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The Araucaria Project. The Distance to the Small Magellanic Cloud from Near-Infrared Photometry of RR Lyrae Variables

We have obtained deep infrared J- and K-band observations of nine 4.9 × 4.9 arcmin fields in the Small Magellanic Cloud (SMC) with the ESO New Technology Telescope equipped with the SOFI infrared camera. In these fields, 34 RR Lyrae stars cataloged by the OGLE collaboration were identified. Using different theoretical and empirical calibrations of the infrared period–luminosity–metallicity relation, we find consistent SMC distance moduli, and find a best true distance modulus to the SMC of 18.97 ± 0.03 (statistical) ±0.12 (systematic) mag, which agrees well with most independent distance determinations to this galaxy, and puts the SMC 0.39 mag more distant than the Large Magellanic Cloud for which our group has recently derived, from the same technique, a distance of 18.58 mag.


Introduction
In our ongoing Araucaria Project (e.g. Gieren et al. 2005a), we are applying a number of different stellar standard candles to independently determine the distances to a sample of nearby galaxies. The systematic differences between the distance results obtained for the individual galaxies from the various stellar candles will be analyzed in forthcoming papers. This analysis is expected to lead to a detailed understanding of how the various stellar techniques, which are fundamental to calibrate the first rungs of the distance ladder, depend on metallicity and age. While the objects we use for the distance determinations are usually detected from optical wide-field imaging surveys of the target galaxies (e.g. Pietrzyński et al. 2002b), the most accurate distance work is then done from follow-up near-infrared images which virtually eliminate reddening as a significant source of error on the results. Examples of this very successful approach are the Cepheid work on the Sculptor galaxies NGC 300 (Gieren et al. 2005b), NGC 55 ) and NGC 247 (Gieren et al. 2009), and the red clump star work on the LMC (Pietrzyński & Gieren 2002). The Araucaria Project has also been developing a new spectroscopic distance indicator, viz. the Flux-weighted Gravity-Luminosity Relationship (FGLR) for blue supergiants (Kudritzki et al. 2003; which is able to yield distances accurate to 5% to galaxies containing massive blue stars out to about 10 Mpc from low-resolution spectra ).
Thanks to several recent theoretical and empirical studies, evidence has been mounting that RR Lyrae (RRL) stars are excellent standard candles in the near-infrared spectral range, providing distance results which are superior to the traditional optical method (e.g. Bono 2003a). Longmore et al. (1986) were the first to show that RRL variable stars follow a period-luminosity (PL) relation in the near-infrared K-band. Their pioneering work was followed by Liu & Janes (1990), Jones et al. (1996), and Skillen et al. (1993), who applied infrared versions of the Baade-Wesselink method to calibrate the luminosities and distances of RRL stars. A very comprehensive analysis of the IR properties of RRL stars was given by Nemec et al. (1994). The first theoretical constraints on the K-band PL relation of RRL stars are based on non-linear convective pulsation models which were presented by Bono et al. (2001). Dall'Ora et al. (2004) later demonstrated that the K-band PL relation for RRL stars appears to have a very small scatter for globular clusters, with small intrinsic spread in metallicity for this type of stars. Further theoretical explorations of the RRL period-mean magnitude-metallicity relations in near-infrared passbands were carried out by Bono et al. (2003b), Catelan et al. (2004), and Cassisi et al. (2004). Most recently, Sollima et al. (2008) analyzed near-infrared K-band data of RRL stars in some 15 Galactic globular clusters and provided the first empirical calibration of the period-luminosity-metallicity (PLZ) relation in the K band. All these existing theoretical and empirical studies have suggested that the RRL star K-band PLZ relation appears to be a superb means to determine accurate distances to galaxies hosting an abundant old stellar population.
We have therefore started to include this tool in the Araucaria Project distance work. In previous papers, we determined the distance to the Sculptor dwarf galaxy , and to the LMC (Szewczyk et al. 2008) from this method. In this paper, we apply the technique to a sample of RR Lyrae stars distributed over the SMC, and determine the distance to the SMC for the first time with the infrared RR Lyrae technique.

