Tidal Evolution and Satellite Formation: Orbital Resonances and Volcanic Activity, Papers of Geology

Download Tidal Evolution and Satellite Formation: Orbital Resonances and Volcanic Activity and more Papers Geology in PDF only on Docsity! P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? Annu. Rev. Astron. Astrophys. 1999. 37:533–602 Copyright 1999 by Annual Reviews. All rights reserved ORIGIN AND EVOLUTION OF THE NATURAL SATELLITES S. J. Peale Department of Physics, University of California, Santa Barbara, California 93117; e-mail: peale@io.physics.ucsb.edu Key Words tides, dissipation, dynamics, solar system, planets 1. INTRODUCTION The natural satellites of planets in the solar system display a rich variety of orbital configurations and surface characteristics that have intrigued astronomers, physi- cists, and mathematicians for several centuries. As detailed information about satellite properties have become available from close spacecraft reconnaissance, geologists and geophysicists have also joined the study with considerable enthu- siasm. This paper summarizes our ideas about the origin of the satellites in the context of the origin of the solar system itself, and it highlights the peculiarities of various satellites and the configurations in which we find them as a motiva- tion for constraining the satellites’ evolutionary histories. Section 2 describes the processes involved in forming our planetary system and points out how many of these same processes allow us to understand the formation of the regular satellites (those in nearly circular, equatorial orbits) as miniature examples of planetary systems. The irregular satellites, the Moon, and Pluto’s satellite, Charon, require special circumstances as logical additions to the more universal method of origin. Section 3 gives a brief description of tidal theory and a discussion of various ap- plications showing how dissipation of tidal energy effects secular changes in the orbital and spin configurations and how it deposits sufficient frictional heat into individual satellites to markedly change their interior and surface structures. These consequences of tidal dissipation are used in the discussions of the evolutions of satellite systems of each planet, starting with the Earth-Moon system in Section 4. Additional processes such as collisional breakup and reassembly of some of the smaller satellites associated with the ubiquitous equatorial rings of small particles are discussed, although ring properties and evolution are excluded (see Nicholson 1999). There is no attempt to include technical details of published explanations of the various phenomenologies. Rather, the explanations are described and the uncertainties in the assumptions and resulting conclusions are emphasized. The 0066-4146/99/0915-0533$08.00 533 P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 534 PEALE outstanding gaps in our understanding of each system are pointed out, where ad- ditional knowledge about a system or one of its parts always seems to generate more interesting problems than it resolves. This is the first time all of the satellite systems have been discussed in any detail in one place, and it is hoped that the reader will find in one or more of these systems problems to which he can apply his own expertise. The Galileo spacecraft has already uncovered some amazing properties of the Jupiter’s satellites, which have shaken some of our long-held beliefs about the satellites; the Cassini spacecraft approaching Saturn promises to do the same. Let us build the context, which this and other new information will alter and refine. 2. ORIGINS The origins of the natural satellites are of course closely linked to the origin of the solar system and the formation of planets therein. So it is appropriate to outline our current understanding and necessary speculations about solar system formation to understand the creation of the satellite systems. The origin of the satellite systems of the major planets can be understood in terms of similar processes and events, sometimes with extreme examples of some events for each system. The Moon and probably Pluto’s satellite, Charon, required special circumstances within the broader set of processes that occurred during the early evolution of our planetary system. It is generally accepted that our planetary system formed from a flat dissipative disk of gas and dust that surrounded the young Sun. The disk is the natural conse- quence of the collapse of a rotating cloud with angular momentum conservation, where dissipation leads to both the flat geometry and the circular orbits of the disk constituents. This general picture is observationally confirmed, since all young stars appear to have such disks of material for some part of their early existence (Strom 1995, Beckwith & Sargent 1998)—consistent with theoretical expecta- tions. As the disk cools, nonvolatile elements and compounds will condense into small particles that settle to the midplane of the disk, where they collect into larger and larger sizes chiefly through collisional coagulation (Weidenschilling 1995). The nonvolatiles will consist of rock and iron-type materials in the terrestrial planet region close to the Sun but will include water and other ices beyond ∼4 AU from the Sun. The latter condensates increase the solids fraction of the nebular disk by a factor of ∼3 compared with the terrestrial zone. All of the details of the early coagulation process are not well understood, but the gravitational instabili- ties in a very thin disk of solid particles thought to dominate the process (Safronov 1969, Goldreich & Ward 1973, Ward 1976) apparently are completely frustrated by shear-induced turbulence and persistent velocity dispersions for the solid bod- ies (Weidenschilling 1995). Both the shear-induced turbulence and the persistent velocity dispersion result from the fact that partial support of the nebular gas by a radial pressure gradient causes it to orbit the Sun at an angular velocity that is P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 537 problem remains in supplying a sufficient amount of heavy elements through later accretion of solid planetesimals. Also, the perturbations by full-sized Jupiter and Saturn would stir the planetesimal disk sufficiently to hinder the runaway growth that created the large embryos needed for the successful modeling of terrestrial planet formation. Some also question whether the nebula could be simultaneously cool enough and dense enough for Boss’s conditions of gravitational collapse to be met. (Toomre’s 1964 Q factor must indicate instability.) We must conclude that there has been no robust method demonstrated for the formation of Jupiter or Saturn within observational constraints. Uranus and Neptune apparently formed their cores more leisurely since the nebula was by then too thin to contribute much gas to their bulk. Still, core forma- tion at this distance from the Sun requires >109 years in a minimum-mass solar nebula (Stewart & Levison 1998), so perhaps these planets should have received less gas than is apparent. The remaining problems with planet formation aside, however the major plan- ets formed—either by the two-stage process or by gravitational instability—the material going into these planets would have had significant angular momentum relative to the center of mass of the forming planet. For Jupiter and Saturn, the ultimate spin of the planet is determined largely by angular momentum contributed by the gaseous component. Neptune and Uranus accreted a relatively small frac- tion of their mass as hydrogen and helium and the terrestrial planets, essentially none. So the planetesimal accretion determined the angular momenta of these planet-satellite systems. Each accreting planetesimal contributes part of its angu- lar momentum to the spin and part to the orbit of the combined planet-planetesimal mass, the relative contributions being determined by the details of the collision. Thus, both the magnitude and the direction of the spin vector undergo random walks as the accretion progresses, where the magnitude of the steps grows with the sizes of the planetesimals (Dones & Tremaine 1993). The wide scatter of plan- etary spin vectors testifies to the fact that the last stages of accretion involved very large planetesimals, although Mercury and Venus have had their spins altered by dissipative processes. This stage is especially dramatic for Uranus, whose spin axis is inclined by 97◦ relative to its orbit normal. For Jupiter and Saturn, the cross-section for continued accretion of solid plan- etesimals will be enhanced by an extended gaseous atmosphere, where those plan- etesimals that do not plunge directly into the planet will be captured by gas drag or collisions into orbits whose eccentricities and inclinations relative to the equator plane are damped by the continued atmospheric gas drag and collisions with other orbiting debris, and the orbit semimajor axes will decrease at rates that depend on the size of the planetesimal. Even after the accretion is nearly complete and the atmosphere has continued its collapse toward the planet as it cools, any remaining debris in orbit will damp down to circular equatorial orbits. This damping can be understood by considering a hypothetical ring of orbiting planetesimals that is inclined to the equator plane of the planet. Because each planetesimal has a slightly different orbit, the orbits will precess at different rates owing to the oblate P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 538 PEALE distribution of mass in the rotating planet. The ring will tend to spread into a cylindrical distribution where collisions in crossing orbits within the distribution will remove the components of velocity perpendicular to the equator plane. Ec- centricities are similarly damped by dissipative collisions. The satellitesimals will then accumulate into satellites on much shorter time scales than for the analogous accretional accumulation of the planets in the solar nebula (Pollack 1985). This rapid accretion into large objects may be crucial to the survival of satellite material as the drag from the waning gaseous component of the disk causes it to spiral into the planet. The debris around Uranus and Neptune relaxed to the oblique equator planes before accumulating into their equatorial satellites. Consistent with this picture of satellite formation, the orbits of the closer satellites of the major planets are nearly circular and nearly equatorial as shown in Table 1. The regular satellites of the major planets thus form naturally by the accretion of debris in a dissipative disk much like the process in the solar nebula leading to the planetary bodies. The irregular satellites were later captured from the remaining planetesimal swarm either by three-body interactions within the planetary sphere of influence, by collision with debris already orbiting the forming planet, or by gas drag in the extended primordial atmosphere of the forming planet. The last capture mechanism may be the least effective, since sufficient gas to capture the satellite is most probably also sufficient to cause the satellite to spiral into the planet before the atmosphere dissipates. The two families of distant irregular satellites orbiting Jupiter (one family in retrograde orbits) could be products of a disintegration in which single parent bodies were shattered either in collisions causing the captures or by later collisions with high-speed cometary bodies. Alternatively, gas drag from the extended atmosphere has been proposed for both the capture and the breakup of weakly bound parent bodies. Although the timing and the rate of atmosphere removal are critical in this scenario if the captured satellites are not to spiral into the planet, the occupancy of orbital resonances by two of the retrograde satellites implies that at least a thin atmosphere was necessarily in place at the time of their capture (Saha & Tremaine 1993). The capture of an intact satellite by tidal torques on the single first pass is possible for such a miniscule volume of the phase space of initial conditions that it most probably did not occur (see Boss & Peale 1986). The origins of the Moon and of Pluto’s satellite, Charon, require special consid- eration related to the large amount of specific angular momentum in each system. The origin and evolution of the Moon has understandably received more attention than has any other satellite. The analysis began with GH Darwin (1879, 1880) when he realized that the consequences of the tidal distortion of the Earth by the Moon would lead to an orbital evolution requiring the Moon to have once been very close to the Earth. Darwin suggested that the Moon formed from the outer layers of the Earth—a variation of the rotational fission type of origin elaborated during this century. Several other theories of origin have been proposed and opposed over the intervening years, including intact capture. Each variation of these theories is criticized in detail by Boss & Peale (1986), and all are rejected except formation from the debris resulting from the giant impact of a Mars-sized body (Hartmann P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 539 & Davis 1975). This idea has grown in popularity because it naturally provides the large angular momentum of the Earth-Moon system, it can easily account for the lack of volatiles and iron on the Moon if the impacting body is differentiated, and it is much more probable than any means of intact capture (Cameron & Ward 1976). The debris from a giant impact does not follow ballistic trajectories, which would automatically reimpact the Earth. Pressure gradients and the distribution of mass in the ejected debris cause accelerations after the impact that leave material in initially stable orbits. Recall that we concluded earlier that the final stages of accretion of the terrestrial planets involved planetary embryos from Moon-to-Mars size, so it is entirely reasonable that such an impact occurred. Although the giant impact origin of the Moon has surfaced as the only viable scheme by a process of elimination, the consequences of the impact depend very much on the choices of many free parameters, such as the initial angular mo- mentum and mass of the impactor. Large regions of the parameter space have been and are being explored numerically with various amounts of condensible silicate-like material being placed in orbit beyond the Roche radius, where it can collect into the Moon (Cameron 1997, Canup & Esposito 1996, Ida et al 1997, Cameron & Canup 1998a,b and references therein). (Inside the Roche radius, (r ∼ 2.5RE (ρP/ρC)1/3 ∼ 2.9RE ) tidal forces would break apart a fluid satellite and therefore inhibit accretion.) The orbiting debris from the impact would damp down to circular equatorial orbits through collisions, where it can acrete into the Moon. For many if not most of the sets of choices of the many parameters, insuf- ficient material ends up in orbit beyond the Roche radius. Details along the route from impact to Moon are still uncertain, so improving the resolution and reliability of the numerical calculations remains an active area of research. These processes will be discussed further in Section 4. A giant impact is also proposed as the origin of the Pluto-Charon binary system (McKinnon 1989, Stern 1991, Tancredi & Fernández 1991), and by the same process of elimation that was applied to the origin of the Moon, this method of origin again emerges as the only viable option (Dobrovolskis et al 1997). The two satellites of Mars are in nearly circular equatorial orbits, which supports the argument of their accretion in situ from a debris disk in the equatorial plane left over from the planet’s formation—like the regular satellites of the major planets. However, the Phobos mean density was estimated to be 2.0 ± 0.5 g/cm3 from Viking orbiter data (Christensen et al 1977, Tolson et al 1978), and that of Deimos 2.0 ± 0.7 g/cm3 (Duxbury & Veverka 1978). The difference from the mean density of Mars of 3.9 g/cm3 lends support to the suggestion that the satellites are composed of material that did not originate in the vicinity of Mars, and albedos near 5% and reflection spectra are consistent with carbonaceous chondritic material (Pang et al 1978, Pollack et al 1978). Intact capture was considered (Pollack & Burns 1977, Mignard 1981b), but Szeto (1983) showed several seemingly insurmountable inconsistencies with the capture hypothesis independent of the impossibility of relaxing the captured satellites into the circular equatorial orbits where they are found. Despite the low density and spectral indications of carbonaceous chondrite P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 542 PEALE SV II I H yp er io n 14 6 21 .2 76 6 C 14 81 .1 0. 