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Bills et al.: Rotational Dynamics of 119

Rotational Dynamics of Europa

Bruce G. Bills NASA Goddard Space Flight Center and Scripps Institution of Oceanography

Francis Nimmo University of California, Santa Cruz

Özgür Karatekin, Tim Van Hoolst, and Nicolas Rambaux Royal Observatory of Belgium

Benjamin Levrard Institut de Mécanique Céleste et de Calcul des Ephémérides and Ecole Normale Superieure de Lyon

Jacques Laskar Institut de Mécanique Céleste et de Calcul des Ephémérides

The rotational state of Europa is only rather poorly constrained at present. It is known to rotate about an axis that is nearly perpendicular to the plane, at a rate that is nearly constant and approximates the mean orbital rate. Small departures from a constant rotation rate and os- cillations of the rotation axis both lead to stresses that may influence the location and orienta- tion of surface tectonic features. However, at present geological evidence for either of these processes is disputed. We describe a variety of issues that future geodetic observations will likely resolve, including variations in the rate and direction of rotation, on a wide range of timescales. Since the external perturbations causing these changes are generally well known, observations of the amplitude and phase of the responses will provide important information about the internal structure of Europa. We focus on three aspects of the rotational dynamics: obliquity, forced librations, and possible small departures from a synchronous rotation rate. Europa’s obliquity should be nonzero, while the rotation rate is likely to be synchronous unless lateral shell thickness variations occur. The tectonic consequences of a nonzero obliquity and true polar wander have yet to be thoroughly investigated.

1. INTRODUCTION state of Europa is that its rotation rate is very close to syn- chronous, and the direction of its spin pole is very close to The primary objective of this chapter is to describe a that of its orbit pole, so the obliquity is very small. variety of rotational phenomena that Europa is expected to As we will discuss below, dynamical arguments suggest exhibit, and that will, when properly observed, provide that the mean rotation rate is indeed equal to the mean or- important diagnostic information about the internal struc- bital rate, but that the obliquity (the angle between orbit pole ture. The rotational state of any or is impor- and spin pole), while small, is nonzero. These assertions are tant to understand for at least three reasons. First, proper at odds with what is usually assumed about the rotation of collation of observations at various epochs and locations Europa in studies attempting to interpret the tectonics of requires a good understanding of the rotation rate and di- the body, and thus need to be carefully explored and sup- rection of the rotation pole. , some aspects of the ported. In addition, gravitational torques from are mean rotation state, and all plausible variations in the rota- expected to cause librations, which are periodic variations in tion state, provide information about the structure of the the direction and rate of rotation. As the position and interior. Third, variations in rotation rate or rotation axis of Jupiter are well known, the amplitude and phase of these orientation lead to global stresses, thus surface tectonic fea- variations are diagnostic of internal structure. tures may constrain the existence of such processes. All that This chapter will consist of three main parts. (1) The first is presently known, from observations, about the rotation part will discuss the obliquity history of Europa, and explain

119 120 Europa how observations of the current orientation of the spin pole about the internal structure of the body. Most of the remain- will constrain the moments of inertia of the body. (2) The der of this section will attempt to explain that connection. second part will discuss forced librations, with primary em- phasis on longitudinal librations. It will also be discussed 2.1. Moments and how observations of the amplitude and phase of the periodic variations in rotation rate will constrain internal structure. Measurements of the mass M and mean radius R of a (3) The third part will discuss arguments for and against satellite yield a mean estimate, which for Europa nonsynchronous rotation (NSR). Most models of tidal dis- is already rather well known (Anderson et al., 1998a) sipation predict that a body like Europa will be close to a synchronous rotation state, but that the rotation rate at which 〈ρ〉 = (2989 ± 46) kg m–3 (1) the tidal torque vanishes differs slightly from exact synchro- nism. From a dynamical perspective, the question is whether The Galilean show an interesting progression of gravitational torques on a permanent asymmetry are large decreasing density with increasing distance from Jupiter enough to “finish the job.” We will also briefly discuss the (Johnson, 2005), but density only rather weakly constrains associated issue of true polar wander (TPW) of the shell. internal structure (Consolmagno and Lewis, 1978). How- ever, the moments of inertia provide additional constraints 2. OBLIQUITY on the radial density structure (Bills and Rubincam, 1995; Sotin and Tobie, 2004). There are several ways to estimate In this section we discuss the obliquity of Europa. The the moments of inertia, and the rotational dynamics pro- obliquity of a planet or satellite is the angular separation vide several options. between its spin pole and orbit pole, or equivalently, the Perturbations of trajectories, either on cap- angle between the equator plane and orbit plane. For , tured or during a close flyby, can be used to infer the current obliquity is 23.439° (Lieske et al., 1977), which the low-degree terms in the gravitational potential. The sets the locations of the tropics of Cancer and Capricorn, coefficients of harmonic degree 2 in the gravitational po- which are the northern and southern limits at which the tential of a body are related to the principal moments of appears directly overhead, and the Arctic and Antarctic inertia (A < B < C) via (Soler, 1984) circles, which are the equatorward limits beyond which the 2 Sun does not rise on the days of the corresponding solstices. J2MR = C – (A + B)/2 (2) 2 Earth’s obliquity is presently decreasing (Rubincam et al., C2,2MR = (B – A)/4 1998), and oscillates between 22.1° and 24.5° with a 41-k.y. period (Berger et al., 1992; Laskar et al., 1993), due to There are, in general, five terms of harmonic degree 2, and lunar and solar torques on Earth’s oblate figure. The associ- six independent terms in the inertia tensor. However, if the ated changes in seasonal and latitudinal patterns of insola- coordinate axes are chosen to coincide with the principal tion have a significant impact upon global climate (Milanko- axes of the inertial ellipsoid, then only these two potential vitch, 1941; Hays et al., 1976; Hinnov and Ogg, 2007). terms remain. Measurements of the gravitational field alone For planetary satellites, the solar radiation cycles can be do not suffice to determine the moments of inertia, as the more complex, depending as they do upon the obliquity of system of equations is underdetermined by 1. the planet, inclination of the satellite orbit, and obliquity One approach to estimating those moments, in the ab- of the satellite itself. Several recent studies of the radiative sence of further constraints, is to assume that the body is in environment of (Flasar, 1998; Roos-Serote, 2005; , and that the degree-2 harmonics of Tokano and Neubauer, 2005) have referred to the angular the field reflect a response to the well-known tidal separation between the spin pole of Titan and the orbit pole and rotational potentials. This approach was developed by of as Titan’s obliquity. This angle, approximately Hubbard and Anderson (1978) and applied to Europa by equal to the 26.73° dynamical obliquity of Saturn (Ward and Anderson et al. (1998a) (see chapter by Schubert et al.). That Hamilton, 2004; Hamilton and Ward, 2004), is certainly the method provides the current best estimates of the moments relevant angle for consideration of radiative input to the of inertia of the Galilean satellites (Anderson et al., 1996a,b, . However, from an orbital and rota- 1998a,b). In terms of the dimensionless polar moment tional dynamics perspective, the important angle is the much smaller separation between the spin pole of Titan and the C c = (3) pole of its own orbit about Saturn. Likewise for Europa, MR2 we are interested in its own dynamical obliquity. As the obliquity of Jupiter is only about 3.1° (Ward and Canup, the Galilean satellite values are {0.379, 0.346, 0.311, 0.355}, 2006), the solar radiative pattern at Europa is simpler than for , Europa, and , respectively (Schu- for Titan. bert et al., 2004). Recall that a homogeneous sphere has The obliquity of Europa is not currently known, other c = 2/5, and smaller values indicate a more centrally con- than that it is certainly quite small (Lieske, 1979). However, densed structure. when measurement accuracies increase sufficiently to allow The hydrostatic assumption can be verified if both J2 and a determination of that value, it will provide information C2,2 can be measured independently, since for a hydrostatic Bills et al.: Rotational Dynamics of Europa 121

