# OGLE-2017-BLG-1038: A Possible Brown-dwarf Binary Revealed by Spitzer
Microlensing Parallax

Amber Malpas University of Canterbury, Department of Physics and Astronomy,
Private Bag 4800, Christchurch 8020, New Zealand  Michael D. Albrow University
of Canterbury, Department of Physics and Astronomy, Private Bag 4800,
Christchurch 8020, New Zealand  Jennifer C. Yee Center for Astrophysics $|$
Harvard & Smithsonian, 60 Garden St.,Cambridge, MA 02138, USA  Andrew Gould
Max-Planck-Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH
43210, USA  Korea Astronomy and Space Science Institute, Daejon 34055, Korea
Andrzej Udalski Astronomical Observatory, University of Warsaw, Al.
Ujazdowskie 4, 00-478 Warszawa, Poland  Antonio Herrera Martin University of
Canterbury, Department of Physics and Astronomy, Private Bag 4800,
Christchurch 8020, New Zealand  Charles A. Beichman  IPAC, Mail Code 100-22,
Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  Geoffery Bryden
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA  Sebastiano Calchi Novati IPAC, Mail Code
100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  Sean Carey
IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125,
USA  Calen B. Henderson IPAC, Mail Code 100-22, Caltech, 1200 E. California
Blvd., Pasadena, CA 91125, USA  B. Scott Gaudi Department of Astronomy, Ohio
State University, 140 W. 18th Ave., Columbus, OH 43210, USA  Yossi Shvartzvald
Department of Particle Physics and Astrophysics, Weizmann Institute of
Science, Rehovot 76100, Israel  Wei Zhu Department of Astronomy, Tsinghua
University, Beijing 100084, China  Sang-Mok Cha Korea Astronomy and Space
Science Institute, Daejon 34055, Republic of Korea  School of Space Research,
Kyung Hee University, Yongin, Kyeonggi 17104, Republic of Korea  Sun-Ju Chung
Korea Astronomy and Space Science Institute, Daejon 34055, Republic of Korea
University of Science and Technology, Korea, (UST), 217 Gajeong-ro Yuseong-gu,
Daejeon 34113, Republic of Korea  Cheongho Han Department of Physics, Chungbuk
National University, Cheongju 28644, Republic of Korea  Kyu-Ha Hwang Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea  Youn
Kil Jung Korea Astronomy and Space Science Institute, Daejon 34055, Republic
of Korea  Dong-Jin Kim  Korea Astronomy and Space Science Institute, Daejon
34055, Republic of Korea  Hyoun-Woo Kim Korea Astronomy and Space Science
Institute, Daejon 34055, Republic of Korea  Seung-Lee Kim Korea Astronomy and
Space Science Institute, Daejon 34055, Republic of Korea  University of
Science and Technology, Korea, (UST), 217 Gajeong-ro Yuseong-gu, Daejeon
34113, Republic of Korea  Chung-Uk Lee Korea Astronomy and Space Science
Institute, Daejon 34055, Republic of Korea  University of Science and
Technology, Korea, (UST), 217 Gajeong-ro Yuseong-gu, Daejeon 34113, Republic
of Korea  Dong-Joo Lee  Korea Astronomy and Space Science Institute, Daejon
34055, Republic of Korea  Yongseok Lee Korea Astronomy and Space Science
Institute, Daejon 34055, Republic of Korea  School of Space Research, Kyung
Hee University, Yongin, Kyeonggi 17104, Republic of Korea  Byeong-Gon Park
Korea Astronomy and Space Science Institute, Daejon 34055, Republic of Korea
University of Science and Technology, Korea, (UST), 217 Gajeong-ro Yuseong-gu,
Daejeon 34113, Republic of Korea  Richard W. Pogge Department of Astronomy,
Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA  Yoon-Hyun
Ryu Korea Astronomy and Space Science Institute, Daejon 34055, Republic of
Korea  In-Gu Shin Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
St.,Cambridge, MA 02138, USA  Department of Physics, Chungbuk National
University, Cheongju 28644, Republic of Korea  Weicheng Zang Department of
Astronomy, Tsinghua University, Beijing 100084, China  Patryk Iwanek
Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478
Warszawa, Poland  Szymon Kozłowski Astronomical Observatory, University of
Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  Przemek Mróz Astronomical
Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
Division of Physics, Mathematics, and Astronomy, California Institute of
Technology, Pasadena, CA 91125, USA  Paweł Pietrukowicz Astronomical
Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
Radoslaw Poleski Astronomical Observatory, University of Warsaw, Al.
Ujazdowskie 4, 00-478 Warszawa, Poland  Krzysztof A. Rybicki  Astronomical
Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
Jan Skowron Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4,
00-478 Warszawa, Poland  Igor Soszyński Astronomical Observatory, University
of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  Michał K. Szymański
Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478
Warszawa, Poland  Krzysztof Ulaczyk Department of Physics, University of
Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK

###### Abstract

We report the analysis of microlensing event OGLE-2017-BLG-1038, observed by
the Optical Gravitational Lensing Experiment, Korean Microlensing Telescope
Network, and Spitzer telescopes. The event is caused by a giant source star in
the Galactic Bulge passing over a large resonant binary lens caustic. The
availability of space-based data allows the full set of physical parameters to
be calculated. However, there exists an eightfold degeneracy in the parallax
measurement. The four best solutions correspond to very-low-mass binaries near
($M_{1}=170^{+40}_{-50}M_{J}$ and $M_{2}=110^{+20}_{-30}M_{J}$), or well below
($M_{1}=22.5^{+0.7}_{-0.4}M_{J}$ and $M_{2}=13.3^{+0.4}_{-0.3}M_{J}$) the
boundary between stars and brown dwarfs. A conventional analysis, with scaled
uncertainties for Spitzer data, implies a very-low-mass brown dwarf binary
lens at a distance of 2 kpc. Compensating for systematic Spitzer errors using
a Gaussian process model suggests that a higher mass M-dwarf binary at 6 kpc
is equally likely. A Bayesian comparison based on a galactic model favors the
larger-mass solutions. We demonstrate how this degeneracy can be resolved
within the next ten years through infrared adaptive-optics imaging with a 40 m
class telescope.

binaries: general — brown dwarfs — gravitational lensing: micro

##  1 Introduction

Microlensing is a phenomenon in which the path of light emitted from a distant
star (the source) is bent by a curve in space-time, caused by a massive object
(the lens). If the source is approximately behind the lens, as seen by an
observer, it brightens as unresolved images of the source are formed about the
Einstein ring that has angular radius

$\theta_{\rm E}=\sqrt{\frac{4GM}{c^{2}}\left(\frac{{\rm au}}{D_{\rm
L}}-\frac{{\rm au}}{D_{\rm S}}\right)}=\sqrt{\kappa M\pi_{\rm rel}},$ (1)

where $\pi_{\rm rel}={\rm au}(D_{\rm L}^{-1}-D_{\rm S}^{-1})$, $M$ is the mass
of the lens system, $D_{\rm L}$ and $D_{\rm S}$ are the distance to the lens
and source, respectively, and $\kappa=4G/(c^{2}{\rm{\rm au}})\sim 8.14\,{\rm
mas}/M_{\odot}$.

For transient alignments, where the closest angular separation of the source
and lens is on the order of $\theta_{\rm E}$ or smaller, photometric
microlensing events can be observed as increasing and decreasing apparent
brightness of the combination of the source star and unresolved neighbors,
including the lens. Because only the source light is magnified, the luminosity
of the lens system does not directly contribute to the event detection rate.
As a result, microlensing is uniquely sensitive to the detection of low-mass,
dim lenses such as brown dwarfs (BD; for example, Gould et al. 2009,
Shvartzvald et al. 2016, and Han et al. 2020) and unbound planetary-mass
objects (for example, Mróz et al. 2020 and Mróz et al. 2019) as proposed by
Paczyński (1986).

A limitation of the microlensing method is that, for most microlensing events,
the light-curve model leaves a degeneracy between the mass and distance of the
lens. This degeneracy can in principle be resolved either by measuring two
other parameters (the Einstein radius $\theta_{\rm E}$, and the microlens
parallax ${\bm{\pi}}_{\rm E}$) or by separately observing the lens and source
some years after the event in high-resolution images. While $\theta_{\rm E}$
has been measured for most planetary and binary events published to date,
${\bm{\pi}}_{\rm E}$ has not.

For events with an extremely dim lens, proper-motion measurement via late time
imaging is not feasible at typical lens distances, given current observing
capabilities. Breaking the mass-distance degeneracy for very faint lens
systems thus requires a measurement of the microlens parallax. The spatial
separation between observers required to detect parallax at a single epoch
depends on characteristics of the microlensing event, such as the distance to
the lens system and duration of the event. Because of the large separation
between Earth and the Spitzer Space Telescope (located more than $1\,{\rm au}$
distant from Earth), microlensing observations from Spitzer, in conjunction
with those from Earth, provide a reliable means of measuring parallax. This
uniquely wide separation is what motivated the Spitzer microlensing project
(Yee et al., 2015).

Microlensing has been used to discover 34 BDs from beyond the local regime
(Chung et al., 2019). So far, this extended population has demonstrated
unusual dynamics, such as an unexpected number of counter-rotating BDs (Chung
et al., 2019; Shvartzvald et al., 2019, 2017). It is unclear to what degree
these extreme kinematics are representative of the population as a whole.

BDs are stellar-like objects that are not massive enough to maintain a
sufficient core temperature for main-sequence hydrogen fusion. Though the more
massive BDs are capable of lithium fusion, and most BDs are capable of
deuterium fusion, these processes do not provide sufficient heat to stop BDs
from gradually cooling as they radiate the heat generated during their
formation. As a result, they are very faint and become fainter as they age.
Deuterium fusion occurs in objects with masses of approximately $>13\,M_{J}$.
This is often adopted as a criterion to distinguish BDs from planets; objects
below this mass are planets, be they bound to a stellar object or free
floating. However this mass definition is sometimes in conflict with the
formation definition: BDs form like stars and planets form in circumstellar
disks.

All but five of the microlensing BDs have been detected as binary systems. The
number of BDs detected in binaries makes up an artificially high proportion of
the total number of detections because binary events have more easily detected
finite-source effects and therefore are more likely to have their associated
masses calculated. Some of these have member masses at about the deuterium
fusion limit (Choi et al., 2013; Han et al., 2017; Albrow et al., 2018),
supporting the arguments of Grether & Lineweaver (2006) and Chabrier et al.
(2014) for a mass overlap between the gas-giant planet and BD regimes.
Deuterium fusion has become an insufficient metric for classification between
BD and gas-giant planets. These populations have distinct formation histories,
which, though difficult to infer, provide a more meaningful way to separate
them in the mass-overlap region.

The upper BD cutoff is defined by sustained hydrogen fusion. Studies
evaluating the hydrogen burning limit are summarised in Table 5 of Dieterich
et al. (2018), from which we deduce that the BD upper limit is in the range of
($\sim 70-95\,M_{J}$). This variance has a large dependence on chemical
composition (e.g., Chabrier & Baraffe 1997). Forbes & Loeb (2019) investigate
the idea of over-massive BDs. These are theoretically formed through Roche
lobe overflow. The result is that, with only the mass information to draw
from, this cutoff is vague.

Little is known about the very low mass end of the stellar initial mass
functions (IMF). The empirical IMFs of Kroupa (2001), Chabrier (2005), Thies &
Kroupa (2007, 2008), and Kroupa et al. (2013) show disparity with the
theoretical IMFs deduced from analytical descriptions of pre-stellar-cloud
core distributions (Padoan & Nordlund, 2002; Hennebelle & Chabrier, 2008;
Henebelle & Chabrier, 2009) at the very-low-mass end, approximately between
$84\,M_{J}$ and $210\,M_{J}$ ($0.08\,M_{\odot}$ and $0.2\,M_{\odot}$).
Empirical IMFs usually require assumptions about age and metalicity in order
to determine the IMF from an observed luminosity function. Observationally,
measuring a mass function across the entire stellar mass range is challenging
because sampling the upper mass range requires massive star clusters, and
sampling the lower mass range requires nearby clusters. With the closest
massive clusters at distances of a few kiloparsecs, observing both ends of the
mass function in one star cluster is not currently possible photometrically
(Elmegreen, 2009). Wegg et al. (2017) shows one way in which microlensing
surveys can be used to probe the IMF of the inner Milky Way, although this
method used an existing dynamical model to infer the masses from the
timescales ($t_{\rm E}$) of $\sim 4000$ events and therefore is not purely
empirical. The timescales considered were $2\,{\rm days}<t_{\rm E}<200\,{\rm
days}$, which relates to the mass via $t_{\rm E}^{2}\propto M$.

