On the forced orbital plane of the Hilda asteroids

Abstract

Hilda-group asteroids librate in Jupiter's interior 3:2 mean motion resonance. We estimate that the Hilda group is observationally complete up to absolute magnitude $H\leq16.3$. This provides a statistically useful sample of thousands of resonant objects, all within a narrow range of semimajor axes, to compare with theoretical expectaitons of their orbital distribution from dynamical theory. We use von Mises-Fisher statistics to calculate the sample mean planes and mean plane uncertainties for the Hilda group and its Hilda, Schubart, and Potomac collisional subfamilies. Although Laplace-Lagrange linear secular theory is considered inapplicable within mean motion resonances, we find that the Laplace plane and the orbital plane of Jupiter are both statistically indistinguishable from the sample mean plane of the Hildas. In future work, we intend to extend this investigation to resonant populations in the Kuiper belt so as to further test the validity of Laplace-Lagrange linear secular theory for the mean planes of resonant populations.

Figures (5)

• Figure 1: Histograms for the inclinations of the Hilda group, and the Schubart, Hilda, and Potomac collisional families, relative to the vMF mean plane for each population.
• Figure 2: Heliocentric osculating orbital elements in the J2000 ecliptic frame for the observationally-complete Hilda-group asteroids with absolute magnitude $H\leq16.3$ as of May 1, 2025. The collisional Hilda, Schubart, and Potomac families are respectively colored red, blue, and goldenrod. All other objects are colored green.
• Figure 3: The present-day sample mean planes and their uncertainty regions. Each subplot contains the invariable plane (purple, I), the Jupiter plane (black, J), the Laplace plane (orange, L), the Hilda group mean plane (green, A), the Hilda family mean plane (red, H), the Schubart family mean plane (blue, S), and the Potomac family mean plane (goldenrod, P), as well as the 68%, 95%, and 99.7% uncertainty regions for the sample mean of the population labeled in the bottom-right corner. Note that this figure is contained well inside the ring of blue dots in Figure \ref{['fig:plot_qp_zoomout_allgroups_15.69']}, and this figure does not show the orbital planes of individual asteroids. In subplot (d), only the 68% and 95% uncertainty regions are shown.
• Figure 4: The time-varying angular separation, $i_\mathrm{rel}$, between the osculating orbit plane of the individual asteroids 1911 Schubart, 190 Ismene, 153 Hilda and 1345 Potomac, and the Laplace plane, Jupiter's orbital plane, and the invariable plane. The straight horizontal lines (in black) indicate the time-averaged values for each object.
• Figure 5: The time-varying angular separation, in degrees, between the mean planes of the indicated groups in each subplot and the invariable plane (purple), Jupiter's (black), and the instantaneous Laplace plane (orange). The red trace is the inclination width of the 95% confidence interval for the mean plane. To make the differences between the traces easier to see, each trace has been smoothed over a 50-kyr moving average.