Geometrical definition of a continuous family of time transformations on the hyperbolic two-body problem

Abstract

This paper is aimed to address the study of techniques focused on the use of a family of anomalies based on a family of geometric transformations depending on a parameter α that includes the true anomaly. This family is an extension of the elliptic geometrical transformation at the hyperbolic case. This family allows to get closed equations for the classical quantities of the hyperbolic two body problem both in the attractive and in the repulsive case. In this paper we obtain the link between hyperbolic funtions of hyperbolic argument H with trigonometric funtions for each temporal variable in the new family. This study includes also the inverse relations. This paper includes in the attractive case the study of the minimization of the errors due to the choice of the a temporal variable include in our family in the numerical integration by an appropriate choice of parameters. This study includes the analysis the dependence on the parameter of integration errors in a great time span for several eccentricities and the study of local truncation errors in the region with true anomaly contained in the intervall [−π/2, π/2] around the primary for several values of the parameter.