Jetting during oblique impacts of spherical impactors
Shigeru Wakita, Brandon Johnson, C. Adeene Denton, Thomas M. Davison
JJetting during oblique impacts of spherical impactors
Shigeru Wakita a, ∗ , Brandon Johnson a , C. Adeene Denton a , Thomas M.Davison b a Department of Earth, Atmospheric, and Planetary Sciences, Purdue University, WestLafayette, IN, 47907, USA b Department of Earth Science and Engineering, Imperial College London, London, UK
Abstract
During the early stages of an impact a small amount material may be jetted andejected at speeds exceeding the impact velocity. Jetting is an important processfor producing melt during relatively low velocity impacts. How impact angleaffects the jetting process has yet to be fully understood. Here, we simulatejetting during oblique impacts using the iSALE shock physics code. Assumingboth the target and impactor have the same composition (dunite), we examinethe jetted material which exceeds the impact velocity. Our results show thatoblique impacts always produce more jetted ejecta than vertical impacts, exceptfor grazing impacts with impact angles < ◦ . A 45 ◦ impact with an impactvelocity of 3 km/s produces jetted material equal to ∼
7% of the impactormass. This is 6 times the jetted mass produced by a vertical impact withsimilar impact conditions. We also find that the origin of jetted ejecta dependson impact angle; for impact angles less than 45 ◦ , most of the jet is composed ofimpactor material, while at higher impact angles the jet is dominated by targetmaterial. Our findings are consistent with previous experimental work. In allcases, jetted materials are preferentially distributed downrange of the impactor. Keywords:
Impact processes, Collisional physics, Asteroids ∗ Corresponding author
Preprint submitted to Elsevier February 5, 2021 a r X i v : . [ a s t r o - ph . E P ] F e b . Introduction Impact jetting occurs during the earliest stage of impact cratering when theimpactor is still making contact with the target (e.g., Birkhoff et al., 1948; Walshet al., 1953). The jet contains the first material ejected, is highly shocked, and isejected at speed exceeding the impact velocity. The total mass of jetted ejecta issmall, typically less than a few percent of the impactor mass. Impact jetting hasbeen proposed as a mechanism for producing tektites (Vickery, 1993; St¨offleret al., 2013), producing melt in lightly shocked material during grain-on-graincollision (Kieffer, 1977), and producing chondrules (Kieffer, 1975; Johnson et al.,2015; Wakita et al., 2017). Jetting has also been proposed as a critical com-ponent of vaporization during massive impacts (Melosh and Sonett, 1986), andto affect the final ice-rock-ratio of icy bodies (McKinnon, 1989a,b). However,jetting, especially during oblique impacts, remains poorly understood.Experimental and theoretical studies have explored jetting during planetaryimpacts using thin plate theory (e.g., Kieffer, 1975; Melosh and Sonett, 1986;Vickery, 1993). Thin plate theory describes jetting during the symmetric col-lision of two thin plates and can predict the jet velocity and the critical angleabove which jetting occurs as a function of plate material and impact velocity(Birkhoff et al., 1948; Walsh et al., 1953). However, thin plate theory fails whenapplied to planetary impacts, which are better described by the impact betweena spherical impactor and a plate than of two thin plates, including predictions ofejecta temperature and the ratio of target to impactor material in jets producedby spherical impactors (Sugita and Schultz, 1999). Numerical models (Johnsonet al., 2014) and laboratory experiments (Kurosawa et al., 2015) show that themaximum velocities of jets produced during the impact of spherical bodies areabout half the velocity predicted by thin plate theory.Vickery (1993) estimated that very oblique impacts could have up to tentimes more jetted material than vertical impacts with the same impact velocity.While these estimates are based on thin plate theory, which does not accu-rately describe the jetting physics during impact of spherical bodies (Yang and2hrens, 1995; Kurosawa et al., 2015), oblique impacts should produce morejetted material than head-on collisions as experimental results of the sphericalimpactor indicate (Sugita and Schultz, 1999). However, without direct simula-tion in three dimensions, we cannot quantify how impact angle will affect theamount of jetted material produced by a given impact.Most impacts are believed to be oblique; a typical impact angle is expectedto be 45 ◦ (Shoemaker, 1962). The effect of impact angle on crater formationhas been studied experimentally (e.g., Gault and Wedekind, 1978; Schultz andGault, 1990; Burchell and Mackay, 1998) and numerically (e.g., Davison et al.,2011; Elbeshausen et al., 2009). Highly oblique impacts can produce elongatedcraters as found on Mars, Venus, and the Moon (Bottke et al., 2000). Evenmoderately oblique impacts have a significant effect on distal impact ejecta,producing a wedge of avoidance in the uprange direction (Gault and Wedekind,1978), and elongated craters produced by grazing impacts exhibit in a butterfly-like distal ejecta pattern (Gault and Wedekind, 1978; Schultz and Gault, 1990;Speyerer et al., 2016). How impact angle affects the distribution of the veryfastest and earliest ejecta remains uncertain. Here, we simulate oblique impactsto determine how impact angle affects the jetting process. This includes deter-mining how the amount and distribution of jetted material depends on impactangle. In Section 2, we describe the setup for our numerical simulations. Wethen present results from our simulations (Section 3) and discuss the implica-tions for our understanding of the unique physics surrounding jetted materialduring impact (Section 4).