Currently, photometric surveys are only capable of probing relatively bright
and very local populations of BDs. For example, Rosell et al. (2019) quote a
distance limit in their Dark Energy Survey catalog of “beyond $400\,{\rm
pc}$”. This selection bias in observability provides a limited view of BDs, in
distance, mass and age. Further detections of very-low-mass objects in binary
systems, will help to clarify our understanding of the dynamical properties of
BD populations and the low-mass end of the IMF, because such systems are
likely to have been formed as part of the very-low-mass end of the IMF, not
like planets in a circumstellar disk.

The following sections in this paper describe our analysis of microlensing
event OGLE-2017-BLG-1038 and how we determined this event to be a BD binary.
§2 describes the observations made of this event, and the data-reduction
methods used. §3 outlines our analysis of the ground-based data and resulting
conclusions about source star characteristics. §4 details our analysis of the
space-based, Spitzer data and our final modeling results. The corresponding
physical parameters for our most likely models are calculated in §5. In §6 we
compare the relative probabilities of our best model solutions and then we
discuss, in §7, how different assumptions of the galactic model, as well as
selection effects, may influence these probabilities.

##  2 Data Collection and Reduction

OGLE-2017-BLG-1038 is located at (R.A., decl.)${}_{\rm
J2000}=(17:58:36.55,\,-27:18:58.4)$, $(l,b)=(2.8536,-1.6382)^{\circ}$. It was
first identified as a microlensing event candidate by the Optical
Gravitational Lensing Experiment early warning system (OGLE; Udalski et al.
1994), on 2017 June 3, from their ongoing survey (mostly in $I$ band) using
the $1.3\,\rm{m}$ Warsaw telescope in the Las Campanas Observatory in Chile.
Repeated OGLE observations of the event took place at an interval of mostly 1
day.

The Korean Microlensing Telescope Network (KMTNet; Kim et al. 2016) also
discovered this event as KMT-2017-BLG-0363 and observed it in the V and I
bands. OGLE-2017-BLG-1038 was observed in two overlapping KMTNet search fields
(BLG03 and BLG43), from each of the three KMTNet telescopes: Cerro Tololo
Inter-American Observatory (KMT-C), South African Astronomical Observatory
(KMT-S), and Siding Springs Observatory (KMT-A). This resulted in a cadence of
$\sim 15$ minutes between successive observations. The KMTNet observations
were also primarily made in the I band. However, occasional V-band
observations were made to provide color information. Therefore, 12 sets of
KMTNet light curves were obtained for this event.

The end of the event was also observed by the Spitzer Space Telescope Infrared
Array Camera (IRAC; Fazio et al. 2004) instrument at an approximately 1 day
cadence. While both the KMTNet and OGLE observations were made as part of
regular survey operations, the Spitzer observations were scheduled for this
event specifically as part of a program to enable space-parallax measurements
for microlensing events (Calchi Novati et al., 2015a; Yee et al., 2015). This
event was selected for Spitzer observations on 2017 June 13 (HJD’ = 7918.11)
and met the objective criteria on 2017 June 19 (HJD’ = 7923.95). Both of these
selections took place before the binary nature of the event was recognized,
i.e., when it was still believed to be a point lens. Members of the Spitzer
Team first noticed that the event was anomalous on 2017 June 20 (HJD’
7925.04).

Kinematic measurements from the source star in this event, as well as
surrounding field stars, were obtained from Gaia Early Data Release 3 (Gaia
Collaboration et al., 2020, 2016).

The ground-based data were reduced using difference imaging (Tomaney & Crotts,
1996; Alard & Lupton, 1998) procedures. The OGLE images were reduced with
their custom difference image procedures (see Wozniak 2000). The KMTNet light
curves were extracted from the images using pyDIA (Albrow, 2017) software, and
the Spitzer light curve was extracted by the methods detailed in Calchi Novati
et al. (2015b).

##  3 Ground-Based Analysis

The light curve of this event (see Figure 1) has a triple-peaked perturbation
over a 5 day period (2017 June 22-27) with the three peaks showing smoothed
curves, indicative of a resolved source crossing a caustic. Caustics are
features of a multiple-lens system. Therefore, we began our modeling with a
binary-lens model, which we ultimately found was sufficient to describe the
light curves for this event.

Figure 1: Magnification curves resulting from the fitted static binary-lens
model.

The binary-lens model is parameterized by ($s$, $q$, $\rho$, $u_{0}$,
$\alpha$, $t_{0}$ , $t_{\rm E}$), where $s$ is the angular separation of the
two lens masses in units of $\theta_{\rm E}$, $q$ is the mass ratio of the
lens objects, $\rho$ is the source angular radius in units of $\theta_{\rm
E}$, $u_{0}$ is the closest line-of-sight point of approach to the lens center
of mass made by the source in its relative trajectory (again in units of
$\theta_{\rm E}$), and $t_{0}$ is the time at which this happens
($\left|u_{0}\right|=u\left(t_{0}\right)$, where ${\mathbf{u}}(t_{i})$ is the
position of the source, projected onto the lens plane, at a given time, ($t$),
$\alpha$ is the angle of the projected rectilinear source trajectory relative
to an axis that passes through the lens masses, and $t_{\rm E}$ is the
Einstein radius crossing time (the time the source takes to travel an angular
distance of $\theta_{\rm E}$). For simplification, the motions in these models
were considered from the reference frame of the lens system. This meant that,
for modeling purposes, the relative velocities of any of the bodies involved
were attributed to the “source velocity”.

Our analysis of the ground-based light curves began by performing a grid
search over a fixed resolution on $s$, $q$, $u_{0}$, and $\alpha$, using
point-source approximations away from the caustics, for their computational
speed, and convolved magnification maps in high-magnification regions, where
finite-source effects were significant. The other model parameters were fitted
by $\chi^{2}$ minimization with $\rho$ values found by interpolating between
grid points with discrete convolutions. These calculations used a modified
version of the Microlensing Observations Rapid Search for Exoplanets code
(McDougall & Albrow, 2016).

The best 20 grid solution regions were further investigated using the Emcee
sampler (Foreman-Mackey et al., 2013). For this process we used the more
accurate Image Centered Inverse RAy Shooting (ICIRAS) (Bennett, 2010) or
contour integration (Bozza, 2010; Bozza et al., 2018) methods to calculate the
model magnification in regions close to caustics, and the hexadecapole
approximation (Pejcha & Heyrovský, 2009; Gould, 2008) otherwise. A fixed limb-
darkening coefficient ($\Gamma=0.53$)111Attempts to include this limb-
darkening coefficient as a free parameter in the model later in the modeling
process did not result in a more likely coefficient being found. was applied
to the source in these calculations. Two of the regions converged to the same,
and significantly most likely solution, while the next most likely solution
had a $\Delta\chi^{2}$ of $\sim 110\,000$, before renormalization. The
geometry of this static, ground-based solution is shown in Figure 2, and the
magnification curve, with ground-based data, is shown in Figure 1. The fitted
model parameters are displayed in Table 1 as the Static model. The solution
corresponds to a source passing over the edges of a large resonant caustic. We
note that this solution corresponds to small negative blending for three of
the data sources, though this is a normal occurrence for microlensing
photometry in a very crowded bulge field (Park et al., 2004), especially for
dim lenses. Table 1 shows $F_{\rm B}/F_{\rm S}$ for the OGLE source, which is
within $2\,\sigma$ of being positive.

Figure 2: Lens system and caustic geometry resulting from the
$(s,q)=(1.0,1.0)$ grid seed, with a projected source trajectory, for the
static binary-lens model, fitted to the ground-based data. Colored circles
show the source position at the times of the data points, where the colors
correspond to those specified in Figures 1 and 4, and the circle size depicts
the source size.

The source fluxes for each data set, were found from a linear fit;

$F_{i}=A_{i}\times F_{\rm S}+F_{\rm B},$ (2)

where $F_{\rm S}$ is the source-star flux, $F_{\rm B}$ is the blended
flux222The blended flux is made up of the nonlensed contributions to the
light-curve flux measurements, from light sources near the line of sight.
Sometimes the largest contributor to this flux component is the lens star,
though this is rarely the case., $A_{i}$ is the magnification at time $t_{i}$,
and $F_{i}$ is the observed total flux at time $t_{i}$. This solution to the
static model was used to renormalize the ground-based data uncertainties (see
Yee et al. 2012), and the solution was then allowed to reconverge.

###  3.1 Lens Orbital Motion or Ground-Based Parallax?

Although the peaks of the light curve are well fitted by this static-lens,
rectilinear-source model, there is a region between dates 7915-7922 where the
model systematically underpredicts the data (Figure 3). In Figure 4 we show
the cumulative $\chi^{2}$ as a function of time for each individual data set.
All curves show significant jumps near 7915-7922, indicating that there is a
real missing feature in our static model. Higher-order effects are required
for the model to provide a good description of these data.

Figure 3: 7915-7922 HJD crop of the magnification model from the best, static
fit (solid black lines) and the corresponding ground-based light-curve data
with renormalized errors. The data show a clear trend above the fit line in
this region. The dotted black lines show the lens-orbital-motion-inclusive
magnification model used in the next step of this event analysis. Outside of
this crop region the two models are visually indistinguishable. The data
colors correspond to those specified in Figures 1 and 4 Figure 4: Cumulative
$\chi^{2}$ plot for the renormalized static model.

Common high-order effects in microlensing light curves are orbital parallax
(motion of Earth during an event) and orbital motion of the binary-lens
system. A known degeneracy exists between these. Suspecting the significance
of one or both of these higher-order effects, we added them to the generative
model, both collectively and separately. We approximated the orbital motion of
the lens objects by allowing $\alpha$ and $s$ to vary linearly with time,
adding the model parameters $\dot{\alpha}$ and $\dot{s}$. Modeling the
parallax effect requires the introduction of two new parameters, $(\pi_{{\rm
E},N},\pi_{{\rm E},E})$, which are components of the vector
${\boldsymbol{\pi}_{\rm E}}$, where $\lVert{\boldsymbol{\pi}_{\rm
E}}\rVert=\frac{\pi_{\rm rel}}{\theta_{\rm E}}$, and its direction is that of
the lens-source relative proper motion. The introduction of measurable
parallax breaks the reflected symmetry of the source trajectory about the lens
axis; a trajectory above the lens axis is not equivalent to a trajectory below
the lens axis (except in the limit that the source lies exactly on the
ecliptic). We therefore modeled both positive and negative $u_{0}$ solutions
in which parallax was considered. For those solutions with both parallax and
lens orbital motion, we calculate $\beta$ (the ratio of the projected kinetic
to potential energy of the lens; An et al. 2002; Dong et al. 2009), where
values less than unity indicate a lens system consistent with a bound orbit;

$\beta=\frac{2({\rm au})^{2}}{c^{2}}\frac{\pi_{\rm E}}{\theta_{\rm
E}}\frac{\left[\left(\frac{1}{s}\frac{ds}{dt}\right)^{2}+\left(\frac{d\alpha}{dt}\right)^{2}\right]s^{3}}{\left[\pi_{\rm
E}+\left(\frac{\pi_{\rm S}}{\theta_{\rm E}}\right)\right]^{3}}.$ (3)

In our investigations of the significance of these two higher-order effects
(Table 1), we find that, alone, lens orbital motion describes the static model
discrepancies better than parallax. Including both higher-order effects yields
only a minor $\chi^{2}$ reduction compared with the purely lens-orbital-motion
model, and the lens-orbital-motion parameters change very little. (The low
$\beta$ values for these models show that the implied orbits are bound.)
Conversely, the posteriors of the parallax model change drastically when lens
orbital motion is added. We therefore conclude lens orbital motion is well
constrained and sufficient to describe the deviation on the static model from
7915-7922. This model is illustrated by the dotted lines in Figure 3.

