# OGLE-2019-BLG-0960Lb: The Smallest Microlensing Planet

Jennifer C. Yee  Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
St.,Cambridge, MA 02138, USA  Weicheng Zang Department of Astronomy, Tsinghua
University, Beijing 100084, China  Andrzej Udalski Astronomical Observatory,
University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  Yoon-Hyun
Ryu  Korea Astronomy and Space Science Institute, Daejon 34055, Republic of
Korea  Jonathan Green  Kumeu Observatory, Kumeu, New Zealand  Steve Hennerley
Kumeu Observatory, Kumeu, New Zealand  Andrew Marmont  Kumeu Observatory,
Kumeu, New Zealand  Takahiro Sumi  Department of Earth and Space Science,
Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Shude Mao  Department of Astronomy, Tsinghua University, Beijing 100084, China
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China  Mariusz Gromadzki 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  Jan Skowron 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  Michał K. Szymański 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  Paweł Pietrukowicz 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  Krzysztof Ulaczyk Department of Physics, University
of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK  Krzysztof A. Rybicki
Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478
Warszawa, Poland  Patryk Iwanek Astronomical Observatory, University of
Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  Marcin Wrona Astronomical
Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
Michael D. Albrow  University of Canterbury, Department of Physics and
Astronomy, Private Bag 4800, Christchurch 8020, New Zealand  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  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  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  Hyoun-Woo Kim  Korea Astronomy and
Space Science Institute, Daejon 34055, Republic of Korea  In-Gu Shin  Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea  Yossi
Shvartzvald  Department of Particle Physics and Astrophysics, Weizmann
Institute of Science, Rehovot 76100, Israel  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
Dong-Jin 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  Etienne Bachelet  Las Cumbres
Observatory, 6740 Cortona Drive, suite 102, Goleta, CA 93117, USA  Grant
Christie  Auckland Observatory, Auckland, New Zealand  Markus P.G. Hundertmark
Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität
Heidelberg (ZAH), 69120 Heidelberg, Germany  Dan Maoz  School of Physics and
Astronomy, Tel-Aviv University, Tel-Aviv 6997801, Israel  Jennie McCormick
Farm Cove Observatory, Centre for Backyard Astrophysics, Pakuranga, Auckland,
New Zealand  Tim Natusch  Auckland Observatory, Auckland, New Zealand
Institute for Radio Astronomy and Space Research (IRASR), AUT University,
Auckland, New Zealand  Matthew T. Penny Department of Astronomy, The Ohio
State University, 140 W. 18th Avenue, Columbus, OH 43210, USA  Rachel A.
Street  Las Cumbres Observatory, 6740 Cortona Drive, suite 102, Goleta, CA
93117, USA  Yiannis Tsapras  Astronomisches Rechen-Institut, Zentrum für
Astronomie der Universität Heidelberg (ZAH), 69120 Heidelberg, Germany
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  B. Scott Gaudi
Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH
43210, USA  Calen B. Henderson  IPAC, Mail Code 100-22, Caltech, 1200 E.
California Blvd., Pasadena, CA 91125, USA  Samson Johnson  Department of
Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA
Wei Zhu  Canadian Institute for Theoretical Astrophysics, University of
Toronto, 60 St George Street, Toronto, ON M5S 3H8, Canada  Ian A. Bond
Institute of Natural and Mathematical Sciences, Massey University, Auckland
0745, New Zealand  Fumio Abe  Institute for Space-Earth Environmental
Research, Nagoya University, Nagoya 464-8601, Japan  Richard Barry  Code 667,
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA  David P. Bennett
Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
Aparna Bhattacharya  Code 667, NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA  Department of Astronomy, University of Maryland, College Park, MD
20742, USA  Martin Donachie  Department of Physics, University of Auckland,
Private Bag 92019, Auckland, New Zealand  Hirosane Fujii  Department of Earth
and Space Science, Graduate School of Science, Osaka University, Toyonaka,
Osaka 560-0043, Japan  Akihiko Fukui  Department of Earth and Planetary
Science, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-0033, Japan  Instituto de Astrofísica de Canarias, Vía
Láctea s/n, E-38205 La Laguna, Tenerife, Spain  Yuki Hirao  Department of
Earth and Space Science, Graduate School of Science, Osaka University,
Toyonaka, Osaka 560-0043, Japan  Stela Ishitani Silva  Department of Physics,
The Catholic University of America, Washington, DC 20064, USA  Code 667, NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA  Yoshitaka Itow
Institute for Space-Earth Environmental Research, Nagoya University, Nagoya
464-8601, Japan  Rintaro Kirikawa  Department of Earth and Space Science,
Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Iona Kondo  Department of Earth and Space Science, Graduate School of Science,
Osaka University, Toyonaka, Osaka 560-0043, Japan  Naoki Koshimoto  Department
of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  National Astronomical Observatory of
Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Man Cheung Alex Li
Department of Physics, University of Auckland, Private Bag 92019, Auckland,
New Zealand  Yutaka Matsubara  Institute for Space-Earth Environmental
Research, Nagoya University, Nagoya 464-8601, Japan  Yasushi Muraki  Institute
for Space-Earth Environmental Research, Nagoya University, Nagoya 464-8601,
Japan  Shota Miyazaki  Department of Earth and Space Science, Graduate School
of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Greg Olmschenk
Universities Space Research Association, Columbia, MD 21046, USA  Clément Ranc
Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA  Nicholas
J. Rattenbury  Department of Physics, University of Auckland, Private Bag
92019, Auckland, New Zealand  Yuki Satoh  Department of Earth and Space
Science, Graduate School of Science, Osaka University, Toyonaka, Osaka
560-0043, Japan  Hikaru Shoji  Department of Earth and Space Science, Graduate
School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Daisuke
Suzuki  Institute of Space and Astronautical Science, Japan Aerospace
Exploration Agency, 3-1-1 Yoshinodai, Chuo, Sagamihara, Kanagawa, 252-5210,
Japan  Yuzuru Tanaka  Department of Earth and Space Science, Graduate School
of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Paul J.
Tristram  University of Canterbury Mt. John Observatory, P.O. Box 56, Lake
Tekapo 8770, New Zealand  Tsubasa Yamawaki  Department of Earth and Space
Science, Graduate School of Science, Osaka University, Toyonaka, Osaka
560-0043, Japan  Atsunori Yonehara  Department of Physics, Faculty of Science,
Kyoto Sangyo University, 603-8555 Kyoto, Japan

###### Abstract

We report the analysis of OGLE-2019-BLG-0960, which contains the smallest
mass-ratio microlensing planet found to date ($q=$1.2–1.6$\times 10^{-5}$ at
$1\sigma$). Although there is substantial uncertainty in the satellite
parallax measured by Spitzer, the measurement of the annual parallax effect
combined with the finite source effect allows us to determine the mass of the
host star ($M_{\rm L}=$ 0.3–0.6$\ M_{\odot}$), the mass of its planet
($m_{p}=$1.4–3.1$\ M_{\oplus}$), the projected separation between the host and
planet ($a_{\perp}=$ 1.2–2.3 au), and the distance to the lens system ($D_{\rm
L}=$ 0.6–1.2 kpc). The lens is plausibly the blend, which could be checked
with adaptive optics observations. As the smallest planet clearly below the
break in the mass-ratio function (Suzuki et al., 2016; Jung et al., 2019), it
demonstrates that current experiments are powerful enough to robustly measure
the slope of the mass-ratio function below that break. We find that the cross-
section for detecting small planets is maximized for planets with separations
just outside of the boundary for resonant caustics and that sensitivity to
such planets can be maximized by intensively monitoring events whenever they
are magnified by a factor $A>5$. Finally, an empirical investigation
demonstrates that most planets showing a degeneracy between $(s>1)$ and
($s<1$) solutions are not in the regime ($|\log s|\gg 0$) for which the
“close”/“wide” degeneracy was derived. This investigation suggests a link
between the “close”/“wide” and “inner/outer” degeneracies and also that the
symmetry in the lens equation goes much deeper than symmetries uncovered for
the limiting cases.

gravitational lensing: micro–planetary systems

##  1 Introduction

Statistical studies of microlensing planets have shown that while the mass-
ratio function for giant planets increases with decreasing mass ratio, $q$,
there appears to be a break in the mass-ratio function at small mass ratios.
Suzuki et al. (2016) presented the first study to fit a broken power-law to
the microlensing planet distribution. They analyzed a sample of 1474 events
with 23 planets from the Microlensing Observations in Astrophysics (MOA)
survey. Combining their sample with planets from other work (Gould et al.,
2010; Cassan et al., 2012), they report a power-law slope for planets with
$q<q_{\rm br}$ of $p=0.6$ in contrast to $p=-0.93$ for larger planets (for
$dN/d\log q\propto q^{p}$) for a fiducial break at $q_{\rm br}=1.7\times
10^{-4}$. Independently, Udalski et al. (2018) later used a $V/V_{\rm max}$
analysis and a complementary sample to confirm this turnover. Jung et al.
(2019) then suggested a much smaller $q_{\rm br}=0.55\times 10^{-4}$ and a
change in $p$ above and below the break of $p(q<q_{\rm br})-p(q>q_{\rm
br})>3.3$. While this $q_{\rm br}$ is substantially smaller than that of
Suzuki et al. (2016), it is consistent within the uncertainty contours of
Suzuki et al. (2016) (see their Figure 15). These studies strongly suggest
that Neptune/Sun mass-ratio planets are the most common planets at separations
of a few au around microlensing host stars.

However, the power-law slope of the mass-ratio distribution below $q_{\rm br}$
and even the exact location of $q_{\rm br}$ are highly uncertain due to a lack
of planet detections below this break. In Suzuki et al. (2016), the smallest
planet had $q=0.58\times 10^{-4}$, while in Jung et al. (2019), the smallest
planet had $q=0.46\times 10^{-4}$. The analysis by Jung et al. (2019) is
consistent with a very sharp break at $q_{\rm br}$, i.e., it suggests the
possibility that smaller planets might not exist. Hence, depending on the
strength of the break, it could be quite difficult to find such planets and so
to measure their distribution.

Since that time, three planets have been discovered with mass ratios
$q<0.55\times 10^{-4}$, largely due to the Korea Microlensing Telescope
Network (KMTNet; Kim et al., 2016). With its $4\ {\rm deg}^{2}$ field of view
and telescopes at three sites (CTIO, SAAO, SSO; i.e. KMTC, KMTS, KMTA), KMTNet
can achieve near-continuous observations of $92\ {\rm deg}^{2}$ of the bulge
at a high cadence. (92, 84, 40, 12 deg2) are covered at cadences of
$\Gamma\geq$ (0.2, 0.4, 1, 4) obs hr-1. This has led to the discovery of
KMT-2018-BLG-0029Lb (Gould et al., 2020) and KMT-2019-BLG-0842Lb (Jung et al.,
2020) with mass ratios $q=0.18\times 10^{-4}$ and $q=0.41\times 10^{-4}$,
respectively. A third planet, in OGLE-2019-BLG-0960, is reported in this work
and was discovered through a combination of KMTNet observations and followup
data.

