# Systematic KMTNet Planetary Anomaly Search, Paper VII: Complete Sample of $q < 10^{-4}$ Planets from the First Four-Year Survey
**Authors**: Weicheng Zang, Youn Kil Jung, Hongjing Yang, Xiangyu Zhang, Andrzej Udalski, Jennifer C. Yee, Andrew Gould, Shude Mao, Michael D. Albrow, Sun-Ju Chung, Cheongho Han, Kyu-Ha Hwang, Yoon-Hyun Ryu, In-Gu Shin, Yossi Shvartzvald, Sang-Mok Cha, Dong-Jin Kim, Hyoun-Woo Kim, Seung-Lee Kim, Chung-Uk Lee, Dong-Joo Lee, Yongseok Lee, Byeong-Gon Park, Richard W. Pogge, Przemek Mróz, Jan Skowron, Radoslaw Poleski, Michał K. Szymański, Igor Soszyński, Paweł Pietrukowicz, Szymon Kozłowski, Krzysztof Ulaczyk, Krzysztof A. Rybicki, Patryk Iwanek, Marcin Wrona, Mariusz Gromadzki, Hanyue Wang, Jiyuan Zhang, Wei Zhu
## Systematic KMTNet Planetary Anomaly Search, Paper VII: Complete Sample of q < 10 -4 Planets from the First Four-Year Survey
WEICHENG ZANG, 1,2 YOUN KIL JUNG, 3,4 HONGJING YANG, 1 XIANGYU ZHANG, 5 ANDRZEJ UDALSKI, 6 JENNIFER C. YEE, 2 ANDREW GOULD, 5,7 AND SHUDE MAO 1,8
(LEADING AUTHORS)
MICHAEL D. ALBROW, 9 SUN-JU CHUNG, 3,4 CHEONGHO HAN, 10 KYU-HA HWANG, 3 YOON-HYUN RYU, 3 IN-GU SHIN, 2 YOSSI SHVARTZVALD, 11 SANG-MOK CHA, 3,12 DONG-JIN KIM, 3 HYOUN-WOO KIM, 3 SEUNG-LEE KIM, 3,4 CHUNG-UK LEE, 3 DONG-JOO LEE, 3 YONGSEOK LEE, 3,12 BYEONG-GON PARK, 3,4 AND RICHARD W. POGGE 7 (THE KMTNET COLLABORATION)
PRZEMEK MR´ OZ, 6 JAN SKOWRON, 6 RADOSLAW POLESKI, 6 MICHAŁ K. SZYMA´ NSKI, 6 IGOR SOSZY ´ NSKI, 6 PAWEŁ PIETRUKOWICZ, 6 SZYMON KOZŁOWSKI, 6 KRZYSZTOF ULACZYK, 13 KRZYSZTOF A. RYBICKI, 6 PATRYK IWANEK, 6 MARCIN WRONA, 6 AND MARIUSZ GROMADZKI 6
(THE OGLE COLLABORATION)
HANYUE WANG, 2 JIYUAN ZHANG, 1 AND WEI ZHU 1 (THE MAP COLLABORATION)
1 Department of Astronomy, Tsinghua University, Beijing 100084, China
2 Center for Astrophysics | Harvard & Smithsonian, 60 Garden St.,Cambridge, MA 02138, USA
3 Korea Astronomy and Space Science Institute, Daejon 34055, Republic of Korea
4 University of Science and Technology, Korea, (UST), 217 Gajeong-ro Yuseong-gu, Daejeon 34113, Republic of Korea
Max-Planck-Institute for Astronomy, K¨ onigstuhl 17, 69117 Heidelberg, Germany
6 Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
7 Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA
8 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
9 University of Canterbury, Department of Physics and Astronomy, Private Bag 4800, Christchurch 8020, New Zealand
10 Department of Physics, Chungbuk National University, Cheongju 28644, Republic of Korea
11 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel
12 School of Space Research, Kyung Hee University, Yongin, Kyeonggi 17104, Republic of Korea
13 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK
## ABSTRACT
We present the analysis of seven microlensing planetary events with planet/host mass ratios q < 10 -4 : KMT-2017-BLG-1194, KMT-2017-BLG-0428, KMT-2019-BLG-1806, KMT-2017-BLG-1003, KMT-2019BLG-1367, OGLE-2017-BLG-1806, and KMT-2016-BLG-1105. They were identified by applying the Korea Microlensing Telescope Network (KMTNet) AnomalyFinder algorithm to 2016-2019 KMTNet events. A Bayesian analysis indicates that all the lens systems consist of a cold super-Earth orbiting an M or K dwarf. Together with 17 previously published and three that will be published elsewhere, AnomalyFinder has found a total of 27 planets that have solutions with q < 10 -4 from 2016-2019 KMTNet events, which lays the foundation for the first statistical analysis of the planetary mass-ratio function based on KMTNet data. By reviewing the 27 planets, we find that the missing planetary caustics problem in the KMTNet planetary sample has been solved by AnomalyFinder. We also find a desert of high-magnification planetary signals ( A 65 ), and a follow-up project for KMTNet high-magnification events could detect at least two more q < 10 -4 planets per year and form an independent statistical sample.
1. INTRODUCTION
Among current exoplanet detection methods, a unique capability of the gravitational microlensing technique (Mao & Paczynski 1991; Gould & Loeb 1992) is to detect lowmass ( M planet 20 M ⊕ ) cold planets beyond the snow line (Hayashi 1981; Min et al. 2011), including Neptune-mass cold planets, which are common (Uranus and Neptune) in
our Solar System and cold terrestrial planets, which are absent in our Solar System. Because the typical host stars of the microlensing planetary systems are M and K dwarfs, detections of q < 10 -4 planets (where q is the planet/host mass ratio) can reveal the abundance of low-mass cold planets and answer how common the outer solar system is.
However, since the first microlensing planet, which was detected in 2003 (Bond et al. 2004), the first 13 years of microlensing planetary detections only discovered six q < 10 -4 planets 1 and none of them had mass ratios below 4 . 4 × 10 -5 . The paucity of detected q < 10 -4 planets led to important statistical implications for cold planets. Suzuki et al. (2016) analyzed 1474 microlensing events discovered by the Microlensing Observations in Astrophysics (MOA) survey (Sako et al. 2008) and formed a homogeneously selected sample including 22 planets. They found that the mass-ratio function of microlensing planets increases as q decreases until a break at q ∼ 1 . 7 × 10 -4 , below which the planetary occurrence rate likely drops. This break suggests that the Neptune-mass planets are likely to be the most common of cold planets. However, the Suzuki et al. (2016) sample only contains two q < 10 -4 and thus may be affected by small number statistics. To examine the existence of the break, a larger q < 10 -4 sample is needed.
After its commissioning season in 2015, the new-generation microlensing survey, the Korea Microlensing Telescope Network (KMTNet, Kim et al. 2016), has been conducting nearcontinuous, wide-area, high-cadence surveys for ∼ 96 deg 2 . The fields with cadences of Γ ≥ 2 hr -1 are the KMTNet prime fields ( ∼ 12 deg 2 ) and the other fields are the KMTNet sub-prime fields ( ∼ 84 deg 2 ). Since 2016, the detections of q < 10 -4 planets have been greatly increased in two ways, and the KMTNet data played a major or decisive role in all detections. First, more than ten q < 10 -4 planets have been detected from by-eye searches, including three with q < 2 × 10 -5 (Gould et al. 2020; Yee et al. 2021; Zang et al. 2021a). Second, Zang et al. (2021b, 2022a) developed the KMTNet AnomalyFinder algorithm to systematically search for planetary signals. This algorithm has been applied to the 2018 and 2019 KMTNet prime fields ( Γ ≥ 2 hr -1 ) and uncovered five new q < 10 -4 planets (Zang et al. 2021b; Hwang et al. 2022; Gould et al. 2022). Moreover, the systematic search opens a window for a homogeneous large-scale KMTNet planetary sample. According to the experience from 2018 and 2019 KMTNet prime fields, we expect to detect 20 planets with q < 10 -4 from 2016-2019 seasons.
1 They are OGLE-2005-BLG-169Lb (Gould et al. 2006), OGLE-2005BLG-390Lb (Beaulieu et al. 2006), OGLE-2007-BLG-368Lb (Sumi et al. 2010), MOA-2009-BLG-266Lb (Muraki et al. 2011), OGLE-2013-BLG0341Lb (Gould et al. 2014b), OGLE-2015-BLG-1670 (Ranc et al. 2019).
This will be an order of magnitude larger than the Suzuki et al. (2016) sample at q < 10 -4 .
To build the first KMTNet q < 10 -4 statistical sample, we applied the KMTNet AnomalyFinder algorithm to the 2016-2019 KMTNet microlensing events. In this paper, we introduce seven new q < 10 -4 events from this search. They are KMT-2017-BLG-1194, KMT-2017BLG-0428, KMT-2019-BLG-1806/OGLE-2019-BLG-1250, KMT-2017-BLG-1003, KMT-2019-BLG-1367, OGLE2017-BLG-1806/KMT-2017-BLG-1021, and KMT-2016BLG-1105. Together with 17 already published and three that will be published elsewhere, the KMTNet AnomalyFinder algorithm found 27 events that can be fit by q < 10 -4 models from 2016-2019 KMTNet data. However, whether a planet can be used for statistical studies requires further investigations, which is beyond the scope of this paper.
The paper is structured as follows. In Section 2, we briefly introduce the KMTNet AnomalyFinder algorithm and the procedure to form the q < 10 -4 sample. In Sections 3, 4 and 5, we present the observations and the analysis of seven q < 10 -4 events. Finally, we discuss the implications from the 2016-2019 KMTNet q < 10 -4 planetary sample in Section 6.
## 2. THE BASIC OF ANOMALYFINDER AND THE PROCEDURE
Section 2 of Zang et al. (2021b) and Section 2 of Zang et al. (2022a) together introduced the KMTNet AnomalyFinder algorithm. The AnomalyFinder uses a Gould (1996) 2dimensional grid of ( t 0 , t eff ) to search for and fit anomalies from the residuals to a point-source point-lens (PSPL, Paczy´ nski 1986) model. Here t 0 is the time of maximum magnification, and t eff is the effective timescale. For our search, the shortest t eff is 0.05 days and the longest t eff is 6.65 days. The parameters that evaluate the significance of a candidate anomaly are ∆ χ 2 0 and ∆ χ 2 flat . See Equation (4) of Zang et al. (2021b) for their definitions. The criteria of ∆ χ 2 0 and ∆ χ 2 flat are the same as the criteria used in Zang et al. (2022a); Gould et al. (2022); Jung et al. (2022), with ∆ χ 2 0 > 200 , or ∆ χ 2 0 > 120 and ∆ χ 2 flat > 60 for the KMTNet prime-field events and ∆ χ 2 0 > 100 , or ∆ χ 2 0 > 60 and ∆ χ 2 flat > 30 for the KMTNet sub-prime-field events. Future statistical studies should use the same criteria. In addition, an anomaly is required to contain at least three successive points ≥ 2 σ away from a PSPL model.
As a result, we found 464 and 608 candidate anomalies from 2016-2019 KMTNet prime-field and sub-prime-field events, respectively. We checked whether the data from other surveys are consistent with the KMTNet-based anomalies and cross-checked with C. Han's modeling. We fitted all the q < 10 -3 candidates with online data and found 13 new
candidates with q < 2 × 10 -4 . Then, we conducted tenderloving care (TLC) re-reductions and re-fitted the 13 events. Of these, eight events unambiguously have q < 10 -4 , three events, KMT-2016-BLG-1307, KMT-2017-BLG-0849, and KMT-2017-BLG-1057, have 10 -4 < q < 2 × 10 -4 , and two events, KMT-2016-BLG-0625 (Shin et al. in prep) and OGLE-2017-BLG-0448/KMT-2017-BLG-0090 (Zhai et al. in prep), have ambiguous mass ratios at 10 -5 q 10 -3 and will be published elsewhere.
Among the eight unambiguous q < 10 -4 events, one event, OGLE-2016-BLG-0007/MOA-2016-BLG-088/KMT2016-BLG-1991, will be published elsewhere because it has the lowestq of this sample. We analyze and publish the remaining seven events in this paper. We note that the planetary signals of the seven events are not strong, although they are confirmed by at least two data sets. We thus further check whether the light curves have other similar anomalies, to exclude the possibility of unknown systematic errors. We applied the AnomalyFinder algorithm to the re-reduction data. For all of the seven events, besides the known planetary signals no anomaly with ∆ χ 2 0 > 20 was detected. Therefore, the light curves of the seven events are stable and planetary signals are reliable.
## 3. OBSERVATIONS AND DATA REDUCTIONS
Table 1 lists the basic observational information for the seven events, including event names, the first discovery date, the coordinates in the equatorial and galactic systems, and the nominal cadences ( Γ ). The seven planetary events were all identified by the KMTNet post-season EventFinder algorithm (Kim et al. 2018a). Of them, KMT2019-BLG-1806/OGLE-2019-BLG-1250 and OGLE-2017BLG-1806/KMT-2017-BLG-1021 were discovered by the KMTNet alert-finder system (Kim et al. 2018b) and the Early Warning System (Udalski et al. 1994; Udalski 2003) of the Optical Gravitational Lensing Experiment (OGLE, Udalski et al. 2015), respectively, during their observational seasons. Hereafter, we designate KMT-2019-BLG-1806/OGLE2019-BLG-1250 and OGLE-2017-BLG-1806/KMT-2017BLG-1021 by their first-discovery name, KMT-2019-BLG1806 and OGLE-2017-BLG-1806. During the 2019 observational season, the KMTNet alert-finder system also discovered KMT-2019-BLG-1367. In addition, OGLE observed the locations of KMT-2019-BLG-1367 and KMT-2016-BLG1105 but did not alert them. We also include the OGLE data for these two events into the light-curve analysis, for which the OGLE data confirm the planetary signals found by the KMTNet. MOA did not issue alerts for any of the seven events, and there were no follow-up data to the best of our knowledge.
KMTNet conducted observations from three identical 1.6 m telescopes equipped with 4 deg 2 cameras in Chile
(KMTC), South Africa (KMTS), and Australia (KMTA). OGLE took data using an 1.3m telescope with 1.4 deg 2 field of view in Chile. For both surveys, most of the images were taken in the I band, and a fraction of V -band images were acquired for source color measurements. Each KMTNet Vband data point was taken one minute before or after one KMTNet I-band data point of the same field.
The KMTNet and OGLE data used in the light-curve analysis were reduced using the custom photometry pipelines based on the difference imaging technique (Tomaney & Crotts 1996; Alard & Lupton 1998): pySIS (Albrow et al. 2009, Yang et al. in prep) for the KMTNet data, and Wozniak (2000) for the OGLE data. For each event, the KMTC data were additionally reduced using the pyDIA photometry pipeline (Albrow 2017) to measure the source color. Except for OGLE-2017-BLG-1806 and KMT-2016-BLG-1105 whose sources are not located in any OGLE star catalog, the I -band magnitudes of the other five events reported in this paper have been calibrated to the standard I -band magnitude using the OGLE-III star catalog (Szyma´ nski et al. 2011).
## 4. LIGHT-CURVE ANALYSIS
## 4.1. Preamble
Because all seven events contain short-lived deviations from a PSPL model, we first introduce the common methods for the light-curve analysis. The PSPL model is described by three parameters, t 0 , u 0 , and t E , which respectively represent the time of lens-source closest approach, the closest lens-source projected separation normalized to the angular Einstein radius θ E , and the Einstein timescale,
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where κ ≡ 4 G c 2 au 8 . 144 mas M , M L is the lens mass, and ( π rel , µ rel ) are the lens-source relative (parallax, proper motion). In addition, for each data set i , we introduce two linear parameters, ( f S ,i , f B ,i ), to fit the flux of the source and any blend flux, respectively.
We search for binary-lens single-source (2L1S) models for each event. A 2L1S model requires four parameters in addition to the PSPL parameters, ( s, q, α, ρ ) , which respectively denote the planet-host projected separation in units of θ E , the planet/host mass ratio, the angle between the source trajectory and the binary axis, and the angular source radius θ ∗ scaled to θ E , i.e., ρ = θ ∗ /θ E .
Although the final results need detailed numerical analysis, some of the 2L1S parameters can be estimated by heuristic analysis. A PSPL fit excluding the data points around the anomaly can yield the three PSPL parameters, t 0 , u 0 , and t E . If an anomaly occurred at t anom , the corresponding lens-
Table 1. Event Names, Alerts, Locations, and Cadences for the six planetary events
| Event Name | Alert Date | RA J2000 | Decl . J2000 | | b | Γ(hr - 1 ) |
|--------------------|--------------|-------------|----------------|------------------|--------|--------------|
| KMT-2017-BLG-1194 | Post Season | 18:17:17.31 | - 25:19:26.18 | +6.63 | - 4.34 | 0.4 |
| KMT-2017-BLG-0428 | Post Season | 18:05:32.46 | - 28:29:25.01 | +2.59 | - 3.55 | 4 |
| KMT-2019-BLG-1806 | 26 Jul 2019 | 18:02:09.01 | - 29:24:53.60 | +1.41 | - 3.35 | 1 |
| OGLE-2019-BLG-1250 | | | | | | 0.3 |
| KMT-2017-BLG-1003 | Post Season | 17:41:38.76 | - 24:22:26.18 | +3.42 | +3.15 | 1 |
| KMT-2019-BLG-1367 | 27 Jun 2019 | 18:09:53.12 | - 29:45:43.96 | +1.93 | - 4.99 | 0.4 |
| OGLE-2017-BLG-1806 | 14 Oct 2017 | 17:46:29.58 | - 24:16:20.17 | +4.09 | +2.26 | 0.3 |
| KMT-2017-BLG-1021 | | | | | | 1 |
| KMT-2016-BLG-1105 | Post Season | 17:45:47.34 | - 26:15:58.93 | +2.30 | +1.16 | 1 |
source offset, u anom , and α can be estimated by
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Because the planetary caustics are located at the position of | s -s -1 | ∼ u anom , we obtain
<!-- formula-not-decoded -->
where s = s + and s = s -correspond to the major-image (quadrilateral) and the minor-image (triangular) planetary caustics, respectively. For two degenerate solutions with similar q but different s , Ryu et al. (2022) suggested that the geometric mean of two solutions satisfies
<!-- formula-not-decoded -->
In addition, Zhang et al. (2022) suggested a slightly different formalism, and Zhang & Gaudi (2022) provided a theoretical treatment of it. For a dip-type planetary signal, Hwang et al. (2022) pointed out that the mass ratio can be estimated by
<!-- formula-not-decoded -->
where ∆ t dip is the duration of the dip, and the accuracy of Equation (5) should be at a factor of ∼ 2 level.
To find all the possible 2L1S models, we conduct twophase grid searches for the parameters, ( log s , log q , α , ρ ). In the first phase, we conduct a sparse grid, which consists of 21 values equally spaced between -1 . 0 ≤ log s ≤ 1 . 0 , 20 values equally spaced between 0 ◦ ≤ α < 360 ◦ , 61 values equally spaced between -6 . 0 ≤ log q ≤ 0 . 0 and five values equally spaced between -3 . 5 ≤ log ρ ≤ -1 . 5 . We use a code based on the advanced contour integration code (Bozza 2010; Bozza et al. 2018), VBBinaryLensing 2 to compute the 2L1S magnification. For each grid point, we search for the minimum χ 2 by Markov chain Monte Carlo (MCMC) χ 2 minimization using the emcee ensemble sampler (Foreman-Mackey et al. 2013), with fixed ( log q , log s ) and free ( t 0 , u 0 , t E , ρ, α ). In the second phase, we conduct a denser ( log s , log q , α , ρ ) grid search around each local minimum (e.g., Zang et al. 2022b). Finally, we refine the best-fit models by MCMC with all parameters free.
For degenerate solutions, Yang et al. (2022) suggested that the phase-space factors can be used to weight the probability of each solution. We follow the procedures of Yang et al. (2022) and first calculate the covariance matrix, C , of ( log s, log q, α ) from the MCMC chain. Then, the phasespace factor is
<!-- formula-not-decoded -->
Because whether a planet and its individual solutions can be used for statistical studies requires further investigations, we provide the phase-space factors for the event with multiple solutions but do not use them to weight or reject solutions.
We also investigate whether the inclusion of two highorder effects can improve the fit. The first is the microlensing parallax effect (Gould 1992, 2000, 2004), which is due to the Earth's orbital acceleration around the Sun. We fit it by two parameters, π E , N and π E , E , which are the north and east components of the microlensing parallax vector π E in equatorial coordinates,
<!-- formula-not-decoded -->
2 http://www.fisica.unisa.it/GravitationAstrophysics/VBBinaryLensing. htm
Table 2. 2L1S Parameters for KMT-2017-BLG-1194
| Parameter | A | B |
|---------------|---------------------|---------------------|
| χ 2 /dof | 928.0/928 | 950.6/928 |
| t 0 ( HJD ′ ) | 7942 . 66 ± 0 . 13 | 7942 . 59 ± 0 . 13 |
| u 0 | 0 . 256 ± 0 . 018 | 0 . 246 ± 0 . 011 |
| t E (days) | 47 . 0 ± 2 . 5 | 47 . 9 ± 1 . 7 |
| ρ (10 - 3 ) | < 2 . 6 | < 1 . 4 |
| α (rad) | 2 . 505 ± 0 . 013 | 2 . 515 ± 0 . 011 |
| s | 0 . 8063 ± 0 . 0103 | 0 . 8055 ± 0 . 0065 |
| log q | - 4 . 582 ± 0 . 058 | - 4 . 585 ± 0 . 074 |
| I S , OGLE | 20 . 28 ± 0 . 08 | 20 . 34 ± 0 . 06 |
NOTE-The upper limit on ρ is 3 σ .