2. Methods
We simulate jetting during oblique impacts using the iSALE-3D shock physicscode, which includes both a strength model and the capability to simulate porouscompaction (Hirt et al., 1974; Collins et al., 2004; Melosh et al., 1992; Ivanovet al., 1997; W¨unnemann et al., 2006; Collins et al., 2011). We assume thatour spherical impactor and flat target have the same dunite composition (Benz3t al., 1989), as used in previous jetting work (e.g., Johnson et al., 2014). Wenote these results are only strictly valid for a dunite impactor and target. Morecompressible materials are known to have a lower jetting efficiency (Johnsonet al., 2014). Jetting during the impact of a projectile with distinct compositionfrom the target will also affect jetting dynamics (Sugita and Schultz, 1999). Weexpect, however, that the downrange momentum of the impactor in oblique im-pacts will still tend to enhance jetting regardless of composition. The materialproperties for dunite used in iSALE-3D are described in Table 1.In our simulations, we vary impact angle ( θ imp ) and impact velocity ( v imp ).Impact angle in our simulations is measured from the horizontal surface of thetarget, such that θ imp = 90 ◦ is a vertical impact. We simulate a range of impactangles from 15 ◦ to 90 ◦ in 15 ◦ increments, with impactor velocities of 2 km/s, 3km/s, and 5 km/s. This range of velocities spans the expected range of impactvelocities during planetesimal accretion, which depends on the size of the targetbodies (i.e., the escape velocity of the system, Kokubo and Ida, 2000; Johnsonet al., 2015; Hasegawa et al., 2016). This is also the velocity range where thejetted mass is highest for vertical impacts of dunite impactors on dunite targets(Johnson et al., 2015; Wakita et al., 2017).We record the behavior of jetted ejecta in iSALE-3D using Lagrangian tracerparticles to track material position, velocity, pressure, and temperature. Notethat the tracer particles, which track the motion of a parcel of material, areinitially placed at the center of each cell and we calculate their mass using theirinitial spacing, locations, and density (e.g., Johnson and Melosh, 2014). Wefocus on tracer particles since they allow us to determine material provenanceas well as mass. High resolution is required to accurately evaluate the velocities,ejection angles, and mass of jetted ejecta. We use a fixed impactor diameterof D imp = 1 km, with a spatial resolution of 5 m or 100 cells per projectileradius. Previous work showed that for 10-km-diameter bodies, jetting occurs ina near-surface zone at most ∼
400 m deep (Wakita et al., 2017). As such, thehigh-resolution zone of our model incorporates the lower half of the impactorand a thin surface layer of target material. We use this setting for most of our4uns. iSALE-3D divides simulations into a high-resolution zone and an extensionzone; in the extension zone cell size increases with distance from the high-resolution zone. To avoid errors introduced by motion through the extensionzone, we only track tracer particles in the high resolution zone. To confirm thatour high-resolution zone appropriately resolves the jetting process, we compareour iSALE-3D results of vertical impacts with iSALE-2D runs using the sameresolution (Appendix A). We also ensure that all jetted material originates fromand is tracked within the high resolution zone. We confirm that a high resolutionzone encompassing half of the impactor resolves jetted material well for mostoblique impact scenarios. For some oblique impact scenarios ( θ imp = 30 ◦ at v imp = 5 km/s and θ imp = 45 ◦ at v imp = 2, 3, and 5 km/s), we enlarge the high-resolution zone in the vertical direction to include all jetted impactor material.5 able 1: iSALE input parameters Description ValuesEquation of state ANEOSBulk material of impactor/target dunite a Solidus temperature 1373 K b Simon approximation constant A 1520 MPa c Simon approximation exponent C 4.05 c Poisson’s ratio 0.25 d Thermal softening parameter 1.1 d Strength model Rock e Cohesion (damaged) 0.01 MPa d Cohesion (undamaged) 10 MPa d Frictional coefficient (damaged) 0.6 d Frictional coefficient (undamaged) 1.2 d Strength at infinite pressure 3.5 GPa d Damage model Ivanov e Minimum failure strain 10 − Damage model constant 10 −
11 d
Threshold pressure for damage model 300 MPa da Benz et al. (1989) b Davison et al. (2016) c Davison (2010) d Kurosawa and Genda (2018) e Collins et al. (2004)
3. Results
Figure 1 shows a time series of our fiducial model, an oblique impact with θ imp = 45 ◦ and v imp = 3 km/s. Each panel represents t = 0.0 t s , 0.25 t s ,6 igure 1: Time series for our fiducial model ( v imp = 3 km/s and θ imp = 45 ◦ ). Colorrepresents tracer particle velocity ( v ej ) normalized by the impact velocity. The top panelsare a cross-sectional view, while the bottom panels are a top view on the impact from above.Each panel represents (a) t = 0.0 t s , (b) t = 0.25 t s , (c) t = 0.51 t s , and (d) 0.76 t s after theimpact, respectively. White arrow in top-left panel shows the direction of impactor. Note theimpactor is spherical, but only contains tracer particles in lower part, due to our setting ofhigh resolution zone (shown as dashed line). t s , and 0.76 t s after the impact, respectively, where t s is the characteristictime for contact and compression defined by t s = D imp / ( v imp sin( θ imp )). Thetracer particles shown in Figure 1 reflect our setup of high-resolution zone,which includes the lower portion of the spherical impactor and the near-surfaceportion of the target. Figure 2 illustrates only the tracer particles whose velocityis higher than the half of the impact velocity with a positive velocity in verticaldirection. Faster ejecta ( v ej /v imp >
1) is focused in the downrange direction andejected at very low angles as evidenced by its nearly horizontal motion. Thehigher velocity ejecta ( v ej /v imp > .
