OGLE-2012-BLG-0563Lb: A SATURN-MASS PLANET AROUND AN M DWARF WITH THE MASS CONSTRAINED BY SUBARU AO IMAGING

The microlensing event OGLE-2012-BLG-0563 occurred on a star located at the equatorial coordinate of (α, δ)_J2000 = (18^h05^m5772, −27°42'432) and the Galactic coordinate of (l, b) = (331, −325). This event was first discovered by the OGLE collaboration on 2012 May 1 UT ($\mathrm{JD}^{\prime} \;\equiv$ JD-2450000 = 6049), during its regular photometric monitoring toward the Galactic bulge by using the 1.3 m Warsaw telescope equipped with the wide field (1.4 deg²) camera OGLE-IV at Las Campanas Observatory in Chile (Udalski et al. 2015). The event was discovered in the field BLG519, which was regularly monitored twice a night with an I-band filter. The same event was independently discovered as MOA-2012-BLG-288 by the MOA collaboration on 2012 May 18 UT (JD' = 6066), using the 1.8 m MOA-II telescope and the wide field (2.2 deg²) camera moa-cam3 (Sako et al. 2008) at Mt. John University Observatory (MJUO) in New Zealand. MOA monitored the event (in field gb14) with a typical cadence of 20 minutes using a custom $R+I$ filter. After these discoveries, both teams have made the light curves public and updated them frequently.

On May 20 (JD' = 6067.9), the μFUN collaboration circulated an alert that the event was peaking at a high magnification, which means that the source star was approaching very close to the lens star on the sky plane. In such a case, there is a high probability that the light curve will show anomalous features around the peak if the lens star hosts a planet, because one of the caustics (central caustic) produced by a planetary system is always created near the host star on the sky plane (Griest & Safizadeh 1998; Rhie et al. 2000; Han & Kim 2001).

After the circulation of the high-magnification alert, the μFUN and RoboNet collaborations started to follow up the event with high cadence photometry. μFUN used the following telescopes (filters): the 1.3 m CTIO telescope in Chile (V, I, and H), the 0.40 m Auckland telescope in New Zealand (R), the 0.36 m Possum telescope in New Zealand (R), the 0.36 m Farm Cove telescope in New Zealand (clear filter), and the 0.30 m PEST in Australia (clear filter). The 1.3 m CTIO's H-band data were simultaneously obtained with the V- or I-band data with the same instrument. RoboNet used three 2.0 m telescopes, including the Liverpool Telescope (LT) in Canary Islands, Spain (i'); the Faulkes Telescope North (FTN) in Hawaii, USA (i'); and the Faulkes Telescope South (FTS) in Australia (i'). In addition, OGLE and MOA increased the observational cadence with their survey telescopes. MOA also utilized the 0.61 m B&C telescope at MJUO with I-band filter to follow up the event. On May 21 13:30 UT (JD' = 6069.06), MOA reported the detection of an anomaly as a result of its high cadence observations. The event peaked at around May 21 13:20 (JD' = 6069.06) with the maximum magnification of more than 600, around which the event was well covered by several telescopes in New Zealand and Australia. As a result, a quick look light curve clearly showed an anomalous asymmetric feature around the peak, meaning that the lens star hosted a companion. Prompt analyses of the light curve by several teams indicated that this was a firm planetary event having the planet–star mass ratio of $\sim {10}^{-3}$. We summarize these photometric observations in Table 1, and show the observed light curve in Figure 1. We note that data obtained by Possum, Farm Cove, and FTN are not used for the analyses due to the following reasons: data from Farm Cove do not constrain the model because they consist of just two short epochs, and data from Possum and FTN are of relatively low quality.

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Table 1.  Summary of Observations

Telescope	Diameter (m)	Filter	Nusea/Nobs b
The OGLE collaboration
OGLE	1.3	I	700/751
The MOA collaboration
MOA-II	1.8	$R+I$	5449/5947
B&C	0.61	I	385/439
The μFUN collaboration
CTIO 1.3 m	1.3	V	12/12
CTIO 1.3 m	1.3	I	30/31
CTIO 1.3 m	1.3	H	110/140
PEST	0.30	clear	163/172
Auckland	0.40	R	44/46
Possum	0.36	R	0/21
Farm Cove	0.36	clear	0/23
The RoboNET collaboration
FTS	2.0	i'	195/218
LT	2.0	i'	20/23
FTN	2.0	i'	0/10

Notes.

^a The number of data points used in the analysis. ^b The number of observed data points.

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In addition to the light curve, a high-resolution spectrum of the source star was obtained by Bensby et al. (2013) with VLT/UVES on 2012 May 19 (JD' = 6067), when the source was magnified by a factor of ∼50. They reported that the source star was a metal-poor G dwarf with ${T}_{\mathrm{eff}}=5907\pm 89$ K, $\mathrm{log}g=4.40\pm 0.10$, and [Fe/H]$\;=-0.66\pm 0.07$.

After the event was over, all the observed photometric data were carefully re-reduced by their respective teams. For the re-reductions, OGLE and MOA used their customized pipelines as described in Udalski et al. (2015) and Bond et al. (2001), respectively; μFUN used the DoPHOT package (Schechter et al. 1993) for the CTIO 1.3 m H-band data and a variant of the PySIS package (Albrow et al. 2009) for the others; and RoboNET used the DanDIA package (Bramich 2008). All the pipelines except for DoPHOT apply the Difference Image Analysis method for photometry in order to achieve good photometric precisions in the star-crowding field. Note that because a bright neighboring star was located 22 away from the target star, the photometry was carefully done to minimize systematics (e.g., by masking the bright star). The time for each data point was converted to heliocentric julian day (HJD) based on UT.

We conducted high-resolution imaging of the event field by using the 8.2 m Subaru telescope equipped with the AO instrument AO188 and Infrared Camera and Spectrograph (IRCS, Kobayashi et al. 2000) at 6:06–7:03 UT on 2012 July 28 (JD' = 6137.8), when the source star was still magnified by a factor of 1.47. We used the "high-resolution" mode of IRCS, which provides a pixel scale of 20.6 mas pixel⁻¹ and a field of view (FOV) of 211 × 211. We used the natural guide star (NGS) mode for AO. As an NGS, we selected a bright neighboring star having R = 11 and being separated by $26^{\prime\prime}$ from the source star. The field was observed through J-, H-, and K'-band filters, each with 30 s exposure×15 times at five dithering points within $2^{\prime\prime}$ (100 pixel) square. The airmass toward the target field was 1.74–1.54 during the observations. The natural seeing was ∼05–06, and the AO-worked seeing was 023–030, 019–024, and 017–025 for J, H, and K', respectively.

All the J-, H-, and K'-band AO images were dark-subtracted and flat-fielded in a standard manner. For the flat fielding, we used sky flat images obtained during evening twilight on the observation night. All the images in each band were combined in average, after the image positions were aligned and sky levels were subtracted. The image-overlapping region was trimmed for further analyses, which resulted in the effective FOV of $18^{\prime\prime} \;\times \;$ $18^{\prime\prime}$.

