11institutetext:  Department of Physics, Chungbuk National University,
Cheongju 28644, Republic of Korea
11email<EMAIL_ADDRESS>22institutetext: Korea Astronomy and
Space Science Institute, Daejon 34055, Republic of Korea 33institutetext:
Center for Astrophysics $|$ Harvard & Smithsonian 60 Garden St., Cambridge, MA
02138, USA 44institutetext: Department of Astronomy and Tsinghua Centre for
Astrophysics, Tsinghua University, Beijing 100084, China 55institutetext:
Auckland Observatory, Auckland, New Zealand 66institutetext: University of
Canterbury, Department of Physics and Astronomy, Private Bag 4800,
Christchurch 8020, New Zealand 77institutetext: Max Planck Institute for
Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany 88institutetext:
Department of Astronomy, The Ohio State University, 140 W. 18th Ave.,
Columbus, OH 43210, USA 99institutetext: Department of Particle Physics and
Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel
1010institutetext: School of Space Research, Kyung Hee University, Yongin,
Kyeonggi 17104, Republic of Korea 1111institutetext: Korea University of
Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon, 34113, Republic
of Korea 1212institutetext: Institute for Radio Astronomy and Space Research
(IRASR), AUT University, Auckland, New Zealand 1313institutetext: National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
1414institutetext: School of Physics and Astronomy, Tel-Aviv University, Tel-
Aviv 6997801, Israel 1515institutetext: Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, LA 70803 USA

# KMT-2021-BLG-2010Lb, KMT-2022-BLG-0371Lb, and KMT-2022-BLG-1013Lb: Three
microlensing planets detected via partially covered signals

Cheongho Han 0101 Chung-Uk Lee 0202 Weicheng Zang 03030404 Youn Kil Jung 0202
Grant W. Christie 0505 Jiyuan Zhang
(Leading authors)
Michael D. Albrow 04040606 Sun-Ju Chung 0202 Andrew Gould 07070808 Kyu-Ha
Hwang 0202 Doeon Kim 0101 Yoon-Hyun Ryu 0202 In-Gu Shin 0303 Yossi Shvartzvald
0909 Hongjing Yang 0404 Jennifer C. Yee 0303 Sang-Mok Cha 02021010 Dong-Jin
Kim 0202 Seung-Lee Kim 02021111 Dong-Joo Lee 0202 Yongseok Lee 02021010
Byeong-Gon Park 0202 Richard W. Pogge
(The KMTNet Collaboration)
Tim Natusch 050505051212 Shude Mao 04041313 Dan Maoz 1414 Matthew T. Penny
1515 Wei Zhu
(The MAP & $\mu$FUN Follow-up Teams) 0404

(Received ; accepted)

###### Abstract

Aims.  We inspect 4 microlensing events KMT-2021-BLG-1968, KMT-2021-BLG-2010,
KMT-2022-BLG-0371, and KMT-2022-BLG-1013, for which the light curves exhibit
partially covered short-term central anomalies. We conduct detailed analyses
of the events with the aim of revealing the nature of the anomalies.

Methods.  We test various models that can give rise to the anomalies of the
individual events including the binary-lens (2L1S) and binary-source (1L2S)
interpretations. Under the 2L1S interpretation, we thoroughly inspect the
parameter space to check the existence of degenerate solutions, and if they
exist, we test the feasibility of resolving the degeneracy.

Results.  We find that the anomalies in KMT-2021-BLG-2010 and
KMT-2022-BLG-1013 are uniquely defined by planetary-lens interpretations with
the planet-to-host mass ratios of $q\sim 2.8\times 10^{-3}$ and $\sim
1.6\times 10^{-3}$, respectively. For KMT-2022-BLG-0371, a planetary solution
with a mass ratio $q\sim 4\times 10^{-4}$ is strongly favored over the other
three degenerate 2L1S solutions with different mass ratios based on the
$\chi^{2}$ and relative proper motion arguments, and a 1L2S solution is
clearly ruled out. For KMT-2021-BLG-1968, on the other hand, we find that the
anomaly can be explained either by a planetary or a binary-source
interpretation, making it difficult to firmly identify the nature of the
anomaly. From the Bayesian analyses of the identified planetary events, we
estimate that the masses of the planet and host are $(M_{\rm p}/M_{\rm
J},M_{\rm h}/M_{\odot})=(1.07^{+1.15}_{-0.68},0.37^{+0.40}_{-0.23})$,
$(0.26^{+0.13}_{-0.11},0.63^{+0.32}_{-0.28})$, and
$(0.31^{+0.46}_{-0.16},0.18^{+0.28}_{-0.10})$ for KMT-2021-BLG-2010L,
KMT-2022-BLG-0371L, and KMT-2022-BLG-1013L, respectively.

###### Key Words.:

Gravitational lensing: micro – planets and satellites: detection

##  1 Introduction

Current planetary microlensing experiments are being carried out with the use
of multiple wide field telescopes. With the high observational cadence
achieved by employing large-format cameras and continuous coverage using
multiple telescopes, the experiments aim to construct a large planet sample
including terrestrial planets. The microlensing planet sample is of scientific
importance in studying the demographic distribution of planets because it
includes planet populations to which the detection efficiency of other major
planet detection methods is low, for example, cold planets with faint host
stars.

Despite the enhanced observational cadence of the current microlensing
surveys, the coverage for a fraction of planet-induced signals can be
incomplete, especially for those with short durations. The most common cause
for the incomplete coverage is a bad weather. In the case of the Korea
Microlensing Telescope Network (KMTNet: Kim et al., 2016) experiment, its
three telescopes are located in the three continents of the Southern
Hemisphere including the Siding Spring Observatory in Australia (KMTA), the
Cerro Tololo Interamerican Observatory in Chile (KMTC), and the South African
Astronomical Observatory in South Africa (KMTS). If one of these telescope
sites is clouded out, the coverage of a signal with a duration $\lesssim 1$
day produced by a low-mass planet would be incomplete. Another cause for the
incomplete anomaly coverage is that a subset of survey fields are observed
with a relatively low cadence. The observational cadence for the peripheral
fields of the KMTNet survey is 4–20 times lower than the 0.25 hr cadence of
the prime fields, and thus the coverage of a short-duration signal detected in
these low-cadence fields can be incomplete, for example, the planetary events
KMT-2017-BLG-0673 and KMT-2019-BLG-0414 (Han et al., 2022a). Besides, the
coverage of signals detected in the early or late phase of a bulge season can
also be incomplete, for example, the two-planet event OGLE-2019-BLG-0468 (Han
et al., 2022b), due to fact that the target, respectively, rises and sets
close to twilight.

In this paper, we present the analyses of 4 microlensing events
KMT-2021-BLG-1968, KMT-2021-BLG-2010, KMT-2022-BLG-0371, and
KMT-2022-BLG-1013, for which the light curves of the events exhibit partially
covered short-term central anomalies. With the aim of revealing the nature of
the anomalies, we conduct detailed analyses of the events under various
interpretations that can give rise to the anomalies of the individual events.

