A new bi-parametric family of temporal transformations to improve the integration algorithms in the study of the orbital motion

Abstract

One of the fundamental problems in celestial mechanics is the study of the orbital motion of the bodies in the solar system. This study can be performed through analytical and numerical methods. Analytical methods are based on the well-known two-body problem; it is an integrable problem and its solution can be related to six constants called orbital elements. To obtain the solution of the perturbed problem, we can replace the constants of the two-body problem with the osculating elements given by the Lagrange planetary equations. Numerical methods are based on the direct integration of the motion equations. To test these methods we use the model of the two-body problem with high eccentricity. In this paper we define a new family of anomalies depending on two param- eters that includes the most common anomalies. This family allows to obtain more compact developments to be used in analytical series. This family can be also used to improve the efficiency of the numerical methods because defines a more suitable point distribution with the dynamics of the two-body problem.