The Astronomical Journal, 128:484–491, 2004 July # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A. SPIN-ORBIT SECONDARY RESONANCE DYNAMICS OF ENCELADUS J. Wisdom Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139; [email protected] Received 2003 November 12; accepted 2004 March 16 ABSTRACT For Enceladus the frequency of small-amplitude oscillations about the synchronous rotation state is approx- imately one-third the orbital frequency. Near this commensurability a period-three bifurcation occurs; on a surface of section a period-three chain of secondary islands appears near the synchronous island center. Forced libration in this resonance may be large and would imply large tidal heating. This paper explores the possibility that Enceladus is or once was locked in this period-three librational resonance, and that the observed resurfacing of Enceladus is a consequence. Key words: celestial mechanics — gravitation — methods: analytical — methods: numerical — planets and satellites: individual (Enceladus, Mimas, Saturn) — planets and satellites: formation 1. INTRODUCTION scenario does not pass the Mimas test. There are also numerous questionable aspects of the dynamics. Enceladus, now there’s a puzzle. A substantial portion of Ross & Schubert (1989) found that estimates of tidal the imaged surface of Enceladus is covered by smooth plains heating with a viscoelastic Maxwell rheology can be sub- that are nearly crater-free. From the crater density, these plains stantially larger than for Darwin models with a bulk effective are estimated to be less than a billion years old, perhaps less tidal Q. They considered both homogeneous ice models and than 100 million years old. Thus, much of the surface of models with a liquid ammonia-water decoupling layer. They Enceladus has been resurfaced. There may be two distinct argue that their models give sufficient heating to explain the smooth plains regions, one of which is lightly cratered. So, observed activity on Enceladus. Ross & Schubert do not Enceladus may have been resurfaced more than once. The discuss why their models do not also predict substantial tidal only viable source of energy for this activity on Enceladus is heating of Mimas. If a homogeneous ice model predicts sub- tidal heating, but the estimated rate of tidal heating is not only stantial tidal heating of Enceladus, should not this also be the too small to initiate melting in Enceladus but probably too case for Mimas? small even to maintain a molten layer produced at some The dichotomy between Enceladus and Mimas suggests that earlier epoch (for which one would still need to identify an some additional heating mechanism is operating on Enceladus adequate heat source for the initial formation). There are many and that this heating mechanism results from some charac- discussions of the problem (Squyres et al. 1983; Schubert teristic that Enceladus possesses that other satellites do not et al. 1986; Morrison et al. 1986; Squyres & Croft 1986; share. One characteristic that is unique to Enceladus is that Peale 2003). A successful explanation must pass the ‘‘Mimas the frequency of small-amplitude oscillations about syn- test’’: Squyres et al. (1983) pointed out that Mimas is com- chronous rotation is nearly commensurate with its orbital parable in size to Enceladus, has a larger orbital eccentricity, frequency: specifically, the frequency of small-amplitude os- and is closer to Saturn, so tidal heating in Mimas should cillations for Enceladus is approximately one-third its orbital be substantially larger (about a factor of 25) than heating frequency. in Enceladus, yet Mimas does not show evidence of tidal In their analysis of Voyager images of Enceladus, Dermott heating. & Thomas (1994) found that the shape of Enceladus was well Squyres et al. (1983) suggested that Enceladus may have represented by a triaxial ellipsoid with principal radii a more ammonia than Mimas. Not only does adding ammonia 256:3 0:3 km, b 247:3 0:3 km, and c 244:6 ¼ lower the melting temperature (and so lower the heating re- 0:3 km.Æ They interpret¼ this figureÆ in terms of hydrostatic¼ Æ quired for resurfacing), but an ammonia-water solution with models and deduce that Enceladus has a density below 1.12 g 3 3 greater than 20% mole fraction of ammonia (perhaps achieved cmÀ , with a preferred value near 1.00 g cmÀ . The ratio of as ice freezes out) is less dense than ice and so might spew out moments of inertia determines the spin-orbit libration fre- onto the surface, perhaps even creating the E ring. But am- quency. Assuming uniform density, the implied (B A)=C is monia has not been detected on the surface. 