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The kinematic Doppler factor of a moving source is defined as

Equation (A1) (A1)

where beta is its bulk velocity in units of the speed of light, gamma = (1 - beta2)-1/2 is the corresponding Lorentz factor, and theta 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° leq theta leq 90°, delta ranges between deltamin = delta(90°) = gamma-1 and deltamax = delta(0°) = (1 + beta)gamma ~ 2gamma for gamma >> 1. Moreover, delta = 1 for thetadelta = arccos sqrt[(gamma - 1) / (gamma + 1)] (e.g., for an angle thetadelta appeq 35° if gamma = 5), and for decreasing thetadelta with increasing gamma (Fig. 20); for angles larger than thetadelta relativistic deamplification takes place.

Figure 20
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, gamma = 15, 10, 5, 2. The expanded scale on the inset shows the angles for which delta = 1.

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

Equation (A2) (A2)

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

Equation (A3) (A3)

which gives a useful upper limit to theta if delta > 1.

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

Equation (A4) (A4)

It can be shown that if beta > 1/sqrt2 appeq 0.7, then for some orientations superluminal motion is observed. The maximum value of the apparent velocity, betaa, max = sqrt[gamma2 -1], occurs when cos theta = beta or sin theta = gamma-1; for this angle, delta = gamma. This implies a minimum value for the Lorentz factor gammamin = sqrt[betaa2 + 1] (see Fig. 21). For example, if one detects superluminal motion in a source with betaa ~ 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 theta ~ 50°, have betaa gtapprox 2 if gamma gtapprox 5, and sources in the plane of the sky (theta = 90°) have betaa = beta ~ 1 for gamma gtapprox 3.

The apparent velocity in terms of gamma and delta is

Equation (A5) (A5)

We find from equations (A1) and (A4), betaa = delta gamma beta sin theta, and for the angle that maximizes the apparent velocity, sin theta = gamma-1, betaa = delta beta = gamma beta = sqrt[gamma2 -1] appeq gamma appeq delta.

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:

Equation (A6) (A6)

Equation (A7) (A7)

Note that gamma reaches its minimum value when delta = sqrt[betaa2 + 1] ident gammamin. If delta is a lower limit (as when it is derived from an SSC calculation) and delta < sqrt[betaa2 + 1], then the gamma estimated from Eq. (A6) is an upper limit (of course always bound to be geq gammamin), while if delta > sqrt[betaa2 + 1], it is a lower limit. For delta >> sqrt[betaa2 + 1], gamma appeq delta / 2, while if delta << sqrt[betaa2 + 1], gamma appeq gammamin/2 delta. As for Eq. (A7), when delta is a lower limit, the inferred theta 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 delta and betaa as

Equation (A8) (A8)
Equation (A9) (A9)

In the simplest cases, p = 2 + alpha or 3 + alpha (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 approx delta2p. From the jet/counter-jet ratio alone [Eq. (A8)], we obtain

Equation (A10) (A10)

from which an upper limit to theta (since beta leq 1) and a lower limit to beta (since cos theta leq 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 theta1 to theta2, the mean orientation angle is

Equation (A11) (A11)
Equation (A12) (A12)

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

Equation (A13) (A13)
Equation (A14) (A14)

Finally, the mean value of cos theta is given by

Equation (A15) (A15)
Equation (A16) (A16)

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