Astron. Astrophys. 354, 537-550 (2000)

3. Preparation of the input data sets

3.1. Phasing the data

The photometric and spectroscopic variability of V711 Tau can be attributed entirely to the primary. The first step in the data preparation process is thus to phase the various data with the stellar rotation period of the primary. Because the rotational and the orbital motions are practically synchronized, we may use the more precise orbital period to link the various data. Following previous investigators, we adopt Fekel's (1983) original spectroscopic ephemeris,

where the zero point is a time of superior conjunction (primary in front) and the period is the orbital period.

3.2. Spectroscopic data preparation

Subtracting the secondary spectrum from the composite spectrum is mandatory in the case of V711 Tau because, firstly, the non-negligible amount of continuum dilution due to the secondary and, secondly, the blending of the primary lines with the secondary lines near conjunction.

We make use of a computer program developed by Huenemoerder & Barden (1984) that combines two template spectra and compares it to an observed (composite) spectrum by searching through a three-dimensional parameter space to find the best estimates for , radial-velocity shift, and relative intensity weight for each of the two components. In this way, we find an average ratio of the relative intensity weights of 2.1680.028 for V711 Tau, i.e. a relative continuum level of 0.3155 for the secondary and 0.6845 for the primary. This value was obtained from nine good-S/N spectra at phases where the two components were completely separated; the quoted error is the standard deviation. This ratio is consistent with the result of 2.1 used by Donati et al. (1992) and with the 2.30.4 originally given by Fekel (1983). We note that the primary's actual contribution to the joint continuum is phase dependent due to the star being a light variable with an amplitude of in V (in 1996). The effective relative intensity weight of the primary then changes by 0.02 during one rotational cycle, i.e. from 0.664 to 0.704 from light minimum to light maximum. This will affect the primary's line equivalent widths - but not the line-profile shape - and is actually used as an additional constraint by our Doppler-imaging code and must not be removed from the data.

For the extraction of the primary's line profiles, we distinguish two cases depending on whether or not the two components were well separated. The well-separated cases by far outnumber the blended cases, and an example is illustrated in Fig. 2. In this case, we use a spectrum of HR 4523 (G5V ) as the template for the secondary, and a spectrum of  Gem (K0III ) as the (dummy) template for the primary star.

Fig. 2. An example of the secondary-star removal at phase 0:p237. The dotted line is the observed (composite) spectrum and the dash-dotted line shows the contribution of the secondary star as a result of a two-component fit to the composite spectrum as described in the text. Note that the relative continuum level of the secondary (=0.3155) was shifted to unity for better visualization. The spectrum of the primary (full line) is then obtained by removing the secondary and re-normalizing it to the continuum level of the primary (i.e. dividing it by 0.6845).

When the two stellar components are close to conjuction the line wings of the secondary start to blend with the wings of the same lines from the primary. Already in this case,  Gem would not be a sufficiently accurate template star for the primary due to the spot induced spectral-line distortions in the V711 Tau profiles. For these phases, we apply the method described by Vogt et al. (1999) in which a template spectrum is constructed out of a properly deblended primary spectrum outside conjunction and closest in phase and time to the observed spectrum in question. The two completely blended spectra around 0:p08 of the times of the two conjunctions were dropped from the analysis due to inevitable uncertainties that occur during the extraction process, e.g. due to spectral-type mismatches of the secondary star.

3.3. Photometric data preparation

Our photoelectric observations through a 30" diaphragm not only includes the variable primary component but also both non-variable companions. Note that the tertiary component is 6" away from the (unresolvable) spectroscopic pair and is not included in the projected slit of the spectroscopic observations.

The additional light from the two stars considerably dilutes the amplitude of the photometric variations of the primary. We remove this dilution effect by applying a simple scaling equation to the observed differential magnitudes:

where the indices obs, pri, sec, ter, and cp, denote the observed magnitude, the value for the primary, the secondary, the tertiary, and the comparison star, respectively. The secondary and tertiary components were found to have the following magnitudes: = (Henry & Hall 1992) and (interpolated for a G5V -star and transformed to the Cousins system from Bessel's (1979) equations); (Eggen 1966) and (as above). The appropriate values for our comparison star, HD 22484, were given in Sect. 2.2.

