Volume 121, Issue 9 p. 1786-1797
Research Article
Free Access

On the possibility of viscoelastic deformation of the large south polar craters and true polar wander on the asteroid Vesta

Saman Karimi

Corresponding Author

Saman Karimi

Department of Earth and Environmental Sciences (MC-186), University of Illinois at Chicago, Chicago, Illinois, USA

Correspondence to: S. Karimi,

[email protected]

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Andrew J. Dombard

Andrew J. Dombard

Department of Earth and Environmental Sciences (MC-186), University of Illinois at Chicago, Chicago, Illinois, USA

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First published: 05 September 2016
Citations: 7

Abstract

The asteroid Vesta, located within the inner asteroid belt, is a differentiated body with a prominent rotational bulge. NASA's Dawn mission revealed the presence of two large, relatively shallow impact craters in the south polar region, one with a high-standing central peak. The shallowness and prominent central peak are reminiscent of large craters on some icy satellites that may have experienced strong topographic relaxation. The location of these basins near the south pole is also unusual and suggests true polar wander, which requires relaxation of the rotational bulge. Thus, we use the finite element method and a viscoelastic rheology to examine the feasibility of relaxation processes operating on Vesta. Given the plausible thermal state of Vesta by the decay of long-lived radioactive elements, we find that the lithosphere is not compliant enough to allow strong relaxation of the large south polar craters, and thus the peculiar morphology is possibly a product of the formation of these large basins at a planetary scale. Additionally, the asteroid has not been warm enough to permit the relaxation of the rotational bulge. Consequently, these craters both happened to form near the south pole, as unlikely as that is.

Key Points

  • The potential for viscoelastic deformation of the large south polar craters and rotational bulge of Vesta is tested
  • The lithosphere of Vesta is not compliant enough to allow strong relaxation of the large south polar craters and rotational bulge
  • Peculiar morphology of Vesta is possibly a product of a planetary scale impact

1 Introduction

The asteroid 4 Vesta is the second most massive asteroid in the solar system and has an average diameter of ~525 km [Russell et al., 2012]. Previous geochemical analyses of howardite-eucrite-diogenite (HED) meteorites have demonstrated that the asteroid is a differentiated body with an iron core and silicate crust and mantle [Coradini et al., 2012; Russell et al., 2012]. The shape of the asteroid is well fit by an ellipsoid with semimajor axes of 285 × 277 × 226 km [Konopliv et al., 2014], and the shortest dimension is aligned with Vesta's spin axis [Russell et al., 2012]. Observations clearly demonstrate a large equatorial bulge for Vesta, which is consistent with a short rotational period of 5.3 h for the asteroid [Reddy et al., 2013] (Figure 1).

Details are in the caption following the image
An image of Vesta captured from near the plane of the asteroid's equator (image date: 11 September 2012). The rotational bulge at the equator and flattening of the poles are evident. NASA/JPL-Caltech/UCLA/MPS/DLR/IDA.

Images taken by the Hubble Space Telescope [Thomas et al., 1997a] revealed a large morphological feature in the south polar region. Thomas et al. [1997a] suggested that the feature is a crater, with a diameter comparable to that of Vesta. More recent images (Figure 2) from the Dawn mission revealed the feature to be two large impact craters [Jaumann et al., 2012], with the younger Rheasilvia (diameter of ~500 km) largely overprinting about half of the older Veneneia (diameter ~400 km). Indeed, these impact basins are the most prominent geologic features on the surface of this asteroid, yet aspects of these craters are unusual.

Details are in the caption following the image
The topographic map of the southern hemisphere of the asteroid Vesta. The two large craters, Rheasilvia and Veneneia, are shown on the map with their centers marked with white and red signs, respectively. Image modified from Jaumann et al. [2012].

Based on Dawn observations, Schenk et al. [2012] determined that the depths of the south polar craters are 19 ± 6 km and 12 ± 2 km for Rheasilvia and Veneneia, respectively. In contrast, an earlier study of Thomas et al. [1997a] used comparative gravity scaling from the Moon to Vesta to show that the average depth of a large south polar crater should be ~27 km. Therefore, a comparison between the initial and current states of these basins shows a difference in their depths. This discrepancy suggests that the basins are currently anomalously shallow, although this could simply be the byproduct of the formation of impact basins on the scale of the asteroid [e.g., Thomas et al., 1997b]. Fujiwara et al. [1993] demonstrated that for impacts whose sizes are comparable to that of the planetary body, the resultant basins are shallower than expected. Furthermore, by applying two-dimensional numerical modeling, Ivanov and Melosh [2013] explored the formation of the two large Vestan impacts, finding generally shallower initial depths of 22–25 km and 21–29 km for Rheasilvia and Veneneia.

