How fast do Galilean satellites spin?

ICARUS 58, 186--196 (1984)

How Fast Do Galilean Satellites Spin?* RICHARD G R E E N B E R G AND STUART J. WEIDENSCHILLING Planetary Science Institute, 2030 East Speedway, Suite 201, Tucson, Arizona 85719 Received September 20, 1983; revised December 12, 1983 Each of the Galilean satellites, as well as most other satellites whose initial rotations have been substantially altered by tidal dissipation, has been widely assumed to rotate synchronously with its orbital mean motion. Such rotation would require a small permanent asymmetry in the mass distribution in order to overcome the small mean tidal torque. Since Io and Europa may be substantially fluid, they may not have the strength to support the required permanent asymmetry. Thus, each may rotate at the unknown but slightly nonsynchronous rate that corresponds to zero mean tidal torque. This behavior may be observable by Galileo spacecraft imaging. It may help explain the longitudinal variation of volcanism on Io and the cracking of Europa's crust.

I. INTRODUCTION

The Galilean satellites have long been assumed to rotate synchronously with their orbital motions around Jupiter. Like the Moon, each is presumed to present one face to the planet, with only slight geometrical libration due to periodic variations in orbital angular velocities. There have been good reasons for belief in such synchronous rotation. First, if as seems plausible the satellites originally spun nonsynchronously, tides raised by the planet would slow rotation toward synchroneity on time scales much less than the age of the solar system (e.g., Peale, 1977). Estimates of spin-down time range from ~103 years for Io to -108 years for Callisto, with considerable uncertainty due to lack of knowledge of the relevant tidal parameters. And second, photometric lightcurves show differences between hemispheres that have remained stable relative to the satellite-Jupiter orientations over the half century of observations (Morrison and Morrison, 1977). Despite this theoretical and observational evidence for synchroneity, it is actually possible that the Galilean satellites, es* Paper presented at the "Natural Satellites Conference," Ithaca, N.Y., July 5-9, 1983.

pecially Io and Europa, rotate at rates slightly, but significantly, different from synchronous. [The possibility was first mentioned for Io by Yoder (1979); see discussion in Section III below.] Consider the theoretical grounds for expected synchroneity. While it is certainly true that tidal torques rapidly drive spin rates toward nearly synchronous values, they promote exact synchroneity only if a satellite's orbit is circular. In the more general case of an eccentric orbit, tides tend to increase the spin rate of a synchronously rotating satellite (Peale and Gold, 1965; Goldreich, 1966). An equilibrium spin rate (zero tidal torque) would be slightly faster than synchronous. Thus, stable synchronous rotation is only to be expected in the case of a perfectly circular orbit or, alternatively, if the satellite has a permanent asymmetry in the distribution of its mass, as in the case of Earth's Moon. By permanent asymmetry, we mean deviation from spherical symmetry (e.g, a bulge or mascon) that is frozen-in on time scales long compared with the dynamical behavior under study. :.Such permanence contrasts with the behavior of a tidal bulge that may constantly be reoriented toward the tide-raising planet. Of course, there also may be distortions of intermedi-

186 0019-1035/84 $3.00 Copyright© 1984by AcademicPress, Inc. All rightsof reproductionin any formreserved.

SATELLITE ROTATION ate permanence, which could have interesting effects, as discussed later in this paper. In synchronous rotation, torque on a permanent asymmetry can balance tidal torque if the axis of minimum moment of inertia (for example, the long axis of an ellipsoid) is oriented slightly off the direction of the planet. The orientation is stable and depends on the strength of the tidal torque. Before 1979, Io and Europa seemed to satisfy both criteria for synchroneity, even though either one alone would have been sufficient. First, most tables of orbital elements showed these satellites to have zero eccentricity, and second, both were believed on the basis of thermal models to be thoroughly solid and thus capable of sustaining frozen-in asymmetries. In fact, the orbits are significantly elliptical. Tabulated eccentricities were misleading in that they generally showed only "proper" (or "free") eccentricities, whereas for these satellites the actual eccentricities are dominated by components forced by the resonant interaction among the satellites. The eccentricities imply, of course, that there would be a nonzero tidal torque in the case of synchroneity and, moreover, that tidal heating might maintain these satellites in a partially fluid state (Peale et al., 1979; Cassen et al., 1982). Such a state may limit the degree of permanent asymmetry possible. Thus, rather than rotating synchronously, each of these satellites might rotate at that slightly faster rate for which the tidal torque does go to zero. In this paper we discuss the possibility of asynchronous rotation and estimate quantitatively the requirements for synchronous rotation. We assume zero inclinations and obliquities throughout. Although it is far from certain that either Io or Europa does rotate asynchronously, such behavior is quite plausible (see Section III) and could explain some fundamental observed geological properties of these satellites (see Section IV), such as the cracking of Europa's surface and the distribution of volcanism on Io. Before making these applications