We present the analyses of the events according to the following organization.
In Sect. 2, we describe the acquisition and reduction procedure of the data
used in the analyses. In Sect. 3, we depict the models tested for the
interpretations of the anomalies, and introduce parameters used in the
modeling. In the subsequent subsections, we present the detailed analyses
conducted for the events KMT-2021-BLG-1968 (Sect. 3.1), KMT-2021-BLG-2010
(Sect. 3.2), KMT-2022-BLG-0371 (Sect. 3.3), and KMT-2022-BLG-1013 (Sect. 3.4).
Based on these analyses, we judge whether the anomalies are of planetary
origin or not. In Sect. 4, we specify the source stars of the events by
estimating the colors and magnitudes, and estimate angular Einstein radii. In
Sect. 5, we estimate the physical parameters of the mass and distance to the
identified planetary system. We summarize results in Sect. 6.

##  2 Observations and data

All the lensing events analyzed this work were detected in the 2021 and 2022
seasons by the KMTNet survey from the monitoring of stars lying toward the
Galactic bulge field. The three telescopes used by the KMTNet survey are
identical, and each 1.6 m telescope is equipped with a camera yielding 4 deg2
field of view. Since July 2020, the MAP & $\mu$FUN Follow-up teams have been
conducting a long-term follow-up program for high-magnification events (Zang
et al., 2021). For the two very high-magnification events KMT-2022-BLG-0371
and KMT-2022-BLG-1013, observations were additionally conducted using the 1 m
telescopes of the Las Cumbres Observatory global network at CTIO (LCOC) and
SAAO (LCOS) for both KMT-2022-BLG-0371 and KMT-2022-BLG-1013, and using the
0.4 m telescope at the Auckland Observatory in New Zealand for
KMT-2022-BLG-1013. KMTNet also went into ”followup mode” and increased its
observational cadence over the peaks of these two high-magnification events.
Images from the KMTNet survey and MAP followup observations were mostly
acquired in the $I$ band, and a fraction of KMTNet images were obtained in the
$V$ band for the source color measurement. The Auckland Observatory data were
taken in the Wratten #12 filter.

Image reductions and source star photometry of the KMTNet, LCO, and Auckland
data were done with the use of the pySIS pipeline developed by Albrow et al.
(2009) and Yang et al. (in prep) on the basis of the difference image method
(Tomaney & Crotts, 1996; Alard & Lupton, 1998). The error bars of data
estimated by the individual photometry codes were readjusted following the Yee
et al. (2012) routine in order that they are consistent with the scatter of
data and $\chi^{2}$ per degree of freedom (dof) for each data set becomes
unity.

##  3 Analyses

The anomalies in the lensing light curves of the events analyzed in this work
were identified from the visual inspection of the microlensing data collected
during the 2021 and 2022 seasons. The common characteristic of the events and
the anomalies appearing in their lensing light curves is that the peak
magnifications of the events are very high, with $A_{\rm peak}\sim 100$, 60,
570, and 280 for KMT-2021-BLG-1968, KMT-2021-BLG-2010, KMT-2022-BLG-0371, and
KMT-2022-BLG-1013, respectively, and the anomalies appear near the peaks of
the light curves. There are three channels that can produce such central
anomalies, including the planetary, binary-lens, and binary-source channels.

A central anomaly through the planetary channel arises when a source star
passes through the central anomaly region formed around the tiny caustic
induced by a planetary companion to the primary lens. A planetary companion
induces two sets of caustics, in which one (central caustic) forms near the
position of the planet host (Chung et al., 2005), and the other (planetary
caustic) forms away from the host at a position $\sim{\bf s}-1/{\bf s}$ (Han,
2006), where ${\bf s}$ denotes the host-planet separation vector with its
length scaled to the angular Einstein radius $\theta_{\rm E}$. Then, the
chance for the peak region of a high-magnification event, resulting from the
close approach of the source to the planet host, being perturbed by a planet
is very high (Griest & Safizadeh, 1998).

Table 1: Model parameters of KMT-2021-BLG-1968  Parameter | 2L1S (close) | 2L1S (wide) | 1L2S
---|---|---|---
$\chi^{2}$/dof | $7356.9/7360$ | $7357.8/7360$ | $7359.9/7360$
$t_{0}$ (HJD′) | $9429.849\pm 0.004$ | $9429.848\pm 0.003$ | $9429.781\pm 0.005$
$u_{0}$ ($10^{-3}$) | $10.15\pm 0.54$ | $10.41\pm 0.50$ | $11.10\pm 0.52$
$t_{0,2}$ (HJD′) | – | – | $9430.245\pm 0.009$
$u_{0,2}$ ($10^{-3}$) | – | – | $4.36\pm 0.41$
$t_{\rm E}$ (days) | $20.65\pm 1.02$ | $20.27\pm 0.93$ | $20.18\pm 0.79$
$s$ | $0.618\pm 0.017$ | $1.662\pm 0.056$ | –
$q$ ($10^{-3}$) | $3.08\pm 0.37$ | $3.14\pm 0.47$ | –
$\alpha$ (rad) | $3.587\pm 0.008$ | $3.592\pm 0.012$ | –
$\rho$ ($10^{-3}$) | $<4$ | $<4$ | –
$\rho_{2}$ ($10^{-3}$) | – | – | –
$q_{F}$ | – | – | $0.123\pm 0.015$

A binary lens with roughly equal mass components can also induce a central
anomaly. In the binary case, a single caustic forms near the barycenter of a
close binary lens with $s\ll 1$, while two caustics form near the positions of
the individual lens components of a wide binary lens with $s\gg 1$. Then, the
chance of the central perturbation of a high-magnification event resulting
from the close approach of the source either to the barycenter of the close
binary or to each of the wide lens components of a wide binary is high (Han &
Hwang, 2009).

A central anomaly can also arise when the closely spaced components of a
binary source successively approach the lens. In this case, the resulting
light curve is the superposition of those of the events involved with the
individual source stars, and the combined magnifications can appear to be
anomalous.

We analyze the individual lensing events by modeling the light curves under
the planetary, binary-lens, and binary-source interpretations. In each
modeling, we search for a lensing solution, which specifies a set of lensing
parameters depicting the lensing light curve. In the simplest case of a
lensing event involving a single lens and a single source (1L1S), the lensing
light curve is described by 3 basic lensing parameters of $(t_{0},u_{0},t_{\rm
E})$, which denote the time of the source star’s closest approach to the lens,
the lens-source separation (impact parameters) at that time, and the Einstein
time scale, respectively. The impact parameter is scaled to $\theta_{\rm E}$,
and the Einstein time scale is defined as the time for the source to transit
the Einstein radius. In the planetary and binary-lens cases, the lens system
is composed of two lens masses and a single source (2L1S), and the inclusion
of an extra lens component requires including additional parameters to depict
the lens binarity. These additional parameters are $(s,q,\alpha)$, and the
first two parameters denote the projected separation (normalized to
$\theta_{\rm E}$) and mass ratio between the lens components, respectively,
and the last one represents the angle between the source trajectory and the
binary-lens axis (source trajectory angle). In the binary-source case, the
lens system comprises a single lens and two source stars (1L2S), and the
addition of the extra source requires including three additional parameters
$(t_{0,2},u_{0,2},q_{F})$, which denote the approach time and impact parameter
of the source companion, and the flux ratio between the two source stars,
respectively (Hwang et al., 2013). In all tested models, we include an
additional parameter $\rho$, which is defined as the ratio of the angular
source radius to $\theta_{\rm E}$ (normalized source radius), for the
consideration of finite-source effects, which may affect the lensing light
curves of high-magnification events when the lens passes over the surface of a
source or a central caustic. In the 1L2S model, there are two source stars,
and we denote the normalized radius of the secondary source as $\rho_{2}$. In
the following subsections, we explain the detailed procedure of modeling
conducted for the individual events.