6) are preferentially distributed downrangeand originate from the impactor.Figure 3 illustrates the relationship between ejecta velocity and azimuthalejection angle as defined with respect to the orientation of the impactor (down-range is 0 ◦ , uprange 180 ◦ ). As Figure 3 indicates, jetted ejecta ( v ej /v imp > ◦ . Azimuthal angleshallows as the ejecta velocity increases. Because the impactor is traveling inthe downrange direction on impact, downrange ejecta acquires higher velocitythan ejecta not aligned with the impactor. Although the distribution of ejectaproduced from the target and the impactor in Figure 2 is similar, the maximumvelocity of ejecta produced from impactor material is v ej /v imp = 2, which ishigher than ejecta produced from the target of v ej /v imp = 1 . θ imp = 90 ◦ , 60 ◦ , 30 ◦ , and 15 ◦ . For a head-on impact( θ imp = 90 ◦ ) ejecta are distributed symmetrically from the impact point (definedas the origin, 0, 0) (Figures 4a, 5a, 6a). Ejecta behavior for θ imp = 60 ◦ is similarto θ imp = 45 ◦ , our fiducial case (Figures 4b, 5b and 6b); however, as impactangle continues to decrease ( θ imp = 30 ◦ ), the high-velocity portion of ejecta( v ej /v imp > ◦ ) or at an angle of ∼ ◦ (Figures 5cand 6c). Note that ejection angles in iSALE-3D have some preferences, such as45 ◦ and 135 ◦ (Figure 6a), due to the use of cartesian coordinates. While sucha bimodal distribution is only observed for a 30 ◦ impact, this behavior may be8 igure 2: Material ejected at (a) t = 0.25 t s , (b) t = 0.51 t s , and (c) t = 0.76 t s for our fiducialmodel ( v imp = 3 km/s and θ imp = 45 ◦ ). Color represents ejecta velocity ( v ej ) normalized bythe impact velocity. Top panels are a cross-sectional view, while the bottom panels are abird’s eye view on the impact from above. Ejecta produced by the target are shown on theleft, ejecta from the impactor in the middle, and the total ejecta produced from the impacton the right, respectively. Angle[ ∘ ] v e j / v i m p Target
Angle[ ∘ ] Impactor
Angle[ ∘ ] Total (Target + Impact r) (7 (6 (5 (4 (3 (2 M e j / M i m p Impact vel city = 3 km/s, Impact angle = 45 ) , t = 0∘76 t s Figure 3: Heatmap of ejecta density as a function of ejection angle and velocity for ourfiducial model ( v imp = 3 km/s and θ imp = 45 ◦ ). Ejection angle is defined with respect to theorientation of the oblique impactor (downrange is 0 ◦ , uprange 180 ◦ ). Note ejection angle ismeasured in the azimuthal direction, not in vertical direction (elevation angle). Gray contoursrepresent mass of ejecta M ej normalized by mass of impactor M imp : each bin of ejection angleand velocity represents the amount of ejecta in the bin (the dark-colored bin has more ejectathan the light-colored bin). Note that angle bins are 5 degree wide and velocity bins havea width of 0.05 v ej /v imp . Ejecta produced by the target are shown on the left, ejecta fromthe impactor in the middle, and the total ejecta produced from the impact on the right,respectively. similar to the distal distribution of ejecta around elongated craters on Moonand Mars, which originate from oblique impacts (Gault and Wedekind, 1978;Schultz and Lutz-Garihan, 1982). The volume of ejecta produced for a grazingimpact ( θ imp = 15 ◦ ) scenario appears minimal (Figures 5d and 6d), and high-velocity ejecta ( v ej /v imp > .
