THE ARAUCARIA PROJECT: THE DISTANCE TO THE CARINA DWARF GALAXY FROM INFRARED PHOTOMETRY OF RR LYRAE STARS^*

The main goal of the Araucaria Project is to improve the calibration of the cosmic distance scale from accurate observations of various stellar distance indicators in nearby galaxies (e.g., Gieren et al. 2005; Pietrzyński et al. 2013b). In a long-term perspective, this analysis is expected to lead to a detailed understanding of the impact of metallicity or/and age on the various standard candles that are fundamental to calibrate the first rungs of the cosmic distance ladder. Errors in distance measurement methods propagate up the distance ladder, so minimization of errors at a local scale can significantly improve the accuracy of secondary distance indicators, and thus improve the determination of the Hubble constant. Significant improvement in the accuracy of distance measurements is possible by employing near-infrared (NIR) photometry, which minimizes the influence of both interstellar extinction and metallicity on the brightness of most stellar distance indicators (e.g., Pietrzyński et al. 2003; Gieren et al. 2009).

Several theoretical and empirical studies of RR Lyrae (RRL) stars demonstrated that these stars are able to provide improved distances from their magnitudes in the NIR domain as compared to traditional optical studies (e.g., Bono et al. 2003). Longmore et al. (1986) were the first to show that RRL stars follow a period–luminosity (PL) relation in the K band. Important contributions were added by Nemec et al. (1994), who made a comprehensive analysis of IR properties of RRL stars, and Bono et al. (2001), who put the first theoretical constrains on the K-band PL relation of RRLs based on nonlinear convective pulsation models.

The continuation of theoretical studies of RRL stars in NIR wavebands carried out by Bono et al. (2003) and Catelan et al. (2004) yielded the period–luminosity–metallicity (PLZ) relation in the NIR domain. They reported that the effect of metallicity on the luminosity of RRLs is noticeably smaller in the IR domain compared to the visual domain. Indeed, the metallicity term enters the PLZ relation with a coefficient of approximately 0.2 mag dex⁻¹ in the K band (Bono et al. 2003), while in the V band the same metallicity coefficient can reach 0.3 mag dex⁻¹ (Di Criscienzo et al. 2004). Following theoretical studies, empirical PLZ relations reported by Sollima et al. (2008), Borissova et al. (2009), and Dékány et al. (2013) showed the metallicity coefficient to be even smaller, ranging from 0.05 to 0.17 mag dex⁻¹. While in the optical domain the effect of metallicity (and even more subtle factors, like the [α/Fe] ratio) on the PLZ relation is extensively studied, truly precise calibration of the PLZ in the NIR requires further theoretical and empirical work, and is beyond the scope of this paper.

In summary, detailed theoretical and empirical studies of RRL variables in the NIR domain demonstrated a decrease in scatter in the PL plane as well as a decrease in the metallicity dependence toward longer wavelengths. These two NIR properties of RRL variables, together with the fact that the interstellar reddening has a marginal effect on the brightness of stars in the IR domain, make the NIR PLZ relation for RRL stars an excellent means for distance determination to galaxies with a large old stellar population. This successful technique has already been used to provide accurate distances to the LMC (Szewczyk et al. 2008; Borissova et al. 2009), the SMC (Szewczyk et al. 2009), and the Sculptor dSph galaxy (Pietrzyński et al. 2008). In this paper, we report the first distance determination to the Carina dSph galaxy from an application of J- and K-band PLZ relations on Carina RRL stars. This study complements the existing distance determinations to Carina (Table 1) with a very competitive (accuracy at the 5% level or better) new measurement.

