The Rheasilvia impact on asteroid 4 Vesta may have been sufficiently large to create disrupted terrains at the impact antipode. This paper investigates the amount of deformation expected at the Rheasilvia antipode using numerical models of sufficient resolution to directly observe terrain modification and material displacements following the arrival of impact stresses. We find that the magnitude and mode of deformation expected at the impact antipode is strongly dependent on both the sound speed and porosity of Vesta's mantle, as well as the strength of the Vestan core. In the case of low mantle porosities and high core strengths, we predict the existence of a topographic high (a peak) caused by the collection of spalled and uplifted material at the antipode. Observations by NASA's Dawn spacecraft cannot provide definite evidence that large amounts of deformation occurred at the Rheasilvia antipode, largely due to the presence of younger large impact craters in the region. However, a deficiency of small craters near the antipodal point suggests that some degree of deformation did occur.
 Hypervelocity impacts are arguably the most violent geologic process known today [Melosh, 1989]. Some impacts are so large they are believed to cause significant deformation on the opposite side of the target body. This is a consequence of the target's ability to focus the impact-induced stress waves to the crater's antipode where the near-surface pressure gradient, intensified by constructive interference, can cause significant material deformation and damage. The archetype of this antipodal morphology is the disrupted terrain found directly opposite and suspected to be related to the Imbrium basin on the Moon [Schultz and Gault, 1974; Hood and Artemieva, 2008], a region characterized as “hilly and lineated.” Similar features have been investigated on Mercury [Murray et al., 1974; Schultz and Gault, 1974], Phobos [Fujiwara, 1991], and various icy satellites of the outer solar system [Watts et al., 1991; Bruesch and Asphaug, 2004]. As a rule of thumb, hilly and lineated terrains are found opposite only the largest craters, as the amount of energy required to cause significant deformation on the opposite side of a large body is considerable [Thomas and Veverka, 1979; Richardson et al., 2005]. In addition to the size, velocity, and incidence angle of the projectile, the extent and mode of deformation expected at an impact antipode relies largely on the properties of the target body. Size, oblateness, internal structure, and the presence/size of a core all play important roles in determining how energy is focused and dissipated following an impact [Meschede et al., 2011]. While some of this parameter space has been explored [Fujiwara, 1991; Watts et al., 1991; Bruesch and Asphaug, 2004], a simple predictive relationship between source crater and antipode morphology cannot yet be made and must be explored numerically on a case-by-case basis.
 The Rheasilvia impact basin on asteroid 4 Vesta has long been known as one of the largest impact features in the solar system with respect to target body size [Thomas et al., 1997]. Centered near Vesta's south pole, the basin has a depth of ~19 km and a diameter of ~500 km, nearly equal to that of the asteroid (~525 km) [Russell et al., 2012; Schenck et al., 2012]. The basin is estimated to be relatively young (~1 Ga), meaning that any related antipodal terrains may remain well preserved [Marchi et al., 2012a; Schenck et al., 2012]. Recent observations by NASA's Dawn spacecraft set rigid constraints on many of the free parameters that could affect the expected mode and degree of deformation at Rheasilvia's antipode. The size and velocity of the impactor, the size and composition of Vesta's core, and the bulk density of the asteroid are all relatively well known or easily estimated [Russell et al., 2012; Jutzi and Asphaug, 2011]. In addition, recently released imagery of Vesta's north pole allows for direct comparison between impact model output and spacecraft observation. This makes Vesta an ideal body on which to examine the effects of antipodal focusing of impact stresses.
 The impact that formed the Rheasilvia basin (hereafter referred to as the Rheasilvia impact) was large enough that some degree of antipodal deformation is expected. Previous numerical simulations [Jutzi and Asphaug, 2011] of the Rheasilvia impact find only weak evidence for antipodal deformation, with surface velocities of ~5 m s−1 opposite the crater. Jutzi and Asphaug  suggest that these low velocities may be a matter of resolution. Due to computational limitations in three dimensions, their work is limited to a resolution of ~3 km, the smoothing length used in their simulations. In addition to resolution issues, Jutzi and Asphaug  assume that Vesta has a bulk porosity of 10%. As the Rheasilvia impact's stress wave passes through Vesta's mantle, it inevitably crushes out some amount of pore space. This process is irreversible, causes heating, and can rapidly reduce the energy of the passing wave, energy that would otherwise be available to deform the surface at the antipode [Davison et al., 2010].
