4 Refining the binary orbit

$\delta$ Sco has been observed with the speckle interferometric technique at several observatories for a long time (1973-1991). In total $\sim$30 observations were obtained. Most of the data are collected in McAlister & Hartkopf (1988), later observations were published by Bedding (1993), McAlister et al. (1990), and Fu et al. (1997). The data analysis has led to two different orbital solutions presented by Bedding (1993) and Hartkopf et al. (1996). As one can see in Fig. 5, the data show a large scatter, suggesting that the orbital solution needs to be tested by other observational techniques. One of the options is spectroscopy, which gives information about the components' RVs.

We were unable to measure RVs of the photospheric lines close to periastron (except for those in the ESO spectrum). However, the circumstellar matter (where the emission lines originate) moves with the star and should reflect the star's motion. Indeed, the orbital motion of a Be star was detected using RVs of the H$\alpha$emission line in such binaries as $\phi$ Per (Bozic et al. 1995) and $\gamma$ Cas (Harmanec et al. 2000). Thus we think it reasonable to use our RV measurements to constrain the orbital solution for $\delta$ Sco. The strongest line in the spectrum, and the one that gives the most reliable RVs, is H$\alpha$. We estimate the mean accuracy of the individual H$\alpha$ RVs to be about 1-2 kms^-1.

We suggest that this line is formed in the disk around the primary component, and its mean RV coincides with that of the star. This suggestion seems to be fairly justified, since the line profile has almost equal peak strengths ( $V/R \simeq 1$). The latter indicates an almost axisymmetric matter distribution in the disk. Moreover, most of the nearly symmetric absorption lines in the ESO spectrum that allow reliable RV measurements (such as Si III 4552, He I 4143, absoprtion wings of H$\beta$, and some others) gave values within $\pm$2 kms^-1 of that from the H$\alpha$ line.

In order to compare the measured RVs with predictions from the orbital solutions, one needs to know the systemic velocity ( $v_{\rm sys}$). The RV measurements for $\delta$ Sco published so far span a range from -26 to +6 kms^-1. Since the orbital eccentricity is large, the RVs obtained within approximately a year around a periastron passage are affected by the components' acceleration and cannot be adopted as a systemic velocity. Analysis of the literature data show that the most accurate value, which is probably not affected by the periastron acceleration, is -7 kms^-1 (Evans 1967). This value turned out to be in good agreement with our independent estimate from the orbital solution fitting described below as well as with that derived from several absorption lines in the 1998 SAAO spectrum (-5 kms^-1).

According to our data, the H$\alpha$ RV minimum took place between 2000 September 7 and 12 (see Table 3). This is about 5 months later than the Bedding (1993) prediction for the periastron passage and about 1.5 months later than the Hartkopf et al. (1996) prediction. Furthermore, both previous orbital solutions predicted a noticeably longer RV minimum (see Fig. 6).

Figure 4: The photometric and spectroscopic variations of $\delta$ Sco during 2000. a) The light curve from Otero et al. (2001). The visual observations are shown by open circles, and the photoelectric data by filled circles. The variation of the H$\alpha$ EWs, RVs (from Table 3), and relative flux are shown in panels b), c), and d), respectively. The Skinakas data are shown by open squares, those obtained by French amateurs by open upward triangles, the BOL data by open downward triangles, the CAO data by open circles, the Ritter data by filled circles, and the ESO data by filled squares. Uncertainties of the RVs are comparable with the symbol size, while that of the intensities is shown in the upper right-hand corner of panel d).

Our observations turned out to provide constraints for most of the orbital parameters. The time coverage constrains the periastron epoch (T₀) within several days. The symmetry of the RV curve suggests that the periastron longitude ($\omega$) is close to 0, while its depth depends on the components' mass ratio (q), the eccentricity (e), and the orbital inclination angle (i). At the same time, e, i, and the node line longitude ($\Omega$) affect the orbit spatial orientation, while $\Omega$ has no impact on the RV curve. The eccentricity also controls the width of the RV curve around periastron. The mass ratio ($q \sim$ 1.5-2.0 depending on the secondary's spectral type) was estimated by Bedding (1993) using the brightness ratio mentioned in Sect. 1. The central depression depth of the H$\alpha$ profiles indicates that the disk inclination angle is intermediate ( ${\sim} 30^{\circ}$-50$^{\circ}$). However, we should note here that line profile shapes do not always give unambiguous information about the disk inclination angle (e.g., Quirrenbach et al. 1997). We assume that the disk plane coincides with the orbital plane, and the primary's rotational axis is perpendicular to this plane. At least, we do not have any evidence that the disk is tilted.

Having in hand our RV data and the speckle interferometry information, we calculated the RV curve and the orbit projection in the plane of the sky for different sets of the orbital parameters. The best solution is listed in the last line of Table 1. It is based on a study of the 6 parameters (e, i, $\omega$, T₀, q, and $v_{\rm sys}$) space, which also provided us with the uncertainty estimates. This study showed that i and q are not independent in the vicinity of the best fit. Their relationship can be expressed as $i=(15.8\pm0.3)\,q + (10.6\pm0.5)$ and gives $i=42^{\circ}$ for q=2.0 and $i=34^{\circ}$ for q=1.5. The best value of $v_{\rm sys}$ is $-6\pm0.5$ kms^-1. It coincides with the mean RV we measured using the 1998 SAAO spectrum taking into account a +1 kms^-1correction, which follows from our orbital solution. There are two orbital parameters we cannot estimate from our data: the orbital period and the semi-major axis. We adopt them from the Hartkopf et al. (1996) solution, since its predicted RV curve is closer to the observed one than that of the Bedding (1993) solution.

Thus, our spectroscopic data improve the orbital solution for the $\delta$ Sco system. The solution is now consistent with both the RV and interferometric data.