Tidal Evolution of the Uranian Satellites

Home , Ariel (moon), Ariel 5, Miranda (moon), Miranda (satellite), Tidal heating, Umbriel (moon)

ICARUS 85, 394--443 (1990)

III. Evolution through the Miranda-Umbriel 3:1, Miranda-Ariel 5:3, and ArieI-Umbriel 2 : 1 Mean-Motion Commensurabilities 1

WILLIAM C. TITTEMORE Department of Planetary Sciences, Lunar and Planetary Laboratory, University of Arizona, Tucson, Arizona 85721

AND

JACK WISDOM Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received January 27, 1989; revised November 20, 1989

Tidal evolution has significantly affected the history of the Uranian satellite system. The presence of large chaotic zones at past mean-motion commensurabilities among the Uranian satellites has resulted in significant changes in the orbital elements of some of the satellites, while allowing them to escape from the commensurabilities and evolve to their present nonresonant configuration. Miranda and Umbriel have probably passed through the 3:1 commensurability, resulting in Miranda's current anomalously high inclination, and constraining the Q of Uranus to be less than 39,000 (W. C. Tittemore and J. Wisdom 1989, Icarus 78, 63-89). The orbits of both satellites become chaotic during evolution through the eccentricity resonances associated with the 3 : 1 commensurability. During this phase of evolution, the orbital eccentricity of Miranda can be driven up to a value of about 0.05. Miranda can then escape from the 3:1 commensurability with a relatively large orbital eccentricity, which can damp to the current value in the time since resonance passage. Tidal friction may have heated the interior of Miranda to a temperature near the eutectic melting point of NH 3 • H20 , but most likely did not result in the melting of significant quantities of water ice. Miranda and Ariel passed through the 5 : 3 mean- motion commensurability if the Q of Uranus is less than about 12,000. During evolution through this commensurability, the semimajor axis ratio (aM/aA) decreased. As the orbits enter a large chaotic zone associated with this commensurability, both the eccentricity and inclination of Miranda's orbit jump to values up to six times higher than the values approaching the resonance. Upon escaping from the resonance, the orbit of Miranda may have retained moderately high eccentricity and inclination, or eM and i M may have decreased back to values comparable to those approaching the resonance. If the Q of Uranus is smaller than about U,000, Ariel and Umbriel would have encountered the 2:1 mean-motion commensurability. However, capture into this resonance is very likely if the eccentricities of the orbits approaching the resonance were comparable to the current values: the probability of escape would not have been significant unless the initial eccentricities were of order 0.03 or larger. It is therefore unlikely that these satellites encountered this resonance, constraining the Q of Uranus to be greater than 11,000. The specific dissipation function of Uranus is therefore well constrained: 11,000 < Q < 39,000. ©1990 Academk Press, Inc.

i Contribution 89-07 of the University of Arizona Theoretical Astrophysics Program. 394 0019-1035/90 $3.00 Copyright 6) 1990 by Academic Press, Inc. All rights of reproduction in any form reserved. DYNAMICS OF PAST URANIAN RESONANCES 395

1. INTRODUCTION of the small oblateness of Uranus was pointed out by Dermott (1984), who sug- The Uranian satellite system contains a gested that since resonances at mean-mo- rich variety of features of interest to plane- tion commensurabilities among the Uranian tary scientists. Voyager 2 revealed that the satellites are not well separated, there might satellites Miranda, Ariel, and Titania have be chaotic behavior. He speculated that cha- had spectacular geological histories (Smith otic behavior at past orbital commensurabil- et al. 1986), despite their small sizes and low ities might explain some of the currently current temperatures. Sources of internal observed dynamical features. Significant heat for these satellites must have been pres- secular interactions between the satellites ent at some point in time to account for the (Greenberg 1975, Dermott and Nicholson exotic features observed on their surfaces. 1986, Laskar 1986) resulting from the rela- In addition, there are some interesting tively large satellite masses may also con- anomalies in the orbits of the Uranian satel- tribute to chaotic behavior. Since there are lites. For one, the satellites are not currently large increases of eccentricity and inclina- involved in any low-order mean-motion res- tion at the asteroidal resonances (Wisdom onances, in contrast to the satellite systems 1982, 1983, 1987b), it is natural to speculate of Jupiter and Saturn. Also, the orbital ec- that similar chaotic behavior among the centricities of the inner large satellites are Uranian satellites might lead to significant anomalously large when the time scale of changes in the orbits and tidally heat the damping due to satellite tides is taken into satellites. consideration (Squyres et al. 1985). Finally, To address this problem in a rigorous the orbital inclination of Miranda is more manner, we have been carrying out a sys- than an order of magnitude larger than those tematic study of mean-motion resonances of the other satellites (Whitaker and Green- which may have been encountered by the berg 1973, Veillet 1983). Uranian satellites as the orbits evolved due Tidal evolution is an important process to tidal friction. We have found significant in planetary satellite systems. Most of the chaotic zones at low-order mean-motion commensurabilities in the Jovian and Satur- commensurabilities between the Uranian nian systems are probably the result of tidal satellites (Tittemore and Wisdom 1987, evolution of the orbits (Goldreich 1965). 1988a,b, 1989, Tittemore 1988). The pres- Tidal heating of Io in the Laplace resonance ence of these chaotic zones significantly af- explains the geological activity on this satel- fects the mechanisms and outcomes of reso- lite (Peale et al. 1979). Tides raised on Ura- nance passage. Satellites may spend a nus by its major satellites tend to increase considerable period of time evolving within in the orbital semimajor axes, as energy is a commensurability, during which orbital dissipated in the planetary interior. Because eccentricities and/or inclinations may vary the major satellites move in eccentric orbits, significantly, and then escape from the reso- the tidal distortion of the satellites caused nant interaction, allowing the satellites to by the planet varies periodically. Internal evolve to their present nonresonant config- friction dissipates energy in the satellites, uration. The standard integrable theory of tending to decrease the orbital semimajor evolution through resonances (Goldreich axes and eccentricities, and heating the sat- and Peale 1966, Counselman and Shapiro ellite interiors. 1970, Yoder 1979a, Henrard 1982, Henrard Two features of the Uranian system dis- and Lemaitre 1983, Borderies and Goldreich tinguish it from the Saturnian and Jovian 1984, Lemaitre 1984) has limited applicabil- systems: the oblateness of the planet is rela- ity to the Uranian satellite system. tively small, and the satellite-to-planet mass Our approach has been to model the most ratios are relatively large. The importance important interactions involving the orbital 396 TITTEMORE AND WISDOM

TABLE I

URANIAN SATELLITES

Satellite R a m/Mu ~ a/Rtj ~ T ~ e b i c (km) (days) (radians)

Miranda 242 + 5 8.63 ± 2.6 × 10 -7 4.96 1.41 0.0014 -+ 0.0002 0.0737 --- 0.0028 Ariel 580 - 5 1.55 ± .28 × 10 -5 7.29 2.52 0.0017 --- 0.0002 0.0054 ± 0.0019 Umbriel 595 ± 10 1.47 ± .28 × 10 -5 10.15 4.15 0.0043 ± 0.0002 0.0063 ± 0.0014 Titania 805 ... 5 4.00 --- .21 × 10 -5 16.65 8.70 0.0025 ± 0.0001 0.0025 - 0.0005 Oberon 775 - 10 3.37 ± .19 × 10 -5 22.27 13.46 0.0003 --- 0.0001 0.0018 ± 0.0004

Stone and Miner (1986). M U = 8.69 x 1025 kg, and Rtj = 26,200 km (French et al. 1985). b Peale (1988). ¢ Veillet (1983). eccentricities or inclinations of satellites to study resonant asteroidal motion, which near a resonance by a Hamiltonian system reduce the amount of computer time needed with a parameter that evolves due to tidal to carry out the numerical simulations. We friction. The parameter changes at a rate have integrated many trajectories through determined primarily by the tidal specific the commensurabilities using the mappings, dissipation function (Q) of the planet. Tidal to determine probabilities of various out- dissipation in the satellites may also affect comes, and have tested the effects of vari- the rate at which the model parameter ous perturbations on our models. changes, and affects the orbital eccentrici- During our program of study of the Ura- ties through direct tidal damping, which in- nian satellite system, we have found that a troduces frictional terms into the equations large chaotic zone exists at the Ariel-Um- of motion. The time scale over which tidal briel 5 : 3 commensurability (Tittemore and effects become important is in general much Wisdom 1988a, henceforth referred to as longer than the dynamical time scale of the TW-I). During evolution through this com- resonant interaction, and therefore tidal ef- mensurability, the orbital eccentricities of fects act as slow changes in the Hamilto- Ariel and Umbriel may have increased mod- nian. As a first approximation, we have only erately within the chaotic zone, before the modeled the effects of planetary tides on the satellites escaped from the resonance. How- resonant interactions. This has allowed us ever, the increases in the orbital eccentricity to explore the dynamical environment of the most important resonances among the Ura- nian satellites, and to determine constraints TABLE II on the extent to which the satellite system TIDAL PARAMETERS has tidally evolved. We have used nominal orbital and physical properties of the satel- Satellite p~ /zb k2c Q (g/cm 3) (dyn/cm2) lites in our models (see Tables I and II), to represent as closely as possible the evolu- Miranda 1.26 ± 0.39 4 x 101° 1.4 × 10 -3 100 tion of the physical system. Ariel 1.65 ± 0.30 l011 4.3 x l0 -3 100 In some cases, a two-degree-of-freedom Umbriel 1.44 ± 0.28 l011 3.3 × 10 -3 100 Hamiltonian model is appropriate, allowing a Stone and Miner (1986). detailed study of the dynamics using sur- b Shear modulus, ~4 × 1010 dyn/cm z (water ice), faces of section. We have developed alge- ~6.5 x l0 II dyrdcm 2 (rock). braic mappings from these models (see, e.g., 3pigiRi c k2 i = ~ __, Tittemore and Wisdom 1988a), similar to 1 + 19P.i/2pigiR i 19P.i those developed by Wisdom (1982, 1983) k2 = 0.104 (Gavrilov and Zharkov 1977). DYNAMICS OF PAST URANIAN RESONANCES 397 of Ariel at this resonance do not appear to be also below), the dynamics of resonance pas- sufficient to significantly affect its thermal sage may be extremely sensitive to the simu- evolution. We have also found that the cur- lated rate of tidal evolution, even if the simu- rent anomalously high orbital inclination of lated rate is orders of magnitude slower than Miranda can be accounted for by passage the rate predicted by the single-resonance through the 3:1 commensurability with adiabatic criterion. In such cases, the rele- Umbriel (Tittemore and Wisdom 1987, vant criterion is based on a "chaotic" adia- Tittemore and Wisdom 1989, henceforth batic invariant (e.g., Brown et al. 1987). In referred to as TW-II). The satellites are tem- the case of chaotic Hamiltonian systems porarily captured into one of the inclination with two degrees of freedom, the adiabatic resonances, until a secondary resonance, invariant is the phase space volume en- or commensurability between the resonant closed by the energy surface containing the libration frequency and the frequency differ- chaotic zone. This is a natural generalization ence between neighboring resonant argu- of the concept of adiabatic invariants in ments, drags the satellites into a large cha- integrable systems (e.g., Lenard 1959, otic zone at large iM, whereupon the Kruskal 1962, Arnold 1963). Approximate satellites can escape from the inclination conservation of the chaotic adiabatic invari- resonance region. The requirement that the ant requires that the chaotic zone be well satellites have passed through this commen- explored by the trajectory on time scales surability constrains the Q of Uranus to be much shorter than the time scale of reso- less than 39,000 [using k2 = 0.104 (Gavrilov nance passage. We are continuing to exam- and Zharkov 1977)]. We also found that the ine the applicability of chaotic adiabatic in- orbital eccentricity of Miranda increases variants to orbital resonances. In some dramatically during evolution through this cases, however, we have found that adia- commensurability (Tittemore and Wisdom batic invariants do not exist even at tidal 1987, TW-II), possibly providing a signifi- evolution rates that are within our con- cant internal heat source. This phase of straints on the physical rate, for example, in lution is further discussed in Section the Miranda-Umbriel 3 : 1 commensurabil- 2.1. ity during evolution through the eccentricity We have also carried out a careful evalua- resonances when the inclination of Miranda tion of the effect of the simulated rate of is high (see Section 2.1). In this case, high- tidal evolution on the dynamical outcome. dimensional chaos is important, and the In the integrable single-resonance theory, dynamics are not well approximated by a the dynamics of resonance passage are inde- two-degree-of-freedom model. pendent of rate if a particular "adiabatic Dermott et al. (1988) have also found cha- criterion" is satisfied, that is, the action of otic variations at the Miranda-Umbriel 3 : 1 a trajectory (area in phase space) is approxi- commensurability, by integrating the full mately invariant except at the point of tran- gravitational equations of motion at the res- sition across the separatrix. This occurs onance for a relatively short time with the when the fractional change in the libration orbital eccentricity and inclination of frequency in one libration period due to tidal Miranda artifically enhanced. Although the evolution is small (see Sections 2.1. I and integration of the full gravitational equations 2.2.1 and TW-II). In cases where the evolu- of motion includes all of the higher-order tion is dominated by quasiperiodic behavior resonant dynamical effects, this approach (TW-II), the single-resonance adiabatic cri- requires significantly longer computational terion may be used to determine the simu- times: direct numerical integrations require lated rate of tidal evolution at which the about three orders of magnitude more com- dynamics become independent of rate. puter time than the algebraic mapping meth- However, in cases where the evolution is ods we have employed (Wisdom 1982). dominated by chaotic behavior (TW-I, see Therefore, using direct numerical integra- 398 TITTEMORE AND WISDOM tions, it is not yet feasible to simulate tidal encountered, because if they had been, Ar- evolution of trajectories fully through a iel and Umbriel must have passed through commensurability at rates within our con- the 2 : 1 commensurability, which we have straints on the physical rate. However, as shown to be highly unlikely (see Section we show (see Section 2.1), at the 3 : 1 Mi- 2.3). randa-Umbriel commensurability, it is nec- Marcialis and Greenberg (1988) have pro- essary to use simulated tidal evolution rates posed an alternate model for the tidal heat- within our constraints on the physical rate, ing of Miranda, in which the damping of the or there will be rate-induced artifacts in the orbital eccentricity during a phase of chaotic dynamics. Even the short numerical integra- rotation at some point in its early history tions carried out by Dermott et al. use too provides the energy source. However, a high a simulated tidal evolution rate (see number of key assumptions made by the Section 2.1), and therefore the dynamics are authors are not well justified: (a) The au- significantly affected by rate-induced arti- thors have not determined the size of the facts. A number of techniques used by those chaotic zone for parameters appropriate to authors to reduce the computing time for Miranda. Their assertion that a large chaotic numerical integrations that evolve all the zone exists is dependent upon Miranda hav- way through the commensurability also in- ing a large orbital eccentricity of order 0. l, troduce artifacts into the dynamics. First, which cannot be primordial, since the or- the authors use the single-resonance adia- bital eccentricity damping time scale of batic criterion to judge whether the simu- Miranda is only of order 108 years (see Sec- lated rate of tidal evolution is slow enough. tion 2.1). (b) The authors do not demon- As we have shown in TW-I and TW-II and strate a mechanism for transferring Miranda here in Section 2.1, this is not a sufficient from the synchronous rotation state to the criterion. The authors also artificially in- chaotic zone. The resonance overlap crite- crease the mass of Umbriel, simultaneously rion (see Wisdom et al. 1984) cannot by itself increasing the J2 of Uranus. The justification be used to determine whether a satellite is given for this is to increase the single-reso- attitude unstable. For example, Phobos and nance adiabatic rate and, thereby, further several other small, irregularly shaped satel- reduce the computing time. However, lites satisfy the resonance overlap criterion changing the masses and J2 significantly for the synchronous and 3/2 rotation states, changes the dynamical characteristics of the and have large chaotic zones in the spin-or- problem. Finally, the authors cause trajec- bit phase space (Wisdom 1987a), but are not tories to ultimately escape from the reso- currently in a chaotic rotation state. (c) For nance by increasing the simulated rate of a body in a chaotic tumbling state, tidal evo- tidal evolution by a factor of 20 in the middle lution of the spin and orbit is determined of the integration, which also significantly by the damping of the "rotational Jacobi alters the dynamics and probably artificially integral" due to tidal dissipation, not just drags the trajectory out of the chaotic zone. the damping of the orbital eccentricity (see Therefore, although the large increases in Wisdom 1987a). These problems must be eccentricity and inclination and eventual addressed before this mechanism can be disruption of the resonance found by the considered viable. authors are interesting, they do not repre- We present here the results of our contin- sent the dynamics of the actual Mi- uing studies of the Uranian satellite system. randa-Umbriel system. The results of their Section 2 explores the evolution of Miranda numerical integrations of Miranda through and Umbriel through the 3 : l mean-motion a series of first-order resonances with Ariel commensurability. Tittemore and Wisdom are also affected by similar problems. How- (1987, see also TW-II) found that during ever, those resonances were probably never evolution through this commensurability, DYNAMICS OF PAST URANIAN RESONANCES 399 the orbital eccentricity of Miranda may also have been encountered if Q is less than have increased dramatically. We more fully 12,000 [using k 2 = 0.104 (Gavrilov and explore this phase of the evolution here. Zharkov 1977)], is discussed in Section 2.1. Following an initial phase of temporary cap- This commensurability is similar to the 3 : 1 ture into one or more of the second-order Miranda-Umbriel commensurability in that eccentricity resonances associated with this the mass of Ariel is much greater than that of commensurability, the satellites enter a Miranda, so the orbital elements of Miranda large chaotic zone. In the planar-eccentric are more strongly affected by the resonance approximation, the maximum and minimum passage than are those of Ariel. It is similar eccentricities in the chaotic zone increase in to the 5 : 3 Ariel-Umbriel commensurability a very regular manner, indicating the exis- (TW-I) in that they involve the same combi- tence of a chaotic adiabatic invariant. How- nations of resonance angles for the two sat- ever, because of the high inclination of Mi- ellites, and because the orbits are relatively randa's orbit that develops during the prior close together, there is a relatively strong passage through the inclination resonances, secular interaction between the satellites there is a strong coupling between the ec- which significantly affects the orbit of centricity and inclination resonances, and Miranda. However, passage through this the actual variations of eccentricity are resonance differs from the previously con- much more irregular than the planar approx- sidered ones in a significant way: for the imation, although the maximum values of nominal masses of Miranda and Ariel (see eccentricity are similar in both models. Dur- Table I), the semimajor axis ratio (aM/aA) ing evolution through the large chaotic zone, decreases due to tidal dissipation in the the orbital eccentricity of Miranda varies planet. Upon encountering this resonance, chaotically from zero to about 0.05. Chaotic the orbital eccentricity and inclination of variations occur over time scales compara- Miranda suddenly increase by about a factor ble to the resonance passage time scale even of 6 as the satellites evolve through a large for simulated tidal evolution rates within our chaotic zone. The changes in the orbit of constraints for the physical rate of evolu- Ariel are relatively insignificant. Miranda tion, due to significant high-dimensional may leave this resonance with an orbital chaotic wandering. This indicates that a cha- eccentricity and inclination comparable to otic adiabatic invariant does not exist in the or somewhat larger than the current values. three-dimensional case at physical rates of The maximum rate of tidal heating of Mi- tidal evolution. Miranda may have retained randa at this resonance was about one-third a relatively large value of eccentricity after the maximum rate at the 3 : 1 commensura- the satellites escaped from the resonant in- bility with Umbriel. Both the structure of teraction. In the time since the satellites en- the phase space and the numerical experi- countered this commensurability, the incli- ments indicate that there is no mechanism nation of Miranda's orbit would have of capture into this resonance, a conclusion remained high, explaining the current anom- also reached by Peale (1988) using argu- alously high value, while the orbital eccen- ments based on the available energy. tricity would have damped to the current Finally, the Ariel-Umbriel 2:1 mean- value. Tidal heating of Miranda may have motion commensurability is discussed in raised its internal temperature to a value Section 2.3. The probability of capture was near the eutectic melting point of ammo- very high if the satellites approached this nia hydrate, but most likely did not result resonance with orbital eccentricities compa- in the melting of large quantities of water rable to the current values. For such small ice. initial eccentricities, the evolution is domi- The 5 : 3 mean-motion commensurability nated by quasiperiodic behavior: although involving Miranda and Ariel, which would secondary resonances among fundamental 400 TITTEMORE AND WISDOM frequencies affect the amplitude of libration 2-1 mean-motion commensurabilities. We in the resonance, the chaotic zones associ- have considered three models for each com- ated with the secondary resonances are mensurability. First, we consider the pla- small. Once captured into the resonance, nar-eccentric approximation, in which the the satellites would have evolved to the resonant motion can be described by a two- equilibrium eccentricity of Ariel without the degree-of-freedom Hamiltonian, and the dy- resonance becoming unstable. For much namics can be studied in detail using sur- larger initial eccentricities, significant cha- faces of section. We have carried out a sys- otic zones are present at this resonance; tematic survey of each commensurability, however, eccentricities of order 0.03 would determining capture probabilities, studying have been required for the probability of the effects of the rate of tidal evolution on escape to be significant. Therefore, it is un- the dynamics, and determining the structure likely that the satellites ever encountered of the phase space at different points in the this commensurability. The 2 : 1 commensu- evolution. This has allowed us to gain a de- rability is probably a dynamical "barrier" tailed understanding of the dynamical mech- to the tidal evolution of the satellite system. anisms affecting the evolution of the orbital Therefore, the specific dissipation function eccentricities at these resonances. For the (Q) of Uranus is constrained to be greater planar-eccentric models, numerical calcula- than 11,000 [using k 2 = 0.104 (Gavrilov and tions have been carded out using the alge- Zharkov 1977)]. braic mappings. The development of the pla- There are significant chaotic zones associ- nar-eccentric Hamiltonian models is given ated with all of the resonances considered in Appendix I, and the development of tidal in this investigation. The dynamics are ex- parameters is given in Appendix II. In calcu- citing. The mechanisms of resonance pas- lating the numerical values of the Hamilto- sage are significantly different than in the nian coefficients and other quantities, the integrable theories of resonance passage. adopted unit of time is one year, of mass is There are some similarities between differ- the mass of Uranus, and of length is the ent resonances, but each has unique proper- radius of Uranus (see Appendix I), unless ties, resulting in a rich variety of dynamical otherwise specified. behavior in the Uranian satellite system. We also consider the effects of perturba- The dynamics of the individual resonances tions on the planar-eccentric dynamical may be used to constrain the evolution of models. There are important resonant terms the system as a whole. Since passage involving the inclinations at each commen- through the 3 : 1 resonance with Umbriel can surability. The inclinations are coupled to explain the current inclination of Miranda's the eccentricities through nonlinear terms. orbit, it is likely that the satellites have tid- We have developed Hamiltonian models for ally evolved at least enough to encounter the study of the three-dimensional resonant this commensurability. This sets an upper behavior (Appendix I) and algebraic map- limit of 39,000 on the specific dissipation pings to carry out the numerical integra- function (Q) of Uranus. Requiring that Ariel tions. and Umbriel did not encounter the 2 : 1 com- In addition, for each commensurability mensurability constrains Q to be greater we determine the effects of perturbations than 11,000. This rather narrow range may due to other satellites on the resonant be- be useful in modeling the interior of Uranus. havior. The secular interactions between the major Uranian satellites result in sig- 2. RESONANCE DYNAMICS nificant variations of the eccentricities and In this section, we describe in detail the inclinations (Dermott and Nicholson 1986, dynamics of the Miranda-Umbriel 3:1, Laskar 1986), and the frequencies are com- Miranda-Ariel 5 : 3, and Ariel-Umbriel parable to the important frequencies at reso- DYNAMICS OF PAST URANIAN RESONANCES 401

