Modeling Global Impact Effects on Middle-Sized Icy Bodies: Applications to Saturn’S Moons

Home , Cassini (lunar crater), Ithaca Chasma, Odysseus (crater), Stickney (crater), Tirawa (crater)

Icarus 168 (2004) 457–466 www.elsevier.com/locate/icarus

Modeling global impact effects on middle-sized icy bodies: applications to Saturn’s moons

Lindsey S. Bruesch ∗ and Erik Asphaug

Department of Earth Sciences, University of California—Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA Received 9 July 2003; revised 27 October 2003

Abstract Disrupted terrains that form as a consequence of giant impacts may help constrain the internal structures of planets, asteroids, comets and satellites. As shock waves and powerful seismic stress waves propagate through a body, they interact with the internal structure in ways that may leave a characteristic impression upon the surface. Variations in peak surface velocity and tensile stress, related to landform degradation and surface rupture, may be controlled by variations in core size, shape and density. Caloris Basin on Mercury and Imbrium Basin on the Moon have disturbed terrain at their antipodes, where focusing is most intense for an approximately symmetric spheroid. Although, the icy saturnian satellites Tethys, Mimas, and Rhea possess giant impact structures, it is not clear whether these structures have correlated disrupted terrains, antipodal or elsewhere. In anticipation of high-resolution imagery from Cassini, we investigate antipodal focusing during giant impacts using a 3D SPH impact model. We first investigate giant impacts into a fiducial 1000 km diameter icy satellite with a variety of core radii and compositions. We find that antipodal disruption depends more on core radius than on core density, suggesting that core geometry may express a surface signature in global impacts on partially differentiated targets. We model Tethys, Mimas, and Rhea according to their image-derived shapes (triaxial for Tethys and Mimas and spherical for Rhea), varying core radii and densities to give the proper bulk densities. Tethys shows greater antipodal values of peak surface velocity and peak surface tensile stress, indicating more surface damage, than either Mimas or Rhea. Results for antipodal and global fragmentation and terrain rupture are inconclusive, with the hydrocode itself producing global disruption at the limits of model resolution but with peak fracture stresses never exceeding the strength of laboratory ice.  2003 Elsevier Inc. All rights reserved.

Keywords: Satellites, general; Impact processes; Interiors; Surfaces, satellite

1. Introduction differentiation (Canup and Ward, 2002). Therefore, variations in the extent of differentiation can speak to condi- The formation of our Solar System has been, and contin- tions at the time of formation. For instance, according to ues to be, one of the pinnacle problems in the field of plan- moment of inertia data, Callisto is the only galilean satel- etary science. Because regular satellite systems around gas lite not completely differentiated (Anderson et al., 2001; giant planets may originate and evolve in a manner closely Sohl et al., 2002). Some speculate that this is due to differ- analogous to planet formation, some researchers have sought ences in the formation time of each galilean satellite (Canup information from these systems to improve understanding of and Ward, 2002), while others believe differences in tidal our own Solar System origin (e.g., Stevenson et al., 1986; heating are responsible (Anderson et al., 1998). Canup and Ward, 2002; Mosqueira and Estrada, 2003). Most of the current knowledge about the internal struc- A key step in understanding the origin of satellite systems tures of the icy satellites of the outer Solar System is based is to understand the internal structures of the satellites. on moment of inertia data and mass estimates for each satellite (cf. Dermott and Thomas, 1988; Thomas and Dermott, The interiors of satellites hold insights into the process 1991; Anderson et al., 2003). These quantities, for all but the of differentiation. Longer formation times produce less ac- galilean satellites, are only loosely constrained until space- cretional heat from impacts which leads to less significant craft flybys allow better parameterization. However, even with more accurate values for mass and moment of inertia, * Corresponding author. we still must make assumptions about the internal structure E-mail address: [email protected] (L.S. Bruesch). such as the size and density of the core. Investigating a satel-

