Two-dimensional numerical modeling of the Rheasilvia impact formation

Authors

B. A. Ivanov,

Corresponding author

Institute for Dynamics of Geospheres, Russian Academy of Sciences, Moscow, Russia

Corresponding author: B. Ivanov, Institute for Dynamics of Geospheres, Russian Academy of Sciences, Leninsky Prospect 38-1, Moscow 119334, Russia. (boris_a_ivanov@mail.ru, baivanov@idg.chph.ras.ru)

[1] We numerically modeled the formation of Rheasilvia crater, an enormous impact basin centered on asteroid 4 Vesta's south pole. Using a trial and error method, our models were adjusted to produce the best possible fit to Rheasilvia's size and shape, as observed during the Vesta orbital stage of the Dawn mission. The final model yields estimates of the shock wave decay, escaped material volume, depth of excavation, and other relevant characteristics, to the extent allowed by the two-dimensional (axially symmetric) approximation of the Simplified Arbitrary Lagrangian Eulerian hydrocode. Our model results permit interpretation of the Dawn data on Vesta's shape, topographic crater profiles, and the origin of the Vestoid asteroid family as escaped ejecta from the Rheasilvia crater.

[2] The Dawn mission to the asteroid 4 Vesta delivered valuable new data about this differentiated planetary body. The largest south pole impact basin Rheasilvia (~500 km in diameter) overlaps the older Veneneia basin (~400 km in diameter). Here we present two-dimensional axisymmetric (2-D) numerical modeling of these basins' formation including self-gravity [Ivanov et al., 2010] and the acoustic fluidization model [Melosh and Ivanov, 1999].

2 Modeling Technique and Target Construction

[3] The Simplified Arbitrary Lagrangian Eulerian code, version B (SALEB) hydrodynamic solver [Ivanov et al., 2010] is used to model the motion of solid material due to impact. We simultaneously solve the Poisson equation for the gravitational potential to compute the components of the gravity acceleration at each computational cell vertex. The boundary conditions associated with the gravitational potential are updated only at every 100th time step (this is the most time-consuming procedure, and more frequent updates are not necessary). The time step is about 0.015 s or less for our finest cell size of 0.93 km, so the gravity update occurs approximately every 1.5 s. This is much less than the transient crater formation time (500 s to reach the maximum crater volume for a 50 km projectile).

2.1 Target Structure

[4] Geochemical analysis and modeling of HED meteorites indicate that Vesta is probably differentiated [Righter and Drake, 1997; Ruzicka et al., 1997; Warren, 1997]. The analysis of Dawn data strongly supports these premission models [Russell et al., 2012]. In our premission models, we followed the choice in Jutzi and Asphaug [2010a, 2010b] and treated Vesta as a spherical body ~540 km in diameter with an iron core of 200 or 240 km in diameter. Vesta's crustal thickness is assumed to be 20 or 40 km. The core size for the model (assuming an iron density of 7850 kg m^{−3}) is close to the range of 214 to 226 km estimated from the Dawn analysis, which assumed a core density of 7100 to 7800 kg m^{−3} [Russell et al., 2012]. The crustal thickness is not as well constrained, and we treat the current model as preliminary, with a limited set of possible target materials. Other authors are beginning to explore a wider range of materials [Jutzi et al., 2012].

[5] In the current model, ANEOS-based [Thompson and Lauson, 1972] tables for basalt, dunite, and pure iron are used to represent the equations of state (EOS) of the crust, mantle, and core, respectively. Details of the EOS usage and the assumed strength and friction properties have been published earlier [Ivanov et al., 2010]. Our strength model assumes a gradual accumulation of damage in the material when shear or tensile stresses exceed the elastic limit. Totally damaged material is assumed to have zero tensile strength. For example, during spallation (see section 6 below), a spall fracture appears in the calculation as a layer of extended cells with a local density below normal. The pressure is set to zero in these cells. A formal void fraction can be computed in such cells by assuming that the solid material retains its zero pressure density.

[6] Two types of the model “Vesta” have been constructed. The spherical target asteroid is first equilibrated with self-gravity active in a special premodeling run without a projectile. This separate preimpact model run is needed to achieve the minimum possible initial internal velocities. Although we begin with a numerical solution for a spherical body with the given density and temperature profiles, the finite cell size of the 2-D computation induces small spurious velocities when self-gravity is turned on, which this premodeling run is designed to eliminate [Ivanov et al., 2010].

[7]

Target 1: Crust thickness of ~20 km and iron core radius of ~120 km. After equilibration in its self-gravity field, the target radius settles down to ~273 km (versus 262.7 km Dawn mean radius) and the total target mass is 3.04 × 10^{20} kg (slightly above the Dawn value of 2.59 × 10^{20} kg [Russell et al., 2012]). The model's surface gravity is about 0.273 m s^{−2}, and the average target density is 3630 kg m^{−3}—slightly above the best current estimate of 3456 kg m^{−3} [Russell et al., 2012].

