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4 Orbital solution

The most extensive and most accurate compilation of radial velocities (RVs) of HDE 228766 in the literature concerns the He II $\lambda $ 4686 emission line. We measured the RVs of this line on our spectra by fitting a Gaussian. Adopting the orbital period of 10.7424 d as derived by MC77, we used our He II $\lambda $ 4686 RVs to determine a new orbital solution for this line (see Rauw et al. 2002). Apart from a small phase-shift, this new solution is in excellent agreement with the results of Hart (1957) and MC77. This finding prompted us to take advantage of the entire RV dataset of He II $\lambda $ 4686 available in the literature to derive a better estimate of the orbital period of HDE 228766. We applied the generalized spectrogram technique of Heck et al. (1985; see also Gosset et al. 2001) and the trial method of Lafler & Kinman (1965) to this timeseries and both methods consistently yield 10.7426 d as the best estimate for $P_{\rm orb}$. Assuming that the uncertainty on this result amounts to one tenth of the natural width of the peak in the periodogram, we estimate an uncertainty of $6\times10^{-4}$ d. Therefore, our new orbital period confirms the previous determinations. In the following, all the orbital solutions are computed adopting our new value of 10.7426 d.

A close inspection of our data reveals that most of the absorption lines of the binary components remain blended over the main part of the orbital cycle (Fig. 2). The only absorption feature that seems reasonably well suited to establish an orbital solution for both stars is the He II $\lambda $ 4542 absorption line. The He I $\lambda $ 4471 and H$\gamma$ absorptions are seldom completely deblended and suffer most probably from a contamination by a variable wind emission.

To determine the radial velocities of the He II $\lambda $ 4542 absorption lines, we have built a template of the line in the spectrum of each star by fitting two Gaussians to the blend around maximum separation. At the other orbital phases, we used these templates within a cross-correlation like technique to deblend the lines of the two components. We define phase 0.0 as the phase of conjunction with the secondary star being in front of the primary and we adopt the 10.7426 d orbital period derived above.

   
Table 2: Journal of observations of HDE 228766 in the blue-violet range obtained with the Carelec and Aurélie spectrograph at OHP. The orbital phases are computed with respect to HJD 2451822.052 with a period of 10.7426 d (see Table 3). The last column yields the equivalent width of the He II $\lambda $ 4686 emission line integrated from 4665 to 4705 Å.
HJD-2 440 000 $\phi$ He II $\lambda $ 4542 III $\lambda $ 4634 III $\lambda $ 4641 He II $\lambda $ 4686
    RV1 (km s-1) RV2 (km s-1) RV (km s-1) RV (km s-1) RV (km s-1) EW (Å)
9917.412 .702 164.2 -299.4 -224.6 -222.0 -190.2 -5.33
11373.492 .245 -91.3 96.7 221.3 196.6 283.8 -5.61
11374.379 .327 -68.2 24.0 178.6 168.6 254.7 -5.72
11375.415 .424 -48.7 -9.3 96.1 67.3 159.7 -5.49
11376.425 .518 20.3 -161.4 -19.9 -47.1 32.5 -5.22
11377.428 .611 119.0 -250.7 -153.0 -171.2 -107.5 -5.60
11378.417 .703 168.3 -293.9 -224.0 -244.5 -187.2 -5.74
11379.494 .803 154.7 -314.1 -220.2 -236.7 -172.6 -5.57
11396.488 .385 -87.6 34.3 142.3 125.5 214.3 -5.49
11397.543 .484 27.5 -124.5 9.9 -14.9 80.6 -4.99
11398.473 .570 116.4 -207.1 -107.2 -120.4 -50.2 -5.66
11399.486 .664 165.6 -253.7 -207.7 -218.1 -159.3 -5.36
11401.537 .855 135.2 -264.2 -185.8 -207.2 -141.1 -5.53
11402.465 .942 49.3 -214.9 -98.6 -118.9 -36.5 -5.34
11403.524 .040 -0.6 -96.5 39.3 15.2 119.0 -5.36
11404.518 .133 -57.0 51.7 151.4 124.8 236.3 -5.43
11405.514 .226 -67.2 94.4 207.7 189.3 298.0 -5.52
11406.500 .317 -64.2 54.5 208.5 190.8 266.5 -5.61
11407.503 .411 -97.5 31.1 106.0 88.3 197.9 -5.24
11408.503 .504 47.4 -150.7 -6.5 -30.5 54.6 -5.21
11409.517 .598 123.0 -236.8 -138.5 -163.3 -101.8 -5.88
11810.440 .919 99.7 -231.7 -122.5 -123.3 -71.4 -5.88
11811.408 .009 -12.6 -132.8 2.5 -3.4 96.1 -4.36
11812.427 .104 -52.5 5.6 122.8 120.8 191.0 -5.81
11813.484 .202 -79.2 77.9 200.9 202.5 281.1 -5.78
11814.469 .294 -66.2 64.5 210.3 210.0 279.2 -5.92
11815.473 .388 -86.2 31.3 142.1 137.8 216.9 -5.77
11819.453 .758 163.8 -306.2 -234.4 -233.3 -202.2 -5.99
11820.460 .852 176.8 -273.4 -194.9 -180.0 -143.2 -5.71
11820.487 .854 143.7 -293.3 -186.9 -181.3 -144.4 -5.43
11821.461 .945 18.2 -201.0 -89.4 -87.7 -20.5 -5.66

