THE PALOMAR TRANSIENT FACTORY AND RR LYRAE: THE METALLICITY–LIGHT CURVE RELATION BASED ON AB-TYPE RR LYRAE IN THE KEPLER FIELD

The Palomar Transient Factory (PTF, 2009–2012, see Law et al. 2009; Rau et al. 2009) and its successor, the intermediate Palomar Transient Factory (iPTF, 2013–2016),⁷ are dedicated wide-field synoptic sky survey projects with aims of detecting various types of transients in the universe. Given the synoptic nature of PTF/iPTF surveys, a large number of known or new RR Lyrae with homogeneous light-curve data is expected to be found in PTF/iPTF data. Since RR Lyrae are population II standard candles at which they have roughly a constant absolute magnitude in V band (M_V), RR Lyrae have been used in various distance-scale studies such as tracing the Galactic halo structure (for example, see Watkins et al. 2009; Sesar et al. 2010). Therefore, we have initiated a program to investigate the properties of RR Lyrae in the PTF/iPTF native R-band photometric system (hereafter R_PTF).

It is well-known that M_V for RR Lyrae is correlated with metallicity (for example, see McNamara 1999; Caputo et al. 2000; Demarque et al. 2000; Sandage & Tammann 2006), where the metallicity is mostly measured or expressed in terms of [Fe/H], then the distance to an RR Lyrae can be deduced by knowing its metallicity and hence its M_V value.⁸ The best way to obtain [Fe/H] is via a spectroscopic technique; however, this can be quite expensive in terms of telescope time. Fortunately, photometric [Fe/H] for RR Lyrae can be estimated via the metallicity–light curve relation, at which the light curves for RR Lyrae can be fitted with a truncated sine-series of Fourier decomposition (for example, see Simon & Lee 1981; Deb & Singh 2009)⁹ :

$$\begin{eqnarray}&&m(t)={m}_{0}+\displaystyle \sum _{i=1}^{n}{A}_{i}\sin \left(\displaystyle \frac{2i\pi t}{P}+{\phi }_{i}\right),\end{eqnarray} \tag{ 1 }$$

where n is the order of fitting, P is pulsation period in days, and t is time of observation. The mean magnitude m₀, amplitude A_i, and phase ${\phi }_{i}$ values at given ith-order can be obtained by fitting the observed light-curve data with Equation (1). The light-curve parameters, or Fourier parameters, can be expressed in terms of A_i and ${\phi }_{i}$:

$$\begin{eqnarray}&&{R}_{{ij}}=\displaystyle \frac{{A}_{i}}{{A}_{j}};\ {\phi }_{{ij}}={\phi }_{i}-i{\phi }_{j}.\end{eqnarray} \tag{ 2 }$$

The metallicity–light curve relation for fundamental mode ab-type RR Lyrae (hereafter RRab) was first quantitatively studied in Simon (1988). Later, such a relation was derived, or fitted, from field RRab in Kovács & Zsoldos (1995), and subsequently extended by Jurcsik & Kovács (1996). In Jurcsik & Kovács (1996), such a relation is given as ${[\mathrm{Fe}/{\rm{H}}]}_{V}=-5.038\mbox{--}5.394P\,+1.345{\phi }_{31}$, where ${\phi }_{31}={\phi }_{3}-3{\phi }_{1}$ is calculated using Equations (1) and (2) based on the V-band light curves. A preliminary updated version of the Jurcsik & Kovács (1996) relation is presented in Martinez-Vazquez et al. (2016), which incorporates RR Lyrae in globular clusters. Other metallicity–light curve relations based on the V-band light curves can be found, for example, in Sandage (2004). Besides the V-band light curves, Smolec (2005) gives the relation based on I-band light curves. Wu et al. (2006) and De Lee (2008) derived a similar relation as in Jurcsik & Kovács (1996) for unfiltered, or white light, CCD observations, and for the g-band SDSS (Sloan Digitized Sky Survey) data, respectively. Watkins et al. (2009), Sesar et al. (2010) and Oluseyi et al. (2012) further developed the metallicity–light curve relation in the SDSS photometric system with additional terms in the relation. Similarly, Nemec et al. (2011) and Nemec et al. (2013) derived such a relation in the Kepler magnitude (K_p) system. Alternatively, Deb & Singh (2010) and Skowron et al. (2016) derived the Fourier interrelations to convert the ${\phi }_{31}$ parameters in I band to V band and then applied the metallicity–light curve relations mentioned. The validity of such a metallicity–light curve relation has been tested and verified, for example, in Jurcsik (2003), Gratton et al. (2004), Kovács (2005), Wu et al. (2006), and Kunder & Chaboyer (2008).

To fully utilize the RR Lyrae found in the PTF/iPTF data for future distance-scale work, it is necessary to derive the metallicity–light curve relation in the native R_PTF-band photometric system. Even though, in principle, it is possible to apply the transformation equation provided in Ofek et al. (2012a) to convert the PTF/iPTF photometry in the g_PTF-band to the Johnson-Cousin V-band and hence to apply the Jurcsik & Kovács (1996) relation; in practice, this is difficult to achieve because (1) this transformation requires the $(V-{R}_{c})$ color curves for the RR Lyrae found in PTF/iPTF to be available, but we generally do not have such data, and (2) the majority of the data taken in PTF/iPTF are in the R_PTF-band. Therefore, direct derivation of the metallicity–light curve relation in the native R_PTF-band is desirable. In this work, we present the derivation of such a relation by using the known RR Lyrae located in the Kepler field, because these RR Lyrae possess very precise and accurate period determination based on the Kepler light curves and spectroscopic measurements of [Fe/H] (Nemec et al. 2013). A brief description of the PTF/iPTF project is presented in Section 2. The PTF/iPTF data for RR Lyrae in the Kepler field were mentioned in Section 3, followed by the construction of the light curves in Section 4. Based on the PTF/iPTF light curves, we derived the ${\phi }_{31}$ Fourier parameters in Section 5. The R_PTF-band metallicity–light curve relation will be derived and tested in Section 6. Finally, a discussion and our conclusions will be presented in Section 7. Throughout the paper, the Fourier parameter ${\phi }_{31}$ is based on the sine-series as shown in Equation (1), which can be converted to the cosine-based ${\phi }_{31}$ by subtracting π (that is, ${\phi }_{31}^{\mathrm{cosine}}={\phi }_{31}^{\mathrm{sine}}-\pi$). Also, as a reminder, throughout the paper, a ±0.25 dex difference at [Fe/H] =−1.5 dex corresponds to roughly a factor of two difference in metal abundance by mass.

The PTF/iPTF project mainly utilizes the 48 inch Samuel Oschin Telescope located at the Palomar Observatory, known as the P48 Telescope, to search for transients. The P48 Telescope is equipped with a wide-field mosaic camera, which consists of eleven $2\,K\times 4\,K$ CCD¹⁰ (Rahmer et al. 2008), for the surveys carried out by both PTF and iPTF. The pixel scale of each CCD is 1.01 arc-second per pixel, hence providing a total field of view (FOV) of ∼7.26-degree² for a single PTF/iPTF image. Observations with the P48 Telescope were mainly done in the Mould R-band filter (i.e., the R_PTF filter), with occasional observations carried out in the g_PTF or Hα filters. The nominal exposure time for PTF/iPTF images is 60 s, which can reach a depth of 20.5 mag in the R_PTF band (with a 3σ detection).

