Dynamics of Local Group Satellite Galaxies in the Era of Precision Astrometry

High-precision astrometric data from the {\em Hubble Space Telescope} (HST) and {\em Gaia} are revolutionizing our ability to study galaxies in the Local Group (LG) in unprecedented detail. Recent breakthroughs, such as high accuracy proper motion (PM) measurements for Andromeda (M31), Triangulum (M33), and many of the dwarf galaxies orbiting the Milky Way (MW) and M31 now make it possible to analyze the interaction histories between satellite galaxies and their hosts. In this dissertation, I explore how the combination of astrometric data with high-resolution cosmological simulations can be used to improve our classical understanding of the LG's dynamical history. Using PMs from HST and the \textit{Very Long Baseline Array} and independently measured PMs with \textit{Gaia}, I show that the most plausible orbital history for M31's most massive satellite galaxy, M33, is a first infall scenario where M33 enters M31's halo in the last 2-4 Gyr, arriving at its closest position relative to M31 today. I also demonstrate that this orbital history is consistent with those of M33 analogs in M31-mass systems in \textit{Illustris-1-Dark}. M33's new orbital history is contrary to its classical orbit (developed before M31's PM was measured) where M33 has a close ($<$ 100 kpc), recent ($<$ 3 Gyr) tidal interaction with M31, resulting in M33's curious morphology. If on first infall into M31's halo, M33 is expected to retain its infall mass and any associated satellite galaxies. Here, I quantify the predicted satellite galaxy population of M33 in a $\Lambda$CDM paradigm, which may provide an alternative explanation for M33's warped morphology and can be directly tested with existing observatories. In the remaining chapters, I develop innovative methods to statistically constrain the precise masses of the MW and M31 by comparing the kinematic and dynamical properties of their satellite galaxies to the properties of simulated analogs in \textit{Illustris-1-Dark}. I show that the orbital angular momenta of satellite galaxies is the most reliable estimator of host halo mass over time. Using the angular momenta of multiple satellites simultaneously narrows the current uncertainty in the MW’s mass by $\sim$25\%, showing promise for improved constraints on the mass of M31 and ultimately the LG