Allowing for an eccentric orbital solution yields $e = 0.06 \pm 0.02$ for the RVs derived from the He II $\lambda $ 4542 absorption lines. However, the radial velocity curves of the He II $\lambda $ 4686 and N III  $\lambda \lambda $ 4634, 4641 emission lines all indicate an eccentricity below 0.03 with correspondingly large errors ($\sim$0.01, $1~\sigma$). Therefore, although we cannot completely rule out a very small eccentricity, in the following we assume e = 0.0.


  \begin{figure}
\par\includegraphics[width=7.3cm,clip]{MS2665fig3.ps}
\end{figure} Figure 3: Radial velocity curve of the HDE 228766 binary system as derived from the He II $\lambda $ 4542 absorption line (see Table 3). Open circles and filled triangles stand for the primary and secondary RVs respectively.


 

 
Table 3: New orbital solutions for HDE 228766. The RVs of the emission lines vary almost perfectly in phase with the RV of the secondary's He II $\lambda $ 4542 absorption line. T0 refers to the time of conjunction with the primary being behind. $R_{\rm RL}$ stands for the radius of a sphere with a volume equal to that of the Roche lobe computed according to the formula of Eggleton (1983).
  He II $\lambda $ 4542 III $\lambda $ 4634 III $\lambda $ 4641 He II $\lambda $ 4686
  Primary Secondary      
P (days) 10.7426 (fixed) (fixed)
e 0.0 (adopted) 0.0 (adopted)
T0 (HJD-2 450 000) $1822.052 \pm 0.305$ $1822.081 \pm 0.044$ $1822.034 \pm 0.100$ $1822.040 \pm 0.072$
$\gamma$ (km s-1) $31.4 \pm 7.3$ $-119.0 \pm 8.2$ $-10.1 \pm 0.7$ $-22.4 \pm 1.5$ $52.0 \pm 1.2$
K (km s-1) $133.7 \pm 7.6$ $200.2 \pm 10.9$ $224.9 \pm 1.0$ $222.1 \pm 2.1$ $244.5 \pm 1.7$
$a\sin i$ ($R_{\odot}$) $28.4 \pm 1.6$ $42.5 \pm 2.3$ $47.7 \pm 0.2$ $47.1 \pm 0.4$ $51.9 \pm 0.4$
q = m1/m2 $1.50 \pm 0.11$      
$m\sin^3 i$ ($M_{\odot}$) $24.8 \pm 1.8$ $16.6 \pm 2.4$      
$R_{\rm RL}~\sin{i}$ ($R_{\odot}$) $29.3 \pm 1.5$ $24.4 \pm 1.2$      


