3. Analysis of the data
3.1. Period search
The Hipparcos data (set C) represent the longest coverage, and although they are seriously sampled, they offer the best impression of the long-term stability of the mean level of brightness (Fig. 1). It can be seen that the brightness changes are dominated by the short-term variations, occurring on the time scale of several days. The linear fit of the Hipparcos data spanning 915 days, displayed in Fig. 1, sets the upper limit of 0.04 mag to a possible secular trend. However, it is quite probable that the slope of the fitting line is purely due to the sampling. The standard deviation of this fit 0.05 mag is quite large and we will show that it is caused by the orbital modulation. The V-band data of B93b (set A2) which are plotted in Fig. 1, too, appear to be shifted by more than 0.1 mag. This shift can easily be explained by the different wavelengths of the filters used. The passband used in Hipparcos has its maximum roughly between the V and B filter. V 1080 Tau is heavily reddened (; ; Walter et al. 1990) and this reddening causes that V 1080 Tau appears systematically brighter in V and fainter in B in comparison with the passband.
In the first step separate period searches were carried out for the data sets A1, B, C using the PDM (Phase Dispersion Minimization) program, based on the method of Stellingwerf (1978) and written by Dr. J. Horn at the Ondejov Observatory. This program enables not only an automatic search for the best period within a given interval, but also an interactive examination of the resultant data foldings for the respective period lengths. The PDM method evaluates the significance of the period by the parameter which lies in the range 0-1. The lower , the better defined period. The period near 8.8 days was found in all three sets. The best value always laid below 0.4, suggesting a well defined period. The Hipparcos data can therefore be interpreted as the constant long-term level of brightness with variations due purely to the modulation with P near 8.8 days, persisting for the whole interval.
We found that the amplitude of the modulation slightly decreases from the R to V and B passbands. Fortunately, the difference of the amplitude in the R and passbands is small enough to allow us to refine the period length using the combined set of data. We proceeded in the following way. Under the reasonable assumption that the secular trend is absent or very small, it is possible to shift the respective data sets to their common zero point. This shifting is quite justified for sets A1 and C. We also carried out this procedure for set B, because we have magnitudes just in the relative system for the Brno data. All three sets were solved for their common zero point using the code SPEL (author J. Horn). This program solves the parameters of the folded curve. It takes the systematic shifts of the subsets of the input data file into account and evaluates them. The mean brightness of set A1 was used as the basic level and the remaining two sets were allowed to converge to it.
First, the folded data of set B were allowed to converge to set A1 in SPEL. The systematic shift, determined this way, proved to be only marginally sensitive to the exact value of the period length. The refined period was determined again from the new set, combined of A1 and the corrected B. In the next step the Hipparcos data were included to the set A1+B and converged in SPEL again to evaluate their shift. In the last step the sets A1, B and C were solved simultaneously in PDM; the final period length of 8.8451741 days was determined this way. The significance parameter still suggests a plausibly defined period. The final values of the shifts of set B and set C with respect to set A1 were determined to be mag(R) and mag(), respectively. This simultaneous solution for the sets A1, B and C allowed to resolve between several closely spaced aliases which were present in the solution just for A1+B. The resultant folding is displayed in Fig. 2a. The course is approximately sinusoidal with the full amplitude about 0.14 mag. We can readily see that the agreement of the respective data sets, shifted to the same mean level of brightness, is very good. It confirms that the shifting procedure is justified.
3.2. Character of the orbital modulation
The binary nature of V 1080 Tau was convincingly proven from spectroscopy by Martín (1993). This system comprises the main-sequence hot star and evolved cool star. The smooth modulation of brightness (Fig. 2), allows us to infer that the light variations may be caused by the proximity effects in the close binary. Only the reflection effect and ellipticity can play a role since no eclipses are apparent in Fig. 2.
The folded light curve of V 1080 Tau displays a scatter which cannot be entirely attributed to the observational inaccuracies. For example the maxima of the light curve observed by B93a,b display unequal height for the respective cycles. As a result, it gives rise to enhanced scatter at some orbital phases of the folded data although this scatter is intrinsic. We will also show that there are some differences between set A1 and set B which are separated by a long gap. Due to the low amplitude of the variations these cycle-to-cycle changes smear the course of the modulation caused by the geometrical effects. Nevertheless, it is still instructive at least to constrain the role of the proximity effects in V 1080 Tau. The light curve can be still analyzed under some reasonable assumptions. The spectroscopic parameters (A3V primary and the evolved secondary of the type G to early K) put the basic constraints. The secondary, less luminous than the primary, is evolved off the main sequence while the primary is still a main-sequence star. This situation is typical for Algols which underwent mass transfer between the components. It can offer a natural explanation for the overluminosity of the secondary in V 1080 Tau. It also yields an additional constraint on the mass ratio because almost all Algols are observed in the phase after the mass ratio reversal. In order to further lower the number of parameters to be searched for, the fractional radius of the secondary can be set equal to the radius of its Roche lobe, as typical for Algols.