10 42 0. 43 SV II Ia pe tu s 71 8 15 .9 1. 02 79 .3 30 2 S 36 51 .3 0. 02 83 7. 79 c, f SI X Ph oe be 11 0 55 0. 48 R 0. 4 12 95 2 0. 16 3 17 5. 3c U ra nu s 25 55 9 8. 66 25 × 10 5 1. 31 8 U V I C or de lia 13 0. 33 50 49 .7 52 0. 00 0 0. 1 U V II O ph el ia 16 0. 37 64 53 .7 64 0. 01 0 0. 1 U V II I B ia nc a 22 0. 43 46 59 .1 65 0. 00 1 0. 2 U IX C re ss id a 33 0. 46 36 61 .7 77 0. 00 0 0. 0 U X D es de m on a 29 0. 47 37 62 .6 59 0. 00 0 0. 2 U X I Ju lie t 42 0. 49 31 64 .3 58 0. 00 1 0. 1 U X II Po rt ia 55 0. 51 32 66 .0 97 0. 00 0 0. 1 U X II I R os al in d 29 0. 55 85 69 .9 27 0. 00 0 0. 3 U X IV B el in da 34 0. 62 35 75 .2 55 0. 00 0 0. 0 U X V Pu ck 77 0. 76 18 86 .0 04 0. 00 0 0. 3 U V M ir an da 23 6 0. 65 9 1. 20 1. 41 3 S 12 9. 8 0. 00 27 4. 22 U I A ri el 57 9 13 .5 3 1. 67 2. 52 0 S 19 1. 2 0. 00 34 0. 31 U II U m br ie l 58 5 11 .7 2 1. 49 4. 14 4 S 26 6. 0 0. 00 50 0. 36 U II I T ita ni a 78 9 35 .2 7 1. 71 8. 70 6 S 43 5. 8 0. 00 22 0. 10 T A B L E 1 Sa te lli te R b (k m ) m (1 02 0 kg ) ρ (g cm −3 ) P o (d ay s) P r (d ay s) a (1 03 km ) e i( de g) (c on ti nu ed ) P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 543 U IV O be ro n 76 1 30 .1 4 1. 63 13 .4 63 S 58 2. 6 0. 00 08 0. 10 U X V I C al ib an ∼3 0 58 0 71 .7 × 10 5 0. 08 23 13 9. 68 d U X V II Sy co ra x ∼6 0 12 90 12 2. 1 × 10 5 0. 50 9 15 2. 67 d N ep tu ne 24 76 4 1. 02 78 × 10 6 1. 68 3 N II I N ai ad 29 0. 29 44 48 .2 27 0. 00 0 4. 74 N IV T ha la ss a 40 0. 31 15 50 .0 75 0. 00 0 0. 21 N V D es pi na 74 0. 33 47 52 .5 26 0. 00 0 0. 07 N V I G al at ea 79 0. 42 87 61 .9 53 0. 00 0 0. 05 N V II L ar is sa 96 0. 55 47 73 .5 48 0. 00 0 0. 20 N V II I Pr ot eu s 20 9 1. 12 23 11 7. 64 7 0. 00 0 0. 55 N I T ri to n 13 53 21 4. 7 2. 05 4 5. 87 69 R S 35 4. 76 15 6. 83 4 N II N er ei d 17 0 36 0. 13 62 55 13 .4 0. 75 12 7. 23 c Pl ut o 11 70 13 1. 5 2. 0 PI C ha ro n 58 6 19 .0 2. 24 6. 38 72 S 19 .4 05 0. 00 76 e 0 a T he se da ta ar e up da te d sl ig ht ly fr om Y od er 19 95 . b T he sa te lli te ra di ia re av er ag es if th e bo dy is ve ry as ym m et ri c. Se e Y od er (1 99 5) fo r m or e co m pl et e de sc ri pt io ns of sa te lli te sh ap es ,f or un ce rt ai nt ie s in th e de te rm in at io ns of al lt ab le en tr ie s, an d fo r th e pr im ar y re fe re nc es . c R el at iv e to th e L ap la ci an pl an e. d R el at iv e to th e ec lip tic pl an e of A D 20 00 (P N ic ho ls on .P ri va te co m m un ic at io n) . e T ho le n an d B ui e (1 99 7) . f V ie nn e & D ur ie z 19 95 .S ,s yn ch ro no us ;C ,c ha ot ic ; R , r et ro gr ad e; T , T ro ja n. P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 544 PEALE material, the coplanar equatorial orbits indicate that these satellites must have formed from a dissipative disk of debris orbiting the planet. If that debris were indeed carbonaceous chondritic (which is not at all certain), one possible way it could have gotten into orbit about Mars would be from the shattering of such a planetesimal that was formed in the asteroid belt region of the nebula when it collided with a denser object already in orbit about Mars. The condition here would be that the pieces would have to be sufficiently small and of sufficient number to make a dissipative disk. Samples of both Phobos and Deimos would tell us if such a contrived origin were necessary. Any scheme to capture these satellites intact and bring them into their current orbits cannot survive close inspection of the assumptions involved. Although consensus on many details is still elusive, we have plausible origins for all of the known natural satellites. It is well known that the current distribution of satellite orbits is not the initial distribution nor are even the masses and perhaps the total number the same. The larger satellites are likely to have remained intact under the bombardment of now high-speed comets or asteroid-type planetesimals, but the smaller ones may have been shattered and possibly recollected into new bodies several times in the history of the solar system. Some of the debris from such collisions remains as rings of smaller particles. The densely cratered surfaces of many of the satellites—the Moon, in particular—are testimonies to the flux of impactors after the satellites were formed. The changes so far described have resulted from stochastic, short-time-scale events characterized by collisions. However, the most interesting changes in the satellite systems involve the orbital modifications and changes in the satellites themselves caused by gravitational tides raised on the primaries by the satellites and raised on the satellites by the primary. Angular momentum is transferred between a spinning planet and its closer satellites, causing either a reduction in the spin rate while expanding the satellite orbits or an acceleration of the spin while the orbit contracts. The direction of the transfer depends on the relative spin and orbital angular velocities. Differential expansion of the satellite orbits leads to the establishment of the many examples of orbital resonances in which the ratio of the orbital mean angular velocities is that of two small integers. The resonances are stable against further tidal evolution up to a point. Dissipation of tidal energy within the satellites has led to striking consequences. From Table 1, we see that all of the closer satellites are rotating synchronously with their orbital motion—a natural consequence of tidal evolution. But sometimes the tidal retardation of a spin leads to chaotic tumbling when synchronous rotation is unstable, as it is for Hyperion. Several of the satellites have surfaces much younger than their age as indi- cated by the paucity of impact craters, which were probably erased by indigenous processes energized by tidal dissipation. Tidal dissipation may have maintained a liquid water ocean under the ice of the Jupiter satellite, Europa. It might have soft- ened the interiors of Ganymede to account for its partial resurfacing and perhaps an interior sufficiently hot to support the observed intrinsic magnetic field (Kivelson et al 1997). The most striking consequence of tidal dissipation in a satellite remains P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 547 liquids, but empirical studies of Q in Earth-like materials yield nearly constant values over many orders of magnitude in frequency (Knopoff 1964). Even though these tests do not approach tidal frequencies, constant Q is often assumed in calculations of time scales for tidal evolution. This assumption is probably a fair approximation if frequencies and properties of the planets and satellites involved do not change significantly over the time of evolution, or if the resulting time scales are so extreme that more elaborate models for Q would not affect the conclusions. We will adopt this practice in most of what follows, where the nece- ssity for more elaborate models for the tides is pointed out where needed. One instance where traditional tidal models will probably have to be abandoned is in the determination of the dissipative properties of the giant gaseous planets, for it is becoming increasingly clear that the full dynamics of atmospheric response to the tide-raising potential must be considered (Ioannou & Lindzen 1993). The simple tide is a poor representation of the tides in the oceans of Earth, where basins defined by the continents have sloshing periods that are comparable with those of the lunar and solar tides. The resultant tides are often far out of phase with the forcing potential. Despite this difficulty, the simple tide has been used for stud- ies of the Earth-Moon history in hopes that the complicating effects of the oceans will average to something close to those of the simple tide. Formulating an evolu- tionary tidal model is difficult, since the effects of the ocean tides on lunar orbit evolution have changed in largely unknown ways as the continental configuration has changed. It is usually assumed that the tidal effective Q was larger in the past when the continents were joined. With the measured tidal effective Q appropriate to current expansion of the lunar orbit, the Moon would have been at the Earth surface less than 2 billion years ago. Attempts to invoke more complicated tidal models for the Earth have not produced qualitatively different plausible evolutions of the lunar orbit (Kaula 1969). The phase lag in the response of the tidally distorted body results in misalign- ment of the tidal bulge with the body raising the tide and leads to the gravitational torque that transfers angular momentum between the spin of the tidally distorted body and the orbit of the body raising the tide (e.g. see Peale 1999). The angular phase lag in a periodic tidal oscillation is 1/Q, and this lag leads to a geometric angle between an ideal tidal bulge and the direction to the tide-raising satellite of 1/2Q since the tide has a 180◦ geometric symmetry. The torque on a satellite in a circular equatorial orbit about a planet deduced from Equation 3 or a Fourier expansion thereof is (e.g. MacDonald 1964) T = 3 2 k2GmS2 RP5 a6 Q P , (5) where the subscripts S and P refer to satellite and planet respectively, and a is the semimajor axis of the orbit. If the satellite is rotating relative to the primary, there will be a similar expression with the indices interchanged for the torque exerted on the planet. The equal and opposite torque on the tidally distorted body results P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 548 PEALE in a change in the spin angular velocity ψ̇ at a rate given for a small satellite with constant Q by dψ̇ dt = −45 76 ρn4 R2 µQ sign(ψ̇ − n), (6) where n is the mean orbital angular velocity (mean motion), the approximate form of Equation 4 has been used, and the moment of inertia, 2mS R2/5, used is that for a homogeneous sphere. Both the torque on the satellite by tides raised on the planet and the torque on the planet by satellite tides affect the orbital motion. However, the tidally evolved satellites will be locked in synchronous rotation, and tides raised on these satellites will not contribute to changes in the orbital angular momentum. For nearly all satellites, the spin angular momentum is such a small fraction of the orbital angular momentum that satellite effects on the orbital motion are negligible in any case. The rate of change of the orbital mean motion (mean orbital angular velocity) from the torque on an equatorial satellite in circular orbit is then (with Q P constant) (Peale 1988) dn dt = −9 2 kP Q P mS m P RP5n16/3 G(m P + mS)]5/3 sign(ψ̇ − n). (7) If mS were also spinning nonsynchronously in the same sense as its orbital motion, but with nonzero obliquity (spin axis inclined relative to the orbit normal), the tidal bulge on mS would be carried out of the orbit plane, and there would be a component of the torque perpendicular to the spin axis. Both the spin magnitude and its direction will thus change as a result of tidal dissipation. The obliquity of the spinning satellite, if initially small, would tend to increase from the tides, if the spin is fast. This behavior can be understood if we resolve the spin angular momentum into components perpendicular and parallel to the orbit plane. The tide is decreasing the magnitude of the perpendicular component at all points in the orbit, whereas it does not reduce the parallel component when that component is pointing toward or away from M. Averaged around the orbit, the fractional change in the perpendicular component exceeds the fractional change in the parallel component, and the obliquity increases. There is an equilibrium obliquity between 0◦ and 90◦ toward which the spin axis is driven where the fractional changes in the two components are the same (Goldreich & Peale 1970). This equilibrium is close to 90◦ for fast spins, but it decreases toward 0◦ as the spin is retarded. If the spin angular velocity ψ̇ < 2n, the equilibrium obliquity is 0◦, so a satellite in a fixed circular orbit should approach synchronous rotation and zero obliquity simultaneously. If the orbit is eccentric, the tidal torque averaged around the orbit vanishes at a spin angular velocity that is slightly larger than the synchronous value. From Equation 5 we see that the tidal torque on the satellite varies as m2/r6. One factor of m/r3 determines the magnitude of the tide, being the difference of 1/r2 forces, whereas the second factor of m/r3 comes from the differential force on the two tidal bulges. As the maximum torque occurs at the orbit periapse, the tides will P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 549 try to synchronize the spin with the instantaneous value of the orbital angular velocity at this point, which is larger than n. Satellites in eccentric orbits maintain synchronous rotation in spite of a slightly faster spin favored by the tides because much larger torques on the permanent, nonaxially symmetric distribution of mass force the long axis of such a satellite to librate about the direction to the planet when the satellite is at periapse. It can only so librate while maintaining a spin that is a half-integer multiple of its orbital mean motion, where synchronous rotation is the overwhelmingly most common example. The spin of Mercury is locked at 1.5n, but no satellite is in such a higher-order spin state. If the permanent deviation from axial symmetry is smaller than that induced by the tidal forces, the tides could win, and the endpoint of the evolution would be a spin slightly faster than synchronous. There is evidence that at least the surface ice layer on Europa may be rotating slightly faster than the synchronous rate (Geissler et al 1998, Greenberg et al 1998). Tidal dissipation will persist in a synchronously rotating satellite in an eccentric orbit at a rate given by (Peale & Cassen 1978) d E dt = 21 2 kS f QS Gm P2n RS5eS2 a6 , (8) where the factor f ≥ 1 has been added to account for an increase in kS if there is a molten core (Peale et al 1979). The tide will oscillate in magnitude as the satellite- planet distance varies, and it will oscillate in a direction relative to the coordinate system fixed in the satellite because the rotation is nearly uniform, whereas the orbital motion is not. This dissipation will tend to reduce the eccentricity e, as e 6= 0 is the cause of the dissipation. The spin angular momentum of the satellite is conserved because of the lock into synchronous rotation. The specific orbital angu- lar momentum [G(m P + mS)a(1 − e2)]1/2 can thus not gain angular momentum from the satellite. The orbital energy, −Gm P mS/2a, must decrease if energy is dissipated in the satellite, and a must thereby decrease. But the conserved angular momentum means that e must also decrease if a decreases. At the same time, the tide raised on the planet by the satellite tends to increase the eccentricity. The greater tidal force on the satellite at the orbit periapse tends to fling the satellite to a greater apoapse distance than it would have reached without the kick–thereby increasing e. The variation in the eccentricity from the two effects is (Goldreich 1963) de dt = 57 8 kP n mS m P RP5 a5 e Q P − 21 2 kSn m P mS RS5 a5 e QS . (9) The dissipation in the satellite wins if 19kP QSmS RSρS2/28kS Q P RPρP2 < 1). This situation is generally true for the giant planets, so free eccentricities of these satellites will tend to damp. For large planets, those satellites for which the decay time constants depending on the satellite parameters (e.g. kS, r, R, Q) are short compared with the age of the solar system, should be in nearly circular orbits and be synchronously rotating with their orbital motions with nearly zero obliquity. Table 1 shows that all the P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 552 PEALE cold-accretion scenario, it appears that a lunar-sized object can accrete outside the Roche radius only for impacts by a body twice the mass of Mars with twice the current angular momentum of the current Earth-Moon system (Canup & Esposito 1996), although this minimum mass for the impactor can be relaxed if the im- pact occurred when Earth’s formation was only ∼50 to 70% complete (Cameron & Canup 1998a,b). However, continued