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

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, according to core accretion theory predictions, planet formation is most efficient (Lissauer 1993; Pollack et al. 1996). The Microlensing Observations in Astrophysics (MOA) Collaboration (Bond et al. 2001; Sumi et al. 2003) 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 to 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 ${q}_{\mathrm{br}}\,=({6.7}_{-1.8}^{+9.0})\times {10}^{-5}$ with power-law slopes of $n=-{0.85}_{-0.13}^{+0.12}$ and $p={2.6}_{-2.1}^{+4.2}$ above and below q_br, respectively. ⁷⁰ 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) revealed 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_⊕ compared to those inferred from microlensing observations. Similar tension is indicated by Atacama Large Millimeter/submillimeter Array (ALMA) observations. Nayakshin et al. (2019) compared the wide-orbit (9–99 au) planet candidates with masses of 0.01M_Jup to a few M_Jup suggested by ALMA protoplanetary disk observations to a population synthesis prediction from the runaway gas accretion scenario. They found that the scenario predicts fewer sub-Jovian planets than the ALMA observations inferred. Three-dimensional hydrodynamical simulations of protoplanetary 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 low-mass ratios are uncertain due to the lack of planets with mass ratios of q < 5.8 × 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 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 in the low-mass regime, p ∼ 0.73, using seven published planets with q < 1 × 10⁻⁴. Jung et al. (2019b) found q_br ≃ 0.55 × 10⁻⁴ using 15 published planets with low-mass ratios (q < 3 × 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 × 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 (D. Suzuki et al. 2021, 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_L, and the distance to the lens star, D_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_L and D_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 θしーた_E from the finite source effect and the microlens parallax πぱい_E from the parallax effect are measured, we can derive two mass–distance relations as follows:

$$\begin{eqnarray}&&{M}_{{\rm{L}}}=\displaystyle \frac{{c}^{2}}{4G}{{\theta }_{{\rm{E}}}}^{2}\displaystyle \frac{{D}_{{\rm{S}}}{D}_{{\rm{L}}}}{{D}_{{\rm{S}}}-{D}_{{\rm{L}}}}=\displaystyle \frac{{c}^{2}}{4G}\displaystyle \frac{\mathrm{au}}{{{\pi }_{{\rm{E}}}}^{2}}\displaystyle \frac{{D}_{{\rm{S}}}-{D}_{{\rm{L}}}}{{D}_{{\rm{S}}}{D}_{{\rm{L}}}},\end{eqnarray} \tag{ 1 }$$

where D_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 θしーた_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_E (typically t_E is ∼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 relatively nearby lens systems, yielding mass measurements in less than half of the 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 to 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 published ⁷¹ planets with satellite parallax measurements from Spitzer: OGLE-2014-BLG-0124Lb (Udalski et al. 2015), 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. 2020), OGLE-2018-BLG-0596Lb (Jung et al. 2019), KMT-2018-BLG-0029Lb (Gould et al. 2020), OGLE-2017-BLG-0406Lb (Hirao et al. 2020), and OGLE-2018-BLG-0799Lb (Zang et al. 2020). Comparison of 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 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 have confirmed this expectation (Shan et al. 2019; Zang et al. 2020). 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. It suggested that this conflict was probably caused by systematic errors in 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 the characterization of systematics in Spitzer photometry for three previously published events (Gould et al. 2020; Hirao et al. 2020; Zang et al. 2020). Those analyses show that systematics in Spitzer photometry can be present at the level of 1–2 instrumental flux units, so observed signals in Spitzer photometry on those scales should be interpreted with caution.

In this paper, we present an analysis of the planetary microlensing event OGLE-2018-BLG-1185, which was simultaneously observed by many ground-based telescopes and the Spitzer Space Telescope. From ground-based light-curve analysis, the planet–host star mass ratio turns out to be very low, q ∼ 6.9 × 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 of hosting the 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.