Our He II $\lambda $ 4542 radial velocity measurements yield a significantly larger mass ratio (q = 1.50) than the value (q = 1.04) obtained by MC77. This larger mass ratio results from a slightly lower K1 (134 vs. 150 km s-1) and a significantly larger K2 (200 vs. 156 km s-1). We find a huge difference of 150 km s-1 between the apparent systemic velocities of the two components of the system with the secondary line being blue-shifted with respect to the primary. Our non-LTE analysis of the spectrum of HDE 228766 (Sect. 6.1 below) confirms that this shift of the apparent $\gamma$-velocities can be ascribed to wind contamination of the secondary's helium lines. This blue-shift of the secondary's absorption was already discovered by Massey & Conti (1977) but these authors reported a difference of "only'' 45 km s-1. MC77 used the mean RVs of several absorption lines to derive their orbital solution. Due to the severe blending problems, these authors cautioned that their RV measurements of the absorption lines on their photographic plates were at least partly subjective. Our RVs obtained through a cross-correlation technique are in principle less subjective, but refer to the sole He II $\lambda $ 4542 line thus rendering a direct comparison of our $\gamma$ velocities with the ones of MC77 difficult.

  \begin{figure}
\par\includegraphics[width=7.3cm,clip]{MS2665fig4.ps}
\end{figure} Figure 4: Radial velocity curves of the N III $\lambda $ 4634 (open squares), N III $\lambda $ 4641 (stars) and He II $\lambda $ 4686 (black dots) emission lines in the spectrum of HDE 228766. The He II $\lambda $ 4542 orbital solutions of the primary (dashed line) and secondary (dotted line) are shown for comparison.

We have measured the radial velocities of the N III  $\lambda \lambda $ 4634, 4641 and He II $\lambda $ 4686 emission lines (Table 2). The lines appear to follow closely the orbital motion of the secondary star without any significant phase lag (Table 3). The semi-amplitude of our N III RV-curves overlap within 3 km s-1 with previous results, whereas the semi-amplitude of our He II $\lambda $ 4686 RV-solution (244 km s-1) is intermediate between the values of 218 and 260 km s-1 obtained respectively by MC77 and Hart (1957). It is worth mentioning that the absorption components of the N V  $\lambda \lambda $ 4604, 4620 lines as well as the N V $\lambda $ 4604 P-Cygni emission also follow the motion of the secondary star with a semi-amplitude of 223-227 km s-1 (see also Table 4 below).

A tricky question that occurs frequently for binary systems harboring Wolf-Rayet or extreme Of stars is which of the spectral lines reflects the actual orbital motion of the WR or Of star. In the case of HDE 228766, it seems likely that the absorption lines of the secondary are formed in the stellar wind rather than in the photosphere. The $\gamma$ velocity of the He II $\lambda $ 4542 absorption is blue-shifted by more than 100 km s-1 with respect to the RV-curve of the N III emission lines. This blue-shift suggests that the absorption forms in the wind, where the outflow has already reached larger velocities than in the formation area of the narrow emission lines. This situation is reminiscent of the case of the WN7ha + O binary WR 22 for which the H9 absorption appears blue-shifted by 80 km s-1 with respect to the N IV $\lambda $ 4058 emission (Rauw et al. 1996). For WR 22, it was found that the narrow N IV emission provides the best indicator of the orbital motion of the WN7ha star. It seems likely that the same holds for the N  III, N IV and N V emission lines of the secondary in HDE 228766. Indeed, the various narrow emission lines yield the same K and their $\gamma$ velocities are in reasonable agreement with that of the primary's He II $\lambda $ 4542 absorption line.

The line formation region of the He II $\lambda $ 4686 emission is probably more extended and hence more affected by the presence of the primary and/or by a wind interaction phenomenon than that of the nitrogen lines.

Therefore, we suggest that the true $K_2 \simeq K_{\rm N~III} \simeq 225$ km s-1. Adopting this value yields a mass ratio of q = 1.67, a projected separation of $(a_1 + a_2)~\sin{i} = 76~R_{\odot}$ and minimum masses of $m_1~\sin^3{i} = 31.7$ and $m_2~\sin^3{i} = 25.5~M_{\odot}$ for the primary and secondary respectively. The rather large minimum masses suggest that the inclination i could be large and HDE 228766 might display photometric variability. Unfortunately, it seems that there has been no photometric investigation of this system so far and no light-curve is available that could allow to constrain i.


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