Mass Production of 2021 KMTNet Microlensing Planets. III. Analysis of Three Giant Planets In-Gu Shin1 , Jennifer C. Yee1 , Andrew Gould2,3, Kyu-Ha Hwang4 , Hongjing Yang5 , Ian A. Bond6 (Leading Authors), Michael D. Albrow7 , Sun-Ju Chung4 , Cheongho Han8 , Youn Kil Jung4,9, Yoon-Hyun Ryu4 , Yossi Shvartzvald10 , Weicheng Zang5 , Sang-Mok Cha4,11, Dong-Jin Kim4, Seung-Lee Kim4 , Chung-Uk Lee4 , Dong-Joo Lee4, Yongseok Lee4,11, Byeong-Gon Park4,9 , Richard W. Pogge3 (The KMTNet Collaboration), and Fumio Abe12, Richard Barry13 , David P. Bennett13,14 , Aparna Bhattacharya13,14, Hirosane Fujii12, Akihiko Fukui15,16 , Yuki Hirao17 , Stela Ishitani Silva13,18, Yoshitaka Itow12 , Rintaro Kirikawa17, Iona Kondo17 , Naoki Koshimoto19 , Yutaka Matsubara12 , Sho Matsumoto17, Shota Miyazaki17 , Yasushi Muraki12 , Arisa Okamura17, Greg Olmschenk13 , Clément Ranc20 , Nicholas J. Rattenbury21 , Yuki Satoh17 , Takahiro Sumi17 , Daisuke Suzuki17, Taiga Toda17, Paul . J. Tristram22, Aikaterini Vandorou13,14, and Hibiki Yama17 (The MOA Collaboration) 1 Center for Astrophysics | Harvard & Smithsonian 60 Garden St., Cambridge, MA 02138, USA 2Max Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany 3 Department of Astronomy, The Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA 4 Korea Astronomy and Space Science Institute, Daejon 34055, Republic Of Korea 5 Department of Astronomy and Tsinghua Centre for Astrophysics, Tsinghua University, Beijing 100084, Peopleʼs Republic of China 6 Institute of Natural and Mathematical Sciences, Massey University, Auckland 0745, New Zealand 7 University of Canterbury, Department of Physics and Astronomy, Private Bag 4800, Christchurch 8020, New Zealand 8 Department of Physics, Chungbuk National University, Cheongju 28644, Republic Of Korea 9 University of Science and Technology, Korea, (UST), 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic Of Korea 10 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel 11 School of Space Research, Kyung Hee University, Yongin, Kyeonggi 17104, Republic Of Korea 12 Institute for Space-Earth Environmental Research, Nagoya University, Nagoya 464-8601, Japan 13 Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 14 Department of Astronomy, University of Maryland, College Park, MD 20742, USA 15 Department of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 16 Instituto de Astrofísica de Canarias, Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain 17 Department of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan 18 Department of Physics, The Catholic University of America, Washington, DC 20064, USA 19 Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 20 Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, F-75014 Paris, France 21 Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand 22 University of Canterbury Mt. John Observatory, P.O. Box 56, Lake Tekapo 8770, New Zealand Received 2022 September 7; revised 2022 October 14; accepted 2022 October 19; published 2022 December 7 Abstract We present the analysis of three more planets from the KMTNet 2021 microlensing season. KMT-2021-BLG- 0119Lb is a∼6MJup planet orbiting an early M dwarf or a K dwarf, KMT-2021-BLG-0192Lb is a∼2MNep planet orbiting an M dwarf, and KMT-2021-BLG-2294Lb is a∼1.25MNep 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0.1 au, which shows a strong preference for small planets. Supporting material: data behind figures 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. (2022a) and Ryu et al. (2022b), we published eight planet detections; 10 other planets from the 2021 season have been published in other work (Han et al. 2022a, 2022b, 2022c, 2022d, 2022e; 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. 2022a). However, because IGS uses a variation on a method that was previously described, we The Astronomical Journal, 165:8 (19pp), 2023 January https://doi.org/10.3847/1538-3881/ac9d93 © 2022. The Author(s). 