 The porosity of Vesta's mantle is estimated based on Dawn measurements to be 5% [Russell et al., 2012], half of the value used by Jutzi and Asphaug . The Dawn estimate is based on radio tracking of the spacecraft and requires assumptions about the composition and shape of Vesta's mantle and core. The mantle is assumed to have a grain density equal to that of measured diogenites [Russell et al., 2012; Britt et al., 2010]. The core is assumed to be in hydrostatic equilibrium, and its density and size are simultaneously solved for by inversion [Russell et al., 2012], which when combined with a grain density for the mantle, a shape model, and an estimate of the total body mass based on spacecraft tracking, yields a porosity. While these assumptions are not unreasonable, the dependency of antipodal deformation on porosity provides another way with which to constrain our estimates of Vesta's mantle properties. However, it is possible that the porosity of Vesta's mantle was different before the Rheasilvia impact occurred and was significantly modified by the impact itself. In section 4 of this paper, we will show that the dependency of antipodal deformation on mantle porosity is so strong that if the mantle has zero porosity, the Rheasilvia impact should produce a topographic peak several kilometers high at its antipodal point, a feature considerably more extreme than observed at impact antipodes on solar system bodies, Vesta included.
 Another important parameter controlling how impact stresses are focused to the antipode of the Rheasilvia crater is the sound speed, or seismic velocity, of the mantle and core. Impact stresses are carried to the antipode as a complex set of waves. Initially, near the impact site, these stresses are carried as a shock wave that moves faster than the speed of sound. As the shock wave decreases in intensity with distance, it decays into several modes: an elastic precursor, which travels at the speed of sound, the “main” plastic wave, which travels slightly slower, and several modes that propagate only near the surface of the target body. The seismic velocity of Vesta's mantle and core controls how quickly each of these modes can transmit stress to the antipode of the impact, how waves interact with one another upon arrival, and the magnitude of the resulting deformation. While the seismic velocity structure of Vesta is not known, sound speed is generally inversely related to square root of density. Consequently, in a similar manner to Earth, the sound speed in Vesta's iron core is likely lower than in its rocky mantle [Stein and Wysession, 2002]. The core acts to focus seismic energy at the antipode similar to the way a convex lens focuses light. This degree of antipodal focusing relies on the relative sound speeds of the mantle and core. Increasing the porosity of a material generally leads to a net reduction in its sound speed [Herrmann, 1969; O'Connell and Budiansky, 1974]. With little certainty regarding the seismic velocity structure of Vesta, and despite the fact that sound speed is a complex function of material properties, we treat mantle sound speed as a free parameter in order to isolate the focusing effect of the core. We demonstrate in section 4 that the sound speed of Vesta's mantle plays an important role in determining both the degree and breadth of deformation expected at the Rheasilvia antipode.
 The amount of energy transmitted through the Vestan core is dependent on its material strength. The core of Vesta is thought to have formed from the differentiation of the parent body [Consolmagno and Drake, 1977; Jaumann et al., 2012] and is likely composed largely of iron. Its strength, however, is unknown. The strength of iron alloys can vary considerably depending on how much nickel is present [Petrovic, 2001] and how much strain and fracturing the core has accumulated over time from previous impacts in the body. In addition to mantle sound speed and porosity, we run Rheasilvia impact simulations to test the dependence of antipodal deformation on core strength. We demonstrate in section 4 that the amount of deformation expected at the Rheasilvia antipode is dependent on what rheological parameters are used for Vesta's core and that a stronger core will lead to stronger deformation than a weaker core that behaves in a ductile manner.
 Previous numerical studies on antipodal focusing required that physical parameters such as peak surface velocities be used as a proxy for deformation. Watts et al.  assume that peak surface pressure is representative of the degree of expected deformation. This is an unfortunate choice, as free surfaces (such as the surface at the antipode) require a zero-pressure boundary condition and the peak pressure in a given cell becomes dependent on the resolution of the computational mesh. Work using smooth particle hydrodynamics codes [Bruesch and Asphaug, 2004] utilizes surface velocity and peak tensional stress to understand how much deformation occurs, but are limited to ~10 km scales of resolution, considerably larger than the observed features in hilly and lineated terrains [Schultz and Gault, 1974]. In this work, we model the Rheasilvia impact using a resolution of 400 m. Our models are capable of resolving individual deformation features such as antipodal peaks. In cases where no significant antipodal feature can be resolved, we use surface velocity (which is less sensitive to mesh resolution) as a proxy for the degree of deformation and displacement expected following the arrival of the impact stress wave.