nances. We have developed Hamiltonian The dynamical properties of the 3 : 1 com- models for the motion at a mean-motion mensurability were determined by integra- commensurability perturbed by the secular ting sets of trajectories through the reso- interactions of the two resonant satellites nance. The initial coordinates for each with a third (see Appendix I). At the trajectory in a set were determined from the Miranda-Umbriel 3 : 1 commensurability, physical parameters aM = 4.8630, we consider the perturbations due to Ariel; a U = 10.1179, e~ = 0.005, and e u = 0.005, at the Miranda-Ariel 5:3 commensurabil- together with the initial angles o-u = ~-/2, ity, those due to Umbriel; and at the and o-u = 3~r/2, from which the coordinates Ariel-Umbriel 2 : 1 resonance, those due to x i = 0.0, YM = 0.0009213, XU = 0.0, and Titania. Numerical integrations of the aver- Yu = -0.004574, and the parameter aged equations of motion were carded out 8 = -0.6153 were computed for an initial using the Bulirsch-Stoer (1966a,b) algo- trajectory. These eccentricities are compa- rithm. rable to the current eccentricities of the Ura- nian satellites. From this initial point, the 2.1. THE MIRANDA--UMBRIEL 3 : 1 coordinates of 199 additional points, spaced COMMENSURABILITY in time by 100 mapping periods (T = 2zr/40 2.1.1. The Planar-Eccentric Model years) were computed using the mapping We consider first the interaction between without tidal dissipation. These points Miranda and Umbriel in the planar-eccen- formed the initial coordinates of the trajec- tric approximation, initially ignoring the fact tories, each starting with the same energy, that it is likely that the orbital inclination of 8, and action (area enclosed by a trajectory Miranda was comparable to its current value in phase space), but with different phases. at the time of encountering the eccentricity These trajectories were then tidally evolved resonances. Although in fact the large or- through the resonance using the mapping. bital inclination of Miranda significantly af- As in TW-I and TW-II, we have carried fects the evolution through the eccentricity out a study to determine the influence of resonances (see Section 2.1.2), consider- the simulated rate of tidal evolution on the ation of the planar model enables us to un- dynamics of the 3 : 1 resonance, by integra- derstand many qualitative features of the ting sets of trajectories through the reso- evolution through the eccentricity resonance at different dissipation rates. The nances. simulated rate of tidal evolution is parame- The tidal parameter is defined to be 8 = terized by the effective specific dissipation 3nu - nM -- t~U -- &tJ (see Appendix I). At function of Uranus, as usual designated by this commensurability, the orbital semima- ~, which is distinguished from the physical jor axis of Miranda increases much more specific dissipation function Q for Uranus, rapidly than that of Umbriel. The time rate which is presumably fixed and unknown. of change of 8 is approximated by (see Ap- We wish to determine a simulated rate of pendix II) tidal evolution sufficiently slow that the dynamics of resonance passage are not af- 3 hM --n" M --- ~nM--. (1) fected by rate-induced artifacts. aM Plotted in Fig. 1 are the mean escape or- The equilibrium eccentricity at this reso- bital eccentricities of Miranda and Umbriel nance is e M = 0.026, if Q = 11,000 for Ura- for each set of trajectories as a function of nus and Q = 100 for Miranda (see Appendix dissipation rate. The mean escape orbital II). However, as we will show, the interac- eccentricity of Miranda is quite high, of or- tion between the satellites at such eccentric- der 0.02, for ~ > 1. For ~ < 0.1, the escape ities is chaotic, and an equilibrium configu- orbital eccentricities of Miranda and ration may not be achieved. Umbriel are the same as the initial values: 402 TITTEMORE AND WISDOM

0.03 , , , , ~ , , since the dynamical time scale is then much shorter than the tidal time scale. The libration frequency of the e~ resonance considered independently is given by 0.02 (see TW-II for details of analogous calculations in the circular-inclined approximation) (e) toE = -- 4H(8 - 80) (2)

0.01 where 60 = 2(C - D) - 16BXM is the value m m m at which libration zones first appear in the s ~ t ~ phase space of this resonance. The change in the libration frequency of the e~ reso- 0 it.0 I I 0 20 310 410 50 610 710 8.0 nance in one period is Aw L -~ -47rHS/to 2, log 10 (11,000 / Q) wherea ~4.1 × 10-4/&. For eM = eu = 0.005 well before the resonance is encoun- FIG. 1. Mean escape orbital eccentricities of Miranda tered, 80 = -0.29, and the trajectory en- (0) and Umbriel ([~) for ensembles of 200 trajectories as a function of tidal dissipation rate, expressed in counters the resonance at 8 = 0.5. There- terms of the effective specific dissipation function of fore, the single-resonance theory predicts Uranus ~. The single-resonance adiabatic rate for the that for the e 2 resonance with e i = e U = e~ resonance is denoted by the vertical bar (9~ = 0.07, 0.005, AtOL/OL ~ 0.07/~. In the single-reso- itM/a M = 9.4 × 10 -9 per orbit period). For simulated nance approximation, then, the results of tidal evolution rates larger than this the trajectories are dragged through the resonance without displaying any numerical simulations should be indepen- interesting behavior. However, the dynamics are sensi- dent of the simulated tidal evolution rate tive to the rate even for rates an order of magnitude if ~ > 0.07. We have denoted the single- slower than this. resonance adiabatic rate in Fig. 1 by the vertical bar in the upper right of the figure. The results shown in Fig. 1 indicate that for simulated tidal evolution rates higher than they have been dragged through the reso- this value, not much of dynamical interest nance without displaying any interesting be- happens. However, the dynamical outcome havior. Between these values of ~, the aver- appears to be sensitive to the simulated rate age escape eccentricity varies significantly. of tidal evolution for rates about an order of As we have done for the circular-inclined magnitude slower than the rate predicted problem (TW-II), we can compare the sim- by the single-resonance adiabatic criterion. ulated tidal evolution rate at which dynami- Therefore, the single-resonance approxima- cal artifacts appear to the adiabatic rate tion is not sufficient to determine the adia- based on the single-resonance model. The batic rate in the planar-eccentric model. A first eccentricity resonance to be encoun- large chaotic zone dominates the evolution tered is the e~ resonance. We express the through the eccentricity resonances, and the simulated rate of tidal evolution as the relevant adiabatic criterion for the planar- change of libration frequency of the e~ eccentric model is based on a chaotic adia- resonance considered independently in one batic invariant. libration period divided by the libration The slowest rate used to integrate many frequency: (AtOL/tOL). In the single-reso- trajectories, ~ = 110 (//M/am ~ 6 × 10 -12 nance model, the action (area enclosed per orbit period), appears to be slow enough by a trajectory in the phase space) is to represent the dynamics of passage approximately conserved if this nondimen- through this commensurability well in the sional quantity is less than about unity, planar-eccentric approximation. Numerical DYNAMICS OF PAST URANIAN RESONANCES 403 simulations of resonance passage at this rate havior of the trajectory. The surfaces of sec- show that the chaotic zone is well explored tion chosen plot YM versus XM when Xu = 0, on time scales much shorter than the tidal which we designate section I, and tr u versus evolution time scale (e.g., Fig. 3, see be- Eu when XM = 0 (O'M = ~'/2), which we low), indicating that a chaotic adiabatic in- designate section II. Initial conditions on variant is approximately conserved. the surface of section are determined by In Fig. 2, the distributions of time-aver- solving a quartic equation (see TW-I). As aged escape eccentricities for Miranda and with the circular-inclined model (TW-II), Umbriel for ~ = 110 are shown. As in the the phase space is simple enough that we circular-inclined problem (see TW-II), the need only study one quartic root family for orbit of Umbriel is also significantly affected each surface of section: the second largest by the resonant interaction, although the root for section I and the largest root for changes in eccentricity are small compared section II. On section I, e M is the radial to those of Miranda. It is quite possible for distance from the origin, while the resonant the orbit of Miranda to attain orbital eccen- argument o-M is measured counterclockwise tricities much larger than the single-reso- from the positive abcissa. On section II, eu nance equilibrium value of about 0.026 dur- is plotted along the right ordinate axis for ing passage through this resonance, even in scale. the planar approximation. The high orbital Figure 3 shows the variations of the or- eccentricity of Miranda remaining after es- bital eccentricities of Miranda and Umbriel cape from the resonance can damp to the for one of the trajectories evolved through current value in the time since resonance the 3:1 commensurability in the planar- passage. eccentric approximation. The maximum Again, for this problem we can "freeze" and minimum orbital eccentricities are plot- the energy and 8 of a trajectory at any point ted in a small interval of 8 (AS ~ 0.0035). in the evolution and study the structure of Well before the resonance is encountered, the phase space by computing surfaces of the motion is regular. Both orbital eccentric- section, to understand the qualitative be- ities are nearly constant: the mutual secular

50.0 60.0 a b

Y2 50.0 40.0

Y//,/.,, 40.0 30.0 "//A

N ,//, ~ Y/9"/" N 30.0 "///Y//:/:/, 20.0 ,,,,,, Y/l, Y/ 20.0

10.0 10.0

0 0 R 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.005 0.01 0.015 0.02 eM eu

FIG. 2. Distributions of escape orbital eccentricities of Miranda (a) and Umbriel (b) for an ensemble of 200 trajectories evolved through the resonance with ~ = 110, or i~M/aM = 6 × 10-12 per orbit period. Both orbits are affected by the resonant interaction. The high orbital eccentricity of Miranda retained after escape from the resonance can damp to the current value in the time since resonance passage. 404 TITTEMORE AND WISDOM

0.015 0.07 u n i n u a b 0.06

0.05 0.01 eM eU j /j

// ~¸ '~" 0.005

O.Ol

01'~----'~/' , ~l , 0 I I I I I 0 20 ,'0 6'o 8'0 100 0 2.0 4.0 60. 8.0 10.0 8 8

FIG. 3. Evolution of the orbital eccentricities of Miranda (a) and Umbriel (b) during passage through the 3 : 1 commensurability in the planar-eccentric approximation, with hM/aM = 6 × l0 -12 per orbit period (~ = 110). The maximum and minimum eccentricities are plotted in intervals A8 = 0.0035. After initially being captured into the mixed (eMeu) resonance, the trajectory enters a large chaotic zone and the orbital eccentricity of Miranda evolves to a large value before the trajectory escapes from the resonance.

