Spin-Orbit Secondary Resonance Dynamics of Enceladus J

Home , Janus (moon)

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 approximately 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 ½mal;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 secondary resonance libration can be enhanced by a factor of 100 to 1000 over the rate of tidal heating without secondary resonance 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 synchronous 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. A typical surface of keeping only the terms linear in e. section for parameters appropriate for Enceladus is shown in The Hamiltonian H is solvable. First make a canonical Figure 1. The distinctive feature of the surface of section for 0 transformation to a rotating frame nt, with conjugate Enceladus is the period-three chain of islands near each of the momentum È p nC. The new Hamiltonian¼ À is synchronous island centers. ¼ À 3. SPIN-ORBIT SECONDARY RESONANCES È2 2 2 È n C H00(t;; ) cos 2: 7 Here we develop an approximate model for the 3:1 sec- ¼ 2C À 4 ð Þ ondary resonances for synchronous rotation in the spin-orbit problem. We use standard techniques of Hamiltonian pertur- This is time-independent and has a single degree of freedom, bation theory (Sussman & Wisdom 2001). and so the solution is reducible to quadratures. The solutions 486 WISDOM Vol. 128 can be expressed in terms of elliptic functions. The pertur- The expression for 2 is of the form bation is also transformed: 2 F1 sin F3 sin 3 ; 16 ¼ þ þÁÁÁ ð Þ 2 2 È n C 7e e where H10(t;; ) cos (2 nt) cos (2 nt) : ¼À 4 2 À À 2 þ 1 F 2J˜ ( 2J˜ )3 ; 8 1 16 ð Þ ¼ pffiffiffiffiffiffi þ pffiffiffiffiffiffi þÁÁÁ 1 ˜ 3 With the goal of approximating the secondary resonances F3 ( 2J ) : 17 ¼ 192 pffiffiffiffiffiffi þÁÁÁ ð Þ between the frequency of oscillation and the orbital frequency, we next make the transformation to action-angle coordinates Making these substitutions in equation (14), writing the result for the Hamiltonian (eq. [7]) in the oscillation region. The as a Poisson series in and nt, and then picking out the terms È with argument 3 nt, which are slow near the 3:1 secondary action J is the area on the phase plane (, ), divided by 2. À The conjugate angle evolves linearly in time. The trans- resonance, the resonant part of the perturbation is formed Hamiltonian depends only on the action: H3:100 (t; ;J ) 1 1 1 2 2 1 2 1 3 2 2 3 H 00(t; ;J ) n C 1 J˜ J˜ J˜ n Ce F1 2F3 cos (3 nt) 0 ¼ 4 À þ À 16 À 256 þÁÁÁ ¼À4 12 þ þÁÁÁ À 1 1 1 1 2 2 3 ˜ 3 2n2C nJ J 2 J 3 ; n Ce ( 2J ) cos (3 nt): 18 ¼À4 þ À 4C À 16nC2 þÁÁÁ ¼À4 32 pffiffiffiffiffiffi À ð Þ 9 We have kept only the leading term. Without the F3 contri- ð Þ bution, the coefficient is 1/12 rather than 3/32. 2 where J˜ (4=)(J=nC ). Note that for small J the frequency, Keeping terms up to J and dropping the constant term, the the coefficient¼ of the linear term in J, is just n, as expected. full 3:1 secondary resonance Hamiltonian is The relation between the action-angle coordinates ( , J ) and 1=2 is 1 2 3 n H3:1(t; ;J ) nJ J e ¼ À 4C À 16 C 1=2 3 1 2 27 3 ; (p2J ) cos (3 nt): 19 2 2J˜ J˜ J˜ sin À ð Þ ¼þ 4 þ 512 þÁÁÁ ffiffiffiffiffiffi Finally, we make a canonical transformation to the resonance 1=2 1 3 9 variable 0 nt=3, with conjugate momentum J 0 J. (2J˜) 1 J˜ sin 3 : 10 ¼ À ¼ þ 192 þ 16 þÁÁÁ þÁÁÁ ð Þ The resonance Hamiltonian is

