Jetting in Oblique, Asymmetric Impacts

ICARUS

134, 163–175 (1998) IS985945

ARTICLE NO.

Jetting in Oblique, Asymmetric Impacts Gregory H. Miller Shock Wave Lab, Department of the Geophysical Sciences, University of Chicago, 5734 South Ellis Avenue, Chicago, Illinois 60637 E-mail: [email protected] Received September 19, 1997; revised March 16, 1998

Experiments were conducted to determine the composition of jets emitted in asymmetric oblique impacts. The jets were sampled indirectly using witness plates, which were quantitatively analyzed by SEM. The results indicate that decreasing the thickness of the inclined target plate increases the relative abundance of projectile in the jet. Increasing the impact angle has the same effect. No systematic dependence of jet composition on impact velocity was observed. Thin plate theory, variations of which have been applied to several problems in planetary geophysics, gives the jet composition as a function of plate thickness and impact angle. This theory gives compositional dependencies on plate thickness and impact angle that are opposite to the experimentally observed trends and predicts compositional abundances that differ in magnitude from experimental values by over 40 wt% in all cases. A critical review of the theory, which is extended to explicitly include shock entropy production, and a comparison of the theory with hydrocode simulations, reveals several assumptions that are not valid in the present application. The revised thermodynamic model predicts peak jet temperatures that are significantly lower than calculated using earlier models. If the experimentally observed compositional trends apply to the collisions of meteoroids with planets, then the relative abundance of meteoroid in the jet will be greater in oblique impacts than in normalincidence ones.  1998 Academic Press Key Words: collisional physics; experimental techniques; impact processes.

I. INTRODUCTION

Impact jetting refers to the emission of a stream of material from two colliding bodies. Its discovery and initial application relate to shaped charges, where the explosive collapse of a thin metal cone results in the formation of a high-velocity columnar jet stream with remarkable penetrating power (Birkhoff et al. 1948). The jet’s velocity is high in comparison to the velocity of the colliding objects, and its peak temperature is significantly higher than would occur in a normal-incidence collision (Kieffer 1977). These properties of anomalously high velocity and temperature distinguish jetted materials from other forms of impact

ejecta and have motivated many planetary geophysics applications. Jetting has been proposed as a mechanism for the formation of impact melts in porous rocks (Kieffer 1975) and the formation of certain chondrules (Kieffer 1975) and of certain impact melts (Kieffer 1977), including tektites (Vickery 1993). Jets may have played a role in building Earth’s pre-biotic organic carbon inventory from exogenous sources (Blank and Miller 1997). Jets may be important in the impact ejection of mass off planetary bodies such as the SNC meteorites off Mars (O’Keefe and Ahrens 1986), the formation of the Moon in a giant Earth impact (Melosh and Sonett 1986), and the formation of Charon in the giant impact of Pluto (McKinnon 1989). Thin plate theory (Walsh et al. 1953, Cowan and Holtzman 1963), developed to describe the symmetric collision of thin plates, provides a very successful criterion for the onset of jetting. According to the theory, a jet will not form when shocks exist in the converging streams, and when these shocks are attached at the point of contact of the streams. However, jets will form if the flow is shockless, or if the shocks are detached. Jetting thus always occurs in incompressible flow (Bachelor 1967, pp. 392–396), and will occur in compressible flow when the flow is subsonic (sonic criterion) or when the Hugoniot equations do not permit a sufficient turning angle (von Neumann criterion). It is important to note that this prediction depends only on wave configurations near the point of contact of the converging streams and not on far-field flow properties or boundary conditions. This prediction has been experimentally verified for symmetric thin plate collisions (Walsh et al. 1953). A version of this theory applicable to asymmetric collisions (Walsh et al. 1953) has also been verified experimentally (Allen et al. 1959). The extension of thin plate theory to predict the onset of jetting when spheres collide has been proposed (Melosh and Sonett 1986, Ang 1990, 1992, Vickery 1993) but remains to be verified experimentally. The mass and momentum fluxes of steady-state jets in symmetric impacts are also given by thin plate theory (Birkhoff et al. 1948, Harlow and Pracht 1966). Godunov et al. (1975) measured jet velocities in symmetric collisions