5) is only distributed downrange of the impact site.We find almost all of high-velocity ejecta originate from the impactor.
We use Lagrangian tracer particles to quantify the amount of jetted ejectafor different impact angle and impact velocity scenarios. We define tracer par-ticles as jetted ejecta when the tracer velocity exceeds the impact velocity, thevertical component of tracer velocity is positive, and the material is locatedabove the surface of the target. Following the evolution of tracer particles thatsatisfy these conditions, our analysis indicates that as the impact progressesmore material is ejected until jetting ceases. The earliest ejecta is the fastest10 igure 4: Behavior of ejected material for θ imp = 90 ◦ , 60 ◦ , 30 ◦ , and 15 ◦ (top to bottom).Same color scheme and viewing geometries as Figure 2. igure 5: Same as Figure 4, but at later time. igure 6: Same color scheme and measurements as Figure 3 produced for θ imp = 90 ◦ , 60 ◦ ,30 ◦ , and 15 ◦ (top to bottom). .0 0.5 1.0 1.5 2.0 time/t s −3 −2 −1 Σ M j e t / M i m p Total(Impactor+Target)TargetImpactor
Figure 7: Evolution of cumulative jetted material mass over time for our fiducial model( v imp = 3 km/s and θ imp = 45 ◦ ). Each line represents jetted mass sourced from the target(dotted-dashed line), the impactor (dotted line), and the total amount of material ejected(solid line), respectively. time/t s −3 −2 −1 Σ M j e t / M i m p Total(Impactor+Target)TargetImpactor
Figure 8: Additional plots of jetted material mass over time as in Figure 7, but for θ imp =90 ◦ . Impact angle [ ∘ ] t j e t / t s v imp = 5 km/sv imp = 3 km/sv imp = 2 km/s Figure 9: Time of jetting ( t jet ) as a function of impact angles. Color represent differentimpact velocities: 2 km/s (blue), 3 km/s (black), and 5 km/s (green). while the latest is ejected just above the impact velocity. To calculate for thecumulative jetted mass, we sum all tracer particles that fit this definition ofjetted ejecta during impact. Figure 7 illustrates the evolution of the cumulativejetted mass M jet normalized by the impactor mass M imp over time for our fidu-cial model ( v imp = 3 km/s and θ imp = 45 ◦ ). The jetted material reaches 90%of the total cumulative mass at t = 0.76 t s . However, for a head-on impact,this peak occurs much earlier as t = 0.33 t s (Figure 8); this delay in jettingshows that impact jetting occurs over longer timescales for oblique impacts(note t s = D imp / ( v imp sin( θ imp ))). Figure 9 represents the time of jetting t jet ,when Σ M jet reaches 90% of its final value, as a function of impact angle. Evenwhen scaled by the contact and compression timescale, the jetting timescaleinitially increases as impact angle decreases at a given impact velocity. This islikely due to the downrange momentum of the impactor and a favorable, lessasymmetric, geometry for jetting. Simulations without material strength showthis trend continues to θ imp = 30 ◦ for all impact velocities, demonstrating thatmaterial strength causes jetting to cease earlier in the case of θ imp = 30 ◦ . Thejetting timescale for θ imp = 15 ◦ is shorter than other oblique impacts (Fig-ure 9). For grazing impacts downward velocity is reduced before the contact15 Impact angle [ ∘ ] −3 −2 −1 Σ M j e t / M i m Target
15 30 45 60 75 90
Im act angle [ ∘ ] Im actor
15 30 45 60 75 90
Im act angle [ ∘ ] Total ∘Target + Im actor) v im = 5 km/sv im = 3 km/sv im = 2 km/sEjecta from targetEjecta from im actorTotal ejecta Figure 10: Mass of jetted material ( M jet ) as a function of impact angles. Color representdifferent impact velocities: 2 km/s (blue), 3 km/s (black), and 5 km/s (green). Each paneldepicts the total mass M jet normalized by M imp from the target (left), impactor (middle),and the total mass (right), respectively. and compression phase is over, which sometimes results in decapitation of theimpactor (Schultz and Gault, 1990; Davison et al., 2011). When this occurs,the impactor does not penetrate the target and jetting ceases earlier than inmoderately oblique impacts.Figure 10 summarizes the trends in cumulative jetted material mass Σ M jet for all impact scenarios. Each panel represents Σ M jet separated by materialorigin: material originating