Expansion of the Hamiltonian of the planetary problem into the Poisson series in elements of the second Poincare system

Perminov, Alexander; Kuznetsov, E. D.

doi:10.1134/s0038094615050081

Cited by 13 publications

(5 citation statements)

References 2 publications

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“…The first stage of our work is the expansion of the planetary system Hamiltonian into the Poisson series in all orbital elements. The algorithm of this expansion is described in detail in [33]. Let us briefly consider it here.…”

Section: The Motion Theorymentioning

confidence: 99%

“…So we have expressed the quantities 1/∆ ij in the Poincare elements. The analysis of the convergence for the quantities 1/∆ ij depending on the maximum included degree of Legendre polynomials are given in [33].…”

Section: The Motion Theorymentioning

confidence: 99%

“…It should be noted the following. In previous works [33,35] the cosines of the angles θ ij between Jacobi radius vectors r i and r j are saved in the Hamiltonian expansion and the motion equations as the symbol variables without inner structure. Therefore the obtained series contain six such variables (for each of six possible pairs of radius vectors) in addition to the orbital elements and the mass parameters.…”

Section: The Motion Theorymentioning

confidence: 99%

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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%

Section: The Motion Theorymentioning

confidence: 99%

Section: The Motion Theorymentioning

confidence: 99%

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

Perminov,

Kuznetsov

2018

Preprint

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“…It allows eliminating rounding errors in the process of the Hamiltonian construction. The algorithm of the Hamiltonian series expansion is described in detail by authors in Perminov & Kuznetsov (2015).…”

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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“…Then it is expanded into the Poisson series in the small parameter and orbital elements of the second Poincaré system. This algorithm is described in (Perminov, Kuznetsov (2015)).…”

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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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Expansion of the Hamiltonian of the planetary problem into the Poisson series in elements of the second Poincare system

Cited by 13 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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