Equilibrium rotational stability and figure of Mars

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1 Icarus 194 (008) Equilibrium rotational stability and igure o Mars Amy Daradich a, Jerry X. Mitrovica a,, Isamu Matsuyama b, J. Taylor Perron c, Michael Manga d, Mar A. Richards d a Department o Physics, University o Toronto, 60 St. George Street, Toronto, M5S 1A7 Canada b Department o Terrestrial Magnetism, Carnegie Institution o Washington, 541 Broad Branch Road NW, Washington, DC 0015, USA c Department o Earth and Planetary Sciences, Harvard University, 0 Oxord St., Cambridge, MA 0138, USA d Department o Earth and Planetary Sciences, University o Caliornia, Bereley, CA , USA Received 0 May 007; revised 4 October 007 Available online November 007 Abstract Studies extending over three decades have concluded that the current orientation o the martian rotation pole is unstable. Speciically, the gravitational igure o the planet, ater correction or a hydrostatic orm, has been interpreted to indicate that the rotation pole should move easily between the present position and a site on the current equator, 90 rom the location o the massive Tharsis volcanic province. We demonstrate, using general physical arguments supported by a luid Love number analysis, that the so-called non-hydrostatic theory is an inaccurate ramewor or analyzing the rotational stability o planets, such as Mars, that are characterized by long-term elastic strength within the lithosphere. In this case, the appropriate correction to the gravitational igure is the equilibrium rotating orm achieved when the elastic lithospheric shell (o some thicness LT) is accounted or. Moreover, the current rotation vector o Mars is shown to be stable when the correct non-equilibrium theory is adopted using values consistent with recent, independent estimates o LT. Finally, we compare observational constraints on the igure o Mars with non-equilibrium predictions based on a large suite o possible Tharsis-driven true polar wander (TPW) scenarios. We conclude, in contrast to recent comparisons o this type based on a non-hydrostatic theory, that the reorientation o the pole associated with the development o Tharsis was liely less than 15 and that the thicness o the elastic lithosphere at the time o Tharsis ormation was at least 50 m. Larger Tharsis-driven TPW is possible i the present-day gravitational orm o the planet at degree has signiicant contributions rom non-tharsis loads; in this case, the most plausible source would be internal heterogeneities lined to convection. 007 Elsevier Inc. All rights reserved. Keywords: Rotational dynamics; Mars 1. Introduction A diverse set o studies have argued that the rotation pole o Mars has been subject to large excursions relative to the surace geography, or true polar wander (TPW), including 90 reorientations nown as inertial interchange events. The observations supporting these studies include equatorial deposits that resemble sediments at the present poles o Mars (Schultz and Lutz, 1988), hydrogen rich equatorial deposits which are inerred to be remnants o ancient polar caps (Wieczore et al., 005) and analyses o crustal magnetic anomalies that sug- * Corresponding author. Fax: +1 (416) address: jxm@physics.utoronto.ca (J.X. Mitrovica). gest an early magnetic ield existed that is not aligned with the current rotation axis (Arani-Hamed and Boutin, 004; Hood et al., 005). Other geological evidence or signiicant TPW on Mars is ound in the spatial distribution o valley networs that appear to be ormed by liquid surace runo (Mutch et al., 1976) and craters ormed by oblique impacts, which may record the demise o ancient equatorial satellites (Schultz and Lutz-Garihan, 198). The most prominent eature in the geology o the martian surace is the Tharsis rise. The igure o Mars, as deined by the gravitational potential ield o the planet, is dominated by the signature o this massive volcanic structure (Smith et al., 1999) and a rotationally-induced equatorial bulge. Previous analyses o this igure have commonly ocused on two questions related to the stability and evolution o the mar /$ see ront matter 007 Elsevier Inc. All rights reserved. doi: /j.icarus

2 464 A. Daradich et al. / Icarus 194 (008) Fig. 1. Schematic illustration highlighting the physics o TPW. (A) The scenario described by Gold (1955) in which an initially hydrostatic planet (A0, A1) is subject to a surace mass load (green beetle, A) which is assumed to remain partially uncompensated. The load will push the pole away (green arrow), while the hydrostatic rotational bulge will, initially, resist this tendency (blue arrow). However, this resistance will disappear as the hydrostatic rotational bulge relaxes completely to the new rotational state (A3), leading to urther load-induced TPW. The process will continue until the load eventually reaches the equator (A4). (B) The rotational stability o an initially spherical and non-rotating planet (B0) with an elastic lithosphere (blue shell) subject to a surace mass loading. Once the model planet is set rotating, it will ultimately achieve the equilibrium orm shown in B1. As in (A), the application o a surace mass load will move the pole away, though this motion is initially resisted by the equilibrium rotational bulge (B). The equilibrium rotational bulge will eventually reorient perectly to the new rotation vector (i.e., the equilibrium rotational bulge in B3 is identical to that o B1) and TPW will proceed without memory o the previous rotational state. Eventually, as in (A), TPW will cease when the load reaches the equator (B4). (C) The scenario described by both Willemann (1984) and Matsuyama et al. (006), in which an initially hydrostatic planet (C0) develops an elastic shell through cooling o the interior (C1). The shell develops without any internal elastic stresses and thus the orm o the planet will be identical to the hydrostatic case in A1 rather than to the (less oblate) equilibrium orm in B1. The appearance o a surace mass load will act to push the pole away, and this will be resisted by the rotational bulge. However, in contrast to the irst two cases, in this scenario the bulge cannot perectly adjust to the new rotation axis (note that the oblate orm in C3 is not symmetric relative to the rotation axis) since the shell had an initially hydrostatic orm (C1), while TPW introduces elastic stresses within the shell. This incomplete adjustment, which introduces a memory o the original rotational state, is reerred to as the remnant rotational bulge (Willemann, 1984), and it acts to stabilize the pole. Ultimately, the load will not reach the equator (C4), but rather a position governed by the balance between the load-induced push and the pull associated with the remnant rotational bulge. Note that i one removed the load in A4 and B4, the pole would not move; however, this removal would cause the pole in C4 to return to its original orientation (C1). For each scenario, the igures are drawn so that the z-axis is ixed to the initial rotation pole in order to be consistent with the mathematical analysis appearing in the appendix. The igure could also have been drawn so that the rotation axis at any time coincided with the z-axis; the evolving geometry in this case would be essentially consistent with the view o an inertial observer. tian rotation vector. First, to what extent did the development o Tharsis, which is now located near the equator o the planet (Zuber and Smith, 1997), cause TPW (Melosh, 1980; Sprene et al., 005)? Second, how stable is the current rotation axis to changes in the surace and internal mass distribution (Bills and James, 1999)? In an important conceptual study, Gold (1955) discussed the rotational stability o a simple, hydrostatic planet subject to a surace mass loading (see Fig. 1A). A surace load will act to push the pole away. In the short term, the rotational bulge will resist this tendency and stabilize the rotation axis (Fig. 1A). However, in the long term the bulge will relax completely (i.e., hydrostatically) to any reorientation o the pole position (Fig. 1A3), thus erasing all memory o past positions, and permitting urther load-induced TPW. Eventually, the load, even one as small as Gold s amous beetle, will reach the equator (Fig. 1A4), the minimum energy state o the system. That is, TPW will continue until the pole aligns with the maximum axis o inertia associated with any surace load (i.e., non-hydrostatic) orcing. [Note that the Gold (1955) scenario includes a physical inconsistency since it assumes that the rotational bulge will relax perectly to a change in centriugal potential but that the surace load is never completely compensated isostatically.] It remains a common view, ollowing Gold s inluential analysis, that the stability o the martian rotation pole both at present day and in response to Tharsis loading is governed by the observed igure o the planet ater correction or a hydrostatic orm (Melosh, 1980; Bills and James, 1999; Sprene et al., 005). In this context, a series o studies since 1980 have reached the conclusion that the present non-hydrostatic orm o Mars (i.e., the orm which results rom removing the igure predicted on the basis o the present-day rotation vector in the case where there are no viscous or elastic stresses) is characterized by a maximum axis o inertia that lies along the current equator, 90 rom Tharsis, while the intermediate axis o inertia is aligned with the present-day pole (Melosh, 1980; Bills and James, 1999). The more recent o these analyses demonstrate that the maximum and intermediate non-hydrostatic moments o inertia are nearly equal (e.g., Bills and James, 1999), and thus

3 Equilibrium rotational stability and igure o Mars 465 conclude that the current rotation vector o Mars is inherently unstable. That is, relatively small surace mass loads can cause large (order 90 ) excursions o the pole along the great circle joining the present-day pole and the maximum non-hydrostatic inertia axis. Furthermore, analysis o the non-hydrostatic orm, ater correction or the signal rom Tharsis loading, has led to a conclusion that the development o this volcanic province must have induced a large (15 90 ) excursion o the rotation pole (Sprene et al., 005). The incomplete isostatic compensation o the ancient Tharsis load implies that Mars is characterized by non-zero longterm strength within the lithosphere (Zuber and Smith, 1997), and estimates o the elastic thicness o this region range up to several hundred ilometers (McGovern et al., 004; Zhong and Roberts, 003; Turcotte et al., 00; Sohl and Spohn, 1997). The question then arises: Is the non-hydrostatic stability theory cited above valid or a planet with an elastic lithosphere? [This question was also posed, without resolution, by Bills and James (1999, p. 9094).] Willemann (1984) recognized that the presence o an elastic shell has a potentially signiicant stabilizing eect on TPW. However, his analysis, recently reined and corrected by Matsuyama et al. (006), has been neglected in subsequent studies o Mars rotational stability (Bills and James, 1999; Sprene et al., 005). There may be two reasons or this neglect. First, Willemann (1984), and also Matsuyama et al. (006), considered the general problem o load-induced TPW, with some emphasis on Mars and Tharsis, but they did not quantitatively address the implications o their results or the present-day igure o the planet. Second, Willemann (1984) and Matsuyama et al. (006) analyzed a speciic scenario in which a lithosphere develops through cooling o an initially hydrostatic planet which is then subject to loading (see below). In any event, the Willemann (1984) study indicates that a nonhydrostatic theory or rotational stability is not appropriate or a planet that has a suiciently thic elastic lithosphere (e.g., Mars; see also the discussion on p. 8,68 o Zuber and Smith, 1997). Accordingly, the main goal o this paper is to derive, using physical arguments supported by standard mathematical analysis, a new, generalized statement o rotational stability that is valid or any planet whether it has an elastic lithosphere or not. With this generalization in hand, we irst reassess the level o present-day rotational stability implied by Mars gravitational igure. Next, we extend the analysis o Matsuyama et al. (006) to quantiy the total contribution o load-induced TPW to the planetary igure. Finally, we compare these new expressions to observational constraints on the igure o Mars in order to set bounds on the range o Tharsis-induced TPW.. The physics o rotating planets In this section we will discuss the physics o load-induced TPW on planetary models that are characterized by the presence or absence o a uniorm elastic lithosphere (Fig. 1). In the ormer case, the thicness o the lithosphere is denoted by LT. Our discussion will be supported by the mathematical analysis appearing in Appendix A, which is based on standard luid Love number theory. That is, we consider time scales long enough such that all viscous stresses associated with the response to surace mass loading and changes in the rotational (i.e., centriugal) potential have relaxed completely. As we discussed in reerence to Fig. 1A, Gold (1955) was concerned with the rotational stability