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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 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 ( )-10 3 or smaller, or the event timescale should be shorter than ∼10 days for a relatively small mass ratio (i.e.,( )-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, the machine-based selection is required to find a well-defined sample of planets. However, there are certain advantages to 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, the 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)23 on (2021 March 25, 2021 March 29, 2021 August 27). The KMTNet observations are made using three identical 1.6 m telescopes with wide-field cameras (i.e., 2°× 2° 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. For selected events (i.e., planet candidates), individual KMTNet data were carefully rereduced using photometry packages that adopted the differential image analysis (DIA) technique called pySIS (Albrow et al. 2009) and pyDIA (Bramich et al. 2013; Albrow 2017). We analyze the light curves using these tender-loving care (TLC) versions of data sets. 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: (t0, u0, tE). These are the time at the peak of the light curve (t0), the impact parameter (u0) in units of the angular Einstein ring radius (θE), and the Einstein timescale (tE), 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, α) are required. These are the projected separation between binary components in units of θE (s), the mass ratio of binary components defined as q≡Msecondary/Mprimary, and the angle between source trajec- tory and binary axis (α). From a localized anomaly, we can predict solution(s) using the unified s† formalism described in Hwang et al. (2022) and Ryu et al. (2022a). From the time of the anomaly, tanom, we obtain the scaled time offset from the peak of the light curve, ( )t º -t t t 1anom anom 0 E and the source position offset from the host, ( )t= +u u . 2anom anom 2 0 2 Then, we can also predict, ( )† a t º +  = s u u u4 2 ; tan , 3anom 2 anom 0 anom where the ± subscript of † s indicates either a major or minor image perturbation, respectively (Gould & Loeb 1992). In general, the major image perturbations ( † +s ) appear as a “bump” shaped anomaly, while the minor image perturbations ( † -s ) Table 1 Observations of 2021 Planetary Events Event KMT-2021-BLG-0119 KMT-2021-BLG-0192 KMT-2021-BLG-2294 R.A. (J2000) 18h16m00 13 17h52m25 19 18h00m14 98 Decl. (J2000) -  ¢ 29 44 38 .00 -  ¢ 30 00 31 .28 -  ¢ 28 36 44 .78 (ℓ, b) (2°.572, − 6°.155) (−0°.158, −1°.821) (1°.908, −2°.597) KMTNet field BLG33 BLG02, BLG42 BLG03, BLG43 Γ (hr−1) 1.0 4.0 4.0 Extinction (AI) 0.38 2.06 1.21 Alert date (YYYY-MM-DD) 2021-03-25 2021-03-29 2021-08-27 Additional Observations L MOA L Note. The MOA alerted MOA-2021-BLG-080 on 2021 April 10, which is the identical event to KMT-2021-BLG-0192. 23 KMTNet Alert System (https://kmtnet.kasi.re.kr/ulens/) 2 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. https://kmtnet.kasi.re.kr/ulens/ appear as a “dip” shaped anomaly. For minor image perturbations, we can additionally predict the mass ratio (to a factor ∼2 level) from the duration of the “dip” anomaly, Δtdip : ⎜ ⎟ ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ∣ ∣ ∣ ∣ ( )a a= D = D q t t s u t t s u4 sin 4 sin . 4 dip E 2 2 anom dip E 2 0 3 The predictions ( † s ) can be compared to the empirical result. In the case of only one solution, s should correspond to one of the values of † s . If there are two solutions (s+, s−), we expect them to be related by ( )† = + -s s s . 5 In that case, it is the value of s† that should correspond to one of the values of † s . 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 “nLmS” 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 (t0, u0, tE, s, q, α, and ρ*), where ρ* is the angular source radius (θ*) scaled by the Einstein radius, i.e., ρ*≡ θ*/θE. 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 ( ) [ ]Î -slog 1.0, 1.010 and ( ) [ ]Î -qlog 5.5, 1.010 . For the remaining parameters (t0, u0, tE, α, and ρ*), we find optimal solutions using the Markov Chain Monte Carlo (MCMC) algorithm (Doran and Müller 2004) to minimize χ2. We start the modeling from the 1L1S parameters for t0, u0, and tE, plus 21 initial values within a range of α= [0.0, 2π] (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 data sets to make each data point contribute χ2∼ 1.0. We follow the procedure described in Yee et al. (2012); i.e., = +e k e enew old 2 min 2 , where enew is the rescaled error, k is the rescaling factor, eold is