# OGLE-2018-BLG-1185b : A Low-Mass Microlensing Planet Orbiting a Low-Mass
Dwarf

Iona Kondo  MOA collaboration  Department of Earth and Space Science, Graduate
School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Jennifer
C. Yee  The SpitzerTeam  The KMTNet Collaboration  Center for Astrophysics $|$
Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA  David P.
Bennett  MOA collaboration  Code 667, NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA  Department of Astronomy, University of Maryland,
College Park, MD 20742, USA  Takahiro Sumi  MOA collaboration  Department of
Earth and Space Science, Graduate School of Science, Osaka University,
Toyonaka, Osaka 560-0043, Japan  Naoki Koshimoto  MOA collaboration
Department of Astronomy, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Ian A. Bond  MOA collaboration
Institute of Natural and Mathematical Sciences, Massey University, Auckland
0745, New Zealand  Andrew Gould  The SpitzerTeam  The KMTNet Collaboration
Max-Planck-Institute for Astronomy, Konigstuhl 17, 69117 Heidelberg, Germany
Department of Astronomy Ohio State University, 140 W. 18th Ave., Columbus, OH
43210, USA  Andrzej Udalski  OGLE collaboration  Warsaw University
Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  Yossi Shvartzvald
The SpitzerTeam  The KMTNet Collaboration  Department of Particle Physics and
Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel  Youn Kil
Jung  The KMTNet Collaboration  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  Weicheng
Zang  The SpitzerTeam  The KMTNet Collaboration  Department of Astronomy,
Tsinghua University, Beijing 100084, China  Valerio Bozza  MiNDSTEp
Collaboration  The ROME/REA project team  Dipartimento di Fisica “E.R.
Caianiello”, Università di Salerno, Via Giovanni Paolo II 132, 84084,
Fisciano, Italy  Istituto Nazionale di Fisica Nucleare, Sezione di Napoli,
Napoli, Italy  Etienne Bachelet  The ROME/REA project team  MiNDSTEp
Collaboration  Las Cumbres Observatory, 6740 Cortona Drive, Suite 102,93117
Goleta, CA, USA  Markus P.G. Hundertmark  The ROME/REA project team  MiNDSTEp
Collaboration  Astronomisches Rechen-Institut, Zentrum für Astronomie der
Universität Heidelberg (ZAH), 69120 Heidelberg, Germany  Nicholas J.
Rattenbury  MOA collaboration  Department of Physics, University of Auckland,
Private Bag 92019, Auckland, New Zealand  F. Abe  Institute for Space-Earth
Environmental Research, Nagoya University, Nagoya 464-8601, Japan  R. Barry
Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA  A.
Bhattacharya  Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771,
USA  Department of Astronomy, University of Maryland, College Park, MD 20742,
USA  M. Donachie  Department of Physics, University of Auckland, Private Bag
92019, Auckland, New Zealand  A. 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  H. Fujii  Institute for Space-
Earth Environmental Research, Nagoya University, Nagoya 464-8601, Japan  Y.
Hirao  Department of Earth and Space Science, Graduate School of Science,
Osaka University, Toyonaka, Osaka 560-0043, Japan  S. 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  Y. Itow  Institute for Space-Earth Environmental Research, Nagoya
University, Nagoya 464-8601, Japan  R. Kirikawa  Department of Earth and Space
Science, Graduate School of Science, Osaka University, Toyonaka, Osaka
560-0043, Japan  M. C. A. Li  Department of Physics, University of Auckland,
Private Bag 92019, Auckland, New Zealand  Y. Matsubara  Institute for Space-
Earth Environmental Research, Nagoya University, Nagoya 464-8601, Japan  S.
Miyazaki  Department of Earth and Space Science, Graduate School of Science,
Osaka University, Toyonaka, Osaka 560-0043, Japan  Y. Muraki  Institute for
Space-Earth Environmental Research, Nagoya University, Nagoya 464-8601, Japan
G. Olmschenk  Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771,
USA  C. Ranc  Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de
Paris, 98 bis bd Arago, 75014 Paris, France  Y. Satoh  Department of Earth and
Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka
560-0043, Japan  H. Shoji  Department of Earth and Space Science, Graduate
School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  D.
Suzuki  Department of Earth and Space Science, Graduate School of Science,
Osaka University, Toyonaka, Osaka 560-0043, Japan  Y. Tanaka  Department of
Earth and Space Science, Graduate School of Science, Osaka University,
Toyonaka, Osaka 560-0043, Japan  P. J. Tristram  University of Canterbury Mt.
John Observatory, P.O. Box 56, Lake Tekapo 8770, New Zealand  T. Yamawaki
Department of Earth and Space Science, Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan  A. Yonehara  Department of
Physics, Faculty of Science, Kyoto Sangyo University, 603-8555 Kyoto, Japan
P. Mróz  Division of Physics, Mathematics, and Astronomy, California Institute
of Technology, Pasadena, CA 91125, USA  R. Poleski  Warsaw University
Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  J. Skowron  Warsaw
University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  M. K.
Szymański  Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa,
Poland  I. Soszyński  Warsaw University Observatory, Al. Ujazdowskie 4, 00-478
Warszawa, Poland  S. Kozłowski  Warsaw University Observatory, Al. Ujazdowskie
4, 00-478 Warszawa, Poland  P. Pietrukowicz  Warsaw University Observatory,
Al. Ujazdowskie 4, 00-478 Warszawa, Poland  K. Ulaczyk  Warsaw University
Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  Department of
Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK  K. A.
Rybicki  Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa,
Poland  P. Iwanek  Warsaw University Observatory, Al. Ujazdowskie 4, 00-478
Warszawa, Poland  M. Wrona  Warsaw University Observatory, Al. Ujazdowskie 4,
00-478 Warszawa, Poland  M. D. Albrow  University of Canterbury, Department of
Physics and Astronomy, Private Bag 4800, Christchurch 8020, New Zealand  S.-J.
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  C. Han  Department of Physics,
Chungbuk National University, Cheongju 28644, Republic of Korea  K.-H. Hwang
Korea Astronomy and Space Science Institute, Daejon 34055, Republic of Korea
H.-W. Kim  Korea Astronomy and Space Science Institute, Daejon 34055, Republic
of Korea  I.-G. Shin  Korea Astronomy and Space Science Institute, Daejon
34055, Republic of Korea  S.-M. 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  D.-J. Kim  Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea  S.-L.
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  C.-U. Lee  Korea Astronomy and
Space Science Institute, Daejon 34055, Republic of Korea  D.-J. Lee  Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea  Y. 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  B.-G. 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  R. W.
Pogge  Department of Astronomy, Ohio State University, 140 W. 18th Ave.,
Columbus, OH 43210, USA  Y.-H. Ryu  Korea Astronomy and Space Science
Institute, Daejon 34055, Republic of Korea  C. A. Beichman  IPAC, Mail Code
100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  G. Bryden
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA  S. Calchi Novati  IPAC, Mail Code 100-22,
Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  S. Carey  IPAC,
Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA
B. S. Gaudi  Department of Astronomy, The Ohio State University, 140 W. 18th
Ave., Columbus, OH 43210, USA  C. B. Henderson  IPAC, Mail Code 100-22,
Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  W. Zhu  Canadian
Institute for Theoretical Astrophysics, University of Toronto, 60 St George
Street, Toronto, ON M5S 3H8, Canada  D. Maoz  School of Physics and Astronomy,
Tel-Aviv University, Tel-Aviv 6997801, Israel  M. T. Penny  Department of
Physics & Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001
M. Dominik  University of St Andrews, Centre for Exoplanet Science, SUPA
School of Physics & Astronomy, North Haugh, St Andrews, KY16 9SS, United
Kingdom  U. G. Jørgensen  Centre for ExoLife Sciences, Niels Bohr Institute,
University of Copenhagen Øster Voldgade 5, 1350 - Copenhagen, Denmark  P.
Longa-Pe ̃na  Centro de Astronomía (CITEVA), Universidad de Antofagasta, Avda.
U. de Antofagasta 02800, Antofagasta, Chile  N. Peixinho  Univ Coimbra,
Instituto de Astrofísica e Ciências do Espaço, OGAUC, R. do Observatório s/n,
3040-004 Coimbra, Portugal  S. Sajadian  Department of Physics, Isfahan
University of Technology, Isfahan, Iran  J. Skottfelt  Centre for Electronic
Imaging, Department of Physical Sciences, The Open University, Milton Keynes,
MK7 6AA, UK  C. Snodgrass  Institute for Astronomy, University of Edinburgh,
Royal Observatory, Edinburgh EH9 3HJ, UK  J. Tregloan-Reed  Instituto de
Investigación en Astronomia y Ciencias Planetarias, Universidad de Atacama,
Copiapó, Atacama, Chile  M. J. Burgdorf  Universität Hamburg, Faculty of
Mathematics, Informatics and Natural Sciences, Department of Earth Sciences,
Meteorological Institute, Bundesstraße 55, 20146 Hamburg, Germany  J.
Campbell-White  SUPA, School of Science and Engineering, University of Dundee,
Nethergate, Dundee DD1 4HN, UK  S. Dib  Centre for ExoLife Sciences, Niels
Bohr Institute, University of Copenhagen Øster Voldgade 5, 1350 - Copenhagen,
Denmark  Max Planck Institute for Astronomy, Königstuhl 17, 69117, Heidelberg,
Germany  Y. I. Fujii  Institute for Advanced Research, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan  Department of Physics, Nagoya
University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan  Niels Bohr Institute
& Centre for Star and Planet Formation, University of Copenhagen Øster
Voldgade 5, 1350 - Copenhagen, Denmark  Centre for ExoLife Sciences, Niels
Bohr Institute, University of Copenhagen Øster Voldgade 5, 1350 - Copenhagen,
Denmark  Graduate School of Human and Environmental Studies, Kyoto University,
Yoshida-Nihonmatsu, Sakyo, Kyoto 606-8501, Japan  T. C. Hinse  Institute of
Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus
University, Grudziadzka 5, 87-100 Torun, Poland  Chungnam National University,
Department of Astronomy and Space Science, 34134 Daejeon, Republic of Korea
E. Khalouei  Department of Physics, Sharif University of Technology, P. O. Box
11365-9161, Tehran, Iran  S. Rahvar  Department of Physics, Sharif University
of Technology, P. O. Box 11365-9161, Tehran, Iran  M. Rabus  Las Cumbres
Observatory Global Telescope, 6740 Cortona Dr, Suite 102, Goleta, CA 93111,
USA  Department of Physics, University of California, Santa Barbara, CA
93106-9530, USA  J. Southworth  Astrophysics Group, Keele University,
Staffordshire, ST5 5BG, UK  Y. Tsapras  Zentrum für Astronomie der Universität
Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr. 12-14, 69120
Heidelberg, Germany  R. A. Street  Las Cumbres Observatory Global Telescope
Network, 6740 Cortona Drive, suite 102, Goleta, CA 93117, USA  D. M. Bramich
Center for Space Science, NYUAD Institute, New York University Abu Dhabi, PO
Box 129188, Saadiyat Island, Abu Dhabi, UAE.  Division of Science, New York
University Abu Dhabi, PO Box 129188, Saadiyat Island, Abu Dhabi, UAE.
Division of Engineering, New York University Abu Dhabi, PO Box 129188,
Saadiyat Island, Abu Dhabi, UAE.  A. Cassan  Institut d’Astrophysique de
Paris, Sorbonne Université, CNRS, UMR 7095, 98 bis bd Arago, 75014 Paris,
France  K. Horne  SUPA, School of Physics & Astronomy, University of St
Andrews, North Haugh, St Andrews KY16 9SS, UK  J. Wambsganss  Zentrum für
Astronomie der Universität Heidelberg, Astronomisches Rechen-Institut,
Mönchhofstr. 12-14, 69120 Heidelberg, Germany  S. Mao  National Astronomical
Observatories, Chinese Academy of Sciences, 100012 Beijing, China  A. Saha
National Optical Astronomy Observatory, 950 North Cherry Ave., Tucson, AZ
85719, USA

