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Sun-Jupiter-Saturn System may exist: A verified computation of quasiperiodic solutions for the planar three body problem

Jordi-Lluís Figueras, Alex Haro

Abstract

In this paper, we present evidence of the stability of a simplified model of the Solar System, a flat (Newtonian) Sun-Jupiter-Saturn system with realistic data: masses of the Sun and the planets, their semi-axes, eccentricities and (apsidal) precessions of the planets close to the real ones. The evidence is based on convincing numerics that a KAM theorem can be applied to the Hamiltonian equations of the model to produce quasiperiodic motion (on an invariant torus) with the appropriate frequencies. To do so, we first use KAM numerical schemes to compute translated tori to continue from the Kepler approximation (two uncoupled two-body problems) up to the actual Hamiltonian of the system, for which the translated torus is an invariant torus. Second, we use KAM numerical schemes for invariant tori to refine the solution giving the desired torus. Lastly, the convergence of the KAM scheme for the invariant torus is (numerically) checked by applying several times a KAM iterative lemma, from which we obtain that the final torus (numerically) satisfies the existence conditions given by a KAM theorem.

Sun-Jupiter-Saturn System may exist: A verified computation of quasiperiodic solutions for the planar three body problem

Abstract

In this paper, we present evidence of the stability of a simplified model of the Solar System, a flat (Newtonian) Sun-Jupiter-Saturn system with realistic data: masses of the Sun and the planets, their semi-axes, eccentricities and (apsidal) precessions of the planets close to the real ones. The evidence is based on convincing numerics that a KAM theorem can be applied to the Hamiltonian equations of the model to produce quasiperiodic motion (on an invariant torus) with the appropriate frequencies. To do so, we first use KAM numerical schemes to compute translated tori to continue from the Kepler approximation (two uncoupled two-body problems) up to the actual Hamiltonian of the system, for which the translated torus is an invariant torus. Second, we use KAM numerical schemes for invariant tori to refine the solution giving the desired torus. Lastly, the convergence of the KAM scheme for the invariant torus is (numerically) checked by applying several times a KAM iterative lemma, from which we obtain that the final torus (numerically) satisfies the existence conditions given by a KAM theorem.
Paper Structure (13 sections, 2 theorems, 43 equations, 3 figures)

This paper contains 13 sections, 2 theorems, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our results and their organization in the paper
  3. The planetary model and the problem
  4. Invariance equation
  5. Continuation from the integrable case with translated tori methods
  6. A translated torus algorithm
  7. Solving Equations \ref{['eq:translated']}
  8. An invariant torus algorithm
  9. Application to the Sun-Jupiter-Saturn problem
  10. Numerical verification of the KAM constants
  11. Computation details
  12. Acknowledgements
  13. KAM Theorem and Iterative Lemma

Key Result

Theorem A.1

Let $h:{\mathcal{U}}\rightarrow {\mathbb C}$ be a real-analytic Hamiltonian, defined in an open set ${\mathcal{U}}\subset {\mathbb T}_{\mathbb C}^m \times {\mathbb C}^m$. Let $K:\bar{{\mathbb T}}^m_\rho\rightarrow {\mathcal{U}}$ be a continuous map, real-analytic in ${\mathbb T}^m_\rho$, whose deri Then, for each $\delta\in ]0,\rho/6[$, there exists constants $\mathfrak{C}, \mathfrak{C}_{\Delta K

Figures (3)

  • Figure 1: Projections of the 3D invariant torus in Delaunay coordinates onto $(\ell_1,L_1)$, $(\ell_2,L_2)$ and $(\hat{g},\hat{G})$ components.
  • Figure 2: Projections of the 3D invariant torus (generating a 4D torus) in Cartesian coordinates onto positions $x_1= (x_{1,1},x_{1,2})$, $x_2= (x_{2,1},x_{2,2})$ and momenta $y_1= (y_{1,1},y_{1,2})$, $y_2= (y_{2,1},y_{2,2})$. Coordinates $x_1,y_1$ correspond to Jupiter and $x_2,y_2$ correspond to Saturn, and are plot in orange and grey, respectively.
  • Figure 3: Fits of Fourier coefficients of the (complexified) components of the parameterization $\tilde{f}_{\ell_1}, \tilde{f}_{\ell_2}$ and $\tilde{f}_{\hat{g}}$, and estimates of analyticity strips.

Theorems & Definitions (3)

  • Remark 3.1
  • Theorem A.1
  • Lemma A.2: The Iterative Lemma