Timing Argument take on the Milky Way and Andromeda past-encounter

The two-body problem of $M31$ and the Milky Way (MW) galaxies with a Cosmological Constant background is studied, with emphasis on the possibility that they experienced Past Encounter (PE). PE are possible only for non-zero transverse velocity and their viability is subject to observations of the imprints of such near collisions. By implementing the Timing Argument (TA) for two isolated point bodies, it is shown that if $M{31}$ and the MW have had PE, then the predicted mass of the Local Group (LG) should be twice larger. Since the predicted mass is too large, the MW and $M31$ galaxies should have collided at $\sim 8 Gys$. Therefore, the TA analysis show that PE is not possible for the Local Group (LG) system.