Additionally, the primary's light curve is folded with the periodic variations from a small ellipticity effect due to the fact that the primary is nearly filling its Roche lobe (Fekel 1983). We correct for this by removing a term ( being the phase) with a full amplitude of in V according to the recipe of Vogt et al. (1999). The corresponding amplitude of is obtained in this paper by applying a similar scaling law as in Eq. (2) assuming an average color index of 0.800.016 (rms). Note that the seasonal average index of the primary star after removal of the secondary and tertiary contributions as well as the ellipticity effect is 0.860.02. Fig. 3 shows the entire 1996/97 photometric data and the successive stages of the data preparation as just described.

Fig. 3a and b. Differential VI-photometry for the 1996/97 observing season and the fit with a time-dependent spot model; a V-band data, b Cousins I-band data. The respective upper light curves (dots) are the original light curves prior to any preparations, while the respective lower light curves (dots) are the light curves after removal of the dilution and the ellipticity effect. The various line styles indicate the spot-model fits with the individual effects removed. Note that the removal of the ellipticity effect has a relatively small impact on the light curves.

Also plotted in Fig. 3 is a spot-model fit from a least-squares solution of the seasonal V and light curves. Time-series photometric spot modeling was described, e.g., by Olàh et al. (1997) and helps to identify the spots' longitudes and areas and, more important in our case, their variation with time. The two-spot model obtained suggests that almost all variations of the light-curve shape are due to the smaller of the two spots at a longitude of 50^o (phase 0:p14). Its angular radius continuously decreased from 14^o at the begin of the observations until it vanished at around JD 2,450,437. It re-appeared in the solutions just two rotations later with a radius of 10^o and then increased up to a radius of 20^o thereafter. During the same time interval the larger of the two spots remained more or less constant with a radius of 30^o at a longitude of 160^o (0:p45) and a latitude of roughly +60^o.

3.4. Radial velocities and orbit determination

The spectrum-synthesis analysis in Sect. 3.2 also yields the relative radial velocities for the two stellar components. Their internal precision is unfortunately no better than 1-2 km s^-1 due to the small wavelength coverage of our spectra. Nightly zero-point corrections were applied from the measurements of the radial-velocity standard  Ari whenever possible. Table 1 lists the radial velocities for both stellar components.

Assuming zero orbital eccentricity (Fekel 1983, Donati et al. 1992), we redetermine an orbit from the data in Table 1. We keep the orbital period fixed to the value originally determined by Fekel (1983) and obtain the phase shift of the superior conjunction, , with respect to Fekel's value. The orbital parameters are listed in Table 2 together with previous determinations and with Donati's (1999) solution for nearly the same epoch as the epoch in this paper. Our radial velocities from the NSO McMath-Pierce stellar spectrograph sometimes show arbitrary shifts of up to 5 km s^-1, most likely due to mechanical motion of the spectrograph components. Such shifts were also noted in the 1988/89 NSO data of Donati et al. (1992) attributed to the same effect and, consequently, our orbital elements are of large external uncertainty. The O-C's in Table 1 are nevertheless comparable to Fekel's original orbit from data taken between 1975 and 1981. Despite the velocity shifts, the velocity differences between the two stellar components should not be affected; see Hynes & Maxted (1998) for a general discussion of the velocity errors from spectrum disentangling. Also note that the large external uncertainties have no effect on the Doppler-imaging process because we subtract the secondary based on its observed radial-velocity difference rather than on the computed absolute velocity. The good agreement between our and that from Donati (1999) seems to support the time variation of the orbital phase that he found. Applegate (1992), and most recently Lanza & Rodonó (1999), explained such variations as a quadrupole-moment variation of the primary due to a periodic exchange between kinetic and magnetic energy driven by the magnetic activity cycle.

Table 2. A comparison of orbital solutions for V711 Tau. and denote the velocity amplitudes of the primary and the secondary, respectively. is the systemic velocity, and is the orbital phase of the superior conjunction (primary in front).

© European Southern Observatory (ESO) 2000

Online publication: February 9, 2000
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