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Astron. Astrophys. 364, 455-466 (2000)

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3. Reddenings

3.1. Reddenings from Strömgren photometry

The classical way to derive reddenings from Strömgren photometry is to use the [FORMULA] index and [FORMULA] which are very well correlated for F-type stars (Crawford 1975). The [FORMULA] index is unaffected by reddening, so the reddening can be determined by comparing the observed [FORMULA] to the value expected from the standard [FORMULA]-[FORMULA] relation. However, in the Magellanic Clouds the F-type main sequence stars are much too faint to be reached by our photometry, and there are additional problems with the [FORMULA] photometry and its calibration because of the non-vanishing radial velocities of the clouds (Knude 1993).

However, the Strömgren system also provides a way to determine reddenings for the more luminous early-type stars (A0 and earlier, Strömgren 1966). Instead of [FORMULA] and [FORMULA], the reddening for early-type stars is derived from the [FORMULA] and [FORMULA] indices, following the "standard iterative procedure" described e.g. by Crawford et al. (1970). Both [FORMULA] and [FORMULA] are temperature indicators for stars of spectral type earlier than A0, and for such stars the de-reddened indices [FORMULA] and [FORMULA] are related to each other through a standard relation in much the same way as are [FORMULA] and [FORMULA] in the case of F stars. The calibration of the [FORMULA] standard relation was empirically established by Crawford (1978), and agrees well with Strömgren indices based on model atmosphere calculations for early-type stars (Grebel, private communication. See also Sect. 3.3 and Fig. 5). The basic assumption is then that any observed deviation from the standard relation is caused by reddening.

The reddening vector in the [FORMULA] diagram is nearly horizontal so the reddening of a star can, to the first order, be estimated simply as the difference between the observed [FORMULA] and the intrinsic value [FORMULA] corresponding to the observed [FORMULA] according to the standard relation:

[EQUATION]

With this initial estimate of the reddening, the [FORMULA] index can be corrected and a new reddening can be derived. Usually only one or two iterations are required.

Due to the smaller wavelength difference between the b and y filters compared to the Johnson B and V filters, [FORMULA] is somewhat larger than [FORMULA], as can be shown from the standard law for interstellar extinction (Savage & Mathis 1979). More specifically, [FORMULA] and [FORMULA] are related through the following expression:

[EQUATION]

A very convenient point regarding the de-reddening of stars earlier than spectral type A0 is that the [FORMULA] index is not required. However, because the [FORMULA] index involves the u filter [FORMULA], it has until recently been a time-consuming process to acquire sufficiently deep photometry for stars in the Magellanic Clouds with available CCD detectors.

3.2. Selection of B stars

In principle, the selection of early-type stars is most effectively carried out in the Strömgren system using the two reddening free indices

[EQUATION]

and

[EQUATION]

According to Strömgren (1966) and Olsen (1979) the [FORMULA] diagram should resemble the one in Fig. 1. The early-type stars are located in the left part of the diagram, on the part where [FORMULA] is increasing as a function of [FORMULA]. The lower, right border of the band containing the early-type stars is defined quite well by a straight line given by the equation

[EQUATION]

(E. H. Olsen, private communication). The exact location of the line is only critical for [FORMULA], since no stars are found immediately to the right of it for lower [FORMULA] values.

[FIGURE] Fig. 1. [FORMULA] diagram of bright galactic stars (Olsen 1979)

The [FORMULA] diagrams for stars in the LMC/SMC fields with [FORMULA] mag (corresponding to [FORMULA]) are shown in Fig. 2. Obviously, they are not quite similar to the diagram shown in Fig. 1. The line separating early-type stars and stars of later types passes right through the distribution of points even for low [FORMULA] values, and many stars are located inside the supposedly empty "loop".

[FIGURE] Fig. 2. [FORMULA] diagrams for the four LMC/SMC fields.

We have made several attempts at understanding this effect. The observations and data reduction procedures have been carefully checked to ensure that the effect is not a manifestation of some error in the stacking procedure, the cosmic ray elimination process or the use of DAOPHOT. This has been done by using the PSF fitting programme DoPHOT (Schechter et al. 1993) on a set of single uvby frames without removal of cosmic rays. Apart from the expected larger scatter, the diagrams look qualitatively identical to the ones based on the full data set.