Figure 1:  Light curve of KMT-2021-BLG-1968. The lower panel shows the whole
view and the upper panel shows the enlarged view of the peak region around the
anomaly. Drawn over the data is the 1L1S model curve.  Figure 2:  Light curve
KMT-2021-BLG-1968 in the region of the anomaly. The curves drawn over the data
points are 2L1S, 1L2S, and PSPL models and the lower four panels show the
residuals from the individual models. The three insets in the top panel shows
the lens system configurations of the close and wide 2L1S solutions and the
1L2S solution. For each 2L1S configuration, the cuspy red figure represents
the caustic and the arrowed line indicates the source trajectory. For the 1L2S
configuration, the small filled dot represents the lens position and the
arrowed lines marked in red and blue represent the trajectories of the primary
and secondary source stars, respectively.  Figure 3:  $\Delta\chi^{2}$ map on
the $(\log\leavevmode\nobreak\ s,\log\leavevmode\nobreak\ q)$ plane obtained
from the 2L1S modeling of KMT-2021-BLG-1968. The right panel shows the
enlargement of the map around the local minima. Colors are assigned to
represent points with $\Delta\chi^{2}\leq 1n\sigma$ (red), $\leq 2n\sigma$
(yellow), $\leq 3n\sigma$ (green), and $\leq 4n\sigma$ (cyan), where $n=3$.

###  3.1 KMT-2021-BLG-1968

The event KMT-2021-BLG-1968 occurred on a source lying at the Equatorial
coordinates (RA, DEC)${}_{\rm J2000}=(17$:52:25.77, -28:09:05.87), which
correspond to the Galactic coordinates
$(l,b)=(1^{\circ}\hskip-2.0pt.443,-0^{\circ}\hskip-2.0pt.878)$. The source
position corresponds to the KMTNet prime fields of BLG02 and BLG42, toward
which observations were conducted with a 0.5 hr cadence for each field, and a
0.25 hr in combination. The extinction toward the field, $A_{I}\sim 4.1$, was
high due to the closeness of the field to the Galactic center. Together with
the faintness of the source, the measurement of the source baseline was
difficult, but the source was registered in the Dark Energy Camera (DECam)
catalog with an $i$-band baseline magnitude of $i_{\rm base}=21.83$. The
KMTNet alert of the event was issued on 2021 August 2, which corresponds to
the abridged heliocentric Julian data of ${\rm HJD}^{\prime}\equiv{\rm
HJD}-2450000=9428$. The event reached its peak a day after the alert at ${\rm
HJD}^{\prime}=9429.8$ with a peak magnitude of $I_{\rm peak}\sim 17.4$. The
event was observed solely by the KMTNet survey without any followup
observations.

The light curve of KMT-2021-BLG-1968 is displayed in Figure 1, in which the
lower panel shows the whole view and the upper panel shows the peak region of
the light curve. The curve drawn over the data points is a point-source point-
lens (PSPL) model. It shows that a short-term anomaly with about 1 day
duration occurred near the peak of the light curve. The residual from the PSPL
model presented in the bottom panel of Figure 2 shows that negative and
positive deviations occurred in the rising and falling parts, respectively.
The time gap between the anomaly regions covered by the KMTC and KMTS data
sets corresponds to the night time of the KMTA site, but observations at the
Australian site could not be done due to bad weather.

For the interpretation of the anomaly, we first conducted a 2L1S modeling of
the light curve. Figure 3 shows the $\Delta\chi^{2}$ map obtained from the
grid searches for the binary lens parameters $s$ and $q$. The map shows a
unique pair of local minima at $(\log s,\log q)\sim(\pm 0.2,-2.5)$ resulting
from the close-wide degeneracy (Dominik, 1999; An, 2005). In Table 1, we list
the full lensing parameters of the close ($s<1$) and wide ($s>1$) 2L1S
solutions obtained after refining the solutions by allowing all parameters to
vary. The degeneracy between the solutions is very severe with
$\Delta\chi^{2}<1$. For both solutions, the estimated mass ratios between the
lens components, $q\sim 3.1\times 10^{-3}$ , are very small, indicating that
the companion to the lens is a planet according to the 2L1S interpretation.
The model curve (solid curve) of the close planetary solution is drawn over
the data points in the top panel of Figure 2, and the residuals of both the
close and wide solutions are presented in the lower panels. The lens system
configurations of the close and wide 2L1S solutions are presented in the left
two insets of the top panel of Figure 2. It shows that the anomaly was
produced by the source passage through the anomaly region extending from the
protruding cusp of the central caustic induced by a planetary companion. The
measured event time scale is $t_{\rm E}\sim 20$ days. The normalized source
radius cannot be accurately measured due to the non-caustic-crossing nature of
the anomaly, and only the upper limit $\rho_{\rm max}\sim 4\times 10^{-3}$ is
constrained.

Figure 4:  Light curve of KMT-2021-BLG-2010. The layout and scheme of the
plots are same as those of Fig. 1.

We check the feasibility of explaining the anomaly with the introduction of a
source companion by additionally conducting a 1L2S modeling. The lensing
parameters of the best-fit 1L2S solution are listed in Table 1. The fact that
$q_{F}\sim 0.12$ indicates that the secondary source is fainter than the
primary source, and the fact that $t_{0}<t_{0,2}$ and $u_{0}<u_{0,2}$
indicates the secondary source trailed the primary source and approached
closer to the lens than the primary source. The model curve (dotted curve) and
the residual of the binary-source solution together with the configuration of
the 1L2S lens system are presented in Figure 1. In principle, binary source
models sometimes show measurable color-dependent effects, for example,
MOA-2012-BLG-486 (Hwang et al., 2013). However, because of the high
extinction, the event is too faint in the $V$-band to carry out such tests.

Table 2: Model parameters of KMT-2021-BLG-2010  Parameter | Close | Wide
---|---|---
$\chi^{2}$/dof | $3817.5/3811$ | $3817.8/3811$
$t_{0}$ (HJD′) | $9428.960\pm 0.007$ | $9428.962\pm 0.008$
$u_{0}$ | $0.017\pm 0.003$ | $0.017\pm 0.003$
$t_{\rm E}$ (days) | $15.10\pm 1.69$ | $15.18\pm 1.67$
$s$ | $0.845\pm 0.024$ | $1.140\pm 0.036$
$q$ (10-3) | $2.77\pm 0.88$ | $2.82\pm 0.98$
$\alpha$ (rad) | $0.925\pm 0.084$ | $0.918\pm 0.093$
$\rho_{1}$ (10-3) | $<5$ | $<5$

We find that the observed anomaly in the lensing light curve is almost equally
well explained with the planetary and binary-source interpretations. From the
comparison of the fits, it is found that the planetary solution is preferred
over the binary-source solution by a mere $\Delta\chi^{2}=3.0$. We note that
resolving the degeneracy between the planetary and binary-source solutions
would be difficult even if the gap between the KMTC and KMTS data sets had
been covered by the KMTA data because the two models in the gap region are
almost identical.

