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 Rybicki & 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):
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, Schmidt 1963), 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 ∼ 4.5, and the true velocity ∼ 0.994 (Lorentz factor Γ ∼ 8.8) which means the velocity of the ‘ordinary’ matter in the AGN jet is extremely close to the speed of light.
The formulas (1) & (2) actually imply that the group velocity of light in the medium surrounding a blob in an AGN jet is equal to one, which is actually a hypothesis. In the following, we will consider a general scheme.
We assume that the medium surrounding a blob is uniform, transparent, isotropicand stationary relative to the AGN core, and the group velocity of light is β g which is not necessarily equal to one (β g may be a function of frequency).
Then we could apply the similar derivation like Rybicki & Lightman ( 2004) and get the ’new’ (modified) apparent velocity and Doppler formula: 2. The plot shows that the apparent velocity and Doppler factor change with n from 1 to 1.000001 (corresponding to the group velocity of light from 0.999999 to 1). We assume z = 0, θ = 0.1 • and Γ = 700.
Where n = 1/β g (if β g is equal to the phase velocity of light in material, then n means refractive index of material).
We apply the formulas (3) & (4) to the 3C 273 jet (we assume that β a = 8, θ = 10 • , n = 1.01 corresponding to β g = 0.99) and obtain the Doppler factor δ ∼ 7.2, the true velocity ∼ 0.984 (Lorentz factor Γ ∼ 5.6) which are clearly different from the ones in the case of n = 1 corresponding to β g = 1.
Fig. 1 and Fig. 2 show that the apparent velocity and Doppler factor change with n (For Fig. 1, we take z = 0, θ = 10 • and Γ = 7; In Fig. 2, we assume that z = 0, θ = 0.1 • and Γ = 700).
As shown, when the viewing angle and Lorentz factor take some appropriate values, the group velocity of light close to one could still seriously affect apparent superluminal phenomena and doppler effect in an AGN jet, which may be verified by the high-sensitivity observations in future.
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