2.1. Ground-based Survey Observations

2.2. Spitzer Observations

2.3. Ground-based Follow-up Observations

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

Table 1. The Number of Data Points in the Light Curves and the Normalization Parameters

Name	Site	Collaboration	Aperture (m)	Filter	k	${e}_{\min }$	Nuse/Nobs
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	RMOA	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 1.3 m	Chile	μみゅーFUN	1.3	I	0.852	0.003	18/18
CTIO 1.3 m	Chile	μみゅー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	⋯	26/26

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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). The μみゅーFUN data were reduced using DoPHOT (Schechter et al. 1993), and the 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 data pipelines can be underestimated (or more rarely overestimated). Various factors, 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

$$\begin{eqnarray}&&\sigma {{\prime} }_{i}=k\sqrt{{\sigma }_{i}^{2}+{e}_{\min }^{2}},\end{eqnarray} \tag{ 2 }$$

where $\sigma {{\prime} }_{i}$ is the ith renormalized error, σしぐま_i is the ith error obtained from DIA, and k and ${e}_{\min }$ are the renormalizing parameters. We set the value of ${e}_{\min }$ to account for systematic errors that dominate at high magnification, and we adjust the value of k to achieve χかい²/dof = 1. The data from Salerno, LCO SAAO by the LCO–Spitzer program, and LCO SSO and SAAO by the ROME/REA project are too few to give any significant constraint or show systematics and disagreement with other data sets. Therefore, we do not use them for the modeling. We list the calculated error bar renormalization parameters in Table 1.

3.1. Binary-lens model

The magnification of the binary-lens model depends on seven parameters: the time of lens–source closest approach, t₀; the Einstein radius crossing time t_E; the impact parameter in units of the Einstein radius, u₀; 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, αあるふぁ; and the ratio of the angular source size to the angular Einstein radius, ρろー. The model flux f(t) of the magnified source at time t is given by

$$\begin{eqnarray}&&f(t)=A(t){f}_{{\rm{S}}}+{f}_{{\rm{b}}},\end{eqnarray} \tag{ 3 }$$

where A(t) is the 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,

$$\begin{eqnarray}&&{S}_{\lambda }(\vartheta )={S}_{\lambda }(0)[1-{u}_{\lambda }(1-\cos (\vartheta ))],\end{eqnarray} \tag{ 4 }$$

where S_λらむだ (ϑ) is the limb-darkened surface brightness. The effective temperature of the source star estimated from the extinction-free source color presented in Section 4 is T_eff ∼ 5662 K (González Hernández & Bonifacio 2009). Assuming a surface gravity $\mathrm{log}g=4.5$ and a metallicity of $\mathrm{log}[{\rm{M}}/{\rm{H}}]=0$, we select limb-darkening coefficients of 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_MOA passband, we use the coefficient for u_Red = 0.5919, which is the mean of u_I and u_R .

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

The parameters of the best-fit models are summarized in Table 2. The light curve and the caustic geometry are shown in Figures 1 and 2. As a result of the grid search, we found that the best-fit binary-lens model is favored over the single-lens model by Δでるたχかい² ∼ 2330. The bottom panels in Figure 1 show the clear deviations of the light curve with respect to the single-lens model from $\mathrm{HJD}^{\prime} \sim 8310.9$ to ∼8311.8, which are well fitted by the approach to the central caustic for the best binary-lens model. Although the additional magnification from the cusp approach to the planetary caustic is small, the asymmetric feature on the right side of the light curve due to the approach to the central caustic shows clear residuals from the single-lens model, which suggest the existence of a companion. The best binary-lens model suggests that the lens system has a very-low-mass ratio, q ∼ 6.9 × 10⁻⁵, with a normalized separation s ∼ 0.96. It is well known that there is a close/wide degeneracy in high-magnitude 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 ∼ 9.2 × 10⁻⁵ and s ∼ 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 Δでるたχかい² ∼ 268. The Δでるたχかい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.

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Table 2. The Best-fit Models for Ground-only Data

Parameters	Unit	2L1S (Close)	2L1S (Wide)	1L2S
χかい2/dof	...	23,221.473/23,252	23,489.306/23,252	23,601.431/23,249
t0,1	HJD'	8310.7772 ± 0.0003	8310.7793 ± 0.0003	8310.7726 ± 0.0003
t0,2	HJD'	⋯	⋯	8311.5874 ± 0.0010
tE	days	15.931 ± 0.133	16.312 ± 0.144	15.730 ± 0.189
u0,1	10−3	6.877 ± 0.063	6.606 ± 0.067	7.777 ± 0.131
u0,2	10−3	⋯	⋯	8.773 ± 1.515
q	10−5	6.869 ± 0.229	9.164 ± 0.552	⋯
s	...	0.963 ± 0.001	1.144 ± 0.003	⋯
αあるふぁ	rad	0.114 ± 0.001	3.261 ± 0.002	⋯
ρろー1	10−3	3.468 ± 0.083	<1.026 a	7.234 ± 0.241
ρろー2	10−3	⋯	⋯	1.613 ± 0.956
qf,I	10−2	⋯	⋯	1.699 ± 0.192
fS (OGLE) b	...	107.777 ± 0.437	106.493 ± 0.448	108.583 ± 0.550
fb (OGLE) b	...	396.165 ± 0.594	397.397 ± 0.440	393.516 ± 0.587

Notes.