Because the light curve of the event clearly shows an asymmetric feature around the peak that cannot be explained by a standard single-lens microlensing model, we introduce a binary-lens microlensing model assuming that the lens consists of two objects. The standard binary-lens model can be described with seven basic parameters: the time of the closest source approach to the binary centroid, t₀; the Einstein radius (R_E) crossing time, t_E; the minimum impact parameter, u₀; the mass ratio of the binary components, q; the projected separation of the binary components in units of R_E, s; the angle between the source star trajectory and the binary-lens axis, α; and the angular source radius in units of ${\theta }_{{\rm{E}}}$, ρ. In addition, the model requires two instrument-dependent parameters of the unmagnified source flux F_S and the blended flux F_B. We model the intensity distribution on the surface of the source star with a linear limb-darkening raw of $I(\theta )=I(0)$ $[1-{u}_{X}(1-\mathrm{cos}\theta )]$, where θ is the angle between the normal to the stellar surface and the line of sight, $I(\theta )$ is the stellar intensity as a function of θ, and u_X is a coefficient for filter X. For u_X, we adopt the theoretical values of Claret et al. (2013) for a G dwarf with the temperature of 5900 K and log g = 4.5, namely, u_V = 0.656, u_R = 0.572, ${u}_{R+I}$ = 0.528, ${u}_{{i}^{\prime }}$ = 0.504, u_I = 0.483, and u_H = 0.290. The ${u}_{R+I}$ value is calculated as the mean of u_R and u_I. For clear filter, we adopt u_R as an approximation. The model calculation is done by using a customized code developed by MOA, which is based on the image-centered ray-shooting method (Bennett & Rhie 1996; Bennett 2010).

The initially calculated flux uncertainties in the light curves are normalized as follows. First, all the light curves are simultaneously fit with the binary-lens models by the Markov Chain Monte Carlo (MCMC) method following Sumi et al. (2010). Then the flux uncertainties in each light curve are scaled by a single factor so that the ${\chi }^{2}$ per degrees of freedom, ${\chi }_{\mathrm{red}}^{2}$, for each light curve becomes unity. This process is iterated until the best-fit light curve model becomes converged. Next, after excluding 4σ outliers, we renormalize the flux uncertainties for each light curve using the following formula,

$$\begin{eqnarray}&&{\sigma }_{i}^{\prime }=k\sqrt{{\sigma }_{i}^{2}+{e}_{\mathrm{min}}^{2}},\end{eqnarray} \tag{ 1 }$$

where ${\sigma }_{i}$ is the initial error bar of the ith data point in magnitude, and k and e_min are coefficients for each data set. Here, the term e_min represents systematic errors that dominate when the stellar brightness significantly increases. We adjust the k and e_min values so that the cumulative ${\chi }^{2}$ distribution sorted by magnitude is close to linear, and ${\chi }_{\mathrm{red}}^{2}$ becomes unity. Finally, all the normalized light curves are fit again and the flux uncertainties are rescaled by a single factor so that ${\chi }_{\mathrm{red}}^{2}$ for all data becomes unity.

In modeling binary microlensing light curves, several models are often degenerate. In particular for central-caustic crossing and approaching events, a severe degeneracy between s and ${s}^{-1}$ often occurs because the central-caustic pattern created by a binary lens with s and that with ${s}^{-1}$ can be very similar (wide-close degeneracy; Dominik 1999). To check for this type of degeneracy and possible other degeneracies, we calculate an ${\chi }^{2}$ map in the $\mathrm{log}q$ and $\mathrm{log}s$ plane by dividing $\mathrm{log}q$ ([−5, 0]) and $\mathrm{log}s$ ([−0.7, 0.7]) into 50 × 40 grids and fitting the light curve while fixing $\mathrm{log}q$ and $\mathrm{log}s$ at each section. These $\mathrm{log}q$ and $\mathrm{log}s$ ranges are chosen such that the usual sensitivity region of microlensing is well covered. The calculated ${\chi }^{2}$ map is shown in Figure 2. The map shows that there are two local minima at ($\mathrm{log}s$, $\mathrm{log}q$) ∼ (−0.4, −3) and (0.4, −3), indicating that the wide-close degeneracy clearly exists. On the other hand, there is no other local minimum in the map, and the q value at the two local minima of ∼10⁻³ is well within the planetary range ($q\lt {10}^{-2}$ for G- and later-type dwarf hosts) with ${\rm{\Delta }}{\chi }^{2}\gt 1000$ over the models of $q\gt {10}^{-2}$. Therefore, the planetary nature of this event is quite robust. Note that one can wonder whether there are any other degenerated binary (small-mass ratio) models outside the searched $\mathrm{log}s$ range. However, a very close binary with $\mathrm{log}s\lt -0.7$ would cause a higher-order effect in the light curve, due to its orbital motion, which we do not detect. In addition, we also check for very wide binary models with $0.7\lt \mathrm{log}s\lt 2$, but find no comparable models with the planetary one.

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To properly derive the best-fit model parameters and their uncertainties for the wide and close models, we re-run the MCMC program by starting with the parameter values at each local minimum and letting all parameters be free. For each model, we conduct 40 independent MCMC runs with 3200 steps each, and create posterior probability distributions of the parameters from the merged ∼10⁵ steps. The best-fit value and its 1σ lower (upper) uncertainties for each parameter are calculated as the median and 15.9 percentile (84.1 percentile) values of the posterior probability distribution, respectively. The best-fit light curve and caustic models are shown in Figure 1, while the resultant parameter values, their uncertainties, and the minimum-${\chi }^{2}$ values for the wide and close models are listed in Table 2. The difference of the minimum-${\chi }^{2}$ values of the two models is only 0.6, meaning that these two models are indistinguishable. Note that all the parameter values except for s are almost identical between the two models, and we proceed with further analyses of only the close model as representative of the two, unless s is relevant.