the original error, and emin 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 tE> 15 days. For the APRX effect, we introduce two additional parameters: πE,N and πE,E, which are north and east components of the microlensing parallax vector (πE) projected onto the sky, respectively. Even if there is not a significant improvement in χ2, πE is often well constrained along one axis, which is roughly aligned with the πE,E direction. If the APRX model significantly improves χ2, 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 dα/dt, where ds/dt is the rate of change of the binary separation (i.e., s) and dα/dt is the rate of change of the α 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 β< 0.8, where ⎜ ⎟⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎡ ⎣⎢ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎤ ⎦⎥  ( ) b k p p q p p q a º = + ´ + ^ M s s ds dt d dt KE PE yr 8 1 , 6 2 2 E E E S E 3 2 2 where κ is a constant defined as κ≡ 4G/c2au= 8.144 mas/Me and πS is the source parallax defined as πS≡ au/DS where DS 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† (or offset) degeneracy, 2L1S models may be subject to a degeneracy in ρ*, which may affect the value of q (Ryu et al. 2022a; 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 ρ* is small in comparison) or the size of the source (so ρ* is  the width of the caustic). Hence, in some cases, high-cadence observations can distin- guish 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 a 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, qflux, and separate values of t0,i, u0,i, and optionally ρ*,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 3 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. Figure 1. Light curve of KMT-2021-BLG-0119 with APRX model curves, geometries, and residuals. (The data used to create this figure are available.) 4 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. types such as the ecliptic degeneracy (Jiang et al. 2004; Poindexter et al. 2005), the ±u0 degeneracy (Smith et al. 2003), and the jerk-parallax degeneracy (Gould 2004).24 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: (u0, α, πE,N)→−(u0, α, πE,N). 4. Analysis Results 4.1. KMT-2021-BLG-0119 4.1.1. Light Curve 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 microlen- sing planets (Han et al. 2021). For the heuristic analysis we have (tanom= 9308.3, t0= 9305.97, u0= 0.067, tE= 62 day), which yields ( † a= = = +u s0.077, 1.039, 60anom ). These values are well matched to the fitted values derived below. 4.1.2. STD Models and the ρ* 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¢ = 9303.47 falls on the caustic entrance. In the “B” family of models, which was discovered while checking for ρ* = 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). 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 ρ* (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 suboptimal, 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 ρ* and small ρ* cases, respectively, although the small ρ* case is consistent with ρ* = 0. We find that the small ρ* case shows better fits at the entrance (i.e., ¢ = ~HJD 9303.5 9304.0), while the large ρ* case shows slightly better fits at the exit (i.e., ¢ = ~HJD 9312.5 9313.0). We also find that the small ρ* model fits better than the large ρ* fit as the source approaches the caustic exit (i.e., ¢ = ~HJD 9311. 9312.5). In total, the Δχ2 between the large ρ* and small ρ* cases is only 1.57. 4.1.3. APRX Models and Solving the ρ* Degeneracy Problem The STD models have long timescales (tE 60 days), and the two caustic crossings separated by ∼9 days give strong timing constraints on the light curve. Thus, we consider the APRX effect. We find χ2 improvement Δχ2= 23∼ 30 between the STD (two ρ* cases) and APRX (u0< 0 and u0> 0 cases) models. In addition, the ARPX contours shown in Figure 3 are well converged and inconsistent with zero at 6σ. We check the improvements using the cumulative Δχ2 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 Table 2 Model Parameters of KMT-2021-BLG-0119 Model STD APRX Parameter Large − ρ* Small − ρ* Local A Local ¢A Local B Local ¢B cground 2 1084.207 1082.637 1060.093 1059.787 1054.530 1054.422 t0 [ ]¢HJD 9305.971 9305.956 9305.868 9305.895 9305.769 