The PTF/iPTF imaging data from the P48 Telescope was reduced and processed with two different pipelines (Law et al. 2009). One of the pipelines, based on the image subtraction technique, is tailored for quick discovery of transients; while another pipeline, hosted at the Infrared Processing and Analysis Center (IPAC), will fully reduce the raw images and provide catalogs of all detected objects in the images. Photometric calibration of the detected objects was also included in the IPAC pipeline. Further details of the IPAC pipeline and the photometric calibration procedure can be found in Ofek et al. (2012a, 2012b), Laher et al. (2014), and Surace et al. (2015), and will not be repeated here. Since the main goals for PTF and iPTF include the detections of transients (such as supernovae) in the local universe and pursue for new discoveries with dedicated and well-designed experiments, hence the cadence carried out in PTF and iPTF varies from 90 s to a few days. Time-series data from PTF/iPTF has not only been used for the search of transients, but also in studies of other time-domain phenomena such as variable stars (for example, see van Eyken et al. 2011) and asteroids (for example, see Chang et al. 2014).

In total, there have been 41 RR Lyrae found in Kepler field, including 21 non-Blazhko RRab stars, 16 Blazhko RRab stars, and 4 first overtone c-type RR Lyrae (hereafter RRc). In this work, we excluded the 4 RRc stars, because the small number of them in the sample did not permit a meaningful statistical analysis of their metallicity–light curve relation. For the RRab stars, Tables 1 and 2 listed out their basic information and summarized the available PTF/iPTF data in the R_PTF band, respectively. Note that a portion of the PTF data on the Kepler field is publicly available, which belongs to the first data release¹¹ of the PTF data. The difference in number of catalogs between the full and publicly available data ranges from 4 (for KIC 6186029) to 79 (for KIC7988343); for the majority of them, the difference is less than 10.

Table 1.  Basic Information for RR Lyrae in the Kepler Field^a

KICb	R.A. (J2000)	Decl. (J2000)	P [days]	to	$\langle {Kp}\rangle$	Typec	Other Name
7198959	19:25:27.912	+42:47:03.72	0.566788	2455278.2263	7.862	RRab-B	RR Lyr
11125706	19:00:58.774	+48:44:42.30	0.6132200	2454981.0658	11.367	RRab-B	KIC 11125706
3733346	19:08:27.228	+38:48:46.19	0.6820264	2454964.7403	12.684	RRab-NB	NR Lyr
6936115	19:10:22.250	+42:27:31.57	0.52739847	2454953.2656	12.876	RRab-NB	FN Lyr
11802860	19:00:48.000	+50:05:31.27	0.6872160	2454954.2160	13.053	RRab-NB	AW Dra
6763132	19:07:48.374	+42:17:54.67	0.5877887	2454954.0702	13.075	RRab-NB	NQ Lyr
9591503	19:33:00.912	+46:14:22.85	0.5713866	2454953.5624	13.293	RRab-NB	V894 Cyg
9947026	19:19:57.958	+46:53:21.41	0.5485905	2454953.7832	13.300	RRab-NB	V2470 Cyg
7030715	19:23:24.527	+42:31:42.34	0.68361247	2454953.8427	13.452	RRab-NB	KIC 7030715
6100702	18:50:37.730	+41:25:25.72	0.4881457	2454953.8399	13.458	RRab-NB	KIC 6100702
7021124	19:10:26.681	+42:33:37.04	0.6224925	2454965.6471	13.550	RRab-NB	KIC 7021124
10789273	19:14:03.905	+48:11:58.60	0.48027971	2455807.9302	13.770	RRab-B	V838 Cyg
10136603	19:20:18.888	+47:07:48.54	0.4337747	2455778.7060	14.066	RRab-NB	V839 Cyg
7505345	18:53:25.903	+43:09:16.45	0.4737027	2455124.7072	14.080	RRab-B	V355 Lyr
7988343	19:59:50.669	+43:42:15.52	0.5811436	2454964.6700	14.494	RRab-NB	V1510 Cyg
5559631	19:52:52.740	+40:47:35.45	0.62070001	2454975.5439	14.643	RRab-B	V783 Cyg
12155928	19:18:00.490	+50:45:17.93	0.43638507	2455120.8363	15.033	RRab-B	V1104 Cyg
4484128	19:45:39.024	+39:30:53.42	0.5478642	2454970.2834	15.363	RRab-B	V808 Cyg
6070714	19:56:22.906	+41:20:23.53	0.5340941	2454964.8067	15.370	RRab-NB	V784 Cyg
5299596	19:51:16.999	+40:26:45.20	0.5236377	2454964.5059	15.392	RRab-NB	V782 Cyg
3864443	19:40:06.963	+38:58:20.35	0.4869538	2454976.3672	15.593	RRab-B	V2178 Cyg
10136240	19:19:45.279	+47:06:04.44	0.5657781	2454964.7551	15.648	RRab-NB	V1107 Cyg
9508655	18:49:08.369	+46:11:54.96	0.5942369	2454964.7820	15.696	RRab-NB	V350 Lyr
9658012	19:41:20.004	+46:23:28.64	0.533206	2455779.9450	16.001	RRab-NB	KIC 9658012
9697825	19:01:58.634	+46:26:45.74	0.5575765	2454988.9332	16.001	RRab-B	V360 Lyr
7742534	19:10:53.403	+43:24:54.94	0.4564851	2454964.7860	16.002	RRab-NB	V368 Lyr
6183128	18:52:50.359	+41:33:49.47	0.561691	2455245.1590	16.260	RRab-B	V354 Lyr
3866709	19:42:07.997	+38:54:42.30	0.47070609	2454964.6037	16.265	RRab-NB	V715 Cyg
8344381	18:46:08.640	+44:23:13.99	0.5768288	2454964.9231	16.421	RRab-NB	V346 Lyr
9578833	19:09:40.637	+46:17:18.17	0.5270283	2455326.1915	16.537	RRab-B	V366 Lyr
7257008	18:47:27.408	+42:49:52.68	0.51177516	2455758.5859	16.542	RRab-B	KIC 7257008
7671081	19:09:36.634	+43:21:49.97	0.5046123	2454996.3226	16.653	RRab-B	V450 Lyr
9001926	18:52:01.805	+45:18:31.61	0.5568016	2455082.6820	16.914	RRab-B	V353 Lyr
9973633	19:58:49.068	+46:50:56.83	0.51075	2455780.3655	16.999	RRab-B	KIC 9973633
9717032	19:38:19.155	+46:27:47.06	0.5569092	2455779.8956	17.194	RRab-NB	KIC 9717032
6186029	18:58:25.692	+41:35:49.45	0.5131158	2455160.5957	17.401	RRab-B	V445 Lyr
7176080	18:49:24.434	+42:44:45.56	0.5070740	2454964.9588	17.433	RRab-NB	V349 Lyr

Notes.

^aInformation taken from Nemec et al. (2013). ^bKepler Input Catalog. ^cRRab-NB: Non-Blazhko RRab stars; RRab-B: Blazhko RRab stars.

Download table as:  ASCIITypeset image

Using the publicly available catalog data for 16 non-Blazhko RRab stars in Kepler field, Ngeow (2015) attempted to derive a preliminary metallicity–light curve relation in the R_PTF band. However, it was found that the PTF light curves for 8 of them displayed a larger scatter in their light curves. These RR Lyrae have a mean R_PTF magnitude of ∼14 mag or brighter, which is close to the saturation limit of PTF data (van Eyken et al. 2011; Ofek et al. 2012a). A provisional metallicity–light curve relation based on these 8 bright RRab stars exhibited a dispersion of 0.76 dex. In contrast, the remaining 8 RRab stars, with mean R_PTF magnitudes fainter than ∼14 mag, showed much tighter PTF light curves, and the dispersion of the derived metallicity–light curve relation is 0.13 dex. One finding of Ngeow (2015) is that the bright RRab stars should not be used in the derivation of the metallicity–light curve relation because their photometry will be affected by saturation limits in PTF/iPTF surveys. Excluding the prototype RR Lyr (KIC 7198959) itself and V808 Cyg (KIC 4484128, because it does not have data in PTF/iPTF), there are ∼15 RRab stars (mixed of both Blazhko and non-Blazhko stars) that have mean R_PTF magnitudes brighter than ∼14 mag, which is almost half of the total 29 RRab stars that have spectroscopic [Fe/H] measurements (Nemec et al. 2013). To maximize usable RRab stars in deriving the metallicity–light curve relation, we launched a dedicated iPTF experiment to re-observe these bright RRab stars with a 10 s exposure time (so their light curves will not suffer from the saturation limit), in opposition to the nominal 60 s exposure time set in the regular PTF/iPTF surveys.