from the target (dotted-dashed lines, left), impactor(dotted line, middle), and the total mass (solid line, right), respectively. We firstfocus on the results of v imp = 3 km/s (black lines), our fiducial velocity. In thisscenario, almost all oblique impacts ( θ imp = 75 ◦ , 60 ◦ , 45 ◦ , and 30 ◦ ) producemore jetted ejecta than the head-on impacts: 1.6%, 3.1%, 6.8%, and 4.9% jettedmass normalized by impactor mass ( M imp ), respectively, in comparison to 1.1% for a vertical impact. Jetted ejecta from our grazing impact ( θ imp = 15 ◦ ) inthis scenario is less than 1% (Figures 5d and 6d). θ imp = 45 ◦ produces the mostjetted mass ( ∼ v imp = 3 km/s)depends on impact angle, which also influences the source of jetted material(Figure 10). As Figure 7 indicates, material from the impactor dominates thecumulative mass of jetted ejecta in our fiducial model ( v imp = 3 km/s and θ imp = 45 ◦ ). In contrast, jetted ejecta is dominated by target material for head-on16 Impact angle [ ∘ ] −2 −1 Σ M j e ∘ T a r g e ) / Σ M i m p ∘ I m p a c o r ) v imp = 5 km/sv imp = 3 km/sv imp = 2 km/s Figure 11: Mass ratio of jetted material from the target to the impactor as a function ofimpact angles. Note that no symbols (e.g., 15 ◦ cases) means no jetted material (see Figure10). impacts (Figure 8); this distinction may result from the downrange momentumof the oblique impactor. The fraction of impactor to the total jetted mass forour nominal impact velocity is 84% for a 45 ◦ impact and 20% for a head-onimpact. This trend is consistent with the results of laboratory experiments inprevious work (Schultz and D’Hondt, 1996; Sugita and Schultz, 1999). Theseexperiments showed that oblique impacts tend to produce more jetted materialfrom the impactor. Our results are consistent with this trend: Figure 11 clearlyillustrates that the ratio of jetted material from the target to the impactordecreases as the impact angle gets shallower.Our results for a lower impact velocity ( v imp = 2 km/s, blue lines in Figure10) indicate that Σ M jet also depends on impact velocity. In this scenario, thejetted mass produced in head-on impacts is 0.6% of the impactor mass, almosthalf of that produced for v imp = 3 km/s. Oblique impacts also produce less jet-ted ejecta at a lower velocity; however, we see similar trends with impact angleand θ imp = 45 ◦ still produces the largest mass of jetted material (4.0% of theimpactor mass). Since the jetting timescale of 45 ◦ impact is much longer thanother impacts with v imp = 2 km/s (Figure 9), jetting in θ imp = 45 ◦ continues17onger then produces more jetted mass. A lower jetted mass at a lower (2 km/s)impact velocity contradicts previous simulations using a strengthless material(Johnson et al., 2014), likely because our incorporation of material strength lim-its the amount of jetted ejecta produced by low velocity impacts. This behavioris supported by previous modeling work (Johnson et al., 2015; Wakita et al.,2017, see also Appendix A).When v imp is increased ( v imp = 5 km/s, green lines in Figure 10), the result-ing behavior of the jetted ejecta can be explained by two competing effects. Asimpact velocity increases, jetting initiates later (e.g., Sugita and Schultz, 1999;Johnson et al., 2014). For θ imp = 90 ◦ and 75 ◦ this effect reduces the amount ofjetted material as compared to our simulations with v imp = 3 km/s. While thetime when jetting ceases is almost the same for θ imp = 90 ◦ and 75 ◦ regardlessof impact velocity, jetting ceases earlier for lower-velocity impacts occurring atlower impact angles (Figure 9). Thus, despite increasing impact velocity, jet-ted mass for θ imp = 45 ◦ and 60 ◦ are similar to the fiducial case. As impactangle is decreased ( θ imp = 30 ◦ ), the time of jetting increases and more jettedmass is produced than our fiducial case (14% compared to 6.8%), which is thehighest value among impact with v imp = 5 km/s. Simulations without materialstrength show that jetted mass decreases as impact velocity increases regardlessof impact angle, which is consistent with previous work (Johnson et al., 2014).Thus, material strength has an important effect on the jetting efficiency duringoblique impacts in the velocity range explored.
