The Stability Boundary of the Distant Scattered Disk

Abstract

Abstract The distant scattered disk is a vast population of trans-Neptunian minor bodies that orbit the Sun on highly elongated, long-period orbits. The orbital stability of scattered-disk objects (SDOs) 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 semimajor 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 exterior 2: j resonances with Neptune. The widths of these resonances increase as the perihelion distance approaches Neptune’s semimajor axis, and their overlap drives chaotic motion. Within the context of this theoretical 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>crit</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">N</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>ln</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mn>24</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>5</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">N</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊙</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mn>5</mml:mn> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> . This expression constitutes an analytic boundary between the “detached” and actively “scattering” subpopulations of distant trans-Neptunian minor bodies. Additionally, we find that within the stochastic layer, the Lyapunov time of SDOs approaches the orbital period, and show that the semimajor axis diffusion coefficient is approximated by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> <mml:mspace width="0.25em"/> <mml:mo>∼</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>8</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>5</mml:mn> <mml:mi>π</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">N</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊙</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊙</mml:mo> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">N</mml:mi> </mml:mrow>

Topics & Concepts