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

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5. The eccentricity, periastron date, and systemic velocity

The eccentricity is the only orbital element that can be determined without difficulty. The five emission lines without absorption blends listed in Table 1 yield consistently [FORMULA]. The average eccentricity resulting from the absorption-line solutions is [FORMULA]. Thus, the value from the absorption lines is consistent with that from the emissions. However, we note that the absorptions yield a factor of 10 higher uncertainty. Therefore, we have adopted the mean eccentricity of the emission lines for the analysis the absorption lines (Table 2 and 3).

From the emission lines we calculate a mean periastron angle of [FORMULA]. The RMS error is [FORMULA] but the distribution of the values is definitely not random but rather bimodal. The individual periastron angles are probably influenced by systematic effects and therefore, we adopt an error that is large enough to include all results except that of the HeII   [FORMULA] line: [FORMULA]. The absorption line solutions yield a large scatter for the periastron angle (cf. Table 2). If we adopt [FORMULA] we obtain RMS deviations that are only marginally larger than by using [FORMULA] as a free parameter. Hence, there is no reason to allow for periastron angle of the O star other than opposite to the WR star. We adopt [FORMULA] in order to calculate the elements of the O star (Table 3).

The mean periastron date of both the emission and the absorption lines, is [FORMULA] (Table 1 & 3). The RMS error is 1.6 d but again, as it was the case for the periastron angle, the distribution is obviously bimodal. We adopt the mean and set an error that includes all values from the well defined solutions. In the next section we discuss the differences between individual solutions in more detail. The ephemeris for the time of periastron passage is thus

[EQUATION]

In this paper all phases are calculated with Eq.  1.

For some applications it is preferable to give phases relative to the conjunction of the stars. The WR star is in front, i.e. mid eclipse if [FORMULA], 48 days after periastron passage, i.e. at phase [FORMULA]

[EQUATION]

The ephemeris given by Eqs.  1and 2differ from those of Moffat et al. (1986) in that their periastron date is 1.3 d late and their conjunction date is 1.9 d early. The periastron date of Niemela & Sahade (1980) is 2.9 d later then the date corresponding to Eq. 1.

In principle, we should be able to calculate the systemic velocity with the same accuracy as the amplitudes. However, it is obvious (Table 3) that this is not the case. We believe that the explanation for the large scatter in the [FORMULA] -velocities is partially due to incorrect normalization of the spectra and partially, due to wrong reference wavelengths. To understand the former we remind the reader that we measure wavelength shifts with absorption troughs that are very broad. The O star rotates with [FORMULA] [FORMULA] (Baade et al. 1990), thus the lines have a width of five times the velocity amplitude with a flat line center. If the underlying continuum has a wrong slope then it is indeed possible to obtain shifts of the order of a few [FORMULA]. The latter explanation is due to line blends. Two of the three lines in Table 3 that have well defined solutions are blends. Since we do not know the ratio of the line strengths of its components we cannot calculate the effective wavelengths of the blends. In order to derive the systemic velocity we simply calculate the mean and adopt RMS deviation of the five lines for the precision: [FORMULA] [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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