# Systematic KMTNet Planetary Anomaly Search, Paper II: Six New $q<2\times
10^{-4}$ Mass-ratio Planets

Kyu-Ha Hwang1, Weicheng Zang2, Andrew Gould3,4, Andrzej Udalski5, Ian A.
Bond6, Hongjing Yang2, Shude Mao2,7
(Lead Authors)
Michael D. Albrow8, Sun-Ju Chung1,9, Cheongho Han10, Youn Kil Jung1, Yoon-Hyun
Ryu1, In-Gu Shin1, Yossi Shvartzvald11, Jennifer C. Yee12, Sang-Mok Cha1,13,
Dong-Jin Kim1, Hyoun-Woo Kim1, Seung-Lee Kim1,9, Chung-Uk Lee1,9, Dong-Joo
Lee1, Yongseok Lee1,13, Byeong-Gon Park1,9, Richard W. Pogge4
(KMTNet Collaboration)
Przemek Mróz14, Radek Poleski5,4, Jan Skowron5, Michał K. Szymański5, Igor
Soszyński5, Paweł Pietrukowicz5, Szymon Kozłowski5, Krzysztof Ulaczyk15,
Krzysztof A. Rybicki5, Patryk Iwanek5, Marcin Wrona5, Mariusz Gromadzki5
(OGLE Collaboration)
Fumio Abe16, Richard Barry17, David P. Bennett17,18, Aparna Bhattacharya17,18,
Hirosame Fujii16 Akihiko Fukui19,20, Yuki Hirao21, Yoshitaka Itow16, Rintaro
Kirikawa21, Iona Kondo21, Naoki Koshimoto22,23, Brandon Munford24, Yutaka
Matsubara16, Shota Miyazaki21, Yasushi Muraki16, Greg Olmschenk17, Clément
Ranc25, Nicholas J. Rattenbury24, Yuki K. Satoh21, Hikaru Shoji21, Stela
Ishitani Silva26,17, Takahiro Sumi21, Daisuke Suzuki27, Paul J. Tristram28,
Atsunori Yonehara29
(The MOA Collaboration)
Xiangyu Zhang2, Wei Zhu2, Matthew T. Penny30, Pascal Fouqué31,32
(The Tsinghua & CFHT Microlensing Group)
1Korea Astronomy and Space Science Institute, Daejon 34055, Republic of Korea
2 Department of Astronomy and Tsinghua Centre for Astrophysics, Tsinghua
University, Beijing 100084, China  3Max-Planck-Institute for Astronomy,
Königstuhl 17, 69117 Heidelberg, Germany  4Department of Astronomy, Ohio State
University, 140 W. 18th Ave., Columbus, OH 43210, USA  5Astronomical
Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
6Institute of Natural and Mathematical Science, Massey University, Auckland
0745, New Zealand  7National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China  8University of Canterbury, Department of
Physics and Astronomy, Private Bag 4800, Christchurch 8020, New Zealand
9Korea University of Science and Technology, Korea, (UST), 217 Gajeong-ro,
Yuseong-gu, Daejeon, 34113, Republic of Korea  10Department of Physics,
Chungbuk National University, Cheongju 28644, Republic of Korea  11Department
of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot
76100, Israel  12 Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
St., Cambridge, MA 02138, USA  13School of Space Research, Kyung Hee
University, Yongin, Kyeonggi 17104, Republic of Korea  14Division of Physics,
Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA
91125, USA  15Department of Physics, University of Warwick, Gibbet Hill Road,
Coventry, CV4 7AL, UK  16Institute for Space-Earth Environmental Research,
Nagoya University, Nagoya 464-8601, Japan  17Code 667, NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA  18Department of Astronomy, University
of Maryland, College Park, MD 20742, USA  19Department of Earth and Planetary
Science, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-0033, Japan  20Instituto de Astrofísica de Canarias, Vía
Láctea s/n, E-38205 La Laguna, Tenerife, Spain  21Department of Earth and
Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka
560-0043, Japan  22Department of Astronomy, Graduate School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  23National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
24Department of Physics, University of Auckland, Private Bag 92019, Auckland,
New Zealand  25Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique
de Paris, 98 bis bd Arago, 75014 Paris, France  26Department of Physics, The
Catholic University of America, Washington, DC 20064, USA  27Institute of
Space and Astronautical Science, Japan Aerospace Exploration Agency, Kanagawa
252-5210, Japan  28University of Canterbury Mt. John Observatory, P.O. Box 56,
Lake Tekapo 8770, New Zealand  29Department of Physics, Faculty of Science,
Kyoto Sangyo University, Kyoto 603-8555, Japan  30Department of Physics and
Astronomy, Louisiana State University, Baton Rouge, LA 70803 USA  31CFHT
Corporation, 65-1238 Mamalahoa Hwy, Kamuela, Hawaii 96743, USA  32Université
de Toulouse, UPS-OMP, IRAP, Toulouse, France

###### Abstract

We apply the automated AnomalyFinder algorithm of Paper I (Zang et al., 2021b)
to 2018-2019 light curves from the $\simeq 13\,{\rm deg}^{2}$ covered by the
six KMTNet prime fields, with cadences $\Gamma\geq 2\,{\rm hr}^{-1}$. We find
a total of 11 planets with mass ratios $q<2\times 10^{-4}$, including six
newly discovered planets, one planet that was reported in Paper I, and
recovery of four previously discovered planets. One of the new planets,
OGLE-2018-BLG-0977Lb, is in a planetary-caustic event, while the other five
(OGLE-2018-BLG-0506Lb, OGLE-2018-BLG-0516Lb, OGLE-2019-BLG-1492Lb,
KMT-2019-BLG-0253, and KMT-2019-BLG-0953) are revealed by a “dip” in the light
curve as the source crosses the host-planet axis on the opposite side of the
planet. These subtle signals were missed in previous by-eye searches. The
planet-host separations (scaled to the Einstein radius), $s$, and planet-host
mass ratios, $q$, are, respectively, $(s,q\times 10^{5})=(0.88,4.1)$,
$(0.96\pm 0.10,8.3)$, $(0.94\pm 0.07,13)$, $(0.97\pm 0.07,18)$, $(0.97\pm
0.04,4.1)$, and $(0.74,18)$, where the “$\pm$” indicates a discrete
degeneracy. The 11 planets are spread out over the range $-5<\log q<-3.7$.
Together with the two planets previously reported with $q\sim 10^{-5}$ from
the 2018-2019 non-prime KMT fields, this result suggests that planets toward
the bottom of this mass-ratio range may be more common than previously
believed.

gravitational lensing: micro

##  1 Introduction

By the end of 2005, four microlensing planets had been discovered, of which
two were in the low planet-host mass-ratio ($q<10^{-4}$) regime (Beaulieu et
al., 2006; Gould et al., 2006). Twelve years later, there were only seven such
discoveries (Udalski et al., 2018) out of a total of more than 60. Moreover,
none of these had mass ratios below $q_{\rm thresh,2018.0}=4.3\times 10^{-5}$.
The absence of low-$q$ planets in a more restricted, but homogeneously
selected sample, led Suzuki et al. (2016) to argue for a break in the mass-
ratio function at about $q_{\rm break}=17\times 10^{-5}$, albeit with a large
error bar. That is, they confirmed the earlier finding by Sumi et al. (2010)
that above the break, the mass-ratio function was a falling power law toward
higher mass ratios. However, below the break, they found that this function
was either flat, or more likely falling sharply toward lower mass ratios.
Based on a larger, though inhomogeneous sample, Jung et al. (2019a) found the
break to be a factor three lower, $q_{\rm break}=5.6\times 10^{-5}$, but still
consistent with Suzuki et al. (2016) within the latter’s error bar.

Since the Udalski et al. (2018) study (and prior to this paper), seven
additional $q\leq 10^{-4}$ planets have been discovered. Five of these seven
have mass ratios below the pre-2018 threshold, $q_{\rm
thresh,2018.0}=4.3\times 10^{-5}$, and these five were discovered in 2018–2020
data. Although these 14 planets were not homogeneously selected, it is
unlikely that this sample is strongly affected by publication bias. That is,
$q<10^{-4}$ planets are relatively rare and are very likely to be published
soon after discovery. Only one of the seven recent low-$q$ planets was
submitted for publication substantially after discovery, OGLE-2015-BLG-1670
(Ranc et al., 2019).

The very different discovery patterns before and after 2018.0 strongly suggest
a systematic difference in the discovery process beginning in 2018. See Table
1. One possibility is the upgrade in the Korea Microlensing Telescope Network
(KMTNet, Kim et al. 2016) online photometry in that year. KMT data played a
major or decisive role in all five 2018–2020 discoveries, and many of these
planets would have been missed with substantially inferior photometry. For
example, KMT-2018-BLG-0029 (Gould et al., 2020) had been the focus of
considerable interest during the season because it was targeted for Spitzer
observations, yet the few-hour planetary signal at the peak was not noticed
until the end-of-year re-reductions, when the new pipeline results were
manually reviewed.

Regardless of the exact cause, the changing pattern of discoveries before and
after 2018.0 calls for re-evaluation of the conclusions regarding the behavior
of the mass-ratio function in the $\log q\lesssim-4$ regime. Ultimately, this
can only be done on the basis of homogeneous selection of planets, which was a
key characteristic of the original Suzuki et al. (2016) study.

Table 1: Summary of Microlensing Planets $q<10^{-4}$ Years | $q<4.3\times 10^{-5}$ | $q>4.3\times 10^{-5}$ | Total
---|---|---|---
2003-2017 | 0 | 8 | 8
2018-2020 | 5 | 1 | 6
All | 5 | 9 | 14

Yee et al. (2021) outlined one such approach: make tender-loving care (TLC)
reductions of all KMT events meeting a magnification threshold, e.g., $A_{\rm
max}>20$, and subject these to automated planet searches. This would require
improved efficiency of the TLC pipeline to re-reduce so many events.

Zang et al. (2021a) outlined a second approach: intensive follow-up
observations of high-magnification events in low-cadence $\Gamma\leq 0.4\,{\rm
hr}^{-1}$ KMT fields. This would be similar to the approach formerly carried
out by the Microlensing Follow Up Network ($\mu$FUN, Gould et al. 2010), but
with much more efficient target selection due to continuous KMT coverage.

Zang et al. (2021b) (Paper I) initiated a third approach: subject the
residuals from single-lens single-source (1L1S) fits to the end-of-year-
pipeline light curves to a search for “bumps” or “dips” based on a modified
version of the KMT EventFinder algorithm (Kim et al., 2018a; Gould, 1996). At
that time, the Zang et al. (2021b) approach could only be applied to 2019 data
because the 2016–2017 online reductions were not of sufficiently high quality
and the 2018 photometry files lacked some auxiliary data needed for this
method. However, the 2018 files have already been updated and work on the
2016-2017 files is in progress. Note that all three methods are complementary
and all could be used simultaneously.

As reported by Zang et al. (2021b), their method recovered the planets in
KMT-2019-BLG-0842 (Jung et al., 2020b) and OGLE-2019-BLG-0960 (Yee et al.,
2021), i.e., the two previous low-$q$ planets from 2019. And they also
reported the discovery of the lowest mass-ratio planetary-caustic111 There
were three planets with comparable mass ratios KMT-2018-BLG-0029 (Gould et
al., 2020), OGLE-2019-BLG-0960 (Yee et al., 2021), and KMT-2020-BLG-0414 (Zang
et al., 2021a). However, all three were detected via the resonant channel,
which, together with the “near-resonant” channel, account for about 3/4 or all
published microlensing planets. event to date: OGLE-2019-BLG-1053. Hence,
there are good prospects for establishing a large homogeneous sample of
$q<10^{-4}$ planets by applying this technique to several years of KMT data.
Here, we begin a program of detailed analysis and publication of all
$q<2\times 10^{-4}$ planets found from application of the Zang et al. (2021b)
method. This will enable a systematic study of the planet mass-ratio function
in the regime of the hypothesized break.

We briefly outline the steps leading up to these publications and the steps
that are expected to follow. As discussed by Zang et al. (2021b), their
automated AnomalyFinder selects potential anomalies based on purely objective
criteria applied to KMT light curves. The criteria used here are identical
except that we extend the effective-timescale range down to $t_{\rm eff}\geq
0.05\,$day. Three operators (W.Zang, H. Yang and W. Zhu) then manually and
independently select genuine anomalies (or anomalies that could be genuine)
based on machine-generated displays. If there are counterparts to the KMT
event on the OGLE and/or MOA web pages, then these light curves are checked to
determine whether they contradict the apparent KMT anomaly. If so, the anomaly
is rejected. If there are no OGLE or MOA data that overlap the anomaly, its
reality is checked by visual inspection of the images. Ultimately, we expect
to publish all planets (defined as $q<0.03$) derived from this search, which
will eventually be extended to all KMTNet fields and at least the 2016–2019
seasons.

The present paper is based on a search of the KMT database that is restricted
to 2018–2019 and to the three pairs of overlapping KMT fields BLG01/BLG41,
BLG02/BLG42, and BLG03/BLG43. Each of these fields has a nominal cadence
$\Gamma=2\,{\rm hr}^{-1}$, so the combined cadence is usually $\Gamma=4\,{\rm
hr}^{-1}$. The first restriction is determined by the current state of the
photometry files (see above). The second is based on maintaining a stable work
flow in what will be a very big project.

In particular, each event that is selected for publication must be subjected
to detailed analysis, so their publication in groups does not substantially
reduce either the amount of work required or the length of the descriptions.

The long-term goal is to measure the mass-ratio function for the full
planetary range $q<0.03$. To do so, we must measure the sensitivity of the
selection process by injecting planets into KMT light curves. Then two
questions will be asked. First, is the anomaly selected by the machine
criteria? Second, would this machine-selected candidate ultimately be
published? The answer to the first question is unambiguous. The second
question is more difficult because it involves several steps that cannot be
mimicked for each simulated event, such as re-reduction of the data and
systematic searches for alternate models. This analysis is currently in
progress (Y.K. Jung et al., in prep).

Here, we present six planets, all of which ultimately satisfy $q<2\times
10^{-4}$. To construct the present sample, we manually review all anomalous
light curves and tentatively classify them by eye into one of five categories
“planet”, “planet/binary”, “binary/planet”, “binary”, and “finite source”.
Among the “planets”, we do preliminary 2L1S modeling of all events that could
plausibly be $q<10^{-3}$, using end-of-season pipeline data. Here, $n$L$m$S
means $n$ lenses and $m$ sources. One of these, OGLE-2018-BLG-0383 (with
$q=2.1\times 10^{-4}$), was separated out at an early stage as part of an
investigation of wide-orbit planets (Wang et al., 2021). For those with
$q<3\times 10^{-4}$ among the remainder, the KMT data are re-reduced with TLC
and then fully analyzed.

All of the other “planets” as well as all of the events in the other four
categories are then cross-matched against events that are published or are
known to us to be in preparation. We then cross match against the very large
personal modeling archive of co-author C. Han. Combined, these cross matches
cover about 75% of all the AnomalyFinder detections. They give us not only
good estimates of $q$ for these events, but also permit us to check the
robustness of our by-eye classifications. Based on this investigation, it is
likely that all of the $q<2\times 10^{-4}$ planets in the AnomalyFinder sample
are contained in the present work. However, we cannot be certain until we
complete our systematic investigation of all the anomalous events, which is
currently underway.

