# The Zwicky Transient Facility Catalog of Periodic Variable Stars

Xiaodian Chen CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China  Shu Wang
CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100101, China  Licai Deng CAS Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, China  Richard de Grijs Department of
Physics and Astronomy, Macquarie University, Balaclava Road, Sydney, NSW 2109,
Australia  Research Centre for Astronomy, Astrophysics and Astrophotonics,
Macquarie University, Balaclava Road, Sydney, NSW 2109, Australia
International Space Science Institute–Beijing, 1 Nanertiao, Zhongguancun, Hai
Dian District, Beijing 100190, China  Ming Yang IAASARS, National Observatory
of Athens, Vas. Pavlou & I. Metaxa, Penteli 15236, Greece  Hao Tian Key
Laboratory of Space Astronomy and Technology, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China

(Received 2019 November 29; Revised 2020 May 18; Accepted 2020 Jun 11)

###### Abstract

The number of known periodic variables has grown rapidly in recent years.
Thanks to its large field of view and faint limiting magnitude, the Zwicky
Transient Facility (ZTF) offers a unique opportunity to detect variable stars
in the northern sky. Here, we exploit ZTF Data Release 2 (DR2) to search for
and classify variables down to $r\sim 20.6$ mag. We classify 781,602 periodic
variables into 11 main types using an improved classification method.
Comparison with previously published catalogs shows that 621,702 objects
(79.5%) are newly discovered or newly classified, including $\sim$700
Cepheids, $\sim$5000 RR Lyrae stars, $\sim$15,000 $\delta$ Scuti variables,
$\sim$350,000 eclipsing binaries, $\sim$100,000 long-period variables, and
about 150,000 rotational variables. The typical misclassification rate and
period accuracy are on the order of 2% and 99%, respectively. 74% of our
variables are located at Galactic latitudes, $|b|<10^{\circ}$. This large
sample of Cepheids, RR Lyrae, $\delta$ Scuti stars, and contact (EW-type)
eclipsing binaries is helpful to investigate the Galaxy’s disk structure and
evolution with an improved completeness, areal coverage, and age resolution.
Specifically, the northern warp and the disk’s edge at distances of 15–20 kpc
are significantly better covered than previously. Among rotational variables,
RS Canum Venaticorum and BY Draconis-type variables can be separated easily.
Our knowledge of stellar chromospheric activity would benefit greatly from a
statistical analysis of these types of variables.

Periodic variable stars (1213); Distance indicators (394); Cepheid variable
stars (218); RR Lyrae variable stars (1410); Galaxy disks (589); Eclipsing
binary stars (444); Long period variable stars (935)

††journal: ApJS

##  1 Introduction

Periodic or quasi-periodic variables encompass mainly eclipsing binary systems
and rotational stars characterized by extrinsic variability, while pulsating
variables are among objects exhibiting intrinsic fluctuations in the
variability tree (Gaia Collaboration et al., 2019). These objects are
interesting, since their periods contain information about their luminosity or
mass. Variable stars have been studied for several hundred years. However,
this field has been advancing significantly in recent years. With increases in
the depth and efficacy of time-domain surveys, the number of newly detected
variables is roughly doubled every year. Gaia’s third Data Release (DR3) is
expected to include 7 million variables, with about one-third to half of
periodic nature. This huge number is of the same order as the number of extant
stellar spectra. By combining the spectra and light curves (LCs) of variable
stars, we will be able to improve our knowledge of a range of fundamental
stellar parameters, which in turn can benefit developments in theory and the
applications of stellar physics. A complete catalog of all-sky periodic
variables can also be used to separate variable and non-variable stars in
spectroscopic surveys—such as those undertaken with the Large Sky Area Multi-
Object Fiber Spectroscopic Telescope (LAMOST; Cui et al., 2012; Deng et al.,
2012) and the Sloan Digital Sky Survey (SDSS; Eisenstein et al., 2011)—to
reduce the systematic effects pertaining to the presence of variables.

Although searches for variables date back hundreds of years, the number of
known variables first increased significantly in the 1990s thanks to the
development of CCDs. The Massive Compact Halo Object (MACHO) survey (Alcock et
al., 1993) found 20,000 variable stars in the Large Magellanic Cloud (LMC).
The Optical Gravitational Lensing Experiment (OGLE) subsequently made
significant progress in this field. Over its 20-year duration, OGLE detected
more than 900,000 variables in the Magellanic Clouds, the Galactic bulge, and
the Galactic plane (Udalski et al., 1992, 2015). The first all-sky variability
survey was carried out by the All Sky Automated Survey (ASAS), which detected
10,000 eclipsing binaries and 8000 periodic pulsating stars (Pojmanski et al.,
2005). Several thousand variables were also found by the Robotic Optical
Transient Search Experiment (ROTSE; Akerlof et al., 2000) and the Lincoln
Near-Earth Asteroid Research (Palaversa et al., 2013, LINEAR) project. The
Catalina Surveys (Drake et al., 2014, 2017) found more than 100,000 variables,
covering almost the entire sky, which allowed significant progress to be made
in studying streams, structures, and the shape of the Milky Way’s halo.
Variable surveys have also been undertaken in infrared passbands, e.g., the
VISTA Variables in the Vía Láctea (VVV) survey (Minniti et al., 2010) and the
VISTA survey of the Magellanic Clouds system (VMC; Cioni et al., 2011). In
2018, a number of variable-star catalogs were published, including the All-Sky
Automated Survey for Supernovae (ASAS-SN; Jayasinghe et al., 2018), the
Asteroid Terrestrial-impact Last Alert System (ATLAS; Heinze et al., 2018),
the Wide-field Infrared Survey Explorer (WISE) catalog of periodic variable
stars (Chen et al., 2018b), and the variable catalog of Gaia DR2 (Clementini
et al., 2019; Mowlavi et al., 2018). The number of known variables is
currently experiencing a second substantial increase, resulting in millions of
known variable objects.

Alongside the increasing numbers of known variables, advances have also been
made as regards the corresponding and ancillary science, e.g., pertaining to
distance measurements, structure analysis, stellar-mass black hole searches,
and multi-messenger astrophysics. As regards distance indicators, detached
eclipsing binaries were used to measure the distance to the LMC to an accuracy
of 1% (Pietrzyński et al., 2019). This accurate benchmark distance is of great
importance to refine the value of the Hubble parameter (Riess et al., 2019).
As to structure analyses, the sample of newly found classical Cepheids led to
the first intuitive 3D map of two-thirds of the Milky Way’s disk (Chen et al.,
2019; Skowron et al., 2019a). This new map will help us understand the
evolution of both the Galaxy’s spiral arms and its outer disk. As regards
long-period binary variables, their unseen companions may include black holes.
A low-mass black hole has been found to orbit a giant star with an orbital
period of $\sim$83 days (Thompson et al., 2019). Searching for variables with
the shortest or the longest periods is attractive (Burdge et al., 2019; Graham
et al., 2015), since such systems contain information about both the
properties of their electromagnetic radiation and possible gravitational-wave
signals that can motivate multi-messenger astrophysics.

In this paper, we search for periodic variables in the recent data release of
the Zwicky Transient Facility (ZTF), the deepest time-domain survey of the
northern sky. Periodicities are analyzed for the full database to identify a
sample of periodic variables that is as complete as possible. Using an
improved method of classification, 781,602 variables belonging to 11 main
types are classified. Our sample’s detection rate of newly identified objects
is about 80%. We construct a large variable-star catalog that will benefit
many follow-up studies. The data are described in Section 2. We explain how we
identify and classify variables in Sections 3 and 4, respectively. The new
variable catalog and a comparison of its consistency with previous catalogs
are covered in Section 5. We provide a detailed discussion of each type of
variable object and their applications in Section 6. Section 7 summarizes our
conclusions.

##  2 ZTF data

The ZTF is a 48-inch Schmidt telescope with a 47 deg2 field of view (Masci et
al., 2019; Bellm et al., 2019). This large field of view ensures that the ZTF
can scan the entire northern sky every night. The ZTF survey started on 2018
March 17. During the planned three-year survey, ZTF is expected to acquire
$\sim$450 observational epochs for 1.8 billion objects. Its main science aims
are the physics of transient objects, stellar variability, and solar system
science (Graham et al., 2019; Mahabal et al., 2019). ZTF DR2 contains data
acquired between 2018 March and 2019 June, covering a timespan of around 470
days. ZTF DR2 photometry is obtained in two observation modes, three-night
cadence for the northern sky and one-night cadence for the Galactic plane,
$|b|\leq 7^{\circ},{\rm Dec.}>-25^{\circ}$. The photometry is provided in the
$g$ and $r$ bands, with a uniform exposure time of 30 s per observation. The
limiting magnitude is $r\sim 20.6$ mag. ZTF $g$ and $r$ photometry is
calibrated with the help of Pan-STARRS1 DR1 (Chambers et al., 2016); there is
only a small offset of $\sim 0.01$ mag between both systems at the bright end
($r<15.5$ mag). ZTF DR2 includes more than 1 billion stars, about half of
which have $>20$ epochs of observations. For the majority of stars located in
the northern Galactic plane, ZTF contains $\sim 150$ epochs of observations.
As such, ZTF can be used to detect numerous variables in the northern Galactic
plane, which has not been well-studied by previous time-domain surveys. We
downloaded the full catalog (comprising a total volume of $\sim$3.4
TB),111https://irsa.ipac.caltech.edu/data/ZTF/lc_dr2/ which is provided in the
form of 856 files, ordered as a function of spatial position.

##  3 Search for periodic variables

To search for periodic variables, we need to impose some conditions to make
the process easier and quicker. First, we only selected objects in the ZTF
data with at least 20 detections. Adoption of this criterion will not cause us
to miss many periodic candidates, since a period’s false-alarm probability
(FAP) based on $<20$ detections is high. Second, poor-quality images and
photometry were excluded by adopting ${\rm INFOBITS}<33,554,432$ and ${\rm
catflags}\neq 32,768$, respectively. Third, preliminary searching for
periodicity for a given object was based on photometry associated with the
same internal product ID. This choice made this process much quicker. Although
a fraction of the objects have multiple internal IDs, since they may have been
observed for different projects, their periodicity is unlikely to be missed
based on analysis of the data from the main project. A redetermination of the
period will be done based on the object’s position (R.A. and Dec.) during the
next steps.

We cut the data into smaller segments to mitigate computational memory
restrictions and ran Lomb–Scargle periodogram (Lomb, 1976; Scargle, 1982)
analysis using the MATLAB code ‘plomb’ to search for variables. The code
returns the Lomb–Scargle power spectral density (PSD) based on the maximum
input frequency. We searched for periods from 0.025 to 1000 days (in steps of
0.0001 day-1 in frequency), a range covering variables from short-period
$\delta$ Scuti to long-period Mira stars. We attempted to exclude aliased
periods—such as one day and its multiples—using a method similar to that of
Jayasinghe et al. (2019b). Some aliased periods were successfully excluded,
although many remained. Many candidate variables with aliased periods are real
long-period variables (LPVs). Their aliased periods are a combination of real
periods, $P_{\rm r}$, and the one-day sampling cadence, $P_{\rm c}$: $1/P_{\rm
a}=1/P_{\rm r}+1/P_{\rm c},P_{\rm c}=0.5,1,2...$. After excluding $P_{\rm a}$,
the real periods of well-sampled variables could be obtained (see also Section
5.3).

A number of parameters were recorded during this process to help with the
selection of variable candidates, including the FAP, which represents the
confidence of the periodicity determination. We selected periodic candidates
by imposing ${\rm FAP}<0.001$. Adoption of this criterion mistakenly excludes
only a few short-period variables. These will be rediscovered with better
sampling in the ZTF’s future DRs. Other parameters of importance are the mean
PSD, $\langle P_{f}\rangle$, and the number of frequencies associated with
${\rm PSD}>0.5{\rm PSD}_{\rm max}$. These parameters are used to avoid
inclusion of false variables produced by abnormal sampling. ZTF is not
affected by problems caused by the inhomogeneous distribution of the data
points, except for LPVs. Therefore, we do not need to record any parameters to
trace the distribution of the data points in the phase-folded LC.
Periodicities were determined in both the $g$ and $r$ bands, and candidates
were accepted if the LC in one band met all conditions.

Running the code took one month on two computers with a total of 18 i7 CPUs. A
grand total of 1.4 million variable candidates were recorded. We redetermined
their periods based on all of the photometric data within $1^{\prime\prime}$
around their spatial position. The LCs were fitted with a fourth-order Fourier
function, $\displaystyle{f=a_{0}+\sum_{i=1}^{4}a_{i}\cos(2\pi
it/P+\phi_{i})}$, which is a suitable choice for survey LCs to avoid missing
important parameters or overfitting. About 90% of these ZTF LCs have $a_{4}$
values less than the typical photometric uncertainties of 0.02 mag. $a_{i}$
and $\phi_{i}$ denote amplitudes and phases for each order, respectively.
Candidates with poorly fitted LCs were excluded using the adjusted $R^{2}$
(which represents how well LCs are fitted by the Fourier function), i.e.,
$R_{g}^{2}<0.4$ and $R_{r}^{2}<0.4$. Objects still characterized by aliased
periods were excluded based on their distributions in the period versus
$R^{2}$ or $\phi_{21}$ ($\phi_{21}=\phi_{2}-2\phi_{1}$) diagram, e.g., objects
with $|P-1|<0.03$ days and $R^{2}<0.8$ or
$|{\phi_{21}}_{r}-{\phi_{21}}_{g}|>0.2$ were excluded. At this point, 1
million candidates remained for further classification.

