The planetesimal bow shock model for chondrule formation: A more quantitative assessment of the standard (fixed Jupiter) case


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Abstract– One transient heating mechanism that can potentially explain the formation of most meteoritic chondrules 1–3 Myr after CAIs is shock waves produced by planetary embryos perturbed into eccentric orbits via resonances with Jupiter following its formation. The mechanism includes both bow shocks upstream of resonant bodies and impact vapor plume shocks produced by high-velocity collisions of the embryos with small nonresonant planetesimals. Here, we investigate the efficiency of both shock processes using an improved planetesimal accretion and orbital evolution code together with previous simulations of vapor plume expansion in the nebula. Only the standard version of the model (with Jupiter assumed to have its present semimajor axis and eccentricity) is considered. After several hundred thousand years of integration time, about 4–5% of remaining embryos have eccentricities greater than about 0.33 and shock velocities at 3 AU exceeding 6 km s−1, currently considered to be a minimum for melting submillimeter-sized silicate precursors in bow shocks. Most embryos perturbed into highly eccentric orbits are relatively large—half as large as the Moon or larger. Bodies of this size could yield chondrule-cooling rates during bow shock passage compatible with furnace experiment results. The cumulative area of the midplane that would be traversed by highly eccentric embryos and their associated bow shocks over a period of 1–2 Myr is <1% of the total area. However, future simulations that consider a radially migrating Jupiter and alternate initial distributions of the planetesimal swarm may yield higher efficiencies.


Chemical, petrologic, and isotopic analyses place a number of constraints on the processes by which chondrules formed and their environment. It is generally inferred that most chondrules formed when particles or aggregates suspended in solar nebula gas were melted by transient heating events (e.g., Boss 1996). Many chondrules have apparently been heated more than once and some contain recycled fragments of older chondrules, implying multiple heating events (e.g., Alexander 1994; Rubin and Krot 1996). The resulting partially or fully molten particles apparently solidified as free-floating objects before being accreted into planetesimals that became the parent bodies of chondritic meteorites. Petrologic evidence from furnace experiments (e.g., Hewins et al. 2005) implies that chondrules cooled more slowly than isolated objects radiating to free space. Their cooling rates, and the lack of isotopic fractionation of relatively volatile elements (Na, K, and Mg) imply that melting occurred in regions of high particle density (>10 m−3) with sizes of at least hundreds of km (Cuzzi and Alexander 2006). The Na contents of olivine crystal cores in chondrules appear to require even higher solids enrichments that may be difficult to accommodate in a normal nebular setting (Alexander et al. 2008; Fedkin et al. 2012).

Isotopic data indicate that most chondrules formed 1–3 Myr after CAIs, which are the oldest surviving solids (Connelly et al. 2008; for a review, see also Scott 2007). Isotopic evidence also indicates that the formation of chondrules in specific chondrite groups may have occurred at somewhat different times. For example, the Pb-Pb ages of CV chondrules are 1.66 ± 0.48 Myr after CAIs while those of CR chondrules are 2.4–3.0 Myr after CAIs (Connelly et al. 2008). One 26Al-26Mg age for an Allende CV chondrule is only about 200,000 yr after CAI formation (Bizarro et al. 2004). This range of ages may suggest that chondrule formation was a short-lived and recurrent process that produced chondrules in different chondrite groups and continued during most of the life span of the protoplanetary nebula (Connelly et al. 2008). In this interpretation, the near-absence of old chondrules with ages < 1 Myr after CAIs could be due to their incorporation into planetesimals that formed early and differentiated by radiogenic heating, effectively removing them from the available collection. The survival of large numbers of CAIs from this early period could then reflect their early and complete removal from the inner nebular disk, followed by temporary storage in another part of the nebula. However, efficient chondrule formation probably did not occur during the CAI formation epoch as many of those putative early chondrules would have survived as did the CAIs. Therefore, although the relative timing of CAI and chondrule formation is still being established, most of the available evidence suggests a significant time delay of about 1 Myr between CAI formation and the onset of efficient chondrule formation.

Transient heating events associated with the formation of the solar nebula (e.g., infall shocks, shocks in spiral density waves) or activity of the early Sun (X-ray flares, X-wind, FU Ori events) would have been more frequent and/or intense at early times, declining with time thereafter. The apparent time delay between CAI and the formation of most chondrules is therefore difficult to explain if heating events associated with these processes were responsible for the formation of most chondrules. One mechanism that can satisfy the latter constraint is the planetesimal nebular shock mechanism, which involves planetesimals perturbed into eccentric orbits in Jovian resonances (Hood 1998; Weidenschilling et al. 1998). Chondrules are hypothesized to have been melted in either bow shocks upstream of the eccentric planetesimals moving supersonically through the nebular gas (Marzari and Weidenschilling 2002) or in localized nebular shocks caused by collisions between resonant and nonresonant bodies. Order-of-magnitude calculations indicate that bow shocks would have dominated over collisional shocks (Hood et al. 2009). The timing of Jupiter’s formation is not well constrained observationally but it must have occurred prior to dissipation of the nebula as it is composed mainly of hydrogen and helium. The median lifetime of protostellar disks is about 3 Myr (Haisch et al. 2001). A formation time as early as 1 Myr after CAIs allowed whether Jupiter is formed by the core accretion mechanism (Alibert et al. 2004; Hubickyj et al. 2005) or by the gravitational instability mechanism (Boss 2000, 2002). Therefore, the bow shock mechanism can potentially satisfy the time delay constraint.