In addition to potentially shallowed depths, the Dawn images revealed that Rheasilvia displays a central peak whose height matches and even exceeds the elevation of the surrounding terrain. This central peak is 180 km wide and up to 25 km high, which is almost as high as Olympus Mons on Mars [Russell et al., 2012]. The apparent shallowness of the basins coupled with a high-standing central peak that is reminiscent of strongly relaxed large craters on some satellites of Saturn [Dombard et al., 2007a; White et al., 2013] suggests that the topography of these basins have been modified by long-term viscoelastic effects after their formation.

We have previously explored the mechanics of viscoelastic deformation of large craters (~200–500 km in diameter) on a layered terrestrial body, namely, Mars [Karimi et al., 2016]. While the near surface does not experience viscous flow because of low temperatures, the development of lower crustal flow is feasible under conditions of a thick crust, a sufficiently high background heat flux, and a pressure gradient due to crustal thickness variations associated with the crater. For the right conditions, lower crustal flow can deform the crustal structure and change the topography at the crust-mantle boundary and, by loss of buoyant support, at the surface. The phenomenon of crater evolution due to lower crustal flow is very sensitive to the temperature structure of a planetary body, and studies have linked the evolution of large craters to the thermal history of these worlds [e.g., Mohit and Phillips, 2006, 2007; Karimi et al., 2016]. Consequently, in the first part of this study, we investigate the possibility of the evolution of the large south polar craters of Vesta by modeling viscoelastic deformation and thus aim to place a constraint on the thermal state of the asteroid. For this explanation to be viable, relaxation needs to result in kilometer-scale vertical displacement of the surface.

Along with their unusual shapes, the locations of these basins in the vicinity of Vesta's south pole are also peculiar. Rheasilvia and Veneneia impacts are centered 15° and 38° from the south pole of Vesta. We calculate the area of the spherical cap poleward of these latitudes (in which these large impacts occurred) and compared it with the total surface area of Vesta. Our calculations show that the ratio of the spherical cap over the total area of Vesta is about 1/10. Considering this small area of the south polar region relative to the global area of the asteroid, the chance of occurrence of an impact at such a high latitude is slim (~10%). Thus, the probability of two large impacts in this region is highly unlikely (~1%). It is conceivable, however, that these basins formed at statistically more favorable lower latitudes.

A large load on the surface of a rotating planetary body can change the orientation of spinning by generating a long-wavelength geoid anomaly. A large positive geoid anomaly tends to reorient toward the equator, whereas a large negative geoid anomaly tends to reorient toward poles. This true polar wander modifies the orientation of the surface relative to the spin axis. In the case of Vesta, the presence of these superimposed large south polar basins produces a long-wavelength negative geoid anomaly. As a result, this integrated mass deficit applies a torque to the lithosphere that might reorient the body relative to the spin axis (Figure 3), thereby placing these basins near the south pole [e.g., Thomas et al., 1997b; Kattoum and Dombard, 2009].

Details are in the caption following the image
A schematic that demonstrates various stages of our hypothesis related to the potential true polar wander for the asteroid Vesta.

In order for this scenario to happen, a deformable lithosphere is required [e.g., Matsuyama et al., 2006] to allow for the near-complete relaxation of the old rotational bulge and concurrent formation of the new (current) rotational bulge. Without a deformable lithosphere, large degrees of true polar wander would be precluded for Vesta, lest the highly oblate shape of Vesta be significantly tilted from the rotational axis. This deformable lithosphere should be relatively thin, and that implies a warm asteroid. Thus, in the second part of this study, we explore the potential for relaxation of the rotational bulge, which would be a critical step for true polar wander, again aiming to provide constraints on the thermal state of the asteroid after the two large impacts. Again, for this explanation to be viable, the bulge must relax almost completely.

2 Methods

In our study, we use the commercially available Marc-Mentat finite element package (http://www.mscsoftware.com), which we have used many times previously [Dombard and McKinnon, 2006; Dombard et al., 2007a, 2007b; Dombard and Phillips, 2010; Dombard and Schenk, 2013; Karimi and Dombard, 2014; Karimi, 2015; Karimi et al., 2016]. We discuss the setup of the simulations below (for more details, see Karimi et al. [2016]). Table 1 lists the various applied values for the simulations.