##  4 Classification

Classification is more challenging than the identification process. The main
idea is based on the varying densities of different variables in multi-
dimensional space. Parameters including the period ($\log P$), phase
difference ($\phi_{21}$), amplitude ratio ($R_{21}=a_{2}/a_{1}$), amplitude
(${\rm Amp.}$), absolute Wesenheit magnitude ($M_{W_{gr}}$), and adjusted
$R^{2}$ were adopted to help with our classification. The period is the most
significant parameter; $\phi_{21}$, $R_{21}$, and ${\rm Amp.}$ are parameters
determined from the LC fits. The absolute Wesenheit magnitude is used to
separate the variables in the overlap region of the LCs’ parameter space. It
is also helpful to narrow down the types of low-amplitude variables. To avoid
significant extinction in the optical bands, we calculated Wesenheit
magnitudes, $m_{W_{gr}}=\langle g\rangle-R_{gr}\times(\langle g\rangle-\langle
r\rangle)$, making use of the $g,r$ photometry. Here, $\langle g\rangle$ and
$\langle r\rangle$ are the mean magnitudes determined from our LC fits. We
adopted the recently derived Galactic extinction law (Wang & Chen, 2019) based
on red clump stars with accurate parameters and obtained
$m_{W_{gr}}=3.712\langle r\rangle-2.712\langle g\rangle$.

Gaia DR2 parallaxes (Gaia Collaboration et al., 2018) were adopted to estimate
the absolute magnitudes. Since many of our objects are located in the Galactic
plane, we used the inverse of the parallaxes to estimate distances. To reduce
the systematic effects of the Lutz–Kelker bias (Lutz & Kelker, 1973), only
parallaxes with $\varpi>0.2,\sigma_{\varpi}/\varpi<0.2$ and
$\sigma_{\varpi}/\varpi<0.5$ were adopted for our detailed classification and
the dwarf–giant separation, respectively. $R^{2}$ contains information to
separate characteristic and non-characteristic LCs, where ‘characteristic’ LCs
have similar patterns, and we can find a rule to classify variable stars based
on their LCs. ‘Non-characteristic’ LCs exhibit arbitrary patterns. In other
words, variables with characteristic LCs form either a clump or a sequence in
the $P$ versus $\phi_{21}$ diagram, while variables with non-characteristic LC
exhibit a random distribution. As $R^{2}$ decreases, the probability of
variables with characteristic LCs—e.g., eclipsing binaries and radially
pulsating stars—decreases.

Note that the relation between the amplitudes in both filters (${\rm
Amp}_{g}=a\times{\rm Amp}_{r}+b$) is different for each variable. The
1$\sigma$ scatter (0.03–0.06 mag) on these relations in the current database
is too large to allow for a robust classification of the majority of our
variables, considering that the amplitude difference between the $g$ and $r$
bands is not significant. As such, we did not adopt this criterion for
automatic classification, but only for a visual check of the easily-confused
variables.

The classification process proceeded based on the auto-manual Density-Based
Spatial Clustering of Applications with Noise (DBSCAN). DBSCAN clusters data
points based on a neighborhood search radius and a minimum number of
neighbors. Both of these parameters were adjusted in each step to achieve an
optimal separation of the different variables. Since classification rules for
variables (in particular for variables with non-characteristic LCs) are not
well-established, we only classified variables with a significant density in
parameter space. The classification order progresses from Cepheids, to
fundamental-mode (RRab) and first-overtone (RRc) Lyrae, $\delta$ Scuti stars,
eclipsing binaries, and finally to variables with non-characteristic LCs.

As our first step, we established the best samples of Cepheids, RRab and RRc
Lyrae, and $\delta$ Scuti stars based on strict constraints on the LC quality
($R^{2}>0.9$, similar periods in both filters), the LC shape, and stellar
luminosity. For each type of variable, we found around 1000 candidates. These
objects were added to the remaining candidates to enhance their density in
parameter space. We then selected Cepheids, RRab and RRc Lyrae, and $\delta$
Scuti stars based on their density of data points. In doing so, candidates
located close to the best-established sample objects were assumed to be of the
same type. This process was repeated until the density of the remaining
candidates was similar to the density of variables with non-characteristic LCs
in their vicinity. After removing these four types of variables, the majority
of the remaining candidates were eclipsing binaries. We selected all eclipsing
binaries with amplitudes larger than 0.08 mag to distinguish them from RS
Canum Venaticorum-type systems (RS CVn), which are eclipsing binaries
exhibiting chromospheric activity.

The LCs of eclipsing binaries are highly symmetric, and their phase
differences do not deviate significantly from $2\pi$
($|\phi_{21}-2\pi|<0.1\pi$) for periods longer than 0.3 days (half periods for
eclipsing binaries222The period determination process produces half-periods
for the majority of eclipsing binaries because the two eclipses over a full
period are too similar to be separated. Only very few eclipsing binaries with
significantly modulated LCs yield full periods.). However, contact (EW-type)
eclipsing binary systems with half-periods shorter than 0.3 days show a more
scattered distribution of $\phi_{21}$. We retained these half-periods until we
had completed the separation of eclipsing binaries from other types of
variables. Since the number density of EW-type eclipsing binaries is higher
than that of the other types of periodic variables (around 0.1–0.4% of the
overall stellar population, depending on environment Rucinski, 2006; Chen et
al., 2016b), more than half of our candidates are eclipsing binaries. After
removal of the eclipsing binaries, the remaining candidates were variables
characterized by non-characteristic LCs. From short to long periods, they are
low-amplitude $\delta$ Scuti (LAD) stars, RS CVn, BY Draconis (BY Dra)-type
variables, semi-regular variables (SRs), and Miras. LADs have shorter periods
and brighter luminosities than their counterpart EW-type eclipsing binaries or
RS CVn.

The LCs of RS CVn show an additional sine signal out-of-eclipse, which is the
result of chromospheric activity. They have very similar luminosities to
eclipsing binaries. BY Dras are K–M dwarf rotational stars also exhibit
chromospheric activity. Their rotational periods range from less than one to a
few days. Since almost all K–M dwarfs show BY Dra LCs, they are second in
terms of their numbers only to EW-type eclipsing binaries in magnitude-limited
samples. The period ranges of BY Dra and RS CVn overlap significantly; they
can only be separated with the help of their luminosities. BY Dra are fainter
than RS CVn. Their absolute magnitudes are concentrated in the range
$2.8<M_{W_{gr}}<4.6$ mag. We also found that the luminosities of BY Dra have
no relation to their periods, which means that stellar mass does not affect
the rate of rotation among these objects. SRs are red giants, red supergiants,
or asymptotic giant branch (AGB) stars with periodic and variable LCs. SR
periods range from 20 to thousands of days, which overlaps significantly with
the Mira period range. Miras are oxygen- or carbon-rich AGB stars. The DBSCAN
results show that SRs and Miras can be separated because of an apparent
difference in amplitude (see also Jayasinghe et al., 2018). Here, we selected
Miras by adopting $\rm{Amp}_{r}>2$ mag or $\rm{Amp}_{g}>2.4$.

The criteria adopted to assist with the DBSCAN process are listed in Table 1.
The $\phi_{21}$–$\log P$ diagrams of all variables are shown in Figures 1 and
2. We find that aliased periods around one day are almost excluded. In Figure
2, variables with non-characteristic LCs exhibit a random distribution of
$\phi_{21}$ between $\pi$ and $3\pi$. A low density in $-0.7<\log P<1.3$
[days] and $1.95\pi<\phi_{21}<2.05\pi$ is due to the difficulty to separate
non-characteristic LCs from eclipsing binaries’ LCs.

Eclipsing binaries can be divided into EW, semi-detached (EB), and detached
(EA) types according to the degree by which they fill their Roche lobes. An
accurate classification requires a detailed orbital analysis, which represents
a major effort. Fortunately, we can also use well-established empirical
relations related to their LCs. We refitted the LCs of our sample of eclipsing
binaries using their full periods. The newly determined Fourier parameters
$a_{2}$ and $a_{4}$ (similar to $a_{1}$ and $a_{2}$ pertaining to their half-
periods) and the difference between the primary and secondary minimum
magnitudes $\Delta_{\rm{min}}=m_{\rm p}-m_{\rm s}$ were obtained to perform
the classification. Compared with EW-type eclipsing binaries, EA-type objects
have a larger amplitude $a_{4}$. To separate EW from EA types, we used the
($a_{4},a_{2}$) diagram, similarly to Chen et al. (2018b).

We also adopted $R_{41}=a_{4}/a_{2}=0.43$ (see Figure 4) as additional
boundary to identify eclipsing binaries that were not well separated based on
the equations of Chen et al. (2018b). The main difference between semi-
detached and contact eclipsing binaries is the difference between both minima.
Both components of contact eclipsing binary systems share a common envelope
and, as a result, they have similar temperatures. However, semi-detached
eclipsing binaries usually exhibit larger $\Delta_{\rm{min}}$ because of the
different temperatures of the two components. $\Delta_{\rm{min}}=0.2$ mag
(corresponding to a 5% difference between the components’ temperatures for an
inclination of $i=90^{\circ}$) is usually adopted as the boundary between
semi-detached and contact eclipsing binaries. This criterion offers a rough
classification of the two subtypes, and it is reliable for higher orbital
inclinations ($i>70^{\circ}$) and good photometric quality. We did not try to
classify and distinguish semi-detached eclipsing binaries from other types of
eclipsing binaries. Instead, we recorded the two-band minimum differences
$\Delta_{\rm{min}}$ in our catalog to facilitate any further classification
based on future DRs.

Figure 1: $\phi_{21}$ versus $\log P$ diagram for all ZTF variables with
characteristic LCs. From short to long periods, these include high-amplitude
$\delta$ Scuti stars (HADs: orange), EW-type eclipsing binaries (red), RRc
Lyrae (dark green), RRab Lyrae (cyan), EA-type eclipsing binaries (black),
first-overtone classical Cepheids (blue triangles), fundamental-mode classical
Cepheids (blue dots), and Type II Cepheids (green circles). Figure 2:
$\phi_{21}$ versus $\log P$ diagram for all ZTF variables with non-
characteristic LCs. From short to long periods, these include low-amplitude
$\delta$ Scuti stars (LADs: dark green), RS CVn (red), BY Dra stars (black),
SRs (magenta), and Miras (cyan). Orange HADs are also shown, for comparison
with the LADs.

The main subtypes of Cepheids among our candidates are classical Cepheids
pulsating in their fundamental and first-overtone modes and Type II Cepheids.
Classical and Type II Cepheids cannot be separated cleanly only based on LC
information. However, classical Cepheids are two or three magnitudes brighter
than Type II Cepheids. In addition, classical Cepheids are young and
associated with the Milky Way’s thin disk. Armed with these two additional
classification criteria, both Cepheid types can be separated more easily. The
fundamental and first-overtone classical Cepheids exhibit different LC shapes.
First-overtone Cepheids usually have smaller amplitudes (${\rm Amp.}~{}<0.4$
mag) and a low amplitude ratio ($R_{21}<0.3$). In Figures 3 and 4, we can see
a clear boundary between fundamental (blue dots) and first-overtone (blue
triangles) classical Cepheids (see also Soszynski et al., 2008). Just like for
Cepheids, RRab and RRc Lyrae pulsating in their fundamental and first-overtone
modes also exhibit a clear boundary.

Figure 3: ${\rm{Amp}}_{r}$ versus $\log P$ diagram for all ZTF variables.
Symbols are as in Figure 1. LADs have been added as gray dots. Figure 4:
$R_{21}$ versus $\log P$ diagram for all ZTF variables. Symbols are as in
Figure 3.

###  4.1 Luminosity Selection

Luminosity is the other important parameter one can use to classify variables.
We present the period–luminosity diagram in Figure 5. Variables with Gaia
parallax accuracy better than 20% are shown, except for the luminous
variables, since most of the latter are distant and exceed our parallax
constraint. Cepheids with $<100$% parallax uncertainty, Miras with $<50$%
parallax uncertainty, and SRs with $<50$% parallax uncertainty are shown.
Variables outside these ranges exhibit more scatter in their absolute
magnitude distributions and a tail at faint magnitudes, which is the result of
the Lutz–Kelker effect.

