Galaxy luminosity function

From a big galaxy redshift survey, we can make a histogram of the number of galaxies with absolute luminosity between $L$ and $L+dL$, where $L$ is either the total (or bolometric) luminosity, or the luminosity in some band, e.g. the B-band, per unit volume. Figure 10.5 shows that the number of faint galaxies increases like a power law, whereas there is an abrupt cut-off toward the brighter galaxies. This behaviour is well captured by the Schechter function

$${dN\over dL/L_\star} = N_\star\,\left({L\over L_\star}\right)^\alpha\,\exp(-L/L_\star)\,.$$ (10.14)

This function has three parameters, $L_\star$, $\alpha\sim -1$ and $N_\star$. For luminous galaxies, it has an exponential cut-off for galaxies brighter than $L_\star$. For less luminous galaxies, the number increases toward fainter galaxies, as $L^\alpha\sim 1/L$. Finally, $N_\star$ is a normalisation constant, which determines the mean number density of galaxies.

Values are $N_\star\approx 0.007$Mpc$^{-3}$, the steepness of the faint-end slope $-1.3\le \alpha \le -0.8$, and the exponential cut-off sets in at $L_\star\approx 2\times 10^{10}\hbox{$L_\odot$}$ in the B-band. From Table 3.1, we find that the total B-band luminosity of the MW is about $1.8\times 10^{10}\hbox{$L_\odot$}$, so the MW is just slightly fainter than an $L_\star$ galaxy.

If the faint-end slope is steeper than $\alpha=-1$, then the total number of galaxies per unit volume, $n=\int_0^\infty (dN/dL)\, dL$ diverges, because there are so many faint galaxies. In reality, this does not happen of course, since there must be some finite number of small galaxies. The mean luminosity density can be found from integrating

$$l=\int_0^\infty L\, {dN\over dL}\, dL\,.$$ (10.15)

This is the mean luminosity per unit volume. Combining this with the mean density of the universe as obtained in section 9.4, we can estimate the mass-to-light ratio for the Universe as a whole!

Figure 10.5: Histogram depicting the number of galaxies per Mpc$^3$ with given luminosity $L$ in a given colour band. At magnitudes fainter than -18, the number increases as a power in $L$, but for magnitudes brighter than -22, the number of galaxies cuts-of exponential. The drawn line is the best-fit Schechter function.

Finally, Figure10.6 shows how the Schechter luminosity function is composed of contributions from the various galaxy types, E, S0 and the different types of spirals, and irregular galaxies. Top panel is for a sample of galaxies away from big clusters, bottom panel refers to the Virgo cluster. Here we recognise the density-morphology once more: most of the brighter galaxies in the top panel are Spirals, whereas in the cluster sample, E an S0s play a much bigger role.

Figure 10.6: This plot shows how the galaxy luminosity function for galaxies in a cluster (bottom panel is for the Virgo cluster) or galaxies not in a cluster (`field galaxies', top panel), is made-up from the mix of galaxy types.