o a planet in purely hydrostatic equilibrium. Such a planet has no elastic strength (LT = 0), and in this case the rotational lattening (or oblateness) o the bacground hydrostatic orm (Fig. 1A0 or 1A1) is a unction o the rotation rate, Ω, and the internal density structure o the planet [Eq. (A.11); the sensitivity to internal structure is embedded within the luid tidal Love number computed or the model with no lithosphere, ]. In this scenario, any non-hydrostatic contributions to the inertia tensor will be associated with the applied surace mass load [Eq. (A.9)]. Thus, diagonalization o the non-hydrostatic inertia tensor will yield a maximum principal axis (or rotation pole) that is oriented 90 rom the load, as in Fig. 1A4. The minimum principal axis will pass through the center o the load, and both the intermediate and minimum axes will pass through the equator. Next, we turn to scenarios in which the planet has an elastic lithosphere, beginning with Fig. 1B. In this somewhat unrealistic, but nevertheless physically instructive case, an initially non-rotating planet with a pre-existing elastic lithosphere (Fig. 1B0) is spun-up to its current rotation rate, and the system is allowed to reach a state in which all sub-lithosphere viscous stresses relax completely (Fig. 1B1). This relaxed orm will not be in hydrostatic equilibrium, since the elastic lithosphere has permanent strength, and thus the oblateness o the orm will be less than the hydrostatic lattening in Fig. 1A (Mound et al., 003). To distinguish the ormer rom the latter we will henceorth use the term equilibrium orm, and note that this orm approaches the hydrostatic igure as the elastic lithospheric thicness approaches zero. Mathematically, the oblateness o the equilibrium orm is a unction o the rotation rate, Ω, and the luid tidal Love number, T [Eq. (A.16)], and the latter is a unction o both the internal density structure o the planet and the thicness o the elastic lithosphere ( T rom below as LT 0) (Mitrovica et al., 005; Matsuyama et al., 006). As in Fig. 1A, a load applied to this model planet will ultimately reach the equator (Fig. 1B4) since the equilibrium rotational bulge which deines the initial rotating state (Fig. 1B1) will eventually reorient perectly to a change in the position o the rotation pole [i.e., it provides no memory o a previous rotational state; see igure caption and the discussion between Eqs. (A.1) and (A.14)]. However, it would be incorrect to analyze the rotational stability o this system using non-hydrostatic stability theory. Speciically, i one were to correct the igure in Fig. 1B1 or a hydrostatic orm, one would be let with a residual, non-hydrostatic orm that was characterized by a deicit in oblateness, or mass excess at the poles, i.e., a prolate spheroid. (Mathematically, this dierence arises because T < when LT 0.) One would thus erroneously introduce a spurious tendency or the entire igure to drive a TPW event that would

4 466 A. Daradich et al. / Icarus 194 (008) move the pole towards a point on the current equator, 90 rom Tharsis. We conclude that the long-term stability o a rotating planet is governed by the terms in the inertia tensor which do not perectly reorient to the contemporaneous rotation axis [Eq. (A.13)]. That is, the hydrostatic orm in the scenario o Fig. 1A [Fig. 1A1; Eq. (A.8)] and the equilibrium orm in the scenario o Fig. 1B [Fig. 1B1; Eq. (A.13)] are irrelevant to the long-term rotational stability. Thus, the stability o the system in Fig. 1B is governed by the non-equilibrium (rather than non-hydrostatic) inertia tensor. This statement provides a undamental extension o the Gold (1955) stability theory to the case o planets, lie Mars, with non-zero (very long-term) elastic strength in the lithosphere. Fig. 1C shows a more complicated and realistic scenario or such planets that has been considered by both Willemann (1984) and Matsuyama et al. (006). An initially hydrostatic, rotating planet (Fig. 1C0) cools and develops a uniorm and unbroen elastic lithosphere (Fig. 1C1). Lithospheric ormation will not disturb the hydrostatic orm since the elastic lithosphere will grow in a ully relaxed state. In contrast to Fig. 1B (orfig. 1A), the rotational bulge cannot reorient perectly to a change in the pole position since there would be no way or the elastic lithosphere to re-establish a hydrostatic orm around the new pole position: TPW will introduce stresses in the previously stress-ree lithosphere. The system thus has a memory o the initial rotational state (Fig. 1C1) and any departures rom this state would be resisted. The inal load position (Fig. 1C4), which is not at the equator, represents a balance between this resistance and the tendency o the load to drive TPW. Unless the load is o the same order o magnitude as the mass associated with the rotational bulge, little TPW can occur. Is our generalized statement that rotational stability is governed by non-equilibrium components o the inertia tensor appropriate to the scenario depicted in Fig. 1C? To answer this requires that we separate the change in shape between the initial (Fig. 1C1) and inal (Fig. 1C4) rotational states into a contribution that perectly reorients as the pole moves around and a residual term. Physically, the latter can be inerred by simply switching o rotation in the case o Fig. 1C1 and determining the departure rom sphericity that would result. This departure would be the dierence between the hydrostatic orm (Fig. 1C1) and the equilibrium orm associated with a planet having the same rotation rate and elastic lithospheric thicness (as in Fig. 1B1). This dierence, which is termed the remnant rotational bulge (Willemann, 1984; Matsuyama et al., 006), is rozen into place relative to the initial pole position. Thus the oblateness in Fig. 1C1 can be decomposed into a component rom the remnant rotational bulge, which stays ixed relative to the initial rotation axis, and an equilibrium rotational orm that will adjust perectly (in the long-time limit) as the pole moves rom the initial to inal (Fig. 1C4) state. Thereore, as in Fig. 1B, the rotational stability o the planet is governed by non-equilibrium components o the inertia tensor. As in previous scenarios, these components include the surace mass load, but, in the case o Fig. 1C they also include a remnant bulge. The same separation o the igure o the planet into: (1) an equilibrium orm that adjusts perectly to the change in the orientation o rotation and thus has no bearing on the rotational stability; and () a non-equilibrium, remnant rotational orm oriented with the initial pole position, is derived mathematically in Appendix A.1.3 [see Eqs. (A.19) to (A.1)]. As a inal point, Matsuyama et al. (007) have analyzed the rotational stability o the scenario in Fig. 1C by writing expressions or the total energy in the system and inding the TPW that minimizes this energy. Their expressions provide an independent conirmation that it is the diagonalization o the non-equilibrium inertia tensor that governs the rotational stability. 