###  3.2 KMT-2021-BLG-2010

The lensing event KMT-2021-BLG-2010 occurred on a source lying at $({\rm
RA},{\rm DEC})_{\rm J2000}=($17:52:02.04, -32:16:11.10),
$(l,b)=(-2^{\circ}\hskip-2.0pt.148,-2^{\circ}\hskip-2.0pt.899)$, toward which
the $I$-band extinction is $A_{I}\sim 2.1$. The source lies in the KMTNet
prime field of BLG41 and the sub-prime field BLG22, toward which the combined
observational cadence is 0.33 hr. The event was observed solely by the KMTNet
group, who issued the alert of the event on 2021 August 03 (${\rm
HJD}^{\prime}=9429$) at around the time when the event reached its peak.

In Figure 4, we present the light curve of KMT-2021-BLG-2010. The light curve
is constructed with the use of only the KMTC and KMTS data sets because the
photometry quality of the KMTA data set is not good. It is found that the
light curve exhibits an anomaly appearing around the peak with an approximate
duration of 1 day, and the deviation from the PSPL model appears both in the
rising and falling parts as shown in the PSPL residual presented in the bottom
panel of Figure 5. Similar to the case of KMT-2021-BLG-1968, the coverage of
the anomaly is incomplete due to the absence of the KMTA data set111Two KMTA
data points were actually taken during the anomaly, but they proved to be
unusable due to bad observing conditions..

Figure 5:  Zoom of the central anomaly region of KMT-2021-BLG-2010. The lower
4 panels show the residuals from the close and wide 2L1S, 1L2S, and PSPL
models. The two insets in the top panel show the lens-system configurations of
the close and wide 2L1S solutions.  Figure 6:  $\Delta\chi^{2}$ map from the
2L1S modeling of KMT-2021-BLG-2010. Notations and color scheme of the plots
are same as those of Fig. 3, except that $n=2$.

We find that the anomaly of the event is uniquely explained by a pair of
planetary models. In Figure 6, we present the $\Delta\chi^{2}$ map obtained
from the grid searches for the binary parameters $s$ and $q$. The estimated
binary parameters are $(\log\leavevmode\nobreak\ s,\log\leavevmode\nobreak\
q)\sim(\pm 0.06,-2.6)$, indicating that the anomaly was produced by a
planetary companion and the two solutions result from the close-wide
degeneracy. The final lensing parameters of the close and wide planetary
solutions are listed in Table 2, and the model curve of the close 2L1S
solution is drawn over the data points in the top panel of Figure 5, and the
residuals from both the close and wide 2L1S solutions are shown in the lower
panels. We also present the lens system configurations of the close and wide
2L1S solutions in the insets of the top panel. According to the solutions, the
anomaly was produced by the source passage through the negative deviation
region formed in the back-end region of the central caustic induced by a
planet. The source did not cross the caustic, and thus only the upper limit of
the normalized source radius, $\rho_{\rm max}\sim 5\times 10^{-3}$, is
constrained.

We find that the planetary and binary-source models can be distinguished with
a high confidence level of $\Delta\chi^{2}=62.7$. The model curve and the
residual of the best-fit 1L2S solution are presented in Figure 5, showing that
the model results in a poor fit especially in the region $9429.2\lesssim{\rm
HJD}^{\prime}\lesssim 9429.5$, that was covered by the KMTS data set.

Table 3: Model parameters of the solutions ”A” and ”B” of KMT-2022-BLG-0371  Parameter | Sol A, close | Sol A, wide | Sol B, close | Sol B, wide
---|---|---|---|---
$\chi^{2}$/dof | $1469.2/1487$ | $1469.9/1487$ | $1469.0/1487$ | $1469.4/1487$
$t_{0}$ (HJD′) | $9689.419\pm 0.002$ | $9689.420\pm 0.002$ | $9689.335\pm 0.003$ | $9689.335\pm 0.003$
$u_{0}$ (10-3) | $2.539\pm 0.045$ | $2.658\pm 0.111$ | $2.089\pm 0.076$ | $2.156\pm 0.081$
$t_{\rm E}$ (days) | $94.89\pm 1.91$ | $89.33\pm 4.12$ | $82.43\pm 2.56$ | $79.93\pm 3.15$
$s$ | $0.123\pm 0.018$ | $9.689\pm 1.211$ | $0.310\pm 0.014$ | $3.299\pm 0.063$
$q$ (10-3) | $46.09\pm 37.69$ | $70.26\pm 21.74$ | $18.44\pm 1.90$ | $19.82\pm 1.44$
$\alpha$ (rad) | $2.794\pm 0.042$ | $2.768\pm 0.049$ | $4.918\pm 0.014$ | $4.914\pm 0.010$
$\rho$ (10-3) | $2.53\pm 0.12$ | $2.59\pm 0.18$ | $2.25\pm 0.08$ | $2.33\pm 0.09$
$\pi_{{\rm E},N}$ | $-0.74\pm 0.31$ | $0.84\pm 0.57$ | $-0.26\pm 0.66$ | $0.34\pm 0.60$
$\pi_{{\rm E},E}$ | $-0.18\pm 0.08$ | $0.04\pm 0.10$ | $-0.04\pm 0.12$ | $0.08\pm 0.11$
Table 4: Model parameters of the solutions ”C” and ”D” of KMT-2022-BLG-0371  Parameter | Sol C, close | Sol C, wide | Sol D, close | Sol D, wide
---|---|---|---|---
$\chi^{2}$/dof | $1470.7/1487$ | $1470.7/1487$ | $1461.3/1487$ | $1462.3/1487$
$t_{0}$ (HJD′) | $9689.418\pm 0.001$ | $9689.418\pm 0.001$ | $9689.363\pm 0.005$ | $9689.365\pm 0.007$
$u_{0}$ (10-3) | $2.887\pm 0.124$ | $3.024\pm 0.092$ | $1.709\pm 0.153$ | $1.869\pm 0.164$
$t_{\rm E}$ (days) | $83.328\pm 3.13$ | $80.19\pm 2.19$ | $103.76\pm 7.56$ | $95.08\pm 7.89$
$s$ | $0.624\pm 0.077$ | $1.714\pm 0.055$ | $0.940\pm 0.005$ | $1.065\pm 0.006$
$q$ (10-3) | $0.72\pm 0.45$ | $0.95\pm 0.10$ | $0.39\pm 0.05$ | $0.42\pm 0.06$
$\alpha$ (rad) | $4.549\pm 0.025$ | $4.571\pm 0.026$ | $5.311\pm 0.063$ | $5.287\pm 0.086$
$\rho$ (10-3) | $3.00\pm 0.16$ | $3.17\pm 0.11$ | $0.79\pm 0.10$ | $0.89\pm 0.11$
$\pi_{{\rm E},N}$ | $0.19\pm 0.63$ | $-0.09\pm 0.58$ | $1.05\pm 0.68$ | $0.86\pm 0.67$
$\pi_{{\rm E},E}$ | $0.03\pm 0.11$ | $-0.03\pm 0.12$ | $0.16\pm 0.127$ | $0.10\pm 0.12$

###  3.3 KMT-2022-BLG-0371

Figure 7:  Light curve of KMT-2022-BLG-0371.

