Chondrule destruction in nebular shocks
Abstract
Chondrules are millimetersized silicate spherules ubiquitous in primitive meteorites, but whose origin remains mysterious. One of the main proposed mechanisms for producing them is melting of solids in shock waves in the gaseous protoplanetary disk. However, evidence is mounting that chondruleforming regions were enriched in solids well above solar abundances. Given the high velocities involved in shock models destructive collisions would be expected between differently sized grains after passage of the shock front as a result of differential drag. We investigate the probability and outcome of collisions of particles behind a 1D shock using analytic methods as well as a full integration of the coupled mass, momentum, energy and radiation equations. Destruction of protochondrules seems unavoidable for solid/gas ratios , and possibly even for solar abundances because of “sandblasting” by finer dust. A flow with requires much smaller shock velocities ( vs 8 km s) in order to achieve chondrulemelting temperatures, and radiation trapping allows slow cooling of the shocked fragments. Initial destruction would still be extensive; although reassembly of mmsized particles would naturally occur by grain sticking afterward, the compositional heterogeneity of chondrules may be difficult to reproduce. We finally note that solids passing through smallscale bow shocks around fewkmsized planetesimals might experience partial melting and yet escape fragmentation.
Subject headings:
protoplanetary disks – shock waves – meteorites, meteors, meteoroids – methods: analytical – methods: numerical.1. Introduction
A primary feature of nearly all primitive meteorites, or chondrites, is the presence of abundant millimetersized silicate spherules known as chondrules. They typically occupy 2090 % of the volume (Brearley & Jones, 1998), and some debris of such objects have even been found in samples returned from comet Wild 2 (e.g. Bridges et al., 2012). Beyond all doubt, the hightemperature mechanism responsible for the formation of chondrules was a pervasive process in the early solar system, and likely other protoplanetary disks as well. Yet, despite two centuries of research since their original discovery (Howard, 1802), no consensus is in sight as to the nature of this mechanism (Boss, 1996; Connolly & Desch, 2004; Ciesla, 2005; Krot et al., 2009; Desch et al., 2012). A crucial firstorder piece in the puzzle of protoplanetary disk physics is obviously missing.
A leading contender in this vexed debate is melting by shock waves (e.g. Wood, 1963; Hood & Horanyi, 1991; Iida et al., 2001; Desch & Connolly, 2002; Ciesla et al., 2004a; Desch et al., 2005; Boss & Durisen, 2005a, b; Morris & Desch, 2010; Morris et al., 2012; Hood & Weidenschilling, 2012; Boley et al., 2013; Nagasawa et al., 2014). The basic picture is that if a portion of the disk is overrun by a sufficiently strong shock (say, 6 km s), the solids embedded in the gas will experience a strong drag, heating them to the point of melting. They rapidly approach thermal equilibrium with the hot (postshock) gas, which cools as the shock front recedes away from them, allowing their eventual solidification. A consensus has not, however, been reached on a compelling mechanism for exposing primitive material to such shocks. Amongst the various possibilities (see e.g. Boss & Durisen (2005b)), the debate has lately largely boiled down to gravitational instabilities (Boss & Durisen, 2005a; Morris & Desch, 2010) versus bow shocks produced by planetesimals (Ciesla et al., 2004a; Hood & Weidenschilling, 2012) or planetary embryos (Morris et al., 2012; Boley et al., 2013) on eccentric orbits.
The shock scenario can arguably boast a high level of theoretical development, with quantitative predictions on the thermal histories of chondrules comparing favorably with constraints from observations and furnace experiments, e.g. as to the 11000 K/h cooling rates inferred for them or the short time spent near the liquidus temperature (Hewins et al., 2005). It must be noted though that recent bow shock simulations with radiative transfer have difficulties in reproducing the protracted cooling of porphyritic chondrules (Boley et al., 2013). As for largescale shocks such as those expected from gravitational instabilities, a “preheating” lasting hours until the shock is normally predicted, at variance with the lack of isotopic fractionation of S (a moderately volatile element) in putative primary troilite (Morris & Desch, 2010; Tachibana & Huss, 2005); also Stammler & Dullemond (2014) questioned whether sideway energy loss would be efficient enough to account for chondrule cooling rates if the shock scale is comparable to the disk thickness.
One important challenge to shock models was pointed out by Nakamoto & Miura (2004) and Uesugi et al. (2005): “protochondrules” (chondrule precursors) of different sizes should decelerate (relative to the gas) at different rates in the postshock region. They would develop relative velocities commensurate with the full shock velocity (a few km s), which would lead to their destruction upon collision. Ciesla (2006) examined the problem in some detail and argued that the molten protochondrules would be able to withstand collision velocities up to a few hundred meters per second and the time span by which relative velocities would have decreased below that threshold would allow few collisions under canonical solid/gas ratios (; Lodders (2003)).