[8]

Target 2: Crust thickness of 40 km and iron core radius of 100 km. After equilibration in its self-gravity field, the target radius is about 271.6 km, the total target mass is 2.83 × 10^{20} kg, the average density is 3375 kg m^{−3}, and the surface gravity is 0.244 m s^{−2}.

[9] Our decision to use ANEOS-based tables limits our ability to vary the initial densities of model materials. In the future, the initial density may be specified more freely by using simplified equations of state such as that of Tillotson [1962]. With more flexibility in the material density, one could reiterate the models to achieve a better fit to the observed target size and average density. For simplicity, we assumed a constant initial temperature of 293 K.

[10] Model basalt projectiles (density 2858 kg m^{−3}) with an impact velocity of 5.5 km/s were given radii ranging from 30 to 96 km. Our spatial resolution varies from 20 to 30 cells per projectile radius (CPPR) with a cell size of 0.93 × 0.93 km in cross section. For selected runs, we use twice as coarse 1.96 × 1.96 km or 2 × 2 km resolution with CPPR ≥ 10. Parameters for each model run are listed in Table 1.

Table 1. Parameters of Model Runs and Main Dimensions of Modeled Craters

Number of computational cells per projectile radius.

^{d}

Spherical projectile diameter.

^{e}

Efficiency of maximum particle velocity conversion into AF model oscillation amplitude. Extreme parameters are in bold.

^{f}

Decay time of AF oscillation amplitude (equation ((1))).

^{g}

Diffusion coefficient for AF oscillation amplitude. Extreme values are in bold.

^{h}

Applicability of the AF model for crust/mantle. 0 = AF is off, 1 = AF is on.

^{i}

Transient crater diameter measured along the spherical preimpact surface.

^{j}

Transient crater maximum depth.

^{k}

Final rim crater diameter measured along the spherical preimpact surface.

