How far away are the galaxies? That is an interesting question whose answer has all sorts of interesting ramifications in cosmology. But, measuring the distances to galaxies is difficult. Measuring distances to anything in astronomy is difficult, but distances to the galaxies is even tougher to measure.
For nearby stars, you can use parallax to measure distances. Stellar parallax is the apparent shift in position of a star due the motion of the Earth around the Sun. The farther the star is away, the smaller the parallax. But, past a few hundred parsecs distance, the parallax is far too small to measure. The galaxies are all vastly farther away than that. A break came when Edwin Hubble discovered Cepheid variable stars in the galaxy M31. Cepheid variables have a relationship between how bright they are and how fast they pulsate. By measuring the pulsation period, he knew how bright the stars really were. Then, he compared that to how bright they appeared. The dimmer they appear, the farther away they are. So, he could compute the distance to the stars, and hence to the galaxies that they were in. The Cepheid variables became a standard candle, or an object of known brightness that can be used to determine distances based upon brightness. But, beyond a certain distance, individual Cepheid variables can not be picked out in a galaxy, so this method is only useful for nearby galaxies. So, how do you measure the distance to galaxies too far away to see Cepheids in them?
That is where the Tully-Fisher relationship comes in. In the 1970’s, astronomers Brent Tully and Richard Fisher conducted studies of rotational velocities in spiral galaxies. What they found was that the brighter the galaxy, the faster that the stars moved in their orbits around the center of the galaxy. Empirically, the relationship is that the luminosity of the galaxy scales approximately as the fourth power of the maximum stellar velocities: L = K v4 (where K is a proportionality constant).Â The proportionality seems to work quite well for each Hubble type of galaxy, though different Hubble classifications have a slightly different proportionality.
Tully and Fisher found the relationship empirically.Â Soon theorists were scrambling to provide an explanation.Â The reasoning for the relationship is simply that the more mass that a galaxy has, the faster the stars will be moving around it.Â The more mass that a galaxy has, presumably the more stars that it will have, and thus the brighter that it will be.Â What complicates the matter is that not all stars are equally bright.Â The brightest stars live the shortest lives, so they are less common in galaxies with low star formation rates.Â On the other hand, galaxies with high star formation rates will have more of the brighter stars, skewing the luminosity function of the whole galaxy.Â But, the star formation rates are related to the amount of interstellar medium in the galaxy, and that is related to the Hubble galaxy classification.Â So, it all relates to give a proportionality slightly different for the different galaxy types.
The Tully-Fisher relationship isn’t exact, and it doesn’t give a luminosity for the galaxy as precisely as one would like for a standard candle, but it isn’t bad.Â If you average the results for several galaxies in a galaxy cluster, then you can use the relation for determining the distance to the cluster.Â Type Ia supernovae are much better standard candles, but they are also fairly rare.Â You are not likely to get one at a convenient time to measure the distance to a particular galaxy of interest.Â But, you can measure the rotational characteristics of a galaxy by observing the Doppler shift of the spectral lines on one side of the galaxy compared with that on the other side of the galaxy.Â For example, in the galaxy NGC 253, shown above, one side of the galaxy is coming towards us and the other is going away.Â Note, that this works best for galaxies that are edge on.Â NGC 253 is not exactly edge on, but it is close.Â For galaxies that are at more oblique angles, the angle of obliquity must be measured, and then the Doppler shifts can be adjusted to give the actual rotational rates.Â For galaxies that are nearly face on, though, then this method fails, and the Tully-Fisher relation is not useful for distance determination.
What about elliptical galaxies?Â Well, they don’t really rotate like the spirals.Â But, still, the more mass that an elliptical galaxy has, the faster that stars can move in them without achieving escape velocity.Â So, a similar relationship to the Tully-Fisher relation can be developed using the dispersion ofÂ stellar velocities in elliptical galaxies.Â Again, the more mass, the brighter the galaxy, so the greater the velocity dispersion, the greater the luminosity.Â This analogous relationship for elliptical galaxies is the Faber-Jackson relation.
Image courtesy NASA, STScI