###### Abstract

We report the analysis of planetary microlensing event OGLE-2018-BLG-1185,
which was observed by a large number of ground-based telescopes and by the
$Spitzer$ Space Telescope. The ground-based light curve indicates a low
planet-host star mass ratio of $q=(6.9\pm 0.2)\times 10^{-5}$, which is near
the peak of the wide-orbit exoplanet mass-ratio distribution. We estimate the
host star and planet masses with a Bayesian analysis using the measured
angular Einstein radius under the assumption that stars of all masses have an
equal probability to host this planet. The flux variation observed by
$Spitzer$ was marginal, but still places a constraint on the microlens
parallax. Imposing a conservative constraint that this flux variation should
be $\Delta f_{\rm Spz}<4$ instrumental flux units indicates a host mass of
$M_{\rm host}=0.37^{+0.35}_{-0.21}\ M_{\odot}$ and a planet mass of $m_{\rm
p}=8.4^{+7.9}_{-4.7}\ M_{\oplus}$. A Bayesian analysis including the full
parallax constraint from Spitzer suggests smaller host star and planet masses
of $M_{\rm host}=0.091^{+0.064}_{-0.018}\ M_{\odot}$ and $m_{\rm
p}=2.1^{+1.5}_{-0.4}\ M_{\oplus}$, respectively. Future high-resolution
imaging observations with HST or ELTs could distinguish between these two
scenarios and help to reveal the planetary system properties in more detail.

Gravitational microlensing (672) — Gravitational microlensing exoplanet
detection (2147) — Satellite microlensing parallax (2148)

##  1 Introduction

The gravitational microlensing method has a unique sensitivity to low-mass
planets (Bennett & Rhie, 1996) beyond the snow line of the host star (Gould &
Loeb, 1992), where the core accretion theory predicts that planet formation is
most efficient (Lissauer, 1993; Pollack et al., 1996). The Microlensing
Observations in Astrophysics (MOA; Bond et al., 2001; Sumi et al., 2003)
collaboration presented the most complete statistical analysis of planets
found by microlensing to date and the best measurement of the planet
distribution beyond the snow line in Suzuki et al. (2016). They found that the
mass-ratio distribution from the $2007-2012$ MOA-II microlensing survey
combined with earlier samples (Gould et al., 2010; Cassan et al., 2012) is
well fitted by a broken power-law model.

Their result shows the mass-ratio distribution peaks at $q_{\rm
br}=(6.7^{+9.0}_{-1.8})\times 10^{-5}$ with power-law slopes of
$n=-0.85^{+0.12}_{-0.13}$ and $p=2.6^{+4.2}_{-2.1}$ above and below $q_{\rm
br}$, respectively111These values are the median and 68% confidence level by
the Markov Chain Monte Carlo analysis with the thirty planet sample, which are
given in Table 5 of Suzuki et al. (2016). So the 1$\sigma$ range of the mass-
ratio distribution peaks is roughly $q_{\rm br}\sim(0.5-2)\times 10^{-4}$. At
the same time, they also show that the best fitting parameters are $q_{\rm
br}=1.65\times 10^{-4}$ with power-law slopes of $n=-0.92$ and $p=0.47$ in
Table 4 of Suzuki et al. (2016).. This result is consistent with previous
microlensing analyses which suggest that Neptune mass-ratio planets are more
common than larger gas giants (Gould et al., 2006; Sumi et al., 2010) and
further indicates that Neptune mass-ratio planets are, in fact, the most
common type of planet (large or small) in wide orbits.

Additionally, Suzuki et al. (2018) reveals a disagreement between the measured
mass ratio distribution in Suzuki et al. (2016) and the predictions of the
runaway gas accretion scenario (Ida & Lin, 2004), which is part of the
standard core accretion theory. Population synthesis models based on core
accretion, including runaway gas accretion, predict too few planets in the
mass range of approximately $20-80M_{\oplus}$ compared to those inferred from
microlensing observations. Similar tension is indicated by ALMA observations.
Nayakshin et al. (2019) compared wide orbit (9-99 au) planet candidates with
masses of $0.01M_{\rm Jup}$ to a few $M_{\rm Jup}$ suggested by ALMA proto-
planetary disk observations to a population synthesis prediction from the
runaway gas accretion scenario. They found that the scenario predicts fewer
sub-Jovian planets than those inferred from the ALMA observation. 3D
hydrodynamical simulations of proto-planetary disks do not support the runaway
gas accretion scenario either (Lambrechts et al., 2019).

The peak position of the mass-ratio function and its slope at small mass
ratios are uncertain due to the lack of planets with mass ratios of
$q<5.8\times 10^{-5}$ in the Suzuki et al. (2016) sample. Udalski et al.
(2018) and Jung et al. (2019b) used samples of published planets to refine the
estimates of the peak and the low mass-ratio slope of the mass-ratio function.
Udalski et al. (2018) confirmed the turnover shown in Suzuki et al. (2016) and
obtained the slope index on the low-mass regime, $p\sim 0.73$, using seven
published planets with $q<1\times 10^{-4}$. Jung et al. (2019b) found $q_{\rm
br}\simeq 0.55\times 10^{-4}$ using 15 published planets with low mass ratio
($q<3\times 10^{-4}$). The Jung et al. (2019b) study was subject to
“publication bias”. That is, the planets were not part of a well-defined
statistical sample. Instead, these planets were selected for publication for
reasons that are not well characterized. Nevertheless, the authors make the
case that this publication bias should not be large enough to invalidate their
results. By contrast, the Udalski et al. (2018) study only made the implicit
assumption that all planets with $q<1\times 10^{-4}$ (and greater than that of
the actual published planet) would have been published. If this is true (which
is very likely), the study is not subject to publication bias.

A more definitive improvement of the Suzuki et al. (2016) mass-ratio function
can be obtained with an extension of the MOA-II statistical sample to include
additional microlensing seasons (Suzuki et al., in preparation). The low mass-
ratio planet analyzed in this paper, OGLE-2018-BLG-1185Lb, will be part of
that extended sample, and it will contribute to an improved characterization
of the low end of the wide orbit exoplanet mass-ratio function.

The statistical analysis of the wide-orbit planet population can also be
improved by including information on the lens physical parameters, such as the
lens mass, $M_{\rm L}$, and the distance to the lens star, $D_{\rm L}$. While
the lens planet-host mass ratios, $q$, are usually well constrained from the
light-curve modeling, we need at least two mass-distance relations in order to
derive $M_{\rm L}$ and $D_{\rm L}$ directly. There are three observables that
can yield mass-distance relations: finite source effects, microlens parallax
effects, and direct detection of the lens flux.

In recent years, lens flux detection by high-resolution imaging follow-up
observations (such as by the Hubble Space Telescope (HST) or Keck) has been
done for several microlens planetary systems after the lens and the source are
separated enough to be detected (Bennett et al., 2006, 2007, 2015, 2020;
Batista et al., 2014, 2015; Bhattacharya et al., 2017, 2018; Koshimoto et al.,
2017; Vandorou et al., 2020). However, the required separation for resolving
the lens and source depends on their relative brightnesses, and even if they
are comparable in brightness, it typically takes a few years for them to
separate sufficiently.

If both the Einstein radius $\theta_{\rm E}$ from the finite source effect and
the microlens parallax $\pi_{\rm E}$ from the parallax effect are measured, we
can derive two mass-distance relations as follows,

$M_{\rm L}=\frac{c^{2}}{4G}{\theta_{\rm E}}^{2}\frac{D_{\rm S}D_{\rm
L}}{D_{\rm S}-D_{\rm L}}=\frac{c^{2}}{4G}\frac{{\rm au}}{{\pi_{\rm
E}}^{2}}\frac{D_{\rm S}-D_{\rm L}}{D_{\rm S}D_{\rm L}},$ (1)

where $D_{\rm S}$ is the distance to the source (Gould, 1992, 2000). Finite
source effects are detected in most planetary-lens events through the
observation of a caustic crossing or a close approach to a caustic cusp, thus
enabling the measurement of $\theta_{\rm E}$.

The most common method for measuring the microlens parallax has been via the
effects of the motion of the observer, which is called the “orbital parallax
effect.” In order to detect the orbital parallax, the ratio of $t_{\rm E}$
(typically $t_{\rm E}$ is $\sim 30$ days) to Earth’s orbital period (365 days)
should be significant. Thus, we only measure the orbital parallax effect for
microlensing events with long durations and/or with relatively nearby lens
systems, yielding mass measurements in less than half of published
microlensing planetary systems.

The most effective method for routinely obtaining a microlens parallax
measurement is via the “satellite parallax effect” (Refsdal, 1966), which is
caused by the separation between two observers. Because the typical Einstein
radius projected onto the observer plane, $\tilde{r}_{\rm E}$, is about 10 au,
the satellite parallax effect can be measured for a wide range of microlenses
provided the separation between Earth and the satellite is about 1 au (as was
the case for Spitzer).

For the purpose of measuring the Galactic distribution of planets and making
mass measurements through the satellite parallax effect, the $Spitzer$
microlensing campaign was carried out from 2014–2019 (Gould, & Yee, 2013;
Gould et al., 2014, 2015a, 2015b, 2016, 2018). During the six-year program,
close to 1000 microlensing events were simultaneously observed from the ground
and by $Spitzer$, and there are 11 published222In addition Yee et al. (2021)
have submitted a paper on OGLE-2019-BLG-0960. planets with satellite parallax
measurements from $Spitzer$: OGLE-2014-BLG-0124Lb (Udalski et al., 2015b),
OGLE-2015-BLG-0966Lb (Street et al., 2016), OGLE-2016-BLG-1067Lb (Calchi
Novati et al., 2019), OGLE-2016-BLG-1195Lb (Shvartzvald et al., 2017),
OGLE-2016-BLG-1190Lb (Ryu et al., 2018), OGLE-2017-BLG-1140Lb (Calchi Novati
et al., 2018), TCP J05074264+2447555 (Nucita et al., 2018; Fukui et al., 2019;
Zang et al., 2020b), OGLE-2018-BLG- 0596Lb (Jung et al., 2019a),
KMT-2018-BLG-0029Lb (Gould et al., 2020), OGLE-2017-BLG-0406Lb (Hirao et al.,
2020), and OGLE-2018-BLG-0799Lb (Zang et al., 2020c). Comparing the planet
frequency in the disk to that in the bulge could probe the effects of the
different environments on the planet formation process.