9305.779 ±0.032 ±0.043 ±0.042 ±0.043 ±0.050 ±0.050 u0 0.066 0.071 −0.077 0.076 −0.081 0.080 ±0.002 ±0.002 ±0.002 ±0.003 ±0.003 ±0.003 tE [days] 64.124 60.705 56.635 58.309 53.348 54.288 ±1.523 ±1.794 ±1.646 ±1.793 ±1.610 ±1.670 s 1.039 1.043 1.049 1.045 1.054 1.053 ±0.003 ±0.003 ±0.003 ±0.003 ±0.003 ±0.003 q (×10−4) 63.607 71.886 82.413 78.592 97.455 93.935 ±3.239 ±5.612 ±5.068 ±5.293 ±6.149 ±6.255 α [rad] 1.045 1.046 −1.076 1.049 −1.120 1.122 ±0.031 ±0.039 ±0.028 ±0.029 ±0.030 ±0.030 ρ* 0.003 L L L L L ±0.001 L L L L L * r ,max L <0.0025 <0.0014 <0.0014 <0.0018 <0.0018 πE,N L L 0.000 −0.244 −0.031 −0.034 L L ±0.234 ±0.232 ±0.222 ±0.220 πE,E L L 0.143 0.132 0.209 0.183 L L ±0.032 ±0.033 ±0.036 ±0.037 Note. ¢ = -HJD HJD 2450000.0. The total number of data points (Ndata) is 1059. 24 The APRX degeneracies are well described/organized in Skowron et al. (2011). 5 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. of the bulge season when Earth is accelerating rapidly to the east, which can produce the strong πE,E signal as is observed. Furthermore, we find that, for the “A” family of solutions, APRX models always favor the small ρ* solutions, even when the fits are initialized at the large ρ* STD solutions. Large ρ* solutions are excluded at the 4σ 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 ρ* case at the entrance but the large ρ* 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 ρ* degeneracy is resolved when the APRX is included. Figure 2. The comparison of STD large ρ*, STD small ρ*, and ARPX models of KMT-2021-BLG-0119. 6 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. 4.1.4. Test of the OBT Effect We find χ2 improvement of Δχ2∼ 16 when we include the OBT parameters in the APRX solutions. However, the OBT parameters show large values, (ds/dt, dα/dt)∼ (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 |KE/PE|⊥< 0.8 and ML< 3.0Me, we find that most of the χ2 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. 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: τanom= 0.00389 and uanom= 0.01073. From the heuristic analysis, we find that † =+s 1.005, † =-s 0.995 and α= 1.200 radians. 4.2.2. STD Models By following the 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± cases) having mass ratios in the planetary event regime, i.e., ( )~ -q 10 4 . The Δχ2 between the s− and s+ solutions is only 0.510, so they are statistically indistinguishable. In Table 3, we present the parameters of the best-fit models. Figure 3. The APRX contours of KMT-2021-BLG-0119. Each color indicates the χ2 difference between the best fit and chains. That is, ( )c c cD º - =- n2 chain 2 best fit 2 2, where n = 1 (red), 2 (yellow), 3 (green), 4 (light blue), 5 (blue), and 6 (purple). 7 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. The geometric mean of these two solutions is s†= 1.006, in good agreement with the † +s prediction from the heuristic analysis. Likewise, the value of α= 1.184 is also in good agreement with the heuristic expectation. 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 ρ*. The extremely dense coverage at the anomaly makes this measurement very secure. 4.2.3. Resolving the ρ* 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. 2022a), which suggests there may be alternate solutions for KMT-2021-BLG-0192 caused by the ρ* degeneracy. We explicitly search for such solutions and present the caustic geometries of the possible large ρ* solutions compared to those of the best-fit solutions in Figure 6. The possible solutions show worse fits with Δχ2= 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 ρ* 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 interpreta- tion. We find a plausible 1L2S model shown in Table 3. Both the flux ratio of binary sources, ºq flux fluxS Sflux 2 1, and * r S, 2 are well measured, but there is only an upper limit on * r S, 1 , which may be either larger or smaller than * r S, 2 . Hence, it is not possible to rule out this solution based on these physical considerations. On the other hand, for this solution θ*,2∼ 0.3 μas, and t*,2= 0.05 d, so μ= θ*/t* = 1.6 mas yr−1, which is somewhat unlikely, though not impossible. In addition, the 1L2S solution fits worse than the 2L1S solutions by Δχ2∼ 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., tE∼ 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 |πE,N|> 2. However, our investigation of the cumulative Δχ2 plots showed that the χ2 improvement mostly came from the baseline data toward the end of the microlensing season. Thus, we exclude data with ¢ >HJD 9360.0 from the modeling for this event. Figure 4. Cumulative Δχ2 plots between STD and APRX models of KMT-2021-BLG-0119. 