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Astron. Astrophys. 327, 1094-1106 (1997)

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3. Photometric analysis

3.1. Observations

The photometric observations were obtained at two different sites: (1) from 1989 to 1991 at PDO (LNA-CNPq, Brasópolis, Brazil), with the ZEISS 60 cm telescope and a single-channel photometer equipped with a photon-counting system and using a diaphragm of [FORMULA] diameter (Cunha et al. 1990); and (2) from 1990 to 1994 at ESO (La Silla, Chile), with the Strömgren Automatic Telescope (SAT) equipped with the six-channel spectrograph-photometer and photon-counting system described by Nielsen et al. (1987). In the measurements taken from 1990 to 1993 with SAT, a circular diaphragm of [FORMULA] diameter was used, but in 1994 observations the [FORMULA] diameter diaphragm was used. All observations are published in a separate paper (Vaz et al. 1997) where the reduction procedure is described.

HD [FORMULA], HD [FORMULA], HD [FORMULA] and HD [FORMULA], all relatively close to V 3903 Sgr (Table 1), were used as comparison stars in all observing runs and observed alternately between the measurements of the variable. All four stars were found to be constant within the observational accuracy throughout the observing periods in both sites.

The PDO light curves, u (478 points), v (532), b (544) and y (537), are shown in Vaz et al. (1997). Typical rms errors of one magnitude difference between the four comparison stars were: [FORMULA], for [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. The SAT y light curve, the [FORMULA] and [FORMULA] colour indices curves (687 observations in each colour) are shown in Fig. 3 and the typical rms errors of one magnitude difference between the comparison stars were: [FORMULA], and [FORMULA]. Note that in Fig. 3 all observations made with SAT are shown, and the effect of changing the size of the diaphragm is surprisingly small, as the region is very close to bright nebulae (Lagoon Nebula and IC 4685) which could contaminate the observations with spurious light. This contamination is perhaps noticed in the observations made in 1990, mainly in the shoulders of the primary eclipse (JD-2 440 000=)8019, covering phases [FORMULA] to [FORMULA], and 8029, covering phases [FORMULA] to [FORMULA]). However, night 8016 ([FORMULA] to [FORMULA]) and the primary eclipse part of 8029 do match the 1994 observations very well (in Vaz et al. 1997).

[FIGURE] Fig. 3. y, [FORMULA] and [FORMULA] magnitude differences V 3903 Sgr-HD 165999 obtained at ESO and the theoretical light curves (Table 10, [FORMULA]: observations made from 1990 to 1993 with a diaphragm of [FORMULA], [FORMULA]: observations made in 1994 with a diaphragm
of [FORMULA]).

3.2. Ephemeris and period analysis

From the present observations, we determined the times of minima given in Table 5, by applying the method of Kwee & van Woerden (KvW, 1956) to all four colours and controlling the results with second and third degrees polynomials. The mean of the four measures was adopted, with an uncertainty derived from their internal rms dispersion.


Table 5. Times of minima for V 3903 Sgr.

With the Lafler and Kinman (1965) period-search method applied to the PDO observations we determine the period [FORMULA] = [FORMULA] [FORMULA] [FORMULA]. By applying the method to (all) SAT observations we find [FORMULA] = [FORMULA] [FORMULA] [FORMULA]. Applying the method to all (PDO and SAT) observations combined, the result is [FORMULA] = [FORMULA] [FORMULA] [FORMULA]. By using the least-squares method and linear ephemeris to minimize the O-C of the prediction of times of minima, given in Table 5, we find the periods: [FORMULA] = [FORMULA] [FORMULA] [FORMULA] (primary minima) and [FORMULA] = [FORMULA] [FORMULA] [FORMULA] (secondary minima), which are in good agreement with each other and with our period search determinations. Introducing a quadratic term in the procedure does not change the numbers significantly and the second order coefficients turn out to be very small (5 [FORMULA] [FORMULA] for the primary minima and 5 [FORMULA] [FORMULA] for the secondary ones) and barely larger than their formal errors. Therefore we adopt the new ephemeris:

Min I at: HJD 2 447 754.4713 + 1 [FORMULA]  E

The most distant reference we have of an orbital phase for the system is the time of maximum radial velocity (at phase [FORMULA], cycle -1469.25, as referred to our new determined ephemeris). If we add this to the analysis above, we again find the period [FORMULA]. However, the rms of the O-C increases by a factor of 3. The two periods are essentially equivalent, considering their errors, and we will use the one given in the ephemeris above.

