# Mass Production of 2021 KMTNet Microlensing Planets III: Analysis of Three
Giant Planets

In-Gu Shin  Center for Astrophysics $|$ Harvard & Smithsonian 60 Garden St.,
Cambridge, MA 02138, USA  Jennifer C. Yee  Center for Astrophysics $|$ Harvard
& Smithsonian 60 Garden St., Cambridge, MA 02138, USA  Andrew Gould  Max
Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany
Department of Astronomy, The Ohio State University, 140 W. 18th Ave.,
Columbus, OH 43210, USA  Kyu-Ha Hwang  Korea Astronomy and Space Science
Institute, Daejon 34055, Republic of Korea  Hongjing Yang  Department of
Astronomy and Tsinghua Centre for Astrophysics, Tsinghua University, Beijing
100084, China  Ian A. Bond  Institute of Natural and Mathematical Sciences,
Massey University, Auckland 0745, New Zealand  Michael D. Albrow  University
of Canterbury, Department of Physics and Astronomy, Private Bag 4800,
Christchurch 8020, New Zealand  Sun-Ju Chung  Korea Astronomy and Space
Science Institute, Daejon 34055, Republic of Korea  Cheongho Han  Department
of Physics, Chungbuk National University, Cheongju 28644, Republic of Korea
Youn Kil Jung  Korea Astronomy and Space Science Institute, Daejon 34055,
Republic of Korea  University of Science and Technology, Korea, (UST), 217
Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea  Yoon-Hyun Ryu  Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea  Yossi
Shvartzvald  Department of Particle Physics and Astrophysics, Weizmann
Institute of Science, Rehovot 76100, Israel  Weicheng Zang  Department of
Astronomy and Tsinghua Centre for Astrophysics, Tsinghua University, Beijing
100084, China  Sang-Mok Cha  Korea Astronomy and Space Science Institute,
Daejon 34055, Republic of Korea  School of Space Research, Kyung Hee
University, Yongin, Kyeonggi 17104, Republic of Korea  Dong-Jin Kim  Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea  Seung-
Lee Kim  Korea Astronomy and Space Science Institute, Daejon 34055, Republic
of Korea  Chung-Uk Lee  Korea Astronomy and Space Science Institute, Daejon
34055, Republic of Korea  Dong-Joo Lee  Korea Astronomy and Space Science
Institute, Daejon 34055, Republic of Korea  Yongseok Lee  Korea Astronomy and
Space Science Institute, Daejon 34055, Republic of Korea  School of Space
Research, Kyung Hee University, Yongin, Kyeonggi 17104, Republic of Korea
Byeong-Gon Park  Korea Astronomy and Space Science Institute, Daejon 34055,
Republic of Korea  University of Science and Technology, Korea, (UST), 217
Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea  Richard W. Pogge
Department of Astronomy, The Ohio State University, 140 W. 18th Ave.,
Columbus, OH 43210, USA  Fumio Abe  Institute for Space-Earth Environmental
Research, Nagoya University, Nagoya 464-8601, Japan  Richard Barry  Code 667,
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA  David P. Bennett
Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
Aparna Bhattacharya  Code 667, NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA  Department of Astronomy, University of Maryland, College Park, MD
20742, USA  Hirosane Fujii  Institute for Space-Earth Environmental Research,
Nagoya University, Nagoya 464-8601, Japan  Akihiko Fukui  Department of Earth
and Planetary Science, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Instituto de Astrofísica de
Canarias, Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain  Yuki Hirao
Department of Earth and Space Science, Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan  Stela Ishitani Silva  Department
of Physics, The Catholic University of America, Washington, DC 20064, USA
Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Yoshitaka Itow  Institute for Space-Earth Environmental Research, Nagoya
University, Nagoya 464-8601, Japan  Rintaro Kirikawa  Department of Earth and
Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka
560-0043, Japan  Iona Kondo  Department of Earth and Space Science, Graduate
School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Naoki
Koshimoto  Department of Astronomy, Graduate School of Science, The University
of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Yutaka Matsubara
Institute for Space-Earth Environmental Research, Nagoya University, Nagoya
464-8601, Japan  Sho Matsumoto  Department of Earth and Space Science,
Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Shota Miyazaki  Department of Earth and Space Science, Graduate School of
Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Yasushi Muraki
Institute for Space-Earth Environmental Research, Nagoya University, Nagoya
464-8601, Japan  Arisa Okamura  Department of Earth and Space Science,
Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Greg Olmschenk  Code 667, NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA  Clément Ranc  Sorbonne Université, CNRS, UMR 7095, Institut
d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France  Nicholas J.
Rattenbury  Department of Physics, University of Auckland, Private Bag 92019,
Auckland, New Zealand  Yuki Satoh  Department of Earth and Space Science,
Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Takahiro Sumi  Department of Earth and Space Science, Graduate School of
Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Daisuke Suzuki
Department of Earth and Space Science, Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan  Taiga Toda  Department of Earth
and Space Science, Graduate School of Science, Osaka University, Toyonaka,
Osaka 560-0043, Japan  Paul . J. Tristram  University of Canterbury Mt. John
Observatory, P.O. Box 56, Lake Tekapo 8770, New Zealand  Aikaterini Vandorou
Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
Hibiki Yama  Department of Earth and Space Science, Graduate School of
Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

###### Abstract

We present the analysis of three more planets from the KMTNet 2021
microlensing season. KMT-2021-BLG-0119Lb is a $\sim 6M_{\rm Jup}$ planet
orbiting an early M-dwarf or a K-dwarf, KMT-2021-BLG-0192Lb is a $\sim 2M_{\rm
Nep}$ planet orbiting an M-dwarf, and KMT-2021-BLG-2294Lb is a $\sim
1.25M_{\rm Nep}$ planet orbiting a very–low-mass M dwarf or a brown dwarf.
These by-eye planet detections provide an important comparison sample to the
sample selected with the AnomalyFinder algorithm, and in particular,
KMT-2021-BLG-2294 is a case of a planet detected by-eye but not by-algorithm.
KMT-2021-BLG-2294Lb is part of a population of microlensing planets around
very-low-mass host stars that spans the full range of planet masses, in
contrast to the planet population at $\lesssim 0.1\,$ au, which shows a strong
preference for small planets.

Gravitational microlensing (672) — Gravitational microlensing exoplanet
detection (2147)

##  1 Introduction

This paper is the third in our series that aims to publish all by-eye planet
detections from the 2021 Korea Microlensing Telescope Network (KMTNet; Kim et
al., 2016) observing season. Previously, in Ryu et al. (2022) and Ryu et al.
(2022b), we published 8 planet detections; 10 other planets from the 2021
season have been published in other work (Han et al., 2022a, b, c, d, e; Yang
et al., 2022). Here we present the analysis of three additional planetary
events: KMT-2021-BLG-0119, KMT-2021-BLG-0192, and KMT-2021-BLG-2294.

The three planetary events were identified by IGS (the first author of this
paper) using the traditional “by-eye” selection (described in Ryu et al.
2022). However, because IGS uses a variation on method that was previously
described, we document that here. One key element of IGS’s selection process
is to use an automatic program to fit single-lens/single-point-source (1L1S)
light curves (Paczynski, 1986) to all events. The 1L1S curves play a key role
in providing a reference for noticing anomalies in the observed light curves.
Once anomaly-like features are found, IGS conducts initial modeling to reveal
what kind of a lens system produces the features. Then, it is possible to
decide the selection based on the initial model parameters of the mass ratio
and event timescale. For planetary events, the mass ratio should be
$\mathcal{O}(10^{-3})$ or smaller, or the event timescale should be shorter
than $\sim 10$ days for a relatively small mass ratio (i.e.,
$\mathcal{O}(10^{-2})$).

The automatic 1L1S–fitting step is almost identical to the first step of the
AnomalyFinder (Zang et al., 2021). Ultimately, for rigorous statistical
analysis, machine-based selection is required to find a well-defined sample of
planets. However, there are certain advantages in by-eye selections. First,
the human decision process can be used to identify advanced criteria to
improve machine-based selection. In addition, because the anomalies are
identified based on the insight and experience of a researcher, by-eye
selection provides an important cross-check of the algorithm, and in
particular in identifying planets that might be missed by an algorithm. In
fact, as we will see, the signal in KMT-2021-BLG-2294 does not meet the
detection criteria of the AnomalyFinder algorithm, which gives us an
opportunity to consider the algorithm’s failure modes.

##  2 Observations

These planetary microlensing events from 2021 (KMT-2021-BLG-0119,
KMT-2021-BLG-0192, KMT-2021-BLG-2294) were first discovered by the Korea
Microlensing Telescope Network (KMTNet: Kim et al., 2016) using the KMT
AlertFinder (Kim et al., 2018)111KMTNet Alert System
(https://kmtnet.kasi.re.kr/~ulens/) on (2021 Mar 25, 2021 Mar 29, 2021 Aug
27). The KMTNet observations are made using three identical $1.6$ m telescopes
with wide-field cameras (i.e., $2^{\circ}\times 2^{\circ}$ field of view). The
telescope network consists of three sites in well-separated timezones, which
are located at the Cerro Tololo Inter-American Observatory in Chile (KMTC),
the South African Astronomical Observatory in South Africa (KMTS), and the
Siding Spring Observatory in Australia (KMTA). In Table 1, we summarize
observational information of each event. We note that, for KMT-2021-BLG-0192,
the Microlensing Observations in Astrophysics (MOA: Bond et al., 2001; Sumi et
al., 2003) independently detected the identical event (i.e., MOA-2021-BLG-080
on 2021 Apr 10). Thus, we incorporate the MOA observations in the analysis.

