A von Mises–Fisher distribution for the orbital poles of the plutinos

Abstract

Small solar system bodies have widely dispersed orbital poles, posing challenges to dynamical models of solar system origin and evolution. To characterize the orbit pole distribution of dynamical groups of small bodies it helps to have a functional form for a model of the distribution function. Previous studies have used the small-inclination approximation and adopted variations of the normal distribution to model orbital inclination dispersions. Because the orbital pole is a directional variable, its distribution can be more appropriately modeled with directional statistics. We describe the von Mises–Fisher (vMF) distribution on the surface of the unit sphere for application to small bodies’ orbital poles. We apply it to the orbit pole distribution of the observed Plutinos. We find a mean pole located at inclination i0 = 3.57○ and longitude of ascending node Ω0 = 124.38○ (in the J2000 reference frame), with a 99.7% confidence cone of half-angle 1.68○. We also estimate a debiased mean pole located 4.6○ away, at i0 = 2.26○, Ω0 = 292.69○, of similar-size confidence cone. The vMF concentration parameter of Plutino inclinations (relative to either mean pole estimate) is κ = 31.6. This resembles a Rayleigh distribution function, with width parameter σ = 10.2○. Unlike previous models, the vMF model naturally accommodates all physical inclinations (and no others), whereas Rayleigh or Gaussian models must be truncated to the physical inclination range 0–180○. Further work is needed to produce a theory for the mean pole of the Plutinos against which to compare the observational results.

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“…We note that each estimate of the concentration parameter is based on a relatively large sample of step-directions and therefore comes with reliability information (e.g., a value based on 20 steps is less reliable than that based on 100 steps). Several methods have been proposed to compute the confidence intervals of κ estimates (Mardia and Jupp, 2000;Matheson et al, 2023), which can be obtained for each fiber.…”