2 ˆ 1 ˆ 2 ˆ 3 Compare these expressions with those for a harmonic (small H3:10 (t; 0; J 0) n C J J ( 2J ) cos 3 0 ; 20 ¼ ½ À 4 À ð Þ ) approximation to the Hamiltonian in equation (7): pffiffiffiffiffiffi ˆ 1 where J J 0=(nC ), 3 ,and (3=16)ep. Apart from È2 2 2 2 the factor¼n2C, the resonance¼ À Hamiltonian¼ is written in terms of È n C ffiffi H00(t;; ) : 11 dimensionless quantities, Jˆ is the dimensionless action, is a 2C þ 2 ð Þ parameter that determines how close the system is to resonance, The canonical transformation to action-angle coordinates is and measures the strength of the resonance. The resonance trajectories are contours of this time-independent, one-degree- (2J= )1=2 sin ; È (2J )1=2 cos ; 12 of-freedom Hamiltonian. Trajectories of the resonance model ¼ ¼ ð Þ are compared with the surface of section in Figure 2. As is varied, there are special points where the phase where nC. The transformed Hamiltonian is ¼ portrait changes qualitatively. We can find these by solving È for the equilibrium points. The equilibrium points have a H00(t;; ) nJ; 13 threefold symmetry, with one set along the axis for which ð Þ 1=2 1=2 sin 0 0. In the variables x (2 Jˆ) cos 0, y (2Jˆ ) sin 0, which again gives n for the frequency of small-amplitude the equilibria¼ for y 0 satisfy¼ the equation ¼ oscillations. Notice that in this approximation the expression ¼ for the angle agrees with the first term of the full expression, 0 (x2=4 3 x ) x: 21 equation (10). ¼ þ À ð Þ Next we write the perturbation in terms of the full action- So, the origin is always an equilibrium and the other equilibria angle coordinates. For small it is appropriate to write the are solutions of a quadratic equation. For 9 2 >0, there þ dependence of H10 as a series, are two nonzero solutions. In terms of this bifurcation point is È 1 2 2 1=3 H10(t;; ) 4 n C(3eC2 cos nt 4eS2 sin nt) 14 1 : 22 ¼À þ ð Þ ¼ 1 (9=16)2e2 ð Þ þ with The second special point is where one of the nonzero roots 1 1 2 passes through zero. This occurs for 0, or 2 3. In the C2 cos 2 1 2 (2) ¼ ¼ ¼ ¼ À þÁÁÁ phase plane the bifurcation pattern is as follows: For <1 1 3 S2 sin 2 2 (2) : 15 there is a single equilibrium point. As is increased to ,a ¼ ¼ À 6 þÁÁÁ ð Þ 1 No. 1, 2004 DYNAMICS OF ENCELADUS 487

Fig. 3.—Amplitude of secondary libration (in radians) at the stable 3:1 ﬁxed point in the full spin-orbit problem (solid curve), compared with the estimated amplitude using perturbation theory. The dashed curve uses both terms in eq. (24); the dotted curve uses just the ﬁrst.

particularly near the point of bifurcation. The truncated estimate is better than the more complete estimate. 4. TIDAL HEATING Time-dependent tidal distortion of a body leads to internal heating. For a synchronously rotating body in an eccentric orbit, the rate of energy dissipation is

