THE ARAUCARIA PROJECT. THE DISTANCE TO THE SMALL MAGELLANIC CLOUD FROM LATE-TYPE ECLIPSING BINARIES

The orbital periods were calculated with the string-length method (Lafler & Kinman 1965; Clarke 2002). The phased light curves were inspected visually to set a preliminary epoch of the primary minimum and to verify the photometric stability of the systems—a lack of spots, flares, the O'Connell effect (O'Connell 1951)—at the precision level of our photometry.

The effective temperature of the primary component of each system (eclipsed during minimum at orbital phase 0) was estimated as follows. We determined the interstellar extinction in the direction of our targets using Haschke et al. (2011) reddening maps—see Section 3.5. Average out-of-eclipse magnitudes were calculated from all observations taken outside photometric minima. In the case of the two systems showing proximity effects between eclipses, we used magnitudes only around both quadratures. All out-of-eclipse V-band magnitudes and (V − I) colors are presented in Table 1. These magnitudes were dereddened using the interstellar extinction law given by Cardelli et al. (1989) and O'Donnell (1994) and assuming R_V = 3.1.

To set the effective temperature scale of each system, we ran the WD code initially assuming a temperature of T₁ = 5000 K for the primary and [Fe/H] = −0.5. The resulting luminosity ratios in the V, I, and K bands were combined with dereddened magnitudes to calculate first guesses for the intrinsic (V − I) and (V − K) colors of both components. Then we utilized several calibrations between colors and effective temperatures (di Benedetto 1998; Alonso et al. 1999; Houdashelt et al. 2000; Ramírez & Meléndez 2005; Masana et al. 2006; González Hernández & Bonifacio 2009; Casagrande et al. 2010; Worthey & Lee 2011) to set new temperatures for the primaries. We iterated these steps until new derived temperatures changed by less than 10 K. The resulting effective temperatures were used in the "preliminary" solution.

We estimated rotational velocities of the components from a BF analysis. Rotation in all cases is consistent with synchronous rotation; thus, we set the rotation parameter F = 1.0 for both components. The albedo parameter was set to 0.5, and the gravity brightening was set to 0.32; both values are appropriate for a cool, convective atmosphere. The limb darkening coefficients were calculated internally by WD code (setting LD = −2) according to the logarithmic law (Klinglesmith & Sobieski 1970) during each iteration of the Differential Correction (DC) subroutine using tabulated data computed by van Hamme (1993).

As free parameters of the WD model, we chose the orbital period P_obs, the semimajor axis a, the orbital eccentricity e, the argument of periastron ω, the epoch of the primary spectroscopic conjunction T₀, the systemic radial velocity γ, the orbital inclination i, the secondary star average surface temperature T₂, the modified surface potential of both components (Ω₁ and Ω₂), the mass ratio q, and the relative monochromatic luminosity of the primary star in two bands (L1_V and L1_I).

For each star, we simultaneously fitted the I-band and V-band light curves and the two radial velocity curves using the DC subroutine from the WD code. The detached configuration (Mode 2) was chosen during all the analyses and a simple reflection treatment (MREF = 1, NREF = 1) was employed. A stellar atmosphere formulation was selected for both stars (IFAT = 1). Level dependent weighting was applied (NOISE = 1), and curve dependent weightings (SIGMA) were calculated after each iteration. The grid size was set to N = 40 on both components. We did not break the adjustable parameters set into subsets, but instead, we adjusted all free parameters at each iteration.

For each system, we used information obtained from spectroscopy to derive the relative intensity of absorption lines of both components from the MIKE spectra. We treat these as approximate light ratios in the blue and red part of the optical spectra. These light ratios are reported in Table 4. We do not use them to fix light ratios in our models, however, but we use them to rule out alias solutions and to verify if the light curve solution is consistent with the spectroscopy.

To constrain the important input parameters of the eclipsing binary components, we decided to disentangle the spectra of the individual components and carry out an abundance analysis. The most important parameter is the effective temperature of each component which scales the luminosity ratio of the components and the limb darkening coefficients. Also, by calculating the intrinsic (V − I) colors from the derived temperatures, we can estimate the reddening of each system.