body the ratio of these two quantities is 10/3 (e.g., Murray The most complex spin pole motion occurs when the and Dermott, 1999). However, determination of J2 requires orbit pole rates and spin pole rate are comparable. In that polar or near-polar flybys, while C2,2 requires equatorial or case, the motion of the spin pole is resonantly enhanced. near-equatorial trajectories, so that it is not always possible These features are extensively discussed in the literature on (as at Callisto) (Anderson et al., 1998a) to verify the hy- obliquity variations (Ward, 1973, 1992; Bills, 1990). drostatic assumption. The orbital precession amplitudes for Earth and Mars are Other approaches to determining internal structure rely similar, and the periods are identical, but Mars has obliq- upon the fact that the rotational dynamics of the body are uity variations that are substantially larger than those for controlled by the moments of inertia. It is often the case Earth because the spin pole precession rate of Earth is too that the applied torques are well known, and that observa- fast for resonance enhancement, whereas Mars does see tions of the rotational response thus constrain the moments. resonant effects. In fact, it has been claimed that the obliq- For a rapidly rotating body, like Earth or Mars, the solar uity variations for Mars are chaotic (Touma and Wisdom, gravitational torque acting on the oblate figure of the body 1993; Laskar and Robutel, 1993). However, even relatively causes it to precess about its orbit pole. If we ignore effects small amounts of dissipation will suppress the chaotic varia- of an eccentric orbit, the precessional motion of the unit tions (Bills and , 2005; Bills, 1994, 1999, 2005). vector sˆ, aligned with the spin pole, is governed by A resonant enhancement of spin pole motion requires orbital precession rates comparable to the spin pole preces- dsˆ sion rate. However, for most bodies, the dif- = α (nˆˆˆ · s) (s × n) ˆ (4) dt ference between polar and equatorial moments is a small fraction of either value, and thus the spin pole precession where nˆ is the orbit pole unit vector, and α is a spin preces- rates are much slower than the spin or orbital rates. How- sion rate parameter given by (Kinoshita, 1977; Ward, 1973) ever, there are often orbit-orbit interactions, so-called secu- lar perturbations, that have periods much longer than the 3 n2 C – (A + B)/2 orbital periods. It is a near commensurability between the α = spin pole precession rate of Mars, and some of its secular 2 ω C (5) orbital variations, which give rise to the large obliquity vari- 3 n2 J ations. = 2 2 ω c The situation at Europa is not particularly well approxi- mated by either Earth or Mars. A somewhat more relevant with n the orbital mean motion, and ω the spin rate. analog is provided by the . The rotational state of the If both of the degree-2 gravity coefficients and the spin Moon is well approximated by three features, first enunci- pole precession rate α can be measured, as has been done ated by G. D. in 1693, which can be paraphrased for Earth (Hilton et al., 2006) and Mars (Folkner et al., as (1) the spin period and orbit period are identical; (2) the 1997), then the polar moment C can be estimated, without spin axis maintains a constant inclination to the ecliptic requiring the hydrostatic assumption. That is, in fact, how pole; and (3) the spin axis, orbit pole, and ecliptic pole re- the moments of inertia of those two bodies were determined. main coplanar. A difficulty with this approach, of directly observing the The first of these had, of course, been known much ear- spin pole precession rate, is that typical rates are very low. lier, and the dynamical importance of the second and third For Earth and Mars, the spin pole precession rates are 50 laws was not fully appreciated until much later. It is now and 10 arcsec/yr, respectively. The challenge of seeing the understood (, 1966; Peale, 1969; Ward, 1975a; spin pole of Europa precess, without a relatively long-lived Gladman et al., 1996) that adjustment of the obliquity to lander, would be formidable. Fortunately, there are better achieve co-precession of the spin and orbit poles about an ways to accomplish the same objective. invariable pole can occur without synchronous locking of the spin and orbit periods. That is, Cassini’s first law is at 2.2. Spin Pole Trajectories least partially decoupled from the other two. In fact, most features of the lunar spin pole motion are reproduced in a We now consider briefly how the spin pole precession model where the lunar gravity field is approximated as axi- trajectory depends on the motion of the orbit pole. In the symmetric (Wisdom, 2006). simplest case, where the orbit plane orientation remains The condition for this coplanar precession, in nearly constant, the spin pole trajectory is along a circular cone circular orbits, can be written as (Ward, 1975a) centered on the orbit pole. In that case, the spin pole main- tains a constant obliquity as it precesses. If the orbit pole (v + (u – v)cos[ε])sin[ε] = sin(i – ε)(6) is itself precessing, as is generally the case, the spin pole trajectory can be quite complex. If the orbit pole is precess- where i is the inclination of the orbit pole to the invariable ing much faster than the spin pole can move, then the spin pole, and ε is the obliquity or separation of spin and orbit pole essentially sees a spin-averaged orbit pole, and pre- poles. The parameters u and v are related to the moments cesses at nearly constant inclination to the invariable pole, of inertia of the body, and the relative rates of orbital mo- which is the pole about which the orbit is precessing. tion and orbital precession. 122 Europa