8 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. Ultimately, we find that the parallax improves the fit by Δχ2∼ 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). 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. 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. (The data used to create this figure are available.) 9 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. Table 3 Model Parameters of KMT-2021-BLG-0192 Model 2L1S (STD) 2L1S (APRX) 1L2S Parameter s− s+ s− (u0+) s− (u0−) s+ (u0+) s+ (u0−) Parameter χ2 4738.844 4739.354 4702.863 4702.032 4702.928 4702.911 χ2 4773.792 t0 [ ]¢HJD 9315.697 9315.697 9315.697 9315.697 9315.697 9315.697 t S0, 1 [ ]¢HJD 9315.686 ±0.001 ±0.001 ±0.001 ±0.001 ±0.001 ±0.001 ±0.001 u0 0.010 0.010 0.010 −0.010 0.010 −0.010 u S0, 1 0.010 ±0.001 ±0.001 ±0.001 ±0.001 ±0.001 ±0.001 ±0.001 tE [days] 32.601 32.257 31.539 31.627 31.204 31.890 t S0, 2 [ ]¢HJD 9315.823 ±0.605 ±0.611 ±0.716 ±0.717 ±0.725 ±0.696 ±0.001 s 0.776 1.303 0.774 0.761 1.321 1.312 u S0, 2 (×10−3) −0.058 ±0.016 ±0.028 ±0.018 ±0.017 ±0.030 ±0.030 ±0.287 q (×10−4) 3.327 3.333 3.541 3.733 3.707 3.544 tE [days] 32.321 ±0.323 ±0.326 ±0.370 ±0.363 ±0.362 ±0.363 ±0.623 α [rad] 1.184 1.184 1.183 −1.186 1.181 −1.184 * r S, ,max1 <0.0087 ±0.005 ±0.004 ±0.005 ±0.005 ±0.005 ±0.005 ρ* (×10−4) 19.607 19.318 20.028 19.212 20.063 19.201 * r S, 2 (×10−4) 16.277 ±1.634 ±1.696 ±1.892 ±1.824 ±1.861 ±1.878 ±0.755 L L L L L L L qflux 0.019 L L L L L L ±0.001 πE,N L L 1.637 2.312 2.896 2.143 L L L L ±2.124 ±2.117 ±2.143 ±2.081 L L πE,E L L 0.272 0.304 0.360 0.285 L L L L ±0.140 ±0.137 ±0.141 ±0.135 L L Note. ¢ = -HJD HJD 2450000.0. The total number of data points (Ndata) is 4676. For the 1L2S model, the angular radius of the first source ( * r S, 1 ) is not measured. The best-fit value of * r S, 1 is 7.690 × 10−4. Figure 6. Geometries of best fit and possible solutions (s± cases) caused by the ρ* degeneracy of KMT-2021-BLG-0192. 10 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. 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 τanom∼ 0 and uanom∼ u0= 0.006. Hence, the heuristic analysis suggests † =-s 0.997 and † =+s 1.003 and α=±π/2 radians. Further- more, because this is a “dip” anomaly, we can predict the mass ratio from Δtdip= 0.06 days and tE= 7.1 days; i.e., q= 7.4× 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 ¢ ~HJD 9451 to ∼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. The degeneracies arise from a combination of the s± 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†= 0.996 and α= 4.777 radians, in good agreement with the heuristic analysis. The Δχ2 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 “RC” (Resonant, s< 1) and “RW” (Resonant, s> 1). These also obey the expectations from the heuristic analysis with s†= 0.998 and α= 4.777. One remarkable aspect of these solutions is that ρ 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 ¢ =HJD 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 (“RC”) and 30.0 (“RW”). 4.3.3. Tests of ARPX and OBT Effects Because of the short timescale of this event (i.e., 7∼ 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¢ = 9452.55. In total, the 1L2S model is disfavored by Δχ2∼ 770 relative to the planetary models. Figure 7. The (πE,E, πE,N) distributions of APRX models of KMT-2021-BLG-0192. The color scheme is identical to Figure 3. 11 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. 5. Source Color and Angular Source Radius When ρ* is measured, it can be used to determine the angular Einstein ring radius (θE= θ*/ρ*, where θ* is the angular source radius). While ρ* was measured for KMT-2021-BLG-0192 and KMT-2021-BLG-2294, for KMT-2021-BLG-0119, we can only measure the ρ* distribution, which can be used to set limits on θE in the Bayesian analysis (Section 6). We measure θ* 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 centroid of the red giant clump 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. (The data used to create this figure are available.) 