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Flux limited radio source catalogs are dominated by elliptical galaxies with high luminosities [phi (L) peaks at L ~ 1025.5 W Hz-1] at very large distances (median z ~ 1). This is because the present luminosity function is quite flat; it is roughly fit by two power laws:

11. Equation 11

Above L appeq 2 x 1026 W Hz-1 it drops off. Evolution of the luminosity function is critical in determining the observed n(s), as discussed by Condon (1984a). Spiral galaxies make up a small fraction (~ 1%) of the radio sources brighter than 1 mJy. Typical spirals detected at this level are nearby, because for these phi (L) peaks near 1021 W Hz-1. The luminosity function for spirals is given by (Condon 1984b):

12. Equation 12

Above L appeq 1021.5 watt Hz-1 the density of spirals drops off, and ellipticals dominate. An alternative form suggested by Hummel (1981, cf. Gavazzi and Jaffe 1986) is a log-normal luminosity function:


with sigma = 0.67. L0, the mean luminosity, is roughly proportional to optical luminosity, with L0 appeq 1021 watt Hz-1 for Mp appeq -20.

To study the radio luminosity function of normal galaxies in the present epoch requires that we sift through a large sample of radio sources, selecting those few associated with optically bright, nearby galaxies. This preselection by optical properties causes a bias for optically brighter objects. To properly include this selection in a statistical treatment requires computation of the bivariate radio luminosity function (BRLF), f(P, M), which gives the fraction of all galaxies with optical magnitudes M to M + dM which have radio luminosity P to P + dP (Auriemma et al. 1977, Hummel et al. 1983). A simpler approach is to compute the radio-optical ratio function, RORF, given by f(R), the fraction of galaxies with radio-optical luminosity ratio in the range R to R + dR, where R is commonly defined as

13. Equation 13

(Condon 1980). Either of these functions can be written as a differential function, f(R), or integral (cumulative) function F>(R).

Results for f(R) are summarized by Gavazzi and Jaffe (1986), who find a log-normal distribution for f(R), with mean value R0 = 10 for Sc's and R0 = 25 for Sb's, and width sigma = 0.67 in log R, i.e., a factor of five in R.

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