Could the peculiar [FORMULA] diagrams result from the lower metallicities in the Magellanic Clouds relative to local stars from which Fig. 1 was derived? In order to test this we obtained theoretical [FORMULA] colours for a set of stellar isochrones from the Padua group (Girardi et al. 2000), transformed into Strömgren colours using 1997 versions of Kurucz model atmospheres (Kurucz 1979 , 1992). In Fig. 3 we show synthetic [FORMULA] diagrams for four different metallicities ([Fe/H] = 0, -0.3, -0.6 and -2.0). Each panel contains 5 isochrones corresponding to ages between [FORMULA] and [FORMULA] years. Only stars in the magnitude interval [FORMULA], roughly corresponding to the range covered by our data, have been included in the plot. Gaussian distributed random errors of [FORMULA] mag, about similar to the typical errors in the observations, have been added to each axis (note that the 0.10 mag error limit in [FORMULA] is an upper limit). The plots in Fig. 3 do suggest a significant change in the appearance of the [FORMULA] diagrams for progressively lower metallicities in the sense that more late-type stars will move in from the right and fill the empty region of the diagram. This effect is enhanced by the photometric scatter, although the early-type stars show only a very small systematic shift and for the HV982 field in particular, the theoretical [FORMULA] diagrams still do not provide a quite satisfactory match to the observations. Compared to Fig. 1, Fig. 2 shows only few early-type stars above the expected location, whereas random photometric errors in Fig. 3 cause equal amounts of stars to scatter upwards and downwards. However, the number of data points in different parts of the [FORMULA] diagram depends on the age distribution of the stars and unless this is taken into account, an exact match between the observed and simulated distributions cannot be expected.

[FIGURE] Fig. 3. Synthetic [FORMULA] diagrams based on Padua stellar models and Kurucz model atmospheres. Gaussian distributed random errors ([FORMULA] mag) have been added to the synthetic [FORMULA] and [FORMULA] values.

We thus conclude that much of the apparently peculiar morphology of the observed [FORMULA] diagrams in the LMC and SMC could be due to a combination of lower metallicities and photometric errors. In any case, for our purpose the [FORMULA] diagram clearly is not a practical tool for selecting B stars.

Therefore, we have eventually chosen the alternative approach of selecting B stars directly as stars brighter than V=19.0 in the LMC and V=19.4 in the SMC and bluer than [FORMULA]. The numbers of B stars in each field selected in this way range between 102 (for the HV12578 field) and 322 (for the HV982 field). The [FORMULA] limit was chosen so that no confusion with red giants occurred, while nearly all main sequence stars would be included. The magnitude limit corresponds to [FORMULA]. A potential problem is that evolved stars of later types than A0 could enter our sample, but these would have relatively high [FORMULA] values, and any significant contamination by stars later than type A0 should therefore be visible in the [FORMULA] diagram as an excess of stars with [FORMULA]. No such excess is observed.

3.3. Foreground reddening

The [FORMULA] diagrams for B stars in each of the four fields are shown in Fig. 4 together with a line corresponding to the standard relation of Crawford (1978). The arrow shows the direction of the reddening vector. Only stars with a photometric error in [FORMULA] of less than 0.015 and less than 0.05 in [FORMULA] according to DAOPHOT were included in the analysis. Stars with a DAOPHOT [FORMULA] estimate larger than 2 in uvby were also rejected regardless of the estimated error. The left-most boundary of the data points is quite well-defined, but offset with respect to the [FORMULA] standard relation (due to foreground reddening), while the right-most boundary is more diffuse. In the SMC HV11284 field we note the presence of a few stars with apparently very low reddenings.

[FIGURE] Fig. 4. [FORMULA] diagrams for B stars in the four fields. The arrow indicates the direction of the reddening vector.

Once again, we used the Padua isochrones combined with Kurucz atmospheres to test if the Crawford (1978) relation applies also to the metal-poorer LMC/SMC stars. The theoretical [FORMULA] diagrams for stars with [FORMULA] are shown in Fig. 5 for four different metallicities together with the Crawford relation. Like in Fig. 3, five different isochrones corresponding to ages between [FORMULA] and [FORMULA] years are included but no random errors are added in Fig. 5. The models and the empirical relation generally agree well for all metallicities and there are no evident trends with metallicity that would affect the reddening determinations, so we conclude that reddenings derived from the [FORMULA] diagram are reliable.

[FIGURE] Fig. 5. Synthetic [FORMULA] diagrams based on Padua stellar models and Kurucz model atmospheres. The plot shows data for four metallicities, offset by steps of 0.07 mag in [FORMULA]. Metallicities (left to right) are [Fe/H] = 0, -0.3, -0.6 and -2.0.

The reddening distributions derived from the data in Fig. 4 as described in the previous section are shown in Fig. 6. The dashed Gaussians represent the average observational scatter. The lower limit of the reddening distribution is interpreted as being caused by Galactic foreground extinction. We estimate that this amounts to [FORMULA] in both of the LMC fields and [FORMULA] in the SMC fields, corresponding to [FORMULA] and [FORMULA], respectively. The estimated errors in these reddenings are largely due to zero-point errors in the b and y photometry, while errors in [FORMULA] do not affect the results significantly.