We used the prescription given by González & Levato (2006) to disentangle individual spectra of the binary components. The method works in the real wavelength domain. It requires rather high signal to noise ratio spectra to work properly, and thus, we were restricted to using only the red MIKE spectra. The red MIKE spectra were shifted and stacked using the radial velocities found by using the BF as described in the previous section. No initial secondary spectrum was used in the disentangling. The range of wavelength used was from 4960 Å to 6800 Å where numerous Fe i and Fe ii lines are present. We emphasise that we do not use the disentangled spectra to derive new radial velocities because they always give higher dispersion than synthetic templates. The reason is that the relatively low S/N of our disentangled spectra reduces the accuracy of radial velocity measurements.

The disentangled spectra have to be renormalized in order to account for the companion's continuum which dilutes the depth of the absorption lines. If the light ratio of the secondary to the primary component at a given wavelength is k(λ) = L₂₁(λ), then we subtract a value k/(1 + k) from the disentangled spectrum of the primary (corresponding to the secondary flux). Then, we again normalize the final spectrum. In the case of the secondary spectrum, we subtract 1/(1 + k). Here, two things must be emphasized: (1) the choice of a proper normalization level before disentangling and (2) the choice of a proper model of a system to calculate an appropriate light ratio function L₂₁(λ). Regarding the first point, all spectra have to be normalized, and for high S/N spectra, the level of normalization is simply the mode value. In the case of lower S/N spectra like those obtained for our binaries in the SMC, the level of normalization is a bit ambiguous, and usually, we set it to a value of ∼90 percentile.

Regarding the second point, we must be aware that there is a large number of possible model solutions to the observed light and radial velocity curves. A model which is the best, i.e., it minimizes the residua of synthetic fits, does not always predict light ratios in agreement with spectroscopic light ratios. This is because our data have limited S/N and because an optical depth where absorption lines are formed is usually different from an optical depth at which we observe stellar photosphere. If we use an "improper" model for a system to predict the spectroscopic light ratios, the resulting absorption line depths in the disentangled spectra are invalid. An easy way to notice an inappropriate model is to check if the absorption line bottoms in the renormalized spectrum of the primary or the secondary are below zero, which is a level signifying an unphysical model. Another check is based on a comparison of equivalent widths (EWs) of the same absorption lines in both components. Usually the hotter component should have lines with smaller equaivalent widths. However such a comparison is not unique because components of the same binary may have different abundances (Folsom et al. 2010). Finally, we can calculate the EW ratios for the components for the same lines and determine that this ratio is not wavelength dependent. We expect that for realistic models, this ratio is approximately constant within the spectral range of disentangling.

Atmospheric parameters and the iron content were obtained from the EWs of the iron spectral lines. See Marino et al. (2008) for a more detailed explanation of the method we used to measure the EWs and Villanova et al. (2010) for a description of the line list that was used. We adopted log (Fe) = 7.50 as the solar iron abundance.

Atmospheric parameters were obtained in the following way. As a first step, model atmospheres were calculated using ATLAS9 (Kurucz 1970) assuming as initial estimations for T_eff, log g, and v_t values typical for giant stars (4800 K, 2.00 dex, 1.80 km s⁻¹) and [Fe/H] =−0.8 dex as a typical value for young stars in the SMC (Dias et al. 2010).

Then T_eff, v_t, and log g were adjusted and new atmospheric models were calculated in an interactive way in order to remove trends in excitation potential (EP) and EWs versus abundance for T_eff and v_t, respectively, and to satisfy the ionization equilibrium for log g. Fe i and Fe ii were used for this purpose. The [Fe/H] value of the model was changed at each iteration according to the output of the abundance analysis. The local thermodynamic equilibrium program MOOG (Sneden 1973) was used for the abundance analysis.

The derived atmospheric parameters for the eclipsing binary components are given in Table 5. For all systems with the exception of SMC101.8 14077, we manage to calculate parameters for both components. For SMC101.8 14077, the iron lines of the primary are too weak to carry out an abundance analysis. Our measurements of abundance are consistent with each component in a binary having the same abundance. The typical accuracy of the parameters are 75 K, 0.4 dex, 0.15 dex, and 0.3 km s⁻¹ for T_eff, log g, [Fe/H] and v_t, respectively.

