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Appendix A: The Wilkin model

In L10, the apparent shape of the bow shock was modelled following the exact analytical solutions of Wilkin (1996), under certain assumptions. In particular, we had assumed that the column density tends to reach its highest value where the bow shock cone intersects with the plane of the sky including the central star. The Monte Carlo simulations of the 3-D structure described below show that this is not the case, and that for non-zero inclinations of the bow shock the surface brightness peaks at a location away from this plane.

Here we present a 3-D Monte Carlo simulation of the case where an isotropic stellar wind interacts with the ISM of homogeneous velocity V_w relative to the star and with a stratified ISM density along the y-axis of the form ρ = ρ₀ + a   y. This more complicated case than Wilkin (1996), where a = 0, can also be described analytically (Wilkin 2000, and Canto et al. 2005; hereafter CRG). Here, we also assume thay a = 0. The coordinate system is defined in Fig. A.1.

The Monte Carlo simulation starts with drawing the angle θ, 0 < θ < θ_max (θ_max = 165° adopted) from a probability density function (A.1)where σ is the mass surface density (Eq. (12) in Wilkin 1996). The azimuthal angle φ is a random value between 0 and 2π. R(θ,φ) can be solved from a third-order equation (Eq. (28) in CRG) for a given ϵ = aR₀/ρ₀, where R₀ is the so-called standoff distance. For a = 0, R is a function of θ only. The velocities in the z and x-direction are given by Eqs. (17), (18), (33), (34), (35) in CRG.

The position and velocities in the cylindrical coordinate system are then transformed to the (x,y,z) system, which is then rotated over specified angles λ, PA, and i to the observers frame. The outline of points can than be compared to the observed location of the bow shock, in order to infer the standoff distance, PA, and inclination (when a = 0 there is no dependence on the angle λ).

The results of the calculations are summarised in Table A.1, and an example of the fit to the observed trace is illustrated in Fig. A.2. For a fixed inclination, the standoff distance R_o and position angle PA were derived from a fit to the trace of the bow shock in the SPIRE 250 μm filter (L10). The reduced χ² () is reported as a measure of the fit. The reduced χ² is quite large and is related to the systematic deviation between observations and the Wilkin model for larger Z-values. This probably indicates the limitations of the analytical model. We note that every simulated point is assumed to be equally “observable”. What is observed in reality is dust emission in the PSW filter, and the effect of changing the dust density and dust temperature along the bow shock is not considered here. However, such effects are likely the reason why the bow shock can not be traced beyond  ~±500′′. Since the procedure fits the trace of the bow shock, this should have little effect.

Fig. A.1 Definition of the right-handed coordinate system for the thin-shell bow shock model. With reference to the plane of the sky, the positive x-axis points east, the positive y-axis points north, while the positive y-axis points towards the observer. θ is the polar angle from the axis of symmetry, as seen from the star at the origin. The azimuthal angle φ (not shown) is counted from the positive z-axis towards the positive y-axis. The coordinate system may be rotated over the x-axis by an angle λ counted in the same way as φ, over the y-axis by an angle PA (the position angle) counted from the positive x-axis towards the negative z-axis (i.e. south-of-east), and over the z-axis by an angle i (the inclination) counted positive from the positive x-axis towards the negative y-axis. Shown is the Wilkin curve for a standoff distance of R0 = 1. The star is at rest and colliding head-on with a wind moving at a velocity Vw.

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	Fig. A.2 Monte Carlo simulation of a bow shock, for a standoff distance R0 = 499″, 0° inclination, and position angle −0.67°. CW Leo is at (0,0), and the units of the axis are in arcseconds. The red crosses indicate the trace of the bow shock as seen with SPIRE at 250 μm (L10 and this paper).
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Although the smallest χ² are found for large inclination angles, the minimum is very shallow and the inclination angle cannot be derived from the Wilkin fitting (the same conclusion is reached by Cox et al. 2012). The error quoted is the formal fit

error. Monte Carlo simulations were performed allowing for a Gaussian error in the position of the trace of 3′′ (half a SPIRE PSW pixel) along the z-axis. The results show that the errors reported for R_o and PA are realistic, but also that the spread in the reduced χ² is large, approximately 1 unit, indicating again that the inclinations angle cannot be derived from the Wilkin fitting alone.

For each combination of i, R_o, and PA and every point inside the apertures shown in Fig. 1, the true distance to the central star is recorded and are reported in Table 3.

© ESO, 2012