# Shortest Microlensing Event with a Bound Planet: KMT-2016-BLG-2605

Yoon-Hyun Ryu1, Kyu-Ha Hwang1, Andrew Gould2,3, Jennifer C. Yee4, Michael
D.Albrow5, Sun-Ju Chung1,6, Cheongho Han7, Youn Kil Jung1, Hyoun-Woo Kim1,8,
In-Gu Shin1, Yossi Shvartzvald9, Weicheng Zang10, Sang-Mok Cha1,11, Dong-Jin
Kim1, Seung-Lee Kim1,6, Chung-Uk Lee1,6, Dong-Joo Lee1, Yongseok Lee1,11,
Byeong-Gon Park1,6, Richard W. Pogge3
1Korea Astronomy and Space Science Institute, Daejon 34055, Republic of Korea
2Max-Planck-Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg,
Germany  3Department of Astronomy, Ohio State University, 140 W. 18th Ave.,
Columbus, OH 43210, USA  4 Center for Astrophysics $|$ Harvard & Smithsonian,
60 Garden St., Cambridge, MA 02138, USA  4Department of Astronomy and Space
Science, Chungbuk National University, Cheongju 28644, Republic of Korea
5University of Canterbury, Department of Physics and Astronomy, Private Bag
4800, Christchurch 8020, New Zealand  6Korea University of Science and
Technology, Korea, (UST), 217 Gajeong-ro, Yuseong-gu, Daejeon, 34113, Republic
of Korea  7Department of Physics, Chungbuk National University, Cheongju
28644, Republic of Korea  8Department of Astronomy and Space Science, Chungbuk
National University, Cheongju 28644, Republic of Korea  9Department of
Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot
76100, Israel  10Physics Department and Tsinghua Centre for Astrophysics,
Tsinghua University, Beijing 100084, China  11School of Space Research, Kyung
Hee University, Yongin, Kyeonggi 17104, Republic of Korea

###### Abstract

KMT-2016-BLG-2605, with planet-host mass ratio $q=0.012\pm 0.001$, has the
shortest Einstein timescale, $t_{\rm E}=3.41\pm 0.13\,$days, of any planetary
microlensing event to date. This prompts us to examine the full sample of 7
short ($t_{\rm E}<7\,$day) planetary events with good $q$ measurements. We
find that six have clustered Einstein radii $\theta_{\rm E}=115\pm 20\,\mu{\rm
as}$ and lens-source relative proper motions $\mu_{\rm rel}\simeq 9.5\pm
2.5\,{\rm mas}\,{\rm yr}^{-1}$. For the seventh, these two quantities could
not be measured. These distributions are consistent with a Galactic-bulge
population of very low-mass (VLM) hosts near the hydrogen-burning limit. This
conjecture could be verified by imaging at first adaptive-optics light on
next-generation (30m) telescopes. Based on a preliminary assessment of the
sample, “planetary” companions (i.e., below the deuterium-burning limit) are
divided into “genuine planets”, formed in their disks by core accretion, and
very low-mass brown dwarfs, which form like stars. We discuss techniques for
expanding the sample, which include taking account of the peculiar “anomaly
dominated” morphology of the KMT-2016-BLG-2605 light curve.

gravitational lensing: micro

##  1 Introduction

Microlensing planets are almost always discovered from short-lived
perturbations on otherwise single-lens single-source (1L1S) bell-shaped
Paczyński (1986) light curves, as predicted by (Mao & Paczyński, 1991), Gould
& Loeb (1992), and Griest & Safizadeh (1998). However, there are occasional
exceptions.

If the planet-host mass ratio $q$ is relatively large and the lens-source
separation (normalized to the Einstein radius, $\theta_{\rm E}$) is close to
unity, $s\sim 1$, then the central and planetary caustics merge into a single,
large, resonant caustic, which can induce a long-term anomaly over the peak of
the event. For example, the $q=10^{-2}$ event MOA-2009-BLG-387 (Batista et
al., 2011) showed strong, continuous anomalies over 9 days. Nevertheless, over
the remainder of the $2\,t_{\rm E}\sim 80\,$days of the event, it appeared as
qualitatively normal. And, indeed, it is on the basis of this normal rising
behavior that the Microlensing Follow Up Network ($\mu$FUN) initiated followup
observations two days before the anomaly. Here, $t_{\rm E}$ is the Einstein
timescale.

A very different counter-example, in this case a “purely anomalous event” is
given by MOA-bin-1 (Bennett et al., 2012), for which the entire observed event
consists of the source crossing the planetary caustic. Because the planet-host
separation is $s\sim 2$ and the source trajectory is nearly perpendicular to
the planet-host axis, $\alpha\sim 90^{\circ}$, the $t_{\rm E}\sim 31\,$day
“host event” leaves barely a trace on the light curve. Thus, essentially all
that remains is the short, $\sim 0.2\,$ day triangular anomaly due to the
$q\sim 5\times 10^{-3}$ planet.

Here we present another rare case for which the light curve is dominated by a
planet-induced anomaly, KMT-2016-BLG-2605. Like MOA-2009-BLG-387, the anomaly
is due to a resonant caustic of a $q\sim 10^{-2}$ planet. And like MOA-bin-1,
the duration of the anomaly is short ($1.5\,$days). Indeed, the observed
portion of the anomaly is only $\sim 0.5\,$ days. However, in contrast to
either of these cases, the underlying timescale is very short, $t_{\rm
E}=3.4\,$days, while the microlensed source is faint, $I_{s}=20.2$, so that
only the anomaly is really noticeable, particularly in the initial reductions
from which the event was discovered. For this reason, the event was not
discovered in the original search carried out by the Korean Microlensing
Telescope Network (KMTNet, Kim et al. 2016) EventFinder system (Kim et al.,
2018a), during which it was misclassified as a cataclysmic variable (CV). It
was recognized as genuine microlensing only as a byproduct of a special search
that was conducted for another purpose (Kim et al., 2020).

KMT-2016-BLG-2605 is one of only seven short-timescale ($t_{\rm E}<7\,$day)
planetary events with a well measured mass ratio $q$ (less than factor 2
difference between competing solutions at $\Delta\chi^{2}<10$). The roughly
comparable properties of this ensemble ($\theta_{\rm E}\sim 0.11\,{\rm mas}$,
$\mu_{\rm rel}\sim 9\,{\rm mas}\,{\rm yr}^{-1}$, where $\mu_{\rm rel}$ is the
lens-source relative proper motion) are consistent with a population of
Galactic-bulge hosts that have masses near the hydrogen-burning limit. For six
of these seven (including KMT-2016-BLG-2605), this assessment can be confirmed
or contradicted at first adaptive-optics (AO) light on next-generation (30m
class) telescopes.

Despite the fact that all seven were detected through resonant or near-
resonant caustics, KMT-2016-BLG-2605 is the only anomaly-dominated event,
which would potentially make it more difficult to recognize during the manual
stage of event selection. We consider this and other factors to evaluate the
challenges in identifying more of these short-$t_{\rm E}$ planetary systems.

##  2 Event Identification

KMT-2016-BLG-2605 was identified as a “clear” microlensing event during a
special search of the 2016 KMTNet data that was conducted as part of a project
to identify all finite-source point-lens (FSPL) giant-source events during the
four year span 2016-2019. The primary objective of this project is to create a
statistically well defined parent sample that will contain a free-floating
planet (FFP) subsample. A special additional search was found to be necessary
because a significant fraction of short FSPL events were missed by the primary
EventFinder (Kim et al., 2018a) searches that are done annually. For example,
some giant-star sources had been eliminated from the search because of
previously cataloged variability, or light-curve artifacts. Of course, it is
known that variable stars can undergo microlensing events that can be
distinguished from their intrinsic variations, and also that many light-curve
artifacts do not repeat from year to year. However, the fraction of real
events that are removed this way is small $(\sim 1\%)$, while the human cost
of manual review is high. Thus, removal of these sources from the regular
search is a rational approach. Nevertheless, when the search is restricted to
giant sources, the cost is reduced by a factor of about 20, making lifting (or
strongly modifying) these criteria worthwhile for the FSPL project.

Another feature of the special search was that the candidates selected by the
machine algorithm were shown to the operator in several additional displays.
This is because the shortest FSPL events can be highly anomalous due to finite
source effects. Thus, when the three data sets are automatically aligned using
Paczyński (1986) or Gould (1996) fits, the joint light curve can appear to be
“clearly not microlensing”. By having multiple displays, by having the
operator spend more time reviewing each candidate, and by adopting somewhat
lower standards on what is a plausible microlensing event, it is much less
likely that these short FSPL events will be rejected at this stage. Of course,
non-microlensing events can still be rejected at a later stage when each
candidate is manually fit to point-source point-lens (PSPL) and FSPL
functional forms. For more details on these special searches, see Kim et al.
(2020) and Ryu et al. (2021).