[FIGURE] Fig. 6. Reddening histograms (B stars) for the four fields; see text for details.

For the LMC fields, we can compare the foreground reddening determinations to the reddening map by Oestreicher et al. (1995), which predicts foreground reddenings of [FORMULA] and [FORMULA] in the HV12578 and HV982 fields. Thus, our estimated foreground reddenings are somewhat higher and agree well with the typical value of [FORMULA] reported by Schlegel et al. (1998). However, their typical SMC value of [FORMULA] is significantly lower than our results.

3.4. The correlation of reddening with position.

The width of the histograms representing the reddening distributions cannot be accounted for by observational uncertainties and must, consequently, represent real scatter in the reddenings of LMC/SMC B stars. This could either mean that the foreground reddening varies by as much as 0.1 in [FORMULA] within the fields, or that the B stars are subject to different amounts of reddening from the interstellar medium in the Clouds themselves, being located at different optical depths. In the first case, we would expect that reddenings are strongly correlated with the position in the image, whereas such a correlation would be weaker or absent if the variations are caused by depth effects. Of course, any intermediate scenario is in principle possible.

If reddening is correlated with position then the difference between the reddenings of two neighbouring stars will, on the average, be smaller than that of two widely separated stars. In order to investigate this effect, we used our measurements of individual B star reddenings to calculate the r.m.s. reddening difference between two stars as a function of their separation in the image. We denote the r.m.s reddening difference between the reddening of a star and other stars at distances [FORMULA] from it [FORMULA] where [FORMULA] is the bin size. We found [FORMULA] pixels to be a reasonable bin size.

If reddening is correlated with position on scales smaller than some characteristic limit, [FORMULA] will decrease as R becomes smaller than the characteristic limit, which could for example depend on the typical size of interstellar clouds. [FORMULA] is not a very critical parameter. It should be chosen sufficiently large that a reasonable number of stars will be located within an annulus of inner radius R and outer radius [FORMULA]. On the other hand, it should not be larger than the typical size of the structure we are looking for.

Fig. 7 shows [FORMULA] for each field, with [FORMULA] set to 10 pixels or roughly 4 arcseconds. There is no significant decrease in C even for quite small R, with a hint of such a decrease only in the SMC fields (HV1433, HV11284). At the same time, Fig. 7 also gives for each field an indication of how large the typical error will be if just the average reddening of the field is used as an estimate of the reddening for a given star. This varies from one field to another, from [FORMULA] in the HV12578 field to 0.07 in the HV11284 field, and it is clear that the error is in no case negligible. Based on the two fields we have observed in each galaxy, the contribution from internal reddening in the Clouds appears to decrease as a function of distance to the centre.

[FIGURE] Fig. 7. The quantity [FORMULA], denoting the rms [FORMULA] difference between B stars located in the distance interval [FORMULA] from each other. [FORMULA] (pixels) corresponds to 38".

3.5. Discussion of reddenings

The previous section has shown a lack of correlation between reddening and position in the fields, even on the smallest scales on which this can be investigated by means of the present Strömgren photometry, i.e. a few arc seconds (for LMC and SMC, 1 pc corresponds to about [FORMULA] and [FORMULA], respectively).

Many stars have reddenings much larger than the Galactic foreground contribution, in some fields up to [FORMULA] ([FORMULA]) or so. Because the reddening is not correlated with the position in the field, we conclude that the variations are most likely due to the stars being located at different depths in the Magellanic Clouds. Therefore, the difference between the maximum and minimum reddening in each field presumably represents the total reddening when looking through the LMC or SMC at the corresponding position.

It is of interest to compare our results with the investigation by Oestreicher & Schmidt-Kaler (1996). Their reddening map shows a lack of stars with high reddenings in the neighbourhood of the HV12578 field, in agreement with our results, whereas a large number of stars with relatively high reddenings are found near the 30 Dor region, again in agreement with our results. Oestreicher & Schmidt-Kaler (1996) did not study the Small Magellanic Cloud. Olsen (1999) only found strong differential reddening in one out of four LMC fields observed with the HST, centered on the cluster NGC 1916 near the centre of the LMC bar.

As we shall see in the following section, the fact that average reddenings are only accurate to within several hundreths of a magnitude is potentially a serious problem for photometric studies of stars where reddenings cannot be directly determined, such as metallicity studies of GK giants in the Strömgren system (Hilker et al. 1995; Grebel & Richtler 1992; Dirsch et al. 2000).

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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