The kinematic Doppler factor of a moving source is defined as
where
Given a value of
When
which gives a useful upper limit to
In the relativistic beaming model, the observed transverse
velocity of an emitting blob, va =
It can be shown that if
The apparent velocity in terms of
We find from equations (A1) and (A4),
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
The predicted jet/counter-jet ratio (i.e., the ratio between the
approaching and receding jets), can be expressed in terms of
In the simplest cases, p = 2 +
from which an upper limit to
It is useful to calculate the angular parameters relevant to tests
of unified schemes. For sources randomly oriented within the angular range
The linear size of extended sources is proportional to sin
Finally, the mean value of cos
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.
Figure 20. The dependence of the Doppler factor on the angle
to the line of sight. Different curves correspond to different Lorentz
factors: from the top down, =
15, 10, 5, 2. The expanded scale on the inset shows
the angles for which = 1.
, a lower
limit to the Lorentz
factor is given by the condition max; that is,
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 ,
if > 1.
a c,
is related to its true velocity, v = c, and the angle to the line of sight by
> 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[a2 + 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.
and is
a =
sin , and for the angle that
maximizes the apparent velocity, sin
= -1, a
= = =
sqrt[2 -1]
.
reaches its minimum
value when = sqrt[a2
+ 1] 
min. If is a lower limit (as when it is
derived from an SSC calculation)
and < sqrt[a2
+ 1], then the estimated
from Eq. (A6) is an upper limit (of course always bound
to be 
min), while if >
sqrt[a2 + 1], it is a
lower limit. For >>
sqrt[a2 + 1],
/ 2, while if <<
sqrt[a2 + 1],
min/2 . As for Eq. (A7),
when is a lower limit, the
inferred is always an upper
limit.
and a as
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
(since 1)
and a lower limit to (since cos 1) can be derived.
1 to 2, the mean orientation angle is
, for which the mean value is
is given by