163 0019-1035/98 $25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

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that were smaller by typically P25% than predicted by the theory. These authors attributed this difference to material viscosity, which is not accounted for in the theory. The extension of thin plate theory to predict the mass and momentum fluxes in asymmetric plate impact is indeterminate (Birkhoff and Zarantonello 1957, p. 49). Nevertheless, several workers have introduced ad hoc assumptions to extend the theory (Wang 1989, Pack and Curtis 1990, Yang and Ahrens 1995). Inferences of jet mass in asymmetric collisions do not agree well with the extended theory (Yang et al. 1991, Yang and Ahrens 1995). Mechanistically, the jet mass and momentum fluxes must depend on far-field boundary and flow field conditions (e.g., according to thin plate theory, the steady state jet thickness is proportional to the thickness of the symmetric incoming streams), which significantly complicates the extension of thin plate theory to the impacting sphere problem (cf., Melosh and Sonett 1986, Vickery 1993). The present study was motivated by a desire to better understand the fluid dynamic and thermodynamic conditions associated with planetesimal and planetary collisions. The colliding sphere problem is characterized by a lack of symmetry, the absence of steady state, and a continuously evolving geometry—conditions at odds with the assumptions of thin plate theory. This study focused on the asymmetric collision of moderately thin plates. The problem lacks symmetry and steady state, but the geometry of the collision is constant. This report summarizes and expands on previous work (Puckett and Miller 1996, Miller 1997a,b) and includes new experimental results. The extended thin plate theory is derived and discussed in Section II. The present development differs from previous versions in its explicit treatment of entropy production. The theory is used to predict trends in jet composition as a function of plate thickness, angle, and velocity. In Section III the stability of the flow field in asymmetric collisions is discussed. New experiments involving the oblique collision of relatively thin plates of Cu and Sn are described in Section IV. The experiments provide indirect evidence for the composition of the jet formed, and these new experimental results disagree with the theoretical predictions in Section II. In Section V hydrocode simulations are used to help elucidate aspects of the flow topology not readily observed in the experiments. Section VI presents some applications to planetary and planetesimal impacts, and concluding remarks are given in Section VII. II. THEORY

The essential characteristics of impact jets, namely their high velocity and temperature, stem from the ‘‘geometrical leverage’’ available in oblique collisions. In a laboratory frame of reference, the point of contact of two obliquely colliding bodies has a much higher velocity than do the

FIG. 1. Generalized oblique impact in lab and stationary frames. Velocity vectors are to scale. The thin dashed lines represent shock waves in the ‘‘regular regime’’ when jets do not form. The shocks are linear as indicated only when the flow is supported on the hachured surfaces.

bodies themselves (Fig. 1). The natural frame of reference is the one in which the collision point is stationary. Figure 1 shows how this reference frame transformation amplifies velocities by roughly a factor of csc(u), where u is the impact angle. If there exists a shock that is stationary in this reference frame, then its strength will be commensurately higher than a shock in a normal-incidence impact (u 5 0) with the same plate-normal velocities. In thin plate theory it is assumed that the stresses encountered are significantly greater than yield strengths, and so the materials may be approximated as isotropic fluids. Moreover, it is assumed that viscosity, thermal conductivity, and other dissipative mechanisms may be neglected. Given these assumptions the problem may be modeled with the compressible Euler equations. This has two consequences for the present study. First, solutions to the compressible Euler equations consist of regions of smooth flow in which entropy is conserved. These regions may be bounded by discontinuities, including shocks across which entropy may jump. Second, Bernoulli’s law, I 1 AsuU u2 5 constant,

(1)

holds on streamlines, where I 5 E 1 PV is the specific enthalpy, E is the specific internal energy, P is the pressure, V 5 1/r is the specific volume, r is the mass density, and U is the velocity in the stationary reference frame. Consider the highly simplified steady flow depicted in Fig. 2. This flow differs from that sketched in Fig. 1 by the existence of a jet stream, whose orientation is approximately opposite the incoming flows U0A and U0B . The part of streams A and B that is downstream of the point of contact, and that continues in the same sense as in Fig. 1, is called the ‘‘slug.’’ To render this already simplified problem tractable analytically it is necessary to further

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implicitly considers a slip interface between materials A and B in the slug and in the jet. The densities of the jet and slug streams, and their specific internal energies, are functions of the equation of state (EOS) and the unspecified entropy jumps. Schematically,

r0A 5 rA(S0A , P 5 0);

r1A 5 rA(S0A 1 DSA , 0)

(2)

r0B 5 rB(S0B , P 5 0);

r1B 5 rB(S0B 1 DSB , 0)

(3)

E0A 5 EA(S0A , P 5 0);

E1A 5 EA(S0A 1 DSA , 0)

(4)

E0B 5 EB(S0B , P 5 0);

E1B 5 EB(S0B 1 DSB , 0).

(5)

A mass balance across the domain shown in Fig. 2 gives a0 r0AU0A 5 (a1 1 a2)r1AU1A

(6)

b0 r0BU0B 5 (b1 1 b2)r1BU1B .

