ICARUS "/4, 172--230 (1988)
Tidal Evolution of the Uranian Satellites
I. Passage of Ariel and Umbriel through the 5:3 Mean-Motion Commensurability
WILLIAM C. TITTEMORE AND JACK WISDOM Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received June 17, 1987; revised November9, 1987
Ariel and Umbriel have passed through the 5 : 3 mean-motion commensurability if the specific dissipation function (Q) for Uranus is less than about 100,000. There is a signifi- cant chaotic zone associated with this resonance. Due to the presence of this chaotic zone, the standard theory describing passage through orbital resonances is not applica- ble. In particular, there are significant changes in the mechanism for and probability of capture into resonance. Tidal evolution within the chaotic zone may have driven orbital eccentricities to relatively high values, with some probability of escape from resonance remaining. In the planar approximation, eccentricities high enough to have a significant effect on the thermal history of Ariel have not been found. © 1988 AcademicPress, inc.
1. INTRODUCTION cantly affected the Uranian satellites war- rants investigation. The major satellites of Uranus are not Dermott (1984) has suggested that reso- presently involved in any mean-motion nant motion of the Uranian satellites may commensurabilities. However, a number of be chaotic. His argument is based on the low-order resonances between various sat- resonance overlap criterion, which pro- ellites may have been encountered in the vides a useful criterion for predicting the past as the orbits of the satellites evolved as onset of large-scale chaotic behavior in a result of tidal friction (see Peale 1988 for a many systems (see Chirikov 1979). For the recent review). The present eccentricities satellite systems of Jupiter and Saturn, the of the inner large satellites are anomalously group of resonances associated with a par- high when the time scales of eccentricity ticular mean-motion commensurability are damping due to tides raised on the satellites well separated by precession induced by by Uranus are considered (Squyres et al. the planetary oblateness, However, the J2 1985), and this cannot be accounted for by of Uranus is small (,/2 -~ 0.0033) and the the mutual perturbations alone (Dermott resonances associated with any particular and Nicholson 1986, Laskar 1986). Also, mean-motion commensurability between the recent observations by Voyager 2 the Uranian satellites are not well sepa- (Smith et al. 1986) of extensive resurfacing rated. Dermott argues that the resonance of Miranda, Ariel, and Titania indicate that overlap criterion then implies that there the thermal histories of these satellites have should be chaotic behavior. While Dermott been spectacular. The importance of tidal was correct in pointing out the importance heating in the thermal balance of satellites of the small J2 for the motion of the Uranian has been demonstrated in the melting of Io satellites, the specific application of the res- due to tides raised by Jupiter on this satel- onance overlap criterion is incorrect. The lite (Peale et al. 1979), and the possibility overlap criterion states that chaotic behav- that analogous phenomena have signifi- ior ensues when the sum of the libration 172 0019-1035/88 $3.00 Copyright © 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 173 half-widths of two neighboring resonances small certainly enhances the likelihood of is larger than their separation. Intuitively, interesting dynamical behavior. two independent combinations of the vari- The theory of passage through isolated ables in the problem cannot simultaneously mean-motion commensurabilities between librate. For example, if o-j = 0j - 02 oscil- satellites is well developed (Allan 1969, Sin- lates about zero while both 01 and 02 are clair 1972, 1974, Yoder 1979, Henrard and circulating, then the possibility of the si- Lemaitre 1983, Lemaitre 1984, Borderies multaneous oscillation of 0-2 = 01 - 202 is and Goldreich 1984). The work of Henrard excluded. If a "good" perturbation theory (1987), in particular, places the theory of suggests that both variables should oscillate passage through mean-motion resonances then chaotic behavior most likely results. in a general framework of passage through For mean-motion resonances the situation isolated resonances. The single resonance is more complicated. Different resonance theory can be used to compute the probabil- variables may correspond to independent ity of capture into a mean-motion commen- degrees of freedom. In this case the libra- surability, and the changes in the eccen- tion of one resonance is independent of the tricity or inclination as the resonance is libration of the other resonance. In the ex- encountered, provided the assumptions ample above, the oscillation or circulation which have been made in deriving the for- of 01 may be completely independent of the mulas are satisfied. Unfortunately, critical oscillation or circulation of 02. The fact that assumptions of the single resonance deriva- a system is in a region of phase phase where tion are not valid for the Uranian satellites. both 01 and 02 oscillate does not indicate the First, the Hamiltonian which is used in the presence of chaotic behavior. In this case derivation of the single resonance formulas the simultaneous libration of independent is assumed to be well represented by the combinations of 01 and 02 is not prohibited. terms from the disturbing function involv- Wisdom (1985a) has shown that "overlap- ing a single resonant argument, and actually ping" resonances belonging to the same more commonly by a single resonant term mean-motion commensurability do not nec- from the disturbing potential. However, essarily lead to chaotic behavior, but also since the J2 of Uranus is small (J2 --~ 0.0033) that chaotic behavior may result from only the resonances are not well separated, and a single mean-motion resonance if accom- the quasiperiodic resonance regions can be panied by the proper secular terms. For ex- significantly different from those predicted ample, the mathematical problem of motion by single resonance approximations. In near a first-order commensurability in the some cases alternate formulas could be de- elliptic restricted three-body problem, tak- rived which are based on the more com- ing into account only the two resonances plete resonance Hamiltonian, though to our which are first order in eccentricity, is inte- knowledge calculations of this sort have not grable, even though there is strong "over- as yet been published (see, however, Titte- lap" of the region of libration in the individ- more and Wisdom 1988). A more funda- ual resonances (see Wisdom 1987 and mental problem with the standard single references therein). No simple argument resonance theory is that the motion at the about overlapping mean-motion resonances point of capture into resonance is assumed can predict whether large-scale chaos ex- to be regular, i.e., quasiperiodic. Whether ists. A proper application of the resonance the trajectory is captured or escapes de- overlap criterion involves checking for the pends on the (generally unknown) phase of overlap of regions of libration of indepen- the resonant argument as the separatrix is dent resonances between the degrees of encountered. The separatrix is the regular freedom. In any case, the proximity of sev- trajectory of the single resonance problem eral resonances in the situation where ./2 is which separates regions where the resonant 174 TITTEMORE AND WISDOM argument librates from regions where it cir- known. Our systematic examination begins culates. The "decision" of capture or es- then with the current configuration and pro- cape is made on a single libration and de- ceeds backward in time. Hopefully, at pends on the phase at which the separatrix some point a barrier will be reached, is encountered. A capture probability is de- through which evolution to the present con- fined by assuming some initial distribution figuration will be impossible or unlikely. of phases and is computed by evaluating The requirement that that configuration certain integrals along the separatrix. The has never been encountered would then assumption of a regular, quasiperiodic sep- provide a constraint on the Q of Ura- aratrix completely permeates the existing nus. theory of passage through mean-motion In this, our first study, we consider only resonances. Generically, though, the mo- the Ariel-Umbriel 5:3 mean-motion com- tion near a separatrix is chaotic, not quasi- mensurability. The 5:3 commensurability periodic (see e.g., Chirikov 1979). Thus the between Ariel and Umbriel would have boundary has a finite width. The motion been the most recently encountered of the near the separatrix is no longer the motion first- and second-order commensurabilities assumed by the isolated resonance theory. between the Uranian satellites. For this The presence of the chaotic zone will be commensurability there are three important unimportant if the chaotic zone is so small mean-motion resonances involving the ec- that it is crossed in a time short compared centricities and three involving the inclina- to a libration period, for then the chaotic tions, as well as a strong secular coupling. dynamics are indistinguishable from the The 5:3 resonance therefore promises to regular dynamics assumed in the standard be quite interesting. theory of passage through resonance. On We consider the Ariel-Umbriel 5 : 3 reso- the other hand, if the region of chaotic be- nant interaction in the planar approxima- havior near the separatrix is large enough tion. The restriction to the planar problem that the trajectory has enough time to make has been made in an effort to get some un- significant chaotic wanderings during the derstanding of the dynamical mechanisms transition then there is no reason to believe involved in the passage through such a that the single resonance predictions have complicated resonance. The planar system any relevance at all. The process of pas- has the advantage of being reducible to two sage through resonance has qualitatively degrees of freedom, which allows the de- changed. As will be demonstrated below tailed study of the phase space using the there are significant chaotic zones near Poincar6 surface of section technique. This mean-motion commensurabilities between technique has been found to be invaluable the Uranian satellites. Thus predictions for understanding the qualitative features of based on the single resonance model are motion of asteroids in the 3 : 1 Kirkwood highly suspect, and most likely irrelevant. gap (Wisdom 1985a). The importance of the chaotic separatrix in In Section 2, the planar resonant Hamil- changing the mechanism of capture was tonian of the system is discussed, the full previously pointed out by Wisdom et al. development being given in Appendix I. (1984). Since the masses of the satellites are com- This paper is the first of a series of papers parable, a treatment of the full resonant which will systematically examine the tidal three-body problem is required. The evolu- evolution of the orbits and spins of the Ura- tion of the system is studied with the aid of nian satellites. The full extent of the tidal an algebraic mapping with the same reso- evolution of the Uranian satellites is un- nant structure as this three-body problem, known, since the magnitude of the specific analogous to that developed by Wisdom dissipation function (Q) for Uranus is un- (1982, 1983) for the restricted problem to PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 175 study resonant asteroid motion. The map of quasiperiodic circulation, the trajectory and its derivation are given in Appen- escapes from the resonance. If, on the dix II. other hand, the trajectory remains in the In Section 3 some aspects of the dy- chaotic zone, then the energy surface even- namics of the 5:3 Ariel-Umbriel mean-mo- tually divides the phase space into two re- tion commensurability are exhibited. Sur- gions in which libration of one or more res- faces of section for this system reveal the onant arguments must occur. At this point presence of significant chaotic zones, even capture into resonance has occurred, even for relatively low eccentricities. A pair of though the trajectory may still be chaotic. islands on the surfaces of section may be Upon further evolution those trajectories associated with the normal modes of the which were chaotic at the point of capture linear secular theory. The development of eventually become quasiperiodic librators. large-scale chaotic zones is associated with The probability of escape depends not only the fixed point associated with one of these on the eccentricities of the satellites far modes becoming unstable. from the resonance, which vary considera- Section 4 explores the evolution of the bly due to the strong secular interaction, system through the resonance with small but also on the initial distribution of energy tidal dissipation included. Results are com- between the two secular modes. The cou- piled for five numerical runs with distinct pled nature of the problem makes compari- families of initial conditions. In each case son with the standard single resonance large numbers of trajectories have been fol- model of evolution through resonances im- lowed to assess the probabilities of various possible. Even qualitative features of pas- outcomes. It is found that the satellites can sage through resonance differ from the pic- be driven to relatively high eccentricities in ture developed for passage through isolated the chaotic zone and still escape from the resonances. For example, the average ec- resonance. The maximum eccentricities centricity of Umbriel after escape from the during passage through resonance are al- resonance can be higher than the average ways found to be higher than the initial val- value before the resonance is encountered. ues and occur during the chaotic phase of In the standard theory, the eccentricity af- the resonance encounter. The mechanism ter escape is always smaller. of capture is markedly different from that of Section 5 discusses physical applications the isolated resonance theory. In the single of these results. Final eccentricities of es- resonance picture capture or escape occurs caping trajectories are consistent with the depending on the phase of the trajectory as present values. Eccentricities high enough the regular separatrix is encountered. For to have had a significant effect on the ther- the 5:3 commensurability the "decision" mal history of Ariel have not been found so of capture or escape occurs while the mo- far in this study. This may be due to limita- tion is chaotic. The trajectory approaches tions of the planar approximation; higher the resonance from a quasiperiodic region eccentricities may be possible in the full of phase space in which the resonant argu- three-dimensional problem (see Wisdom ments circulate. There is another quasipe- 1983, 1987). riodic circulation region on the other side of Since we are concerned with the determi- the resonance, into which trajectories may nation of the probabilities of various out- escape. In the resonance region the phase comes of tidal evolution in a complicated space is dominated by a large chaotic zone. dynamical environment we have as a check The chaotic zone acts as a bridge between verified that our methods properly repro- these two quasiperiodic regions. If, during duce the known capture probabilities in the the period of chaotic evolution, the trajec- single resonance approximation. Appendix tory becomes trapped in the second region III discusses these results. 176 TITTEMORE AND WISDOM
2. THE RESONANT HAMILTONIAN TABLE I