We also fit the u 0 > 0 and u 0 < 0 solutions to consider the 'ecliptic degeneracy' (Jiang et al. 2004; Poindexter et al. 2005). For four cases in this paper, the parallax contours take the form of elongated ellipses, so we report the constraints on the minor axes of the error ellipse, ( π E , ‖ ), which is approximately parallel with the direction of the Earth's acceleration. For the major axes of the parallax contours, π E , ⊥ ∼ π E , N , we only report it when the constraint is useful.
The second effect is the lens orbital motion effect (Batista et al. 2011; Skowron et al. 2011), and we fit it by the parameter γ = ( ds/dt s , dα dt ) , where ds/dt and dα/dt represent the instantaneous changes in the separation and orientation of the two components defined at t 0 , respectively. To exclude unbound systems, we restrict the MCMC trials to β < 1 . 0 . Here β is the absolute value of the ratio of projected kinetic to potential energy (An et al. 2002; Dong et al. 2009),
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and where π S is the source parallax estimated by the mean distance to red clump stars in the direction of each event (Nataf et al. 2013).
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Figure 1 shows the observed data together with the best-fit PSPL and 2L1S models for KMT-2017-BLG-1194. There is a dip centered on HJD ′ ∼ 7958 . 9 (HJD ′ = HJD -2450000) , i.e., t anom ∼ 7958 . 9 , with a duration of ∆ t dip ∼ 1 . 05 days. The dip and the ridge around the dip are covered by three KMTNet sites, so the anomaly is secure. A PSPL fit yields ( t 0 , u 0 , t E ) = (7942.7, 0.26, 46), and using the heuristic formalism of Section 4.1, we obtain
<!-- formula-not-decoded -->
Figure 1. The observed data and the 2L1S (the black and orange solid lines) and 1L1S models (the grey dashed line) for KMT-2017BLG-1194. The data taken from different data sets are shown with different colors. The bottom panels show a close-up of the dip-type planetary signal and the residuals to the 2L1S models.
<details>
<summary>Image 1 Details</summary>
### Visual Description
## Time-Series Light Curve Analysis: KMT-2017-BLG-1194
### Overview
The image presents a multi-panel technical analysis of an astronomical light curve, likely from a microlensing event (KMT-2017-BLG-1194). It includes three primary data series (KMTA31, KMTC31, KMTS31), a fitted curve, residuals, and model comparisons (2L1S A/B vs. 1L1S). The data spans a time range of ~7920–7980 HJD-2450000, with magnitude measurements in I-band (I-Mag).
---
### Components/Axes
- **Main Chart (Top Panel)**:
- **X-axis**: HJD-2450000 (Heliocentric Julian Date, offset by 2450000).
- **Y-axis**: I-Mag (I-band magnitude, inverted scale: lower values = brighter).
- **Data Series**:
- **KMTA31** (green): Scatter plot with error bars.
- **KMTC31** (red): Scatter plot with error bars.
- **KMTS31** (blue): Scatter plot with error bars.
- **Fitted Curve**: Black solid line (model fit to data).
- **Legend**: Located on the right, with color-coded labels for KMTA31, KMTC31, KMTS31.
- **Residuals (Second Panel)**:
- **X-axis**: Same as main chart (HJD-2450000).
- **Y-axis**: Residuals (I-Mag - Model Fit), ranging from -0.25 to +0.25.
- **Data Series**: Same as main chart (green, red, blue).
- **Model Comparison (Third Panel)**:
- **X-axis**: HJD-2450000 (same as main chart).
- **Y-axis**: I-Mag (same as main chart).
- **Data Series**:
- **2L1S A** (black solid line).
- **2L1S B** (orange solid line).
- **1L1S** (dashed gray line).
- **Legend**: Located on the right, with color-coded labels for 2L1S A/B and 1L1S.
- **Residuals for Models (Bottom Panels)**:
- **Panel A**: Residuals for 2L1S A (blue) and 2L1S B (red).
- **Panel B**: Residuals for 1L1S (gray).
---
### Detailed Analysis
#### Main Chart Trends
- **Peak at HJD-7940**: All three data series (KMTA31, KMTC31, KMTS31) show a sharp brightness increase (I-Mag ~18.2–18.4) centered at HJD-7940, followed by a gradual decline.
- **Symmetry**: The light curve is approximately symmetric around HJD-7940, with a rise time of ~20 days and a fall time of ~30 days.
- **Fitted Curve**: The black solid line closely follows the data points, suggesting a good model fit. The residuals (second panel) are small and randomly distributed, indicating minimal systematic errors.
#### Model Comparison
- **2L1S A/B**:
- **2L1S A** (black) and **2L1S B** (orange) show a double-peaked structure, with a secondary peak at HJD-7958.5. This suggests a binary lens system or a secondary source.
- **Residuals**: Panel A shows residuals for 2L1S A/B, with most points within ±0.1 I-Mag, indicating a reasonable fit.
- **1L1S** (dashed gray):
- A single-peaked model with a peak at HJD-7940. Residuals (Panel B) are larger than for 2L1S models, suggesting a poorer fit.
#### Statistical Metric
- **χ² Difference**: The text explicitly states **χ²₁LIS - χ²₂LIS = 135.6**, indicating that the 2L1S model (A/B) provides a significantly better fit than the 1L1S model.
---
### Key Observations
1. **Peak Timing**: The brightness peak occurs at HJD-7940, consistent across all data series.
2. **Model Fit**: The 2L1S model (A/B) outperforms the 1L1S model by a large margin (χ² difference = 135.6).
3. **Residuals**: All residuals are small and randomly distributed, suggesting no significant systematic errors in the data or models.
4. **Double-Peaked Structure**: The 2L1S models reveal a secondary peak at HJD-7958.5, which is not visible in the 1L1S model.
---
### Interpretation
- **Astronomical Significance**: The light curve likely corresponds to a microlensing event, where a foreground object (e.g., a star or planet) temporarily magnifies the light of a background source. The double-peaked structure (2L1S) suggests a binary lens system or a secondary source contributing to the magnification.
- **Model Validation**: The 2L1S model’s superior fit (χ² difference = 135.6) supports the hypothesis of a binary lens or complex source configuration. The 1L1S model, while simpler, fails to capture the secondary peak, indicating it is insufficient for this event.
- **Data Consistency**: The three data series (KMTA31, KMTC31, KMTS31) show nearly identical trends, confirming the robustness of the observations across different instruments or observations.
---
### Spatial Grounding
- **Legend Position**: Right-aligned, with clear color-coded labels for all data series and models.
- **Axis Labels**: X-axis (HJD-2450000) and Y-axis (I-Mag) are explicitly labeled, with numerical ranges provided.
- **Residuals**: Plotted directly below the main chart, with separate panels for model-specific residuals.
---
### Content Details
- **HJD Values**:
- Peak: 7940.0
- Secondary Peak (2L1S): 7958.5
- Range: 7920.0–7980.0
- **I-Mag Values**:
- Minimum (brightest): ~18.2–18.4
- Maximum (dimmed): ~19.4
- **Residuals**: All residuals are within ±0.25 I-Mag, with most points clustered near zero.
---
### Final Notes
The image provides a comprehensive analysis of a microlensing event, with strong evidence for a binary lens system (2L1S) based on the double-peaked structure and statistical metrics. The residuals confirm the reliability of the data and models, while the χ² difference highlights the importance of using the more complex 2L1S model for accurate interpretation.
</details>
The grid search yields one solution. Its parameters are presented in Table 2 and are in good agreement with the heuristic estimates. The top left panel of Figure 2 displays the caustic structure and the source trajectory, for which the two minor-image planetary caustics are located on both sides of the source trajectory. We label the solution as the solution 'A'. To further investigate the parameter space and check whether the event has the inner/outer solutions (Gaudi & Gould 1997), for which the source passes inside (the 'Inner' solution) the two planetary caustics (closer to the central caustic) or outside (the 'Outer' solution), we follow the procedures of Hwang et al. (2018a). First, we conduct a 'hotter' MCMC with the error bar inflated by a factor of √ 3 . 0 . Second, we make a scatter plot of log q versus ∆ ξ from the 'hotter' MCMC chain. Here ∆ ξ represents the offset between the source and the planetary caustic as the source crosses the binary axis,
<!-- formula-not-decoded -->
The resulting scatter plot is shown in Figure 3, from which we find another local minimum at ∆ ξ ∼ 0 . 02 . We label this solution as the 'B' solution. As shown in the top right panel
<details>
<summary>Image 2 Details</summary>
### Visual Description
## Scatter Plots: KMT-BLG Event Data Analysis
### Overview
The image contains six scatter plots arranged in a 3x2 grid, each representing data from different KMT-BLG events (e.g., KMT-2017-BLG-1194, KMT-2017-BLG-0428, etc.). Each plot features two axes (x_s and y_s) with directional arrows, red and green geometric shapes (triangles and circles), and blue text labels ("Inner" and "Outer"). The plots are labeled with event identifiers and include directional indicators.
---
### Components/Axes
- **Axes**:
- **x_s**: Horizontal axis, labeled with numerical values (e.g., -0.46 to -0.36 in the top row).
- **y_s**: Vertical axis, labeled with numerical values (e.g., -0.02 to 0.02).
- **Legends**:
- **Red**: Labeled "Inner" (triangles).
- **Green**: Labeled "Outer" (circles).
- **Text Labels**:
- "Inner" and "Outer" in blue, positioned near the plots.
- Event identifiers (e.g., "KMT-2017-BLG-1194") at the top of each plot.
- **Arrows**: Black arrows indicating directional trends (e.g., from lower-left to upper-right).
---
### Detailed Analysis
#### KMT-2017-BLG-1194 (Top-Left)
- **Data Points**: Red triangles (Inner) clustered around x_s ≈ -0.45 to -0.40, y_s ≈ -0.02 to 0.02.
- **Trend**: Arrow points from lower-left to upper-right, suggesting a directional flow.
- **Legend**: Red = Inner, Green = Outer (no green data points here).
#### KMT-2017-BLG-0428 (Top-Right)
- **Data Points**:
- Red triangles (Inner) in upper-right (x_s ≈ -0.20 to -0.18, y_s ≈ 0.02).
- Green circle (Outer) in lower-left (x_s ≈ -0.26, y_s ≈ -0.02).
- **Trend**: Arrow points from lower-left to upper-right.
- **Legend**: Red = Inner, Green = Outer.
#### KMT-2019-BLG-1806 (Middle-Left)
- **Data Points**: Red triangles (Inner) in lower-left (x_s ≈ -0.15 to -0.10, y_s ≈ -0.04 to 0.02).
- **Trend**: Arrow points from lower-left to upper-right.
- **Legend**: Red = Inner, Green = Outer (no green data points here).
#### KMT-2017-BLG-1003 (Middle-Right)
- **Data Points**:
- Red triangles (Inner) in upper-right (x_s ≈ -0.20 to -0.18, y_s ≈ 0.02).
- Green circle (Outer) in lower-left (x_s ≈ -0.24, y_s ≈ -0.02).
- **Trend**: Arrow points from lower-left to upper-right.
- **Legend**: Red = Inner, Green = Outer.
#### KMT-2019-BLG-1367 (Bottom-Left)
- **Data Points**: Red triangles (Inner) in upper-right (x_s ≈ -0.14 to -0.10, y_s ≈ -0.02 to 0.02).
- **Trend**: Arrow points from lower-left to upper-right.
- **Legend**: Red = Inner, Green = Outer (no green data points here).
#### KMT-2019-BLG-1367 (Bottom-Right)
- **Data Points**: Red triangles (Inner) in upper-right (x_s ≈ -0.08 to 0.00, y_s ≈ 0.02).
- **Trend**: Arrow points from lower-left to upper-right.
- **Legend**: Red = Inner, Green = Outer (no green data points here).
---
### Key Observations
1. **Directional Arrows**: All plots show arrows pointing from lower-left to upper-right, indicating a consistent directional trend across events.
2. **Data Distribution**:
- "Inner" (red) data points are consistently in the lower-left or upper-right regions.
- "Outer" (green) data points appear only in specific plots (e.g., KMT-2017-BLG-0428, KMT-2017-BLG-1003).
3. **Anomalies**:
- A green circle (Outer) in KMT-2017-BLG-1003 is positioned near the center, distinct from other data points.
- Some plots (e.g., KMT-2019-BLG-1806) lack green data points, suggesting variability in Outer region representation.
---
### Interpretation
The plots likely represent spatial or temporal data from KMT-BLG events, with "Inner" and "Outer" regions differentiated by color. The directional arrows suggest a systematic flow or progression (e.g., movement, expansion, or interaction). The presence of green circles in specific plots may indicate unique events or outliers in the Outer region. The absence of green in some plots could reflect event-specific conditions or data limitations. The consistent directional trend across all plots implies a shared underlying mechanism or process influencing the data.
</details>
Xs
Figure 2. Geometries of the five 'dip' planetary events. In each panel, the red lines represent the caustic, the black solid line represents the source trajectory, and the line with an arrow indicates the direction of the source motion. For the outer solution of KMT2017-BLG-1003, ρ is constrained at the > 3 σ level, so the radius of the green dot represents the source radius. For other solutions, ρ only has weak constraints with < 3 σ , so their source radii are not shown.
of Figure 2, the 'B' solution corresponds to the 'Inner' solution. Its parameters from MCMC are given in Table 2 and it is disfavored by ∆ χ 2 = 22 . 6 compared to the 'A' solution. In Figure 1, the 'B' solution cannot fit the anomaly well and all three KMTNet data sets contribute to the ∆ χ 2 . The ratio of the phase-space factors is p A : p B = 1 : 0 . 54 , which also prefers the 'A' solution. Thus, we exclude the 'B' solution. In addition, the models, which have the geometry of the 'Outer' solution, do not form a local minimum and are disfavored by ∆ χ 2 > 60 compared to the 'A' solution.
For the 'A' solution a point-source model is consistent within 1 σ and the 3 σ upper limit is ρ < 0 . 0026 . The inclusion of higher-order effects yields a constraint on π E , ‖ , and with the other 2L1S parameters being almost unchanged. We obtain π E , ‖ = -0 . 18 ± 0 . 35 and adopt the constraints on π E and ρ in the Bayesian analysis of Section 5. This is a new
Figure 3. Scatter plot of log q vs. ∆ ξ for KMT-2017-BLG-1194, where ∆ ξ = u 0 csc( α ) -( s -s -1 ) represents the offset between the source and the center of the planetary caustic at the moment that the source crosses the binary axis. The distribution is derived by inflating the error bars by a factor of √ 3 and then multiplying the resulting χ 2 by 3 for the plot. Red, yellow, magenta, green, blue and black colors represent ∆ χ 2 < 2 × (1 , 4 , 9 , 16 , 25 , ∞ ) . 'A' and 'B' represent two local minima and the corresponding parameters are given in Table 2.
<details>
<summary>Image 3 Details</summary>
### Visual Description
## Scatter Plot: KMT-2017-BLG-1194
### Overview
The image is a scatter plot visualizing the relationship between two variables, Δξ (delta xi) and log(χ), for a dataset labeled "KMT-2017-BLG-1194." The plot includes two distinct regions labeled **A** (left) and **B** (right), with a color gradient indicating data point density. The axes are logarithmic, and the distribution suggests clustering in specific parameter spaces.
---
### Components/Axes
- **X-axis (Δξ)**: Ranges from **-0.02** to **0.03** (linear scale).
- **Y-axis (log(χ))**: Ranges from **-5.0** to **-4.0** (logarithmic scale).
- **Legend**: No explicit legend is present, but the color gradient (red to blue) represents data point density, with red indicating the highest density and blue the lowest.
- **Labels**:
- Region **A** (left cluster, marked in red).
- Region **B** (right cluster, marked in red).
---
### Detailed Analysis
- **Region A**:
- Dominates the left side of the plot.
- Contains a high-density core (red) at Δξ ≈ 0.00 and log(χ) ≈ -4.4.
- Density decreases outward, transitioning to blue at the edges.
- Approximately **80% of data points** are concentrated in this region.
- **Region B**:
- Located on the right side, separated from A by a gap.
- Lower overall density compared to A.
- Contains a smaller high-density sub-region (red) at Δξ ≈ 0.02 and log(χ) ≈ -4.6.
- Approximately **20% of data points** are in this region.
- **Color Gradient**:
- Red → Yellow → Green → Blue indicates decreasing density.
- No explicit colorbar is present, but the gradient is visually consistent.
---
### Key Observations
1. **Bimodal Distribution**: Two distinct clusters (A and B) suggest two populations or phases in the dataset.
2. **Density Gradient**: The red-to-blue gradient highlights a central peak in Region A, with a secondary peak in Region B.
3. **Separation**: The gap between A and B implies a potential dynamical or physical distinction between the two groups.
4. **Outliers**: A few isolated points exist outside the main clusters, particularly near Δξ = -0.02 and log(χ) = -5.0.
---
### Interpretation
The plot likely represents a parameter space for a binary system or stellar population, where:
- **Region A** corresponds to a dominant population (e.g., primary stars or close binaries).
- **Region B** may represent a secondary population (e.g., distant companions or unbound objects).
- The separation between A and B could indicate differences in orbital separation, mass ratios, or evolutionary stages.
- The density gradient suggests that the most probable configurations (red) are tightly clustered around specific parameter values, while less probable configurations (blue) are more dispersed.
The absence of a legend for the color gradient limits quantitative interpretation of density values, but the visual trend confirms a clear bimodal structure. Further analysis (e.g., statistical tests or dynamical modeling) would be needed to confirm the physical significance of the separation.
</details>
microlensing planet with q ∼ 2 . 62 × 10 -5 ; i.e., about nine times the Earth/Sun mass ratio.
## 4.2.2. KMT-2017-BLG-0428
Table 3. 2L1S Parameters for KMT-2017-BLG-0428
| Parameter | Inner | Outer |
|---------------|----------------------|----------------------|
| χ 2 /dof | 9952.0/9952 | 9952.1/9952 |
| t 0 ( HJD ′ ) | 7943 . 976 ± 0 . 030 | 7943 . 978 ± 0 . 031 |
| u 0 | 0 . 205 ± 0 . 009 | 0 . 205 ± 0 . 009 |
| t E (days) | 44 . 4 ± 1 . 5 | 44 . 3 ± 1 . 5 |
| ρ (10 - 3 ) | < 6 . 4 | < 6 . 1 |
| α (rad) | 1 . 890 ± 0 . 005 | 1 . 889 ± 0 . 005 |
| s | 0 . 8819 ± 0 . 0044 | 0 . 9146 ± 0 . 0050 |
| log q | - 4 . 295 ± 0 . 072 | - 4 . 302 ± 0 . 075 |
| I S , OGLE | 20 . 43 ± 0 . 05 | 20 . 43 ± 0 . 05 |
Figure 4 shows a ∆ I ∼ 0 . 12 mag dip at t anom ∼ 7947 . 00 , with a duration of ∆ t dip ∼ 0 . 74 days. The dip is defined by the KMTA and KMTC data, and the subtle ridges are sup-
Figure 4. The observed data and models for KMT-2017-BLG-0428. The symbols are similar to those in Figure 1. In the top panel, the black arrow indicates the position of the planetary signal.
<details>
<summary>Image 4 Details</summary>
### Visual Description
## Line Graph with Residuals and Model Comparisons: KMT-2017-BLG-0428
### Overview
The image presents a multi-panel chart analyzing the light curve of the astronomical object KMT-2017-BLG-0428. It includes three panels:
1. **Top Panel**: Observed data (blue, green, red) with a model fit (black line).
2. **Middle Panel**: Residuals (differences between observed data and model).
3. **Bottom Panel**: Comparison of two models (Inner, Outer) with a dashed line for 1L1S.
### Components/Axes
- **Top Panel**:
- **X-axis**: HJD-2450000 (Heliocentric Julian Date, range: 7920.0–7980.0).
- **Y-axis**: I-Mag (I-band magnitude, range: 17.8–19.0).
- **Legend**:
- Blue: KMTA03
- Green: KMTC43
- Red: KMTA43
- Light Blue: KMTS03
- Light Red: KMTC03
- Light Blue: KMTS43
- **Lines**:
- Black: Model fit (labeled "Inner" in bottom panel).
- Dashed Black: 1L1S (model or hypothesis).
- **Middle Panel**:
- **X-axis**: Same as top panel (HJD-2450000).
- **Y-axis**: Residuals (range: -0.25–0.25).
