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Article · Universe · October 10, 2025

Secular Evolution of a Two-Planet System of Three Bodies with Variable Masses

Alexander N. Prokopenya, M. Zh. Minglibayev, Aiken Kosherbayeva

Mukhtar Minglibayev
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Abstract

A classical three-body problem with two planets moving around a central star of variable mass on quasi-periodic orbits is considered. The bodies are assumed to attract each other according to Newton’s law of universal gravitation. The star loses its mass anisotropically, and this leads to the appearance of reactive forces. The problem is analyzed in the framework of Newtonian’s formalism, and equations of motion are derived in terms of the osculating elements of aperiodic motion on quasi-conic sections. As equations of motion are not integrable, the perturbation theory is applied with the perturbing forces expanded into power series in terms of eccentricities and inclinations, which are assumed to be small. Averaging these equations over the mean longitudes of the planets in the absence of mean-motion resonances, we obtain the differential equations describing the long-term evolution of orbital elements. Numerical solutions to the evolution equations are obtained and analyzed for three different three-body systems. The obtained results demonstrate clearly that variability of masses may influence essentially the secular evolution of the orbital elements. All the relevant symbolic and numerical calculations are performed with the computer algebra system Wolfram Mathematica.

Stellar, planetary, and galactic studiesSpacecraft Dynamics and ControlAstro and Planetary Sciencethree-body problemsecular evolutionorbital elements

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