3. Results The non-equilibrium theory described above provides a generalized ramewor or assessing the rotational stability o a planet on the basis o its gravitational igure or, equivalently, its inertia tensor. In this section we reassess two issues that were previously investigated by applying a non-hydrostatic rotation theory to the igure o Mars. First, how stable is the present-day rotation vector o the planet? Second, what level o TPW was driven by the development o the Tharsis volcanic province? 3.1. The present-day rotational stability o Mars We can write the total inertia tensor o Mars as a δ hyd a δ nhyd a b = δ hyd b + δ nhyd b, (1) c δ hyd c δ nhyd c where a, b, and c are the non-dimensional moments (nondimensionalized by the mass and mean radius o Mars) in the principal axis system (a <b<c) and the superscripts hyd and nhyd denote hydrostatic and non-hydrostatic contributions. On Mars, these three moments reer to axes on the equator at the same longitude as Tharsis, on the equator 90 rom Tharsis, and the current rotation axis, respectively. Embedded within the hydrostatic component o the total inertia tensor is a spherical term which results in a trace, or this contribution, that is non-zero. The non-hydrostatic moment increments are commonly expressed in terms o the observed harmonic (Stoes) coeicients o the gravitational potential at degree two, J and J,inthe same principal axis system (Bills and James, 1999): δ nhyd a 1/3 δ nhyd b = 1/3 ( J J hyd ) () δ nhyd + J, c /3 0 where the parameter J hyd is a correction to the observed J harmonic associated with the hydrostatic orm o the planet. Satellite-based measurements (Smith et al., 1999) have yielded the estimates: J = (1.960 ± 0.0) 10 3 and J = (6.317 ± 0.003) This decomposition maes no assumption regarding a connection between the rotational stability and the inertia tensor. However, let us proceed by assuming that the non-hydrostatic

5 Equilibrium rotational stability and igure o Mars 467 inertia tensor governs the rotational stability. In this case, once the observed harmonics J and J are speciied, assessing the rotational stability reduces to estimating J hyd in Eq. (). Bills and James (1999) combined a satellite-based estimate o J with a constraint on the spin-axis precession rate (or precession constant) based on radio tracing rom the Pathinder mission (Folner et al., 1997), to estimate the non-dimensional polar moment o inertia as c = ± They then used the Darwin Radau relationship to convert this value to an estimate or J hyd = 1.840± [see their Eq. (88)]. Using this value in Eq. () yields: δ nhyd a ± 7.0 δ nhyd b = δ nhyd c ± ± (3) Bills and James (1999) concluded, since δ nhyd b>δ nhyd c, that the current rotation pole o Mars is 90 rom where it should be on the basis o the non-hydrostatic stability theory o Gold (1955). Moreover, since δ nhyd b δ nhyd c, they also concluded that the pole is unstable; small mass loads would be capable o moving the pole along a great circle joining the maximum and intermediate axes o inertia (i.e., along the great circle 90 rom Tharsis). Yoder et al. (003) have derived a more recent estimate o c = ± We have repeated the Bills and James (1999) analysis or this range o values and the result is shown in Fig.. [One dierence in our analysis is that we correct the polar moment c or a small non-hydrostatic contribution beore we apply the Darwin Radau relation. Under the assumption that these non-hydrostatic contributions to the inertia tensor are axisymmetric about a point on the equator at the same longitude as Tharsis, the correction is simply 4J /3(Bills and James, 1999).] Over this entire range o c values δ nhyd b δ nhyd c, and thereore one would again conclude on the basis o a nonhydrostatic theory that the martian rotation pole is unstable. Note that the c value adopted by Bills and James (1999) alls at the high end o the range considered in Fig., and that δ nhyd c>δ nhyd b when c< As we noted in the last section, this non-hydrostatic theory is not appropriate or the analysis o Mars rotational stability i the martian lithosphere is characterized by non-zero elastic strength. Accordingly, we need to analyze the non-equilibrium, rather than non-hydrostatic orm to properly assess the stability. Simply put, the hydrostatic correction removes too much lattening rom the observed orm and will thus imply a signiicantly less stable planet than the correct, non-equilibrium approach. Let us begin by re-writing Eq. (1) in the orm: a δ eq a δ ne a b = δ eq b + δ ne b, (4) c δ eq c δ ne c where the superscripts eq and ne denote the equilibrium and non-equilibrium contributions to the total inertia tensor. The second o these contributions can be written in terms o J Fig.. Non-hydrostatic moments o inertia, computed using Eq. (), as a unction o the total polar moment o inertia, where the latter is varied within the uncertainty (0.3650±0.001) cited by Yoder et al. (003). The moments δ nhyd c and δ nhyd a reer to axes in the direction o the current rotation pole and the equatorial location o Tharsis. The axis associated with the moment δ nhyd b is aligned with a point on the equator 90 rom Tharsis. All moments are normalized by Ma,whereM and a are the mass and mean radius o Mars. We adopt the observed values o J = and J = citedinthe main text. and J using a modiied orm o Eq. (): δ ne a 1/3 δ ne b = 1/3 ( J J hyd ) δ ne c /3 1/3 + 1/3 ( J hyd J eq ) + J. /3 0 The second term on the right-hand side o this equation is the dierence between the hydrostatic and equilibrium contributions to the inertia tensor and it can be written in the orm [Eq. (A.17)]: J hyd J eq = Ω a 3 [ T ], (6) M where a and M are the radius and mass o the planet, respectively, and G is the gravitational constant. The quantity within square bracets represents the dierence between luid Love numbers computed or no lithosphere (i.e., the hydrostatic orm) and a lithosphere o thicness LT (the equilibrium orm). In Fig. 3 we use Eqs. (5) and (6) to compute the nonequilibrium moments δ ne a, δ ne b and δ ne c, as a unction o the elastic lithospheric thicness. The Love numbers were computed using the Mars model o Sohl and Spohn (1997) (see (5)

