### APPENDIX C: Ratio of Core- to Extended-Flux

The ratio between core and extended flux is an important beaming indicator.
We use the observed luminosity ratio, *R* = *L*_{core} /
*L*_{ext}, which is
related to the observed flux ratio via a *K*-correction:

(C1)

(C2)

We do not correct for the fact that *F*_{ext} diminishes
faster with redshift than *F*_{core}, as this depends on
the source morphology. [In
any case it is not a large effect. The corrections calculated by
Perlman and Stocke 1993
for 14 BL Lac objects with redshifts from 0.2 to 0.5 are
roughly proportional to (1 + *z*)^{1.2}.]

We associate the core with the relativistically beamed jet and the extended
power with the unbeamed emission. In terms of the beaming formalism,

(C3)

where *L*_{j} is the observed jet luminosity [Eq. (B7)],
_{u} is the
unbeamed luminosity, *f*
_{j} / _{u} is the ratio of intrinsic jet to
unbeamed luminosity, is the
Doppler factor, and *p* is the
appropriate exponent (Appendix B). We
call sources ``beamed'' when *R* >
*R*_{min} ~ 1.

The largest angle between the jet and the line of sight for beamed objects is
a critical angle, _{c}, defined by the condition
*R*_{min}
*f* ^{p}_{min}, where _{min} =
(_{c}). That is,

(C4)

Conversely, the largest ratio, *R*_{max}
*f* ^{p}_{max},
will occur at the smallest angle, _{min}. If
_{min} =
0°, as is usually the case, then

(C5)

If there is a range of Lorentz factors, *R*_{max} is evaluated at
_{max}. If
_{min} 0°, as is the case for
SSRQ, *R* will be maximum for = 1 / sin _{min}.

For large angles (
arccos [0.5 / ]),
emission from the
receding jet is no longer negligible and Eq. (C3) is replaced by

(C6)

The relationship between
our parameter *f* and *R*_{T} *R*(90°) used
by other authors (e.g.,
Orr and Browne 1982)
is given by

(C7)

For _{min} = 0°,
*R*_{max} = *f* [(1-)]^{-p}
*f* (2)^{p}, so that
*R*_{max} / *R*_{T} 2^{p-1}
^{2p}.
This implies

(C8)

Combining the maximum value of *R* for a set of beamed objects,
*R*_{max, b},
and the minimum value of *R* for the parent population,
*R*_{min, unb}, gives
a lower limit to the value of
(since *R*_{max} /
*R*_{T}
*R*_{max, b} / *R*_{min, unb}). In the case of
a distribution of s,
the Lorentz factor derived from this argument is the largest one,
since that will be responsible both for *R*_{max} (
^{p}) and
*R*_{T} (
^{-p}).