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to specific satellites, we first (in Section II) discuss criteria for synchronous rotation as given by standard tidal models. II. CRITERIAFOR SYNCHRONOUSROTATION In order to determine the criteria for synchronous rotation, we first need an expression for the magnitude of the tidal torque on a satellite's spin in the synchronous state. Goldreich (1966) derived such an expression (T) = ~ \

R / a

~-7 ( 4 e c o s ~ -

8)

(1) where sin 6 = (oJ - n)/(2en) - 3e/4, R is satellite radius, a is orbital semimajor axis, e is orbital eccentricity, co is satellite rotation rate, n is orbital mean motion, M is the planet's mass, and Q' is the tidal dissipation parameter Q with the correction factor for "tidal effective rigidity." The brackets around T indicate the torque averaged over an orbit. This expression is based on the assumption of a "MacDonald tide" (MacDonald, 1964): The tidal bulge, with amplitude dependent on the instantaneous satellite-planet distance r, is always offset from the direction of the planet by an angle, generally of constant magnitude - I / Q ' . The sign of the offset angle is given by the sign of the difference between o~ and instantaneous orbital angular velocity, such that the bulge lags behind the direction of the planet as it moves over the surface of the satellite. This lag represents the response of a medium, the body of the satellite, that dissipates energy as it is periodically tidally distorted. For synchronous rotation (~0 = n), Eq. (1) becomes (/9 = - ~

~

~-; e.

(2)

This average torque is positive because around pericenter, where the tidal bulge is biggest, the lag orientation is such that the torque is positive. ff there is no other torque acting, Eql (2) shows that tidal torque will accelerate the

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spin, making it faster than synchronous. For such faster rates, the bulge is oriented in the sense that gives positive torque for a smaller portion of each orbit around pericenter. Eventually, when the spin is fast enough (in practice only slightly faster than synchronous), the bulge orientation is in the sense that gives negative torque over enough of the orbit (centered around apocenter) that the average torque goes to zero. This equilibrium is reflected in Eq. (1) by the fact that the right side becomes zero for a value of to a bit greater than n. Thus, under the influence of tidal torque alone, the equilibrium spin rate is slightly faster than synchronous. While the "MacDonald tide" is the form of tidal behavior most often assumed in studies of evolution of rotation, a very different form, the "Darwin tide," is usually used in computing tidal effects on orbits (e.g., Darwin, 1880, 1908; Jeffreys, 1961; Goldreich, 1963; Kaula, 1964). In this latter model, the tide-raising potential is Fourier decomposed into components that move at various rates relative to the body of a satellite. For example, there is a component of zeroth order in eccentricity that represents an elongation oriented toward the direction of the planet if the planet moved at uniform rate. Superimposed are additional components of order e that travel around the satellite and periodically enhance the main bulge (at pericenter) and diminish it (at apocenter) and others that add to the main bulge so as to reorient it toward the actual instantaneous direction of the planet. Due to imperfect elasticity, each of the tidal components in the satellite has a lag in its response to the planet's tidal-raising potential. If these lags were appropriate functions of frequency, amplitude, and time, a Darwin tide could closely simulate MacDonald tidal behavior, or any other assumed model; however, in general, the behavior of the tidal bulge relative to the tide-raising potential in the Darwin model may be quite different from that in the MacDonald model, and it is not obvious which

is the closest approximation to reality. Certainly the MacDonald tide must represent an idealization that differs to some extent from actual behavior. For example, at that point in the orbit where orbital angular velocity instantaneously equals the rotation rate, the behavior assumed by Goldreich would involve an instantaneous discrete shift in bulge orientation. For generality, therefore, we consider the tidal torque derived on the basis of the Darwin tide and obtain plausible alternatives to Eqs. (1) and (2), which were based on the MacDonald tide. As we show in Section III, an assumed MacDonald tide leads to more stringent requirements on the magnitude of frozen-in asymmetry required to maintain synchroneity. Moreover, if rotation is not synchronous, the MacDonald form leads to rotation rates generally much faster than given by Darwin tides, in fact too fast to be consistent with the constraints of photometric lightcurves for Io and Europa (see Section III). In order to evaluate the effects of a Darwin tide, we consider the potential of the tidal bulge as evaluated by Jeffreys (1961):