3.1. The Dedicated iPTF Experiment

Catalog data from the IPAC pipeline for RRab stars in our sample were downloaded from the PTF/IPAC data archive hosted at the NASA/IPAC Infrared Science Archive (IRSA).¹² These R_PTF-band PTF/iPTF SExtractor (Bertin & Arnouts 1996) catalog data, including both of the catalogs from regular PTF/iPTF surveys (with a nominal 60 s exposure time) and the dedicated iPTF experiment as mentioned previously, were stored in FITS binary table format. A python script¹³ was used to extract the R_PTF-band light curves for our RRab stars. This was done by matching the detected sources in the catalogs to the input RRab stars with a match radius of two arc-seconds. The heliocentric Julian date (HJD), photometric magnitude, and magnitude error of the matched sources were saved into python arrays. The PTF R_PTF-band magnitudes were constructed by adding the MAG_AUTO and ZEROPOINT in PTF catalogs (for more details, see Ofek et al. 2012a).

Since none of the images involved in this work were taken under photometric conditions as defined in Ofek et al. (2012a, i.e., the corresponding flag is PHTCALFL = 0), the extracted "raw" light curves displayed numerous outliers. The left panel of Figure 1 presents examples of the "raw" light curve for a bright and faint non-Blazhko RR Lyrae. For the 10 s light curves, plots in the left panels of Figure 1 show that one (or two) night from the dedicated iPTF experiment might be affected by weather. Furthermore, the upper-left panel of Figure 1 displayed a vertical shift between the light curves taken with the 10 s and 60 s exposure time for the bright RR Lyrae. As discussed in Ngeow (2015), the 60 s data was affected by saturation, hence some fluxes were lost when using the aperture photometry. In contrast, the faint RR Lyrae shown in lower left panel of Figure 1 does not have this problem.

Zoom In Zoom Out Reset image size

To remedy the problem of large scatter shown in the "raw" light curves taken under the non-photometric condition, we employed a differential photometric technique (e.g., see Honeycutt 1992) to construct differential light curves for both the 10 s and 60 s data. In addition to the reduction of nightly data, the IPAC pipeline also created stacked reference images and the associated SExtractor reference catalogs (see Laher et al. 2014, for more details on how the reference images were created). We selected a subset of reference stars given in the SExtractor reference catalogs for the RR Lyrae listed in Table 1 ¹⁴ to determine the mean magnitude differences, or relative zero-points Δm, between the reference stars and the nightly reduced catalogs. These reference stars have to meet with the following selection criteria: (1) exclude the targeted RR Lyrae stars in the reference catalogs; (2) FLAGS = 0 in the SExtractor catalog; (3) SExtractor parameter ${CLASS}\_{STAR}\gt 0.95$ (for stars-galaxies separation); (4) $15\lt {MAG}\_{AUTO}\lt 17$ in the reference catalogs for the 60 s data (or $13\lt {MAG}\_{AUTO}\lt 15$ for the 10 s data) such that stars with good enough signal-to-noise ratios were retained; and (5) more than 20 detections in nightly single-epoch catalogs (only for a few RR Lyrae stars, do the number of detections need to be tuned to a smaller value). For each of the RR Lyrae stars, we then loop over the nightly single-epoch catalogs and cross-matched to the corresponding reference stars using a 2 arc-second search radius. Furthermore, we removed reference stars that might be variables using the following procedure: (1) calculate the mean variance of the photometric errors based on the "raw" light curves, $\langle {\sigma }_{m}^{2}\rangle ;$ (2) calculate the variance of the "raw" light curves using the median absolute deviation (MAD) algorithm, ${\sigma }_{{LC}}^{2};$ and (3) remove stars with $| {\sigma }_{{LC}}^{2}-\langle {\sigma }_{m}^{2}\rangle | \gt 0.1$. For the remaining reference stars (for each of the RR Lyrae), the Δm values are taken to be the median difference between the magnitudes from reference stars and the magnitudes in each single-epoch catalogs. The final adopted Δm were then applied to the "raw" light curves to construct the differential light curves. The middle panels of Figure 1 present the improvement of the light curves based on our procedures for the two example RR Lyrae.

Several RR Lyrae in our sample still displayed few obvious outliers in the refined differential light curves and/or small offsets between the 60 s and 10 s light curves (see the middle panels of Figure 1). Those outliers were manually removed, and a small magnitude offset (which is smaller than 0.05 mag) is added to the 10 s light curves if needed. The right panels of Figure 1 present the final refined light curves for the two exampled RR Lyrae. The final adopted differential light curves for the Blazhko and non-Blazhko RRab stars in our sample will be displayed in the next section. Note that the magnitudes in these light curves are not in absolute scale, as the a common zeropoint of ${R}_{\mathrm{PTF}}=27.0$ was adopted when constructing the reference catalogs (Laher et al. 2014). Nevertheless, this would not affect the determination of Fourier parameters from these differential light curves, because Fourier parameters are independent of the global photometric calibration.

4.1. Excluded RRab Stars

We excluded the following RRab stars from our sample due to various reasons described below.

KIC 3733346: this bright RR Lyrae star only has seven data points in the R_PTF-band light curve (see top panel of Figure 2) and hence does not permit a meaningful fitting of the Fourier parameter.

Zoom In Zoom Out Reset image size

KIC 7198959: the prototype RR Lyr is too bright to be included in the 10 s observation, and the photometry (even after applying the differential light-curve technique) is severely affected by saturation. The light curve of this RR Lyrae is shown in the bottom panel of Figure 2.

KIC 4484128: this RR Lyrae is located within the footprint of CCD 03—the only CCD chip that is out of function at the beginning of PTF/iPTF surveys, and hence no data collected from the PTF and iPTF observations.

KIC 7021124: photometry of this RR Lyrae was affected by the diffraction spike from a nearby bright star and a very close star with similar brightness, as shown in the left panel of Figure 3. Therefore, we excluded this RR Lyrae in our sample. Differential light curves for this RR Lyrae were displayed in the right panel of Figure 3.

Zoom In Zoom Out Reset image size

KIC 3864443 and KIC 6186029: based on the long-term and almost continuous observations from Kepler, Nemec et al. (2013) identified these two RR Lyrae as extreme Blazhko stars because they exhibit large amplitudes and phase modulations when compared to other Blazhko RR Lyrae in the Kepler field. Figure 4 shows their R_PTF-band light curves, which also displayed obvious Blazhko modulations. Note that Nemec et al. (2013) excluded them in their analysis; therefore, we also excluded them in our sample. Further analysis and discussion on KIC 6186029 (V445 Lyr) can be found in Guggenberger et al. (2012).

Zoom In Zoom Out Reset image size

5.1. For Non-blazhko RRab Stars

Differential light curves for the remaining 19 non-Blazhko RR Lyrae stars were fitted with the truncated Fourier decomposition as given in Equation (1). We fit the 60 s and the 10 s light curves separately, as well as to the combined light curves. The best-fit orders n of the Fourier decomposition were chosen based on visual inspection of the fitted light curves. For the majority of the 10 s light curves, a relatively large gap was seen in the phased light curves, which affected the fitted light curves when applying the Fourier decomposition technique. To remedy this, we added a data point near the mid-point of the phased gap either taken from the 60 s light curves (for faint RR Lyrae) or derived from a cubic spline interpolation function (for bright RR Lyrae). Note that we only applied this additional data point to the 10 s light curves and not to the 60 s light curves. The only exception is the 10 s light curve for KIC 7176080, at which the phased gap is too large to apply a meaningful Fourier fit.

