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CHONDRULE FORMATION IN BOW SHOCKS AROUND ECCENTRIC PLANETARY EMBRYOS

Recent isotopic studies of Martian meteorites by Dauphas & Pourmand have established that large (∼3000 km radius) planetary embryos existed in the solar nebula at the same time that chondrules—millimeter-sized igneous inclusions found in meteorites—were forming. We model the formation of chondrules by passage through bow shocks around such a planetary embryo on an eccentric orbit. We numerically model the hydrodynamics of the flow and find that such large bodies retain an atmosphere with Kelvin–Helmholtz instabilities allowing mixing of this atmosphere with the gas and particles flowing past the embryo. We calculate the trajectories of chondrules flowing past the body and find that they are not accreted by the protoplanet, but may instead flow through volatiles outgassed from the planet's magma ocean. In contrast, chondrules are accreted onto smaller planetesimals. We calculate the thermal histories of chondrules passing through the bow shock. We find that peak temperatures and cooling rates are consistent with the formation of the dominant, porphyritic texture of most chondrules, assuming a modest enhancement above the likely solar nebula average value of chondrule densities (by a factor of 10), attributable to settling of chondrule precursors to the midplane of the disk or turbulent concentration. We calculate the rate at which a planetary embryo's eccentricity is damped and conclude that a single planetary embryo scattered into an eccentric orbit can, over ∼105 years, produce ∼1024 g of chondrules. In principle, a small number (1–10) of eccentric planetary embryos can melt the observed mass of chondrules in a manner consistent with all known constraints.


Introduction
Chondrules are submillimeter to millimeter-sized, mostly Fe-Mg silicate spherules found within chondritic meteorites. Chondrule precursors were heated to high temperatures and melted, and then slowly cooled and crystallized to form chondrules, while they were independent, free-floating objects in the early solar nebula. A wealth of elemental and isotopic compositional measurements, as well as mineralogical and petrological information, potentially place important constraints on the melting and cooling of chondrules. The physical conditions and processes acting in the early solar nebula could be deciphered, if the chondrule formation mechanism could be understood.
Various chondrule formation models have been proposed. These are assessed against their ability to match the following meteoritic constraints: (1) Models must explain why most chondrules appear to have formed only millions of years after the oldest solids formed in the solar nebula (calcium-rich, aluminum-rich inclusions, otherwise known as CAIs).
Radiometric (Al-Mg and Pb-Pb) dating shows that nearly all extant chondrules formed ≈ 1.5 − 4 Myr after CAIs, with the majority forming 2-3 Myr after CAIs (Kurahashi et al. 2008;Villeneuve et al. 2009). (2) Models must explain the high number density of chondrules in the formation region. About 5% of chondrules are compound chondrules, chondrules that fused together while still partially molten (Gooding & Keil 1981;Ciesla et al. 2004a). Based on this frequency and an assumed relative velocity 100 cm s −1 , a number density of chondrules ∼ 1 − 10 m −3 has been inferred (Gooding & Keil 1981;Ciesla et al. 2004a). Cuzzi & Alexander (2006) also require a density > 10 m −3 so that the partial pressure of volatiles following chondrule melting remains high enough to suppress substantial loss of further volatiles. This density is significantly higher than the average density of chondrules in the solar nebula. Based on a silicate-to-gas mass fraction 0.5% (Lodders 2003) and the observation that chondrules comprise 75% of the mass of an ordinary chondrite (e.g., Grossman 1988), we expect the chondrules-to-gas ratio to be 0.375% (although this value may be lower if a large percentage of the available solids are locked up in planetary embryos). Assuming a chondrule mass m c = 3.7 × 10 −4 g, consistent with a chondrule of radius 300 µm and internal density 3.3 g cm −3 , and a nebula gas density 1 × 10 −9 g cm −3 (Desch 2007), we would expect a chondrule number density 0.01 C m −3 , where the chondrules-to-gas mass ratio is defined to be 3.75 × 10 −3 C, with C representing an additional concentration factor. Here, the chondrule precursor radius of 300µm is chosen to be consistent with chondrules found in most chondrites. It is reasonable, however, to assume that there was a size distribution between matrix-sized grains and large chondrule precursors in the solar nebula prior to a chondrule-forming event. Morris & Desch (2012) find that cooling rates are increased slightly for simple size distributions, but not enough to affect the general results of the shock model. We therefore consider only a single size for simplicity. In the chondrule-forming region, chondrules appear to have been concentrated by large factors, up to 100 times their background density (for a relative velocity of 100 cm s −1 ).
(3) Models must allow for simultaneous chondrule formation over large regions (> 10 3 km in size) to limit diffusion of volatiles away from chondrules, in order to keep the partial pressures of the volatiles high enough to suppress isotopic fractionation of these volatiles (Cuzzi & Alexander 2006). Likewise, the existence of compound chondrules argues for regions larger than the mean free path of a chondrule through a cloud of chondrules, which is 100 -1000 km for the densities quoted above. (4) Models must explain the relatively slow cooling of chondrules over hours. Comparison of chondrule petrologic textures with experimental analogs show that the 80% of chondrules in ordinary chondrites that have porphyritic textures (Gooding & Keil 1981) cooled from their peak temperatures at rates ∼ 10 − 10 3 K hr −1 and crystallized in a matter of hours (Desch & Connolly 2002;Hewins et al. 2005;Connolly et al. 2006;Lauretta et al. 2006). (5) The partial pressure of Na may have been considerably higher than is easily explained by volatile loss from chondrules. Alexander et al. (2008) recently reported high mass fractions ≈ 0.010 − 0.015 wt% of Na 2 O in the cores of olivine phenocrysts of Semarkona chondrules, indicating that the olivine melt contained substantial Na. The implied partial pressures of Na vapor in the chondrule-forming region are ∼ 10 −5 bar. If this vapor is supplied by the chondrules themselves, chondrule concentrations C > 10 5 would be required. Such a high chondrule concentration would have significant and even incredible physical consequences, but it has been difficult to explain the high Na concentrations in the chondrule olivine by any other method (Alexander et al. 2008). Many additional constraints on chondrule formation exist as well (see Desch et al. 2012, accepted). The model that appears most consistent with the constraints on chondrule formation is the nebular shock model, in which chondrules are melted by passage through shock waves in the solar nebula gas (Desch et al. 2005(Desch et al. , 2012. Of the sources of shocks considered to date, two were found to be most plausible by Desch et al. (2005): large-scale shocks driven by gravitational instabilities (e.g., Boss & Durisen 2005;Boley & Durisen 2008), and bow shocks driven by planetesimals on eccentric orbits (e.g., Hood 1998;Ciesla et al. 2004b;Hood et al. 2009;Hood & Weidenschilling 2011).
A planetesimal or other body on an eccentric orbit will inevitably drive a bow shock in front of it, because its orbital velocity necessarily differs from that of the gas, which remains on circular orbits with nearly the Keplerian velocity v K . A body with eccentricity e will have a purely azimuthal velocity at aphelion or perihelion, but one that differs from the local Keplerian velocity by an amount ∼ ev K /2. At other locations in its orbit the protoplanet also has a radial velocity. If the body's orbit is inclined, this also enhances the relative velocity. The combined effect is that the body's velocity relative to the gas can remain a substantial fraction of the Keplerian velocity (v K ∼ 20 km s −1 at 2 AU) over its entire orbit. As one plausible example, which we discuss further in section 6.1, we show in Figure 1 the velocity difference V s between the gas and solid body on an eccentric orbit with semi-major axis a = 1.25 AU, eccentricity e = 0.2 and inclination i = 15 • . The relative velocity is seen to vary over the orbit but it remains > 3 km s −1 at all times. As the typical sound speed in the gas is < 1 km s −1 , a bow shock is the inevitable result of an eccentric planetary body. The authors above (Hood 1998;Ciesla et al. 2004b;Hood et al. 2009;Hood & Weidenschilling 2011) have argued that planetesimals, bodies up to several ×10 2 km in radius, on eccentric orbits, would be common enough to drive sufficient shocks to melt the observed mass of chondrules, which is 2 × 10 −4 M ⊕ (Grossman 1988).