4. Discussion
Our results show that the efficiency and dynamics of the jetting processare sensitive to impact angle and impact velocity. In general, oblique impactstend to produce less shock-heated material than vertical impacts (Pierazzo andMelosh, 2000; Davison et al., 2014). However, this trend does not hold whenconsidering only jetted material. Previous work suggests that the fraction of jet-ted material that reaches high temperatures (i.e., may experience melting and18
00 1000 1500 2000 2500
Peak Tempera ure [K] −3 −2 M e j / M i m p Targe
500 1000 1500 2000 2500
Peak Tempera ure [K]
Impac or
500 1000 1500 2000 2500
Peak Tempera ure [K]
To al (Targe + Impac or) θ imp = 90 ∘ θ imp = 60 ∘ θ imp = ∘5 ∘ θ imp = 30 ∘ θ imp = 15 ∘ Figure 12: Histogram of peak temperature of ejecta shown in Figures 2(c) and 5 ( v ej /v imp > . t jet for v imp = 3 km/s). Bin size of peak temperature is 100 K. Mass of ejecta ( M ej ) isnormalized by mass of impactor M imp . vaporization) will also increase as the impact becomes more oblique (Meloshand Sonett, 1986; Vickery, 1993). Figure 12 represents the histogram of peaktemperature of ejecta shown in Figures 2(c) and 5 ( v ej /v imp > . t jet for v imp = 3 km/s). The spatial distribution of fast ejecta colored according to peak tem-perature (Figure 13) demonstrates that the fastest ejecta, preferentially directeddownrange, also reaches the highest temperatures. The peak of each histogram(Figure 12) shifts to higher temperature as the impact angle increases. Thisseems broadly consistent with laboratory experiments, which showed that thetemperature of jetted material increases with impact angles (i.e., vertical im-pacts produce the hottest jet) (Sugita et al., 1998; Sugita and Schultz, 1999). Totrack the evolutional history of their temperature (i.e., they have experiencedmelting), we expand the analyze and quantify the potential amount of jettedmelt from our simulations. We define jetted melt as jetted ejecta that exceedsthe solidus temperature of dunite (1373K) after the pressure has dropped below1 bar (10 Pa). When jetted ejecta have reached high pressure, we track themuntil their pressure becomes lower than this threshold, then examine their tem-perature as a post-shock temperature to compare with the solidus temperature.Figure 14 shows the total amount of jetted melt M meltjet normalized by M imp asa function of the impact angle. Although the melted ejecta of most cases for v imp = 3 km/s fail to reach 1% of M imp , likely due to the lower resolution of19 igure 13: Behavior of ejected material for θ imp = 60 ◦ , 45 ◦ , and 30 ◦ (top to bottom). Sameviewing geometries and time as Figures 2(c) and 5, but color represents peak temperature inKelvin. Impact angle [ ∘ ] −3 −2 −1 Σ M m e l t j e t / M i m Target
15 30 45 60 75 90
Im act angle [ ∘ ] Im actor
15 30 45 60 75 90
Im act angle [ ∘ ] Total ∘Target + Im actor) v im = 5 km/sv im = 3 km/sv im = 2 km/sEjecta from targetEjecta from im actorTotal ejecta Figure 14: Same as Figure 10, but for Σ M meltjet . Note that Σ M meltjet of 2 km/s is less than10 − and cannot be seen here. the model run, melted material for v imp = 5 km/s exceeds 1% of M imp (greenlines in Figure 14), which we consider to be a significant melt mass. We findthat melt fraction in jetted ejecta depends on the impact angle, and that meltfraction decreases as impact angles get shallower (Figure 15), which directlyconflicts with previous work based on thin plate theory (Vickery, 1993) suggest-ing that jets produced by grazing impacts have a higher melt content than thejet produced by vertical impacts. However, because the jetted mass producedby moderately oblique impacts is larger than that of vertical impacts (Figure10), so is the amount of melt. Comparison to high resolution axisymmetricsimulations shows that jetted melt in our v imp = 5 km/s is underestimated bya factor of 1.2 similar to the resolution of jetted mass. The same comparisonfor v imp = 3 km/s, however, shows the jetted melt is underestimated by a fac-tor of 2.1 (Appendix A). This result is not surprising as previous simulationsof Johnson et al. (2015) show that the early fastest-moving portion of the jet,which is more difficult to resolve, is dominated by melt.If jetted ejecta are melt-dominated (Figures 14 and 15), this may increasethe amount of chondrules produced during planetary accretion (e.g., Kieffer,1975; Johnson et al., 2015; Wakita et al., 2017). Chondrules are millimeter-sized spherical materials found in primitive meteorites (e.g., Scott and Krot,2014). While the processes that produce chondrules are still debated, impactjetting during accretionary impacts is one proposed formation mechanism. Inthis scenario, melt jetted during a relatively low velocity collision may break up21 Impact a gle [ ∘ ] Σ M m e l t j e t / Σ M j e t Target
15 30 45 60 75 90
Impact a gle [ ∘ ] Impactor
15 30 45 60 75 90
Impact a gle [ ∘ ] Total (Target + Impactor) v imp = 5 km/sv imp = 3 km/sv imp = 2 km/sEjecta from targetEjecta from impactorTotal ejecta Figure 15: Same as Figure 10, but for Σ M meltjet / Σ M jet . Note that no symbols for 2km/s casesmeans no jetted melt. and cool, ultimately becoming chondrules (Johnson et al., 2015; Johnson et al.,2018). The amount of jetted melt from our three-dimensional vertical impactsimulations is smaller than previous two-dimensional calculations, likely due toour lower resolution (see also Appendix A). Our work, however, indicates thatoblique impacts produce more jetted melt than vertical impacts (Figure 14).Given that oblique impacts occur more frequently (Shoemaker, 1962), they mayhave enhanced efficiency in chondrule formation in the early Solar System.Our results also show that the amount of jetted ejecta originating from theimpactor increases as impact angles become shallower (Figure 10); this