Among the 19 non-Blazhko RR Lyrae stars, three RR Lyrae do not have 10 s light curves and their 60 s light curves are displayed in Figure 5. The 60 s and 10 s phased light curves for the rest of the 16 non-Blazhko RR Lyrae were shown in the top and middle panels for each of the sub-figures in Figure 6. The dashed curves displayed in Figures 5 and 6 are the fitted light curves based on the Fourier decomposition technique. We compared the low-order Fourier parameters, R₂₁, R₃₁, ${\phi }_{21}$, and ${\phi }_{31}$, calculated with Equation (2), between the 60 s and 10 s light curves. The difference of the Fourier parameters as a function of mean R_PTF-band magnitudes is given in Figure 7. As can be seen from this figure, light curves with mean magnitudes fainter than ∼14 mag show a smaller dispersion of the difference in Fourier parameters than those with mean magnitudes brighter than ∼14 mag. Therefore, we merged the 60 s and 10 s light curves for the eight faint non-Blazhko RR Lyrae. For other eight bright non-Blazhko RR Lyrae, we added few data points from the 60 s light curves that have FLAG = 0 in the SExtractor catalogs, as these data points follow the light curve shapes defined by the 10 s light curves. We referred the merged 60 s and 10 s light curves as the combined light curves, as shown in bottom panels in each sub-figure of Figure 6. The final adopted ${\phi }_{31}$ Fourier parameters, based on these combined light curves, for the non-Blazhko RRab stars are listed in Table 3.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 3.  Fitted Results for Fourier Parameter ${\phi }_{31}$ from PTF Light Curves and the Derived Photometric Metallicity

KIC	P [days]a	${\phi }_{31}$	${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{spec}}$ a	${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$
Non-Blazhko RR Lyrae
6936115	0.52739847	4.796 ± 0.019	−1.98 ± 0.09	−1.82 ± 0.05
11802860	0.6872160	5.644 ± 0.007	−1.33 ± 0.09	−1.91 ± 0.07
6763132	0.5877887	5.207 ± 0.010	−1.89 ± 0.10	−1.74 ± 0.04
9591503	0.5713866	5.178 ± 0.012	−1.66 ± 0.12	−1.66 ± 0.04
9947026	0.5485905	5.946 ± 0.040	−0.59 ± 0.13	−0.51 ± 0.06
7030715	0.68361247	5.971 ± 0.023	−1.33 ± 0.08	−1.47 ± 0.07
6100702	0.4881457	5.803 ± 0.017	−0.16 ± 0.09	−0.25 ± 0.05
10136603	0.4337747	5.764 ± 0.013	−0.05 ± 0.14	0.10 ± 0.07
7988343	0.5811436	5.130 ± 0.005	⋯	−1.79 ± 0.04
6070714	0.5340941	6.210 ± 0.029	−0.05 ± 0.10	−0.06 ± 0.07
5299596	0.5236377	5.949 ± 0.016	−0.42 ± 0.10	−0.32 ± 0.05
10136240	0.5657781	5.406 ± 0.019	−1.29 ± 0.23	−1.33 ± 0.04
9508655	0.5942369	5.215 ± 0.008	−1.83 ± 0.12	−1.78 ± 0.04
9658012	0.533206	5.250 ± 0.006	−1.28 ± 0.14	−1.29 ± 0.03
7742534	0.4564851	4.886 ± 0.028	−1.28 ± 0.20	−1.19 ± 0.06
3866709	0.47070609	4.876 ± 0.033	−1.13 ± 0.09	−1.31 ± 0.06
8344381	0.5768288	5.260 ± 0.033	⋯	−1.59 ± 0.05
9717032	0.5569092	5.340 ± 0.016	−1.27 ± 0.15	−1.35 ± 0.04
7176080	0.5070740	4.954 ± 0.039	⋯	−1.47 ± 0.06
Blazhko RR Lyrae
11125706	0.6132200	6.098 ± 0.021	−1.09 ± 0.08	−0.79 ± 0.05
10789273	0.48027971	5.068 ± 0.015	−1.01 ± 0.10	−1.13 ± 0.04
7505345	0.4737027	5.027 ± 0.005	−1.14 ± 0.17	−1.13 ± 0.04
5559631	0.62070001	5.967 ± 0.011	−1.16 ± 0.11	−1.01 ± 0.05
12155928	0.43638507	4.983 ± 0.005	−1.23 ± 0.15	−0.92 ± 0.05
9697825	0.5575765	4.947 ± 0.016	−1.50 ± 0.29	−1.85 ± 0.05
6183128	0.561691	5.137 ± 0.019	−1.44 ± 0.16	−1.64 ± 0.04
9578833	0.5270283	5.069 ± 0.043	−1.16 ± 0.09	−1.47 ± 0.07
7257008	0.51177516	5.163 ± 0.051	⋯	−1.24 ± 0.07
7671081	0.5046123	5.134 ± 0.050	−1.51 ± 0.12	−1.22 ± 0.07
9001926	0.5568016	5.220 ± 0.020	−1.50 ± 0.20	−1.50 ± 0.04

Note.

^aValues are taken from Nemec et al. (2013).

Download table as:  ASCIITypeset image

5.2. For Blazhko RRab Stars

For the remaining 12 Blazhko RRab stars in our sample, seven of them have both 10 s and 60 s light curves, and only KIC 11125706 has a mean magnitude brighter than ∼14 mag. Therefore, we only fit the 10 s light curve of this star, and merged the 10 s and 60 s light curves for the other six Blazhko RRab stars. The differential light curves for these 12 Blazhko RRab stars were displayed in upper panels of Figure 8. Due to the amplitude and/or phase modulation, these light curves are "noisier" than the light curves of non-Blazhko RRab stars. Based on a similar approach presented in Smolec (2005), these modulated light curves were fitted with the following expression (for examples, see Kovács 1995; Alcock et al. 2003; Benkő et al. 2011):

$$\begin{eqnarray}&&m(t)={F}_{0}(t,{f}_{0})+{F}_{m}(t,{f}_{0},{f}_{m}),\end{eqnarray} \tag{ 3 }$$

where F₀ is the same as in Equation (1):

$$\begin{eqnarray*}&&{F}_{0}(t,{f}_{0})={m}_{0}+\displaystyle \sum _{i=1}^{n}{A}_{i}\sin (2\pi {{if}}_{0}t+{\phi }_{i}),\end{eqnarray*}$$

and F_m includes the modulated components:

$$\begin{eqnarray*}{F}_{m}(t,{f}_{0},{f}_{m}) & = & \displaystyle \sum _{j=0}^{r}{A}_{j}^{m}\sin (2\pi {{jf}}_{m}t+{\phi }_{j}^{m})\\ & & +\displaystyle \sum _{k=0}^{q}{A}_{k}^{-}\sin [2\pi ({{kf}}_{0}-{f}_{m})t+{\phi }_{k}^{-}]\\ & & +\displaystyle \sum _{k=0}^{q}{A}_{k}^{+}\sin [2\pi ({{kf}}_{0}+{f}_{m})t+{\phi }_{k}^{+}].\end{eqnarray*}$$

In Equation (3), ${f}_{0}=1/P$ is the fundamental frequency, ${f}_{m}=1/{P}_{{BL}}$ is the modulated frequency, and ${A}_{0}^{m}={A}_{0}^{-}\ ={A}_{0}^{+}=0$. The Blazhko periods, P_BL, of these Blazhko RRab stars have already been derived in Nemec et al. (2013), and we adopted their values in this work.¹⁵ Various combinations of Fourier order $(n,r,q)$ were visually inspected, and the best-fit combinations were adopted. Following an approach similar to that of Smolec (2005), we removed the F_m components that are associated with the modulated frequency f_m after fitting the light curves with Equation (3). The resulted light curves were shown in lower panels of Figure 8, and were used to determine the ${\phi }_{31}$ Fourier parameters of these Blazhko RRab stars. Except for KIC 9973633, these light curves resemble the light curves for RR Lyrae stars pulsating in the fundamental frequency (and its harmonics) only.

