THE CARINA PROJECT. IX. ON HYDROGEN AND HELIUM BURNING VARIABLES

The left panels of Figure 5 display the Bailey diagram—amplitude versus period—for RRLs. From top to bottom the different panels show amplitudes in the B (top), V (middle) and I (bottom) bands. The Bailey diagram is a very good diagnostic for splitting fundamental and first overtones. The transition period between low-amplitude, short-period RR_c and large-amplitude, long-period RR_ab is located around log P ∼ −0.35/−0.30 (P ∼ 0.45–0.5 days). The RR_ab display a steady decrease in amplitude when moving toward longer periods. On the other hand, the RR_c show a "bell-shaped" (Bono et al. 1997a) or a "hairpin" (Kunder et al. 2013) distribution. The Bailey diagram is also used to constrain the Oosterhoff class (see, e.g., Bono et al. 1997b) of stellar systems hosting sizable samples of RRLs. The middle panel shows the comparison between RR_ab in the V-band with the empirical relations provided by (Clement & Shelton 1999, dashed lines) and by Cacciari et al. (2005, black solid line). The former relations are based on both OoI and OoII GCs, while the latter are only for OoI GCs. The comparison indicates that Carina can be classified either as an OoI or as an Oosterhoff intermediate system, since for periods longer than log P > −0.2 there is a group of RR_ab that, at fixed period, attains larger luminosity amplitudes. On the other hand, the amplitudes typical of RR_c variables in OoII GCs (Kunder et al. 2013, blue solid line) agree quite well the current data.

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In Paper VI we classified Carina, on the basis of the mean period of fundamentalized RRLs, as an OoII system. Indeed, we found $<mml:math> <mml:mo stretchy="true">&lang;</mml:mo> <mml:msub> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>RR</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="true">&rang;</mml:mo> </mml:math>$ = 0.60 ± 0.01 days (σ = 0.07). Including the new discovered RR_ab variables we find that the mean period of fundamental RRLs is $<mml:math> <mml:mo stretchy="true">&lang;</mml:mo> <mml:msub> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>RR</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="true">&rang;</mml:mo> </mml:math>$ = 0.637 ± 0.006 days (σ = 0.05). This estimate suggests that Carina is closer to an OoII stellar systems, since OoII GCs show mean fundamental periods of $<mml:math> <mml:mo stretchy="true">&lang;</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="true">&rang;</mml:mo> </mml:math>$ ∼ 0.65 days, while the OoI GCs show shorter periods, namely $<mml:math> <mml:mo stretchy="true">&lang;</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="true">&rang;</mml:mo> </mml:math>$ ∼ 0.55 days. We have also computed the ratio between the number of RR_c and the total number of RRLs, and we found $<mml:math> <mml:msub> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="italic">ab</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&plus;</mml:mo> <mml:msub> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="true">)</mml:mo> <mml:mo>&sim;</mml:mo> <mml:mn>0.14</mml:mn> <mml:mo>,</mml:mo> </mml:math>$ i.e., a fraction of RR_c variables that is more typical of OoI (∼17%) than OoII (∼44%) GCs. The above evidence indicates that the Oosterhoff classification of Carina depends on the adopted diagnostic. This means that the Oosterhoff classification should be cautiously treated, since it depends on the adopted diagnostic on the completeness and size of the RRL sample and on the morphology of the HB (Fiorentino et al. 2015a).