accretion by the Moon as the Earth completes its own accretion could lead to excessive siderophile or volatile con- tamination of the lunar surface layers that is not observed (Stewart & Canup, 1998). The diversity of the accretion scenarios demonstrates the inadequacy of theoretical models of the impact-generated disk. In contrast to our ignorance of many fundamental parameters for all of the other satellites, we have detailed information on the geochemistry and ages of rocks from six sites on the Moon thanks to landings during the Apollo program. The ages of the surfaces all over the Moon can then be estimated from the cali- bration of the crater densities at each of the visited sites. The youngest surfaces range from 2.9 to 3.9 billion years old (Warren 1985), where repeated basaltic lava flows have covered vast areas called maria—usually the interiors of giant impact basins such as Mare Imbrium or Mare Serenitatis. The maria occupy only ∼17% of the lunar surface; the remainder (lunar highlands) is a thick layer (thickness is estimated from assumed isostacy of lunar highlands floating on dense mantle basalts and from the depth of sampling by large impact craters) of low density rock that is >75% plagioclase (Ca, Na-aluminosilicate). So much plagioclase on the surface is thought to result from differential crystallization in a deep (∼250-km) magma ocean with the less dense plagioclase crystals floating to the top. The min- imum depth of the ocean is controversial, but it is estimated as that necessary to provide the observed plagioclase if all the available plagioclase has floated to the surface. There are additional, more subtle geochemical indications of the exis- tence of such a magma ocean on the early Moon (Warren 1985). Very few of the returned lunar rocks are older than 4.0 billion years, where older rocks are thought to have been destroyed by a continued heavy bombardment (late heavy bombard- ment) that persisted until ∼500 million years after the Moon formed (Mottman 1977). In the sections that follow, we invoke tidal dissipation (only sometimes success- fully) to account for resurfacing of icy satellites with no other apparent sources of internal energy. However, one investigation of the contribution of tidal dissipation to lunar thermal history (Peale & Cassen 1978) yielded negative results. This study was motivated by the realization that the obliquity of the Moon would undergo large variations as it changed from equilibrium Cassini state 1 to a precession about and dissipative relaxation toward a still very inclined state 2 as the Moon passed through ∼34RE from the Earth during the tidally induced growth of the semima- jor axis of the Moon (Ward 1975). The obliquity reached as high as 77◦ during this process, which led to significant tidal dissipation within the Moon. However, the growth of the semimajor axis was sufficiently rapid during this transition that P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 553 the contribution to the lunar thermal budget was negligibly small (Peale & Cassen 1978). There is a possibility discussed below that tidal dissipation caused profound changes in the lunar interior during passage through an orbital resonance when the Moon was very close to the Earth. Several authors (Darwin 1879, 1880, Gerstenkorn 1955, MacDonald 1964, Kaula 1964, Goldreich 1966) have attempted to constrain the origin of the Moon by integrating the motion approximated by Equation 7 backwards in time until the Moon was close to the Earth. The actual motion is much more complicated than implied by Equation 7. The lunar orbit has a variable inclination relative to the Earth’s equator. The noncircular orbit precesses while maintaining a nearly constant inclination to the Laplacian plane, where that plane is nearly coincident with the Earth’s equator plane when the Moon is close, but is now nearly coinci- dent with the ecliptic. The Earth’s spin rate is decreasing as angular momentum is transferred to the Moon and the Sun. The Earth’s obliquity is changing in response to lunar and solar torques. The Earth’s spin axis itself is precessing due to the torques on its oblate figure. The current rate of regression of the Moon was de- duced long ago by the determination that the length of an Earth day was increasing 0.0016 s/century from the timing of solar eclipses. That rate has since been made more precise from lunar laser ranging to the Moon over the past 25 years (daM/dt = 3.82± 0.07 cm year−1) (Dickey et al 1994). The first thing learned from the calculations of the Earth-Moon evolution was that the Moon would have been at the surface of Earth <2 billion years ago if the tidal-effective QE was maintained at that value yielding the current rate of lunar orbit expansion. As the major fraction of the Earth’s tidal dissipation is in the oceans and as that dissipation depends on the configuration of continental shore- lines, a reasonable solution to this problem is that the changes in the continental configurations have led to a tidal effective Q today that is less than its value in the distant past. The second result was that the Moon returned to an inclination relative to the Earth’s equator of ∼10◦ instead of to the equatorial plane. As the then popular coaccretion model of lunar formation had the Moon forming in the Earth’s equator plane from a debris disk, this is referred to as the “inclination problem.” This result has been confirmed in a modern Hamiltonian formulation of the problem and by a symplectic integration including the complete chaotic solar system, but with the same accelerated evolution rates and no dissipation in the Moon (Touma & Wisdom 1994). The currently popular giant impact origin of the Moon also leads to formation in an equatorial disk, so the problem remains. However, it takes only another large impact on the Earth or Moon after the Moon has formed to change the orbit inclination relative to the Earth equator and thereby provide a consistent tidal evolution (e.g. Stevenson 1987). But Touma & Wisdom (1998) have another solution. In their 1994 paper, Touma & Wisdom note that the artificially accelerated rate of tidal evolution would drag the Moon through any orbital resonances it might have encountered such that the consequences of capture in such a resonance would P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 554 PEALE be missed. In addition, the evolution of the Moon is not time reversible because the Moon might be captured into an orbital resonance when approaching it from one direction, but not the other. Any viable representation of the Earth-Moon history must then involve a rate of tidal evolution reasonably close to the physical rate and must progress forward in time. In an integration that produces the current configuration when the Moon has reached its current distance from the Earth, Touma & Wisdom (1998) start the Moon in the equatorial plane of the Earth at a separation of 3.5RE with an ec- centricity of 0.01, which is consistent with current expectations from the giant impact origin (e.g. Ida et al 1997). The initial obliquity of the Earth is 10◦, and the initial Earth rotation period is five h. Realistic rates of tidal evolution are used in the symplectic integrations that include the entire chaotic solar system, and dissipation in the Moon is included at a variety of dissipation rates. A tidal model was used with the constant time lag as discussed in Section 3 but with the Mignard (1981a) formulation. The first strong resonance is encountered when the Moon is ∼4.5RE , where the period of the periapse motion of the lunar orbit relative to an inertial reference is ∼1 year. This resonance is called the evection resonance because the same term in the disturbing function gives rise to the 1.3◦ amplitude, 31.8-day periodic variation in the Moon’s mean longitude called the evection. Capture into the evection resonance is certain if the eccentricity is<0.07 as the resonance is approached and if the rate of tidal evolution is sufficiently slow. With the assumed parameters, capture occurs and the eccentricity grows rapidly to large values, where the maximum value reached before the system escapes the resonance is determined by the value of A = kM kE 1tM 1tE ( m E mM )2( RM RE )3 , (10) where k,1t,m, and R refer to Love number, the constant tidal time lag, mass, and radius with subscripts referring to Earth and Moon, respectively. A is a measure of the relative rates of energy dissipation in the Earth and Moon. The current value of A from the lunar laser-ranging experiment is∼1.1 (Dickey et al 1994). For A = 0 (no dissipation in the Moon), the maximum eccentricity is ∼0.5 before escape, and the eccentricity continues to climb after escape from the resonance because of tides raised on Earth and no dissipation in the Moon (Goldreich 1963). For A = 10 (high dissipation in the Moon), the maximum eccentricity is only ∼0.15. For 1 ≤ A ≤ 10, the energy dissipated in the Moon in∼8000 years is in the range from∼2×1035 to 1.5×1036 ergs, which could lead to substantial melting (Touma & Wisdom 1998). After escape from the evection resonance, the continued expansion of the orbit further decreases the prograde motion of the orbit periapse and twice the time derivative of the evection resonance variable plus the retrograde motion of the lunar orbit node approaches zero. The term in the Hamiltonian corresponding to this resonance has eM2iM in the coefficient, but this resonance affects the inclination more than the eccentricity. Touma & Wisdom name this resonance the eviction— P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 557 5. MARS SYSTEM From Table 1, we see that Phobos is located well inside the corotation radius of ∼5.9 Martian radii (RM ), and Deimos is just outside this radius. The tides raised on Mars thus cause Phobos to be spiraling toward Mars and Deimos to be spiraling away. In fact Phobos is inside the Roche radius for a density of 1.9 g cm−2 and would be torn apart by tidal forces if it were a fluid. It needs only a shear strength of 105 dynes cm−2 to resist disruption (Yoder 1982)—a loose rubble pile would survive (Soter & Harris 1977, Dobrovolskis 1982). Phobos and Deimos are synchronously rotating. Diemos would have reached this state in<108 years from an unlikely small initial spin period of 4 h, if a rigidity of 5 × 1011 dynes cm−2 and Q = 100 is assumed. Under the same assumptions, Phobos would have reached this state in <105 years at its current separation from Mars and in <107 years at its likely initial separation near the corotation radius. [See Peale (1977) for a detailed discussion of the rotation histories of all of the satellites known at that time.] We have already dismissed an intact capture origin for the satellites of Mars as untenable, given the regularity of the current orbits. A formation from accretion in a debris disk then implies that the initial orbits of Phobos and Diemos were also regular with near-zero eccentricities and inclinations to the equator plane of Mars. The fact that Phobos and Diemos deviate considerably from spherical symme- try leads to special circumstances as they approach synchronous rotation. Both enter chaotic zones in the phase space surrounding that of stable libration about synchronous rotation where the spin axis is attitude unstable (Wisdom 1987b). This means the satellites will tumble chaotically with time scales comparable with the orbital periods until they are trapped into stable libration about the synchronous state. Tidal dissipation under these circumstances is like that for a nonsynchronous rotation, and the eccentricity will be damped at a much higher rate than it would have if the satellite were in synchronous rotation. Hence, the process of synchro- nizing the rotation of the satellites with the orbital motion will also effectively damp any eccentricity remaining from the accretion process. This damping pre- cludes high eccentricities in the past and concern about collisions between Phobos and Diemos as discussed by Szeto (1983). The evolution of the system to the cur- rent configuration would then seem to require only the expansion of the orbit of Diemos from tides raised on Mars and the shrinkage of the orbit of Phobos from similar tides. Although the effect of tides on the inclinations of the orbits to the Martian equator plane is negligible, it is of interest that the initial equatorial orbits will remain equatorial in spite of the chaotic, large amplitude variations in the obliquity of Mars (Ward 1979, Laskar & Robutel 1993, Touma & Wisdom 1993) and despite the precession of the spin axis of Mars (Goldreich 1965b). The solid angle described by the orbit normal as the satellite orbit precesses due to Mars’s oblateness is an adiabatic invariant (Goldreich 1965b), as these precession rates for the Martian satellites [periods of 2.3 and 57 years for Phobos and Deimos, respectively (Peale P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 558 PEALE 1977)] are fast compared with rates of change of Mars’ spin axis direction relative to inertial space [timescales O (105 years), Touma & Wisdom 1993]. Determinations of the Q of Mars lie between about 66 and 144 (Shor 1975, Sinclair 1978, Duxbury & Callahan 1981) from observations of the secular acceler- ation of Phobos’ orbital mean motion. If we choose a constant value of QM = 100 with kM = 0.14, the orbit of Deimos could have expanded by <200 km in 4.6 × 109 years. The initial semimajor axis of Phobos would have been ∼5.6RM under the same assumptions. The rotation period of Mars would be essentially unaffected by the exchange of angular momentum with the satellites and would have been only ∼10 min longer due to solar tides. Deimos has essentially its initial orbit; Phobos, having started inside the corotation radius, is consistent with the measured current value of QM ≈ 100. If we insist that both satellites started with nearly circular orbits, how then can we explain the current eccentricity of Phobos’ orbit eP = 0.0151? If the orbital motion is integrated backwards in time, this eccentricity grows to large values and collisions with Deimos would have been likely (Yoder 1982, Szeto 1983), even if there were no tidal dissipation in Phobos. Significant dissipation in Phobos reduces the time scale for a crossing orbit with Deimos to <109 years in the past (Yoder 1982). The current eccentricity cannot therefore be a remnant from tidal decay beginning 4.6× 109 years ago. Yoder (1982) has identified three commensurabilities (defined when two characteristic periods in the description of the motion are in the ratio of small whole numbers) that Phobos has passed through within the past 109 years that provide likely gravitational excitations of the Phobos eccentricity during its inward spiral. The commensurabilities are encountered at a = 3.8, 3.2, and 2.9RM , where the earliest resonance was encountered only 5× 108 years ago. The first and third are 2:1 and 3:1 commensurabilities between the rotation of Mars and the orbital mean motion, where the resonant interaction is with the axial asymmetry of Mars. At 3.2RM , the 2:1 commensurability is between the apparent mean motion of the Sun and the periapse of the orbit of Phobos, where the secular motion of the latter is caused by the oblate figure of Mars. This resonance is like the evection resonance for the Moon. There is also a 3:2 spin-orbit resonance excitation of the eccentricity when a = 4.6RM , but this excitation happened so long ago that there would be no contribution to the current eccentricity. The eccentricity would have decayed after each excitation, and it plausibly arrives at the current eccentricity after the series of kicks and subsequent decays (Yoder 1982). Orbital inclination can also be excited, and, even though the resonance interaction is not as strong as it is for the eccentricities, the excited inclinations do not decay. Still, the current inclinations of the orbits are consistent with the resonance passages (Yoder 1982). There is a condition on the dissipation in Phobos for this scenario to work. Yoder (1982) has calculated the dissipation in the satellite accounting for both the tidal dissipation caused by the eccentric orbit as discussed earlier and that caused by the forced libration of the very asymmetric satellite. This libration has an amplitude of 3.9◦ (Duxbury & Callahan 1981, Yoder 1982) and causes twice P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 559 the tidal dissipation in Phobos that would occur if Phobos were nearly axially symmetric in the same eccentric orbit (Yoder 1982). Both the dissipation in Phobos and that in Mars from tides raised by Phobos damp the eccentricity. There cannot be too much damping since the series of eccentricity excitations or the current eccentricity would be less than that observed. Because the dissipation in Mars can be presumed known from the measurement of QM , and the magnitude of the probable excitations can be reasonably estimated from the resonance passage analysis, the current value of eP limits the contribution by Phobos. Yoder finds that µP Q P > 3 to 6× 1012 dynes cm−2 or, if Q P ∼ 100, µP & 1010 dynes cm−2, which is about that of ice. The properties of Phobos are not sufficiently well known for one to be sure that the rigidity could satisfy this constraint, but this rigidity is not unreasonable. During the spiral of Phobos toward Mars, it is likely that it passed through the 2:1 orbital mean motion commensurability with Deimos. Such a passage would excite an eccentricity of ∼0.002 in the orbit of Deimos if the eccentricity of Deimos were much smaller than this before resonance passage. The time of this commensurability is known if the current dissipative properties of Mars have not changed substantially since the resonance encounter. This places a lower bound on the dissipation in Deimos if the current eccentricity is the tidal remnant from an initial value of 0.002 excited by the resonance passage. Yoder (1982) finds µD Q D(1 − αD)2/αD2 . 