Numerical analysis of Lyapunov Times for Trans-Neptunian Objects and Main-Belt Asteroids: stability, accuracy, and methodological comparisons

TL;DR

This study compares numerical methods for calculating Lyapunov times of TNOs and MBAs, highlighting the robustness of ensemble-based estimates over single-orbit calculations for understanding long-term orbital stability.

Contribution

It introduces a systematic comparison of variational and renormalization techniques for Lyapunov time estimation, emphasizing the benefits of ensemble methods for more reliable stability analysis.

Findings

Ensemble-based Lyapunov times are more robust than single-orbit estimates.

TNOs show greater consistency across methods than MBAs.

Method-dependent biases are reduced using clone ensembles.

Abstract

We computed Lyapunov times ($T_L$) for a sample of trans-Neptunian objects (TNOs) and outer main-belt asteroids (MBAs) using three numerical approaches: the variational method and two implementations of the renormalization technique. For each object, $T_L$ was derived both from the nominal orbit and from ensembles of 1001 orbital clones, enabling direct comparison between single-orbit and ensemble-based estimates. Across the sample, the methods generally produced consistent results, though larger discrepancies were observed for some MBAs. TNOs, in contrast, displayed greater consistency across methods, likely due to fewer overlapping resonances. Importantly, clone ensembles provided more robust and reliable stability indicators than nominal-orbit computations. Median values from clone populations reduced method-dependent biases and revealed dynamical behaviors that would remain hidden in single-orbit analyses, especially for objects with poorly constrained orbits or evolving in resonant regions. While our study focused on a limited but diverse set of objects, the methodology can be directly extended to larger populations, offering a systematic framework for exploring the long-term stability and dynamical evolution of main-belt asteroids, trans-Neptunian objects or other classes of objects in the Solar System.