Arches of chaos, heteroclinic connections of first-order MMRs and the chaotic transport of small bodies in the Sun-Jupiter system

Abstract

We investigate the heteroclinic connections between stable and unstable manifolds of unstable periodic orbits associated with the most important mean motion resonances (MMRs) in the Sun-Jupiter planar restricted three-body problem. We explicitly compute the stable and unstable manifolds of the unstable periodic orbits associated with the first order interior MMRs 2:1, 3:2, and the exterior MMR 2:3. We also compute short-time FLI maps showing the chaotic saddle structure created by the manifolds of several interior or exterior MMRs other than the 1:1 (co-orbital) resonance. Transits of particles from the exterior to the interior of Jupiter's orbit and vice versa are allowed for Tisserand parameter lesser than 3, and are shown to exist through a variety of heteroclinic channels. Besides the classical ones by Koon et al., we find heteroclinic connections between manifolds of short-period orbits around L3 and periodic orbits of interior or exterior first order MMRs, as well as direct connections between interior and exterior MMR manifolds not involving co-orbital periodic orbits. Through these manifolds and the corresponding FLI ridges, we explain the 'arches-of-chaos' in the asteroid orbital plane (a,e). Chaotic orbits shadowing heteroclinic trajectories exhibit resonance hopping, suggesting links to quasi-Hildas and Jupiter-family comets. Results are obtained in the circular RTBP but persist in the elliptic problem.

Figures (7)

• Figure 1: Comparison of the phase portrait and the manifold structure for the Jacobi energy $E_J=-1.48$, corresponding to the Tisserand parameter $T_J = 2.96$. Top left: The phase portrait in the Poincaré section $\mathcal{P}_{E_J=-1.48}$ obtained by numerical integration of several trajectories with initial conditions in the square $\varphi \in [0,2\pi]$ and $p_\varphi \in [9,16]$. Top right: color-scaled FLI map for the backward-integrated trajectories with initial conditions in a $300\times 300$ grid in the above square and integration time down to $t=-100~yrs$. Bottom left: FLI map (same as above) but for forward-integrated trajectories up to the time $t=100~yrs$. Bottom right: Superposition of the FLI maps for the forward and backward-integrated trajectories.
• Figure 2: Poincaré sections (left) and superposition of the FLI maps for forward and backward-integrated trajectories (right) at the Jacobi energies $E_J=-1.42$ (corresponding to $T_J=2.84$, top), or $E_J=-1.52$ (corresponding to $T_J=3.04$, bottom).
• Figure 3: Superposition of the (backward/forward integrated) FLI maps with the unstable/stable manifolds of the unstable periodic orbits of various MMRs $q:p$ at the Jacobi energy $E_J = -1.48$ ($T_J=2.96$). Top left: Unstable/stable manifolds of the 2:1 resonance (red/green) superposed to those of the periodic orbit PL3 of the 1:1 resonance (blue/black). Top right: Manifolds of the orbit PL3 of the 1:1 resonance (blue/black) superposed to those of the 3:2 resonance (magenta/white). Bottom left: Manifolds of the orbit PL3 of the 1:1 resonance (blue/black) superposed to those of the 2:3 resonance (magenta/yellow). Top right: Manifolds of the 2:1 resonance (red/green) superposed to those of the 2:3 resonance (magenta/yellow).
• Figure 4: Overlap of FLI maps with 2:1 resonance stable-unstable manifolds. Left: $T_J = 3.04$ (red-unstable, green-stable); Right: $T_J = 2.84$ (red-unstable, black-stable).The FLI values are shown on a logarithmic scale following the color bar on the right.
• Figure 5: Top-left The FLI map for $E_J=-1.48$ and forward integrated trajectories (same as in the bottom-left panel of Fig.\ref{['fig:portraitfli148']}). The points A,B,C,D,and E are chosen at local ridges of the FLI map intersecting the vertical line $\varphi=\pi/3$. Top-middle The FLI color map along the line $\varphi=\pi/3$ is mapped to a curve the $(a,e)$ representation of the section (see text). Top-right Superposition of 40 curves (and the corresponding FLI color maps) computed through the mapping $(\varphi,p_\varphi)\rightarrow(a,e)$ for $\varphi=\pi/3$ and various equidistant values of the Jacobi energy ranging from $E_J=-1.52$ to $E_J=-1.40$. Bottom-left The stable manifolds of the 2:3 exterior periodic orbit superposed to the FLI map of the top-left panel. The points $A',B',C'$ are taken on the manifold. Bottom-middle same as in the top middle panel, but showing the mapping of the points $A',B',C'$ in the $(a,e)$ plane for the adopted section. Bottom-right The entire FLI map for forward integrated trajectories with initial conditions in a grid in the $(a,e)$ plane for the section $\varphi=\pi/3$. The 'arches of chaos' are produced by the union of all curves (and their corresponding ridge-located points like A,B,C,D, and E) as those of the top-right panel, but for continous variation of the Jacobi energy $E_J$.