interactions are quite weak. As the reso- otic zone at the bottom of the figure. The nance is approached, the amplitude of ec- trajectory displayed in Fig. 3 has just been centricity variation increases slightly. At captured into the eMeu resonance, and is in 8 ~ 0.5, the trajectory encounters the e 2 the libration region in the lower left of the resonance, which it is not captured into, figure. resulting in the slight decrease of average e U As this trajectory evolves within the eMetj visible in Fig. 3b. Shortly thereafter, the resonance, the orbital eccentricities of both trajectory encounters the mixed (eMeu) res- satellites increase. Shortly after being cap- onance, at 8 = 0.8. Figure 4a displays the tured, though, the trajectory encounters a phase space (section II) through which the chain of four islands associated with a sec- trajectory has evolved at this point in its ondary resonance between the libration fre- evolution. The two large islands and narrow quency of the eMeu resonance and the fre- chaotic zone visible in the center of the fig- quency of circulation of the e~j resonant ure are associated with the e 2 resonance. At argument. Figure 4b shows the phase space the bottom left of the figure is the libration inhabited by the trajectory displayed in Fig. region associated with the eMeu resonance, 3 just after capture into this secondary reso- surrounded by a chaotic zone. Between the nance. The trajectory from Fig. 3 generates two primary resonant libration regions are two loops in the chain of four islands within smaller chains of islands and narrow chaotic the zone of libration of the eMeU resonance zones, associated with low-order commen- in the bottom left of the figure. The chaotic surabilities between the circulation frequen- zone surrounding the libration region has cies of the e 2 and eMeu resonant arguments. grown, having "swallowed" more neigh- Near the eMeU resonance, the spacing be- boring secondary resonant island chains tween these secondary resonant zones is that were separate in Fig. 4a. smaller than the widths of the islands, and The trajectory is "dragged" out of the gives a complicated appearance to the cha- eMeu libration zone by the secondary reso- DYNAMICS OF PAST URANIAN RESONANCES 405

. ._ 0.0060 1.59e-05..... 0.0062

1.322e-05 0.0056 0.0058

eu eu

1.122e-05~ 0.0052 119• 05 ~~t 0.0053

9 22e-06 "~"" 0.0047 99 .o6 -x -~/2 0 ~/2 rc -n -~/2 0 ~/2 x UU OU

1'78e-051 ~'~ ' . /0.0065

-.\

1.38e-05 0.0057

1.18e-05 0,0053 -n -n/2 0 n~ x OU

FIG. 4. Surfaces of section II showing the phase space available to the trajectory displayed in Fig. 3. (a) 8 = 0.804. The trajectory has escaped from the e~ resonance (center), and has just been captured into the eMeu libration zone at lower left. A large chaotic zone is forming at the eMeo resonance. (b) 8 = 1.01. The trajectory has been captured in a secondary resonance in the eMeo libration zone, and generates the two loops in the chain of four islands. The chaotic zone has increased in extent. (c) 8 = 1.385. The trajectory has been dragged into the large chaotic zone. The eMeU resonance has become unstable. nance, and enters the chaotic region. This increases. Figure 4c shows section II after mechanism of escape from resonance the system has entered the chaotic zone. through secondary resonance capture was The emeU libration zone has become unsta- first identified in our study of the ble due to a period 3 secondary resonance. Miranda-Umbriel 3 : 1 inclination reso- The period 3 secondary resonance visible in nances (TW-II). However, in contrast to the Figs. 4a and b between the primary libration inclination resonances, the trajectory zones has been absorbed into the large cha- spends a considerable period of time in the otic zone. Eventually, the chaotic separatrix chaotic zone before escaping from the ec- associated with the e 2 resonance also centricity resonances. Both eccentricities merges with the large chaotic zone. Figure continue to increase, and as they do, the 5 shows section I at the same point in the extent of the chaotic region in phase space evolution as Fig. 4b. The heavy solid curve 406 TITTEMORE AND WISDOM

0.02 shorter than the time scale over which 8 evolves significantly due to tides. This in turn indicates that while the trajectory is in 0.01 the chaotic zone, a chaotic adiabatic invariant is being approximately conserved. Note also that at values of 8 of approximately 2 and 4.5, the trajectory is temporarily recap-

© tured into a quasiperiodic resonant region. For small initial eccentricities, the loca- -0.01 tions of secondary resonances are well predicted by the analytical techniques we employed in our study of the inclination prob-

-0.02 i i I lem (see TW-II), allowing us to analytically -0.02 -0.01 0 0.01 0.02 determine the point of transition from quasi- eMCOS(CM) periodic to chaotic behavior.

FIG. 5. Surface of section I showing the phase space 2.1.2. The Eccentric-Inclined Model available to the trajectory displayed in Fig. 3 at 8 = 1.385 (see also Fig. 4c). The eMeu resonance zone is at In TW-II, we introduced the eccentric- the bottom of the figure. The e 2 resonance zone has inclined Miranda-Umbriel 3 : 1 resonance appeared at the center of the figure. model, and showed that the inclination of Miranda's orbit remains high during chaotic evolution through the eccentricity reso- is the energy surface boundary of the sur- nances at the 3 : 1 commensurability. (See face of section, within which trajectories are Note at end of Appendix I.) We further ex- constrained. The large chaotic zone fills plore the dynamics of this resonance here. much of the displayed region of the phase The tidal evolution parameter 8 is defined space, but remnants of the individual sec- to be the nonresonant contributions to ond-order eccentricity resonances are still 3nu - nM -- ½(¢~M d- ~0U -I- tiM "~ flU)(see visible at this point in the evolution. The Appendix I). Initial conditions for trajector- unstable libration region of the eMeu reso- ies to be numerically evolved through the nance is visible at the bottom of the figure. resonance were generated as follows. From The quasiperiodic zone in the center of the an initial set of coordinates, calculated from figure is the region into which trajectories the orbital elements a M = 4.86345, av = eventually escape from the eccentricity res- 10.1179, e M = e M = 0.005, i M = i U = 0.005 onances. The two quasiperiodic zones radian, o-M = o-c = 0, ~bM = ~bu = 0, points above and below center are libration regions were computed without tidal dissipation, associated with the eL resonance. The cha- spaced in time by 100 mapping iterations, or otic zone becomes much more uniform in 15.7 years (l~ = 40). structure as the system continues to evolve. This model comprises three eccentricity As the trajectory evolves within the large resonances, three inclination resonances, chaotic zone, the eccentricities vary be- and secular interactions. The eccentricities tween well-defined maximum and minimum are coupled to the inclinations through non- values, forming a smooth "envelope" of linear terms; this model has four degrees eccentricities visible in Fig. 3. At a particu- of freedom. For chaotic systems with two lar value of 8, these "envelopes" corre- degrees of freedom, such as the planar- spond to the boundaries of the chaotic zones eccentric case described above, chaotic on the surfaces of section of the frozen Ham- zones are isolated from each other in phase iltonian. This indicates that the chaotic zone space by quasiperiodic zones, and thus the is being well explored on a time scale much variations possible within a chaotic zone are DYNAMICS OF PAST URANIAN RESONANCES 407

0.35 0.3 a b 0.3 D'

0.29 0.25 ___..#

0.2 0.28 0.15 ..=

0.1 0.27

0.05 /...~, J 0 i , : , , , , o26 , , , , , , 1.0 1.5 2 0 2 5 3.0 3.5 4.0 4 5 1.0 1 5 2 0 2.5 3.0 3.5 4.0 4.5 8 8

0.035 i i i i i i i 0.01 C d 0.03 /. 0.008 0.025 /- . \

0.006 0.02 eM eu 0.015 0.004

0.01

------}/ 0.002 0.005

0 1.0 1.5 2.0 2.5 3 0 3.5 4 0 4.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

FiG. 6. Eccentricity and inclination variations of a trajectory evolved through the 3 : 1 Miranda-Um- briel mean-motion commensurability (hu/a u = 6 × 10-12 per orbit period). (a) Orbital inclination of Miranda. (b) Orbital inclination of Umbriel. (c) Orbital eccentricity of Miranda. (d) Orbital eccentricity of Umbriel. The inclination resonances are encountered first, and the satellites escape from them, leaving both satellites with low inclinations. Evolution through the large chaotic zone associated with the eccentricity resonances is qualitatively similar to that in the planar-eccentric approximation (compare with Fig. 3). restricted. In systems with more than two of these chaotic wanderings will depend on degrees of freedom, though, chaotic zones the diffusion time scale compared to the are not isolated. (see, e.g., Chirikov 1979, tidal evolution time scale. Since this higher- Hrnon 1983). Therefore, where this higher- dimensional diffusion occurs generally over dimensional chaos is important, the trajec- a much longer time scale than the fundamen- tory may wander far from its initial state, tal dynamical periods in the problem (Chiri- over a long enough time scale, exploring a kov 1979), a chaotic adiabatic criterion much larger volume of phase space than the which is based on the trajectory exploring two-degree-of-freedom system. The extent all of the chaotic zone before significant tidal 408 TITTEMORE AND WISDOM evolution takes place may not exist for 0.035 , , higher-dimensional systems. Therefore, where this higher-dimensional chaos is sig- 0.03 nificant, the dynamical outcome may be sen- 0.025 sitive to the simulated tidal evolution rate, even for rates implied by our constraints on 0.02 / the physical Q of Uranus. eM In the planar-eccentric two-degree-of- 0.015 freedom model, the conservation of a chaotic adiabatic invariant is indicated by the 0.01 / smooth "envelope" of maximum and minimum eccentricities in the chaotic zone, 0.005 / which show that the chaotic zone is well 0 i I I I explored by the trajectory on a time scale 1.0 1 5 2.0 25 310 315 4.0 4.5 much shorter than the tidal evolution time 8 scale. Figure 6 shows the inclination and Fro. 7. Eccentricity variations of Miranda in the cha- eccentricity variations of Miranda and otic zone of the 3 : 1 commensurability with Umbriel, Umbriel for a trajectory that escapes from at a simulated rate of tidal evolution of ~tM/a M = 6 × all the inclination resonances, thus encoun- 10 -14 per orbit period (Q = 11,000) within our con- tering the eccentricity resonances with both straints on the physical Q of Uranus. With both inclina- inclinations small. In each plot, the maxi- tions small, the boundaries of the chaotic zone vary smoothly. mum and minimum eccentricity or inclination is plotted in intervals of A8 = 0.0033. For this trajectory, the nonlinear coupling between inclination and eccentricity reso- tion of a chaotic adiabatic invariant at physi- nances is weak, because both orbital inclina- cal rates. The planar-eccentric model is tions are small. All of the elements vary therefore a good approximation if the orbital chaotically during evolution through the ec- inclinations are small. centricity resonances, but we still see a However, with a high inclination of smooth "envelope" of the maximum and Miranda's orbit resulting from temporary minimum eccentricities in the chaotic zone capture into the inclination resonances, we (Figs. 6c and d), as we did in the planar see markedly different behavior. Figure 8 approximation (Fig. 3). This indicates that shows the inclination and eccentricity varia- an adiabatic invariant is being nearly con- tions of Miranda and Umbriel evolved served. through the 3 : 1 commensurability with the The rate of tidal evolution used for the three-dimensional mapping with ~ = trajectory in Fig. 6 was i~M/a i = 6 × 10-12 l 1,000. The maximum and minimum eccen- per orbit period (~ = 110), the same rate tricities and inclinations are plotted in inter- used for the trajectory displayed in Fig. 3 in vals of AS -- 0.0037, and the region between the planar approximation. To test whether the maximum and minimum envelopes has the behavior is similar at physical rates, we been shaded, to better show how the ele- have taken the state variables of the trajec- ments vary. We have not artifically en- tory shown in Fig. 6 in the chaotic zone hanced the simulated rate of tidal evolution (8 = 2.649), and continued the integration in this calculation; 9~ is within our con- with ~ = 11,000. The orbital eccentricity straints on the physical Q of Uranus set by variations of Miranda for this experiment the dynamics. In this example, the trajec- are shown in Fig. 7. The maximum and mini- tory is captured into the i~ resonance. The mum boundaries of the chaotic zone are still evolution through the inclination reso- well defined, confirming the near conserva- nances is not significantly affected by the DYNAMICS OF PAST URANIAN RESONANCES 409

6.0 ~ ~ ~ ~ ~ ~ ~ 0.5 "'"r a b

5.0 0.4

4.0 0.3 • •

"~ ..~ 0.2 2.0

0.1 1.0

0 0 1.

0.04 0.012

c cl 0.01

0.03

0.008 eM 0.02 eu 0.006

0.01 0.004

0.002

0 0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 8 8

FIG. 8. Eccentricity and inclination variations of a trajectory evolved through the 3 : 1 Miranda- Umbriel mean-motion commensurability with ~ = 11,000 (/~M/aM = 6 × 10- t4 per orbit period), within our constraints on the physical Q of Uranus. (a) Orbital inclination of Miranda. (b) Orbital inclination of Umbriel. (c) Orbital eccentricity of Miranda. (d) Orbital eccentricity of Umbriel. The maximum and minimum values are plotted in intervals of A8 = 0.0066. This figure represents about 10II orbit periods of Miranda. In this case, the trajectory is captured into the i~ resonance, and iM evolves to a high value. With the orbital inclination of Miranda high at the point of encounter of the eccentricity resonances, the orbital eccentricity of Miranda varies extremely irregularly within this chaotic zone, even on timescales comparable to the resonance passage timescale. The orbital inclinationof Miranda remains high.

nonresonant variations in eccentricity, as argument and the circulation frequency of we previously showed in TW-II. At 8 ~ 8.0, the i 2 resonant argument (see TW-II). the trajectory in Fig. 8 is captured into the This 2 : 1 secondary resonance eventually 2:1 secondary commensurability between drags the trajectory displayed in Fig. 8 into the libration frequency of the i 2 resonant the chaotic zone associated with the inclina- 410 TITTEMORE AND WISDOM tion resonances. During evolution through Even though this trajectory was com- this chaotic zone, both the inclinations and puted with a simulated tidal evolution rate the eccentricities vary chaotically. With a within the physical bonds, the orbital ele- high orbital inclination of Miranda, the non- ments vary extremely irregularly on time linear coupling between the eccentricity and scales comparable to the resonance passage inclination resonances is strong and causes time scale: there is not a smooth "enve- significant variations in the orbital eccen- lope" to the chaotic eccentricity variations tricity of Miranda. The trajectory escapes as seen in the previous example.There is from the inclination resonances, becomes no evidence for an adiabatic invariant when briefly quasiperiodic, and then enters the both the orbital inclination and eccentricity large chaotic zone associated with the ec- of Miranda are large, even at this very slow centricity resonances, which dominates the simulated tidal evolution rate. The dynam- rest of the evolution through the commensu- ics are sensitive to the simulated rate, even rability. The variations of the eccentricities if the Q of Uranus is within our constraints. and inclinations of both satellites are very Therefore, to avoid dynamical artifacts, irregular in this chaotic zone. Note, how- simulated tidal evolution rates must be ever, that the variations in orbital inclination within the physical constraints. The reso- of Miranda are relatively small: it maintains nant mapping methods first developed by a large average value. As the trajectory Wisdom (1982, 1983) have made this compu- evolves further in the chaotic zone, the max- tationally feasible. imum eccentricity of Miranda's orbit be- Although the eccentric-inclined model comes larger, reaching a maximum value has too many degrees of freedom to study of about 0.04 before the trajectory escapes using surfaces of section, we can verify the from the commensurability. After escape, presence of the secondary resonances. In the orbital eccentricity of Miranda is about Fig. 9a, the i~j resonant argument, d/u = 0.011. The chaotic variations in the orbital ½(3hu - h M - 2l}u), is plotted against the eccentricity and inclination of Umbriel are i 2 resonant argument, tkM = ½(3hu -- hM -- not large compared to those of Miranda. 21}M) for the trajectory shown in Fig. 8 at

360.0 360.0 a b /i' 270.0 ) 270.0 ?