The first of these parameters has the form (1969), the usual numbering of these separate Cassini states ˆ {S1, S2, S3, S4} is that S1 is sˆ near to k and on the same ˆ u = U p (7) side as nˆ; S2 is somewhat farther from k, and on the opposite side from nˆ; S3 is retrograde, and thus nearly antiparallel ˆ where the moment dependent factor is to nˆ; and S4 is on the same side of k as S1, but farther from nˆ and kˆ. These spin states represent tangential intersections of a sphere (possible orientations of the spin pole) and a 3 C – A 3 J + 2C U = = 2 2,2 (8) parabolic cylinder representing the Hamiltonian. 2 C 2 c If the radius of curvature of the parabola is too large, there are only two possible spin states, otherwise there are The relative rates of orbital motion and orbit plane preces- four. At the transition point, states 1 and 4 coalesce and sion is vanish. In the axisymmetric case, for which v = 0, the tran- sition occurs at (Henrard and Murigande, 1987; Ward and n p = (9) Hamilton, 2004) dΩ/dt where n is the mean motion, and Ω is the longitude of the u = –(sin[i]2/3 + cos[i]2/3)3/2 (16) ascending node of the orbit. For most bodies, the node re- gresses and this ratio is thus negative. The second param- and eter has a similar factorization tan[ε] = –tan[i]1/3 (17) v = V p (10) If the magnitude of the parameter u is larger than the value with given by equation (16), then all four Cassini states exist. All four of the Cassini states represent equilibrium con- figurations. That is, if the spin pole sˆ is placed in such a 3 B – A 3 C V = = 2,2 (11) state, it will precess in such a way as to maintain a fixed 8 C 2 c ˆ orientation relative to nˆ and k. The states S1, S2, and S3 are stable, in the sense that small departures from equilibrium When these substitutions are made, the constraint equa- will lead to finite amplitude librations. Each of these states tion (6) can be written in either of the alternative forms is the dynamical center of a domain of stable librations, and these three domains cover the entire sphere. In contrast, S4 ε ε] ε] 2c sin(i – ) = 3p (C2,2 + (J2 + C2,2) cos[ ) sin[ (12) is unstable. On longer timescales, when tidal effects are included, only states S1 and S2 appear as secularly stable 8C sin(i – ε) = (13) (Peale, 1974), and Gladman et al. (1996) have further ar- ε] ε] 3p (B – A + (4C – B – 3A) cos[ ) sin[ gued that whenever S1and S2 both exist, S1 will be favored. Returning briefly to consideration of the Moon, it is the These constraint equations are linear in polar moment, but only body in the solar system known to occupy Cassini state nonlinear in obliquity. Thus, if both gravitational coeffi- S2. Ward (1975) has argued that the Moon initially occu- cients and the inclination and obliquity can be measured, pied S1, but during its orbit evolution outward to the present we could rather trivially solve for the polar moment as distance from Earth, the states S1 and S4 merged and dis- appeared, forcing the Moon to transition to state S2. ε ε 3p (C2,2 + (J2 + C2,2) cos[ ]) sin[ ] c = (14) 2.3. Application to Europa 2 sin[i – ε] From the perspective of obliquity dynamics, there are two important ways in which Europa differs from the Moon. 3p (B – A – (3A + B) cos[ε]) sin[ε] C = (15) The orbit precession for Europa is not steady, because its ε ε 2 4sin[i – ] – 3p sin[2 ] orbit is significantly perturbed by Io, Ganymede, and Cal- listo (Lieske, 1998; Lainey et al., 2004a,b) and the presumed If only one of J2 and C2,2 are known, the hydrostatic as- presence of an icy shell decoupled from the underlying sumption can be used (see section 2.1). When solving these material implies that the moments of inertia of the shell constraint equations for obliquity, the situation is somewhat itself need to be considered. The first effect, as discussed more subtle. In general, there are either two or four distinct below, can be included by considering precession effects real solutions for obliquity, depending upon the values of on a mode-by-mode basis. The latter effect is considered the input parameters. In all cases, the spin pole sˆ, orbit pole in some detail in the following discussion on forced libra- nˆ, and invariable pole kˆ are coplanar. It is also convenient to tions (section 3). define a signed obliquity, with positive values correspond- Obliquity variations for dissipative bodies in nonuni- ing to sˆ and nˆ on opposite sides of kˆ. Following Peale formly precessing orbits can be easily accommodated via Bills et al.: Rotational Dynamics of Europa 123

a linearized analysis of the torque balance. Similar linear damped forced spin evolution takes the form analyses of spin pole precession have been constructed γ previously, in the context of studying obliquity variations S[t] =∑ sj (exp[ifjt]) exp[i j] (24) of the Earth (Miskovitch, 1931; Sharaf and Boudnikova, j 1967; Vernekar, 1972; Berger, 1976), Mars (Ward, 1973, The second of the terms in square brackets in the original 1992), (Ward and de Campli, 1979; Yoder and Ward, equation for forced response is removed by dissipation. To 1979; Yoder, 1995, 1997), (Bills and Comstock, obtain this expression, we allow a finite value of β, take 2005), and the Galilean satellites (Bills, 2005). the limit as t → ∞, and then set β back to zero. It is evi- The first step in that process is to represent the unit vec- dent that the orbit pole and spin pole trajectories are char- tors sˆ and nˆ, which point along the spin pole and orbit pole, acterized by identical frequencies and phases, but different in terms of complex scalars S and N, by projecting each of amplitudes. them onto the . In the present context, that This solution can be viewed as a rough generalization will be approximated by Jupiter’s equator plane. That is, of the Cassini state for the case of nonuniform orbit preces- we are ignoring the slow precession of Jupiter’s spin pole, sion. In the case of a single orbit precession frequency, the since it is much slower than the Galilean satellite preces- expected end-state for dissipative spin evolution is a spe- sion rates. If we also ignore the variations in satellite or- cial situation in which the obliquity has adjusted to a value bital eccentricity values, the governing equation for spin at which the system maintains a constant relative geometry. pole precession can now be written in the simple linear form That is, the spin pole and orbit pole remain coplanar with the invariable pole as the spin pole precesses about the or- ddS = –iα (N – S) (18) bit pole and the orbit pole precesses about the invariable dt pole (Colombo, 1966; Peale, 1969; Ward, 1975b; Henrard where α is the spin precession rate parameter (equation (5)). and Murigande, 1987). If the orbit pole evolution is represented via the series If the orbit pole precession is not steady, no such copla- nar configuration is attainable. However, the motions of the γ N[t] =∑ nj exp[i (fjt + j)] (19) orbit and spin poles can achieve a mode-by-mode equiva- j lent of the Cassini state. The solution above is such that each γ where fj is an orbit pole precession rate and j represents a mode of the orbit pole precession, with amplitude nj, rate γ phase offset, then the corresponding solution for the spin fj, and phase j, has a corresponding mode of spin pole pole can be written simply as precession with rate and phase identical to the orbit mode values, and with an amplitude proportional to the orbit am- α S[t] = Sfree + Sforced (20) plitude. The constant of proportionality is just the ratio / α ( + fj) of the spin precession rate to the relative spin-orbit where the free pole motion, which depends only on the ini- precession rate. tial condition, is The angular separation between spin pole and orbit pole has a simple expression α Sfree = S[0] exp(i t) (21) Δ ≡ Δ γ S[t] S[t] – N[t] =∑ sjexp[i(fjt + j)] (25) and the forced motion is j The amplitude of each term is just the difference in ampli- α γ Sforced =s∑ j [exp[ifjt] – exp[i t]] exp[i j] (22) tudes of the spin and orbit solutions j with amplitudes given by α –f Δs = s – n = – 1 n = j n (26) j j j α + f j α + f j α j j s = n (23) j α j + fj The magnitude of this complex quantity is the obliquity. It has the same frequencies as the orbital inclination, but dif- Each term in the series describing the orbit pole has a corre- ferent amplitudes. sponding term in the forced spin pole series. The spin rate Application of this theory to Europa has been made α parameter is positive, and all the orbit pole rates fj are (Bills, 2005), using the orbit model of Lieske (1998) and α negative. If one of the sums + fj is close to zero, then the moment estimates from Anderson et al. (1998a). The pre- corresponding amplitude in the spin trajectory will be am- dicted spin pole precession rate parameter α and main or- plified. bital inclination rates are given in Table 1. The average in- Dissipation can be easily introduced by simply making clination of Europa’s orbit to Jupiter’s equator plane is ~0.5° the spin precession parameter complex: α → α + i β. When and, based on the measured gravitational moments, the pre- included this way, the dissipation completely damps the free dicted average obliquity is ~0.1°. term and somewhat modifies the forced terms. Assuming If the ice shell is fully decoupled from the deeper inte- that the damping term is small, the resulting model for rior, then the moments of inertia of the shell will determine 124 Europa

TABLE 1. Precession rates. the other factors are known, the polar can be determined directly. The analysis is similar but more Rate Period complicated if, as at Europa, dissipation or non-uniform Parameter (10–3°/) (yr) precession occur. Although Europa’s current obliquity is not α 191 5.16 known, it is likely to be ~0.1° unless the ice shell is de-

f1 –133 7.42 coupled from the interior, in which case it is likely to be f2 –32.6 30.2 somewhat larger. f3 –7.18 137 f4 –1.75 560 3. LIBRATIONS