V = 9 (GmR~{h~(GMR2~ F

×

12.o sin(2X + 2tot -

+ 2 -~

2nt)

e2sin(2h + 2 t o t -

4

nt) nt] (3)

where h is the tidal Love number, m is the satellite's mass, g is its surface gravity, and the ~'s are the angular lags of the components. Equation (3) gives the potential of the tidal bulge at a point in the equatorial (and orbital) plane a distance rl from the center of the satellite, at longitude h in a satellite-body-fixed coordinate system with h defined as zero at the subplanet point at pericenter passage (t = 0). The phase lags

SATELLITE ROTATION 80, e l , 82, and 83 correspond to the frequen-

cies (and hence have the signs of) 2oJ - 2n, 2to - 3n, 2to - n, and n, respectively. [Note a typographical error in Goldreich (1963) gives "3/2n" for the last frequency.] The biggest terms in V have been neglected in Eq. (3); they are terms independent of the 8's, so they are symmetrical with respect to the planet and contribute nothing to the torque. The torque exerted by the tidal bulge on the planet is OV T -- M - ~ .

(4)

The partial derivative is evaluated at the planet, where rl ~ a(1 - e cos nt) and )t 2e sin nt - (oJ - n)t, and is averaged over an orbital period, yielding

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(1) equal to zero and solving for spin rate to. Equivalently, for the Darwin tide we can set (5) equal to zero. However, in our ignorance of the dependence of the phase lags (e's) on to, we cannot solve for co. Most critical is the dependence of e0 on o. In synchronous rotation, e0 = 0. As soon as the satellite spins faster than synchronously, the phase lag e0 becomes nonzero and positive. If this change in e0 with frequency is a step function such that e0 - -8~ for any value to - n > 0, then Eq. (5) shows that the torque changes sign instantaneously. The equilibrium spin rate would be only infinitesimally faster than n. However, in any real system, e0 must increase continuously, rather than stepwise, with frequency. This unknown dependence of 80 on frequency determines the equilibrium spin rate, which would be slightly greater than r/.

×

[

49

e2

1

e0 + -~- ezel + ~- ~:2

(5)

where the sign has been reversed from (4) to give the torque on the bulge due to the planet. In the case of synchronous rotation, 80 = 0 and --81 = 82 ~ 8 > 0, SO (7") = +

108 "~ (RS/a6)hGM2eEe.

(6)

By definition of Q' (e.g., Goldreich, 1966), we have h e = 5/(4Q'), so (6) becomes ( I ) = 27(RS/a6)GM2e2/Q'.

(7)

We can compare this expression (7) with the equivalent expression (2) for a MacDonald tide. It has the same sign, which yields the same crucial result that in the synchronous state tides tend to accelerate spin. However, for a given value of Q' this torque is much smaller in the Darwin case by a factor of about 4e, which is 41 for most cases of interest. In the case of a Darwin tide, what is the equilibrium spin rate at which (7) goes to zero? Recall that for the MacDonald model this is obtained by setting the right side of

In order to maintain synchronous rotation, there must be some frozen-in asymmetry, which provides the additional torque on the satellite needed to counteract the tidal spin-up. This additional torque is given by T = - ~ ( B - A ) ( G M / a 3) sin 2~?