The data plotted in Figure 5 also show mixed-mode pulsators, the so-called RR_d variables. These are radial variables oscillating simultaneously in at least two different pulsation modes. The RR_d typically oscillate in the first overtone and in the fundamental mode, and the former mode is usually stronger than the latter (e.g., Smith 2006), but there are exceptions (Clementini et al. 2004). On the other hand, Classical Cepheids display a wide range of mixed-mode pulsators among the overtones and fundamental mode (Soszynski et al. 2008; Soszyñski et al. 2010), suggesting that surface gravity and effective temperature might play fundamental roles in driving the occurrence of such a phenomenon. Figure 6 shows the comparison in the Bailey diagram between the current observations and predicted amplitudes. Pulsation prescriptions rely on a large set of RRL models recently provided by M2015. The black solid and dotted lines display predicted amplitudes for the sequence of metal-poor (Z = 0.0001, Y = 0.245) models constructed by assuming a stellar mass of 0.80 M_⊙ and two different luminosity levels. The black solid line shows predictions for the Zero-Age-Horizontal-Branch (ZAHB) luminosity level ($<mml:math> <mml:mi>log</mml:mi> <mml:mo stretchy="true">(</mml:mo> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mo form="prefix">&CircleDot;</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="true">)</mml:mo> </mml:math>$ = 1.76), while the dotted line for a brighter luminosity level ($<mml:math> <mml:mi>log</mml:mi> <mml:mo stretchy="true">(</mml:mo> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mo form="prefix">&CircleDot;</mml:mo> </mml:mrow> </mml:msub> </mml:math>$) = 1.86). The purple lines display the same predictions, but for a slightly more metal-rich chemical composition (Z = 0.0003, Y = 0.245; M = 0.716 M_⊙, log(L/L_⊙) = 1.72 and 1.82).

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The predicted amplitudes appear to be slightly larger, at fixed pulsation period, when compared with observations. This is a limit in the current theoretical framework, since the amplitudes are tightly correlated with the efficiency of convective transport. In passing we note that the comparison between predicted and observed luminosity amplitudes is hampered by the current theoretical uncertainties in the treatment of the time dependent convective transport. We current adopt a mixing length parameter of α = 1.5 (see Marconi et al. 2011). Larger values cause a steady decrease in the luminosity amplitude (see Di Criscienzo et al. 2004). Indeed, observed amplitudes attain, at fixed period smaller amplitudes. This applies to both fundamental and first overtone variables. Moreover, we are assuming that observed light curves have a good sampling around the phases of minimum and maximum light. However, the comparison shows two interesting features. (i) The predicted amplitudes for FO pulsators show a larger dependence on metal content than fundamental pulsators. (ii) The regular pulsators display, at fixed pulsation period, a small spread in amplitudes. The current predictions suggest that their evolutionary status is quite homogenous, since they appear to be located to the ZAHB luminosity level.

Carina was previously known to host six RR_d variables: V11, V26, V89, V192, V198, V207 (D03); in the current analysis we discovered other three double-mode pulsators: V74, V210, and V225. To further understand their nature, and in particular to properly define the location of RR_d pulsators in the Bailey diagram, we estimated both primary (first overtone) and secondary (fundamental) periods and decomposed their light curves. Indeed, the quality of the photometry allowed us to estimate not only the "global luminosity amplitude," but also the amplitude of both fundamental (open orange triangles) and first overtone mode (filled orange triangles). Table 6 gives from left to right their periods, mean magnitudes and amplitudes in the BVI bands. To our knowledge this is the first time in which we can associate to the two modes of double-mode variables their individual luminosity amplitudes. The data plotted in the left panels of Figure 5 indicate that the primary components (first overtone) are located in the long period tale (log P ∼ −0.4) of single mode first overtone variables. Moreover, the secondary components (fundamental) are located in the short period tale (log P ∼ −0.25) of single mode fundamental variables. This evidence is further supported by their mean colors. The mean color of RR_d is systematically redder (B − V ∼ 0.33 mag) than the color range covered by RR_c variables (0.2 ≤ B − V ≤ 0.35 mag) and systematically bluer than the typical color range of RR_ab variables (0.3 ≤ B − V ≤ 0.5 mag).

