GEOPHYSICAL RESEARCH LETTERS, VOL. 38, L14205, doi:10.1029/2011GL047967, 2011
Reorientation of Vesta: Gravity and tectonic predictions I. Matsuyama1 and F. Nimmo2 Received 28 April 2011; revised 3 June 2011; accepted 11 June 2011; published 29 July 2011.
[ 1 ] Vesta’s large southern hemisphere impact basin is likely to have caused reorientation. However, because the basin is not centred at the south pole, Vesta likely also has a remnant rotational figure. Reorientation of ≈6° is predicted to have occurred based on the dimensions of the basin. Existing measurements of Vesta’s shape are consistent with ≈20° or less reorientation, and ≈20% or less despinning. Both the remnant rotational figure and the basin contribute to the degree‐2 gravity coefficients, which will be measured by the Dawn mission and will provide a test of the reorientation hypothesis. Reorientation and despinning also give rise to stresses. Vesta’s stress state is likely to be dominated by isotropic contraction due to cooling (which does not affect the gravity coefficients). However, the orientation of the resulting thrust features will be controlled by the amount of reorientation and despinning, providing another observational test. Citation: Matsuyama, I., and F. Nimmo
Here, we quantify the expected TPW taking into account the rotational stability provided by the elastic lithosphere and the effect of despinning [Dobrovolskis and Burns, 1984]. These processes have implications for the global gravity field and tectonic pattern, and we make predictions in anticipation of forthcoming observations by the Dawn mission [Russell et al., 2007].
2. Impact Driven Reorientation [4] The average depth and rim heights of the large impact crater near the south pole are ∼6 and ∼7 km respectively [Thomas et al., 1997a]. We assume axisymmetric crater floor and rim profiles in spherical coordinates (r, , ) given by [Melosh, 1989, chap. 6] 8 2 > < R hc 1 c r¼ 3 > :R þ h r c
(2011), Reorientation of Vesta: Gravity and tectonic predictions, Geophys. Res. Lett., 38, L14205, doi:10.1029/2011GL047967.
1. Introduction [2] The shape of Vesta has been characterized using Hubble Space Telescope (HST) limb profile observations [Thomas et al., 1997a, 1997b]. The most dramatic feature is a large impact crater near the south pole, with a radius comparable to the mean radius of Vesta. Thomas et al. [1997a], Kattoum and Dombard [2009], and Schmidt and Moore [2010] all noted that the mass redistribution associated with this impact could have caused a reorientation of the rotation pole relative to the surface, or true polar wander (TPW). However, these previous studies did not quantify the expected reorientation, nor the observable consequences. [3] Reflectance spectra suggest a basaltic magma composition for the surface of Vesta [McCord et al., 1970]. Thus, Vesta appears to have melted and differentiated, most likely due to 26Al decay [Bizzarro et al., 2005]. Furthermore, tungsten isotope anomalies [Kleine et al., 2004] and siderophile element depletion patterns [Ruzicka et al., 1997] in howardite‐eucrite‐diogenite (HED) meteorites, which are thought to be fragments of Vesta, provide evidence for core formation. An elastic lithosphere will develop as Vesta cools [e.g., Ghosh and McSween, 1998], and is probably required to support the topography of the large impact crater. The presence of an elastic lithosphere stabilizes the rotation pole by preserving a record of the original rotational figure before the impact [Willemann, 1984; Matsuyama et al., 2006]. 1
Department of Earth and Planetary Science, University of California, Berkeley, California, USA. 2 Department of Earth and Planetary Sciences, University of California, Santa Cruz, California, USA. Copyright 2011 by the American Geophysical Union. 0094‐8276/11/2011GL047967
