We study the systematic bias introduced when selecting the spectroscopic redshifts of brighter cluster galaxies to estimate the velocity dispersion of galaxy clusters from both simulated and observational galaxy catalogues. We select clusters with Ngal > 50 at five low redshift snapshots from a semi-analytic model galaxy catalogue, and from a catalogue of SDSS DR8 groups and clusters across the redshift range 0.021<z<0.098. We employ various selection techniques to explore whether the velocity dispersion bias is simply due to a lack of dynamical information or is the result of an underlying physical process occurring in the cluster, for example, dynamical friction. The velocity dispersions and stacked particle velocity distributions of the parent dark matter (DM) halos are compared to the corresponding cluster dispersions and galaxy velocity distribution. We find a clear bias between the halo and the semi-analytic galaxy cluster velocity dispersion on the order of sigma gal / sigma DM = 0.87-0.95 and a distinct difference in the stacked galaxy and DM particle velocity distribution. We identify a systematic underestimation of the velocity dispersions when imposing increasing absolute I-band magnitude limits. This underestimation is enhanced when using only the brighter cluster members for dynamical analysis on the order of 5-35%, indicating that dynamical friction is a serious source of bias when using galaxy velocities as tracers of the underlying gravitational potential. In contrast to the literature we find that the resulting bias is not only halo mass-dependent but that the nature of the dependence changes according to the galaxy selection strategy. We make a recommendation that, in the realistic case of limited availability of spectral observations, a strictly magnitude-limited sample should be avoided to ensure an unbiased estimate of the velocity dispersion.
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