Zoom In Zoom Out Reset image size

The differential light curve for KIC 9973633 exhibits strong amplitude and phase modulation, similar to KIC 3864443 and KIC 6186029 (as shown in Figure 4). After experimenting with various combinations of Fourier order $(n,r,q)$ to fit the combined 10 s and 60 s light curves with Equation (3), we still could not remove the modulated components in the combined light curve (see lower panel in Figure 8). This implies that additional frequency terms, such as ${{kf}}_{0}\pm {{lf}}_{m}$ (where l is an integer), might need to be included in Equation (3), or this Blazhko star exhibits complex modulations as in the case of KIC 6186029 (V445 Lyr, Guggenberger et al. 2012). Nevertheless, a detailed investigation of the power spectrum of KIC 9973633 is beyond the scope of this work, and it is obvious that it should be excluded from our sample.

The ${\phi }_{31}$ Fourier parameters derived in the previous section were listed in the third column of Table 3. Among these RRab stars, 26 of them have [Fe/H] values, listed in the forth column of Table 3, based on high-resolution spectroscopic observations taken from the 3.6 m Canada–France–Hawaii Telescope and the 10 m Keck I Telescope (Nemec et al. 2013). To derive the R_PTF-band metallicity–light curve relation, we adopted the well established regression function as in Jurcsik & Kovács (1996) and Wu et al. (2006): $[\mathrm{Fe}/{\rm{H}}]={b}_{0}+{b}_{1}P+{b}_{2}{\phi }_{31}$. We did not adopt the five-parameter regression function from Nemec et al. (2013) because it did not improve the dispersion (σ) of the fitted relation. The initial fit to the 26 RRab stars given in Table 3 returns a $\sigma \sim 0.20$ dex, which is much higher than the typical dispersion based on this technique (∼0.13 dex to ∼0.14 dex, Jurcsik & Kovács 1996; Wu et al. 2006; Ngeow 2015). After removing outliers that deviate more than $2\times 0.13$ dex from the regression, we derived the following relation,¹⁶ in the native R_PTF-band photometric system:

$$\begin{eqnarray}{[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}} & = & -4.089(\pm 0.339)-7.346(\pm 0.439)P\\ & & +1.280(\pm 0.062){\phi }_{31},\end{eqnarray} \tag{ 4 }$$

with a dispersion of $\sigma =0.118$ dex. Uncertainty on the photometric [Fe/H] based on the above equation, which incorporates the covariance matrix, can be calculated with the following expression.

$$\begin{eqnarray}{\sigma }_{[\mathrm{Fe}/{\rm{H}}]}^{2} & = & 0.115+0.193{P}^{2}\\ & & +0.004{\phi }_{31}^{2}-0.103P-0.031{\phi }_{31}\\ & & -0.020P{\phi }_{31}+1.638{\sigma }_{{\phi }_{31}}^{2}+53.965{\sigma }_{P}^{2}.\end{eqnarray} \tag{ 5 }$$

The fifth column in Table 3 listed the photometric [Fe/H] in the native R_PTF-band and the associated uncertainties calculated from Equation (4) and (5). Note that we assume ${\sigma }_{P}=0$ for the RRab stars in the Kepler field as their very precise and accurate periods were determined from nearly continuous Kepler light curves (Nemec et al. 2013).

In Figure 9, we compare the photometric metallicities derived from Equation (4) to the spectroscopic metallicities given in Nemec et al. (2013). A clear outlier, KIC 11802860, can be seen from the top panel of this figure. We will discuss this outlier further in the next section. Figure 9(b) shows the difference between the photometric and spectroscopic metallicities (Δ) as a function of spectroscopic metallicity. Excluding KIC 11802860, these RRab stars have a $| {\rm{\Delta }}|$ less than ∼0.35 dex (or $3\sigma$, where $\sigma =0.118$ is the dispersion of Equation (4)), with a mean Δ of −0.028 dex. Out of the 26 RRab stars, 20 and 13 of them fall within the $2\sigma$ and 1σ boundaries, respectively. When comparing two quantities, it is customary in astronomy for these quantities to be considered in agreement if their absolute difference is within two to three times the quadrature sum of their errors. In Figure 9(c), we present the ratios of absolute difference of the two metallicities and their quadrature sum errors, at which the majority of them fall within the ratio of approximately three.

Zoom In Zoom Out Reset image size

The Blazhko RRab stars in the Kepler field provide an opportunity to test the applicability of using their modulated light curves in estimating the ${\phi }_{31}$ Fourier parameter and hence the photometric metallicity. For the 11 Blazhko RRab stars in our Kepler sample (excluding KIC 9973633), we fit the light curves of the Blazhko RRab stars as presented in the upper panels of Figure 8 using Equation (1) only, without removing the modulated components (i.e., the F_m term as done in Section 5.2). For differential comparison, we adopted the same order n in the Fourier decomposition as in the case of including the modulated components. We found that several Blazhko RRab stars show a small difference in the ${\phi }_{31}$ Fourier parameter (KIC 9001926: 0.001; KIC 10789273: 0.013; KIC 12155928: 0.015) with and without removing the modulated components, while few others the differences are much larger (KIC 5559631: 0.233; KIC 11125706: 0.295; KIC 7671081: 0.666). The averaged difference of 0.102 translates to a difference of 0.13 dex in ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$ from Equation (4), which is comparable to the dispersion of the metallicity–light curve relation. Excluding KIC 7671081, the averaged difference in the ${\phi }_{31}$ Fourier parameter is reduced to 0.045 or a difference of 0.06 dex in ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$. Therefore, except for a few cases, our test suggested the Blazhko RRab stars can be included in the estimation of photometric ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$, given that their Blazhko periods can be well determined.

In the following sub-sections, we test and verify our metallicity–light curve relation using the new spectra data from a low-resolution spectroscopic observation of few RRab stars in the Kepler field (Section 6.1), as well as several publicly available data taken from the literature (Sections 6.2–6.3).

6.1. P200 Observations on Selected RR Lyrae in the Kepler Field

Spectra of several RRab stars listed in Table 3 were obtained with the P200 Telescope, using the available low-resolution spectrograph DBSP (the Double-Beam Spectrograph; Oke & Gunn 1982, with $R\sim 1360$), on 2015 June 20 and August 08. These spectra were reduced using a pyRAF-based reduction pipeline¹⁷ tailored for the P200/DBSP spectrograph (Bellm & Sesar 2016; Bellm et al. 2016), including the bias subtraction, flat-field correction, spectral extraction, and wavelength calibration. Spectroscopic metallicity were then measured on these reduced spectra using the pseudo-equivalent widths of Balmer lines and [Ca ii] K lines, following the procedures outlined in Sesar et al. (2012, hereafter S12) and Sesar et al. (2013, hereafter S13). To avoid line broadening due to velocity gradients and shock waves, Nemec et al. (2013) only took the spectra at pulsational phases between ∼0.2 and ∼0.5. Since only a small fraction of the P200/DBSP spectra fall within this range of pulsational phases, we adopted the criterion given in Sesar et al. (2013) to retain those spectra taken at the pulsational phases between 0.10 and 0.85 in order to increase usable P200/DBSP spectra in our sample. The six RRab stars and their measured metallicities from the P200/DBSP observations were listed in Table 4, and their spectra are presented in Figure 10. Two of the six RRab stars were observed on both nights. Also, three of the RRab stars have spectroscopic metallicities obtained from high-resolution spectra (Nemec et al. 2013), which can be used to compare to the spectroscopic metallicities taken from P200/DBSP.

