ICARUS 58, 137-152 (1984)
The Chaotic Rotation of Hyperion*
JACK WISDOM AND STANTON J. PEALE Department of Physics, University of California, Santa Barbara, California 93106
AND
FRANGOIS MIGNARD Centre d'Etudes et de Recherehes G~odynamique et Astronomique, Avenue Copernic, 06130 Grasse, France
Received August 22, 1983; revised November 4, 1983
A plot of spin rate versus orientation when Hyperion is at the pericenter of its orbit (surface of section) reveals a large chaotic zone surrounding the synchronous spin-orbit state of Hyperion, if the satellite is assumed to be rotating about a principal axis which is normal to its orbit plane. This means that Hyperion's rotation in this zone exhibits large, essentially random variations on a short time scale. The chaotic zone is so large that it surrounds the 1/2 and 2 states, and libration in the 3/2 state is not possible. Stability analysis shows that for libration in the synchronous and 1/2 states, the orientation of the spin axis normal to the orbit plane is unstable, whereas rotation in the 2 state is attitude stable. Rotation in the chaotic zone is also attitude unstable. A small deviation of the principal axis from the orbit normal leads to motion through all angles in both the chaotic zone and the attitude unstable libration regions. Measures of the exponential rate of separation of nearby trajectories in phase space (Lyapunov characteristic exponents) for these three-dimensional mo- tions indicate the the tumbling is chaotic and not just a regular motion through large angles. As tidal dissipation drives Hyperion's spin toward a nearly synchronous value, Hyperion necessarily enters the large chaotic zone. At this point Hyperion becomes attitude unstable and begins to tumble. Capture from the chaotic state into the synchronous or 1/2 state is impossible since they are also attitude unstable. The 3/2 state does not exist. Capture into the stable 2 state is possible, but improbable. It is expected that Hyperion will be found tumbling chaotically.
I. INTRODUCTION some time that Iapetus rotates synchro- nously (Widorn, 1950). Since the time scale The rotation histories of the natural satel- for the despinning of Hyperion is somewhat lites have been summarized by Peale less than that for Iapetus, it is likely that (1977). Most of the natural satellites fall Hyperion has significantly evolved as well. into one of the two well-defined categories: As a satellite tidally despins, it may be those which have evolved significantly due captured in a variety of spin-orbit states to tidal interactions and those which have where the spin rate is commensurate with essentially retained their primordial spins. the orbital mean motion. Mercury, how- The exceptions are Hyperion and Iapetus, ever, is the only body in the solar system for which the time scales to despin to spin which is known to have a nonsynchronous rates which are synchronous with their re- yet commensurate spin rate (see Goldreich spective orbital mean motions are esti- and Peale, 1966). Among the tidally mated to be on the order of one billion evolved natural satellites, where the spin years. However, it has been known for rates are known the satellites are all in syn- chronous rotation, and in those cases * Paper presented at the "Natural Satellites Confer- where the spin rate is not known the proba- ence," Ithaca, N.Y., July 5-9, 1983. bility of capture in a nonsynchronous state 137 0019-1035/84 $3.00 Copyright © 1984 by Academic Press, Inc. All rights of reproduction in any form reserved. 138 WISDOM, PEALE, AND MIGNARD
is small. Most of the tidally evolved satel- that the spin axis is normal to the orbit lites are expected to be synchronously ro- plane. In Section IV the stability of this ori- tating. Hyperion is the only remaining pos- entation is examined for the spin-orbit sibility for an exotic spin-orbit state (see, states, where it is shown that for principal e.g., Peale, 1978). We shall see that it may moments appropriate for Hyperion the syn- indeed be exotic. chronous and 1/2 spin-orbit states are atti- In their original paper on spin-orbit cou- tude unstable! In Section V rotation in the pling, Goldreich and Peale (1966) derive a chaotic zone is also shown to be attitude pendulum-like equation for each spin-orbit unstable. The resulting three-dimensional state by rewriting the equations of motion tumbling motions are considered in Section in terms of an appropriate resonance vari- VI, and shown to be fully chaotic. Conse- able and eliminating the nonresonant, high- quences of these results for the tidal evolu- frequency contributions through averaging. tion of Hyperion are discussed in Section The strength of each resonance depends on VII and it is concluded that Hyperion will the orbital eccentricity and the principal probably be found to be chaotically tum- moments of inertia through (B - A)/C. As bling. A summary follows in Section VIII. long as (B - A)/C ~ 1, averaging is a good approximation and the resulting spin states are orderly. However, the figure of Hyper- II. SPIN-ORBIT COUPLING REVISITED ion has been determined from Voyager 2 Consider a satellite whose spin axis is images (Smith et al., 1982; T. C. Duxbury, normal to its orbit plane. The satellite is 1983, personal communication) and (B - assumed to be a triaxial ellipsoid with prin- A)/C -~ 0.26. This is significantly larger than cipal moments of inertia A < B < C, and C the hydrostatic value assumed in Peale is the moment about the spin axis. The orbit (1978), and averaging is no longer an appro- is assumed to be a fixed ellipse with semi- priate approximation. In fact, the reso- major axis a, eccentricity e, true anomaly f, nance overlap criterion (Chirikov, 1979) instantaneous radius r, and longitude of pe- predicts the presence of a large zone of cha- riapse o3, which is taken as the origin of otic rotation. longitudes. The orientation of the satellite's In this paper, we reexamine the problem long axis is specified by O and thus 0 - f of spin-orbit coupling for those cases measures the orientation of the satellite's where averaging is not applicable, with spe- long axis relative to the planet-satellite cen- cial emphasis on parameters appropriate ter line. This notation is the same