(7)

To apply conservation of momentum and energy it is convenient to locate the control surfaces far from the collision point where the pressures of the incoming streams, the jet streams, and the slug streams will be equal with P 5 0. Conservation of momentum in the y direction ( y is perpendicular to the orientation of the slug, Fig. 2) then gives 2a0 r0AU 20A sin(u 2 f) 1 b0 r0BU 20B sin(f) 2 a2 r1AU 21A sin a 2 b2 r1BU 21B sin a 5 0,

(8)

and conservation of momentum in the x direction (parallel the slug) gives

FIG. 2. Simplified geometry of a jet-forming oblique impact.

suppose that there exists only one shock in the upstream limb of flow A and another shock in the upstream limb of flow B. These shocks turn the incoming flows and cause the specific entropies of the jet and slug streams to be greater than those of the incoming streams by amounts DSA and DSB . One must also assume that the shocks are linear and so affect each stream uniformly: all streamlines in material A will experience the same entropy jump, and likewise for material B. With these assumptions, the farfield P 5 0 densities of the jet and slug streams will be equal, r 1A 5 r 2A and r 1B 5 r2B , and by application of Bernoulli’s law the jet and slug velocities far downstream of the collision point will also be equal, U1A 5 U2A and U1B 5 U2B (the velocities U are scalars with orientations indicated in Fig. 2). The A and B slug streams are parallel to each other by hypothesis, but the slug velocities U1A and U1B may differ in the general case. Likewise, the jet streams are considered parallel but may have different velocities. Thin plate theory

a0r0AU 20A cos(u 2 f) 1 b0 r0BU 20B cos(f) 2 a1r1AU 21A (9) 2 b1r1BU 21B 1 a2r1AU 21A cos a 1 b2r1BU 21B cos a 5 0. If the interface is stationary then the streams do no work on each other and conservation of energy applies separately to each stream: AsU 20A 1 E0A 5 AsU 21A 1 E1A

(10)

AsU 20B 1 E0B 5 AsU 21B 1 E1B .

(11)

The 10 Eqs. (2)–(11) constitute a four-parameter model for the 14 variables r1A , U1A , E1A , a1 , a2 , r1B , U1B , E1B , b1 , b2 , f, DSA , DSB , and a. The EOS and the parameters r0A , E0A , U0A , a0 , r0B , E0B , U0B , b0 , and u are given. The amount of entropy produced in the collision cannot be predicted, a priori, but with the postulated shock configuration it can be bounded. The shock is strongest when it is normal to the incoming stream, i.e., when U0 5 Us , where Us is the velocity of the shock with respect to the unshocked, upstream flow. If the shock is inclined by angle b then the shock will have velocity Us 5 U0 sin(b). As b decreases from f/2, the maximum strength case, the zero-

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strength shock Us R c0 is approached. Entropy is generally monotonic in shock velocity; thus the minimum DS is zero when Us 5 c0 , and the maximum value is specified by the thermodynamic state of the Hugoniot point Us 5 U0 (S1 in Fig. 4). In practice we do not need to evaluate entropy explicitly, which would require a full thermal equation of state model. Instead, it is sufficient to compute rarefaction isentropes departing from the Hugoniot between the centering point (e.g., r0A , E0A) and the maximum shock state. Possible downstream thermodynamic states (e.g., r1A , E1A) lie on these release isentropes at P 5 0. Their calculation requires only a P(r, E ) equation of state. Picking DSA and DSB as two formal free parameters of the model, a relationship between the variables f and a is established, and one of these angles must be chosen as the third of four free parameters. This requires that one of the variables a1 , a2 , b1 , and b2 be selected as the fourth free parameter. The solution of this system of equations thus requires in effect that the composition of the jet be partially specified, hence use of this model to determine jet composition is circular. If one assumes that the jet and slug are collinear (Wang 1989, Pack and Curtis 1990, Yang and Ahrens 1995), a 5 0, then the x momentum conservation equation may be written separately for each material:

r

a0 0AU 20A

cos(u 2 f) 5 (a1 2 a2)r

2 1AU 1A

b0 r0BU 20B cos(f) 5 (b1 2 b2)r1BU 21B .

(12) (13)

The problem then consists of 13 variables and 11 equations, and it is possible to solve for each of the four variables a1 , a2 , b1 , and b2 using only DSA and DSB as free parameters. The solution to this system of equations is explored below. It should be noted that the assumption a 5 0 is motivated by calculation necessity and not by physical reasoning or experimental observation except in the trivial case where it is required by symmetry. Equation (8) may be solved immediately for the orientation f of the jet and slug: cot(f) 5

a0 r0AU 20A cos(u) 1 b0 r0BU 20B . a0 r0AU 20A sin(u)

(14)

Given a particular choice of DSA and DSB , Eqs. (2)–(5) determine r1A , r1B , E1A , and E1B . Next, Eqs. (10) and (11) may be solved for U1A and U1B . Finally, Eqs. (6) and (12) give a1 and a2 , a1 5 a2 5

S S

D D

a0 r0AU0A U 1 1 0A cos (u 2 f) 2 r1AU1A U1A

(15a)

a0 r0AU0A U 1 2 0A cos (u 2 f) , 2 r1AU1A U1A

(15b)

FIG. 3. Experimental setup f is the horizontal coordinate of first contact of projectile and target. t is the projection of the target front surface.

while Eqs. (7) and (13) give b1 and b2 :

S S

D D

b1 5

b0 r0BU0B U 1 1 0B cos (f) 2 r1BU1B U1B

b2 5

b0 r0BU0B U 1 2 0B cos (f) . 2 r1BU1B U1B

(16a) (16b)