~/U 180.0 ! q/U 180.0 / 90.0 90.0

,9 I Y , '~IX , 0 ) i I 45.0 90.0 135.0 180.0 0 45.0 90.0 135.0 180.0 ~M ~M

FIG. 9. Variation of the resonant arguments ~bM = ½(3hu - hM -- 21~M) and ~0U = ~(3'hu - hM -- 2flu) for the trajectory displayed in Fig. 8 at 8 = 9.147. (a) Computed using the mapping. (b) Computed using unaveraged differential equations. DYNAMICS OF PAST URANIAN RESONANCES 411

= 9.147. Points were computed using the displayed in Fig. 8 at 8 = 10.397,just before mapping with an iterative time step of 57.37 it enters the chaotic zone associated with days. qJM is librating in this figure as Ou circu- the inclination resonances. From these co- lates. In addition, the trajectory is nearly ordinates, we have generated three new sets periodic: it is librating in the 2 : 1 secondary of coordinates, by integrating the mapping resonance region (see Fig. 12 of TW-II forward in time, without tidal dissipation, for a surface of section of this region of spaced in time by 100 mapping iterations phase space in the circular-inclined (15.7 years). This provides a set of initial model). conditions with identical values of 8 and the As a further check of the secondary reso- Hamiltonian, but with different angular nance mechanism for escape from this reso- phases. These different initial conditions nance, we have explored the same region of have each been integrated forward in time phase space using direct unaveraged numer- through the eccentricity resonances using ical integrations. Our Hamiltonian models the mapping, with ~ = 11,000, to provide utilize Jacobi coordinates (see Appendix I). some idea of the possible range of behavior We have taken the Jacobi coordinates of the of trajectories at this resonance. Figure 10 trajectory shown in Fig. 9a, and converted shows the orbital eccentricity variations of them to the following Uranus-centered or- Miranda for the three trajectories, displayed bital elements: a M = 4.863473, a U = in the same format as Fig. 8c. For each case, 10.117934, e M = 0.00497385, e U = the variations of eccentricity are extremely 0.00500281, iM = 4.416421 degree, iU = irregular, with the trajectory in Fig. 10a 0.279794 degree, XM = h u = 0, &M = reaching a maximum value of about 0.05. 45.013320 degree, &u = 21.380209 degrees, Upon escape from the resonance, Miranda f~M = 271.650075 degrees, and f~u = may have a relatively large orbital eccentric- 282.078623 degrees. We carried out a direct ity (Fig. 10b), or a value comparable to the unaveraged numerical integration of the value before the resonance was encountered gravitational equations of motion for the res- (Fig. 10a). The mechanism by which trajec- onant system, in Uranus-centered coordi- tories enter the large chaotic zone appears nate, including planetary oblateness terms to depend on the behavior during passage of order J2. The integration was carried out through the chaotic zone associated with the using the Bulirsch-Stoer integrator, with a inclination resonances. For example, in Fig. relative precision of 10 -9 per step size of 10a, the eccentricity varies chaotically be- approximately 0.069 day (about 1/20th of an tween 10.5 < 8 < 11.4, but leaves the chaotic orbital period of Miranda), and points were zone with a relatively low value. It then plotted in intervals of approximately 57.37 becomes temporarily trapped in the eMeu days. Figure 9b shows the variations of ~M resonance between 11.5 < 8 < 12.3, exhibit- and tktj for Miranda and Umbriel using unav- ing behavior similar to what is seen in the eraged differential equations. The variations planar-eccentric model. The trajectory dis- of the resonant arguments on this plot are played in Fig. 10b shows a similar temporary virtually identical to those in Fig. 9a, indi- capture phase. However, the trajectories cating that the eccentric-inclined Hamilto- displayed in Figs. 10c and 8c do not show nian model and mapping represent the ac- the temporary capture phase. There appear tual dynamics of the 3 : 1 Miranda-Umbriel to be other mechanisms for entering the cha- resonance well. otic zones in these cases. For each of the We have further explored the evolution trajectories displayed in Fig. 10, the orbital of Miranda through the 3 : 1 mean-motion inclination of Miranda remains large during commensurability. We have generated evolution through the chaotic zones, and the starting conditions for trajectories with large variations in the orbital elements of Umbriel iM, using the state variables of the trajectory are relatively small. 412 TITTEMORE AND WISDOM

0.06 I a I I I I I 0.04 I I a b 0.05 0.03 0.04 eM 0.03 eM 0.02

0.02 0.01 0.01

0~ I I 11.0 12.0 13.0 14.0 15.0 16.0 17.0 11.0 12.0 13.0 14.0 8 8

0.03

0.02

0.01

11.0 12.0 13.0 14.0

FIG. 10. Eccentricity variations of Miranda for three trajectories evolved through the 3 : 1 Mi- randa-Umbriel eccentricity resonances with iM having a high initial value (~ = 11,000). The maximum and minimum values are plotted in intervals of A8 = 0.0033. As in Fig. 8, the orbital eccentricity of Miranda varies extremely irregularly within the chaotic zone.

2.1.3. Secular Perturbations = 0, t3A = - Ir/2, and ~u = zr/2. In Fig. 11, the maximum and minimum eccentricities Figure 11 shows an example of the evolu- are plotted in the intervals of A8 = 0.0033, tion of the orbital eccentricity of Miranda with the resulting envelope shaded to better during evolution through the 3 : 1 mean-mo- display the variations. The behavior of e M is tion commensurability with Umbriel, in- qualitatively similar to that in the planar- cluding secular perturbations due to Ariel, eccentric case, with the average value in- with ~tM/aM = 6 × 10 -13 per orbit period. creasing during evolution through the cha- The initial orbital parameters for this trajec- otic zone. The orbital eccentricities of both tory were a M = 4.8631, a g = 7.0846, au = other satellites vary chaotically, but these 10.1179, era = eA = eu = 0.005, ~b = 0, &M variations are relatively small. However, as DYNAMICS OF PAST URANIAN RESONANCES 413

0.04 Because aM/a A is decreasing, there is not an equilibrium eccentricity configuration for this resonance in the single-resonance ap- 0.03 proximation (see also Peale 1988). The initial coordinates of the trajectories to be numerically evolved through the com- eM 0.02 mensurability were determined from an initial state with the following orbital parameters: a M = 4.6361, a A =6.5110, eM = 0.005, e A = 0.005, o"M = 90 °, and O'A = 90 °. From 0.01 this initial state, the 19 other states in the ensemble were determined by integrating the equations of motion forward in time with 0 0 0.5 1.0 1.5 2.0 the mapping without tidal dissipation, spac- 5 ing successive states 66 mapping iterations ([l = 40.0 year -l, At = 66 × 2zr/[l = 10.4 FIG. 11. Eccentricity variations of Miranda for a years). Each of these initial states was then trajectory evolved through the 3 : 1 Miranda-Umbriel integrated forward in time with tidal dissipa- eccentricity resonances, perturbed by the secular variations of Ariel. The maximum and minimum eccentrici- tion included. ties are plotted in intervals of A8 = 0.0033. The eccen- Ensembles of 20 trajectories were tricity variations are more irregular than in the evolved through the resonance at different unperturbed case. simulated rates of tidal evolution, to determine the effect of the simulated rate of tidal evolution on the dynamics. Figure 12 shows in the eccentric-inclined model, the maxi- the results of this study of the effect of the mum and minimum boundaries of the cha- rate of tidal evolution on the dynamics. Plot- otic zone are not well defined, indicating ted are the averages of the maximum orbital that there is no chaotic adiabatic invariant. eccentricities in the chaotic zone as a func- This has been verified by integrating a tra- tion of the rate expressed in terms of ~. jectory through the chaotic zone with /~M/ There is a dependence of the maximum ec- aM = 6 × 10 -14 per orbit period (~ = centricity of Miranda on ~ for ~ < 11 (//A/aA 11,000). The initial coordinates were those 2.4 x 10 -l° per orbit period); for slower of the trajectory displayed in Fig. 11 at 8 = dissipation rates (larger ~), the dynamics in 1.374. For this trajectory, the maximum and the chaotic zone appear to become indepen- minimum eccentricity envelopes were very dent of the rate of evolution. Therefore, the irregular. trajectory explores the chaotic region well on time scales much shorter than the evo- 2.2. THE MIRANDA-ARIEL 5 : 3 lution time scale, and there appears to COMMENSURABILITY be a chaotic adiabatic invariant for ~ > 2.2.1. The Planar-Eccentric Model ll. At this resonance, the mean motion of At this resonance, the single-resonance Ariel decreases more rapidly than that of adiabatic criterion is insufficient to deter- Miranda. The time rate of change of 8 = mine where the dynamics become indepen- 5hA -- 3riM -- t~u - t~ A (see Appendix I) is dent of the simulated rate of tidal evolution. approximated by Because the orbit of Ariel is evolving out- ward more rapidly than that of Miranda, the ~ _ 0.492(5hA) first eccentricity resonance to be encoun- -- 0.492 15 nA aA. (3) tered is the e 2 resonance. We compute 2 a A (A(.OL/t.OL) for this resonance• The libration 414 TITTEMORE AND WISDOM

0.025 , , , , , , , right of Fig. 12. However, Fig. 12 reveals that the chaotic zone is not fully explored in the tidal evolution time scale until the 0.02 ! simulated rate of evolution is approximately three orders of magnitude slower than the 0.015 criterion based on the single-resonance ap- (e) proximation. Figure 13 shows the evolution of this same 0.01 ensemble of trajectories in 8, AE parameter • t space at a simulated tidal evolution rate of 0.005 a [] tl a a g i ! [] i~ CIA/a A = 2.4 × I0 -11 per orbit period (~ = ll0). AE is defined as AE = e - E 2 (5) 0 0 1.0 , 2 i 0 3.0 ' 4.0 ' 5 i 0 6 i 0 7.0 ' 8.0 log10(11,000/Q) where e is the energy of the Hamiltonian, FIG. 12. Maximum eccentricities of Miranda (~) and and Ariel ([J) in the chaotic zone of the 5 : 3 commensurability, related to the rate of tidal evolution expressed in E2 = ~ 0, 8< -2(C- D) + 4F terms of the effective specific dissipation function of L -(8 + 2(C - D) - 4F)Z/64B, Uranus ~. For ~ > 10, trajectories explore the full extent of the chaotic region before escaping from the 8-> -2(C- D) + 4F (6) resonance; for higher rates, the dynamics are not well represented. The single-resonance adiabatic rate for (see TW-I). If the Hamiltonian is equal to the e L resonance is denoted by the vertical bar (9~ = E2(AE = 0), the quartic equation that deter- 0.0063, i~A/aA ~ 4 X 10 -7 per orbit period). For simulated tidal evolution rates larger than this, the trajectories are dragged through the resonance without displaying any interesting behavior. Note that the dynamics are sensitive to the rate for rates more than three orders of magnitude slower than this. Clearly, the single-resonance adiabatic criterion is not sufficient for the planar- eccentric case.

frequency of this resonance is given by Z~E to2 = - 4F(8 - 80) (4)

where 80 = 2(C - D) - 16B~u. The change in the libration frequency of the (e 2) resonance in one libration period due to tidal evolution of the orbits is Ato L --~ -.~vv -4zrFS/to~, where ~ ~ -1.8 × 10-3/~ in 15.0 II.0 7.0 3.0 -1.0 our units, at low eccentrities. For e M = e U 8 = 0.005 well before the resonance is en- FIG. 13. Twenty trajectories in 8, AE space with countered, 80 = 7.03. When the resonance = 110. The mean initial eccentricities of the orbits is encountered at 8 = 8.46, AtOL/tOL ~- are e M = 0.0045 and e x = 0.0050. Solid lines indicate quasiperiodic behavior; dashed lines indicate chaotic -- 0.0063/~. In the single-resonance approximation, dynamical artifacts should appear behavior. Evolution of the energy during the chaotic phase of evolution is quite regular, indicating the exis- if~ < 0.0063, or hA/a A ~ 4 X 10-7 per orbit tence of a chaotic adiabatic invariant. Trajectories es- period. The single-resonance adiabatic rate cape from the resonance via the chaotic zone at irregu- is denoted by the vertical bar in the upper lar intervals, evolving to large negative AE. DYNAMICS OF PAST URANIAN RESONANCES 415 mines initial conditions on the surface of then enter a large chaotic zone. Solid lines section has no solutions at zero eccentricity indicate quasiperiodic behavior, and dashed on the section. If the Hamiltonian is greater lines indicate chaotic behavior in this figure. than EE(AE > 0), there is a finite area on the The evolution of the energies is also very surface of section near the origin in which regular during the chaotic phase, until the the quartic equation has no solutions. trajectory escapes to a quasiperiodic region Within this area, points cannot be generated of phase space and evolves to large negative on the surface of section. AE. The regularity of the evolution of AE In Fig. 13, 8 decreases with time, and while the trajectories are chaotic is further trajectories evolve toward the right. Trajec- evidence for the approximate conservation tories are excluded from the heavily shaded of a chaotic adiabatic invariant. There is not region in the upper fight of the figure. The an obvious pattern to the escape process: lightly shaded region represents the range trajectories "drop out" of the chaotic zone of values of Hamiltonian and the parameter in a seemingly random manner. 8 where large-scale chaos is present in the Figure 14 shows the distribution of aver- phase space. The extent of this region was age orbital eccentricities of Miranda after numerically determined using surfaces of escape from the resonance. For reference, section. The surfaces of section used for this the time-averaged initial value of about purpose plot x M versus YM when XA = 0. 0.0045 is shown by the vertical bar in the For different values of AE and 8, the initial upper part of the figure. For most trajector- conditions of 20 trajectories were deter- ies, the average orbital eccentricity of mined, with initial coordinates equally Miranda is higher than the value approach- spaced on the ordinate axis of the surface ing the resonance, which is qualitatively in of section. For each trajectory, the rate of agreement with the predictions of the single- growth of distance between nearby trajecto- resonance theory. However, a few trajectories (Lyapunov exponent) was evaluated. Where one or more trajectories indicate a 10.0 positive Lyapunov exponent, large-scale chaotic behavior is present. In the small unshaded region near the 8.0 I boundary of the excluded zone, the resonant arguments o-M = ½(5hA -- 3hM -- 2&M)and orA = ½(5hA - 3h M - 2tbA)must librate 6.0 about +-~-/2, because of the shape of the energy surface boundary. This region disap- 4.0 pears as 8 decreases, indicating that perma- nent capture into quasiperiodic libration does not occur during passage through this commensurability via the capture mechanism described in TW-I. Tidal evolution 0 "'"" ' ' drives trajectories into the unshaded region 0.004 0.008 0.012 0.016 of parameter space in the lower right of this eM figure, where there is no known mechanism Fie. 14. Distribution of time-averaged orbital eccen- of capture into resonance. [Peale (1988) also tricities of Miranda for the trajectories displayed in Fig. concludes that capture into this resonance 13. The vertical bar at the top of the figure indicates cannot occur, using arguments based on the the average eccentricity prior to resonance encounter. Most trajectories escape from the resonance with an available energy.] average eccentricity larger than the initial value, but in The trajectories shown in Fig. 13 initially a few cases the average eccentricity decreases during evolve to positive AE in a regular fashion, resonance passage. 416 TITTEMORE AND WISDOM

i i a 0.02 b 0.008

0.015 0.00 eM 0.01 eA 0.004

0.005 0.002

0 12.0, 9 i0 6i 0 3i 0 0' 0 12.0, 9 i0 6.0' 3i 0 0

FIG. 15. Variations of the orbital eccentricities of Miranda (a) and Ariel (b) during evolution through the 5 : 3 mean-motion commensurability, t~A/a A = 2.4 x 10-11 per orbit period (& = 110). The maximum and minimum values are plotted in intervals of A8 = 0.005. Upon entering the chaotic zone, the maximum orbital eccentricity of Miranda suddenly jumps to relatively high values. The final eccentricities of both orbits are larger than the initial values. ries evolve to lower eccentricity, which can- There is a correlation between the orbital not happen in the single-resonance theory. eccentricities upon escape from the reso- Also, the mean value of the time-averaged nance and the length of time spent in the orbital eccentricities of Ariel decreases from chaotic zone. Figures 15 and 16 show the about 0.005 prior to resonance encountered evolution of the orbital eccentricities of two to about 0.0048 after escape, with a scatter different trajectories in this ensemble. In of about 10% about this value. these figures, the maximum and minimum

a 0.02 b 0.008

0.015 0.006 eM 0.01 eA 0.004

0.005 0.002

, I "i, :"-i:,:~'L*~ ~ I I 0 I I I I 12.0 9.0 6.0 3.0 0 12.0 8.0 4.0 0 8

FIG. 16. Variations of the orbital eccentricities of Miranda (a) and Ariel (b) during evolution through the 5 : 3 mean-motion commensurability. ~lA/a A = 2.4 × 10- H per orbit period (~ = 110). The maximum and minimum values are plotted in intervals of A8 = 0.005. The final eccentricities are smaller than the initial values. DYNAMICS OF PAST URANIAN RESONANCES 417

eccentricities in a small interval of 8 (AS with passage through secondary reso- 0.005) are plotted versus 8. In each case, the nances. eccentricity "envelopes" are smooth and To better understand the qualitative fea- well defined well before the resonance is tures of the dynamics of resonance passage, encountered. The orbital eccentricities ini- we have studied the evolution of the trajec- tially oscillate with amplitudes determined tory shown in Fig. 16 using surfaces of sec- by the strength of the secular coupling. As tion. The surfaces of section chosen for the trajectories approach the resonance, the study plot YM versus XM when XA = 0 amplitude of oscillation increases. There (defined as section I) and YA versus XA when may be a sudden small change in the oscilla- x M = 0 (defined to be section II). The section amplitude, just before the large chaotic tion condition x,. = 0 corresponds to 2 or 4 zone is entered, when the trajectory crosses values of the conjugate Yi, due to the quartic a secondary commensurability (see below). nature of the Hamiltonian. These values of When the trajectory enters the chaotic zone, Yi are obtained numerically by solving a the maximum orbital eccentricity of Mi- quartic equation, and are designated a-d in randa suddenly jumps to a value approxi- descending numerical order (see TW-I). For mately four times the maximum prior to res- section I, there are two high-eccentricity onance encounter. The orbital eccentricity quartic root "families" and two low-eccen- of Miranda varies irregularly between about tricity root families. Unfortunately, to dis- 0.002 and 0.02. Meanwhile, the orbital ec- play all of the features of the phase space, centricity of Ariel also varies chaotically, it is necessary to plot surfaces of section but through a much smaller range. As the corresponding to at least two of these root trajectories tidally evolve through the cha- families, as was the case in the Ariel-Um- otic zone, the eccentricities continue to vary briel 5:3 resonance problem (TW-I). For in an irregular manner, but within well- section II, the phase space is even more defined limits which both decrease as 8 complicated due to the appearance of an decreases. The presence of these well- excluded region near the origin for AE > 0. defined "envelopes" further indicates that We have chosen here to display sections the trajectory fully explores the chaotic Ia and Ib. On these plots, e M is the radial zone on time scales much shorter than the distance from the origin. time scale of tidal evolution, and that there- Figure 17 shows sections I just after the fore a chaotic adiabatic invariant is con- trajectory in Fig. 16 has entered the chaotic served. zone, at 8 = 8.337. The heavy solid curve Eventually, the trajectories escape from enclosing each plot is the energy surface the resonance. The trajectory shown in Fig. boundary, which constrains where points 15, after only a short time in the chaotic can be generated on the section. On both zone, escapes with an average orbital eccen- sections Ia and Ib there is a quasiperiodic tricity of Miranda approximately twice the zone near the origin and a large chaotic zone value prior to resonance encounter. The or- which extends to the energy surface bound- bital eccentricity of Ariel also increases a ary. The trajectory has entered the chaotic tiny amount. For a trajectory that spends a zone from the quasiperiodic zone on section long time in the chaotic zone, such as the Ib (Fig. 17b), and will eventually escape one shown in Fig. 16, the average eccentrici- from the resonance into the quasiperiodic ties of both orbits are lower after escape zone on section Ia (Fig. 17a). The chaotic from the resonance than before the reso- zone acts as a bridge between these two nance was encountered. In both cases, es- regions. By studying this figure, it is possible cape from the chaotic zone is accompanied to understand the origin of the sudden large by slight sudden changes in the oscillation jump in orbital eccentricity experienced by amplitude of the eccentricities associated Miranda as it enters the chaotic zone. Prior 418 TITTEMORE AND WISDOM

