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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.

A Study about the Integration of the Elliptical Orbital Motion Based on a Special One-Parametric Family of Anomalies