1010 dynes cm−2, where αD = 3(B − A)/C,with A < B < C being the principal moments of inertia of Deimos. This limit may be unreasonably low, but the dissipation in Deimos may be increased if the forced libration is nearly resonant with the free libration. The enhanced amplitude of libration would lead to higher dissipation and would relax the constraint onµD Q D . The free libration period could be better constrained by an estimate of αD from a more accurate determination of the shape of Deimos along with an accurate measure of its physical libration amplitude. In any case, the Yoder hypothesis (that the satellite orbits have always been reg- ular and current properties of the system then attributed to the effects of resonance passages by Phobos) is well supported. This hypothesis is consistent with our pre- sumed origin from a dissipative disk of small particles. However, the necessary approximations in the developments and still uncertain dynamical properties of the satellites warrant a more thorough numerical exploration of the phase space of the system with the value of Phobos’ rate of tidal orbit decay comparable with the real value. 6. JUPITER SYSTEM The Jupiter system has four classes of satellites. The dynamical evolutions of the four small satellites closest to Jupiter, after the disk of gas and solid particles from which they formed had disappeared, are limited to the tidal retardation of their spins to synchronous rotation, unless there have been episodes of breakup P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 562 PEALE Ganymede-Callisto surface dichotomy can be obtained. If these implications are not sufficiently puzzling, analysis of variability of an induced magnetic field in Callisto is consistent with a conducting layer of liquid, salty water below the in- sulating ice crust (Khurana et al 1998). Sustaining such a layer of liquid water on Callisto over any significant time span seems totally implausible. The apparent incomplete differentiation of Callisto is another mystery (McKinnon 1997), but it is significant that the last Callisto flyby by the Galileo spacecraft has considerably reduced the deduced CC/mC RC2 from∼0.4 from the two earlier encounters to 0.358 ± 0.004 (Anderson et al 1998a). McKinnon (1997) has pointed out that even a homogeneous Callisto will have CC/mC RC2 = 0.38 because of the compression of deeper layers and polymorphism of ice, so the lat- est value of CC/mC RC2 would indicate some differentiation. Next, the flyby data were interpreted under the assumption of hydrostatic equilibrium, and McKinnon (1997) finds that nonhydrostatic contributions to the gravitational harmonic coef- ficients J2 and C22 could mimic an undifferentiated Callisto and by inference lead to CC/mC RC2 = 0.358, which still is too high. If one allows for a 3σ error as well, it seems that the conclusion that Callisto is only partially differentiated is not that secure. If there were to be no differentiation, the internal temperature and, in particular, the surface temperature could never exceed 273 K. If one assumes that all of the accretional energy must be radiated away at surface temperature T , the accretion time t = 4πGρC2 RC3/(9σT 4) is ∼4.5× 105 years for T = 273 K (DJ Stevenson, private communication), where σ is the Stefan-Boltzman constant and where only the gravitational binding energy is accounted for. This million-year accretion time scale is much longer than that obtained by McKinnon & Parmentier (1986), who find that both Ganymede and Callisto should have been substantially melted during accretion. A more detailed mapping of the Callisto gravitational field seems appropriate before theorists devote too much time to explaining incomplete differentiation. The most striking characteristic of the Galilean satellite system is the set of orbital resonances where the orbital mean motions satisfy the relations nI − 3nE + 2nG = 0, nI − 2nE + $̇I = 0, nI − 2nE + $̇E = 0, nE − 2nG + $̇E = 0, (11) which lead to the following constraints on the mean longitudes: λI − 3λE + 2λG = 180◦, λI − 2λE +$I = 0◦, λI − 2λE +$E = 180◦, λE − 2λG +$E = 0◦. (12) P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 563 The subscripts refer to Io, Europa, and Ganymede, respectively, and $i are the longitudes of pariapse with the dot indicating time differentiation. The Laplace relation is the first of both sets of equations. If we define ω1 = nI − 2nE and ω2 = nE − 2nG, the Laplace resonance can be thought of as a 1:1 commensurabi- lity between ω1 and ω2 whose current value is 0.74◦ per day. The combination λE−2λG + $G is not constrained. At the conjunction of Io and Europa, Io is thus at its periapse and Europa is at its apoapse, whereas at the conjunction of Europa and Ganymede, Europa must be at its periapse and Io must be on the opposite side of Jupiter. The longitude of Ganymede’s periapse is not constrained at conjunction. Because the phase-space volume for the Laplace relation is so small compared with that available, it has been long assumed that the resonances were assem- bled from initially random orbits through differential tidal expansion of the orbits (T Gold, personal communication, 1962; Goldreich 1965a). Yoder (1979), with elaboration in Yoder & Peale (1981), was the first to develop a consistent analysis of tidal evolution arriving at the current configuration, where the high tidal dissipation in Io was shown to be a vital consideration in damping the amplitude of libration of the Laplace angle to the current remarkably low value of 0.066 ± 0.013◦ (Lieske 1987). The damping of this libration to such a small value during evolution within the resonance was not possible without the high tidal dissipation in Io (Sinclair 1975). Substitution of parameter values from Table 1 into Equation 7 shows that Io’s orbit will expand more rapidly than Europa’s and Europa’s more rapidly than Ganymede’s. Thus Yoder starts Io inside the 2:1 mean motion commensurability with Europa, so that it approaches this commensurability from a direction where capture into each of two eccentricity-type resonances corresponding to libration of the second and third angles defined by Equation 12 is certain if the respective eccentricities far from resonance are sufficiently small (e.g. Peale 1986). This condition for certain capture will surely prevail, since the secular decrease of the eccentricity e of a synchronously rotating satellite as a consequence of tidal dissipation will dominate the secular increase induced by tides raised on the planet and quickly reduce e to negligibly small values before resonance encounter (Peale et al 1980). (Time constant for decay of free eccentricity for a cold Io would be 8.2 × 104 QI years with µ = 6.5 × 1011 dynes cm−2 from Equation 9). Just after capture into the two eccentricity-type resonances at the 2:1 mean motion commensurability, ω1 = nI − 2nE = −$̇I = −$̇E is considerably larger than the current value. The retrograde periapse motions are a resonance effect that dominates the or- dinary prograde motion from Jupiter’s oblateness and from the solar perturbation. The contribution to $̇ from the resonance term is inversely proportional to the eccentricity (Peale et al 1979) such that as tides raised on Jupiter by Io con- tinue to reduce Io’s mean motion, the decrease in ω1 forces a similar decrease in the retrograde motions of $̇I and $̇E if the resonance is to be maintained. But since this rate is inversely proportional to the eccentricites, both eccentricities are forced to higher values as tides push Io deeper into the resonance, that is, P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 564 PEALE closer to exact commensurability of the mean motions. But higher eccentricities result in higher tidal dissipation in the satellites according to Equation 8, where 1 ≤ f ≤ 13 in this equation for a two-layer Io model with kI ≈ 0.035 for a homogeneous, solid Io with rigidity between those of Earth rocks and the outer layers of the Moon (Peale et al 1979). This dissipation decreases the eccentricity at a rate given by Equation 9. Thus, tides on Jupiter are forcing Io deeper into the resonance and thereby increasing the eccentricity while tidal dissipation in Io and (much less so) in Europa from the forced eccentricities are tending to reduce the ec- centricites. An equilibrium is approached where the two effects balance with eI and eE , ω1, $̇I , and $̇E essentially constant as the locked pair of satellites move out- ward together as Io transfers angular momentum to Europa through the resonance interaction. It is easy to see that the values of the equilibrium eccentricities where the two dissipative effects balance determine the ratio kJ QI /kI f I Q J (Yoder & Peale 1981). The locked pair of satellites continue to move away from Jupiter, leading to the eventual encounter of the 2:1 mean motion commensurability between Europa and Ganymede. Capture into the three-body Laplace resonance (ω1/ω2 = 1) has a probability of∼0.9 (Yoder 1979; Yoder & Peale 1981) with simultaneous capture into the eccentricity type 2:1 resonance involving Europa’s periapse longitude but not that of Ganymede. Initially there are free eccentricities induced that manifest themselves as large amplitudes of libration of the angles defined in Equations 12 about their mean resonant values. Subsequent evolution of the set from continued application of the torque from the tide raised on Jupiter by Io (The torques on Europa and Ganymede from their Jupiter tides are negligible by comparison.) forces larger eccentricities for the orbits of Io and Europa asω1is pushed to smaller values, while the tidal dissipation of orbital energy in Io damps the libration of the Laplace angle. The evolution is such that the amplitude of libration of the Laplace angle approaches zero as eI and eE approach new equilibrium values. In this state, the three satellites move out together while maintaining fixed ratios of their mean motions. Angular momentum acquired from Jupiter by Io is transferred to Europa and from Europa to Ganymede through the resonant interactions to maintain the configuration. The miniscule amplitude of libration of the Laplace angle is consistent with the current values eI and eE being equilibrium values. Consequently, the balance of the dissipation effects in Io (trying to decrease eI ) to those in Jupiter (trying to increase eI ) leads to kI f I /QI ≈ 900kJ/Q J (Yoder & Peale, 1981). The 1600 Q J year age for the Laplace resonance, deduced by Yoder & Peale if the amplitude is an evolutionary remnant, was increased to 2100Q J years in a refined analysis by Henrard (1983) and to possibly an even greater age under circumstances to be explored below (Malhotra 1991). The age could be greater than any of these constraints if the amplitude was not a remnant of the Laplace resonance evolution but had been excited by an impact on one of the satellites or if it is a libration forced by the proximity of the (2074 ± 10)-day period of free libration to the 2076-day period of a term in the solar perturbation. It may be the case at the time P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 567 the above evolutionary scheme and current configuration of the satellites (Yoder & Peale 1981). The lower bound on the average Q J results from the proximity of Io to Jupiter after expanding from just outside the synchronous orbit with the Laplace resonance assumed to exist for 4.6× 109 years. The upper bound depends on the current eccentricity of Io being nearly the equilibrium value leading to (Yoder 1979, Yoder & Peale 1981) kI kJ ( RI RJ )5(m J MI )2 Q J fI QI = 4200, (13) from which, Q J fI /QI = 1.24× 104. From Equation 8, the tidal heating exceeds the radiogenic heating rate of Io of 6×1018 ergs/s if QI / f I < 370 (Yoder & Peale 1981). This inequality is almost certainly satisfied since the Moon most probably has radiogenic heating comparable with that of Io but is largely unmelted, so the high temperature of Io needs considerable tidal heating. This bound on QI / f I then imposes Q J < 4.6 × 106. More likely QI / f I < 100 by comparison with other rocky bodies such as Mars and the solid Earth. This latter upper bound on QI / f I leads to Q J < 1.2× 106, which we nudge up to 2× 106 to be conservative. The upper bound on Q J is lower than several estimates of the Q J to be ex- pected from turbulent viscosity in Jupiter’s gaseous interior—the most extreme being Q J ≈ 1013 (Goldreich & Nicholson 1977). A value of Q J much larger than the above upper bound means there would be insufficient torque from Jupiter to as- semble the resonances and to maintain the current hypothesized equilibrium. This insufficiency would mean that the dissipation in Io would be currently decreasing the eccentricity, increasingω1,2, and dissassembling the Laplace relation and asso- ciated two-body resonances. Could the Laplace relation simply be decaying from an original state much deeper in the resonance? Libration of the Laplace angle ∼180◦ becomes unstable if eI > 0.012, ω1 < 0.14◦/day, and the time to decay from eI = 0.012 to eI = 0.0041 is only a few tens of millions of years with the current dissipation in Io (Yoder & Peale 1981). However, there is another stable stationary state with libration of the Laplace angle ∼0◦ with ω1 < 0 (Sinclair 1975). Greenberg’s (1987) attempt to store the system here for subsequent decay to the current configuration requires a series of improbable events, not the least of which is the establishment of the Laplace relation at the time of satellite formation within the small amount of phase space allowed by the resonances. The unlikeli- ness of this scenario is in spite of possible paths between the two stationary states (ω1 > 0 and ω1 < 0, respectively), along which stable libration could apparently be maintained (Greenberg 1987). In support of the Yoder (1979) hypothesis of an equilibrium configuration and the route thereto or modifications of that route by Malhotra (1991), two theoretical determinations of the tidal Q J are well below the upper bound established by the observed dissipation in Io. Stevenson (1983) invokes hysteresis in the tidally induced condensation and evaporation of Helium raindrops to obtain the necessary dissipation, whereas Ioannou & Lindzen (1993) abandon the equilibrium tides used almost exclusively in the past and treat the dynamic response of Jupiter’s P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 568 PEALE fluid atmosphere and interior to the tidal forcing by Io to obtain Q J as low as 103. This latter result depends on some parts of Jupiter being stably stratified with a Brunt-Väisälä frequency, the frequency of adiabatic oscillations of an element in the stably stratified region, the same as the tidal forcing frequency 2(ψ̇ J − nI ) at some depth, where ψ̇ J is Jupiter’s rotational angular velocity and nI is Io’s mean motion. If Io’s eccentricity is indeed very close to the equilibrium value, it can be used in place of the unknown interior properties of the satellite to express