2 5 2 dE 21 ks f GmpnRs e 6 25 dt ¼ 2 Qs a ð Þ

(Peale & Cassen 1978; Peale et al. 1979; Peale 1999), where ks is the satellite potential Love number, Qs is the satellite effective tidal Q, f > 1 is an enhancement factor to account Fig. 2.—Top, surface of section for 0:336 and e 0:0045, showing the ¼ ¼ for a partially molten interior, G is the gravitational constant, a 3:1 secondary islands; bottom, plot showing trajectories of the perturbative is the orbit semimajor axis, m is the mass of the host planet, resonance Hamiltonian, on the surface of section. The agreement of the per- p turbative model with the surface of section is good. n is the orbital mean motion (and rotation rate), which is 3 1/2 approximately (Gmp /a ) , Rs is the satellite radius, and e is the orbital eccentricity. The derivation of this formula assumes cusp appears at a finite distance away from the origin, from that the body is incompressible, the rotation is uniform and which a stable and unstable point emerge for larger . The cusp synchronous, and the body is small enough that the dis- appears at x1 3 =2, y1 0. The equilibrium farther from placement Love number hs is 5ks /3. We may view this dissi- the origin is¼À stable. As is¼ increased to zero, the unstable pation as arising from two distinct sources of time dependence point goes through zero. As (and ) is increased further, the in the tide: time variation in the distance to the tide-raising stable and unstable equilibria move away from the origin. The planet, and the optical libration (the relative rocking motion of action of the stable equilibrium is a uniformly rotating satellite relative to the planet that results from the nonuniform motion in the elliptic orbit). Here this ˆ 2 1=2 formula is rederived and generalized to include obliquity, 2J 6 2(9 ) : 23 forced libration, and forced secondary libration. pffiffiffiffiffiffi ¼ þ þ ð Þ Let UT be the tide-raising potential. The satellite may be For parameters relevant to Enceladus, the dependence on thought of as consisting of a myriad of small constituent mass is weak, and to a good approximation, at the stable equilib- elements. The force on each mass element is the negative rium (2Jˆ)1=2 2p. The corresponding amplitude of libra- gradient of the potential energy, where the potential energy is ¼ tion can be determinedffiffiffi using equation (10). The amplitude is the tidal potential multiplied by the mass of the constituent. approximately The rate at which work is done on each constituent is the scalar product of this force with the velocity of the constituent. 1 ˜ 1 ˜ 2 1=2 Integrating over the volume of the satellite gives the rate at S 2 (2J 4 J ) : 24 ¼ þ ð Þ which work is done on the satellite: Figure 3 compares the perturbative estimate of the amplitude dE of libration at the stable fixed point with the actual amplitude v =999999999UT dV; 26 of libration (determined numerically). The agreement is good, dt ¼ÀZbody ð Þ 488 WISDOM Vol. 128 where is the density and dV is the volume element. To a a surface element in the unrotated satellite (or at the initial good approximation the satellite may be assumed to be time) are incompressible, 999999999= v 0 (Peale & Cassen 1978). The chain rule gives ¼ s0 (R sin cos k; R sin sin k; R cos ): 35 ¼ ð Þ