Table 1.  Summary of Recent Distance Determinations for the Carina dSph Galaxy, Obtained with Different Stellar Indicators, in the Optical and Near-infrared Domains

${(m-M)}_{0}$	Methoda	Band	Reference
19.87 ± 0.11	CMD	V, I	Mighell (1997)
20.24 ± ...	CMD	V	Monelli et al. (2003)
19.96 ± 0.08	RC	I	Udalski (2000)
19.96 ± 0.06	RC	I	Girardi & Salaris (2001)
20.165 ± 0.015	RC	K	Pietrzyński et al. (2003)
20.05 ± 0.09	TRGB	I	Smecker-Hane et al. (1994)
19.94 ± 0.08	TRGB	I	Udalski (2000)
20.09 ± 0.03 ± 0.12	TRGB	J	Pietrzyński et al. (2009)
20.14 ± 0.04 ± 0.14	TRGB	K	Pietrzyński et al. (2009)
20.06 ± 0.12	DC	V	Mateo et al. (1998)
20.17 ± 0.10	DC	B, V	Vivas & Mateo (2013)
20.01 ± 0.05	SXP	V	McNamara (1995)
20.22 ± 0.05	DS	V	McNamara (2011)
20.12 ± 0.08	HB/RRL	I	Smecker-Hane et al. (1994)
19.93 ± 0.08	RRL	V	Udalski (2000)
20.10 ± 0.12	RRL	V	Dall'Ora et al. (2003)
20.18 ± 0.05	RRL	V	McNamara (2011)
20.09 ± 0.07 ± 0.10	RRL	B, V	Coppola et al. (2013)
20.118 ± 0.017 ± 0.11	RRL	J, K	This paper

Note.

^aDistance determinations are based on: color–magnitude diagram (CMD), red clump stars (RC), tip of red giant branch stars (TRGB), dwarf Cepheids (DC), SX Phoenicis variables (SXP), δ Scuti variables (DS), horizontal branch stars (HB), and RR Lyrae variables (RRL).

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NIR observations were conducted between 2008 November 11 and 2008 December 5 with the High Acuity Wide-field K-band Imager (HAWK-I) installed at the Nasmyth focus of the UT4/VLT ESO 8 m telescope at the Paranal observatory in Chile. The dates and coordinates of target fields are given in Table 2. The location of six target fields in the Carina dSph galaxy was purposely chosen to overlap the Wide Field Imager (WFI) field from Dall'Ora et al. (2003), as shown in Figure 1. The HAWK-I field of view of four 2048 × 2048 pixel detectors was about 75 × 75 with a scale of 0106 pixel⁻¹ (a detailed description of the HAWK-I instrument is available in Kissler-Patig et al. 2008 and on the ESO website⁴ ). Six fields were observed in the J and K_s (further denoted as K) wavebands, under photometric conditions (seeing range 04–07). Our objects, having a minimal brightness higher than 20 mag in K and 21 mag in J, were well above the limiting HAWK-I magnitudes, i.e., 23.9 mag (J) and 22.3 mag (K) (Kissler-Patig et al. 2008). In order to minimize the influence of sky background, the observations were made in jitter mode, where the telescope was offset randomly between consecutive exposures, at a distance of 20'' from the original pointing. Total integration times were 31 and 15 minutes for the K and J bands, respectively.

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2.1. HAWK-I Reduction Pipeline

2.2. Photometry and Calibration

2.3. Identification of Variables

The list of variables from Dall'Ora et al. (2003) served as a reference to cross-identify 33 RRL (29 RRab + 4 RRc) stars in our fields. Although our six HAWK-I fields overlapped the WFI field only 30% of the time (see Figure 1), the sample of identified RRL stars constitutes 61% of the sample of Dall'Ora. The position of our RRLs on the K, J–K color–magnitude diagram (CMD) is shown in Figure 2. All RRLs from our final sample have at least one measurement collected during photometric conditions. In the case of nine RRL stars, which were observed twice or found in two overlapping fields, we took a straight average of the random-phased magnitudes, which is expected to lead to a better approximation of their mean magnitudes. In case of single-measurement RRL stars, we took advantage of RRL infrared light curves (nearly sinusoidal and small-amplitude) and accepted single-phased measurements of J and K magnitudes as reasonably accurate approximations of their mean magnitudes in these bands. Indeed, since the RRL NIR amplitude is not larger than 0.4 mag (Marconi et al. 2003; Szabó et al. 2014), random-phased single measurements may cause deviations from the mean magnitude of a maximum of 0.2 mag for a single star. This error is extensively reduced by taking a large sample of stars. The J- and K-band random-phased magnitudes of the final RRL sample together with formal photometry errors from DAOPHOT and pulsational periods from the reference list (Dall'Ora et al. 2003) are presented in Table 4. In the case of nine RRL stars observed more than once, individual observations were put in separate rows.