Table 1: Observations of 2021 Planetary Events

Event | KMT-2021-BLG-0119 | KMT-2021-BLG-0192 | KMT-2021-BLG-2294
---|---|---|---
R.A. (J2000) | $18^{h}16^{m}00^{s}.13$ | $17^{h}52^{m}25^{s}.19$ | $18^{h}00^{m}14^{s}.98$
Dec (J2000) | $-29^{\circ}44{{}^{\prime}}38{{}^{\prime\prime}}.00$ | $-30^{\circ}00^{{}^{\prime}}31^{{}^{\prime\prime}}.28$ | $-28^{\circ}36^{{}^{\prime}}44^{{}^{\prime\prime}}.78$
$(\ell,b)$ | $(2^{\circ}.572,-6^{\circ}.155)$ | $(-0^{\circ}.158,-1^{\circ}.821)$ | $(1^{\circ}.908,-2^{\circ}.597)$
KMTNet field | BLG33 | BLG02, BLG42 | BLG03, BLG43
$\Gamma$ (${\rm hr}^{-1}$) | 1.0 | 4.0 | 4.0
Extinction ($A_{I}$) | 0.38 | 2.06 | 1.21
Alert date (YYYY–MM–DD) | 2021–03–25 | 2021–03–29 | 2021–08–27
Additional Observations |  $\cdots$ | MOA |  $\cdots$

Note. — The MOA alerted MOA-2021-BLG-080 on 2021–04–10, which is the identical
event to KMT-2021-BLG-0192.

For selected events (i.e., planet candidates), individual KMTNet data were
carefully re-reduced using photometry packages that adopted the differential
image analysis (DIA) technique called pySIS (Albrow et al., 2009) and pyDIA
(Albrow, 2017; Bramich et al., 2013). We analyze the light curves using these
tender-loving care (TLC) versions of datasets. The MOA data were reduced by
their pipeline adopting the DIA method, which is described in Bond et al.
(2001).

##  3 Light Curve Analysis Methodology

###  3.1 Heuristic Analysis

A planetary microlensing event usually shows a short-term/localized anomaly in
the 1L1S light curve. A 1L1S light curve can be described using three
parameters: ($t_{0},u_{0},t_{\rm E}$). These are the time at the peak of the
light curve ($t_{0}$), the impact parameter ($u_{0}$) in units of the angular
Einstein ring radius ($\theta_{\rm E}$), and the Einstein timescale ($t_{\rm
E}$), i.e., the travel time of the source through the angular Einstein ring
radius. To explain the anomaly induced by a planet, three additional
parameters ($s$, $q$, $\alpha$) are required. These are the projected
separation between binary components in units of $\theta_{\rm E}$ ($s$), the
mass ratio of binary components defined as $q\equiv M_{\rm secondary}/M_{\rm
primary}$, and the angle between source trajectory and binary axis ($\alpha$).

From a localized anomaly, we can predict solution(s) using the unified
$s^{\dagger}$ formalism described in Hwang et al. (2022) and Ryu et al.
(2022). From time of the anomaly, $t_{\rm anom}$, we obtain the scaled time
offset from the peak of the light curve,

$\tau_{\rm anom}\equiv\frac{t_{\rm anom}-t_{0}}{t_{\rm E}}$ (1)

and the source position offset from the host,

$u_{\rm anom}=\sqrt{\tau_{\rm anom}^{2}+u_{0}^{2}}\quad.$ (2)

Then, we can also predict,

$s_{\pm}^{\dagger}\equiv\frac{\sqrt{u_{\rm anom}^{2}+4}\pm u_{\rm
anom}}{2};~{}~{}\tan\alpha={\pm}\frac{u_{0}}{\tau_{\rm anom}},$ (3)

where the $\pm$ subscript of $s_{\pm}^{\dagger}$ indicates either a major or
minor image perturbation, respectively (Gould & Loeb, 1992). In general, the
major–image perturbations ($s_{+}^{\dagger}$) appear as a “bump”–shaped
anomaly, while the minor–image perturbations ($s_{-}^{\dagger}$) appear as a
“dip”–shaped anomaly. For minor–image perturbations, we can additionally
predict the mass ratio (to a factor $\sim 2$ level) from the duration of the
“dip” anomaly, $\Delta t_{\rm dip}$ :

$q=\left(\frac{\Delta t_{\rm dip}}{4t_{\rm
E}}\right)^{2}\frac{s\sin^{2}\alpha}{u_{\rm anom}}=\left(\frac{\Delta t_{\rm
dip}}{4t_{\rm E}}\right)^{2}\frac{s}{|u_{0}|}|\sin^{3}\alpha|.$ (4)

The predictions ($s_{\pm}^{\dagger}$) can be compared to the empirical result.
In the case of only one solution, $s$ should correspond to one of the values
of $s_{\pm}^{\dagger}$. If there are two solutions ($s_{+}$, $s_{-}$), we
expect them to be related by

$s^{\dagger}=\sqrt{s_{+}s_{-}}\quad.$ (5)

In that case, it is the value of $s^{\dagger}$ that should correspond to one
of the values of $s_{\pm}^{\dagger}$. The theoretical origins of such
degeneracies are discussed in Gaudi & Gould (1997), Griest & Safizadeh (1998),
and Zhang & Gaudi (2022).

###  3.2 Basic Modeling

We start the modeling procedure from a static 2L1S model (we treat the static
case, i.e., motions of lenses and source are not considered, as a “standard
(STD)” model. Also, the “$n$L$m$S” indicates there are $n$ lenses and $m$
sources), including the finite-source effect, to find the best-fit model
describing the observed light curve. Thus, the STD model requires seven
parameters ($t_{0}$, $u_{0}$, $t_{\rm E}$, $s$, $q$, $\alpha$, and
$\rho_{\ast}$), where $\rho_{\ast}$ is the angular source radius
($\theta_{\ast}$) scaled by the Einstein radius, i.e.,
$\rho_{\ast}\equiv\theta_{\ast}/\theta_{\rm E}$. The procedure consists of two
basic steps, which may be repeated several times, if necessary.

First, we conduct a grid search to find all possible solutions (i.e., local
minima). For the search, we explore $s-q$ parameter space on a grid that spans
the values $\log_{10}(s)\in[-1.0,1.0]$ and $\log_{10}(q)\in[-5.5,1.0]$. For
the remaining parameters ($t_{0}$, $u_{0}$, $t_{\rm E}$, $\alpha$, and
$\rho_{\ast}$), we find optimal solutions using the Markov Chain Monte Carlo
(MCMC) algorithm (Doran & Müller, 2004) to minimize $\chi^{2}$. We start the
modeling from the 1L1S parameters for $t_{0}$, $u_{0}$, and $t_{E}$, plus $21$
initial values within a range of $\alpha=[0.0,\,2\pi]$ (radians). We compute
the magnification of the 2L1S model using the inverse ray-shooting technique
with the “map-making” method (Dong et al., 2006, 2009). Once we find local
minima, we explore restricted regions that contain the (possible) local
minima, if necessary.

Second, we refine the possible solutions by setting all parameters to vary
freely within (physically) possible ranges. Thus, we obtain fine-tuned model
parameters with errors based on the distributions of MCMC chains. During the
process of refining the solutions, we rescale the errors of the datasets to
make each data point contribute $\chi^{2}\sim 1.0$. We follow the procedure
described in Yee et al. (2012); i.e., $e_{\rm new}=k\sqrt{e_{\rm
old}^{2}+e_{\rm min}^{2}}$, where $e_{\rm new}$ is the rescaled error, $k$ is
the rescaling factor, $e_{\rm old}$ is the original error, and $e_{\rm min}$
is the systematics term.

###  3.3 Higher-order Effects

The STD models assume a static lens system with rectilinear motion relative to
the source. However, we should also check for effects from the observer’s
motion (i.e., Earth’s orbit) or the orbital motion of the binary lens system.

First, we check signals of the annual microlensing parallax (APRX: Gould,
1992), which is caused by the acceleration of Earth. In general, we check the
APRX if $t_{\rm E}>15$ days. For the APRX effect, we introduce two additional
parameters: $\pi_{{\rm E},N}$ and $\pi_{{\rm E},E}$, which are north and east
components of the microlensing parallax vector ($\mbox{\boldmath$\pi$}_{\rm
E}$) projected onto the sky, respectively. Even if there is not a significant
improvement in $\chi^{2}$, $\pi_{\rm E}$ is often well-constrained along one
axis, which is roughly aligned with the $\pi_{{\rm E},E}$ direction. If the
APRX model significantly improves $\chi^{2}$, we investigate the origin of the
improvement to check whether or not the APRX measurement is reasonable and not
caused by systematics.