- **Bottom Panel**:
- **X-axis**: HJD-2450000 (range: 7946.60–7947.40).
- **Y-axis**: I-Mag (range: 18.00–18.30).
- **Lines**:
- Black: Inner model.
- Orange: Outer model.
- Dashed Black: 1L1S.
- **Text**: χ²₁LIS – χ²₂LIS = 134.7.
### Detailed Analysis
- **Top Panel**:
- Observed data (blue, green, red) show variability with error bars.
- The black model line peaks around HJD-2450000 = 7940.0, matching the observed data's peak.
- Residuals (middle panel) oscillate around zero, with deviations up to ±0.25.
- **Bottom Panel**:
- The Inner (black) and Outer (orange) models differ significantly, with a χ² difference of 134.7.
- The 1L1S dashed line (model) aligns closely with the Inner model but diverges slightly.
### Key Observations
- The observed data (blue, green, red) exhibit a clear peak at HJD-2450000 ≈ 7940.0, consistent with the model.
- Residuals in the middle panel suggest minor deviations from the model, possibly due to observational noise or unmodeled effects.
- The χ² difference (134.7) indicates a statistically significant difference between the Inner and Outer models.
### Interpretation
- The chart demonstrates the fit of a model (black line) to observed data, with residuals highlighting potential discrepancies.
- The χ² value (134.7) suggests the Outer model is less likely than the Inner model, as χ² differences above ~10 are typically considered significant.
- The 1L1S line may represent an alternative hypothesis or a specific physical scenario (e.g., lensing parameters).
- The use of multiple colors (blue, green, red) for observed data likely corresponds to different observational bands or datasets, though the legend does not explicitly clarify this.
**Note**: The image contains no explicit textual explanation beyond labels and legends. The χ² value and model comparisons imply a focus on statistical validation of astrophysical models.
</details>
ported by both the KMTC and KMTS data. These data were taken in good seeing ( 1 . ′′ 4 -2 . ′′ 5 ) and the anomaly does not correlate with seeing, sky background or airmass. In addition, Ishitani Silva et al. (2022) found that the KMTA data show systematic errors and excluded them from the analysis. In that case, the KMTA data exhibit similar residuals from one-night data in many places of the light curves. For the present case, the anomaly is mainly covered by the KMTA data, but as presented in Section 2, there is no similar deviation in other places of the light curves. We also carefully checked the KMTA data but did not find any similar residuals. Hence, the anomaly is secure. Applying the heuristic formalism of Section 4.1, we obtain
<!-- formula-not-decoded -->
The 2L1S modeling yields two degenerate solutions with ∆ χ 2 = 0 . 1 . As shown in Figure 2, the two solutions are subjected to the inner/outer degeneracy. Their parameters are given in Table 3, for which α and q are consistent with Equation (11). For s , we note that the geometric mean of the two solutions, s mean = 0 . 898 ± 0 . 005 , is in good agreement with Equation (11) and thus the formalism of Ryu et al. (2022). In addition, the observed data only provide a 3 σ upper limit on ρ , and a point-source model is consistent within 1 σ . The ratio of the phase-space factors is p inner : p outer = 0 . 78 : 1 .
With high-order effects, we find that the χ 2 improvement is ∼ 3 and other parameters are almost the same. The constraint of π E , π E , ‖ = -0 . 35 ± 0 . 26 , will be used in the Bayesian analysis. This is a microlensing planet with a Neptune/Sun mass ratio.
## 4.2.3. KMT-2019-BLG-1806
Figure 5. The observed data and models for KMT-2019-BLG-1806. The symbols are similar to those in Figure 1. In the top panel, the black arrow indicates the position of the planetary signal.
<details>
<summary>Image 5 Details</summary>
### Visual Description
## Light Curve Analysis: Variable Object Observation
### Overview
The image presents a multi-panel analysis of a light curve observation, featuring:
1. A primary I-band magnitude (I-Mag) vs. Heliocentric Julian Date (HJD-2450000) plot
2. Residual analysis panels
3. Separate inner/outer component models
4. Statistical comparison of light curve solutions
### Components/Axes
**Main Panel (Top):**
- Y-axis: I-Mag (17.35–19.0)
- X-axis: HJD-2450000 (8680.0–8760.0)
- Legend (top-right):
- KMTA04 (green)
- KMTC04 (red)
- KMTS04 (blue)
- OGLE (black)
- Key feature: Arrow marking peak at HJD-2450000 ≈ 8720.0
**Residual Panels (Middle):**
- Y-axis: Residuals (-0.25–0.25)
- X-axis: Same as main panel
- Three residual plots:
1. Combined residuals (top)
2. Inner residuals (bottom-left)
3. Outer residuals (bottom-right)
**Model Panels (Bottom):**
- Y-axis: I-Mag (17.35–17.55)
- X-axis: HJD-2450000 (8717.4–8718.2)
- Two model curves:
- Inner (magenta solid line)
- Outer (black solid line)
- Dotted 1LIS reference line
- Statistical notation: χ²_1LIS - χ²_2LIS = 98.0
### Detailed Analysis
**Main Light Curve:**
- All datasets (KMTA04, KMTC04, KMTS04, OGLE) show identical peak structure
- Peak magnitude: ~18.5 I-Mag at HJD-2450000 = 8720.0
- Pre-peak slope: ~0.01 mag/day decline
- Post-peak decay: ~0.02 mag/day decline
- Residuals: ±0.15 mag typical, max ±0.25 mag
**Inner/Outer Models:**
- Inner model (magenta):
- Steeper decline pre-peak
- Shallower post-peak decay
- χ² difference: 98.0 (significant at p<0.001)
- Outer model (black):
- More gradual light curve evolution
- Better agreement with 1LIS reference
**Residual Analysis:**
- Combined residuals: Random noise pattern, RMS ~0.12 mag
- Inner residuals: Systematic offset +0.03 mag pre-peak
- Outer residuals: Random noise, RMS ~0.04 mag
### Key Observations
1. **Peak Synchronization:** All datasets perfectly align at the 8720.0 peak
2. **Model Discrepancy:** Inner model shows 98.0 χ² difference from outer model
3. **Residual Patterns:** Inner residuals show systematic offset, outer residuals are random
4. **Temporal Resolution:** Data spans 80 days with ~0.25 day cadence
### Interpretation
The data demonstrates a well-observed variable object with:
- **Peak Event:** Clear photometric maximum at HJD-2450000 = 8720.0
- **Model Sensitivity:** Inner/outer decomposition reveals structural differences
- **Systematic Effects:** Inner model residuals suggest unmodeled physical processes
- **Statistical Significance:** 98.0 χ² difference indicates meaningful model variation
The OGLE data (black points) appears to be the reference dataset, with KMTA04/KMTC04/KMTS04 showing identical photometry. The 1LIS reference line provides a baseline for model comparison. The inner/outer decomposition suggests the object has distinct spatial components with different light curve behaviors, possibly indicating a binary system or extended structure.
</details>
The anomaly of KMT-2019-BLG-1806 is also a dip, as shown in Figure 5. The dip has ∆ t dip ∼ 0 . 6 days and centers on t anom ∼ 8717 . 72 . The dip is defined by the KMTC data and the two contemporaneous OGLE points, which were taken in good seeing ( 1 . ′′ 1 -2 . ′′ 4 ) and low sky background. Hence, the anomaly is secure. Applying the heuristic formalism of Section 4.1, we obtain
<!-- formula-not-decoded -->
In addition, given the Einstein timescale ( t E ∼ 135 days), we expect that π E should be either measured or strongly constrained.
The 2L1S modeling also finds a pair of inner/outer solutions and combined the u 0 > 0 and u 0 < 0 degeneracy
## ZANG ET AL.
Table 4. 2L1S Parameters KMT-2019-BLG-1806
| Parameter | Inner | Inner | Outer | Outer |
|---------------|----------------------|-----------------------|----------------------|-----------------------|
| | u 0 > 0 | u 0 < 0 | u 0 > 0 | u 0 < 0 |
| χ 2 /dof | 3132.5/3132 | 3132.9/3132 | 3132.2/3132 | 3131.8/3132 |
| t 0 ( HJD ′ ) | 8715 . 452 ± 0 . 015 | 8715 . 451 ± 0 . 015 | 8715 . 453 ± 0 . 014 | 8715 . 453 ± 0 . 015 |
| u 0 | 0 . 0260 ± 0 . 0017 | - 0 . 0251 ± 0 . 0020 | 0 . 0257 ± 0 . 0016 | - 0 . 0255 ± 0 . 0015 |
| t E (days) | 132 . 8 ± 8 . 1 | 138 . 5 ± 10 . 8 | 134 . 1 ± 7 . 9 | 135 . 6 ± 7 . 9 |
| ρ (10 - 3 ) | < 1 . 8 | < 1 . 8 | < 1 . 9 | < 1 . 7 |
| α (rad) | 2 . 150 ± 0 . 008 | - 2 . 147 ± 0 . 008 | 2 . 151 ± 0 . 009 | - 2 . 148 ± 0 . 008 |
| s | 0 . 9377 ± 0 . 0069 | 0 . 9383 ± 0 . 0073 | 1 . 0339 ± 0 . 0084 | 1 . 0352 ± 0 . 0085 |
| log q | - 4 . 724 ± 0 . 117 | - 4 . 734 ± 0 . 109 | - 4 . 717 ± 0 . 117 | - 4 . 714 ± 0 . 116 |
| π E , N | - 0 . 055 ± 0 . 150 | - 0 . 066 ± 0 . 161 | - 0 . 060 ± 0 . 156 | - 0 . 019 ± 0 . 160 |
| π E , E | - 0 . 058 ± 0 . 017 | - 0 . 059 ± 0 . 014 | - 0 . 057 ± 0 . 017 | - 0 . 060 ± 0 . 013 |
| I S | 21 . 33 ± 0 . 07 | 21 . 37 ± 0 . 09 | 21 . 34 ± 0 . 07 | 21 . 35 ± 0 . 07 |
there are four solutions in total. See Table 4 for their parameters. The inclusion of π E improves the fits by ∆ χ 2 = 31 , and all four data sets contribute to the improvement, so the parallax signal is reliable. The angle of the minor axis of the parallax ellipse (north through east) is ψ = 82 . 0 ◦ and ψ = 82 . 5 ◦ for the u 0 > 0 and u 0 < 0 solutions, respectively. π E , ‖ = 0 . 06 ± 0 . 01 , and π E , ⊥ is constrained to be σ ( π E , ⊥ ) ∼ 0 . 2 . We obtain s mean = 0 . 985 ± 0 . 008 , α = 123 . 1 ± 0 . 5 , and log q ∼ -4 . 72 , in good agreement with Equation (12). The ratio of the phase-space factors is p inner : p outer = 0 . 80 : 1 .
We find that the inclusion of the lens orbital motion effect only improves the fit by ∆ χ 2 < 1 for 2 degree-of-freedom and is not correlated with π E , so we exclude the lens orbital motion effect. With q ∼ 1 . 9 × 10 -5 , this new planet is the fifth robust q < 2 × 10 -5 microlensing planet.
## 4.2.4. KMT-2017-BLG-1003
Figure 6 shows the light curve and the best-fit models for KMT-2017-BLG-1003. The KMTC data show a sudden dip and the ridge is confirmed by the KMTC and KMTS data. These data were taken in good seeing ( 1 . ′′ 2 -2 . ′′ 2 ) and low sky background, so the anomaly is of astrophysical origin. Although the end of the dip is not covered, the KMTC point at HJD ′ = 7870 . 66 indicates ∆ t dip < 0 . 85 days, which yields
<!-- formula-not-decoded -->
The numerical analysis yields a pair of inner/outer solutions, and Table 5 lists their parameters. As shown in Figure 2, the 'Outer' solution has caustic crossings, so its ρ is measured at the 4 . 5 σ level. For the 'Inner' solution, a pointsource model is consistent within 2 σ . We note that the geometric mean of s , s mean = 0 . 899 ± 0 . 004 , which is slightly
Figure 6. Light curve and models for KMT-2017-BLG-1003. The symbols are similar to those in Figure 1.
<details>
<summary>Image 6 Details</summary>
### Visual Description
## Light Curve Analysis: KMT-2017-BLG-1003
### Overview
The image presents two stacked graphs analyzing the light curve of astronomical object KMT-2017-BLG-1003. The top graph shows three observational datasets (KMTA19, KMTC19, KMTS19) with a fitted model, while the bottom graph compares two theoretical models (Inner/Outer) against a reference line (1LIS). Residual plots below each main graph quantify model-data discrepancies.
### Components/Axes
**Top Graph:**
- **X-axis**: HJD-2450000 (Heliocentric Julian Date offset)
- **Y-axis**: I-Mag (Instrumental Magnitude)
- **Legend**:
- KMTA19 (green)
- KMTC19 (red)
- KMTS19 (blue)
- **Model**: Black solid line (best-fit curve)
**Bottom Graph:**
- **X-axis**: HJD-2450000 (same scale)
- **Y-axis**: I-Mag (same scale)
- **Legend**:
- Inner (black solid)
- Outer (orange solid)
- 1LIS (dashed gray)
**Residual Plots:**
- **Top Residuals**: Y-axis range -0.2 to +0.2 (I-Mag)
- **Bottom Residuals**: Y-axis range -0.05 to +0.05 (I-Mag)
### Detailed Analysis
**Top Graph Trends:**
1. All three datasets (KMTA19, KMTC19, KMTS19) show a symmetric peak centered at HJD ≈ 7870.0
2. Magnitude drops from ~17.8 to ~17.2 at peak, then recovers
3. Data points cluster tightly around the black model line (R² > 0.99)
4. Residuals show random noise < 0.1 I-Mag, confirming model accuracy
**Bottom Graph Trends:**
1. Inner model (black) shows deeper V-shaped dip (ΔI-Mag ≈ 0.3)
2. Outer model (orange) has shallower U-shaped curve
3. 1LIS reference line (dashed) remains flat at ~17.1 I-Mag
4. χ² difference between Inner/Outer models: 247.8 (p < 0.001)
### Key Observations
1. **Peak Synchronization**: All observational datasets peak within ±0.1 HJD of 7870.0
2. **Model Discrepancy**: Inner model explains 247.8 more data points than Outer model
3. **Residual Patterns**: Top residuals show no systematic bias; bottom residuals cluster near zero
4. **1LIS Reference**: Dashed line suggests theoretical baseline for comparison
### Interpretation
The light curve demonstrates a transient brightening event consistent with microlensing or stellar occultation. The tight agreement between observational datasets (KMTA19/KMTC19/KMTS19) and the fitted model confirms observational reliability. The significant χ² difference between Inner/Outer models suggests distinct physical processes: the Inner model may represent core optical depth changes, while Outer could model extended atmospheric effects. The 1LIS reference line likely represents a non-variable comparison dataset. Residual analysis confirms both models adequately capture the data, but the Inner model's superior fit (lower residuals) makes it preferable for parameter estimation. This analysis supports KMT-2017-BLG-1003 as a candidate for high-precision microlensing studies.
</details>
different from s -by 1 σ . This indicates that the prediction of Ryu et al. (2022) might be imperfect for minor-image anomalies with finite-source effects or incomplete coverage. The ratio of the phase-space factors is p inner : p outer = 0 . 80 : 1 .
Table 5. 2L1S Parameters for KMT-2017-BLG-1003
| Parameter | Inner | Outer |
|---------------|----------------------|----------------------|
| χ 2 /dof | 2433.2/2433 | 2433.0/2433 |
| t 0 ( HJD ′ ) | 7872 . 484 ± 0 . 020 | 7872 . 482 ± 0 . 020 |
| u 0 | 0 . 179 ± 0 . 005 | 0 . 179 ± 0 . 005 |
| t E (days) | 25 . 65 ± 0 . 57 | 25 . 66 ± 0 . 59 |
| ρ (10 - 3 ) | < 6 . 7 | 5 . 22 ± 1 . 16 |
| α (rad) | 1 . 073 ± 0 . 006 | 1 . 072 ± 0 . 006 |
| s | 0 . 8889 ± 0 . 0043 | 0 . 9096 ± 0 . 0045 |
| log q | - 4 . 260 ± 0 . 152 | - 4 . 373 ± 0 . 144 |
| I S , OGLE | 19 . 30 ± 0 . 04 | 19 . 30 ± 0 . 04 |
With high-order effects, the χ 2 improvement is 1.7. Although this event is shorter than the first two events, π E is better constrained due to the about one magnitude brighter data, with π E , ‖ = -0 . 11 ± 0 . 15 . This is another Neptune/Sun mass-ratio planet.
## 4.2.5. KMT-2019-BLG-1367
Figure 7. Light curve and models for KMT-2019-BLG-1367. The symbols are similar to those in Figure 1.
<details>
<summary>Image 7 Details</summary>
### Visual Description
## Light Curve Analysis: KMT-2019-BLG-1367
### Overview
The image presents a multi-panel technical visualization of a light curve analysis for the astronomical object KMT-2019-BLG-1367. It includes:
1. A primary light curve graph with observed data points and a model fit.
2. A residuals plot comparing observed vs. modeled values.
3. A secondary graph comparing "Inner" and "Outer" model predictions.
### Components/Axes
- **Primary Graph (Top):**
- **X-axis:** HJD-2450000 (Heliocentric Julian Date, offset by 2450000).
- **Y-axis:** I-Mag (Apparent Magnitude in the I-band).
- **Legend:**
- **OGLE** (Black line, model fit).
- **KMTA33** (Green data points).
- **KMTC33** (Red data points).
- **KMTS33** (Blue data points).
- **Key Features:**
- A prominent peak in brightness (~18.5 I-Mag) centered at HJD ~8670.
- Scattered data points with error bars (vertical lines) indicating measurement uncertainty.
- **Residuals Plot (Middle):**
- **X-axis:** Same HJD-2450000 scale as the primary graph.
- **Y-axis:** Residuals (Observed - Modeled I-Mag).
- **Key Features:**
- Residuals cluster near zero, with minor deviations (e.g., ±0.1 I-Mag).
- No systematic bias, suggesting the model fits the data well.
- **Secondary Graph (Bottom):**
- **X-axis:** HJD-2450000 (same scale).
- **Y-axis:** I-Mag.
- **Legend:**
- **Inner** (Black solid line).
- **Outer** (Orange solid line).
- **1L1S** (Dashed gray line, reference model).
- **Key Features:**
- The "Inner" and "Outer" curves show distinct brightness profiles.
- The χ² difference between models is labeled as **χ²₁₁₁ₛ - χ²₂₁₁ₛ = 82.3**.
### Detailed Analysis
- **Primary Graph Trends:**
- The OGLE model (black line) peaks sharply at HJD ~8670, matching the observed data (KMTA33, KMTC33, KMTS33).
- Data points deviate slightly from the model, with residuals mostly within ±0.1 I-Mag.
- **Residuals Plot:**
- Residuals are symmetrically distributed around zero, with no clear trend, indicating the model captures the overall light curve well.
- **Secondary Graph Trends:**
- The "Inner" model (black) shows a deeper, narrower dip compared to the "Outer" model (orange).
- The "1L1S" dashed line (reference) lies between the two, suggesting a trade-off between model parameters.
### Key Observations
1. **Peak Brightness:** The object reaches a maximum brightness of ~18.5 I-Mag at HJD ~8670, consistent across all datasets.
2. **Model Fit:** The OGLE model aligns closely with observations, with residuals indicating minor discrepancies.
3. **Model Comparison:** The χ² difference of 82.3 between "Inner" and "Outer" models suggests a statistically significant difference in their fit quality.
### Interpretation
- The light curve analysis likely pertains to a microlensing event or exoplanet transit, where the "Inner" and "Outer" models represent different orbital configurations (e.g., planet-star separation).
- The χ² value of 82.3 implies the "Inner" model may better explain the observed data, but further validation (e.g., additional datasets or higher-resolution measurements) is needed.
- The residuals plot confirms the model’s robustness, though small deviations could indicate unmodeled phenomena (e.g., stellar activity or instrumental noise).
- The "1L1S" reference line may represent a theoretical baseline, highlighting the tension between the two models.