as that of for Hyperion. In the next section, the prob- Goldreich and Peale (1966). Without exter- lem is recalled and the qualitative features nal tidal torques, the equation of motion for of the nonlinear spin-orbit problem are dis- O (Danby, 1962; Goldreich and Peale, 1966) cussed. One mechanism for the onset of is chaos, the overlap of first-order reso- nances, is briefly reviewed and the reso- ~o + o,o~ (1) nance overlap criterion is used to predict dt 2 ~ sin 2(0 - 30 = 0, the critical value of (B - A)/C above which there is large-scale chaotic behavior. In where to20 = 3(B - A)/C and units have been Section III, the spin-orbit phase space is chosen so that the orbital period of the sat- numerically explored using the surface of ellite is 27r and its semimajor axis is one. section method. The existence of the large, Thus the dimensionless time t is equal to chaotic zone is verified, and the critical the mean longitude. Since the functions r value for the onset of chaos is compared to and fare 2~r periodic in the time, the second the prediction of the resonance overlap cri- term in Eq. (1) may be expanded in a terion. In Section II and III it is assumed Fourier-like Poisson series giving THE CHAOTIC ROTATION OF HYPERION 139
of the resonance is characterized by the dt 2 + -~ ~ H -~, e sin(20 - rot) = O. maximum value of dyJdt on the separatrix. rn = -~ When Ye librates Idyp/dt[ is always less than The coefficients H(m/2,e) are proportional this value which is equal to O~o~/]H(p,e)[. to e 21(m/2)-ll and are tabulated by Cayley Averaging is most useful for studying the (1859) and Goldreich and Peale (1966). motion near a resonance when the reso- When e is small, H(m/2,e) ~- -½e, 1, ½e for nance half-widths are much smaller than m/2 = ½, 1, ~, respectively. The half-integer their separation. In this case, most solu- m/2 will be denoted by the symbol p. tions of the actual equation of motion differ Resonances occur whenever one of the from those of the averaged equations by arguments of the sine functions is nearly only small regular oscillations resulting stationary, i.e., whenever I(dO/dt) - Pl ~ ½. from the nonresonant, high-frequency In such cases it is often useful to rewrite the terms. An important exception occurs for equation of motion in terms of the slowly motion near the infinite-period separatrix varying resonance variable yp = 0 - pt, which is broadened by the high-frequency terms into a narrow chaotic band (Chirikov, d2yp + 0~2 dt 2 -~ H(p,e) sin 2yp 1979). While the band is present for all val- ues of too it is extremely narrow for small £02 w0. Chirikov has given an estimate of the e) sin(27p - nt) =0. + "-2 ,~o H (p +5' half-width of this chaotic separatrix, which (2) is expressed in terms of the chaotic varia- tions of the integral Ip, viz., ff oJ0 is small enough the terms in the sum
will oscillate rapidly compared to the much Ip -- lp s = 47reh3e(_lrx)/2 ' slower variation of yp determined by the we = lpS first two terms and consequently will give where e is the ratio of the coefficient of the little net contribution to the motion. As a nearest perturbing high-frequency term to first approximation for small w0, then, these the coefficient of the perturbed term, and h high-frequency terms may be removed by = l~/oJ is the ratio of the frequency differ- holding yp fixed and averaging Eq. (2) over ence between the resonant term and the an orbital period. The resulting equation is nearest nonresonant term (f~) to the fre- dZT~ oj2 quency of small-amplitude librations (oJ). + ~- H(p,e) sin 2yp = 0 dt 2 x. For the synchronous spin-orbit state per= turbed by the p = ~ term, e = H(~,e)/H(l,e) and is equivalent to that for a pendulum. = (7e)/2 and h = 1/oJ0. Thus The first integral of this equation is 11 - Ii s 147ree_(~/2,o0)" (3) 1 (dr.)2 ,oi wl - ~ I~ - oJ~- Ip = ~ \ dt / - ~ H(p'e) c°S 2yp; For Mercury, for instance, where co0 = yp librates for I~, < Ip s and circulates for lp > 0.017 (for (B - A)/C = 10 -4) and e = 0.206, ips, where the separating value Ip s = (~o2/4) the width of the chaotic region associated ]H(p,e)]. For Ip < 1~,s and H(p,e) > O, 7p with the synchronous state is wl = 1.4 × librates about zero; while for Ip < Ip s and 10-34, and a similar estimate for the width of H(p,e) < O, yp librates about ~-/2. In both the p = 3/2 chaotic band gives w3/2 = 5.4 × cases the frequency of small-amplitude os- 10-43! Averaging is certainly a good approx- cillations is O~oV]H(p,e) I. For Ip = Ip s, yp imation for Mercury. On the other hand, follows the infinite period separatrix which the width of the chaotic layer depends ex- is asymptotic forward and backward in time ponentially on o~0, and as too increases the to the unstable equilibrium. The half-width size of the chaotic separatrix increases dra- 140 WISDOM, PEALE, AND MIGNARD
matically. Now most of the natural satel- section, where the spin-orbit phase space lites are expected, on the basis of hydro- is investigated numerically using the sur- static equilibrium, to have values of o~0 face of section method. larger than that expected for Mercury, in several cases approaching unity (Peale, 1977). At such large a)0 chaotic separatrices III. THE SPIN-ORBIT PHASE SPACE are a major feature in the phase space. In Most Hamiltonian systems display both studying the rotations of the natural satel- regular and irregular trajectories. The lites caution must be exercised when using phase space is divided; there are regions in the averaging method. which trajectories behave chaotically and The widths of the libration regions also regions where trajectories are quasiperiodic grow as too increases. At some point their (Hrnon and Heiles, 1964). The simplest and widths, as calculated above using the aver- most intuitive method of determining aging method, are so large that the reso- whether a trajectory is chaotic or quasipe- nances begin to overlap. Analyzed sepa- riodic is the surface of section method. The rately, libration would be expected in each spin-orbit problem, as defined in the last of two neighboring resonances. However, section, is 27r periodic in the dimensionless simultaneous libration in two spin-orbit time. A surface of section is obtained by states is impossible. The result is wide- looking