This study looks at the system of Uranus PHYSICAL PARAMETERS and its satellites. The planet is considered Units: to be the dominating mass, which is sym- Mass: mass of Uranus: GM = 5.794 × 106 km 3 metric about its rotation axis, and the satel- sec -1 (Stone and Miner 1986) lites are considered to be n - 1 point Distance: radius of Uranus: R = 26,200 km masses. The gravitational potential of this (French et al. 1985) Time: years system can be written rn--3AM= 0.0000155, -~ = 0.0000147 n-lGMmi I ~ (R) 1 ] U = - ~'~ 1 + Jt P/(sin ~i) (Stone and Miner 1986) i=1 ri l=2 RA = 0.0221R, Rv = 0.0227R _ ~ Gmimj, (1) (Stone and Miner 1986) e^ = 0.0017, eu = 0.0043 (Peale 1988) i The secular interactions of the Uranian resonant terms in the disturbing function satellites are relatively strong (Dermott and which influence the evolution of the eccen- Nicholson 1986, Laskar 1986). However, tricities and pericenter longitudes are of or- the largest contributions to the secular vari- der e~. The lowest order secular terms are ations of Ariel and Umbriel arise from their also of this order (see Appendix I). The mutual interaction. In this first paper we Hamiltonian is now expanded in powers of have ignored the secular perturbations due ~,i/Fi to this order, and constant terms are to the other satellites. With this approxima- removed. The resulting expression is tion the problem remains a two degree of freedom problem, allowing a detailed study = 2A(Y~A + Y~U) + 4B(~A + ~u) 2 of the complicated dynamics with the sur- + 2C~A + 2D~Zu face of section technique. + 2EX,/]~A~U COS(O"A -- O'U) Near the resonance, the evolution of e; + 2FS~A COS(2OrA) and o)i is dominated by the low frequency + 2GX/~A~U COS(O'A + O'U) perturbations, with frequencies associated + 2H~u cos(2o-u). (6) with changes in the resonant combination of longitudes 57,v - 3hA and the longitudes Expressions for the coefficients are given of pericenter o3i. The high-frequency contri- in Appendix I. For low eccentricities, the butions associated with the motions of XA coefficient A -~ l(5nu - 3hA). Near the 5 : 3 and htj, and with other nonresonant combi- mean-motion resonance, this quantity is nations of the mean longitudes, are re- near zero. Tidal dissipation in the planet moved in first order by averaging. causes the orbits of the satellites to expand As resonance coordinates we have cho- differentially (Goldreich 1965). Because of sen (see Appendix I) this differential expansion, nA decreases 1 relative to nu (see Section 4.1). Therefore, OrA = ~ (5hU -- 3h A -- 2O3A) the quantity (5nu - 3hA) increases as en- 1 ergy is dissipated in the planet and changes tru = ~ (5AU -- 3hA -- 2o30) (3) sign when the mean motions are exactly commensurate. This provides a convenient which together with ha and hu form a com- measure of distance from the resonance. plete set of generalized coordinates. The We define the parameter 3 = 4A + 2(C + D) momenta conjugate to o'A and o'u are, in to be the resonance parameter in this prob- terms of the Delaunay momenta Li -~ lem. At small eccentricities it is propor- mini and Gi = LiX/I -- e 2 (see Plum- tional to the nonresonant contribution to mer 1960): 5ntj - 3nA -- ~A -- o~U, and it changes sign ~A = LA -- GA in the middle of the resonance region (see ~u = Lu - Gu. (4) Appendix III). The other coefficients are proportional to the semimajor axis ratio as This choice of variables results in two inte- it appears in the Leverrier coefficients (see grals of motion, the momenta conjugate to Appendix 1). The change in semimajor axis YA = hA and Yu = Xu, since in the new ratio as the resonance is traversed is of or- variables the resonant Hamiltonian is cyclic der one part in a thousand, while the frac- in these variables. The resonance integrals tional change in 3 is of order unity. The are 3 changes in the numerical values of the coef- FA =LA +~(~;A + ~U) ficients are therefore small compared to the changes in 3 and consequently will be ne- 5 glected. Fu = Lu - ~ (~A -~ ~U)- (5) The transformation to canonical coordi- Note that E,- ~ Fie2~2. The lowest order nates 178 TITTEMORE AND WISDOM Yi = ~ sin(o-i) ~ eiV~i sin(o-i) (7) speed between qualitatively different meth- and the conjugate momenta ods. In practice, though, the mapping makes a MicroVAX competitive with con- xi = ~ cos(o'e) -~ ei~ii cos(o'i) (8) ventional methods used on supercom- is made, resulting in the final form of the puters. The gain in this case is approxi- Hamiltonian: mately a factor of 10 over the analytically 1 averaged differential equations with a =~(8-2(C+D))(x 2 + y2A + xzU + y2) relative precision of 10 -9 per time step, + B(X2A + y2A + X~j + y2)2 which is on the order of one year. The gain + C(XZA + y2) + D(x~j + y2u) over direct integration of unaveraged differ- + E(XAXU + YAYU) + F(X2A -- Y2A) ential equations would be at least one or two orders of magnitude. One could argue + G(XAXu -- YAYU) + H(X~j -- y~). (9) that since the relative precision of the map- ping is the full numerical precision of the This Hamiltonian has two degrees of free- computer, the real gain in speed of compu- dom. Note the similarity between this and tation is much greater. This gain is signifi- the Hamiltonian studied by H6non and cant for long numerical runs involving Heiles (1964)--essentially a pair of har- slow tidal evolution. It is essential that the monic oscillators with nonlinear coupling. simulated tidal evolution be slow enough H6non and Heiles studied a cubic interac- to avoid artifacts (see Section 4.1). tion; in this case the nonlinear terms are 3. RESONANCE DYNAMICS quartic. Wisdom (1982, 1983) has derived an alge- 3.1. Surfaces of Section braic map of phrase space onto itself for the In problems such as that considered by Hamiltonian describing the motion of aster- H6non and Heiles the presence of large- oids near the 3 : 1 Kirkwood gap, analogous scale chaos is difficult to predict with the to the approach used by Chirikov (1979) for resonance overlap criterion or other meth- systems with pendulum-like Hamiltonians. ods. The Hamiltonian (9) is quite similar to This method has had considerable success the H6non-Heiles Hamiltonian in this re- in predicting the behavior of asteroidal mo- spect; it is a bit too complicated to rely on a tion near the Jovian mean-motion commen- simple criterion such as the resonance surabilities. In a similar way, the Ariel- overlap criterion. In order to securely de- Umbriel 5:3 resonance problem can be termine the extent of chaotic behavior, it is approximated by a mapping (see Appendix necessary to resort to numerical methods. II). However, since this problem is not a For a system with two degrees of freedom, restricted three-body problem, it requires a it is possible to study the structure of phase different development. The basic principle space using the Poincar6 surface of section involves changing the form of the high-fre- technique (see H6non and Heiles 1964). quency terms in the Hamiltonian. These The basic idea of the surface of section is to terms are first removed by averaging, and study the intersections of a trajectory with then new high-frequency terms are intro- a two-dimensional plane through the phase duced in such a way that periodic data func- space, rather than the full four-dimensional tions are formed. The resulting equations phase space. These intersections reveal the are "locally integrable" across the delta qualitative character of the trajectories, functions and between them. Integration of i.e., whether they are chaotic or quasipe- the equations of motion now involves only riodic. For quasiperiodic trajectories suc- the evaluation of simple functions. The cessive intersections will fall on smooth main advantage of the algebraic mapping is curves; for chaotic trajectories successive speed of computation. Of course, if is diffi- intersections appear to fill an area in an ir- cult to make meaningful comparisons of regular manner. PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 179 Two sections have been chosen for study rameter space in which these root families in this paper: plotting YA VS XA when xu = 0, may be found. For energies less than zero, which we designate section I, and plotting only two real roots of the quartic equation yu vs xu when XA = 0, which we designate exist. This is the outer pair. For energies section II. On these plots, the radial dis- greater than zero and 8 > -2(C - D) + 4F, tance from the origin is proportional to ec- the inner pair of root families exists for centricity, and the resonance variables tri points on both sections near the origin. are polar angles. The choice of these sec- For energies greater than tion conditions simplifies the determination of initial conditions on the section. 0, 6 < 2(C - D) + 4H, A point on a section defined by a condi- E1 = -(8 - 2(C - D) - 4H)Z/64B, tion such as x; = 0 does not necessarily cor- 6->2(C-D) + 4H, (10) respond to a unique trajectory. It is useful to add further conditions on the section so there are no real roots for points at the ori- that this will be the case. For instance, in gin of sections I. The boundary of the en- the Hrnon-Heiles (1964) problem, the ergy surface on the section divides it into usual surface of section is to plot py versus two separate regions, and O'A must librate y whenever x = 0. Since the Hrnon-Heiles on sections I. In this case libration on the Hamiltonian is quadratic in the momenta, section corresponds to an actual libration of each point on the section corresponds to or A in the full phase space. two possible values ofpx. Points on the sec- Similarly, for energies greater than tion will correspond to unique trajectories if an additional condition, say that px be non- 0, 6 < -2(C - D) + 4F, negative, is added. Our problem is more complicated. The Hamiltonian is quartic in E2 = -(6 + 2(C - D) - 4F)2/64B, the xi and yi. Due to the quartic nature of 8-> -2(C-D) + 4F, (11) the Hamiltonian, for given values of 8, the energy, and the coordinates on the sub- trtj must librate on sections II and in the full space defining the section, there can be two phase space (see Fig. 6). Note that E~ > E2. or four values of yg conjugate to that xi for The most interesting dynamical behavior which the section condition xi = 0 has been occurs for energies near E~ and E2. To more chosen. It is desirable, therefore, to further easily display the results of our calculations constrain the surface of section as belong- we introduce the new parameter AE = % - ing to one of four root "families." When E2, where % is the numerical value of the there are four roots these families are la- Hamiltonian. AE naturally reflects the beled a, b, c, d in order of decreasing nu- types of motion possible on sections. For merical value. When there are only two AE ___ 0, o-tj must librate; for AE > E1 - E2, roots they are labelled a and d in order of both O"n and tru must librate. For 0 > AE > decreasing numerical value. As indicated -E2, the resonance variables may circulate by the nomenclature these two families join or librate, and there are four root families. continuously the corresponding families in For AE < -E2 (equivalently % < 0), there the four root case. Root families a and d are only two root families, and the reso- will be referred to as the "outer" pair and nance variables circulate. This is summa- root families b and c will be referred to rized in Fig. 1. The shaded regions in the as the "inner" pair. The sections for upper left corners of each plot are regions these particular root families will be re- of 8, AE in which there are no real roots for ferred to as Ia, Ib, Ic, Id and lla, lib, IIc, either subspace, and are therefore forbid- and IId. den regions. Inspection of the Hamiltonian (9) allows Far from the resonance, the resonant us to determine regions in the &energy pa- combination of mean longitudes 5Xu - 3hA 180 TITTEMORE AND WISDOM 4e-05 E 1 2e-05 kid E 2 -2e-05 S 2 S 1 -4e-05 -2.0 0 2.0 4.0 6.0 8.0 10.0 8 2e-06 E 1 le-06 <~ E 2 -le-06 S 2 S I -2e-06 -0.4 -0.2 0 0.2 0.4 5 FIG. 1. (a,b) ~, AE parameter space at two scales. Crosses indicate parameters of the surfaces of section shown in this paper. Various types of possible motion: QC, quasiperiodic circulation; QL, quasiperiodic libration; CC, chaotic circulation; CL, chaotic libration. $1 is the energy of the separatrix associated with the Ariel (e2A) resonance, and Sz is the energy of the separatrix associated with the Umbriel (e~) resonance. For energies greater than E2, o-u must librate. For energies greater than E~, both O'u and O'A must librate. PASSAGE THROUGH 5:3 URANIAN RESONANCE 181 circulates rapidly, and the interaction is xA = AAj cos(ajt + /~r) dominated by the secular interaction. The terms in (6) containing the angles 20"A, 2tru, YA = AAr sin(art + fir) and O'A + O'U oscillate rapidly; the angle xv = Aur cos(ajt + fir) O'A -- O'U = --(O3A -- O3U) varies slowly. If a Yu = Aur sin(art +/3j). (14) canonical transformation to new variables which are the sum and difference of trA and If the section condition is, for instance, o-u is made, the combination ZA + ~v is Xu = 0, then art + flj = 7r/2 or 37r/2, and conjugate to the rapidly circulating sum and therefore XA = 0 also. There are two fixed is therefore approximately conserved. Af- points on the section, with :cA = 0 and YA -- ter removing the high-frequency resonant +AAr- The "roots," the values of the coor- terms by averaging, the new Hamiltonian dinate conjugate to the section variable becomes xu(=0) for the fixed points on the section, are Yu -- -+Aur, one root in family a and one 1 root in family d. There are two such fixed ~' = ~ (8 + 2(C - D))(x 2 + y~) points associated with the other eigen- 1 mode. We therefore expect, that for a given + ~ (8 - 2(C - D))(xZu + yZu) section condition, there are four fixed points on the y-axis, two of which belong to (12) + E(xaxu + YaYu) root family a and two to root family d. Thus sections Ia, Id, IIa, and IId each have two which is the linear secular Hamiltonian for fixed points in the linear secular problem. these two satellites in the resonance coordi- Representative surfaces of section for the nates. For a given value of the Hamilto- full 5 : 3 resonance problem before the reso- nian, a point on a surface of section defined nance is encountered are shown in Figs. 2 by x; = 0 can have either of two values of and 3. Figures 2a and b display sections the conjugate coordinate yi, which may be Ia,d, respectively, and Figs. 3a and b dis- found by solving a quadratic equation. The play sections IIa, d, respectively. The same points on the section are made to corre- trajectories are plotted on all figures. As ex- spond to unique trajectories by specifying pected, on each section there are two fixed which of the two root families is being plot- points corresponding to each pure mode, ted. These two root families correspond to each surrounded by concentric invariant the outer pair of roots in the more complete curves. Note that the regions dominated by resonance problem. each mode are not periodic islands associ- This Hamiltonian can be diagonalized ated with a resonance phenomenon, and and normal modes found, yielding the stan- they are not divided by an infinite period dard Lagrange solution of the secular prob- separatrix with an unstable equilibrium. lem (see, e.g., Brouwer and Clemence The position of a curve on the section de- 1961). The solution is of the form pends on the relative strengths of the two eigenmodes. For a particular trajectory Xi = Ail cos(o~lt ~- •1) + Ai2 cos(o~2t -~ •2) points alternately appear on the sections corresponding to the two root families. For Yi = Ail sin(alt + flO + Ai2 sin(o~2t +/~2), instance the points forming the small loop (13) at the top of Fig. 3a and the small loop at where a's are the eigenfrequencies. the bottom of Fig. 3b were generated by the Periodic orbits are found wherever the same trajectory. Similarly, the big loops on amplitude of one of the modes is zero, and Figs. 3a and 3b were generated by the same the eccentricities are constant. In these trajectory. Each section is dominated by cases, the solutions are of the form one of the modes. The other mode appears 182 TITTEMORE AND WISDOM 0.016 I I I I I I I 0.008 t) "4 -0.008 a -0.016 I | I I I I I -0.016 -0.008 0 0.008 0.016 c A COS(CA ) 0.016 I I I I i I I 0.008 ? .g 0 -0.008 \ x, / b -0.016 i i , m l i i -0.016 -0.008 0 0.008 0.016 e, cos(a, ) FIG. 2. Surfaccs of section showing quasipcriodic sccular normal modes, at 8 = 1.4263, AE = -2.1626 × 10 -5. (a) Section la (largest quartic root family). (b) Section Id (smallcst quartic root family). The Modc I fixed point occupics thc central region of both figurcs. The Mode II fixcd point occupies the small regions close to the energy surface boundary, ncar the bottom of section Ia and near the top of section Id. PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 183 0.016 I I I I I I I 0.008 t9 "3 0 O -0.008 -0.016 I I I I | I I -0.016 -0.008 0 0.008 0.016 e u c°S(°u ) 0.016 I I I I I I I 0.008 ? .