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Astron. Astrophys. 345, 949-964 (1999)

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Appendix A: a model for the SiO jet

Here we present a simple model to reproduce the maximum and minimum velocities of the SiO line emission observed along the jet axis. An estimate of the line intensity goes beyond the purpose of the present treatment. The goal is to determine the kinematical and geometrical parameters of the jet by fitting the shape of the position-velocity plot of Fig. 3 bottom.

We assume that the gas traced by the SiO line is ejected inside a cone with aperture angle [FORMULA] (see Fig. A1), up to a maximum distance [FORMULA] from the centre. The gas is moving along straight lines passing through the apex of the cone into two opposite directions, with velocity proportional to the angular distance from the apex, R. The velocity drops to zero at [FORMULA]. The axis of the cone is inclined by an angle [FORMULA] with respect to the plane of the sky. Under this assumption only the radial component of the velocity vector is non-zero and can be written as

[EQUATION]

where [FORMULA] is the maximum velocity reached by the ejected gas.

[FIGURE] Fig. A1. Sketch of the model used to describe the SiO jet. The observer lies at [FORMULA]. The grey area indicates the region filled by the jet. The dotted line corresponds to the axis of the jet, which is assumed to lie in the x-z plane

In the following we define a coordinate system with z axis along the line of sight and x axis coincident with the projection of the jet axis on the plane of the sky. We also assume that the cone representing the jet intersects the plane of the sky, i.e. that [FORMULA]. It is then easy to see that, for a fixed value of x, the maximum and minimum velocities projected along the line of sight are attained on the two lines indicated with (1) and (2) in Fig. A1 and are given by the following expressions:

[EQUATION]

where [FORMULA] is the maximum velocity and [FORMULA] the minimum for [FORMULA], and vice versa for [FORMULA].

In the x-[FORMULA] plane, these equations correspond to two straight lines and an ellipse, as shown in Fig. 13. Our goal is to find a reasonable match between these curves and the lowest contour level of the SiO emission in the position-velocity plane of Fig. 13. The slope of the straight lines and the size and eccentricity of the ellipse depend on the input parameters of the model, namely [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. In order to fix these, one can proceed as follows.

First of all, we know that the outflow intersects the plane of the sky, i.e. [FORMULA]: therefore, [FORMULA] can be directly measured in the map as the maximum distance along the axis of the flow between the centre and the farthest point from the centre where SiO emission is detected. This gives [FORMULA][FORMULA]12" to a very good approximation (see e.g. Fig. 12 bottom).

The other three parameters can be obtained by fixing the values of the minimum and maximum velocities for [FORMULA] (or, equivalently, for [FORMULA]). For [FORMULA], for example, these are reached respectively for [FORMULA] and [FORMULA], and are given by the following expressions:

[EQUATION]

After some algebra, these take the form

[EQUATION]

where we have posed [FORMULA]. We need a third condition to determine [FORMULA], [FORMULA], and [FORMULA]. This is given by the expression relating the aperture angle of the cone, [FORMULA], to the projection of it on the plane of the sky ([FORMULA]), which can be written as

[EQUATION]

The three Eqs. (A6), (A7), and (A8) can be solved to determine [FORMULA], [FORMULA], and [FORMULA]. For this we need an estimate for [FORMULA], [FORMULA], and [FORMULA]. The first two parameters can be measured from Fig. 12 bottom, whereas [FORMULA] can be estimated from the map of the SiO jet (see Fig. 3 right). Plausible values are [FORMULA]=32.5-44 km s-1, [FORMULA]=14-16.5 km s-1, and [FORMULA]=8:O5-22o.

We thus conclude that the three unknowns are constrained in the following ranges: [FORMULA]=2:O8-11:O6, [FORMULA]=8:O5-22o, [FORMULA]=63-206 km s-1. The best fit has been obtained by varying the input parameters in these intervals and comparing the result with the position-velocity SiO map. The fit is shown by the dashed curves in Fig. 13, for [FORMULA]=12" (i.e. 0.099 pc), [FORMULA]=100 km s-1, [FORMULA]=21o, and [FORMULA]=9o.

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999
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