999999999= (UT v) v =999999999UT UT 999999999= v; 27 Assuming uniform synchronous rotation about the z-axis ¼ þ ð Þ (perpendicular to the orbit plane), the rectangular components so with the assumption of incompressibility, the rate of energy of this element at time t are dissipation is s Rz(nt)s0 ¼ dE k k 999999999=( UT v)dV: 28 (R sin cos ( nt); R sin sin ( nt); R cos ); 36 dt ¼ÀZbody ð Þ ¼ þ þ ð Þ where Rz() is an active rotation about the z-axis by the angle If we ignore any variation of density in the body, Gauss’s . The scalar product of the surface element with the orbital theorem allows us to write the rate of energy dissipation as a position gives o = s rR cos . This completes the expression surface integral, for the tidal potential¼ as a function of time and location on the surface. The delayed tidal potential UT0 is found by replacing dE Á UT v = n dS; 29 nt by nt in the expression for UT. dt ¼À Zsurface ð Þ The averageþ rate of energy dissipation is found by carrying out the surface integral, equation (31), and averaging over an where n is the normal to the surface and dS is the surface area v = n orbital period. These integrals are straightforward and will not element. Now, is the rate at which the height of the be shown. The result is surface changes. The height of the tide at any point on the surface is approximately dE 21 3h Gm2R5ne2 s p s sin Á: 37 dt ¼À 2 5 a6 ð Þ Ár hsU 0 =g; 30 ¼À T ð Þ The contribution due to the optical libration can be separated where hs is the displacement Love number for the satellite, g is from that due to radial variation. Of the factor 21/2, 9/2 can the local acceleration of gravity, and the prime on UT indicates be attributed to the radial variation, and 12/2 to the optical that the tidal potential is given a phase delay because the libration. The optical libration is more important by a factor dissipative tidal response lags the forcing. So, of 4/3. For small homogeneous spherical elastic bodies, the Love dE hs d numbers are approximately given by UT (UT0 )dS: 31 dt ¼À g Zsurface dt ð Þ 5=2 5gR hs 38 The tide-raising gravity gradient potential is ¼ 1 19=(2gR) 19 ð Þ þ 2 GmpR 3 UT P2(cos ); 32 (Thompson 1863; Love 1944) and ks 5 hs, where is ¼À r3 ð Þ the rigidity. The latter approximation is for¼ the case where the rigidity is large (compared with gR). This formula and the where P is the second Legendre polynomial, is the angle at 2 relationship between ks and hs are not valid for inhomoge- the center of the satellite between the planet-to-satellite line neous bodies, such as planetary bodies for which the density and the point in the satellite where the potential is being varies with radius. Estimates of k2 and h2 for the Moon evaluated, R is the distance from the satellite center to the from lunar laser ranging (Williams et al. 2003) are k2 ¼ evaluation point, and r is the planet-to-satellite distance. 0:0257 0:0025 and h2 0:029 0:013. These are nomi- Consider motion in a ﬁxed elliptical orbit, with small ec- nally inconsistentÆ with the¼ 5:3 ratioÆ expected for a small centricity e. Choosing rectangular coordinates with the x-axis homogeneous body, but barely consistent given the large aligned with the orbit pericenter and the orbit in the (x, y)- estimated error in h2. Nevertheless, assuming ks 3hs=5, plane, the orbital position is ¼ replacing sin Á by 1/Qs, and adding the enhancement factor f, we have derivedÀ equation (25). Using the approximation o (r cos f ; r sin f ; 0) 33 to equation (38) for large rigidity, the rate of tidal heating ¼ ð Þ becomes with true anomaly f. For small eccentricity, dE 42 2R7n5e 2 1 1 ; 39 rÀ aÀ (1 e cos nt); dt ¼ 19 Q ð Þ ¼ þ cos f cos nt e(cos 2nt 1); in agreement with the corrected expression in Peale (2003). ¼ þ À Next we assume the rotation is locked to the center of a sin f sin nt e sin 2nt; 34 threefold secondary resonance. To lowest order in the libration ¼ þ ð Þ amplitude S, the location of the surface element is now all to ﬁrst order in e. These expressions are easily generalized (generalizing eq. [36]) to higher order in eccentricity. In terms of the planetocentric s R s longitude k and colatitude , the rectangular components of 0 z(nt S sin (nt=3)) 0 : 40 ¼ À ð Þ No. 1, 2004 DYNAMICS OF ENCELADUS 489

For small S, we may expand

s0 s SR(sin sin (k nt)sin(nt=3); ¼ þ þ sin cos (k nt)sin(nt=3); 0): 41 À þ ð Þ The rest of the calculation proceeds as before. We assume the tidal response is delayed by replacing nt by nt Á. In this case, the averaging is over three orbital periods, toþ complete a secondary cycle, instead of one. For completeness we include two other sources of tidal dissipation: the forced libration and satellite obliquity. To add the forced libration, we take

s0 Rz(nt F sin nt S sin (nt=3))s0: 42 ¼ À À ð Þ

The amplitude of the forced libration is approximately 1 : Fig. 4.—Rate of energy dissipation (ergs sÀ ) in Enceladus in the 3 1 synchronous secondary resonance. The curves mark the dissipation using the 2e2 v F 43 amplitude determined numerically (solid cur e) and using the approximate ¼ 1 2 ð Þ perturbative expression (dashed curve). The dotted horizontal line indicates À the dissipation in Enceladus without the secondary resonance. The bars at the (Wisdom et al. 1984). And then, with obliquity I the expres- bottom indicate the range of formally allowed by the triaxial ellipsoid ﬁt to the Voya er images. The inner range is 1 ; the outer is 2 . sion for the surface element is g