The interstellar extinction in the direction to each of our target stars was derived in three ways. First, we utilized Magellanic Cloud reddening maps published by Haschke et al. (2011). We calculate an average E(V − I) value from all of the reddening estimates within a 2 arc min radius of our stars and divided this average by a factor of 1.28 to get a (B−V) color excess for each system. We then added ΔE(B−V) = 0.037 mag, the mean foreground Galactic reddening in the direction of the SMC as derived from the Schlegel et al. (1998) maps. The second column of Table 7 gives the resulting reddening to each eclipsing binary. These were used to set the temperature scale of each system and to derive the "preliminary" solution (see Section 3.1).

The second method is based on a calibration of the EW of the interstellar Na i D1 line (5890.0 Å) and reddening (Munari & Zwitter 1997). The calibration works best for relatively small values of reddening, E(B−V) < 0.4 mag. Figure 2 presents the spectrum of SMC101.8 14077 around the sodium doublet. Narrow interstellar absorption lines from the Galaxy and the SMC can be identified together with wider stellar absorption from both stars. We separately measured the EWs of both interstellar components and derived the reddening for each. The results for all our stars is presented in Table 6. In the system SMC126.1 210, we could not detect any SMC component of the Na i D1 line because of strong blending with stellar lines.

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The final method uses the effective temperatures we derived from the atmospheric analysis described in Section 3.4. From the effective temperature—(V − I) color calibrations of Houdashelt et al. (2000) and Worthey & Lee (2011), we estimated the intrinsic (V − I) colors of each component. These colors were compared with the observed colors of the components obtained from the preliminary solution to directly derive E(V − I) color excesses. The reddening to each system was calculated as the mean value of the two components with the exception of the system SMC101.8 14077 where we have at our disposal a reddening estimate from only the secondary star.

Each of the three methods has an accuracy of approximately 0.03–0.04 mag. We calculated an average reddening for each system from the three estimates, and used this new reddening estimate to update the temperature scale of the components. We then calculated new models and repeated the reddening derivation using the third method. The fourth column of Table 7 gives the reddening estimates after two such iterations. The fifth column presents the adopted E(B−V) for each system as used in the "final" solution. We assigned a statistical error of 0.02 mag and an additional 0.02 mag systematic error for each estimate of the reddening.

This variable star is the faintest and bluest analyzed in this paper. The best solution for this system was obtained with a third light contribution included. If we assume that the presence of the third light is real and is not connected with some imperfections in the absolute calibration of the OGLE photometric data, then this suggests the presence of a red companion or a blended star with an almost negligible contribution to the optical portion of the spectrum. Indeed we could not detect any additional source of absorption lines in our spectra. For the purpose of the determination of the distance, we assumed that the third light contribution in the K band is at least equivalent to that in the I band. The uncertainty in the amount of third light in the K band is the largest source of statistical error of the absolute stellar radii and the distance determination to the system.

This eclipsing binary has the identifier SC4 192903 in OGLE-II database. It was previously analyzed by Graczyk (2003) who used only the photometric data from the second phase of the OGLE project (Udalski et al. 1998) without any spectroscopic observations. A comparison of the photometric parameters presented in our Table 8 and in Table 2 from Graczyk (2003) shows very similar model solutions with the only difference resulting from our inclusion of a contribution from third light. This results in different measured orbital inclinations. Although the stellar mass estimate in Graczyk (2003) seems to be too low (see Table 3 in that paper), the stellar radii, temperatures, and luminosities show good agreement.

The system contains the most luminous and massive late-type giants in the Magellanic Clouds analyzed by our team. The system was analyzed before by Graczyk (2003) using data from OGLE-II. As in the case of SMC101.8 14077, our photometric solution is almost identical with that reported by Graczyk (2003), but a difference appears in the absolute dimensions which are significantly underestimated in the previous study. The results of our preliminary work on this system, reported by Graczyk et al. (2013), are not significantly different from those reported in this paper, and based on more extensive observational data, we refine the previous distance estimate to this binary.

An interesting feature of this system is the difference of the center-of-mass radial velocities between the components. The hotter primary has systematically blueshifted radial velocities with respect to the secondary star amounting to 0.78 km s⁻¹—see Figure 3 in Graczyk et al. (2013). We accounted for this effect by subtracting the difference and solving the radial velocities equating the systemic velocity of the secondary with the barycenter velocity of the system. Some possible reasons for the difference are discussed in Torres et al. (2009) for Capella, a physically similar but lower mass binary system.

