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Laplace-Legendre expansion of the planar planetary Hamiltonian

Aya Alnajjarine, Jacques Laskar, Federico Mogavero

Abstract

We explore a hybrid expansion of the disturbing function in planetary dynamics that combines elements of the classical Laplace and Legendre developments. This formulation retains the structure of the Laplace expansion, but expresses the inverse of the mutual distance as a series whose terms keep an exact dependence on both the eccentricity and the semi-major axis ratio. We use it to construct the first-order secular Hamiltonian of the planar 3-body problem, relevant for modeling the long-term evolution of planetary systems. We assess the convergence of the new expansion numerically and compare it with that of the Laplace and Legendre series across a range of orbital configurations. The results show that the new expansion provides consistent performance across diverse dynamical regimes, bridging the domains of applicability of the two classical approaches.

Laplace-Legendre expansion of the planar planetary Hamiltonian

Abstract

We explore a hybrid expansion of the disturbing function in planetary dynamics that combines elements of the classical Laplace and Legendre developments. This formulation retains the structure of the Laplace expansion, but expresses the inverse of the mutual distance as a series whose terms keep an exact dependence on both the eccentricity and the semi-major axis ratio. We use it to construct the first-order secular Hamiltonian of the planar 3-body problem, relevant for modeling the long-term evolution of planetary systems. We assess the convergence of the new expansion numerically and compare it with that of the Laplace and Legendre series across a range of orbital configurations. The results show that the new expansion provides consistent performance across diverse dynamical regimes, bridging the domains of applicability of the two classical approaches.
Paper Structure (12 sections, 41 equations, 3 figures, 3 tables)

This paper contains 12 sections, 41 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Hamiltonian formulation and expansion of the disturbing function
  3. Legendre expansion of the direct part
  4. Laplace expansion of the direct part
  5. Laplace-Legendre expansion
  6. Laplace-Legendre expansion of order 0
  7. Laplace-Legendre expansion of order 1
  8. Generalization to arbitrary order
  9. Convergence of the Laplace-Legendre expansion
  10. Conclusion
  11. Laplace-Legendre expansion of order 2
  12. Number of terms in the secular expansions

Figures (3)

  • Figure 1: Relative error in the computation of $\langle U_1 \rangle$ as a function of the truncation order $N$, comparing the Legendre (black), Laplace (red), and Laplace–Legendre (blue) expansions. Each panel corresponds to a specific configuration of eccentricity and semi-major axis ratio. Columns show increasing eccentricities $e = 0.01$, $0.1$, and $0.2$ (left to right), while rows correspond to increasing semi-major axis ratios $\alpha = 0.001$, $0.01$, $0.1$, and $0.2$ (top to bottom). The value of $\Delta \omega$ is fixed at $0^\circ$. All the numerical evaluations are performed to double precision.
  • Figure 2: Same as Fig. \ref{['fig:convergence_grid_phi0']}, but for a fixed value of $\Delta \omega = 180^\circ$.
  • Figure 3: Relative error in the computation of $\langle U_1 \rangle$ as a function of the truncation order $N$, comparing the Legendre (black), Laplace (red), and Laplace–Legendre (blue) expansions, for $\Delta \omega = 0^\circ$. The left column corresponds to the nearly circular compact regime ($e = 0.01$, $\alpha = 0.4, 0.5$), while the right column shows the hierarchical eccentric one ($e = 0.6$, $\alpha = 0.001, 0.01$).