Second, we also check the lens-orbital (OBT) effect. In reality, the signal
caused by the OBT effect is rarely detected. The OBT signal is most often
detected in cases with well covered and well-separated (in time) caustic
crossings. Thus, for planetary events (that usually have relatively short
anomalies), the OBT effect is hard to detect from the light curve. On the
other hand, the OBT effect can affect the APRX measurement because both
effects can bend the source trajectory. Hence, we test the OBT effect to see
if it affects the APRX signal. For the OBT effect, we introduce two additional
parameters: $ds/dt$ and $d\alpha/dt$, where $ds/dt$ is the rate of change of
the binary separation (i.e., $s$) and $d\alpha/dt$ is the rate of change of
the $\alpha$ parameter. We also constrain the unphysical solutions using the
absolute ratio of transverse kinetic to potential energy (An et al., 2002;
Dong et al., 2009). That is, by requiring $\beta<0.8$, where

$\beta\equiv\left|\frac{\rm KE}{\rm PE}\right|_{\perp}=$ (6)
$\left(\frac{\kappa M_{\odot}{\rm yr^{2}}}{8\pi^{2}}\right)\frac{\pi_{\rm
E}}{\theta_{\rm E}}\left(\frac{s}{\pi_{\rm E}+\pi_{\rm S}/\theta_{\rm
E}}\right)^{3}\left[\left(\frac{1}{s}\frac{ds}{dt}\right)^{2}+\left(\frac{d\alpha}{dt}\right)^{2}\right],$

where $\kappa$ is a constant deinfed as $\kappa\equiv 4G/c^{2}{\rm
au}=8.144\,{\rm mas}/M_{\odot}$ and $\pi_{\rm S}$ is the source parallax
deinfed as $\pi_{\rm S}\equiv{\rm au}/D_{\rm S}$ where $D_{\rm S}$ is the
distance to the source.

###  3.4 Degenerate Solutions

We also explicitly check for several types of known degeneracies to be sure we
have found all of the relevant 2L1S models and competing solutions.

In addition to the $s^{\dagger}$ (or offset) degeneracy, 2L1S models may be
subject to a degeneracy in $\rho_{\ast}$, which may affect the value of $q$
(Ryu et al., 2022; Yang et al., 2022). Typically, the degeneracy between the
two solutions arises because the observed duration of a “bump” anomaly may be
controlled either by the width of the caustic (so $\rho_{\ast}$ is small in
comparison) or the size of the source (so $\rho_{\ast}$ is $\gtrsim$ the width
of the caustic). Hence, in some cases, high-cadence observations can
distinguish between the two cases, e.g., by demonstrating whether or not the
caustic entrance is resolved from the exit.

We also check the 2L1S/1L2S degeneracy (Gaudi, 1998), which Shin et al. (2019)
demonstrated can exist in wider range of cases than those presented in Gaudi
(1998). This is especially true for light curves that are sparsely covered.
For the 1L2S models, we adopt the parameterization described in Shin et al.
(2019) (A–type; see their Appendix), which uses the ratio of the second source
flux to the first, $q_{\rm flux}$, and separate values of $t_{0,i}$,
$u_{0,i}$, and optionally $\rho_{\ast,i}$ for each source as necessary. Then,
we compare the 1L2S model with the best-fit 2L1S solution to see if the
2L1S/1L2S degeneracy can be resolved.

Finally, if we detect the APRX effect, then we check the degenerate APRX
solutions, which can be caused by several types such as the ecliptic
degeneracy (Jiang et al., 2004; Poindexter et al., 2005), the ${\pm}u_{0}$
degeneracy (Smith et al., 2003), and the jerk-parallax Degeneracy (Gould,
2004).222The APRX degeneracies are well described/organized in Skowron et al.
(2011). In practice, the most effective way to find degenerate APRX solutions
is to undertake trial searches using different seeds by switching the signs of
the parameters: $(u_{0},\,\alpha,\,\mbox{$\pi_{{\rm
E},N}$})\rightarrow-(u_{0},\,\alpha,\,\mbox{$\pi_{{\rm E},N}$})$.

##  4 Analysis Results

###  4.1 KMT-2021-BLG-0119

####  4.1.1 Light Curve

Figure 1: Light curve of KMT-2021-BLG-0119 with APRX model curves, geometries,
and residuals.

In Figure 1, we present the observed light curve of KMT-2021-BLG-0119
(hereafter, KB210119) with the best-fit models (i.e., APRX models) and their
caustic geometries. The light curve shows two bump-shaped anomalies. The
anomalies are likely induced by crossings of a central/resonant caustic, which
is a potential channel for discovering microlensing planets (Han et al.,
2021a). For the heuristic analysis we have $(t_{\rm
anom}=9308.3,t_{0}=9305.97,u_{0}=0.067,t_{\rm E}=62\,$ d) , which yields
$(u_{\rm anom}=0.077,s^{\dagger}_{+}=1.039,\alpha=60\,$ deg). These values are
well-matched to the fitted values derived below.

####  4.1.2 STD models and the $\rho_{\ast}$–degeneracy

For KMT-2021-BLG-0119, we find two degenerate families of models. The ‘A’
family of models was found in the standard grid search and the KMTS point at
HJD${}^{\prime}=9303.47$ falls on the caustic entrance. In the ‘B’ family of
models, which was discovered while checking for $\rho_{\ast}=0$ solutions, the
caustic entrance occurs before this KMTS point. These two families of models
have slight differences in the values of microlensing parameters, including
$s$ and $q$ (see Table 2).

Table 2: Model Parameters of KMT-2021-BLG-0119

Model | STD | APRX
---|---|---
Parameter | large$-\rho_{\ast}$ | small$-\rho_{\ast}$ | Local A | Local A′ | Local B | Local B′
$\chi^{2}_{\rm ground}$ | $1084.207$ | $1082.637$ | $1060.093$ | $1059.787$ | $1054.530$ | $1054.422$
$t_{0}$ [${\rm HJD^{\prime}}$] | $9305.971$ | $9305.956$ | $9305.868$ | $9305.895$ | $9305.769$ | $9305.779$
| $\pm 0.032$ | $\pm 0.043$ | $\pm 0.042$ | $\pm 0.043$ | $\pm 0.050$ | $\pm 0.050$
$u_{0}$ | $0.066$ | $0.071$ | $-0.077$ | $0.076$ | $-0.081$ | $0.080$
| $\pm 0.002$ | $\pm 0.002$ | $\pm 0.002$ | $\pm 0.003$ | $\pm 0.003$ | $\pm 0.003$
$t_{\rm E}$ [days] | $64.124$ | $60.705$ | $56.635$ | $58.309$ | $53.348$ | $54.288$
| $\pm 1.523$ | $\pm 1.794$ | $\pm 1.646$ | $\pm 1.793$ | $\pm 1.610$ | $\pm 1.670$
$s$ | $1.039$ | $1.043$ | $1.049$ | $1.045$ | $1.054$ | $1.053$
| $\pm 0.003$ | $\pm 0.003$ | $\pm 0.003$ | $\pm 0.003$ | $\pm 0.003$ | $\pm 0.003$
$q$ ($\times 10^{-4}$) | $63.607$ | $71.886$ | $82.413$ | $78.592$ | $97.455$ | $93.935$
| $\pm 3.239$ | $\pm 5.612$ | $\pm 5.068$ | $\pm 5.293$ | $\pm 6.149$ | $\pm 6.255$
$\alpha$ [rad] | $1.045$ | $1.046$ | $-1.076$ | $1.049$ | $-1.120$ | $1.122$
| $\pm 0.031$ | $\pm 0.039$ | $\pm 0.028$ | $\pm 0.029$ | $\pm 0.030$ | $\pm 0.030$
$\rho_{\ast}$ | $0.003$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
| $\pm 0.001$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
$\rho_{\ast,{\rm max}}$ |  $\cdots$ | $<0.0025$ | $<0.0014$ | $<0.0014$ | $<0.0018$ | $<0.0018$
$\pi_{{\rm E},N}$ |  $\cdots$ |  $\cdots$ | $0.000$ | $-0.244$ | $-0.031$ | $-0.034$
|  $\cdots$ |  $\cdots$ | $\pm 0.234$ | $\pm 0.232$ | $\pm 0.222$ | $\pm 0.220$
$\pi_{{\rm E},E}$ |  $\cdots$ |  $\cdots$ | $0.143$ | $0.132$ | $0.209$ | $0.183$
|  $\cdots$ |  $\cdots$ | $\pm 0.032$ | $\pm 0.033$ | $\pm 0.036$ | $\pm 0.037$

Note. — ${\rm HJD^{\prime}=HJD-2450000.0}$. The total number of data points
(${\rm N_{data}}$) is $1059$.

Figure 2: The comparison of STD large–$\rho_{\ast}$, STD small–$\rho_{\ast}$,
and ARPX models of KMT-2021-BLG-0119.

In addition, for the ‘A’ family of models, we find two STD models with very
similar values of $s$ and $q$, but different values for $\rho_{\ast}$ (see
Table 2). In Figure 2, we present the caustic geometry and the zoom-in on the
light curve of each case. The geometries of the two cases are almost
identical. However, the observational coverage at the caustic entrance and
exit is sub-optimal, so models with both strong finite source effects and no
finite source effect fit the data almost equally well. We refer to these as
the large–$\rho_{\ast}$ and small–$\rho_{\ast}$ cases, respectively, although
the small–$\rho_{\ast}$ case is consistent with $\rho_{\ast}=0$. We find that
the small–$\rho_{\ast}$ case shows better fits at the entrance (i.e., ${\rm
HJD^{\prime}=9303.5\sim 9304.0}$), while the large–$\rho_{\ast}$ case shows
slightly better fits at the exit (i.e., ${\rm HJD^{\prime}=9312.5\sim
9313.0}$). We also find that the small–$\rho_{\ast}$ model fits better than
the large–$\rho_{\ast}$ fit as the source approaches the caustic exit (i.e.,
${\rm HJD^{\prime}=9311.\sim 9312.5}$). In total, the $\Delta\chi^{2}$ between
the large-$\rho_{\ast}$ and small-$\rho_{\ast}$ cases is only $1.57$.

####  4.1.3 APRX Models and Solving the $\rho_{\ast}$–degeneracy Problem

The STD models have long timescales ($t_{\rm E}\gtrsim 60$ days), and the two
caustic crossings separated by $\sim 9$ days give strong timing constraints on
the light curve. Thus, we consider the APRX effect. We find $\chi^{2}$
improvement $\Delta\chi^{2}=23\sim 30$ between the STD (two $\rho_{\ast}$
cases) and APRX ($u_{0}<0$ and $u_{0}>0$ cases) models. In addition, the ARPX
contours shown in Figure 3 are well converged and inconsistent with zero at
$\gtrsim 6\sigma$.

Figure 3: The APRX contours of KMT-2021-BLG-0119. Each color indicates the
$\chi^{2}$ difference between the best-fit and chains. That is,
$\Delta\chi^{2}\left(\equiv\chi^{2}_{\rm chain}-\chi^{2}_{\rm best-
fit}\right)=n^{2}$, where $n=1$ (red), $2$ (yellow), $3$ (green), $4$ (light
blue), $5$ (blue), and $6$ (purple).  Figure 4: Cumulative $\Delta\chi^{2}$
plots between STD and APRX models of KMT-2021-BLG-0119.

We check the improvements using the cumulative $\Delta\chi^{2}$ plots shown in
Figure 4. From this investigation, we find that the improvements mostly come
from KMTC, which had a higher effective cadence and was taken under better
observational conditions than KMTS and KMTA ($3.5$ and $6$ times lower
effective cadence). As a result, the contributions of KMTS and KMTA are minor
in this case. In addition, the main improvement comes from the left wing,
during the beginning of the bulge season when Earth is accelerating rapidly to
the East, which can produce the strong $\pi_{{\rm E},E}$ signal as is
observed.

Figure 5: The light curve of KMT-2021-BLG-0192 with STD model curves and
residuals. The geometries of 2L1S models are presented in Figure 6.

Furthermore, we find that, for the ‘A’ family of solutions, APRX models always
favor the small–$\rho_{\ast}$ solutions, even when the fits are initialized at
the large–$\rho_{\ast}$ STD solutions. Large–$\rho_{\ast}$ solutions are
excluded at the $4\sigma$ level. Indeed, we have a clue about this behavior
from STD fits at the caustic–crossings shown in Figure 2. The STD models
prefer the small–$\rho_{\ast}$ case at the entrance but the
large–$\rho_{\ast}$ case at the exit. However, the APRX fits are better than
STD fits at both the caustic entrance and exit, including the part approaching
the exit. Hence, the $\rho_{\ast}$–degeneracy is resolved when the APRX is
included.

####  4.1.4 Test of the OBT effect

We find $\chi^{2}$ improvement of $\Delta\chi^{2}\sim 16$ when we include the
OBT parameters in the APRX solutions. However, the OBT parameters show large
values, $(ds/dt,\,d\alpha/dt)\sim(0.455,\,-5.973)$, which implies the lens
system is unbound or the lens is a very massive object, such as a stellar-mass
black hole. If we apply the constraints ${\rm|KE/PE|}_{\perp}<0.8$ and $M_{\rm
L}<3.0\,M_{\odot}$, we find that most of the $\chi^{2}$ improvement is
eliminated. In addition, the OBT parameters are not strongly constrained and
are not correlated with the APRX parameters, so we can neglect the OBT in our
modeling.

####  4.1.5 2L1S/1L2S Degeneracy

For KB210119, the two planetary anomalies on the light curve are induced by a
resonant caustic. Thus, a 1L2S model cannot describe both anomalies. Hence, we
do not test the 2L1S/1L2S degeneracy for this event.

Figure 6: Geometries of best–fit and possible solutions ($s_{\pm}$ cases)
caused by the $\rho_{\ast}$ degeneracy of KMT-2021-BLG-0192.

###  4.2 KMT-2021-BLG-0192

####  4.2.1 Heuristic Analysis

In Figure 5, we present the observed light curve of KMT-2021-BLG-0192
(hereafter, KB210192) with the best-fit STD model. The light curve exhibits a
bump anomaly at the peak, which was densely covered by KMTC observations. The
localized anomaly has the properties: $\tau_{\rm anom}=0.00389$ and $u_{\rm
anom}=0.01073$. From the heuristic analysis, we find that
$s_{+}^{\dagger}=1.005$, $s_{-}^{\dagger}=0.995$ and $\alpha=1.200$ radians.

####  4.2.2 STD Models

By following procedures described in Section 3.2, we conduct STD modeling to
find the best-fit models and possible degenerate solutions. We find that there
exist two solutions (i.e., $s_{\pm}$ cases) having mass ratios in the
planetary event regime, i.e., $q\sim\mathcal{O}(10^{-4})$. The
$\Delta\chi^{2}$ between the $s_{-}$ and $s_{+}$ solutions is only $0.510$, so
statistically indistinguishable. In Table 3, we present parameters of the
best-fit models. The geometric mean of these two solutions is
$s^{\dagger}=1.006$, in good agreement with the $s_{+}^{\dagger}$ prediction
from the heuristic analysis. Likewise, the value of $\alpha=1.184$ is also in
good agreement with the heuristic expectation.

Table 3: Model Parameters of KMT-2021-BLG-0192

Model | 2L1S (STD) | 2L1S (APRX) | 1L2S
---|---|---|---
Parameter | $s_{-}$ | $s_{+}$ |  $s_{-}$ ($u_{0}+$) |  $s_{-}$ ($u_{0}-$) |  $s_{+}$ ($u_{0}+$) |  $s_{+}$ ($u_{0}-$) | Parameter |
$\chi^{2}$ | $4738.844$ | $4739.354$ | $4702.863$ | $4702.032$ | $4702.928$ | $4702.911$ | $\chi^{2}$ | $4773.792$
$t_{0}$ [${\rm HJD^{\prime}}$] | $9315.697$ | $9315.697$ | $9315.697$ | $9315.697$ | $9315.697$ | $9315.697$ |  $t_{0,{S_{1}}}$ [${\rm HJD^{\prime}}$] | $9315.686$
| $\pm 0.001$ | $\pm 0.001$ | $\pm 0.001$ | $\pm 0.001$ | $\pm 0.001$ | $\pm 0.001$ |  | $\pm 0.001$
$u_{0}$ | $0.010$ | $0.010$ | $0.010$ | $-0.010$ | $0.010$ | $-0.010$ | $u_{0,{S_{1}}}$ | $0.010$
| $\pm 0.001$ | $\pm 0.001$ | $\pm 0.001$ | $\pm 0.001$ | $\pm 0.001$ | $\pm 0.001$ |  | $\pm 0.001$
$t_{\rm E}$ [days] | $32.601$ | $32.257$ | $31.539$ | $31.627$ | $31.204$ | $31.890$ |  $t_{0,{S_{2}}}$ [${\rm HJD^{\prime}}$] | $9315.823$
| $\pm 0.605$ | $\pm 0.611$ | $\pm 0.716$ | $\pm 0.717$ | $\pm 0.725$ | $\pm 0.696$ |  | $\pm 0.001$
$s$ | $0.776$ | $1.303$ | $0.774$ | $0.761$ | $1.321$ | $1.312$ |  $u_{0,{S_{2}}}$ ($\times 10^{-3}$) | $-0.058$
| $\pm 0.016$ | $\pm 0.028$ | $\pm 0.018$ | $\pm 0.017$ | $\pm 0.030$ | $\pm 0.030$ |  | $\pm 0.287$
$q$ ($\times 10^{-4}$) | $3.327$ | $3.333$ | $3.541$ | $3.733$ | $3.707$ | $3.544$ |  $t_{\rm E}$ [days] | $32.321$
| $\pm 0.323$ | $\pm 0.326$ | $\pm 0.370$ | $\pm 0.363$ | $\pm 0.362$ | $\pm 0.363$ |  | $\pm 0.623$
$\alpha$ [rad] | $1.184$ | $1.184$ | $1.183$ | $-1.186$ | $1.181$ | $-1.184$ | $\rho_{\ast,{S_{1}},{\rm max}}$ | $<0.0087$
| $\pm 0.005$ | $\pm 0.004$ | $\pm 0.005$ | $\pm 0.005$ | $\pm 0.005$ | $\pm 0.005$ |  |
$\rho_{\ast}$ ($\times 10^{-4}$) | $19.607$ | $19.318$ | $20.028$ | $19.212$ | $20.063$ | $19.201$ |  $\rho_{\ast,{S_{2}}}$ ($\times 10^{-4}$) | $16.277$
| $\pm 1.634$ | $\pm 1.696$ | $\pm 1.892$ | $\pm 1.824$ | $\pm 1.861$ | $\pm 1.878$ |  | $\pm 0.755$
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $q_{\rm flux}$ | $0.019$
|  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  | $\pm 0.001$
$\pi_{{\rm E},N}$ |  $\cdots$ |  $\cdots$ | $1.637$ | $2.312$ | $2.896$ | $2.143$ | $\cdots$ |  $\cdots$
|  $\cdots$ |  $\cdots$ | $\pm 2.124$ | $\pm 2.117$ | $\pm 2.143$ | $\pm 2.081$ | $\cdots$ |  $\cdots$
$\pi_{{\rm E},E}$ |  $\cdots$ |  $\cdots$ | $0.272$ | $0.304$ | $0.360$ | $0.285$ | $\cdots$ |  $\cdots$
|  $\cdots$ |  $\cdots$ | $\pm 0.140$ | $\pm 0.137$ | $\pm 0.141$ | $\pm 0.135$ | $\cdots$ |  $\cdots$

Note. — ${\rm HJD^{\prime}=HJD-2450000.0}$. The total number of data points
(${\rm N_{data}}$) is $4676$. For the 1L2S model, the angular radius of the
first source ($\rho_{\ast,{S_{1}}}$) is not measured. The best-fit velue of
$\rho_{\ast,{S_{1}}}$ is $7.690\times 10^{-4}$.