12 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. (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 dereddened 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 dereddened color and magnitude of the source. Based on the source color, we determine θ* using the surface brightness–color relation of Kervella et al. (2004). 25 We note that, for KMT-2021-BLG-0119, the red giant stars in the KMTNet CMD are too sparse to precisely 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. Table 4 Model Parameters of KMT-2021-BLG-2294 Parameter Close (C) Resonant (RC) Resonant (RW) Wide (W) c Nground 2 data 8219.429/8254 8243.348/8254 8248.935/8254 8218.934/8254 t0 [ ]¢HJD 9452.558 ± 0.001 9452.558 ± 0.001 9452.558 ± 0.001 9452.558 ± 0.001 u0 0.006 ± 0.001 0.005 ± 0.001 0.005 ± 0.001 0.006 ± 0.001 tE [days] 7.074 ± 0.253 8.067 ± 0.290 8.038 ± 0.297 7.144 ± 0.258 s 0.935 ± 0.009 0.993 ± 0.001 1.003 ± 0.001 1.062 ± 0.010 q (×10−3) 0.567 ± 0.041 0.468 ± 0.018 0.466 ± 0.018 0.559 ± 0.040 α [rad] 4.777 ± 0.006 4.776 ± 0.005 4.777 ± 0.005 4.777 ± 0.005 ρ* 0.003 ± 0.001 0.003 ± 0.001 0.003 ± 0.001 0.003 ± 0.001 Note. ¢ = -HJD HJD 2450000.0. 25 Because 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) 13 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. 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 crossmatching of field stars. For the other events, the RGC can be determined from the KMTNet CMDs. However, for con- sistency, 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. Table 5 CMD Analysis of Three Events Event KMT-2021-BLG-0119 KMT-2021-BLG-0192 KMT-2021-BLG-2294 (V − I, I)cl (1.40, 14.70) (3.05, 16.62) (2.02, 15.68) (V − I, I)0,cl (1.06, 14.36) (1.06, 14.45) (1.06, 14.38) Solution Local A & ¢A STD (s−), APRX (s±, u0 + ) C, W (V − I, I)S (1.11 ± 0.02, 19.34 ± 0.01) (2.57 ± 0.01, 19.85 ± 0.01) (1.84 ± 0.01, 20.72 ± 0.01) (V − I, I)0,S (0.77 ± 0.05, 19.00 ± 0.01) (0.58 ± 0.05, 17.68 ± 0.01) (0.88 ± 0.05, 19.42 ± 0.01) (V − I, I)B (1.77 ± 0.09, 19.81 ± 0.02) (2.98 ± 0.02, 18.61 ± 0.01) (1.58 ± 0.02, 19.18 ± 0.01) θ* (μas) 0.53 ± 0.03 0.80 ± 0.04 0.50 ± 0.03 θE (mas) > 0.38 0.40 ± 0.05 0.15 ± 0.02 Solution Local B & ¢B STD (s+), APRX (s±, u0 − ) RC, RW (V − I, I)S (1.11 ± 0.02, 19.26 ± 0.01) (2.57 ± 0.01, 19.84 ± 0.01) (1.85 ± 0.01, 20.90 ± 0.01) (V − I, I)0,S (0.77 ± 0.05, 18.92 ± 0.01) (0.58 ± 0.05, 17.68 ± 0.01) (0.89 ± 0.05, 19.60 ± 0.01) (V − I, I)B (1.89 ± 0.12, 19.95 ± 0.02) (2.98 ± 0.02, 18.61 ± 0.01) (1.59 ± 0.02, 19.14 ± 0.01) θ* (μas) 0.55 ± 0.03 0.80 ± 0.04 0.47 ± 0.03 θE (mas) > 0.31 0.42 ± 0.05 0.17 ± 0.02 Table 6 Coefficients of the ρ* 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 ρ*,limit 0.002344 0.002512 0.002344 0.002042 ρ*,break 0.000066 0.000054 L L 14 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. 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. the matter density profile of the disk from Robin et al. (2003) and Bennett et al. (2014), 2. the matter density profile of the bulge from Han & Gould (1995) and Dwek et al. (1995), 3. the mean velocity and the velocity dispersion of bulge stars from GAIA proper motion information (Collabora- tion et al. 2018), and 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 4×107 events for each case) and apply available constraints from the microlensing light curve. For all cases, tE are well measured, so we use a simple Gaussian weight. Depending on the particular event, we may also have priors from θE or πE. For KMT-2021- BLG-0192 and KMT-2021-BLG-2294, we measure ρ*, so we apply a Gaussian weight based on θE (see Section 5). In addition, for KMT-2021-BLG-0192, apply the 2D πE con- straint 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 πE. Then, because ρ* was not clearly measured, for each solution, we construct a weight function (W(ρ*)) by fitting the distribution ofΔχ2 as a function of ρ*. For the Local A and ¢A cases, we use a piecewise function ⎧ ⎨ ⎪ ⎩⎪ * * * * * * * *  ( ) ( ) ( ) ( ) ( )( )r r r r r r r r = < + > +W a e constant if 0.6 if 0.0 if , 7c x b ,break ,break ,limit ,limit 2 where * ( )rºx log10 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, ρ* cannot be zero because there is a point during the caustic entrance. However, it is increasingly difficult to probe models with ρ* < ρ*,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 θE (θE(ρ*,break= 6.6× 10−5)∼ 80 mas), so our assumption of a constant weight below ρ*,break has little effect on the Bayesian estimates. For the Local B and ¢B cases, we use ⎧ ⎨⎩ * * * * *  ( ) ( ) ( ) ( )r r r r r = + > - W a e 1.0 if 0.0 if , 8,limit ,limit x b c where * ( )rºx log10 , (a, b, c) are coefficients, and 1.0 is the normalization factor. We present the coefficients for all models, ρ*,limit, and ρ*,break in Table 6. 