s0 Rz( )Rx(I)Rz(nt F sin nt S sin (nt=3))s0; 44 The shape of Enceladus is approximately hydrostatic ¼ À À ð Þ (Dermott & Thomas 1994). If a hydrostatic shape is main- where is the ascending node of the equator relative to the tained as Enceladus tidally evolves away from Saturn, then pericenter. there will be a corresponding change in . So tidal evolution, The generalization of equation (25) is found to be with hydrostatic shape, can drive the system across the secondary resonance. Unfortunately, normal tidal evolution is away from the planet, and so the shape would evolve from dE 9 2 3 2 1 2 3 2 e (2e F) S (sin I) more out of round to less out of round (larger to smaller). dt ¼ 2 þ 2 þ þ 2 þ 2 This is the wrong direction for capture into the secondary fGm2nR5 ; 3hs 1 p s resonance to occur. 6 : 45 5 Qs a ð Þ Peale & Lee (2002) have recently examined scenarios for the capture of the Galilean satellite system into the Laplace The first term is due to the radial variation of the tidal am- resonance that involve contracting orbits. The starting point plitude. The second term combines the optical and forced for their deliberations is the Canup & Ward (2002) model for libration. The amplitude of the optical libration is approxi- the formation of the Galilean satellites. In this model the mately 2e; F is the amplitude of the forced libration. The sum Galilean satellites form slowly at large radii from material of the optical and radial terms gives equation (25). The third streaming through the Lagrange points and spiraling onto term is due to the forced secondary libration, presuming the Jupiter. The inward migration assumed by Peale & Lee results system is in the secondary resonance. The amplitude of the from interaction of the satellites with this same material. A forced secondary libration is approximately similar scenario for Enceladus would provide the inward evolution that is required for Enceladus to be captured into 1=3 1=2 the secondary resonance, if the inward evolution were slow S 2 À : 46 enough that the shape of Enceladus remained approximately ¼ ð Þ hydrostatic. Capture into the secondary resonance would have : 1 to have occurred after the orbital resonance with Dione was The 3 1 secondary resonance is only present for >3. The last term in equation (45) gives the heating due to satellite established, because the strength of the secondary resonance obliquity. depends on the orbital eccentricity. In this scenario, Enceladus The energy dissipated in Enceladus as a function of the would have been in the secondary resonance for most of the age of the solar system. Why then would the resurfacing of out-of-roundness parameter is shown in Figure 4. For this plot we assume a rigidity appropriate for ice of 4 ; 1011 dyn Enceladus be episodic? Though there is considerable uncertainty, with recent estimates of cratering rates on the Saturnian cm 2, Q 20, a density of 1:12 g cm 3, R 249:35 km, À À satellites it is estimated that there is a nonnegligible prob- and e 0¼:0045, and we neglect¼ the forced libration¼ and any obliquity.¼ ability that Enceladus has suffered a disruptive impact (Lis- sauer et al. 1988; Zahnle et al. 2003). Could Enceladus have avoided smaller impacts that might have knocked the system 5. PLACEMENT INTO SECONDARY RESONANCE out of the secondary resonance? We see that the tidal heating in Enceladus can be enhanced The fact that the shape of Enceladus is nearly hydrostatic is by 2 to 3 orders of magnitude, depending on physical pa- interesting. Weisskopf (1985) argued that there is a charac- rameters, over previous estimates if the rotation is in the 3 :1 teristic radius, on the order of 300 km for a rocky body, below secondary resonance. Here we discuss some possibilities for which the body should be significantly out of round, and above placing Enceladus in the secondary resonance. which the body should be nearly spherical. The argument uses 490 WISDOM Vol. 128 an order-of-magnitude estimate of the energy of plastic de- 20 km; they estimate that 30–100 craters in this size range formation. The result is roughly consistent with an observed would form on Mimas from Herschel ejecta. Perhaps a dis- transition to irregular shape below a radius of about 200 km ruption event of some former satellite near Enceladus placed among the natural satellites (Thomas et al. 1986). However, an numerous multikilometer-sized objects in Enceladus-crossing order-of-magnitude estimate for the timescale for relaxation of orbits. The timescale for Enceladus to despin to synchronous an ellipsoidal distortion is 6=(gR), where is the vis- rotation is about 60,000 yr, for Q 100 (Peale 1977). As long cosity, is the density, g is the¼ surface