6 468 A. Daradich et al. / Icarus 194 (008) TPW driven by Tharsis loading An important, outstanding issue in the long-term evolution o Mars concerns the extent to which the development o Tharsis changed the orientation o the rotation vector. For example, there have been numerous inerences o Tharsis-driven TPW based on tectonic patterns, geomorphologic eatures, magnetic anomalies and grazing impacts (see introduction and also Sprene et al., 005). In addition, there have been theoretical predictions o polar wander driven by a surace mass loading consistent with the size and current location o Tharsis (Melosh, 1980; Willemann, 1984; Matsuyama et al., 006, 007). These theoretical analyses admit both small and large polar wander solutions [TPW angle, δ, o 10 or 80, respectively; see discussion below Eq. (A.30)]. Sprene et al. (005) analyzed the observed igure o Mars using a non-hydrostatic stability theory and concluded that Tharsis induced a polar wander o Speciically, they began by adopting the non-hydrostatic orm given by Eq. (3) and then corrected this orm or Tharsis using the load model o Zuber and Smith (1997). They next perormed a search through all possible (pre-tharsis) pole positions that satisied the ollowing stability equation (Bills and James, 1999): Fig. 3. Non-equilibrium moments o inertia, computed using Eqs. (5) and (6), as a unction o the adopted thicness o the elastic lithosphere, LT (m). The moments δ ne a, δ ne c,andδ ne b reer to axes in the direction o Tharsis, the current rotation pole and a point on the equator 90 rom Tharsis, respectively. All igures or the moments are normalized by Ma,whereM and a are the mass and mean radius o Mars. We adopt the observed values o J = and J = cited in the main text. The luid Love numbers required in Eq. (6) are given in Table 1 or various values o LT. Table 1 Eects o lithospheric thicness (LT) ontheluid Love numbers o Mars. We adopt the 5-layer model o martian structure described by Sohl and Spohn (1997). The tidal luid Love number or LT = 0is = LT (m) L T Table 1). Furthermore, in evaluating the irst term on the righthand side o Eq. (5), we adopt the hydrostatic correction o Bills and James (1999). ForLT = 0, the non-equilibrium theory collapses to the old non-hydrostatic case [ = T in Eq. (6) and thereore the second term on the RHS o Eq. (5) vanishes] and the results suggest an unstable rotation pole (i.e., δ ne b δ ne c, as noted in Fig. ). However, the estimate o δ ne c increases rapidly relative to δ ne b as LT is increased above zero. Indeed, a value o LT = 100 m yields a moment dierence δ ne c δ ne b which is comparable to the dierence δ ne b δ ne a, and a highly stable rotation pole. We thereore conclude that the current orientation o the martian rotation pole is stable or values o LT that are consistent with widely cited estimates (e.g., McGovern et al., 004; Zhong and Roberts, 003; Turcotte et al., 00; Sohl and Spohn, 1997). J nhyd > J nhyd, (7) where the superscript denotes the residual non-hydrostatic ield ater correction or Tharsis. The collection o acceptable pole positions deined a pre-tharsis stability ield, and the range o TPW angles that moved the pole rom within this stability ield to the present position yielded the inerence o δ = (Sprene et al., 005). To understand the origin o Eq. (7), we can consider a special case where the correction or Tharsis does not alter the principal axis orientation determined rom the igure o Mars. In this case, we could revise Eq. () to remove the Tharsis load using: δ nhyd a 1/3 δ nhyd b = 1/3 ( J J hyd J ) (8) δ nhyd c /3 + ( J J ), 0 where J and J are the Tharsis contributions to these coeicients. I we deine J nhyd = J J hyd J nhyd and J = J J, then the stability equation (7) is seen simply as a condition that δ nhyd c>δ nhyd b. There are various assumptions inherent to the procedure adopted by Sprene et al. (005). First, that the non-hydrostatic orm governs the rotational stability (see their discussion on p. 488). Second, related to the irst, that the remnant bulge dynamics discussed by Willemann (1984) may be ignored. This assumption is implied by the procedure o searching through possible pre-tharsis pole positions. In the physics o Fig. 1C, each reorientation o the pole in this manner would introduce a remnant bulge contribution to both the J and J that should

7 Equilibrium rotational stability and igure o Mars 469 be accounted or. A urther assumption o Sprene et al. (005) is that the igure o Mars at spherical harmonic degree two has not changed, with the exception o a simple rotation, subsequent to the end o the development o Tharsis [i.e., the values o J and J used in Eq. (7) are present-day values]. In this section, we revisit the inerence o Tharsis-driven TPW based upon the observed igure o Mars by using the non-equilibrium rotation theory appropriate to the physics o Fig. 1C. In Appendix A. we derive expressions or the total Stoes coeicients J and J (in the principal axis system) arising rom the loading o a planet within this scenario [Eqs. (A.31) and (A.3)]. For the beneit o the reader, we repeat these equations here: J = T Ω a 3 M + ( 1 + L ) Q Ω a 3 M )Ω a 3 [ 1 3 cos δ ( T M [ 1 3 cos ] θ L ], 4J = ( 1 + L ) Q Ω a 3 M sin θ L ( T )Ω a 3 M sin δ, where the TPW angle δ is given by [Eq. (A.9)]: δ = 1 arcsin[ Q α sin(θ L ) ] and α is a parameter which depends on the planetary model and the adopted lithospheric thicness [Eq. (A.30)]: α = 1 + L. 1 T / These expressions yield the Stoes coeicients or the inal state given by Fig. 1C4. In these equations, θ L denotes the inal colatitude o the Tharsis load, which we tae to be 83 (Zuber and Smith, 1997). For a given model o Mars density structure, which yields (Table 1), there are two ree parameters on the RHS o these equations. The irst is the lithospheric thicness. Speciying LT sets the values o the luid tide and load Love numbers, T and L, respectively (Table 1), as well as the parameter α. The second is the uncompensated size o the Tharsis load, Q [Eq. (A.6)], deined as the ratio o the gravitational potential perturbation at degree due to the direct eect o the load and the hydrostatic bulge [the latter, together with LT,sets the angle δ in Eq. (A.9)]. Our procedure or determining the range o acceptable TPW angles δ driven by the Tharsis load is as ollows. First, we speciy some tolerance within which the predictions should it the observed values o the Stoes coeicients J and J.Next, we choose a value o LT. For this lithospheric thicness, we then search through a wide range o Q values and note all predictions that it the Stoes coeicients. We then repeat this procedure or dierent choices o LT. For each Q, this provides a