# Study of the Blazhko type RRc stars in the Stripe 82 region using SDSS and
ZTF

Vaidehi Varma<EMAIL_ADDRESS>Graduate Institute of Astronomy,
National Central University, No. 300, Zhongda Rd., Zhongli District, Taoyuan
City 320317, Taiwan  József M. Benkő  Konkoly Observatory, Research Center for
Astronomy and Earth Sciences, HUN-REN; MTA Centre of Excellence,
Konkoly Thege M. u. 15-17., H-1121 Budapest, Hungary  Chow-Choong Ngeow
Graduate Institute of Astronomy, National Central University, No. 300, Zhongda
Rd., Zhongli District, Taoyuan City 320317, Taiwan

###### Abstract

RR Lyrae stars are pulsating stars, many of which also show a long-term
variation called the Blazhko effect which is a modulation of amplitude and
phase of the lightcurve. In this work, we searched for the incidence rate of
the Blazhko effect in the first-overtone pulsating RR Lyrae (RRc) stars of the
Galactic halo. The focus was on the Stripe 82 region in the Galactic halo
which was studied by Sesar et al using the Sloan Digital Sky Survey (SDSS)
data. In their work, 104 RR Lyrae stars were classified as RRc type. We
combined their SDSS light curves with Zwicky Transient Facility (ZTF) data,
and use them to document the Blazhko properties of these RRc stars. Our
analysis showed that among the 104 RRc stars, 8 were rather RRd stars, and
were excluded from the study. Out of remaining 96, 34 were Blazhko type, 62
were non-Blazhko type, giving the incidence rate of 35.42% for Blazhko RRc
stars. The shortest Blazhko period found was $12.808\pm 0.001$ d for SDSS
747380, while the longest was $3100\pm 126$ d for SDSS 3585856. Combining SDSS
and ZTF data sets increased the probability of detecting the small variations
due to the Blazhko effect, and thus provided a unique opportunity to find
longer Blazhko periods. We found that 85% of RRc stars had the Blazhko period
longer than 200 d.

##  1 Introduction

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, non-radial 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 ($P_{\mathrm{B}}$) 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 analysed 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 declination
limits of $\pm 1\degree 27$, and extending from R.A. $\sim$ 20h to R.A. $\sim$
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 multi-band and multi-epoch 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 $\sim 23$ year-long LCs, which were
long enough to explore the long Blazhko cycles.

##  2 Data and analysis

S10 catalogued 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 RR Lyrae LCs
are known to be higher in $g$-band than in the other bands, and that gives us
the best opportunity to detect the smallest changes produced by the Blazhko
effect.

The ZTF LCs were retrieved also in $g$-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:

$g_{\mathrm{ps1}}-g_{\mathrm{SDSS}}=-0.012-0.139(g-r)_{\mathrm{SDSS}}$ (1)

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

Period04 package (P04, Lenz & Breger, 2004) was used to analyse 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 a
LC using the Fourier series:

$y=A_{0}+\sum_{k}A_{k}sin(2\pi f_{k}t+\phi_{k})$ (2)

where $A_{0}$ is the mean magnitude of the LC, and $f_{k}$, $A_{k}$,
$\phi_{k}$ are the k-th frequency, its amplitude and phase, respectively. Only
the frequency peaks with $A_{k}/\sigma_{k}>4.5$ 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 we will present
P04.

There was a large gap of about a decade between SDSS and ZTF observations
which produced frequency peaks at around 4000 d 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 analysed separately,
i.e. all 104 stars were analysed three times.

Figure 1: BRRc star SDSS ID 3585856: Fourier analysis of individual LCs from
SDSS (left panels) and ZTF (right panels). In the left panels, (a) shows
$f_{1}$ detected from SDSS LC, and (b) shows no Blazhko peaks but just the
background noise. Similarly in the right panels, (c) shows $f_{1}$ detected
from ZTF LC, and (d) shows no Blazhko peaks but just the background noise.