We consider the physical librations (i.e., periodic vari- the spin pole precession rate. The ice shell thickness esti- ations in the rotation of Europa) due to the gravitational mates from Ojakangas and Stevenson (1989a) imply mo- torque of Jupiter on Europa’s nonspherical shape. The other ments of inertia of solar system bodies also exert torques on Europa but the Jupiter torque is at least three orders of magnitude larger A 1 –58.24 than the torque by any other body because of Jupiter’s mass and distance. Physical librations can be in longitude (in the B = 0.610M R2 1 + +48.76 × 10–3 (27) s equatorial plane) as well as in latitude (normal to the equa- C 1 +9.78 torial plane). The former corresponds to changes in rotation rate (see section 4 below) and the latter to variations in polar where Ms is the satellite mass, and a corresponding spin orientation. Among the primary parameters of interest are pole precession rate parameter of the deviation from spherical symmetry given by the mo- 2 ments of inertia difference B – A [or C2,2 = (B – A)/4MR ] α 2 s = 0.0218 n = 2.21°/day (28) for the longitudinal librations and C – (A + B)/2 (=MR J2) for the latitudinal librations. Orbital parameters such as the or roughly 12 times the solid body result. This is much eccentricity, e, and the obliquity, ε, are also important for higher than any of the secular orbital rates, but a value in the longitudinal and latitudinal librations, respectively. that range could produce a resonant interaction with short- Because of Europa’s likely small obliquity (see section 2 period terms, and thereby produce a larger obliquity. above), the latitudinal librations are at least one order of Two aspects of these moment estimates deserve com- magnitude smaller (Henrard, 2005). Secular changes in lati- ment. First is that B > C, contrary to the stated convention tude (true polar wander) may arise under some circum- (just above equation (2)). This implies an unstable rotational stances, and have important geological effects (Ojakangas configuration, as is discussed at length by Ojakangas and and Stevenson, 1989b) (see also section 4), but here we Stevenson (1989b). Second is that the fractional departures consider only the longitudinal libration (Comstock and Bills, from spherical symmetry are substantially larger than the 2003). values cited by Ojakangas and Stevenson (1989a,b). How- ever, their estimates include isostatic compensation of the 3.1. Rigid Librations ice shell by the underlying fluid layer, which is certainly relevant to the body as a whole, but we are interested in the In the absence of a subsurface ocean, Europa’s response moments of the shell itself. How appropriate these moment can be approximated as that of a rigid triaxial body de- estimates are for Europa’s spin state is difficult to determine, formed from sphericity by centrifugal forces as well as the but we note that a recent determination of the obliquity of permanent tides of Jupiter. In this case, the spin angular mo- Titan (Stiles et al., 2007), suggests that it is larger than ex- mentum H of Europa can be given as pected for a solid body, and can be interpreted as indicative of a decoupled shell (Bills and Nimmo, 2008). dθ H = C (29) In the absence of direct detection of the obliquity of Eu- dt ropa, we might hope to place constraints on that parameter via its influence on the tidal stress field and resulting tec- where θ is the angle of rotation of Europa (see Fig. 1). By tonic patterns on the surface. Some progress has been made neglecting tidal deformations over the period of interest in that direction (e.g., Hurford et al., 2006) (see also chap- (i.e., the ), the change in rotation rate is pro- ter by Kattenhorn and Hurford). portional to the gravitational torque L of Jupiter on Europa.

2.4. Summary dH d2θ = C = L (30) dt dt2 In simple cases, a satellite’s spin precession rate depends on its gravitational moments, its orbital inclination and obliq- Europa, assumed to be in synchronous rotation, does not uity, and its polar moment of inertia (equation (14)). If all have a uniform orbital motion due to its noncircular orbit Bills et al.: Rotational Dynamics of Europa 125

in spin-orbit resonance. The separatrix, which separates li- brational from circulatory behavior, is shown with a bold line. Figure 2 shows three equilibrium points. The points with coordinates (+π/2,1) and (–π/2,1) are unstable hyper- bolic points, whereas the point at the center (0,1) corre- sponds to an elliptical point that is stable. In the following, we describe the librational motion close to the elliptical point where the amplitude of libration is small. In this case, the time taken for θ to move along one librational curve π ω ω is equal to 2 / f where f = n 3(B – A)/C is the free li- bration frequency, which is determined by solving equa- tion (32) using the average value of the second term. For small angular difference θ – f, equation (32) can be Fig. 1. Geometry of Europa’s spin-orbit motion. solved by expressing the time-dependent parameters (f, r) in terms of series in eccentricity e and mean anomaly M (see Murray and Dermott, 1999). The main periodic forc- (e = 0.0094). Europa’s long axis hence is not always di- ing term has a period equal to the orbital period (3.55 days). rected exactly toward Jupiter. The gravitational torque of The first order solution in e yields the libration amplitude as Jupiter on Europa’s nonspherical figure tends to modify the satellite’s rotation, resulting in the longitudinal librations. ω2 2 f e (B – A) This torque depends on the differences of the equatorial mo- Aθ = – ≈ 6e (33) 2 ω2 ments of inertia B – A (e.g., Goldreich and Peale, 1966) n – f e C

3 α 3 Since the moment of inertia ratio (B – A)/C is usually 2 θ ω L = – n (B – A) sin[2( – f)] (31) much smaller than unity, the free libration frequency f 2 r ω ≈ compared to the orbital frequency n is small ( f 0.067n). The above equation shows that the amplitude of libra- where n is the mean motion of Europa, a the semimajor tion depends linearly on the difference of equatorial mo- axis, f the true anomaly, and r the distance between the mass ments of inertia B – A and the eccentricity e and is inversely centers of Jupiter and Europa (Fig. 1). proportional to the polar moment of inertia C. Values for The governing equation describing the rotation of Europa the moment of inertia ratio (B – A)/C = 0.0015 and the po- becomes lar moment of inertia factor C/MR2 = 0.3479 can be de- –6 duced by using the measured C2,2 = 131.5 ± 2.5 10 from d2θα3 3 the Galileo flyby (Anderson et al., 1998b) and by assum- C + n2 (B – A) sin[2(θ – f)] = 0 (32) dt2 2 r ing that Europa is in hydrostatic equilibrium (Hubbard and Anderson, 1978). The resulting amplitude of forced libra- Equation (32) is a differential equation of the second tions is then 8.52 × 10–5 rad, which corresponds to an equa- order in time t analogous to the classical pendulum equation torial surface displacement of ±133 m over an orbital pe- with the exception that the second term on the lefthand side riod. contains time-dependent variables r and f due to the eccen- tric orbit. Europa can librate, circulate, or tumble chaoti- cally depending on the energy .of the system, or equivalently θ θ on the initial conditions ( 0, 0). The qualitative behavior of the librational motion of Europa can be best described by using the Poincaré surface of sections (Poincaré, 1892). Figure 2 shows a surface of section plot resulting from. the numerical integration of equation (32) where θ and θ/n are plotted at each periapse passage. The global behavior of the rotation of Europa is regular and the chaotic zones are not visible at this scale. The central area of the graph with closed curves represents the libration zone where the mean rotational velocity is equal to n and Europa is in spin-orbit resonance. Outside this region, Europa is out of resonance, 2 θ the system has more energy than 3n (B – A)/4, and is. un- bounded. The width of the libration area is equal to θ/n = 0.134 and represents twice the maximum increment in the instantaneous velocity of Europa for which Europa remains Fig. 2. Phase diagram for spin-orbit dynamics of a rigid Europa. 126 Europa

the mantle (Peale, 1976; Peale et al., 2002; Rambaux et al., 2007). Future spacecraft missions to Europa will certainly be able to detect radial tidal motions (Wahr et al., 2006), and may also resolve libration effects.