Nonetheless, evidence is mounting that the chondruleforming regions were considerably enriched in condensible elements relative to solar abundances. First, the FeO content of olivine in type I and especially type II chondrules, which is unlikely to have been inherited from nebular condensates (Grossman et al., 2012), appears to record oxygen fugacities calling for solid/gas ratio of 101000 solar (e.g. Schrader et al., 2013; Fedkin & Grossman, 2013). Second, the significant proportion of compound chondrules—that is, pairs (or multiplets) of chondrules stuck together— as well as the thickness of igneous rims, if interpreted to be accreted during chondrule formation (Jacquet et al., 2013), also point to solid densities typically a few orders of magnitude above those of “standard” solar nebula model (Hayashi, 1981; Desch, 2007), although this somewhat depends on the assumed collision velocities and cooling timescales (Gooding & Keil, 1981b; Wasson et al., 1995; Ciesla et al., 2004b; Akaki & Nakamura, 2005). Third, the retention of Na in chondrules despite their hightemperature history (Alexander et al., 2008; Hewins et al., 2012) indicates high partial pressures of Na, which, if ascribed to partial volatilization from the chondrules themselves, would require solid concentrations more than five orders of magnitude above Minimum Mass Solar Nebula expectations (Hayashi, 1981). Fourth, in the specific framework of the largescale shock models, an overabundance of fine dust may be needed to shorten the aforementioned preheating (Morris & Desch, 2014). In fact, some enhancement above solar may be theoretically expected anyway as a result of settling to the midplane or turbulent concentration (Morris et al., 2012).
However, enhanced solid/gas ratios would increase the probability of highvelocity collisions and may jeopardize anew the survival of chondrules. It is thus important to assess what maximum enrichment of the solid abundance is allowed in the shock model if wholesale destruction of chondrules is to be avoided. To that end, we investigate the dynamics and collisional evolution of solids through a gasdominated shock, using analytic methods. Our focus is on the deceleration stage, independently of the source or largescale structure of the shock. We are able to express the probability of collision between chondrules as a function of the solid/gas ratio, independently of the gas density. We likewise quantify the collisions of chondrules with finer dust, which may collectively lead to their neartotal erosion, a process we will refer to as “sandblasting”. This is depicted in Figure 1.
The difficulty we find in maintaining macroscopic particles behind the shock motivates a consideration of substantially different shock scenarios. The settling of particles to the disk midplane can in some circumstances create large ratios of solids to gas. We consider how the shocked flow is modified in such a situation and briefly consider the prospects for the reassembly of mmsized particles after solids and gas equilibrate, and for their longer term survival. Chondrule survival may also be enhanced in smallscale shocks.
The outline of the paper is as follows: In Section 2, we describe the dynamical and thermal evolution of a single particle, before envisioning the probability, velocity, and outcome of collisions in Section 3. These two sections consider a gasdominated shock. We will then consider soliddominated shocks (Section 4) and small planetesimal bow shocks where deceleration is incomplete (Section 5). In Section 6, we summarize our results and discuss their implications on the viability of the shock model. Table 1 lists the symbols used in the text.
Symbol  Meaning 

Particle radius 

Speed of light  
,  Isothermal, adiabatic gas sound speed 
Sound speed of dustloaded gas  
Heat capacity of solids  
Particle emissivity  
Radiative energy density  
Boltzmann constant  
Preshock separation between two particles  
Critical value of for postshock collision  
Postshock cooling distance  
Particle mass 

Mach number  
Planetesimal mass  
Particle number density 

Pressure  
Catastrophic collision probability  
Planetesimal radius  
Collision time  
Postshock cooling time  
,  Time of chondrule mergers, compound formation 
Particle heating timescale  
Temperature 

Recovery temperature (from gas molecular heating)  
Peak particle temperature  
Asymptotic postshock temperature  
Particle velocity (in shock rest frame) 

Gas velocity (in shock rest frame)  
Particle/gas velocity upstream of shock  
Gas velocity jump across shock  
Asymptotic speed of equilibrated flow  
Particle velocity relative to gas  
Velocity constant entering ejection yield  
Thermal speed  
upon collision  
Abscissa of test particle  
Compound chondrule frequency  
Fraction of mass flux in solids,  
Ejection yield  
Power law exponent of ejection yield  
Adiabatic exponent of gas  
Collision velocity  
Solid/gas mass ratio 

Stopping time correction factor  
Thermal exchange correction factor  
Molten chondrule viscosity  
Mean molecular mass  
Mass density 

Internal particle density  
Surface tension of molten chondrules  
Surface density 

StefanBoltzmann constant  
Stopping time 

Keplerian angular velocity 
2. Singleparticle Dynamics
We consider a onedimensional normal shock in the gas, with subscripts “1” and “2” referring to the pre and postshock regions respectively. In what follows, is the density of the gas, and and its pressure and temperature, respectively. We denote by the gas velocity in the frame where the shock is stationary, and by the combined velocity of gas and particles upstream of the shock. Then the gas velocity jump is . In this section we investigate the postshock fate of one spherical solid particle of internal density and radius , originally comoving with the gas, by studying its motion and then its thermal evolution.
Fragmentation already has important effects in gasdominated flows, where the solid/gas mass ratio is . We therefore set aside the backreaction of solids on the gas flow in the analytic approach of Sections 2 and 3. The the figures show numerical results based on the full twofluid equations developed in Appendix A and Section 4.