The first step of the analysis in P04 was to determine the dominating
pulsation frequency $f_{1}$. For many BRRc stars the expected pulsation
frequency peak was above the Nyquist frequency which was in the range: 0.158 -
2.729 d-1, except SDSS 376465 which had Nquist frequency of 218.208 d-1. In
many 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 (SNR)
$>4.5$ as computed by P04. The main frequency $f_{1}$ 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.

Figure 2: SDSS 3585856: Panels (a) and (b) show the raw data and folded LC,
respectively. Magenta points are data from SDSS and blue points are from ZTF.
Panels (c), (d), and (e) are the steps of prewhitening the LC shown as the
Fourier spectrum. (c) shows the raw spectrum around the main frequency
$f_{1}$, (d) shows the position of the Blazhko peaks $f_{1}\pm f_{m}$ in the
residual spectrum after we prewhitened with the main frequency and its
harmonics, and (e) is the noise after prewhitening the LC with all the
previous frequencies.

When a Blazhko RRc LC is analysed using Fourier decomposition, the different
Blazhko modulations appear as close frequency component(s) in the vicinity of
radial mode frequency and its harmonics $kf_{1}$ ($k=1,2,\dots$) as
$kf_{1}-f_{\mathrm{m}}$ and $kf_{1}+f_{\mathrm{m}}$. The modulation frequency
$f_{\mathrm{m}}$ is called the Blazhko frequency, and its inverse is the
Blazhko period $P_{\mathrm{B}}$. Thus longer the $P_{B}$, shorter the
$f_{\mathrm{m}}$, which implies that the LC needs to be extended over a long
time span, and has enough data points to detect long modulations. The side
peaks, $kf_{1}-f_{\mathrm{m}}$ and $kf_{1}+f_{\mathrm{m}}$, can be equidistant
or non-equidistant from $kf_{1}$, 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 $f_{1}$, and
are difficult to detect in the presence of $f_{1}$. After prewhitening the LCs
with $kf_{1}$, 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 $f_{\mathrm{m}}$. If only one of the side peaks was
observed, then that value was used to calculate $f_{\mathrm{m}}$.

When the individual LCs from SDSS and ZTF were analysed, they did not always
yield $P_{\mathrm{B}}$, or, if they did, the corresponding error was be high.
One such example is seen in Fig 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 (Fig. 2).

The steps of the analysis in different cases are illustrated in Figs. 2-5. The
top panels show the combined SDSS and ZTF data series. The second panels of
the figures contain the LCs folded with the main period. The subsequent panels
represent the Fourier spectra around the main frequencies 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 (Figs. 2-3), two
(Fig. 4), and three modulations (Fig. 5). In all cases, the bottom panels show
the noise after pre-whitening with all significant pulsation and modulation
frequencies.

The error associated to each frequency was computed using Monte Carlo
simulation in P04. Frequency error is in the range $9.5\times 10^{-7}$ to
$1.00\times 10^{-5}$ d-1. Standard errors were computed for each
$f_{\mathrm{m}}$ originated from the side peak(s), and were propagated when
taking the average for $P_{\mathrm{B}}$.

Figure 3: SDSS 747380: Panels (a) and (b) show the raw data and folded LC,
respectively. Magenta points are data from SDSS, and blue ones are from ZTF.
Panels (c), (d), and (e) are the steps of prewhitening. (c) shows the main
spectrum around the main pulsation frequency $f_{1}$, (d) shows left hand side
Blazhko peak $f_{1}-f_{\mathrm{m}}$ after we prewhitened the LC with $f_{1}$
and its significant harmonics, and (e) is the noise at the end of the
prewhitening process.

Figure 4: SDSS 2863787: panels (a) and (b) are similar to that of the Fig 3.
(c) shows the Fourier spectrum around the main frequency, (d) shows the first
pair of the Blazhko side peaks after we prewhitened the LC with $f_{1}$ and
its significant harmonics, (e) shows the second pair after we prewhitened the
LC with the first Blazhko pair, and (f) shows the noise at the end of the
prewhitening process.