Despite their presumed ubiquity, planetesimal bow shocks have not been favored as the chondrule-forming mechanism. Because chondrules are associated with asteroids and formation in the asteroid belt, only planetesimals < ∼ 10 3 km in diameter have been considered so far. For such bodies, the size of the heated region, which is necessarily comparable to the size of the planetesimal itself, is probably too small to satisfy the above constraints on the chondrule-forming region. More importantly, the small size of the heated region means chondrules are never far from cold, unshocked gas to which they can radiate and cool. The optically thin geometry is expected to lead to very fast cooling rates, and matching the cooling rates necessary to reproduce the dominant porphyritic texture in particular, < ∼ 10 3 K hr −1 , requires physical conditions that appear extreme (e.g., Ciesla et al. 2004b;Morris et al. 2010a,b). Intuitively, suppression of the radiation that cools the chondrules requires requires the optical depth τ between the chondrule-forming region and the unshocked gas to be > ∼ 1. In their treatment of the problem, Morris et al. (2010a,b) found that optical depths τ > ∼ 0.1 led to cooling rates ∼ 10 3 K hr −1 , just barely consistent with the constraints. Our treatment of chondrule cooling here (see §4) suggests slightly slower cooling, but the cooling rates ∼ 100 K hr −1 consistent with porphyritic textures require optical depths τ ∼ 0.1. The optical depth is a function of the opacity and the size of the region. Dust grains that might otherwise increase the opacity are found to vaporize in shocks strong enough to melt chondrules . The opacity of chondrules themselves, at their background levels, does not yield τ > 0.1 unless the size of the region is l 3 × 10 4 km. Even if chondrules are concentrated by the considerable factor of 100 times their background level, so that n c = 1 m −3 , planetesimals at least several hundred kilometers in radius are required to yield marginally consistent cooling rates (Ciesla et al. 2004b; Morris et al. 2010a,b). To the extent that mass is locked up in large planetary bodies rather than dust and chondrule-sized objects, even higher concentrations of chondrules above background levels are required.
Another difficulty of the planetesimal bow shock model, one that has not been discussed previously, is that due to the small size of the planetesimal, chondrule precursors passing close enough to the body to be melted as chondrules are likely to be accreted by the planetesimal. Unlike the gas, whose velocity is immediately altered at the shock front to flow around the body, chondrules have sufficient inertia that they retain their pre-shock velocity until they have collided with their own mass of gas. Chondrules can be deflected around the body driving the bow shock only by moving a minimum length through the gas past the shock front. For typical parameters this length is ≈ 300 km. The stand-off distance between the bow shock and a 500 km-radius planetesimal driving it is typically < ∼ 100 km (see §2), meaning that accretion onto the planetesimal is probable. This is problematic because chondrules are expected to mix with cold nebular components before accretion, and a given chondrite will contain chondrules with a wide variety of ages.
Chondrules melted in bow shocks around bodies larger than planetesimals may conform better to the constraints on chondrule cooling rates and the size of the chondrule forming region. The terrestrial planets have long been recognized to have formed from planetary embryos, bodies > 1000 km in diameter or even much larger, and it has been recognized that these embryos could have formed in a few Myr or less (Wetherill & Stewart 1993;Weidenschilling 1997). Mars itself is considered a starved planetary embryo (Chambers & Wetherill 1998) show that the protoplanet's orbit remains eccentric and can drive shocks for ∼ 10 5 yr. We find that a single scattered planetary embryo can process up to 10 24 g of chondrules.

Hydrodynamics of Planetary Embryo Bow Shocks
We have numerically investigated the hydrodynamics of a bow shock around a planetary embryo using the adaptive-mesh, Eulerian hydrodynamics code, FLASH (Fryxell et al. 2000). We consider a stationary protoplanet on a cylindrical (axisymmetric) 2-D grid. We simulate the relative motion of the protoplanet through the gas by imposing a supersonic inflow of gas along the symmetry axis, originating at the −z boundary. Our assumed boundary conditions allow for inflow on the −z boundary, reflection on the symmetry axis, and diode (zero-gradient) boundary conditions on the other boundaries. We consider two physical cases: that of a planetesimal and one of a planetary embryo.
In the first case of a planetesimal, we consider a body with mass M p = 2 × 10 24 g, a radius of R p = 500 km, yielding a density of 3.80 g cm −3 (similar to Mars's density).
The computational domain has radius 16,000 km and vertical extent 32,000 km. The simulation is run with 7 levels of refinement, resulting in our best grid resolution of 32 km = X size /[N X block · 2 (levels of refinement−1) ], where N X block = 8. In the second case of a planetary embryo, motivated by the results of Dauphas & Pourmand (2011), we initialize our simulations with a protoplanet of mass M p = 3.20 × 10 26 g (half of Mars's mass) and a radius R p = 2720 km (80% of Mars's radius), with density 3.80 g cm −3 .
Once again, the computational domain has radius 16,000 km and vertical extent 32,000 km, and the simulation is run with 8 levels of refinement. In both cases the body is embedded in a non-radiating H 2 / He gas with uniform initial density ρ 0 = 1.0×10 −9 g cm −3 , pressure P 0 = 5.3 × 10 −6 atm, temperature T 0 = 150 K, and assumed adiabatic index γ = 7/5, appropriate for the solar nebula at around 2 AU (Desch 2007). The gravitational pull of the protoplanet was simulated using a point-mass gravity module.
Before we initialize the inflow, we allow the planetary embryo to acquire a primary atmosphere of H 2 and He. It is straightforward to show that accretion of adiabatic gas will lead to a spherically symmetric profile where r is the distance from the protoplanet center, r g = (γ − 1)(GM p ρ 0 /P 0 )/γ is akin to a scale height. For the case of the planetesimal, r g ≈ 71 km, and the atmosphere that develops is very thin and hardly resolveable by our best grid resolution of 32 km.
Physically it behaves as part of the planetesimal surface in our runs. For the case of the planetary embryo, r g ≈ 11, 500 km. At the protoplanet surface the density is expected to reach 6.2 × 10 −8 g cm −3 , the pressure to reach 1.74 × 10 −3 bar, and the temperature to reach 780 K, values that are 62.5, 330.0, and 5.2 times their background values, respectively.
The effective scale height near the protoplanet surface is ≈ 1000 km, and is easily resolved by our best grid resolution of 32 km. The primary atmosphere acquired by the planetary embryo in our simulations matched this solution within a few percent. The mass of the atmosphere inside 1000 km of the protoplanet surface is very nearly 3 × 10 18 g, also in accord with analytical estimates. We do not directly simulate the effects of a secondary atmosphere of species outgassed from the protoplanet's magma ocean, which would be orders of magnitude denser, but we expect it would behave very similarly dynamically.
We begin our simulation of the bow shock with the protoplanet and primary atmosphere, and introduce gas streaming up from the lower z boundary with density ρ 0 = 1 × 10 −9 g cm −3 and upward velocity V s = 8 km s −1 . We chose this density to represent conditions of the chondrule-forming region, presumed to be at 2-3 AU (but see §6).
However, disk models differ on gas densities in the protoplanetary disk. For example, the density ρ 0 = 1 × 10 −9 g cm −3 is achieved in a minimum-mass solar nebula model (Hayashi 1981) at 1.25 AU (depending on the scale height), or at several (up to 5) AU in more massive disk models (Desch 2007). We chose the relative velocity of 8 km s −1 based on the results of , who showed that this speed was necessary to achieve chondrule melting. Such a large relative velocity may appear to be extreme, but such values are expected to be met episodically throughout planet formation, even though the typical relative velocity is expected to be much lower. A velocity difference of 8 km s −1 between the body and the gas can occur for a range of eccentricities and inclinations, which will be explored in more detail in Section 6. Here, consider the example of a body on an orbit with a semi-major axis a = 1.25 AU and e = 0.29. This body will produce a ∼ 8 km/s bow shock at quadrature (a ∼ 1.1 AU), which is a plausible site for the formation of some chondrules.
Compared with the total population of embryos, such high relative velocities may be rare, but when they do occur, they can be very significant, as we will show.