2 Numerical Model
 We use the iSALE shock hydrodynamics code [Wünnemann et al., 2006], an extension of the SALE code [Amsden et al., 1980], to simulate the Rheasilvia impact and its antipodal effects. SALE was built to model shock processes in gaseous materials, and iSALE extends this work to include sophisticated constitutive models and equations of state and is capable of modeling shock processes in geologic materials [Melosh et al., 1992; Ivanov et al., 1997; Collins et al., 2004]. Our models are run in two dimensions using cylindrical symmetry. Simulations are run on an Eulerian (fixed cell) computational grid with 400 m resolution (Table 1). We utilize central gravity, which is computationally faster than self-gravity and sufficient in accuracy for this investigation. We model Vesta based on Dawn observations as a 520 km diameter spherical dunite mantle surrounding a 220 km diameter solid iron core [Jaumann et al., 2012; Russell et al., 2012]. Due to model limitations, we do not consider the effects of a Vestan crust, but its inclusion may significantly affect results, and should be included in future work. We used the ANEOS equations of state for dunite and iron to represent Vesta's mantle and core, respectively [Benz et al., 1989; Thompson, 1990]. The impactor is treated as a 36.8 km diameter dunite sphere impacting the surface of Vesta at 5.5 km s−1 and at a 90° incidence angle. Our impact parameters are derived from simulations that reproduce the topography of the Rheasilvia basin [Ivanov and Melosh, 2013]. These values are consistent with pre-Dawn estimates of the impactor velocity [Asphaug, 1997; Jutzi and Asphaug, 2011], suggesting robustness.
Table 1. Simulation Parameters
Number of cells in x direction
Number of cells in y direction
Cell size in x direction
Cell size in y direction
Radius of projectile
Radius of core
Thickness of mantle
Radius of impactor
5.5 km s−1
Initial surface temperature
Internal temperature profile of the target
 We use the strength model of Collins et al.  to accurately represent the rheology of geologic materials, which assumes that the strength of both the mantle and core is dependent on cohesion, internal friction, and pressure. While the damage of material plays its most important role adjacent to the impact site, where strains are very large, damage weakening of rocks can also play a role in determining final morphologies at the antipode. We adopt the damage model of Ivanov et al. , which takes into account the pressure dependence of material damage. We incorporate the weakening of rock with increasing temperature using the thermal weakening model of Ohnaka  (Table 2), although our target's initial condition is constant temperature with depth.
 As described above, the degree of antipodal deformation is strongly dependent on mantle porosity φ. We use the ε-α porosity model [Wünnemann et al., 2006; Collins et al., 2010]. The ε-α model relates the volume strain to the distension α = 1/(1 − φ) in different total strain regimes. These regimes include an elastic regime [Collins et al., 2010], where the matrix material is allowed to compress but no pore space is permanently crushed out, and a crushing regime [Wünnemann et al., 2006], where the amount of total volume strain is sufficient to permanently reduce the porosity of the material. In the elastic regime, the rate at which porosity changes with respect to total volume strain is related to χ, the ratio of porous to nonporous sounds speeds and α0, the initial distension
 Once a critical strain threshold is reached, distension decreases monotonically as
where κ is a prescribed parameter controlling the rate of compaction. If the stress wave induces strains sufficient to enter the compaction regime (), then pore space crushing begins and energy is lost from the wave. The most important parameters of this model for our work are the initial porosity φ, corresponding to an initial distension α, and the elastic strain threshold . The parameter κ = 0.98 which controls the rate of compaction is the same as used in Wünnemann et al.  and is not varied within this study.
3 Impact Simulations
 We seek to explore the effects of three free parameters: the porosity of the mantle, which determines how the Rheasilvia stress wave is dissipated; the mantle sound speed, which determines how effectively stresses are focused at the Rheasilvia antipode; and the strength of the Vestan core, which controls where stresses are localized in the target body. To look at how variations in mantle porosity affect antipodal deformation, we give the elastic strain threshold () a physical basis by equating it to the crushing strength of rock (Ycrush) as
where E is Young's modulus [Güldemeister et al., 2013]. A Young's modulus of 70 GPa and a crushing strength of ~700 MPa, similar to that of lunar rock [Stephens and Lilley, 1971], yield a strain threshold of , a reasonable value. We then vary the initial porosity from 0 to 10%, covering a fairly large range around the Dawn estimate of 5% [Russell et al., 2012].
 Such as iSALE, the rate at which elastic waves propagate (the bulk sound speed) comes directly from the equation of state, which relates the density and energy of a material to its pressure and temperature. In reality, the speed of sound in material is a complex function of density, strength, mineral properties, porosity, etc. The speed of sound of the Vestan mantle plays a critical role in the focusing of the impact stress wave to the antipode. We are interested in isolating the effects of sound speed on antipodal deformation but are unable to directly assign a seismic velocity profile to the Vestan mantle. Instead, we utilize the ability of the porosity routine to alter the effective sound speed (equation ((1))). This is done by raising the strain threshold for crushing to (a crushing strength of ~7 GPa) and varying the sound speed ratio of porous to non-porous material χ. This, in effect, forces the porosity routine to remain in the elastic compaction regime while altering the sound speed of the mantle around the reference velocity (~6.5 km s−1) determined by the equation of state [Pierazzo et al., 1997]. Even at an unrealistically high crushing strength of ~7 GPa, some pore space will inevitably be lost near the point of impact. We choose an initial porosity φ = 0.5%, a value high enough that the pore space is not completely crushed out in most of the mantle, but low enough that whatever crushing that does occur does not significantly dissipate energy from the wave. In other words, the choice of a low initial porosity and a high crushing strain strikes a balance between minimizing the effects of pore space crushing (which will still occur near the impact site, where strains are very high) and allowing the porosity model to vary the effective bulk sound speed. We vary the sound speed ratio χ from 0.7 to 1.2, corresponding to bulk sound speeds of ~4.5 to 7.8 km s−1 [Pierazzo et al., 1997]. These variations in sound speed are most likely much wider than should be expected for a realistic Vestan mantle, but as we are interested in isolating the focusing effects of mantle sound speed, we choose to vary the parameter over a wide range. For each of our simulations, we choose to compare deformation states 1500 s after impact. This is well after the mass movement of material ceases at the antipode, but before the fallback of ejecta.