Table 1. Parameters of Our Finite Element Simulations
Parameters Values
Geometry
Crater radius 250 km
Initial total depth of the crater 22, 25 km
Rim height 7 km
Initial topographic relief of the crust-mantle boundary 25–45 km
Crustal thickness 25–80 km
Mesh width (for the planar mesh) 750 km
Mesh depth (for the planar mesh) 750 km
Thermal Analysis
Surface temperature 180 K
Thermal conductivity, crust 1–2.5 W m−1 K−1
Thermal conductivity, mantle 4 W m−1 K−1
Applied background heat flux 3.5–200 mW m−2
Mechanical Analysis
Gravitational acceleration 0.25 m s−2
Density, crust 2800–2900 kg m−3
Density, mantle 3200–3300 kg m−3
Poisson's ratio ~0.5a
Young's modulus, crust 52 GPaa
Young's modulus, mantle 112 GPaa
Crustal creep rheology Hydrous/anhydrous diabaseb, c
Mantle creep rheology Hydrous/anhydrous olivined
Minimum viscosity 1021 Pa s
  • a Turcotte and Schubert [2014].
  • b Caristan [1982].
  • c Mackwell et al. [1998].
  • d Karato and Wu [1993].

2.1 Evolution of the South Polar Craters

Our simulations begin with building the finite element meshes, which possess 103–104 elements; we have tested to ensure that our results are not sensitive to how the meshes are drawn. We have two approaches to model the possible deformation of the large south polar craters: (i) a planar mesh and (ii) a spherical mesh (Figure 4). The planar geometry is simpler to implement, although it ignores planetary curvature. Consequently, we explore the parameter space primarily with the planar geometry and use the application of the spherical geometry to test the effects of curvature.

Details are in the caption following the image
Two example schematics of our planar and spherical finite element meshes. (a) The planar mesh is three crater radii deep and wide, while (b) the spherical mesh is a full hemisphere. Both spherical and planar meshes are axisymmetric and have surface and subsurface topography, and both meshes are finer underneath the crater depression and coarser farther from the crater. These meshes are notional; the actual meshes use far more elements. The arrows show the axes of symmetry, and the dotted line shows the spherical surface. In both cases of Figures 4a and 4b, the crustal thickness is 80 km. The “rollers” demonstrate free-slip boundary condition, and “triangles” stand for fixed boundary condition (see sections 4 for boundary conditions).

In the first approach, we apply a two-layer axisymmetric geometry, with a crust on top of the mantle, both with a viscoelastic rheology. In order to eliminate the effect of the far edge boundaries on the process of crater deformation, we set the size of the planar mesh to be three crater radii deep and wide; tests in Karimi et al. [2016] confirmed that these distances are sufficiently far from the crater to affect negligibly the deformation process. Although the crustal thickness of the asteroid is not precisely constrained, many studies have determined the crustal thickness to be in the range of 20–80 km [Zuber et al., 2011; Jutzi et al., 2013; Konopliv et al., 2014]. Due to its role in controlling the flow channel width and in changing the basal crust temperatures, the crustal thickness can contribute to the lateral flow of the lower crustal material and to the reduction of topography. Thus, in our study, we test various crustal thickness values in the mentioned range. For the initial crater shape, we use a fourth-order polynomial for the crater depression and an inverse third power law to define the shape of the rim and ejecta blanket [cf. Dombard and McKinnon, 2006; Karimi et al., 2016]. Crater depth and rim height (listed in Table 1) are defined in other studies [e.g. Kattoum and Dombard, 2009; Schenk et al., 2012].

In addition to the surface topography, we also model an initially uplifted crust-mantle boundary, because such uplift is a characteristic of large craters on terrestrial worlds [e.g., Melosh, 1989; Neumann et al., 2004]. In our finite element simulations, the mantle is not exposed on the surface, following the studies of McFadden et al. [2015] and Ammannito et al. [2013] who demonstrated that mantle materials (e.g., olivine) exist on the surface of Vesta although in the northern hemisphere and not in the south polar region. The subsurface topography (i.e., mantle uplift) follows the shape of a Gaussian-like curve, is narrower than the diameter of the crater, and lacks the rim and ejecta blanket topography (Figure 5; see Karimi et al. [2016]). This shape for the crust-mantle boundary is consistent with the initial shape found for large craters on terrestrial worlds. We consider two cases of (i) central isostatic compensation and (ii) isostatic undercompensation. The first case (i) explores the idea that the central part of this uplift is in a full isostatic compensation with the center of the surface depression, whereas the second case (ii) explores the idea that this uplift amplitude is a fraction of that of a full isostatic compensation state (e.g., 20%).