0.02 , 0.02 , , , , , ,

"t0.01 I t b0.01

t~ 0

-0.01 -0.01

-0.02 -0.02 ~ i i t ~ ) -0.02 -0.01 0 0.01 0.02 -0.02 -0 01 0 0.01 0.02 eMCOS(GM) e MCOS(GM)

FIG. 17. Sections I showing the phase space available to the trajectory in Fig. 15 just after entering the chaotic zone, at 8 = 8.337. The chaotic zone extends from the edge of the quasiperiodic region on section Ib (b) to the energy surface boundary on section Ia (a), explaining the sudden jump in e M when the chaotic zone is entered.

to resonance encounter, the trajectory is Figs. 15 and 16. Clearly, to fully explore the confined to the quasiperiodic region in Fig. chaotic zone, a tidally evolving trajectory 17b. The evolution in this region involves must reach the maximum eccentricity in the the secular interaction between the satel- chaotic zone as defined by the energy sur- lites, and the orbital eccentricity of Miranda face boundary of the surface of section. This never gets above about 0.005. When the tra- explains the fact that the mean maximum jectory enters the large chaotic zone, a much eccentricity reached by trajectories levels more extensive region of phase space is off for slow simulated tidal evolution rates available to the trajectory, including the at a value corresponding to the maximum chaotic zone in Fig. 17a. The chaotic zone distance of the energy surface boundary extends all the way out to the energy surface from the origin on the phase plane (see Fig. boundary, which constrains the extent of 12). motion. On Section Ia, the energy surface Note that there is a chain of four islands boundary extends to an orbital eccentricity associated with a secondary resonance visi- of Miranda of about 0.02, about four times ble in the quasiperiodic region in Fig. 17b. the maximum in the quasiperiodic zone be- The appearance of this and other secondary fore resonance encounter. In the chaotic resonances at the edge of the chaotic zone zone, the orbital eccentricity varies irregu- may result in the transfer of trajectories into larly between a value corresponding to the the chaotic zone. maximum extent of the energy surface As a trajectory evolves through the com- boundary on Fig. 17a and a minimum value mensurability, surfaces of section generated corresponding to the outer edge of the quasi- for the "frozen" Hamiltonian at successive periodic region on Fig. 17b. Because the points in the evolution qualitatively resem- time scale of evolution within the chaotic ble Fig. 17, but occupy a smaller region of zone is much shorter than the tidal evolution the phase plane, and the maximum extent time scale, we see the sudden jump in the of the chaotic zone decreases. maximum orbital eccentricity of Miranda in We have also studied the evolution of tra- DYNAMICS OF PAST URANIAN RESONANCES 419 jectories with higher initial eccentricities expected, based on the results of Section 2, (eM = eA "~ 0.01), and similar relative in- that higher-dimensional chaos may be im- creases in the orbital eccentricity of Mi- portant, and therefore the dynamical out- randa have been found. come is sensitive to the simulated rate of evolution. Therefore, we have used a simu- 2.2.2. The Eccentric-Inclined Model lated rate of tidal evolution corresponding The tidal evolution parameter 8 is defined to ~ = 11,000, within our constraints on to be the nonresonant contributions to the physical Q or Uranus. The initial orbital 5hA -- 3nM -- ½(¢~M q- (@A + tiM -~- fiA)(see elements used in the numerical experiments Appendix I). Because both the orbital ec- are a M = 4.6361, aA = 6.5110, e M = e A = centricity and the inclination of Miranda are 0.005, iM = iA = 0.005 radian, o-M = ~r/2, significantly affected by this resonance, it is OrA ~- 37r/2, ~/M = 37r/2, and tkA = 7r/2. In

2.0 , , , 0.5 a

0.4 1.5

t .3 . . ~ .. - , ,, ~..,. 1.0 ..~ 0.2

0.5 0.1

i , ~11~ i • " i " i i ~ I , I 15.0 14.0 13.0 12.0 II.0 15.0 14,0 13.0 12.0 11.0 8 8

0.04 0.01 C d

0.008 0.03

0.006

~M 0.02 ¢A

0.004

0.01 0.002

0 -' "" .... ' ...... ':" ~ '' ~: 0 15.0 14.0 13.0 12.0 11.0 15.0 14.0 13.0 12.0 11.0

FIG. 18, Variations of the orbital inclinations (a, b) and eccentricities (c, d) of Miranda and Ariel. The rate of evolution is ak/a A = 2.4 × 10-13 per orbit period (9~ = 11,000), within our constraints on the physical Q of Uranus. The maximum and minimum values are plotted in intervals of A8 = 0.0033. Both e M and i M suddenly jump to moderately large values upon encountering the large chaotic zone, and e M reaches larger values than in the planar-eccentric approximation. Note that the limits of eccentricity and inclination appear to evolve in a fairly regular manner in the chaotic zone, in contrast with the Miranda-Umbriel 3 : 1 problem (see Figs. 8, 10). 420 TITTEMORE AND WISDOM

Fig. 18, the maximum and minimum eccen- of 2 larger than the initial value. The changes tricities and inclinations of Miranda and Ar- in orbital eccentricity of both Ariel and Um- iel are displayed in intervals of AS = 0.0033. briel, though chaotic, are not significant. When the trajectory enters the large chaotic zone, the variations in the orbital inclination 2.3. THE ARIEL-UMBRIEL 2:1 and eccentricity of Miranda are significant. COMMENSURABILITY e M reaches a maximum value of about 0.03, 2.3.1. The Planar-Eccentric Model compared to a value of 0.02 reached in the At this commensurability, the semimajor planar-eccentric model for a comparable ini- axis of Ariel is increasing much more rapidly tial eccentricity. The inclination reaches a than that of Umbriel. The rate at which maximum value of about 1.5 °. Note that at 8 = 4n v - 2n A - t~A - t~U increases due this simulated rate of tidal evolution, there to planetary tides is given by (see Appendix appear to be relatively smooth "envelopes" II) of maximum and minimum eccentricity and inclination in the chaotic zone: perhaps ~ - 2hA ~- 3hA aM. (7) there may be an adiabatic invariant for this a M model at sufficiently slow rates. It is inter- For the Ariel-Umbriel 2 : 1 resonance, the esting to note that the maximum orbital ec- equilibrium eccentricity of Ariel is approxi- centricity reached by Miranda is only about mately 0.02 (see Appendix II) if Q = 11,000 a factor of 2 lower than the maximum for Uranus and Q = 100 for Ariel. reached at the 3 : 1 commensurability with To determine capture probabilities for Umbriel (see Fig. 9). this resonance, we numerically integrated six sets of 20 trajectories through the com- 2.2.3. Secular Perturbations mensurability using the mapping, each with To take into consideration the effects of different eccentricities approaching the res- other satellites on the Miranda-Ariel 5:3 onance. Orbital parameters used to calcu- resonance, we used a model that includes late the initial states of the trajectories in secular perturbations due to Umbriel. Fig- each run are given in Table III. From the ure 19 shows an example of the evolution of initial state in the table, 19 other initial con- the orbital eccentricities of Miranda, Ariel, ditions were computed by integrating the and Umbriel for a trajectory evolved equations of motion forward in time using through the resonance using this model with the mapping without tidal dissipation, with = 22 (/~M/aM = 1.2 × l0 -i° per orbit a time interval of 66 mapping iterations period). The initial orbital parameters for (~ = 40.0 year -i , At = 66 x 27r/f~ = 10.4 this trajectory were a M = 4.6394, aA = years), yielding a set of trajectories with 6.5157, a U = 10.056, eM = eg = eu = 0.005, identical initial values of 8 and the Hamilto- ~b = 0, and &u = &A = &U = --7r/2. Shown nian, but with different angular phases. in the figure are the maximum and minimum Each of the initial states in the run was then eccentricities in intervals of A8 = 0.005. integrated forward in time with tidal dissipa- The evolution of Miranda's orbital eccen- tion included. tricity is not markedly different than in the These numerical experiments were car- planar-eccentric case involving only Mi- tied out using simulated rates of tidal evolu-

randa and Ariel: the maximum eccentricity tion between ~lA/a A ~--- 2.7 x 10 -s per orbit jumps suddenly when the evolution be- period (~ = 0.00011) and ~lA/a A = 2.7 × comes chaotic• There still appear to be well- 10 -ll per orbit period (~ = 110). At rates defined limits to the chaotic variations in e M slower than about ~lA/a A = 2.7 × l0 -7 per as the satellites evolve through the reso- orbit period (~ -- 0.011), all trajectories in nance. The average orbital eccentricity after runs 1-4 were captured into resonance. escape from the resonance is about a factor Only trajectories from runs 5 and 6 had mea- DYNAMICS OF PAST URANIAN RESONANCES 421

i i a 0.02 b 0.008

0.015 0.006 eM 0.01 eA 0.004

0.005 0.002

I. i 0 I I I I 12.0 9.0 60 3 0 0 12.0 9.0 6.0 3.0

0.01 0

0.008

0.006

0.004

0.002

0 I I I I I 12.0 9.0 6.0 3.0 0

FIG. 19. Variations of the orbital eccentricities of Miranda (a), Ariel (b), and Umbriel (c) during evolution through the Miranda-Ariel 5 : 3 mean-motion commensurability. The maximum and minimum values are plotted in intervals of A8 = 0.005. The orbital elements of all three satellites vary chaotically. surable probabilities of escape at slow tidal partly a consequence of the fact that at ec- rates (~ = 110). centricities comparable to the current val- This indicates that the probability of es- ues, the evolution is dominated by quasipe- cape from the Ariel-Umbriel 2 : I resonance riodic behavior, and trajectories do not is very small if the orbital eccentricities are cross separatrices associated with the pri- smaller than about 0.03, much larger than mary mean-motion resonances. Rather, the current values. The capture mechanism capture occurs via evolution of the energy for trajectories with low initial eccentricities surface on which a trajectory evolves (see is very robust: the rate at which artifacts below), analogous to the capture mecha- appear is two to three orders of magnitude nism at the Ariel-Umbriel 5:3 resonance larger than the rates at which artifacts ap- (TW-I). The region of escape does not appeared in the dynamics of the previously pear until eccentricities are much higher. considered resonances. This appears to be Figure 20 is a plot of the maximum and 422 TITTEMORE AND WISDOM

TABLE III

ARIEL--UMBRIEL 2; 1 COMMENSURABILITY: INITIAL PARAMETERS

Run AE 8 aA eA O'A XA YA n au eu O'u Xu Yu

1 -7.7095 x 10 -5 -7.8351 6.3100 0.005 90.0 0.0 4.1764 x 10 -3 10.0400 0.005 90.0 0.0 4.5651 × 10 -3 20 2 -2.7344 x l0 -4 -6.3464 6.3100 0.01 90.0 0.0 8.3527 × 10 -3 10.0400 0.01 270.0 0.0 -9.1302 × 10 -3 20 3 -7.3931 × 10 -4 -6.5488 6.3050 0.015 90.0 0.0 1.2529 × 10 -2 10.0400 0.015 270.0 0.0 -1.3695 × 10 -2 20 4 -1.4568 x 10 -3 -5.7580 6.3000 0.02 90.0 0.0 1.6705 × 10 -2 10.0400 0.02 270.0 0.0 -1.8260 x 10 -2 20 5 -1.0299 x 10 -3 7.1446 6.3055 0.03 90.0 0.0 2.5058 x 10 -2 10.0400 0.03 270.0 0.0 -2.7391 x 10 -2 20 6 -9.2686 x 10 -4 21.2018 6.3057 0.04 90.0 0.0 3.3411 x 10 -2 10.0400 0.04 270.0 0.0 -3.6521 x 10 -2 20

minimum orbital eccentricities of Ariel and tricities, for example, at values of 8 of about Umbriel in intervals of A8 = 0.008 for a 0.6, 2.7, 4.7, and 7.5. The orbital eccentric- trajectory in run 1, with a simulated tidal ity of Ariel reaches its equilibrium value evolution rate corresponding to 9~ = 110. (0.02) without disruption of the resonance. Both orbits are captured into resonance To understand this complicated behavior, without displaying significant chaotic be- we have computed surfaces of section for havior. However, there are sudden changes the frozen Hamiltonian at various values of in the amplitudes of oscillation of the eccen- 8. The section conditions chosen for study

0.05 , , , , , 0.05 ~ , ,

0.04 0.04

0.03 0.03

CA eu

0.02 ~'//~ 0.02

0.01 0.01 U/ J

0 I ~"~.~ I / I I I -5.0 0 5.0 10.0 15.0 -5.0 0 5.0 10.0 15.0 5

FIG. 20. Orbital eccentricity variations of Ariel (a) and Umbriel (b) of a trajectory from run 1 during evolution through the 2 • 1 commensurability. Shown are the maximum and minimum eccentricities in intervals of AS = 0.008. The evolution is primarily quasiperiodic, but with modulations of the oscillation amplitudes ofe A and e U due to secondary resonances. The trajectory attains the equilibrium eccentricity e A = 0.02 without escaping from the resonance. DYNAMICS OF PAST URANIAN RESONANCES 423 are plotting YA versus x A when Yu = 0, curves: quasiperiodic behavior dominates defined to be section I, and plotting Yu ver- the evolution. The trajectory displayed in sus xtj when YA = 0, defined to be section II. Fig. 21 has been temporarily captured into As with the second-order resonant models, a secondary resonance, which is why it oc- initial conditions on the surface of section cupies the chain of islands. The chaotic sep- are determined by solving a quartic equa- aratrix associated with the secondary res- tion. Only one quartic root (the one with the onance is also displayed; however, it lowest value) need be plotted to show the occupies only a small region of the displayed important features of the phase space. phase space. During evolution within the Figure 21 shows section II at 8 = 2.786, secondary resonance, the orbital eccentric- during one of the aforementioned jumps in ity of Umbriel increases slightly. However, the oscillation amplitude of the eccentricity. evolution within the secondary resonance In this plot, the orbital eccentricity of Um- does not lead to escape from the resonance: briel is the radial distance from the origin; the trajectory does not cross a chaotic sepa- the value of the resonant argument tro = ratrix associated with the primary reso- ½(2hu - hA -- &u) is measured counter- nance. Capture into resonance for low ec- clockwise from the positive abscissa. The centricities occurs via a mechanism heavy solid curve is the energy surface analogous to the mechanism found at the boundary of the surface of section: points Ariel-Umbriel 5:3 resonance (see TW-I): generated on the section according to the evolution of the energy surface boundary on above criteria are constrained to lie within the surface of section constrains the reso- this boundary. At this point in its evolution, nant argument to librate. In Fig. 21, the en- the trajectory displayed in Fig. 21 generates ergy surface boundary still encloses the ori- points that form the three quasiperiodic gin, which means that some trajectories pass loops in the chain of three resonant islands through all values ofo-U. However, note that shown on the figure. Points generated by the center of the region of motion is offset neighboring trajectories have also been plot- from the origin. Eventually, the energy sur- ted to display the structure of the phase face boundary no longer encloses the origin, space. Most of the points appear to lie on and for all trajectories, o-o librates about 7r on the surface of section and in the full phase space. °° For large initial eccentricities, chaotic behavior becomes more important, and the evolution is more complicated. The secondary resonant zones widen and eventually overlap, resulting in a large chaotic sea. At sufficiently large initial eccentricities, the : i chaotic zones associated with the primary resonances also appear. The initial eccentricities must be quite large, of order 0.03, for the probability of escape from the resonance to become significant. Interestingly, the eccentricity required for escape in our model is similar to the eccentricity below -0.015 -0.01 -0.005 0 0.005 which capture is certain in the single-reso- 001I/le u cos(% ) ;¢ nance model (see Peale 1988), even though the behavior is qualitatively quite different FIG. 21. Section II at 8 = 3.769. The trajectory in Fig. 20 generates the three resonant islands. Quasiperiodic when the resonance is encountered at such behavior dominates the phase space. large values of eccentricity. 424 TITTEMORE AND WISDOM

0.06 0.06 a b : :?" 0.05 0.05

...... ~ • 0.04 0.04

eA 0.03 eu 0.03

. 0.02 0.02

0.01 0.01

:,!-:;~. )7: . :

0 0 I I I I 10.0 15.0 20.0 25.0 10.0 15.0 20.0 25.0

FIG. 22. Orbital eccentricity variations of Ariel (a) and Umbriel (b) during evolution through the 2 : 1 commensurability for a trajectory from run 5. Shown are the maximum and minimum eccentricities in intervals of 48 = 0.0066. The trajectory is chaotic for a significant interval of time during evolution through the resonance, then escapes from the resonance, with e U > eA.

