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A Non-Mainstream Viewpoint on Apparent Superluminal Phenomena in AGN Jet

The group velocity of light in material around the AGN jet is acquiescently one (c as a unit), but this is only a hypothesis. Here, we re-derive apparent superluminal and Doppler formulas for the general case (it is assumed that the group velocity of light in the uniform and isotropic medium around a jet (a beaming model) is not necessarily equal to one, e.g., Araudo et al. (2010) thought that there may be dense clouds around AGN jet base), and show that the group velocity of light close to one could seriously affect apparent superluminal phenomena and Doppler effect in the AGN jet (when the viewing angle and Lorentz factor take some appropriate values).

Concerning the apparent superlunimal motion, the most popular and classic viewpoint is the relativistic beaming model (Rees 1966). The Fig. S.6 in Ribicki & Lightman (2004) showed the simple geometry scenario of emission for a moving source, the observed transverse velocity of separation of a blob relative to the speed of light c (in this paper, all velocities are at c as a unit) is : where β is the true velocity, and θ is the angle to the line of sight. The corresponding Doppler factor δ (considering the effect of redshift) is : where Γ = (1 − β 2 ) −1/2 is the Lorentz factor, z is the redshift of this AGN.
For the 3C 273 jet (z = 0.158), VLBI observations have detected apparent superluminal motions in the parsec-scale jet with apparent velocities 6 ∼ 10 (e.g., Unwin et al. 1985). We assume the apparent velocity is 8, and the angle to the line of sight is 10 • . Then, based on the formula (1) and (2), we could obtain the Doppler factor δ ∼ 5, and the true velocity ∼ 0.994 (Lorentz factor Γ ∼ 9) which means the velocity of the 'ordinary' matter in the AGN jet is extremely close to the speed of light (this requires a very high energy and an effective acceleration mechanism, which is unimaginable on Earth).
The formula (1) & (2) actually imply that the refractive index of the medium surrounding a blob in the AGN jet is equal to 1, which is actually a hypothesis. In the following, we will consider another scheme.
We assume that the unknown transparent medium surrounding a blob is uniform, stationary relative to the AGN core, and the equivalent refractive index is n which is not equal to 1.

Then we could apply the similar derivation like Ribicki & Lightman (2004) and get the 'new' (modified) apparent velocity & Doppler formula :
We apply the formula (3) & (4) to the 3C 273 jet (we assume that β a = 8, θ = 10 • , n = 10 4 ) and obtain the Doppler factor δ ∼ 4×10 5 , the true velocity ∼ 0.0001 (Lorentz factor Γ ∼ 1). For synchrotron emission, the observed fluxes are proportional to δ 3+α (Dermer 1995, α is the spectral index). If α = α radio ∼ 0.8 (e.g., Stawarz 2004) and δ ∼ 4 × 10 5 , then the observed fluxes of the 3C 273 jet are 2 × 10 21 times the local ones, which may reduce the singularity of AGN jet. The high observed luminosity of AGN may also be attributed to the unknown surrounding medium which of the equivalent refractive index is not equal to 1. The equivalent refractive index may be related to the gravitational effect or optical properties of the surrounding medium. The physical meaning of the equivalent refractive index needs to be further studied. This work was supported by the Scientific Research Foundation of Civil Aviation University of China (09QD15X).