The special search for 2016 identified 281 candidates, of which 37 had not
been found in the regular EventFinder search111Of these 37, 10 had previously
been identified by other teams, including seven by OGLE, two by MOA and one by
both.. KMT-2016-BLG-2605 was number 17 on this list of 37 new candidates.
Following the convention of Mróz et al. (2020) and Ryu et al. (2021), it was
assigned the sequential label “2605” ($=2588+17$) because there were 2588
events discovered in the original search.

KMT-2016-BLG-2605 is neither an FSPL event nor does it have a giant-star
source. Nevertheless, it passed the various selection criteria imposed to
obtain a sample of just seven new events222Two of these seven proved to be
FSPL events. from 2016 that would then be subjected to manual FSPL fitting.
The machine search of the KMTNet database is restricted to “giant” source
stars, defined as having dereddened baseline magnitudes $I_{\rm base,0}=I_{\rm
cat}-A_{I}<16.2$, where $I_{\rm cat}$ is the magnitude of the catalog entry,
$A_{I}=A_{K}/7$, and $A_{K}$ is derived from Groenewegen (2004). This led to
$I_{\rm base,0}=18.63-2.74=15.89$. Being a very short event, it then easily
passed the machine-search criterion that the effective timescale be less than
5 days. When it was displayed to the operator, it did not look like 1L1S
microlensing (neither PSPL nor FSPL), but at this stage the only criterion is
that the variation is plausibly due to microlensing. As described by Kim et
al. (2020), all events (including these 37 from the 2016 special search) are
selected for manual review by two criteria. The first is
$I_{s,0}=I_{s}-A_{I}<16$, where $I_{s}$ is the source magnitude from the
pipeline fit to the event. As we will show in Section 4, the true value of
$I_{s,\rm kmt}=20.0$, for which case $I_{s,0}=17.3$, i.e., failing this
criterion by more than a magnitude. Nevertheless, due to the extremely
anomalous form of the light curve, the pipeline fit assigned $I_{s}\rightarrow
I_{\rm cat}$, which allowed the event to pass this criterion. Second, it
easily passed the criterion meant to select plausibly FSPL (as opposed to
almost certainly PSPL) events, $\mu_{\rm thresh}=3\,{\rm
mas}\,10^{(16-I_{s,0})/5}/u_{0}t_{\rm E}>1\,{\rm mas}\,{\rm yr}^{-1}$, where
$u_{0}$ is the impact parameter (normalized to $\theta_{\rm E}$) of the
pipeline fit. Given that $u_{0}t_{\rm E}=0.147\times 1.59\,{\rm day}=5.6\,$hr,
it had $\mu_{\rm thresh}=5\,{\rm mas}\,{\rm yr}^{-1}$.

It was only in the course of fitting the event to 1L1S models by hand that it
became clear that the slope of the light curve showed a discontinuous change
at HJD${}^{\prime}={\rm HJD}-2450000=7565.3$, indicating a caustic crossing,
which could plausibly be explained by a planetary system.

In brief, KMT-2016-BLG-2605 came to our attention by a most unlikely and
circuitous path, a point to which we will return in Section 7. For example, if
the source star were not blended with another star that was several times
brighter, it would not have even been selected for machine fitting at the
first step.

##  3 Observations

KMT-2016-BLG-2605 is at equatorial coordinates
(R.A.,Decl)${}_{J2000}=$(17:59:17.54, $-$26:58:55.20), corresponding to
galactic coordinates $(l,b)=(+3.22,-1.60)$. It therefore lies in KMTNet field
BLG03. KMTNet consists of three 1.6m telescopes, each equipped with a $4\,{\rm
deg^{2}}$ camera, and located in Chile (KMTC), South Africa (KMTS), and
Australia (KMTA). At the time of the event, BLG03 was observed with a cadence
$\Gamma=2\,{\rm hr}^{-1}$ from each observatory, primarily in the $I$ band. In
2016, every tenth $I$-band observation at KMTC was complemented by one in the
$V$ band, as was every twentieth observation at KMTS. There were no $V$-band
observations from KMTA333Based on experience with OGLE-2015-BLG-1459 (Hwang et
al., 2019), it was realized that KMTA $V$-band observations were potentially
very important, and ultimately both KMTS and KMTA were observed in the $I$ and
$V$ bands at a ratio of $10:1$. Unfortunately, this was not the case in 2016..

The data were initially reduced using pySIS (Albrow et al., 2009), which is a
specific implementation of difference image analysis (Tomaney & Crotts, 1996;
Alard & Lupton, 1998). Moreover, the original light-curve analysis was based
on tender loving care (TLC) pySIS re-reductions. However, as we describe in
Section 4, it was ultimately necessary to use a related package, pyDIA
(Albrow, 2017) because it returns field-star photometry on the same system as
the light curve. To avoid confusion, we present the entire investigation of
the light curve using pyDIA photometry.

##  4 Light Curve Analysis

We fit the data to binary-lens single-source (2L1S) models, which are
characterized by seven parameters $(t_{0},u_{0},t_{\rm E},s,q,\alpha,\rho)$,
where $t_{0}$ is the time of closest lens-source approach and
$\rho=\theta_{*}/\theta_{\rm E}$ is the source radius normalized to the
Einstein radius. As is almost always done, we begin with a grid search over an
$(s,q,\alpha)$ grid, in which $(s,q)$ are held fixed and $(t_{0},u_{0},t_{\rm
E},\alpha,\rho)$ are allowed to vary in a Monte Carlo Markov chain (MCMC). The
three Paczyński (1986) parameters $(t_{0},u_{0},t_{\rm E})$ are seeded at the
PSPL fit, while $\rho$ is seeded at $\rho=0.003$.

However, in contrast to typical experience, we find a plethora of quite
distinct solutions. After refitting each local minimum of the grid with all
seven parameters allowed to vary, we find seven distinct solutions within the
range of $\Delta\chi^{2}<100$. However, only Locals 1–3 are potentially
viable, with $\Delta\chi^{2}<11$, while Locals 4–7 have $\Delta\chi^{2}>50$.
Closer examination of Local 1 shows that it breaks up into three nearby
minima. See Table 1. Figure 1 shows the models and data for the three
principal Locals, while Figure 2 shows the principal Local 1 and its two
satellite solutions.

Comparing Locals 1 and 1b in Figure 2, we see that the peak of the former
(specifically, the second KMTS point at HJD${}^{\prime}=7565.28$) is 0.25 mag
fainter. This is unusual for competing microlensing solutions. When presenting
microlensing models, one data set (in this case, KMTC) is chosen as the
“anchor”. Its data values are exactly reproduced in the figure. All other data
sets are aligned, by linear regression of the fluxes to the model, to this
anchor. Because this alignment is usually based on several nights of data
during which the event is evolving in a regular way, the alignment
coefficients are normally the same for different models. Hence, the different
data sets are rigidly aligned to the same fiducial scale, independent of
model. For this reason, all models can be shown by curves that are superposed
on data points whose positions are fixed.

However, in the present case, the KMTS and KMTA data are strongly magnified on
only one night. Hence the alignment is not fully constrained by other nights.
The lack of a rigid constraints is reflected in the range of values in the
quantity $\Delta I=I_{S,,\rm kmts}-I_{S,\rm kmtc}$ that is shown in Table 1.

At most one of these $\Delta I$ values can be correct. That is, these offsets
represent the relative transparency and throughput of the detectors at the two
sites. And this quantity can be measured from field stars (Gould et al., 2010;
Yee et al., 2012).

To make this comparison precise, we have carried out the MCMCs with the
source-flux and blend-flux parameters from each observatory treated as chain
variables, That is, normally one writes

$F_{i}(t_{i,j})=f_{s,i}A(t_{i,j};t_{0},u_{u},t_{\rm
E},s,q,\alpha,\rho)+f_{b,i}\ ,$ (1)

where $F_{i}(t)$ is the observed flux from observatory $i$ at time $t$, and
$(f_{s,i},f_{b,i})$ are the source-flux and blend-flux parameters for
observatory $i$. At each step on the MCMC, one inserts the trial values for
the seven parameters $(t_{0},u_{u},t_{\rm E},s,q,\alpha,\rho)$, but one
determines $(f_{s,i},f_{b,i})$ from a linear fit to the model magnifications
$A$. For the vast majority of cases, the errors that would be induced in these
parameters due to the flux errors at a given model is tiny compared their
error due to variation between different models. Hence, this approach is
usually appropriate.

