# Simulations of Triple Microlensing Events I: Detectability of a scaled Sun-
Jupiter-Saturn System

Renkun Kuang (匡仁昆) 1,2, Weicheng Zang (臧伟呈) 1,3, Shude Mao (毛淑德)1,4, Jiyuan
Zhang (张纪元)1, Haochang Jiang (蒋昊昌) 1,5
1Department of Astronomy, Tsinghua University, Beijing 100084, China
2Department of Engineering Physics, Tsinghua University, Beijing 100084, China
3Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden St.,Cambridge,
MA 02138, USA 4National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China 5European Southern Observatory, Karl-
Schwarzschild-Str 2, 85748 Garching, Germany E-mail:
<EMAIL_ADDRESS>0000-0003-2337-0533 0000-0001-6000-3463
0000-0003-2948-5614

(Accepted 2023 February 7. Received 2023 February 3; in original form 2022
November 23)

###### Abstract

Up to date, only 13 firmly established triple microlensing events have been
discovered, so the occurrence rates of microlensing two-planet systems and
planets in binary systems are still uncertain. With the upcoming space-based
microlensing surveys, hundreds of triple microlensing events will be detected.
To provide clues for future observations and statistical analyses, we initiate
a project to investigate the detectability of triple-lens systems with
different configurations and observational setups. As the first step, in this
work we develop the simulation software and investigate the detectability of a
scaled Sun-Jupiter-Saturn system with the recently proposed telescope for
microlensing observations on the “Earth 2.0 (ET)” satellite. With the same
$\Delta\chi^{2}$ thresholds of detecting a single planet and two planets, we
find that the detectability of the scaled Sun-Jupiter-Saturn analog is about
1% and the presence of the Jovian planet suppresses the detectability of the
Saturn-like planet by $\sim$13% regardless of the adopted detection
$\Delta\chi^{2}$ threshold. This suppression probability could be at the same
level as the Poisson noise of future space-based statistical samples of
triple-lenses, so it is inappropriate to treat each planet separately during
detection efficiency calculations.

###### keywords:

gravitational lensing: micro – planets and satellites: detection

††pubyear: 2022††pagerange: Simulations of Triple Microlensing Events I:
Detectability of a scaled Sun-Jupiter-Saturn System–References

##  1 Introduction

With complementary exoplanet detection methods such as the transit, radial
velocity, microlensing, and direct imaging, there are more than 5,000
confirmed exoplanets111https://exoplanetarchive.ipac.caltech.edu/. The fast
pace of exoplanet detection triggers studies in various aspects from
statistics (to capture the main characteristics of the known planetary sample)
to planet formation and evolution theories (to understand and reproduce the
main properties of the known planetary sample), see e.g., Zhu & Dong (2021)
for a recent review. With these studies, we have a better understanding of
both the solar system and exoplanet systems. For example, multi-body systems
are common. Both the planet multiplicity rate (e.g., Zhu, 2022) and the
stellar multiplicity rate of planet-host stars (e.g., Wang et al., 2014) are
substantial. These systems are valuable for studies on planet formation and
evolution theories.

Among different exoplanet detection methods, the microlensing method (Mao &
Paczyński, 1991; Gould & Loeb, 1992) allows us to detect exoplanets beyond the
snowline and covers an important parameter space (e.g., Gaudi, 2012; Mao,
2012). Although some statistical studies (Gould et al., 2010; Cassan et al.,
2012; Shvartzvald et al., 2016; Suzuki et al., 2016) on microlensing planets
showed that microlensing planets are abundant, with a frequency of order $\sim
50\%$–100% (with planetary masses down to several Earth-mass), no specific
study has been conducted to investigate the statistical properties of
microlensing planets in multi-body systems. One reason is that the sample size
of triple-lens systems still being small. There are 13 firmly established
triple microlensing events, including five two-planet systems and eight
circumbinary (or circumstellar) planets (see Table 7 for the list). A larger
sample can be built with the upcoming space missions such as the Nancy Grace
Roman Space Telescope (Roman, formerly WFIRST, Spergel et al., 2015), the
Chinese Space Station Telescope (CSST, Yan & Zhu, 2022), and the “Earth 2.0”
(ET, Gould et al., 2021; Ge et al., 2022) satellite222The ET satellite
consists of seven telescopes (with pupil diameter $\sim$30 cm), of which six
will be used for transit observations and one for microlensing observations..
For example, the joint survey with the telescope for microlensing observations
on the ET satellite and the Korea Microlensing Telescope Network (KMTNet, Kim
et al. 2016) is expected to discover several hundred planetary events, among
them $\sim 5\%$333From the simulation results of Zhu et al. (2014a), as well
as from the apparent multi-planetary fraction among all microlensing planets
from observations. may show multiple-planet signatures (Ge et al., 2022).

Apart from the forecast that we will discover a large number of triple-lens
systems, currently we know little about the properties of these systems. Many
factors are involved such as the intrinsic property of the lenses and sources,
the instruments, the observing strategies, and the detectabilities of triple-
lens systems. To provide some clues for future observations, we initiate a
project based on simulations to investigate the detectabilities of different
triple-lens systems with different configurations and observation setups. In
addition, we note that the magnification pattern and light curves from
multiple planets may be degenerate with those from single-planet lensing
(e.g., Gaudi et al., 1998; Zhu et al., 2014b). Some of the previous
microlensing planet-detection sensitivity calculations ignore multiple planets
or treat each planet as being independent (e.g., Suzuki et al., 2018). Here,
we investigate to what extent the detectability of a planet will be suppressed
or enhanced by the presence of a second planet. Due to the large parameter
space, in this paper we focus on building a pipeline from light curve
calculation to planetary signal detection. And as the first step, we take a
scaled Sun-Jupiter-Saturn system as the lens and take the telescope for
microlensing observations on the ET satellite as an example survey. Below we
discuss why we choose a scaled Sun-Jupiter-Saturn system as the lens.

Despite the discovery of many exoplanets, an outstanding question remains to
be answered from both theoretical and observational sides is how unique the
solar system is (e.g., Beer et al., 2004; Martin & Livio, 2015; Portegies
Zwart, 2019; Raymond et al., 2020), i.e., what is the frequency of solar
system analogs?

The formation and evolution of the solar system involves many intertwined
processes, especially when multiple planets cover a wide range of orbital
distances. The in-situ formation of outer giant planets is difficult due to
their longer formation timescale compared with the lifetime of protoplanetary
disks. In addition, the solar system has some peculiarities, such as the
Earth/Mars mass ratio being exceptionally large, and the asteroid and Kuiper
belts being low mass yet dynamically excited (Raymond et al., 2020). Some
studies found that the instability of giant planets may have a significant
impact on the inner solar system (e.g., Clement et al., 2019). The Grand Tack
model (Walsh et al., 2011) suggested that Jupiter and Saturn underwent a two-
stage, inward-then-outward migration, which can truncate the inner disk of
rocky material and can explain the large Earth/Mars mass ratio.

Due to the important role of Jupiter and Saturn in the formation of the solar
system, and the dimensions of a planetary system may scale with the snowline
radius (Min et al., 2011; Kennedy & Kenyon, 2008), throughout this paper we
define the solar system analogs as systems that have both Jovian and Saturn-
like planets at similar locations in units of the snowline radius of the host
star. Under this definition, the determination of the frequency of solar
system analogs from the observational side is still challenging due to the
difficulties in detecting long-period planets. For example, a decade-long
radial velocity survey would be required to detect Jupiter orbiting the Sun,
while Saturn, Uranus, and Neptune are too distant (Raymond et al., 2020).

Among the five firmly established two-planet systems detected with
microlensing, OGLE-2006-BLG-109L has a pair of Jupiter/Saturn planets and was
regarded as a “scaled version” of our solar system (Gaudi et al., 2008;
Bennett et al., 2010). It is not clear whether this detection is due to a
coincidence or due to the intrinsic high fraction of such systems. This leads
to one of the aims of this paper: to investigate the detectability of scaled
Sun-Jupiter-Saturn systems with a space telescope, such as ET.

There are some other reasons for us to use a scaled Sun-Jupiter-Saturn system
as the lens. First, in most cases only two planets are detectable although
there are multiple planets in the lenses (Shvartzvald & Maoz, 2012; Zhu et
al., 2014a). Second, the inner and the outer planetary systems are correlated
that outer cold Jupiters almost always have inner planetary companions (Zhu &
Dong, 2021), so the frequency of Jupiter/Saturn analogs could be regarded as a
rough approximation to the frequency of solar system analogs. Third, the
multiplicity rate of giant planets is about 50% (Bryan et al., 2016; Zhu,
2022). Lastly, the light curve calculations for multiple lenses are
computationally expensive.

Ever since the microlensing planet detection method has been proposed,
parallel with observational efforts, there were numerical simulations to
investigate the detection probabilities and to predict possible outcomes of
observational experiments. Due to it being computationally expensive in both
light curve (or magnification map) calculations and analyses (to identify
planetary / multiplanetary events), previous simulations generally investigate
either with small samples of lenses (e.g., Gaudi et al., 1998) or source
trajectories (e.g., Zhu et al., 2014a, b), or with simplified “detection
criteria” (e.g., Ryu et al., 2011). We survey these works below.

Ryu et al. (2011) investigated the detection probability of a low-mass planet
from high-magnification events caused by triple-lens systems composed of a
star, a Jovian mass planet, and a low-mass planet. They use the Gould & Loeb
(1992) criterion to calculate the fractional deviation in magnification maps
at the central region. However, the detection of deviation signals for a low-
mass planet may not be sufficient for the discovery of such a planet.

Shvartzvald & Maoz (2012) used scaled solar system analogs as the lenses and
took sampling sequences and photometric error distributions from a real
observing experiment. They found that a generation-II microlensing network can
find about 50 planetary events, among them $1/6$ reveal two-planet anomalies.
However, they did not fit their light curves with a binary-lens model to
identify the two-planet events. Instead, they use a “running” local $\chi^{2}$
estimator with 31 consecutive points to detect short-timescale deviations in
the light curve relative to a point-lens model (see their §3.5 and §4.3 for
more details). Furthermore, they did not investigate how the detectability of
a planet will be inhibited or promoted by other planets in the same system.

Zhu et al. (2014a, b) conducted a simulation in the context of the core
accretion planet formation model (Ida & Lin, 2004). Zhu et al. (2014a)
generated 10 light curves for each of the 669 selected planetary systems and
found the fraction of planetary events is 2.9%, out of which 5.5% show
multiple-planet signatures. Among the 23 two-planet event candidates, they
confirmed 16 triple-lens events with the criterion $\Delta\chi^{2}>300$, where
$\Delta\chi^{2}$ is the $\chi^{2}$ difference between the best-fit binary-lens
model and the input multiplanetary model.

