15.6.1. The Angular Size-Redshift Relation
Even though radio sources are not good "standard rods," some statistic describing their size distribution might be stable enough to permit direct measurements of size evolution in populations of optically identified radio sources with known redshifts.
Since the apparent size of a double radio source is always reduced by projection onto the sky, the maximum m of the distribution of angular sizes will minimize projection errors. Miley (1971) plotted the angular sizes of strong radio galaxies (nearly all with z < 0.3) and quasars (nearly all with z > 0.3) versus redshift and found that their upper bound obeys the "static Euclidean" relation m 1 / z. In any expanding Friedmann model, this result indicates either size evolution or an inverse correlation between luminosity and size for radio quasars.
These two effects can be distinguished by comparing radio quasars in a narrow luminosity range but widely separated in redshift; i.e., quasars found in surveys complete to different flux-density limits. Such a comparison of 3CR (S 10 Jy at = 178 MHz) and 4C (S 2.5 Jy at = 178 MHz) quasars (Hooley et al. 1978) was inconclusive because m depends on small numbers of sources and hence is a fairly insensitive statistic. No size evolution was needed to fit the data, but size evolution of the form d = d0(1 + z)-N with N < 1.5 could not be excluded.
A recent comparison of the median angular sizes <> of radio galaxies with luminosities near L 1026 W Hz-1 found in three 1.4-GHz samples complete to S = 2, 0.55, and 0.01 Jy indicates size evolution in the range 1 < N < 2 (Kapahi 1985). Although the formal statistical significance of this result is high, there remain two possible selection effects. (1) Only the faint-source sample was obtained from an aperture-synthesis survey that discriminates against sources larger than 12". (2) Flux-limited surveys conducted at the same frequency but covering different redshift ranges are effectively complete at different emitted frequencies (1 + z). A correlation between linear size and either spectral index or frequency in the source frame will mimic size evolution. This angular size-redshift relation for median angular sizes has recently been extended to include quasars (Kapahi 1986), as shown in Figure 15.12.
Figure 15.12. Median angular sizes of samples of galaxies (open circles), galaxies in a narrow luminosity range (crosses), and quasars (filled circles) (Kapahi 1986). The relation <> 1/z suggests size evolution in Friedmann models. Abscissa: redshift. Ordinate: median largest angular size (arcsec).