The classification of astronomical objects is more than a semantic pursuit. Words matter. The terminology we choose to describe our research affects how appealing and accessible it sounds to funding agencies and the public. Words also dramatically affect the research community's ability to draw global conclusions from diverse sets of astronomical objects. For example, massive star cluster Cen may be the remnant of a stripped, nucleated dwarf galaxy (Lee et al. 1999, Bekki & Freeman 2003). However, it is cataloged as a GC and thus is not considered in studies of the MW's dwarf galaxy population. Well-defined, well-chosen classification schemes therefore improve our understanding of the universe by facilitating meaningful comparisons between models and observations. Conversely, ill-defined, ill-chosen classification schemes can muddy our understanding of astrophysical phenomena.
Although no single definition of galaxy will be optimal for all purposes, we propose a physically motivated definition that will facilitate studies of galaxies both in and out of a cosmological context:
A galaxy is a gravitationally bound collection of stars whose properties cannot be explained by a combination of baryons and Newton's laws of gravity.
In a dark matter context (whether cold, warm, self-interacting, or other), this definition loosely translates to measuring whether an object contains dark matter. Alternatively, the definition can be interpreted to delineate those objects for which non-standard theories of gravity could be relevant (e.g., Milgrom 1983, Sotiriou & Faraoni 2010). For such theories, our definition would require distinct observable consequences of non-standard gravity to be imprinted on an astrophysical system for it to be classified as a galaxy. In the interest of simplicity, we focus Sections 3 and 4 of this paper on diagnosing our proposed galaxy definition in a dark matter-based context. However, we encourage others to more explicitly explore these diagnostics in alternative contexts.
We refer to Newton's laws on a macroscopic scale; this part of the definition should be considered to include objects that require general relativity to be understood. We recognize that galaxies can have a significant amount of baryonic mass in non-stellar forms such as gas and dust. It is uncertain whether the local universe harbors any "dark galaxies" that contain gas but are entirely free of stars (Minchin et al. 2005, Duc & Bournaud 2008). Whether these objects should be classified as galaxies is an open question that we do not confront here but that deserves further debate.
A purely descriptive astronomical classification (such as relaxation time) may be relatively straightforward for either observers or theorists to implement. A weakness of the definition proposed here thus lies in finding adequate diagnostics to measure whether the properties of the lowest luminosity and most compact objects are explicable with baryons and Newton's Laws (see Section 3 for a discussion of possible diagnostics).
Because galaxy formation itself is hierarchical in a cold dark matter universe, there is no trivial distinction between a single galaxy with satellite galaxies (such as the MW) and a galaxy group or cluster. As a diagnostic, (Busha et al. 2012) propose that a bound collection of galaxies is a "galaxy" rather than a galaxy cluster if at least 50% of the stellar light is associated with one central object. This diagnostic is ultimately driven by the decreasing efficiency of galaxy formation in more massive dark matter halos.
The Busha et al. (2012) diagnostic helps guide intuition when exercising reasonable common sense in applying our definition to astrophysical systems. For example, the intracluster star population of a galaxy cluster is composed of a gravitationally bound collection of stars whose dynamics cannot be explained by orbits within a Newtonian potential well dominated by cluster baryons. Such a system should not be classified as a galaxy because it is physically associated with a galaxy cluster. Similarly, the Milky Way's stellar halo is a merely a component of the Milky Way - not its own galaxy.