2.1. Motion in the postshock gas
Since the hydrodynamic jump (a few molecular mean free paths in width) is much narrower than either the particle collisional mean free path or the stopping length, the particle arrives in the postshock region at the velocity of the preshock gas, that is, with a velocity relative to postshock gas equal to the jump velocity . The jump velocity is transsonic in the postshock region: from the Hugoniot Rankine relations (e.g. Desch et al., 2005), for evaluated immediately after the shock:
(1)  
with the ratio of specific heats (taken to be 7/5 in numerical applications) and the Mach number of the shock, with kg the mean molecular weight, and where the last equation holds in the limit (as we will also adopt in numerical applications).
From now on, we place ourselves in the frame of the postshock gas, which will turn out to be quite convenient to calculate the dynamics of the particles. In this frame, the drag force is:
(2) 
with the velocity of the particle relative to the gas, the particle mass and the stopping time given by:
(3) 
with the thermal speed and a correction factor. For a perfect conductor
Thus, taking to correspond to the position of the shock front at when the particles crosses it, the velocity of the particle is:
(4) 
and the abscissa (increasing in the direction of the flow):
(5) 
hence a stopping length of .
2.2. Thermal evolution
A particle entering the postshock region is heated by photons and gas molecules. The energy evolution of the particle temperature is governed by (Gombosi et al., 1986; Desch et al., 2005):
(6)  
Here is the particle emissivity, the radiative energy density, the specific heat capacity (Morris & Desch, 2010), the “recovery temperature” and a correction factor normalized to the subsonic regime. For strong shocks with , we have and (see Gombosi et al. (1986)).
For penetration distances shorter than the photon mean free path, the radiation field is not significantly different from that in the preshock region and heating will be dominated by molecular collisions. Hence the heating timescale may be expressed as:
(7) 
Its ratio to the stopping time is then
(8) 
that is 0.15 for our shock parameters. The particles thus reach peak temperature after a few tenths of their stopping time from passage through the shock (as is demonstrated in the numerical calculation shown in Figure 10 and discussed further in Section 4). If this temperature is in the chondruleforming range, then the particles are at least partly molten from that point onward.
The peak temperature may be obtained by balancing gasparticle heating and emission of radiation,
(9)  
where the last two equalities refer to the strong shock limit.
2.3. Evaporation
With such high temperatures, evaporation may threaten the survival of the smaller particles (Desch & Connolly, 2002; Miura & Nakamoto, 2005; Morris & Desch, 2010). In order to assess their possible collisional role (in Section 3.4), we need to investigate how much mass they lose to evaporation during chondrule deceleration.
We first note that while the actual peak temperature of the grain would correspond to a enhancement relative to the normalization in Equation (9), that temperature would hold for a timescale . The thickness evaporated during that particular time would thus be proportional to the radius (for grain sizes larger than the wavelength of emission m). So if the shock event is modeled as to allow chondrule survival during deceleration, dust should withstand that phase as well, and thus its survival on the chondrule stopping time should be assessed with the temperature estimates corresponding to . We conclude that if the peak temperatures were in the range 14001850C, as inferred for chondrules (Hewins et al., 2005), the temperatures to consider should be .
What are then the constraints on the evaporation rates, say for forsterite (the magnesian endmember of olivine, MgSiO), a major chondruleforming mineral? The experimentally calibrated model of Tsuchiyama et al. (1999) implies that the evaporation rate remains below at temperatures , given a total pressure and solar abundances. The corresponding linear rates are m s, increasing by an order of magnitude if the temperature bound is taken to be 2000 K instead. Dust larger than a micron should therefore survive deceleration on the timescale corresponding to . The same conclusion holds at smaller densities as well since the increase in stopping time is partly compensated by the decrease of the evaporation rate (Tsuchiyama et al., 1999).
Moreover, in a medium enriched in condensibles (as may be required by chondrule data, see Introduction), the evaporation enters the HO/H bufferdominated regime. Here the evaporation rate is suppressed by a factor (see equation 9 of Tsuchiyama et al. (1999)). In fact forsterite becomes stable over a wider temperature range, with e.g. complete evaporation at K instead of 1400 K at bar for dust/gas ratio of 1000 x solar (Tsuchiyama et al., 1999)). So while submicronsized dust, being a poorer radiator of heat than its bigger counterparts, would likely evaporate (Morris & Desch, 2010), supermicronsized dust may actually survive during the deceleration phase. This would not prevent complete evaporation before significant cooling has taken place further downstream: for example, Miura & Nakamoto (2005) found that grains smaller than m would eventually completely evaporate.
3. Collisions
Having studied the fate of an isolated solid particle, we now consider the possibility of collisions between populations of grains of different sizes. A given population (labelled “”) has a mean density and a solid/gas ratio (which will vary across the shock), and we designate by the overall solid/gas ratio. We will first study the collision probability and velocity for a general twosize population. Following a discussion of the outcome of the collisions, we evaluate the extent of catastrophic fragmentations before turning to the possibility of progressive “sandblasting” of protochondrules by dust. We finally assess the possibility of recoagulation further downstream.