 Simulations in which we treat mantle porosity and mantle sound speed as a free parameter assume a core rheology that is “rock-like,” in which material strength is a function of cohesion, friction, and pressure. This rheology can be considered a “strong” end-member, one in which much of the core reacts elastically as the impact stress wave passes. An alternative type of strength model applicable to metals is one in which material is ductile, and strength dominantly a function of accumulated strain [Johnson and Cook, 1983]. In order to test the effects of different types of core strength on antipodal deformation, we ran simulations with both rock-like and ductile models. In the former, the strength of intact core material is pressure dependent as
where Y0 is the material's cohesion, μ is the coefficient of internal friction, Ym is the limiting strength at high pressure, and P is pressure. Strength reduction due to material damage is taken into account. In the latter, strength is dependent on both strain and strain rate as
where ε is the accumulated strain of the material, is the strain rate, and A, B, C, and N are material-dependent parameters. This model also takes into account temperature-dependent weakening of material.
 We treat our ductile core as Armco iron [Ivanov et al., 2010], where A = 100 MPa (corresponding to a compressional strength of ~170 MPa; Table 2). However, laboratory tests suggest that the strength of iron meteorites can be considerably higher than that of Armco iron [Petrovic, 2001]. Measured strengths vary as a function of nickel and carbon content in the meteoritic alloy, and nickel concentrations of ~15% can yield compressive strengths of up to 1 GPa. Additionally, because the strength of metal can increase with accumulated strain, it is possible that Vesta's core could have been considerably hardened by strains from previous large impacts, such as the one that formed the Veneneia basin [Jaumann et al., 2012].
 The details of how the impact stress wave propagates through the Vestan mantle and core vary based on the porosity and sound speed of the mantle and the strength of the iron core (Figure 1). In the “control” case (0% mantle porosity, strong rock-like core, χ = 1), the wave passes through most of the core and mantle with a minimal amount of decay. The waves in the mantle, especially near the core mantle boundary, move faster than in the core, and are focused around it like a convex lens. These waves then interfere to cause very high surface velocities at and near the antipode. In simulations with a weak, ductile core (0% mantle porosity, weak ductile core, χ = 1), the wave speed in the mantle is considerably larger than in the core, changing the amount of antipodal focusing that occurs. The wave that does pass directly through the core is also reduced somewhat in amplitude. This is likely because energy is used to deform the core itself, causing the wave to attenuate more quickly. In simulations with high mantle porosity (5% mantle porosity, strong rock-like core, χ = 1), the stress wave is considerably weakened in its passage from the impact site to the core, as well as the rest of its passage through the mantle. The wave speed in the mantle is reduced throughout the body, and the core is less effective at focusing the wave to the antipode. In simulations with a significantly reduced mantle sound speed (0% mantle porosity, strong rock-like core, χ = 0.7), the waves passage through the mantle is both slowed and attenuated in magnitude. This affects both the magnitude of the incident wave at the antipode as well as its focusing.
 We examine the amount of deformation at our model's antipode by directly comparing model output surface topography to initial conditions. The arrival of the impact stress wave causes very large near-surface pressure gradients, and assuming that the tensile strength of the rock is exceeded in some cases (especially when material is damaged), large amounts of material are spalled upward (Figure 2). In the case of 0% mantle porosity, this material is lifted over 10 km above the surface of Vesta before falling back and accumulating. During this period of free flight, material is uplifted from depth and translated horizontally toward the symmetry axis. Much of this collected material is completely damaged and fragmented. The spalled and uplifted materials collect at or near the antipode. For initial porosities less than ~1%, the net result of this fairly extreme surface modification is the formation of an antipodal peak (Figure 3). In the case of no porosity, this peak is nearly 6 km tall, with a half width of ~25 km. The inclusion of porosity rapidly reduces the expected size of the antipodal feature. At the Dawn-estimated mantle porosity of 5%, no discernable antipodal peak is found after the period of deformation ends.
