Evaluating planetesimal bow shocks as sites for chondrule formation

Transcription

1 Meteoritics & Planetary Science 39, Nr 11, (2004) Abstract available online at Evaluating planetesimal bow shocks as sites for chonrule formation Fre J. CIESLA, 1* Lon L. HOOD, 2 an Stuart J. WEIDENSCHILLING 3 1 NASA Ames Research Center, MS 245-3, Moffett Fiel, California 94035, USA 2 Lunar an Planetary Laboratory, University of Arizona, 1629 East University Boulevar, Tucson, Arizona 85721, USA 3 Planetary Science Institute, 1700 East Fort Lowell, Tucson, Arizona 85719, USA * Corresponing author. ciesla@cosmic.arc.nasa.gov (Receive 17 November 2003; revision accepte 28 May 2004) Abstract We investigate the possible formation of chonrules by planetesimal bow shocks. The formation of such shocks is moele using a piecewise parabolic metho (PPM) coe uner a variety of conitions. The results of this moeling are use as a guie to stuy chonrule formation in a oneimensional, finite shock wave. This moel consiers a mixture of chonrule-size particles an micron-size ust an moels the kinetic vaporization of the solis. We foun that only planetesimals with a raius of ~1000 km an moving at least ~8 km/s with respect to the nebular gas can generate shocks that woul allow chonrule-size particles to have peak temperatures an cooling rates that are generally consistent with what has been inferre for chonrules. Planetesimals with smaller raii ten to prouce lower peak temperatures an cooling rates that are too high. However, the peak temperatures of chonrules are only matche for low values of chonrule wavelength-average emissivity. Very slow cooling (<~100s of K/hr) can only be achieve if the nebular opacity is low, which may result after a significant amount of material has been accrete into objects that are chonrule-size or larger, or if chonrules forme in regions of the nebula with small ust concentrations. Large shock waves of approximately the same scale as those forme by gravitational instabilities or tial interactions between the nebula an a young Jupiter o not require this to match the inferre thermal histories of chonrules. INTRODUCTION Chonrules are millimeter-size, silicate spheres that are abunant in most chonritic meteorites. The textures of these objects suggest that they forme before being incorporate into their respective meteorite parent boies (for a etaile review of our current unerstaning of chonrules an their formation, see Jones et al. [2000]). The exact metho by which these chonrule melts were forme an kept warm as crystals grew has not yet been ientifie. Recently, shock waves within the solar nebula have been shown to be capable of explaining the complex thermal histories that have been inferre for chonrules an possibly allow for the formation of other meteoritic components (Iia et al. 2001; Desch an Connolly 2002; Ciesla an Hoo 2002; Ciesla et al. 2003). While such work supports the hypothesis that shock waves were the ominant chonrule forming mechanism in the solar nebula, the source of the shocks still remains to be etermine. In the stuies mentione above, large scale shock waves (>10 5 km) were moele, suggesting that shocks ue to gravitational instabilities (Boss 2002) or tial interactions of Jupiter with the nebula (Bryen et al. 1999; Rafikov 2002) may have been where chonrules forme. Other possible sources are planetesimals, the orbits of which cause them to attain supersonic velocities with respect to the nebular gas (Hoo 1998; Weienschilling et al. 1998). Such velocities coul be attaine if the planetesimals were in highly eccentric or incline orbits aroun the Sun. While the existence of planetesimals before the formation of the very primitive chonrules (an thus the chonrite parent boies) seems paraoxical, it may have been possible given the inferre extene lifetime of the nebula before chonrule formation. Amelin et al. (2002) use Pb-Pb ating to show that chonrules forme 2.5 ± 1.2 Myr after the formation of calcium-aluminum-rich inclusions (CAIs), the first objects to have forme in the solar nebula. This time is much longer than the expecte time neee to accrete large boies in a laminar solar nebula (~10 4 yr), an possibly long enough to accrete these same boies in a weakly turbulent nebula (Weienschilling 1988; Weienschilling an Cuzzi 1993). In fact, it is possible that this age of chonrules overlappe the global ifferentiation of the parent boy of the HED meteorites, thought to be the 500 km-iameter asteroi 1809 Meteoritical Society, Printe in USA.