Table 1: Comparison of the Highest Likelihood Fit Parameters for Binary-lens Models with and without the Higher-order Effects of Parallax and Lens Orbital Motion, Fit to Ground-based Data, with Renormalized Errors | Static | Parallax | LOM | Parallax + LOM
---|---|---|---|---
${\hat{u}}_{0}$ | - | + | - | - | + | -
$s$ | $0.9833^{+0.0006}_{-0.0005}$ | $0.9757^{+0.0003}_{-0.0009}$ | $0.9978^{+0.0011}_{-0.0005}$ | $0.9932^{+0.0002}_{-0.0009}$ | $0.9916^{+0.010}_{-0.0009}$ | $0.9888^{+0.015}_{-0.0004}$
$q$ | $0.621\pm 0.002$ | $0.616^{+0.003}_{-0.002}$ | $0.590^{+0.002}_{-0.003}$ | $0.607\pm 0.003$ | $0.609^{+0.004}_{-0.003}$ | $0.615^{+0.002}_{-0.005}$
$\log_{10}\rho$ | $-1.6027^{+0.0006}_{-0.0011}$ | $-1.6094^{+0.0011}_{-0.0008}$ | $-1.5818^{+0.0017}_{-0.0008}$ | $-1.5842^{+0.0004}_{-0.0012}$ | $-1.5859^{+0.0018}_{-0.0005}$ | $-1.5911^{+0.0026}_{-0.0004}$
$u_{0}$ | $-0.5693^{+0.0008}_{-0.0006}$ | $0.468^{+0.003}_{-0.009}$ | $-0.430^{+0.012}_{-0.005}$ | $-0.5551^{+0.0004}_{0.0012}$ | $0.552\pm 0.005$ | $-0.647^{+0.029}_{-0.004}$
$\alpha$ | $-2.9702^{+0.0005}_{-0.0007}$ | $2.845^{+0.003}_{-0.011}$ | $-2.853^{+0.011}_{-0.004}$ | $-2.9710^{+0.0007}_{-0.0006}$ | $2.965^{+0.007}_{-0.006}$ | $-3.064^{+0.030}_{-0.004}$
$t_{0}$ | $7926.900^{+0.007}_{-0.006}$ | $7924.47^{+0.07}_{-0.16}$ | $7927.33^{+0.06}_{-0.04}$ | $7927.009^{+0.003}_{-0.011}$ | $7926.83^{+0.10}_{-0.15}$ | $7927.37^{+0.03}_{-0.22}$
$t_{\rm E}$ | $11.852^{+0.015}_{-0.018}$ | $13.57^{+0.10}_{-0.07}$ | $10.55^{+0.02}_{-0.08}$ | $11.855^{+0.023}_{-0.007}$ | $12.02^{+0.15}_{-0.10}$ | $12.33^{+0.11}_{-0.16}$
$\pi_{{\rm E}N}$ | 0 | $-11.5^{+0.3}_{-1.0}$ | $12.6^{+1.2}_{-0.5}$ | 0 | $-0.6^{+0.6}_{-0.5}$ | $-9.6^{+3.1}_{-0.4}$
$\pi_{{\rm E}E}$ | 0 | $10.7\pm 0.5$ | $-10.0^{+0.2}_{-0.8}$ | 0 | $1.1^{+1.1}_{-0.7}$ | $1.9^{+0.8}_{-0.7}$
$\dot{s}$ | 0 | 0 | 0 | $0.30\pm 0.04$ | $0.31^{+0.08}_{-0.04}$ | $0.24^{+0.07}_{-0.04}$
$\dot{\alpha}$ | 0 | 0 | 0 | $-1.27\pm 0.04$ | $1.20^{+0.06}_{-0.05}$ | $-1.57^{+0.10}_{-0.06}$
$\beta$ |  |  |  |  | 0.13 | 0.01
$\chi^{2}_{min}$ | 12592.83 | 11946.44 | 12047.65 | 11468.38 | 11466.26 | 11441.77
$\Delta\chi^{2}_{min}$ | 0 | -646.40 | -545.19 | -1124.46 | -1126.57 | -1151.07
$N$ | $12607$
$I_{\rm S,OGLE}$ | 16.4 |  |  |  |  |
$F_{\rm B,OGLE}/F_{\rm S,OGLE}$ | -0.0075 $\pm$ 0.0041 |  |  |  |  |

###  3.2 Source Color

Color-magnitude Diagrams (CMDs) were created for each KMTNet observation site
and field with $I$ and $V$ data (KMTC-03; Figure 5, KMTC-43, KMTS-03, KMTS-43,
KMTA-03, and KMTA-43). We use the normal KMT practice of adopting magnitude
zero points of $I_{ZP}=28$ and $V_{ZP}=28.65$. The source-star fluxes,
obtained from fitting the magnification model to each light curve, were used
to find the source star’s position on the corresponding CMDs. The source
fluxes for the highest likelihood solution (ground based) are given in Table
2.

Table 2: Source fluxes, for Each Observation Source and Band Source | $F_{{\rm S},I}$ | $F_{{\rm S},V}$ | $F_{{\rm S},L}$
---|---|---|---
KMTC-03 | 52595.17 | 5000.67 |
KMTC-43 | 34761.72 | 5074.91 |
KMTS-03 | 50862.79 | 4125.91 |
KMTS-43 | 52803.33 | 4511.05 |
KMTA-03 | 40084.08 | 4813.62 |
KMTA-43 | 38048.48 | 4800.27 |
OGLE | 43537.75 |  |
Spitzer |  |  | 56.09

The red clump in each CMD was centroid fitted, and acted as a calibration for
obtaining the intrinsic colors and magnitudes of the field. The galactic bulge
red clump can be used to calibrate the CMD because its intrinsic color and
magnitude are known to high precision. The intrinsic color of the red clump is
$(V-I)_{{\rm RC},0}=1.06$ (Bensby et al., 2011). The intrinsic I-magnitude of
the red-clump was found by interpolating the extinction correction table from
(Nataf et al., 2013) for the target’s galactic longitude
$(l=2.85^{\circ},\,b=-1.64^{\circ})$; $I_{{\rm RC},0}=14.35\pm 0.04$. Assuming
that the source is obscured by the same amount of dust as the average red
clump star in this field, $(V-I)_{{\rm RC},0}$ and $I_{{\rm RC},0}$ provide an
absolute color and magnitude calibration to the CMDs.

Figure 5: Color-magnitude diagram from the KMTC-03 field with the fitted
centroid of the red clump and the source position indicated by the red “+” and
blue “+”, respectively.

Using the mean calibrated color and magnitude, the intrinsic magnitude and
color of the source was found to be $(I_{0},(V-I)_{0})=(14.01\pm 0.05,1.11\pm
0.04)$, averaged over all six CMDs. These values are very similar for each of
the possible solutions for the final model.

This source color information was also used to infer the Spitzer source flux
and a color-color relation between KMTC-03 and Spitzer using the method of
Calchi Novati et al. (2015b). The expected Spitzer Source flux is $F_{{\rm
S},L}=56.1\pm 1.7$ and the optical-infrared source color is $(I-L)_{\rm
S}=-4.43\pm 0.03$, with an $L$-magnitude zero point of 25.

##  4 Inclusion of Satellite Data

Having a Spitzer light curve for this event meant that, despite there being
very inconclusive orbital parallax signals in the ground-based data, parallax
could still be measured (Refsdal, 1966). In this section we describe our
analysis of the space-based Spitzer data using typical error renormalization
methods, discuss concerns over systematics errors in the data, and present an
alternate approach to coping with such systematics.

###  4.1 Satellite Parallax Degeneracies

Figure 6: Raw Spitzer light curve and model light curve resulting from the
static binary-lens model (no parallax), fitted to the ground-based data, and
transforming to the Spitzer flux system assuming $F_{\rm L}\equiv F_{{\rm
S},L}A$, where $F_{{\rm S},L}=56.1$ and $(I-L)_{\rm S}=-7.4$ . The ground-
based observations have also been scaled to the Spitzer flux system. The
residuals between the model and the data are depicted in black for ground-
based data and red for Spitzer data. These show a dramatic difference for the
$t<7935$ data.

Figure 6 shows the raw Spitzer data and a corresponding magnification curve
from estimating $F_{\rm S}=56.1$ (as is suggested by the color comparisons of
§3.2), $F_{\rm B}=0$, and adopting the ground-based model. In this figure, we
can see a clear, decreasing signal that has $\Delta F>30$ Spitzer flux units.
The Spitzer data are inconsistent with very small parallax, as the shape of
the magnification curve is not well represented by the static ground-based
model, and no alternative values of $F_{\rm S}$ and $F_{\rm B}$ could bring
them into agreement. At the time of the first Spitzer observation, the ground-
based light curve is still exiting the cusp while the Spitzer data are clearly
not. This is strong evidence for a parallax effect. At the same time, the
required magnification change as seen from Spitzer ($\Delta A\sim 1.6$)
indicates that the parallax cannot be too large.

When viewed from Spitzer, the angular source trajectory across the lens plane
is offset by a vector $(\Delta\beta,\Delta\tau)/\theta_{\rm E}$, in directions
(perpendicular, parallel) to ${\bm{D}}_{\perp}$, the separation between
Spitzer and Earth projected onto the lens plane. This vector is related to the
parallax measurement, but can be more useful in understanding the parallax
likelihood space in comparison with the caustic diagram representation of the
event. The two parameters $(\Delta\beta,\Delta\tau)$ can be mapped onto
$\pi_{{\rm E},E}$ and $\pi_{{\rm E},N}$, via ${\boldsymbol{\pi}_{\rm
E}}=\frac{{\rm au}}{D_{\perp}}\left(\Delta\tau,\Delta\beta\right).$ The
parallel offset is simply

$\Delta\tau=\frac{t_{0,{\rm Spitzer}}-t_{0,{\rm Earth}}}{t_{\rm E}}.$ (4)

In the case of a single lens, the perpendicular offset suffers from a four-
fold satellite parallax degeneracy,

$\Delta\beta=\pm u_{0,{\rm Spitzer}}-\pm u_{0,{\rm Earth}},$ (5)

due to the exact circular symmetry of the magnification field about the lens
(Refsdal, 1966), as illustrated in Gould (1994). (The sign convention we adopt
here is that a positive value of $u_{0}$ indicates that, during its projected
trajectory, the source approaches the lens center of mass on its right hand
side.) In general, this fourfold degeneracy usually reduces to twofold with
the addition of a second lens body, as the resulting caustic features break
the symmetry of the magnification field. However, for binary-lens events in
which the trajectory runs approximately parallel to the lens axis (such as the
current case), trajectories reflected about the lens axis result in similar
magnification curves, in which case the four-fold degeneracy is retained (Zhu
et al., 2015).

A grid-search approach was used to determine the most likely parallax-solution
regions. With the inclusion of space-based data, the two parallax parameters
($\pi_{{\rm E},N}$ and $\pi_{{\rm E},E}$) were added to the model.

When performing the parallax grid search, the ground-based model parameters
(including lens orbital motion) were fixed, and a maximum-likelihood search
was performed for the Spitzer light curve over a large range of discrete
$\pi_{{\rm E},N}$ and $\pi_{{\rm E},E}$ values.

This grid search indicated that there were four solution regions for the given
ground-based model, with the two outer regions having much higher likelihoods
(i.e, lower $\chi^{2}$) than the two inner regions (Figure 7a). These four
solutions regions represent the $\pm u_{0,{\rm Spitzer}}$ degenerate
trajectories relating to two distinct solution families. We refer to these
families as close (c) and wide (w). The four solutions regions result from
only $-u_{0,Earth}$ and indicate that, including the $+u_{0,Earth}$
trajectory, we have an eightfold degeneracy for this particular geometry.

Because the Spitzer data only cover the falling part of the light curve and
cover no caustic feature, the light curve alone does not contribute very
strong constraints on the parallax measurement. We have thus implemented in
the modeling an additional $\chi^{2}$ penalty term that weighted the fit
toward a source-flux ratio (between KMT-C03 and Spitzer $L$) matching that
inferred by the calculated $(I-L)_{0}$ source color, found in §3.2. This
color-constraint (Shin et al., 2017) term was of the form

$\chi^{2}_{constraint}=\frac{\left(2.5\log_{10}\left(\frac{\left(\frac{F_{I}}{F_{L}}\right)_{model}}{\left(\frac{F_{I}}{F_{L}}\right)_{constraint}}\right)\right)^{2}}{\sigma^{2}_{constraint}}.$
(6)

The constraint changed the likelihood space of the parallax model. The four
solutions-regions from the unconstrained grid, remained as features in the
constrained grid. However, the close set of solutions have more comparable
likelihoods to the wide set than in the unconstrained grid.