range o acceptable δ values (this range can be zero). Equations (A.31) and (A.3) assume that the only contributors to the non-equilibrium planetary orm are the Tharsis load Fig. 4. The ull range o TPW angles, δ, as a unction o the uncompensated size o Tharsis, Q, that yield acceptable its to the observed Stoes coeicients J and J or the present-day gravitational igure o Mars. The calculations are based on the non-equilibrium stability theory summarized in Fig. 1C and by Eqs. (A.9) (A.3). These predictions adopt a inal Tharsis colatitude o 83 (Zuber and Smith, 1997), and a lithospheric thicness, LT, which varies rom 30 to 00 m. The range o solutions includes all predictions which it the observed J and J coeicients to within 10 and 5%, respectively. The color contours in (A) show the variation in the J misit across the range o acceptable solutions. (B) is the analogous result or the J misit. (represented by Q ) and the remnant bulge (whose contribution depends on the level o TPW). Although Tharsis dominates the observed gravitational orm o Mars, not including other loading contributions and their associated TPW will introduce some error in our predictions o both J and J, and the misit tolerance we discuss above is an attempt to explore the sensitivity o our inerences to this error. In Fig. 4 we show all acceptable solutions or δ as a unction o the uncompensated size o the load when we allow a misit o up to 10% o the observed value o J and up to 5% o the value o J. [These dierent values relect the act that the bacground rotational bulge dominates the J observation; as an example, the non-hydrostatic igure o Mars inerred by Bills and James (1999) is 6% o the observed value.] Fig. 4A maps out the variation in the misit to the J coeicient (as indi-

8 470 A. Daradich et al. / Icarus 194 (008) cated by the color bar) within the range o acceptable solutions, while Fig. 4B is the analogous map or the J harmonic. Embedded within these calculations are elastic lithospheric thicnesses ranging rom 30 to 00 m. No solutions exist below a Q value o Moreover, in our calculations we adopt an upper bound Q value o 3.0, which is signiicantly larger than the bound cited by Willemann (1984). As discussed in Appendix A., there are, in theory, two possible true polar wander solutions or a given load size and inal colatitude when Q α>1; in this case, i δ is a solution o Eq. (A.9), then 90 δ is also a solution. However, both solutions are not necessarily able to reconcile the additional constraint we have imposed in regard to the it to the observed Stoes coeicients. For example, high TPW solutions (i.e., greater than 45 )doexist, but only or Q > and a mismatch to the observed value o J > 4% (or Q < 3). As we allow a progressively greater mismatch to the J coeicient, high TPW solutions are ound or progressively lower values o Q. A comparison o Figs. 4A and 4B indicates that the J coeicient provides a more stringent constraint on the acceptable range o TPW than J. As an example, while high TPW solutions only exist or a misit tolerance greater than 4% o J, these solutions span a wide range o J misits (i.e., rom less than a percent upwards). We conclude, under the assumptions inherent to the present analysis, that the development o Tharsis could only have driven a large excursion o the pole i a signiicant raction o the present-day J observation is due to signals rom sources other than Tharsis and its associated remnant bulge reorientation. The size o this required contribution, which reaches 10% or Q = 1.7(Willemann, 1984), is larger than any surace load ound on Mars (Smith et al., 1999). This suggests that the only plausible source would be related to internal, convectively-driven dynamics. We can rephrase this by concluding that the J signal associated with a large TPW event (and remnant bulge reorientation) driven by Tharsis diers signiicantly rom the present-day observation o this harmonic. The relationship between the lithospheric thicness and misit within the suite o acceptable TPW solutions shown in Fig. 4 is plotted in Fig. 5. That is, or a given Q, Fig. 5 provides the range in LT embedded within the solutions or δ on the associated rame o Fig. 4. Note that or both the low and high TPW solutions, the elastic thicness o the lithosphere which produced a solution or the Stoes coeicients within the speciied misit tolerances tends to decrease as Q is increased. This trend reveals some interesting physics. Consider, irst, the small TPW branch o solutions. The results in Fig. 4 indicate that a small Tharsis load (Q 0.5) emplaced at a colatitude very close to its inal colatitude (i.e., δ o a ew degrees) will yield a good it to both the J and J observations. Increasing the uncompensated size o the load will increase the TPW angle δ (Fig. 4). To maintain a similar it to the gravitational ield as Q is increased, that is, to maintain a similar eective load size and remnant bulge signal, requires, in this case, that the lithospheric thicness be reduced. Next, within the high TPW branch, decreasing Q leads to a higher, not smaller, level o TPW (Fig. 4). Once again, one can maintain a similar contribution to the Stoes coeicients rom the surace load by Fig. 5. The range o lithospheric thicness, as a unction o the uncompensated size o Tharsis, that yield acceptable its to the observed Stoes coeicients J and J or the gravitational igure o Mars. The associated range in TPW angles, δ, isgiveninfig. 4. The details o the calculation are discussed in the caption to Fig. 4 andinthetext.(a)and(b)showthevariationinthej and J misit across this range o acceptable solutions. increasing LT as Q is decreased since this will reduce the level o isostatic compensation. However, in this case, the signal rom remnant bulge reorientation will not remain the same; rather it will increase because both the TPW angle and the size o the remnant bulge will increase. The result is an increasing level o J misit as Q is decreased in the high TPW branch. Under the assumptions inherent to the analysis, Figs. 4 and 5 may be used to constrain the thicness o the martian lithosphere at the time o the ormation o Tharsis and the TPW driven by this ormation. As discussed above, i we accept an upper bound on the (uncompensated) size o Tharsis cited by Willemann (1984), Q = 1.7, then high TPW solutions are ruled out unless the misit to J is well over 10%. In this particular case, the TPW angle is limited to less than 15 (Willemann, 1984; Matsuyama et al., 006) and the minimum elastic thicness o the martian lithosphere at Tharsis ormation would be 45 m. For a Q value o 1, the TPW angle would be less than 10, and LT > 90 m.