Figure 22 shows the evolution of the or- upon escape from the resonance (see Fig. bital eccentricities for a trajectory from run 22b). The oval quasiperiodic region near the 5 that escapes from the 2 : 1 commensurabil- origin of the figure is a region into which ity. The maximum and minimum orbital ec- trajectories may escape from the eu rest- centricities of Ariel (Fig. 22a) and Umbriel (Fig. 22b) are plotted in intervals of A8 =

0.0066. Well before the resonance is en- 0.06 l , , , , countered the evolution is quasiperiodic and dominated by the secular interaction among 0.04 the satellites. At 8 ~ 12.3, the system enters a large chaotic zone, which dominates the evolution until the satellites escape from the 0.02 resonance at 8 ~ 20.7. Figure 23 shows section II at 8 = 20.64, -4 0 just before the trajectory escapes from the o = resonance. This picture looks quite different -0,02 from the surfaces of section for lower initial eccentricities (Fig. 21). At this point in the -0.04 evolution, the trajectory displayed in Fig. 22 is still in the large chaotic zone, which -0.06 i ~ ~ i -0.06 -0.04 -0.02 0 0.02 0.04 0.06 occupies most of the phase space in the fig- e u cos(au ) ure. The small crescent-shaped quasiperiodic region to the right of the origin of the FIG. 23. Section II at 8 = 20.640. The trajectory figure is the region into which trajectories displayed in Fig. 22 is still in the large chaotic zone. Trajectories may escape from the resonance into the escape from the resonance. Note that this is small crescent-shaped quasiperiodic region at the right well away from the origin: hence, the orbital of center. The quasiperiodic region near the origin is eccentricity of Umbriel is relatively high associated with capture into the e A resonance only. DYNAMICS OF PAST URANIAN RESONANCES 425 nance, but are captured into the e A reso- order secular interactions, and the eccen- nance. tricities are coupled to the inclinations through nonlinear terms. ~i is defined to be 2.3.2. The Eccentric-Inclined Model the nonresonant contributions to 4n v - At the Ariel-Umbriel 2:1 mean-motion 2hA -- ½(t~A + t~u + ~A + ['~U)" commensurability, there are two first-order Figure 24 displays the evolution of the eccentricity resonances, three second-or- orbital eccentricities and inclinations of Ar- der eccentricity resonances, three second- iel and Umbriel during evolution through order inclination resonances, and second- the resonance with & = ll,000 (fiA/aA =

6.0 i ¢ i i i i 6.0 fl

5.0 5.0

4.0 4.0 t ~m 3.0 3.0

._< • v:~, '~ • 2.0 2.0

1.0 1.0

,' , '~r i I ' I:, ''~i I I 0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 8

0,05 t i ~ i i , i i 0.05 d

°10.04' t,, 0.04

0.03 0.03

@A eu

0.02 0.02

0.01 0.01

0 0 .... ~::~, ~.~vt~ , i/- 0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 8 8

FIG. 24. Variations in orbital inclination [Ariel (a), Umbriel (b)] and eccentricity [Ariel (c), Umbriel (d)] for a trajectory integrated through the 2 : 1 commensurability using the mapping with ~ = 11,000. The maximum and minimum values are plotted in intervals of A8 = 0.033. The qualitative behavior is initially similar to the planar-eccentric problem, but at 8 = 12.6 the system is captured into the i2 resonance. A period of chaotic behavior at large values of e A and iA does not lead to escape from the resonance. 426 TITTEMORE AND WISDOM

2.7 × 10-13 per orbit period). The initial or- a trajectory evolved through the resonance bital parameters for this trajectory were using this model with ~ = 220 (aA/a A = a A = 6.3100, au = 10.0400, eA = eu = 1.4 × 10-11 per orbit period). The initial or- 0.005, i A = itj = 0.005 radian, orA = o-u = bital parameters for this trajectory were

7r/2, and qJA = qJU = ~r/2. The mapping was a A = 6.3100, a u = 10.0400, a x = 16.633, used, with l~ = 40 year-1. Maximum and e A = e U = e T = 0.005, q~ = 0.0, and 6~A minimum eccentricities are plotted in inter- = ~u = ~x = --zr/2" The maximum and vals of A8 = 0.033. minimum eccentricities are plotted in inter- The evolution through the resonance is vals of A8 = 0.01. The presence of Titania qualitatively similar to that in the planar- does not affect the evolution much: both eccentric approximation with similar initial resonant arguments are captured into libra- eccentricities (run 1, see Fig. 20), until 8 = tion. As the trajectory evolves, it encoun- 13.7. At this point, the trajectory is captured ters various secondary resonances which into the i 2 resonance. As i A increases, e A only change the oscillation amplitudes of the and e u maintain constant average values. eccentricities. The chaotic zones at these Eventually, the system becomes chaotic at secondary resonances are perhaps some- iA ~ 1.3 °. However, this does not lead to what larger than in the two-satellite problem escape from the 2 : 1 resonance. Eventually with similar initial eccentricities (run 1), as the trajectory becomes quasiperiodic again, can be seen from the burst of chaotic behav- remaining captured in the resonance. ior in the orbital eccentricities at 8 just Thirteen other trajectories, generated greater than zero. However, the qualitative from the above orbital elements, were inte- behavior of the trajectories does not change grated through the resonance, using a simu- much: e A reaches the equilibrium value lated tidal evolution rate corresponding to without escaping from the resonance. Two 2~ = 110. All of the trajectories displayed other trajectories, with similar initial condi- qualitative behavior similar to that dis- tions, and simulated rates of tidal evolution played in Fig. 20, and all were captured into corresponding to ~ = 110, showed similar resonance. qualitative behavior. The numerical results indicate that for low We have also explored the small chaotic initial eccentricities, the interaction be- zone, at 8just above zero, using a simulated tween the inclination and eccentricity reso- tidal evolution rate corresponding to ~ = nances does not lead to a significantly en- 11,000. The initial coordinates of the trajec- hanced probability of escape. Because the tory were taken from the trajectory dis- inclination resonances are not encountered played in Fig. 25 at 6 = -0.0608, and the until after Ariel reaches its equilibrium ec- trajectory was integrated to 8 = 0.9392. The centricity, these interactions would proba- trajectory did not escape from the resonance bly never be important. Furthermore, even during the chaotic phase of evolution, and if the high-inclination chaotic zone were to eventually became quasiperiodic again. lead to escape, it would most likely leave Ariel with a substantial orbital inclination, 3. DISCUSSION which we do not observe today. At the 3:1 commensurability with Umbriel, both the orbital inclination and ec- 2.3.3. Secular Perturbations centricity of Miranda undergo large in- To take into consideration the effects of creases. Miranda escapes from the com- other satellites on the Ariel-Umbriel 2:1 mensurability with a high inclination which resonance, we used a model that includes is observed today, as we found in TW-II in secular perturbations due to Titania. Figure the circular-inclined approximation. 25 shows the evolution of the orbital eccen- Miranda displays many features that indi- tricities of Ariel, Umbriel, and Titania for cate that it has been strongly heated at some DYNAMICS OF PAST URANIAN RESONANCES 427

0.05 0.05 b

0.04 0.04

0.03 0.03 eA eu

0.02 0.02

0.01 0.01

-5.0 0 5 0 10.0 15.0 -5.0 0 5.0 10.0 15.0 8 6

0.05 O

0.04

0.03

0.02

0.01

I i i I -5.0 0 5 0 10.0 15.0 6

FIG. 25. Orbital eccentricity variations of Ariel (a), Umbriel (b), and Titania (c). The maximum and minimum eccentricities in intervals of A8 = 0.01 are plotted. The trajectory does not escape from the resonance before e A reaches its equilibrium value.

point in its evolution. The surface of pears to be the youngest in the Uranian sat- Miranda is very complex (Smith et al. 1986). ellite system (Plescia 1988). Could tidal In addition to cratered terrain on the sur- heating of Miranda during passage through face, there are regions referred to as "trape- the 3:1 resonance have driven the pro- zoids," "banded ovoids," and "ridged cesses that altered the surface of this sat- ovoids" by Smith et al. (1986), and named ellite? "coronae" by Strobell and Masursky We can estimate the rate at which the (1987). There are possibly solid-state volca- interior of Miranda was heated during the nic flow features on the surface (Jankowski high-eccentricity phase of evolution through and Squyres 1988).The cratered terrain ap- the 3 : 1 commensurability with Umbriel, us- 428 TITTEMORE AND WISDOM ing (Peale and Cassen 1978) rate of 1.3 × 1016 ergs sec-1 over this time 5 3 period, the internal thermal energy of the dE _ 21 k2M MR Mn~ e2 satellite would increase by approximately dt 2 a~lQ~ 1.2 × l03° ergs, most of which would be ~7.4 × 1018e2ergssec-I (8) retained over such a short time scale. If the heat capacity of the interior of Miranda has using parameters appropriate for this satel- a value comparable to that of water ice lite (Tables I and II). Tidal heating due to (~-~2 × 107 ergs g- 1 sec- l), this would result the nonzero obliquity of Miranda in Cassini in a global temperature increase of about state 1 would have been negligible, since I°K. The tidal heating of a body is not uni- the obliquity would have been small (see formly distributed throughout the interior: Tittemore 1988). Based on the results of the the rate of heating at the center of a homoge- numerical integrations using the eccentric- neous, incompressible body is about three inclined model, the maximum instantaneous times the globally averaged heating rate tidal heating rate is about 1.9 × 1016 ergs (Kaula 1964, Peale and Cassen 1978), so sec-1 at eu ~ 0.05. However, this tidal heat- near the center of the satellite the increase in ing rate is not sustained for long periods. temperature might be a few degrees. Several Figure 26 shows a short section of Fig. such bursts of tidal heating are possible dur- 10a during the period over which the orbital ing evolution through the chaotic zone; eccentricity of Miranda reaches its largest however, it is not expected that the increase values. The time interval represented by this in interior temperature would be large if this plot is about 4.9 × 107 years (~ = 11,000). were the only heating mechanism. Over this time interval, the eccentricity re- The time scale of evolution through the mains at large values for relatively short pe- commensurability is comparable to the time riods. For example, at 8 ~ 16.5, the average scale for the eccentricity of Miranda's orbit value of e M is about 0.042 for a period of to damp due to dissipation of tidal energy about 3 million years. At an average heating in the satellite (Appendix II). Satellite tides affect both the values of the eccentricities 0.06 i i i i f and the rate of evolution through the resonance. Because the excursions to large val- 0.05 ues of eccentricity occur on time scales much shorter than the eccentricity damping time scale, the effects of tidal dissipation in 0.04 the satellites on the chaotic variations of eccentricity are probably not important. It is eM 0.03 possible, however, that a quasi-stable state might be reached by the system: chaotic 0.02 dynamical systems may mimic regular behavior for significant periods of time (see 0"01I[ I~' I I I ~' ~ I' [I~ Wisdom 1982, 1983). The equilibrium eccentricity of Miranda in the 3 : 1 resonance is 0~15.4 15.7 16.0 16.3 16.6 16.9 17.2 0.026 (see Appendix II). It is not clear 8 whether an equilibrium eccentricity could be maintained in the chaotic zone; however, FIG. 26. Orbital eccentricity variations of Miranda it provides an upper limit to the sustained during evolution through the 3:1 commensurability rate of heating of the interior of Miranda of with Umbriel. This figure is a portion of Fig. 10a with an expanded scale. The time interval represented is about 5 × 1015 ergs sec-1 for Q = 11,000. about 4.9 × 107 years. Miranda spends relatively short The central temperature of a uniformly heat- periods of time at large eccentricity. ing spherical body in conductive equilibrium DYNAMICS OF PAST URANIAN RESONANCES 429 and with constant thermal conductivity is The largest escape eccentricity found in the given by (e.g., Stevenson 1984) eccentric-inclined model was only 0.02 (Fig. Hv 10b); however, this was based on a more Te = Ts + ~-R 2 (9) limited set of runs. Tidal damping of the eccentricity following escape from the reso- where R is the radius of the satellite, H v is nance may have significantly heated the volumetric heating rate, k is the thermal Miranda. Dermott et al. (1988) estimate that conductivity, and Ts is the surface tempera- the damping of an eccentricity of 0.07 is ture, which is about 70°K for Miranda. The required to raise the internal temperature thermal conductivity k of water ice at low of Miranda to the eutectic melting point of temperatures is about 5.3 x 105 erg cm -1 ammonia hydrate. This value is comparable sec -1 deg -1 at 100°K (Hobbs 1974). If an to the largest escape eccentricity we found equilibrium tidal heating state were possi- in our numerical simulations. Their estimate ble, this model predicts that sustained tidal is really a lower limit, though, since it was heating of Miranda at the equilibrium eccen- calculated assuming that all of the tidally tricity in the 3 : 1 resonance with Umbriel generated heat is retained in the interior of can raise the central temperature of Miranda Miranda, and it does not take into consider- by only about 16°K. As mentioned above, ation the possible importance of internal however, the central tidal heating rate in a heat transport processes over the eccentric- homogeneous body is about three times the ity damping time scale of order 108 years. In global average rate, so the temperature in- addition, this value of eccentricity is much crease near the center may be somewhat larger than the equilibrium eccentricity of larger. In addition, the presence of bubbles 0.026, so it is not clear whether the satellite and imperfections in the ice, silicate parti- could reach such a large value before escap- cles, and hydrates and clathrates with rela- ing from the resonance. However, it is likely tively low conductivities may lower the that Miranda left the 3 : 1 resonance with an thermal conductivity of the interior of Mi- orbital eccentricity significantly larger than randa by about a factor of 2 compared to the preencounter value, and that some tidal pure water ice (Stevenson 1982), resulting heating occurred as the eccentricity damped in a steeper temperature gradient. If these to its current value. effects are important, a stable equilibrium In summary, our numerical experiments tidal heating configuration could increase indicate that the orbital eccentricity of the central temperature of Miranda to a Miranda may have reached a large enough value comparable to the eutectic melting value to have affected its thermal evolution. temperature of NH3 H20, about 175°K. A combination of the heating mechanisms Whether or not a stable equilibrium state is discussed above may have raised the inter- likely depends on the dynamical effects of nal temperature to a value comparable to the tidal dissipation in the satellites, which in- eutectic melting point of ammonia hydrate. troduce direct damping terms into the equa- Nevertheless, it is not likely that tidal heat- tions of motion. Further work on this prob- ing of Miranda could have caused significant lem is currently underway. melting of water ice in the interior of this Miranda can escape from the 3 : 1 reso- satellite. nance with an orbital eccentricity signifi- There is a large chaotic zone associated cantly larger than the value prior to reso- with the Miranda-Ariel 5:3 mean-motion nance encounter. Our numerical commensurability, even in the planar experiments in the planar-eccentric approx- approximation. The orbital eccentricities of imation show that the escape eccentricity of both satellites may vary chaotically for a Miranda in the absence of satellite dissipa- considerable period. After a sudden in- tion can be larger than 0.06 (see Fig. 2a). crease at the point of entering the chaotic 430 TITTEMORE AND WISDOM zone, the maximum orbital eccentricity of For initial eccentricities approaching the Miranda gradually decreases until the tra- 2 : 1 commensurability which are compara- jectory escapes from the resonance. The ble to or up to about a factor of 10 higher final average orbital eccentricity and incli- than the current values, capture into this nation of Miranda are most likely, though resonance is very likely. Although even at not necessarily, somewhat higher than the low eccentricities a series of important sec- initial values. On average, the average final ondary resonances between the frequencies eccentricity of Ariel is lower than the initial of libration of (rU and o-g are encountered, value. they affect only the details of resonance pas- Since the rate of tidal heating depends sage, i.e., the oscillation amplitudes of the strongly on the distance to the planet eccentricities. Interactions between these terms and terms associated with the inclina- dE 1 tion-type resonant arguments or with secu- -- oc (10) dt a 15/2 lar terms involving the other Uranian satellites may affect the resonant interaction, but a given value of e M will cause the interior of do not appear to increase the likelihood of Miranda to be heated about 40% more at the escape from the eccentricity-type reso- 5 : 3 resonance with Ariel than at the 3 : 1 nances. resonance with Umbriel. The maximum Could the satellites have encountered the eccentricities achieved at the 5:3 reso- resonance at high eccentricity and escaped? nance are about half as high as those The eccentricities required are of order 0.03, achieved at the 3:1 resonance with Um- according to the numerical simulations. Are briel, implying a maximum tidal heating rate these likely initial values? We can estimate about a third as large. Considering the un- the equilibrium eccentricity of Ariel in the certainties about the interior of Miranda dis- absence of the resonant interaction, using cussed above, if tidal heating of Miranda Eqs. (42) and (51) in Appendix II and QA = was important at the 3:1 resonance with 100, Q = 11,000: Umbriel, it may have been significant here, too. Another possibility is that Miranda may eA = ~ 0.038. (11) have left the 5 : 3 resonance with an orbital inclination of order one degree. The rela- At this eccentricity, the secular increase in tively rapid increase in inclination possible semimajor axis due to tides on the planet during chaotic evolution through this reso- would just be counterbalanced by the de- nance (e.g., Fig. 19) may have left Miranda crease due to dissipation in the satellite; with a substantial rotational obliquity; how- hence, the orbit would not be evolving to- ever, this would have quickly damped to ward the resonance. Therefore, for the sat- the equilibrium Cassini state (see Tittemore ellites to have evolved into the resonance, 1988). Therefore, a relatively high inclina- e A must have been less than this. Since this tion of Miranda's orbit could have been re- is comparable to the initial eccentricity retained over the time interval between the quired to escape from the resonance, it is 5 : 3 resonance involving Ariel and the 3 : 1 not likely that Ariel and Umbriel ap- resonance involving Umbriel, possibly af- proached the 2 : 1 resonance with eccentrici- fecting the evolution through the latter com- ties large enough to escape from the reso- mensurability. However, we have shown nance. that even if Miranda approached the 3:1 We conclude that it is unlikely that Ariel commensurability involving Umbriel with a and Umbriel ever encountered the 2:1 relatively high orbital inclination, i M still mean-motion commensurability. This im- may have jumped to its present value (see plies that the specific dissipation function of Fig. 15 of TW-II). Uranus (Q) is greater than about 11,000. DYNAMICS OF PAST 'URANIAN RESONANCES 431