Obvious correlated noise in the $Spitzer$ photometry was first noted by
Poleski et al. (2016) and Zhu et al. (2017), but those works did not expect
the systematic errors would have a significant effect on the parallax
measurements. Indeed, two comparisons of small, heterogeneous samples of
published $Spitzer$ microlensing events confirmed this expectation (Shan et
al., 2019; Zang et al., 2020a). However, a larger study (Koshimoto & Bennett,
2020) of the 50-event statistical sample of Zhu et al. (2017) indicated a
conflict between the $Spitzer$ microlensing parallax measurements and Galactic
models. They suggested that this conflict was probably caused by systematic
errors in the $Spitzer$ photometry. Based, in part, on the Koshimoto & Bennett
(2020) analysis, the $Spitzer$ microlensing team has made a greater effort to
understand these systematic errors, including obtaining baseline data in 2019
for many of the earlier planetary events. These additional baseline data
proved very useful in characterizing systematics in the $Spitzer$ photometry
for three previously published events (Gould et al., 2020; Hirao et al., 2020;
Zang et al., 2020c). Those analyses show that systematics in the Spitzer
photometry can be present at the level of 1–2 instrumental flux units, so
observed signals in the Spitzer photometry on those scales should be
interpreted with caution.

In this paper, we present the analysis of planetary microlensing event
OGLE-2018-BLG-1185, which was simultaneously observed by many ground-based
telescopes and by the $Spitzer$ Space Telescope. From the ground-based light-
curve analysis, the planet-host star mass ratio turns out to be very low,
$q\sim 6.9\times 10^{-5}$, which is thought to be near the peak of the wide-
orbit exoplanet mass-ratio distribution in Suzuki et al. (2016), Udalski et
al. (2018), and Jung et al. (2019b). Section 2 explains the observations and
the data reductions. Our ground-based light-curve modeling method and results
are shown in Section 3. In Section 4, we derive the angular Einstein radius
from the source magnitude and color and the finite source effect in order to
constrain the physical parameters of the planetary system. In Section 5, we
estimate the physical properties such as the host star and planet masses based
on the ground-based light curve alone by performing a Bayesian analysis using
the measured angular Einstein radius under the assumption that stars of all
masses have an equal probability to host this planet. We present our parallax
analysis including the $Spitzer$ data in Section 6. Finally, we discuss the
analysis and summarize our conclusions in Section 7.

##  2 Observations and Data Reductions

###  2.1 Ground-based Survey Observations

The microlensing event OGLE-2018-BLG-1185 was first discovered on 2018 July 7
(${\rm HJD^{\prime}}$ $=$ ${\rm HJD}-2450000$ $\sim$ $8306$), at the J2000
equatorial coordinates $({\rm RA,Dec})$ = ($17^{h}$ $59^{m}$
$10^{s}$.$26,-27^{\circ}$ $50^{\prime}$ $06^{\prime\prime}.3$) corresponding
to Galactic coordinates $(l,b)$ = $(2.465^{\circ},-2.004^{\circ})$ by the
Optical Gravitational Lensing Experiment (OGLE; Udalski, 2003) collaboration.
The OGLE collaboration conducts a microlensing survey using the 1.3m Warsaw
telescope with a 1.4 ${\rm deg^{2}}$ field-of-view (FOV) CCD camera at Las
Campanas Observatory in Chile and distributes alerts of the discovery of the
microlensing events by the OGLE-IV Early Warning System (Udalski et al., 1994;
Udalski, 2003). The event is located in the OGLE-IV field BLG 504, which is
observed with a cadence of one observation per hour.

The event was also discovered independently on 2018 July 9 by the MOA
collaboration and identified as MOA-2018-BLG-228 by the MOA alert system (Bond
et al., 2001). The MOA collaboration conducts a microlensing exoplanet survey
toward the Galactic bulge using the 1.8m MOA-II telescope with a 2.2 ${\rm
deg^{2}}$ wide FOV CCD-camera, MOA-cam3 (Sako et al., 2008) at the University
of Canterbury’s Mt. John Observatory in New Zealand. The MOA survey uses a
custom wide-band filter referred to as $R_{\rm MOA}$, corresponding to the sum
of the Cousins $R$\- and $I$-bands and also uses a Johnson $V$-band filter.
The event is located in the MOA field gb10, which is observed at a high
cadence of one observation every 15 minutes.

The Korea Microlensing Telescope Network (KMTNet; Kim et al. 2016)
collaboration conducts a microlensing survey using three 1.6m telescopes each
with a 4.0 ${\rm deg^{2}}$ FOV CCD camera. The telescopes are located at Cerro
Tololo Interamerican Observatory in Chile (KMTC), South African Astronomical
Observatory in South Africa (KMTS), and Siding Spring Observatory in Australia
(KMTA). This event is located in an overlapping region between two fields
(KMTNet BLG03 and BLG43) and was identified by the KMTNet EventFinder (Kim et
al., 2018) as KMT-2018-BLG-1024.

###  2.2 Spitzer Observations

In order to construct statistical samples from the $Spitzer$ microlensing
campaign, Yee et al. (2015) established detailed protocols for the selection
and observational cadence of $Spitzer$ microlensing targets. On 2018 July 8
(${\rm HJD^{\prime}}\sim 8308.25$), OGLE-2018-BLG-1185 was selected as a
“Subjective, Immediate” (SI) target to be observed with the “objective”
cadence by the $Spitzer$ microlensing team. The selection as SI meant that
this event was observed even though it never met the objective criteria
established in Yee et al. (2015). The $Spitzer$ Space Telescope began to
observe this event on 2018 July 14 (${\rm HJD^{\prime}}\sim 8313.83$), which
was three days after the peak observed from the ground-based telescopes. The
“objective” cadence resulted in approximately one observation per day for the
remainder of the observing window (27 days total). These observations were
taken with the IRAC camera in the $3.6$ $\mu$m ($L$) band.

###  2.3 Ground-based Follow-up Observations

After the event was selected for Spitzer observations, some ground-based
follow-up observations were conducted. Microlensing Network for the Detection
of Small Terrestrial Exoplanets (MiNDSTEp) used the 1.54m Danish Telescope at
La Silla Observatory in Chile and the 0.6m telescope at Salerno University
Observatory in Italy. The Microlensing Follow Up Network ($\mu$FUN) used the
1.3m SMARTS telescope at CTIO in Chile. Las Cumbres Observatory (LCO; Brown et
al., 2013) used the 1.0m telescopes at CTIO in Chile, at SSO in Australia, and
at SAAO in South Africa, as a part of LCO-$Spitzer$ program. The ROME/REA team
(Tsapras et al., 2019) also used the 1.0m LCO robotic telescopes at CTIO in
Chile, at SSO in Australia, and at SAAO in South Africa. A summary of
observations from each telescope is given in Table 1.

###  2.4 Data Reduction

The OGLE, MOA, and KMTNet data were reduced using the OGLE Difference Image
Analysis (DIA) photometry pipeline (Udalski, 2003), the MOA DIA photometry
pipeline (Bond et al., 2001), and the KMTNet pySIS photometry pipeline (Albrow
et al., 2009), respectively. The MiNDSTEp data were reduced using DanDIA
(Bramich, 2008; Bramich et al., 2013). $\mu$FUN data were reduced using DoPHOT
(Schechter et al., 1993), and LCO data from the LCO-Spitzer program were
reduced using a modified ISIS package (Alard & Lupton, 1998; Alard, 2000; Zang
et al., 2018). The LCO data obtained by the ROME/REA team were reduced using a
customized version of the DanDIA photometry pipeline. The $Spitzer$ data were
reduced using the photometry algorithm described in Calchi Novati et al.
(2015).

It is known that the photometric error bars calculated by the data pipelines
can be underestimated (or more rarely overestimated). Various reasons, such as
observational conditions, can cause systematic errors. In order to get proper
errors of the parameters in the light-curve modeling, we empirically normalize
the error bars by using the standard method of Bennett et al. (2008). We use
the formula,

$\sigma^{\prime}_{i}=k\sqrt{\sigma^{2}_{i}+e^{2}_{\rm min}},$ (2)

where $\sigma^{\prime}_{i}$ is the $i$th renormalized error, $\sigma_{i}$ is
the $i$th error obtained from DIA, and $k$ and $e_{\rm min}$ are the
renormalizing parameters. We set the value of $e_{\rm min}$ to account for
systematic errors which dominate at high magnification, and we adjust the
value of $k$ to achieve $\chi^{2}{\rm/dof}=1$. The data from Salerno, LCO SAAO
by the LCO-Spitzer program, LCO SSO and SAAO by the ROME/REA project are
either too few to give any significant constraint or show systematics and
disagreement with other datasets. Therefore, we do not use them for the
modeling. We list the calculated error-bar renormalization parameters in Table
1.