if
c;
if >
c
ð1Þ
respectively, where we assume a reference frame in which the crater center is at the north pole ( = 0). Here, R is the mean radius of Vesta, hc is the maximum depth of the crater, hr is the maximum rim height, and yc is the angular radius of the crater. Ignoring mass loss during the impact, mass conservation requires R hr ¼ hc
0
c
2 d sin 1 c ; 3 R r d sin c c
ð2Þ
where yr is the outer angular radius of the rim. We assume R = 258 km [Thomas et al., 1997b], hc = 10 km, yc = 60°, and yr = 120° in our fiducial model. [5] The mass redistribution associated with a large impact can cause significant TPW [Melosh, 1975; Nimmo and Matsuyama, 2007]. Because the rotation pole is stabilized by the remnant rotational figure preserved by the elastic lithosphere, it is useful to define a dimensionless load size
3 Z r 1 þ k2L 6G Q d sin P2 ðcos Þ hr k2T * k2T w2i R c c " 2 #! Z c ; d sin P2 ðcos Þ 1 hc 0
c
ð3Þ
given by the ratio between the degree‐2 gravity contributions of the load and the remnant rotational figure [Willemann, 1984; Matsuyama et al., 2006]. Here, G is the gravitational constant, wi is the rotation rate at the time the remnant rotational figure is established, r is the density of the excavated material, and P2(cos ) is a Legendre polynomial. The first term on the right‐hand‐side in equation (3)
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fiducial model we find kT2 = 0.03 and kT* 2 = 1.04. Figure 1a shows the normalized basin size, Q, as a function of the crater angular radius, yc, for different maximum basin depths, hc, and assuming mass conservation and an outer rim radius twice the crater radius (yr = 2yc). For our fiducial model with yc = 60°, hc = 10 km, and no despinning (wi = w); Q = −0.32. Since Q / w−2 i (equation (3)), if we assume the same parameters and 15% despinning, Q = −0.24. Similarly, if we assume mass conservation, Q / hc (equations (2) and (3)); therefore, Q = −0.48 for the same parameters of our fiducial model except for hc = 15 km. [6] The possible TPW solutions are given by B = R and Q sinð2B Þ ¼ sinð2R Þ;
Figure 1. Normalized basin size, Q, as a function of (a) the basin angular radius and (b) the reorientation angle. In Figure 1a we assume mass conservation and an outer rim radius twice the crater radius. The solid, dotted, and dashed lines correspond to maximum basin depths of 10, 15 and 20 km respectively, as labeled. In Figure 1b, the solid, dotted, and dashed lines correspond to basin center latitudes of 70°S, 65°S, and 75°S respectively, as labeled. (c) Misfit between the predicted and observed shape as a function of the reorientation angle. The solid, dotted, and dashed lines correspond to 0, 15, and 50% despinning respectively. depends on the secular degree‐2 load and tidal Love numbers, kL2 and kT2 respectively, and the secular degree‐2 tidal Love number for the case without an elastic lithosphere, kT* 2 . These dimensionless numbers describe the response to loading and tidal forcings and depend on the interior structure of Vesta. We adopt a three‐layer internal structure, with a core 135 km in radius and density 7.5 g cm−3, a mantle with density 3.3 g cm−3, and a 50 km thick elastic lithosphere with density 2.9 g cm−3 and rigidity 40 GPa [Kattoum and Dombard, 2009]. We use the method of Sabadini and Vermeersen [2004] to calculate the corresponding secular Love numbers, and discuss the effect of varying the elastic lithosphere thickness below. For our
ð4Þ