3.3. Photometric analysis

The light curves were solved initially with the WINK (Wood 1971, Vaz 1986) model and the final solutions were found with the more realistic WD (Wilson 1979, 1993) model. Both models were improved with the modifications described by Vaz et al. (1995). The system turned out to be detached, with both components still well inside their Roche Lobes. Models using simpler approximations for the geometric figure of the components, like EBOP (Popper & Etzell 1981), are not adequate for the analysis of this system, which presents moderate proximity effects (Fig. 3).

Only PDO light curves were used to find initial solutions, because they were completed before the ESO ones.

3.3.1. Starting values and initial solutions

We studied the possible values for the orbital inclination, i, so that the system would be detached, but close to the contact configuration (a starting hypothesis, Cunha 1990), concluding that i should be close to [FORMULA]. For normal O7V and O9V stars (Niemela & Morrison 1988) Popper's (1980) work indicates that the ratio of radii should be close to 0.8, which we used as our initial value for [FORMULA]. The photometric standard indices (Table 1) give a mean system temperature close to 35 000 K (Lester et al. 1986). The unreddened combined [ [FORMULA] ] index is consistent with an O8V spectral type for the system, while [ [FORMULA] ] disagrees with any calibration. The system is far too hot for the use of the calibrations by Napiwotzki et al. (1993) and Schönberner & Harmanec (1995). We then took the temperature starting values to be 38 000 K and 33 000 K as adequate for the spectral types O7V and O9V, respectively, according to the works of Conti (1973) and Lamers (1981).

The intrinsic [FORMULA] index for these temperatures is -0.31 (Morton 1969, Popper 1980). There is a well defined empirical relation between [FORMULA] and ([FORMULA]) for O stars with small interstellar reddening (Crawford 1975 using his own observations and measurements by Hiltner & Johnson 1956): [FORMULA]. Using this relation and the observed [FORMULA] from Table 1 we find [FORMULA], yielding [FORMULA]. By using a normal total to selective extinction, [FORMULA] (e.g. Seaton 1979), we find [FORMULA] (the error is dominated by the uncertainty in [FORMULA] and by the dispersion in the [FORMULA] versus ([FORMULA]) relation above).

We fixed [FORMULA] at 38 000 K and left [FORMULA] to be adjusted by the least-squares method. The reflection albedos were fixed at 1.0 and the gravity darkening exponents were chosen to follow von Zeipel's (1924) work, as appropriate for atmospheres in radiative and hydrostatic equilibrium.

The limb-darkening coefficients were initially taken from Wade and Ruci[FORMULA] ski (1985) and then from the tables of Van Hamme (1993), both for Kurucz (1979) stellar model atmospheres and calculated by bi-linear interpolation at the current values of [FORMULA] and [FORMULA]. As Van Hamme's (1993) tables do not cover the region with [FORMULA] for temperatures above 35 000 K, we extrapolated the values of limb-darkening in this region, always taking into account the general pattern of variation of the limb-darkening coefficient for the listed values of [FORMULA] and [FORMULA]. For these hot temperatures, this procedure should be sufficiently precise (Van Hamme 1995).

Even though V 3903 Sgr is in a rich field close to bright nebulae, no other star could be detected inside the diaphragm with an image intensifier and no third light was assumed in the initial solutions. At first we assumed synchronous rotation for both components. Their rotation velocities (Sect. 3.2) were then used to calculate the rotation rates relative to the orbital movement, on which the sizes and deformations of both components depend.

Starting with these initial values and the y light curve which presents the least dispersion, we applied the WINK model to a normal curve (39 points strategically distributed along the orbital phases), obtained from the observations with a spline interpolation curve. As soon as a physically plausible solution was achieved for the y colour, we added the bvu normal light curves to the analysis.

