The Tides of Titan

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Science  27 Jul 2012:
Vol. 337, Issue 6093, pp. 457-459
DOI: 10.1126/science.1219631


We have detected in Cassini spacecraft data the signature of the periodic tidal stresses within Titan, driven by the eccentricity (e = 0.028) of its 16-day orbit around Saturn. Precise measurements of the acceleration of Cassini during six close flybys between 2006 and 2011 have revealed that Titan responds to the variable tidal field exerted by Saturn with periodic changes of its quadrupole gravity, at about 4% of the static value. Two independent determinations of the corresponding degree-2 Love number yield k2 = 0.589 ± 0.150 and k2 = 0.637 ± 0.224 (2σ). Such a large response to the tidal field requires that Titan’s interior be deformable over time scales of the orbital period, in a way that is consistent with a global ocean at depth.

Since its gravitational capture by Saturn on 1 July 2004, the spacecraft Cassini has flown by Titan more than 80 times, carrying out extensive observations of the surface and the atmosphere by means of particle and remote sensing instruments. In contrast, information on the moon’s deep interior is scarce. Lacking a detectable internally generated magnetic field, constraints on the interior of Titan come from gravity, topography, and rotation measurements. Titan’s main deviations from spherical symmetry are caused by centrifugal and tidal forces, associated respectively with the rotation about its spin axis and the gradient of Saturn’s gravity. The moon responds to the centrifugal and tidal potentials with deformations that (to the lowest order) change its quadrupole field. In a body-fixed frame with the prime meridian pointing to the central planet at pericenter and the z axis along the instantaneous rotation axis (coinciding with the orbit normal), only the J2 and C22 quadrupole coefficients are different from zero for a relaxed, synchronous satellite. They are bound by the constraint J2/C22 = 10/3. The satellite’s static response to the external fields is usually characterized by a single parameter, the fluid Love number kf, which reaches its maximum value of 3/2 for an incompressible fluid body. Previous determinations of Titan’s gravity (1) yielded kf =1.0097 ± 0.0039, implying a relaxed shape, very close to hydrostatic equilibrium. The value smaller than 3/2 revealed a significant concentration toward the center, with a moment of inertia factor C˜=0.3414±0.0005 (inferred from the Radau-Darwin equation). However, the nonnegligible eccentricity of Titan’s orbit causes a variation with time of the quadrupole tidal field [proportional to 1/r3 (r, distance between Titan and the Saturn barycenter)]. These short-term variations change the satellite’s physical shape and gravity. Titan’s linear response to the periodic tidal field entails a corresponding periodic change in its own quadrupole potential. The ratio between the perturbed and the perturbing potentials is known as the k2 Love number. It is an indication of the mass redistribution inside the body in response to the forcing potential. k2, like kf, reaches its theoretical upper limit of 3/2 for an incompressible liquid body, whereas for a perfectly rigid body, k2 = 0. If Titan hosts a global subsurface ocean, then k2 must differ substantially from zero. We have detected the signature of the tidal forcing in Cassini data and derived a value of k2.

Our observational strategy entailed gravity determinations near the pericenter and apocenter of Titan’s orbit. For k2 = 0.4 (a typical value if an ocean is present), the expected peak-to-peak variations of the quadrupole coefficients are about 4% for J2 and 7% for C22 (2, 3). The corresponding change in the spacecraft acceleration, about 0.2 mgal in the most favorable geometry, is measurable by the Cassini tracking system. We inferred the gravity field of Titan and its tidal variations from precise range-rate measurements of the spacecraft, enabled by microwave radio links with the ground antennas of NASA’s Deep Space Network (4).

In order to increase the confidence in the results, and following the same approach adopted in the previous determination of Titan’s gravity (1), we carried out two independent analyses. In the first one (SOL1), all Doppler data from the six gravity flybys (table S1) were processed in a multi-arc fit estimating a full 3 × 3 static gravity field; the (real) Love number (5); Titan’s state vector (position and velocity) at a reference epoch; and, for each flyby, the state vector of the spacecraft.