Table 1: The number of data points in the light curves and the normalization parameters Name | Site | Collaboration | Aperture(m) | Filter | $k$ | $e_{\rm min}$ | $N_{\rm use}/N_{\rm obs}$
---|---|---|---|---|---|---|---
OGLE | Chile | OGLE | 1.3 | $I$ | 1.660 | 0.003 | 3045/3045
OGLE | Chile | OGLE | 1.3 | $V$ | 1.301 | 0.003 | 68/68
MOA | New Zealand | MOA | 1.8 | $R_{\rm MOA}$ | 1.650 | 0.003 | 7277/7509
MOA | New Zealand | MOA | 1.8 | $V$ | 1.321 | 0.003 | 240/240
KMT SSO f03 | Australia | KMTNet | 1.6 | $I$ | 1.900 | 0.003 | 2087/2706
KMT SSO f43 | Australia | KMTNet | 1.6 | $I$ | 1.824 | 0.003 | 2080/2658
KMT CTIO f03 | Chile | KMTNet | 1.6 | $I$ | 1.579 | 0.003 | 2304/2486
KMT CTIO f43 | Chile | KMTNet | 1.6 | $I$ | 1.443 | 0.003 | 2195/2363
KMT SAAO f03 | South Africa | KMTNet | 1.6 | $I$ | 2.444 | 0.003 | 1813/2096
KMT SAAO f43 | South Africa | KMTNet | 1.6 | $I$ | 1.900 | 0.003 | 1846/2078
Danish | Chile | MiNDSTEp | 1.54 | $Z$ | 1.015 | 0.003 | 139/154
Salerno | Italy | MiNDSTEp | 0.6 | $I$ |  |  | 0/5
LCO SSO | Australia | LCO-$Spitzer$ | 1.0 | $i^{\prime}$ | 2.528 | 0.003 | 31/44
LCO CTIO | Chile | LCO-$Spitzer$ | 1.0 | $i^{\prime}$ | 1.129 | 0.003 | 17/17
LCO SAAO | South Africa | LCO-$Spitzer$ | 1.0 | $i^{\prime}$ |  |  | 0/19
CTIO 1.3m | Chile |  $\mu$FUN | 1.3 | $I$ | 0.852 | 0.003 | 18/18
CTIO 1.3m | Chile |  $\mu$FUN | 1.3 | $V$ | 0.566 | 0.003 | 3/3
LCO SSO | Australia | ROME/REA | 1.0 | $g$ |  |  | 0/25
LCO SSO | Australia | ROME/REA | 1.0 | $i^{\prime}$ |  |  | 0/74
LCO SSO | Australia | ROME/REA | 1.0 | $r$ |  |  | 0/29
LCO CTIO | Chile | ROME/REA | 1.0 | $g$ | 1.110 | 0.003 | 33/33
LCO CTIO | Chile | ROME/REA | 1.0 | $i^{\prime}$ | 1.589 | 0.003 | 61/61
LCO CTIO | Chile | ROME/REA | 1.0 | $r$ | 1.337 | 0.003 | 31/31
LCO SAAO | South Africa | ROME/REA | 1.0 | $g$ |  |  | 0/17
LCO SAAO | South Africa | ROME/REA | 1.0 | $i^{\prime}$ |  |  | 0/19
LCO SAAO | South Africa | ROME/REA | 1.0 | $r$ |  |  | 0/45
$Spitzer$ | Earth-trailing orbit | $Spitzer$ | 0.85 | $L$ | 2.110 |  $\cdots$ | 26/26

##  3 Ground-based Light Curve Analysis

###  3.1 Binary-lens model

The magnification of the binary lens model depends on seven parameters: the
time of lens-source closest approach $t_{0}$, the Einstein radius crossing
time $t_{\rm E}$, the impact parameter in units of the Einstein radius
$u_{0}$, the planet-host mass ratio, $q$, the planet-host separation in units
of the Einstein radius, $s$, the angle between the trajectory of the source
and the planet-host axis, $\alpha$, and the ratio of the angular source size
to the angular Einstein radius, $\rho$. The model flux $f(t)$ of the magnified
source at time $t$ is given by,

$f(t)=A(t)f_{S}+f_{b},$ (3)

where $A(t)$ is a magnification of the source star, and $f_{S}$ and $f_{b}$
are the unmagnified flux from the source and the flux from any unresolved
blend stars, respectively.

We also adopt a linear limb-darkening model for the source star,

$S_{\lambda}(\vartheta)=S_{\lambda}(0)[1-u_{\lambda}(1-\cos(\vartheta))],$ (4)

where $S_{\lambda}(\vartheta)$ is a limb-darkened surface brightness. The
effective temperature of the source star estimated from the extinction-free
source color presented in Section 4 is $T_{\rm eff}\sim 5662$K (González
Hernández & Bonifacio, 2009). Assuming a surface gravity $\log g=4.5$ and a
metallicity of $\log[M/H]=0$, we select the limb-darkening coefficients to be
$u_{I}=0.5494$, $u_{V}=0.7105$, $u_{R}=0.6343$, $u_{Z}=0.6314$,
$u_{g}=0.7573$, $u_{r}=0.6283$ and $u_{i}=0.5389$ from the ATLAS model (Claret
& Bloemen, 2011). For the $R_{\rm MOA}$ passband, we use the coefficient for
$u_{Red}=0.5919$, which is the mean of $u_{I}$ and $u_{R}$.

We first conducted the light-curve fitting with only ground-based data. We
employed the Markov Chain Monte Carlo algorithm (Verde et al., 2003) combined
with the image-centered ray-shooting method (Bennett & Rhie, 1996; Bennett,
2010). We conducted the grid search analysis following the same procedure in
Kondo et al. (2019). First, we performed a broad grid search over
$(q,s,\alpha)$ space with the other parameters free. The search ranges of
$q,s,$ and $\alpha$ are $-6<\log q<0$, $-0.5<\log s<0.6$, and $0<\alpha<2\pi$,
with 11, 22, and 40 grid points, respectively. Next, we refined all parameters
for the best 100 models with the smallest $\chi^{2}$ to search for the global
best-fit model.

As a result of the grid search, we found that the best-fit binary-lens model
is favored over the single-lens model by $\Delta\chi^{2}\sim 2330$. Although
the additional magnification from the cusp approach to the planetary caustic
is small, the asymmetric feature in the right side of the light curve due to
the approach to the central caustic shows clear residuals from the single-lens
model, which suggests the existence of a companion. The best binary-lens model
suggests that the lens system has a very low mass ratio, $q\sim 6.9\times
10^{-5}$, with a normalized separation $s\sim 0.96$. It is well known that
there is a close/wide degeneracy in high-mag binary-lens events (Griest &
Safizadeh, 1998; Dominik, 1999; Chung et al., 2005), which is due to the
similar shape and size of the central caustic between $s$ and $s^{-1}$. From
the grid search, we found the best wide binary-lens model ($s>1$) has $q\sim
9.2\times 10^{-5}$ and $s\sim 1.14$. The separation of this wide model is
slightly different from the reciprocal of the separation of the close model
($s<1$), yielding a different shape and size for the central caustic from
those of the best close model. We ruled out the wide model because the best
close binary-lens model is favored over the wide model by $\Delta\chi^{2}\sim
268$. The $\Delta\chi^{2}$ is large because the source trajectory is parallel
to the lens axis and approaches not only the central caustic but also the
planetary caustics.

The parameters of the best-fit models are summarized in Table 3.3. The light
curve and the caustic geometry are shown in Figure 1 and Figure 2.

###  3.2 Binary-source model

We checked the possibility that the observed light-curve can be explained by
the binary-source (1L2S) model because it is known that there is a possible
degeneracy between single-lens binary-source (1L2S) model and binary-lens
single-source (2L1S) model (Griest & Hu, 1993; Gaudi, 1998). For the 1L2S
model, the total effective magnification of the source stars $A$ is expressed
as follows,

$A=\frac{A_{1}f_{1}+A_{2}f_{2}}{f_{1}+f_{2}}=\frac{A_{1}+q_{f}A_{2}}{1+q_{f}},$
(5)

where $A_{1}$ and $A_{2}$ are the magnification of the two sources with model
flux $f_{1}$ and $f_{2}$, respectively, and $q_{f}$ is the flux ratio between
the two sources ($=f_{2}/f_{1}$). In order to explain the magnification of the
second source, we introduce the additional parameters: the time of lens-source
closest approach $t_{0,2}$, the impact parameter in units of the Einstein
radius $u_{0,2}$, and the ratio of the angular source size to the angular
Einstein radius, $\rho_{2}$. We found the best-fit 1L2S model is disfavored
relative to the best-fit 2L1S model by $\Delta\chi^{2}\sim 380$, and we
excluded the 1L2S model. The parameters of the best-fit 1S2L model are
summarized in Table 3.3. The light curve of the 1L2S model is shown in Figure
1.

###  3.3 Ground-Based Parallax

The magnification of the binary lens model with parallax effects need two
additional parameters: the North and East components of parallax vector
${\mbox{\boldmath$\pi$}}_{\rm E}$ in equatorial coordinates, $\pi_{\rm E,E}$
and $\pi_{\rm E,N}$ (Gould, 2004). The orbital parallax effects are caused by
Earth’s orbital motion. In the case of OGLE-2018-BLG-1185, the timescale,
$t_{\rm E}\sim 15.9$ days, is small compared to Earth’s orbital period, which
makes it less likely to measure the parallax effects. The best-fit parallax
model improves the fit slightly by $\Delta\chi^{2}\sim 20$, but there is
disagreement of the $\chi^{2}$ improvement between the datasets. The parallax
information such as the direction and the value is easily influenced by the
systematics in each telescope dataset. Considering these facts, we concluded
that we should disregard the parallax information from the ground-based data.

The best-fit models for ground-only data Parameters Unit 2L1S(close)
2L1S(wide) 1L2S $\chi^{2}/{\rm dof}$ 23221.473/23252 23489.306/23252
23601.431/23249 $t_{0,1}$ HJD′ 8310.7772 $\pm$ 0.0003 8310.7793 $\pm$ 0.0003
8310.7726 $\pm$ 0.0003 $t_{0,2}$ HJD′ … … 8311.5874 $\pm$ 0.0010 $t_{\rm E}$
days 15.931 $\pm$ 0.133 16.312 $\pm$ 0.144 15.730 $\pm$ 0.189 $u_{0,1}$
$10^{-3}$ 6.877 $\pm$ 0.063 6.606 $\pm$ 0.067 7.777 $\pm$ 0.131 $u_{0,2}$
$10^{-3}$ … … 8.773 $\pm$ 1.515 $q$ $10^{-5}$ 6.869 $\pm$ 0.229 9.164 $\pm$
0.552 … $s$ 0.963 $\pm$ 0.001 1.144 $\pm$ 0.003 … $\alpha$ radian 0.114 $\pm$
0.001 3.261 $\pm$ 0.002 … $\rho_{1}$ $10^{-3}$ 3.468 $\pm$ 0.083 $<$1.026a
7.234 $\pm$ 0.241 $\rho_{2}$ $10^{-3}$ … … 1.613 $\pm$ 0.956 $q_{f,I}$
$10^{-2}$ … … 1.699 $\pm$ 0.192 $f_{S}$ (OGLE) b 107.777 $\pm$ 0.437 106.493
$\pm$ 0.448 108.583 $\pm$ 0.550 $f_{b}$ (OGLE) b 396.165 $\pm$ 0.594 397.397
$\pm$ 0.440 393.516 $\pm$ 0.587

  * a

The value is the $3\sigma$ upper limit.

  * b

All fluxes are on a 25th magnitude scale, e.g., $I_{\rm S}=25-2.5\log(f_{\rm
S})$.

Figure 1: The light curve and models with the ground-based data for
OGLE-2018-BLG-1185. Top panel shows the light curve, models, and residuals
from the best-fit close binary-lens (2L1S) model. The blue line shows the
best-fit close binary-lens (2L1S) model. The red, orange, and green dot lines
show the single-lens (1L1S) model, the wide 2L1S model, and the binary-source
(1L2S) model, respectively. The left bottom panel and the right panel show the
zoom-in of the light curve, where we can find clear deviations of data points
from the 1L1S and 1L2S models. Figure 2: Caustic geometry of the best-fit
model. The caustics are shown in red lines. The blue line shows the source
trajectory on the lens plane and the arrow indicates the direction of the
source/lens relative proper motion. The blue open circle indicates the source
size and position at $t_{0}$.

