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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:

Equation (C1) (C1)
Equation (C2) (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,

Equation (C3) (C3)

where Lj is the observed jet luminosity [Eq. (B7)], curlyLu is the unbeamed luminosity, f ident curlyLj / curlyLu is the ratio of intrinsic jet to unbeamed luminosity, delta 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, thetac, defined by the condition Rmin ident f deltapmin, where deltamin = delta(thetac). That is,

Equation (C4) (C4)

Conversely, the largest ratio, Rmax ident f deltapmax, will occur at the smallest angle, thetamin. If thetamin = 0°, as is usually the case, then

Equation (C5) (C5)

If there is a range of Lorentz factors, Rmax is evaluated at gammamax. If thetamin neq 0°, as is the case for SSRQ, R will be maximum for gamma = 1 / sin thetamin.

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

Equation (C6) (C6)

The relationship between our parameter f and RT ident R(90°) used by other authors (e.g., Orr and Browne 1982) is given by

Equation (C7) (C7)

For thetamin = 0°, Rmax = f [gamma(1-beta)]-p appeq f (2gamma)p, so that Rmax / RT appeq 2p-1 gamma2p. This implies

Equation (C8) (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 gamma (since Rmax / RT gtapprox Rmax, b / Rmin, unb). In the case of a distribution of gammas, the Lorentz factor derived from this argument is the largest one, since that will be responsible both for Rmax (propto gammap) and RT (propto gamma-p).

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