# Tully-Fisher relation

In astronomy, the Tully-Fisher relation, published by astronomers R. Brent Tully and J. Richard Fisher in 1977, is an empirical relationship between the intrinsic luminosity (proportional to the stellar mass) of a spiral galaxy and its velocity width (the amplitude of its rotation curve). The luminosity is the amount of light energy emitted by the galaxy per unit time; it can be measured using the galaxy's apparent brightness when the distance to the galaxy is known. The velocity width is measured via the width or shift of spectral lines and the Doppler effect.

The quantitative relationship between luminosity and velocity width is a function of the wavelength at which the luminosity is measured, but roughly speaking, luminosity is proportional to velocity to the fourth power.

The relation connects the directly (and relatively easily) observable velocity width to the difficult-to-observe intrinsic luminosity. Because the luminosity is related to the (easily observed) apparent brightness by the distance (squared), the Tully-Fisher relation can be used as a distance measurement, or, in the astronomical parlance a "secondary standard candle".

Internal dynamics of stars in galaxies are driven by gravity. For this reason, the amplitude of the galaxy rotation curve is related to the galaxy's mass; the Tully-Fisher relation is a direct observation of a close relationship between galaxy stellar mass (which sets the luminosity) and total gravitational mass (which sets the amplitude of the rotation curve).

The relation is measured and calibrated using primary standard candles.

Used to measure the distance to a spiral galaxy:

* Measure the red and blue shifts of the rotation curve

* Calculate the speed of the stars orbiting the center of the galaxy

* Calculate the gravitational force acting on the stars

* Take a 10th of the mass because 90% of the mass if the galaxy is made of dark matter

* Find the luminosity and combine it with the apparent magnitude to finally get the distance.

Works out to about 10 Mpc (10,000,000 parsecs).

Does not work for elliptical galaxies which are in general not rotationally supported. Still similar methods exist for them: the Faber-Jackson relation and the Fundamental plane (elliptical galaxies).

* Distance modulus

* Modified Newtonian dynamics

* Standard candle

* Fundamental plane (elliptical galaxies)

* Faber-Jackson relation

References

• zebu.uoregon.edu/~imamura/209/apr7/TF.html
• Kuhn, Karl F., In Quest of the Universe. ISBN 0-314-02393-3.
• 2003 C-level Astronomy Presentation for Science Olympiad (Microsoft PowerPoint format, openable in OpenOffice.org)
• Tully, R. B.; Fisher, J. R., A new method of determining distances to galaxies. (pdf) Astronomy and Astrophysics, vol. 54, no. 3, Feb. 1977, p. 661-673. (abs)
• Macri, L. M.; K. Z. Stanek & D. Bersier et al. (2006), "A New Cepheid Distance to the Maser-Host Galaxy NGC 4258 and Its Implications for the Hubble Constant", Astrophysical Journal 652 (2): 1133-1149, <http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2006ApJ...652.1133M>