The source of the lensing event KMT-2022-BLG-0371 lies at $({\rm RA},{\rm
DEC})_{\rm J2000}=($17:41:26.86, -34:41:55.21),
$(l,b)=(-5^{\circ}\hskip-2.0pt.372,-2^{\circ}\hskip-2.0pt.267)$. The $I$-band
extinction toward the field is $A_{I}=2.18$. The event was found on 2022 April
11 (${\rm HJD}^{\prime}=9680$), when the source became brighter than the
baseline magnitude of $I_{\rm base}=19.72$ by 0.45 mag. The source location
corresponds to the KMTNet sub-prime field BLG37, toward which observations
were carried out with a 2.5 hr cadence. The event reached its peak at ${\rm
HJD}^{\prime}=9689.4$ with a very a high magnification of $A_{\rm max}\sim
570$. Four days before the event reached the peak (${\rm
HJD}^{\prime}=9685.27$), a high-magnification alert was issued by the KMTNet
HighMagFinder system (Yang et al., 2022). In response to this alert, two days
before the peak, follow-up observations were conducted using the LCOC and LCOS
telescopes and the cadence of KMTNet observations was increased. The
magnification of the source flux lasted more than 100 days, which comprises a
significant fraction of the Earth’s orbital period, that is, 1 yr, and thus we
consider microlens-parallax effects in modeling the lensing light curve.

Figure 8:  Zoom of the central anomaly region of KMT-2022-BLG-0371. The lower 5 panels show the residuals from the 4 degenerate 2L1S (close A, B, C, and D solutions) and FSPL models.  Figure 9:  $\Delta\chi^{2}$ map constructed from the 2L1S modeling of KMT-2022-BLG-0371. Notations and color scheme of the plots are same as those of Fig. 3, except that $n=2$. The locations of the 4 sets of local solutions are marked by ”A”, ”B”, ”C”, and ”D”, and the subscripts ”c” and ”w” denote the close and wide solutions, respectively.  Table 5: Proper motion probability for the 4 local solutions of KMT-2022-BLG-0371  Solution |  $\rho$ (10-3) |  $\theta_{*}$ ($\mu$as) |  $\theta_{\rm E}$ (mas) |  $t_{\rm E}$ (day) |  $\mu_{\rm obs}$ (mas/yr) | $p(\mu<\mu_{\rm obs})$
---|---|---|---|---|---|---
Sol A | 2.53 | 0.337 | 0.133 | 95 | 0.51 | $7\times 10^{-3}$
Sol B | 2.25 | 0.370 | 0.164 | 82 | 0.73 | $15\times 10^{-3}$
Sol C | 3.00 | 0.367 | 0.122 | 83 | 0.54 | $8\times 10^{-3}$
Sol D | 0.79 | 0.326 | 0.419 | 104 | 1.45 | $59\times 10^{-3}$

The light curve of KMT-2022-BLG-0371 as constructed from the combination of
the survey and followup data is presented in Figure 7, in which the curve
drawn over the data is a finite-source point-lens (FSPL) model. It shows that
the peak region of the light curve exhibits an anomaly that lasted for about 1
day. While the falling side of the anomaly was densely covered by the
combination of the KMTS and KMTC survey and the LCOS followup data, the rising
part of the peak region, which corresponds to the night time in Australia, was
not covered because site containing KMTA and LCOA was clouded out. Likewise,
it was not possible to observe from New Zealand due to clouds. Despite the
partial coverage, the peak region of the light curve clearly displays an
anomaly feature with a 0.05 mag deviation level as shown in the FSPL residual
presented in the bottom panel of Figure 8. Without the increased cadence of
observations over the peak, the standard KMTNet cadence of 2.5 hr ($\sim 0.1$
days) for the BLG37 field would not have been sufficient to accurately
characterize the anomaly.

Figure 10:  Lens system configurations of the four pairs of the local
solutions of KMT-2022-BLG-0371. In each panel, the empty circle on the source
trajectory represent the source size in units of the Einstein radius as given
by the tick marks.  Figure 11:  Scatter plot of points in the MCMC chain on
the $\pi_{{\rm E},N}$–$\pi_{{\rm E},E}$ parameter plane of KMT-2022BLG-0371.
The right panel shows the enlargement of the region around the origin. Color
scheme of the plots are same as those of Fig. 3, except that $n=1$.

We find that the anomaly in the lensing light curve is described by multiple
sets of local solutions with different combinations of
$\log\leavevmode\nobreak\ s$ and $\log\leavevmode\nobreak\ q$, as shown in the
$\Delta\chi^{2}$ map presented in Figure 9. We identify 4 close-wide pairs of
solutions, which we designate as ”Sol A”, ”Sol B”, ”Sol C” and ”Sol D”. The
refined lensing parameters of the individual solutions are presented in Tables
4 (for solutions A and B) and 5 (for solutions C and D). For the solutions A
and B, the mass ratio between the lens components are $q_{\rm A}\sim 0.05$ and
$q_{\rm B}\sim 0.02$, respectively, and thus the companion would likely be a
brown dwarf according to these solutions. On the other hand, the mass ratios
of the solutions C and D are $q_{\rm C}\sim 0.8\times 10^{-3}$ and $q_{\rm
D}\sim 0.4\times 10^{-3}$, respectively, and thus the companion is a planet
according to these solutions. In Figure 8, we present the model curve of the
best-fit 2L1S model (close Sol D solution) and the close model residuals of
all local solutions. In Figure 10, we present the lens system configurations
of the individual local solutions. We reject the 1L2S interpretation of the
anomaly because the 1L2S model yields a poorer fit than the 2L1S solution by
$\Delta\chi^{2}=176.6$.

We find that the solution D is strongly favored over the other 2L1S solutions
by a combination of two arguments. First, the solution D yields a better fit
by $\Delta\chi^{2}=7.8$, 7.7, and 9.5 than the solutions A, B, and C,
respectively. Second, the solution D results in a higher probability of the
relative lens-source motion $\mu$ than those of the other solutions. According
to Eq. (22) of Gould (2022), the relative probability for a proper motion of a
lensing event to be less than an observed value $\mu_{\rm obs}$ is
$p(\mu<\mu_{\rm obs})=(\mu_{\rm obs}/6\leavevmode\nobreak\ {\rm
mas}\leavevmode\nobreak\ {\rm yr}^{-1})^{2}$ in the regime of low proper
motions. In Table 5, we summarize the values of the normalized source radius
$\rho$, angular source radius $\theta_{*}$, angular Einstein radius
$\theta_{\rm E}=\theta_{*}/\rho$, event time scale $t_{\rm E}$, proper motion
$\mu=\theta_{\rm E}/t_{\rm E}$, and the resulting probabilities
$p(\mu<\mu_{\rm obs})$ for the individual local 2L1S solutions. The detailed
procedure of estimating $\theta_{*}$ is described in Sect. 4. From the
comparison of the probabilities, it is found that the solution D is at least 4
times more probable than any of the other solutions. While each of the two
arguments based on $\chi^{2}$ and $\mu$ is not compelling by itself, together
they strongly favor the solution D, although the other solutions are not
completely ruled out. We note that direct measurement of the proper motion by
resolving the lens and source from future high-resolution followup
observations can ultimately lift the degeneracy among the local solutions.