3.2. Differential Librations

If an ocean exists under the icy shell, Europa can no longer be assumed to perform rigid rotations. The interior [likely composed of a metallic core and a silicate mantle (Anderson et al., 1998b)] and the icy shell perform differen- tial librations under the gravitational torque of Jupiter, with amplitudes proportional to their respective equatorial mo- ment of inertia differences. If their motions are decoupled, the libration amplitudes would be inversely proportional to Fig. 3. Librational response of a rigidly rotatating Europa, with their respective polar moments of inertia (Ci and Cs where both free and forced components. the subscripts i and s stand for the interior and the shell, respectively). The amplitude of the libration on Europa’s surface would then be amplified by a factor C/Cs in com- Figure 3 illustrates this forced motion and also shows the parison with the rigid librations, provided that the equato- presence of a long-period oscillation that corresponds to the rial moment of inertia difference (B – A) of the shell and free libration frequency of the dynamical system. The li- rigid satellite are similar. Depending on the thickness of the brational response can thus be decomposed into an angular icy shell (1 km < h < 100 km), C/Cs varies approximately motion at the orbital period, i.e., forced libration, and a ro- between 10 and 1000, suggesting periodic equatorial dis- tation rate variation at a longer period, i.e, free libration. It placements up to hundreds of kilometers for a decoupled is a characteristic of the free libration period that the associ- thin icy shell. Such displacements would generate large sur- ated amplitude is arbitrary. However, it is expected to be face stresses, likely visible as tectonic features, as well as close to zero since tidal dissipation within the satellite damps significant dissipation within the icy shell. the free librations (as with the free obliquity term in the pre- In reality, it is more likely that the motions of the icy ceding section). The free libration frequency depends on the shell and the interior are coupled, through either the grav- internal structure, whereas the period of the forced libra- itational force or through pressure components in the tion is determined by the period of the Jupiter torque, i.e., ocean caused by the rotational potential [so-called Poincaré 3.55 days. Note that for a sphere or an oblate spheroid with torques; see Ojakangas and Stevenson (1989b)]. For lon- B – A = 0, the free libration period goes to infinity. With gitudinal librations, only the former are likely to be impor- the estimated value of (B – A)/C = 0.015, the free libration tant, and we focus on them here. period of Europa is 52.7 days. An internal gravitational torque between the solid inte- Equation (33) shows that if the libration amplitude can rior and the outer icy shell arises when the principal axes be measured along with C2,2, then the polar moment of iner- of the icy shell and the interior are not aligned (see Fig. 4). tia can be determined directly. Investigations of Mercury The gravitational coupling has been studied in the context using radar speckle interferometry have detected a forced of the rotation variations of the Earth, in addition to other libration component (Margot et al., 2007). The amplitude couplings due to pressure on topography, and viscous and of that libration is indicative of decoupling of the core from electromagnetic torques (see, e.g., Buffett, 1996; Szeto and Xu, 1997). For terrestrial , the principal gravitational coupling occurs between the large mantle and the small solid inner core. In the case of Europa, the massive inte- rior with a much larger moment of inertia than the thin outer icy shell has a very significant effect on the rotation of the surface layer. Taking into account gravitation as the princi- pal internal coupling, the libration of Europa can be ex- pressed in terms of two equations for the two solid layers, the icy shell and the interior (Van Hoolst et al., 2008)

d2θ α 3 s 3 2 Cs + (Bs – As) n dt2 2 r (34) θ θ θ Fig. 4. Schematic geometry of differential librations of Europa. sin[2( s – f)] = –K sin[2( s – i)] Bills et al.: Rotational Dynamics of Europa 127

that of the interior. The gravitational coupling between the d2θ α 3 i 3 2 massive interior and the thin outer shell does not allow the Ci + (Bi – Ai) n dt2 2 r (35) surface to librate with an amplitude larger than 140–150 m sin[2(θ – f)] = +K sin[2(θ – θ )] for reasonable ice shell thicknesses (several tens of kilo- i s i meters). This is only ≈10% larger than the rigid body libra- tion amplitude. These equations are coupled by the gravitational cou- Because the libration amplitude is a function of the in- pling parameter ternal structure, its observations can yield information about the interior of Europa. If Europa reacts as a rigid body to 4πG 8π the gravitational torque of Jupiter, the measurements pro- K = ρ β ((ρ – ρ )β r5 + (ρ – ρ )β r5) (36) 5 15 s s m s m m c m c c vide information principally on the moment of inertia ratio (B – A)/C of the satellite. Europa’s surface performs larger where β is the equatorial flattening; ρ is the density; r is librations in the presence of a subsurface ocean. If the gravi- the radius; and the subscripts s, m, and c stand for the icy tational torque between the shell and the interior is the dom- shell, silicate mantle, and iron core, respectively. Here, the inant coupling mechanism, the existence of a subsurface internal ocean is assumed to have the same density as the ocean can be determined if a libration measurement preci- icy shell. In the above equations the dynamics of the ocean sion on the order of several meters can be achieved. With a are neglected; in particular, the effect of pressure on the el- better accuracy, information on the thickness of the icy shell lipsoidal boundaries is not included (cf. Ojakangas and Ste- can be obtained. As the position of a spacecraft in orbit will venson, 1989b). be perturbed mostly by the librations of the solid interior The polar moments of inertia and the equatorial flattening (Wu et al., 2000), simultaneous observations of the libra- of Europa’s inner layers can be calculated from Clairaut’s tions of both the interior and the surface could enhance the theory by taking into account both rotational and tidal defor- knowledge of Europa’s internal structure, in particular the mations (e.g., Murray and Dermott, 1999). Note that gravi- density and radius of the solid interior. tational coupling would be small if Europa were not com- ρ ≈ ρ ≈ ρ pletely differentiated ( s m c), or if internal flattening 4. NONSYNCHRONOUS ROTATION β ≈ β ≈ is negligible ( m c 0). The internal gravitational coupling strongly reduces the We now consider the mean rotation rate of Europa. It is libration amplitude with respect to decoupled shells to clear that Europa is very nearly in a synchronous state, simi- maximum surface displacement of about 140–150 m (Van lar to that of the Moon, in which the rotation period and Hoolst et al., 2008). The libration amplitude increases lin- orbit period are exactly equal. However, both theoretical and early proportional to the ice shell thickness for h > 10 km. observational arguments have been presented for Europa This is not surprising considering the linear relationship deviating slightly from a state of exact synchronous rota- between Cs and h. On the other hand, the equatorial dis- tion. This question is important because nonsynchronous placement of the interior remains constant at about 140 m, rotation (NSR) results in tectonic stresses that can be up to independent of the thickness of the icy shell, because the 2 orders of magnitude larger than the diurnal tidal stresses influence of the thin outer layer on the massive interior is thought to be responsible for many of the surface tectonic small. A particular consequence of the inclusion of the features (Greenberg et al., 1998) (see also chapters by Sotin gravitational coupling between the shell and the interior is et al. and Kattenhorn and Hurford). However, we will dif- that the libration amplitude is resonantly amplified at h ≈ fer from most previous works and argue below that NSR is 1 km. The resonance occurs when one of the two free li- unlikely on dynamical grounds, unless lateral variations in bration periods of the angular momentum equations (34) icy shell thickness occur, and that a small but finite obliquity and (35) approaches the orbital period. might instead be responsible for the geological observations.