$\delta$ Scuti stars, RRab and RRc Lyrae, and classical Cepheids that are
located in the instability strip follow tight PLRs. The scatter in their
absolute magnitudes is mainly caused by parallax uncertainties. We adopted
$M_{W_{gr}}-M_{W_{gr}}({\rm PL})<3\sigma_{\rm DM}$ to exclude dwarf
contamination. Here, $M_{W_{gr}}$ and $M_{W_{gr}}({\rm PL})$ are the observed
and predicted absolute magnitudes, respectively, and $\sigma_{\rm DM}$ is the
error in the distance modulus, which has been propagated from the parallax
uncertainty. The $M_{W_{gr}}({\rm PL})$–$P$ relations for these variables were
calculated using objects with parallax uncertainties less than 10%. Since
these PLRs were roughly determined and only used to exclude contamination, we
do not report the corresponding coefficients here. For more accurate PLRs, we
refer to individual papers discussed in Section 6.

In Figure 5, we find that both HADs and LADs follow similar PLRs, and their
scatter seems smaller than that pertaining to the RR Lyrae and Cepheids. This
smaller scatter is driven by the high number of nearby $\delta$ Scuti stars.
Except for these variables, EW-type eclipsing binaries, Type II Cepheids, SRs,
and Miras also follow PLRs characterized by moderate scatter. EW-type
eclipsing binaries have been found to obey PLRs with 6–7% accuracy in infrared
bands (Chen et al., 2016a, 2018a). SRs and Miras are LPVs that obey different
sequences in the period–luminosity diagram (Lebzelter et al., 2019). Without
access to accurate parallaxes and well-determined periods, their sequences are
rather unclear.

At the bottom of the the period–luminosity diagram, we find that BY Dra
variables are vertically distributed in the range of $2.8<M_{W_{gr}}<4.6$ mag,
and they are fainter than other variables. This BY Dra sequence is always
clear in a parallax-limited sample, which means that this is a real
distribution rather than one caused by selection effects associated with the
limit imposed on the Gaia parallaxes. Since the absolute magnitude boundary
for BY Dra variables is unknown, we adopted these rough luminosity cuts to
classify BY Dra stars. The rotational periods of BY Dra variables range from
0.25 to 20 days, with a peak around a few days. RS CVn occupy the same
distribution as eclipsing binaries. They are not shown in the diagram. The
PLRs or luminosity ranges were determined for each variable type to help with
our classification. For example, candidates 2 mag fainter than classical
Cepheids have a low probability of being classical Cepheids..

Figure 5:  $\log P$ versus $M_{W_{gr}}$ diagram for variables with better than
20% (all variables except giants), 50% (Miras and SRs), and 100% (Cepheids)
Gaia parallax accuracy. $M_{W_{gr}}$ are the absolute Weisenheit magnitudes
estimated from a combination of $g,r$ photometry and Gaia parallaxes. Symbols
are as in Figure 4. Miras, SRs, and BY Dra variables have been added using
blue plus signs, and pink and green dots, respectively.

For candidates in the overlap region in parameter space, such as RRc and EW-
type eclipsing binaries, Cepheids and rotational variables, LADs and EW-type
eclipsing binaries, we performed a visual check of their LCs to assist with
and confirm the classification process. We thus classified 781,602 periodic
variables from our master sample of 2.1 million candidates. The remaining
candidates were recorded as suspected variables. They will be better
classified based on enhanced data from future ZTF DRs.

Table 1: Criteria adopted to classify different types of variables Type | Selection criteria
---|---
LAD |  $R^{2}>0.4$, ${\rm Amp.}<0.2$, ${\rm FAP}<0.001$, $M_{W_{gr}}-M_{W_{gr}}({\rm PL})<3\sigma_{\rm DM}$, $P<0.2$ days
HAD |  $R^{2}>0.4$, ${\rm Amp.}>0.2$, ${\rm FAP}<0.001$, $M_{W_{gr}}-M_{W_{gr}}({\rm PL})<3\sigma_{\rm DM}$
RRc |  $R^{2}>0.6$, ${\rm Amp.}>0.08$, ${\rm FAP}<0.001$, $M_{W_{gr}}-M_{W_{gr}}({\rm PL})<3\sigma_{\rm DM}$, $\phi_{21}<1.9$
RRab |  $R^{2}>0.6$, ${\rm Amp.}>0.08$, ${\rm FAP}<0.001$, $M_{W_{gr}}-M_{W_{gr}}({\rm PL})<3\sigma_{\rm DM}$
Cepheid |  $R^{2}>0.6$, ${\rm Amp.}>0.08$, ${\rm FAP}<0.001$, $M_{W_{gr}}-M_{W_{gr}}({\rm PL})<3\sigma_{\rm DM}$, $P<40$ days
Eclipsing binary |  $R^{2}>0.4$, ${\rm Amp.}>0.08$, ${\rm FAP}<0.001$
Mira |  $R^{2}>0.4$, ${\rm Amp.}>=2$, ${\rm FAP}<0.001$ , $P>80$ days
SR |  $R^{2}>0.4$, ${\rm Amp.}<2$, ${\rm FAP}<0.001$, $P>20$ days
BY Dra |  $R^{2}>0.4$, ${\rm FAP}<0.001$, $2.8<M_{W_{gr}}<4.6$, $0.25<P<20$ days
RS CVn |  $R^{2}>0.4$, ${\rm FAP}<0.001$, $M_{W_{gr}}~{}=M_{W_{gr}}$(Eclipsing binary), $P<20$ days
EW | $a_{4}<=a_{2}(a_{2}+0.125)$
|  $a_{2}(a_{2}+0.125)<a_{4}<a_{2}(a_{2}+0.375)$, $a_{4}/a_{2}<=0.43$
EA | $a_{4}>=a_{2}(a_{2}+0.375)$
|  $a_{2}(a_{2}+0.125)<a_{4}<a_{2}(a_{2}+0.375)$, $a_{4}/a_{2}>0.43$
Table 2: ZTF Variables CatalogaaThe entire table is available in the online journal; 50 lines are shown here for guidance regarding its form and content.. ID | R.A. (J2000) | Dec. (J2000) | Period | $R_{21}$ | $\phi_{21}$ | $T_{0}$ | $\langle g\rangle$ | $\langle r\rangle$ | … |  Ampg |  Ampr | Type
---|---|---|---|---|---|---|---|---|---|---|---|---
| ∘ | ∘ | days |  |  |  HJD$-$2400000.5 | mag | mag | … | mag | mag |
ZTFJ000000.13+620605.8 | 0.00056 | 62.10163 | 1.9449979 | 0.198 | 5.189 | 58386.2629478 | 17.995 | 16.571 | … | 0.113 | 0.078 | BYDra
ZTFJ000000.14+721413.7 | 0.00061 | 72.23716 | 0.2991500 | 0.263 | 6.308 | 58388.2555794 | 19.613 | 18.804 | … | 0.540 | 0.438 | EW
ZTFJ000000.19+320847.2 | 0.00080 | 32.14645 | 0.2870590 | 0.010 | 8.024 | 58280.4780813 | 15.311 | 14.610 | … | 0.219 | 0.197 | EW
ZTFJ000000.26+311206.3 | 0.00109 | 31.20176 | 0.3622166 | 0.132 | 6.281 | 58283.4619944 | 16.350 | 15.844 | … | 0.233 | 0.226 | EW
ZTFJ000000.30+233400.5 | 0.00125 | 23.56682 | 0.2698738 | 0.193 | 6.302 | 58437.2686640 | 17.890 | 16.944 | … | 0.373 | 0.352 | EW
ZTFJ000000.30+711634.1 | 0.00125 | 71.27616 | 0.2685154 | 0.160 | 5.236 | 58657.4235171 | 19.144 | 17.875 | … | 0.173 | 0.154 | EW
ZTFJ000000.39+605148.8 | 0.00163 | 60.86358 | 0.5591434 | 0.424 | 6.322 | 58338.4702019 | 19.965 | 19.002 | … | 0.797 | 0.841 | EW
ZTFJ000000.51+583238.7 | 0.00215 | 58.54409 | 2.9797713 | 0.143 | 9.013 | 58663.4378649 | 16.586 | 15.368 | … | 0.082 | 0.068 | BYDra
ZTFJ000001.00+612832.8 | 0.00420 | 61.47580 | 0.3643510 | 0.285 | 6.400 | 58385.2702714 | 20.640 | 19.627 | … | 0.970 | 0.681 | EW
ZTFJ000001.37+561504.9 | 0.00571 | 56.25137 | 0.2557214 | 0.312 | 6.256 | 58471.1392038 | 19.416 | 18.559 | … | 0.674 | 0.623 | EW
ZTFJ000001.74+614940.9 | 0.00729 | 61.82803 | 1.0158276 | 0.370 | 6.366 | 58349.3156217 | 19.094 | 18.013 | … | 0.289 | 0.228 | EW
ZTFJ000001.75+594739.4 | 0.00732 | 59.79428 | 112.0533126 | 0.390 | 6.950 | 58320.7237689 | 18.676 | 17.393 | … | 0.062 | 0.055 | SR
ZTFJ000001.92+554800.7 | 0.00804 | 55.80020 | 112.3366840 | 0.209 | 5.763 | 58389.1259060 | 12.664 | 0.0 | … | 0.114 | 0.0 | SR
ZTFJ000002.02+540550.8 | 0.00844 | 54.09746 | 7.6299744 | 0.677 | 3.277 | 58472.1363770 | 18.176 | 17.187 | … | 0.187 | 0.151 | RSCVN
ZTFJ000002.20+480720.8 | 0.00918 | 48.12246 | 0.3810135 | 0.227 | 3.701 | 58282.4549819 | 17.196 | 16.037 | … | 0.107 | 0.093 | BYDra
ZTFJ000002.21+385226.4 | 0.00921 | 38.87401 | 0.4103992 | 0.168 | 6.512 | 58274.4798066 | 15.265 | 14.853 | … | 0.209 | 0.211 | EW
ZTFJ000002.27+331411.5 | 0.00947 | 33.23655 | 0.1161226 | 0.069 | 3.714 | 58367.3685172 | 18.781 | 17.726 | … | 0.238 | 0.197 | RSCVN
ZTFJ000002.28+592423.4 | 0.00954 | 59.40651 | 0.3286918 | 0.063 | 5.511 | 58314.4511431 | 20.860 | 19.268 | … | 0.280 | 0.255 | EW
ZTFJ000002.61+560944.8 | 0.01091 | 56.16245 | 0.2780012 | 0.095 | 5.197 | 58644.4715685 | 16.017 | 15.664 | … | 0.074 | 0.074 | RSCVN
ZTFJ000002.69+640614.8 | 0.01123 | 64.10413 | 0.4084968 | 0.250 | 6.617 | 58307.4094573 | 21.241 | 19.953 | … | 0.471 | 0.440 | EW
ZTFJ000002.99+611736.0 | 0.01246 | 61.29334 | 0.5534820 | 0.269 | 6.289 | 58431.2049383 | 17.362 | 16.649 | … | 0.241 | 0.228 | EW
ZTFJ000003.05+410045.6 | 0.01271 | 41.01267 | 0.2701826 | 0.236 | 6.311 | 58296.3618793 | 16.796 | 16.055 | … | 0.365 | 0.354 | EW
ZTFJ000003.10+722838.4 | 0.01292 | 72.47736 | 0.2000857 | 0.014 | 5.677 | 58364.3618372 | 17.146 | 16.473 | … | 0.069 | 0.054 | RSCVN
ZTFJ000003.13–022548.2 | 0.01306 | -2.43006 | 0.4936916 | 0.431 | 3.906 | 58456.1669642 | 17.510 | 17.347 | … | 1.301 | 0.902 | RR
ZTFJ000003.17+582138.3 | 0.01325 | 58.36065 | 331.9141625 | 0.334 | 6.273 | 58400.7947631 | 13.180 | 0.0 | … | 0.073 | 0.0 | SR
ZTFJ000003.23+543605.4 | 0.01347 | 54.60151 | 0.7863971 | 0.105 | 8.826 | 58389.2748730 | 17.639 | 16.527 | … | 0.166 | 0.112 | BYDra
ZTFJ000003.24+692214.2 | 0.01353 | 69.37062 | 0.3728018 | 0.169 | 6.048 | 58282.4524423 | 15.855 | 14.678 | … | 0.398 | 0.359 | EW
ZTFJ000003.33+550558.3 | 0.01390 | 55.09955 | 0.2513022 | 0.377 | 6.044 | 58646.4653817 | 20.761 | 19.701 | … | 0.765 | 0.773 | EW
ZTFJ000003.40+623837.5 | 0.01419 | 62.64377 | 0.2627338 | 0.288 | 6.245 | 58362.3199849 | 21.027 | 19.922 | … | 0.573 | 0.605 | EW
ZTFJ000003.74+393925.9 | 0.01559 | 39.65722 | 0.2394832 | 0.050 | 7.755 | 58325.4868272 | 17.943 | 16.731 | … | 0.192 | 0.158 | EW
ZTFJ000003.74+623637.4 | 0.01560 | 62.61040 | 0.3755206 | 0.344 | 6.383 | 58343.3639966 | 20.429 | 19.378 | … | 0.653 | 0.652 | EW
ZTFJ000003.76+532917.1 | 0.01568 | 53.48811 | 0.5921959 | 0.408 | 7.638 | 58352.3259739 | 16.446 | 15.747 | … | 0.164 | 0.159 | BYDra
ZTFJ000003.88+553759.9 | 0.01619 | 55.63331 | 0.2652778 | 0.341 | 6.266 | 58348.4098300 | 19.632 | 18.711 | … | 0.747 | 0.661 | EW
ZTFJ000003.88+673554.8 | 0.01620 | 67.59858 | 7.8057274 | 0.096 | 6.155 | 58386.3788529 | 20.419 | 18.262 | … | 0.498 | 0.426 | EW
ZTFJ000003.92+614456.1 | 0.01636 | 61.74894 | 0.3473332 | 0.272 | 6.092 | 58257.4187066 | 20.679 | 19.451 | … | 0.521 | 0.511 | EW
ZTFJ000003.96+182425.2 | 0.01654 | 18.40700 | 0.4851359 | 0.456 | 3.909 | 58295.4504510 | 15.268 | 15.096 | … | 1.126 | 0.879 | RR
ZTFJ000004.14+570616.8 | 0.01725 | 57.10468 | 246.2392553 | 0.319 | 3.591 | 58347.9920985 | 15.589 | 13.508 | … | 1.600 | 1.459 | SR
ZTFJ000004.34+554359.8 | 0.01809 | 55.73328 | 0.3470882 | 0.272 | 6.349 | 58274.4650397 | 19.871 | 19.026 | … | 0.602 | 0.610 | EW
ZTFJ000004.34+522031.5 | 0.01809 | 52.34209 | 4.0833915 | 0.189 | 7.602 | 58320.3673808 | 17.703 | 16.745 | … | 0.087 | 0.095 | BYDra
ZTFJ000004.34+510505.0 | 0.01810 | 51.08474 | 0.3332412 | 0.034 | 4.458 | 58635.4032937 | 17.188 | 16.681 | … | 0.192 | 0.191 | EW
ZTFJ000004.41+581846.9 | 0.01840 | 58.31303 | 0.4418413 | 0.126 | 3.603 | 58278.4683106 | 18.842 | 17.921 | … | 0.110 | 0.088 | BYDra
ZTFJ000004.49+514615.4 | 0.01871 | 51.77095 | 0.0918632 | 0.187 | 4.174 | 58428.2433174 | 17.392 | 17.104 | … | 0.161 | 0.115 | DSCT
ZTFJ000004.56+453022.7 | 0.01902 | 45.50633 | 6.6168334 | 0.084 | 6.243 | 58490.2445564 | 16.378 | 15.673 | … | 0.047 | 0.052 | RSCVN
ZTFJ000004.98+524230.9 | 0.02076 | 52.70860 | 7.1420745 | 0.359 | 6.024 | 58471.1790394 | 17.357 | 15.933 | … | 0.080 | 0.072 | BYDra
ZTFJ000005.00+555613.9 | 0.02087 | 55.93720 | 0.8834472 | 0.278 | 6.004 | 58332.3958118 | 18.238 | 16.738 | … | 0.141 | 0.131 | EW
ZTFJ000005.13+503833.5 | 0.02138 | 50.64264 | 6.8118535 | 0.046 | 4.986 | 58316.3336707 | 16.984 | 16.128 | … | 0.125 | 0.072 | BYDra
ZTFJ000005.21+650537.8 | 0.02174 | 65.09385 | 0.1251307 | 0.066 | 8.199 | 58353.4356974 | 14.071 | 13.139 | … | 0.107 | 0.070 | DSCT
ZTFJ000005.34+543148.7 | 0.02228 | 54.53020 | 0.0602120 | 0.101 | 4.100 | 58305.4711181 | 14.514 | 14.305 | … | 0.080 | 0.055 | DSCT
ZTFJ000005.74+573228.7 | 0.02393 | 57.54131 | 3.5412731 | 0.215 | 5.945 | 58304.4028037 | 16.206 | 15.416 | … | 0.060 | 0.059 | BYDra
ZTFJ000005.94+555900.1 | 0.02479 | 55.98338 | 0.4009322 | 0.071 | 6.156 | 58354.4293181 | 17.257 | 16.647 | … | 0.167 | 0.175 | EW
… | … | … | … | … | … | … | … | … | … | … | … | …
… | … | … | … | … | … | … | … | … | … | … | … | …
… | … | … | … | … | … | … | … | … | … | … | … | …

