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To compound the ambiguities inherent to classifying objects such as extreme ultra-faint MW dwarfs and UCDs, observers have neither agreed upon a definition of galaxy nor reached consensus on how to interpret observations in hand. To facilitate comparisons between dwarf galaxy predictions and the increasingly complex sets of observations of candidate dwarf galaxies, the field needs an agreed-upon definition for galaxy. We have accordingly proposed a physically motivated definition that does not insist on a cold dark matter interpretation of data: 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.

We have explored possible diagnostics of this galaxy definition (primarily in the context of a cold dark matter dominated universe), primarily kinematic studies and [Fe/H] spread. Although kinematic studies generally provide the most direct way to infer a galaxy definition, it can be difficult to measure the dynamical mass of low luminosity and/or low velocity dispersion (< few km s-1) systems. Even once robustly measured, interpreting relatively modest dynamical M / L ( 10) may face several stumbling blocks: those that could have generated overestimates of dynamical mass (e.g., binary stars, contaminants in spectroscopic sample) and those could generate underestimates of stellar mass (e.g., sparse sampling of the stellar luminosity function, an overabundance of stellar remnants). While these effects do not appear to be a major problem for objects currently classified as galaxies, including the Milky Way's dwarfs, they should be carefully considered as discoveries at the extremes of the cosmic zoo continue. Systems such as UCDs and massive GCs also may not bear a kinematic signature of dark matter or non-Newtonian gravity, even if present, because their baryons are so densely packed.

sigma[Fe / H] provides an complimentary means to diagnose a galaxy definition for systems less luminous than MV = -10. Using public spectroscopic [Fe/H] measurements, we recalculated the average systemic [Fe/H] and associated dispersions for 24 Milky Way GCs and 16 Milky Way dwarf galaxies. All dwarf galaxies show spectroscopic [Fe/H] spreads of ~ 0.3 dex or more. No GC less luminous than MV = -10 shows a notable ( 0.1 dex) [Fe/H] spread. The sigma[Fe / H] diagnostic has already been applied to the Segue 1 (Simon et al. 2011) and Willman 1 dwarf galaxies (Willman et al. 2011). One possible caveat with the sigma[Fe / H] diagnostic is the possibility that the mergers of multiple star clusters could yield an iron abundance spread. This merging star cluster hypothesis, which would produce a multimodal [Fe/H] distribution, should be carefully considered when classifying objects by [Fe/H] dispersion alone.

The Fundamental Plane and its variants do not presently provide an alternative means to diagnose a galaxy definition for low luminosity systems. However, these scaling relations do provide a useful benchmark against which to compare ambiguous objects. For example, an outlier from known scaling relations may signal a problem with its calculated velocity dispersion or estimated stellar mass (such as the issues discussed in Section 3.1.1.) Well behaved scaling relations can also help rule out pathological explanations for sets of objects, especially when metallicity is included. For example, the metallicity-luminosity relation followed by the Milky Way's lowest luminosity dwarfs helps rules out alternative hypotheses for their existence as a population, such as tidal tails at apocenter and clumps in streams.

There are some classes of objects not discussed in this paper, but which would be worth new consideration in the context of our proposed galaxy definition. For example, dwarf ellipticals (dEs) do not typically show strong kinematic evidence for non-baryonic mass in their central regions (e.g. Wolf et al. 2010, Forbes et al. 2011). However, recent kinematic studies of stars and globular clusters in their outer regions (Beasley et al. 2009, Geha et al. 2010) have consistently suggested that M / L increases with radius and that stars alone cannot account for the observations. Therefore, the current data favors the classification of dEs as galaxies by our definition. If future observations of dEs do not support this emerging consensus, then this classification should be revisited.

After examining massive globulars, UCDs and tidal dwarfs in detail, we find that they can not yet conclusively be classified given existing diagnostics of our galaxy definition. Their ultimate classification must be guided by future observational data. If UCDs and tidal dwarfs are inconsistent with a galaxy definition, this does not mean that they should automatically be classified as star clusters. Both of these classes of objects are interesting and stand on their own as worthy to investigate, given their unique properties and possibly formation channels. For tidal dwarfs, in particular, we advocate that these objects are not lumped in with galaxies or clusters but that they remain their own distinct class of objects.

We have suggested several measurements, some of which are possible now, that could facilitate the classification of these, and other, extreme objects:

Basic cold dark matter plus galaxy formation models predict a dichotomy between systems that form in the centers of dark matter halos and systems that form in the monolithic collapse of gas clouds that are not the primary baryonic components of dark matter halos. Our best present interpretation of the observations in this context reveals that systems forming at the center of a dark matter halo bear an observable imprint of this formation channel, such as the kinematic or chemical diagnostics discussed here. This observable imprint would translate to a galaxy classification by our proposed definition. The fact that systems classified as galaxies may be equivalent to the set of astrophysical systems that formed in dark matter halos can be used as strong guidance to theorists when selecting against which astrophysical systems to compare their predictions. However, even in a dark matter context, we cannot take for granted that systems classified as galaxies by our definition are inclusive of all systems that formed in dark matter halos and exclusive of systems that formed otherwise.

As our understanding of the universe grows, it may be possible for systems that formed inside of dark matter halos to fail the galaxy diagnostics discussed here. For example, it would not be unreasonable to conceive that a very low-luminosity fossil galaxy could form all of its stars over a sufficiently short timescale that no opportunity for self-enrichment by supernovae could occur, leading to a minimal spread in [Fe/H]. If this is the case, then alternative diagnostics need to be identified for such "first" galaxies. No objects meeting our definition of a galaxy via kinematics, but without a spread in [Fe/H], have yet been discovered, but it is plausible they exist. Cosmological globular clusters, if they exist, may be such objects (Griffen et al. 2010). Conversely, it may be possible for stellar systems that formed inside of a dark matter halo to lose most or all of their dark matter. In this scenario, a(n almost) stellar-only system may exist with the chemical imprint of formation within a dark matter halo. Although simulations of dwarf galaxies in both cuspy and cored halos show that this is unlikely (see Section 3.1), simulations of globular clusters within dark matter halos have shown that it may be possible to remove most of the dark matter in systems close to disruption (e.g. Mashchenko & Sills 2005).

Our proposed galaxy definition is itself independent of our observational knowledge and currently favored theories for structure formation; it can thus remain unchanged even as our understanding of the complex universe evolves. However, the particular diagnostics of this definition as investigated in this polemic may indeed need to be revisited as our knowledge of extreme objects grows - both observationally and theoretically. For example, the possible use of spreads in elements other than iron (such as calcium) to diagnose a galaxy classification is something that should continue to be scrutinized as our knowledge of such abundance patterns grow.

BW acknowledges support from NSF AST-0908193. BW also thanks NYU's Center for Cosmology and Particle Physics and Drexel University's Physics department for hosting her during the writing of of this paper. We thank Dr. Pierre-Alain Duc, the referee, for comments that helped improve the quality and clarity of this paper. We thank Michele Bellazzini, Joerg Dabringhausen, Ross Fadely, Duncan Forbes, Amanda Ford, Marla Geha, Amina Helmi, David Hogg, Evan Kirby, Pavel Kroupa, George Lake, Erik Tollerud, and Enrico Vesperini for stimulating conversations and emails leading up to and during the preparation of this paper. This research has made use of NASA's Astrophysics Data System Bibliographic Services.

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