Both components were found to be well inside their Roche lobes, and the preliminary solutions given in Table 6 were found. One can see that the solutions agree well in all four colours. The limb-darkening coefficients for the u colour, however, was increased by 0.1 for both components, in order to get a better agreement of the orbital inclination for this colour with those found for the other colours. This tendency has already been noted in other works on hot (B) stars, such as Giménez et al. (1986), Vaz et al. (1995), and references cited in these works.


Table 6. Initial solution for V 3903 Sgr, obtained with the WINK model and PDO observations.

3.3.2. Final solutions with the WD model

The initial solutions indicate that V 3903 Sgr is moderately distorted, and that WINK probably generates a good representation for the components. However, due to its more accurate geometric approximation for the figure of the components, we decided to apply the WD model, first to the normal curves and then to all SAT observations, to find the final solutions. PDO observations were also analysed with the WD model, but not used in the final solution, as explained below. The code of the WD model was modified as described by Vaz et al. (1995), where they discuss in some detail the usefulness of the improvements.

Starting from the WINK solutions (Table 6) and using a set of UNIX scripts and small FORTRAN programs (Vaz et al. 1995), developed to make sure that all of the model parameters were integrally consistent both with the observed quantities and with themselves internally, the detached configuration was confirmed by the WD model, applied to SAT observations. No contact or semi-detached configuration could reproduce the observed light curves so well as the detached configuration. The solutions were performed on different sets, shown in Table 7: (1) applying WD simultaneously to all 4 colours (uvby), (2) to the vby, (3) and (4) only to the u colour (solutions 1 to 4 used only SAT observations). Solutions (5, uvby) and (6, vby) were done with PDO observations, and correspond to solutions (1) and (2), done with SAT observations, respectively. Solution (7) solves simultaneously all SAT uvby light curves and the 2 radial velocity curves. The model input parameters [FORMULA] are [FORMULA] steradian luminosities ([FORMULA] is automatically calculated from the input temperatures and radii), while [FORMULA] is in units of the apparent light from the eclipsing stars integrated in the observer's direction.


Table 7. Final solution for V 3903 Sgr, obtained with the WD model applied to all SAT (solutions 1, 2, 3, 4 and 7) and to PDO observations (solutions 5 and 6). Parameters marked (*) were kept consistent during iterations. The errors quoted are the least-squares formal errors.

The WD model was used with the atmosphere tables of Kurucz (1979), clearly a better approximation than the normal possibilities offered by the WD model: Carbon-Gingerich model atmospheres and the blackbody radiation approximation (see Vaz et al. 1995). Even though WINK uses the same set of atmosphere model tables, the better geometric approximation of the WD model gives an effective temperature of the secondary systematically higher (by more than 2 000 K, [FORMULA] 6%) and an orbital inclination [FORMULA] [FORMULA] higher than those found in the initial solutions.

As happened with WINK (Sect. 3.3.1) the solution for the u colour (3) does not agree with the solution for the vby colours (2), with the main differences in the sizes of the stars, but also in the orbital inclination. These differences can be diminished by increasing the limb-darkening coefficient for the u colour, interpolated from Van Hamme's (1993) tables, by 0.2 (Giménez et al. 1986, Vaz et al. 1995), as shown in solution (4). However, unlike the solution for LZ Cen (B1III, Vaz et al. 1995) and EM Car (O8V, Andersen & Clausen 1989), the discrepancy could not be completely removed with this procedure. Separate solutions were done for the individual vby (SAT) colours, which fully agree with solution (2).