Details are in the caption following the image
Crustal profile of the south polar craters used in the simulations. Crustal thickness is 80 km, while depth and rim height are 15 and 7 km, respectively.

In the second approach using a spherical mesh, we apply a similar axisymmetric geometry at the surface and subsurface and wrap it around a sphere, effectively creating a full hemisphere (see Figure 4b). The rest of the geometric and material properties of the spherical mesh such as crustal thickness, crust and mantle rheology, and the state of isostatic compensation are similar to those of the planar mesh. We only model the crust and mantle and do not simulate the core. Because of its separation from the region where the lower crustal flow might occur, the core-mantle boundary is fixed in our simulations.

2.2 Relaxation of the Rotational Bulge

A critical step for true polar wander to occur is that Vesta's old rotational bulge must relax contemporaneously with growth of a new bulge. In this part, we explore whether the shape of Vesta can suitably deform by examining the possibility of the rotational bulge relaxation. We use an axisymmetric oblate spheroidal mesh with symmetry across the equator and with the equatorial radius 60 km larger than the polar radius (Figure 6), and we thus simulate a body that relaxes under its own weight (i.e., transitions from this initial oblate shape to a more spherical one under the force of gravity). The basic geometries of the spherical meshes in sections 2 and 3 are almost identical with two simple differences: (1) there is no topography related to the large craters at the surface and subsurface when modeling the relaxation of rotational bulge and (2) there is no bulge when modeling the crater relaxation. Because we seek to understand whether Vesta's lithosphere can relax a bulge, the details of the shape of the core are secondary, and we can assume a spherical core with a radius of 110 km [Russell et al., 2012]. Crustal thickness is constant, although we test a couple of plausible values for its thickness (25 and 45 km), and consequently, the values for the thickness of the mantle vary from pole to equator.

Details are in the caption following the image
An example of a two-layer spherical mesh used for testing the potential relaxation of a rotational bulge. The crust (gray) is sitting on top of the mantle. The equatorial radius of 285 km is 60 km larger than the polar radius. The core has the constant radius of 110 km, and the crustal thickness is constant. The arrow shows the axis of symmetry, while rollers and triangles demonstrate boundary conditions (see sections 4).

2.3 Thermal and Mechanical Simulations

We follow the basic methodology of Karimi et al. [2016]. The temperature structure, which largely controls the material viscosity, plays a crucial role in the thickness of a lithosphere and in its possible deformation [e.g., Kusznir and Park, 1987; Nimmo and Stevenson, 2001; Burov, 2011]. Here we perform a thermal simulation that finds a steady state equilibrium between the average surface temperature (nominally constant at 180 K), zero heat flux from the sides of the mesh, and a specified basal heat flux. For the case of simulating the south polar craters' relaxation, we also consider a proxy for the effects of remnant impact heat by locking the temperature of the uplifted crust-mantle boundary to the value this boundary has away from the crater. Karimi et al. [2016] demonstrated the role of the impact heat on the crater relaxation process, who also found that the thermal anomaly for a large crater diffuses in 10 Myr time scales. Thus, because of the billion year age difference between these two craters [Marchi et al., 2012; Schenk et al., 2012; Schmedemann et al., 2014], we do not consider the effect of the Veneneia thermal anomaly on the potential relaxation of Rheasilvia.

Since the thermal conductivity of Vesta's crust has a significant role in determining the temperature structure, we test a range of conductivity values of 1–2.5 W m−1 K−1, with lower values likely arising from impact-generated porosity that could extend to a significant depth in the crust such as has been found for the Moon [Wieczorek et al., 2013; Han et al., 2014]. Once determined, this steady state thermal solution, which determines the thermal state of the body, is input into the mechanical simulation.

To run a thermal simulation, an understanding of the thermal budget of the asteroid is required. We calculate the surface heat flow of Vesta produced by the decay of long-lived radioactive elements in the crust and mantle assuming chondritic abundances [see Turcotte and Schubert, 2014, Section 4.5]. Various crater-counting studies have estimated the age of the south polar craters in the range of 1 to 3.8 Gyr [Marchi et al., 2012; Schenk et al., 2012; Schmedemann et al., 2014], values that we use in our calculation of the surface heat flux. We therefore predict an upper limit of the surface heat flow as 3.5 mW m−2. In order to examine a broader range of the possibilities, we also consider a case with the concentration of heat-generating elements being twice chondritic. Thus, the largest surface heat flow we consider reasonable is 7 mW m−2. These values are applied to the base of the planar meshes but need further adjustment for the spherical meshes. When heat propagates through a thick shell, surface heat flow decreases with decreasing depth because of geometric spreading. Thus, in the case of a spherical mesh, the applied basal heat flux at the core-mantle boundary is larger than the surface heat flux by the factor of (r1/r2)2 in which r1 and r2 are the radial distances from the center of the asteroid to the surface and core-mantle boundary, respectively.