It is obvious from Eqs. (15) and (16) that if there were no entropy production, i.e., if DS R 0, whence U1 R U0 and r1 R r0 , then for any angle u either the A component of the jet, a2 , or the B component of the jet, b2 , or both, will be nonzero. Thus, the experimentally observed jet-free regular regime owes its existence to the irreversible character of the flow sketched in Fig. 1. Effect of Plate Thickness on Jet Composition These equations suggest circumstances under which the jet will consist predominately of a single material. Suppose, for example, that plate A were thick; a0 @ b0 . Then, according to Eq. (14) f R u and cos(u 2 f) R 1. Allowing some small entropy production DSA . 0, then E1A . E0A , and U0A /U1A . 1 by Eq. (10). As the term cos(u 2 f)U0A / U1A R 1, the A component of the jet, a2 , approaches 0 according to Eq. (15b). At the same time, f being large favors a large contribution of component B in the jet by Eq. (16b). Thus, everything else being equal, as the thickness of a given stream increases, its relative contribution to the jet should decrease. This prediction is contradicted by the experimental results presented in Section IV. Effect of Impact Angle on Jet Composition Consider the asymmetric oblique collision depicted in Fig. 3. The velocity of the inclined target (say A) in the

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167

steady state reference frame (Fig. 1) will always be greater than that of the projectile (B) by a factor of sec(u). This will tend to increase f, which will in turn reduce the contribution of A in the jet as described above. Moreover, the larger velocity of A can support stronger shocks and so the potential for entropy production is greater in the target. The ratio U0A /U1A (target) will therefore potentially be greater than U0B /U1B (projectile), reducing the contribution of target to the jet. Thus, everything else being equal, lowering the angle u should increase the contribution of projectile to the jet. This prediction is also contradicted by the new experimental results. Effect of Impact Velocity on Jet Composition Consider again the experiment shown in Fig. 3. The target is initially stationary, and the projectile has an initial laboratory-frame velocity of Vimp (i.e., with reference to Fig. (1) we have VA 5 0 and VB 5 Vimp). For impact angle u, the stationary-frame velocities are U0A 5 Vimp csc(u) (target) and U0B 5 Vimp cot(u) (projectile). If the flow is reversible, DSA 5 DSB 5 0. Then, for any impact velocity Vimp one obtains r1A 5 r0A and r1B 5 r0B from Eqs. (2) and (3), respectively, and U1A 5 U0A and U1B 5 U0B from Eqs. (10) and (11), respectively. The jet angle f will not depend on Vimp (Eq. (14)), and the jet stream thicknesses will not depend on Vimp (Eqs. (15b) and (16b)), and so jet composition will not depend on Vimp . The effect of velocity on composition must therefore be through the relative entropy production in streams A and B, which will increase as Vimp increases in a manner dependent upon the equations of state and the orientation of shocks. To the extent that entropy production is modest in the experiments to be described below, changing velocity at fixed angle is expected to have little effect on jet composition. No systematic dependence of jet composition on impact velocity was observed in the experiments. III. STABILITY

Although the analysis presented above assumes a stationary material interface, fluid dynamic considerations suggest that this cannot be correct. In particular, since U1A and U1B will be different, a Kelvin–Helmholtz instability is expected in both jet and slug streams (Melosh and Sonett 1986). One consequence of this instability is that the material interface will not be flat or stationary, and the A and B streams can exchange energy through work. Therefore, the A and B energy conservation equations must combine to form a single equation. The most general problem, with a ? 0, is then underspecified by 5 degrees of freedom. In computations described in Section V, this Kelvin– Helmholtz instability is strong, and the fluctuations of the material interface drive shocks. The material streams are therefore multiply shocked, which raises their specific en-

FIG. 4. Thermodynamic model for the stagnation point pressure (Puckett and Miller 1996). H is the Hugoniot, S0 and S1 are isentropes (S1 . S0), and Isp is the stagnation isenthalp. 0 is the centering point, and 1 is the strongest possible single shock state for the given stream velocity. Increasing entropy lowers the stagnation point pressure: Pa . Pb . Point c, the intersection of the Hugoniot with the stagnation isenthalp, is not physically realizable. c is Kieffer’s (1977) model for the stagnation point.

