Study of the Blazhko Type RRc stars in the Stripe 82 Region Using SDSS and ZTF

RR Lyrae stars (RRLs) are low-mass pulsating giant stars that have evolved away from the main sequence, and are on the intersection of the horizontal branch and the classical instability strip of the Hertzsprung–Russell diagram. They pulsate mostly radially but on occasion, nonradial modes can be observed, which make them a good source to study the stellar evolution and stellar pulsation (for a review see e.g., Kolenberg 2012; Marconi 2017). Their pulsation modes and the shape of their lightcurves (LCs) are used to classify them into three different types: RRab type pulsates in the fundamental mode and has an asymmetric sawtooth shaped LC; RRc type pulsates in the first-overtone mode and has more sinusoidal LC than RRab; and RRd type pulsates in both the modes and also has a sinusoidal LC.

In addition to the pulsation, there is a long-term variation observed in all types of RRLs. This is called the Blazhko effect which is a quasi-periodic modulation of amplitude and phase of a LC (Blazhko 1907; Shapley 1916). Blazhko period (P_B) is longer than the main pulsation period (P), and can be in the range of a few to a hundred days (Smith 2004), and for some, it is even longer than 1000 days. About 50% of RRab stars show the Blazhko effect (Benkő et al. 2014; Jurcsik et al. 2017), while the incidence rate is lower for RRc stars (4%–19%, Netzel et al. 2018), and it is even lower for RRd stars (Soszyński et al. 2016). Despite the lower Blazhko incidence rate than RRab stars, Blazhko type RRc (BRRc) stars have been analyzed and published in many stellar systems e.g., Galactic Bulge (Moskalik & Poretti 2003; Mizerski 2003; Netzel et al. 2018; Netzel & Smolec 2019), Large Magellanic Cloud (Nagy & Kovács 2006), and Galactic globular clusters (Clement et al. 2001; Jurcsik et al. 2015). Though, no such systematic study is known for the RRc stars in the Galactic halo. Since the physical origin of the Blazhko effect is still unknown, in spite of its relatively high abundance and long history, a study of any Galactic subsystems could contribute to understanding the effect.

In the halo region of the Milky Way, an equatorial strip with decl. limits of ±1°27, and extending from R.A. ∼20h to R.A. ∼4h is known as the Sloan Digital Sky Survey (SDSS; York et al. 2000) Stripe 82 region. Sesar et al. (2010; hereafter S10) studied RRLs of the area using SDSS's multiband and multiepoch observations. They published only the basic pulsation properties of RRLs, with no information about the Blazhko effect. We attempt to study BRRc stars using the data of S10 and Zwicky Transient Facility (ZTF; Bellm et al. 2019; Graham et al. 2019). ZTF is a time-domain survey of the northern sky, exploring the transients and variable stars using the Palomar 48 inch Schmidt telescope that began its operation in 2018. Combining ZTF and SDSS data sets gave us ∼23 yr long LCs, which were long enough to explore the long Blazhko cycles.

S10 cataloged 483 RRLs in total. Among them, 104 were classified as RRc type. The details pertaining to them can be found in their paper. These LCs are available for the public, and can be accessed using astroML (VanderPlas et al. 2012) package gatspy (VanderPlas 2015; VanderPlas & Ivezić 2015). In the present study, we used the g-band data. The amplitudes of RRL LCs are known to be higher in g band than in the other bands, and that gave us the best opportunity to detect the smallest changes produced by the Blazhko effect.

The ZTF LCs were retrieved also in the g band using the data from the DR15. The g filters of SDSS and ZTF are not identical but both can be calibrated to Pan-STARRS magnitude system. ZTF pipeline calibrates ZTF magnitudes to that of Pan-STARRS system (Masci et al. 2019), and Tonry et al. (2012) proved a way to calibrate SDSS magnitudes to that of Pan-STARRS. We used the linear form of their proposed formula in order to ease the error propagation:

$$\begin{eqnarray}&&{g}_{\mathrm{ps}1}-{g}_{\mathrm{SDSS}}=-0.012-0.139{(g-r)}_{\mathrm{SDSS}}.\end{eqnarray} \tag{ 1 }$$

Finally, the combined LCs had been constructed that extended over a period of about 23 yr, of which 13 yr were covered by observations. The number of data points varied between 139 and 1316, with an average of 376.