 The effect of sound speed on the height of the antipodal topographic peak is readily apparent (Figure 4). The peak height is maximal when the mantle sound speed is set to 90% of the reference determined by the equation of state (approximately 6.5 km s−1). Sound speed also helps determine the width of the antipodal peak. As the mantle and core work in concert to focus or defocus the impact stress wave, the energy of that wave can alternately be concentrated in a very small area or spread out broadly. In the former case, the horizontal extent of deformation should be smaller, but the magnitude should be large. In the latter case, the impact stress wave is never focused sufficiently to cause extreme levels of deformation, but what deformation does occur is spread out over a much larger area.
 At porosities higher than ~1% our models are not capable of directly resolving deformation features at the Rheasilvia antipode, as the magnitude of material displacement is smaller than the model resolution. As an alternative proxy, following Bruesch and Asphaug , we rely on surface velocities at the antipode to inform our understanding of how much deformation occurs (Figures 5 and 6). Our results suggest that across the range of initial mantle porosities used in this study, significant antipodal deformation can be expected. At 5% porosity, the case that corresponds best with Dawn estimates of Vesta's mantle porosity [Jaumann et al., 2012], velocities reach values of ~35 m s−1. Even at a porosity of 10%, the antipode experiences surface velocities of ~25 m s−1. In an environment where surface gravity is only 0.25 m s2, velocities this high can be expected to result in significant deformation, with materials lofted as high as ~1 km above the surface.
 The effect of sound speed on model surface velocities (Figures 7 and 8) is similar to the effect noticed on the size and breadth of antipodal peaks. The horizontal range over which high surface velocities are found seems to be a function of the mantle sound speed. This reflects how well the impact stress wave is focused to a single point at the antipode. When the wave is well focused, the peak surface velocity is high but only extends to a small region around the antipodal point. When the wave is less focused, the peak surface velocity is not as high, but modest velocities are spread over a much larger area. Below a critical sound speed threshold (in this study ~0.7csound), the core instead acts to defocus the impact stress wave, and the highest surface velocities are found at some distance from the antipode. This is similar to core-size dependencies found in Watts et al. .
 The dependence of antipodal surface velocities on core strength is considerable (Figure 9). Simulations with a ductile Armco iron core result in ~5 times lower than simulations with a strong rock-like core. The ductile core also concentrates high velocities in a smaller region near the antipodal point. An increase in the compressive strength of the ductile core leads to a corresponding increase in antipodal surface velocity. The rock-like core acts as a “strong” end-member with the highest surface velocities. It should be noted that on a larger scale, the inclusion of a ductile core can change the post-impact stress distribution across the entire target body and may play a fundamental role in localizing other Rheasilvia-related morphologies such as the Divalia Fossae [Jaumann et al., 2012; Buczkowski et al., 2012; Bowling et al., 2013].
 Dilation (the volume bulking of porous material with increasing strain) is not modeled in the current version of iSALE. Because of the high strains induced throughout much of the Vestan mantle following the Rheasilvia impact, the dilatant bulking of material could play a considerable role in the production of antipodal topography. We can estimate the volume bulking from the total strain of material following Henkel et al. , which is work based on the shear flow of material underneath terrestrial craters, and relate the change in porosity to the volumetric strain as
 We choose a maximum porosity of 15%, which is consistent with that seen in terrestrial craters [Henkel et al., 2010] as well as the lunar crust [Wieczorek et al., 2013]. The change in porosity for a given computational cell becomes
where is the amount of volumetric strain within a given cell and φf is the final model porosity in a given cell (after crushing from the passage of the impact stress wave has occurred). We assume that volume bulking is equal in all dimensions, meaning that the increase in length of a cell in the “upward” direction at the antipode is equal to 1/3 of the total volume increase of that cell. We then sum the change in length of each cell between the surface and the core mantle boundary
where ΔH is the change of topography due to dilatant bulking and Δx is the size of a computational cell (400 m), It should be noted, however, that if the Vestan crust is already considerably “bulked” due to impacts pre-dating Rheasilvia, the contribution due to Rheasilvia induced dilatancy should be minimized.
 The contribution of dilatant bulking in our simulations can be considerable (generally leading to a topographic increase of ~1 km) and is somewhat dependent on both initial mantle porosity (Figure 10) and mantle sound speed (Figure 11). The former suggests that even at the Dawn-estimated mantle porosity of 5%, it can be expected that a detectable antipodal topographic high could be produced by the Rheasilvia impact. The dilatant contribution to antipodal topography is also, in general, shallower and wider than from antipodal spallation and uplift alone. Antipodal dilatancy also varies with mantle sound speed, but is at a maximum when the mantle sound speed is equal to the reference sound speed as determined by the equation of state.