Table 2.  Results of the MCMC Analysis

Parameter	Units	Close	Wide
t0	HJD-2450000	6069.02790 ± 0.00026	6069.02811 ± 0.00027
tE	days	77.5 ± 2.2	77.7 ± 2.1
u0	10−3	1.405 ± 0.040	1.403 ± 0.039
q	10−3	1.086 ± 0.040	1.085 ± 0.039
s	${\theta }_{{\rm{E}}}$	0.4134 ± 0.0032	2.425 ± 0.019
α	rad	5.7843 ± 0.0013	5.7841 ± 0.0013
ρ	10−4 ${\theta }_{{\rm{E}}}$	3.27${}_{-0.26}^{+0.24}$	3.27${}_{-0.26}^{+0.24}$
${\chi }_{\mathrm{min}}^{2}$		7068.6	7069.2

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In order to know the absolute masses and projected separation of the planetary system instead of q and s, one needs to derive the total mass of the planetary system, m_L, and the distance from the Earth to the system, D_L. One way to do this is to detect the microlens parallax effect in the light curve, which is seen as a slightly asymmetric distortion over the non-parallax light curve. The distortion depends on m_L, D_L, the vectors of the Earth's orbital acceleration, and the lens-source relative proper motion (Gould 1992). From this effect, one can obtain an additional parameter ${\pi }_{{\rm{E}}}$, which is defined as the ratio of 1 AU to the projected Einstein radius onto the Earth's plane (Gould 2000). From ${\pi }_{{\rm{E}}}$ and the angular Einstein radius, ${\theta }_{{\rm{E}}}$, which will be measured in Section 4, one can derive m_L and D_L as

$$\begin{eqnarray}&&{m}_{{\rm{L}}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{\kappa {\pi }_{{\rm{E}}}}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{D}_{{\rm{L}}}=\displaystyle \frac{\mathrm{AU}}{{\pi }_{{\rm{E}}}{\theta }_{{\rm{E}}}+1/{D}_{{\rm{S}}}},\end{eqnarray} \tag{ 3 }$$

where κ $\equiv \;4G/{c}^{2}$ $\simeq \;8.144$ $\mathrm{mas}\;{M}_{\odot }^{-1}$ and D_S is the source distance. Here G is the gravitational constant and c is the speed of light.

To search for the microlens parallax effect, we fit the light curve by freeing two additional parameters of ${\pi }_{{\rm{E}},{\rm{E}}}$ and ${\pi }_{{\rm{E}},{\rm{N}}}$, which are the east and north components of ${\boldsymbol{}}{\pi }_{{\rm{E}}}$, respectively, where ${\boldsymbol{}}{\pi }_{{\rm{E}}}$ is a vector whose length is ${\pi }_{{\rm{E}}}$ and direction is the same as the lens direction relative to the source.

As a result, the parallax model fit gives the best-fit values of ${\pi }_{{\rm{E}},{\rm{E}}}$ = −0.024 ± 0.058 and ${\pi }_{{\rm{E}},{\rm{N}}}$ = 0.62 ± 0.17 with the ${\chi }^{2}$ of 7054.1, which has the improvement of 14.5 compared with that for the non-parallax model. Statistically speaking, this is a marginal (3.8σ) detection of the parallax signal, however, we consider it suspect because of the following reasons.

The difference between the best-fit parallax and non-parallax models emerges around the wings of the light curve, where the MOA-II and OGLE data are dominant. In Figure 3, we show the MOA-II and OGLE light curves along with the best-fit parallax (green) and non-parallax (cyan) models in the top panel, the residuals of the observed light curves against the non-parallax model in the middle panel, and the difference of cumulative ${\chi }^{2}$ between the two models in the bottom panel. The residual plot shows that the difference of the two models is quite small compared to the error bars of the data points. In such a case, systematics in a small number of data points can lead to a false positive detection. In fact, as indicated by the Δcumulative-${\chi }^{2}$ plot, the major ${\chi }^{2}$ improvement comes from the MOA-II data in the limited range of 6070 ≲ HJD-2450000 ≲ 6110, although the model difference emerges more widely over several hundred days. Moreover, the Δcumulative-${\chi }^{2}$ plots for the MOA-II and OGLE data are anti-correlated rather than correlated, indicating that the detected parallax signal is suspicious. Therefore, the detected signal is probably a false positive, and we only calculate an upper limit on ${\pi }_{{\rm{E}}}\lt 1.53$ (3σ). We note that other effects that can mimic the parallax signal, such as the orbital motion of the source and/or lens systems, can also be rejected for the same reasons.

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We first derive the apparent I- and V-band magnitudes of the source star, I_S and V_S, respectively, from the light-curve fitting. We derive I_S from the two light-curve data sets obtained by the OGLE and CTIO 1.3 m telescopes. Calibrating the instrumental magnitude to the standard (Landolt) one via the OGLE-III catalog (Szymański et al. 2011), we obtain ${I}_{{\rm{S}}}=20.147\pm 0.031$ and ${I}_{{\rm{S}}}=20.082\pm 0.031$ from the respective data sets. By taking the mean of the two values, we obtain a final value of I_S = 20.1150.031. Note that the uncertainty of I_S is conserved in this calculation because this uncertainty is dominated by the light-curve model uncertainty and therefore the two I_S values are correlated. V_S is derived from the V-band light curve obtained by the CTIO 1.3 m telescope. In the same way as for the I band, we derive the calibrated source magnitude of ${V}_{{\rm{S}}}=21.595\pm 0.032$. Then, from I_S and V_S, we obtain the apparent source color of ($V-I$)_S = 1.480 ± 0.032.

Next, we estimate the extinction and reddening for the source star using red clump stars in the Galactic bulge. In Figure 4, we plot the color–magnitude diagram (CMD) toward the event coordinate (∼28 × 28 area) created from the OGLE-III catalog, along with the measured I_S and ($V-I$)_S. The source position is largely consistent with, but a bit fainter than, the region of MS stars in the Galactic bulge, which is consistent with the fact that the source is located behind the center of the bulge along the line of sight (see Section 4.3). The centroid of the red clump stars on the CMD is measured as ($V-I$)${}_{\mathrm{RC}}=1.883\pm 0.013$ and ${I}_{\mathrm{RC}}=15.361\pm 0.015$, respectively. These values are then compared to the intrinsic color and magnitude of the red clump stars of ($V-I$)${}_{\mathrm{RC},0}=1.06\pm 0.06$ (Bensby et al. 2011) and ${I}_{\mathrm{RC},\ 0}=14.34\pm 0.04$ (Nataf et al. 2013), leading to the reddening and extinction magnitudes for the red clump stars of $E(V-I)=0.82\pm 0.06$ and ${A}_{I}=1.02\pm 0.04$, respectively. Assuming that this reddening and extinction can also be applied for the source star, we derive the intrinsic color and magnitude of the source star as ($V-I$)${}_{{\rm{S}},0}=0.66\pm 0.07$ and ${I}_{{\rm{S}},0}=19.10\pm 0.05$, respectively.

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Finally, we estimate the angular radius of the source star ${\theta }_{*}$ from ${I}_{{\rm{S}},0}$ and ($V-I$)${}_{{\rm{S}},0}$ using the following equation:

$$\begin{eqnarray}&&\mathrm{log}{\theta }_{\mathrm{LD}}=0.5014+0.4197(V-I)-0.2I,\end{eqnarray} \tag{ 4 }$$

where ${\theta }_{\mathrm{LD}}$ is the limb-darkened stellar angular diameter. This linear equation is derived from a subset of the interferometrically measured stellar radii presented in Boyajian et al. (2014), restricting stars with 3900 K $\lt \;{T}_{\mathrm{eff}}\;\lt$ 7000 K to improve the fit for FGK stars. This gives ${\theta }_{*}\equiv {\theta }_{\mathrm{LD}}/2=0.454\pm 0.033$ μas.

The derived source properties are summarized in Table 3.