Zoom In Zoom Out Reset image size

Table 4.  Metallicity for Selected RRab Stars in the Kepler Field with P200/DBSP Observations

KIC	Typea	Observed Date	Exposure timeb	${[\mathrm{Fe}/{\rm{H}}]}_{S12}$	${[\mathrm{Fe}/{\rm{H}}]}_{S13}$	${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{spec}}$ c	${[\mathrm{Fe}/{\rm{H}}]}_{{Kp}}$ c	${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$
6763132	RRab-NB	2015 Jun 20	240	−1.48	−1.54	−1.89 ± 0.10	−1.81 ± 0.03	−1.74 ± 0.04
6763132	RRab-NB	2015 Aug 08	300	−1.83	−1.90	−1.89 ± 0.10	−1.81 ± 0.03	−1.74 ± 0.04
9591503	RRab-NB	2015 Aug 08	300	−1.39	−1.48	−1.66 ± 0.12	−1.74 ± 0.03	−1.66 ± 0.04
10136603	RRab-NB	2015 Aug 08	300	0.72	0.43	−0.05 ± 0.14	−0.06 ± 0.05	0.10 ± 0.07
7176080	RRab-NB	2015 Jun 20	300	0.37	0.15	⋯	−1.63 ± 0.04	−1.47 ± 0.06
8344381	RRab-NB	2015 Jun 20	300	−1.65	−1.75	⋯	−1.82 ± 0.03	−1.59 ± 0.05
8344381	RRab-NB	2015 Aug 08	300	−1.28	−1.40	⋯	−1.82 ± 0.03	−1.59 ± 0.05
7257008	RRab-B	2015 Jun 20	300	0.29	0.31	⋯	−1.02 ± 0.03	−1.24 ± 0.07
Averaged ${[\mathrm{Fe}/{\rm{H}}]}_{S12}$ and ${[\mathrm{Fe}/{\rm{H}}]}_{S13}$ values for KIC 6763132 and KIC 8344381
6763132	RRab-NB	⋯	⋯	−1.66	−1.72	−1.89 ± 0.10	−1.81 ± 0.03	−1.74 ± 0.04
8344381	RRab-NB	⋯	⋯	−1.47	−1.58	⋯	−1.82 ± 0.03	−1.59 ± 0.05

Notes.

^aTypes are the same as in Table 1. ^bExposure time in seconds for the P200/DBSP observations. ^cAdopted from Nemec et al. (2013).

Download table as:  ASCIITypeset image

Since the coefficients in the transformation of pseudo-equivalent widths were slightly different in S12 and S13, we measured the metallicity using both transformations. They are listed in columns 5 and 6 in Table 4 as ${[\mathrm{Fe}/{\rm{H}}]}_{S12}$ and ${[\mathrm{Fe}/{\rm{H}}]}_{S13}$, respectively. We assume an uncertainty of ∼0.15 dex on these metallicities, which is a typical value based on the method of using pseudo-equivalent widths (Sesar et al. 2012). As shown in Table 4, the two prescriptions give consistent spectroscopic metallicities (except for KIC 10136603, with the largest difference of 0.29 dex), with ${[\mathrm{Fe}/{\rm{H}}]}_{S13}$ being more metal-poor than ${[\mathrm{Fe}/{\rm{H}}]}_{S12}$ by ∼0.1 dex on average. When compared to the metallicity from high-resolution spectra (${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{spec}}$), excellent agreement was found for KIC 6763132 taken on August 08, followed by marginal agreements for KIC 9591503 and KIC 6763132 taken on June 20. For KIC 10136603, even though the values for both ${[\mathrm{Fe}/{\rm{H}}]}_{S12}$ and ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{spec}}$ indicate a metal-rich RRab star, they are not in agreement with each others. In contrast, the value from ${[\mathrm{Fe}/{\rm{H}}]}_{S13}$ is closer to ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{spec}}$ for this RRab star. In short, values from ${[\mathrm{Fe}/{\rm{H}}]}_{S13}$ are in better agreement with those from ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{spec}}$ than those from ${[\mathrm{Fe}/{\rm{H}}]}_{S12}$, with a mean difference of ∼0.25 dex and ∼0.38 dex, respectively.

When comparing the metallicities from P200/DBSP spectra to photometric metallicities based on PTF/iPTF light curves (${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$), we found that good agreements can be seen in three RRab stars: KIC 6763132, KIC 9591503, and KIC 8344381. Based on the five measurements of low-resolution spectroscopic metallicities for these three RRab stars, the ${[\mathrm{Fe}/{\rm{H}}]}_{S13}$ values are again in better agreement with ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$ than those from ${[\mathrm{Fe}/{\rm{H}}]}_{S12}$, the mean difference becomes ∼0.05 dex and ∼0.14 dex, respectively. For KIC 10136603, ${[\mathrm{Fe}/{\rm{H}}]}_{S12}$ disagrees with ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$ and yet ${[\mathrm{Fe}/{\rm{H}}]}_{S13}$ marginally agrees with the latter value. Finally, both KIC 7176080 and KIC 7257008 show large discrepancies between the metallicities from P200/DBSP spectra and PTF/iPTF light curves, by more than 1.5 dex. We suspected that this might be caused by noisier spectra toward the short-wavelength ends. Figure 11 compares the transformed pseudo-equivalent widths for the eight spectra, those from KIC 7176080, KIC 7257008, and KIC 10136603 appeared to be outliers in this figure. Another possibility is that the observations of KIC 7176080 and KIC 7257008 were affected by weather because their spectra were taken within 10 minutes of each other on June 20. A similar result and conclusion can also be found when comparing ${[\mathrm{Fe}/{\rm{H}}]}_{S12/S13}$ to ${[\mathrm{Fe}/{\rm{H}}]}_{{Kp}}$.

Zoom In Zoom Out Reset image size

For the two RRab stars, KIC 6763132 and KIC 8344381, that have two P200/DBSP observations, the arithmetic average of the ${[\mathrm{Fe}/{\rm{H}}]}_{S12/S13}$ values are listed in the last two rows of Table 4. These averaged values were in good agreement with those from ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$, in particular, excellent agreements can be seen between the values of ${[\mathrm{Fe}/{\rm{H}}]}_{S13}$ and ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$. Note that the difference of the measured spectroscopic metallicities at the two different pulsation phases is $\sim \pm 0.35$ dex for these two RRab stars.

6.2. RR Lyrae Samples in K2E2 Field

Molnár et al. (2015) presented the light-curve analysis for 27 RRab stars toward Pisces. These light-curve data were taken under the K2 Two-wheel Engineering Test (hereafter K2E2) after the failure of the second reaction wheel on board Kepler. Based on the 8.9-day light curves, Molnár et al. (2015) classified 13 of them as non-Blazhko RRab stars, and the remaining 14 of them are Blazhko RRab stars.¹⁸ These authors also derived the photometric metallicity ${[\mathrm{Fe}/{\rm{H}}]}_{{Kp}}$ based on the K2E2 light curves with the relation presented in Nemec et al. (2013). Therefore, the RRab stars in the K2E2 Field provide a sizable sample to test our metallicity–light curve relation. We retrieved and constructed the PTF light curves for 24 of these RRab stars using the same approaches as described in Section 4, the remaining three (EPIC 60018663, 60018669, and 60018779) either fell on the inoperational CCD 03 or on the gap between the CCD chips. Number of data points per light curves for these RRab star ranges from ∼30 to ∼240. Similar to Section 5, these light curves were fitted with Equation (1) to determine the Fourier parameters ${\phi }_{31}$ and hence the photometric metallicity ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$ with Equation (4). We did not remove the F_m components for the Blazhko RRab stars because the majority of them do not have modulated period determined in Molnár et al. (2015). Nevertheless, this also provides an opportunity to test our metallicity–light curve relation in the absence of modulated period. Finally, pulsation periods are taken from Molnár et al. (2015), and assume that ${\sigma }_{P}\sim 0$.

