METAL ABUNDANCES, RADIAL VELOCITIES, AND OTHER PHYSICAL CHARACTERISTICS FOR THE RR LYRAE STARS IN THE KEPLER FIELD^*

Pulsation periods, P_puls, times of maximum light, t₀, and total amplitudes on the Kepler photometric system, A_tot, are given for the program RR Lyrae stars in Columns 4–6 of Table 1. All values are newly derived based on analyses of the available Kepler Q0–Q11 LC and SC data, which in most cases comprises almost 1000 days of quasi-continuous high-precision photometry.

The P_puls are mean values of the dominant oscillation frequencies, ignoring the secondary periods that are present in some non-Blazhko stars (V350 Lyr, KIC 7021124) and in all the Blazhko stars, ignoring changing periods, and averaging over many modulation cycles for the Blazhko stars. The periods were computed with the "Period04" package of Lenz & Breger (2005) using the Monte Carlo mode for error estimation, with Stellingwerf's (1978) "PDM2" program¹¹ updated to deal with rich data sets, and with the period-finding program of Kolaczkowski (see Moskalik & Kołaczkowski 2009).

The accuracy of the pulsation periods reported in Table 1 is ∼few units in the last decimal place given, with the periods for the Blazhko and RRc stars tending to be less accurate than those for the non-Blazhko stars. The uncertainties depend on several factors: the particular data set (or subset) that was analyzed, the search method, the constancy of the primary pulsation period, and whether or not amplitude or frequency modulations are present. In general the periods agree well with those given by B10 for 15 non-Blazhko and 14 Blazhko stars, and with the more precise values derived by N11 for 19 non-Blazhko stars. Since the new periods were computed with longer time baselines than these earlier studies, and are based on both LC and SC data, they are the best available values yet reported for these stars. In general, the accuracy of the P_puls values is more than sufficient for estimating the phases at which the spectroscopic observations were made.

The A_tot in Table 1 are the trough-to-peak Kp amplitudes. For the non-Blazhko stars they are time invariant values obtained by inspection of the complete light curve and by averaging A_tot calculated from individual light curves for successive time segments (see Figure 2). For the Blazhko variables and the four RRc stars, all of which are multiperiodic and show amplitude variations, A_tot is the time-averaged value. According to the transformation given in Equation (2) of N11 the A_tot(Kp) are ∼0.14 mag smaller than the A_tot(V), where V is for the Johnson UBV system.

Zoom In Zoom Out Reset image size

The zero points t₀ were estimated from the mean light curves and correspond to times of zero-slope at maximum light. For the non-Blazhko stars they were usually chosen to approximately coincide with the time of the first Kepler observations (i.e., just after BJD−2,454,953), and for the Blazhko variables the t₀ generally coincide with a time of maximum light occurring during a period of maximum amplitude. Thus, t₀ serves as the zero point for both the pulsation phase, ϕ_puls, defined to be the fractional part of (t − t₀)/P_puls, and for the Blazhko phase, ϕ_BL, defined to be the fractional part of (t − t₀)/P_BL.

The most noticeable signature of the Blazhko effect is variation in the light curve shape, in particular amplitude modulations, over time scales ranging from several days to several years. The amplitude that is usually considered is the total amplitude (i.e., from trough to peak) through the V or B filter, but may also be one of the Fourier amplitude coefficients. In the absence of BV photometry for most of the Kepler-field RR Lyr stars we focus in this paper on the A_tot and A₁ amplitudes derived from the Kp photometry. The A₁ values are from Fourier decompositions of the Kp data performed using the following sine-series sum:

$$\begin{equation} m(t) = A_0 + \sum \limits ^{N}_{k=1} A_k \sin (k 2 \pi t / P + \phi _k), \end{equation} \tag{ 1 }$$

where m(t) is the apparent Kp magnitude as a function of time, A₀ is the mean Kp magnitude (assumed to be the value given in the KIC catalog), N is the adopted number of terms in the Fourier series, the A_k and ϕ_k are the Fourier amplitudes and phases, and P is the pulsation period.

Blazhko stars are also known to exhibit period (frequency, or phase) modulations, although usually these are less noticeable than the amplitude modulations. Frequency modulations were detected previously in the Kepler-field Blazhko stars studied by B10, in RR Lyrae by K10 and K11, and in V445 Lyr by G12. The 16 Kepler-field Blazhko stars studied here all exhibit both amplitude and frequency modulations.

While the detection of amplitude and frequency modulations is usually straightforward, the physical mechanism responsible remains unknown, and Blazhko stars are almost as enigmatic now (see Kovács 2009) as when they were discovered over 100 yr ago (Blazhko 1907). The problem in recent years has taken on new significance with the discovery that almost half of all RRab stars are Blazhko variables (Jurcsik et al. 2009c), and that possibly all RRc stars are multiperiodic (Moskalik et al. 2012). These discoveries raise the question whether all RR Lyrae stars might be amplitude and frequency modulators.

The measured amplitude and frequency modulations for the Blazhko RRab stars in our sample are summarized in Table 2, where the stars have been ordered from largest to smallest amplitude modulations, ΔA₁ (i.e., from the most extreme Blazhko variability to the least variability). All the quantities are newly derived from both the LC and SC Q0–Q11 photometry, and the uncertainties are at the level of the precisions given. Columns 2–6 contain, respectively, the measured ranges (minimum to maximum values) for: the total amplitude in the Kp-passband, ΔA_tot; the pulsation period, ΔP_puls; the Fourier amplitude coefficient, ΔA₁; the first Fourier phase coefficient, Δϕ₁; and the Fourier phase-difference parameter $\Delta \phi _{\rm 31}^s$, where $\phi _{\rm 31}^s = \phi _3 - 3\phi _1$ (adjusted to the value closest to the mean value of 5.8) and the superscript "s" indicates that a sine (rather than a cosine) series was used for the Fourier decomposition (see Equation (1)).

Table 2. Amplitude and Period (Phase) Modulations of 16 Blazhko Stars in the Kepler Field

Star	ΔAtot	ΔPpuls	ΔA1	Δϕ1	$\Delta \phi _{\rm 31}^s$
	(mag)	(days)	(mag)	(rad)	(rad)
(1)	(2)	(3)	(4)	(5)	(6)
V445 Lyr	0.80 (0.20–1.00)	0.013 (0.509–0.522)	0.380 (0.018–0.398)	3.12 (1.48–4.60)	6.28 (0.00–6.28)
V2178 Cyg	0.84 (0.30–1.14)	0.0014 (0.4859–73)	0.279 (0.162–0.441)	1.09 (3.91–5.00)	3.55 (3.41–6.96)
V450 Lyr	0.56 (0.57–1.13)	0.0007 (0.5042–49)	0.240 (0.207–0.447)	0.44 (4.74–5.18)	1.12 (4.53–5.65)
KIC 7257008	0.68 (0.44–1.12)	0.0030 (0.5105–35)	0.211 (0.179–0.390)	0.60 (3.00–3.60)	1.05 (4.77–5.82)
V354 Lyr	0.60 (0.49–1.09)	0.0003 (0.5616–19)	>0.20 (<0.19–0.39)	1.26 (2.51–3.77)	0.70 (4.85–5.55)
V360 Lyr	0.21 (0.54–0.75)	0.0010 (0.5570–80)	0.162 (0.176–0.338)	0.37 (0.99–1.36)	0.82 (4.78–5.60)
V808 Cyg	0.35 (0.71–1.06)	0.0020 (0.5467–87)	0.130 (0.237–0.367)	0.67 (0.28–0.95)	0.64 (4.96–5.60)
RR Lyrae	0.23 (0.59–0.82)	0.0024 (0.5657–81)	0.127 (0.186–0.313)	0.65 (4.77–5.42)	1.21 (4.85–6.06)
V366 Lyr	0.18 (0.77–0.95)	0.0003 (0.5269–72)	0.127 (0.248–0.375)	0.14 (1.21–1.35)	0.98 (4.75–5.73)
KIC 9973633	0.39 (0.61–1.00)	0.0009 (0.5105–14)	0.119 (0.225–0.344)	0.22 (4.00–4.22)	0.78 (4.80–5.58)
V355 Lyr	0.15 (0.87–1.02)	0.0008 (0.4733–41)	0.110 (0.313–0.423)	0.15 (1.05–1.20)	0.62 (4.67–5.29)
V353 Lyr	0.24 (0.71–0.95)	0.0004 (0.5566–70)	0.082 (0.250–0.332)	0.16 (0.59–0.75)	0.24 (5.01–5.25)
V1104 Cyg	0.17 (1.03–1.20)	0.0003 (0.4363–66)	0.062 (0.365–0.427)	0.070 (0.165–0.235)	0.16 (4.78–4.94)
V783 Cyg	0.08 (0.76–0.84)	0.0005 (0.6204–09)	0.033 (0.253–0.286)	0.08 (1.82–1.90)	0.14 (5.44–5.58)
KIC 11125706	0.03 (0.45–0.48)	0.0004 (0.6128–32)	0.008 (0.176–0.184)	0.031 (5.452–5.483)	0.112 (5.772–5.884)
V838 Cyg	0.02 (1.09–1.11)	0.0002 (0.4801–03)	0.0016 (0.3871–87)	0.0010 (3.3577–87)	0.003 (4.855–4.858)

Notes. The columns contain: (1) star name; (2) range of total amplitude (Kp-system), with minimum and maximum values in parentheses; (3) range of the pulsation period (min and max in parentheses); (4) range of the Fourier A₁ amplitude coefficient (min and max in parentheses)—note that ΔA₁ here is the A₁ range, which is not the same as the ΔA₁ defined by B10; (5) range of the first Fourier phase coefficient, ϕ₁ (min and max in parentheses); (6) range of the Fourier phase-difference parameter $\phi _{\rm 31}^s$ (min and max in parentheses).

