APPENDIX B: DOPPLER ENHANCEMENT

The Doppler factor, (Appendix A), relates intrinsic and observed flux for a source moving at relativistic speed v = c. For an approaching source, time intervals measured in the observer frame are shorter than in the rest frame (even allowing for time dilation) because the emitter ``catches up'' to its own photons:

(B1)

where primed quantities refer to the rest frame of the source. Since the number of wavefronts per unit time is constant, the emission is blue-shifted (essentially the inverse relation):

(B2)

The intensity enhancement (``Doppler boosting'') is an even more dramatic effect. Because I / ³ is a relativistic invariant, the transformation of specific intensity is:

(B3)

(Rybicki and Lightman 1979). One power of comes from the compression of the time interval [Eq. (B1)] and two come from the transformation of the solid angle, d = ² d '.

If the emission is isotropic in the source rest frame (i.e., I'_^' is not a function of angle), the flux density, F, transforms in the same way as the specific intensity. For a power-law spectrum of the form F'_^' (' )^-, Eq. (B3) becomes

(B4)

is just the ratio of the intrinsic power-law fluxes at the observed and emitted frequencies.

Broad-band fluxes are obtained from integrating Eq. (B3) over frequency, and since d = d' [Eq. (B2)], these are boosted by another factor of :

(B5)

The degree of variability in AGN is frequently measured by the change in flux over a given period of time, which from Eqs. (B1) and (B5) is:

(B6)

Equations (B4), (B5), and (B6) assume that the emission comes from a moving source, so they apply to the case of discrete, essentially point-like, components. For a smooth, continuous jet, the observed emitting volume is decreased by one power of the Doppler factor because of Lorentz contraction, so that the exponents in Eqs. (B4), (B5), and (B6) become 2 + , 3, and 4, respectively (Begelman et al. 1984; Cawthorne 1991; Ghisellini et al. 1993).

Relaxing the assumption of isotropic emission in the rest frame can also change the relationship between intrinsic and observed flux. If the source is an optically thin jet with magnetic field parallel to its axis, then in the rest frame F'_'(' ) (sin ' )¹⁺. Since sin ' = sin , Eq. (B4) becomes F() = ⁽³⁺²⁾ (sin )¹⁺ F'_'() (Cawthorne 1991; Begelman 1993).

So far we have assumed power-law emission in the rest-frame. Strictly speaking, for synchrotron emission this would mean the jet is either completely optically thin (F ^-, 0.5) or completely optically thick (F ^5/2). Real jets are probably inhomogeneous and have flat spectra caused by the superposition of individual synchrotron components with different self-absorption frequencies. Relativistic beaming distorts these components differentially because of the dependence of optical depth and F on , so the overall spectral shape should change. For a standard conical jet with tangled magnetic field, the integrated flux transforms approximately as ²⁺, where refers to the observed integrated spectrum rather than the spectral index of the local emission (Cawthorne 1991).

Additional complications in the evaluation of the amplification factor include the lifetimes of the emitting components, the radial dependence of the their emissivities, and the presence of shocks (Lind and Blandford 1985). In the following we will hide all these possibilities in a single parameter, p, by assuming that the observed luminosity, L_j, of a relativistic jet is related to its intrinsic luminosity, _j, via

(B7)

with p = 3 + for a moving, isotropic source and p = 2 + for a continuous jet (other values are certainly possible).

The recent detection of superluminal motion within our own Galaxy (Mirabel and Rodríguez 1994) permits for the first time a direct estimate of p. Since proper motions are measured for both the approaching and receding components, cos is uniquely determined to be 0.323 ± 0.016 (where refers to the pattern speed; Mirabel and Rodríguez 1994). Using the observed jet/counter-jet ratio, F_j / F_cj = 8 ± 1, and assuming the bulk speed in Eq. (A8) is equal to the pattern speed, we find p = 3.10 ± 0.25. One might expect p = 3 + because the components are discrete blobs, whereas our estimate implies p 2 + since the measured spectral index is = 0.8. Alternatively, p ~ 3.8 is allowed if the bulk speed is actually lower than the pattern speed, with a ratio _bulk / _pattern ~ 0.8 (cf. Bodo and Ghisellini 1995).

Regardless of the precise value of p, relativistic beaming has a very strong effect on the observed luminosity. A relativistic jet has 2 for arcsin ^-1 (Appendix A), meaning a modest bulk Lorentz factor of = 10 amplifies the intrinsic power by 2-5 orders of magnitude (depending on p). The Doppler boosted radiation is strongly collimated and sharply peaked: at ~ 1/ ~ 6°l, the observed jet power is already ~ 4-16 times fainter than at = 0° (for p = 2-4). At 90°, the reduction is huge, a factor of ~ 10⁴ - 10⁸. Although this is a very large ratio, it is actually much smaller than the inferred extinction at optical wavelengths caused by an obscuring torus, which can be up to a factor of 10²⁰ (Djorgovski et al. 1991).