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Astron. Astrophys. 341, 857-866 (1999)

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3. Effective temperature determination

3.1. Strömgren data

Among the approximately sixty SB2 systems gathered in Lastennet (1998), only 20 have both individual uvby Strömgren photometric indices and uncertainties for each component. Uncertainties are a key point in the calculation presented later (Sect. 3.2). The photometry used for the 20 systems of our working sample (see Table 1) is from the recent Table 5 of Jordi et al. (1997), who have taken the individual indices directly from the literature but have also added their own results for three systems (YZ Cas, WX Cep, and IQ Per).

3.2. Methodology

To compute synthetic colours from the BaSeL models, we need effective temperature ([FORMULA]), surface gravity ([FORMULA]), and metallicity ([FORMULA]). Consequently, given the observed colours (namely, b-y, m1, and c1), we are able to derive [FORMULA], [FORMULA], and [FORMULA] from a comparison with model colours. As the surface gravities can be derived very accurately from the masses and radii of the stars in our working sample, only two physical parameters have to be derived ([FORMULA] and [FORMULA]).

This has been done by minimizing the [FORMULA]-functional, defined as


where n is the number of comparison data, colour(1)= (b-y)0, colour(2)= m0, and colour(3) = c0. The best [FORMULA] is obtained when the synthetic colour, colour[FORMULA], is equal to the observed one.

Reddening has been taken into account following Crawford (1975): (b-y)0 = (b-y) - E(b-y), m0 = m1 + 0.3 [FORMULA] E(b-y), c0 = c1 - 0.2 [FORMULA] E(b-y), in order to derive the intrinsic colours from the observed ones. With [FORMULA] observational data (b-y, m1, c1) and [FORMULA] free parameters ([FORMULA] and [FORMULA]), we expect to find a [FORMULA]-distribution with [FORMULA] degree of freedom. Finding the central minimum value [FORMULA], we form the [FORMULA]-grid in the ([FORMULA], [FORMULA])-plane and compute the boundaries corresponding to 1 [FORMULA], 2 [FORMULA], and 3 [FORMULA] respectively. As our sample contains only stars belonging to the Galactic disk, we have explored a restricted range of metallicity, -1.0 [FORMULA] [FORMULA] [FORMULA] +0.5.

Fig. 1 illustrates the different steps of the method, here for BW Aqr A. All possible combinations of observational data (as indicated on the top of each panel) are explored, hence varying the number of degrees of freedom for minimizing the [FORMULA]. The top panels show the results obtained for matching uniquely one colour index (b-y, m1, or c1). In these cases, [FORMULA], which simply means that it is impossible to fix both [FORMULA] and [FORMULA] with only one observational quantity, as indeed is illustrated by the three top panels of Fig. 1. From (b-y) only, the effective temperature boundaries appear to be very similar across the whole metallicity range, highlighting the fact that this index is traditionally used to derive [FORMULA]. Alternatively, the m1 index only provides constraints on the metallicity of the star. Used together (lower central panel), these two indices outline restricted "islands" of solutions in the ([FORMULA], [FORMULA])-plane, and hence offer a good combination to estimate these parameters. The c1 index has been originally designed to estimate the surface gravity, but it also appears to be a good indicator of temperature in the parameter range explored for BW Aqr A (upper right panel). On the lower right panel, all the available observational information (b-y, m1, c1, and [FORMULA]) can be exploited. The range of [FORMULA] values that we then derive for BW Aqr A agree well with previous estimates (as indicated by the vertical dotted lines), and the same is true for its metallicity, which is compatible with the Galactic disk stars. Finally, in order to take full advantage of all the observational information available for the stars in our sample, we choose to estimate [FORMULA] and [FORMULA] from a [FORMULA] minimization performed on the three colour indices.

[FIGURE] Fig. 1. Simultaneous solutions of [FORMULA] and [FORMULA] for BW Aqr A (assuming [FORMULA] = 3.981): matching (b-y) (upper left ), m1 (upper central ), c1 (upper right ), (b-y), and c1 (lower left ), (b-y), and m1 (lower central ), (b-y), m1, and c1 (lower right ). Best fit (black dot ) and 1-[FORMULA] (solid line ), 2-[FORMULA] (dashed line ), and 3-[FORMULA](dot-dashed line ) confidence levels are also shown. Previous estimates of [FORMULA] from Clausen (1991) are indicated as vertical dotted lines in all panels.