Table 3.  Properties of the Source Star

Parameter	Value	Value
	(w/o spec. info)	(w/spec. info)
Teff (K)	⋯	5907 ± 89a
[Fe/H]	⋯	−0.66 ± 0.07a
$\mathrm{log}g$ (cgs)	⋯	4.40 ± 0.10a
VS	21.595 ± 0.032	⋯
IS	20.115 ± 0.031	⋯
HS	18.569 ± 0.032	⋯
${(V-I)}_{{\rm{S}},0}$	0.66 ± 0.07	0.660 ± 0.027
${(V-H)}_{{\rm{S}},0}$	⋯	1.41 ± 0.06
${V}_{{\rm{S}},0}$	19.76 ± 0.08	19.73 ± 0.07
${I}_{{\rm{S}},0}$	19.10 ± 0.05	19.14 ± 0.08
${H}_{{\rm{S}},0}$	⋯	18.314 ± 0.041
$E(V-I)$	0.82 ± 0.06	0.820 ± 0.042
$E(J-{K}_{{\rm{s}}})$	0.244 ± 0.109b	0.266 ± 0.019
AV	1.84 ± 0.07	1.80 ± 0.09
AI	1.02 ± 0.04	0.98 ± 0.08
AJ	⋯	0.423 ± 0.024
AH	⋯	0.255 ± 0.029
${A}_{{K}_{{\rm{s}}}}$	⋯	0.157 ± 0.017
${\theta }_{*}$ [μas] (I, $V-I$)	0.454 ± 0.033	0.446 ± 0.023c
${\theta }_{*}$ [μas](H, $V-H$)	⋯	0.444 ± 0.014
MV	⋯	4.99${}_{-0.14}^{+0.26}$
MI	⋯	4.28${}_{-0.19}^{+0.25}$
MH	⋯	3.54${}_{-0.16}^{+0.24}$
t (Gyr)	⋯	13${}_{-5}^{+0}$
DS (kpc)	⋯	9.1${}_{-1.1}^{+0.9}$

Notes.

^a From Bensby et al. (2013). ^b From the BEAM calculator (Gonzalez et al. 2012). ^c The value adopted for the rest of analyses.

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4.2.1. From $(V-I)$ and I

The spectroscopically measured effective temperature ${T}_{\mathrm{eff}}=5907\pm 89$ K and metallicity [Fe/H] = −0.66 ± 0.07 (Bensby et al. 2013) of the source star can be directly converted to the source's intrinsic color. Using the color-metallicity-temperature relation of Casagrande et al. (2010), we derive ($V-I$)${}_{{\rm{S}},0}=0.660\pm 0.027$. This value is in good agreement with the photometrically derived ($V-I$)${}_{{\rm{S}},0}$, but has a smaller uncertainty by a factor of ∼2.5. Subtracting this value from the apparent source color of ($V-I$)${}_{{\rm{S}}}=1.480\pm 0.032$, we derive the reddening magnitude up to the source star of $E(V-I)=0.820\pm 0.042$. Then, assuming that the extinction law

$$\begin{eqnarray}{A}_{I} & = & 1.217\\ & & \times E(V-I)[1+1.126\times ({R}_{{JKVI}}-0.3433)]\end{eqnarray} \tag{ 5 }$$

(Nataf et al. 2013) applies along the line of sight, and adopting ${R}_{{JKVI}}\equiv E(J-{K}_{{\rm{s}}})/E(V-I)=0.3249$ for the event coordinate from the online Extinction Calculator⁴⁸ (Nataf et al. 2013), we obtain ${A}_{I}=0.98\pm 0.08$ (the error includes the uncertainty of Equation (5), for which we adopt 0.06 mag). Consequently, we derive the intrinsic source magnitude of ${I}_{{\rm{S}},0}={I}_{{\rm{S}}}-{A}_{I}=19.14\pm 0.08$. This value is also consistent with the previous estimation but has a bit larger uncertainty, which mainly comes from the uncertainty of the estimation of A_I. Note, however, that this A_I estimation is not based on any assumption about the absolute extinction or reddening, but on the extinction law of Equation (5). The source angular radius is then calculated using Equation (4) as

$$\begin{eqnarray}&&{\theta }_{*}=0.446\pm 0.023\ \mu \mathrm{as},\end{eqnarray} \tag{ 6 }$$

whose uncertainty is 1.4 times smaller than the previous one. Using this ${\theta }_{*}$ and ρ, we derive the angular Einstein radius of

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}={1.36}_{-0.12}^{+0.14}\ \mathrm{mas}.\end{eqnarray} \tag{ 7 }$$

We also derive the geocentric source-lens relative proper motion from ${\theta }_{{\rm{E}}}$ and t_E as

$$\begin{eqnarray}&&{\mu }_{\mathrm{geo}}={6.4}_{-0.5}^{+0.6}\ \mathrm{mas}\;{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 8 }$$

4.2.2. From $(V-H)$ and H

We recalculate ${\theta }_{*}$ from $(V-H)$ and H of the source star in order to check for the robustness of the previous estimation. In general, one can obtain a better estimation of ${\theta }_{*}$ from $(V-H)$ and H rather than from $(V-I)$ and I (c.f., Table 4 of Kervella et al. 2004).

Table 4.  Parameters of the Planetary System

Parameters	Units		Values
Contamination fractiona	%	0 (<10)	20 (10–30)	40 (30–50)
Stellar mass (mhost)	${M}_{\odot }$	$0.{34}_{-0.20}^{+0.12}$	0.19${}_{-0.07}^{+0.21}$	0.13${}_{-0.02}^{+0.17}$
Planetary mass (mp)	MJup	${0.39}_{-0.23}^{+0.14}$	0.22${}_{-0.08}^{+0.24}$	0.15${}_{-0.02}^{+0.19}$
	${M}_{\oplus }$	${123}_{-73}^{+44}$	70${}_{-26}^{+75}$	47${}_{-7}^{+60}$
Distance (DL)	kpc	${1.3}_{-0.8}^{+0.6}$	0.85${}_{-0.34}^{+0.93}$	0.63${}_{-0.12}^{+0.83}$
Projected separation (${r}_{\perp }$)	AU
Close model		${0.74}_{-0.42}^{+0.26}$	0.45${}_{-0.17}^{+0.44}$	0.33${}_{-0.06}^{+0.39}$
Wide model		${4.3}_{-2.5}^{+1.5}$	2.6${}_{-1.0}^{+2.6}$	1.9${}_{-0.3}^{+2.3}$
Semi-major axis (acirc)	AU
Close model		${0.90}_{-0.50}^{+0.52}$	0.67${}_{-0.34}^{+0.55}$	0.45${}_{-0.14}^{+0.53}$
Wide model		${5.2}_{-2.9}^{+3.1}$	3.9${}_{-2.0}^{+3.2}$	2.6${}_{-0.8}^{+3.1}$
Contamination probability	%
Chance-alignment star		92.9	5.2	1.9
Companion to the source		89.7	6.7	3.5
Companion to the lens		99.62	0.22	0.16
Total		83.2	11.4	5.4

Note.