~, 0 -0.008 b -0.016 i i i , i i i -0.016 -0.008 0 0.008 0.016 e u cos(% ) FIG. 3. Surfaces of section showing quasiperiodic secular normal modes, at 8 = 1.4263, AE = -2.1626 × 10-~. (a) Section IIa (largest quartic root family). (b) Section IId (smallest quartic root family). The curves shown in these figures are generated by the same trajectories as generated the curves in Fig. 2. The Mode II fixed point occupies the central region of both figures. The Mode I fixed point occupies the small regions close to the energy surface boundary, near the top of section Ia and near the bottom of section Id. 184 TITTEMORE AND WISDOM 0.02 , , ~ , 0,02 0.01 0.01 < o" -0.01 -0.01 a b -0.02 i I I i -0.02 i I I I I -0.02 -0.I0 1 0 0.01 0,02 -0.02 -0,01 0 0.01 0.02 e,, cos(o A ) e Acos(o^ ) 0.02 [ 0.02 I / \ / \ 0.01 \ 0,01 / \ / < 0 \ .... / o" -0.01 -0.01 d c -0.02 I I I I i -0.02 i i I I i -0.02 -0.01 0 0.01 0.02 -0.0: -0.01 0 0.01 0.02 e A COS(OA ) e A cos(°;, / FIG. 4. Sections 1 in the region of parameter of space in which large-scale chaotic behavior is possible, at 8 = 2.258, AE = -8.78 x 10-6. Quartic root families are ordered (a)-(d) according to numerical value. Much of the region surrounding the Model II fixed point on sections Ia and Id in Fig. 2 has become chaotic. Escape from the resonance occurs in the large quasiperiodic zone in roots (b) and (c). The fixed point of this large quasiperiodic region becomes Mode I far from the resonance. The small quasiperiodic curve near the bottom of section Ia and the small quasiperiodic curve near the bottom of section Ib are generated by a trajectory for which crA librates. on the sections near the boundary of the The surfaces of section in Figs. 4 and 5 energy surface. Note that even though the show more interesting behavior. The same resonance angle cri may not encircle the ori- trajectories on both sections Ia-d (Fig. 4) gin on the section for a particular trajec- and sections IIa-d (Fig. 5) are shown. For tory, the resonance variable is not librating. these parameters all four root families ex- The apparent libration on the section is sim- ist. All sections for all root families con- ply a matter of one resonance variable be- tinue to show the bimodal structure, but ing strobed by the other. We designate the now the modes are separated by a large mode dominating sections I as Mode I, and chaotic region. The fixed point at the center that dominating sections II as Mode II. of the large quasiperiodic island on sec- PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 185 0.02 0.02 ~\\ 0.01 / \ 0.01 / \ ( "1 -0.01 \ / -0.01 0 \ a -0.02 .lO1 I i i I -0.02 i i i i i i -0,02 -0 0 0.01 0.02 -0.02 -0.01 0 0.01 0.02 e ~cos(% ) e u c°s(°v ) 0.02 0.02 0.01 0.01 "3D 0 -0.01 -0.01 cl -0.02 i i i i i -0.02 i I i i i i -0.02 -0.01 0 0.01 0.02 -0,02 -0.01 0 0.01 0.02 e ucos(% ) e ucos(% ) FIG. 5, Sections II in the region of parameter space in which large-scale chaotic behavior is possible, at ~ = 2.258, AE = -8,78 × 10 -6. Quartic root families are ordered (a)-(d) according to numerical value. The curves shown in these figures are generated by the same trajectories as generated the curves in Fig. 4. Escape from the resonance occurs in the large quasiperiodic zone in roots (b) and (c). The small quasiperiodic curves in the midst of the chaotic zone on sections IIc and lid are generated by the same trajectory that generates the small quasiperiodic curves in Figs, 4a and 4b, for which O'A librates. tions Ia and Id is part of a continuous fam- which trajectories escape from the reso- ily of periodic orbits which becomes Mode I nance. We continue then to refer to the is- far inside the resonance region (6 ~ 0). Sim- lands surrounding these fixed points as ilarly, the fixed point at the center of the Mode I. Trajectories in Mode I on sections large quasiperiodic island on sections Ib Ia and Id alternately visit each of these sec- and Ic is part of a continuous family of peri- tions. In a similar manner, Mode I trajecto- odic orbits which becomes Mode I far out- ries on sections Ib and Ic alternately visit side the resonance region (6 >> 0). The each of the sections. No Mode I trajectory quasiperiodic zone surrounding these latter generates points on all the sections corre- islands is the region of phase space into sponding to all four root families. For cha- 186 TITTEMORE AND WISDOM 0.02 0.02 /" \ / \ 0.01 / \ 0.01 / \ /, 0 -0.01 / -0.01 \ / -0.02 , i i i i i -0.02 i0 I I I I -0.02 -0.01 0 0.01 0.02 -0.02 -0 1 0 O.Ol 0.02 e u cos(% ) e gcos(% ) 0.02 0.02 0.01 o o .~ 0 g • . , . . il •.... -0.01 ...... " " , -0.01 / e -0.02 .tO ...... 0.02 .tO .... -0.02 -0 1 0 0.01 0.02 -0.02 -0 1 0 0.01 0.02 e u cos(% e u cos(% ) FIG. 6. Sections II in the region of parameter space in which large-scale chaotic behavior is possible, at ~ = 2.258, AE = -8.78 x 10-6, computed using differential equations for the averaged planar Hamiltonian. Quartic root families are ordered (a)-(d) according to decreasing numerical value. The curves shown in these figures have the same initial coordinates at the curves in Fig. 5. otic trajectories successive points appear on the figure. Trajectories on these small on the sections corresponding to all four islands librate in O" A and may librate in tru. root families in a typically chaotic manner. Mode I trajectories do not librate. The sur- The small islands in the lower parts of Figs. faces of section in Fig. 5 show the same 4a and 4b (sections Ia and Ib) are alter- features as in Fig. 4, but in Umbriel vari- nately visited by the trajectories which gen- ables (sections II). The large features in erate them. The periodic orbits at the cen- Fig. 5 were generated by the same trajecto- ter of these islands are associated with ries as generated the large features in Fig. Mode II, but in a more complicated way 4. The small islands in both Figs. 4 and 5 than was the case for Mode I (see next sec- were generated by the same trajectories. tion). A similar part of small islands exists The surfaces of section in Fig. 6 show the on sections Ic and Id, but trajectories be- same trajectories as in Fig. 5, but this time longing to these islands were not computed computed using the averaged Hamiltonian PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 187 0.02 0.01 ID -0.01 a -0.02 I I I I I -0.02 -0.01 0 0.01 0.02 e ~ cos(o~ ) 0.02 I I I I I 0.01 -0.01 b -0.02 t I I t I I i -0.02 -0.01 0 0.01 0.02 e ~ cos(o~ ) FIG. 7. Sections II at ~ = 3.5, AE = 0, on the boundary in parameter space above which capture into the resonance occurs. (a) Section IIa (larges quartic root family). (b) Section lid (smallest quartic root family), tr U librates in the full phase space for all trajectories on these plots. All trajectories shown, even chaotic trajectories, have been captured into the resonance. 188 TITTEMORE AND WISDOM (9) with a relative precision of 10 -9 per time confined to one region or the other. At still step. The correspondence between trajec- higher AE, the two regions move apart, and tories in excellent, indicating that the map- the large chaotic zone disappears. The ping represents the Hamiltonian (9) well. forced libration of chaotic trajectories is a The slight difference in appearance of the new feature of this problem. More usually, chaotic zone is expected, since using a dif- the chaotic zone occurs at the boundary be- ferent integrator is equivalent to beginning tween libration and circulation, and thus a with a slightly different initial set of coordi- chaotic trajectory alternately librates and nates. In a chaotic region, two trajectories circulates. This new feature provides a new with slightly different initial coordinates di- mechanism for capture into resonance. Tra- verge exponentially with time. While the jectories may be captured into resonance detailed evolution within the chaotic zone while they are still chaotic. will differ, the qualitative behavior is the Figure 8 shows results of a computation same. In the long run the same region of of the maximum Lyapunov characteristic phase space will be filled. exponent h (see, e.g., Bennettin et al. The surface of section in Fig. 7 is at AE = 1976). The Lyapunov characteristic expo- 0, computed with the mapping. The energy nent provides a measure of the exponential surface has split the section into two inde- divergence of trajectories in chaotic zones. pendent regions, and O-u librates for trajec- Trajectories from two different regions of tories in each region, independently of the parameter space have been represented: quasiperiodic or chaotic character of the one in the large chaotic zone shown in Figs. trajectory. The libration is forced by the to- 4-6 and one in the large chaotic zone in the pology of the energy surface. Successive librating region shown in the lower half of appearances of points on the section are Fig. 7. In both cases, h approaches a value -1.0 -2.0 ] -3.0 0 1.0 2.0 3.0 ~glo(t) FIG. 8. Maximum Lyapunov characteristic exponents for trajectories in the chaotic zones in the previous figures. The unit of time is one year. The upper curve at low t was computed for a trajectory in the chaotic zone in the lower libration region shown in Fig. 7, and the lower curve at low t was computed for a trajectory in the chaotic zone in Figs. 4 and 5. PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 189 ~0.03 year -1. This corresponds to an e- Mode II stable fixed point before the bifur- folding time for the distance function of cation (Fig. 9a) alternate between sections about 30 years, which is comparable to the IIb and IIc, and trA and o-tj circulate. As the secular evolution time scale. parameters evolve toward larger 8 and AE, the region of stability near the Mode II fixed point shrinks, the extent of the cha- 3.2. Onset of Chaos otic zone in phase space increases, and The lightly shaded regions in the lower eventually the Mode II fixed point becomes right of Fig. 1 show the parameter regions unstable (Fig. 9b). After the first bifurca- where "macroscopic" chaotic zones are tion, a stable island appears around the accessible some place on the surface of sec- fixed point (Fig. 9c), but now the points al- tion. The extent of the region was deter- ternate between sections IIa and lib. O'A mined by evaluating the rate of growth of librates for trajectories within these islands. distance between two nearby trajectories A second period-3 bifurcation occurs (Figs. for 20 initial conditions evenly spaced along 9d-9e). Note that the orientation of the the y-axis between the energy boundary roughly triangular trajectory surrounding limits on Sections II for a given 8 and AE--- the stable fixed point has inverted, a char- if one of these was found to indicate chaos, acteristic feature of this type of bifurcation the section is considered to have a "macro- (see, e.g., Hrnon 1966a,b, 1969, 1970). scopic" chaotic zone. Note that the two Again, the points alternate between sec- single resonance separatrix energies, de- tions IIa and IIb, and O'A librates. The tra- noted $1 and $2, run through the middle of jectories generating these islands librate in the chaotic zone (see Appendix III). trh and, as tidal evolution continues, even- The onset of large-scale chaos is associ- tually in trtj around ~-/2 or 3~r/2. (A similar ated with an instability of the Mode II fixed pair of islands appears on sections IIc and point. A tidally evolving system with in- IId.) Note that throughout the sequence of creasing 8 will encounter the region in 8- Figs. 8a-8e the region surrounding the AE parameter space in which large-scale Mode I fixed point remains stable. chaos is present on the surfaces of section Another qualitative change in phase from the lower left of Fig. 1. Near the space concerns the accessibility of the root boundary of this region of large-scale chaos families. Before the bifurcation, trajecto- the fixed point of Mode II goes through two ries generate points alternately on the sec- period-3 bifurcations (see, e.g., Meyer tions corresponding to the "outer" pair of 1970, Hrnon 1970). These bifurcations are root families (a and d) or the "inner" pair associated with a 3:1 resonance between (b and c). As the first bifurcation occurs, the degrees of freedom in the problem. Af- the large chaotic zone appears around the ter these bifurcations the nature of the fixed fixed point on the section corresponding to points has changed, with the result that trA the periodic orbit associated with Mode II, librates in the small quasiperiodic island which has become unstable. Successive surrounding the Mode II fixed point. Before points generated by a chaotic trajectory ap- the bifurcations irA circulated in Mode II. pear on the sections corresponding to all Figure 9 illustrates these bifurcations on four root families. Therefore, the appear- section IIb. The changes in 8 and AE in the ance of the chaotic zone supplies a dynami- series 9a-9e simulate possible variations cal mechanism by which trajectories which during tidal evolution (see next Section). generate points on sections corresponding These surfaces of section therefore illus- to one pair of root families gain access to trate the changes in the structure of the the sections corresponding to the other pair phase space available to a trajectory as it of root families. This is important in the crosses the boundary. Successive appear- mechanism of escape from the resonance, ances of points on the section near the as seen in the next Section. 190 TITTEMORE AND WISDOM 0.008 , , , , , , , 0.008 f ' ' ' ' ' ' 000, r 0.004 % to 0 0 ...... o= -o.oo4! -0.008 I i I I I I I b t I -0.008 -0.004 0 0,004 0.008 -0.008 -0.004 0 0.004 0.008 e u cos(% ) e u cos(o. ) 0,008 0.008 , , , , , , 0.004 0.004 to o= \_ -0.004 -0.004 J f c d -0.008 i i i t t I i -0.008 i , , i i i -0.008 -0.004 0 0.004 0.008 -0.008 -0.004 0.004 0.008 e o cos(% ) e u cos(% ) 0.008, , , , , _.~ , , 0.004 t~ to E 0 r o= £. _3 -0.004 e -0.00 i i L i i , , -0.008 -0.004 0 0.004 0.008 e u cos(% ) FIG. 9. Period-3 bifurcation on section IIb. (a) 8 = 2.066, AE = -9.34 × 10-6; (b) 8 = 2.09, AE = -9.1 × 10 -6, (c) 8 = 2.11, AE = -8.9 × 10 6; (d) 8 = 2.13, AE = -8.7 × 10 6; (e) 8 = 2.15, AE = -8.5 × 10 -6. This corresponds to a 3 : 1 resonance between the degrees of freedom. PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 191 0.0012 I I ! I ~ I I I I I .:..;,.;:. '!%l 0.0008 bll!, L\/r 0.0004 i? r,,: ..'~..., s if.:! ;.~;:.. 0 t 0 --0.0004 2i~?,: -0.0008 a -0.0012 ' I I I I I I -0.0012 -0.0008 -0.0004 0 0.0004 0.0008 0.0012 e o cos(o U ) 0.0012 , , I I I ~ I I I I I • ~ i~ i[~ F- 0.0008 ! - ! 'I! • ! I -': i t" 0.0004 ;) f: 0 O -0.0004 -0.0008 7[0/I • an,', b -0.0012 ' ' I I I I I I -0.0012 -0.0008 -0.0004 0 0.0004 0.0008 0.0012 e u cos(o~ ) FIG. 10. Sections II for low eccentricities, ~ = -0.32, AE = -1.0 x 10 -9. (a) Section IIa (largest quartic root family). (b) Section lid (smallest quartic root family). The chaotic zone is characteristic of a perturbed separatrix. 