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<p>In this paper, we study the limits of application of canonical perturbation theory, specifically the validity of the normal form approximations, for the study of the mean motion resonances of the planar circular restricted three-body problem in the domain of orbits which cross, or are close to cross, the orbit of the secondary body. First, we discuss analytic issues related to the definition of a MMR normal form within a domain of the phase-space to which these orbits belong; then, we compute with a numerical method the canonical transformations defining the MMR normal forms. The limits of validity of the averaging method are studied by supplementing the information provided by the phase-portrait representations of the normal form Hamiltonian with analyses concerning the preservation of the quasi-integrals of motion obtained through one step of perturbation theory, and the computation of fast Lyapunov indicators regularized with respect to the close encounters with the secondary body. We provide numerical explorations using as model examples the external 1:2 and 5:6 MMRs for values of the mass ratios which are representative of the Sun–Jupiter and Sun–Neptune cases. For the 1:2 MMR, the efficacy of the averaging method is established also for chaotic orbits of eccentricity up to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10569_2024_10232_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">\(e\sim 0.55\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>e</mi> <mo>∼</mo> <mn>0.55</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On the limits of application of mean motion resonant normal forms of the three-body problem for crossing orbits and close encounters

  • 期刊论文

  • 发表日期：

    2024年12月31日

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  • Xiang Liu,
  • Massimiliano Guzzo

摘要

In this paper, we study the limits of application of canonical perturbation theory, specifically the validity of the normal form approximations, for the study of the mean motion resonances of the planar circular restricted three-body problem in the domain of orbits which cross, or are close to cross, the orbit of the secondary body. First, we discuss analytic issues related to the definition of a MMR normal form within a domain of the phase-space to which these orbits belong; then, we compute with a numerical method the canonical transformations defining the MMR normal forms. The limits of validity of the averaging method are studied by supplementing the information provided by the phase-portrait representations of the normal form Hamiltonian with analyses concerning the preservation of the quasi-integrals of motion obtained through one step of perturbation theory, and the computation of fast Lyapunov indicators regularized with respect to the close encounters with the secondary body. We provide numerical explorations using as model examples the external 1:2 and 5:6 MMRs for values of the mass ratios which are representative of the Sun–Jupiter and Sun–Neptune cases. For the 1:2 MMR, the efficacy of the averaging method is established also for chaotic orbits of eccentricity up to \(e\sim 0.55\) e ∼ 0.55 .