##  5 The periodic variables catalog

The periodic variables catalog333The full catalog and LCs can be accessed at
http://variables.cn:88/ztf/ or doi.org/10.5281/zenodo.3886372. of ZTF DR2,
containing 781,602 periodic variables, is included in Table 2. It contains the
source ID, position (J2000 R.A. and Dec.), period, mean magnitudes ($\langle
g\rangle$, $\langle r\rangle$), number of detections, LC parameters, and type
of variable star. The LC parameters include the amplitude, amplitude ratio
$R_{21}$, phase difference $\phi_{21}$, Heliocentric Julian Date (HJD) of the
minimum $T_{0}$, $R^{2}$ of the LC fit, two-band minimum differences for
eclipsing binaries, and the period’s FAP. Candidates without a good
classification are collected in a suspected variables catalog (Table 6). The
single-exposure photometry catalog is available online, including the objects’
SourceID, R.A. (J2000), Dec. (J2000), HJD, gmag, rmag, e_gmag, and e_rmag.
Figure 6 shows the density distribution of these periodic variables in
Galactic coordinates. This density distribution is similar to the distribution
of the number of detections. 74% of the variables are located in the northern
Galactic plane ($|b|<10^{\circ}$), where variables have thus far not been
recorded systematically.

Figure 6:  Distribution map (in Galactic coordinates) of the 781,602 periodic
variables classified in ZTF DR2.

###  5.1 New variables

We cross-matched our variables catalog with several other large variables
catalogs to check the accuracy of our classification and of the derived
periods, as well as the completeness of our catalog. The reference catalogs
include the ASAS-SN Catalog of Variable Stars in the Northern Hemisphere, the
ASAS-SN Catalog of Variable Stars with a reclassification of 412,000 known
variables, the ATLAS catalog of variables, the WISE catalog of periodic
variable stars, the Catalina catalog of periodic variable stars, the Gaia
catalog of RR Lyrae and Cepheids, the OGLE variables catalog, and other
variables collected in the General Catalogue of Variable Stars (GCVS). We
assembled these catalogs and then used an angular radius of $2^{\prime\prime}$
to match the nearest known variables to our ZTF variables. Almost all objects
were detected within $1^{\prime\prime}$, which means that the recent variable
surveys all have good positional accuracy. In addition, a $1^{\prime\prime}$
matching radius can significantly reduce sample confusion, which is a
potentially serious problem in dense fields. 621,702 of the 781,602 variables
in the ZTF database are newly detected variables. The total number and the
number of newly detected variables of each type are listed in Table 3. The new
detection rate of these variables is high, ranging from 10% for RRab Lyrae to
more than 90% for $\delta$ Scuti and BY Dra variables. This high new detection
rate is not only the result of the database’s high photometric accuracy
($\sim$0.02 mag accuracy down to $r=18$ mag) and the faint limiting magnitude
of the ZTF database (about 1.5 mag deeper than ATLAS in the corresponding $r$
band) but also of the improvements made in our method of variable-star
identification and classification. The relatively low new detection rate of
RRab Lyrae is explained easily, because they were already well-covered by the
Catalina survey and Gaia DR2. However, both the purity and the period accuracy
of the RRab Lyrae are significantly improved in our ZTF catalog.

###  5.2 Classification Accuracy

The classification accuracy pertaining to each type of variable was tested by
comparison with the four variable-star catalogs: see Table 4. The coincidence
rates range from 95% to 99%, with any disagreement usually found at the faint
end for each catalog. In addition, different sampling strategies may also
result in somewhat inconsistent classifications. We note that a few (sub)types
have relatively low coincidence rates of around 80%. This means that the
standard of classification for these variables is not well-established. Among
the four catalogs we compared our results with, the ASAS-SN catalog has the
best classification accuracy overall.

In our catalog, RRab Lyrae have the best classification standard. Their purity
is around 99%. The main contaminants are unsolved RRc Lyrae and EW-type
eclipsing binaries. RRc Lyrae exhibit a purity of 91%. Their more symmetric
LCs can easily mask them as EW-type eclipsing binaries. The purity of the
eclipsing binaries is not as high as that of the RRab Lyrae. This is because
of the difficulty in unequivocal classifications among the three binary
subtypes. For lower LC amplitudes the classification becomes rather difficult.
In this paper, we classify all eclipsing binaries into EW and EA types.
Contamination by non-binaries is only 1%. This means that our separation of
binaries from pulsators is efficient. EW-type eclipsing binaries have a purity
of 96%; their main contaminants are EA-type eclipsing binaries. EA-type
eclipsing binaries have a coincidence rate of 84%.

The coincidence rate of classical Cepheids is around 92%, based on a
comparison of $\sim$200 known Cepheids. The percentage of disagreement for
Type II Cepheids is around 10%. LPV purity is around 98%. Here, the main
misclassification occurs between SRs and Miras, even though all reference
catalogs adopted a similar boundary, $\rm{Amp.}=2.0$ mag. These different
classifications can be explained by the different limiting magnitudes
pertaining to the reference catalogs: a shallow limiting magnitude will lead
to underestimated Mira amplitudes. For example, the ASAS-SN catalog has a
limiting magnitude of 17, so that for Miras fainter than 15 mag, the amplitude
determined is likely less than 2 mag. These Miras will hence be classified as
SRs and, as a result, about 46% of matched Miras are mistaken for SRs in the
ASAS-SN catalog. In turn, 99.2% of ASAS-SN Miras are correctly classified in
our catalog. Toward the faint magnitude limit of ZTF, this type of
misclassification also occurs. In our catalog, 1694 SRs with $m_{g,r}>19$ mag
and ${\rm Amp}_{r}>1.0$ (${\rm Amp}_{g}>1.2$) mag run the risk of
misclassification.

The short-period $\delta$ Scuti variables have a purity of 88%. However,
$\delta$ Scuti stars are not well classified in previously published catalogs.
In our catalog, the purity of HADS is as high as those of RR Lyrae and
eclipsing binaries, while the LADS purity is relatively lower. As regards
rotational stars, the purity of BY Dra (93.7%) is high since they were
strictly selected based on their Gaia parallaxes. However, without a
luminosity criterion, it is not easy to separate RS CVn and eclipsing binaries
with poor-quality photometry. To obtain a cleaner sample of eclipsing
binaries, many eclipsing binaries with relatively poor-quality LCs have been
classified as RS CVn, which leads to a systematically lower purity level of RS
CVn (75.2%). Given that we have double checked the LCs of all variables where
we noticed disagreement with other catalogs, our classification is more
reliable for most of these objects. This is reasonable, since ZTF is deeper,
has better-quality photometry, and has been obtained in two passbands.
However, toward the faint end of the ZTF, the ZTF sample purity will also
decrease.

Table 3: All and newly detected ZTF variables. Type | Total | New (fraction)
---|---|---
Cep-I | 1262 | 565 (44.8%)
Cep-II | 358 | 154 (43.0%)
RRab | 32,518 | 3034 ( 9.3%)
RRc | 13,875 | 2178 (15.7%)
$\delta$ Scuti | 16,709 | 15,396 (92.1%)
EW | 36,9707 | 306,375 (82.9%)
EA | 49,943 | 40,201 (80.5%)
Mira | 11,879 | 4,997 (42.1%)
SR | 119,261 | 97,737 (82.0%)
RS CVn | 81,393 | 70,957 (87.2%)
BY Dra | 84,697 | 80,108 (94.6%)
Total number | 781,602 | 621,702 (79.5%)
Table 4:  Variable purity comparison between ZTF and other catalogs. Type | ATLAS | ASAS-SN | Catalina | WISE | Total sample
---|---|---|---|---|---
RRab | 99.8% | 98.6% | 98.8% | 96.2% | 99.0%
RRc | 89.0% | 90.4% | 93.9% |  | 91.2%
EW | 95.1% (99.8%)bbThe purities in brackets only account for contamination by non-binary variables. | 95.1% (97.1%) | 97.1% (97.7%) | 96.3% (98.4%) | 95.6% (98.7%)
EA | 82.0% (99.8%) | 93.6% (96.0%) | 70.5% (98.3%) | 95.6% (98.5%) | 84.3% (99.2%)
Cep-I | 98.5% | 84.1% |  | 96.1% | 91.7%
Cep-II | 96.9% | 82.8% | 94.5% | 88.1% | 90.1%
SR | 97.7% | 98.0% |  |  | 97.9%
Mira | 88.5% | 53.3% |  |  | 69.3%
$\delta$ Scuti | 83.5% | 94.6% | 85.5% |  | 87.6%
RS CVn |  | 75.2% |  |  | 75.2%
BY Dra |  | 93.7% |  |  | 93.7%

###  5.3 Period Accuracy

ZTF DR 2 covers a timespan of 470 days, so the periods in our catalog are most
accurate for $P\lesssim 100$ days. This limitation causes problems for the
periods of SRs and Miras. Period problems pertaining to short-period variables
are usually caused by the limited sampling cadence, e.g., a daily cadence is a
major problem for ground-based time-domain surveys. Except for the WISE
catalog, the other three previously published catalogs and the ZTF catalog are
all affected by this problem. This period problem is not easily eliminated by
increasing the sampling using the same cadence, but it can be reduced based on
multi-passband information and information about the physical properties of
the variables. Figure 7 shows a comparison of the periods determined from the
$g$ and $r$ LCs. One of the two periods is the real period ($P_{\rm r}$),
while the other is the aliased period ($P_{\rm a}$). Both periods are found to
mainly follow three relations: $1/P_{\rm a}=|1/P_{c}\pm 1/P_{\rm
r}|,P_{c}=0.5,1,2$ (corresponding to the red dotted, dashed, and solid lines
in Figure 7, respectively). We can justify which period is more reliable,
e.g., the bright variables are usually LPVs ($M_{W_{gr}}<-2$ mag), or
variables with $\phi_{21}<4$ are likely associated with RRab rather than RRc
periods. However, for eclipsing binaries and $\delta$ Scuti stars, there is no
significant feature to validate the real period. For example, assuming that
$\delta$ Scuti stars have real and aliased periods of 0.08 and 0.074 days,
respectively, it is hard to justify which period is real, even with access to
their luminosities and $\phi_{21}$ phases. Therefore, we visually compared the
folded LCs pertaining to the two periods individually for those variables.
This process can determine the correct period for most variables if the LC
quality is good.