While the KMTNet data were crucial to the discovery of the planet in
OGLE-2019-BLG-0960, the initial discovery of the event by the Optical
Gravitational Lens Experiment (OGLE) was critical for identifying this event
as a potential Spitzer target. The early OGLE alert triggered reduction of
KMTNet real-time data almost three weeks before the event was discovered by
KMTNet. As a result, OGLE-2019-BLG-0960 was under consideration as a potential
Spitzer target prior to its discovery by KMTNet. Later, the event was
identified as high-magnification, which triggered additional followup
observations.

A detailed timeline of the observations is given in Section 2. The analysis of
the light curve including the parallax is described in Sections 3 and 4. We
derive the properties of the source and the physical parameters of the lens
including the planet in Section 5. Then, in Section 6, we consider whether or
not the planet was detectable in survey data alone (Section 6.1), and we
consider how the sample of small planets can be increased to enable a precise
measurement of $q_{\rm br}$ and $p$ for planets with $q<q_{\rm br}$ (Section
6.2). We also explore events with both $(s>1)$ and $(s<1)$ solutions and the
relationship to the “close”/“wide” degeneracy of Griest & Safizadeh (1998)
(Section 6.3). We conclude in Section 7.

##  2 Observations

###  2.1 Ground-Based Survey Observations

OGLE-2019-BLG-0960 was alerted by the OGLE-IV Early Warning System (Udalski,
2003; Udalski et al., 2015) on 23 June 2019 at UT 21:04
(HJD${}^{\prime}=$HJD$-2,450,000=8658.38$). Its coordinates were ${\rm R.A.}=$
18:16:03.43, ${\rm Decl.}=$ -25:46:28.8 corresponding to
$(\ell,b)=(6.10,-4.30)$. It is in OGLE-IV field BLG522, which was observed in
the $I$ band with a cadence of $\Gamma\sim 2\,{\rm night}^{-1}$. This event
was independently discovered as KMT-2019-BLG-1591 by KMTNet (Kim et al., 2018)
on 11 Jul 2019 at UT 03:05 (HJD${}^{\prime}=8675.63$) and as MOA-2019-BLG-324
by Microlensing Observations in Astrophysics (MOA; Bond et al., 2001) on 17
July 2019 at UT 16:07 (HJD${}^{\prime}=8682.17$). This field is observed by
KMTNet at a cadence of $\Gamma\sim 0.4\,{\rm hr}^{-1}$. Most of the KMTNet
observations were taken in the $I$ band with about 10 in the $V$ band for the
purpose of measuring colors. MOA observed the field containing
OGLE-2019-BLG-0960 using a custom $R$ filter (which covers standard $R$ and
$I$ bands) at a nominal cadence of $\Gamma\sim 0.6\ {\rm hr}^{-1}$, but see
below (Section 2.3).

###  2.2 Spitzer Observations

OGLE-2019-BLG-0960 was selected for Spitzer observations and announced as a
Spitzer target on HJD${}^{\prime}=8676.23$ with the condition that it must
reach $I<17.45$ prior to HJD${}^{\prime}=8679.5$, i.e., the decision point for
the next Spitzer target upload. Thus, it was initially selected as a
“subjective, conditional” target, and observations began at
HJD${}^{\prime}=8685.21$. The following week HJD${}^{\prime}\sim 8687$, the
event met the “objective” criteria for selecting Spitzer targets. Thus,
observations taken after HJD${}^{\prime}=8691.5$ may be considered
“objective”. See Yee et al. (2015) for a detailed explanation of different
types of selection and their implications and criteria.

###  2.3 Ground-Based Followup Observations

Once OGLE-2019-BLG-0960 was selected for Spitzer observations, we began to
monitor it with the dual-channel imager, ANDICAM (DePoy et al., 2003) on the
SMARTS 1.3m telescope at CTIO in Chile (CT13). Observations were taken
simultaneously in both the $I$ and $H$ bands for the purpose of tracking its
evolution and obtaining an $I-H$ color for the source. Starting at
HJD${}^{\prime}\sim 8683.5$, we increased the cadence to a few points per
night to supplement the survey data in the hopes of better characterizing any
planetary perturbations. We also began monitoring the event with the
telescopes in the LCO 1-m network at SAAO (South Africa; LCOS) and SSO
(Australia; LCOA01, LCOA02).

On 20 Jul 2019, the Spitzer team realized that the event was consistent with
high magnification ($A>100$) and sent an alert to the Microlensing Follow-Up
Network (MicroFUN) at UT 13:06 (HJD${}^{\prime}=8685.05$) encouraging dense
followup observations to capture the peak of the event. Auckland, Farm Cove,
and Kumeu Observatories in New Zealand (AO, FCO, and Kumeu, respectively) all
responded to the alert and began intensive observations of the event. MOA also
increased its cadence in response to the alert. At the same time, we scheduled
dense observations with CT13 and the LCO 1-m telescopes at SAAO and SSO. Two
days later at HJD${}^{\prime}=8687.05$, we called off the alert because the
event was declining and no significant perturbations had been observed over
the peak. At this point, most followup observations ceased. However, we
continued to observe this event at a cadence of a few per night with CT13,
while Kumeu Observatory took an additional hour of normalizing data, which
would prove crucial for recognizing and characterizing the planetary
perturbation.

On 23 July 2019 (HJD${}^{\prime}=8688.06$), a routine inspection of the CT13
data led to the discovery of a $\sim 0.5$ mag outlier in the light curve. The
outlier was confirmed by a data point in the KMTC light curve taken nearly
simultaneously. This triggered additional dense observations with the LCO
network. We did not realize the existence of Kumeu observations that
characterized the falling side of the perturbation until HJD${}^{\prime}\sim
8689.05$.

###  2.4 Data Reduction and Error Renormalization

The data were reduced by each collaboration and the photometric error bars
were renormalized according to the prescription in Yee et al. (2012). Table 1
gives the specific reduction method, the error renormalization factors, and
other properties for each data set. In the case of KMTNet data from CTIO and
SAAO, we restricted the data to points taken after HJD${}^{\prime}=8616$ to
control for possible effects of systematics on the parallax measurement.
Likewise, data from KMTA were restricted to $\pm 5$ days from the peak of the
event.

##  3 Ground-Based Light-Curve Analysis

Figure 1: Light curve of OGLE-2019-BLG-0960 with best-fit models. Although the
primary evidence for the planet is a strong bump in the light curve at
HJD${}^{\prime}\sim 8687.8$, the points before and after the bump also show a
negative deviation relative to the best-fit point lens model (dashed gray
line), which contributes to the detection. Note that KMTC and CT13 each have a
datapoint at HJD${}^{\prime}=8687.796$, but because they are contemporaneous,
they do not appear as distinct points on the plot. The two best-fit planetary
models are shown as the solid black and cyan lines. Although one model has
$s>1$ and one has $s<1$, this degeneracy is far from the regime of the
“close”/“wide” degeneracy identified by Griest & Safizadeh (1998, see Section
6.3).

Figure 2: Results of the $s$-$q$ grid-search for the best-fit solutions to the
light curve of OGLE-2019-BLG-0960. Left: the fits including all ground-based
data yield two distinct minima, one with $s<1$ and one with $s>1$, but not
centered around $s=1$. Right: the fits to just the ground-based survey data
(OGLE, MOA, KMT) give an additional pair of minima because the caustic
crossing is less well-characterized (see Section 6.1).  Figure 3: Caustics
(red dashed lines) and source trajectory (solid black line and arrow) for the
two degenerate solutions for OGLE-2019-BLG-0960. The top two panels show the
“$s>1$” solution, and the bottom two panels show the “$s<1$” solution. The top
panel in each pair shows the full view of the caustic, while the bottom panel
in the pair shows a close up of the caustic crossing. The locations of the
source at the times of the observations are indicated by the dots/circles; the
size of each circle is set by $\rho$.

The full light curve of this event is shown in Figure 1. With the exception of
a brief, 0.2 day, bump that occurs 1.4 days after the peak, the light curve is
well described by a standard single-lens single-source (1L1S) Paczynski (1986)
fit, which is specified by three geometric parameters, the lens-source closest
approach $t_{0}$, the impact parameter $u_{0}$ (normalized to the Einstein
radius, $\theta_{\rm E}$), and the Einstein radius crossing time $t_{\rm E}$.
In principle, the bump could be explained by either an additional lens (2L1S)
or additional source (1L2S), but we begin by focusing on the former. This
requires three additional parameters: the companion-host separation $s$
(normalized to $\theta_{\rm E}$), the companion-host mass ratio $q$, and the
companion-host orientation $\alpha$ (relative to the direction lens-source
relative motion). Because the source transits (or comes very close to) a
caustic structure generated by the lens system, we must also specify
$\rho\equiv\theta_{*}/\theta_{\rm E}$, where $\theta_{*}$ is the angular
radius of the source.

###  3.1 Static Analysis

We search for the best-fit models following the method of Yang et al. (2020).
In brief, we initially conduct a grid search over ($\log s,\log q$) to locate
the solutions. For each set of ($\log s,\log q$), we fix $\log s,\log q$, and
free $t_{0},u_{0},t_{\rm E},\alpha$ and $\rho$. For the initial values of
$t_{0}$, $u_{0}$, and $t_{\rm E}$, we use the values from 1L1S fitting to the
light curve excluding the planetary anomaly. The initial values of $\alpha$
and $\rho$ are estimated from the 1L1S parameters as follows:

$\alpha=\tan^{-1}\frac{t_{\rm eff}}{t_{0,{\rm
anom}}-t_{0}}=0.266~{}(15.2^{\circ});\qquad t_{\rm eff}=u_{0}t_{\rm E}$ (1)

where $t_{0,{\rm anom}}=8687.78$. Using the method of Street et al. (2016),
$\rho$ can be estimated by

$\rho\sim\frac{t_{0,{\rm anom}}-t_{\rm cc}}{t_{\rm
E}}\times\sin{\alpha}=3.2\times 10^{-4},$ (2)

where $t_{\rm cc}\sim 8687.70$ is the time of caustic entry. Then, we search
for the best-fit parameters using Markov chain Monte Carlo (MCMC) $\chi^{2}$
minimization as implemented in the emcee ensemble sampler (Foreman-Mackey et
al., 2013). We use the advanced contour integration code VBBinaryLensing
(Bozza,
2010)111http://www.fisica.unisa.it/GravitationAstrophysics/VBBinaryLensing.htm
to compute the binary-lens magnification. This code uses multiple rings to
account for the limb-darkening effect and sets the number of rings
automatically based on the required light curve precision. We use linear limb-
darkening coefficients for a $T_{\rm eff}=5250$K star: $\Gamma_{I}=0.43$,
$\Gamma_{R}=0.52$, $\Gamma_{L}=0.15$ (Claret & Bloemen, 2011).