Target 1

Vesta15rn

0.6

2

10

40.00

0

0/0

252

68

330.0

Vesta_01

0.6

2

13

52.45

0

0/0

348

95

451.5

Vesta_09

0.6

2

20

80.25

0

0/0

530

138

650.0

t111s3

0.5

1.86

10

37.30

0.1

900

1 × 10^{6}

0/1

248

76

470

t111s4

0.5

1.86

12

44.40

0.1

900

1 × 10^{6}

0/1

302

94

500

V53_805q

0.5

0.93

24

44.56

0.1

800

1 × 10^{6}

0/1

308

96

488

V53_800q

0.6

0.93

24

44.56

0.1

800

1 × 10^{6}

0/1

304

88

480

V54_700

0.5

0.93

26

48.40

0.1

700

1 × 10^{6}

1/1

342

103

572

t111s2

0.5

1.86

13

48.78

0.1

900

1 × 10^{6}

0/1

329

103

520

t111s1

0.5

1.86

14

52.07

0.1

900

1 × 10^{6}

0/1

352

111

550

V55_800vsm

0.4

0.93

28

52.18

0.1

800

1 × 10^{6}

1/1

372

110

670

V55_800v

0.4

0.93

30

55.81

0.1

800

1 × 10^{6}

1/1

364

127

700

V55_700v

0.4

0.93

30

55.81

0.1

800

1 × 10^{6}

1/1

400

127

728

b112_1d8

0.6

1.86

15

56.17

0.05

600

1 × 10^{8}

0/1

374

100

450

b_12

0.6

1.86

15

56.17

0.1

750

0

0/1

376

118

600

b11

0.6

1.86

15

56.17

0.1

900

0

0/1

386

120

680

b111_1e4

0.6

1.86

15

56.17

0.1

900

1 × 10^{4}

0/1

374

116

560

b111_1e5

0.6

1.86

15

56.17

0.1

900

1 × 10^{5}

0/1

374

118

600

b111_1e6

0.6

1.86

15

56.17

0.1

900

1 × 10^{6}

0/1

374

117

580

b111_800

0.6

1.86

15

56.17

0.1

800

1 × 10^{6}

0/1

374

117

570

t111

0.5

1.86

15

56.17

0.1

900

1 × 10^{6}

0/1

374

122

550

Vesta_N52

0.5

0.93

32

59.62

0.1

600

5 × 10^{5}

1/1

420

129

680

Vest13so

0.6

2

15

60.40

0.1

500

0

0/1

400

110

564.0

b_1

0.6

1.86

20

74.63

0.1

500

0

0/1

500

133

680.0

Vesta12rn

0.6

2

20

80.25

0.1

500

0

0/1

536

145

728.0

b_3

0.6

1.86

22

82.04

0.1

500

0

0/1

550

133

760

b_4

0.6

1.86

24

89.11

0.1

500

0

0/1

594

158

860

b_2

0.6

1.86

25

93.30

0.1

600

0

0/1

620

162

938

Vesta14rn

0.6

2

24

95.82

0.2

700

0

1/1

630

163

1290.0

Target 2

Ven10052

0.5

0.93

16

29.20

0.1

800

1 × 10^{6}

0/1

204

62

242

Ven100_4

0.4

0.93

17

31.70

0.1

800

1 × 10^{6}

1/1

242

71

400

B4010045

0.5

0.93

20

37.32

0.1

800

1 × 10^{6}

1/1

286

82

508

B40_10052

0.6

0.93

23

42.80

0.1

800

1 × 10^{6}

1/1

300

95

540

B40_1007

0.7

0.93

24

44.56

0.1

800

1 × 10^{6}

0/1

300

89

480

B40_10072

0.7

0.93

24

44.56

0.1

800

1 × 10^{6}

1/1

320

87

550

B40_106

0.6

0.93

27

50.24

0.1

800

1 × 10^{6}

1/1

356

109

610

Secondary Impacts, Target 2

S4010045

0.5

0.93

20

37.32

0.1

800

1 × 10^{6}

1/1

~200

78

560

S4010045L

0.5

0.93

24

44.56

0.1

800

1 × 10^{6}

1/1

~200

90

560

2.2 Acoustic Fluidization Model

[11] A trial model run with a standard SALE-friendly mechanical rock model [Collins et al., 2004] creates a simple bowl-shaped crater, in dramatic contrast to the observed morphology of Rheasilvia. To fit Rheasilvia's actual morphology, we assumed a temporary reduction of friction in the rocks surrounding the growing crater. This assumption is also required to explain complex crater formation on other rocky planetary bodies. The model of acoustic fluidization (AFM) in the “block model” version [Melosh and Ivanov, 1999] has been successfully used to fit complex crater formation on Earth [Ivanov, 2005] and Mars [Parker et al., 2010]. The most important AFM parameter is the decay time that defines how fast normal dry friction is restored after the impact.

[12] The AFM model implemented into SALEB computes the maximum particle velocity of a Lagrangian particle; that is, one that moves with the material (typically, this is a particle at the shock front) and assumes that a fraction (C_{vib}) of this velocity (typically 10%) is converted into a “vibrational velocity,” v_{vib}. Dry friction is decreased depending on the ratio of a “vibrational strength” ρ × c × v_{vib} (ρ is the local density, c is the local sound speed) to the local pressure [Melosh and Ivanov, 1999]. The value of v_{vib} is advected through the grid as a Lagrangian variable and decays in time following a standard (for oscillations with damping) exponential law:

vvib=vvibvmaxexpâˆ’t/Tdec,(1)

where t is the time and T_{dec} is the characteristic decay time.

[13] Despite some previously published discussions about scaling of AFM parameters [Ivanov and Artemieva, 2002], the much lower surface gravity on Vesta precludes the direct use of any scaling from much larger planetary bodies. Here we treat the decay time of AFM as a free parameter and find the model with the best possible reproduction of the crater shape within a set of model runs. The starting value of the AFM decay time is chosen close to the transient cavity collapse in a liquid target [Ivanov and Artemieva, 2002].

3 Transient Crater

[14] Our models follow the standard sequence of impact crater formation: projectile penetration, transient crater growth, and finally crater collapse or modification [Ivanov et al., 2010]. We employ the transient crater radius to compare our results with previously published experiments and models. As in Ivanov et al. [2010], this is defined as the radius where the inner crater surface crosses the initial target surface at the moment when the radial component of the particle velocity changes its direction from outward (ejection) to downward (collapse) [see Ivanov et al., 2010, Figure A7].

[15] For impacts on a flat target surface, we usually use pi-scaling [Schmidt and Housen, 1987] to estimate transient crater diameters. In pi-scaling, the projectile diameter, mass, and velocity (D_{p}, m, and U, respectively) and the target gravity and density (g and ρ) are combined into nondimensional variables defined as π_{2} = (1.61 g D_{p})/U^{2} and π_{D} = D (ρ/m)^{1/3}. Pi-scaling collapses a relatively wide range of data on both experimental and numerically modeled crater sizes into an exponential relation with a constant exponent n: π_{D} = k × π_{2}^{−n}. Fitting this relation to data for impacts at 6.5 km s^{−1} on flat targets composed of simple nonporous Tillotson EOS material with a constant friction coefficient [Elbeshausen et al., 2009; Wuennemann et al., 2010], we find for vertical impacts and a friction coefficient of 0.4 the approximate relation