0019-1035/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2003.11.007 458 L.S. Bruesch, E. Asphaug / Icarus 168 (2004) 457–466 lite’s cratering record may provide independent clues to its interior. The internal structure of Earth was not known until the advent of seismology. The extensive global seismic network on Earth has led to many discoveries such as the fluid outer core (Oldham, 1906; Jeffreys, 1926; Lehmann, 1936) and discontinuities within the mantle (Engdahl and Flinn, 1969). The internal structure of the Moon was only realized follow- ing the deployment of seismometers during Apollo missions from 1969 to 1972 (Goins et al., 1981). Seismic networks have even been contemplated for Mars (Lognonne et al., Fig. 1. (a) Odysseus Crater on Tethys. Voyager 2 Image ID 0290S2-001. 2000). However, for most planetary bodies, especially small (b) Ithaca Chasma on Tethys. Voyager 2 Image ID 1221S2-001. (c) Herschel Crater on Mimas. Voyager 1 Image ID 1194S1-001. (d) Tirawa Crater on satellites in the outer Solar System, it is not yet feasible to Rhea. Rendered from Voyager 1 Image ID 1156S1-000 by Paul Schenk use seismology to determine internal structure, and we must (LPI). consider other means of studying their interior structures. The method considered here is to examine global effects Saturn’s satellites Tethys (DT = 1048 km), Mimas (DM = from large cratering events, beginning with a general exami- 394 km) and Rhea (DR = 1528 km) have giant impact struc- nation of how internal structure affects the outcome (such as tures Odysseus (D = 400 km; D/DT = 0.38), Herschel seismic shaking of the antipode, or the point opposite to an (D = 135 km, D/DM = 0.34) and provisionally named impact) and concluding with a specific analysis of three of Tirawa (D = 350 km, D/DR = 0.23), as shown in Fig. 1. the middle-sized icy satellites of Saturn. The antipode of Odysseus on Tethys is a relatively smooth Large impacts are relatively well-understood phenom- planar region. Ithaca Chasma, a trough system that extends ena that have occurred on all satellites in the Solar Sys- some 270◦ around Tethys, is hypothesized to be the result tem. For a number of planetary bodies, large impacts have of the impact event that formed Odysseus because they are been invoked to explain observed antipodal terrain disrup- of similar age (Moore and Ahern, 1983) and because Ithaca tion (e.g., Schultz and Gault, 1975; Moore and Ahern, 1983). conforms closely to the impact equator (Moore et al., sub- Because an impact generates both internal and surface seis- mitted for publication). That is, instead of being the result mic waves that are capable of leaving widespread surface of antipodal disruption, Ithaca Chasma may be the result of expressions, the modeling of large impact events and the whole body strain circumferential to the impact axis (Moore analysis of their global effects may be used as a proxy for and Ahern, 1983) or reflection of stress channeling from a seismic data to determine the internal structures of impacted relatively shallow core-mantle boundary. bodies (Chapman and McKinnon, 1986; Watts et al., 1991; On Mimas, antipodal disruption from Herschel is not Asphaug and Melosh, 1993; Asphaug et al., 2002). detectable due to low resolution in the Voyager images In this paper we focus on impact events that may be (Schenk, 1989). There is, however, some evidence for global large enough to cause observable damage at the antipode. scale fracturing (McKinnon, 1985). The antipode of Tirawa On Earth, large impacts have been hypothesized to cause on Rhea is a region of rolling hills, which is consistent with flood basalt and hotspot activity (Rampino and Stothers, impact degraded terrain (e.g., Schultz and Gault, 1975). 1988; Hagstrum and Turrin, 1991; Rampino and Caldeira, The highest quality images of both Tethys and Mimas 1992) through the focusing of seismic energy (Boslough have resolutions of 2 km/line-pair from the Voyager space- and Chael, 1994). For a large terrestrial planet like Earth crafts, resulting in fewer pixels across the much smaller where distances are great, this process may only apply in Mimas. Although the highest quality images of Rhea have the largest impacts, despite the low attenuation of surface resolutions of ∼ 1 km/line-pair, from the Voyager 1 space- wave energy (Boslough and Chael, 1994). For smaller bod- craft, they do not depict Tirawa clearly (Smith et al., 1981, ies, antipodal disruption appears to be a part of the geologic 1982). record. Disrupted antipodal terrains have been identified op- Building on the work of Hughes et al. (1977) and Schultz posite to Caloris Basin on Mercury (Murray et al., 1974), and Gault (1975), Watts et al. (1991) modeled antipodal Imbrium Basin on the Moon (Wilhems and McCauley, 1971) pressures and surface accelerations caused by large impacts and Stickney on Phobos (Fujiwara, 1991). Because seismic into several medium-sized bodies in the Solar System, in- waves reflect and refract differently in different media, we cluding Tethys, Mimas, and Rhea. Using the Los Alamos may be able to use the degree of surface damage at the an- Simplified Arbitrary Lagrangian Eulerian (SALE) 2D hy- tipode to determine the size and density of a body’s core, if drocode (Amsden et al., 1980), Watts et al. (1991) found that one exists. The first step in understanding this process is to the modeled surface accelerations and pressures were greater accurately model the generation of impact shock energy, its than the assumed surface material strength in most cases; conversion to seismic energy (surface and body waves) and therefore, giant impacts were sufficient to account for the the propagation of this energy globally, including focusing observed disrupted antipodal terrain. However, many have effects at the antipode. been skeptical of these conclusions because a 2D axisym- Global impact effects 459 metry brings all wave energy to a perfect focus at the an- por. (See Section 2.4 for a description of how SPH handles tipode (Boslough and Chael, 1994). Fujiwara (1991) found shock.) evidence for very different kinds of focusing in elastic ellip- Our models consisted of a target body and an impactor, soids. In this study, he solved the exact elastic equations and as described below, undergoing hypervelocity collision. To derived surface stress states, which he correlated with ex- investigate the global impact effects in our simulations, we pressions of antipodal fracture manifest on Phobos from the calculated the peak surface velocity (as a proxy for mechani- Stickney impact. In particular, he observed stress trajectories cal devastation of a landscape) and peak surface tensile stress consistent with the locus of fracture grooves on Phobos, for (as a proxy for surface rupture) for a particular location a particular value of the Poisson coefficient. Even for nearly during an entire simulation. Direct calculation of fracture spherical bodies such as Mimas and Tethys, whose major damage (as per Melosh et al., 1992; Asphaug et al., 1996) and minor diameters differ by only a few tens of kilometers, is possible but not particularly meaningful when shock res- wave arrivals for a ∼ 10 km impactor are typically going to olution is > 10 kilometers at best and when fragmentation be out of phase at the antipode of a given impact and in phase depends on extrapolation of the ice fracture constants, which where axisymmetric modeling would not predict. It is there- are poorly established even for laboratory ices (see Table I fore critical to model giant impacts and their effects using in Asphaug et al., 2002), resulting in fracture stress extrapo- a 3D code capable of simulating non-spherical bodies and lations (size dependent strengths) that are not meaningfully non-symmetric interior structures. constrained. For this reason we implement the fragmentation algorithm of Benz and Asphaug (1995) but note that for the chosen fracture constants, all of the impacts modeled here 2. Method result in total damage of the target lithosphere by the end of the calculation. Effectively this means that cracks permeate the lithosphere at the model resolution (∼ 10 km). Because We modeled giant impact events on a fiducial middle- of the uncertainty in the ice fracture constants, when extrap- sized icy test satellite, as well as on Tethys, Mimas, and olated from laboratory scales to satellite scales, and to the Rhea, using a 3D smooth particle hydrodynamics (SPH) much lower strain rates associated with large-scale impact code (Benz and Asphaug, 1995). We first explore how core fragmentation, the degree to which the target lithosphere size and density affect the focusing of seismic energy from actually disrupts is better answered by examining the mag- an impact. Ultimately of interest is whether, and to what nitude of the peak tensile stress and peak surface velocity degree, large-scale features like those described above for during the course of the calculation. Tethys, Mimas, and Rhea may be correlated to giant impact Peak surface velocity indicates the magnitude of seismic events and what such correlation implies for each satellite’s shaking and the degree of terrain degradation. For degra- interior. dation of terrains held together by gravity (deeply faulted mountains, ejecta deposits) the magnitude of degradation 2.1. SPH (expressed in terms of distance thrown, normalized to target radius) scales with vej/vesc,wherevej is the peak surface SPH is a three-dimensional gridless Lagrangian algo- velocity and vesc is the escape velocity at the surface. rithm capable of incorporating realistic shape models (cf. Peak surface tensile stress is a measure of material fail- Asphaug et al., 1998). (For a detailed review of the the- ure, faulting and spallation. Given the evidence (e.g., Hoppa ory behind hydrodynamics codes, see Benz (1991); and et al., 1999), however controversial, that ice lithospheres for a detailed review of the SPH method, see Benz, 1990; are very weak in their response to diurnal strains, we can Monaghan, 1992.) Our version of SPH has a unique fracture only qualitatively apply our model results at present as implementation (Benz and Asphaug, 1994, 1995), making it an indicator of the propensity for surface rupture. In most ideal for modeling geologic surface damage in solid body cases, our peak tensile stresses are greater than the very collisions. Although SPH is capable of using various equa- low (∼ 104 dyne cm−2) stresses required by Hoppa et al. tions of state, only a limited number of these are available (1999) but strictly lower (though not always by much) for realistic icy satellite material. Appropriate liquid water than the stresses required to fracture laboratory ice (∼ 107 models exist (cf. Gisler et al., 2003) but for Solar System dyne cm−2; Hawkes and Mellor, 1972; Lange and Ahrens, ices, including water ice, good shock equations of state re- 1983). The SPH code shows fracture damage at the modeled main in development (Stewart and Ahrens, 2003; E. Pier- resolution owing to the rate- and size-dependence of strength azzo and H.J. Melosh, private communication, 2002). We, (Housen and Holsapple, 1999), which remains poorly quan- therefore, modeled all materials using Tillotson equations tified for geologic ices. The use of a damage rheology in of state (Tillotson, 1962), which are adequate for models the simulations provides a more realistic response to impact in which the greatest importance is the treatment of shock than allowing the bodies to ring like elastic bells, although Hugoniot compression and adiabatic release, elastic-plastic future models in the post-Cassini era will use what is learned behavior of the solid phase including associated fracture about the detailed effects of these impacts in order to derive and fragmentation and expansion of shock-generated va- damage models that coincide with observations. 460 L.S. Bruesch, E. Asphaug / Icarus 168 (2004) 457–466