One potential problem with the bow shock model is that the size of the heated region is comparable to that of the supersonic planetesimal that produces the shock, and the characteristic time scale for the heating is this length divided by the velocity. Cooling rates through crystallization temperatures inferred from furnace experiments are only 10–1000 K h−1 (Hewins et al. 2005). Such slow inferred cooling rates can be matched only if the planetesimals producing the shocks were large—more than 1000 km in diameter (Ciesla et al. 2004; Morris et al. 2012). However, Marzari and Weidenschilling (2002) showed that individual planetesimals as small as about 50 km can become trapped in Jovian resonances and could, in principle, have attained eccentricities large enough to produce strong shocks. If chondrules were produced by bow shocks, the lack of chondrules with very short cooling times implies that asteroid-sized bodies did not attain supersonic velocities, and/or that the eccentric population was dominated by large (possibly Moon- to Mars-sized) planetary embryos. For this reason, it is important to investigate whether planetesimal orbital evolution and accretion simulations naturally predict that primarily the largest bodies are perturbed into orbits eccentric enough to produce chondrules during resonance passages.

Regions of enhanced number densities of chondrules (or their precursors) with sizes and densities sufficient to prevent volatile depletion and isotopic fractionation (about 10 m−3) could have been produced by concentrations in eddies in a turbulent nebula (Cuzzi et al. 2008; Chambers 2010) or by settling to the midplane (e.g., Weidenschilling 2010). Settling to the midplane would imply a very low degree of turbulence in the nebula. An early turbulent stage may be inferred from the observed presence of CAIs in both chondrites and comets (Yang and Ciesla 2012), but that turbulence may have decayed approximately 1 Myr before the onset of efficient chondrule production. A setting for chondrule formation in turbulent eddies is allowed in the context of the planetesimal bow shock model but a setting close to the midplane may have been more efficient as the maximum gas-planetesimal relative velocities, and the strongest bow shocks, occur during midplane passage (e.g., Hood 1998). In any case, neither turbulent eddies nor settling to the midplane is likely to produce solids enrichments sufficient to prevent the evaporation of Na and other alkalis during chondrule formation (e.g., Fedkin et al. 2012b). Morris et al. (2012) have recently proposed that chondrule formation in large planetesimal (planetary embryo) bow shocks could be consistent with the latter constraint provided that the embryo was large enough to have an outgassed (e.g., magma ocean) atmosphere. Thus, it is again important to investigate whether mainly large (Moon- to Mars-sized) embryos were preferentially perturbed into highly eccentric orbits during Jovian resonance passages.

A second potential problem for the bow shock model is that strong bow shocks produced by highly eccentric planetesimals may not have been numerous or widespread enough to explain the high abundance of chondrules in chondrites. In a previous assessment (Hood et al. 2009), this question was investigated using the planetesimal orbit simulations of Weidenschilling et al. (1998) and Marzari and Weidenschilling (2002), which considered only individual test bodies evolving sunward in the presence of gas drag and Jupiter’s perturbations. No collisions or close encounters with other planetesimals were assumed to occur and planetesimal diameters were a few hundred km or less. For bodies of this size, gas drag was effective in moving them gradually inward with low eccentricities until they reached the nearest commensurability resonance inside their starting orbit. This condition is favorable for capture into resonance, so nearly all bodies became trapped in resonances and attained high eccentricities. Moreover, Hood et al. (2009) assumed a minimum shock velocity of about 5 km s−1, corresponding to planetesimal eccentricities at 3 AU of e > 0.3, would be sufficient for melting of chondrule precursors. Using this methodology, it was estimated that 20–30% of bodies in the primordial asteroid belt between 2 and 4 AU would have been perturbed into orbits sufficiently eccentric to produce chondrule-forming shocks at any given time. Assuming a mean planetesimal radius of 500 km, a total planetesimal mass in the primordial belt of three Earth masses, a total precursor particle mass in the nebula of about 1 Earth mass, and an approximately 2 Myr chondrule formation period, it was estimated that resonant planetesimal bow shocks could have processed a substantial fraction of the nebula, producing up to 1027 g of chondrules. For comparison, it was estimated in the final section of the Hood et al. (2009) paper that the minimum mass of chondrules in the primordial belt was in the range of 1024 to 1025 g. On this basis, it was considered possible that the bow shock model was efficient enough to explain the observed abundance of chondrules in chondrites.

Even if chondrule formation in nebular shocks (e.g., bow shocks) is ultimately found to be the main mechanism for producing most chondrules, it is clear that some chondrules were probably formed by other mechanisms. For example, chondrules in CB chondrites were apparently completely melted during a single heating event in the absence of nebular small-scale dust and have relatively young isotopic ages of about 5 Myr after CAIs (Krot et al. 2005). It has therefore been proposed that these chondrules formed directly from droplets in an impact-generated vapor-melt plume after the nebula had largely dissipated (Campbell et al. 2002; Rubin et al. 2003; Krot et al. 2005). In addition, collisions between partially molten planetesimals may have led to the formation of some chondrules (Asphaug et al. 2011). However, the vast majority of chondrules have characteristics consistent with incomplete, multiple heating in a dusty nebular environment and are therefore most easily explained in the context of a shock-wave model regardless of the source of such shocks.