2 1810 F. J. Ciesla et al. 4 Vesta, which must have occurre no later than 4 Myr after CAI formation (Wahwa an Russell 2000). If this were the case, then it is reasonable to think that other such boies existe at the same time but were subsequently remove from the asteroi belt (Chambers an Wetherill 2001). This scenario requires further stuy ue to the epenence on unconstraine parameters use in solar nebula an accretion/ coagulation moels. For the purposes of this stuy, we assume that chonrule precursors existe in a nebula where boies up to 1000 km in size ha accrete. These precursors coul either have been material that ha not yet been accrete by planetesimals or material ejecte from the collisions of planetesimals. In aition, this scenario requires Jupiter to have forme fairly rapily so that it coul stir up planetesimals in time to make chonrules. A rapi formation of Jupiter is possible if it forme via the gravitational instability mechanism (e.g., Boss 2002; Mayer et al. 2002). In aition, recent work suggests that the core accretion mechanism for forming Jupiter can work on much shorter timescales (~1 Myr) than previously calculate, provie that the core migrate through the nebula over time (Alibert et al. 2004). Thus, having Jupiter present at the time of chonrule formation is possible through either giant planet formation mechanism. Weienschilling et al. (1998; see also Marzari an Weienschilling [2002]) foun that, in the solar nebula, planetesimals larger than 10 km coul get trappe in resonances with Jupiter an attain velocities of 5 km/s or higher with respect to the nebular gas. Hoo (1998) performe hyroynamic simulations to investigate the characteristics of bow shocks that woul be create by such supersonic planetesimals. He foun that the shocks create by these planetesimals woul be strong enough to melt silicates in a few secons, which is consistent with how chonrules are thought to have forme, provie that the planetesimals ha relative velocities of at least km/s with respect to the gas. However, no explicit calculations have been carrie out to stuy the cooling histories of the silicate particles heate by these small-scale (<100 km) shocks. Hoo an Ciesla (2001) foun that cooling rates of particles in a relatively small (<1000 km) shocke region of the nebula that was fille significantly with micron-size ust particles coul be similar to those of chonrules if certain generous assumptions were mae. Some of these assumptions were that no relative motion between the chonrule-size particles an ust particles existe an that all of the particles were initially at the same temperature throughout the region of interest. If a shock wave passe through a region of the nebula with ust ( = 1 µm) an chonrules ( = 1 mm), the particles woul move through the gas until they lost their velocity with respect to the gas. The stopping istance of a particle is equal to the istance it must travel before encountering a mass of gas equal to itself. This istance is equal to x stop = 2ρ/ρ g, where ρ is the mass ensity of the particle an ρ g is the mass ensity of the gas. For typical values of ρ an ρ g (1 g/cm 3 an 10 9 g/cm 3 respectively), x stop is ~10 9 (10 8 in shocke gas where the gas ensity increases by roughly an orer of magnitue over the ambient value). Thus, a millimeter-size chonrule woul have a stopping istance three orers of magnitue greater than that of the micron-size ust particle. Therefore, the assumption that no relative velocity between the two soli particles woul exist is not vali until the larger of these istances has been traverse (~200 km for typical values). The relative velocity between these particles coul cause the chonrules an ust to be at very ifferent temperatures over a large fraction of the region of interest, which was not consiere by Hoo an Ciesla (2001). In this paper, the moel of Hoo (1998) is extene to investigate whether planetesimal bow shocks coul have thermally processe silicates in a manner similar to how chonrules are believe to have been processe. In particular, the cooling rates of the silicates are investigate to see if those preicte in the moel are consistent with those expecte for chonrules. In the next section, we present new simulations of the bow shocks that woul be create by supersonic planetesimals an escribe the structure of these shock waves in etail. Using these results as a guie, we evelop a oneimensional moel to track the heating an cooling of a particle-gas suspension with a mixture of chonrule-size an micron-size particles that is overrun with prouce shock waves iscusse in the Shock Moel section. This moel allows for the vaporization an reconensation of the silicates. We iscuss the results of this moeling in the Results section. In the Discussion section, we iscuss what implications this work has for unerstaning chonrule formation. PLANETESIMAL BOW SHOCKS We have use the publicly available Virginia Hyroynamics-1 coe evelope by John Blonin an colleagues that solves the equations of gas ynamics using a Lagrangian remap version of the piecewise parabolic metho (PPM). The coe is base on the methos outline in Colella an Woowar (1984) an has been use to simulate a variety of astrophysical problems (e.g., Richars an Ratliff 1998). For this work, we use the coe to stuy the flow of gas aroun a planetesimal of arbitrary size an velocity in the solar nebula. In our runs, the equation of state for the gas is the ieal gas law, where effects such as issociation an raiation were not consiere. Therefore, the gas behaves aiabatically in these simulations. In our simulations, we use a two-imensional, Cartesian gri with the planetesimal centere in the mile of the gri. The planetesimal was treate as a highensity, low-pressure flui with no velocity. The surrouning nebular gas was set at a given temperature an pressure an given an initial, arbitrary velocity with respect to the planetesimal. The coe was run until a steay state was