Our first aim is to explore how peak surface velocity and Table 1 peak surface tensile stress change as core properties are var- Satellite properties ied for each satellite considered; our post-Cassini efforts Tethys Mimas Rhea − will examine, explicitly and at sufficiently high resolution, g(cms 2)a 18.57.928.5 −1 b whether strain localization and faulting occur in specific ter- vesc (m s ) 436 161 659 rains of the saturnian satellites and how this relates specifi- D (km) 1048 394 1528 c cally to their internal differentiation. Dcrater (km) 400 135 350 a From Satellites (1986). 2.2. Input parameters for test satellite impact scenario b Calculated from values found in Satellites (1986). c Final crater diameters. Our fiducial target is a 1000 km diameter differentiated sphere, with a mantle composed of pure water ice (ρ = by holding the bulk density of each satellite constant for all 0.917 g cm−3) overlying a silicate core. The size and den- runs. All impacts are modeled in the true geodetic frame in sity of the core are free parameters within this model. We order to facilitate application of the satellite 3D shape and for ran nine simulations, permuting the size and density of the future use in evolving the ultimate fate of the Saturn-bound test satellite’s core. Three simulations use a baseline core ejecta. − = density of basalt (ρ = 2.7gcm 3), three simulations use a The diameter of the cometary impactor in each case (di − = low-density (ρ = 2.0gcm 3) silicate core and three simula- 12.7 km to form Odysseus, di 2.9kmtoformHerschel, − = tions use a high-density (ρ = 3.97 g cm 3) silicate core. For di 10.9 km to form Tirawa) was calculated using gravity the high-density silicate core, we use the equation of state scaling for water ice (cf. Melosh, 1989) and assumed tran- for gabbroic anorthosite as computed from laboratory exper- sient crater diameters dt = 246, 95, and 198 km, respectively iments by Ahrens and O’Keefe (1977). For the low-density (Moore et al., submitted for publication; W.B. McKinnon, silicate core, we use the equation of state of basalt (Ahrens private communication, 2003). Impact velocity in all cases −1 and O’Keefe, 1977) but substituting 2.0 g cm−3 as the zero- was 20 km s and impact angle was allowed to vary be- ◦ ◦ pressure reference density. For all three silicate types, rheol- tween 0 and 80 , measured from the surface normal, pri- ogy is equivalent to that used by Benz and Asphaug (1995) marily in order to break any axisymmetry that might result in their modeling of laboratory disruption experiments by from an unlikely vertical impact. We introduce the impactors Nakamura and Fujimura (1991). As mentioned in the pre- in true geometry relative to the satellite geoid; on Tethys the ◦ ◦ ceding section, the materials we have chosen to use for these impact occurred at the location of Odysseus, 30 N, 130 W; simulations are limited by the available equations of state. on Mimas the impact occurred at the location of Herschel, ◦ ◦ The impactor is a 13 km diameter undifferentiated spher- 11 N, 115 W; and on Rhea the impact occurred at the loca- ◦ ◦ oid, representing a comet composed of pure water ice. In all tion of Tirawa, 36 N, 150 W (Burns and Matthews, 1986, cases the impact velocity is 20 km s−1 and the impact angle Map Section). is 0◦. 2.4. Model resolution 2.3. Input parameters for saturnian satellite impact scenarios In all our model runs, we used ∼ 141,300 particles for the target body and 700 particles for the impactor, far be- Tethys, Mimas and Rhea have normalized dimensionless yond the requirement for resolution convergence established moments of inertia (γ = I/(MR2)) less than 0.4 (Dermott for catastrophic disruption of spherical targets (Benz and As- and Thomas, 1988; Thomas and Dermott, 1991; Anderson phaug, 1995) or for the impact origin of the Moon (Canup et al., 2003), indicating they are differentiated. All of our and Asphaug, 2001) but not sufficient to meaningfully re- model explorations of these satellites thus also consist of solve discrete fragments in the satellite lithosphere. While a differentiated target body composed of a pure water ice fracture damage is an important part of the satellite response mantle and a silicate core. We represent Tethys and Mi- to impact, including how it propagates and limits stress en- mas with triaxial ellipsoidal shapes. For Tethys, principal ergy, at the modeled resolution about six geographical de- axes are a = 535.9km,b = 527.5km,andc = 526.3km grees (three particles) would define the smallest resolved dis- (Thomas and Dermott, 1991). For Mimas, principal axes are crete fragment plus its boundary, even if artificial viscosity a = 210.3km,b = 197.4km,andc = 192.6km(Dermott (the means of modeling shocks in a hydrocode; Von Neu- and Thomas, 1988). We represent Rhea as a spheroid with mann and Richtmyer, 1950) did not smooth out the shocks, radius r = 764 km (Davies and Katayama, 1983). (See Ta- which lead to the powerful elastic stresses whose reverber- ble 1 for more properties of these satellites.) ations are ultimately responsible for fragmentation. More- The core size and density and impact angle varied over over, it is not very useful at present to extrapolate ice fracture nine simulations for each satellite. The variety of core mate- statistics out to hundred-kilometer scales, when the strength rial is the same as that for the test satellite runs. However, the scaling exponent m remains poorly characterized (Table I in core size in each of the satellites’ simulations is determined Asphaug et al., 2002). Global impact effects 461