0:03573 0:0017. More to the point, the impliedÀ ratio of Lissauer et al. (1984) proposed that Enceladus was recently the librationalÆ frequency to the orbital frequency is locked in an orbital resonance with Janus, which was disrupted 3(B A)=C 1=2 0:3274 0:0077. The error is purely for-¼ in the last 10 million years by an impact or when Enceladus mal;½ À the real error¼ is probablyÆ larger. became locked in the current Enceladus-Dione resonance. If This near commensurability has very interesting dynamical 1 locked in this resonance with Janus, Enceladus would have had consequences. For near 3 there is period-three bifurcation in a larger forced eccentricity and thus been subject to larger tidal the spin-orbit phase space. On a surface of section, a chain of heating. But in their model the eccentricity only reaches values three secondary islands appears near the center of the syn- near 0.01, compared with the present 0.0045, so the rate of tidal chronous island. Now, if Enceladus is locked in this period- heating is only enhanced by a factor of about 5. Thus, their three island, then there is a forced libration of the figure of 484 DYNAMICS OF ENCELADUS 485 Enceladus relative to Saturn. Depending on the system parameters, the rate of tidal heating due to this forced sec- ondary resonance libration can be enhanced by a factor of 100 to 1000 over the rate of tidal heating without secondary res- onance libration. A number of natural satellites are significantly out of round, 1 with larger than 3 . For instance, Dermott & Thomas (1988) measured the elliptical radii for Mimas. From these, for Mimas is estimated to be near 0.44. Such satellites also dis- play secondary resonances on their surfaces of section, but because is not near a low-order commensurability, these secondary islands are not near the synchronous island center and so are unlikely to play any role in the dynamics of these satellites. This paper examines the dynamics of the 3 :1 secondary spin-orbit resonance, computes the consequent rate of tidal heating, and discusses mechanisms that might have placed Enceladus into this unique dynamical state. 2. SPIN-ORBIT DYNAMICS The Hamiltonian governing the rotational dynamics of an Fig. 1.—A surface of section for 0:34 and e 0:0045. The dominant out-of-round satellite in a fixed elliptical orbit with the spin feature is the synchronous island. The¼ 3:2 and 1: 2 resonances¼ are above and axis perpendicular to the orbit plane is below the synchronous island, respectively. The synchronous island is sur- rounded by a small chaotic zone. Near each of the synchronous equilibria there is a chain of three islands, in which the libration about synchronous p2 2n2C a 3 rotation is in a 3:1 resonance with the orbital motion. H(t;;p) cos 2 f (t) 1 ¼ 2C À 4 r(t) ½ À ðÞ For a fixed orbit the planet-to-satellite distance and the true anomaly are periodic. Expanding these as Fourier series, the (Wisdom et al. 1984; Wisdom 1987; Sussman & Wisdom spin-orbit Hamiltonian is 2001), where p2 2n2C H(t;;p) Hk (e) cos (2 knt) 3 3(B A)=C: 2 ¼ 2C À 4 À ð Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀ ð Þ Xk The moments of inertia are A < B < C, n is the orbital fre- with coefficients H (e) 1 5e2=2, H (e) 7e=2, and quency, measures the orientation of the axis of minimum 2 3 H (e) e 2, to second¼ orderÀ in e. ¼ moment from the line to pericenter, p is the angular momen- 1 = We consider¼À the terms that depend on the eccentricity as a tum conjugate to , a is the semimajor axis of the orbit, r is the perturbation and write planet-to-satellite distance, and f is the true anomaly (the angle from the pericenter to the satellite). For nonzero eccentricity, the spin-orbit problem exhibits the H H0 H1; 4 full suite of phenomena generally found in nonlinear Hamil- ¼ þ ð Þ tonian systems, such as nonlinear resonances and chaos. For where small eccentricity, the phase space is dominated by the syn- chronous resonance in which the angle nt oscillates about 0 or . There is a chaotic zone surrounding¼ À the syn- p2 2n2C H0(t;;p) cos (2 2nt); 5 chronous resonance. Other primary resonances are present, the ¼ 2C À 4 À ð Þ : : two largest are the 3 2and1 2, in which the body rotates three 2n2C 7e e or one times, respectively, for every two orbits. Mercury is in H1(t;;p) cos (2 3nt) cos (2 nt) ; 4 2 2 the 3: 2 resonance. A surface of section is generated by plot- ¼À À À À ting the rate of change of the orientation (˙) versus the ori- 6 entation () at each pericenter for a number of orbits for a ð Þ representative set of initial conditions.