However, for the present case, the value of $f_{s,\rm kmtc}$ is basically
determined by just 16 data points on the night after the peak. These
individually have fractional scatter $\sigma(F)/[(A-1)f_{s}\sim 0.075$, which
implies that the standard error of the mean should be of order $(2.5/\ln
10)\times 0.075/\sqrt{16}\sim 0.02$ mag. This is too large to be ignored in
the present context. Hence, we also treat $f_{s}$ and $f_{b}$ as chain
parameters.

Finally, we match the field-star photometry of the KMTS03 and KMTC03
reductions, and we plot the differences as a function of magnitude in Figure
3. For this purpose, we only include stars with $(V-I)_{\rm kmts}>3.0$, which
includes the colors of the source and the red clump. We plot the predicted
offsets from the five models as horizontal lines with error ranges.

We see that $(I_{s,\rm kmts}-I_{s,\rm kmtc})_{\rm Local-1}$ is consistent with
the field stars at $1\,\sigma$, whereas $(I_{s,\rm kmts}-I_{s,\rm kmtc})$ is
inconsistent at $\geq 2.3\,\sigma$ for all the other solutions. Keeping in
mind that Local 1 was already favored over Locals 2 and 3 by
$\Delta\chi^{2}\gtrsim 10$ (and by slightly less compared to its satellite
solutions), we regard this as clear confirmation of Local 1.

Finally, we impose the flux constraint within the MCMC as a $\chi^{2}$
penalty,

$\chi^{2}_{\rm flux}=\biggl{(}{-2.5\log(f_{S,\rm kmts}/f_{S,\rm
kmtc})--0.033\over 0.005}\biggr{)}^{2}.$ (2)

We find that Locals 1a and 1b disappear as separate minima, while Locals 2 and
3 are each disfavored by $\Delta\chi^{2}\sim 16$. See Table 2. We adopt the
Local 1 microlensing parameters in this table as our final result. Figure 4
shows the best-fit model and data after imposing this constraint. It also
shows the caustic topology, which is resonant.

##  5 Source Properties

As with most other microlensing events, we measure $\theta_{*}$ using the
method of Yoo et al. (2004). This requires that we first find the offset
$\Delta[(V-I),I]=[(V-I),I]_{s}-[(V-I),I]_{\rm cl}$ of the source star relative
to the clump. Adopting $[(V-I),I]_{\rm cl,0}=(1.06,14.34)$ (Bensby et al.,
2013; Nataf et al., 2013), we would then derive the dereddened source color
and magnitude, $[(V-I),I]_{s,0}=\Delta[(V-I),I]+[(V-I),I]_{\rm cl,0}$, convert
from $V/I$ to $V/K$ photometry using the $VIK$ color-color relations of
Bessell & Brett (1988), and finally, use the color/surface-brightness relation
of Kervella et al. (2004) to derive the angular radius of the source star,
$\theta_{*}$.

Unfortunately, the first step in this procedure, determining
$\Delta[(V-I),I]$, poses substantially greater challenges for
KMT-2016-BLG-2605 than it does for typical events because the color-magnitude
diagram (CMD) positions of both the source star and the clump are more
difficult to determine.

In the Appendix, we delineate the steps to measure the clump position, which
we show is well determined. However, we show that while the source magnitude
is also well determined, the source color remains somewhat ambiguous. The key
issue is that the source color derived from the light curve, which rests on a
single magnified $V$-band data point, is in formal conflict with Bayesian
expectations based on the well-determined magnitude (together with the
morphology of the CMD). After weighing all the evidence, we conclude that,
most likely, the discrepancy is due to a relatively large $(\lesssim
3\,\sigma)$ statistical error in the single $V$-band data point, and we adopt

$(V-I)_{s,0}=1.00\pm 0.05$ (3)

Then, following the steps outlined in the first paragraph of this section, we
find

$\theta_{*}=1.38\pm 0.10\,\mu{\rm as},$ (4)

where we have added 5% in quadrature to the error bar to take account of
systematics that are inherent to the method. From this we then derive,

$\theta_{\rm E}={\theta_{*}\over\rho}=0.116\pm 0.009\,{\rm mas};\qquad\mu_{\rm
rel}={\theta_{*}\over t_{*}}=12.3\pm 1.0\,{\rm mas}\,{\rm yr}^{-1},$ (5)

where $t_{*}\equiv\rho t_{\rm E}$.

However, in the Appendix, we also keep track of the possibility that the
light-curve color measurement is actually correct, in which case,
$\theta_{*}=1.77\pm 0.23\,\mu{\rm as}$, $\theta_{\rm E}=1.49\pm 0.19\,{\rm
mas}$, and $\mu_{\rm rel}=15.8\pm\,1.3\,{\rm mas}\,{\rm yr}^{-1}$.

##  6 Physical Parameters

We make a standard Bayesian analysis to derive physical parameters. That is,
we draw events randomly from a Galactic model, and we weight each simulated
event by how well it conforms to Equation (5). We additionally weight by the
event rate, $\Gamma\propto\theta_{\rm E}\mu_{\rm rel}$, although this has very
little effect because these parameters are very similar for all simulated
events that satisfy Equation (5). The Galactic model follows that of Jung et
al. (2018a) as modified by Jung et al. (2021). The results are reported in
Table 4, and they are illustrated in Figure 5, which shows that the lens
probably lies in the bulge, but even if not, it most likely lies in
overlapping regions of the disk. The median host mass estimate, $M_{\rm
host}=0.064_{-0.032}^{+0.099}$ is very close to the hydrogen-burning limit,
i.e., there is a roughly 50% probability that it is a brown dwarf (BD). If we
adopt a snow-line scaling $a_{\rm snow}=2.7{\rm AU}(M_{\rm host}/M_{\odot})$,
then the planets projected separation is $a_{\perp}\sim 4\,a_{\rm snow}$.

Kim et al. (2021) have shown that, unless $\mu_{\rm rel}>10\,{\rm mas}\,{\rm
yr}^{-1}$ (or there is additional information such as a microlens parallax
measurement), the Bayesian mass estimate depends only on $\theta_{\rm E}$.
While this condition does not apply to KMT-2016-BLG-2605, if we nevertheless
input $\theta_{\rm E}=0.116\,{\rm mas}$ into their Figures 7, we obtain
$M=0.08_{-0.04}^{+0.06}\,M_{\odot}$. The difference between this and our
Bayesian estimate is accounted for by the “bend” in the median trajectories
above $\mu=10\,{\rm mas}\,{\rm yr}^{-1}$ in their Figure 6.

##  7 Discussion

###  7.1 Ensemble of Short-timescale Planets

At $t_{\rm E}=3.41\,$days, KMT-2016-BLG-2605 has the shortest timescale of any
planetary microlensing event. There are eight previous binary-lens
microlensing events with $3.7<t_{\rm E}/{\rm day}<7$ that are listed by the
NASA Exoplanet Archive444https://exoplanetarchive.ipac.caltech.edu/. We chose
“7 days” as the upper limit, without foreknowledge of the sample that it would
produce, because 7 is the closest integer to $2\times t_{\rm E}/$day of
KMT-2016-BLG-2605.. For the present discussion, we restrict attention to the
subset with (1) unambiguous measurement of $q$ (specifically, no solutions
with $\Delta\chi^{2}<10$ and $q$ values differing by a factor $>2$); and (2)
“verifiable planet”, specifically $q<M_{\rm D-burn}/M_{\rm H-burn}=0.16$. That
is, we accept the formal definition of a “planet” as having mass $m_{p}$ below
the deuterium burning limit, $m_{p}<M_{\rm D-burn}$. Systems with $q$ above
this limit can be ruled out as planets if their hosts are stars $M_{\rm
host}>M_{\rm H-burn}$ because these can be imaged at late times. But a non-
detection would leave the status of the companion ambiguous. However, for
systems satisfying this condition, even a non-detection would prove the
companion had planetary mass. Two events fail criterion (1): MOA-bin-29 (Kondo
et al., 2019) and MOA-2015-BLG-337 (Miyazaki et al., 2018). One event fails
criterion (2): KMT-2016-BLG-2124 (Jung et al., 2018b).