Table 6: Model parameters of KMT-2022-BLG-1013  Parameter |  Sol Ac |  Sol Aw | Sol B | Sol C
---|---|---|---|---
$\chi^{2}$/dof | $1702.7/2047$ | $1704.3/2047$ | $1688.9/2047$ | $1659.9/2047$
$t_{0}$ (HJD′) | $9737.168\pm 0.002$ | $9737.164\pm 0.003$ | $9737.197\pm 0.003$ | $9737.145\pm 0.002$
$u_{0}$ (10-3) | $7.60\pm 0.36$ | $6.96\pm 0.33$ | $4.02\pm 0.23$ | $3.58\pm 0.16$
$t_{\rm E}$ (days) | $22.95\pm 0.97$ | $24.36\pm 1.02$ | $36.69\pm 1.61$ | $40.17\pm 1.75$
$s$ | $0.518\pm 0.011$ | $1.908\pm 0.043$ | $0.968\pm 0.002$ | $1.072\pm 0.002$
$q$ (10-3) | $14.08\pm 0.61$ | $13.21\pm 0.59$ | $0.23\pm 0.02$ | $1.60\pm 0.14$
$\alpha$ (rad) | $0.593\pm 0.016$ | $0.555\pm 0.019$ | $3.209\pm 0.004$ | $6.027\pm 0.007$
$\rho$ (10-3) | $7.89\pm 0.35$ | $7.16\pm 0.32$ | $4.52\pm 0.29$ | $4.93\pm 0.24$

It is found that reliable measurements of the parallax parameters $(\pi_{{\rm
E},N},\pi_{{\rm E},E})$ are difficult despite the long time scale of the
event. This is shown in the scatter plot of $\Delta\chi^{2}$ on the $\pi_{{\rm
E},N}$–$\pi_{{\rm E},E}$ plane presented in Figure 11. The scatter plot shows
that the parallax solution is not only consistent with a zero-$\pi_{\rm E}$
model but also the uncertainty of $\mbox{\boldmath$\pi$}_{\rm E}$, especially
the north component, is very big.

Figure 12:  Light curve of KMT-2022-BLG-1013.

###  3.4 KMT-2022-BLG-1013

The event KMT-2022-BLG-1013, occurred on a bulge star with coordinates $({\rm
RA},{\rm DEC})_{\rm J2000}=($17:38:50.22, -28:21:20.41),
$(l,b)=(-0^{\circ}\hskip-2.0pt.291,1^{\circ}\hskip-2.0pt.570)$, was detected
in its early stage by the KMTNet survey on 2022 May 31 (${\rm
HJD}^{\prime}\sim 9729.7$). The baseline magnitude registered in the DECam
catalog is $i_{\rm base}=21.31$, and the extinction toward the field is
$A_{I}\sim 3.1$. The source was in the KMTNet BLG14 field, toward which
normal-mode observations were done with a 1.0 hr cadence. At ${\rm
HJD}^{\prime}=9736.89$, the KMTNet HighMagFinder system issued an alert that
this event was peaking at a high magnification. Then, the peak region of the
light curve was covered by the KMTS data with an intensified cadence of 0.12
hr during the period of $9737.30\leq{\rm HJD}^{\prime}\leq 9737.65$, and
additional data were acquired from followup observations conducted using the
LCOC, LCOS, and Auckland telescopes.

Figure 13:  Zoom of the central anomaly region of KMT-2022-BLG-1013. The lower
5 panels show the residuals from the 3 degenerate 2L1S (solutions A, B, and
C), 1L2S, and FSPL models. The three insets in the top panel show the 2L1S
lens-system configurations of the solutions A, B, and C.

The light curve of KMT-2022-BLG-1013 is presented in Figure 12. For the
construction of the light curve, we did not use the KMTA data set due to its
poor photometric quality. In any case, there were no KMTA data taken near
peak. The peak region appears to exhibit deviations caused by finite-source
effects, and thus we first fitted the light curve with a FSPL model. It is
found that the FSPL model approximately describes the peak, but it leaves
substantial positive residuals of $\sim 0.1$ mag level in both the rising
($9734\lesssim{\rm HJD}^{\prime}\lesssim 9736$) and falling ($9738\lesssim{\rm
HJD}^{\prime}\lesssim 9740$) sides, as shown in the bottom panel of Figure 13.

From the through inspection of the parameter space, it is found that the
residuals from the FSPL model are explained by a unique planetary model.
Figure 14 shows the $\Delta\chi^{2}$ map obtained from the 2L1S grid searches
for the binary-lens parameters $s$ and $q$. We identify 4 local solutions,
designated as ”Sol Ac”, ”Sol Aw”, ”Sol B”, and ”Sol C”, and the locations of
the individual locals are marked in the $\log\leavevmode\nobreak\
s$–$\log\leavevmode\nobreak\ q$ parameter plane presented in Figure 14. We
note that the locals Sol Ac and Sol Aw are the pair of solutions resulting
from the close-wide degeneracy. In Table 6, we list the refined lensing
parameters of the individual local solutions, and the corresponding lens-
system configurations are presented in the insets of the top panel of Figure
13. We note that the Sol Ac and Sol Aw solutions result in similar
configurations, and thus we present the configuration of the Sol Ac solution
as a representative one. The residuals from the individual models are
presented in the lower 4 panels of Figure 13. From the comparison of the
residuals and the $\chi^{2}$ values of the models, it is found that the Sol C
model yields a substantially better fit to the data than the other models, by
$\Delta\chi^{2}=42.8$, 44.4, and 29.0 with respect to Sol Ac, Sol Aw, and Sol
B models, respectively. We also find that the Sol C model is preferred over
the 1L2S model by $\Delta\chi^{2}=50.7$.

Figure 14:  $\Delta\chi^{2}$ map obtained from the 2L1S modeling of KMT-2022-BLG-1013. Notations and color scheme are same as those of Fig. 3, except that $n=2$.  Table 7: Source stars  Quantity | KMT-2021-BLG-1968 | KMT-2021-BLG-2010 | KMT-2022-BLG-0371 | KMT-2022-BLG-1013
---|---|---|---|---
$(V-I)_{\rm S}$ | $2.640\pm 0.097$ | $3.063\pm 0.234$ | $2.375\pm 0.050$ | $4.090\pm 0.136$
$I_{\rm S}$ | $22.072\pm 0.186$ | $22.830\pm 0.009)$ | $22.269\pm 0.006$ | $21.652\pm 0.003$
$(V-I,I)_{\rm RGC}$ | $(2.950,18.500)$ | $(2.920,16.900)$ | $(2.697,16.900)$ | $(4.260,17.590)$
$(V-I,I)_{{\rm RGC},0}$ | $(1.060,14.396)$ | $(1.060,14.559)$ | $1.060,14.617$ | $(1.060,14.460)$
$(V-I)_{{\rm S},0}$ | $0.750\pm 0.097,$ | $1.203\pm 0.234$ | $0.738\pm 0.050$ | $0.890\pm 0.136$
$I_{{\rm S},0}$ | $17.968\pm 0.186$ | $20.489\pm 0.009$ | $19.987\pm 0.006$ | $18.523\pm 0.003$
Source type | G6V | K5V | G4V | K2V
$\theta_{*}$ ($\mu$as) | $0.839\pm 0.101$ | $0.445\pm 0.109$ | $0.326\pm 0.028$ | $0.759\pm 0.116$

According to the best-fit model, the mass ratio between the lens components is
$q\sim 1.6\times 10^{-3}$, indicating that the companion to the lens is a
planet. The projected separation, $s\sim 1.07$, is very close to unity, and
thus the planet induces a resonant caustic. The anomaly was produced by the
successive caustic crossings of the source, which entered the caustic by
crossing the upper fold of the caustic at ${\rm HJD}^{\prime}\sim 9734.6$,
passed through the inner caustic region, and then exited the caustic by going
through the strong lower cusp of the caustic at ${\rm HJD}^{\prime}\sim
9738.9$. According to the solution, the positive deviation in the rising part
of the anomaly is explained by the source crossing over the upper fold
caustic, and the positive deviation in the falling part is explained by the
source passage through the positive anomaly region extending from the strong
caustic cusp.

