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When reading about the recent release of data from the Dark Energy Survey, and the 'pictures' of well over 200 million distant galaxies, I got to wondering:
How can they know that a small, faint, distant smudge is not, in fact, a much-less-distant globular cluster of stars?
Other answers have provided general ideas about how to confirm that individual sources are distant galaxies and not clusters, so I'll focus on the question of how astronomers in the Dark Energy Survey (DES) might be confident about galaxy identifications for the hundreds of millions of sources in their catalogs -- particularly as the overwhelming majority of those sources do not have direct distance measurements or spectroscopy.
I'm going to concentrate on globular clusters (GCs), since these are the ones that might be confused with galaxies (they are bright and can be found outside the interiors of nearby galaxies). The simple answer is that GCs are small, and thus won't be mistaken for galaxies unless they are quite nearby. But the number of nearby GCs is vastly smaller than the number of objects classified as "galaxies" in the DES catalogs, so there will be very, very little confusion.
The identification of "galaxies" in the DES -- as described in the Data Release 2 paper -- is based on whether or not they are big enough (in angular size) to not be stars. Stars (and also active galactic nuclei) are point sources, and will be "unresolved", having the same size in the images as the point-spread function (PSF) of the observations (full-width-half max $\sim 1$ arc second for DES). Galaxies, being extended objects, will be somewhat broader in the images (they will be "partially resolved"). As the paper describes, they have a classification for each object; there are $\sim 543$ million objects with "EXTENDED_COADD" $> 1$, which they refer to as a "benchmark galaxy" classification.
GCs are small -- a typical half-light radius (the radius within which half the total stellar light is found) is about 4 parsecs. (Bright galaxies, on the other hand, have half-light radii of 1000 parsecs or more.) This means that they will become indistinguishable from stars once they are more distant than, say, 5 or 10 megaparsecs.
So only GCs that are close enough to be partially resolved (but not so close they are actually resolved into individual stars) could be mistaken for galaxies, and this means there will only be a few thousand or tens of thousands of possible cases. Given that DES has hundreds of millions of partially resolved sources in their catalogs, you can be pretty confident that most of those are not GCs.
Moving forward, it's possible to discriminate between galaxies and the few partially resolved GCs via photometric redshifts. This is an approach that uses the brightness in different filters (DES uses five) to estimate the redshift of a source. (This is inferior to using actual spectroscopy, but doesn't require time-consuming additional observations.) GCs can then be identified as sources with a redshift of $\sim 0$, unlike the vast majority of galaxies in the catalog.
We measure the distance.
A cluster is a group of stars in close proximity to each other. For relatively near stars you can directly measure the distance via the parallax. That's what the Gaia observatory has been doing with unprecedented accuracy during the last years. Thus anything which we can measure a parallax for is known - anything we don't measure a parallax for is at least as distant as the limit of Gaia. Accuracy is around 10 micro-arcseconds, thus measures distances up to 100.000pc (300kLY). That covers most, if not all of our own Milky Way (something which will quite change the way we view and understand our Milky Way).
For objects further away we climb the cosmic distance ladder: From observations and theory we know the typical brightness variation and a brightness-period relation for certain types of variable stars (delta Cepheids, RR Lyrae, delta Scuti, etc.). The delta Cepheids are also particularly bright. Thus if we observe a star of a known variable type (we know that when we look at the spectrum and the brightness variation), we can simply measure its apparent brightness and calculate its distance using the inverse square law, knowing how bright it really is. This works for close galaxies.
For galaxies further we measure the redshift by looking at the spectra and identifying well-known lines like the hydrogen lines. That's what Hubble did first. We calibrate the redshift on the nearby galaxies for which we established their distances via the known variable stars. Hubble's law shows a clear trend that the further away, the larger the red shift - and this way we get to as distant galaxies as we want.
I don't know.
However, almost all globular star clusters have globular shapes and thus look circular from every angle they are viewed. Many galaxies are also spherical, and so look circular from every angle. But the majority of galaxies are either ellipical or disc shaped, and so look circular only when viewed from a narrow range of directions.
So most galaxies look rather oval when viewed from Earth. There are also many irregular galaxies which look irregular and not circular from any angle.
Astronomers have identified well over a hundred globular clusters in our galaxy and hundreds more in nearby galaxies whose distances are known rather accurately. From studing the globular star clusters with known distances, astronomers know the range of actual diameters and absolute luminosities of globular star clusters.
Thus they can calculate the range of apparent diameters and apparent magnitudes globular star clusters in the outer halo of our galaxy would have. If a circular looking object in a sky photograph is much narrower and/or much dimmer than a globular star cluster in the galactic halo would look, astronomers will know it can't be a globular star cluster.
Furthermore, astronomers have observed nearby galaxies from the outside, and thus have cataloged all of their globular star clusters. They know it is normal for large spiral galaxies like the Milky Way to have hundreds of globular star clusters, not thousands or millions or billions of them.
So if a photo of a tiny patch of sky shows hundreds or thousands of objects which are probably distant galaxies, enough that it can be calculated that there are billions of objects like them in the whole sky, statistically the vast majority of them must not be globular clusters in our own galaxy.
And of course if astronomers take the spectra of those objects and see that the spectra are strongly red shifted, that means that those objects are too distant to be nearby globular clusters and must be distant galaxies.