In the present study, we use a more realistic approach to estimate the chondrule production efficiency by bow shocks and also investigate whether there is any tendency for the largest and most massive planetesimals to be perturbed into the most eccentric orbits during resonance passages. For this purpose, we make use of an improved planetesimal accretion and orbital evolution numerical code developed by S. Weidenschilling and F. Marzari. Unlike earlier studies that integrated orbits of individual bodies, the present study follows the simultaneous orbital evolution of a population of planetary embryos that interact by collisions and gravitational encounters. We also account for the fact that most of the mass of the primordial belt was likely in bodies more than a few hundred km in diameter, which would have been less susceptible to gas drag. Finally, the accreted mass of each embryo is tracked as a function of time in the simulations, which allows a more quantitative estimate for the mean impact rate with smaller planetesimals. We also take into consideration recent advances in gas dynamic shock modeling for solar nebula conditions (Morris and Desch 2010; Morris et al. 2012), which lead to slightly higher minimum shock velocities to melt mm-sized precursors in bow shocks in a cool nebula, i.e., about 6 km s−1, corresponding to minimum eccentricities at 3 AU of about 0.33. In the next section, the standard version of the planetesimal nebular shock model (with Jupiter in its current orbit) is tested using the improved numerical code. In particular, the chondrule formation efficiency, defined as the fraction of the nebular midplane region between 2 and 4 AU that would be processed by shocks strong enough to melt chondrule-sized particles, is estimated for both bow shocks and impact-generated shocks. In the Dependence of Eccentricity on Embryo Mass section, the mass and eccentricity of each remaining body is evaluated after several integration times for two different simulations to investigate whether only the largest embryos may have had eccentricities large enough to produce chondrules. In the Future Improvements to the Orbital Simulations section, the need to modify the standard model to consider a radially migrating Jupiter and to more self-consistently model the accretion of the planetesimals is emphasized. Conclusions and discussion are given in the final section.

Simulations for the Standard (Fixed Jupiter) Case

The improved numerical code, developed by S. Weidenschilling and F. Marzari, resembles that used by Weidenschilling (2011) to model accretion in the asteroid region but includes also the effects of gravitational perturbations from Jupiter, which is assumed to have formed. It includes a statistical continuum of small bodies that interact with a population of larger planetary embryos whose orbital evolution is treated individually, including gas drag, collisional damping, and gravitational scattering. The embryos are subject to mutual gravitational scattering; the continuum of small bodies damps their eccentricities and inclinations by collisions and dynamical friction. Gas drag also damps eccentricities and inclinations, and causes secular decay of their semimajor axes; the damping terms are modeled with the expressions of Adachi et al. (1976), including terms to the third power of eccentricity. For simplicity, the background population does not collide with itself, but is allowed to collide with the embryos. The small-body population is stirred gravitationally by the embryos and Jupiter. The rate of stirring of the background population is calculated by the changes in orbital elements of massless test particles during a timestep. The background population is also damped by gas drag. The earlier accretion code (Weidenschilling 2011) treated encounters between embryos as stochastic scattering events without explicitly integrating their orbits. The new code used in this work uses a version of the SyMBA symplectic integrator (Duncan et al. 1998) to compute the orbital evolution of the embryos, with drag terms added. This allows inclusion of long-range gravitational perturbations, including resonances with Jupiter. Jupiter’s eccentricity and semimajor axis are set at fixed values. Collisions between embryos are rare during the Myr time scale of a simulation; those that occur are assumed to result in coalescence. During the integration, some of the embryos may be lost from the system, ejected by encounters with Jupiter or colliding with it. All simulations were performed using the Planetary Science Institute Beowulf computer cluster.

To test the nebular shock model with a population of embryos and planetesimals accreting and evolving in the asteroid belt region, we choose a standard case with Jupiter at its present semimajor axis of 5.2 AU and fixed eccentricity of 0.05, approximately equal to present-day values. Note that in the present solar system, Jupiter’s eccentricity varies continuously on time scales of tens of thousands of years due to perturbations by the other planets (mainly Saturn), and ranges between approximately 0.025 and 0.06. Although there is no general consensus on what Jupiter’s initial eccentricity was, according to one popular model (the Nice model), it may have been nearly zero (e.g., O’Brien et al. 2007), which would result in reduced perturbations of planetesimals into highly eccentric orbits. Although a relatively large eccentricity of 0.05 is adopted to produce a favorable simulation (best case scenario), a separate simulation for a much smaller eccentricity (0.01) is also performed for comparison purposes (see below). As Jupiter’s initial orbit may have been coplanar with the solar nebula, we set its inclination at zero. Also, although Saturn is not included in the simulations reported here, we have carried out separate simulations that include Saturn and find that the basic conclusions of the analysis regarding chondrule formation efficiency and sizes of highly eccentric embryos do not change significantly.

For the nominal case studied here, the surface density of the solar nebula is assumed to vary inversely with heliocentric distance R. The region modeled is divided into 24 radial zones of width 0.1 AU spanning the range 2–4.4 AU. The initial population of small planetesimals is modeled by a series of logarithmic bins in diameter from 1 to 350 km with a size distribution characteristic of collisional equilibrium (a power law with incremental diameter slope of 3.5). The 1/R surface density implies equal numbers of bodies per interval of heliocentric distance. The number of bodies is chosen to give a total mass of 1.1 Earth mass for the background population. An initial population of 240 planetary embryos is assigned semimajor axes evenly spaced at intervals of 0.01 AU (10 embryos per radial zone). Their sizes are chosen randomly from a power law distribution, also with equilibrium slope, with diameters between 500 and 5000 km. This range yields a total mass of the embryo population of 1.1 Earth mass, equal to that of the background of small bodies. The stochastic variation in embryo masses means that the total mass in a given radial zone of 0.1 AU varies by about a factor of 2, depending mainly on the mass assigned to its largest embryo, but the total population is consistent with the trend of 1/R in surface density. All bodies are assumed to have density 2.5 g cm−3, regardless of size.

The nominal gas density is 1.3 × 10−10 g cm−3 at 3 AU, which is a factor of 8 less than has been assumed in most evaluations of chondrule formation and thermal histories in bow shocks (e.g., Ciesla et al. 2004; Morris and Desch 2010). The gas has a fractional deviation from Keplerian rotation of 0.0019, but we note that for significant eccentricity, the dominant term in the orbital decay rate is independent of that value (Adachi et al. 1976). Also, inward migration into resonances for bodies that are several hundred km in size or larger is mainly caused by orbital diffusion associated with gravitational encounters and collisions rather than gas drag. Therefore, the orbital evolution simulations for resonant bodies reported here should not be sensitive to this choice of gas density.

Figure 1 shows eccentricities plotted versus semimajor axis for the nominal case at model times 0.1, 0.5, and 1 Myr (where t = 0 corresponds to the formation of Jupiter). The vertical dashed lines show the principal Jovian commensurability resonances. Discrete bodies that are fortuitously placed near these locations initially, or scattered there at later times, typically attain high eccentricities before escaping from resonance; they then experience simultaneous decay of their semimajor axes and eccentricities due to gas drag. The horizontal dashed lines indicate an eccentricity of 0.33, corresponding to a shock velocity of approximately 6 km s−1 during passage through the nebular midplane at 3 AU (see fig. 2 of Hood et al. 2009). Velocities in this range would produce bow shocks that could partially or completely melt silicate precursors with diameters less than 1 mm. As noted in the Introduction, this minimum shock velocity is based on a recent reassessment of chondrule thermal histories in bow shocks of large embryos (Morris et al. 2012). As seen in Fig. 1, fewer than 10 embryos have eccentricities e ≥ 0.33 at a given time; most of these are among the largest bodies with diameters greater than 1000 km. Semimajor axes for these bodies are mostly between 2 and 3.5 AU. These are only three snapshots in time during the simulation, but they are representative of typical results after the initial transient stage. Inclinations (sin i) of remaining discrete bodies after 1 Myr of integration time are shown in Fig. 2, plotted against eccentricity. Inclinations are small, remaining below 2.5o for all but a few bodies.

Figure 1.

 Eccentricities versus semimajor axes (AU) of protoplanetary embryos surviving after three integration times up to 1 Myr. The largest circles have diameters > 1500 km; intermediate circles have diameters between 700 and 1500 km; small filled circles have diameters < 700 km. The positions of the strongest Jovian commensurability resonances are shown and horizontal dotted lines indicate the approximate minimum eccentricity (0.33) to produce shock velocities during midplane passages capable of melting mm-sized silicate precursors at 3 AU.

Figure 2.

 Inclinations versus eccentricity of remaining embryos after an integration time of 1 Myr. Note: sin i = 0.05 corresponds to i ≈ 2.9o.

To allow a direct comparison with the analysis of Hood et al. (2009), which was based on the orbital simulations of Marzari and Weidenschilling (2002), we note that eight embryos have e > 0.3 after 0.1, 0.5, and 1 Myr, or about 4–5% of the surviving bodies at each time. Although the number of bodies with e > 0.3 is coincidentally the same at each of these times, they mostly represent different individuals; only two embryos appear in two of these groups, and none in all three.

Figure 3 shows the semimajor axes of embryos for the same three times as shown in Fig. 1, plotted versus their initial semimajor axes. At t = 0, all bodies were on the diagonal line. The width of the distribution shows the extent of radial dispersion. Note that most of the migration is diffusion due to gravitational scattering with no preferred direction, rather than systematic inward migration due to gas drag. The few bodies in the lower right corner of each diagram have migrated inward due to passage through resonances, gas drag, and/or scattering by close approaches to Jupiter. Only 171 bodies (of the original 240) remain outside of 2 AU after 500,000 yr. By 1 Myr, another dozen have been lost by scattering, collisional coagulation, or migration to <2 AU, and there are more bodies whose semimajor axes have decreased substantially. Figure 4 plots each body’s velocity relative to the local gas during nodal crossings of Jupiter’s orbit plane (approximately the nebular midplane) for the remaining discrete bodies after the same three integration times. Each body crosses the nebular midplane twice per orbit at different radial distances; its velocity relative to the local circular velocity generally differs slightly between the ascending and descending nodes. Note that while the bodies with higher eccentricities have semimajor axes near the principal commensurability resonances, the nodal passages are distributed over a larger range of heliocentric distance. This figure confirms that gas-body relative velocities sufficient to produce shocks strong enough to melt submillimeter-sized silicate particles (about 6 km s−1) are achieved for a small fraction (<5%) of the remaining discrete bodies.