Our comparison of the four catalogs shows that our period accuracy is as high
as 99%, except for LPVs and $\delta$ Scuti stars. The inconsistency rates for
RRab, RRc, EW-type eclipsing binaries, and for classical Cepheids, range from
0% to 2% (Table 5). This is a common accuracy for large variable-star
catalogs. The inconsistency rate of $\delta$ Scuti stars is a little higher,
since their periods are much shorter than the sampling cadence. Complementary
short-cadence observations can resolve this aliased period problem. The full-
versus half-period problem occurred for a few EA-type eclipsing binaries with
insignificant secondary eclipses. If we do not consider the full- versus half-
period problem, the period accuracy of EA-type eclipsing binaries is as high
as those for other short-period variables. We have visually checked the LCs of
EA-type eclipsing binaries to ensure that they all show the full periods. As
to LPVs, around 11% of Miras and 32% of SRs have period differences larger
than 20% by comparison with the ASAS-SN and Catalina catalogs. The precision
of Mira periods is limited by the observational timespan. SRs usually contains
several short- and long-term periods, so the low inconsistency derived here is
not surprising. The accuracy of the LPV periods will be refined with the
availability of future DRs.

Figure 7:  Comparison of periods determined from the $g$\- and $r$-band LCs. The red lines are the predicted relations between real and aliased periods. Table 5:  Period comparison between ZTF and other catalogs. Type | ATLAS | ASAS-SN | Catalina | WISE | Total sample
---|---|---|---|---|---
RRab | 99.7% | 99.7% | 98.0% | 99.6% | 99.6%
RRc | 98.9% | 98.4% | 97.8% |  | 98.3%
EW | 99.8% | 98.6% | 95.3% | 99.9% | 98.7%
EA | 92.9% (98.7%)ccThe values in brackets do not take into account the full- versus half-period problem. | 93.8% (99.4%) | 95.1% (98.1%) | 93.1% (99.7%) | 93.2% (98.9%)
Cep-I | 100% | 97.9% |  | 100% | 99.2%
SR |  | 68.3% | 93.5% |  | 68.4%
Mira |  | 88.2% | 94.6% |  | 88.7%
$\delta$ Scuti | 98.8% | 94.8% | 91.5% |  | 96.6%

###  5.4 Completeness

We estimate the completeness of our catalog by comparison with previously
published catalogs based on known variables in the ZTF magnitude range deemed
reliable ($r>12.5$ mag), as well as its spatial coverage
($\rm{Dec.}>-13^{\circ}$). The completeness for the full sample is 71.1%.
Since sampling of the ZTF is not homogeneous across the entire northern sky,
the completeness varies with spatial position (see Figure 6). For regions at
high declinations ($\rm{Dec.}>40^{\circ}$), the completeness can be as high as
76%. Some 24% of our variables are not classified because of insufficient
sampling. The ZTF DR2 contains $\sim 150$ detections for each object, a
smaller number than the equivalent numbers of detections in the ASAS-SN and
Catalina catalogs. As the declination decreases, the completeness gradually
decreases to 50%. For different magnitude ranges, the completeness is always
around 71.1%, which means that the completeness is not affected by the
objects’ apparent magnitudes or photometric uncertainties. With future DRs,
ZTF will be able to detect and classify more than 1 million periodic variables
in the northern sky, assuming current completeness levels. Gaia may be able to
help classify 3 million periodic variables across the full sky, considering
their enhanced density in the Galactic bulge and the Magellanic Clouds in the
southern sky.

Except for spatial sampling, time sampling issues may also cause
incompleteness. Since our catalog only covers a timespan of 470 days, its
completeness for LPVs is not as high as that for short-period variables. We
compare the LPVs in the ASAS-SN catalog with our sample to estimate their
completeness, since ASAS-SN covers both a longer timespan and a larger number
of LPVs. Globally, 42% of LPVs across the entire sky were rediscovered in our
catalog. This fraction is some 30% lower than that for our full sample of
variables.

###  5.5 Cepheids and RR Lyrae

In addition to our comparisons with the four variables catalogs, we also
evaluated the periods and completeness levels of the Cepheids and RR Lyrae in
our catalog by comparison with the OGLE and Gaia catalogs, both of which have
similar depths as ZTF. OGLE IV recently released their variables catalog of
Cepheids and RR Lyrae in the Galactic disk (Skowron et al., 2019b; Soszyński
et al., 2019). Although the overlap region between OGLE and ZTF is limited, a
comparison is still feasible. Gaia DR2 contains about 140,000 RR Lyrae across
the full sky, as well as 500 classical Cepheids in the Milky Way.

As for the classical Cepheids, we compared our catalog with the 2682 high-
probability Cepheids collected by the OGLE team (Pietrukowicz et al., 2013;
Udalski et al., 2018). The number of objects in common is 691, and the
completeness of our ZTF Cepheids is 63.3% given that 1092 of the full sample
of Cepheids are located in the ZTF’s detection region. This relatively low
completeness level compared with the global completeness of our variables
catalog suggests that the completeness of variable types with intermediate
periods is lower than that of short-period variables. Only two of the 691
confirmed objects have inconsistent periods. Having double checked these
periods, we conclude that our periods are more reliable.

OGLE IV contains 10,000 RR Lyrae in the Galactic disk, and two of their fields
overlap with the field covered by the ZTF catalog. We found that 1327 of the
1854 (71.6%) OGLE RR Lyrae in these two fields have been rediscovered in our
catalog; 99.5% of these 1327 RR Lyrae have consistent periods. Both the period
accuracy and completeness are in agreement with the equivalent values
pertaining to the full catalog. For RR Lyrae contained in Gaia DR2, 39,301 are
located in the ZTF’s detection region. Some 80.1% (31483) of Gaia RR Lyrae
have been rediscovered in our catalog. This completeness level is 9% higher
than the catalog’s global completeness of 71.1%, which is mainly driven by the
incompleteness levels characteristic of Gaia RR Lyrae. About 400 of the Gaia
RR Lyrae are contaminated by EW-type eclipsing binaries and $\delta$ Scuti
stars.

##  6 Discussion

In this section, we discuss for each type of variable their LCs, PLRs, and
their applications as potential distance tracers.

###  6.1 Cepheids

Figure 8:  Milky Way disk in 3D (left) based on $\sim$3300 Cepheids and their
projection onto the $(X,Y)$ plane (right). Red: new Cepheids in the ZTF
catalog; green: other Cepheids in ZTF catalog; gray: Cepheids from OGLE, WISE,
ATLAS, ASAS-SN, and other catalogs. The grid in the left-hand panel represents
the warp model of Chen et al. (2019). The triangle in the right-hand panel
denotes the position of our Sun.

Classical Cepheids are the most important primary distance tracers. They are
widely used for estimating distances to galaxies in the Local Volume and
measuring the near-field Hubble constant (Riess et al., 2019). Because of
their young age, classical Cepheids are also the best stellar tracer of the
Galactic disk. Compared with extragalactic Cepheids, Cepheids in our Milky Way
are rare. Before 2018, only around 1000 Milky Way Cepheids had been detected
(Pojmanski et al., 2005; Berdnikov, 2008). These Cepheids were distributed
within 3 kpc of the Sun and showed few features of our Milky Way’s structure.
In 2018, several times the number of previously known classical Cepheids were
found. The WISE catalog of variables listed about 1000 Cepheids that were
homogeneously distributed across the Galactic plane, except for the Galactic
Center direction. OGLE detected another one thousand Cepheids in the southern
disk. ALTAS, ASAS-SN, and Gaia also detected several hundred Cepheids.

This rapid increase in the number of Cepheids motivated studies of the Milky
Way’s disk. Samples of 1339–2500 carefully selected disk Cepheids revealed the
first intuitive 3D map of our Galaxy’s stellar disk (Chen et al., 2019;
Skowron et al., 2019a). The outer disk was found to be warped, and the warp is
now known to be precessing (Chen et al., 2019). Recently, 640 new Cepheids in
the southern disk affected by heavy extinction were discovered by the VVV
survey (Dékány et al., 2019). In our ZTF catalog, we found an additional 565
new classical Cepheids, thus further enlarging the sample in the northern warp
and around the edge of the disk. Figure 8 shows the distribution of 3300 well-
determined disk classical Cepheids ( satisfying the Gaia parallax and LC
criteria), which agree with the warp model of Chen et al. (2019) to within
1–2$\sigma$. This new sample of Cepheids is sufficient to trace the detailed
morphology of the disk beyond Galactocentric distances of 15 kpc. In the
$(X,Y)$ plane, the Cepheid distribution is not homogeneous, but it shows
bubble and filament features. A significant bubble is found at a distance of
about 6 kpc from the Sun in the Galactic Anticenter direction. With a detailed
study of these features, we could potentially infer the dynamical evolution
history of the Galactic disk.

In the future, new Cepheids are expected to be found in the Galactic Center
direction (Matsunaga et al., 2011; Dékány et al., 2015). In addition, low-
amplitude or long-period Cepheids will be classified once more accurate
parallaxes become available. Our larger sample of Cepheids will also be
helpful in determining the zero points and slopes of their PLRs (Breuval et
al., 2019), particularly of infrared PLRs (Chen et al., 2017; Wang et al.,
2018). In addition to Cepheids in the Milky Way, we also found 21 Cepheids
(one new Cepheid) associated with M31 (Kodric et al., 2018), one Cepheid in
M33 (Hartman et al., 2006), and one in IC 1613 (Udalski et al., 2001). ZTF DR2
contains only about 50 detections in the M31 direction. With increased
sampling in future DRs, several hundred Cepheids will likely be found in M31.

Figure 9:  Example LCs for (a) fundamental-mode classical Cepheids, (b) first-
overtone classical Cepheids, and (c) Type II Cepheids. The blue dots and red
plus signs are LCs in the $g$ and $r$ bands, respectively. From top to bottom,
phase differences $\phi_{21}$ increase. The mean magnitudes are fixed at
$0.6i+0.2$, $0.3i$, and $0.8i$ ($i=0,1,...,9$) in the $g$ band and $0.6i$,
$0.3i$, and $0.8i$ in the $r$ band, for fundamental-mode classical Cepheids,
first-overtone classical Cepheids, and Type II Cepheids, respectively.

Cepheid LCs in $g$ and $r$ are shown in Figure 9. From their LCs, the majority
of classical Cepheids can be divided into fundamental-mode (Figure 9a) and
first-overtone Cepheids (Figure 9b). First-overtone Cepheids have lower
amplitudes and shorter periods than the corresponding fundamental-mode
Cepheids. At a given period, first-overtone Cepheids have slightly brighter
luminosities. This magnitude difference is around 0.5 mag in the LMC (Inno et
al., 2016), Small Magellanic Cloud (SMC Ripepi et al., 2017), and M31 (Kodric
et al., 2018), while the situation in the Milky Way is less clear (Ripepi et
al., 2019). With an increasing number of first-overtone Cepheids and better
parallaxes, the PLRs of Milky Way Cepheids will be better constrained.

Compared with classical Cepheids, the LC shapes of Type II Cepheids are more
highly variable (Figure 9c). Type II Cepheids are older, fainter, and follow
PLRs characterized by a somewhat larger scatter (Groenewegen & Jurkovic, 2017;
Bhardwaj et al., 2017). Type II Cepheids are mostly located away from the disk
and associated with the Galactic bulge and halo. As distance indicators, Type
II Cepheids can be used to study the structure of the bulge (Braga et al.,
2018a) and measure the distances to globular clusters and dwarf galaxies. We
found 154 new Type II Cepheids, which represents a significant increase and
will be useful to anchor their PLRs and explore old Galactic structures.