Ï€D=1.73Ï€2âˆ’0.19.(2)

[16] Figure 1 shows a comparison between pi-scaling results for our Vesta model runs, flat surface data, and previously published models for Mars and the Moon [Ivanov et al., 2010]. Most of the Vesta model runs were conducted for a friction coefficient f ~ 0.5 in the mantle material (Table 1; friction is varied from 0.4 to 0.7). However, our pi-values for the transient crater diameter plot above the flat surface models with f = 0.4. We are presently uncertain about the reason for the slightly larger craters in Vesta model runs. The reason for this (minor) discrepancy may be the acoustic fluidization implementation or the large ratio of the crater diameter to the diameter of the target. Previously published results for larger planetary bodies—Mars and the Moon [Ivanov et al., 2010]—have smaller crater/planet diameter ratios than for Vesta. Moreover, most of the model runs for the Moon and Mars were conducted for hot planetary interiors (i.e., thermally softened dry friction that vanished near the melting point). With all these caveats, we can tentatively assume that with the increase of the impact scale (measured by π_{2}), the transient crater scaling tends to deviate from the dry friction “cold” case toward frictionless (“hydrodynamic”) scaling. Another possible factor is the impact velocity variation—the Moon and Mars modeling use two to fourfold larger impact velocities of 10 to 18 km s^{−1}.

[17] We do not see a significant difference between transient craters for Target 1 and Target 2 cases.

[18] The maximum transient crater depth in the present work is shown in Figure 2. Until the transient crater reaches the iron core, the transient crater depth is approximately proportional to the transient crater diameter (the d_{tc}/D_{tc} ratio is about 0.31). Note that the transient crater depth is reached much earlier in time than the transient crater diameter.

4 Final Crater

[19] The evolution of a transient cavity toward the final complex crater with a central mound is shown in Figure 3 for one of the model runs. Here we use a mapping-like style: “horizontal” distances are measured along a reference surface (here we use an initial spherical shape; fitting the final target shape with an ellipse will change the profile). Vertical distances represent the altitude along radial directions, normal to the preimpact spherical surface of the target. In this representation, the diameter is larger than if it were measured horizontally from rim crest to rim crest.

[20] We study the influence of a temporary frictional strength decrease, as parameterized with the AF model, where the most important parameter is found to be the decay time (see subsection 2.2). We find a systematic change of the impact crater profile for various projectile diameters and impact velocities for the variation of T_{dec} from 400 s to 1400 s. The variation of the decay time shows that the close (but not exact, of course) fit of the model final crater profile and the Dawn imaging and mapping [Jaumann et al., 2012; Schenk et al., 2012] is reached for T_{dec} values in the range of 800 to 900 s, provided that the assumed dry friction coefficient for crushed rocks is about 0.5 ± 0.1. Selected modeled profiles are shown in Figure 4. Model runs shown in Figures 4a and 4b resulted in crater rim-to-rim diameters close to 500 km and a maximum final depth of 22 to 25 km in the ring trough. The model run shown in Figure 4c results in a slightly smaller diameter (470 km), but the depth of 19 km seems to be closer to published data on Rheasilvia's shape [Schenk et al., 2012].

[21] Within the limits of the model, we find that the “best” (in terms of the rim diameter) final crater has a slightly larger maximal crater depth in the ring trough. This indicates that the mechanical model we have used is not perfect. The most likely inadequacy is the lack of initial porosity and shear bulking (dilatancy) which should be added to the model for a future reiteration.

5 Shock Wave Effects in the Target

[22] Once we have determined the model parameters that provide a reasonably correct reproduction of the observed crater shape, we can be more confident of the shock wave parameters that are computed and recorded in each model run. The shock loading pattern is constructed in the standard way: a massless tracer is initially embedded in each grid cell, after which it moves through the grid as a Lagrangian particle. The maximum shock pressure is recorded for each tracer, allowing us to map the shock pressure load onto either the initial or the final target geometry [Pierazzo et al., 1997].

[23] Figure 5 shows typical results for the model impact of a 48 km basaltic asteroid (this projectile diameter is larger than the “best fit”). Due to the relatively small impact velocity (assumed to be 5.5 km s^{−1}), the maximum shock pressure at the contact point is only slightly above 45 GPa (Figure 5a). The ANEOS-basalt model predicts that incipient shock melting occurs at 96 GPa [Pierazzo et al., 2005]; hence, in a vertical impact, there is no chance for voluminous impact melting (in reality, melting may occur for selected minerals and be enhanced with the material porosity). The zone of material displaced during the crater formation is compressed to shock pressures between 3 and ~40 GPa. Consequently, low-grade shock metamorphism (below the stage of bulk melting) is likely to occur in ejecta.