(8)

where A is the minimum moment of inertia (e.g., about the long axis), B is the moment of inertia about the axis in the equatorial plane perpendicular to the axis of A, and is the orientation of the axis of A relative to the direction of the planet. The torque is a maximum at ~ = +45 °. Thus, the minimum requirement on (B - A) for synchroneity is obtained by setting sin 2~7 = 1 and equating (8) with the tidal torque (2) for MacDonald tide or (7) for Darwin tide. III. SOME REAL SATELLITES Based on the criteria for synchronous rotation established at the end of Section II, Table I shows the required values o f B - A, expressed in terms of the maximum moment C, for several satellites that might be considered candidates for asynchronous ro-

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GREENBERG AND WEIDENSCHILLING TABLE/

Satellite

I0 Europa Ganymede Cal[isto Mimas Enceladus Titan

Required ( B - A ) / C ~

Hydrostatic

MacDonald tide

Darwin tide

(B - A)/C b

10 -5 10 5 10 -s 3 x 10-8 3 x 10 5 3 X 10 -6 10 7

10 -7 10-6 10-11 10-9 3 X 10 -6 10 7 10 8

l0 -2 10-3 10- 3 10-4 l0 -I 10-2 10-4

a Values of frozen-in ( B - A ) / C required in order to maintain synchronous rotation. b Hydrostatic values from Peale (1977) are shown for comparison; the latter are not necessarily "'frozeni n . " All values are given to order-of-magnitude precision.

tation. Values are given for the assumptions of both MacDonald and Darwin tides, with Q' assumed to be - 1 0 in all cases. For comparison, we show the hydrostatic values of ( B - A ) / C given by Peale (1977). This is the value that would pertain if the satellite were of uniform density, conformed hydrostatically to the tidal potential, and then froze preventing further redistribution of mass. (A denser interior or core would yield slightly lower values, but within a factor - 2 of those shown.) Obviously according to Table I, such a permanent asymmetry would be far more than adequate to maintain synchronous rotation for any of the satellites. However, for a satellite that may be largely fluid, like Io or Europa, the hydrostatic figure may not be frozen-in. Instead, it may simply be the temporary tidal bulge whose behavior gives the synchoneity-destroying tidal torque [Eq. (2) or (7)]. Even if a satellite is essentially solid, over enough time its figure may creep to a new configuration. Suppose a satellite's hydrostatic figure is frozen-in, and the rotation becomes locked into synchroneity with the frozen-in bulge oriented so that the torque balances the tidal torque. In this state, the frozen figure is somewhat out of

hydrostatic equilibrium because r/ ~ 0. If the frozen figure gradually creeps to conform to the new tidal potential, the satellite must rotate slowly and continually in order to keep ~ at its equilibrium value. Thus, even a satellite with an asymmetry frozenin on a long time scale and large enough to satisfy the requirement for synchronous lock may rotate slightly asynchronously if gradual creep takes place. We can make a rough estimate of the time for a "frozen-in" bulge to migrate by viscous relaxation in the following manner: If the offset angle ~ is small, it is roughly equal to the degree of strain needed to bring the bulge back into alignment. We assume that the nonhydrostatic stress due to misalignment is proportional to ~. Its magnitude is - p g A R 7 1 , where p is the satellite's density, and A is the difference between the equatorial semiaxes of the hydrostatic equilibrium figure. The relaxation time r is viscosity x strain/stress, or, for viscosity b',

r ~ v/pgAR.

(9)

Since both the stress and the required strain are proportional to r/, r is independent of'0. It is the time for the bulge to shift by an angle - ~ , so the rotation period, P, relative to synchroneity, is ~2~-r/r/. Yoder (1979) gives another expression for r which is much smaller. This expression is, in effect, the relaxation time of the entire bulge if the tidal force were suddenly removed (cf. Johnson and McGetchin, 1973). It neglects the fact that the nonhydrostatic stress vanishes for ~7 = 0. For a numerical example of the magnitude of rotation rates due to viscous creep of a frozen-in bulge, consider a hypothetiCal satellite with all the parameters of Io, but with its hydrostatic bulge frozen-in. If Q' - 10 (Cassen e t a l . , 1982), Eqs. (7) and (8) yield ~ - 3 × 10-4. This value is the offset of the frozen-in bulge that counteracts the tidal spin-up effect. According to the discussion in the previous paragraph, viscous creep of the frozen-in bulge allows

