New Two-Body Regularization

Abstract

We present a new scheme to regularize a three-dimensional two-bodyproblem under perturbations. It is a combination of Sundman's timetransformation and Levi-Civita's spatial coordinate transformationapplied to the two-dimensional components of the position and velocityvectors in the osculating orbital plane. We adopt a coordinate triadspecifying the plane as a function of the orbital angular momentumvector only. Since the magnitude of the orbital angular momentumis explicitly computed from the in-the-plane components of the positionand velocity vectors, only two components of the orbital angularmomentum vector are to be determined. In addition to these, we selectthe total energy of the two-body system and the physical time asadditional components of the new variables. The equations of motionof the new variables have no singularity even when the mutual distanceis extremely small, and therefore, the new variables are suitableto deal with close encounters. As a result, the number of dependentvariables in the new scheme becomes eight, which is significantlysmaller than the existing schemes to avoid close encounters: twoless than the Kustaanheimo-Stiefel and the Bürdet-Ferrandiz regularizations,and five less than the Sperling-Bürdet/Bürdet-Heggie regularization.