Figure 3.

 Semimajor axes (AU) of surviving embryos plotted versus starting values after three integration times up to 1 Myr. The diagonal dashed line shows the locus of starting values. Symbols represent the sizes of the embryos as in the preceding figures.

Figure 4.

 Gas-body relative velocities of the embryos at the nodal crossings of Jupiter’s orbit plane (approximately the nebular midplane) plotted versus radial distance in AU after three integration times up to 1 Myr. The dotted lines indicate the approximate minimum shock velocity of 6 km s−1 needed to melt chondrule precursors at 3 AU. There are two nodal crossings per embryo, at the ascending and descending nodes.

To allow an estimate of the efficiency of chondrule production by planetesimal nebular shocks for this simulation, we track the cumulative area, expressed as a fraction of the total area of the nebular midplane between 2 and 4 AU, that has been intercepted by all discrete bodies at velocities exceeding 5 and 8 km s−1 relative to the local gas. A velocity of 5 km s−1 is the minimum for partial melting of small (about 0.1 mm diameter) chondrule precursors while 8 km s−1 would be sufficient to completely melt large (about 1 mm diameter) precursors. To first order, this cumulative area is also the cumulative volume fraction of a layer with finite thickness (e.g., a solids-rich zone near the midplane). To provide a lower bound on the actual area fraction subject to bow shock melting, only the geometric cross-section of each body is considered in the calculation. As seen in Fig. 5, the fraction swept by the embryos at >5 km s−1 is approximately 10−3 after 106 yr, and about an order of magnitude lower for >8 km s−1. For the same swarm parameters, but with Jupiter’s eccentricity set at 0.01, the corresponding curves for each velocity are similar in shape, but lower by about a factor of 5.

Figure 5.

 The cumulative fractional area of the nebular midplane between 2 and 4 AU that has been swept at a given time by embryos with gas-body velocities greater than 5 and 8 km s−1. These values include only the physical cross-section of the embryos. The size of the bow shock wave and other effects would increase the fraction capable of producing chondrules; see the text.

In principle, it would be possible to modify the numerical code to provide a more direct estimate for the area fraction swept by the associated bow shocks at melting speeds (rather than by the geometric cross-sections of the bodies as in Fig. 5). For example, one could compute the effective melting cross-sectional area of the bow shock for each resonant embryo at each time step of the integration. This could be done using a parameterization of the effective shock velocity as a function of position along the shock (see, e.g., fig. 2 of Ciesla et al. 2004) and scaled with the actual gas-body relative velocity at each time step. A full calculation would also need to account for (1) the dependence of the minimum shock velocity for melting on the sizes of the chondrule precursors, which will be a function of position in the nebula; and (2) transient increases in the effective obstacle size of the embryos due to the production of impact-generated dust and debris clouds (see below). However, such a full calculation would likely increase the computational requirements (already 2–3 months for a 1 Myr simulation on the PSI Beowulf cluster). In the present work, we therefore take the simpler approach of adopting the lower bound results of Fig. 5 and then consider several factors that would increase the effective melting area fraction.

First, it is likely that the effective obstacle size of the embryos will be larger than the geometric cross-section due to the presence of impact-generated dust and debris clouds. It is well known that impacts of smaller bodies on larger bodies in the present-day asteroid belt produce dust and debris clouds that are partially gravitationally bound and can persist for several months (e.g., Jewitt et al. 2011). (As discussed later in this section, these same impacts in the primordial belt would also produce short-lived impact vapor plume shocks in the surrounding nebular gas.) Collisions in the primordial belt would have been much more frequent than for the present-day belt; the fastest-moving resonant bodies would have had the highest impact rate due to collisions with smaller, nonresonant bodies, especially during midplane passages. The resulting dust and debris clouds could easily have increased the effective obstacle size of a given resonant body to the nebular gas flow by a factor of as much as 2 or 3. Moreover, the impact-generated solids enrichments around such chondrule-forming resonant embryos could provide an alternate explanation for the Na contents of olivine crystal cores in chondrules (Alexander et al. 2008; Fedkin et al. 2012).

