On the evolution of Cold Classical Kuiper Belt Objects

At a region between 39.4 AU and 47.8 AU, the Kuiper Belt contains planetesimals thought to remnants of the protoplanetary disk in the early Solar System. There are 3 types of object which occupy the Kuiper Belt: Scattered Objects, Hot Classicals, and Cold Classicals.

Scattered Objects have highly eccentric and inclined orbits, not constrained to the 40-50 AU region, believed to be due to Neptune’s early migration outwards when Jupiter and Saturn entered a 2:1 resonance, derived from the Nice Model. This migration scattered the primordial Kuiper Belt which had occupied the region where Neptune orbits today.

Hot Classicals are objects which were believed to have been scattered by Neptune but not to the extent of the Scattered Objects. These objects would’ve had their eccentricities and inclinations increased, but ultimately would find stable orbits within the region bounded by 2:3 and 1:2 resonances with Neptune, thus being deemed “Classical”

Cold Classicals are objects which are believed to be unaffected by Neptune’s early migration, being formed between 39.4 AU and 47.8 AU with low eccentricities and inclinations (with \(e\approx0.02, i\approx0.02\)), and remaining stable within the region bounded by 2:3 and 1:2 resonances with Neptune, with perturbations in orbits being due to self-scattering.

Since the Cold Classical Kuiper Belt Objects (CCKBOs) presumably are only affected by self-scattering, by understanding that process it would be possible to simulate these objects over the age of the Solar System to see if that alone can reproduce the observed orbital properties of CCKBOs. The process in question is the combined effect of Viscous Stirring and Dynamical Friction.

Viscous Stirring involves multiple encounters of multiple objects over time, which lead to a wider spread of velocity distribution, which leads to an increase of eccentricity and inclination of objects. Dynamical Friction involves gravitational encounters with masses of different velocities. Large mass objects are dragged by low mass objects, while low mass objects are gravitationally assisted by large mass objects. This leads to a general decrease of velocity for large masses and an increase for low masses.

Ohtsuki & Stewart (2002, Icarus 155, 436-453, hereafter O&S) obtained evolution rates of eccentricity and inclination using Viscous Stirring and Dynamical Friction rates where the coordinates are referred to a reference point that moves in a circular orbit with semimajor axis \(a_{0}\) at the Keplerian angular velocity \(\Omega = (GM_\odot/{a_0^3})^{1/2}\), and described them for a system of 2 planetesimal components, with object masses \(m_1\) and \(m_2\)

Eqn 1: (O&S Eqn 6)

where \(N_{sj}\) is the surface number density of component \(j\) planetesimals, and \(h_{ij}\) is a scaling factor described by,