Figure 7: _Left_ : contour maps demonstrating the results of the parallax grid
searches over discretely varied $\pi_{{\rm E},E}$ and $\pi_{{\rm E},N}$ for
the $-u_{0,Earth}$ configuration, including only the Spitzer $\chi^{2}$
components. The dashed contours show the $\chi^{2}$ landscape without a color
constraint and the solid lines with the constraint. Note that the $\pi_{{\rm
E},N}$-axis of this figure is reversed from the usual orientation so that the
two figures approximately align. _Right_ : caustic diagram with projected
ground-based and Spitzer-based trajectories (black and red, respectively). The
four Spitzer trajectories are the result of minimization from the local
$\chi^{2}$ minima from the left figure, with all modeling parameters free to
evolve. The data points are represented by colored circles on the
trajectories, where the colors correspond to the observation site and field,
as specified in Figures 1 and 4. The caustics change with the orbiting of the
lens bodies and are depicted here at the instances of the first and last
Spitzer data points, specifically for the c -/+ solution (all four
$u_{0,Earth}<0$ solutions looks very diagrammatically similar to the one
shown). These epochs are represented on grounds-based trajectory (also from
the c -/+ solution) with colours matching their corresponding caustics. Figure
8: Caustic diagram with projected ground-based trajectory for the c -/+
solution. The ground-based data points are represented by colored circles on
the trajectories, where the colors correspond to the observation site and
field, as specified in Figures 1 and 4. The caustics are depicted here at
three instances corresponding to the start of the “problem region” (7915 =
$t_{0}-1.01t_{\rm E}$), $t_{0}$ (7926.91), and the time of the first Spitzer
data point ($7931.47=t_{0}+0.38t_{\rm E}$)). These epochs are represented on
the grounds based trajectory with colors matching their corresponding
caustics.

When comparing Figure 7a and Figure 7b, the reason for the four lobes in the
likelihood space becomes apparent. For this event, $\Delta\beta$ approximately
aligns with $\pi_{{\rm E},N}$ and $\Delta\tau$ with $\pi_{{\rm E},E}$.
Simplistically, changing $\Delta\tau$ moves the Spitzer-data nodes backward or
forward in time along the Spitzer trajectory, whereas $\Delta\beta$ shifts the
“parallel” space-based trajectory closer to or farther away from the ground-
based trajectory. The lobes and connective contours in Figure 7a result from
solutions for which the Spitzer data hug the leftmost cusps of the caustic of
Figure 7b.

Figure 8 shows a more restricted view of the ground-based trajectory for this
set of solutions, and the caustics at key epochs in the light curve, which
change over the course of the event due to the orbital motion of the lenses.

Within each wide or close set, the pairs are the previously predicted $\pm
u_{0,{\rm Spitzer}}$ degenerate solutions. A further four degenerate solutions
are obtained by reflecting all trajectories in Figure 7b (ground and Spitzer)
about the lens axis.

The eight degenerate solution regions were further investigated using emcee
with both ground-based and Spitzer data, renormalized errors, and both
parallax and lens orbital motion included in the model. All model parameters
were left free to evolve for all instances. Model parameters for the resulting
solutions are given in Table 3. They are all somewhat similar in likelihood
with an overall range in $\Delta\chi^{2}\leq 87$. The best solution found was
the c -/+ geometry. All close solutions were favored over the wide by a margin
of $\chi^{2}_{w}-\chi^{2}_{c}\geq 8.96$. The nonfavored close solutions have a
range $12.48<\Delta\chi^{2}<28.06$.

Table 3: Final Model Parameters and Physical Parameters for the Eight
Degenerate Solutions Utilizing Fixed Spitzer Error-bar Scaling ($S_{{\rm
Spitzer}}=3.93$)

| c -/- | c -/+ | c +/- | c +/+ | w -/- | w -/+ | w +/- | w +/+
---|---|---|---|---|---|---|---|---
$s$ | $0.9923^{+0.0007}_{-0.0004}$ | $0.9934^{+0.0001}_{-0.0011}$ | $0.9920\pm 0.0006$ | $0.9924\pm 0.0006$ | $0.9929^{+0.0005}_{-0.0007}$ | $0.9933^{+0.0005}_{-0.0006}$ | $0.9919^{+0.0007}_{-0.0005}$ | $0.9930^{+0.0006}_{-0.0005}$
$q$ | $0.609\pm 0.003$ | $0.609^{+0.004}_{-0.002}$ | $0.607\pm 0.003$ | $0.608\pm 0.003$ | $0.607\pm 0.003$ | $0.607^{+0.002}_{-0.003}$ | $0.604^{+0.003}_{-0.002}$ | $0.607^{+0.004}_{-0.002}$
$\log_{10}\rho$ | $-1.5852^{+0.0015}_{-0.0004}$ | $-1.5835^{+0.0006}_{-0.0013}$ | $-1.5858\pm 0.0010$ | $-1.5851^{+0.0013}_{-0.0007}$ | $-1.55844^{+0.0006}_{-0.0011}$ | $-1.5835^{+0.0008}_{-0.0011}$ | $-1.5860^{+0.0009}_{-0.0007}$ | $-1.5843^{+0.0012}_{-0.0008}$
$u_{0}$ | $-0.5552^{+0.0011}_{-0.0006}$ | $-0.5455^{+0.0001}_{-0.0015}$ | $0.5472\pm 0.0008$ | $0.5552\pm 0.0008$ | $-0.5580^{+0.0006}_{-0.0010}$ | $-0.5431^{+0.0008}_{-0.0007}$ | $0.5439^{+0.0009}_{-0.0007}$ | $0.5577\pm 0.0008$
$\alpha$ | $-2.9698^{+0.0006}_{-0.0008}$ | $-2.9612^{+0.0006}_{-0.0009}$ | $2.9597^{+0.0004}_{-0.0010}$ | $2.9695^{+0.0009}_{-0.0005}$ | $-2.9733^{+0.0006}_{-0.0008}$ | $-2.9589^{+0.0011}_{-0.0003}$ | $2.9557\pm 0.0007$ | $2.9738^{+0.0005}_{-0.0008}$
$t_{0}$ | $7926.991^{+0.010}_{-0.004}$ | $7926.973^{+0.003}_{-0.012}$ | $7926.930^{+0.006}_{-0.009}$ | $7926.990^{+0.008}_{-0.007}$ | $7927.034\pm 0.007$ | $7926.982^{+0.004}_{-0.011}$ | $7926.925^{+0.009}_{-0.006}$ | $7927.041^{+0.007}_{-0.008}$
$t_{\rm E}$ | $11.883^{+0.015}_{-0.021}$ | $11.825^{+0.029}_{-0.006}$ | $11.868^{+0.021}_{-0.014}$ | $11.881^{+0.018}_{-0.017}$ | $11.845^{+0.024}_{-0.012}$ | $11.800^{+0.013}_{-0.022}$ | $11.840^{+0.017}_{-0.018}$ | $11.847\pm 0.018$
$\pi_{{\rm E},N}$ | $0.121^{+0.009}_{-0.017}$ | $1.015^{+0.007}_{-0.015}$ | $-1.028\pm 0.010$ | $-0.107^{+0.007}_{-0.016}$ | $-0.277^{+0.007}_{-0.010}$ | $1.295^{+0.013}_{-0.003}$ | $-1.357^{+0.008}_{-0.007}$ | $0.256^{+0.005}_{-0.011}$
$\pi_{{\rm E},E}$ | $0.020^{+0.013}_{-0.024}$ | $-0.230^{+0.027}_{-0.009}$ | $-0.073^{+0.017}_{-0.021}$ | $0.020^{+0.026}_{-0.009}$ | $-0.136^{+0.018}_{-0.023}$ | $-0.457^{+0.019}_{-0.013}$ | $-0.293^{+0.017}_{-0.021}$ | $-0.160^{+0.022}_{-0.014}$
$\dot{s}$ | $0.31^{+0.05}_{-0.03}$ | $0.32^{+0.05}_{-0.03}$ | $0.25^{+0.04}_{-0.03}$ | $0.31^{+0.04}_{-0.03}$ | $0.29^{+0.04}_{-0.03}$ | $0.33^{+0.01}_{-0.07}$ | $0.22^{+0.03}_{-0.05}$ | $0.31\pm 0.04$
$\dot{\alpha}$ | $-1.25^{+0.02}_{-0.05}$ | $-1.25^{+0.05}_{-0.02}$ | $1.21^{+0.05}_{-0.02}$ | $1.27^{+0.03}_{-0.04}$ | $-1.25^{+0.02}_{-0.05}$ | $-1.23^{+0.03}_{-0.04}$ | $1.22^{+0.04}_{-0.03}$ | $1.28\pm 0.04$
$\chi^{2}_{min}$ | 11555.55 | 11529.44 | 11541.92 | 11537.49 | 11566.46 | 11601.31 | 11615.94 | 11570.26
$\Delta\chi^{2}_{min}$ | 26.12 | 0 | 12.48 | 28.06 | 37.02 | 71.87 | 86.50 | 40.83
$\beta$ | 0.40 | 0.19 | 0.17 | 0.39 | 0.43 | 0.13 | 0.12 | 0.45
$s\,({\rm au})$ | $1.77^{+0.13}_{-0.12}$ | $0.67\pm 0.05$ | $0.68\pm 0.05$ | $1.81^{+0.12}_{-0.14}$ | $1.33\pm 0.10$ | $0.55\pm 0.05$ | $0.54\pm 0.04$ | $1.34\pm 0.10$
$M_{tot}\,(M_{\odot})$ | $0.29^{+0.05}_{-0.03}$ | $0.034\pm 0.002$ | $0.035\pm 0.002$ | $0.33^{+0.03}_{-0.05}$ | $0.116^{+0.009}_{-0.010}$ | $0.0259\pm 0.0017$ | $0.0258\pm 0.0017$ | $0.118^{+0.010}_{-0.008}$
$m_{1}\,(M_{J})$ | $188^{+29}_{-14}$ | $22.30^{+0.39}_{-0.14}$ | $22.6\pm 0.2$ | $213^{+9}_{-32}$ | $76^{+3}_{-4}$ | $16.90^{+0.07}_{-0.13}$ | $16.85^{+0.11}_{-0.10}$ | $77^{+4}_{-2}$
$m_{2}\,(M_{J})$ | $115^{+18}_{-9}$ | $13.53^{+0.25}_{-0.05}$ | $13.73\pm 0.14$ | $130^{+5}_{-20}$ | $46\pm 2$ | $10.26^{+0.03}_{-0.09}$ | $10.18^{+0.07}_{-0.05}$ | $47^{+3}_{-1}$
$D_{\rm L}\,(\rm{kpc})$ | $6.12^{+0.21}_{-0.14}$ | $2.33\pm 0.11$ | $2.34\pm 0.11$ | $6.28^{+0.10}_{-0.23}$ | $4.61^{+0.14}_{-0.16}$ | $1.90^{+0.09}_{-0.10}0$ | $1.88\pm 0.09$ | $4.65^{+0.16}_{-0.13}$
$\mu_{rel,hel}(N,E)\,({\rm mas\,yr^{-1}})$ | $(8.8,1.7)$ | $(8.73,-0.11)$ | $(-8.96,1.22)$ | $(-8.8,1.9)$ | $(-8.0,-3.4)$ | $(8.46,-0.53)$ | $(-8.81,0.60)$ | $(7.6,-4.2)$
$\delta\mu_{rel,hel}(N,E)\,({\rm mas\,yr^{-1}})$ | $(^{+0.1}_{-0.2},^{+0.9}_{-1.8})$ | $(^{+0.05}_{-0.07},^{+0.25}_{-0.15})$ | $(^{+0.03}_{-0.02},^{+0.19}_{-0.22})$ | $(^{+0.5}_{-0.1},^{+1.8}_{-0.8})$ | $(^{+0.3}_{-0.2},^{+0.4}_{-0.5})$ | $(^{+0.05}_{-0.02},^{+0.21}_{-0.18})$ | $(\pm 0.03,^{+0.20}_{-0.21})$ | $(^{+0.3}_{-0.2},^{+0.5}_{-0.3})$
$\mu_{{\rm L},hel}(N,E)\,({\rm mas\,yr^{-1}})$ | $(3.1,-5.9)$ | $(3.1,-7.7)$ | $(-14.6,-6.4)$ | $(-14.5,-5.8)$ | $(-13.7,-11.0)$ | $(2.8,-8.2)$ | $(-14.5,-7.0)$ | $(1.9,-11.8)$
$\delta\mu_{{\rm L},hel}(N,E)\,({\rm mas\,yr^{-1}})$ | $(^{+0.2}_{-0.3},^{+1.0}_{-1.8})$ | $\pm(0.2,0.5)$ | $\pm(0.2,0.5)$ | $(^{+0.6}_{-0.2},^{+1.8}_{-0.9})$ | $(\pm 0.3,^{+0.5}_{-0.6})$ | $\pm(0.2,0.5)$ | $\pm(0.2,0.5)$ | $(\pm 0.3,^{+0.6}_{-0.5})$
$v_{{\rm L},hel}(l,b)\,({\rm km\,s^{-1}})$ | $(-6,195)$ | $(-13,91)$ | $(-176,-19)$ | $(-458,-66)$ | $(-380,59)$ | $(-15,76)$ | $(-143,-10)$ | $(-93,247)$
$\delta v_{{\rm L},hel}(l,b)\,({\rm km\,s^{-1}})$ | $(^{+13}_{-25},^{+50}_{-28})$ | $(^{+5}_{-4},\pm 5)$ | $\pm(9,4)$ | $(^{+52}_{-17},^{+20}_{-37})$ | $(^{+15}_{-14},\pm 11)$ | $\pm(4,5)$ | $\pm(8,3)$ | $(^{+12}_{-11},\pm 9)$
$-2\Delta\ln z_{galactic}$ | $3.73$ | $12.42$ | $25.81$ | $0$ | $10.64$ | $14.63$ | $31.45$ | $1.86$
$-2\Delta\ln z$ | $13.48$ | $0$ | $25.81$ | $15.53$ | $35.35$ | $74.13$ | $105.43$ | $30.24$