The remaining five events are MOA-2011-BLG-262 (Bennett et al., 2014),
OGLE-2015-BLG-1771 (Zhang et al., 2020), KMT-2018-BLG-0748 (Han et al., 2020),
OGLE-2018-BLG-0677 (Herrera-Martin et al., 2020), and KMT-2016-BLG-1820 (Jung
et al., 2018b). We note that, strictly speaking, the first of these events has
an ambiguous measurement of $\rho$, i.e., $\rho=3.44\times 10^{-3}$ or
$\rho=5.73\times 10^{-3}$, with the first preferred by $\Delta\chi^{2}=3$.
However, the first solution would imply a geocentric proper motion555The
authors quote $19.6\pm 1.6\,{\rm mas}\,{\rm yr}^{-1}$, which may reflect a
posterior result after applying unstated Bayesian priors. $\mu_{\rm
rel}=21.6\pm 2.3\,{\rm mas}\,{\rm yr}^{-1}$. As the authors note, their OGLE-
III-based measurement of the source proper motion
${\mbox{\boldmath$\mu$}}_{s}(l,b)\sim(-2.4,-0.4)\pm(2.7,2.7)\,{\rm mas}\,{\rm
yr}^{-1}$ implies that the high proper-motion solution is inconsistent with
bulge lenses. It would imply that the lens must lie far in the foreground,
e.g., at $D_{L}\lesssim 1\,{\rm kpc}$. In this case, the host mass would be
$M\lesssim 6\,M_{J}$, with “planet” (aka “moon”) mass $m\lesssim
0.9\,M_{\oplus}$. The “novelty” of this putative system, combined with the
small number of potential lenses in the nearby observational cone, renders
this solution highly unlikely. Therefore, for this purpose, we adopt the
higher-$\rho$ solution.

To this sample, we add KMT-BLG-2019-BLG-0371 (Kim et al., 2021), which is not
listed at NASA Exoplanet Archive because it has not yet been accepted for
publication.

Table 3 shows the observed characteristics of the six previous systems666The
values in this table have been somewhat compressed and simplified in order to
aid visual assimilation of the patterns. The reader should consult the
original papers for the exact parameter values and error bars., together with
those of KMT-2016-BLG-2605. Excluding for the moment OGLE-2018-BLG-0677 (for
which $\rho$ is not measured), the remaining six events all have Einstein
radii $\theta_{\rm E}$ in the range $115\pm 20\,\mu{\rm as}$, and three have
proper motions $\mu_{\rm rel}\sim 9.5\,{\rm mas}\,{\rm yr}^{-1}$, with the
other three deviating by 2–3 ${\rm mas}\,{\rm yr}^{-1}$.

These characteristics are consistent with expectations for a population of
planet-bearing hosts near the star-BD boundary and lying in the Galactic
bulge. That is, the total mass of these systems (dominated by the host) is
given by

$M={\theta_{\rm E}^{2}\over\kappa\pi_{\rm
rel}}=0.10\,M_{\odot}\biggl{(}{\theta_{\rm E}\over 115\,\mu{\rm
as}}\biggr{)}^{2}\biggl{(}{\pi_{\rm rel}\over 16\,\mu{\rm
as}}\biggr{)}^{-1};\qquad\kappa\equiv{4G\over c^{2}{\rm AU}}\simeq 8.14\,{{\rm
mas}\over M_{\odot}},$ (6)

where we have scaled to a typical relative parallax, $\pi_{\rm
rel}=16\,\mu{\rm as}$, for bulge-bulge microlensing. Under conditions that the
“mass function” (i.e., in this case, the mass function of stars/BDs that host
planets) has a “hard floor”, the shortest-timescale events will be generated
by systems near this floor, and with source-lens separations $D_{LS}$ near the
“edge” of the bulge distribution. In fact, there is no real “edge”, but there
is a rapid fall-off. It is also unlikely that there a “hard floor” to the mass
function, but it is plausible that there is, again, a rapid fall-off.

The same picture naturally explains the high proper motions. For an isotropic
proper-motion distribution with Gaussian width $\sigma=2.9\,{\rm mas}\,{\rm
yr}^{-1}$ (which approximately characterizes the bulge), the mean and standard
deviation of the lens-source relative proper motion is (Gould et al., 2021a),

$\mu_{\rm
rel}=\biggl{(}{4\over\sqrt{\pi}}\pm\sqrt{6-{16\over\pi}}\biggr{)}\sigma\rightarrow
6.5\pm 2.8\,{\rm mas}\,{\rm yr}^{-1}.$ (7)

Thus, we expect that if there is a “floor” on the mass function (whether
“hard” or “soft”), the proper motions of the shortest events will tend toward
the upper range where the distribution is falling off rapidly, roughly
$1\,\sigma$ above the mean, which is $9.3\,{\rm mas}\,{\rm yr}^{-1}$ in the
present case.

These are just plausibility arguments, and no more is really possible at this
point because of the inhomogeneous selection of the sample. However, it will
be straightforward to test this conjecture by imaging the systems at first AO
light on next-generation (30m class) telescopes, in roughly 2030. In all
cases, the sources are dwarf stars, turnoff stars or subgiants, and hence have
$M_{K}\gtrsim 2$, compared to $M_{K}\sim 10$ of stars at the bottom of the
main sequence, i.e., contrast ratios of $\lesssim 8$ magnitudes. Bowler et al.
(2017) achieved contrast ratios of (5, 10) magnitudes at separation
$\Delta\theta\sim(150,320)\,{\rm mas}$ using AO on the Keck 10m telescope.
Scaling to 25m (for the Giant Magellan Telescope, GMT) to 30m (for the Thirty
Meter Telescope, TMT), and to 39m (for the European Extremely Large Telescope,
EELT), these correspond to $\Delta\theta\sim(60,130)\,{\rm mas}$,
$\Delta\theta\sim(50,110)\,{\rm mas}$, and $\Delta\theta\sim(40,80)\,{\rm
mas}$, respectively. All but the last two events in Table 3 will have
$\Delta\theta\gtrsim 110\,{\rm mas}$ by 2030, making them accessible down to
the hydrogen burning limit at either TMT or EELT, with a few requiring several
additional years for access from GMT. KMT-2019-BLG-0371 would only be fully
accessible from EELT in 2030.

OGLE-2018-BLG-0677 presents a special case because the $3\,\sigma$ lower limit
on its proper motion (derived from Figure 8 of Herrera-Martin et al. 2020) is
only $3.7\,{\rm mas}\,{\rm yr}^{-1}$. It is quite plausible that the Einstein
radius of this system is like the others in Table 3, i.e., $\theta_{\rm E}\sim
115\,\mu{\rm as}$, in which case $\mu_{\rm rel}\sim 10\,{\rm mas}\,{\rm
yr}^{-1}$, implying that it would be feasible to image this system in 2030,
like the others. This could be tried, but a non-detection would not clearly
establish that the host was a BD.

Some progress is possible using present telescopes. For example,
MOA-2011-BLG-262 is already separated by $\Delta\theta\sim 130\,{\rm mas}$,
and so it should be possible to probe companions to a contrast ratio of $\sim
5$ magnitudes on Keck. However, a non-detection would yield only an upper
limit on the host mass that would be well within the stellar range. Note that
Bennett et al. (2014) have presented a first epoch for comparison.

If the lens is detected in these observations, then its mass can be reliably
inferred from the $K$-band flux, together with the improved $\mu_{\rm rel}$
determination (and so improved $\theta_{\rm E}=\mu_{\rm rel}t_{\rm E}$
determination) from the lens-source separation measurement. Hence, the planet
mass can also be determined. Non-detection of the lens would imply that the
host is a BD, and would also give an upper limit on the mass of the planet,
i.e., $m_{p}<qM_{\rm H-burn}$. The relative fraction of BD and stellar hosts
would constrain the “mass function”, i.e., the mass function of low-mass stars
and BDs that host planets. This could then be compared to the mass function of
(apparently) isolated stars and BDs, which can also be obtained from
microlensing.

Assuming that future AO observations confirm that the hosts of the planets in
Table 3 lie close to the star-BD boundary, such objects host a wide variety of
planets. Adopting $M_{\rm host}=M_{\rm H-burn}=0.075\,M_{\odot}$ for
illustration, the six planets would have (in order of mass)
$m_{p}=[(2,12,51,135,300)M_{\oplus},(7.8,8.8)M_{J}]$. This distribution
already hints at two populations of “planetary” companions of very low mass
objects, genuine planets $m_{p}\lesssim M_{j}$ formed by core-accretion and
much more massive objects $m_{p}\gg M_{J}$, drawn from the tail of BDs that
are formed by gaseous collapse in a manner similar to stars.

###  7.2 Patterns of Short-timescale Planetary Events

There are several features of this sample that are important for understanding
the detectability of these systems.

One key feature is that in only one of these seven events did followup
observations play a role. Indeed, in this case (MOA-2011-BLG-262), follow-up
observations (including auto-followup by MOA) were essential in the
interpretation of the anomaly. The remaining six cases were survey-only
detections, and the KMTNet survey (which began in 2015) was crucial in all
six.