##  4 Angular Einstein Radius

We can estimate the angular Einstein radius $\theta_{\rm
E}=\theta_{\rm*}/\rho$ because $\rho$ can be derived by the light-curve
fitting and the angular source radius $\theta_{\ast}$ can be derived by using
an empirical relation between $\theta_{\ast}$, the extinction-corrected source
color $(V-I)_{\rm S,0}$, and the magnitude $I_{\rm S,0}$ (e.g., Boyajian et
al., 2014).

We derived the OGLE-IV instrumental source color and magnitude from the light-
curve fitting and then converted them to the standard ones by using the
following color-color relation from Udalski et al. (2015a):

$\displaystyle I_{\rm O3}-I_{\rm O4}$ $\displaystyle=$ $\displaystyle(0.182\pm
0.015)+(-0.008\pm 0.003)(V-I)_{\rm O3},$ (6) $\displaystyle V_{\rm O3}-V_{\rm
O4}$ $\displaystyle=$ $\displaystyle(0.257\pm 0.015)+(-0.074\pm
0.004)(V-I)_{\rm O3}.$ (7)

The apparent color and the standard magnitude of the source star are
$(V-I,I)_{\rm S,O4,calib}=(2.344\pm 0.031,20.082\pm 0.012)$.

We also derived the apparent source color and magnitude from the CT13
measurements in the $I$\- and $V$-bands from the light-curve fitting, and then
converted them to the standard ones following the procedure explained in Bond
et al. (2017). We cross-referenced isolated stars in the CT13 catalog reduced
by DoPHOT (Schechter et al., 1993) with the stars in the OGLE-III map within
$120^{\prime\prime}$ of the source star and obtained the color-color relation:

$\displaystyle I_{\rm O3}-I_{\rm CT13}$ $\displaystyle=$
$\displaystyle(-0.880\pm 0.005)+(-0.042\pm 0.005)(V-I)_{\rm CT13},$ (8)
$\displaystyle V_{\rm O3}-I_{\rm CT13}$ $\displaystyle=$
$\displaystyle(1.290\pm 0.004)+(-0.036\pm 0.004)(V-I)_{\rm CT13}.$ (9)

The apparent color and magnitude of the source star are $(V-I,I)_{\rm
S,CT13,calib}=(2.335\pm 0.025,20.105\pm 0.013)$. This color is consistent with
$(V-I)_{\rm S,O4,calib}$ within $1\sigma$ and the magnitude is consistent with
$I_{\rm S,O4,calib}$ within $2\sigma$. Because the light curve was well
covered by the OGLE observations, while it was highly magnified, we adopted
$(V-I,I)_{\rm S,O4,calib}$ as the source color and magnitude.

To obtain the extinction-corrected source color and magnitude, we used the
standard method from Yoo et al. (2004). The intrinsic color and magnitude are
determined from the source location relative to the color and magnitude of the
red clump giant (RCG) centroid in the color-magnitude diagram (CMD). In Figure
3, the red point shows the RCG centroid color and magnitude, $(V-I,I)_{\rm
RCG}=(2.720,16.325)\pm(0.009,0.032)$ for the field around the source star.
Assuming that the source star suffers the same reddening and extinction as the
RCGs, we compared these values to the expected extinction-corrected RCG color
and magnitude for this field, $(V-I,I)_{\rm
RCG,0}=(1.060,14.362)\pm(0.070,0.040)$ (Bensby et al., 2013; Nataf et al.,
2013). As a result, we obtained an extinction of $A_{I}=1.963\pm 0.051$ and a
color excess of $E(V-I)=1.660\pm 0.071$. Finally, the intrinsic source color
and magnitude were derived,

$(V-I,I)_{{\rm S},0}=(0.684,18.119)\pm(0.077,0.053).$ (10)

As a reference for the later discussion of the future follow-up observations,
we also estimated the intrinsic source magnitudes in $H$\- and $K$-bands from
the color-color relation in Kenyon & Hartmann (1995), including a 5%
uncertainty. Then, we applied the extinction in the $H$\- and $K$-bands, which
were derived from the extinction in the $I$\- and $V$-bands of the RCGs
according to Cardelli et al. (1989).

Figure 3 shows that the source is consistent with being part of the standard
bulge sequence of stars, i.e., it falls within the distribution of stars from
(Holtzman et al., 1998) after they have been shifted to the same reddening and
extinction as the field for OGLE-2018-BLG-1185. However, the source also has a
similar color to the Sun. Thus, it would also be consistent with being a
similar absolute magnitude to the Sun but somewhat in the foreground, e.g., at
$\sim 6$ kpc. Thus, we also checked how a different assumption about the
source would affect our results. If the source was more in the foreground, it
would then suffer less extinction and reddening than the RCGs. However, even
if we assume 10% less extinction and reddening than the RCGs, the value of
$\theta_{\rm E}$ increases by only 7%, which is still consistent within
$1\sigma$ with values obtained assuming the same extinction and reddening as
the RCGs. We summarize the source color and magnitudes in Table 4.

Applying the empirical formula, $\log(\theta_{\rm
LD})=0.501414+0.419685(V-I)-0.2I$ (Boyajian et al., 2014), where $\theta_{\rm
LD}\equiv 2\theta_{*}$ is the limb-darkened stellar angular diameter, (Fukui
et al., 2015), we found the angular source radius,

$\theta_{\rm*}=0.730\pm 0.059\ {\rm\mu as}.\\\ $ (11)

Finally, we obtained the source angular radius and the lens–source relative
proper motion in the geocentric frame,

$\displaystyle\theta_{\rm E}$ $\displaystyle=$
$\displaystyle\frac{\theta_{*}}{\rho}=0.211\pm 0.018\ {\rm mas},$ (12)
$\displaystyle\mu_{\rm rel,geo}$ $\displaystyle=$
$\displaystyle\frac{\theta_{\rm E}}{t_{\rm E}}=4.832\pm 0.410\ \mathrm{mas\
yr}^{-1}$ (13)

This $\theta_{\rm E}$ value is relatively small, which suggests that the lens
is a low-mass star and/or distant from the observer.

Figure 3: Color magnitude diagram (CMD). The stars in the OGLE-III catalog
within $120^{\prime\prime}$ of the source star are shown with black dots. The
green dots indicate the $HST$ CMD of (Holtzman et al., 1998), which is
transformed to the same reddening and extinction of the field of the event.
The red dot shows the centroid of the red clump giant distribution. The colors
and magnitudes of the source star and the blend are shown with blue and pink,
respectively.

The source color and magnitudes Parameters Unit Source Source Source
(apparent) (intrinsic)a (intrinsic)b $I$ mag 20.082 $\pm$ 0.012 c
18.119$\pm$0.053 18.315$\pm$0.053 $V-I$ mag 2.344 $\pm$ 0.031 c
0.684$\pm$0.077 0.850$\pm$0.077 $H$ d mag 18.012$\pm$0.143 17.444$\pm$0.095
$\cdots$ $K$ d mag 17.756$\pm$0.145 17.394$\pm$0.095 $\cdots$

  * a

Extinction-corrected magnitudes assuming that the source star suffers the same
reddening and extinction as the RCGs.

  * b

Extinction-corrected magnitudes assuming that the source star suffers the
reddening and extinction of 0.9 times as much as that for the RCGs.

  * c

The magnitude and color are measured from the light-curve fitting.

  * d

The magnitudes are estimated from the color-color relation in Kenyon &
Hartmann (1995) and the extinction law in Cardelli et al. (1989).

##  5 Lens Physical Parameters by Bayesian Analysis

If we can measure both the finite source effects and the parallax effects, the
lens physical parameters such as the host mass $M_{\rm host}$ and the distance
to the lens $D_{\rm L}$ are calculated directly, following the equations:

$M_{\rm host}=\frac{\theta_{\rm E}}{(1+q)\kappa\pi_{\rm E}};\ \ D_{\rm
L}=\frac{\rm au}{\pi_{\rm rel}+\pi_{\rm S}};\ \ \pi_{\rm rel}=\theta_{\rm
E}\pi_{\rm E};\ \ {\bm{\mu}}_{\rm rel}=\frac{\theta_{\rm E}}{t_{\rm
E}}\frac{{\bm{\pi}}_{\rm E}}{\pi_{\rm E}},$ (14)

where $\kappa\equiv 4G/(c^{2}{\rm au})=8.1439\ {\rm mas}/M_{\odot}$, and
$\pi_{\rm S}={\rm au}/D_{\rm S}$ is the source parallax. From the ground-based
light-curve alone, we are only able to measure $\theta_{\rm E}$ (via finite
source effects), but no meaningful constraint on $\pi_{\rm E}$ (see Section
3.3).

In order to estimate the probability distributions of $M_{\rm L}$ and $D_{\rm
L}$, we conducted a Bayesian analysis with the Galactic model of Koshimoto et
al. (in preparation) as a prior. We randomly generated 50 million simulated
microlensing event samples. Then we calculated the probability distributions
for the lens physical parameters by weighting the microlensing event rate by
the measured $t_{\rm E}$ and $\theta_{\rm E}$ likelihood distribution. It is
important to note that we conducted the Bayesian analysis under the assumption
that stars of all masses have an equal probability to host this planet.

We calculated some parameters in addition to the lens physical parameters,
$M_{\rm L}$ and $D_{\rm L}$. For instance, the lens-source proper motion in
the geocentric frame, ${\bm{\mu}}_{\rm rel}$, is converted to that in the
heliocentric frame,

${\bm{\mu}}_{\rm rel,hel}={\bm{\mu}}_{\rm
rel}+{\bm{v}}_{\oplus,\perp}\frac{\pi_{\rm rel}}{\rm au},$ (15)

where ${\bm{v}}_{\oplus,\perp}=({v}_{\oplus,N},{v}_{\oplus,E})=(-0.78,27.66)$
km/s is the projected velocity of Earth at $t_{0}$.

We also calculated the $I$\- and $V$-band magnitudes of the lens from the
mass-luminosity relations of main-sequence stars (Kenyon & Hartmann, 1995),
and the 5 Gyr isochrone for brown dwarfs from Baraffe et al. (2003). Then we
estimated $H$\- and $K$-band magnitudes of the lens from the color-color
relation in Kenyon & Hartmann (1995), including a 5% uncertainty. In order to
estimate the extinction in the foreground of the lens, we assumed a dust scale
height of $h_{\rm dust}=0.10\pm 0.02$ kpc (Bennett et al., 2015),

$A_{\lambda,{\rm L}}=\frac{1-e^{-|D_{\rm L}/(h_{\rm
dust}\sin{b})|}}{1-e^{-|D_{\rm S}/(h_{\rm dust}\sin{b})|}}A_{\lambda,\rm{S}},$
(16)

where the index $\lambda$ refers to the passband: $V$-, $I$-, $H$-, or
$K$-band. We obtained the extinction in the $I$\- and $V$-band magnitudes of
the source from the RCGs in Section 4, and then we converted them to the
extinction in the $H$\- and $K$-bands according to Cardelli et al. (1989).

