## Tully-Fisher Relation

Tully-Fisher relation is a correlation for spiral galaxies between their luminosity and how fast they are rotating. The idea is that the bigger the galaxy is, faster it is rotating. That means that if you know the rotation velocity of the spiral galaxy, you can tell by using this Tully-Fisher relation its intrinsic brightness (that is, how bright that galaxy really is). By comparing the intrinsic brightness with the apparent magnitude (what you actually observe -- because the further the galaxy, the dimmer it "appears"), you can calculate its distance. For a cluster of galaxies where you can observe tens of galaxies, you can measure the distance to each galaxy via Tully-Fisher relation. Then take the average to calculate the distance to that cluster. (Because these clusters are so far away, you can assume that the actual size of the cluster itself can be ignored with respect to its distance from us). The figure below shows some examples of Tully-Fisher relations. The further the cluster of galaxies is located, more the Tully-Fisher relation is shifted downward on this diagram.

Here, Tully-Fisher relations of two clusters at different distances are shown. The one shown in the lower part represent a cluster called Abell 1367 which is much farther than the Fornax cluster which is shown in the upper part of the diagram. The relative difference between the distances of two clusters is estimated by measuring Delta D as indicated.