Another irregularity is a discrepancy between the temperature ratio T₂/T₁ as obtained from the analysis of the atmospheric parameters (see Section 3.4 and Table 5) compared to that arising from the light curve analysis (Table 8). The light curve solution and the spectroscopic light ratios suggest a temperature difference of about ΔT = 720 K, while the atmospheric analysis suggests a difference of ΔT = 150 K. We repeated the atmospheric analysis by fixing the gravities of both stars to reliable values obtained from the dynamical solution reported in Table 9. However, we obtained the same small difference of the effective temperatures. We also performed additional analysis by forcing the same metallicity for both components. However, the difference in effective temperatures do not vary again. Probably the problem with this system is the low S/N of the spectrum of the secondary star. While the primary has a nice spectrum and by consequence the parameters are well established, for the secondary both T_eff and v_t are uncertain, and by consequence the metallicity. It is worth noting that the mean temperature of the system (T₁ + T₂)/2 = 5275 K obtained from the atmospheric analysis is the same as the mean temperature calculated from color calibrations (i.e., 5315 K) to within the errors.

This system contains two stars with very similar masses and surface temperatures but with quite different radii. It is the only eccentric system in our sample. The system is very well-detached with no proximity effects visible in the light curve, not surprising given its long orbital period. The best solution is found for a logarithmic law of limb darkening, a marginal improvement to that found using a linear law of limb darkening.

An analysis of the broadening of the absorption lines provides an estimate of the ratio of the projected rotational velocities, v₁/v₂ ≈ 1.1. This value is consistent with the ratio of the radii and signifies similar periods of rotation for the components (tidal locking of both stars), in spite of their relatively large separation.

This is the only system in our sample with total eclipses. This allows us to determine the stellar radii with a precision of better than 1% for both components. This eclipsing binary shows quite large proximity effects because of the relatively large secondary component. An interesting feature is the reversed luminosity ratio of the components: the less massive secondary seems to be more evolved and much brighter than the hotter primary component. Taking into account the relative proximity of the stars, we cannot exclude that there has been an episode of mass transfer in the system when the present secondary star filled its Roche-lobe as it evolved along the red giant branch. The present system is detached, and such past mass exchange does not influence our distance determination.

The main contributions to the statistical uncertainties for all systems are presented in Table 10 (columns 3–7). The error in the semimajor axis is given as σA and an uncertainty of 1% in the absolute scale of the system translates to a 0.022 mag error in the distance modulus. We performed Monte Carlo simulations as described in Pietrzyński et al. (2013) for each of our binaries—see Figure 8. The resulting uncertainty of the distance is reported in column 4. This uncertainty takes into account the error in the determination of the relative radii, magnitude disentangling, and correlations between model parameters in the WD code. The main contribution to the uncertainty returned by the Monte Carlo simulations is the error in the sum of the relative radii r₁ + r₂. The statistical uncertainty in the extinction was assumed to be 0.020 mag in each case. Columns 6 and 7 give the standard deviations of the mean brightness at maximum light in the V and K bands, respectively. Remaining sources of statistical uncertainty, i.e., model atmosphere approximations, disentangling of individual magnitudes, the adopted limb darkening law, contribute at a level of 0.001 mag (Graczyk et al. 2012; Pietrzyński et al. 2013), and these are insignificant in the total statistical error budget.

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The sources of systematic uncertainty in our method are the following: the uncertainty of the empirical calibration of surface brightnesses by di Benedetto (2005) (0.040 mag), uncertainty in the extinction (0.020 mag, translating to a 0.010 mag error in distance), metallicity dependence on the surface brightness–color relation (0.004 mag), and zero point errors of V- and K-band photometry (0.010 mag and 0.015 mag, respectively). Combining these in quadrature, we obtain a systematic error of 0.048 mag in the distance modulus of each eclipsing binary.