The pulsation and evolutionary status of the RR_d variables depends on their evolutionary direction and on their position in the so-called OR region (Bono et al. 1997b). In this context it is worth mentioning that RRLs in dwarf spheroidal galaxies appear to lack High Amplitude Short Period (HASP) fundamental variables (Stetson et al. 2014b; Fiorentino et al. 2015a), thus resembling Oosterhoff II globular clusters in the Milky Way (Bono et al. 1997b).

The occurrence of a good sample of RR_d variables in Carina seems to suggest that this region of the Bailey diagram might be populated by mixed-mode variables. Obviously, the occurrence of RR_d variables depends on the topology of the IS, but also on the evolutionary properties (extent in temperature of the so-called blue hook) and, in particular, on the occurrence of the hysteresis mechanism (van Albada & Baker 1971; Bono et al. 1995; Fiorentino et al. 2015a; Marconi et al. 2015) when moving from more metal-poor to more metal-rich stellar structures.

The Blazhko RRLs plotted in Figure 5 also appear to be located in a very narrow period range. No firm conclusion can be reached concerning the distribution in the Bailey diagram of Blazhko RRLs, since the sample size is quite limited and also because the Blazhko cycle is poorly sampled.

The above findings further support the crucial role played by cluster and galactic RRLs to constrain the topology of the IS and to investigate the evolutionary and pulsation status of exotic objects like RR_d and Blazhko RRLs.

The right panels of Figure 5 show the distribution of Carina ACs in the same Bailey diagrams as the RRLs. Their properties have already been discussed in Paper VI. We confirm the separation at log P ∼ −0.1 between long-period and high-amplitude with short-period and low-amplitude ACs. In passing we note that three out of the 20 ACs have periods around one day, thus further supporting the need for data sets covering large time intervals to remove the one-day alias.

Note that in the current analysis of evolved variables we did not include the six RRLs (three RR_c, three RR_ab) and the three ACs recently detected by VM13 outside the tidal radius of Carina. The reasons are the following. The three RR_c variables attain periods that are systematically shorter (−1.0 ≲ log P ≲ −0.25) that typical RR_c variables (see Table 7), but their mean B − V colors are typical of RR_ab variables. The same outcome applies to the three RR_ab variable, and indeed they are located in the short period range of fundamental pulsators, but their B − V colors are typical of objects located close to the red edge of the IS. Moreover, the newly identified ACs have periods that are systematically shorter (−0.8 ≲ log P ≲ −0.7) than the typical Carina ACs.

The top panel of Figure 7 displays the position of the Carina RR_d variables in the Petersen diagram, i.e., the first-overtone-to-fundamental period ratio (P₁/P₀) versus the fundamental period. The data in this panel also show the comparison with RR_d variables identified in Galactic globulars (see labeled names) and in the Galactic field (Halo: blue squares; Bulge: small green circles). Table 8 gives from left to right the name of the stellar system, the number of RR_d variables, the reference for the RR_d data, the mean metallicity, and the reference for the metallicity estimate. The data plotted in this figure display several interesting features.

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Table 8.  Number of RR_d Hosted in the Stellar Systems Plotted in Figure 7 and Their References

System	N	Reference	[Fe/H]	Reference
NGC 2419	1	Clement et al. (1993)	−2.20 ± 0.09	Carretta et al. (2009)
NGC 6426	1	Clement et al. (1993)	−2.33 ± 0.15	Hatzidimitriou et al. (1999)
LMC	985	Soszyński et al. (2009)	−0.33 ± 0.13	Romaniello et al. (2008)a
SMC	257	Soszyñski et al. (2010)	−0.75 ± 0.08	Romaniello et al. (2008)a
Bulge	173	Soszyński et al. (2014)	−1.5/–0.5	Zoccali et al. (2008)b
Halo (NSV 09295)	1	Garcia-Melendo & Clement (1997)	−1.5	Layden (1994)
Halo (AQ Leo)	1	Clement et al. (1991)	−1.5	Layden (1994)
Halo (VIII–10, VIII–58)	2	Clement et al. (1993)	−1.5	Layden (1994)
Halo (CU Com)	1	Clementini et al. (2000)	−1.5	Layden (1994)
Halo (ASAS)	32	Pojmanski (2002)	−1.5	Layden (1994)
M3	8	Clementini et al. (2004)	−1.50 ± 0.05	Carretta et al. (2009)
IC4499	16	Walker & Nemec (1996)	−1.62 ± 0.09	Carretta et al. (2009)
M68	12	Walker (1994)	−2.27 ± 0.04	Carretta et al. (2009)
M15	14	Nemec (1985b)	−2.33 ± 0.02	Carretta et al. (2009)
Draco	10	Nemec (1985a)	−1.98 ± 0.01	Kirby et al. (2013)
Sculptor	18	Kovács (2001)	−1.68 ± 0.01	Kirby et al. (2013)
Sagittarius	40	Cseresnjes (2001)	−0.62 ± 0.2	Carretta et al. (2010)

