*6.1.3 Beamed LFs of High-Luminosity Radio Sources*

The beaming calculation is as follows. We start with the derived parent
luminosity function, then calculate the effect of beaming it
[Eqs. (4)-(6)]) adjusting free parameters
(the Lorentz factor) and
*f* (the fraction of luminosity intrinsic to
the jet) to match the observed luminosity function of FSRQ. It was
not possible to fit the observed FSRQ luminosity function with a single
Lorentz factor; it required instead a distribution in the range 5
40, weighted toward low
values: *n* ()
^{-2.3}, with a mean
value < > 11 and
*f* 5 x 10^{-3}.
[Note that this last parameter is fixed by the
largest and by the maximum
value of the ratio between beamed and unbeamed radio flux, *R*;
Eq. (C5).]

Figure 14 shows the
beamed (solid line) and observed (filled circles) radio LFs of FSRQ, which
are in very good agreement. The ratio between FSRQ and parents, integrated
over the full luminosity function assuming that of the FR IIs cuts off at the
low luminosity end, is ~ 2%. (Because of the flat LF slope at low
luminosities, this percentage is not too sensitive to the cutoff.)
The critical angle separating FSRQ from SSRQ and FR IIs ^{(15)} is _{c}(_{1}) ~
14°. The fitted parameters of
this beaming model are summarized in Table 3.

According to the beaming hypothesis, the steep-spectrum radio quasars are
supposed to be at intermediate angles and their intrinsic properties - in
particular, the value(s) of
and *f* - must be identical to those of
the FSRQ. Therefore, these parameters are already fixed.
The method of calculation is similar to the one used in the previous section;
however, since SSRQ are supposed to be misaligned objects, _{min}
0°. We take _{min} = 14°, and
the observed value of *R*_{min} ~ 0.002 is
used to constrain _{max} for SSRQ [Eq. (C4)]).
With no free parameters, then,
we calculate the beamed luminosity function of SSRQ.

The comparison between observations (open triangles) and the beaming model (dashed line) for SSRQ is shown in Fig. 14. The agreement is quite good, especially considering we did not adjust the parameters to optimize the FSRQ and SSRQ fits jointly. The angle dividing SSRQ from FR IIs is ~ 38° (Table 3).

The same distribution of Lorentz factors produces a good fit to both the
luminosity functions of FSRQ and SSRQ. The value of ~ 38°
for the angle separating SSRQ from FR II galaxies is in reasonable agreement
with the angle derived from the number density ratio in the observed range
of overlapping luminosity, =
arccos (1 + 1/6.4)^{-1} 30°.
The latter estimate is valid only when dealing with unbeamed
luminosities or (in
an approximate way) when the effect of beaming is not very strong because the
objects are viewed off-axis, as in this case.
For the 3CR sample, in which beaming is unimportant,
Barthel (1989)
found
= 44°.4 from the ratio of
quasars to radio galaxies in the interval 0.5 < *z* < 1.

Table 3 summarizes the beaming parameters for the
radio band for different
classes of objects (see Sec. 6.2 for a
discussion of the
parameters for the BL Lac class). Note that we have used *p* = 3 +
(see
Appendix B); using instead *p* = 2
+ results in the Lorentz factors
extending to higher values
(Padovani and Urry
1992),
and in a slightly larger critical angle.

From the observed values of *R* we can estimate a lower limit to
the maximum Lorentz factor
(Appendix C). The FR II galaxy OD -159 is the most
lobe-dominated source in the 2 Jy sample. The most core-dominated FSRQ known
is 0400+258, which does not actually belong to the 2 Jy sample (its 2.7 GHz
flux ~ 1.5 Jy); its measured *R* is comparable to the lower limits (when
no extended emission was detected) for some of the 2 Jy FSRQ. For these two
objects, the *R*-values, *K*-corrected [Eq. (C2)] and extrapolated
(when necessary)
to 2.7 GHz rest frequency assuming _{core} - _{ext} = -1,
are *R*_{min, FR IIs} < 6 x 10^{-5}
(OD -159;
Morganti et al. 1993)
and *R*_{max, FSRQ} 1000
(0400+258;
Murphy et al. 1993).
Using Eq. (C8),
we find _{max} > (1.7
x 10^{7} 2^{1-p})^{1/2p} ~ 13 for *p* = 3.

For these observed *R* values, small values of *p* imply high
values for the
largest Lorentz factor (here, for *p* = 2, _{max} 54). More
precisely, _{r} -0.3 for the 2 Jy FSRQ
(Padovani and Urry
1992)
so *p* 1.7 - 2.7
if *p* ranges between 2 +
and 3 + .
Then _{max} 17 (for
*p* = 2.7) or 120
(for *p* = 1.7). The
need for quite high values of the largest s for smaller values of *p*
was noted by
Urry et al. (1991a)
and
Padovani and Urry
(1992)
from their fits to the observed LFs.

While the SSRQ-FSRQ-only scheme (ignoring the radio galaxies;
Orr and Browne 1982)
can not be ruled out, it is much harder to reconcile with the available
data, mainly because there seem to be too few SSRQ for them to be the parents
of FSRQ
(Padovani and Urry
1992).
The FR II-SSRQ-FSRQ scheme, illustrated by
the curves in Fig. 14, can be
tested further via the predicted
radio counts of flat- and steep-spectrum quasars, which converge at easily
accessible levels
(Padovani and Urry
1992).