Observations, Data Reduction and Calibration
During several observing runs of the Araucaria Project, we have obtained all the data analyzed and presented in this paper with the SOFI infrared camera attached to the ESO New Technology Telescope. The Large Field setup was used to yielding a 4.9 × 4.9 arcmin field of view and a scale of 0.288 arcsec per pixel.
It was possible to collect deep Js and Ks images of six distinct SMC fields during 3 nights in 2007, and of three more fields over 2 nights in 2008. All but one nights were found to have photometric conditions. Detailed information on each observed field can be found in Table 1. Figure 1 illustrates the location of the observed fields on an image of the SMC. In order to take into account the rapid sky level variations in the IR passbands, we have been using a dithering technique during the observations. Total integration times per field were 60 min in the Ks band, and 15 min in the Js band.
For all the reductions and calibrations, the pipeline developed in the course of the Araucaria Project was used. As a first step, the subtraction of sky level subprocess (including the masking of stars with the IRAF xdimsum package) was applied (Pietrzyński & Gieren 2002). Later, each single image was flatfielded and stacked into the final deep field. PSF (point spread-function) photometry with aperture corrections was then performed in the same way as described in Pietrzyński et al. (2002b).
The photometry was calibrated onto the standard system using observations of 15 standard stars from the UKIRT list (Hawarden et al. 2001). All of them were observed under photometric conditions at different airmasses spread in between the regular target fields acquisition. The data obtained during the non-photometric night was compared with nearby photometric data and the zero-point difference was estimated. Thanks to the large number of standard stars observed, the accuracy of our photometric zero points was estimated to be as good as 0.02 mag. The results were compared with the Two Micron All Sky Survey (2MASS) catalog for common stars. This way we could determine the zero point differences which are given in Table 2. The calibrated NIR magnitudes for all RRL stars identified in our science target fields are presented in Table 3. All uncertainties given in the Tables are standard deviations.

Near-Infrared Period-Luminosity Relations
The final sample of 34 RRL stars analyzed in our SOFI/NTT fields was chosen by rejecting all stars which were blended with or contaminated by other close-by objects in the frames. Then it was cross-identified with the Optical Gravitational Lensing Experiment (OGLE) catalog of RRL stars in the SMC (Soszyński et al. 2002). The positions of the RRL stars in the K, J − K color-magnitude diagram are shown in Figure 2. All variables have at least one measurement collected during photometric conditions. In the case of RRL stars which were observed more than once, we took a straight average of the random-phase magnitudes, which is expected to lead to a better approximation of their mean magnitudes.
Following the identification from the OGLE catalog (Soszyński et al. 2002), we fun-damentalize the RRc periods by adding log(P ) = 0.127 in order to combine the RRab and RRc stars. We have corrected the observed magnitudes for extinction, adopting the reddening values published by Udalski et al. (1999) which are shown in Table 1. Adopting the reddening law from Schlegel et al (1998), we calculate the following selective extinctions in the different bands: . The final photometric data for the sample of RRL stars are presented in Table 4. The PL relations for the J and K bands derived from our magnitudes are shown in Figure 3. We compare the PL relations derived from the observed RRL stars against the existing theoretical (Bono et al. 2003;Catelan et al. 2004) and empirical (Sollima et al. 2008) relations. Table 5 lists our results for the slope and zero-point values of the PL relations and as well as those from the theoretical and empirical ones.

