk isochron.
The OC glass data have been combined in Fig. 7 and an isochron fitted using ISOPLOT (Ludwig 2003)—it was decided to only use the data from a single group to minimize the possibility of systematic inter-laboratory differences. The error-weighted fit to the OC glass/plagioclase data gave a 26Al/27Al = 0.72 ± 0.08 × 10−5 and a normal initial 26Mg/24Mg ratio (δ26Mg = 0.07 ± 0.31‰), although the error is large (all errors are 1σ). The bulk 26Al/27Al ratio given by all the glass/plagioclase analyses is consistent with the average 26Al/27Al ratio for all the individual isochrons (Fig. 6). This lends support to the suggestion that all the OC chondrules have similar Al-Mg ages.
For the COs, a weighted fit to all the plagioclase data gave a 26Al/27Al = 0.44 ± 0.05 × 10−5 and a slightly 26Mg enriched initial 26Mg/24Mg ratio (δ26Mg = 0.61 ± 0.15‰). Unlike for the OCs, the 26Al/27Al ratio inferred from the individual analyses is significantly lower than the average of the individual chondrule isochrons (Fig. 6). The lower 26Al/27Al and the positive δ26Mg for the intercept, is suggestive of a resetting of the Al-Mg systems in the CO plagioclases after some 26Al decay but while 26Al was still alive. Any redistribution of the Mg isotopes could not have involved significant exchange with the ferromagnesium minerals in the chondrules or with the matrix, otherwise the isochron intercept would have been unresolvable from the normal Mg isotopic composition.
If the Al-Mg systems in CO chondrule plagioclases were reset, is it possible that the OC chondrules have also been reset while 26Al was still alive? There is certainly evidence that the higher the petrologic grade of the meteorite, the lower the 26Al/27Al ratios tend to be (Huss et al. 2001). Even in Semarkona, there is abundant evidence that the chondrules did not always remain closed systems during parent body processing—chondrule glasses appear to have exchanged alkalis, halides, H/OH, Ca and even O with the matrix (Hutchison et al. 1987; Grossman et al. 2000, 2002; Alexander and Grossman 2005; Kita et al. 2008, 2010). If Ca was mobilized in some Semarkona chondrule glasses, it seems likely that Mg was too. Kita et al. (2010) showed that except when relict grains are present, the minerals within a chondrule (olivine, low-Ca pyroxene, and high-Ca pyroxene) have O isotopic compositions that are within error of one another. The glass, on the other hand, always has higher δ18O and Δ17O values, which Kita et al. (2010) attributed to O isotopic exchange with a fluid during parent body alteration/metamorphism. Parent body processes could also initiate the crystallization of plagioclase, albite, or other secondary phases in the glass. Exclusion of Mg during formation of feldspar would lead to the alteration of the Al/Mg ratios in the feldspar and the residual glass, but would not modify the Mg isotopes.
Three of the chondrules (CH4 and CH36 from Semarkona, and K27 from Krymka) whose phenocrysts and mesostases were measured for their O isotopes by Kita et al. (2010) had previously been measured for their Al-Mg systematics (Tachibana et al. 2003; Kita et al. 2005). All three had measurable 26Mg excesses in their glasses, and Kita et al. (2010) concluded that despite the considerable O isotope exchange, the Al-Mg systematics in the glass had not been disturbed. If true, this would be a remarkable result since, while the O and Mg isotope diffusion rates in chondrule glasses are not known, O generally diffuses more slowly than Mg in silicates. Indeed, there is a tendency for the inferred 26Al/27Al ratios for these chondrules (CH4 = 9.0 ± 1.6 × 10−6, K27 = 6.1 ± 1.7 × 10−6, CH36 = 6.6 ± 1.9 × 10−6) to decrease as the difference in δ18O between minerals and glass increases (respectively, 1.3–2.8‰, 7.9‰, and 13.1‰). An alternative way to look at the extent of O isotope exchange is to compare the Δ17O of the glasses (CH4 = 1.6‰, K29 = 3.3‰, and CH36 = 4.8‰) with the magnetites that formed from the altering fluid in Semarkona (∼5‰—Choi et al. 1998). For chondrule CH36 in particular (average Δ17O of pyroxene+olivine is 0.6‰), there appears to have been almost complete O isotope exchange with the fluid. However, the small number of chondrules studied and the uncertainties in the measurements means that the existence of an inverse correlation between the 26Mg excesses and the O isotopes of the glass remains to be definitively demonstrated. Clearly, more extensive correlated petrologic, and O and Al-Mg isotopic studies of chondrules in the most primitive meteorites are needed to resolve these issues.