To remedy the small sample and loose detection critetia of previous studies,
in this work, we simulate and analyse $4\times 10^{5}$ events to investigate
the detectability of a scaled Sun-Jupiter-Saturn system, the suppression and
enhancement effects, and their dependence on parameters, e.g., the impact
parameter. The paper is structured as follows. In §2, we introduce the
microlensing basics, and details about how we generate light curves and detect
planets from the light curves. In §3, we present the results. We then discuss
the results and future works in §4 and conclude in §5.

##  2 Simulation and Light-curve Analysis

###  2.1 Microlensing basics

A microlensing event occurs when a lens object is close to the line of sight
from the observer to the background source. The light rays from the background
source are deflected by the lens, forming multiple images as seen by the
observer. The projected position of the source, lens, and images are related
by the lens equation (Witt, 1990),

$z_{s}=z-\sum_{j=1}^{N}\frac{m_{j}}{\overline{z}-\overline{z_{j}}},$ (1)

where $z_{s},z,z_{j}$ are respectively the complex positions of the source,
the image, and the $j$-th lens, $\overline{z}$ and $\overline{z_{j}}$ are
respectively the complex conjugates of $z$ and $z_{j}$, and $m_{j}$ is the
fractional mass of the $j$-th lens, with $\sum_{j}m_{j}=1$. The positions are
represented in units of the angular Einstein radius $\theta_{\rm E}$ of the
lensing system,

$\displaystyle\theta_{\rm E}$
$\displaystyle=\sqrt{\frac{4GM}{c^{2}}\frac{D_{\rm S}-D_{\rm L}}{D_{\rm
L}D_{\rm S}}}$ (2) $\displaystyle=0.55\;\mathrm{mas}\sqrt{\frac{1-D_{\rm
L}/D_{\rm S}}{D_{\rm L}/D_{\rm S}}}\left(\frac{8\;\mathrm{kpc}}{D_{\rm
S}}\frac{M}{0.3\;M_{\odot}}\right)^{1/2},$

where $M$ is the total mass of the lens system, $D_{\rm L}$ and $D_{\rm S}$
are the distances from the observer to the lens and the source, respectively,
$c$ is the speed of light, and $G$ is the gravitational constant.

In Equation (1), for a given source position $z_{s}$, multiple solutions for
$z$ exist. For an extended source, multiple distorted images are formed, with
separations of the order of $\mathrm{mas}$. In most cases, these images cannot
be resolved. Instead, the observable is the change of the total flux due to
the changing magnification. The time scale of a microlensing event is related
to the lens-source relative proper motion $\mu_{\rm rel}$ by

$t_{\rm E}=\frac{\theta_{\rm E}}{\mu_{\rm{rel}}}.$ (3)

Throughout the paper, we denote a triple-lens system with five parameters
($s_{2}$, $q_{2}$, $s_{3}$, $q_{3}$, $\psi$). Here $q_{2}$ and $q_{3}$ are
respectively the mass ratios of the second and third lens to the first lens,
$s_{2}$ and $s_{3}$ are respectively the separations normalized to
$\theta_{\rm E}$ of the second and third lens from the first lens, and $\psi$
represents the orientation angle of the third lens. The relations between
($s_{2}$, $q_{2}$, $s_{3}$, $q_{3}$, $\psi$) and $m_{j}$, $z_{j}$ are,

$\displaystyle m_{1}=1/(1+q_{2}+q_{3}),\quad m_{2}=q_{2}m_{1},\quad
m_{3}=q_{3}m_{1},$ (4) $\displaystyle z_{1}=-q_{2}s_{2}/(1+q_{2})+i\;0,$
$\displaystyle z_{2}=s_{2}/(1+q_{2})+i\;0,$ $\displaystyle
z_{3}=-q_{2}s_{2}/(1+q_{2})+s_{3}\cos{\psi}+i\;s_{3}\sin{\psi},$

where $i$ is the imaginary unit. Note that the first and second lenses are on
the $x$-axis.

###  2.2 Lens and source properties

In reality, the lens physical parameters vary from event to event. For
simplicity, we fix the lens mass as $M=0.5\,M_{\odot}$, the lens distance as
$D_{\rm L}=5\;\mathrm{kpc}$, the source distance as $D_{\rm
S}=8\;\mathrm{kpc}$. So the angular Einstein radius is $\theta_{\rm E}=0.55$
$\mathrm{mas}$, corresponding to an Einstein radius of $R_{\rm E}=D_{\rm
L}\theta_{\rm E}=2.75$ au.

In the Solar system, the mass ratios of Jupiter and Saturn to the Sun are
$q_{\rm Jupiter}=9.546\times 10^{-4}$, and $q_{\rm Saturn}=2.857\times
10^{-4}$, respectively. The distances from Jupiter and Saturn to the Sun are
$4.965\;\mathrm{au}$ and $9.905\;\mathrm{au}$, respectively444The distances
are calculated based on real-time locations at the epoch $\rm{MJD}=59731$ with
the Python package solarsystem (Nasios, 2020).. We adopt $q_{2}=q_{\rm
Jupiter}$ and $q_{3}=q_{\rm Saturn}$. The planet-host distances are scaled
such that the distances remain the same in units of the snowline distance555We
assume the snowline distance scales as $a_{\rm SL}=2.7(M/M_{\odot})$ au,
(Kennedy & Kenyon, 2008)., so (regardless of the projection effect)

$s_{2,\rm{true}}=0.903,\quad s_{3,\rm{true}}=1.801.$ (5)

We randomly generate 200 triple-lens systems by using the above lens
parameters but with different planet orbital phases and viewing angles. We
place the planets on their orbit666Jupiter and Saturn are close to the 5 : 2
resonance (so-called mean-motion near resonance). They obey the relation
$5n_{\rm{Saturn}}-2n_{\rm{Jupiter}}\approx 0$, where $n$ is the inverse of
orbital period. But an exact resonance does not exist for Jupiter and Saturn
(Michtchenko & Ferraz-Mello, 2001). So we do not consider the mean motion
resonance and treat the two planets as having random orbital phases. by
sampling the position angle $\psi_{\rm{true}}$ from a uniform distribution
$\mathcal{U}(0,2\pi)$, and the orbital plane of the triple-lens system at
random directions by using two directional angles, $\beta$ and $\delta$. They
define the direction of the source as seen from the orbital plane of the lens
system. Figure 1 shows the geometry. We randomly choose $\beta$ from a uniform
distribution $\mathcal{U}(0,2\pi)$, and $\sin\delta$ from a uniform
distribution $\mathcal{U}(-1,1)$. For each combination of ($\psi_{\rm{true}}$,
$\beta$, $\delta$), we can calculate the projected positions of $m_{2}$ and
$m_{3}$, and convert their positions to triple-lens parameters ($s_{2}$,
$s_{3}$, $\psi$) which follow our convention defined in Equation (4).

Figure 1: A Schematic diagram shows the geometry of the lens orbital plane and
the direction of the source. The host star $m_{1}$ and the two planets $m_{2}$
and $m_{3}$ are located in the same plane, where $m_{1}$ and $m_{2}$ are on
the $x$-axis. The two angles $\beta$ and $\delta$ define the direction of the
line-of-sight (LOS) to the source.

For a given triple-lens system, we then generate 2000 source trajectories with
random impact parameter $u_{0}$ and direction angle $\alpha$, so there are
$200\times 2000=4\times 10^{5}$ lens-source pairs, with the same $t_{0}$ and
$t_{\rm{E}}$, where $t_{0}$ is the time of closest approach of the source to
the origin, $u_{0}$ is the closest distance of the source to the origin in
units of $\theta_{\rm E}$. We use $t_{0}=9000$ (in units of
$\rm{HJD}^{\prime}\equiv\rm{HJD}-2450000$) and $t_{\rm E}=30$ days,
corresponding to a lens-source relative proper motion
$\mu_{\rm{rel}}=6.7\,\rm{mas}/\rm{yr}$. We sample the trajectory angle
$\alpha$ from a uniform distribution $\mathcal{U}(0,2\pi)$. Because events
with small $u_{0}$ (i.e., high-magnification events) are more sensitive to
planets (Griest & Safizadeh, 1998), we use the importance sampling method by
drawing $\log u_{0}$ from a uniform distribution of $\mathcal{U}(-3,0.3)$,
rather than directly drawing $u_{0}$ from a uniform distribution of
$\mathcal{U}(10^{-3},2)$. Then, for $M$ detectable events from $N(=2000)$
simulated light curves, the detectability is

$P=\frac{\Sigma_{j=1}^{M}u_{0,j}}{\Sigma_{i=1}^{N}u_{0,i}},$ (6)

where $i=1,\cdots,N$ represents the index for all source trajectories and
$j=1,\cdots,M$ represents the index for all source trajectories that allow the
detection of two planets.

We adopt a source star with a radius $R_{*}=0.95\,R_{\odot}$, i.e., the scaled
source size $\rho=\theta_{*}/\theta_{\rm E}=0.001$. We assume that the source
has a uniform surface brightness profile, i.e., we do not consider the limb
darkening effect. Furthermore, for simplicity, we do not account for the lens
orbital motion (Batista et al., 2011; Skowron et al., 2011) and the microlens
parallax effect (Gould, 2000) in this work, although these two effects have
been detected in the two-planet event, OGLE-2006-BLG-109 (Gaudi et al., 2008;
Bennett et al., 2010).

###  2.3 Generating light curves

The first goal of this work is to investigate the detectability of
Jupiter/Saturn analogs. The second goal is to find out how the two planets
would affect each other’s detectability. The influence of $m_{3}$ on the
magnification pattern generated by $m_{1}$-$m_{2}$ is non-negligible, see
Figure 3 for an example map of magnification excess (see also Chung et al.,
2005). The planet $m_{3}$ with mass ratio $q_{3}=2.857\times 10^{-4}$ perturbs
the magnification pattern of the $m_{1}$-$m_{2}$ system not only along the
$m_{1}$-$m_{3}$ axis, but also along the $m_{1}$-$m_{2}$ axis (horizontal). In
turn, the magnification pattern of the $m_{1}$-$m_{3}$ system is also
influenced by the presence of $m_{2}$, as shown in Figure 4. The overall
procedures used to achieve the above two goals are shown as a flowchart in
Figure 2.

Figure 2: A flowchart shows the overall procedures used to achieve the main
goals of this work. We follow the convention in the microlensing field to
represent the lensing object of an event. The host star is represented with
capital letter “A”, followed by small letters representing the planets. See §3
for the detailed definitions of the symbols in this flowchart.

