Parametric approximations of fast close encounters of the planar three-body problem as arcs of a focus-focus dynamics

IF 1.6 2区 数学 Q2 MATHEMATICS, APPLIED Nonlinearity Pub Date : 2024-09-09 DOI:10.1088/1361-6544/ad72c6
Massimiliano Guzzo
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Abstract

A gravitational close encounter of a small body with a planet may produce a substantial change of its orbital parameters which can be studied using the circular restricted three-body problem. In this paper we provide parametric representations of the fast close encounters with the secondary body of the planar CRTBP as arcs of non-linear focus-focus dynamics. The result is the consequence of a remarkable factorisation of the Birkhoff normal forms of the Hamiltonian of the problem represented with the Levi–Civita regularisation. The parameterisations are computed using two different sequences of Birkhoff normalisations of given order N. For each value of N, the Birkhoff normalisations and the parameters of the focus-focus dynamics are represented by polynomials whose coefficients can be computed iteratively with a computer algebra system; no quadratures, such as those needed to compute action-angle variables of resonant normal forms, are needed. We also provide some numerical demonstrations of the method for values of the mass parameter representative of the Sun–Earth and the Sun–Jupiter cases.
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平面三体问题快速近距离相遇的参数近似,作为焦点-焦点动力学的弧线
小天体与行星的引力近距离相遇可能会使其轨道参数发生重大变化,这可以利用环形受限三体问题进行研究。在本文中,我们以非线性聚焦-聚焦动力学弧线的形式提供了与平面 CRTBP 次级天体快速近距离相遇的参数表示。这一结果是对用 Levi-Civita 正则化表示的问题的哈密顿的 Birkhoff 正则形式进行显著因式分解的结果。对于每个 N 值,伯克霍夫正则表达式和焦点-焦点动力学参数都用多项式表示,其系数可以用计算机代数系统迭代计算;不需要二次方程,如计算共振正则表达式的作用角变量所需的二次方程。我们还提供了该方法在太阳-地球和太阳-木星情况下质量参数值的一些数值演示。
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来源期刊
Nonlinearity
Nonlinearity 物理-物理:数学物理
CiteScore
3.00
自引率
5.90%
发文量
170
审稿时长
12 months
期刊介绍: Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest. Subject coverage: The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal. Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena. Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.
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