I am trying to do some analysis of galaxy velocities within cluster environments. This involves finding the velocity of that galaxy with respect to the cluster-frame, or the "proper velocity", $v_i$. It would make sense to simply take the difference in redshift between the galaxy, $z_i$ and the cluster, $z$ (where $z$ is obtained through some average estimator).

The equation for a galaxy proper velocity given in Ruel et al. (2014) is

$v_i = c(z_i - z) / (1+z)$

The numerator makes obvious sense. But why are we dividing by $1$ plus the reference redshift, $z$? I know that

$1 + z = 1/a(t)$

where $a$ is the scale factor of the universe, so it would seem that we are taking the difference in velocities between the galaxy and the cluster, derived from their observed redshifts, and then multiplying by the scale factor at the time that the observed light was emitted. I don't understand why this is.


By definition, the total redshift $z_i$ of a galaxy is given by $$ 1 + z_i = \frac{\lambda_\text{ob}}{\lambda_\text{em}}, $$ where $\lambda_\text{em}$ is an emitted wavelength, and $\lambda_\text{ob}$ is the observed wavelength. This ratio can by factorized in two parts: $$ 1 + z_i = \frac{\lambda_\text{ob}}{\lambda_\text{cl}}\frac{\lambda_\text{cl}}{\lambda_\text{em}}, $$ where $\lambda_\text{cl}$ is the wavelength in the restframe of the cluster. The first ratio is the cosmological redshift of the cluster, while the second ratio is the (non-relativistic) relative velocity of the galaxy in the cluster frame: $$ 1 + z_i = (1 + z)(1 + v_i/c), $$ so that $$ v_i = c\frac{z_i - z}{1+z}. $$


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