This grid search yields only two local minima (left panel of Figure 2). We
then further refine these minima by allowing all seven parameters to vary in
an MCMC. The results are shown in Table 2, and the caustics and source
trajectories are shown in Figure 3. As with many events, the two solutions
appear to be related by the famous $s\leftrightarrow s^{-1}$ (a.k.a.
“close”/“wide”) degeneracy (Griest & Safizadeh, 1998; Dominik, 1999) because
one solution has $s<1$ and the other $s>1$. In fact, this is not the case, as
we discuss in Section 6.3.

Regardless, the two solutions are very similar in their physical implications.
First, the mass ratio is $q=1.27\pm 0.07$ or $q=1.45\pm 0.15\times 10^{-5}$,
making this the smallest mass-ratio planet discovered by microlensing to date.
Second, although the two solutions are distinct, $s$ is very close to 1, so
the separation of the planet from the host star is similar in the two cases.
Third, the relatively long Einstein timescale, $t_{\rm E}\sim 62\,$days,
suggests that it may be possible to measure the annual parallax effect for
this event.

###  3.2 Ground-based Parallax Analysis

Figure 4: Cumulative distribution of $\Delta\chi^{2}$ as a function of time
between the best-fit parallax and static models. For clarity, we only show the
$\Delta\chi^{2}$ contributions from the survey data (and we exclude KMTA
because only the data over the peak and anomaly are used in the modeling). The
top panel shows the model light curve for reference.  Figure 5:  (1, 2, 3, 4,
5, 6, 7)$\sigma$ contours (black, red, yellow, green, cyan, blue, purple) for
the parallax vector. The large ellipses in the top panels show the measurement
from the annual parallax effect in the ground-based data. These data give a
strong constraint on $\pi_{\rm E,\parallel}$, the component of the parallax
parallel to the projected position of the Sun, but a relatively weak
constraint on $\pi_{\rm E,\perp}$, resulting in highly elliptical error
contours. The arcs in the bottom panels are the constraint from the
Spitzer-“only” parallax analysis and are overlaid on the annual parallax
contours. These show tension with the annual parallax measurement at the $\sim
3\sigma$ level; limits from the blended light are also in tension with the
Spitzer-“only” parallaxes. Together, these suggest systematics in the Spitzer
light curve may be affecting the parallax measurement.

Thus, we are immediately led to introduce two additional parameters
${\mbox{\boldmath$\pi$}}_{\rm E}=(\pi_{{\rm E},N},\pi_{{\rm E},E})$, i.e., the
north and east components of the parallax vector (Gould, 2004) in equatorial
coordinates. Because the effect of Earth’s motion can be partially mimicked by
lens orbital motion (Batista et al., 2011; Skowron et al., 2011), we initially
introduce two further additional parameters
${\mbox{\boldmath$\gamma$}}=((ds/dt)/s,d\alpha/dt)$, i.e., the instantaneous
angular velocity at $t_{0}$. We find that $\gamma$ is poorly constrained, so
we impose a limit on the ratio of projected kinetic to potential energy (Dong
et al., 2009)

$\beta=\bigg{|}{\frac{\rm KE_{\perp}}{\rm PE_{\perp}}}\bigg{|}={\frac{\kappa
M_{\odot}{\rm yr}^{2}}{8\,\pi^{2}}}{\frac{\pi_{\rm E}}{\theta_{\rm
E}}}\gamma^{2}\biggl{(}{\frac{s}{\pi_{\rm E}+\pi_{s}/\theta_{\rm
E}}}\biggr{)}^{3},$ (3)

where $\pi_{s}$ is the source parallax. We only accept links on the chain for
which $\beta<0.8$.222Strictly speaking, only $\beta>1$ solutions are ruled out
by the requirement of a bound system. However, $0.8<\beta<1$ would require a
system with an extremely high eccentricity that was observed at a rare epoch
and in a rare orientation. We find that $\gamma$ is neither significantly
constrained in the fit nor significantly correlated with
${\mbox{\boldmath$\pi$}}_{\rm E}$, so we suppress these two degrees of
freedom.

The final parallax solutions are summarized in Table 2. Figure 4 shows the
cumulative $\Delta\chi^{2}$ diagram between the best-fit parallax and static
solutions. All of the survey datasets that span the event (OGLE, MOA, KMTC,
KMTS) contribute positively and consistently to the parallax signal. This
meets our expectation that a real parallax signal should affect data from all
observatories and makes it extremely unlikely that the signal has a non-
astrophysical origin.

We show in Figure 5 the $\chi^{2}$ contours projected on the
${\mbox{\boldmath$\pi$}}_{\rm E}$ plane, which are in the form of an ellipse
with a large axis ratio $a/b\sim 8$ . Such “1-D parallax” measurements were
predicted by Gould et al. (1994) and explored in greater detail by Smith et
al. (2003) and Gould (2004) for moderately long microlensing events. They
arise because $\pi_{{\rm E},\parallel}$, the component of
${\mbox{\boldmath$\pi$}}_{\rm E}$ that is parallel to the projected position
the Sun at $t_{0}$, is directly constrained by the asymmetry in the light
curve that is induced by Earth’s instantaneous acceleration near the peak of
the event. The first clear example of this in a planetary event was
OGLE-2005-BLG-071 (Udalski et al., 2005; Dong et al., 2009). Indeed, the
orientations (north through east) of the short axes of the ellipses for
OGLE-2019-BLG-0960 are close to the projected position of the Sun. Hence,
$\pi_{{\rm E},\parallel}$ is both relatively large and well-constrained from
the ground-based photometry even though there are substantial uncertainties in
the total value of $\pi_{{\rm E}}$.

###  3.3 Binary Source Models

Figure 6: OGLE-2019-BLG-0960 cannot be explained by a point-lens, binary-
source (1L2S) model, which is shown in two different bands ($R$, red line,
corresponding to the Kumeu Observatory data; $I$, black line, corresponding to
CT13 and KMT data). The dotted gray line shows the best-fit point-
source/point-lens model.

In principle, a short duration bump on an otherwise point-lens–like light
curve can also be caused by a binary source with a large flux ratio (Gaudi,
1998). Hence, we also search for 1L2S solutions. For this search, we introduce
the additional parameters $t_{0,2}$ and $u_{0,2}$ to describe the trajectory
of the second source and $\rho_{2}$ for its radius. We also introduce $q_{\rm
F,I}$, the flux ratio between the two sources, as a fit parameter. The results
of these fits are given in Table 3 and show that such solutions are disfavored
by $\Delta\chi^{2}>1000$. Figure 6 shows that the best-fit 1L2S model fails to
reproduce the decrease in magnification (relative to a point lens) seen before
and after the bump.

##  4 Spitzer Parallax Analysis

Figure 7: Spitzer light curve (red points) in instrumental-flux units,
together with the best-fit Spitzer-“only” model in black. The fact that the
event is clearly faint from Spitzer indicates a significant parallax effect. A
rising light curve is also consistent with the annual parallax measurement.
That is, $\pi_{{\rm E},E}<0$ implies that the event peaked later as seen from
Spitzer, which lay to the west of Earth.

According to the original conception of Refsdal (1966), the microlens parallax
(${\mbox{\boldmath$\pi$}}_{\rm E}$ in modern notation) could be determined up
to a four-fold degeneracy by combining information from the ground-based and
space-based Paczyński parameters $(t_{0},u_{0},t_{\rm E})$ for the event,

${\mbox{\boldmath$\pi$}}_{\rm E}={\frac{{\rm
AU}}{D_{\perp}}}\biggl{(}{t_{0,\rm sat}-t_{0,\oplus}\over t_{\rm E}},u_{0,\rm
sat}-u_{0,\oplus}\biggr{)},$ (4)

where ${\bf D_{\perp}}$ is the Earth-satellite separation projected on the
plane of the sky and $t_{\rm E}$ is approximated as being the same for both
Earth and the satellite. The two terms in Equation (4) are then the components
of ${\mbox{\boldmath$\pi$}}_{\rm E}$ that are, respectively, parallel and
perpendicular to ${\bf D_{\perp}}$. A four-fold degeneracy arises because
$u_{0}$ is a signed quantity, but generally only its magnitude can be
determined from the light curve. See Figure 1 of Gould (1994).

Figure 7 shows the Spitzer light curve in instrumental-flux units. Because the
Spitzer observations cover the peak of the event (which was very bright,
$I=14.3$), the lack of a strong microlensing signal (i.e., a substantial
change in the flux) indicates a significant parallax effect. In fact, the
Spitzer light curve is rising throughout the observations, which begin
approximately at $t_{0,\oplus}$. Because Spitzer was almost due west of Earth
at these times, this rising light curve implies that $\pi_{{\rm E},E}<0$, in
good agreement with the ground-based results.

However, the total Spitzer flux variation of OGLE-2019-BLG-0960 is extremely
small (only 1.5 flux units). Recent experience with KMT-2018-BLG-0029 (Gould
et al., 2020), OGLE-2017-BLG-0406 (Hirao et al., 2020), and OGLE-2018-BLG-0799
(Zang et al., 2020) has shown that there can be systematics in the Spitzer
parallax measurements for Spitzer light curves with weak signals. Hence, while
the Spitzer light curve for OGLE-2019-BLG-0960 shows a clear parallax signal,
we must be cautious in interpreting it.

We carry out a quantitative investigation of the Spitzer parallax signal
following the method described in Gould et al. (2020). Spitzer-“only” light
curves are obtained by fitting only the Spitzer data, while fixing the non-
parallax ground-based parameters $(t_{0},u_{0},t_{\rm E},s,q,\alpha,\rho)$ at
their best-fit, ground-based values. In particular, because Spitzer-“only”
fits are used primarily to analyze information flow rather than to obtain
final results, it is best to keep them as simple as possible without tracking
finer details. In addition, one must specify the Spitzer source flux (and
error bar), both of which can be derived from a Spitzer-ground color-color
(e.g., $IHL$) relation together with the $I$-band source flux.

For this color-constraint, we cross-match stars from the $6^{\prime}\times
6^{\prime}$ CT13 $I$-band field with VVV (Minniti et al., 2017) to construct
an $I-H$ CMD. While the VVV $H$-band is calibrated, CT13 is not, so the
resulting colors are instrumental rather than absolute. In this system, we
find the source color is $I_{\rm CT13}-H_{\rm VVV}=3.433\pm 0.013$ (see
Section 5.1). When cross-matched to stars in the Spitzer field, this yields a
color-color relation:

$I_{\rm CT13}-L=2.681\pm 0.019.$ (5)

The resulting parallax contours are shown in Figure 5. This figure shows that
the ground-only and Spitzer-“only” contours are inconsistent at the $\sim
3\sigma$ level. Furthermore, we will show in Section 5.4 that the best-fit
value for the Spitzer-“only” parallax, $\pi_{\rm E}=0.25$, is in tension with
constraints on the lens due to the blended light (unless the host is a stellar
remnant).