tropy. This lowers the stagnation point pressure, as indicated in Fig. 4, and lowers the density of the jet and slug streams per Eqs. (2) and (3), raises the internal energy per Eqs. (4) and (5), and lowers the jet and slug velocities per Eqs. (10) and (11). Moreover, since the entropy production jetward and slugward of the stagnation streamline may be different, these thermodynamic changes will affect the jet and slug differently. The computational results suggest that the specific entropy in the jet is much greater than that in the slug. If one assumed that the entropy changes to the jet and slug streams were identical, the effect of entropy on velocity via Eqs. (15) and (16) would be to increase the dimension of the jet (a2 , b2) and increase the mass fluxes (a2 r1AU1A , b2 r1BU1B) of the jet streams. The existence of interfacial instabilities in asymmetric jetting impacts has long been recognized, particularly in the explosive welding literature. Explosive welding occurs when obliquely impacted plates form a jet that scours their surfaces and removes layers that inhibit adhesion (e.g., surface oxidation) (Crossland and Williams 1970, Carpenter and Wittman 1975). Photomicrographs of explosive welds (e.g., Bahrani et al. 1967) show periodic waves typical of Kelvin–Helmholtz instabilities. Similar features are seen in the hydrocode computations presented in Section V. IV. EXPERIMENTS

A number of experiments of the type illustrated in Fig. 3 were conducted under the conditions indicated in Fig. 5. Cu ‘‘flyer plates’’ (6.35 mm thick, 70 mm diameter) were accelerated with the University of Chicago 80-mm singlestage gun, and impacted against 6.35-mm- and 1.59-mm-

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FIG. 5. Experimental conditions. In field a it is possible to find stationary shocks compatible with the flow illustrated in Fig. 1, and according to thin plate theory no jets will form. In fields b and c the flow is supersonic, but the attached shocks cannot turn the flow: jets are expected to form according to the von Neumann criterion. In fields d and e one or both streams are subsonic and jets are expected to form according to the sonic criterion. In fields b and d the mechanistically feasible range of stagnation point pressures for the target and the range for the projectile overlap. It is possible that a single stagnation point exists on the material interface. In fields c and e the feasible stagnation point pressure ranges do not overlap. Experiments indicated by numbered circles use 6.35-mm-thick targets and projectiles. Experiments indicated by numbered squares use 6.35-mm-thick projectiles and 1.59-mm-thick targets.

thick, 50.8-mm-wide, 76.2-mm-long inclined Sn target plates. Al or Fe witness plates (12.5 mm thick) placed beneath the point of impact recorded ejecta emitted from the Cu–Sn collision. From each witness plate a single 203mm-diameter circular piece was cut for chemical analysis in a JEOL JSM-5800LV scanning electron microscope. For each plate, 1000 raster measurements were made, each measurement sampling an area of approximately 200 3 200 em. These measurements were made in the neighborhood of the inferred point of contact of the jet with the witness plate. Direct capture of the jet using the polystyrene catch box technique (Yang and Ahrens 1995) was also attempted. A steel box containing a 152-mm-thick high-density polystyrene foam laminate was placed beneath the collision point. At the conclusion of each such experiment the polystyrene was found to contain a significant quantity of debris, such as fragments of a plywood momentum trap originally located P2 m downrange of the impact, that were obviously not due to jetting. It was therefore impossible to determine whether Cu–Sn particles found in the box were derived from a jet or whether they also included fortuitously captured fragments of the slug or surface spall fragments. Consequently, witness plates were used in subsequent experiments. Stray rebounded particles are not likely to become embedded on witness plates, and spall fragments

may be distinguished using geometric constraints described below. The witness plates (Fig. 6) contain evidence of two ejecta patterns. Most pronounced is a radially scoured region which is interpreted to be formed by impact of a Cu–Sn jet striking the plate in a diffuse circular region (arrow) then flowing outward radially across the plate. The location of the center of this radial pattern, between the projection of the projectile and target plates, is consistent with this interpretation. Surrounding the radiating pattern is a halo of distinct millimeter-diameter craters. These are concentrated in the downrange direction and are almost exclusively downrange of the projection of the inclined target surface (Figs. 3 and 6, line t). Their formation by jets is thus ruled out. These craters may have been created by the impact of spall fragments emitted from the rear free surface of the Sn target (Andriot et al. 1982). Figure 7 (see also Table I) summarizes the witness plate chemical analyses for experiments with equal plate thickness (6.35 mm). Witness plate 39 is cold rolled steel, the

FIG. 6. Witness plate from shot 38.

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169

FIG. 7. Compositional analyses in weight fraction for experiments with 6.35-mm-thick Cu and Sn plates. Plots are arranged in the order of Fig. 5.

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GREGORY H. MILLER

TABLE I Experimental Conditions and Calculated Flow Properties Shot

Vimpact [km/s]

u [deg]

fa [deg]

U a1,Sn [km/s] (max/min)

ra1,Sn [g/cc] (max/min)

U a1,Cu [km/s] (max/min)

ra1,Cu [g/cc] (max/min)

f eCu Theory (max/min)

38c 39c 40c 42c 43c 44c 45c 45d 50d

1.622 1.903 1.341 1.907 1.621 1.325 2.132 1.620 1.694

18.0 18.0 18.0 22.5 22.5 22.5 22.5 18.0 22.5

8.5 8.5 8.5 11.0 11.0 11.0 11.0 3.3 4.3

5.249/5.191 6.158/6.076 4.340/4.311 4.983/4.933 4.236/4.210 3.462/3.457 5.571/5.503 5.242/5.184 4.427/4.395

