3. Physical parameters
3.1. Epoch and period
The Hipparcos data suggested that the light variation of HV UMa can be described with a single period of (ESA 1997). We observed only one moment of minimum (Hel. JD = 2451346.388), but the consecutive minimum appeared to be slightly fainter, therefore, we adopted a doubled Hipparcos period as a first approach () and shifted the observed time of minimum with to obtain the final epoch Hel. .
The next step was to refine the period. This was done by phasing Hipparcos epoch photometry with the newly determined epoch and the doubled Hipparcos-period. The resulting phase diagram showed a shift of (=). That shift was eliminated by recalculating the period until correct phase diagrams for both our and Hipparcos data (Fig. 3) were reached. The resultant period is . The fact that earlier Hipparcos data agree very well with our data suggests a quite stable period of HV UMa.
The shape of the light curve, i.e. the continuous light variation and the very deep secondary minimum (almost as deep as the primary one), the absence of significant colour variation, the appearance of the secondary line in the spectra at the quadrature phases all suggest that HV UMa is probably an eclipsing contact binary. This is confirmed by its low mass ratio and a consistent model of the light curve (see below).
Comparing the light curve phased with the final epoch and period (Fig. 1) with the line profiles observed at phases of maximum light (Fig. 2) it is visible that the secondary line appears on the blue side at the quadrature phase that follows the deeper minimum, while this bump is redshifted at . This indicates that the smaller companion star approaches us after the primary minimum, therefore that minimum is due to an occultation eclipse. The weak point of this analysis is that the light curve is not very well covered around the minima either by the Hipparcos data or our observations. The data obtained at Sierra Nevada have better inner precision (less scatter) than those provided by Hipparcos, and these data indicate that the minimum at is slightly deeper. Therefore, we adopted this eclipse as primary minimum, but this needs further confirmation. If the deeper minimmum is really due to an occultation eclipse then HV UMa is a so-called W-type contact binary.
Contact binaries can contain early (O-B) or late (G-K) spectral type stars. The latter group is referred to as the W UMa stars, while the former is known as early-type contact systems. The colours of HV UMa indicate early F spectral type, therefore HV UMa is an "intermediate" type contact binary between the W UMa stars and the OB-type contact systems. The surface temperature and the line profile of the HV UMa system makes it similar to the known contact systems UZ Leo and CV Cyg (Vinkó et al. 1996).
3.3. Radial velocities and spectroscopic mass ratio
Since the secondary component is only partly resolved, the radial velocities must be determined by modelling the individual line profiles in order to avoid blending effects (e.g. systematic decrease of the velocity amplitude). For this purpose we chose those spectra that were obtained around the quadratures. These show the presence of the secondary most clearly. One spectrum around light minimum was also modelled to test the applied method.
Because the H profile is strongly affected by the Stark broadening and shows wide non-gaussian wings, we normalized the profiles to the surrounding continuum, and selected the lower part of the profiles below the 0.9 intensity value. We fitted two individual Gaussian profiles to the line cores adjusting the amplitudes, FWHM values and line core positions. The initial values of these parameters were estimated from two spectra very close to the quadratures (=0.25 and 0.77). The FWHM converged very quickly to the final values, being 7.6 Å and 4.0 Å for the primary and secondary components, respectively. Line depths changed slightly from spectrum to spectrum, as the contributions are phase-dependent, resulting in 0.29-0.30 for the primary and 0.07-0.10 for the secondary (note, that these values mean line depths below 0.9 normalized intensity). The fitted line core positions resulted in the radial velocity variations for both components. Sample spectra with the fitted profile are shown in Fig. 4, while the radial velocities are presented in Table 2. The estimated accuracy of the individual velocities is about 5 km s-1 for the primary, and 10 km s-1 for the secondary, which is mainly determined by the resolution of the line core in wavelength. The velocity amplitudes resulting from this method are 471.5 km s-1 and 25410 km s-1, where the uncertainties are due to the random errors caused by the observational scatter. The corresponding mass-ratio is .
Table 2. The observed heliocentric radial velocities obtained by the Gaussian fit. The velocity resolution is about 5 km s-1.
However, as was also pointed out by the referee, this kind of velocity determination may contain a large amount of systematic error, mainly due to the assumed Gaussian shape of the individual line profiles. The intrinsic H profiles of the components of HV UMa are probably quite different from Gaussian, therefore this approach can be considered as only the first approximation for extracting the radial velocities from the H profiles. The major part of the systematic error is governed by the shape of the wing of the primary component's model profile on the side where the secondary star appears (blueward at and redward at phases). It is well visible in Fig. 4 that the position of the secondary line is shifted toward larger velocities with respect to the position of the "hump" on the observed profile, due to the increasing contribution of the primary line toward the main minimum of the combined line. If the primary line was steeper on the side where the secondary line exists, overlapping the secondary by a smaller amount, then the secondary line would be less shifted, thus, its position would be closer to the local hump on the observed profile, resulting in a smaller radial velocity of the secondary. On the other hand, a shallower secondary profile would give us systematically higher velocities due to the same reason.
In order to estimate the amount of this kind of systematic error, we simply determined the positions of the two local minima (the main minimum and the secondary's hump) on the profiles observed around quadratures (four spectra around and two around ) when the presence of the hump appeared to be most prominent. This was done interactively, by eye, plotting the line profiles on the computer screen, which again introduced some subjectivity into the procedure, but it is stressed that this is done only for estimating the errors of the velocities and not for obtaining their actual values. Of course, the velocities of the secondary measured in this way were systematically smaller than those obtained by the Gaussian fitting. The velocities of the primary were almost the same, as could be expected. The total amplitude turned out to be km s-1, while the mass ratio changed to . Comparing these values with the results of the Gaussian fitting, we conclude that the errors of the radial velocity amplitude and the spectroscopic mass ratio (both random and systematic) are approximately km s-1 and , respectively. The finally adopted parameters determined spectroscopically, together with their errors are collected in Table 3. It is important to note that the mass ratio can be refined by modelling the light curve (Sect. 3.4), but the total velocity amplitude is tied only to the spectroscopic data, thus, its uncertainty will directly appear in the absolute parameters of the system.
