The Stability Boundary of the Scattered Disk: Octupole and Beyond

Scattered Disk Objects (SDOs) are distant minor bodies outside of the Kuiper Belt on highly eccentric orbits, frequently with perihelia near Neptune’s orbit. Their interactions with Neptune can lead to interesting dynamics, with the orbital stability of any given SDO determined by its perihelion distance. Batygin et al. (2021) developed a perturbative approach for scattered disk dynamics, finding that to leading order in the semi-major axis ratio, an infinite series of 2:j resonance chains can model for the dynamics of the scattered disk, with overlaps between resonances driving chaotic motion. In this work we address the limitations of the 2:j resonance model for shorter-period orbits by taking the spherical harmonic expansion for Neptune’s gravitational potential to octopole order. We find that while 2:j resonances dominated long-period orbits, for smaller semi-major axes, a chain of 1:j resonances emerges as the primary driver of orbital evolution, with 3:j resonances facilitating a small correction. Applying the Chirikov Criterion, we derive a more complete analytic form for the stability boundary of the scattered disk. Additionally, we determine the boundary between the 1:j and 2:j resonance chains and discuss the behavior at transition between the two regimes.