3.1. Probability and speed of collision between two grains of different sizes
In this subsection, we calculate the probability of collision of a given big grain (henceforth with subscript “”) with a member of a population of small grains (subscript “”) originally comoving with it. To that end, we seek the maximum initial separation of a small grain downstream of a big grain that will allow it to be overrun by the big grain in the postshock region. Knowing the collisional cross section and the density of the small particles then allows us to obtain the collision probability for the big grain in closed form. The only restriction made here is that the shock is gasdominated, i.e. .
We adopt the convention that and corresponds to the entry of the smaller particle in the postshock region (remember we work here in the frame of the postshock gas). Taking into account the fact that the bigger grain enters the postshock region at time and at abscissa (as the shock front is receding backward in the frame of the postshock gas), the time of collision — if it does take place — may be calculated by setting the abscissas equal:
This may be rewritten in terms of the variable as
(11) 
This equation has a (unique) solution – that is, collision will actually occur – if and only if:
(12) 
This simply amounts to saying that the separation of the entry points in the postshock region (in its rest frame) must be smaller than the difference of stopping lengths between the big and the small particle. Introducing the collisional mean free path in the preshock region with the number density of small grains, and given that the number of grains in a given volume obeys a Poisson distribution, we obtain a collision probability
(13) 
with
(14) 
The first factor works out to 1.5 given our shock parameters.
If the condition (12) is met, what are the collision speeds? The collision velocity is obtained from Equations (4) and (5),
(15) 
where we have used Equation (11) in the second equality. Expressing as a function of and inserting in Equation (11), we obtain a direct relationship between and :
(16) 
We can work out useful asymptotic solutions in two complementary regimes:
1. For , one finds (that is, the smaller particle has not completely decelerated). The collision velocity is given by
(17)  
The factor with the square root in the last equality evaluates to 4.2 for our fiducial shock parameters.
2. For , one finds , which is negligibly small (that is, the small particle has essentially been stopped by the gas). Then
(18) 
The general collision velocity as a function of is plotted in Figure 2. It reaches a maximum at and decreases both for short (the two particles have had less time to develop relative velocities downstream of the shock), and for long (both particles are then increasingly coupled to the gas). We also plot the mean first collision speed in Figure 3 and the collision speed averaged over all collisions in Figure 4, as a function of the size ratio for different solid/gas ratios. Clearly, is typically within one order of magnitude of the full jump velocity.
3.2. Outcome of collisions
Collisions between similarsize silicate aggregates at speeds in excess of m s will lead to fragmentation (Güttler et al., 2010); the critical speed may rise to m s in the case of less porous targets and/or stickier material like water ice or organics (Güttler et al., 2010) or to m s for entirely coherent bodies (Vedder et al., 1974). The latter is however unlikely to apply to chondrule precursors (except in the case of remelting of preexisting chondrules) given the absence of such coherent bodies in chondrites – apart from the chondrules themselves and rare and compositionally distinctive igneous CAIs and other clasts. In the case of unequal size collisions (a small projectile hitting a bigger target), fragmentation of the target would not be necessarily catastrophic. Using dimensional analysis (Holsapple, 1993), the ratio of the ejected mass to the projectile mass, which we will call the “ejection yield” may be cast in the form:
(19) 
Here is a constant with the dimensions of velocity, and the index is expected to lie between 1 and 2 (limits corresponding to ejected mass proportional to the incident momentum and energy, respectively: Housen et al. 1983; Holsapple 1993; Poelchau et al. 2013). Hypervelocity impact experiments using a variety of target materials (wet soil, soft rock, hard rock) yield in the strength regime, with km s; whereas for sand and s (Holsapple 1993; see also the recent MEMIN experiments: Poelchau et al. 2013). In either case, we would have at collision speeds of order a few km s.
For collisions occurring after melting was complete, new  albeit less well understood physics – would come into play (Qian & Law, 1997; Planchette et al., 2012). Under the assumption that breakup is controlled by some critical value () of the Weber number with the surface tension, Kring (1991) derived critical fragmentation velocities commensurate with those for solids ( m s). Ciesla (2006) then argued that viscosity, which may vary by orders of magnitude during cooling, may make droplets more robust against disruption. It must be commented though that inasmuch as ferromagnesian chondrules have basaltic compositions, they should have relatively low viscosities around liquidus temperatures: bulk chondrule viscosity estimates by Gooding & Keil (1981a) only exceed 1 Pa.s around 1700 K. The viscosity would be increased in the presence of crystals, both by physical (Roscoe, 1952) and meltcompositional (Gooding & Keil, 1981a) effects, but this would be initially limited ( 12 orders of magnitude). Also, experiments of drop impact on liquid films suggest that the critical splashing velocity has a weak dependence on viscosity (Walzel, 1980; Rein, 1993; Yarin, 2006). Using the expression given by Walzel (1980), the splashing condition is:
(20) 
with the (dynamic) viscosity. So it would seem that collision velocities above a few m s should lead to disruption—a somewhat lower threshold range than envisioned by Ciesla (2006).