The second analysis (SOL2) combined all available radiometric tracking and optical navigation data from the Cassini mission, data from the Pioneer and Voyager Saturn encounters, and astronomical observations of Saturn and its satellites, in a global solution for the planet and satellite ephemerides and the gravitational parameters of the bodies in the saturnian system.

Both solutions provide consistent values of k2 (Table 1) and gravity harmonics (Table 2) at 3σ for quadrupole gravity coefficients and 1σ for k2. The large k2 value (~0.6) indicates that Titan is highly deformable over time scales of days. The central value exceeds the range previously expected for a subsurface ocean and a silicate core (6). Although the consistency of the two analyses strengthens the result, we have further tested the robustness of the solutions to perturbations of the dynamical model (3). The estimate of k2 survived to an increase of the solution rank by using a 4 × 4 gravity field (SOL1b in Table 1) and to changes of the a priori uncertainties in the state vectors of Titan, Cassini, and the gravity coefficients. Estimating a complex k2 leaves the real part statistically unchanged (5). The inclusion of a nonzero k2 and two new flybys in the solution does not modify in any statistically significant way the gravity field previously determined (1). Titan is confirmed to be in a relaxed shape with a moment of inertia factor (inferred from C22 using the Radau-Darwin equation) of 0.3431 ± 0.0004 in SOL1a and 0.3438 ± 0.0005 in SOL2. The updated geoid and associated uncertainties (from SOL1a) are shown in Fig. 1 and fig. S2.

Table 1

Titan’s k2 Love number, estimated from different data analysis procedures (supplementary materials) and representations of the gravity field: multi-arc analysis and 3 × 3 gravity field (SOL1a); multi-arc analysis and 4 × 4 gravity field (SOL1b); and global solution with 3 × 3 gravity field (SOL2). SOL1 and SOL2 were produced independently by the Cassini Radio Science Team and the Navigation Team.

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Table 2

Coefficients of the unnormalized spherical harmonics (3 × 3 gravity field, static component), estimated using two different approaches.

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Fig. 1

Geoid heights (in meters) from SOL1a, over the reference ellipsoid defined by the gravitational parameter (GM) of Titan, J2, C22, and the rotational velocity ω. The ground tracks of the six gravity flybys [T11 (green), T22 (orange), T33 (blue), T45 (red), T68 (magenta), and T74 (purple)] show the unequal coverage of the satellite: Although the spacecraft repeatedly probed the equatorial regions, no measurements at high latitudes are available. The geoid height uncertainties reach the maximum value of 2.2 m over the poles (fig. S3).

The physically likely range of k2 for Titan is bounded by the purely elastic value of ~0.04 (6) and the perfectly hydrostatic (fluid) value of about 1.0 (1) for the measured permanent tide (a higher formal value for k2 of 1.5 is obtained for a fluid body of uniform density). Our result is closer to the perfectly hydrostatic value and clearly excludes the elastic value. It is physically unrealistic for Titan to be entirely elastic yet have a very much lower rigidity than that of solid ice or rock, given its size; therefore our result strongly suggests that some global layer within Titan is behaving like a fluid on orbital time scales. There are two ways in which this can arise: by the presence of a very low-viscosity layer (an ocean) beneath an outer ice shell or a low-viscosity deep interior (or some combination) (6). In the former model, the outer ice shell must be sufficiently thin that the resulting increase in hoop stress allows for a nearly fluidlike strain for the tidal forcing (a state also postulated for Io and Europa). In that thin-shell limit, the value of the shell elasticity has little effect on the Love number. For example, doubling the rigidity decreases k2 by ~10%. The thickness of the underlying ocean is unimportant. The resulting predicted k2 is then about 0.42 to 0.48 for a shell thickness ranging from 100 to 0 km, assuming that the portion of the interior below the ocean (the deep interior) is effectively rigid; i.e., contributes negligibly to the external time-dependent tidal gravity.