In the following, we introduce the detail of the light curve calculations
within the context of the ET satellite. The telescope for microlensing
observations on the ET satellite will have a 4 $\mathrm{deg}^{2}$ field of
view, and will monitor the Galactic bulge with a cadence of 10 minutes from
March 21 to September 21 every year. We take 26.8 as the $I$-band magnitude
zero-point (1 count/s). For simplicity, we adopt the background magnitude due
to surrounding stars as $I_{\rm{blend}}$ = 18.9 and take 20.9 as the apparent
$I$-band magnitude of the source star. We assume the noise is dominated by
Possion noise, and the standard deviation of each photometric measurement is
the square root of the flux measurement in photon counts. To be conservative,
the chosen star is quite faint and thus has fairly large noise. We list the
simulation-related parameters in Table 1.

For a given set of lensing parameters $(s_{2},q_{2},s_{3},q_{3},\psi)$ we
generate three light curves for each source trajectory $(u_{0},\alpha)$. For
the first light curve, the lensing object is a binary-lens system with lensing
components $m_{1}$-$m_{2}$ (designated as the 2Lb1S light curve). For the
second light curve, the lensing components are $m_{1}$-$m_{3}$ (designated as
the 2Lc1S light curve). For the third light curve, the lensing object is the
triple-lens system, $m_{1}$-$m_{2}$-$m_{3}$ (designated as the 3L1S light
curve). Note that when calculating the three light curves, we take into
account the fact that $\theta_{\rm E}$ changes as the total mass of the lens
changes.

Figure 3: An example shows the influence of $m_{3}$ on the magnification
pattern generated by the binary-lens system $m_{1}$-$m_{2}$. The left panel
shows the magnification map corresponding to the triple-lens system
($\mu_{\rm{3L1S}}$). The middle and right panels show the fractional
difference (magnification excess)
$(\mu_{\rm{3L1S}}-\mu_{\rm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S}}})/\mu_{\rm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S}}}$,
the dashed black line connects $m_{1}$ and $m_{3}$, the solid magenta curve
shows the caustics of the triple-lens system. The crosses in the middle panel
show the locations of the three lenses. In the right panel, the solid cyan and
black lines show the $\pm 1\%$ contours for the dotted region showed in the
middle panel. The mass ratios and separations (in units of $\theta_{\rm{E}}$)
of these two planets to the host star are $(s_{2},q_{2})=(0.903,9.55\times
10^{-4})$, $(s_{3},q_{3})=(1.80,2.86\times 10^{-4})$, respectively. The
position angle of $m_{3}$ is $\psi=-33.4^{\circ}$. Figure 4: The same as
Figure 3, except that now we are showing the influence of $m_{2}$ on the
magnification pattern generated by the binary-lens system $m_{1}$-$m_{3}$. The
mass ratios and separations of these two planets are the same as in Figure 3.
The right panel shows a magnified view of the dotted region in the left panel.

We use the VBBinaryLensing (Bozza, 2010; Bozza et al., 2018) and
triplelens777We used version 1.0.8 with the Github commit ID:
bb348cfad1ea865ba5f533f5dbba71c78a66dc81. (Kuang et al., 2021) to calculate
the magnifications for binary-lens and triple-lens, respectively. Then the
light curves are calculated by multiplying the magnifications with the
baseline source flux and adding the constant blend flux and measurement
noises. So for each light curve, we can calculate the
$\chi^{2}_{\rm{theoretical}}$ by using the theoretical model.

Table 1: The simulation-related parameters. Parameters | Value | Parameters | Value
---|---|---|---
$q_{2}(10^{-4})$ | $9.546$ | $\psi_{\rm{true}}$ | $\mathcal{U}(0,2\pi)$
$q_{3}(10^{-4})$ | $2.857$ | $\beta$ | $\mathcal{U}(0,2\pi)$
$s_{2,\rm{true}}$ | $0.903$ | $\sin\delta$ | $\mathcal{U}(-1,1)$
$s_{3,\rm{true}}$ | $1.801$ | $\log u_{0}$ | $\mathcal{U}(-3,0.3)$
$I_{\rm{blend}}$ | $18.9$ |  $t_{\rm E}$ (days) | $30.0$
$I_{\rm{source}}$ | $20.9$ | $\rho(10^{-3})$ | $1.0$
$I_{\rm{zeropoint}}$ | $26.8$ | $\alpha$ | $\mathcal{U}(0,2\pi)$

Figure 5 shows an example of caustic structures and source trajectories. From
left to right, the red curves are the caustics corresponding to the lens
$m_{1}$-$m_{2}$-$m_{3}$, $m_{1}$-$m_{2}$, and $m_{1}$-$m_{3}$, respectively.
The dotted orange line shows the $m_{1}$-$m_{3}$ axis. The black, magenta, and
blue dashed lines with arrows show the source trajectories corresponding to
each lens system. The corresponding model light curves are shown in the upper
panel of Figure 6 with the same colours. The green points with error bars in
the middle and bottom panels of Figure 6 are the simulated data points
corresponding to the triple-lens.

Figure 5: An example case of caustic structures and source trajectories. From
left to right, the black, magenta, and blue solid curves are the caustics
corresponding to the lens $m_{1}$-$m_{2}$-$m_{3}$, $m_{1}$-$m_{2}$, and
$m_{1}$-$m_{3}$, respectively. The dotted black line shows the $m_{1}$-$m_{2}$
axis, and the dotted orange line shows the $m_{1}$-$m_{3}$ axis. The black
plus signs represent the lensing objects. In the inset of the right panel, the
filled blue circle represents the source. The black, magenta, and blue dashed
lines with arrows show the source trajectories corresponding to each lens
system. The corresponding model light curves are shown in the upper panel of
Figure 6 with the same colour. Figure 6: The corresponding light curves for
the example cases as shown in Figure 5. The upper panel shows the three model
light curves with different colours. The green points with error bars in the
middle and bottom panels are the simulated data points corresponding to the
triple-lens.

###  2.4 Planet Finder

The aim of this step is to determine from each of the three light curves,
whether there are detectable planetary signatures.

We fit each of the three light curves with the single-lens-single-source
(1L1S) model. The free (non-linear) parameters are $t_{0}$, $u_{0}$, $t_{\rm
E}$, and $\rho$, their initial values are taken from the values used to
generate the corresponding light curves. We use the Nelder-Mead simplex
algorithm (Nelder & Mead, 1965; Gao & Han, 2012) to find the best-fitting
parameters.

For clarity, we use the following notation to represent the $\chi^{2}$
differences between different models. For $\Delta\chi^{2}_{\rm
A,B}=\chi^{2}_{\rm A}-\chi^{2}_{\rm B}$, we use model B to generate data
points in the light curve and calculate the theoretical $\chi^{2}$
$(\chi^{2}_{\rm B})$, and fit the data points with model A to obtain
$\chi^{2}_{\rm A}$.

The criterion we used to evaluate whether a light curve contains planetary
signal is that
$\Delta\chi_{\rm{1L1S{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0},theoretical}}}^{2}=\chi_{\rm{1L1S}}^{2}-\chi^{2}_{\rm{theoretical}}>200$,
here “theoretical” can be 2L1S or 3L1S models. We note that the
$\Delta\chi_{\rm{1L1S,2L1S}}^{2}$ thresholds vary between different
statistical studies, simulations, and planetary searches. The current four
microlensing planetary statistical studies used
$\Delta\chi_{\rm{1L1S,2L1S}}^{2}$ thresholds from 100 to 500 (Gould et al.,
2010; Cassan et al., 2012; Shvartzvald et al., 2016; Suzuki et al., 2016). The
simulation conducted by Zhu et al. (2014a) and Henderson et al. (2014) adopted
$\Delta\chi_{\rm{1L1S,2L1S}}^{2}=200$ or 500, respectively. The lowest
$\Delta\chi_{\rm{1L1S,2L1S}}^{2}$ of the systematic KMTNet planetary anomaly
search is 60 (Zang et al., 2021b, 2022). We adopt 200 as a preliminary
threshold to investigate the related probabilities and will explore the
dependence of the results on the $\Delta\chi^{2}$ threshold in §3.4.3.

###  2.5 Triple-lens Finder

This step aims to evaluate whether $m_{2}$ and $m_{3}$ can be simultaneously
detected from a 3L1S light curve, i.e., whether a 2L1S model is sufficient to
explain the 3L1S light curve.888We do not consider the scenario that other
models (e.g., binary-lens binary-source, Suzuki et al., 2018; Ryu et al.,
2020; Han et al., 2020b, 2022c) could be degenerate with triple-lens model. If
1L1S and 2L1S models cannot explain the light curve, we regard this as the
case that both planets are detectable. We fit the 3L1S light curve with a 2L1S
model, the initial guess for the 2L1S model parameters is determined by the
following steps. 1). For the 2L1S model consists of $m_{1}$-$m_{2}$, we
calculate the initial chi-squared $\chi^{2}_{\rm{init,m_{2}}}$. 2). For the
2L1S model consists of $m_{1}$-$m_{3}$, we calculate the initial chi-squared
$\chi^{2}_{\rm{init,m_{3}}}$. 3). We choose the 2L1S model with the smaller
$\chi^{2}$ as the initial 2L1S parameters for further minimisation with the
Nelder-Mead algorithm.

The criterion we used to evaluate whether a light curve contains two-planet
signals is
$\Delta\chi_{\rm{1L1S{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0},3L1S}}}^{2}>200$
and
$\Delta\chi_{\rm{2L1S{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0},3L1S}}}^{2}>200$,
i.e., the planetary signal can be detected with the “Planet Finder”, in the
meanwhile the two-planet signal can be detected with the “Triple-lens Finder”
from the triple-lens light curve. We note that there are fewer 2L1S events
than 1L1S events. For example, the 2L1S rate of the systematic KMTNet
planetary anomaly search is about 11% (Zang et al., 2022; Gould et al., 2022;
Jung et al., 2022). Thus, statistically the $\Delta\chi_{\rm{2L1S,3L1S}}^{2}$
threshold should be lower than the $\Delta\chi_{\rm{1L1S,2L1S}}^{2}$
threshold. However, in practice as pointed out by Zhu et al. (2014a), with a
much lower occurrence of 3L1S than that of 2L1S, the deviations after
subtraction of the first planet may be explained by systematics in the data
(e.g., Gould et al. 2014), stellar variabilities or high-order effects (the
lens orbital motion and microlens parallax effect). Thus, we may need a higher
$\Delta\chi_{\rm{2L1S,3L1S}}^{2}$ threshold. Because this question is still
uncertain, in this work we adopt
$\Delta\chi_{\rm{1L1S,2L1S}}^{2}=\Delta\chi_{\rm{2L1S,3L1S}}^{2}$.

