Flux limited radio source catalogs are dominated by elliptical
galaxies with high luminosities [
(L) peaks at L ~ 10^{25.5} 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:

Above L 2 x 10^{26} 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 (L) peaks near 10^{21} W Hz^{-1}. The
luminosity function for spirals is given by
(Condon 1984b):

Above L 10^{21.5} 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 = 0.67. L_{0},
the mean luminosity, is roughly proportional to
optical luminosity, with L_{0} 10^{21} watt Hz^{-1} for M_{p} -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

(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
R_{0} = 10 for Sc's and R_{0} = 25 for Sb's, and width = 0.67 in log R, i.e., a factor
of five in R.