Figure 5: SDSS 1482164: The meaning of panels a)-e) are the same as in Fig. 4,
panel f) shows the result of one more prewhitening step due to the third
Blazhko modulation. Panel g) shows the end of the prewhitening.

##  3 Results and Discussion

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 summarised in Table 2. It lists information on
the newly identified BRRc stars including the main pulsation frequency
$f_{1}$, its Fourier amplitude $A_{1}$ and phase $\phi_{1}$; the Blazhko
period $P_{\mathrm{B}}$ and its $1\sigma$ error $\sigma(P_{\mathrm{B}})$. 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 d 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.

SDSS ID | $f_{0}$ | $f_{1}$ | ratio
---|---|---|---
| (d-1) | (d-1) |
1078860 | 1.886082 | 2.528637 | 0.746
3478713 | 2.047186 | 2.746912 | 0.745
850835 | 1.826386 | 2.449997 | 0.745
3214909 | 1.977810 | 2.655536 | 0.745
2488976 | 1.842614 | 2.468807 | 0.746
2291937 | 1.923916 | 2.577745 | 0.746
2464128 | 1.878133 | 2.514649 | 0.747
2506078 | 2.060011 | 2.769921 | 0.744
Table 1: RRd stars discovered in the Stripe 82 region which were misclassified
by S10 as RRc stars. The columns are: SDSS ID, frequency of the fundamental
mode $f_{0}$, the first overtone $f_{1}$, and their ratio.

The $P_{\mathrm{B}}$ is in the range of 12.808 and 3088 d. 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 d (Netzel et al., 2018). Out of 34 BRRc stars, 25 (74%) show single, and
9 (26%) show multiple Blazhko modulations. The distribution of
$P_{\mathrm{B}}$ is shown in Fig 6, where black and gray colors correspond to
BRRc stars with single and multiple Blazhko effect, respectively. For stars
with multiple Blazhko effect, $P_{\mathrm{B}}$ was greater than 200 d. We did
not find any BRRc stars with multiple short Blazhko periods.

Figure 6: Histogram of the Blazhko period $P_{\mathrm{B}}$. Black and gray
colors represent BRRc stars with single and multiple $P_{\mathrm{B}}$,
respectively.

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 $P_{\mathrm{B}}$, show very long period Blazhko effect
($P_{\mathrm{B}}>1000$ d) which can be also seen in Fig 6. This fact might be
the key to why we obtained a much higher incidence rate from out long data
series than in previous works.

Figure 7: Plot of the main pulsation period P to the Blazhko period
$P_{\mathrm{B}}$ in log-log scale. Green and red points show the BRRc stars
with single and multiple Blazhko effect, respectively.

We plotted $P$ against $P_{B}$ in Fig 7 to check the presence of any
correlation. Jurcsik et al. (2005) found a tendency between the two
quantities, such that RRLs with periods $P<0.4$ d had comparatively shorter
$P_{\mathrm{B}}$ 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 $\sim 10$ and $\sim
1500$ days, and there is a gap between the two groups (between $\sim$20 and
$\sim$300 d) 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 Fig. 6 in
Netzel et al., 2018). Our data series are mainly sensitive to the long
periods, so in Fig. 7 we have mostly obtained the long-period part of the
bimodal distribution.