The fact that in Semarkona chondrules there has been O isotopic exchange in the glass but some 26Mg excesses remain, could be explained if the isotopic exchange was incomplete or if it took place while 26Al was still alive. In other words, as with the CO chondrules, the Al-Mg age of the OC chondrules in Figs. 6 and 7 is a reset age recording the time of alteration and not the time of formation. Unlike the COs, the chondrule glasses must have exchanged with a large Mg reservoir, such as matrix, to explain an intercept that is within error of the origin. As noted by Alexander (2005), if the chondrule ages have been reset to varying degrees and/or the spread in individual chondrule ages largely reflects the errors in the internal isochrons, the apparent conflict between the need to keep OC and CO chondrules separate for 1.5–2 Myr and the necessity of having a turbulent nebula that will, by its very nature, rapidly mix the disk goes away. It also means that the OC and CO chondrules probably formed earlier than currently recognized, and possibly at quite different times. The latter would also ease, although not fully resolve, the mystery of why the CAI abundances in OC and CO chondrites are so different.
How much earlier the CO chondrules may have formed is hard to constrain, but there are some constraints for the OC chondrules. The three outlier OC chondrules not included in Fig. 6 had 26Al/27Al ratios of about 2.3–1.4 × 10−5, or about 0.8–1.3 Myr after CAIs, and may provide a lower limit on the formation ages of OC chondrules. No chondrules with significantly older ages than those in Fig. 6 have been reported for the COs. Kleine et al. (2008) estimated the ages for H chondrite chondrules to be 1.7 ± 0.7 Myr after CAIs based on Hf/W systematics. This would suggest 26Al/27Al ratios when they formed of about 1 + 0.9/−0.5, which is consistent with both the highest initial 26Al/27Al ratios among OC chondrules and the average/bulk age of the chondrules (Figs. 6 and 7), and thus does not provide a very useful constraint. To try to match various estimates of metamorphic cooling rates at different temperatures, Trieloff et al. (2003) and Harrison and Grimm (2010) estimated that the H chondrite parent body must have accreted with an average 26Al/27Al ratio of approximately 4 × 10−6 and approximately 7.5 × 10−6, respectively. These values are close to the average ratio inferred for OC chondrules in Figs. 6 and 7. However, at these 26Al abundances, an OC parent body would not undergo even partial melting and differentiation (Trieloff et al. 2003; Kunihiro et al. 2004; Harrison and Grimm 2010). Yet the IIE and IVA irons have been linked to the H and LL chondrites, respectively. Kunihiro et al. (2004) argued that 60Fe could not have provided the extra energy to drive melting and differentiation, so they suggested that impacts were the additional heat source. On the other hand, if most chondrule 26Al/27Al ratios have been disturbed, the 26Al/27Al ratios at the time of accretion of the OC parent bodies would have been significantly higher, perhaps as high as 1–2 × 10−5. If 26Al/27Al ratios were this high at the time of accretion, at least partial melting and differentiation could have occurred, although the details of the thermal histories will have depended on the sizes of the parent bodies amongst other poorly known parameters. Nevertheless, the absence of achondrites that are obviously related to the OCs is problematic to any model that invokes much higher 26Al/27Al ratios in OC chondrules than implied in Figs. 6a and 7a.
Villeneuve et al. (2009) reported considerably higher precision Al-Mg data than in previous studies for 14 chondrules from the OC Semarkona. Because of the higher precision that they were able to achieve, they focused on chondrule mesostases with much less extreme Al/Mg ratios than in previous studies. From their data, Villeneuve et al. (2009) concluded that Semarkona chondrules formed in several distinct events that occurred between about 1.2 and 4 Myr after CAI formation. Despite the higher precision, the distribution of initial 26Al/27Al ratios that they inferred for Semarkona chondrules is very similar to that of the earlier, less precise studies (Fig. 6a). If correct, the Villeneuve et al. study calls into question our present interpretation of the earlier studies and poses a more severe challenge to astrophysical models that require an even modestly turbulent nebula.