##  2 Observations

As outlined in Section 1, all six planets described in this paper were
identified in searches of KMT events from its “prime fields”, with nominal
cadences $\Gamma\geq 2\,{\rm hr}^{-1}$, either by the AlertFinder (Kim et al.,
2018b) or the post-season EventFinder (Kim et al., 2018a). KMTNet observes
from three identical 1.6m telescopes, each equipped with a $(2^{\circ}\times
2^{\circ})$ camera at CTIO in Chile (KMTC), SAAO in South Africa (KMTS), and
SSO in Australia (KMTA). KMTNet observes primarily in the $I$ band, but in
2018 and 2019, every tenth such observation was complemented by one in the $V$
band.

Five of the six events were independently discovered by the Optical
Gravitational Lensing Experiment (OGLE, Udalski et al. 1994; Udalski 2003),
using its 1.3m telescope with $1.4\,{\rm deg}^{2}$ camera at Las Campanas
Observatory in Chile. OGLE also observed primarily in the $I$ band, with some
$V$ exposures as well.

Two of the six events were independently discovered by the Microlensing
Observations in Astrophysics (MOA, Bond et al. 2004) collaboration, using
their 1.8m telescope with $2.2\,{\rm deg}^{2}$ camera at Mt. John in New
Zealand, using their $R_{\rm MOA}$ filter, which is roughly the sum of the
Cousins $R$ and $I$ bands.

Table 2 gives the catalog names in order of discovery for each event in order
to permit information to be easily traced. However, after that, we generally
use the first-discovery name. Table 2 also presents the observational cadences
$\Gamma$, as well as the discovery dates and the sky locations.

In addition, OGLE observed the location of KMT-2019-BLG-0953 and MOA observed
the location of the OGLE-2019-BLG-1492 field, although they did not issue
alerts for these events. We therefore also incorporate these OGLE and MOA data
into the analysis.

Although all six events were alerted in real time, to the best of our
knowledge, there were follow-up observations for only one: OGLE-2018-BLG-0977.
These were from the 3.6m Canada-France-Hawaii Telescope (CFHT) and were at a
cadence of about one per night.

The data were reduced using variants of difference image analysis (DIA,Tomaney
& Crotts 1996; Alard & Lupton 1998), as realized by Albrow et al. (2009)
(KMT), Woźniak (2000) (OGLE), Bond et al. (2001) (MOA), and Zang et al. (2018)
(CFHT).

##  3 Light-curve Analyses

###  3.1 Preamble

There are several features of the light-curve analyses that are common to all
six events, which we present in this preamble.

All six events exhibit short perturbations on otherwise standard Paczyński
(1986) 1L1S light curves, which are characterized by three parameters (in
addition to two flux parameters for each observatory). These are
$(t_{0},u_{0},t_{\rm E})$, i.e., the time of lens-source closest approach, the
impact parameter (in units of the Einstein radius, $\theta_{\rm E}$), and the
Einstein timescale.

In each case, the anomaly can be localized fairly precisely by eye at $t_{\rm
anom}$, which then yields the offset from the peak, $\tau_{\rm anom}$ and the
offset from the host, $u_{\rm anom}$, both in units of $\theta_{\rm E}$, as
well as the angle $\alpha$ of the source-lens motion relative to the binary
axis (Gould & Loeb, 1992),

$\tau_{\rm anom}={\Delta t_{\rm anom}\over t_{\rm E}}\equiv{t_{\rm
anom}-t_{0}\over t_{\rm E}};\qquad u_{\rm anom}^{2}=\tau_{\rm
anom}^{2}+u_{0}^{2};\qquad\tan\alpha=-{u_{0}\over\tau_{\rm anom}}.$ (1)

Note that this also implies that $u_{\rm anom}=|u_{0}/\sin\alpha|$.

We define the quantities $s^{\dagger}_{\pm}$ by

$s^{\dagger}_{\pm}={\sqrt{u_{\rm anom}^{2}+4}\pm u_{\rm anom}\over 2}.$ (2)

Note that $s^{\dagger}_{-}=1/s^{\dagger}_{+}$ and $u_{\rm
anom}=s^{\dagger}_{+}-s^{\dagger}_{-}$. If the source crosses the minor-image
(triangular) planetary caustics, then we expect $s\simeq s^{\dagger}_{-}$, and
if it crosses the major image (quadrilateral) planetary caustic, then we
expect $s\simeq s^{\dagger}_{+}$. Here, $s$ is the projected planet-host
separation scaled to $\theta_{\rm E}$. However, only one of the six events
analyzed here (OGLE-2018-BLG-0977) has such caustic-crossing features. For the
rest, we may expect that the event may be subject to the inner/outer
degeneracy identified by Gaudi & Gould (1997). That is, for minor-image
perturbations, which are generally characterized by a “dip”, roughly the same
anomaly can in principle be produced by the source passing “inside” the
planetary caustics (closer to the central caustic) or “outside”. Then we may
find two solutions, which obey,

$s^{\dagger}_{-}={s_{\rm inner}+s_{\rm outer}\over 2}\qquad{\rm(minor\
image)}.$ (3)

For major-image perturbations, which are generally characterized by a “bump”,
a similar formula applies for $s^{\dagger}_{+}$. However, in the present
paper, all six planets give rise to minor-image perturbations (“dips”).

Gould & Loeb (1992) also show how to make by-eye estimates of $q$ for events
in which the source crosses a planetary caustic. Here, we present an extension
of their method to estimate $q$ for the case of non-caustic-crossing minor-
image perturbations caused by low mass-ratio planets.

We first focus on caustic-crossing events. In this case, there is a de-
magnification trough between the two flanking caustics, with separation (using
notation from Han 2006),

$\Delta u\simeq 2\eta_{c,1}-\Delta\eta_{c}=4\sqrt{q(s^{-1}-s)\over
s}=4\sqrt{q{u_{\rm anom}\over s}}.$ (4)

Therefore, the duration of the dip in the light curve between the two caustics
is $\Delta t_{\rm dip}=(\Delta u/\sin\alpha)t_{\rm E}$, implying

$q=\biggl{(}{\Delta t_{\rm dip}\over 4\,t_{\rm
E}}\biggr{)}^{2}{s\sin^{2}\alpha\over u_{\rm anom}}=\biggl{(}{\Delta t_{\rm
dip}\over 4\,t_{\rm E}}\biggr{)}^{2}{s\over|u_{0}|}|\sin^{3}\alpha|.$ (5)

For the case of source trajectories that miss the caustic, but still
experience a dip (by which the planet is detected), we apply the same formula,
but with $s\rightarrow s^{\dagger}_{-}$, i.e., implicitly assuming that the
width of the dip is approximately the same as the separation between the
caustics. As shown by Figure 1 of Chung & Lee (2011), this approximation holds
best for low mass-ratio lenses $q\lesssim 2\times 10^{-4}$, which is the main
focus of interest here. We do not expect this mass-ratio estimate to be
extremely accurate simply because the dip does not itself have precisely
defined edges (except in the region of the trough that lies between the two
triangular caustics). Moreover, $\Delta t_{\rm dip}$ enters quadratically into
Equation (5). Thus, by contrast with Equations (1) and (2), which should be
quite precise, we expect that Equation (5) should be accurate at the factor
$\sim 2$ level.

Nevertheless, Equation (5) can be quite powerful. Note in particular that for
events in which the anomaly occurs at magnifications of at least a few,
$A_{\rm anom}\gtrsim 5$ and for which the blending is small compared to the
magnified flux, we have $\sin\alpha=u_{0}/u_{\rm anom}\simeq A_{\rm
anom}/A_{\rm max}\simeq 10^{-0.4\Delta I}$, where $\Delta I$ is the magnitude
offset between the peak and the anomaly. For such cases
$s^{\dagger}_{-}\rightarrow 1-0.5/A_{\rm anom}\simeq 1$, and so Equation (5)
can be approximated by

$q\rightarrow{10^{-1.2\Delta I}\over u_{0}}\biggl{(}{\Delta t_{\rm dip}\over
4\,t_{\rm E}}\biggr{)}^{2};\qquad{\rm(limiting\ form)}.$ (6)

We generally search for seven-parameter “static” (i.e., without lens orbital
motion or microlens parallax effects) 2L1S solutions on an $(s,q)$ grid,
characterized by $(t_{0},u_{0},t_{\rm E},s,q,\alpha,\rho)$, where $\rho$ is
the angular source radius scaled to $\theta_{\rm E}$, i.e.,
$\rho=\theta_{*}/\theta_{\rm E}$. In the grid search, $(s,q)$ are held fixed
at the grid values while $(t_{0},u_{0},t_{\rm E},\rho)$ are allowed to vary in
a Markov chain Monte Carlo (MCMC). We use the advanced contour integration
code VBBinaryLensing (Bozza, 2010; Bozza et al., 2018) to calculate the
magnification of the 2L1S model, and we identify the best-fit solution via the
Markov chain Monte Carlo (MCMC) method emcee (Foreman-Mackey et al., 2013).
The three Paczyński parameters $(t_{0},u_{0},t_{\rm E})$ are seeded at their
1L1S values, and $\rho$ is seeded using the method of Gaudi et al. (2002) at a
value near $\rho=\theta_{*,\rm est}/\mu_{\rm typ}t_{\rm E}$, where $\mu_{\rm
typ}\equiv 6\,{\rm mas}\,{\rm yr}^{-1}$ is a typical value of the lens-source
relative proper motions, $\mu_{\rm rel}$, and $\theta_{*,\rm est}$ is
estimated based on the value of $I_{s}$ from the 1L1S fit and the KMT-website
extinction. For example, $\theta_{*,\rm est}=0.6\,\mu{\rm as}$ for
$I_{s}-A_{I}=18.65$ (Sun-like star) and $\theta_{*,\rm est}=6\,\mu{\rm as}$
for $I_{s}-A_{I}=14.5$ (clump-like star). The final parameter, $\alpha$ is
seeded at a grid of values around the unit circle, and it is either held fixed
or allowed to vary with the chain, depending on circumstances.

After finding one or more local minima on the $(s,q)$ plane, each local is
further refined by allowing all seven parameters to vary in an MCMC.

If it is suspected that microlens parallax effects can be detected or
constrained, then four additional parameters are initially added,
${\mbox{\boldmath$\pi$}}_{\rm E}=(\pi_{{\rm E},N},\pi_{{\rm E},E})$ and
${\mbox{\boldmath$\gamma$}}=((ds/dt)/s,d\alpha/dt)$. Here,
${\mbox{\boldmath$\pi$}}_{\rm E}$ parameterizes the effects of Earth’s orbital
motion on the light curve (Gould, 1992, 2000, 2004),

${\mbox{\boldmath$\pi$}}_{\rm E}={\pi_{\rm rel}\over\theta_{\rm
E}}\,{{\mbox{\boldmath$\mu$}}_{\rm rel}\over\mu_{\rm rel}};\qquad\theta_{\rm
E}=\sqrt{\kappa M\pi_{\rm rel}};\qquad\kappa\equiv{4G\over c^{2}{\rm
au}}\simeq 8.14{{\rm mas}\over M_{\odot}},$ (7)

where $M$ is the lens mass and $(\pi_{\rm rel},{\mbox{\boldmath$\mu$}}_{\rm
rel})$ are the lens-source relative (parallax, proper motion), while
$(ds/dt,d\alpha/dt)$ are the first derivatives on lens orbital motion. The
four parameters must be introduced together, at least initially, because the
orbital motion of the lens can mimic the effects on the light-curve of orbital
motion of Earth (Batista et al., 2011; Skowron et al., 2011). Lens orbital
motion can be poorly constrained by the light curve, so that we generally
impose a constraint $\beta<0.8$ where $\beta$ is the absolute value of the
ratio of transverse kinetic to potential energy (An et al., 2002; Dong et al.,
2009),

$\beta=\bigg{|}{\rm KE\over PE}\bigg{|}_{\perp}={\kappa M_{\odot}{\rm
yr}^{2}\over 8\pi^{2}}{\pi_{\rm E}\over\theta_{\rm
E}}\gamma^{2}\biggl{(}{s\over\pi_{\rm E}+\pi_{S}/\theta_{\rm
E}}\biggr{)}^{3},$ (8)

and where $\pi_{S}$ is the source parallax. Strictly speaking $\beta$ is only
physically constrained to be $\beta<1$, but combinations of orbits and viewing
angles that would generate $\beta>0.8$ are extremely rare.

When (as usual) the parallax vector ${\mbox{\boldmath$\pi$}}_{\rm E}$ is
evaluated in the geocentric frame, it often has elongated, nearly straight
contours whose minor axis is aligned with the projected position of the Sun at
$t_{0}$, i.e., the (negative of the) direction of the Sun’s instantaneous
apparent direction of acceleration on the plane of the sky. This is because
$\pi_{{\rm E},\parallel}$, the component of ${\mbox{\boldmath$\pi$}}_{\rm E}$
that is aligned to this acceleration, induces an asymmetry in the light curve,
which leads to strong constraints, whereas $\pi_{{\rm E},\perp}$ induces more
subtle, symmetric distortions (Gould et al., 1994; Smith et al., 2003; Gould,
2004). Because the right-handed coordinate system $(\pi_{{\rm
E},\parallel},\pi_{{\rm E},\perp})$ is rotated with respect to the (North,
East) equatorial coordinates, the tightness of the error ellipse is often not
fully reflected when $(\pi_{{\rm E},N},\pi_{{\rm E},E})$ and their errors are
tabulated in papers. In this paper, we therefore also tabulate $(\pi_{{\rm
E},\parallel},\pi_{{\rm E},\perp})$ and their errors in addition to those of
$(\pi_{{\rm E},N},\pi_{{\rm E},E})$. For this purpose, we define $\psi$ as the
angle of the minor axis of the parallax ellipse (North through East) rather
than the angle of the projected position of the Sun. Of course, there are two
such directions, separated by $180^{\circ}$. We choose the one closer to the
Sun’s apparent acceleration on the sky. In all four events for which the
parallax is measurable in the current work, $\psi$ is the same as the Sun’s
acceleration within one or two degrees. See, e.g., Figure 3 of Park et al.
(2004) for an explicit example of the coordinate system.

Even when these so-called “one-dimensional (1-D) parallax” measurements fail
to constrain the magnitude $|{\mbox{\boldmath$\pi$}}_{\rm E}|=\pi_{\rm E}$,
which is what directly enters the mass and distance estimates, they can
interact with the Bayesian priors from a Galactic model to constrain the final
estimates of these quantities substantially better than
${\mbox{\boldmath$\pi$}}_{\rm E}$ or the Galactic priors do separately.

More generally, ${\mbox{\boldmath$\pi$}}_{\rm E}$ measurements can provide
valuable information even when they are consistent with zero. As discussed in
the Appendix to Han et al. (2016), no “evidence” of parallax is required to
introduce the two ${\mbox{\boldmath$\pi$}}_{\rm E}$ parameters because it is
known a priori that $\pi_{\rm E}$ is strictly positive. Hence, measurements
that constrain it to be small provide additional information, even if they are
formally consistent with zero.

Another important application of 1-D parallax measurements is that they can
yield a very precise mass estimate when the source and lens are separately
resolved many years after the event (Ghosh et al., 2004; Gould, 2014). That
is, such imaging generally yields a very precise measurement of
${\mbox{\boldmath$\mu$}}_{\rm rel}$, which has the same direction as
${\mbox{\boldmath$\pi$}}_{\rm E}$. Thus, if $\pi_{{\rm E},\parallel}$ is well
constrained (and assuming that the ${\mbox{\boldmath$\pi$}}_{\rm E}$ error
ellipse is not by chance closely aligned to ${\mbox{\boldmath$\mu$}}_{\rm
rel}$), both $\pi_{\rm E}$ and $\theta_{\rm E}=\mu_{\rm rel}t_{\rm E}$ can be
precisely determined from such imaging. See Dong et al. (2009) and Bennett et
al. (2020) for a case in which the original 1-D parallax measurement was
turned into lens mass and distance measurements by successively better
determinations of ${\mbox{\boldmath$\mu$}}_{\rm rel}$.