3 Evaluating planetesimal bow shocks as sites for chonrule formation 1811 forme an the ynamics of the flow were well establishe. Since a Cartesian gri is use, the calculate flow is essentially equal to that aroun an infinite cyliner rather than a spherical planetesimal. Since we ignore the expansion of the gas in the z-irection (into an out of the page), the ensities presente here may be slightly higher than they woul be in the case of a spherical planetesimal. Thus, we may be overestimating the gas ensity in the vicinity of a planetesimal an issue we iscuss in etail below. We consiere 10 km, 100 km, an 1000 km raii planetesimals with gas velocities of 4 km/s, 6 km/s, an 8 km/s relative to the planetesimal. These velocities were foun to be representative of those foun in the moel runs of Weienschilling et al. (1998). Significantly higher velocities woul require much greater eccentricities or significant inclinations of the planetesimal s orbit (Hoo 1998). An example of one of our runs is shown in Fig. 1, where we consier a 1000 km planetesimal moving through nebular gas at a velocity of 8 km/s (we focus on this velocity because it creates the strongest shocks an, therefore, the greatest amount of heating of solis in the nebula). This velocity woul correspon to a planetesimal with an eccentricity of ~0.5 at 3 AU, near the maximum value as foun by Marzari an Weienschilling (2002). The gas is assume to initially be at a ensity of 10 9 g/cm 3 an at a temperature of 400 K upstream of the planetesimal (same values use in Hoo [1998]), creating a shock of about Mach 4. In this figure, the ensity of the gas is shown as it flows aroun the planetesimal (the ense white object in the center). The gas initially flows from left to right at a velocity of 8 km/s until the flow is isturbe by the planetesimal. The moele smaller planetesimals prouce similar shocks that scale with the size of the planetesimal. For example, the 100 km-raius planetesimal prouce results similar to that shown in Fig. 1, except the istances on the two axes were smaller by a factor of 10. The shock front that forms in this scenario is the parabolic iscontinuity between the upstream gas an the enser region behin it. For lower velocities of the upstream gas, the curvature of the shock front is not as great. The shape of this shock front is important to consier in stuying the processing of particles suspene in the nebula because the strength of the shock wave is etermine by the component of the gas velocity that is perpenicular to the shock front. Since the incient gas is moving irectly left to right in Fig. 1, the effective velocity with which it passes through the shock front is less than the initial upstream velocity. Figure 2 shows how the effective velocity of the shock varies with istance from the center of the planetesimal for the ifferent velocities that were consiere. Here, the istance from the center of the planetesimal refers to the impact parameter, or istance perpenicular to the initial gas flow from the center of the planetesimal to the point of interest. The effective velocity was foun by tracing a trajectory parallel to the velocity of the upstream gas through the shock an fining Fig. 1. A simulation of gas flow aroun a planetesimal using the piecewise parabolic metho to solve the equations of flow. The planetesimal is the white object at the center. The ensity of the gas is shown. The gas is assume to be owing from the left bounary at 8 km/s at a temperature of 400 K an a ensity of 10 9 g/cm 3. The planetesimal is assume to have a raius of 1000 km. Fig. 2. The effective shock velocities are plotte as a function of istance from the center of the planetesimal for a number of planetesimal-gas relative velocities (V p ). The effective velocity ecreases with istance from the planetesimal ue to the curvature of the bow shock. The plot is truncate at ~1.5 R p ue to the complex flow near the surface of the planetesimal. the maximum ensity of the gas along this trajectory. Using the ratio of this maximum ensity to the ensity of the upstream gas, we calculate what velocity of shock woul be neee to prouce that ensity jump. This treatment etermines the shock strength everywhere, accounting for the fact that the gas velocity is not perpenicular to the shock front. Note that the results shown in Fig. 2 are inepenent of

4 1812 F. J. Ciesla et al. planetesimal size, as they are plotte in units of planetesimal raii (in the runs of planetesimals of ifferent sizes the results i not vary). The bumps an wiggles of the plot are the result of the finite resolution of the simulations. The plots are truncate at ~1.5 planetesimal raii because the flow of the gas aroun the surface of the planetesimal is complex an oes not allow for simple consierations as those use here. This is because large pressure graients an lateral flow evelop in the gas as it flows aroun the planetesimal. Particles that enter the bow shock insie of this istance woul experience large eflections ue to this extreme change in gas flow. Since the region interior to this istance makes up a small volume of space, we feel that neglecting it will not change our results significantly. For the purposes of this stuy, we focus on the bow shock exterior to 1.5 planetesimal raii where the lateral gas flow is low. We will iscuss the likely effects of chonrules entering the shocks insie of this istance at the en of this paper. Overall, it is important to note that the effective shock velocity rops off with increasing istance from the planetesimal an the rop off is steeper with increasing planetesimal velocity. Figure 3 shows the ensity profile of the gas along the horizontal trajectory at 1.6 planetesimal raii (1600 km) from the center of the planetesimal in the simulation shown in Fig. 1, where the effective shock velocity is near its maximum value (~6 km/s). The istance is measure from the left ege of the simulation. The gas ensity increases sharply at the shock front, but then slowly ecreases with istance. The ensity actually reaches a value below the original gas ensity in the wake of the planetesimal (~3 planetesimal raii behin the shock) as is foun in other simulations of flow aroun a rigi boy. Similar profiles woul be foun at larger istances from the planetesimal with the maximum ensity ecreasing with increasing istance. Thus, this figure illustrates the structure of the oneimensional slice of the shock create by the planetesimal an how the gas ensity (as well as temperature an pressure) is elevate over a finite istance. SHOCK MODEL The structure of the shock in the one-imensional moel use here is iagramme in Fig. 4, where the temperatures of the gas an particles are sketche at ifferent points in the shock. In the region upstream from the shock, particles are heate by the raiation from the hot particles behin the shock front. These particles in turn heat the gas as the shock front approaches. Upon passing through the shock front, the gas properties change as given by the Rankine-Hugoniot relations: T T 1 [ 2γM 2 ( γ 1) ][( γ 1)M 2 + 2] = [( γ + 1) 2 M 2 ] (1) n n 1 ( γ + 1)M 2 = [( γ 1)M 2 + 2] v v 1 = n n 2 where T, n, an v represent the gas temperature, number ensity of gas molecules, an gas velocity with respect to the shock front, an the subscripts 1 an 2 represent the values immeiately before an after the shock, respectively, γ is the ratio of specific heats for the gas, an M is the Mach number (the ratio of the spee of the gas with respect to the shock front to the spee of soun in that gas immeiately before passage through the front). Upon passing through the shock front, the gas begins to cool rapily ue to the issociation of hyrogen molecules. The particles pass through the shock front unaffecte an, ue to energy an momentum exchange with the gas, are rapily heate as their velocity with respect to the gas ecreases. This rapi heating is what causes the spike in the temperature profile behin the shock front. After reaching their peak temperatures, the particles begin to cool, rapily at first, before reaching a slower an roughly constant rate (represente by the gentle slope immeiately before the en of the shock in the iagram). This slow cooling region is where the raiative energy loss of the particles is only slightly higher than the heat input ue to thermal collisions with the gas molecules an the absorbe raiative energy from the surrouning hot particles. This region is calle the relaxation zone in this moel. After some istance, the gas will start to approach its original, unisturbe, state. This likely happens after the gas expans an cools an the nebula erases any evience of the shock. In our moel, upon passing through the en of the shock (assume to be 3 planetesimal raii behin the shock front), the gas properties return to what they were before crossing through the shock front. The istance of 3 planetesimal raii was chosen to be consistent with Fig. 3 (over this istance, the gas ensity is at a higher value than in the ambient state). The particles are, again, unaffecte when crossing through this region an exchange energy an momentum with the gas as the system relaxes back to its original state (same temperature an velocity as far upstream from the shock). This immersion of the particles back in the cool gas helps quench the particles, causing them to cool very rapily. In aition, since the gas begins to expan an carry the solis with it, the ecrease in concentration of the solis weakens the raiation fiel that the particles will feel, an thus the solis will not be able to absorb as much energy. Previous shock moels that focuse on large scale shocks (Iia et al. 2001; Desch an Connolly 2002; Ciesla an Hoo 2002; an Ciesla et al. 2003) i not consier what happene to the gas an particles as they exite the relaxation zone of (2) (3)

5 Evaluating planetesimal bow shocks as sites for chonrule formation 1813 system are similar to those use by Ciesla an Hoo (2002). Along with chonrule-size particles, we consier the presence of micron-size ust particles. The ust particles interact with the gas in the same manner as the chonrules. In aition, the chonrules an ust particles are allowe to vaporize as they are heate. This process is hanle in the same manner that Moses (1992) treate silicate vaporization. Thus, the equations which govern the evolution of the solis are: Mass Evolution m chon = m chon t v chon (4) Fig. 3. The gas ensity is plotte along a horizontal trajectory in Fig. 1 that crosses the bow shock approximately 1.5 R p from the center of the planetesimal. The istance is measure from the left bounary of Fig. 1. The ensity of the gas ecreases immeiately behin the shock front, reaching its original, pre-shock value after traveling a istance of approximately 3 R p. m ust Density Evolution = m ust t v ust (5) the shock. The etails likely epen on the source of the shocks, but shoul be consiere in future work. In treating the system this way, a number of simplifying assumptions have been mae. In the simulations escribe in the Planetesimal Bow Shocks section, the gas properties began to relax back to their pre-shock values as they move away from the shock front rather than being maintaine at the shocke values. In the moel use here, the gas expansion is ignore. This means that the gas in the present moel is kept at a higher temperature an higher ensity than woul be expecte in a real situation. Thus, this allows the gas to transfer a higher amount of energy to the particles (an keep them warm) for a longer perio of time. Also, the shocke region is assume to be infinite in the irections perpenicular to the velocity of the shock for the raiative transfer calculations. Both of these treatments likely lea to unerestimating the cooling rates of the particles in our moel. These assumptions will be iscusse later. The equations which govern the evolution of the Energy Evolution ρ chon m chon m ( ρ chon v chon ) = v chon chon ρ ust m ust m ( ρ ust v ust ) = v ust ust Momentum Evolution 2 ρ ( ρ chon v chon ) = F chon D chon v 2 m chon chon m chon 2 ρ ( ρ ust v ust ) = F ust D ust v 2 m ust ust m ust (6) (7) (8) (9) C chon T chon + --v 2 2 chon ρchon v Q Q Q F chon g chon p chon chon ra D chon L ρ chon m v chon chon n 3 = chon v chon m chon m chon ρ C chon T chon m + chon v chon chon m chon (10) C ust T ust + --v 2 2 ust ρust v ust = Q g ust Q p ust Q ust ra F D ust L ρ ust m v ust ust n ust v ust m ust m ust ρ C ust T ust m + ust v ust ust m ust (11)

6 1814 F. J. Ciesla et al. where x is the one-imensional spatial coorinate, m chon an m ust are the mass of a chonrule an ust particle, v chon an v ust are the velocities of the chonrules an ust particles with respect to the shock front, ρ chon an ρ ust are the mass ensities of the chonrules an ust particles suspene in the gas, an T chon an T ust are the temperatures of the chonrules an ust particles. Since we assume that the chonrules an ust particles are compose of the same material, the heat capacities, C chon an C ust, an latent heat of vaporization, L, are assume to be equal for the two particles. The terms F Dchon an F D ust represent the gas rag force exerte on the particles per unit volume, while Q g chon an Q g ust represent the rate at which thermal energy is transferre from the gas to the particles per unit volume. These terms are calculate in the same manner as escribe in Ciesla an Hoo (2002). The terms Q p chon, Q p ust, Q chon ra, an Q ust ra represent the rate of raiative heating an cooling the particles experience an are calculate using the principles of raiative transfer as escribe in Desch an Connolly (2002). The rate at which the particles lose mass is calculate in a manner similar to that employe in Moses (1992). The rate at which the chonrule woul lose mass is given by: m chon 2 µ = π t chon ( P eq P feel ) πkT chon (12) with a similar equation escribing the mass loss for the ust particles. In the equation above, chon is the iameter of the chonrule, P eq represents the equilibrium vapor pressure of the silicate particles (in ynes/cm 2 ), P feel is the vapor pressure the particles feel on their surfaces, µ is the mean