Download table as:  ASCIITypeset image

The amplitude modulations, ΔA_tot, range from as large as ∼0.82 mag (V445 Lyr, V2178 Cyg) to as small as ∼0.02 mag (V838 Cyg, KIC 11125706), corresponding to 2%–80% of A_tot(max). The stars with the largest amplitude variations tend also to exhibit the largest period variations. The average period range ΔP_puls is 0.0010 days. Plots of ΔA_tot versus period, and $\Delta \phi _{\rm 31}^s$ versus period, show an approximately normal distribution with the largest modulations occurring for those stars with pulsation periods around 0.53 days. In the period–$\Delta \phi _{\rm 31}^s$ diagram the extreme Blazhko stars V445 Lyr and V2178 Cyg (see Figure 3) are clearly outliers with $\Delta \phi _{\rm 31}^s$ values considerably larger than for all the other Blazhko stars. Estimates of ΔP_puls and ΔA₁ of RR Lyrae by K11, and V445 Lyr by G12, agree well with the values reported in Table 2.

Zoom In Zoom Out Reset image size

3.2.1. V838 Cyg and KIC 11125706

When the amplitude modulations of Blazhko stars are very small careful analysis is required to ensure that the fluctuations are not artificial, perhaps caused by the data reduction methods. Such borderline Blazhko stars among the Kepler-field RR Lyr stars include KIC 11125706, V838 Cyg and V349 Lyr (discussed by N11). Previously, KIC 11125706 was found by B10 (page 1588) to have "the lowest amplitude modulation ever detected in an RR Lyrae star," even smaller than the small amplitude modulations seen in RR Gem (Jurcsik et al. 2005) and in SS Cnc (Jurcsik et al. 2006); and V838 Cyg was classified as non-Blazhko by both B10 and N11. We confirm the low amplitude modulations of KIC 11125706, but also find that V838 Cyg exhibits amplitude and frequency modulations and that these are even smaller than those of KIC 11125706 (ΔA₁ = 0.0016 mag versus 0.008 mag, and ΔP_puls = 0.0002 days versus 0.0004 days). The two stars are considerably different in that V838 Cyg has a much higher amplitude and shorter period than KIC 11125706 (A_tot = 1.10 mag versus 0.47 mag; P_puls = 0.48 days versus 0.61 days; see Figure 4 below).

Zoom In Zoom Out Reset image size

Figure 2 shows for V838 Cyg (left) and KIC 11125706 (right) four Fourier variables plotted as time series. The plotted variables are (from bottom to top): the amplitude A₁ and the phase ϕ₁, both from the first term in the Fourier sine series; the ratio R₂₁ of the second and first Fourier amplitudes; and the Fourier phase difference parameter, $\phi _{\rm 31}^s$. The four variables were obtained by dividing the LC data from Q1 to Q11 into time segments, each segment corresponding to three pulsation periods (cf. Figure 5 of N11 where the Kepler stars NQ Lyr and V783 Cyg are plotted). Approximately 40,000 data points were analyzed for each star, and 15 terms were used for the Fourier decompositions.

For both stars the mid-times and Blazhko phases of the respective CFHT spectra are indicated (labeled red vertical lines), as are the adopted t₀ zero points (labeled blue vertical lines). For KIC 11125706 the large gaps at t = 253–323 and at t = 600–688 are because no observations were made in the second two months of Q4 and in Q8. For the same star the blue open squares result from analysis of the 126,955 SC photometric measurements made in Q7—agreement of the SC and LC results is excellent.¹²

The modulation periods are obvious and seen to be close to 55 days for V838 Cyg, and 40 days for KIC 11125706—these are the basic Blazhko periods. In addition to the amplitude modulations seen in the A₁ and R₂₁ panels, the variation of ϕ₁ is indicative of frequency modulation. That the time-series plots for ϕ₁ and $\phi _{\rm 31}^s$ (which measures the pulse shape) are mirror images of each other is not uncommon (but not always the case, as is seen for V2178 Cyg in Figure 3 below).

On relatively short time scales, such as the ∼90 day baseline of the available SC:Q10 data for V838 Cyg (not plotted), the modulations look sinusoidal; however, closer inspection shows that they are not. That the A₁ waves are compressed around time t = 200 and 550 but are wider around t = 400 and 750 proves that the modulation period for V838 Cyg is varying. Thus the amplitude and frequency modulations are complex. A "Period04" frequency analysis of the LC data identified multiple Blazhko periods of 54, 64, and 47 days, the first being the most significant. Finally, the cycle-to-cycle amplitude variations seen in the bottom right panel suggest that additional amplitude modulations may be present for KIC 11125705.

Many other Kepler-field Blazhko stars in addition to V838 Cyg and KIC 11125706 have been found to exhibit interesting and complex modulation behaviors. In the next subsection two quite different Blazhko variables (V1104 Cyg and V2178 Cyg) are used to illustrate the time- and frequency-domain methods that were used for deriving the Blazhko periods and Blazhko types.

3.2.2. Blazhko Periods and Types from Power Spectra

3.2.3. Blazhko Phases

The early studies by Oosterhoff (1939, 1944), Arp (1955), and Preston (1959) established that in the period–amplitude diagram for RR Lyrae stars there is a separation by metal abundance, with metal-poor RRab stars tending to have longer pulsation periods at a given amplitude than metal-rich RRab stars. This effect has been used to better understand the Oosterhoff dichotomy and to derive metallicities for RR Lyrae stars found in different environments, such as in globular clusters (Sandage 1981, 1990, 2004; Cacciari et al. 2005), in different fields of our Galaxy, and in other galaxies. The effect is now known to be due to the lower metal abundance stars having greater luminosities and therefore longer periods (see Sandage 2010; Bono et al. 2007), a result which is consistent with the recent Warsaw convective pulsation hydro-models (see Figure 14 of N11).

In the 1980s Simon and his collaborators introduced Fourier decomposition techniques for describing the shapes of RR Lyr light curves, and established useful correlations between various Fourier parameters and [Fe/H], mass, luminosity, and other physical parameters (Simon & Lee 1981; Simon & Teays 1982; Simon 1985, 1988; Simon & Clement 1993). Since the mid-1990s Kovács and Jurcsik and their collaborators (Kovács & Zsoldos 1995; Jurcsik & Kovács 1996, hereafter JK96; Kovács & Jurcsik 1996; Kovács & Walker 2001; Kovács 2005) have expanded upon these Fourier ideas and have produced useful empirical equations that describe the relationships between RRab light curve parameters and physical characteristics. In particular, JK96 derived a P–$\phi _{\rm 31}^s$–[Fe/H] relation (their Equation (3)), which is often used to estimate the metallicities of RRab stars. More recently, Morgan et al. (2007, hereafter M07) derived analogous equations for RRc stars.

In Figure 4 the log P–A_tot and log P–$\phi _{\rm 31}^s$ diagrams are plotted for the Kepler-field RRab stars. The pulsation periods, A_tot(Kp) and $\phi _{\rm 31}^s$(Kp) are the new values given in Table 1, and the diagrams include both non-Blazhko (black squares) and Blazhko stars (red open squares). By reversing the direction of the $\phi _{\rm 31}^s$ axis one observes directly the similarity of the two diagrams (see Sandage 2004) and that those stars with the largest amplitudes tend to have the most asymmetric light curves, i.e., the smallest $\phi _{31}^s$ values. In both diagrams, all but a few of the most crowded points have been labeled with the star names. For the two most extreme Blazhko stars, V445 Lyr and V2178 Cyg, the $\phi _{\rm 31}^s$ ranges exceed the range of the plotted y-axis. The RRc stars in the Kepler field are off-scale to the left of both graphs, with small amplitudes, large $\phi _{\rm 31}^s$ values, and periods shorter than 0.4 days.

Several trends are apparent in Figure 4, the most obvious being the similarity of the two diagrams. There is also an apparent separation of the stars into two groups, discriminated better by P_puls and $\phi _{\rm 31}^s$ than by P_puls and A_tot. Most of the stars are observed to have A_tot > 0.8 mag and $\phi _{31}^s < 5.4$ rad. According to the metallicity correlations mentioned above these are metal-poor stars with [Fe/H] < −1.0 dex. The other group, those shorter-period stars with lower amplitudes and higher $\phi _{\rm 31}^s$ values, are expected to be more metal-rich, with [Fe/H] > −1.0 dex. For a fixed metallicity as the pulsation period increases the stars tend to have smaller amplitudes and more sinusoidal light curves (i.e., smaller $\phi _{31}^s$ values).

When all the stars in a given sample are suspected of having the same or similar metallicities (e.g., in globular clusters with narrow RGBs), or when there exist independent spectroscopic metal abundances for the sample stars, then diagonal lines can be fit to the observational data (e.g., Sandage 2004; Cacciari et al. 2005; N11). Such diagonal trends are evident in the Figure 4 log P–$\phi _{\rm 31}^s$ diagram, especially when the spectroscopic standard stars (blue crosses) are included in the diagram. This diagram also suggests that the new non-Blazhko star V839 Cyg, and the low-amplitude-modulation star KIC 11125706 (Figure 2) are metal rich. Indeed, we show below that V839 Cyg is metal-rich, with [Fe/H] = −0.05 ± 0.14 dex; however, the CFHT spectra of KIC 11125706, with [Fe/H] = −1.09 ± 0.08 dex (see Table 7), indicate that it probably belongs to the metal-poor group.

The Blazhko stars require special consideration. They are represented in Figure 4 by their mean pulsation periods and their mean A_tot or $\phi _{\rm 31}^s$ values (red open squares), with vertical lines spanning the measured ranges of A_tot and $\phi _{\rm 31}^s$. If metal abundances are derived by substituting the mean $\phi _{\rm 31}^s$ values given in Table 1 into the JK96 P–$\phi _{\rm 31}^s$–[Fe/H] relation, then the resulting photometric [Fe/H] values suggest that almost all of the Blazhko variables are metal-poor; indeed, this is supported by the spectroscopic metallicities (next section), and runs contrary to the suggestion by Moskalik & Poretti (2003) that the incidence of Blazhko variables increases with [Fe/H]. Also, since the Blazhko stars appear to have, at a given period, lower mean A_tot values and higher mean $\phi _{31}^s$ values than the non-Blazhko stars, it appears that the Kepler Blazhko stars must be more metal-rich, on average, than the non-Blazhko stars. The validity of these conclusions depends on the applicability of the JK96 formula, and on the appropriateness of the assumed $\phi _{\rm 31}^s$ values, the latter being particularly true for the most extreme Blazhko stars (V445 Lyr, V2178 Cyg) where the $\phi _{\rm 31}^s$ show large modulations. These topics will be discussed further in Section 5.

