Applications of Computer Algebra in the Study of the Two-Planet Problem of Three Bodies with Variable Masses

Prokopenya, Alexander N.; Minglibayev, M. Zh.; Shomshekova, S. A.

doi:10.1134/s0361768819020087

Cited by 11 publications

(2 citation statements)

References 20 publications

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“…These are very cumbersome and time-consuming algebraic calculations that are performed using computer algebra. In work [8,11,12], such calculations were performed for the two-planet problem of three bodies with variable masses, with precision to second orders including relatively small quantities (3.20).…”

Section: The Expansion Of the Perturbing Functionsmentioning

confidence: 99%

Equations of Planetary Systems Motion

Minglibayev¹,

Kosherbayeva²

2020

OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHST

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The study of the dynamically evolution of planetary systems is very actually in relation with findings of exoplanet systems. free spherical bodies problem is considered in this paper, mutually gravitating according to Newton's law, with isotropically variable masses as a celestial-mechanical model of non-stationary exoplanetary systems. The dynamic evolution of planetary systems is learned, when evolution's leading factor is the masses' variability of gravitating bodies themselves. The laws of the bodies' masses varying are assumed to be known arbitrary functions of time. When doing so the rate of varying of bodies' masses is different. The planets' location is such that the orbits of planets don't intersect. Let us treat this position of planets is preserve in the evolution course. The motions are researched in the relative coordinates system with beginning in the center of the parent star, axes that are parallel to corresponding axes of the absolute coordinates system. The canonical perturbation theory is used on the base aperiodic motion over the quasi-canonical cross-section. The bodies evolution is studied in the osculating analogues of the second system of canonical Poincare elements. The canonical equations of perturbed motion in analogues of the second system of canonical Poincare elements are convenient for describing the planetary systems dynamic evolution, when analogues of eccentricities and analogues of inclinations of orbital plane are sufficiently small. It is noted that to obtain an analytical expression of the perturbing function main part through canonical osculating Poincare elements using computer algebra is preferably. If in expansions of the main part of perturbing function is constrained with precision to second orders including relatively small quantities, then the equations of secular perturbations will obtained as linear non-autonomous system. This circumstance meaningful makes much easier to study the non-autonomous canonical system of differential equations of secular perturbations of considering problem.

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Section: The Expansion Of the Perturbing Functionsmentioning

confidence: 99%

Equations of Planetary Systems Motion

Minglibayev¹,

Kosherbayeva²

2020

OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHST

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“…we have widely used as an unperturbed motion in studying the dynamics of unsteady gravitating systems [11,12,13].…”

Section: Aperiodic Motion Along a Quasiconical Sectionmentioning

confidence: 99%

New Forms of the Perturbed Motion Equation

Minglibayev

Omarov

Ibraimova

2020

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Real celestial bodies are neither spherical nor solid. Celestial bodies are unsteady, in the process of evolution their masses, sizes, shapes and structures are changes. The paper considers a model problem proposed as an initial approximation for the problems of celestial mechanics of bodies with variable mass. Based on this model problem, perturbation theory methods are developed and new forms of the perturbed motion equation are obtained. The model problem as the problem of two bodies with variable mass in the presence of additional forces proportional to speed and mutual distance is a class of intermediate motions. This class of intermediate motions describes an aperiodic motion along a quasiconical section. In this paper, on the basis of this class of aperiodic motion over a quasiconical section, various new forms of the perturbed motion equation in the form of Newton's equations are obtained. Based on the known equations of perturbed motion for the osculating geometric elements p, e, , i, , in the form of the Newton equation, we obtained the equations of perturbed motion for the following system of osculating elements p, e, i, , , and a, e, i, , , . Oscillating variables involving a dynamic element are suitable in the general case. A system of variables, where instead of the dynamic element is introduced - the average longitude in orbit is used in the quasielliptic case . The obtained new forms of the equation of perturbed motion, in the form of Newton's equations, in various systems of osculating variables can be effectively used in the study of the dynamics of non-stationary gravitating systems.

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