3.3. Discussion 4.1. Geological Evidence

The libration amplitude of Europa depends on the pres- Direct determination of the rotation rate of Europa’s ice ence of a global ocean between the icy shell and the sili- shell should be possible by comparing the position of vari- cate mantle. In its absence, Europa performs rigid librations ous surface features as imaged at different times, but up to with an amplitude of ≈133 m. In the presence of a subsur- now, no image has combined a sufficient resolution and/ face ocean, the icy shell and the interior (likely composed or a precise location to be used in such a way. Hoppa et al. of a silicate mantle and metallic core) perform differential (1999a) identified a pair of images each taken at the same librations. If their motions are decoupled, the amplitude of resolution (1.6 km/pixel) by the and Galileo the libration would be amplified by a factor of 10 to 1000 spacecrafts 17 years apart, which show the same region rela- with respect to the rigid libration, resulting in periodic equa- tive to the terminator. They found that the longitudinal varia- torial surface displacements up to hundreds of kilometers. tion of the terminator is within the error bar (estimated to However, the rotation of the icy shell is likely coupled with be ~0.5°), suggesting that the period of rotation relative to 128 Europa the direction of Jupiter must be greater than 12,000 years, formation and propagation of such features have been ex- with no compelling evidence of NSR. plained by diurnally varying tidal stresses (Hoppa et al., Observations and interpretations of tectonic features in 1999c). Again, accounting for the additional tidal stress due the icy shell have furnished other lines of evidence for a to NSR allows a better fit between observed and modeled NSR. Some classes of tectonic features on Europa are cycloid orientations (Hoppa et al., 1999c; Hurford et al., thought to be the result of diurnal tidal stresses of around 2007). Additional constraints on Europa’s rotation rate have 100-kPa amplitude caused by Europa’s eccentric orbit (e.g., been investigated by Hoppa et al. (2001). They analyzed a Hoppa et al., 1999c). The evolution and propagation of set of cycloidal ridges having cross-cutting relationships and some tectonic features is thought to follow the ever-chang- located in the Astypalaea region. Hoppa et al. (2001) con- ing stress field, in both amplitude and orientation. The ad- cluded that the cycloids probably formed at different lon- dition of a long-period component associated with a small gitudes from their current location and during different NSR supersynchronous rotation of the shell modifies the ampli- cycles, lending credence to the idea that only a few such tude of the tidal stress and induces an increasingly clock- ridges form over a single NSR period. If only a few such wise reorientation of tectonic features as the surface re- ridges form over one such period, determining the amount orients relative to fixed global patterns of tidal stress. The or timing of reorientation from lineament orientations is first data-based argument for NSR was the observation of very challenging (see above). systematic longitudinal and azimuthal shifts in the crack lo- Another observational argument suggesting that Europa’s cations from those predicted by an infinitesimal amount rotation is nonsynchronous is implicit in the work of Shoe- of NSR (e.g., McEwen, 1986). Three types of tectonic fea- maker and Wolke (1982) and Passey and Shoemaker (1982). tures are generally interpreted to constrain the rotation rate These authors predicted that leading-trailing hemisphere (Greenberg et al., 2002): crater would be found on synchronously rotating 1. Orientations of tectonic lineaments: The cross-cut- satellites. Galileo images show a weak apexantapex crater ting sequences of global and regional lineaments exhibit a density variation on Ganymede (Zahnle et al., 2001; Schenk rotation of azimuthal orientation over time, which is consis- et al., 2004), suggesting that NSR has smeared out much tent with predictions based on tidal stress calculations (e.g., of the expected signal. Europa’s surface is much less heavily Geissler et al., 1998). More detailed analyses have sug- cratered than that of Ganymede, but the global distribution gested that correlations are better if both diurnal and non- of 1-km craters on Europa shows no sign of a statistically synchronous tidal stresses are taken into account (Greenberg significant asymmetry (Schenk et al., 2004). This result sug- et al., 1998). The observation that lineaments in the south- gests a NSR period much less than the surface’s age. ern hemisphere show similar cross-cutting sequences but an In the next section, we discuss the dynamical arguments opposite sense of azimuth rotation to those in the northern concerning the possibility of a NSR of Europa. hemisphere is also consistent with the theory. However, it has been shown that NSR in the opposite sense to that pre- 4.2. Dynamical Considerations dicted provides an equally good fit to the observations (Sarid et al., 2004). This work, and subsequent work by the As for many satellites close to their primary, tidal torques same authors (e.g., Sarid et al., 2006), suggest that the ob- tend to slow down the rotation of an initially fast rotating servational evidence for NSR from lineament orientations satellite. Using the reasonable value Q ~ 100 for the tidal is weak, at best. These investigations do not provide con- quality factor leads to a rough estimate of Europa’s despin- straints on Europa’s rotation rate. One reason is that sev- ning timescale of about 105 yr (e.g., Goldreich and Peale, eral sets of older lineaments do not exhibit a monotonic 1966; Murray and Dermott, 1999), suggesting that Europa change in azimuthal orientation, suggesting that they may has currently reached its equilibrium mean rotation rate. have formed over several rotational periods and indicating The corresponding timescale for the eccentricity damping that there are various scales over which fracture interpreta- (~108 yr) is also lower than the age of the system, but the tion can be carried out (e.g., Kattenhorn, 2002). In this con- Laplace resonance between Europa, Io, and Ganymede text, it was concluded that the rotation period must be sig- maintains a nonzero orbital eccentricity e close to 0.01 (e.g., nificantly lower than the age of the surface (e.g., Greenberg Peale and Lee, 2002). et al., 2002), estimated to be ~108 yr (Zahnle et al., 2003). As a consequence of its nonzero eccentricity, Europa will 2. Strike-slip faults: Strike-slip displacements, in which experience periodically reversing torques due to the gravi- one part of the surface has sheared past another along a tational influence of Jupiter if its figure (as measured by fault, are common at Europa’s surface and have been ex- B – A) contains permanent asymmetries. These torques give plained as a result of “tidal walking” driven by diurnal tides rise to forced librations (see section 3), and the averaged (e.g., Hoppa et al., 1999b). According to the theory, the torque favors stable librations about spin rates of p × n sense of the shear depends on location and crack azimuth where p is an integer or an half-integer corresponding to but the fit to tidal stress patterns is better if one allows an spin-orbit resonances. However, if tidal dissipation occurs additional slow NSR of several tens of degrees (Hoppa et in the satellite, there are additional torques that arise due al., 1999b). to its tidally deformed figure. These tidal torques will tend 3. Cycloidal cracks: Many of the lineaments on Europa to drive the satellite to a rotation state slightly faster than appear in the form of arcuate cracks called cycloids. The synchronous (Greenberg and Weidenschilling, 1984). It is Bills et al.: Rotational Dynamics of Europa 129