SDSS ID | $f_{1}$ | $P_{\mathrm{B}}$ | $\sigma(P_{\mathrm{B}})$ | $A_{1}$ | $\phi_{1}$ | $f_{+}$ | $A_{+}$ | $\phi_{+}$ | $f_{-}$ | $A_{-}$ | $\phi_{-}$
---|---|---|---|---|---|---|---|---|---|---|---
| (d-1) | (d) | (d) | (mag) | (deg) | (d-1) | (mag) | (deg) | (d-1) | (mag) | (deg)
747380* | 3.536821 | 12.808 | 0.001 | 0.251 | 0.616 | - | - | - | 3.458743 | 0.084 | 0.706
562035 | 2.922886 | 22.889 | 0.003 | 0.230 | 0.500 | 2.966577 | 0.070 | 0.860 | 2.879197 | 0.122 | 0.668
4614863* | 3.831802 | 39.28 | 0.03 | 0.249 | 0.459 | 3.857263 | 0.050 | 0.244 | 3.806343 | 0.055 | 0.103
2659801 | 3.033801 | 182 | 1 | 0.320 | 0.836 | 3.03928 | 0.016 | 0.393 | 3.028298 | 0.018 | 0.147
638540 | 2.972634 | 192 | 1 | 0.245 | 0.138 | 2.977831 | 0.014 | 0.896 | 2.967441 | 0.008 | 0.634
2609442 | 3.231722 | 426 | 3 | 0.269 | 0.427 | 3.234067 | 0.013 | 0.939 | - | - | -
3681504 | 3.076957 | 428 | 4 | 0.279 | 0.756 | 3.079215 | 0.016 | 0.832 | 3.074537 | 0.022 | 0.701
288008* | 2.628844 | 798 | 10 | 0.194 | 0.734 | 2.630027 | 0.113 | 0.596 | 2.627511 | 0.065 | 0.270
3348773 | 2.381611 | 858 | 15 | 0.238 | 0.839 | 2.382785 | 0.038 | 0.174 | 2.380455 | 0.037 | 0.853
1217801 | 2.939625 | 1095 | 32 | 0.243 | 0.536 | 2.940526 | 0.012 | 0.226 | 2.9387 | 0.015 | 0.790
959802 | 2.641075 | 1219 | 22 | 0.247 | 0.885 | 2.642081 | 0.042 | 0.110 | 2.640383 | 0.045 | 0.746
376465 | 2.312534 | 1288 | 26 | 0.239 | 0.961 | 2.313314 | 0.016 | 0.239 | 2.311762 | 0.015 | 0.733
4383295 | 2.700236 | 1292 | 19 | 0.266 | 0.489 | 2.701011 | 0.050 | 0.386 | 2.699463 | 0.047 | 0.939
2314306 | 3.081313 | 1341 | 18 | 0.272 | 0.509 | 3.082061 | 0.106 | 0.146 | 3.08057 | 0.105 | 0.083
3358190 | 2.388038 | 1711 | 29 | 0.229 | 0.303 | 2.388621 | 0.052 | 0.915 | 2.387453 | 0.034 | 0.327
1478867 | 2.970986 | 1870 | 77 | 0.257 | 0.748 | 2.971514 | 0.028 | 0.191 | 2.970444 | 0.017 | 0.696
3397977 | 2.757541 | 1884 | 83 | 0.223 | 0.019 | 2.758071 | 0.028 | 0.875 | 2.757009 | 0.047 | 0.711
3031571 | 2.731217 | 1949 | 42 | 0.306 | 0.753 | 2.731697 | 0.040 | 0.967 | 2.730667 | 0.041 | 0.144
3308790 | 2.662176 | 1985 | 38 | 0.225 | 0.041 | 2.66268 | 0.078 | 0.132 | 2.661672 | 0.066 | 0.427
765345 | 3.811996 | 2011 | 87 | 0.153 | 0.930 | 3.812496 | 0.015 | 0.392 | 3.811501 | 0.022 | 0.659
1091627 | 3.201902 | 2119 | 41 | 0.250 | 0.220 | 3.202372 | 0.051 | 0.614 | 3.201428 | 0.053 | 0.624
1986301 | 3.242536 | 2150 | 35 | 0.298 | 0.716 | 3.242995 | 0.082 | 0.175 | 3.242065 | 0.093 | 0.567
2213142 | 3.313901 | 2514 | 112 | 0.248 | 0.505 | 3.314263 | 0.028 | 0.268 | 3.313459 | 0.024 | 0.081
429508 | 2.85003 | 2906 | 148 | 0.245 | 0.523 | 2.850376 | 0.059 | 0.151 | 2.849688 | 0.034 | 0.113