Notes. Columns 4 and 5 give the iron abundances of the hosting stellar system and their references.

^aMean iron abundances and standard deviations of classical Cepheids based on high spectral resolution spectra. Metal abundances for 98 LMC RRLs were provided by Gratton et al. (2004) using low-resolution spectra and found $<mml:math> <mml:mo stretchy="true">[</mml:mo> <mml:mi>Fe</mml:mi> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">H</mml:mi> <mml:mo stretchy="true">]</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>&minus;</mml:mo> <mml:mn>1.48</mml:mn> </mml:mrow> <mml:mo>&PlusMinus;</mml:mo> <mml:mn>0.03</mml:mn> <mml:mo>&PlusMinus;</mml:mo> <mml:mn>0.06</mml:mn> <mml:mo>.</mml:mo> </mml:math>$ Metal abundances for SMC RRLs are not available. The iron abundance of the single SMC GC (NGC 121) is [Fe/H] = −1.19 ± 0.12 provided by Da Costa & Hatzidimitriou (1998). ^bRange in iron abundance covered by red giants in the Galactic bulge. The iron abundance of Bulge RRLs was provided by Walker & Terndrup (1991) [Fe/H] = −1.00 ± 0.16 using low-resolution spectra.

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The period ratio shows a steady decrease when moving from more metal-poor to more metal-rich systems (see also Bragaglia et al. 2001). The largest values in the period ratio are attained in the Halo and in very metal-poor GCs, while the smallest values are attained in the Bulge. The trend with the metallicity was also suggested on both theoretical and empirical bases by Soszyński et al. (2011) and more recently by Soszyński et al. (2014). Note that the fraction of double-mode variables appears to be anti-correlated with the mean metal abundance, and indeed it increases from 0.5% in the Bulge to 4% in the LMC and to 10% in the SMC (Soszyński et al. 2011). The RR_d variables in Carina appear to follow a similar trend, since the current fraction is ∼10%. Two RR_d variables (V192, V210) display period ratios that are systematically larger then period ratios in Galactic globulars, in dwarf galaxies, and in the Bulge. In passing we note that one RR_d in M68 (V36) and two in M15 (V51, V53) display period ratios that systematically smaller than the bulk of RR_d variables. The referee suggested the difference on a more quantitative basis. To avoid spurious fluctuations in the period range covered by the different data sets, we ranked the entire sample (398) as a function of the fundamental period. Then we estimated the running average by using a box including the first 40 objects in the list. We estimated the mean period ratio, the mean fundamental period, and the standard deviations of this sub-sample. We estimated the same quantities by moving one object in the ranked list until we took account of the last 40 objects in the sample. We performed several tests, changing both the number of objects included in the box and the number of stepping stars. The current findings are minimally affected by plausible variations. We found that the quoted five RR_dstars are located within 3σ of the mean. The current statistics is too limited to claim solid evidence of discrepancy. Note that we double checked the photometric quality and coverage of the above five objects and we found that they are similar to the other canonical RR_dstars.