7.299/6.422 7.299/5.646 7.299/6.985 7.299/6.618 7.299/7.030 7.299/7.257 7.299/6.159 7.229/6.424 7.299/6.943

4.992/4.986 5.857/5.834 4.127/4.127 4.604/4.602 3.913b 3.199b 5.147/5.139 4.986/4.980 4.090/4.090

8.931/8.901 8.931/8.779 8.931/8.931 8.931/8.923 8.931b 8.931b 8.931/8.886 8.931/8.901 8.931/8.931

0.84/0.46 0.99/0.38 0.65/0.49 0.68/0.50 0.60/0.51 0.53/0.51 0.72/0.49 0.08/0.02 0.07/0.06

f fCu Exp. 0.18 0.24 0.18 0.26 0.31 0.36 0.30 0.47 0.53

6 6 6 6 6 6 6 6 6

0.02 0.04 0.03 0.01 0.03 0.04 0.02 0.02 0.02

a

Calculated. No steady state shock is possible in this case. c Cu and Sn plates are 6.35 mm thick. d Cu plate is 6.35 mm thick, Sn plate is 1.59 mm thick. e Calculated using Eq. (17). f Weight fraction of Cu, [Cu]/[Cu 1 Sn], measured on witness plates. b

others are cast Al. It is not known whether the jet plated onto the witness plates faithfully reflects its initial composition, or whether some chemical fractionation occurred. It is remarkable that the Cu content recorded on the witness plates is apparently uniform at P10 wt%, with variation in the Cu weight fraction Cu/(Cu 1 Sn) due principally to variation in Sn. It is also remarkable that the measured Cu weight fraction in this series of experiments is lower by 40–370 wt% than the theoretical abundances presented in Table I, which were calculated using fCu 5

b2U1B r1B , a2U1A r1A 1 b2U1B r1B

(17)

with the equations developed in Section II, and equations of state described in the following section. Many of the calculated quantities are sensitive to the shock entropy production parameters DSA and DSB . Table I gives the range in these quantities calculated from the range in permitted shock entropy production (Section II). The data show a trend of increasing fCu with increasing angle u for fixed impact velocity, and it seems unlikely that this trend is influenced by chemical fractionation in the witness plate capture process. In the context of the twoparameter (DSA , DSB) model used to compute Table I, the observed trend (but not the observed abundances) can be explained by maximizing entropy production in the Cu projectile while minimizing entropy production in the Sn plate, but computations do not support this hypothesis. The data do not show an unambiguous trend in the dependence of composition on velocity at fixed angle and plate thickness. The calculated results (Table I) permit either increasing or decreasing trends, depending on the

relative entropy production in the Cu and Sn streams. However, the magnitude of the velocity effect is small, in accordance with the theory. Figure 8 (also Table I) shows the effect of plate thickness on jet chemistry. Experiments at approximately constant velocity but differing angle and Sn plate thickness (6.35 and 1.59 mm) are compared. According to the theory developed in Section II, the Cu fraction in experiments 49 and 50, with the thinner Sn targets, should be smaller than in experiments 38 and 43, with the thicker Sn targets, respectively. Instead, the thin Sn plate experiments show a distinct increase in Cu jet fraction as inferred from the witness plate analyses. The theoretical Cu abundances for the two thin Sn experiments are low by 82–96 wt% of the measured abundances. V. COMPUTATIONS

Several of the experiments illustrated in Fig. 3 were simulated using a computational method described by Miller and Puckett (1996). The simulation used a Mie– Gru¨neisen EOS with a Hugoniot reference curve constructed of piece-wise linear shock velocity Us 2 particle velocity Up segments, Us 5 c0 1 sUp (Ahrens and Johnson 1995). The parameters for Cu were r0 5 8.931 g/cc, c0 5 3.982 km/s, s 5 1.460, with Gru¨neisen parameter c 5 1.99. For Sn r0 5 7.299 g/cc, (c0 in km/s, s) 5 (2.60, 2.2), (3.33, 20.14), (2.48, 1.57), (3.43, 1.205), and c 5 2.11. For both materials, rc 5 constant. The piece-wise linear Hugoniot reference state is not strictly necessary for these simulations because the EOS of Sn is well-approximated with a single linear Hugoniot over the pressure range of interest. Numerical results using

JETTING IN OBLIQUE, ASYMMETRIC IMPACTS

171

FIG. 8. Compositional analyses in weight fraction for experiments with 6.35-mm-thick Cu and 6.35- or 1.59-mm-thick Sn. The nominal impact velocity is 1.6 km/s.