at the system stroboscopically with spread chaotic behavior. An estimate of the period 2rr. A natural choice for the section too at which this happens is provided by the is to plot d~9/dt versus t9 at every periapse Chirikov resonance overlap criterion. This passage. The successive points define criterion states that when the sum of two smooth curves for quasiperiodic trajecto- unperturbed half-widths equals the separa- ries; for chaotic trajectories the points ap- tion of resonance centers, large-scale chaos pear to fill an area on the section in an ap- ensues. In the spin-orbit problem the two parently random manner. It is a remarkable resonances with the largest widths are the p property of Hamiltonian systems that these = 1 and p = 3/2 states. For these two states two types of behavior are usually readily the resonance overlap criterion becomes distinguishable and that they are generally both present on any surface of section. 1 co0ROM'IH(I,e)] + O~oR°X/IH(3/2,e)] = Because of the symmetry of a triaxial el- lipsoid, the orientation denoted by va is or equivalent to that denoted by O + rr. Conse- quently, O may be restricted to the interval 1 from 0 to rr. The spin-orbit states found in °~0RO = 2 + k/~e" (4) the previous section by the averaging method are states where a resonance vari- For e = 0.1, the mean eccentricity of Hype- able yp = O - pt librates. For each of these rion, the critical value of too above which states dO/dt has an average value precisely large-scale chaotic behavior is expected is equal to p, and O rotates through all values. o0R° = 0.31. This is well below the actual If attention is restricted, however, to the value of Hyperion's COo which has been de- times of periapse passage, i.e., t = 27rn, termined from Voyager 2 images to be o~0 = then each yp taken modulo ~- is simply O. A 0.89 ___ 0.22 (T. C. Duxbury, 1983, personal libration in yp becomes a libration in O on communication). It is expected then that the surface of section. For quasiperiodic li- for Hyperion there is a large chaotic zone bration successive points trace a simple surrounding (at least) the p = 1 and p = 3/2 curve on the section near dO/dt = p which states, and possibly more. covers only a fraction of the possible inter- These predictions are verified in the next val from 0 to 7r. For nonresonant quasipe- THE CHAOTIC ROTATION OF HYPERION 141
I I I given by Eq. (I). For Hyperion, e = 0.1, and to0R° = 0.31. Numerically, we find that the p = 1 and p = 3/2 chaotic zones merge between tOo = 0.25 and tOo = 0.28. The pre- diction of the resonance overlap criterion is in excellent agreement with the numerical results, especially considering that tOo var- ies over two orders of magnitude for the natural satellites. As tOo is further increased the simplicity
O I I I of the picture developed for small tOo disap- o 7r/2 pears. The now large chaotic zone sur- 0 rounds more and more resonances, and the FIG. 1. Surface of section for tOo = 0.2 and e = 0.1. sizes of the principal quasiperiodic islands dO/dt versus 0 at successive periapse passages for ten decrease. Figure 2 illustrates the main fea- separate trajectories: three illustrating quasiperiodic tures of the surface of section for e = 0.1 libration, three illustrating the surrounding chaotic lay- ers and four illustrating that quasiperiodic rotation and tOo = 0.89, values appropriate for Hype- separates the chaotic zones. rion. The chaotic sea is very large, sur- rounding all states from p = 1/2 to p = 2. Notice the change in scale from Fig. 1. The riodic trajectories, all Tp rotate, and succes- tiny remnant of the p = 1/2 island is in the sive points on the surface of section will lower center of the chaotic sea; the p = 3/2 trace a simple curve which covers all values island has disappeared altogether. The sec- of O. For small tOo, resonant states are sepa- ond-order p = 9/4 island in the top center of rated from nonresonant states by a narrow the chaotic zone is now one of the major chaotic zone, for which successive points features of the section. A total of 17 trajec- fill a very narrow area on the surface of tories of Eq. (1) were used to generate this section. The surface of section displayed in figure: eight quasiperiodic librators, illus- Fig. 1 illustrates these various possibilities trating the p = 1/2, 1, 2, 9/4, 5/2, 3, and 7/2 for tOo = 0.2 and e = 0.1. Equation (1) was states, five nonresonant quasiperiodic rota- numerically integrated for ten separate tra- tors, and four chaotic trajectories (one for jectories and dO/dt was plotted versus 0 at every periapse passage. Three trajectories illustrate quasiperiodic libration in the p = 1 (synchronous), p = 1/2, and p = 3/2 ,,,,,1~. ":" ,',-,.~ ...... ,.,.4- ~'"""" states. Three trajectories illustrate the cha- otic separatrices surrounding each of these resonant states, and four trajectories show dO 2 that each of these chaotic zones is sepa- dt rated from the others by impenetrable non- resonant quasiperiodic rotation trajecto- ries. Five hundred successive points are plotted for each quasiperiodic trajectory, and 1000 points for each chaotic trajectory. o 7r/a 7r As tOo is increased both the resonance 0 widths and the widths of the chaotic separa- trices grow. The resonance overlap crite- FIc. 2. Surface of section for tOo = 0.89 and e = 0.1. Hyperion's spin-orbit phase space is dominated by a rion predicts that the chaotic zones will be- chaotic zone which is so large that even the p = 1/2 gin to merge when tOo > tO0R°, where tO0R° is and p = 2 states are surrounded by it. 142 WISDOM, PEALE, AND MIGNARD