The period ratios display a larger spread when moving into the metal-poor regime (long fundamental periods). To define the trend on a more quantitative basis we adopted several sets of nonlinear, convective pulsation models (M2015). The adopted chemical compositions are labeled. The new set of RRL models relies on the theoretical framework outlined in Di Criscienzo et al. (2004) and Marconi et al. (2011), but on new evolutionary prescriptions for low-mass He burning models provided by (Pietrinferni et al. 2004) (BASTI data base, http://albione.oa-teramo.inaf.it). The pulsation predictions plotted in this panel cover the so-called OR region, i.e., the region in which RRLs show a stable limit cycle in both the fundamental and the first overtone mode. This region is located between the blue edge of the fundamental mode and the red edge of the first overtone. The comparison between theory and observations indicates that an increase in the luminosity (log L/L_⊙ = 1.76, 1.86, dotted and dashed red lines) mainly causes, at fixed chemical composition (Z = 0.0001) and stellar mass (M/M_⊙ = 0.85), a steady increase in the fundamental period, and in turn a decrease in the period ratio. Moreover, a decrease in stellar mass (M/M_⊙ = 0.80, solid red line) at fixed chemical composition and luminosity level (log L/L_⊙ = 1.76) causes a systematic decrease in the period ratio and a moderate increase of the fundamental period. The above trends take into account a significant fraction of RR_d pulsators located in metal-poor GCs and in Carina dSph. This indicates that RR_d in Carina have a metallicity of the order of Z = 0.0001 and a mean stellar mass close to 0.85 M_⊙. The pulsation masses are slightly larger than predicted by evolutionary models, but are within the current empirical and theoretical uncertainties.

The above findings further support the evidence that the old stellar component in Carina is quite metal-poor. This evidence is soundly supported by recent photometric and spectroscopic results by Monelli et al. (2014) and Fabrizio et al. (2015) suggesting a mean metal abundance for the old stellar component of [Fe/H] = (−2.13 ± 0.03 ± 0.28) dex.

In this context it is worth mentioning that theoretical predictions for more metal-rich pulsation models (see the black lines and the labeled values) provide a sound explanation of the steady decrease in the period ratio of Bulge RR_d variables, i.e., the stellar systems with the broader metallicity distribution.

To further define the pulsation and evolutionary properties of Carina RR_d variables, the bottom panel of Figure 7 shows the same Petersen diagram, but the comparison is now extended to RR_d in nearby dwarf spheroidals (Draco, Sculptor, Sagittarius), in dwarf irregulars (LMC; Small Magellanic Cloud, SMC) and in the Bulge. The data plotted in this panel bring forward several interesting new findings.

The range in period ratios covered by LMC RR_d is on average larger (0.740 ≤ P₁/P₀ ≤ 0.749) than the range of SMC RR_d variables. This evidence supports spectroscopic measurements of LMC RRLs suggesting metal abundances ranging from [Fe/H] = −2.12 to −0.27 dex (Gratton et al. 2004). The SMC RR_d cover a slightly narrower period ratio range (0.741 ≤ P₁/P₀ ≤ 0.747 days) but according to recent studies, and within current uncertainties affecting metallicity estimates for RRLs in these systems (Haschke et al. 2012) the metallicity spread for the old stellar populations in the two Clouds is similar.