##  3 Results

For a given set of $(s_{2},q_{2},s_{3},q_{3},\psi)$, different source
trajectory parameter $(u_{0},\alpha)$ determines the shape of the
corresponding light curves and determines whether planet $m_{2}$ and/or
$m_{3}$ can be detected. Some source trajectories allow the detection of both
planets in the triple-lens light curve, we denote these events as “Abc”. The
flowchart in Figure 2 is an overview on how “Abc” events and other type of
events (“Ab”, “Ac” and “Ap”) are found with “Planet Finder” and/or “Triple-
lens Finder”. We explain them in more detail as follows,

  * •

“Ab”: The Jovian planet is detectable in the 2Lb1S light curve (generated with
the lensing components $m_{1}$-$m_{2}$, see §2.3 for details) The “Planet
Finder” is used to find these events by fitting the 1L1S model to the 2Lb1S
light curve to check whether $\Delta\chi_{\rm
1L1S,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S}}^{2}>200$.

  * •

“Ac”: The Saturn-like planet is detectable in the 2Lc1S light curve. The
“Planet Finder” is used to find these events.

  * •

“Ab $\cap$ Ac”: Both planets are detectable from the corresponding 2L1S light
curves. These events are selected as the intersection of the above two
samples. These were considered two-planet event candidates in Zhu et al.
(2014b), however, planets can be individually detected does not guarantee that
they can be simultaneously detected in the same event.

  * •

“Ap”: The planet signal is detectable in the 3L1S light curve. The “Planet
Finder” is used to find these events by fitting the 1L1S model to the 3L1S
light curve to check whether $\Delta\chi_{\rm 1L1S,3L1S}^{2}>200$. If the
detectable planet is $m_{2}$, we denote the event as “$\rm{Ap,b}$”. Both
“Planet Finder” and “Triple-lens Finder” are used to select “$\rm{Ap,b}$”
events. The condition is $\Delta\chi_{\rm 1L1S,3L1S}^{2}>200$ and
$\Delta\chi_{\rm{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S},3L1S}^{2}<200$.
This means that there is planetary signal in the 3L1S light curve
($\Delta\chi_{\rm 1L1S,3L1S}^{2}>200$), but the third mass $m_{3}$ is not
required to explain the 3L1S light curve
($\Delta\chi_{\rm{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S},3L1S}^{2}<200$).
Similarly, we denote the event as “$\rm{Ap,c}$” if the detectable planet is
$m_{3}$.

  * •

“Abc”: Both planets are detectable in the 3L1S light curve. Both “Planet
Finder” and “Triple-lens Finder” are used to select these events. The
condition is $\Delta\chi_{\rm 1L1S,3L1S}^{2}>200$ and
$\Delta\chi_{\rm{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2L(best)1S},3L1S}^{2}>200$,
where $\rm 2L(best)1S$ is either 2Lb1S or 2Lc1S, depends on which one has
smaller initial $\chi^{2}$ for fitting the 3L1S light curve.

With the above definitions, we investigate the detectability of the
Jupiter/Saturn analog (§3.1) as well as the enhancement (§3.2) and suppression
(§3.3) effects. We summarise the results in Tables 2-6.

Table 2: The detectability of two-planet events and the enhancement effect.
See §3.1 and §3.2 for details.

* | $\rm{Ab}\cap\rm{Ac}$ | $\rm{Ab}\cap\overline{\rm{Ac}}$ | $\overline{\rm{Ab}}\cap\rm{Ac}$ | $\overline{\rm{Ab}}\cap\overline{\rm{Ac}}$ | Total
---|---|---|---|---|---
$P(\rm{*})$ | 1.29% | 6.65% | 1.94% | 90.1% | 100%
$P(\rm{Abc}\cap\rm{*})$ | 1.10% | 0.0387% | 0.0299% | $3.18\times 10^{-6}$ | 1.17%

  * •

Note: “*” has different meaning for different columns. For example, for the
second column, “*” represents the event set {$\rm{Ab}\cap\rm{Ac}$}, so $P(*)$
represents $P(\rm{Ab}\cap\rm{Ac})$ and $P(\rm{Abc}\cap*)$ represents
$P(\rm{Abc}\cap\rm{Ab}\cap\rm{Ac})$.

Table 3: The probabilities related to the suppression effect. See §3.3 for details. * | $\rm{Ap,c}$ | $\rm{Ap,b}$ | $\overline{\rm{Ap}}$ | Total ($\overline{\rm{Abc}}$)
---|---|---|---|---
$P(*|(\rm{Ab}\cap\rm{Ac}))$ | 1.72% | 13.0% | 0.119% | 14.9%
Table 4: The detectability of both of the two planets given one of the planets is already detectable (see §3.4.1 for details). * | $\rm{Ab}$ | $\rm{Ac}$
---|---|---
$P(\rm{*})$ | 7.94% | 3.23%
$P(\rm{Abc}|\rm{*})$ | 14.7% | 36.1%
$P(\rm{Abc}|(\rm{*}\cap\log u_{0}\leq-1))$ | 26.9% | 67.2%
$P(\rm{Abc}|(\rm{*}\cap\log u_{0}\leq-2))$ | 81.5% | 83.8%
Table 5: The dependence of the detectability of two-planet events on the
impact factor (see §3.4.1 for details).

* | $\log u_{0}\leq-1$ | $\log u_{0}\leq-1.63$ | $\log u_{0}\leq-2$
---|---|---|---
$P(\rm{*})$ | 4.96%a | 1.13% | 0.454%
$P(\rm{Abc}|\rm{*})$ | 18.7% | 50.2% | 76.9%

  * a

We note that we draw $u_{0}$ between [0.001, 2] with the importance sampling
method. Thus we have $P(\log u_{0}\leq-1)=(0.1-0.001)/(2-0.001)\sim 4.95\%$
and $P(\log u_{0}\leq-2)=(0.01-0.001)/(2-0.001)\sim 0.450\%$. These
probabilities can also be calculated with Equation (6).

Table 6: The dependence of the detectability of two-planet events on whether the two planets are inside the lensing zone $(0.6<s<1.6)$. See §3.4.2 for details. * | $s_{2,\rm{in}}\cap s_{3,\rm{in}}$ | $s_{2,\rm{in}}\cap\overline{s_{3,\rm{in}}}$ | $\overline{s_{2,\rm{in}}}\cap s_{3,\rm{in}}$ | $\overline{s_{2,\rm{in}}}\cap\overline{s_{3,\rm{in}}}$
---|---|---|---|---
$P(\rm{*})$ | 37.2% | 40.9% | 12.9% | 8.99%
$P(\rm{Abc}|\rm{*})$ | 2.35% | 0.482% | 0.623% | 0.152%

###  3.1 Detectability of the scaled Sun-Jupiter-Saturn system

With the “Planet Finder” and “Triple-lens Finder”, we can evaluate whether the
two planets $m_{2}$ and $m_{3}$ are both detectable for a given pair of
triple-lens system and source trajectory. By averaging the results from all
simulated events, we obtain the overall detection probability $P(\rm{Abc})\sim
1.17\%$ (the last column of Table 2). The set of events {Abc} can be divided
into four categories (see other columns of Table 2),

  * •

“Abc $\cap$ (Ab $\cap$ Ac)”, which means that $m_{2}$, $m_{3}$ are detectable
from the corresponding 2L1S light curves with the “Planet Finder”, and the
two-planet signal is detectable in the 3L1S light curve with the “Triple-lens
Finder”. We find $P(\rm{Abc}\cap\rm{Ab}\cap\rm{Ac})=1.10\%$, i.e., the
majority ($1.10\%/1.17\%\approx 94\%$) of events in set {Abc} are contributed
by the set {Ab $\cap$ Ac}.

  * •

“Abc $\cap$ (Ab $\cap$ $\overline{\rm{Ac}}$)”, which means that $m_{3}$ is
undetectable from the 2L1S light curve corresponds to the lens system
$m_{1}$-$m_{3}$ with the “Planet Finder”, but it can be detected in the 3L1S
light curve. We find $P(\rm{Abc}\cap\rm{Ab}\cap\overline{\rm{Ac}})=0.0387\%$.

  * •

“Abc $\cap$ ($\overline{\rm{Ab}}$ $\cap$ Ac)”, which means that $m_{2}$ is
undetectable from the 2L1S light curve corresponds to the lens system
$m_{1}$-$m_{2}$ with the “Planet Finder”, but it can be detected in the 3L1S
light curve. We find $P(\rm{Abc}\cap\overline{\rm{Ab}}\cap\rm{Ac})=0.0299\%$.

  * •

“Abc $\cap$ ($\overline{\rm{Ab}}$ $\cap$ $\overline{\rm{Ac}}$)”, which means
that $m_{2}$, $m_{3}$ are undetectable from the corresponding 2L1S light
curves with the “Planet Finder”. However, the two-planet signal is detectable
in the 3L1S light curve with the “Triple-lens Finder”. This category is very
rare. We find
$P(\rm{Abc}\cap\overline{\rm{Ab}}\cap\overline{\rm{Ac}})=3.18\times 10^{-6}$.

Around $6\%$ of the detectable two-planet events are from the set
{$\overline{\rm{Ab}\cap\rm{Ac}}$}, i.e., the detectability of two-planet
events is enhanced (see §3.2 for more details). On the other hand, not all
events in the set {$\rm{Ab}\cap\rm{Ac}$} contribute to the set {Abc}, i.e.,
the detectability of two-planet events is suppressed. We find
$P(\rm{Abc}|(\rm{Ab}\cap\rm{Ac}))={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1.10\%/1.29\%\approx}85\%$
(see §3.3 for more details).

###  3.2 Enhancement effect

Among the detectable triple-lens events, $\sim$94% come from the set {Ab
$\cap$ Ac}, i.e., $m_{2}$ and $m_{3}$ are detectable from the corresponding
2L1S light curves with the “Planet Finder” (see Table 2, $1.10/1.17=0.94$).
Apart from this, about $1-0.94=6\%$ of {Abc} come from the cases where $m_{2}$
or/and $m_{3}$ are undetectable in the 2L1S light curves. We regard these
cases as the enhancement effect. Figure 7 shows three example cases of the
enhancement effect. The upper panel corresponds to the set
{$\rm{Abc}\cap\rm{Ab}\cap\overline{\rm{Ac}}$}, where the detectability of
$m_{3}$ is enhanced by the presence of $m_{2}$. For the 2L1S light curve with
lenses $m_{1}$-$m_{3}$, the best-fitting 1L1S model in the “Planet Finder” has
$\Delta\chi_{\rm{1L1S,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lc1S}}}^{2}=122.7$,
i.e., the planet $m_{3}$ is undetectable. However, with the presence of
$m_{2}$, the two-planet signal is detectable with
$\Delta\chi_{\rm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S},3L1S}}^{2}=489.3$
after applying the “Triple-lens Finder” to the 3L1S light curve. This is due
to the magnification pattern of the $m_{1}$-$m_{2}$ system being perturbed by
$m_{3}$ in the region near the central caustics (see Figure 3), so the signal
of $m_{3}$ might be detectable under favourable source trajectories.