###  6.2 RR Lyrae

Figure 10:  Example LCs for (a) RRab and (b) RRc Lyrae. The blue dots and red
plus signs are LCs in the $g$ and $r$ bands, respectively. From top to bottom,
phase differences $\phi_{21}$ decrease.

RR Lyrae are the most numerous distance indicators in old environments. Their
PLRs or PL–metallicity relations can be as accurate as those for the classical
Cepheids (Catelan, 2009, and references therein). In the Milky Way halo, RR
Lyrae are used to trace dwarf galaxies and distant substructures (Drake et
al., 2013; Sesar et al., 2017; Medina et al., 2018; Martínez-Vázquez et al.,
2019; Erkal et al., 2019). RR Lyrae can be used to reveal structures out to
distances of some 120 kpc. Better identification of RR Lyrae in the Dark
Energy Survey can help to reach distances greater than 200 kpc (Stringer et
al., 2019). RR Lyrae at moderate distances are usually used to determine
accurate distances to globular clusters (Braga et al., 2018b; Bono et al.,
2019; Palma et al., 2019). With a more complete sample, RR Lyrae are perhaps
the best tracers to trace the Milky Way halo’s shape (Iorio et al., 2018;
Iorio & Belokurov, 2019). In the Galactic inner regions, featuring numerous RR
Lyrae, the shape of the bulge can also be constrained (Pietrukowicz et al.,
2015; Gran et al., 2016). In the ZTF catalog, we detected more than 5,000 new
RR Lyrae out to distances of 125 kpc. We detected several RR Lyrae associated
with the Draco dwarf galaxy at a distance of around 76 kpc (Bonanos et al.,
2004). Unlike previous RR Lyrae samples, the new sample may reveal unknown
structures at low Galactic latitudes ($|b|<20^{\circ}$).

Studies of the physical properties of RR Lyrae will also benefit from large
and complete samples. In recent years, many metal-rich RR Lyrae have been
found (Chadid et al., 2017; Sneden et al., 2018), which challenge the
traditional theory explaining these metal-poor and old variables. Both the
WISE and ZTF catalogs focus on the Galactic disk; RR Lyrae in the disk are,
with high probability, metal-rich. With LAMOST or SDSS-SEGUE spectra, the
physical properties of a large sample of RR Lyrae can be investigated. In
Figure 3, the distribution of RR Lyrae in the period–amplitude diagram is more
scattered than dichotomous, which suggests that the well-known Oosterhoff
dichotomy problem of RR Lyrae may just be due to a lack of intermediate-
metallicity RR Lyrae (Fabrizio et al., 2019).

Similarly to classical Cepheids, most RR Lyrae can be divided into
fundamental-mode (RRab) and first-overtone pulsators (RRc). Example LCs of
RRab and RRc Lyrae are shown in Figure 10. The amplitudes of RRab Lyrae are
larger, decreasing as the LCs become symmetric (from top to bottom in the
left-hand panel of Figure 10). This trend is the result of the
period–amplitude relation and is only significant for RRab LCs. The PLRs or
PL–metallicity relations of Milky Way RRab Lyrae can be studied with Gaia DR2
parallaxes (Muraveva et al., 2018; Layden et al., 2019; Neeley et al., 2019).
The accuracy of these PLRs is mainly limited by parallax uncertainties. RRc
Lyrae also follow PLRs, which are about 0.5 mag brighter than for RRab Lyrae
at a given period. In Figure 5, RRc Lyrae seem to exhibit more scatter in
their luminosities than RRab Lyrae. If we can better constrain the
luminosities of RRc Lyrae, they may be potentially viable distance tracers. A
number of RRab and RRc Lyrae exhibit multiple periods, and about one-third of
the RRab Lyrae show period modulation. These special RR Lyrae will be analyzed
with enhanced data from future DRs.

###  6.3 Eclipsing binaries

Figure 11:  Example LCs for (a) EA- and (b) EW-type eclipsing binaries. The
blue dots and red plus signs are LCs in the $g$ and $r$ bands, respectively.
From top to bottom, the variation amplitudes increase.

The three types of eclipsing binaries are distinguished by their evolutionary
stage. From EA- to EW-type eclipsing binaries, the period distribution
gradually shifts to shorter periods. Statistically, EW-type eclipsing binaries
tend to be older, with a peak in their age distribution around 4 Gyr. EA-type
eclipsing binaries are young, with a peak age around 1 Gyr. The basic
evolution of eclipsing binaries has recently become clear. Nuclear evolution
and angular momentum loss are the main mechanisms (Hilditch et al., 1988;
Nelson & Eggleton, 2001; Yakut & Eggleton, 2005; Stepien, 2006; Jiang, 2020).
Both mechanisms run in parallel. Driven by angular momentum loss, the period
decreases and both components become closer. When one component evolves to or
away from the terminal main sequence, its radius increases. Until one
component fills its Roche lobe, an EA-type eclipsing binary evolves to become
an EB-type eclipsing binary. Then, with material and energy transferring from
the primary to the secondary component, combined with possible mass reversal,
the secondary component also fills its Roche lobe. The resulting system is an
EW-type eclipsing binary. In Figure 1, the significant cut-off period of EW-
type eclipsing binaries around 0.19 days is driven by the nuclear evolution
timescale. The progenitor of the primary component of this 0.19-day EW-type
eclipsing binary is a low-mass star with a nuclear evolution timescale very
close to the age of the Galaxy.

Since their binary fraction is higher, eclipsing binaries account for a large
proportion of variables in magnitude-limited samples. The fraction of EW-type
eclipsing binaries was estimated at around 0.1% in the field (Rucinski, 2006).
It could be as high as 0.4% in a $\sim 4$ Gyr environment (Chen et al.,
2016b). This fraction does not correct for the low-inclination problem and
only counts EW-type eclipsing binaries seen at higher inclinations
($i>40$–$60^{\circ}$). In our catalog, the fraction of EW-type eclipsing
binaries identified is 0.07%, which is a lower limit. If we consider the
number of missing EW-type eclipsing binaries owing to insufficient sampling
(see Section 5.4), that fraction can be as high as 0.1%.

Detached eclipsing binaries evolve on timescales as long as or even longer
than those of EW-type eclipsing binaries. However, depending on the spatial
separation and radius ratio of both components, we can only observe eclipses
of detached eclipsing binaries with $i\sim=90^{\circ}$. Consequently, the
fraction of EA-type eclipsing binaries is lower than that of EW types. EB-type
eclipsing binaries have the shortest evolution timescale and account for the
lowest fraction among the three types of eclipsing binaries.

Figure 11 shows LCs of eclipsing binaries. Their shapes and amplitudes in both
bands are almost the same and differ from those of pulsating stars. In the
left-hand panel, we can identify when the eclipses start in the LCs of EA-type
eclipsing binaries. In the right-hand panel, the eclipse onset in EW(EB)-type
eclipsing binary LCs is hard to identify. In the period range of 0.25–0.56
days, EW-type eclipsing binaries follow tight infrared PLRs (Chen et al.,
2018a) and can be potential distance indicators. The viability of these PLRs
has been validated by eclipsing binaries from the ASAS-SN catalog (Jayasinghe
et al., 2019a).

In Figure 5, about 90,000 EW-type eclipsing binaries with $<20$% Gaia parallax
uncertainties are found to also follow a PLR. In theory, the PLR of EW-type
eclipsing binaries is driven by both Roche lobe theory and the mass–luminosity
relation. Based on this PLR and a sample of several 100,000 EW-type eclipsing
binaries, a better study of the detailed structure of the Milky Way will be
possible. Except for distance determination, EW-type eclipsing binaries can
also provide age information based on their periods. In addition, kinematic
properties derived from Gaia data can provide constraints on the binaries’
ages (Hwang & Zakamska, 2019). We expect that the eight-dimensional parameter
space pertaining to EW-type eclipsing binaries can be established and used to
constrain the Milky Way’s evolution. Without strict constraints on the
secondary components, the PLR of EB-type eclipsing binaries exhibits more
significant scatter. Less evolved, EA-type eclipsing binaries are the best
objects to obtain absolute parameters from the orbital equations. By analyzing
both their LCs and radial velocity curves, late-type EA-type eclipsing
binaries can yield distances with 1% accuracy (Pietrzyński et al., 2019).

Figure 12:  Example LCs for (a) SRs and (b) Miras. The blue dots and red plus
signs are LCs in the $g$ and $r$ bands, respectively. From top to bottom, the
variation amplitudes increase. The mean magnitudes are fixed at $1.0i$ and
$4.0i$ ($i=0,1,...,9$) in the $g$ and $r$ bands, for SRs and Miras,
respectively.

###  6.4 SRs and Miras

SRs and Miras are most likely red giants, red supergiants, or AGB stars
exhibiting long-period luminosity variations. They are important distance
tracers in the context of the cosmological distance scale since they are
brighter than classical Cepheids in infrared bands. However, LPVs follow
multi-sequence PLRs, and each PLR sequence exhibits larger scatter (Lebzelter
et al., 2019) than the classical Cepheid PLRs. To become a reliable distance
indicator, LPV properties must be studied in more detail, particularly in the
Milky Way and M31. The PLRs of red supergiants belonging to the LMC, SMC, M31,
and M33 have been well-studied, and their 1$\sigma$ accuracy is 10–15% in
infrared bands (Yang & Jiang, 2011, 2012; Ren et al., 2019). However, in the
Milky Way, the number of red supergiants with reliable distances is too small
to establish reliable PLRs (Chatys et al., 2019).

Oxygen-rich Miras also follow tight PLRs. In infrared bands, the 1$\sigma$
scatter about their PLRs is around 10% (Yuan et al., 2017, 2018). Compared
with red supergiants, Miras are 0–4 mag fainter but several times more
numerous. All of these LPVs are potential tracers to measure the Hubble
parameter (Huang et al., 2019). In the Milky Way, LPVs are young and
intermediate-age structure tracers, complementary to Cepheids and RR Lyrae.
Similarly to eclipsing binaries, the ages of each LPV subtype can be inferred
from their periods (Mowlavi et al., 2018). As such, LPVs are also tracers that
can constrain the evolution history of our Galaxy.

LCs of SRs and Miras are shown in the left- and right-hand panels of Figure
12, respectively. In the ZTF, several Miras are found with $r$-band amplitudes
exceeding 6 mag. Since the total timespan of the ZTF DR2 is 470 days, many
Miras only have LCs available for at most one period. We detected about
130,000 LPVs, with the majority belonging to our Galaxy or to nearby dwarf
galaxies. To better understand the physics of these LPVs, we also show them in
color–magnitude and color–color diagrams (Figure 13a,b). WISE $W_{1}$,
$W_{1}-W_{2}$, and $W_{2}-W_{3}$ magnitudes with good photometric signal-to-
noise ratios (${\rm SNR}_{W1}>5,{\rm SNR}_{W2}>5,{\rm SNR}_{W3}>10$) and
quality were adopted to indicate LPV magnitudes and colors. This choice
reduces the effects of the significant extinction in the Galactic plane. By
comparison with massive stars in the LMC in color–absolute magnitude space
(Yang et al., 2018), we find that SRs (blue dots in Figure 13) are composed of
red giants, red supergiants, oxygen- and carbon-rich AGB stars, as well as
extremely carbon-rich AGB stars. Miras (red dots) contain a few oxygen-rich, a
few extremely carbon-rich, and many carbon-rich AGB stars.

AGB populations are characterized by an absolute magnitude of about $-8$ mag
in the mid-infrared, and the bright end of the red supergiant population is
around $-12$ mag. Therefore, data points with magnitudes brighter than $-12$
mag in the left-hand panel ($W_{1}-W_{2}<0$) or brighter than $-9$ mag in the
right-hand panel ($W_{1}-W_{2}>0$) are stars that may have larger parallax
uncertainties. In Figure 13c, the amplitudes grow monotonously as the periods
increase. In particular, some sequences are found in the SR distribution,
referred to as the A, B, and C′ (third, second, and first overtone) sequences
(Trabucchi et al., 2017). Miras and some SRs form sequence C, which is the
location of high-amplitude fundamental pulsators. In Figure 13d, SRs and Miras
are separated by a boundary, with Miras being high-amplitude and red. The
detailed subtype classification of these LPVs is not further discussed in this
paper as it requires access to better periods and parallaxes.

Figure 13:  LPVs in the ZTF catalog. (a) Color–absolute magnitude diagram. (b)
Color–color diagram. (c) Period–amplitude diagram. (d) Color–amplitude
diagram. SRs and Miras are shown as red and blue dots, respectively. Figure
14:  Example LCs for (a) HADs and (b) LADs. The blue dots and red plus signs
are LCs in the $g$ and $r$ bands, respectively. From top to bottom, phase
differences $\phi_{21}$ increase. The mean magnitudes are fixed at $0.4i$ and
$0.2i$ ($i=0,2,...,9$) in the $g$ and $r$ bands, for HADs and LADs,
respectively.

