A new set of integrals of motion to propagate the perturbed two-body problem

Abstract

A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Pelaez et al. (Celest Mech Dyn Astron 97(2): 131-150, 2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Pelaez's method for near-circular motion under the J2 perturbation is transformed into linear. Moreover, the method reveals to be competitive with two very popular element methods derived from the Kustaanheimo-Stiefel and SperlingBurdet regularizations.

Time Elements for Enhanced Performance of the Dromo Orbit Propagator

“…This interesting feature, which is not observed in the K-S regularization, suggests that a time element can be very effective in the B-F method, at least when the perturbation does not explicitly depend on the physical time. This is in line with results shown in Baù et al (2013) in relation to the Dromo(P) method.…”

“…In its most recent version (Baù et al 2013), referred to as Dromo(P), a generalized Sundman transformation relates the time t to the new independent variable φ:…”

“…As detailed in Baù et al (2013), c can be obtained from the angular momentum definition by formally replacing the Keplerian energy ε K with the total energy…”

“…In this paper, we define and test two time elements for a special perturbation method (Baù et al 2013) that relies on a generalized Sundman time transformation of the form shown in Equation (1) with the exponent α set equal to 2. The method, called Dromo(P), represents a generalization of the original formulation published by Peláez et al (2007) for taking advantage of perturbations deriving from a potential.…”

“…Both conservative and time-dependent perturbations are applied with different initial eccentricities. As a first goal, we propose showing the improvement caused in Dromo(P) when the spatial elements are equipped with the time elements described in this paper as opposed to the physical time as in Baù et al (2013). Finally, we compare the performance of the eight generalized orbital elements of Dromo(P) with respect to other efficient and popular sets of elements.…”