The radio core-to-lobe ratio, 
, is often used in studies of radio-loud AGN as a
relative measure of orientation (e.g. Orr & Browne 1982).
In the absence of high quality interferometric maps showing full details of the
radio structure, this ratio can be approximated as 
. The
distribution of 
(Fig. 3 (click here)a) will differ in detail from
that of the true R, both because the VLA and GB observations were not
simultaneous and because the low resolution GB measurement may contain some
emission from unrelated sources. Nevertheless, we believe these effects should
not introduce any significant biases into the distribution of
.
For comparison we show in Fig. 3 (click here)b the distribution of core-to-lobe ratios for a large sample of FRI and FRII radio galaxies compiled by Zirbel & Baum (1995). (See Zirbel & Baum 1995 for a discussion of the assumptions used to derive R and the upper limits on R for those sources without a measured radio core.) To illustrate the properties of an extremely core-dominated population, we show in Fig. 3 (click here)c the core-to-lobe ratio for BL Lacertae objects, with the radio-selected objects (RBLs) represented by the hatched histogram. The objects shown are the X-ray-selected BL Lacs (XBLs) from the HEAO-1 Large Area Sky Survey (Kollgaard et al. 1996) and the Einstein Extended Medium Sensitivity Survey (Morris et al. 1991), and the radio-selected BL Lacs in the 1 Jansky sample (Stickel et al. 1991). The radio flux densities used to derive the core-to-lobe ratios were taken from Kollgaard et al.
Figure 3 (click here) shows the RGB sample is more core-dominated (40% of the
sources have 
) than the radio galaxy sample of Zirbel &
Baum (1995; 3% with 
), but is less core-dominated than the BL
Lacertae objects (82% with 
). We used the Astronomy SURVival
(ASURV) data analysis software (Rev. 1.2; LaValley et al.
1992) to compute the Kaplan-Meier estimator of the R
distributions. This properly takes into account the upper limits in the
radio galaxy sample (Feigelson & Nelson 1985). The
median R of each distribution is given in Table 5 (click here). We find that
both classes of BL Lac objects are significantly more core-dominated than
the RGB sample. The median of the radio galaxy sample, however, is 27 times
less core-dominated than the RGB sources.
The differences discussed above are clearly due to the type of object which dominates each of the samples. Although >70% of the RGB catalog is optically unidentified, most of the identified sources are quasars (B95). A comparison of the optically identified and unidentified sources shows that while the identified sources generally exhibit higher radio and X-ray fluxes, other properties (e.g. their optical colors) are not statistically different (B96). This suggests the unidentified sources are also primarily quasars. The differences in the distribution of R therefore indicate the RGB catalog consists primarily of quasars whose radio emission is moderately beamed.
Within the framework of the unified scheme scenario which hypothesizes flat and
steep spectrum quasars are radio galaxies seen close to the line-of-sight (e.g.
Barthel 1989), we use a simple beaming model and the
core-to-lobe ratio distributions to constrain the jet speed and orientation
characteristic of objects in the RGB sample. The dependence of R on jet
speed and orientation are given by (e.g. Urry &
Padovani 1995):
where f is the intrinsic core-to-lobe ratio, p is the beaming index, and
is the Doppler factor:
Here 
, where v is the bulk velocity, 
, and 
is the angle to the
line-of-sight. We assume p = 2.7, applicable to a jet consisting of a
single sphere with a spectral index 
=0.3 (
; e.g. Pearson & Zensus 1987). We make the
further assumption that the Zirbel & Baum (1995) sample of FRI and FRII
radio galaxies is characteristic of the parent population of RGB sources,
although we examine this hypothesis more carefully at the end of this
section.
Table 5: Median core-to-lobe ratios
Kollgaard et al. (1996), analyzing the same population of radio galaxies and
BL Lacertae objects, found that the relative core enhancement of these
populations implied that 
> 4.5 and probably exceeded 
= 6.
We
therefore assume initially 
= 6 for all three populations and adopt
= 
for the radio galaxies.
(See Kollgaard et
al.) For a sample like the RGB catalog which consists largely of radio-loud
quasars (B95; B96; Laurent-Muehleisen et al., in
preparation), the assumption 
= 6 is a reasonable lower
limit to the jet speed (Urry & Padovani 1995). Using
these assumptions and the median R values in Table 5 (click here), this
implies that the average angle to the line-of-sight for the RGB sample
(
) is approximately 
, significantly
larger than that obtained for the BL Lac objects where
and
(Kollbaard et al. 1996).
Figure 4: The predicted line-of-sight orientation for the RGB sample,
, derived from the relative median core-to-lobe
ratios of the RGB and radio galaxy samples. 
is assumed and the horizontal line is the observed ratio of
The assumption that 
is a single value is most likely incorrect in
detail since a range of jet speeds probably characterizes any given population
of objects. Assuming 
= 
and
= 
, but allowing both Lorentz
factors to
vary over the range 2 
20, constrains the average angle
to the
line-of-sight for the RGB sample to lie within a fairly small range,
20
< 
< 32
(Fig. 4 (click here)). If we further constrain 
5, which is a
reasonable minimum based on studies of the observed luminosity function of
flat and steep spectrum quasars (Urry & Padovani
1995), then 
is narrowly
confined to be about 31
.
Finally, we consider the possibility that the population of FRI and FRII radio
galaxies used here is not characteristic of the parent population of objects
in the RGB catalog. Assuming some form of a unified scheme is not
unreasonable, but it is possible that the RGB sample exhibits an average jet
speed substantially different than that characteristic of the Zirbel & Baum
(1995) radio galaxy sample. This could be the case if the RGB catalog is
biased toward objects with a larger 
. Using the results of
Kollgaard et al. (1996), we fix 
= 6 and
= 
. As before, we constrain
to be larger than 5. The average angle to the
line-of-sight for the RGB sample is then
20
35
. We also
consider the case where the intrinsic core-to-lobe ratio (f in Eq.
(4)) of the Zirbel & Baum (1995) radio galaxies is different than that of
the ``true'' parent population. Since the RGB catalog is likely dominated
by radio- and X-ray-loud quasars, if the FRII-quasar unified scheme is
correct (Barthel 1989), the true parent population of
RGB objects will have extended radio powers approximately two to three
orders of magnitude higher than those objects in Zirbel & Baum (1995).
Because the core-to-lobe ratio decreases with increasing extended radio
power (Kollgaard et al. 1996), our ratio of the
core-to-lobe parameters would be too low by a factor of 
4. However,
the effect on the average angle to the line-of-sight is fairly modest,
decreasing it to 
25
. These
considerations indicate that 
is relatively
insensitive to assumptions about the detailed characteristics of the parent
population.