On tidal energy in Newtonian two-body motion with infinite initial separation

In $\href{https://doi.org/10.48550/arXiv.1708.04307}{[8]}$ we have studied the dynamics of tidal energy in Newtonian two-body motion and how it affects the center-of-mass orbit of two identical gravitating fluid bodies. It is shown in $\href{https://doi.org/10.48550/arXiv.1708.04307}{[8]}$ that for a class of initial configuration, the tidal energy caused by the deformation of boundaries of two fluid bodies can be made arbitrarily large relative to the positive conserved total energy of the entire system. This reveals the possibility that the center-of-mass orbit, which is unbounded initially, may become bounded during the evolution. This result in $\href{https://doi.org/10.48550/arXiv.1708.04307}{[8]}$ is based on a quantitative relation between the tidal energy and the distance of two bodies. However, this relation only holds when the two-body distance are within multiples of the first closest approach, due to the fact that initially the tidal energy vanishes but the two-body distance is finite. In this work, based on the a priori estimates established in $\href{https://doi.org/10.48550/arXiv.1708.04307}{[8]}$, we construct a solution to the same two-body problem as in $\href{https://doi.org/10.48550/arXiv.1708.04307}{[8]}$ but with infinite initial separation. Therefore the above mentioned quantitative relation holds during the entire evolution up to the first closest approach.