SATELLITE ROTATION rotation with P - 1 0 - 1 2 v years, where v is in poise. A value of v comparable to that of the Earth's mantle (1021 to 1022Poise) would give P comparable to the age of the solar system. For the real, solid satellites listed in Table I, P would be longer (i.e., viscous creep a negligible process) according to (9) because those satellites are smaller, have smaller AR, and possibly higher v than the hypothetical satellite. For the actual Io and Europa, which may be significantly molten, effective viscosities are much lower. A viscosity of -1013 Poise, for example, would give rotation faster (P - 10 years) than allowed by observational constraints, according to the formalism of Eq. (9). For a largely molten satellite, however, the formalism that treats the hydrostatic bulge as a creeping frozen-in asymmetry becomes less meaningful. Instead, the hydrostatic bulge is essentially the everchanging tidal bulge treated in the MacDonald or Darwin theories, which tends to spin-up synchronous rotation according to Eq. (2) or (7). In this case we must consider what other permanent mass asymmetries on these satellites might act to maintain synchroneity. The permanent asymmetries required are actually very small compared with hydrostatic figures (Table I). We can consider what scale of local topography or mass concentrations would correspond to the requirement, for comparison with observable surface features. The required mass is - m ( B - A ) / C . In the case of Io, the required (B - A ) / C for a MacDonald tide could be achieved by a circular slab of material of radius 100 km, thickness 10 km, and density 1 g/cm 3 stuck on the surface of an otherwise spherical satellite. (The effect on moment of inertia is computed by taking the slab's mass times R2.) For the Darwin tide model, a smaller but comparable appendage would do the job. This size is comparable to the mountains observed on Io (Schaber, 1982). However, given the probable thin crust of Io, these mountains are almost certainly isostatically compensated

191

by roots extending into the mantle. The relatively low density of the roots would largely compensate for the effect of the topography on (B - A ) / C ; their net contribution is diminished by a factor - d / R , where d is the depth of compensation. Moreover, the mountains may be sprinkled fairly uniformly over the surface of Io (Schaber, 1982), so the effect of any one mountain range on (B - A ) / C would be largely neutralized by the others. Perhaps the required slab of mass is not expressed as topography, but rather as a 100-km-sized mass concentration in the crust with density differing from ordinary crust by - 1 g/cm3; however, such a mascon would likely sink into the fluid mantle. If there is an asymmetrical density distribution deep within Io, because of the smaller moment arm it would have to be bigger than the size needed near the surface in order to give the required (B - A ) / C . An asymmetry of the required magnitude might be dynamically supported by convection within Io (S. Peale, private communication, 1983). If one of the above sources of asymmetry produced synchronous locking, this state would be only temporary, due to ongoing mountain building, changes in the convection pattern, or other tectonic processes that might alter Io's mass distribution. If such effects were dominant, Io would reorient itself at irregular intervals, between episodes of synchronous rotation. Whether the required asymmetry can exist is problematic, so it seems at least plausible that Io may well be rotating nonsynchronously. Europa would require topography or mascons comparable to the requirement for Io. Europa may well have a liquid-water mantle, so all the arguments given above against the asymmetry for Io apply at least as well for Europa. Moreover, Europa's surface is very smooth; it has no mountains. Of course, one can still hypothesize several ways in which Europa might support a permanent asymmetry: Perhaps improved heat flow models will again melt away belief in the liquid-water mantle (Cas-

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GREENBERG AND WEIDENSCHILLING

sen e t al., 1982); perhaps hypothetical convection can give a sufficient bulge (Ransford e t al., 1981; note that Ransford e t al. fallaciously give the "observed" synchronous rotation of Europa as evidence for their convection theory); or perhaps the solid material below the liquid mantle has sufficient asymmetry frozen-in. In the latter case, only the solid core would be locked in synchronous rotation. Tidal torque on the crust would cause it to drift relative to the core. Viscous coupling would transmit the torque through the liquid layer, allowing it to be counteracted to some degree by the tidal torque on the core's permanent asymmetry. While these possibilities remain open, it seems at least as likely that Europa rotates nonsynchronously as that Io does. Ganymede and Callisto require much smaller values of permanent asymmetry for synchronous locking (Table I). These satellites are believed to be completely solid, and thus able to support such asymmetry over geologic time. Hence, they are probably synchronous at present. However, they may have been extensively melted during accretion. In that case their ice crusts might have rotated nonsynchronously, as suggested above for Europa, until their mantles froze. Later solid-state convection in the mantles might redistribute enough mass to cause each satellite to reorient itself episodically. These effects might explain the smaller than expected apex-antapex variations in crater density on these bodies. We know with great certainty that Mimas is capable of supporting the required degree of frozen asymmetry. The scale of the big crater alone (Smith e t al., 1981) is much greater than the scale of any required topography or mascon. There are undoubtedly other mass features in and on Mimas that affect gravitational moments as well. However, it is probably no coincidence that the big crater is 90° from Saturn; the crater along with other topography may have helped determine the axis of minimum moment of inertia. Mimas probably rotates synchronously. For Enceladus, the re-