In short, stress propagation and its attenuation due to antipode is between 91 and 160 cm s−1 (see Fig. 2a, simula- brittle fragmentation is modeled in detail by our code, but tion III). discrete fissuring of the lithosphere is poorly resolved. (Plas- Peak tensile stress is also highest at the impact site and tic yielding is included using the simple von Mises treatment decreases with distance from the impact with some antipo- which also reduces shear stresses to zero at the melting tem- dal increase in all cases. The degree of antipodal focusing perature; see Benz and Asphaug, 1995.) We thus focus on in peak tensile stress is greater than for peak velocity. Peak two model outcomes that are particularly robust: the peak tensile stress is enhanced by at least an order of magnitude particle velocity achieved at any time during the simula- in simulations using the largest core size (see Fig. 3a, sim- tion (for particles within 5% of the target’s radius from the ulations III, VI, and IX), while there is much less antipodal surface) and the peak tensile stress responsible for rupture. focusing for smaller cores. The dependence of global pat- Future calculations on faster computers, with the availability terns in peak tensile stress on core radius is strong. of Cassini data, can usefully explore more detailed outcomes For all test satellite simulations, the maximum value of such as flexure, opening of grabens, sliding along thrust peak surface tensile stress is between 7 × 105 and 1 × 106 − faults and peak acceleration. At present, maximum surface dyne cm 2 (see Fig. 3a, e.g., simulations VI and VII). Al- velocity is the most useful expression of time-integrated ac- though the global peak tensile stress is higher for smaller celeration and of the maximum distance that surface debris core sizes, the peak surface tensile stress never exceeds the can be expected to be thrown in a global-scale impact (cf. rupture strength measured for ice in a laboratory, 1 × 107 − Asphaug and Melosh, 1993). Until fracture properties of dyne cm 2 (Hawkes and Mellor, 1972; Lange and Ahrens, these ices are better understood, we view maximum tensile 1983) except at the impact site itself. These stresses are, stress as a good qualitative measure of the local degree of however, greater than the rupture strength (40 kPa or 4 × − surface rupture. 105 dynes cm 2) required for Hoppa et al.’s (1999) Europa For the Tethys impact scenario, and for the similarly sized model of cycloidal ridge propagation by diurnal tides. test satellite, the SPH smoothing length of each particle in the target is ∼ 10 km. (Smoothing length is the width of the 3.2. Rhea interpolation kernel used to express properties of the material, such as density, in terms of values at a set of disordered Antipodal focusing is weak or non-existent in peak ve- points or particles. For more information on smoothing locity (see Fig. 2c, e.g., simulations I and IV). Antipodal lengths and kernels, see reviews by Benz, 1990, and Mon- focusing in peak tensile stress is present in two cases and aghan, 1992.) For the Mimas and Rhea impact scenario, the is enhanced by one order of magnitude at most (see Fig. 3c, smoothing lengths of each particle in the target are ∼ 4and simulations V and VIII). There are no clear trends of either ∼ 14 km, respectively. Particles belonging to the impactors peak velocity or peak tensile stress on impact angle or core have much smaller smoothing lengths, although the calcu- properties. lations are not sensitive to this choice since the impactor For all Rhea simulations, the maximum value of peak sur- −1 is mainly used to introduce impact energy and momentum. face velocity at the antipode is between 0.23 and 0.75 cm s The principal limitation to these runs is the comparative lack (see Fig. 2c, simulations IV and VI). A velocity of 0.75 −1 of resolution in the target bodies. Runs utilizing a modified cm s is sufficient to throw material on Rhea a distance of ∼ = −1 parallel version of the code would require ∼ 1000 times as only 0.1 mm, given g 28.5cms . many particles and almost 10,000 times as many cpu hours For all Rhea simulations, the maximum value of peak × 3 to attain a 10-fold enhancement in 3D model resolution. surface tensile stress at the antipode is between 6 10 and 2 × 104 dyne cm−2 (see Fig. 3c, e.g., simulation V). These values of peak tensile stress do not exceed the rup- 3. Results ture strength of laboratory ice and are less than the Hoppa et al. (1999) fracture threshold. 3.1. Test satellite 3.3. Mimas Peak velocity is highest at the impact site in all cases and decreases with distance from the impact with a slight in- There is slight peak velocity focusing at the antipode in crease at the antipode in two cases (see Fig. 2a, simulations some cases (see Fig. 2b, simulations II, III, and IV). Chang- IV and VIII). In these cases, the degree of focusing is small ing the core density and radius does not make a significant (no more than a factor of ∼ 2 increase at the antipode). The change in the global pattern of peak velocity. Antipodal peak minimum and maximum peak velocities are similar in all velocity values decrease as the impact angle becomes more cases. However, the average global peak velocity is highest grazing. An asymmetric gradation is observed in nearly all for the cases with the largest core radius (see Fig. 2a, simu- cases (see Fig. 2b, e.g., simulation V). This asymmetry is lations III, VI, and IX). Varying the core density makes little likely due to the oblate nature of the shape model used difference in the peak velocity. For all test satellite simu- for Mimas. For all Mimas simulations, the maximum value lations, the maximum value of peak surface velocity at the of peak surface velocity at the antipode is between 6 and 462 L.S. Bruesch, E. Asphaug / Icarus 168 (2004) 457–466

