The Hori–Deprit method for averaged motion equations of the planetary problem in elements of the second Poincaré system

Perminov, Alexander; Kuznetsov, E. D.

doi:10.1134/s0038094616060022

Cited by 9 publications

(3 citation statements)

References 2 publications

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“…It is characterized by efficiency and very ease for the computer implementation. In more detail it is described in [14,34]. The variables of the problem can be divide into the slow variables x and the fast λ.…”

Section: The Motion Theorymentioning

confidence: 99%

The construction of averaged planetary motion theory by means of computer algebra system Piranha

Perminov,

Kuznetsov

2018

Preprint

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The application of computer algebra system Piranha to the investigation of the planetary problem is described in this work. Piranha is an echeloned Poisson series processor authored by F. Biscani from Max Planck Institute for Astronomy in Heidelberg. Using Piranha the averaged semi-analytical motion theory of four-planetary system is constructed up to the second degree of planetary masses. In this work we use the algorithm of the Hamiltonian expansion into the Poisson series in only orbital elements without other variables. The motion equations are obtained analytically in time-averaged elements by Hori-Deprit method. Piranha showed high-performance of analytical manipulations. Different properties of obtained series are discussed. The numerical integration of the motion equations is performed by Everhart method for the Solar system's giantplanets and some exoplanetary systems.

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Section: The Motion Theorymentioning

confidence: 99%

The construction of averaged planetary motion theory by means of computer algebra system Piranha

Perminov,

Kuznetsov

2018

Preprint

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“…This approach allows us to sufficiently increase the integration step of the equations of motion in averaged elements. The algorithm of construction of the averaged Hamiltonian and the equations of motion in averaged elements is considered in Perminov & Kuznetsov (2016). The equations of motion are constructed as the Poisson brackets of the averaged Hamiltonian with the corresponding orbital elements.…”

Section: Introductionmentioning

confidence: 99%

The Investigation of the Dynamical Evolution of Extrasolar Three-planetary System GJ 3138

Perminov¹,

Kuznetsov²

2022

Res. Astron. Astrophys.

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This article is devoted to studying the dynamical evolution and orbital stability of compact extrasolar three-planetary system GJ 3138. In this system, all semimajor axes are less than 0.7 au. The modeling of planetary motion is performed using the averaged semi-analytical motion theory of the second order in planetary masses, which the authors construct. Unknown and known with errors orbital elements vary in allowable limits to obtain a set of initial conditions. Each of these initial conditions is applied for the modeling of planetary motion. The assumption about the stability of observed planetary systems allows to eliminate the initial conditions leading to excessive growth of the orbital eccentricities and inclinations and to identify those under which these orbital elements conserve moderate values over the whole modeling interval. Thus, it becomes possible to limit the range of possible values of unknown orbital elements and determine their most probable values in terms of stability.

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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.…”

mentioning

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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The Hori–Deprit method for averaged motion equations of the planetary problem in elements of the second Poincaré system

Cited by 9 publications

References 2 publications

The construction of averaged planetary motion theory by means of computer algebra system Piranha

The construction of averaged planetary motion theory by means of computer algebra system Piranha

The Investigation of the Dynamical Evolution of Extrasolar Three-planetary System GJ 3138

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

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