Evolution of the orbital-plane orientations in the two-protoplanet three-body problem with variable masses

In this article will be considered many spherical bodies problem with variable masses, varying non-isotropic at different rates as celestial-mechanical model of non-stationary planetary systems. In this article were obtained differential equations of motions of spherical bodies with variable masses to reach purpose exploration of evolution planetary systems. The scientific importance of the work is exploration to the effects of masses’ variability of the dynamic evolution of the planetary system for a long period of time. According to equation of Mescherskiy, we obtained differential equations of motions of planetary systems in the absolute coordinates system and the relative coordinates system. On the basis of obtained differential equations in the relative coordinates system, we derived equations of motions in osculating elements in form of Lagrange's equations and canonically equations in osculating analogs second systems of Poincare's elements on the base aperiodic motion over the quasi-canonical cross- section.

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“…Let us consider the planetary system, which consists of Let rates of masses variation are different [7][8] , 0,.., , 0,... , ,…”

Section: Differential Equations Of Motions Of Problem 21 Statement mentioning

confidence: 99%

Differential Equations of Planetary Systems

Minglibayev

Kosherbayeva

2020

Reports

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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

Reports

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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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“…Solving of systems of differential equations of secular perturbations (9)-(10) can be obtained numerically and analytically [1][2][3][4].…”

Section: Equations Of the Secular Perturbationsmentioning

confidence: 99%

“…The equations of motion in the Jacobian coordinates [1][2][3][4][5] can be written as (3), is constructed the perturbation theory for aperiodic motion on quasiconic section [1].…”

Section: Introductionmentioning

confidence: 99%

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Protoplanetary three-body problem with variable masses and its solutions

Mayemerova¹,

Minglibayev²,

Prokopenya³

2015

INT J MATH PHYS

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Abstract. Two protoplanetary three-body problem is considered in the case when the masses of all three bodies vary isotropically in different ratesis assumed that the mass of the protoplanets much less than the mass of the proto-Sun. Laws of change masses of bodies are known. The bodies are assumed as material points. On the basis of motion equations in the Jacobi coordinates the problem is described in the analogues of the Poincare second system. The perturbation function is symbolically computed in the system of the analytical calculations Mathematica up to the second degree of small quantities , , 2 1 e e 2 1 , i i inclusive. It is obtained the equations of secular perturbations and its solutions for the protoplanetary three-body problem with masses changing isotropically in different rates.

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Evolution of the orbital-plane orientations in the two-protoplanet three-body problem with variable masses

Cited by 15 publications

References 3 publications

Differential Equations of Planetary Systems

Differential Equations of Planetary Systems

New Forms of the Perturbed Motion Equation

Protoplanetary three-body problem with variable masses and its solutions

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