Note. — Those solutions indicated to by “LOM” refer to the models in which
lens orbital motion was included. The source magnitude uses a zero point of
$I_{ZP}=28$. N is the total number of light-curve data points. Solutions with
$\beta<1$ are consistent with a bound orbit, but can only be calculated for
models including both lens orbital motion and parallax.

Note. — These values were calculated using an orbiting, binary-lens model, for
each of the ground-based sources. The Spitzer source flux is an estimate based
on comparative CMDs between the Spitzer field and the KMTC-03 field.

Note. — These models included both ground-based and space-based data, with
parallax and energy-constrained lens orbital motion included in the model. The
parameter values quoted in this table are those corresponding to the minimum
$\chi^{2}$ samples from the posterior. The uncertainties recover the 16th and
84th percentiles and are asymmetric because the posteriors are not entirely
Gaussian and parameter values corresponding to the minimum $\chi^{2}$ solution
differ from the mode of the same parameter’s samples. The physical parameters
have additional uncertainty components, added in quadrature to the percentile
uncertainties, based on the fixed uncertainties of values used in the
calculations of these parameters. We use $-2\Delta\ln z$ as an effective
$\chi^{2}$ value. $-2\Delta\ln z$ incorporates both the fit likelihood and the
detection probability, based on our galactic model.

###  4.2 Spitzer Systematic Errors

Before we can have faith in these Spitzer parallax measurements, we must first
address concerns of systematics in the Spitzer light curve.

Yee et al. (2021), Gould et al. (2020), Hirao et al. (2020), and Zang et al.
(2020) include detailed investigations into Spitzer systematics. These
investigations point to poorly determined positions of nearby blend stars in
combination with the seasonal rotation of the Spitzer camera. This has
resulted in variable blended levels ($F_{\rm B}$) seen over timescales on the
order of tens of days. These works conclude that Spitzer systematics are at
the level of $\sim 1$ Spitzer flux unit where, for a typical event, $F_{\rm
B}\approx 3$. Concerns have been raised for previous events (Zhu et al. 2017;
Koshimoto & Bennett 2019) where the flux levels were $F_{\rm S}<5$ and thus
$F_{{\rm S}}\sim F_{\rm B}$, in which case systematics on the order of 1 could
be considered fractionally significant.

We now consider whether systematics in the Spitzer data are significant for
this event. The Spitzer magnification curve has a bump between $t=7936$ and
$t=7941$ (corresponding to $\Delta F\simeq 5$ Spitzer flux units; see Figure
6) that is not produced by any of our best generative model solutions that
incorporate satellite parallax (Section 4.1). This implies a systematic error
and demonstrates the scale to which we can expect them in this specific
Spitzer data set; a few flux units over timescales of around 5 days. This is a
higher $\Delta F$ perturbation than is expected for Spitzer systematic on a
smooth curve (typically $\Delta F\simeq 1$ Spitzer flux units)

The parallax terms in the model are sensitive to small contiguous
perturbations in the data, especially for those data after $t=7955$, where
flux changes of a few units change the shape of the slope enough to result in
different parallax measurements, which affect the resulting physical
solutions.

For this event, we have a Spitzer source flux much larger than the expected
blend flux, a light curve with clearly and significantly decreasing flux, and
baseline observations. Therefore, we would not ordinarily expect systematics
to play a major role in this case. However, this event is somewhat sensitive
to systematics in the baseline and shows evidence of similar systematics
elsewhere in the light curve. We are therefore cautious of the effects
systematic error in the Spitzer data may have on our conclusions.

###  4.3 Modeling Spitzer Errors

In an attempt to properly consider the apparent systematic errors in the
Spitzer data, we have included in our model an error-bar renormalization
parameter and two Gaussian process (GP) parameters.

Gaussian processes were first introduced in microlensing event analysis by Li
et al. (2019). In this paper they used a GP to model source variability,
rather than systematics, as well as a traditional inflated-error-bar scaling
method. The GP method achieved better results in their case, as evidenced by
the residuals in their Figure 1. However, they adopt their inflated-error-bar
scaling model due to multiple practical and theoretical concerns. The
practical issues they raise are how to cope with different blending effects
between observation sources and how to perform error re-scaling. The blending
issue is not relevant in our case because we only apply a GP model on the
Spitzer data set. The theoretical issues they raise are in regards to choice
of GP kernel and the possibility of degeneracies between the microlensing and
GP parameters, for which they saw no evidence in their posterior
distributions. We also saw no evidence of degeneracies between microlensing
and GP parameters in our posterior distributions. In regards to the choice of
GP kernel we tested both the exponential (described below) and Matern 3/2
kernels and found no significant difference between the results. We did not
test the kernel used in Li et al. (2019) as it is meant for modeling quasi-
periodic variations.

The degenerate solutions of Section 4.1 have reduced $\chi^{2}$ values that
imply that the Spitzer flux uncertainties have been underestimated by factors
of between 2 and 5 times before renormalization. Because these factors change
for each solution we include a multiplicative Spitzer error renormalization as
a free parameter and consequently the likelihood must change to include the
penalty

$\ln P_{{\rm S}}=-N\ln{S},$

where $S$ is the Spitzer error scaling factor and $N$ refers here to the
number of Spitzer data points.

Simultaneously we included an exponential GP model to fit the systematic
features in the Spitzer light curve using the Celerite package (Foreman-Mackey
et al., 2017). This replaces the vector of data variances with a data
covariance matrix,

$K_{nm}=\sigma^{2}_{n}\delta_{nm}+k(t_{n},t_{m}).$

We use a GP kernel

$k(\tau_{nm})=a\exp({-c\tau_{nm}}),$

where $\tau_{nm}=|t_{n}-t_{m}|$, and $a$ and $c$ are the GP model parameters.

The GP likelihood is then

$\ln{P_{GP}}=-\frac{1}{2}{\mathbf{r}}^{T}{\mathbf{K}}^{-1}{\mathbf{r}}-\frac{1}{2}\ln{\det{\mathbf{K}}}-\frac{N}{2}\ln{2\pi},$

where $\mathbf{r}$ is the vector of (data - model) residuals.

The results of this modeling are displayed in Table 4. We find that inclusion
of these three new model parameters has little effect on the microlensing
parameters of all eight solutions, although the spread of likelihood values
between solutions does change. With the GP parameters included in the model,
our best solution is no longer the c -/+ but the c +/-, although, by a very
small margin. The light curve for this model is shown in Figure 9. The full
family of close solutions are all similarly likely,
$-2\Delta\ln\mathcal{L}<2.3$, where we consider $-2\Delta\ln\mathcal{L}$ as an
effective $\Delta\chi^{2}$.

Figure 9: Spitzer magnification curve for the c +/- solution. The blue lines
show the fitted magnification curve for 100 samples from the posterior with
the GP effects shown. The red line is the magnification curve matching the
parameters in Table 4. The error-bar scaling in this figure corresponds to the
red line ($S_{{\rm Spitzer}}=2.49$), and the size of the error bars is not
necessarily the scaling used for each of the blue samples. Left: shows the
magnification curve over the same time period as Figure 1. Right: covers only
the Spitzer data set. Table 4: Final Model Parameters, Physical Parameters and
Likelihood Components for the Eight Degenerate Solutions with Three Additional
Parameters, Compared with the Models of Table 3, for the Spitzer Error-bar
Scaling ($S_{{\rm Spitzer}}$) and Gaussian Process Modeling ($a$, $c$).