where (B, B) and (R, R) are the spherical coordinates of the basin center and the rotation pole prior to reorientation [Willemann, 1984; Matsuyama et al., 2006]. We estimate a basin center at 70°S, 260°E based on the observed topography [Thomas et al., 1997b, 1997a]. If there were no stabilization provided by a remnant rotational figure, the basin center would be aligned with the south pole. Thus, the offset between the basin center and the south pole provides evidence for a remnant rotational figure. The normalized basin size is constrained by the present colatitude of the basin center since equation (4) implies ∣Q∣ < 1/∣sin(2B)∣. Assuming B = 160° yields ∣Q∣ < 1.56, which is consistent with Q = −0.32 for our fiducial model and any of the cases considered in Figure 1a. Figure 1b shows the normalized basin size as a function of the reorientation angle for basin center latitudes in the range 65°S–75°S. If the basin size is larger than the size of the remnant rotational figure (Q < −1), there are two possible paleopole locations given the present location of the basin. For our fiducial model without despinning, Q = −0.32 and the amount of reorientation R = 6.0°. For comparison, if we assume the same parameters except for a maximum basin depth of 15 km, Q = −0.48 and R = 9.1°; and if we assume the same parameters except for 15% despinning, Q = −0.24 and R = 4.5°. [7] The remnant rotational figure affects the shape of Vesta; therefore, we can use the shape estimated using HST images [Thomas et al., 1997a] to constrain the amount of reorientation. We expand Vesta’s radius at a point with spherical coordinates (r, , ) in spherical surface harmonics as r ¼RþR
∞ X ‘ X
P2m ðÞ½c2m cosðmÞ þ s2m sinðmÞ:
ð5Þ
‘¼1 m¼0
We adopt the following definition for the associated Legendre functions [e.g., Arfken and Weber, 1995]: P‘m(x) = (1 − x2)m/2d mP‘(x)/dx m, where P‘ is a Legendre polynomial. The shape coefficients have both equilibrium and remnant RR rotational figure contributions, cEQ 2m and c2m, respectively [Matsuyama and Nimmo, 2009]. In a reference frame aligned with the present rotation axis, the only non‐zero equilibrium T 2 3 coefficient is cEQ 20 = −h2 w R /(3GM), where w is the present T rotation rate and h2 is the secular degree‐2 tidal displacement Love number that describes the deformation in response to the present rotational potential (hT2 = 0.01 for our nominal model). We assume a present rotation period of 5.34 hours [Drummond et al., 1988]. The remnant rotational
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the misfit significantly compared with the best‐fit solution. For example, if we ignore changes in rotation rate, the misfit is 0.52 for our fiducial model with a reorientation of 6.0°, and 0.66 for a 20° reorientation. The smallest misfit corresponds to 15% despinning; however, the misfit is only marginally smaller (0.48 and 0.62 for reorientation angles of 6.0° and 20° respectively). A 50% despinning increases the misfit to 0.86 and 1.00 for reorientation angles of 6.0° and 20° respectively.
3. Predictions [9] We expand the gravitational potential of Vesta at a point with spherical coordinates (r, , ) in spherical surface harmonics as ∞ X ‘ ‘ GM GM X R þ F¼ P‘m ðÞ½C‘m cosðmÞ þ S‘m sinðmÞ; r r ‘¼1 m¼0 r
ð8Þ
where G is the gravitational constant, M is the asteroid mass, and C‘m and S‘m are harmonic coefficients. We assume M = 2.59 × 1020 kg [Kuzmanoski et al., 2010]. The degree‐2 gravity field is expected to be dominated by contributions arising from the equilibrium and remnant rotational figures, and the large impact basin near the south pole. The degree‐2 gravity coefficients can be written as [Matsuyama and Nimmo, 2009; Matsuyama and Manga, 2010]
Figure 2. Predicted degree‐2 gravity coefficients (J2 ≡ −C20, C22, and S22) as a function of the reorientation angle. The solid and dotted lines correspond to 0 and 15% despinning respectively. figure contributions can be written as [Matsuyama and Manga, 2010]
cRR 2m sRR 2m
¼ cRR′ 20 ð2 m0 Þ
ð2 mÞ! cosðmR Þ ; P2m ðR Þ sinðmR Þ ð2 þ mÞ!
