## 3. Photometric analysis## 3.1. ObservationsThe 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 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 diameter was used, but in 1994 observations the diameter diaphragm was used. All observations are published in a separate paper (Vaz et al. 1997) where the reduction procedure is described. HD , HD , HD and HD , 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,
## 3.2. Ephemeris and period analysisFrom 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.
With the Lafler and Kinman (1965) period-search method applied to the PDO observations we determine the period = . By applying the method to (all) SAT observations we find = . Applying the method to all (PDO and SAT) observations combined, the result is = . 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: = (primary minima) and = (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 for the primary minima and 5 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
E The most distant reference we have of an orbital phase for the system is the time of maximum radial velocity (at phase , cycle -1469.25, as referred to our new determined ephemeris). If we add this to the analysis above, we again find the period . 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 analysisThe 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 solutionsWe studied the possible values for the orbital inclination,
The intrinsic index for these temperatures is -0.31 (Morton 1969, Popper 1980). There is a well defined empirical relation between and () for O stars with small interstellar reddening (Crawford 1975 using his own observations and measurements by Hiltner & Johnson 1956): . Using this relation and the observed from Table 1 we find , yielding . By using a normal total to selective extinction, (e.g. Seaton 1979), we find (the error is dominated by the uncertainty in and by the dispersion in the versus () relation above). We fixed at 38 000 K and left 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 Ruciski (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 and . As Van Hamme's (1993) tables do not cover the region with 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 and . 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 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
## 3.3.2. Final solutions with the WD modelThe 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
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, 6%) and an orbital inclination higher than those found in the initial solutions. As happened with WINK (Sect. 3.3.1) the solution for the 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 Only solutions with the PDO 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 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 and 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 All sets of solutions of Table 7 do reproduce the observed light
curves quite well, excepting PDO
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