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).