Contrary to the previous case, the middle panel of Figure 7 corresponds to the
set {$\rm{Abc}\cap\overline{\rm{Ab}}\cap\rm{Ac}$}, where the detectability of
$m_{2}$ is enhanced by the presence of $m_{3}$. The best-fitting 1L1S model
for the 2L1S light curve with lenses $m_{1}$-$m_{2}$ has
$\Delta\chi_{\rm{1L1S,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S}}}^{2}=166.8$.
With the presence of $m_{3}$, the two-planet signal is detectable with
$\Delta\chi_{\rm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lc1S},3L1S}}^{2}=233.5$.
This is due to the magnification pattern in region around the planetary
caustics of the $m_{1}$-$m_{3}$ system is perturbed by the presence of $m_{2}$
(see Figure 4).

Finally, the bottom panel of Figure 7 corresponds to the set
{$\rm{Abc}\cap\overline{\rm{Ab}}\cap\overline{\rm{Ac}}$}, where the
detectability of $m_{2}$, $m_{3}$ are mutually enhanced. Each of the two
planets are undetectable individually with the “Planet Finder” in the
corresponding 2L1S light curves, with
$\Delta\chi_{\rm{1L1S,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S}}}^{2}=143.5$
and
$\Delta\chi_{\rm{1L1S,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lc1S}}}^{2}=168.4$
for $m_{2}$ and $m_{3}$, respectively. However, when applied to the 3L1S light
curve, the “Planet Finder” has
$\Delta\chi_{\rm{1L1S{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0},3L1S}}}^{2}=367.8$,
and the “Triple-lens Finder” has
$\Delta\chi_{\rm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S},3L1S}}^{2}=203.9$.
The two anomalies corresponding to individual planets are both present in the
triple-lens light curve, which leads to the enhancement effect. The “Triple-
lens Finder” uses the parameters of the 2L1S $m_{1}$-$m_{2}$ model parameters
as the initial guess. This is because the anomaly caused by the
$m_{1}$-$m_{2}$ system has higher magnification and smaller error bars in the
data points (during $\rm{HJD}-2450000=8999$ \- 9002) than the anomaly caused
by $m_{1}$-$m_{3}$ system (during $\rm{HJD}-2450000=9014$ \- 9022). If we take
the 2L1S $m_{1}$-$m_{3}$ model parameters, the initial $\chi^{2}$ is larger by
$\sim$110.

Figure 7: Example cases of the enhancement effect. The left panels show the
model light curves, the right panels show the caustics and source
trajectories. The upper panel corresponds to the set
{$\rm{Abc}\cap\rm{Ab}\cap\overline{\rm{Ac}}$}, i.e., the detectability of
$m_{3}$ is enhanced by the presence $m_{2}$. The middle panel corresponds to
the set {$\rm{Abc}\cap\overline{\rm{Ab}}\cap\rm{Ac}$}, i.e., the detectability
of $m_{2}$ is enhanced by the presence $m_{3}$. The bottom panel corresponds
to the set {$\rm{Abc}\cap\overline{\rm{Ab}}\cap\overline{\rm{Ac}}$}, i.e., the
detectability of $m_{2}$ and $m_{3}$ are mutually enhanced by each other, the
green points with error bars are the triple-lens light curve. The solid curves
with different colours show the light curves corresponding to different
lenses. In each panel, we labelled the best-fitting $\Delta\chi^{2}$ of the
“Triple-lens Finder”. In the inset of the middle right panel, we show the
caustics (in blue) of the $m_{1}$-$m_{3}$ lens system on top of the caustics
(in black) of the triple-lens system, the tiny difference in the caustics
leads to the difference in the model light curves around
$\rm{HJD}-2450000=9036$-9038.

We find that (see Table 2):

  * •

The detectability of $m_{3}$ is enhanced by the presence $m_{2}$:
$P(\rm{Abc}|(\rm{Ab}\cap\overline{\rm{Ac}}))$ $=0.0387\%/6.65\%$ $=0.582\%$,
where $P(\rm{Ab}\cap\overline{\rm{Ac}})=6.65\%$ (see the third column of Table
2).

  * •

The detectability of $m_{2}$ is enhanced by the presence $m_{3}$:
$P(\rm{Abc}|(\overline{\rm{Ab}}\cap\rm{Ac}))=1.54\%$, where
$P(\overline{\rm{Ab}}\cap\rm{Ac})=1.94\%$ (see the fourth column of Table 2).

  * •

The detectabilities of $m_{2}$, $m_{3}$ are mutually enhanced:
$P(\rm{Abc}|(\overline{\rm{Ab}}\cap\overline{\rm{Ac}}))=3.52\times 10^{-6}$,
where $P(\overline{\rm{Ab}}\cap\overline{\rm{Ac}})=90.1\%$ (see the fifth
column of Table 2).

###  3.3 Suppression effect

Not all events in set {$\rm{Ab}\cap\rm{Ac}$} correspond to detectable two-
planet events. Figure 8 shows three example cases of the suppression effect.
The upper panel corresponds to the set {$\rm{Ap,b}\cap\rm{Ab}\cap\rm{Ac}$},
where the detectability of $m_{3}$ is suppressed by the presence of $m_{2}$.
The “Planet Finder” has
$\Delta\chi_{\rm{1L1S,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lc1S}}}^{2}=306.2$
when finding the planet signal in the 2L1S light curve corresponding to lens
$m_{1}$-$m_{3}$, i.e., $m_{3}$ is detectable with the absence of $m_{2}$.
However, with the presence of $m_{2}$, the 3L1S light curve can be explained
by a 2L1S model (with lenses $m_{1}$-$m_{2}$) with
$\Delta\chi_{\rm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S},3L1S}}^{2}=180.1$,
i.e., $m_{3}$ is no longer “detectable” in the 3L1S light curve, since a 2L1S
model with $m_{1}$-$m_{2}$ (2Lb1S) is already able to explain the 3L1S light
curve where
$\Delta\chi_{\rm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S},3L1S}}^{2}$
falls below our threshold of 200.

The middle panel of Figure 8 corresponds to the set
{$\rm{Ap,c}\cap\rm{Ab}\cap\rm{Ac}$}, where the detectability of $m_{2}$ is
suppressed by the presence of $m_{3}$. The best-fitting 1L1S model for the
2L1S light curve with lenses $m_{1}$-$m_{2}$ has
$\Delta\chi_{\rm{1L1S,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S}}}^{2}=235.9$,
because of anomaly at 8997.5, i.e., close to the primary peak. However, with
the presence of $m_{3}$, the two-planet signal is undetectable with
$\Delta\chi_{\rm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lc1S},3L1S}}^{2}=194.9$,
the 3L1S light curve can be explained by a 2L1S model (with lenses
$m_{1}$-$m_{3}$).

Finally, the bottom panel of Figure 8 corresponds to the set
{$\overline{\rm{Ap}}\cap\rm{Ab}\cap\rm{Ac}$}, where the detectability of
$m_{2}$ and $m_{3}$ are both suppressed. Each of the two planets are
detectable individually with the “Planet Finder” in the corresponding 2L1S
light curves, with
$\Delta\chi_{\rm{1L1S,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lb1S}}}^{2}=207.5$
and
$\Delta\chi_{\rm{1L1S,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2Lc1S}}}^{2}=379.9$
for $m_{2}$ and $m_{3}$, respectively. However, when applied to the 3L1S light
curve, the “Planet Finder” has
$\Delta\chi_{\rm{1L1S{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0},3L1S}}}^{2}=190.1$,
below our threshold of 200.

Figure 8: Similar to Figure 7, but for the suppression effect. The upper panel
corresponds to the set {$\rm{Ap,b}\cap\rm{Ab}\cap\rm{Ac}$}, where the
detectability of $m_{3}$ is suppressed by the presence of $m_{2}$. The middle
panel corresponds to the set {$\rm{Ap,c}\cap\rm{Ab}\cap\rm{Ac}$}, where the
detectability of $m_{2}$ is suppressed by the presence of $m_{3}$. The bottom
panel corresponds to the set {$\overline{\rm{Ap}}\cap\rm{Ab}\cap\rm{Ac}$},
where the detectability of $m_{2}$ and $m_{3}$ are both suppressed.

We summarize different probabilities as the following (see Tables 2 and 3):

  * •

Overall suppression rate:
$P(\overline{\rm{Abc}}|(\rm{Ab}\cap\rm{Ac}))=1-P(\rm{Abc}|(\rm{Ab}\cap\rm{Ac}))=1-1.10\%/1.29\%\approx
15\%$ (see the second column of Table 2).

  * •

The detectability of $m_{2}$ is suppressed by $m_{3}$:
$P(\rm{Ap,c}|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}(}\rm{Ab}\cap\rm{Ac}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0})})=1.72\%$
(see the second column of Table 3).

  * •

The detectability of $m_{3}$ is suppressed by $m_{2}$:
$P(\rm{Ap,b}|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}(}\rm{Ab}\cap\rm{Ac}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0})})=13.0\%$
(see the third column of Table 3). This means $m_{3}$ is “overshadowed” by
$m_{2}$ more significantly.

  * •

The detectability of $m_{2}$ and $m_{3}$ is mutually suppressed by each other:
$P(\overline{\rm{Ap}}|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}(}\rm{Ab}\cap\rm{Ac}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0})})=0.119\%$
(see the fourth column of Table 3).

The suppression effect is mainly contributed by cases where the detectability
of $m_{3}$ is suppressed by the presence of $m_{2}$, i.e., the signal of
$m_{3}$ is “buried” in the signal of $m_{2}$. We note that the suppression
effect is related to the relative separations and mass ratios of the two
planets. In our simulation, the intrinsic separations of $m_{2}$ and $m_{3}$
are fixed as $s_{\rm{2,true}}$ and $s_{\rm{3,true}}$, respectively. Among the
200 random projections, the ratio $s_{3}/s_{2}$ peaks at $\sim$2, and about
half of all triple-lens systems have $s_{3}/s_{2}\in(1.7,2.3)$. We leave the
investigation of the detailed dependence of the suppression effect on the
separations and mass ratios to a future work.

We note that the net effect of $m_{2}$ on the detectability of $m_{3}$ is
suppression, i.e.,
$P(\rm{Ap,b}|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}(}\rm{Ab}\cap\rm{Ac}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0})})\times
P(\rm{Ab}\cap\rm{Ac})-P(\rm{Abc}|(\rm{Ab}\cap\overline{\rm{Ac}}))\times
P(\rm{Ab}\cap\overline{\rm{Ac}})=0.129\%>0$. While
$P(\rm{Ap,c}|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}(}\rm{Ab}\cap\rm{Ac}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0})})\times
P(\rm{Ab}\cap\rm{Ac})-P(\rm{Abc}|(\overline{\rm{Ab}}\cap\rm{Ac}))\times
P(\overline{\rm{Ab}}\cap\rm{Ac})=-8\times 10^{-5}\approx 0$, i.e., $m_{3}$ has
a negligible net effect on the detectability of $m_{2}$.

