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15.6.2. The Angular Size-Flux Density Data

Most steep-spectrum extragalactic sources can be resolved with sensitive aperture-synthesis telescopes, and their angular sizes theta are often by-products of deep surveys. Thus, fairly unbiased statistics describing the angular-size distributions of radio sources as a function of flux density can easily be obtained and used to estimate the linear size evolution of extragalactic sources. Unfortunately, the angular sizes of radio sources with differing morphologies are not easy to define precisely, the theta - S relation is rather insensitive to size evolution, and size evolution is either weak or nonexistent.

Radio sources at cosmological distances have angular sizes ranging from < 0."001 for the most compact flat-spectrum sources to several minutes of arc, so no one instrument can detect and resolve all of them. However, the majority of steep-spectrum sources stronger than S approx 1 mJy are larger than theta approx 10 arcsec, making the median <theta> of their broad angular-size distribution accessible to the aperture-synthesis telescopes used to make deep radio surveys.

A suitable definition of theta is complicated by the variety of source structures encountered. Ideally, theta should be a metric diameter, which is more sensitive to redshift and differences between world models than the isophotal diameters used in optical astronomy (Sandage 1961). A good working definition of theta should be insensitive to details of the source brightness distribution and observational limitations, low dynamic range and limited resolution especially. The traditional definition of theta is the component separation of a double source because most strong sources selected in low-frequency surveys have this morphology and also because their component separations are easy to measure directly from contour maps. Unfortunately, the measured value of theta depends on the component intensity ratio if the double source is barely resolved or observed with low dynamic range, and this definition must be generalized to the "largest angular size" before it can be applied to sources with core jet or more complex radio morphologies. However, the largest angular size is generally larger on maps of strong sources and can lead to an apparent change of theta with S. For example, the strong low-redshift quasar 3C273 is listed as having an angular size theta = 21 arcsec (Kapahi et al. 1987) because its jet was mapped with high dynamic range. If 3C273 were moved to a high redshift and discovered as a faint source in an aperture-synthesis survey, its jet would not be distinguishable from the bright compact core, and its quoted angular size would be very small. Since the angular variances of the observing beam and the source brightness distribution B(phi) add under convolution, the angular diameter defined by theta* ident 2[integ phi2 B(phi) dphi / integ B(phi) dphi]1/2 can be measured even for sources just large enough to broaden the beam (cf. Coleman 1985). It can be applied to any source morphology and it is independent of map dynamic range, except at very low signal-to-noise ratios where it may be overestimated by Gaussian fitting. Its main drawback is that it is more difficult to determine from contour plots.

Even with a good definition of theta, there are biases that affect the angular-size distribution of faint sources found in aperture-synthesis surveys. The survey maps are complete only above some peak flux density, so weak sources significantly larger than the synthesized beam (typically 10" to 20" FWHM) will be missed and must be corrected for. The sky density of sources fainter than a few mJy is so high that distinguishing physically associated components of double sources from unrelated projected pairs of compact sources becomes a problem (Condon et al. 1982b).

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