**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 (_{a}^{2} + 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.