Astron. Astrophys. 325, 19-26 (1997)

2. A beaming statistics

The kinematic effect of a relativistically moving jet can be characterized by the Doppler factor

where is the speed of the jet flow in units of the speed of light, is the Lorentz factor and is the angle between the observer and the axis of jet. The relativistic Doppler effect on the radiation from the jet is beaming and boosting, i.e. the superluminal effect. The relation between the luminosity L in the observer's frame and the intrinsic luminosity in the rest frame of jet can be quantified through the Doppler factor ,

where is the power law index of the differential energy spectrum of the radiation. is a geometrical parameter, for point sources and for one-dimensional continuous sources (e.g. Lind & Blandford 1985). Here the coefficient is defined as beaming factor. The above relation is valid only for the case in which both the projectiles and targets are assumed to be isotropic in the jet comoving frame.

For the case in which the isotropic target photons (in the observer's frame) are external to the jet comoving frame, Dermer (1995) has derived the beaming factor for a blob (point source) in which gamma rays are produced via inverse Compton scattering,

where is the gamma ray energy spectral index and the average observational value is about 1.0; and takes a larger value . It can be seen clearly that the beaming effect is more dramatic than the previous case. There is also an angular factor in the Dermer's formulation. We neglect it in our formulation for simplicity. It is useful to mention that the beaming factor is still of the same order of magnitude if the external photons are not isotropic. For example, for the case of external photons coming from the accretion disk, Dermer, Schlickeiser and Mastichiadis (1992) have demonstrated the beaming factor to be where s is the spectral index of relativistic electrons and their preferred value is 3 (cf. the canonical value of is 1). The effect caused by anisotropy in the external photon field is just a different angular profile of the gamma ray beams.

Now we use the difference in beaming effect to discriminate between the two classes of gamma ray emission models, statistically. First of all, we need to specify the key parameter which enables the theoretical models to be related to the observed statistics. The parameter is the index in the beaming factor, specifically the power-law index of radiation spectrum. Observationally, radio and gamma ray emissions have different indices and thus their flux ratio will retain the Doppler factor. Then we choose the K-corrected flux ratio of gamma ray to radio as the random variable to formulate our statistical test. The radio emission is generally believed to be synchrotron emission along the jet and its beaming factor is thus fixed. We consider only a special combination of emission characters in which gamma ray emission is from a moving blob and radio emission is from a continuous jet. This choice is consistent with some theoretical models of gamma ray emission (e.g. Ghisellini & Madau 1996; Böttcher, Mause & Schlickeiser 1996). The results from simultaneous multiwavelength observations of gamma ray blazars 3C 273 (Lichti et al. 1995) and 3C 279 (Hartman et al. 1996) show little time variability in radio emission and no radio flares co-occurring with gamma ray flares. Thus we may conclude that the radio emission in blazars is from a continuous jet rather than from a moving blob. It might not hold for the radio cores seen by VLBI. Other combinations can be formulated in a similar way.

Two assumptions are made here to simplify the formulation. The first one is that there is a proportionality between the intrinsic luminosities of gamma ray and radio emission in blazars, and that its dispersion is small. The second one is the same Lorentz factor for both gamma ray and radio emissions of all blazars. The observational justification and theoretical argument will be made later in Sect. 4. With these two assumptions, we define x to be the observed flux density ratio of gamma ray to radio (after the K-corrections) and thus x is approximately proportional to the beaming factor ratio of gamma ray to radio,

where and depends on the gamma ray models. For our choice of radio emission model, . For the SIP gamma ray model, and . For the SEP gamma ray model, and .

For a fixed , is the sole function of as defined in Eq. (1). So is x. Here we ascribe the variation of to the change in . If a gamma ray emitting jet is isotropically oriented in 3-dimensional space, i.e. is uniformly distributed, it is easy to derive the distribution of which follows a power law (e.g. Urry & Shafer 1984; Chi & Young 1996),

A power law distribution has the property of scaling-invariance with respect to its variable. As , we have the same power law distribution for x

The index depends on the model of gamma ray emission. For the choice of and , it equals 1.5 for the SIP model and 1.25 for the SEP model. If the angular factor is included in the SEP model, the index will become even smaller and thus the index difference between the two models will become larger. As argued by Dermer (1995), the effect of this angular factor is relatively small and thus can be neglected in our formulation.

© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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