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15.4.2. Source Counts

Because of the wide range of flux density and source density involved, no individual radio telescope can provide complete data, even at a single frequency (Figure 15.4). Pencil-beam instruments, large steerable dishes, and phased arrays are typically used to survey large regions of the sky to obtain statistically significant counts for the stronger sources with relatively low surface densities. Separate surveys made from the northern and southern hemispheres are necessary to cover the whole sky, and all-sky catalogues at nu = 0.408, 2.7, and 5 GHz have been complied from large-scale radio surveys (Robertson 1973, Wall and Peacock 1985, and Kühr et al. 1981, respectively). A number of pencil-beam surveys go much deeper than the all-sky surveys over limited areas. Synthesis instruments provide the most sensitive surveys, but only in very small regions of the sky, typically 10-5 to 10-3 sr. Counts of very faint sources based on only a few such small fields may be subject to error if there is significant clustering. Nevertheless, in contrast to the early radio source surveys (cf. Jauncey 1975), modern data obtained by different observers using very different kinds of radio telescopes are in good agreement with respect to individual source positions and flux densities as well as surface densities.

Figure 4

Figure 15.4. Profile plots of the sky near the north galactic pole mapped with (a) the NRAO 91-m telescope (beamwidth approx 12') (Condon and Broderick 1985) and (b) the VLA (beamwidth 17".5) (Mitchell and Condon 1985) illustrate the range of source intensities and sky densities that go into the nu = 1.4 GHz source count. (From Condon 1984.)

The number n(S | nu) dS of sources per steradian with flux densities S to S + dS found in a survey made at frequency nu is called the differential source count; the total number per steradian stronger than S, integSinfty n(s | nu) ds, is called the integral source count. Integral counts are rarely used any more because they smear rapid changes of source density with flux density and the numbers are not statistically independent from one flux-density level to the next (Jauncey 1967, Crawford et al. 1970). The steep slopes of the differential source counts tend to obscure features in graphical presentations, so the counts are usually either weighted (simply multiplied by S5/2) or normalized [divided by the count n0(S | nu) = k0 S-5/2 expected in a static Euclidean universe; the constant k0 is usually set so that n(S | nu) / n0(S | nu) approx 1 at S = 1 Jy] before plotting. Historically, this normalization has been used to facilitate comparisons with the static Euclidean count - level portions of the actual normalized counts are said to have a "Euclidean slope," for example. Such comparisons can be misleading, however, because the static Euclidean approximation has surprisingly little relevance to the actual source counts except at the very highest flux densities (cf. Section 15.9). In particular, a Euclidean slope does not signify that the sources in that flux-density range have low redshifts or are not evolving.

Source counts covering a wide range of flux densities are currently available at nu = 0.408, 0.61, 1.4, 2.7, and 5 GHz (cf. Condon 1984b). The most extensive is at nu = 1.4 GHz and is shown in Figure 15.5. The NRAO 91-m telescope was used to measure the flux densities of sources stronger than S = 2 Jy at nu = 1.4 GHz (Fomalont et al. 1974) and also in the 0.175 leq S < 2 Jy range (Machalski 1978). The fainter levels are based on VLA "snapshot" surveys (Condon et al. 1982b, Mitchell 1983; Coleman et al. 1985), the WSRT deep survey of the Lynx area (Oort 1987), and the deepest VLA survey (Mitchell and Condon 1985). The sky densities of sources too faint to be detected and counted individually in the latter survey were estimated statistically from their contribution to the map fluctuation or "P(D)" distribution (Scheuer 1957, 1974, Condon 1974) and are indicated by the shaded region extending down to S = 10 µJy. The integrated emission from extragalactic sources can be used to constrain the source count at even fainter levels. After subtracting galactic emission, Bridle (1967) obtained T approx 30 K at nu = 178 MHz, corresponding to T approx 0.1 K at nu = 1.4 GHz. The contribution T(S) = [c2 / (2k nu2)] integSinfty sn(s) ds from sources stronger than S = 10 µJy is about 0.08 K, so the bulk of the extragalactic background can be accounted for by known source populations.

Figure 5

Figure 15.5. Weighted source count at nu = 1.4 GHz. Abscissa: flux density (Jy). Ordinate: weighted source count S5/2 n(S) (Jy1.5 sr-1).

Most sources found in low-frequency surveys have power law spectra with spectral indices near alpha approx + 0.8, but some have the more complex spectra and lower spectral indices (alpha approx 0) indicative of synchrotron self-absorption in compact (theta < 0."01) high-brightness (T approx 1011 K) components. These two source types are effectively distinguished by the simple criterion alpha geq 0.5 ("steep-spectrum" source) or alpha < 0.5 ("flat-spectrum" source). The flat-spectrum sources can usually be identified with quasars, while most steep-spectrum sources are associated with galaxies (or empty fields if the galaxies are too distant). Many flat-spectrum sources vary in both intensity and structure on time scales of years, and their apparent luminosities may be affected by relativistic beaming (see Chapter 13). The evolutionary histories of these two source types may also differ. Being so compact, flat-spectrum sources are probably less sensitive than extended steep-spectrum sources to changes in the average density of the intergalactic medium or in the energy density of the microwave background radiation with cosmological epoch (Rees and Setti 1968). Finally, flat-spectrum sources can be seen at greater redshifts because they are not so strongly attenuated by the (1 + z)1+alpha Doppler term in Equation (15.1). For these reasons, it is worthwhile to separate the steep- and flat-spectrum sources and count them independently when possible. The numbers of flat-spectrum (alpha < 0.5) and steep-spectrum (alpha geq 0.5) sources are comparable in high-frequency samples, and their counts at 5 GHz are plotted separately in Figure 15.6. The data were taken from Pauliny-Toth et al. (1978), Condon and Ledden (1981), Owen et al. (1983), and Fomalont et al. (1984).

Figure 6

Figure 15.6. Weighted counts of steep-spectrum (alpha geq 0.5) (filled symbols) and flat-spectrum (alpha < 0.5) (open symbols) sources found at nu = 5 GHz, along with model predictions. (From Condon 1984b) (dashed and dot-dashed lines, respectively). Since the weighted counts of these two spectral populations are comparable but peak at slightly different flux densities, their sum (solid line) has a very broad peak.

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