Figure 5 shows that the best-fit solutions do not have caustic-crossing
geometries. However, for this event, the source’s proximity to the cusp along
the binary axis means that it passes over a relatively sharp magnification
“ridge” that allows a measurement of $\rho_{\ast}$. The extremely dense
coverage at the anomaly makes this measurement very secure.

####  4.2.3 Resolving the $\rho_{\ast}$ degeneracy

The caustic geometry of KMT-2021-BLG-0192 is similar to the cases of
KMT-2021-BLG-1391Lb and KMT-2021-BLG-1253Lb (Ryu et al., 2022), which suggests
there may be alternate solutions for KMT-2021-BLG-0192 caused by the
$\rho_{\ast}$ degeneracy. We explicitly search for such solutions and present
the caustic geometries of the possible large–$\rho_{\ast}$ solutions compared
to those of the best-fit solutions in Figure 6. The possible solutions show
worse fits with $\Delta\chi^{2}=23.4$ and $21.7$ for $s_{-}$ and $s_{+}$ the
cases, respectively. The caustic-crossing feature cannot describe the
observations at the anomaly very well. Thus, because of the extremely dense
coverage, we can resolve the $\rho_{\ast}$ degeneracy for this event.

####  4.2.4 Resolving the 2L1S/1L2S degeneracy

Localized bump-shaped anomalies, like that seen in KMT-2021-BLG-0192, may also
be explained by a 1L2S interpretation. We find a plausible 1L2S model shown in
Table 3. Both the flux ratio of binary sources, $q_{\rm flux}\equiv{\rm
flux}_{S_{2}}/{\rm flux}_{S_{1}}$, and $\rho_{\ast,S_{2}}$ are well-measured,
but there is only an upper limit on $\rho_{\ast,S_{1}}$, which may be either
larger or smaller than $\rho_{\ast,S_{2}}$. Hence, it is not possible to rule
out this solution based on these physical considerations. On the other hand,
for this solution $\theta_{\ast,2}\sim 0.3\,\mu$as, and $t_{\ast,2}=0.05$ d,
so $\mu=\theta_{\ast}/t_{\ast}=1.6\,\mathrm{mas\,yr}^{-1}$, which is somewhat
unlikely, though not impossible. In addition, the 1L2S solution fits worse
than the 2L1S solutions by $\Delta\chi^{2}\sim 35$ (more relative to APRX, see
below), so it can be ruled out on that basis.

####  4.2.5 Tests of APRX and OBT effects

Because the timescale of this event is about $1$ month (i.e., $t_{\rm E}\sim
32$ days), it is worth testing the detection of the APRX signal. In our
initial fits, we found an extreme value of the parallax with
$|\mbox{$\pi_{{\rm E},N}$}|>2$. However, our investigation of the cumulative
$\Delta\chi^{2}$ plots showed that the $\chi^{2}$ improvement mostly came from
the baseline data toward the end of the microlensing season. Thus, we exclude
data with ${\rm HJD^{\prime}}>9360.0$ from the modeling for this event.

Ultimately, we find that the parallax improves the fit by $\Delta\chi^{2}\sim
37$. We present the APRX distributions in Figure 7. While the magnitude of the
parallax is not well-constrained, the vector is well-constrained along one
axis (as expected).

Figure 7: The $(\mbox{$\pi_{{\rm E},E}$},\mbox{$\pi_{{\rm E},N}$})$
distributions of APRX models of KMT-2021-BLG-0192. The color scheme is
identical to Figure 3.

In addition, we check for the OBT effect. The APRX+OBT models strongly prefer
unphysical values for the OBT parameters (implying unbound orbits). However,
including the OBT parameters does not affect the parallax constraints.
Therefore, we suppress OBT effects in our modeling.

###  4.3 KMT-2021-BLG-2294

####  4.3.1 Heuristic Analysis

The light curve of KMT-2021-BLG-2294 shows a clear anomaly at the peak of the
light curve (see Figure 8). Because the anomaly occurs at the peak of the
event $\tau_{\rm anom}\sim 0$ and $u_{\rm anom}\sim u_{0}=0.006$. Hence, the
heuristic analysis suggests $s^{\dagger}_{-}=0.997$ and
$s^{\dagger}_{+}=1.003$ and $\alpha=\pm\pi/2$ radians. Furthermore, because
this is a “dip” anomaly, we can predict the mass ratio from $\Delta t_{\rm
dip}=0.06\,$ days and $t_{\rm E}=7.1\,$ days; i.e., $q=7.4\times 10^{-4}$.

####  4.3.2 STD Models

The KMTA images have extremely low S/N for the source and did not cover the
anomaly or other magnified parts of the light curve (there are no data from
${\rm HJD^{\prime}}\sim 9451$ to $\sim 9454$) Thus, we do not include KMTA
data in the modeling.

From the detailed modeling, we find that there exist four degenerate
solutions. Figure 9 shows four solutions in $s-q$ parameter space and also
presents the caustic geometry of each solution. Their best-fit model
parameters are given in Table 4.

Table 4: Model Parameters of KMT-2021-BLG-2294

Parameter | Close (C) | Resonant (${\rm R_{C}}$) | Resonant (${\rm R_{W}}$) | Wide (W)
---|---|---|---|---
$\chi^{2}_{\rm ground}/{\rm N_{data}}$ | $8219.429/8254$ | $8243.348/8254$ | $8248.935/8254$ | $8218.934/8254$
$t_{0}$ [${\rm HJD^{\prime}}$] | $9452.558\pm 0.001$ | $9452.558\pm 0.001$ | $9452.558\pm 0.001$ | $9452.558\pm 0.001$
$u_{0}$ | $0.006\pm 0.001$ | $0.005\pm 0.001$ | $0.005\pm 0.001$ | $0.006\pm 0.001$
$t_{\rm E}$ [days] | $7.074\pm 0.253$ | $8.067\pm 0.290$ | $8.038\pm 0.297$ | $7.144\pm 0.258$
$s$ | $0.935\pm 0.009$ | $0.993\pm 0.001$ | $1.003\pm 0.001$ | $1.062\pm 0.010$
$q$ ($\times 10^{-3}$) | $0.567\pm 0.041$ | $0.468\pm 0.018$ | $0.466\pm 0.018$ | $0.559\pm 0.040$
$\alpha$ [rad] | $4.777\pm 0.006$ | $4.776\pm 0.005$ | $4.777\pm 0.005$ | $4.777\pm 0.005$
$\rho_{\ast}$ | $0.003\pm 0.001$ | $0.003\pm 0.001$ | $0.003\pm 0.001$ | $0.003\pm 0.001$

Note. — ${\rm HJD^{\prime}=HJD-2450000.0}$.

Figure 8: Light curve of KMT-2021-BLG-2294 with model curves and their
residuals. The geometries of 2L1S models are presented in Figure 9.  Figure 9:
The 2L1S model geometries of the best–fit and possible solutions (right-side
panels) of KMT-2021-BLG-2294. The left-side panels show the locations of the
solutions in the $(s,q)$ parameter space. The color scheme of the space is
identical to Figure 3.

The degeneracies arise from a combination of the $s_{\pm}$ degeneracy and an
unexpected resonant caustic degeneracy. We refer to the preferred set of
solutions as “C” (close) and “W” (wide). The “W” solution has a resonant
caustic, but the “C” solution does not. For these solutions,
$s^{\dagger}=0.996$ and $\alpha=4.777$ radians, in good agreement with the
heuristic analysis. The $\Delta\chi^{2}$ between the best-fit (i.e., “W” case)
and the “C” case is only $0.5$. The close and wide cases produce almost
identical light curves and so are completely degenerate.

The second pair of solutions both have resonant caustics, so we refer to them
as “${\rm R_{C}}$” (Resonant, $s<1$) and “${\rm R_{W}}$” (Resonant, $s>1$).
These also obey the expectations from the heuristic analysis with
$s^{\dagger}=0.998$ and $\alpha=4.777$. One remarkable aspect about these
solutions is that $\rho$ is very similar to the “C” and “W” solutions.
Examining the source trajectory and caustic structure in Figure 9 suggests
that there should be four distinct caustic crossings even though only two
bumps are seen in the light curve. In fact, due to the source location at the
outer edges of the caustics, those crossings (which would occur at ${\rm
HJD}^{\prime}=9452.39$ and $9452.71$) are so weak as to produce almost no
change in magnification relative to a point lens. Nevertheless, these slight
differences lead to these solutions being disfavored relative to the “W” case
by $24.4$ (“${\rm R_{C}}$”) and $30.0$ (“${\rm R_{W}}$”).