the energy dissipation in Io in terms of Q J alone. If the three satellites have reached the equilibrium configuration, ω1, ω2, the ratio of the semimajor axes and all the forced eccentricities are nearly fixed as the system continues its expansion from the tides raised on Jupiter by Io. Conservation of angular momentum and energy requires d dt (L I + L E + LG) = T, (14) d dt (EI + EE + EG) = nI T − H, (15) where T is the torque on Io and H is the energy dissipation rate in Io. We have neglected the tidal torques on Europa and Ganymede as well as the energy dissipa- tion therein as these are small compared with these parameters for Io. The second equation follows from the fact that dissipation in the satellites must come from the orbits because of the fixed synchronous spins. With L = m √ Gm J a(1− e2), E = −Gm J m/2a, and a−1I daI /dt = a−1E daE/dt = a−1G daG/dt from the constancy of the semimajor axis ratios, we have (Lissauer et al 1984) H = nI T ( 1− 1+ m E aI m I aE + mG aIm I aG 1+ m Em I √ aE aI + mGm I √ aG aI ) , (16) for the rate of energy dissipation in Io, where we have neglected e 2i . If we use the lower bound on the averaged Q J = 6.6 × 104 in Equation 5 for T, H in Equation 16 or H = d E/dt in Equation 8 corresponds to a surface flux density of heat on Io of ∼1000 ergs cm−2 sec−1. The comparison of this surface flux density with a measured value of 2500 ergs cm−2 sec−1 (Veeder et al 1994) is the source of the discrepancy between the maximum dissipation rate in Jupiter and that necessary to account for the measured energy dissipation in Io. This discrepancy has led several authors to propose that the tidal heating of Io or the release of the energy from the surface may be episodic with the current values of the heat flux near a maximum of the fluctuations (e.g. Greenberg 1982, Ojakangas & Stevenson 1986). However, this maximum rate of Jupiter dissipation is based on a minimum averaged Q J over all of history, where the minimum is derived from the proximity of Io to Jupiter. But notice that the discrepency would be removed if the current Q J were only a factor of 2.5 smaller than the minimum averaged P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 569 value, and there is no real reason for excluding a change in Q J with time. Indeed, Ioannou & Lindzen (1993) find that the current Q J could be even much lower than this, and their work suggests a possible mechanism for the current Q J being much lower than it was in the ancient past. According to Ioannou & Lindzen, a low Q J requires that some of the outer layers of Jupiter be stably stratified. It is probable that the early Jupiter was fully convective, but as it aged and cooled some layers may have become stably stratified leading to a much higher rate of dissipation of tidal energy. In principle, Q J can be determined by measuring the secular acceleration of Io’s mean motion. There is a possibility of doing so because reasonably precise observations of eclipses of the Galilean satellites data back >300 years. Unfortu- nately, neither of the current determinations of dnI /dt is consistent with the ob- served heating of Io in an equilibrium configuration. In this state, (dnI /dt)/nI ≈ −7.4 × 10−11/year (daI /dt ≈ 2.1 cm/year)—if we assume Q J = 6.6 × 104, its minimum averaged value, and 2.5 times this value if Q J is lowered to 2.6×104— to be consistent with the current measured heat flux from Io. Lieske (1987) finds (dnI /dt)/nI = −0.74 ± 0.87 × 10−11/year—more than 1 order of magnitude too small to account for the dissipation in Io in an equilibrium configuration. Gold- stein & Jacobs (1995) find (dnI /dt)/nI = 4.54 ± 0.95 × 10−10/year, which would imply that the Laplace relation is rapidly being destroyed. In fact with zero torque from Jupiter, this rapid increase in Io’s mean motion would require the surface heat flux density from Io to be ∼6000 erg cm−2 s−1 in a steady state—2.4 times the measured value! Any torque from Jupiter would require even more dis- sipation in Io. It would seem that any attempt to resolve these large discrepancies of observationally estimated fractional rates of change in nI from a value consis- tent with the observed dissipation in Io must await another analysis of the ancient eclipse data and the timing thereof. Perhaps nothing in solar system science causes as much excitement today as the possibility of a current liquid ocean under the ice of Europa. The images from Voyager 1 revealed cracks and blocks of ice that were displaced and rotated, resem- bling the patterns on terrestrial ice flows (Smith et al 1979). Many more examples of disrupted surfaces viewed with much higher resolution by the Galileo spacecraft show lateral displacements and rotations of blocks that retain the groove patterns of the undisrupted surface, and the blocks can thereby be reassembled into orig- inal relative locations (Carr et al 1998). Gravity experiments from recent Galileo flybys of Europa imply a differentiated satellite with a low-density layer (ice and water) perhaps 150 km thick (Anderson et al 1998b), whereas the properties of the blocks discussed by Carr et al (1998) imply an ice thickness of only a few kilometers at the time of the surface disruption. Estimates of the surface age are controversial, but the disrupted surface may be no more than 108 years old and could be much younger—implying that the processes leading to breakup of the surface and displacement of the blocks may be ongoing (Carr et al 1998). The global patterns and superpositions of large cracks in the surface are consistent with fracture perpendicular to tidal stress fields in an ice layer that slowly shifts P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 572 PEALE Ward points out that the position of the Laplacian plane is influenced greatly by the mass in the disk and that a relatively rapid dissipation of the disk that violated the adiabatic condition, thereby rotating the Laplacian plane, could leave Iapetus with its present inclination while preserving its origin in a dissipative disk, as its small eccentricity seems to imply. The leading hemisphere of Iapetus is very dark (albedo ∼0.04), whereas the trailing hemisphere is very bright (albedo ∼0.5) (Smith et al 1982). It has been suggested that this dichotomy came about because dust from Phoebe that was spiraling in due to Poynting-Robertson drag would impact only on the leading hemisphere of Iapetus because of the retrograde motion of Phoebe (Soter 1974, Hamilton 1997). Smith et al (1982) point out that the transitions between the bright and dark regions are often sharp and that craters on the bright side have dark interiors—characteristics that are not consistent with a dusting from Phoebe debris. However, Smith et al (1982) also point out difficulties with an endogenic origin of the dichotomy. Hamilton (1997) would have Iapetus being dusted on all sides during a billion-year period as it slowed to synchronous rotation. Iapetus would have then been coated with frost but with only the leading side kept dark with continued Phoebe contamination. Dark floored craters on the bright side are then the excavation down to the dark Phoebe material collected during the first phase. But it would take meticulous cratering indeed to just go down to a relatively thin layer of Phoebe dust without penetrating to the ice below. The frost layer would have to be kilometers thick as well, and the boundaries between light and dark are still too sharp for the dusting hypothesis. These caveats and the lack of a model for endogenic origin leave the origin of the albedo dichotomy on Iapetus still not understood. Six sets of orbital mean motion resonances between satellites persist in the Saturn system, and there is a secular resonance where the line of apsides of the Rhea orbit librates about that of the Titan orbit (Greenberg 1975; Pauwels 1983). The 2:1 mean motion resonance of Mimas (SI) and Tethys is the only inclination (i-type) resonance in the solar system in which the coefficient in the term in the disturbing function controlling the resonant motion contains the product of the two orbital inclinations instead of an eccentricity (e-type), and the argument of that term contains the longitudes of the ascending nodes instead of a periapse longitude. Enceladus (SII) and Dione are in a 2:1 e-type mean motion resonance, and Titan and Hyperion are in a 4:3 e-type resonance. We have already pointed out the satellites in 1:1 coorbital resonances: Janus-Epimetheus, Telesto-Tethys-Calypso, and Helene-Dione. In the Janus-Epimetheus resonance, Epimetheus describes a horseshoe orbit in a frame rotating with the average of the two mean motions; Janus, the more massive, follows a shorter loop. Yoder et al (1983) give a clear analysis of this resonance and show how the mass of each member can be determined by the distance of closest approach during the reversal of the relative motions (see also Yoder et al 1989). Telesto and Calypso librate about the L4 and L5 Lagrange equilibrium points in the Tethys orbit in the frame rotating with the mean motion of Tethy, and Helene librates about the L4 point in the Dione orbit. There are many P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 573 descriptions of such librations in tadpole-shaped trajectories about the equilibria, where the restricted three-body problem is the basis for analysis (e.g. Brown & Shook 1964). Libration in eccentricity-type resonances also has many description, but see Peale (1976) for a heuristic description of the physical mechanism of stability. Conjunctions of Enceladus and Dione always occur near the periapse of the Enceladus orbit with a corresponding forced eccentricity in the Enceladus orbit like that discussed for the Jupiter Galilean satellites. Conjunctions of Titan and Hyperion always occur near the Hyperion apoapse, and the resonance forces the Hyperion eccentricity. The mixed i-type resonance of Mimas and Tethys is considerably more complicated than the simple e-type resonances, in which both orbital inclinations are forced [but see Greenberg (1973a) for a lucid description of the physical mechanism of libration and stability]. Conjunctions of Mimas and Tethys librate about the average of the ascending node longitudes of the two orbits on the equator plane of Saturn. It is the goal of analyses of evolutionary schemes to understand how these three resonances among the somewhat larger, classical satellites came to be and at the same time to account for the exotic properties of the satellites involved. We shall see that both Mimas-Tethys and Enceladus-Dione resonances could have been assembled from initially random orbits from orbit- expanding torques, but that such an origin for the Titan-Hyperion resonance is far from robust. We consider first the Mimas-Tethys resonance. From Equation 7, we find dnM/dt − 2dnT /dt = −3.7 × 10−19/QS = −2.2 × 10−23 rad sec−2, where the fluid Love number for Saturn, kS = 0.317 (Yoder 1995), and QS = 1.7× 104, the minimum average value that would bring Mimas from the edge of the A-ring to its present position in 4 × 109 years, are used. The negative value means the Mimas orbit is approaching that of Tethys sufficiently quickly to be captured and driven deeper into a resonance at the 2:1 commensurability—a necessary condition for a tidal origin of the resonance (e.g. Peale 1986). This conclusion assumes the same value of QS for both satellites. There are several slowly varying frequencies at the 2:1 commensurability of mean motions where those with the lowest order coefficients are (2nM − 4nT + 2̇M), (2nM − 4nT + ̇M + ̇T ), (2nM − 4nT + 2̇T ), (nM − 2nT + $̇T ), (nM − nT + 2$̇M). Mimas and Tethys are locked in the mixed i-type resonance corresponding to the second frequency. The time variations in the node and periapse positions are sufficiently rapid due to Saturn oblateness that these frequencies are actually well separated. Compare the 78.8-year libration period of the Mimas-Tethys resonance variable with the separation of the first two frequencies ̇M − ̇T = −293◦ year−1. This separation motivates the treatment of each resonance as isolated. But why did the system choose to occupy the second resonance? As the Mimas-Tethys system approaches the 2:1 commensurability, nM − 2nT is decreasing, and the resonances are encountered in the order given above if only the secular perturbations of the node and periapse are considered. (e.g. The first slowly varying frequency is the first to vanish.) However, from the Lagrange Planetary equations (e.g. Danby 1988) applied to a disturbing function that selects P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 574 PEALE the resonant arguments, an approach to resonance induces the variations( d$M dt ) res = nM mT mS aM aT C1 1 eM ; ( dM dt ) res = nM mT mS C2 aM aT iT iM , (17) for the simple lowest-order e and mixed i-type resonances, where C1,C2 are negative constants. For small e, d$/dt will be large and negative. The node can have a large negative motion if iM  iT , but the presence of iT in the numerator means that the node motion will almost always be dominated by the nonresonant secular perturbations. It is much more likely that a small e will induce such a large retrograde motion in$ that a simple e-type resonance can be encountered before any of the inclination resonances and the system can automatically enter into the e-type resonance libration. However, from Equation 17 eM < 1.3× 10−6 in order for |d$M/dt | > |dM/dt |. Since such a small average eM is highly unlikely, it appears safe to assume that the Mimas-Tethys system first encountered the well separated inclination resonances in the order given (Yoder 1973; Peale 1976). We need but account for its avoidance of the first i-type resonance and capture into the second. The current amplitude of libration of the resonance variable 2λM − 4λT + M + T corresponding to the second frequency (λs are mean longitudes) is 97◦. By numerically integrating the resonance evolution backwards in time, Allen (1969) determined iM = 0.42◦ and iT = 1.05◦ at the time when the libration amplitude was 180◦, that is, at the time of capture into the existing resonance about 2.2×108 years ago. With this value of iM , Sinclair (1972, 1974) numerically calculated a capture probability of only 4.3% into the existing resonance, which was obtained analytically by Yoder (1973). The value of iM = 0.42◦ now applies, as the Mimas-Tethys system passed through the first encountered, simple i-type resonance, which yields a capture probability into this first resonance of 7.3%, numerically and analytically by the respective authors. We can thus account for Mimas-Tethys skipping the first resonance encountered and stopping in the second because the captures are probabilistic. Although this scenario makes a nice, self-consistent story of the evolution of the Mimas-Tethys system into and within the resonance, it was developed without the benefit of modern nonlinear dynamics and high-speed numerical computations. Champenois and Vienne (1999a,b) find that secondary resonances between the libration frequency and newly discovered long-period terms in the mean longitude of Mimas introduce chaos into the system that may have been important at the time of capture. The inclination of the Mimas orbit at the time of capture may have been quite different than the above value calculated by Allen (1969). If the Tethys eccentricity before capture was much larger than it is today, the calculation of the capture probabilities is more complicated than the single, isolated resonance theory used above. Although Champenois and Vienne find that a moderate eccen- tricity in the Tethys orbit could increase the capture probability into the current inclination resonance, tidal damping of that eccentricity would appear to preclude much enhancement. The full richness of the Mimas-Tethys dynamical history is P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 577 (Yoder & Peale 1981), the latter eccentricity of nearly 0.09 to initiate melting of a solid Enceladus could never occur unless QS were reduced by a factor of 10 from the minimum average used above to determine the equilibrium-forced eccentricity of 0.03. If the thermal conductivity was reduced by the inclusion of clathrate hydrates in the icy material, a significant enhancement of the present eccentricity would still be required to initiate melting, but it might be possible to maintain a molten interior and allow geologic activity with the present eccentricity. However, the conditions are