In this study, we run a steady state thermal simulation to modulate the temperature structure. The actual thermal state, however, is not a steady state and changes through time due to secular cooling of Vesta and, for the case of the impact craters, diffusion of impact heat. Karimi et al. [2016] demonstrated that the difference between the final topography of a crater after applying both a steady state and a transient thermal structure is insignificant, because the bulk of the deformation occurs in the first few tens of millions of years before substantial dissipation of the thermal anomaly. This phenomenon arises because in these earliest times postimpact, the lateral pressure gradients are at a maximum and only decay with lower crustal flow. Consequently, we simulate the potential viscoelastic deformation of the south polar craters over a time frame of 100 Myr, a long enough period to cover the substantial deformation at the surface and subsurface (if any). Karimi et al. [2016] showed that 100 Myr is a long enough time to capture the appreciable deformation at the surface and subsurface caused by the lower crustal flow, and similar tests for Vesta lead to the same conclusion.

For the mechanical simulations, the nodes at the bottom of the mesh are fixed, whereas the nodes on the two sides of the mesh have free-slip boundary conditions. A downward gravitational acceleration of 0.25 m s−2, equal with that at the surface of Vesta, is applied to all the mesh elements. Because we are concerned with buoyancy loads from the near surface (i.e., the surface and crust-mantle boundary), we ignore the variations in the strength of gravity that arise with great depth.

2.4 Material Properties

For both suites of simulations, we apply a viscoelastic rheology. For the elastic behavior, we use parameters typical for the basaltic crusts and peridotitic mantles of rocky bodies [e.g., Turcotte and Schubert, 2014]. The nominal values of the Young's moduli for the crust and mantle are 65 and 140 GPa, respectively, with a Poisson's ratio of 0.25 for both. With this value of Poisson's ratio, the materials are compressible and will undergo gravitational self-compaction; to combat this phenomenon, we increase the Poisson's ratio very close to the incompressibility limit (0.5). The elastic response is primarily one of lithospheric flexure; we thus scale the nominal Young's moduli by a factor of 0.8 in order to keep the flexural rigidity constant (for validation of this approach, see Dombard et al. [2007b] and Karimi et al. [2016]). For the viscous behavior, we test our models with the viscous creep parameters for both hydrous and anhydrous crust and mantle [Caristan, 1982; Karato and Wu, 1993; Mackwell et al., 1998]. The viscosity spans many orders of magnitude, with higher values in the lithosphere (> 1030 Pa s at the surface) and lower values beneath (<1021 Pa s in the deep mantle). The evolution of the lithosphere controls the time scales of deformation, but the time steps of the simulations are controlled by the warmer interior that simply needs to move and to accommodate the deforming lithosphere. Thus, we limit the minimum viscosity in the mesh to 1021 Pa s [see Karimi et al., 2016] to keep the simulation run time reasonable (up to 1 week). Test simulations using a lower minimum viscosity yielded very similar results.

In our simulations, the thermal conductivities of the crust and mantle are 1–2.5 and 4 W m−1 K−1, respectively. Studies have explored the thermal properties of the Vestan crust by working on the HED classes of meteorites [e.g. Opeil et al., 2012; Capria et al., 2014]. Thermal conductivities of most of the samples are determined to be in the range of ~1 to 3 W m−1 K−1. In addition, several studies have estimated the density values of the crust and mantle in the range of 2800–2900 kg m−3 and 3200–3300 kg m−3, respectively [e.g., Zuber et al., 2011; Russell et al., 2012; Jutzi et al., 2013], though we will explore a wider range of values in order to test a large density contrast between the crust and mantle. Doing so leads to widening the flow channel in the lower crust by reducing the initial mantle uplift. In turn, a larger amount of the lower crust can flow, thus increasing the probability of topography relaxation. Exploring higher density contrasts enables us to maximize the potential of lower crustal flow in changing the topography of the surface and subsurface.

