Mission Analysis
12 Feb 2025

Analytical propagator for lunar satellite orbits

Background

The exploration of the Moon and the deployment of satellites in lunar orbit are essential milestones in advancing space exploration. Over time, the study of satellite motion around the Moon has evolved significantly. The presence of lunar mascons ([1]) leads to substantial complexity as regards the modelling of lunar satellite orbits, which requires taking into account high order terms of the multipole expansion of the lunar gravitational potential. Recent studies, (see [2] and references therein), have highlighted the difficulty in constructing a viable secular model yielding the long term variations in a satellites orbital elements at altitudes below 100 km. On the other hand, the forces due to the lunar potential harmonics of degree as high as n = 10 become smaller than the Earth's tidal forces only at altitudes above ~500 km. These facts suggest a natural division of the cislunar area into zones characterized by distinct features of the secular dynamics, as discussed in [6]. The simplest semi-analytical models of lunar satellite orbits stem from averaging the dynamics with respect to the satellite’s mean anomaly. The averaging can take place either through series expansions of the Hamiltonian of motion in the orbital eccentricity, or in 'closed-form' (see [2]-[10]). Such models offer insight into particular aspects of the secular dynamics (see, for example, [7],[8],[9]), but require, in general, numerical integration to obtain the long-term evolution of the mean orbital elements via the secular equations of motion (see ([2], [5], [6]). A fully analytical theory to account for the secular variations has been developed recently in [10], which, however, encompasses substantial simplifications as regards the adopted gravity model as well as numerical integral evaluations during the procedure to obtain a 'closed-form' variant for the Hamiltonian terms describing the effect of lunar tesseral harmonics.

Project goal

The goal of the present project is to produce a fully analytical propagator for the long-term dynamics of a lunar satellite. To this end, a precise secular Hamiltonian model is explored, including i) all important zonal and tesseral harmonics of the lunar potential averaged in closed form (the latter harmonics require averaging by a variant of the relegation algorithm, see [2]), ii) a quadripolar model for the Earth's tide using the exact orbit of the Earth in the Principal Axis Lunar Reference Frame. After representing all the basic frequencies of the problem through the introduction of suitable canonical pairs of action-angle variables, a 'secular normal form' is constructed removing the dependence of the equations of motion on the `secular' (slow) angles g (argument of perilune), h (longitude of the satellite's ascending node), as well as the canonical angles associated with the Moon's rotation, precession of the perigee and precession of the line of nodes in the ecliptic plane. The analytical process leads to a complete time-dependent analytical representation of the secular variations of a satellite's mean orbital elements. Combined with the averaging transformations from osculating to mean elements in closed form, the provided set of algebraic formulas serves to construct a fully analytical propagator for lunar satellite orbits. These formulas are validated through precision tests quantifying the level of error in comparison with a fully numerical propagation of the Cartesian equations of motion. The project aims to the provision of a complete library of computer routines serving as the core for user-independently written analytical propagators of lunar satellite orbits.

Example of comparison between the numerical and analytical results. On the left, the evolution of the eccentricity in time. On the right, the inclination. In black, the result of the numerical integration. In red, the semi-analytical one. For all pictures we fix, as initial conditions g(0) = h(0) = M (0) = 0◦, e(0) = 0.111556, altitude a = 2438 [Km] and final time of integration tf = 1000 [days]. On the top row, we consider i(0) = 11.2508◦. Bottom row, i(0) = 33.75◦.
Example of comparison between the numerical and analytical results. On the left, the evolution of the eccentricity in time. On the right, the inclination. In black, the result of the numerical integration. In red, the semi-analytical one. For all pictures we fix, as initial conditions g(0) = h(0) = M (0) = 0◦, e(0) = 0.111556, altitude a = 2438 [Km] and final time of integration tf = 1000 [days]. On the top row, we consider i(0) = 11.2508◦. Bottom row, i(0) = 33.75◦.

References

  1. P. M. Muller, W. L. Sjogren, Mascons: Lunar mass concentrations, Science 161 (3842) (1968) 680–684 doi: 10.1126/science.161.3842.680
  2. Christos Efthymiopoulos, Kleomenis Tsiganis, Ioannis Gkolias, Michalis Gaitanas, Carlos Yanez: SELENA: Semi-analytical theory for the motion of lunar artificial satellites, ArXiv preprint (2023) https://arxiv.org/abs/2309.11904
  3. Bernard De Saedeleer: Complete zonal problem of the artificial satellite: generic compact analytic first order in closed form, CM&DA 91 (2005) 239—268 https://doi.org/10.1007/s10569-004-1813-6
  4. Bernard De Saedeleer, Henrard Jacques: Orbit of a lunar artificial satellite: Analytical theory of perturbations, Proceedings of IAU (2004) doi:10.1017/S1743921305001432
  5. Giacaglia Giorgio E.O., Murphy James P., Felsentreger Theodore L.: A semi-analytic theory for the motion of a lunar satellite, Cel. Mech. 3 (1970) 3—66 https://doi.org/10.1007/BF01230432
  6. Edoardo Legnaro, Christos Efthymiopoulos: Secular dynamics and the lifetimes of lunar artificial satellites under natural force-driven orbital evolution, Acta Astronautica 225 (2024) 768-787https://www.sciencedirect.com/science/article/pii/S0094576524005538
  7. Martin Lara, Sebastián Ferrer, Bernard De Saedeleer: Lunar Analytical Theory for Polar Orbits in a 50-Degree Zonal Model Plus Third-Body Effect, Journal of the Astronautical Sciences 57 (2009) 561—577 https://doi.org/10.1007/BF03321517
  8. Martin Lara, Bernard De Saedeleer, Sebastián Ferrer: Preliminary design of low lunar orbits, Proceedings of the 21st International Symposium on Space Flight Dynamics 1—15 (2009) https://issfd.org/ISSFD_2009/InterMissionDesignII/Lara.pdf
  9. Bharat Mahajan, Kyle T. Alfriend: Analytic orbit theory with any arbitrary spherical harmonic as the dominant perturbation, CM&DA 131 (2019) 45 https://doi.org/10.1007/s10569-019-9923-3
  10. Bharat Mahajan, Lunar satellite analytic theory with complete gravity and third-body perturbations Part-I: Gravity harmonics and their coupling effects, preprint (2024) https://assets-eu.researchsquare.com/files/rs-5721259/v1_covered_7210893c-a37e-436d-a9e3-894c5abfe9ee.pdf
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