15.5.1. Source Distribution Equations
The actual spectra of radio sources are normally approximated by power laws so that the spectral luminosity function at all frequencies is determined by the spectral luminosity function at any one frequency _{0} from
(15.17) |
for a measured between any two frequencies. Models based on this approximation generally work well at = 408 MHz and higher frequencies, but they overestimate the 178-MHz source count significantly (Peacock and Gull 1981, Condon 1984b). Both 178-MHz flux-density scale errors and genuine spectral curvature caused by synchrotron self-absorption may contribute to this discrepancy.
The total number of sources with luminosities L to L + dL spectral indices to + d, at frequency , and lying in the spherical shell with comoving volume dV at redshift z is
(15.18) |
The number of sources in this shell equals the total number (S, , z | ) dS d dz of sources with flux densities S to S + dS spectral indices to + d, and redshifts z to z + dz found in a survey of the whole sky (4 sr) at frequency . Weighting by S^{5/2} and eliminating both dV and dL / dS yields
(15.19) |
Integrating over redshift and dividing by 4 gives the weighted (spectral) source count S^{5/2} n(S, | ), where n(S, | ) dS d is the number of sources per steradian with flux densities S to S + dS and spectral indices to + d found at frequency :
(15.20) |
The distribution equations (15.19) and (15.20) can be integrated numerically to give the observables described in Section 15.4. Using the weighted luminosity function to calculate the weighted source count directly minimizes the interpolation errors that can be significant in numerical integrations of the more rapidly varying luminosity function _{m} (cf. Danese et al. 1983, Peacock 1985) to obtain the unweighted source count.
The weighted differential source count S^{5/2} n(S | ) at frequency (Section 15.4.2) is
(15.21) |
The (unnormalized) spectral-index distribution N( | S, ) (Section 15.4.3) is obtained by integrating the differential spectral count:
(15.22) |
The redshift/spectral-index diagram (Section 15.4.4) shows the values of N(, z | S, ) given by
(15.23) |
The (unnormalized) integral redshift distribution N(z| S, ) (Section 15.4.5) is found by integrating over :
(15.24) |
Most radio sources in any flux-limited sample have flux densities only slightly higher than the flux-density limit, so redshift and luminosity are strongly correlated. Thus the integral luminosity distribution N(L| S, ) can be used instead of the integral redshift distribution N(z| S, ). Let n(L, S, | ) d[log(L)]dS d, be the differential number of sources per steradian with luminosities log(L) to log(L) + d[log(L)], flux densities S to S + dS and spectral indices to + d at frequency . Then, 4n(L, S, | ) d[log(L)] = (S, , z | ) dz and
(15.25) |
where
(15.26) |
and
(15.27) |
Integrating Equation (15.25) over flux density and spectral index yields the (unnormalized) integral luminosity distribution
(15.28) |