^a The number in parenthesis is for the probability listed in the bottom part of this table.

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In the same way as for ($V-I$)${}_{{\rm{S}},0}$, we calculate the intrinsic ($V-H$) source color of ($V-H$)${}_{{\rm{S}},0}=1.41\pm 0.06$, where H is in the 2MASS system. Then, the intrinsic H-band source magnitude ${H}_{{\rm{S}},0}$ is estimated as follows. First, the apparent H-band source magnitude is measured from the microlensing fit to the H-band light curve obtained by the CTIO 1.3 m telescope, yielding ${H}_{{\rm{S}}}=18.569\pm 0.032$ in the 2MASS system. Next, the H-band extinction A_H is estimated by combining the $(J-{K}_{{\rm{s}}})$ reddening of $E(J-{K}_{{\rm{s}}})$ = ${R}_{{JKVI}}\times E(V-I)=0.266\pm 0.019$ (adopting 0.01 for the uncertainty of R_JKVI) and two extinction coefficients of ${A}_{J}/{A}_{{K}_{{\rm{s}}}}$ and ${A}_{H}/{A}_{{K}_{{\rm{s}}}}$. We adopt ${A}_{J}/{A}_{{K}_{{\rm{s}}}}=2.70\pm 0.15$ and ${A}_{H}/{A}_{{K}_{{\rm{s}}}}=1.63\pm 0.05$, which are the mean values for the Galactic bulge estimated by Chen et al. (2013), but have conservative uncertainties taking into account the non-uniformity of these coefficients toward the Galactic bulge. We solve these equations for A_H and derive ${A}_{H}=0.255\pm 0.029$. As a by-product, we also obtain ${A}_{J}=0.423\pm 0.024$ and ${A}_{{K}_{{\rm{s}}}}=0.157\pm 0.017$, which will be used in Section 5.3. Finally, we derive ${H}_{{\rm{S}},0}={H}_{{\rm{S}}}-{A}_{H}=18.314\pm 0.041$.

The derived ($V-H$)${}_{{\rm{S}},0}$ and ${H}_{{\rm{S}},0}$ values are then converted to ${\theta }_{*}$ using the following equation:

$$\begin{eqnarray}\mathrm{log}{\theta }_{\mathrm{LD}} & = & 0.53598+0.07427(V-H)\\ & & +0.04511[\mathrm{Fe}/{\rm{H}}]-0.2\;H,\end{eqnarray} \tag{ 9 }$$

where H is in the Johnson magnitude system. This equation is derived in the same way as for Equation (4), but includes the metallicity term because a small metallicity dependence appears in this relation. We adopt [Fe/H] = −0.66 ± 0.07 from Bensby et al. (2013). We convert ($V-H$) in the 2MASS magnitude to that in the Johnson system via the transformation of ($V-{H}_{\mathrm{Johnson}}$) $=\;0.28302\times {10}^{-2}+1.0021$ $(V-{H}_{2\mathrm{MASS}})$ $+\;0.36618\times {10}^{-2}{(V-{H}_{2\mathrm{MASS}})}^{2}-0.17906\times {10}^{-2}$ ${(V-{H}_{2\mathrm{MASS}})}^{3}+0.16113\times {10}^{-3}{(V-{H}_{2\mathrm{MASS}})}^{4}$, which is derived by combining Equations (B5) and (B8) from Carpenter (2001) and a color–color relation of Table A5 from Kenyon & Hartmann (1995). As a result, we derive ${\theta }_{*}=0.444\pm 0.014\ \mu \mathrm{as}$, which is quite consistent with the previous estimations, and even has the smallest uncertainty. However, this value is derived partly by relying on the two extinction coefficients of ${A}_{J}/{A}_{{K}_{{\rm{s}}}}$ and ${A}_{H}/{A}_{{K}_{{\rm{s}}}}$, which can vary depending on the line of sight (Chen et al. 2013), although we adopt the mean values for the galactic bulge. That could produce some systematics that are difficult to asses. Therefore, we adopt the ${\theta }_{*}$ value derived in the previous section for the rest of analyses to be conservative.

The distance to the source star D_S can be measured by estimating the absolute magnitude of the source star in addition to the intrinsic one. To this end, we use the isochrone models of PARSEC version 1.2S (Bressan et al. 2012). Through the web interface CMD 2.7,⁴⁹ we obtain a grid of isochrone tables for the stellar age in the range of 8.0 $\lt \;\mathrm{log}t[\mathrm{yr}]\;\lt$ 10.13 with a step size of $\mathrm{log}t$ = 0.02, and for the metallicity in the range of −0.88 < [M/H] < −0.45 with a step size of 0.035. For each table, we further grid the table interpolatively by $\mathrm{log}{T}_{\mathrm{eff}}$ with a step size of 0.001. Then, for each grid, we calculate the following ${\chi }^{2}$ value

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \frac{{({T}_{\mathrm{eff}}-T)}^{2}}{{\sigma }_{{T}_{\mathrm{eff}}}^{2}}+\displaystyle \frac{{([\mathrm{Fe}/{\rm{H}}]-M)}^{2}}{{\sigma }_{[\mathrm{Fe}/{\rm{H}}]}^{2}}+\displaystyle \frac{{(\mathrm{log}g-G)}^{2}}{{\sigma }_{\mathrm{log}g}^{2}},\end{eqnarray} \tag{ 10 }$$

where T, M, and G are the model temperature, metallicity, and surface gravity, respectively. We then find the best-fit values of the absolute V, I, and H magnitudes M_V, M_I, and M_H, as well as the stellar age t, by minimizing the ${\chi }^{2}$ value. In addition, 1σ uncertainties of these parameters are calculated by searching for the region where ${\rm{\Delta }}{\chi }^{2}=1$. The resultant values and uncertainties are ${M}_{V}={4.99}_{-0.14}^{+0.26}$, ${M}_{I}={4.28}_{-0.19}^{+0.25}$, ${M}_{H}={3.54}_{-0.16}^{+0.24}$, and $t={13}_{-5}^{+0}$ Gyr. We note that Bensby et al. (2013) also estimated M_V, M_I, and t by using the Yonsei-Yale isochrone models as 4.89, 4.23, and ${10.2}_{-4.5}^{+1.8}$ Gyr, respectively. Our estimations are consistent with theirs.

Combining these absolute magnitudes with the intrinsic ones derived in Section 4.2, we calculate the source distance to be D_S = 8.8${}_{-1.0}^{+0.7}$, 9.4${}_{-1.1}^{+0.9}$, and 9.0${}_{-1.0}^{+0.7}$ kpc for V, I, and H, respectively. By taking the mean of the three values while conservatively keeping the uncertainties, we obtain a final value of D_S = 9.1${}_{-1.1}^{+0.9}$ kpc. On the other hand, the distance to the centroid of the red clump stars toward the event coordinate is estimated using the Extinction Calculator of Nataf et al. (2013) to be ${D}_{\mathrm{RC}}={7.9}_{-0.8}^{+0.9}$ kpc, which indicates that the source star is likely located at the far side of the Galactic bulge. This result is in agreement with the fact that the microlensing event rate is higher for far-side source stars because of the higher stellar density per unit solid angle.

