Transformation of orientation and rotation angles of synchronous satellites: Application to the Galilean moons

TL;DR

The paper develops a rigorous second-order analytical method to transform Galilean satellites’ orientation and rotation angles between the Laplace plane and the ICRF reference frames, preserving the physical meaning of forcing frequencies. By expressing spin-axis orientation in Cartesian components and projecting onto a fixed reference plane, the authors derive exact and approximate transformations to Earth-equatorial coordinates, and connect spin precession to obliquity, node longitude, and Cassini-plane offset. They generate detailed spin orientation series for solid satellites (Io–Callisto) under two orbital theories (JUP387 and NOE), quantify transformation accuracy (down to arcseconds), and show how interior structure (solid vs liquid) alters amplitudes through resonant effects. The study proposes concrete updates to the IAU WGCCRE model, advocating non-zero obliquity and a streamlined set of terms with Laplace-plane mean values, and demonstrates the method’s utility for JUICE, Europa Clipper, and Earth-based radar data interpretation. Overall, the approach offers a transparent, physically interpretable path from internal geophysics and orbital dynamics to observable rotational elements, with ready-to-use series and SPICE kernels for practical implementation.

Abstract

The orientation and rotation of a synchronous satellite can be referred to both its Laplace plane and the ICRF equatorial plane, in terms of Euler angles or spin axis Cartesian coordinates and Earth equatorial coordinates, respectively. We computed second-order analytical expressions to make the transformation between the two systems and applied them to the Galilean satellites (Io, Europa, Ganymede, and Callisto). If one term of the spin axis Cartesian coordinates series is dominant, trigonometric series can be generated for the inertial and orbital obliquities, node longitude and offset with respect to the Cassini plane. Since the transformation does not require any fit of amplitudes and frequencies on numerical series, the physical meaning of the frequencies is preserved from the input series and the amplitudes can be directly related to the geophysical parameters of interest. We provide tables for the coordinates and angles' series assuming that the satellites are entirely solid, and considering two different orbital theories. The possible amplitude ranges for the main terms are also examined in the case where a liquid layer is assumed in the interior model. We use our transformation method to propose an updated IAU WG solution which would result in an improvement with respect to zero obliquity models used so far. This method will also be useful for the interpretation of future Earth-based radar observations or JUICE data.

Figures (11)

• Figure 1: Planes and angles. Angles are not drawn to scale but are exaggerated for the purpose of illustration.
• Figure 2: Temporal evolution of the spin axis coordinates in the Laplace plane $s_x$ and $s_y$ for the four Galilean satellites for a solid satellite. The time coverage is different for each satellite. The purple point is the Laplace pole, for which by definition $s_x=s_y=0$. The black dots in the right graphs are the J2000 spin positions.
• Figure 3: Temporal evolution of the orbital ($\varepsilon$) and inertial ($\theta$) obliquities and of the orbital inclination ($i$) for the four Galilean satellites, assuming a solid interior model.
• Figure 4: Node longitude of the spin ($\psi$) and of the orbit ($\Omega$) as a function of time for the four Galilean satellites, assuming a solid interior model.
• Figure 5: Offset $\zeta$ of the spin axis with respect to the Cassini plane for the four Galilean satellites, assuming a solid interior model.