Second, the melting cross-section of a resonant body can be larger than its effective obstacle size because the shock velocity along the bow shock surface will remain larger than the minimum needed to melt chondrule precursors out to some distance away from the nose of the shock (e.g., Ciesla et al. 2004; Morris et al. 2012). The combination of these factors could increase the fraction swept at >8 km s−1 to as much as approximately 4 or 5 × 10−3, and the fraction at >5 km s−1 to approximately 0.01 after 106 yr. Finally, if the primordial belt consisted of four Earth masses rather than two Earth masses (allowed by minimum mass nebula models; see, e.g., Hood et al. 2009), with a correspondingly larger number of embryos, the fraction swept would increase by approximately 2 (or possibly more, if the efficiency increases with the mass of the swarm; see below). The total fraction swept by bow shocks at 6 km s−1 after 1 or 2 Myr might therefore be approximately 1%. However, the swept fraction also depends on Jupiter’s eccentricity. As noted above, corresponding simulations with a lower value of Jupiter’s eccentricity of 0.01 yield cumulative fractions swept by the shocks lower by about a factor of 5. In principle, one might assume a higher eccentricity than our nominal value of 0.05 for Jupiter, which could increase these values; however, this causes a more rapid depletion of the embryo population by scattering into Jupiter-crossing orbits.

To allow an assessment of the nebular fraction that would have been processed by impact-generated shocks for this simulation, we make use of the code output which keeps track of the mass of each embryo as it accretes mass from collisions with the background continuum of small planetesimals. The output shows that bodies with e > 0.2 and diameters of 1000–3000 km typically gained mass at rates of 1016 to 1017 g yr−1 during the interval from 200 to 250 kyr. The corresponding impact interval for (assumed) 1 km diameter bodies (mass approximately 1015 g) is days to weeks. Resonant bodies in orbits with e > 0.4 would have experienced impacts with other bodies in near-circular orbits at velocities greater than approximately 7 km s−1. According to the impact simulations of N. Artemieva in Hood et al. (2009) (see fig. 3 of that paper), a 1 km diameter silicate body impacting at 8 km s−1 in the presence of a nebula with mass density 10−7 g cm−3 produces a vapor-melt cloud that expands outward into the nebula to about 50 projectile diameters with a mean shock velocity of about 7–9 km s−1. After this expansion, the shock velocity decreases substantially so that chondrule-sized precursors would not reach melting temperatures. However, the assumed nebular mass density for these simulations was rather high; this was necessary to obtain a reliable numerical solution. A more realistic nebular mass density near 3 AU would be about 10−9 g cm−3 (e.g., Hood 1998; Morris and Desch 2010). Expansion of the vapor-melt cloud into such a nebula might continue out to as much as approximately 5000 projectile diameters if the expansion distance scales approximately inversely with ambient nebula mass density. This would occur in a period of the order of 10 min. Afterward, the residual vapor-melt cloud would be rapidly swept away due to the high velocity of the planetesimal relative to the nebular gas, which is also of order 7 km s−1. Therefore, although a substantial volume of the local nebular region around the embryo would be thermally processed by a given impact shock, the processing would be short-lived.

If the impact interval for 1 km diameter bodies is 2 days and a spherical volume of radius 5000 km is processed immediately following each impact, then the total nebular volume that is processed over a 10-day period would be about 3 × 1012 km3. During the same 10-day period, the bow shock for a 1000 km diameter embryo moving at 7 km s−1 relative to the nebular gas (assuming an effective melting radius of three times the embryo radius) would process a total volume of about 3 × 1013 km3. Thus, the nebular volume processed by impact-generated shocks is small in comparison to that processed by the bow shocks, consistent with the conclusions of Hood et al. (2009). The total fraction of the nebula midplane region that would be thermally processed by impact-generated shocks occurring on resonant embryos over a 1 Myr period would be <0.1%.

Dependence of Eccentricity on Embryo Mass

Figure 6 plots eccentricities versus mass of the remaining embryos for the standard case after the same three integration times as for Figs. 1 and 3. At 0.1 Myr, there is no clear dependence of eccentricity on mass. At 0.5 and 1 Myr, however, there is an increasing tendency for the largest bodies to have the highest eccentricities. The largest bodies with eccentricities near or above 0.4 (mass about 2 × 1026 g) are almost as large as Mars and three or four others are Moon-sized or larger (diameters >2000 km). Such a tendency could be a consequence of collisional damping and impact disruption, which are accounted for in this simulation, and should have greater effects on smaller bodies. Very small bodies should be damped even more by gas drag, precluding highly eccentric orbits, but the size at which this occurs is not determinable from this simulation.

Figure 6.

 Eccentricities versus mass in grams of remaining embryos for the standard case after three integration times up to 1 Myr. The dotted lines indicate the approximate minimum eccentricity to produce chondrule-forming shocks at 3 AU.

To test further whether collisional damping and impact disruption could produce a dependence of eccentricity on mass, a separate simulation was performed using an initial population of 1000 bodies with diameters between approximately 70 and approximately 4000 km and a total mass of 1 Earth mass, and a comparable mass of continuum bodies smaller than approximately 50 km, distributed between 2 and 4 AU. This larger population has a mass range spanning more than five orders of magnitude, and should show more clearly any tendency for mass dependence of eccentricity. This case also assumed a fixed eccentricity of 0.05 for Jupiter. Because of the larger number of bodies, this case was only computed to 0.3 Myr model time. Figure 7 plots the eccentricities of remaining embryos versus mass after simulation times of 0.1, 0.2, and 0.3 Myr. A tendency for more massive bodies to have higher eccentricities is more clearly seen in this simulation, supporting the hypothesis. Eccentricities greater than 0.4 are almost entirely confined to embryos larger than approximately 1024 g, with diameters >1000 km. Figure 8 plots the eccentricities of the same bodies versus semimajor axis. It is seen that most of the bodies with higher eccentricities have semimajor axes between 2.7 and 3.4 AU. The body with the highest eccentricity at 100,000 yr (0.570) has a diameter about 2100 km and semimajor axis of 2.65 AU. At 300,000 yr it has decayed to = 0.29, = 2.16 AU.