^a The value is the 3σしぐま upper limit. ^b All fluxes are on a 25th-magnitude scale, e.g., ${I}_{{\rm{S}}}=25-2.5\mathrm{log}({f}_{{\rm{S}}})$.

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3.2. Binary-source model

We checked the possibility that the observed light curve can be explained by the binary-source model because it is known that there is a possible degeneracy between the single-lens binary-source (1L2S) model and the 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:

$$\begin{eqnarray}&&A=\displaystyle \frac{{A}_{1}{f}_{1}+{A}_{2}{f}_{2}}{{f}_{1}+{f}_{2}}=\displaystyle \frac{{A}_{1}+{q}_{f}{A}_{2}}{1+{q}_{f}},\end{eqnarray} \tag{ 5 }$$

where A₁ and A₂ are the magnification of the two sources with model flux f₁ and f₂, respectively, and q_f is the flux ratio between the two sources (= f₂/f₁). In order to explain the magnification of the second source, we introduce 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, ρろー₂. We found the best-fit 1L2S model is disfavored relative to the best-fit 2L1S model by Δでるたχかい² ∼ 380, and we excluded the 1L2S model. The parameters of the best-fit 1L2S model are summarized in Table 2. The light curve of the 1L2S model is shown in Figure 1.

3.3. Ground-based Parallax

We can estimate the angular Einstein radius θしーた_E = θしーた_*/ρろー because ρろー can be derived by the light-curve fitting and the angular source radius θしーた_* can be derived by using an empirical relation between θしーた_*, the extinction-corrected source color (V − I)_S,0, and the magnitude I_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. (2015):

$$\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{O}}3}-{I}_{{\rm{O}}4} & = & (0.182\pm 0.015)\\ & & +\,(-0.008\pm 0.003){\left(V-I\right)}_{{\rm{O}}3},\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}\begin{array}{rcl}{V}_{{\rm{O}}3}-{V}_{{\rm{O}}4} & = & (0.257\pm 0.015)\\ & & +\,(-0.074\pm 0.004){\left(V-I\right)}_{{\rm{O}}3}.\end{array}\end{eqnarray} \tag{ 7 }$$

The apparent color and the standard magnitude of the source star are (V − I, I)_S,O4,calib = (2.344 ± 0.031, 20.082 ± 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'' of the source star and obtained the color–color relations

$$\begin{eqnarray}\begin{array}{rcl}{I}_{{\rm{O}}3}-{I}_{\mathrm{CT}13} & = & (-0.880\pm 0.005)\\ & & +\,(-0.042\pm 0.005){\left(V-I\right)}_{\mathrm{CT}13},\end{array}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}\begin{array}{rcl}{V}_{{\rm{O}}3}-{I}_{\mathrm{CT}13} & = & (1.290\pm 0.004)\\ & & +\,(-0.036\pm 0.004){\left(V-I\right)}_{\mathrm{CT}13}.\end{array}\end{eqnarray} \tag{ 9 }$$

The apparent color and magnitude of the source star are (V − I, I)_S,CT13,calib = (2.335 ± 0.025, 20.105 ± 0.013). This color is consistent with (V − I)_S,O4,calib within 1σしぐま and the magnitude is consistent with I_S,O4,calib within 2σしぐま. Because the light curve was well covered by the OGLE observations, while it was highly magnified, we adopted (V − I, I)_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)_RCG = (2.720, 16.325) ± (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)_RCG,0 = (1.060, 14.362) ± (0.070, 0.040) (Bensby et al. 2013; Nataf et al. 2013). As a result, we obtained an extinction of A_I = 1.963 ± 0.051 and a color excess of E(V − I) = 1.660 ± 0.071. Finally, the intrinsic source color and magnitude were derived:

$$\begin{eqnarray}&&{\left(V-I,I\right)}_{{\rm{S}},0}=(0.684,18.119)\pm (0.077,0.053).\end{eqnarray} \tag{ 10 }$$

As a reference for later discussion of future follow-up observations, we also estimated the intrinsic source magnitudes in the 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 was derived from the extinction in the I and V bands of the RCGs according to Cardelli et al. (1989).