molecular mass of the silicate vapor, an k is Boltzmann s constant (note Moses [1992] assume that P feel was negligible an, thus, this term was not inclue in that work). The equilibrium pressure is calculate by: P eq = 10 B A T chon (13) where A = an B = 24,605 (Moses 1992). The pressure felt by the particle is given by: 1 P feel = n SiV KT g + --ρ 3 SiV ( v chon v g ) 2 (14) where n SiV is the number ensity of silicate vapor molecules present, ρ SiV is the mass ensity of those molecules, an v g is the velocity of the gas with respect to the shock front (Miura et al. 2002). Following Moses (1992), we o not consier the ecomposition of the silicate particles into ifferent components (e.g., Mg 2 SiO 4 + 3H 2 2Mg + SiO + 3H 2 O), but instea treat it as losing a single molecule of mean molecular mass µ. Since chonrules are not pure substances an, therefore woul ecompose into a number of ifferent of species, tracking each species woul be ifficult. Since the amount of vapor release by vaporization is small compare the amount of hyrogen or helium present, it likely has little effect on the results of this work. We ignore the effects of collisions between particles in our moel an assume that the thermal an ynamical histories of the particles woul not be significantly affecte by such consierations. Unless collisions woul lea to large changes in the size istribution of the particles, which woul affect the opacity of the nebula, this assumption shoul be vali. The most significant outcome of such collisions may be the seeing of chonrule melts by collisions with small ust particles (Connolly an Hewins 1995). This coul affect the textures of the chonrule-size particles processe in these shocks. The nebular gas also evolves as escribe by the following equations: Molecular Hyrogen Evolution Atomic Hyrogen Evolution Helium Evolution -----n ( H2 v g ) = η Silicate Vapor Evolution H n ( H v g ) = 2η H n ( He v g ) = 0 (15) (16) (17) -----n ( SiV v g ) = 1 m n chon chon v chon + m ust n ust v ust m SiV (18) Gas Momentum Evolution 2 P ρ ( ρ g v g ) F D chon F chon 2 m D ust v chon 2 m = chon v chon ust m chon ρ ust m ust (19)

7 Evaluating planetesimal bow shocks as sites for chonrule formation 1815 Gas Energy Evolution e H2 + e H + e He + e SiV + --ρ 2 g v g vg Q 1ρ = g chon Q g ust F D chon v chon F D ust v ust Q chon iss v 3 m chon chon m chon m v ust ust C SiV T ρ p chon m chon v chon chon C SiV T ρ p ust m ust v ust ust ρ ust 3 2m ust m chon m ust (20) where η H2 is the rate of issociation per unit volume of hyrogen molecules, Q iss is the rate of energy loss ue to issociation per unit volume both calculate as escribe in Iia et al. (2001), n H2 an n H are the number ensities of hyrogen molecules an atoms, n He is the number ensity of helium atoms, an ρ g is the mass ensity of the nebular gas. The terms e H2, e H, e He, an e SiV represent the internal energies of the hyrogen molecules, atoms, an silicate vapor, respectively. They can be calculate using: e i = ρ i C i p T g (21) where i = H 2, H, He, or SiV, C i p is the specific heat at constant pressure for the corresponing species, an ρ i is the corresponing mass ensity. In this moel, we neglect line emission cooling of the nebular gas. This effect has been consiere in some shock wave moels for chonrule formation (Iia et al. 2001). The importance of this cooling is strongly epenent on the amount of coolant molecules present in the gas, with H 2 O being the ominant cooling species. The concentration of water in the nebular gas in the asteroi belt region (where we are assuming chonrules forme) likely varie significantly over time. J. Cuzzi (personal communication) has shown that if there were a major planetesimal sink which accrete the solis aroun it, such as a young Jupiter or a Jovian core near the snow line, the region of the nebula interior to this sink woul be eplete in water by iffusion on timescales on the orer of ~ yr. The moel investigate here requires that Jupiter ha grown to significant mass such that the planetesimals near orbital resonances with the planet coul be stirre up to supersonic spees. The results of Amelin et al. (2002) suggest that chonrules forme ~2.5 Myr after the formation of the solar nebula. If Jupiter forme over this time (by either core accretion or gravitational instability), then this provies more than enough time for iffusion to remove water from the nebular gas (Stevenson an Lunine 1988; Cyr et al. 1998). In fact, if Jupiter forme by core accretion, the removal of water from the inner nebula may have been require to form the core (Stevenson an Lunine 1988). Thus, neglecting the cooling by line emission from water molecules woul be a vali approximation. If water vapor was not iffusively remove from the inner nebula before the stage of nebula evolution consiere here, then it may have playe a role in cooling the gas behin the shock waves as iscusse by Iia et al. (2001). The main effect woul be causing the gas to cool more quickly, which woul also increase the gas ensity slightly. This likely woul cause the particles to reach slightly higher peak temperatures (ue to the increase in gas ensity an, thus, in gas rag heating immeiately behin the shock front) but woul also likely increase the cooling rates of the particles ue to loss of thermal energy to the gas. For the cases presente, the nebula is assume to be originally at some temperature, T i, far upstream of the shock, which sets the forwar bounary conition for the raiative transfer calculations. Far behin the shock (thousans of kilometers, etermine by the integration), the suspension returns to this same original temperature an, therefore, the en bounary conition for the raiative transfer is the same. In solving the equations that escribe the evolution of the system, the temperature of the chonrules is initially calculate after they pass through the shock front base on the other moes of heat transfer. The calculations are then repeate, an the heat transferre ue to raiation from the other particles is foun using the previously calculate temperature profile as a function of istance. The proceure is repeate until the temperature of the chonrules oes not change by more than 1 K. RESULTS In all cases presente here, the region of the nebula uner consieration has an initial temperature of 400 K, a hyrogen molecule number ensity of cm 3, an a helium number ensity of cm 3 (ρ g = 10 9 g/cm 3 ), for a total pressure of ~10 5 bars. This gas ensity is roughly equal to that expecte at 2.5 AU in a minimum mass solar nebula, the type use in previous stuies of planetesimal bow shocks (Hoo 1998; Weienschilling et al. 1998). The properties of the chonrule-size particles are the same as those use in Ciesla et al. (2003) (unless otherwise specifie): iameter of 0.1 cm, mass ensity of 3.3 g/cm 3, a wavelength average emissivity of 0.9, a heat capacity of 10 7 erg g 1 K 1, an a latent heat of melting of erg g 1 sprea over the temperature range K to account for incongruent melting of chonrules. The latent heat of vaporization was assume to be erg g 1 close to the values use by Moses (1992) an Miura et al. (2002). The ust particles are assume to have the same properties as the chonrules, ue to their assume similar composition, except they are originally treate as spheres 1 µm in iameter an have an assume (constant)

