The orbital motion around a central body is an interesting problem that involves the theory of artificial satellites and the planetary theories in the solar system. Nevertheless some difficult situations appear while studying this apparently simple problem, depending on each particular case. The real problem consists of searching the perturbed solution from a basic two-body motion problem. In addition, the perturbed problem must be solved using a numerical method and its efficiency depends on the selected coordinate system and the corresponding time. In fact, local and global errors are not necessarily homogeneously distributed over the orbit. In other words, there is a strong relationship between the spatial distribution of the selected points and the temporal independent variable. This is particularly dramatic in specially difficult cases. This issue leads us to consider different anomalies as temporal variables, searching for both precision and efficiency. Therefore, we are interested in the study of techniques to integrate the orbital motion equations using different anomalies as temporal variables which are functions of one or more parameters. The final aim of this paper is the minimization of the integration errors using an appropriate choice of the parameter depending on the eccentricity value in the family of the generalized Sundman anomalies.

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.

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One of the important challenges that Gaia imposes on the Astrometric Catalogs, is a careful study in everything affected by parallax. A particularly important case is the necessary linkage Gaia - HCRF - ICRF2, which require methods of analysis that are accurate enough so that the provided results are at the same precision level as the work data.