This lecture is part of the course "physics of active galactic nuclei" offered to graduate students in astrophysics by Rodrigo Nemmen and Joao Steiner at IAG USP.
sees disk Ghisellini Difference between intensity and ﬂux inside the shell. to F nsity I and the total ﬂux F must account for the fact t of the surface is seen under a different angle θ. See sider the projected area, and introduce a cos θ term in F = I cosθdΩ (1.15)
total optical depth τν ≫ 1 is observed face-on (left) and from an angle θ from its normal (right). The two observers receive always the emission produced in the shell of unit optical depth. But an optical depth of unity corresponds to the length AB (left) or CD (right). The two lengths are equal (AB = CD), but one is inclined, therefore the two volumes are different (by the factor DH/AB = cosθ) Ghisellini At an inclined angle of observation, the emitting volume is less by a cosθ factor
a photon without interacting Distance for which ⌧⌫ = 1 Path (for a photon) is the average distance ℓ traveled by a photon t corresponds to a distance for which τν = 1: τν = 1 → σνnℓν = 1 → ℓν = 1 nσν (1.44)
1.10 The electric ﬁeld produced by a charge initially in uniform rectilinear motion that is suddenly stopped. At large distances, the electric ﬁeld points to where the charge would be if it had not been stopped. At closer distances, the electric ﬁeld had time to “adjust” and points to where the charge is. There is then a region of space where the electric ﬁeld has to change direction. This Electric ﬁeld produced by an accelerated charge Ghisellini
1.10 The electric ﬁeld produced by a charge initially in uniform rectilinear motion that is suddenly stopped. At large distances, the electric ﬁeld points to where the charge would be if it had not been stopped. At closer distances, the electric ﬁeld had time to “adjust” and points to where the charge is. There is then a region of space where the electric ﬁeld has to change direction. This Electric ﬁeld produced by an accelerated charge Ghisellini electromagnetic wave (radiation) c t Run Mathematica demonstration
Proc. Camb. Philos. Soc. 55, 137 (1959) Terrel, J. Invisibility of the Lorentz contraction. Phys. Rev. 116, 1041 (1959) We ﬁgured out apparent shapes of relativistic objects >50 years after special relativity was published
in the BLR of a radio loud AGN) Ghisellini actor Γ . In the rest of the blob the oming from 90° in are seen to come at 1/Γ . The energy seen by the blob is by a factor ∼Γ 2 ll of broad line clouds. For simplicity, assume that the broad line photons uced by the surface of a sphere of radius R and that the jet is within it. As- so that the radiation is monochromatic at some frequency ν0 (in frame K). moving (in frame K′) observer will see photons coming from a cone of semi- 1/Γ (the other half may be hidden by the accretion disk): photons coming 3 Beaming S frame S’ frame U F U' F'
Useful relativistic transformations ν = ν′δ frequency t = t′/δ time V = V ′δ volume sinθ = sinθ′/δ sine cosθ = (cosθ′ + β)/(1 + β cosθ′) cosine I(ν) = δ3I′(ν′) speciﬁc intensity I = δ4I′ total intensity j(ν) = j′(ν′)δ2 speciﬁc emissivity κ(ν) = κ′(ν′)/δ absorption coefﬁcient TB = T ′ B δ brightn. temp. (size directly measured) TB = T ′ B δ3 brightn. temp. (size from variability) U′ = (1 + β + β2/3)Γ 2U radiation energy density within an emisphere Dividing by ta = te(1 − β cosθ) we have the measured apparent velocity as βc t sinθ β sinθ