values of 0. Whenever the island was sur- rounded by the large chaotic sea, the line has been broadened.The resonance curves in Fig. 3 thus give a clear picture of which ...... }~._~.. states may be visited by traveling in the dO chaotic sea for any particular too. It is inter- d-T['°°------'~. esting to note that for too within the range 1.27 < tOo < 1.36 two different synchronous islands are simultaneously present. For the large tOo and e appropriate for Hy- perion the forced librations are not well ap- proximated by the linear theory that was I I I 0.4 0.8 1.2 1.6 used for Phobos. However, good approxi- Wo mations are obtained from the nonlinear Fro. 3. Major island centers versus to0. Each island method of Bogoliubov and Mitropolsky may be identified by the value of dO/dt at too = 0, (1961). If we define tOp = vq - pfand Eq. (1) except for the second synchronous branch which ap- is rewritten with the true anomaly as the pears in the upper right quadrant. A broad line indi- independent yariab!e, the equation of mo- cates that the island is surrounded by the large chaotic zone. These resonance curves summarize the surfaces tion for tOp becomes of section by showing which states may be reached by d% traveling in the chaotic sea for any particular too. The (1 + e cos f) ~ - 2e sinf(p + df/ dotted lines show the usual linear approximation for the forced librations in the synchronous state, while tOg the dashed lines show a much superior nonlinear ap- + ~- sin 2(top + (p - 1)f) = 0. (5) proximation. The island centers are fixed points on the surface of section, thus top(f) is 2¢r periodic. each chaotic zone). Two thousand points This suggests that tOp bewritten as a Fourier are plotted in the large chaotic sea. While the average value of dO/dt is pre- series, tOp(f) = tOp0+ ~] tOpk sin kf. tOpo as- k=l cisely p for a quasiperiodic librator in state sumes a value of 0 or 7r/2 depending on p, on the surface of section the island cen- whether libration is about O equal to 0 or ters are displaced from these values. This 7r/2, respectively, and only sine terms are displacement results from a forced libration included since the equation of motion is in- with the same period as the orbital period variant under a simultaneous change in sign and amplitude (in the variations of 0 and of tOp and f. A first approximation to the dO~dO equal to the displacement. This phe- solution is obtained by retaining only the nomenon is familiar from the forced libra- first term (k = 1) in this Fourier series. Sub- tion of Phobos (Duxbury, 1977; Peale, stituting this into Eq.,(5), multiplying by 1977). A convenient way to summarize the sin(f) and integrating from 0 to 27r yields an results of the surfaces of section for various implicit equation for the amplitude tOp' tOo is to plot the location of all the major island centers. The resonance centers oc- tO2[J3_zp(2tOp' ) A- (-1)2pJ2p_l(2tOp')] = cur at 0 = 0 or O = 7r/2, so it is sufficient to (2tOp I + 4ep)(-1)~2% °)/~ plot only dO/dt versus tOo. This plot is pre- where the Ji are the usual Bessel functions. sented in Fig. 3 for an orbital eccentricity of The dashed line in Fig. 3 shows the solution 0.1. Curves with different symmetry may to this equation for p = 1, where cross. For example, the p = 1 and p = 1/2 curves cross, yet the islands are always dis- dO dtop) df (p + top,) (f(1+ --e~/2 e) 2 tinct since their centers occur at different d---7 = p+ df /--~ = THE CHAOTIC ROTATION OF HYPERION 143
on the surface of section (at periapse). Evi- study. A more convenient set of angles has dently, even one term in this Fourier series therefore been chosen and is specified rela- is a much better representation of the full tive to an inertial coordinate system xyz solution than the usual linear solution (see which is defined at periapse. The x axis is Peale, 1977) which is drawn as a dotted line chosen to be parallel to the planet to satel- in Fig. 3. Though they are not drawn in Fig. lite radius vector, the y axis parallel to the 3 the nonlinear approximations forp = 1/2 orbital velocity, and the z axis normal to the and p = 3/2 are also quite good. orbit plane so as to complete a right-handed Up to now it has been assumed that the coordinate system. Three successive rota- spin axis is perpendicular to the orbit plane. tions are performed to bring the abc axes to However, Fig. 2 bears little resemblance to their actual orientation from an orientation the picture of spin-orbit coupling devel- coincident with the xyz set of axes. First, oped for small tOo. In the next section this the abc axes are rotated about the c axis by question of attitude stability is reevaluated an angle O. This is followed by a rotation in this now strongly nonlinear regime. about the a axis by an angle ~p. The third rotation is about the b axis by an angle tO. ' IV. ATTITUDE STABILITY OF ROTATION AT The first two rotations are the same as the THE ISLAND CENTERS Euler rotations, but their names have been JConsider now -the fully three-dimensional interchanged. In terms of these angles the motion of a triaxial ellipsoid in a fixed ellip- three angular velocities are tical orbit, which is specified as before. Let a, b, and c denote a right-handed set of axes dO ds0 toa = - d--t-cos ~o sin tO + ~- cos tO, fixed in the satellite, formed by the princi- pal axes of inertia with moments A < B < dO dtO C, respectively. In this case Euler's equa- COb = ~ sin ~o + d---t-' tions are (Danby, 1962) dO d~ A dtOa 3 toc = -~- cos ~ cos tO + ~ sin tO, dt - tObtOc(B - C) = - -fi fly(B ~- C), and the three direction cosines are dtob 3
B -d-i- - toctoa(C -- A) = - --fi ya(C - A), c~ = cos tO cos(O -f)
- sin tO sin ¢ sin(O - f), (6) /3 = - cos ~ sin(O - f), dtoc 3 C--d- t - - toatob(A -- B) = - -fi otfl(A - B), y = sin tO cos(O - f) + cos tO sin ~o sin(O - f). where toa, tob, and to~ are the rotational an- gular velocities about the-three axes a, b, The equations of motion for O, ~,, and tO are and ¢, respectively, and ~, t, and ~/are the then derived in a straightforward manner. direction cosines of the planet to satellite For reference, the three canonically conju- radius vector on the same three axes. gate momenta are To solve these equations, a set of gener- Po = -Atoacos ~p sin tO + Btobsin ~o alized coordinates to specify the orientation + Ctoccos ~p cos tO, of the satellite must be chosen. The Euler angles (as specified in Goldstein, 1965) are p, = Ato~cos tO + Ctocsin tO, not suitable for this purpose because the p~ = Btob. resulting equations have a coordinate sin- gularity when the spin axis is normal to the When 9, tO; P~,: and p, are set equal to orbit plane, which is just the situation under zero, they remain equal to zero. In this 144 WISDOM, PEALE, AND MIGNARD equilibrium situation the spin axis is normal B>C to the orbit plane and O is identical to the O ,.oI used in the two previous sections. At the island centers O(t) is periodic and the stabil- ~o + ,,~. +~,,~k~%~O¢,,o o / ity of this configuration may be determined O.Bi + e' + # + ~\~\/(\~\ff/ o / by the method of Floquet multipliers (see, ,~ ,/~k~*# o ~ + ,1~,_X¢~)o / B/C ; ~o~,; .\x.'~l'~.#,o,'%* ~o +,,%o'7~o+~ o ~ o I/ A,B e.g., Poincarr, 1892; Cesari, 1963; Kane, / ",;/ .toSt o ~ o ~¢ o / 1965). A trajectory near this periodic trajec- 0.6 ..,gU...~%w., o ~ o / tory is specified by O' = O + 80, 9' = 9 + t