###  3.2 KMT-2019-BLG-0253

In the original automated search for anomalies (Zang et al., 2021b), the
deviation in KMT-019-BLG-0253 from the 1L1S model (Figure 1) was entirely due
to KMTC data forming a 0.23 day, roughly linear rising track that lay below
the model, centered on HJD${}^{\prime}\equiv$ HJD - 2450000 = 8588.81, i.e.,
$\Delta t_{\rm anom}=-1.8$ days before the peak. The method aggressively
removes bad-, or even questionable-seeing and high-background points, which
very frequently produce such “features”. Nevertheless, because the algorithm
examines ${\cal O}(10^{2})$ nights of data from three observatories on ${\cal
O}(10^{3})$ events, there are numerous such single-observatory deviations even
after this aggressive automated culling of the data. Hence, they must be
treated with caution. However, after the TLC re-reductions, the previously
excluded KMTS points on the same night could be re-included, and these
confirmed the dip, whose duration can be estimated $\Delta t_{\rm
dip}=0.6\,$days. Moreover, there are five OGLE points on this night, which
generally track the KMTC deviation, including three at the beginning of the
night when the dip is pronounced. Finally, dense MOA coverage confirms the
subtle “ridge” feature immediately following the dip. Thus, there is no
question that the deviation is of astrophysical origin. See Figure 1.

####  3.2.1 Heuristic Analysis

The 1L1S model yields parameters $t_{\rm E}\simeq 60\,$days and $u_{0}\simeq
0.06$. The midpoint of the dip is at $\Delta t_{\rm anom}=-2.0\,$days, and
$\Delta t_{\rm dip}=0.6\,$days. Applying the heuristic formalism of Section
3.1, these yield,

$\alpha=61^{\circ};\qquad s\simeq s_{-}^{\dagger}\pm\Delta s;\qquad
s_{-}^{\dagger}=0.97;\qquad q\sim 7\times 10^{-5},$ (9)

where $\pm\Delta s$, which is induced by the offset of the source trajectory
from the caustic, cannot be evaluated from the general appearance of the light
curve.

####  3.2.2 Static Analysis

Notwithstanding this seemingly secure reasoning, we conduct a systematic
search for 2L1S solutions, as described in Section 3.1. As expected, we find
two solutions, which we then seed into two MCMCs in which all parameters are
allowed to vary. The resulting parameters are shown in Table 3. As can be seen
from Figure 1. the two model light curves are virtually indistinguishable. A
classic case of this inner/outer degeneracy for the minor image perturbation
is presented by Calchi Novati et al. (2019). However, in contrast to that
case, the “outer” solution does not have a planetary caustic (outside of which
the source would pass). Rather the planetary and central caustics have merged
into a single resonant caustic (e.g., Gaudi 2012), and the “outer” solution
passes outside the “planetary wing” of this resonant caustic. See Figure 2.
Within the context of the heuristic treatment of Section 3.2.1, we find from
Table 3 that the two solutions can be represented as
$(s^{\dagger}_{-}\pm\Delta s)=(0.969\pm 0.040)$, while $\alpha=59^{\circ}$.
Both are in good agreement with Equation (9). By contrast, the static-model
mass-ratio, $q=4.1\times 10^{-5}$, is almost a factor two smaller than the
heuristic estimate.

We note that the $(s,\alpha)$ geometry, caustic topology, and inner/outer
degeneracy of OGLE-2018-BLG-0677 (Herrera-Martin et al., 2020) are the same as
for KMT-2019-BLG-0253, except that the anomaly comes just after the peak,
while it comes just before peak for KMT-2019-BLG-0253. In that case,
$(s^{\dagger}_{-}\pm\Delta s)=(0.95\pm 0.03)$, which is also very similar to
the present case.

The case of OGLE-2016-BLG-1195 (Bond et al., 2017; Shvartzvald et al., 2017)
is also closely related. It likewise has an inner/outer degeneracy with the
outer solution having a resonant caustic. However, in that case, the source
intersected the binary axis on the planet’s side of the host rather than the
opposite side. Hence, $s^{\dagger}_{+}>1$, i.e., $(s_{+}^{\dagger}\pm\Delta
s)=(1.03\pm 0.05)$, compared to $s_{-}^{\dagger}<1$ for KMT-2019-BLG-0253 and
OGLE-2018-BLG-0677. OGLE-2019-BLG-0960 (Yee et al., 2021) is yet another
example of a major-image anomaly induced by a similar caustic topology, with
$(s^{\dagger}_{+}\pm\Delta s)=(1.012\pm 0.015)$.

####  3.2.3 Parallax Analysis

In the static solution, the event timescale is a substantial fraction of a
year, which implies that it may be possible to measure the annual-parallax
effect due to Earth’s orbital motion. As mentioned in Section 3.1, we must at
least initially also introduce the linearized orbital-motion parameters
$\gamma$. However, we find that the orbital-motion parameters are neither
significantly constrained nor significantly correlated with the parallax
parameters. Therefore, we suppress these two degrees of freedom (dof).

We initially find a seemingly strongly constrained and large value of
$\pi_{\rm E}$. However, after conducting several tests, we conclude that this
signal is the result of systematics in the KMT data. First, by examining the
cumulative distribution, $\Delta\chi^{2}(t)=\chi^{2}(t;{\rm
static})-\chi^{2}(t;{\rm parallax})$, we find that the great majority of the
“signal” comes from late in the light curve, $t>t_{0}+2\,t_{\rm E}$. It is
nearly impossible to arrange a physical situation for which this would be the
case. Moreover, this “signal” appears only in the KMT data and not in the OGLE
data. With fewer data points, we would expect the signal to be weaker in OGLE,
but still present. Finally, two KMT observatories (KMTS and KMTA) show
strongly increasing $\Delta\chi^{2}(t)$ during these late times, while the
third is strongly decreasing. We also considered that this “signal” might be
due to “xallarap”, i.e., orbital motion of the source about an unseen
companion, rather than orbital motion of Earth. Indeed, there was a stronger
xallarap signal than parallax signal. However, the same inconsistencies
between data sets persisted. And this continued when we fit for parallax and
xallarap simultaneously. We attempted to eliminate only these late-time KMT
data, but similar signatures of systematics remained, albeit at a lower level.
We are not certain what is the cause of these systematics, which affect all
three KMT telescopes, but in different directions. We note that there are
several bright variable stars in the field, though none closer
$20^{\prime\prime}$. These, in principle, could affect KMT photometry at a
very low level, without affecting OGLE photometry, which is overall better.
Then, the very large number of these points could amplify the small “signal”
in each to create strong trends.

We therefore measure ${\mbox{\boldmath$\pi$}}_{\rm E}$ by fitting only to OGLE
data, but with the planet parameters $(s,q,\alpha,\rho)$ held fixed at those
of the static solution. The results are shown in Table 3. There are two points
to note. First, the axis ratio of the error ellipse, $\sigma(\pi_{{\rm
E},\perp})/(\sigma(\pi_{{\rm E},\parallel})$, is large, ranging from 15 to 20,
depending on the solution. That is, this is a 1-D parallax measurement.
Second, $|\pi_{{\rm E},\parallel}|/\sigma(\pi_{{\rm E},\parallel})\simeq 3$,
indicating that the parallax is robustly detected. In fact, we find a very
similar value for this parameter from the full data set but with much smaller
errors $\pi_{{\rm E},\parallel}\simeq-0.100\pm 0.009$. As $\pi_{{\rm
E},\parallel}$ is much less prone to systematics than $\pi_{{\rm E},\perp}$,
we take this as qualitative confirmation of the correction of the $\pi_{{\rm
E},\parallel}$ measurement.

We note that $\rho$ is not constrained to be different from zero at the
$3\,\sigma$ level. Rather, there is only an upper limit on $\rho$, which
arises because if $\rho$ were sufficiently large, the “dip” would be more
rounded than is actually the case. In fact, this will prove to be the case for
all six events presented here. Nevertheless, this upper limit on $\rho$ (so
lower limit on $\theta_{\rm E}=\theta_{*}/\rho$) can help constrain the
physical interpretation of the event, as we discuss in Section 5.

For completeness, we report the static solutions for the OGLE-only data set:
$(\chi^{2},t_{0},u_{0},t_{\rm E},I_{S})=(1653.7,8590.5587\pm 0.0134,0.0558\pm
0.0018,56.35\pm 1.53,19.771\pm 0.036)$ and $(1652.7,8590.5686\pm
0.0144,0.0576\pm 0.0018,54.91\pm 1.46,19.735\pm 0.035)$, for the inner and
outer solutions, respectively. Because there are of order 9 times fewer OGLE
points compared to OGLE+KMT, we expect that the errors will be of order
$\sqrt{9}=3$ times larger than those of the static solutions from the full
data set, and this expectation is confirmed by comparison to Table 3. Within
these errors, the static and parallax solutions are consistent for these
parameters.

Comparing the OGLE-only static and parallax fits, we see that the errors in
the latter are of order two times larger for $(u_{0},t_{\rm E},I_{S})$. This
is partially explained by correlations between each of these three parameters
and $\pi_{{\rm E},\perp}$. That is, to leading order, $\pi_{{\rm
E},\parallel}$ and $\pi_{{\rm E},\perp}$ give rise to light-curve distortions
that are, respectively, anti-symmetric and symmetric in $(t-t_{0})$ (Smith et
al., 2003; Gould, 2004). Because the Paczyński (1986) formula is even in
$(t-t_{0})$ for $(u_{0},t_{\rm E},I_{S})$, these three parameters are
correlated with $\pi_{{\rm E},\perp}$, which has a large uncertainty. We note
that this same effect is present in the three other events with parallax
solutions that are analyzed below, although it is strongest in the case of
KMT-2019-BLG-0253.

###  3.3 OGLE-2018-BLG-0506

Figure 3 shows a clear $\Delta I\sim 0.06$ mag dip in the KMTC data at $\Delta
t_{\rm anom}\sim 0.54\,$days after the peak, with a duration $\Delta t_{\rm
dip}=0.4\,$days. The depression defined by these 26 data points, which are
taken in good seeing ($1.2^{\prime\prime}$–$1.6^{\prime\prime}$) and low
background is confirmed by the two contemporaneous OGLE points. Hence, the
anomaly is secure.

####  3.3.1 Heuristic Analysis

The 1L1S model yields parameters $t_{\rm E}\simeq 24\,$days and $u_{0}\simeq
0.09$. As noted above, $\Delta t_{\rm anom}=+0.54\,$days and $\Delta t_{\rm
dip}=0.4\,$days. Applying the heuristic formalism of Section 3.1, these yield,

$\alpha=104^{\circ};\qquad s\simeq s_{-}^{\dagger}\pm\Delta s;\qquad
s_{-}^{\dagger}=0.95;\qquad q\sim 17\times 10^{-5}.$ (10)

Again, $\pm\Delta s$, which is induced by the offset of the source trajectory
from the caustic, cannot be evaluated from the general appearance of the light
curve.

####  3.3.2 Static Analysis

The grid search described in Section 3.1 returns two local minima, which we
then further refine by allowing all parameters to vary in the MCMC. The
resulting solutions are given in Table 4 and illustrated in Figure 3. The two
solutions can be represented as $(s^{\dagger}_{-}\pm\Delta s)=(0.961\pm
0.099)$, while $\alpha=106^{\circ}$, both in good agreement with Equation
(10). The mass ratios of the two solutions are $q_{\rm inner}=(7.8\pm
2.3)\times 10^{-5}$ and $q_{\rm outer}=(7.6\pm 2.3)\times 10^{-5}$,
respectively. Note that these measurements are in only qualitative agreement
with the estimate in Equation (10).

Figure 2 shows that the caustic topologies are identical to the case of
KMT-2019-BLG-0253. That is, for the inner solution, the source traverses the
trough that extends along the minor-image axis between the central and
planetary caustics, while in the outer solution, these two triangular caustics
have merged into a resonant caustic.

####  3.3.3 Parallax Analysis

The timescale of this event is relatively short, and the source is faint.
Therefore, we do not expect to be able to measure the microlens parallax
${\mbox{\boldmath$\pi$}}_{\rm E}$. Nevertheless, as a matter of due diligence,
we explore this possibility. We first find after including both
${\mbox{\boldmath$\pi$}}_{\rm E}$ and the lens orbital parameters $\gamma$,
that the latter are not meaningfully constrained (relative to the physical
condition $\beta<0.8$). Hence, we suppress these two degrees of freedom. As in
the case of KMT-2019-BLG-0253 correlated errors in the KMT baseline data
induced spurious parallax signals, but in contrast to that case, there was no
evidence of spurious signals in the body of this event. Hence, most KMT
baseline data were removed, i.e., roughly 70% of the full-year KMT data set.
For consistency, the same cuts were applied for the static fits. The results
are given in Table 4. These show that both $\pi_{{\rm E},\parallel}$ and
$\pi_{{\rm E},\perp}$ are consistent with zero at the $1\,\sigma$ level, which
is reflected by the fact that $\chi^{2}$ is essentially unchanged by the
addition of two degrees of freedom. Nevertheless, the fact that $\pi_{{\rm
E},\parallel}$ is constrained to be near zero within relatively small errors
can be a significant constraint in the Bayesian analysis of Section 5.
Moreover, none of the other parameters are significantly affected by the
inclusion of ${\mbox{\boldmath$\pi$}}_{\rm E}$ in the fit. We therefore adopt
the parallax solutions for our final result.

###  3.4 OGLE-2018-BLG-0516

Again, there is clear dip in the light curve, which has $\Delta t_{\rm
anom}=2.2\,$days and duration $\Delta t_{\rm dip}=0.5\,$days. It is most
clearly defined by KMTC data, but these are supported by contemporaneous OGLE
data. Moreover, the beginning of the dip is traced by KMTS data, while the end
of the dip is supported by both MOA and KMTA data. Hence, the anomaly is
secure.

####  3.4.1 Heuristic Analysis

The 1L1S model yields parameters $t_{\rm E}\simeq 25\,$days and $u_{0}\simeq
0.10$. As noted above, $\Delta t_{\rm anom}=2.2\,$days and $\Delta t_{\rm
dip}=0.5\,$days. Applying the heuristic formalism of Section 3.1, these yield,

$\alpha=131^{\circ};\qquad s\simeq s_{-}^{\dagger}\pm\Delta s;\qquad
s_{-}^{\dagger}=0.93;\qquad q\sim 10\times 10^{-5}.$ (11)

Again, $\pm\Delta s$ cannot be determined at this level of analysis.

####  3.4.2 Static Analysis

The grid search described in Section 3.1 returns two local minima, which we
then further refine by allowing all parameters to vary in the MCMC. The
resulting solutions are given in Table 5 and illustrated in Figure 4. The two
solutions can be represented as $(s^{\dagger}_{-}\pm\Delta s)=(0.936\pm
0.069)$, while $\alpha=130^{\circ}$, in good agreement with Equation (11). The
mass ratios of the two solutions are $q_{\rm inner}=(12.9\pm 1.4)\times
10^{-5}$ and $q_{\rm outer}=(13.2\pm 1.4)\times 10^{-5}$, respectively, which
are also in good agreement with Equation (11).

Figure 2 shows that the caustic topologies are again identical to the cases of
KMT-2019-BLG-0253 and OGLE-2018-BLG-0506.

