15.4.1. The Local Luminosity Function
The simplest form of the radio luminosity function m(L | z, ) specifies the comoving space density of all sources per unit of logm(L) [often m dex(0.4) = 1 "magnitude"] in luminosity L at frequency (in the source frame) and redshift z. The vertical bar notation is used to distinguish distribution variables from parameters. The "local" luminosity function m(L| z = 0, ) of low-luminosity sources can be obtained directly from flux-limited radio observations of nearby galaxies in optically complete samples. Let Vm be the maximum volume in which an optically selected galaxy would be brighter than both the radio flux density-and optical magnitude limits. If the sample galaxies are distributed uniformly in space, the density contributed by the N radio-detected galaxies with luminosities in the luminosity bin of width m centered on L is
with an rms statistical uncertainty
The actual errors in m, are larger at the lowest luminosities because clustering in the accessible volumes Vm associated with intrinsically faint galaxies is significant. Since radio sources of intermediate luminosities have space densities too low for them to be numerous in the small volumes covered by most optically complete samples, sources identified with bright low-redshift galaxies found in complete radio surveys must be used (e.g., Auriemma et al. 1977).
At the very highest luminosities the source density is so low that the nearest sources are already at cosmological distances and lookback times affected by evolution; their "local" luminosity function can only be estimated with the aid of evolutionary models. The density of evolving sources is uniform in volume elements dV' that have been weighted by the evolution function E(L, z) (Equation 15.8). The weighted volume V'm replacing Vm in Equation (15.9) is
for a source of luminosity L that could be seen out to a redshift zm.
The distribution of
should be uniform in the interval (0,1) if the sources in any sample have a constant (comoving) density (cf. Schmidt 1968). Incompleteness is usually revealed by a deficit of V / Vm values near unity, and monotonically increasing evolution by <V / Vm> > 0.5 for the whole sample.
The local luminosity function is best determined at = 1.4 GHz. The local luminosity function of spiral galaxies has recently been derived from sensitive VLA observations (Condon 1987) of all spirals north of = - 45° and brighter than BT = + 12 mag, the completeness limit of the Revised Shapley-Ames Catalog (Sandage and Tammann 1981). The corresponding luminosity function for low-redshift E and SO galaxies was obtained by Auriemma et al. (1977) from both optical and radio samples as described above. There are essentially no local radio-selected quasars. The contributions of Seyfert galaxies and optically selected quasars to the local luminosity function fall within the high-luminosity extension of the spiral galaxy component (Meurs and Wilson 1984), and the radio-loud quasars are presumed to be in elliptical galaxies. The resulting 1.4-GHz local luminosity function m(L | z = 0, = 1.4 GHz) is plotted in Figure 15.3(a) for a Hubble parameter H0 = 100 km s-1 Mpc-1 (the value used throughout this chapter).
Figure 15.3. (a) Local luminosity function m(L | z = 0, = 1.4 GHz). Open circles represent radio sources associated with spiral galaxies (Condon 1987), and the filled circles are based on the Auriemma et al. (1977) luminosity function for E and S0 galaxies. Abscissa: log luminosity (W Hz-1). Ordinate: log comoving density (mag-1 Mpc-3). (b) Local weighted luminosity function (L | z) = 0, = 1.4 GHz). Abscissa: log luminosity (W Hz-1). Ordinate: log weighted luminosity function (Jy1.5).
Other forms of the luminosity function are sometimes useful. One is the co-moving density (L | z, ) dL of sources with luminosities L to L + dL Since m(L | z, ) d[logm(L)] (L| z, ) dL
The weighted luminosity function, or "visibility function,"
introduced by von Hoerner (1973) emphasizes the contributions of sources in different luminosity ranges to the weighted source count S5/2 n(S | ) (Section 15.4.2). In the low-redshift (static Euclidean) limit L = 4 D2 S and dV = 4 D2 dD, the number n(L, D | ) dL dD of sources per steradian with luminosities L to L + dL and distances D to D + dD is related to the local luminosity function by n(L, D | ) dL dD = (L | ) dL × D2 dD. The corresponding number n(L, S | ) dL dS in the flux-density range S to S + dS is given by n(L, S | ) = n(L, D | )| dD / DS|. The total number n(S | ) of sources per steradian with flux densities S to S + dS is
This can be rearranged to yield
The local weighted luminosity function (L | z = 0, = 1.4 GHz) is plotted in Figure 15.3(b). With Equation (15.16) it shows that, in the low-redshift limit, most radio sources selected at = 1.4 GHz have luminosities L 1024 to 1027 W Hz-1 and are found in elliptical galaxies. Spiral galaxies contribute only about 1% of the sources; and radio-selected spiral galaxies have luminosities L 1022 W Hz-1, about an order of magnitude higher than the typical radio luminosity of an optically selected spiral galaxy.