The location of RR_d of Carina and Draco is the same in the Petersen diagram. Indeed current spectroscopic estimates, based on medium resolution spectra, provide a very metal-poor iron abundance ([Fe/H] = −1.92 ± 0.01 dex, see Table 8) also for Draco. The steady decrease in period ratio of RR_d in Sculptor is strongly supported by the recent spectroscopic measurements suggesting an iron abundance of [Fe/H] = −1.68 ± 0.01 dex (see Table 8). The empirical evidence concerning Sagittarius needs to be discussed in detail, because the periods and period ratios attain values that are on average smaller than for RR_d in other dSphs. Spectroscopic estimates based on high-resolution spectra by Carretta et al. (2010), suggest for Sagittarius a mean iron abundance, based on 27 RGs, of [Fe/H] = −0.62 and individual values ranging from −1.0 to above solar. A smaller spread in iron abundance was also suggested by (Kunder & Chaboyer 2008) using RR Lyrae properties. The spread in period ratios and the range in fundamental periods (0.45 ≤ P ≤ 0.49 days) showed by Sagittarius RRLs soundly support the spectroscopic measurements and the similarity with LMC RRLs. This indicates that Sagittarius is a fundamental nearby laboratory to constrain the pulsation properties of metal-rich RRL in gas poor systems.

The RR_d in the Bulge (small green circles) display a clear overdensity for P₀ ∼ 0.46 days. This overdensity was explained by Soszyński et al. (2011) as the relic of a former dwarf galaxy that was captured by the Milky Way. More recent investigations based on a larger sample (28 versus 16) indicate that they are distributed along a stream that crosses the Galactic bulge almost vertically. Note that the comparison with theoretical predictions suggests, for the above stellar system, a metal-intermediate chemical composition (Z = 0.001–0.002).

The RR_d also provide a unique opportunity to validate the current approach to fundamentalizing the first overtones. Whenever the sample of RRLs hosted in a stellar system is limited, fundamental and first overtone variables are treated as a single sample by transforming the periods of the first overtones into "equivalent" fundamental periods, using the relation log P_F = log P_FO + 0.127. This assumption dates back to almost half a century and relies on the few RR_d variables known at that time (Sandage et al. 1981; Cox et al. 1983; Petersen 1991). The above constant period shift is further supported by the new theoretical scenario by M2015 and by the sizable sample of RR_d variables recently identified by large, dedicated photometric surveys.

In passing we note that this issue is far from being an academic dispute, since the same fix is also used to improve the precision of distance determinations based on both RRLs (Braga et al. 2015) and classical Cepheids (e.g., Marengo et al. 2010). Figure 8 shows fundamental versus first overtone periods (P₀ versus P₁) for the entire sample of RR_d plotted in Figure 7. The red line shows the fit to the empirical data. We found:

$$<mml:math> <mml:msub> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.0096</mml:mn> <mml:mo>&plus;</mml:mo> <mml:mn>1.317</mml:mn> <mml:mo>&lowast;</mml:mo> <mml:msub> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>.</mml:mo> </mml:math> \tag{ 1 }$$

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The new estimate of both the slope and the zero-point soundly supports the old fix, and indeed the green line, showing the classical fix, agrees quite well with the new observations.

To further constrain the impact that the period ratios of RR_d variables have in constraining the pulsation properties of RRLs, we plotted in the same plane theoretical predictions for the "OR" regions adopted in Figure 7. The black solid line was estimated by considering only models more metal-poor than Z = 0.001, i.e., a metallicity range similar to the observed one. The agreement is quite good over the period range. In passing we also note that when moving to more metal-rich models (Z > 0.001, blue circles) the period increase is significantly larger in the fundamental period than in the first overtone one. We still lack firm empirical evidence for such objects and it is not clear whether it is an observational bias or the consequence of an evolutionary property connected to the dependence of the HB morphology on the metal content.

One of the most important tools for deriving distances from pulsating stars is the so called Wesenheit relation (see for example, van den Bergh 1975; Madore 1982) that is independent of reddening by definition, assuming that the ratio of total-to-selective absorption is fixed. This is a period–luminosity relation that includes a color term whose coefficient is the ratio between the total and the selective extinction coefficients. Figure 9 shows the observed PW relations and the empirical fit to the data (solid black lines) obtained by fundamentalizing the first overtone pulsators by using Equation (1). Dashed lines depict the dispersion of the above inferred relations. Results of these fits are listed in Table 9, where the zero-points, the slopes, and the dispersions of the relations are reported in the first three columns, respectively. In the fit determination we excluded stars outside 3σ of the inferred empirical BV Wesenheit relation. These stars are the two peculiar pulsators V158 and V182 and the stars V170 and V171 for which we have uncertain parameters (red open symbols).

