Small Body Dynamics with SBDynT: Proper Elements and Chaos Analysis

Abstract

The Small Body Dynamics Tool (SBDynT) is software written for the community of Solar System small body researchers to perform dynamical classification, characterization, and investigation. SBDynT provides advanced simulation analysis capabilities that make it straightforward to determine mean motion resonance occupation, proper orbital elements, and a variety of stability indicators. These calculations can be performed for small bodies that are known, newly discovered, or simulated; observational uncertainties can be incorporated through the use of dynamical clones. In this paper, we describe the methods for producing proper orbital elements and stability indicators, which serve as essential tools for characterizing dynamical stability and long-term evolution. Through extensive validation, we demonstrate that this code offers a robust open-source framework for investigating the dynamics of Solar System small bodies with high accuracy. We also aim for computational efficiency allowing SBDynT to provide dynamical information for the several-fold increases in small bodies expected in the LSST era.

Figures (24)

• Figure 1: The transformed $\accentset{\rightharpoonup}{e}$ and $\accentset{\rightharpoonup}{I}$ vectors in frequency space for Asteroid (46) Proserpina before (top panels) and after (bottom panels) filtering. In the top panels, vertical lines indicate forcing frequencies from the 7 planets included in the simulation as well as the peak proper frequency of the asteroid itself. Peaks corresponding to these frequencies are removed by the filter, as desired for the computation of proper elements. In the bottom panels, the proper terms corresponding to the free motion of Proserpina are shown; these terms are preserved in our filtering. There are many additional linear and non-linear forced terms which are also handled are not shown explicitly in this figure.
• Figure 2: Eccentricity and Inclination time arrays for Asteroid (46) Proserpina, before (blue) and after (orange) SBDynT filtering. Black lines indicate the 2.5% cutoff for producing the proper element from the mean of the filtered value. The red and green horizontal lines indicate the mean of the osculating elements and the proper elements, respectively.
• Figure 3: TNO (47171) Lempo's eccentricity and inclination evolution over time; the chaotic evolution is the result of Lempo's occupation within 3:2 mean motion resonance with Neptune. Separating the proper elements from the forced elements is therefore noticeably difficult, with perturbations from Neptune dominating the orbit. The final reported proper element resulting from our filtering routine is nearly equal to the simple mean of the unfiltered time array. The individual computed proper element for each of the five time windows is shown, highlighting how chaos affects the uncertainty measurement.
• Figure 4: TNO 2019 QQ110's eccentricity and inclination evolution over a 150 Myr integration. The $g+s-g_8-s_8$ resonance produces an $\approx 300$ Myr long-period term in the evolution of this object. While technically periodic, the growth in eccentricity and inclination in this resonance is so slow, it only appears periodic on timescales comparable to the age of the Solar System. Objects even closer to the center of a resonance would have longer period terms, producing apparent constant adiabatic growth in eccentricity and inclination over the age of the Solar System. Even with an additional filter applied to the inclination time-array, this long-term variation cannot be fully filtered out since it is not resolved by the full integration, artificially increasing the computed proper element uncertainties. Increasing the integration length can better handle this long-period term.
• Figure 5: Top: Residuals comparing the Asteroid Families Portal Proper Elements against the Nesvorny:2024 Catalog of the first 10,000 numbered asteroids. Bottom: Residuals comparing SBDynT Proper Elements against the Nesvorny:2024 Catalog of the first 10,000 numbered asteroids. The error-bars represent the uncertainties reported by Nesvorny:2024 in both figures. The unweighted $\sigma$ is simply the standard deviation in the proper element residuals, and the error-Weighted RMS is described in Equation \ref{['wRMS']}.