The kinematic Doppler factor of a moving source is defined as

where is its bulk
velocity in units of the speed of light,
= (1 - ^{2})^{-1/2} is the corresponding Lorentz
factor, and
is the angle between the velocity vector and the line of sight. The Doppler
factor has a strong dependence on the viewing angle (as shown in
Fig. 20), which gets stronger for larger
Lorentz factors. For
0° 90°, ranges between
_{min} = (90°) =
^{-1} and _{max} =
(0°) = (1 + ) ~
2 for >> 1.
Moreover, = 1 for
_{} = arccos sqrt[( - 1) / ( + 1)]
(e.g., for an angle _{}
35° if = 5), and for
decreasing _{} with increasing
(Fig. 20); for angles larger
than _{}
relativistic *deamplification* takes place.

Given a value of , a lower
limit to the Lorentz
factor is given by the condition _{max}; that is,

When is a lower limit, as in the SSC case, this expression is valid only for > 1, since for < 1, + 1/ decreases for increasing . It can also be shown that for any value of ,

which gives a useful upper limit to if > 1.

In the relativistic beaming model, the observed transverse
velocity of an emitting blob, *v _{a}* =

It can be shown that if > 1/2 0.7, then for some
orientations superluminal motion is observed. The maximum value of the
apparent velocity, _{a, max} = sqrt[^{2} -1], occurs when cos
= or sin = ^{-1}; for this angle, =
. This implies a minimum value
for the Lorentz factor _{min}
= sqrt[_{a}^{2} + 1] (see
Fig. 21). For
example, if one
detects superluminal motion in a source with _{a} ~ 5,
the Lorentz factor responsible for it has to be at least 5.1. It is
also apparent from Fig. 21 that
superluminal speeds are
possible even for large angles to the line of sight; sources oriented at
~ 50°, have
_{a}
2 if
5, and sources in the
plane of the sky ( = 90°) have
_{a} = ~ 1 for
3.

The apparent velocity in terms of and is

We find from equations (A1) and (A4), _{a} =
sin , and for the angle that
maximizes the apparent velocity, sin
= ^{-1}, _{a}
= = =
sqrt[^{2} -1]
.

With a measurement of superluminal velocity and an independent estimate of the Doppler factor (for example from an SSC calculation), one can combine Eqs. (A1) and (A4) to obtain two equations in four unknowns. That is, under the hypothesis that the ``bulk'' and ``pattern'' speeds are the same, one can derive the value of the Lorentz factor and the angle to the line of sight:

Note that reaches its minimum
value when = sqrt[_{a}^{2}
+ 1]
_{min}. If is a lower limit (as when it is
derived from an SSC calculation)
and < sqrt[_{a}^{2}
+ 1], then the estimated
from Eq. (A6) is an upper limit (of course always bound
to be
_{min}), while if >
sqrt[_{a}^{2} + 1], it is a
lower limit. For >>
sqrt[_{a}^{2} + 1],
/ 2, while if <<
sqrt[_{a}^{2} + 1],
_{min}/2 . As for Eq. (A7),
when is a lower limit, the
inferred is always an upper
limit.

The predicted jet/counter-jet ratio (i.e., the ratio between the
approaching and receding jets), can be expressed in terms of and _{a} as

In the simplest cases, *p* = 2 + or 3 +
(Appendix B).
Figure 22 shows the
jet/counter-jet
ratio as a function of the viewing angle for various values of the Lorentz
factors and for *p* = 2 (which minimizes the effect, as *p* is
likely to be
larger). The dependence on orientation is very strong,
since *J* ^{2p}. From the
jet/counter-jet ratio alone [Eq. (A8)], we obtain

from which an upper limit to (since 1) and a lower limit to (since cos 1) can be derived.

It is useful to calculate the angular parameters relevant to tests
of unified schemes. For sources randomly oriented within the angular range
_{1} to _{2}, the mean orientation angle is

The linear size of extended sources is proportional to sin , for which the mean value is

Finally, the mean value of cos is given by