If you look at any cluster of galaxies, you will see a wide variety in galaxy luminosities, from the very faint to the very bright. The luminosity function (LF) specifies the relative number of galaxies per unit of luminosity. An observed luminosity function looks like this: Fig 2 of Schechter (1976). This fits a function to 13 Abell clusters.

We’ve got luminosity on the x-axis, in either luminosities on the top or absolute magnitudes on the bottom. The number of galaxies in each luminosity bin is given on the y-axis. Note that there are more faint galaxies than bright galaxies and the luminosity function has to display that.

In 1976, Paul Schechter wrote down an analytic expression for the LF:

$\varphi \left(L\right)\phantom{\rule{0.167em}{0ex}}dL={\varphi }_{*}{\left(\frac{L}{{L}_{*}}\right)}^{\alpha }exp\left(-\frac{L}{{L}_{*}}\right)\phantom{\rule{0.167em}{0ex}}d\left(\frac{L}{{L}_{*}}\right)$

This function has three parameters. The characteristic luminosity, L*, is the luminosity that separates the function into the high- and low-luminosity parts; it forms the “knee” of the function. Current estimates for the characteristic luminosity is around 1010 L h-2. The variable α controls the slope of the faint-end. This tends to vary around -0.8 to -1.3 depending on the population being studied. Note that α=-1 is referred to as “flat”, with the slope getting steeper the more negative α is. Finally, φ* is the normalization factor. This quantity varies due to underlying systematics, but for a total galaxy population, it is around 0.02 h3 Mpc-3.

There are two parts to the Schechter function as well. At low luminosity, L < L*, we have a power-law,

$\varphi \left(L\right)\phantom{\rule{0.167em}{0ex}}dL\propto {L}^{\alpha }$

so that low-luminosity galaxies are more common. At high luminosity, L > L*, we have an exponential cutoff,

$\varphi \left(L\right)\phantom{\rule{0.167em}{0ex}}dL\propto {e}^{-L}$

so that high-luminosity galaxies are more rare. It is more common to find the Schechter function written in terms of observable magnitudes rather than luminosities. The Schechter function can be relatively easily changed into magnitudes by utilizing the conversion between luminosities and absolute magnitudes (1).

We also need a differential dL/dM conversion as well, given by (2). Substituting these into the Schechter function gives

Here, M* takes the role of the characteristic magnitude where values tend to hover around -21 + 5log(h).

There are two useful integrations of the Schechter function. If we integrate over number, we get the Gamma function:

Here, the upper incomplete Gamma function is defined as

$\Gamma \left(s,x\right)={\int }_{x}^{\infty }{t}^{s-1}{e}^{-t}\phantom{\rule{0.167em}{0ex}}dt$

If we integrate from zero to infinity, we get the total number of galaxies,

${N}_{\mathrm{\text{tot}}}={\varphi }_{*}\Gamma \left(\alpha +1\right).$

Note that for α≤-1, the number of galaxies diverges. In reality, the function must turn over at some luminosity to avoid this. Integration over luminosity reveals a similar use of the Gamma function:

Once again, if we integrate from zero to infinity, we get the total luminosity:

${L}_{\mathrm{\text{tot}}}={\varphi }_{*}{L}_{*}\Gamma \left(\alpha +2\right).$

Although the luminosity does diverge for α≤-2, these are not typical values for alpha, so in practice the luminosity does not diverge (and given a mass-to-light ratio, neither does the mass).

There are a few methods of getting a LF depending on the sample. If you are investigating a galaxy cluster, it is relatively straightforward in concept:

1. Bin all the galaxies by apparent magnitude, down to some limit, to get φ(m).
2. Use the cluster distance/redshift to convert apparent magnitudes to absolute magnitudes, giving you φ(M).
3. Fit a Schechter function to φ(M) using a chi-square technique to obtain M* and α.

The primary complications in this method is eliminating foreground and background contaminating galaxies. Radial velocity measurements are sometimes useful, but dwarf galaxies are too dim to get good measurements. Dwarfs also have low surface brightness, while background galaxies usually have high surface brightness, making it easy to confuse the two groups. You also have to apply statistical corrections to N(m) using galaxies from the field.

Methods for galaxies in the field are more complicated. I will refer you here for a summary of the methods.

It has become increasingly clear that the LFs of different galaxy types are different regardless of environment. It is the relative difference in number between the galaxies types that changes because of environment.

Some basic plots: