# Hardware Trojan Threats to Cache Coherence in Modern 2.5D Chiplet Systems

Gino A. Chacon, Charles Williams, Johann Knechtel, Ozgur Sinanoglu, and Paul
V. Gratz G. A. Chacon, C. Williams, and P. V. Gratz are with Texas A&M
University (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>pgratz@gratz1.com).J. Knechtel and O. Sinanoglu are with New York University
Abu Dhabi (e-mail<EMAIL_ADDRESS>ozgursin@nyu.edu).

###### Abstract

As industry moves toward chiplet-based designs, the insertion of hardware
Trojans poses a significant threat to the security of these systems. These
systems rely heavily on cache coherence for coherent data communication,
making coherence an attractive target. Critically, unlike prior work, which
focuses only on malicious packet modifications, a Trojan attack that exploits
coherence can modify data in memory that was never touched and is not owned by
the chiplet which contains the Trojan. Further, the Trojan need not even be
physically between the victim and the memory controller to attack the victim’s
memory transactions. Here, we explore the fundamental attack vectors possible
in chiplet-based systems and provide an example Trojan implementation capable
of directly modifying victim data in memory. This work aims to highlight the
need for developing mechanisms that can protect and secure the coherence
scheme from these forms of attacks.

##  1 Introduction

Computing systems are moving toward 2.5D designs that source various hard IPs,
called chiplets, from multiple vendors and integrate them using an interposer.
Industry has demonstrated that 2.5D designs lower manufacturing costs,
enabling further scaling post-Moore’s Law [1]. Future 2.5D designs may
leverage standards such as Compute Express Link [2] to interoperate via a
shared memory system. While 2.5D designs provide many benefits, we show they
also increase the risk of Trojan attacks, specifically targeting the coherence
system. Here, we demonstrate several novel Trojans attacking cache coherence
in 2.5D designs. We illustrate the risks for these systems and hope to excite
the architecture community to address these risks. Though we focus on 2.5D
integrated systems, note that these attacks also apply to general cache-
coherent systems integrating closed-source or hard IP blocks from various
vendors.

_Hardware Trojans_ , or Trojans for short [3], are a threat in which an
attacker infiltrates some level of the design or fabrication process to insert
malicious circuitry into a design. Trojans can cause disastrous system
failures via confidentiality, integrity, and/or availability violations. Prior
work has shown that Trojans can leak data from memory [4], disrupt
cryptographic security features [5], and induce denial-of-service attacks [6].
As industry moves towards 2.5D designs integrating multiple vendor chiplets,
specific chiplets used in building these systems may be untrustworthy. Even if
the IP vendor is trustworthy, the manufacturing process may not be, leading to
infiltration and the insertion of Trojans.

In 2.5D designs, memory coherence is crucial to allow each component and
chiplet to maintain an up-to-date view of the system’s memory. We identify
this system as an ideal target for Trojan attacks as coherence mechanisms
control how all components communicate data updates. Existing coherence
schemes do not enforce existing virtual/physical memory permissions, thus, a
Trojan connected to the coherence scheme can directly manipulate any memory
region in the full system regardless of memory permissions or physical
location. Unlike prior packet-level NoC attacks, Trojans on cache coherence do
not need to be physically on the path between the victim and the memory
controller to launch effective attacks. Despite this attractiveness, there is
a lack of works deeply exploring coherence exploits and their defense in 2.5D
systems or otherwise.

Here, we propose several new Trojan attacks that leverage the coherence system
protocol to maliciously manipulate the victim process’ memory. We first
describe a set of new fundamental attacks that a Trojan can mount on coherence
systems, _passive reading_ , _masquerading_ , _modifying_ , and _diverting_
attacks, according to Basak _et al._ ’s taxonomy [7]. Here we assume an
attacker implements these coherence system attacks in hardware through
compromised design or manufacturing. While each of these attacks individually
violates a system’s security, we further show that adversaries can orchestrate
them together to perform complex _Forging_ attacks that modify _any_ process’
memory. These purely hardware attacks cannot be thwarted by contemporary
software defense mechanisms since all exploited coherence interactions are
transparent to software and legal within the coherence protocol. Further, no
prior work considers such attacks on coherence systems, neither in the context
of 2.5D systems with chiplets nor traditional 2D systems.

_Contributions._ This work provides new insights into how Trojans can
manipulate coherence systems to violate the security of a chiplet system. We
present a simulated example of a substantial attack that can directly
manipulate memory in an address space other than that of the compromised
chiplet. This work makes the following contributions:

  * •

We present potential attack stages available to a Trojan designers exploiting
coherence systems.

  * •

We demonstrate how to use these fundamental stages to orchestrate a complex
Trojan attack in a chiplet-based system.

  * •

We provide suggestions for future work on hardening modern chiplet designs.

##  2 Background

###  2.1 Hardware Trojans

Hardware Trojans are malicious hardware inserted by an attacker during a
device’s design or manufacturing process. Chiplet-based devices have a complex
multistage design flow that can be compromised at many levels, especially as
multiple 3rd party vendors emerge to provide chiplets from separate foundries
and design teams. This design flow makes verification much more difficult and
expensive than traditional System-on-Chip (SoC) manufacturing.

Detecting Trojans is challenging as chiplet-based systems contain multiple
complex IPs sourced from various vendors. Testing a chiplet’s functionality
may occur during the manufacturing or integration stage, which requires a
reference model or device [3]. However, if the 3rd party IP’s source is
untrustworthy, the reference model itself may incorporate the Trojan, or the
IP may camouflage the Trojan as a correct implementation. Attackers can
conceal a Trojan by only allowing it to trigger under specific conditions. For
example, the Trojan we describe in Sec. 3.2 only activates when it observes
references to a specific address. These properties make the Trojan difficult
to detect by simply testing the functionality of the chiplet. For this work,
we assume that an attacker infiltrates some stage within the design or
manufacturing process to target the system’s coherence mechanisms.

Prior art focuses on infiltrating the NoC of a target design to cause
deadlocks [6], leak information [4], or disrupt security features [5].
However, NoC-based attacks require the Trojan to directly attack NoC packets,
limiting the Trojan to only packets traversing a particular path in the NoC.
Prior attacks would not work in a 2.5D integrated system because attacking
chiplets are not on the path between victim chiplets and the memory
controller.

###  2.2 Coherence Protocols

Multi-processor systems incorporate cache coherence protocols to ensure data
coherency across processors’ private caches. The coherence system has a
complete vantage point and control over the memory system. All communication
between cores and main memory follows the coherence protocol, making it an
ideal target for a Trojan co-located with a processor’s private caches. At
this location, a Trojan can snoop on coherence messages produced by other
processors, manipulate those messages, or generate messages without incurring
exceptions and remaining invisible to software running on the system. Further,
coherence schemes do not enforce virtual/physical memory permissions, thus any
Trojan connected to the coherence scheme can directly manipulate any memory
region in the full system, without regard to memory permissions or physical
location.

We target the _MOESI Hammer_ coherence protocol, a hybrid broadcast-directory
system used in many AMD processors [8]. Though our focus is MOESI Hammer, our
attack scenarios can easily be ported to other coherence schemes. In schemes
without broadcast, the Trojan simply needs to register “S” or “X” state with
the directory to ensure that update requests on the given line are seen.

##  3 Trojans Targeting Coherence Systems

Here we propose a new set of “basic” Trojan attacks on the coherence scheme
that follows the general, not coherence specific, taxonomy for Trojan attacks
by Basak _et al._ [7]: _passive reading_ , _masquerading_ , _modifying_ , and
_diverting_. These basic attacks can adversely affect the system but cannot
provide an attacker with control the memory system. We then propose a novel,
more sophisticated and powerful ‘ _‘Forging”_ attack to modify data belonging
to another core, even on a different chiplet than that holding the Trojan.

Target system: We demonstrate our attacks on a 64-core processor with eight
chiplets, eight cores per chiplet, based on the Rocket-64 architecture [9].
Each core has private L1 and L2 caches with a unified cache controller. An NoC
connects each chiplet and four memory controllers that maintain a portion of
the global state directory. The cache controllers generate coherence messages
that are injected into the NoC as network packets.

###  3.1 Basic Trojan Attacks on the Coherence Systems

Figure 1 illustrates our proposed basic coherence attacks. We assume a Trojan
is placed at a core’s cache controller and can intercept coherence messages
from the network interface ahead of the state directory. While these attacks
target the the MOESI Hammer hybrid protocol, the basic principles of the
attacks, are consistent with any coherence protocol.

(a) Passive Reading: Trojan passively observes write traffic for other
chiplets. (1) Misses from Chiplet A cause (2) broadcast invalidations to all
chiplets; (3) Trojan snoops invalidation addresses.

(b) Masquerading: Trojan acts as another core. (1) Miss causes GETX to
directory; (2) broadcast invalidations to each chiplet; (3) Trojan blocks
local observation, replies with different core ID; (4) requesting core
proceeds, leaving local caches incoherent.

(c) Modifying: Trojan modifies a message to achieve incoherent state. (1)
Chiplet A sends GETS to directory; (2) directory forwards request to Trojan’s
core which has line in ‘E’ state. Trojan blocks GETS and (3) replies with GETX
to requestor, (4) invalidating Chiplet A’s cache entry, leaving attacker in
control of another cache’s contents.

(d) Diverting: Trojan diverts invalidation requests. (1) Chiplet A sends GETX
to the directory; (2) directory broadcasts invalidations. (3) Trojan blocks
message and diverts a request to another core, (4) which responds with a
negative-acknowledge or acknowledgment resulting in (5) the directory allowing
original requestor to continue.

Figure 1: Coherence Trojan attacks in interposer-based systems in which
Chiplet A is the victim of a Trojan attack from Chiplet B.

Passive Reading (Fig. 1(a)): Trojans passively reading (snooping) observe
incoming coherence messages from the chiplet’s network-on-chip (NoC) sub-
system as they reach the L2’s state directory. The Trojan may buffer messages,
identify specific request patterns, and facilitate a covert communication
channel. The Trojan does not affect the system’s state but may trigger a more
complex Trojan.

Masquerading (Fig. 1(b)): Masquerading (spoofing) occurs when a Trojan
modifies the packet’s sender field such that the packet appears to originate
from a different core. If the target packet is a request, such an attack can
result in a deadlock since all responses from the directory or other cores are
sent to the incorrect core. If the target packet is a response, the Trojan may
block it and respond with an acknowledgment that appears to be from a
different core, resulting in an incoherent memory state.

Modification (Fig. 1(c)): Such attacks occur when the Trojan directly modifies
the message type of a coherence message. This attack may result in a deadlock
since the Trojan may cause the memory controller’s directory to assume the
data is in one state, due to a modified packet, while the local directory
holds the data in a different—incorrect—state.

Diverting (Fig. 1(d)): Trojans can launch diverting attacks by blocking the
local state directory from observing a request and then resending the request
with a different destination field. This results in the compromised core and
the original requestor becoming incoherent with respect to the rest of the
memory system.

Limitations of Basic Attacks: Any of the above attacks can individually result
in incoherence or deadlocks but cannot directly manipulate a victim’s data.
Only combining these attacks allows for a more complex set of attack vectors
that would enable a Trojan to pose a significant security threat.

###  3.2 Trojan Design

In the remainder of this paper, we propose the _Forging Attack_ , a novel
attack that manipulates legal coherence transactions to allow a Trojan to
write to a target address in a different process operating in a different
chiplet. The compromised chiplet containing the Trojan does not have access to
the victim process’ address space but can observe coherence interactions
broadcasted by the MOESI Hammer protocol. Since the Trojan resides between the
network interface and a core’s state directory in the compromised chiplet
(Figure 2), the Trojan has a complete view of incoming or outgoing coherence
messages, enabling it to block the core from observing specific interactions.
The Trojan holds few registers to track the target data’s current state
relative to the Trojan. These registers imitate the core’s state directory to
ensure the Trojan correctly responds to the global directory.

Figure 2: Our proposed Trojan attack on the coherence system that forges
messages to gain control and modify specific addresses accessed by a process
operating in a different chiplet. The attack executes in two phases. The first
phase (top) allows the Trojan to gain control of the target address and the
second phase (bottom) enables it to mimic the steps required to write back
maliciously formed data to main memory.

###  3.3 Forging Attack Demonstration

Here we assume the Trojan has a predefined target address. In a real-world
scenario, the Trojan can observe coherence messages broadcasted to the
compromised chiplet of the network to select its target. The coherence
protocol requires that the global directory sends invalidation messages each
time a core sends a write request, or GETX, to a line that it does not own.
The invalidation broadcast removes all copies in other cores before updating
the line with new data.

The Trojan operates in two phases. During the first phase, the Trojan deceives
the global directory into giving the Trojan access to the data. During the
second phase, the Trojan follows the protocol’s required transactions to write
to the target address, which the victim will later read. The interactions
caused by the Trojan in both phases are legal from the perspective of the
global directory. Furthermore, they are transparent to the software executing
in the victim process and all other security software in the system.

Phase 1, Acquiring Access to Target Data: Figure 2(top) illustrates the
initial steps the Trojan must take to gain access permissions to the target
address before it can maliciously write to it. 1 The Trojan observes coherence
requests, waiting for a specific address to trigger the attack. 2 The Trojan
generates a malicious GETX packet for the target address. 3 The directory
receives the GETX request, broadcasts an invalidation to all cores, and waits
for all cores to send acknowledgments (ACKs). 4 The directory forwards the
data and all ACKs to the compromised core. 5 The Trojan blocks the local
directory from seeing any response from the directory or cores, waiting to
receive all ACKs. 6 Once all ACKs are received, the Trojan can access the
data, since the directory considers the compromised core as the data owner.

Phase 2, Writing Malicious Data: Once access permissions are acquired, the
global directory assumes that the Trojan’s core is the exclusive owner of the
data. Figure 2(bottom) illustrates Phase 2 of the attack. This phase allows
the Trojan to mimic the legal operations that enable writing to main memory as
if the core was evicting the data after modifying it. The steps of the attack
are as follows: 1 Once the Trojan receives the final ACK, the requests to the
target address are unblocked. 2 The Trojan immediately sends a PUTX to the
directory to indicate that it is “evicting” modified data. 3 The directory
responds with a WRITEBACK_ACKNOWLEDGEMENT, allowing the Trojan to proceed with
“evicting” the maliciously changed dirty data. 4 The Trojan responds to the
WRITEBACK_ACKNOWLEDGEMENT with a WRITEBACK_EXCLUSIVE_DIRTY response containing
the malicious data. 5 The data is written to memory.

###  3.4 Results

We evaluate the Trojans in gem5, targeting a victim which iterates over an
array to set each value to ’1’ or ’0’ and then reads the array to compute a
sum. Figure 3(a) shows the data the victim process observes without the Trojan
enabled. The victim writes ‘0’ or ‘1’ to various locations in its data array
and then re-reads these locations, seeing the expected data values. Figure
3(b) shows the data the victim receives when it attempts to read the data
array after writing to all indexes. The _Forging Attack_ successfully modifies
the data array’s first value, which the victim then reads unknowingly of the
manipulation. This demonstrates our Trojan can manipulate the coherence system
to modify data that another application is operating on, even without
requiring shared memory access.

Unlike prior work, which focuses on Trojans modifying packets [10, 11, 12,
13], we leverage the coherence mechanism itself to modify data in memory that
is _never touched_ and _not owned_ by the chiplet containing the Trojan. Our
attack does not require the data to be in the compromised core’s caches.
Generating and blocking specific coherence messages allows the Trojan to
mislead the global directory about the state and ownership of the targeted
data.

(a) Data received by the victim when the Trojan is not activated. The
application reads an alternating sequence of ‘1’ and ‘0.’

(b) Data received after the Trojan has completed its attack. The first entry
in the array is now set to ‘5,’ instead of the expected ‘1.’

Figure 3: Data values as seen by the victim.

##  4 Conclusion and Future Defenses

As industry moves toward chiplet-based designs, hardware Trojans pose a
significant threat to security. These systems will rely heavily on coherence
to ensure that data remains up-to-date in all components, making the coherence
protocol an attractive target. Critically, unlike prior work, which focuses
only on packet modifications, we show that a coherence-centric Trojan attack
can modify memory that is not even owned by the compromised chiplet. We
provide an example of a complex Trojan implementation that modifies memory
without relying on malicious software components. This work highlights the
need for mechanisms to protect the coherence scheme from these novel attacks.

Detecting Trojans during chiplet manufacturing is challenging considering the
complexity of individual IPs. Defenses against hardware Trojan exploiting a
system’s coherence mechanisms could implement runtime monitoring to identify
malicious behavior originating from a particular chiplet. A benefit of 2.5D
integration is that the components are usually sourced from vendors and then
integrated onto an interposer layer at a separate foundry than each IP’s
manufacturing [9]. Requiring an interposer’s manufacturing and integration by
a trustworthy facility could allow the 2.5D interposer to act as a hardware
root of trust that can embed security features. Embedding the security
features into the interposer layer could allow defenders to observe coherence
packets and securely control data flow freely. We plan to explore these themes
in our future work.

## References

  * [1] S. Naffziger, N. Beck, T. Burd, K. Lepak, G. H. Loh, M. Subramony, and S. White, “Pioneering chiplet technology and design for the AMD EPYC and Ryzen processor families : Industrial product,” in ACM/IEEE ISCA, pp. 57–70, 2021.
  * [2] “Compute Express Link (CXL), www.computeexpresslink.org.”  Accessed: 2022-05-18.
  * [3] S. Bhunia and M. Tehranipoor, The Hardware Trojan War.  Springer, 2018.
  * [4] M. N. I. Khan, A. De, and S. Ghosh, “Cache-out: Leaking cache memory using hardware Trojan,” IEEE TVSLI, vol. 28, no. 6, pp. 1461–1470, 2020\.
  * [5] M. Bidmeshki, G. R. Reddy, L. Zhou, J. Rajendran, and Y. Makris, “Hardware-based attacks to compromise the cryptographic security of an election system,” in IEEE ICCD, pp. 153–156, 2016.
  * [6] M. Kim, S. Kong, B. Hong, L. Xu, W. Shi, and T. Suh, “Evaluating coherence-exploiting hardware Trojan,” in IEEE DATE, pp. 157–162, 2017\.
  * [7] A. Basak, S. Bhunia, T. Tkacik, and S. Ray, “Security assurance for system-on-chip designs with untrusted IPs,” IEEE TIFS, vol. 12, no. 7, pp. 1515–1528, 2017.
  * [8] P. Conway, N. Kalyanasundharam, G. Donley, K. Lepak, and B. Hughes, “Cache hierarchy and memory subsystem of the AMD Opteron processor,” IEEE Micro, vol. 30, no. 2, pp. 16–29, 2010.
  * [9] J. Kim et al., “Architecture, chip, and package co-design flow for 2.5D IC design enabling heterogeneous IP reuse,” in ACM/IEEE DAC, pp. 1–6, 2019.
  * [10] D. M. Ancajas, K. Chakraborty, and S. Roy, “Fort-NoCs: Mitigating the threat of a compromised NoC,” in ACM/EDAC/IEEE DAC, pp. 1–6, 2014.
  * [11] T. Boraten and A. K. Kodi, “Mitigation of denial of service attack with hardware Trojans in NoC architectures,” in IEEE IPDPS, pp. 1091–1100, 2016.
  * [12] M. H. Khan, R. Gupta, J. Jose, and S. Nandi, “Dead flit attack on NoC by hardware Trojan and its impact analysis,” in ACM NoCArc, pp. 10–15, 2021\.
  * [13] N. Prasad, R. Karmakar, S. Chattopadhyay, and I. Chakrabarti, “Runtime mitigation of illegal packet request attacks in networks-on-chip,” in IEEE ISCAS, pp. 1–4, 2017.


11institutetext: Anton Pannekoek Institute for Astronomy, University of
Amsterdam, 1090 GE Amsterdam, The Netherlands

# A systematic search for double eclipsing binaries in Zwicky Transient
Facility data

T. Vaessen<EMAIL_ADDRESS>J. van Roestel<EMAIL_ADDRESS>
(Received 31 October, 2023; accepted 11 December, 2023)

###### Abstract

Context. Double eclipsing binaries are gravitationally bound quadruple systems
in a ‘2+2’ configuration where both of the binaries are eclipsing. These
systems are interesting objects to better understand stellar formation, to
investigate the dynamical interaction between the two binary systems or to
study certain stages of stellar evolution, such as common-envelope events or
Type Ia Supernovae.

Aims. With this work, we aim to determine if double eclipsing binaries can be
found using ZTF data and what the difficulties are in doing so. Secondly, we
aim to significantly increase the number of known double eclipsing systems and
determine how this sample differs from samples of double eclipsing binaries
found with other telescopes.

Methods. We develop a new method to systematically search for double eclipsing
binaries in sparsely sampled light curves. For this we use box-least-squares
(BLS) to search for the period of the first binary in the system. We then
remove that signal from the light curves, and search the residual light curve
again with BLS to find the second period. We applied this method to ZTF light
curves of 575 526 eclipsing binaries known in the Gaia eclipsing binary
catalogue.

Results. We report the discovery of 198 new double eclipsing binary systems.
The shortest and longest orbital periods of the newly detected systems are
0.11 days to 323 days respectively.

Conclusions. We successfully implemented a method that systematically searches
for double eclipsing binary systems in sparsely sampled data. In total 198 new
double eclipsing binary systems have been found in 575 526 light curves
($\approx$ 0.034$\%$). The ZTF sample typically contains more short period
binaries compared to the TESS sample, but is also able to find systems with
longer periods than what is currently known. We expect that at least three to
four times more quadruples can be found by applying this method to all ZTF
stellar light curves, by increasing the number of data points as a result of
longer observations, and by implementing an automatic detection mechanism that
replaces visual inspection.

###### Key Words.:

(Stars:) binaries: eclipsing - (Stars:) binaries (including multiple): close -
Methods: data analysis

##  1 Introduction

Most stars in the Universe can be found in gravitationally bound binaries or
higher-order hierarchies (Tokovinin, 2014). It is estimated that approximately
$4\%$ of all solar-type stars can be found in quadruple systems (Tokovinin,
2021), either in a 3+1 or 2+2 configuration. A small fraction of 2+2 quadruple
systems are double eclipsing binaries, in which two eclipsing binaries also
orbit each other (Zasche et al., 2019). If the motion of the inner pair is not
strongly perturbed by the outer companions, the motions of the stars are
approximated by stable Keplerian orbits and can survive for a long time
(Tokovinin, 2014).

Compact double eclipsing binaries are interesting objects of study to better
understand stellar formation or to investigate the dynamical interaction
between the two binary systems. An important example of such a dynamical
interaction is the Kozai-Lidov mechanism where, on long timescales, a periodic
exchange between the binaries’ eccentricities and inclination can take place
(Kozai, 1962; Lidov, 1962). Furthermore, these objects serve as an
astrophysical laboratory when studying certain stages of stellar evolution,
such as common-envelope events or Type Ia Supernovae (Kostov et al., 2022a).
Therefore, the identification of more double eclipsing binaries will lead to a
larger sample that can be used to study the origin and life cycle of these
objects. Quadruples are intrinsically rare and require a favourable alignment
of both binaries with the observer in order to be detected. However, in the
case of fortunate alignment, the objects can be identified by periodic
eclipses in the light curve. Zasche et al. (2019) systematically searched the
OGLE-LMC data by visually inspecting the light curves of eclipsing binary
systems and found 72 systems of double eclipsing binaries. More recently,
light curves of the Transient Exoplanet Survey Satellite (TESS, Ricker et al.
2015) have been used to identify double eclipsing binaries in this way. Zasche
et al. (2022b) identified 116 double eclipsing binaries using TESS, and Kostov
et al. (2022a, b) detected in total 199 double eclipsing binaries, 18 of which
overlapped with Zasche et al. (2022b).

In this paper, we present the search for double eclipsing binaries in Zwicky
Transient Facility light curves. In Sect. 2 we present the target preselection
and the ZTF data. In Sect. 3 we discuss the method we used to identify double
eclipsing binaries. In Sect. 4 we present the analysis of the data. The
results are presented in Sect. 5 and we discuss them further in Sect. 6. We
end the paper with a summary and recommendations for future work in Sect. 7.

##  2 Data

As part of the Zwicky Transient Facility (ZTF), the Palomar 48-inch (P48)
telescope has been imaging the sky every night since 2018 (Graham et al.,
2019; Bellm et al., 2019; Dekany et al., 2020). Most of the time, ZTF uses the
$g$ and $r$ bands, but a small fraction of the observations are also made in
the $i$ band. The exposure times are predominantly 30 seconds for $g$ and $r$
and 60 or 90 seconds for $i$ exposures. The median limiting magnitude,
averaged over the lunar cycle, is $\approx 20.5$ in all three bands. We used
the PSF-photometry-based light curves which are automatically generated for
all persistent sources detected in the ZTF reference images (for a full
description see Masci et al., 2019) and are publicly available.

In order to avoid searching billions of ZTF light curves, we make a
preselection of objects. We process only light curves of objects that have
been identified as eclipsing binaries by Gaia (Mowlavi et al., 2023). Although
this preselection inevitably means that we will miss any system that has not
been identified as an eclipsing star by Gaia first, the advantage is that it
strongly reduces the number of light curves we need to search.

The Gaia eclipsing binary catalogue contains 2 184 477 sources distributed
over the whole sky and down to magnitude $G\approx 20.5$. While ZTF goes
slightly deeper than Gaia, ZTF only covers the sky from the celestial north
pole to a declination of $-30\deg$. This means there are 1 210 001 sources in
the Gaia eclipsing binary catalogue that are also in the ZTF footprint and are
fainter than $G=12$ (the approximate ZTF saturation limit). To download ZTF
light curves for these objects, we used
ZTFquery111https://github.com/MickaelRigault/ztfquery (Rigault, 2018).

##  3 Method

Light curves of double eclipsing binaries contain the signal of two eclipsing
binaries with different periods. If the light curve is well sampled (e.g. the
continuous light curves of TESS), multiple epochs sample each individual
eclipse and the individual eclipses are easy to identify. This makes the
identification of double eclipsing binaries relatively straightforward (Zasche
et al., 2019; Kostov et al., 2022a; Zasche et al., 2022a). However, if a light
curve is sparsely sampled (the typical time between observations is much
larger than the duration of an eclipse) individual eclipses are sampled by a
few or just a single epoch only and the light curve needs to be phase-folded
in order to identify eclipses. Therefore, searching for double eclipsing
binaries requires a slightly different approach for sparsely sampled light
curves like ZTF. Instead of identifying individual eclipses, we use the fact
that both eclipse signals are periodic and use period-finding methods to
identify both periods.

We implemented this method using box-least-squares (BLS, Kovács et al., 2002)
which is available through the Python package Astrobase (Bhatti et al., 2017).
First the data from the different bands ($g$ and $r$) were combined by
normalising the data in each band to its corresponding mean. Then we performed
the BLS period search on the light curve to find the first period, which we
define as period A. In our experience, this period was often half the orbital
period and thus the primary and secondary eclipse overlapped. To avoid this
overlap, we phase-folded the light curve to twice period A. To remove the
signal of period A, we binned the phase-folded light curve using an
empirically determined number of 100 bins. We then divided each flux
measurement in the light curve by the mean of its corresponding bin. After the
removal of the signal at period A, we applied the BLS method again to find the
second period; period B. At each point in this process, we saved the phase-
folded and binned light curves for visual inspection. An example of a (phase-
folded) light curve can be seen in Fig. 1.

##  4 Analysis

To ensure that the method worked correctly, we used a test dataset consisting
of 68 double eclipsing binaries found in prior research (Zasche et al., 2022b)
for which ZTF data is available. Furthermore, this test sample also helped to
establish a method for identifying double eclipsing binaries quantitatively.
For this, the relative peak height (RPH) was introduced as

$\mathrm{RPH}\equiv\frac{\mathrm{peak\
periodpower}-\overline{\mathrm{periodpower}}}{\sigma_{\mathrm{periodpower}}}$
(1)

where max periodpower is the BLS-periodogram peak associated with the best
period, $\overline{\mathrm{periodpower}}$ is the mean of powers and
$\sigma_{\mathrm{periodpower}}$ is the standard deviation. In Astrobase these
periodpowers are referred to as lspvals (Bhatti et al., 2017). For readers not
familiar with the BLS algorithm, an example of a periodogram is shown in the
appendix in Fig. 8. Here we see the periodogram power value of each orbital
period. Additionally, Astrobase offers a function to calculate the signal-to-
noise ratio (S/N) for each periodic signal (Bhatti et al., 2017). From the
test sample we empirically determined that a RPH ¿ 10 and S/N ¿ 10 may
indicate the presence of a _double_ eclipsing binary. As it was not feasible
to visually inspect all objects in the sample, we selected only candidates
with RPH ¿ 10 and S/N ¿ 10. This significantly reduced the number of light
curves that needed visual inspection to identify double eclipsing binary
candidates. These selection criteria are discussed further in Sect. 6.

Some objects were, based on a high relative peak height and S/N, initially
identified as quadruple candidate but appeared to exhibit a sinusoidal rather
than an eclipsing signal. These objects were associated with relatively short
periods $\mathrm{P_{B}}<0.30$ days, which could be an indication of a variable
star in the binary system that lies on the instability strip, such as $\delta$
Scuti stars (Pietrukowicz et al., 2020). Another common false positive is
associated with the imperfect removal of signal A. This would result in in
period B being the same as period A, or an integer multiple of period A. We
therefore removed any candidate for which period B was a multiple of period A.

When testing the method, it appeared that light curves containing high cadence
observations (nights with more than 30 epochs) had a negative impact on the
ability to find the right periods. This is probably caused by correlated noise
introduced by the calibration of the data. Therefore, observations with more
than 30 epochs per night were removed. A similar effect was observed by
outliers in high and low flux. For this reason, outliers in normalised flux
were removed using the interquartile range with a cut-off value of 5. Other
unreliable data, flagged by the ZTF data process pipeline, was also removed.
After this data pre-processing and cleaning, a minimum threshold of 500 data
points per light curve was set for analysis. This decreased the sample size
from 1 210 001 to 575 526 objects.

Based on the objects in the test sample, we set the period search range for
new objects between 0.1 and 150 days with a stepsize determined by the minimum
transit duration and observational period. We chose this lower limit as it is
not expected to find any objects with shorter orbital periods (Drake et al.,
2014). The upper limit was based on the five-year observational period of the
ZTF and the decreasing likelihood of finding longer periods. For the transit
duration, the BLS performed a search for a periodic signal lasting between
0.01 and 0.2 in phase.

Finally, we applied the pipeline to the 575 526 ZTF light curves. We used the
HELIOS computer cluster to process the light curves. The average processing
time per object is approximately 10 seconds.

We used the BLS output statistics to selected 5350 promising candidates. These
were visually inspected in order to confirm the double eclipsing nature of the
objects. At this point, we also determined, to the best of our abilities, what
the two orbital periods are, since the BLS algorithm typically finds half the
orbital period. Finally, we also checked if the photocenter position was
correlated with the eclipse periods to identify chance alignments of two
unrelated eclipsing binaries. The typical RMS positional accuracy is $\approx
0.1\arcsec$. No chance alignments were found in this manner.

Figure 1: ZTF light curves of Gaia dr3 source ID 2216420454386370176. Upper
panel: Normalised flux as a function of time. It shows two sets of eclipses
with $\mathrm{P_{A}}=1.072780$ days and $\mathrm{P_{B}}=0.404213$ days. Middle
panel: Phase folded light curve for binary A with $\mathrm{P_{A}}=1.072780$
days. We also see that not all data points fall nicely on the curve,
indicating the presence of another signal. Lower panel: Phase folded light
curve for binary B, after removal of the signal at period A, with
$\mathrm{P_{B}}=0.404213$ days.

##  5 Results

Of the 575 526 light curves analysed (500 data points or more), 203 ($\approx$
0.035$\%$) are identified as double-eclipsing binaries candidates. They are
summarised in the appendix in Table 1, and an example is shown in Fig. 1.
Almost all candidates for double eclipsing binaries are not currently known in
any scientific literature (i.e. they are not listed as such on SIMBAD; Wenger
et al., 2000). Only four of these 203 objects have been found in prior
research and marked in Table 1 by an asterisk. One of these four objects, Gaia
dr3 source ID 2003428628134264448, has previously been identified as a triple
star system (TIC 388459317; Borkovits et al., 2021) but the ZTF data suggests
that it is a double-eclipsing binary. The other three objects, Gaia dr3 source
ID’s 2060949991271681664 (Zasche et al., 2022a), 407849617690288128 (TIC
354314226; Kostov et al., 2022b) and 158123726424621824 (TIC 150055835; Kostov
et al., 2022b) have been identified as double eclipsing binary systems by
previous studies that used TESS data. Using the ZTF data, we find that the
periods of these objects agree between both studies.

The periods of the double eclipsing binaries in the ZTF sample range from 0.11
days to 323.20 days and the mean of periods A and B are 1.21 days and 10.52
days respectively, see Fig. 2. The figure shows that most objects have both
periods shorter than a few days. Using a kernel density estimate, this
distribution seems to be the same for the candidates and the test objects.

Figure 2: Periods A and B of 13 test objects and 203 quadruple candidates.
Here $P_{short}$ is defined as the shortest period of A and B and $P_{long}$
as the longest period of A and B. The figure shows that most objects have both
periods shorter than a few days. Using a kernel density estimate, this
distribution seems to be the same for the candidates and test objects.

To understand if there is any bias in our search, all 1 210 001 binaries in
the sample were plotted in an HR-diagram, see the left panel of Fig. 3. Here
we see that the double eclipsing binaries, plotted separately as dots, are
scattered throughout the diagram and do not seem to have any preferred
location.

Since it is interesting to not only compare the double eclipsing binaries to
the input sample but also to a general star population within the Milky Way,
we refer to the HR-diagram of all stars in the Gaia dr3 archive within 200 pc
of the sun. This can be seen in the right panel of Fig. 3. The quadruple
candidates and test objects are systematically above the single stars in this
diagram and are mostly Sun-like stars. We note, however, that quadruple
systems on the HR-diagram have a lower magnitude of $2.5\log_{10}(4)=1.505$
compared to single stars, assuming all stars in the quadruple are similar and
equally bright.

Figure 3: Left: HR-diagram of all 1 210 001 binary objects in the sample. The
double eclipsing binary candidates and test objects, separately plotted as
dots, are scattered throughout the diagram and do not seem to have any
preferred location. Using the sun as a reference with $M_{G}=5$ and
$G_{BP}-G_{RP}=1$, a larger number of objects seem to be similar to the sun.
Right: An HR-diagram of all stars in the Gaia dr3 archive within 200 pc of the
Sun. The candidates and test objects are systematically above the single stars
in this diagram and are mostly sun-like stars. The arrow with a length of
1.505 magnitude indicates where one star in the quadruple system would lie,
assuming all stars in the system are equally bright.

##  6 Discussion

###  6.1 Confusion with triple eclipsing systems

As we already noted; Gaia dr3 source ID 2003428628134264448 was previously
identified as a triply eclipsing triple system (Borkovits et al., 2021) (a
multistar system where all three stars eclipse each other). We initially
identified this system as a quadruple because we detected a primary and
secondary eclipse at a period of 88.8 days. However, as is shown by Borkovits
et al. (2021) using well-sampled TESS and ground-based follow-up light curves;
the eclipse shapes are complex and only consistent with a tertiary star that
eclipses, and is being eclipsed, by a short period binary. Triple eclipsing
stars are also extremely interesting for reasons similar to those of quadruple
systems. For recent work on triple eclipsing systems, see Carter et al.
(2011); Derekas et al. (2011); Hajdu et al. (2017); Mitnyan et al. (2020);
Borkovits et al. (2021); Powell et al. (2022); Rappaport et al. (2022, 2023);
Czavalinga et al. (2023)

The fact that one of our double eclipsing binary candidates turned out to be a
triply eclipsing triple system, prompted us to reconsider the long period
systems that we recovered. We focus on long period systems since, for triple
systems, the outer orbit has to be at least 5 times the inner orbit, while a
factor of 10 is required for slightly eccentric orbits (Mardling & Aarseth,
2001; Borkovits et al., 2021). With just the ZTF data, individual eclipses are
not well-resolved, which makes an unambiguous identification of triples
difficult. However, an eclipse of a triple eclipsing system is more noisy in a
folded light curve because of the complex and changing shape of individual
eclipses. We use this to determine whether some systems might be triple-
eclipsing systems instead of double-eclipsing binaries.

Objects 2020970648976206208, 2210495392378660864, and 2038100043701450112 show
somewhat irregular, long-duration eclipses in ZTF data and could be eclipsing
triple systems. Objects 2058085351143518464, 2003428628134264448,
2201723866572302976, and 413930359370384384 show a primary and a shallow
secondary eclipse at period B, with the primary eclipse somewhat irregular,
suggesting that they could also be triply eclipsing systems. Object
2041659197183314304 shows well-behaved eclipses, and we judge that this system
is unlikely to be a triply eclipsing system.

Objects 207943285475761280 and 2032118077650924672 are two systems for which
the eclipse of the long period signal is much deeper in $g$ than $r$ and maybe
a shallow secondary eclipse can be seen. This suggests that a large, cold star
is eclipsed by a smaller, hotter star. We suspect that these are double
eclipsing systems where one of the components has evolved off the main-
sequence. For 260945106052343296 and 2012994482376100736 the period is too
uncertain to draw any conclusions.

Since we cannot definitively classify these objects as triple eclipsing
systems, we consider all these objects double eclipsing binary candidates in
the rest of this paper. Follow-up light curve observations of the eclipses are
needed for each object to definitively determine their nature.

###  6.2 Orbital period distribution

In general, binaries with shorter periods seem easier to detect than binaries
with longer periods. One explanation is that stars in binaries with longer
orbital periods are further apart from each other. As the orbital distance
increases, the eclipse probability

$\text{Eclipse probability}=\frac{R_{1}+R_{2}}{a}\propto P^{-2/3}$ (2)

decreases since this requires a more precise alignment with the observer for
the eclipse to be detected. In addition, longer orbital periods thus lead to
shorter eclipse durations as fraction of the orbit. Some light curves showed
that the eclipse duration of binary B was either very short (¡ 0.01 in phase),
or contained very few data points, which makes it difficult for the algorithm
to identify the eclipse. This agrees with the results shown in Fig. 2, where
most double eclipsing binary candidates have both periods shorter than a few
days.

Figure 4: The orbital periods of double eclipsing binaries of the three
largest samples: the sample from OGLE (Zasche et al., 2019), the sample from
TESS by Kostov et al. (2022b, a) and Zasche et al. (2022b) and the ZTF sample
from this work. Here $P_{short}$ is defined as the shortest period of A and B
and $P_{long}$ as the longest period of A and B

The orbital periods of the candidates found in this research seem fairly
similar to those found by Kostov et al. (2022a); Zasche et al. (2022b), see
Fig. 4. However, one difference is that we find some shorter periods in this
research. Whereas previous research shows that most objects show a period
ratio of $\mathrm{P_{B}}/\mathrm{P_{A}}$ between 1 and 2 (Kostov et al.,
2022a), in Fig. 5 we see that here also much smaller ratios are found,
indicating a larger difference between period A and period B. An explanation
for this can be the data cadence of TESS which is 26 days, meaning it most
likely does not find any significant periods larger than approximately 13
days. Furthermore, short 30 minute exposures by TESS may lead to the spreading
out of short eclipses, which is why TESS might not find very short periods
either. We also note that all stars in our sample are listed in the Gaia
catalogue. Any bias of Gaia towards shorter periods may therefore reflect in
our results.

Figure 5: Period ratios $\mathrm{P_{short}}/\mathrm{P_{long}}$ of each object.
Here $P_{short}$ is defined as the shortest period of A and B and $P_{long}$
as the longest period of A and B. The vertical lines indicate where potential
harmonics would fall. We note that integer multiple harmonics were removed in
the object search. However, non integer harmonics were not removed. Here we
see that most objects do not fall on any of the harmonics lines indicating
that there does not seem to be any particular correlation between the periods.

###  6.3 Detection efficiency and completeness

In this section, we briefly discuss the detection efficiency and completeness
of our search. Of the 575 526 analysed objects, at least 203 ($\approx$
0.035%) passed the visual inspection tests and showed to be a double eclipsing
binary candidate rather than a binary system. This is approximately a factor
4.5 lower success rate than Zasche et al. (2022b) (116 out of 70.000) and a
factor 7 lower than Pawlak et al. (2013) (15 out of 6138). However, we note
that OGLE-III has observed all stars in a dense area in the Small Magellanic
Cloud for a longer period of time (Pawlak et al., 2013). Zasche et al. (2022b)
have, like this research, only looked at eclipsing binaries, but a difference
with this research is the data cadence, which for ZTF typically is one
observation per night only for five years. TESS, on the other hand, provides
continuous undisturbed photometry for 26 days (Zasche et al., 2022b). Lastly,
the angular resolution of ZTF, OGLE, and TESS are very different. This,
combined with the new method used here, makes it difficult to compare success
rates, but could explain the difference.

In the rest of this section, we briefly discuss a few causes that limit the
completeness of our search. First, we consider the efficiency of our quadruple
detection method. Our method was able to retrieve the correct periods for most
known objects in the test dataset. For periods that could not be recovered,
further investigation revealed why this was not possible. In some cases, many
data points were flagged as unreliable, leaving only very few data points in
the light curve for period finding. In other instances, the eclipse depth of
one binary was very small, on the order of a few percent, making it difficult
for the algorithm to detect any periodic signal. Other reasons why the
algorithm was not successful included that some light curves showed a very
small trend in flux over time, resulting in the cadence of the telescope being
found as significant period; this typically being one or two days. Even light
from the Moon coming in as background noise could lead to the period of the
Moon being found to be a significant period. For compact double eclipsing
binaries, eclipse-timing variations (ETV) can smear periodic signals which may
affect the period finding. However, when comparing our results with known
systems with ETV, we found this not to be the case. We therefore expect this
method to be suitable for finding compact double eclipsing binaries.

The completeness of our search for double eclipsing binaries is also limited
by our use of the Gaia eclipsing binary catalogue. When we look at the results
of other studies (Kostov et al., 2022a, b; Zasche et al., 2022b), we see that
approximately 30% of the found quadruples are in the Gaia catalogue. In this
research we have only looked at ZTF objects that are known in the Gaia
catalogue. This means that potentially three times more quadruples can be
found in the ZTF data that are not in the Gaia catalogue.

As it was not practical to look at 575 526 light curves, we used automatically
calculated statistics to significantly decrease the number of light curves to
inspect which could also affect the completeness of our search of the ZTF
data. We used the test objects to empirically determine cuts; relative peak
height $>10$ and S/N $>10$. As can be seen in Fig. 9, the wide range of values
of relative peak height and S/N cannot reliably indicate whether an object
might be a quadruple candidate. This suggests that more double eclipsing
binaries can be found for values of relative peak height and S/N below 10. It
can also be seen in the figure that some objects did have a high relative peak
height and S/N but were not identified as a quadruple candidate. This could
have several reasons. In some cases, the algorithm was not able to completely
remove the first periodic signal, which means that the signal, or an harmonic,
was returned again for period B. Other explanations include, but are not
limited to, an outlier in the light curve, the cadence of the telescope, or
the influence of the Moon. Sometimes the algorithm did not find an eclipsing
signal but a sinusoidal signal, possibly due to the pulsation of one of the
stars in the eclipsing binary. Even though this is not a double eclipsing
binary, a strong periodic signal was still found, resulting in a high relative
peak height and S/N.

Figure 6: Left: Cumulative histogram for the number of data points per light
curve. After passing a threshold of approximately 1000 data points there is a
significant increase in the quadruple detection efficiency. Note that the
plateau and discontinuity at around 2000 data points is a result of the ZTF
survey strategy. Right: Cumulative histogram for the magnitude of stars for
the complete sample and quadruple candidates in the magnitude range 13-20.5.
We see that for stars fainter than $G\gtrsim 18$ the quadruple detection
efficiency rapidly approaches zero. The sharp increase at $G\approx 15$ is not
fully understood.

We suspected that the detection efficiency of double eclipsing binary stars is
also strongly affected by the number of epochs in the light curve. If we look
at the left panel in Fig. 6 we see a cumulative histogram for the number of
data points per light curve. It shows that, only after passing a threshold of
approximately 1000 data points, the quadruple detection efficiency
significantly increases. After that, there are more double eclipsing binaries
as the number of data points increases. The difference, however, is small.
This means that, as ZTF keeps collecting data (or ZTF is combined with
external data), we can expect the number of detectable double eclipsing
binaries to double.

We also briefly consider to what brightness we can reliably find double
eclipsing binaries in ZTF data. On the right panel in Fig. 6 we see a
cumulative histogram for the magnitude of stars, both for the complete sample
and for the quadruple candidates. The ZTF has looked at stars in the magnitude
range 13-20.5. Since the curve of the quadruples is constantly above the
overall sample curve, this indicates that it is easier to find quadruples for
bright stars compared to faint stars. Almost all quadruples have a magnitude
of $G\lesssim 18$, while only half of the input sample is fainter than this
limit. We conclude that the light curve signal-to-noise ratio is too low for
sources fainter than $G\approx 18$. As can be seen in Masci et al. (2019), the
RMS precision is $\approx 1\%$ for sources brighter than 18 magnitude, but the
precision degrades rapidly for fainter sources.

To summarise, we expect that future research can find at least three to four
times more quadruples by including the following considerations. First, the
search should be not limited to the Gaia eclipsing binary catalogue but the
entire ZTF database can be searched instead. Secondly, systems with a relative
peak height and S/N lower than 10 should also be considered. Thirdly, longer
observation time of the ZTF will lead to more data points which has a positive
effect on quadruple finding. Lastly, by using an automatic detection method
that can replace visual detection, for instance machine learning, all the
light curves can be analysed more efficiently.

##  7 Conclusions and future work

We have developed a method to systematically search for double eclipsing
binaries in sparsely sampled data. Using this method we found 198 new double
eclipsing binaries candidates in the ZTF data. The periods of the objects
ranged from 0.11 days to 323 days with a mean of periods A and B of 1.21 days
and 10.52 days respectively. By searching the entire ZTF database, increasing
the number of data points per light curve, and implementing an automatic
detection mechanism that replaces visual inspection, we expect that at least
three to four times more quadruples can be found using this method. Other
recommendations for future work are; to apply this method to data collected by
other telescopes such as Gaia. In addition, we recommend that the multi star
systems are included in the target lists of large multiplex spectroscopic
surveys such as SDSS, DESI, and WEAVE in order to obtain phase-resolved
spectra of the quadruple candidates to further investigate their properties.
Lastly, detection of eclipse-timing variations in the double eclipsing
binaries can definitively prove their quadruple nature.

###### Acknowledgements.

We thank Silvia Toonen for a useful discussion about multi-star systems. This
publication is part of the project ”The life and death of white dwarf binary
stars” (with project number VI.Veni.212.201) of the research programme NWO
Talent Programme Veni Science domain 2021 which is financed by the Dutch
Research Council (NWO). 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 No. 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. Based on observations obtained with the Samuel
Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar Observatory
as part of the Zwicky Transient Facility project. ZTF is supported by the
National Science Foundation under Grants No. AST-1440341 and AST-2034437 and a
collaboration including current partners Caltech, IPAC, the Weizmann Institute
for Science, the Oskar Klein Center at Stockholm University, the University of
Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO
Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity
College Dublin, Lawrence Livermore National Laboratories, IN2P3, University of
Warwick, Ruhr University Bochum, Northwestern University and former partners
the University of Washington, Los Alamos National Laboratories, and Lawrence
Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW.
The ztfquery code was funded by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement n°759194 - USNAC, PI: Rigault). This research has made use of the
VizieR catalogue access tool, CDS, Strasbourg, France. This research made use
of NumPy (Harris et al., 2020) This research made use of matplotlib, a Python
library for publication quality graphics (Hunter, 2007) This research made use
of Astroquery (Ginsburg et al., 2019) This research made use of Astropy, a
community-developed core Python package for Astronomy (Astropy Collaboration
et al., 2022)

## References

  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167
  * Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, Publications of the Astronomical Society of the Pacific, 131, 018002
  * Bhatti et al. (2017) Bhatti, W., Bouma, L. G., & Wallace, J. 2017, Astrobase
  * Borkovits et al. (2021) Borkovits, T., Mitnyan, T., Rappaport, S. A., et al. 2021, Monthly Notices of the Royal Astronomical Society, 510, 1352
  * Carter et al. (2011) Carter, J. A., Fabrycky, D. C., Ragozzine, D., et al. 2011, Science, 331, 562
  * Czavalinga et al. (2023) Czavalinga, D. R., Borkovits, T., Mitnyan, T., Rappaport, S. A., & Pál, A. 2023, MNRAS, 526, 2830
  * Dekany et al. (2020) Dekany, R., Smith, R. M., Riddle, R., et al. 2020, Publications of the Astronomical Society of the Pacific, 132, 038001, arXiv:2008.04923 [astro-ph, physics:physics]
  * Derekas et al. (2011) Derekas, A., Kiss, L. L., Borkovits, T., et al. 2011, Science, 332, 216
  * Drake et al. (2014) Drake, A. J., Djorgovski, S. G., Garcia-Alvarez, D., et al. 2014, The Astrophysical Journal, 790, 157, arXiv:1406.4504 [astro-ph]
  * Ginsburg et al. (2019) Ginsburg, A., Sipőcz, B. M., Brasseur, C. E., et al. 2019, AJ, 157, 98
  * Graham et al. (2019) Graham, M. J., Kulkarni, S. R., Bellm, E. C., et al. 2019, Publications of the Astronomical Society of the Pacific, 131, 078001, arXiv:1902.01945 [astro-ph]
  * Hajdu et al. (2017) Hajdu, T., Borkovits, T., Forgács-Dajka, E., et al. 2017, MNRAS, 471, 1230
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90
  * Kostov et al. (2022a) Kostov, V. B., Powell, B. P., Rappaport, S. A., et al. 2022a, The Astrophysical Journal Supplement Series, 259, 66, arXiv:2202.05790 [astro-ph]
  * Kostov et al. (2022b) Kostov, V. B., Powell, B. P., Rappaport, S. A., et al. 2022b, The Astrophysical Journal Supplement Series, 259, 66
  * Kovács et al. (2002) Kovács, G., Zucker, S., & Mazeh, T. 2002, Astronomy & Astrophysics, 391, 369, arXiv:astro-ph/0206099
  * Kozai (1962) Kozai, Y. 1962, AJ, 67, 591
  * Lidov (1962) Lidov, M. 1962, Planetary and Space Science, 9, 719
  * Mardling & Aarseth (2001) Mardling, R. A. & Aarseth, S. J. 2001, Monthly Notices of the Royal Astronomical Society, 321, 398
  * Masci et al. (2019) Masci, F. J., Laher, R. R., Rusholme, B., et al. 2019, Publications of the Astronomical Society of the Pacific, 131, 018003, arXiv: 1902.01872 [astro-ph.IM] tex.adsnote: Provided by the SAO/NASA Astrophysics Data System tex.adsurl: https://ui.adsabs.harvard.edu/abs/2019PASP..131a8003M
  * Mitnyan et al. (2020) Mitnyan, T., Borkovits, T., Rappaport, S. A., Pál, A., & Maxted, P. F. L. 2020, MNRAS, 498, 6034
  * Mowlavi et al. (2023) Mowlavi, N., Holl, B., Lecoeur-Taïbi, I., et al. 2023, A&A, 674, A16
  * Pawlak et al. (2013) Pawlak, M., Graczyk, D., Soszynski, I., et al. 2013, Eclipsing Binary Stars in the OGLE-III Fields of the Small Magellanic Cloud, arXiv:1310.3272 [astro-ph]
  * Pietrukowicz et al. (2020) Pietrukowicz, P., Soszynski, I., Netzel, H., et al. 2020, Acta Astronomica, 70, 241, arXiv:2103.10436 [astro-ph]
  * Powell et al. (2022) Powell, B. P., Rappaport, S. A., Borkovits, T., et al. 2022, ApJ, 938, 133
  * Rappaport et al. (2022) Rappaport, S. A., Borkovits, T., Gagliano, R., et al. 2022, MNRAS, 513, 4341
  * Rappaport et al. (2023) Rappaport, S. A., Borkovits, T., Gagliano, R., et al. 2023, MNRAS, 521, 558
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Rigault (2018) Rigault, M. 2018, ztfquery, a python tool to access ZTF data, Zenodo
  * Tokovinin (2014) Tokovinin, A. 2014, The Astronomical Journal, 147, 87
  * Tokovinin (2021) Tokovinin, A. 2021, Universe, 7, 352
  * Wenger et al. (2000) Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS, 143, 9
  * Zasche et al. (2022a) Zasche, P., Henzl, Z., & Ká ra, J. 2022a, A&A, 659, A8
  * Zasche et al. (2022b) Zasche, P., Henzl, Z., & Masek, M. 2022b, Astronomy & Astrophysics, 664, A96, arXiv:2205.03934 [astro-ph]
  * Zasche et al. (2019) Zasche, P., Vokrouhlický, D., Wolf, M., et al. 2019, A&A, 630, A128

##  Appendix A Additional figures

For the ZTF, the periods of objects that lie in the Galactic plane can be more
easily retrieved. This is can also be seen in Fig. 7 where the location of all
quadruple candidates and test objects are plotted. These results agree with
the results found in earlier research (Zasche et al. 2022b; Kostov et al.
2022a). It is most likely that the number of double eclipsing binaries is
higher in the Galactic plane as a result of higher star density.

Figure 7: Right ascension (RA) and declination (DEC) of the double eclipsing
binary candidates and test objects in this research. As expected, most
candidates lie in the Galactic plane. No objects are seen below a declination
of -30° as this is not visible by the ZTF. Figure 8: A periodogram of Gaia dr3
source ID 2216420454386370176 showing the lspvals as a function of orbital
period. Left: The periodogram power values for period A. Right: The periodgram
power values for period B after removing the signal of period A. Figure 9: The
relative peak height and S/N of all objects including double eclipsing binary
candidates and test objects. Left: Relative peak heights A and B for all
objects. Middle: S/N A and S/N B results for all objects. Right: S/N B and
relative peak height B of all objects. We see that quadruples can have a wide
range of RPH and S/N values. However, this also means that more can be found
at values of RPH ¡ 10 & S/N ¡10.

##  Appendix B Summary of double eclipsing binary candidates

Table 1 in this appendix lists all new double eclipsing binaries we found in
ZTF.

Table 1: Summary of the double eclipsing binary candidates. For each candidate the location (RA & DEC), Gaia dr3 source ID, periods ($\mathrm{P_{A}[d]}$ & $\mathrm{P_{B}[d]}$) and eclipse depths ($\mathrm{D_{A}}$ & $\mathrm{D_{B}}$) are given. Four objects have been found in previous research and are marked here with an asterisk. The object marked with a ”T” was initially identified as a quadruple, but further investigation revealed that this is, in fact, a triple system.  RA [J2000.0] | DEC [J2000.0] | Gaia dr3 source ID | $\mathrm{P_{A}[d]}$ | $\mathrm{D_{A}}$ | $\mathrm{P_{B}[d]}$ | $\mathrm{D_{B}}$
---|---|---|---|---|---|---
1.461502 | 44.447714 | 386474080852740992 | 2.599973 | 0.178 | 0.114405 | 0.017
2.702797 | 43.426465 | 384686137507666944 | 0.561156 | 0.057 | 0.30431 | 0.02
3.255083 | 66.065778 | 528173641885393792 | 1.377078 | 0.007 | 1.958498 | 0.067
5.530266 | 63.915121 | 431071367690614016 | 1.993434 | 0.207 | 2.83458 | 0.12
6.850741 | 42.275447 | 382118159381427840 | 0.302332 | 0.045 | 11.331499 | 0.063
8.806599 | 48.41824 | 390942874067575936 | 0.254808 | 0.066 | 100.03476 | 0.159
13.007138 | 60.631402 | 427296611137250432 | 1.39752 | 0.108 | 1.233086 | 0.034
13.938753 | 62.946494 | 523702890174595456 | 0.620434 | 0.129 | 0.766964 | 0.071
14.457916 | 46.742013 | 377967807928167552 | 1.557556 | 0.059 | 0.569134 | 0.181
17.516555 | 60.901287 | 522445255030705920 | 0.357662 | 0.183 | 0.653228 | 0.098
18.801672 | 57.606248 | 413446544200880896 | 0.951178 | 0.076 | 7.547427 | 0.073
20.433859 | 58.575703 | 413930359370384384 | 1.745764 | 0.211 | 74.595949 | 0.184
23.839866 | 48.216903 | 399464707656254976 | 6.934429 | 0.026 | 8.655612 | 0.066
24.594285 | 62.368258 | 511572184543165184 | 1.111992 | 0.017 | 0.893415 | 0.099
25.46978 | 48.453185 | 405343761970536320 | 3.214925 | 0.181 | 0.475366 | 0.051
26.105725 | 57.363963 | 505854376178706048 | 0.395542 | 0.283 | 0.613647 | 0.075
26.99833 | 58.484187 | 506382008620308352 | 0.469946 | 0.117 | 1.607418 | 0.082
27.345582 | 51.957608 | 407610302112852992 | 0.27613 | 0.067 | 4.742124 | 0.105
27.478311 | 61.054208 | 511202954794627072 | 2.522958 | 0.088 | 2.118876 | 0.037
27.619643 | 53.266914 |  407849617690288128∗ | 1.388514 | 0.182 | 1.842243 | 0.133
29.369827 | 36.938288 | 330442487264141952 | 2.165996 | 0.148 | 3.201325 | 0.069
32.660282 | 56.361212 | 457045959812411520 | 2.907525 | 0.249 | 1.766785 | 0.083
35.274278 | 59.921344 | 507367789513371392 | 4.733592 | 0.119 | 1.111969 | 0.06
35.979435 | 63.411351 | 514335841379134336 | 1.718536 | 0.264 | 1.220503 | 0.075
36.881203 | 63.214977 | 514291963992396928 | 3.006382 | 0.093 | 0.212537 | 0.019
38.031578 | 62.343023 | 513987498047116544 | 1.019208 | 0.086 | 0.948252 | 0.071
38.292867 | 59.484742 | 465028723463664384 | 3.460204 | 0.192 | 0.92533 | 0.045
39.195205 | 60.984813 | 465274395594682240 | 3.887904 | 0.148 | 1.896361 | 0.083
42.417307 | 58.265701 | 461012963401936256 | 1.060832 | 0.048 | 10.593118 | 0.034
44.301404 | 59.454303 | 461553472144934656 | 0.270063 | 0.053 | 1.385304 | 0.049
49.178341 | -10.308834 | 5166225811703486464 | 0.275808 | 0.248 | 2.432133 | 0.098
54.711333 | 44.047482 | 244408588611828096 | 0.178865 | 0.145 | 0.243028 | 0.022
57.458651 | 56.117922 | 445534966419218560 | 1.292364 | 0.246 | 1.088451 | 0.051
59.624908 | 56.534336 | 469545650369412864 | 0.224785 | 0.178 | 1.551227 | 0.05
64.168902 | 50.317876 | 271381051952218496 | 2.245528 | 0.108 | 1.40276 | 0.05
67.213782 | 42.115243 | 228205115816360832 | 1.125492 | 0.13 | 2.201828 | 0.02
67.217417 | 48.572915 | 258090121033397248 | 5.48996 | 0.156 | 25.643685 | 0.035
68.629624 | 48.698441 | 257683439171736576 | 0.81849 | 0.107 | 4.499209 | 0.08
69.925315 | 29.134575 |  158123726424621824∗ | 1.58676 | 0.128 | 13.783777 | 0.06
72.308262 | 52.265466 | 260945106052343296 | 3.070683 | 0.241 | 37.120622 | 0.061
73.106774 | 45.534572 | 206454920393197440 | 0.583875 | 0.365 | 0.715742 | 0.089
76.053644 | 42.807356 | 202201601392851072 | 0.624014 | 0.05 | 14.132366 | 0.087
77.972018 | 51.672042 | 262779881722935424 | 1.637614 | 0.193 | 2.266959 | 0.101
80.461508 | 37.37017 | 184468231186516352 | 2.345842 | 0.133 | 6.86879 | 0.09
80.679986 | 37.664808 | 184497570107735296 | 2.3771 | 0.094 | 2.839768 | 0.055
82.738969 | 44.803406 | 207943285475761280 | 52.829036 | 0.274 | 1.184305 | 0.066
83.858615 | 33.004921 | 3448886540015107584 | 2.989964 | 0.115 | 0.52143 | 0.06
86.591414 | 28.923839 | 3443402279093860096 | 0.53534 | 0.144 | 0.541528 | 0.079
87.5282 | 21.330875 | 3400120484904001408 | 3.572446 | 0.166 | 1.759766 | 0.066
88.00886 | 8.805691 | 3335190578070549632 | 0.36877 | 0.083 | 1.890454 | 0.068
96.842245 | 21.72127 | 3376140681762631168 | 0.39992 | 0.093 | 0.842531 | 0.049
98.587002 | 7.745466 | 3325963098535865728 | 0.584642 | 0.307 | 0.6136 | 0.053
Table 2: Continued RA [J2000.0] | DEC [J2000.0] | Gaia dr3 source ID | $\mathrm{P_{A}[d]}$ | $\mathrm{D_{A}}$ | $\mathrm{P_{B}[d]}$ | $\mathrm{D_{B}}$
---|---|---|---|---|---|---
101.872246 | -0.444575 | 3113400013097881600 | 0.86539 | 0.121 | 2.182992 | 0.048
103.057739 | -1.125556 | 3112366227355971328 | 0.948312 | 0.155 | 0.632336 | 0.069
104.069404 | 15.809035 | 3355042672829484032 | 2.502663 | 0.177 | 0.530042 | 0.06
105.184549 | 32.479562 | 890716293806690816 | 0.263998 | 0.105 | 3.144128 | 0.136
105.242128 | -16.522209 | 2935716391431248256 | 0.424388 | 0.076 | 7.412474 | 0.067
109.571295 | 1.765508 | 3111928351141425792 | 0.274758 | 0.174 | 0.144463 | 0.054
109.612639 | -10.149447 | 3047190824491991168 | 0.45911 | 0.163 | 0.25037 | 0.05
111.089388 | -2.032067 | 3061770932787038336 | 0.422212 | 0.082 | 0.966995 | 0.054
111.47435 | -19.281677 | 2930398740886024064 | 0.347152 | 0.219 | 2.294894 | 0.144
117.697495 | 28.467895 | 875709132614498560 | 0.442358 | 0.158 | 1.466414 | 0.069
118.096034 | -5.697484 | 3044312990239139968 | 1.452632 | 0.176 | 1.779009 | 0.072
202.102553 | 61.876817 | 1663741485747221248 | 2.770754 | 0.15 | 8.163296 | 0.08
223.42518 | 52.715832 | 1594082407606370176 | 2.700472 | 0.074 | 1.488501 | 0.043
242.649013 | 34.62108 | 1323756753679794048 | 7.041158 | 0.114 | 0.128904 | 0.014
257.277242 | 49.948796 | 1414515322519168640 | 0.301544 | 0.197 | 0.231617 | 0.019
259.250732 | -20.297582 | 4115999620890627968 | 0.621141 | 0.24 | 0.773402 | 0.114
263.775976 | 38.311866 | 1343353448205456256 | 3.048348 | 0.095 | 11.792556 | 0.033
273.743299 | 39.207529 | 2109307852667550464 | 2.205617 | 0.187 | 4.373204 | 0.029
276.008132 | -5.774128 | 4161007815803011968 | 3.070536 | 0.211 | 1.371504 | 0.102
278.026086 | 54.402996 | 2147458221793949184 | 3.109478 | 0.097 | 1.853905 | 0.079
279.117985 | -8.525243 | 4156431820198005376 | 1.407928 | 0.281 | 1.331908 | 0.085
279.719282 | 13.163625 | 4508288602094506112 | 0.399492 | 0.061 | 1.528379 | 0.102
280.635591 | -5.252461 | 4256607950976483968 | 1.907376 | 0.222 | 0.903688 | 0.036
280.635808 | 13.120684 | 4505600296166989312 | 4.870614 | 0.097 | 4.70215 | 0.019
280.713417 | 34.65467 | 2091761983553949696 | 0.271486 | 0.204 | 0.372731 | 0.028
281.121588 | 11.13593 | 4504043245957168640 | 10.015578 | 0.077 | 1.042907 | 0.022
282.145741 | 30.399305 | 2041659197183314304 | 0.283664 | 0.061 | 55.568657 | 0.06
282.312661 | 13.469479 | 4505497904155844992 | 0.28221 | 0.303 | 3.150768 | 0.103
283.996449 | 28.282368 | 2040256804464235648 | 5.739774 | 0.104 | 0.139815 | 0.012
284.492451 | -11.193995 | 4201927347951103488 | 0.513114 | 0.405 | 0.453093 | 0.05
285.268359 | 18.064422 | 4517289341731165568 | 1.056042 | 0.07 | 0.545708 | 0.031
286.391765 | 16.558561 | 4513882917294832384 | 0.875846 | 0.043 | 2.838248 | 0.022
286.904493 | -9.457319 | 4204113619406564480 | 0.33327 | 0.097 | 0.867417 | 0.047
287.114664 | -8.543055 | 4204326997705570304 | 0.272314 | 0.3 | 0.170204 | 0.061
287.445354 | 19.597742 | 4516377056329377664 | 1.035768 | 0.171 | 6.325886 | 0.082
287.648865 | 16.355321 | 4513263857861420672 | 0.45636 | 0.166 | 2.129335 | 0.1
287.884621 | 46.028573 | 2130268255146656128 | 0.422266 | 0.269 | 1.029215 | 0.035
288.026347 | 23.298684 | 4521240642906912128 | 3.00593 | 0.011 | 1.616295 | 0.071
288.248394 | 0.080909 | 4264206469687918848 | 3.166976 | 0.208 | 1.852653 | 0.103
288.471817 | -1.094539 | 4263034768239285632 | 1.862082 | 0.16 | 0.592354 | 0.053
288.648651 | 22.157332 | 4520164564651728128 | 0.355088 | 0.09 | 0.32902 | 0.045
288.825262 | 4.312846 | 4293109022582320640 | 0.390062 | 0.277 | 0.164066 | 0.11
289.517888 | 15.160579 | 4320727174167431296 | 2.163692 | 0.033 | 1.607176 | 0.042
290.184501 | 28.541396 | 2038100043701450112 | 0.325552 | 0.122 | 79.87107 | 0.122
290.604938 | 9.66816 | 4308597705466532224 | 5.097978 | 0.173 | 1.412602 | 0.057
290.991691 | 35.958626 | 2050141310916611584 | 3.877374 | 0.021 | 4.766349 | 0.047
291.640789 | 4.331223 | 4292639977810529152 | 1.64215 | 0.063 | 1.061804 | 0.058
