We combine Gaia EDR3 astrometry with accurate photometry and utilize a probabilistic mixture model to measure the systemic proper motion of 52 dwarf spheroidal (dSph) satellite galaxies of the Milky Way (MW). For the 46 dSphs with literature line-of-sight velocities we compute orbits in both a MW and a combined MW + Large Magellanic Cloud (LMC) potential and identify Car II, Car III, Hor I, Hyi I, Phx II, and Ret II as likely LMC satellites. 40% of our dSph sample has a >25% change in pericenter and/or apocenter with the MW + LMC potential. For these orbits, we Monte Carlo sample over the observational uncertainties for each dSph and the uncertainties in the MW and LMC potentials. We predict that Ant II, Boo III, Cra II, Gru II, and Tuc III should be be tidally disrupting by comparing each dSph's average density relative to the MW density at its pericenter. dSphs with large ellipticity (CVn I, Her, Tuc V, UMa I, UMa II, UMi, Wil 1) show a preference for their orbital direction to align with their major axis even for dSphs with large pericenters. We compare the dSph radial orbital phase to subhalos in MW-like N-body simulations and infer that there is not an excess of satellites near their pericenter. With projections of future Gaia data releases, we find dSph orbital precision will be limited by uncertainties in the distance and/or MW potential rather than proper motion precision. Finally, we provide our membership catalogs to enable community follow-up.
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