Table 3. Physical parameters of HV UMa.
3.4. Light curve modelling
The V-light curve was synthesized with the computer code BINSYN described briefly in Vinkó et al. (1996). This code is based on the usual Roche-model characterized by the geometric parameters (photometric mass-ratio), F (fill-out) and i (orbital inclination). The relative depth of the eclipses were modelled introducing the relative temperature excess of the secondary (hot-secondary model). Because the primary minimum turned out to be due to occultation, the phases were shifted by 0.5 assigning to the transit eclipse (built-in default in BINSYN ).
First, the effective temperature of the primary component was estimated based on synthetic colour grids by Kurucz (1993) and the observed mean Strömgren colour indices ( mag, mag, mag), resulting in K and dex (assuming and solar chemical abundance). The interstellar reddening in the direction and distance of HV UMa is expected to be small, because this variable lies far from the galactic plane. TU UMa, an RR Lyr variable lying 17o SE from HV UMa also has a negligible colour excess (Liu & Janes 1989).
Other parameters necessary for modelling the binary star were as follows. A linear limb-darkening law with coefficient was adopted from tables of Al-Naimiy (1977). The gravity darkening exponent and the bolometric albedo were chosen at their usual values for radiative atmospheres: and . All these parameters were kept fixed during the solution for the best light-curve model.
The light-curve fitting was computed using a controlled random search method, the so-called Price algorithm (Barone et al. 1990; Vinkó et al. 1996). The optimized parameters were , F, i and X. The best solution was searched for in the following parameter domains: , , and . The fit quickly converged to low inclination and low mass ratio values that were expected from the shallow eclipses ( mag) and the small spectroscopic mass ratio (). Also, it turned out that there are strong correlations between the optimized parameters. Due to this correlation, the physical parameters determined from the light curve fitting cannot be considered as a unique solution: certain parameter combinations describe the light curve almost equally well. In these cases the combination of photometric and spectroscopic information is very important: one can use the spectroscopic data to determine a consistent set of physical parameters that gives the best model fitting all the available data.
In order to combine the photometric and spectroscopic information and find a consistent description of the system, we modelled the observed H line profiles using the parameters from the light curve fitting. The model line profiles were computed by convolving an intrinsic H profile of a non-rotating star with the Doppler-broadening profile of the contact binary. The broadening profiles were calculated with the WUMA4 code (Rucinski 1973). For the determination of the intrinsic H profile we used Kurucz's ATLAS9 code modified by John Lester. This approach, however, has some limitations, because the H line is strongly affected by NLTE-mechanisms, therefore the ATLAS9-model profile will be somewhat different from the real intrinsic profile, especially near the line core. Thus, only a crude comparison between the modelled and observed line profiles was possible, neglecting the differences in the line core.
However, it turned out that the originally adopted effective temperature K was too high. With this temperature, the Stark-wings of the H line are so strong that they overwhelm the rotational broadening, producing much wider line profiles than observed. Thus, we reduced the value of the effective temperature until satisfactory agreement was found between the observations and the broadened model profiles. The resulting fit is plotted in Fig. 5 together with the observed line at . The residuals of three model profiles are also shown. It can be seen that at K the wings can be fitted quite well, but the computed line core is too shallow. On the other hand, if K is used, the computed line core agrees better, but the wings are wider. Thus, K was accepted as a compromise. The Strömgren-colours of the Kurucz-grid for this temperature are almost the same as for the originally adopted 7300 K, therefore this temperature is still consistent with the photometric colours.
The light curve modelling was recomputed with K. The optimized parameters changed only slightly, resulted in a slightly smaller mass ratio and a slightly higher temperature excess. Table 3 shows the final set of physical parameters, while the fitted light curve is visible in Fig. 6. The distribution of the random points around the -minimum in the parameter space is plotted in Fig. 7. The structure of the sub-spaces in this diagram indicates the correlation between the different parameters.
Another light curve model was computed in order to test the effect of the gravity darkening and reflection parameters that were originally fixed as if the atmosphere was radiative. The model with their "convective" values and resulted in an even larger temperature excess than in the radiative case. Because the temperature excess of the secondary is only a "correction" parameter in the light curve solution and it may not mean real temperature difference, it would be difficult to explain physically a very large value of the temperature excess that still does not cause significant colour variation. Thus, we adopted the model with radiative atmospheric parameters as our final solution, and this model is listed in Table 3. Note that the errors of the fitted parameters (3rd column) are difficult to estimate, because of the parameter correlation. We monitored the behaviour of the function during the optimization and assigned uncertainties to each parameter according to the spread of the random points for which . This criterion defines those solutions when the fitted curve runs well within the error bar at each measured normal point, giving a feasible fit to all observations. The parameter correlation means that the uncertainties are also not independent of each other: e.g. slightly decreasing q forces increasing F or X (see Fig. 7). The uncertainties of the calculated parameters (3rd panel in Table 3) were estimated assuming a km s-1 error in the radial velocity amplitude K.
Because of the correlation between the optimized parameters, it is very important to check whether the radial velocities calculated from the model match the observed velocities. This comparison is plotted in Fig. 8 where an almost perfect agreement can be seen. The low mass ratio results in the distortion of the sinusoidal velocity curves, which is a well-known effect in close binaries.
© European Southern Observatory (ESO) 2000
Online publication: April 10, 2000