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The classical problem of (n+1) bodies with variable masses attracting each other according to Newton's law of universal gravitation is investigated. The central body P0 is assumed to lose mass isotropically, while the masses of the bodies P1,…, Pn can change anisotropically with different rates, which leads to the appearance of reactive forces. The laws of the masses change are arbitrary twice differentiable given functions of time. Since the differential equations of motion are not integrable, the problem is studied using perturbation theory methods. An exact solution to the two-body problem with variable masses which describes the aperiodic motion of the bodies along quasi-conical sections is used as the first approximation. It is assumed that the bodies P1,…, Pn move around the central body P0 along quasi-elliptical orbits in such a way that their orbits do not intersect and the mean-motion resonances are absent in the system.
Differential equations determining the perturbed motion of the bodies are obtained in terms of the osculating elements of aperiodic motion on quasi-conical sections in the framework of Newton’s formalism. In the case of small eccentricities and inclinations of the orbits, the perturbing forces are expanded into power series in small parameters and contain the secular terms and the short-period perturbations due to the orbital motion of the bodies. Averaging the equations of the perturbed motion over mean longitudes of the bodies P1,…, Pn , we eliminate the short-period oscillations and obtain the evolutionary equations which describe the orbital elements behavior on long-time intervals. These equations are solved numerically for different laws of body mass variations and numerical parameters corresponding to the exoplanetary system TOI-700 composed of five bodies. All the relevant computations are carried out using Wolfram Mathematica.
