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

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.

💡 Research Summary

The paper presents a rigorous analytical framework for converting the orientation and rotation angles of synchronously rotating satellites between two reference frames: the Laplace plane (LP), which is the mean plane about which the satellite’s orbital plane precesses, and the International Celestial Reference Frame (ICRF) equatorial plane, which is the Earth‑centric inertial reference used by the IAU Working Group on Cartographic Coordinates and Rotational Elements (WGCCRE). The authors derive second‑order transformation formulas that relate the satellite’s spin‑axis Cartesian components (sₓ, s_y, s_z) expressed in the LP frame to the right ascension (α), declination (δ), and prime‑meridian angle (W) defined in the ICRF frame. By retaining terms up to the second order in small parameters such as the obliquity, the transformation achieves arc‑second level accuracy, a substantial improvement over first‑order approximations that are insufficient for the Galilean moons whose node precession periods range from a few to several hundred years.

The methodology begins with a clear definition of the body‑fixed frame (BF) attached to the satellite and the inertial frame (IF) aligned with the ICRF equator. Two equivalent rotation matrices are constructed: one based on Euler angles (θ, ψ, φ) referenced to the LP, and another based on the ICRF angles (α, δ, W). The authors then expand the transformation matrix in a Taylor series, keeping second‑order terms, which yields explicit expressions for α and δ as functions of the LP‑based spin components and the LP orientation angles (α_LP, δ_LP). This approach preserves the physical meaning of the frequencies and amplitudes present in the input series, avoiding the need for numerical fitting of the output series.

To apply the theory, the authors use two modern ephemerides for the Galilean satellites: JUP387 (Jacobson, 2026) and the latest version of NOE (Lainey et al., 2009). They fit the orbital normal vector projected onto the LP (nₓ, n_y, n_z) with quasi‑periodic Fourier series containing at least ten terms. The fitting employs a Levenberg‑Marquardt algorithm and yields amplitudes, frequencies (integer combinations of the fundamental Laplace‑resonance arguments), and phases with residuals below 2 × 10⁻⁴ °. The LP orientation for each moon is determined from the zero‑frequency term, and the derived α_LP and δ_LP values are listed in Table 1.

With the orbital forcing series in hand, the authors construct a dynamical model for a solid satellite. The model solves the torque‑balance equations for the spin axis, treating the obliquity θ and node ψ as the primary variables and incorporating physical librations φ driven by orbital eccentricity and inclination. The resulting spin‑axis Cartesian series are then transformed into ICRF coordinates using the second‑order formulas, producing trigonometric series for α, δ, and W. These series retain the original frequencies, allowing a direct link between observable quantities (e.g., radar speckle tracking, spacecraft imaging) and interior parameters such as the moment of inertia.

The paper also explores the effect of a subsurface liquid layer (e.g., an ocean) on the rotational response. Introducing a fluid layer reduces the effective moment of inertia, amplifying the amplitudes of the forced librations and obliquity variations. The authors quantify this amplification for each Galilean moon, finding increases of 10–30 % for Ganymede and Callisto, which are the most sensitive due to their larger distances from Jupiter and weaker tidal coupling.

A major outcome is the proposal of an updated IAU WGCCRE rotation solution that incorporates a non‑zero obliquity (θ on the order of 0.1°–0.3°) and the full set of periodic terms derived from the analytical transformation. This solution supersedes the traditional “zero‑obliquity” models, reducing systematic errors in the predicted right ascension and declination to a few tens of micro‑arcseconds—well within the precision of upcoming Earth‑based radar observations and the JUICE and Europa Clipper missions.

In summary, the paper delivers (1) a second‑order analytical transformation between Laplace‑plane‑based Euler angles and ICRF‑based celestial coordinates, (2) high‑fidelity Fourier representations of the orbital forcing for the Galilean moons, (3) a solid‑body rotational model linked directly to observable angles, (4) an assessment of liquid‑layer effects on rotational amplitudes, and (5) a concrete, non‑zero‑obliquity IAU rotation solution ready for implementation. The methodology is general and can be applied to other synchronously rotating bodies such as Titan or the Uranian satellites, providing a valuable tool for interpreting future high‑precision rotational measurements.