192 TITTEMORE AND WISDOM For very low eccentricities, the coupling where between resonances is weakened, and one [ lhi ~i1 might expect to observe behavior charac- ~/-~Xi -Sni + 2~ . (17) teristic of a perturbed single resonance problem. Figure 10 illustrates this for val- Tidal friction results in a secular varia- ues of 8 and AE near those where the Um- tion of the mean motions (e.g., Goldreich briel resonance is first encountered in the 1965) single resonance approximation (see Ap- 9 m, (R]' 1 pendix III). There is a relatively narrow ti i = -- "~ k2 n2 ~ \~/// ~ (18) chaotic zone with the characteristic "figure 8" shape of the separatrix in the single res- and a damping of the orbital eccentricities onance case. (Goldreich 1963) 4. EVOLUTION THROUGH THE RESONANCE 21 M(R)Sei ki- 2 k2ini- -- 4.1. Tidal Dissipation mi ~ Qi 57 mi (Ri] 5 _~, (19) The results of the previous section illus- d- -~ k2ni --~ \ai/ trate the behavior of the Hamiltonian (9) for fixed values of the parameters 8 and AE. where k2 and Q are, respectively, the po- This section discusses the tidal evolution tential Love number and specific dissipa- through the resonance. Tidal friction enters tion function for Uranus, and the kzi's and in two distinct ways. The principal modifi- Oi's are the same quantities for the satel- cation of the model is that the coefficients lites. The first term in the expression for ~;i of the Hamiltonian become time depen- is the contribution due to tidal dissipation in dent. This is primarily due to the differen- the satellite, and the second term is due to tial tidal expansion of the orbits. There is tidal dissipation in the planet. The contribu- also the direct action of tidal friction in the tion due to tidal dissipation in the satellites satellites on the degrees of freedom. Thus dominates the contribution due to tidal dis- tidal friction gives rise to time-dependent sipation in the planet by approximately a coefficients as well as explicit friction terms factor of 100. Using numerical values from in the equations of motion. Table I with Q = 6600 to evaluate relative Of the various coefficients in the reso- contributions to ~, nance Hamiltonian, only 8 depends strongly on the resonant combination of 5hu 5mu (aA] 8 mean motions; during passage through the 3ri----~ -- 3m~ \a---u/ "~ 0. I (20) resonance, the fractional change in the pa- so the orbit of Ariel expands relative to that rameter 8 is large compared to the frac- of Umbriel. The ratio of the eccentricity de- tional changes in the other coefficients. The pendent terms is other coefficients will therefore be taken to be constant. In addition, the value of 8 is ~u 0.1 ev. (21) influenced by the damping of eccentricities ~A eA due to tidal dissipation in the satellites. The The most significant terms in (16) are there- time rate of change of 8 is fore = 4A. (15) = --3hA- 16B~a (22) From the expansion of A to first order in I~ and the relative contributions from these d terms are expressed t$ = ~ [(5no -- 3hA) -- 16B(~A + ~U)] 16B~__.___~A ~ --1.5 x 103e~. (23) = (Stiu -- 3tiA) -- 16B(~A + ~U), (16) 3tiA PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 193 For the current eccentricity of Ariel, the of formation of the Solar System. This above expression -~-0.004, so the hA term study is only relevant if the Q of Uranus is is clearly dominant in the expression for 8. less than about 100,000, since this Q places The ~A term becomes comparable to the hA Ariel and Umbriel at the 5:3 resonance at term only for eccentricities of approxi- the time of formation of the Solar System. mately 0.025. Since we are studying the Higher values of Q are possible, but then evolution for systems with initial eccentric- Ariel and Umbriel would never have en- ities much less than this, we have only re- countered the 5 : 3 resonance. tained the hA term in the expression for 8. Whether or not ki is important depends However, if large increases in the eccen- on Q. The maximum time of resonance pas- tricity result from resonant interactions be- sage would occur for the maximum Q = tween the satellites, this approximation will 100,000. We have assumed that during the require some reconsideration. In our ap- passage through resonance 8 varies linearly proximation, the expression for ~ is with time due to the decrease of nA. The change in 8 during resonance passage is of ~- 9GZMZm3 ~iA (24) order 1-3 year -1. This indicates that the 2F 3 aA" maximum time of resonance passage is less Tidal friction also introduces explicit fric- than approximately 3.1 × 108 years. As dis- tional terms into the equations of motion. cussed above, the tidal damping time scales The most important frictional terms arise of the satellites are approximately 108 years from the damping of the orbital eccentrici- for Ariel and 109 years for Umbriel. These ties from dissipation within the satellites. time scales are comparable to the maximum From Eq. (19), with rigidities /XA = /zU time of resonance passage. For the maxi- 10 N dynes/cm2 (see Table I), the eccentric- mum value of Q, then, direct tidal damping ity damping time scale for Ariel is about of eccentricities will be important. The rate 1.5 × 106QA years, and that for Umbriel is of evolution is inversely proportional to the about 1.4 x 107Qu years. For QA -~ Qu value of Q assumed for Uranus. Therefore, 100, these time scales are on the order of for values of Q near the minimum allowed 108 and 109 years, respectively. Whether Oi Q, the time scale of resonance passage will is important in our study depends on the be much shorter (less than ~ 107 years), and time scale of resonance passage compared damping of eccentricities should have a to the eccentricity damping time scale. The much smaller effect on the dynamics of the time scale of resonance passage depends in system. The direct tidal damping of ei could turn on the rate of orbital evolution, which be important, but we neglect it in this initial is determined by the specific dissipation study. function Q of Uranus, as discussed below. The principal effect of tidal friction is that The value of Q can be constrained by the parameter 8 varies secularly. The rate considering reasonable scenarios for the or- of secular variation of 8 depends primarily bital evolution of the satellites of Uranus on the unknown value of the Q of Uranus (Peale 1988). The rate at which the system (see Eqs. (18)-(24)). evolves is inversely proportional to Q. By The numerical study of this system is extrapolating the evolution backward in constrained by finite computer resources. time, and eliminating unreasonable configu- Unfortunately, even the lower limit of Q rations of the satellites from the present to discussed above requires enormous expen- the time of formation of the Solar System, it ditures of computer time to make a satisfac- is possible to find upper and lower limits to tory study of the evolution of trajectories, the possible values Q can take. A lower even with the algebraic mapping. It is nec- limit Q ~ 6600 places Miranda and Ariel at essary, therefore, to artificially increase the the same distance from Uranus at the time rate of tidal evolution of the system above 194 TITTEMORE AND WISDOM the maximum limit imposed by the dy- resonance and hope that it is adequately namics of the Uranian satellite system. In slow. The burden of showing that the the single, integrable resonance model of results of numerical experiments of passage passage through resonance the outcome of through resonance reflect the true dy- the resonance encounter does not depend namics rests with the experimenter. As we on the rate of passage through the reso- illustrate below, the rate of passage through nance as long as the passage is sufficiently the 5 : 3 Ariel-Umbriel resonance must be slow. The single resonance formulas only extremely slow in order to avoid artifacts. require that the fractional change in the li- We parameterize the rate of evolution in bration frequency over a libration period be our numerical model by the effective spe- much less than unity. For the problem un- cific dissipation function of Uranus. We der consideration here, where the dynamics designate this effective specific dissipation and mechanism of passage through reso- function by the script 9~, in order to empha- nance are different than those of the single size that it is the parameter in our numerical resonance theory, it is no longer clear that model and not the physical Q of Uranus. the outcome of resonance passage will be- With limited computer resources we must come independent of the rate of passage for increase the rate of evolution determined sufficiently slow passage. It is however by 9~ as much as possible without affecting plausible that this should be the case, when- the outcome of the resonance passage. In ever the time scale for resonance passage order to draw conclusions about the physi- becomes much greater than all dynamical cal system we must show that the dynamics time scales. In any case, in numerical stud- of passage through resonance become inde- ies of passage through complicated reso- pendent of 9~ for sufficiently large 9~. nances it is not satisfactory to simply To illustrate the effect of ~ on the dy- choose an arbitrary rate of passage through namics of the full resonant problem, Fig. 11 1.0 l i 0.8 0.6 P 0.4 0.2 0 i i I I I le 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 log ]o (6600 / Q) FIG. 11. Measured probability of escape versus tidal dissipation rate for the full resonance problem, in terms of the effective specific dissipation function of Uranus ~. Circles, f~ = 20 year-~; diamond, 1) = 80 year ~; square, differential equations for analytically averaged Hamiltonian. PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 195 displays the probability of escape from the facts. This is even more evident when the resonance against dissipation rate. The energy evolution of the trajectories is com- probability of escape was determined by in- pared: the trajectories in Fig. 12 have the tegrating sets of 100 trajectories through the same initial conditions as those in Fig. 15 resonance with different dissipation rates (see next subsection), but the dissipation (see Section 4.2 for more details). The rate rate is 1000 times higher (t~Aa A ~ 1.4 × 10 -8 of dissipation increases to the right in the per orbit period). The faster evolution gives figure, and the value of Q decreases. The a more exciting picture of the evolution minimum physically reasonable value of through resonance, with large increases in Q = 6600 corresponds to the minimum dis- eccentricity, but it is wrong. sipation rate shown. The choice of ~ appar- ently can have a significant effect on the 4.2. Numerical Runs results: there is a dependence of escape Figures 13 through 17 show results of nu- probability on rate, especially for high dissi- merical runs in which, for each of five initial pation rates (small ~). Most of the runs de- values of 8 and AE, 100 trajectories were scribed in this paper have used 9~ = 110, or evolved through the resonance using the 60 times the minimum physical dissipation mapping (see Appendix II) with slow tidal rate. This is the minimum dissipation rate evolution (9~ = 1 I0). The initial values of 8 for which a data point is shown in Fig. 11. and AE, and the physical parameters corre- This value of 9~ corresponds to aA/aA sponding to these values, are given in Table 1.4 × 10 -11 per orbit period during passage II. The 100 initial points for Run 3 were on through the resonance. For somewhat an invariant curve of sections II. For the larger dissipation rates, the measured es- other runs each set of 100 initial points was cape probability is similar to that for ~ = obtained by mapping the trajectory forward l l0, indicating that changes in 9. near this in time 7 (Runs 1, 2, and 4) or 10 (Run 5) value do not greatly affect the dynamics. It mapping iterations from the initial point is plausible that the numerical experiments with no tidal dissipation, in order to thor- at this 9~ reflect the dynamics at the physical oughly mix the phases or n and cry, but keep Q. Limitations on available computer time the "actions" the same. Since the trajecto- have not allowed us to further verify this ries are initially quasiperiodic there is, in numerically. However, trajectories evolved principle, a transformation to action-angle through the resonance with 9~ = 3300 do variables. Under slow variation of parame- show features similar to these with 9~ = 110 ters these actions are to first order con- (see next section). As a further check of the served; the actions are first-order adiabatic mapping, we have computed runs with a invariants. For slightly different initial con- higher mapping frequency ~ = 80 year -1 ditions the dynamical frequencies are (diamond in Fig. l l) and with differential slightly different, and the initial angles will equations of the averaged Hamiltonian with rapidly be spread over all values. (Motion a relative precision of 10 -9 per time step on the n-torus is ergodic.) Thus the essen- (square) at 9~ = 6.6. These have similar es- tial parameters which determine the proba- cape probabilities to those runs with fl = 20 bilities of various outcomes are the actions; year -1 (circles in Fig. 11). the angles are physically unknowable and Figure 11 emphasizes that for much considered to be uniformly distributed in higher dissipation rates, the dynamics are the calculation of probabilities. In the single strongly affected by the choice of 9~. It is resonance theory the phase corresponding apparent that a run with ~ = 3.3, or aA/aA to these angles is also assumed to be uni- 4.8 × 10 -1° per orbit period, will produce formly distributed in the calculation of results significantly different from the phys- probabilities. ical system, i.e., there are significant arti- The trajectories in 8, AE parameter space 196 TITTEMORE AND WISDOM 4e-05 2e-05 0 -2e-05 -4e-05 -2.0 0 2.0 4.0 6.0 8.0 10.0 5 FIG. 12. Trajectories evolved through the resonance with a high tidal dissipation rate (~ = 0.11). Solid lines indicate quasiperiodic behavior, dashed lines indicate chaotic behavior. The types of behavior possible for each orbit may be determined by comparing this figure with Fig. la. Evolution of these trajectories is strongly affected by the rate of tidal dissipation: the extremely wide range of outcomes is an artifact of evolving the system too rapidly through resonance. These trajectories have the same initial conditions as the trajectories in Run 3 (Fig. 15), but the dissipation rate is 1000 times higher. This picture is more exciting than that presented in Fig. 15, but this picture is wrong. are represented by solid lines when the evo- the distance between nearby trajectories lution is quasiperiodic and by dashed lines (Lyapunov exponent) at discrete intervals when the evolution is chaotic, as deter- in 8 along the trajectories. mined by evaluation of the rate of growth of The trajectories appear to have nearly TABLE II INITIAL PARAMETERS Run AE 8 aA eA O'A XA YA au eu O'u Xu Yu 1 -9.4511 x 10 7 -0.1781 7.1633 0.0017 270.0 0.0 -1.4649 x 10 3 10.0703 0.0043 270.0 0.0 -3.9289 x 10 -3 2 -5.6281 x 10 -6 -0.1869 7.1625 0.005 90.0 0.0 4.3084 x 10 3 10.0703 0.005 90.0 0.0 4.5685 × 10 3 3 -1.9623 x 10 -5 -0.0283 7.1611 0.0102 90.0 0.0 8.7907 x 10 3 10.0703 0.0031 45.0 0.002 0.002 4 -2.1626 × 10 -s 1.4263 7.1610 0.01 0.0 8.6168 × 10 -3 0.0 10.0703 0.01 0.0 9.1371 × 10 -3 0.0 5 -6.0706 × 10 -s 2.0000 7.1588 0.0125 0.0 1.0771 × 10 2 0.0 10.0703 0.0125 0.0 1.1421 x 10 2 0.0 PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 197 2e-06 le-06 <~ -le-06 -2e-06 -0.4 -0.2 0 0.2 0.4 FIr. 13. Trajectories in 8,AE parameter space for Run 1 with ~ = 110. Solid lines indicate quasipe- riodic behavior, dashed lines indicate chaotic behavior. The types of behavior possible for each orbit may be determined by comparing this figure with Fig. lb. None of the trajectories in this set escaped from the resonance. 4e-05 2e-05 0 -2e-05 -4e-05 -2.0 0 2.0 4.0 6.0 8.0 10.0 FIG. 14. Trajectories in ~,~E parameter space for Run 2 with ~ = 110. Solid lines indicate quasipe- riodic behavior, dashed lines indicate chaotic behavior. The types of behavior possible for each orbit may be determined by comparing this figure with Fig. la. Only one of these trajectories escaped from the resonance (see Table III), closely following the curve of E = 0.0. 4e-05 2e-05 t13 <~ 0 -2e-05 -4e-05 -2.0 0 2.0 4.0 6.0 8.0 10.0 5 FIG. 15. Trajectories in 8, AE parameter space for Run 3 with 9~ = 110. Solid lines indicate quasipe- riodic behavior, dashed lines indicate chaotic behavior. The types of behavior possible for each orbit may be determined by comparing this figure with Fig. la. Of the 100 trajectories, 29 escaped from the resonance. 4e-05 2e-05 0 -2e-05 -4e-05 -2.0 0 2.0 4.0 6.0 8.0 10.0 8 FIG. 16. Trajectories in 8,AE parameter space for Run 4 with 9~ = 110. Solid lines indicate quasipe- riodic behavior, dashed lines indicate chaotic behavior. The types of behavior possible for each orbit may be determined by comparing this figure with Fig. la. All of the trajectories which did not escape by the end of the run displayed were eventually captured into the resonance. Of the 100 trajectories, 28 escaped from the resonance. The trajectories which appear linear and quasiperiodic between 8 ~ 2.5 and 8 ~ 4.5 are temporarily captured into a quasiperiodic region of phase space in which O'A librates. 