In order to use satellite galaxies to probe the potentials of host galaxies, one needs to define an appropriate sample of host and satellite galaxies. Unlike cosmology simulators who are blessed with full 6-dimensional phase space information, observers are, of course, limited to 3 dimensions (RA, DEC, and redshift). Given this limited information, then, one must base the selection criteria on projected radii (evaluated at the redshift of the host) and relative radial velocities, dv, of the candidate hosts and satellites. To guarantee that the dynamics of the satellites are determined solely by their host galaxy, the hosts must be determined to be "isolated" in some sense. That is, if another large, bright galaxy is too close to a candidate host galaxy to guarantee that the satellite orbits are affected solely by the candidate host, that candidate host and its satellites are rejected from the sample. Satellites must, necessarily, be fainter than their host, be found within some reasonable projected radius of the host, and have some reasonable line of sight velocity with respect to the host.
There are a number of different selection criteria that have been used in the recent literature, and three sets of selection criteria that have been used in more than one investigation are summarized below:
Hosts must be at least 8 times brighter than any other galaxy that is within rp < 500 kpc and |dv| < 1000 km sec-1. In addition, hosts must be at least 2 times brighter than any other galaxy that is within rp < 1 Mpc and |dv| < 1000 km sec-1. Satellites must be at least 8 times fainter than their host, must be found within rp < 500 kpc, and must have |dv| < 500 km sec-1. Here h = 0.7 has been adopted (, ).
Hosts must be at least 2 times brighter than any other galaxy that falls within rp < 2.86 Mpc and |dv| < 1000 km s-1. Satellites must be at least 4 times fainter than their host, must be found within rp < 714 kpc and must have |dv| < 1000 km s-1. Here h = 0.7 has been adopted (, , , ).
Hosts must be at least 2.5 times brighter than any other galaxy that is within a projected radius of rp < 700 kpc and a relative radial velocity difference of |dv| < 1000 km sec-1. Satellites must be at least 6.25 times fainter than their host, must be found within rp < 500 kpc, and the host-satellite velocity difference must be |dv| < 500 km sec-1. Here h = 0.7 has been adopted (, ).
Although the above criteria may seem lax or even somewhat arbitrary, in the case of the first two sets of criteria, both the Milky Way and M31 would be excluded from the sample of hosts. That is, these particular selection criteria give rise to samples of unusually isolated host galaxies. In addition, both Prada et al.  and Brainerd  adopted a number of different selection criteria in their investigations of the satellites of SDSS galaxies and concluded that there were no statistical differences between results that were obtained with different selection criteria. In other words, provided sufficiently "reasonable" criteria are adopted for selecting isolated hosts and their satellites, the results of the investigations are stable to modest differences in the details of those selection criteria.
No matter what selection criteria are adopted, however, there will always be "interlopers" in the satellite data. Interlopers are galaxies that are falsely identified as satellites; that is, they pass the formal selection criteria, but they are not, in fact, dynamically associated with the host galaxy. The presence of interlopers will artificially inflate any measurement of the velocity dispersion of genuine satellites, and recent investigations of satellite dynamics (, , , ) have corrected for the effects of interlopers by modeling the distribution of host-satellite velocity differences as the sum of a Gaussian distribution (due to the genuine satellites) and a constant offset (due to the interlopers). Prada et al.  used numerical simulations to show that this is a sensible way in which to correct for the effects of interlopers. Moreover, both Brainerd & Specian  and Prada et al.  have pointed out that an accurate determination of the velocity dispersion profile, v(rp), for satellite galaxies depends on a proper determination of the interloper fraction as an explicit function of the projected radius. That is, by purely geometrical effects, the interloper fraction is necessarily an increasing function of rp. An example of fitting a "Gaussian plus offset" to the distribution of velocity differences for late-type galaxies and early-type galaxies in the 2dFGRS is shown in Figure 4. One can clearly see from this figure that the velocity dispersion of the satellites is a function of the morphology of the host galaxy (being larger for early-type hosts than late-type hosts), and that the interloper fraction increases with projected radius.
Figure 4. Points with error bars show the observed distribution of velocity differences, N(|dv|), for a subset of host-satellite systems in the 2dFGRS for which the host morphologies have been visually classified. Solid lines show the best-fitting "Gaussian plus offset" function, from which the velocity dispersion of the satellites, v, and the fraction of interlopers, fi, is determined. Left panels: late-type hosts. Right panels: early-type hosts. Top panels: satellites located close to the host in projection on the sky. Bottom panels: satellites located far from the host in projection on the sky. A substantially larger value of v is obtained for the satellites of early-type hosts than for the satellites of late-type hosts. Note, too, that the fraction of interlopers increases significantly with the projected radius, r, of the satellites.
The above "Gaussian plus offset" fit to the distribution of host-satellite velocity differences accounts for the fact that the number of interlopers is a function of projected radius, and it assumes a priori that the number of interlopers at a given projected radius is constant with |dv|. Recently, however, van den Bosch et al.  used simulations of galaxy redshift surveys to investigate this and found a sharp increase in the number of interlopers for small relative velocities. van den Bosch et al.  note, however, that the value of v that is determined from a simple "Gaussian plus offset" fit is not strongly affected by the fact that the number of interlopers varies with |dv|. This is because the best-fitting value of v is rather insensitive to the precise value of the interloper fraction. Brainerd  also finds that the number of interlopers is larger for small values of |dv| than it is for large values of | dv|, but that the effect is not nearly as pronounced as found by van den Bosch et al. . Given the size of the error bars on the distribution of host-satellite velocity differences in the current observational samples, then, it would appear that the simple "Gaussian plus offset" fit to the distribution of velocity differences is more than adequate to the task of estimating v(rp).