3 Results

3.1 Evolution of South Polar Craters

We show simulation results for a range of cases and Table 2 lists the details of the studied cases. Case 1 shows our nominal situation of a thick crust (80 km) with an isostatic compensated structure, surface temperature of 180 K, crustal thermal conductivity of 2.5 W m−1 K−1, and a background heat flux of 3.5 mW m−2. To avoid exposure of the underlying mantle, a full isostatic compensated structure requires a large crustal thickness. For this case, vertical displacements are minimal, far less than 1 km needed for relaxation to be a viable explanation for the unusual shapes of these craters (Figures 7 and 8). We can reduce crustal thickness somewhat before exposure of the mantle; however, simulations testing thinner crusts show less deformation because of lower temperatures at shallower depths for a given heat flow.

Details are in the caption following the image
Topographic profile of Rheasilvia and simulated results for Case 1 and Case 2. Case 1 shows a thick crust (80 km) with an isostatic compensated structure, surface temperature of 180 K, crustal thermal conductivity of 2.5 W m−1 K−1, and a background heat flux of 3.5 mW m−2. Case 2 is identical to Case 1 except that the applied background heat flux is 30 mW m−2, which is ~8 times larger than the plausible thermal budget of Vesta.
Details are in the caption following the image
Maximum vertical simulated displacements for various cases (Cases 1–8) in the central part of the crater at the surface.

In short, the thermal states arising from the above conditions are simply not warm enough to permit lower crustal flow. Appreciable deformation, consistent with a relaxation origin for the unusual characteristics of these basins, is only realized by pushing the input parameters to their extremes or beyond. Case 2 is identical to Case 1, except that the applied background heat flux is 30 mW m−2, which requires abundances of radiogenic nuclides to be more than 8 times chondritic. Here vertical displacements do exceed 1 km (Figures 7 and 8).

We apply a proxy for remnant impact heat by adjusting the uplifted mantle temperature to that of mantle far from the impact [cf. Karimi et al., 2016]. The presence of remnant impact heat lowers the local viscosity underneath the craters, which facilitates the development of lower crustal flow. In Case 1, the temperature of the crust-mantle boundary beneath the impact is fixed at the temperature of crust-mantle boundary far from impact (see Table 2). Case 3 is identical to Case 1, except that we double the uplifted mantle temperature. The results show that despite the changes made in Case 3, the crustal deformation is not substantial (Figure 8). The local viscosity may be lower, but the region surrounding this fairly restricted pocket is too stiff to permit appreciable flow.

Table 2. The Details of the Cases (1–8) Are Listed
Case # Isostatic Compensation (%) Crustal Thickness (km) Tsa (K) Tmb (K) Kc (W m−1 K−1) qd (mW m−2)
1 100 80 180 ~300 2.5 3.5
2 100 80 180 1140 2.5 30
3 100 80 180 ~700 2.5 3.5
4 20 45 180 450 2.5 15
5 20 45 180 720 2.5 30
6 20 45 180 630 1.5 15
7 20 45 180 855 1 15
8 20 45 240 510 2.5 15
  • a Surface temperature.
  • b Crust-mantle temperature (proxy for impact heat).
  • c Crustal thermal conductivity.
  • d Background heat flux.

Our method is a generalized approach and incorporates both the effects of lower crustal flow and isostatic rebound. In fact, a simulation in which the structure is isostatically undercompensated can enable us to evaluate the role of isostatic rebound in the deformation of the south polar craters. Thus, we investigate a case in which the structure is isostatically undercompensated. For the smaller values of the crustal thickness (e.g., 25 and 45 km), we consider cases that the structure is isostatically undercompensated (e.g., 20% isostatic compensation). In Case 4, an isostatic undercompensated structure with crustal thickness of 45 km is considered. In this case, the thermal conductivity of the crust is 2.5 W m−1 K−1, surface temperature is 180 K, and background heat flux is 15 mW m−2. Case 5 is identical to Case 4, except that the applied background heat flux is 30 mW m−2. Our results show that as expected, in Case 5 under a warmer temperature structure, the structure goes through more deformation than Case 4 (Figure 8).

We test our simulations with smaller values of the crustal thermal conductivity (up to 2.5 times smaller than the thermal conductivity of a rocky crust). Cases 6 and 7 are identical to Case 4, except that the crustal thermal conductivities are 1.5 and 1 W m−1 K−1. Applying a reduced value of thermal conductivity noticeably enhances the lower crustal flow by increasing the temperature in the lower crust. This phenomenon, however, is only possible under very high background heat fluxes (15 mW m−2, Figure 8), which is higher than allowed by the thermal budget of asteroid Vesta.