##  4 Source stars and Einstein radii

In this section, we specify the source stars of the events. The main purpose
of the source specification is estimating the angular Einstein radius of the
planetary lens event from the measured normalized source radius by the
relation

$\theta_{\rm E}={\theta_{*}\over\rho},$ (1)

where the angular source radius $\theta_{*}$ is deduced from the source type.
We are able to measure the Einstein radii of the two events KMT-2022-BLG-0371
and KMT-2022-BLG-1013, for which the lenses are identified as planetary
systems and the $\rho$ values are measured. Although the lens of
KMT-2021-BLG-2010 is a planetary system, the Einstein radius cannot be
measured because the $\rho$ value is not measured. In the case of
KMT-2021-BLG-1968, neither is the lens uniquely identified as a planetary
system nor is the $\rho$ value measured. Nevertheless, we specify the source
stars of all events for fully characterizing the lensing events.

Figure 15:  Source locations (filled blue dots) of the events in the
instrumental color-magnitude diagrams of stars around the source stars. In
each panel, the filled red dot denotes the centroid of the red giant clump.

We specified the source star of each event by measuring its de-reddened color
$(V-I)_{{\rm S},0}$ and magnitude $I_{0}$ using the Yoo et al. (2004) routine.
Following the routine, we first measured the instrumental color and magnitude,
$(V-I,I)_{\rm S}$, of the source, placed the source in the color-magnitude
diagram (CMD) of stars around the source, measured the offset $\Delta(V-I,I)$
of the source position from the centroid of the red giant clump (RGC), with
$(V-I,I)_{\rm RGC}$, in the CMD, and then estimate the de-reddened source
color and magnitude as

$(V-I,I)_{{\rm S},0}=(V-I,I)_{{\rm RGC},0}+\Delta(V-I,I).$ (2)

Here $(V-I,I)_{{\rm RGC},0}$ denote the known values of the de-reddened color
and magnitude of the RGC centroid from Bensby et al. (2013) and Nataf et al.
(2013), respectively. The instrumental source color were measured by
regressing the $I$\- and $V$-band photometry data with respect to the lensing
magnification predicted by the model. In the cases of KMT-2021-BLG-1968 and
KMT-2021-BLG-2010, for which the source stars are very faint, it was difficult
to securely measure the $V$-band magnitudes. In this case, we utilized the
Holtzman et al. (1998) CMD constructed from the observations of bulge stars
using the Hubble Space Telescope (HST), and interpolate the source color from
the main-sequence branch on the HST CMD using the measured $I$-band magnitude
difference between the source and RGC centroid.

Figure 16:  Bayesian posteriors of the mass of the lens system for the planetary events KMT-2021-BLG-2010 and KMT-2022-BLG-1013. In each panel, the distributions drawn in red and blue represent the contributions of the bulge and disk lenses, respectively, and the distribution drawn in black is the sum of the contributions by the two lens populations. The solid vertical line indicates the median and the dotted vertical lines represent the 16% and 84% of the distribution.  Table 8: Einstein radii and relative lens-source proper motions  Event |  $\theta_{\rm E}$ (mas) |  $\mu$ (mas/yr)
---|---|---
KMT-2021-BLG-2010 | $>0.089$ | $>2.15$
KMT-2022-BLG-0371 | $0.413\pm 0.067$ | $1.45\pm 0.24$
KMT-2022-BLG-1013 | $0.154\pm 0.024$ | $1.40\pm 0.22$

Figure 15 shows the positions of the source stars and RGC centroids in the
CMDs of the individual events. In Table 7, we list the values of $(V-I,I)_{\rm
S}$, $(V-I,I)_{{\rm S},0}$, $(V-I,I)_{\rm RGC}$, and $(V-I,I)_{{\rm RGC},0}$
together with the stellar types determined from the de-reddened color and
magnitude. For the estimation of the angular source radius from $(V-I,I)_{{\rm
S},0}$, we first converted the $V-I$ color into $V-K$ color, and then deduce
$\theta_{*}$ from the Kervella et al. (2004) relation between $(V-K,V)$ and
$\theta_{*}$. The estimated source radii of the individual events are listed
in Table 7.

With the measured source radius, we estimated the angular Einstein radii using
the relation in Equation (1). We also estimated the relative lens-source
proper motion from the combination of the Einstein radius and event time scale
by

$\mu={\theta_{\rm E}\over t_{\rm E}}.$ (3)

In Table 8, we list the $\theta_{\rm E}$ and $\mu$ values of the events for
which the planetary nature of the lenses are identified.

Figure 17:  Bayesian posteriors of the distance to the lens system for the
events KMT-2021-BLG-2010 and KMT-2022-BLG-1013. Notations are same as in Fig.
16.

##  5 Physical parameters

In this section, we estimate the physical parameters of the three identified
planetary lens systems of KMT-2021-BLG-2010L, KMT-2022-BLG-0371L, and
KMT-2022-BLG-1013L. For none of these events, the lensing observables $(t_{\rm
E},\theta_{\rm E},\pi_{\rm E})$ are not fully measured, making it difficult to
uniquely determine the mass $M$ and distance $D_{\rm L}$ to the planetary
systems via the analytic relations

$M={\theta_{\rm E}\over\kappa\pi_{\rm E}};\qquad D_{\rm L}={{\rm
AU}\over\pi_{\rm E}\theta_{\rm E}+\pi_{\rm S}}.$ (4)

Here $\kappa=4G/(c^{2}{\rm AU})$, $\pi_{\rm S}={\rm AU}/D_{\rm S}$ denotes the
parallax of the source, and $D_{\rm S}$ represents the distance to the source.
We, therefore, estimate the physical lens parameters by conducting Bayesian
analyses based on the measured observables of the individual events.