The results are shown in Table 2 and Figure 4. According to Figure 4, the lens
system is likely to be a super-Earth with a mass of $m_{\rm
p}=8.1^{+7.6}_{-4.4}\ M_{\oplus}$ orbiting a late M-dwarf with a mass of
$M_{\rm host}=0.36^{+0.33}_{-0.19}M_{\odot}$ at a projected separation of
$a_{\perp}=1.54^{+0.18}_{-0.22}\ {\rm au}$. The system is located $D_{\rm
L}=7.4^{+0.5}_{-0.9}$ kpc from Earth. For reference, we also plot the source
magnitudes in $V$-, $I$-, $H$-, and $K$-band as the red lines; the $H$\- and
$K$-band magnitudes were estimated in Section 4. We also show the parallax
contour derived from the Bayesian analysis in Figure 5.

Figure 4: Probability distribution of lens properties derived from the Bayesian analysis with a Galactic prior and constrained by $t_{\rm E}$ and $\theta_{\rm E}$. The vertical blue lines show the median values. The dark-blue and the light-blue regions show the 68.3% and 95.4% confidence intervals. The red vertical lines in the probability distributions of $I$-, $V$-, $H$-, and $K$-band magnitudes show the magnitudes of the source star with extinction. Figure 5: Left: The parallax contours for OGLE-2018-BLG-1185 expected from the Galactic model of Koshimoto et al. (in preparation) after imposing the two observational constraints of the angular Einstein radius, $\theta_{\rm E}$ and the Einstein radius crossing time, $t_{\rm E}$, on the event rate. The colorbar corresponds to the logarithm of the event rate and the red region indicates higher probability. Center: Including the constraint that $\Delta f_{\rm Spz}<4$. Right: Including the full constraint from the Spitzer-“only” parallax. Table 2: The lens physical parameters |  | Bayesian | Naive Spitzer-“only”
---|---|---|---
Parameters | Unit | Ground-only | Ground + $\Delta f_{\rm Spz}$ | Ground + ${\mbox{\boldmath$\pi$}}_{\rm E,Spz}$ | $(u_{0}>0)$ | $(u_{0}<0)$
$M_{\rm host}$ | $M_{\sun}$ | $0.355^{+0.334}_{-0.192}$ | $0.366^{+0.345}_{-0.207}$ | $0.091^{+0.064}_{-0.018}$ | $0.073\pm 0.011$ | $0.070\pm 0.010$
$m_{\rm p}$ | $M_{\oplus}$ | $8.123^{+7.642}_{-4.400}$ | $8.377^{+7.884}_{-4.730}$ | $2.089^{+1.459}_{-0.414}$ | $1.68\pm 0.26$ | $1.61\pm 0.22$
$D_{L}$ | kpc | $7.396^{+0.507}_{-0.852}$ | $7.403^{+0.505}_{-0.875}$ | $5.446^{+1.697}_{-0.657}$ | $4.96\pm 0.74$ | $4.89\pm 0.66$
$a_{\perp}$ | au | $1.535^{+0.181}_{-0.215}$ | $1.537^{+0.182}_{-0.219}$ | $1.141^{+0.317}_{-0.147}$ | $1.01\pm 0.18$ | $0.99\pm 0.16$
$\pi_{\rm E}$ |  | $0.075^{+0.087}_{-0.036}$ | $0.073^{+0.093}_{-0.035}$ | $0.292^{+0.066}_{-0.120}$ | $0.354\pm 0.042$ | $0.369\pm 0.037$
$\mu_{{\rm rel,hel}}$ | mas/${\rm yr}$ | $5.042^{+0.434}_{-0.439}$ | $5.057^{+0.430}_{-0.437}$ | $4.858^{+0.441}_{-0.443}$ |  $\cdots$ |  $\cdots$
$V$ | mag | $29.377^{+2.919}_{-2.561}$ | $29.259^{+3.108}_{-2.563}$ | $34.057^{+5.220}_{-1.628}$ |  $\cdots$ |  $\cdots$
$I$ | mag | $24.688^{+2.260}_{-2.001}$ | $24.596^{+2.405}_{-2.004}$ | $28.217^{+3.349}_{-1.188}$ |  $\cdots$ |  $\cdots$
$H$ | mag | $21.263^{+1.743}_{-1.562}$ | $21.192^{+1.853}_{-1.565}$ | $23.897^{+2.623}_{-0.859}$ |  $\cdots$ |  $\cdots$
$K$ | mag | $20.822^{+1.672}_{-1.502}$ | $20.754^{+1.778}_{-1.505}$ | $23.332^{+2.885}_{-0.814}$ |  $\cdots$ |  $\cdots$

##  6 Analysis including Spitzer data

Figure 6: The light curve and models with the $Spitzer$ data. The blue dotted
line shows the Spitzer flux predicted by the 2L1S best-fit model derived from
the ground-based analysis for ${\mbox{\boldmath$\pi$}}_{\rm E}=(0,0)$
evaluated at the central value of the color-constraint. The black and gray
shaded regions show models derived from the $Spitzer$-“only” parallax
analysis. Each color (black, dark gray, light gray) represents
$\Delta\chi^{2}<$ (1,4,9). Figure 7: Same as Figure 4, but with the addition
of the constraint $\Delta f_{\rm Spz}<4$.

We measure the microlens parallax vector ${\mbox{\boldmath$\pi$}}_{\rm E}$ via
the “satellite parallax effect”, which can be approximated as:

${\mbox{\boldmath$\pi$}}_{\rm E}=\frac{\rm au}{D_{\perp}}\left(\frac{t_{0,{\rm
sat}}-t_{0,\oplus}}{t_{\rm E}},u_{0,{\rm sat}}-u_{0,\oplus}\right),$ (17)

where $D_{\perp}$ is the Earth-satellite separation projected on the plane of
the sky, and $t_{0,{\rm sat}}$ and $u_{0,{\rm sat}}$ are the time of lens-
source closest approach and the impact parameter as seen by the satellite. The
Einstein timescale $t_{\rm E}$ is assumed to be the same for both Earth and
the satellite. In practice, we fully model Spitzer’s location as a function of
time.

The Spitzer light curve for OGLE-2018-BLG-1185 shows a very weak decline of
$\sim 1$ instrumental flux unit over the four-week observation period (see
Figure 6). The magnitude of the decline is comparable to the level of
systematics seen in a few other events (Gould et al., 2020; Hirao et al.,
2020; Zang et al., 2020c) and, thus, should be treated with caution. However,
even this weak decline indicates a significant parallax effect for the event
as seen from Spitzer. We derive a color constraint for the Spitzer data by
measuring the $IHL$ color-color relation for clump stars in CT13 $I$ and $H$,
and Spitzer $L$. Evaluating this relation at the measured $(I-H)$ color of the
source gives the constraint on the Spitzer source flux:

$I_{\rm CT13}-L=-4.518\pm 0.028,$ (18)

which gives an expected source flux from Spitzer of $f_{\rm S,Spz}=0.6254$
flux units for the best-fit value of $I_{\rm CT13}$. This constraint and the
best-fit ground-based model (Table 2) together imply some tension with the
observed Spitzer light curve unless there is a significant parallax effect.
They predict that the observed Spitzer flux should have been substantially
brighter at the start of the Spitzer observations ($\sim 6$ flux units) and
declined by a total of $\Delta f_{\rm Spz}\sim 3.3$ flux units as compared to
the observed $\Delta f_{\rm Spz}\sim 1$ flux unit. This tension can be seen in
Figure 6 and suggests that, due to the parallax effect, the event peaked at a
lower magnification and/or earlier as seen from Spitzer.

We can use limits on the change in the Spitzer flux to place conservative
constraints on the physical properties of the lens. Suppose that systematics
affect the Spitzer light curve at the level of 1–2 flux units, i.e., at the
level seen in previous work. If the true signal is $\sim 4$ flux units, it is
very unlikely that systematics would cause us to measure $\Delta f_{\rm
Spz}=1$ flux unit. Therefore, we repeat the Bayesian analysis imposing the
constraint $\Delta f_{\rm Spz}<4$, where $\Delta f_{\rm Spz}$ is calculated
from Equation (18). The parallax effect can produce a degeneracy in the sign
of $u_{0}$. In this case, because $u_{0}$ is small, the effect of this
degeneracy is much smaller than the uncertainties (Gould, & Yee, 2012), so we
only carry out this calculation for the $u_{0}>0$ case.

The results are given in Table 2 (as “Ground + $\Delta f_{\rm Spz}$”), Figure
7, and the center panel of Figure 5. This constraint rules out higher-mass,
more distant solutions for the lens, and instead suggests a $M_{\rm
host}=0.37^{+0.35}_{-0.21}\ M_{\odot}$ host with a $m_{\rm
p}=8.4^{+7.9}_{-4.7}\ M_{\oplus}$ planet at a projected separation
$a_{\perp}=1.54^{+0.18}_{-0.22}\ {\rm au}$. We adopt these values as our
conservative Bayesian estimate of the properties of the lens system.

###  6.1 Spitzer-“only” Parallax

Figure 8: Parallax contours from Spitzer-“only” analysis (see text). Colors
(black, red, yellow, green, cyan, blue, magenta) indicate (1, 2, 3, 4, 5, 6,
7)$\sigma$ from the minimum, respectively. Left panel is for $(u_{0}>0)$ and
right panel is for $(u_{0}<0)$.

If we take the Spitzer light curve at face value, we can derive stronger
constraints on the parallax using the Spitzer-“only” parallax method. This
method has been used in several previous analyses (starting with Gould et al.,
2020) to show how the Spitzer light curve constrains the parallax. For this
analysis, we hold the microlensing parameters $t_{0}$, $u_{0}$, and $t_{\rm
E}$ fixed at values found by fitting the ground-based data and make the
assumption that the Spitzer light curve is in the point lens regime.333In
principle, we should calculate the Spitzer magnification using the full
planetary model, but in practice, this makes almost no difference because the
Spitzer observations start well after the planetary perturbation. Then, for a
grid of parallax values, we fit for the Spitzer flux while applying the color-
constraint from Equation (18). We repeat the analysis for $-u_{0}$, which
produces an indistinguishable ground-based light curve and, as expected, only
slight variations in the parallax.

The resulting parallax contours are shown in Figure 8. The four minima
correspond to the well-known satellite parallax degeneracy (Refsdal, 1966;
Gould, 1994) and the overall arc shape follows the expectation from the Gould
(2019) osculating circles formalism. The values for the magnitude of the
microlens parallax vector are $\pi_{\rm E}=0.35\pm 0.04$ for the $(u_{0}>0)$
case and $\pi_{\rm E}=0.37\pm 0.04$ for the $(u_{0}<0)$ case. The $3\sigma$
ranges are $\pi_{\rm E}=[0.18,0.50]$ and $\pi_{\rm E}=[0.20,0.48]$,
respectively.