ð6Þ
where (R, R) are the spherical coordinates of the paleoT* T 2 3 pole. Here, cRR 20 ′ = −(h2 − h2 )wi R /(3GM) is the only non‐ zero coefficient for the case of a paleopole aligned with the present rotation pole (R = 0) and hT* 2 is the secular degree‐2 tidal displacement Love number for the case without an elastic lithosphere (hT* 2 = 2.04 for our nominal model). [8] We define the misfit between the expected shape and the observed shape, rOBS, as misfit
2 N ri riOBS 1 X ; N 1 i¼1 2
ð7Þ
where N = 2557 is the number of available observations and we assume s = 15 km. We use the shape model of Thomas et al. [1997b] constructed from HST images of Vesta, available at http://sbn.psi.edu/pds/resource/oshape.html. Figure 1c shows the misfit as a function of the reorientation angle for different initial rotation rates, and demonstrates that reorientation angles smaller than about 20° do not increase
w2 R3 w2 R3 1 T * i C20 ¼ k2T k2 k2T GM
3GM 6 2 Q 1 3 cos B 1 3 cos2 R w2 R3
1 T* i C22 ¼ Q sin2 B cosð2B Þ sin2 R cosð2R Þ k2 k2T GM 12 w 2 R3
1 T* i k2 k2T Q sin2 B sinð2B Þ sin2 R sinð2R Þ ; S22 ¼ 12 GM ð9Þ
where the normalized basin size, Q, is given by equation (3). Although both the basin and the remnant rotational figure have contributions to C21 and S21, these contributions must cancel each other under the assumption of principal axis rotation. Figure 2 shows the predicted degree‐2 gravity coefficients as a function of the reorientation angle. For our fiducial model without despinning, Q = −0.32, R = 6.0°, and the predicted gravity coefficients are J2 ≡ −C20 = 4.44 × 10−2, C22 = −3.95 × 10−4, and S22 = 1.44 × 10−4. For our fiducial model with 15% despinning, Q = −0.24, R = 4.5°, and the predicted gravity coefficients are J2 = 5.56 × 10−2, C22 = −3.73 × 10−4, and S22 = 1.36 × 10−4. Dawn gravity measurements will recover the degree‐2 coefficients (A. Konopliv et al., The dawn gravity investigation at Vesta and Ceres, submitted to Space Science Reviews, 2011) and thus test our predictions. Note that these coefficients are not sensitive to likely variations in elastic lithosphere thickness (see below). [10] Radial displacements due to changes in the rotational figure produce stresses that can in turn generate observable tectonic patterns. We calculate the stresses generated due to despinning and reorientation, and the corresponding expected tectonic patterns using the method of Matsuyama and Nimmo [2008]. Figure 3 shows the expected tectonic
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Figure 3. Predicted tectonic patterns assuming (a) no despinning, (b) no despinning and a 10 m global contraction, (c) 15% despinning, and (d) 15% despinning and a 10 m global contraction. The red, blue, and gray lines indicate the strike of the expected normal, thrust, and strike‐slip faults, and black lines separate these tectonic regions. Solid circles indicate the location of the initial rotation poles, and contours correspond to the maximum shear stress ((s+ − s−)/2, where s+ and s− are the maximum and minimum principal stresses) in units of 0.4 MPa. Figures 2a and 2b for the cases without despinning assume the parameters of our fiducial model with Q = −0.32 and R = 6.0°. Figures 2c and 2d for the case with 15% despinning assume Q = −0.24 and R = 4.5°. patterns for our fiducial models assuming 0 and 15% despinning. If we ignore despinning (Figure 3a), TPW produces provinces of normal, thrust, and strike‐slip faulting [Melosh, 1980]. The maximum shear stresses are modest (∼1 MPa); thus, rather than creating new fractures, TPW is likely to reactivate optimally‐oriented pre‐existing fractures created by other processes (such as impacts or global cooling). Despinning decreases the size of the thrust faulting provinces, and a 15% despinning is large enough for the despinning stresses to dominate the reorientation stresses. In this case, the thrust faulting provinces disappear, and the predicted tectonic pattern is characterized by provinces of normal faulting around the paleopoles and provinces of strike‐slip faulting in a band around the paleoequator (Figure 3c). Despinning alone can produce an equatorial band of thrust faulting if the elastic lithosphere thickness is ∼5–50% of the asteroid radius [Melosh, 1977]. However, we ignore this effect since it is small when additional stresses due to contraction are taken into account (see below). [11] Vesta likely experienced global contraction as it cooled and solidified, on timescales comparable to the likely despinning and reorientation timescales. For a volumetric thermal expansivity of 3 × 10−5 K−1 [Turcotte and Schubert, 2002], a modest 4 K mean global cooling would generate contraction of order 10 m. Figures 3b and 3d illustrate that this amount of global contraction is large enough to completely modify the expected tectonic pattern in Figures 3a and 3c for our fiducial models with 0 and 15 % despinning respectively. In this case, the compressional stresses due to contraction dominate the extensional stresses due to reorientation and despinning, and thrust faulting on the entire surface is expected. However, although contraction is
responsible for the style of faulting, the orientation of these faults is determined by reorientation and despinning.
