The Local Group Barycentre: Present-Day Uncertainty Region, Membership Dependence, and Uncertainty Growth

The Local Group (LG) barycentre is used as a kinematic reference, but its positional uncertainty and sensitivity to adopted membership are rarely quantified together. Observationally motivated uncertainties for the Milky Way, M31, M33, and the Large Magellanic Cloud are propagated through 1000 Monte Carlo realizations. The present-day distribution has principal-axis scales of (54.8, 7.7, 0.2) kpc and a median Galactocentric distance of 452 kpc. Relative to MW+M31, adding M33, the LMC, or both yields median barycentric displacements of 53, 27, and 33 kpc; the effects are not additive. Ballistic propagation in the MW rest frame increases the radius enclosing 68% of realizations from 55 to 103 kpc after 5 Gyr and 202 kpc after 10 Gyr. These values describe uncertainty propagation under an isolated-system assumption, not physical spreading of the centre of mass.