Asteroid mean elements: Higher order and iterative theories

Mean orbital elements are obtained from osculating ones by removing the sho rt periodic perturbations. Large catalogues of asteroid mean elements need to be computed, as a first step in the computation of proper elements, used to study asteroid families. The algorithms for this purpose available so f ar are only accurate to first order in the masses of the perturbing planets ; the mean elements have satisfactory accuracy for most of the asteroid bel t, but degraded accuracy in the neighbourhoods of the main mean motion reso nances, especially the 2:1. We investigate a number of algorithms capable o f improving this approximation; they belong to the two classes of Breiter-t ype methods and iterative methods. The former are obtained by applying some higher order numerical integration scheme, such as Runge-Kutta, to the dif ferential equation whose solution is a transformation removing the fast ang ular variables from the equations; they can be used to compute a full secon d order theory, however, only if the full second order determining function is explicitly computed, and this is computationally too cumbersome for a c omplicated problem such as the N-body. The latter are fixed point iterative schemes, with the first order theory as an iteration step, used to compute the inverse map from mean to osculating elements; formally the method is f irst order, but because they implement a fixed frequency perturbation theor y, they are more accurate than conventional single iteration methods; a sim ilar method is already in use in our computation of proper from mean elemen ts. Many of these methods are tested on a sample of asteroid orbits taken f rom the Themis family, up to the edge of the 2:1 resonance, and the dispers ion of the values of the computed mean semimajor axis over 100 000 years is used as quality control. The results of these tests indicate that the iter ative methods are superior, in this specific application, to the Breiter me thods, in accuracy and reliability This is understood as the result of the cancellations occurring between second order perturbation terms: the incomp lete second order theory, resulting from the use of a Breiter method with t he first order determining function only, can be less accurate than complet e, fixed frequency theories of the first order. We have therefore computed new catalogues of asteroid mean and proper elements, incorporating an itera tive algorithm in both steps (osculating to mean and mean to proper element s). This new data set, significantly more reliable even in the previously d egraded regions of Themis acid Cybele, is in the public domain.