291.714347 | 18.228635 | 4323257219102940288 | 1.588804 | 0.186 | 2.302579 | 0.077
291.820676 | 35.97582 | 2049954600104034432 | 4.030792 | 0.113 | 14.062535 | 0.062
293.462031 | 19.846805 | 1825483906846429056 | 6.45557 | 0.145 | 2.467705 | 0.047
293.504188 | 27.432478 | 2025279046615280512 | 2.759184 | 0.134 | 0.863998 | 0.026
293.592507 | 11.006839 | 4314657904341880960 | 3.349092 | 0.109 | 1.32064 | 0.027
293.657387 | 19.881442 | 1825578567909317760 | 4.770348 | 0.152 | 2.118271 | 0.038
293.823697 | 25.752582 | 2021549301357843328 | 0.417266 | 0.168 | 0.182446 | 0.089
Table 3: Continued RA [J2000.0] | DEC [J2000.0] | Gaia dr3 source ID | $\mathrm{P_{A}[d]}$ | $\mathrm{D_{A}}$ | $\mathrm{P_{B}[d]}$ | $\mathrm{D_{B}}$
---|---|---|---|---|---|---
294.156333 | 14.316215 | 4318042922678224768 | 1.840298 | 0.023 | 0.915486 | 0.087
294.332664 | 27.158424 | 2025146250574308480 | 0.430934 | 0.184 | 0.698726 | 0.062
294.518815 | -7.404433 | 4207090654517833344 | 3.426032 | 0.186 | 0.315937 | 0.04
294.832597 | 26.871716 | 2025081478136260480 | 3.072568 | 0.066 | 2.212912 | 0.052
294.839444 | 35.975154 | 2048268017993738624 | 0.234828 | 0.043 | 4.944031 | 0.052
294.933937 | 20.202009 | 1825745006505362048 | 7.807304 | 0.051 | 3.649778 | 0.042
295.309706 | 42.822037 | 2077913565886105344 | 0.373692 | 0.344 | 0.322663 | 0.03
295.479516 | 34.635703 | 2047323949824485376 | 2.346872 | 0.18 | 2.481432 | 0.055
295.527859 | 22.054362 | 1827708802964917888 | 0.545902 | 0.187 | 0.96728 | 0.047
295.87929 | 25.881643 | 2021774220180193536 | 1.357318 | 0.0 | 2.09345 | 0.026
296.235173 | 29.583425 | 2031977619399032448 | 8.147016 | 0.064 | 8.083514 | 0.023
296.585034 | 23.395937 | 2020056405032433152 | 0.812118 | 0.18 | 0.753871 | 0.086
296.895517 | 25.636448 | 2020970648976206208 | 6.375142 | 0.168 | 323.201568 | 0.109
296.955946 | 30.241753 | 2032118077650924672 | 171.076078 | 0.28 | 2.335024 | 0.089
297.00612 | 24.058537 | 2020490437267187712 | 1.35866 | 0.148 | 1.729201 | 0.093
297.034812 | 30.08806 | 2031925182082532608 | 3.590404 | 0.001 | 0.789163 | 0.032
297.056926 | 25.787105 | 2026976864358731904 | 1.425158 | 0.0 | 0.89166 | 0.105
298.002327 | 31.866931 | 2033828574151981184 | 1.768354 | 0.231 | 0.571495 | 0.122
298.172845 | 26.864218 | 2027154955201119488 | 1.647592 | 0.093 | 1.191313 | 0.026
298.530047 | 24.682368 | 1834469768675202560 | 2.881026 | 0.081 | 4.353022 | 0.036
298.820248 | 32.01454 | 2033656156987971840 | 7.548416 | 0.327 | 0.159107 | 0.049
298.881534 | 29.89156 | 2030312580776346240 | 0.462184 | 0.179 | 7.651529 | 0.06
298.908525 | 32.746538 | 2034277144881784704 | 2.452324 | 0.245 | 0.558699 | 0.026
299.522067 | 30.246542 | 2030426861313716224 | 1.305256 | 0.061 | 1.607371 | 0.019
299.620393 | 33.527522 | 2034419218049672704 | 1.411226 | 0.099 | 1.614413 | 0.072
299.976835 | 29.83971 | 2030212490900354176 | 1.189566 | 0.022 | 3.314311 | 0.025
301.355473 | 34.989363 | 2058617549147113728 | 2.08353 | 0.045 | 15.704131 | 0.051
301.37899 | 35.97206 | 2059117002318116096 | 1.741608 | 0.114 | 1.064947 | 0.033
301.391172 | 42.318524 | 2074882590295669376 | 1.127028 | 0.197 | 2.46579 | 0.132
302.250175 | 21.436492 | 1829670778385777920 | 5.414028 | 0.109 | 3.970943 | 0.085
302.637062 | 38.400325 | 2061735248734886400 | 1.079346 | 0.226 | 0.468297 | 0.029
302.98334 | 36.463072 | 2059178162649303808 | 5.993378 | 0.266 | 0.563785 | 0.056
303.229266 | 27.908953 | 1836995514370597120 | 0.374674 | 0.076 | 3.126105 | 0.05
303.523133 | 38.382872 |  2060949991271681664∗ | 1.144666 | 0.129 | 0.481849 | 0.017
304.151811 | 41.930677 | 2068658632915468672 | 4.363316 | 0.078 | 1.242846 | 0.052
304.756649 | 37.920034 | 2061050802729418496 | 1.325726 | 0.172 | 0.813421 | 0.097
304.795015 | 30.560399 | 1861612037845705728 | 5.302642 | 0.194 | 0.323736 | 0.037
306.408352 | 37.894974 | 2058085351143518464 | 1.152476 | 0.193 | 85.498447 | 0.139
308.681181 | 48.20847 | 2167733048719069312 | 2.17693 | 0.17 | 0.584749 | 0.086
308.779214 | 35.826622 | 2056704914305732096 | 5.152764 | 0.146 | 0.910183 | 0.069
309.435623 | 30.852742 | 1862347439319475072 | 0.920364 | 0.145 | 1.417808 | 0.094
311.55134 | 31.682545 | 1859824197571796736 | 0.283018 | 0.133 | 0.394832 | 0.064
311.97923 | 34.771038 | 1869489385821202816 | 0.488582 | 0.115 | 2.110629 | 0.109
312.564502 | 12.412033 | 1760943570684334208 | 3.047312 | 0.209 | 15.580245 | 0.076
312.603366 | 47.202077 | 2166511040319614208 | 0.38052 | 0.147 | 1.264149 | 0.127
313.727051 | 39.848684 | 1872913471188045184 | 2.26941 | 0.054 | 0.678956 | 0.023
314.203035 | 66.204173 | 2245787855903677952 | 1.313048 | 0.21 | 1.250967 | 0.228
315.998617 | 47.984506 | 2165439879783209344 | 3.822778 | 0.095 | 2.650119 | 0.037
316.499668 | 36.761395 | 1868409321799840768 | 0.543036 | 0.063 | 0.725643 | 0.036
316.935639 | 47.04712 | 2165141946503725056 | 1.272084 | 0.103 | 3.086064 | 0.035
317.167097 | -8.177796 | 6897094092939104768 | 0.223992 | 0.027 | 0.622656 | 0.041
317.354427 | 55.485175 | 2176903937763785088 | 3.446506 | 0.256 | 3.158925 | 0.236
318.20909 | 56.26039 | 2177329891137899904 | 0.90333 | 0.253 | 0.515231 | 0.087
318.442107 | 51.071054 | 2166229226047556096 | 15.0598 | 0.169 | 3.180452 | 0.086
Table 4: Continued RA [J2000.0] | DEC [J2000.0] | Gaia dr3 source ID | $\mathrm{P_{A}[d]}$ | $\mathrm{D_{A}}$ | $\mathrm{P_{B}[d]}$ | $\mathrm{D_{B}}$
---|---|---|---|---|---|---
321.274386 | 49.322575 | 2170863908089217280 | 3.714414 | 0.154 | 2.643588 | 0.078
322.785455 | 37.809557 | 1952168090372267008 | 1.158128 | 0.035 | 1.25026 | 0.023
322.795044 | 41.61673 | 1967117802081173504 | 4.672324 | 0.102 | 3.584245 | 0.036
324.064461 | 40.289056 | 1966045232784781952 | 0.264192 | 0.119 | 0.779663 | 0.061
325.003123 | 56.109079 | 2178057359827382144 | 3.296832 | 0.225 | 1.291317 | 0.036
325.081161 | 48.44538 | 1978029359778790016 | 0.678389 | 0.01 | 0.746095 | 0.054
325.126003 | 39.737305 | 1953928099249228032 | 1.068532 | 0.22 | 1.123977 | 0.027
325.523023 | 58.222165 | 2178581380197081216 | 1.810504 | 0.115 | 0.707697 | 0.062
325.791627 | 27.217748 | 1800509737128120320 | 0.85369 | 0.239 | 3.215801 | 0.166
326.593483 | 49.781513 | 1978958103506727424 | 0.421342 | 0.068 | 1.667499 | 0.039
327.020592 | 62.78985 | 2216420454386370176 | 2.14556 | 0.228 | 0.404213 | 0.082
327.271647 | 58.943552 | 2202597664775578112 | 1.06056 | 0.143 | 2.05528 | 0.107
328.521806 | 57.784601 | 2199284050276697216 | 0.433052 | 0.121 | 1.459645 | 0.106
329.000459 | 52.050747 | 1981103972253289088 | 0.685961 | 0.114 | 0.739756 | 0.027
329.356711 | 45.001855 | 1973338392191011584 | 0.397442 | 0.112 | 6.522002 | 0.093
329.904956 | 43.718061 | 1961000256819235200 | 0.473181 | 0.095 | 0.964035 | 0.103
330.10459 | 57.341933 | 2199104211409698816 | 0.979846 | 0.194 | 0.443703 | 0.067
330.629145 | 55.984692 | 2198180415480756096 | 1.328192 | 0.0 | 22.393498 | 0.086
332.366702 | 56.166443 | 2197966251234148352 | 2.705704 | 0.168 | 2.822163 | 0.04
332.720979 | 59.64723 | 2199925099913441920 | 2.15344 | 0.175 | 1.57226 | 0.067
332.943294 | 54.42099 | 2005474711889658624 | 3.234134 | 0.101 | 0.617324 | 0.135
333.801196 | 53.749768 | 2004590532743751424 | 0.340078 | 0.058 | 2.522543 | 0.034
334.655366 | 55.538438 | 2005911802117955968 | 1.508362 | 0.197 | 0.947192 | 0.097
335.103223 | 36.014911 | 1905994992912248448 | 0.834723 | 0.19 | 0.422142 | 0.116
335.115083 | 24.563415 | 1878929101147587200 | 2.107929 | 0.256 | 2.068346 | 0.048
337.436575 | 54.279883 | 2001849763802035840 | 2.263908 | 0.235 | 0.545817 | 0.03
337.832153 | 60.857037 | 2201723866572302976 | 2.627162 | 0.107 | 101.237102 | 0.125
339.088072 | 47.538959 | 1986716807300365696 | 6.451769 | 0.218 | 0.164641 | 0.043
339.963254 | 54.98005 |  2003428628134264448∗T | 2.184788 | 0.145 | 88.853882 | 0.143
340.921211 | 57.261299 | 2007240188275431936 | 1.625996 | 0.064 | 1.379002 | 0.038
341.302946 | 56.012542 | 2003907912126462592 | 0.404544 | 0.102 | 2.944626 | 0.058
341.422596 | 53.230019 | 2002122850696945664 | 3.484958 | 0.063 | 2.227952 | 0.036
342.74 | 58.714103 | 2007474143733297152 | 1.63917 | 0.116 | 1.617598 | 0.023
344.803515 | 48.306854 | 1984790600366568576 | 0.699564 | 0.452 | 1.513966 | 0.034
347.564145 | 65.090617 | 2208692193312331904 | 5.095412 | 0.059 | 1.198208 | 0.03
349.093506 | 66.228219 | 2210495392378660864 | 2.568998 | 0.241 | 285.579588 | 0.218
350.55902 | 56.248535 | 1997226175657082496 | 0.351738 | 0.091 | 1.855745 | 0.043
351.050509 | 59.819433 | 2010769070836151424 | 1.312382 | 0.17 | 0.89549 | 0.057
352.521122 | 57.484618 | 1998851258144250112 | 0.280512 | 0.094 | 1.619602 | 0.047
352.593979 | 48.961381 | 1942119756681849216 | 4.303021 | 0.4 | 0.149352 | 0.039
353.524226 | 36.870485 | 1918839728265382912 | 0.366704 | 0.208 | 4.201864 | 0.116
358.069196 | 62.459358 | 2012994482376100736 | 1.950456 | 0.256 | 34.404796 | 0.119
358.221987 | 61.525403 | 2012695312133566208 | 3.21884 | 0.179 | 2.533716 | 0.056