89, to' = tO + 8to, p~ = Po + Spa, p~, = p~ + _ ,~ .o/ 8p~, and p; = p, + 8p,. The equations of A+B
B>C the p = 9/4 state is similar to that for the p 1.0 = 2 state, and mainly indicates stability. The p = 3/2 state is likewise mainly stable, but exists only for ~0 < 0.56. To summarize, for principal moments
o ~ + within the error ellipsoid for Hyperion the B/C + ~ + synchronous (p = 1) and p = 1/2 states are mainly attitude unstable, while the p = 2 and p = 9/4 states are stable. Except for ~2" ,9 / certain moments of inertia near the edge of 0.4 - A+B They may be calculated from a characteris- In this section we are concerned with at- tic equation determined from a numerical titude stability. In all cases the reference integration over one orbit period because trajectory has its spin axis normal to the the reference trajectory is periodic and the orbit plane. An attitude instability is indi- variations are being subjected to the same cated if the spin axes of neighboring trajec- "forces" over and over again. When there tories exponentially separate from the equi- is no underlying periodicity, the Floquet librium orientation. If the reference method is not useful. Rather, a measure trajectory is quasiperiodic then one pair of of exponential separation is provided by LCEs must be zero, and instability is indi- the Lyapunov characteristic exponents cated if any other LCE is nonzero. On the (LCEs). The Lyapunov characteristic ex- other hand, if the reference trajectory is ponents play a dual role in this paper. In chaotic, one LCE must be positive (see this section, they are defined and used to Section VI), and attitude instability is indi- determine the attitude stability of the large cated if two or more LCEs are positive. In chaotic zone, while in the next section they cases where the reference trajectory is peri- are used as indicators of chaotic behavior. odic there is a correspondence between the (See Wisdom (1983) for more discussion of Floquet multipliers, ai, and the LCEs, h~, these exponents.) viz., for every i there is a j such that X] = The LCEs measure the average rate of (lnlail)/T, where T is the period of the refer- exponential separation of trajectories near ence trajectory. For every Floquet multi- some reference trajectory. They are defined plier with modulus greater than one, there as is a Lyapunov exponent greater than zero. The calculation of the largest LCE is not X = lira y(t) = lim ln[d(t)/d(to)] difficult, but to determine the attitude sta- t~ t--,~ t - to ' (7) bility of the large chaotic zone it is neces- where sary to determine at least the two largest LCEs, as one exponent must be positive to d(t) = reflect the fact that the reference trajectory "~/8192 + 8(02 + 8to2 + 8po 2 + 8p¢ 2 + 8ptk2, is chaotic. Because several different rates 5 of exponential growth are simultaneously the usual Euclidean distance between the present, the numerical determination of reference trajectory and some neighboring more than the largest LCE is not a trivial trajectory. The variations 80, 8¢, 8tO, etc. task: The algorithm used here is that de- satisfy the same six linear first-order differ- vised by Benettin et al. (1980b). ential equations as in the last section. The The infinite limit in Eq. (7) is of course difference is that now the reference trajec- not reached in actual calculations. If)~ = 0, tory need no longer be periodic. In general, then d(t) oscillates or grows linearly and as the direction of the initial displacement y(t) approaches zero as ln(t)/t. If, however, vector is varied, h may take at most N val- h 4:0 then y(t) approaches this nonzero ues, where N is the dimension of the sys- limit. These two cases are easily distin- tem. In Hamiltonian systems, the hi are ad- guished on a plot of log y(t) versus log t, ditionally constrained to come in pairs: for where the ln(t)/t behavior appears roughly every hi > 0 there is a hj < 0, such that hi + as a line with slope - L This is illustrated in X] = 0. Thus in the spin-orbit problem only Fig. 7 where calculations are presented of three LCEs are independent; it is sufficient the three largest LCEs for the synchronous to only study those which are positive or island center with moments appropriate for zero. (For a review of the mathematical Hyperion (A/C = 0.5956 and B/C = 0.8595). results regarding LCEs see Benettin et al. Two LCEs are approaching a (positive) (1980a)). nonzero limit and one has the behavior ex- THE CHAOTIC ROTATION OF HYPERION 147 0 0.1. The e-folding time is thus of order 10 or only two orbital periods! These attitude in- -1 stabilities are very strong. log107 VI. CHAOTIC TUMBLING -3 Almost all trajectories initially near a -4 chaotic reference trajectory separate from it exponentially on the average, while al- -5 8 8 most all trajectories initially near a quasipe- log lOt riodic reference trajectory separate from it roughly linearly. Chaotic behavior can thus FiG. 7. Lyapunov characteristic exponents for the synchronous island center with moments appropriate be detected by examining the behavior of for Hyperion. Two exponents are positive and the neighboring trajectories. The rate of expo- third has the behavior expected of a zero exponent. nential divergence of nearby trajectories is quantified by the Lyapunov characteristic exponents which were introduced in the pected of a zero exponent. This verifies the last section. A nonzero LCE indicates that result of the Floquet analysis that with the reference trajectory is chaotic. If all of these moments rotation in the synchronous the LCEs are zero then the reference tra- island is attitude unstable. As a further jectory is quasiperiodic. More generally, check, the attitude stability of all the syn- every pair of zero exponents indicates the chronous island centers previously deter- existence