The Stability Boundary of the Distant Scattered Disk

Abstract

The scattered disk is a vast population of trans-Neptunian minor bodies that orbit the sun on highly elongated, long-period orbits. The stability of scattered disk objects is primarily controlled by a single parameter - their perihelion distance. While the existence of a perihelion boundary that separates chaotic and regular motion of long-period orbits is well established through numerical experiments, its theoretical basis as well as its semi-major axis dependence remain poorly understood. In this work, we outline an analytical model for the dynamics of distant trans-Neptunian objects and show that the orbital architecture of the scattered disk is shaped by an infinite chain of $2:j$ resonances with Neptune. The widths of these resonances increase as the perihelion distance approaches Neptune's semi-major axis, and their overlap drives chaotic motion. Within the context of this picture, we derive an analytic criterion for instability of long-period orbits, and demonstrate that rapid dynamical chaos ensues when the perihelion drops below a critical value, given by $q_{\mathrm{crit}}=a_{\mathrm{N}}(\ln ((24^2/5)(m_{\mathrm{N}}/M_{\odot })(a/a_{\mathrm{N}}{)}^{5/2}){)}^{1/2}$. This expression constitutes a boundary between the "detached" and actively "scattering" sub-populations of distant trans-Neptunian minor bodies. Additionally, we find that within the stochastic layer, the Lyapunov time of scattered disk objects approaches the orbital period, and show that the semi-major axis diffusion coefficient is approximated by ${\mathcal{D}}_a\sim (8/(5\pi ))(m_{\mathrm{N}}/M_{\odot })\sqrt{\mathcal{G}{M}_{\odot }{a}_{\mathrm{N}}}\exp [-(q/a_{\mathrm{N}}{)}^2/2]$. We confirm our results with numerical simulations and highlight the connections between scattered disk dynamics and the Chirikov Standard Map. Implications of our results for the long-term evolution of the distant solar system are discussed.

arXiv comment14 pages, 3 figures, published in ApJ

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