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Astron. Astrophys. 362, 673-682 (2000)

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3. Simultaneous photometric and spectroscopic least-squares solution

In agreement with the line-ratio discussion in Sect. 4.1 and the discussion of nomenclature in Kallrath & Milone (1999, Sect. 2.7), we refer to the primary component labelled by index 1 as the photometric primary (the star eclipsed at primary minimum). The photometric primary corresponds to the spectroscopic secondary star (the star with the weaker spectral lines). The spectroscopic primary is eclipsed at secondary minimum.

To solve the light curves, we used the software package WD98, a successor of WD95 (Kallrath et al. 1998) based on the 1993 version of the WD code, and a further improved version, called WD98, based on the 1998 version of the WD code. The least-squares problem of parameter estimation is treated in the usual way described in Kallrath & Milone (1999, Chapter 4).

We used all observations described in Sect. 2, i.e. , 18 individual data points in the radial velocity curve, 387, 342, and 320 in U , B , and V with the Levenberg-Marquardt scheme. The individual weights [FORMULA] are computed as


with the following meaning:

  • [FORMULA] is a flux-dependent weight chosen proportional to the phase-dependent inverse flux [FORMULA] of the binary system [for details, see Kallrath & Milone (1999, p.103], as appropriate for photon statistics [FORMULA] in the nomenclature of Linnell & Proctor 1970); and

  • [FORMULA] is a curve-dependent weight accounting for the standard deviation of the data points in that curve.

The factors [FORMULA] and [FORMULA] are described in detail in Wilson (1979) and Kallrath & Milone (1999). Calculations were performed either in mode 2 (cf. Wilson 1988), i.e. , [FORMULA] is coupled to [FORMULA] through the Planck function, while no geometrical constraints are imposed, or in mode 5 (which forces the secondary component to fill its Roche lobe). In WD98 all configurations possible in the Roche model (detached, semi-detached, contact, over-contact) may be realized within mode 2. That means, if one or both stars overfill their Roche lobe, the corresponding lobe-filling constraint is automatically applied. We fixed the temperature of the primary at [FORMULA] K estimated from its spectral type B5V (Popper 1980).

The strong correlation of the limb-darkening coefficients with other light-curve parameters and its negative influence on the numerical properties of the light curve is a well-known problem (see, e.g., Wilson & Devinney 1971, or Twigg & Rafert 1980). Only under exceptional circumstances (i.e. total eclipses) will it be possible to extract meaningful limb-darkening coefficients. In view of these difficulties it is perhaps the wisest to trust the coefficients derived from the best available model atmospheres. Therefore, we adopted the square root and, alternatively, in some additional test runs the logarithmic limb-darkening coefficients given by Van Hamme (1993a), which are based on the model-atmosphere grid of Kurucz (1979). For BF Aur as an early type system the square-root coefficients, [FORMULA] and [FORMULA], seem to be more appropriate and we used


Van Hamme's (1993a) interpolation software was used to derive these values. If the temperature of the secondary component changed in the course of the iterations, the limb-darkening coefficients were adopted accordingly.

The albedos were fixed at the values appropriate for radiative envelopes, [FORMULA], corresponding to full reradiation. Gravity darkening exponents [FORMULA] were chosen, corresponding to von Zeipel's law.

Since the new observations are of high quality and are consistent in time, a simultaneous analysis of both radial-velocity curves and the U, B and V curves seemed warranted. However, at first we analyzed the radial-velocity curves separately which gave a mass ratio q almost identical to unity. Fig. 2a shows the radial velocity curves and the fits when proximity effects were excluded from the model, i.e., only the semi-major axis, a, the system velocity, [FORMULA], and the mass ratio, q, were free parameters. Fig. 2b shows the fit when proximity effects are included. The results are quantified in Table 2.

[FIGURE] Fig. 2. Radial velocity curves (solid line is star 1, dashed line is star 2) and fits obtained when proximity effects are excluded (top panel) and included (bottom panel). The fit is improved significantly when proximity effects are included. The standard deviation of the fits decreased from 11.693 to 10.008 km s-1 as shown in Table 2. We estimate the internal precision of a single measurement to [FORMULA] km s-1 at quadrature and to [FORMULA] km s-1 near conjunction based on a comparison with other data taken during the same nights.