(a) (b)

Fig. 2. Peak surface velocity for (a) test satellite, (b) Mimas, (c) Rhea, and (d) Tethys. Shown are cylindrical projections of each satellite’s surface. In all cases the impact occurs in the upper right-hand quadrant, with the antipode at the lower-left. Because the near-ﬁeld target response is identical in all simulations of a given series, the area of impact in each series looks almost identical while the antipodal hemispheres vary with target structure and associated antipodal response. The white region in the center of each crater is empty space where particles were ejected from the crater and are, therefore, not included in our cylindrical projections. Mottled texture in these ﬁgures indicates either a boundary of two color-scale values or overprinting of one value above another (since several particles deep are being plotted as a surface). Both values could be present in these locations.

18 cm s−1 (see Fig. 2a, simulation II, III, and IV). A sur- Saturn system. Antipodal focusing is observed in peak ve- − face velocity of 18 cm s 1 is sufﬁcient to throw material on locity for all cases. No clear trends exist for peak velocity − Mimas a distance of only ∼ 20 cm, given g = 7.9cms 1. dependence on either core properties or impact angle. The Antipodal focusing in peak tensile stress is weak or non- maximum value for peak surface velocity at the antipode is existent in all cases. No clear trends are apparent for peak between 50 and 110 cm s−1 (see Fig. 2d, e.g., simulations tensile stress dependence on varying core properties. As II and VI). With a surface gravity of 18.5 cm s−1, a velocity the impact angle increases, antipodal values of peak ten- of 110 cm s−1 would throw loose materials up to ∼ 3.3m sile stress decrease. For all Mimas simulations the maximum and severely degrade loosely consolidated terrains (such as value for peak surface tensile stress at the antipode is be- crater rims) at scales of tens to hundreds of meters. tween 4 × 104 and 1 × 105 dyne cm−2 (see Fig. 3b, sim- There is antipodal peak tensile stress focusing in some ulation I and III). This peak tensile stress does not exceed cases, with peak tensile stress enhancing by more than an the rupture strength of laboratory ice and is lower than the tensile stress required by Hoppa et al. (1999) as a fracture order of magnitude in the strongest cases (see Fig. 3d, sim- threshold on Europa. ulations I and IV). Varying core properties changes peak tensile stress more than varying the impact angle. The max- 3.4. Tethys imum value for peak surface tensile stress at the antipode is between 1×106 and 3×106 dyne cm−2 (see Fig. 2d, simula- Odysseus is 0.38 times the diameter of Tethys—larger, tion I). This would almost certainly result in ice rupture (see relative to the satellite diameter, than any other crater in the discussion section). The manifestation of this rupture awaits Global impact effects 463

(a) (b)

Fig. 3. Peak surface tensile stress for (a) test satellite, (b) Mimas, (c) Rhea, and (d) Tethys. See caption of Fig. 2 for detailed explanation of plots. higher resolution runs that can realistically express the fault Tirawa impact event on Rhea reaches escape velocity. More geometry, linear extent and depth. rigorous tracking of ejecta will be studied in a later paper. Although simulations of all the satellites we modeled indicate relatively weak antipodal focusing, Tethys shows 3.6. Model time both more antipodal focusing and greater values of peak surface velocity and peak surface tensile stress in the antipodal All test satellite simulations were run for 800 s or three to hemisphere than either Mimas or Rhea, as one might expect four wave-crossing times. Simulations of the saturnian satel- given its larger crater diameter relative to target diameter. An lites were run for a minimum of two wave-crossing times or assessment of the observed surface geology coupled to the ∼ 600 s for Tethys, ∼ 250 s for Mimas and ∼ 650 s for Rhea giant impacts on Tethys, Mimas and Rhea is forthcoming in (see Fig. 4). In two wave-crossings the initial compressive Moore et al. (submitted for publication). (pressure) wave travels from the point of impact to the antipode and is reﬂected as a tensile wave, traveling back to the 3.5. Ejecta point of impact. Most of the surface damage occurs within two wave-crossing times, as the wave energy begins to at- We estimate ejecta loss by considering how much mass tenuate after this; certainly the peak surface velocities have reaches or exceeds escape velocity from each satellite sur- been achieved by this time. face and how much mass reaches or exceeds escape velocity for the Saturn system. Ejecta from all impact events on the saturnian satellites modeled stays in Saturn-bound or- 4. Discussion bit. Some ejecta from the Odysseus impact event on Tethys and the Herschel impact event on Mimas reaches each satel- The results of modeling in 3D demonstrate relatively lite’s respective escape velocity; however, no ejecta from the weak antipodal focusing of the peak surface velocity and 464 L.S. Bruesch, E. Asphaug / Icarus 168 (2004) 457–466