trendsuggests that the amount of melted material sufficient to form chondrules willhave a greater component of impactor material than head-on impacts at thesame impact velocity. Chondrule precursor material is thought to be pristineundifferentiated material (Taylor et al., 1983). While smaller impactors tend toremain undifferentiated, the larger target might be differentiated (Gail et al.,2014; Lichtenberg et al., 2018; Wakita et al., 2018), though such larger bod-ies might have an undifferentiated surface layer (e.g., Weiss and Elkins-Tanton,2013). Our findings that jetted material produced during oblique impacts isdominated by impactor material may suggest that an undifferentiated impactorcan produce chondrules with primitive compositions even if the target is com-pletely differentiated.Our results may also have implications for the origin of unique distal ejecta22atterns observed around craters produced by oblique impacts. Distal distribu-tion of ejecta, known as butterfly patterns, are found around elongated craterson Moon and Mars (Gault and Wedekind, 1978; Schultz and Lutz-Garihan,1982). As Figures 5c and 6c indicate, there is a discontinuous distribution ofjetted material for θ imp = 30 ◦ (the gap around ∼ ◦ of (azimuthal) ejectionangle in Fig. 6c, which corresponds to around (x, y) = (-2 km, 1 km) in Fig.5c). Slower ejecta released at later stages would be the primary source of anasymmetric ejecta pattern, rather than the earlier jetted ejecta. It is unclearfrom our work if the asymmetric pattern for fastest ejecta would affect thedistribution of later slower ejecta. Additionally, we note the impact velocitiesexplored here are significantly lower than typical impact velocities on the Moonand other terrestrial planets so these comparisons may be limited. s Our resultsmay also explain differences in reflectance in distal ejecta on the Moon as a dif-ference in material composition between target and impactor (Speyerer et al.,2016). While the impact velocity on the Moon ranges over 6 km/s to 40 km/s(Chyba, 1991; Yue et al., 2013), the escape velocity of the Moon is 2.4 km/s.Thus, jetted ejecta as defined here will exceed the escape velocity of the Moon.As previously noted, the fastest portions of the jet eject at very low angles andgraze the surface. Considering a realistic target with topography and slopessome of this material will impact the surface and be deposited even though itis ejected above escape velocity. Our results for grazing impacts ( θ imp = 15 ◦ )record a deficit in ejecta velocity, (around 0 . v imp ) resulting from the differencein ejecta velocity between ejecta originating from target and impactor (Figures5d and 6d). If the composition of the impactor differs from that of the target,ejecta of oblique impacts originating from the impactor might produce distalejecta with different reflectance characteristics than proximal ejecta, which mayoriginate from the target (Speyerer et al., 2016).Lastly, we assess the influence of resolution on our results by comparing ourthree dimensional simulations with two-dimensional calculations. Our three-dimensional results are in good agreement with two-dimensional simulations atthe same resolution of 5 m or 100 cells per projectile radius (CPPR); however,23igher resolution is required to produce nearly same amount of jetted material asfully resolved two-dimensional calculations (see Appendix A): The jetted massin our simulations with 100 CPPR are 60%-70% less than that produced in our1000 CPPR resolution simulations. This finding is consistent with previous workreporting that resolution >
400 CPPR is necessary to resolve high-velocity jettedmaterial based on two-dimensional simulations (Johnson et al., 2014; Wakitaet al., 2017; Kurosawa et al., 2018). However, because the cumulative jettedmass produced by oblique impacts is more than twice the amount producedin head-on impacts, our simulations should adequately resolve the influence ofimpact angle on the mass and the distribution of impactor and target materialsin jetted ejecta. It is difficult and computationally expensive to run the three-dimensional simulations in high resolution, however, limiting the high-resolutionzone as done in this paper will help to tackle this. Our results provide animportant step toward understanding jetting during oblique impacts.
5. Conclusions
Our simulations of oblique impacts illustrate that the amount of jetted mate-rials and their origin strongly depends on impact angle. For an impact velocityof 3 km/s, a 45 ◦ impact produces six times the mass of jetted materials relativeto a vertical impact. The source of jetted material varies; for < ◦ impacts,most jetted material originates from the impactor; however, target material be-gins to be the primary source of jetted ejecta at shallower impact angles (75 ◦ and 90 ◦ ). While jetted ejecta from oblique impacts is distributed downrange ofthe impactor (Figures 2 and 5), grazing impacts (impact angles < ◦ ) have amore discontinuous distribution. Such distribution by grazing impacts may helpexplain the distal ejecta patterns observed around elongated craters. Additionalhigher resolution simulations of oblique impacts can provide more accurate es-timates of both the amount of jetted ejecta and its melt fraction, a potentialsource for chondrules. 24 cknowledgments We gratefully acknowledge the developers of iSALE-3D, including GarethCollins, Kai W¨unnemann, Dirk Elbeshausen, Boris Ivanov and Jay Melosh.Numerical computations were carried out on the PC cluster and the analysisservers at the Center for Computational Astrophysics, National AstronomicalObservatory of Japan. This work was supported by grant 80NSSC20K0422from the NASA Emerging Worlds program. T. M. D. was funded by STFCGrant ST/S00615/1. We thank anonymous referees whose comments helped toimprove this manuscript.
