The Tully-Fisher Relation, discovered in 1977 by R.B. Tully and J.R. Fisher, uses the correlation between the maximum rotational velocity of a Spiral galaxy and its absolute magnitude to calculate distances to galaxies. Because the arms of the Spiral are rotating, emission lines from one side is more red shifted, while the ohter side is more blue shifted from the overall redshift due to the expansion of the universe. This causes the emission line to broaden, and the rotational velocity, after subtracting off the radial velocity, follows:
where is the shift of the peak of the emission line from (the wavelength of the emission line in its rest frame), and is the radial velocity of the galaxy.
The Tully-Fisher relation is calculated emperically, following:
by plotting the log of the velocity width against the absolute magnitude (calculated using distances derived from Cepheids or a similar method). The relation varies with the type of galaxy (Hubble Classification) as well as the bandwidth used in the observation.
Tully-Fisher Relations for Different Bandwidths
The T-F relations for the B, R, I, and H bands found by Tully-Fisher:
are the inclination corrected absolute magnitudes. is the inclination-independent measure of the rotationa rate of the spiral galaxy, and and are correction factors formulated by Pierce & Tully in 1992 when they found inconsistencies from different bands.