Interpreting the librations of a synchronous satellite – How their phase assesses Mimas’ global ocean
Introduction
The Cassini space mission has given us invaluable data on the satellites of Saturn. In particular, we now dispose of measurements of their shapes as triaxial ellipsoids (Thomas, 2010, e.g.), and the rotation of Janus, Epimetheus (Tiscareno et al., 2009), Mimas (Tajeddine et al., 2014), Enceladus (Thomas et al., 2016), and Titan (Meriggiola, Iess, Stiles, Lunine, Mitri, 2016, Stiles, Kirk, Lorenz, Hensley, Lee, Ostro, Allison, Callahan, Gim, Iess, Persi Del Marmo, Hamilton, Johnson, West, 2008) have been measured. An issue is to get clues on the interior from these informations.
Most of the natural satellites of the giant planets are expected to have reached a state of synchronous rotation, known as Cassini State. This is a dynamical equilibrium from which small departures are signatures of the internal structure. These small departures are longitudinal and latitudinal librations, the latter ones translating into an obliquity. The main part of the longitudinal librations is a periodic diurnal oscillation, named physical librations. For a rigid body, they read (Murray and Dermott, 2000, Eq. 5.123, e.g.) e being the orbital eccentricity of the satellite, n its orbital frequency, and ω0 the frequency of the small proper oscillations around the equilibrium. We have: where the quantities Ixx are diagonal elements of the tensor of inertia of the satellite, and G200(e) is an eccentricity function made popular by Kaula (1966): This function is also known as the Hansen function H(1, e), and is present in Cayley (1861).
The quantity sometimes written as represents the equatorial ellipticity, or the triaxiality, of the distribution of mass inside the satellite. This formula assumes a rigid shape, i.e. the tensor of inertia is constant.
A seminal paper by Goldreich and Mitchell (2010) has shown that if the satellite is not strictly rigid, but viscoelastic, then a restoring torque tends to counterbalance the tidal torque of the parent planet, to lower the amplitude of the physical librations. This has motivated recent studies (Makarov, Frouard, Dorland, 2016, Richard, Rambaux, Charnay, 2014, Van Hoolst, Baland, Trinh, 2013), which revisit the theory of the librations in including a tidal parameter, k2 or h2, which characterizes the amplitude of variation of the gravity field, or of the surface, at the diurnal, or orbital, frequency n. In these studies, the body is assumed to be at the hydrostatic equilibrium.
The theory of the hydrostatic equilibrium tells us that the mass distribution in the body, i.e. its inertia, is shaped by its rotation and the tidal torque of its parent planet, while the inertia rules its rotation. However, in most of the studies, the inertia is mainly composed of a constant component which has no chance to be shaped by the rotation, while being assumed to correspond to the hydrostatic equilibrium.
In this paper I go further, in assuming that part of the mass distribution of these bodies is frozen, while part of it is still being shaped by the rotational and tidal deformation. For that, I first show from the measured radii that the departures from the hydrostatic equilibrium are ubiquitous in the system of Saturn (Section 2). Then I express the tensor of inertia of a satellite orbiting a giant planet, in considering a frozen triaxiality superimposed with elastic deformation (Section 3). I then deduce the librational dynamics of the satellite (Section 4), which I apply to the specific cases of Epimetheus and Mimas, for which longitudinal librations have been measured, and which are assumed to have rigid structure. Finally, I introduce the dissipative part of the tides (Section 7), to investigate their influence on the measurements. The reader can refer to Table 6 for the notations.
Section snippets
Departures from the hydrostatic equilibrium
Usually the shape of such a body is assumed to be the signature of an hydrostatic equilibrium. This means that the mass distribution is an equilibrium between the gravity of the body, and the rotational and tidal deformations it is subjected to. For a natural satellite orbiting a giant planet, it is assumed that the tidal deformation is only due to the planet, and that its spin rate is equal to its orbital rate, i.e. it rotates synchronously. This gives the following relations (Correia,
The tensor of inertia
I consider a homogeneous, triaxial and synchronous satellite. Its orbital dynamics comes from the oblate two-body problem, i.e. the semimajor axis a and the eccentricity e are constant, and the mean longitude λ and longitude of the pericentre ϖ have a uniform precessional motion, the associated frequencies being n and respectively. Moreover, its orbit is assumed to lie in the equatorial plane of the planet, as a consequence the satellite has no obliquity. I also neglect the polar motion,
The librational equations
The gravitational torque of the parent planet, which acts on the satellite, is collinear to the polar axis, since we consider a planar orbit. Its non-null component Γ reads (Williams et al., 2001, e.g.):
After expansion with respect to the orbital elements, I get, up to the second order in the eccentricity:
The Maxwell model
The classical Maxwell model (Karato, 2008, e.g.) gives a pretty good estimation of the frequency-dependency of k2. It depends on one parameter, the Maxwell time η being the viscosity and μ the rigidity. The complex Love number reads with and J* is the complex compliance defined as: ν being the tidal frequency, and g the surface gravity of the body. k2 is the real part of i.e.
An improvement at high frequencies
At high frequencies,
Application to Epimetheus and Mimas
Diurnal librations have been measured for Epimetheus and Mimas, thanks to Cassini data (Tajeddine, Rambaux, Lainey, Charnoz, Richard, Rivoldini, Noyelles, 2014, Tiscareno, Thomas, Burns, 2009). These two bodies are a priori assumed to be solid bodies, which legitimates the use of this model to explain their librations.
Influence of the creep
I now introduce the imaginary part of the Love number, that is often denoted as where Q is a dissipation function. This imaginary part introduces a dissipation as a lag between the gravitational excitation of the parent planet and the response of the satellite. This lag depends on the tidal frequency, this is why this effect will appear as .
Influence of the rheology
The Maxwell–Andrade model gives: the quantitites A2, and having already been defined ((59), (65) and 66). As shown in the Fig. 10, this model gives a higher dissipation than the Maxwell model at high frequencies.
I here compare the phase shifts given by the Maxwell and Andrade–Maxwell rheologies, for different Maxwell times (Figs. 11 and 12).
The phase shifts σ0 and ϕ1 have been plotted for Epimetheus for i.e. consistently with the measured libration. We
Conclusion
In this study I have proposed a novel theory of librations of a synchronous satellite, in which the role of the frozen component of the inertia of a triaxial satellite is differentiated from the one of the secular elastic deformation. The frozen component should not be addressed as a result of the hydrostatic equilibrium, a recent study by Van Hoolst et al. (2016) mentions this issue. I have shown that the measured physical librations of Epimetheus and Mimas are consistent with the presence of
Acknowledgments
This study took benefit from the financial support of the contract Prodex CR90253 from the Belgian Science Policy Office (BELSPO), and is part of the activities of the ISSI Team Constraining the dynamical timescale and internal processes of the Saturn System from astrometry. I am indebted to Rose–Marie Baland, Marie Yseboodt, and two anonymous reviewers, for their careful readings and valuable suggestions.
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