Because of these tensions, for the remainder of the paper, we focus on what
can be derived from the ground-based data alone. In the future, we will carry
out a systematic study of low-amplitude Spitzer light curves. Once the impact
of systematics on the Spitzer parallax measurements is better understood, we
can reassess the constraints from the Spitzer data on this event and assess
whether or not it can be included in the Spitzer statistical sample of
planets. Ultimately, these tensions can be resolved by separately resolving
the source and the lens (assuming that the lens is luminous). This would allow
both a measurement of the lens flux and a measurement of the direction of the
lens-source relative proper motion (e.g., Batista et al., 2015; Bennett et
al., 2015; Bhattacharya et al., 2020; Vandorou et al., 2020). This would both
independently constrain the mass of the lens and the direction of the parallax
vector (because $\hat{\pi}_{\rm E}\equiv\hat{\mu}_{\rm rel}$).

##  5 Physical Properties

###  5.1 CMD & Source Properties

Figure 8: Left: OGLE CMD (calibrated to OGLE-III) of a $120^{\prime\prime}$
square centered on OGLE-2019-BLG-0960. Right: CT13-VVV CMD of a
$240^{\prime\prime}$ square centered on the event. In each panel, the red
asterisk and blue dot show the centroid of the red clump and the position of
the microlens source, respectively. In the left plot, the green and magenta
dots show the positions of the blend and the baseline object, respectively.
The cyan lines show a naive estimate (see Section 5.4) for the lens color and
magnitude calculated for $\theta_{\rm E}\pm 1\sigma$ (solid and dashed lines,
respectively); the cyan dot shows the position for $M_{\rm L}=0.5\ M_{\odot}$.

We measure the intrinsic color and magnitude of the source by finding its
location in a color-magnitude diagram (CMD) relative to the red clump (Yoo et
al., 2004). We do this in two ways. First, we use the OGLE-IV photometry to
find $(V-I,I)_{\rm S}=(1.53,19.81)\pm(0.04,0.03)$, and we find the centroid of
the red clump is $(V-I,I)_{\rm cl}=(1.73,15.24)\pm(0.02,0.04)$ (see Figure 8).
The intrinsic color and magnitude of the red clump along this line of sight is
$(V-I,I)_{0,\rm cl}=(1.06,14.27)$ (Bensby et al., 2011; Nataf et al., 2013).
This implies that $A_{I}=0.97$ and $E(V-I)=0.67$ for this field. Hence ,
assuming the source experiences the same reddening and extinction as the
clump, we find $(V-I,I)_{0,\rm S}=(0.86,18.84)\pm(0.05,0.05)$. The OGLE
magnitudes reported here and elsewhere have all been calibrated from the OGLE-
IV system to the standard OGLE-III Johnson-Cousins system. Using the
color/surface-brightness relation of Adams et al. (2018), we obtain
$\theta_{*}=0.628\pm 0.040\ \mu{\rm as}$.

As a check, we also derive the $(V-I)$ color from the KMTC data. We obtain
$(V-I)_{0,\rm S}=0.81$, in good agreement with the OGLE-IV color.

We also construct an $I-H$ versus $I$ CMD by cross-matching the CT13 $I$-band
and the VVV $H$-band stars within a $4^{\prime}\times 4^{\prime}$ square
centered on the event (see Figure 8). We measure the centroid of the red giant
clump as $(I-H,I)_{\rm cl}=(3.81\pm 0.01,17.04\pm 0.03)$, and find the
intrinsic centroid of the red giant clump to be $(I-H,I)_{0,\rm
cl}=(1.30,14.27)$ (Nataf et al., 2013, 2016). For the source color, which is
independent on any model, we first get $(I-H)_{\rm CT13,S}=-1.766\pm 0.012$ by
regression of CT13 $H$ versus $I$ flux as the source magnification changes. We
then measure the offset between CT13 $H$-band stars and VVV stars to be
$H_{\rm CT13}$ \- $H_{\rm VVV}=5.199\pm 0.004$, yielding $(I-H)_{\rm
S}=3.433\pm 0.013$. Therefore, we obtain the intrinsic color and brightness of
the source $(I-H,I)_{0,\rm S}=(0.92\pm 0.03,18.85\pm 0.05)$. Then, from Adams
et al. (2018), we obtain $\theta_{*}=0.623\pm 0.037~{}\mu{\rm as}$.

As our final value, we adopt the weighted mean of the OGLE and CT13
measurements, $\theta_{*,0}=0.625\pm 0.028$. For the source apparent
brightness, we note that the source magnitude depends on the model. For
simplicity, we have explicitly derived results for $I_{\rm OGLE,S}=19.81$.
Then, for different source magnitude, one can derive $\theta_{*}$ of a
solution with a particular $I_{\rm S}$ by $\theta_{*}=\theta_{*,0}\times
10^{-0.2(I_{\rm S}-19.81)}$. We use this formula to account for slight
variations in $f_{{\rm S},I_{\rm OGLE}}$ for different degenerate solutions.
Finally, the measured value of $\rho$ implies $\theta_{\rm E}=1.9-2.1\ {\rm
mas}$ depending on the solution (see Table 4).

###  5.2 Lens Constraints from Blend and Baseline Object

Figure 9: Constraints on the lens mass and distance. Magenta lines show the
mass-distance relation calculated from $\theta_{\rm E}$ (solid) with the
$1\sigma$ limits (dotted) for the $(s>1,u_{0}<0)$. Blue lines show the mass-
distance relation and $1\sigma$ limits from the ground-based parallax for that
solution (solid and dotted, respectively). Red, dashed line shows the mass-
distance relation for $\pi_{\rm E}=0.25$, i.e., the best-fit value for the
Spitzer-“only” parallax. The black lines are the $3\sigma$ upper limits
derived from the blended light in I band. The cyan lines are the $3\sigma$
limits derived from the color of the baseline object. These give upper limits
on the lens mass if it is behind all of the dust (solid) or in front (dot-
dashed). Lenses below these lines are allowed.

Independent of any other constraints, the observed flux constrains the mass of
the lens. First, there is a constraint from the blend flux. Because the blend
consists of light from stars other than the lensed source (i.e., the lens and
any other stars blended into the PSF), the lens cannot be brighter than the
blend. In principle, because the background in the bulge consists of
unresolved stars rather than blank sky, OGLE-2019-BLG-0960 could be sitting on
a “hole” in the background (Park et al., 2004) and, therefore, the blend could
be slightly brighter than measured. To place a $3\,\sigma$ upper limit on the
flux from the lens, we take account of this effect in addition to the ordinary
photon-based error in the measurement of the flux from the baseline object.
These must be handled separately because the photon noise is Gaussian but the
effect of the mottled background, including a possible “hole”, is highly non-
Gaussian.

For the photon noise, we estimate the photometric error in the baseline object
to be 0.020 mag. Subtracting the measured source flux (taking accont of its
very small error), and propagating the errors, we find that this term
contributes 0.0285 mag at $1\,\sigma$ and so 0.085 mag at $3\,\sigma$.

To account for the mottled background, we follow the method of Gould et al.
(2020). We model the bulge stellar distribution using the Holtzman et al.
(1998) luminosity function, scaled down by a factor 0.50 according the bulge
projected clump-star density map of Nataf et al. (2013) and D. Nataf (2019,
private communication). We estimate that OGLE-IV imaging captures field stars
$I<20$ but leaves fainter stars unresolved, and that this map is constructed
from FWHN$\sim 0.9^{\prime\prime}$ images. We then evaluate the impact of
random field stars on the baseline-object (hence, blend) photometry by Monte
Carlo. We find that at $(1,2,3)\sigma$ [i.e., (84.1,97.7,99.9) percentiles],
the blend could be brighter than its naive value by (0.08,0.12,0.13) mag.
Hence, the combined three sigma limit due to both photon noise and mottled
background is 0.16 mag. Hence, the $3\,\sigma$ limit on the brightness of the
lens is $I_{L}>18.38$.

Figure 9 shows the resulting limits on the lens mass and distance. For this
figure, we have converted the limit on $I_{\rm L}$ to a limit on $M_{\rm L}$
using Pecaut & Mamajek (2013). To show the effect of dust, we have calculated
these limits for both $I=18.38$ and $I_{0}=I-A_{I}=17.41$.

Second, the color of the baseline object provides an additional constraint on
the lens. The baseline object (measured when the source is unmagnified)
consists of light from the source, the lens, companions to those stars, and
ambient stars blended into the PSF. We have measured the astrometic offset
between the baseline object in higher-resolution CFHT imaging and the KMTNet
microlensing event. We find $\Delta\theta_{\rm CFHT}=(23.1,7.6)\pm(12.2,18.4)\
$ mas, which is consistent with zero offset. Hence, all or most of the light
from the baseline object (and blend) is likely to be associated with the event
as either the source, the lens, or a companion to one or both of them.

In this case, the baseline object in OGLE is very red: $(V-I)_{\rm
base}=2.18\pm 0.13$. In the KMTC pyDIA CMD, the baseline object is the same,
very red, color within errors. Because the source is blue relative to the
clump, this necessarily implies that the blended light is at least as red or
even redder than the baseline object. Hence, either the lens is also very red
or it is much fainter than the blend. Even if the lens is behind all of the
dust, this implies $(V-I)_{0,\rm L}>1.12$ at $3\sigma$, assuming
$E(V-I)=0.67$. Then, using Pecaut & Mamajek (2013) to convert this color to a
mass limit gives a limit of $M_{\rm L}\lesssim 0.73M_{\odot}$. This limit is
shown in Figure 9 as is the limit for $(V-I)_{\rm L}>1.79$, which would imply
the lens is in front of all of the dust.

###  5.3 Tension with Spitzer-“only” Parallax

If we combine the flux constraints on the mass of the lens with the
measurement of $\theta_{\rm E}$, we can then place a constraint on the
microlens parallax and the distance to the lens:

$M_{\rm L}=\frac{\theta_{\rm E}}{\kappa\pi_{\rm E}};\quad\pi_{{\rm
rel}}=\frac{\theta_{\rm E}^{2}}{\kappa M}=\left(\frac{{\rm AU}}{D_{\rm
L}}-\frac{{\rm AU}}{D_{\rm S}}\right)$ (6)

where we assume that the source is at the mean distance to the red clump,
i.e., $D_{\rm S}=7.56\ $ kpc. Given that $\theta_{*,0}\simeq 0.625\ \mu$as,
the Einstein radius is quite large $\theta_{\rm E}=\theta_{*}/\rho\simeq
1.9\,{\rm mas}$. Hence, a mass limit of $M_{\rm L}<0.73\ M_{\odot}$ (from
$(V-I)_{\rm base}$) implies $\pi_{\rm E}>0.32$ and $D_{\rm L}<1.35\ $ kpc.
However, this estimate is not fully self-consistent because, even if the lens
were behind all of the dust, a $0.73\ M_{\odot}$ star at 1.35 kpc would be
brighter than the observed blend (i.e., in Figure 9 the intersection of the
cyan and magenta lines is above the black lines). Combining the blended light
constraint with the $\theta_{\rm E}$ measurement results in an upper limit on
the lens mass of $\sim 0.55\ M_{\odot}$, so $\pi_{\rm E}>0.42$ and $D_{\rm
L}\lesssim 1\ $kpc (intersection of the black and magenta lines in Figure 9).