One possible reason for this discrepancy is the difference in material involved in the impact. Mercury and the Moon have silicate mantles, while Tethys, Mimas, and Rhea have icy mantles. The difference in mantle material may dictate how efficient an impact is at damaging the antipode. The composition of the impactors in all of these cases is unknown. In the saturnian system, the impactors are assumed to be comets; while for Mercury and the Moon the impactors could be either comets or asteroids. Perhaps the difference in impactor composition and velocity makes a difference in energy transmission to the target body during an impact. No researchers to date have modeled the formations of Caloris and Imbrium basins and the resulting global effects in 3D; therefore, the formation of their antipodal terrains is not well understood at present. In particular, the cause of the observed antipodal terrains (e.g., seismic shaking, tensile fracturing, etc.) is unknown. We plan to investigate this further in a future paper. Besides the impact-induced shock wave, the reimpact of ejected materials (both prompt secondaries and those that go into temporary orbit about Saturn) may also explain the formation of disrupted antipodal terrain. For this reason we want to further investigate the trajectories of impact ejecta. Schultz and Gault (1975) showed that for large impact events on the Moon, surface (Rayleigh and Love) waves would not arrive at the antipode prior to the reimpact of secondaries. Fig. 4. This time sequence follows a pressure wave traveling through the in- Although surface waves do more damage (due to their rel- terior of our baseline (basalt core) Tethys simulation. The model sequence atively low attenuation with distance and the constructive starts at t = 25 s and ends at t = 300 s, with snapshots every 25 s of model time. The Tethys basalt core case was chosen as an illustrative ex- interference of the waves at the antipode), the damage due to ample because the ratio of core to mantle material and physical properties the surface waves could not be distinguished from the dam- was such that this case had the longest wave-crossing time on any of the age due to reimpact of secondaries. Future simulations will Tethys runs. Here we verify that 600 s is adequate time to follow two or include a ballistics calculation for ejecta, and we hope to in- more wave-crossings and the accompanying wave attenuation for this case. vestigate the arrival times of surface waves versus ejecta at Because the simulations of Mimas and Rhea use identical materials with identical wave propagation speeds, 250 and 650 s, respectively, should be the antipode. adequate time to follow at least two wave-crossings through these bodies. Though we have concentrated on antipodal damage, it is possible for impact-induced terrains to appear elsewhere on a body. In fact, the most interesting feature on Tethys is peak surface tensile stress for middle-sized icy bodies. This Ithaca Chasma, which is not at the antipode to the Odysseus is in marked contrast to the 2D simulations of Watts et al. impact structure. In none of our simulations do the peak (1991), suggesting that focusing found in their study is an surface tensile stresses exceed the rupture strength of lab- artifact of a 2D axisymmetric code. Three-dimensional mod- oratory ice. However, fracture damage as computed by the eling of the kind presented here was not feasible twelve years Benz and Asphaug (1995) model, which uses the Weibull ago. Although it still faces many challenges, namely limita- (1951) fracture coefficients to extrapolate to size dependent tions on model resolution, its ability to incorporate realistic strengths, is global throughout the lithosphere since static geometries makes it an important tool for modeling antipo- strength decreases with roughly the cube root of size for dal processes. masses of pure ice. Strictly speaking, a 1000 km diameter Our model results indicate a possible difference between piece of ice should be hundreds of times weaker than a lab- the antipodal terrains of Caloris (Mercury) and Imbrium oratory sample. It is obviously important to study in more (the Moon) and those of Odysseus (Tethys) and Herschel detail any genetic correlation between Odysseus and Ithaca (Mimas). Although the ratio of impact crater diameter to Chasma. To resolve geometry of fissuring around the im- parent body diameter is less for Caloris and Imbrium than pact’s equator, one needs to resolve surface elements to the for Odysseus and Herschel, the antipodal terrains of Caloris extent that strain localization can become manifest—that is and Imbrium have been plausibly correlated to giant impact to say, so that the widespread lithospheric rupture seen in the events. Odysseus and its antipodal terrain have not been ab- present simulations is resolved into discrete crack propaga- solutely correlated on Tethys and, in fact, disturbed terrain tion events that release stress in neighboring zones. At that antipodal to Herschel has not yet been identified on Mimas. point, one can explore which combination of ice fracture co- Global impact effects 465 efficients and interior structural models best fits the genesis References of Ithaca Chasma. Core shape may also affect the focusing of seismic en- Ahrens, T.J., O’Keefe, J.D., 1977. Equation of state and impact-induced ergy. It is possible that this could not only explain the pres- shock-wave attenuation on the Moon. In: Roddy, D.J., Pepin, R.O., Mer- ence of Ithaca Chasma but its location, offset from the true rill, R.B. (Eds.), Impact and Explosion Cratering. Pergamon Press, New York, pp. 639–656. antipode of Odysseus. In all of our simulations, each target Amsden, A.A., Ruppel, H.M., Hirt, C.W. 1980. SALE: simplified ALE body had a spherical core. However, Dermott and Thomas computer