A fundamental test of the proposed unification of blazars and radio galaxies is whether the number statistics of the populations agree with the relativistic beaming hypothesis. The total number of beamed objects (here, blazars) must be small compared to the number of parent objects (radio galaxies), as they are presumably oriented at small angles to the line of sight. This ratio depends only on the critical angle dividing blazars and radio galaxies, which in turn depends on the amount of beaming (essentially, the Lorentz factor and the relative luminosities of beamed and unbeamed components). The critical angle is therefore central to unification.
In recent years it has become possible to test unified schemes via the number statistics of complete samples of blazars and radio galaxies. This section describes separately the statistics of the unification of quasars and FR II radio galaxies (Sec. 6.1) and of BL Lac objects and FR I radio galaxies (Sec. 6.2). Since relativistic beaming is best constrained by radio observations (thanks to VLBI), we compare in Sec. 6.3 the Lorentz factors from superluminal motion, SSC calculations, and jet/counter-jet ratios with those derived from number statistics.
6.1 Unification of Radio Quasars and FR II Galaxies
We start by deriving the luminosity function of FR II radio galaxies, then we
beam it according to the prescription outlined in
Sec. 5.6 and compare the beamed and
observed radio luminosity functions for quasars. The free parameters are
,
the bulk Lorentz factor of the jet, and f, the fractional
luminosity of the
jet. One can further constrain these two parameters from R, the ratio
of beamed to unbeamed flux (core to extended flux in high resolution radio
maps; Appendix C).
6.1.1 Content of the 2 Jy Sample of Radio Sources
As in
Padovani and Urry (1992),
we derive quasar and radio galaxy luminosity functions from the 2 Jy sample
(Wall and Peacock
1985),
a complete flux-limited sample of 233 sources with flux at 2.7 GHz
F2.7 
2 Jy.
Redshifts and optical spectroscopic identifications have been updated using
the latest version of the 1 Jy catalog
(Stickel et al. 1994)
and new optical spectra for sources with declination d < 10°
(di Serego Alighieri
et al. 1994b).
Quasars (and BLRG) are distinguished from radio galaxies (NLRG) by whether
they have broad optical/ultraviolet emission lines or not. This criterion
might not be as straightforward to apply as it sounds: recent papers
(Laing et al. 1994;
Economou et al. 1995;
Hill et al. 1995)
have pointed out that some objects
classified as narrow-line galaxies actually show broad H
emission.
This is particularly likely for high-redshift sources classified on the basis
of an H
line, for
which the broad wings can be extinguished by
modest reddening, thus introducing a redshift dependence (in practice)
to the definition of Type 1 versus Type 2 AGN.
(11)
Only 3 of the 2 Jy sources (1% of the sample) have no optical counterpart, while fourteen more (6%) have no redshift information. The great majority of the objects without redshifts are classified as galaxies, i.e., they appear extended, and based on their visual magnitudes they have estimated redshifts larger than 0.7.
Morphological classifications as FR I or FR II for the
galaxies in the 2 Jy sample have been updated according to new radio maps for
objects with d < 10° and z < 0.7
(Morganti et al. 1993),
and using additional information as available
(Zirbel and Baum 1995,
and references
therein). Fifteen galaxies are classified as compact or unresolved while 7 more
are, to the best of our knowledge, unclassified. The division between steep-
and flat-spectrum quasars was made at a radio spectral index between 2.7 and 5
GHz of = 0.5.
These updates have essentially no effect on the FSRQ and SSRQ luminosity functions. For FR II radio galaxies, the luminosity function was originally derived (Padovani and Urry 1992) from a different sample, the 3CR catalog (Laing et al. 1983), because at the time a large fraction (nearly one-third) of FR II galaxies in the 2 Jy sample had uncertain redshift estimates and their estimated evolution differed from that of radio galaxies in other samples. (12) While neither the different selection frequency nor the different flux limit of the 3CR appeared to affect the resulting luminosity function substantially, the 2 Jy FR II sample is now much better defined so we revisit the question of using it.
The new 2 Jy FR II galaxies still exhibit less evolution
than the 3CR FR II galaxies, although the evolutionary properties of the two
samples are consistent at the 1 
level. Using only the definite FR II
radio galaxies with z < 0.7 (the completeness limit
of the spectroscopic subsample of
di Serego Alighieri
et al. 1994b),
one obtains < V/Vmax > = 0.55 ± 0.04, where the quoted
error is 1/sqrt(12N), appropriate for a uniform distribution.
The best-fit evolution parameter
(13)
is 
= 0.26 and the associated 1
interval is [0.16, 1.0]. This
evolution is consistent with the earlier estimate
(Padovani and Urry
1992)
and is within 1 uncertainty of the
value for the 3CR FR II galaxies, for which the corresponding number is
= 0.17 with 1 interval of
[0.15, 0.19]. If there is significant
contamination of the 3CR NLRG sample with BLRG, as mentioned above, this could
make their observed evolution too high (more like quasars). The two FR II
luminosity functions, for the 2 Jy and 3CR galaxies, are in good agreement,
although due to the smaller evolution the 2 Jy one is somewhat flatter at
higher powers. For consistency with the FSRQ and SSRQ samples,
we now use the 2 Jy FR II LF (with =
0.26) in the calculations that follow.
The 2 Jy sample includes, as do other radio
samples, a sizeable fraction of compact steep-spectrum (CSS) and gigahertz
peaked-spectrum (GPS) sources (~ 20% CSS;
Morganti et al. 1993).
We exclude galaxies classified by
Morganti et al. (1993)
as compact or with unresolved radio
structures, so no CSS or GPS objects should be in our FR II (or FR I)
samples. However, since 2 Jy objects with broad optical line emission
and unresolved radio emission are classified as quasars (FSRQ or SSRQ depending
on radio spectral index), some SSRQ could actually be CSS
sources and some FSRQ could actually be GPS sources.
The place of CSS and GPS sources in the unified scheme is discussed
further in Sec. 8.2.4. Even if they should
be excluded from the unified
scheme, which is not necessarily the case, they are unlikely to
have ``contaminated'' the quasar sample
because the fraction of such sources is not large; the derived luminosity
functions will be distorted only if there are systematic trends with luminosity
or redshift.
in the rest frame but
excluding the Paschen lines).
Based on spectra meeting these criteria, all AGN could be categorized
definitely as Type 1 or Type 2. One would then have to devise other
names to represent objects with similar intrinsic (as opposed to
observed) properties. For example, 3C 234 would be called a Type 1 object
because it has strong broad lines but at the same time it would belong
with NGC 1068 (a Type 2 AGN) in some new category because
both have hidden broad-line regions. Back
12 The evolutionary properties of a sample can be characterized by the mean value of the ratio between V, the volume enclosed by an object, and Vmax, the maximum accessible volume within which the object could have been detected above the flux limit of the sample. In a Euclidean geometry, then, V / Vmax = (F/Flim)-3/2, where F is the observed flux of an object and Flim is the flux limit of the survey. In the absence of evolution V / Vmax has the property of being uniformly distributed between 0 and 1, with a mean value of 0.5 (Rowan-Robinson 1968; Schmidt 1968). When the survey is made up of separate fields with different flux limits, as is the case for the EMSS, it is more appropriate to use Ve / Va, that is the ratio between enclosed and available volume (Avni and Bahcall 1980). Back.
13 Here we characterize evolution in terms of
exponential luminosity evolution, P(z) = P(0)
exp[T (z) / ], where
T (z) is
the look-back time and is the time
scale of evolution in units of the Hubble time. Back.