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This discussion of radio spectra is far from exhaustive: it sets out to serve two purposes. One is related to the K-corrections, the correction for spectral form which must be used to derive luminosities at rest-frame frequencies. Getting these corrections right is essential in determining space density. The second issue concerns relating source counts at different frequencies, and in particular modelling the poorly-determined high-frequency counts from the well-defined low-frequency counts. This limited discussion thus ignores some aspects of radio spectral measurements which are critical to the astrophysics of radio AGNs, such as variability and monitoring, mentioned briefly in the Introduction.

We have noted that spectra of radio sources are frequently represented as simple power-laws, S propto nu-alpha, with the spectral index, alpha ~ 0.8 for steep-spectrum sources and ~ 0 for the flat-spectrum ones. However, all radio galaxies deviate from this simple behaviour. Various physical mechanisms contribute to shaping the emission spectrum. At low rest-frame frequencies spectra generally show a sharp decline with decreasing frequency, attributed to synchrotron self-absorption; a low energy cut-off to the spectrum of relativistic electrons may also have a role (Leahy et al. 1989). The decline is mostly observed at rest frequencies of tens of MHz, but the absorption turnover frequency can be orders of magnitude higher, as in GPS and ADAF/ADIOS sources.

In the optically-thin regime, the spectral index of synchrotron emission, the dominant radiation mechanism encountered in classical radio astronomy, reflects the index of the energy distribution of relativistic electrons. This distribution is steepened at high energies by synchrotron losses as the source radiates, and by inverse Compton losses on either the synchrotron photons themselves or on photons of the external environment (Krolik & Chen 1991). Inverse Compton losses off the cosmic microwave background (CMB) increase dramatically with redshift since the radiation energy density grows as (1 + z)4. As a consequence, a decline with increasing redshift of the frequency at which the spectral steepening occurs can be expected.

While inverse Compton losses are most important to sources with weak magnetic fields, powerful sources may possess more intense magnetic fields enhancing the synchrotron emission. The faster electron energy losses yield a more pronounced steepening, correlated with radio power (P). Disentangling the effects of radio power and redshift is difficult because in flux-limited samples the more powerful sources are preferentially found at higher redshifts. A further complication arises because a convex spectral shape means that redshifting produces an apparent systematic steepening of the spectrum between two fixed observed frequencies as redshift increases. Since the redshift information is frequently missing, K-corrections cannot be applied and a P-alpha correlation may arise from any combination of these three causes.

This is the situation for the correlation reported by Laing & Peacock (1980). Employing (a large proportion of) redshift estimates for a sample drawn from the 38 MHz 8C survey, Lacy et al. (1993) showed that the high-frequency (2 GHz) spectral index correlated more closely with redshift than with luminosity. While at first sight this may suggest the dominant importance of inverse Compton losses on high-frequency spectra, Lacy et al. (1993) pointed out that the correlation between spectral index and redshift weakens when the radio K-correction is applied. This means that such correlation may be induced, at least in part, by the spectral curvature due to self absorption at the very low selection frequency. In fact, magnetic fields in the very luminous Lacy et al. sources should be strong enough for synchrotron losses to dominate inverse Compton losses. Blundell et al. (1999), studying a number of complete samples of radio sources selected at frequencies close to 151 MHz, with a coverage of the P-z plane (see Section 5) substantially improved over previous studies, concluded that:

  1. The rest-frame spectral index at low frequency depends on the source luminosity (P-alpha correlation), but also on physical size (D-alpha correlation) in the sense that sources with larger physical sizes D have steeper spectra.

  2. The rest-frame spectral index at high frequency (GHz) depends on the source redshift.

Simple expressions for the average rest-frame spectra of FRI and FRII radio galaxies as a function of radio power and Fanaroff-Riley type Fanaroff & Riley (1974) were derived by Jackson & Wall (2001).

With regard to the second issue, relating source counts at different frequencies, relevant aspects are to what extent a power-law approximation of the source spectra may be viable, i.e. to what extent low-frequency self-absorption, electron ageing effects at high frequencies etc. can be neglected; up to what frequencies do blazars have "flat" spectra; are source spectra correlated with other parameters (luminosity, redshift) and, if so, how can these correlations be described?

In this regard we have noted that surveys at frequencies of 5 GHz and higher are dominated (at least at the higher flux densities) by `flat-spectrum' sources. The spectra of these sources are generally not power-laws, but have complex and individual behaviour, showing spectral bumps, flattenings or inversions (i.e. flux increasing with increasing frequency), frequently bending to steeper power-laws at higher frequencies. Examples are shown in Fig. 3. The complex behaviour results from the superposition of the peaked (self-absorption) spectra of up to several components. These components are generally beamed relativistically with the object-axis close to the line of sight; they are the parts of jet-base components racing towards the observer at highly relativistic speeds.

The dominant populations of flat-spectrum sources are BL Lac objects and flat-spectrum radio quasars (FSRQs), collectively referred to as "blazars". Their spectral energy distributions (i.e. the distributions of nu Lnu) show two broad peaks. The low-energy one, extending from the radio to the UV and sometimes also to X-rays, is attributed to synchrotron emission from a relativistic jet, while the high-energy one, in the gamma-ray range, is interpreted as an inverse-Compton component arising from upscattering of either the synchrotron photons themselves (synchrotron self-Compton process, SSC, e.g. Maraschi et al. 1992, Bloom & Marscher 1993) or photons produced by the accretion disc near the central black-hole and/or scattered/reprocessed in the broad-line region (Blandford 1993, Sikora et al. 1994).

Gear et al. (1994) investigated the radio to sub-millimeter spectra of a random sample of very luminous BL Lacs and radio-loud violently variable quasars. They found generally flat or slowly rising 5-37 GHz spectra (median alpha537 appeq 0 for both populations), and declining 150-375 GHz spectra, with a statistically significant difference between BL Lacs and quasars, the former having flatter spectra (median alpha150375 appeq 0.43) than the latter (median alpha150375 appeq 0.73).

Indication of strong spectral curvature was reported by Jarvis & Rawlings (2000) for a quasi-complete sample drawn from the 2.7 GHz Parkes half-Jansky flat-spectrum sample (Drinkwater et al. 1997). It should be noted however that the data used to construct the radio spectra are heterogeneous, and the bias which this introduces is very serious (Wall et al. 2005). An extrapolation of this spectral behaviour to frequencies above 20 GHz would be in conflict with WMAP finding of a median spectral index alpha appeq 0 (Bennett et al. 2003).

Fossati et al. (1998) found evidence for an anticorrelation between the frequency of the synchrotron peak nup and the blazar luminosity, and proposed a scenario, dubbed "the blazar sequence", for a unified physical understanding of the vast range of blazar properties. The scenario was further extended by Ghisellini et al. (1998). The idea is that blazars indeed constitute a spectral sequence, the source power being the fundamental parameter. The more luminous sources are "redder", i.e. have both the synchrotron and the inverse Compton components peaking at lower frequencies than the lower-luminosity "blue" blazars. If so, the sub-mm steepening could be a property of only the brightest sources. The validity of the scenario has been questioned, however (e.g., Giommi et al. 1999, Padovani et al. 2003, Caccianiga & Marchã 2004, Antón & Browne 2005, Nieppola et al. 2006, Landt et al. 2006, Padovani 2007, Nieppola et al. 2008).

On the whole, the spectral curvature question is still subject to dispute. Substantial progress is expected from the forthcoming surveys by the Planck mission (The Planck Collaboration 2006), covering the range 30-857 GHz, that will provide the first complete samples allowing unbiased studies of the high frequency behaviour.

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