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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 having an absolute magnitude similar to that of the Sun but being somewhat in the foreground, e.g., at ∼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 those of the RCGs, the value of θしーた_E increases by only 7%, which is still consistent within 1σしぐま with values obtained assuming the same extinction and reddening as those of the RCGs. We summarize the source colors and magnitudes in Table 3.

Applying the empirical formula $\mathrm{log}({\theta }_{\mathrm{LD}})=0.501414\,+\,0.419685(V-I)-0.2I$ (see Fukui et al. 2015 but also Boyajian et al. 2014), where θしーた_LD ≡ 2θしーた_* is the limb-darkened stellar angular diameter, we found the angular source radius,

$$\begin{eqnarray}&&{\theta }_{\mathrm{LD}}=1.461\pm 0.109\,\mu \mathrm{as},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\theta }_{* }=0.730\pm 0.059\ \mu \mathrm{as}.\end{eqnarray} \tag{ 12 }$$

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

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}=\displaystyle \frac{{\theta }_{* }}{\rho }=0.211\pm 0.018\ \mathrm{mas},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\mu }_{\mathrm{rel},\mathrm{geo}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{{t}_{{\rm{E}}}}=4.832\pm 0.410\ \mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 14 }$$

This θしーた_E value is relatively small, which suggests that the lens is a low-mass star and/or distant from the observer.

If we can measure both the finite source effects and the parallax effects, the lens physical parameters such as the host mass M_host and the distance to the lens D_L are calculated directly, following the equations

$$\begin{eqnarray}\begin{array}{rcl}{M}_{\mathrm{host}} & = & \displaystyle \frac{{\theta }_{{\rm{E}}}}{(1+q)\kappa {\pi }_{{\rm{E}}}};\ {D}_{{\rm{L}}}=\displaystyle \frac{\mathrm{au}}{{\pi }_{\mathrm{rel}}+{\pi }_{{\rm{S}}}};\\ {\pi }_{\mathrm{rel}} & = & {\theta }_{{\rm{E}}}{\pi }_{{\rm{E}}};\ {{\boldsymbol{\mu }}}_{\mathrm{rel}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{{t}_{{\rm{E}}}}\displaystyle \frac{{{\boldsymbol{\pi }}}_{{\rm{E}}}}{{\pi }_{{\rm{E}}}},\end{array}\end{eqnarray} \tag{ 15 }$$

where κかっぱ ≡ 4G/(c²au) = 8.1439 mas/M_⊙, and πぱい_S = au/D_S is the source parallax. From the ground-based light curve alone, we are only able to measure θしーた_E (via finite source effects), and find no meaningful constraint on πぱい_E (see Section 3.3).

In order to estimate the probability distributions of M_L and D_L, we conducted a Bayesian analysis with the Galactic model of Koshimoto et al. (2021a). ⁷² We randomly generated a 50 million simulated microlensing event sample. Then we calculated the probability distributions for the lens physical parameters by weighting the microlensing event rate by the measured t_E and θしーた_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 of hosting the planet.

We calculated some parameters in addition to the lens physical parameters, M_L and D_L. For instance, the lens–source proper motion in the geocentric frame, μみゅー _rel, is converted to that in the heliocentric frame,

$$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{\mathrm{rel},\mathrm{hel}}={{\boldsymbol{\mu }}}_{\mathrm{rel}}+{{\boldsymbol{v}}}_{\oplus ,\perp }\displaystyle \frac{{\pi }_{\mathrm{rel}}}{\mathrm{au}},\end{eqnarray} \tag{ 16 }$$

where v _⊕,⊥ = (v_⊕,N, v_⊕,E) = (−0.78, 27.66) km s⁻¹ is the projected velocity of Earth at t₀.

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 the 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_dust = 0.10 ± 0.02 kpc (Bennett et al. 2015):

$$\begin{eqnarray}&&{A}_{\lambda ,{\rm{L}}}=\displaystyle \frac{1-{e}^{-| {D}_{{\rm{L}}}/({h}_{\mathrm{dust}}\sin b)| }}{1-{e}^{-| {D}_{{\rm{S}}}/({h}_{\mathrm{dust}}\sin b)| }}{A}_{\lambda ,{\rm{S}}},\end{eqnarray} \tag{ 17 }$$

where the index λらむだ refers to the passband: the 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 it to the extinction in the H and K bands according to Cardelli et al. (1989).