In both the planetesimal and planetary embryo cases, the evolution of the gas is marked by a reverse shock bouncing off the body surface and multiple shocks reflecting off each other at the symmetry axis. Within a few crossing times (i.e., a few ×10 3 s) for the planetesimal case, and a few × ∼ 10 4 s for the planetary embryo case), a quasi steady-state bow shock structure is reached. An animation showing the case for the protoplanet is shown in Figure 2. Contour plots of gas densities 4 × 10 4 s after the introduction of supersonic material on the grid is shown in Figures 3a and 3b. In both panels the gas density is depicted in blue, with the darkest levels referring to densities of 2.5 × 10 −10 g cm −3 and the brightest levels to 2.5 × 10 −8 g cm −3 . At the bow shock itself the density increases by a factor of 6 at the shock front, from 1 × 10 −9 g cm −3 to 6 × 10 −9 g cm −3 , as expected for a γ = 7/5 gas, in a strong shock.
We observed that the bow shock around the planetesimal quickly achieved steady state, but as is evident from Figure 3b, steady-state is not formally achieved around the protoplanet. This is observed by the varying density in the wake of the protoplanet, and the slightly irregular structure of the bow shock (e.g., the kink at x = 6000 km). We attribute this to the boundary layer that exists between the primary atmosphere and the post-shock flow of gas past the protoplanet. Inside this boundary, evident at a distance ≈ 1000 km from the protoplanet surface, the densities are considerable higher (> 10 −8 cm −3 ) than the densities ≈ 6 × 10 −9 g cm −3 between this boundary and the bow shock. This boundary is unstable to the Kelvin-Helmholtz (KH) instability, and we observed frequent shedding of vortices and rolls of material at this interface; these rolls are evident in Figure 3b at regular (≈ 2000 km) intervals. Reflections of sound waves off of these rolls is responsible for the irregular structure at the bow shock.
Eventually these KH rolls will strip away the atmosphere surrounding the protoplanet.
Numerical simulations of the stripping of gravitationally bound gas by KH rolls at the surfaces of protoplanetary disks by Ouellette et al. (2007) suggest that the efficiency of stripping is about 1%, meaning that the mass of gas that is stripped is about 1% of the mass of the impacting gas. Based on the inflow of gas ρ 0 πR 2 p V s , we estimate a stripping rate 2 × 10 12 g s −1 . This suggests that loss of the primary atmosphere would take > 10 6 s, hundreds of times longer than the duration of our simulation. We were able to quantify this stripping rate via a "dye" calculation in which we kept track separately of material accreted before the supersonic flow of gas on the grid. We found that after an initial loss of gas, the primary atmosphere was being lost on timescales ∼ 10 5 s, suggesting a stripping rate > 10 13 g s −1 . The existence of this interface between an atmosphere and the post-bow shock flow is a fundamental difference between the planetesimal and protoplanet cases.

Chondrule Trajectories in Planetary Embryo Bow Shocks
Having established the dynamics of the gas and the structure of the bow shock surrounding the planetary embryo, we now turn our attention to the trajectories of chondrules and their precursors in the flow. In general, the particles are dynamically coupled to the gas, except in the vicinity of the bow shock, when the gas suddenly changes velocity over a few molecular mean free paths (i.e., over a few meters). Particles will dynamically recouple with the gas on a timescale comparable to the aerodynamic stopping time, where ρ s ≈ 3 g cm −3 is the particle internal density, a c ≈ 300 µm is its radius, ρ g the post-shock gas density, and c s the post-shock gas sound speed. For ρ g = 6 × 10 −9 g cm −3 and T g ≈ 2500 K, the stopping time is 50 seconds. If the planetary body is traveling through the gas at a speed V s = 8 km s −1 , particles will move past the shock front several hundred km before they dynamically couple to the gas. Based on the numerical simulations above, this lengthscale is less than or comparable to the distance between the shock front and the planetary bodies themselves. The expectation is that particles passing through a bow shock will be accreted onto a planetesimal and will flow around a planetary embryo, although clearly a careful calculation of the trajectories is required to determine their fate.
We are also motivated to calculate the trajectories of particles carefully to determine if they will melt and cool in a manner consistent with chondrules. The maximum heating of the chondrules is set by their velocity relative to the gas after passing through the shock front, which sets their degree of melting. Their cooling after melting is set by the optical depth between the particles and the unshocked gas, which in turn is set by the geometry and spatial distribution of the particles. These must be calculated numerically.
We have computed the trajectories of particles by integrating the particles' motions through the gas, assuming the gas flow around the planetary body is in a steady state. We assume the particles are tracers only, with no dynamical effect on the gas. Particles are accelerated by the gravity of the planetary embryo (with mass M p ) and by the drag force due to gas of density ρ g : where r is the position vector of the chondrule (with respect to the planet center), and V c and V g are the chondrule and gas velocities. Particles are in the Epstein limit (much smaller than the mean free paths of gas molecules), and the drag force is given by where we will assume the particle temperature T c locally equals the gas temperature T g , and where wherem = 2.33m H is the mean molecular weight of the gas and k is Boltzmann's constant (Probstein 1968). Gas velocities V g , temperatures T g and densities ρ g are found from the output of the hydrodynamic simulations, using bilinear interpolation on a grid with spatial resolution of 32 km. Particle velocities are integrated using a simple explicit scheme that is first-order in time, with timesteps of 0.05 s. Particles move no more than 0.4 km during each timestep, which is significantly smaller than the spatial resolution of the hydrodynamics simulations.
The trajectories of chondrules around both the planetesimal and the planetary embryo are depicted in Figures 3a and 3b. These are seen to be significantly affected by the size of the region between the shock front and the body. At the shock front the gas undergoes a nearly instantaneous shift in velocity. Chondrules regain dynamical equilibrium with the gas only after they move through the gas by a stopping length V rel t stop , where t stop is the stopping time (Equation 2) and V rel is the velocity of a chondrule with respect to the gas. This quantity depends only on the gas density ρ g , not the radius of the planetary body. For both the planetesimal and planetary embryo cases, the stopping time t stop ≈ 50 s (assuming a gas density ρ g = 1 × 10 −9 cm −3 ), over which time the chondrules slow from a velocity (relative to the planet) of 8 km s −1 to about 2 km s −1 . Assuming an average velocity ≈ 5 km s −1 , chondrules will typically move about 250 km past the shock front. This stopping length is to be compared to the size of the post-shock region, specifically the standoff distance between the shock front and the planetary body itself. For the R p = 500 km planetesimal case, this standoff distance between the shock front and the planetesimal is ≈ 110 km ≈ 0.2 R p . It is therefore not surprising that all chondrules with impact parameter b < 460 km are accreted onto the planetesimal. In contrast, for the R p = 2720 km planetary embryo case, the standoff distance to the planet is ≈ 2000 km ≈ 0.8 R p , or ≈ 1400 km ≈ 0.5 R p to the atmosphere, much greater than the stopping length. Thus, the vast majority of chondrules dynamically recouple with the gas before hitting the planet.
Only those chondrules with b < ∼ 400 km might be accreted, and chondrules that escape accretion flow around the body. Along the way, their trajectories are affected by the KH instabilities between the shocked gas and the primary atmosphere. This makes it difficult to predict their exact trajectories, but it is clear that they will escape accretion onto the body.
Moving forward it is important to distinguish between particles that pass through the shock front more directly and are melted as chondrules, and those particles that pass through the shock front far from the body and are not melted as chondrules. Only particles that pass through the shock front with sufficient velocity V rel relative to the gas will melt as chondrules. The heating rate of chondrules after they pass through the shock front scales as V 3 rel (up to the liquidus temperature). As we discuss in §4, a shock speed of 6 km s −1 is required to ensure melting of chondrules, which is equivalent to V rel > ∼ 5 km s −1 . The relative velocity between chondrules and gas is a function of the impact parameter b, and is easily calculated for the case b = 0 km (chondrules and gas impact the body directly).
Just past the shock front, chondrules retain their velocity but gas is immediately slowed by a factor of (γ + 1)/(γ − 1) ≈ 6. The velocity difference between gas and chondrules is thus ≈ (5/6)(8 km s −1 ) ≈ 6.7 km s −1 . As the impact parameter b is increased, this relative velocity decreases. This is because only the component of the velocity normal to the shock front is reduced, whereas the tangential component of velocity is conserved. As b increases and the shock becomes increasingly oblique, the relative velocity is decreased, and therefore the chondrule heating rate is decreased. We therefore anticipate a maximum impact parameter for precursors to be melted as chondrules.
