Terminator periodic orbits around binary asteroids: an application to the ESA Hera mission, Second-order modeling of the Cassini states of large satellites

Research in celestial mechanics addresses challenges in spacecraft mission design and the dynamics of planetary bodies. One study investigated terminator periodic orbits (TPOs) for the ESA Hera mission in the Didymos–Dimorphos binary asteroid system. Using analytic continuation, eight distinct TPO families were identified in a non-rotating frame, classified into unstable (Group I, with resonant angle ϕ_R = 0) or quasi-stable/stable (Group II, with ϕ_R = π) configurations. In a rotation model, the Coriolis force caused slow orbital drift, but characteristic terminator properties (inclination ≈90°, ascending node ≈±90°) were preserved for approximately 200 days. Optimized initial conditions limited orbital deviations to below 25 m for 30 days and 40 m for two months, demonstrating suitability for Hera's parking. A separate study developed a second-order dynamical model for Cassini states in large satellites, specifically Cassini state 1 for the Galilean moons. This model incorporated the full gravitational torque and coupled obliquity with polar motion, quantifying nutations in obliquity and longitude. Inertial obliquity exhibited primarily semidiurnal nutation, while body-frame obliquity showed predominantly diurnal nutation. For Callisto, diurnal nutations reached 400 milliarcseconds (40 m on surface), and second-order polar motion was resonantly amplified by factors up to 60. The presence of a subsurface ocean was shown to significantly influence Cassini state 1, causing resonant amplification of obliquity and polar motion for certain ocean thicknesses (e.g., 100-500 km for Ganymede), offering constraints for internal structures. Terminator periodic orbits around binary asteroids: an application to the ESA Hera mission Terminator orbits are near-circular polar orbits around small bodies, stabilized by the balance between gravity and solar radiation pressure, making them suitable as parking orbits for spacecraft near asteroids. Here, we extend this concept to binary asteroid systems. We first characterize terminator orbits as a family of polar periodic orbits around a single oblate body and compute their stability. Introducing a secondary companion, we apply analytic continuation to derive terminator orbits for the binary system, focusing on orbits whose period is in a 3:1 resonance with the binary's orbital period. Accounting for the binary's elliptic motion around the Sun, we numerically investigate the role of inertial forces. Results are presented for physical parameters corresponding to the ESA Hera mission to the Didymos–Dimorphos binary asteroid system. Second-order modeling of the Cassini states of large satellites We use an angular momentum approach to study the Cassini states (CS) of large natural satellites such as the Galilean satellites and Titan. Unlike classical approaches where obliquity is the solution of a trigonometric equation, our approach allows us to identify not only the mean obliquity of satellites, but also their nutation in space as well as their polar motion (PM) with respect to the solid surface. We show that triaxiality has a significant effect on the mean obliquities of CSI (up to 55% for Titan), CSII and CSIV (up to 22 degrees for Callisto), but no effect on CSIII. We assess the stability of the Cassini states over a wide range of free and forced precession frequency ratios and find that CSI and CSIII are always stable. CSII and CSIV are only stable for relatively fast orbital precession, and are therefore unstable for the current orbital parameters of the Galilean satellites and Titan. By solving the dynamic equations governing CSI and CSIII for a fully rigid satellite without averaging the external torque over the mean anomaly, we find mean obliquities close to their time-constant classical counterparts. CSI nutations are at quasi semi-diurnal and diurnal periods, with amplitudes ranging from tens to thousands of mas, while the nutations for CSIII are at quasi diurnal, semi-diurnal or quater-diurnal periods. We analytically describe one diurnal and one long-period CSI polar motion, the latter induced by the precession of the pericenter. A third, ter-diurnal PM also appears in CSIII. CSI diurnal and long-period PMs are of the same order of magnitude for Io, Europa and Ganymede, while the long-period PM dominates the solution for Callisto and Titan. The analytical solution for the long-period PM obtained here by second-order developments correctly describes the Moon's actual behavior and could help explain future observations of the rotation of large satellites. By extending the system of equations governing CSI, including gravitational and pressure couplings between misaligned layers, we predict the orientation of the spin axes of the outer shell, internal ocean and solid interior for an ocean-bearing body. We find five eigenmodes, including a free ocean nutation (FON) and an inner Chandler wobble (ICW). Far from resonance with an eigenmode, obliquity asymptotically tends towards values close to or slightly higher than that of a fully solid satellite. A large amplitude of nutation and polar motion can be obtained through a node precession/FON resonance, but also through resonance between the pericenter precession and the ICW. Depending on the thickness of the shell and the ocean, polar motion is dominated by its diurnal or long-period term, with amplitudes that can reach tens or even hundreds of meters.