  A growing number of low luminosity and low surface brightness astronomical
objects challenge traditional notions of both galaxies and star clusters. To
address this, we propose a definition of galaxy independent of the cold dark
matter model of the universe: A galaxy is a gravitationally bound set of stars
whose properties cannot be explained by a combination of baryons and Newton's
laws of gravity. After exploring observational diagnostics of this definition,
we examine the classification of ultra-faint dwarfs, globular clusters,
ultra-compact dwarfs, and tidal dwarfs. While kinematic studies provide an
effective galaxy diagnostic in many regimes, they can be less useful for
compact or faint systems. To explore the use of [Fe/H] spread as a
complementary diagnostic, we use published spectroscopic [Fe/H] measurements of
16 Milky Way dwarfs and 24 globular clusters to calculate their [Fe/H] spreads
and uncertainties. Our principal results are: (i) no old star cluster less
luminous then M_V = -10 has a significant (> 0.1 dex) spread in iron; (ii)
known ultra-faint dwarfs can be classified with a combination of kinematics and
[Fe/H]; (iii) the observed [Fe/H] spreads in massive (> 10^6 M_Sun) globular
clusters do not necessarily imply they are the stripped nuclei of dwarfs, nor a
need for dark matter; and (iv) if ultra-compact dwarfs reside in dark matter
halos akin to those of ultra-faint dwarfs of the same half-light radii, then
they will show no clear dynamical signature of dark matter. We suggest
measurements that may assist future classification of massive globular
clusters, ultra-compact dwarfs, and ultra-faint galaxies. Our galaxy definition
is designed to be independent of the details of current observations and
models, while our proposed diagnostics can be refined or replaced as our
understanding of the universe evolves.
