2.1 Blazar Properties and Relativistic Beaming

The main properties of blazars can be summarized as follows:

• rapid variability (high L / t);
• high and variable optical polarization (Popt > 3%);
• compact, flat-spectrum radio emission (fcore >> fextended);
• superluminal motion in sources with multiple-epoch Very Large Baseline Interferometry (VLBI) maps.

The last property might require some explanation. The term ``superluminal motion'' describes proper motion of source structure (traditionally mapped at radio wavelengths) that, when converted to an apparent speed vapp, gives vapp > c. This phenomenon occurs for emitting regions moving at very high (but still subluminal) speeds at small angles to the line of sight (Rees 1966). Relativistically moving sources ``run after'' the photons they emit, strongly reducing the time interval separating any two events in the observer's frame and giving the impression of faster than light motion.

Analytically, 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

(1)

It can be shown that if > 1/ 2 0.7, then for some orientations superluminal motion (that is, a > 1) is observed. The maximum value of the apparent velocity, a, max = sqrt(2 - 1), where = (1 - 2)-1/2 is the Lorentz factor, occurs when cos = or sin = -1. This implies a minimum value for the Lorentz factor min = sqrt(a2 + 1). 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.

All these properties are consistent with relativistic beaming, that is with bulk relativistic motion of the emitting plasma towards the observer. There are by now various arguments in favor of relativistic beaming in blazars, summarized for example by Urry and Padovani (1995). Beaming has enormous effects on the observed luminosities. Adopting the usual definition of the relativistic Doppler factor = [ (1 - cos )]-1 and applying simple relativistic transformations, it turns out that the observed luminosity at a given frequency is related to the emitted luminosity in the rest frame of the source via

(2)

with p = 2 + or 3 + respectively in the case of a continuous jet or a moving sphere (Urry and Padovani 1995; being the spectral index), although other values are also possible (Lind and Blandford 1985). For ~ 0°, ~ 2 (Fig. 2) and the observed luminosity can be amplified by factors of thousands (for ~ 5 and p ~ 3, which are typical values). That is, for jets pointing almost towards us we can overestimate the emitted luminosity typically by three orders of magnitude. Apart from this amplification, beaming also gives rise to a strong collimation of the radiation, which is larger for higher (Fig. 2): decreases by a factor ~ 2 from its maximum value at ~ 1/ and consequently the inferred luminosity goes down by 2p. For example, if ~ 5 the luminosity of a jet pointing ~ 11° away from our line of sight is already about an order of magnitude smaller (for p = 3) than that of a jet aiming straight at us.

 Figure 2. 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 (solid line), = 10 (dotted line), = 5 (short-dashed line), = 2 (long-dashed line).

All this is very relevant to the issue of -ray emission from blazars. In fact, if blazars were not beamed, we would not see any -ray photons from them! The qualitative explanation is relatively simple: in sources as compact as blazars all -ray photons would be absorbed through photon-photon collisions with target photons in the X-ray band. The end product would be electron-positron pairs. But if the radiation is beamed then the luminosity/radius ratio, which is the relevant parameter, is smaller by a factor p+1 and the -ray photons manage to escape from the source. More formally, it can be shown (Maraschi et al. 1992) that the condition that the optical depth to photon-photon absorption (x) is less than 1 implies (under the assumption that the X-ray and -ray photons are produced in the same region)

(3)

where L48 L / (1048 erg/s), td is the -ray variability time scale in days (which is used to estimate the source size), x h / me c2, x is the X-ray spectral index, and C is a numerical constant 10. In other words, transparency for the -ray photons requires a relatively large Doppler factor for most blazars (Dondi and Ghisellini 1995) and therefore relativistic beaming.