198 4e-05 2e-05 0 i i -2e-05 -4e-05 I I I I I -2.0 0 2.0 4.0 6.0 8.0 10.0 8 FIG. 17. Trajectories in 8,AE parameter space for Run 5 with ~ = 110. Solid lines indicate quasipe- riodic behavior, dashed lines indicate chaotic behavior. The types of behavior possible for each orbit may be determined by comparing this figure with Fig. la. All of the trajectories which did not escape by the end of the run displayed were eventually captured into the resonance. Of the 100 trajectories, 39 escaped from the resonance. 4e-05 2e-05 -2e-05 -4e-05 -2.0 0 2.0 4.0 6.0 8.0 10.0 8 FIG. 18. This plot shows the evolution in ~, AE parameter space of selected trajectories from Runs 2 to 5 as they evolve through the 5 : 3 mean-motion commensurability. The corresponding behavior of the eccentricities of Ariel and Umbriel for these trajectories during resonance passage can be seen in Figs. 19-22. 199 200 TITTEMORE AND WISDOM the same evolution in AE until they first eccentricity of Ariel is relatively high while become chaotic. This can be understood in the mean eccentricity of Umbriel is rela- terms of the first-order adiabatic invariance tively low. There are relatively sudden of the actions of the trajectories while they changes in the eccentricities at the bound- are quasiperiodic. The paths begin to di- aries of this range in 8. Outside of this range verge in the chaotic region. of 8, the mean eccentricity of Ariel is rela- Sample trajectories from each of Runs 2- tively low, while the mean eccentricity of 5 have been plotted in Fig. 18. Figure 19-22 Umbriel is relatively high. This behavior is show the eccentricity variations of Ariel typical for trajectories in the chaotic zone: and Umbriel for these trajectories. The there are time intervals during which the maximum and minimum eccentricities in mean eccentricity of Ariel is relatively high small intervals of 8 (Fig. 19:A8 ~ 0.0007, while that of Umbriel is relatively low, dur- Fig. 20:A8 ~ 0.001, Fig. 21:A8 ~- 0.003, ing other time intervals the mean eccentric- and Fig. 22: A~ ~ 0.002) are plotted versus ity of Umbriel is high while that of Ariel is 8. These "envelopes" are smooth when the low, with relatively sudden transitions be- trajectory is quasiperiodic, but the maxima tween these intervals. This behavior results and minima vary in an irregular fashion in in the erratic alternations between high and the chaotic zone. In each case, the maxi- low maximum eccentricity seen in Figs. 19- mum eccentricity in the chaotic zone is 23. higher than the initial eccentricity. Note Since the tidal dissipation is small, the that in some cases, the average eccentricity change in energy over a short time interval of Umbriel after escape from the resonance is also small, and the system can be consid- is higher than the average eccentricity be- ered approximately Hamiltonian. It is pos- fore the resonance is encountered (e.g., sible to study the qualitative behavior of a Fig. 20b), a phenomenon not possible in the trajectory by "freezing" the value of 8 and standard theory of evolution through iso- computing surfaces of section. This allows lated mean-motion resonances us to determine which regions of phase Figure 23 shows the eccentricity varia- space are accessible to a trajectory at par- tions of a trajectory from Run 5 integrated ticular values of 8 and AE, and the types with 9~ = 3300, which is close to the maxi- of behavior that are possible in each re- mum dynamically allowed tidal dissipation gion. rate in Uranus. The plot was produced in As discussed in Section 4.1, for chaotic the same manner as Figs. 19-22, showing trajectories points appear in an irregular the maximum and minimum eccentricities manner on the sections corresponding to all in intervals of A8 ~ 0.003. The decrease of four root families. The trajectories are ob- effective dissipation rate by a factor of 30 served to spend some time with low values does not dramatically affect the dy- of both eu and eA, during which time points namics-features which are seen in this fig- appear on sections Ib, Ic, IIb, and IIc. Tra- ure are similar to those seen in the previous jectories may jump back and forth between figures. This trajectory was captured into regions near to and far from the origin, dur- the resonance. ing which time points appear on all sections Figure 24 shows the chaotic variations in corresponding to all four root families. eccentricity of part of the trajectory shown They may also spend time with high values in Fig. 22, with the scale of 8 greatly en- of both eu and eA, during which time points larged. The eccentricities of the satellites appear on sections Ia, Id, IIa, and IId. vary in an irregular manner. This behavior During the periods of time when points is similar to that observed for asteroid mo- appear on sections corresponding only to tion in the chaotic zone associated with the either the outer root families or the inner 3:1 Kirkwood gap (Wisdom 1982, 1983). root families, there is a correlation between Note that for 6.008 < ~ < 6.0t7 the mean the eccentricities of Ariel and Umbriel. At PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 201 0.01 , , , , 0.0075 e A 0.005 0.0025 0 , ~T ~ I ' : ...... 0 0.5 1.0 1.5 2.0 0.01 I I I 0.0075 cu 0.005 0.0025 b 0 I , I I I 0 0.5 1.0 1.5 2.0 FIG. 19. Eccentricity variations of the trajectory from Run 2 which escaped from the resonance. Shown are the maximum and minimum eccentricities of Ariel (a) and Umbriel (b) in intervals of A6 0.0007. 202 TITTEMORE AND WISDOM 0.015 0.01 CA 0.005 I ", -I I 0 l.O 2.O 3.O 0.015 0.01 e U 0.005 ". ';?i :! i" !' b I I 0 1.0 2.0 3.0 FIG. 20. Eccentricity variations of a trajectory from Run 3 which escapes from the resonance. Shown are the maximum and minimum eccentricities of Ariel (a) and Umbriel (b) in intervals of 46 0.001. In this example, the average eccentricity of Umbriel after escape from the resonance is higher than the average value before the resonance is encountered. PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 203 0.03 , , , , 0.02 .,~;".'....,.:.:' ! ..... eA 0.01 • . , - . . :: • . • a ,:; I I ~ ,]~ ','~..-~-: ,: ~q ~. - " \ ---"- - 2.0 4.0 6.0 8.0 10.0 0.03 ¢.,. :. :... 0.02 ,~ ~: :i :11 .,, : : , :. :: e U i. 0.01 i . 2.0 4.0 6.0 8,0 10.0 FIG. 21. Eccentricity variations of a trajectory from Run 4 which is temporarily captured into a resonant island. Shown are the maximum and minimum eccentricities of Ariel (a) and Umbriel (b) in intervals of A6 -~ 0.003. Between 8 = 2.5 and 8 ~ 4.5, the eccentricity of Ariel increases during quasiperiodic libration of trg. After entering the chaotic region again, the trajectory is ultimately captured into the resonance. 204 TITTEMORE AND WISDOM 0.03 0.02 e^ 0.01 0 2.0 4.0 6.0 8.0 8 0.03 . t," ~" ! ~ ~Y" .~-,. ,. . • 0.02 &~-., ..:., : ~. ,~ .. e U 0.01 J=;5ii /:il,, i,: L,i;i 2.0 4.0 6.0 8.0 8 FIG. 22. Eccentricity variations of a trajectory from Run 5. Shown are the maximum and minimum eccentricities of Ariel (a) and Umbriel (b) in intervals of A8 ~ 0.002. For this trajectory, the average eccentricity of Umbriel after escape from the resonance is also higher than the average value before the resonance is encountered. PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 205 0.03 , , , 0.02 J J f :• ! 0.01 • . • , ++•. ++.'• • .... + + ,,.: a ~I ::,.. v:. +~ . :... • +. /....% 0 2.0 4.0 6.0 8.0 I0.0 0.03 0.02 ¢ J C o 0.01 •. '? 2.0 4.0 6.0 8.0 10.0 8 FIG. 23. Eccentricity variations of a trajectory with the same initial conditions as the trajectory shown in Fig. 22, but with the tidal dissipation rate 30 times smaller. Shown are the maximum and minimum eccentricities of Ariel (a) and Umbfiel (b) in intervals of AS ~ 0.003. Between 8 ~ 4 and 8 4.4, O'A was captured into temporary quasiperiodic libration. This trajectory is ultimately captured into the resonance. 206 TITTEMORE AND WISDOM 0.03 0.02 e A 0.01 a 0 I ~ I I 6.0 6.005 6.01 6.015 8 0.03 I I I 0.02 e U I 0.01 b 0 t m I 6.0 6.005 6.01 6.015 8 FIG. 24. The eccentricity variations of Ariel (a) and Umbriel (b) in the chaotic zone for the trajectory shown in Fig. 22 are shown at a greatly expanded scale. The time resolution of this plot is one mapping period (At = 2~r/20 years). The eccentricities of both satellites vary in an irregular manner. At ~ 6.008, the average eccentricity of Ariel increases suddenly, while the average eccentricity of Umbriel decreases. At 8 ~ 6.017, the average eccentricity of Ariel decreases suddenly, while the average eccentricity of Umbriel increases. PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 207 such times, a relatively high eccentricity for suiting in maximum eccentricities more like Ariel requires a relatively low eccentricity those of the system shown in Fig. 22, which for Umbriel, and vice versa. This results in has higher initial eccentricities. The energy the interesting behavior of the eccentrici- evolution of these two trajectories in the ties displayed in Fig. 24: over relatively chaotic zone further indicates the similar- short time scales, the mean eccentricity of ity. This particular trajectory is eventually one satellite is high while that of the other captured, but examination of the trajectory satellite is low. The eccentricities alternate in Fig. 18 indicates that escape is possible between high and low mean values in an for similar trajectories. The trajectory from irregular manner. Eccentricities tend to be Run 5 in Fig. 18, which escapes to large higher when points are appearing on the negative AE, overlaps the trajectory from outer root sections. Run 4, indicating a period in which evolu- While the trajectory is chaotic, AE tion through the chaotic zone occurred in evolves in a surprisingly regular manner to the same region of phase space. This mech- higher values with increasing 8, provided anism of temporary quasiperiodic libration that the rate of tidal evolution is slow. Qual- allows for the possibility that trajectories itative changes in the evolution of AE have approaching the resonance with relatively been associated with temporary capture low eccentricities could be driven to much onto the librating island surrounding the higher eccentricities than they would with Mode II fixed point, or with the trajectory the evolution in the chaotic zone alone, and "sticking" to a quasiperiodic island for a the satellites might escape from the reso- period of time. nance with higher eccentricities than they Figure 21 shows some of this interesting had approaching the resonance. The trajec- behavior: after becoming chaotic, the tra- tory shown in Fig. 23 also shows this phe- jectory apparently becomes quasiperiodic nomenon of temporary capture. again, during which time the eccentricity of Trajectories may escape from the reso- Ariel increases considerably. It then be- nance almost immediately, or may spend a comes chaotic again, exhibiting eccentric- considerable period of time in the chaotic ity behavior similar to that of the other tra- zone before they escape, during which time jectories plotted. By plotting surfaces of the maximum eccentricities can be driven section along this "quasiperiodic inter- to relatively high values. lude," it was found that the trajectory was Eventually, the trajectories become qua- on one of the islands in the chaotic zone siperiodic again. The final states of the tra- and on which trA librates in the full phase jectories fall into two major categories: tra- space. The trajectory became unstable jectories which escape from the resonance again at a later time, possibly due to a and trajectories which are captured into change in the size of the island. It may also resonance. For trajectories which are cap- have been a case of "sticking," where the tured AE increases and ends up in the upper trajectory spent time near an island, and half of the parameter space; for trajectories exhibited behavior which only appeared to which escape AE decreases to large nega- be quasiperiodic (e.g., Wisdom 1983). The tive values. fact that two of the trajectories in Fig. 16 Those trajectories which evolve to large which exhibit this behavior become unsta- negative AE escape into the large quasipe- ble at similar values of 8 indicates that the riodic region on the sections corresponding phenomenon is not extremely sensitive to to root families b and c, in which both reso- initial conditions, suggesting that the trajec- nant arguments circulate. This requires that tory actually was captured onto the quasi- both eccentricities have low mean values in periodic island and not stuck to it. Tempo- the chaotic zone (with points appearing rary quasiperiodic libration has served as a only on sections Ib,c and IIb,c) for a period conduit to higher eccentricity for Ariel, re- of time long enough to allow evolution into 208 TITTEMORE AND WISDOM this quasiperiodic zone. Then, as 8 evolves is inherently unpredictable (though still de- to higher values, the system moves away terministic), due to the extreme sensitivity from the resonance region, and the dy- of the motion to initial conditions. Two tra- namics of the system are dominated by the jectories entering the chaotic zone with secular interactions between the satellites, even very similar phases will diverge expo- as they were before the system encoun- nentially, making it impossible to predict tered the commensurability. their comparative evolutions over the long Examination of section II for AE _> 0 term. The width of the chaotic zone when a (see Fig. 7) indicates that capture into reso- trajectory enters it may be important. The nance occurs when the two regions of the probability of escape may be much larger section defined by the energy surface for a trajectory if the chaotic zone is narrow boundary separate. Trajectories appear to and the quasiperiodic region of escape on be confined to one region of phase space or sections corresponding to root families b the other, and o-u must librate. We there- and c occupies a majority of the phase fore have capture into resonance through space. This is inferred from observations evolution of the energy surface. There is a that most escapes occur very shortly after relatively small region of parameter space the trajectories become chaotic, when the for AE just larger than zero where the tra- chaotic zone occupies a relatively small re- jectory can still be chaotic (see Fig. 1). A gion of phase space. trajectory can therefore be captured into The trajectories in each of the runs de- resonance, yet still be chaotic. This phe- scribed so far in this paper originally all nomenon is a novel feature of our problem. have the same actions. The actions depend As the system continues to evolve, the two on the positions of the invariant curves on a "bubbles" move apart on sections II, and section. There appears to be a dependence the eccentricity of Umbriel increases. In of the probability of escape on the initial addition, the large chaotic zone disappears, distribution of energy between the normal and trajectories become quasiperiodic, in- modes, and therefore on the actions. Figure dicating permanent capture into the reso- 25 shows the energy evolution of trajecto- nance involving the longitude of pericenter ries with the same 8's and z~E's as the five of Umbriel's orbit. If further evolution of runs in Fig. 13-17, but the trajectory on the trajectory results in AE > E1 - E2, the which the initial points are computed gener- energy surface also divides Sections I into ates invariant curves on sections I and II two regions in which O'A must librate (see which are now very close to the fixed point Section 3.1). As the system continues to associated with Mode I. Ten trajectories evolve, these two "bubbles" also move were integrated for each run. For these tra- apart on sections I, and the eccentricity of jectories, the evolution of energy is mark- Ariel increases. The orbits of both satellites edly different from the evolution of the cor- are trapped into quasiperiodic libration. responding trajectories in Figs. 13-17: the The island surrounding the Mode II fixed trajectories all evolve to much higher val- point could provide another mechanism of ues of AE before becoming chaotic. All of capture, although the ultimate stability of the trajectories were captured into reso- the island is unknown. nance. The details of the secular interac- The observed spread in outcomes con- tions between the satellites before the reso- trasts with the behavior of trajectories in nance is encountered have a significant the single resonance approximation (see, effect on the outcome of the passage e.g., Borderies and Goldreich 1984), in through the resonance. which the outcomes are probabilistic due to an unknown phase, but are otherwise quite 4.3. Statistics of Escape predictable. The presence of a chaotic zone Overall escape statistics are given in Ta- in this problem indicates that the outcome ble III. There is a trend toward higher prob- PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 209 4e-05 liii:.:::'. ~:~:i:.::!:~~:~:~:~:~ :~:".':'~.