The Tully-Fisher Relation

Astronomers use many different techniques to determine the distance between the Earth and the stars. Many of these techniques rely on finding a “standard candle.” This is the term for being able to determine the intrinsic brightness of an object. If that brightness is known and we compare it to the apparent brightness as seen on Earth, then we can determine how far away an object is. The Tully-Fisher relation is an attempt do this but this time for galaxies instead of stars. The problem is the same, determine how intrinsicly bright the galaxy is and then measure how bright it appears on Earth. In this case, the intrinsic brightness is inferred from a measurement of the rotational speed of the galaxy. The logic goes something like this: faster galactic rotation -> larger galactic mass -> more (or larger) stars -> brighter galaxy.

I attempted to see how well this relationship held up by looking at a database of galactic statistics, the LEDA database. I plotted the measured rotation versus the absolute magnitude for those galaxies that have this information.The relation is apparent in the graph on the left. The line shows the general direction – down indicates brighter. The slope of the line is given by the number before the x variable and in this case is -4.49. We will see below that we can make the relation stronger, that is, we can get more steeply angled slopes. You will also notice considerable scatter in the data. The scatter is quantified by computing the standard deviation of this data. The lower the standard deviation, the less scattering of the data and the more the data will resemble the line itself. The graph shows the data from more than 10,000 galaxies – the 1% of all galaxies in LEDA that have rotational measurements.

I looked at the galaxies by type. Spirals, ellipticals and irregular galaxies all fit the relation but not all in the same way. Spiral galaxies with well-defined arms fit the relation more strongly (-4.7) than spirals with hard to detect arms (-2.7). Ellipticals and irregulars fit the relation somewhere in the middle of this range. Galaxies that are “face-on” to us fit the relation more strongly (up to -5.63) but that’s because it is easier to measure the rotation of these galaxies.

The LEDA database does not have absolute magnitudes in all parts of the spectrum. However, we found a journal article with a number (21) of galaxy magnitudes in the I (infrared), R (red), V (green) and B (blue) bands. The relation showed a clear preference toward the red band and the I band was strongest of all - increasing the strength of the relation to -7.1. A more detailed version of this article is available online.