We also test the effects of the elevated surface temperature on crater relaxation on Vesta. This test is motivated by the possibility of a higher effective surface temperature under a thin insulating regolith. Case 8 is identical to Case 4, except that the surface temperature is 240 K. The enhancement of relaxation is minimal. Similarly, we have tested the role of the surface temperature change in relaxation process of Martian craters [Karimi et al., 2016] and showed that a change of ±30 K does not have significant effect on crater relaxation.

For all the results listed in Figure 8, we have applied the ductile rheological parameters for rocks with Earth levels of hydration. In these simulations, the crust is considered to be Maryland diabase—a fine-grain basaltic rock [Caristan, 1982], and the mantle is olivine [Karato and Wu, 1993]. Whether Vesta's interior is hydrous to that degree is equivocal (see below), so we also test our model with parameters for anhydrous materials. For the anhydrous crust, we use dry Maryland diabase and dry Columbia diabase; these samples were heated up to 1000°C for 50 h to be nominally water free [Mackwell et al., 1998]. For the mantle, we use an anhydrous olivine that follows the flow rule of the water-free sample from Karato and Wu [1993]. These flow laws are significantly stiffer, so the displacement is restricted, and relaxation is reduced more than 4 times (and thus are not shown in the figures).

The above results are derived from simulations using the planar mesh. Since the size of the asteroid and the south polar craters are comparable, we also ran a series of simulations in which we use a spherical finite element mesh. As mentioned earlier, the spherical mesh takes into account the curvature of the asteroid. In contrast to the planar cases, these simulations using a spherical geometry require a still higher surface heat flow, because of the extra resistance afforded by membrane support of the lithosphere. Specifically, the ~1 km uplift that needed 25 mW m−2 in the planar case needs slightly more than 35 mW m−2 in the spherical geometry.

In summary, we only see appreciable deformation at the surface of the craters (vertical changes on the scale of kilometers) when applying a hydrous rheology, a relatively thick crust, an isostatic undercompensated structure, and an unreasonably high heat flux that exceeds the thermal budget of asteroid Vesta.

3.2 Possibility of True Polar Wander

Our results show that under plausible thermal states (surface heat flux of 3.5–7 mW m−2), relaxation of the equatorial bulge does not occur. Figure 9 demonstrates the degree of relaxation of the rotational bulge versus the applied heat flux. These results are for simulations run over 100 Myr; simulations over 200 Myr are negligibly different. The model has a crustal thickness of 45 km and an initial flattening of 21% (cf. Figure 4). As is evident, the heat flux required to produce relaxation is implausibly high; a heat flux of ~ 200 mW m−2 only experiences ~30% relaxation, far shy of the near-complete relaxation needed for the hypothesized true polar wander scenario. Our simulations also show that the crustal thickness of 25 km does not noticeably change the results. Considering a low value for the thermal conductivity of the crust (e.g., 1 W m−1 K−1) increases the relaxation rate by a factor of ~2–3, yet even with this enhancement in the efficiency of relaxation, the required heat flux exceeds what is probable for Vesta. Again, these results assume a hydrous rheology; application of an anhydrous rheology only exacerbates the heat flow requirements.

Details are in the caption following the image
A demonstration of rotational bulge relaxation (percent) versus applied heat flux.

4 Discussion

The inefficient relaxation of Rheasilvia and Veneneia under plausible conditions suggests that the unexpectedly shallow depths and high-standing central peak cannot be a consequence of postimpact viscoelastic deformation. The thermal state of the asteroid has the most important role in the control of lithospheric deformation, and Vesta's thermal state has been too cool over the last 4 Gyr to permit change of the surface topography. Instead, these anomalous aspects of the craters' shapes could be the product of impact on a planetary scale [e.g., Fujiwara et al., 1993; Ivanov and Melosh, 2013].

Our investigation on the potential relaxation of the rotational bulge also shows that the Vestan lithosphere is incapable of a high degree of relaxation required for true polar wander and moving the basins toward the pole, because the requisite heat flows are 1–2 orders of magnitude higher than what is plausible for Vesta. As a result of the relatively cool temperature structure of Vesta's crust and mantle, and a corresponding high viscosity, the lithosphere supporting a bulge is too rigid to relax. This notion means that the formation of the observed rotational bulge was only likely possible in early stages of the solar system, when Vesta was warmer because of heat of accretion and intense heating of the asteroid in the presence of the short-lived radioactive elements [cf., Fu et al., 2014], notably 26Al and 60Fe (with pervasive melting and basaltic crustal formation). Consequently, the phenomenon of the true polar wander following the occurrence of two large impacts in a nonpolar region is an untenable scenario for Vesta. As unlikely as it is, it would appear that these two large craters both formed near the south pole.