4. Summary and Conclusions [12] The mass redistribution associated with the large impact basin near the south pole of Vesta is expected to cause TPW. The offset between the basin center and the south pole indicates a rotational state governed by a balance between the impact mass redistribution and a remnant rotational figure. The remnant rotational figure arises due to TPW and/or despinning, and affects the shape of Vesta. We compare the predicted shape with the shape estimated using HST images [Thomas et al., 1997a] to constrain these processes. The misfit between the predicted and observed shapes increases with the amount of TPW; however, the misfit change is small for reorientation angles in the range ∼0–20° (Figure 1c). Similarly, the misfit change is small for despinning in the range ∼0–20%. Improved shape models and observations of tectonic patterns after the Dawn mission may allow us to place tighter constraints on the amount of TPW and despinning. [13] TPW and despinning have implications for the degree‐2 gravity field and the global tectonic pattern, and we have made predictions which may be tested by Dawn observations [Russell et al., 2007]. The amplitude of the degree‐2 gravity coefficients increases with the amount of TPW and despinning (Figure 2). The effective size of the basin and the remnant rotational figure depends on the compensation state, and thus the elastic lithosphere thickness, Te. However, somewhat counter intuitively, the degree‐2 gravity coefficients are not sensitive to the thick-
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ness of the elastic lithosphere if Te ^ 10 km. This insensitivity arises because a ∼10 km thick elastic lithosphere is already large enough (due to membrane stresses) for the basin and the remnant rotational figure to be nearly fully elastically supported (i.e., ∣kL2 ∣ 1 and kT2 kT* 2 for Te ^ 10 km). Figure 6 of Ghosh and McSween [1998] shows that an elastic lithosphere 10 km thick will develop within 30– 100 My of Vesta’s formation, assuming that the base of the lithosphere is defined by the 700 K isotherm. [14] The global tectonic pattern is sensitive to the amount of TPW, despinning, and contraction due to cooling. A global isotropic contraction can be characterized as a degree‐0 perturbation, and thus it does not affect the degree‐2 gravity field or fault orientation. TPW and despinning produce both extensional and compressional stresses, while contraction produces only compressional stresses. Since a global cooling of 4 K produces ∼10 m of contraction ‐ large enough to dominate TPW and despinning ‐ we expect thrust faulting on the entire surface of the asteroid. However, the orientation of the faults will be controlled by the TPW and despinning stresses. [15] The topography is commonly referenced to an ellipsoid describing the expected shape assuming hydrostatic equilibrium. However, as we discussed above, non‐hydrostatic perturbations associated with the basin and the remnant rotational figure are expected. The remnant rotational figure dominates the equilibrium rotational figure if Te ^ 10 km because the remnant rotational figure is nearly fully uncompensated in this case. Thus, the remnant rotational figure provides a natural reference frame for the observed topography. [16] We have argued that the present‐day figure, degree‐2 gravity field and surface tectonics of Vesta are likely to be dominated by the effects of impact‐induced reorientation, and thus provide clues to Vesta’s thermal and spin evolution. Forthcoming data from Dawn will test whether these predictions are correct. [17] Acknowledgments. We acknowledge support from the Miller Institute for Basic Research. [ 18] The Editor thanks Marianne Greff‐Lefftz and an anonymous reviewer for their assistance in evaluating this paper.
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