program for fluid flow at all speeds. Los Alamos National (1988) postulated that while Mimas has components of both Laboratory LA-8095. ice and silicate, it may not be totally differentiated. It may be Anderson, J.D., Schubert, G., Jacobson, R.A., Lau, E.L., Moore, W.B., Sjo- that Mimas has pockets of rock scattered throughout its in- gren, W.L., 1998. Distribution of rock, metals, and ices in Callisto. Science 280, 1573–1576. terior rather than a single rocky core. Although the same has Anderson, J.D., Jacobson, R.A., McElrath, T.P., Moore, W.B., Schubert, not been said for Tethys, a fragmented core is something we G., Thomas, P.C., 2001. Shape, mean radius, gravity field, and interior are able to model. Similarly, it may be possible to explain structure of Callisto. Icarus 153, 157–161. Ithaca Chasma by a variable porosity mantle, something that Anderson, J.D., Rappaport, N., Giampieri, G., Schubert, G., Moore, W.B., is also readily incorporated into hydrocodes. 2003. Gravity field and interior structure of Rhea. Phys. Earth Planet. In. 136, 201–213. Asphaug, E., Melosh, H.J., 1993. The stickney impact of phobos: a dynam- 5. Conclusion ical model. Icarus 101, 144–164. Asphaug, E., Moore, J.M., Morrison, D., Benz, W., Nolan, M.C., Sullivan, R.J., 1996. Mechanical and geological effects of impact cratering on The main result of this study is that, in multiple simula- Ida. Icarus 120, 158–184. tions, 3D modeling finds only weak evidence for antipodal Asphaug, E., Ostro, S.J., Hudson, R.S., Scheeres, D.J., Benz, W., 1998. Dis- focusing of seismic energy, in contrast to previous 2D cal- ruption of kilometre-sized asteroids by energetic collisions. Nature 393, culations. In addition, for the test satellite simulations, an- 437–440. tipodal damage is more dependent on core geometry (radius) Asphaug, E., Ryan, E.V., Zuber, M.T., 2002. Asteroid interiors. In: Bottke, B.F., Cellino, A., Paolicchi, P., Binzel, R. (Eds.), Asteroids III. Univ. of than core density, indicating that incomplete or asymmetric Arizona Press, Tucson, AZ, pp. 463–484. differentiation can result in a unique response to large im- Benz, W., 1990. Smooth particle hydrodynamics: a review. In: Buchler, J.R. pact events. The simulations of Tethys, Mimas, and Rhea (Ed.), The Numerical Modeling of Nonlinear Stellar Pulsations. Kluwer show that Tethys has larger values of peak surface velocity Academic, The Netherlands, pp. 269–288. and peak surface tensile stress than either Mimas or Rhea, Benz, W., 1991. An introduction to computation methods in hydrodynamics. In: de Loore, C.B. (Ed.), Late Stages of Stellar Evolution: Com- as one would expect for the larger crater diameter relative to putational Methods in Astrophysical Hydrodynamics: Proceedings of target diameter. This indicates that Tethys is more damaged the Astrophysics School II. Springer-Verlag, Ponte de Lima, Portugal, during its giant impact event than either Mimas or Rhea. pp. 259–312. Results for antipodal and global fragmentation and terrain Benz, W., Asphaug, E., 1994. Impact simulations with fracture. I. Method rupture are inconclusive, with the hydrocode itself (using and test. Icarus 107, 98–116. size- and rate-dependent strength scaling) producing global Benz, W., Asphaug, E., 1995. Simulations of brittle solids using smooth particle hydrodynamics. Comput. Phys. Commun. 87, 253–265. disruption at the limits of model resolution (tens of km) but Boslough, M.B., Chael, E.P., 1994. Axial focusing of seismic energy from with peak fracture stresses never exceeding the strength of a large impact on earth: preliminary numerical simulations. In: Proc. laboratory ice. This problem awaits better characterization Lunar Planet. Sci. Conf. 25th, pp. 145–146. of the fracture properties of ice and much higher (107 parti- Burns, J.A., Matthews, M.S., 1986. Satellites. Univ. of Arizona Press, Tuc- cles or more) model resolution in 3D. son, AZ. Canup, R.M., Asphaug, E., 2001. Origin of the Moon in a giant impact near Antipodal focusing is not simple or straightforward. We the end of the Earth’s formation. Nature 412, 708–712. need to better understand what magnitude of seismic shak- Canup, R.M., Ward, W.R., 2002. Formation of the galilean satellites: con- ing can account for a given degree of terrain degradation. ditions of accretion. Astron. J. 124, 3404–3423. We also need better constraints for the behavior of ice at Chapman, C.R., McKinnon, W.B., 1986. Cratering of planetary satellites. the temperatures and pressures experienced during giant im- In: Burns, J.A., Matthews, M.S. (Eds.), Satellites. Univ. of Arizona Press, Tucson, AZ, pp. 492–580. pact events on icy satellites in the outer Solar System. De- Davies, M.E., Katayama, F.R., 1983. The control network of Rhea. tailed answers to these issues must await better images from Icarus 56, 603–610. Cassini, laboratory studies and future modeling. Dermott, S.F., Thomas, P.C., 1988. The shape and internal structure of Mi- mas. Icarus 73, 25–65. Engdahl, E.R., Flinn, E.A., 1969. Seismic waves reflected from discontinu- Acknowledgments ities within Earth’s upper mantle. Science 163, 177–179. Fujiwara, A., 1991. Stickney-forming impact on PHOBOS—crater shape We thank Jeff Moore, Erin Kraal, and John Chambers and induced stress distribution. Icarus 89, 384–391. for insightful conversations and Eileen Ryan and Akira Fu- Gisler, G., Weaver, R.P., Mader, C.L. Gittings, M.L. 2003. Two- and three- dimensional simulations of asteroid ocean impacts. Workshop on Im- jiwara for their careful reviews. This work was supported by pact Cratering No. 8054. NASA’s Planetary Geology and Geophysics Program Grant Goins, N.R., Dainty, A.M., Toksoz, M.N., 1981. Lunar seismology: the in- #NAG5-8914. ternal structure of the Moon. J. Geophys. Res. 86, 5061–5074. 466 L.S. Bruesch, E. Asphaug / Icarus 168 (2004) 457–466