This variation of relative velocity with impact parameter is indeed seen in our numerical simulations, as illustrated in Figures 4a (planetesimal case) and 4b (planetary embryo case). In both cases, the maximum relative velocity at b = 0 km approaches ≈ 6.7 km s −1 , and decreases with increasing impact parameter. In the R p = 500 km case, the relative velocity drops below the critical threshold for melting, 5 km s −1 , for impact parameters > ∼ 400 km. For the R p = 2720 km planetary case, on the other hand, particles experience V rel > 5 km s −1 for impact parameters as high as b ≈ 4300 km. If the planetary embryo had even higher speed with respect to the gas, this maximum impact parameter would be expanded. Because of the slightly irregular shape of the shock front, some regions are not as oblique as in others. In the snapshot of Figure 3b, for example, particles with b ≈ 6000 km encounter the shock more directly and also may experience thermal histories consistent with chondrules. Note that even particles with b = 10, 000 km will be significantly heated (by a shock with speed 2.5 km s −1 ), but will not be melted as chondrules. Comparing the cross sections, the vast majority (> 80%) of particles encountered by the bow shock will be heated, but not melted.
There is generally a maximum impact parameter for which particles are melted as and will produce shocks that stand off farther from the body, decreasing the fraction of particles accreted.
One important caveat to the results stated above is that the chondrule trajectories were calculated under the assumption that the gas flow was in steady state. This assumption is valid for gas as it flows to the shock, and even in the immediate post-shock region, but is manifestly not valid at the boundary where the post-shock gas flows past the planetary atmosphere, since this boundary is strongly affected by KH rolls. Figure 3b makes clear that where chondrules intercept a KH roll (e.g., at impact parameters 2000 km and 5000 km), the gas flow drives chondrules out of the KH roll. Likewise, at impact parameter ≈ 4000 km, characterized by the absence of KH rolls, the gas is flowing from the shock front inward, driving chondrules into the void. Thus, there is a tendency for chondrules to "skim" the boundary surface between the atmosphere and the post-shock gas, although it is difficult to confirm this tendency under the assumption of a steady-state flow. Chondrules in the vicinity of the flow-atmosphere boundary have a range of velocities ≈ 1.8 − 5.6 km s −1 (see below), while the KH rolls themselves have velocities (pattern speeds) on the order of 3.5 km s −1 , so that chondrules will probably move in and out of KH rolls, but not at the rates they do so in this calculation, in which the gas flow is kept in steady state.
The trajectories of chondrules determine their density in the chondrule-forming region.
In what follows, we specifically focus on particles that achieved temperatures high enough to melt as chondrules. To be specific, we consider the particles processed by the planetary embryo bow shock, in Figure 2b, which will pass through a slice with Y = +3000 km near the shock front, with X ≈ 7000 − 10000 km. Considering just the subset of those particles that melted as chondrules, they have impact parameters b = 400 − 4300 km, and pass through X = 7180 − 8060 km. Particles with the lower end of impact parameter (b = 400 km) move most slowly, at speeds ≈ 1.8 km s −1 , and take 2-4 hours to reach this region. Chondrules with the higher impact parameter (b = 4300 km) tend to have speeds ≈ 5.6 km s −1 and take just under 1 hour to reach this region. On average, chondrules' velocities are reduced from their pre-shock values (8 km s −1 ) by factors ≈ 2.3, implying that chondrule densities will increase by a factor 2.3 as they slow. In addition, the chondrules are also seen to be geometrically concentrated as they are funneled into a smaller conical volume in past the bow shock. This increases their density by a factor (4300 2 − 400 2 )/(8060 2 − 7180 2 ) = 1.37.
Combined, the chondrule density is raised above the background value by a factor of 3.2.
A similar calculation for those particles with impact parameters 4300 km < b < 10000 km shows the density of these particles is increased by a similar factor. These particles will not have achieved the same peak temperatures as those particles with smaller impact parameters, but they will be almost as hot. For example, if particles with b = 0 km achieved peak temperatures of 2000 K, and peak temperature scales as V 3/4 rel , then Figure 3b suggests that particles with b = 8000 km will have peak temperatures ≈ 1400 K.
The chondrule densities derived above bear on the frequency of compound chondrules.
The mean free path of a chondrule in a sea of chondrules is l mfp = (n c 4πa 2 c ) −1 . For a c = 300 µm and assuming a compression of a factor of 3.2 in the post-shock region, we find l mfp ≈ 2.7 × 10 4 C −1 km. For C = 10, the mean free path is 2700 km (about the radius of the planet). Within the part of the flow containing chondrules, their velocities range between 1.8 and 5.6 km s −1 over a distance 800 km, shorter than the mean free path. Chondrules therefore will collide with other chondrules at relative velocities ∆V up to and above that needed to shatter chondrules, which has been estimated to lie in the range 1 m s −1 (Kring 1991) to 100 m s −1 (Gooding & Keil 1981). We assume a median threshold for shattering of 10 m s −1 , assert that collisions with chondrules up to (and beyond) this speed will occur, and that these will dominate the production of compound chondrules. The compound chondrule frequency f we predict is where t plast is the time for which chondrules remain plastic enough to stick after a collision.
The observed frequency of compound chondrules is reproduced in a planetary embryo bow shock assuming chondrules are concentrated by factors C ≈ 10, although the exact percentage will depend on the maximum speed at which chondrules can collide and stick without shattering.
Finally, the density of chondrules in the post-shock region will also bear on the optical depths between chondrules and the cool, unshocked gas, which we estimate as follows. The optical depth is τ = n c ( πa 2 c ) l, where l is the distance between the locations of chondrules and the bow shock, and πa 2 c is the wavelength-integrated absorption cross-section, with ≈ 0.8 in the near-infrared (Li & Greenberg 1997). Dust evaporates at the shock front, so the opacity is due solely to chondrules . The number density of chondrules in the background nebular gas is where we have assumed a chondrule radius a c = 300 µm, an internal density ρ s = 3.3 g cm −3 , and defined C ≡ (ρ c /ρ g )/(3.75 × 10 −3 ). We estimate l by considering those chondrules (objects with b < 4300 km) with most probable impact parameter, b ≈ 3000 km. These chondrules pass Y = +3000 km at X ≈ 7800 km. The particles with the largest impact parameter to be chondrules (b = 4300 km) pass through Y = +3000 km at X = 8060 km.
Importantly, though, even particles with b = 8000 km will achieve near-chondrule-like temperatures of 1400 K, and pass through Y = +3000 km at about X = 9000 km. The unshocked gas lies at X > 10000 km at Y = +3000 km. Thus the typical chondrule lies 250 km from the edge of the cloud of particles shocked enough to melt as chondrules, but 2100 km from the bow shock (after correcting for the tilt of shock front). We assume chondrules lie 2200 km from the bow shock and consider chondrules to be concentrated above their background value by a factor of 3.2, to find The gradual spatial transition from shocked chondrules to unshocked gas makes it difficult to define l more exactly. Nevertheless, this calculation demonstrates that if chondrules are concentrated by factors of about 10, then τ ∼ 0.16 for most chondrules melted by planetary embryo bow shocks.

Thermal Histories of Chondrules in Planetary Embryo Bow Shocks
Having calculated the trajectories of chondrules past the planetary embryo, an important goal is to calculate their thermal histories. These are influenced strongly by the absorption of infrared radiation emitted by nearby chondrules, and by the proximity of a cooler region into which the chondrules can radiate (without receiving a return of radiation).
In the previously studied cases of large-scale shocks driven by disk-wide gravitational instabilities (e.g., , the shock front is planar and the geometry is 1-D. The cooler region into which chondrules radiate is the pre-shock gas (ahead of a radiation front). Chondrule trajectories are parallel to each other and normal to the shock front, and chondrule properties do not vary in the lateral direction. In contrast, following melting by a bow shock around a planetesimal or planetary embryo, the cooler region to which chondrules radiate is in the lateral direction, and the shock clearly has a 2-D geometry.
The only correct way to calculate chondrule thermal histories is to compute the fully 2-D radiation field in cylindrical geometry, but the chondrule trajectories displayed in Figure 3b suggest that a perturbation to the 1-D chondrule thermal histories may yield an adequate solution.
