Relative distance estimators.IV. 

The  relation

Elliptical galaxies also follow scaling laws which relate their size, velocity dispersion and luminosity (or surface brightness).

The most useful relation to measure distances to elliptical galaxies is the

The parameter 
Dn is the diameter within which the mean surface brightness is
In= 20.75  mag/arcsec2 in the blue.  is the central velocity dispersion.

The relation represented by a straight line in the above (log-log) plot is

To obtain the distance we only need to compare the angular size of 
Dn to the scale predicted by this equation.

There is a 15% scatter in the relation from galaxy to galaxy, which will imply a 15% intrinsic uncertainty in the distance. However, if one is interested in measuring the distance to a cluster of galaxies, the uncertainty to the cluster distance can be reduced by observing large numbers of elliptical galaxies.