| c -/- | c -/+ | c +/- | c +/+ | w -/- | w -/+ | w +/- | w +/+
---|---|---|---|---|---|---|---|---
$s$ | $0.9927^{+0.0006}_{-0.0005}$ | $0.9933^{+0.0004}_{-0.0008}$ | $0.9919^{+0.0007}_{-0.0004}$ | $0.9926\pm 0.0006$ | $0.9932^{+0.0002}_{-0.0009}$ | $0.9934^{+0.0005}_{-0.0007}$ | $0.9920^{+0.0007}_{-0.0005}$ | $0.9933^{+0.0004}_{-0.0007}$
$q$ | $0.6061^{+0.0049}_{-0.0010}$ | $0.6078^{+0.0027}_{-0.0033}$ | $0.6074^{+0.0015}_{-0.0042}$ | $0.6094^{+0.0013}_{-0.0045}$ | $0.6077^{+0.00020}_{-0.0036}$ | $0.6078^{+0.0016}_{-0.0041}$ | $0.6035^{+0.0045}_{-0.0013}$ | $0.6065^{+0.0038}_{-0.0022}$
$\log_{10}\rho$ | $-1.5840^{+0.0005}_{-0.0013}$ | $-1.5837^{+0.0009}_{-0.0010}$ | $-1.5858^{+0.0011}_{-0.0009}$ | $-1.5846^{+0.0001}_{-0.0009}$ | $-1.5842^{+0.0006}_{-0.0013}$ | $-1.5831^{+0.0005}_{-0.0014}$ | $-1.5868^{+0.0019}_{-0.0002}$ | $-1.5843^{+0.0013}_{-0.0007}$
$u_{0}$ | $-0.5546^{+0.0005}_{-0.00013}$ | $-0.5457^{+0.0009}_{-0.0010}$ | $0.5471^{+0.0011}_{-0.0009}$ | $0.5551^{+0.0001}_{-0.0009}$ | $-0.5582^{+0.0006}_{-0.0013}$ | $-0.5431^{+0.0005}_{-0.0014}$ | $0.5442^{+0.0019}_{-0.0002}$ | $0.5575^{+0.0010}_{-0.0007}$
$\alpha$ | $-2.9703^{+0.0013}_{-0.0002}$ | $-2.9612^{+0.0006}_{-0.0008}$ | $2.9591^{+0.0011}_{-0.0004}$ | $2.9695^{+0.0008}_{-0.0007}$ | $-2.9738^{+0.0008}_{-0.0006}$ | $-2.9590^{+0.0010}_{-0.0004}$ | $2.9562^{+0.0006}_{-0.0008}$ | $2.9737^{+0.0006}_{-0.0008}$
$t_{0}$ | $7926.992^{+0.006}_{-0.010}$ | $7926.969^{+0.006}_{-0.010}$ | $7926.924\pm 0.008$ | $7926.987^{+0.007}_{-0.010}$ | $7927.067^{+0.001}_{-0.016}$ | $7926.973^{+0.005}_{-0.011}$ | $7926.911^{+0.015}_{-0.004}$ | $7927.068^{+0.005}_{-0.023}$
$t_{\rm E}$ | $11.881^{+0.015}_{-0.020}$ | $11.834^{+0.020}_{-0.017}$ | $11.876^{+0.014}_{-0.022}$ | $11.884^{+0.016}_{-0.020}$ | $11.811^{+0.029}_{-0.006}$ | $11.811^{+0.016}_{-0.021}$ | $11.853^{+0.018}_{-0.022}$ | $11.811^{+0.037}_{-0.009}$
$\pi_{{\rm E},N}$ | $0.117^{+0.045}_{-0.020}$ | $1.020^{+0.013}_{-0.023}$ | $-1.037^{+0.030}_{-0.013}$ | $-0.112^{+0.022}_{-0.037}$ | $-0.331^{+0.021}_{-0.006}$ | $1.291^{+0.004}_{-0.010}$ | $-1.3208^{+0.0005}_{-0.0175}$ | $0.274^{+0.002}_{-0.031}$
$\pi_{{\rm E},E}$ | $0.061^{+0.027}_{-0.016}$ | $-0.193^{+0.032}_{-0.025}$ | $-0.050^{+0.052}_{-0.007}$ | $0.075^{+0.034}_{-0.015}$ | $-0.373^{+0.066}_{-0.008}$ | $-0.350^{+0.024}_{-0.029}$ | $-0.135^{+0.005}_{-0.065}$ | $-0.393^{+0.175}_{-0.021}$
$\dot{s}$ | $0.30^{+0.05}_{-0.03}$ | $0.33^{+0.03}_{-0.05}$ | $0.25^{+0.03}_{-0.05}$ | $0.32^{+0.03}_{-0.05}$ | $0.29^{+0.03}_{-0.05}$ | $0.348^{+0.004}_{-0.074}$ | $0.21^{+0.05}_{-0.03}$ | $0.29^{+0.05}_{-0.03}$
$\dot{\alpha}$ | $-1.27^{+0.04}_{-0.03}$ | $-1.26^{+0.06}_{-0.02}$ | $1.24^{+0.03}_{-0.04}$ | $1.27^{+0.02}_{-0.05}$ | $-1.27^{+0.06}_{-0.02}$ | $-1.24^{+0.05}_{-0.02}$ | $1.20^{+0.06}_{-0.02}$ | $1.28^{+0.04}_{-0.03}$
$S_{{\rm Spitzer}}$ | $2.2^{+0.9}_{-0.6}$ | $1.9^{+0.8}_{-0.7}$ | $2.0^{+1.0}_{-0.6}$ | $2.3^{+0.6}_{-0.8}$ | $1.9^{+1.0}_{-0.6}$ | $1.9^{+1.3}_{-0.8}$ | $1.3^{+1.3}_{-0.9}$ | $1.4^{+1.3}_{-0.4}$
$a$ | $1.6^{+10.4}_{-0.6}$ | $0.8^{+2.2}_{-0.4}$ | $1.1^{+4.8}_{-0.4}$ | $2.0^{+10.7}_{+0.9}$ | $4.0^{+12.9}_{-2.3}$ | $3.4^{+18.5}_{-1.2}$ | $4.2^{+23.1}_{-1.5}$ | $3.0^{+15.4}_{-0.9}$
$c$ | $0.12^{+0.11}_{-0.10}$ | $0.27^{+0.12}_{-0.24}$ | $0.13^{+0.14}_{-0.11}$ | $0.11\pm 0.10$ | $0.08^{+0.13}_{-0.07}$ | $0.14\pm 0.12$ | $0.13\pm 0.11$ | $0.14^{+0.08}_{-0.12}$
$-2\ln{\mathcal{L}}$ | 11559.391 | 11559.801 | 11557.491 | 11558.619 | 11564.373 | 11588.644 | 11587.596 | 11572.086
$-2\Delta\ln{\mathcal{L}}$ | 1.90 | 2.31 | 0 | 1.13 | 6.88 | 31.15 | 30.11 | 14.60
$\beta$ | 0.42 | 0.19 | 0.18 | 0.43 | 0.36 | 0.14 | 0.12 | 0.37
$s\,({\rm au})$ | $1.74^{+0.14}_{-0.18}$ | $0.67^{+0.06}_{-0.05}$ | $0.67^{+0.06}_{-0.05}$ | $1.73^{+0.14}_{-0.18}$ | $1.06^{+0.11}_{-0.08}$ | $0.56\pm 0.05$ | $0.56\pm 0.05$ | $1.08^{+0.23}_{-0.08}$
$M_{tot}\,(M_{\odot})$ | $0.27^{+0.05}_{-0.08}$ | $0.034\pm 0.002$ | $0.035\pm 0.002$ | $0.27^{+0.06}_{-0.07}$ | $0.072^{+0.011}_{-0.005}$ | $0.0266\pm 0.0018$ | $0.0270^{+0.0018}_{-0.0019}$ | $0.074^{+0.035}_{-0.006}$
$m_{1}\,(M_{J})$ | $180^{+30}_{-50}$ | $22.4^{+0.6}_{-0.3}$ | $22.5^{+0.7}_{-0.3}$ | $170^{+40}_{-50}$ | $46.6^{+6.6}_{-0.7}$ | $17.38^{+0.19}_{-0.09}$ | $17.65^{+0.08}_{-0.41}$ | $48.6^{+22.7}_{-1.7}$
$m_{2}\,(M_{J})$ | $110^{+20}_{-30}$ | $13.6^{+0.4}_{-0.2}$ | $13.7^{+0.4}_{-0.2}$ | $110^{+20}_{-30}$ | $28.3^{+4.0}_{-0.5}$ | $10.53^{+0.10}_{-0.08}$ | $10.65^{+0.02}_{-0.22}$ | $29.5^{+13.9}_{-1.0}$
$D_{\rm L}\,(\rm{kpc})$ | $6.04^{+0.25}_{-0.50}$ | $2.33^{+0.12}_{-0.11}$ | $2.33^{+0.12}_{-0.11}$ | $6.00^{+0.27}_{-0.47}$ | $3.67^{+0.29}_{-0.13}$ | $1.94\pm 0.10$ | $1.94\pm 0.10$ | $3.75^{+0.76}_{-0.15}$
$\mu_{rel,hel}(N,E)\,({\rm mas\,yr^{-1}})$ | $(7.9,4.4)$ | $(8.79,0.20)$ | $(-8.97,1.44)$ | $(-7.4,5.2)$ | $(-5.97,-5.82)$ | $(8.64,0.05)$ | $(-8.97,1.48)$ | $(5.14,-6.51)$
$\delta\mu_{rel,hel}(N,E)\,({\rm mas\,yr^{-1}})$ | $(\pm 0.4,^{+0.8}_{-0.9})$ | $(^{+0.05}_{-0.04},^{+0.25}_{-0.23})$ | $(\pm 0.02,^{+0.42}_{-0.13})$ | $(^{+0.7}_{-0.4},^{+1.0}_{-0.6})$ | $(^{+0.04}_{-0.45},^{+0.34}_{-0.07})$ | $(\pm 0.05,^{+0.21}_{-0.24})$ | $(^{+0.07}_{-0.01},^{+0.01}_{-0.41})$ | $(^{+1.53}_{-0.22},^{+1.12}_{-0.13})$
$\mu_{{\rm L},hel}(N,E)\,({\rm mas\,yr^{-1}})$ | $(2.2,-3.3)$ | $(3.1,-7.4)$ | $(-14.6,-6.2)$ | $(-13.1,-2.4)$ | $(-11.6,-13.5)$ | $(3.0,-7.6)$ | $(-14.6,-6.2)$ | $(-0.5,-14.1)$
$\delta\mu_{{\rm L},hel}(N,E)\,({\rm mas\,yr^{-1}})$ | $(\pm 0.5,^{+0.9}_{-1.0})$ | $\pm(0.2,0.5)$ | $(\pm 0.2,^{+0.6}_{-0.4})$ | $(^{+0.8}_{-0.4},^{+1.0}_{-0.7})$ | $(^{+0.2}_{-0.5},^{+0.5}_{-0.4})$ | $\pm(0.2,0.5)$ | $(\pm 0.2,^{+0.5}_{-0.6})$ | $(^{+1.5}_{-0.3},^{+1.2}_{-0.4})$
$v_{{\rm L},hel}(l,b)\,({\rm km\,s^{-1}})$ | $(9,113)$ | $(-11,88)$ | $(-174,-21)$ | $(-358,-126)$ | $(-292,102)$ | $(-11,74)$ | $(-145,-18)$ | $(-133,213)$
$\delta v_{{\rm L},hel}(l,b)\,({\rm km\,s^{-1}})$ | $(\pm 10,^{+29}_{-31})$ | $\pm(5,5)$ | $(\pm 9,^{+3}_{-5})$ | $(^{+49}_{-26},\pm 14)$ | $(^{+12}_{-27},^{+6}_{-7})$ | $\pm(4,5)$ | $\pm(8,^{+5}_{-3})$ | $(^{+16}_{-9},^{+21}_{-10})$
$-2\Delta\ln z_{galactic}$ | $1.64$ | $10.41$ | $25.46$ | $2.73$ | $13.69$ | $12.37$ | $31.65$ | $0$
$-2\Delta\ln z$ | $0.01$ | $8.92$ | $21.83$ | $0$ | $16.97$ | $39.77$ | $57.95$ | $11.25$

Note. — These models included both ground-based and space-based data, with
parallax and energy-constrained lens orbital motion included in the model. The
parameter values quoted in this table are those corresponding to the minimum
$\ln\mathcal{L}$ samples from the posterior. The uncertainties recover the
16th and 84th percentiles and are asymmetric because the posteriors are not
entirely Gaussian and parameter values corresponding to the minimum $\chi^{2}$
solution differ from the mode of the same parameter’s samples. We use
$-2\Delta\ln\mathcal{L}$ and $-2\Delta\ln z$ as effective $\chi^{2}$ values.
$-2\Delta\ln\mathcal{L}$ quantifies the light-curve-fit likelihood and
$-2\Delta\ln z$ incorporates both the fit likelihood and the detection
probability, based on our galactic model.

###  4.4 Model Comparison

When we compare the likelihoods using the standard analysis approach (Table 3)
and GP analysis (Table 4), both favour the close solutions. For the close
solutions, the range of $\Delta\chi^{2}<28$ using the standard approach and
$\Delta\chi^{2}_{\rm eff}<3$ using GP.333Here we refer to effective
$\Delta\chi^{2}$ values, which are calculated via
$\Delta\chi^{2}_{eff}=-2\Delta\ln{\mathcal{L}}$. This is in order to provide
an equivalent scale to the regular $\Delta\chi^{2}$ values we use to appraise
solutions in the standard approach. For the standard approach
$\Delta\chi^{2}=-2\Delta\ln{\mathcal{L}}$ because the extra likelihood
components are the same for all solutions. We use these two parameters to
compare spreads between methods, as they are on the same scale, but we do not
directly compare solutions between methods, as these values are not
equivalent. The physical properties $M_{tot}$ and $D_{\rm L}$ of all four
close solutions are in agreement between the standard and GP approaches to
within $1.2\,\sigma$. Therefore, the physical interpretation is the same in
both cases.

This is not true of all the degenerate wide-family solutions. While the large-
parallax solutions remain most disfavoured between approaches, with matching
$M_{tot}$ and $D_{\rm L}$ values (masses at or below the deuterium fusion
limit, 2 kpc away), the small-parallax solutions tell a different story. Using
the standard approach, all wide solutions are disfavoured by a
$\Delta\chi^{2}>37$. However, using the GP approach the small-parallax
solutions have $\Delta\chi^{2}_{\rm eff}$ values of 7 and 15, within the
$\Delta\chi^{2}$ range of close solutions using the standard approach. The
physical interpretation is also different for these two solutions. The
physical properties $M_{tot}$ and $D_{\rm L}$ differ between approaches by
$<4\sigma$.

Our interpretation is that the physical solutions are not equally sensitive to
systematic errors in the Spitzer data. The posteriors of $\pi_{{\rm E},E}$ are
wider using the GP approach than the standard approach, especially the wide
solutions for which the extrapolated trajectories do not cross caustics. It
appears that the parallax measurement (particularly $\pi_{{\rm E},E}$) is
proportionally more affected for smaller parallax solutions, making them more
sensitive to systematic errors, but that the affect this has on the close
solutions is limited by the nearness to a caustic crossing, which has a
dominating effect on the likelihood space. Whether these conclusion are true
in general is an interesting thought for future work.

While inflating error bars may be the correct approach for accommodating noise
in data that is approximately Gaussian, it is appropriate to use a correlated
noise approach where there are obvious systematic trends. The apparent
perturbations in our Spitzer data are not represented by any of our best model
solutions and therefore show that the errors in this data set are clearly
correlated on time scales of a few days. However, the importance of using a
correlated noise approach varies for our different solutions families and we
believe that the importance of such modelling in other Spitzer events would
also be dependent on many event-specific-properties.