8 1816 F. J. Ciesla et al. wavelength average emissivity of 0.1. This lower emissivity is use to account for the ifficulty that small particles have raiating away energy in wavelengths that are large compare to the size of the particle. We assume that the silicate vapor has a mean molecular mass of 50 m H an a specific heat capacity at constant pressure of 9k/2m SiV, while the values for the hyrogen molecules an atoms are 7k/2 m H2 an 5k/2m H, respectively, an the value for helium is 5k/2m He. The ust was consiere to have totally vaporize if its mass roppe below m ust 0, where m ust 0 is the original mass of a ust particle. In none of the cases presente below i the ust completely vaporize, espite reaching temperatures over 2000 K. This is because as the ust began to lose mass, the particles were coole ue to latent heat effects. Stronger shocks (effective velocity >8 km/s) may have provie enough energy to totally vaporize the particles. The solis in the moel are suspene at a solar ratio with respect to the nebular gas such that (ρ chon + ρ ust )/ρ g = The mass of the solis is istribute such that 75% of the mass is in chonrule-size particles an 25% as micron-size ust (Desch an Connolly 2002). We investigate other possible ratios of chonrules to ust an non-solar solis to gas mass ratios in the section title Various Chonrules: Dust an Soli-Gas Mass Ratios. The solar values give a chonrule number ensity of cm 3 an a ust number ensity of 0.85 cm 3. Fig. 4. An approximate thermal profile of the particles (gas) is iagramme using the shock moel. The particles are heate by raiation that leaks out from the shock front as the shock approaches (shock moves from right to left in this figure). Upon passing through the shock front, the particles are heate rapily by gas rag before cooling as they lose their velocity with respect to the gas. The particles then cool, rapily at first, an if the shock is wie enough the cooling rate ecreases when the particles raiate away slightly more energy than they absorb from the ambient raiation fiel. Upon passing through the en of the shock, the particles are immerse in gas, the properties of which are ientical to what they were immeiately upstream of the shock front. The particles then cool very rapily ue to being quenche in col gas an a weaker raiation fiel. 10-km Planetesimal The top panel in Fig. 5 shows the temperature evolution of the chonrules with time as they are encountere by a 6 km/s, one-imensional shock that is 30 km in with, where the with is efine as the istance between the shock front an the shock en (Fig. 4). Time is measure from the time in which the shock front encounters the chonrules. The chonrules are heate by raiation as they approach the shock front, reaching a temperature of ~800 K before crossing the shock front. The chonrules then reach a peak temperature of ~1440 K before cooling rapily (>10 5 K/hr) below the solius. The cooling rate of the chonrules while they are above their assume solius (1400 K) is shown in the bottom panel of Fig. 5. It is generally thought that chonrules forme by being heate to temperatures between K an then cooling uring crystallization (temperatures above the solius) at rates below 1000 K/hr, with some chonrules possibly neeing to cool at rates below 100 K/hr at these temperatures (Jones et al. 2000). It has been suggeste by some authors that more rapi cooling coul also lea to chonrule formation we return to this issue later in this stuy. The relatively low peak temperature (<1450 K) an very rapi cooling (>10 5 K/hr) of the particles in this simulation o not match those generally inferre for chonrules. Fig. 5. The top panel shows the temperature evolution of a chonrulesize particle that encounters a one-imensional, 30-km-wie shock moving at 6 km/s. The bottom panel shows the cooling rate of the same particle at temperatures above the assume solius (1400 K). The cooling rate is zero at the peak temperature of the chonrule but quickly increases to much higher rates. 100-km Planetesimal The panels in Fig. 6 show the same thermal evolution as escribe above, except in this case the shock with is 300 km. The solis are again heate by raiation upstream of the shock, this time with the chonrules reaching a

9 Evaluating planetesimal bow shocks as sites for chonrule formation km Planetesimal Fig. 6. Same as Fig. 5, except the shock is 300 km wie. The slight increase in temperature after the rapi cooling in the top panel is where the particle passes through the en of the shock an is immerse in the cool but fast moving gas of the nebula. The increase in the particle s temperature is ue to the collisions with the fast moving gas. Again, the cooling rate is nearly zero at the peak temperature of the chonrule, but quickly increases to >10 4 K/hr before slowly ecreasing at lower temperatures. temperature of ~1300 K before crossing the shock front. This higher temperature is ue to the fact that the raiation emitte from behin the shock front is more intense ue to the larger volume of hot particles within the shock. The increase with of the shock also allows the chonrules to reach a higher peak temperature (~1580 K) than in the case previously consiere. This is because the stopping istance of the chonrules (~200 km) is less than the with of the shock, an thus they are heate by gas rag over this istance an subjecte to a higher raiation fiel. Since the chonrules completely lose their velocity with respect to the gas, they begin to cool while still within the shocke gas. The higher temperature an ensity of the gas in this region allow more heat to be transferre to the chonrules through thermal collisions, an therefore they cool at a slower rate than in the previous case. Upon exiting the shocke region, the particles are moving at a velocity with respect to the shock front that is much less than when they entere the shock region. The gas, as it exits the shock front, returns to the high value it ha right before entering the shock front (ue to the assumptions mae in this moel), an therefore there