###  3.4 Dependence on parameters

####  3.4.1 Impact parameter

Figure 9: Cumulative probabilities as a function of $\log u_{0}$. The solid
magenta line in the upper panel corresponds to the detectability of the “Abc”
events. The solid red line in the middle panel corresponds to the probability
of the suppression effect. In the lower panel, the dashed red line corresponds
to the probability that the detectability of $m_{3}$ is enhanced by the
presence of $m_{2}$, while the dashed blue line corresponds to the probability
that the detectability of $m_{2}$ is enhanced by the presence of $m_{3}$.

High magnification events are favoured in multiple-planet detections (e.g.,
Griest & Safizadeh, 1998; Gaudi et al., 1998; Zhu et al., 2014a). Among the
five firmly established two-planet events listed in Table 7, four have
$|u_{0}|\lesssim 10^{-2}$. Figure 9 shows the cumulative probabilities as a
function of $\log u_{0}$. The solid magenta line in the upper panel
corresponds to the detectability of the “Abc” events. The solid red line in
the middle panel corresponds to the probability of the suppression effect. In
the lower panel, the dashed red line corresponds to the probability that the
detectability of $m_{3}$ is enhanced by the presence of $m_{2}$, while the
dashed blue line corresponds to the probability that the detectability of
$m_{2}$ is enhanced by the presence of $m_{3}$.

Both the detection probability of two-planet events and the suppression effect
show strong dependence on the impact parameter. In the upper panel of Figure 9
(see also Table 5), more than half of the events with $\log u_{0}$ smaller
than $-1.63$ are detectable as two-planet events. The importance of high
magnification events in detecting multiple-planet systems can be seen from
another perspective (Table 4). The probability that $m_{3}$ is detectable
given that the $m_{2}$ is detectable, i.e., $P(\rm{Abc}|\rm{Ab})$, is $14.7\%$
(the third row of Table 4). If we consider only high magnification events,
e.g., for $\log u_{0}\leq-2$, this probability is $P(\rm{Abc}|\rm{Ab},\log
u_{0}\leq-2)=81.5\%$. For comparison, $P(\rm{Abc}|\rm{Ac})=36.1\%$ and
$P(\rm{Abc}|\rm{Ac},\log u_{0}\leq-2)=83.8\%$. This implies there is a high
probability of detecting a second planet in a high magnification event if the
first planet is detectable, which is consistent with the result of Gaudi et
al. (1998).

Figure 10: The dependence of the probability density for the suppression
effect on the impact parameter. The black bars show the probability density of
the triple-lens event candidates {$\rm{Ab}\cap\rm{Ac}$}. The red bars show the
probability density of the suppressed candidates {$\overline{\rm{Abc}}$} among
{$\rm{Ab}\cap\rm{Ac}$}.

We show the dependence of the suppression probability density on the impact
parameter in Figure 10. The black bars show the probability density of the
triple-lens event candidates {$\rm{Ab}\cap\rm{Ac}$}. The red bars show the
probability density of the suppressed candidates {$\overline{\rm{Abc}}$} among
{$\rm{Ab}\cap\rm{Ac}$}. There is no simple relation between the suppression
probability and the impact parameter. The suppression probability is small
($\lesssim 0.2\%$) when $\log u_{0}$ is small ($\lesssim-2.5$). This is
reasonable since small impact parameter allows the source trajectory to
interact with the central perturbations caused by both planets (e.g., see the
right panel of Figure 3). So the planet signatures are hard to be suppressed.
The suppression probability peaks at $\log u_{0}\approx-2$. A possible reason
is that at such $u_{0}$, the signal of the Jovian planet is still strong, but
the signal of the Saturn-like planet is not strong enough, so it can be buried
more easily (see the upper panel of Figure 8). For $\log u_{0}\gtrsim-2$, the
suppression probability decreases as $u_{0}$ increases. The reason may be that
a larger $u_{0}$ allows the source trajectory to interact with the planetary
caustics corresponding to the Saturn-like planet.

####  3.4.2 Planet-host separation

For about $37.2\%$999Given two planets with intrinsic true separations
$s_{2,\rm{true}}=0.903$ and $s_{3,\rm{true}}=1.801$, with random orbital
inclination and orbital phases, the theoretical probability that they have
projected separations both inside the standard lensing zone is $34\%$. In this
work we obtain a different value (37%) due to the numerical noise caused by a
limited number (200) of projection realisations. of our simulated events, the
two planets are simultaneously projected inside the standard lensing zone
($0.6<s_{2},s_{3}<1.6$, Griest & Safizadeh, 1998), among them $2.35\%$ are
detected as two-planet events. In comparison, only $0.464\%$ of the events
where the two planets are not simultaneously inside the standard lensing zone
are detected as two-planet events, indicating multiple planets are more likely
to be simultaneously detected if all of them are located inside the lensing
zone (Table 6).

We note that among the five firmly established two-planet microlensing events
as listed in Table 7, almost all planets are located inside the lensing zone,
except for OGLE-2018-BLG-1011, where the larger planet has separation
$s_{2}=0.582$, just outside the inner lensing zone (0.6).

####  3.4.3 $\Delta\chi^{2}$ detection threshold

To investigate how the chosen $\Delta\chi^{2}$ threshold of the “Planet
Finder” and “Triple-lens Finder” will influence the results, we re-calculate
probabilities by using different $\Delta\chi^{2}$ thresholds. Figure 11 shows
the result. As expected, the detection probability of both two-planet event
candidates ({$\rm{Ab}\cap\rm{Ac}$}) and two-planet events ({$\rm{Abc}$}) both
decrease as the $\Delta\chi^{2}$ threshold increases. However, the suppression
probability ($\sim$15%) is not greatly affected.

Figure 11: The detection probability of two-planet event candidates
({$\rm{Ab}\cap\rm{Ac}$}) and two-planet events ({$\rm{Abc}$}) as a function of
$\Delta\chi^{2}$ threshold of the “Planet Finder” and “Triple-lens Finder”.
The lower panel shows the level of the suppression effect.

##  4 Discussions

The microlensing magnification excess caused by planets makes them detectable
(Gould & Loeb, 1992; Chung et al., 2005). The pattern of the magnification
excess map (e.g., Figures 3 and 4) is complex, especially when there are
multiple planets in the same system (Gaudi et al. 1998; Han 2005, see also
Daněk & Heyrovský 2015, 2019 for the complex caustics caused by triple
lenses). When there are multiple planets, the magnification excess caused by
each planet influence each other, leading to the enhancement or suppression of
their detectability. Furthermore, the magnification excess pattern changes as
the projected positions of the lenses change, making it hard to evaluate the
detection probability of a given triple-lens system under different
instrumentations and observing strategies. In this work, we simulated a large
number of events with cadence $\Gamma=6\,\mathrm{hr}^{-1}$ to assess the
detection probability of scaled Sun-Jupiter-Saturn systems with the telescope
for microlensing observations on the ET satellite. We discuss the implications
of our results in §4.1, and future works in §4.2.

###  4.1 The implications of our results

A few previous studies estimated the occurrence rate of multiplanetary systems
in the microlensing field. Gould et al. (2010) conducted a detailed
sensitivity analysis of 13 $\mu$FUN planetary events with magnification larger
than 200, including a two-planet event (OGLE-2006-BLG-109, Gaudi et al.,
2008). They investigated that if every microlensed star possessed a “scaled
version” of the solar system, they would have detected 6.1 systems with two or
more planet detections. They thus estimated that the frequency of solar-like
systems is 1/6. Suzuki et al. (2018) estimated that the occurrence rate of
systems with two cold gas giants is $6\%\pm 2\%$ based on two triple-lens
events (OGLE-2006-BLG-109 and OGLE-2014-BLG-1722) and assuming that the
detection efficiency of two-planet systems can be approximated as the product
of detection efficiencies for each planet.

The suppression probability ($15\%$) found in this work has important
implications for the calculations of detection efficiency and occurrence rate
of microlensing multi-planetary systems. The upcoming microlensing surveys are
expected to discover $\sim$100 two-planet systems. The occurrence rate
estimation of two-planet systems based on these discoveries would have
$\sim$10% Poisson noise, which may be at the same level or even below the
suppression probability. If we assume the detection efficiency of a two-planet
system can be approximated as
$\hat{\epsilon}_{\rm{Abc}}=\epsilon_{\rm{Ab}}\times\epsilon_{\rm{Ac}}$, the
estimated value $\hat{\epsilon}_{\rm{Abc}}$ could deviate substantially from
the real value.

We note that there is no microlensing study on the estimation occurrence rate
of planets in binary systems. A recent study noticed that all planets in
binary systems published by the KMTNet survey are located inside the resonant
caustics with mass ratio $\gtrsim 2\times 10^{-3}$ (Kuang et al., 2022),
implying the incompleteness of the KMTNet sample for planets in binary
systems, or the suppression effect for the planet may be more severe. More
detailed studies are needed to gain a better understanding.

###  4.2 Future works

The parameter space covered in this work is small. The known triple
microlensing events (Table 7) and the results from exoplanet demographic
surveys have shown the diversity of planetary systems (Gaudi et al., 2021).
Further improvements can be made in several ways.

The first is to investigate the detectability properties across a wider range
of parameter space related to the lens, i.e., $(s_{2},q_{2},s_{3},q_{3}$), as
upcoming surveys will be sensitive to a broad range of masses and semi-major
axes. For example, on the low mass end, the Roman telescope has the
sensitivity to the mass of Ganymede (Penny et al., 2019); several works have
investigated the detectability of extrasolar moons with the microlensing
method (Han & Han, 2002; Liebig & Wambsganss, 2010; Bachelet et al., 2022).