We are using the two sets of observations we have in their own instrumental system so it is natural to expect some differences between solutions produced with SAT or PDO data, due to differences in the equipment, photomultipliers, filters, diaphragms and sites, even though the comparison stars were the same in both data sets. One consequence of these differences is that the light curve u from PDO was the only one which required the use of "third light", of around 1.4% of the eclipsing components' contribution at quadrature (according to WINK definition, implemented in WD as an option), in order to better reproduce the geometric elements for solutions of the vby colours. When this parameter, [FORMULA], was adjusted by the least-squares method for the vby light curves, it remained zero or became slightly negative, which is not physically reasonable. The diaphragm used in PDO observations was larger than both those used in SAT runs (Sect. 3.1, Vaz et al. 1997). Therefore, if there was any necessity of considering extra light in the solutions, the PDO solutions should use larger values than SAT ones, due to sky background contamination. However, it seems that the third light for u is an artifact created by the combination of instruments and site at PDO. By analysing the u filter transmission (both sites) it becomes clear that PDO u filter is approximately 15% broader than that used in SAT photometer. As V 3903 Sgr is of a much earlier spectral type than the comparison stars used (especially [FORMULA], Table 1), this difference in the bandwidth alone could be the reason for this "artificial" third light in the u light curve.

Only solutions with the PDO u colour showed differences with respect to solutions with SAT data, as can be seen in Table 7, solutions (5) and (6) corresponding to the simultaneous solutions (1, uvby) and (2, vby), respectively. The vby light curve solutions are essentially the same for both data sets, considering the formal errors of the parameters. The inclusion of the u colour in solution (5) really produced another configuration (i, [FORMULA], [FORMULA] and [FORMULA]), presenting a worse fit with systematic deviations, reflected by [FORMULA] (mag) in Table 7.

There are many possible reasons for this, ranging from the difference in the site (PDO is only at 1860 m above sea level and in a much more humid region than SAT at ESO) to the different filters and photomultipliers used. As discussed in Vaz at al. (1997), we believe that this problem is mostly due to the transmission curve of the PDO u filter, which does not match that for the SAT instrumental system. On the other hand, the u light curve solutions are almost always problematic, often not agreeing with the solutions for longer wavelengths. Supported by the fact that there is a good agreement between solutions for SAT data and for the vby light curves from PDO, and that there is a large number of high-quality solutions based on SAT observations (the Copenhagen Group project), we use only SAT data in our final solution and in the discussion below.

Both components seem to be rotating synchronously with the orbit (Table 7) and are inside their Roche lobes, with the fill-out factors (Mochnacki 1984) being [FORMULA] and [FORMULA] and the system being detached. Since there is no evidence of mass exchange (no period changes), the components should be representative for evolutionary studies of single stars.

Solution (7) of Table 7 was done with the "spectroscopic" parameters a (semi-major axis), [FORMULA] (center of mass velocity) and q (mass ratio) free to be adjusted by the least-squares method together with the "photometric" parameters i (orbital inclination), [FORMULA], [FORMULA] (gravitational potentials), [FORMULA] (temperature of the secondary), and [FORMULA] (model luminosity of the primary). It agrees very well with solutions (1), (2), (4) and (6), considering that the formal errors quoted are a lower limit for the uncertainty in the parameters. Solution (7) is also in excellent agreement with the spectroscopic solutions of Table 4 (Sect. 2).

All sets of solutions of Table 7 do reproduce the observed light curves quite well, excepting PDO u light curve, and the rms scatter of the observations from the solutions are comparable to the typical rms errors of the observations (Sect. 3.2). The largest uncertainty is for the values for the luminosity ratios, but the values in Table 7 agree with the spectroscopic determination of Sect 2.2 from the equivalent widths of He I lines. We adopt solution (7), performed for all the 4 colours simultaneously with the radial-velocity curves for both components (Table 3), as our final solution and Table 8 gives the final mean elements for V 3903 Sgr. The O-C residuals for the four colours (SAT observations) and the final solution are shown in Fig. 4, where no systematic trends can be noticed. The O-C between the final solution (adjusted only in the model normalization parameters, magnitude at quadrature and central phase of the primary minimum) for PDO observations show larger and systematic deviations for the u colour, evident in Fig. 1 of Vaz et al. (1997).

[FIGURE] Fig. 4. Residuals of the SAT uvby observations from the theoretical light curves (last column of Table 7 and Table 8). The dotted lines delimit the start and end of both eclipses.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998