Another feature of Table 3 is that the planetary signals for all seven events
are generated by resonant or “near-resonant” caustics. Six come from resonant
caustics, i.e., the six-sided caustics formed by the “merger” of central and
planetary caustics that occurs as $s\rightarrow 1$. One comes from a “near-
resonant” caustic structure, which was defined by Yee et al. (2021) as
topologically disjoint caustic structures that have ridges (or valleys) of
excess magnification of at least 10% that connect the central and planetary
caustics. However, this is not surprising. Yee et al. (2021) showed that the
great majority of microlensing planets are found in events from these two
caustic topologies in roughly the proportion 3:2. Hence, from binomial
statistics, the probability that one or fewer from a sample of seven would be
near-resonant is $p=16\%$. Nevertheless, this feature is important in that it
means that these systems are detected from relatively short-lived anomalies
near the peak of relatively high-magnification events.

Finally, all seven events have faint source-star magnitudes, $I_{s}>19.2$.
This is mainly explained by the fact that faint sources are much more common
than bright ones. However, it does emphasize that in typical real cases, the
source has only marginally brightened two days before peak, and has hardly
brightened one day before peak. That is, for a typical $t_{\rm E}\sim 4\,$day
event on an $I_{s}=19.5$ source, the “difference star” on subtracted images is
just $I_{\rm diff}=19.5$ two days before peak and $I_{\rm diff}=18.3$ one day
before peak. The OGLE EWS system (Udalski et al., 1994; Udalski, 2003) rarely
alerts on single-night excursions at this level, and the MOA system (Bond et
al., 2001) never does. Moreover, OGLE alerts are usually issued about 10 hours
after the end of the night. This explains why there were no such detections
based on follow-up observations of OGLE alerts.

By contrast, MOA attempts to issue alerts shortly after a fast-rising event is
detected. The MOA threshold of detection is much brighter than OGLE, but for
fast-rising events, this is more than compensated by this quick response. In
the case of MOA-2011-BLG-262, MOA issued its alert about 6 hr after the first
observation of the night, and just 50 min after three observations confirmed a
rapid rise. This enabled the first followup observations less than 30 minutes
later, allowing full coverage of the anomaly. Without this alert, there would
have been only one or two data points over the anomaly. Nevertheless, this is
truly a unique example from 14 years of the MOA-II experiment. MOA did not
issue alerts for any of the other events in Table 3, except for
KMT-2019-BLG-0371, which it alerted at about the mid-point of the
anomaly777Using online MOA and OGLE data that covered only the caustic
entrance of KMT-2019-BLG-0371, Valerio Bozza issued an anomaly alert for this
event at UT 08:30 on 19 April, which gave a basically correct estimate of the
event parameters. However, the anomaly had just ended at the time this alert
was issued..

MOA did issue an alert for MOA-2015-BLG-337 on HJD${}^{\prime}=7214.02$, which
would have been plenty of time to initiate intensive observations from Chile
at HJD${}^{\prime}\sim 7214.7$, which could have distinguished the two models
with $q$ differing by a factor $\sim 20$. See Figure 1 from Miyazaki et al.
(2018). However, the main team that could have carried out such observations,
$\mu$FUN, had discontinued intensive followup observations at this time in
order to focus on Spitzer microlensing candidates (Yee et al., 2015). There
were survey observations from KMTC in Chile, but these commissioning-year data
were of insufficient quality. We note that MOA-bin-29 (Kondo et al., 2019) was
not discovered in real time, so there was no possibility of followup
observations during the 2006 season, and hence there were substantial gaps in
the light-curve coverage. Moreover, it is not completely clear that the
“Wide-1”/“Wide 3” degeneracy (with different $q$ by a factor 2.7) could have
been resolved by additional coverage.

In brief, all six of the survey-only short-$t_{\rm E}$ planets in Table 3
occurred after the start of KMTNet observations in 2015, and KMTNet data were
essential to all six. During the nearly two decades of microlensing planet
detections, there has been only one short-$t_{\rm E}$ planet detected by means
of survey-plus-followup observations. The above discussion shows that these
patterns are reasonably well understood.

Thus, if the currently very small sample of these important systems is to be
increased, the most likely path is to improve the harvest from KMT survey.

###  7.3 Path to Additional Short-$t_{\rm E}$ Planetary Events

There are two obvious paths to finding more planetary anomalies in archival
short-$t_{\rm E}$ KMT events. First, as noted by Zang et al. (2021), the
online data reductions were substantially improved starting in 2018. Simply
applying the same algorithms to 2016 and 2017 data would make it much easier
to spot anomalies by eye, or to find them by the automated technique described
by Zang et al. (2021). We note that of the six survey-only detections in Table
3, three were from prior to 2018. Of these three, one was not discovered by
KMT (OGLE-2015-BLG-1771), one was part of the special 2016 search and so was
reduced using the new algorithm (KMT-2016-BLG-2605), and one was a massive
planet with a huge, easily discernible anomaly (KMT-2016-BLG-1820). Hence,
updating the 2016-2017 reductions, which is currently underway, may well
increase the detectability of moderate mass-ratio planets for these
seasons888It is not clear that it will be possible to improve the pipeline
light curves for 2015, due to the lower quality of this commissioning-year
data. There are currently no plans to do so..

A second path would require a small alteration of the program outlined by Yee
et al. (2021) to make TLC reductions for all “high-magnification” events,
defined as perhaps $A_{\rm max}>20$ or $A_{\rm max}>10$. Subtle anomalies,
like the one seen in OGLE-2018-BLG-0677 (Herrera-Martin et al., 2020), will
only appear convincing (or may only be noticed) in high-quality TLC
reductions. Subtle anomalies may reflect very low-mass planets (as in that
case), or somewhat higher mass planets in events for which the source passes
farther from the caustics. Excluding the two high-$q$ events
(KMT-2019-BLG-0371 and KMT-2016-BLG-1820), whose pronounced anomalies are
easily recognizable without TLC reductions, the remaining survey-only events
have peak magnifications (as judged by $A_{\rm max}=1/u_{0}$) of $A_{\rm
max}=(9,10,20,29)$. And machine PSPL fits could easily underestimate the peak
magnification, depending on how these fits were affected by the anomaly.
Therefore, the $A_{\rm max}$ criterion for TLC reductions could be loosened
for short-$t_{\rm E}$ events.

The problems posed by anomaly-dominated events like KMT-2019-BLG-2605 are more
challenging. While this event constitutes only 14% of the current sample and
may therefore appear relatively inconsequential, it arrived in the sample by a
quite accidental route. Hence, it could be under-represented. It would be
impractical to repeat the EventFinder searches of archival KMT data, but going
forward, the human reviews of the machine-selected EventFinder and AlertFinder
(Kim et al., 2018b) candidates could be more aggressive for short events. In
particular, when there are magnified data from only one night for each
observatory and the event is anomalous, the machine alignment of the data can
be radically incorrect, and one or more data sets can even be eliminated from
the fit. Recognition of these issues could enable more potentially-anomalous,
short events to be conditionally selected at this stage.

For the same reason, it is possible that anomalous EventFinder events that
have been selected are being overlooked in manual reviews of the KMTNet
webpage. That is, the poor machine alignment of the different data sets can
make the event look like “not microlensing”, leading to it not being selected
for further analysis. Simple recognition of this possibility, based on the
experience of KMT-2016-BLG-2605, may lead to a revised preliminary assessment
of such events.

Here, it should be pointed out that archival events are, in some sense, more
productive than prospective ones, because they will become eligible for AO
imaging sooner.

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 (2020R1A4A2002885 and 2019R1A2C2085965).

##  Appendix A Assessment of Source Color

Figure 6 shows OGLE-III (Szymański et al., 2011) stars within a
$180^{\prime\prime}$ circle, centered on the lensing event. The clump is
easily visible, but it is extended upper-left to lower-right, which is a
standard signature of differential reddening. Hence, we should be cautious
about identifying the centroid of the clump feature in this diagram with the
center of the clump at the position of the event. Figure 7 shows OGLE-III
stars in a $60^{\prime\prime}$ circle centered on the event. The clump is less
visible, but guided by Figure 6, it can be recognized, and its center is
marked by a red circle, $[(V-I),I]_{\rm cl}=(3.42,16.98)\pm(0.02,0.04)$. This
same position is marked by a circle in Figure 6, which demonstrates that the
centroid of the clump feature has indeed shifted fainter and redder from the
first to the second figure.

The source magnitude is well measured from the microlensing fit in the KMTS
pyDIA system, $I_{s,{\rm kmts}}=20.06\pm 0.04$. By comparing field-star
photometry from OGLE-III with that of the KMTS pyDIA reductions, we find
$I_{\rm kmts}-I_{\rm ogle-iii}=-0.15\pm 0.01$, implying $I_{s,{\rm ogle-
iii}}=20.21\pm 0.05$. Hence, the offset in brightness of the source relative
to the clump is

$\Delta I=I_{s}-I_{\rm cl}=3.23\pm 0.07.$ (A1)

However, the color offset $\Delta(V-I)$ is substantially more difficult to
determine. There is only one substantially magnified $V$-band data point. This
would make it difficult to measure the source color under any circumstances
because there would be no internal check on the measurement. In addition, as
we report below, the image quality of the one magnified $V$-band point
exhibits some problems. We therefore begin by asking what can be deduced about
the source color without a measurement from the light curve.