First, the particles are seen to flow in a more or less laminar fashion, without crossing paths. This suggests that chondrule thermal histories are a function of distance past the shock front, as in the 1-D example. Second, chondrules on parallel trajectories do not differ from each other significantly. As stated above, it takes about 3 hours for chondrules with the lowest initial impact parameter (b = 400 km), and about 1 hour for chondrules with the highest impact parameter (b = 4300 km), to reach a certain representative point downstream (Y = +3000 km). We show below that typical chondrule cooling rates are roughly 60 K hr −1 , so that at this location all chondrules are within about 100 K of each other. Compared to the typical chondrule temperature at this location (≈ 1600 K), this difference is not significant. Third, these properties hold even as the particles move downstream by roughly 20,000 km (not shown in Figure 3b). Further downstream, rarefaction of the shocked gas is presumed to take place. Indeed, some hints of this are seen in the slightly diverging chondrule trajectories in Figure 3b, but this does not affect the gross geometry. Certainly the flow remains basically the same for at least the 5 hours it takes for chondrules in our simulations to cool below their solidus temperatures. These facts suggest that a simple alteration to the 1-D code of , to allow for loss of energy by radiation into the unshocked gas, may allow a useful estimate of chondrule thermal histories.
We accommodate for radiative losses in an approximate way using the formulation of Morris et al. (2010a,b). Specifically, we conduct a 1-D simulation as in , based on Desch & Connolly (2002), including emission, absorption, and transfer of continuum radiation from solids, and the molecular radiation from H 2 O, dissociation and recombination of H 2 molecules, and evaporation of small particles. However, we replace the 1-D radiation field J seen by chondrules with one that is a mixture of the local 1-D radiation field and the radiation field they see from the cool gas on the other side of the bow shock. To calculate this mixed radiation field, we consider chondrules to lie in one of two semi-infinite spaces separated by a plane, which represents the bow shock. As seen in Figure 3b, chondrules tend to move very close to, and parallel to, the shock front. We assume they all remain at a fixed optical depth τ from the bow shock. On the unshocked side of the bow shock, we assume the temperature is that of the background nebula, T bkgrnd ≈ 150 K. On the other side lies heated chondrules, and the hot (T ∼ 10 3 K) gas in the planetary embryo's wake. The column density across the wake region is ∼ 10 g cm −2 , which for a solar-composition gas yields a column density of water molecules N H2O ∼ 10 21 cm −2 , which is optically thick in the infrared (Morris et al. 2009). In addition, gas that might contain dust is mixed into the wake region, especially at Y > +8000 km. This justifies the assumption of a semi-infinite space at high temperature, which we assume is the local temperature of chondrules, T ch . Making these assumptions, it is straightforward to show that chondrules see a radiation field with mean intensity where B is the Planck radiation function and E 2 is the second exponential integral.
We replace the radiation field seen by chondrules with this quantity, where τ is an input parameter describing the optical depth between the chondrules and the gas. We note that the previous work of Morris et al. (2010a,b) differed in two ways. In that work, E 2 (τ ) was approximated by exp(−τ ), which is a reasonably good assumption. That work also assumed the wake of the planetesimal was not optically thick, so that exp(−τ ) was used instead of the more appropriate E 2 (τ ) used here. Because of these differences, cooling rates are roughly an order of magnitude slower here than found by Morris et al. (2010a,b). As discussed above in §3, we consider τ = 0.16 to be a typical value for those particles that melt as chondrules (impact parameters from 400 to 4300 km), provided C ≈ 10. Based on this, E 2 (0.16)/2 = 0.313 = exp(−1.16), meaning that the radiation field seen by chondrules is roughly 69% the radiation field they would normally see in the 1-D shock, mixed with 31% of the background radiation field. Chondrules are warmed by the radiation field from nearby chondrules, but not as much as they would be in a 1-D shock.
We now consider the possible effects of H 2 O line cooling in this case.  found that molecular line cooling due to H 2 O was negligible in large scale shocks, due to the combined effects of high column densities of water seconds after the shock, backwarming, and, most importantly, H 2 recombinations. However, we must consider whether line cooling would become significant in the geometry of a bow shock.
Recall from above that the column density of water molecules across the wake region is N H2O ∼ 10 21 cm −2 , so the region is optically thick to line photons shortly after the shock (Morris et al. 2009). Not only is the wake region optically thick to line photons, but the high column density of water also tends to block continuum radiation escaping through the wake region. Although the effects of backwarming may be reduced somewhat due to the bow shock geometry, buffering by H 2 recombinations will be unaffected. The combination of high column densities of water and buffering by H 2 recombinations will dominate over any reduction in the efficiency of backwarming due to the geometry. Therefore, we still consider H 2 O line cooling negligible in bow shocks.
In Figure 5, the thermal histories of chondrules are shown for four possible values of the shock speed: V s = 5 km s −1 , 6 km s −1 , 7 km s −1 and 8 km s −1 . In each case chondrules are heated by absorbing infrared radiation emitted by other chondrules in the vicinity, by thermal exchange with the gas in the post-shock region, and by frictional heating in the minute or so past the shock where the chondrules move at supersonic speeds V rel , with respect to the gas. As described above, the radiation field seen by chondrules has been artifically lowered from the radiation field they would normally see, using the factor τ = 0.1 (τ = 1). The peak temperature is sensitive to the supersonic heating rate, which scales as V 3 rel (up to the liquidus temperature), and strongly correlates with shock speed. We find T peak ≈ 1430 K for V s = 5 km s −1 , T peak ≈ 1800 K for V s = 6 km s −1 , T peak ≈ 1820 K for V s = 7 km s −1 , and T peak ≈ 1820 K for V s = 8 km s −1 . Note that thermal histories become increasingly uncertain for times > 5 hours (i.e., for temperatures < 1400 K) because the assumption of a 1-D geometry becomes increasingly less valid.
Chondrule peak temperatures must exceed 1770 -2120 K to produce porphyritic textures (Hewins & Connolly 1996;Desch & Connolly 2002). Formation of chondrules therefore requires V s > ∼ 6 km s −1 , or V rel > ∼ 5 km s −1 . For the runs where peak temperatures were sufficient to melt the chondrules, the cooling rates through the crystallization temperature range (≈ 1400 − 1820 K, Desch & Connolly 2002) were < 10 2 K hr −1 (the cooling rate for chondrules in the shock with V s = 6 km s −1 did not begin to cool slowly until their temperatures dropped below 1400 K). If we repeat our calculations with C > ∼ 100, we recover the optically thick, 1-D limit for which cooling rates are < 10 2 K hr −1 . If we reduce C below 10, the chondrule cooling rates increase. They may not increase significantly if the wake region remains optically thick, but if it does not, then C ∼ 1 leads to cooling rates ∼ 10 3 K hr −1 (Morris et al. 2009). From all this, we conclude that both peak temperatures and cooling rates are consistent with formation of porphyritic textures of chondrules, provided shocks speeds are V s > 6 km s −1 (V rel > 5 km s −1 ), and C > 10. Lower particle concentrations may also be consistent with cooling rates < 10 3 K hr −1 , although we have not tested this. The required shock speeds are obtained so long as the planetary embryo has sufficient orbital eccentricity and/or inclination.
Because cooling rates are sensitive to chondrule concentrations C, it is worth considering likely values of C as well as the lengthscales over which such concentrations are achieved.
One mechanism that may lead to a high concentration of chondrules is settling to the midplane, which is inferred observationally in certain young protoplanetary disks (Furlan et al. 2011). In this scenario, the level of turbulence at the midplane is low enough that chondrules have a scale height h c significantly smaller than the scale height H of the gas. where Ω is the Keplerian orbital frequency, t stop = ρ s a c /ρ g c s is the chondrules' aerodynamic stopping time as defined in Section 3, ν = αc s H is the turbulent viscosity, c s is the sound speed of the gas, and H = c s /Ω. The chondrule scale height is where h/H = (γ + 1) −1/4 S 1/2 . If α Ω t stop , then h H and chondrules may be concentrated at the disk midplane. Assuming ρ g = 1 × 10 −9 g cm −3 , C = 0.7 km s −1 (T = 150 K) and Ω = 7 × 10 −8 s −1 (r = 2 AU), t stop = 1400 s and Ω t stop = 1 × 10 −4 . If α 10 −4 then chondrules will concentrate to the midplane.