 It should be noted that the type of dilatant bulking we calculate is induced by shear strains in a flowing continuum of material. An alternate effect may occur at the Rheasilvia antipode where large amounts of material are spalled upward from the surface. Upon reaccumulation at the surface, spalled and fragmented materials can collect in a disordered manner sufficient to resist gravitational collapse but with fragment arrangement leaving considerable void space. This effect should lead to an unquantified net increase in volume, contributing additionally to antipodal topography.
 Our use of mantle sound speed variations is primarily targeted at understanding how body stress waves (namely the main shock wave, elastic precursor, and plastic wave) are focused at the antipode, and the deformation that results. This neglects the contribution of surface waves such as Love and Raleigh waves. Surface waves can play an important role in causing deformation at far distances from impacts because their amplitude decreases as 1/r, compared to body waves, which decay as 1/r2 (where r is the distance from the impact point). While not strongly affected by focusing due to core/mantle sound speed ratios, surface wave modes do occur in hydrocodes simulations and their effect is taken into account in our final measure of antipodal surface topography. The magnitude of surface wave effects should be dependent on arrival time and the presence and thickness of a crust. However, we currently cannot distinguish how the antipodal effects of surface waves compare to antipodal effects due to body waves alone, except to say that the surface waves should arrive sufficiently late so as not to affect the peak surface velocities shown in Figures 5, 7, and 9.
 We compare different rheological models for the Vestan core, one “rock-like” in which material strength is dependent on friction and pressure (equation (4)), the other “metal-like” in which material strength is dependent on accumulated strain (equation (5)). We refer to the former as a “strong” core and the latter a “weak” core. However, the rock-like core is only stronger under considerable confining pressure and before damage has accumulated (which reduces the coefficient of internal friction) (Figure 12). If enough strain has accumulated in the core when the metal-like strength model is used, the strength of the core can actually exceed that of the rock-like model.
 The effects of model resolution on shock wave pressures and decay can be considerable. In most shock hydrocodes, results do not converge until the model resolution reaches ~20 cells per projectile radius (CPPR) [Pierazzo et al., 2008]. Our suite of models is run at 46 CPPR, which should be sufficient to accurately simulate shock effects. In order to test this, we compared several of our runs to simulations run at 18 CPPR. The differences between antipodal surface velocities in our control runs and models at 18 CPPR are very small. This suggests that our suite of models is accurately reproducing shock-induced effects at the impact antipode.
 The use of two-dimensional axisymmetric models has several shortcomings. We cannot model oblique impacts, which intrinsically preclude symmetry. The dependence of antipodal deformation on impact angle has been investigated [Bruesch and Asphaug, 2004], but the authors were unable to establish a clear relationship between the two phenomena at the limits of model resolution (~10 km). Jutzi and Asphaug  investigate angular dependence in the work on the characteristics of the Rheasilvia basin but find that their oblique (45°) impact scenario only induced slight asymmetry to basin and ejecta geometries (when spin is neglected). The authors do not comment on how impact angle affects antipodal surface velocities in their models. Simulations of basin-forming impacts on Martian-sized bodies [Bierhaus et al., 2012; Bierhaus et al. 2013], which look at temperature increases at the impact antipode, suggest that impact angle does indeed play an important role in governing antipodal effects. Dawn observations [Jaumann et al., 2012], as well as numerical modeling [Jutzi et al., 2013] of the Rheasilvia ejecta blanket, reveal an asymmetry, suggesting that the impact did indeed occur at an oblique angle. This issue should be the subject of further investigation using fully three-dimensional modeling.
 Two-dimensional cylindrically symmetric calculations can result in numerical instabilities and nonphysical effects at or near the axis of symmetry. A common criticism of two-dimensional simulations of antipodal deformation is that because the antipode lies at the symmetry axis, constructive interference of the stress wave and the resulting amount of deformation can be exaggerated. Three-dimensional simulations are preferable for this reason but are severely limited in resolution, as they are computationally expensive. Previous numerical work on antipodal deformation relied on proxies such as peak tensile stress in order to imply whether or not deformation has occurred. Because considerably higher resolution is required to directly resolve deformation at the antipode of an impact, two-dimensional axisymmetry is necessary. In order to try and estimate how much of an effect the axis of symmetry has in the focusing of waves at the impact antipode, we ran a simulation in 3-D geometry as a comparison. Because of the increased computational costs of 3-D simulations, our simulation was run on a much coarser grid with a resolution of 24 km. Qualitatively, the same effect is observed, with the impact stress wave being focused around the iron core and superposing upon itself at the antipode, leading to high velocities. However, the magnitude of surface velocities at the antipode in this simulation is ~3 times lower than in our previous 2-D calculations. As pointed out above, the decay of shock waves in these simulations is dependent on the resolution of the numerical mesh. As such, we ran a 2-D simulation with the same grid size as the 3-D simulation for comparison. The velocities were similarly reduced, suggesting that the effects of the axis of symmetry are minimal.