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Table 4.  Random-phase J and K Magnitudes of the Complete Sample of 33 RR Lyrae Stars. In Case of Nine RRL Stars Observed More than Once, Individual Observations are Placed in Separate Rows

Star IDa	HAWK-I field	R.A.	Decl.	Period	Type	J	σJ	K	σK	$J-K$
		(J2000.0)	(J2000.0)	(d)		(mag)	(mag)	(mag)	(mag)	(mag)
V30	CAR-FIII	06:42:24.16	−51:02:39.1	0.619	ab	19.601	0.010	19.318	0.026	0.286
V31	CAR-FIII	06:42:22.98	−50:59:16.1	0.651	ab	19.528	0.010	19.203	0.019	0.327
V40	CAR-FIV	06:42:15.63	−51:06:59.9	0.394	c	19.923	0.013	19.600	0.021	0.324
V47	CAR-FV	06:42:09.04	−50:53:53.4	0.324	c	19.851	0.020	19.741	0.028	0.111
V49	CAR-FIII	06:42:08.85	−51:01:11.8	0.686	ab	19.566	0.012	19.260	0.020	0.308
V57	CAR-FV	06:42:02.80	−50:52:56.5	0.612	ab	19.552	0.015	19.293	0.022	0.259
V60	CAR-FIV	06:41:59.75	−51:06:38.2	0.615	ab	19.769	0.014	19.377	0.022	0.392
V65	CAR-FV	06:41:55.77	−50:55:34.9	0.642	ab	19.918	0.021	19.539	0.025	0.378
V67	CAR-FII	06:41:52.66	−51:05:19.3	0.613	ab	19.774	0.012	19.441	0.022	0.335
V67	CAR-FII	06:41:52.66	−51:05:19.3	0.613	ab	19.758	0.012	19.392	0.022	0.368
V67	CAR-FII	06:41:52.66	−51:05:19.3	0.613	ab	19.786	0.012	19.486	0.023	0.300
V73	CAR-FIV	06:41:46.91	−51:07:06.9	0.571	ab	19.819	0.014	19.496	0.022	0.323
V74	CAR-FV	06:41:43.79	−50:54:08.3	0.403	c	19.733	0.017	19.485	0.031	0.248
V77	CAR-FII	06:41:39.26	−51:05:39.5	0.605	ab	19.797	0.014	19.437	0.021	0.361
V77	CAR-FII	06:41:39.26	−51:05:39.5	0.605	ab	19.871	0.014	19.454	0.022	0.417
V85	CAR-FV	06:41:35.66	−50:50:07.1	0.644	ab	19.800	0.017	19.405	0.025	0.393
V91	CAR-FII	06:41:29.47	−51:04:28.4	0.720	ab	19.754	0.010	19.339	0.020	0.415
V91	CAR-FII	06:41:29.47	−51:04:28.4	0.720	ab	19.754	0.012	19.386	0.018	0.367
V92	CAR-FII	06:41:28.77	−51:06:50.3	0.620	ab	19.568	0.012	19.233	0.022	0.335
V92	CAR-FII	06:41:28.77	−51:06:50.3	0.620	ab	19.758	0.013	19.539	0.024	0.220
V105	CAR-FII	06:41:16.57	−51:09:02.8	0.630	ab	19.548	0.012	19.243	0.020	0.305