From the ground-based parallax alone, we have a robust measurement of
$\pi_{\rm E,\parallel}$. This gives a lower limit on the microlens parallax of
$\pi_{\rm E}\gtrsim 0.3$. Combined with $\theta_{\rm E}$, $\pi_{\rm
E,\parallel}$ alone implies $M_{\rm L}\lesssim 0.78M_{\odot}$. These limits
are consistent with our expectations based on $\theta_{\rm E}$ and the blended
light. The mass-distance relation calculated from the full ground-based
$\pi_{\rm E}$ for the $(s>1,u_{0}<0)$ solution is shown in Figure 9 with
$1\sigma$ limits. This shows that there are regions of overlap between this
relation and the $\theta_{\rm E}$ relation that are consistent with the
constraints from the blended light.

However, the constraints from the blend and baseline object support the
conclusion that the low-amplitude signal in the Spitzer light curve should be
treated with caution. The preferred value of the Spitzer-“only” parallax gives
$\pi_{\rm E}=0.25$ (red line in Figure 9). This value is in tension with the
ground-based parallax values at $\sim 3\sigma$, but, when combined with
$\theta_{\rm E}$, it also implies $M_{\rm L}\simeq 0.93\ M_{\odot}$, which is
too blue and too bright for the blended light333One exception to this limit
would be if the lens were a white dwarf and the blend were instead a lower-
mass companion to that lens.. At the same time, the full $7\sigma$ arc from
the Spitzer-“only” parallaxes includes regions that overlap with the ground-
based parallax contours at $<2\sigma$ (see Figure 5) and which is also more
compatible with the blended light. Thus, this tension may be resolved in the
future once the constraints from the Spitzer light curve are better
understood.

###  5.4 Physical Properties of the Lens

For now, we focus on constraints on the lens properties that may be derived
from the ground-based analysis. Table 4 gives the physical properties of the
lens derived from each solution to the microlensing light curve.

These values are both consistent with the constraints from the blended light
and baseline object and suggest that the lens is responsible for most of the
blend. For example, for the $(s>1),(u_{0}<0)$ solution, we have $M_{L}=0.50\pm
0.12M_{\odot}$ and $D_{\rm L}=0.98\pm 0.2$ kpc. For a 0.5 $M_{\odot}$ star,
Pecaut & Mamajek (2013) give $M_{\rm I}=7.6$ and $(V-I)_{0}=2.0$. Hence, for a
lens at 0.98 kpc,

$\displaystyle I_{\rm L}$ $\displaystyle=$ $\displaystyle 17.6+A_{I}f_{\rm
dust},$ (7) $\displaystyle(V-I)_{\rm L}$ $\displaystyle=$ $\displaystyle
2.0+E(V-I)f_{\rm dust},$ (8)

where $f_{\rm dust}$ is the fraction of the dust in front of the lens. We make
the naive assumption that the dust follows an exponential profile such that
$f_{\rm dust}=1-\exp(-D_{\rm L}/(1.334\ {\rm kpc}))$. Then, this yields
$I_{\rm L}=18.07$ and $(V-I)_{\rm L}=2.35$, which is shown as the cyan point
(“naive lens”) in Figure 8. The solid cyan line shows the values of $I_{\rm
L}$ and $(V-I)_{\rm L}$ calculated for $\theta_{\rm E}=1.90\ $ mas for the
$1\sigma$ ranges for $\pi_{\rm E}$. The dotted lines show the results
evaluated at $\theta_{\rm E}\pm 1\sigma$. Because of the simplified assumption
about the dust, these calculations are simply meant to be illustrative.
Nevertheless, they indicate that, within the uncertainties, the lens mass and
distance are consistent with the blended light constraints within $1\sigma$
and suggest that the lens could be responsible for most or all of the blended
light.

The hypothesis that the blend is the lens could be tested immediately with
adaptive optics observations. These observations could give a stronger
constraint associating the blended light with the event as well as a more
precise measurement of that light by resolving out unrelated stars. Given the
high lens-source relative proper motion ($\mu_{\rm rel}=11.4\pm 0.8\,{\rm
mas\,yr^{-1}}$ or $\mu_{\rm rel}=12.2\pm 0.7\,{\rm mas\,yr^{-1}}$), it will be
possible by $\sim 2025$ to separately resolve the source and lens (e.g., Terry
et al., 2020), provided that the lens is responsible for a substantial
fraction of the blended light. Such observations would confirm that the
blended light is moving with the lens, i.e., that the separation matches the
expectations from the lens-source relative proper motion given in Table 4. As
discussed in Section 4, this would also give an independent measurement of the
direction of that motion, which could be compared to the constraints from the
parallax.

##  6 Discussion

OGLE-2019-BLG-0960Lb is the smallest mass-ratio microlensing planet ever
found. With its discovery, there are now nineteen known planets with mass
ratios below the fiducial break proposed by Suzuki et al. (2016), and four
planets with mass ratios below the break found by Jung et al. (2019).
Measuring the slope of the mass-ratio distribution below $q_{\rm br}$ requires
a statistically robust sample, and while the current sample does not meet that
criterion, it does offer insight into how such a sample might be obtained.

First, in Section 6.1, we consider whether or not the planetary perturbation
can be characterized with only the survey data. Then, we note in Section 6.2
that most of the known microlensing planets with $q<10^{-4}$ have $s\sim 1$.
We explore the implications of this for designing a search strategy for such
planets, e.g., in either existing data or with targeted followup observations.
Finally, we discuss in Section 6.3 the fact that $s\sim 1$ is far from the
regime for which the $s\rightarrow 1/s$ degeneracy was derived, how the
persistence of an $(s<1)$ vs. $(s>1)$ degeneracy suggests a deeper symmetry in
the lens equation, and how the departure from the original ideal affects our
ability to measure and interpret the mass-ratio function.

###  6.1 Characterization with Survey Data

Figure 10: Zoom of the planetary perturbation for best-fit models to only the
ground-based survey data. We recover the two solutions found when including
all the data (“$s>1$, Single” and “$s<1$, Single”, black and cyan,
respectively). However, the sparseness of the data allows two additional
solutions to the light curve, with a much smaller source and two distinct
caustic crossings (“$s>1$, Double” and “$s<1$, Double”, yellow and magenta,
respectively). Nevertheless, the mass-ratio of the planet remains small for
these solutions. The 1L2S solution (dashed red line) is still strongly ruled
out by the survey data that are de-magnified relative to the 1L1S model (gray
dashed line).

Naively, we would not expect such a small mass-ratio planet to be found in
such a low-cadence survey field unless, as in this case, there are additional
followup observations. Such planetary perturbations would typically last

$t_{\rm p}\sim\sqrt{q}t_{\rm E}=4.8\
\mathrm{hr}\left(\frac{q}{10^{-4}}\right)^{1/2}\left(\frac{t_{\rm E}}{20\
\mathrm{d}}\right).$ (9)

By contrast, all of the surveys observe this field at a cadence of less than
one observation per hour (see Section 2). The highest survey cadence is for
the MOA survey at $0.6\ \mathrm{hr}^{-1}$, which was not able to observe the
night of the anomaly. Hence, the mass-ratio of this planet is an order of
magnitude smaller than the expected minimum mass ratio for this field if we
require 3 observations during the anomaly. Indeed, there is only a single
survey observation over the main bump caused by the planet. However, because
we know that the planet is real, we can ask whether this planet can be
recovered and characterized with the sparse survey data (Yee et al., 2012).
Then, we can consider whether or not this is evidence that short duration
signals with only one to a few observations during the anomaly (the most
likely case for the smallest planets) in general can be reliably and
meaningfully recovered from sparse survey fields.

Therefore, we repeat the search for solutions using only the survey data
(i.e., the data from OGLE, MOA, and KMTNet). The grid search over $s$, $q$,
and $\alpha$ finds the previous two solutions but also a second pair of
degenerate solutions (see Table 5 and Figure 2). The new set of solutions
corresponds to the case in which the source is much smaller than the caustic
and the single KMTC datapoint at 8687.796 falls in the trough between two
caustic crossings (see Figure 10). Hence, this is an observational degeneracy
that could be resolved with the addition of more data. Nevertheless, all four
solutions lead to the same conclusions regarding the planetary parameters: the
planetary perturbation is created by an extremely small mass-ratio planet at
$s\sim 1$. We also find that the 1L2S solution is still strongly ruled out (by
$\Delta\chi^{2}\sim 300$, see Table 3).

In some ways, it is not surprising that the planet parameters are so well
constrained. Because observations are taken every $\sim 2.5\ $hr, the maximum
duration of the bump is $\sim 5\ $ hr. Then, given Equation (9) and $t_{E}\sim
62\ $ days, we would expect $q\lesssim 1.2\times 10^{-5}$. Then, the fact that
the event does not produce a perturbation over the peak, despite the extremely
high magnification, suggests a planetary caustic-like perturbation. At the
same time, the perturbation occurs only 1.4 days after the peak, which places
the “planetary” caustic very close to the central caustic, i.e., in a resonant
or close-to-resonant configuration.

However, if the planetary perturbation were found and characterized based a
single outlier, this would raise substantial questions as to the believability
(or publishability) of the planet. In this case, we are fortunate to have
extensive follow-up data, so the existence of the perturbation is well-
supported. However, the detection of this planet in the survey data is
strengthened by several other factors. First, this event has a longer
timescale than average ($t_{\rm E}=62$ days), which enhances the duration of
the perturbation via Equation (9), and hence, the likelihood of obtaining
observations during the anomaly. Second, similar to KMT-2019-BLG-0842 (Jung et
al., 2020), the source trajectory passes at an oblique angle with respect to
the binary axis, which further lengthens the duration of the perturbation. As
a result, we see from Figure 10 that there are five additional points that are
significantly demagnified relative to the point-lens light curve. Hence, the
perturbation lasts for almost a day and the detection does not, in fact, rely
solely on a single outlier.

In terms of the broader implications for characterizing planetary
perturbations based on sparse survey data, it is interesting to compare this
event to other cases. The first such case was described in Dominik & Hirshfeld
(1996), who showed that a number of different 2L1S models could explain two
outliers in MACHO-LMC-001. Another case was described in Gaudi & Han (2004),
who found multiple 2L1S models that fit a single outlier in OGLE-2002-BLG-055.
One major contrast between these two events and OGLE-2019-BLG-0960 is the
cadence of observations relative to the Einstein timescale. In both MACHO-
LMC-001 and OGLE-2002-BLG-055, observations were only obtained once per few
days during the perturbation, which allows for a much broader range of
potential models. For example, the perturbation could always be much shorter,
resulting in a relatively weak constraint on $q$ using Equation (9). Hence,
even though the survey observations of OGLE-2019-BLG-0960 were relatively
sparse compared to the duration of the perturbation, they are dense compared
to $t_{\rm E}$, leading to good constraints on the planet. However, further
investigation of short and sparsely-observed perturbations (but that might
have well-constrained models) is needed to determine whether or not they are
publishable in general.