the simpler Hugoniot Us 5 2.59 1 1.49Up give substantially identical results. The more complicated EOS was used as part of a program to extend the code to more general equations of state and to model phase changes in particular. It is interesting to note that the piece-wise linear Sn EOS exhibits a phase change, and because the slope s in the mixed-phase region is negative, 2D and 3D planar shocks are not everywhere dynamically stable (Menikoff and Plohr 1989). This instability is seen in the two-phase region (Fig. 9). The computational algorithm used here associates a single velocity to each Eulerian grid cell, whether that cell contains one or two material phases. Consequently, in these computations the interfaces are more nearly ‘‘stick’’ than ‘‘slip,’’ and this is different than is assumed in the thin plate theory. Figure 9 is a pressure contour plot of a 2D computational approximation to experiment 42 shown 12.5 es after impact. The most striking feature of this computational result is the Kelvin–Helmholtz interface instability. As a consequence of the instability the jet undulates, and so the angles f and a (Fig. 2) are poorly defined. This is compatible with the region of contact of the jet with the witness plate being diffuse (Fig. 6). A sequence of circular shocks can be seen emanating from the Sn–Cu interface near the point of emergence of the jet. The proximity of the jet to these instabilities results

in the jet being affected preferentially by these secondary shocks. The approach to steady state is slow in this system. One measure of this approach is the orientation of the leading shock in the Sn target. Figure 10 shows the 5 GPa pressure contour (which lies within the leading stock) at 1-es intervals from 1 to 13 es. The duration of the experiment is P15 es; thus the approach to steady state is not rapid in comparison to the duration of the experiment, and in this sense 6.35 3 76.2-mm plates are not thin. As the steady state condition develops the shock approaches the flownormal orientation, maximizing entropy production and minimizing the stagnation point pressure (Fig. 4). If this steady state shock orientation occurs generally, it suggests a way to specify the free parameters DS in the theory described above. The free surface streamlines in the Sn and Cu slug streams diverge in Fig. 4. This suggests that the approximation that the slug streams are parallel, U1AiU1B , implicit in Fig. 2, and the theory of Section II is inaccurate far from the collision point. VI. PLANETARY AND PLANETESIMAL COLLISIONS

Jet Phase As remarked in Section II, the amount of entropy production in a jetting collision cannot be determined a priori,

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FIG. 9. Simulation of experiment 42. Problem domain is 600 by 680, giving 254 cells across the plate; 12.5 es after impact. The thin dashed and continuous contours indicate pressure in 10-GPa intervals. The bold lines indicate the material interfaces.

but it may be bounded if one assumes the existence of a single shock. Likewise, the peak thermodynamic state of the jet cannot be predicted a priori, but it too may be bounded. In a steady state flow, the most extreme conditions of pressure and temperature will occur at the stagnation point where the initial kinetic energy of the stream is converted entirely to enthalpy. The flow revealed by the numerical simulations is not in steady state, and hence this limit is only approximate. The stagnation point isenthalps (Isp in Fig. 4) may be constructed for each stream. They will be different, in general, for streams A and B, because of differences in velocity. The thermodynamic state of the stagnation point for a given stream is a point on curve Isp . One way to pick this point is to determine how much entropy production occurs along the stagnation streamline and seek the intersection of the stagnation isenthalp with the appropriate isentrope. For typical equations of state, the relative orientation of isenthalps, isentropes, and the Hugoniot, is as indicated in Fig. 4. Note in particular that as entropy increases along the isenthalp, pressure decreases. Therefore, the highest pressure possible at the stagnation point occurs if there is no entropy production (no shock) on the stagnation streamline. The minimum stagnation

point pressure occurs with the maximum entropy production, which occurs when there is a stationary shock normal to the stagnation point streamline. The stagnation isenthalp does intersect the Hugoniot in Fig. 4, but this intersection does not represent a thermodynamic state accessible to the system described here (Puckett and Miller 1996). The reason is, in essence, that the flow emerging from any stationary shock much have kinetic energy. It will therefore have less enthalpy than it would if it had stagnated (Eq. (1)). The stagnation streamline passes from the shock to the stagnation point along an isentrope (e.g., S1 in Fig. 4). Kieffer (1977) estimated the stagnation point thermodynamic state as the intersection of the stagnation point isenthalp with the shock Hugoniot. This assumption underestimates the stagnation point pressure (Fig. 4) and overestimates its temperature. Figure 11a illustrates this point with the specific example of fully compacted dunite with an initial velocity U0 of 8 km/s. If there is no shock the stagnation point will have a pressure of 145 GPa and a temperature of 472 K. If there is a shock normal to the stagnation streamline the stagnation point pressure will be slightly lower at 144 GPa, but its temperature will be significantly higher at 636 K. The intersection of the Hugoniot with the isenthalp, not a physically realizable condition, gives a pressure of 131 GPa, which is too low by only 9%. However, the temperature of 2630 K calculated by this approximation is high by 300%. When

FIG. 10. Simulation of experiment 42. Orientation of shock in Sn at 1-es intervals.