The CFHT spectra were acquired from 2010 July–December using the ESPaDOnS prism cross-dispersed échelle spectrograph in the "star+sky" non-polarimetric mode. Two optical fibers fed the star and sky images from the Cassegrain focus to the Coudé focus, then onto a Bowen–Walraven image slicer at the entrance of the spectrograph. The resolving power was R = λ/Δλ ∼ 65, 000, which corresponds to a spectral resolution of Δλ ∼ 0.008 nm at λ = 517 nm (Mg triplet). The spectra are spread over 40 grating orders (on an EEV1 CCD chip), with wavelength coverage from 370 to 1048 nm, an average reciprocal dispersion of 0.00318 nm pixel⁻¹, and ∼3 pixels per resolution element.

Eighteen of the brightest RR Lyrae stars in the Kepler field were observed at CFHT (Service Observing). The total observing time amounted to 20 hr. The observations were made under photometric skies for all but three of the stars (V355 Lyr, V838 Cyg, KIC 7030715), and at various hour angles and airmasses. The seeing was rarely better than 1 arcsec. Multiple spectra were taken for most stars, usually with exposure times of 4 × 900 s; for the faintest observed stars (14.5 < Kp < 15.0) the exposure times were increased to 4 × 1200 s.

The "Upena" software, which uses "Libre-ESpRIT,"¹³ was used to pre-process the CFHT spectra—i.e., perform bias subtraction and flat fielding, subtract the sky and scattered light, identify the echelle orders and perform optimal image extraction, make the wavelength calibrations (using spectra of a Th–Ar lamp), calculate the signal-to-noise ratios (S/Ns per pixel), and make the necessary heliocentric corrections for Earth's motion. The individual spectra were normalized and the overlapping échelle orders merged using the VWA "rainbow" widget (Bruntt et al. 2010a, 2010b). For those stars with multiple observations the normalized spectra were co-added to increase the S/Ns. Details of the individual spectra are given in Table 4. The S/N per pixel values (Column 5) were measured at 570 nm¹⁴ and range from 7–96, with typical values ∼20–30. The FWHM values (Column 6) were measured using the Fe ii lines at 450.8, 451.5, and 452.0 nm, and range from ∼100–400 mÅ.¹⁵

Co-added spectra for the 16 CFHT stars for which metallicities could be derived are plotted in Figure 6, where the wavelength range 515 < λ < 520 nm contains the green Mg i triplet lines. The spectra have been corrected for Doppler shifts, and, to illustrate the effect of increasing metallicity, are ordered by [Fe/H]. The spectra of V1104 Cyg (Kp = 15.03) and V808 Cyg (Kp = 15.36) were of insufficient quality to permit the derivation of metal abundances (but were sufficient for deriving RVs for V1104 Cyg) and are not shown; both stars were re-observed with the Keck telescope.

Zoom In Zoom Out Reset image size

The faintest Kepler-field RR Lyrae stars were observed on the nights of 2011 August 4/5, 5/6 and 6/7 with the HIRES echelle spectrograph (Vogt et al. 1994) mounted on the Keck-I 10 m telescope (see Cohen & Melendez 2005a, 2005b; Cohen & Huang 2009)—see Table 5 for details. The spectrograph was used in the HIRES-r configuration and thus optimized for long wavelength observations (QE > 60% between 400 and 900 nm). A slit of width 1.15 arcsec and length 7 arcsec (C5 decker, no filters) was used, resulting in resolving power R ∼ 36, 000 (Δλ ∼ 0.014 nm at 517 nm) over the range 389–836 nm. The average reciprocal dispersion is 0.00230 nm pixel⁻¹ (140 nm spread over 60,882 pixels) and the number of pixels per resolution element is ∼6. The spectra were spread over a mosaic of three MIT/Lincoln Labs 2048×4096 CCD chips.¹⁶ The λ-scale was calibrated using ThAr lamp exposures taken at the beginning and end of each night; measurements of the widths of typical emission lines suggest an instrumental FWHM of 5.6 ± 0.3 pixels, corresponding to 129 ± 7 mÅ. To reduce phase-smearing and potential cosmic ray problems exposure times for the program stars were never longer than 1200 s, and usually multiple exposures were taken and co-added. The Mauna Kea Echelle Extraction (MAKEE¹⁷) reduction package was used to pre-process the data.

In addition to the program stars we observed nine bright RR Lyr standard stars studied by Layden (1994, hereafter L94) and Clementini et al. (1995, hereafter C95), and three red giant stars in the very metal-poor globular cluster M92 (mean [Fe/H] = −2.33 dex, with little within-cluster variation; Cohen 2011). All of the RR Lyr standard stars are RRab pulsators and include RR Lyrae itself (Kp = 7.862), which happens to be located in the Kepler field and has been the subject of several recent detailed investigations (K10; K11; Benedict et al. 2011). The M92 stars served primarily as RV standards and the bright RR Lyr stars as metallicity standards. The most metal poor and metal rich standard stars are X Ari and SW And, respectively, for which [Fe/H] = −2.74 ± 0.09 dex and +0.20 ± 0.08 dex (see Table 7), and the corresponding Kepler-field RR Lyr stars are NR Lyr and V784 Cyg, for which [Fe/H] = −2.54 ± 0.11 dex and −0.05 ± 0.10 dex. Although X Ari and NR Lyr are very metal poor, and have metallicities similar to those of the most metal-poor RR Lyr stars known in the LMCs and SMCs (see Haschke et al. 2012), they are not the most metal poor stars known in our Galaxy (see Schörck et al. 2009).

Sample Keck spectra are plotted in Figure 7, for the same wavelength range (515–520 nm) as Figure 6. The three panels, each of which includes the high S/N spectrum of RR Lyrae, show program stars more metal-poor than RR Lyrae (Figure 7(a)), program stars more metal-rich than RR Lyrae (Figure 7(b)), and Keck RR Lyr standard stars (Figure 7(c)). The high-resolution, S/N ∼ 1000, solar spectrum supplied with VWA (Figure 7(c)) was acquired with the KPNO Fourier Transform Spectrometer (Kurucz et al. 1984; Hinkle et al. 2000).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Typical FWHM values were measured for individual and co-added Keck spectra using the same three Fe ii spectral lines that were used for the CFHT spectra. The measured FWHM values for the individual spectra (see Table 5) and for the co-added spectra (see Table 7) range from 150 to 470 mÅ, which is similar to the range for the CFHT spectra. As expected the measured FWHM values tend to be largest for phases away from maximum radius. This effect is illustrated in the lower panel of Figure 5, where the FWHM for the three Fe ii lines near 450 nm and for the Ti ii line at 450.1 nm are plotted as a function of the pulsation phase for the nine bright RR Lyr standard stars. The narrowest lines are seen to occur at the earliest phases (∼0.3–0.4), and at all phases the Fe ii lines (solid symbols) are significantly narrower than the Ti ii line (crosses). The same pattern for these four lines is also seen for the Kepler stars (where phase range and scatter are larger), for RR Lyrae (Figure 4 of K10), and for XZ Apr (Figure 18 of For et al. 2011).

It is well known that the RVs derived from spectra of RR Lyr stars are phase dependent and are not necessarily the same as the velocities that the stars would have if they were not pulsating (Oke 1966; Liu & Janes 1989, 1990a, 1990b). Furthermore, the derived RVs depend on the spectral lines used for their measurement: RV curves calculated with metal lines tend to have smaller amplitudes than those calculated using Balmer-series lines (Hα, Hβ, Hγ)—a difference that has recently been quantified by Sesar (2012). A nice illustration of both effects is given in Figure 2 of Preston (2011), which shows multiple RV curves for the southern RRab stars Z Mic and RV Oct.

In general, the RVs of RR Lyr stars are most negative near maximum light (i.e., at ϕ_puls ∼ 0) and most positive near minimum light (which, depending on the risetime, occurs for RRab stars around ϕ_puls ∼ 0.80–0.95 and at earlier phases for RRc stars). Fundamental-mode pulsators also tend to show larger RV variations than first-overtone pulsators (as is also the case for the light variations), with double-mode (RRd) and other multi-mode pulsators showing the most complex variations. If a star exhibits the Blazhko effect then the RV curve will reflect the amplitude and phase variations of such stars, and if pulsational velocities are required (e.g., Baade–Wesselink studies) then the observed RV must be corrected for projection effects (see Nardetto et al. 2004, 2009).

For our spectra the heliocentric RVs given in Tables 4 and 5 were derived using metal lines and the IRAF "fxcor" routine. The heliocentric corrections were made during standard pipeline reductions, by Upena for the CFHT data and by MAKEE for the Keck data. For the ESPaDOnS spectra only the limited wavelength range 511–518 nm was used for the cross-correlations and a typical RV uncertainty is ∼ ± 0.2 km s⁻¹. The HIRES flux spectra were normalized using MAKEE and very accurate pixel shifts were calculated by cross-correlating all pairs of spectra that were taken over the three night run. The results were used to derive very precise and accurate relative velocities that were then shifted to the mean of the RVs computed relative to the known wavelengths.

The derived heliocentric RVs for the three M92 red giant standard stars (see Table 5) are all close to the cluster mean of −120 km s⁻¹, suggesting that all are cluster members (see Roederer & Sneden 2011; Cohen 2011). Comparison with the RVs derived by Drukier et al. (2007) gives a median difference of ∼0.5 km s⁻¹ and an rms difference of 0.2 km s⁻¹.

The γ-velocities for the nine bright RR Lyrae standard stars were computed using Liu's RV template and are given in Table 6. The last column contains the differences between our estimates and those of Layden (1994), where the mean difference is 1.3 km s⁻¹.