Figure 8 shows the Hubble Space Telescope (HST) CMD for a Baade Window field
constructed by Holtzman et al. (1998). The red circle shows the clump centroid
$[(V-I),I]_{\rm BW}=(1.62,15.15)$ as determined by Bennett et al. (2008). The
two magenta lines are displaced $\pm 0.1$ (i.e., $1.5\,\sigma$) from the best
estimate of the offset (Equation (A1)) for KMT-2016-BLG-2605, $\Delta I=3.23$.
Based on the stars between these two lines, we can make three different
characterizations of the stars at this offset:

$-0.47<\Delta(V-I)<-0.01;\quad\langle\Delta(V-I)\rangle=-0.31\pm
0.09;\quad\Delta(V-I)_{\rm median}=-0.32_{-0.07}^{+0.08}.$ (A2)

The first is the full “reasonably populated” region of the strip. The second
is the mean and standard deviation of this populated region. The third is the
median and (16,84)th percentiles of the full distribution. If there were
absolutely no other information about the source color, one would take either
the mean or median estimator, which in the present case are almost identical.

We will next consider the color measurement based on the light curve, i.e., on
the single magnified $V$-band measurement. However, before proceeding, we
should “predict” the KMTS $V$-band flux measurement at
HJD${}^{\prime}=7565.4446$ based on the contemporaneous $I$-band flux
measurement ($F_{I}=35688\pm 325$), and the range of “reasonably populated”
$\Delta(V-I)$ given by Equation (A2). To do so, we take note of the offset
(measured from field stars) $(V-I)_{\rm kmts}-(V-I)_{\rm ogle-iii}=0.27\pm
0.02$, the KMTS instrumental photometric zero points $(V_{\rm zero}=28.65$ and
$I_{\rm zero}=28.00)$, and the OGLE-III clump centroid $(V-I)_{\rm cl}=3.42\pm
0.02$. That is,

$F_{V,\rm predicted}=0.0608F_{I}\times 10^{-0.4\Delta(V-I)}\rightarrow
2170\times 10^{-0.4\Delta(V-I)},$ (A3)

and hence, for the full “reasonable range” of $\Delta(V-I)$, we predict,
$2200<F_{V,\rm predicted}<3360$, which should be compared to the observed
$V$-band difference flux returned by the photometry program, $F_{V}=1545\pm
238$. That is, the observed flux lies $2.75\,\sigma$ below the “reasonable
range”. This could mean that the source is a very rare, exceptionally red
star, that the error bar has been substantially underestimated, or that the
measured value is the outcome of a rare statistical fluctuation.

We find no evidence that the photometry program has generally underestimated
the error bars on the $V$-band light-curve measurements. In particular, we
look at the distribution of $\sigma_{i}/F_{i}$ of the 87 measurements apart
from the well-magnified one and the one on the previous night at modest
magnification (for which the predicted difference flux is $<1\,\sigma$). For
these 87, the expected difference flux is zero to high precision. We find that
this distribution is consistent with a Gaussian of zero mean and unit
variance.

We examine the original and subtracted images for the magnified point and
compare these to several images for unmagnified points. In the original
images, the source generally appears isolated, and there are only very faint
stars within a few arcseconds. With the exception of the magnified point, the
subtracted images generally appear “blank” at, and for several arcseconds
around, the source. Hence, there is no obvious cause for difficulty in
performing the photometry, in line with the fact (just reported) that the
normalized error distribution is a unit Gaussian.

The magnified image is taken seven days after passage of the full moon through
the bulge, so that the background is about 2.8 times the dark-time level. As a
result of this higher background, the subtracted image appears substantially
more mottled than for dark-time images. Nevertheless, the background level
(453 ADU per pixel) is by no means high. Similarly, the seeing has a
FWHM${}_{\rm see}\sim 2.75^{\prime\prime}$, which is higher than the median
($2.39^{\prime\prime}$), but hardly unusual (66th percentile). And also
similarly, the transparency is about 88% relative to typical good nights,
which is hardly out of the normal range.

Finally, we consider the general possibility that the program has
underestimated the error bar for some “unknown reason”. The program makes its
estimate by varying the fit parameters and finding the change of $\chi^{2}$
that results. This should be robust, but for any relatively complex program,
one can imagine that it confronts some unexpected condition and makes a
catastrophic error. As a sanity check, we make a naive estimate of the error
as being proportional to [FWHM${}_{\rm see}\times$(FWHMback/transparency)1/2],
where FWHMback is the full width at half maximum of the difference-flux pixel-
count distribution of the subtracted image. For images that are well below
sky, this scaling should be close to accurate. We normalize this estimator to
an image with low background (164), good seeing ($1.51^{\prime\prime}$) and
100% relative transparency and find only a 19% difference in predicted versus
reported error bars. This is an order of magnitude below what would be
required to explain the apparent discrepancy (and also goes in the wrong
direction).

In brief, the source location is isolated, the program overall evaluates the
$V$-band errors correctly, the seeing and background of the magnified image
are slightly worse than average but by no means unusual, and a simple sanity
check confirms the program’s evaluation of the error bar.

If the $2.75\,\sigma$ discrepancy between prior expectations and the observed
data point are to be explained within the context of Gaussian statistics, then
$p_{\rm gauss}=0.0031$. Therefore, before accepting this explanation, we
should consider various others that are of such low probability that they
would normally be dismissed without detailed investigation.

First, the source may actually be drawn from the extremely red population that
is the reflected in the HST CMD. Of the 414 stars shown between the magenta
lines, two are within the $1\,\sigma$ range of the magnified point,
$\Delta(V-I)=+0.37\pm$0.17, and one other is redward of this range. This
fraction, $3/414=0.007$, is greater than $p_{\rm gauss}$, and so this
possibility should be considered. However, from the morphology of Figure 8,
these very red stars appear to be part of the disk red-dwarf population that
lives “above” the bulge main sequence in this diagram. As such, the red stars
within the magenta bands lie about 3 mag in front of the bulge in distance
modulus, i.e., at about $D_{S}=2\,{\rm kpc}$. In addition to being extremely
rare (as just noted), the optical depth to microlensing of such nearby disk
sources is two orders of magnitude lower than for bulge sources. Thus, we
regard this potential explanation as highly improbable.

Bensby et al. (2017) provide some corroboration of this assessment. They
obtained 91 high-resolution spectra of highly-magnified “dwarf and subgiant”
sources. These were almost all selected solely on source-brightness relative
to the clump (i.e., not giants) and observability (magnified enough to obtain
a good spectrum), which in practice essentially produced an unbiased sample of
turnoff stars and subgiants. None of these 91 had spectroscopic temperatures
cooler than the clump ($\sim 4750\,$K). See upper panel of their Figure 7.
While 0/91 does not place restrictions at the level of $p_{\rm gauss}$, it
does demonstrate that such extremely red microlensed sources lying $3\,$mag
below the clump are very rare.

Another possibility is that the microlensing model is incorrect, so that the
source is actually brighter (relative to the clump) than the magenta band. For
example, the source is 0.8 mag brighter for Local 3 than for Local 1. However,
the HST CMD is even less populated 0.8 mag above the red end of the magenta
band than in the band itself. One might posit that there is another solution
with an even brighter source that we failed to discover. However, the source
cannot be much brighter999It could be slightly brighter because the source
might be projected against a “hole” in the mottled background due to
unresolved field stars (Park et al., 2004). However, this effect is far too
small to be relevant here. than the baseline object, which is only
$-2.5\log(1+f_{B}/f_{S})=-1.5\,$mag brighter than the magenta bar. This is
still far below the region of the CMD that is populated by upper giant-branch
stars.

Yet another possibility is that the source is actually a giant in the far side
of the disk. There would be extremely few such stars in the HST CMD because it
lies in the Baade Window at $b\sim-4$, so that the line of sight intersects
the bulge about $z_{\rm bulge}\sim-550\,$pc from the Galactic plane. Far-disk
sources are more plausible for KMT-2016-BLG-2605 for which $b=-1.6$, so that
$z_{\rm bulge}\sim-210\,$pc. For example, at $D_{S}=12\,{\rm kpc}$, the line
of sight passes $z_{\rm 12\,{\rm kpc}}\sim-320\,$pc from the plane, where
potential source stars remain plentiful. Nevertheless, in order to access the
red upper-giant-branch stars, the source would have to have a distance modulus
at least 2.5 larger than the bulge, i.e., $D_{S}>25\,{\rm kpc}$ or $\sim
2R_{0}$ from the Galactic center on the far side of the Galaxy, with $z_{\rm
25\,{\rm kpc}}\sim-700\,$pc from the plane. This is a very thinly populated
region of the Galaxy. While we do not exclude this possibility, and we report
its implications further below, we consider it less likely than a statistical
error in the $V$-band measurement.