Little seems to be known about the velocityscaling of the ejection yield at small ratios of projectile to target mass. Micronsized grains impacting on a melt surface will produce splash at speeds above km s according to Equation (20). In the gravitydominated regime (irrelevant here), the behavior of water is fairly similar to that of solid substrates, though crater volumes are more than one order of magnitude larger at the same impact energy (Holsapple, 1993). We will continue to adopt the general form of Equation (19) for the ejection yield, but it should be borne in mind that experimental data are still lacking for droplet collisions, as previously emphasized by Ciesla (2006).
A complication which we have ignored is the dependence of collisions on angle (or equivalently on impact parameter). If we were to consider that the general ejection yield is given by the same expression as Equation (19) but with replaced by its normal component (e.g. Housen & Holsapple, 2011), the yield averaged over the whole cross section would be its maximum value divided by . Whatever the angular dependence may actually be, it would likely only yield a small correction compared to the global uncertainty in .
3.3. Critical solid abundances for extensive catastrophic collisions
What solid/gas thresholds for extensive catastrophic collisions do we obtain? From Section 3.1, we have seen that collisions, if they take place, occur at velocities comparable to . A particle of size would thus typically incur catastrophic disruption of the big particle upon collision. If we take this translates into a critical , and from Equation (13), such (catastrophic) collisions will be frequent for , a threshold lower than the solar total solid/gas ratio.
Yet individual components of such size are a minority in chondrites, with observed chondrule size distributions being quite narrow (typically within a factor of 2; e.g. Cuzzi et al. 2001; Nelson & Rubin 2002). The size distribution of solids may, of course, have been different in the chondruleforming region: for instance, Morris & Desch (2014) considered a population of 10 m grains—eventually evaporated—to alleviate the problem of preheating in largescale shocks—this however would, as we see, exacerbate that of collisional destruction. But a conservative approach may be to restrict attention to the presently observed size distribution. If we consider the size distribution of metal grains measured by Guignard (2011) for the Forest Vale H4 chondrite, generalizing Equation (13) to a polydisperse population (by replacing the argument of the exponential with an integral over the size distribution), we obtain , which, considering that metal only makes up 5.37 wt% of this meteorite, amounts to a critical total solid/gas ratio of 0.1. A similar threshold can be obtained from considering collisions with chondrules (which, although representing the majority of the mass, offer smaller size contrasts); if we take and , we obtain . So catastrophic collisions, while rare for solar abundances, as seen by Ciesla (2006), should be prevalent for solid/gas ratios .
Given that the solid/gas ratio is high enough to induce catastrophic disruptions of mmsized particles, might the chondrules we observe derive from fragments of much bigger precursors? There are several arguments against such a possibility. First, it is difficult theoretically to achieve sizes larger than mm or cm by coagulation in the disk (e.g. Brauer et al., 2008; Birnstiel et al., 2010; Zsom et al., 2010), except perhaps via the destruction of a prior generation of kmsize gravitationally bound objects, and indeed inclusions larger than mm are rare in chondrites. Second, fragmentation would also be expected to produce a range of sizes, which is at variance with the narrow size distribution exhibited by chondrules. Some narrowing of the size distribution may be possible following chondrule formation (e.g. Cuzzi et al., 2001), but Jacquet (2014) has argued against such a process on compositional grounds. Catastrophic disruption during the shock, if prevalent, would thus be difficult to reconcile with meteoritic evidence (see also Section 3.5).
3.4. Sandblasting
Although collisions between protochondrules are violent, most collisions experienced by a given protochondrule will be with small dust particles, which could cumulatively have a stronger effect. The dust in question may originally be (molten) dust surviving from the preshock region (even if restricted to its most refractory components in case of preheating), the debris of protochondruleprotochondrule collisions (representing a proportion of the solids; see Equation (13)); or the result of ram pressure stripping of large droplets (Susa & Nakamoto, 2002; Kadono et al., 2008).
Small particles need contribute only a modest fraction (14) of the mass flux in order to maintain frequent collisions with larger grains. We will therefore model fine dust by a continuous fluid (identified with subscript “d”), which is coupled tightly to the gas on the short timescale (3). Collisions of dust grains with protochondrules (considered as a population of identical particles, identified with subscript “p”) thus take place at the full speed .
Under the influence of erosion, the mass of the protochondrule evolves as:
(21) 
Dividing by Equation (2) (where we neglect momentum transferred by the impinging dust as well as recoil from erosion) yields:
(22) 
The dusttogas ratio also increases at the expense of the eroded protochondrules
(24) 
Equation (3.4) is equivalent to:
(26) 
So quite insensitively to the seed dust/protochondrule ratio and the threshold below which a chondrule is conventionally considered destroyed, for a critical erosion velocity , a destruction condition may be extracted as essentially
(27) 
Yields , as deduced in Section 3.2, allow at best a limited enrichment of solids above solar composition (). This constraint is more stringent than the one obtained from protochondruleprotochondrule collisions (Section 3.3).
Given the uncertainties in the rheology of liquid droplets (Section 3.2), one may resort to a conservative approach considering only the initial phase where protochondrules are solid. (We assume that a liquid state of the impinging droplets does not qualitatively change their effect on the solids, as in a “point source” picture of the impact, e.g. Holsapple 1993.)