The high mass end is also interesting to explore. One can investigate the
detectability of planets in binary systems (e.g., Luhn et al. 2016; Han et al.
2017b) and how the binary star system will influence the detectability of the
planet. It is known that stellar binarity affects planet formation and
evolution (e.g., Moe & Kratter, 2021). For example, most circumbinary planets
detected by the Kepler satellite are located near the stability limit such
that they would be dynamically unstable if they were in a slightly closer
orbit (Holman & Wiegert, 1999; Ballantyne et al., 2021). A possible
explanation is that these planets formed further out, then migrated to the
cavity edge (close to the location of the stability limit) truncated by the
binary (Kley & Haghighipour, 2014). However, Quarles et al. (2018) conducted
$N$-body simulations and did not find strong evidence for a pile-up in the
circumbinary planet systems detected by the Kepler satellite. In addition,
Bennett et al. (2016) reported the first microlensing circumbinary planet with
an orbit well beyond the stability limit. To test the degree to which stellar
binarity inhibits or promotes planet formation, one needs more samples as well
as a better understanding of the detectability properties, e.g., how the
binary stars suppress or enhance the detectability of planets. Eggl et al.
(2013) estimated the detectability of a terrestrial planet in coplanar S-type
binary configurations by using the radial velocity, astrometry, and transit
photometry methods. They found that the gravitational interactions between the
second star and the planet can facilitate the planet’s detection. There is no
such study for the microlensing method.

The second is to carefully take into account the properties of the stellar
populations in the galactic disk and bulge and their dynamics, which influence
the related parameters such as $t_{\rm E}$, $\rho$, $I_{\rm{blend}}$, and
$I_{\rm{source}}$. Although larger source radii increase the probability that
the sources interact with the caustic structure, the finite-source effect will
eliminate small deviations caused by low-mass objects. Han & Han (2002) found
that detecting signals from satellites will be very difficult due to the
signals being seriously smeared out by the finite-source effect. Gaudi &
Sackett (2000) found that the finite-source effect can significantly influence
the detection efficiency for an object with mass ratio $\lesssim 10^{-3}$. In
addition, they found that the fraction of blended light is an important factor
such that higher blend fractions imply a smaller $u_{0}$ is needed for
detection.

The third is to investigate the detectability properties with different
instruments and observing strategies. Shvartzvald & Maoz (2012) quantified the
dependence of detected planet yield on observation cadence and duration and
found that the yield is doubled when the baseline cadence increases from
$1/3\;\mathrm{hr}^{-1}$ to $4\;\mathrm{hr}^{-1}$, or when the observation
duration increases from 80 days to 150 days. The upcoming microlensing surveys
with different instruments and observing modes will lead to different yields.
In addition, space $+$ space based (e.g., Roman $+$ Euclid, Bachelet & Penny,
2019; Bachelet et al., 2022), and space $+$ ground based (e.g., ET + KMTNet,
Gould et al., 2021; Ge et al., 2022) joint microlensing surveys have been
proposed to better constrain the properties of microlensing events.

Finally, the degeneracy in triple microlensing is more complex and needs to be
considered carefully (e.g., Song et al., 2014). The larger parameter space
will result in many pairs of degenerate triple-lens solutions in reality,
especially when the data coverage is not dense enough. On the other hand, the
triple-lens model can be degenerate with other models, such as the binary-
lens-binary-source (2L2S) model, as demonstrated in several events (shown as
triple-lens event candidates in Table 7). In addition, it would be more
complicated if high-order effects are included, such as the microlens parallax
effect (Gould, 2000), xallarap effect (Poindexter et al., 2005), and the lens
orbital motion (Batista et al., 2011; Skowron et al., 2011; Penny et al.,
2011). The suppression effect might be more severe when 2L1S $+$ high-order
effects are considered to explain 3L1S light curves. For example,
MACHO-97-BLG-41 (Bennett et al., 1999) and OGLE-2013-BLG-0723 (Udalski et al.,
2015) were once interpreted as 3L1S events, but later studies found 2L1S model
with lens orbital motion is also feasible (Albrow et al., 2000; Jung et al.,
2013; Han et al., 2016). A recent event shows severe degeneracy between 2L1S
$+$ high-order effects and 3L1S interpretations, which implies that the
degeneracy can be quite common (KMT-2021-BLG-0322, Han et al., 2021c).
Therefore, it may be necessary to explore the results with
$\Delta\chi_{\rm{1L1S,2L1S}}^{2}\neq\Delta\chi_{\rm{2L1S,3L1S}}^{2}.$

Table 7: List of triple-lens events.

Event name | $q_{2}(10^{-3})$ | $s_{2}$ | $q_{3}(10^{-3})$ | $s_{3}$ | $|\psi|(^{\circ})$ | $|u_{0}|(10^{-3})$ |  $t_{\rm E}$ (days)
---|---|---|---|---|---|---|---
| Firmly established two-planet systems
OGLE-2006-BLG-109 (Gaudi et al., 2008; Bennett et al., 2010) | 1.36 | 0.627 | 0.506 | 1.04 | $13.5$ | 3.48 | 127
OGLE-2012-BLG-0026 (Han et al., 2013) | 0.784 | 1.25 | 0.130 | 1.03 | $137$ | 9.20 | 93.9
OGLE-2018-BLG-1011 (Han et al., 2019) | 15.0 | 0.582 | 9.84 | 1.28 | $82.0$ | 53.0 | 12.4
OGLE-2019-BLG-0468 (Han et al., 2022a) | 10.6 | 0.717 | 3.54 | 0.853 | 48.0 | 12.0 | 75.2
KMT-2021-BLG-1077 (Han et al., 2022b) | 1.75 | 0.973 | 1.56 | 1.31 | 31.5 | $10.6$ | 24.9
| Firmly established planets in binary systems
OGLE-2006-BLG-284 (Bennett et al., 2020) | 284 | 0.798 | 1.16 | 0.764 | $177$ | 77.5 | 39.7
OGLE-2007-BLG-349 (Bennett et al., 2016) | 869 | 0.020 | 0.638 | 0.815 | 21.2 | 1.98 | 118
OGLE-2008-BLG-092 (Poleski et al., 2014) | 220 | 17.0 | 0.241 | 5.26 | 158 | 1545 | 38.6
OGLE-2013-BLG-0341 (Gould et al., 2014) | 1211 | 12.9 | 0.0480 | 0.814 | 170 | 23.3 | 33.4
OGLE-2016-BLG-0613 (Han et al., 2017a) | 29.0 | 1.40 | 3.27 | 1.17 | $54.5$ | 21.0 | 74.6
OGLE-2018-BLG-1700 (Han et al., 2020a) | 297 | 0.274 | 10.0 | 1.18 | 37.7 | 6.70 | 41.9
KMT-2019-BLG-1715 (Han et al., 2021a) | 246 | 0.551 | 4.01 | 1.05 | 77.2 | 56.0 | 44.0
KMT-2020-BLG-0414 (Zang et al., 2021a) | 57.8 | 0.0940 | 0.0113 | 0.999 | 41.9 | 0.688 | 94.4
| Triple-lens event candidates
OGLE-2014-BLG-1722 (Two-planets, Suzuki et al., 2018) | 0.639 | 0.851 | 0.447 | 0.754 | 126 | 131 | 23.8
OGLE-2018-BLG-0532 (Two-planets, Ryu et al., 2020) | 3.08 | 0.364 | 0.0975 | 1.01 | 2.06 | 8.23 | 139
KMT-2019-BLG-1953 (Two-planets, Han et al., 2020b) | 8.65 | 4.92 | 1.91 | 2.30 | 119 | 2.36 | 16.2
OGLE-2019-BLG-0304 (Planet in binary, Han et al., 2021b) | 1.82 | 0.885 | 0.363 | 3.73 | 138 | 0.540 | 17.8
OGLE-2019-BLG-1470 (Planet in binary, Kuang et al., 2022) | 359 | 0.439 | 3.47 | 1.11 | 73.9 | 187 | 42.6
KMT-2021-BLG-0240 (Two-planets, Han et al., 2022c) | 1.83 | 2.72 | 0.640 | 0.954 | 97.5 | 3.03 | 42.3

  * •

Note: We show the solution with smallest $\chi^{2}$ when there are multiple
degenerate triple-lens solutions. We unify the coordinate system used by
different authors according to Equation (4) and sort the masses as
$q_{2}>q_{3}$.

##  5 Conclusions

The upcoming microlensing surveys are expected to discover a larger sample of
triple microlensing events. We investigate the detectability of a scaled Sun-
Jupiter-Saturn system within the context of the telescope for microlensing
observations on the ET satellite. The key findings of our work are as follows,

  1. 1.

The probability that a scaled Sun-Jupiter-Saturn system being detectable with
the telescope for microlensing observations on the ET satellite is about
$1\%$.

  2. 2.

The presence of a Saturn-like planet has a negligible effect on the detection
probability of the Jovian planet. While the presence of the Jovian planet
suppresses the detectability of the Saturn-like planet by about $13\%$. This
level of suppression is nearly constant regardless of the adopted
$\Delta\chi^{2}$ threshold and is about the same order as the result reported
in Zhu et al. (2014a), although their lenses are different from ours,
indicating that the suppression effect is prevalent and non-negligible. Future
statistical works on the sensitivity of multi-planetary microlensing events
should be cautious in this regard. The assumption that the detection
efficiency of multi-planetary systems is approximately the product of
detection efficiency for each planet could lead to substantial errors.

  3. 3.

There is no simple relation between the suppression probability and the impact
parameter. The suppression probability peaks at $u_{0}\approx 0.01$ in the
present work. For events with $-2.5\lesssim\log u_{0}\lesssim-1.0$, it is not
safe to assume that the detection efficiency of multiple planets can be
calculated by treating each of the planets as being independent.

  4. 4.

High magnification events are important in discovering two-planet events. For
the scaled Sun-Jupiter-Saturn lens considered here, more than half of the
events with $\log u_{0}\leq-1.63$ are detectable as two-planet events with the
telescope for microlensing observations on the ET satellite. The probability
that $m_{3}$ (Saturn-like) is detectable given that $m_{2}$ (Jovian) is
detectable, i.e., $P(\rm{Abc}|\rm{Ab})$, is $14.7\%$. While for high
magnification events, e.g., for $\log u_{0}\leq-2$, this probability is
$P(\rm{Abc}|\rm{Ab},\log u_{0}\leq-2)=81.5\%$. In turn,
$P(\rm{Abc}|\rm{Ac})=36.1\%$ and $P(\rm{Abc}|\rm{Ac},\log
u_{0}\leq-2)=83.8\%$.

  5. 5.

If two planets are simultaneously located inside the lensing zone
($0.6<s<1.6$), the probability of discovering triple-lens events is higher.
For about $37\%$ of our simulated events, the two planets are simultaneously
inside the lensing zone, among these $2.4\%$ are detected as two-planet
events. However, if the two planets are not simultaneously inside the lensing
zone, this probability (0.46%) becomes five times smaller.