Estimation of metal abundances and other atmospheric parameters from high dispersion spectra depends on numerous assumptions regarding the stellar atmosphere where the spectral lines were formed. In our analyses, which were performed using the VWA, MOOG, and Spectroscopy Made Easy (SME) packages, the lines were assumed to form in a one-dimensional plane-parallel atmosphere under local thermodynamic equilibrium (LTE) conditions. For RR Lyr stars (absolute magnitudes ∼0.5) the atmospheric extensions are sufficiently small (less than 1% according to Figure 1 of Heiter & Eriksson 2006) that the plane-parallel assumption was judged to be adequate for the derivation of iron-to-hydrogen ratios.

The observed Kepler-field RR Lyr stars are a mixture of halo and old-disk stars, with distances ranging from 265 ± 11 pc for RR Lyrae (Benedict et al. 2011) to ∼25 kpc for the faintest stars (N11); therefore they are expected to be old and to have a wide range of chemical and kinematic characteristics. In the H-R diagram they are horizontal branch stars at various stages of evolution away from the ZAHB, and over a pulsation cycle L and T_eff trace out a loop in the instability strip. The masses are expected to be in the range 0.5–0.8 $\cal M_{\odot }$, and most of the stars are expected to be near the ZAHB-level appropriate for the mass and metal abundance of the star (see Figure 13 of N11).

The Blazhko and non-Blazhko RR Lyr stars in the Kepler field occur in approximately equal proportions. The pulsations of the non-Blazhko stars are mainly fundamental or first-overtone radial modes. To date no "classical" double-mode RR Lyrae stars (i.e., RRd stars) pulsating simultaneously in both modes have been identified; however, Molnár et al. (2012) recently detected a very weak first-overtone pulsation in RR Lyrae. When averaged over a pulsation cycle the mean values of T_eff are expected to range from 6000–7500 K, with the ab-type stars having cooler mean surface temperatures than the c-type stars. The ranges of the mean values of log g and L are expected to be 2–3 and 40–60 L_☉. The instantaneous values for these quantities, which are phase dependent, span much wider ranges—see, for example, Jurcsik et al. (2009a, 2009b, 2009c), who suggested that the T_eff range might be as large as 3500 K (5500–9000 K), the range of log g ∼ 1.5 to 4, and the range of L ∼ 30–90 L_☉. For all three physical characteristics the maxima occur slightly before ϕ_puls = 1. Because the CFHT and Keck spectra were taken at pulsation phases away from maximum light the RRab stars were expected to have "instantaneous" T_eff values in the considerably smaller range 6000–6500 K (see Figures 7–10, 36 and 37 of For et al. 2011), with log g between 2 and 3. An exception is V839 Cyg, which was observed just after maximum light, at ϕ_puls = 0.12, and therefore expected to have a rather higher temperature. Unlike solar-like stars which typically have a microturbulent parameter ξ_t ∼ 1 km s⁻¹, the ξ_t for RR Lyr stars generally range from 2.5 to 4.0 km s⁻¹, with instantaneous values at maximum radius tending to occur at the lower end of this range (i.e., ξ_t ∼ 2.5–3.0 km s⁻¹ for phases between 0.2 and 0.6—see Figures 13, 14, and 17 of For et al. 2011). Other information is available for some of the brighter stars. For example, for RR Lyrae at ϕ_puls = 0.54 the log g and T_eff are expected to be ∼2.75 and 6250 ± 100 K (see Table 5.2 of Kolenberg 2002), both of which vary from one Blazhko cycle to the next.

For the solar photosphere the abundance of iron relative to the number of hydrogen atoms per unit volume was assumed to be log (Fe) = 7.50, where log (Fe) ≡ log (N_Fe/N_H) + 12.0 and log N_H = 12.0. This value is consistent with the iron abundance of 7.45 ± 0.08 recommended by Holweger (2001), the value of 7.52 adopted by MOOG (Sneden 2002), and with the most recent value of 7.50 ± 0.04 given in the review by Asplund et al. (2009). For iron abundances relative to hydrogen and relative to the Sun we use [Fe/H]_* = log (N_Fe/N_H)_* − log (N_Fe/N_H)_solar. When abundances are expressed relative to the total number of atoms per unit volume (e.g., the values output from VWA and Kurucz's ATLAS9 program), the photospheric "abundance" of Fe is denoted A_Fe ≡ log (N_Fe/N_total), where N_Fe is the number of Fe atoms per unit volume and N_Fe/N_total is the corresponding fraction of Fe relative to the total number density of atoms. If the number densities for hydrogen and helium are assumed to be log N_H = 12.0 and log N_He = 10.99 then these two elements are the only significant contributors to N_total and we have log N_total = 12.04. Thus, the solar number fractions of H, He, and Fe are, respectively, N_H/N_total = 91.1%, N_He/N_total = 8.9%, and N_Fe/N_total = 0.0029% (i.e., the log of the Fe abundance relative to the total number density is log (N_Fe/N_tot) = −4.54, which follows from A_Fe = log (Fe) − log N_total = 7.50 − 12.04).

4.4.1. Parameter Estimation with VWA and MOOG

The bulk of the spectroscopic data analyses of both the CFHT and Keck data was carried out with the spectral synthesis program VWA¹⁸ (Bruntt et al. 2002, 2008, 2010a, 2010b). This all-purpose program has the capability of normalizing and co-adding spectra, computing synthetic spectra, selecting suitable lines for abundance determination, measuring EWs, calculating expected EWs for the assumed model parameters, and calculating abundances for a wide range of elements/ions. The abundances follow from iteratively varying the model parameters, calculating synthetic spectra, and then comparing the EWs calculated from the spectra with the measured EWs.

For each RR Lyrae star the following photospheric parameters were derived: the effective temperature, T_eff (K); the surface gravity, log  g (cm s⁻²); the microturbulent velocity, ξ_t (km s⁻¹); the macroturbulent velocity, v_mac; the projected rotational velocity, vsin i; and the spectroscopic iron-to-hydrogen ratio, [Fe/H]_spec. Because the RR Lyr stars are undergoing (mainly) radial pulsations, these parameters are "instantaneous" quantities, specific for the times (phases) at which the spectroscopic observations were made. The metal abundances, while depending on the assumed atmospheric conditions, should be independent of the observed pulsation phases (see Figures 13 and 14 of For et al. 2011, which show the variation with pulsation phase of their derived T_eff, log g, [Fe/H] and ξ_t values).

The synthetic spectra were calculated within VWA using the SYNTH code (Piskunov 1992; Valenti & Piskunov 1996), with atomic parameters (e.g., log gf values) and line-broadening coefficients retrieved from the Vienna atomic line database (VALD; Kupka et al. 1999).¹⁹ Two sets of model stellar atmospheres were used. For all the stars the model atmosphere grid of ATLAS9 models provided by Heiter et al. (2002) was used. For this grid the step-increments in effective temperature, surface gravity, and metal abundance are, respectively, ΔT_eff = 250 K, Δlog g = 0.5, and Δ[Fe/H] = 0.25 dex. For the bright RR Lyrae stars both ATLAS9 and MARCS models were used.

The derivation of [Fe/H]_spec values followed from analysis of the normalized flux spectra. For each star an initial set of values for the atmospheric parameters was assumed, and a synthetic spectrum was calculated. The usual starting values were: T_eff = 6500 K, log g = 2.50, ξ_t = 2.50 km s⁻¹ and v_mac = 2 km s⁻¹. After adjusting the projected rotational velocity, vsin i and the resolving power (R = 65, 000 for the CFHT spectra and 36,000 for the Keck spectra) for each analyzed spectrum, the spectrum was Doppler shifted according to the RV (see Tables 4 and 5) and aligned with the synthetic template spectrum.

The final set of fitted lines was inspected and poor fits rejected. The microturbulent velocity, ξ_t, was determined by requiring that the correlation between the derived Fe abundances and the reduced EWs (i.e., EW/λ) was near zero; T_eff was estimated by requiring that the correlation of Fe abundance and the lower excitation potential (EP) was near zero; and the surface gravities were calculated by requiring good agreement between the iron abundances determined from lines of the Fe i and Fe ii ionization stages (which depended on the assumed T_eff and log g values). Although LTE conditions are assumed by VWA, the effect of departures from LTE on the [Fe/H] values derived using the Fe i lines were considered, and where appropriate the corrections calculated by Rentzsch-Holm (1996) were applied; these were usually smaller than ∼0.10 dex. The Fe ii lines were assumed to be unaffected by deviations from LTE.

Table 7 summarizes the basic spectroscopic information derived from the VWA analyses of the CFHT and Keck spectra, where the stars are sorted by type (non-Blazhko, Blazhko, RRc, etc.), and within a given type by spectroscopic metal abundance. The pulsation phases (Column 3) are mid-exposure values (i.e., average values of the mid-times when multiple spectra were taken), and the S/N per pixel (Column 4) are for the co-added normalized spectra.²⁰ The FWHM values (Column 5) are for the co-added spectra and the T_eff, log g, and ξ_t values (Column 6) are either those assumed for the adopted VWA runs, or are interpolated values for two similar atmosphere models. The value v_mac was arbitrarily set to either 2 or 3 for all the stars, while the projected rotational velocity, vsin i, was set to 2 km s⁻¹ for the Keck spectra and 7–10 km s⁻¹ for the CFHT spectra; the impact on these choices on the derived abundances seems to have been minimal. The iron-to-hydrogen abundances (Columns 7 and 8) are for the neutral and singly ionized ions, the weighted average of which, [Fe/H], is given in Column 9. The derived metallicities range from a low of −2.54 ± 0.11 dex (for NR Lyr) to a high of −0.05 ± 0.10 dex (for V784 Cyg). The same two stars were the extremes identified by N11 in their analysis of the Q1–Q5 Kepler photometry of non-Blazhko variables, and the four non-Blazhko stars that were suspected (from their locations in Figure 9 of N11) of being metal rich (V782 Cyg, V784 Cyg, KIC 6100702, and V2470 Cyg) are confirmed as such. Not included in Table 7 are four stars for which the VWA analysis of the co-added spectrum proved inconclusive or impossible: (1) V1510 Cyg, a non-Blazhko star (Kp = 14.5 mag); (2) the faint non-Blazhko star V346 Lyr (Kp = 16.4 mag); (3) the faint Blazhko star KIC 9973633 (Kp = 17.0 mag), for which the S/N of the co-added Keck spectrum (not plotted in Figure 7) was insufficient for estimating [Fe/H]; and (4) the faint (Kp = 17.4 mag) extreme-Blazhko star V445 Lyr (see Guggenberger et al. 2012). In all cases except KIC 9973633 we were able to derive [M/H] values using SME (see Table 9).

