6.3 Independent Estimates of Relativistic Beaming Parameters
Independent estimates of Lorentz factors and angles to the line of sight are available, mostly from VLBI data. In VLBI observations of twelve FSRQ in the 2 Jy sample (Vermeulen and Cohen 1994, and references therein), all but one FSRQ show evidence of superluminal motion (a > 1). The apparent separation of radio components gives a lower limit to the Lorentz factor, = sqrt (a2 + 1) (Appendix A); the distribution of for the fastest superluminal components in FSRQ covers the range 5-35, roughly the same as that necessary to explain the observed luminosity function (Sec. 6.1).
The observed a distribution can in principle be compared with that predicted using the LF-derived beaming parameters. Vermeulen and Cohen (1994) determined that a distribution of Lorentz factors peaked at high values best fit the observed a distribution for a small heterogeneous sample defined by a post-facto flux limit. (They calculated the expected distribution of a for naked jets without extended emission, which is appropriate to their sample of core-dominated objects and which is straightforward to calculate.) This is quite different from the steep distribution of Lorentz factors we found from fitting luminosity functions (Sec. 6.1.3). Whether these two results can be reconciled remains to be seen but we note that our calculation is most sensitive to low values of , which yield the largest number of beamed objects, while superluminal motion statistics (particularly for samples of ``favorite objects'' studied in the past) are most sensitive to high values. Alternatively, one can compare the results from the recent VLBI surveys of well-defined flux limited samples (Pearson and Readhead 1988; Polatidis et al. 1995; Thakkar et al. 1995; Taylor et al. 1994) to the a distribution predicted by a more-difficult calculation incorporating selection effects appropriate to jets plus extended emission, but this calculation has not yet been done. As is the case for emission-line blazars, all thirteen 1 Jy BL Lac objects observed more than once with VLBI techniques appear to show superluminal motion (Vermeulen and Cohen 1994; Gabuzda et al. 1994). For all but two objects, is higher than ~ 4, and reaches at least ~ 15. These results agree roughly with the distribution in r required to fit the luminosity function and the core-halo ratios of radio-selected BL Lacs (Sec. 6.2).
Since two BL Lac objects have as low as 1.5, we tried a fit to the radio LF of BL Lacs with 1 = 2, leaving all other parameters unchanged. The resulting beamed LF is still in good agreement with the observed one, although the Lorentz factors are now lower, 2 r 20 with an average value < r > ~ 3, corresponding to a ratio between BL Lacs and FR Is of 5% (Table 3). This shows that a scenario where the Lorentz factors are lower for BL Lacs than for FSRQ (Gabuzda et al. 1994; Morganti et al. 1995), although not required by our fits, is consistent with the luminosity function and VLBI data.
The SSC formalism (Sec. 4.4) can be used to estimate the Doppler factor independent of superluminal motion or luminosity functions, via comparison of the predicted and observed X-ray fluxes. For a sample of ~ 100 radio sources with published VLBI measurements of the core angular size, lower limits to the Doppler factor correlate well with the apparent velocity a obtained from multi-epoch VLBI maps (Ghisellini et al. 1993; Sec. 4.2). This result suggests superluminal motion is related to bulk motion of the emitting plasma and is not simply an illusion. The Doppler factor, , can be combined with the measured superluminal speed, a, where available, to constrain and the angle to the line of sight, [Eqs. (A6) and (A7)]; the mean values of these parameters for the different classes (BL Lacs, FSRQ, and SSRQ; Ghisellini et al. 1993) are in good agreement with those derived from the radio luminosity function studies (Secs. 6.1, 6.2).
Similarly, the jet velocities and orientations of radio galaxies, derived from the jet to counter-jet brightness ratio [Eq. (A10)] and the SSC formalism, are broadly consistent with those derived from luminosity function studies (Giovannini et al. 1994). Their Doppler factors are also much lower than for BL Lacs and FSRQ, as expected if the radio jets are closer to the plane of the sky.