Galaxy infall models for arbitrary velocity directions

For most galaxies in the cosmos, our knowledge of their motion is limited to line-of-sight velocities from redshift observations. To determine the radial velocity between two galaxies the minor and major infall models were established by Karachentsev & Kashibadze (2006). Regardless of the background cosmology, our derivations reveal that these infall models approximate the total radial velocity between two galaxies by two different projections employing different information about the system. For galaxies having small angular separations $\theta$, all infall models agree that the radial velocity is the difference of their line-of-sight components. Applying these models to ca. $500$ halos of the Illustris-3 simulation, we find the perpendicular and tangential velocity parts to be non-negligible for more than 90% of all, more than 5000 infalling subhalos. Thus, even for $\theta < 10$ deg, the infall-model velocities deviate from the true radial velocity. Only for 30% we found the true one lay between the minor and major infall velocity. However, the infall models yield robust upper and lower bounds to the true radial velocity dispersion. Observed under $\theta < 10$ deg the velocity dispersion inferred from the sole difference of line-of-sight velocity components even coincides with the true one, justifying this approach for high-redshift groups and clusters. Based on these findings, we predict the radial velocity dispersion of the M81-group from the minor infall model (upper bound) $\sigma_{\mathrm{r,min}} = (180 \pm 42)~\mbox{km}/\mbox{s}$, from the major infall model (lower bound) $\sigma_{\mathrm{r,maj}} = (142 \pm 64) ~\mbox{km}/\mbox{s}$ and $\sigma_\mathrm{r,\Delta v} = (99 \pm 36)~\mbox{km}/\mbox{s}$ from the line-of-sight-velocity difference.