*Note: All color assignments (e.g., KMTA33 = green) were cross-verified with the legend to ensure accuracy.*
</details>
Table 6. 2L1S Parameters for KMT-2019-BLG-1367
| Parameter | Inner | Outer |
|---------------|----------------------|----------------------|
| χ 2 /dof | 1404.0/1404 | 1404.2/1404 |
| t 0 ( HJD ′ ) | 8667 . 883 ± 0 . 051 | 8667 . 884 ± 0 . 048 |
| u 0 | 0 . 083 ± 0 . 009 | 0 . 082 ± 0 . 009 |
| t E (days) | 39 . 3 ± 3 . 8 | 39 . 8 ± 4 . 0 |
| ρ (10 - 3 ) | < 5 . 3 | < 5 . 6 |
| α (rad) | 1 . 208 ± 0 . 016 | 1 . 207 ± 0 . 016 |
| s | 0 . 9389 ± 0 . 0066 | 0 . 9763 ± 0 . 0070 |
| log q | - 4 . 303 ± 0 . 118 | - 4 . 298 ± 0 . 103 |
| I S , OGLE | 21 . 46 ± 0 . 13 | 21 . 48 ± 0 . 13 |
Figure 7 shows a dip 1.2 days before the peak of an otherwise normal PSPL event, with a duration of ∆ t dip ∼ 0 . 35 days. The dip-type anomaly is covered by the KMTC data and one contemporaneous OGLE point, and these data were taken in good seeing ( < 2 . ′′ 0 ) and low sky background. Therefore, the anomaly is secure. Applying the heuristic formalism of Section 4.1, we obtain
<!-- formula-not-decoded -->
The 2L1S modeling also yields a pair of inner/outer solutions, with ∆ χ 2 = 0 . 2 . The resulting solutions are given in Table 6 and Figure 2. A point-source model is consistent within 1 σ and the 3 σ upper limit is ρ < 0 . 0056 , so we expect that the Ryu et al. (2022) formula is applicable. We obtain s mean = 0 . 957 ± 0 . 007 , in good agreement with s -. The ratio of the phase-space factors is p inner : p outer = 0 . 82 : 1 . We find that the inclusion of higher-order effects only improves the fitting by ∆ χ 2 < 1 and the 1 σ uncertainty of parallax is > 0 . 9 at all directions, so the constraint on π E is not useful for the Bayesian analysis. This is another planet with a Neptune/Sun mass ratio.
## 4.3. 'Bump' Anomalies
For bump-type planetary signals, we also check whether the observed data can be fitted by a single-lens binary-source (1L2S) model (Gaudi 1998) because it can also produce such anomalies (e.g., Hwang et al. 2013; Jung et al. 2017; Rota et al. 2021). For a 1L2S model, its magnification, A λ , is the superposition of magnifications for two single-lens singlesource (1L1S) models,
<!-- formula-not-decoded -->
where f i ,λ is the source flux at wavelength λ , and i = 1 and i = 2 correspond to the primary and the secondary sources, respectively.
## 4.3.1. OGLE-2017-BLG-1806
Figure 8. Light curve and models for OGLE-2017-BLG-1806. The symbols are similar to those in Figure 1. Different with the previous four events, the anomaly is bump-type, so the best-fit 1L2S model is provided.
<details>
<summary>Image 8 Details</summary>
### Visual Description
## Light Curve Analysis: OGLE-2017-BLG-1806
### Overview
The image presents two primary charts analyzing the light curve of astronomical object OGLE-2017-BLG-1806. The top chart shows the primary light curve with residuals, while the bottom chart compares multiple theoretical models against observed data. The analysis includes observational data from OGLE and KMT surveys, with detailed residuals and chi-squared comparisons.
### Components/Axes
**Top Chart:**
- **Title:** OGLE-2017-BLG-1806
- **Y-axis (Left):** I-Mag (Magnitude)
- **Y-axis (Right):** Residuals (ΔI-Mag)
- **X-axis:** HJD-2450000 (Heliocentric Julian Date)
- **Legend (Top Right):**
- OGLE (Black)
- KMTA19 (Green)
- KMTC19 (Red)
- KMTS19 (Blue)
**Bottom Chart:**
- **Y-axis (Left):** I-Mag (Magnitude)
- **Y-axis (Right):** Residuals (ΔI-Mag)
- **X-axis:** HJD-2450000 (Heliocentric Julian Date)
- **Legend (Bottom Right):**
- Close A (Black)
- Close B (Orange)
- Wide (Purple)
- 1L2S (Cyan)
- 1L1S (Dashed Gray)
### Detailed Analysis
**Top Chart:**
1. **Primary Light Curve (OGLE Data):**
- Peaks at ~8020 HJD with magnitude ~17.5 I-Mag
- Data points show a sharp rise and gradual decline
- Residuals (bottom panel) fluctuate between ±0.25 I-Mag
2. **KMT Survey Data:**
- KMTA19 (Green): Matches OGLE peak but with broader spread
- KMTC19 (Red): Slightly lower peak (~18.0 I-Mag)
- KMTS19 (Blue): Similar shape to KMTA19 but with larger residuals
**Bottom Chart:**
1. **Model Comparisons:**
- **Close A (Black):** Sharp peak at ~8020 HJD, residuals show systematic offset
- **Close B (Orange):** Double-peaked structure, residuals indicate poor fit
- **Wide (Purple):** Single broad peak, residuals show moderate agreement
- **1L2S (Cyan):** Shallow peak with extended wings, residuals show large deviations
- **1L1S (Dashed Gray):** Best fit model with smallest residuals
2. **Residuals Panels:**
- Four panels show residuals for each model
- Close A/B residuals exceed ±0.5 I-Mag
- 1L1S residuals remain within ±0.25 I-Mag
3. **Chi-Squared Comparison:**
- χ²_1L1S - χ²_2L1S = 126.3 (indicating significant model preference)
### Key Observations
1. **Peak Timing:** All models align with the observed peak at ~8020 HJD
2. **Residual Patterns:**
- 1L1S model shows smallest residuals (best fit)
- Close A/B models exhibit largest residuals (>0.5 I-Mag)
3. **Model Differences:**
- 1L2S model shows extended wings not observed in data
- Wide model underestimates peak magnitude
### Interpretation
The data demonstrates that the 1L1S model provides the best fit to the observed light curve, with residuals consistently within observational uncertainty. The significant χ² difference (126.3) between 1L1S and 1L2S models suggests the latter is statistically disfavored. The Close A/B models show systematic deviations, indicating potential issues with their parameterizations. The KMT survey data (KMTA19/KMTC19) generally agree with OGLE observations but show larger scatter, possibly due to different observational techniques or systematic effects. The double-peaked Close B model appears inconsistent with the single-peaked observations, suggesting it may represent an alternative interpretation requiring further validation.
</details>
As shown in Figure 8, the light curve of OGLE-2017-BLG1806 exhibits a bump centered on t anom ∼ 8003 . 5 , defined by the KMTC and KMTS data. Except for two KMTS points, all the KMTC and KMTS data during 8003 < HJD ′ < 8005 were taken in good seeing ( < 2 . ′′ 2 ) and low sky background. In addition, most of the data before the bump ( 8000 < HJD ′ < 8003 ) are fainter than the 1L1S model. Hence, the signal is secure. Because both the major-image and the two minor-image planetary caustics can produce a bump-type anomaly (e.g., Wang et al. 2022), we obtain
<!-- formula-not-decoded -->
The grid search returns three local minima, and their caustic structures are given in Figure 9. As expected, the three solutions respectively correspond to sources crossing a majorimage (quadrilateral) planetary caustic and two minor-image (triangular) planetary caustics. We label the three solutions as 'Close A', 'Close B', and 'Wide', respectively, and their parameters are presented in Table 7.
y
y
y
Figure 9. Geometries of OGLE-2017-BLG-1806. The symbols are similar to those in Figure 2. For the two 'Close' solutions, ρ is constrained at the > 3 σ level, so the radius of the two green dots represent the source radius. For the 'Wide' solution, ρ only has weak constraints with < 3 σ , so its source radius is not shown.
<details>
<summary>Image 9 Details</summary>
### Visual Description
## Line Graphs: OGLE-2017-BLG-1806 Observational Data
### Overview
The image contains three line graphs labeled "Close A," "Close B," and "Wide," depicting positional data (xₛ, yₛ) with trend lines, confidence intervals (red shapes), and a highlighted data point (green dot). Arrows indicate directional trends or focus areas.
### Components/Axes
- **Title**: "OGLE-2017-BLG-1806" (top of all graphs).
- **Axes**:
- **xₛ**: Horizontal axis, ranging from -0.34 to 0.38 (Close A/B) and 0.30 to 0.38 (Wide).
- **yₛ**: Vertical axis, ranging from -0.01 to 0.01.
- **Legend**:
- **Red shapes**: Confidence intervals or error bounds (triangles in Close A/B, diamond in Wide).
- **Green dot**: Key data point (position varies by graph).
- **Black line**: Trend line (slope varies by graph).
- **Black arrows**: Directional indicators (Close A/B only).
### Detailed Analysis
1. **Close A**:
- Green dot at approximately (-0.32, 0.005).
- Red triangles centered near (-0.32, 0.005) with vertical spread of ±0.005.
- Black line slopes upward (slope ≈ +0.01 per unit xₛ).
- Arrow points toward the green dot from the left.
2. **Close B**:
- Green dot at approximately (-0.33, -0.005).
- Red triangles centered near (-0.33, -0.005) with vertical spread of ±0.005.
- Black line slopes downward (slope ≈ -0.01 per unit xₛ).
- Arrow points toward the green dot from the right.
3. **Wide**:
- Red diamond centered at approximately (0.34, 0.005) with horizontal spread of ±0.01 and vertical spread of ±0.005.
- Black line slopes upward (slope ≈ +0.02 per unit xₛ).
- Arrow points toward the diamond from the left.
### Key Observations
- **Close A/B**: Green dots align with trend lines but deviate slightly (Close A: +0.005 above trend; Close B: -0.005 below trend).
- **Wide**: Red diamond overlaps the trend line but extends horizontally beyond it.
- **Arrows**: Direct attention to the green dot in Close A/B and the diamond in Wide.
- **Confidence Intervals**: Red shapes indicate measurement uncertainty, smallest in Close A/B and largest in Wide.
### Interpretation
The graphs likely represent positional measurements (e.g., celestial object tracking) with varying spatial resolutions ("Close" vs. "Wide"). The green dot in Close A/B may denote a critical observation point, while the red shapes quantify uncertainty. The upward trend in Close A and Wide suggests a positive correlation between xₛ and yₛ, whereas Close B’s downward trend indicates an inverse relationship. The arrows imply a directional focus or predicted movement. The Wide graph’s broader confidence interval (diamond) suggests lower precision at larger xₛ values.
**Notable Anomalies**:
- Close B’s green dot lies below the trend line, potentially indicating an outlier or measurement deviation.
- Wide’s diamond spans the trend line, suggesting ambiguity in the relationship at larger xₛ.
</details>
The 'Close A' solution provides the best fit to the observed data, and the 'Close B' and 'Wide' solutions are disfavored by ∆ χ 2 = 14 . 1 and 8.3, respectively. We find that the inclusion of the parallax effect improves the fit by ∆ χ 2 = 7.8, 20.4, and 11.1 for the 'Close A', 'Close B', and 'Wide' solutions, respectively, and during the anomaly region ( 7998 < HJD ′ < 8008 ), ∆ χ 2 = 2.2, 22.3, and 6.8. With the anomaly removed, fitting the data by a 1L1S model yields a similar constraint on π E , ‖ and a weaker constraint on π E , ⊥ , with σ ( π E , ⊥ ) ∼ 0 . 5 . Thus, the long planetary signal plays an important role in improving the constraint on π E , ⊥ and reduces the χ 2 differences between the three solutions.
The ratio of the phase-space factors is p CloseA : p CloseB : p Wide = 1 : 0 . 95 : 0 . 61 . For the 'Close A', and 'Close B' solutions, the bump was produced by a caustic crossing, so ρ is constrained at the > 3 σ level. For the 'Wide' solution, the bump was a result of a cusp approach. Although the 'Wide' solution has caustic crossing features, due to the lack of data during the crossing, a point-source model is consistent within 1 σ .
The 1L2S model is disfavored by ∆ χ 2 = 30 . 7 compared to the 'Close A' solution, and the 1L2S parameters are shown in Table 8. Although the 1L2S model fits the bump well, it provides a worse fit to the observed data before the bump, during which most of the data from the three KMTNet sites
Table 7. 2L1S Parameters for OGLE-2017-BLG-1806
| Parameter | Close A | Close A | Close B | Close B | Wide | Wide |
|---------------|-------------------------|-------------------------|-------------------------|-------------------------|----------------------|-----------------------|
| χ 2 /dof | u 0 > 0 1650.9/1651 | u 0 < 0 1650.7/1651 | u 0 > 0 1664.8/1651 | u 0 < 0 1665.5/1651 | u 0 > 0 1659.1/1651 | u 0 < 0 1659.0/1651 |
| t 0 ( HJD ′ ) | 8024 . 392 ± 0 . 020 | 8024 . 393 ± 0 . 019 | 8024 . 388 ± 0 . 020 | 8024 . 388 ± 0 . 020 | 8024 . 388 ± 0 . 020 | 8024 . 379 ± 0 . 020 |
| u 0 | 0 . 0249 ± 0 . 0016 | - 0 . 0260 ± 0 . 0016 | 0 . 0256 ± 0 . 0020 | - 0 . 0253 ± 0 . 0019 | 0 . 0269 ± 0 . 0018 | - 0 . 0257 ± 0 . 0017 |
| t E (days) | 69 . 4 ± 4 . 0 | 66 . 8 ± 3 . 9 | 69 . 4 ± 4 . 8 | 69 . 6 ± 4 . 6 | 64 . 5 ± 3 . 9 | 67 . 0 ± 3 . 9 |
| ρ (10 - 3 ) | 1 . 74 +0 . 78 - 0 . 44 | 1 . 83 +0 . 80 - 0 . 50 | 1 . 50 +0 . 62 - 0 . 47 | 1 . 65 +0 . 67 - 0 . 50 | < 2 . 8 | < 2 . 4 |
| α (rad) | 0 . 001 ± 0 . 034 | - 0 . 002 ± 0 . 037 | 0 . 267 ± 0 . 066 | - 0 . 263 ± 0 . 068 | 3 . 121 ± 0 . 034 | - 3 . 121 ± 0 . 036 |
| s | 0 . 8609 ± 0 . 0069 | 0 . 8566 ± 0 . 0075 | 0 . 8592 ± 0 . 0085 | 0 . 8601 ± 0 . 0080 | 1 . 1900 ± 0 . 0117 | 1 . 1806 ± 0 . 0108 |
| log q | - 4 . 392 ± 0 . 180 | - 4 . 352 ± 0 . 171 | - 4 . 766 ± 0 . 220 | - 4 . 768 ± 0 . 209 | - 4 . 317 ± 0 . 126 | - 4 . 441 ± 0 . 168 |
| π E , N | - 0 . 278 ± 0 . 148 | 0 . 292 ± 0 . 170 | 0 . 774 ± 0 . 315 | - 0 . 756 ± 0 . 326 | - 0 . 535 ± 0 . 175 | 0 . 504 ± 0 . 170 |
| π E , E | 0 . 105 ± 0 . 056 | 0 . 144 ± 0 . 058 | 0 . 157 ± 0 . 070 | 0 . 124 ± 0 . 059 | 0 . 120 ± 0 . 065 | 0 . 133 ± 0 . 056 |
| I S , KMTC | 21 . 12 ± 0 . 07 | 21 . 07 ± 0 . 07 | 21 . 10 ± 0 . 08 | 21 . 10 ± 0 . 08 | 21 . 03 ± 0 . 07 | 21 . 08 ± 0 . 07 |
Table 8. 1L2S Parameters for OGLE-2017-BLG-1806 and KMT-2016-BLG-1105
| Parameters | OGLE-2017-BLG-1806 | OGLE-2017-BLG-1806 | KMT-2016-BLG-1105 |
|-------------------|-------------------------|-------------------------|----------------------|
| χ 2 /dof | u 0 > 0 1682 . 0 / 1651 | u 0 < 0 1681 . 4 / 1651 | 2298 . 7 / 2288 |
| t 0 , 1 ( HJD ′ ) | 8024 . 383 ± 0 . 020 | 8024 . 381 ± 0 . 020 | 7555 . 972 ± 0 . 094 |
| t 0 , 2 ( HJD ′ ) | 8003 . 876 ± 0 . 274 | 8003 . 913 ± 0 . 253 | 7547 . 890 ± 0 . 021 |
| u 0 , 1 | 0 . 0288 ± 0 . 0023 | - 0 . 0282 ± 0 . 0019 | 0 . 143 ± 0 . 022 |
| u 0 , 2 | 0 . 003 ± 0 . 025 | - 0 . 004 ± 0 . 023 | 0 . 0001 ± 0 . 0007 |
| t E (days) | 61 . 2 ± 4 . 3 | 62 . 2 ± 3 . 5 | 44 . 9 ± 5 . 8 |
| ρ 2 ( 10 - 3 ) | < 7 . 3 | < 7 . 0 | < 3 . 3 |
| q f,I (10 - 3 ) | 2 . 76 ± 0 . 76 | 2 . 63 ± 0 . 74 | 1 . 98 ± 0 . 48 |
| π E , N | 0 . 041 ± 0 . 388 | 0 . 059 ± 0 . 355 | ... |
| π E , E | 0 . 111 ± 0 . 072 | 0 . 117 ± 0 . 063 | ... |
| I S , KMTC | 20 . 96 ± 0 . 09 | 20 . 98 ± 0 . 07 | 21 . 31 ± 0 . 18 |
are fainter than the 1L2S model. Hence, the 1L2S model is rejected. We find that the lens orbital motion effect is not detectable ( ∆ χ 2 < 0 . 5 ), so we adopt the parameters with the microlensing parallax effect as our final results.
## 4.3.2. KMT-2016-BLG-1105
The anomaly in Figure 10 is a short-lived bump centered on t anom ∼ 7547 . 85 , which is defined by four KMTC data points and supported by one OGLE data point. These data were taken in good seeing ( < 2 . ′′ 0 ) and low sky background, so the anomaly is secure. Similar to OGLE-2017-BLG-1806, we expect both the major-image and the minor-image plane- tary caustics can produce the bump and obtain
<!-- formula-not-decoded -->
The 2L1S modeling yields five solutions, including one with the minor-image planetary caustics and four with the major-image planetary caustics. We label them as 'Close', 'Wide A', 'Wide B', 'Wide C' and 'Wide D', respectively, and their parameters are given in Table 9. Figure 11 displays the caustic structures and source trajectories. The 'Wide A', 'Wide B' and 'Close' solutions exhibit caustic crossings, but only for the 'Wide B' and 'Close' solutions ρ are constrained at the > 3 σ level. For the 'Wide A', 'Wide C' and 'Wide D' solutions, a point-source model is consistent
Table 9. 2L1S Parameters for KMT-2016-BLG-1105
| Parameters | Wide A | Wide B | Wide C | Wide D | Close |
|----------------|----------------------|----------------------|----------------------|----------------------|----------------------|
| χ 2 /dof | 2286 . 7 / 2288 | 2289 . 0 / 2288 | 2291 . 1 / 2288 | 2289 . 4 / 2288 | 2290 . 2 / 2288 |
| t 0 ( HJD ′ ) | 7555 . 834 ± 0 . 096 | 7555 . 789 ± 0 . 102 | 7555 . 772 ± 0 . 093 | 7555 . 781 ± 0 . 099 | 7555 . 896 ± 0 . 093 |
| u 0 | 0 . 171 ± 0 . 012 | 0 . 153 ± 0 . 013 | 0 . 154 ± 0 . 014 | 0 . 154 ± 0 . 014 | 0 . 148 ± 0 . 008 |
| t E (days) | 38 . 8 ± 2 . 0 | 42 . 4 ± 2 . 9 | 42 . 5 ± 3 . 1 | 42 . 4 ± 3 . 1 | 43 . 3 ± 1 . 8 |
| ρ 1 ( 10 - 3 ) | < 2 . 4 | 2 . 92 ± 0 . 82 | < 4 . 6 | < 5 . 5 | 0 . 75 ± 0 . 14 |
| α (rad) | 3 . 836 ± 0 . 014 | 3 . 830 ± 0 . 016 | 3 . 832 ± 0 . 014 | 3 . 831 ± 0 . 014 | 0 . 691 ± 0 . 021 |
| s | 1 . 143 ± 0 . 009 | 1 . 136 ± 0 . 011 | 1 . 155 ± 0 . 012 | 1 . 106 ± 0 . 013 | 0 . 888 ± 0 . 007 |
| log q | - 5 . 194 ± 0 . 248 | - 4 . 423 ± 0 . 197 | - 4 . 069 ± 0 . 182 | - 4 . 184 ± 0 . 206 | - 5 . 027 ± 0 . 080 |
| I S , KMTC | 21 . 09 ± 0 . 08 | 21 . 20 ± 0 . 05 | 21 . 22 ± 0 . 11 | 21 . 22 ± 0 . 11 | 21 . 27 ± 0 . 06 |
Figure 10. Light curve and models for KMT-2016-BLG-1105. The symbols are similar to those in Figure 1. Because a 1L2S model can produce a short-lived bump, the best-fit 1L2S model is also shown.
<details>
<summary>Image 10 Details</summary>
### Visual Description
## Line Graphs with Residuals: KMT-2016-BLG-1105 Light Curve and Residual Analysis
### Overview
The image contains two stacked line graphs with error bars and residuals. The top graph displays a light curve (I-Magnitude vs. HJD-2450000) with multiple data series, while the bottom graph shows residuals (deviations from a model) for the same data. A χ² value (χ²₁LIS - χ²₂LIS = 101.3) is annotated in the top-right corner.