###  6.5 $\delta$ Scuti stars

Among the variables in the ZTF, $\delta$ Scuti stars have the shortest
periods. $\delta$ Scuti stars are main-sequence pulsators located in the
instability strip, so they follow a PLR similar to RR Lyrae and Cepheids.
Different from RR Lyrae and Cepheids, a fraction of $\delta$ Scuti stars is
thought to exhibit multiple periods. The PLRs of $\delta$ Scuti stars are not
well studied, even for HADs, because of their fainter magnitudes and limited
sampling in the past. In the LMC, the magnitude range of $\delta$ Scuti stars
ranges from 20 to 22 mag in the $R$ band (Garg et al., 2010), which is close
to the limiting magnitude of many time-domain surveys. PLRs of $\delta$ Scuti
stars are therefore best studied in the Milky Way. Ziaali et al. (2019) tried
to establish a $V$-band PLR based on 1100 $\delta$ Scuti stars in the Kepler
field with good Gaia DR2 parallaxes; however, their PLR is not as tight as the
equivalent PLRs pertaining to other tracers. Recently, Jayasinghe et al.
(2020b) derived multi-band PLRs for fundamental and overtone pulsators, based
on $\sim$ 8400 $\delta$ Scuti stars from ASAS-SN, and found a 1$\sigma$
scatter around 10%.

Our ZTF sample is several times larger and more homogeneously distributed
across the northern sky, which paves the way to improve the $\delta$ Scuti
PLRs. In Figure 5, we show that the Wesenheit PLR of $\delta$ Scuti stars is
tight, with few outliers. A simple estimation shows that the 1$\sigma$ scatter
of the HADs is around 10%. This means that the $\delta$ Scuti PLR is tighter
in infrared bands. This is similar to the situation for other variables that
follow PLRs. A forthcoming paper will investigate the multi-band PLRs of a
large sample of $\delta$ Scuti stars. As distance indicators, $\delta$ Scuti
stars cover a wide age range (according to their masses). Old $\delta$ Scuti
(SX Phoenicis) stars can be used to measure the distances to globular clusters
and nearby dwarf galaxies (McNamara, 2011). Young or intermediate-age $\delta$
Scuti stars are found in open clusters (Wang et al., 2015), and they are
potential tracers for studies of the Milky Way disk’s structure.

In Figure 14, HADs exhibit tooth-saw-shaped LCs similar to RR Lyrae and
Cepheids (left-hand panel), while LADs show various types of LCs. Based on
their LCs (Figures 3 and 4) alone, it is difficult to separate fundamental-
mode and first-overtone $\delta$ Scuti stars. However, similarly to RR Lyrae
and classical Cepheids, first-overtone $\delta$ Scuti stars are generally
brighter than fundamental-mode $\delta$ Scuti stars. The boundary between HADs
and LADs is not clear. In the period–luminosity diagram (Figure 5), LADs also
follow a PLR and are a little brighter than HADs. All these features imply
that HADs and LADs are first-overtone and fundamental-mode $\delta$ Scuti
stars, respectively, while their boundary is rather obscure. More properties
of $\delta$ Scuti stars will be revealed by high-order frequency analysis with
increased sampling in the next ZTF DR.

Figure 15:  Example LCs for (a) RS CVn and (b) BY Dra stars. The blue dots and
red plus signs are LCs in the $g$ and $r$ bands, respectively. From top to
bottom, the variation amplitudes increase. The mean magnitudes are fixed at
$0.3i$ ($i=0,1,...,9$) in the $g,r$ bands for both RS CVn and BY Dra stars.

###  6.6 RS CVn and BY Dra

Both RS CVn and BY Dra exhibit periodic but non-characteristic LCs (Figure
15). Their amplitudes or luminosities can be used to classify them. The LCs of
RS CVn and BY Dra in the two different bands are similar. However, the overall
shapes of their LCs are no longer symmetric about phase $\phi=0.5$ or
$\phi=1.5$ (in most cases). A number of these variables have been detected in
the ASAS-SN Catalog of the Southern Hemisphere (Jayasinghe et al., 2020a),
where the authors treat them as rotational stars. Iwanek et al. (2019) also
studied 12,660 spotted variable stars along the line of sight to and inside
the Galactic bulge, based on OGLE data. Most of their variables are giant
stars. In this paper, we divided these variables based on their different
luminosities. RS CVn are F–G-type eclipsing binaries with strong chromospheric
activity. Their spectra show Ca ii H and K emission (Hall, 1976). The LCs of
RS CVn are a combination of eclipses and spot modulation. RS CVn have thus far
only been studied based on small sample sizes (Zhang et al., 2014). Therefore,
our catalog of $\sim 80,000$ RS CVn is the best sample to investigate these
objects’ properties statistically. A new discovery already resulting from this
enlarged sample is that RS CVn have a wider period distribution than
previously inferred (Hall & Kreiner, 1980; Dempsey et al., 1993; Donati et
al., 1997). In Figure 2, for all periods, eclipsing binaries have the highest
probability of having RS CVn features. Nevertheless, the peak of the RS CVn
$\log P$ distribution occurs around a few days, which is much longer than that
for eclipsing binaries (0.3–0.4 days).

BY Dra stars are also variables featuring chromospheric activity, but they are
of K–M type and fainter than RS CVn. Several papers have studied BY Dra and
some RS CVn stars based on samples of a few hundred objects (Strassmeier et
al., 1993; Eker et al., 2008; Hartman et al., 2011). However, criteria to
separate the two types of variables are not well-established. In our sample,
with the help of Gaia parallaxes, BY Dra stars are predominantly distributed
in a narrow magnitude range, $2.8<M_{W_{gr}}<4.6$ mag (Figure 5). The period
range of BY Dra stars is narrower than that of RS CVn variables. As for the
overall distributions of periods and amplitudes, both types of variables
exhibit similar features. A better study of RS CVn and BY Dra variables will
be achieved with a combination of LCs and spectra.

Table 6: ZTF Suspected Variables Catalog. ID | R.A. (J2000) | Decl. (J2000) | $\langle g\rangle$ | $\langle r\rangle$ |  Periodg |  Periodr |  Ampg |  Ampr |  Numg |  Numr |  $\log$ (FAP)g
---|---|---|---|---|---|---|---|---|---|---|---
| ∘ | ∘ | mag | mag | days | days | mag | mag |  |  |
ZTFJ000000.06+583327.2 | 0.00026 | 58.55756 | 12.974 | 12.324 | 1.00157 | 0.05415 | 0.059 | 0.057 | 118 | 137 |  $-$5.865
ZTFJ000000.09+632559.3 | 0.00041 | 63.43315 | 15.137 | 14.451 | 339.5882 | 344.55948 | 0.075 | 0.038 | 124 | 146 |  $-$6.685
ZTFJ000000.14+552150.6 | 0.00060 | 55.36408 | 17.081 | 15.919 | 0.49772 | 0.09674 | 0.049 | 0.024 | 121 | 143 |  $-$5.067
ZTFJ000000.15+540322.4 | 0.00066 | 54.05624 | 18.144 | 16.776 | 0.49768 | 0.10488 | 0.087 | 0.023 | 122 | 144 |  $-$5.566
ZTFJ000000.16+621059.7 | 0.00069 | 62.18326 | 16.454 | 15.585 | 0.4979 | 0.08946 | 0.046 | 0.018 | 125 | 146 |  $-$5.494
ZTFJ000000.17+592103.3 | 0.00073 | 59.35093 | 17.342 | 16.391 | 0.99392 | 0.13364 | 0.055 | 0.026 | 124 | 147 |  $-$4.545
ZTFJ000000.25+605628.0 | 0.00108 | 60.94112 | 17.745 | 15.993 | 0.99385 | 0.44865 | 0.127 | 0.019 | 125 | 146 |  $-$3.811
ZTFJ000000.27+665503.0 | 0.00116 | 66.91750 | 16.711 | 15.505 | 251.12632 | 0.05878 | 0.069 | 0.021 | 169 | 186 |  $-$6.459
ZTFJ000000.28+583810.1 | 0.00119 | 58.63614 | 15.121 | 14.359 | 295.46337 | 0.33318 | 0.068 | 0.059 | 118 | 138 |  $-$5.534

##  7 Conclusions

ZTF DR2 contains about 0.7 billion objects with more than 20 exposures each.
As such, ZTF DR2 represents a highly suitable database for the detection and
exploration of new variable stars. In this paper, we have adopted the
Lomb–Scargle periodogram approach to search for periodic variables. We
obtained a sample of 1.4 million candidates with $\rm{FAP}<0.001$. Following
exclusion of candidates with aliased periods and poor-quality LCs, $\sim 1.0$
million candidates remained for classification. Classification was done using
the DBSCAN method based on the distributions of the objects’ periods, LC
parameters, and luminosities. We classified 781,602 periodic variables into
$\delta$ Scuti, EW- and EA-type eclipsing binaries, RRab and RRc Lyrae,
classical Cepheids, Type II Cepheids, SRs, and Miras. After cross-matching
with known variables, 621,702 of the 781,602 objects turned out to be newly
detected variables. This high new detection rate highlights both the high
quality of the ZTF photometry and the improvement of our classification
method.

Checks of the periods and types contained in our catalog show that they are
better than or as good as the same parameters contained in previously
published catalogs. The sampling completeness is not homogeneous in the
northern sky due to the observation mode adopted, ranging from 76% at high
declinations to 50% in low declination regions ($\rm{Dec.}<0^{\circ}$). The
completeness of LPVs is 30% lower than that of the full sample. For each
variable type, our catalog represents a significant increase in sample size.
The new variables will aid studies of the physics of variable stars as such,
of the Milky Way’s structure and evolution, and of the cosmological distance
scale, particularly based on a combination of LCs and spectra.

Future ZTF DRs will cover both increased timespans and larger numbers of
exposures. These aspects will help improve the determination of the LPV
periods and increase the completeness of the catalog. LCs associated with
double modes or amplitude (period) modulation will be re-analyzed when
enhanced sampling becomes available. In addition to the main types of
variables discussed here, there are many cataclysmic variables, low-amplitude
pulsating stars, binaries with compact objects, and stars with exoplanets in
the ZTF database. These variables will be identified to construct a large
sample for further study based on future ZTF DRs.

We thank the anonymous referee for comments that helped us improve the paper.
This publication is based on observations obtained with the Samuel Oschin
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 AST-1440341 and a collaboration including Caltech, IPAC, the Weizmann
Institute for Science, the Oskar Klein Center at Stockholm University, the
University of Maryland, the University of Washington, Deutsches Elektronen-
Synchrotron and Humboldt University, Los Alamos National Laboratories, the
TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, and
Lawrence Berkeley National Laboratories. Operations are conducted by COO,
IPAC, and UW. We are grateful for research support from the National Natural
Science Foundation of China through grants 11903045, U1631102, and 11633005.
This work was supported by the National Key Research and Development Program
of China through grant 2017YFA0402702. We also thank Dr. Changqing Luo and Dr.
Fan Yang for providing computing time.