Figure 7.

 Eccentricities versus mass in grams of remaining embryos for the 1000 body simulation described in the text. Embryos have a wider range of sizes, from approximately 65 to approximately 5000 km in diameter. Results are shown after three integration times up to 300,000 yr. Same format as Fig. 6.

Figure 8.

 Eccentricities versus semimajor axes for the 1000-body case shown in Fig. 7. The different symbols refer to the sizes of the embryos as defined in the caption to Fig. 1.

The present simulations therefore indicate that orbital evolution factors alone in the presence of Jovian resonances will favor higher eccentricities for the largest planetesimals. This tendency would produce chondrules in bow shocks with slower cooling rates that may be more compatible with furnace experiment results. However, it is important to note an alternative or additional factor that might also tend to produce chondrules with longer cooling time scales. In the accretion simulations of Weidenschilling (2011), which began with subkilometer planetesimals in the absence of Jovian perturbations, the size distribution of embryos grown in the asteroid region was not a simple power law. Rather, runaway growth produced an “oligarchy” dominated by a small number of large bodies, ranging in size from about that of Ceres to Mars on time scales of less than 1 Myr. Thus, any tendency for large bodies to attain higher velocities through orbital evolution may have been augmented by a mass distribution weighted to favor such bodies.

Future Improvements to the Orbital Simulations

Models for planet formation and orbital evolution have long predicted a substantial radial migration of the giant planets owing to tidal interactions with the nebular disk (e.g., Lin and Papaloizou 1979, 1986; Goldreich and Tremaine 1980; Ward 1997). This theoretical expectation has been confirmed by the discovery of Jupiter-sized extrasolar planets within a few AU or less of their central stars. The fact that our planetary system has Jupiter and Saturn at much larger radial distances implies that inward migration must be inhibited in at least some cases. One possible mechanism for reversing tidally forced inward migration involves co-evolution of two massive planets (e.g., Jupiter and Saturn). Specifically, some dynamical models have found that inward migration of the larger body can be reversed at some minimum distance when the smaller body becomes trapped in a 2:3 mean motion resonance (Masset and Snellgrove 2001; Morbidelli and Crida 2007; Pierens and Nelson 2008). As an extreme example, one recent simulation has proto-Jupiter migrating inward to 1.5 AU over a 100,000 yr period followed by outward migration to its present orbital radius over the next 500,000 yr (Walsh et al. 2011).

Any substantial radial migration of Jupiter immediately following its formation during the solar nebula epoch would cause radial sweeping of resonance locations, which would in turn increase the number of embryos that are scattered into eccentric orbits at a given time. For example, the simulation of Walsh et al. (2011) starts with a disk containing several Earth masses of volatile-poor (“S-type”) planetesimals interior to Jupiter, with assumed diameters of 100 km. After 500,000 yr of outward Jovian migration from 1.5 AU to about 5 AU, the S-type planetesimals have been scattered into eccentric orbits with semimajor axes ranging from about 0 to 4 AU; a significant fraction (>10%) have eccentricities exceeding 0.4 and inclinations exceeding 10o with semimajor axes ranging from about 1.2 to 3 AU (see their Figs. 3c and 3d). We note that production of chondrules by bow shocks might be favored at smaller heliocentric distances, due to both the higher orbital velocities and expected higher gas densities, which would yield stronger bow shocks. There would, of course, be questions as to how chondrules produced in the terrestrial region were transported to their resting place in the present asteroid belt. Although the model of Walsh et al. may be an extreme example of Jovian radial migration, it shows that a more accurate assessment of the efficiency of chondrule production by the bow shock mechanism must await planetesimal orbit simulations in the presence of a radially migrating Jupiter. Initial simulations of this type are currently in progress and results will be reported in a future publication.

We have assumed a standard case with one Earth mass of material in the form of embryos with diameters approximately 500–5000 km between 2 and 4.4 AU. Bodies of such size are placed into Jovian resonances and removed from them not by gas drag, but mainly by gravitational encounters with other bodies of comparable size. The fraction of embryos with velocities large enough to produce chondrules by bow shocks may therefore depend on the total mass of the swarm and its size distribution. Thus, the rate of chondrule production may not be simply proportional to the mass of the swarm. Preliminary results from simulations in progress suggest that the chondrule formation efficiency increases with both the swarm mass and the size of the largest embryos. As emphasized above, the chondrule formation efficiency may also have been increased if Jupiter migrated radially during the chondrule formation epoch. In such cases, the entry into and removal from resonances will be a more complex interplay between gravitational scattering among the embryos and the rate of migration of those resonances. Finally, some recent models for the assembly of terrestrial planets and S-type asteroids begin with all of the mass initially inside of 1.0 AU (e.g., Hansen 2009; Walsh et al. 2011). A more comprehensive study of all of these possibilities is needed.