###  6.2 Finding Small Mass-Ratio Planets

Figure 11: Known microlensing planets with $q<10^{-3}$ (NASA Exoplanet
Archive, accessed 10/27/20). Solid points show planets with a single solution.
Planets with multiple degenerate solutions are shown as open circles (one for
each solution) connected by dotted lines (excludes planets with mass ratios
that are ambiguous by more than a factor of 2). The two degenerate solutions
for OGLE-2019-BLG-0960Lb are shown as the red stars. The gray lines show the
fiducial $q_{\rm br}=1.7\times 10^{-4}$ as proposed by Suzuki et al. (2016)
(dotted) and the value proposed by Jung et al. (2019) (solid). Black solid
lines show the boundary between resonant and non-resonant caustics, and the
dashed lines show $(3,1.8)\log s_{\rm resonant}$ for $(s<1)$ and $(s>1)$
caustic structures. The vast majority of planets are found between the dashed
lines (so in resonant or close-to-resonant caustics, see Figure 12), and two
are found at $\log s\sim 0.2$ in Hollywood events.  Figure 12: Left: The
horizontal extent of the caustics; for non-resonant caustics, the extent is
the sum of the widths of the planetary and central caustics. Center: The total
extent of the region(s) along the binary axis that is perturbed by at least
10% relative to a point lens. The extents are measured as fractions of an
Einstein ring. Right: Ratio of the 10% width to the caustic width. The black
lines are as in Figure 11. The dashed lines were chosen to roughly correspond
to the value of $s$ for which the extent of the 10% region is maximized at
fixed $q$ (they are the same as those in Figure 11.). The range of $s$ over
which the source plane is significantly perturbed by the planet is
substantially larger than the range of $s$ for which the caustic size is
maximized. Thus, the cross-section for a planetary perturbation to occur is
substantially enhanced over the raw size of the caustics.  Figure 13: Maps
showing the difference between the planetary magnification and the
magnification of a point lens: $\log q=-4.85$ and $\log s=1.8\log s_{\rm
resonant,wide}=0.0278$ (top) and $\log s=3\log s_{\rm resonant,close}=-0.0238$
(bottom). Planets just outside of resonance produce significant deviations to
the point lens magnification extending between the two caustic structures
(magenta), which may be either positive (white) or negative (black).  Figure
14: Black (dashed, solid): maximum radial extent “max u” of the $(s<1)$ and
$(s>1)$ resonant caustic structures for a given value of $\log q$. Blue:
maximum value of $|x|$ (which we use as a proxy for $u$) along the binary axis
for which the magnification pattern is perturbed by at least 10% relative to a
point lens. The value of $s$ is chosen such that the extent of the region is
maximized for a given $q$ (see text, Figure 12). Magenta: same as blue but for
a 20% perturbation. Gray lines as in Figure 11. Fully probing the region
perturbed by the planet requires continuously monitoring the light curve
through $\pm$ “max u”.

Given that there appear to be a number of factors that make this planetary
perturbation “special,” we now consider how detections of small planets in
general can be enhanced.

Previously, Abe et al. (2013) had suggested that the sensitivity of
microlensing events to small planets was enhanced for moderate magnification
events ($A>50$). Indeed, examining the published microlensing planets444NASA
Exoplanet Archive, accessed 10/27/20. with $q<q_{\rm br}=1.7\times 10^{-4}$
supports this conclusion. Here, we include events with multiple degenerate
solutions if all solutions are planetary and at least one solution has
$q<q_{\rm br}$. Seven out of nineteen planets with $q<q_{\rm br}$ have been in
events with $u_{0}\leq 0.02$ (i.e., $A\geq 50$). Another seven are found in
events with $0.02<u_{0}\leq 0.1$ and two more in events with $0.1<u_{0}\leq
0.2$. The remaining three planets were all found in “Hollywood” events, in
which the source was a giant.

The second thing to notice about the known small planets is that about half of
them are found in events with resonant caustics and many of the others have
separations that are close to resonance (see Figure 11 in which we exclude
planets with ambiguous mass ratios, see caption). On the one hand, resonant
caustics are larger than planetary caustics (the size of the resonant caustic
scales as $q^{1/3}$ whereas the size of the planetary caustic scales as
$q^{1/2}$; Dominik, 1998; Han, 2006), enhancing the probability that the
images of the source are perturbed. On the other hand, resonant caustics are
generally expected to produce weaker perturbations (Gaudi, 2012), which might
not be significant enough to be detected. Empirically, Figure 11 shows that
the majority of the detected planets are in resonant or near-resonant
configurations, suggesting that the enhanced cross-section is the dominant
effect and extends to configurations outside of resonance. It also suggests
that the “enhancement” for detectability extends to planets with separations
$s$ well outside the formal range for resonant caustics.

The enhanced cross-section for a perturbation for caustics close to, but
outside of resonance, comes about because the perturbed magnification pattern
extends out from and stretches all the way between the planetary and central
caustics. Figure 12 compares the width of the caustics and the width of the
region for which the magnification of a star+planet lens deviates by at least
10% compared to the star alone. For this exercise, we choose a 10% deviation
as a reasonable threshold for producing a detectable perturbation. The “width
of the caustics” is measured as the horizontal extent of the caustics (i.e.,
measured along the binary axis) and, if the caustic is not resonant, is the
sum of the widths of the planetary and central caustics. The width of the
deviation from a point lens is measured along a slice through the binary axis.

We usually think of the cross-section for detecting planetary perturbations as
being equal to (or proportional to) the size of the caustics. However, Figure
12 shows that when measured in terms of a 10% deviation, the cross-section is
largest for separations just outside of resonance and is significantly larger
at that boundary than one would expect from strict proportionality. The
decreased width of the 10% perturbation region for $\log s=0$ reflects the
relative “weakness” of true resonant caustics555The form of the right panel in
Figure 12 also has a similarity to the detection sensitivity diagram for
OGLE-2008-BLG-279 (see Figure 7 of Yee et al., 2009) arising from the same
effect.. The maximum size of a 10% deviation for fixed $q$ peaks at $\sim
3\log s_{\rm resonant}$ for $s<1$ and $\sim 1.8\log s_{\rm resonant}$ for
$s>1$, where $s_{\rm resonant}$ is the boundary between resonant and non-
resonant caustics (i.e., Equations (57) and (58) from Dominik, 1998). Even
though caustics in this region are not formally resonant, they are clearly
influencing each other. Thus, we refer to caustics in this range as “semi-
resonant.”

Abe et al. (2013) noted this effect for caustic structures with $s<1$ and
referred to it as a “cooperative effect.” For such caustics, there is an
extended “trough” (negative magnification deviation) along the binary axis
connecting the planetary and central caustics (bottom panel of Figure 13).
However, we also see this effect for $s>1$ caustics but in the form of a
narrow “ridge” (positive magnification deviation) connecting the planetary and
central caustics (top panel of Figure 13). These effects can also be seen in
Figure 10 of Gaudi (2010). In terms of detectability, it may be that $s<1$
caustic perturbations are easier to detect because the vertical extent of the
10% deviation region is larger666Compared to Figure 13, the separation between
the planetary caustics for $s<1$ will grow much faster with $q$ than the
height of the planetary caustic for $s>1$ (Han, 2006)., and thus, a typical
perturbation will last longer. Empirically, we do see that there are more
$s<1$ planets than $s>1$ planets for $q<1.7\times 10^{-4}$, but the total
numbers are still too small for this to be a statistically meaningful result.
By contrast, Jung et al. (2020) noted that published perturbations crossing
“ridges” found in $s>1$ events tend to have source trajectories with oblique
angles. Such trajectories will result in proportionally longer perturbations,
but represent only a small fraction of possible trajectories, suggesting many
such perturbations are missed due to insufficient cadence.

In terms of detection strategy, Figure 14 shows the maximum value of $u$ for
which the magnification pattern due to a planet is perturbed by at least 10%
relative to a point lens. Because the orientation of the caustic structure
relative to the source trajectory is random, this suggests that events should
be monitored continuously for $\tau=\pm u$ in order to probe this full
structure. In the case of OGLE-2019-BLG-0960, we called off the alert at
$\tau\sim+0.01$, and thus, the planet was nearly missed. This investigation
suggests that to search for planets with $\log q<-4.5$, events should be
densely monitored for $\tau=\pm 0.2$, i.e. $A>5$. Indeed, Han et al. (2020)
report the detection of a planet in KMT-2018-BLG-1025 with $q\sim 10^{-4}$ in
which the perturbation occurred at $\tau\sim 0.05$.

###  6.3 “Close”/“Wide” Degeneracy for Semi-Resonant Caustics?

Figure 15: Left: microlensing planets with two degenerate solutions (NASA
Exoplanet Archive, accessed 10/27/20). Dotted line at $\log s=0$, solid and
dashed lines as in Figures 11 and 12. Points are colored by impact parameters,
$u_{0}$, and the symbol types indicate whether the caustics are near-
resonant/resonant (triangles) or non-resonant (circles), i.e., inside or
outside the dashed lines. OGLE-2019-BLG-0960 is marked by black stars. Right:
difference from a “true” “close”/“wide” degeneracy. Events with a perfect
$s\leftrightarrow s^{-1}$ degeneracy should have mean$(\log s)\rightarrow 0$.
By contrast, most of the events shown, including OGLE-2019-BLG-0960, have
resonant or semi-resonant caustics (triangles), i.e., are not in the $|\log
s|\gg 0$ regime, and many do not have mean$(\log s)\rightarrow 0$. For these
events, the different degenerate solutions can have significant fractional
deviations in $q$, in contrast to the expectation that $q$ is invariant to the
“close”/“wide” degeneracy for the limit as $|\log s|\gg 0$.

Figure 11 also shows that many planets have multiple, degenerate solutions.
These degenerate solutions usually have one solution with $s<1$ and another
with $s>1$ and are often referred to as arising from the “close”/“wide”
degeneracy. However, we will see that this degeneracy is more complex than the
degeneracy that was first described by Griest & Safizadeh (1998) and expanded
upon by Dominik (1998) and An (2005).

As an example, the two solutions for OGLE-2019-BLG-0960 nominally have the
hallmarks of the “close”/“wide” degeneracy for central caustics. First, the
event is in the “high-magnification” regime ($u_{0}<0.01$). Second, one of the
solutions has $s<1$ while the other has $s>1$. However, the two solutions are
not centered around $\log s=0$, i.e., $s_{\rm close}\neq s_{\rm wide}^{-1}$.
Furthermore, the “close”/“wide” degeneracy was derived by Griest & Safizadeh
(1998) and Dominik (1998) from the lensing equation in the regime where $|\log
s|$ is large, i.e., the regime in which the central and planetary caustics are
well-separated from each other. Griest & Safizadeh (1998) even state “We
expect the formula to break down when $q\rightarrow 1$ or $x_{p}\rightarrow
1$” (where they use $x_{p}$ in place of $s$). OGLE-2019-BLG-0960 is clearly
not in this regime: in both solutions, $s\sim 1$ and the caustic structure is
resonant. Thus, it is interesting to consider whether this is in fact a case
of the “close”/“wide” degeneracy.