[24] Even the submelting shock loading produces considerable heating of the material. However, most of highly shocked (and heated) material is displaced outward from the impact point. Close to the moment when the transient cavity reaches its maximum depth, this hot material covers the transient crater walls as a relatively thin layer (Figure 5b). The rest of the target is shocked mechanically but not affected thermally except for the central part of the crater—the inverted surface of the transient cavity.

6 Spallation

[25] The most prominent mechanical effect visible in all model runs is a kind of spallation: the near surface layers inside the crust are detached from the main body due to stress wave passage. After some free flight (of comparatively long duration in a small gravity field), spalled material falls back to the surface. The analysis of this effect in the model shown here has been done in a very preliminary manner due to (i) the low spatial resolution of the model and (b) a poor representation of “meridional” fractures given by 2-D modeling.

[26] Spallation in the crust can be documented by any of several variables recorded in each model run. Figure 6 plots the density for selected output frames. The presence of basalt in a computational cell with a density below normal (~2800 kg m^{−3}) indicates that the solid material is separated into fragments which have been launched in a temporary free flight before the gravity returns them to the surface. The rarefied material (also indicated by zero pressure in a computational cell) is seen near the growing crater and near the target's “equator.” Within ~200 s, the spalled layers return back to the main target body.

[27] With an initial spallation velocity of ~30 m s^{−1}, crustal material can rise to ~1 km altitude before falling back. A radius increase of 1 km (or ~0.3% of the target radius) stresses the material well above the elastic limit, mechanically disrupting the crust. Unfortunately, 2-D modeling cannot immediately predict the orientation of fractures in the crust (meridional or equatorial) that would preferentially grow during this uplift. We can only estimate the scale of displacement of the upper crust relative to the center of gravity.

[28] Figure 7 gives some impression of how far the spalled material travels and its final displacement. For this illustration, we plot the final displacement of all tracers initially located near the “equatorial” plane of the target (which assumes the impact point is at the “north pole”). In this plane, the core is slightly deformed: The core/mantle interface moved ~100 m relative to its initial position. A displacement of 100 m for a 100 km core radius implies a strain of the order of 10^{−3}. The mantle's displacement is of the same order of magnitude. In contrast, crustal material (the outer part of the equatorial slice) is displaced “considerably” (i.e., more than one cell size). The upper half of the crust suffers an increase of displacement toward the free surface of up to 10 to 15 km, which is 10 times larger than the cell size. More detailed models are needed to resolve the effect well enough to compare it with the observed groove geometry.

7 Ejected and Escaped Material

[29] The maximum depth of excavation (of material thrown beyond the transient cavity) is shown in Figure 8. For small craters in Target 2, the depth of excavation is about 1.05 of the projectile diameter (at an impact velocity of 5.5 km/s). At the depth of mantle (40 km for Target 2), the depth of excavation stops growing when the excavation flow encounters more competent mantle material. Computed models for Target 1 penetrate into the mantle, and the maximum depth of ejecta grows as 1.0*D_{p} − 10 km.

[30] The relatively smooth dependence of the excavation depth versus the projectile diameter exhibits more scatter when it is plotted as a function of the final rim crater diameter (Figure 9). This is expected, because the expansion of the transient crater rim during its collapse phase is sensitive to variations of the target layering (Target 1 versus Target 2) and other model parameters (Table 1). The possible range of modeled excavation depth for Rheasilvia (here created by a vertical impact) is from 30 to 45 km.

[31] For all model runs, we performed a postprocessing analysis to estimate the amount of material leaving the computational grid with a velocity above Vesta's escape velocity. The velocity of ejected material was evaluated where it reached the limits of the computational grid. The modeled volume of escaped ejecta is shown in Figure 10.

[32] The enhanced escaped volume for a ~40 km projectile into a crust of 40 km thickness (Figure 10a) is the result of interference between the subcrater material flow and the more resistant mantle. For a projectile diameter around 40 km (which gives the best available to date reproduction of the Rheasilvia profile), the amount of ejected material varies from ~6 to ~10 projectile volumes for a vertical impact. Plotting the same data in absolute terms of the escaped material volume versus the modeled rim crater diameter (Figure 10b), we find for the crater of 500 km diameter a possible range of 200,000 to 800,000 km^{3} of escaped material.