9 Equilibrium rotational stability and igure o Mars Conclusions Numerous analyses o the rotational stability o Mars, either at present day or in response to loading by Tharsis, have been based on the assumption that this stability is governed by the non-hydrostatic gravitational igure o the planet. We have demonstrated that such a treatment is incorrect. In particular, or any planet with long-term elastic strength within the lithosphere, the stability o the rotation vector is governed by the gravitational igure ater correction or an equilibrium, rather than hydrostatic orm. The ormer is deined as the shape achieved by an initially non-rotating planet with an elastic outer shell ater all viscous stresses below the shell have relaxed subsequent to the onset o rotation (e.g., Fig. 1B1). The equilibrium orm depends on the thicness o the elastic plate as well as on the rotation rate and the internal density structure o the planet, and it is the component o the gravitational igure which will perectly reorient (in the luid limit) to a change in the rotation vector; thus, it provides no long-term memory o any previous rotational state. The non-equilibrium theory provides the necessary extension o the ot-cited nonhydrostatic theory o Gold (1955) to the case o planets with lithospheres that maintain elastic strength over very long time scales. The observed igure o Mars, ater correction or the equilibrium orm, indicates that the present-day rotation axis o the planet is stable or adopted elastic thicnesses o the martian lithosphere well below current estimates (Fig. 3). This counters previous conclusions, based on a non-hydrostatic theory o planetary rotation, that the present-day orientation o the pole is unstable and will move easily on a great circle deined by the arc joining the current pole and a point 90 rom Tharsis on the martian equator. Finally, we have used a version o the non-equilibrium theory valid or the scenario o planetary evolution considered by Willemann (1984) and Matsuyama et al. (006), in which a lithosphere develops on an initially hydrostatic orm, to estimate the range o possible Tharsis-driven TPW. Our analysis, based on a comparison o predictions o the Stoes coeicients J and J with present-day observational constraints, suggests that Tharsis drove less than 15 o polar motion. These calculations also indicate that LT at the time o Tharsis ormation was at least 50 m, though a more liely lower bound (i we accept Q 1) is 100 m. This bound suggests that the ormation o Tharsis did not maredly reduce the strength o the martian lithosphere. This inerence o Tharsis-driven TPW assumes that the igure o Mars has been altered by a relatively minor amount, deined by an imposed misit tolerance, since the end o Tharsis ormation. We can clearly not rule out that other loads, in particular internal heterogeneity related to convection, provide a signiicant contribution to the present-day orm. However, our intent was to speciically reassess the conclusions o previous wor that had assumed that these contributions were small. In this regard, our results demonstrate that arguments that Tharsisinduced TPW was at least 15, based on a non-hydrostatic theory o martian rotational stability, are not robust. Acnowledgments This wor was supported by CIFAR, NSERC, NSF, the Carnegie Institution o Washington, a Daly Fellowship (to T.P.), the NASA Astrobiology Institute (NNA04CC0A) and the Miller Institute. Appendix A. Mathematical treatment o Fig. 1 In this appendix we use standard (luid) Love number theory to derive expressions or the inertia tensor appropriate to each o the scenarios in Fig. 1. In each case, we consider the connection between these expressions and the J and J harmonics within the principal axis system. The reader is ased to note that prior to Eq. (A.9), the symbol δ reers to the Kronecer delta and not the TPW angle. In the time domain (t), the viscoelastic load and tidal (or tidal-eective) Love numbers at spherical harmonic degree two have the orm (Peltier, 1974): J L (t, LT) = L,E δ(t) + r j exp( s j t) (A.1) and T (t, LT) = T,E δ(t) + j=1 J r j exp( s j t). j=1 (A.) These Love numbers yield the gravitational potential perturbation, at degree two, arising rom the deormation o a spherically symmetric, Maxwell viscoelastic planetary model subject to an impulsive surace mass load and gravitational potential (tidal) orcing, respectively. The irst term on the right-hand side (henceorth RHS) o each expression represents the immediate elastic response to the loading (hence the superscript E), while the second term is a non-elastic response comprised o aseriesoj normal modes o viscoelastic decay. The modes or the load and tidal Love numbers have a common set o decay times (s j ), but distinct modal amplitudes (r j and r j ). These Love numbers are dependent on the viscoelastic structure o the planetary model. For our purposes the dependence on the elastic thicness o the lithosphere (LT) is most important, and thus we mae this dependence explicit. In the Laplace transorm domain, these Love numbers have the orm: J L (s, LT) = L,E r j + (A.3) s + s j and T (s, LT) = T,E + j=1 J j=1 r j s + s j. (A.4) The so-called luid Love numbers represent the response o the planetary model ater all viscous stresses have relaxed. These may be derived rom the above expressions either by taing the s = 0 limit o the Laplace-domain equations (A.3) and (A.4), or by convolving the time-domain equations (A.1)

