### APPENDIX A: RELATIVISTIC BEAMING PARAMETERS

The kinematic Doppler factor of a moving source is defined as

(A1)

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.

 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.

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

(A2)

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 ,

(A3)

which gives a useful upper limit to if > 1.

In the relativistic beaming model, the observed transverse velocity of an emitting blob, va = a c, is related to its true velocity, v = c, and the angle to the line of sight by

(A4)

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[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.

The apparent velocity in terms of and is

(A5)

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:

(A6)

(A7)

Note that 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.

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

(A8)
(A9)

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

(A10)

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

(A11)
(A12)

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

(A13)
(A14)

Finally, the mean value of cos is given by

(A15)
(A16)