Whether or not we consider the expense of a GP approach necessary, in our
case, depends on the $\Delta\chi^{2}_{\rm eff}$ ranges we are prepared to
accept. If we accept solutions at the $\Delta\chi^{2}_{\rm eff}\lesssim 90$
level, all eight degenerate solutions are valid, whether or not a GP is
included. However, at the $\Delta\chi^{2}_{\rm eff}\lesssim 50$ level, we
would reject the w -/+ and w +/- solutions using the standard approach. Using
the GP approach, we would accept all of these solutions, with w -/- and w +/+
converging into significantly different physical lens compositions.

##  5 Physical Parameters

###  5.1 Angular Einstein Radius

There exist empirical relations for determining the angular size of a star
from its intrinsic color and magnitude. According to Kervella & Fouqué (2008),
the most appropriate of these relations for non-M-type giants are those found
in Nordgren et al. (2002) and van Belle (1999).444Depending on the selection
of surface brightness relation, the implied $\theta_{\rm E}$ differs by around
$8\%$, in this case. We use the Nordgren et al. (2002) surface brightness
relation, specifically for non-variable giant stars (their Equation 12),

$\log_{10}(2\theta_{*})=0.5522+0.246\left(V-K\right)_{0}-V_{0}/5.$ (7)

Using the empirical color-color relations of Bessell & Brett (1988) for giant
stars we find the $(V-K)_{0}$ equivalent of the intrinsic source color,
$(V-I)_{0}$, that was calculated from the CMDs, $(V-K)_{{\rm S},O}=2.57\pm
0.09$. The solutions for the models including higher-order effects have
effectively identical $\theta_{*}=7.6\pm 0.5\,\mu{\rm as}$.

$\theta_{\rm E}$ was calculated using the fitted $\rho$ value for each
solution, where $\theta_{\rm E}=\theta_{*}/\rho$. The light-curve data
provided good coverage of the caustic crossing and therefore $\rho$ was well
constrained, and almost identical, in our models. The calculated value of
$\theta_{\rm E}$ for all solutions is

$\theta_{\rm E}=0.29\pm 0.02\,{\rm mas}.$

Knowing $\theta_{\rm E}$ gives an angular scale to the geometric models.

###  5.2 Mass, Distance and Separation

The intrinsic I-band magnitude of the source star was previously calculated by
comparing its fitted $I$-band magnitude to the mean red-clump magnitude on a
CMD. By assuming the intrinsic red-clump magnitude and that the source star is
at the distance of the average red-clump star in the CMD field, we find
$D_{\rm S}=7.85\pm 0.06\,\rm{kpc}$.

With values for $D_{\rm S}$, $\theta_{\rm E}$ and $\pi_{\rm E}$, the
degeneracy in Equation 1 is broken, and the mass and distances can be
calculated for each solution. Given the fitted parameters $\pi_{{\rm E},E}$
and $\pi_{{\rm E},N}$, $\theta_{E}$, and $D_{\rm S}$, the distance to the lens
was found, using

$\frac{1}{D_{\rm L}[\rm{kpc}]}=\pi_{\rm rel}[{\rm mas}]+\frac{1}{D_{{\rm
S}}[{\rm kpc}]},$

where $\pi_{\rm rel}=\theta_{\rm E}\pi_{\rm E}$. Knowing the distance to the
lens system and $\theta_{\rm E}$ in angular units, the lens geometry can be
calculated in absolute terms. The masses for each of the lens components,
their projected separations, and the distances to the lens system are given in
Tables 3 and 4. All large-parallax solutions, and both small-parallax wide
solutions, are consistent with BD binary lenses of varying masses. However,
the small-parallax close solutions are consistent with M-dwarf binaries, where
the mass of the smaller of the binary objects ($m_{2}=110^{+20}_{-30}\,M_{J}$)
is very near the BD upper cut-off ($\sim 70-95\,M_{J}$) and therefore may or
may not be large enough for hydrogen fusion, depending mostly on its chemical
composition.

###  5.3 Proper Motion and Velocity

The relative lens-source heliocentric proper motion was determined via

${\bm{\mu}}_{rel,hel}=\frac{\theta_{\rm E}}{t_{\rm E}}\hat{\bm{\pi}}_{\rm
E}+\frac{\pi_{\rm rel}}{{\rm au}}{\bm{v}}_{\oplus,\perp},$ (8)

for each solution, where ${\bm{v}}_{\oplus,\perp}$ is the projected velocity
of Earth at $t_{0}$, parallel to the lens plane,
${\bm{v}}_{\oplus,\perp}(N,E)=(-0.104,29.296)\,{\rm km\,s}^{-1}$, and
$\hat{\bm{\pi}}_{\rm E}={\bm{\pi}_{\rm E}}/\left|\pi_{\rm E}\right|$ is a unit
vector in the microlensing parallax direction.

The $\mu_{rel,hel}$ values for each solution are shown in Tables 3 and 4. All
of the degenerate solutions have high relative proper motions,
$\mu_{rel}>8\,{\rm mas}\,{\rm yr}^{-1}$. A proper motion of $\mu_{rel}\lesssim
10\,{\rm mas}\,{\rm yr}^{-1}$ does not innately give any information on the
location of the lens i.e. disk vs. bulge. However, if one adds knowledge of
the source proper motion, $\mu_{\rm S}$, then $\mu_{rel}=8\,{\rm mas}\,{\rm
yr}^{-1}$ may give such information. For example, if $\mu_{\rm S}$ were at the
center of the bulge distribution then a bulge lens would be very unlikely
because a proper motion of $8\,{\rm mas}\,{\rm yr}^{-1}$ from the centroid is
extreme compared with the bulge dispersion of $\sigma(l,b)=(3.0,2.5)\,{\rm
mas}\,{\rm yr}^{-1}$. Therefore this hypothetical case would favor a disk
lens.

Figure 10: Proper motions of the lens solutions and source star. For context,
we also include contour representations of the disk (blue) and bulge (red)
distributions. The bulge contours are from histograms of the red-clump stars
from Gaia EDR3, selecting stars within a $0.2^{o}$ radius cone centred on the
lens. The distribution of red-clump stars is the results of a Gaussian fit to
the red clump on the field’s CMD. The innermost thicker line of the red-clump
distribution contains approximately 68% of the population samples. The
outermost thicker line contains approximately 95% of the population samples.
The blue contours depict the theoretical distribution of the disk stars used
in our galactic model. The solid ellipses correspond to the 1 and 2 $\sigma$
proper-motion dispersions of disk stars at $D=6$ kpc. The dotted ellipses show
the same for disk stars at $D=2.3$ kpc.

The source star for this event was observed by Gaia (EDR3
4063557344313009920), and hence its heliocentric proper motion is precisely
measured as $\mu_{{\rm S},hel}(N,E)=(-5.7,-7.7)\pm(0.2,0.3)\,{\rm
mas\,yr^{-1,\,}}$555Here we have doubled the published errors, as recommended
by Rybizki et al. (2021). relative to quasars in the distant universe. The
source is $\sim 1\sigma$ due west of the centroid (see Figure 10). This means
that a bulge lens is more easily accommodated, provided that direction of
$\mu_{rel}$ is roughly east. Similarly, the $\mu_{rel}$ direction most
consistent with a disk lens is northeast, although this direction is also very
plausible for a bulge lens.

The heliocentric lens proper motion is calculated via

$\mu_{{\rm L},hel}=\mu_{{\rm S},hel}+\mu_{rel,hel}.$ (9)

The unexpected outcome of our $\mu_{\rm L}$ calculations is that none of the
eight degenerate solutions align well with the disk or bulge dispersions, as
shown in Figure 10. However, this demonstrates a misleading aspect of proper
motion comparisons in that closer objects have higher proper- motions given
the same tangential velocity.

The lens proper motion relates to the heliocentric lens velocity via

$v_{{\rm L},hel}=4.74\times D_{\rm L}\mu_{{\rm L},hel},$ (10)

where distance is expressed in kiloparsecs, $\mu_{{\rm L},hel}$ is in
miliarcseconds per year, and 4.74 is a conversion factor so that $v_{{\rm
L},hel}$ is in kilometers per second. These physical parameters for each
solution can also be found in Tables 3 and 4.

From Figure 10 we can see that the source is a fairly kinematically typical
bulge star, lying on the $1\sigma$ contour of the Gaia field bulge dispersion.

Comparisons of the lens velocities, from each of the eight degenerate
solutions, with disk and bulge dispersions from Gaia EDR3 are shown in Figure
11. These empirical dispersions are used for demonstrative purposes only. All
eight lens solutions have unusual velocities when compared to typical disk
stars, with the w +/+ and both +/- lens solutions rotating about the galactic
center more slowly than typical disk stars, the w -/- and c +/+
counterrotating, and the -/+ and c -/- solutions seemingly moving through the
disk, with large $b$ velocities. The solutions are all less exceptional when
compared with bulge kinematics, although only the small-parallax solutions
have distances that allow for the lens to be a bulge member according to
current galactic density models (e.g., Han & Gould 2003). The velocities of
the w -/-, c +/+, w +/-, and c +/- solutions also appear consistent with the
retrograde microlensing group.

Figure 11: Heliocentric velocity of the most likely lens star solutions. As
previously, we also include contour representations of the bulge (red) and
disk (blue) distributions. The bulge contours are from histograms of the red-
clump stars from Gaia EDR3. The red-clump distribution was selected as in
Figure 10. Due to the unreliable nature of Gaia distances obtained from
parallax, especially at large distances, $D_{\rm RC}=D_{\rm S}=7.85\,\rm{kpc}$
was used to estimate the red-clump velocities from the Gaia proper motion
measurements. The outermost thicker line contains approximately 95% of the
population samples. The blue contours depict the theoretical distribution of
disk stars, used in our galactic model. The solid ellipses correspond to the 1
and 2 $\sigma$ velocity dispersions. Left: The small-parallax solutions and
both distributions. Right: Only the disk distribution, with the large-parallax
solutions, as these solutions are at distances not compatible with a lens
belonging to the bulge.

##  6 Solution Probabilities

The somewhat uncommon physical parameters compel us to look at our solution
probabilities more cautiously and holistically than a purely likelihood-based
comparison. One problem with the likelihood calculation is that, formally, it
relies on the assumption that our data are Gaussian distributed, with accurate
uncertainties. Practically, this is never true for microlensing photometry.
However, for this analysis, we apply Bayes theorem as though they were
Gaussian.

The probability of a system having the solution-specific proper motion or
velocity, mass, and distance is also an important factor. We therefore
calculate the probability factor $\ln{z}$ that determines the relative
detection probability of each solution given a galactic model, with a bias to
incorporate their relative light-curve-fit likelihoods.

We compute the galactic probability (Equation 15 of Gould (2020)) using a
modified version of the Galactic Bayesian code described in Herrera-Martín et
al. (2020). This model is based on the stellar Milky Way density model of Han
& Gould (2003) and mass functions of Chabrier (2003) using the prescription of
Dominik (2006).

There is a common wisdom in microlensing analysis that small-parallax events
are more probable than their large-parallax degenerate counterparts. This is
known as the Rich argument, as detailed in Calchi Novati et al. (2015a). For
single-lens events and binary-lens events for which the lens axis and source
trajectory are approximately parallel (as in this case), if the true parallax
solution is the smaller parallax solution it will always generate a large-
parallax degenerate counterpart. The reverse, however, is not always true. The
ratio of these probabilities (Rich factor) is implicitly accounted for in our
galactic models (Gould, 2020).

Our calculated $-2\Delta\ln{z}$ values for each degenerate solution, and both
error approaches, are displayed in Tables 3 and 4. Kass & Raftery (1995)
interpret $-2\Delta\ln{z}<(2.3,4.6,9.2)$ as (“substantial”,“strong”,
“decisive”) evidence favoring one solution over another. By their metric, we
would decisively consider the c -/+ the best solution given a standard Spitzer
error approach. Using this approach, the probability of the c -/+ solution,
compared with the next most probable solution (c -/-), is
$-2\Delta\ln{z}$=13.48. However, using the GP error approach with the Kass &
Raftery (1995) interpretation, the evidence supporting the equally favoured,
small-parallax, close solutions (c -/-, c +/+; $-2\Delta\ln{z}$=0.01) over the
c -/+ is “strong” ($-2\Delta\ln{z}$=8.9).