####  3.4.3 Parallax Analysis

The overall situation here is similar to that of OGLE-2018-BLG-0506 (Section
3.3.3). However, in contrast to that case, there were no spurious signals from
the KMT baseline data, so the full data set was included in the fit. Again we
test for, then suppress the orbital-motion parameters $\gamma$. In this case,
we find that $\pi_{{\rm E},\parallel}$ is consistent with zero at the $\simeq
1.5\,\sigma$ level, so the $\chi^{2}$ improvement is $\simeq 2$ for 2 degrees
of freedom. Again, we find little difference for the other parameters between
the static and parallax analyses. And again, we adopt the parallax solution
for our final result, noting that the $\pi_{{\rm E},\parallel}$ constraint can
play some role in the Bayesian analysis of Section 5.

###  3.5 OGLE-2019-BLG-1492

Again, there is clear dip in the light curve, which has $\Delta t_{\rm
anom}=+1.6\,$days and duration $\Delta t_{\rm dip}=0.8\,$days. It is most
clearly defined by KMTC data, but these are supported by contemporaneous OGLE
data. Moreover, there are “ridges” around the dip, which are supported by all
observatories.

####  3.5.1 Heuristic Analysis

The 1L1S model yields parameters $t_{\rm E}\simeq 50\,$days and $u_{0}\simeq
0.05$. As noted above, $\Delta t_{\rm anom}=1.6\,$days and $\Delta t_{\rm
dip}=0.8\,$days. Applying the heuristic formalism of Section 3.1, these yield,

$\alpha=123^{\circ};\qquad s\simeq s_{-}^{\dagger}\pm\Delta s;\qquad
s_{-}^{\dagger}=0.97;\qquad q\sim 18\times 10^{-5}.$ (12)

Again $\pm\Delta s$ cannot be determined at this level of analysis.

####  3.5.2 Static Analysis

The grid search described in Section 3.1 returns two local minima, which we
then further refine by allowing all parameters to vary in the MCMC. The
resulting solutions are given in Table 6 and illustrated in Figure 5. The two
solutions can be represented as $(s^{\dagger}_{-}\pm\Delta s)=(0.971\pm
0.073)$, while $\alpha=122^{\circ}$, in good agreement with Equation (12). The
mass ratios of the two solutions are $q_{\rm inner}=(19.1\pm 5.8)\times
10^{-5}$ and $q_{\rm outer}=(17.6\pm 5.4)\times 10^{-5}$, respectively, which
are in good agreement with Equation (12).

Figure 2 shows that the caustic topologies are again identical to the cases of
KMT-2019-BLG-0253, OGLE-2018-BLG-0506, and OGLE-2018-BLG-0516.

####  3.5.3 Parallax Analysis

The situation here is similar to that of OGLE-2018-BLG-0506 (Section 3.4.3).
Spurious parallax signals from the KMT baseline force us to remove most of
these data points, corresponding to about half of the full-year KMT data set.
Again we test for, then suppress the orbital-motion parameters $\gamma$. Again
we find that ${\mbox{\boldmath$\pi$}}_{\rm E}$ is only weakly constrained (and
only in the $\pi_{{\rm E},\parallel}$ direction). Again, we find little
difference for the other parameters between the static and parallax analyses.
And again, we adopt the parallax solution for our final result.

###  3.6 OGLE-2018-BLG-0977

The KMTS data show a strong and sudden dip ending in a flat minimum, which is
characteristic of the source passing into the trough between the two
triangular minor-image caustics, with the entrance close to one of these
caustics. One therefore expects the source to pass over the other caustic, an
expectation that is confirmed by the rapid drop of the KMTA data during the
first three points of the night, which span 69 minutes. Thus, KMTS and KMTA
give independent, and completely consistent evidence for a minor-image caustic
crossing.

The center of the dip occurs $\Delta t_{\rm anom}=+4.0\,$days after peak.
Because the source crosses the two triangular caustics, the $\Delta t_{\rm
dip}=0.5\,$day depression is a much more clearly defined structure than in the
four previous cases.

####  3.6.1 Heuristic Analysis

The 1L1S model yields parameters $t_{\rm E}\simeq 20\,$days and $u_{0}\simeq
0.15$. As noted above, $\Delta t_{\rm anom}=+4.0\,$days and $\Delta t_{\rm
dip}=0.5\,$days. Applying the heuristic formalism of Section 3.1, these yield,

$\alpha=143^{\circ};\qquad s=s_{-}^{\dagger}=0.88;\qquad q\simeq 5\times
10^{-5}.$ (13)

That is, because the source crosses the planetary caustic, there is only one
solution. In such cases, we expect the mass-ratio estimate to be quite
accurate.

####  3.6.2 Static Analysis

The grid search described in Section 3.1 returns only one local minimum, which
we then further refine by allowing all parameters to vary in the MCMC. We find
that, due to the extreme faintness of the source $I_{S}\sim 21$ and shortness
of the event ($t_{\rm E}=20\,$days), correlations in the KMT baseline data
impact even the static fit. We therefore eliminate these data, keeping only
$8260<{\rm HJD}^{\prime}<8296$. The resulting solution is given in Table 7 and
illustrated in Figure 6. In particular, $(s,q,\alpha)=(0.879,4.15\times
10^{-5},144^{\circ})$ in good agreement with Equation (13).

Figure 2 shows that, as expected, and in contrast to the previous four events,
the source crosses the caustic.

####  3.6.3 Parallax Analysis

When we incorporate ${\mbox{\boldmath$\pi$}}_{\rm E}$ and $\gamma$ into the
fits, we find that neither is meaningfully constrained. Therefore, we adopt
the parameters of the static model.

###  3.7 OGLE-2019-BLG-0953

In contrast to the previous five events, the amplitude of the anomaly (a dip
at HJD${}^{\prime}\simeq 8643.7$, i.e., $\Delta t_{\rm anom}=+5.4\,$days) is
not much larger than the individual KMT error bars. See Figure 7. However,
this event lies in a small $(0.4\,{\rm deg}^{2})$ zone that is covered by four
fields and so is observed at $\Gamma=8\,{\rm hr}^{-1}$. All four fields from
two observatories (KMTC and KMTS) participate in the same overall trend during
the night of the dip. Moreover the dip is confirmed by eight OGLE points.
Hence, it is secure.

####  3.7.1 Heuristic Analysis

The 1L1S model yields parameters $t_{\rm E}\simeq 11.5\,$days and $u_{0}\simeq
0.5$. We estimate $\Delta t_{\rm dip}=0.9\,$days, and, as noted above, $\Delta
t_{\rm anom}=5.4\,$days. Applying the heuristic formalism of Section 3.1,
these yield,

$\alpha=137^{\circ};\qquad s\simeq s_{-}^{\dagger}\pm\Delta s;\qquad
s_{-}^{\dagger}=0.71;\qquad q\sim 17\times 10^{-5}.$ (14)

Again $\pm\Delta s$ cannot be determined at this level of analysis.

####  3.7.2 Static Analysis

In sharp contrast to the other four non-caustic crossing events (all the above
except OGLE-2018-BLG-0977), the grid search described in Section 3.1 returns
only one local minimum. We then further refine this by allowing all parameters
to vary in the MCMC. The resulting solution is given in Table 8 and
illustrated in Figure 7.

The physical reason that the inner-outer degeneracy is broken in this case can
be assessed by comparing the caustic topology of KMT-2019-BLG-0953 to the
other four non-caustic-crossing cases in Figure 2. For this event, the
(unique) solution has the “outer” geometry, but with isolated planetary
caustics, whereas in the other four cases, the outer solution had a resonant
caustic. Hence, if there were two solutions, they would lie on opposite sides
of isolated planetary caustics. This alternate topology follows from the fact
that $u_{\rm anom}\sim 0.7$, compared to 0.07–0.13 for the other four cases,
which means that it is far from the resonant regime (for low-$q$ companions).
Because minor-image isolated caustics have substantial structure in the
direction perpendicular to the binary axis, the inner-outer degeneracy can
usually be broken when both trajectories pass isolated caustics, unless the
trajectories are themselves close to perpendicular, as in the case of
OGLE-2016-BLG-1067 (Calchi Novati et al., 2019).

Because there is only one solution, $s^{\dagger}$ is expected to be merely
offset from the (unique) value of $s$, rather than being nearly equal to the
average of two values, as in the other four cases. Nevertheless, this offset
is small: $s-s^{\dagger}=0.03$, while the fitted values of both $q$ and
$\alpha$ are all in good agreement with Equation (14).

Note that there is only a weak constraint on the source size, $\rho$, which
corresponds to $t_{*}<0.33\,$day.

####  3.7.3 Parallax Analysis

Similarly to several other cases, we find that the baseline data would
generate a spurious parallax signal. We therefore restrict the analysis to
$8600<t<8670$, i.e., roughly $t_{0}\pm 3\,t_{\rm E}$, which we then also apply
to the static solution. After doing so, we find that the microlens parallax
${\mbox{\boldmath$\pi$}}_{\rm E}$ is not meaningfully constrained, and we
therefore adopt the static solution.

##  4 Source Properties

Overall, our principal objective in this section is to measure the source-star
angular radius, $\theta_{*}$, and then to either make measurements of, or
obtain lower limits upon,

$\theta_{\rm E}={\theta_{*}\over\rho};\ \qquad\mu_{\rm rel}={\theta_{\rm
E}\over t_{\rm E}}.$ (15)

depending on whether $\rho$ is measured, or itself only has an upper limit.
For the first step, we use the method of Yoo et al. (2004). That is, we first
measure the color-magnitude offset between the source and the centroid of the
red clump on a color-magnitude diagram (CMD),

$\Delta[(V-I),I]=[(V-I),I]_{s}-[(V-I),I]_{\rm cl}.$ (16)

Next we adopt $(V-I)_{\rm cl,0}=1.06$ from Bensby et al. (2013), and we
evaluate $I_{\rm cl,0}$ from Table 1 of Nataf et al. (2013), based on the
Galactic longitude of the event. We then find the dereddened color and
magnitude of the source

$[(V-I),I]_{s,0}=[(V-I),I]_{\rm cl,0}+\Delta[(V-I),I].$ (17)

We then apply the $VIK$ color-color relations of Bessell & Brett (1988) to
transform from $V/I$ to $V/K$, and finally, we apply the color/surface-
brightness relations of Kervella et al. (2004) to obtain $\theta_{*}$. After
propagating the measurement errors, we add 5% to the error in quadrature to
take account of systematic effects in the method as a whole.

Figure 8 shows the CMDs for all six events. For one of these
(OGLE-2018-BLG-0516), the CMD field stars are drawn from the pyDIA (Albrow,
2017) reduction of a $2^{\prime}\times 2^{\prime}$ square centered on the
event. The source color is determined from regression of the $V$-band and
$I$-band source fluxes during the event. The source magnitude is calculated
from the value in Table 5 added to the offset between the pyDIA and pySIS
systems as determined by regression of their respective light curves.

For the remaining four events (KMT-2019-BLG-0253, OGLE-2018-BLG-0516,
OGLE-2019-BLG-1492, and OGLE-2018-BLG-0977), the CMD field stars are drawn
from the calibrated OGLE-III catalog (Szymański et al., 2011), with respective
radii of $80^{\prime\prime}$, $120^{\prime\prime}$, $100^{\prime\prime}$, and
$90^{\prime\prime}$. These values were chosen by balancing the competing
demands of attaining sufficient density to determine the position of the clump
and minimizing the effects of differential extinction. To determine the source
position on the OGLE-III CMD, the same steps were first taken as for
OGLE-2018-BLG-0516. Then the pyDIA CMD was calibrated to OGLE-III, allowing
the source position to be transformed.

Table 9 shows the measurements and logical train described above for all six
events. In addition, we comment on specific aspects of particular events,
below.

###  4.1 KMT-2019-BLG-0253

KMT-2019-BLG-0253 has significant blended light, which raises the question of
whether or not this blend could be due to the lens. The CMD position of the
baseline-object $[(V-I),I]_{\rm base}=(3.30,18.84)\pm(0.18,0.06)$ was
determined by transforming the KMTC02 pyDIA field-star photometry at the
position of the event to the OGLE-III system222Note that the CMD position of
the baseline object is below the threshold of detection (roughly $V\sim 21.5$)
on the OGLE-III CMD, but still has only moderate color error in the KMT field-
star photometry. This is because the $V$-band photometry is carried out by
different methods. For OGLE-III, sources on the $V$ and $I$ templates are
found independently and then matched based on astrometry. For KMT sources are
identified only in $I$ band. Then the $V$-band measurement is derived from the
$V$-band flux at this $I$-band location, averaged over all available images..
We also measured the SDSS $i$ flux at this position from seven images taken
with the Canada-France-Hawaii Telescope (CFHT) in $0.5^{\prime\prime}$ seeing,
transformed to the OGLE-III system. This confirmed the KMTC02 $I_{\rm base}$
measurement within errors. Subtracting the source flux from the baseline flux
yields the blended flux $[(V-I),I]_{b}=(3.67,19.42)\pm(0.46,0.10)$. The
baseline object and blended light are shown in Figure 8 as green and magenta
points, respectively.

We transform the source position (as determined from KMTC02 difference images)
to the CFHT $0.5^{\prime\prime}$ images and measure its offset from the
baseline object, finding,

$\Delta{\mbox{\boldmath$\theta$}}(E,N)={\mbox{\boldmath$\theta$}}_{\rm
base}-{\mbox{\boldmath$\theta$}}_{x}=(-31,-6)\pm(7,10)\,{\rm mas}.$ (18)

This offset is formally inconsistent with zero at more than $4\,\sigma$, which
would nominally rule out the lens as the origin of the blended light. In fact,
as we show below, there are physical effects that can account for this offset,
even if the blended light is due to the lens.

However, first we note that this small offset can be compared with the overall
surface density of field stars that are brighter than the blend,
$I_{b}=19.42$, which we find from the OGLE CMD to be $N=0.012\,{\rm
arcsec}^{-2}$. Taking Equation (18) at face value and noting that the
blend:source flux ratio is 1.75, the separation of the source and blend is 50
mas. Thus, the probability of such a chance superposition is $p=\pi
N((11/7)\Delta\theta)^{2}\sim 1\times 10^{-4}$, which is to say, virtually
ruled out. Hence, the blended light is due either to the lens itself, a
companion to the lens, or a companion to the source.

One possibility for the offset in Equation (18), just mentioned, is that it is
due to a companion to the source or the lens. For the source (at $D_{S}\sim
8\,{\rm kpc}$), the projected separation would correspond to $\sim 250\,{\rm
au}$. For the lens, it would be proportionately closer, e.g., $\sim 150\,{\rm
au}$ at $D_{L}\sim 5\,{\rm kpc}$. Such $\log(P/{\rm day})\sim 6$ stellar
companions are among the most common. See Figure 7 of Duquennoy & Mayor
(1991). On the other hand, if the blend were a companion to the source, it
would be a subgiant, i.e., of almost identical mass to the source. From Table
7 of Duquennoy & Mayor (1991), this would be quite rare. It is more plausible
that the blend is a companion to the lens, simply because there are no
independent constraints on its mass ratio or distance.

However, another possibility is that an ambient star (contributing some, but
not all, of the blended light) is corrupting the measurement of
${\mbox{\boldmath$\theta$}}_{\rm base}$. Suppose, for example, that 10% of the
baseline flux were due to an ambient star at $\sim 300\,{\rm mas}$ from the
source (and lens). This would be too faint and too close to be separately
resolved, even in 500 mas seeing. Therefore, the fitting program would find
the flux centroid displaced by $0.1\times 300\,{\rm mas}=30\,{\rm mas}$, which
is what is observed. We can estimate the surface density of stars that are
$2.5\pm 0.5$ mag fainter than the baseline object (i.e., $M_{I}=4.5\pm 0.5$)
using the Holtzman et al. (1998) luminosity function, based on Hubble Space
Telescope (HST) observations of Baade’s Window. We must first multiply by a
factor 3 based on the relative surface density of clump stars in the
KMT-2019-BLG-0253 field compared to Baade’s Window (Nataf et al. 2013, D.Nataf
2019, private communication). We then find a surface density $N=1.0\,{\rm
arcsec}^{-2}$. Hence, the probability that one such star falls within
$0.3^{\prime\prime}$ of the lens is about 25%.