This result, combined with the results from randa. Passage through this resonance in- TW-II, constrains Q to within less than a volves interesting new dynamical features: factor of 4, or more tightly than the Q of any escape from the resonance occurs via sec- other giant planet has been constrained so ondary resonances, which drag trajectories far [see Burns (1986) for a recent review]. into a chaotic region (Tittemore and Wis- We have found the Uranian satellite sys- dom 1987, 1989). tem to have a rich variety of dynamical be- • There are large increases in the orbital havior. To summarize, the principal results eccentricity of Miranda during chaotic evo- of our research on the Uranian satellites to lution through the 3:1 Miranda-Umbi'iel date are the following: commensurability. This may have affected the thermal evolution of Miranda, if low • Due to the low oblateness of Uranus temperature melting occurred. (J2 = 0.0033), there are significant chaotic • Passage through the 5:3 Miranda-Ariel zones at the low-order mean-motion com- resonance may have modestly increased the mensurabilities among the Uranian satel- eccentricity and inclination of Miranda. lites. There are chaotic variations in the ec- • Escape from the 2:1 Ariel-Umbriel centricities and/or inclinations for commensurability is not likely at typical sat- significant intervals of time during evolution ellite eccentricities; therefore, this probably through these commensurabilities. provides a dynamical barrier to the evo- • These chaotic zones have a significant lution. effect on the mechanism of passage through • Since the anomalously high orbital in- resonance. For example, at both the 5:3 clination of Miranda is a natural outcome of and 2 : 1 commensurabilities involving Ariel passage through the 3 : 1 commensurability and Umbriel, trajectories may be captured with Umbriel, the requirement that the sat- into resonance while they are still chaotic. ellites encountered this resonance places a • Simulated evolution through the reso- lower limit on the specific dissipation func- nances must be extremely slow to avoid dy- tion (Q) of Uranus of 39,000. namical artifacts: slower than (1/a)(da/dt) ~- • Since capture into resonance at the 2 : 1 10-1, to 10-10 per orbit period where chaotic Ariel-Umbriel commensurability is likely at adiabatic invariants are present, and slower typical satellite orbital eccentricities, the re- than (1/a)(da/dt) ~- 10-14 to 10-13 per orbit quirement that the satellites did not encoun- period where higher-dimensional chaotic ter this resonance constrains Q to be greater diffusion is important. For example, to than 11,000. properly represent the dynamics of Miranda • The requirement that the orbital eccen- and Umbriel during evolution through the tricity of Miranda damps to its current value 3 : 1 eccentricity resonances, the simulated since passage through the 3 : 1 commensura- tidal evolution rate must be within the physi- bility with Umbriel indicates that Q is closer cal constraints, 2 × 10 -14 < (1/a)(da/dt) < to 11,000. 6 × 10-14 per orbit period. • At the 5:3 Ariel-Umbriel resonance the eccentricities in the chaotic zone are APPENDIX I: THE AVERAGED typically a factor of 2 to 3 higher than the RESONANT HAMILTONIAN preencounter values. Eccentricities large In this appendix, we develop the averaged enough to have a significant effect on the Hamiltonian models used in this paper. thermal evolution of Ariel have not been Tables IV to VII give expressions and nu- found so far at this resonance (Tittemore merical values of the Hamiltonian coeffi- and Wisdom 1988a). cients. For calculations of the numerical val- • Evolution through the 3:1 Mi- ues of the coefficients and other quantities, randa-Umbriel commensurability explains the adopted unit of time is one year, of mass the anomalously high inclination of Mi- is the mass of Uranus, GM u = 5.794 × 106 432 TITTEMORE AND WISDOM

TABLE IV

PLANAR-ECCENTRIC MODEL COEFFICIENTS: EXPRESSIONS

Coefficient Miranda-Umbfiel 3 : I Miranda-Ariel 5 : 3 ArieI-Umbriel 2 : 1

a - 4 r~ + 4 r~ - 4 ' r'. 4 r~, 2 r------~-, r~

- 3a2M2m~m~l _ 27GZM2m~mu - 2_._TO2M2mhm},l 75GZM2rn2Am'n 3GZM2mZAm'A12G2M2m~mb B 32 F 4 32 F~j 32 F 4 32 F 4 8 F 4 8 F~j

14 2 [9m4"'m'~_3mMm'M] 1 4 [ 15m~m~ 6m~m~] lG4M4RZJ2[6ra~7~_9m~m'~] c 2r v r;, J 2-if, - r;, J , 2aJ

G2MmMmt,,rab [ 3 ~ G2MmMm2 m'A 5 0 + G2Mm Am~m~J [~blr2(~2 o )

+ _.d.dj,0 . ~ mu mb/FM 3_~ ~ d m^mk/3r M 5F~ ~_dho, mUm{j[FA 2r~'~

(2)-'°'1 (2),0q FM J ru J FA J

I G,M~R2J [3m~m~ 3mhm~] 1 [6m[m2 9m~lm~.ll I G,M~R2" f9m~m~ 3m~Am~] o 2 2[ r~ - 2r~ 3 2~'M~2"~2[ r~ - 2r 7 J 2 -2[ 2r 7 - r~ J

~--.-~ r±o G2MmMm2Am'Ar' o , + G2MmAm~mlJ[~u bl%(O) + 4r2u [ru bl/2(a) + 4F--~A [~A b'a(~' 4r~

+ 7b.(a)----/--d 0 m o m~ [F M + ~/3F2u'~ + ...... d . m^ mk [3FM 5F~\ + 7a/,%(,0 mu'~' (FA + 2r~ ] act "" m. mh\r~j r~: dab'a(ct)rnMmh~r~+ r~) ...... w~ r~,/ (3),oq ( o!1 (3),0,] Fu J F^ J Fu J

- G2MmMm~:mb (21)(- I) G2MmMmZAm'A (21)(- 0 _ GZMmAm~j m[~ (21)(- I} 4r2x/r--~u 4r2Vrur^ 4r2VrAru

2 2 , _ G MmMm,,mv U (172)()3 G2MmMm2Am^(172)(5)' _ GZMmAm2,,m~v (172)(~) F 4r~rM 4r~rM 4r~r^

_ G 2Mm~.~.mu 2 , (182)(2) G2MrnMm~AmA(182)(4), - G2MmAm2m{j%-.Z.- (182)(3) G 4FuV FMFu 4F2~/VMF^ 4F2V FAro

_ G2MmMm~mu(192)(1) G2MmMm2Am'A(192)0) G2MmAm2mu H 4F~ 4F3 - 4r 3 (192)(2)

_ G2MmAm~mu (50)(2) 1 2r~Vr^

GZMmAm~mb I j - 2~uu (70)()

km 3 sec-~ (Stone and Miner 1986), and of nate system, the coordinates of the inner length is the radius of Uranus, R o = 26,200 satellite are referred to the center of the km (French et al. 1985). The Hamiltonians planet, and the coordinates of the outer sat- are developed in Jacobi coordinates (see, ellite are referred to the center of mass of the e.g., Plummer 1960). In the Jacobi coordi- inner satellite and the planet. The reduced DYNAMICS OF PAST URANIAN RESONANCES 433

TABLE V

PLANAR-ECCENTRIC MODEL COEFFICIENTS: NUMERICAL VALUES

Coefficient Miranda-Umbriel 3 : 1~ Miranda-Ariel 5 : 3 b Ariel-Umbriel 2 : V

B -5164.7 -49209 - 1617.0 C - 0.31244 - 0.51831 - 0.13026 D - 0.14744 - 0.35891 - 0.073054 E 0.00039381 0.0051029 0.0054908 F - 0.0070667 - 0.064822 - 0.017678 G 0.0052590 0.037080 0.047364 H - 0.00017401 - 0.0052682 - 0.031349 1 0.010364 J - 0.0034123

a aM = 4.8659, a U = 10.1179. b aM = 4.6361, a A = 6.5157. c aA = 6.3255, a o = 10.0400.

TABLE VI

ECCENTRIC-INCLINED MODEL COEFFICIENTS: EXPRESSIONS

Coefficient Miranda-Umbriel 3 : 1 Miranda-Ariel 5 : 3 Ariel-Umbriel 2 : 1

K I-G4M4R2Js[~] 1~4'~4°21 r-15m4Am~ ± 3,~mk]] !c,~Rsj [~_ , ,3 + 2 [ 2F~ J 2 u,.,,,~2 [ 2F A7 T--F7M a/ 2 2 [ F7 2F 7 j + GSMmMm~mlJ4F~ [F~ b°l/2(ct) @ GSMmMm2m'A~4FA LF~r5--b~ts(ct)" + GSMmAmmu, L~r'-

d mumij/F M 3F~ d~ mAm'A/3FM 5F~,i\ 9dho,,mum~[FA 2FS~ + ~ biO/2(°0~ ~ CF~"U + '-~U7 + do~br/2(a) m"~Mm"~MM t F-'~"A + F'~'-A7 +" da ~'"2'='1 m r m~ tF~s -i- F 3 ]

( 1ll)(°' ] (11)<°' ] (11) '°' ] FM J "~M "J FA J

• r6m4m,3 3m4 ,31 i 4--2 r-9m4m~ ~] I--. r-15mf,m;? 3m~m~] L ±G4M4RSj2 / U U _ /draM| + - G'M'R'J ~-...... --v-..~ + __. 2 t rb 2r7 j 26 M'RJs[----~A- 2r7 j 2 ~[ 2r~ 2r~ J G2Mmum2,m~2 r 3 G2MmMm2m'A ~ 0 G2MmAm~mb r 4 0 + 4ru2~ t-u/F-b°a(a) + 4r~ [FA bin(a) + 4r~ [~ bi'2(~)

__d. o mumD/rM 3r~ + d bo a mAm'A/3FM 5_.~ ~ d~o, mum{j/F ^ 2r2\ + de°li2(a)mld m~I tF~s + F~-~u) ~ ,:2( )'~u'~ut'~A + r~ ] + "~ %2'<~,m"-~Am--~Atr---~u + r-~u] (11)`0) ] (11)(0!] (11)(°) ] ru 1 r^ J ~ j

GSMmMm~m~:(ll)(O ) GSMmMm2m,A (11)(0) G2 MmAm~m 2 U, (11)(0) M 2r~ rvrf~ur% 2r2x/rurA 2r2vr^ru

- G2MmMm~mb (212)(3) GSMmMm2m'A (212)0) G2MmAm~mb (212)(4) N 4r2ru 4rsru - 4rsr^

G2MmMm~m~ 3 GSMmMm2m'A 5) GSMmAm~m~2 (212)(4) o 2r~ rx/'f~Mr~ (212)() 2r~rx/-f-~urA (212)( 2r~Vr-':~

_ C-2MmMm~lm{1(212)(3) G2MmMm2Am'A(212)(5) G2MmAm~mb (212)(4) P 4r~ 4r~ - 41"4 434 TITTEMORE AND WISDOM

TABLE VII

ECCENTRIC-INCLINED MODEL COEFFICIENTS: NUMERICAL VALUES

Coefficient Miranda-Umbriel 3 : 1 ° Miranda-Ariel 5 : 3 b Ariel-Umbriel 2 : I c

B - 5166.14 - 49209 - 1617.03 C -0.31263 -0.51823 -0.13030 D - 0.14783 - 0.35883 - 0.073083 E 0.00039358 0.005106 0.0054893 F - 0.0070635 - 0.064858 - 0.017674 G 0.0052572 0.037098 0.047355 H - 0.00017391 - 0.0052705 - 0.031344 I 0.010362 J • -0.0034115 - K 0.044366 -0.070840 0.018914 L - 0.12043 - 0.23024 - 0.038308 M - 0.00067590 - 0.0062127 - 0.0073917 N - 0.00097703 - 0.0072991 - 0.0021365 O 0.00039358 0.0031582 0.0039091 P - 0.000039637 - 0.00034162 - 0.0017881

a aM = 4.8652, a v = 10.1179. b a M = 4.6361, a A = 6.5152. c aA = 6.3256, a u =~ 10.0400.

masses are m I' = ml (Mu/(Mu + mO)and be m~ = m2((Mu + mO/(M U + m I + m2)) for 0-1 = ½(iX2 --JXl -- 2cbl) (al < a2). A development of the planar second-order resonant mapping is given in 0" 2 = ½(iX 2 --jX 1 -- 2t~2) (12) TW-I; the other mappings used here are and the conjugate momenta are similar in form. L1 2 Z, = -~-- e, Planar-Eccentric Models In this section, we summarize the reso- L2 2 ~2 ~ -2- e2 (13) nant Hamiltonians for the planar-eccentric problems. These can be approximated by whereL i -~ m i ~. two-degree-of-freedom models. We give a The resonant part of the Hamiltonian for brief description of the resonant coordinates a second-order resonance is given by (see, and the form of the resonant Hamiltonians. e.g., Leverrier 1855): A detailed derivation of the Ariel-Umbriel 5:3 resonance planar-eccentric model is Gmlmz ~R-- given in TW-I. The Hamiltonians of the a 2 other resonances are derived in an analo- e 2 gous way, and only the expressions and values of the coefficients are given here. We consider second-order resonances involving the combination of mean longitudes -I- (182) (i-I) -~-~ Cos(i~ 2 -- J~l -- ~)1 -- ~)2) ih 2 - jh I, i - j = 2, where subscripts 1 and 2 refer to the inner and outer satellites, + (192)(i-z)(2)e 2 COS(iX2--JXl-- 2&2)] respectively. The resonance coordinates are defined to (14) DYNAMICS OF PAST URANIAN RESONANCES 435 where nian in the following terms:

1 (172) (/) = ~ (-51 + 412)b~/2(a) + (-I + 2l) I"V~-~ ! cos 0-1 -[- JV~2 cos 0"2. (19) In terms of the Cartesian elements 1 °t 2 d 2 ot d b~/E(a) + 2 ~a 2 b~/E(a) Yi = ~ sin(o"i) ~ eiV~i sin(o-i) or (182) a) = (-2 - 41)lb~/2(ct) + (-2 - 4/) iiX/-£ii sin(o-i). (20) d E and the conjugate momenta a ~ b~/E(Ct) - o~2 ~2 b~/2(a) xi = V~i cos(o'i) ~ eiX/L/i cos(o-i) or (192)a) = 1(4 + 91 + 412)b~/E(a) + (3 + 2/) 6X/~ sin(o-i) (21) the Hamiltonian may be expressed d b~/E(Ot) + 1 °t 2 d E l a -d-da ~ bl/2(a) (15) = ¼(8 - 2(C + D))(x 2 + y2 + x 2 + y22)2 where l is an integer, a = aJa2 is the ratio + B(x + + + y )2 of semimajor axes, and b~(a) are Laplace + C(x~ + y2) + D(x~ + y2) coefficients. For the 3 : 1 resonance, there must also be included the indirect term + E(XlX2 + YlY2) -(27/2) a in the expression for (192) m. + F(x 2 _ y2) + G(xlx2 - YlY2) The form of the Hamiltonian is + H(x 2 _ y2) (22) = 2A(EI 4- ~2) + 4B(~1 + ~2) 2 with the additional terms + 2C~ 1 + 2D~ 2 Ix I + Jx 2 (23) "4- 2EV~I~ 2 COS(O-I -- 0-2) for the 2 : 1 resonance, and where

+ 2F~ I cos(20-1) 8--- 4A + 2(C + D) ~ in 2 -jn l - t~l - ~2. (24) + 2GX/~lY~ 2 cos(o-1 + 0"2) This definition of 8 is chosen so that the + 2H~2 cos(20-2). (16) coefficients are evaluated in the middle of the resonance region. Numerical values of For the 2 : 1 resonance, there are the addi- the coefficients were calculated at zero ec- tional first-order resonance terms in Ha: centricity and 8 = 0. Expressions for the coefficients are given in Table IV, and nu- (50)(2)(-~) COS(2hU -- ~.A -- ~ A) merical values in Table V. In the evaluation of the coefficients, the unit of mass is the mass of Uranus, the unit of time is one year, + (70)(1) (~-~) COS(2hu -- hA -- 6~u) (17) and the unit of length is the radius of Uranus. During evolution through the resonance, the where quantity in2 - jn~ changes by a large amount relative to the other coefficients, which are (50) 2 = _4b~n(a) _ a ~aad b2/2(a) therefore approximated to be constant.