Period04 package (P04, Lenz & Breger 2005) was used to analyze the combined LCs. We also coded a Python script for an alternative analysis. Using the lombscargle package of SciPy (Virtanen et al. 2020), we searched for the dominating frequencies from the LCs. These frequencies were then fitted to the LCs using the Fourier series:

$$\begin{eqnarray}&&y={A}_{0}+\displaystyle \sum _{k}{A}_{k}\sin (2\pi {f}_{k}t+{\phi }_{k})\end{eqnarray} \tag{ 2 }$$

where A₀ is the mean magnitude of the LC, and f_k , A_k , ϕ_k are the kth frequency, its amplitude and phase, respectively. Only the frequency peaks with A_k /σ_k > 4.5 were accepted. The fitted series was subtracted from the LCs for prewhitening, and the residual was searched for the next dominating frequencies. The process was repeated until the noise remained. P04 and our code gave similar results but here we will present results obtained using P04.

There was a large gap of about a decade between SDSS and ZTF observations which produced frequency peaks at around 4000 days for a few BRRc stars. Considering their origin, we excluded these peaks from the results. In addition to the combined LCs, both SDSS and ZTF LCs were analyzed separately, i.e., all 104 stars were analyzed three times.

The first step of the analysis in P04 was to determine the dominating pulsation frequency f₁. For many BRRc stars the expected pulsation frequency peak was above the Nyquist frequency which was in the range: 0.158–2.729 day⁻¹, except SDSS 376465 which had Nquist frequency of 218.208 day⁻¹. In such cases we had no choice but to search for frequencies above the Nyquist frequency; we can do this with due caution (Murphy et al. 2013). In this work, the frequencies were included if their single-to-noise ratio >4.5 as computed by P04. The main frequency f₁ and its significant number of (generally three) harmonics were subtracted from the LCs to prewhiten them. The second step was to search for the other significant frequency peaks in the prewhitened LCs. Each time, prewhitening included all the previously detected peaks subtracted simultaneously, to reduce the error.

When the LC of a BRRc is analyzed using Fourier decomposition, the different Blazhko modulations appear as close frequency component(s) in the vicinity of radial mode frequency and its harmonics kf₁ (k = 1, 2,...) as kf₁ − f_m and kf₁ + f_m. The modulation frequency f_m is called the Blazhko frequency, and its inverse is the P_B. Thus longer the P_B , shorter the f_m, which implies that the LC needs to be extended over a long time span, and should have enough data points to detect these long modulations. The side peaks, kf₁ − f_m and kf₁ + f_m, can be equidistant or nonequidistant from kf₁, and can have asymmetric amplitudes (Benkő et al. 2011; VanderPlas 2018). For some Blazhko RRLs, there may be only one of the side peaks, while for some, multiple Blazhko modulations can be observed (Jurcsik et al. 2008; Chadid et al. 2010). The amplitudes of Blazhko peaks are around a few tenths of magnitudes, much smaller than that of f₁, and are difficult to detect in the presence of f₁. After prewhitening the LCs with kf₁, these Blazhko side peaks emerge in the Fourier spectrum. In the current work, if both the side peaks were detected, an average was taken as the final value of f_m. If only one of the side peaks was observed, then that value was used to calculate f_m.

When the individual LCs from SDSS and ZTF were analyzed, they did not always yield P_B, or, if they did, the corresponding error was high. One such example is seen in Figure 1. The left panels are Fourier spectra originated from SDSS data, while the right panels are that from ZTF data. No Blazhko peaks were detected in either of them but from the combined LC, we could detect the Blazhko peaks (Figure 2).