the balance between the tidal torques and the permanent from mathematical simplifications than reliable physical torques that determines the rotational state of the satellite. arguments (see Greenberg and Weidenschilling, 1984). The Since the orbital eccentricity of Europa is small, only the simplest model of tidal response is generally called the “vis- possibility of capture into a synchronous state is discussed cous” model as described in Mignard (1980). He assumed here (Goldreich and Peale, 1966; Correia and Laskar, 2004). a constant time lag for any frequency component of the tidal In all the following discussion, we also suppose that Eu- perturbation. In other words, the tidally deformed surface ropa’s obliquity is zero (although see section 2 above). We of the satellite always assumes the equipotential surface it will consider additional effects that can arise when Europa’s would have formed a constant time lag Δt ago, in the ab- shell thickness varies laterally at the end of this section. sence of dissipation. In this case, the ratio 1/Q is propor- Around the resonance, it is convenient to introduce the tional to the frequency of the tides. The expression for the γ γ ω ω angular variable such as d /dt = – n, where is Eu- average tidal torque can be thus written as (Goldreich and ropa’s rotational velocity and n is the orbital mean motion. Peale, 1966; Correia and Laskar, 2004) The time-averaged evolution of the satellite’s rotation over Γ an orbital period is then given by (Goldreich and Peale, tidal = (41) 1966) K 1 + 3e2 + 3e4/8 ω 1 + 15e2/2 + 45e4/8 d2γ 3 –2 – 2 γ Γ 2 9/2 2 6 C = – H[1,e](B – A) n sin[2 ] + tidal (37) nn(1 – e ) (1 – e ) dt2 2 Γ where tidal is the mean tidal torque acting to brake the spin with of the satellite, H[1,e] is the Hansen coefficient for the syn- 2 chronous resonance, and A ≤ B ≤ C are the satellite’s prin- 3 k GM2 M R 6 K = 2 s J s n (42) cipal moments of inertia. For very small eccentricity, it is 2 Q Rs Ms a reasonable to consider that H(1,e) = 1 – 5e2/2 = 1. The synchronous resonance has a finite angular frequency of where k2 is Europa’s Love number of degree 2, Rs is Eu- libration and the width of the resonance is ropa’s radius, MJ and Ms the respective of Jupiter and Europa, and a is the orbit semimajor axis. The equilib- B – A rium rotational state in which the tidal torque vanishes is Δω = n 3 H[1,e] (38) C then given by β The oblateness parameter = (B – A)/C is related to the 2 4 ω 1 + 15e /2 + 45e /8 second-degree gravitational coefficient C2,2 by eq = n (43) (1 – e2)3/2(1 + 3e2 + 3e4/8)

B – A 4MR2 4MR2 β = = C (39) which can be reasonably approximated by ω = n(1 + 6e2) 2 2.2 eq 4MR CC for e ≤ 0.4. Although the equilibrium spin rate is slightly faster than an exactly synchronous rate, this particular –6 2 Using C2,2 = (131.5 ± 2.5) 10 and C/MR = 0.346 ± model for Q does not support the rotational properties in- 0.005, which are values determined from measure- ferred from the interpretation of tectonic features: ments of the Galileo spacecraft (Anderson et al., 1998b), 1. The small but nonzero value of Europa’s eccentric- β –3 Δω ω we have = (B – A)/C = 1.52 × 10 so that = 0.0675 n. ity provides a pseudo-synchronous rotation rate of eq = As a comparison, β = 10–4 for Mercury and the Moon. 1.0006 n, implying that the subjovian point would travel To determine whether Europa has some chance of be- around Europa’s equator with a synodic period of approxi- ing trapped into this resonance requires an estimate of the mately 15 years. This clearly disagrees with the minimum Γ 4 magnitude of the tidal torque tidal. Synchronous rotation ~10 -yr synodic period estimated by Hoppa et al. (1999a). will be achieved only if the maximum restoring torque ex- 2. The deviation from the synchronous state (~6 e2n ~ ceeds the net tidal torque at the synchronous rate. If the tidal 0.0006 n) is lower than the resonance width (0.0675 n), indi- ω torque yields a stable equilibrium rotational state eq, in cating that the tidal torque is not large enough to offset the which the tidal torque vanishes, it is easy to show that an rigid torque due to the permanent asymmetry of Europa’s equivalent condition for the capture into synchronous reso- mass distribution, as described by C2,2. Equations (38) and nance (i.e., ω = n) is that the equilibrium rotation rate is (40) shows that the “viscous” model would require an ec- located within the resonance, that is centricity larger than

ω Δω | eq – n| < (40) 1/4 1 B – A = ec = 0.1 (44) Conversely, if the latter condition is not satisfied, the final 12 C ω rotation state is given by eq. Mechanisms of tidal dissipation are poorly constrained to escape from synchroneity, which is not possible in Eu- in the solar system and current models of tides result more ropa’s current orbital configuration. 130 Europa