It is worth exploring whether there are any avenues that could permit the viscoelastic deformation needed for both processes. To that end, we run test simulations that span a range of density values of the crust and mantle, predicting that a larger density contrast across the crust-mantle boundary would facilitate more deformation (see section 5). Our simulations show, however, that reasonable changes in density value do not affect the results of relaxation process remarkably (less than 5%).

The thermal conductivity of the crust has a significant effect on the temperature structure of the asteroid. Thermal conductivity and porosity have a direct relationship, such that an increase in porosity leads to a decrease in the thermal conductivity. Results derived from Dawn mission data indicate that the porosity of the surface is ~6% [Russell et al., 2013]. Increasing pressure and temperatures with depth will eventually squeeze out this porosity, so a lower thermal conductivity is likely only applicable for the shallow regions of the crust (e.g., 10 MPa pressures are achieved ~15 km deep in Vesta). But even applying a reduced conductivity throughout the entire crust, our simulations show that heat flows need to exceed 10 mW m−2 for the large craters to have kilometer-scale relaxation. Such high-heat flows are improbable for Vesta over the last 4 Gyr. Additionally, relaxation of the bulge remains improbable.

We also test surface temperature, as small values for the thermal inertia for Vesta (~30 J m−2 s−0.5 K−1) suggest that the crust is covered by a fine-grain regolith [Capria et al., 2014] that would act as insulating blanket and increase the effective surface temperature. Indeed, Damptz and Dombard [2011] showed the importance of the surface temperature in lithospheric deformation of icy satellites. In contrast, Karimi et al. [2016] tested the effects of surface temperature on the deformation of Martian craters, and their results showed an insignificant role of temperature changes of ± 30 K on the final topography of the surface and subsurface. Here we test the effects of the surface temperature on lithospheric deformation, with our simulations demonstrating that a 60 K increase in the surface temperature (Case 8) does not appreciably change the final topography at the surface and subsurface, even when high basal heat flux (15 mW m−2) is applied (Figure 8).

In simulating the potential relaxation of south polar craters and rotational bulge, we apply various crustal thickness values. Karimi et al. [2016] showed that as the crustal thickness increases the temperature at the base of the crust elevates and thus drops the viscosity. Consequently, there is enhanced lower crustal flow and subsequent relaxation. This study, similarly, shows that as the Vestan crustal thickness increases, there is more deformation associated with the lower crustal flow; however, requisite heat flow values are still implausibly high for relaxation to explain the topographic characteristics of these craters.

Our simulations require an understanding of the viscous rheological behavior of Vesta's crust and mantle. Vesta has been considered as a dry (anhydrous) body for a long time. New studies, however, indicate that hydrated minerals might have existed within Vesta [e.g., Scully et al., 2015]. The above statements necessitate the application of both dry and wet rheologies. As mentioned previously, appreciable deformation starts to occur only after applying the unreasonably high heat fluxes and a wet rheology. When we apply a dry rheology [Karato and Wu, 1993; Mackwell et al., 1998], virtually no deformation results at the surface and subsurface, even with the application of very high heat fluxes. This result is an expected outcome since a material with a dry viscous rheology (as opposed to wet rheology) is harder to deform.

5 Conclusions

Using plausible thermal conditions arising from the decay of the long-lived radioactive elements within the crust and mantle, we test the possibility of viscoelastic deformation of Vesta's lithosphere. We find that it is unlikely for the large central peak and the anomalously shallow depths of the south polar craters to have formed by viscoelastic deformation due to lower crustal flow. Instead, these characteristics are likely the products of impact in a planetary scale. By testing the possibility of relaxation of the rotational bulge, we also examine the feasibility of true polar wander for the asteroid Vesta after the formation of Rheasilvia and Veneneia, possibly driving these craters toward the south pole. Our study reveals that the asteroid has been too cool over the bulk of its lifetime to allow relaxation of the rotational bulge. The corollary is that Vesta's large rotational bulge is primordial feature dating from its earliest history.

Acknowledgments

This study was supported by NASA grant NNX12AO38G to A.J.D. All data presented in this manuscript are properly cited and referred to in the reference list. Also, the data associated with Figures 7-9 are provided in the supplementary information. We thank the Associate Editor and two anonymous reviewers whose comments/suggestions improved the manuscript.