Hagstrum, J.T., Turrin, B.D., 1991. Is ﬂood basalt volcanism a seismically- Nakamura, A., Fujimura, A., 1991. Velocity distribution of fragments induced response to large antipodal bolide impacts? EOS 72, 516. formed in a simulated collisional disruption. Icarus 92, 132–146. Hawkes, I., Mellor, M., 1972. Deformation and fracture of ice under uniax- Oldham, R.D., 1906. The constitution of the Earth. Quart. J. Geol. Soc. ial stress. J. Glaciol. 11, 103–131. London 62, 465–472. Hoppa, G.V., Tufts, B.R., Greenberg, R., Geissler, P., 1999. Formation of Rampino, M.R., Stothers, R.B., 1988. Flood basalt volcanism during the cycloidal features on Europa. Science 285, 1899–1902. past 250 million years. Science 241, 663–668. Housen, K.R., Holsapple, K.A., 1999. Scale effects in strength-dominated Rampino, M.R., Caldeira, K., 1992. Antipodal hotspot pairs on the Earth. collisions of rocky asteroids. Icarus 142, 21–33. Geophys. Res. Lett. 19, 2011–2014. Hughes, H.G., App, F.N., McGetchin, T.R., 1977. Global seismic effects of Schenk, P.M., 1989. Mimas grooves, the Herschel impact, and tidal stresses. basin-forming impacts. Phys. Earth Planet. In. 15, 251–263. In: Proc. Lunar Planet. Sci. Conf. 40th, pp. 960–961. Jeffreys, H., 1926. The rigidity of the Earth’s central core. Mon. Not. R. Astron. Soc. Geol. Suppl. 1, 371–383. Schultz, P.H., Gault, D.E., 1975. Seismic effects from major basin forma- Lange, M.A., Ahrens, T.J., 1983. The dynamic tensile strength of ice and tions on the Moon and Mercury. Moon 12, 159–177. ice-silicate mixtures. J. Geophys. Res. 88, 1197–1208. Smith, B.A., 26 colleagues, 1981. Encounter with Saturn: Voyager 1 imag- Lehmann, I., 1936. P’. Bureau Central Seismoloque International Travaux ing science results. Science 212, 163–191. Scientiﬁque A 14, 87–115. Smith, B.A., 28 colleagues, 1982. A new look at the Saturn system: the Lognonne, P., 28 colleagues, 2000. The NetLander very broad band seis- Voyager 2 images. Science 215, 504–537. mometer. Planet. Space Sci. 48, 1238–1302. Sohl, F., Spohn, T., Breuer, D., Nagel, K., 2002. Implications from Galileo McKinnon, W.B., 1985. Geology of icy satellites. In: Klinger, J., Benest, D., observations on the interior structure and chemistry of the galilean satel- Dollfus, A., Smoluchowski, R. (Eds.), Ice in the Solar System. Reidel, lites. Icarus 157, 104–119. Boston, MA, pp. 829–856. Stevenson, D.J., Harris, A.W., Lunine, J.I., 1986. Origins of satellites. In: Melosh, H.J., 1989. Impact Cratering: A Geologic Process. Oxford Univ. Burns, J.A., Matthews, M.S. (Eds.), Satellites. Univ. of Arizona Press, Press, New York, NY. Tucson, AZ, pp. 39–88. Melosh, H.J., Ryan, E.V., Asphaug, E., 1992. Dynamic fragmentation in im- Stewart, S.T., Ahrens, T.J., 2003. Hugoniots and shock-melting criteria for pacts: hydrocode simulation of laboratory impacts. J. Geophys. Res. 97, solid and porous H2O ice. In: Proc. Lunar Planet. Sci. Conf. 34th. Ab- 14735–14759. stract 1622. Monaghan, J.J., 1992. Smoothed particle hydrodynamics. Annu. Rev. As- Thomas, P.J., Dermott, S.F., 1991. The shape of Tethys. Icarus 94, 391–398. tron. Astrophys. 30, 543–574. Tillotson, J.H. 1962. Metallic equations of state for hypervelocity impact. Moore, J.M., Ahern, J.L., 1983. The geology of Tethys. Proceedings of General Atomic Report GA-3216. the 13th Lunar and Planetary Science Conference. J. Geophys. Res. 88, Von Neumann, J., Richtmyer, R.D., 1950. A method for the numerical cal- A577–A584. Mosqueira, I., Estrada, P.R., 2003. Formation of regular satellites of giant culation of hydrodynamic shocks. J. Appl. Phys. 21, 232–237. planets in an extended gaseous nebula I: subnebula model and accretion Watts, A.W., Greeley, R., Melosh, H.J., 1991. The formation of terrains of satellites. Icarus 163, 198–231. antipodal to major impacts. Icarus 93, 159–168. Murray, B.C., Belton, M.J.S., Danielson, G.E., Davies, M.E., Gault, D.E., Weibull, W., 1951. A statistical distribution function of wide applicability. Hapke, B., O’Leary, B., Strom, R.G., Suomi, V., Trask, N., 1974. Mer- J. Appl. Mech. 18, 293–297. cury’s surface: preliminary description and interpretation from Mariner Wilhems, D.E., McCauley, J.F. 1971. Geologic map of the near side of the 10 pictures. Science 185, 169–179. Moon. US Geol. Survey Map I-703.