3.3. Surface gravity accuracy and influence of reddening

3.3.1. Surface gravity

As our results depend not only on the accuracy of the photometric data, but also on that of the surface gravity determination, we analysed the effect of a variation of [FORMULA] upon the predicted [FORMULA] and [FORMULA] values. We investigated this "[FORMULA] effect" for the AR Aur system, for which the known value of [FORMULA] has the largest uncertainties in our working sample: for instance, the surface gravity of the coolest component of AR Aur (AR Aur B) is [FORMULA] [FORMULA] 4.280[FORMULA]0.025.

For [FORMULA] [FORMULA] 4.280, the central [FORMULA] value predicted is about 10,500K. If we consider [FORMULA]-0.025 dex (left panel in Fig. 2) or [FORMULA]+0.025 dex (right panel in Fig. 2), neither the central [FORMULA] value nor the pattern of contours change significantly.

[FIGURE] Fig. 2. Influence of [FORMULA] on the simultaneous solution of [FORMULA] and [FORMULA] for AR Aur B. Two different [FORMULA] values are considered: [FORMULA] [FORMULA] 4.255 (left panel ), [FORMULA] [FORMULA] 4.305 (right panel ). These values are 0.025 dex higher or lower than the true [FORMULA] (4.280).

This example shows that our results for ([FORMULA],[FORMULA]) will not change due to variations of surface gravity within the errors listed in Table 1.

3.3.2. Interstellar reddening

Interstellar reddening is of prime importance for the determination of both [FORMULA] and [FORMULA]. A great deal of attention was therefore devoted to the E(b-y) values available in the literature, for each star of our sample. We explore different reddening values (as described in Sect. 3.2), and we compare their resulting [FORMULA]-scores. For the following systems, we adopted the published values, in perfect agreement with our results: BW Aqr, AR Aur, [FORMULA] Aur, GZ Cma, EM Car, CW Cep, GG Lup, TZ Men, V451 Oph, AI Phe, [FORMULA] Phe, VV Pyx, and DM Vir. As we did not find any indication about the interstellar reddening of YZ Cas, we kept E(b-y) [FORMULA] 0 as a quite reasonable hypothesis. We have neither found any data on interstellar reddening for the WX Cephei system. But the hypothesis of no significant reddening for WX Cep is ruled out by the very high [FORMULA]-value obtained in reproducing simultaneously the quadruplet (b-y, m1, c1, [FORMULA]) of the observed data. From the different reddening values explored in Table 2, we find that E(b-y) [FORMULA] 0.32 for WX Cep A and E(b-y) [FORMULA] 0.28 for WX Cep B provide the best solutions.


Table 2. Influence of reddening on the [FORMULA] of the components of the WX Cephei system. Best [FORMULA] are in bold characters. The hypothesis of no reddening is definitively ruled out.

The influence of reddening variations is illustrated in Fig. 3. While for the system, WX Cep AB, an average value E(b-y) [FORMULA] 0.30 appears justified from the results of Table 2 - and will indeed be adopted in the remainder of this paper -, Fig. 3 shows how for the individual component, WX Cep A, small changes [FORMULA]E(b-y)[FORMULA]0.02 away from its own optimum value E(b-y) [FORMULA] 0.32 induce significant changes in the possible solutions of the ([FORMULA], [FORMULA])-couples. In particular, going to E(b-y) [FORMULA] 0.30 (upper right panel) implies a dramatic jump in predicted [FORMULA] from a (plausible) metal-normal (lower left) to a (rather unlikely?) metal-poor composition at or near the lower limit ([FORMULA] [FORMULA]) of the exploration range.

[FIGURE] Fig. 3. Influence of reddening on the simultaneous solutions of [FORMULA] and [FORMULA] for WX Cep A. Different reddening values are considered: E(b-y) [FORMULA] 0.00 (upper left ), E(b-y) [FORMULA] 0.30 (upper right ), E(b-y) [FORMULA] 0.32 (lower left ), and E(b-y) [FORMULA] 0.34 (lower right ). Previous determination of [FORMULA] from Popper (1987) (using Popper's 1980 calibrations) is also shown for comparison (vertical dotted lines ).

