5.6 Effect of Relativistic Beaming on Number Statistics
AGN with anisotropic emission patterns will be boosted into and shifted out of flux-limited samples according to their orientation. For some radiation patterns, including those caused by relativistic beaming (Urry and Shafer 1984) and thick accretion disks (Urry et al. 1991b), a narrow distribution in intrinsic luminosity is broadened into a flat distribution over a wide range of observed luminosities (which follows from the probability function for flux enhancement.) This leads to a distortion in the measured shape of the luminosity function of the boosted AGN relative to their intrinsic luminosity function (Urry and Shafer 1984; Urry and Padovani 1991).
More specifically, under the basic assumption that AGN are randomly oriented on the sky, and assuming a radiation pattern, we can predict the exact numbers of AGN with a given observed luminosity relative to their intrinsic luminosity (summed over all angles). Given the luminosity function (LF) of the misaligned AGN, then, one can predict the LF of the aligned AGN, subject to the form of the radiation pattern. In practice, the pattern for relativistic beaming depends on the assumed distribution of Lorentz factors and on whether one assumes a uni-directional jet or a fan beam; the pattern for obscuration depends on the size and optical depth of the torus; the pattern for thick disks depends primarily on the funnel geometry.
To evaluate the number statistics of radio-loud unification schemes we incorporate the effect of relativistic beaming on the observed LFs (Urry and Shafer 1984). Consider an ensemble of emitters all having the same intrinsic luminosity () and all moving with the same relativistic bulk speed () at random angles () to the line of sight. Given that L = ^{p} [Eq. (B7)], the probability of having a particular Doppler factor = [(1 - cos)]^{-1} (Appendix A) is P() = d (cos) / d = (^{2})^{-1}. The probability of observing luminosity L given intrinsic (emitted) luminosity is
For fixed , the distribution of observed luminosities is a flat power law (with index in the range 1-1.5 for likely values of p; Appendix B) extending from L ~ (2)^{-p} to L ~ (2)^{p} . The observed luminosity distributions are illustrated in Fig. 13(a) (thick dashed lines) for three different intrinsic luminosities (thin dashed lines). The low-luminosity cutoff corresponds to an emitter moving directly away from us ( = 180°) and the high-luminosity cutoff to an approaching emitter perfectly aligned ( = 0°). The normalization of this flat power law decreases with increasing beaming (higher , , or p) because the beaming cone angle gets smaller.
For a distribution of intrinsic luminosities (i.e., a luminosity function), the observed LF is just the integral of the intrinsic luminosity distribution times the conditional probability function in Eq. (4):
This integral [thick solid line in Fig. 13(a)] is basically the envelope of the beamed LFs for fixed intrinsic luminosities (the intrinsic LF is shown as a thin solid line). For simple power law luminosity functions, Eq. (5) can be integrated analytically (Urry and Shafer 1984).
In practice, one expects an unbeamed component (e.g., radio lobes) to be present in addition to a beamed component (jet). We use the simple parametrization that the intrinsic luminosity in the jet is a fixed fraction of the unbeamed luminosity, _{j} = f _{u}, so the total observed luminosity is
This takes account of the approaching jet only; it can be shown that if jets come in oppositely directed pairs of similar intrinsic power, the receding beamed component (with Doppler factor equal to = [ (1 + cos )]^{-1}) will have a negligible contribution for likely values of the Lorentz factor and approaching jets within ~ 60° of the line of sight (Appendix C). As before, the conditional probability is derived from Eq. (6) but in this case Eq. (5) is integrated numerically. A (likely) distribution of Lorentz factors can also be included (Urry and Padovani 1991). If we define the critical angle, _{c}, to be where the beamed and unbeamed luminosities are comparable (i.e., f ^{p} = 1), then for < _{c} the luminosity will be dominated by beamed emission and we can identify these sources as blazars. The observed parent and beamed LFs for the case of a single-power-law intrinsic LF are shown in Fig. 13(b), for four values of f.
The key point is that the luminosity function of the beamed population has a characteristic broken-power-law form, flat at the low luminosity end and steep at the high luminosity end (Urry and Shafer 1984). This remains approximately the case even when the intrinsic luminosity function has a more complicated form, without sharp cutoffs (Urry and Padovani 1991). This means that the comparison of the number densities of beamed and parent populations is a strong function of luminosity, and for samples biased by relativistic beaming can be evaluated only by measuring the luminosity functions for each.
Since radio maps of blazars show the presence of a diffuse component we consider the two-component model for which L_{T} = (1 + f ^{p}) (Eq. 6). The assumption that the intrinsic jet power is linearly proportional to the extended power may not be consistent with observations. Observed core power and total radio power are well correlated but apparently not linearly (Feretti et al. 1984; Giovannini et al. 1988; Jones et al. 1994). Although results vary, all seem to find that observed core power has a less than linear dependence on total radio power. (Since their samples include mainly radio galaxies, total power is essentially the same as extended power.) For example, Jones et al. (1994) find that P_{core} P^{0.8}_{total}. The range of observed slopes, generally calculated taking upper limits into account, is 0.4-0.8. Calculating the regression with P_{core} as the dependent parameter will lead to a systematically different slope and indeed treating the variables symmetrically would be the appropriate approach (Isobe et al. 1990).
We note that the observed nonlinearity does not immediately conflict with the assumption that _{jet} = f _{ext}. First, beaming will cause large scatter in the observed P_{c} - P_{ext} plane due to the spread in viewing angles and possibly Lorentz factors. Second, selection effects could influence the slope of the correlation, although we do not immediately see why it should be flatter than unity. If the intrinsic relation is nonlinear (e.g., _{jet} = f ^{x}_{ext}), then the calculation described here [represented by Eqs. (5) and (6)] would need to be modified. This would change the derived parameter values but as long as x is reasonably close to 1, it should not be a major effect.
Radio-loud unification schemes, which involve primarily relativistic beaming, have been verified with the luminosity function approach just described. Radio-quiet schemes have not been tested this way because the radiation patterns are still very uncertain and the radiation anisotropy is probably much more Draconian at the usual (blue) selection wavelengths (Pier and Krolik 1992; Ward et al. 1991; Djorgovski et al. 1991). For the high-power radio-loud scheme, obscuration is needed to explain the optical properties (namely, to allow for the fact that we do not have a direct view of the broad line region in FR II radio galaxies) but it is not important for statistical arguments based on radio luminosities alone. In the next section we outline the quantitative evaluation of the observed and beamed luminosity functions for radio-loud AGN.