Then sandblasting would be restricted to velocities , and the threshold for would need to be divided by . The minimum solid fraction would be raised by almost an order of magnitude, to for . But if is indeed near the upper limit as seems the case for ’s of several km s (Holsapple, 1993; Poelchau et al., 2013), that would again only allow a limited margin of variation above solar values to avoid erosional destruction.
3.5. Recoagulation downstream?
Assuming the protochondrule destruction condition is met, one could nevertheless expect that further downstream, when collision velocities allow sticking rather than disruption (below m s), the dispersed droplets grow again. One may then wonder whether millimetersize chondrules may be produced that way.
In his theoretical review on chondrule size, Jacquet (2014) found that this would require high particle densities. Indeed the density required to obtain a given size increase during a timescale (during which colliding droplets merge rather than bounce or freeze as compound chondrules) is (Jacquet, 2014):
Here is a measure of the (random)
Note however that this would not necessarily prevent the shock models from accounting for the observed proportion of compound chondrules (Ciesla, 2006; Morris et al., 2012) if destruction was actually not extensive at the shock. The chondrule density corresponding to a compound fraction is (Jacquet et al., 2013):
(29)  
with the time during which compound chondrule formation took place and where we have normalized to the fraction found in ordinary chondrites (Gooding & Keil, 1981b; Wasson et al., 1995). If we adopt the higher end of conceivable sticking velocities and compound formation times as in the above normalizations, this would be comparable to the solid densities in relatively dense disk models, with the help of the compression due to the shock itself (Boley et al., 2013), although all these parameters are uncertain at the orderofmagnitude level.
In principle, the emergence of the chondrules from the shocked zone may lead to further fragmentation. Indeed a pressure gradient over a lengthscale would incur a gasgrain drift (in the terminal velocity approximation of Youdin & Goodman (2005))
(30)  
Here is the isothermal sound speed. Equation (30) implies collisions at speeds well above the fragmentation limit as long as these collisions are not too frequent. The number of collisions works out to
(31) 
which is for our (gasdominated) shock parameters, hence little further fragmentation is to be expected during escape.
Our general conclusion is that gasdominated shocks, with heating regions extending further than a stopping length, would be generally detrimental to the integrity of chondrules. In the two sections to follow, we relax some of our assumptions to explore alternative regimes, namely soliddominated shocks (Section 4) and then smallscale bow shocks (Section 5).
4. Structure of the Postshock Flow with High Particle Density
Previous chondrule shock models have not considered large relative solid abundances. However, exploring such a regime is well motivated both by chondrule studies (as described in the Introduction)
and by our developing understanding of protoplanetary disks. These
disks typically have a layered structure: an outer magnetically active layer extends to a column kg m below the disk surface (Gammie, 1996). Particles that are initially suspended within a fully turbulent disk
(e.g. during an initial massive and gravitationally active phase) will mostly settle to the midplane once the
magnetorotational instability becomes the primary source of flow irregularities.
Mass transfer in the disk may have the effect of raising the density of settled particles with respect to the gas, and solid abundances approaching the Roche density become possible once the vertically averaged abundance ratio increases by a factor over solar values (e.g. Youdin & Shu, 2002). Beyond this point, planetesimal formation becomes possible, and the particle layer is stirred by gravitationally driven modes. A planetesimal, or a largescale shock, interacting with such a settled particle layer therefore encounters relatively welldefined conditions at a distance of 2.5 AU from the Sun: a total density approaching kg m and a solid/gas density ratio with the Keplerian angular velocity.
In this section, we examine soliddominated shocks taking into account the exchange of momentum and heat between particles and gas, the effects of fragmenting collisions, and the emission and absorption of radiation by the particles. The equations describing these interactions, and the approximations made, are described in Appendix A. It is recalled that all plots in the paper, unless otherwise noted, have been produced using these more complete equations, with .
It is worth emphasizing that the processes described here do not depend on a flow speed exceeding the gas sound speed. That is because the effective sound speed of waves with a period larger than the Epstein stopping time (3) of the embedded particles is
(32) 
where is the usual adiabatic sound speed in the gas. A shocklike disturbance can form in a flow with a speed exceeding . When , the details of the flow structure, and the characteristic fragmentation lengths, will differ from the case in which a gas shock does form. For example, at large , the effective thickness of the shock increases from the mean free path of gas particles to the mean free path for collisions between solid particles. Nonetheless, we find in practice that chondrulemelting temperatures generally require shock speeds larger than the gas sound speed at K. We therefore maintain our focus on the flow behind a gas shock.
4.1. Flow and particle behavior as a function of
The flow structure for a single species of particle is shown in Figure 5, for several values of , in the rest frame of the shock. Here collisions are not possible. The horizontal axis in the figure is normalized to the postshock stopping length.