The Distance Determination
In order to derive the apparent distance moduli to the SMC from our data, we used the following calibratons of the NIR PL relations of mixed population RRL stars: We recall that the calibration of Sollima et al. (2008) was constructed for the 2MASS photometric system, while the calibrations of Catelan et al. (2004) and Bono et al. (2003) are usable for the Glass or Bessel & Brett systems, respectively. Therefore we transformed our own data, calibrated onto the UKIRT system (Hawarden et al. 2001), to the Glass and Bessel & Brett systems using the transformations presented by Carpenter (2001) before calculating distances using the calibrations of Catelan et al. (2004) and Bono et al. (2003). Since there is no signifficant difference between the K-band of 2MASS and UKIRT systems (Carpenter 2001), we did not apply any transformations to our data while using the Sollima et al. (2008) calibration.
Assuming the mean metallicity of RRL stars in the SMC to be [F e/H] = −1.7 dex (e.g. Udalski 2000) we have calculated the K-and J-band distance moduli for our sample of RRL stars using the final photometric magnitudes as given in Table 4. The fits for the relations (1) − (4) to both sets of data are displayed in Figure 4.
The true distance moduli of the SMC obtained from the different PLZ calibrations are summarized in Table 6. As our best distance determination, we adopt the average value from the different calibrations, yielding 18.97 mag. The uncertainty on this value will be discussed in the next section.

Discussion
The distance moduli obtained from several independent theoretical and empirical calibrations of the infrared RR Lyrae PLZ relation are consistent. The maximum difference of 0.06 mag between the results from the calibrations of Sollima et al. (2008) and Bono et al. (2003b) is certainly not significant taking into account all the uncertainties, which affect the whole process of constructing these calibrations. It is interesting to note that a very similar difference between the distance moduli derived using these two calibrations was recently obtained by Pietrzyński et al. (2008) for the Sculptor galaxy ([F e/H] = −1.83 dex). Therefore, perhaps there is just a zero point offset in the sense that the distances from the calibration of Sollima et al. (2008) are slightly shorter compared to those from the calibration of Bono et al. (2003b).
Taking into account the errors associated with the adopted calibrations, mean metallicity, photometric zero point and absorption correction, we estimate the systematic error of our distance determination to be of 0.12 mag. Therefore our best distance modulus determination to the SMC is: 18.97 ± 0.03 (statistical) ±0.12 (systematic) mag. This value agrees very well with the value of 18.967 ± 0.018 mag derived from K-band photometry of red clump stars (Pietrzyński, Gieren & Udalski 2003). It also agrees, within the combined 1σ uncertainties, with the true SMC distance modulus derived from Cepheid variables (e.g. Udalski 2000; Barnes et al. 1993). The distance modulus of the SMC found in this paper is 0.39 mag larger than the value for the LMC we found from the same technique (Szewczyk et al. 2008). Again, this difference is very much in line with modern values found from other techniques, like Cepheid variables (0.5 mag; Udalski 2000), and red clump stars (0.47 mag; Pietrzyński, Gieren & Udalski 2003).
We do not observe any significant difference in the dispersions of the NIR P-L relations defined by the RR Lyrae variables in the SMC (this paper), and the LMC (Szewczyk et al. 2008), arguing against a more significant distance spread in the line of sight in the SMC than in the SMC, as usually found for younger tracers like Cepheids.

Summary and Conclusions
The results of our deep infrared imaging and photometry of 34 RR Lyrae stars spread over nine 4.9 × 4.9 arcmin fields in the SMC are presented. Our data show two clear sequences in the period luminosity plane, which correspond to the RRc and RRab stars. After fundamentalizing the RRc periods to the period of the RRab stars by adding log P = 0.127, both groups were merged, and the distance moduli to the SMC were determined using different theoretical and empirical calibrations. Our final adopted distance agrees very well with modern results obtained from a number of different techniques.
Our results confirm that the RR Lyrae period-luminosity-metallicity relations in the near-infrared passbands are potentially a very good tool for accurate distance measurements. In future programs, particularly for the LMC, we want to reduce the systematic uncertainty of our distance determinations with this method by providing multi-phase JK observations of the RR Lyrae variables to determine their mean magnitudes with better precision.
WG and GP gratefully acknowledge financial support for this work from the Chilean Center for Astrophysics FONDAP 15010003, and from the BASAL Center for Astrophysics and Related Tecnologies (CATA 2007 PFB 06). Support from the Polish grant N203 002 31/046 and the FOCUS subsidy of the Fundation for Polish Science (FNP) is also acknowledged. It is a special pleasure to thank the support astronomers at ESO-La Silla for their expert help in the observations, and the ESO OPC for the generous amounts of observing time at the NTT allocated to our programme.