## References

  * Akerlof et al. (2000) Akerlof, C., Amrose, S., Balsano, R., et al. 2000, AJ, 119, 1901
  * Alcock et al. (1993) Alcock, C., Akerlof, C. W., Allsman, R. A., et al. 1993, Nature, 365, 621
  * Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Barlow, T., et al. 2019, PASP, 131, 068003
  * Berdnikov (2008) Berdnikov, L. N. 2008, VizieR Online Data Catalog, II/285
  * Bhardwaj et al. (2017) Bhardwaj, A., Rejkuba, M., Minniti, D., et al. 2017, A&A, 605, A100
  * Bonanos et al. (2004) Bonanos, A. Z., Stanek, K. Z., Szentgyorgyi, A. H., Sasselov, D. D., & Bakos, G. Á. 2004, AJ, 127, 861
  * Bono et al. (2019) Bono, G., Iannicola, G., Braga, V. F., et al. 2019, ApJ, 870, 115
  * Braga et al. (2018a) Braga, V. F., Bhardwaj, A., Contreras Ramos, R., et al. 2018a, A&A, 619, A51
  * Braga et al. (2018b) Braga, V. F., Stetson, P. B., Bono, G., et al. 2018b, AJ, 155, 137
  * Breuval et al. (2019) Breuval, L., Kervella, P., Arenou, F., et al. 2019, arXiv e-prints, arXiv:1910.04694
  * Burdge et al. (2019) Burdge, K. B., Coughlin, M. W., Fuller, J., et al. 2019, Nature, 571, 528
  * Catelan (2009) Catelan, M. 2009, Ap&SS, 320, 261
  * Chadid et al. (2017) Chadid, M., Sneden, C., & Preston, G. W. 2017, ApJ, 835, 187
  * Chambers et al. (2016) Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560
  * Chatys et al. (2019) Chatys, F. W., Bedding, T. R., Murphy, S. J., et al. 2019, MNRAS, 487, 4832
  * Chen et al. (2016a) Chen, X., de Grijs, R., & Deng, L. 2016a, ApJ, 832, 138
  * Chen et al. (2017) —. 2017, MNRAS, 464, 1119
  * Chen et al. (2018a) Chen, X., Deng, L., de Grijs, R., Wang, S., & Feng, Y. 2018a, ApJ, 859, 140
  * Chen et al. (2019) Chen, X., Wang, S., Deng, L., et al. 2019, Nature Astronomy, 3, 320
  * Chen et al. (2018b) Chen, X., Wang, S., Deng, L., de Grijs, R., & Yang, M. 2018b, ApJS, 237, 28
  * Chen et al. (2016b) Chen, X., Deng, L., de Grijs, R., et al. 2016b, AJ, 152, 129
  * Cioni et al. (2011) Cioni, M. R. L., Clementini, G., Girardi, L., et al. 2011, A&A, 527, A116
  * Clementini et al. (2019) Clementini, G., Ripepi, V., Molinaro, R., et al. 2019, A&A, 622, A60
  * Cui et al. (2012) Cui, X.-Q., Zhao, Y.-H., Chu, Y.-Q., et al. 2012, Research in Astronomy and Astrophysics, 12, 1197
  * Dékány et al. (2019) Dékány, I., Hajdu, G., Grebel, E. K., & Catelan, M. 2019, ApJ, 883, 58
  * Dékány et al. (2015) Dékány, I., Minniti, D., Majaess, D., et al. 2015, ApJ, 812, L29
  * Dempsey et al. (1993) Dempsey, R. C., Linsky, J. L., Fleming, T. A., & Schmitt, J. H. M. M. 1993, ApJS, 86, 599
  * Deng et al. (2012) Deng, L.-C., Newberg, H. J., Liu, C., et al. 2012, Research in Astronomy and Astrophysics, 12, 735
  * Donati et al. (1997) Donati, J. F., Semel, M., Carter, B. D., Rees, D. E., & Collier Cameron, A. 1997, MNRAS, 291, 658
  * Drake et al. (2013) Drake, A. J., Catelan, M., Djorgovski, S. G., et al. 2013, ApJ, 765, 154
  * Drake et al. (2014) Drake, A. J., Graham, M. J., Djorgovski, S. G., et al. 2014, ApJS, 213, 9
  * Drake et al. (2017) Drake, A. J., Djorgovski, S. G., Catelan, M., et al. 2017, MNRAS, 469, 3688
  * Eisenstein et al. (2011) Eisenstein, D. J., Weinberg, D. H., Agol, E., et al. 2011, AJ, 142, 72
  * Eker et al. (2008) Eker, Z., Ak, N. F., Bilir, S., et al. 2008, MNRAS, 389, 1722
  * Erkal et al. (2019) Erkal, D., Belokurov, V., Laporte, C. F. P., et al. 2019, MNRAS, 487, 2685
  * Fabrizio et al. (2019) Fabrizio, M., Bono, G., Braga, V. F., et al. 2019, ApJ, 882, 169
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Gaia Collaboration et al. (2019) Gaia Collaboration, Eyer, L., Rimoldini, L., et al. 2019, A&A, 623, A110
  * Garg et al. (2010) Garg, A., Cook, K. H., Nikolaev, S., et al. 2010, AJ, 140, 328
  * Graham et al. (2015) Graham, M. J., Djorgovski, S. G., Stern, D., et al. 2015, Nature, 518, 74
  * Graham et al. (2019) Graham, M. J., Kulkarni, S. R., Bellm, E. C., et al. 2019, PASP, 131, 078001
  * Gran et al. (2016) Gran, F., Minniti, D., Saito, R. K., et al. 2016, A&A, 591, A145
  * Groenewegen & Jurkovic (2017) Groenewegen, M. A. T., & Jurkovic, M. I. 2017, A&A, 604, A29
  * Hall (1976) Hall, D. S. 1976, in Astrophysics and Space Science Library, Vol. 60, IAU Colloq. 29: Multiple Periodic Variable Stars, ed. W. S. Fitch, 287
  * Hall & Kreiner (1980) Hall, D. S., & Kreiner, J. M. 1980, Acta Astron., 30, 387
  * Hartman et al. (2011) Hartman, J. D., Bakos, G. Á., Noyes, R. W., et al. 2011, AJ, 141, 166
  * Hartman et al. (2006) Hartman, J. D., Bersier, D., Stanek, K. Z., et al. 2006, MNRAS, 371, 1405
  * Heinze et al. (2018) Heinze, A. N., Tonry, J. L., Denneau, L., et al. 2018, AJ, 156, 241
  * Hilditch et al. (1988) Hilditch, R. W., King, D. J., & McFarlane, T. M. 1988, MNRAS, 231, 341
  * Huang et al. (2019) Huang, C. D., Riess, A. G., Yuan, W., et al. 2019, arXiv e-prints, arXiv:1908.10883
  * Hwang & Zakamska (2019) Hwang, H.-C., & Zakamska, N. 2019, arXiv e-prints, arXiv:1909.06375
  * Inno et al. (2016) Inno, L., Bono, G., Matsunaga, N., et al. 2016, ApJ, 832, 176
  * Iorio & Belokurov (2019) Iorio, G., & Belokurov, V. 2019, MNRAS, 482, 3868
  * Iorio et al. (2018) Iorio, G., Belokurov, V., Erkal, D., et al. 2018, MNRAS, 474, 2142
  * Iwanek et al. (2019) Iwanek, P., Soszyński, I., Skowron, J., et al. 2019, ApJ, 879, 114
  * Jayasinghe et al. (2018) Jayasinghe, T., Kochanek, C. S., Stanek, K. Z., et al. 2018, MNRAS, 477, 3145
  * Jayasinghe et al. (2019a) Jayasinghe, T., Stanek, K. Z., Kochanek, C. S., et al. 2019a, arXiv e-prints, arXiv:1911.09685
  * Jayasinghe et al. (2019b) —. 2019b, MNRAS, 486, 1907
  * Jayasinghe et al. (2020a) —. 2020a, MNRAS, 491, 13
  * Jayasinghe et al. (2020b) —. 2020b, MNRAS, 493, 4186
  * Jiang (2020) Jiang, D. 2020, MNRAS, 492, 2731
  * Kodric et al. (2018) Kodric, M., Riffeser, A., Hopp, U., et al. 2018, AJ, 156, 130
  * Layden et al. (2019) Layden, A. C., Tiede, G. P., Chaboyer, B., Bunner, C., & Smitka, M. T. 2019, AJ, 158, 105
  * Lebzelter et al. (2019) Lebzelter, T., Trabucchi, M., Mowlavi, N., et al. 2019, A&A, 631, A24
  * Lomb (1976) Lomb, N. R. 1976, Ap&SS, 39, 447
  * Lutz & Kelker (1973) Lutz, T. E., & Kelker, D. H. 1973, PASP, 85, 573
  * Mahabal et al. (2019) Mahabal, A., Rebbapragada, U., Walters, R., et al. 2019, PASP, 131, 038002
  * Martínez-Vázquez et al. (2019) Martínez-Vázquez, C. E., Vivas, A. K., Gurevich, M., et al. 2019, MNRAS, 490, 2183
  * Masci et al. (2019) Masci, F. J., Laher, R. R., Rusholme, B., et al. 2019, PASP, 131, 018003
  * Matsunaga et al. (2011) Matsunaga, N., Kawadu, T., Nishiyama, S., et al. 2011, Nature, 477, 188
  * McNamara (2011) McNamara, D. H. 2011, AJ, 142, 110
  * Medina et al. (2018) Medina, G. E., Muñoz, R. R., Vivas, A. K., et al. 2018, ApJ, 855, 43
  * Minniti et al. (2010) Minniti, D., Lucas, P. W., Emerson, J. P., et al. 2010, New A, 15, 433
  * Mowlavi et al. (2018) Mowlavi, N., Lecoeur-Taïbi, I., Lebzelter, T., et al. 2018, A&A, 618, A58
  * Muraveva et al. (2018) Muraveva, T., Delgado, H. E., Clementini, G., Sarro, L. M., & Garofalo, A. 2018, MNRAS, 481, 1195
  * Neeley et al. (2019) Neeley, J. R., Marengo, M., Freedman, W. L., et al. 2019, MNRAS, 2412
  * Nelson & Eggleton (2001) Nelson, C. A., & Eggleton, P. P. 2001, ApJ, 552, 664
  * Palaversa et al. (2013) Palaversa, L., Ivezić, Ž., Eyer, L., et al. 2013, AJ, 146, 101
  * Palma et al. (2019) Palma, T., Minniti, D., Alonso-García, J., et al. 2019, MNRAS, 487, 3140
  * Pietrukowicz et al. (2013) Pietrukowicz, P., Dziembowski, W. A., Mróz, P., et al. 2013, Acta Astron., 63, 379
  * Pietrukowicz et al. (2015) Pietrukowicz, P., Kozłowski, S., Skowron, J., et al. 2015, ApJ, 811, 113
  * Pietrzyński et al. (2019) Pietrzyński, G., Graczyk, D., Gallenne, A., et al. 2019, Nature, 567, 200
  * Pojmanski et al. (2005) Pojmanski, G., Pilecki, B., & Szczygiel, D. 2005, Acta Astron., 55, 275
  * Ren et al. (2019) Ren, Y., Jiang, B.-W., Yang, M., & Gao, J. 2019, ApJS, 241, 35
  * Riess et al. (2019) Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, ApJ, 876, 85
  * Ripepi et al. (2019) Ripepi, V., Molinaro, R., Musella, I., et al. 2019, A&A, 625, A14
  * Ripepi et al. (2017) Ripepi, V., Cioni, M.-R. L., Moretti, M. I., et al. 2017, MNRAS, 472, 808
  * Rucinski (2006) Rucinski, S. M. 2006, MNRAS, 368, 1319
  * Scargle (1982) Scargle, J. D. 1982, ApJ, 263, 835
  * Sesar et al. (2017) Sesar, B., Hernitschek, N., Dierickx, M. I. P., Fardal, M. A., & Rix, H.-W. 2017, ApJ, 844, L4
  * Skowron et al. (2019a) Skowron, D. M., Skowron, J., Mróz, P., et al. 2019a, Science, 365, 478
  * Skowron et al. (2019b) —. 2019b, Acta Astron., 69, 305
  * Sneden et al. (2018) Sneden, C., Preston, G. W., Kollmeier, J. A., et al. 2018, AJ, 155, 45
  * Soszynski et al. (2008) Soszynski, I., Poleski, R., Udalski, A., et al. 2008, Acta Astron., 58, 163
  * Soszyński et al. (2019) Soszyński, I., Udalski, A., Wrona, M., et al. 2019, Acta Astron., 69, 321
  * Stepien (2006) Stepien, K. 2006, Acta Astron., 56, 199
  * Strassmeier et al. (1993) Strassmeier, K. G., Hall, D. S., Fekel, F. C., & Scheck, M. 1993, A&AS, 100, 173
  * Stringer et al. (2019) Stringer, K. M., Long, J. P., Macri, L. M., et al. 2019, AJ, 158, 16
  * Thompson et al. (2019) Thompson, T. A., Kochanek, C. S., Stanek, K. Z., et al. 2019, Science, 366, 637
  * Trabucchi et al. (2017) Trabucchi, M., Wood, P. R., Montalbán, J., et al. 2017, ApJ, 847, 139
  * Udalski et al. (1992) Udalski, A., Szymanski, M., Kaluzny, J., Kubiak, M., & Mateo, M. 1992, Acta Astron., 42, 253
  * Udalski et al. (2015) Udalski, A., Szymański, M. K., & Szymański, G. 2015, Acta Astron., 65, 1
  * Udalski et al. (2001) Udalski, A., Wyrzykowski, L., Pietrzynski, G., et al. 2001, Acta Astron., 51, 221
  * Udalski et al. (2018) Udalski, A., Soszyński, I., Pietrukowicz, P., et al. 2018, Acta Astron., 68, 315
  * Wang et al. (2015) Wang, K., Deng, L., Zhang, X., et al. 2015, AJ, 150, 161
  * Wang & Chen (2019) Wang, S., & Chen, X. 2019, ApJ, 877, 116
  * Wang et al. (2018) Wang, S., Chen, X., de Grijs, R., & Deng, L. 2018, ApJ, 852, 78
  * Yakut & Eggleton (2005) Yakut, K., & Eggleton, P. P. 2005, ApJ, 629, 1055
  * Yang & Jiang (2011) Yang, M., & Jiang, B. W. 2011, ApJ, 727, 53
  * Yang & Jiang (2012) —. 2012, ApJ, 754, 35
  * Yang et al. (2018) Yang, M., Bonanos, A. Z., Jiang, B.-W., et al. 2018, A&A, 616, A175
  * Yuan et al. (2017) Yuan, W., Macri, L. M., He, S., et al. 2017, AJ, 154, 149
  * Yuan et al. (2018) Yuan, W., Macri, L. M., Javadi, A., Lin, Z., & Huang, J. Z. 2018, AJ, 156, 112
  * Zhang et al. (2014) Zhang, L.-Y., Pi, Q.-f., & Yang, Y.-G. 2014, MNRAS, 442, 2620
  * Ziaali et al. (2019) Ziaali, E., Bedding, T. R., Murphy, S. J., Van Reeth, T., & Hey, D. R. 2019, MNRAS, 486, 4348