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It has been nearly a century since the term "galaxy" (de Sitter 1917, Crommelin 1918, Shapley 1919) was first applied to the spiral nebulae that were later found to be true extra-galactic stellar systems (Slipher 1917, Hubble 1926, see also Graham 2011 and references therein). The term "cluster" has been used to refer to Milky Way (MW) open and globular cluster star systems since their initial discoveries more than 200 years ago (Messier 1781). In the early 20th century, the primary differences between objects classified as galaxies and star clusters were (i) the milky 1 appearances of galaxies versus the grainy appearances of star clusters, and (ii) the "island universe" environments of galaxies versus the association of star clusters with the MW system. In the intervening years, galaxies and star clusters have largely been classified based on their physical sizes: galaxies have typical sizes of hundreds of pc to tens of kpc whereas star clusters have typical sizes of a few pc, with a scatter to tens of pc.

The lion's share of known star clusters and galaxies can be classified by this simple "I know it when I see it" size-based distinction. However, there are a growing number of astronomical objects that are not so easily classified - those at extreme low luminosities and surface brightnesses, and those filling gaps previously observed in size-luminosity space (Misgeld & Hilker 2011). These objects currently hold the most promise to shed new light on galaxy formation at the bottom of the hierarchy and the distribution of dark matter to the smallest possible size and mass scales. For example, ultra-compact dwarfs (UCDs) have luminosities (-13 < MV < -9) similar to those of both dwarf spheroidal galaxies and luminous globular clusters (GCs), but sizes (10 pc < rhalf < 100 pc) intermediate to both populations and dynamical mass-to-light ratios of ~ 2-5 2, larger than those typical of GCs (e.g., Hilker et al. 1999, Drinkwater et al. 2003, Hasegan et al. 2005, Evstigneeva et al. 2007, Mieske et al. 2008, Chilingarian et al. 2011, Brodie et al. 2011). Some ultra-faint MW satellites (e.g., Segue 1, Segue 2, Boötes II, Willman 1) have also challenged our notion of galaxies. These objects have luminosities (-3 < MV < -1) lower than those of nearly any known old star cluster or dwarf galaxy, physical sizes (20 pc < rhalf < 40 pc) between those of most star clusters and dwarf galaxies, and dynamical mass-to-light ratios as high as 3000 (Simon et al. 2011). While tidal dwarfs 3 may provide a piece to this puzzle, they have proved difficult to study and classify themselves (see Duc 2012 for a review).

The origin and properties of systems such as UCDs, extreme MW dwarf satellites, and tidal dwarfs are fundamental to many open questions in galaxy formation and cosmology. They might hold unique clues to relationships between different classes of hot stellar systems (e.g., giant elliptical galaxies, dwarf elliptical galaxies, dwarf spheroidal galaxies, UCDs, nuclear star clusters, GCs; Dabringhausen et al. 2008, Wolf et al. 2010, Misgeld & Hilker 2011, Zaritsky et al. 2011). They might be our best luminous tracers of sub-galactic dark matter. Having a well-defined classification scheme will be essential to these studies, since imminent and upcoming wide-field surveys, including Pan-STARRS 1 (Kaiser et al. 2002), the Southern Sky Survey (Keller et al. 2007), the Dark Energy Survey (The Dark Energy Survey Collaboration 2005) and LSST (Ivezic et al. 2008), are expected to reveal large numbers of previously unseen low surface-brightness systems.

As the rate of discoveries and the diversity of the known cosmic zoo increases, the question "What is a galaxy?" is being discussed at conferences and in the literature (e.g., Gilmore et al. 2007, Kroupa 2008, van den Bergh 2008, Forbes & Kroupa 2011). The most common distinction currently made between galaxies and star clusters is the presence of dark matter - galaxies reside at the centers of dark matter halos and star clusters do not (e.g., Simon et al. 2011, Tollerud et al. 2011, Willman et al. 2011). One strength of this definition is that it facilitates studies of dwarf galaxies in a cosmological context: it allows a straightforward connection between the set of objects classified as galaxies and a comparison with the predictions of dark matter plus galaxy formation models. One weakness of this physically motivated definition is the fact that the cold dark matter model is a theory rather than a physical law.

Other recently proposed definitions for a galaxy have included rhalf > 100 pc, a relaxation time longer than a Hubble time, or complex stellar populations (Forbes & Kroupa 2011). Although each of these definitions have their own strengths (namely that they are straightforward to diagnose), they each also have shortcomings 4. For example, size-based classifications are becoming increasingly arbitrary as size-luminosity space is becoming continuously populated with objects (Misgeld & Hilker 2011). The concept of "complex" stellar populations has also become ill-defined now that light element abundance spreads have been identified in a large fraction of MW GCs (e.g., Gratton et al. 2004, Cohen & Meléndez 2005, D'Antona et al. 2005, Carretta et al. 2009b), which were once thought to be pristine examples of simple stellar populations.

In this paper, we tweak past definitions of the term "galaxy" with the aim of appealing to a sufficiently broad cross-section of astronomers that consensus might be reached. In Section 2, we motivate the importance of having a clear definition of galaxy within the astronomical community, and then present a physically motivated definition. In Section 3, we consider kinematics and complementary indirect diagnostics such as [Fe/H] spread and population-based diagnostics that might be used to test whether an object is a galaxy. In Section 4 we examine the known properties of ultra-faint dwarfs, UCDs, GCs, and tidal dwarfs in the context of our proposed definition.

1 The word "galaxy" derives directly from the Greek word for "milky". Back.

2 All mass-to-light ratios are in V unless otherwise stated. Back.

3 We use "tidal dwarf" rather than "tidal dwarf galaxy" throughout this paper, to avoid presupposing a galaxy definition for this class of objects. Back.

4 See also the related, informal discussion linked at the URL Back.

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