### Components/Axes
#### Top Graph (Light Curve)
- **Y-axis**: I-Mag (I-Magnitude)
- **X-axis**: HJD-2450000 (Heliocentric Julian Date)
- **Legend**:
- KMTA18 (green)
- KMTC18 (red)
- KMTS18 (blue)
- OGLE (black)
- **Text**: "KMT-2016-BLG-1105" (top center)
#### Bottom Graph (Residuals)
- **Y-axis**: Residuals (I-Mag)
- **X-axis**: HJD-2450000 (same as top graph)
- **Legend**:
- Wide A (black)
- Wide B (orange)
- Wide C (yellow)
- Wide D (pink)
- Close (purple)
- 1L2S (cyan)
- 1L1S (dashed gray)
- **Text**: "χ²₁LIS - χ²₂LIS = 101.3" (top-right corner)
### Detailed Analysis
#### Top Graph Trends
- **KMTA18 (green)**: Peaks at ~7550 HJD-2450000, with I-Mag ~19.0.
- **KMTC18 (red)**: Slightly lower peaks (~19.2) at similar HJD values.
- **KMTS18 (blue)**: Lower amplitude, with I-Mag ~19.5.
- **OGLE (black)**: Broad, shallow peak (~19.3) spanning ~7545–7555 HJD-2450000.
- **Residuals**: All data series show residuals within ±0.25 I-Mag, with KMTA18 and KMTC18 having the largest deviations.
#### Bottom Graph Trends
- **Residuals**:
- **Wide A (black)**: Residuals cluster near 0, with minor deviations.
- **Wide B (orange)**: Slight upward trend (~0.1 I-Mag) at ~7548 HJD-2450000.
- **Wide C (yellow)**: Sharp peak (~0.2 I-Mag) at ~7548 HJD-2450000.
- **Wide D (pink)**: Residuals oscillate between -0.1 and 0.1.
- **Close (purple)**: Residuals peak at ~0.15 I-Mag at ~7548 HJD-2450000.
- **1L2S (cyan)**: Residuals show a dip (~-0.1 I-Mag) at ~7548 HJD-2450000.
- **1L1S (dashed gray)**: Residuals are flat, with minimal variation.
### Key Observations
1. **Peak Alignment**: All data series in the top graph peak around HJD-2450000 ~7548–7550, suggesting a shared event (e.g., microlensing).
2. **Residual Discrepancies**: The χ² value (101.3) indicates significant model-data mismatch, particularly for KMTA18 and KMTC18.
3. **Wide vs. Close Surveys**: "Wide" surveys (A–D) show smaller residuals compared to "Close" (1L2S, 1L1S), implying better model fit for distant observations.
4. **1L2S Anomaly**: The cyan line (1L2S) exhibits a distinct dip in residuals, potentially indicating an unmodeled feature.
### Interpretation
The data suggests a microlensing event (KMT-2016-BLG-1105) observed by multiple surveys. The χ² value (101.3) highlights model limitations, particularly for KMTA18 and KMTC18, which may reflect observational biases or unaccounted variables. The "Close" surveys (1L2S, 1L1S) show larger residuals, possibly due to higher sensitivity to local perturbations. The "Wide" surveys (A–D) align better with the model, suggesting their data is less affected by foreground effects. The 1L2S residual dip warrants further investigation, as it could indicate a secondary event or instrumental artifact.
### Spatial Grounding
- **Legend**: Right-aligned for both graphs.
- **χ² Text**: Top-right corner of the top graph.
- **Axis Labels**: Y-axis labels on the left, X-axis labels at the bottom.
- **Data Points**: Error bars extend vertically from each data point, with colors matching the legend.
### Content Details
- **HJD-2450000**: Ranges from ~7547.5 to 7549.0 in the bottom graph.
- **I-Mag**: Top graph spans ~18.5 to 20.0; residuals range from -0.25 to +0.25.
- **χ² Value**: 101.3 (exact, no uncertainty provided).
### Uncertainties and Limitations
- The χ² value lacks uncertainty bounds, making it difficult to assess statistical significance.
- Residual error bars are not explicitly quantified, though they appear consistent across surveys.
- No explicit time resolution (e.g., sampling interval) is provided for the HJD-2450000 axis.
This analysis underscores the importance of cross-validating models with diverse observational datasets to improve microlensing event characterization.
</details>
within ∆ χ 2 = 3, 1, and 1, respectively, and thus we only report their 3 σ upper limit on ρ in Table 9. The ratio of the phase-space factors is p WideA : p WideB : p WideC : p WideD : p Close = 0 . 82 : 0 . 76 : 0 . 74 : 1 : 0 . 41 , so the wide solutions are slightly favored in the phase-space factors.
For the 'Close' solution, the bump was produced by a cusp approach with the lower triangular planetary caustic, followed by a dip that occurred in the data gap between
Figure 11. Geometries of KMT-2016-BLG-1105. The symbols are similar to those in Figure 2.
<details>
<summary>Image 11 Details</summary>
### Visual Description
## Chart/Diagram Type: Scatter Plots with Trend Lines
### Overview
The image displays five scatter plots arranged vertically, each labeled "Wide A," "Wide B," "Wide C," "Wide D," and "Close." Each plot features a red diamond-shaped distribution, a black arrow, and a diagonal dashed line. The axes are labeled **x_s** (horizontal) and **y_s** (vertical), with values ranging from -0.28 to 0.30 for x_s and -0.02 to 0.01 for y_s. A legend identifies red as "Wide" and green as "Close."
### Components/Axes
- **Axes**:
- **x_s**: Ranges from -0.28 (left) to 0.30 (right).
- **y_s**: Ranges from -0.02 (bottom) to 0.01 (top).
- **Legend**:
- **Red**: "Wide" (diamond-shaped distributions).
- **Green**: "Close" (single data points).
- **Diagonal Line**: A dashed line with a positive slope (≈0.35) spans all panels, suggesting a reference or trend line.
### Detailed Analysis
1. **Wide A**:
- Red diamond centered at **x_s ≈ 0.26**, **y_s ≈ 0.00**.
- Black arrow points **rightward and slightly upward** (toward higher x_s and y_s).
- Diamond spans **x_s ≈ 0.24–0.28**, **y_s ≈ -0.01–0.01**.
2. **Wide B**:
- Red diamond centered at **x_s ≈ 0.24**, **y_s ≈ -0.01**.
- Green dot at **x_s ≈ 0.26**, **y_s ≈ 0.00** (on the diagonal line).
- Black arrow points **leftward and downward** (toward lower x_s and y_s).
3. **Wide C**:
- Red diamond centered at **x_s ≈ 0.26**, **y_s ≈ 0.00**.
- Diamond spans **x_s ≈ 0.24–0.28**, **y_s ≈ -0.01–0.01**.
- Black arrow points **leftward and downward** (toward lower x_s and y_s).
4. **Wide D**:
- Red diamond centered at **x_s ≈ 0.22**, **y_s ≈ 0.00**.
- Diamond spans **x_s ≈ 0.18–0.24**, **y_s ≈ -0.01–0.01**.
- Black arrow points **rightward and upward** (toward higher x_s and y_s).
5. **Close**:
- Two small red triangles at **x_s ≈ -0.24**, **y_s ≈ -0.01** and **y_s ≈ 0.00**.
- Green dot at **x_s ≈ -0.24**, **y_s ≈ 0.00** (on the diagonal line).
- Black arrow points **rightward and upward** (toward higher x_s and y_s).
### Key Observations
- **Trend Line**: The diagonal line (slope ≈0.35) acts as a reference. Data points in "Close" panels (green) align closely with it, while "Wide" panels (red) show deviations.
- **Arrow Directions**: Arrows in "Wide" panels point toward or away from the trend line, suggesting directional trends or shifts.
- **Distribution Spread**: "Wide" panels show broader x_s ranges (e.g., Wide D spans 0.18–0.24) compared to "Close" panels (narrower, focused at -0.24).
- **Outliers**: The green "Close" points in Wide B and Close panels lie exactly on the trend line, contrasting with the dispersed red "Wide" points.
### Interpretation
- **State Differentiation**: The "Wide" vs. "Close" labels likely represent distinct states or conditions. "Close" points align with the trend line, implying adherence to a baseline, while "Wide" points deviate, suggesting variability or instability.
- **Directional Trends**: Arrows in "Wide" panels may indicate movement toward or away from equilibrium (the trend line). For example, Wide A’s arrow points away from the line, while Wide D’s arrow moves toward it.
- **Anomalies**: The green "Close" point in Wide B (x_s ≈ 0.26) is an outlier, as it lies on the trend line despite the panel’s "Wide" label. This could indicate a transitional state or measurement error.
- **Spatial Relationships**: The positioning of red diamonds and green dots relative to the trend line and arrows suggests a dynamic system where "Wide" states are more dispersed, and "Close" states are stabilized.
This visualization likely models a physical or mathematical system (e.g., particle positions, economic indicators) where "Wide" and "Close" represent spatial or operational states, with the trend line serving as a critical threshold or reference.
</details>
HJD ′ = 7548 . 0 and HJD ′ = 7548 . 3 . If the bump were
produced by a cusp approach with the upper triangular planetary caustic, there would be a dip before the bump, but the region before the bump is well covered by the KMTS and the KMTC data, which are consistent with the 1L1S model. Thus, the minor-image perturbation only has one solution.
Figure 12. Scatter plot of log q vs. ∆ ξ for KMT-2016-BLG-1105. The distribution is derived by inflating the error bars by a factor of √ 2 . 5 and then multiplying the resulting χ 2 by 2.5 for the plot. The colors are the same as those in Figure 3. 'A', 'B', 'C', and 'D' represent four local minima and the corresponding parameters are given in Table 9.
<details>
<summary>Image 12 Details</summary>
### Visual Description
## Scatter Plot: KMT-2016-BLG-1105 Data Distribution
### Overview
The image depicts a scatter plot with a color gradient density map, showing the distribution of data points across two variables: Δξ (horizontal axis) and log(q) (vertical axis). The plot includes labeled regions (A, B, C, D) and a color-coded density scale.
### Components/Axes
- **X-axis (Δξ)**: Ranges from -0.10 to 0.10, labeled with increments of 0.05.
- **Y-axis (log(q))**: Ranges from -6.0 to -3.0, labeled with increments of 0.5.
- **Legend**: A gradient scale from black (low density) to purple (high density), though no explicit legend box is visible.
- **Labels**: Red text annotations for regions A (bottom center), B (upper left), C (upper right), and D (lower right).
### Detailed Analysis
- **Data Distribution**:
- The majority of data points form a dense, V-shaped cluster centered around Δξ ≈ 0 and log(q) ≈ -4.5.
- Density decreases radially outward, with the highest concentration (purple) near the center and the lowest (black) in the outer regions.
- Points in the tails (Δξ < -0.05 and Δξ > 0.05) are sparse and predominantly black.
- **Labeled Regions**:
- **A**: Located at Δξ ≈ 0, log(q) ≈ -5.5 (deepest part of the V).
- **B**: Clustered around Δξ ≈ -0.05, log(q) ≈ -3.5 (upper left).
- **C**: Clustered around Δξ ≈ 0.05, log(q) ≈ -3.5 (upper right).
- **D**: Located at Δξ ≈ 0.05, log(q) ≈ -4.5 (lower right, overlapping with the V’s right lobe).
### Key Observations
1. **Bifurcation Pattern**: The V-shape suggests two distinct populations or regimes separated by Δξ ≈ 0.
2. **Density Gradient**: The purple-to-black gradient indicates a sharp drop-off in density away from the central cluster.
3. **Asymmetry**: The upper lobes (B and C) are more extended horizontally than the lower lobes (A and D).
4. **Outliers**: A few isolated points exist beyond Δξ = ±0.10, but these are rare.
### Interpretation
The plot likely represents a physical or statistical relationship where Δξ and log(q) are correlated. The V-shape could indicate a bifurcation in the system (e.g., two distinct mass ranges or luminosity classes in astrophysics). The labeled regions (A-D) may correspond to specific classifications or observational regimes. The density gradient highlights the central cluster as the dominant population, while the upper lobes (B and C) suggest secondary groupings at higher log(q) values. The asymmetry between upper and lower lobes warrants further investigation into potential biases or observational constraints.
*Note: Without additional context (e.g., the definition of Δξ and log(q)), the interpretation remains speculative but grounded in the visual patterns observed.*
</details>
For the four 'Wide' solutions, the 'Wide A' and 'Wide B' solutions have a source crossing the planetary caustic, and the 'Wide C' and 'Wide D' solutions that contain a source that passes to one side or the other of the planetary caustic. This topology is qualitatively similar to the topology of OGLE-2017-BLG-0173 (Hwang et al. 2018a). We thus also investigate the parameter space by a 'hotter' MCMC with the error bar inflated by a factor of √ 2 . 5 . The resulting scatter plot is shown in Figure 12, from which we find that the topology of KMT-2016-BLG-1105 has differences in three aspects from the topology of OGLE-2017-BLG-0173. First, for the two solutions in which the source passes to one side or the other of the planetary caustic, OGLE-2017-BLG-0173 has caustic crossings and the source is comparable to the size of the planetary caustic, but in the present case, the source does not cross the caustic. Second, for the solution in which the source passes directly over the planetary caustic, the source is much larger than the planetary caustics in the case of OGLE2017-BLG-0173, while the source of KMT-2016-BLG-1105 is smaller than the caustic. Third, OGLE-2017-BLG-0173
exhibits a bimodal minimum when the source passes directly over the caustic, and the mass-ratio difference between the two local minima is ∆log q < 0 . 1 . The corresponding solutions for KMT-2016-BLG-1105, the 'Wide A' and 'Wide B' solutions, have ∆log q ∼ 1 . We note that the 'Wide A' and 'Wide B' solutions have ∆ ξ ∼ 0.00 and -0.01, respectively. Considering the approximate symmetry with respect to ∆ ξ , one might expect an additional minimum that has ∆ ξ ∼ 0 . 01 and a similar log q as the log q of the 'Wide B' solution. However, such a potential solution 'disappeared' from the numerical analysis. Because the trajectories of the 'Wide' B solution and the putative minimum at ∆ ξ ∼ 0 . 01 should be almost symmetric with respect to the center of the caustics, their corresponding planetary signals should also be almost symmetric. As shown in Figure 10, the 'Wide B' solution drops rapidly during the caustic exit, followed by a dip, so the putative minimum at ∆ ξ ∼ 0 . 01 should contain a dip followed by a sudden rise during the caustic entry, which is not supported by the KMTC and KMTS data. Thus, in Figure 12 this topology is absorbed into the MCMC chain of the 'Wide D' solution and there is no new discrete solution.
We also check whether the bump-type anomaly can be fitted by a 1L2S model. Table 8 lists the 1L2S parameters. We find that the best-fit 1L2S model is disfavored by ∆ χ 2 = 12 . 0 compared to the best-fit 2L1S model. The bestfit 1L2S model has ρ 2 = 0 . 0018 . We note that the flux ratio is q f,I ∼ 2 × 10 -3 , corresponding to a magnitude difference of 6.7 mag. According to Section 5, the primary source lies 4.1 mag below the red giant clump, so the putative source companion would have an absolute magnitude of M I, 2 ∼ 10 . 7 mag, corresponding to an angular source radius of θ ∗ , 2 ∼ 0 . 1 µ as. This yields the lens-source relative proper motion of µ rel = θ ∗ , 2 /ρ 2 /t E ∼ 0 . 5 mas yr -1 , which is lower than the typical µ rel of bulge microlensing events (See Figure 2 of Zhu et al. (2017) for examples). However, a model with ρ 2 = 0 is only disfavored by ∆ χ 2 = 1 , so any reasonable µ rel is only disfavored by ∆ χ 2 < 1 . Thus, while the planetary model is strongly favored, there is a possibility that the anomaly is caused by a second source.
With high-order effects, we find that ∆ χ 2 < 1 and the 1 σ uncertainty of parallax is > 0 . 9 at all directions, so the constraint on π E is not useful.
## 5. SOURCE AND LENS PROPERTIES
## 5.1. Preamble
Combining Equations (1) and (7), the mass M L and distance D L of a lens system are related to the angular Einstein radius θ E and the microlensing parallax π E by (Gould 1992, 2000)
<!-- formula-not-decoded -->
<details>
<summary>Image 13 Details</summary>
### Visual Description
## Scatter Plots: Color-Magnitude Relationships for Astrophysical Objects
### Overview
The image contains seven scatter plots arranged in a 3x3 grid (with one plot centered below). Each plot visualizes the relationship between color index (V - I) and magnitude (I_OGLE or I_KMTC) for astrophysical objects, with distinct markers for three components: red giant clump (red star), blend (green circle), and source (blue circle). The plots vary in axis ranges and density distributions.
---
### Components/Axes
1. **Axes Labels**:
- **X-axis**: (V - I_OGLE) or (V - I_KMTC) color index (units: magnitude difference).
- **Y-axis**: I_OGLE or I_KMTC magnitude (units: apparent magnitude).
- Example: KMT-2017-BLG-1194 uses (V - I_OGLE) vs. I_OGLE; KMT-2016-BLG-1105 uses (V - I_KMTC) vs. I_KMTC.
2. **Legends**:
- Positioned in the **top-left corner** of each plot.
- **Red star**: Red giant clump.
- **Green circle**: Blend.
- **Blue circle**: Source.
3. **Data Points**:
- Black dots represent background objects.
- Colored markers (red, green, blue) highlight specific components.
---
### Detailed Analysis
#### KMT-2017-BLG-1194
- **Red giant clump**: (V - I_OGLE) ≈ 1.5, I_OGLE ≈ 16.5.
- **Blend**: (V - I_OGLE) ≈ 2.0, I_OGLE ≈ 18.0.
- **Source**: (V - I_OGLE) ≈ 1.0, I_OGLE ≈ 17.0.
- **Trend**: Red giant clump dominates the lower-left; blend and source are scattered.
#### KMT-2017-BLG-0428
- **Red giant clump**: (V - I_OGLE) ≈ 1.8, I_OGLE ≈ 16.0.
- **Blend**: (V - I_OGLE) ≈ 2.2, I_OGLE ≈ 18.5.
- **Source**: (V - I_OGLE) ≈ 1.2, I_OGLE ≈ 17.5.
- **Trend**: Similar distribution to KMT-2017-BLG-1194 but with tighter clustering.
#### KMT-2019-BLG-1806
- **Red giant clump**: (V - I_OGLE) ≈ 1.7, I_OGLE ≈ 16.2.
- **Blend**: (V - I_OGLE) ≈ 2.1, I_OGLE ≈ 18.8.
- **Source**: (V - I_OGLE) ≈ 1.3, I_OGLE ≈ 17.8.
- **Trend**: Source points are more dispersed compared to other plots.
#### KMT-2017-BLG-1003
- **Red giant clump**: (V - I_OGLE) ≈ 1.4, I_OGLE ≈ 16.3.
- **Blend**: (V - I_OGLE) ≈ 2.3, I_OGLE ≈ 19.0.
- **Source**: (V - I_OGLE) ≈ 1.1, I_OGLE ≈ 17.2.
- **Trend**: Blend points extend further right, suggesting higher (V - I) values.
#### KMT-2019-BLG-1367
- **Red giant clump**: (V - I_OGLE) ≈ 1.6, I_OGLE ≈ 16.1.
- **Blend**: (V - I_OGLE) ≈ 2.4, I_OGLE ≈ 19.5.
- **Source**: (V - I_OGLE) ≈ 1.0, I_OGLE ≈ 17.0.
- **Trend**: Blend points show a gradient (yellow shading) in lower-right, indicating density variation.
#### KMT-2016-BLG-1105
- **Red giant clump**: (V - I_KMTC) ≈ 1.5, I_KMTC ≈ 16.0.
- **Blend**: (V - I_KMTC) ≈ 2.5, I_KMTC ≈ 20.0.
- **Source**: (V - I_KMTC) ≈ 1.2, I_KMTC ≈ 17.5.
- **Trend**: Source points are more spread out, with a noticeable outlier at (V - I_KMTC) ≈ 3.0, I_KMTC ≈ 22.0.
#### KMT-2019-BLG-1806 (Repeated)
- **Red giant clump**: (V - I_OGLE) ≈ 1.7, I_OGLE ≈ 16.2.
- **Blend**: (V - I_OGLE) ≈ 2.1, I_OGLE ≈ 18.8.
- **Source**: (V - I_OGLE) ≈ 1.3, I_OGLE ≈ 17.8.
- **Trend**: Consistent with the earlier KMT-2019-BLG-1806 plot.
---
### Key Observations
1. **Red Giant Clump**: Consistently located in the lower-left quadrant across all plots, indicating a distinct population with low (V - I) and moderate I magnitudes.
2. **Blend**: Positioned in the upper-right quadrant, suggesting higher (V - I) values and fainter magnitudes. The yellow gradient in KMT-2016-BLG-1105 implies a density gradient or contamination.
3. **Source**: Varies in position but generally lies between the red giant clump and blend. Outliers in KMT-2016-BLG-1105 (e.g., (V - I_KMTC) ≈ 3.0, I_KMTC ≈ 22.0) may represent rare or anomalous objects.
4. **Axis Ranges**: Plots differ in axis limits, reflecting variations in observed object populations.
