Gaia early DR3 systemic motions of Local Group dwarf galaxies and orbital properties with a massive Large Magellanic Cloud

We perform a comprehensive determination of the systemic proper motions of 74 dwarf galaxies and dwarf galaxy candidates in the Local Group based on Gaia early data release 3. The outputs of the analysis for each galaxy, including probabilities of membership, will be made publicly available. The analysis is augmented by a determination of the orbital properties of galaxies within 500 kpc. We adopt the flexible Bayesian methodology presented by McConnachie \& Venn (2020), which takes into account the location of the stars on the sky, on the colour-magnitude diagram and on the proper motion plane. We apply some modifications, in particular to the way the colour-magnitude diagram and spectroscopic information are factored in, e.g. by including stars in several evolution phases. The bulk motions are integrated in three gravitational potentials: two where the Milky Way is treated in isolation and has a mass 0.9 \& 1.6 $\times 10^{12}$M$_{\odot}$ and the time-varying potential by Vasiliev et al. (2021), which includes the infall of a massive Large Magellanic Cloud (LMC). We are able to determine bulk proper motions for 73 systems, and we consider reliable 66 of these measurements. For the first time, systemic motions are presented for galaxies out to a distance of 1.4 Mpc, in the NGC~3109 association. The inclusion of the infall of a massive LMC significantly modifies the orbital trajectories of the objects, with respect to orbit integration in static Milky Way-only potentials, and leads to 6 galaxies being likely associated to the LMC, 3 possibly associated and 1 recently captured object. We discuss the results of the orbit integration in the context of the relation of the galaxies to the system of Milky Way satellites, implications for the too-big-to-fail problem, impact on star formation histories, and tidal disruption.