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

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.

Figures (7)

• Figure 1: Distribution of the analysed TNOs in the $a-e$ plane, with the main dynamical classes indicated: Centaurs, scattered disc, classical, detached, and resonant (Plutinos). Grey points illustrate the overall distribution of the TNO population, based on data retrieved from the Minor Planet Center webpage on 14 January 2026.
• Figure 2: Time evolution of $T_L(t)$ for trans-Neptunian objects, showing the nominal orbit (black line) and four randomly selected virtual objects, calculated using the variational method.
• Figure 3: Statistical distribution of Lyapunov time calculated for the swarm of VAs after 15 Myr for GR orbits representing TNO objects analysed in this study. The histograms show the calculation results performed by two methods: variational (black line) and neighbour trajectories method in two variants (blue and red lines). The $T_L$ for the nominal orbit is represented by the vertical dashed lines.
• Figure 4: Distribution of the analysed Main Belt asteroids in the $a$–$e$ plane, with the main mean motion resonances. These resonances are represented on the plot by a vertical dashed green lines. Grey points illustrate the overall distribution of numbered Main Belt asteroids only, based on data retrieved from the AstDys-2 webpage on 14 January 2026.
• Figure 5: Time evolution of $T_L(t)$ for outer Main Belt objects, showing the nominal orbit (black line) and four randomly selected virtual objects, calculated using the variational method.