On the spin-orbit problem for highly elliptical orbits and recursive excitation

TL;DR

The paper tackles spin-orbit coupling for satellites in highly elliptical orbits by modeling the gravity-gradient torque as instantaneous periapsis excitations, and derives a recursive discrete map that updates attitude and spin at each periapsis. It compares this Dirac-pulse impulsive model to the full spin-orbit problem (SOP) and uses Fast Lyapunov Indicators to explore the phase space, identifying initial conditions that lead to unbounded spin-up (in-phase) or bounded growth (counterphase). The discrete map serves as a tractable first-pass tool to understand high-eccentricity attitude dynamics, providing insights for mission design (e.g., NRHOs and Lunar Gateway) while highlighting differences from the continuous SOP and the role of chaos near resonances. Overall, the work offers a practical framework for preliminary attitude analysis in Helically Elliptical Orbits, with a clear path for refinement and deeper dynamical investigation.

Abstract

Examining the spin-orbit coupling effects for highly elliptical orbits is relevant to the mission design and operation of cislunar space assets, such as the Lunar Gateway. In high-eccentricity orbits, the gravity-gradient moment is here modelled as an instantaneous excitation at each periapsis passage. By approximating the gravity-gradient moment through Dirac pulses, we derive a recursive discrete map describing the rotational state of the satellite at the periapsis passage. Thanks to the recursive map, we are able to find the initial attitude corresponding to an unbounded growth of angular velocity, and to identify initial conditions whose evolution is such that the pulses have the same sign (in-phase condition) or the alternate sign (counterphase condition) at successive periapsis passages. In the recursive map, we perform the numerical analysis up to ten periapsis passages. In order to justify the introduction of the discrete map, we compare the results of the discrete map with those found in the spin-orbit problem. Because of numerical errors due to the high eccentricity, we restrict the investigation up to three periapsis passages in the spin-orbit problem. Moreover, we apply the Fast Lyapunov Indicators method to draw the phase portrait and detect the initial conditions fulfilling the counterphase condition.

Figures (20)

• Figure 1: Illustration of the SOP.
• Figure 2: Representation of $\overline{A}(\sigma)$ for different orbital eccentricities and for one orbit (at $\sigma=0$ the satellite is at the periapsis). The values of the inertia ratio used are $\kappa = 0.25$ (top panel), $\kappa = 0.5$ (central panel), and $\kappa = 0.75$ (bottom panel).
• Figure 3: Set of initial conditions ($\alpha_0$, $\alpha_0'$), represented in white color, for which the in-phase impulse condition (with $\sin(2\alpha_{0}) > 0$) is satisfied for two successive periapsis passages ($r=1,2$) for $e= 0.9$ and $\kappa = 0.25$ (top panel), $\kappa = 0.5$ (central panel), $\kappa = 0.75$ (bottom panel).
• Figure 4: Set of initial conditions ($\alpha_0$, $\alpha_0'$), represented in white color, for which the in-phase impulse condition (with $\sin(2\alpha_0) > 0$) is satisfied for three (left panels) and four (right panels) periapsis passages for $e= 0.9$ and $\kappa =0.25$ (top panels), $\kappa = 0.5$ (central panels), and $\kappa = 0.75$ (bottom panels).
• Figure 5: Set of initial conditions ($\alpha_0$, $\alpha_0'$), represented in white color, for which the in-phase impulse condition (with $\sin(2\alpha_0) > 0$) is satisfied for five (left panels) and ten (right panels) periapsis passages for $e= 0.9$ and $\kappa =0.25$ (top panels), $\kappa = 0.5$ (central panels), and $\kappa = 0.75$ (bottom panels).