Table 7. Spectroscopic Iron-to-Hydrogen Abundances (VWA) for the Kepler-field RR Lyrae Stars

Star	Obs.	ϕpuls	S/N	FWHM	Teff/log g/ξt	[Fe/H]spec
						Fe i Lines	Fe ii Lines	Adopted
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
(a) 17 Non-Blazhko RRab-type stars
NR Lyr	CFHT	0.29	37	0.163 (02)	6500/2.6/2.5	−2.54 ± 0.13 (6)	−2.53 ± 0.17 (5)	−2.54 ± 0.11
FN Lyr	CFHT	0.28	32	0.209 (19)	6300/2.44/2.5	−2.00 ± 0.13 (10)	−1.97 ± 0.12 (12)	−1.98 ± 0.09
NQ Lyr	CFHT	0.32	30	0.197 (15)	6000/1.80/2.5	−1.88 ± 0.19 (11)	−1.90 ± 0.10 (3)	−1.89 ± 0.10
V350 Lyr	Keck	0.53	34	0.293 (11)	6180/2.9/2.6	−1.84 ± 0.15 (12)	−1.81 ± 0.14 (10)	−1.83 ± 0.12
V894 Cyg	CFHT	0.39	28	0.193 (08)	6200/2.46/2.0	−1.65 ± 0.18 (11)	−1.66 ± 0.16 (9)	−1.66 ± 0.12
AW Dra	CFHT	0.30	38	0.202 (05)	6540/2.40/2.5	−1.33 ± 0.13 (9)	−1.33 ± 0.12 (6)	−1.33 ± 0.09
KIC 7030715	CFHT	0.26	19	0.26  (04)	6500/2.64/2.5	−1.32 ± 0.07 (6)	−1.34 ± 0.15 (3)	−1.33 ± 0.08
V1107 Cyg	Keck	0.48	21	0.220 (17)	6300/2.8/3.0	−1.23 ± 0.42 (33)	−1.31 ± 0.18 (11)	−1.29 ± 0.23
V368 Lyr	Keck	0.43	20	0.183 (02)	6300/2.8/3.0	−1.33 ± 0.31 (22)	−1.25 ± 0.24 (10)	−1.28 ± 0.20
KIC 9658012	Keck	0.84	37	0.413 (16)	6500/2.8/3.0	−1.22 ± 0.26 (26)	−1.31 ± 0.12 (11)	−1.28 ± 0.14
KIC 9717032	Keck	0.23	19	0.262 (06)	6620/2.8/3.5	−1.27 ± 0.17 (14)	−1.26 ± 0.17 (13)	−1.27 ± 0.15
V715 Cyg	Keck	0.51	21	0.241 (13)	6400/3.0/3.0	−1.13 ± 0.12 (25)	−1.14 ± 0.14 (12)	−1.13 ± 0.09
V2470 Cyg	CFHT	0.32	26	0.262 (11)	6400/2.44/2.6	−0.59 ± 0.14 (45)	−0.59 ± 0.21 (14)	−0.59 ± 0.13
V782 Cyg	Keck	0.42	34	0.290 (12)	6525/3.2/2.9	−0.40 ± 0.15 (102)	−0.43 ± 0.14 (19)	−0.42 ± 0.10
KIC 6100702	CFHT	0.26	23	0.281 (10)	6500/2.50/2.5	−0.19 ± 0.14 (36)	−0.13 ± 0.12 (6)	−0.16 ± 0.09
V839 Cyg	Keck	0.12	70	0.375 (03)	7200/3.1/4.0	+0.07 ± 0.18 (137)	−0.09 ± 0.14 (16)	−0.05 ± 0.14
V784 Cyg	Keck	0.47	27	0.323 (12)	6400/3.3/4.3	−0.05 ± 0.15 (107)	−0.06 ± 0.13 (10)	−0.05 ± 0.10
(b) 12 Blazhko RRab-type stars (excluding RR Lyrae)
V2178 Cyg	Keck	0.43	41	0.211 (09)	6450/2.8/3.0	−1.63 ± 0.24 (17)	−1.67 ± 0.09 (11)	−1.66 ± 0.13
V450 Lyr	Keck	0.57	21	0.236 (17)	6450/3.0/3.0	−1.52 ± 0.13 (13)	−1.50 ± 0.21 (9)	−1.51 ± 0.12
V353 Lyr	Keck	0.32	20	0.205 (17)	6300/2.8/3.0	−1.48 ± 0.20 (15)	−1.50 ± 0.35 (4)	−1.50 ± 0.20
V360 Lyr	Keck	0.52	29	0.378 (20)	6400/2.8/10:	−1.46 ± 0.37 (72)	−1.54 ± 0.44 (26)	−1.50 ± 0.29
V354 Lyr	Keck	0.26	30	0.298 (14)	6400/2.8/3.0	−1.44 ± 0.24 (20)	−1.44 ± 0.20 (13)	−1.44 ± 0.16
V1104 Cyg	Keck	0.42	33	0.174 (03)	6300/2.8/3.0	−1.25 ± 0.25 (51)	−1.21 ± 0.15 (17)	−1.23 ± 0.15
V808 Cyg	Keck	0.70	19	0.473 (16)	6300/2.8/3.0	−1.20 ± 0.27 (15)	−1.18 ± 0.25 (7)	−1.19 ± 0.18
V783 Cyg	CFHT	0.58	16	0.23  (01)	6400/2.1/2.6	−1.19 ± 0.18 (13)	−1.13 ± 0.14 (2)	−1.16 ± 0.11
V366 Lyr	Keck	0.69	21	0.332 (24)	6430/3.0/3.0	−1.13 ± 0.13 (16)	−1.19 ± 0.10 (10)	−1.16 ± 0.09
V355 Lyr	CFHT	0.19	17	0.247 (16)	6900/3.28/2.5	−1.13 ± 0.14 (10)	−1.17 ± 0.30 (1)	−1.14 ± 0.17
KIC 11125706	CFHT	0.33	69	0.232 (04)	6200/2.35/2.25	−1.09 ± 0.13 (57)	−1.09 ± 0.10 (17)	−1.09 ± 0.08
V838 Cyg	CFHT	0.36	13	0.236 (28)	6850/2.12/2.4	−1.02 ± 0.19 (4)	−1.00 ± 0.10 (3)	−1.01 ± 0.10
(c) Five RRc-type stars (including the RRc/HADS? star KIC 3868420)
KIC 9453114	CFHT	0.28	29	0.278 (27)	6500/2.5/2.5	...	−2.13 ± 0.12 (4)	−2.13 ± 0.12
KIC 4064484	Keck	0.59	33	0.288 (11)	6500/2.8/3.0	−1.65 ± 0.22 (9)	−1.54 ± 0.12 (6)	−1.58 ± 0.13
KIC 3868420	CFHT	0.47	96	0.359 (09)	7480/3.9/2.0	−0.35 ± 0.14 (13)	−0.29 ± 0.08 (6)	−0.31 ± 0.08
KIC 8832417	CFHT	0.29	30	0.349 (16)	7000/3.30/2.5	−0.22 ± 0.19 (27)	−0.29 ± 0.09 (7)	−0.27 ± 0.11
KIC 5520878	Keck	0.39	40	0.319 (02)	7250/3.9/2.4	−0.18 ± 0.18 (76)	−0.19 ± 0.12 (18)	−0.18 ± 0.10
(d) Nine bright RR Lyrae metal abundance standards (including RR Lyrae)
X Ari	Keck	0.31	223	0.188 (05)	6075/2.4/2.0	−2.75 ± 0.08 (9)	−2.74 ± 0.17 (11)	−2.74 ± 0.09
VY Ser	Keck	0.45	120	0.232 (03)	6170/2.8/2.5	−1.73 ± 0.12 (29)	−1.69 ± 0.08 (16)	−1.71 ± 0.07
ST Boo	Keck	0.36	135	0.198 (02)	6313/2.6/1.8	−1.66 ± 0.12 (39)	−1.60 ± 0.04 (15)	−1.62 ± 0.06
UU Cet	Keck	0.38	66	0.226 (03)	6150/2.6/1.6	−1.34 ± 0.14 (68)	−1.32 ± 0.07 (19)	−1.33 ± 0.08
RR Lyr	Keck	0.54	252	0.276 (06)	6400/2.8/3.0	−1.27 ± 0.22 (95)	−1.27 ± 0.09 (15)	−1.27 ± 0.12
VX Her	Keck	0.32	97	0.202 (02)	6690/2.9/2.2	−1.22 ± 0.17 (51)	−1.23 ± 0.17 (17)	−1.23 ± 0.12
RR Cet	Keck	0.37	184	0.212 (03)	6420/2.9/1.9	−1.17 ± 0.15 (65)	−1.19 ± 0.08 (17)	−1.18 ± 0.09
V445 Oph	Keck	0.44	106	0.315 (05)	6200/2.8/1.8	+0.15 ± 0.14 (218)	+0.11 ± 0.13 (16)	+0.13 ± 0.10
SW And	Keck	0.56	151	0.367 (06)	6660/3.6/2.2	+0.19 ± 0.14 (225)	+0.21 ± 0.09 (19)	+0.20 ± 0.08

Notes. The columns contain: (1) star name; (2) source of the spectra, either "Keck I 10 m + HIRES" or "CFHT 3.6 m + ESPaDOnS"; (3) pulsation phase at mid-exposure; (4) signal-to-noise ratio per pixel for the co-added spectra, measured at ∼570 nm and calculated using the VWA "rainbow" widget; (5) average FWHM value (Å) for the three Fe ii lines at 4508, 4515, and 4520 Å, derived from the spectrum (either single or co-added exposures) analyzed by VWA—the uncertainty (given in parentheses) is approximated by the standard deviation of the mean FWHM for the three lines; (6) the assumed values for the model effective temperature T_eff (K), surface gravity log g (cm s⁻²), and microturbulent velocity ξ_t (km s⁻¹)—in general the T_eff have uncertainties ∼ ± 150 K, the log g are ±0.10 to 0.20, and the ξ_t are ±0.3 to 0.5 km s⁻¹; (7) [Fe/H] abundance based on the Fe i lines (number of lines given in parentheses); (8) [Fe/H] abundance based on the Fe ii lines (number of lines given in parentheses); (9) adopted VWA spectroscopic metal abundance, equal to the weighted (in inverse proportion to uncertainty) average of the abundances in Columns 7 and 8.

