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:

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):

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

(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

^{} 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
:

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:

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,

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}* /

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 ~ 10^{4} - 10^{8}. 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 10^{20}
(Djorgovski et
al. 1991).