If the turbulent viscosity of the disk is dominated by the magnetorotational instability, it is possible, but not certain, that α may be low enough at the midplane to allow chondrules The molecular viscosity ν m = η/ρ g , where η is given by Sutherland's formula. Following the discussion in Desch (2007), and assuming a solar composition gas with T = 150 K and the parameters above, we calculate η = 59 µP and ν m = 5.9 × 10 4 cm 2 s −1 . For α = 10 −4 we determine the size of particles necessary for concentration is a c = 280 µm. Chondrule precursors with radius 300 µm, as we have assumed in our model, have almost the exact size to be optimally concentrated. Particle concentrations C ∼ 10 are easily achieved (Cuzzi & Hogan 2003).
Even if a significant fraction of the mass of solids is locked up in large planetesimals, there will be many regions in the nebula that have these concentrations of chondrule

Chemical Environment of Planetary Embryo Bow Shocks
We now consider the formation environment experienced by chondrules in bow shocks around planetary embryos, which we expect to be quite different from the environment in planetesimal bow shocks. Dauphas & Pourmand (2011) have suggested that Mars accreted roughly half its mass within 1.8 +0.9 −1.0 Myr after the formation of CAIs. It therefore must have contained abundant live 26 Al when it formed, sufficient to melt the planet (Dauphas & Pourmand 2011;Grimm & McSween 1993). It is presumed that such a body would also form a magma ocean, would convect, and would rapidly outgas volatile species. Zahnle et al. (2007) have considered the outgassing of volatiles from the Earth following the Moon-forming impact, when the Earth was in its magma ocean state, and concluded that an atmosphere in chemical equilibrium with the magma could rapidly develop in < 10 3 yr. Elkins-Tanton (2008) showed that the mantle of an Earth-sized planet takes 5 Myr to become 98% solidified. This suggests that planetary embryos could maintain a magma ocean and outgas volatile species for several Myr, perhaps aided by the continuous decay of 26 Al.
The composition of the planetary embryo's atmosphere probably was similar to the Earth's outgassed atmosphere during its magma ocean stage. Zahnle et al. (2007) considered the atmosphere outgassed from the proto-Earth's magma ocean. They calculated that a fraction of the Earth's CO 2 would be outgassed, equivalent to a partial pressure P CO2 ≈ 100 bar, and that a substantial fraction of the H 2 O would be outgassed too, equivalent to a partial pressure P H2O ≈ 100 bar. Other moderately volatile species such as S, Na, Zn, Cl and K, are also expected to be outgassed during the magma ocean stage (Zahnle et al. 2007;.
We estimate the abundances of these species in the atmosphere of the planetary embryo by assuming that while outgassing may continue for many Myr, at any one instant in time the magma ocean is in chemical equilibrium with the atmosphere. The mass fraction of CO 2 dissolved in the magma scales linearly with the partial pressure in the atmosphere, as x CO2 = 4.4 × 10 −7 P CO2 1 bar (12) (Stolper & Holloway 1988). It is straightforward to show that if the planetary embryo has the same total abundance of CO 2 (C) that P CO2 will scale as g 2 , where g is the gravitational acceleration at the planet surface. We accordingly estimate P CO2 ≈ 10 bar.
The solubility behavior of N 2 is similar to that of CO 2 (Fricker & Reynolds 1968) and we assume P N2 ∼ 0.1 bar. The mass fraction of water dissolved in the magma, x H2O , is related to the partial pressure of water vapor in the atmosphere, P H2O , by the relation x H2O = 6 × 10 −7 P H2O 1 dyn cm −2 0.54 (13) (Fricker & Reynolds 1968 Dann et al. 2001). We take a median value, x H2O = 0.2wt%, as a representative value (a mass fraction equivalent to the Earth having 8 oceans, in line with current estimates by Mottl et al. 2007), so that the total mass of H 2 O in the planetary embryo is 6.6 × 10 23 g, and P H2O = 3.3 bar. Assuming M p = 3.31 × 10 26 g, R p = 2720 km, and a gravitational acceleration g = 300 cm s −2 , we determine an equilibrium mass of H 2 O in the atmosphere of 1.0 × 10 22 g, which is a small fraction of the total mass. The equilibrium abundance of Na in the atmosphere is complicated by the fact that it may dissolve in the magma either as Na 2 O or as NaOH, in which case its solubility is affected by the water content. van Limpt et al. (2006) considered Na solubility for silicate glasses and found the following relationship to hold when Na dissolves as NaOH (i.e. in the presence of H 2 O): For P H2O = 3.3 bar, we find P NaOH ≈ 2 × 10 −3 bar. Altogther, we infer that if the bulk abundance of water on the planetary embryo is 0.2%, then the planetary embryo's atmosphere during the magma ocean stage is mostly (88wt%) CO 2 with partial pressure 10 bar, some (12wt%) H 2 O with partial pressure 3.3 bar, and alkalis and volatiles being trace species, with Na having a partial presure ≈ 2 × 10 −3 bar. The total mass of the atmosphere is ≈ 4 × 10 22 g. This atmosphere will be continuously stripped by KH instabilities at the interface between the atmosphere and the post-shock gas streaming by the planet. We did not include such an outgassed atmosphere, but the primary atmosphere accreted by the planetary embryo in our simulations plays the same role. The stripping by KH rolls at the interface between the planetary atmosphere and the post-shock gas, in particular, is expected to follow the same general trends. We find using dye calculations (see §2) that the atmosphere is lost at a rate ≈ 2 × 10 12 g s −1 , about 1% of the rate at which the planetary embryo intercepts nebular gas. This is in line with similar rates of KH stripping in other examples of strongly bound gas (e.g., around protoplanetary disks; Ouellette et al. 2007). This is sufficiently rapid to deplete the atmosphere after about 600 years, but outgassing can continue to replenish the atmosphere as the magma ocean convects (on shorter timescales) and bring new volatiles to the surface. The planetary inventory of water, in particular, in this example could persist for 8 × 10 4 yr. The reservoirs of Na and other species would persist for many Myr.
Not only are gases from the planetary embryo's atmosphere stripped by KH rolls at the interface with the shocked gas, as seen in Figure 3; chondrules also make excursions in and out of these rolls and this planetary gas. Chondrules will be exposed, at least some of the time, to volatiles such as H 2 O and Na outgassed from the planetary embryo's magma ocean. The scale height of the atmosphere proper (which has been shocked to ≈ 2500 K) is approximately 185 km, so even at the 800 km distance between the planetary surface and the boundary layer at the top of the atmosphere, gas pressures should be ∼ 1% of their values at the surface. We estimate P H2O ∼ 3 × 10 −2 bar and P Na ∼ 2 × 10 −5 bar. Chondrules passing through KH rolls will experience approximately these partial pressures of volatile species.
It is important to note that when we calculated approximate chondrule thermal histories in §4, we assumed the only gas to which chondrules were exposed was the shocked nebula gas, with pressure ∼ 10 −3 bar. If chondrules pass in and out of the plumes of planetary atmosphere gas in KH rolls, they will see much higher presssures, ∼ 10 −1 bar.
This could potentially affect chondrule thermal histories in two ways: during the first minute past the shock front, higher gas densities can more rapidly decelerate the chondrule and heat it by friction; and thermal exchange with the hot gas will become more efficient.
Neither of these effects is likely to drastically alter the thermal histories of the chondrules considered here, because nearly all of the chondrule deceleration takes place in the shocked nebula gas during the first minute, outside of the planetary atmosphere (see Figure 3b).
Also, the denser plumes of gas are not likely to be at significantly different temperatures, and chondrule temperatures will be dominated by radiative effects. A full treatment of chondrule thermal histories in future work must more fully consider these effects.