Download table as:  ASCIITypeset image

The EWs derived using VWA are in good agreement with the values measured by hand using the IRAF "splot" routine, with values calculated with ARES, and with the values calculated from the synthetic spectra. The EWs of the standard stars also agree (to within the uncertainties arising from the observations being made at different phases) with the values listed in Table 3(b) of C95.²¹ To minimize the effects of line saturation only Fe i and Fe ii lines with EW <90 mÅ were used in our analysis. A table of measured and calculated EWs (mÅ) for the Fe i and Fe ii lines (program and standard stars), including the assumed lower EPs and log gf values (from the VALD database), is available from the first author.

As a check on the VWA results [Fe/H] abundances for several of the stars were also derived using the LTE Stellar Line Analysis program MOOG (Sneden 1973, 2002; Sneden et al. 2011).²² In this case EWs were calculated using the highly automated program ARES (Sousa et al. 2007, 2008). A comparison of the EWs derived by ARES, by VWA, and by the IRAF "splot" routine, showed excellent agreement for EWs between 5 and 100 mÅ. The WEBMARCS grid of model atmospheres (Gustafsson et al. 1975, 2008)²³ was used for the MOOG calculations. These models used the Grevesse et al. (2007) compositions for the solar photosphere, and the available grid had T_eff values up to 7000 K with a step size of 250 K. Unfortunately the smallest surface gravities available were for log g = 3, which is larger than the value appropriate for most RR Lyrae stars.

The MOOG results are summarized in Table 8. The assumed model atmospheric parameters are given in Column 2, where the uncertainties are ∼ ± 150 K for T_eff, ±0.2 cm s⁻² for log g, and ±0.5 km s⁻¹ for ξ_t. Columns 3 and 4 give log  values (relative to hydrogen) for the Fe i and Fe ii ions (standard output from the "ABFIND" routine in MOOG). Also given are the number of lines used to derive the abundance. Column 5 contains the average iron abundance, log (Fe), weighted inversely by the Fe i and Fe ii uncertainties, and the last column gives [Fe/H] values, derived by adding 12.50 ± 0.05 to the Column 5 abundances. Since only a limited number of MOOG runs were made the uncertainties are larger than for the VWA results, the largest discrepancies occurring for V782 Cyg and SW And.

Table 8. Spectroscopic Iron-to-Hydrogen Abundances (MOOG)

Star	Teff/log g/ξt/[Fe/H]	log (Fe i)	log (Fe ii)	log (Fe)	[Fe/H]spec
(1)	(2)	(3)	(4)	(5)	(6)
X Ari	6000/3.0/2.0/−2.5	4.73 ± 0.12 (105)	5.19 ± 0.18 (24)	4.91 ± 0.07	−2.59 ± 0.12
VY Ser	6250/3.0/2.0/−1.5	5.83 ± 0.12 (147)	6.01 ± 0.12 (26)	5.92 ± 0.06	−1.58 ± 0.11
ST Boo	6250/3.0/2.0/−1.5	5.78 ± 0.12 (177)	6.09 ± 0.12 (29)	5.94 ± 0.06	−1.57 ± 0.11
RR Lyr	6000/3.0/2.0/−1.5	5.96 ± 0.14 (145)	6.25 ± 0.09 (19)	6.14 ± 0.05	−1.36 ± 0.10
VX Her	6750/3.0/2.0/−1.5	6.24 ± 0.12 (169)	6.35 ± 0.13 (24)	6.29 ± 0.06	−1.21 ± 0.11
SW And	6250/3.0/2.0/+0.0	7.35 ± 0.15 (177)	7.42 ± 0.15 (11)	7.39 ± 0.08	−0.12 ± 0.13
V782 Cyg	6250/3.0/2.0/+0.0	7.41 ± 0.30 (162)	7.45 ± 0.28 (11)	7.43 ± 0.14	−0.07 ± 0.19
V445 Oph	6250/3.0/2.0/+0.0	7.45 ± 0.11 (188)	7.61 ± 0.13 (11)	7.52 ± 0.06	+0.02 ± 0.11
V784 Cyg	6250/3.0/2.0/+0.0	7.86 ± 0.35 (142)	7.92 ± 0.34 (7)	7.89 ± 0.17	+0.39 ± 0.22

Notes. The columns contain: (1) the star name; (2) the assumed values for the (WebMARCS) model effective temperature, T_eff (K), surface gravity, log g (cm s⁻²), microturbulent velocity, ξ_t (km s⁻¹), and metal abundance, [Fe/H]; (3) the [Fe i/H] abundance (based on the number of Fe i lines given in parentheses); (4) the [Fe ii/H] abundance (based on the number of Fe ii lines given in parentheses); (5) the weighted average of the [Fe i/H] and [Fe ii/H] abundances; (6) the adopted MOOG spectroscopic iron abundance.

Download table as:  ASCIITypeset image

4.4.2. Parameter Estimation with SME

4.4.3. Comparison of Atmospheric Parameters

In Table 10 the VWA, MOOG, and SME iron abundances are compared. Also included for comparison with our values are the [Fe/H] values derived by L94, Suntzeff et al. (1994), C95, Lambert et al. (1996, hereafter L96), Feast et al. (2008), and Wallerstein & Huang (2010). Because C95 assumed log (Fe) = 7.56, whereas we adopted the value log (Fe) = 7.50, the C95 metallicity values (from their Table 12) have been increased by 0.06 dex. L96 report log (Fe ii) values, to which we have added 7.50 to give the abundances in Column 5. The L94 metallicities are based on measurements of the Ca ii K-line.

Table 10. Spectroscopic Iron-to-Hydrogen Abundances for the Keck Standard Stars

Star	[Fe/H] (Previous Papers)						[Fe/H] (This Paper)
	L94	SKK94	C95	L96	Fer98	Wal10	VWA	MOOG	SME
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
X Ari	−2.40	−2.16	−2.44	−2.53	−2.43	−2.68	−2.74 ± 0.09	−2.59 ± 0.12	−2.71 ± 0.10
VY Ser	−1.82	−1.83	−1.65	−1.75	−1.79	...	−1.71 ± 0.07	−1.58 ± 0.11	−1.71 ± 0.10
ST Boo	−1.86	...	−1.62	...	−1.76	−1.77	−1.62 ± 0.06	−1.57 ± 0.11	−1.61 ± 0.10
RR Cet	−1.52	−1.55	−1.32	...	−1.45	...	−1.27 ± 0.12	...	−1.22 ± 0.10
UU Cet	...	−1.88	−1.32	...	−1.28	...	−1.33 ± 0.08	...	−1.33 ± 0.10
RR Lyr	−1.37	−1.37	−1.32	−1.47	−1.39	...	−1.27 ± 0.12	−1.36 ± 0.10	−1.33 ± 0.10
VX Her	...	−1.37	−1.52	...	−1.58	−1.48	−1.23 ± 0.12	−1.21 ± 0.11	−1.38 ± 0.10
SW And	−0.38	−0.41	−0.00	−0.23	−0.24	−0.16	+0.20 ± 0.08	−0.12 ± 0.13	+0.19 ± 0.10
V445 Oph	−0.23	−0.30	+0.23	...	−0.19	+0.24	+0.13 ± 0.10	+0.02 ± 0.11	+0.14 ± 0.10

Notes. The columns contain: (1) the star name; (2) [Fe/H] from Table 9 of L94, where [Fe/H] is based on the Ca II K-line; (3) [Fe/H] from Table 3 of Suntzeff et al. (1994), based on ΔS measurements; (4) [Fe/H] from Table 12 of C95, made more metal rich by 0.06 since Clementini et al. (1995) assumed for the Sun log (Fe) = 7.56, whereas we are assuming 7.50; (5) [Fe/H] from Table 5 of Lambert et al. (1996)—the log  given, which are the Fe ii values, have had the assumed solar Fe abundance (7.50) subtracted; (6) [Fe/H] from Table 1 of Fernley et al. (1998)—these same values were adopted by Feast et al. (2008); (7) [Fe/H] from Table 2 of Wallerstein & Huang (2010), based on analysis of Apache Point Observatory spectra; (8–10) spectroscopic [Fe/H] values derived here using VWA, MOOG, and SME.

Download table as:  ASCIITypeset image

X Ari, with [Fe/H] ∼ −2.65 dex, is probably more metal deficient than NR Lyr, the most metal-poor RR Lyr star in the Kepler field, which we estimated to have [Fe/H] = −2.54 ± 0.11 dex (Table 7). Recently Haschke et al. (2012) derived a spectroscopic [Fe/H] = −2.61 dex for X Ari, a value consistent with our estimate and with the value derived by Wallerstein & Huang (2010). It is unclear whether the metal-rich end is defined by the standard star V445 Oph or by SW And. Our VWA and SME analyses suggest that the [Fe/H] values for both stars are greater than that of the Sun, while the MOOG analyses suggest that SW And may be more metal-poor and V445 Oph more metal rich than the Sun. Most previous analyses lean toward V445 Oph being the more metal rich, as does our photometric [Fe/H] (see Figures 4 and 12). In any case, both have metallicities similar to that of the most metal-rich star in the Kepler field, V784 Cyg.