###  6.2 Physical Lens Properties from Spitzer Parallax

We can derive the physical properties of the lens by combining the measurement
of the parallax from the Spitzer-“only” analysis with the measurement of
$\theta_{\rm E}=0.211\pm 0.019$ mas from fitting the ground-based light curve.
These estimates and their uncertainties are derived from Equation (14) using
simple error propagation, and so are the “naive” values of these quantities.
For the $(u_{0}>0)$ solution, this yields a lens mass of $M_{\rm L}=0.073\pm
0.011M_{\odot}$ and $D_{\rm L}=4.96\pm 0.74$ kpc for $D_{\rm S}=7.88$ kpc.
This would then imply that the mass of the planet is $m_{\rm p}=1.7\pm 0.3\
M_{\oplus}$ and that it is separated from the host by $a_{\perp}=1.01\pm 0.18$
au. The values for the $(u_{0}<0)$ solution are comparable. See Table 2.

In order to estimate the lens magnitude, we also performed a Bayesian analysis
including the $\pi_{\rm E}$ constraint derived from the $Spitzer$-“only”
parallax analysis. First, we took the average of the $\chi^{2}$ values for the
two $(u_{0}>0)$ and $(u_{0}<0)$ solutions for each value of $\pi_{\rm E,E}$
and $\pi_{\rm E,N}$. Then, the event rate was weighted by
$\exp(-{\Delta\chi^{2}}/2)$ and the measured $t_{\rm E}$ and $\theta_{\rm E}$
constraints to calculate the probability distribution. Table 2 and Figure 9
show the results. The distributions for some of the parameters in Figure 9 are
bimodal. In addition to the expected peak for lenses at $D_{\rm L}\sim 5$ kpc,
there is a second peak for lenses with $D_{\rm L}\sim 7.5$ kpc. This second
peak corresponds to events with lenses in the bulge and sources in the far-
disk, which were not considered in our naive calculations. For the bimodal
distributions, the central values and confidence intervals reported in Table 2
are not a complete description of the distributions and should be considered
in context with Figure 9. However, the mass distribution is not subject to
this issue. We find that the lens system is likely a terrestrial planet with a
mass of $m_{\rm p}=2.1^{+1.5}_{-0.4}\ M_{\oplus}$ orbiting a very-low-mass
(VLM) dwarf with a mass of $M_{\rm host}=0.091^{+0.064}_{-0.018}\ M_{\odot}$.

Figure 9: Same as Figure 4, but with the addition of the $\pi_{\rm E}$
constraint from the $Spitzer$-“only” parallax measurement.

###  6.3 Implications

Figure 10: Test of source proper motion predicted by the Spitzer-“only”
parallax. Black points: derived source proper motions for $\pi_{\rm E}$ within
$1\sigma$ of the minimum for the Spitzer-“only” contours (based on $\mu_{\rm
rel,hel}$). Black cross: mean proper motion for disk stars assuming a distance
of $D_{\rm L}=4.9\ $ kpc. Dashed circle: centered on black cross with a radius
$\mu_{\rm rel,geo}=4.832\ \mathrm{mas\ yr}^{-1}$. Note that the black cross
and dashed circle are merely reference points. Red: $1\sigma$ error ellipse
for the bulge stars as derived from Gaia. Blue: $1\sigma$ error ellipse for
the disk stars derived from $(\sigma_{v,\phi},\sigma_{v,z})$. Dotted black
contours: 1, 2, 3$\sigma$ contours adding the dispersions of the bulge and
disk in quadrature. The observed constraints are consistent with a lens in the
disk and a source in the bulge.

Hence, if the Spitzer-“only” parallax is correct, this would be the second
detection of a terrestrial planet orbiting a VLM dwarf from the Spitzer
microlensing program. The first was OGLE-2016-BLG-1195Lb (Bond et al., 2017;
Shvartzvald et al., 2017), which is a $m_{\rm p}={1.43}_{-0.32}^{+0.45}\
M_{\oplus}$ planet orbiting an $M_{\rm L}=0.078_{-0.012}^{+0.016}\ M_{\odot}$
VLM dwarf at a separation of $a_{\perp}=1.16_{-0.13}^{+0.16}$ au. The distance
to the OGLE-2016-BLG-1195L system is also comparable: $D_{\rm
L}=3.91_{-0.46}^{+0.42}$ kpc. One curiosity about OGLE-2016-BLG-1195L is that
the lens-source relative proper motion suggests that the lens could be moving
counter to the direction of Galactic rotation, which would be unusual for a
disk lens.

Therefore, we also consider the implications of the Spitzer-“only”
${\mbox{\boldmath$\pi$}}_{\rm E}$ for constraining the lens motion in
OGLE-2018-BLG-1185. First, we note that there is no independent information on
the proper motion of the source $\mu_{\rm S}$ because there is no evidence
that the blend, which dominates the baseline object, is associated with the
event (see Appendix A). Second, given $D_{\rm L}\sim 4.9\ $ kpc, we assume
that the lens is in the disk, and therefore, has a proper motion similar to
other disk stars. The velocity model of Koshimoto et al. (in preparation) is
based on the Shu distribution function model in Sharma et al. (2014), but the
mean velocity and velocity dispersion in the disk are fitted to the Gaia DR2
data (Gaia Collaboration et al., 2018) as a function of the Galactocentric
distance, $R$ and the height from the Galactic plane, $z$. The velocity of
disk stars at 4.9 kpc is
$(v_{\phi},v_{z})=(207.6^{+42.7}_{-44.0},-0.4^{+38.8}_{-39.6})\ \mathrm{km\
s}^{-1}$. Hence, for the velocity dispersion, we use
$(\sigma_{v,\phi},\sigma_{v,z})=(43.4,39.2)\ \mathrm{km\ s}^{-1}$. Table 3
summarizes the disk star velocities and proper motions expected from the
Galactic model at $D=4.9\pm 0.7$ kpc. The values in the table are derived from
the Bayesian analysis with a Galactic prior and constrained by $\theta_{\rm
E}$ and $t_{\rm E}$. For the Sun’s motion, we use $(v_{R},v_{\phi},v_{z})_{\rm
Sun}=(-10,243,7)\ \mathrm{km\ s}^{-1}$ (for $(R_{\odot},z_{\odot})=(8160,25)$
pc). We combine the two velocities to estimate the proper motion of disk
stars. Finally, by applying Equation 15, we can derive the expected source
proper motion $\mu_{\rm S}=\mu_{\rm L}-\mu_{\rm rel,hel}$ for a given value of
the parallax. Figure 10 shows the results for values of
${\mbox{\boldmath$\pi$}}_{\rm E}$ out to the $1\sigma$ Spitzer-“only” contours
for the $(u>0)$ solution (the results for the $(u<0)$ solution are nearly
identical). The properties of bulge stars are derived from Gaia stars within
$5^{\prime}$ of the target: $\mu_{\rm
bulge}(\ell,b)=(-6.310,-0.163)\pm(0.088,0.076)\ \mathrm{mas\ yr}^{-1}$ and
$\sigma_{\rm bulge}(\ell,b)=(3.176,2.768)\pm(0.062,0.054)\ \mathrm{mas\
yr}^{-1}$. To account for the uncertainty in the lens motion, we add the
proper motion dispersions of the disk and bulge in quadrature. One of the two
Spitzer minima suggests a source more than $2\sigma$ from the bulge
distribution, but the other minimum is consistent with a bulge source at $\sim
1.5\sigma$. Therefore, there is no reason to believe that the Spitzer
${\mbox{\boldmath$\pi$}}_{\rm E}$ requires a lens proper motion in tension
with the motion of typical disk stars.

Finally, in order to be included in the statistical samples for the study of
the Galactic distribution of planets, Zhu et al. (2017) proposed the criteria:

$\sigma(D_{8.3})<1.4{\rm kpc};\ \ D_{8.3}\equiv\frac{\rm kpc}{\pi_{\rm
rel}/{\rm mas}+1/8.3}.$ (19)

We find $D_{8.3}=5.15\pm 0.28$ kpc for the ($u_{0}>0$) case and
$D_{8.3}=5.04\pm 0.28$ kpc for the ($u_{0}<0$) case by combining the
measurement of ${\mbox{\boldmath$\pi$}}_{\rm E}$ from the Spitzer-“only”
analysis with the measurement of $\theta_{\rm E}$ from fitting the ground-
based light curve. The small $\sigma(D_{8.3})$ is consistent with the
expectation for the high magnification event as investigated by Gould, & Yee
(2012), Shin et al. (2018), and Gould (2019). They show that accurate parallax
measurements are possible even if there are only a few observations taken by
the Spitzer when the Earth-based magnification is high ($A_{\oplus}\geq 100$).
Therefore, in terms of $\sigma_{\rm D8.3}$ (Zhu et al., 2017), the
Spitzer-“only” parallax suggests that the apparent signal is good enough to
include OGLE-2018-BLG-1185Lb in the statistical sample of Spitzer events.
However, the systematics need be studied and understood before membership in
the sample can be definitively evaluated.

Table 3: Disk star velocities and proper motions at $D=4.9\pm 0.7$ kpc Star Component | Velocity Component | Unit | $-2\sigma$ | $-1\sigma$ | Median | $+1\sigma$ | $+2\sigma$
---|---|---|---|---|---|---|---
Thin Disk | $v_{l}$ | $\mathrm{km\ s}^{-1}$ | 110.7 | 163.6 | 205.9 | 242.4 | 280.7
| $v_{b}$ | $\mathrm{km\ s}^{-1}$ | -95.0 | -48.8 | -13.7 | 22.4 | 71.1
| $\mu_{{\rm hel},l}$ | $\mathrm{mas\ yr}^{-1}$ | -5.615 | -3.349 | -1.577 | -0.024 | 1.656
| $\mu_{{\rm hel},b}$ | $\mathrm{mas\ yr}^{-1}$ | -4.364 | -2.388 | -0.884 | 0.653 | 2.690
Thick Disk | $v_{l}$ | $\mathrm{km\ s}^{-1}$ | 60.7 | 125.2 | 181.4 | 236.5 | 293.7
| $v_{b}$ | $\mathrm{km\ s}^{-1}$ | -147.8 | -86.4 | -12.1 | 63.3 | 128.6
| $\mu_{{\rm hel},l}$ | $\mathrm{mas\ yr}^{-1}$ | -7.662 | -4.995 | -2.602 | -0.275 | 2.177
| $\mu_{{\rm hel},b}$ | $\mathrm{mas\ yr}^{-1}$ | -6.577 | -3.987 | -0.808 | 2.379 | 5.102
All | $v_{l}$ | $\mathrm{km\ s}^{-1}$ | 103.6 | 161.0 | 204.9 | 242.2 | 281.3
| $v_{b}$ | $\mathrm{km\ s}^{-1}$ | -101.1 | -50.5 | -13.6 | 24.4 | 77.4
| $\mu_{{\rm hel},l}$ | $\mathrm{mas\ yr}^{-1}$ | -5.878 | -3.457 | -1.620 | -0.034 | 1.685
| $\mu_{{\rm hel},b}$ | $\mathrm{mas\ yr}^{-1}$ | -4.611 | -2.462 | -0.883 | 0.737 | 2.960

##  7 Discussion and Summary

We analyzed the microlensing event OGLE-2018-BLG-1185, which was
simultaneously observed from a large number of ground-based telescopes and the
$Spitzer$ telescope. The ground-based light-curve modeling indicates a small
mass ratio of $q=(6.9\pm 0.2)\times 10^{-5}$, which is close to the peak of
the wide orbit exoplanet mass-ratio distribution derived by Suzuki et al.
(2016) and investigated further by Udalski et al. (2018) and Jung et al.
(2019b). Suzuki et al. (2016) derived the wide orbit planet occurrence rate
using a sample of thirty planets, primarily from the MOA-II microlensing
survey during $2007-2012$. The planet presented here, OGLE-2018-BLG-1185Lb,
will be included in an extension of the MOA-II statistical analysis (Suzuki et
al., in preparation), and its low mass ratio will help to define the mass
ratio function peak.