V105	CAR-FII	06:41:16.57	−51:09:02.8	0.630	ab	19.693	0.012	19.388	0.022	0.308
V116	CAR-FI	06:41:05.60	−51:00:28.2	0.685	ab	19.777	0.012	19.497	0.023	0.280
V122	CAR-FI	06:40:56.95	−51:00:46.0	0.627	ab	19.539	0.010	19.230	0.020	0.309
V125	CAR-FI	06:40:53.33	−50:58:53.4	0.597	ab	19.796	0.013	19.443	0.023	0.353
V126	CAR-FI	06:40:52.31	−50:58:56.7	0.560	ab	19.896	0.014	19.543	0.025	0.353
V127	CAR-FI	06:40:52.19	−50:59:20.9	0.625	ab	19.692	0.011	19.349	0.020	0.343
V135	CAR-VI	06:40:46.44	−51:05:23.6	0.590	ab	19.575	0.011	19.266	0.019	0.311
V135	CAR-VI	06:40:46.44	−51:05:23.6	0.590	ab	19.800	0.012	19.481	0.021	0.319
V143	CAR-VI	06:40:38.25	−51:11:29.6	0.601	ab	19.665	0.013	19.335	0.024	0.330
V159	CAR-VI	06:40:12.75	−51:09:35.6	0.578	ab	19.557	0.012	19.332	0.020	0.229
V159	CAR-VI	06:40:12.75	−51:09:35.6	0.578	ab	19.879	0.014	19.586	0.024	0.293
V179	CAR-FV	06:41:49.33	−50:54:10.6	0.665	ab	19.523	0.017	19.232	0.024	0.291
V188	CAR-VI	06:40:12.37	−51:07:06.1	0.600	ab	19.621	0.012	19.358	0.021	0.263
V188	CAR-VI	06:40:12.37	−51:07:06.1	0.600	ab	19.646	0.012	19.374	0.021	0.272
V189	CAR-VI	06:40:09.97	−51:08:05.1	0.700	ab	19.544	0.010	19.313	0.018	0.232
V197	CAR-FI	06:41:00.26	−51:02:39.7	0.295	c	20.117	0.014	19.910	0.032	0.207
V199	CAR-FII	06:41:19.65	−51:10:23.3	0.784	ab	19.402	0.009	19.126	0.018	0.279
V199	CAR-FII	06:41:19.65	−51:10:23.3	0.784	ab	19.506	0.009	19.142	0.018	0.364
V201	CAR-FIII	06:42:10.49	−50:59:07.5	0.692	ab	19.646	0.010	19.315	0.018	0.332
V202	CAR-FIII	06:42:22.61	−50:59:18.4	0.620	ab	19.528	0.010	19.287	0.021	0.244
V204	CAR-FIII	06:42:04.17	−51:01:51.3	0.629	ab	19.560	0.018	19.222	0.023	0.340
V206	CAR-FIV	06:42:04.21	−51:09:40.9	0.585	ab	19.826	0.013	19.545	0.025	0.282

Note.

^aStar IDs from Dall'Ora et al. (2003).

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In order to derive the apparent distance moduli to the Carina galaxy from our data, we used the following calibrations of the NIR PLZ relation for RRL stars:

