CMMSE: Study of a new symmetric anomaly in the elliptic,hyperbolic, and parabolic Keplerian motion

Abstract

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 antifo-cal anomaly,𝑓′; Fukushima defined𝑓′as the angle between the periapsis andthe secondary around the empty focus. In this first part of the paper, we takean approach to the study of the semifocal anomaly in the hyperbolic motionand in the limit case corresponding to the parabolic movement. From here, wefind a relation between the semifocal anomaly and the true anomaly that holdsindependently of the movement type. We focus on the study of the two-bodyproblem when this new anomaly is used as the temporal variable. In the secondpart, we show the use of this anomaly—combined with numerical integrationmethods—to improve integration errors in one revolution. Finally, we analyzethe errors committed in the integration process—depending on several values ofthe eccentricity—for the elliptic, parabolic, and hyperbolic cases in the apsidalregion.