quired (B - A ) / C is equivalent to a surface slab of 20 km radius, 1 km thick. Yet, this satellite is very smooth. If the process that warmed and relaxed the surface also relaxed the internal mass distribution, Enceladus may rotate nonsynchronously, or may have done so at some time in the past, if heating was episodic. Synchronous rotation of Titan would require a slab of about 100 km radius, 1 km thick. Is there such topography? Sagan and Dermott (1982) argue against features much bigger than that on the grounds of their effects on tidal dissipation rates in Titan's " o c e a n . " We cannot be certain about Titan's spin rate. The best candidates for nonsynchronous rotation are Io and Europa. Yet, as discussed earlier, photometric lightcurves indicate that these satellites have each kept a fixed longitude (at least to within - 2 0 °) toward Jupiter for over 50 years. Calculating the equilibrium nonsynchronous spin rate for MacDonald tides by the method described earlier yields a rotation period, 27r/ (to - n), relative to the direction of the planet of - 5 0 years for Io and - 2 0 years for Europa, both seemingly inconsistent with observations. However, Io is resurfaced on a time scale shorter than 50 years, so the photometric constraint on rotation only applies if albedo and coloring distributions remain fixed relative to the body of the satellite. If instead the shading is controlled by processes that maintain the distribution fixed relative to the direction of Jupiter, then a rotation period as short as 50 years could not be ruled out by the lightcurves. The periods for Darwin tides could be much longer, in which case there would be no question of agreement with photometry. Recall that for Darwin tides the equilibrium rotation rate depend on how the lag e0 behaves near zero frequency. Periods of thousands of years or more seem plausible, and would be perfectly consistent with the lightcurves. Such long periods might also pertain if there is a bulge frozen-in on time scales long compared with the orbital period so it resists the tidal torque, but that

SATELLITE ROTATION

193

and Reynolds, 1983) suggest that frost is deposited over the surface at a rate of > 10 cm in 106 years. Thus, complete repainting may occur in only -103 years. Squyres et al. believe that continuing sulfur deposition IV. SOME IMPLICATIONS OF on the trailing hemisphere maintains the NONSYNCHRONOUS ROTATION photometric asymmetry. Even if the resurWhile the possibility that Io, Europa, and facing is a million times slower, albedo feaother satellites may rotate nonsynchro- tures associated with the cracks produced nously is exciting in its own right, it has by the long-term stress load would have several even more interesting implications. been erased; at the rate computed by SquyConsider the geology of Europa. The most res et al., the topography associated with striking geological features of that satellite, those old cracks would have been buried as in fact practically the only geological fea- well. The idea of tidal cracking can be resurtures, are the mysterious cracks that show rected in the face of those objections by as dark markings and cover the surface. considering the effects of nonsynchronous The global scale of the crack system sugrotation. In this case, cracks may represent gests that they are the result of global response to relatively recent stresses due to stress. Helfenstein and Parmentier (1983) the ever-changing orientation of the crust argue that the crack patterns are consistent relative to the tidal bulge. Suppose frost rewith tidal stress induced by a combination surfacing takes place in a fraction of a rotaof effects. First, spin-down from an initially tion period (as usual measured relative to more rapid rotation would have reduced the the direction of Jupiter), for example, 103 satellite's oblateness, stressing a previously relaxed crust. Second, orbital evolution years frosting compared with 10 4 years ro(due to tides raised on the planet by the tation. Then the cracks would represent the satellite) has slowly increased Europa's stressing due to the past 10 3 years (-40 ° of mean distance from Jupiter over the age of rotation). These stresses are much greater the solar system, decreasing the amplitude (by a factor -1/e - 100) than the cyclical of the hydrostatic bulge. If the crust early stresses that are dependent on e. With this on had relaxed to the hydrostatic figure and mechanism, there is plenty of stress to do if rotation has been synchronous, reducing the cracking. Helfenstein and Parmentier the bulge in this way would induce stress in find that cracks do not align perfectly with the crust. And third, variations in the tidal theoretical stress orientations for tidal bulge due to orbital eccentricity would con- bulges at 0° and 180° longitude. Our cracking model based on nonsynchronous rotatinue to stress the crust periodically. Voyager imagery did not give sufficiently tion would give offset orientations. Qualitacomplete coverage to establish unequivo- tively, one can predict a region of tension at cally that crack patterns correlate with low-to-middle latitudes at longitudes just these global stresses. And there are two se- west of 0° (sub-Jupiter) and of 180° longirious theoretical problems with the idea. tudes. Here the crust is moving toward the First, the cyclical tidal stresses associated bulge and must be stretched over it. East of with orbital eccentricity are a few times 0° and of 180°, the crust is coming off the smaller than the probable tensile strength of bulge and is thus going through a compresthe crust (Cassen et al., 1982). And second, sion stage. Perhaps the indications of asymcracks due to the earlier stress load (spin- metry in the crack patterns relative to the down and orbital recession) may have been sub-Jupiter longitude (Helfenstein and Parerased (or "painted over") by rapid resur- mentier, 1983) are due to this stress pattern. These ideas can be tested by looking for facing. Squyres et al. (1983; also Squyres