We conclude that the most plausible resolution is that the source color is
toward the red end of the “reasonable range” from Equation (A2) and that the
very red light-curve measurement is the result of a relatively large
statistical fluctuation. We therefore adopt

$\Delta(V-I)_{s}=-0.06\pm 0.05\Rightarrow(V-I)_{s,0}=1.00\pm 0.05$ (A4)

However, we also consider the possibility that the light-curve measurement is
actually correct (due, e.g., to a very distant far-disk source), i.e.,
$(V-I)_{0}=1.43\pm 0.17$, and thus we trace the consequences of this
possibility.

###  A.1 Effects of Alternate Color Estimate

We have adopted a source color $(V-I)_{s}=3.37\pm 0.05$ (equivalently,
$(V-I)_{s,0}=1.00\pm 0.05$) by combining prior information from the Holtzman
et al. (1998) CMD with the KMTS color measurement. Here, we consider the
consequences if the source color is actually given by the KMTS measurement,
i.e., $(V-I)_{s}=3.89\pm 0.17$ (equivalently, $(V-I)_{s,0}=1.43\pm 0.17$).

The first point is that the true source color can eventually be determined by
high-resolution imaging, and indeed this may already be possible with
10m-telescope class AO imaging. Using Bessell & Brett (1988) to convert from
$(V-I)$ to $(I-K)$, and adopting $E(I-K)=2.35$ from Section 2, we find
$K_{s}=16.48\pm 0.08$ or $K_{s}=15.96_{-0.19}^{+0.35}$, for the two scenarios.
These values can be compared to the $K$-band magnitude of the baseline object
from the VVV survey (Minniti et al., 2017) of $K_{\rm base}=14.89\pm 0.07$.
That is, roughly 23% or 37% of the baseline-object $K$-band light comes from
the source.

There are logically only four possibilities for the remainder of the $K$-band
light: the lens, a companion to the lens, a companion to the source, or an
ambient star (or some combination). It is very unlikely that an ambient star
would lie within the $\sim 55\,{\rm mas}$ point-spread function (PSF) of a 10m
telescope. If the blended light were due to the lens or a companion to the
lens, then by 2021, it would have already separated from the source by
$\Delta\theta=\mu_{\rm rel}\Delta t=61\pm 5\,{\rm mas}$ (or $79\pm 7\,{\rm
mas}$). In either case, the source and lens could be separately resolved. See
Figure 1 of Bennett et al. (2020) for a separate resolution of a source and
lens with flux ratio 3.15 at $\Delta\theta=55\,{\rm mas}$, and Figure 1 of
Bhattacharya et al. (2019) for an unambiguous distinction between a source and
lens with flux ratio 1.46 at $\Delta\theta=34\,{\rm mas}$, both based on
$K$-band observations with the Keck telescope. Thus, unless the blended light
is due to a companion to the source (which would then be a lower-giant-branch
star, which is a priori unlikely due to its short lifetime), the source color
could almost certainly be determined by observations in 2021.

Such immediate observations might also resolve the lens, and, even if not,
would give a definite prediction as to when the lens could be resolved. For
example, suppose that these observations found that $K_{s}=15.96$ (with small
error). One could then conclude that $\mu_{\rm rel}=15.8\pm 0.8\,{\rm
mas}\,{\rm yr}^{-1}$ so that the annulus of possible lens positions (at 1.5
FWHM) could be predicted with precision. Hence, one could already detect the
lens, or place strong constraints on its brightness. If the source proved to
be substantially fainter in $K$, this would imply a smaller $\theta_{\rm E}$
and hence a smaller $\mu_{\rm rel}$. However, it would still be possible to
use this information to predict when the lens would be observable. As
discussed in Section 7.1 non-detection of the lens in relatively shallow
imaging would indicate the need for deeper imaging, either on 10m or future
30m class telescopes.

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Table 1: Best Solutions without Flux Constraint Parameters | Local 1 | Local 1a | Local 1b | Local 2 | Local 3
---|---|---|---|---|---
$\chi^{2}/\rm{dof}$ | 3596.308/3598 | 3603.450/3598 | 3605.222/3598 | 3606.825/3598 | 3606.218/3598
$t_{0}-2457560$ | 5.451 $\pm$ 0.021 | 5.457 $\pm$ 0.010 | 5.473 $\pm$ 0.010 | 5.643 $\pm$ 0.023 | 5.473 $\pm$ 0.013
$u_{0}$ | 0.049 $\pm$ 0.004 | 0.046 $\pm$ 0.004 | 0.047 $\pm$ 0.004 | 0.097 $\pm$ 0.006 | 0.085 $\pm$ 0.007
$t_{\rm E}$ $(\rm{days})$ | 3.370 $\pm$ 0.139 | 3.291 $\pm$ 0.123 | 3.023 $\pm$ 0.146 | 2.237 $\pm$ 0.057 | 2.319 $\pm$ 0.104
$s$ | 0.939 $\pm$ 0.011 | 0.924 $\pm$ 0.006 | 0.914 $\pm$ 0.004 | 1.827 $\pm$ 0.064 | 0.797 $\pm$ 0.014
$q$ | 0.012 $\pm$ 0.002 | 0.013 $\pm$ 0.002 | 0.007 $\pm$ 0.001 | 0.242 $\pm$ 0.074 | 0.019 $\pm$ 0.003
$\alpha$ $(\rm{rad})$ | 0.101 $\pm$ 0.019 | 0.180 $\pm$ 0.018 | 0.057 $\pm$ 0.031 | 2.580 $\pm$ 0.040 | -0.042 $\pm$ 0.065
$\rho$ $(10^{-2})$ | 1.203 $\pm$ 0.125 | 1.068 $\pm$ 0.079 | 1.145 $\pm$ 0.156 | 3.345 $\pm$ 0.262 | 1.651 $\pm$ 0.269
$f_{S}$ [KMTC] | 0.151 $\pm$ 0.016 | 0.163 $\pm$ 0.011 | 0.206 $\pm$ 0.024 | 0.320 $\pm$ 0.023 | 0.336 $\pm$ 0.032
$f_{B}$ [KMTC] | 0.388 $\pm$ 0.016 | 0.376 $\pm$ 0.011 | 0.332 $\pm$ 0.024 | 0.220 $\pm$ 0.023 | 0.203 $\pm$ 0.032
$f_{S}$ [KMTS] | 0.151 $\pm$ 0.010 | 0.141 $\pm$ 0.009 | 0.170 $\pm$ 0.014 | 0.389 $\pm$ 0.023 | 0.314 $\pm$ 0.026
$f_{B}$ [KMTS] | 0.436 $\pm$ 0.010 | 0.446 $\pm$ 0.009 | 0.417 $\pm$ 0.014 | 0.198 $\pm$ 0.023 | 0.273 $\pm$ 0.026
$f_{S}$ [KMTA] | 0.096 $\pm$ 0.006 | 0.090 $\pm$ 0.005 | 0.103 $\pm$ 0.009 | 0.234 $\pm$ 0.013 | 0.200 $\pm$ 0.016
$f_{B}$ [KMTA] | 0.270 $\pm$ 0.006 | 0.276 $\pm$ 0.005 | 0.263 $\pm$ 0.009 | 0.133 $\pm$ 0.013 | 0.167 $\pm$ 0.016
$I_{S,{\rm kmts}}-I_{S,{\rm kmtc}}$ | -0.002 $\pm$ 0.066 | 0.152 $\pm$ 0.062 | 0.208 $\pm$ 0.048 | -0.213 $\pm$ 0.049 | 0.072 $\pm$ 0.045
$t_{*}$ $(\rm{days})$ | 0.041 $\pm$ 0.004 | 0.035 $\pm$ 0.002 | 0.035 $\pm$ 0.003 | 0.075 $\pm$ 0.005 | 0.038 $\pm$ 0.005
Table 2: Best Solutions with Flux Constraint Parameters | Local 1 | Local 2 | Local 3
---|---|---|---
$\chi^{2}/\rm{dof}$ | 3597.963/3599 | 3622.808/3599 | 3613.757/3599
$t_{0}-2457560$ | 5.451 $\pm$ 0.005 | 5.553 $\pm$ 0.039 | 5.500 $\pm$ 0.008
$u_{0}$ | 0.049 $\pm$ 0.004 | 0.091 $\pm$ 0.012 | 0.081 $\pm$ 0.007
$t_{\rm E}$ $(\rm{days})$ | 3.405 $\pm$ 0.128 | 2.207 $\pm$ 0.071 | 2.402 $\pm$ 0.100
$s$ | 0.940 $\pm$ 0.005 | 1.756 $\pm$ 0.093 | 0.787 $\pm$ 0.008
$q$ | 0.012 $\pm$ 0.001 | 0.175 $\pm$ 0.092 | 0.023 $\pm$ 0.002
$\alpha$ $(\rm{rad})$ | 0.104 $\pm$ 0.010 | 2.483 $\pm$ 0.097 | 0.053 $\pm$ 0.043
$\rho$ $(10^{-2})$ | 1.192 $\pm$ 0.083 | 3.683 $\pm$ 0.283 | 1.298 $\pm$ 0.203
$f_{S}$ [KMTC] | 0.145 $\pm$ 0.009 | 0.354 $\pm$ 0.031 | 0.291 $\pm$ 0.025
$f_{B}$ [KMTC] | 0.393 $\pm$ 0.009 | 0.186 $\pm$ 0.031 | 0.248 $\pm$ 0.025
$f_{S}$ [KMTS] | 0.150 $\pm$ 0.009 | 0.364 $\pm$ 0.032 | 0.300 $\pm$ 0.025
$f_{B}$ [KMTS] | 0.437 $\pm$ 0.009 | 0.223 $\pm$ 0.032 | 0.287 $\pm$ 0.025
$f_{S}$ [KMTA] | 0.095 $\pm$ 0.006 | 0.223 $\pm$ 0.018 | 0.190 $\pm$ 0.016
$f_{B}$ [KMTA] | 0.271 $\pm$ 0.006 | 0.144 $\pm$ 0.018 | 0.176 $\pm$ 0.016
$I_{S,{\rm kmts}}-I_{S,{\rm kmtc}}$ | -0.033 $\pm$ 0.005 | -0.032 $\pm$ 0.005 | -0.033 $\pm$ 0.005
$t_{*}$ $(\rm{days})$ | 0.041 $\pm$ 0.002 | 0.081 $\pm$ 0.005 | 0.031 $\pm$ 0.004
Table 3: 7 Planetary Events with $t_{\rm E}<7\,$days

