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Astron. Astrophys. 328, 219-228 (1997)

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7. The inclination

St-Louis et al. (1987) determined [FORMULA] for the inclination of the [FORMULA]  Vel system. Their value is based on polarimetric observations but their analysis was based on an adapted eccentricity of [FORMULA]. Since this value differs from our result, we have re-analyzed their polarization data using [FORMULA]. In Fig. 7 we compare calculated Stokes parameters with the observed Q and U values. As St-Louis et al. (1987) we adopt the model of Brown et al. (1982) that assumes a point light source at the place of the O star and a scattering region localized at the WR star's position. We fit 7 free parameters by minimizing [FORMULA], where [FORMULA] and the corresponding expression for [FORMULA]. The best fit solution yields [FORMULA], [FORMULA], [FORMULA] ([FORMULA]), [FORMULA] ([FORMULA]), [FORMULA], [FORMULA] %, and [FORMULA] % polarization. The epoch of the periastron passage, [FORMULA], is in excellent agreement with the value derived by the radial velocity analysis. The fit is very sensitive to this parameter and we estimate an internal precision of about one day. The periastron angle is [FORMULA], which differs from the value [FORMULA] derived from the radial velocities. However, the fit to the polarization data is not sensitive to that parameter and therefore, the difference is not significant.

[FIGURE] Fig. 7. Comparison of the observed Stokes parameters Q and U (St-Louis et al. 1987) with solutions calculated with a model assuming a localized scattering region associated with the WR star. The full drawn line marks the overall best solution with an inclination [FORMULA] and the dashed curve represents the solution with [FORMULA].

We find that the changed eccentricity does not affect the resulting inclination. A comparison of Fig. 7 with Fig. 7 of St-Louis et al. (1987) reveals that our solution looks almost identical to their fit. The only apparent differences are the smaller amplitudes of our curves. The amplitude is parameterized by [FORMULA], and consequently our value is considerably smaller than [FORMULA] determined by St-Louis et al. (1987). The main reason that we obtain a different solution is that we do not include other polarization observations that are of inferior quality compared to the data of St-Louis et al. (1987).

The amplitude of the variation of the polarization depends on the number and distribution of the scatterers, i.e. the free electrons in the wind of the WR star. With the theory of St-Louis et al. (1988), that bases on the scattering model of Brown et al. (1978), it is possible to derive a mass loss rate. From our polarization fit we obtain a semi-major axis in the Q-U plane of [FORMULA] % polarization. This value yields [FORMULA] [FORMULA] (Eq. 6 of St-Louis et al. 1988). The interesting aspect of this number is that it is derived with a method that is proportional to the density. In contrast, the mass loss rate based on the interpretation of the radio flux depends on the density squared. Schaerer et al. (1997) calculate [FORMULA] [FORMULA] with the HIPPARCOS distance, the mm and radio flux of Leitherer & Robert (1991) and Hogg (1985), and the terminal velocity of Eenens & Williams (1994). The ratio between the two mass loss determinations yields an estimate of the clumping in the WR wind. The numbers above imply a clumping factor of the order of 4, which agrees well with the factor 3 derived by Moffat & Robert (1994) from interpretations of line profile variations.

If it were only for the values given above it would be hardly worthwhile to report them. However, there is an important difference between our results and those of St-Louis et al. (1987). While they give a range [FORMULA] we find that the published polarization data do not confine the inclination. In Fig. 8 we display the RMS-deviation of the best fits for given inclination. For all possible inclinations our solutions have a smaller RMS deviation than [FORMULA] % polarization reported by St-Louis et al. (1987) for their best fit.

[FIGURE] Fig. 8. Resulting RMS deviations of least squares fits to the observed Stokes parameters Q and U (see Fig. 7) as a function of the inclination.

Although we have improved the added squared deviations by 40% compared to the solution of St-Louis et al. (1987), our fit is still far beyond what we would expect for randomly distributed data points with the instrumental uncertainty. St-Louis et al. (1987) claim [FORMULA] % polarization which seems to be correct judging from the accuracy they have obtained in the cases of measurements of other WR stars. With 82 measurements and 7 free parameters we expect [FORMULA] %. Thus, there is an effect of unknown origin that influences the observations. St-Louis et al. (1987) suspect a relation to the non-radial pulsations of the O star that are reported by Baade et al. (1990). Nevertheless, if we assume these deviations to be of random nature with [FORMULA] % polarization, which is the RMS deviation of our best fit, then we can also calculate a confidence range for the RMS deviation, [FORMULA] % polarization. For the most unlikely inclination, [FORMULA], we calculate an RMS deviation of 0.039 % polarization. Thus, the fit with this inclination differs only by 2 [FORMULA] from the best solution and has still a probability of 15% to be correct. This is demonstrated in Fig. 7 where we have also drawn the solution with [FORMULA]. For an inspection by eye, both curves fit the data equally well. Obviously, the polarization observations are not accurate enough for a determination of the system's inclination.

A restriction of the system's inclination is possible from other considerations. Moffat (1977) did not find an eclipse in the continuum light curve. Therefore, we can set an upper limit to the inclination. With the radii 2 [FORMULA] [FORMULA] and [FORMULA] [FORMULA] (Schaerer et al. 1997) and the separation of the two stars, [FORMULA] km (Table 4), we find [FORMULA].


Table 4. Physical parameters of [FORMULA]  Velorum

Van der Hucht et al. (1997) and Schaerer et al. (1997) used the distance to [FORMULA]  Vel measured by HIPPARCOS, the period, and the observed angular separation of the binary components (Hanbury Brown et al. 1970) to derive the total mass of the system. They find [FORMULA] [FORMULA]. The inclination resulting from the comparison with [FORMULA] derived here (Table 4) is [FORMULA]. Because of the large uncertainty of the total mass, this inclination is not very precise. However, we obtain a lower limit [FORMULA].

From a spectral analysis of the O star Schaerer et al. (1997) derived its luminosity and then, by using single star evolutionary models, they obtain [FORMULA] [FORMULA]. The combination with our value for [FORMULA] (Table 4) yields [FORMULA]. The quoted error is an internal precision and does not include an uncertainty for the evolutionary tracks.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998