$$\begin{eqnarray}&&{M}_{K}=-1.07-2.38\mathrm{log}\;P+0.08\;{[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{CG}}\\ &&\qquad \qquad \qquad \quad {\rm{(Sollima}}\;{\rm{et}}\;{\rm{al.}}\;{\rm{2008)}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{M}_{K}=-0.77-2.101\mathrm{log}P+0.231\;{[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{CG}}\\ &&\qquad \qquad \qquad \quad {\rm{(Bono}}\;{\rm{et}}\;{\rm{al.}}\;{\rm{2003)}}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{M}_{K}=-0.597-2.353\mathrm{log}P+0.175\mathrm{log}Z\\ &&\qquad \qquad \qquad \quad {\rm{(Catelan}}\;{\rm{et}}\;{\rm{al.}}\;{\rm{2004)}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{M}_{J}=-0.141-1.773\mathrm{log}P+0.190\mathrm{log}Z\\ &&\qquad \qquad \qquad \quad {\rm{(Catelan}}\;{\rm{et}}\;{\rm{al.}}\;{\rm{2004)}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{M}_{K}=-0.6365-2.347\mathrm{log}P+0.1747\mathrm{log}Z\\ &&\qquad \qquad \qquad \quad {\rm{(D\acute{e}k\acute{a}ny}}\;{\rm{et}}\;{\rm{al.}}\;{\rm{2013)}}.\end{eqnarray} \tag{ 5 }$$

We recall that the calibration of Sollima et al. (2008) was constructed for the 2MASS photometric system;⁶ the calibrations of Bono et al. (2003) and Catelan et al. (2004) are valid for the Bessel, Brett, and Glass system (BBG); and Dékány et al. (2013) calibrated his formula onto the VISTA photometric system. Therefore, we transformed our own data and calibrated them onto UKIRT (Hawarden et al. 2001), the BBG and 2MASS systems using the transformations presented by Carpenter (2001), and the VISTA system through the equations provided by CASU.⁷ The above equations also account for first-overtone pulsators, once their period has been "fundamentalized" by adding a value of 0.127 to the logarithm of their periods.

We calculated K- and J-band photometric zero points and slopes for our sample of 33 RRLs using photometric magnitudes derived from our data and pulsational periods excerpted from Dall'Ora et al. (2003), as given in Table 4. The linear least squares fitting to our data yielded the following slopes: −1.79 ± 0.13 and −2.35 ± 0.08 in the J and K bands, respectively. We compared the obtained slopes with corresponding slopes from theoretical (Bono et al. 2003; Catelan et al. 2004) and empirical (Sollima et al. 2008; Dékány et al. 2013) PLZ relations (1)–(5). As seen in Figure 3, a large number of data points is concentrated within a small period range, which leads to an increase in uncertainties of both the slope and the zero point of our fit. In order to reduce this effect and gain accuracy of the zero point, our further calculations relied on the slopes from formulae (1)–(5), provided that our slope values exhibit no virtual difference from the "reference" ones. We derived the apparent distance moduli by fitting the zero point for the fixed slopes of relations (1)–(5), assuming the mean metallicity of Carina [Fe/H] = −1.72 ± 0.25 dex (Koch et al. 2006) in the Carretta & Gratton (1997) metallicity scale.

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In order to correct obtained apparent distance moduli for interstellar extinction, we adopted a foreground reddening calculated toward the Carina galaxy from the Galactic extinction maps of Schlegel et al. (1998) along with the reddening law R_V = 3.1 (Cardelli et al. 1989). We assumed the value $E(B-V)=0.06$ mag to be an average of our six target fields (for discussion see Section 4.2). We obtained the following values of extinction in the NIR bands: A_J = 0.054 mag, A_K = 0.022 mag. The resulting true distance moduli for the Carina dSph galaxy with associated uncertainties are summarized in Table 5. We adopt an averaged value of 20.118 ± 0.017 mag as the true distance modulus to the Carina galaxy. The systematic uncertainties are discussed in Section 4.1.

Table 5.  True Distance Moduli Determined from Different Calibrations

Filter	K	K	K	K	J
Calibration	Sollima	Bono	Dékány	Catelan	Catelan
${(m-M)}_{0}$	20.078	20.142	20.115	20.116	20.140
Statistical error	0.016	0.016	0.016	0.016	0.019
Systematic error	0.090	0.110	0.100	0.100	0.100

Note. The average foreground reddening $E(B-V)=0.06$ mag toward the Carina galaxy was calculated using reddening maps from Schlegel et al. (1998).

