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 new set of anomalies based on geometric continuous transformations, depending on a parameter α , that includes the true anomaly. This family is an extension of the elliptic geometrical transformation to the hyperbolic case. This transformation allows getting 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 functions of hyperbolic argument H to trigonometric functions for each temporal variable in the new family, including also the inverse relations. We also carry out the study, in the attractive case, of the minimization of the errors due to the choice of a temporal variable included in our family in the numerical integration by an appropriate choice of parameters. This study includes the analysis of the dependence on the parameter of integration errors in a great time span for several eccentricities as well as the study of local truncation errors along the region with true anomaly contained in the interval [ − π / 2 , π / 2 ] around the primary for several values of the parameter.