*..':.~" -:~::~:`.~:.``~:.:.~.~:..~::::::::~.~`..:~.:..~.`~.%%%~x~:.`.~..::::.`..::~.~.~..x~.~;~...... ~..~.~.~¢~...... `..~,...~$;~.~.:.~¢.~.~.`.`*$~`.~.~`....`..~.~..~i~`.:.~e:.`:.~ x '~- .,~e, :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ~':~ 4~::" ~.::'.'i~.,'.:.~.,¢~:~'~4:.:~,~'.::':'~¢'x~:~:i.,:x:~ :e'.~-.':~.~':~'~¢: %~,.~¢~-:.x.~ .... .,,:.~.. :.:.,~.~.::.~,:~,:~.:.~ ~.~., ...... ~@~!*':~'::'..,~j ": .. ~i !g"~'t ~!~.'."~:::-'-':~i ~ 2e-05 :...~ <] 0 -2e-05 -4e-05 ~ , , ~ ~ , J :K ::::t ~ ...... -2.0 0 2.0 4.0 6.0 8.0 10.0 8 FIo. 25. Trajectories in &AE parameter space with initial coordinates near the Mode I fixed point. The initial values of 6 and AE for each set of 10 trajectories are the same as the initial values of 6 and AE for the corresponding sets of trajectories in Runs 1-5 (Figs. 13-17). Note that the trajectories become chaotic much closer to AE = 0 than the corresponding trajectories in Figs. 13-17. None of the trajectories shown escaped from the resonance. TABLE III ESCAPE STATISTICS Run 6o (CA0)a (rest) (¢amax) Pesob 6j {eat} (¢U0) (eUma×) (eUl) +0.00066 2 -0.187 0.00493 0.412 0.0078 0.01 3.000 0.00034 -0.00108 +0.00120 0.00503 0.0074 0.00123 -0.00073 +0.00064 3 -0.028 0.00960 1.182 ± 0.301 0.01170 ± 0.00052 0.29 + 0.046 10.000 0.00345 + 0.00101 -0.00110 +0.00172 0.00417 0.01071 ± 0.00143 0.00365 +- 0.00220 0.00117 +0.00168 4 1.426 0.00837 2.841 ± 0.463 0.01546 ± 0.00139 0.28 ± 0.036 8.114 0.00378 + 0.00073 0.00288 +0.00218 0.01116 0.01577 + 0.00100 0.00937 +- 0.00161 0.00122 +0.00215 5 2.000 0.01037 4.757 -+ 0.812 0.01916 + 0.00216 0.39 -+ 0.048 10.000 0.00541 ± 0.00092 -0.00315 +0.00223 0.01401 0.01953 + 0.00136 0.01285 + 0.00210 -0.00153 Time average and range of eccentricity on initial trajectory. b Error for Peso is standard deviation of the mean. All other standard deviations for single value. 210 TITTEMORE AND WISDOM ability of escape from the resonance as the mean of the time-averaged final eccentrici- mean eccentricities of the satellites before ties for each satellite after escape from the the resonance is encountered are increased. resonance (last column in Table III) is com- For both satellites, the maximum eccen- parable to or lower than the mean initial tricities in the chaotic zone are higher than eccentricity (third column). Figure 26 the initial maximum eccentricities. The shows the distribution of time-averaged fi- 14.0 14.0 t b 12.0 12.0 ! 10.0 10.0[ I 8.0 8.0 N oo! l 6.0 4.0 4.0 2.0 2.0 0 0 i i i 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 CA CO 14.0 14.0 12.0 12.0 10.0 10.0 I 8.0 8.0 N 6.0 6.0 4.0 4.0 2.0 2.0 0 i I 0 0.005 0.01 0.015 0.02 0.025 0.005 0.01 0.015 0.02 0.025 eA eu 14.0 14.0 12.0 12.0 10.0 10.0 I i 8.0 8.0 N 6.0 6.0 4.0 2.0 2.0 0 0 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 e A eu FIG. 26. Average final escape eccentricities. (a) and (b) refer to Run 3, (c) and (d) to Run 4, and (e) and (f) to Run 5. The average final eccentricity of Umbriel tends to be higher than the average final eccentricity of Ariel. Compare with average initial eccentricities, given in Table III. It is possible for the average final eccentricity of Umbriel to be higher than the average value before the resonance is encountered. PASSAGE THROUGH 5:3 URANIAN RESONANCE 211 20.0 a tered. This may be contrasted with the sin- gle resonance model in which the eccentric- ity after escape is always lower than the 15.0 initial eccentricity. There is a tendency for the average final 10.0 eccentricity of Ariel to be lower than that of Umbriel. This can be understood by exam- ining the structure of the sections corre- 5.0 sponding to root families b and c (see Figs. 4b,c and 5b,c). The trajectories escape into i the quasiperiodic regions surrounding the 0 0.5 1.0 1.5 210 215 310 3.5 A8 fixed point associated with Mode I, which is near the origin in Ariel variables and near 20.0 the boundary in Umbriel variables deter- mined by the energy integral. 15.0 [b Figure 27 shows the distribution of times spent in the chaotic zone in terms of 8 for the escaping trajectories. For Q = 6600, i0.0 A6 = 1.0 corresponds to about 6.9 × 10 6 years. There is a trend, with increasing ini- 5.0 tial eccentricity, for the distribution to spread out and for escaping trajectories to spend more time in the chaotic zone. 0.5 1.0 1.5 210 215 310 3.5 Figure 28 shows the distributions of max- A8 imum eccentricities in the chaotic zone for 20.0 the escaping trajectories. There is a correla- e tion between the length of time spent in the chaotic zone and the maximum eccentricity 15.0 achieved--corresponding essentially to the upper "envelope" of eccentricity in the 10.0 chaotic zones in Figs. 19-23. The trajecto- ries travel up the chaotic zone to higher and higher eccentricities. The largest eccentric- 5.0 ity reached in any of the escaping trajecto- ries was the maximum of the trajectory in o ~'1~ ~ ~ Fig. 22, approximately 0.023. 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Note that some of the trajectories in Runs 4 and 5 (see Figs. 16 and 17) are still FIG. 27. A8 spent in the chaotic zone for escaping chaotic at the end of the run. When further trajectories. (a)-(c) refer to Runs 3-5, respectively. A8 = 1.0 corresponds to 6.9 x 106 years of evolution evolved, none of these trajectories es- for Q = 6600. Trajectories with higher initial eccentric- caped. ities tend to spend more time in the chaotic zone. 5. DISCUSSION nal eccentricities after escape from the res- Well before they encounter the 5 : 3 reso- onance. Note that in some cases the aver- nance, the orbital evolution of Ariel and age final eccentricity of Umbriel after Umbriel is dominated by their mutual secu- escape from the resonance is higher than lar perturbations. The eccentricities of the the value before the resonance was encoun- satellites oscillate between minimum and 212 TITTEMORE AND WISDOM 24.0 24.0 20.0 20.0 16.0 16.0 N 12.0 12.0 8.0 8.0 4.0 4.0 a 0 I 0 I I 0 0.005 0.01 0.015 0.02 0.025 0.005 0.01 0.015 0.02 0.025 eA eU 24.0 24.0 20.0 20.0 16.0 16.0 12.0 N 12.0 8.0 g.0 4.0 4.0 c 0 I 0 L 0.0{)5 0.01 0.015 0.02 0.025 0.005 0.01 0.015 0.02 0.025 eA Co 24.0 24.0 20.0 20.0 16.0 16,0 12.0 N 12,0 8.0 8,0 4,0 4.0 0 0 L i 0 0.005 0.01 0.015 0.02 0.025 0.005 0.01 0.015 0.02 0.025 eA elj FIG. 28. Maximum eccentricities in chaotic zone. (a) and (b) refer to Run 3, (c) and (d) to Run 4, and (e) and (f) to Run 5. The maximum eccentricities of the two satellites in the chaotic zone tend to be comparable. maximum values determined by the relative rapidly than that of Umbriel, and as they strengths of the two normal modes. The approach the 5:3 mean-motion commen- variations are quasiperiodic, with two fun- surability resonant perturbations become damental frequencies associated with the stronger. One of the secular modes eventu- modes, and with high-frequency perturba- ally becomes unstable, and the system en- tions superimposed. The semimajor axes ters a large chaotic zone. Further tidal dis- slowly increase due to tidal dissipation in sipation within Uranus may drive the Uranus. The orbit of Ariel expands more eccentricities up the chaotic zone to rela- PASSAGE THROUGH 5:3 URANIAN RESONANCE 213 tively high values. They may then either The probability of escape is higher for escape or be captured into resonance. larger initial average eccentricities, but also When the system enters the chaotic depends on which secular mode is stronger. zone, a new low-eccentricity region of Since the satellites are not presently in the phase space is made accessible to the sys- resonance, they must have escaped. It ap- tem. While in the chaotic zone, the system pears that in order for the probability of es- spends time in both regions of phase space, cape to have been significant, the average and the eccentricities irregularly alternate eccentricities before the resonance was en- between high and low values. If the system countered were probably higher than cur- is caught in the circulating quasiperiodic re- rent values. The largest increases in ec- gion in the low eccentricity region of phase centricity occur when the trajectory is space, it escapes from the resonance, and temporarily captured into quasiperiodic li- the orbits continue to differentially expand bration during evolution through the reso- due to dissipation in Uranus. The average nance. eccentricities after escape from the reso- The relatively high current eccentricities nance tend to be lower than the values be- must be explained. For the highest dynami- fore the resonance is encountered, but in cally allowed rate of tidal dissipation in some cases the average eccentricity of Um- Uranus (Q = 6600), the time since reso- briel increases, a phenomenon not pre- nance passage is about 3 × l08 years. For dicted by the standard single resonance the- the lowest rate of tidal dissipation in ory of evolution through resonances. It Uranus which allows Ariel and Umbriel to may also be possible for the system to be encounter this commensurability, the time temporarily captured into a librating island since resonance passage is of order 4 × 109 during passage through the resonance (see years. As discussed in Section 4. l, the time Section 4.2), with the result that the aver- scales of direct tidal damping of the satellite age eccentricities after escape from the res- eccentricities are about l08 years for Ariel onance could be significantly higher than and about 10 9 years for Umbriel. After es- the values before the resonance is encoun- cape from the resonance, the eccentricity tered. After escaping from the resonance, of Ariel tends to be smaller than that of the evolution of the orbits is again domi- Umbriel, which is also the case at the nated by their mutual secular perturbations, present time. If the rate of tidal dissipation and the eccentricities slowly damp due to is near the maximum dynamically allowed tidal dissipation within the satellites. value, the damping time scale for Umbriel If the system does not escape, the energy is about a factor of 3 longer than the time surface divides into two separate regions. since resonance passage, and that of Ariel When this happens, the system is captured is about a factor of 3 smaller. This indicates into resonance, even though the trajectory that the eccentricity of Umbriel has proba- may still be chaotic. Initially, only o'u li- bly not changed significantly since reso- brates, and tidal dissipation drives the or- nance passage, and that the eccentricity of bit of Umbriel to higher eccentricity. The Ariel, which was probably lower than that trajectory becomes quasiperiodic at some of Umbriel immediately after escape from point. Eventually, O'A may also be trapped the resonance, has damped to still lower into a librating state, and the eccentricity values, but not necessarily to negligible val- of Ariel is also driven to high values. Both ues, which is consistent with the current orbits are then captured into the resonance. observations. The run with outcomes most The outcome of resonance passage does closely resembling the current physical not depend simply on the mean eccentrici- configuration is Run 3 (Fig. 15). The current ties; the individual amplitudes of the secu- relatively high eccentricities, then, are rem- lar modes appear to play a significant role. nants of relatively high eccentricities after 214 TITTEMORE AND WISDOM passage through the resonance. Of course, dE k4rrR2a AT the average eccentricities of the satellites dt - AR' (25) for the initial conditions of Run 3 (eA ----- 0.01 and eu -~ 0.003) are quite high, so the prob- where k is the thermal conductivity of ice, lem is not really solved. We still require RA is the radius of Ariel, A T is the differ- some mechanism to provide the satellites ence in temperature between the base of the with high eccentricities before the reso- mantle and the surface, and AR is the man- nance is encountered, unless the temporary tie thickness. If the bottom of the mantle is capture into a quasiperiodic librating state maintained just below the melting tempera- occurred. In this case a trajectory ap- ture of 273 ° K, conduction through a rigid proaching the resonance with low eccen- ice mantle can remove approximately 1017 tricities could escape from the resonance ergs/sec of thermal energy. This is approxi- with significantly higher eccentricities. It is mately the thermal energy that would be difficult to determine if this happened to released by radioactive decay of materials Ariel and Umbriel, but it is a tantalizing in the core with chondritic abundances, possibility. On the basis of our numerical with an energy production of about l0 -7 experiments, temporary quasiperiodic cap- ergs g-~ sec -1 (Kaula 1968). It is also equiv- ture seems to occur with relatively low alent to the tidal heating produced if the probability. We point out, though, that our eccentricity of Ariel is about 0.03, using the single run with 9~ = 3300 showed this phe- expression for the energy dissipation in a nomenon. synchronously rotating satellite (e.g. Peale 1988): If the rate of tidal dissipation in Uranus is near the minimum value which allows pas- dEi _ 21 k2iMniRi3 5 e2i (26) sage through this resonance (Q = 100,000), dt 2 a 3 Qi then the damping time scale for Umbriel is about a factor of 4 smaller than the time with reasonable physical parameters for since resonance passage, and that of Ariel Ariel (see Table I). If the energy input to is about a factor of 40 smaller. In this case, the interior exceeds this rate, the tempera- the eccentricities would have damped sig- ture at the base of the mantle can exceed nificantly since passage through the reso- the melting temperature. The mantle can nance. Since the present eccentricities are become unstable to solid state convection high, this seems to argue for a Q of Ura- (see Reynolds and Cassen 1979). The nus closer to the minimum allowed steady state configuration for the mantle in value. this case is a convecting solid layer topped Although Ariel and Umbriel have similar by a rigid surface layer. This configuration masses and radii (see Table I), the surfaces is capable of removing heat at a signifi- of the two satellites are quite different. cantly higher rate than the conducting con- Ariel has had a geologically active past figuration, and can still remain mostly solid. (Smith et al. 1986, Plescia 1987), with Peale (1988) estimates that the solid state faults, fractures, and flow features remain- convection process could remove up to ap- ing on its surface. Umbriel presents a proximately 7 × 1017 ergs/sec of thermal en- bland, heavily cratered, primitive surface. ergy from the interior of Ariel. The equiva- The density of Ariel is 1.66 -+ 0.30 g cm 1 lent tidal dissipation rate in the satellite (Stone and Miner 1986). If the interior is would result if the eccentricity of Ariel was differentiated, it will have a rocky core about 0.09 at the 5 : 3 resonance, not taking (p ~ 3.0 g cm -l) about 0.7RA in radius, sur- into consideration possible enhancement rounded by an icy mantle. If the mantle is due to radioactive decay in the interior. A solid, the rate of heat conduction through higher eccentricity would result in large- the mantle can be approximated by scale melting of the interior of Ariel. The PASSAGE THROUGH 5:3 URANIAN RESONANCE 215 maximum eccentricity reached by Ariel in through the 5:3 mean-motion commensu- this study for an escaping trajectory is ap- rability is significantly affected by the pres- proximately 0.023. Considering our results ence of a large chaotic zone. The mecha- in the planar approximation, it seems un- nism of capture is qualitatively different likely that passage through the 5:3 reso- from the isolated resonance capture mecha- nance alone could have melted the interior nism. Relatively large eccentricities are ob- of Ariel. Furthermore, the rate