gravity, and R is the as significant debris is swept up on¼ shorter timescales, the spin radius (Schubert et al. 2001). The viscosity depends very rate and obliquity of Enceladus would evolve stochastically strongly on temperature T: using the expression for viscosity (Dones & Thomas 1993). The secondary resonance might given by Schubert et al. (1986), the relaxation timescale varies then be reached by a multiple-impact process. The heavily from 1018 yr for T 100 K to 106 yr for T 200 K. Because cratered regions of Enceladus might be due to the large Enceladus is small,¼ it is expected to be cold¼ (T < 100 K) for number of impacts of smaller debris, as is argued to be the most of its history (Ellsworth & Schubert 1983). The observed case for Mimas. Though detailed modeling might allow a hydrostatic shape must have been achieved during formation, stronger case to be made for the plausibility of this scenario, when the required temperature could have been reached by there are so many possibilities and physical uncertainties that accretional heating, or during some later special heating epi- such modeling would be of questionable value. An advantage sode, such as tidal heating during resonance encounter. The of this scenario is that Enceladus need not have or once have scenarios that involve gradual hydrostatic shape change with had a large crater. tidal evolution appear unlikely. An advantage of the impact scenarios is that they might Could an impact have knocked Enceladus into the sec- allow for multiple resurfacings. Indeed, the rate of tidal dis- ondary resonance? Consider the case 0:336 illustrated in sipation in the secondary resonance depends primarily on the Figure 2. The angular velocity difference¼ between the center out-of-roundness parameter , and only weakly on the orbital of the synchronous island and the secondary resonance is eccentricity. An interesting possibility is that after Enceladus about 5%. The orbital velocity of Enceladus is about 12.6 km is placed in the secondary resonance by an impact or multiple 1 sÀ ; a parabolic impactor (with respect to the planet) has a impacts, the eccentricity decays because of the large tidal 1 velocity at the orbit of Enceladus of about 17.9 km sÀ . In the dissipation. This would cause Enceladus to temporarily retreat model of Zahnle et al. (2003), the typical impact velocity on from the Enceladus-Dione resonance. Once the eccentricity 1 Enceladus from ecliptic comets is 24 km sÀ . But if the orbits had decayed sufficiently, the secondary resonance, whose 1 are aligned the impact velocity could be as little as 5.3 km sÀ . strength depends on the eccentricity, might become unstable. For an ideal (but unrealistic) grazing collision that imparts all Enceladus would then tidally evolve outward into the Dione its momentum to the rotation, these impact velocities imply an resonance, reestablishing its eccentricity and, so, effectively 3 impactor (with density 1 g cmÀ ) with radius in the range 6 km resetting the trap until another impact again takes the system 1 1 (for 24 km sÀ impact velocity) to 11 km (for 5 km sÀ impact into secondary resonance. velocity). For a typical 45 incidence angle these impactor There are other possibilities. We have only considered the radii are increased to 7 and 12 km, respectively. These impacts simplest dynamical model here. More complete models might would produce large craters, with diameters on the order of exhibit a richer dynamical structure. For instance, if additional 225 and 175 km, respectively, using the crater scaling of perturbations broadened the 3 :1 secondary resonance separa- Zahnle et al. (2003). To give an adequate kick at smaller ve- trix into a chaotic zone, it might be possible to evolve into this locity requires a larger impactor, but as the energy scales with chaotic zone or the vicinity of the secondary resonance the square of the velocity, the larger slower impactor can leave without appealing to impacts or reverse tidal evolution. And if a smaller crater. These craters would have to be in an unim- the structure is rich enough, there might be multiple intervals aged region of Enceladus or erased by the resurfacing (and not of chaotic evolution, as we observed in our studies of the tidal in one of the old, heavily cratered regions). Actually, if there evolution of the orbits of the Uranian satellites (Tittemore & are large craters on Enceladus they are more likely to be found Wisdom 1988, 1989, 1990). But these possibilities must await on the unimaged leading hemisphere, as the leading hemi- further study. sphere is typically more heavily cratered by heliocentric Consider the following scenario: Perhaps at some point comets than is