Table 9: Physical lens parameters  Parameter | KMT-2021-BLG-2010 | KMT-2022-BLG-0371 | KMT-2022-BLG-1013
---|---|---|---
$M_{\rm p}$ ($M_{\rm J}$) | $1.07^{+1.15}_{-0.68}$ | $0.26^{+0.13}_{-0.11}$ | $0.31^{+0.46}_{-0.16}$
$M_{\rm h}$ ($M_{\odot}$) | $0.37^{+0.40}_{0.23}$ | $0.63^{+0.32}_{-0.28}$ | $0.18^{+0.28}_{-0.10}$
$D_{\rm L}$ (kpc) | $7.09^{1.18}_{-1.53}$ | $7.14^{+1.07}_{-1.34}$ | $7.71^{+1.01}_{-1.09}$
$a_{\perp}$ (AU) (close) | $1.79^{+0.30}_{-0.38}$ | $3.02^{+0.45}_{-0.56}$ | $1.38^{+0.18}_{-0.20}$
(wide) | $2.42^{+0.40}_{-0.52}$ | $3.42^{+0.51}_{-0.64}$ | –

The Bayesian analysis of each lensing event was carried by conducting a Monte
Carlo simulation of Galactic lensing events. From the simulation, we produced
a large number ($10^{7}$) of lensing events, for which the locations of the
lens and source and the relative proper motions were assigned based on a prior
Galactic model, and the lens masses were allocated based on a prior mass
function of Galactic objects. In the simulation, we adopted the Galactic model
of Jung et al. (2021) and the mass function model of Jung et al. (2018). For
each simulated event, we computed lensing observables by

$t_{\rm E}={\theta_{\rm E}\over\mu};\qquad\theta_{\rm E}=(\kappa M\pi_{\rm
rel})^{1/2};\qquad\pi_{\rm E}={\pi_{\rm rel}\over\theta_{\rm E}},$ (5)

where the relative parallax is defined as $\pi_{\rm rel}=\pi_{\rm L}-\pi_{\rm
S}={\rm AU}(1/D_{\rm L}-1/D_{\rm S})$. We then construct posterior
distributions of the physical parameters by imposing a weight
$w_{i}=\exp(-\chi^{2}/2)$ to each simulated event. Here the $\chi^{2}$ value
is computed as

$\chi_{i}^{2}=\left({t_{{\rm E},i}-t_{\rm E}\over\sigma_{t_{\rm
E}}}\right)^{2}+\left({\theta_{{\rm E},i}-\theta_{\rm
E}\over\sigma_{\theta_{\rm E}}}\right)^{2},$ (6)

where $(t_{{\rm E},i},\theta_{{\rm E},i})$ represent the time scale and
Einstein radius of each simulated event, and $(t_{\rm E},\theta_{\rm E})$
denote the measured values. In the case of KMT-2021-BLG-2010, for which only
the lower limit of $\theta_{\rm E}$ is constrained, we set $w_{i}=0$ for
simulated events with $\theta_{{\rm E},i}<\theta_{\rm E,min}$. Although the
constraint given by the measured microlens parallax parameters is weak in the
case of KMT-2022-BLG-0371, we imposed the $\pi_{\rm E}$ constraint by adding
an additional term

$\sum_{j=1}^{2}\sum_{k=1}^{2}b_{j,k}(\pi_{{\rm E},j,i}-\pi_{\rm E})(\pi_{{\rm
E},k,i}-\pi_{\rm E})$ (7)

to the right side of Equation (6). Here $b_{jk}$ represents the inverse matrix
of $\mbox{\boldmath$\pi$}_{\rm E}$, $(\pi_{{\rm E},1},\pi_{{\rm
E},2})_{i}=(\pi_{{\rm E},N},\pi_{{\rm E},E})_{i}$ denote the parallax
parameters of each simulated event, and $(\pi_{{\rm E},N},\pi_{{\rm E},E})$
are the measured parallax parameters.

We present the posteriors of the mass and distance to the three planetary
systems in Figures 16 and 17, respectively. In Table 9, we list the estimated
masses of the host, $M_{\rm h}$, and planet, $M_{\rm p}$, distance, $D_{\rm
L}$, and projected separation of the planet from the host, $a_{\perp}=sD_{\rm
L}\theta_{\rm E}$. For each parameter, we present the median as a
representative value and the uncertainties are estimated as the 16% and 84% of
the posterior distribution. For the events with degenerate close and wide
solutions, we present a pair of the projected separations resulting from the
close and wide solutions. According to the estimated planet and host masses,
the planetary system KMT-2021-BLG-2010L is composed of a Jovian-mass planet
and an M-dwarf host, the system KMT-2022-BLG-0371L consists of a sub-Jovian-
mass planet and a K-dwarf host, and the system KMT-2022-BLG-1013L is composed
of a sub-Jovian-mass planet and an M-dwarf host.

##  6 Summary and conclusion

We conducted analyses of 4 anomalous microlensing events detected in the 2021
and 2022 seasons, including KMT-2021-BLG-1968, KMT-2021-BLG-2010,
KMT-2022-BLG-0371, and KMT-2022-BLG-1013. The light curves of the events
commonly exhibit partially covered short-term central anomalies appearing near
the highly magnified peaks. In order to reveal the nature of the anomalies, we
tested various models that could give rise to the anomalies of the individual
events including the binary-lens and binary-source interpretations. Under the
2L1S interpretation, we thoroughly inspected the parameter space to check the
existence of degenerate solutions, and if they exist, we tested the
feasibility of resolving the degeneracy.

We found that the anomalies of the two events KMT-2021-BLG-2010 and
KMT-2022-BLG-1013 were uniquely explained by planetary-lens interpretations,
for which the planet-to-host mass ratios of $q\sim 2.8\times 10^{-3}$ and
$\sim 1.6\times 10^{-3}$, respectively. For KMT-2022-BLG-0371, we found
multiple sets of local 2L1S solutions with different combinations of binary
parameters, but a planetary solution with a mass ratio $q\sim 4\times 10^{-4}$
was strongly preferred over the other degenerate solutions based on the
$\chi^{2}$ and relative proper motion arguments, and a 1L2S solution was
clearly ruled out. For KMT-2021-BLG-1968, on the other hand, it was found that
the anomaly could be explained either by a planetary or a binary-source
interpretation, making it difficult to firmly identify the nature of the
anomaly.

From the Bayesian analyses of the identified planetary events, we estimated
that the masses of the planet and host were $(M_{\rm p}/M_{\rm J},M_{\rm
h}/M_{\odot})=(1.07^{+1.15}_{-0.68},0.37^{+0.40}_{-0.23})$,
$(0.26^{+0.13}_{-0.11},0.63^{+0.32}_{-0.28})$, and
$(0.31^{+0.46}_{-0.16},0.18^{+0.28}_{-0.10})$ for KMT-2021-BLG-2010L,
KMT-2022-BLG-0371L, and KMT-2022-BLG-1013L, respectively.

###### Acknowledgements.

Work by C.H. was supported by the grants of National Research Foundation of
Korea (2020R1A4A2002885 and 2019R1A2C2085965). J.C.Y., I.G.S, and S.J.C.
acknowledge support from NSF Grant No. AST-2108414. W.Z. and H.Y. acknowledge
the support by the National Science Foundation of China (Grant No. 12133005).
Y.S. acknowledges support from BSF Grant No. 2020740. This research has made
use of the KMTNet system operated by the Korea Astronomy and Space Science
Institute (KASI) at three host sites of CTIO in Chile, SAAO in South Africa,
and SSO in Australia. Data transfer from the host site to KASI was supported
by the Korea Research Environment Open NETwork (KREONET). W.Zang, J.Z., H.Y.,
S.M., and W.Zhu acknowledge support by the National Natural Science Foundation
of China (Grant No. 12133005). W.Zang acknowledges the support from the
Harvard-Smithsonian Center for Astrophysics through the CfA Fellowship. This
research uses data obtained through the Telescope Access Program (TAP), which
has been funded by the TAP member institutes. W.Zhu acknowledges the science
research grants from the China Manned Space Project with No. CMS-
CSST-2021-A11. 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.

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