The Araucaria Project. The Distance to the Sculptor Group Galaxy NGC 7793 from Near-Infrared Photometry of Cepheid Variables

Following the earlier discovery of classical Cepheid variables in the Sculptor Group spiral galaxy NGC 7793 from an optical wide-field imaging survey, we have performed deep near-infrared $J$- and $K$-band follow-up photometry of a subsample of these Cepheids to derive the distance to this galaxy with a higher accuracy than what was possible from optical photometry alone, by minimizing the effects of reddening and metallicity on the distance result. Combining our new near-infrared period-luminosity relations with the previous optical photometry we obtain a true distance modulus to NGC 7793 of $(27.66 \pm 0.04)$ mag (statistical) $\pm 0.07$ mag (systematic), i.e. a distance of $(3.40 \pm 0.17)$ Mpc. We also determine the mean reddening affecting the Cepheids to be $E(B-V)=(0.08 \pm 0.02)$ mag, demonstrating that there is significant dust extinction intrinsic to the galaxy in addition to the small foreground extinction. A comparison of the new, improved Cepheid distance to earlier distance determinations of NGC 7793 from the Tully-Fisher and TRGB methods yields agreement within the reported uncertainties of these previous measurements.


Introduction
The Araucaria Project is an international project focusing on precise calibration of the cosmic distance scale, using a variety of stellar distance indicators (Gieren et al. 2005a). The calibration of the first rungs of the distance ladder still resembles the most important problem in the determination of the Hubble constant (e.g. Freedman et al. 2001;Riess et al. 2016). Nearby galaxies like those in the Local Group or the Sculptor Group provide perfect environments for improving stellar methods of distance measurement and determine their dependence on metallicity. Important results have been achieved by our project in the recent past; for instance, we measured the distance to the Magellanic Clouds with an unprecedented accuracy using eclipsing binaries, 2% in the case of the LMC (Pietrzynski et al. 2013), and 3% in the case of the SMC (Graczyk et al. 2014). With these results, the Magellanic Clouds, and in particular the LMC, are now firm anchors for the extragalactic distance scale. Our project has also shown that combined optical and near-infrared photometry of classical Cepheid variables provides an excellent way to measure the distances to nearby spiral or irregular galaxies with a typical precision of 3% (e.g. NGC 300, Gieren et al. 2005b;NGC 55, Gieren et al. 2008;NGC 247, Gieren et al. 2009;IC 1613, Pietrzynski et al. 2006NGC 3109, Soszynski et al. 2006 andM 33, Gieren et al. 2013).
In a previous study, we reported on the discovery of a sample of classical Cepheids in the Sculptor Group spiral galaxy NGC 7793 and derived a first Cepheid distance to this galaxy based on these objects to be 27.68 ± 0.05 mag (statistical) ±0.08 mag (systematic) (Pietrzynski et al. 2010;hereafter Paper I). The obvious drawback of this distance determination from optical (V,I) data was its relatively high sensitivity to dust extinction, including dust extinction produced in the galaxy itself which must affect the Cepheids' luminosities. In addition, evidence has been mounting in recent years that -6the effect of metallicity on Cepheid absolute magnitudes is smaller in near-infrared than in optical bands, making a distance measurement from near-infrared photometry very desirable. We therefore obtained near-infrared follow-up photometry of the Cepheids, as reported in section 2 of this paper. An accurate distance to NGC 7793 from Cepheids is of high importance because this galaxy has previous distance determinations from the TRGB and Tully-Fisher methods whose accuracy can be checked by comparing with the Cepheid distance. Moreover, the Type IIP supernova SN 2008bk has been observed in the galaxy and an improved Cepheid distance will lead to a more accurate determination of its absolute peak magnitude. Also, the previous Cepheid distance measurement from optical data indicated that NGC 7793 might be the most distant spiral galaxy in the Sculptor Group, arguing for a significant extension of the Group in the line of sight, a conclusion that we wanted to confirm with an improved Cepheid distance from infrared data.