However, the Villeneuve et al. (2009) study is open to the same potential problems as the earlier studies. The large range in mesostasis Al/Mg ratios that they report for some chondrules suggests the inclusion of CPX crystallites in the analyzed areas. Indeed, some measurements they reported and included in their isochron fits had ratios of less than one. In the detailed study of 28 Semarkona chondrules, involving approximately 180 electron probe analyses of, where possible, microcrystallite-free chondrule glasses (Alexander et al. 2008), no glass analyses had Al/24Mg ratios of less than 1.6. The Villeneuve et al. isochrons were also calculated with the inclusion of multiple olivine and low-Ca pyroxene analyses. Table 2 compares the initial 26Al/27Al ratios and the initial Mg isotopic compositions inferred by Villeneuve et al. for the analyzed chondrules with those obtained by only fitting to the mesostasis analyses and only including analyses with 27Al/24Mg>1.5. As can be seen, the uncertainties for the mesostasis-only fits are much larger and for many chondrules the 26Al/27Al ratios are not significant or just barely significant at the 95% confidence level. This clearly shows that the precisions of most of the isochrons reported by Villeneuve et al. (2009) are primarily determined by the inclusion of the olivine/pyroxene data, which is valid only if the glass has remained undisturbed since chondrule crystallization ceased. If disturbance of the glass was a closed system, then an isochron that included the olivine/pyroxene and an accurate bulk glass composition would give the age of the chondrule. However, obtaining an accurate bulk glass composition may be difficult and, as discussed above, there is evidence that the disturbance of the OC chondrule glasses was open system.
Table 2. The initial 26Al/27Al ratios, initial Mg isotopic compositions, and mean square of weighted deviates (MSWD) values for chondrules analyzed by Villeneuve et al. (2009). The left hand set of results are from the ISOPLOT fits reported by Villeneuve et al. (2009) that included olivine and pyroxene analyses. The right hand set of data are for ISOPLOT fits to the chondrule mesostases only, and included only those mesostasis analyses with 27Al/24Mg >1.5. All uncertainties are 2σ (95% confidence).
26Al/27Al (1 × 10−6)
26Al/27Al (1 × 10−6)
5.07 ± 1.8
−0.00 ± 0.01
2.9 ± 21
0.03 ± 0.41
11.1 ± 1.4
−0.01 ± 0.01
14.1 ± 8.1
−0.06 ± 0.12
7.2 ± 2.0
−0.01 ± 0.01
16.5 ± 17
−0.17 ± 0.34
4.8 ± 1.1
−0.01 ± 0.03
5.1 ± 2.7
−0.04 ± 0.27
6.6 ± 1.2
−0.01 ± 0.01
5.6 ± 2.1
0.06 ± 0.14
3.4 ± 1.2
−0.01 ± 0.02
2.1 ± 1.1
0.06 ± 0.04
7.9 ± 1.1
−0.01 ± 0.01
6.2 ± 5.0
0.06 ± 0.19
3.0 ± 1.2
0.00 ± 0.01
0.9 ± 6.7
0.06 ± 0.17
7.2 ± 1.3
−0.01 ± 0.02
9.9 ± 3.2
−0.22 ± 0.22
8.9 ± 0.9
0.00 ± 0.01
10.3 ± 4.1
−0.11 ± 0.34
4.8 ± 1.0
0.00 ± 0.01
4.6 ± 3.3
0.01 ± 0.07
7.9 ± 2.9
0.00 ± 0.02
3.3 ± 12
0.09 ± 0.21
5.8 ± 1.0
0.00 ± 0.02
5.9 ± 1.7
−0.03 ± 0.17
16.2 ± 1.7
−0.02 ± 0.01
9.7 ± 8.4
0.15 ± 0.22
Also included in Table 2 are the mean square of weighted deviates (MSWD) values for the various fits. If the scatter in the data about a fitted line is that expected from the measurement uncertainties, the MSWD is unity. If the scatter about the fitted line is greater than expected from the measurement uncertainties, the MSWD > 1 and the larger the MSWD the greater the scatter. If the MSWD >> 1, either the measurement errors have been underestimated or the data do not belong to a single isochron (e.g., it is disturbed). If the MSWD << 1, either the measurement errors have been underestimated or some other process is causing a correlated variation in the data. As can be seen from Table 2, few fits to the Villeneuve data, whether they include the olivine/pyroxene data or not, have MSWD values that are close to one—they scatter to both significantly larger and smaller values. The range of MSWD values suggests that other factors, not just in situ 26Al decay, are responsible for the scatter in the data.
Thus, it is not clear that the higher precision data of Villeneuve et al. (2009) definitively demonstrate that there was a protracted period of OC chondrule formation. In support of this conclusion, Fig. 8 compares the Villeneuve et al. (2009) mesostasis results with the isochron calculated in Fig. 7 from the mesostasis data collected in earlier studies. As can be seen, most of the Villeneuve et al. (2009) points plot parallel to but slightly below the isochron. A small adjustment to the isochron that is well within the uncertainties would bring most points in Fig. 8 within error of the line. A fit to all the OC data was not carried out, partly because of the great disparity between the uncertainties of Villeneuve et al. and those from earlier studies. A fit to only the Villeneuve et al. mesostasis analyses (27Al/24Mg>1.5) is shown in Fig. 9. Most data plot within error of the line, but there is sufficient scatter to give a MSWD of 5.5. Such scatter is perhaps not surprising if, as argued above, the OC chondrules could have had a single Al-Mg age, or narrow range of ages, and they have subsequently experienced variable degrees of disturbance during parent body alteration.