The first step to extract the excess flux is to identify the microlensing target (source+lens superposition) on the Subaru/ IRCS images. To do so, we use an OGLE-IV I-band image obtained near the brightness peak with the source magnification of more than 400 times (upper right panel in Figure 5), which overwhelms any blended fluxes so that the source position can unambiguously be measured. Because the target position is almost identical to the source position at the time of the Subaru observation, the measured source position on the OGLE peak image can be used to identify the target on the IRCS image once the instrumental coordinate of the OGLE image is calibrated to that of the IRCS one. The coordinate calibration is done by constructing a calibration ladder via an OGLE reference image (upper left panel in Figure 5) obtained under a better seeing condition (10) compared to the OGLE peak image (12). Specifically, we first calibrate the instrumental coordinate of the OGLE peak image to that of the OGLE reference one using the centroid positions of 18 bright common stars (first calibration), and then calibrate the instrumental coordinate of the OGLE reference image to that of the IRCS/J-band one using the centroid positions of nine well-isolated common stars (second calibration). All the stellar centroids are measured using the DoPHOT package. The coordinate calibrations are done with the IRAF⁵⁰ GEOMAP algorithm with free parameters of xy shifts and rotation for the first calibration, and xy shifts, xy pixel-scale magnifications, and rotation for the second calibration. The rms values for the first and second calibrations are 38 mas for 18 stars and 44 mas for nine stars, respectively, meaning that the total 1σ calibration error is 17 mas. We measure the source centroid position on the OGLE peak image with an adopted 1σ uncertainty of 0.1 pixel, or 26 mas, which is then calibrated to the coordinate on the IRCS image (marked as cyan lines in Figure 5) with the total uncertainty of 31 mas. On the IRCS image, we find that there is one stellar object close to the source position with a separation of only 44 mas, which is comparable with the measurement uncertainty. We therefore conclude that this object is the microlensing target.

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The next step is to measure the brightness of the microlensing target. We perform aperture photometry of the target object on the IRCS J-, H-, and K'-band images, by using a customized code with an aperture radius of 10 pixels (020). As a bright star is located 108 pixels (22) west of the target and its point-spread function (PSF) tail spreads toward the target, we carefully select a region to estimate the sky level such that the distance from the centroid of the bright star is the same as for the target and there is no other noticeable flux contamination. The selected region, a box with the size of 20 × 20 pixels, is indicated in yellow in Figure 5. We calculate a median sky value from this region, and subtract it from the fluxes in the target's aperture. The measured target flux is then calibrated to the 2MASS magnitude system. For this calibration, we construct a photometric calibration ladder using J-, H-, and K_s-band archive images of VISTA Variables in the Via Lactea (VVV; Minniti et al. 2010), because there are only three overlapping stars between the IRCS images and the 2MASS catalog. The VVV images are three times finer in pixel scale and four times deeper in limiting magnitude compared to the 2MASS catalog. We here approximate that the IRCS J, H, and K' bands are identical to the 2MASS J, H, and K_s bands, respectively. On the VVV images, we perform stellar extraction and PSF-fitting photometry on a 25 × 25 subarea around the target using the DoPHOT package. Among 40–60 common stars with the 2MASS catalog, 30 bright-end stars are used to for photometric calibration in order to avoid the effect of blending on the fainter objects. For the Subaru/IRCS images, we carefully select 12, 11, and 10 calibration stars for J, H, and K_s, respectively, such that they are well isolated on the IRCS images and also detected on the VVV images. Aperture photometry of these stars is done by the same manner as for the target object, but this time the sky levels are estimated from the stellar-centroid annulus with an inside and outside radii of 70 and 80 pixels (14 and 16), respectively. The constructed calibration ladders are shown in Figure 6. As a result, we measure the target brightness in J, H, and K_s as

$$\begin{eqnarray}&&{J}_{\mathrm{target}}=17.704\pm 0.034\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{H}_{\mathrm{target}}=17.207\pm 0.039\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{K}_{{\rm{s}},\mathrm{target}}=17.071\pm 0.044.\end{eqnarray} \tag{ 13 }$$

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The final step to extract the excess flux is to subtract the apparent source fluxes from the measured target ones.

Although the apparent H-band magnitude of the source star is measured from the light curve fitting as ${H}_{{\rm{S}}}=18.569\pm 0.032$, no J- or K_s-band light curves were obtained. So, we estimate the apparent J- and K_s-band source magnitudes J_S and ${K}_{{\rm{s}},{\rm{S}}}$ as follows. First, using the color-metallicity-temperature relation of Casagrande et al. (2010), we obtain the intrinsic source colors in $(V-J)$ and $(V-{K}_{{\rm{s}}})$ as ${(V-J)}_{{\rm{S}},0}=1.13\pm 0.05$ and ${(V-{K}_{{\rm{s}}})}_{{\rm{S}},0}=1.48\pm 0.06$, respectively. Next, substituting these values from ${V}_{{\rm{S}},0}={H}_{{\rm{S}},0}-{(V-H)}_{{\rm{S}},0}=19.73\pm 0.07$, we obtain the intrinsic source magnitudes of ${J}_{{\rm{S}},0}=18.60\pm 0.09$ and ${K}_{{\rm{s}},\ {\rm{S}},\ 0}=18.25\pm 0.09$, respectively. Finally, the apparent J and K_s source magnitudes are derived as ${J}_{{\rm{S}}}=19.02\pm 0.09$ and ${K}_{{\rm{s}},{\rm{S}}}=18.40\pm 0.10$ by adding A_J and ${A}_{{{\rm{K}}}_{{\rm{s}}}}$ to ${J}_{{\rm{S}},0}$ and ${K}_{{\rm{s}},{\rm{S}},0}$, respectively.

The best-fit light curve models suggest that the source star was still magnified by 1.474 at the time of Subaru observation, meaning that the source magnitudes in J, H, and K_s at the time of the Subaru observation were 18.60 ± 0.09, 18.15 ± 0.032, and 17.98 ± 0.10, respectively. All of these magnitudes are significantly fainter than the measured target ones, indicating that excess flux at the source position is clearly detected. Subtracting the source fluxes from the target ones, we obtain the J-, H-, and K_s-band magnitudes of the excess flux:

$$\begin{eqnarray}&&{J}_{\mathrm{excess}}=18.33\pm 0.09\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{H}_{\mathrm{excess}}=17.80\pm 0.07\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{K}_{s,\mathrm{excess}}=17.69\pm 0.11.\end{eqnarray} \tag{ 16 }$$

In principle, there are three possible scenarios for the origin of the detected excess flux: (1) it solely comes from the lens star, (2) it is a combination of fluxes from the host star and from other astronomical sources (such as an unrelated chance-alignment star, a companion to the source star, and/or a companion to the lens star), and (3) it entirely comes from other astronomical sources mentioned previously.