Observations, Data Reduction and Calibration
We collected our J-and K-band data under photometric conditions using the High Acuity Wide field K-band Imager (HAWK-I) installed on the 8.2m ESO Very Large Telescope at Paranal as a part of a program 087.D-0425(B). The field of view of HAWK-I is 7.5 x 7.5 arcmin and the pixel scale is 0.106 arcsec/pix. The detector of HAWK-I consists of four chips with a 15 arcsec cross-shaped gap between them. We observed one field during two photometric nights on September 5 and September 18, 2011. The location of this field is shown in Figure 1. It was chosen such as to cover as many of the previously known Cepheids as possible (13 out of 17). The periods of the Cepheids and a preliminary distance from the optical data were published in Paper I.
From the 13 Cepheids in NGC 7793 located in our HAWK-I field, we were able to identify eleven objects. One Cepheid fell into a gap between the different chips of the -7 -HAWK-I imager, and another Cepheid, located closer to the center of the galaxy than any other of the variables, could not be clearly identified because the field in that region was too crowded in the near-infrared.
The HAWK-I observations in the J-and K-bands were performed using a dithering technique with a jitter radius of 30 -40 arcsec in order to subtract the sky. For each band, 30 exposures were taken resulting in 30 minutes of total net exposure. To avoid saturation, each exposure performed in the J (K) filter consisted of six (four) sub-integrations of 10 seconds (15 seconds) each.
In order to reduce our data, we used the dedicated ESO HAWK-I pipeline implemented in Esorex software and corresponding optional Gasgano graphic interface. It includes all necessary procedures such as basic reduction, sky subtraction, correction for distortion of the detector and stacking into the final images. We performed point-spread function (PSF) photometry of our images using DAOPHOT and ALLSTAR, as described in Pietrzynski et al. (2002) after dividing each multi-extension fit into four ordinary fits corresponding to each chip of the detector. We carefully selected about 7-20 isolated stars to calculate the PSF models on each chip.
The calibration of the photometry onto the standard system was based on the observations of 14 standard stars from the United Kingdom Infrared Telescope ( UKIRT ) list (Hawarden et al. 2001) observed on 2 photometric nights. All of them were observed along with the target fields in NGC 7793 at different air masses and under photometric conditions. We obtained the absolute photometric zero points with an accuracy close to 0.03 mag, in both filters. This is demonstrated in Figure 2 which shows the histograms of the measured magnitude differences for constant stars in our fields between the two nights.
In addition we analysed IR images of NGC 7793 which we obtained with the ISAAC near-infrared camera on the ESO VLT as part of ESO proposal 171.D-0004(A). These -8observations were obtained under photometric conditions but the integration times for both J and K bands were too short to obtain accurate photometry of the Cepheids in this galaxy.
However, they are very well suited for checking the zero points of our HAWK-I photometry.
In order to do that we reduced the ISAAC data in the same manner as reported in our previous papers (e.g. Gieren et al. 2005b). The calibration to the UKIRT system was based on 7 standard stars. The photometric zero points were determined with an accuracy better than 0.03 mag for both J and K filters.
The observed ISAAC field overlaps with all HAWK-I chips. We cross-matched common stars in both photometric lists and calculated the mean difference between ISAAC and HAWKI photometry. In all cases the differences in the zero points in both J and K filters, and for all HAWK-I chips were smaller than 0.03 mag, a result which strongly reinforces both calibrations.

Near-infrared photometry and mean magnitudes of the Cepheids
In the resulting near-infrared photometric catalog we were able to identify, as stated above, 11 Cepheids previously discovered by Pietrzynski et al. (2010). These variables span a period range from 26 to 62 days. Figure 3 shows their location in the K versus J − K color-magnitude diagram obtained from our data. As one can appreciate, the Cepheids fall into the expected region of the HRD and do fill the instability strip rather homogeneously, showing that no significant selection bias affects our period-luminosity relations. It is also worthwhile noticing that the faintest Cepheid in our sample is still about 3 mag brighter than the faintest stars measured in the field. Therefore we do not expect that our near-infrared period-luminosity relations are significantly affected by a Malmquist bias.