of an "integral" of the motion. mined by the Floquet method were redeter- The trajectory is "integrable" if all LCEs mined with the LCEs. In all cases the two are zero. methods agreed, both qualitatively and In the previous two sections several quantitatively. cases of attitude instability were found. Finally, the stability diagram for the large However, the methods used are only indi- chaotic zone is shown in Fig. 8. For all val- cators of linear instability since the equa- ues of A/C and B/C which were studied, tions of motion for the variations were lin- reference trajectories in the large chaotic earized. It is possible that when the zone with axes perpendicular to the orbit orientation of the spin axis normal to the plane have three positive LCEs. This indi- cates attitude instability; small displace- ments of the spin axis from the orbit normal ,.ol B >C grow exponentially for all trajectories in the large chaotic sea! The LCEs also provide a time scale for the divergence of the spin axis from the or- bit normal. Since neighboring trajectories .\.y / separate from the reference trajectory as B,c 1F " ,g~'~X.* ~,,- * ,. ~ m'XA,o/ e xt, the e-folding time for exponential diver- o.o Y gence is 1/h. For quasiperiodic (or periodic) I ~9 .h;"/ V reference trajectories the appropriate X to r _ " +YC,<>X use is the largest LCE; for chaotic refer- ence trajectories the second-largest LCE is " o'2 ' "o'~ ' o!6 ' o!8 ' ,!o appropriate since at least one of the first A/C two is associated with attitude instability. Fio. 8. Attitude stability diagram for the large cha- For values of the principal moments near otic zone. In all cases studied the large chaotic zone is those of Hyperion these k's are both near attitude unstable. 148 WISDOM, PEALE, AND MIGNARD o i i When ~PE = 0, the Oe and the toe rotations are parallel.) Figure 9 shows the results for -1 two trajectories with initial conditions ~ = -2 0.1, to = 0.01, d~o/dt = O, dto/dt = 0, and dO~ l°glO~ dt = 1. One of them began with 0 - ~r/2 and -3 the other at 0 = ~r/2 + 1/2. The principal -4 moments are A/C = 0.5956 and B/C = 0.8595, values appropriate for Hyperion. -5 2 3 4 The initial conditions are such that the O loglot motion by itself would be chaotic, i.e., the trajectory with the axis of rotation fixed FIG. 9. Three Lyapunov characteristic exponents for two trajectories whose axes were initially slightly perpendicular to the orbit plane would lie in displaced from the orbit normal. The resulting tum- the large chaotic zone. The three LCEs for bling motion is fully chaotic; there are no zero expo- the two trajectories are approaching nents. The LCEs for the two trajectories are approach- roughly the same limits. The results show ing similar values. clearly that the tumbling motion is fully chaotic; none of the LCEs is zero. orbit plane is unstable, the spin axis never- theless remains near that orientation or pe- riodically returns to it. This turns out not to VII. TIDAL EVOLUTION be the case. Wherever the linear analyses We have seen many qualitatively new indicated instability a trajectory slightly features in the rotational behavior of satel- displaced from the equilibrium orientation lites with large oJ0 in eccentric orbits. New was numerically integrated. In every case, features also appear in their tidal evolution. the spin axis subsequently went through In general, tidal dissipation tends to drive large variations and the c body axis went the spin rate of a satellite to a value near more than 90 ° from its original orientation synchronous (e.g., Peale and Gold, 1965). perpendicular to the orbit plane. Still, these In this process the satellite has most likely large excursions could be of a periodic or passed through several stable spin-orbit quasiperiodic nature. states where libration of its spin angular ve- A calculation of the LCEs for one of locity about a nonsynchronous value could these trajectories which originates near the be stabilized against further tidal evolution equilibrium has been made to answer this by the gravitational torque on the perma- question. The algorithm of Benettin et al. nent asymmetry of the satellite's mass dis- (1980b) was again used to avoid the numeri- tribution. Whether or not the satellite will cal difficulties in calculating several LCEs. be captured as it encounters one of these However, an additional difficulty was en- spin-orbit states depends on the spin angu- countered, namely, the equations of motion lar velocity as the resonance variable yp en- as described in Section IV become singular ters its first libration. If the spin rate is be- when ~o = zr/2. At this point the 0 rotation low a critical value capture results, and and the to rotation become parallel. Since otherwise the satellite passes through the all of the angles go through large variations resonance. In most situations there is not this singularity is frequently encountered. enough information to determine if this To navigate past this coordinate singular- condition is satisfied and capture probabili- ity, a change of coordinates was made to ties may be calculated by introducing a suit- the usual Euler angles, which are singular able probability distribution over the initial at ~oE = 0. (The first two Euler rotations are angular velocity. For instance, the capture the same as those used here, but the third is probability may be defined as the ratio of a rotation about the c axis by the angle t0e. range of the first integral Ip which leads to THE CHAOTIC ROTATION OF HYPERION 149 capture to the total range of Ip allowing a for Mercury this inequality is satisfied for first libration (Goldreich and Peale, 1966). all oJ0 > 0.075. This critical o~0 is only a little This standard picture of the capture pro- more than four times Mercury's actual ~o0 cess implicitly assumes that the behavior as assumed in Goldreich and Peale (1966)! near the separatrix is regular and thus well The chaotic separatrix should not be described by the averaged equations of mo- blithely ignored. tion. In general, though, the motion near a If o0 is much larger a trajectory may separatrix in a nonlinear dynamical system spend a considerable amount of time in a is not regular but chaotic. The calculation chaotic zone before