The Implementation of Hori–Deprit Method to the Construction Averaged Planetary Motion Theory by Means of Computer Algebra System Piranha

Perminov, Alexander; Kuznetsov, E. D.

doi:10.1007/s11786-019-00441-4

Cited by 6 publications

(2 citation statements)

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“…In article [2] integrability of the N body problem was described. In [3] the problem of deriving theory of motion of four planet around center star was considered. Here Hamiltonian was given in the Poisson series in the osculating elements of the second Poincare systems.…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

Evolution equations of multi-planet systems with variable masses

Minglibayev

Kosherbayeva²

2022

JMMCS

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“…The averaged Hamiltonian is constructed by the Hori-Deprit method. The implementation of the Hori-Deprit algorithm is considered in (Perminov, Kuznetsov (2016) and Perminov, Kuznetsov (2020)). The first-order terms of the averaged Hamiltonian are constructed up to 6 th degree in eccentric and oblique Poincaré elements, the second-order and third-order terms are constructed up to 4 th and 2 nd degrees correspondingly.…”

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confidence: 99%

The semi-analytical motion theory of the third order in planetary masses for the Sun – Jupiter – Saturn – Uranus –Neptune’s system

Perminov

Kuznetsov

2019

Proc. IAU

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The averaged four-planetary motion theory is constructed up to the third order in planetary masses. The equations of motion in averaged elements are numerically integrated for the Solar system’s giant planets for different initial conditions. The comparison of obtained results with the direct numerical integration of Newtonian equations of motion shows an excellent agreement with them. It suggests that this motion theory is constructed correctly. So, we can use this theory to investigate the dynamical evolution of various extrasolar planetary systems with moderate orbital eccentricities and inclinations.

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