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
( =
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
(0°
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:
 |
(1) |
 |
(2) |
where D is the distance from the observer to the source. These
two equations can be transformed to the equivalent pair of equations:
 |
(3) |
 |
(4) |
 |
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):
 |
(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
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
= 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.