The results are shown in Table 4 and Figure 4. According to Figure 4, the lens system is likely to be a super-Earth with a mass of ${m}_{{\rm{p}}}={8.1}_{-4.4}^{+7.6}\ {M}_{\oplus }$ orbiting a late M dwarf with a mass of ${M}_{\mathrm{host}}={0.36}_{-0.19}^{+0.33}{M}_{\odot }$ at a projected separation of ${a}_{\perp }={1.54}_{-0.22}^{+0.18}\ \mathrm{au}$. The system is located at ${D}_{{\rm{L}}}={7.4}_{-0.9}^{+0.5}$ kpc from Earth. For reference, we also plot the source magnitudes in the V, I, H, and K bands as 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.

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Table 4. The Lens Physical Parameters

		Bayesian			Naive Spitzer-only
Parameters	Unit	Ground-only	Ground + ΔでるたfSpz	Ground + πぱい E, Spz	(u0 > 0)	(u0 < 0)
Mhost	M⊙	${0.36}_{-0.19}^{+0.33}$	${0.37}_{-0.21}^{+0.35}$	${0.091}_{-0.018}^{+0.064}$	0.073 ± 0.011	0.070 ± 0.010
mp	M⊕	${8.1}_{-4.4}^{+7.6}$	${8.4}_{-4.7}^{+7.9}$	${2.1}_{-0.4}^{+1.5}$	1.7 ± 0.3	1.6 ± 0.2
DL	kpc	${7.40}_{-0.85}^{+0.51}$	${7.40}_{-0.88}^{+0.51}$	${5.45}_{-0.66}^{+1.70}$	4.96 ± 0.74	4.89 ± 0.66
a⊥	au	${1.54}_{-0.22}^{+0.18}$	${1.54}_{-0.22}^{+0.18}$	${1.14}_{-0.15}^{+0.32}$	1.01 ± 0.18	0.99 ± 0.16
πぱいE	...	${0.075}_{-0.036}^{+0.087}$	${0.073}_{-0.035}^{+0.093}$	${0.292}_{-0.120}^{+0.066}$	0.354 ± 0.042	0.369 ± 0.037
μみゅーrel, hel	mas yr−1	${5.04}_{-0.44}^{+0.43}$	${5.06}_{-0.44}^{+0.43}$	4.86 ± 0.44	⋯	⋯
V	mag	${29.4}_{-2.6}^{+2.9}$	${29.3}_{-2.6}^{+3.1}$	${34.1}_{-1.6}^{+5.2}$	⋯	⋯
I	mag	${24.7}_{-2.0}^{+2.3}$	${24.6}_{-2.0}^{+2.4}$	${28.2}_{-1.2}^{+3.4}$	⋯	⋯
H	mag	${21.3}_{-1.6}^{+1.7}$	${21.2}_{-1.6}^{+1.9}$	${23.9}_{-0.9}^{+2.6}$	⋯	⋯
K	mag	${20.8}_{-1.5}^{+1.7}$	${20.8}_{-1.5}^{+1.8}$	${23.3}_{-0.8}^{+2.9}$	⋯	⋯

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We measure the microlens parallax vector πぱい _E via the satellite parallax effect, which can be approximated as

$$\begin{eqnarray}&&{{\boldsymbol{\pi }}}_{{\rm{E}}}=\displaystyle \frac{\mathrm{au}}{{D}_{\perp }}\left(\displaystyle \frac{{t}_{0,\mathrm{sat}}-{t}_{0,\oplus }}{{t}_{{\rm{E}}}},{u}_{0,\mathrm{sat}}-{u}_{0,\oplus }\right),\end{eqnarray} \tag{ 18 }$$

where D_⊥ is the Earth–satellite separation projected on the plane of the sky, and t_0,sat and u_0,sat are the time of lens–source closest approach and the impact parameter as seen by the satellite. The Einstein timescale t_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 Δでるたf_Spz ∼ 1 flux unit over the four-week observation period (see Figure 6). This change (rather than, e.g., the value of the flux at the start of observations) is the most robust constraint because it is independent of the unknown blended light. However, 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. 2020) and thus should be treated with caution. At the same time, 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 a constraint on the Spitzer source flux:

