Tully-Fisher and Fundamental plane relations as standard candles

The Tully-Fisher relation Eq. (10.1) relates the absolute luminosity $L$ of a galaxy to its circular velocity. Now suppose we were able to measure the proportionality constant, by determining the absolute luminosity for galaxies with known distance. Then by just measuring the circular velocity of a spiral galaxy, we could infer its absolute luminosity, and hence from its apparent magnitude obtain the distance, since

$$m-M=5\log(r) .$$ (10.13)

Therefore, the Tully-Fisher relation can be used as a standard candle, since it allows us to determine the absolute luminosity from the distance independent quantity $V_c$.

The fundamental plane relation plotted in Figure 10.4 can also be used as a standard candle: all we need to do is measure the velocity dispersion of the stars $\sigma$ from a spectrum, and determine the surface brightness (recall that the surface brightness $I$ is distance independent). By putting the E galaxy onto Figure 10.4, we then find $R_e$. And so from the apparent size of the galaxy, we can estimate its distance.

Both these methods are widely used, since measuring velocities is relatively easy from a good quality spectrum, and can be done even for faint and/or distant galaxies. And since the scatter in the relations is small, we can obtain a relatively accurate distance estimate.