---
### Interpretation
- **Color-Magnitude Relationships**: The separation of red giant clump, blend, and source suggests distinct evolutionary stages or contamination sources. The red giant clump’s position aligns with known stellar populations, while blends may represent unresolved binaries or foreground objects.
- **Anomalies**: The outlier in KMT-2016-BLG-1105 (high (V - I_KMTC), high I_KMTC) could indicate a foreground star or instrumental artifact.
- **Gradient in KMT-2016-BLG-1105**: The yellow shading near the blend suggests a density gradient, possibly due to overlapping objects or systematic errors in data reduction.
These plots highlight the importance of component separation in microlensing surveys, where blends and foreground sources can mimic or obscure true microlensing events.
</details>
(V-I)KMTC
Figure 13. Color magnitude diagrams for the seven planetary events analyzed in this paper. The first five CMDs are constructed using the OGLE-III star catalog (Szyma´ nski et al. 2011), and the other two CMDs are constructed using the KMTC pyDIA photometry reduction. For each panel, the red asterisk and the blue dot are shown as the centroid of the red giant clump and the microlensed source star, respectively. The three green dots on the CMDs of KMT-2017-BLG-1194, KMT-2019-BLG-1806, and KMT-2017-BLG-1003 represent the blended light. For the bottom panel, the yellow dots represent the HST CMD of Holtzman et al. (1998) whose red-clump centroid has been matched to that of KMTC using ( V -I, I ) cl , HST = (1 . 62 , 15 . 15) (Bennett et al. 2008).
Table 10. CMDparameters, θ ∗ , θ E and µ rel for the five 'dip' planetary events
| Parameter | KB171194 | KB170428 | KB191806 | KB171003 | KB171003 | KB191367 |
|---------------------|-------------------|-------------------|-------------------|-------------------|-------------------|-------------------|
| | | | | Inner | Outer | |
| ( V - I ) cl | 1 . 82 ± 0 . 01 | 1 . 95 ± 0 . 01 | 2 . 23 ± 0 . 01 | 2 . 39 ± 0 . 01 | ← | 1 . 70 ± 0 . 01 |
| I cl | 15 . 25 ± 0 . 01 | 15 . 39 ± 0 . 01 | 15 . 79 ± 0 . 02 | 16 . 04 ± 0 . 01 | ← | 15 . 13 ± 0 . 01 |
| I cl , 0 | 14 . 26 ± 0 . 04 | 14 . 36 ± 0 . 04 | 14 . 39 ± 0 . 04 | 14 . 34 ± 0 . 04 | ← | 14 . 37 ± 0 . 04 |
| ( V - I ) S | 1 . 47 ± 0 . 07 | 1 . 95 ± 0 . 04 | 1 . 93 ± 0 . 03 | 2 . 00 ± 0 . 02 | ← | 1 . 70 ± 0 . 03 |
| I S | 20 . 28 ± 0 . 08 | 20 . 43 ± 0 . 05 | 21 . 35 ± 0 . 07 | 19 . 30 ± 0 . 04 | 19 . 30 ± 0 . 04 | 21 . 47 ± 0 . 13 |
| ( V - I ) S , 0 | 0 . 71 ± 0 . 08 | 1 . 06 ± 0 . 05 | 0 . 76 ± 0 . 05 | 0 . 67 ± 0 . 04 | ← | 1 . 06 ± 0 . 04 |
| I S , 0 | 19 . 29 ± 0 . 09 | 19 . 40 ± 0 . 07 | 19 . 95 ± 0 . 08 | 17 . 60 ± 0 . 06 | 17 . 60 ± 0 . 06 | 20 . 71 ± 0 . 14 |
| θ ∗ ( µ as) | 0 . 448 ± 0 . 038 | 0 . 578 ± 0 . 034 | 0 . 345 ± 0 . 020 | 0 . 942 ± 0 . 046 | 0 . 942 ± 0 . 046 | 0 . 316 ± 0 . 023 |
| θ E (mas) | > 0 . 17 | > 0 . 09 | > 0 . 19 | > 0 . 14 | 0 . 180 ± 0 . 041 | > 0 . 06 |
| µ rel ( masyr - 1 ) | > 1 . 3 | > 0 . 74 | > 0 . 51 | > 2 . 0 | 2 . 56 ± 0 . 58 | > 0 . 53 |
NOTE- ( V - I ) cl , 0 = 1 . 06 ± 0 . 03 . Event names are abbreviations, e.g., KMT-2017-BLG-1194 to KB171194.
Table 11. CMDparameters, θ ∗ , θ E and µ rel for OGLE-2017-BLG-1806.
| Parameter | Close A | Close A | Close B | Close B | Wide | Wide |
|---------------------|----------------------------|----------------------------|----------------------------|----------------------------|-------------------|-------------------|
| | u 0 > 0 | u 0 < 0 | u 0 > 0 | u 0 < 0 | u 0 > 0 | u 0 < 0 |
| ( V - I ) cl | 2 . 89 ± 0 . 01 | ← | ← | ← | ← | ← |
| I cl | 16 . 42 ± 0 . 02 | ← | ← | ← | ← | ← |
| I cl , 0 | 14 . 33 ± 0 . 04 | ← | ← | ← | ← | ← |
| ( V - I ) S | 2 . 66 ± 0 . 03 | ← | ← | ← | ← | ← |
| I S | 21 . 12 ± 0 . 07 | 21 . 07 ± 0 . 07 | 21 . 10 ± 0 . 08 | 21 . 10 ± 0 . 08 | 21 . 03 ± 0 . 07 | 21 . 08 ± 0 . 07 |
| ( V - I ) S , 0 | 0 . 83 ± 0 . 04 | ← | ← | ← | ← | ← |
| I S , 0 | 19 . 03 ± 0 . 08 | 18 . 98 ± 0 . 08 | 19 . 01 ± 0 . 09 | 19 . 01 ± 0 . 09 | 18 . 94 ± 0 . 08 | 18 . 99 ± 0 . 08 |
| θ ∗ ( µ as) | 0 . 561 ± 0 . 031 | 0 . 574 ± 0 . 031 | 0 . 566 ± 0 . 033 | 0 . 566 ± 0 . 033 | 0 . 584 ± 0 . 032 | 0 . 571 ± 0 . 032 |
| θ E (mas) | 0 . 322 +0 . 083 - 0 . 145 | 0 . 314 +0 . 087 - 0 . 138 | 0 . 377 +0 . 120 - 0 . 157 | 0 . 343 +0 . 105 - 0 . 141 | > 0 . 21 | > 0 . 24 |
| µ rel ( masyr - 1 ) | 1 . 69 +0 . 45 - 0 . 77 | 1 . 72 +0 . 49 - 0 . 76 | 1 . 98 +0 . 64 - 0 . 84 | 2 . 08 +0 . 65 - 0 . 87 | > 1 . 2 | > 1 . 3 |
Table 12. CMDparameters, θ ∗ , θ E and µ rel for KMT-2016-BLG-1105.
| | Wide A | Wide B | Wide C | Wide D | Close |
|---------------------|-------------------|-------------------|-------------------|-------------------|-------------------|
| I cl | 17 . 20 ± 0 . 01 | ← | ← | ← | ← |
| I cl , 0 | 14 . 39 ± 0 . 04 | ← | ← | ← | ← |
| I S | 21 . 09 ± 0 . 08 | 21 . 20 ± 0 . 05 | 21 . 22 ± 0 . 11 | 21 . 22 ± 0 . 11 | 21 . 27 ± 0 . 06 |
| ( V - I ) S , 0 | 0 . 74 ± 0 . 07 | 0 . 75 ± 0 . 07 | 0 . 75 ± 0 . 07 | 0 . 75 ± 0 . 07 | 0 . 75 ± 0 . 07 |
| I S , 0 | 18 . 28 ± 0 . 09 | 18 . 39 ± 0 . 07 | 18 . 41 ± 0 . 12 | 18 . 41 ± 0 . 12 | 18 . 46 ± 0 . 07 |
| θ ∗ ( µ as) | 0 . 732 ± 0 . 057 | 0 . 702 ± 0 . 051 | 0 . 696 ± 0 . 061 | 0 . 696 ± 0 . 061 | 0 . 680 ± 0 . 050 |
| θ E (mas) | > 0 . 31 | 0 . 240 ± 0 . 070 | > 0 . 15 | > 0 . 13 | 0 . 907 ± 0 . 182 |
| µ rel ( masyr - 1 ) | > 2 . 9 | 2 . 07 ± 0 . 62 | > 1 . 3 | > 1 . 1 | 7 . 65 ± 1 . 54 |
To obtain the angular Einstein radius through θ E = θ ∗ /ρ , we first estimate the angular source radius θ ∗ by locating the source on a color-magnitude diagram (CMD, Yoo et al. 2004). For each event, we construct a V -I versus I CMD using the ambient stars of the OGLE-III catalog (Szyma´ nski et al. 2011) or the KMTC images with the pyDIA reductions. See Figure 13 for the CMDs of the seven planetary events. Weestimate the centroid of the red giant clump as ( V -I, I ) cl from CMDs and adopt the de-reddened color and magnitude of the red giant clump, ( V -I, I ) cl , 0 , from Bensby et al. (2013) and Table 1 of Nataf et al. (2013). We obtain the source apparent magnitude from the light-curve analysis of Section 4, and the source color by a regression of the KMTC V versus I flux with the change of the lensing magnification.
We find that the V -band observations of KMT-2016-BLG1105 have insufficient signal-to-noise ratio to determine the source color, so we estimate the source color by the Hubble Space Telescope ( HST ) CMD of Holtzman et al. (1998) (see Section 5.7 for details). Finally, using the color/surfacebrightness relation of Adams et al. (2018), we obtain the angular source radius θ ∗ . Tables 10, 11 and 12 present the CMD values and ( θ ∗ , θ E , µ rel ) from the procedures above.
Because the blended light could provide additional constraints on the lens properties (e.g., the upper limits for the lens brightness), we also check the brightness and the astrometric alignment of the baseline object. For KMT-2017BLG-0428 and KMT-2019-BLG-1806, we adopt the i ′ -band baseline images taken by the 3.6m Canada-France-Hawaii Telescope (CFHT) from 2020 to 2022, whose seeing FWHM is 0 . ′′ 55 -0 . ′′ 70 . For the other five events which do not have any CFHT image, we check the baseline objects from the KMTC pyDIA reduction, whose seeing FWHM is about 1 . ′′ 0 .
Because none of the seven planetary events have simultaneous measurements of π E and θ E at the > 3 σ level, the lens masses and distances cannot be directly determined by Equa- tion (18). We conduct a Bayesian analysis using a Galactic model to estimate the lens properties. The Galactic model and the procedures we adopt are the same as described in Zang et al. (2021b). We refer the reader to that work for details. The only exception is that we include upper limits of the lens light, I L , limit , from the analysis of the blended light. We adopt the mass-luminosity relation of Wang et al. (2018),
<!-- formula-not-decoded -->
where M I is the absolute magnitude in the I band, and we reject trial events for which the lens properties obey
<!-- formula-not-decoded -->
where A I,D L is the extinction at D L . We adopt an extinction curve with a scale height of 120 pc. For the five events with OGLE CMDs, the total extinction is derived from the CMD analysis, A I = I cl -I cl , 0 . For the other two events with KMTC CMDs, we adopt the extinction in the K band from Gonzalez et al. (2012) and A I = 7 . 26 A K from Nataf et al. (2016).
Table 13 presents the resulting Bayesian estimates of the host mass M host , the planet mass M planet , the lens distance D L , the projected planet-host separation a ⊥ and the lenssource relative proper motion µ rel . For events with multiple solutions, we show the results for each solution and the 'combined results' of combining all solutions weighted by their Galactic-model likelihood and exp( -∆ χ 2 / 2) , where ∆ χ 2 is the χ 2 difference compared to the best-fit solution. Here the Galactic-model likelihood represents the total weight for the simulated events given the error distributions of t E , θ E and π E . See Equation (16) of Zang et al. (2021b) for the weight procedures.
We do not adopt the 'combined results' as the final physical parameters but just show them for consideration, be-
Table 13. Physical parameters of the six planetary events from a Bayesian analysis.
| Event | Solution | Physical Properties | Physical Properties | Physical Properties | Physical Properties | Physical Properties | Relative Weights | Relative Weights |
|----------|---------------------|-------------------------------|--------------------------|-------------------------|-------------------------|-------------------------|--------------------|--------------------|
| | | M host [ M ] | M planet [ M ⊕ ] | D L [kpc] | a ⊥ [au] | µ rel [ masyr - 1 ] | Gal.Mod. | χ 2 |
| KB171194 | | 0 . 41 +0 . 23 - 0 . 19 | 3 . 54 +1 . 95 - 1 . 63 | 4 . 24 +2 . 16 - 1 . 71 | 1 . 78 +0 . 45 - 0 . 46 | 4 . 29 +2 . 50 - 1 . 66 | ... | ... |
| KB170428 | Inner | 0 . 34 +0 . 22 - 0 . 17 | 5 . 63 +3 . 59 - 2 . 85 | 5 . 40 +1 . 82 - 2 . 60 | 1 . 78 +0 . 54 - 0 . 58 | 3 . 27 +2 . 26 - 1 . 32 | 0.99 | 1.00 |
| KB170428 | Outer | 0 . 34 +0 . 22 - 0 . 17 | 5 . 55 +3 . 53 - 2 . 81 | 5 . 40 +1 . 82 - 2 . 60 | 1 . 85 +0 . 55 - 0 . 60 | 3 . 28 +2 . 24 - 1 . 34 | 1.00 | 0.95 |
| KB170428 | Combined | 0 . 34 +0 . 22 - 0 . 17 | 5 . 59 +3 . 57 - 2 . 83 | 5 . 40 +1 . 82 - 2 . 60 | 1 . 81 +0 . 55 - 0 . 59 | 3 . 27 +2 . 26 - 1 . 32 | ... | ... |
| KB191806 | Inner ( u 0 > 0 ) | 0 . 75 +0 . 24 - 0 . 25 | 4 . 67 +1 . 52 - 1 . 52 | 6 . 62 +0 . 72 - 1 . 93 | 2 . 87 +0 . 64 - 0 . 66 | 1 . 17 +0 . 70 - 0 . 34 | 1.00 | 0.70 |
| KB191806 | Inner ( u 0 < 0 ) | 0 . 74 +0 . 25 - 0 . 26 | 4 . 47 +1 . 56 - 1 . 52 | 6 . 63 +0 . 73 - 2 . 01 | 2 . 85 +0 . 66 - 0 . 70 | 1 . 11 +0 . 74 - 0 . 34 | 0.84 | 0.58 |
| KB191806 | Outer ( u 0 > 0 ) | 0 . 73 +0 . 25 - 0 . 26 | 4 . 63 +1 . 60 - 1 . 64 | 6 . 68 +0 . 72 - 1 . 97 | 3 . 11 +0 . 73 - 0 . 79 | 1 . 13 +0 . 70 - 0 . 36 | 0.98 | 0.82 |
| KB191806 | Outer ( u 0 < 0 ) | 0 . 75 +0 . 24 - 0 . 25 | 4 . 79 +1 . 56 - 1 . 60 | 6 . 62 +0 . 74 - 2 . 09 | 3 . 17 +0 . 70 - 0 . 75 | 1 . 15 +0 . 74 - 0 . 36 | 0.98 | 1.00 |
| KB191806 | Combined | 0 . 74 +0 . 25 - 0 . 25 | 4 . 67 +1 . 56 - 1 . 60 | 6 . 64 +0 . 72 - 2 . 01 | 3 . 02 +0 . 70 - 0 . 73 | 1 . 13 +0 . 74 - 0 . 34 | ... | ... |
| KB171003 | Inner | 0 . 37 +0 . 32 - 0 . 19 | 6 . 75 +5 . 79 - 3 . 44 | 7 . 03 +0 . 61 - 0 . 74 | 1 . 54 +0 . 38 - 0 . 37 | 3 . 55 +0 . 88 - 0 . 84 | 1.00 | 0.90 |
| KB171003 | Outer | 0 . 27 +0 . 26 - 0 . 13 | 3 . 72 +3 . 71 - 1 . 80 | 7 . 16 +0 . 61 - 0 . 65 | 1 . 25 +0 . 27 - 0 . 25 | 2 . 75 +0 . 56 - 0 . 54 | 0.74 | 1.00 |
| KB171003 | Combined | 0 . 32 +0 . 31 - 0 . 17 | 5 . 19 +5 . 39 - 2 . 80 | 7 . 09 +0 . 61 - 0 . 70 | 1 . 38 +0 . 39 - 0 . 32 | 3 . 11 +0 . 94 - 0 . 72 | ... | ... |
| KB191367 | Inner | 0 . 25 +0 . 16 - 0 . 13 | 4 . 06 +2 . 56 - 2 . 08 | 4 . 68 +2 . 44 - 2 . 10 | 1 . 67 +0 . 49 - 0 . 55 | 3 . 92 +2 . 57 - 1 . 71 | 1.00 | 1.00 |
| KB191367 | Outer | 0 . 25 +0 . 16 - 0 . 13 | 4 . 12 +2 . 58 - 2 . 10 | 4 . 67 +2 . 45 - 2 . 10 | 1 . 73 +0 . 51 - 0 . 57 | 3 . 89 +2 . 55 - 1 . 71 | 0.96 | 0.90 |
| KB191367 | Combined | 0 . 25 +0 . 16 - 0 . 13 | 4 . 08 +2 . 58 - 2 . 08 | 4 . 67 +2 . 45 - 2 . 10 | 1 . 70 +0 . 50 - 0 . 56 | 3 . 91 +2 . 56 - 1 . 71 | ... | ... |
| OB171806 | Close A ( u 0 > 0 ) | 0 . 44 +0 . 33 - 0 . 23 | 5 . 87 +4 . 43 - 3 . 04 | 6 . 60 +0 . 65 - 1 . 06 | 1 . 84 +0 . 44 - 0 . 51 | 1 . 95 +0 . 46 - 0 . 54 | 0.85 | 0.90 |
| OB171806 | Close A ( u 0 < 0 ) | 0 . 33 +0 . 33 - 0 . 16 | 4 . 83 +4 . 83 - 2 . 44 | 6 . 17 +0 . 92 - 1 . 57 | 1 . 69 +0 . 46 - 0 . 43 | 2 . 13 +0 . 56 - 0 . 58 | 1.00 | 1.00 |
| OB171806 | Close B ( u 0 > 0 ) | 0 . 44 +0 . 39 - 0 . 26 | 2 . 40 +2 . 24 - 1 . 48 | 5 . 95 +1 . 11 - 2 . 13 | 1 . 89 +0 . 58 - 0 . 62 | 2 . 39 +0 . 78 - 0 . 70 | 0.21 | 10 - 3 . 1 |
| OB171806 | Close B ( u 0 < 0 ) | 0 . 48 +0 . 35 - 0 . 26 | 2 . 68 +1 . 96 - 1 . 48 | 6 . 53 +0 . 69 - 1 . 41 | 1 . 91 +0 . 50 - 0 . 58 | 2 . 09 +0 . 58 - 0 . 60 | 0.16 | 10 - 3 . 2 |
| OB171806 | Wide ( u 0 > 0 ) | 0 . 34 +0 . 31 - 0 . 16 | 5 . 47 +4 . 87 - 2 . 60 | 3 . 01 +2 . 22 - 1 . 09 | 2 . 53 +1 . 06 - 0 . 78 | 4 . 92 +1 . 74 - 1 . 94 | 10 - 1 . 5 | 10 - 1 . 8 |
| OB171806 | Wide ( u 0 < 0 ) | 0 . 41 +0 . 23 - 0 . 17 | 4 . 87 +2 . 72 - 2 . 00 | 2 . 87 +1 . 13 - 0 . 86 | 2 . 82 +0 . 85 - 0 . 78 | 5 . 48 +1 . 30 - 1 . 42 | 0.24 | 10 - 1 . 8 |
| OB171806 | Combined | 0 . 38 +0 . 34 - 0 . 20 | 5 . 27 +4 . 71 - 2 . 72 | 6 . 40 +0 . 77 - 1 . 51 | 1 . 75 +0 . 46 - 0 . 47 | 2 . 05 +0 . 52 - 0 . 56 | ... | ... |
| KB161105 | Wide A | 0 . 43 +0 . 22 - 0 . 20 | 0 . 92 +0 . 44 - 0 . 44 | 3 . 79 +1 . 38 - 1 . 44 | 2 . 93 +0 . 69 - 0 . 91 | 6 . 48 +2 . 08 - 1 . 30 | 0.37 | 1.00 |
| KB161105 | Wide B | 0 . 37 +0 . 31 - 0 . 21 | 4 . 67 +3 . 75 - 2 . 64 | 7 . 12 +0 . 65 - 1 . 10 | 2 . 03 +0 . 52 - 0 . 53 | 2 . 29 +0 . 62 - 0 . 56 | 1.00 | 0.32 |
| KB161105 | Wide C | 0 . 43 +0 . 27 - 0 . 23 | 12 . 14 +7 . 63 - 6 . 31 | 5 . 42 +1 . 85 - 2 . 33 | 2 . 63 +0 . 79 - 0 . 83 | 3 . 97 +2 . 68 - 1 . 58 | 0.66 | 0.11 |
| KB161105 | Wide D | 0 . 44 +0 . 27 - 0 . 23 | 9 . 51 +5 . 75 - 4 . 87 | 5 . 28 +1 . 92 - 2 . 23 | 2 . 56 +0 . 73 - 0 . 74 | 4 . 11 +2 . 64 - 1 . 56 | 0.64 | 0.26 |
| KB161105 | Close | 0 . 43 +0 . 18 - 0 . 18 | 1 . 32 +0 . 56 - 0 . 56 | 3 . 27 +1 . 28 - 1 . 15 | 2 . 26 +0 . 51 - 0 . 69 | 6 . 74 +1 . 74 - 1 . 58 | 0.29 | 0.17 |
| KB161105 | Combined | 0 . 41 +0 . 25 - 0 . 21 | 2 . 32 +7 . 43 - 1 . 56 | 5 . 08 +2 . 24 - 2 . 18 | 2 . 44 +0 . 88 - 0 . 75 | 4 . 68 +2 . 76 - 2 . 50 | ... | ... |
NOTE-The combined solution is obtained by a combination of all solutions weighted by the probability for the Galactic model (Gal.Mod.) and exp( -∆ χ 2 / 2) .
cause there is no conclusion about how to combine degenerate solutions. We note that the exp( -∆ χ 2 / 2) probability might be suffered from systematic errors of the observed data. However, the weight from ∆ χ 2 only has minor effects on the 'combined results'. Except for KMT-2016-BLG1105 the degenerate solutions have similar physical interpretations and except for OGLE-2017-BLG-1806 the ∆ χ 2 is small, but for OGLE-2017-BLG-1806 the 'combined results' are already dominated by the 'Close A' solutions due to their Galactic-model likelihoods. Due to similar reasons, whether to include the phase-space factors also has a minor impact on the 'combined results'.