For VY Ser a metal abundance [Fe/H] = −1.77 ± 0.10 dex was derived by Carney & Jones (1983). The good agreement between this value and our VWA, MOOG, and SME estimates is undoubtedly due to their spectra and ours having been taken at similar phases (0.61 versus our 0.48). Furthermore, their log g = 2.3 ± 0.3 and ξ_t = 4.2 ± 0.5 km s⁻¹ agree with our SME values of 2.1 (cgs units) and 4.4 km s⁻¹; and their T_eff = 6000 ± 150 K is in accord with our SME estimate of 6167 K.

In Figure 8 the SME estimates of the atmospheric parameters T_eff, log g, ξ_t, and [M/H] are compared (solid squares) with the values derived using VWA. While the results are similar, the temperatures from SME tend to be smaller than the VWA values (upper left panel), the mean difference being ∼100 K, with an rms scatter ∼100 K. The surface gravities from SME are also smaller on average than the VWA values (lower left panel)—mean Δlog g ∼ −0.5 with an rms scatter of 0.3. The trend seen in ξ_t (upper right panel) implies that the SME-derived values are systematically larger than the VWA values in most cases, where the difference increases as ξ_t increases (not shown in the diagram is the extreme outlier V360 Lyr, where Δξ_t = −4.7 km s⁻¹ is due primarily to the large ξ_t (=10 km s⁻¹) adopted for the VWA analysis). Despite these differences the lower right panel shows that there is reasonably good agreement of the [M/H] abundances derived with SME and the [Fe/H] values derived using VWA (and MOOG)—the outliers at the bottom of the panel are V1107 Cyg, V839 Cyg, and VX Her (a standard star). For the VWA analyses, the v_mac values were generally set to 2–3 km s⁻¹ and vsin i values of 2 km s⁻¹ and 7–10 km s⁻¹ were adopted for the Keck and CFHT spectra, respectively. These values differ considerably from the v_mac values derived by SME, which have a mean of 13.0 km s⁻¹ (standard deviation σ = 5.4 km s⁻¹, N = 43 stars), and from the vsin i values (see bottom panel of Figure 9) derived by SME, which have a mean of 4.3 km s⁻¹ (σ = 1.2 km s⁻¹, N = 40, excluding the three stars with vsin i > 9 km s⁻¹).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Also plotted in Figure 8 (as crosses) are the differences between our SME-derived parameters for the nine Keck RR Lyrae standard stars and the corresponding values derived by C95. Since the pulsation phases for our Keck spectra differ from those of the Palomar 1.5 m spectra analyzed by C95 the effective temperatures are expected to differ. On the other hand, because the RRab stars lie on the red side of the instability strip and both studies used spectra taken away from maximum light the T_eff should be similar; in fact, ΔT_eff(SME−C95) (upper left panel) does not exceed 700 K and is typically ∼200 K. The largest differences occur for VX Her (558 K) and for SW And (659 K). The difference for VX Her is readily explained by the different ϕ_puls values: the average pulsation phase for the five C95 spectra is ϕ_puls = 0.61 and therefore they measured T_eff = 5978 K; on the other hand the Keck spectrum was taken at ϕ_puls = 0.32 and the average T_eff = 6660 ± 70 K from the VWA, MOOG, and SME analyses is hotter. The large temperature difference for SW And is problematic. The average ϕ_puls for the three C95 spectra is 0.82, which is to be compared with ϕ_puls = 0.56 for our Keck spectrum. According to For et al. (2011, their Figures 8–10) both phases are near minimum temperature and therefore T_eff ∼ 6200 K is expected in both cases. The T_eff = 6337 K derived by C95 is consistent with this prediction, as is the T_eff from our MOOG analysis; however, both the VWA and SME analyses suggest a higher temperature, and as a consequence a slightly higher [Fe/H] value.

The comparison of surface gravities (Figure 8, lower left panel) is intriguing, with the differences for the nine standard stars lying along the line Δlog g = 0.89log g(SME) − 2.46, i.e., the gravities derived by SME tend to be smaller than the C95 values when log g is less than 2.76 and larger when log g is greater than 2.76. A linear trend is also seen when our VWA ξ_t estimates for the standard stars (crosses in the upper right panel) are compared with the C95 microturbulent parameter, where the former values are offset by ∼2 km s⁻¹. Despite these atmospheric parameter differences, the metallicities derived here and by C95 are all within ∼0.2 dex of each other (bottom right panel), with a difference <0.1 dex in most cases.

4.4.4. Correlations of Atmospheric Parameters

Figure 9 shows the SME-derived atmospheric parameters v_mac, ξ_t, and vsin i, plotted as a function of T_eff. Each panel includes the values derived for the program RR Lyr stars, for the Sun, for the candidate HADS star KIC 3868420, and for the bright Keck RR Lyrae standard stars. As expected, the RRc stars, KIC 5520878 and KIC 8832417 (both of which are metal-rich) are the hottest among the RR Lyr stars. The candidate HADS star KIC 3868420 (also metal-rich) is even hotter. Differences between KIC 3868420 and the bona fide hot RRc stars appear to be comparatively small. While this observation lends support to the possibility that it may be an unusual metal-rich short-period RRc star, possibly an RRe (second overtone?) pulsator (Walker & Nemec 1996; Kovács 1998), identification as a hot HADS star of 1.6 to 2 solar masses seems more likely. Balona & Nemec (2012) found that δ Sct stars hotter than log T_eff = 3.88 are quite common and suggested that KIC 3868420, based on its photometric and kinematic properties, is not an SX Phe star (i.e., a halo population δ Sct star such as are found among the blue stragglers in globular clusters). The metal-poor RRc star KIC 9453414 has the highest derived macroturbulent velocity of all the program stars, i.e., v_mac = 29 km s⁻¹. The three stars with the greatest projected rotation velocities are the non-Blazhko RRab stars KIC 9658012, KIC 9717032, and KIC K7030715—all the other stars have vsin i values smaller than 7 km s⁻¹. The RRab star with the coolest T_eff is V1510 Cyg, a non-Blazhko star for which we were unable to perform a successful VWA analysis of the CFHT spectra. V808 Cyg (a Blazhko star) also has a high v_mac and V1104 Cyg (another Blazhko star) has the extremely low value of v_mac = 0.4 km s⁻¹ (see top panel of Figure 9) For the standard stars (all with high S/N spectra) and the RRc stars, in Figure 9 three linear trends are apparent: the hottest stars tend to have higher v_mac, lower ξ_t, and higher vsin i velocities than the cooler stars. Equations of the fitted trend lines are given on the graphs. It is interesting to note that the v_mac and vsin i trends are similar to those for the 23 bright solar-type stars studied by Bruntt et al. (2010a) and summarized in their Figure 11, although at a given T_eff the RR Lyr stars appear to have higher v_mac values than the solar-type stars.

In Figure 10 the atmospheric parameters derived with SME for the RR Lyr stars are compared with the same quantities (also derived using SME) for the 253 halo metal-poor red giants and red horizontal branch (RHB) stars investigated by Barklem et al. (2005) as part of the HERES survey. In all four panels KIC 3868420 (dotted triangle) appears to be similar to the two metal-rich RRc stars KIC 5520878 and KIC 8832417, suggesting again that it might be a short period RRc star and not a HADS star. The ξ_t versus log g diagram (upper left panel) shows that the RR Lyrae stars have systematically higher microturbulent velocities than the HERES metal-poor red giants and RHB stars with the same log g. The log g versus T_eff diagram (upper right panel), where the log g scale is reversed, closely resembles an H-R diagram. At a given surface gravity the RR Lyrae stars are hotter than the RHB stars, which in turn are hotter than the metal-poor red giants. As before the hottest stars are the RRc stars (and KIC 3868420), and the two metal-rich RRc stars are seen to have higher surface gravities and temperatures than the two metal-poor RRc stars. The v_mac versus projected rotational velocity diagram (lower left panel) identifies those stars with the most extreme values of v_mac and vsin i (see earlier discussion). The mean v_mac, excluding the three outliers with v_mac greater than 20 km s⁻¹, equals 12.1 km s⁻¹ (σ = 4.2 km s⁻¹, 40 RR Lyr stars), and the corresponding mean vsin i (after eliminating the three vsin i outliers) is 4.3 km s⁻¹ (σ = 1.2 km s⁻¹, 40 RR Lyr stars). The surface gravity versus metallicity diagram (lower right panel) reveals a clear trend between log g and [M/H]: the metal-rich RR Lyrae stars have larger surface gravities (hence smaller luminosities) than the more metal-poor RR Lyrae stars, which was explained by Sandage's (1958) "stacked horizontal-branch luminosity levels" model (see Bono et al. 1997, and discussion in N11). No significant differences between the Blazhko and non-Blazhko stars are evident in any of the diagrams of Figure 10.

Zoom In Zoom Out Reset image size

Several models were evaluated for the Kepler-field RRab stars and the optimum fit was achieved with the following nonlinear model:

$$\begin{equation} [{\rm Fe/H}] = b_0 + b_1 P + b_2 \phi _{31}^s + b_3 \phi _{31}^s P + b_4 \left(\phi _{31}^s \right)^2, \end{equation} \tag{ 2 }$$

where $\phi _{\rm 31}^s$ is the mean $\phi _{\rm 31}^s$(Kp) value and [Fe/H] is the VWA spectroscopic value (see Table 7). Excluded from the final calibration were five Blazhko stars with large residuals: V445 Lyr and V2178 Cyg, both of which exhibit extremely large amplitude and frequency modulations (see Table 2, and right side of Figure 3), and KIC 11125706, V838 Cyg, and V1104 Cyg, which have small amplitude and frequency modulations (see Table 2, Figures 2 and 3 and Section 3.2.1). Also excluded from the calibration were two non-Blazhko stars with unreliable metallicities, V346 Lyr and V1510 Cyg. The resulting estimated coefficients and their standard errors are: b₀ = −8.65 ± 4.64, b₁ = −40.12 ± 5.18, b₂ = 5.96 ± 1.72, b₃ = 6.27 ± 0.96, and b₄ = −0.72 ± 0.17. The rms error of the fit was 0.084 dex, with adjusted R² = 0.98. Seventeen of the 26 calibration stars were non-Blazhko variables and nine were Blazhko stars. The photometric [Fe/H] values derived from the fitted model are given in Column 9 of Table 1, where the estimated standard errors were calculated using the usual Gaussian theory formula for nonlinear least squares regression (analogous to Equations (4) and (5) of JK96, except that we have ignored the errors in $\phi _{\rm 31}^s$ and P owing to the high accuracy of the Kepler photometry).