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 (Mplanet∼ 6MJup) and an early M-type or a K-type dwarf host star (Mhost∼ 0.56 or∼ 0.69Me, for the A and B families of solutions, respectively). The planet orbits the host with a projected separation of ∼2.9 or ∼3.2 au beyond its snow line (∼1.5 or∼1.9 au). The planetary system is located at a distance of∼3–4 kpc from us; i.e., halfway 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 MI= 7.2. Assuming a distance of 3.0 kpc and that it is behind all of the dust (AI∼ 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 (MI∼ 6.0 and DL∼ 4.9 kpc), the observed magnitude would be also well Table 7 Lens Properties of Three Events Event Constraints Case Mhost Mplanet Mplanet DL a⊥ asnow μrel (Me) (MJup) (MNep) (kpc) (au) (au) (mas yr−1) KB210119 tE + ρ* + πE 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 tE + θE 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 tE + θE + πE s−, u0 + - +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−, u0 − - +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+, u0 + - +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+, u0 − - +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 tE + θE 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 RC - +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 RW - +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 15 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. 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 Δθ(N, E)= (64, 1.5) mas. Given that the uncertainties in such measurements are on the order of tens of mas, this measure- ment 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 follow-up 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 (Mplanet∼ 2MNep) and an M dwarf host star (Mhost∼ 0.3Me). Without the parallax constraint, the values Figure 11. The (DL, Mhost) and (a⊥, Mplanet) 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, 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−, u0−) 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 RC (red). 16 The Astronomical Journal, 165:8 (19pp), 2023 January Shin et al. are somewhat larger but consistent at 1σ. The planet is a typical microlensing planet located beyond the snow line. The planetary system is located at DL∼ 5 kpc. For completeness, we note that the baseline object appears to be offset from the microlensing event by Δθ(N, E)∼ (−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 (Mp∼ 1.2MNep) orbiting a late M dwarf host (Mh∼ 0.1Me). The system is located in or near the bulge at DL∼ 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., tE∼ 7–8 days). In this case, the baseline object appears to be offset from the microlensing event by Δθ(N, E)∼ (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 ΔK= (2.2, 2.8, 3.5) mag for a lens mass of Mlens= (0.4, 0.3, 0.2)Me (Bessell & Brett 1988; Baraffe et al. 2015). Given the magnitude of the lens source relative proper motion (∼7.6 mas yr−1), it should be possible to either measure or place strong upper limits on the lens flux at the first light of 30 m 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 pla- nets 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 ∼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 cD = 370 2 for teff= 0.05, and cD = 590 2 for teff= 0.025. By contrast, the algorithm has a default threshold of at least Δχ2> 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 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: ( ) < -m Mlog 2p J , red: ( )- < < -m M2 log 1.5p J , yellow: ( )- < < -m M1.5 log 1p J , green: ( )- < < -m M1 log 0.5p J , cyan: ( )- <