We conclude that the blend is most likely either the lens itself or a
companion to the lens. It could be a companion to the source with much lower
probability. And it is very unlikely to be an ambient star.

###  4.2 OGLE-2018-BLG-0506

We note that the pyDIA reductions of the KMTC data are only approximately
calibrated. However, this has no practical importance because the filters are
near standard and the only quantity entering the calculations that follow is
the offset between the source and the clump. Moreover, this offset is small.
See Figure 8. For reference, we note that in the other other four events, the
KMT$\rightarrow$OGLE-III correction was found to be $[(V-I),I]_{\rm
s,kmt}-[(V-I),I]_{\rm s,ogle-iii}=$ [0.06, 0.22], [0.17, 0.12], [0.14, 0.01],
and [0.03 ,0.22], for KMT-2019-BLG-0253, OGLE-2018-BLG-0516,
OGLE-2019-BLG-1492, and OGLE-2018-BLG-0977, respectively.

We find a best estimate for the blended light $I_{b}=21.08$ on the OGLE-IV
system and $I_{b}=21.11$ on the KMTC41 pyDIA system. Even ignoring the
uncertainties in this estimate due to the mottled background light, which can
be considerable (e.g., Gould et al. 2020), this measurement would play very
little role when treated as an upper limit on lens light. For example, even at
$M=1\,M_{\odot}$ (the $1\,\sigma$ upper limit derived in Section 5), and at
$D_{L}=5.6\,{\rm kpc}$, the lens magnitude would be $I_{L}\sim 22.2$.
Therefore, we do not pursue a detailed investigation of this blend.

###  4.3 OGLE-2018-BLG-0516

The KMTC01 pyDIA analysis formally yields a negative blended flux that, after
conversion to the OGLE-III system, corresponds to an $I_{b}=21.1$ “anti-star”.
This level of negative blending would be easily explained by errors in the
PSF-fitting photometry of faint stars in the baseline images, even without
taking account of the unresolved mottled background. Hence, it is of no
concern in itself, and can be regarded as consistent with zero. The field has
extinction $A_{I}=1.87$. Hence, a lens with $M=0.84\,M_{\odot}$ (the
$1\,\sigma$ upper limit derived in Section 5) at $D_{L}\sim 7.0\,{\rm kpc}$
would have brightness $I_{L}=20.8$. This is comparable to the error in the
measurement of the blended flux (as demonstrated by its negative value).
Hence, no useful limits on the lens flux can be derived.

###  4.4 OGLE-2019-BLG-1492

For four of the six events, the position of the source on the CMD corresponds
to what is expected for a reasonably common class of bulge star. These are
turnoff stars (KMT-2019-BLG-0253 and OGLE-2018-BLG-0516), clump giant
(OGLE-2018-BLG-0506), and mid-G dwarf (OGLE-2018-BLG-0977). However, the
source position for OGLE-2019-BLG-1492 is somewhat puzzling and deserves
further investigation.

The source has a similar color to the Sun, but is about 1.45 mag fainter than
the Sun would be if it were at the mean distance of the clump. This could, in
principle, be resolved by it being 1.45 mag dimmer than the Sun or 1.45 mag
more distant than the clump, or some combination. For example, for the first,
it could be that the source is a relatively unevolved metal-poor star. Another
potential explanation is that the source color has not been correctly
measured. For example, if the source were 0.2 mag redder, it would be expected
to be 1 mag or more dimmer than the Sun. However, the color is determined from
regression of about a dozen well-magnified $V$-band points that lie along the
regression line. Hence this explanation appears unlikely. Regardless of the
exact explanation of the discrepancy, it has little practical effect. That is,
even if $\theta_{*}$ were double the value that we have estimated in Table 9,
the constraints provided by the resulting limits on $\theta_{\rm E}$ and
$\mu_{\rm rel}$ would still be too weak to matter.

The blended light has $I_{b}=19.47$ on the calibrated OGLE-III scale. However,
the baseline $V$-band photometry is not precise enough to reliably determine
$(V-I)_{b}$. In principle, this blended light might be due primarily to the
lens. However, we find from CFHT baseline images in $0.45^{\prime\prime}$
seeing that the blend is displaced from the event by $0.33^{\prime\prime}$.
Therefore, it cannot be the lens, and so must be either a companion to the
lens or to the source, or an ambient star that is unrelated to the event. We
find that surface density of stars that are brighter than the blend
$(I<I_{b})$ is $0.33\,{\rm arcsec}^{-2}$. Thus, the probability of an ambient
star within a circle of radius equal to the separation is 11%. This is the
most likely explanation for the blended light.

###  4.5 OGLE-2018-BLG-0977

In Table 9, we only list a $3\,\sigma$ upper limit on $\rho$ because the light
curve is formally consistent with $\rho=0$ at this level. However, it should
be noted that at the $1\,\sigma$ and $2\,\sigma$ levels, $\rho$ is constrained
by $\rho=1.9^{+0.3}_{-0.6}\times 10^{-3}$ and $\rho=1.9^{+0.6}_{-1.6}\times
10^{-3}$, respectively. These tighter limits on $\rho$ result primarily from
the source crossing a caustic. When, evaluating the physical parameters in
Section 5, we will make full use of the $\chi^{2}$ fit values as a function of
$\rho$. The blended light is a relatively bright star on the foreground main
sequence $[(V-I),I]_{b,OGLE-III}=(17.39,1.98)$, which raises the possibility
that this is the lens. However, we find from examining the CFHT baseline image
that this blended light is a resolved star with a separation of $\sim
0.5^{\prime\prime}$. Thus, it is definitely not the lens, nor can it be a
companion to the source. This leaves two possibilities: companion to the lens
or ambient star. The surface density of foreground main-sequence stars
brighter than the blend is $N=0.0085\,{\rm arcsec}^{-2}$. Hence, the
probability of such an ambient star within $0.5^{\prime\prime}$ is $p=1/150$.
This is relatively low, but the prior probability that a given star has a
substantially more massive companion at separation $a_{\perp}\gtrsim$1000–2000
au (for distance 2–4 kpc) is also of order 1% or less. Thus, no information
about the lens system can be inferred from the presence of this bright blend.

###  4.6 KMT-2019-BLG-0953

There was significant difficulty measuring the source color of
KMT-2019-BLG-0953. As with the other four events that lie in the OGLE-III
catalog (Szymański et al., 2011) footprint, our approach is to first measure
the source color in the KMT system by regression and then to transform this
color to the OGLE-III system by comparison of field stars. The results for all
events are shown as blue points in Figure 8. However, for KMT-2019-BLG-0953,
we applied this procedure five times, using the independent data sets from
KMTC02, KMTC42, KMTC03, KMTC43, and KMTS02. Strikingly, these five
determinations are comprised of two groups: (KMTC02, KMTC43) were closely
grouped near $(V-I)_{s,\rm OGLE-III}\simeq 1.94$, while (KMTC03, KMTC42,
KMTS02) were closely grouped near $(V-I)_{s,\rm OGLE-III}\simeq 2.05$, The
errors in each case were near 0.04. One approach would simply be to take the
average of these five measurements, which would yield a mean and standard
error of the mean of $2.006\pm 0.018$. This joint fit has $\chi^{2}=9.1$ for
four degrees of freedom, which has a Gaussian probability of
$p=(1+\chi^{2}/2)\exp(-\chi^{2}/2)=6\%$. Under normal circumstances, this
value would not be low enough to raise serious concerns.

However, there are two further issues that must be taken into account. First,
using the clump centroid from Table 9, this would lead to a dereddened source
color $(V-I)_{s,0}=0.51\pm 0.02$. Such blue stars are extremely rare in the
bulge. Given that the source is 2.3 magnitudes below the clump, it would have
to be either a blue straggler or a relatively young star. Alternatively it
could be an F dwarf in the disk, relatively close to the Galactic bulge. This
would also be unexpected.

The second issue is that the color measurement may be impacted by the presence
of a bright neighbor at $1.0^{\prime\prime}$, which is about 2 mag brighter
than the source. DIA works by first subtracting the reference image convolved
with a kernel and then multiplying the resulting difference image by the point
spread function (PSF). If the subtraction were perfect, the only impact of the
neighbor would be to add photon noise, which is already taken into account in
the measurement process. However, the PSF varies over the field, particularly
near the edge of the field (recall that this event lies in the overlap of two
neighboring fields), so that the PSF (and resulting kernel) are inevitably
less than perfect. Then, residuals from an imperfectly subtracted, bright,
overlapping neighbor can impact the photometry. Thus, the difficulties induced
by this neighbor might plausibly account for the relatively high
$\chi^{2}/$dof of the joint fit, in which case the high scatter would be due
to systematics, so that the errors could not be treated as Gaussian.

We resolve these challenges as follows. First, we derive the color from the
three redder measurements and ignore the two bluer measurements as plausibly
due to systematics. Second, we adopt the standard deviations of these
measurements, rather then the standard error of the mean, in order to reflect
the non-Gaussian errors. Third, with these assumptions, the source is very
plausibly a bulge star, and so we assign it bulge kinematics in the Bayesian
analysis of Section 5.

However, we also briefly discuss the implications for both present and future
work, if the source is later proved by followup observations to be bluer than
we have assumed.

First, such a bluer source would yield a smaller $\theta_{*}$. In particular,
it would be $\sim 10\%$ smaller if the source were 0.11 mag bluer. Under
normal circumstances, this would affect the Bayesian estimates by reducing
both $\theta_{\rm E}$ and $\mu_{\rm rel}$ by 10%. However, in the present
case, we have only weak lower limits on these quantities (see Table 9), so
there is essentially no impact. Furthermore, if the lens and source are
separately resolved by future adaptive optics (AO) observations, then
$\mu_{\rm rel}$ (and so $\theta_{\rm E}=\mu_{\rm rel}t_{\rm E}$) will be
directly determined from these observations, so that the current $\theta_{*}$
estimate will become moot.

The main impact would be that a blue source would almost certainly imply that
the source is in the disk, which then implies that the lens (which must lie in
front of the source) is also in the disk. Thus, adopting a blue color would
affect the Bayesian estimates. The error bars of these estimates are already
quite large, and we regard the probability of a blue source to be sufficiently
small to ignore this possibility. Hence, the main role of this section will be
to alert workers carrying out future followup observations to the difficulties
of the color measurement, so that they can properly take these into account in
their analyses.

##  5 Physical Parameters

When both $\theta_{\rm E}$ and $\pi_{\rm E}$ are measured, one can directly
determine the lens mass and lens-source relative parallax by inverting
Equation (7), and one can then infer the lens distance $D_{L}$ by estimating
the source parallax $\pi_{S}$, which also yields an estimate of the projected
planet-host separation, $a_{\perp}$

$M={\theta_{\rm E}\over\kappa\pi_{\rm E}};\qquad D_{L}={{\rm
au}\over\theta_{\rm E}\pi_{\rm E}+\pi_{S}};\qquad a_{\perp}={s\over\pi_{\rm
E}+\pi_{S}/\theta_{\rm E}}{\rm au}.$ (19)

Unfortunately, we do not measure both $\theta_{\rm E}$ and $\pi_{\rm E}$ for
any of the six events. For four of the six events, we obtain a constraint on
${\mbox{\boldmath$\pi$}}_{\rm E}$ (a “one-dimensional parallax” measurement),
while for the other two (OGLE-2018-BLG-0977 and KMT-2019-BLG-0953), we do not
obtain any constraint of ${\mbox{\boldmath$\pi$}}_{\rm E}$. For each of the
six events, we obtain only a constraint on $\rho$ (and so on $\theta_{\rm
E}=\theta_{*}/\rho$). We therefore must estimate the physical parameters from
a Bayesian analysis using Galactic-model priors.

We apply the procedures of Zang et al. (2021b) to each of the six planetary
events. In each case, we consider an ensemble of simulated events drawn from a
Galactic model. We weight these events by the product of five terms,

$\Gamma\propto\theta_{\rm E}\mu_{\rm rel}\chi^{2}(t_{\rm
E})\chi^{2}({\mbox{\boldmath$\pi$}}_{\rm E})\chi^{2}(\rho).$ (20)

The first two terms are simply the cross section and speed from the very
general rate formula “$\Gamma=n\sigma v$”. The third term reflects how well
the Einstein timescale of the simulated event matches that of the actual
event, within the latter’s measurement error. Similarly, the fourth term
reflects the match of ${\mbox{\boldmath$\pi$}}_{\rm E}$ between the simulated
and observed events. The only difference is that, because
${\mbox{\boldmath$\pi$}}_{\rm E}$ is a two-dimensional (2-D) quantity, the
comparison is made via a 2-D error ellipse rather than a 1-D error bar. For
OGLE-2018-BLG-0977 and KMT-2019-BLG-0953, the best value of $t_{\rm E}$ and
its error bar come from Table 7. For the remaining four events, the parallax
values and error ellipses come from $(\pi_{{\rm E},\parallel},\pi_{{\rm
E},\perp},\psi)$ in Tables 3–6, while the $t_{\rm E}$ and error bars come from
the corresponding solutions.

The only element of Equation (20) that is somewhat less familiar is
$\chi^{2}(\rho)$. Normally, one measures $\rho$ and infers $\theta_{\rm
E}=\theta_{*}/\rho$ (with some error bar) and so this term is usually
expressed in the form $\chi^{2}(\theta_{\rm E})$. However, lacking $\rho$
measurements, we proceed as follows.

From the MCMC, we evaluate the lower envelope of $\chi^{2}$ as a function of
$\rho$. For example, for the case of KMT-2019-BLG-0253, this is similar to a
Heaviside step function, e.g., $\chi^{2}(\rho)=100\Theta(0.0035)$, in which
any value $\rho<0.0035$ would be permitted with no $\chi^{2}$ penalty, and all
others would be strictly excluded. By contrast, for the case of
OGLE-2018-BLG-0977, $\chi^{2}(\rho)$ has a clear minimum at $\rho\sim 0.0019$,
i.e., well below the $3\,\sigma$ limit $\rho<0.0031$. However, even $\rho=0$
is formally disfavored at only the $\sim 2\,\sigma$ level.

For each simulated event, $i$, we evaluate $\rho_{i}=\theta_{*}/\theta_{{\rm
E},i}$ where $\theta_{*}$ is the value determined in Section 4 and
$\theta_{{\rm E},i}$ is the Einstein radius of the simulated event.