In Figure 11 the fitted photometric metallicities are plotted against the spectroscopic metallicities (top panel). Also shown (bottom panel) are the differences, Δ[Fe/H] = [Fe/H]_phot − [Fe/H]_spec, plotted against [Fe/H]_spec. Included in the graphs are the five Blazhko stars that were excluded from the final calibration (labeled in the bottom panel). All of the calibrating stars lie close to the 1:1 line, with ∣ Δ[Fe/H] ∣ < 0.13 dex for all 17 non-Blazhko stars.

Zoom In Zoom Out Reset image size

The largest metallicity differences correspond to the five Blazhko stars excluded from the calibration. It is perhaps not surprising that the two most extreme Blazhko stars (V445 Lyr, V2178 Cyg) showed the greatest disagreement. Since both stars exhibit considerable variation in $\phi _{\rm 31}^s$ over a Blazhko cycle, and since the calibration model (Equation (2)) is nonlinear, it is questionable (Jensen's inequality) whether substitution of mean ϕ₃₁ rather than averaging the "instantaneous" [Fe/H] values is the appropriate way of calculating [Fe/H]_phot. The [Fe/H]_phot given in Table 1 were calculated using the first (i.e., substitution of mean-$\phi _{\rm 31}^s$) method. To investigate the second method we computed [Fe/H]_phot for the five Blazhko stars by averaging (over complete Blazhko cycles) the metal abundances evaluated (Equation (2)) at each "instant" along the $\phi _{\rm 31}^s$ time series. As expected, the two methods gave similar results for the three low amplitude/frequency modulators. However, for V445 Lyr and V2178 Cyg averaging over the instantaneous abundances gave [Fe/H]_phot = −1.69 and −1.65 dex, respectively, both of which are much closer to the spectroscopic abundances: −1.78 dex for V445 Lyr (which is from the SME analysis since the VWA analysis was inconclusive), and −1.66 ± 0.13 dex for V2178 Cyg.

Of course the spectroscopic metallicities are also uncertain and contribute to the differences. Four of the five Blazhko stars seem to have reliable (and consistent) metal abundances from the VWA and SME analyses. The largest difference, that for V445 Lyr (Figure 11), was calculated assuming the SME abundance, −1.78 dex (which appears to be consistent with the partial spectrum shown in Figure 7(a)). Consideration of the pulsation phases at the times of the spectroscopic observations, and of the Blazhko properties of the stars, reveals no obvious patterns: the pulsation phases range from 0.31 to 0.43; four of the five stars have Blazhko periods ∼50 days, while V2178 Cyg has a considerably longer P_BL = 234 ± 10 days; and the Blazhko phases range from 0.32 to 0.82.

We conclude that since the large metallicity differences found for extreme Blazhko stars are reduced when "instantaneous" [Fe/H]_phot values are averaged instead of calculated using the mean-ϕ₃₁ method, and that the two methods give the same result for the non-Blazhko and less extreme Blazhko stars, that this method might be preferable for future [Fe/H]_phot calculations.

In Figure 12 two sets of isometallicity curves have been added to the P–ϕ₃₁ diagram (see Figure 4). The solid curves (black) were calculated by solving our nonlinear relation (Equation (2)) for five [Fe/H] values (ranging from −2.0 to −0.1 dex), and the dashed lines (blue) were calculated using the well-known JK96 formula (their Equation (3)) and solving for [Fe/H] = −2.0, −1.0 and 0.0 dex. Since our photometry is on the Kp system it was necessary to transform the JK96 formula to the Kp system, which was done using Equation (2) of N11, i.e., $\phi _{\rm 31}^s$(V) = $\phi _{\rm 31}^s$(Kp)−0.151(± 0.026), resulting in

$$\begin{equation} {\rm [Fe/H]} = -5.241 - 5.394 P + 1.345 \, \, \phi _{31}^s({Kp}). \end{equation} \tag{ 3 }$$

The original version of this relation was established using a calibration data set consisting of 81 galactic-field RR Lyr stars with reliable V light curves and HDS abundances. It was considered optimal after various linear and nonlinear models were investigated, and gives photometric abundances on the HDS metallicity scale established by Jurcsik (1995). It is applicable for stars meeting the compatibility condition, D_m < 3, a criterion that fails to hold for stars with peculiar light curves, such as certain Blazhko variables and RR Lyr stars in advanced evolutionary stages.

Zoom In Zoom Out Reset image size

In general our nonlinear model and the JK96 model agree for [Fe/H] < −1.0 dex, with progressively larger discrepancies occurring as [Fe/H] decreases. It is well-known that for the lowest [Fe/H] stars the JK96 formula gives metallicities too metal-rich by ∼0.3 dex, the problem having been identified by JK96 themselves and discussed further by Nemec (2004) and Smolec (2005). Our model appears to correct this problem (compare the [Fe/H] = −2.0 dex curves in Figure 12), in part due to the wider inclusion in our sample of stars more metal-poor than −2.0 dex, as well as our use of a nonlinear rather than linear model. It should also be noted that JK96 adopted a spectroscopic [Fe/H] of −2.10 dex for X Ari, whereas the spectroscopic metallicities in Table 10 suggest a much lower value, [Fe/H] ∼ −2.65 dex.

The Kepler-field RR Lyr stars divide into two groups, those with $\phi _{\rm 31}^s$(Kp) ∼ 5 having low metallicities, [Fe/H] < −1 dex, and a high metallicity group with $\phi _{\rm 31}^s$(Kp) ∼ 5.7. Four of the five Kepler stars comprising the latter group were identified as metal-rich by N11 (V782 Cyg, V784 Cyg, KIC 6100702 and V2470 Cyg); to this group we can now add V839 Cyg (and the Keck standard stars V445 Oph and SW And).

The P–ϕ₃₁–[Fe/H] relation for RRc stars was investigated previously by M07. Two nonlinear formulas were derived, one for metallicities on the ZW84 scale (Zinn & West 1984) and the other for the CG97 scale (Carretta & Gratton 1997). For calibration purposes 106 stars in 12 galactic globular clusters were used, with cluster mean metallicities ranging from −2.2 to −1.0 dex.

With only four RRc stars in our sample it was not possible to derive independently a similar relation for the Kepler-field RRc stars. We therefore added the Kepler stars to the M07 calibration data set (kindly sent to us by Dr. Morgan) and derived a new relation. The inclusion of the Kepler stars extends the [Fe/H] range to considerably higher metallicities.

Since most of the data were from the M07 sample we chose to work with $\phi _{\rm 31}^c$(V) Fourier phase parameters. The $\phi _{\rm 31}^s$(Kp) values for the Kepler stars were transformed to $\phi _{\rm 31}^c(V)$ values using Equation (2) of N11, which, while probably not optimal for the hotter RRc stars (since it was derived for non-Blazhko RRab stars) is all that is presently available; the sine-to-cosine transformation was made using $\phi _{\rm 31}^c = \phi _{\rm 31}^s - \pi$.

The metallicities assumed for the Kepler stars were those given in Table 7, and the [Fe/H] values assumed for the globular cluster stars were the cluster means given in Table A.1 of C9. The latter are very similar to the metal abundances adopted by W. Harris in the latest edition (2010) of his online catalog of globular cluster parameters (Harris 1996).

Using the revised data set and the $\phi _{\rm 31}^c$(V) phase parameters we refit the M07 model,

$$\begin{equation} [{\rm Fe/H}] = b_0 + b_1 P + b_2 \phi _{31}^c + b_3 \phi _{31}^c P + b_4 P^2 + b_5 \left(\phi _{31}^c \right)^2, \end{equation} \tag{ 4 }$$

where the metallicities are now on the C9 scale. After removing nine outliers (V17 in M9; V26 in NGC 6934; V27 and V19 in M2; V50 in M15; V3, V11, and V14 in NGC 4147; and KIC 9453114) the estimated model coefficients and their standard errors were found to be: b₀ = 1.70 ± 0.82, b₁ = −15.67 ± 5.38, b₂ = 0.20 ± 0.21, b₃ = −2.41 ± 0.62, b₄ = 18.00 ± 8.70, and b₅ = 0.17 ± 0.04. The rms error of the fit was 0.13 dex, the adjusted R² = 0.94, and the calibration sample size N = 101 stars. The fitted [Fe/H]_phot (and their standard errors) for the Kepler-field RRc stars are given in Column 9 of Table 1.

The metallicity differences for the four Kepler-field RR Lyr stars (blue triangles) and for the globular cluster stars (small open circles) are plotted in the bottom panel of Figure 11. For the three Kepler stars that were included in the calibration the differences are small, all under 0.18 dex; for KIC 9453114 the derived [Fe/H]_phot is 0.43 dex more metal rich than the [Fe/H]_spec, −2.13 ± 0.12 dex. For the cluster stars the groups correspond to the individual clusters, with the vertical variation reflecting the range of the P and ϕ₃₁ values within each cluster (for clarity the uncertainties in the [Fe/H]_spec have been omitted).

In Figure 13 the P–$\phi _{\rm 31}^c$ diagram is plotted for the RRc stars in the Kepler-field and in the M07 globular clusters (the symbols are the same as in Figure 11). Also shown are two sets of isometallicity curves for [Fe/H] ranging from −2.0 dex to −0.5 dex. The solid curves (black) were calculated using the new P–$\phi _{\rm 31}^c(V)$–[Fe/H] relation (Equation (4)) and solving for the four different [Fe/H] values; and the dotted curves (red) are the same curves transformed (using the transformation equation given in C9) to the ZW84 scale.

Zoom In Zoom Out Reset image size