Appendix A. Head-on impacts in two-dimensional calculation
Here, we compare the results for the mass of jetted material produced inhead-on impacts for iSALE-3D to those using the iSALE-2D shock physics code.iSALE-2D is based on the SALE hydrocode (Amsden et al., 1980; W¨unnemannet al., 2006; Collins et al., 2016, version iSALE-Dellen). The highest resolutionwe used in iSALE-3D was 5m, with 100 cells per projectile radius (CPPR).For our iSALE-2D runs we include the entire spherical impactor in the highresolution zone, instead of the lower part used in iSALE-3D (see Section 2 andFigure 1). The comparison below demonstrates that the smaller high resolutionzone used in our 3D simulations does not affect jetted mass. Figure A.1 showsthe evolution of cumulative jetted mass produced from head-on impacts usingiSALE-2D with 100 CPPR and v imp = 3 km/s. Although the details differ be-tween Figures 8 (iSALE-3D) and A.1 (iSALE-2D), the cumulative jetted mass is1.1 % of impactor mass for both scenarios. Our results for Σ M jet produced fromhead-on impacts in iSALE-2D as a function of resolution are are summarizedin Figure A.2. Our results for 100 CPPR from iSALE-2D and iSALE-3D with v imp = 2 km/s and 5 km/s give similar results (Σ M jet /M imp is 0.68 % and 0.46% for iSALE-2D, and 0.62 % and 0.48 % for iSALE-3D, respectively).Our results show that our simulations using iSALE-2D are well convergedat 400 CPPR. The jetted mass fraction compared to that produced from 100025 .0 0.5 1.0 1.5 2.0 time/t s −3 −2 −1 Σ M j e t / M i m p Total(Impactor+Target)TargetImpactor
Figure A.1: Same evolution of jetted material over time as in Figure 8, but using iSALE-2D(CPPR=100).
CPPR are shown in Table A.1. Taking our 1000 CPPR results as fully con-verged, our 3D head-on impact results with 100 CPPR underestimate the jettedmass by a factor of 1.4 to 1.6, depending on impact velocity. In simulations with100 CPPR, there is little difference between the amount of jetted melt producedin iSALE-2D and iSALE-3D, but they underestimate jetted melt relative to 1000CPPR simulations by a factor of 2.1 at 3 km/s and 1.2 at 5 km/s (Figure A.3).Note that jetted melt at 2 km/s with 800 and 1000 CPPR exceeds 0.1% of M imp .It remains unclear whether oblique impact simulations with higher resolutionwould produce more jetted melt than simulations at 100 CPPR. As Figure 10illustrates, oblique impacts generate more jetted material than head-on impacts.Therefore, our setting of 100 CPPR may be sufficient to characterize the jettedmass produced by oblique impacts, though not vertical impacts. Regardless, weexpect the trends in jetted mass as a function of impact angle and velocity forhigher resolution simulations to be similar to those presented here.26 CPPR −3 −2 −1 Σ M j e t / M i m p Target
CPPR
Impactor
CPPR
Total (Target + Impactor) v imp = 5 km/sv imp = 3 km/sv imp = 2 km/sEjecta from targetEjecta from impactorTotal ejecta3D results Figure A.2: Cumulative mass of jetted material (Σ M jet ) as a function of CPPR for the head-on impacts. Color represents different impact velocities: 2 km/s (blue), 3 km/s (black), and5 km/s (green). The panels depict the total mass M jet normalized by M imp from the target(left), the impactor (middle), and the total mass (right), respectively. The head-on impactresults from iSALE-3D are plotted as circles. Note that jetted ejecta from the impactor inthe case of 5 km/s is less than 10 − (middle panel). CPPR −3 −2 −1 Σ M m e l t j e t / M i m p Target
CPPR
Impactor
CPPR
Total (Target + Impactor) v imp = 5 km/sv imp = 3 km/sv imp = 2 km/sEjecta from targetEjecta from impactorTotal ejecta3D results Figure A.3: Same as Figure A.2, but for Σ M meltjet . Note that jetted melt from the target at 2km/s and the impactor at 2 km/s and 5 km/s are less than 10 − . able A.1: The jetted mass fraction ratio to that from CPPR=1000 * v imp CPPR = 100 (3D) 100 (2D) 200 400 500 800 10002 km/s 0.63 0.68 0.87 0.98 0.99 1.1 1.0 (9.9 × − )3 km/s 0.72 0.72 0.87 0.95 0.96 0.98 1.0 (1.6 × − )5 km/s 0.67 0.65 0.83 0.94 0.93 1.0 1.0 (7.1 × − ) * These values are respected to the jetted mass fraction of CPPR=1000 (see FigureA.2). Absolute values of CPPR=1000 are shown in bracket.
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