In the first case, the lens star is almost certainly an MS star rather than a giant or stellar remnant, given the brightness of the excess flux. In the second case, as we show in Section 6.2, the amount of the contamination flux from the extra sources is constrained to $\lesssim 50$% of the measured excess flux by the ${\theta }_{{\rm{E}}}$ measurement and the upper limit of ${\pi }_{{\rm{E}}}$; with this limitation, the lens star is also most likely an MS star.

In the final case, the lens star must not be a MS star, but rather a stellar remnant such as a white dwarf (WD), a neutron star (NS), or a black hole (BH). This could happen in principle, however we ignore this possibility here because of their relatively low population in the Galaxy (MS:WD:NS:BH ∼ 1:0.18:0.021:0.0031; Sumi et al. 2011) and the presumably low planetary abundance around them (no planet has yet been detected around a WD or BH). In particular, these objects are created after disruptive stellar evolutions, including radius inflation for all cases and subsequent violent explosions for the case of an NS and BH, which could reduce planetary abundance around them. Secondary planet formation after the evolutions might be possible, however, its efficiency is not yet known.

Therefore, hereafter we simply assume that the lens star is an MS star, and consider only scenarios (1) and (2). We note that the stellar remnant scenario can be tested by obtaining high-resolution images in the near future when the source and lens stars will be separated enough; we would not see any object at the expected separation if the lens were a remnant.

The absolute physical parameters of the lens system can be derived by combining the lens flux and ${\theta }_{{\rm{E}}}$, both of which are independent functions of the host star's mass, m_host, and the distance to the lens system from the Earth, D_L.

Using ${\theta }_{{\rm{E}}}$, m_host is expressed by

$$\begin{eqnarray}&&{m}_{\mathrm{host}}=\displaystyle \frac{1}{1+q}\displaystyle \frac{{\theta }_{{\rm{E}}}^{2}}{\kappa {\pi }_{\mathrm{rel}}},\end{eqnarray} \tag{ 17 }$$

where ${\pi }_{\mathrm{rel}}\equiv \mathrm{AU}(1/{D}_{{\rm{L}}}-1/{D}_{{\rm{S}}})$. In Figure 7, we plot the m_host–D_L relation for the ${\theta }_{{\rm{E}}}$ value derived in Section 4.2, fixing D_S at 9.1 kpc.

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The other m_host–D_L relation can be derived from the lens flux. We first assume that the observed excess flux comes solely from the lens flux (i.e., there is no contamination from other sources). Because the excess brightness is measured most precisely in the H band among the three bands, we use H_excess to derive the physical parameters. The absolute H-band magnitude of the host star, ${M}_{H,{\rm{L}}}$, is calculated using the H-band lens flux H_L, here ${H}_{{\rm{L}}}={H}_{\mathrm{excess}}$, as a function of D_L by the following equation,

$$\begin{eqnarray}&&{M}_{H,{\rm{L}}}={H}_{{\rm{L}}}-{A}_{H,{\rm{L}}}-5\mathrm{log}\displaystyle \frac{{D}_{{\rm{L}}}}{10\;\mathrm{pc}},\end{eqnarray} \tag{ 18 }$$

where ${A}_{H,{\rm{L}}}$ is the H-band extinction up to D_L, for which we simply assume ${A}_{{\rm{H}},{\rm{L}}}={A}_{{\rm{H}}}{D}_{{\rm{L}}}/{D}_{{\rm{S}}}$. For a given D_L, ${M}_{H,{\rm{L}}}$ can be converted to m_host via an isochrone model. We use the PARSEC isochrones version 1.2S (Bressan et al. 2012; Chen et al. 2014), which was improved for low-mass stars over the previous versions. We assume the host star's age of 4 Gyr and a solar metallicity. In Figure 7, we include a plot of this m_host–D_L relation. We also plot the 3σ excluded area from the ${\pi }_{{\rm{E}}}$ upper limit calculated in Section 3.4 as the cyan hatched region. As a result, we find that the two m_host–D_L relations cross each other at $0.4\lesssim {D}_{{\rm{L}}}$/kpc $\lesssim \;2.6$ and $0.1\lesssim {m}_{\mathrm{host}}$/M${}_{\odot }\lesssim 0.55$ within 2σ, indicating that the host star is likely a nearby M dwarf.

To properly estimate m_host and D_L as well as other related physical parameters including the planetary mass, m_p, and the projected star–planet separation, ${r}_{\perp }$, we calculate probability distributions of these parameters by means of the Monte Carlo technique. Specifically, we solve for m_host, D_L, m_p, and ${r}_{\perp }$ from a given set of observed and assumed parameters: X = {ρ, q, s, ${\theta }_{*}$, H_L, A_H, D_S, [M/H]}, where [M/H] is the metallicity of the host star. We repeat this calculation about 10⁵ times by randomly samplingX. For the random distributions, we use the posterior distributions obtained from the MCMC analysis in Section 3.3 for ρ, q, and s, and use a Gaussian distribution for ${\theta }_{*}$, H_L, A_H, D_S, and [M/H]. We assume [M/H] = −0.05 ± 0.2, which is consistent with the metallicity distribution of nearby M dwarfs (e.g., Gaidos et al. 2014). Note that we fix the stellar age at 4 Gyr, because the age dependence on the isochrone models is negligible for moderately mature M dwarfs (≳0.5 Gyr).

The probability distributions of the lens-system parameters are shown in Figure 8. Each distribution shows a bimodal shape, which comes from the fact that the two m_host–D_L relations form a Y-like shape. If ${\theta }_{{\rm{E}}}$ takes an upper-side value (above the black solid line), then the stellar mass will most likely be very low ($\sim 1.2{M}_{\odot }$), which forms the left-hand sharp peak. On the other hand, if ${\theta }_{{\rm{E}}}$ takes a lower-side value, then the stellar mass can take a value in a wider range of ∼0.3–0.5 ${M}_{\odot }$, which forms the right-hand loose peak. Therefore, this bimodal shape does not come from two distinct solutions, but from one solution. For this reason, for a final value of each parameter and its uncertainties, we just take the median and 68.3% confidence intervals of each probability distribution. As a result, we obtain ${m}_{\mathrm{host}}\;=$ 0.34${}_{-0.20}^{+0.12}$ M${}_{\odot }$, ${D}_{{\rm{L}}}\;=\;$ 1.3${}_{-0.8}^{+0.6}$ kpc, ${m}_{{\rm{p}}}\;=\;$ 0.39${}_{-0.23}^{+0.14}$M_Jup (123 ${}_{-73}^{+44}$M_Earth), and ${r}_{\perp }\;=\;$ 0.74${}_{-0.42}^{+0.26}$ AU for the close model and 4.3${}_{-2.5}^{+1.5}$ AU for the wide model. In addition, we calculate the probability distribution of the de-projected semimajor axis a, assuming a circular orbit, uniform distribution of a, and random distributions of orbital inclination and phase (Gould & Loeb 1992), resulting in the estimated semimajor axis of ${a}_{\mathrm{circ}}={0.90}_{-0.50}^{+0.52}$ AU for the close model and ${a}_{\mathrm{circ}}={5.2}_{-2.9}^{+3.1}$ AU for the wide model. All the resultant parameters are listed in Table 4.