## Acknowledgements

We thank the anonymous referee for a valuable report that improved the paper.
R.K., W.Z., S.M., and J.Z. acknowledge support by the National Science
Foundation of China (Grant No. 12133005). The authors acknowledge the Tsinghua
Astrophysics High-Performance Computing platform at Tsinghua University for
providing computational and data storage resources that have contributed to
the research results reported within this paper.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Albrow et al. (2000) Albrow M. D., et al., 2000, ApJ, 534, 894
  * Bachelet & Penny (2019) Bachelet E., Penny M., 2019, ApJ, 880, L32
  * Bachelet et al. (2022) Bachelet E., et al., 2022, A&A, 664, A136
  * Ballantyne et al. (2021) Ballantyne H. A., et al., 2021, MNRAS, 507, 4507
  * Batista et al. (2011) Batista V., et al., 2011, A&A, 529, A102
  * Beer et al. (2004) Beer M. E., King A. R., Livio M., Pringle J. E., 2004, MNRAS, 354, 763
  * Bennett et al. (1999) Bennett D. P., et al., 1999, Nature, 402, 57
  * Bennett et al. (2010) Bennett D. P., et al., 2010, ApJ, 713, 837
  * Bennett et al. (2016) Bennett D. P., et al., 2016, AJ, 152, 125
  * Bennett et al. (2020) Bennett D. P., et al., 2020, AJ, 160, 72
  * Bozza (2010) Bozza V., 2010, MNRAS, 408, 2188
  * Bozza et al. (2018) Bozza V., Bachelet E., Bartolić F., Heintz T. M., Hoag A. R., Hundertmark M., 2018, MNRAS, 479, 5157
  * Bryan et al. (2016) Bryan M. L., et al., 2016, ApJ, 821, 89
  * Cassan et al. (2012) Cassan A., et al., 2012, Nature, 481, 167
  * Chung et al. (2005) Chung S.-J., et al., 2005, ApJ, 630, 535
  * Clement et al. (2019) Clement M. S., Raymond S. N., Kaib N. A., 2019, AJ, 157, 38
  * Daněk & Heyrovský (2015) Daněk K., Heyrovský D., 2015, ApJ, 806, 99
  * Daněk & Heyrovský (2019) Daněk K., Heyrovský D., 2019, ApJ, 880, 72
  * Eggl et al. (2013) Eggl S., Haghighipour N., Pilat-Lohinger E., 2013, ApJ, 764, 130
  * Gao & Han (2012) Gao F., Han L., 2012, Comput. Optim. Appl., 51, 259–277
  * Gaudi (2012) Gaudi B. S., 2012, ARA&A, 50, 411
  * Gaudi & Sackett (2000) Gaudi B. S., Sackett P. D., 2000, ApJ, 528, 56
  * Gaudi et al. (1998) Gaudi B. S., Naber R. M., Sackett P. D., 1998, ApJ, 502, L33
  * Gaudi et al. (2008) Gaudi B. S., et al., 2008, Science, 319, 927
  * Gaudi et al. (2021) Gaudi B. S., Meyer M., Christiansen J., 2021, in Madhusudhan N., ed., , ExoFrontiers; Big Questions in Exoplanetary Science. pp 2–1, doi:10.1088/2514-3433/abfa8fch2
  * Ge et al. (2022) Ge J., et al., 2022, arXiv e-prints, p. arXiv:2206.06693
  * Gould (2000) Gould A., 2000, ApJ, 542, 785
  * Gould & Loeb (1992) Gould A., Loeb A., 1992, ApJ, 396, 104
  * Gould et al. (2010) Gould A., et al., 2010, ApJ, 720, 1073
  * Gould et al. (2014) Gould A., et al., 2014, Science, 345, 46
  * Gould et al. (2021) Gould A., Zang W.-C., Mao S., Dong S.-B., 2021, Research in Astronomy and Astrophysics, 21, 133
  * Gould et al. (2022) Gould A., et al., 2022, A&A, 664, A13
  * Griest & Safizadeh (1998) Griest K., Safizadeh N., 1998, ApJ, 500, 37
  * Han (2005) Han C., 2005, ApJ, 629, 1102
  * Han & Han (2002) Han C., Han W., 2002, ApJ, 580, 490
  * Han et al. (2013) Han C., et al., 2013, ApJ, 762, L28
  * Han et al. (2016) Han C., Bennett D. P., Udalski A., Jung Y. K., 2016, ApJ, 825, 8
  * Han et al. (2017a) Han C., et al., 2017a, AJ, 154, 223
  * Han et al. (2017b) Han C., Shin I.-G., Jung Y. K., 2017b, ApJ, 835, 115
  * Han et al. (2019) Han C., et al., 2019, AJ, 158, 114
  * Han et al. (2020a) Han C., et al., 2020a, AJ, 159, 48
  * Han et al. (2020b) Han C., et al., 2020b, AJ, 160, 17
  * Han et al. (2021a) Han C., et al., 2021a, AJ, 161, 270
  * Han et al. (2021b) Han C., et al., 2021b, AJ, 162, 203
  * Han et al. (2021c) Han C., et al., 2021c, A&A, 655, A24
  * Han et al. (2022a) Han C., et al., 2022a, A&A, 658, A93
  * Han et al. (2022b) Han C., et al., 2022b, A&A, 662, A70
  * Han et al. (2022c) Han C., et al., 2022c, A&A, 664, A114
  * Henderson et al. (2014) Henderson C. B., Gaudi B. S., Han C., Skowron J., Penny M. T., Nataf D., Gould A. P., 2014, ApJ, 794, 52
  * Holman & Wiegert (1999) Holman M. J., Wiegert P. A., 1999, AJ, 117, 621
  * Ida & Lin (2004) Ida S., Lin D. N. C., 2004, ApJ, 604, 388
  * Jung et al. (2013) Jung Y. K., Han C., Gould A., Maoz D., 2013, ApJ, 768, L7
  * Jung et al. (2022) Jung Y. K., et al., 2022, AJ, 164, 262
  * Kennedy & Kenyon (2008) Kennedy G. M., Kenyon S. J., 2008, ApJ, 673, 502
  * Kim et al. (2016) Kim S.-L., et al., 2016, Journal of Korean Astronomical Society, 49, 37
  * Kley & Haghighipour (2014) Kley W., Haghighipour N., 2014, A&A, 564, A72
  * Kuang et al. (2021) Kuang R., Mao S., Wang T., Zang W., Long R. J., 2021, MNRAS, 503, 6143
  * Kuang et al. (2022) Kuang R., et al., 2022, MNRAS, 516, 1704
  * Liebig & Wambsganss (2010) Liebig C., Wambsganss J., 2010, A&A, 520, A68
  * Luhn et al. (2016) Luhn J. K., Penny M. T., Gaudi B. S., 2016, ApJ, 827, 61
  * Mao (2012) Mao S., 2012, Research in Astronomy and Astrophysics, 12, 947
  * Mao & Paczyński (1991) Mao S., Paczyński B., 1991, ApJ, 374, L37
  * Martin & Livio (2015) Martin R. G., Livio M., 2015, ApJ, 810, 105
  * Michtchenko & Ferraz-Mello (2001) Michtchenko T. A., Ferraz-Mello S., 2001, Icarus, 149, 357
  * Min et al. (2011) Min M., Dullemond C. P., Kama M., Dominik C., 2011, Icarus, 212, 416
  * Moe & Kratter (2021) Moe M., Kratter K. M., 2021, MNRAS, 507, 3593
  * Nasios (2020) Nasios I., 2020, solarsystem, https://github.com/IoannisNasios/solarsystem
  * Nelder & Mead (1965) Nelder J. A., Mead R., 1965, Computer Journal, 7, 308
  * Penny et al. (2011) Penny M. T., Mao S., Kerins E., 2011, MNRAS, 412, 607
  * Penny et al. (2019) Penny M. T., Gaudi B. S., Kerins E., Rattenbury N. J., Mao S., Robin A. C., Calchi Novati S., 2019, ApJS, 241, 3
  * Poindexter et al. (2005) Poindexter S., Afonso C., Bennett D. P., Glicenstein J.-F., Gould A., Szymański M. K., Udalski A., 2005, ApJ, 633, 914
  * Poleski et al. (2014) Poleski R., et al., 2014, ApJ, 795, 42
  * Portegies Zwart (2019) Portegies Zwart S., 2019, A&A, 622, A69
  * Quarles et al. (2018) Quarles B., Satyal S., Kostov V., Kaib N., Haghighipour N., 2018, ApJ, 856, 150
  * Raymond et al. (2020) Raymond S. N., Izidoro A., Morbidelli A., 2020, in Meadows V. S., Arney G. N., Schmidt B. E., Des Marais D. J., eds, , Planetary Astrobiology. p. 287, doi:10.2458/azu_uapress_9780816540068
  * Ryu et al. (2011) Ryu Y.-H., Chang H.-Y., Park M.-G., 2011, MNRAS, 412, 503
  * Ryu et al. (2020) Ryu Y.-H., et al., 2020, AJ, 160, 183
  * Shvartzvald & Maoz (2012) Shvartzvald Y., Maoz D., 2012, MNRAS, 419, 3631
  * Shvartzvald et al. (2016) Shvartzvald Y., et al., 2016, MNRAS, 457, 4089
  * Skowron et al. (2011) Skowron J., et al., 2011, ApJ, 738, 87
  * Song et al. (2014) Song Y.-Y., Mao S., An J. H., 2014, MNRAS, 437, 4006
  * Spergel et al. (2015) Spergel D., et al., 2015, arXiv e-prints, p. arXiv:1503.03757
  * Suzuki et al. (2016) Suzuki D., et al., 2016, ApJ, 833, 145
  * Suzuki et al. (2018) Suzuki D., et al., 2018, AJ, 155, 263
  * Udalski et al. (2015) Udalski A., et al., 2015, ApJ, 812, 47
  * Walsh et al. (2011) Walsh K. J., Morbidelli A., Raymond S. N., O’Brien D. P., Mandell A. M., 2011, Nature, 475, 206
  * Wang et al. (2014) Wang J., Fischer D. A., Xie J.-W., Ciardi D. R., 2014, ApJ, 791, 111
  * Witt (1990) Witt H. J., 1990, A&A, 236, 311
  * Yan & Zhu (2022) Yan S., Zhu W., 2022, Research in Astronomy and Astrophysics, 22, 025006
  * Zang et al. (2021a) Zang W., et al., 2021a, Research in Astronomy and Astrophysics, 21, 239
  * Zang et al. (2021b) Zang W., et al., 2021b, AJ, 162, 163
  * Zang et al. (2022) Zang W., et al., 2022, MNRAS, 515, 928
  * Zhu (2022) Zhu W., 2022, AJ, 164, 5
  * Zhu & Dong (2021) Zhu W., Dong S., 2021, ARA&A, 59, 291
  * Zhu et al. (2014a) Zhu W., Penny M., Mao S., Gould A., Gendron R., 2014a, ApJ, 788, 73
  * Zhu et al. (2014b) Zhu W., Gould A., Penny M., Mao S., Gendron R., 2014b, ApJ, 794, 53