Conclusions and Discussion

The current simulations for the standard (fixed Jupiter) case with all bodies initially distributed between 2 and 4 AU show that the fraction of planetary embryos that are perturbed into highly eccentric orbits during resonant passages is smaller than previously estimated. Specifically, as shown in Fig. 1, only about 5% of remaining embryos have eccentricities exceeding 0.3 at a given time after 0.5–1 Myr of orbital integration following the formation of Jupiter. For comparison, 20–30% of planetesimals were estimated to have eccentricities exceeding this value by Hood et al. (2009) using results from the Marzari and Weidenschilling (2002) orbital evolution code. This reduction in the number of highly eccentric embryos is due to the fact that most of the mass of the initial population is in bodies with diameters greater than a few hundred km, which are less susceptible to inward orbital decay into resonances by gas drag. Instead, most bodies in the present simulations are transported radially by orbital diffusion due to collisions and gravitational scattering during close encounters. Only a small fraction of embryos spend enough time in resonances to attain shock velocities exceeding 7–8 km s−1.

On the other hand, one encouraging result from the present simulations is that only bodies of substantial size (half as large as the Moon or larger) may be able to attain eccentricities >0.4 sufficient to easily melt chondrule precursors (Figs. 1, 6, 7, and 8). This is apparently because of collisional damping and impact disruption, which preferentially reduce eccentricities for smaller bodies. This tendency may be further augmented if the initial embryo size distribution were more strongly weighted toward larger bodies than that described by the simple power law dependence assumed here (Weidenschilling 2011). As chondrule cooling rates inferred from furnace experiments can be matched by the bow shock model only for bodies of this size (Ciesla et al. 2004; Morris et al. 2012), it is conceivable that the bow shock model may eventually be found to be consistent with this constraint.

The approach toward estimating the efficiency of chondrule formation in bow shocks, while simplified, is more quantitative than that applied by Hood et al. (2009). It tracks the total midplane area swept by embryos (geometrical cross-sections only) during nodal passages as a function of gas-planetesimal relative velocity and integration time. Extrapolating the results of Fig. 5 to about 2 Myr and accounting roughly for the increase in swept midplane fraction due to the larger area of the bow shock relative to the geometrical cross-section of the embryos, the total fraction of the midplane swept by chondrule-forming shocks is still no more than approximately 1%. If Jupiter’s orbital eccentricity is set to 0.01 rather than 0.05, then the swept fraction is reduced by about a factor of 5. The efficiency of chondrule formation in impact vapor plume shocks is small in comparison to that in bow shocks, as found by combining previous simulations of impact vapor plume expansion into a nebula (Hood et al. 2009) with results of the present simulations, which yield an estimate for the impact rate on resonant embryos.

Although these results for the standard (fixed Jupiter) case yield a rather low chondrule formation efficiency, it should be noted that, even at 1% efficiency, if an Earth mass of chondrule precursors were available, then the total mass of chondrules produced would exceed the present mass of the asteroid belt (about 3 × 1024 g) by a factor of 20. The later dynamical depletion of asteroids may have been only about one order of magnitude by number, if most of the mass was in large embryos (e.g., Weidenschilling 2011). As only a small fraction of the precursor material would have been converted into chondrules, they would have had to be preferentially accreted by asteroid-sized parent bodies. Embryos would make repeated nodal passages through the same region of the disk on successive orbits, raising the probability of multiple heating events.

Several factors may have led to more effective accretion of chondrules by asteroid-sized bodies. First, because the dynamical behavior of chondrule-sized and smaller particles is controlled mainly by gas drag forces, their accretion would be dominated by the geometrical area of those bodies, rather than their gravitational cross-sections. This would favor low-speed accretion of chondrules by asteroid-sized bodies rather than planetary embryos. In addition, aerodynamic sorting during later encounters with nonresonant asteroid-sized bodies may have further enhanced chondrule accretion by those bodies, as suggested originally by Whipple (1971, 1972). Specifically, smaller (e.g., <0.1 mm sized) particles, would have been preferentially swept around the accreting planetesimal by the subsonic nebular gas flow while larger chondrule-sized particles would have accreted. Production of chondrules by bow shocks may therefore be at least marginally possible from a quantitative standpoint even for the standard case.

Finally, as emphasized in the previous section, the chondrule formation efficiency may also have been increased significantly if Jupiter migrated radially during the chondrule formation epoch (Walsh et al. 2011). Furthermore, the chondrule formation efficiency may not be a simple linear function of the initial swarm mass and the swarm may not have been initially confined to the 2–4 AU region as assumed in the present work. A final assessment of the ability of the bow shock mechanism to explain the abundance of chondrules in chondrites must therefore await future simulations that consider the combined effects of the swarm mass and radial distribution, the embryo sizes, and the rate and extent of Jovian radial migration.

Acknowledgments—  Supported by the NASA Origins program under Grant No. NNX09AF70G. We thank the two reviewers and the associate editor (E. Scott) for useful criticisms that led to significant improvements of the first manuscript.

Editorial Handling—  Dr. Edward Scott