The mild superheating that the solids experience during their deceleration (Section 2.2) disappears for . One also observes a larger net compression of the flow for larger values of , as a larger fraction of the heat is stored in the solids, which contribute negligible pressure to the flow. The asymptotic temperature is also larger as a result of the lower specific heat of the solids. Far from the shock, when gas and solids have equilibrated again both thermally and dynamically, the temperature and flow speed relax to the values shown in Figure 6. The asymptotic expressions
(33) 
apply when the solids dominate the specific heat (). Temperatures in the chondruleforming range can then be achieved for shock speeds as low as km s. By comparison, for gasdominated shocks, even a high gas density of kg m would require a of about 8 km s (see Equation (9)), amounting to some 40 % of the orbital speed at 2.5 AU. Moreover, the reduction in would allow longer cooling times for smallscale shocks such as those produced by planetary embryos.
The differential deceleration of large and small particles is shown in Figure 7 (see also Figures 8, 9 and 10). Here we include the effect of fragmenting collisions using the prescription for erosion given in Equation (19), and consider an initial particle size ratio . The smaller ‘dust’ particles are assumed to be fixed in size. The larger particles experience faster deceleration as they are ablated, leading to a sharp destruction layer.
Radiative energy losses are negligible on the stopping length of the particles in the case of small particle loadings; but they become significant for large . The optical depth in dust also rises sufficiently (Appendix A) to begin to trap heat in the flow, as the incident particles are breaking down into smaller grains. This means that only a modest fraction of the incident kinetic energy flux is lost to radiation back across the shock (see Figure 11 in Appendix A). The strong compression of the flow occuring at larger is somewhat enhanced by radiative energy losses.
4.2. Chondrule survival
Here we assess the survival of chondrules in soliddominated shocks as compared to gasdominated shocks. As may be seen in Figure 9, soliddominated shocks also lead to wholesale destruction of protochondrules by sandblasting, as may have been intuitively expected. This is despite the reduced time () for dynamical equilibration (because of feedback on the gas and recoil during collisions) which does not reverse the effect of an increasing collision rate ().
In contrast with gasdominated shocks, there is now a good prospect for recoagulation. If the total preshock density is kg m, as limited by gravitational stability, and we take into account the compression behind the shock as enhanced by heat exchange between solids and gas (see above), downstream densities up to kg m may be considered. This would provide rapid enough collisions to reproduce chondrule size as may be judged from Equation (3.5).
The collision rate would also be enhanced during escape (see Equation (31)), but now the collision speed between particles would be limited by the high frequency of collisions. By balancing the acceleration driven by gas pressure gradients against the effect of mutual collisions, we have
(34) 
This gives
where we have set and made use of the asymptotic Equations (33). Coherent solids (as opposed to aggregates) would have fragmentation velocities of order 10100 m s (Vedder et al., 1974) and the destructive effect of collisions between solid chondrules would also be reduced by the repeated accretion and ejection of dusty envelopes.
A problem remains with having chondrules form by recoagulation: their composition would be expected to be homogenized, inconsistent with observations (Hezel & Palme, 2007; Jacquet, 2014). Some variation could be reproduced if the final assembly of the reformed chondrules involved particles of intermediate size, which recorded stochastically different ranges of temperature and density in the flow, and therefore experienced variable exchange with the gas. A significant challenge for such a scenario is however provided by the measured dispersion in the abundance of refractory elements like rare earths (Jacquet, 2014).
5. Flash heating in small planetesimal bow shocks
The preceding calculations are predicated on the assumption that the shocked region is larger than the particle stopping length. One could envision smallscale shocks, specifically bow shocks around planetesimals, that are smaller than and yet thermally affect solids. Since the heating timescale is only one order of magnitude below the stopping time, this would essentially require the whole thermal episode to last .
In this case, solid destruction would be diminished from the estimates of the previous sections. Collisions between protochondrules would be somewhat rarer, in spite of being also possible after exiting the shocked region because of the relative velocities developed by then (see appendix B). Sandblasting would also be reduced in the shocked region (see last paragraph in Section 3.4), and also further downstream since the ejection yield there would be reduced (see Equation (22)).
Such a thermal episode would be quite short. If we eliminate using the expression for in Equation (9), we have:
(36)  
This is much shorter than cooling timescales (hoursdays) generally assigned to chondrules from furnace experiments, trace element and crystallographic systematics (Hewins et al., 2005; Jacquet et al., 2012, and references therein). Rubin (2000); Wasson & Rubin (2003) nevertheless argue for such short cooling times to avoid loss of Na, which would be complete within s under canonical nebular conditions (e.g. Fedkin & Grossman, 2013). While a single thermal pulse sufficiently intense to melt the precusor solids would produce textures very different from known chondrules (Hewins et al., 2005), Wasson & Rubin (2003) argue for a succession of less intense flash heatings (at least tens) to achieve gradual growth of crystals. We note that, in that case, the loss of Na would be the integrated sum of that of each of those events and would actually still overpredict the observations. While chondrules may not then have been produced by such flash heating events, we may still consider the formation of agglomeratic olivine objects (Weisberg & Prinz, 1996; Ruzicka et al., 2012) or igneous rims around chondrules (Krot & Wasson, 1995) by such a process.
How important would such planetesimalinduced flash heating be in the early solar system? Since the extent of the bow shock is comparable to the planetesimal radius (about 3 times in the calculations of Ciesla et al. (2004b)), the planetesimal size required would be of order:
(37)  
Interestingly, the cross sectional area of the presentday asteroid belt appears dominated by asteroids around 6 km in diameter (Ivezić et al., 2001), about the appropriate size range, although the primordial planetesimal size distribution may have been different (e.g. Morbidelli et al., 2009).