We generated a set of synthetic orbital light curves for the above-mentioned parameters using the code Binary Maker 2.0 (Bradstreet 1993). This program, which uses Roche geometry, enables one to calculate theoretical light curves and their comparison with the observations. It is reasonable to suppose that the modulation in V 1080 Tau is due to the large tidally distorted secondary because the main-sequence primary lies deep inside its lobe; its departure from a sphere is therefore very small. We therefore modeled the R-band data (set A1 and set B) first because the contribution of the cool secondary is larger here. We attempted to match the single-wave modulation ( days) with the synthetic light curve first. It became quite clear that no match, fulfilling the constraints given above, is possible. The single-wave modulation implies that only the reflection effect is prominent, it means that the luminosity of the primary is much higher than that of the secondary. This would be in contradiction with the spectroscopic evidence for the lines of the secondary in the red region, confirming that the luminosities of both components cannot be largely divergent here.
Further we examined the possibility that the modulation is double-wave with the real orbital period days (Fig. 2b) and obtained much better agreement (Fig. 3). The system parameters are summarized in Table 2. The three models brake the interval of the possible temperatures of both components and also the remaining parameters in model 1 and model 3 were chosen so that they act to minimize the amplitude of the modulation in the former and maximize it in the latter case . The values of q brake the typical range seen in Algols. The radii of the stars in Table 2 are fractional, i.e. the distance a between the centers of the two stars is defined to be unity and the radii of the components of the binary are expressed as fractions of a. The fractional radii of the primary correspond to the main sequence or a slightly evolved star. We only matched i of each set because it just scales the amplitude of the modulation in a simple way. The limb-darkening coefficients were taken from the table of Al-Naimiy (1978). Set A1 and set B are resolved by the different symbols in Fig. 3a. The set A1 suffers from a larger scatter than set B, mainly at phases 0.25 and 0.5. This effect may be due to the intrinsic activity in the binary (see below). The light curves with the geometrical parameters and temperatures from Table 2 were also generated for the B-band (set A3) and are plotted in Fig. 3b. The course of the modulation is plausibly reproduced for model 2 (Table 2), despite an increased scatter of the observations at phase 0.25. The standard deviations of the residuals for the respective models in the R-filter are 0.06, 0.04 and 0.04 mag(R) while for the B band we obtain 0.04, 0.03 and 0.05 mag(B).
Table 2. Parameters used for generation of the respective synthetic light curves of V 1080 Tau. and refer to the effective temperature of the primary and the secondary while and denote the fractional radii of the components. The mass ratio and the inclination angle are abbreviated as q and i, respectively. See Sect. 3.2 for details.
The relation of the brightness variations in the red spectral region, in which the cool secondary is prominent, and in the blue region, where the hot primary dominates, can be emphasized by the color index (Fig. 4). The curve, folded according to Eq. (1), displays a double-wave course with the minimum at phase 0.5 deeper than that at phase 0.0. The synthetic color indices, corresponding to models 1, 2, 3, are superposed. It can readily be seen that while model 1 and 2 give the same sense of the color variations as the observed data (apart from the phases 0.4-0.6, see below), model 3 is ruled out because it yields an opposite sense. In conjunction with the residuals of the respective models, mentioned above, model 2 best represents the observations of V 1080 Tau .
We note that the U-band data (set A4) display a very large scatter and the course of the modulation largely differs from the R and B bands. Notice mainly that while we obtain a clearly double-wave modulation for the R and B filters, the maximum at phase 0.75 is just barely visible in U. An intrinsic activity may at least partly account for these effects (see below).
Because the double-wave light curve (Figs. 2b and 3) does not enable to resolve unambiguously the superior and the inferior conjunction of the primary, an additional constraint is needed. Three measurements of the radial velocities (RVs) of the cool secondary (88, 94 and -64 km s-1) are available (Martín 1993). They set the lower limit of 160 km s-1 to the full amplitude of the RV variations. As we will show below the real amplitude is not expected to be much larger than this and we can therefore state that the systemic velocity of V 1080 Tau is not very different from 0 km s-1. It means that the velocities 88 and 94 km s-1 must occur in the part of the orbital cycle when the secondary is receding from the observer. Calculation of the orbital phases for these RVs enabled to adjust the proper moment of the superior conjunction of the primary (Eq. (1)). This ephemeris is used in Figs. 2b - 4.
© European Southern Observatory (ESO) 2000
Online publication: August 17, 2000