####  4.3.3 Tests of ARPX and OBT effects

Because of the short timescale of this event (i.e., $7\sim 8$ days), we do not
attempt to place limits on APRX or OBT effects.

####  4.3.4 2L1S/1L2S Degeneracy

The feature at the peak might seem to be explainable by a 1L2S interpretation.
However, we find that the 1L2S models cannot describe the peak of the light
curve, and especially not the KMTC03 point at HJD${}^{\prime}=9452.55$. In
total, the 1L2S model is disfavored by $\Delta\chi^{2}\sim 770$ relative to
the planetary models.

##  5 Source Color and Angular Source Radius

When $\rho_{\ast}$ is measured, it can be used to determine the angular
Einstein ring radius ($\theta_{\rm E}=\theta_{\ast}/\rho_{\ast}$, where
$\theta_{\ast}$ is the angular source radius). While $\rho_{\ast}$ was
measured for KMT-2021-BLG-0192 and KMT-2021-BLG-2294, for KMT-2021-BLG-0119,
we can only measure the $\rho_{\ast}$ distribution, which can be used to set
limits on $\theta_{\rm E}$ in the Bayesian analysis (Section 6).

We measure $\theta_{\ast}$ for all events using the conventional method
described in Yoo et al. (2004). In Figure 10, we present the $V$/$I$ CMD of
each event with the of the centroid of the red giant clump (RGC), source, and
blend. In Table 5, we present the results of the CMD analyses. The intrinsic
color of the RGC is adopted from Bensby et al. (2011). The de-reddened
magnitude of the RGC is adopted from Nataf et al. (2013) according to the
galactic longitude of each event. Under the assumption that the source and RGC
experienced the same stellar extinction, we can obtain the de-reddened color
and magnitude of the source. Based on the source color, we determine
$\theta_{\ast}$ using the surface brightness–color relation of Kervella et al.
(2004)333Because Kervella et al. (2004) provide the relation based on the
($V-K$) color, we convert the source color from ($V-I$) to ($V-K$) using the
color conversion of Bessell & Brett (1988).

Table 5: CMD analysis of Three Events  Event | KMT-2021-BLG-0119 | KMT-2021-BLG-0192 | KMT-2021-BLG-2294
---|---|---|---
$(V-I,I)_{\rm cl}$ | $(1.40,14.70)$ | $(3.05,16.62)$ | $(2.02,15.68)$
$(V-I,I)_{0,\rm cl}$ | $(1.06,14.36)$ | $(1.06,14.45)$ | $(1.06,14.38)$
Solution | Local A & A′ | STD ($s_{-}$), APRX ($s_{\pm},u{0}+$) | C, W
$(V-I,I)_{\rm S}$ | $(1.11\pm 0.02,19.34\pm 0.01)$ | $(2.57\pm 0.01,19.85\pm 0.01)$ | $(1.84\pm 0.01,20.72\pm 0.01)$
$(V-I,I)_{0,\rm S}$ | $(0.77\pm 0.05,19.00\pm 0.01)$ | $(0.58\pm 0.05,17.68\pm 0.01)$ | $(0.88\pm 0.05,19.42\pm 0.01)$
$(V-I,I)_{\rm B}$ | $(1.77\pm 0.09,19.81\pm 0.02)$ | $(2.98\pm 0.02,18.61\pm 0.01)$ | $(1.58\pm 0.02,19.18\pm 0.01)$
$\theta_{\ast}$ ($\mu{\rm as}$) | $0.53\pm 0.03$ | $0.80\pm 0.04$ | $0.50\pm 0.03$
$\theta_{\rm E}$ (mas) | $>0.38$ | $0.40\pm 0.05$ | $0.15\pm 0.02$
Solution | Local B & B′ | STD ($s_{+}$), APRX ($s_{\pm},u{0}-$) |  ${\rm R}_{\rm C}$, ${\rm R}_{\rm W}$
$(V-I,I)_{\rm S}$ | $(1.11\pm 0.02,19.26\pm 0.01)$ | $(2.57\pm 0.01,19.84\pm 0.01)$ | $(1.85\pm 0.01,20.90\pm 0.01)$
$(V-I,I)_{0,\rm S}$ | $(0.77\pm 0.05,18.92\pm 0.01)$ | $(0.58\pm 0.05,17.68\pm 0.01)$ | $(0.89\pm 0.05,19.60\pm 0.01)$
$(V-I,I)_{\rm B}$ | $(1.89\pm 0.12,19.95\pm 0.02)$ | $(2.98\pm 0.02,18.61\pm 0.01)$ | $(1.59\pm 0.02,19.14\pm 0.01)$
$\theta_{\ast}$ ($\mu{\rm as}$) | $0.55\pm 0.03$ | $0.80\pm 0.04$ | $0.47\pm 0.03$
$\theta_{\rm E}$ (mas) | $>0.31$ | $0.42\pm 0.05$ | $0.17\pm 0.02$

We note that, for KMT-2021-BLG-0119, the red giant stars in the KMTNet CMD are
too sparse to precisely determine the RGC. Thus, we use the OGLE-III CMD
(Szymański et al., 2011) to determine the RGC. The instrumental color and
magnitude of KMTNet are aligned to the OGLE instrumental scales using the
cross–matching of field stars. For the other events, the RGC can be determined
from the KMTNet CMDs. However, for consistency, we present the results of the
CMD analyses scaled to OGLE-III.

Figure 10: The color–magnitude diagrams of three events. The color and
magnitudes of the KMTNet CMD (black dot) are aligned to the OGLE–III (gray
dot) instrumental scales. The colored circles indicate the positions of RGC
(red), source (blue/cyan), and blend(green/dark green) shown in Table 5.

##  6 Characteristics of Planets

###  6.1 Bayesian Analysis

For the Bayesian analysis, we adopt the formalism described in Shin et al.
(2021), except that we adopt initial and present-day mass functions from
Chabrier (2003). In brief, we adopt the other Galactic priors from several
studies:

  1. 1.

the matter density profile of the disk from Robin et al. (2003) and Bennett et
al. (2014),

  2. 2.

the matter density profile of the bulge from Han & Gould (1995) and Dwek et
al. (1995),

  3. 3.

the mean velocity and the velocity dispersion of bulge stars from GAIA proper
motion information (Gaia Collaboration et al., 2018), and

  4. 4.

the mean velocity and the velocity dispersion of disk stars from the modified
model of Han & Gould (1995), which is described in Han et al. (2020).

We then generate artificial microlensing events (total $4\times 10^{7}$ events
for each case) and apply available constraints from the microlensing light
curve. For all cases, $t_{\rm E}$ are well measured, so we use a simple
Gaussian weight. Depending on the particular event, we may also have priors
from $\theta_{\rm E}$ or $\pi_{\rm E}$. For KMT-2021-BLG-0192 and
KMT-2021-BLG-2294, we measure $\rho_{\ast}$, so we apply a Gaussian weight
based on $\theta_{\rm E}$ (see Section 5). In addition, for KMT-2021-BLG-0192,
apply the 2D $\pi_{\rm E}$ constraint following the formalism described in Ryu
et al. (2019).

For KMT-2021-BLG-0119, we also use the 2D APRX distributions as a constraint
for $\pi_{\rm E}$. Then, because $\rho_{\ast}$ was not clearly measured, for
each solution, we construct a weight function ($W(\rho_{\ast})$) by fitting of
the distribution of $\Delta\chi^{2}$ as a function of $\rho_{\ast}$. For the
Local A and A′ cases, we use a piece-wise function

$W(\rho_{\ast})=\begin{cases}{\rm constant}&{(\rm
if~{}\rho_{\ast}<\rho_{\ast,{\rm break}})}\\\ a\,e^{c(x+b)^{2}}+0.6&{(\rm
if~{}\rho_{\ast,{\rm break}}\leq\rho_{\ast}\leq\rho_{\ast,{\rm limit}})}\\\
0.0&{(\rm if~{}\rho_{\ast}>\rho_{\ast,{\rm limit}})}\end{cases},$ (7)

where $x\equiv\log_{10}(\rho_{\ast})$ and ($a$, $b$, $c$) are coefficient set
for fitting, and $0.6$ is the normalization factor for making unity weight at
the best-fit value. For the Local A and A′ cases, $\rho_{\ast}$ cannot be zero
because there is a point during the caustic entrance. However, it is
increasingly difficult to probe models with $\rho_{\ast}<\rho_{\rm\ast,break}$
through an MCMC (which tends toward the preferred B and B′ cases). At the same
time, these values are increasingly unlikely because they imply ever larger
values of $\theta_{\rm E}$ ($\theta_{\rm E}(\rho_{\rm\ast,break}=6.6\times
10^{-5})\sim 80\,$ mas), so our assumption of a constant weight below
$\rho_{\rm\ast,break}$ has little effect on the Bayesian estimates. For the
Local B and B′ cases, we use

$W(\rho_{\ast})=\begin{cases}a\,e^{\frac{x-b}{c}}+1.0&{(\rm
if~{}\rho_{\ast}\leq\rho_{\ast,{\rm limit}})}\\\ 0.0&{(\rm
if~{}\rho_{\ast}>\rho_{\ast,{\rm limit}})}\end{cases},$ (8)

where $x\equiv\log_{10}(\rho_{\ast})$, ($a$, $b$, $c$) are coefficients, and
$1.0$ is the normalization factor. We present the coefficients for all models,
$\rho_{\ast,{\rm limit}}$, and $\rho_{\ast,{\rm break}}$ in Table 6.

