Pericentre times in gigayears versus pericentre distances in kiloparsecs for M33 close passages around M31. Time increases from today to early cosmic times. In panel a we show the standard case for M31 and M33 mass distributions, with M33 mass larger in the past. In panel b we consider the low-mass model, in panel c the highvar mass model. In panels a, b, c the most likely values for today’s distances of M31 and M33 from the MW are considered. Points are colour coded according to today’s M31 velocity along the x-axes in the Galactocentric frame, , shown by the colour bar in units of km s⁻¹. In panel d we consider variations of M31 and M33 distances around their mean values and the standard model for M31 and M33 mass distributions. Regions are colour coded according to the mean M33 distance from the MW in kpc.