Applying Liouville's Theorem to Gaia Data
  The Milky Way is filled with the tidally-disrupted remnants of globular
clusters and dwarf galaxies. Determining the properties of these objects -- in
particular, initial masses and density profiles -- is relevant to both
astronomy and dark matter physics. However, most direct measures of mass cannot
be applied to tidal debris, as the systems of interest are no longer in
equilibrium. Since phase-space density is conserved during adiabatic phase
mixing, Liouville's theorem provides a connection between stellar kinematics as
measured by observatories such as Gaia and the original mass of the disrupted
system. Accurately recovering the phase-space density is complicated by
uncertainties resulting from measurement errors and orbital integration, which
both effectively inject entropy into the system, preferentially decreasing the
measured density. In this paper, we demonstrate that these two issues can be
overcome. First, we measure the phase-space density of the globular cluster M4
in Gaia data, and use Liouville's theorem to derive its mass. We then show
that, for tidally disrupted systems, the orbital parameters and thus
phase-space density can be inferred by minimizing the phase-space entropy of
cold stellar streams. This work is therefore a proof of principle that true
phase-space density can be measured and the original properties of the star
cluster reconstructed in systems of astrophysical interest.