[33] Our estimates of mass escaping from a spherical target are limited in 2-D modeling by its restriction to vertical impacts. To put our results in a wider context, we compare our work with recent 3-D modeling that examined a range of impact angles [Artemieva and Shuvalov, 2008; Svetsov, 2011]. For the Moon, these authors estimated the escaped mass of material ejected by asteroid impacts with an impact velocity U of 18 km s^{−1}, Jupiter family comets (U = 25 km s^{−1}), and long periodic comets (U = 55 km s^{−1}) [Artemieva and Shuvalov, 2008]. Corresponding ratios to the escape velocity v_{esc}, U/v_{esc} are 7.5, 10.4, and 22.9. For our assumed average impact velocity on Vesta of ~5.5 km s^{−1} and its escape velocity of the order of 0.36 km s^{−1}, the ratio U/v_{esc} = 15.3. Svetsov [2011] published model estimates for 10 km/s impacts for a wide range of impact conditions. Postponing a discussion of the differences between asteroid and comet impacts as well as the difference between ejection from a flat surface target and from a finite spherical body, we compare the relative escaped mass in our current 2-D modeling and the 3-D modeling by Artemieva and Shuvalov [2008] and Svetsov [2011] in Figure 11. At large U/v_{esc}, the efficiency of ejecta escape decreases only slightly for impact angles from 90° to about 45°, and we can speculate that about 6 to 10 projectile masses escaped after the Rheasilvia impact even for an oblique impact. Moreover, an oblique impact demands a larger projectile to create the same diameter crater, so for a given crater diameter, the escaped target mass may be even larger for an oblique than for a vertical impact. Note that both Artemieva and Shuvalov [2008] and Svetsov [2011] only modeled impacts on flat horizontal target surfaces.

[34] The largest known body of the Vesta asteroid family (“Vestoids”) is about 10 km in diameter. Estimates of the family population vary from an earlier number of 200 [Tanga et al., 1999] to a more complete count of 8000 [Nesvorny et al., 2008] bodies >1 km in diameter. A direct integration of the size frequency distribution for the Vesta family gives a total volume (assuming spherical bodies) ranging from 10,000 to 50,000 km^{3}, or 1/4 to 1/10 of the “best-to-date” Rheasilvia model estimates (Figure 10b). This disparity opens a few lines of speculation:

[35] The large volume is (was) concentrated in small fragments with diameters below 1 to 2 km. This assumes a “steep” continuation of the size frequency distribution (SFD) below 1 km—similar to the case of cratering on a 100 km modeled asteroid [Durda et al., 2007].

[36] Vesta's family was once more massive but has lost many members [Marzari et al., 1996, 1999].

[37] Vesta's family was created by a smaller projectile not connected to the Rheasilvia impact event. This last speculation openly contradicts the observation of “mantle” members in the phase space of Vestoids (see the next subsection).

[38] The recognition of real Vestoids against the background of other basaltic asteroids is still under discussion [Moskovitz et al., 2008; Roig et al., 2008]. Moskovitz et al. [2008, Figure 9] presents the most detailed number of Vestoids, binned by magnitude. The conversion of magnitudes into diameters, using a typical albedo of 0.4, allows us to estimate asteroid diameters and volumes (assuming a spherical shape). The total volume of all observed Vestoids is estimated as about 57,000 km^{3}, starting with a single asteroid in the H = 12.2 bin. Moskovitz et al. [2008] assume a debiased population and get a total volume of 16,000 km^{3} in the integration interval 11.9 < H < 18 (4.8 × 10^{16} kg with a density of 3000 kg m^{−3}). These estimates give the total volume of the modern Vestoids as ~0.5 to ~1.7 V_{proj} (D = 40 km).

[39] If Vestoids were ejected in a single impact event, the size frequency distribution might differ from a simple power law. For a trial, we fit the volume distribution of Vestoids (restored from Moskovitz et al.[ 2008, Figure 9]) with the Weibull distribution [Weibull, 1951], which is often used in explosion fragmentation analysis [Grady and Kipp, 1987; Ivanov and Hartmann, 2007]. The cumulative form of the Weibull SFD is

V>x=Voexpâˆ’x/x*n,(3)

where V_{o} is the total volume of fragments, x* expresses the “characteristic size” in a population of fragments, and n is the so-called Weibull exponent. In its incremental form, the Weibull SFD replicates the R value widely used in impact crater counts

dVdx=x3dNdx=R=V0x*Ã—xx*nâˆ’1expâˆ’xx*n.(4)

[40] Fitting equation (4) to the observational data yields an approximate estimate of V_{o}, x*, and n. One of these possible fits is shown in Figure 12.

[41] The Weibull fit may work for Vestoids better than a power law because it reflects the difference between the collision cascade evolution in the main belt and a single cratering event, resembling in some respects a large-scale surface explosion. If so, modern Vestoids may contain less than one half (or even one third) of Rheasilvia's escaped ejecta volume.

[42] The depletion of Vestoids has been discussed elsewhere [see, e.g., Marzari et al., 1996, 1999; Nesvorny et al., 2008].