of energy tained while the trajectory is chaotic even if dissipation in Ariel equals the energy input the trajectory ultimately escapes. Peak ec- to the orbit due to dissipation in Uranus for centricities for escaping trajectories are an eccentricity of about 0.05 (for Q = 6600). typically of order two to three times the This provides an upper limit on the eccen- initial eccentricities. Eccentricity increases tricity that can be maintained through tidal large enough to have a significant affect on evolution of the orbit, assuming that an the thermal history of Ariel have not been equilibrium configuration is achieved by the found in this study. system. Since this eccentricity is lower 6. APPENDIX I than that estimated for the melting of the interior of Ariel, the eccentricity of Ariel 6.1. Averaged Planar Hamiltonian may never reach the value required for The problem to be considered is the pla- melting. Umbriel is clearly not thermally af- nar Ariel-Umbriel system near the 5:3 fected by passage through the resonance-- mean-motion commensurability. The Ham- it requires an eccentricity of about four iltonian of this system can be written times that of Ariel for an equivalent tidal = ~K + ~o + ~s + ~R, (27) thermal input. The Hamiltonian studied in this project where ~K is the sum of the unperturbed neglects dissipation in the satellite during Keplerian terms, ~o is the perturbation due the relatively brief period of resonance pas- to the oblateness of Uranus, ~s is the per- sage. The damping of the eccentricities can turbation due to the secular interaction be- slow down the rate of evolution of 8, as well tween the satellites, and ~R is the per- as directly affecting the xi and y;. A possible turbation due to the resonant interaction consequence is that the system could have between the satellites. In the following de- remained in the chaotic zone for a longer velopment, subscript A refers to Ariel and period of time and may have reached higher subscript U to Umbriel. eccentricities before escaping. Expressing the contributions in terms of The inclinations were also neglected. Keplerian elements, were ai is the semima- Lowest order terms in the disturbing func- jor axis, ei is the eccentricity, hi is the mean tion for inclination involve terms of order longitude, and o3i is the longitude of perihe- sin 2 (I/2), where I is the mutual inclination lion, of the two orbits. For current values of the ~K = - GMm-----b-A- GMmu. (28) inclinations, this quantity is of the same or- 2aA 2au der as e 2. Therefore, the inclination terms could have contributed significantly to the Keeping only the J2 terms to second or- evolution of the system. It has been found der in eccentricity in the expression of the (Wisdom 1983, 1987) that in the case of res- potential for an oblate planet, onant asteroid motion, including inclination GMmAj2(R)Z[1 +3e~] terms can lead to much higher eccentrici- ~/~O -- 2aA ties than in the planar case. A similar quali- tative change in the behavior of this system GMmu R 2 3 e~ ],j may result. The passage of Ariel and Umbriel (29) 216 TITTEMORE AND WISDOM and from the expression for the disturbing and a is the ratio of semimajor axes, and function, b](a) are Laplace coefficients. Near the resonance, the evolution of ei Gmhmu[(1)(o)+(2)(o)(2)2 and o3i is dominated by the low-frequency ~s- au perturbations, with frequencies associated with changes in the resonant combination of mean longitudes 5Xu - 3hA and the longi- tudes of pericenter o3i. The high-frequency eA etJ ] + (21) (-1) -~- -~- cos (O3A -- O3u) (30) contributions associated with the motions of the mean anomalies and with other and nonresonant combinations of the mean anomalies are removed by averaging. GmAmu ~tf~R -- The Delaunay momenta Li ~ mi GVG-M-a~ au and Gi = LiX/1 - e 2 are conjugate to the mean anomalies and arguments of pericen- x [(172)(5) (2) 2 cos(5hu - 3hA-- 203A) ter, respectively. Canonical Poincar6 ele- ments are defined as the coordinate angles CA dU + 082) (4) ~--~-cos(5hu - 3hA -- OSA -- a3U) hi = li + ~i, which are conjugate to L~ = Li, and o~ji = -a3~, which are conjugate to pli = Li - Gi. We use the generating function +(192) t3) (2) 2 cos(5h U - 3hA-- 2O5u)], 1 F = ~ (5hE - 3hA + 2CO1A)~A (31) 1 + ~ (5hu- 3hA + 2Oglu)~u where (see Leverrier 1855) + hAF A + JkuF U (33) 1 (1)(°) = 5 b°/2(a) to obtain the new resonance coordinates d 2 1 {2)(0) = d + 1 °t 2 bO/2(ot) O'A = ~ (5hU -- 3hA + 2091A) (3)(0) = (2)(0) 1 Cru = ~ (5hu - 3hA + 2(olU) (34) d (21) (-l) = 2b[/z(a) - 2a ~ bl/2(oO which together with YA = hA and Yu = hu d 2 form a complete set of generalized coordi- -- 0~2 ~2 bl/2(°/) nates. The momenta conjugate to era and cru are (172) (5) = b~/2(a) + 9a ~ b~/2(a) ~A ~--- PlA = L~ - GA d E 1 a2 b~/2(a) Eu = plu = L{j - Gu. (35) The momenta conjugate to Ya and Yu d form two integrals of motion, since in the (182) (4) = -72b4/z(Ot) - 18a ~ b4/2(a) new variables the resonant Hamiltonian is d E cyclic in these variables. These resonance __ a2 ~2 bl4/2(°0 integrals are 3 (192) (3) = b3/2(ot)+ 9a ~aa b~/2(00 FA = L~ + ~ (EA + EU) I d 2 5 OtE -- b3/2(o0 (32) Fu = L{j - ~ (EA + Eu). (36) + ~ dot 2 PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 217 Note that Ei ~ F;e~/2. The lowest order = 2A(EA + Eu) + 4B(EA + Eu) 2 terms in the disturbing function which influ- + 2C]~A + 2DEu ence the evolution of the eccentricities and pericenter longitudes are of order e 2. The + 2EX/~,AEU COS(OrA -- O'U) Hamiltonian is now expanded in powers of + 2F.~A COS(20"A) ~,i/Fi to this order, and constant terms are removed. The resulting expression in terms + 2GV',~AvZU COS(O'A + O-U) of the new coordinates is + 2HEu COS(2O'u), (37) where 3 GZMZm~m'A 5 G2MZm~m{j A- + 4 F 3 4 F 3 27 G2M2mZm'A 75 G2MZm~jm{j B _ 32 F 4 32 r 4 1 [.15m4m~3 6m4m~,3] C = -~ G4M4RZj2 I_ 2r~, I'~ J G2MrnAm2orn~j [ 5 + 4r~ ~ b°/2(~) d mu m~ (3FA 5F~ (2) (°)] + ~ b°/z(u) m~ m-~a \~,2,-u + F 3 / - ---F-A-aJ I [6m4m~ 3 9m4m~ 3] D = ~ G4M4R2J2 L G ~ j G2Mmam2mu I 5 + 4F~j Fu b°/2(°0 d mu m{s (3FA 5F~ (3) (°)] + ~ b°/2(c0 m---~ m--~A\~-U + F 3 / - -~uJ G2MmAm~jm{; E = _ (21)(-~) 4r~ FvT-2-~AF% GZMmAmZm{j F = _ 4F~FA (172)(5) GZMmAm~m{j G = _ 4F2uFV/-~AF~ (182)(4) H = - GZMmAm2mo 4F 3 (192) (3) (38) where the m" are reduced masses in the Yi = ~ sin(o-i) ~ ei ~ii sin(o-i) (39) Jacobi coordinate system (see, e.g., Plum- mer 1960). and the conjugate momenta Finally, the transformation to canonical coordinates Xi : ~ COS(O'i) ~ ei ~ii COS(O'i) (40) 218 TITTEMORE AND WISDOM is made, resulting in the final form of the B = -2632.82 Hamiltonian, C = -0.0803237 I D = -0.0477658 =~(3--2(C+D))(X2A+yeA+x 2 + y~) E = 0.0109693 9 9 9 + B(xeA + Y'A + X 2 + Yb)" F = -0.0319666 + C(XA + yeA) + D(x~ + y2) G = 0.0797229 + E(XAXU + YAYu) + F(x 2 - YeA) H = -0.0493825 (42) + G(XAXU -- YAYu) + H(x 2 - Yu), and the integrals of motion are (41) FA = 0.742492 which has 2 degrees of freedom. The units chosen are as follows: distance Fu = 0.834860. (43) is measured in units of the radius of Uranus The state of a system is determined by its B, mass is measured in units of the mass of coordinates xi and Yi and the two parame- Uranus M, and time is measured in years. ters 8 and the numerical value of the Hamil- The coefficients were evaluated at 8 = 0 for tonian %. ei = 0 and av = 10.0703R. (The semimajor axis at resonance encounter is au = 10.1353. This should properly have been 7. APPENDIX II used in the evaluation of the coefficients; the differences are not significant.) The nu- PLANAR MAPPING merical values for the coefficients thus ob- In developing the algebraic mapping, Eq. tained are (37) is rewritten = 2A(EA + Eu) + 4B(EA + ,~.~U)2 q- 2C,~,A + 2DEu + 2E X/~AvZU COS(O'A -- O'U) ~ COS n(~t -- thl) + 2F.~A COS(20"A) ~ COS n(l)t -- oh2) + 2G X/~A~ U COS(irA + tru) ~ COS n(f~t -- ~b3) n=--z¢ + 2H~;u cos(2cru) ~ cos n(~t - 4~4)- (44) Expressing the cosine terms using the Fourier representation of the Dirac delta function yields ~t~ = 2A(]~A + ]~U) + 4B(~A + ~U) 2 + 2C~A + 2D~u + 2E V'~£A~U COS(CA - tru)2"n'ST(l)t -- ~bl) + 2F~A COS(2trA)27r3T(l)t -- ~b2) + 2G ~'/,~A~-~U COS(O"A q- O'U) 27r3T(D,t -- qb3) + 2H~,u COS(2tru)2~'3T(D,t -- 4)4). (45) PASSAGE THROUGH 5:3 URANIAN RESONANCE 219 This form allows us to integrate the an- the secular frequencies in this problem. We gle-dependent terms at singular points, and have chosen ~bl = 0, ~b2 = 6])4 = 7]'/4 (since integrate the contributions from the other these two are independent), and ~bs = 7r/2. terms between them. The delta functions A full mapping cycle involves the follow- are periodic in T = 2rr/l~, which becomes ing steps, starting with coordinates -~A~0) , YA. ~0) , one cycle of the mapping. During one map- x~ ), and y~): ping cycle, the delta functions are inte- Step 1. Integrate across first delta func- grated at times related to the phases ~bi, and tion att = t~ between them the secular terms are inte- grated. These integrations are all performed 1~) : x~) cos (-~-~---E) - y~) sin (-~ --E) analytically. Since the new high-frequency terms are y(~)= y~)cos (2__~_)+ x~ ) sin (_~___E) not directly related to those removed by av- eraging, the choice of the mapping fre- x~) : x~' cos (-~--E) - y~) sin (-~ E) quency 12 and the phases ~b~ is largely arbi- trary. However, for the mapping to be valid, 12 must be much higher than the long- y~)=y~)cos + sin(~ --E) period frequencies, which are of order 10 -4 (46) of the orbital frequencies. An ~ = 20 year -j has been found to be suitable for most pur- Step 2. Integrate secular part from t = t~ poses. This is of order a few hundred times to t = t2 = tl + ¢~2/~~ (C - D) + 4B(x~ )2 + y(~)2 + XiJ)2 + -y~) sin {[~ + (C - D) + 4B(x~ )2 + y~)2 + x~j)2 y~) = y~) cos {[~ + (C - D) + 4B(x~ )2 + y~)2 + x~j)2 + - x~) sin {[ ~ + (C - D) + 4B(x~ )2 + y~)Z + x~)2 + y~)2)](~2~ ~bl)) x~:x~'cos{[~- (C - D) + 4B(x(~ )2 + y~)2 + X~)2 "1- y~'2)](562 ~ ~b')] - y~) sin (I-~ - (C - D) + 4B(x~ )2 + y~)2 + x~})2 + y~)2)](~2~ q~|)} y~) = y~J) COS {[~ - (C - D) + 4B(x~ )z + y~)2 + x}J)z + y~)2)l(~b2 ~ ~b')} + x~, sin IF~- ~c-"+ 4,~x~,2 + y~,2+ x~,2+ y~,2,](~2 - ~,)) ~47~ Step 3. Integrate across second and fourth delta function at t = t2 = ~/l) X(~) = X(~) ( e4rrr/o + e-4~rF/n) + YA. (2) ~1 (e4~rF/lI e_4,rF/fl ) y~) = y(~) 21 (e4wF/~ + e-4~rF/Yt) + -~A'(2)21 (ea~.F/l) _ e_4~F/l.~) X~ ) : X~ ) "21 (e4~rH/~ + e_4rrH/l~) + y~) ~l (e47rH/fl _ e_4~H/~) y~) = y~).~1 (e4~S.s/~+ e_4~,H/n) + x~ ) ~I (e4rrH/sq - e_4rrH/O). (48) 220 TITTEMORE AND WISDOM Step 4. Integrate secular part between t2 Step 5. Integrate across third delta func- and t = t3 = t2 + (~b3 - ~bz)/fL tion at t = t3 = ~b3/ft X(SA) = X(~) ~l (e4~rG/f ~ + e_4~rG/ll) + y~j) -21 (e47rG/f ~ _ e_4rrG/ll) Y(~) = Y(~) 5I (e4rrG/ft + e_4rrG/~ ) + X~j) 1 (ea~.G/f~ _ e_4~G/fz ) X~j) = X~j) ~1 (e4~rG/~l + e_47rG/ll) + y(~) -21 (e47rG/lI _ e 4,,a/n) y~j) = y~)_~l (e47rG/lI + e_4~rG/O ) + X(~) "21 (e4rG/f~ _ e_47rG/a). (49) Step 6. Integrate secular part between t3 by removing the new high-frequency per- and t = t4 = to + 2~-/11. This full cycle of the turbations (see Wisdom and Tittemore mapping iterates all coordinates forward in 1987). time by one mapping period T = 27r/~. We use a Von Zeipel transformation to If tidal dissipation is included, the map- go from mapping coordinates to averaged ping becomes explicitly time dependent coordinates, using the generating function through 8(0 and also through a slightly more complicated secular evolution. The F = ~. "£1o'i + l~S(Y.~,o'i,t) (50) secular evolution is integrated assuming a i linear dependence of 8 on time. and the transformation to first order in/~ Since the mapping includes high-fre- quency perturbations, the averaged Hamil- OS tonian is no longer a conserved quantity. We require a conserved quantity corre- OS sponding to the energy for the analysis of ~r~ = o-i + tx ~ (51) Section 4.2. In addition, if the mapping is used to compute a surface of section ac- The Hamiltonian (44) can be expressed cording to the standard criteria for the aver- ~(~ = ~0(~i) + #~1(~i,O'i) + tx~fCl(~'i,O'i,t), aged Hamiltonian, there will be a slight dis- (52) tortion due to the difference between the mapping coordinates and the averaged co- where ~ contains time-dependent terms ordinates. For these reasons, we have used (n 4= 0) in Eq. (44). perturbation theory to transform from map- To first order in/,, this Hamiltonian be- ping coordinates to averaged coordinates, comes OS ~' = ~(~0(~i(~/)) q- //,~1(~/,0"/) q- p~3~l(Y2i,o'i,t)" t t + tx-~ OS = ~o(YZ) + t~1(~2'~,o-- -') + tXTel('£i,o-i,t)- t r + t~ if~ +~ ~+(C-D)+8B(X;+Xb) 1 +~ ~-(C-D)+88(:~;+Xb) 1• (53) S is chosen to eliminate time-dependent terms, so that PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 221 OS #~l(Xi,~r~,t)- t ! + jz + /x acr-----~a 2 + (C - D) + 8B(E~, + E{j) + /~ ~-(C-D) + 8B(E~+E~) =0. (54) Since ~l = ~ Aij Cos(io'A + jCru) ~ COS n(l)t - ¢~ij) ij n4:O = ~ Aij~ Cos(i(rA + jtru) + n(l)t -- d~ij)) (55) O n4:0 then S = - ~, Aij ~ sin(io'A + jo'u + n(~)t + rhij)) (56) ij .4-0 Z + n~ or S = - 2EVEkEb [K(1,-1) sin(~k - ~b) + L(1,-1) cos(~k - ~b)] - 2F£~ [K(2,0) sin(2~) + L(2,0) cos(2w~)] - 2G~/E~Ef2 [K(I,1) sin(~ + ~f~) + L(1,1) cos(w~ + ~)] - 2HE~ [K(0,2) sin(2~) + L(0,2) cos(2~f~)] (57) with the functions 7r cos(z/f~)(l~t - ~bii - ¢r) 1 K(i,j) = ~ sin(Z/~)Tr Z Ir sin(z/f~)(l~t - chij - rr) L(i,j) = - ~ sin(Z/l~)Tr (58) evaluated at t = 27r/fL where Z= i ~+ (C-D) + 8B(Y~,+ E~) -j[~-(C-D) + 8B(~£~, + Eb)] (59) and i andj refer, respectively, to the coeffi- To first order in /~, the transformation cients of CrA and tru. thij are the phases from from map coordinates to averaged coordi- Eq. (44). nates is just 1 as/a~q[x? xV = ( l - ~ ~---ET-, / cos (/z ~-~.)- ym sin (/x ~-~i)] 1 as/a<)[ m yaV = (1 -- ~/.6T/Ly i cos (/~ ~)+ xT' sin (/x ~)] (60) 222 TITTEMORE AND WISDOM which is a slight rotation and change of angle conjugate to the action: scale. ~(p,q;8) ~ ~(J;8). (62) 8. APPENDIX III Evolution of the parameter 8 at some slow rate ~ = e can be considered a pertur- STANDARD THEORY OF EVOLUTION bation on the motion. By transforming to THROUGH RESONANCES action-angle coordinates and averaging In this appendix we shall examine evolu- over the period of motion on the trajectory, tion into two of the second-order reso- which is in general much shorter than the nances at the 5 : 3 Ariel-Umbriel commen- time scale of the evolution of 8, it can be surability. We treat for the moment these shown that the action is to first order invari- resonances as isolated resonances in order ant (see, e.g., Born 1960, Arnold 1978, Lan- to verify that our numerical methods are in dau and Lifshitz 1978). It has also been satisfactory agreement with the results of shown that there is an adiabatic invariant to the isolated single resonance model when all orders in e, if e is sufficiently small the model is applicable. We emphasize that (Lenard 1959, Kruskal 1962, Arnold 1963). this is being undertaken only to verify our To first order then the evolution of the en- methods; the actual dynamics near the 5 : 3 ergy depends only on the form in which 8 commensurability are much more compli- appears in the Hamiltonian in action angle cated, with the presence of large chaotic variables: zones in the phase space, and the single ~(t) = ~(6(t)). (63) resonance model is not applicable. The evolution of trajectories defined by the Hamiltonian as 8 varies can be viewed as 8.1. Adiabatic Invariance slow changes in the level curves of ~, The theory of adiabatic invariance can be maintaining constant area on the phase used to completely specify the evolution of plane. a one degree of freedom Hamiltonian prob- The standard theory of evolution through lem with a slowly varying parameter 6, in resonances (Goldreich and Peale 1966, the absence of infinite period orbits (separa- Yoder 1979, Henrard 1982, 1987, Henrard trices) and/or chaotic zones. (Because of and Lemaitre 1983, Borderies and the time dependence of the parameter 8 the Goldreich 1984, Lemaitre 1984, Peale 1986) problem is nonautonomous and separa- treats the resonances arising from a com- trices and chaotic zones are strictly speak- mensurability independently. This approxi- ing practically everywhere, though most mation may be valid for mean-motion reso- are too small to be important (see Chirikov nances, for instance, if the Jz of the planet 1979).) For a one degree