For all the events except OGLE-2018-BLG-0977 and KMT-2019-BLG-0953, there are
multiple solutions. For these cases, we combine the results from the different
solutions, with the simulated events from each solution weighted by the
$\Delta\chi^{2}$ (as given in Tables 3–8) for that solution. Table 10 shows
the results. For more details on the Bayesian method, see Zang et al. (2021b)

In Section 4.1, we assessed that blended light in KMT-2019-BLG-0253 (magenta
point in Figure 8) was most likely due to the lens itself or a companion to
the lens. In either case, it would be at the distance to the lens, which, from
Table 10, corresponds to distance modulus $(m-M)\simeq 13.45\pm 0.75$. The
blend lies $2.47\pm 0.10$ mag below the clump, which has $I_{\rm cl,0}=14.44$.
At $D_{L}\sim 5\,{\rm kpc}$, the lens is almost certainly behind essentially
all the dust. Thus, if the blend is either the lens or its companion, then its
absolute magnitude is $M_{I,\rm bl}=3.46\pm 0.75$. This range is somewhat too
luminous for a middle K dwarf, which is indicated by the blend’s dereddened
color being somewhat redder than the clump. On the other hand a K dwarf would
be quite consistent with the mass range shown in Table 10. One possibility
that would account for all the constraints would be that the lens is part of a
roughly equal-mass K-dwarf binary. Alternatively, the lens could be somewhat
closer than indicated by the $1\,\sigma$ Bayesian range, in which case it
would be intrinsically dimmer (at the same observed brightness). These issues
can only be resolved by AO followup observations. In many cases, fruitful AO
observations must wait for the lens and source to move apart sufficiently to
permit them to be separately resolved, and for this event the time required
for such separation is highly uncertain because there is only a lower limit on
$\mu_{\rm rel}$. However, in the present case, immediate AO observations would
be highly informative because the blend is brighter than the source, and the
observations would allow one to confirm (or rule out) the conjecture that the
astrometric offset observed from the ground was due to low-level contamination
by an ambient star.

In Section 3.1, we argued that 1-D parallax measurements can play a role in
constraining the Bayesian estimates of the physical parameters, even when the
$\chi^{2}$ improvement is small (or even zero). We illustrate that impact in
the present case by repeating the Bayesian analysis without the parallax
constraint and reporting the resulting mass estimates for the first four
events in Table 10: $M_{\rm
host}=(0.63_{-0.34}^{+0.38},0.53_{-0.30}^{+0.41},0.42_{-0.26}^{+0.40},0.58_{-0.33}^{+0.40})M_{\odot}$.
Thus, by including the parallax constraint, the mass estimates were changed by
$(+11,+19,+12,+21)\%$, which is notable but below $1\,\sigma$ in all cases.
This is not surprising given that the fractional Bayesian ranges
$\delta_{M}\equiv(M_{+1\,\sigma}-M_{-1\,\sigma})/2M_{\rm median}$ in Table 10
are relatively large: $\delta_{M,\rm parallax}=(0.46,0.56,0.78,0.50)$.
However, another important impact is that these fractional ranges are
themselves reduced by including the parallax constraint: $\delta_{M,\rm
parallax}/\delta_{M,\rm static}=(0.81,0.84,0.99,0.79)$. That is, the
fractional range was substantially reduced in all but one case.

##  6 Discussion

We have applied the method of Paper I (Zang et al., 2021b) to the six KMT
prime fields (BLG01/02/03/41/42/43) for the 2018-2019 seasons, with the aim of
obtaining a homogeneously selected sample of planets with $q<2\times 10^{-4}$.
Specifically, we obtained TLC reductions of all events that had $q<3\times
10^{-4}$ based on pipeline reductions, and we thoroughly investigated these
events using the TLC reductions. While a full analysis must await the further
application of this method to more seasons (at least 2016-2017) and more
fields, it is of considerable interest to review the properties of this
initial sample.

Table 11 shows all the previously known planets that were recovered by our
approach, regardless of $q$. Table 12 shows the 11 events with planets
$q<2\times 10^{-4}$ (7 discovered and 4 recovered) by our approach, rank
ordered by $\Delta\chi^{2}_{0}$ of anomaly detection (see Paper I). A striking
feature of this table is that all the new discoveries are at the top, i.e.,
they all have lower $\Delta\chi^{2}_{0}$ than any of the recoveries. This
implies that, broadly speaking, machine selection of anomaly candidates is
more robust than by-eye selection. Of course, it is still necessary for humans
to vet these candidates, whether initially selected by machine or by eye.

In fact, there are two exceptions to this pattern that are not reflected in
Table 12. First, OGLE-2018-BLG-0677Lb (Herrera-Martin et al., 2020), with
$q=9.1\times 10^{-5}$, was not found by our approach because it was below our
$\Delta\chi^{2}_{0}=75$ threshold, yet it was found by eye. Second,
KMT-2018-BLG-1025Lb (Han et al., 2021a), with $q_{\rm best}=8\times 10^{-5}$,
is not shown in Table 12 because it has a second solution at $q=16\times
10^{-5}$, with $\Delta\chi^{2}=8.4$, and so it is not suitable for
investigating the mass-ratio function. Nevertheless, because it was recovered
at $\Delta\chi^{2}_{0}=140$, it would break the simple pattern of Table 12 if
it were included. Both examples show that humans are competitive with machines
in some individual cases. Nevertheless, the overall message from Table 12
remains the same: many subtle planetary signatures escape recognition in by-
eye searches. Note also from Table 12 that $\Delta\chi^{2}_{0}$ is hardly
correlated with $q$, if at all.

###  6.1 Eleven Discovered/Recovered $q<2\times 10^{-4}$ Planets

Our approach yielded a total of 11 planets with $q<2\times 10^{-4}$ in the KMT
prime fields during 2018-2019, including OGLE-2019-BLG-1053Lb, which was
reported by Zang et al. (2021b), the six reported here, and four previously
discovered planets333Not including KMT-2018-BLG-1025Lb (Han et al., 2021a),
which, as mentioned above, is excluded from the sample because it has two
degenerate solutions that are well-separated in $q$. , OGLE-2018-BLG-0532Lb
(Ryu et al., 2020), OGLE-2018-BLG-0596Lb (Jung et al., 2019b),
OGLE-2018-BLG-1185Lb (Kondo et al., 2021), and KMT-2019-BLG-0842Lb (Jung et
al., 2020b).

There are four known $q<2\times 10^{-4}$ planets from 2018-2019 that were not
found in this search: KMT-2018-BLG-0029Lb (Gould et al., 2020),
OGLE-2019-BLG-0960Lb (Yee et al., 2021), KMT-2018-BLG-1988Lb (Han et al.,
2021d), and OGLE-2018-BLG-0677Lb (Herrera-Martin et al., 2020). The first
three of these do not lie in prime fields, while the last failed our
$\Delta\chi^{2}_{0}>75$ cut in the initial automated anomaly search. Note that
KMT-2018-BLG-1988Lb has a very large (factor 2) $1\,\sigma$ error in $q$, so
that it will almost certainly not be included in mass-ratio studies.

Finally, although it is not directly germane to the present study, we note
that our approach failed to recover (or fully recover) three known planets
with $q>2\times 10^{-4}$. The first is KMT-2019-BLG-1715LABb (Han et al.,
2021c) which is a 3L2S event. The automated AnomalyFinder identified the
anomaly generated by the $q_{2}=0.25$ binary companion, which dominated (and
so suppressed recognition of) the planetary companion $q_{3}=4\times 10^{-3}$.
The AnomalyFinder would have to be modified to take account of the possibility
of planet-binary systems to have detected this planet. The second case is that
of KMT-2019-BLG-1953Lb (Han et al., 2020c), which is a $q\sim 2\times 10^{-3}$
“buried planet” (Dong et al., 2009), i.e., its signature is submerged in the
finite-source effects near the peak of a high-magnification event, $A_{\rm
max}\sim 1000$. It was identified in our search as a finite-source-point-lens
(FSPL) anomaly. In principle, all such events should be systematically
searched for planets because of their overall strong sensitivity, particularly
to Jovian mass-ratio planets. However, this stage was not pursued in the
present context because of the weak sensitivity of such events to very low-$q$
planets, which is the focus of the present effort.

The third case is OGLE-2018-BLG-1011Lb,c (Han et al., 2019). This is a two-
planet system for which the AnomalyFinder reports a single anomaly that
contains both planets. However, in contrast to the case of the binary+planet
system KMT-2019-BLG-1715LABb that was mentioned above, this is not really a
shortcoming. That is, the AnomalyFinder identifies the principal anomaly, but
does not itself classify the anomaly as “planetary” or “binary”. Rather, the
operator must make the decision that an anomaly is “potentially planetary”.
All such events must be thoroughly investigated before they can be published,
and such detailed investigations, which are carried out using TLC reductions,
are far more sensitive to multiplicity of planets than any potential machine
search based on pipeline reductions. By contrast, the detection of a binary
system will not, of itself, trigger such a detailed investigation. Because
OGLE-2018-BLG-1011 was a well-known planetary system at the time that the
AnomalyFinder was run on 2018 data, no investigation was needed. However, it
would have been triggered if the planets were not already known.

For completeness, we note that all three of these planetary events, i.e.,
OGLE-2018-BLG-1011, KMT-2019-BLG-1715, and KMT-2019-BLG-1953, were designated
as “anomalous” by the AnomalyFinder. However, because there were multiple
anomalies in the three cases (either a third body or finite source effects
that dominated over the planet), the AnomalyFinder did not identify all of the
planetary anomalies.

The cumulative mass-ratio distribution of these 11 planets is presented in
Figure 9, which shows that they span the range $-5<\log q<-3.7$. We have not
yet measured the efficiency of our selection, but the automatic selection
should be less sensitive to lower-$q$ planets.

In particular, the sensitivity $\xi(q)$ of any wide-angle ground-based survey
that depends primarily on a $\chi^{2}$ threshold to detect planets will be
approximately a power law, which gradually steepens toward lower $q$. First,
in the regime where the size of the sources (dominated by upper-main-sequence
and turnoff stars) is smaller than the caustic size, the cross section for the
source to interact with the caustics, or with the magnification structures
that extend from them, scales as either $q^{1/2}$ or $q^{1/3}$ for resonant
and planetary caustics, respectively. Second, as $q$ falls, the duration of
the anomaly declines according to the same power laws, which for fixed source
brightness $I_{s}$, proportionately reduces $\chi^{2}$. Hence, for otherwise
similar events, the $\chi^{2}$ threshold restricts lower-$q$ planets to the
brighter end of the luminosity function. Thus, if the luminosity function were
a strict power law (as it roughly is for $0.5<M_{I}<3$, Figure 5 of Holtzman
et al. 1998), then the efficiency would also be a power law, $\xi(q)\propto
q^{\gamma}$. In fact, the luminosity becomes much shallower for $M_{I}\gtrsim
3$, which makes $\xi(q)$ become gradually shallower toward higher $q$.
Finally, for low mass ratios ($\log q\lesssim-4$) where turnoff sources become
comparable to or larger than the caustic size, two new effects take hold. The
first is that the cross section of interaction is fixed by the source size,
rather than declining with $q$ as the caustic size. By itself, this would make
the slope shallower toward lower $q$. However, the second effect is that the
flux deviation due to the caustic is washed out by the extended source, which
at fixed $I_{s}$ reduces $\chi^{2}$ and so restricts detections to the
brighter end of the luminosity function, i.e., more so than for the case
analyzed above of small sources. The net effect is to maintain a steepening
$\xi(q)$ in this regime. A corollary of this logic is that if one
overestimates (or underestimates) the source sizes in the efficiency
calculation, then one will derive a $\xi(q)$ that is too steep (or too
shallow) in the $\log q\lesssim-4$ regime. However Figure 7 of Suzuki et al.
(2016) shows that this is a relatively modest effect.

Therefore, while no rigorous conclusions can yet be drawn, a declining
sensitivity function suggests that the low mass-ratio planets in this
homogeneously selected sample represents a substantial underlying population.
This raises questions about the previously inferred paucity of planets at low
$q$ (Suzuki et al., 2016; Jung et al., 2019a).

We note that it is already known that the low-$q$ 2018-2019 planets
KMT-2018-BLG-0029Lb ($\log q=-4.74$) and OGLE-2019-BLG-0960Lb ($\log q=-4.90$)
will be recovered when we apply our approach to the non-prime KMT fields. See
Figure 9. While we cannot include these in the homogeneously selected
distribution because we do not yet know what new planets will be discovered in
the non-prime fields, these detections prove that such planets exist. So, it
will not be the case that only larger mass-ratio planets are found in these
fields. Therefore, it would require many such discoveries with $\log
q\gtrsim-4.3$ to maintain a strong case for a mass-ratio function that
declines with declining $q$ in the $q<2\times 10^{-4}$ regime. We briefly
discuss the reasons that we do not consider this to be likely in Section 6.3.

###  6.2 Angular Distribution

An interesting difference between the recovered versus discovered low-$q$
planets is that the former all have source trajectories that are closer to the
planet-host axis than the latter. The two groups have angular offsets
($6^{\circ}$, $8^{\circ}$, $15^{\circ}$, $27^{\circ}$), and ($31^{\circ}$,
$36^{\circ}$, $45^{\circ}$, $50^{\circ}$, $58^{\circ}$, $59^{\circ}$,
$74^{\circ}$), respectively. In particular, the two $q<10^{-4}$ recovered
planets have trajectories very close to the planet-host axis ($6^{\circ}$ and
$8^{\circ}$). Although we have excluded KMT-2018-BLG-1025, we note that if the
favored $q<10^{-4}$ solution is indeed correct, then it also has a source
trajectory close to the binary axis ($5^{\circ}$). This probably means that it
is much easier to spot by eye the extended anomalies due to such oblique
encounters, compared to the short dips or bumps in the events containing the
newly discovered planets. This higher sensitivity of oblique trajectories was
already noted by Jung et al. (2020b), Yee et al. (2021), and Zang et al.
(2021a).

###  6.3 Sensitivity as a Function of $\Gamma$

It is overall less likely that the seven newly discovered (as opposed to
recovered) planets would have been discovered if exactly the same event had
occurred in lower cadence fields. All seven planets were newly discovered
because a short-lived, low-amplitude perturbation was densely covered in the
prime fields. Their short duration and low amplitude severely diminished444We
say “diminished” rather than “prevented” because OGLE-2018-BLG-0677 (Herrera-
Martin et al., 2020) was recognized by eye despite the fact that it was below
the threshold of our machine search. the chance that they would be noticed in
by-eye searches, while their dense coverage enabled relatively high
$\Delta\chi^{2}_{0}$ during the small, short deviation. The same event would
likely still escape by-eye detection in a lower-cadence field, while the
$\Delta\chi^{2}_{0}$ would be lower by the same factor as the cadence. Four of
the seven discovered, planets had nominal cadences of $\Gamma=4\,{\rm
hr}^{-1}$, two (OGLE-2018-BLG-0516 and OGLE-2018-BLG-0977) had $\Gamma=2\,{\rm
hr}^{-1}$, and one (KMT-2019-BLG-0953 had $\Gamma=8\,{\rm hr}^{-1}$. If we
simply scale $\Delta\chi^{2}_{0}$ by $\Gamma$, then for KMTNet’s seven
$\Gamma=1\,{\rm hr}^{-1}$ fields, the seven $q<2\times 10^{4}$ discoveries
listed in Table 12 would have $\Delta\chi^{2}_{0}=(22,56,41,98,65,35,75)$. For
these lower-cadence fields, the detection threshold will be set to
$\Delta\chi^{2}_{0}=50$ (rather than 75) because the problem of contamination
by low-level systematics is substantially reduced. Hence, four of the seven
planetary events would have been investigated, and plausibly published. On the
other hand, for KMTNet’s 11 $\Gamma=0.4\,{\rm hr}^{-1}$ fields, the
corresponding numbers are $\Delta\chi^{2}_{0}=(9,22,16,39,26,14,30)$, which
are clearly hopeless. The same applies, ipso facto, to KMTNet’s three
$\Gamma=0.2\,{\rm hr}^{-1}$ fields.

Regarding higher mass-ratio planets, $q>2\times 10^{-4}$, these are generally
much easier to identify by eye. Our preliminary review of the higher mass-
ratio sample from the high-cadence fields confirms this assessment. The ratio
of recovered-to-discovered planets that unambiguously have mass ratios
$q<2<10^{-4}$ is 4:7. For comparison there are 16 recovered planets with
$2\times 10^{-4}<q<0.03$ in Table 11. If the recovered-to-discovered ratio
were the same as for low-$q$ planets, then we would expect 28 high-$q$
discoveries. While we have not yet completed our investigation, we are
confident that the actual number of high-$q$ discoveries will be well below
this number.