continually creeps toward hydrostatic alignment with Jupiter, as described in the first part of this section.

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GREENBERG AND WEIDENSCHILLING

differences in frost coverage and crack structure created during the decade between Voyager and Galileo imaging of Europa. However, a lack of significant differences would still be consistent with the theory of crack formation by nonsynchronous rotation; it would simply mean that the time scales for frosting and rotation are too long for significant change in this interval. A more promising test will be provided by more complete mapping of the global crack system from Galileo imaging, which will permit comparison with the expected stress field. The decade between Voyager and Galileo visits to the Jovian system may permit direct measurement of nonsynchronous rotation of Io and Europa. With rotation periods (relative to synchronous) of -1000 years, surface features observed by Galileo may be >100 km from the positions expected on the assumption of synchronous rotation. Such a shift should be readily observable by Galileo. In contrast, the interval between the two Voyagers was too short to detect nonsynchronous rotation. Periods of - 3 0 0 years, which is about the shortest that would be consistent with the 50-year history of photometric lightcurves (assuming as mentioned above the resurfacing does not invalidate the lightcurve constraints), would correspond to positional shifts of-< 10 km between the encounters by Voyagers 1 and 2. Any detection of such shifts on existing images would be marginal. Resolution at Io was 1 km from Voyager 1, but 20 km from Voyager 2; at Europa, it was 4 km from Voyager 2, but 33 km from Voyager 1. Photogrammetric determinations of Io's figure from Voyager imaging have led to rather surprising results. Early results (Davies, 1980) suggested that Io's elongation along the Jupiter axis was only about onefourth of the hydrostatic amplitude. While reasonable rigidity could maintain such a figure for a while, in synchronous rotation long-term creep would have to bring the figure closer to hydrostatic equilibrium (T. V.

Johnson, private communication; s e e a t s o our discussion of viscous relaxation in Section III). This result might have been taken as evidence for nonsynchronous rotation. However, Davies (1983) has recently reported an updated evaluation of the figure of Io. He now finds that the elongation is about one-half the hydrostatic value, and oriented 30° away from Jupiter (longitude about 330°W of the sub-Jupiter point). This elongation is in the direction that a tidal bulge would be carried by nonsynchronous rotation, but the 30° lag is much greater than any tidal lag could be expected to be. If this bulge is real, the torque exerted on it by Jupiter [Eq. (8)] would yield libration with amplitude >30 ° and a period of only a few weeks, assuming the bulge is frozen-in for time scales longer than that. Such positional variation would probably have been seen by the Voyagers. Davies' work is based predominantly on Voyager 1 data, so his assumption of synchronous rotation is probably not the cause of any systematic error. McEwen and Soderblom (1983) report that the most active volcanic longitudes are around 300° ± 60° W. If rotation is nonsynchronous, these regions are now moving east, away from recent passage through the sub-Jupiter longitudes where a tidal bulge would have stretched the satellite's surface. Perhaps such stretching promoted the volcanism, while a lagging response time carried the region eastward. The longitudes of enhanced volcanic activity are oriented similarly to the long axis given by Davies' results, so that it is tempting to speculate about a causal relationship. However, the fundamental concerns about the figure obtained by Davies, expressed in the previous paragraph, must be resolved before such relations between global figure and geological processes can be interpreted. In contrast, the connections between volcanism and tidal processes are obviously important. In this context, we ask, is there significantly more tidal heating in a satellite rotating nonsynchronously than synchro-

SATELLITE ROTATION nously? Consider a Darwin tidal model. The zeroth-order (in e) component is fixed relative to the body when to = n (synchroneity), and hence there is no corresponding di~ssipation. But in nonsynchronous rotation, this component, which is much larger than those dependent on e, does move over the body, and there is corresponding dissipation. However, in stable nonsynchronous rotation, the torque due to the zerothorder component exactly counterbalances the other torques. This requirement places a specific constraint on the phase lag e0 relative to the phase lag e of higher order terms: e0 - 12e2e. Such a small value of e0 means that dissipation is negligibly enhanced in the nonsynchronous case. Another area where nonsynchronous rotation might have implications is the interpretation of longitudinal variations in the cratering record. Concentration of craters on a satellite's leading hemisphere could be evidence of an impacting population from heliocentric orbits rather than from within a given planet's system. On the other hand, lack of such cratering asymmetries [as may be the case in the Saturn system (McKinnon and Chapman, 1983)] does not necessarily imply the absence of a significant heliocentric contribution. Nonsynchronous rotation during the bombardment period would yield a similar result. During accret i o n (or shortly after), icy satellites may have been partially melted, and hence less likely to sustain asymmetries, even if they can at present. Uniform rotation at a rate faster than orbital mean motion is not the only kind of nonsynchronous rotation possible for a satellite. Even if there is a frozen-in asymmetry capable of sustaining synchronous rotation on average, there may be significant, observable libration about such behavior. For the Galilean satellites, variations in orbital longitude with periods -480 days are driven by mutual gravitational interactions among the satellites (Lieske, 1977). Correspondingly, oscillations in the apparent position of Jupiter as viewed from a satellite

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could pump rotational libration through the torque on the asymmetry [Eq. (8)]. According to Lieske (1977), the amplitude of orbital changes is greatest for Europa (-10 - 3 radians). For a frozen-in (13 - A ) / C of - 2 × 10-5, Eq. (8) shows that Europa's libration would resonate with the driving orbital variations. This plausible value of ( B - A ) / C is in the range adequate to maintain synchroneity (Table I). Such libration might govern surface cracking if one face of Europa does prove to be locked to the direction of Jupiter. ACKNOWLEDGMENTS Conversations with D. R. Davis, C. R. Chapman, T. V. Johnson, and Mark Parmentier helped elucidate many of the ideas in this paper. Comments by referees, including Stanton Peale, were very helpful. This work was supported by NASA's Planetary Geophysics and Geochemistry Program, Contract NASW3524. This is Planetary Science Institute Contribution No. 193. PSI is a division of Science Applications, Inc. REFERENCES CASSEN, P. M., S. J. PEALE, AND R. T. REYNOLDS (1982). Structure and thermal evolution of the Galilean satellites. In Satellites o f Jupiter (D. Morrison, Ed.), pp. 93-128. Univ. of Arizona Press, Tucson. DARWIN, G. H. (1880). On the secular change in the elements of the orbit of a satellite revolving about a tidally distorted planet. Phil. Trans. R. Soc. London 171, 713-891. DARWIN, G. H. (1908). Tidal friction and cosmogony. In Scientific Papers, Vol. 2. Cambridge Univ. Press, New York. DAVIES, M. E. (1980). Coordinates o f Features on the Galilean Satellites. Presented at IAU Colloquium 57 (Satellites of Jupiter). DAVIES, M. E. (1983). The Shape o f Io. Presented at IAU Colloquium 77 (Natural Satellites). GOLDREICH, P. (1963). On the eccentricity of satellite orbits in the solar system. Mon. Not. R. Astron. Sci. 126, 257-268. GOLDREICH, P. (1966). Final spin states of planets and satellites. Astron. J. 71, 1-7. HELFENSTEIN, P., AND E. M. PARMENTIER (1983). Patterns of fracture and tidal stresses on Europa. Icarus 53, 415-430. JEEFREYS, H. (1961). The effect of tidal friction on eccentricity and inclination, Mon. Not. R. Astron. Soc. 122, 339-343. JOHNSON, T. V., AND T. R. MCGETCHIN (1973). Topography on satellite surfaces and the shape of asteroids. Icarus 18, 612-620.

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