Relaxing the assumption of a rigid deep interior yields k2 values all the way up to 1.0 for sufficiently low viscosity, even when an ocean is absent. In (6), such a model is formulated as a Maxwell viscoelastic body, but in the low-viscosity limit (where Maxwell time is less than the orbital period) the elastic part of the response is unimportant. For a homogeneous and entirely viscous body, the relevant viscosity is determined by a dimensionless number introduced by Darwin (7)

ε=arctan[19ηω2ρgR]where η is the dynamic viscosity, ω is the orbital angular velocity, ρ is the density, g is gravitational acceleration, and R is radius. Darwin found that the tidal amplitude is reduced by cos(ε) relative to the hydrostatic tide and that the ratio of the imaginary to real parts of the Love number is tan(ε). For Titan, ε is ~ 0.6(η/1014 Pa · s). This model predicts a small imaginary part of the Love number (<0.1) and satisfies the real part for a viscosity ~1 to a few 1013 Pa · s (8), which is several times less than the value often attributed to ice near its freezing point. The correct value for water ice depends on grain size, which is not an independent variable but is determined by the stress level. Unlike Europa, the tidal stress in Titan’s deep interior is lower than the stress expected for convection; this means that the correct viscosity to use may be the same as that needed to transport the heat out by convection. However, the nature of the rheology at tidal frequencies is still imperfectly understood. A rocky core made weak by active dehydration and hence fracturing and the circulation of water through cracks (9) could also contribute to raising the value of k2.

Alternatively, the value of k2 contributed by the ocean itself would be enhanced if the ocean were more massive; that is, denser than the value of 1 g/cm3 assumed for an ammonia-doped liquid water composition (10). A model in which the ocean has a large amount of sulfur; for example, in the form of ammonium sulfate (NH4)2SO4; increases the ocean density by 35% (11), raising the value of k2 contributed by the ocean alone to 0.57 (2). The value of k2 owing to the ocean could be even higher were it not for the very thick shell required to accommodate the presence of additional ammonium sulfate (11).

Our results do not distinguish between the various models because of the large range of admissible values of k2. Were one to demand that any given model provide a value of k2 closer to the central value, a model with a low-viscosity deep interior or a sulfur-rich ocean would be required. However, these models have other problems. A low viscosity for the high-pressure ice phases of Titan’s deep interior is difficult to reconcile with the values compatible with reasonable heat flows (10), as well as with a largely or fully differentiated rock core. The sulfur-rich ocean model (11) requires extensive leaching of sulfur from the core to a near-surface ocean, which is yet to be quantitatively demonstrated. A further objection to the high-density ocean model is that there is no evidence for the expected sulfur deposits on the surface (12). A simple water or ammonia-doped water ocean overlain by a thin (<100 km) shell is favored on various physical grounds quantified in almost all published evolution models (13), indirectly by the Huygens electric-field measurements (14), and consistent with the lowest end of the range of k2 derived here.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S3

Tables S1 to S4

References (1620)

References and Notes

  1. See the supplementary materials for additional discussion.
  2. Thanks to the use of X- and Ka-band (8.4 and 32.5 GHz) frequencies and state-of-the-art instrumentation, range-rate accuracies were in the range from 2 × 10−5 to 9 × 10−5 m/s at integration times of 60 s, depending on the solar elongation angle. The dependence is due to interplanetary plasma noise, which dominates the Doppler error budget (15).
  3. A determination of the complex Love number yields a value for Im(k2) compatible with zero at the 1σ level. Because the flybys and their geometry were selected to maximize the sensitivity to the real part of k2, the estimate of Im(k2) is less stable than for the real counterpart.
  4. Acknowledgments: L.I., M.D., P.R., and P.T. acknowledge support from the Italian Space Agency. The work of R.A.J., J.W.A., S.W.A., and N.J.R. was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. The Doppler data used in this analysis are archived in NASA’s Planetary Data System.
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