Mathematical modeling of the orbital motion of an artificial satellite of the Moon using Delaunay variables

Objectives. This work aims to derive and study the system of equations of orbital motion of an artificial satellite of the Moon (ASM) in the gravitational field of an attracting planet using Delaunay variables. This will ensure a reduction in computational complexity when modeling long-term trajectories, as well as provide an analysis of stationary orbits of the Moon taking into account the gravitational influence of the Earth as a third body.

Methods. The study uses analytical mechanics, asymptotic methods, in particular, the averaging method, methods of stability theory, numerical methods for integrating systems of ordinary differential equations.

Results. The Hamiltonian and equations of motion of the ASM in canonical Delaunay variables are obtained. Averaged and non-averaged systems of equations of motion of the ASM are derived in the form of autonomous systems of ordinary differential equations with respect to the following orbital parameters: semi-major axis, eccentricity, inclination, longitude of the ascending node, longitude of the pericenter from the ascending node, and true anomalies. A closed system of differential equations of the second order with respect to the orbital eccentricity and the pericenter longitude from the ascending node is obtained. Its stationary solutions are found, their stability is investigated, and conditions for the existence of stationary motions are determined depending on the value of the constant of the first integral of the averaged system of equations. Integral curves and phase portraits were constructed to demonstrate the interrelationship of orbital parameters. A comparative analysis was conducted using JPL Horizons data and previously published works.

Conclusions. The methoddeveloped enablesthedesignof trajectories for future lunar missions tobeoptimized (e.g., Artemis, Luna-Glob), thus providing a balance between accuracy and computational efficiency. The results confirm the prospects of using Delaunay variables for analyzing long-term orbital dynamics in gravitational fields of complex configuration.

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