the trailing hemisphere (L. Dones 2003, private Enceladus developed a figure that was more axisymmetric communication). Using the estimated ecliptic comet cratering than hydrostatic, perhaps as the result of an impact or non- rates on Enceladus from Zahnle et al., Enceladus has probably synchronous reaccretion after a disruption event. Tidal evo- only suffered a few impacts in this size range. It would be lution (perhaps assisted by a resonance lock with Janus, fortuitous for one of these few impactors to have hit Enceladus perhaps just due to Saturn tides) then established the orbital in just the right way to knock it into the secondary resonance. resonance lock with Dione. The resulting forced eccentricity Of course, there is considerable uncertainty in the estimate of might then provide tidal heating to raise the internal temper- cratering populations and rates. We mention that if the im- ature of Enceladus to the point at which the relaxation time for pactor changes the angular momentum of Enceladus too the figure of Enceladus is short compared with the age of the much, then there is a possibility that the system is captured solar system (only a few million years if 200 K is reached). As into the secondary resonance as the libration amplitude damps. the figure relaxed to the observed near-hydrostatic ellipsoidal However, no definitive estimate of the capture probability can shape, the system could pass through the secondary resonance be made, because it depends strongly on the tidal model. in the correct direction for capture to occur. As the resonance Another possibility is that Enceladus could have suffered a with Dione has already been established, the eccentricity is number of impacts from planetocentric debris. Zahnle et al. forced, so the secondary resonance is strong. So Enceladus argue that most of the impacts on Mimas are from planeto- would naturally evolve into the secondary resonance. centric debris ejected from the impact that created Herschel. Is Enceladus currently in the secondary spin-orbit reso- There are about 70 craters on Mimas with diameter larger than nance? The presence of the E ring, with its short lifetime, No. 1, 2004 DYNAMICS OF ENCELADUS 491 suggests that it might be. The Dermott & Thomas (1994) appears on the surface of section. We have developed a per- analysis of the Voyager data presumed exact synchronous turbative model for this secondary resonance. We have cal- rotation. Perhaps a less restrictive reanalysis of the Voyager culated the rate of tidal dissipation presuming Enceladus is data could place limits on the amplitude of the forced sec- locked in this secondary resonance and find that it can be 2 ondary libration, but new data from Cassini should answer to 3 orders of magnitude larger than given by the usual tidal this question. Even if Enceladus is not currently in the sec- heating mechanism for a body in synchronous rotation. We ondary resonance, the observed features might still be have discussed a number of possibilities for placing Enceladus explained if Enceladus was recently in the resonance. in this secondary resonance. Many of these possibilities re- quire further study. Perhaps the most appealing involves the 6. CONCLUSION relaxation of an initially more axisymmetric Enceladus toward a hydrostatic shape and consequent capture into the secondary Enceladus is a puzzle. It shows evidence of recent resurfacing, yet the standard estimates of tidal heating, the only resonance. The fact that this resonance is unique to Enceladus, that significant enhancement of tidal heating would result, and viable source of heating, are too small to account for the ob- that Enceladus appears to have some significant previously served activity. Mimas does not show evidence of internal heating, yet the corresponding estimates of tidal heating in unexplained additional source of heating suggests that this secondary resonance is part of the explanation of the mystery Mimas are 25 times larger than for Enceladus. So something of Enceladus. must be special about Enceladus. One of the main goals of this paper is to point out that there is something special about Enceladus: the moments of inertia of Enceladus happen to have the right values so that the frequency of small-amplitude This research was supported in part by NASA through the libration in the synchronous resonance is approximately one- Planetary Geology and Geophysics Program. I thank Robin third the orbital frequency. As a consequence there is a period- Canup, Luke Dones, Angela Flynn, Brad Hager, and Bill three bifurcation, and a chain of three secondary islands Ward for helpful conversations.

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