Cassini states and librations of the Galilean satellites: A numerical approach in Euler angles

Introduction:  The problem of determining the correct initial conditions for the numerical integration of the rotational motion of a celestial body is a fundamental one. Indeed, even a slightly off initial guess can excite free rotational modes (i.e., free precession and free libration) with amplitudes much larger  than the forced amplitudes. Tidal dissipation effects over long timescales have most likely damped all free modes, although small amplitudes at the free frequencies are still possible due to recent excitation mechanisms. Often, in the literature the problem of the initial conditions for the librations is avoided entirely by using linearized models, which allow to separate the solution to the differential equations as the superposition of a free libration term, that depends on the initial conditions,  and a forced libration term that is independent of the initial conditions, allowing one to determine an approximate solution in closed form. A similar linearized approach can be used to model obliquity.  We employ the full triaxial rotational equations for the Euler angles that define the orientation of the Galilean satellites' principal axes in an inertial frame. To solve the problem of the initial conditions we propose an approach that makes use of the differential state transition matrix to determine initial conditions that lead to zero amplitudes of the free modes.  We determine a set of three libration parameters for each of the four Galilean satellites assuming a rigid body model. These results are useful as an accurate reference rotation model for a solid satellite, and can serve as a basis for comparison with the libration and obliquity estimates that will be obtained from the Juice and Europa Clipper data in the 2030s, which will study in detail Ganymede and Europa, respectively.

Fig. 1: Schematic representation of the relations between the Euler angles, Laplace frame P, principal axis frame B (left). Representation of the relationship between the orbit, and the orientation of these reference frames (right).

We define the perturbations to the Euler angles relative to the exact Cassini laws as three parameters τ, σ and ρ, such that (see, e.g., [6-7])

θ′ = ι + ρ, φ′ = ¯Ω + σ, ψ′ = ¯u + π − σ + τ,      (1)

where, ι is the mean inclination of the satellite’s equator with respect to the Laplace plane normal (p). Then, the corresponding librational parameter (ρ) is the departure of θ′ from ι. We denoted by ¯u the mean argument of latitude of the satellite u. We denoted by ¯Ω the mean longitude of the ascending node of the satellite’s orbit.

Fig.2: Schematic representation of the relative configuration of the Laplace plane normal  ⃗p, orbit normal  ⃗n, and spin axis  ⃗ω in Cassini states I and II. The obliquity ϵ is the angle betwee n ⃗nand ⃗ω. The circles represent the precession paths. The inclination i is the angle between ⃗n and ⃗p. The angles are exaggerated for visualization purposes.

 

Free modes: Even without Jupiter exerting a time-variable torque, the satellites would oscillate regardless at discrete frequencies with amplitudes dependent on the initial conditions [6]. There are three free modes which correspond to the eigenfrequencies of the linearized equation [7]. These are a free longitudinal libration mode, a free wobble, and a free precession (e.g., [8]). The amplitudes of the three modes depend on the initial conditions and are expected to be small due to dissipative effects, although they may have been excited in the recent past due to impacts, or other effects not modelled here.  We aim to determine the initial conditions corresponding to the forced equilibrium, that is, the state in which the amplitudes of the free modes are minimized.

Results and conclusions: We determined the initial conditions needed to numerically integrate the triaxial rotational equations for all four Galilean satellites corresponding to the equilibrium Cassini state (see Fig. 3). Three parameters, corresponding to the latitude and longitudinal librations, that is the deviation from the Cassini state I (Fig. 2a), are determined (Fig. 4) and a frequency analysis is performed.   Future work should focus on extending the current framework to incorporate different plausible interior structure models to properly evaluate the effects of coupling mechanisms between the solid and liquid layers. Nevertheless, the results presented here provide an accurate rigid-body reference model that can be used to interpret future high-precision measurements from missions like Juice and Europa Clipper.

Fig.3: Projection of the spin axis  ⃗ω (blue) and orbit normal  ⃗n (red) of Europa and Ganymede on their respective Laplace planes defined by  ⃗p. Units are in radians. Circle markers indicate the initial time at January 10th 1600, while a diamond marker denotes the final time January 9th 2200. The precession is retrograde relative to the orbital revolution and rotation.

Fig. 4: Evolution of the obliquity (ϵ), orbital inclination (i) relative to the Laplace plane, and libration parameters ισ, τ, and θ′ = ι+ ρ, of Ganymede. Compare with [12].

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