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Article · Axioms · May 30, 2026

Secular Perturbations of Translational–Rotational Motion of an Axisymmetric Body in the Restricted Three-Body Problem with Variable Masses

Alexander N. Prokopenya, M. Zh. Minglibayev, Balnur Assan

Mukhtar Minglibayev
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Abstract

We consider the translational–rotational motion of a small axisymmetric body in the gravitational field of two stars of variable masses moving under the influence of their mutual gravitational attraction. The two stars are assumed to lose their masses isotropically with different rates and their total mass decreases according to the joint Meshcherskii law. The relative motion of the stars is described by the corresponding exact solution to Gyldén’s equation and is considered to be given. The small axisymmetric body may change its mass, size and shape while its initial dynamic structure is retained. The problem is analyzed in the framework of Lagrange’s formalism, and equations of translational–rotational motion are derived in terms of the osculating elements of aperiodic motion on quasi-conic sections and the Andoyer variables. As equations of motion of the small body are not integrable, the perturbation theory is applied with the perturbing function expanded into power series in terms of eccentricity and inclination, which are assumed to be small. Averaging these equations over the mean longitudes of the two bodies and two Andoyer angles in the absence of commensurability of frequencies, we obtain the differential equations describing the long-term evolution of the orbital elements and Andoyer variables which may be investigated numerically for different laws of the system parameters’ variation. All the relevant symbolic calculations are performed with the computer algebra system Wolfram Mathematica.

Spacecraft Dynamics and ControlStellar, planetary, and galactic studiesSpace Satellite Systems and Controlrestricted three-body problemvariable massaxisymmetric bodies

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