Because the two solutions are not centered around $\log s=0$, they may in fact
be more analogous to the “inner”/“outer” degeneracy described by (Calchi
Novati et al., 2019) in which the source trajectory may pass either inside or
outside the planetary caustic relative to the central caustic. An examination
of the caustic structures for OGLE-2019-BLG-0960 (see Figure 3), shows that
the “close” solution passes to the outside the caustic whereas the “wide”
solution passes over the bridge in the resonant caustic created in between
what would be the central and planetary caustics. Two other small planets,
OGLE-2016-BLG-1195Lb and KMT-2019-BLG-0842Lb, also show this degeneracy (Bond
et al., 2017; Shvartzvald et al., 2017; Jung et al., 2020).

The “inner”/“outer” degeneracy in these four events appears to result from the
planetary caustic degeneracy that was first derived in the Chang-Refsdal limit
(Chang & Refsdal, 1979) by Gaudi & Gould (1997). This degeneracy is intrinsic,
i.e. rooted in mathematical symmetries in the lens equation rather than
“accidental,” which refers to degeneracies arising due to insufficient
observational coverage of the perturbation. However, again it was derived in
the limit that $|\log s|$ was large, and it was expected to break down as
$|\log s|\rightarrow 0$. None of these events is in the limit that $|\log s|$
is large, yet a degeneracy persists. This suggests that while the
“close”/“wide” degeneracy of Griest & Safizadeh (1998) and the “inner”/“outer”
degeneracy of Gaudi & Gould (1997) were derived in the limit that $|\log s|$
is large, as $|\log s|\rightarrow 0$, instead of breaking down, the two
degeneracies merge.

To investigate this possibility in more detail, we review the known
microlensing planets with two degenerate “geometric” solutions. These may
refer to either two distinct caustic topologies or two distinct source
trajectories relative to the caustic. As shown in Figure 15, of the planets
with one $s<1$ and one $s>1$ solution, very few can be described as having
well-separated caustics, i.e., are in the “close”/“wide” regime as described
by Griest & Safizadeh (1998). The vast majority of planets with degenerate
solutions are either in the resonant or semi-resonant regime, similar to
OGLE-2019-BLG-0960.

Events in the resonant or semi-resonant regime share the characteristics that
the degenerate solutions tend not to be perfectly symmetric about $|\log
s|=0$. Also, in contrast to the Griest & Safizadeh (1998) limit, the
degenerate solutions tend to have slightly different values of $q$. This is
especially true for events in the moderate magnification regime
($0.01<u_{0}<0.1$), which is precisely the region where we expect the smallest
planets to be found. Fortunately, because $|\log s|$ is very close to 0 in
both cases, this degeneracy does not meaningfully affect the interpretation of
the projected separation between the planet and its host star. However, the
small differences in $q$ will have to be taken into account when interpreting
statistical samples of such planets.

##  7 Conclusions

With a mass ratio of $q\sim 1.27\pm 0.07$ or $\sim 1.45\pm 0.15\times
10^{-5}$, OGLE-2019-BLG-0960Lb is the smallest mass-ratio microlensing planet
ever found. The annual parallax effect combined with the finite source effect
indicate the host star is an M-dwarf at $D_{\rm L}\lesssim 1\,$ kpc with a
super-Earth planet orbiting between 1 and 2 au (see Table 4 for exact values
for each solution). The lens is plausibly responsible for all or most of the
blended light, a hypothesis that can be tested immediately by taking adaptive
optics or HST observations to determine whether the blend is associated with
the event.

OGLE-2019-BLG-0960Lb is the 19th microlensing planet with a mass-ratio below
the fiducial power-law break in the mass-ratio distribution, $q_{\rm
br}=1.7\times 10^{-4}$, posited by Suzuki et al. (2016). It is the fourth
planet below the revised break of $q_{\rm br}=0.55\times 10^{-4}$ from Jung et
al. (2019). The three smallest planets (including this one) have all been
discovered since the advent of continuous survey observations from KMTNet.
This indicates that the current generation of microlensing experiments is now
capable of measuring both the precise location of $q_{\rm br}$ and the power-
law slope, $p$, of the mass-ratio distribution below $q_{\rm br}$.

By comparing OGLE-2019-BLG-0960 with other published planets below $q_{\rm
br}$ (Section 6.2), we found that they are primarily found in moderate
magnification and “Hollywood” (Gould, 1997) events. Moderate magnification
events are the primary source of small planets because the cross-section for a
planetary perturbation is largest when the planetary caustics are near
resonance. In fact, the planet sensitivity is maximized for planets just
outside resonance ($|\log s|>|\log s_{\rm resonant}|$) because significant
perturbations to the magnification field extend well beyond the caustic
structures. However, the planet sensitivity decreases rapidly for $|\log
s|\gtrsim{\rm few}|\log s_{\rm resonant}|$.

Three of the nineteen planets with mass ratios smaller than $q=1.7\times
10^{-4}$ were found in “Hollywood” events, for which the cross-section for
light-curve anomalies is set by the source size rather than the caustic size.
For these planets $s\gg s_{\rm resonant}$. The expected yield for such events
is likely to be much lower than for moderate magnification events. Typical
source sizes are $\rho\sim 0.01$, and thus, the cross-section is a factor of
$\sim 10$ smaller than for moderate magnification events. At the same time,
events with giant sources represent only a fraction of all microlensing
events, making intensive work on such events tractable.

Hence, the focus for discovering planets with $\log q\lesssim-4$ should be on
moderate magnification events and events with giant sources. This search could
be conducted within existing survey data or be supplemented by a followup
campaign. Even though OGLE-2019-BLG-0960Lb could be recovered from survey data
alone (Section 6.1), this recovery was aided by the special geometry of the
light curve (Jung et al., 2020) and suggests that similar, but shorter and
potentially more numerous, perturbations would be missed in the low-cadence
survey fields. For followup observations, Abe et al. (2013) previously
suggested that the focus for small planets should be events with $50<A_{\rm
max}<200$. Our investigation shows that events should be monitored for the
full time that they have $A>5$, which suggests this focus should be extended
to events with peak magnification $A_{\rm max}>10$ or even smaller. If
additional resources are available, followup observations could be further
extended to include events with giant sources. This strategy will maximize the
number of small planets found and enable a robust measurement of the mass-
ratio distribution of microlensing planets with $q<q_{\rm br}$.

Finally, OGLE-2019-BLG-0960 and many of the other planets in moderate
magnification events suffer from a degeneracy that results in two solutions
for the light curve: one with $s>1$ and one with $s<1$. For this event (and
most small planets), such degeneracies have little effect on the
interpretation of the separation of the planet from the host star, because
$s\sim 1$. However, we showed in Section 6.3 that it does have some effect on
the value of $q$. This is contrary to the expectation of Griest & Safizadeh
(1998) that $q$ should be very similar under the “close”/“wide” degeneracy,
likely because $s\sim 1$ is very far from the regime in which this degeneracy
was derived. It seems likely that the origin of these degenerate solutions
arises from some previously undiscovered symmetry in the lens equation of
which the classical “close”/“wide” degeneracy may be a limiting case. However,
truly understanding this symmetry requires going beyond the purely empirical
investigation carried out in this paper.

Work by JCY was supported by JPL grant 1571564. W.Z. and S.M. acknowledge
support by the National Science Foundation of China (Grant No. 11821303 and
11761131004). This research has made use of the KMTNet system operated by the
Korea Astronomy and Space Science Institute (KASI) and the data were obtained
at three host sites of CTIO in Chile, SAAO in South Africa, and SSO in
Australia. The OGLE has received funding from the National Science Centre,
Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU. This research uses data
obtained through the Telescope Access Program (TAP), which has been funded by
the National Astronomical Observatories of China, the Chinese Academy of
Sciences, and the Special Fund for Astronomy from the Ministry of Finance. The
MOA project is supported by JSPS KAKENHI Grant Number JSPS24253004,
JSPS26247023, JSPS23340064, JSPS15H00781, JP16H06287, and JP17H02871. This
work is based (in part) on observations made with the Spitzer Space Telescope,
which is operated by the Jet Propulsion Laboratory, California Institute of
Technology under a contract with NASA. Support for this work was provided by
NASA through an award issued by JPL/Caltech. Work by AG was supported by
AST-1516842 and by JPL grant 1500811. AG received support from the European
Research Council under the European Unions Seventh Framework Programme (FP 7)
ERC Grant Agreement n. [321035]. Work by CH was supported by the grants of
National Research Foundation of Korea (2019R1A2C2085965 and 2020R1A4A2002885).
Wei Zhu was supported by the Beatrice and Vincent Tremaine Fellowship at CITA.
YT acknowledges the support of DFG priority program SPP 1992 “Exploring the
Diversity of Extrasolar Planets” (WA 1047/11-1). This research has made use of
the NASA Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program.

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Table 1: Data with corresponding data reduction method and rescaling factors

Collaboration | Site | Filter | Coverage (${\rm HJD}^{\prime}$) | $N_{\rm data}$ | Reduction Method | $k$ | $e_{\rm min}$
---|---|---|---|---|---|---|---
OGLE |  | $I$ | 7799.9 – 8763.6 | 488 | Woźniak (2000) | 1.27 | 0.000
OGLE |  | $V$ | 7851.9 – 8723.6 | 26 | Woźniak (2000) | … | …
MOA |  | Red | 8537.2 – 8785.9 | 341 | Bond et al. (2001) | 1.14 | 0.005
KMTNet | SSO | $I$ | 8682.0 – 8692.0 | 19 | pySIS1 | 0.80 | 0.000
| CTIO | $I$ | 8616.7 – 8777.6 | 312 | pySIS | 1.33 | 0.007
| SAAO | $I$ | 8616.4 – 8777.3 | 223 | pySIS | 1.46 | 0.000
LCO | SSO01 | $i$ | 8684.0 – 8690.2 | 147 | ISIS2 | 2.03 | 0.005
| SSO02 | $i$ | 8688.1 – 8691.9 | 46 | ISIS | 1.04 | 0.000
$\mu$FUN | CTIO | $i$ | 8684.6 – 8692.7 | 23 | ISIS | … | …
$\mu$FUN | SAAO | $i$ | 8685.4 – 9690.2 | 91 | ISIS | 0.86 | 0.004
$\mu$FUN | CT13 | $I$ | 8678.5 – 8692.7 | 65 | DoPHOT3 | 1.12 | 0.000
$\mu$FUN | CT13 | $H$ | 8678.5 – 8692.7 | 343 | DoPHOT | … | …
$\mu$FUN | AO | 540–700 nm | 8685.9 – 8686.9 | 51 | DoPHOT | 1.29 | 0.000
$\mu$FUN | FCO | unfiltered | 8685.9 – 8686.9 | 75 | DoPHOT | 0.82 | 0.000
$\mu$FUN | Kumeu | 540–700 nm | 8685.9 – 8687.9 | 108 | DoPHOT | 1.29 | 0.000
Spitzer |  | $L$ | 8685.2 – 8712.0 | 25 | Calchi Novati et al. (2015) | 1.98 | 0.000