$$\begin{eqnarray}&&{I}_{\mathrm{CT}13}-L=-4.518\pm 0.028,\end{eqnarray} \tag{ 19 }$$

which gives an expected source flux from Spitzer of f_S,Spz = 0.6254 flux units for the best-fit value of I_CT13. This constraint and the best-fit ground-based model (Table 4) 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 (${f}_{\mathrm{Spz}}(\mathrm{HJD}^{\prime} =8313.83)\sim 6$ flux units) and declined by a total of Δでるたf_Spz ∼ 3.3 flux units as compared to the observed Δでるたf_Spz ∼ 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.

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We can use limits on the change in Spitzer flux (Δでるたf_Spz) 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 Δでるたf_Spz ∼ 4 flux units, it is very unlikely that systematics would cause us to measure Δでるたf_Spz = 1 flux unit. Therefore, we repeat the Bayesian analysis imposing the constraint Δでるたf_Spz < 4, where Δでるたf_Spz is calculated from Equation (19). The parallax effect can produce a degeneracy in the sign of u₀. In this case, because u₀ 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 u₀ > 0 case.

The results are given in Table 4 (as "Ground + Δでるたf_Spz"), Figure 7, and the center panel of Figure 5. This constraint suggests an ${M}_{\mathrm{host}}={0.37}_{-0.21}^{+0.35}\ {M}_{\odot }$ host with an ${m}_{{\rm{p}}}={8.4}_{-4.7}^{+7.9}\ {M}_{\oplus }$ planet at a projected separation ${a}_{\perp }={1.54}_{-0.22}^{+0.18}\ \mathrm{au}$. We adopt these values as our conservative Bayesian estimates of the properties of the lens system.

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6.1. Spitzer-only Parallax

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 t₀, u₀, and t_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. ⁷³ Then, for a grid of parallax values, we fit for the Spitzer flux while applying the color constraint from Equation (19). We repeat the analysis for −u₀, 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 circle formalism. The values for the magnitude of the microlens parallax vector are πぱい_E = 0.35 ± 0.04 for the (u₀ > 0) case and πぱい_E = 0.37 ± 0.04 for the (u₀ < 0) case. The 3σしぐま ranges are πぱい_E = [0.18, 0.50] and πぱい_E = [0.20, 0.48], respectively.

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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 θしーた_E = 0.211 ± 0.019 mas from fitting the ground-based light curve. These estimates and their uncertainties are derived from Equation (15) using simple error propagation, and so are the naive values of these quantities. For the (u₀ > 0) solution, this yields a lens mass of M_L = 0.073 ± 0.011 M_⊙ and D_L = 4.96 ± 0.74 kpc for D_S = 7.88 kpc. This would then imply that the mass of the planet is m_p = 1.7 ± 0.3 M_⊕ and that it is separated from the host by a_⊥ = 1.01 ± 0.18 au. The values for the (u₀ < 0) solution are comparable. See Table 4.

In order to estimate the lens magnitude, we also performed a Bayesian analysis including the πぱい_E constraint derived from the Spitzer-only parallax analysis. First, we took the average of the χかい² values for the two (u₀ > 0) and (u₀ < 0) solutions for each value of πぱい_E,E and πぱい_E,N. Then, the event rate was weighted by $\exp (-{\rm{\Delta }}{\chi }^{2}/2)$ and the measured t_E and θしーた_E constraints to calculate the probability distribution. Table 4 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_L ∼ 5 kpc, there is a second peak for lenses with D_L ∼ 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 4 are not a complete description of the distributions and should be considered in the context of Figure 9. However, the mass distribution is not subject to this issue. We find that the lens system is likely to be a terrestrial planet with a mass of ${m}_{{\rm{p}}}={2.1}_{-0.4}^{+1.5}\ {M}_{\oplus }$ orbiting a very low mass (VLM) dwarf with a mass of ${M}_{\mathrm{host}}={0.091}_{-0.018}^{+0.064}\ {M}_{\odot }$.