Table 6: Coefficients of the $\rho_{\ast}$ weight functions for KMT-2021-BLG-0119 Coefficient | Local A | Local A′ | Local B | Local B′
---|---|---|---|---
a | 0.385439 | 0.402486 | -59.035155 | -55.573560
b | 3.811082 | 3.899674 | -0.497434 | -0.688004
c | -4.760185 | -3.133795 | 0.414940 | 0.393204
$\rho_{\ast,{\rm limit}}$ | 0.002344 | 0.002512 | 0.002344 | 0.002042
$\rho_{\ast,{\rm break}}$ | 0.000066 | 0.000054 |  $\cdots$ |  $\cdots$
Table 7: Lens Properties of Three Events  Event | Constraints | Case | $M_{\rm host}$ | $M_{\rm planet}$ | $M_{\rm planet}$ | $D_{\rm L}$ | $a_{\perp}$ | $a_{\rm snow}$ | $\mu_{\rm rel}$
---|---|---|---|---|---|---|---|---|---
|  |  | ($M_{\odot}$) | ($M_{\rm Jup}$) | ($M_{\rm Nep}$) | (kpc) | (au) | (au) | ($\rm mas\,yr^{-1}$)
KB210119 | $t_{\rm E}+\rho_{\ast}+\mbox{\boldmath$\pi$}_{\rm E}$ | Local A | $0.69_{-0.30}^{+0.34}$ | $5.97_{-2.60}^{+2.94}$ | $110.8_{-48.2}^{+54.6}$ | $3.51_{-1.13}^{+1.72}$ | $3.23_{-0.80}^{+0.76}$ | $1.87_{-0.80}^{+0.92}$ | $5.86_{-2.55}^{+3.15}$
|  | Local A′ | $0.69_{-0.30}^{+0.34}$ | $5.67_{-2.53}^{+2.87}$ | $105.2_{-46.9}^{+53.2}$ | $3.69_{-1.20}^{+1.75}$ | $3.24_{-0.87}^{+0.78}$ | $1.86_{-0.82}^{+0.93}$ | $5.39_{-2.34}^{+3.07}$
|  | Local B | $0.55_{-0.23}^{+0.31}$ | $5.58_{-2.47}^{+3.16}$ | $103.4_{-45.8}^{+58.5}$ | $3.05_{-0.91}^{+1.29}$ | $2.87_{-0.67}^{+0.67}$ | $1.47_{-0.63}^{+0.83}$ | $6.29_{-2.56}^{+3.18}$
|  | Local B′ | $0.56_{-0.24}^{+0.32}$ | $5.52_{-2.40}^{+3.12}$ | $102.4_{-44.5}^{+57.9}$ | $3.13_{-0.91}^{+1.30}$ | $2.92_{-0.68}^{+0.67}$ | $1.51_{-0.64}^{+0.85}$ | $6.11_{-2.46}^{+3.09}$
KB210192 | $t_{\rm E}+\theta_{\rm E}$ | $s_{-}$ | $0.55_{-0.28}^{+0.26}$ | $0.19_{-0.10}^{+0.09}$ | $3.55_{-1.80}^{+1.75}$ | $6.66_{-1.41}^{+0.91}$ | $2.07_{-0.47}^{+0.35}$ | $1.48_{-0.74}^{+0.71}$ | $4.54_{-0.54}^{+0.54}$
|  | $s_{+}$ | $0.55_{-0.28}^{+0.26}$ | $0.19_{-0.10}^{+0.09}$ | $3.59_{-1.82}^{+1.75}$ | $6.62_{-1.43}^{+0.91}$ | $3.51_{-0.80}^{+0.60}$ | $1.50_{-0.75}^{+0.70}$ | $4.68_{-0.57}^{+0.57}$
| $t_{\rm E}+\theta_{\rm E}+\mbox{\boldmath$\pi$}_{\rm E}$ | $s_{-},\,u_{0}{+}$ | $0.27_{-0.09}^{+0.12}$ | $0.10_{-0.03}^{+0.05}$ | $1.83_{-0.62}^{+0.84}$ | $5.26_{-1.01}^{+1.01}$ | $1.62_{-0.31}^{+0.33}$ | $0.72_{-0.24}^{+0.31}$ | $4.69_{-0.60}^{+0.59}$
|  | $s_{-},\,u_{0}{-}$ | $0.27_{-0.09}^{+0.12}$ | $0.11_{-0.04}^{+0.05}$ | $1.99_{-0.69}^{+0.85}$ | $5.12_{-0.99}^{+1.00}$ | $1.62_{-0.30}^{+0.33}$ | $0.74_{-0.24}^{+0.31}$ | $4.87_{-0.61}^{+0.62}$
|  | $s_{+},\,u_{0}{+}$ | $0.27_{-0.09}^{+0.11}$ | $0.10_{-0.04}^{+0.04}$ | $1.91_{-0.67}^{+0.83}$ | $5.23_{-1.00}^{+0.99}$ | $2.76_{-0.51}^{+0.55}$ | $0.72_{-0.24}^{+0.30}$ | $4.76_{-0.60}^{+0.60}$
|  | $s_{+},\,u_{0}{-}$ | $0.27_{-0.09}^{+0.12}$ | $0.10_{-0.03}^{+0.05}$ | $1.89_{-0.64}^{+0.85}$ | $5.14_{-1.00}^{+1.01}$ | $2.79_{-0.52}^{+0.57}$ | $0.74_{-0.25}^{+0.32}$ | $4.81_{-0.62}^{+0.62}$
KB212294 | $t_{\rm E}+\theta_{\rm E}$ | C | $0.11_{-0.06}^{+0.17}$ | $0.07_{-0.03}^{+0.10}$ | $1.24_{-0.64}^{+1.84}$ | $6.86_{-1.06}^{+0.97}$ | $0.94_{-0.16}^{+0.16}$ | $0.30_{-0.16}^{+0.45}$ | $7.63_{-0.77}^{+0.79}$
|  | W | $0.11_{-0.06}^{+0.17}$ | $0.07_{-0.03}^{+0.10}$ | $1.23_{-0.64}^{+1.81}$ | $6.86_{-1.06}^{+0.98}$ | $1.07_{-0.19}^{+0.19}$ | $0.31_{-0.16}^{+0.45}$ | $7.58_{-0.77}^{+0.79}$
|  | ${\rm R}_{\rm C}$ | $0.14_{-0.07}^{+0.20}$ | $0.07_{-0.04}^{+0.10}$ | $1.27_{-0.65}^{+1.79}$ | $6.80_{-1.08}^{+0.97}$ | $1.13_{-0.20}^{+0.19}$ | $0.38_{-0.19}^{+0.53}$ | $7.65_{-0.77}^{+0.79}$
|  | ${\rm R}_{\rm W}$ | $0.14_{-0.07}^{+0.20}$ | $0.07_{-0.04}^{+0.10}$ | $1.26_{-0.65}^{+1.79}$ | $6.80_{-1.08}^{+0.97}$ | $1.15_{-0.20}^{+0.20}$ | $0.38_{-0.19}^{+0.53}$ | $7.69_{-0.78}^{+0.81}$
Figure 11: The ($D_{L},\,M_{\rm host}$) and ($a_{\perp},\,M_{\rm planet}$)
contours with probability distributions of the lens properties of each event.
The blue contour shows the best-fit case of each event. We present an
alternate solution as the red contour for comparison. In the histograms, the
dark gray indicates a $68\%$ confidence interval. The black dashed line
indicates the median value of each property. For KMT-2021-BLG-0119 (upper
panels), we present the Local B family (blue) and Local A family (red). For
KMT-2021-BLG-0192 (middle panels), we compare the APRX ($s_{-}$, $u_{0}-$)
case (blue; the best fit) with the STD $s_{+}$ case (red). For
KMT-2021-BLG-0192 (bottom panels), we present the W case (blue; the best fit)
and ${\rm R}_{C}$ (red).

###  6.2 Lens Properties of Three events

In Table 7, we present the lens properties derived from the Bayesian
posteriors for each event. In Figure 11, we present the contours of the lens
properties with probability distributions of each event. We present the
best–fit cases and selected cases for comparison. The plots visualize the
possible ranges of the lens properties shown in Table 7.

For KMT-2021-BLG-0119, the lens system consists of a super-Jupiter-mass planet
($M_{\rm planet}\sim 6\,M_{\rm Jup}$) and an early M-type or a K-type dwarf
host star ($M_{\rm host}\sim 0.56$ or $\sim 0.69\,M_{\odot}$, for the A and B
families of solutions, respectively). The planet orbits the host with a
projected separation of $\sim 2.9$ or $\sim 3.2$ au beyond its snow line
($\sim 1.5$ or $\sim 1.9$ au). The planetary system is located at the distance
of $\sim 3-4$ kpc from us; i.e., half way to the Galactic bulge.