Similar to Figure 9, we compare the photometric metallicities for these RRab stars based on PTF light curves and K2K2 light curves in Figure 12. After removing the two clear outliers shown in Figure 12(a), the remaining 22 RRab stars show a mean Δ of −0.093 dex, which is consistent with the accuracy of this technique (Kovács 2005). Separating the sample into non-Blazhko RRab stars and Blazhko RRab stars, we obtained a mean Δ of −0.094 dex and −0.092 dex, respectively, suggesting that similar results can be achieved without removing the modulated component for Blazhko RRab stars. Besides the two outliers, there are only two other RRab stars with $| {\rm{\Delta }}| \gt 0.35$ dex. For the remaining 22 RRab stars, there are 15 and 8 located within the $2\sigma$ and $1\sigma$ boundaries, respectively, as shown in Figure 12(b). Similarly, Figure 12(c) demonstrates that all of the RRab stars have a $| {\rm{\Delta }}| /{\sigma }_{T}\lt 3$, except for the two extreme outliers. Our test suggested that Equation (4) can be used to provide reliable metallicity estimation for the majority of RRab stars. Figure 13 displays three examples of the R_PTF-band light curves with the best (right panel) and the worst (left panel) agreements of the photometric metallicities, respectively.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

6.3. RR Lyrae Samples from Sesar et al. (2012, 2013)

As part of the study to search for substructures and tidal streams in the Galactic halo, S12 and S13 searched for the distant RRab stars using the PTF and other survey data. S12 reported the finding of 10 RR Lyrae with spectroscopic follow-up observations using two low-resolution spectrographs, the P200/DBSP and the Low Resolution Imaging Spectrometer (LRIS; Oke et al. 1995, with $R\sim 1760$, equipped on the Keck I Telescope). S13 presented 94 RRab stars of which only 50 of them have spectroscopic observations with P200/DBSP. Spectroscopic metallicities of these RRab stars were then measured from the low-resolution spectra. We retrieved PTF/iPTF light curves for the majority of these RRab stars and fit with Equation (1) following the procedures described in Sections 4 and 5. Pulsation periods of these RRab stars were adopted from S12 and S13.

Since 2012, more data have become available from the iPTF project for the 10 RRab stars listed in S12. The number of data points per light curve increased from merely $\sim 1 \%$ (from 271 to ∼290 for S12_RR6) to $\sim 180 \%$ (from 82 to ∼235 for S12_RR9) as compared to S12, with a range from ∼190 to ∼660. Besides a larger number of data points, these 10 RRab stars are also much fainter than those found in the K2E2 Field shown in the previous subsection, with mean magnitudes fainter than ∼19.5 mag. Consequently, their light curves exhibit a much larger scatter than the K2E2 RRab stars, as displayed in Figure 14. Nevertheless, these RRab stars provide an opportunity to test our metallicity–light curve relation for the faint RRab stars. Comparison of the photometric metallicities and spectroscopic metallicities of the nine RRab stars shown in Figure 14 are given in Figure 15 as filled symbols. The mean Δ of these nine RRab stars is 0.31 dex, after removing the three outliers in Figure 15 this mean value drops to 0.24 dex. Five of the six remaining faint RRab stars have a $| {\rm{\Delta }}|$ value less than the $3\sigma$ (i.e., ∼0.35 dex) boundary. Hence, the performance of our metallicity–light curve relation is acceptable given the faintness, large scatter of light curves, and small number of RRab stars in this sample.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For the 50 RRab stars in S13, we only retained the PTF/iPTF light curves for 32 of them, the rest of the RRab stars either do not have data in PTF/iPTF (i.e., fall into the CCD 03), with only a small number of data points per light curve (less than 15 data points), or the light curves do not exhibit RRab-like light curves. The number of data points for these 32 RRab stars include ∼20 to ∼70 for 26 of them, ∼100 to ∼300 for 5 of them, and ∼580 for 1 of them. These RRab stars are brighter than those in S12, with mean magnitudes ranging from ∼17 mag. to ∼19 mag. Figure 15 compares the photometric and spectroscopic metallicities for these 32 RRab stars as open symbols, which exhibit a much larger scatter than the RRab stars in the K2E2 Fields (Figure 12). Without removing any outliers, the mean value of Δ for this sample is 0.23 dex.

Figure 16 presents R_PTF-band light curves for the selected RRab stars in this sample. Left panels of Figure 16 display examples of three light curves that give a good agreement between the photometric and spectroscopic metallicities. In contrast, the middle panels of Figure 16 show the light curves of three RRab stars with the largest Δ, ranging from 0.84 dex (for S13_RR6) to 1.31 dex (for S13_RR23). However, these light curves do not show any "abnormality" when compared to those on the left panels. The upper-right panel of Figure 16 is the light curve for the only RRab star with ∼580 data points in this sample, which also display a large scatter as those in the S12 sample (see Figure 14). The middle-right panel of Figure 16 is the light curve for the most metal-poor RRab star within the 50 RR Lyrae in S13 sample, with a measured spectroscopic metallicity of −2.73 dex. We obtained a ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}=-3.22\pm 0.64$ dex that is consistent with the spectroscopic metallicity. Finally, the lower-right panel of Figure 16 shows the light curve of an RRab star with the largest $| {\rm{\Delta }}| /{\sigma }_{T}$ ratio. The errors on metallicities for this RRab star are ∼0.15 dex, hence ${\sigma }_{T}\sim 0.21$ dex. However, this RRab star has a Δ of ∼0.79 dex, and a resulting a large value of $| {\rm{\Delta }}| /{\sigma }_{T}$ ratio. After removing the five outliers (three in the middle panel of Figure 16, and the two in middle-right and lower-right panel of Figure 16), the mean Δ reduces to ∼0.15 dex for this sample, which is comparable to the accuracy of both methods.

Zoom In Zoom Out Reset image size

In this work, we derived the metallicity–light curve relation in the native R_PTF-band photometric system using the RRab stars found in the Kepler field. The main reasons for selecting this sample of RRab stars include the availability of accurate pulsation periods (based on Kepler light curves) and spectroscopic metallicities derived from high-resolution spectra (Nemec et al. 2013). Since about half of the RRab stars in the Kepler field are brighter than the saturation limit of ${R}_{\mathrm{PTF}}\sim 14$ mag, we re-observed a number of them with a 10 s exposure time from a dedicated iPTF experiment. Our derived metallicity–light curve relation is presented in Equation (4). When we tested our metallicity–light curve relation for six RRab stars in the Kepler field with low-resolution P200/DBSP observations, we obtained mixed results with good agreements and discrepancies. The later cases might be due to problems in observed spectra rather than our relation. We further tested our relation with three samples taken from the literature and obtained overall good agreements with our derived [Fe/H] to the published values. Specifically, after removing outliers, we obtained the mean difference between our photometric metallicities and published metallicities with the following values: ∼−0.09 dex for the K2E2 RRab stars, ∼0.15 dex for the halo RRab stars in S13, and ∼0.24 dex for a few faint RRab stars in S12 (mainly due to the large scatters of the light curves). When we applied our relation to the only RRab star in the Boötes 3 dSphs galaxy, we derived ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}=-2.15\pm 0.28$ dex, which is consistent with the spectroscopic measurement given in Sesar et al. (2014, −2.0 ± 0.1 dex).

As a demonstration of the applicability of our metallicity–light curve relation, we derived the photometric metallicities for the RRab stars listed in Table 2 of Sesar et al. (2013) that did not have spectroscopic observations. Out of the 44 RRab stars listed in that table, we could only retrieved the PTF/iPTF light curves for 33 of them (for the same reasons as those mentioned in Section 6.3). The distribution of ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$ for this sample is similar to the 32 RRab stars taken from Table 1 of Sesar et al. (2013, for which their photometric metallicities have already been derived in Section 6.3), as demonstrated in Figure 17. In the near future, we can use our relation to select RR Lyrae candidates in the Galactic halo found from the Zwicky Transient Facility (ZTF, Bellm 2014; Smith et al. 2014) before requesting spectroscopic observations with large aperture telescopes for confirmation.¹⁹ The ZTF project is using the same P48 Telescope and almost the same R-band filter as the PTF/iPTF projects, but with an upgraded mosaic CCD camera that fills out the focal plane of the P48 Telescope. With much improved survey rates, ZTF can accumulate a much larger number of data points per light curves for the RR Lyrae candidates, and their ${\phi }_{31}$ Fourier parameters and hence the photometric metallicity can be better constrained.

