In the present work, we define a new anomaly, Ψ, termed semifocal anomaly.It is determined by the mean between the true anomaly, 𝑓 , and the antifocal anomaly, 𝑓 ′ ; Fukushima defined 𝑓 ′ as the angle between the periapsis and the secondary around the empty focus. In this first part of the paper, we take an approach to the study of the semifocal anomaly in the hyperbolic motion and in the limit case corresponding to the parabolic movement. From here, we find a relation between the semifocal anomaly and the true anomaly that holds independently of the movement type. We focus on the study of the two-body problem when this new anomaly is used as the temporal variable. In the second part, we show the use of this anomaly-combined with numerical integration methods-to improve integration errors in one revolution. Finally, we analyze the errors committed in the integration process-depending on several values of the eccentricity-for the elliptic, parabolic, and hyperbolic cases in the apsidal region.

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 parameters 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.

This paper aimed to address the study of a new family of anomalies, called natural anomalies, defined as a one-parameter convex linear combination of the true and secondary anomalies, measured from the primary and the secondary focus of the ellipse, and its use in the study of analytical and numerical solutions of perturbed two-body problem. We take two approaches: first, the study of the analytical development of the basic quantities of the two-body problem to be used in the analytical theories of the planetary motion and second, the study of the minimization of the errors in the numerical integration by an appropriate choice of parameters in our family for each value of the eccentricity. The use of an appropriate value of the parameter can improve the length of the developments in the analytical theories and reduce the errors in the case of the numerical integration.

The main goal of this paper is to define a new one-parametric family of symmetric temporal transformations with respect to the ellipse. This new family contains as a particular case the eccentric anomaly, the regularized length of arc, and the elliptic anomaly. This family is a particular case of the biparametric family of anomalies introduced by the authors in 2016. The biparametric family comprises the most common anomalies used in the study of the two-body problem. Two approaches of this work have been taken. The first one involves the study of the analytical properties of the symmetric family of anomalies. The second approach explores the improvement of the numerical integration methods when the natural time is replaced by an anomaly of this family.

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 that includes the true anomaly f , the eccentric anomaly g and the secondary anomaly f ′ defined as the polar angle with respect to the secondary focus of the ellipse.This family is constructed using a natural generalization of the eccentric anomaly. The use of this family allows closed equations for the classical quantities of the two body problem that extends the classic, which are referred to eccentric, true and secondary anomalies.In this paper we obtain the exact analytical development of the basic quantities of the two body problem in order to be used in the analytical theories of the planetary motion. In addition, this paper includes the study of the minimization of the errors in the numerical integration by an appropriate choice of parameters in our selected family of anomalies for each value of the eccentricity