10 47 A. Daradich et al. / Icarus 194 (008) and (A.) with a Heaviside step loading and considering the ininite time response. In either case, we would then obtain: L (LT) = L,E + and T (LT) = T,E + J r j s j j=1 J j=1 r j s j. (A.5) (A.6) The luid Love numbers are dependent on the density proile o the planetary model, as well as the thicness o the elastic lithosphere. The latter is, o course, subject to no viscous relaxation. For the sae o brevity, we will suppress the dependence on LT in equations that require the luid Love numbers (e.g., L, T ). However, in the case o a purely hydrostatic planet, i.e., one which has no long-term elastic strength (LT = 0), we will adopt the notation L, and. A.1. The inertia tensor: Three case studies We assume a co-ordinate system oriented so that the z-axis is ixed to the rotation pole o the planet just prior to loading (e.g., Figs.1A1,1B1,or1C1). The initial angular velocity vector will be denoted by (0, 0,Ω). At any subsequent time, the rotation vector will be given by ω i (t), i = 1,, 3, with magnitude ω (t). Finally, a and M are the radius and mass o the planet, respectively, while G is the gravitational constant. A.1.1. Case 1: Gold (1955) We begin with the scenario in Fig. 1A, which is the physics treated by Gold (1955). In this case, the total inertia tensor is (Mun and MacDonald, 1960; Ricard et al., 1993): I ij (t) = I 0 δ ij + a5 [ω i (t)ω j (t) 13 ] ω (t)δ ij + Iij L (t), (A.7) where I 0 is the spherical term and Iij L (t) is the contribution to the inertia tensor rom the combined eect o the surace mass load and the direct planetary deormation it induces. The second term on the right-hand side is related to the rotationally-induced lattening o the hydrostatic model adopted by Gold (1955). The irst two terms on the RHS constitute the inertia tensor or a hydrostatic planet with angular velocity ω i (t). Thus,we can write [ω i (t)ω j (t) 13 ] ω (t)δ ij. (A.8) I hyd ij (t) = I 0 δ ij + a5 This represents the component o the inertia tensor in Eq. (A.7) that perectly adjusts (in the luid limit) to any change in the rotation axis (i.e., as in the ully relaxed cases shown in Figs. 1A1, 1A3, and 1A4). As discussed by Gold (1955), this component is not relevant to the long-term rotational stability o the (hydrostatic) planet since it provides no memory o any previous orientation o the rotation vector. Thus, or this planetary model, the reorientation o the pole is governed by the non-hydrostatic inertia tensor: I nhyd ij (t) = Iij L (t). (A.9) In particular, the rotation vector is aligned with the maximum principal axis o I nhyd ij. Thus, the adjustment in Fig. 1A will continue until the load has moved to the equator (Fig. 1A4). It will be instructive to consider the inertia tensor in the initial (t = t 0 ), unloaded state o the system. Applying the initial rotation vector, (0, 0,Ω), into the expression [ω i (t)ω j (t) 1 3 ω (t)δ ij ] within Eq. (A.8) yields: I hyd ij (t 0 ) = I 0 δ ij + Ω a 5 ( δ i3 1 ) δ ij. 3 (A.10) The orientation in Fig. 1A1 is a principal axis orientation, and thus we may use this expression to derive a ormula or the hydrostatic component o the J harmonic (Bills and James, 1999): J hyd = I hyd (A.11) Ma = Ω a 3 M. As a inal point we note an inherent inconsistency in the Gold (1955) scenario. In a purely hydrostatic planet, the contribution to the inertia tensor rom the load and direct deormation, Iij L, will be zero, since the load will be perectly isostatically compensated. In the terminology o luid Love number theory, the uncompensated raction o an applied load is given by 1 + L, but as LT approaches zero L L, = 1. Thus, the scenario in Fig. 1A only holds i one maes the ad-hoc assumption that the rotational bulge will be perectly relaxed in the luid limit, but that the load will not be perectly compensated. 33 I hyd 11 hyd +I A.1.. Case : The equilibrium orm Next, we turn to the scenario in Fig. 1B. The relevant expression or the total inertia tensor is trivially derived rom Eq. (A.7) by replacing the hydrostatic (LT = 0) luid tidal Love number with the more general case T. This yields I ij (t) = I 0 δ ij + a5 T [ω i (t)ω j (t) 13 ] ω (t)δ ij + Iij L (t). (A.1) In this case, the component o the inertia tensor that perectly reorients to a change in the rotation vector (as in Figs. 1B1, 1B3, and 1B4) is given by the irst two terms on the right-hand side o Eq. (A.1). These terms represent the equilibrium (i.e., relaxed) orm achieved by a rotating planet with an elastic shell (Mound et al., 003). We will denote this equilibrium orm as I eq ij (t) = I 0δ ij + a5 T [ω i (t)ω j (t) 13 ] (A.13) ω (t)δ ij. Once again, this component o the total inertia tensor does not play a role in the long-term rotational stability o the planet since it provides no memory o any previous rotational state. Thereore, the reorientation o such a planet is governed by the