Thus, we expect that when our method is applied to lower-cadence fields, it
will generally recover the by-eye detected events, but the fraction of new
discoveries will likely be smaller.

###  6.4 Conclusions

The KMTNet AnomalyFinder was applied to 2018-2019 prime-field data and
returned 11 low mass-ratio ($q<2\times 10^{-4}$) planets, of which four were
previously discovered by eye, one was previously reported in Paper I (Zang et
al., 2021b), and six are reported here for the first time. This 7:4 ratio
suggests that many low mass-ratio planets are missed in by-eye searches. By
contrast there was only one by-eye discovery that was missed by AnomalyFinder,
and this because it was below the $\chi^{2}$ threshold of the algorithm.

While the investigation of the higher mass-ratio ($q>2\times 10^{-4}$)
detections is not yet complete, a preliminary review suggests that by-eye
searches are much more effective in this regime.

All six of the newly discovered low-$q$ planets have higher angular offsets
($31^{\circ}<\alpha^{\prime}<74^{\circ}$) relative to the planet-host axis
compared to the four previous discoveries
($6^{\circ}<\alpha^{\prime}<27^{\circ}$), which suggests that the “dilated”
anomalies induced by acute trajectories are easier to discover by eye in the
low-$q$ regime (Jung et al., 2020b).

Combined, these results suggest that low mass-ratio planets may be more common
than previously believed. However, systematic application of the AnomalyFinder
to the lower-cadence fields and to additional seasons, combined with an
efficiency analysis currently underway (Y.K. Jung et al., in prep), will be
required to make firm statements about this possibility.

Of the six planets reported in this paper, one could benefit from immediate AO
observations. We found that the centroid of the blended light in
KMT-2019-BLG-0253 lies within about 50 mas of the source and is most likely
dominated by the lens or a companion to the lens. AO observations could
greatly clarify or resolve the nature of this blend.

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.
Work by C.H. was supported by the grants of National Research Foundation of
Korea (2019R1A2C2085965 and 2020R1A4A2002885). The OGLE project has received
funding from the National Science Centre, Poland, grant MAESTRO
2014/14/A/ST9/00121 to AU. The MOA project is supported by JSPS KAKENHI Grant
Number JSPS24253004, JSPS26247023, JSPS23340064, JSPS15H00781, and JP16H06287.
W.Z., S.M., X.Z. and H.Y. acknowledge support by the National Science
Foundation of China (Grant No. 11821303 and 11761131004). This research uses
data obtained through the Telescope Access Program (TAP), which has been
funded by the TAP member institutes. We are very grateful to the
instrumentation and operations teams at CFHT who fixed several failures of
MegaCam in the shortest time possible, allowing its return onto the telescope
and these crucial observations.

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Table 2: Event Names, Cadences, Alerts, and Locations Name | $\Gamma\,({\rm hr}^{-1})$ | Alert Date | RAJ2000 | DecJ2000 | $l$ | $b$
---|---|---|---|---|---|---
KMT-2019-BLG-0253 | 4.0 | 2 Apr 2019 | 17:51:31.82 |  $-29$:33:55.7 | +0.13 | $-1.43$
OGLE-2019-BLG-0410 | 1.0 |  |  |  |  |
MOA-2019-BLG-127 | 4.0 |  |  |  |  |
OGLE-2018-BLG-0506 | 4.0 | 30 Mar 2018 | 17:50:31.16 |  $-31$:55:26.6 | $-2.01$ | $-2.45$
KMT-2018-BLG-0835 | 0.3 |  |  |  |  |
OGLE-2018-BLG-0516 | 0.3 | 1 Apr 2018 | 17:58:33.63 |  $-31$:15:44.6 | $-0.57$ | $-3.59$
MOA-2018-BLG-107 | 1.0 |  |  |  |  |
KMT-2018-BLG-0808 | 2.0 |  |  |  |  |
OGLE-2019-BLG-1492 | 3.0 | 6 Oct 2019 | 18:00:23.15 |  $-28$:37:52.1 | +1.91 | $-2.63$
KMT-2019-BLG-3004 | 4.0 |  |  |  |  |
OGLE-2018-BLG-0977 | 0.8 | 3 Jun 2018 | 17:54:01.47 |  $-30$:36:17.3 | $-0.49$ | $-2.42$
KMT-2018-BLG-0728 | 2.0 |  |  |  |  |
KMT-2019-BLG-0953 | 8.0 | 27 May 2019 | 17:57:21.26 |  $-28$:40:28.7 | $+1.54$ | $-2.08$
Table 3: 2L1S Parameters for KMT-2019-BLG-0253 Parameters | Static | Parallax
---|---|---
| Inner | Outer | Inner $u_{0}>0$ | Inner $u_{0}<0$ | Outer $u_{0}>0$ | Outer $u_{0}<0$
$\chi^{2}$ | 14661.2 | 14660.9 | 1640.0 | 1640.2 | 1642.0 | 1641.9
dof | 14661 | 14661 | 1640 | 1640 | 1640 | 1640
$t_{0}$ | 8590.5774 | 8590.5814 | 8590.5618 | 8590.5552 | 8590.5645 | 8590.5599
| 0.0053 | 0.0055 | 0.0183 | 0.0162 | 0.0172 | 0.0154
$u_{0}$ | 0.0555 | 0.0559 | 0.0538 |  $-$0.0532 | 0.0565 |  $-$0.0566
| 0.0006 | 0.0006 | 0.0034 | 0.0035 | 0.0026 | 0.0027
$t_{\rm E}$ (days) | 57.01 | 56.66 | 58.77 | 59.60 | 56.08 | 55.76
| 0.55 | 0.52 | 4.13 | 4.13 | 2.74 | 2.80
$s$ | 0.9289 | 1.0092 | … | … | … | …
| 0.0074 | 0.0088 | … | … | … | …
$q$ ($10^{-5}$) | 4.07 | 4.10 | … | … | … | …
| 0.75 | 0.72 | … | … | … | …
$\alpha$ (rad) | 1.0230 | 1.0223 | … | … | … | …
| 0.0043 | 0.0041 | … | … | … | …
$\rho$ |  $<$0.0045 |  $<$0.0045 | … | … | … | …
$\pi_{\rm E,N}$ | … | … | $-0.550$ | $0.509$ | $-0.362$ | $0.198$
| … | … | $0.609$ | $0.490$ | $0.482$ | $0.445$
$\pi_{\rm E,E}$ | … | … | $0.028$ | $0.114$ | $0.053$ | $0.102$
| … | … | $0.074$ | $0.038$ | $0.062$ | $0.036$
$\pi_{{\rm E},\parallel}$ | … | … | $-0.089$ | $-0.089$ | $-0.094$ | $-0.092$
| … | … | $0.030$ | $0.030$ | $0.029$ | $0.029$
$\pi_{{\rm E},\perp}$ | … | … | $-0.544$ | $0.514$ | $-0.354$ | $0.203$
| … | … | $0.613$ | $0.491$ | $0.485$ | $0.445$
$\psi$ (deg) | … | … | $276.4$ | $272.8$ | $276.5$ | $272.8$
$I_{\rm S,KMTC02}$ | 19.780 | 19.771 | 19.665 | 19.680 | 19.611 | 19.607
| 0.012 | 0.012 | 0.071 | 0.074 | 0.051 | 0.054
Table 4: 2L1S parameters for OGLE-2018-BLG-0506 Parameters | Static | Parallax
---|---|---
| Inner | Outer | Inner $u_{0}>0$ | Inner $u_{0}<0$ | Outer $u_{0}>0$ | Outer $u_{0}<0$
$\chi^{2}$ | 2998.2 | 2997.8 | 2996.0 | 2995.9 | 2995.7 | 2995.5
dof | 2998 | 2998 | 2996 | 2996 | 2996 | 2996
$t_{0}$ | 8224.1580 | 8224.1584 | 8224.1577 | 8224.1577 | 8224.1581 | 8224.1580
| 0.0026 | 0.0028 | 0.0046 | 0.0042 | 0.0046 | 0.0041
$u_{0}$ | 0.0884 | 0.0884 | 0.0878 |  $-$0.0878 | 0.0879 |  $-$0.0878
| 0.0012 | 0.0012 | 0.0014 | 0.0013 | 0.0013 | 0.0014
$t_{\rm E}$ (days) | 23.86 | 23.87 | 23.84 | 23.81 | 23.85 | 23.84
| 0.28 | 0.28 | 0.35 | 0.33 | 0.35 | 0.37
$s$ | 0.8612 | 1.0594 | 0.8564 | 0.8548 | 1.0673 | 1.0653
| 0.0176 | 0.0214 | 0.0153 | 0.0148 | 0.0188 | 0.0189
$q$ ($10^{-5}$) | 7.78 | 7.63 | 8.22 | 8.48 | 8.46 | 8.16
| 2.26 | 2.34 | 2.33 | 2.32 | 2.35 | 2.33
$\alpha$ (rad) | 1.8536 | 1.8532 | 1.8546 |  $-$1.8541 | 1.8544 |  $-$1.8554
| 0.0065 | 0.0067 | 0.0077 | 0.0084 | 0.0078 | 0.0080
$\rho$ |  $<$0.012 |  $<$0.014 |  $<$0.012 |  $<$0.012 |  $<$0.014 |  $<$0.014
$\pi_{\rm E,N}$ | … | … | $0.621$ | $-0.794$ | $0.495$ | $-0.752$
| … | … | $1.636$ | $1.637$ | $1.645$ | $1.683$
$\pi_{\rm E,E}$ | … | … | $0.018$ | $-0.094$ | $0.006$ | $-0.090$
| … | … | $0.145$ | $0.125$ | $0.147$ | $0.128$
$\pi_{{\rm E},\parallel}$ | … | … | $0.035$ | $0.036$ | $0.037$ | $0.035$
| … | … | $0.034$ | $0.034$ | $0.035$ | $0.034$
$\pi_{{\rm E},\perp}$ | … | … | $0.620$ | $-0.799$ | $0.494$ | $-0.757$
| … | … | $1.642$ | $1.641$ | $1.652$ | $1.687$
$\psi$ (deg) | … | … | $274.9$ | $274.2$ | $275.0$ | $274.2$
$I_{\rm S,KMTC01}$ | 19.163 | 19.163 | 19.170 | 19.169 | 19.169 | 19.170
| 0.015 | 0.015 | 0.017 | 0.017 | 0.016 | 0.017
Table 5: 2L1S parameters for OGLE-2018-BLG-0516 Parameters | Static | Parallax
---|---|---
| Inner | Outer | Inner $u_{0}>0$ | Inner $u_{0}<0$ | Outer $u_{0}>0$ | Outer $u_{0}<0$
$\chi^{2}$ | 5074.4 | 5067.7 | 5072.4 | 5072.7 | 5065.7 | 5065.8
dof | 5068 | 5068 | 5066 | 5066 | 5066 | 5065
$t_{0}$ | 8227.6483 | 8227.6514 | 8227.6515 | 8227.6513 | 8227.6541 | 8227.6542
| 0.0055 | 0.0054 | 0.0057 | 0.0058 | 0.0061 | 0.0057
$u_{0}$ | 0.1042 | 0.1050 | 0.1034 |  $-$0.1034 | 0.1044 |  $-$0.1042
| 0.0013 | 0.0014 | 0.0015 | 0.0016 | 0.0015 | 0.0015
$t_{\rm E}$ (days) | 24.96 | 24.85 | 25.47 | 25.12 | 25.06 | 24.97
| 0.25 | 0.25 | 0.39 | 0.39 | 0.37 | 0.36
$s$ | 0.8674 | 1.0055 | 0.8679 | 0.8676 | 1.0059 | 1.0063
| 0.0041 | 0.0048 | 0.0042 | 0.0043 | 0.0047 | 0.0048
$q$ ($10^{-5}$) | 12.90 | 13.18 | 12.80 | 12.92 | 12.99 | 13.13
| 1.35 | 1.40 | 1.34 | 1.44 | 1.46 | 1.42
$\alpha$(rad) | 2.2775 | 2.2754 | 2.2733 |  $-$2.2764 | 2.2718 |  $-$2.2749
| 0.0039 | 0.0037 | 0.0087 | 0.0093 | 0.0077 | 0.0081
$\rho$ |  $<$0.011 |  $<$0.011 |  $<$0.011 |  $<$0.011 |  $<$0.011 |  $<$0.011
$\pi_{\rm E,N}$ | … | … | $-0.461$ | $0.078$ | $-0.510$ | $0.070$
| … | … | $1.049$ | $1.094$ | $0.938$ | $0.999$
$\pi_{\rm E,E}$ | … | … | $-0.108$ | $-0.054$ | $-0.111$ | $-0.053$
| … | … | $0.113$ | $0.092$ | $0.105$ | $0.084$
$\pi_{{\rm E},\parallel}$ | … | … | $0.062$ | $0.060$ | $0.058$ | $0.058$
| … | … | $0.041$ | $0.039$ | $0.041$ | $0.041$
$\pi_{{\rm E},\perp}$ | … | … | $-0.469$ | $0.074$ | $-0.519$ | $0.066$
| … | … | $1.054$ | $1.097$ | $0.943$ | $1.002$
$\psi$ (deg) | … | … | $275.7$ | $274.4$ | $275.9$ | $274.2$
$I_{\rm S,KMTC01}$ | 19.420 | 19.411 | 19.429 | 19.429 | 19.419 | 19.420
| 0.014 | 0.015 | 0.017 | 0.018 | 0.016 | 0.017
Table 6: 2L1S parameters for OGLE-2019-BLG-1492 Parameters | Static | Parallax
---|---|---
| Inner | Outer | Inner $u_{0}>0$ | Inner $u_{0}<0$ | Outer $u_{0}>0$ | Outer $u_{0}<0$
$\chi^{2}$ | 6792. | 6793.4 | 6789.0 | 6789.6 | 6790.2 | 6790.6
dof | 6791 | 6791 | 6789 | 6789 | 6789 | 6789
$t_{0}$ | 8763.137 | 8763.142 | 8763.120 | 8763.125 | 8763.131 | 8763.129
| 0.027 | 0.026 | 0.032 | 0.032 | 0.032 | 0.033
$u_{0}$ | 0.0507 | 0.0508 | 0.0473 |  $-$0.0480 | 0.0480 |  $-$0.0478
| 0.0032 | 0.0031 | 0.0043 | 0.0042 | 0.0045 | 0.0042
$t_{\rm E}$ (days) | 50.1 | 50.1 | 52.8 | 52.4 | 52.4 | 52.1
| 2.7 | 2.7 | 4.2 | 4.0 | 4.1 | 3.9
$s$ | 0.898 | 1.044 | 0.903 | 0.904 | 1.043 | 1.045
| 0.015 | 0.018 | 0.014 | 0.015 | 0.018 | 0.019
$q$ ($10^{-5}$) | 19.1 | 17.6 | 18.0 | 17.8 | 17.3 | 17.6
| 5.8 | 5.4 | 4.9 | 5.4 | 4.9 | 5.0
$\alpha$ (rad) | 2.123 | 2.120 | 2.140 |  $-$2.135 | 2.134 |  $-$2.139
| 0.023 | 0.022 | 0.026 | 0.030 | 0.030 | 0.031
$\rho$ |  $<$0.009 |  $<$0.009 |  $<$0.009 |  $<$0.009 |  $<$0.009 |  $<$0.009
$\pi_{\rm E,N}$ | … | … | $-0.342$ | $0.143$ | $-0.210$ | $0.575$
| … | … | $0.858$ | $0.957$ | $0.922$ | $0.815$
$\pi_{\rm E,E}$ | … | … | $-0.062$ | $-0.026$ | $-0.042$ | $-0.001$
| … | … | $0.093$ | $0.102$ | $0.106$ | $0.108$
$\pi_{{\rm E},\parallel}$ | … | … | $-0.047$ | $-0.034$ | $-0.030$ | $-0.042$
| … | … | $0.085$ | $0.088$ | $0.092$ | $0.091$
$\pi_{{\rm E},\perp}$ | … | … | $0.344$ | $-0.141$ | $0.212$ | $-0.573$
| … | … | $0.859$ | $0.959$ | $0.923$ | $0.817$
$\psi$ (deg) | … | … | $92.5$ | $93.1$ | $93.3$ | $94.1$
$I_{\rm S,KMTC01}$ | 21.01 | 21.00 | 21.09 | 21.07 | 21.07 | 21.07
| 0.07 | 0.07 | 0.10 | 0.10 | 0.10 | 0.10
Table 7: 2L1S Parameters for OGLE-2018-BLG-0977 Parameter | Value | Error
---|---|---
$\chi^{2}$/dof | $1553.0/1553$ |
$t_{0}$ | 8276.600 | 0.018
$u_{0}$ | 0.1470 | 0.0097
$t_{\rm E}$ (days) | 20.37 | 1.15
$s$ | 0.8793 | 0.0066
$q$ ($10^{-5}$) | 4.15 | 0.43
$\alpha$ (rad) | 2.5063 | 0.0079
$\rho$ |  $<$0.0033 |
$I_{\rm S,KMTC01}$ | 20.268 | 0.077
Table 8: 2L1S Parameters for KMT-2019-BLG-0953 Parameter | Value | Error
---|---|---
$\chi^{2}/dof$ | $909.0/9092$ |
$t_{0}$ | 8638.149 | 0.015
$u_{0}$ | 0.491 | 0.023
$t_{\rm E}$ (days) | 11.62 | 0.39
$s$ | 0.736 | 0.010
$q$ ($10^{-5}$) | 17.8 | 3.4
$\alpha$ (rad) | 2.3601 | 0.0077
$\rho$ |  $<$0.028 |
$I_{\rm S,KMTC02}$ | 18.474 | 0.078
Table 9: CMD Parameters for Six $q<2\times 10^{-4}$ Events

