Annu. Rev. Astron. Astrophys. 1999. 37: 409-443
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4. SPECIAL RELATIVITY EFFECTS

4.1. Parameters of the Ejection

The main characteristics of the superluminal ejections can be understood in terms of the simultaneous ejection of a pair of twin condensations moving at velocity beta (beta = v/c), with v being the velocity of the condensations and c the speed of light), with the axis of the flow making an angle theta (0° leq theta leq 90°) with respect to the line of sight of a distant observer (Rees 1966; see Figure 6). The apparent proper motions in the sky of the approaching and receding condensations, µa and µr, are given by:

Equation 1 (1)

Equation 2 (2)

where D is the distance from the observer to the source. These two equations can be transformed to the equivalent pair of equations:

Equation 3 (3)

Equation 4 (4)

Figure 6

Figure 6. Geometry of the two-sided ejection. The emission is symmetric, but when the emitting clouds move at relativistic speeds the approaching component of the pair appears to move faster and to be brighter than the receding component.

If only the proper motions are known, an interesting upper limit for the distance can be obtained from Equations (3) and (4):

Equation 5 (5)

In all equations we use cgs units and the proper motions are in radians s-1. In the case of the bright ejection event of 1994 March 19 for GRS 1915+105, the proper motions measured were µa = 17.6 ± 0.4 mas day-1 and µr = 9.0 ± 0.1 mas day-1. Using Equation (5), we derive an upper limit for the distance, D leq 13.7 kpc, confirming the galactic nature of the source.

The distance to GRS 1915+105 is found to be, from HI absorption studies, 12.5 ± 1.5 kpc (Rodríguez et al 1995, Chaty et al 1996). Then, the proper motions of the approaching and receding condensations measured with the VLA in 1994 and 1995 imply apparent velocities on the plane of the sky of va = 1.25 c and vr = 0.65 c for the approaching and receding components respectively. The ejecta move with a true speed of v = 0.92 c at an angle theta = 70° with respect to the line of sight (Mirabel & Rodríguez 1994). The faster proper motions of 24 mas day-1 measured with MERLIN (Fender et al 1999) and the VLBA (Dhawan et al 1999) in 1997 would imply a true speed of 0.98c at an angle of 66° to the line of sight.

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