For the other four systems for which interstellar reddening has also been previously neglected in the literature, we found small, but not significant, E(b-y) values: RZ Cha (0.003), KW Hya (0.01), V1031 Ori (0.05), and PV Pup (0.06).

E(b-y) [FORMULA] 0.11 has been adopted for IQ Per by comparing different [FORMULA] solutions. This value is consistent with E(b-y) [FORMULA] 0.10[FORMULA]0.01, estimated from the published value of E(B-V) [FORMULA] 0.14[FORMULA]0.01 (Lacy & Frueh 1985), assuming E(b-y) [FORMULA] 0.73[FORMULA]E(B-V) after Crawford (1975). Adopted reddening values for stars of our sample are listed in Table 1.

3.4. General results and discussion

In Figs. 4 and 5 we show the full results obtained (from b-y, m1, and c1) for all the stars of the sample in ([FORMULA], [FORMULA]) planes. All the ([FORMULA],[FORMULA])-solutions inside the contours allow to reproduce, at different confidence levels, both the observed Strömgren colours (b-y, m1, and c1) and the surface gravity with the BaSeL models. As a general trend, it is important to notice that our [FORMULA] ranges do not provide estimates systematically different from previous ones (vertical dotted lines). Furthermore, the 3-[FORMULA] confidence regions show that most previous [FORMULA] estimates are optimistic, except for some stars (e.g., GG Lup A, TZ Men A, and V451 Oph A) for which our method gives better constraints on the estimated effective temperature. At a 1-[FORMULA] confidence level (68.3%), our method often provides better constraints for [FORMULA] determination. However, it is worth noticing that for a few stars the match is really bad (see [FORMULA]-values labelled directly on Fig. 4 and 5). As already mentioned, with 3 observational data (b-y, m1, c1) and 2 free parameters ([FORMULA] and [FORMULA]), we expect to find a [FORMULA]-distribution with 3-2[FORMULA]1 degree of freedom and a typical [FORMULA]-value of about 1. For some stars (e.g. VV Pyx, DM Vir and KW Hya A), [FORMULA] is greater than 10, a too high value to be acceptable because the probability to obtain an observed minimum [FORMULA]-square greater than the value [FORMULA][FORMULA]10 is less than 0.2%. For this reason, the results given for a particular star should not be used without carefully considering the [FORMULA]-value.

[FIGURE] Fig. 4. Simultaneous solution of [FORMULA] and [FORMULA] matching (b-y)0, m0, c0, and [FORMULA]. The name of the star and the [FORMULA] are labelled directly in each panel. When available, effective temperature determinations from previous studies (vertical lines ) and observational indications of metallicity (horizontal lines ) are also shown.

[FIGURE] Fig. 5. Same as Fig. 4.

One of the most striking features appearing in nearly all panels of Figs. 4 and 5 is the considerable range of [FORMULA] accepted inside the confidence levels. This is particularly true for stars hotter than [FORMULA] 10,000 K (as, for instance, EM Car A & B and GG Lup A & B), for which optical photometry is quite insensitive to the stellar metal content. For these stars, a large range in [FORMULA] gives very similar [FORMULA] values. In contrast, for the coolest stars in our sample, our method provides straight constraints on their metallicity. Actually, when observational metallicity indications are available ([FORMULA] Aur, YZ Cas, RZ Cha, AI Phe, and PV Pup), the contour solutions are found in good agreement with previous estimated [FORMULA] ranges (labelled as horizontal lines in Figs. 4 and 5).