The solid mass that is processed in one orbit by a given planetesimal of radius and mass is given by
(38)  
Here is the relative increase in the processing cross section over the geometrical one, is the surface density of solids in the local disk annulus, and a order unity number
Over a few Ma, depending on the fraction of adequately excited few kmsized planetesimals in the planetesimal belt, some significant processing could be achieved. Independently of damping timescales (Morris et al., 2012; Hood & Weidenschilling, 2012), the “flashheating” activity of a given excited planetesimal would be limited by erosion by millimetersize solids. Indeed, any such solid with impact parameter would impact the planetesimal as gas drag would not have time to significantly alter their trajectories (Morris et al., 2012) and they would do so at almost full velocity , with high ejection yields (Poelchau et al., 2013). The resulting lifetime works out to be a (as can be seen by multiplying the righthand side of Equation (38) by ).
6. Summary and conclusions
We have investigated, analytically and numerically, the fate of solids in shocks during the deceleration phase.
We found that, in the gasdominated regime, the deceleration of solids could be reasonably well described with a constant stopping time , lending itself to analytic treatment. Melting of the protochondrules would occur at about one tenth this stopping time, but evaporation of supermicronsize dust grains would not be completed within the deceleration timescale, especially for supersolar solid/gas ratios. We argued that collisions at velocities greater than 10100 m s will lead to fragmentation and/or splashing.
We found that solid/gas ratios would lead to highspeed disrupting collisions for protochondrules with moderate size dispersions (factor of ) comparable to those observed for chondrules in individual chondrites (e.g. Nelson & Rubin, 2002; Teitler et al., 2010). Sandblasting of protochondrules by dust (rapidly coupled to the gas), whether inherited from the preshock region or produced through fragmentation in the postshock region, may be even more efficient at destroying protochondrules. Indeed the ejecta produced by such collisions enhance the dust concentration, generating a positive feedback. Melting of chondrule surfaces may alter the effects of sandblasting; but even so large particles have generally been generally broken down before melting due to rapid collisions.
As a result, the threshold solid/gas ratio for wholesale destruction scales inversely with the ejection yield at the postshock gassolid velocity, and might be as small as the solar value. In the standard gasdominated shock models, there is little prospect that once protochondrules are destroyed, recoagulation further downstream can reconstitute mmsized particles, although collisions would be frequent enough to reproduce the statistics of compound chondrules for reasonable parameters, if chondrules did escape wholesale destruction.
We then considered soliddominated shocks. Such a setting, which may more easily satisfy the empirical constraints from chondrules, may be astrophysically conceivable in the framework of layered accretion where solids remain settled in a dead zone and thus gradually concentrate relative to the gas. Soliddominated shocks would require smaller velocities than gasdominated shocks to achieve chondrulemelting temperatures, and would more easily produce long cooling times (for smallscale shocks) and retention of volatiles. While destruction of chondrules past the shock would be extensive, subsequent recoagulation would be possible because of the higher densities – although assembling chondrules that way may conflict with their observed compositional diversity.
We also considered smallscale bow shocks due to few kmsized planetesimals where deceleration is incomplete. While destruction of solids would be limited compared to the above, the hereby flashheated products (timescales of a few seconds) would likely not resemble chondrules but these may possibly account for slightly melted features in chondrites such as agglomeratic olivine objects or some igneous rims.
So it would appear that existing nebular shock wave models for chondrule formation are inconsistent with significant enhancements of the solid/gas above solar values. Nonetheless, such enrichments are suggested by several chondrule observations such as the frequency of compound chondrules (Gooding & Keil, 1981b; Ciesla et al., 2004b), FeO content (Grossman et al., 2012; Schrader et al., 2013), and Na retention (Alexander et al., 2008). Such observations would need to be explained in another way for the shock model to remain viable. We have already seen that there is some room to adjust the parameters to account for the compound chondrule fraction. Such may also be marginally the case for the FeO constraint of type I (FeOpoor) chondrules, and type II (FeOrich) chondrules could be envisioned to have formed by a mechanism qualitatively different from that of their type I counterparts. One option, especially with regard to the Na retention issue, is to invoke a nonsolar chemistry for the gas component e.g. due to the outgassing of a planetary atmosphere, as proposed by Morris et al. (2012) for the bow shock model. (This conclusion has yet to be verified quantitatively as the atmosphere in question would be far below the shock front.) At any rate, such a solution cannot be applied to largescale shocks such as those expected from gravitational instabilities—in that it would seem that bow shocks are favored (see also Stammler & Dullemond, 2014). But unless such alternatives to a global enrichment of the chondruleforming environment can be found, and ejection yields during erosion are shown to be at several km s, then destruction of chondrules remains a serious issue for shock wave scenarios.
Appendix A Calculating Postshock Flow with Finite
Here we outline the method used to calculate the flow profiles shown in Section 4, as well as the mean collision speeds in Figures 3, 4.
In the frame of the shock, the speed of a particle is written as