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6.3. Implications

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 an ${m}_{{\rm{p}}}={1.43}_{-0.32}^{+0.45}\ {M}_{\oplus }$ 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 πぱい _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 μみゅー_S because there is no evidence that the blend, which dominates the baseline object, is associated with the event (see Appendix). Second, given D_L ∼ 4.9 kpc, we assume that the lens is in the disk and therefore has a proper motion similar to that of other disk stars. The velocity model of Koshimoto et al. (2021a) 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 $({v}_{\phi },{v}_{z})=({207.6}_{-44.0}^{+42.7},-{0.4}_{-39.6}^{+38.8})\ \mathrm{km}\,{{\rm{s}}}^{-1}$. Hence, for the velocity dispersion, we use (σしぐま_v,ϕ , σしぐま_v,z ) = (43.4, 39.2) km s⁻¹. Table 5 summarizes the disk star velocities and proper motions expected from the Galactic model at D = 4.9 ± 0.7 kpc. The values in the table are derived from the Bayesian analysis with a Galactic prior and constrained by θしーた_E and t_E. For the Sun's motion, we use ${({v}_{R},{v}_{\phi },{v}_{z})}_{\mathrm{Sun}}=(-10,243,7)\ \mathrm{km}\,{{\rm{s}}}^{-1}$ (for (R_⊙, z_⊙) = (8160, 25) pc). We combine the two velocities to estimate the proper motion of disk stars. Finally, by applying Equation (16), we can derive the expected source proper motion μみゅー_S = μみゅー_L − μみゅー_rel,hel for a given value of the parallax. Figure 10 shows the results for values of πぱい _E out to the 1σしぐま 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' of the target: μみゅー_bulge(ℓ, b) = (−6.310, −0.163) ± (0.088, 0.076) mas yr⁻¹ and σしぐま_bulge(ℓ, b) = (3.176, 2.768) ± (0.062, 0.054) 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 2σしぐま from the bulge distribution, but the other minimum is consistent with a bulge source at ∼1.5σしぐま. Therefore, there is no reason to believe that the Spitzer πぱい _E requires a lens proper motion in tension with the motion of typical disk stars.

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Table 5. Disk Star Velocities and Proper Motions at D = 4.9 ± 0.7 kpc

Star Component	Velocity Component	Unit	−2σしぐま	−1σしぐま	Median	+1σしぐま	+2σしぐま
Thin Disk	vl	km s−1	110.7	163.6	205.9	242.4	280.7
	v b	km s−1	−95.0	−48.8	−13.7	22.4	71.1
	μみゅーhel,l	mas yr−1	−5.615	−3.349	−1.577	−0.024	1.656
	μみゅーhel,b	mas yr−1	−4.364	−2.388	−0.884	0.653	2.690
Thick Disk	vl	km s−1	60.7	125.2	181.4	236.5	293.7
	v b	km s−1	−147.8	−86.4	−12.1	63.3	128.6
	μみゅーhel,l	mas yr−1	−7.662	−4.995	−2.602	−0.275	2.177
	μみゅーhel,b	mas yr−1	−6.577	–3.987	−0.808	2.379	5.102
All	vl	km s−1	103.6	161.0	204.9	242.2	281.3
	v b	km s−1	−101.1	−50.5	−13.6	24.4	77.4
	μみゅーhel,l	mas yr−1	−5.878	−3.457	−1.620	−0.034	1.685
	μみゅーhel,b	mas yr−1	−4.611	−2.462	−0.883	0.737	2.960

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Finally, in order to include the event in the statistical sample for the study of the Galactic distribution of planets, Zhu et al. (2017) proposed the criteria

$$\begin{eqnarray}&&\sigma ({D}_{8.3})\lt 1.4\mathrm{kpc};\ {D}_{8.3}\equiv \displaystyle \frac{\mathrm{kpc}}{{\pi }_{\mathrm{rel}}/\mathrm{mas}+1/8.3}.\end{eqnarray} \tag{ 20 }$$

We find D_8.3 = 5.15 ± 0.28 kpc for the (u₀ > 0) case and D_8.3 = 5.04 ± 0.28 kpc for the (u₀ < 0) case by combining the measurement of πぱい _E from the Spitzer-only analysis with the measurement of θしーた_E from fitting the ground-based light curve. The small σしぐま(D_8.3) is consistent with the expectation for a 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 Spitzer when the Earth-based magnification is high (A_⊕ ≥ 100). Therefore, in terms of σしぐま_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 to be studied and understood before membership in the sample can be definitively evaluated.

