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

TL;DR

This paper models the spin-orbit coupling for highly elliptical orbits using a recursive map based on Dirac pulses, enabling analysis of rotational dynamics and initial conditions leading to unbounded angular velocity growth.

Contribution

It introduces a novel recursive discrete map for high-eccentricity spin-orbit problems, linking periapsis excitations with rotational state evolution and initial condition analysis.

Findings

Identified initial conditions for unbounded angular velocity growth.

Compared discrete map results with traditional spin-orbit models.

Used Fast Lyapunov Indicators to analyze phase space and initial conditions.

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.