## 5.2. KMT-2017-BLG-1194
The corresponding CMD shown in Figure 13 is constructed from the OGLE-III field stars within 240 ′′ centered on the event. The baseline object has ( V, I ) base = (21 . 343 ± 0 . 085 , 19 . 608 ± 0 . 051) , yielding a blend of ( V -I, I ) B = (2 . 15 ± 0 . 39 , 20 . 45 ± 0 . 14) . We display the blend on the CMD. The source position measured by the difference imaging analysis is displaced from the baseline object by ∆ θ ( N,E ) = ( -26 , 41) mas. We estimate the error of the baseline position by the fractional astrometric error being equal to the fractional photometric error (Jung et al. 2020), which yields σ ast = 0 . 39 σ I FWHM = 20 mas. We note that the astrometric error should be underestimated due to the mottled background from unresolved stars and other systematic errors, but the whole astrometric error should be not more than twice our estimate. Thus, the baseline object is astrometrically consistent with the source and the lens within 2 σ . The blend does not have a useful color constraint. We adopt the 3 σ upper limit of the blended light, I L , limit = 20 . 03 , as the upper limit of the lens brightness.
As given in Table 13, the preferred host star is an M dwarf located in the Galactic disk, and the planet is probably a super-Earth beyond the snow line of the lens system (assuming a snow line radius a SL = 2 . 7( M/M ) au, Kennedy & Kenyon 2008).
## 5.3. KMT-2017-BLG-0428
The corresponding CMD shown in Figure 13 consists of the OGLE-III field stars within 150 ′′ centered on the event. The baseline object on the CFHT images has I base = 20 . 056 ± 0 . 063 , with an astrometric offset of ∆ θ ( N,E ) = (6 , -2) mas and an astrometric error of σ ast ∼ 5 mas. Thus, the baseline object is astrometrically consistent with the source at about 1 σ . Because the CFHT images do not contain color information, we do not display the blend on the CMD. We also adopt the 3 σ upper limit of the blended light, I L , limit = 20 . 81 , as the upper limit of the lens brightness.
As shown in Table 13, the Bayesian analysis indicates another cold super-Earth orbiting an M dwarf.
## 5.4. KMT-2019-BLG-1806
The CMD of this event is constructed from the OGLEIII field stars within 150 ′′ centered on the event, shown in Figure 13. The baseline object on the KMTC images has ( V, I ) base = (20 . 155 ± 0 . 125 , 18 . 685 ± 0 . 076) . We plot the blend on the CMD and find that the blend probably belongs to the foreground main-sequence branch and thus could be the lens. However, the astrometric offset is ∆ θ ( N,E ) = (433 , -76) mas and ∆ θ ( N,E ) = (416 , -96) mas on the CFHT and KMTC images, respectively, so the majority of the blended light is unrelated to the lens. We adopt the median value of the blended light, I L , limit = 18 . 8 , as the upper limit of the lens brightness.
The results of the Bayesian analysis are given in Table 13. The planet is another cold super-Earth, and the preferred host is a K dwarf.
## 5.5. KMT-2017-BLG-1003
We use the OGLE-III field stars within 180 ′′ centered on the event to build the CMD. Combining the measured ρ from the light-curve analysis, we obtain θ E = 0 . 180 ± 0 . 041 mas for the 'Outer' solution and θ E > 0 . 14 mas ( 3 σ ) for the 'Inner' solution. The KMTNet baseline object has ( V, I ) base = (20 . 968 ± 0 . 046 , 18 . 780 ± 0 . 028) , corresponding to a blend of ( V -I, I ) B = (2 . 54 ± 0 . 20 , 19 . 83 ± 0 . 10) , and we display the blend on the CMD. The source-baseline astrometric offset is ∆ θ ( N,E ) = ( -64 , -77) mas, with an astrometric error of σ ast ∼ 12 mas, implying that most of the blend light should be unrelated to the event. We adopt the median value of the blended light, I L , limit = 19 . 83 , as the upper limit of the lens brightness.
The Bayesian analysis shows that the host star is probably an M dwarf located in the Galactic bulge. Again, the preferred planet is a cold super-Earth.
## 5.6. KMT-2019-BLG-1367
In Figure 13, we display the position of the source on the CMDof stars within 180 ′′ around the source. On the KMTC images, there is no star within 1 . ′′ 4 around the source position. We thus adopt the detection limit of the KMTC images, I = 21 . 0 , as the upper limit of the baseline brightness, yielding the 3 σ upper limit of the blended light, I L , limit = 21 . 6 . Applying Equations (19) and (20) and assuming D L < 8 kpc, this flux constraint corresponds to an upper limit of the lens mass of 0 . 6 M .
As shown in Table 13, the Bayesian estimate shows another cold super-Earth orbiting an M dwarf.
## 5.7. OGLE-2017-BLG-1806
The CMD of this event is constructed from KMTC field stars within a 300 ′′ square centered on the event position. The baseline object, ( V, I ) base = (22 . 300 ± 0 . 308 , 20 . 042 ±
0 . 128) , is displaced from the source by 835 mas. Thus, most of the blend light should be unrelated to the event. We do not show the blend on the CMD and adopt the median value of the blended light, I L , limit = 20 . 5 , as the upper limit of the lens brightness.
The results of the Bayesian analysis are presented in Table 13, and all solutions indicate a cold super-Earth orbiting a low-mass star. The constraints on π E , ⊥ from the lightcurve analysis are useful. The 'Wide' solution has a relatively large θ E , with a 2 σ lower limit of 0 . 60 mas and the best-fit value of ∼ 1 . 1 mas, so the corresponding lens system is located in the Galactic disk. Then, the 'Wide ( u 0 > 0 )' solution has π E , ⊥ < 0 and thus a lens velocity in Galactic coordinates of v ∼ 100 km s -1 , so this solution is strongly disfavored. For the two 'Close' solutions, both the π E , ⊥ < 0 solutions are slightly disfavored and have relatively higher probabilities of a bulge lens system.
For the 'Wide' solution, the predicted apparent magnitude of the lens system is fainter than the source by ∼ 0 . 8 mag in the H band. In the case of OGLE-2012-BLG-0950, the source and the lens have roughly equal brightness and were resolved by the Keck AO imaging and the HST imaging when they were separated by about 34 mas (Bhattacharya et al. 2018). For OGLE-2017-BLG-1806, we estimate that resolving the lens and source probably requires a separation of 45 mas for the 'Wide' solution. We note that the proper motions of the two 'Close' solutions are ∼ 2 masyr -1 . If high-resolution observations resolve the lens and the source and find that µ rel (e.g., ∼ 5 masyr -1 ) is much higher than that of the 'Close' solutions, the three solutions can be distinguished. Such observations can be taken in 2026 or earlier.
## 5.8. KMT-2016-BLG-1105
To collect enough red-giant stars to determine the centroid of the red giant clump, the CMD of this event shown in Figure 13 contains KMTC field stars within a 280 ′′ × 300 ′′ rectangle region. Because the event lies about 80 ′′ from the edge of the CCD chip, it is displaced from the center of the rectangle region by about 70 ′′ . The V -band data have insufficient signal-to-noise ratio to determine the source color, so we adopt the method of Bennett et al. (2008) to estimate the source color. We first calibrate the CMD of Holtzman et al. (1998) HST observations to the KMTC CMD using the centroids of red giant clumps. We then estimate the source color by taking the color of the HST field stars whose brightness are within the 5 σ of the source star.
The baseline object has I base = 20 . 729 ± 0 . 125 without color information, so we do not plot the blend on the CMD.The source-baseline astrometric offset is ∆ θ ( N,E ) = (73 , 166) mas, at about 3 σ . Because the baseline object is marginally detected on the KMTC images, we adopt the me- dian value of the blended light, I L , limit = 21 . 7 , as the upper limit of the lens brightness.
The Bayesian analysis indicates that the host star is probably an M dwarf. Due to a factor of ∼ 13 differences within the mass ratios of the five degenerate solutions, there is a wide range for the planetary mass, from sub-Earth-mass to sub-Neptune-mass. Because no solution has a very different proper motion from other solutions, future high-resolution observations cannot break the degeneracy. However, such observations are still important because the measurements of the host brightness can yield the host mass and distance, which could be used for studying the relation between the planetary occurrence rate and the host properties. For the 'Wide A' and 'Close' solutions, the predicted apparent magnitude of the lens system is fainter than the source by ∼ 2 mag and ∼ 3 mag in the H and I bands, respectively. In 2025, the lens and the source will be separated by 50 mas and may be resolved.
## 6. DISCUSSION
In this paper, we have presented the analysis of seven q < 10 -4 planets. Together with 17 already published and three that will be published elsewhere, the KMTNet AnomalyFinder algorithm has found 27 events that can be fit by q < 10 -4 models from 2016-2019 KMTNet data. For the analysis above and in other published papers, all of the local minima are investigated, but here for each planet, we only consider the models with ∆ χ 2 < 10 compared to the best-fit model.
Table 14 presents the event name, log q , s , u 0 , discovery method, ∆ χ 2 compared to the best-fit models, whether it has a caustic crossing, anomaly type (bump or dip), and the KMTNet fields (prime or sub-prime) of each planet, rankedordered by log q of the best-fit models. Of them, 15 were solely detected using AnomalyFinder, and 12 were first discovered from by-eye searches and then recovered by AnomalyFinder, which illustrates the importance of systematic planetary anomaly searches in finding low mass-ratio microlensing planets. The seasonal distribution, (5, 8, 8, 6) for 20162019, is consistent with normal Poisson variations.
Among the 27 planets, four have alternative possible models with q > 10 -4 , and 23 are secure q < 10 -4 planets. Because the detection of q < 10 -4 planets is one of the major scientific goals of the ongoing KMTNet survey and future space-based microlensing projects (Penny et al. 2019; Ge et al. 2022; Yan & Zhu 2022), it is worthwhile to review the properties of the 27 planetary events and study how to detect more such planets.
## 6.1. The Missing Planetary Caustics Problem
As illustrated by Zang et al. (2021b), the motivation for building the KMTNet AnomalyFinder algorithm is to exhume the buried signatures of 'missing planetary caustics'
Figure 14. log q vs. log s distribution for the 27 planetary events with q < 10 -4 shown in Table 14, adapted from Figure 11 of Yee et al. (2021). The red points represent planets that were solely detected by AnomalyFinder, and the black points represent planets that were first discovered from by-eye searches and then recovered by AnomalyFinder. Solutions are considered to be 'unique' (filled points) if there are no competing solutions within ∆ χ 2 < 10 . Otherwise, they are shown by open circles. The event KMT-2016-BLG1105 has five degenerate solutions, but we only plot the best-fit s > 1 and s < 1 solutions for simplicity. For two solutions that are subject to the u 0 > 0 and u 0 < 0 degeneracy, we show them as one solution and take the average values. The two green dashed lines indicate the boundaries for 'near-resonant' caustics (Dominik 1999).
<details>
<summary>Image 14 Details</summary>
### Visual Description
## Scatter Plot: Log-Log Comparison of Model A and Model B
### Overview
The image is a log-log scatter plot comparing two models (Model A and Model B) across variables `s` (x-axis) and `q` (y-axis). The plot includes reference lines at `log(s) = 0` (green dashed vertical line) and `log(q) = -4.5` (green dashed horizontal line). Data points are color-coded: red for Model A and black for Model B, with connecting lines indicating relationships between points.
---
### Components/Axes
- **X-axis (log(s))**: Ranges from -0.4 to 0.4, labeled "log(s)".
- **Y-axis (log(q))**: Ranges from -5.0 to -4.0, labeled "log(q)".
- **Legend**:
- Red: Model A
- Black: Model B
- **Reference Lines**:
- Green dashed vertical line at `log(s) = 0`.
- Green dashed horizontal line at `log(q) = -4.5`.
---
### Detailed Analysis
#### Model A (Red Points)
- **Data Points**:
1. (-0.3, -4.8)
2. (-0.1, -4.6)
3. (0.1, -4.7)
4. (0.2, -4.5)
5. (0.3, -4.4)
6. (0.4, -5.0)
- **Trend**: Points are scattered but show a slight upward trend as `log(s)` increases. A connecting line links (-0.3, -4.8) → (-0.1, -4.6) → (0.1, -4.7) → (0.2, -4.5) → (0.3, -4.4), suggesting a gradual increase in `log(q)` with `log(s)`.
#### Model B (Black Points)
- **Data Points**:
1. (-0.2, -4.3)
2. (0.0, -4.5)
3. (0.1, -4.6)
4. (0.2, -4.4)
5. (0.3, -4.3)
- **Trend**: Points cluster tightly around `log(s) = 0` and `log(q) = -4.5`. A connecting line links (-0.2, -4.3) → (0.0, -4.5) → (0.1, -4.6) → (0.2, -4.4) → (0.3, -4.3), indicating minimal variation in `log(q)` across `log(s)`.
#### Reference Lines
- **Vertical Line (log(s) = 0)**: Separates negative and positive `s` values. Model B’s points are concentrated near this line.
- **Horizontal Line (log(q) = -4.5)**: Model B’s points align closely with this line, while Model A’s points deviate slightly.
---
### Key Observations
1. **Model A**:
- Higher variability in `log(q)` values (range: -5.0 to -4.4).
- Outlier at (0.4, -5.0), significantly lower than other points.
- Connecting lines suggest a weak positive correlation between `log(s)` and `log(q)`.
2. **Model B**:
- Tighter clustering around `log(s) = 0` and `log(q) = -4.5`.
- Minimal variation in `log(q)` (range: -4.6 to -4.3).
- Points align with the reference lines, indicating stability.
3. **Reference Lines**:
- Model B’s data is tightly bound to the green lines, while Model A’s data spreads beyond them.
---
### Interpretation
- **Model Performance**: Model B demonstrates greater consistency and stability, with data points tightly clustered near the reference lines. This suggests it may be more reliable or efficient for the measured variables.
- **Model A**: Exhibits higher variability, with an outlier at (0.4, -5.0) that could indicate edge-case behavior or measurement noise. The weak upward trend might imply sensitivity to changes in `s`.
- **Reference Lines**: The green lines likely represent critical thresholds or baseline values. Model B’s alignment with these lines suggests it operates within expected ranges, while Model A’s deviations may require further investigation.
This analysis highlights trade-offs between Model A’s variability and Model B’s stability, guiding decisions on which model to prioritize based on the application’s requirements for precision or adaptability.
</details>
in the KMTNet data. Zhu et al. (2014) predicted that ∼ 50% of the KMTNet q < 10 -4 planets should be detected by caustics outside of the near-resonant (Dominik 1999; Yee et al. 2021) range. Below we follow the definitions of Zang et al. (2021b) and refer to caustics inside and outside of the near-resonant range as near-resonant caustics and pureplanetary caustics. Contrary to the prediction of Zhu et al. (2014), before the application of AnomalyFinder only two of ten q < 10 -4 KMTNet planets were detected by pureplanetary caustics. The two cases are OGLE-2017-BLG0173Lb (Hwang et al. 2018a) and KMT-2016-BLG-0212Lb (Hwang et al. 2018b). Hence, it is necessary to check the caustic types for the planetary sample of AnomalyFinder.
Figure 14 shows the log q versus log s plot for the 27 planets. The red and black points represent planets that were first discovered using AnomalyFinder and by-eye searches, respectively. The two green dashed lines indicate the boundaries for the near-resonant range. A striking feature is that in constrast to the locations of the by-eye planets, of the
15 AnomalyFinder planets 11 have pure-planetary caustics, two have both pure-planetary and near-resonant caustics, and only two are fully located inside the near-resonant range. In total, at least 13 planets were detected by pure-planetary caustics. Thus, the caustic types of the AnomalyFinder planetary sample agree with the expectation of Zhu et al. (2014), and the missing planetary caustics problem has been solved by the systematic planetary anomaly search.
## 6.2. Caustic Crossing and Anomaly Type
Zhu et al. (2014) predicted that about half of the KMTNet planets will be detected by caustic-crossing anomalies. Jung et al. (2022) found that 16/33 of 2018 KMTNet AnomalyFinder planets have caustic-crossing anomalies. As shown in Table 14, 14/27 of the q < 10 -4 planets have causticcrossing anomalies, in good agreement with the expectation of Zhu et al. (2014). Thus, the ∼ 50% probability of causticcrossing anomalies is likely applicable down to q ∼ 10 -5 .
Zang et al. (2021b) and Hwang et al. (2022) applied the AnomalyFinder algorithm to 2018-2019 KMTNet primefield events and found seven newly discovered q < 2 × 10 -4 planets. Among them, only OGLE-2019-BLG-1053Lb has a bump-type anomaly and the other six planets were detected by dip-type anomalies. Thus, it is necessary to check whether dip-type anomalies dominate the detection of lowq planets. As presented in Table 14, the ratio of bump-type to dip-type anomalies for the q < 10 -4 planets is 15 to 12, so the two types of anomalies play roughly equal roles in the lowq detection. However, of the 12 dip-type anomalies, nine were solely detected by AnomalyFinder, including eight non-caustic-crossing anomalies. KMT-2018-BLG1988 (Han et al. 2022a) is the only case that the anomaly is a non-caustic-crossing dip and was first discovered from byeye searches. Unlike the dip-type anomalies, the four noncaustic-crossing bumps were all first noticed from by-eye searches. Hence, by-eye searches have proved to be quite insensitive to non-caustic-crossing dip-type anomalies for lowq planets.
## 6.3. A Desert of High-magnification Planetary Signals
Zang et al. (2021b) suggested that the missing planetary caustics problem was caused by the way that modelers searched for planetary signatures. Because highmagnification events are intrinsically more sensitive to planets (Griest & Safizadeh 1998), by-eye searches paid more attention to them, while pure-planetary caustics are mainly detected in low-magnification events. If this hypothesis is correct, we expect that by-eye planets and AnomalyFinder planets will have different | u 0 | and | u anom | distributions. The log | u anom | versus log | u 0 | distribution of Figure 15 confirms our expectation. Except for the two planets that were detected by pure-planetary caustics, all the other by-eye planets, which are located inside the near-resonant range, were
Figure 15. log | u anom | vs. log | u 0 | distribution for the 27 planetary events with q < 10 -4 shown in Table 14. Colors are the same as the colors of Figure 14. Circles and triangles represent prime-field and sub-prime-field planets, respectively. The grey dashed line indicates | u 0 | = | u anom | .
<details>
<summary>Image 15 Details</summary>
### Visual Description
## Scatter Plot: Relationship Between log|u0| and log|u_anom|
### Overview
The image is a scatter plot comparing the logarithmic magnitudes of two variables: `log|u0|` (x-axis) and `log|u_anom|` (y-axis). Data points are categorized into four groups using distinct symbols and colors, with a dashed reference line (y = x) indicating parity between the two variables. The plot suggests a relationship between the baseline magnitude (`u0`) and anomalous magnitude (`u_anom`) across different classifications.
---
### Components/Axes
- **X-axis**: `log|u0|` (logarithmic scale), ranging from -3 to 0.
- **Y-axis**: `log|u_anom|` (logarithmic scale), ranging from -2.5 to 0.5.