The effective temperatures derived from our minimization procedure cannot be easily presented in a simple table format, as they are intrinsically related to metallicity. We nonetheless provide in Table 3, as an indication of the estimated stellar parameters for all the stars in our sample, the best ([FORMULA]) simultaneous solutions ([FORMULA],[FORMULA]) for the three following cases: by using b-y and m1 (Case 1), b-y and c1 (Case 2) and by using b-y, m1, and c1 (Case 3). In Case 1 and Case 2, a typical [FORMULA]-value close to zero is theoretically expected, and in Case 3, as previously mentioned, one expects a typical [FORMULA]-value of about 1. There are quite a few stars for which [FORMULA] increases dramatically between Case 1 or 2 and Case 3 to a clearly unacceptable value (most notably AI Phe A between Case 1 and Case 3). This point means that although a good fit is obtained with two photometric indices, no acceptable [FORMULA]-value is obtained by adding one more index in Case 3. Consequently, Case 1 or Case 2-solutions have to be chosen in such cases. For these stars, even if the [FORMULA] solutions shown in Figs. 4 and 5 are not reliable, it is interesting to notice that the contours derived are however still in agreement with previous works.


Table 3. Best simultaneous ([FORMULA],[FORMULA]) solutions using (b-y) and m1 (Case 1), (b-y) and c1 (Case 2) or (b-y), m1, and c1 (Case 3).

The surprising result in Table 3 is that many solutions are very metal-poor. This in fact means that the [FORMULA] solutions are not necessarily the most realistic ones. We must, therefore, emphasize that the values presented in Table 3 should not be used without carefully considering the confidence level contours shown in Figs. 4 and 5. For most stars in our sample, [FORMULA] and [FORMULA] do not appear strongly correlated (i.e. the confidence regions do not exhibit oblique shapes), but there are a few cases for which the assumed metallicity leads to a different range in the derived effective temperature (EM Car B, CW Cep A & B, GG Lup A, [FORMULA] Phe A). These results point out that the classical derivation of [FORMULA] from calibration without exploring all [FORMULA] values is not always a reliable method, even for hot stars.

3.5. Comparison with Hipparcos parallax

Very recently, Ribas et al. (1998) have computed the effective temperatures of 19 eclipsing binaries included in the Hipparcos catalogue from their radii, Hipparcos trigonometric parallaxes, and apparent visual magnitudes corrected for absorption. They used Flower's (1996) calibration to derive bolometric corrections. Only 8 systems are in common with our working sample. The comparison with our results is made in Table 4. The [FORMULA] being highly related with metallicity, a direct comparison is not possible because, unlike the Hipparcos-derived data, our results are not given in terms of temperatures with error bars, but as ranges of [FORMULA] compatible with a given [FORMULA]. Thus, the ranges reported in Table 4 are given assuming three different hypotheses: [FORMULA][FORMULA]0.2, [FORMULA] [FORMULA] 0, and [FORMULA] [FORMULA] 0.2. The overall agreement is quite satisfactory, as illustrated in Fig. 6.
The disagreement for the temperatures of CW Cephei can be explained by the large error of the Hipparcos parallax ([FORMULA]). For such large errors, the Lutz-Kelker correction (Lutz & Kelker 1973) cannot be neglected: the average distance is certainly underestimated and, as a consequence, the [FORMULA] is also underestimated in Ribas et al. 's (1998) calculation. Thus, the agreement with the results obtained from the BaSeL models is certainly better than it would appear in Fig. 6 and Table 4. Similar corrections, of slightly lesser extent, are probably also indicated for the [FORMULA] of RZ Cha and GG Lup, which have [FORMULA] (11.6% and 11.4%, respectively). Finally, it is worth noting that the system with the smallest relative error in Table 4, [FORMULA] Aur, shows excellent agreement between [FORMULA] (Hipparcos) and [FORMULA] (BaSeL), which underlines the validity of the BaSeL models.


Table 4. Effective temperatures from Hipparcos (after Ribas et al. 1998) and from BaSeL models matching (b-y)0, m0, c0, and [FORMULA] for the three following metallicities: [FORMULA][FORMULA]0.2, [FORMULA] [FORMULA] 0 and [FORMULA] [FORMULA] 0.2.

[FIGURE] Fig. 6. Hipparcos- versus BaSeL-derived effective temperatures for [FORMULA] Aur, YZ Cas, CW Cep, RZ Cha, KW Hya, GG Lup, TZ Men, and [FORMULA] Phe. The errors are not shown on the Hipparcos axis for CW Cephei (the hottest binary in these figures). See text for explanation.

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© European Southern Observatory (ESO) 1999

Online publication: December 16, 1998