1 Albrow et al. (2009)
2 Alard & Lupton (1998); Alard (2000); Zang et al. (2018)
3 Schechter et al. (1993)

Table 2: Parameters for 2L1S models using all ground-based data

Model | $\chi^{2}/dof$ | $t_{0}$ | $u_{0}$ | $t_{\rm E}$ | $s$ | $q$ | $\alpha$ | $\rho$ | $\pi_{\rm E,N}$ | $\pi_{\rm E,E}$ | $f_{\rm S,I}$ | $f_{\rm B,I}$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (${\rm HJD}^{\prime}$) |  | (d) |  | $(\times 10^{-5})$ | (rad) | $(\times 10^{-4})$ |  |  |  |
$(s>1)$ : |  |  |  |  |  |  |  |  |  |  |  |
Static | $2099.2/1935$ | $8686.4484$ | $0.0060$ | $61.5$ | $1.028$ | $1.41$ | $0.270$ | $3.20$ | … | … | $0.1781$ | $0.5391$
|  | $0.0005$ | $0.0001$ | $1.4$ | $0.001$ | $0.14$ | $0.001$ | $0.17$ | … | … | $0.0042$ | $0.0037$
Parallax, | $1934.1/1933$ | $8686.4490$ | $0.0061$ | $61.8$ | $1.028$ | $1.43$ | $0.273$ | $3.23$ | $0.395$ | $-0.393$ | $0.1773$ | $0.5385$
($u_{0}>0$) |  | $0.0006$ | $0.0001$ | $1.4$ | $0.001$ | $0.14$ | $0.001$ | $0.18$ | $0.155$ | $0.039$ | $0.0041$ | $0.0036$
Parallax, | $1932.0/1933$ | $8686.4496$ | $-0.0061$ | $61.4$ | $1.029$ | $1.48$ | $-0.272$ | $3.29$ | $-0.351$ | $-0.313$ | $0.1784$ | $0.5380$
($u_{0}<0$) |  | $0.0006$ | $0.0001$ | $1.4$ | $0.001$ | $0.14$ | $0.001$ | $0.16$ | $0.173$ | $0.027$ | $0.0041$ | $0.0036$
$(s<1)$ : |  |  |  |  |  |  |  |  |  |  |  |
Static | $2101.2/1935$ | $8686.4485$ | $0.0060$ | $61.9$ | $0.997$ | $1.23$ | $0.269$ | $2.97$ | … | … | $0.1769$ | $0.5401$
|  | $0.0005$ | $0.0001$ | $1.3$ | $0.001$ | $0.06$ | $0.001$ | $0.09$ | … | … | $0.0040$ | $0.0040$
Parallax, | $1933.9/1933$ | $8686.4487$ | $0.0059$ | $63.0$ | $0.996$ | $1.27$ | $0.272$ | $2.97$ | $0.464$ | $-0.405$ | $0.1736$ | $0.5418$
($u_{0}>0$) |  | $0.0005$ | $0.0001$ | $1.4$ | $0.001$ | $0.07$ | $0.001$ | $0.10$ | $0.146$ | $0.039$ | $0.0042$ | $0.0037$
Parallax, | $1933.0/1933$ | $8686.4494$ | $-0.0060$ | $62.0$ | $0.997$ | $1.27$ | $-0.271$ | $3.01$ | $-0.440$ | $-0.304$ | $0.1766$ | $0.5397$
($u_{0}<0$) |  | $0.0005$ | $0.0001$ | $1.3$ | $0.001$ | $0.07$ | $0.001$ | $0.10$ | $0.161$ | $0.027$ | $0.0039$ | $0.0034$

Note: Uncertainties for each parameter are given in the second line for each
solution.

Table 3: Parameters for 1L2S models using all ground-based data and ground-based survey data Model | $\chi^{2}/dof$ | $t_{0,1}$ | $t_{0,2}$ | $u_{0,1}$ | $u_{0,2}$ | $t_{\rm E}$ | $\rho_{1}$ | $\rho_{2}$ | $\pi_{\rm E,N}$ | $\pi_{\rm E,E}$ | $q_{F,I}$ | $f_{\rm S,I}$ | $f_{\rm B,I}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  (${\rm HJD}^{\prime}$) |  (${\rm HJD}^{\prime}$) |  |  | (d) | $(\times 10^{-4})$ | $(\times 10^{-4})$ |  |  |  |  |
All, | $2952.0/1931$ | $8686.4425$ | $8687.8064$ | $0.0056$ | $0.0003$ | $66.1$ | $9.5$ | $3.1$ | $0.341$ | $-0.308$ | $0.0054$ | $0.1645$ | $0.5501$
($u_{0}>0$) |  | $0.0005$ | $0.0012$ | $0.0004$ | $0.0002$ | $1.4$ | $33.2$ | $92.0$ | $0.174$ | $0.059$ | $0.0028$ | $0.0024$ | $0.0028$
All, | $2951.4/1931$ | $8686.4431$ | $8687.8076$ | $-0.0056$ | $-0.0003$ | $65.6$ | $6.4$ | $3.0$ | $-0.234$ | $-0.233$ | $0.0053$ | $0.1656$ | $0.5499$
($u_{0}<0$) |  | $0.0005$ | $0.0011$ | $0.0004$ | $0.0002$ | $1.3$ | $30.2$ | $82.1$ | $0.370$ | $0.079$ | $0.0025$ | $0.0022$ | $0.0026$
Survey, | $1616.0/1323$ | $8686.4406$ | $8687.7951$ | $0.0065$ | $0.0000$ | $66.3$ | $61.4$ | $0.1$ | $-0.051$ | $-0.224$ | $0.0001$ | $0.1641$ | $0.5510$
($u_{0}>0$) |  | $0.0017$ | $0.0116$ | $0.0003$ | $0.3024$ | $0.5$ | $13.4$ | $273.3$ | $0.299$ | $0.053$ | $0.0009$ | $0.0166$ | $0.0172$
Survey, | $1614.9/1323$ | $8686.4386$ | $8687.7953$ | $-0.0057$ | $-0.0000$ | $67.2$ | $33.4$ | $0.1$ | $-0.622$ | $-0.244$ | $0.0001$ | $0.1614$ | $0.5545$
($u_{0}<0$) |  | $0.0018$ | $0.0126$ | $0.0003$ | $0.3401$ | $0.5$ | $17.1$ | $265.0$ | $0.387$ | $0.070$ | $0.0008$ | $0.0171$ | $0.0180$

Table 4: Physical parameters Model | $\theta_{\rm E}$ | $\pi_{\rm E}$ | $\mu_{\rm rel}$ | $M_{\rm L}$ | $D_{\rm L}$ | $m_{p}$ | $a_{\perp}$
---|---|---|---|---|---|---|---
| (mas) |  | (${\rm mas\,yr^{-1}}$) | ($M_{\odot}$) | (kpc) | ($M_{\oplus}$) | (AU)
$(s>1)$ : |  |  |  |  |  |  |
($u_{0}>0$) | $1.93\pm 0.13$ | $0.557\pm 0.125$ | $11.4\pm 0.8$ | $0.43\pm 0.10$ | $0.83\pm 0.17$ | $2.0\pm 0.5$ | $1.65\pm 0.35$
($u_{0}<0$) | $1.90\pm 0.12$ | $0.470\pm 0.114$ | $11.3\pm 0.8$ | $0.50\pm 0.12$ | $0.98\pm 0.21$ | $2.5\pm 0.6$ | $1.90\pm 0.43$
$(s<1)$ : |  |  |  |  |  |  |
($u_{0}>0$) | $2.09\pm 0.11$ | $0.616\pm 0.126$ | $12.1\pm 0.7$ | $0.42\pm 0.09$ | $0.70\pm 0.13$ | $1.8\pm 0.4$ | $1.45\pm 0.28$
($u_{0}<0$) | $2.07\pm 0.11$ | $0.535\pm 0.121$ | $12.2\pm 0.7$ | $0.48\pm 0.11$ | $0.81\pm 0.17$ | $2.0\pm 0.5$ | $1.67\pm 0.37$

Table 5: Best-fit parameters for the four solutions using only ground-based survey data Model | $\chi^{2}/dof$ | $t_{0}$ | $u_{0}$ | $t_{\rm E}$ | $s$ | $q$ | $\alpha$ | $\rho$ | $\pi_{\rm E,N}$ | $\pi_{\rm E,E}$ | $f_{\rm S,I}$ | $f_{\rm B,I}$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (${\rm HJD}^{\prime}$) |  | (d) |  | $(\times 10^{-5})$ | (rad) | $(\times 10^{-4})$ |  |  |  |
$(s>1)$ : |  |  |  |  |  |  |  |  |  |  |  |
Single | $1325.3/1324$ | $8686.4504$ | $0.0059$ | $62.6$ | $1.027$ | $1.41$ | $0.271$ | $3.12$ | $0.093$ | $-0.352$ | $0.1747$ | $0.5408$
|  | $0.0022$ | $0.0002$ | $1.5$ | $0.007$ | $0.16$ | $0.002$ | $0.28$ | $0.288$ | $0.046$ | $0.0043$ | $0.0037$
Double | $1323.8/1324$ | $8686.4513$ | $0.0060$ | $62.0$ | $1.028$ | $1.69$ | $0.269$ | $1.75$ | $0.032$ | $-0.334$ | $0.1767$ | $0.5390$
|  | $0.0021$ | $0.0002$ | $1.1$ | $0.004$ | $0.32$ | $0.003$ | $0.63$ | $0.293$ | $0.047$ | $0.0033$ | $0.0030$
$(s<1)$ : |  |  |  |  |  |  |  |  |  |  |  |
Single | $1325.2/1324$ | $8686.4506$ | $0.0059$ | $62.6$ | $0.994$ | $1.43$ | $0.269$ | $3.26$ | $0.081$ | $-0.351$ | $0.1748$ | $0.5407$
|  | $0.0016$ | $0.0002$ | $1.4$ | $0.007$ | $0.20$ | $0.002$ | $0.25$ | $0.351$ | $0.042$ | $0.0042$ | $0.0037$
Double | $1323.3/1324$ | $8686.4522$ | $0.0060$ | $61.3$ | $0.994$ | $1.89$ | $0.264$ | $0.78$ | $0.077$ | $-0.332$ | $0.1787$ | $0.5373$
|  | $0.0027$ | $0.0001$ | $1.1$ | $0.007$ | $0.37$ | $0.004$ | $0.30$ | $0.301$ | $0.046$ | $0.0031$ | $0.0028$