6.2.4 Population Statistics for Radio Samples
The radio data are better suited than the X-ray data for testing the beaming
hypothesis for BL Lac objects. One reason is that the radio sample of FR I
galaxies is larger than the X-ray sample. In addition, the higher spatial
resolution available with radio interferometry allows estimates of the ratio
between beamed and unbeamed radio flux
(Antonucci and
Ulvestad 1985),
thus
adding an important constraint to the combination of the parameters
and f (Appendix C).
The radio luminosity function of FR I radio galaxies was derived by
Urry et al. (1991a)
from the 2 Jy catalog (and converted to 5 GHz) and then extended to
lower radio powers using the radio LF of elliptical galaxies of
Franceschini et
al. (1988).
Here we choose not to include the low-luminosity radio
ellipticals but restrict ourselves only to sources classified explicitly as
FR Is (some of the lowest luminosity ellipticals could in fact
have radio emission dominated by thermal emission rather than the nonthermal
BL Lac nucleus;
Phillips et al. 1986).
Our results, however, are basically unchanged, since the beamed LF
for BL Lacs at P5 5 x 1025 W Hz-1 (i.e., where it
overlaps with the observed LF) is largely unaffected by the behavior of the
FR I LF at such low powers. The new optical identifications
(di Serego Alighieri
et al. 1994b)
and radio maps
(Morganti et al. 1993)
also have little effect on the FR I luminosity function;
the fit to the new LF used here is consistent with the one
in Urry et al. (1991a).
The evolutionary properties of FR I radio galaxies are
consistent with no evolution, with < V/Vmax > =
0.42 ± 0.05.
We can compare our FR I luminosity function, derived from the 2 Jy sample
(selected at 2.7 GHz), with that of
de Ruiter et al. (1990),
based on the B2
sample (selected at 408 MHz) plus nearby (z < 0.2) 3CR radio galaxies
(selected at 178 MHz). Converting our FR I LF to H0 =
100 km s-1
Mpc-1 and = 408 MHz
(assuming
r = 0.7),
we find
excellent agreement from P408 ~ 6 x 1023 W
Hz-1 up to the
break at P408 ~ 3 x 1025 W Hz-1 (see
Fig. 13 of
de Ruiter et al. 1990).
Above the break our LF is steeper, not surprisingly, since they did not
exclude FR II sources. The two lowest luminosity bins in our LF
(P5
2
x 1023 W Hz-1), lie a factor of 2-3
above the de Ruiter et al. LF. This disagreement is not surprising, as our two
points are highly uncertain, with only one object in each bin. Also, the two
objects, M82 (16)
and M84, are nearby (z = 0.0014 and 0.0028,
respectively) and
so sample the local overdensity, whereas the B2 sample, being deeper than the
2 Jy, averages over a larger volume of space.
(The equivalent flux limit of the B2 survey translated to 2.7 GHz with
= 0.7 is 130 mJy.) In any case,
excluding the first two bins does not alter our fitted beamed LF
significantly.
Using a two-power-law approximation to our FR I luminosity function,
we fitted a beamed LF to the observed LF for the 1 Jy BL Lacs. The latter was
obtained as described in
Stickel et al. (1991)
with the addition of S5 0454+844,
which had no redshift at the time. It is impossible to fit the data with a
single Lorentz factor; instead, an acceptable fit to the LF and the observed
R-values is obtained for Lorentz factors distributed in the range
5
r
32. While the form of the
distribution is not
well constrained, it is weighted toward lower values; e.g., for a power law
of the form N (
r)
rG, the best-fit index
is G ~ -4.
Figure 17 shows the excellent agreement between beamed
(solid line) and observed (filled circles) radio luminosity functions for the
1 Jy sample of BL Lacs. The mean Lorentz factor, which is approximately
independent of the exact shape of the distribution, is < r
> ~ 7, corresponding to a ratio between BL Lacs and FR Is of ~
1:50, about an order of magnitude smaller than in the X-ray case
(Sec. 6.2.3). The total number of BL Lacs
is most sensitive to the
lowest value of
r,
while the maximum ratio of beamed to unbeamed
flux is sensitive to the highest value. Our fitted model predicts that
radio-selected BL Lacs are aligned within
c ~ 12° of
the line of sight. The fitted beaming parameters for radio-selected BL Lacs
are summarized in Table 3.
![]() |
Figure 17. The local differential radio luminosity functions of radio-selected BL Lac objects (filled circles; Stickel et al. 1991) and FR I radio galaxies (open squares; Sec. 6.2.4), compared to the fitted beaming model (solid line; first set of RBL parameters in Table 3). Error bars correspond to 1 sigma Poisson errors (Gehrels 1986). |
As we did for quasars, we can estimate a lower limit to the maximum Lorentz
factor from the observed maximum and minimum values of R for BL Lacs and
FR Is, respectively (Appendix C). OJ 287
is the most
core-dominated BL Lac in the 2 Jy sample (and in the 1 Jy sample),
excluding those cases where no extended emission has
been detected, and Fornax A is the most lobe-dominated FR I. The
corresponding
values of R, K-corrected [Eq. (C2] and extrapolated (when
necessary) to 5 GHz rest frequency assuming core -
ext = -1, are
Rmin, FR I
4
x 10-4 (Fornax A;
Morganti et al. 1993)
and Rmax, BL Lac
780 (OJ 287;
Kollgaard et al. 1992).
Using Eq. (C8)
as before,
max
(2.0 x 106
21-p)1/2p ~ 9 for p = 3.
As was the case for the quasars, small values of p imply high
values for the
largest Lorentz factor (for p = 2, max
30). More
precisely, since
r
-0.1 for the 1 Jy BL Lacs
(Stickel et al. 1991),
p
1.9 - 2.9 for
p between 2 +
and 3 +
. Then
max
10
(for p = 2.9) or
38 (for p = 1.9).
The value of H0 does affect the luminosity functions
but has no direct
effect on the derived beaming parameters. The fitted Lorentz factor, which is
inversely proportional to the normalization of the beamed luminosity function,
depends only on the ratio between the total numbers of objects (the integrals
of the parent and beamed LFs). H0 enters only because
we constrain
to some extent with the
observed values of superluminal motion (for BL Lacs and for quasars
separately; see Sec. 6.3). A higher
H0 would mean lower superluminal velocities, and in
general, fits with
lower
1 are
possible. We did not explore parameter space
exhaustively but we show an example of a low-
fit for RBL in
Table 3.
It is interesting that the luminosity functions of FR I and FR II
galaxies overlap smoothly, as do those of BL Lac objects and FSRQ
(Figs. 14 and 17). Possibly they represent
different manifestations of the same basic central engine, in which case
the different radio morphologies and emission line strengths would have to
be closely linked to observed radio power (Maraschi and Rovetti 1994).