 In an attempt to reduce the number of free parameters in this work, we assume in all cases an impact velocity of 5.5 km s−1. This is based on work [Ivanov and Melosh, 2013; Jutzi et al., 2013] that attempts to match the topographic profile of the Rheasilvia crater itself. The actual relative velocity, impactor size, and impact angle at which the Rheasilvia forming impact occurred are unknown. All of these parameters may significantly change our results, with a faster impactor causing more antipodal deformation and a slower impactor causing less. This should be the subject of a future investigation.
 The use of central gravity in planetary scale impact simulations can result in unrealistic effects if not given proper consideration. When the impactor collides with the target in central gravity simulations, conservation of linear momentum causes that target's center of mass to displace from and oscillate about the center point of the gravity field. In the case of our impact simulations, the surface gravity of the target is relatively low, the size of the impactor is small compared to the size of the target, and the time scales at which antipodal deformation occurs (generally < 1000 s post impact) are short. Forces induced at the antipode by the movement of the surface through the prescribed gravity field are miniscule compared to forces imparted by the arrival of the impact stress wave, and central gravity is adequate.
 We treat our target as a spherical body, a simplification from Vesta's 286 × 279 × 223 km triaxial ellipsoid [Russell et al., 2012]. Ellipticity can change how impact stress waves constructively interfere because, with different distances to travel, waves can arrive at the antipode out of phase [Fujiwara, 1991]. This effect, however, is minimal when the impact and antipode lie on an axis of symmetry. The center of the Rheasilvia basin is currently very close to the south pole of Vesta (at ~75° south latitude), and as such, a spherical approximation for our Vesta target should not significantly alter our results.
 We do not consider the rotation of Vesta in our simulations. Vesta has a relatively fast rotation period of 5.4 h [Jaumann et al., 2012]. Jutzi and Asphaug  find that rapid spin coupled with oblique impact angle can significantly alter the profile of both the Rheasilvia crater and its ejecta blanket. The effect of spin on antipodal deformation, as such, may be significant but is outside of the purview of this work. Mass redistribution due to large impact craters is thought to have possibly altered the rotation axis of other solar system bodies, in some cases significantly [Melosh, 1975; Nimmo and Matsuyama, 2007; Wieczorek and Le Feuvre, 2009]. Initial calculations in which a Rheasilvia-size basin's worth of material is excavated from a Vesta-type ellipsoid and deposited as an ejecta blanket [following Melosh, 1975] suggest that the Rheasilvia basin changed Vesta's mean moment of inertia by ~1/10. More sophisticated Rheasilvia-specific calculations [e.g., Matsuyama and Nimmo, 2011] suggest the possibility that the Rheasilvia basin and its antipode may have had a different relationship with Vesta's spin axis than they do today.
 Earlier simulations of the Rheasilvia impact [Jutzi and Asphaug, 2011] did not resolve ejecta deposition at the impact antipode, likely a product of model resolution. Our models, which have significantly higher resolution, suggest that ejected material does reaccumulate on the opposite side of the target body. However, we choose to examine antipodal deformation states prior to Rheasilvia ejecta fallback. This is because estimates of the amount of ejecta fallback at the antipode can be unreliable, a result of the aforementioned symmetry axis problem. When ejecta arrives above the antipode (at the symmetry axis), it collides with its mirrored self (akin to a ring of material collapsing into a single point). The material then loses its horizontal momentum, falls back on to the surface of the target, and accumulates directly on the antipode. This geometric problem can lead to unreliable estimates of ejecta thickness at and near the Rheasilvia antipode, so we opt to measure antipodal deformation states before the fallback of ejecta.
6 Comparison to Dawn Observations
 Due to seasonal lighting, the Dawn spacecraft was only able to image the north pole of Vesta (near which the Rheasilvia antipode lies) at the end of its encounter. As a result, much of the imagery is lower resolution and less well illuminated than observations of the south pole, and the ability to directly observe deformation features, such as those seen in “hilly and lineated” terrains, is limited. However, because many of our simulations predict uplifted topography, it is possible to compare model output to observed topography in order to estimate the degree of deformation that has occurred due to the Rheasilvia impact. In doing so, we can attempt to place limits on the internal material properties of Vesta, such as mantle porosity and core strength.