Parameter | KB190253 | OB180506 | OB180516 | OB191492 | OB180977 | KB190953
---|---|---|---|---|---|---
$(V-I)_{\rm s}$ | 2.92$\pm$0.01 | 4.59$\pm$0.07 | 2.07$\pm$0.02 | 1.48$\pm$0.04 | 2.17$\pm$0.05 | 2.05$\pm$0.04
$(V-I)_{\rm cl}$ | 3.31$\pm$0.02 | 4.47$\pm$0.02 | 2.58$\pm$0.02 | 1.84$\pm$0.02 | 2.45$\pm$0.03 | 2.56$\pm$0.03
$(V-I)_{\rm cl,0}$ | 1.06 | 1.06 | 1.06 | 1.06 | 1.06 | 1.06
$(V-I)_{\rm s,0}$ | 0.67$\pm 0.03$ | 1.18$\pm$15.10 | 0.55$\pm$0.03 | 0.70$\pm$0.05 | 0.78$\pm$0.05 | 0.55$\pm$0.05
$I_{\rm s}$ | 19.80$\pm$0.02 | 19.37$\pm$0.02 | 19.45$\pm$0.02 | 21.10$\pm$0.10 | 20.20$\pm$0.08 | 18.46$\pm$0.08
$I_{\rm cl}$ | 16.95$\pm$0.03 | 18.82$\pm$0.04 | 16.35$\pm 0$0.03 | 15.40$\pm$0.04 | 16.12$\pm$0.03 | 16.16$\pm$0.03
$I_{\rm cl,0}$ | 14.44 | 14.55 | 14.48 | 14.38 | 14.47 | 14.38
$I_{\rm s,0}$ | 17.29$\pm$0.03 | 15.10$\pm$0.05 | 17.58$\pm$0.04 | 20.08$\pm$0.11 | 18.55$\pm$0.10 | 16.68$\pm$0.09
$\theta_{*}$ ($\mu{\rm as}$) | 1.05$\pm$0.07 | 5.11$\pm$0.35 | 0.81$\pm$0.05 | 0.30$\pm$0.04 | 0.66$\pm$0.08 | 1.22$\pm$0.08
$\theta_{\rm E}$ (mas) | $>0.23$ | $>0.43$ | $>0.08$ | $>0.03$ | $>0.20$ | $>0.044$
$\mu_{\rm rel}$ (${\rm mas}\,{\rm yr}^{-1}$) | $>1.5$ | $>6.5$ | $>1.1$ | $>0.2$ | $>3.6$ | $>1.4$

Note. — For parallax solutions, $(s,q,\rho,\alpha)$ are held fixed at their
static values, and only the OGLE data are included in the fit. The upper
limits on $\rho$ are at $3\sigma$.

Note. — The upper limits on $\rho$ are at $3\sigma$.

Note. — The upper limits on $\rho$ are at $3\sigma$.

Note. — The upper limits on $\rho$ are at $3\sigma$.

Note. — The upper limit on $\rho$ is at $3\sigma$.

Note. — The upper limits on $\rho$ are at $3\sigma$.

Note. — Event names are abbreviations for, e.g., OGLE-2018-BLG-0506 and
KMT-2019-BLG-0253. Lower limits on $\theta_{\rm E}$ and $\mu_{\rm rel}$ are
$3\,\sigma$.

Table 10: Bayesian Estimates of Physical parameters Event Name | $M_{\rm host}[M_{\odot}]$ | $M_{\rm planet}[M_{\oplus}]$ |  $D_{\rm L}$[kpc] |  $a_{\perp}$[au] |  $\mu_{\rm rel}$[mas]
---|---|---|---|---|---
KMT-2019-BLG-0253 | $0.70_{-0.31}^{+0.34}$ | $9.2_{-4.1}^{+5.0}$ | $4.9_{-1.6}^{+1.9}$ | $3.1_{-0.9}^{+0.8}$ | $4.7_{-2.1}^{+2.8}$
OGLE-2018-BLG-0506 | $0.63_{-0.32}^{+0.37}$ | $16.3_{-8.5}^{+12.0}$ | $5.6_{-1.6}^{+1.4}$ | $3.0_{-0.8}^{+0.7}$ | $8.1_{-1.5}^{+2.5}$
OGLE-2018-BLG-0516 | $0.47_{-0.25}^{+0.38}$ | $20.1_{-10.7}^{+16.4}$ | $7.0_{-2.2}^{+0.9}$ | $2.2_{-0.7}^{+0.8}$ | $5.0_{-1.9}^{+2.9}$
OGLE-2019-BLG-1492 | $0.68_{-0.34}^{+0.34}$ | $37.1_{-19.5}^{+25.1}$ | $5.6_{-2.2}^{+1.7}$ | $2.7_{-0.9}^{+0.9}$ | $3.8_{-1.7}^{+3.2}$
OGLE-2018-BLG-0977 | $0.47_{-0.27}^{+0.38}$ | $6.4_{-3.7}^{+5.2}$ | $6.5_{-2.2}^{+1.2}$ | $2.0_{-0.5}^{+0.6}$ | $6.9_{-1.8}^{+3.0}$
KMT-2019-BLG-0953 | $0.28_{-0.17}^{+0.32}$ | $15.5_{-9.7}^{+19.0}$ | $7.2_{-1.4}^{+0.7}$ | $1.1_{-0.4}^{+0.5}$ | $7.0_{-2.5}^{+3.3}$

Table 11: Recovered Planets in KMT Prime Fields for 2018-2019

Event Name | KMT Name | $\log q$ | $s$ | Reference
---|---|---|---|---
OB181185 | KB181024 | $-4.17$ | 0.96 | Kondo et al. (2021)
KB181025a | KB181025 | $-4.03$ | 0.95 | Han et al. (2021a)
OB180532 | KB181161 | $-4.01$ | 1.01 | Ryu et al. (2020)
OB180596 | KB180945 | $-3.74$ | 0.51 | Jung et al. (2019b)
OB181269 | KB182418 | $-3.24$ | 1.12 | Jung et al. (2020a)
OB180932 | KB182087 | $-3.15$ | 0.74 | in prep
OB180567 | KB180890 | $-2.91$ | 1.81 | Jung et al. (2021)
KB180748 | KB180748 | $-2.69$ | 0.94 | Han et al. (2020b)
OB180962 | KB182071 | $-2.62$ | 1.25 | Jung et al. (2021)
OB180100a | KB182296 | $-2.58$ | 1.30 | in prep
OB181011b,c | KB182122 | $-2.02$ | 0.75 | Han et al. (2019)
OB181700b | KB182330 | $-2.00$ | 1.01 | Han et al. (2020a)
OB181647b | KB182060 | $-1.95$ | 1.27 | in prep
OB181011c | KB182122 | $-1.82$ | 0.58 | Han et al. (2019)
OB181544 | KB180787 | $-1.60$ | 0.50 | Han et al. in prep
KB190842 | KB190842 | $-4.39$ | 0.98 | Jung et al. (2020b)
KB191953d | KB191953 | $-2.72$ | 2.30 | Han et al. (2020c)
OB191180 | KB191912 | $-2.28$ | 1.90 | Chung et al. in prep
KB191552 | KB191552 | $-2.17$ | 0.74 | in prep
OB190954 | KB193289 | $-1.77$ | 0.74 | Han et al. (2021b)
OB190825 | KB191389 | $-1.63$ | 3.78 | Sato et al. in prep
KB193301 | KB193301 | $-1.40$ | 1.47 | Han et al. in prep
KB190371e | KB190371 | $-1.10$ | 0.83 | Kim et al. (2021b)

Note. — Event names are abbreviations for, e.g., OGLE-2018-BLG-1185 and
KMT-2018-BLG-1024. All $(s,q)$ from “in prep” events should be regarded as
preliminary. a: large $q$ degeneracy at $\Delta\chi^{2}<10$, or very large $q$
error. b: $s$ degeneracy. c: Two-planet system. d: identified as FSPL, but not
investigated to find planet. e: inner/outer degeneracy; with factor 1.5 higher
$q$.

Table 12: $\Delta\chi^{2}_{0}$ for $q<2\times 10^{-4}$ Sample Planets

Event Name | KMT Name | $\Delta\chi^{2}_{0}$ | $\log q$ | Method
---|---|---|---|---
OB191492 | KB193004 | 87 | $-3.75$ | Discovery
OB180977 | KB180728 | 117 | $-4.38$ | Discovery
OB180506 | KB180835 | 163 | $-4.09$ | Discovery
OB180516 | KB180808 | 197 | $-3.93$ | Discovery
KB190253 | KB190253 | 260 | $-4.39$ | Discovery
KB190953 | KB190953 | 283 | $-3.75$ | Discovery
OB191053 | KB191504 | 302 | $-4.90$ | Discovery
KB190842 | KB190842 | 585 | $-4.39$ | Recovery
OB181185 | KB181024 | 945 | $-4.17$ | Recovery
OB180596 | KB180945 | 1828 | $-3.74$ | Recovery
OB180532 | KB181161 | 7657 | $-4.01$ | Recovery

Note. — Event names are abbreviations for, e.g., OGLE-2018-BLG-1185 and
KMT-2018-BLG-1024.

Figure 1: Light curve and models of KMT-2019-BLG-0253. The main evidence for
the “inner” and “outer” planetary model (which are virutally identical) is the
“dip” in KMTC data from Chile, near 8588.6. This dip is supported by
contemporaneous OGLE data, also from Chile, and also by two points from KMTS
in South Africa. The dip is the signature of a minor-image perturbation,
passing either “inside” or “outside” the two triangular caustics associated
with the minor image. See Figure 2.  Figure 2: Caustic topologies for all 6
events. For the first 4, the degenerate pairs have identical topologies. In
each case, for the “inner” topology, the source passes between the central
caustic and the two triangular caustics associated with the minor image
(opposite side of host from the planet). The “dip” is due to a magnificatoin
trough that runs along the planet-host axis, between these caustics. For the
“outer” topology, these caustics have merged with the central caustic to form
a resonant caustic. The “dip” is then due to the magnification trough that
extends out from between these two wings of the resulting resonant caustic.
For OGLE-2018-BLG-0977, the source intersects the triangular caustics, so
there is no degeneracy. For KMT-2019-BLG-0953, there is only an “outer”
solution. See Section 3.7.2.  Figure 3: Light curve and models of
OGLE-2018-BLG-0506. The main evidence for an anomaly comes from the dip at
about 8224.7 as seen in 26 KMTC points. This is confirmed by two OGLE points.
The caustic topology is the same as for KMT-2019-BLG-0253. See Figure 2.
Figure 4: Light curve and models of OGLE-2018-BLG-0516. The main evidence for
an anomaly comes from the dip at about 8229.85 . This is confirmed by two OGLE
points. The KMTA and MOA data trace the end of the dip, while the KMTS data
are consistent with the beginning of the dip. The caustic topology is the same
as for KMT-2019-BLG-0253 and OGLE-2018-BLG-0506. See Figure 2.  Figure 5:
Light curve and models of OGLE-2019-BLG-1492. The main evidence for the
anomaly comes from the dip in KMTC data at about 8764.65, which is confirmed
by two OGLE points. It is also supported by the post-dip “ridge” in KMTS data
at about 8665.25. The caustic topology is the same as for KMT-2019-BLG-0253,
OGLE-2018-BLG-0506, and OGLE-2018-BLG-0516. See Figure 2.  Figure 6: Light
curve and model of OGLE-2018-BLG-0977. The KMTS data near 8280.5 briefly rise
and then sharply fall, which indicates entrance into the trough that threads
the two minor-image triangular caustics, near (but not over) one of these
caustics. The KMTA data near 8281.0 show a sharp decline, followed by
flattening, consistent with an exit from the triangular caustic on the
opposite side of the trough. The caustic-topology diagram (Figure 2) confirms
this qualitative light-curve analysis.  Figure 7: Light curve and model of
KMT-2019-BLG-0953. The KMTS and KMTC data show a low-amplitude dip
$8643.4\lesssim{\rm HJD}^{\prime}<8643.9$, which is not much larger than the
individual error bars, but because the event lies in a small region with very
high cadence, there are $>100$ data points during this 12 hour interval. Eight
OGLE points near the trough confirm the dip. The dip indicates that the source
passed near, but not over the minor image due to the host. See Figure 2.
Figure 8: CMDs for each of the six $q<2\times 10^{-4}$ planets reported here.
The source positions (blue) and clump-giant centroids (red) are shown for all
events. For KMT-2019-BLG-0253, the baseline object (green) and blended light
(magenta) are also shown. Photometry is in the calibrated OGLE-III system for
all except OGLE-2018-BLG-0506 (lower left), which is in the KMTC41 pyDIA
system.  Figure 9: Cumulative distribution of recovered and discovered KMT
prime-field planets from 2018-2019 (black) compared to the mass-ratio-function
breaks proposed by Suzuki et al. (2016) and Jung et al. (2019a) (magenta). The
cyan dashed line simply connects the first and last KMT detections. If the
underlying distribution of mass ratios (after multiplying by the detection
efficiency) were uniform in $\log q$, the cumulative distribution would
approximately follow this line. The red event names at left indicate the mass
ratios of the only known $q<2\times 10^{-4}$ planets from 2018-2019 non-prime
fields. If no other such planets are found when the AnomalyFinder is applied
to these fields, the cumulative distribution will be as shown in red.