Another tidal model that is commonly used is the con- Europa’s rotation when its rotational velocity ω is slightly Γ → – stant-Q model where the tidal quality factor Q is assumed larger than n ( tidal 0 ) and speed up Europa’s rotation ω Γ → + to be independent of frequency (cf. Greenberg and Weiden- when is slightly smaller than n ( tidal 0 ). Conversely, schilling, 1984). This model is not appropriate close to spin- at the approach of the resonance, the tidal frequency ap- orbit resonances because it gives rise to nonphysical dis- proaches zero when ω → n, corresponding to very long continuities in the expression of the tidal torque (Kaula, forcing periods for which the mantle’s response is close to 1964). Furthermore, for any eccentricity lower than 0.235, that of a perfect fluid. In that case, no tidal dissipation oc- the equilibrium spin rate is the synchronous state (e.g., curs and the average tidal torque also tends to vanish. Such Kaula, 1964; Goldreich and Peale, 1966) so that tidal dis- an analysis suggests that the sign of the tidal torque must sipation will ultimately drive Europa to this state indepen- change at the synchronous rotation and that this point cor- dent of the asymmetry of its mass distribution. responds to a stable equilibrium state for the viscoelastic More realistic assumptions can be made concerning the tidal torque. deformation of Europa’s interior due to the tidal perturb- All the above models of tidal dissipation operate such ing potential. A common assumption for planetary or sat- that they will drive a body like Europa into the synchro- ellite interiors is that they behave like viscoelastic bodies, nous resonance, either because the gravitational torque on having a Maxwell rheology. This implies that elastic and Europa’s mass asymmetry distribution is large enough to viscous effects are combined in series such that a body’s overcome the tidal torque, or because the tidal endpoint is response is that of a purely elastic material at short periods exact synchronism. but that of a Newtonian viscous material at long periods. A However, there is at least one possible mechanism by key parameter of the viscoelastic response is the Maxwell which NSR can still occur. It was pointed out by Ojakangas relaxation time τ, which is defined as the ratio between the and Stevenson (1989a) that if the floating icy shell is con- effective Newtonian viscosity η and the elastic shear modu- ductive, lateral variations in shell thickness, and thus dif- lus µ of the material: τ = η/µ. Here we examine configu- ferences in A, B, and C for the shell, will arise. If B ≠ C rations in which the net average tidal torque could vanish then the gravitational influence of Jupiter acting on the shell assuming the simplest hypotheses for Europa’s internal will rotate the shell slightly until this torque is balanced by structure. Extended work and discussions are in progress the tidal torque. However, in this configuration the shell will on that topic (see Levrard et al., 2007; Levrard, 2008; Wis- no longer be in thermal equilibrium, because is dom, 2008). If an internal ocean decouples the outer icy symmetrical about the subjovian point. The icy shell thick- shell from an inner mantle, the icy shell is expected to fol- ness distribution will thus change with time, leading to fur- low the fluid deformation of the ocean so that its deforma- ther rotation, and so on. In this manner, a steady-state NSR tion is significantly greater than if an ocean were not present of the shell may still arise. (Moore and Schubert, 2000). In that case, the tidal response Of course, it is not currently clear whether the required of Europa’s surface depends mainly on the properties of the lateral shell thickness variations exist. Various surface fea- ice shell. Rheological and thermal properties of ice at low tures on Europa have been attributed to convection in the stress levels are not well constrained and we adopt the val- ice shell (see chapter by Barr and Showman), in which case ues µ = 3 × 109 Pa and η = 1014 Pa s as is currently assumed the shell thickness is certainly uniform. Even in the absence for internal structure models of Europa and Titan (e.g., Tobie of convection, lateral thickness variations can be smoothed et al., 2005). These parameter choices correspond to a Max- out over timescales depending primarily on the background well time τ ~ 9 h, about 1 order of magnitude lower than shell thickness (e.g., Nimmo, 2004). Existing topographic Europa’s orbital period (~3.55 days); higher viscosities or measurements are insufficient to determine whether or not lower rigidities would result in Maxwell times comparable shell thickness variations exist (Nimmo et al., 2007). None- to the orbital period. For τ much less than the orbital period, theless, the mechanism proposed by Ojakangas and Steven- Europa would behave almost as a viscous fluid. In this case, son (1989a) remains a viable way of generating NSR for a the viscoelastic model matches the “viscous” model previ- floating, conductive icy shell. ously described (e.g., , 1908), leading to the same conclusions. The tidal torque would vanish at a slightly 4.3. Secular Polar Motion supersynchronous rotation rate, but Jupiter’s torque on the permanent departure from spherical symmetry (presumably Nonsynchronous rotation, as proposed by Greenberg and due to irregularities in the silicate interior) is large enough Weidenschilling (1984), and further elaborated by Ojakangas to maintain synchronicity. and Stevenson (1989a), is rather similar in many regards Alternatively, if an internal ocean is not present, Europa’s to secular polar motion or true polar wander (TPW). The tidal deformation is expected to follow the contribution of slightly peculiar phrase “true polar wander” indicates that the deep rocky interior. Europa is most likely differentiated the geographic location of the rotation axis is changing, and into a metallic core surrounded by a rock mantle (e.g., Schu- was needed in the terrestrial literature to distinguish from bert et al., 2004). We consider a silicate mantle with physi- “apparent polar wander,” which is simply due to changes cal properties (µ = 7 × 1010 Pa and η = 1020 Pa s) close to in the orientation of the . Earth’s values (e.g., Tobie et al., 2005) such that the Max- In both NSR and TPW the concern is not with the rota- well relaxation time is ~104 yr, much longer than the or- tional dynamics of a rigid body, but rather the slow defor- bital period. The tidal torque is expected to slow down mation and reorientation of the material within the body, Bills et al.: Rotational Dynamics of Europa 131 while the departure from spherical symmetry of the mass Assuming that the effective viscous torque is propor- distribution remains essentially constant. True polar wan- tional to the between the core and the der is driven by the application of internal or surface loads mantle, Correia (2006) provided complete analytical solu- (e.g., shell thickness variations, impact basins, convective tions of the average motion of the core and mantle spins. diapirs) and resisted by the elastic portion of the tidal and He found that the general formulation of the mantle’s spin rotational bulges. The most important effect of TPW is that, evolution noticeably depends on the efficiency of the vis- as with NSR, it generates large (up to several MPa) stresses cous coupling. If the coupling between the two layers is and can thus give rise to observable global tectonic patterns strong (compared to the amplitude of the gravitational (e.g., Leith and McKinnon, 1996). torque), the entire planet (or satellite) is expected to par- The circumstances under which TPW may occur on sat- ticipate in the librations so that the moments of inertia used ellites, and the resulting stresses, have been recently re- in equation (37) are still appropriate and the width of the viewed by Matsuyama and Nimmo (2007, 2008). If Europa’s resonance is not affected. If the coupling is weak, Goldreich icy shell does show lateral thickness variations, it may be and Peale (1967) and Correia (2006) pointed out that near rotationally unstable and undergo large TPW events, unless a spin-orbit resonance, the liquid layer will not respond in- stabilized by dissipation within the ice shell (Ojakangas and stantaneously to the periodic librations of the solid shell, Stevenson, 1989b). resulting in an additional source of dissipation. As a con- Leith and McKinnon (1996) found only minor evidence sequence, this mechanism greatly increases the chances of for TPW from a global survey of lineations as seen by Voy- capturing a planet or satellite into resonance. One reason ager. More recently, Sarid et al. (2002) argued that the style is that the appropriate moment of inertia in equation (37) of strike-slip offsets could be consistent with a TPW of 30°, would be now that of the icy shell alone Cs, leading to an while Schenk et al. (2008) suggested that two large, antipo- increase of the width of the resonance by a factor C/Cs dal cir-cular features were caused by roughly 90° of TPW. as indicated by equation (38). Considering a reasonable There is currently no consensus on whether significant TPW 100-km-thick icy shell and using the ratio C/MR2 = 0.346, has occurred, but it is a dynamically plausible mechanism we estimate the width of the resonance to be about three that needs to be taken into account when considering how times larger than previously, i.e., Δω ~ 0.2 n. Hence, this global lineament patterns may have formed. clearly makes synchronous rotation even more likely in the If Europa does exhibit TPW, as has been recently claimed present Europa configuration. (Schenk et al., 2008), then it would seem likely that it could We have only considered the coupling between the icy also exhibit NSR. In both cases, the critical issue is whether shell and a hypothetical subsurface ocean. It is important the rheology of the icy shell allows it to maintain long-term to note that all the previous effects are enhanced if Europa elastic strength. If the deformation required for NSR or has a metallic core that is also decoupled from the silicate TPW were to occur on Europa, but were mainly accommo- mantle. dated by viscous flow, rather than brittle failure, it might The arguments presented here show that dynamical mod- have only subtle expression in surface features (Harada and els neglecting shell thickness variations predict quite dif- Kurita, 2006, 2007). ferent scenarios for the current rotational state of Europa to those based on geological analyses. Although tidal mod- 4.4. Discussion els are still not sufficiently well developed to furnish con- clusive information on Europa’s rotation, we stress that there The rotational motion of a solid shell decoupled from a may also be other ways to interpret the formation and evo- liquid one has been widely investigated to model the dy- lution of surface tectonic features. In particular, most mod- namics of planets having a mantle overlying a liquid core, els of tidal stresses currently suppose that the tidal forcing like Mercury or Earth (e.g., Goldreich and Peale, 1967; is only due to the nonzero eccentricity of Europa and that Correia, 2006), and most of the corresponding results can this latter is a constant value. be used to provide some insights about the rotation of the Variations of Europa’s orbital and rotational parameters external icy shell of Europa. In particular, because of their may also affect the geometry of tidal stress. First, the in- different shape and densities, a water ocean and the icy shell fluence of longitudinal and latitudinal librations should be do not have the same permanent departure from the spheri- tested in current models of external tidal forcing. Second, cal symmetry and the two parts tend to precess and librate the eccentricity of Europa varies significantly (e.g., Lainey at different rates. et al., 2006), but these variations can occur over a wide This tendency is probably counteracted by different in- range of timescales: from a few days (corresponding to teractions acting at the interface: the torque of nonradial orbital periods) to 105–106 yr (corresponding to secular inertial pressure forces of the solid shell over the ocean changes in Jupiter’s orbit). Intermediate periods are also provoked by the nonspherical shape of the interface and the expected to result from mutual gravitational perturbations torque of the viscous (or turbulent) friction between the solid between Galilean satellites (e.g., Hussmann and Spohn, and liquid layers (e.g., Ojakangas and Stevenson, 1989b). 2004). Third, although Europa’s forced obliquity is small, Note that in the specific case of Europa, significant varia- time-dependent evolution of the precessional motion and tions or irregularities of the shell thickness would add a of the obliquity of Europe occur on similar timescales (e.g., “topographic” torque, but its contribution is poorly con- Bills, 2005), leading to a complex history of the tidal ge- strained. ometry at the surface of Europa (see section 2 above). A 132 Europa promising investigation has been recently made by Hurford nomical frequencies over the Earth’s history. Science, 255, et al. (2006). 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