Sandage & Tammann (1968) used observations of classical Cepheids to unambiguously show that the Magellanic Clouds have different distance moduli, corresponding to a distance difference of about 11 kpc. This difference is small enough that in practice almost all methods of distance determination can be employed simultaneously to both objects. In fact, the relative distance between the galaxies obtained from the application of a particular method is usually better constrained than the individual distances determined by that method. This is because distance determination methods are dominated by systematic uncertainties which mostly cancel out when used to obtain the distance to each galaxy with the same observational setup and similar quality data.

In Table 11, we summarize recent determinations of distance moduli differences between the Magellanic Clouds (Δμ = μ_SMC − μ_LMC) by authors who used exactly the same method for both galaxies. Distance moduli based on old population stars are metallicity corrected. In the case of our method, we adopted the distance modulus to the LMC from Pietrzyński et al. (2013), μ_LMC = 18.493 ± 0.006 mag. The interpretation of Table 11 is hindered by the fact that some authors do not clearly separate statistical and systematic uncertainties. The average from the young "metal-rich" population containing classical Cepheids and eclipsing binary systems is Δμ = 0.472 mag and the average from the old "metal-poor" population is Δμ = 0.453 mag. We summarize these distributions in Figure 9. The resulting distribution is slightly bimodal with a main peak at the position of the young population average. A Gaussian fit to the distribution gives Δμ = 0.458 ± 0.068 mag.

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The extended structure of the SMC in the line of sight was suggested by Johnson (1961). Over the last 30 yr, substantial evidence has been accumulated about the intricate and extended structure of the SMC from the studies of early-type stars and H ii regions (Martin et al. 1989), stellar clusters (Crowl et al. 2001), classical Cepheids (Welch et al. 1987; Groenewegen 2000; Haschke et al. 2012), RR Lyr stars (Subramanian & Subramaniam 2012; Haschke et al. 2012), red clump stars (Gardiner & Hawkins 1991; Hatzidimitriou et al. 1993; Subramanian & Subramaniam 2009, 2012), eclipsing binary stars (Hilditch et al. 2005; North et al. 2010), and red giant branch stars (Lah et al. 2005). The northeast part of the SMC was shown to be closer to us and have a substantial degree of complexity, and it is reported also to have the largest geometrical depth in line of sight (Subramanian & Subramaniam 2012). The apparent difference in the distance to the eclipsing binary SMC113.3 compared to the rest of our sample may confirm the existence of substructure in the front of the galaxy. We can interpret that the remaining eclipsing binaries are part of the main body of the galaxy.

The small dispersion in the distance moduli of our four binaries (0.05 mag, corresponding to ∼1.4 kpc) may signify a small geometrical depth of the main part of the SMC. We can investigate this more by an analysis of the distribution of differences in the distance moduli (Figure 9). The cumulative distribution has a dispersion of 0.068 mag. Because most of the systematic effects do not influence Δμ, and the geometrical depth of the central part of the LMC is small (Pietrzyński et al. 2013), we can interpret this spread as an imprint of the geometrical depth of the main part of the SMC. At a distance of 62 kpc, this dispersion corresponds to a 1σ line-of-sight depth of 4 kpc. Haschke et al. (2012) gave a review of recent line-of-sight depth estimates for the SMC, which we can use for a comparison. They concluded that the 1σ line-of-sight depth is between 4 and 5 kpc for the old population of stars (RR Lyr and red clump stars). For the Cepheids, the estimated depth is larger by up to ∼8 kpc, and in the case of intermediate-age clusters the depth reaches ∼10 kpc.

Graczyk et al. (2012, 2013) compiled some recent distance determinations to the SMC as reported up to 2011. Their conclusion was that the canonical value of the distance modulus to the SMC (μ_SMC = 18.90 mag, (Westerlund 1997)) remained marginally consistent with most recent determinations (see also Section 6.1). However, modern determinations prefer a slightly larger distance moduli of μ_SMC = 18.95 ± 0.07 mag. The inclusion of the most recent distance determinations (Haschke et al. 2012; Groenewegen 2013; Inno et al. 2013) does not affect this finding. In fact, if we adopt our estimate of the relative distance modulus between the Magellanic Clouds and we assume μ_LMC = 18.493 mag, we derive μ_SMC = 18.493 + 0.458 = 18.951 mag. This result remains in perfect agreement with our distance modulus measured with late-type eclipsing binaries, and being based on a number of independent standard candles, we advocate here its adoption as the "canonical" distance modulus to the SMC.