At the low galactic latitude of our event, and especially given the calculated
distances to the lens of the large-parallax solutions, one would expect lens
bodies to be members of the galactic disk. However, at a distance of $\sim
6\rm{kpc}$ (as in the c -/- and c +/+ cases), it is possible that the lens is
a member of the bulge population. Our galactic modeling of c +/+ showed that
it is on the order of 100 times more likely to be a member of the bulge than
the disk, whereas for c -/+ this was more like 1400 times more likely to be a
member of the disk than the bulge. Currently, our galactic model most highly
disfavours the counter-rotating BD solutions, with disk-like distances (c +/-
and w +/- with $-2\Delta\ln z$, without a light-cure likelihood bias, of 24.45
and 34.12, respectively) .

It is worth noting that $\ln z$ is based on a galactic model and therefore
implicitly favors solutions matching our expectation of kinematic, mass, and
density dispersions. Even the kinematic dispersions displayed in Figures 10
and 11 are informed by mostly bright stars and may not be truly representative
of the dispersions of much dimmer objects, of which we know very little. Some
healthy skepticism needs to exist around the model’s completeness, especially
considering the high proportion of microlensing BDs with unusual proper
motions.

To determine how representative these retrograde detections are of the BD
population as a whole, we must must first have a good understanding of the
innate selection biases in microlensing events, for or against these extreme
proper motions. However, if we were to downweight the light-curve likelihood
based on the knowledge that our errors are not Gaussian, we will generally
favour the low parallax solutions.

##  7 Discussion

In our analysis of event OGLE-2017-BLG-1038, we fit a binary lens model
including higher order effects: lens orbital motion and parallax. We include
space-based data from Spitzer and model systematic errors in these data. We
have a resulting eightfold solution degeneracy in this event. These solutions
have total lens masses ranging from $0.027-0.27M_{\odot}$. We also included in
our probability comparison a galactic probability for each lens configuration.
After these processes we find that our most probable solutions are the c +/+
and c -/-, both with masses of $m_{1}\simeq 170\,M_{J}$ and
$m_{2}=110^{+20}_{-30}M_{J}$ ($0.16$ and $0.11\,M_{\odot}$), separated by
$1.7\,{\rm au}$, at a distance of $6.0\,{\rm kpc}$. The companion masses for
these solutions are near the upper limit for BDs (the hydrogen burning limit).
The lens systems for the c +/+ and c -/- solutions have tangential velocities
of $v_{{\rm L},hel}(l,b)=(-358,-126)\,{\rm km}\,{\rm s}^{-1}$ and $v_{{\rm
L},hel}(l,b)=(9,113)\,{\rm km}\,{\rm s}^{-1}$, respectively.

The c -/- solution has a minutely higher galactic probability than c +/+ with
$-2\Delta\ln{\mathcal{L}}=1.09$. They are equally likely when considered in
the context of both the light-curve fit and the galactic model.

Favouring these solutions over the large-parallax, close-family solutions
($m_{1}\simeq 22.5$ and $m_{2}\simeq 13.7$; $D_{\rm L}=2.33$; $v_{{\rm
L},hel}(l,b)=(-11,88)\,{\rm km}\,{\rm s}^{-1}$ and $v_{{\rm
L},hel}(l,b)=(-174,-21)\,{\rm km}\,{\rm s}^{-1}$ for c -/+ and c +/-,
respectively) relies on our being confident in the galactic model for very-
low-mass objects. Evidence from other microlensing events suggest that we do
not understand the kinematic structure of BDs at distances of $D<4\,{\rm
kpc}$. To date, three BD systems have been discovered using microlensing that
appear to be counterrotating with respect to the disk (Chung et al., 2019;
Shvartzvald et al., 2019, 2017). These microlensing members lie very much in
the plane of the disk and explanations for their characteristics, which we
consider here, are that they are members of the disk with extreme motions;
they are halo members with a coincidental disk alignment; they are members of
a counterrotating population of very-low-mass objects (as suggested by
Shvartzvald et al. 2019); or, they are evidence of an oversimplified galactic
model. The physical parameters of the lens of this event raise the question as
to whether or not OGLE-2017-BLG-1038 is another member of this group.

One explanation for extreme kinematics for a low-mass disk lens is that the
disk may have a larger velocity dispersion for lower mass objects. If we
assume that the lens was born in a cluster, it may have received a kick from
an interaction with a star, and a binary will have a higher scattering cross
section for such an interaction. Cluster dissolution has been extensively
modeled (e.g., Hurley et al. 2005; Wang et al. 2016). Most stars are believed
to come from open clusters, however expulsion velocities from open clusters
are small compared to Galactic rotation. For example, the open cluster
simulations of Jørgensen & Church (2020) show most stars escaping with
velocities $<10\,{\rm km\,s^{-1}}$, relative to their parent cluster. It is
therefore very unlikely that such an escapee would be travelling $\sim
100\,{\rm km\,s^{-1}}$, or more, opposed to the disk. For globular clusters,
higher mass objects preferentially wind up in tight binaries, whose members
can be expelled at very high velocities (Hut et al., 1992a, b), but such
expulsions are likely to account for a tiny fraction of all stars. This
appears an unlikely origin for these counterrotating low-mass objects.

Another aspect of the galactic model that may be misunderstood is the bulge
density model. We propose that a mass dependent spatial cutoff could explain
the observed abundance of counterrotating BDs. If we consider that the bulge
extends further for lower mass objects, then at $D<4\,{\rm kpc}$ the mass
independent model would significantly underrepresent lower mass objects
belonging to the bulge population and therefore having extreme (when compared
to neighbouring disk stars) kinematics. Density models are fit to
observational data and therefore are specifically fit to objects much larger
than our inferred lens and those of the aforementioned retrograde BDs.

Another explanation may be that the lens is a halo star. Halo stars are known
to have a much larger velocity dispersion, and their mean galactic rotation is
much smaller than the disk (Du et al., 2018; Posti et al., 2018). While this
large velocity dispersion could explain the kinematics of the other retrograde
BD stars, it is a leap to make that assumption here, when it is not unlikely
that the lens belongs to the bulge.

Are these retrograde BD detections the first members of a new class of object?
At this stage, the characterization of these events as an independent
population is speculative. Their existence as a discrete population affects
the way we view the galactic probability of this solution, because such a
population is not represented in the galactic model. Even if a misunderstood
selection effect or aspect of the galactic model is responsible for their
overabundance in detection, such an effect is not included in our current
probability calculations. More needs to be known about this retrograde group
before the significance of this solution can be truly understood.

The analysis of more low-mass lens events will provide new insights into the
very-low-mass end of the mass function and its density and kinematics. There
is little observational evidence to constrain any of these distributions at
present. It is always possible that low-mass BDs are far more numerous than
currently known and are currently represented by our galactic model.

Whatever the case, for low-mass lenses, we believe that selection of a
solution based on typical disk kinematic arguments is unlikely to be valid.
The same reasoning leads us to believe that we cannot categorically claim this
lens as a member of either a bulge, halo, or retrograde BD population. A more
complex consideration of selection biases and possible population dynamics
(beyond the scope of this paper) would be required.

A more empirical means of confirming the small parallax configuration would be
to observe the lens photometrically. The hydrogen burning host and likely
hydrogen-burning companion, corresponding to the small-parallax, close-family
solutions, are bright enough to be visible at their implied lens distances
($D_{\rm L}=6$ kpc). Given the relative proper motions of these solutions
($\mu_{rel,hel}=9.0\,{\rm mas\,yr^{-1}}$), we could expect the separation of
source and lens to be sufficient for them to be resolved with the advent of
infrared adaptive optics imaging from the coming generation of 40 m class
telescopes. This is not true of the solutions near the planet-BD boundary,
which are too dim to be resolved, no matter the angular separation between
source and lens.

We expect first light for Multi-AO Imaging Camera for Deep Observations
(MICADO) on the $39\,{\rm m}$ European Extremely Large Telescope (EELT) to be
2030. Kim et al. (2021) have argued, by scaling the work of Bowler et al.
(2015) with the Keck coronograph, that an EELT coronograph could achieve
$\Delta K=11$ contrast at 77 mas. By 2030 the angular separation of the lens
and source will be $\sim 115$ mas. Using the mass-luminosity function of Just
et al. (2015) and the previously calculated source-star K magnitude, we
estimate $\Delta K=9.2$ between the source star and the primary lens body for
the M-dwarf solutions (c -/- and c +/+). Therefore the composition of this
lens, be it BD or M-dwarf, can be verified with astrometric follow-up at the
expected first light of MICADO on EELT.

##  8 Summary

In this paper we report our analysis of microlensing event OGLE-2017-BLG-1038,
with data from KMTNet, OGLE, and Spitzer. Ground-based data show the event is
due to a giant source passing across a fold and cusp of a resonant caustic,
due to a rotating binary lens. The analysis of the combined Spitzer, KMT, and
OGLE light-curve data resulted in eight degenerate satellite-parallax
solutions. With a GP model fit to the Spitzer data to account for systematic
effects, the best solutions are the four belonging to the close family. Of
these solutions the small- parallax solutions both have masses of $M_{1}\simeq
170^{+40}_{-50}M_{J}$ (an M-dwarf) and $m_{2}=110^{+20}_{-30}\,M_{J}$ (at the
BD/M-dwarf cutoff). The large-parallax solutions are both comprised of a BD
binary with $m_{1}=22\pm 2\,M_{J}$ and $m_{2}=14\pm 1\,M_{J}$. Inclusion of a
detection probability based on a galactic model favors the small-parallax
solutions. However, this approach to appraising solutions may be biased by an
incomplete description of the distribution of very-low-mass objects in the
galaxy and should not rule out solutions with similar light-curve-fit
likelihoods. Late-time imaging could be used to reject these low-mass BD
solutions, since an M dwarf should be visible given sufficient lens-source
separation, but a low-mass BD binary will not.

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 Cheongho Han was supported by the grants of National Research
Foundation of Korea (2020R1A4A2002885 and 2019R1A2C2085965). This work has
made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. Software: matplotlib
(Hunter, 2007), numpy (Harris et al., 2020), scipy (Virtanen et al., 2020)

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insulator states at $\nu=\pm 3$. They may provide an explaination to the
absence of Chern insulator at zero magnetic field. We leave HF calculations of
such states for future studies.


$1-p<1/d$. Suppose that $r_{i}(c,s)\triangleq f_{i}(s)-cg_{i}(s)$.

We find that:

(162)
$p^{-(s-1)}\Big{(}f_{i}(s+1)-f_{i}(s)\Big{)}=\Big{(}1-d(1-p)\Big{)}s+l_{i}$

where $l_{i}$ is not related to $c,s$ and is described in Table 2. Also,

(163) $p^{-(s-1)}\Big{(}g_{i}(s+1)-g_{i}(s)\Big{)}=o_{i}.$

where $o_{i}$ is not related to $c,s$ and are described in Table 2 as well.
Thus, (162) and (163) give a nice property:

(164) $\begin{split}&p^{-(s-1)}(r_{i}(c,s+1)-r_{i}(c,s))\\\
&=\Big{(}1-d(1-p)\Big{)}s+l_{i}-co_{i}.\end{split}$

Note that (164) holds for $i\in\\{1,2,3,4\\}$. Since $1-(1-p)d>0$, thus (39)
(for $i=1,3,4$) and (40) (for $i=2$) is the minimum point of $r_{i}(c,s)$.
Thus we complete the proof.

##  Appendix T proof of Corollary 43

Corollary 43(a) comes directly from Theorem 4 (a).

For Corollary 43 (b), there are only two different kinds of Markov chains:
$s\in\\{1,2...d\\}$ or $s\geq d+1$. If $s\in\\{1,2...d\\}$, then
$\mu(1,1,0)=1$ and $\mu(d,1,0)=2$, this implies the Markov chain is the same
with always choosing Channel $2$, thus $3/2d-1/2$. If $s\geq d+1$,
$u(1,1,0)=u(d,1,0)=1$, so by Section 7.3, average cost with threshold $s$ is
$f_{1}(s)/g_{1}(s)$. So we get a simplified and similar form of (52):

(165)
$\bar{\Delta}_{\text{opt}}=\Big{[}\min_{d<s}\frac{f_{1}(s)}{g_{1}(s)}\Big{]}_{\frac{3}{2}d-\frac{1}{2}}.$

From (55), the corollary 43 is solved.