too specific to be likely, and the supposition does not pass the Mimas test as pointed out by Squyres et al (1983). Mimas is comparable in size with Enceladus, it has a much larger eccentricity, and it is close to Saturn. Yet Mimas does not show any sign of tidal heating. If Enceladus contains a significant amount of ammonia along with water, the heating required is a few times less than 1017 ergs/s (Stevenson 1982; Squyres et al 1983). Could Enceladus have more NH3 than Mimas, and thereby decrease, the heating requirement? Maybe, but no NH3 has been detected on the surface. In another attempt to account for the geologic activity, Lissauer et al (1984) noticed that Janus (SX) was just outside the 2:1 commensurability with Enceladus. Spiral density waves generated by Janus in the A-ring lead to torques on the satellite that would place it at the 2:1 commensurability only ∼15 MY ago. If Janus were locked in the 2:1 eccentricity resonance that forced the Enceladus eccentricity, the relatively strong ring torques might force sufficient eccentricities on Enceladus for a melting episode. If we insert T = 8.78 × 1020 g cm2 sec2 (Lissauer & Cuzzi 1982), for an assumed ring-surface density of 50 g cm−2, into Equation 16 along with the parameters for the Janus-Enceladus-Dione system assumed to be in a Laplace-type resonance, H = 6.8 × 1016 ergs/s for the three-body case and 4.5 × 1016 ergs/s if Dione is not involved. These values are close to those required by Squyres et al (1983) for melting Enceladus. Only Enceladus would have a significant forced eccentricity in either the two-body or the three-body cases (Lissauer et al 1984), so all the energy is deposited in Enceladus. There are numerous problems with this scenario. First, the small mass of Janus means the 2:1 resonance is not very stable. The small width of the resonance means that fluctuations from other perturbations (e.g. from Titan) could disrupt the resonance. A Janus-Enceladus resonance is not likely to survive the encounter with Dione since the latter would induce fluctuations that were larger than the resonance width, while Enceladus was still far from the 2:1 resonance with Dione. Since the time to freeze Enceladus after the eccentricity has damped is at most ∼5× 107 years (Squyres et al 1983), the resonance must have been as its peak a shorter time into the past if Enceladus is to be still active in keeping itself white and in supplying E-ring particles. Lissauer et al (1984) point out that the ring torques are so strong that it is a puzzle that the small satellites inside Mimas have remained so close. All of these satellites would have been at the outer edge of the A ring only 50 MYA if the torques are correctly determined. Having some of the small satellites trapped in resonances with the intermediately sized satellites farther out would allow transfer P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 578 PEALE of angular momentum that would permit their continued proximity to the rings, but we have already seen some of the difficulties with this scenario. Having these satellites recently formed from the debris of a catastrophic disruption of a larger body is another possibility as discussed above. But a more serious problem is that the amount of angular momentum available in the A-ring could not supply the small satellites for more than ∼108 years (Lissauer et al 1984). This situation poses another potential problem for the Janus-Enceladus resonance in providing the energy for resurfacing, since Squyres et al (1983) argue that the resurfacing has been going on for much longer than this. It is clear that the means by which tidal dissipation could have resurfaced Enceladus has not been secured. A careful numerical analysis of the system from the modern dynamics point of view should be undertaken with a thorough ex- ploration of the phase space. There may be chaotic behavior and/or secondary resonances that could significantly increase the dissipation beyond that deduced by Equation 8. At the same time, there are no obvious dynamical means to account for the resurfacing of relatively small parts of Dione, Rhea, and Tethys as pointed out by Smith et al (1981, 1982). In the Titan-Hyperion 4:3 e-type orbital resonance, conjunctions librate about the Hyperion apoapse with an amplitude of 36◦ and a period of 2 years. The distance of close approaches is thereby maximized, and Hyperion owes its con- tinued existence to the resonance. The evolution of the Hyperion shape is easy to understand given the local dynamics. Impacts repeatedly chip away at all small satellites, but normally much of the material escaping from the surface is recol- lected in a relatively short time or reassembled into new satellites as must happen in the small satellite-ring region. However, anything escaping from Hyperion is likely to have sufficient initial velocity to escape the protective resonance with the giant Titan. No longer protected from close approaches to Titan, the escaped ma- terial is quickly eliminated from the region and relatively little of it reaccretes onto Hyperion (Farinella et al 1990). The remnant of this process is the flattened ham- burger shape (Smith et al 1982). The large gravitational torques on the asymmetric satellite coupled with the highly eccentric orbit do not permit the normal tidal evolution to synchronous rotation. Tides slow the satellite until it enters a chaotic zone where it is condemned to tumble chaotically for its remaining lifetime in the resonance (Wisdom et al 1984). Analytic descriptions of the Titan-Hyperion 4:3 orbital resonance are generally not adequate representations of the motion (Sinclair & Taylor 1985). The close proximity of the orbits and the high eccentricities cause very slow convergence of the series expansions. Still, it was used as the model for a simple e-type resonance, which led to an understanding of capture into such resonances as the inner orbit was expanded by tides (Greenberg 1973b). Colombo et al (1974) demonstrated capture as the Titan orbit was expanded by tides, but their conclusions are suspect because they accelerated the evolution by 10 orders of magnitude to allow the numerical computation to proceed. Such accelerations are known to introduce artifacts into the calculations (Tittemore & Wisdom 1988), although artificially P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 579 high evolution rates are more likely to frustrate capture rather than to cause it. Tidal origin of the Titan-Hyperion resonance would seem to be precluded, since QS for Titan’s tides must be >1 order of magnitude smaller than the minimum average tidal QS = 1.7 × 104 established by the proximity of Mimas to Saturn (Colombo et al 1974). However, our limited knowledge of dissipative processes in gaseous planets does not prohibit QS having sufficient amplitude and/or frequency dependence to be low enough for Titan for significant tidal expansion of the Titan orbit while being high enough for Mimas to keep the latter close to Saturn. A careful investigation of the dynamics of the Saturn atmosphere as perturbed by the tide-raising potentials would seem appropriate (see Ioannou and Lindzen 1993). Peale (1995), while assuming that significant tidal expansion of the Titan or- bit occurred, has shown that Hyperion would suffer close approaches to Titan long before the orbits approached the 4:3 commensurability unless the satellites were trapped in a secular resonance that kept the lines of apsides nearly aligned during the tidal approach ($H − $T librates about 0◦). Moreover, the initial value of aT /aH must be &0.806 to prevent capture into other resonances before the 4:3 resonance was encountered. The current characteristics of the resonance are produced for initial conditions (aH , eH , fH ,$H ) = (1.171, 0.02, 36◦, 0◦) and (aT , eT , fT ,$T ) = (0.944, 0.025, 0◦, 0◦), where the semimajor axes are in units of the current aT and f is the true anomaly. To move Titan from aT = 0.944 to 1.0, in 4.6× 109 years, QS ≈ 900. This QS necessary for the 5.9% expansion in the Titan orbit is smaller than that obtained by Colombo et al (1974), who used 1.5 instead of 0.317 for the Saturn Love number. Capture into the 4:3 resonance is always certain once Hyperion is in the secular resonance with Titan. Hyperion ec- centricities large enough for probabilistic capture lead to circulation of$H − $T with inevitable destruction of Hyperion or escape to solar orbit. The resonance could also be approached from initially nonresonant orbits if Titan experienced little or no tidal evolution, but Hyperion spiraled in from a larger orbit due to a nebular drag. This drag would act the same way as the solar nebular drag acts on planetesimals because of the differential orbital velocities between the gas and solid bodies. As a Saturn nebula would be rather short-lived, this drag evolution would have to be fairly rapid. But a few numerical integra- tions with nebular drag on Hyperion show that the libration is almost completely damped. Only if the current libration amplitude of 36◦ could be regenerated from the cometary collisions that chipped away at the original Hyperion could this origin of the resonance be viable. This possibility has not been investigated. An alternative to a tidal or nebular drag origin of the 4:3 resonance would be for Hyperion to have accreted from material previously trapped within the secular and/or the 4:3 mean motion resonances. If Titan formed by runaway accretion, it would have been in existence before the smaller Hyperion could have formed and would have cleaned out all the material that was not protected by libration within the resonances. But accretion within the 4:3 resonance, in addition to having less material available, might be prevented because of the increased velocity disper- sion for the nonresonant bodies induced by Titan. These bodies would crash into P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 582 PEALE Saturn and Jupiter. But the resurfacing of the satellites, as well as the anomalously high 4.2◦ inclination of the Miranda orbit, has motivated extensive investigations of possible past orbital resonances, their tidal dissipation within, and their ultimate disruption (Peale 1988, Tittemore & Wisdom 1988, 1989, 1990, Dermott et al 1988, Malhotra & Dermott 1990, Tittemore 1990). Most have used the nominal masses for the satellites in these studies, but Peale (1988) demonstrates a variety of evolutions that depend on the satellite masses within the errors of their deter- minations. The 1 σ errors for the satellite masses have been reduced by about a factor of 2 (factor of 3 for Miranda) since the Peale study (Jacobson et al 1992), and the nominal masses of Miranda and Ariel would now have to be changed by ∼3σ in opposite directions to change the current divergence of the orbits under differential tidal expansion to convergence. Although this mass dependence of possible evolutions should be kept in mind as refined mass determinations are made, we shall assume the nominal masses adopted by the authors of tidal evo- lution scenarios yield the correct directions of approach to the resonances. The evolutionary scenarios have been at best only marginally or speculatively success- ful in obtaining sufficient tidal dissipation in the satellites to account for the resur- facing of Ariel or Miranda, but they have been so successful in accounting for the high orbital inclination of Miranda due to the 3:1 Miranda-Umbriel orbital mean motion commensurability that a reasonably robust upper bound of QU/kU . 3.9× 105 is determined to ensure that the system passed through this resonance (Titte- more & Wisdom 1989). Dermott (1984) remarked that since the gravitational coefficient J2 = 0.003343 (Yoder 1995) for Uranus is small, the several resonances at each of the commen- surabilities may not be separated sufficiently in frequency to be treated under the single resonance theory discussed by several authors (e.g. Yoder 1973; Henrard 1982; Henrard & Lemaitre 1983; Peale 1986, 1988) and used above for the Saturn and Jupiter satellite orbital resonances. The resulting possibility of widespread chaos during resonance passage meant that the predictions of the single-resonance theory discussed by these authors are not valid for the Uranian satellites. The single-resonance theory was indeed found not to apply to most of the Uranus res- onances discussed, when modern numerical and analytical techniques were used (Tittemore & Wisdom 1988, 1989, 1990; Dermott et al 1988; Malhotra & Dermott 1990; Tittemore 1990). Two fundamentally new dynamical results came from the analysis of the Uranus satellite resonances: first, the rate at which a chaotic system could be carried through a resonance (i.e. the magnitude of the tidal torque) without introducing artifacts into the results of the integrations was 10 to 1000× slower than the rate that satisfied the adiabatic invariance of the action integral in single- resonance theory (Tittemore & Wisdom 1988, 1989, 1990); second, a librating system in a stable resonance could be trapped into a secondary resonance be- tween the libration frequency and other nearby frequencies, such as the circulation frequency of another resonance variable associated with the same mean motion commensurability, and dragged during continued tidal evolution into a chaotic zone thereby disrupting the original resonance (Tittemore & Wisdom 1989). This P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 583 secondary resonance evolution can be displayed dramatically in the circular in- clined or planar elliptic approximations where two degrees of freedom allow the construction of surfaces of section. But the secondary resonance capture persists in the full problem with its several degrees of freedom (Tittemore & Wisdom 1989). Malhotra & Dermott (1990) and Malhotra (1990) elaborated the theory of secondary resonances. For quasiperiodic motion, it is clear that the results of an integration will not represent a real resonance encounter and the action of the librational motion will not be adiabatically conserved if the rate of evolution is so high that there is a significant change in the libration frequency during the time of a single libration. Hence, ω̇τ/ω  1 is a necessary condition for adiabatic invariance of the action integral, and the results of the integration would normally be expected to be invariant as long as the rate of evolution was slow enough for this inequality to be satisfied. Here ω is the frequency of libration, τ is its period, and the dot indicates time differentiation. However, if the motion is chaotic from the interaction with nearby resonances, invariance of the integration results are attained for rates slower than some maximum only if a chaotic adiabatic invariant exists that is defined for a two- degree-of-freedom problem as the phase space volume enclosed by the energy surface containing the chaotic zone (Brown et al 1987). The chaotic adiabatic invariant exists if the trajectory in phase space has time to thoroughly explore the chaotic zone before there is significant change in the configuration (Tittemore & Wisdom, 1988). The rates of evolution consistent with this criterion, where results become independent of the rate of evolution, were shown to be 10 to 1000 times slower (depending on the resonance), than those allowed by the criteria for adiabatic invariance in the single resonance theory. The maximum rate of evolution in this case can be determined only by numerical experiment. In some resonances, the results of the integrations remained dependent on the rate of integration even at rates less than those allowed by the physical constraints (Tittemore & Wisdom 1990). We now investigate the consequences of differential tidal evolution of the or- bits where the Uranus QU is assumed the same constant for all of the satellites. Ariel’s orbit expands faster than that of Miranda in spite of its greater distance from Uranus since its mass is so much larger. This expansion requires the av- erage QU/kU & 66,000, since the two orbits would be coincident 4.6 × 109 years ago for QU/kU at the lower bound. Peale (1988) has shown the important first- and second-order resonances that could have been encountered for this max- imal evolution of the system. Miranda would have passed through the 4:3, 3:2, and 5:3 commensurabilities with Ariel, but close approaches between Miranda and Ariel inside the 4:3 resonance would have eliminated Miranda from the sys- tem, so the lower bound on the average QU/kU would have to be increased over the above value to start the system outside the 4:3 resonance. Because the dif- ferential tidal expansions cause the orbits to diverge, all of the Miranda-Ariel resonances are approached from the wrong direction for capture (e.g. Peale 1986, Tittemore & Wisdom 1989). Still, passage through the large chaotic zone of the 5:3 P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 584 PEALE commensurability leads to chaotic variations of the eccentricities and inclinations of both satellites with maxima eM . 