-9 -In order to calculate the Cepheid mean magnitudes, we simply took the average of the two observations per Cepheid obtained in this study (except the object cep012 which has only one measurement in J and K each, which we adopted as its respective mean magnitudes). Table 1 presents the detailed journal of individual observations of our 11 Cepheids. As can be seen, the four Cepheids with short periods (26-28 days) and two observations are brighter on September 18 than on September 5. It should be noted that for these variables the phase progression between the first and second observing night was just about half a pulsation cycle, which can lead to a large difference in their magnitudes on these nights, a fact we have confirmed by simulations. This shows that even having a reasonably large sample of Cepheids it is important to obtain more than one observation in order to determine accurate mean magnitudes for the individual variables.
In principle one could follow the procedures described by Soszynski et al. (2005) and Inno et al. (2015) which allow to calculate the mean magnitude of a Cepheid in near-infrared bands from a single random-phase NIR magnitude measurement if accurate optical light curves and period of the Cepheid are available. The method however relies critically on well-known phase shifts between the optical and near-infrared observations. In our case, the time interval between our previous optical datasets and the current near-infrared follow-up observations of the Cepheids is too large (6-8 years) to apply this technique, so we decided to adopt the average the of the two measured magnitudes to estimate the mean magnitudes.

Near-infrared period-luminosity relations and distance determination
In order to calculate the distance moduli of NGC 7793 relative to the LMC, we had to determine the shifts between the Cepheid PL relations in NGC 7793 and in the LMC in the J and K bands. To do this, we converted our magnitudes from the UKIRT to the Near-Infrared Camera and Multi-Object Spectrometer (NICMOS) system on which the zero-points of Persson et al. (2004) were calibrated. According to Hawarden et al. (2001) there are only small and constant zero-point differences between the UKIRT and NICMOS systems: 0.034 mag and 0.015 mag for J and K, respectively.
To obtain the absolute distance moduli to NGC 7793, we adopted the very accurate (2%) LMC distance modulus of 18.493 mag measured by Pietrzynski et al. (2013) which is in excellent agreement with the value of 18.50 we had adopted in our previous Araucaria work. Thanks to this very accurate distance measurement to the LMC we are also able to improve the NGC 7793 distance moduli from the V -and I-bands given originally in Paper I.
The least-squares fit shown in Figure 5 yields for the true distance modulus of NGC 7793, and the total reddening affecting its Cepheids, the following results: corresponding to distance to NGC 7793 of d = 3.40 Mpc.
The last line in Table 2

Discussion and conclusions
There are several factors contributing to the systematic uncertainty of the distance determination to NGC 7793 derived in this paper. These include the accuracy of the zero point of the photometry, the accuracy of the distance to the LMC which we have adopted as the fiducial galaxy to which the Cepheid PL relation in NGC 7793 is tied, the uncertainty related to a metallicity correction due to a possible metallicity dependence of the Cepheid PL relation in tandem with a possible systematic metallicity difference between the sample -12of Cepheids in NGC 7793 selected for this study, and the sample of calibrating Cepheids in the LMC from the OGLE project.
