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.