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The distance moduli obtained from several independent theoretical and empirical calibrations of the infrared RRL PLZ relations are consistent. The maximum difference of 0.06 mag between the results from the calibration of Sollima et al. (2008) and Bono et al. (2003) is not significant taking into account all uncertainties. A similar discrepancy was recently reported for the SMC (Szewczyk et al. 2009), the LMC (Szewczyk et al. 2008), and the Sculptor galaxy (Pietrzyński et al. 2008). One possible explanation is the underestimation of the effect of metallicity in Equation (1) relative to the rest of the calibrations (2)–(5). The other possible explanation is the zero-point offset, in the sense that the distance modulus from the calibration of Sollima et al. (2008) is slightly shorter compared to those from the calibration of Bono et al. (2003). The earlier zero-point calibration of the PLZ relation of Sollima et al. (2006), based on globular clusters, showed even larger offset (increased by 0.03 mag), and so we consider the present difference of 0.06 mag an improvement in consistency between Bono et al. (2003) and Sollima et al. (2008).

A shorter distance modulus of 19.94 mag, calculated by Udalski (2000), was tied to the underestimated LMC distance modulus (18.24 mag) derived by him from optical photometry of the red clump stars, and served as a point of reference for the distance determination to Carina. Providing a corrected distance modulus to the LMC of 18.493 mag (Pietrzyński et al. 2013a), Udalski's result can be recalculated, yielding a value in close agreement with this work.

4.1. Uncertainties

4.2. Extinction

4.3. Metallicity

We determined the distance to the Local Group Carina dwarf galaxy from single-phase NIR observations in the J and K bands of 33 RRL stars. The distance moduls to Carina was calculated using different theoretical and empirical calibrations of the NIR PLZ relation for RRL stars. We advocate an averaged value of 20.118 ± 0.017 (statistical) ± 0.11 (systematic) mag as a true, reddening-corrected distance modulus to the Carina galaxy. Our results are consistent and agree with other distance determinations obtained from a number of different and independent techniques and in different wavebands, especially with red clump stars and the tip of the RGB examined in the NIR domain in the course of the Araucaria Project (Pietrzyński et al. 2003, 2009). The method presented in this paper was successfully used to measure distances to four members of the Local Group (LMC, SMC, Sculptor, and Carina) and was proved to be an excellent tool for accurate distance determination, particularly for galaxies and clusters that lack young standard candles, like Cepheids. Distances to the LMC and the SMC, derived from the NIR PLZ relation for RRLs, are in excellent agreement with the results obtained from late-type eclipsing binaries (Pietrzyński et al. 2013a; Graczyk et al. 2014). The main asset of using eclipsing binaries for distance determination is that the distance is determined in an almost geometrical way, and thus is very weakly affected by population effects. The consistency of the results from these two methods—eclipsing binaries and NIR RRL variables—proves the reliability of the latter. The NIR PLZ relation for RRL stars is a robust method for accurate distance measurements, being only little influenced by metallicity and reddening within a fair range of values. This study complements existing distance measurements to the Carina galaxy with a very competitive (accuracy at the 5% level or better) new determination.

We greatly appreciate constructive remarks from an anonymous referee that helped to improve this paper. We thank the staff of the ESO Paranal and La Silla observatories for their support during the observations. We also gratefully acknowledge financial support for this work from the Polish National Science Centre grant PRELUDIUM 2012/07/N/ST9/04246 and the TEAM subsidy from the Foundation for Polish Science (FNP). W.G., G.P., and D.G gratefully acknowledge financial support for this work from the BASAL Centro de Astrofisica y Tecnologias Afines (CATA) PFB-06/2007. W.G., M.G., and D.G. also gratefully acknowledge support from the Chilean Ministry of Economy, Development and Tourism's Millennium Science Initiative through grant IC 120009 awarded to the Millennium Institute of Astrophysics (MAS). This research has made use of the 2MASS Database.