Upper panel: the radial distances to M31 (thick red), LMC (thin green), and SMC (dotted black) for the past 14 Gyr for the model parameters in the first line of Table 1 and an external field of 0.03a₀. We note the MW-M31 encounter 10.2 Gyr ago and the presence of the LMC and SMC close to pericenter at that epoch. The Sagittarius dwarf (dotted blue) orbit is also plotted for reference. Lower panel: planar shape of the orbits with respect to the MW. The thick red line is the orbit of M31. The first part of the orbit is indicated as a dotted red line, and becomes a solid red line from the time of the encounter 10.2 Gyr (marked by crosses) ago. All orbits are assumed to be on independent meridional RZ planes with the horizonal axis showing the offset from the MW rotation axis Z.