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Table 9.  Results of the Empirical Fit of Period–Wesenheit Relations $<mml:math> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:mi>&agr;</mml:mi> <mml:mo>&plus;</mml:mo> <mml:mi>&beta;</mml:mi> <mml:mi>log</mml:mi> <mml:mspace width="0.25em"/> <mml:msub> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">F</mml:mi> </mml:mrow> </mml:msub> </mml:math>$

α	β	rms
$<mml:math> <mml:mi>W</mml:mi> <mml:mo stretchy="true">(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="true">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&minus;</mml:mo> <mml:mn>3.06</mml:mn> <mml:mo stretchy="true">(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>&minus;</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:math>$
(18.98 ± 0.03)	(−2.7 ± 0.1)	0.07
$<mml:math> <mml:mi>W</mml:mi> <mml:mo stretchy="true">(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy="true">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&minus;</mml:mo> <mml:mn>0.78</mml:mn> <mml:mo stretchy="true">(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>&minus;</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:math>$
(18.84 ± 0.03)	(−2.6 ± 0.1)	0.06
$<mml:math> <mml:mi>W</mml:mi> <mml:mo stretchy="true">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy="true">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&minus;</mml:mo> <mml:mn>1.38</mml:mn> <mml:mo stretchy="true">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>&minus;</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:math>$
(18.80 ± 0.03)	(−2.5 ± 0.1)	0.08
$<mml:math> <mml:mi>W</mml:mi> <mml:mo stretchy="true">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="italic">BI</mml:mi> <mml:mo stretchy="true">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&minus;</mml:mo> <mml:mn>1.34</mml:mn> <mml:mo stretchy="true">(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>&minus;</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:math>$
(18.88 ± 0.03)	(−2.6 ± 0.1)	0.06

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Thanks to the use of the Fine Guidance Sensor on board the Hubble Space Telescope (HST), Benedict et al. (2011) provided accurate estimates of the trigonometric parallaxes for five field RRLs: SU Dra, XZ Cyg, RZ Cep, XZ Cyg, and RR Lyr. Using their data in Table 2, we derived the mean magnitude in the B and V bands from a fit with a spline under tension. We then calculated the absolute Wesenheit parameter for each star, ($<mml:math> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="true">&lang;</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="true">&rang;</mml:mo> <mml:mrow> <mml:mo>&minus;</mml:mo> <mml:mn>3.06</mml:mn> </mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>&minus;</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="true">)</mml:mo> <mml:mo>&minus;</mml:mo> <mml:mi>&mgr;</mml:mi> </mml:math>$), where μ is the individual distance modulus based on the HST parallax. Individual mean magnitudes of the calibrating RRLs and their distances are listed in Table 10. We applied the individual calibrating RRL to the empirical PW relations (see Figure 9 and Table 9). Note that the calibrating RRLs cover a limited range in metallicity (from −1.80 to −1.41 dex, Benedict et al. 2011). The current theoretical predictions (see Section 4.2) suggest a mild dependence on the metal content. Therefore, we neglected the metallicity dependence of the calibrating RRLs. The data plotted in Figure 10 show in the W-log P plane the calibrating RRLs together with the Carina RRLs. The error bars of the calibrating RRLs take into account both the photometric errors and the uncertainties of the trigonometric parallaxes. A glance at the data discloses that only for RR Lyr is the precision of the absolute distance better than 1%. Using all the calibrating RRLs we found a true distance modulus for Carina of μ = (20.02 ± 0.02 ± 0.05) mag. Note that for the above reasons the accuracy of the distance mainly depends on the accuracy of the data for RR Lyr.