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The steps of the analysis in different cases are illustrated in Figures 2–5. The top panels show the combined SDSS and ZTF data series. The second panels of the figures contain the LCs folded with P. The subsequent panels represent the Fourier spectra around f₁ from top to bottom, starting with the spectra of the original data series, followed by the spectra of the residual LCs prewhitened by the frequencies of one (Figures 2–3), two (Figure 4), and three modulations (Figure 5). In all cases, the bottom panels show the noise after prewhitening with all significant pulsation and modulation frequencies.

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The error associated to each frequency was computed using Monte Carlo simulation in P04. Frequency error was in the range 9.5 × 10⁻⁷ to 1.00 × 10⁻⁵ day⁻¹. Standard errors were computed for each f_m originated from the side peak(s), and were propagated when taking the average for P_B.

For the 104 RRc stars published by S10 in the Stripe 82 region, our analysis showed that eight of them were rather RRd type. These stars are listed in Table 1. Their detailed analysis will be presented in a separate paper. Blazhko side peaks were found for 34 of the remaining 96 stars of S10. The main results on these stars are summarized in Table 2. It lists information on the newly identified BRRc stars including the main pulsation frequency f₁, its Fourier amplitude A₁ and phase ϕ₁; the Blazhko period P_B and its 1σ error σ(P_B). The frequencies, amplitudes, and phases of the right- and left-hand side Blazhko peaks are also given. The period P of BRRc stars were found to be in the range of 0.261 to 0.432 days which sampled the instability strip rather well. For five stars, P values published in S10 were incorrect because their data were suffering the one day alias problem. Using our extended LCs, the correct periods were determined, which can be confirmed via folding the LCs. These stars are marked with an asterisk in Table 2.

The P_B is in the range of 12.808 and 3088 days. The latter value (although with a large error) is the longest Blazhko period ever detected for an RRc star. The previous record holder was OGLE RRLYR-02478 with a period of 2954 days (Netzel et al. 2018). Out of 34 BRRc stars, 25 (74%) show single, and nine (26%) show multiple Blazhko modulations. The distribution of P_B is shown in Figure 6, where black and gray colors correspond to BRRc stars with single and multiple Blazhko effect, respectively. For stars with multiple Blazhko effect, P_B was greater than 200 days. We did not find any BRRc stars with multiple short Blazhko periods.

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Our 34 stars showing the Blazhko effect represent a 35.42% incidence rate. It is higher than any previously documented rates. Netzel et al. (2018) compiled the incidence rates of the Blazhko effect in RRc stars for different stellar systems, and found it between 4% and 19% (see their Table 4). Recently, Benkő et al. (2023) added to the list the missing rate of the Galactic field as 10.4%. It is noteworthy that the higher incidence rates belong to metal-poor globular clusters (M3: 12%; Jurcsik et al. 2014; NGC 6362: 19%; Smolec et al. 2017). On the other hand, Smolec (2005) showed that the different Blazhko incidence rates of different stellar systems are not a simple function of metallicity. Still, for the Galactic halo with the metallicity similar to that of the globular clusters, a higher incidence rate was to be expected but such a high value was surprising. However, 46% of all BRRc stars with single or multiple P_B, show very long-period Blazhko effect (P_B > 1000 days), which can be also seen in Figure 6. This fact might be the key to why we obtained a much higher incidence rate from our long data series than in previous works.

We plotted P against P_B in Figure 7 to check the presence of any correlation. Jurcsik et al. (2005) found a tendency between the two quantities, such that RRLs with periods P < 0.4 days had comparatively shorter P_B of order of some days. Szczygieł & Fabrycky (2007) detected a bimodal distribution for the modulation period of the RRc stars in the ASAS Survey. The two maxima of the distribution are around ∼10 and ∼1500 days, and there is a gap between the two groups (between ∼20 and ∼300 days) where there are hardly any stars. It has been suggested that there may be two different physical origins behind the variations in the two groups. If it is true and the stars with "classical" Blazhko effect are only in the group of 10 days, and the long-period group is something else, then results of Szczygieł & Fabrycky (2007) do not necessarily contradict the finding of Jurcsik et al. (2005). A slight bimodality can also be recognized in the combined OGLE I-IV data, but due to the significantly long data series available for only a few stars, the short part is much stronger (see Figure 6 in Netzel et al. 2018). Our data series are mainly sensitive to the long periods, so in Figure 7 we have mostly obtained the long-period part of the bimodal distribution.

