4 Relativistic effects - Doppler shift and boosting

We now consider relativistic Doppler effects due to the motion of the jet volumes toward the observer. The relativistic Doppler factor is $D = (\gamma(1 - \beta \cos\theta))^{-1},$where $\gamma$ is the Lorentz factor, $\beta$the plasma velocity in units of the speed of light and $\theta$ the angle between the trajectory of the volume and the l.o.s. The observed energies $E_{\rm o}$ and luminosities $L_{\rm o}$ of each volume element are shifted and boosted to the rest frame values (index e),

$$% E_{\rm o} = D\,E_{\rm e} \quad {\rm and} \quad L_{\rm o}(E_{\rm o}) = D^3 \, L_{\rm e}(E_{\rm e}).$$ (2)

Note that the Doppler factor depends on both $\gamma$ and $\theta$ (see also Urry & Padovani 1995) and is unity for $\theta = \arccos \left(\sqrt{(\gamma-1)/(\gamma+1)}\right)$. For larger angles, relativistic de-amplification takes place due to the time lapse in the moving frame of reference. This is the reason for D<1 in our jet (Table 1). Also known as second order Doppler effect, this was first observed in SS 433 (Margon 1984). De-boosting is also present in the asymptotic radio jets (different from the collimation region investigated here) of GRS1915+105 inclined by $70\hbox {$^\circ $ }$ to the l.o.s., actually providing a distance indicator (Mirabel & Rodr $\acute{\rm {\i}}$guez 1994).

4.1 Shifted and boosted spectra

Figures 3a,b show the effect of boosting and shifting of the rest frame spectra. For an angle between the l.o.s. and the jet axis of $40\hbox{$^\circ$ }$, the maximum boosting D³ = D_-40³ = 6.7 is obtained for the volume with T = 10^6.64K (see also Table 1). The maximum de-boosting is for the volume at the opposite side of the cone, D₊₄₀³ = 0.15. As in the rest frame, the "hot'' spectra are flatter.

To obtain a total shifted and boosted spectrum we need to interpolate the single volume luminosity values since they are shifted to different energies. Considering the case where the jet axis is along the l.o.s. ( $\theta _{\parallel }$, see Fig. 3c), we have only a weak effect of shifting, in fact, we are looking almost perpendicular to an uncollimated flow. For a larger jet inclination the Doppler effects become larger. In this case, one should take into account the fact that the angle between the velocity and the l.o.s. ( $\theta _{\parallel }$) varies along the jet-torus. However, we have considered it reasonable to divide the jet-tori in two regions, one third containing volume elements for which the Doppler effect has been calculated using the minimum angle between the plasma velocity and the l.o.s., and two thirds containing volume elements for which the Doppler effect has been calculated using the maximum angle between the plasma velocity and the l.o.s.

The total spectra have been calculated by first considering the blue-shifted and red-shifted parts of the flow and then summing up all the luminosities in each energy bin, where blue and red shifted luminosities are available. The result is shown in Fig. 3d with the luminosity rescaled in order to compare the total spectrum with its components.

Note that the iron line features are considerably shifted also after the interpolation. The change in the line shape is due to the fact that for each of the 5000 volumes along the jet a different Doppler factor must be considered. For a larger jet inclination (D_-40 , D₊₄₀) the lines are spread out widely because of the larger Doppler shift (not shown). The de-boosting contribution of the receding counter-jet has not been taken into account.

 

 

Table 1: Dynamical parameters for four example volume elements. Quoted are temperature T, mass M, particle density $\rho$ and the Lorentz factor $\gamma$. The angle $\theta _{\parallel }$ is the angle between the plasma velocity and the l.o.s., if the l.o.s. is parallel to the jet axis. The corresponding Doppler factor is $D_{\parallel }$. If the l.o.s. is inclined $20^{\circ }$ to the jet axis, the minimum (maximum) angle between the plasma velocity and the l.o.s. is $\theta _{\parallel } - 20^{\circ }$ ( $\theta _{\parallel } + 20^{\circ }$) with a corresponding Doppler factor D_-20 (D₊₂₀) and similarly for an inclination of $40^{\circ }$.

T (K)	109	108	107	106.64
M (gr)	$7 \times 10^7$	$2 \times 10^7$	$1.1 \times 10^7$	$0.97 \times 10^7$
$\rho$ (cm-3)	$6 \times 10^{13}$	$2 \times 10^{12}$	$6 \times 10^{11}$	$2 \times 10^{11}$
$\gamma$	1.014	1.179	1.428	1.494
$\theta _{\parallel }$ $(^\circ)$	82	77	72	70
$D_{\parallel }$	1.010	0.960	0.898	0.899
D-20	1.07	1.19	1.25	1.28
D+20	0.96	0.79	0.68	0.67
D-40	1.12	1.47	1.77	1.88
D+40	0.91	0.68	0.55	0.53