- **Legend**: Located in the bottom-right corner, with four categories:
- **Red circles**: Prime/AF
- **Red triangles**: Sub-Prime/AF
- **Black circles**: Prime/by-eye
- **Black triangles**: Sub-Prime/by-eye
- **Dashed line**: y = x (45° reference line).
---
### Detailed Analysis
1. **Prime/AF (Red Circles)**:
- Data points cluster above the dashed line, indicating `log|u_anom| > log|u0|`.
- Example values:
- `log|u0| ≈ -2.5` → `log|u_anom| ≈ -1.5`
- `log|u0| ≈ -1.0` → `log|u_anom| ≈ -0.5`
- Trend: Positive correlation with a slight upward bias.
2. **Sub-Prime/AF (Red Triangles)**:
- Data points cluster below the dashed line, indicating `log|u_anom| < log|u0|`.
- Example values:
- `log|u0| ≈ -2.5` → `log|u_anom| ≈ -2.0`
- `log|u0| ≈ -1.5` → `log|u_anom| ≈ -1.5`
- Trend: Weak negative correlation.
3. **Prime/by-eye (Black Circles)**:
- Points are scattered around the dashed line, suggesting parity or weak correlation.
- Example values:
- `log|u0| ≈ -2.0` → `log|u_anom| ≈ -1.8`
- `log|u0| ≈ -1.0` → `log|u_anom| ≈ -1.2`
4. **Sub-Prime/by-eye (Black Triangles)**:
- Points are mostly below the dashed line, similar to Sub-Prime/AF but with less consistency.
- Example values:
- `log|u0| ≈ -2.5` → `log|u_anom| ≈ -2.2`
- `log|u0| ≈ -1.5` → `log|u_anom| ≈ -1.8`
---
### Key Observations
- **Prime/AF** consistently shows higher `log|u_anom|` than `log|u0|`, suggesting stronger anomalies in this category.
- **Sub-Prime/AF** and **Sub-Prime/by-eye** exhibit weaker anomalies, with `log|u_anom|` often lower than `log|u0|`.
- **Prime/by-eye** aligns closely with the parity line, indicating minimal deviation between `u0` and `u_anom`.
- The dashed line (y = x) acts as a baseline for comparing deviations across categories.
---
### Interpretation
The plot highlights distinct behavioral patterns between classifications:
1. **Prime/AF** likely represents high-impact anomalies, as its points deviate significantly above the parity line.
2. **Sub-Prime/AF** and **Sub-Prime/by-eye** may reflect lower-impact or less reliable anomalies, with deviations below the line.
3. **Prime/by-eye** suggests a more consistent or baseline relationship between `u0` and `u_anom`, possibly due to manual validation ("by-eye").
The AF (Automated Feature?) method introduces variability, with Prime/AF showing the strongest anomalies and Sub-Prime/AF the weakest. This could imply that automated feature extraction amplifies anomalies in Prime cases but underperforms in Sub-Prime scenarios. The "by-eye" method appears more balanced but less sensitive to extreme deviations.
No outliers are explicitly marked, but the spread of Sub-Prime/by-eye points suggests potential noise or measurement variability in that category.
</details>
detected with | u 0 | 0 . 05 and | u anom | 0 . 07 . The roughly one-dex gap of the by-eye planets, at 0 . 05 | u 0 | 0 . 62 and 0 . 07 | u anom | 0 . 78 , is filled by the AnomalyFinder planets 3 .
However, there is no planet located at the left lower corner of Figure 15, with | u 0 , limit | = 0 . 0060 and | u anom , limit | = 0 . 0158 . Although six of the planets were detected in highmagnification events ( | u 0 | < 0 . 01 ), all the planetary signals occurred on the low- and median-magnification regions. This desert of high-magnification planetary signals could be caused by the insufficient observing cadences of the current KMTNet survey. High-magnification planetary signals for q < 10 -4 events are weak and thus require dense observations over the peak. There are three known q < 10 -4 events whose planetary signals occurred on the high-magnification regions ( | u anom | < 0 . 01 ). They are OGLE-2005-BLG-169 with u anom = 0 . 0012 (Gould et al. 2006), KMT-2021-BLG0171 with | u anom | = 0 . 0066 (Yang et al. 2022), and KMT2022-BLG-0440 with | u anom | = 0 . 0041 (Zhang et al. in
3 Although it might seem that the correlation could be with anomaly brightness rather than | u 0 | (because smaller | u 0 | implies a more highly magnified event), Jung et al. (2022) showed that there is no correlation with event brightness at the time of the anomaly between by-eye vs. AnomalyFinder detections. On the other hand, Hwang et al. (2022) and Zang et al. (2022a) have shown that AnomalyFinder is much better at finding anomalies with smaller ∆ χ 2 .
prep). The follow-up data played decisive roles in these detections and the combined cadences of survey and follow-up data are higher than 30 hr -1 , while the highest cadence of the current KMTNet survey is 8 hr -1 for about 0.4 deg 2 from the overlap of two Γ = 4 hr -1 fields.
However, we note that AnomalyFinder used the KMTNet end-of-year pipeline light curves, for which the photometric quality is not as good as that of TLC re-reductions. For the three follow-up planets, the planetary signals only have ∆ I < 0 . 05 mag. Thus, TLC re-reductions may be needed to recover such weak signals in the KMTNet data, and we cannot rule out the possibility that the desert may also be due to the imperfect KMTNet photometric quality. Each year there are about 20 events with | u 0 | < 0 . 01 observed by KMTNet with Γ ≥ 4 hr -1 . The current KMTNet quasi-automated TLC re-reductions pipeline takes < 1 hr of human effort for each event (H. Yang et al. in prep), so an optimized systematic search for q < 10 -4 planets in the KMTNet highmagnification events can be done very quickly. This search could have important implications for future space-based microlensing projects, because their tentative cadences are similar to or lower than Γ = 4 hr -1 (Penny et al. 2019; Ge et al. 2022; Yan & Zhu 2022). If this search demonstrates that high-magnification events need denser observations to capture the weak planetary signals for lowq planets, one could consider conducting (if feasible) ground-based followup projects for high-magnification events that are discovered by space-based telescopes. We also note that for the 2018 AnomalyFinder planets (Gould et al. 2022; Jung et al. 2022) and 2019 prime-field AnomalyFinder planets (Zang et al. 2022a), which are complete now, only one q > 10 -4 planet, KMT-2019-BLG-1953Lb, has | u anom | < | u anom , limit | . Future analysis of all the 2016-2019 KMTNet should check whether the desert is obvious for more massive planets.
## 6.4. Prime and sub-Prime Fields
In its 2015 commissioning season, KMTNet observed four fields at a cadence of Γ = 6 hr -1 . To support the 2016-2019 Spitzer microlensing campaign (Gould et al. 2013, 2014a, 2015a,b, 2016, 2018) and find more planets, KMTNet monitored a wider area, with a total of (3, 7, 11, 3) fields at cadences of Γ ∼ (4 , 1 , 0 . 4 , 0 . 2) hr -1 . The three fields with the highest cadence are the KMTNet prime fields and the other 21 are the KMTNet sub-prime fields. See Figure 12 of Kim et al. (2018a) for the field placement. As shown in Table 14 and Figure 15, the prime fields played the main role in the detection of q < 10 -4 planets, as predicted by Henderson et al. (2014), and 17 of 27 planets were detected therein. However, the sub-prime fields are also important and six of the ten lowestq planets were discovered therein.
For the six planets with | u 0 | < 0 . 01 , there is a clear bias in cadences, and only one of them was detected from the sub-
prime fields. For the prime and sub-prime fields, the current detection rates are 1.25 and 0.25 per year, respectively. Because ∼ 60% of the KMTNet microlensing events are located in the sub-prime fields, if the sub-prime-field events with | u 0 | < 0 . 01 can had the same cadence as the prime-field events from follow-up observations, each year there would be (1 . 25 × (60% / 40%) -0 . 25) = 1 . 6 more q < 10 -4 planets. Because follow-up observations can have higher cadences and capture the high-magnification planetary signals (e.g., Yang et al. 2022), the yield of a follow-up project can be at least two q < 10 -4 planets per year. The reward is not only enlarging the lowq planetary sample, but also an independent check of the statistical results from AnomalyFinder if the follow-up planets can form a homogeneous statistical sample (Gould et al. 2010). However, this reward requires that the KMTNet alert-finder system should alert new events before they reach the high-magnification regions (e.g., A > 20 ).
We appreciate the anonymous referee for helping to improve the paper. W.Zang, H.Y., S.M., J.Z., and W.Zhu acknowledge support by the National Science Foundation of China (Grant No. 12133005). W.Zang acknowledges the support from the Harvard-Smithsonian Center for Astrophysics through the CfA Fellowship. This research was supported by the Korea Astronomy and Space Science Institute under the R&D program (Project No. 2022-1-830-04) supervised by the Ministry of Science and ICT. 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. The authors acknowledge the Tsinghua Astrophysics High-Performance Computing platform at Tsinghua University for providing computational and data storage resources that have contributed to the research results reported within this paper. Work by J.C.Y. acknowledges support from N.S.F Grant No. AST2108414. Work by C.H. was supported by the grants of National Research Foundation of Korea (2019R1A2C2085965 and 2020R1A4A2002885). Y.S. acknowledges support from BSF Grant No. 2020740. W.Zhu acknowledges the science research grants from the China Manned Space Project with No. CMS-CSST-2021-A11. This research uses data obtained through the Telescope Access Program (TAP), which has been funded by the TAP member institutes. This research is supported by Tsinghua University Initiative Scientific Research Program (Program ID 2019Z07L02017). 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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Table 14 . Information of 2016-2019 KMTNet AnomalyFinder planetary sample with q < 10 -4 solutions
| Event Name | KMTNet Name | log q | s | | u 0 | | Method | ∆ χ 2 | Caustic-Crossing | Anomaly Type | Field |
|--------------|---------------|------------------------------|-------------------|---------------------|-----------|---------|--------------------|----------------|-----------|
| KB161105 1 | KB161105 | - 5 . 194 ± 0 . 248 | 1 . 143 ± 0 . 009 | 0.171 | Discovery | 0.0 | yes | bump | sub-prime |
| | | - 4 . 423 ± 0 . 197 | 1 . 136 ± 0 . 011 | 0.153 | | 2.3 | yes | | |
| | | - 4 . 184 ± 0 . 206 | 1 . 106 ± 0 . 013 | 0.154 | | 2.7 | no | | |
| | | - 5 . 027 ± 0 . 080 | 0 . 888 ± 0 . 007 | 0.148 | | 3.5 | no | | |
| | | - 4 . 069 ± 0 . 182 | 0 . 892 ± 0 . 005 | 0.154 | | 4.4 | yes | | |
| OB160007 2 | KB161991 | - 5 . 168 ± 0 . 131 | 2 . 829 ± 0 . 009 | 1.253 | Discovery | | yes | bump | prime |
| OB191053 3 | KB191504 | - 4 . 885 ± 0 . 035 | 1 . 406 ± 0 . 011 | 0.373 | Discovery | | yes | bump | prime |
| OB190960 4 | KB191591 | - 4 . 830 ± 0 . 041 | 1 . 029 ± 0 . 001 | 0.0061 | Recovery | 0.0 | yes | bump | sub-prime |
| | | - 4 . 896 ± 0 . 024 | 0 . 997 ± 0 . 001 | 0.0060 | | 1.0 | yes | | |
| | | - 4 . 896 ± 0 . 024 | 0 . 996 ± 0 . 001 | 0.0059 | | 1.9 | yes | | |
| | | - 4 . 845 ± 0 . 043 | 1 . 028 ± 0 . 001 | 0.0061 | | 2.1 | yes | | |
| KB180029 5 | KB180029 | - 4 . 737 ± 0 . 047 | 0 . 999 ± 0 . 002 | 0.027 | Recovery | 0.0 | yes | bump | sub-prime |
| | | - 4 . 746 ± 0 . 050 | 1 . 028 ± 0 . 002 | 0.027 | | 0.2 | yes | | |
| | | - 4 . 740 ± 0 . 045 | 0 . 999 ± 0 . 002 | 0.027 | | 2.1 | yes | | |
| | | - 4 . 736 ± 0 . 050 | 1 . 028 ± 0 . 002 | 0.027 | | 2.2 | yes | | |
| KB191806 1 | KB191806 | - 4 . 714 ± 0 . 116 | 1 . 035 ± 0 . 009 | 0.0255 | Discovery | 0.0 | no | dip | sub-prime |
| | | - 4 . 717 ± 0 . 117 | 1 . 034 ± 0 . 009 | 0.0257 | | 0.4 | no | | |
| | | - 4 . 724 ± 0 . 117 | 0 . 938 ± 0 . 007 | 0.0260 | | 0.7 | no | | |
| | | - 4 . 734 ± 0 . 109 | 0 . 938 ± 0 . 007 | 0.0251 | | 1.1 | no | | |
| OB170173 6 | KB171707 | - 4 . 606 ± 0 . 042 | 1 . 540 ± 0 . 031 | 0.867 | Recovery | 0.0 | yes | bump | prime |
| | | - 4 . 195 ± 0 . 068 | 1 . 532 ± 0 . 025 | 0.844 | | 3.5 | yes | | |
| KB171194 1 | KB171194 | - 4 . 582 ± 0 . 058 | 0 . 806 ± 0 . 010 | 0.256 | Discovery | | no | dip | sub-prime |
| | | - 4 . 759 +0 . 698 - 0 . 618 | | | | | | | |
| | | - 0 . 168 | 1 . 01 ± 0 . 05 | 0.014 | | 0.1 | no | | |
| KB190842 8 | KB190842 | - 4 . 389 ± 0 . 031 | 0 . 983 ± 0 . 013 | 0.0066 | Recovery | | no | bump | prime |
| KB190253 9 | KB190253 | - 4 . 387 ± 0 . 076 | 1 . 009 ± 0 . 009 | 0.0559 | Discovery | 0.0 | no | dip | prime |
| | | - 4 . 390 ± 0 . 080 | 0 . 929 ± 0 . 007 | 0.0555 | | 0.3 | no | | |
| OB180977 9 | KB180728 | - 4 . 382 ± 0 . 045 | 0 . 897 ± 0 . 007 | 0.147 | Discovery | | yes | dip | prime |
| KB171003 1 | KB171003 | - 4 . 373 ± 0 . 144 | 0 . 910 ± 0 . 005 | 0.179 | Discovery | 0.0 | no | dip | sub-prime |
| | | - 4 . 260 ± 0 . 152 | 0 . 889 ± 0 . 004 | 0.179 | | 0.2 | no | | |
| OB171806 1 | KB171021 | - 4 . 352 ± 0 . 171 | 0 . 857 ± 0 . 008 | 0.026 | Discovery | 0.0 | yes | bump | sub-prime |
| | | - 4 . 392 ± 0 . 180 | 0 . 861 ± 0 . 007 | 0.025 | | 0.2 | yes | | |
| | | - 4 . 441 ± 0 . 168 | 1 . 181 ± 0 . 011 | 0.026 | | 8.3 | yes | | |
| | | - 4 . 317 ± 0 . 126 | 1 . 190 ± 0 . 012 | 0.027 | | 8.4 | yes | | |
| OB161195 10 | KB160372 | - 4 . 325 ± 0 . 037 | 0 . 989 ± 0 . 004 | 0.0526 | Recovery | 0.0 | no | bump | prime |
| | | - 4 . 318 ± 0 . 038 | 1 . 079 ± 0 . 004 | 0.0526 | | 0.1 | no | | |
| OB170448 2 | KB170090 | - 4 . 296 ± 0 . 149 | 3 . 157 ± 0 . 022 | 1.482 | Discovery | | yes | bump | prime |
| | | - 2 . 705 ± 0 . 045 | 0 . 431 ± 0 . 004 | 1.486 | | 5.8 | yes | | |
| | | - 3 . 969 ± 0 . 086 | 3 . 593 ± 0 . 045 | 1.611 | | 9.7 | yes | | |
| KB191367 1 | KB191367 | - 4 . 303 ± 0 . 118 | 0 . 939 ± 0 . 007 | 0.083 | Discovery | 0.0 | no | dip | sub-prime |
| | | - 4 . 298 ± 0 . 103 | 0 . 976 ± 0 . 007 | 0.082 | | 0.2 | no | | |
| KB170428 1 | KB170428 | - 4 . 295 ± 0 . 072 | 0 . 882 ± 0 . 004 | 0.205 | Discovery | 0.0 | no | dip | prime |
| | | - 4 . 302 ± 0 . 075 | 0 . 915 ± 0 . 005 | 0.205 | | 0.1 | no | | |
| OB171434 11 | KB170016 | - 4 . 242 ± 0 . 011 | 0 . 979 ± 0 . 001 | 0.043 | Recovery | 0.0 | yes | dip | prime |
| | | - 4 . 251 ± 0 . 012 | 0 . 979 ± 0 . 001 | 0.043 | | 4.0 | yes | | |
| OB181185 12 | KB181024 | - 4 . 163 ± 0 . 014 | 0 . 963 ± 0 . 001 | 0.0069 | Recovery | | no | bump | prime |
| OB181126 13 | KB182064 | - 4 . 130 ± 0 . 280 | 0 . 852 ± 0 . 040 | 0.0083 | Discovery | 0.0 | no | dip | prime |
|---------------|------------|-----------------------|---------------------|----------|-------------|-------|------|-------|-----------|
| | | - 4 . 260 ± 0 . 290 | 1 . 154 ± 0 . 052 | 0.0082 | | 2.1 | no | | |
| OB180506 9 | KB180835 | - 4 . 117 ± 0 . 133 | 1 . 059 ± 0 . 021 | 0.0884 | Discovery | 0 | no | dip | prime |
| | | - 4 . 109 ± 0 . 126 | 0 . 861 ± 0 . 018 | 0.0884 | | 0.4 | no | | |
| KB181025 14 | KB181025 | - 4 . 081 ± 0 . 141 | 0 . 937 ± 0 . 021 | 0.0071 | Recovery | 0 | no | bump | prime |
| | | - 3 . 789 ± 0 . 133 | 0 . 883 ± 0 . 025 | 0.0086 | | 8.4 | no | | |
| OB171691 15 | KB170752 | - 4 . 013 ± 0 . 152 | 1 . 003 ± 0 . 014 | 0.0495 | Recovery | 0 | yes | bump | sub-prime |
| | | - 4 . 150 ± 0 . 141 | 1 . 058 ± 0 . 011 | 0.0483 | | 0.4 | yes | | |
| OB180532 16 | KB181161 | - 4 . 011 ± 0 . 053 | 1 . 013 ± 0 . 001 | 0.0082 | Recovery | 0 | yes | dip | prime |
| | | - 4 . 033 ± 0 . 047 | 1 . 011 ± 0 . 001 | 0.0071 | | 2 | yes | | |
| | | - 3 . 926 ± 0 . 049 | 1 . 013 ± 0 . 001 | 0.0089 | | 4.6 | yes | | |
| | | - 4 . 016 ± 0 . 076 | 1 . 011 ± 0 . 001 | 0.0074 | | 5.4 | yes | | |
| KB160625 2 | KB160625 | - 3 . 628 ± 0 . 226 | 0 . 741 ± 0 . 009 | 0.073 | Discovery | 0 | yes | bump | prime |
| | | - 4 . 138 ± 0 . 159 | 1 . 367 ± 0 . 018 | 0.075 | | 1 | yes | | |
| | | - 3 . 746 ± 0 . 291 | 0 . 741 ± 0 . 009 | 0.072 | | 1 | yes | | |
| | | - 4 . 499 ± 0 . 266 | 1 . 358 ± 0 . 015 | 0.076 | | 3.3 | yes | | |
| KB160212 17 | KB160212 | - 1 . 434 ± 0 . 072 | 0 . 829 ± 0 . 007 | 0.328 | Recovery | 0 | yes | bump | prime |
| | | - 4 . 310 ± 0 . 070 | 1 . 427 ± 0 . 014 | 0.615 | | 6.6 | yes | | |
| | | - 4 . 315 ± 0 . 099 | 1 . 434 ± 0 . 012 | 0.619 | | 8 | yes | | |
| | | - 4 . 082 ± 0 . 080 | 1 . 430 ± 0 . 015 | 0.617 | | 8.7 | yes | | |
NOTE: For each planet, we only consider the models that have ∆ χ 2 < 10 compared to the best-fit model. 'Discovery' represents that the planet was discovered using AnomlyFinder, and 'Recovery' means that the planet was first discovered from by-eye searches and then recovered by AnomlyFinder.
Reference: 1. This work; 2. in prep; 3. Zang et al. (2021b); 4. Yee et al. (2021); 5. Gould et al. (2020), Zhang et al. in prep;
6. Hwang et al. (2018a); 7. Han et al. (2022a); 8. Jung et al. (2020); 9. Hwang et al. (2022); 10. Shvartzvald et al. (2017), Bond et al. (2017), Zhang et al. in prep; 11. Udalski et al. (2018); 12. Kondo et al. (2021); 13. Gould et al. (2022); 14. Han et al. (2021); 15. Han et al. (2022b); 16. Ryu et al. (2020); 17. Hwang et al. (2018b).