3585856 | 2.915149 | 3088 | 126 | 0.244 | 0.039 | 2.915485 | 0.031 | 0.203 | 2.914836 | 0.031 | 0.008
4455741 | 2.752802 | 962 | 8 | 0.215 | 0.724 | - | - | - | 2.751762 | 0.041 | 0.590
|  | 2412 | 39 |  |  | 2.753207 | 0.099 | 0.011 | 2.752377 | 0.071 | 0.026
2396176* | 2.762629 | 1027 | 4 | 0.179 | 0.632 | - | - | - | 2.761655 | 0.128 | 0.486
|  | 431 | 3 |  |  | 2.765078 | 0.048 | 0.306 | 2.760429 | 0.062 | 0.643
2639854 | 3.353221 | 1534 | 23 | 0.138 | 0.314 | 3.353902 | 0.062 | 0.944 | 3.352596 | 0.071 | 0.856
|  | 910 | 13 |  |  | 3.354286 | 0.035 | 0.926 | 3.352084 | 0.048 | 0.648
2445511 | 2.536248 | 1182 | 18 | 0.259 | 0.438 | 2.537088 | 0.095 | 0.152 | 2.535396 | 0.059 | 0.980
|  | 712 | 13 |  |  | 2.537599 | 0.046 | 0.642 | 2.534785 | 0.060 | 0.226
2175525 | 2.514798 | 1158 | 11 | 0.220 | 0.676 | 2.515677 | 0.062 | 0.832 | 2.513949 | 0.083 | 0.286
|  | 706 | 12 |  |  | 2.516243 | 0.030 | 0.722 | 2.513409 | 0.037 | 0.796
4455741* | 2.752802 | 2412 | 39 | 0.215 | 0.724 | 2.753207 | 0.099 | 0.011 | 2.752377 | 0.071 | 0.026
|  | 962 | 8 |  |  | - | - | - | 2.751762 | 0.041 | 0.590
1747387 | 2.872223 | 1722 | 13 | 0.140 | 0.519 | 2.872804 | 0.069 | 0.328 | - | - | -
|  | 1124 | 22 |  |  | 2.873118 | 0.039 | 0.257 | 2.871339 | 0.020 | 0.952
2863787 | 2.728025 | 2291 | 71 | 0.289 | 0.571 | 2.728464 | 0.036 | 0.119 | 2.72759 | 0.044 | 0.287
|  | 750 | 14 |  |  | 2.729367 | 0.022 | 0.658 | 2.726701 | 0.026 | 0.988
1482164 | 2.529674 | 610 | 9 | 0.276 | 0.472 | 2.531294 | 0.031 | 0.649 | 2.528014 | 0.031 | 0.864
|  | 2426 | 46 |  |  | 2.530091 | 0.119 | 0.274 | 2.529266 | 0.077 | 0.688
|  | 244 | 1 |  |  | 2.53327 | 0.045 | 0.075 | 2.524887 | 0.038 | 0.995
Table 2: Blazhko RRc stars discovered in the Stripe 82 region. The columns
are: SDSS ID; $f_{1}$ is the main frequency; $P_{\mathrm{B}}$ is the Blazhko
period; $\sigma(P_{\mathrm{B}})$ is the error associated with
$P_{\mathrm{B}}$; $A_{1}$ and $\phi_{1}$ are the amplitude and phase of
$f_{1}$; $f_{+}$ is the right hand side peak ($f_{+}=f_{1}+f_{m}$); $A_{+}$
and $\phi_{+}$ are the amplitude and phase of $f_{+}$; $f_{-}$ is the left
hand side peak ($f_{-}=f_{1}-f_{m}$); $A_{-}$ and $\phi_{-}$ are the amplitude
and phase of $f_{-}$. If any side peak is not detected, ’-’ is written
instead. * after the SDSS ID indicates stars whose main periods were one day
alias in S10. Figure 8: Ratio of $P$ to $P_{\mathrm{B}}$ plotted against
$\log(P_{\mathrm{B}})$. The green and red points represent BRRc stars with
single and multiple $P_{\mathrm{B}}$, respectively, and the silver points are
Netzel et al. (2023) data. The black curve shows the fitting.