Finally, the chondrule compositions of Villeneuve et al. (2009) are consistent, within the errors, with the evolution of their Mg isotopes in systems with approximately solar Al/Mg ratios and with an initial 26Al/27Al ≈ 5 × 10−5. The Mg isotopic composition of a phenocryst records the evolution of the Al-Mg system up to the time of the last melting event. A partial to complete resetting of chondrule glasses in Semarkona, would probably not alter the Mg isotopic compositions of the olivine and pyroxene phenocrysts because diffusion rates in the olivine are likely to be much slower than in an altering glass. If a chondrule formed at the same time or soon after CAIs, its olivine should have a resolvable negative 26Mg anomaly. Given the typical uncertainty of the olivine/pyroxene measurements, the Semarkona chondrules could not have formed with 26Al/27Al ≥ 2–3 × 10−5. This upper limit for the initial 26Al/27Al ratios of the measured Semarkona chondrules is entirely consistent with the highest inferred OC chondrule 26Al/27Al ratios and with the limits set by the thermal histories of the OC parent bodies (see previous).
Summary and Conclusions
Here we show that a range of geochemical indicators point to very high number densities and solid/gas ratios during chondrule formation. The low number of compound chondrules in chondrites seems inconsistent with this finding. However, it is argued that compound chondrules will only have formed at temperatures near the glass transition of the chondrule mesostases. At higher temperatures, the largely molten chondrules would have merged. At temperatures near the glass transition, chondrules are more likely to have collided with one another but not stuck, rather than to have formed compound chondrules. As the mesostases will be very viscous at these low temperatures, colliding chondrules are likely to have preserved some “memory” of any distortions they suffered in these collisions. The number densities implied by the abundance of nonspherical chondrules are high, although at present the estimates are very uncertain. Ultimately, if nebular models cannot account for such high chondrule number densities and the frequencies of compound and nonspherical chondrules, it will be necessary to explore in more detail other formation mechanisms, such as planetesimal impacts, that intuitively should make chondrules in very high density environments and with low relative velocities.
At the number densities implied by the geochemical constraints, if a chondrule-forming region survived a formation event the region is likely that it would have been gravitationally bound and would have collapsed quite rapidly to form a chondritic body. Estimates of the sizes of chondrule-forming regions are much larger than estimates of the diffusion distances in the gas that would have been possible on formation time scales. Hence, it is possible that the diversity of chondrule compositions and textures found in a single chondrite group could have been produced in a single formation event in subregions that were unable to chemically communicate. This diversity of chondrules could subsequently have been mixed by turbulence prior to collapse of the region to form a chondritic parent body. This mixing could also have brought in matrix material and CAIs that would not have survived chondrule formation. However, the evidence for recycling of chondrules and igneous rims around them requires that there was more than one chondrule-forming event prior to the formation of each parent asteroid.
Finally, we point out that there are a number of issues that must be resolved before the Al-Mg isochrons that have been reported for OC and CO chondrules can be used to estimate their formation ages. These issues include evidence for elemental and isotopic exchange with matrix, the possibility of secondary formation of feldspar in chondrule glasses, a generally poor understanding of the causes of elemental variations in chondrule glasses, and the possibly unjustified inclusion of olivine, pyroxene, and CPX-glass mixtures in the calculation of isochron ages. We argue that OC and CO chondrule Mg-Al ages are, at present, each consistent with a single or a narrow range of ages, and that the CO chondrule age, and possibly the OC chondrule age, could date parent body alteration rather than chondrule formation. If correct, this would remove the inconsistency between the astrophysical requirement that the nebula was turbulent, and the similar about 1–2 Myr range of 26Al ages of the otherwise physically and chemically distinct chondrules in CO and OC chondrites. A turbulent nebula would have transported material radially quite rapidly, which apparently has been confirmed by the discovery of chondrules- and CAI-like objects in at least one comet. If OC and CO chondrules formed in even a moderately turbulent asteroid belt over the same 1–2 Myr, they should have been well mixed.
Acknowledgments–– The authors thank A. Boss, F. Ciesla, J. Cuzzi, S. Desch, J. Grossman, H. Palme, and, most particularly, R. Hewins for their help, advice, and many stimulating discussions on this topic over the years. The experiments and ideas put forth by R. Hewins and his group have hugely influenced all attempts to constrain the processes involved in chondrule formation. The manuscript was significantly improved by the reviews of S. Russell and A. Davis.