 Topographic maps of Vesta's north pole show a broad region of high elevation near the location of the Rheasilvia antipode (Figure 13), ~5–10 km higher than the surrounding plains. However, the antipodal point itself lies within an ~63 km diameter fresh impact crater (Pomponia) that is likely younger than the Rheasilvia basin. Much of the topographic uplift predicted by our models would have been obliterated during the formation of this crater. In addition, there is another ~90 km diameter crater (Albana) of uncertain age in proximity to the antipodal point. As a result, it is unclear what portion of the present antipodal topography is a product of Rheasilvia and what portion is due to later impacts. In addition, because the antipode lies near the north pole, the observed region of high elevation could be a product of the chosen reference ellipsoid (285 × 229 × 229 km). If elevations near the antipode are in fact a product of the Rheasilvia impact, very large amounts of deformation are implied, suggesting that Vesta has both a low porosity mantle and a strong core. Unfortunately, post-Rheasilvia modification of the antipodal region makes it difficult to draw any firm conclusions.
 Another metric that can give insight into the amount of deformation induced by the Rheasilvia impact is crater density at the antipode. Because surface deformation should degrade and erase small craters more effectively than large craters [Richardson et al., 2005], a small crater deficiency would serve as evidence that some degree of modification took place. Preliminary crater counts based on data from the Dawn spacecraft's High Altitude Mapping Orbit (HAMO) (Figures 13 and 14) suggest that at diameters larger than ~10 km, the crater density near the Rheasilvia antipode is similar in nature to the highly cratered terrains farther to the south [Marchi et al., 2012a, 2012b]. However, there is a deficiency of craters with diameters in the range of 3–9 km. To further investigate this point, crater size frequency distributions were produced using higher-resolution Dawn Low Altitude Mapping Orbit (LAMO) imagery. At small crater sizes, the size frequency distribution matches that of the Rheasilvia ejecta blanket. This deficiency can only be partially explained as a result of infilling of craters by ejecta from the basin in which the antipode lies, suggesting that the Rheasilvia impact may have completely erased craters up to ~500 m in size at its antipode and significantly degraded craters several kilometers in size. This provides perhaps the best evidence available that significant deformation did occur near the Vestan north pole following the Rheasilvia impact.
 Mapping of geological features near Rheasilvia's antipode (D. T. Blewett et al., Vesta's North Pole Quadrangle Av-1 (Albana): Geologic Map and the Nature of the South Polar Basin Antipodes, submitted to Journal of Geophysical Research – Planets, 2013) revealed sets of linear depressions ~0.25–1 km wide and ~1–10 km long that may be the remnants of a “hilly and lineated” terrain that has been largely obliterated by ejecta from the Albana and Pomponia craters, but beyond this, evidence of large-scale deformation features is ambiguous at best. Analysis of stratigraphic cross sections in the region suggests that the high topography that is characteristic of the north polar region pre-dates the Rheasilvia impact. These findings are tempered by the fact that the accuracy of the mapping was limited by extensive seasonal shadows and poor lighting conditions at the time of observation.
 We have numerically simulated the antipodal effects of very large impacts on solar system bodies at resolutions sufficient to directly observe deformational features. When impact stresses are large enough to cause very large deformations at the impact antipode, the resulting morphology should take the form of a peak composed of uplifted strata and spalled, reaccumulated surface material. In the case of Vesta, the formation of such a feature may precede the arrival of impact ejecta sourced at the impact site.
 The magnitude and breadth of deformation at an impact antipode are strongly dependent on the porosity of the target's mantle and the strength of the target's core. This suggests that disrupted antipodal terrains will be found only on opposite major impact basins on bodies with relatively low mantle porosities and relatively strong cores. Additionally, the magnitude of the difference in sound speed between the mantle and core controls how well impact stresses become focused at the antipode. This consequently determines whether deformation will be highly localized or widely distributed. Finally, dilatant bulking of material by shear strain can further enhance topography already disrupted and uplifted opposite major impacts basins.
 Comparing observed features at the Rheasilvia antipode to model output constitutes a crude type of seismology in which we can hope to constrain the internal properties of Vesta. Observations by the Dawn spacecraft, while somewhat ambiguous, suggest that Vesta may have mantle porosity lower than previous estimates and a core of considerable strength.
 There are several other solar system bodies with impact craters that are large enough to have perhaps caused deformation features at their antipodes, such as Herschel crater on Mimas. Bruesch and Asphaug  use a technique similar to ours to try and discern the core size and density of bodies such as Mimas by simulating the antipodal effects of the major impacts on those bodies. Now that we have shown that parameters such as porosity and core strength also play an important role in governing deformation at impact antipodes, it may be worthwhile to revisit such work and attempt to understand even more internal properties of such bodies.
 We would like to thank the creators of iSALE including Kai Wünnemann, Gareth Collins, Dirk Elbeshausen, Jay Melosh, and Boris Ivanov. We would like to thank the members of NASA's Dawn science team for their invaluable feedback and access to data. We would like to thank Kai Wünnemann and Katarina Miljkovic for their most helpful reviews of this manuscript.