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If the measured excess flux is contaminated by extra sources, the actual lens flux would decrease by the contaminated flux, pushing down the lens mass. The amount of this decrement, however, is limited due to the existence of the upper limit of ${\pi }_{{\rm{E}}}$. In other words, there is an upper limit of the contamination flux; we estimate ∼50% of the excess flux as this limit above which the two mass–distance relations, from H_L and from ${\theta }_{{\rm{E}}}$, do not cross each other outside the ${\pi }_{{\rm{E}}}$ excluding region within the uncertainties. We look into the effects and probabilities of contaminations in the following sections.

6.2.1. Effects of Contamination

6.2.2. Chance-alignment Star

6.2.3. Companion to the Source Star

6.2.4. Companion to the Lens Star

The last case of the contamination source is a companion to the M-dwarf lens star. In this case, the distance to the lens system and the masses of the hypothetical-binary components change depending on the contamination level f. We therefore use the following equation to calculate the probability that the lens star has a companion:

$$\begin{eqnarray}&&P(f)={F}_{\mathrm{binary}}\times \displaystyle \sum _{f}{F}_{{q}_{{\rm{c}}}}(f)\times {F}_{{a}_{{\rm{c}}}}(f),\end{eqnarray} \tag{ 19 }$$

where F_binary is the fraction of M dwarfs that have a companion, and ${F}_{{q}_{{\rm{c}}}}$ and ${F}_{{a}_{{\rm{c}}}}$ are the fractions of M-dwarf companions that have the mass ratio q_c and semimajor axis a_c, respectively.

To calculate ${F}_{{q}_{{\rm{c}}}}$, we estimate the companion mass using Equation (18) and an isochrone model of the PHOENIX/AMES-dusty model (Allard et al. 2001) for a given f. The companion mass varies in the range of 0.10–0.11 ${M}_{\odot }$ for $0.1\lt f\lt 0.5$. Then, the host star's mass for a given f is calculated in the same way as in Section 6.2.1 to derive q_c. The q_c value varies in the range of 0.35–0.94 depending on f. For the q_c distribution, we simply assume a uniform distribution (Janson et al. 2014). We note that because the mass distribution of the M-dwarf companion is not yet clear, the choice of this simple distribution might be systematics in the probability estimation. However, as shown below, the probability of contamination from a lens companion is much smaller than the other contamination sources, and therefore the choice of mass distribution should not affect the total contamination probability.

To calculate ${F}_{{a}_{{\rm{c}}}}$, we set the limits on the binary separation. We set the same upper limit of 016 as the previous cases, which is converted to a_c = 229 AU for f = 0.1 and a_c = 120 AU for f = 0.5. For a lower limit, we consider a caustic induced by a hypothetical companion by which additional anomalies could be produced in the microlensing light curve. We adopt an upper limit on the full width of the hypothetical caustic as $w=4{q}_{{\rm{c}}}/{s}_{{\rm{c}}}^{2}\lt {u}_{0}\sim 1.4\times {10}^{-3}$, where s_c is the projected separation of the companion and the host star normalized by ${\theta }_{{\rm{E}}}$. This inequality provides the lower limits of the angular separation of 46 mas for f = 0.1 and 75 mas for f = 0.5, which correspond to a_c = 53–65 AU depending on f. For the distribution of a_c, we adopt a log-normal distribution with the mean of $\mathrm{log}{a}_{{\rm{c}}}$ (AU) = 0.80 and a standard deviation of ${\sigma }_{\mathrm{log}{a}_{{\rm{c}}}}=0.48$ (Janson et al. 2014). By summing up Equation (19) with a step size of ${\rm{\Delta }}f=5\%$, we calculate the probabilities that the lens star has a companion as 0.22% and 0.16% for f = 10%–30% and 30%–50%, respectively.

6.2.5. Total Probabilities of Contamination

Because the J- and K_s-band excess fluxes are also measured, the ($J-{K}_{{\rm{s}}}$) and ($H-{K}_{{\rm{s}}}$) colors of the host star can be estimated, which can be used as an independent check of whether the host star's mass derived in Section 6.1 is consistent with an M dwarf. We estimate the reddening for the host star by interpolating the online tables of Schultheis et al. (2014) for the Galactic coordinate and distance of the lens system, yielding $E{(J-{K}_{{\rm{s}}})}_{{\rm{L}}}=0.045\pm 0.024$ and $E{(J-H)}_{{\rm{L}}}=0.011\pm 0.007$, where the allowed range of the distance is taken into account. Then, assuming that the excess fluxes come solely from the host star, we calculate the de-reddening colors as ($J-{K}_{{\rm{s}}}$)_L = 0.60 ± 0.14 and ($H-{K}_{{\rm{s}}}$)_L = 0.10 ± 0.13. We plot it on the color–color diagram in Figure 9, along with the distribution of the metallicity-measured nearby M dwarfs presented in Newton et al. (2014) and a main-sequence track of Bessell & Brett (1988). The position of the host star is not exactly at the majority of M dwarfs, but is more consistent with the region for K dwarfs. However, considering the large error bars, the measured colors are still marginally consistent with metal-poor mid-to-early M dwarfs.

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We note that the spectral-type estimation from these color measurements is more sensitive to systematics rather than the mass measurement from ${\theta }_{{\rm{E}}}$ and H_L. Although a systematical change of as large as 0.4 mag in H_L would keep the stellar mass within the M-dwarf range, only a 0.2 mag systematical shift in colors would easily change the inferred spectral type. In addition, potential systematics in the color measurements are larger than that in H_L. Because there is no J- and K_s-band light curves, J_S and ${K}_{{\rm{s}},{\rm{S}}}$ are estimated via several calibration processes, including the estimation of A_J and ${A}_{{K}_{{\rm{s}}}}$, which could be a source of systematics in the colors (see Section 4.2.2). On the other hand, H_S is directly measured from the light curve, which minimizes the uncertainty in H_L. Therefore, it is more likely that the slight discrepancy in the inferred spectral type of the host star comes from systematics, or simply statistical errors, in the measured colors rather than those in the measured mass. We further note that it could also be explained by a contamination of a distant early-type dwarf or a disk WD, which would cause a bluer shift in the measured colors. However, the probabilities of these scenarios should be much lower than those calculated in Section 6.2.2, because the majority of possible chance-alignment stars are bulge dwarfs. Instead of identifying the cause of this discrepancy, we will discuss the prospects of improving the color estimation in Section 7.