0.03, eA . 0.007, iM . 1.5◦, iA . 0.35◦, where values of the inclinations and eccentricities before resonance encounter for both satellites were 0.005 radians and 0.005, respectively (Tittemore & Wisdom 1990). The system leaves the 5:3 resonance region with eM and iM about twice the initial values and eA and i A slightly below their initial values. There is insignificant heating of Miranda or Ariel either during or after this resonance passage as the eccentricities damp according to Equation 9 (Tittemore & Wisdom 1990). It is noteworthy that there is no chaotic adiabatic invariant for this resonance down to an evolution rate within the physical constraints, so the integrations were carried out at a rate corresponding to QU/kU = 1.1 × 105, the tentative lower bound justified below. After leaving the 5:3 commensurability with Ariel, Miranda passes through the 3:1 commensurability with Umbriel with profoundly important consequences. For inclinations and eccentricities before resonance encounter at 0.005 radians and 0.005, respectively, for both satellites, the resonances are encountered in the order (λM − 3λU + 2M), (λM − 3λU +M +U ), (λM − 3λU + 2U ), (λM − 3λU + 2$U ), (λM − 3λU +$M +$U ), (λM − 3λU + 2$M). The most important event in the passage through this series of resonances at the 3:1 commensurability is capture into either the iM2 or the iMiU resonance corresponding to the first two resonance variables involving the node of the Miranda orbit. In either of these resonances the Miranda inclination is driven to large values as tidal evolution of the orbits continues. As the inclination grows, the frequency of libration within the resonance increases and approaches low-order commensurabilities with the circu- lation frequency of the iU2 resonance variable third in the above list. The system can be trapped in one of these secondary resonances where subsequent evolution drags the trajectory into the chaotic zone where the system ultimately escapes the primary resonance involving the Miranda node. The value of iM after escape from the resonance depends on the particular trajectory through the phase space, but it is always comparable with the observed large value of 4.22◦ (Tittemore & Wisdom 1989, 1990). There appears to be a correlation of the peak of the distribution of remnant inclinations of the Miranda orbit with the particular secondary resonance that drags the trajectory into the chaotic zone. The 2:1 secondary resonance pro- duces inclinations that tend to be too high, the 4:1 too low, and the 3:1 close to the observed value (Malhotra & Dermott 1990). This natural explanation for the anomalously high inclination of the Miranda orbit leads one to infer that the system must have passed through this resonance. This requirement places a reasonably robust upper bound on QU/kU . 3.9× 105 (Tittemore & Wisdom 1989). After escape from the inclination resonances at the 3:1 commensurability of mean motions, the Miranda-Umbriel system passes into a large chaotic zone as- sociated with the eccentricity resonances corresponding to the three resonance variables that include the longitudes of periapse. The Miranda eccentricity may reach values as large as 0.05 or 0.06 (Tittemore & Wisdom 1990) during the chaotic fluctuations, but the short time scale of these excursions leads to P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 587 time to establish the densely cratered surface on Miranda before the resurfacing took place. But the time scale for damping the eccentricity and simultaneously softening Miranda to relax to a nearly spherical shape could be as short as 6000 years (Greenberg et al 1991)—much too short for all of the geologic scenarios to have taken place. We are left without an acceptable means to account for the bizarre surface of Miranda. If Miranda and Umbriel necessarily passed through the 3:1 mean motion com- mensurability, then Ariel and Umbriel passed through the 5:3, which is the most recent first- or second-order resonance to have been traversed (e.g. Peale 1988). The only treatment of passage through this resonance is the planar approxima- tion by Tittemore & Wisdom (1988). The eccentricity resonances at the 5:3 mean motion commensurability involve a large chaotic zone separating circulation and libration of the resonance variables. The probability of not being captured in the resonance is no longer determined by a uniform distribution of random phases as in the single resonance theory (e.g. Peale 1986) because the trajectory in phase space can spend a considerable amount of time in the chaotic zone. The numeri- cally determined probability of escape from this resonance is ∼30% in the planar approximation (Tittemore & Wisdom 1988), where significant remnant orbital ec- centricities might account for the somewhat high current eccentricities of the orbits of Umbriel and Ariel. However, at no time were eccentricities maintained in either the Ariel or the Umbriel orbit during and after the resonance passage sufficient for significant tidal heating. Including dissipation in the satellites as well as inclination terms in the analy- sis could significantly change the evolutionary results of this study—the first by keeping the eccentricities at lower values than those obtained by Tittemore and Wisdom and the second by possibly forcing higher eccentricities (Tittemore & Wisdom 1988). Given the results of the planar problem, it is probably the case that a complete three-dimensional treatment of passage of Ariel and Umbriel through the 5:3 commensurability, including dissipation in the satellites, will not alter the conclusion of Tittemore and Wisdom that Ariel could not have been heated suffi- ciently to account for its resurfacing. But we have been surprised many times in the past, so it would be prudent to carry out the calculations to be sure. One last attempt to resurface Ariel through tidal dissipation involves a possi- ble 2:1 resonance between Ariel and Umbriel (Peale 1988; Tittemore & Wisdom 1990). If this resonance is to be encountered QU/kU < 1.1 × 105. If the reso- nance is approached with small eccentricities in both orbits, as is likely, the motion is dominated by quasiperiodic behavior, and capture into libration for both reso- nance variables, ((λA − 2λU + $A), (λA − 2λU + $U ), is apparently certain (Tittemore & Wisdom 1990). No chaotic separatrices (regions in phase space separating circulation from libration) are crossed in this capture for small eccen- tricities. Noncapture into the resonance becomes increasingly likely for approach eccentricities >0.03 for Ariel, where the now interacting resonances at the 2:1 commensurability create substantial chaotic zones. However, approach at such a high value of eccentricity is very unlikely (Tittemore & Wisdom 1990). Upon P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 588 PEALE capture into the resonance, the eccentricities grow as tides raised on Uranus force Ariel deeper into the resonance until an equilibrium eccentricity is approached that remains constant thereafter because of dissipation in the satellite. This process was discussed above for the Jupiter satellite Io. If we include the torque TU on Umbriel in equations analogous to Equations 14 and 15 but with only two satellites, the energy dissipated in the two satellites in an equilibrium configuration is (Peale 1988) H = n ATA [ 1− 1+ (mU/m A)(aA/aU ) 1+ (mU/m A) √ aU/aA ] + nU TU [ 1− 1+ (m A/mU )(aU/aA) 1+ (m A/mU ) √ aA/aU ] = 0.249n ATA, (18) where the symbols are analogous to those in Equation 16 and where the final form uses TU/TA = (mU/m A)2(aA/aU )6 = 0.0469. Nearly all of this maximum energy dissipation is in Ariel (Peale 1988). From Equations 5, 8, and 18, the equilibrium eccentricity eA = 0.018 for QU/kU = 1.1 × 105. Thus, the maximum rate of energy dissipation in Ariel at the time of the necessary disruption of the resonance is 7.7× 1016 ergs sec−1 or 5.69× 10−8 ergs g−1 sec−1 averaged over the mass of Ariel. The resonance must be disrupted when n A = 1.234n A0, with n A0 being the current value, if the system is to reach the current configuration (Peale 1988). If Ariel and Umbriel spent a considerable time in the resonance, then QU/kU would have to be smaller to allow the resonance to begin somewhat before the nec- essary disruption, n A and TA would correspond to smaller semimajor axes, and the corresponding dissipation would have been larger. With a density of 1.67 g cm−3, a water ice (1 g cm−3) mantle and a rocky core (2.8 g cm−3) comprising 61% of the mass would yield a radiogenic heat production rate of ∼ 0.61 × 1.6 × 10−7 ergs g−1 sec−1 averaged over the Ariel mass, where a time average radio- genic heating rate estimated for the Moon is used (Peale 1988). The tidal heating given above, the minimum at the end of the resonance existence, is about half of the averaged radiogenic heating rate. Although tidal heating would have been larger earlier in the resonance existence, it appears inadequate to account for any melting and resurfacing of Ariel, even if the 2:1 resonance with Umbriel had persisted. There is a much more serious problem with the 2:1 Ariel-Umbriel resonance— there is no known way to disrupt the resonance once established (Tittemore & Wisdom 1990). If the eccentricities were allowed to increase to large values within the resonance, increasing chaos offers the possibility of escape, although capture in secondary resonances (so important in the 3:1 Miranda-Umbriel resonance) do not appear to drag the system into the chaotic zone. Still, such continued growth of the eccentricities would almost certainly result in disruption of the resonance, except dissipation in the satellites places a modest upper bound on eA . 0.02. Eccentricity would increase to the equilibrium value and sit there indefinitely as the system librates for the remaining existence of the solar system. At least the secular P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 589 perturbations (terms in the Hamiltonian with $i − $ j in the arguments) from Titania do not appear to disturb the resonance (Tittemore & Wisdom 1990). Perhaps something has been missed that will appear with an integration of the complete system in all its degrees of freedom, and that could disrupt this 2:1 resonance after it was established. For now we must assume that the system never encountered the resonance or it would still be locked within. The almost certain avoidance of this resonance and the almost certain encounter of the 3:1 commensurability between Miranda and Umbriel that so nicely accounts for the large orbital inclination of the former means 1.1 × 105 < QU/kU < 3.9 × 105 are apparently rigorous bounds on the dissipative properties of Uranus (Tittemore & Wisdom 1990). Although a convincing argument for using tidal dissipation to resurface the Uranian satellites may ultimately be constructed, we have so far failed to account for any of the young surfaces on these satellites in a rigorous way. It is significant that Titania has extensive resurfacing but sits between Umbriel and Oberon whose ancient surfaces are completely undisturbed. Titania could have occupied no or- bital resonances of first or second order (Peale 1988), although Tittemore (1990) looked at a possible passage of Ariel and Titania through the 4:1 mean motion commensurability. Here, the Ariel eccentricity could grow to large values that, however, could raise its temperature only∼20 K. Titania was still unaffected ther- mally by this resonance. Higher-order resonances are weaker because of additional factors of e or i in the coefficients of the resonance terms, and one needs to check the stability of such a resonance to perturbations by the other satellites before embracing its consequences. Still, Titania remains untouched by tidal dissipation even if it had occupied third-order resonances with Ariel (Tittemore 1990). Titania may be telling us something about our difficulties in obtaining sufficient tidal dissipation to resurface those satellites that did occupy or pass through orbital resonances. If new surfaces on Ariel are indeed 2.6×109 years old (Plescia 1987b) and if internal activity can persist as long as one billion years after the initial heat pulse from disruption and reaccretion, perhaps the resurfacing is more due to the accretional heating than to tidal dissipation. Even this scheme may require an H2O–NH3 eutectic to lower the melting point, and it seems to fail in any case for Miranda (Squyres et al 1988). On the other hand, it is hard to believe that what we see on the surfaces of the Uranian satellites is independent of the evolution caused by the tides. 9. NEPTUNE SYSTEM The dominant characteristic of the Neptune satellite system is the existence of the large satellite Triton (NI) in a close, circular, retrograde orbit (obliquity 156.8◦). Neptune also has relatively few known satellites compared with the other major planets, and all but two of those, Triton and Neried (NII), were unknown until the Voyager spacecraft observations (Smith et al 1989). Except for Neried, there are no satellites outside the Triton orbit, and the Neried orbital eccentricity of 0.75 P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 592 PEALE Banfield & Murray (1992) thus apply Equation 7 only to Proteus to estimate the lower bound on the average Neptune QN = 12,000 (4× 109 years/TC), where TC is the time before the present that the Triton orbit circularized. This follows from the condition that Proteus could not have started inside the current corotation radius of 3.25RN . The evolution of the distribution of masses and orbits of the inner five satel- lites is completely speculative with the exception of the 4.7◦ inclination of the orbit of innermost satellite, Naiad. All five inner satellites are inside the corotation radius and are therefore spiraling toward Neptune at rates determined by Equa- tion 7. Naiad is so small that its semimajor axis is decreasing from tides raised on Neptune at a rate that is slower than that of any of the other four in spite of its closer proximity to Neptune. Therefore, various mean motion resonances between Naiad and any of the other four satellites are approached from a direction (orbits approaching each other) that allows capture into and evolution within the mean motion resonances. The strongest inclination-type mean motion resonances have i21 , i1i2, or i 2 2 in the coefficient of the appropriate term in the disturbing function, where i refers to the respective orbital inclinations to the Neptune equator plane for the two satellites. The importance of these resonances is that orbital inclinations are forced to grow while the system is forced deeper into the resonance by the Neptune tides, which could account for the Naiad inclination of 4.7◦ (Banfield & Murray 1992), as for the Uranus satellite Miranda (Section 8). Only those reso- nances with the Naiad inclination in the coefficient need be considered, since such a resonance increases only that inclination and not that of the other resonance mem- ber. The other satellites still have orbital inclinations near their initial very small values. Banfield & Murray (1992) have determined capture probabilities for 35 pos- sible inclination-type resonances between Naiad and the next four satellites. Al- though the individual capture probabilities are small, the probability that Naiad was captured in one of these particular resonances is ∼76%. The resonances are disrupted when a secondary resonance between an adjacent primary resonance and the libration drags the system into a chaotic zone (Section 8). The inclination established within the resonance remains after the resonance is disrupted. Three resonances were found to be disrupted when the orbital inclination of Naiad was near the observed 4.7◦—NIII-NV 12:10, NIII-NV 11:9, NIII-NV 10:8. The first has the greatest probability of occurrence, and it was chosen by Banfield & Murray (1992) as the best candidate for accounting