Elevated partial pressures of volatiles may explain puzzling features of the chondrule formation environment. An enhanced abundance of water vapor is potentially consistent with indicators of elevated oxygen fugacity during chondrule formation (Krot et al. 2000;Fedkin & Grossman 2006;Grossman et al. 2011). Partial pressures P H2O > 10 −3 bar have been suggested to explain the fayalite content of chondrule olivine (Fedkin & Grossman 2006). Likewise, the high partial pressure of Na vapor may potentially explain the finding by Alexander et al. (2008) that olivine phenocrysts in chondrules from Semarkona contain ∼ 0.010 − 0.015wt% Na 2 O, even at their cores, strongly implying that Na was dissolved in the melt as the olivine crystallized. Alexander et al. (2008) expressed the needed partial pressures in terms of a density of solids, assuming that the Na derived from the chondrule melts and that 10% of this Na entered the gas phase. Their inferred chondrule densities ranged from low values up to at least ≈ 400 g m −3 . Assuming the chondrule is 0.01wt% Na 2 O, this implies a Na vapor density 3 × 10 −9 g cm −3 and a partial pressure of Na ≈ 2 × 10 −5 bar. Remarkably, the volatiles escaping from the planetary embryo's magma ocean and mixing with the post-shock gas may provide the partial pressures of H 2 O and Na assumed to have been experienced by chondrules.
We have not modeled in detail the production of a secondary atmosphere, nor quantified in detail how that atmosphere is stripped, or how often chondrules encounter this stripped gas. We have demonstrated that chondrules do make excursions into gas that is strongly bound to the planet, and that this gas may contain elevated abundance of species outgassed during a planet's magma ocean stage, including H 2 O, CO 2 , Na, and K vapor, in particular. Our best estimates of the partial pressures of water vapor and Na vapor are consistent with values needed to provide an environment with elevated oxygen fugacity and the ability to suppress evaporation of alkalis, although our results depend on the assumption x H2O = 0.2wt%. Further tests are required to better test these ideas quantitatively.

Masses of Chondrules Formed by Planetary Embryo Bow Shocks
A very large number of chondrules can be produced by a single planetary embryo on an eccentric and/or inclined orbit. Chondrules will be produced as long as the velocity difference between the body and the gas is V s > ∼ 6 km s −1 , which requires the eccentricity e and inclination i of the body to be sufficiently large. Over time, however, the interaction of a planetary embryo with nebular gas will cause its eccentricity and inclination to damp to low values. The total number of chondrules produced depends on the e and i damping timescales. In this section we construct a model for estimating the orbital conditions, duration, and mass processed during a single scattering event. The model includes e and i damping due to gas drag and disk torques, as well as semi-major axis evolution due to energy dissipation from drag. Migration due to torques is ignored for convenience and to reflect the fact that the magnitude and direction of migration can differ depending on the equation of state (e.g., Paardekooper & Mellema 2006) and on whether radiative transfer is included or not (e.g., Bitsch & Kley 2010). Below, we discuss the methods for including torques and drag, in turn, and then combine them to estimate the orbital evolution and mass of chondrules processed.

Eccentricity Damping of Planetary Embryos
We first consider eccentricity damping by torques. Recent simulations by Cresswell et al. (2007) and Bitsch & Kley (2010) demonstrate that eccentric planets in radiatively cooling disks follow analytic limits. When the eccentricity e is high, de/dt| h = −K e e −2 (Papaloizou & Larwood 2000), where K e is a constant. When e is low, de/dt follows an exponential decay that is well characterized by de/dt| = −e/t ecc = −e 0.78 q (Σa 2 /M * ) h −4 Ω, where q is the planet-to-star mass ratio, a is the planet's semi-major axis, M * is the mass of the star, h = H/a for scaleheight H(a) at heliocentric distance a, Σ(a) is the total surface density at heliocentric distance a, and Ω(a) is the average angular speed of the planet (Tanaka & Ward 2004). The low-e limit is valid for e 0.1. Using these results, we explore parameter space without using costly hydrodynamics simulations by making the ansatz that the total rate is the harmonic sum of the limits, such that de/dt = −K e e/(K e t ecc + e 3 ). We select a family of solutions by varying only K e and assuming that K e t ecc is a constant over the parameter space of interest. Based on Cresswell et al. (2007) we set K e t ecc = 0.00253, which allows us to match their eccentricity evolutions.
For the inclination evolution, the rates follow the same limiting behavior as in Kley (2011), we make the same assumptions for the total inclination damping as we have made for the eccentricity damping, and we vary K i while keeping constant K i t inc = 0.00075.
Strictly speaking, the inclination and eccentricity damping are coupled, but they are treated independently for the estimates we present here.
Now we consider the effects of gas drag on large solids. Planetesimals and planets are much larger than the mean free path of gas molecules λ, so the drag is in the Stokes regime, i.e., the Knudsen number Kn ≡ 2R p /λ 1. In this regime the gas drag takes the form where δv is the velocity difference, k d is a coupling coefficient, and ζ = (8/π) 1/2 (ρ g c a )/(ρ s R p ) = (8γ/π) 1/2 t −1 stop , where c a is the adiabatic sound speed and t stop and other quantitites are defined as before. The coupling coefficient k d depends on the Reynolds number Re = 3 (π/8) 1/2 M/Kn, where M = δv/c s . In the regime of interest (Re > 1500), k d = 0.11 Re (Paardekooper 2007). This yields d(δv)/dt ≈ −(δv)/t stop , in accord with Adachi et al. (1976).
The evolution of the planetary embryo's orbit can now be determined if δv can be expressed as a function of e and i. In general δv will vary along the body's orbit, depending on the value of the true anomaly φ. We model the damping by solving Kepler's equation for following the orbit of a planet, and by using a predictor-corrector for modifying the eccentricity and inclinations over a timestep. The time derivatives of the radial and tangential velocity components for a Keplerian orbit are where µ = G(M * + M p ). We find an expression for the change in eccentricity by keeping only the terms with de/dt, as the other terms represent the unaltered Keplerian orbit. This The change in eccentricity over time can thus be expressed in terms of the change of the relative velocity over a computational time step. A similar approach can be taken with the inclination, where d(δv z )/dt ≈ (µ/r cyl ) 1/2 di/dt, where r cyl is the projected radial separation.
We have assumed small i here because drag predominantly takes place within one scale height of the disk, and we in fact only include drag when |z| < H. Finally, we use the energy gained or dissipated in a timestep, δE, to computed an updated semi-major axis a = −(µ/2)(E + δE) −1 .
We now apply these formulae to conditions in protoplanetary disks. In our orbital damping calculations we use a disk model with gas surface density Σ(r) = 149 (r/5.2 AU) −1.5 g cm −2 , a gas scale height H = 0.05r, a midplane density ρ 0 = Σ/(2H), and a stellar mass of 1 M . To illustrate which damping terms are most important, we plot in Figure 6 the eccentricity damping timescale e (de/dt) −1 as a function of planetary mass M p , for our model conditions and with e = 0.4 and i = 0 • . The internal density in this example was set to be ρ s = 5.7 g cm −3 , appropriate for Earth-like bodies but slightly denser than our proto-Mars planetary embryo. Each curve represents damping at pericenter for different a. In Figure 6 it is seen that bodies with small mass (M p < 0.1M ⊕ ) are in the drag-dominated regime, with damping timescales that increase with increasing mass as

Mass of Chondrules Produced Per Embryo
We now calculate the mass of chondrules, M c , produced by a single planetary embryo.
The rate at which chondrules are processed over time is dM c /dt = σ V s ρ c , where σ is the cross section, which we set to π(4300 km) 2 for the planetary embryo described in §2. The density of chondrule (precursors) is defined to be ρ c = f Cρ g , with f = 3.75 × 10 −3 . The concentration factor of chondrules, C, is a local value that can vary spatially throughout the disk due to settling or turbulence. The instantaneous velocity difference between the planetary embryo and the gas is V s , which is also the shock speed. If V s < 6 km s −1 (or greater than 8 km s −1 ) it is assumed that chondrules are not melted (or are completely vaporized), and we set dM c /dt = 0 for that part of the protoplanet's orbit. We calculate the total mass produced by integrating dM c /dt over individual orbits, and over the entire damping history of the protoplanet.
Tables 1-6 list the masses of chondrules produced in bow shocks around planetary embryos with a variety of initial semi-major axes a, eccentricities e and inclinations, i. A lower solids-to-gas ratio would also result in higher chondrule cooling rates, due to lower optical depth, but probably not significantly so; we expect them to remain within the range consistent with chondrule formation (10 − 10 3 K hr −1 ).