escaping or being cap- of capture probabilities is a well-developed tured. Motion in a chaotic zone depends art (see Henrard, 1982; Borderies and extremely sensitively on the initial condi- Goldreich, 1984), but the effect of the cha- tions. Capture will occur if, by chance, the otic separatrix has never been mentioned. trajectory spends enough time near the bor- Of course, when COo is very small the cha- der of the libration zone for the tidal dissi- otic separatrix is microscopically small, pation to take it out of the chaotic region; and even a very small tidal torque can escape occurs if, by chance, the trajectory sweep the system across the chaotic zone spends enough time near the border of reg- so quickly that it has essentially no effect. ular circulation for the tidal dissipation to On the other hand, for larger oJ0 where the move it into the regular region. For a value chaotic zones are sizable, the simple cap- of o0 as large as that of Hyperion, the pic- ture process described above is qualita- ture is even more complicated since many tively incorrect. While still fundamentally islands are accessible to a traveler in the deterministic, the capture process now in- large chaotic sea. Once a trajectory has en- volves the randomness inherent in determi- tered the large chaotic zone, it may repeat- nistic chaos. Probabilities still arise from edly visit each of the accessible states be- unknown initial conditions, but now the fore finally being captured by one of them. outcome is an extremely sensitive, essen- In numerical experiments, this odyssey fre- tially unpredictable function of these initial quently takes a very long time, as com- conditions. The capture process is more pared to capture without a chaotic zone properly viewed as a random process. where capture or escape is decided perma- Following Goldreich and Peale (1966), let nently on a single pass through a reso- AE denote the change in the integral Ip over nance. These experiments were performed one cycle of the resonance variable yp due with principal moments appropriate for Hy- to the tidal torque. When AE is much perion, the spin axis normal to the orbit smaller than the width of the chaotic se- plane and a tidal torque given by paratrix 2wplp the chaotic character of the separatrix may be expected to have a signifi- C d(O - j~ cant effect on the capture process. For the T - r 6 dt p = 3/2 state of a nearly spherical body this condition is which is appropriate when the tidal phase lag is simply proportional to the frequency 15k2R 3 ( 2 ]3/2 1 e ICi e , (Goldreich and Peale, 1966). The constant 8 < 0'O C was chosen for computational conven- ience to be of order 10 -3 . Capture in each where k2 is the Love number, 1/Q is the accessible state appears to be possible, specific dissipation function, R is the ratio though the synchronous state was the most of the radius of the body to the orbit semi- common endpoint. major axis, and/z is the satellite to planet Normally, tidal dissipation not only mass ratio. With parameters appropriate drives the spin rate toward a value near 150 WISDOM, PEALE, AND MIGNARD synchronous, but also drives the spin axis which are separated by times longer than to an orientation normal to the orbit plane. the period of variation and the period is de- Thus, in Goldreich and Peale (1966), for in- termined by folding these observations stance, this orientation for the spin axis is back on each other with an assumed period simply assumed. The discussion of tidal which is varied until the scatter of points evolution is now complicated by the rather about a smooth curve is minimized. This surprising results of Sections IV, V, and VI method will not yield meaningful results if where it was found that in many cases the the period of the observed object varies orientation of the spin axis perpendicular to markedly on a time scale which is short the orbit plane is unstable. In particular, the compared to the time spanning the observa- large chaotic zone is attitude unstable (Sec- tions. Hence the determination of a chaotic tion V). So as soon as the large chaotic light curve requires many magnitude obser- zone is entered the spin axis leaves its pre- vations per orbit period carried out over ferred orientation and begins to tumble cha- several orbit periods. otically through all orientations (Section VI). Capture into one of the attitude-stable islands is still a possibility. However, for VIII. SUMMARY Hyperion, the only attitude-stable end- Hyperion's highly aspherical mass distri- points which are accessible, once the large bution and its large, forced orbital eccen- chaotic sea has been entered, are the p = 2 tricity renders inapplicable the usual theory and p = 9/4 states. The synchronous and p of spin orbit coupling which relies on the = 1/2 states are attitude unstable for most averaging method, In fact, for much smaller values of the principal moments within the (B - A)/C the resonance overlap criterion error ellipsoid of Hyperion and the p = 3/2 predicts the presence of a large chaotic state does not exist for oJ0 above 0.56 (Sec- zone in the spin-orbit phase space, and nu- tion IV). Occasionally, the tumbling satel- merical exploration using the surface of lite may come near one of the attitude-sta- section method has verified its presence. ble islands with its spin axis perpendicular For Hyperion, this chaotic zone is so large to the orbit plane. If it lingers long enough it that it engulfs all states from the p = 1/2 may be captured. However, the chaotic state to the p = 2 state. The p = 3/2 state zone is strongly chaotic (X ~ 0. I) and the has disappeared altogether, and the second- tidal dissipation is very weak (the time order p = 9/4 island is a prominent feature scale for the despinning of Hyperion is of on the surface of section. the order of the age of the Solar System). It Hyperion could stably librate in the syn- may take a very long time for this tumbling chronous spin-orbit state if the spin axis satellite to enter an orientation favorable were able to remain normal to the orbit for capture to occur. Judging from the long plane. However, for most values of the times required in the numerical experi- principal moments within the error ellipsoid ments for