Event | $t_{\rm E}$ | $q$ | $\ln s$ | $\theta_{*}$ | $\theta_{\rm E}$ | $\mu_{\rm rel}$ | $I_{s}$ | Caustic Type
---|---|---|---|---|---|---|---|---
KMT-2016-BLG-2605 | 3.41 | 0.0120 | $-0.06$ | 1.38 | 116 | 12.3 | 20.21 | Resonant
MOA-2011-BLG-262 | 3.87 | 0.00047 | $\pm 0.05$ | 0.78 | 136 | 12.9 | 19.34 | Resonant
OGLE-2015-BLG-1771 | 4.28 | 0.00538 | 0, +0.18 | 0.49 | 111 | 9.5 | 21.77 | Resonant
KMT-2018-BLG-0748 | 4.38 | 0.00203 | $-0.06$ | 1.21 | 111 | 9.2 | 19.21 | Resonant
KMT-2016-BLG-1820 | 4.81 | 0.11300 | $+0.15$ | 0.81 | 123 | 9.3 | 19.38 | Resonant
OGLE-2018-BLG-0677 | 4.94 | 0.00008 |  $-0.09$,$-0.02$ | 0.79 | $>49$ | $>3.6$ | 19.32 | Near-Resonant
KMT-2019-BLG-0371 | 6.53 | 0.08,0.12 |  $-0.19$,$+0.45$ | 0.92 | 135 | 7.6 | 19.76 | Resonant

Note. — $t_{\rm E}$ is in days, $\theta_{*}$ and $\theta_{\rm E}$ are in
$\mu{\rm as}$, and $\mu_{\rm rel}$ is in ${\rm mas}\,{\rm yr}^{-1}$.

Table 4: Physical parameters Quantity | Local 1
---|---
$M_{\rm host}$ $[M_{\odot}]$ | $0.064_{-0.033}^{+0.099}$
$M_{\rm planet}$ $[M_{J}]$ | $0.771_{-0.401}^{+1.183}$
$a_{\bot}$ [au] | $0.681_{-0.110}^{+0.102}$
${\it D_{L}}$ [kpc] | $6.421_{-0.991}^{+0.878}$
Figure 1: Light curve and models for Locals 1–3 for KMT-2016-BLG-2605. Upper
Panel: In contrast to the great majority of microlensing events, the data
points for competing models are offset from one another for both of the non-
anchor observatories (KMTS and KMTA). By construction, they are perfectly
aligned for the anchor (flux-reference) observatory (KMTC). These offsets
occur because each observatory has only one night of strongly magnified data.
See Section 2. The middle and bottom rows of panels show the individual models
and their residuals, respectively. By eye, Local 1 better matches the data.
Formally, the fit is better by $\Delta\chi^{2}\gtrsim 10$. See Table 1. Figure
2: Similar to Figure 1, but for Local 1 and its two satellite solutions, Local
1a, and Local 1b. By eye, Local 1 is favored over the other two, but less
decisively than for Figure 1. See Table 1. Figure 3: Offset between KMTS and
KMTC $I$-band photometry as determined from field stars (open circles),
compared to the predictions of the five different models shown in Figures 1
and 2. See the penultimate row of Table 1. There is $1\,\sigma$ agreement for
Local 1 and $\geq.3\,\sigma$ disagreement for all the others. After
incorporating the flux constraint (black dashed band) into the MCMC, Locals 1a
and 1b are eliminated as distinct minima, while Locals 2 and 3 become
disfavored by $\Delta\chi^{2}\gtrsim 16$. See Table 2.  Figure 4: Light curve
and models for Local 1 for KMT-2016-BLG-2605, after incorporating the flux
constraint shown in Figure 3. Also shown is the caustic topology for the
event.  Figure 5: Bayesian estimates of the host mass, planet mass, system
distance, and planet-host projected separation for KMT-2016-BLG-2605. The red
and blue histograms show the relative contributions of bulge and disk lenses,
respectively, with the total are shown as black histograms. The median host
mass is very close to the star-BD boundary.  Figure 6: OGLE-III (Szymański et
al., 2011) CMD for stars within $180^{\prime\prime}$ of KMT-2016-BLG-2605. The
clump is clearly visible, but is extended from upper left to lower right,
indicating strong differential reddening. The red circle is the clump center
as determined from the $60^{\prime\prime}$ CMD in Figure 7, which is clearly
not aligned with the center of the clump “feature” in this diagram.  Figure 7:
OGLE-III CMD for stars within $60^{\prime\prime}$ of KMT-2016-BLG-2605. The
clump is less clearly visible than in Figure 6 but with the aid of that figure
it can be identified. The red circle is the clump centroid. The magenta circle
represents the CMD position of the source as determined from the light curve
alone. The black circle is the adopted CMD source position after incorporating
information from the HST CMD shown in Figure 8. The implications of these two
different source positions and how they can eventually be distinguished is
discussed in the Appendix. The green circle represents the baseline-object
position $[(V-I),I]_{\rm base}$, where $I_{\rm base}=18.63$ comes directly
from the OGLE-III catalog (Szymański et al., 2011) and $(V-I)_{\rm base}=3.28$
is derived by combining $I_{\rm base}$ with $K_{\rm base}=14.89\pm 0.07$ from
the VVV catalog (Minniti et al., 2017), and then transforming from $(I-K)$ to
$(V-I)$ using matched stars between OGLE-III and VVV. The $K$ magnitude of the
baseline object can be helpful in future AO imaging. See the Appendix.  Figure
8: HST CMD from the observations of the Baade Window by (Holtzman et al.,
1998). The clump centroid, marked by a red circle, is at $[(V-I),I]_{\rm
cl}=(1.62,15.15)$ (Bennett et al., 2008). The magenta lines, lying $\Delta
I=3.23\pm 0.10$ below the clump represent the $1.5\,\sigma$ range for the
source brightness relative to the clump The source color, based on a single
KMTS $V$-band measurement is $\Delta(V-I)=0.37\pm 0.17$ redward of the clump.
So, on this diagram it would be at $(V-I)_{HST,\rm BW}=1.99\pm 0.17$, i.e., at
the extreme right of the magenta band. However, it is not shown to avoid
clutter. See the Appendix for why this is most likely due to a large
statistical fluctuation. The adopted color offset is $\Delta(V-I)_{s}=-0.06\pm
0.05$, consistent with the subgiants at $(V-I)_{HST,\rm BW}=1.56\pm 0.05$ in
this diagram. See black circle in Figure 7