As Tables 1-6  Tables 1-6 assume a planetary embryo mass 0.055 M ⊕ . This may be close to the optimal mass for producing chondrules. Provided the body is large enough to actually produce chondrules (i.e., it is not a planetesimal for which chondrules are accreted, but is a large planetary embryo), the cross section σ scales roughly as R 2 p . The product of this cross section and t damp yields the total mass of chondrules produced. For small bodies whose damping is dominated by gas drag, the mass of chondrules that is produced scales linearly with M p . For large bodies whose damping is dominated by disk torques, the mass of chondrules that is produced scales roughly as M −1/3 p . In fact, the mass of chondrules that can be produced per embryo is optimal for Mars-sized bodies with M p ∼ 0.1 M ⊕ .
Nevertheless, as long as the body is larger than about 1000 km in radius, a comparable amount (to within factors of 3) of chondrules will be produced.
The chondrule masses produced per embryo are intriguingly close to the mass 2 × 10 −4 M ⊕ , and in principle a single scattered planetary embryo could produce the entire present-day inventory of chondrules. Nevertheless, to produce the observed mass of chondrules several planetary embryos probably would be required. This is because some of the solids mass is locked up in large bodies, making fewer chondrule precursors available for processing. Also, the present-day asteroid belt may have been depleted since it formed, meaning there were more chondrules originally, potentially by a factor of 100 (Weidenschilling 1977). On the other hand, eccentric planetary embryos may be a common event. Even for the minimum-mass solar nebula mass distribution assumed above, the annulus between 1.5 and 2.5 AU would have contained ≈ 1.5 M ⊕ of solids. If half of the mass was in small (chondrule-like) particles and half locked up in large planetary embryos, potentially ≈ 25 planetary embryos could exist in that one annulus alone.
Our calculations show that, given the right orbital parameters, chondrules can form over a range of heliocentric distances 1.0 -2.5 AU. Chondrules embedded in a disk with even low levels of turbulence may be spread over lengthscales ∆r = 1 AU in 10 5 yr, assuming an alpha viscosity law ν = αH 2 Ω (where H is the gas scale height, Ω the Keplerian orbital frequency and α a dimensionless constant ≈ 10 −2 ; Hartmann et al. 1998). Chondrules produced at 1.0 -2.5 AU will relatively quickly be spread throughout the present-day asteroid belt to be incorporated into chondrite parent bodies.

Discussion and Summary
Following the recent discovery that a planetary embryo, the half-formed Mars, existed at the same time chondrules were forming (Dauphas & Pourmond 2011), we have been motivated to study chondrule formation in the bow shocks of planetary embryos on eccentric orbits. We have calculated the hydrodynamics of gas flowing through the bow shock of a large (radius 2720 km) planetary embryo. In contrast to bow shocks around smaller planetesimals, an embryo acquires a substantial primary atmosphere from the solar nebula.
A boundary layer exists between this atmosphere and the shocked gas flowing past the planetary embryo, and this layer is unstable to KH instabilities. The planetary embryo is likely to have possess a magma ocean (Dauphas & Pourmand 2011), and the secondary atmosphere outgassed from the magma ocean would behave similarly dynamically.
We have calculated the trajectories of chondrules through the planetary embryo bow shock, subject to gravitational and gas drag forces. We find that the relative velocity between chondrules and gas is sufficient to melt chondrules (> 5 km s −1 , equivalent to shocks with speed > 6 km s −1 ) only for chondrules with impact parameter b < 400 km around a 500 km radius planetesimal, or impact parameter b < 4300 km around a 2720 km radius planetary embryo. We find that the dynamical coupling between chondrules and gas is not acheived until the chondrules have passed about 400 km past the bow shock. For a small planetesimal the bow shock stands only ≈ 100 km from the body, and chondrules are not well coupled to the gas. All chondrules with b < 460 km, including all those particles achieving peak temperatures of chondrules and melted as chondrules, are accreted onto the planetesimal. For a larger planetary embryo, the bow shock stands ≈ 1000 km off of the boundary with the atmosphere, and very few chondrules are accreted by the protoplanet, only those with b < 400 km (< 1% of the chondrules formed in the bow shock).
We have calculated the thermal histories of chondrules in a planetary embryo bow shock. Our calculation of thermal histories builds on the 1-D algorithm of  with the substitution of Morris et al. (2010a,b) for the radiation field, to allow the chondrules to radiate to the cold, unshocked gas. We find that shock speeds > 6 km s −1 and optical depths between chondrules and the unshocked gas > 0.1 are sufficient to melt chondrules and yield cooling rates through the crystallization temperature range < 100 K hr −1 , consistent with formation of porphyritic textures. These optical depths, we calculate, require concentrations of chondrules C > 10 above a presumed background density 3.75 × 10 −3 ρ g for a gas density ρ g = 10 −9 g cm −3 . These concentrations are achievable by chondrules settling to the midplane or turbulent concentration. Moreover, a concentration factor of C = 10 yields a compound chondrule frequency in accord with observations. To achieve the same optical depths in bow shocks around planetesimals requires much higher concentrations of chondrules.
We find that chondrules formed in a planetary embryo bow shock pass through the KH rolls at the boundary between the shocked gas and the planetary atmosphere, and will be exposed to the atmosphere gas. This atmosphere is composed of nebular gas but will be dominated by volatiles outgassed from the protoplanet's magma ocean. We have estimated the composition of the secondary atmosphere outgassed from the protoplanet to be roughly 10 bars of CO 2 , 3 bars of H 2 O, and 2 × 10 −3 bars of Na vapor, along with other trace species. We estimate a partial pressure of water vapor ∼ 3 × 10 −2 bar and a partial pressure of Na vapor ∼ 2 × 10 −5 bar at the boundary region through which chondrules traverse.
The partial pressure of water is consistent with the abundance of fayalite in chondrules (Fedkin et al. 2006) and indicators of variable oxygen fugacity in the chondrule formation environment (Krot et al. 2000). The high partial pressures of Na may resolve the mystery of how olivines crystallized form chondrule melts were able to incorporate substantial Na (Alexander et al. 2008 (Kurahashi et al. 2008;Villeneuve et al. 2009). A diversity of chondrule sizes, compositions and textural types could result from the evolution of the solar nebula between chondrule-forming events involving protoplanets with different a, e 0 and i 0 .
Yet another consideration is what type of object might be produced outside of the impact parameter necessary to produce chondrules. It is likely that solids will be fully or partially melted in these regions, and experience rapid cooling rates, resulting in objects that do not fit the classical definition of chondrules. For example, Miura et al. (2010) proposed that silicate crystals found in meteorites could be produced by supercooling of a silicate vapor in bow shocks. Alexander et al. (2007) (Alexander et al. 2007), and overgrowths on chondrules .
Further detailed modeling of the thermal histories of objects formed in bow shocks around planetary embryos is necessary to address this intriguing possibility.
Finally, we speculate that there is also likely to be a critical size for a body that produces chondrules with the proper thermal histories, that are not subsequently accreted.
Based on the standoff distances found for the two bodies considered in this study ( §3), we anticipate that this critical size is ∼ 1000 km in radius. In future work, we plan a parameter study of bodies ranging in size from radii of 500 -2720 km in order to test this hypothesis.
In this study, we have found that chondrule formation in planetary embryo bow shocks is fundamentally different from formation in planetesimal bow shocks, but on the basis of our preliminary calculations this model appears to fit all the known constraints on  Table 1: Masses of chondrules produced by planetary embryo bow shock over the orbital evolution of the body, assuming an initial semi-major axis of a = 1.0 AU and different initial values eccentricity e and inclination i. We assume chondrule formation occurs only when the instantaneous velocity difference V s between the embryo and the gas is in the range 6 − 8 km s −1 . Max V s is the maximum shock velocity during the orbital evolution. The duration signifies the time period beginning with the first episode of chondrule formation and ending with the last episode of chondrule formation before damping of the embryo's e and i.
A concentration factor C = 1 (implying chondrules-to-gas mass ratio 0.00375) was assumed. Only material that is within H/C, for disk gas scale height H is included in M s . The notation a(−b) means a × 10 −b .