Table 2. Parameters derived from the radial velocities. The fitted parameters are the semi-major axis, a, the systemic velocity, [FORMULA] and the mass ratio, q.

The fit including proximity effects (turned on by the control flags ICORR1 and ICORR2 in the WD program set to 1) requires that the shapes of the stars and the surface flux distribution are known. This was achieved by adopting some of the parameters obtained in the simultaneous analysis. Note that the fit and its standard deviation is better and that the standard deviations of the estimated parameters are smaller. Therefore, we continued to model the radial velocities with the proximity effects. Note that the uncertainty in the mass ratio [FORMULA] prevents us from deciding whether q is smaller or larger than 1. A definite decision requires much more accurate radial velocity data than our 5 km s-1 rms. Unfortunately, as has already been shown by KK the photometric data (in the case of BF Aur) put no further constraints on q as long as [FORMULA]. This fact is consistent with the experience that the mass ratio q, although it may in principle be obtained for close binary systems from light curve synthesis, is often a quite weakly determined parameter, for near-contact binaries (see, e.g., the discussions in Breinhorst et al. 1989or Kaluzny & Semeniuk 1984) and indeterminate for well-detached ones. A semi-detached solution constraint may often give a quite definitive photometric mass ratio, being based more on the size of the lobe filling star than its distortion; however, can we be sure that the system is really semi-detached?

Since q is so weakly defined, we used a grid approach (see Kallrath & Milone 1999, Appendix B, or KK) to trace the quality of the fit as a function of q. Table 3 gives a sequence of solutions obtained with the Levenberg-Marquardt scheme for fixed mass ratio. The solutions in this table have been produced by following a homotopy track, i.e., the solution for [FORMULA] served as the initial guess for [FORMULA] and so on. The table shows that q is probably very close to unity; and it is very difficult to decide whether we have mass reversal or not. Note that all solutions refer to detached solution as is indicated by [FORMULA] and [FORMULA]. A similar run has been performed enforcing that the secondary component is lobe filling (mode 5 in Wilson's program). The results were slightly worse as shown in Table 4.


Table 3. Parameters derived from the radial velocities and the light curves with the Levenberg-Marquard algorithm. Units are as in Table 2. In addition, [FORMULA] and [FORMULA] denote masses in solar mass units, [FORMULA] and [FORMULA] denote the radii in solar radius units, [FORMULA] is the temperature in units of 10,000 Kelvin, and [FORMULA], [FORMULA] and [FORMULA] are the luminosities as defined in the WD program, [FORMULA] and [FORMULA] denote the Roche potentials, [FORMULA] and [FORMULA] the filling factors indicating that BF Aur is detached. The parameters [FORMULA], [FORMULA] and [FORMULA] are the ratios of [FORMULA] in the specific passbands. The parameter, r, in the last column of this table is defined as the maximum of all ratio "parameter correction over standard deviation".


Table 4. Parameters derived from the radial velocities and the light curves with the Levenberg-Marquard algorithm under the additional constraint that the secondary fills its Roche lobe. [FORMULA] and [FORMULA] denote the Roche potentials; the filling factors [FORMULA] and [FORMULA] indicate that BF Aur is semi-detached with the secondary component filling its Roche lobe. Otherwise as in Table 3.

For completeness, we give in Table 7 those values which can be derived from the finally adopted solution (surfaces [FORMULA], volumes [FORMULA], mean radii [FORMULA]. Fig. 3 and Fig. 4 show the final solutions of the light curves and the radial velocity curves, respectively.

[FIGURE] Fig. 3. Differential UBV light curves showing normalized flux in the specific passbands versus phase, and their respective fits. The lower panels show the residuals.

[FIGURE] Fig. 4. Radial-velocity fits from the simultaneous solution (Table 6) of photometry and spectroscopy. The upper panels show the observations (dots) of the primary (left panel) and the secondary (right panel) and their respective fits.

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Online publication: October 24, 2000