8 Mantle Disturbance and Escape of Mantle Material

[43] Astronomical observations show that some of the Vesta family asteroids may represent deep-seated (mantle) material of the differentiated Vesta [Miyamoto and Takeda, 1994; Reddy et al., 2011]. During an early stage of our research [Ivanov et al., 2011], when only Hubble telescopic data were available, we modeled impacts over a wide range of projectile diameters and at a constant impact velocity of 5.5 km s^{−1}. We only examined target models with 20 km thick basaltic crust. Modeling shows (Figure 13) that vertical impacts result in the escape of deep-seated target layers (from below 20 km) only if the basaltic spherical projectile has a diameter above ~50 km. Above this threshold, the volume of escaped deep-seated mantle grows linearly with the projectile diameter.

[44] This result contradicts the idea that mantle Vesta family asteroids are formed by the impact that created the Rheasilvia crater. The best fit of the crater shape to the model (Figure 3) argues in favor of a ~40 km basaltic projectile (assuming a vertical impact). In this case, no material from below ~20 km is ejected faster than the escape velocity, supposing that the crust/mantle material is arranged in spherical shells. This contradiction may be resolved in various ways. Some obvious possibilities include (a) formation of mantle chips from impacts on bodies other than Vesta and (b) the deviation of Vesta's shell geometry from the simple spherical shells assumed in the model. This deviation could be in the form of undulations of the “crust”/mantle boundary. The possible cause of such an undulation may be related to an impact scenario. For example, if previous impacts (seen and unseen large craters older than Rheasilvia) uplifted mantle material into a central mound, the subsequent impact may eject this prelifted material with escape velocities. The other reason for the discrepancy may be the vertical impact in our model—oblique impacts in 3-D modeling [Jutzi et al., 2012] could answer this question.

[45] A preliminary three-layer model was derived by the Dawn team to analyze Vesta's gravity field [Asmar et al., 2012; Raymond et al., 2012]. Examination of the cross section of Vesta [Asmar et al., 2012, Figure 4] suggests that the Dawn “initial” model includes a smooth crust/mantle boundary. Publication of more refined analysis is just beginning [Raymond et al., 2012]. The numerical modeling clearly shows that the crust/mantle boundary is highly deformed by the formation of Rheasilvia. Figure 14 illustrates the prominent uplift of the mantle material for two of the model runs which yield close fits of the crater profile (Figure 3).

9 Veneneia—A Possible Older Impact Crater

[46] The most natural problem to discuss in this context is the presence of the Veneneia crater, which is partially overlapped by the younger Rheasilvia crater [Schenk et al., 2012]. Veneneia's diameter is estimated as about 400 km. We have completed reconnaissance modeling of Veneneia's formation with the same model adjustment to fit the crater rim diameter.

[47] We conducted two model runs in which the projectile size varies only mildly (29.9 to 31.7 km, mass from 4.0 × 10^{16} to 4.8 × 10^{16} kg) with a friction coefficient of 0.4 for the variant with a slightly larger projectile (run Ven10052) and 0.6 (crust)/0.5(mantle) for the slightly smaller projectile (run Ven10052). The main difference is that in the model run Ven100_4, the acoustic fluidization is applied only to the mantle material. These variations resulted from our enduring attempt to attain better similarity of the model crater geometry to the observed one.

[48] The model crater profiles are shown in Figure 15 in comparison to the single available radial profile for Veneneia [Schenk et al., 2012]. The model run with smaller projectile and larger friction results in a crater of ~280 km in diameter and maximum depth in the circular trough of ~29 km (dashed curve in Figure 15), while the second model run produces a crater of ~400 km diameter and depth of ~21 km (solid curve with black dots in Figure 15). The latter crater is more similar in size to Veneneia and has a similar crater wall slope. However, the model does not reproduce the observed shape well. This may be partially explained by the modification of Veneneia's original shape during its geological evolution and, especially, by its partial filling with ejecta from the overlapping younger Rheasilvia.

[49] Despite these concerns about the visible crater shape, the model shows a definite mantle uplift. Even if Vesta's crust is more complicated than our simple layered model, one can definitely say that a complex crater of Veneneia's size should have a central uplift which delivered material from ~40 km depth to the near surface. The horizontal diameter of the deep rock plug is about 60 km, a few kilometers below the crater floor, and 70 km if contoured at 10 km below the crater floor. The visible Rheasilvia center is located 150 to 170 km from the assumed Veneneia center, that is, beyond the diameter of the modeled Veneneia central uplift.

