Comparison between the N-body simulations and the semianalytic model in the four-body configuration. The initial conditions for the galaxies’ masses, concentrations, positions, and velocities are set to the central values of the fiducial priors listed in Table 1. Left panel: projected mass density at the present-day snapshot (t = 0) in the center-of-mass (COM) frame for N-body simulations. The MW lies at the bottom right, with the LMC near the MW’s center; M31 and M33 are in the top left. Middle panel: projected mass density at 10 Gyr in the future in the COM frame, by which time the MW has merged with M31. Right panel: MW–M31 distance as a function of time from the N-body simulations (solid red line) and the semianalytic integration (dashed blue line). Overall, the two approaches yield a consistent merger outcome.