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 low-mass ratio of q = (6.9 ± 0.2) × 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 30 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. 2021, in preparation), and its low-mass ratio will help define the mass-ratio function peak.

From the ground-based light-curve modeling, only the 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 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 of hosting the planet. This analysis indicates a host mass of ${M}_{\mathrm{host}}={0.37}_{-0.21}^{+0.35}\ {M}_{\odot }$ and a planet mass of ${m}_{{\rm{p}}}={8.4}_{-4.7}^{+7.9}\ {M}_{\oplus }$ located at ${D}_{{\rm{L}}}={7.4}_{-0.9}^{+0.5}$ kpc. By contrast, the Spitzer data favor a larger microlensing parallax, which implies a VLM host with a terrestrial planet (${M}_{\mathrm{host}}={0.091}_{-0.018}^{+0.064}\ {M}_{\odot }$, ${m}_{{\rm{p}}}={2.1}_{-0.4}^{+1.5}\ {M}_{\oplus }$) that is either in the disk at D_L ∼ 5 kpc or in the bulge at D_L ∼ 7.5 kpc (these values include a Galactic prior but are not significantly different from the values without the prior; see Table 4).

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 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.

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However, the result that this is a terrestrial planet orbiting a 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_E, θしーた_E, and the Galactic priors. At the same time, a VLM dwarf + terrestrial planet is still within the 2σしぐま range of possibilities from the Bayesian analysis, especially once the constraint on Δでるたf_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 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 + Δでるたf_Spz constraints indicates the lens K-band magnitude with extinction should be $K={20.8}_{-1.5}^{+1.8}$ mag, which is about 3 mag fainter than the source. By contrast, if the Spitzer-only parallax is correct and the lens is a VLM dwarf, it should be $K={23.3}_{-0.8}^{+2.9}$ mag and therefore, much fainter and possibly undetectable. The Bayesian estimate of the heliocentric relative proper motion, μみゅー_rel,hel = 5.0 ± 0.4 mas yr⁻¹, predicts that the angular separation between the source and the lens will be ∼30 mas around mid-2024. Thus, the lens can be resolved from the source by future follow-up observations with Keck or the Extremely Large Telescope. If such resolved measurements were made (and the lens were luminous), they would also lead to a direct measurement of μみゅー . The observed magnitude of μみゅー can serve as a check on θしーた_E. Additionally, the direction of μみゅー 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 for measuring 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 number. 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 (principal investigator: Takahiro, Sumi) and the Roman Space Telescope (Spergel et al. 2015; Penny et al. 2019) in the mid-2020s.

Work by I.K. was supported by JSPS KAKENHI grant No. 20J20633. Work by J.C.Y. was supported by Jet Propulsion Laboratory grant 1571564. Work by D.P.B., A.B., and C.R. was ⁵ supported by NASA through grant NASA-80NSSC18K0274. T.S. acknowledges financial support from the JSPS (JSPS23103002, JSPS24253004, and JSPS26247023). Work by N.K. is supported by JSPS KAKENHI grant No. 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). T.C.H. acknowledges financial support from the National Research Foundation (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 Nos. 11821303 and 11761131004). Work by C.H. was supported by grants of the 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 KAKENHI grant Nos. 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えーゆー. This research has made use of the KMTNet system operated by the Korea Astronomy and Space Science Institute and the data were obtained at the 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 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 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.

Software: OGLE DIA pipeline (Udalski 2003), MOA DIA pipeline (Bond et al. 2001), KMTNet pySIS pipeline (Albrow et al. 2009), DanDIA (Bramich 2008; Bramich et al. 2013), DoPHOT (Schechter et al. 1993), ISIS (Alard & Lupton 1998; Alard 2000; Zang et al. 2018), image-centered ray-shooting method (Bennett & Rhie 1996; Bennett 2010).

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 0175. 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 (∼1000 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 yr of OGLE survey data and find μみゅー _base(R.A., decl.) = (−6.00 ± 0.26, −4.25 ± 0.16) mas yr⁻¹. Because the blend is much brighter than the source, its motion should dominate the measured u _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. 2021) reports there is a G = 20.1 mag star 0177 from the OGLE coordinates for the baseline star (17:59:10.26–27:50:06.3). The reported proper motion of this source is u (R.A., decl.) = (−12.173 ± 1.247, −9.714 ± 0.870) 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 u (R.A., decl.) = (−8.475 ± 2.234, −4.039 ± 1.985) 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.