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 $f_{61}$ and
$f_{63}$ non-radial mode frequencies (see their definition in the cited work)
appear simultaneously, typically showed long-period semi-regular 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 $\sim$5%, and the number of RRc stars where the two
frequencies of the non-radial mode are excited simultaneously, would be
$\sim$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 $f_{61}$ and $f_{63}$ in RRab stars. This hypothesis can be tested
in the future if TESS provides data series of sufficient length (10–15 years)
and quality to detect additional frequencies.

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

$\frac{P}{P_{B}}=\left(\frac{k}{P_{B}}\right)^{\epsilon}$ (3)

where $k=0.22\pm 0.024$ d, and $\epsilon=0.93\pm 0.002$. We gave k the
dimension of days, and kept $\epsilon$ dimensionless to satisfy the dimension
equation. Fitting of this equation is shown as the black curve in Fig 8. We do
not associate this equation with any physical phenomena occurring in RRc
stars. Netzel et al. (2018) (see their fig. 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).

##  4 Summary

S10 classified 104 RRLs as RRc stars. Our study shows 8 of them are RRd stars.
In the remaining 96 RRc stars, 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
($P_{\mathrm{B}}$) found is $12.808\pm 0.001$ d for SDSS 747380, while the
longest is $3100\pm 126$ d for SDSS 3585856. The latter is the longest
$P_{\mathrm{B}}$ ever detected. The vast majority (85%) of detected
$P_{\mathrm{B}}$ are above 200 days. This illustrates well the potential of
the extreme length of the combined SDSS and ZTF data we use. The 8 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 $P$
due to a large number of long $P_{\mathrm{B}}$, which indicates that $P$ is
inversely proportional to $P_{B}$.

##  5 Acknowledgments

The author thanks the reviewers for their comments. We are thankful for
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 Ofﬁce (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.

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The homotopy colimit over refinements of all covers is equivalent to the
homotopy colimit over some truncation below of refinements over all covers,
i.e. we can always assume that our covers are as fine as we wish when
computing the homotopy colimit. But, by taking a fine enough cover
$\mathcal{U}$, every object of
$\mathsf{Green}_{\nabla,0}(X_{\bullet}^{\mathcal{U}})$ restricted to each
$U_{\alpha}$ is free, and so, as in the proof of Lemma 4.2.7, we can invert
quasi-isomorphisms whose target is in
$\mathsf{Green}_{\nabla,0}(X_{\bullet}^{\mathcal{U}})$. This means that, given
some morphism in the localisation
$\operatorname{L}{\mathsf{Green}_{\nabla,0}(X_{\bullet}^{\mathcal{U}})}$,
expressed as a chain of roofs with the left-legs all quasi-isomorphisms, we
can invert the quasi-isomorphisms and compose the resulting morphisms to
obtain a single morphism in
$\mathsf{Green}_{\nabla,0}(X_{\bullet}^{\mathcal{U}})$ which is equal to that
in the localisation with which we started. This means that
$\operatorname{L}{\mathsf{Green}_{\nabla,0}(X_{\bullet}^{\mathcal{U}})}$ is
equivalent to
$\operatorname{L}{\underline{\mathsf{Green}}_{\nabla,0}(X_{\bullet}^{\mathcal{U}})}$,
and so their homotopy colimits agree. ∎

######  Corollary 4.2.14.

There is an equivalence of $(\infty,1)$-categories

$\operatorname{hocolim}_{\mathcal{U}}\operatorname{L}{\underline{\mathsf{Green}}_{\nabla,0}(X_{\bullet}^{\mathcal{U}})}\simeq\operatorname{L}{\mathsf{CCoh}(X)}.$

###### Proof.

This is a consequence of Theorems 4.2.9, 4.2.10, 4.2.11, and 4.2.13. ∎

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