for the inclination of Naiad. The total probability of occurrence of 4% includes capture into the primary resonance, cap- ture into a 2:1 secondary resonance, and escape at the right inclination. Although this probability is not large, the fact that the probability of getting captured into 1 of 35 primary inclination resonances is 76% means that Naiad probably got caught in one, and this one is as likely as any of the others–and it matches the observations (Banfield & Murray 1992). If the NIII-NV 12:10 resonance did indeed cause the in- clination in the Naiad orbit, an upper bound, QN ≤ 330,000 (4×109 years/TC ), is established. This upper bound follows from the necessity that the NIII-NV system P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 593 has passed through the 12:10 mean motion resonance. Although Thalassa (NIV) was also likely captured into inclination resonances that increase its inclination, there are numerous ways in which the likely inclination of escape was not signifi- cantly different from that observed (∼0.21◦). Capture into resonances at the same commensurabilities that could affect the other satellite inclinations apparently did not occur. Of course there is an assumption here that the distribution of masses for the inner satellites has been as it is now for most of the ∼4 × 109 years since the Triton orbit circularized. Any cometary fragmentation and redistribution of mass among reformed satellites after their initial formation following Triton circulariza- tion must have been confined to reasonably early times. The existence of satellites within the Roche radius is not limited to Neptune. Banfield & Murray (1992) hy- pothesize that they could either have formed outside the Roche radius but inside the corotation radius and be transported inwards by tidal friction or that they accreted there in spite of the opposing tidal forces through the pieces sticking together by nongravitational forces. Neither of these hypotheses has been investigated in detail to establish a self-consistent scenario. Given the uncertainty in the collisional and reformation history, any attempt to refine the history of the inner satellites must always remain nondefinitive, and limited scientific return from such an excercise is probably not worth the considerable effort involved. The conjectured possibil- ities for the collisional and dynamical history of the inner satellites of Neptune constructed by Banfield & Murray (1992) are representative of what might have happened. Triton is spiraling into Neptune in its retrograde orbit. Chyba et al (1989) conjectured two possible Cassini states (Peale 1969), which fix the Triton obliquity in the frame precessing with the orbit. In Cassini state 1, Triton obliquity would be nearly zero, whereas in state 2, the obliquity would be∼100◦ with vastly different rates of tidal dissipation in Triton and, hence, different rates of orbital decay for the two cases. Voyager 2 data (Smith et al 1989) revealed Triton to have a very small obliquity consistent with occupancy of state 1. In this case, Chyba et al (1989) find that Triton will reach the Roche radius in ∼3.6 × 109 years. If Triton is mostly solid at that time, it can continue to smaller orbital radii still intact, although it probably would not survive all the way to the surface. The breakup of Triton within the Roche radius would lead to a spectacular set of rings with initially much more total mass than those of Saturn. 10. PLUTO-CHARON SYSTEM It was highly fortuitous that the Pluto satellite Charon was discovered sufficiently far in advance of our passing through the Pluto-Charon orbit plane that a well organized series of observations allowed a remarkably rich characterization of such a distant and otherwise obscure system. Mutual eclipses and occultations during the orbit plane passage allowed radii, masses and albedo distributions to be P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 594 PEALE determined (Binzel and Hubbard 1997). Subsequent observations with the Hubble Space telescope indicate a deviation from the circular orbit expected from tidal evolution (Tholen and Buie 1997). Yet we still know relatively little about this system compared with all the others, since we have not had the benefit of close observations by spacecraft instrumentation. As a result there are perhaps fewer constraints that we must respond to in the context of the evolutionary history of the system. Still, the system displays its own unique features, and increasing knowledge of the system may provide needed constraints on the nature of the objects in the furthest reaches of the solar system. If we assume the Pluto-Charon system is the consequence of a giant impact by a planetesimal whose mass is comparable with the initial Pluto mass (Farinella et al 1979, Dermott 1978 (unpublished); Dobrovolskis 1997), some of the debris from the impact will escape the system and some will fall back onto Pluto, but sufficient debris must end up outside the Roche radius in order to collect into the observed satellite (e.g. Cameron 1997 and references therein). The debris will settle to the equatorial plane and the orbits of individual particles will be circularized through collisional dissipation. Charon will accrete most of its mass from this disk within a few hundred years (Thompson and Stevenson 1988), and it should then end up with a nearly circular orbit with nearly zero inclination relative to the equatorial plane of Pluto. We assume, therefore, that Charon began its existence as a satellite in circular, equatorial orbit at 3RP (Pluto radii). The system will be assumed to have its current angular momentum for this initial condition with the total mass and mass ratio derived from Table 1. From this initial configuration it tidally evolves to its current state of dual synchronous rotation, which has been observationally confirmed (Buie et al 1997). From Equation 6, Charon would reach synchronous rotation at a separation of 3RP from a 4-h initial period in about 25 years (µ = 4× 1010 dyne cm−2, QC = 100). The actual Q should be much less than this, as Charon will have just accreted and may be partially melted. Regardless of assumptions, Charon should be locked into permanent synchronous rotation almost immediately after formation. Torques from tides raised on Charon should thus be unimportant in the subsequent orbital evolution except for helping to keep the eccentricity damped. Integration of Equation 7 for Q P = constant from an initial orbital period of 11.6 h at time ti to the current period of 6.39 days at time t f yields t f − ti = 1.6 × 103 Q P/kP ≈ 1.7 × 107 years, where µ = 1011 and Q p = 100 were assumed. The time to reach the dual synchronous rotation state is short compared with the age of the solar system. Although we expect the Pluto-Charon dual syn- chronous system to have a circular orbit, recent observations with the Hubble Space Telescope have indicated an orbital eccentricity, between 0.003 and 0.007 (Tholen and Buie 1997), although this determination has not been confirmed (Tholen, pri- vate communication, 1998). Two means of exciting such an eccentricity in the face of tidal damping have been proposed—direct collision of a Kuiper belt ob- ject with Charon or Pluto (Tholen and Buie 1997) and differential perturbations by passing Kuiper belt objects (Levinson and Stern 1995). The latter study finds collisional excitation very unlikely but the KBO perturbations can be sufficient, P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 597 Neptune satellite Triton seems to yield a self-consistent history leading to the cur- rent configuration and surface properties of the satellites. The small inner satellites are second or higher generations—the products of repeated breakup and reaccre- tion. The high orbital inclination of Naiad in spite of its accumulation in a dissipa- tive equatorial disk is nicely accounted for by capture into and evolution within an inclination type orbital resonance—like Miranda except here the orbits are spiral- ing toward the primary rather than away. Pluto-Charon have reached the endpoint of tidal evolution of dual synchronous rotation. The corresponding relaxation to circular orbit may be slightly frustrated by the perturbations of passing Kuiper belt objects. We have observed the striking uniqueness of each satellite system within the solar system, and we have had several successes in understanding the origin of current configurations and properties of the several systems. However, we have also pointed out a significant number of remaining interesting problems that will occupy clever minds for years to come as they are resolved one by one. Greater understanding will uncover even more problems as our knowledge of the satellites is refined. ACKNOWLEDGMENTS It is a pleasure to thank the following colleagues for commenting on particular sec- tions of the manuscript: R Greenberg, A Harris, H Levison, R Malhotra, N Murray, A Stern, D Stevenson, J Wisdom, and C Yoder. They detected errors and omissions and generally offered good advice. Special thanks are due MH Lee, who read the entire manuscript and offered many suggestions for improving the clarity and con- sistency. The author received support for this work from the Planetary Geology and Geophysics Program in NASA under grant NAG5 3646. Visit the Annual Reviews home page at http://www.AnnualReviews.org LITERATURE CITED Acuna MH, Neubauer FM, Ness NF. 1981. J. Geophys. Res. 86:8513–21 Allen RR. 1969. Astron. J. 174:497–506 Alterman Z, Jarosch H, Pekeris CL. 1959. Proc. R. Soc. London Ser. A 252:80–95 Anderson JD, Jacobson RA, Lau EL, Sjogren WL, Schubert G, Moore WB. 1998a. Science 280:1573–76 Anderson JD, Lau EL, Sjogren WL, Schubert G, Moore WB. 1996. Nature 384:541–43 Anderson JD, Lau EL, Sjogren WL, Schubert G, Moore WB. 1997. Nature 387:264–66 Anderson JD, Schubert G, Jacobson RA, Lau EL, Moore WB, et al. 1998b. Science 281: 2019–22 Banfield D, Murray N. 1992. Icarus 99:390– 401 Beckwith SVW, Sargent AI. 1998. Nature 383:139–44 Benz W, Cameron AGW, Melosh HJ. 1989. Icarus 81:113–31 Benz W, Slattery WI, Cameron AGW. 1986. Icarus 66:515–35 Binzel RP, Hubbard WB. 1997. In Pluto and Charon, ed. SA Stern, WB Hubbard. Tuc- son: Univ. Arizona Press. pp. 85–102 P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? 598 PEALE Boss AP. 1997. Science 276:1836–39 Boss AP. 1998. Astrophys. J. 503:923–37 Boss AP, Peale SJ. 1986. In Origin of the Moon, ed. WK Hartmann, RJ Phillips, GJ Tay- lor, pp. 59–101. Houston, TX: Lunar Planet. Inst. Brown EW, Shook CA. 1964. Planetary The- ory, pp. 250–88. Mineola, NY: Dover Brown R, Ott E, Grebogi C. 1987. J. Stat. Phys. 49:511–50 Buie MW, Tholen DJ, Wasserman LH. 1997. Icarus 125:233–44 Cameron AGW. 1997. Icarus 125:126–37 Cameron AGW, Benz W. 1991. Icarus 92:204– 16 Cameron AGW, Canup RM. 1998a. In Lun. Planet. Sci. Conf. 29:1062 (Abstr.) Cameron AGW, Canup RM. 1998b. In paper presented at Orig. Earth Moon Conf. Mon- terey, CA. Dec. 1–3 Cameron AGW, Ward WR. 1976. In Proc. Lun. Planet. Sci. Conf. 7:120–22 Canup RM, Esposito LW. 1995. Icarus 113:331–52 Canup RM, Esposito LW. 1996. Icarus 119:427–46 Carr MH, Belton MJS, Chapman CR, Davies ME, Geissler P, et al. 1998. Nature 391:363– 65 Cassen P, Peale SJ, Reynolds RT. 1979. In Geo- phys. Res. Lett. 6:731–34 Cassen P, Peale SJ, Reynolds RT. 1980a. Geo- phys. Res. Lett. 7:987–88 Cassen P, Peale SJ, Reynolds RT. 1980b. Icarus 41:232–39 Champenois S, Vienne A. 1999a. Icarus. In press Champenois S, Vienne A. 1999b. Celest. Mech. Dyn. Astron. In press Christensen EJ, Born GH, Hildebrand CD, Williams BG. 1977. Geophys. Res. Lett. 4:555–57 Chyba CF, Jankowski DG, Nicholson PD. 1989. Astron. Astrophys. 219:L23–26 Colombo G. 1966. Astron. J. 71:891–96 Colombo G, Franklin FA, Shapiro II. 1974. As- tron. J. 79:61–71 Colwell JE, Esposito LW. 1992. J. Geophys. Res. 97:10227–41 Croft SK, Soderblom LA. 1991. In Uranus, pp. 561–628. Tucson: Univ. Arizona Press Cynn HC, Boone S, Koumvakalis A, Nicol M, Stevenson DJ. 1988. Lun. Plan. Sci. Conf. Abs. 19:433 Danby JMA. 1988. Fundamentals of Celestial Mechanics. Richmond VA: Willman-Bell, 466 pp. Darwin GH. 1879. Philos. Trans. R. Soc. 170: 447–530 Darwin GH. 1880. Philos. Trans. R. Soc. 171: 713–891 Degewij J, Zellner B. Andersson LE. 1980. Icarus 44:520–40 Dermott SF. 1978. Pluto, Herculina, Mercury, and Venus: Their real and imaginary satel- lites. Unpublished Dermott SF. 1984. In Uranus and Neptune, ed. J Bergstralh, pp. 377–404. NASA Conf. Publ. 2330 Dermott SF, Malhotra R, Murray CD. 1988. Icarus 76:295–334 Dickey JO, Bender PL, Faller JE, Newhall XX, Ricklefs RL, et al. 1994. Science 265:482– 87 Dobrovolskis AR. 1982. Icarus 52:136–48 Dobrovolskis AR, Peale SJ, Harris AW. 1997. In Pluto and Charon, ed. AS Stern, DJ Tholen, pp. 159–90. Tucson: Univ. of Arizona Press Dones L, Tremaine S. 1993. Icarus 103:67–92 Drake MJ. 1986. J. Geophys. Res. 92:E377–86 Duxbury TC, Callahan JD. 1981. Lun. Planet. Sci. Conf. Abstr. 13:191 Duxbury TC, Veverka J. 1978. Science 201:812–14 Farinella P, Milani A, Novili AM, Valsecchi GB. 1979. Moon Plan. 20:415–21 Farinella P, Milani A, Nobili AM, Valsecchi GB. 1980. Icarus 44:810–12 Farinella P, Paolicchi P, Strom RG, Kargel JS, Zappala V. 1990. Icarus 83:186–204 Geissler PE, Greenberg R, Hoppa G, Helfens- tein P, McEwin A, et al. 1998. Nature 391:368–70 Gerstenkorn H. 1955. Z. Astrophys. 26:245–74 P1: fne/FGM/FGG/FGP P2: FDR/fgm QC: FDR/Arun T1: Fhs September 9, 1999 20:54 Annual Reviews AR088-13 ? SATELLITES 599 Gladman BJ, Nicholson PD, Burns JA, Kave- laars JJ, Marsden BG, et al. 1998. Nature 392:897–99 Goldreich P. 1963. Mon. Not. R. Astron. Soc. 126:257–68 Goldreich P. 1965a. Mon. Not. R. Astron. Soc. 130:159–81 Goldreich P. 1965b. Astron. J. 70:5–9 Goldreich P. 1966. Rev. Geophys. 5:411–39 Goldreich P, Lyndon-Bell D. 1969. Astrophys. J. 156:59–78 Goldreich P, Murray N, Longaretti PY, Banfield D. 1989. Science 245:500–4 Goldreich P, Nicholson PD. 1977. Icarus 30:301–4 Goldreich P, Peale SJ. 1970. Astron. J. 75:273– 84 Goldreich P, Ward WR. 1973. Astrophys. J. 183:1051–61 Goldstein SJ, Jacobs KC. 1995. Astron. J. 110:3054–57 Greeley R, Sullivan R, Klemaszewski J, Homan K, Head JW, et al. 1998. Icarus 135:4–24 Greenberg R. 1973a. Mon. Not. R. Astron. Soc. 165:305–11 Greenberg R. 1973b. Astron. J. 78:338–46 Greenberg R. 1975. Mon. Not. R. Astron. Soc. 170:295–303 Greenberg R. 1982. In Satellites of Jupiter, ed. D Morrison, pp. 65–92. Tucson: Univ. of Arizona Press Greenberg R. 1987. Icarus 70:334–47 Greenberg R, Croft SK, Janes DM, Kargel JS, Lebofsky LA, et al. 1991. In Uranus, pp. 693–735. Tucson: Univ. of Arizona Press Greenberg R, Geissler P, Hoppa G, Tufts BR, Durda DD, et al. 1998. Icarus 135:64–78 Greenberg R, Weidenschilling SJ. 1984. Icarus 58:186–96 Greenzweig Y, Lissauer JJ. 1990. Icarus 87:40– 77 Greenzweig Y, Lissauer JJ. 1992. Icarus 100:440–63 Hamilton DP. 1997. Iapetus: Bull. Am. Astron. Soc. 29:1010 Harris AW, Young JW, Dockweiler T, Gibson J, Poutanen M, et al. 1992. Icarus 95:115–47 Hartmann WK, Davis DR. 1975. Icarus 24: 504–14 Henrard J. 1982. Celest. Mech. 27:3–22 Henrard J. 1983. Icarus 53:55–67 Henrard J, Lemaitre A. 1983. Celest. Mech. 30:197–218 Horanyi M. 1996. Bull. Amer. Astron. Soc. 28:1126 Horanyi M, Burns JA, Hamilton DP. 1992. Icarus 97:248–59 Ida S, Canup RM, Stewart GR. 1997. Nature 389:353–57 Ioannou PJ, Lindzen RS. 1993. Astrophys. J. 406:266–78 Jacobson RA, Campbell JK, Taylor AH, Syn- nott SP. 1992. Astron. J. 103:2068–78 Kary DM, Lissauer JJ. 1993. Icarus 106:288– 307 Kaula WM. 1964. Rev. Geophys. 2:661–85 Kaula WM. 1969. Astron. J. 74:1108–14 Khurana KK, Kivelson MG, Stevenson DJ, Schubert G, Russel CT, et al. 1998. Nature 395:777–80 Kivelson MG, Khurana KK, Coroniti FV, Joy S, Russell CT, et al. 1997. Geophys. Res. Lett. 24:2155–61 Kivelson MG, Warnecke J, Bennett L, Joy S, Khurana KK, et al. 1998. J. Geophys. Res. 103:19963–72 Knopoff L. 1964. Q. Rev. Geophys. 2:625–60 Kokubo E, Ida S. 1998. Icarus 131:171–78 Laskar J, Robutel P. 1993. Nature 361:608–12 Lemmon MT, Karkoschka E, Tomasko M. 1995. Icarus 113:27–38 Levinson HF, Stern SA. 1995. Lunar Planet. Sci. 26:841 (Abstr.) Lieske JH. 1987. Astron. Astrophys. 82:340–48 Lissauer JJ. 1987. Icarus 69:249–65 Lissauer JJ, Cuzzi JN. 1982. Astron. J. 87:1051– 58 Lissauer JJ, Peale SJ, Cuzzi JN. 1984. Icarus 58:159–68 Love AEH. 1944. A Treatise on the Mathemati- cal Theory of Elasticity. Mineola, NY: Dover Luu J. 1991. Astron. J. 102:1213–25 MacDonald GJF. 1964. Rev. Geophys. 2:467– 541