As argued in section 2 of this paper, we estimate the uncertainty of our photometric calibration and the resulting zero points accuracy of around 0.03 mag. Regarding the distance to the LMC we have adopted, it was shown by Pietrzynski et al (2013) that its total uncertainty, including any systematics, is 2.2%, or 0.05 mag. While the effect of metallicity on Cepheid absolute magnitudes is still under dispute, evidence has been mounting that the effect is very small, and possible vanishing at near-infrared wavelengths (Freedman and Madore 2011;Storm et al. 2011;Romaniello et al. 2008). This conclusion is supported by a recent comparison of the distance difference between the two Magellanic Clouds (whose Cepheid populations exhibit a metallicity difference of about 0.4 dex (Luck et al. 1998)) derived from the near-geometrical, metallicity-independent eclipsing binary method (Pietrzynski et al. 2013, Graczyk et al. 2014, and the distance difference between SMC and LMC as determined from the magnitude offsets of the observed Cepheid K-band PL relations in the two galaxies: the two determinations agree to within 1% (Wielgorski et al. 2017). We also note that the metallicities of the young stellar populations in NGC 7793 and the LMC seem to agree to within 0.1 dex (Van Dyk et al. 2012 andStanghellini et al. 2015), minimizing a metallicity correction to our distance result even if the metallicity effect in the J and K bands were much larger than expected according to the studies cited above.
Another factor which might affect the Cepheid magnitudes is the effect of crowding and blending in our images. In an earlier study of our project, we made an exhaustive test of how blending affects the distance to another spiral member of the Sculptor Group, NGC 300, and concluded that its distance modulus was affected by less than 0.04 mag (Bresolin et al. 2005). While NGC 7793 is more distant than NGC 300, the HAWK-I imager used in the present study has a higher resolution (seeing for our HAWK-I data is about 0.5 -13arcsec) than the ISAAC near-infrared camera we used in the NGC 300 NIR work (Gieren et al. 2005b), leading to the expectation that the blending effect on the current NGC 7793 Cepheids should be of a similar size as the one we measured in NGC 300, given also the fact that both NGC 300 and NGC 7793 have very similar inclinations (nearly face-on) with respect to the line-of-sight. We adopt 0.04 mag as a contribution of the effect of crowding and blending to the systematic uncertainty of our current distance determination to NGC 7793. The total systematic uncertainty on our current distance measurement of NGC 7793 is therefore estimated to be about 0.07 mag, or 3%.
Apart from using Cepheids in this work, and previously in Paper I for a distance determination to NGC 7793, the distance to NGC 7793 has also been determined by applying the TRGB method (Karachentsev et al. 2003;Jacobs et al. 2009) and the  ), is consistent with our current Cepheid determination but has a much larger uncertainty.
Our measurement confirms the large difference between the distances to the Sculptor Group galaxies NGC 300 -(m − M) 0 = 26.37 ± 0.04 ± 0.03 mag or d = 1.88 ± 0.06 Mpc (Gieren et al. 2005b), NGC 55 -(m − M) 0 = 26.434 ± 0.037 mag or d = 1.88 ± 0.03 Mpc ) and NGC 7793, confirming the elongated structure of the Sculptor Group in the line of sight. It might even be possible that NGC 7793 does not belong to the Group, but a discussion of this possibility is beyond the scope of this study.
As the principal result of this study, we confirm, and significantly improve the accuracy -14of the distance determination to NGC 7793 obtained in Paper I. Extending our wavelength baseline to 2.2 micron we confirm that there is significant intrinsic reddening in NGC 7793 which dominates the total reddening affecting the Cepheids in the galaxy, and which will need to be taken into account in any other photometric distance determination to this galaxy. The current distance measurement to NGC 7793 from optical and near-infrared photometry of Cepheid variables is estimated to be accurate to about 5%.
We would like to thank the anonymous referee for constructive and helpful comments. averaging two random-phase observations. The respective slopes on these diagrams were adopted from the LMC Cepheids (Persson et al. 2004). plotted against the ratio of total to selective extinction as adopted from the Schlegel et al. (1998) reddening law. The intersection and slope of the best-fitting line give the true distance modulus and reddening, respectively. The extremely small scatter of the reddened distance moduli in the different bands around the best-fitting line is spurious.
-22 -Table 1. Journal of J-and K-band individual observations of NGC 7793 Cepheids.
Names of objects and periods (accurate to 10 −3 × P) were adopted from Pietrzynski et al.