[50] Overlapping the geometry of the escaped material in one of our Rheasilvia model onto the modeled Veneneia profile (rotated at about 30° latitude to approximately reproduce the distance between the two crater centers) allows us to evaluate the interference between these two crater formation events (Figure 16). The escape of some mantle material from the central uplift of the older crater appears probable in this case.

[51] The direct study of the situation shown in Figure 16 is impossible within the limitations of 2-D modeling. To gain some illustrative insights into the problem, we have modeled a second impact into the center of a previously formed crater (no other variants are possible with a 2-D axisymmetric model).

[52] Figure 17 illustrates that, except for the lower angle of the ejecta plume, the second impact produced a structure very similar to the crater caused by the first impact. The main difference is a dramatically larger volume of mantle material. Because of the central uplift, the impact site subsequent to the first impact has hardly any thick crust (40 km initially). Consequently, the fastest ejecta consists of excavated mantle material. Due to the density difference between basaltic projectile and dunitic mantle, the total escaped volume decreases to ~6.34 V_{proj} (diamond sign in Figure 9), but 5 V_{proj} of the mantle material escapes.

[53] Crater formation due to a secondary impact seems to be largely controlled by the geometry of the primary crater. Two model runs with projectiles of 37.3 and 44.6 km diameter (the last two rows in Table 1) produced very similar final craters, shown in Figure 18.

10 Discussion and Conclusion

[54] Within the limits of the present model, as detailed above, the best fit to Rheasilvia impact crater's profile allows us to estimate that it was created by a basaltic projectile size of 37 km in diameter with an impact velocity of 5.5 km s^{−1}. The projectile mass was ~7.8 × 10^{16} kg, and its kinetic energy was ~1.2 × 10^{24} J. However, a strictly vertical impact has very low probability. At the most probable impact angle of 45°, the projectile size would be larger. Unfortunately, detailed modeling of complex crater formation by an oblique impact is presently only beginning [Elbeshausen and Wünnemann, 2011]. Approximate estimates of the effects of oblique impacts may be done by extrapolating published experimental data [Gault and Wedekind, 1978], assuming that the transient cavity volume of an oblique impact at an angle α (measured from the horizon) decreases as sin α. This dependence may slightly vary for targets with different dry friction coefficients [Elbeshausen et al., 2009]. We estimate that a 45° impact of a basaltic projectile with the diameter of 37/sin (45°) ~ 52 km would create the volume equivalent transient cavity as our best fit model.

[55] For these projectile diameters (assuming a weak angle dependence of the escaped mass above 30° above the horizon relative to the projectile mass—see Figure 11 and 3-D modeling by Svetsov [2011]), we compute a total volume of escaping material in the range of 200,000 to 800,000 km^{3}. At a density of 3000 kg m^{−3}, the total mass of the initial Vesta family would be 7 to 25 × 10^{17} kg. This predicted range is dramatically larger than observational estimates of the total Vestoid mass, which lie in the range of 0.5 to 3 × 10^{17} kg. These estimates are derived from astronomical data, supplemented by various downward extrapolations of the complete Vestoid size frequency distribution [Moskovitz et al., 2008, Figure 9]. The idea that the Vesta family is genetically related to the Rheasilvia crater formation thus needs more study of the family's population evolution [Marzari et al., 1996, 1999; Nesvorny et al., 2008].

[56] The analysis of visible surface features related to large impacts (such as Vesta's prominent arcuate grooves) must take into account the almost global spallation of the upper crust (Figure 6) and ~10 km flight of spalled layers outward from the point of impact. However, at the current best resolution of 930 m per cell, no visible features diagnostic of spallation are seen in the areas of observed arcuate grooves.

[57] The crustal thickness is a critical control on the amount of upper mantle material ejected from a spherically layered target. However, the ejection of mantle material is greatly facilitated if previous large impacts on Vesta produced central uplifts, creating spots of thin or vanished crust at the crater center. The partial overlap of Rheasilvia on Veneneia crater may provide a prime example of the interaction of younger and older craters.

[58] The data collected by Dawn and its analysis create a solid basis for tuning future models to align better with observational data. The most straightforward goals for the next step in 2-D modeling are (1) implementation of centripetal acceleration in the model to study the general effects of Vesta's rapid rotation and (2) reiteration of the model with a broader set of materials for the core, crust, and mantle. As the main mechanical events in the impact-generated stress wave occur at pressures well below solid-solid phase transitions in rock-forming minerals, it seems practice to use less complicated equations of state than ANEOS. It appears that Tillotson's approximate equation of state (properly tuned for initial density and compressibility) will give reasonably robust results with a much larger flexibility to meet constraints from the geophysical modeling of Vesta's interior.

Acknowledgments

[59] B.A.I. is supported within the Program 22 of the Russian Academy of Sciences. H.J.M. is supported by PGG grant NNX10AU88G.