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What could be the reason for the bimodality for RRc stars (there is no such evidence for RRab stars)? From one year of continuous measurements of the TESS, Benkő et al. (2023) concluded that RRc stars in which the f₆₁ and f₆₃ nonradial mode frequencies (see their definition in the cited work) appear simultaneously, typically showed long-period semiregular phase variations. These phase variations are indistinguishable from the long-period Blazhko variations found in the present work, or even in the ASAS or OGLE surveys. If this hypothesis is correct, then the Blazhko incidence rate found would actually be ∼5%, and the number of RRc stars where the two frequencies of the nonradial mode are excited simultaneously, would be ∼30%. The difference in the distributions of RRab and RRc stars would naturally be explained by the fact that we have never found frequencies similar to f₆₁ and f₆₃ in RRab stars. This hypothesis can be tested in the future if TESS provides data series of sufficient length (10–15 yr) and quality to detect additional frequencies.

Figure 8 shows the ratio of P to P_B plotted against P_B in log scale. The green points are BRRc stars with single P_B, and the red points are those with multiple P_B, both showing a decreasing trend which can be interpreted as P being inversely proportional to P_B. Using the curve fitting method, and keeping the dimensions in mind, we fit an empirical equation:

$$\begin{eqnarray}&&\displaystyle \frac{P}{{P}_{B}}={\left(\displaystyle \frac{k}{{P}_{B}}\right)}^{\epsilon }\end{eqnarray} \tag{ 3 }$$

where k = 0.22 ± 0.024 days, and = 0.93 ± 0.002. We gave k the dimension of days, and kept dimensionless to satisfy the dimension equation. Fitting of this equation is shown as the black curve in Figure 8. We do not associate this equation with any physical phenomena occurring in RRc stars. Netzel et al. (2018; see their Figure 7) observed such ratios to be in a wide range of values but their scatter hid the decreasing trend observed here. We added the most recent data from Netzel et al. (2023) to the plot as the silver points, and conclude that the same trend was followed, with Netzel et al. (2023) data falling on the left side, while our data is on the right. This illustrates that we have indeed been able to capture the long-period part of the bimodal distribution of Szczygieł & Fabrycky (2007).

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S10 classified 104 RRLs as RRc stars. Our study shows eight of them are RRd stars. In the remaining 96 RRc, 34 display Blazhko effect giving the incidence rate of 35.42%, which is higher than any previously published values. We briefly discussed the possible reasons for this. The shortest Blazhko period (P_B) found is 12.808 ± 0.001 days for SDSS 747380, while the longest is 3088 ± 126 days for SDSS 3585856. The latter is the longest P_B ever detected. The vast majority (85%) of detected P_B are above 200 days. This illustrates well the potential of the extreme length of the combined SDSS and ZTF data we use. The eight BRRc stars show two Blazhko modulations, and one, SDSS 1482164, shows three modulations. Period ratio P to P_B shows a steep decreasing trend with increasing ${P}_{B}$ due to a large number of long P_B, which indicates that P is inversely proportional to P_B .

The author would like to thank the reviewers for their valuable comments. We are thankful for the funding from the National Science and Technology Council (Taiwan) under the contract 109-2112-M-008-014-MY3. JMB was partially supported by the SeismoLab KKP-137523 Élvonal and NN-129075 grants of the Hungarian Research, Development and Innovation Ofce (NKFIH). Based on observations obtained with the Samuel Oschin Telescope 48 inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under grant No. AST1440341 and AST-2034437, and a collaboration including current partners Caltech, IPAC, the Weizmann Institute of Science, the Oskar Klein Center at Stockholm University, the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National Laboratories, IN2P3, University of Warwick, Ruhr University Bochum, Northwestern University and former partners the University of Washington, Los Alamos National Laboratories, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW.