Fractal Geometry of Angular Momentum Evolution in Near-Keplerian Systems

TL;DR

This paper introduces a fractal dimension-based method to analyze resonant relaxation in near-Keplerian systems, enabling precise measurement of angular momentum evolution and revealing long-term correlations.

Contribution

It presents a novel technique using fractal dimensions to study angular momentum dynamics and develops a toy model to replicate observed behaviors.

Findings

Angular momentum growth is faster than a random walk but slower than linear.

Long-term correlations exist due to bounds on angular momentum.

The method reliably determines resonant relaxation timescales.

Abstract

In this paper, we propose a method to study the nature of resonant relaxation in near-Keplerian systems. Our technique is based on measuring the fractal dimension of the angular momentum trails and we use it to analyze the outcome of N-body simulations. With our method, we can reliably determine the timescale for resonant relaxation, as well as the rate of change of angular momentum in this regime. We find that growth of angular momentum is more rapid than random walk, but slower than linear growth. We also determine the presence of long term correlations, arising from the bounds on angular momentum growth. We develop a toy model that reproduces all essential properties of angular momentum evolution.