Tully–Fisher relation

The Tully–Fisher relation for spiral and lenticular galaxies

In astronomy, the Tully–Fisher relation is an empirical relationship between the intrinsic luminosity of a spiral galaxy and the width of its emission lines (a measure of its rotation velocity). It was first published in 1977 by astronomers R. Brent Tully and J. Richard Fisher.[1] The luminosity is calculated by multiplying the galaxy's apparent brightness by 4πd2, where d is its distance from us, and the spectral line width is measured using long-slit spectroscopy.

Several different forms of the TFR exist, depending on exactly what quantities one takes it to relate. Tully and Fisher used optical luminosity, but subsequent work showed the relation to be tighter when defined in terms of K-band light (a good proxy for stellar mass), and even tighter when luminosity is replaced by the galaxy's total baryonic mass (the sum of its mass in stars and gas).[2] This latter form of the relation is known as the Baryonic Tully-Fisher Relation, and states that baryonic mass is proportional to velocity to the power of roughly 3.5-4.[3]

The TFR can be used to estimate the distance to a spiral galaxy. By assuming the galaxy to lie on the relation, one can calculate its luminosity from its directly-measurable line-width. The distance can then be found by comparing the luminosity to the apparent brightness. Thus the TFR constitutes a rung of the cosmic distance ladder, where it is calibrated using more direct distance measurement techniques and used in turn to calibrate methods extending to larger distance.

In the dark matter paradigm, a galaxy's rotation velocity (and hence line width) is primarily determined by the mass of the dark matter halo in which it lives, making the TFR a manifestation of the connection between visible and dark matter mass. In Modified Newtonian Dynamics (MOND), the BTFR (with power-law index exactly 4) is a direct consequence of the gravitational force law effective at low acceleration.[4]

The analogues of the TFR for non-rotationally-supported galaxies, such as ellipticals, are known as the Faber-Jackson relation and the Fundamental Plane.