is a relative velocity between it an the chonrules. This causes the chonrules to be heate somewhat, which can be seen as the little rise in temperature after the chonrules have coole rapily an before the more graual cooling begins (~0.05 hr after crossing the shock front). As in the previous case, the preicte peak temperature (<1600 K) in this simulation an the cooling rates (>10 4 K/hr) o not match those generally expecte for chonrules. The top panel in Fig. 7 shows the temperature evolution of the chonrules in the same situations escribe, except that the shock is 3000 km in with. The chonrules are heate to an even higher temperature (~1400 K) upstream of the shock front ue to the larger volume of hot particles leaing to a more intense raiation fiel. This larger volume again helps the chonrules reach a higher peak temperature than in the previous cases. The cooling of the particles in this case (bottom panel in Fig. 7) is substantially slower than in the cases previously consiere. In fact, as the temperature of the chonrules approaches the solius, the cooling rate is ~10 3 K/hr, which is the upper limit of expecte chonrule cooling rates. Thus, a shock of this size an strength seems to allow for cooling rates comparable to the cooling rates that chonrules are thought to have experience. While the peak temperature of the chonrules is only ~1600 K, it is not far off from temperatures that chonrules are thought to have reache. In Fig. 8, we show a similar moel run as just escribe, except that the chonrule average emissivity is assume to be 0.1 (as oppose to 0.9). In this case, the chonrules reach a significantly higher peak temperature (~2020 K) ue to the fact that the chonrules cannot raiate as much energy away uring the initial perio of heating by gas rag. The particles then cool, rapily at first, before again reaching a cooling rate of ~10 3 K/hr as the temperature approaches the solius. The cooling rate is not greatly affecte by the change in chonrule emissivity. Since the emissivity was assume to be 0.1, the absorptivity was also assume to be 0.1 to be consistent with Kirchoff s law. Thus, the amount of raiation emitte by the chonrules ecreases by the same factor as the amount absorbe (the raiation fiel is ominate by the ust particles). The interaction between the gas an the ust control how quickly the system cools, an therefore a change in the emissivity has little effect on the cooling of the chonrules. The ust will etermine the cooling rate of the system as long as it is the major source of opacity. If the ust were to be completely vaporize (which was not the case in any of our runs), the cooling rate woul likely be much lower. This woul require the shocks to have greater velocities than consiere here. Various Chonrules:Dust an Soli:Gas Mass Ratios The ratio of solis to gas in a given location of the nebula may have varie greatly ue to the settling of material into a layer at the miplane or ue to the accretion of large planetesimals. Similarly, the size istribution of solis may have varie ue to the level of turbulence in the nebula, the etails of the accretion process, an the location where chonrule formation took place. Thus, we also consiere a number of cases where these ratios were ifferent from those

10 1818 F. J. Ciesla et al. Fig. 7. Same as Fig. 5, except the shock is 3000 km wie. The cooling rate near the solius is approximately consistent with the fastest cooling rates inferre for chonrules though the peak temperature is slightly below what has been inferre. Fig. 8. Same as Fig. 7, except that the particle is assume to have a much lower wavelength-average emissivity. This prevents the particle from raiating as much energy as it is heate by gas rag immeiately behin the shock front, an therefore it reaches a higher peak temperature. The particle then cools at a cooling rate of ~1000 K/hr, which is consistent with what has been inferre for some chonrules. escribe above. All other conitions are the same as escribe for the 1000 km-raius planetesimal case escribe above. The results of some of these cases are qualitatively escribe below. We consiere one case where the istribution of chonrule mass to ust mass was 50:50. The cooling rates of the chonrule-size particles were foun to be much higher than in the cases escribe previously. This was ue to the fact that as the chonrules rift away from the shock front, the optical epth between them an the hot ust immeiately behin the shock front increases more rapily than in the previous cases. While they are heate initially by rag, the heating ue to raiation rapily ecreases. This loss of heating becomes significant as the chonrules are surroune by cooler (compare to the temperature behin the shock front) ust particles, which still heat the chonrules but to a much lesser extent. The opposite was true for the case in which the chonrule to ust mass ratio was 90:10. Due to the ecrease in opacity, the chonrules are expose to the raiation from the hot particles immeiately behin the shock front for a longer perio of time. This helps keep the chonrules from cooling too rapily so the cooling rate is as low as ~10 K/hr. The cooling rate increase rapily (to ~100 K/hr) as the chonrules rifte towar the en of the shock where their raiation is not trappe any longer an escapes to the unshocke regions of the nebula. A similar result was foun when we consiere cases in which the mass ratio of solis to gas suspene in the nebula was not at the solar value. For those cases that resulte in significant increases in the opacity of the nebula (larger values of fine-graine ust), the cooling rates of the chonrules increase. This is ue to the fact that as the chonrules rifte through the shocke gas, the optical epth between them an the hot chonrules immeiately behin the shock front increase at a greater rate. Because of this, the chonrules woul be shiele from a major heat source, an therefore cool at a greater rate. For cases when the opacity of the nebula ecrease (either ue to vaporization of finegraine ust or its epletion ue to accretion), the raiation from the hot particles was able to travel further an help keep the chonrules warm for longer perios of time. This correlation of slower cooling rates an lower opacity of the solis was also foun in Desch an Connolly (2002) an Ciesla an Hoo (2002). In summary, if chonrule-size particles ha emissivities significantly less than 0.9 (Desch an Connolly [2002] suggest a value of 0.8, which ecreases when the chonrules are near their peak temperatures), which may be possible given the uncertainties of the measure optical constants, both the peak temperature an cooling rates coul be consistent with what has been inferre for chonrules if they were heate in a 3000 km-wie shock of the type escribe here. However, cooling rates of K/hr, which most chonrules may have experience (Jones et al. 2000), can be reache only if the amount of ust present in the nebula is small or if much of it is vaporize upon passing through the shock front. DISCUSSION In all but one of the cases presente here, the cooling rates an peak temperatures of the particles were not consistent with what has been generally inferre for chonrules. It has been argue that chonrules coul have forme by multiple instances of low level heating (peak