of freedom prob- is large, and has been successfully applied lem the adiabatic invariant is, to first order, to J : J + 1 resonant motions in the satellite the action J, which is defined for fixed 8 in system of Saturn (Yoder 1979, Peale 1976, terms of generalized canonical coordinates Henrard and Lemaitre 1983). p and q as We consider here two of the second-or- /, der eccentricity resonances at the 5:3 J = ~ pdq (61) Ariel-Umbriel commensurability: the reso- or the area on the phase plane enclosed by nance proportional to the square of the ec- the trajectory. For fixed 8 the action is centricity of Ariel and the resonance pro- strictly conserved since trajectories follow portional to he square of the eccentricity of level curves of the Hamiltonian for a one Umbriel. The Hamiltonians for these two degree of freedom system. In action angle resonances, artificially treated as though variables, the Hamiltonian is cyclic in the they are isolated, are PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 223 ~A = ~+ (C-D) EA+ 4BE 2+ 2FEACOS(20"A) =-~ ~ + (C-D) (x 2 + y~) + B(x 2 +y2)2 + F(x 2_y2) (64) and ~u = E~- (C-D) 1 Xu + 4BX 2 + 2HXucos(2o-u) =-~ ~- (C- D) (x~ + y~) + B(x~ + y~)2 + H(x~ - y~) (65) These can be derived by removing the cou- cal values of 3 and coordinates of stable and pling terms from the full Hamiltonian (6) unstable fixed points are summarized in Ta- and (9). These are both one degree of free- ble IV. dom problems, and are therefore integra- For the physical system trajectories en- ble. The mixed second-order resonance counter the resonance as 8 passes from could of course be treated in a similar man- large negative values through values near ner, but will not be considered here. zero. The semimajor axis of the inner satel- The structure of the level curves of the lite Ariel is increasing more rapidly than Hamiltonian for second-order resonances is that of Umbriel. The ratio of their semima- well known. Figure 29 illustrates the level jor axes, aA/au is increasing. For this direc- curves of these systems for various fixed tion of passage through an isolated second- values of 3, computed with the mapping. order resonance the trajectory may be For large negative 3, or before the reso- either captured into resonance or escape nance is encountered, the curves in xi, yi space are approximately concentric circles surrounding a stable fixed point at the ori- TABLE IV gin, and the eccentricity is approximately SINGLE RESONANCES constant. As the value of 3 is increased, curves near the origin become distorted in Ariel resonance the y-direction. At a critical value of 3cj a 8cl : -2(C - D) + 4F = -0.065 period-2 bifurcation of the fixed point oc- 8¢2 = -2(C - D) - 4F = 0.193 Unstable equilibria: curs, and it becomes unstable. There are x~ = 0.0, 8c1 -< 8 < 8~2 now two librating islands surrounding sta- x~ = - [8 + 2(C-D) + 4F] ble fixed points on the y-axis, bounded by a 8B ' ~ => 8c2 separatrix with a "figure 8" shape. As 3 Stable equilibria: y2 = _ [8 + 2(C- D) - 4F] increases, these islands become larger. At a 8B t 8 2> 8c I second critical value of G2, another bifurca- tion occurs at the origin. The fixed point at Umbriel resonance the origin becomes stable again and is sur- 8c~ = 2(C - D) + 4H = -0.263 8c2 = 2(C - D) - 4H = 0.133 rounded by concentric level curves. Two Unstable equilibria: unstable equilibria are emitted from the ori- x~=0.0, Gl<-8<8~2 gin. The separatrix now resembles two "ba- x22 = _ [8 - 2(C- D) + 4H] nanas" joined at the "ends." Further evo- 8B , 8 --> 8c2 lution of 3 results in these islands moving Stable equilibria: y~ = _ [8- 2(C- D)- 4H] further from the origin, and the trajectories 8B , 8 --> 8~t near the origin become more circular. Criti- 224 TITTEMORE AND WISDOM 0.01 is infinite, and it is no longer proper to average the evolution over the libration /.. \\ 0.005 i" ..--...... "\ period. The action (area inside a trajec- /: \,, tory) changes discontinuously on passage through the separatrix, which can be seen YA 0 on inspection of Fig. 29. Trajectories which escape have lower eccentricities than they -0.005 had approaching the resonance. The eccen- tricities of captured trajectories increase a -0.01 I I i until the damping rate due to tidal dissipa- -0,01 -0.005 0 0.005 0.01 XA tion in the satellite becomes comparable to the input of energy to the orbits. 0.01 The evolution of the energy (the numeri- cal value of the Hamiltonian) of a trajectory 0.005 can be estimated from the eccentricity be- havior of the various cases, summarized in Table IV. Interior to the resonance (8 ~ 0), YA 0 the level curves are roughly circular, and adiabatic invariance of the area enclosed -0.005 by the curves results in the eccentricities remaining approximately constant as 8 •0.01 I i , changes. Therefore ~A and ~v are linearly -0.01 -0.005 0 0.005 0.01 XA proportional to 8, and the energies increase linearly with time. The same is true for tra- 0.01 jectories which escape. The change in ec- 1., ~" ...... ~-~. centricity at the point of escape causes the 000, slope of ~i to decrease discontinuously at the value of 8 where the separatrix is crossed. For captured trajectories, the ec- centricity increases approximately as The functions ~A and ~u are quartic in ec- centricity. Therefore, the energy increase is roughly quadratic in time. -0.01 , , t Evolution of trajectories in 8, AE space -0.01 -0.005 0 0.005 0.01 XA for the single resonance Hamiltonians are much simpler than for the full resonant FIG. 29. Level curves for the Ariel (e~) resonance, computed with the uncoupled mapping. (a) ~ = -0.5, Hamiltonian. Since only two fates are pos- (b) 8 = 0.15, (c) 8 = 0.5. sible, trajectories will follow one of only two branches in 8, bE space for values of 6 larger than that at which the separatrix is from resonance. If the separatrix is encoun- crossed. If the trajectory is captured into tered with 8cl -< 8 -< ~c2 then capture is cer- the resonance, AE will increase linearly. If tain. This occurs for trajectories with small it escapes, A E will decrease quadratically. eccentricity. If the separatrix is encoun- The energies of the separatrices for ~A tered with larger values of 8 then the trajec- and ~u are shown in Fig. 1. Expressions tory may either be captured or escape. The for these energies have been obtained by adiabatic invariance theory breaks down as substituting the coordinates of the unstable the trajectory approaches the separatrix, fixed points as a function of 8 (see Table IV) since the period of motion on the separatrix in Eqs. (64) and (65). PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 225 8.2. Capture Probabilities made• Probabilities are assigned under the The determination of the fate of a given assumption that the energy of the physical trajectory upon an encounter with an iso- system is equally likely to have been any- lated resonance has been developed in a where within the balance of energy. Equiv- probabilistic approach based on the "bal- alently, probabilities are assigned by as- ance of energy" as the trajectory crosses suming the actual system belongs to an the separatrix (Goldreich and Peale 1966, ensemble of similar systems whose ener- Counselman and Shapiro 1970, Yoder 1979, gies are distributed uniformly over a range Henrard 1982, 1987, Henrard and Lemaitre of energies comparable to the balance of 1983, Borderies and Goldreich 1984, Le- energy. Probability of capture is then maitre 1984). The balance of energy is de- Bo + Bi fined as the difference between the change Pc = --, (66) B0 in energy of the trajectory and the change in energy of the separatrix in one cycle of the where B0 and Bi are estimated at a given trajectory. value of/5 by approximating the motion as The analytic theory of capture into reso- motion on the separatrix. By distributing nance assumes a uniform distribution of the energies of trajectories uniformly trajectory energies just before separatrix through the range B0, one should obtain a crossing. These energies lie within a range representative sample of trajectories with corresponding to the energy balance of the both fates. outer separatrix B0. The probabilities of The balance of energy has been approxi- capture and escape depend on the magni- mated by Henrard (1987) as the rate of in- tude of the energy balance before transition crease of the area in phase space enclosed across the separatrix Bi. A trajectory loses by the separatrix at the value of 8 at which energy relative to the separatrix as it ap- the trajectory crosses the separatrix: proaches, since the separatrix gains energy quadratically with 8 and the trajectory gains • OJ* energy only linearly with 8. Therefore, B0 is B =-8 a---~' (67) less than zero. Bi can be either positive or negative. If it is negative, the total relative where J* is the area enclosed by the sep- change in energy is negative, and the trajec- aratrix. The probability of capture, then, tory continues losing energy with respect to is the ratio of the rate of increase of the the separatrix, and escapes. If Bi > 0, and combined area of the libration islands to the Bi + Bo < 0, the same thing happens. How- rate of increase of the area enclosed by the ever, if Bi + B0 > 0, and there is a net gain outer boundary of the separatrix. This has a in relative energy, capture will occur. The nice intuitive interpretation in terms of outcome will depend on both the phase and Liouville's theorem. energy of the trajectory relative to the sep- When the standard theory is applied to aratrix just before crossing. Since these the two J : J + 2 resonances in our problem are generally unknown, only an estimate of treated individually, the probabilities of the likelihood of various outcomes can be capture are 2 PA = (68) 1 + zr/(2 sin -l (-2F/X/-F(8 + 2(C - D)))) for the Ariel resonance and 2 Pu = (69) 1 + 7r/(2 sin -1 (-2H/'X/-H(8 - 2(C - D)))) 226 TITTEMORE AND WISDOM for the Umbriel resonance. The value of 8 probability function for a number of sets of is that at which the trajectory crosses the 100 trajectories, the trajectories in each set separatrix. having the same initial eccentricities, but For-2(C-D)+4F_8_<-2(C-D) different phases. Different sets have differ- --4F, PA= 1, andfor2(C-D)+4H~8 ent initial eccentricities. The mapping (44) --< 2(C - D) - 4H, Pu = 1. For larger val- with all coupling terms set to zero was ues of 8, the dependence of probability on used. We have taken a number of steps to eccentricity far from resonance can be ensure that various sources of error are un- found. The action of a trajectory is to first der control. order adiabatically invariant until just be- The values predicted for B0 are esti- fore it crosses the separatrix. At this point, mates. In order to avoid possible "edge ef- the action is approximately equal to the fects" by under- or oversampling ranges of area enclosed by the outer separatrix, for B0 which result in capture or escape, and to which analytic expressions can be found, ensure that the distribution of phases just before crossing the separatrix is uniform, ,l-F(8 + 2(C - D) - 4F 2) J~ 2 we have randomly distributed initial ener- 16B 2 gies and phases far from resonance such _(8 + 2(C- D)) [Tr that near the separatrix the spread in en- 4B L2 ergy is uniform across approximately 100 times the energy balance of the outer sep- + sin-I X/-F(8 + 2(C - D)) , (70) aratrix at the value of 8 at which it is pre- dicted to cross. This has taken into consid- for the Ariel resonance and eration the approximately linear spread in energies of trajectories with slight initial ,/-/-/(8 - 2(C - D) - 4H 2) J~j 2 differences over the range of 8 between the 16B z initial value and the value at separatrix (8 - 2(C - D)) [~r crossing. - 4B [ 2 This initial spread in energies corre- sponds to a spread of initial eccentricities. + sin l ( X/-H(8 + 2(C- D)) )] ' (71) Because of the variation of capture proba- bility with initial eccentricity in the single for the Umbriel resonance. resonance model, trajectories at opposite By considering the trajectories far from ends of the range of initial eccentricity have resonance to be approximately circular slightly different capture probabilities. This with actions Jr,' -- 2~rEio, and equating these contributes some error to the estimate of with the above expressions, the depen- the probability: dence of capture probability on eccentricity I far from resonance can be determined. Us- APc = Aeo × °Pc l ing the above expressions, eA < 0.00404 0---~-I 0.08 , , , , 0.06 0.04 t I I I I I 0.02 x ~ ,-o. _ _ ,.o 0 I I I I 0 0.005 0.01 0.015 0.02 0.025 go 0.08 9 t l t 0.06 l l I l ~' 0.04 I I t t 0.02 x 0 I I I 0 0.005 0.01 6.015 0.02 0.025 e o FIG. 30. Expected probability variations in single resonance capture probabilities. (a) Ariel (e~,) resonance, (b) Umbriel (e~) resonance. Solid line: expected statistical fluctuation for 100 trajectories. Points: initial spread of eccentricities multiplied by the slope of the probability function. 228 TITTEMORE AND WISDOM I I I I 1.0 0.8 0.6 P 0.4 0.2 0 0 0.005 0.01 0.015 0.02 0.025 eo I I I I 1.0 0.8 0.6 P 0.4 0.2 b 0 I I I I 0 0.005 0.01 0.015 0.02 0.025 e0 FIG. 31. Single resonance capture probabilities. (a) Ariel (e~) resonance, (b) Umbriel (e~j) resonance. Solid lines are theoretical predictions from the single-resonance theory. Data points are measured capture probabilities for ensembles of 100 trajectories evolved through the resonance with the uncou- pled mapping. The mapping method properly reproduces the capture probabilities within the range of eccentricities where the approximations concerning the tidal friction are valid (see Section 4.1). proportional to 12 through the K and L func- pected probability errors have been esti- tions (58). 1~ = 80 year -~ has been chosen to mated. These are plotted in Fig. 30 for the keep the error less than about 1%. variation over the probability functions and Based on the above discussion, the ex- for the statistical fluctuations. The energy PASSAGE THROUGH 5 : 3 URANIAN RESONANCE 229 error is less than about 1%. The statistical CHIRIKOV, B. V. 1979. A universal instability of many fluctuations dominate over most of the dimensional oscillator systems. Phys. Rep. 52, 263- range except at the steepest parts of the 379. COUNSELMAN, C. C., AND I. I. SHAPIRO, 1970. Spin- probability functions. orbit resonance of Mercury. Symposia Mathe- Numerical measurements of the capture matica 3, 121-169. probabilities for both resonances were com- DERMOTT, S. F. 1984. Origin and evolution of the Ura- puted by running sets of 100 trajectories nian and Neptunian satellites: Some dynamical con- through the resonance using the map, each siderations. In Uranus and Neptune (J. Bergstralh, Ed.), NASA Conference Publication 2330, pp. 377- with the aforementioned energy and phase 404. spreads and (e0) = 0.0025, 0.005, 0.0075, DERMOTT, S. F., AND P. D. NICHOLSON 1986. Masses 0.01, 0.0125, 0.015, 0.0175, and 0.02. of the satellites of Uranus. Nature 319, 115-120. Results are plotted in Fig. 31, along with FRENCH, R. G., J. L. ELLIOT, AND S. L. LEVlNE 1985. the predictions of the analytic theory. The Structure of the Uranian rings. II. Perturbations of the ring orbits and widths. Icarus 67, 134-163. agreement with the analytical predictions is GAVRILOV, S. V., AND V. N. ZHARKOV 1977. Love very good. Measured errors are somewhat numbers of the giant planets. Icarus 32, 443-449. less than expected on the steep part of the GOLDREICH, P. 1963. On the eccentricity of satellite probability function. orbits in the Solar System. Mon. Not. R. Astron. The agreement between our numerical Soc. 126, 257-268. GOLDREICH, P. 1965. An explanation of the frequent results for these single resonance problems occurrence of commensurable mean motions in the and the expectations from the single reso- Solar System. Mon. Not. R. Astron. Soc. 130, 159- nance model give us confidence that our 181. methods can be used to reliably examine GOLDREICH, P., AND S. J. PEALE 1966. Spin-orbit the passage through the more complicated coupling in the Solar System. Astron. J. 71, 425- 438. resonances considered in this paper. H~NON, M. 1966a. Exploration numtrique du pro- blame restreint. III. Masses 6gales, orbites non pt- ACKNOWLEDGMENTS riodiques. Bull. Astron. Paris 1, fasc. 1, 57. 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