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FIG. 11. Stagnation point thermodynamics for dunite.

porosity is present (Fig. 11b) the thermodynamically allowed stagnation point pressure range expands to 108–145 GPa, and the temperature range becomes 472–6155 K. The intersection of Isp with the Hugoniot gives 96 GPa and 7864 K. Applications of jetting to impact melt formation in low velocity collisions, which were favored by the high stagnation point temperature calculated using Kieffer’s approximation, must be reevaluated in light this thermodynamic argument. Jet Composition A normal-incidence Earth impact is one in which the velocity vector is parallel to Earth’s surface norm. By analogy to asymmetric thin plate experiments (Fig. 3), in a normal impact Earth plays the role of the projectile (Fig. 12a). Conversely, in an oblique impact Earth plays the role of the target (Fig. 12b). The geometry at the point of the

FIG. 12. Generalization of plane impact to spheres.

collision evolves with time because of the curvature of the meteoroid (Melosh and Sonett 1986, Ang 1990, 1992, Vickery 1993). Ignoring complications such as the unknown effects of far-field flow and boundary conditions, the experimentally observed compositional trends may be used to infer compositional trends when spheres collide. Figure 12a shows schematically the normal impact of a small meteroid with Earth in normal-incidence impact. As the impact progresses, the angle of contact increases from u 5 0 through u 5 f/2. The experiments indicate that as u increases, the fraction of projectile increases. Thus the terrestrial component of the jet might be expected to increase with time. In the first moment of contact, u 5 0. Under this condition there is complete symmetry, and if a jet were formed (it will not be) then symmetry would determine the terrestrial fraction to be 50% for identical meteoroid and Earth equations of state. The terrestrial fraction increases from this 50% point as the collision progresses; thus the jet will be terrestrially dominated in normal impact. Figure 12b illustrates schematically an oblique collision where the velocity vector is inclined with respect to the Earth norm. Again, the geometry will evolve with time from an initially parallel configuration, u 5 0, through u 5 f/2. The correspondence between this scenario and the asymmetric impact of thin plates is complicated by a component of shear velocity that does not occur in the normal incidence case. Neglecting this shear component, as the impact progresses the fraction of projectile, meteoroid in this case, will increase from an initial value of 50%. If the shear contribution may be ignored, then oblique collisions should produce predominately meteoritic jets.

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The shear component of velocity will initially be oriented away from the collision point, which will lower the stationary-frame velocity of the meteoroid stream and lower its stagnation pressure. This ought to promote the onset of jetting, possibly lower the flux of meteoroid in the jet, and lower the peak thermodynamic conditions experienced by jetted meteoroid. Further speculation is unwarranted given the poor agreement of theory with experiment. The trend predicted above, based on an interpretation of our experimental results, was predicted by Melosh and Sonnett (1986) in their theoretical study (see their Table II). This trend is also supported by the experimental results of Shultz and Sugita (1997). In their laboratory experiments spheres were collided obliquely against flat plate targets, and the jet was spectroscopically analyzed for composition. They observed that ‘‘the impactor signature increases as impact angle [from horizontal] decreases’’; i.e., the more oblique the impact, the more meteoritic the jet, as predicted above. Computations of the collision of ice spheres against rock targets also support this compositional trend (Blank and Miller, 1997). In normal impacts jets are observed containing both ice and rock, but in highly oblique impacts the jet-like flow consists only of ice. Again, the more oblique the impact, the more meteoritic the jet.

anonymous reviewer for a number of helpful comments. This work was supported in part by NSF EAR-9304263 and EAR-9614179 and by Los Alamos National Laboratory subcontract C51170016-3L under DOE Contract W-7405-ENG-36.

VII. CONCLUSIONS

Birkhoff, G., D. P. MacDougall, E. M. Pugh, and G. Taylor 1948. Explosives with lined cavities. J. Appl. Phys. 19, 563–582.

New experiments presented here give jet compositions and compositional trends that are inconsistent with solvable analytical models of oblique collisions based on macroscopic balance equations. Numerical models of these experiments reveal a Kelvin–Helmholtz instability, with attendant consequences for the momentum and energy balances of each material stream. The computations also suggest that the slug streams diverge, in contrast to idealized model. Peak temperatures experienced by jetted matter were calculated to be significantly lower than one would calculate by approximating the stagnation point as lying on the shock Hugoniot. Jets are not as likely to produce melt or vapor in low velocity collisions as previously hypothesized. The experiments show a trend of increasing projectile abundance in the jet with increasing impact angle. If this result may be generalized to impacting spheres, where critical assumptions such as steady state do not apply, then the terrestrial component of jets will be greater in normal impact than in highly oblique impact.

Blank, J. G., and G. H. Miller 1997. The fate of organic compounds in cometary impacts. In Proceedings of the 21st International Symposium on Shock Waves, in press.

ACKNOWLEDGMENTS Insightful conversations with V. Barcilon are gratefully acknowledged. Compositional analysis of the witness plates was performed by A. Davis. The hydrocode computations were done on SGI Origin 200 computers courtesy of D. Archer and L. Grossman. Thanks to H. J. Melosh and an

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