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Subsections

5 Bow-shock models

We have calculated spectral line profiles expected from three bow-shock models:

1) bow shocks generated by a high-velocity jet impacting on the ambient medium according to the model of Hartigan et al. (1987) with the modification made by Dutrey et al. (1997);

2) bow shocks surrounding UC HII regions generated by the movement of the central star through the ambient medium according to the analytical model of Wilkin (1996); and

3) turbulent wakes behind bow shocks in protostellar jets according to the model of Raga & Cabrit (1993).

It is assumed that the bow-shock is unresolved by the telescope beam. The calculations performed here are qualitative and merely used to test whether any of these models could be utilized in a more detailed analysis of the SiO profiles. The lines are assumed to be optically thin. In models 1) and 2) the shocked gas is furthermore assumed to be isothermal.

In Fig. 12 spectra from an approximate analysis are shown. In all the models the basic form of the spectrum depends strongly on the viewing angle $\phi$, which is defined as the angle between the symmetry axis of the bow (going through its apex) and the line of sight (see Fig. 1 in Hartigan et al. 1987). $\phi=0$ means the situation where the apex is pointing towards the observer. The velocity is expressed in terms of the shock ($v_{\rm s}$), central star ($v_\star$) or jet velocity ($v_{\rm j}$) in models 1, 2 and 3, respectively. The parameter $\sigma_{\rm v}$ is the chaotic velocity dispersion in the mentioned units.

  
\begin{figure*}
\ \ \ \ \ \ \ 
\psfig {file=fig12.eps,width=18cm,bbllx=50pt,bblly=50pt,bburx=550pt,bbury=800pt,angle=270}\end{figure*} Figure 12: Simulated spectra from a bow-shock according to the models (1), (2) and (3) (see text) with different viewing angles. Inclination $0^\circ$ means the case where the shock is seen "head on'', and $90^\circ$means the "edge on'' case. The angular size of the object is assumed to be smaller than the telescope beam size. The velocity has been expressed in units of the shock velocity ($v_{\rm s}$, model 1), stellar velocity ($v_\star$,model 2) or jet velocity ($v_{\rm j}$, model 3). $\sigma_{\rm v}$ is the turbulent velocity dispersion. The meaning of the other parameters can be found in the referred papers and Appendix A

Provided that the density and temperature distributions are not sharply peaked towards the apex, the emission line profile from a bow-shock is dominated by the `tail' since the apex occupies a relatively small volume. In all models discussed here high velocities occur close to the apex whereas the velocity in the tail approaches the ambient velocity. Therefore the velocity of the peak intensity is always close to the ambient cloud velocity, whereas the "wings'' are seen at high velocities. The adopted models are described in some detail in Appendix A.

5.1 Comparison with the observed profiles

In a comparison between the obtained SiO profiles and the predictions of the kinematical bow shock models the following observational facts are noteworthy:

1) the peak intensities appear always close to the ambient cloud velocity;

2) the full widths above two sigma of the SiO lines are between 2 and 60 km s-1, except for Ori KL with a FW of about 100 km s-1;

3) the main isotope SiO lines never exhibit double peaked profiles.

The fact that the SiO lines always peak close to the ambient cloud velocity implies that the emission does not come from plane parallel shocks or from shocked cloudlets exposed to stellar wind or jets. Both in a planar shock and in a shocked cloudlet the intensity maxima occur at the projected shock velocity (see Schilke et al. 1997), and probably we sometimes would see such a shock "face-on'' (i.e. at a small viewing angle), causing the SiO emission peak to be clearly shifted with respect to the ambient gas. Therefore models where a jet or the exciting star itself, when penetrating through the cloud, accelerates a small portion of gas to high velocities, are more plausible. In terms of the Hartigan-Dutrey model (1) this observation means that the ambient gas is not considerably pre-accelerated i.e. the parameter $\gamma$ is small.

In models 1 and 2 the full width above zero intensity (FWZI) of the line is roughly equal to the jet or stellar velocity, irrespective of the viewing angle $\phi$. The absolute line widths in model 2 should be markedly smaller than in model 1. The velocity of a star with respect to the ambient cloud is of the order of 10 km s-1 (Churchwell 1991), whereas jet velocities can be as high as 100-300 km s-1. The distribution of full widths above two sigma (Fig. 7) clearly indicates that a major part of the SiO emission regions are associated with high velocity outflows driven by jets. It is likely that in some cases the observed SiO profile is a result of the superposition of several jet working surfaces or UC HII regions. The cases where both blue- and redshifted linewings are present may represent bipolar distribution, which together with successive jet acceleration (Dutrey et al. 1997) provides an explanation for very large velocity dispersions.

The large variation in the full widths and absence of thermal double peaked profiles suggest that bow-shocks (models 1 and 2) do not provide a universal model for the SiO emission. Otherwise one would expect that the full widhts are always 10 km s-1 or more, and that at least in some cases we see the shock edge-on causing the observed profile to be double peaked.

In contrast, the triangular line shapes and changing line widths predicted by the turbulent wake model of Raga & Cabrit (1993) fit the observed profiles rather well. However, the observed distribution of the asymmetry parameter P (Fig. 8) is not what one would expect by assuming an isotropic orientation of wakes. In the model of Raga & Cabrit the asymmetry parameter P as a function of the viewing angle $\phi$ is simply $-\cos{\phi}$. In reality the extreme velocities would not be detected due to noise in the spectra, and the cosine function would be "flattened'' i.e. |P| would never reach the value 1. Nevertheless, in a histogram like the one presented in Fig.  8 the largest number of sources should occur in the range where $P(\phi)$ changes least i.e. close to the extrema, and the pattern should be double peaked which apparently is not the case. Similarly, the normalized peak position (NPP) for the H2O maser sample of Wouterloot et al. (1995, see their Fig. 3) never shows a two-peaked distribution. This was in contradiction with the earlier results of Palagi et al. (1993), and lead Wouterloot et al. to the conclusion that NPP is not useful to characterize the emission. The fact that maser emission is known to often arise from separate spots makes the difficulty of its characterization understandable. Presumably the situation with thermal SiO emission is less complex. However, if the emission does not come from a single region the asymmetry parameter fails to describe the orientation of the shock. For example if the source is bi-polar but unresolved $P(\phi)$ is always 0.

The determination of the expected full width distribution in the model of Raga & Cabrit is difficult since the jet velocity $v_{\rm j}$ probably changes from source to source. The full width is approximately $\Delta v_{\rm turb}
+ v_{\rm j} \, \vert\cos{\phi}\vert$ and, on the basis of similar arguments as in the case of P, values of the full width which are close to $v_{\rm j}$ should occur more frequently. One would expect the FW distribution to be peaked towards larger values since the jet velocities are likely to be at least a few times 10 km s-1.

Taken the relatively large number of narrow SiO lines it is doubtful if one can reproduce both P- and FW-distributions at the same time with the aid of Raga & Cabrit model, even by combining mono- and bipolar sources. Therefore it seems likely that at least in some sources SiO emission arises from the quiescent gas component, due to processes discussed in the next section.


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