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Table 10.  Periods, B and V Mean Magnitudes, True Distance Moduli and Metallicity for the Five Field RRLs Stars by Benedict et al. (2011)

Name	log P	$<mml:math> <mml:mo stretchy="true">&lang;</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="true">&rang;</mml:mo> </mml:math>$	$<mml:math> <mml:mo stretchy="true">&lang;</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="true">&rang;</mml:mo> </mml:math>$	μ	[Fe/H]
RR Lyr	−0.24655	8.07 ± 0.01	7.75 ± 0.01	7.14 ± 0.07	−1.41 ± 0.13
SU Dra	−0.18018	10.23 ± 0.01	9.95 ± 0.01	9.35 ± 0.24	−1.80 ± 0.2
UV Oct	−0.26552	9.71 ± 0.01	9.35 ± 0.01	8.87 ± 0.13	−1.47 ± 0.11
XZ Cyg	−0.33107	10.13 ± 0.01	9.84 ± 0.01	8.98 ± 0.22	−1.44 ± 0.2
RZ Cep	−0.38352	9.92 ± 0.01	9.46 ± 0.01	8.03 ± 0.16	−1.77 ± 0.2

Note. The distance moduli in this column are slightly different than those in Table 8 of Benedict et al. (2011) due to typographical errors in the paper.

Iron abundances are on the Zinn & West (1984) metallicity scale according to Benedict et al. (2011; see their Table 1).

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To fully exploit the multiband photometry of Carina RRLs we also decided to use predicted PW relations. The new theoretical framework derived by M2015 indicates that the PW (BV) relation is independent of the metal content. This is a very positive feature for two different reasons: (a) the PW (BV) can be applied to estimate individual distances of field RRLs for which the metal content is not available; (b) the PW (BV) is particularly useful to estimate distances of RRLs in nearby dwarf galaxies, since they typically cover a broad range in iron content. We applied the predicted relation to Carina RRLs and we found a true distance modulus of μ = (20.08 ± 0.007 ± 0.07) mag. The distance determination has been estimated using the predicted relation for fundamental pulsators. The number of RR_c variables in Carina is modest (12) and they have been fundamentalized. As noted in the previous section in the current distance determination we did not consider stars outside 3σ of the inferred empirical Wesenheit (BV) relation. The above distance agrees, within the errors, quite well with the true Carina distance based on empirical calibrators. Moreover, the new distance determination also agrees with Carina distances available in the literature that are based on robust standard candles (see Table 11).

To take advantage of the multiband photometry for Carina RRLs, we estimated the distance using the PWZ (BI), the PWZ (VI), and the triple band PWZ (BIV) relations (see Figure 11). The current pulsation predictions suggest a mild dependence on the metal content. Indeed, the coefficients of the metallicity term are 0.106 (BI), 0.150 (VI), and 0.075 (BVI). In passing we note that the metallicity dependence of the optical PW relation for RRL stars displays a different trend when compared with similar relations for classical Cepheids. For the latter objects the PW (VI) relation is almost independent of the metallicity, while the PW (BV) shows a strong dependence on the iron content. The reader interested in a detailed discussion of the difference between RRLs and classical Cepheids is referred to the recent paper by M2015.

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To estimate the distance from the PWZ (BI), the PWZ (VI) and the PWZ (BVI) we adopted a mean iron abundance for Carina RRLs of (−2.13 ± 0.03 ± 0.28) dex (Fabrizio et al. 2015). Using this value we found the following true distance moduli: μ = (20.086 ± 0.006 ± 0.06) mag (BI), μ = (20.07 ± 0.008 ± 0.08) mag (VI), and μ = (20.06 ± 0.006 ± 0.06) mag (VBI). Note that the uncertainties in the distance modulus account for the photometric errors, the standard deviation of the theoretical PWZ relations and the intrinsic spread in iron abundance. Once again the above distance determinations agree quite well with the empirical distance, with the distance based on the PW (BV), and with distances available in the literature.