Zoom In Zoom Out Reset image size

Figure 18 presents the low-order Fourier parameters for all of the RRab stars that have PTF/iPTF light curves and have been studied in this work. This figure also includes those outliers, shown in black symbols, when comparing the derived ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$ to published metallicities (see Section 6). In terms of Fourier parameters, these outliers are mostly confined within the parameter space defined by other RRab stars. Therefore, these outliers do not exhibit abnormality in terms of the PTF/iPTF light curves. We note that in previous studies, outliers were seen in the comparison of metallicities from (low-resolution) spectroscopic measurements and from the metallicity–light curve relation (see, for example, Kovács 2005; Wu et al. 2006). The causes of the majority of the outliers can be traced back to various reasons, including, stars exhibiting Blazhko modulation (without removing the modulated components), problems in the photometry and/or light curves (e.g., noisy light curves, gaps in phased light curves, issues due to photometric calibrations, etc.), inaccurate spectroscopic metallicity, or even the wrong pulsational period being adopted. However, there still exist few outliers that cannot be explained, for example, V341 Aql and DG Hya in Kovács (2005) and V341 Aql, UY Boo, DG Hya, RZ Cam, and BK And in Wu et al. (2006). In the following, we briefly discuss the possible causes of the 11 outliers marked in Figure 18. A detailed investigation of them is beyond the scope of this work.

Zoom In Zoom Out Reset image size

KIC 11802860: this RRab star in the Kepler field is not a Blazhko variable, and its PTF/iPTF light curve did not show any obvious peculiarities. We do not believe the spectroscopic measurements are inaccurate because they were obtained with CFHT and this RRab star is quite bright (∼13 mag). We note that the ${\phi }_{31}$ Fourier parameter in Johnson-Cousin R-band derived from other dense ground-based observations is 5.639 for this RRab star (Jeon et al. 2014), which is in good agreement with the value listed in Table 3 (5.644 ± 0.007). Hence, there are no obvious reasons to explain why this RRab star appears to be an outlier.

EPIC 60018743 and EPIC 60018755: one of these RRab stars is a Blazhko variable (EPIC 60018743), and we did not remove its modulated component as in other Blazhko variables in the Kepler field (see Section 6.2). This might explain why this RRab star is an outlier. We do not think the photometric metallicities ${[\mathrm{Fe}/{\rm{H}}]}_{{Kp}}$ are inaccurate because those measurements are based on almost continuous Kepler light curves. Nevertheless, the quality of the PTF/iPTF light curves for both of them are similar to other RRab stars in the same sample, as demonstrated in left and right panels of Figure 13. Their Fourier parameters also agree with other RRab stars at a similar period. The relatively small number of data points per light curve, ∼40 for both of them, might incorrectly estimate the ${\phi }_{31}$ Fourier parameters. In the near future, the accumulation of a large number of data points from ZTF could assist in resolving the outlier status of these two RRab stars.

S12_RR2, S12_RR3, and S12_RR4: as shown in Figure 14, PTF/iPTF light curves for these three RRab stars, as well as others in the same sample, are noisy because they are faint RRab stars and hence their photometric measurements are less accurate. This could partially explain why these three RRab stars are outliers. However, their Fourier parameters also agree with other RRab stars, except S12_RR3 in ${\phi }_{21}$ plot (upper-left panel in Figure 18). Sesar et al. (2012) noted that the spectroscopic metallicities for S12_RR3 were differed by 0.5 dex from two measurements with different instruments. In contrast, the other two RRab stars (S12_RR1 and S12_RR5) with three spectroscopic observations show very good agreement of measured metallicity (within 0.2 dex). The low-resolution spectroscopic observation could also partially contribute to the outlier status of these three RRab stars, especially for S12_RR2 and S12_RR4 at which they only have one measurement taken from P200/DBSP. Our observations with P200/DBSP, presented in Section 6.1, demonstrated that sometimes the P200/DBSP spectra could lead to an inaccurate measurement of metallicity. Finally, we pointed out that Kovács (2005) declared a spectroscopic metallicity is inaccurate if the number of measurements is less than three.

S13_RR6, S13_RR21, S13_RR23, and S13_RR35: Fourier parameters for these four RRab stars are located within the parameter space defined by all other RRab stars, as shown in Figure 18. Their PTF/iPTF light curves (see Figure 16) also did not exhibit any abnormality, except for S13_RR6, which is noisier. We therefore believe their ${\phi }_{31}$ values are reasonably estimated. Nevertheless, with additional accumulated data points per light curves from the ZTF, the accuracy of ${\phi }_{31}$ values, and hence their photometric metallicities, could be improved. As discussed previously, the possibility of inaccurate spectroscopic metallicity based on single observations from P200/DBSP could not be ruled out either.

S13_RR17: this RRab star has the largest errors on R₂₁ and R₃₁ Four parameters, and the second largest errors on ${\phi }_{21}$ and ${\phi }_{31}$ Four parameters. Its phased PTF/iPTF light curve also has a gap around the phase of ∼0.4. Furthermore, the minimum light for these RRab stars is near ${R}_{\mathrm{PTF}}\sim 19$ mag, hence the photometric measurements are less accurate around phases of minimum light. Combining these reasons, we believe the problems on the light curve lead to the inaccurate measurement of photometric metallicity, which is hence displayed as an outlier in Figure 15. This most metal-poor RRab star worth the collection of additional light curve data in the era of ZTF.

Finally, we examined the influence of a gap in phased light curves when determining the ${\phi }_{31}$ Fourier parameters, which translate to ${[\mathrm{Fe}/{\rm{H}}]}_{\mathrm{PTF}}$ via Equation (4). We took several of the well-sampled light curves from the non-Blazhko RRab stars in the Kepler field and artificially removed some data points to mimic a phase gap in the light curves. Figure 19 presents the test results for the phase gap, with a width of 0.05 and 0.1, located at different parts of the phased light curve. In the case of a phase gap that has a width of 0.05, Figure 19 reveals that such a phase gap will not alter the determination of the ${\phi }_{31}$ Fourier parameters, regardless of the location of the phase gap. When the width of the phase gap is increased to 0.1, the ${\phi }_{31}$ Fourier parameters could be affected by the presence of the phase gap near the maximum or minimum light, as is indicated by the the two most deviated (filled circle) points in Figure 19. This result reiterates the finding in Wu et al. (2006), who suggested that the photometric metallicity can still be estimated from light curves with insufficient phase coverage when there are data points around the maximum or minimum light. In other phases around the ascending and descending branch of the light curve, the phase gap with width of 0.1 does not greatly affect the determination of the ${\phi }_{31}$ Fourier parameters. Certainly, the ${\phi }_{31}$ Fourier parameters would be less accurate when the width of the phase gap increases, and auxiliary techniques such as adding an interpolated data point in the gap (as employed in this work) or applying a polynomial fit (Jurcsik & Kovács 1996) are needed to recover the ${\phi }_{31}$ Fourier parameters. We anticipate that the problem of the phase gap presented in light curves will be diminished within the ZTF project because a much larger number of data points per light curve will be collected in the era of ZTF.

Zoom In Zoom Out Reset image size

We thank the referee for valuable input that improved the manuscript. We acknowledge Wee Siang Edmund Yuen, a 2015 summer student at the National Central University, for carrying out the preliminary analysis of the PTF data for RR Lyrae in the Kepler field. C.C.N. is thankful for the funding from the Ministry of Science and Technology (Taiwan) under grants 104-2112-M-008-012-MY3 and 104-2119-M-008-024. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

Facility: PO:1.2m.