Derivation of Evolutionary Equations in the Many-Body Problem with Isotropically Varying Masses Using Computer Algebra

Prokopenya, Alexander N.; Minglibayev, M. Zh.; Kosherbayeva, A.B.

doi:10.1134/s0361768822020098

Cited by 4 publications

(5 citation statements)

References 16 publications

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“…The evolutionary equations of n planets with variable masses in dimensionless variables (2.2)-(2.5) in the non-resonant case have the form in our work Prokopenya et al (2022)…”

Section: Evolutionary Equations Of N Planets With Variable Massesmentioning

confidence: 83%

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Evolution equations of the multi-planetary problem with variable masses

Kosherbayeva

Minglibayev

2021

Proc. IAU

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We investigated the influence of the variability of the masses of planets and the parent star on the dynamic evolution of n planetary systems, considering that the masses of bodies change isotropically with different rates. The methods of canonical perturbation theory, which developed on the basis of aperiodic motion over a quasi-conical cross section and methods of computer algebra were used. 4n evolutionary equations were obtained in analogues of Poincare elements. As an example, the evolutionary equations of the three-planet exosystem K2 − 3 were obtained explicitly, which is a system of 12 linear non-autonomous differential equations. Further, the evolutionary equations will be investigated numerically.

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“…The evolutionary equations of n planets with variable masses in dimensionless variables (2.2)-(2.5) in the non-resonant case have the form in our work Prokopenya et al (2022)…”

Section: Evolutionary Equations Of N Planets With Variable Massesmentioning

confidence: 83%

“…) and the Laplace coefficients retain their form, but they are already dimensionless quantities. All notations are given in the article Prokopenya et al (2022).…”

Section: Evolutionary Equations Of N Planets With Variable Massesmentioning

confidence: 99%

Evolution equations of the multi-planetary problem with variable masses

Kosherbayeva

Minglibayev

2021

Proc. IAU

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“…In work [16] a scheme for expressing perturbing functions via osculating elements was presented (4). In the article [18] obviously expansion of the perturbing function in analogues of the second system of canonical Poincare variables were obtained up to the second power of small parameters including, for n -planetary systems with variable masses. The equations of secular perturbations in the general case are also obtained…”

Section: The Problem Statement and Differential Equations Of Motionmentioning

confidence: 99%

“…The explicit form of non-autonomous coefficients of equations ( 8) -( 9) are cumbersome, for internal and external perturbing planets they are written separately. They are described in detail and given in the work [18]. These coefficients, in turn, depend on the Laplace coefficients.…”

Section: The Problem Statement and Differential Equations Of Motionmentioning

confidence: 99%

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Evolution equations of multi-planet systems with variable masses

Minglibayev

Kosherbayeva²

2022

JMMCS

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In celestial mechanics and astrodynamics, the study of the dynamical evolution of exoplanetary systems is the relevant topics. For today more than 3,000 exoplanetary systems are known. In this paper, we study the dynamic evolution of extrasolar systems, when the leading factor of evolution is the variability of the masses of gravitating bodies. The problem of n + 1 spherically symmetric bodies with variable masses is considered in a relative coordinate system, this bodies inter-gravitating according to Newton's law. The quasi-elliptical motions of planets whose orbits do not intersect during evolution are investigated. It is believed that the mass of bodies under consideration varies isotropically by various known laws with different velocities. The mass of the parent star is considered to be the most massive than its planets and the origin of the relative coordinate system is in the center of the parent star. Due to the variability of the masses, the differential equations of motion become non-autonomous and the task is difficult. The problem is investigated by methods of perturbation theory. The canonical perturbation theory based on a periodic motion over a quasi-canonical section is used. Canonical equations of motion are obtained in analogues of the second Poincare system, which are effective in the case when the analogues of eccentricities and the analogues of the inclination of the orbital plane of planets are sufficiently small. The secular perturbations of the planets, which determine the behavior of the orbital parameters over long time intervals, are studied. The evolutionary equations of many planetary systems with isotropically varying masses in analogues of the second system of Poincare variables are derived in an analytical form which are obtained using the Wolfram Mathematica computer algebra system. This takes into account the effects of the decreasing mass of the parent star and the growth of the masses of the planets due to the accretion of matter from the remnants of the protoplanetary disk. For the three-planetary problem of four bodies with variable masses, the evolutionary equations in dimensionless variables are obtained explicitly. In the future, these results will be used to study the dynamics of the threeplanet system K2-3 in the non-stationary stage of its evolution.

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