### 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 = Lcore / Lext, which is related to the observed flux ratio via a K-correction:

(C1)
(C2)

We do not correct for the fact that Fext diminishes faster with redshift than Fcore, 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 Lj 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 > Rmin ~ 1.

The largest angle between the jet and the line of sight for beamed objects is a critical angle, c, defined by the condition Rmin f pmin, where min = (c). That is,

(C4)

Conversely, the largest ratio, Rmax f pmax, 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, Rmax 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 RT R(90°) used by other authors (e.g., Orr and Browne 1982) is given by

(C7)

For min = 0°, Rmax = f [(1-)]-p f (2)p, so that Rmax / RT 2p-1 2p. This implies

(C8)

Combining the maximum value of R for a set of beamed objects, Rmax, b, and the minimum value of R for the parent population, Rmin, unb, gives a lower limit to the value of (since Rmax / RT Rmax, b / Rmin, 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 Rmax ( p) and RT ( -p).