From the ground-based light-curve modeling, only finite source effect is
detected, yielding a measurement of the angular Einstein radius. However, the
physical properties of the lens as derived from the light curve are unclear
because the observed flux variation of the $Spitzer$ light curve is marginal.
Using only the constraint from the measured angular Einstein radius and a
conservative constraint on the change in the Spitzer flux, we estimate the
host star and planet masses with a Bayesian analysis under the assumption that
stars of all masses have an equal probability to host this planet. This
analysis indicates a host mass of $M_{\rm host}=0.37^{+0.35}_{-0.21}\
M_{\odot}$ and a planet mass of $m_{p}=8.4^{+7.9}_{-4.7}\ M_{\oplus}$ located
at $D_{L}=7.4^{+0.5}_{-0.9}$ kpc. By contrast, the $Spitzer$ data favor a
larger microlensing parallax, which implies a very low-mass host with a
terrestrial planet ($M_{\rm host}=0.091^{+0.064}_{-0.018}\ M_{\odot}$, $m_{\rm
p}=2.1^{+1.5}_{-0.4}\ M_{\oplus}$) that is either in the disk at $D_{\rm
L}\sim 5$ kpc or in the bulge at $D_{\rm L}\sim 7.5$ kpc (these values include
a Galactic prior but are not significantly different from the values without
the prior, see Table 2).

Figure 11 compares the Bayesian estimates from the conservative Spitzer flux
constraint and the full Spitzer parallax measurement of the host and planet
mass for OGLE-2018-BLG-1185 to those of other planetary systems. The pink
circles show the microlens planets without mass measurements, and the red
circles show the microlens planets with mass measurements from ground-based
orbital parallax effects and/or the detection of the lens flux by high
resolution follow-up observations. The red squares represent microlens planets
with mass measurements from the satellite parallax effect observed by
$Spitzer$. Figure 11 indicates that if the Spitzer parallax is correct, this
is one of the lowest mass planets discovered by microlensing.

However, the result that this is a terrestrial planet orbiting a very-low-mass
(VLM) dwarf in the disk should be treated with caution, because the amplitude
of the Spitzer signal is at the level of systematics seen in other events. A
comparison of these properties to the Bayesian posteriors (Figure 4)
demonstrates that a higher-mass system is preferred given $t_{\rm E}$,
$\theta_{\rm E}$, and the Galactic priors. At the same time, a VLM-dwarf +
terrestrial planet is still within the $2\sigma$ range of possibilities from
the Bayesian analysis, especially once the constraint on $\Delta f_{\rm Spz}$
is imposed (Figure 7). Furthermore, Shvartzvald et al. (2017) suggest that
such planets might be common. Nevertheless, further investigation is needed in
order to assess whether or not the fitted parallax signal (and so the inferred
mass) is real.

Adaptive optics observations are one way to test the Spitzer parallax signal.
The Bayesian analysis with ground-based + $\Delta f_{\rm Spz}$ constraints
indicates the lens $K$-band magnitude with extinction should be
$K=20.8^{+1.8}_{-1.5}$ mag, which is about 3 magnitudes fainter than the
source. By contrast, if the Spitzer-“only” parallax is correct and the lens is
a VLM dwarf, it should be $K=23.3^{+2.9}_{-0.8}$ mag and therefore, much
fainter and possibly undetectable. The Bayesian estimate of the heliocentric
relative proper motion, $\mu_{{\rm rel,hel}}=5.0\pm 0.4$ mas ${\rm yr}^{-1}$,
predicts that the angular separation between the source and the lens will be
$\sim 30$ mas around mid-2024. Thus, the lens can be resolved from the source
by the future follow-up observations with Keck or ELTs. If such resolved
measurements were made (and the lens were luminous), it would also lead to a
direct measurement of $\mu$. The observed magnitude of $\mu$ can serve as a
check on $\theta_{\rm E}$. Additionally, the direction of $\mu$ is the same as
the direction of the microlens parallax vector, which could clarify how the
Spitzer-“only” parallax contours should be interpreted in the presence of
systematics.

If the Spitzer parallax is verified, this event confirms the potential of
microlensing to measure the wide-orbit planet frequency into the terrestrial
planet regime. Although the number of microlens planets with mass measurements
is small for now, observing the satellite parallax effect can continue to
increase the numbers. In particular, this effect can be measured for
terrestrial planets by simultaneous observations between the ground and L2
(Gould et al., 2003). This can be achieved with the PRIME telescope (PI:
Takahiro, Sumi) and $Roman$ Space Telescope (Spergel et al., 2015; Penny et
al., 2019) in the mid-2020s.

Figure 11: The mass distribution of the detected exoplanets as of 2021
February 25 from http://exoplanetarchive.ipac.caltech.edu. The purple stars
indicate OGLE-2018-BLG-1185. The pink circles show the microlens planets
without mass measurements, and the red circles show the microlens planets with
mass measurements from ground-based orbital parallax effects and/or the
detection of the lens flux by the high resolution follow-up observations. The
red squares represent the microlens planets with mass measurements from
satellite parallax effects by $Spitzer$. The blue, yellow, and black dots
indicate planets found by the transit, direct imaging, and radial velocity
methods, respectively.

Work by I.K. was supported by JSPS KAKENHI Grant Number 20J20633. Work by
J.C.Y. was supported by JPL grant 1571564. D.P.B., A.B., and C.R. were
supported by NASA through grant NASA-80NSSC18K0274. T.S. acknowledges the
financial support from the JSPS, JSPS23103002, JSPS24253004, and JSPS26247023.
Work by N.K. is supported by JSPS KAKENHI Grant Number JP18J00897. A.S. is a
University of Auckland Doctoral Scholar. Y.T. acknowledges the support of DFG
priority program SPP 1992 “Exploring the Diversity of Extrasolar Planets” (WA
1047/11-1). TCH acknowledges financial support from the National Research
Foundation (NRF; No. 2019R1I1A1A01059609) U.G.J. acknowledges support from
H2020-MSCA-ITN-2019, grant no.860470 (CHAMELEON) and the NovoNordisk
Foundation grant no. NNF19OC0057374. W.Z. and S.M. acknowledge support by the
National Science Foundation of China (Grant No. 11821303 and 11761131004).
Work by CH was supported by the grants of National Research Foundation of
Korea (2020R1A4A2002885 and 2019R1A2C2085965). Funding for B.S.G. was provided
by NASA grant NNG16PJ32C and the Thomas Jefferson Chair for Discovery and
Space Exploration. The MOA project is supported by JSPS KAK-ENHI Grant Number
JSPS24253004, JSPS26247023, JSPS23340064, JSPS15H00781, JP16H06287,17H02871
and 19KK0082 . The OGLE project has received funding from the National Science
Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU. 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. This research uses
data obtained through the Telescope Access Program (TAP), which has been
funded by the TAP member institutes. 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. 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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##  Appendix A Constraints on the Blended Light & Discrepancy with Gaia

The blended light in this event is roughly four times brighter than the
source. In principle, the blend could be the lens itself or a companion to
either the lens or the source. If so, it could constrain the flux and proper
motion of the lens or the proper motion of the source.

From the KMTNet images, we measure the astrometric offset between the source
and the baseline object and find an offset of $0\farcs 175$. This offset is
larger than the astrometric uncertainties. Therefore, if it is a companion to
the lens or source, it must be a very wide separation companion ($\sim 1000\
{\rm au}$). However, the large separation also suggests that it could be an
ambient star unrelated to the microlensing event.

We measure the proper motion of the baseline object based on 10 years of OGLE
survey data and find ${\mbox{\boldmath$\mu$}}_{\rm base}({\rm RA},{\rm
Dec})=(-6.00\pm 0.26,-4.25\pm 0.16)\ \mathrm{mas\ yr}^{-1}$. Because the blend
is much brighter than the source, its motion should dominate the measured
${\mbox{\boldmath$\mu$}}_{\rm base}$. The measured value is very consistent
with typical proper motions for normal bulge stars, but not unreasonable for
the proper motion of a disk star. Hence, it does not rule out the possibility
that the blend is a wide-separation companion to the source or the lens, but
it also shows that the blend could easily be an unrelated bulge star.

For completeness, we note that the OGLE measurement of the proper motion of
the baseline object is inconsistent with the reported Gaia proper motion of
the nearest Gaia source (4062756831332827136; Gaia Collaboration et al.,
2016). Gaia EDR3 (Gaia Collaboration et al., 2020) reports there is a G = 20.1
mag star $0\farcs 177$ from the OGLE coordinates for the baseline star
(17:59:10.26 -27:50:06.3). The reported proper motion of this source is
${\mbox{\boldmath$\mu$}}({\rm RA},{\rm Dec})=(-12.173\pm 1.247,-9.714\pm
0.870)\ \mathrm{mas\ yr}^{-1}$, which is an outlier relative to the typical
proper motions for stars in this field. Gaia DR2 (Gaia Collaboration et al.,
2018) reports an only slightly less extreme proper motion of
${\mbox{\boldmath$\mu$}}({\rm RA},{\rm Dec})=(-8.475\pm 2.234,-4.039\pm
1.985)\ \mathrm{mas\ yr}^{-1}$. The nature of this discrepancy is unknown, but
because the Gaia proper motion is highly unusual (and the OGLE proper motion
is typical), and the Gaia measurement varies significantly between DR2 and
EDR3, this suggests a problem with the Gaia measurement.