capture to occur even when the for Hyperion, Floquet stability analysis in- spin axis was fixed in the required direction dicates that rotation within the synchro- it seems to us unlikely that Hyperion has nous island is attitude unstable. A small ini- been captured. We expect that Hyperion tial displacement of the spin axis from the will be found to be tumbling chaotically. orbit normal grows exponentially and the Preliminary observations of a 13-day pe- axis appears to pass through all orienta- riod (Thomas et aL, 1984; Goguen, 1983) tions. Likewise, the p = 1/2 state is attitude support this conclusion that capture has not unstable for most principal moments near occurred. We should point out that the tra- those estimated for Hyperion. The only at- ditional method of determining periods titude-stable islands in the large chaotic sea from light variations involves observations are the p = 2 and p = 9/4 states. THE CHAOTIC ROTATION OF HYPERION 151 The orientation of the spin axis perpen- tional data before publication and to Donald Hitzl who dicular to the orbit plane is likewise unsta- pointed out some important references. This work was ble for trajectories in the large chaotic supported in part by the NASA Geochemistry and zone. This is indicated by the Lyapunov Geophysics Program under Grant NGR 05 010 062. characteristic exponents which measure the rate of exponential separation of neigh- REFERENCES boring trajectories. Small displacements of BENETTIN, G., L. GALGANI, A. GIORGILLI, AND J.-M. the spin axis from the orbit normal lead to STRELCYN (1980a). Lyapnnov characteristic expo- large displacements. Lyapunov character- nents for smooth dynamical systems and for Hamil- istic exponents for the resulting tumbling tonian systems; a method for computing all of them: motion indicate that it is fully chaotic; there Part 1. Theory. Meccanica March, 9-20. BENETTIN, G., L. GALGANI, A. GIORGILLI, AND J.-M. are no zero exponents. STRELCYN (1980b). Lyapunov characteristic expo- Over the age of the solar system, tidal nents for smooth dynamical systems and for Hamil- dissipation can drive Hyperion's spin to a tonian systems; a method for computing all of them: near synchronous value. The probability of Part 2. Numerical application. Meccanica March, the spin being captured into any of the 21-30. spin-orbit states with p > 2 is negligibly BORDERIES, N., AND P. GOLDREICH (1984). A simple derivation of capture probabilities for the J + 1 : J small, and Hyperion will have necessarily and J + 2:J orbit-orbit resonance problems. Sub- entered the large chaotic zone. At this mitted for publication. point, Hyperion's spin axis becomes atti- BOGOLIUBOV, N. N., AND Y. A. MITROPOLSKY (1961). tude unstable, and Hyperion begins to tum- Asymptotic Methods in the Theory of Non-Linear Oscillations. Hindustan, Delhi. ble chaotically with large, essentially ran- CAYLEY, A. (1859). Tables of the developments of dom variations in spin rate and orientation. functions in the theory of elliptic motion. Mere. Tidal dissipation may lead to capture if Hy- Roy. Astron. Soc. 29, 191-306. perion's spin comes close enough to one of CESARI, L. (1963). Asymptotic Behavior and Stability the attitude-stable islands with its spin axis Problems in Ordinary Differential Equations. perpendicular to the orbit plane. However, Springer-Vedag, Berlin/New York. CHIRIKOV, B. V. (1979). A universal instability of judging from the long times required in nu- many-dimensional oscillators systems. Phys. Rep. merical experiments for capture to occur 52, 263-379. even when the spin axis was fixed in the DANBY, J. M. A. (1962). Fundamentals of Celestial required orientation and the fact that the Mechanics. Macmillan, New York. DuxatrRV, T. C. (1977). Phobos and Deimos: Geod- tidal dissipation is very weak (the time esy. In Planetary Satellites (J. Burns, Ed.), pp. 346- scale for the despinning of Hyperion is on 362. Univ. of Arizona Press, Tucson. the order of one billion years), it seems to GOGOEN, J. (1983). Paper presented at IAU Collo- us unlikely that capture has occurred. We quium 77, Natural Satellites, Ithaca, N.Y., July expect that Hyperion will be found to be 5-9. GOLDREICr~, P., AND S. J. PEALE (1966). Spin-orbit tumbling chaotically as more extensive ob- coupling in the solar system. Astron. J. 71, 425- servations conclusively define its rotation 438. state. If this chaotic tumbling is confirmed, GOLDSTEIN, H. (1965). Classical Mechanics. Addi- Hyperion will be the first example of cha- son-Wesley, Reading. otic behavior among the permanent mem- H~NON, M. AND C. HEILES (1964). The applicability of the third integral of motion: Some numerical ex- bers of the solar system. periments. Astron. J. 69, 73-79. HENRARD, J. (1982). Capture into resonance: an ex- tension of the use of adiabatic invariants. Celest. ACKNOWLEDGMENTS Mech. 27, 3-22. We gratefully acknowledge informative discussions KANE, T. R. (1965). Attitude stability of Earth-point- with Dale Cruikshank, Mert Davies, Tom Duxbury, ing satellites. AIAA J 3, 726-731. Jay Goguen, Michael Nauenburg, Peter Thomas, and PEALE~ S. J. (1977). Rotation histories of the natural Charles Yoder. Special thanks are due D. Cruikshank satellites. In Planetary Satellites (J. Burns, Ed.), pp. and T. Duxbury who kindly provided their observa- 87-112. Univ. of Arizona Press, Tucson. 152 WISDOM, PEALE, AND MIGNARD PEALE, S. J. (1978). An observational test for the ori- THOMAS, P., J. VEVERKA, D. WENKERT, E. DANIEL- gin of the Titan-Hyperion orbital resonance. Icarus SON, AND M. DAVIES (1984). Voyager photometry of 36, 240-244. Hyperion: Rotation rate. Paper presented at IAU PEALE, S. J., AND T. GOLD (1965). Rotation of the Colloquium 77, Natural Satellites, Ithaca, N.Y., planet mercury. Nature 206, 1240-1241. July 5-9. Submitted for publication. POINCAI~, H. (1892). Les Mdthodes Nouvelles de la WIDORN, T. (1950). Der lichtwechsel des Saturnsatel- Mdcanique Cdleste. Gauthier-Villars, Paris. liten Japetus in jahre 1949. Sitzungsber. Osterr. SMITH, B., et al. (1982). A new look at the Saturn Akad. Wiss., Wiem, Abt lla, 159, 189-199. system: The Voyager 2 images. Science 215, 504- WlSDOM, J. (1983). Chaotic behavior and the origin of 537. the 3/1 Kirkwood gap. Icarus 56, 51-74.