(70) 0) = 3bl/2(a ) + ot ~ bl/2(ot) - 4a. (18) Eccentric-Inclined Models To include the effects of three-dimen- The last term in the expression for (70) o) is sional motion on the interactions at a com- the indirect term. The first-order resonance mensurability, we can develop a resonant terms are included in the resonant Hamilto- Hamiltonian similar to that developed for 436 TITTEMORE AND WISDOM the planar-eccentric case, but generalized to where include the inclination resonant terms. We (212)(0 = ½ab~t/~ 1) (a). (29) define a is the ratio of semimajor axes, and bls(a) 0-1 = ½(ih2 -- jhl - 2&1) are Laplace coefficients. In terms of these coordinates, the Hamil- 0-2 = ½(iX2 -- jhl - 2t~2) tonian is FI 2 Y~I =Te~ = 2A(~1 + ~2 + xI/1 + xlt2) + 4B(~l + ~2 + ~1 + xi/2)2 F2 2 ~2 ~" T e2 + 2C~ 1 + 2DY~2 @1 = ½(ih2 -- Jhl -- 2~1) + 2E~v/~1~2 cos(o- 1 - 0-2) @2 = ½(ih2 -- Jhl -- 2~2) + 2F~ 1 cos(20-1)

"1 ~Tll + 2GX/~1~]2 cos(o-1 + 0-2) + 2H~] 2 cos(2o-2) F2 2 xIt2 ~ ~- i2 + 2K~ 1 + 2Lqt2 + 2MX/~lxIt 2 cos(~bI - t0z) F1 = L1 + ~ (El + E2 --F ~Xtl --F ~tI/2) + 2N~1 cos(2tkl) i F2 = L2 - ~ (ZI + ~2 + Xttl + xtt2)" (25) + 2OV~lXtt2 COS(~bl + I]/2)

The inclination part of the secular contri- + 2PXIt2 cos(2t02). (30) bution to the Hamiltonian is given by (see, For the 2:1 resonance, there are also the e.g., Leverrier, 1855) terms

~s - Gmlm2 {(1) (°) (11)(°)[(2)2 IX/~ 1 cos O'l + JX/~2cos 0-2. (31) Expressions for the coefficients A through + (~)2- 2 (~)(2)COS (~-~1- ~"~2)]} (26) J are the same as in the planar-eccentric model, with the redefined Fi given above. 8 where is defined to be the nonresonant contribu- (1)(o) 1 o tions to in 2 -jn 1 - ½(~oI + ~o2 + (]1 + = 2otbl/2(ot) ~2) = 4A + (C + D + K + L). This defini- (I 1)(°) = _ ½ab~/2(a) (27) tion is chosen so that the coefficients are evaluated in the middle of the resonance and the inclination part of the resonant con- region. Expressions for the coefficients are tribution is given by given in Table VIII, and numerical values of the coefficients, evaluated at 8 = 0 and ~a = Gmlm-~z (212) (0 zero eccentricity and inclination, are given a2 in Table IX. [Note: For the Miranda-Ariel 5 : 3 resonance, 8 was defined incorrectly to be i~ i 2 5hA -- 3riM -- (¢~M -- O-)A -- ~'~M -- ~A) = 4A - 2 ~- ~ COS(ih 2 -- jh~ - ~'1 - ~)-2) + 2(C + D + K + L). The main effect of this is to slightly change the values of the coefficients in Table VII; however, the dif- (28) ferences are not significant.] DYNAMICS OF PAST URANIAN RESONANCES 437

TABLE VIII

SECULAR PERTURBED MODEL COEFFICIENTS; EXPRESSIONS

Coefficient Miranda-Umbriel 3 : 1 Miranda-Ariel 5 : 3 ArieI-Umbriel 2 : 1

tl 3 G4M4m~rn~lR2J2 3 G4M4m4m~tR2J2 3 G4M4m4m'A3R2J2 C 4 L 7 4 L 7 4 L 7 G2Mmum2 m'A (2)(~.A G2MmMm2 m'A (2)~A G2MmArn2mb (2)~b 4L 2 L M 4L 2 L M 4L 2 L A GEMmum2 m~ (2)~u G2MmMm2 m~ (2)~tj GEMm Am2m÷ (2)~ 4L 2 LM 4L 2 LM 4L2 LA 3 G4M4m4m{~REJ2 3 G4M4m4m'A3R2J2 3 GgM4momu4 t3R 2J2 D" 4 L 7 4 L 7 4 L7u G2MmMm~m~j (3)~u G2MmMm2m'A (3)~A GEMmAm2m{j (3)~{j 4L~ Lu 4L 2 LA 4L~ Lc GZMmAm2m~ (3)~ C.3MmAmEm{j (2)~ GEMmum2m~ (2)~ 4L 2 L U 4L 2 L A 4L 2 L U

3 G4Mgm4Am'A3R2J2 3 G4M4m~m{jaR2J2 3 G4M4m4m~3R2J2 U 4 L 7 4 L 7 4 L 7 GEMmMm2m'A (3)~A G 2MmMmum 2 U, (3)MU(0) GEMmAmEm~ (3)~ 4L 2 LA 4L~ L~ 4L~ LT

2 , (0) 2 , (0) GEMmAmumg (2)AU G2MmAmumu (3)AU GEMmomEm~ ra~(0)~/UT 4L 2 LA 4L 2 Lu 4L2 L T G2MmMm2rn'A (21 )(M-~) G2MmMm2mu (21)~tj- l ) G2MmArn2m÷ (21 )(AT') 4L 2AX/ LMLA 4L 2u~/ LMLu 4L2"~v/LALT G2MmArn~mu (21)~-0 G2MmAm2mu (21)(A-JI W G2Mmum2mT " 1" 4L~X/LALU 4L2uVLALu - --2 - (21)UT' '~LTVLuLT

Secular Perturbations (L l - Lm), where the resonant value ofL m To include the secular perturbations due is defined by setting 0~0/0qb = 0 and where to other satellites, the development of the ~0 is the nonresonant part of the Hamil- Hamiltonian is somewhat different, in order tonian. to keep secular variations distinct from reso- The Hamiltonian for the two-body resonant variations (see Wisdom 1982). We de- nance problem can then be expressed as fine as coordinates th =jhl -- ih2 + C'(p 2 + q~) + D'(p 2 + q~)

Pi ~ eiN/-Lii cos( - &i) or ii~V~i cos( - fli) + E(plPz + qJq2) qi ~- eiN/-Lii sin( - &i) or + F((p~ - q2) cos th + 2plql sin ~b) iiV'-Lii sin( - fl i) (32) 4- G((plpz - qlqz) cos th and expand the Hamiltonian about the resonant value of L1 (pendulum approximation). 4- (Plq2 4- qlP2)sin ~b) The momentum conjugate to ~b is qb = (l/j) + H((p~ - q2) cos 4) 4- 2PEq 2 sin ~b) (33) 438 TITTEMORE AND WISDOM

TABLE IX

SECULAR PERTURBED MODEL COEFFICIENTS: NUMERICAL VALUES

Coefficient Miranda-Umbriel 3 : 1a Miranda-Ariel 5 : 3b Ariel-Umbriel 2 : 1c

a -165280 -1.5747 × 106 -5.1742 × 104 C" - 0.18896 - 0.22521 - 0.075839 D" - 0.019399 - 0.068809 - 0.021761 E 0.00039380 0.0051031 0.0054912 F - 0.0070660 - 0.064824 - 0.017679 G 0.0052585 0.037081 0.047366 H - 0.00017399 - 0.052683 - 0.031350 1 0.010364 J - 0.0034125 U - 0.054396 - 0.017831 - 0.0039395 V 0.0036101 0.00033903 0.00056777 W 0.0098431 0.0063445 0.0033443

a aM = 4.8658, a A = 7.0846, a U = 10.1179. b aM = 4.6361, a A = 6.5157, a U = 10.0560. c aA = 6.3256, au = 10.0400, a T = 16.6330.

where a ~ 32B, and where C' andD' contain where the perturbing satellite is designated only precession terms due to planetary ob- by subscript 3. For the 2 : 1 resonance, there lateness and satellite interactions. For the are the additional term 2:1 resonance, there are the additional I(X 1 COS ~ + Yl sin th) terms + J(x zcosth + Y2sin~b). (36) I(x I COS t~ + Yl sin ~b) The expressions for the coefficients E-H + J(x 2 cos qb + Y2 sin ~b). (34) (and I and J for the 2 : 1 resonance) are the To include the perturbations due to a third same as those in the two-satellite problem, satellite, we add the terms in the secular part and a ~ 32B, except that in the expressions of the disturbing function involving the third for the coefficients, L i should be substituted satellite and its interactions with the first for F,.. Numerical values of the coefficients two. For the Hamiltonian we end up with were evaluated at zero eccentricity. Expres- the expression sions for the coefficients C", D", U, V, and W are given in Table VI, and numerical val- = ½Ot(I)2 ues of all the coefficients are given in Table + C"(p 2 + q~) + D"(p~ + q2) VII. Note. In TW-II (Tittemore and Wisdom + E(PlP2 + qlq2) 1989), there is an error in the definition of + F((p~ - q~)cos ~b + 2plql sin ~b) the coefficient of the e 2 resonance in the Miranda-Umbriel 3 : 1 eccentric-inclined + G((PlP2 - qlq2) COS ~b resonance model. The Leverrier coefficient (192) ~1) contains an extra indirect term + (Plq2 + qlP2)sin ~b) -3/2a 2 which should not have been in- + H((p2 _ q2) cos th + 2P2q2 sin ~b) cluded. The incorrect definition of this coef- + U(p2 + q2) + V(pjp3 + qlq3) ficient introduced errors into the numerical integrations of the eccentric-inclined model + W(P3P2 + q3q2) (35) only, and did not affect the main body of DYNAMICS OF PAST URANIAN RESONANCES 439 the paper dealing with the circular-inclined ratio of the two satellites (at/ao) remains problem. The error does not affect the constant as the orbits expand due to tides mechanism that leads to the high orbital in- (see, e.g., Yoder 1979b, Peale 1988). clination of Miranda. The main conclusions We present the calculation of the equilib- drawn from the numerical integrations of rium eccentricity for the 2 : 1 Ariel-Umbriel the eccentric-inclined model, that the orbital case in detail; equilibrium eccentricities for inclination of Miranda remains large during the other commensurabilities may be ob- evolution through the chaotic zone associ- tained in an analogous manner. Note, how- ated with the eccentricity resonances and ever, that the Miranda-Ariel 5 : 3 commen- that the orbital eccentricity of Miranda in- surability does not have an equilibrium creases significantly during passage through eccentricity because the semimajor axis ra- the eccentricity resonances, also remain un- tio (aM/aA) is decreasing. At the end of this changed. section, we summarize the effective tidal evolution rates due to planetary tides, the equilibrium eccentricities at the resonances, APPENDIX II: TIDAL EVOLUTION and the eccentricity damping time scales. In planetary satellite systems, tidal evolu- Of the various coefficients in the reso- tion is caused by tides raised on the planet nance Hamiltonian, only 8 depends strongly by the satellites and tides raised on the satel- on the resonant combination of mean mo- lites by the planet. In the Uranian satellite tions; during passage through the reso- system, the former tend to the orbital semi- nance, the fractional change in the parame- major axes, while the latter act to decrease ter 8 is large compared to the fractional the semimajor axes and eccentricities. changes in the other coefficients. The other In the absence of an orbital resonance, coefficients will therefore be taken to be the equilibrium eccentricity of a satellite is constant. In addition, the value of 8 is influ- the value at which there is no net change in enced by the damping of eccentricities due the semimajor axis due to tides, because the to tidal dissipation in the satellites. For the rate of energy dissipation due to tides raised Ariel-Umbriel 2:1 commensurability, the on the satellite is equal to the rate of energy time rate of change of 8 is input to the orbit due to tides raised on the planet. = 4 A. (37) For satellites involved in an eccentricity From the expansion of A to first order in resonance, the equilibrium eccentricity has ~i, a somewhat different meaning. At the Ar- iel-Umbriel 5 : 3, Miranda-Umbriel 3 : 1, -~ (4ha -- 2hM) -- 16B(~M + ~A)- (38) and Ariel-Umbriel 2:1 resonances, planet tides tend to expand the inner satellite orbit The first two terms are due to the tidal (I) relative to the outer orbit (O); therefore, expansion of the orbits, and the other terms the semimajor axis ratio (aI/ao) tends to in- are due to the exchange of angular momen- crease. However, the forced orbital eccen- tum between the satellites resulting from tricities resulting from the resonant interac- dissipation of energy in the satellites (Yoder tion also increase, so some of the energy 1979b). Note that gained by the orbits is tidally dissipated in one or both of the satellites. At each of the ~i = E i [ -- 43 4n---~.h/"{- 2~]'e;d (39) above resonances in the Uranian system, most of the tidal dissipation occurs in the Tidal friction results in a secular variation inner satellite. Under these conditions, the of the mean motions equilibrium eccentricity of the inner satellite is the value at which the semimajor axis hi = hip + his (40) 440 TITTEMORE AND WISDOM where the first term is due to tides raised on Using numerical values from Table I to the planet (e.g., Goldreich 1965) evaluate relative contributions to 8,

9, 2mi[R'~ 5 1 4hu -~ 4mu (a--3-A/8~ 0.047 (47) hip = -- 5 (41) r2ni -M ~: -Q" 2hA 2mA kaU/

The second term is due to tidal dissipation so the semimajor axis ratio increases. The in the satellite,which for a synchronous ro- ratio of the eccentricity-dependent terms is tation state must conserve the orbital angular momentum: Gi = d/dtV'GMai(I - e 2) = ~'U k2umAnu(go~5(a__.3_AtsaA~'u 0, so h ~- 2aiei~i. Therefore, ~A k2A mu nA \RA/ \ao/ ao ~'A h i = hip - 3niei~ i. (42) keA/- (48) Tidal friction also changes the orbital eccentricities (Goldreich, 1963) The most significant terms in ~ are therefore p.i = _ 21 k2iniM (Ril5 ei = -2h A - 16B~ n. (49) 2 mi \all Qi 578 . mi [R'~5 ei Note that (see Appendix I) + k2ni-~ ~) ~ (43) 16B= --6hA+LA [1 4m--AA(a--AAI2]mU\au/J (50) where k2 and Q are, respectively, the poten- tial Love number and specific dissipation and function for Uranus, and the k2; and Q,. val- en 7 ea -7=gDa - (51) ues are the same quantities for the satellites. nA nA The first term in the expression for ~i is the contribution due to tidal dissipation in the Therefore satellite, and the second term is due to tidal dissipation in the planet. The relative contributions of these components is --2hA{I-- 14DAe2A[l+2mA(aAt2]l'mUkaU/_lJ bs 28 (52) ep - ~ Di (44) The tidal parameter 8 will be stationary at where the equilibrium eccentricity

Di=-~2 -~ik2i(M)'(-~)5Q~ (45) e A = . (53) 14DA 1 + 2 m---A a__&A expresses the ratio of the satellite tidal scale mu \au/ d factor to that of the planet (e.g., Yoder This expression is in agreement with the 1979b). For Ariel, D n = 100.7 (Q = 11,000 expressions for the equilibrium eccentricity and QA = 100), and the satellite dissipation of the 2 : 1 two-body resonances in the Jov- term dominates the planet dissipation term ian satellite system found by Yoder (1979b). by about two orders of magnitude. This is For the Ariel-Umbriel 2:1 resonance, the also the case for the other satellites. Since equilibrium eccentricity of Ariel is hp/n and ~p/e are of the same order, this means that 1 e A = ~/2 5.~D A - 0.02 (54) • ei ~i ~ 2•i7 = Lieiei. (46) ei ifQ = 11,000. DYNAMICS OF PAST URANIAN RESONANCES 441

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