11 Evaluating planetesimal bow shocks as sites for chonrule formation 1819 temperatures <1700 K) followe by rapi cooling (cooling rates of about a few thousans of K/hr an higher) (Wasson 1996; Wasson an Rubin 2003). If this type of thermal processing i result in textures an chemical zoning similar to those of chonrules, then planetesimal bow shocks coul be possible sites for chonrule formation. Experiments investigating these rapi cooling rates are neee. There was one case presente in which the calculate peak temperature an cooling rate of the silicates matche those inferre for chonrules: the case of a 1000 km planetesimal with low wavelength-average emissivities for the chonrule-size particles. In other cases we stuie, the cooling rates in our runs were foun to ecrease as the amount of ust present ecrease. Thus, it is possible that shocks forme by large planetesimals coul form chonrules if there were only a small amount of material present as fine-graine ust. However, it is unclear how such a scenario may be achieve in the nebula. One possibility is that the larger particles (millimeter-size) coul settle to the miplane much faster than the ust particles. This woul be possible over long perios of time if nebula turbulence works such that only very small (<<1 mm) particles are prevente from settling to the miplane. This woul be possible if the turbulent velocity (α 1/2 c s, where α is the turbulence parameter an c s is the local spee of soun) is greater than the settling velocity of the fine particles (Weienschilling an Cuzzi 1993). Thus, this provies a natural way to separate particles by size. Another possibility is that if the nebula is turbulent, Cuzzi et al. (2001) showe that chonrule-size particles can be locally concentrate by turbulent eies. These eies can locally increase the concentration of chonrule-size particles but not particles of other sizes. More work is neee to investigate this possibility. It shoul be remembere that the assumptions mae in this moel favore slow cooling rates for the particles. A more complex moel, which accounts for three-imensional raiative transfer an relaxation (cooling an expansion) of the gas in the relaxation zone, woul likely preict more rapi cooling rates than those foun here. If one imagines a cube raiating to space but only in one imension (say, the x- irection), it will raiate energy at the rate of 2A x σt 4, where A x is the area of one cube face that is perpenicular to the x- axis, T is the temperature of the cube, σ is the Stefan- Boltzmann constant, an the factor of 2 is ue to the secon cube face that is perpenicular to the x-axis. If the cube is allowe to raiate energy away in all three irections it will lose energy at the rate of 2A x σt 4 + 2A y σt 4 + 2A z σt 4, where A y an A z are the areas of the cube faces that are perpenicular to the y- an z-axes, respectively. Since a cube by efinition has A x = A y = A z, the total rate of energy loss woul be 6A x σt 4. Since we only consier a one-imensional moel, we likely are unerestimating the cooling rates by a factor of ~3, though the exact factor woul epen on what the total volume (an imensions) of the material processe by the bow shock was (whether A x iffers greatly from A y an A z, for example). To fully etermine the factor it woul be necessary to etermine the intensity of raiation in the other two imensions (perpenicular to the initial gas velocity). In aition, as shown in Fig. 3, the gas ensity in a bow shock begins to ecrease immeiately behin the shock front. As the ensity ecreases, the number of collisions between the gas an the solis woul also ecrease. Once at rest with respect to the gas, the rate of heat input into a chonrule by thermal collisions with the gas is given by (Hoo an Horanyi 1991): π 2 q ρ g ( T g T c ) γ k = γ 1 8 π m 1 2 T g (22) where ρ g is the gas ensity, T g is the temperature of the gas, T c is the chonrule temperature, γ is the ratio of specific heat temperatures of the gas, m is the mean molecular mass of a nebular gas molecule, an k is Boltzmann s constant. Taking the typical values of these variables behin a shock wave once the chonrule has lost its relative velocity with respect to the gas (ρ g = 10 8 g/cm 3, T g = 1600 K, T c = 1500 K, γ = 1.4, an m = g), we fin that the rate of heat input to the chonrule is ~ erg/s. If the gas ensity ecreases by a factor of 10, as it oes over a istance of ~5 planetesimal raii in these simulations, then q also ecreases by a factor of 10 (making the generous assumption that the gas an chonrule temperatures remain the same), an thus the rate of heat input woul ecay to erg/s. Assuming that the chonrule is g an has a heat capacity of 10 7 erg/g/k, the loss of this heat input woul correspon to an increase in cooling rate of ~18 K/s (> K/hr). This increase in cooling rate woul take place as the particle move through the istance over which the gas ensity ecrease. Looking at the case of a 1000 km planetesimal, this happens after a istance of ~5000 km. If the particle is moving at an average velocity (relative to the planetesimal) of 1 km/s (a typical spee in the post shock region), this means it takes the chonrule 5000 sec to traverse this istance. Thus, the cooling rate of the chonrule woul increase by K/s/s or ~13 K/hr/s as the gas ensity ecreases. Taking a conservative estimate that chonrules ha to cool through a temperature range of just 50 K at a rate of 1000 K/hr or lower to form the textures an zoning observe in meteorites, this means that the chonrules woul have to have coole for a perio of at least 0.05 hr or 180 sec. Over this time perio, the cooling rate woul have increase by 2300 K/hr using the numbers note above. This again means that the cooling rates presente here likely are unerestimates of the real cooling rates that particles woul have experience in the situations stuie here. Thus, it is likely that even for the case in which the calculate cooling rates are within the expecte range for chonrules, in a more realistic moel, the cooling rates woul be significantly higher than those presente here. The increase