We note that the blend of KMT-2021-BLG-0119 is compatible with the lens
posteriors (see Table 5). For example, if the lens is an M-dwarf, it would
have an absolute magnitude of $M_{I}=7.2$. Assuming a distance of $3.0$ kpc
and that it is behind all of the dust ($A_{I}\sim 0.34$), its observed
magnitude would be well matched to the observed blend, which has $I=19.9$ mag.
In the case of the K-dwarf lens ($M_{I}\sim 6.0$ and $D_{\rm L}\sim 4.9$ kpc),
the observed magnitude would be also well matched to the observed blend, which
has $I=19.8$ mag. We use the pyDIA reductions to check for an offset between
the magnified source and the baseline object, which could show that the blend
is not associated with the event. We find
$\Delta\mbox{\boldmath$\theta$}(N,E)=(64,1.5)\,$ mas. Given that the
uncertainties in such measurements are on the order of tens of mas, this
measurement does not rule out the possibility that the blend is the lens;
i.e., it is not strongly inconsistent with zero. Regardless, because the blend
is about $40\%$ of the light, immediate AO followup observations could confirm
that the blend is closely aligned to the source. Because the properties of the
various solutions are so similar, such observations would not resolve the
degeneracy, but they could result in a better characterization of the lens
flux.

For KMT-2021-BLG-0192, when including the parallax constraint, the lens system
consists of a planet slightly larger than Neptune ($M_{\rm planet}\sim
2\,M_{\rm Nep}$) and an M-dwarf host star ($M_{\rm host}\sim 0.3\,M_{\odot}$).
Without the parallax constraint, the values are somewhat larger, but
consistent at 1-$\sigma$. The planet is a typical microlensing planet located
beyond the snow line. The planetary system is located at $D_{\rm L}\sim 5$
kpc. For completeness, we note that the baseline object appears to be offset
from the microlensing event by
$\Delta\mbox{\boldmath$\theta$}(N,E)\sim(-430,100)\,$ mas so it is not likely
to be associated with the event.

For KMT-2021-BLG-2294, the lens consists of a Neptune-mass planet ($M_{\rm
p}\sim 1.2\,M_{\rm Nep}$) orbiting a late M-dwarf host ($M_{\rm h}\sim
0.1\,M_{\odot}$). The system is located in or near the bulge at $D_{\rm L}\sim
6.8\,$ kpc. One interesting point is that the posterior for the host mass
significantly overlaps the brown-dwarf regime. This small host mass arises
from the short timescale of this event (i.e., $t_{\rm E}\sim$ $7$–$8$ days).
In this case, the baseline object appears to be offset from the microlensing
event by $\Delta\mbox{\boldmath$\theta$}(N,E)\sim(160,410)\,$ mas. This offset
implies that the blended light is not due to the lens and that it could be
easily resolved from the microlensing target with high-resolution
observations. The source itself is reasonably faint ($I=20.8$ mag), which
suggests a contrast ratio of $\Delta K=(2.2,2.8,3.5)\,$ mag for a lens mass of
$M_{\rm lens}=(0.4,0.3,0.2)M_{\odot}$ (Bessell & Brett, 1988; Baraffe et al.,
2015). Given the magnitude of the lens-source relative proper motion ($\sim
7.6\mathrm{\,mas\,yr}^{-1}$), it should be possible to either measure or place
strong upper limits on the lens flux at first light of 30m-class AO systems.

##  7 Discussion and Conclusion

We have presented three microlensing planets discovered by the KMTNet survey
in $2021$: KMT-2021-BLG-0119Lb, KMT-2021-BLG-0192Lb, and KMT-2021-BLG-2294Lb.
These planets range in mass from close to a Neptune mass to Super–Jupiter-
sized. As is typical of microlensing events, the planet hosts are all likely
to be low-mass dwarfs and the systems are $\sim 3$ – $7\,$ kpc from us. See
Table 7.

Of these three planets, KMT-2021-BLG-2294 is the most interesting. First, this
event fails the criteria for selection by the AnomalyFinder algorithm (Zang et
al., 2021, 2022). For the AnomalyFinder algorithm, the planet has only
$\Delta\chi_{0}^{2}=37$ for $t_{\rm eff}$ = 0.05, and $\Delta\chi_{0}^{2}=59$
for $t_{\rm eff}=0.025$. By contrast, the algorithm has a default threshold of
at least $\Delta\chi^{2}>120$. At the same time, the planetary signal is
clearly seen by eye in Figure 8. Hence, it would be interesting to consider
how the algorithm might be modified to detect such signals, although any
changes must then be weighed against the potential increase in false
positives.

Second, the Bayesian analysis for KMT-2021-BLG-2294 suggests that the host is
an extremely low-mass M-dwarf. The planet population at this end of the
stellar mass function is particularly interesting because of the extreme
nature of the hosts. Several studies have suggested that it is more difficult
to form giant planets around M-dwarfs via core accretion due to the longer
dynamical times (Laughlin et al., 2004; Ida & Lin, 2005). For example, the
work of Kennedy & Kenyon (2008) show how giant planet formation varies with
stellar mass and suggest that there may be a lower limit on the host mass for
giant planets.

Figure 12: Distributions of host mass and system distance for confirmed
exoplanets with low-mass host stars. The point shape corresponds to the
detection method. The color of each point is set by the (log) mass of the
planet (black: $\log(m_{p}/M_{J})<-2$, red: $-2<\log(m_{p}/M_{J})<-1.5$,
yellow: $-1.5<\log(m_{p}/M_{J})<-1$, green: $-1<\log(m_{p}/M_{J})<-0.5$, cyan:
$-0.5<\log(m_{p}/M_{J})<0$, blue: $0<\log(m_{p}/M_{J})<0.5$, magenta:
$0.5<\log(m_{p}/M_{J})$). For radial velocity planets, $m_{p}\sin i$ is
plotted if $m_{p}$ is not available, and for microlensing planets $a_{\perp}$
(the projection of the semi-major axis on the sky) is plotted in place of $a$.
Reference lines are drawn at $a=0.1\,$ au and $M_{\rm host}=0.3\,M_{\odot}$.
The location of KMT-2021-BLG-2294 in this plane is indicated by the arrow. The
upper left panel shows all planets together, while the other panels show
subsets by planet mass, as indicated in the titles. Data from the NASA
Exoplanet Archive, accessed 2022 July 20.

To place KMT-2021-BLG-2294Lb in better context with other planets around low-
mass host stars, in Figure 12, we plot it together with transit, radial
velocity, and other microlensing planets from the NASA Exoplanet Archive. This
figure shows clear evidence of selection effects, which lead to the appearance
of two distinct groups of planets. One group consists of very short-period
planets with $a\lesssim 0.1\,$ au whose detections are dominated by the
transiting planets, and another group with $a\gtrsim 0.1\,$ au, which is
dominated by microlensing planet detections. There is also a trend in the
microlensing planets that reflects the fact that $M_{\rm host}\propto
a_{\perp}^{2}$ at fixed $\theta_{\rm E}$.

However, in spite of these selection effects, there is a clear lack of giant
planets in the close-in planet population, despite the fact that they should
be readily detected. On the other hand, giant planets are abundant in the
microlensing sample, which shows planet discoveries at a continuous range of
masses. This suggests that there is no particular challenge to forming giant
planets around M dwarfs, but there is a challenge for either getting or
keeping them in close orbits.

On the other hand, the majority of the microlensing host masses are derived
from Bayesian estimates, although there are a few cases with $M_{\rm
L}<0.3\,M_{\odot}$ for which the lens mass is measured through a combination
of $\theta_{\rm E}$ and $\mbox{\boldmath$\pi$}_{\rm E}$ [c.f.,
OGLE-2017-BLG-1140Lb (Calchi Novati et al., 2018), OGLE-2017-BLG-1434Lb
(Udalski et al., 2018), OGLE-2018-BLG-0532Lb (Ryu et al., 2020), and
OGLE-2018-BLG-0596Lb (Jung et al., 2019)]. In particular, OGLE-2017-BLG-1140L
hosts a giant planet that has a well-measured host mass of $0.21\pm
0.03\,M_{\odot}$, demonstrating that such planetary systems exist. However,
for those events with only Bayesian mass estimates, there is usually a
possibility of a more massive host star. As we discussed in Section 6, future
adaptive optics or other high-resolution imaging of KMT-2021-BLG-2294 could
confirm that the host mass is indeed $<0.3M_{\odot}$. More secure host mass
measurements or limits for the microlensing planet population would allow for
a study of how the planet distribution varies with host mass, which could then
be linked back to planet formation theory (e.g., Kennedy & Kenyon, 2008) and
compared to radial velocity studies (e.g., Bonfils et al., 2013).

This research has made use of the KMTNet system operated by the Korea
Astronomy and Space Science Institute (KASI), and the data were obtained at
three host sites of CTIO in Chile, SAAO in South Africa, and SSO in Australia.
I.-G.S., S.-J.C., and J.C.Y. acknowledge support from N.S.F Grant No.
AST–2108414. Work by C.H. was supported by grants of the National Research
Foundation of Korea (2017R1A4A1015178 and 2019R1A2C2085965). Y.S. acknowledges
support from BSF Grant No. 2020740. The MOA project is supported by JSPS
KAKENHI Grant Number JSPS24253004, JSPS26247023, JSPS23340064, JSPS15H00781,
JP16H06287, and JP17H02871. The computations in this paper were conducted on
the Smithsonian High Performance Cluster (SI/HPC), Smithsonian Institution
(https://doi.org/10.25572/SIHPC). This research has made use of the NASA
Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program.

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