MOA-2015-BLG-337: A Planetary System with a Low-mass Brown Dwarf/Planetary Boundary Host, or a Brown Dwarf Binary

Exoplanets and brown dwarfs (BDs) are usually inferred from indirect methods owing to their intrinsic faintness. Each method has its own unique sensitivity range and can probe different regions of the planet parameter space. Gravitational microlensing has an unequaled sensitivity to companions with masses ranging from stellar mass to even Earth-mass planets orbiting beyond the snow line (Mao & Paczynski 1991; Gould & Loeb 1992). According to the core accretion theory (Pollack et al. 1996; Kokubo & Ida 2002), massive Jovian planets could be efficiently formed outside the snow line. The radial velocity (RV; Mayor & Queloz 1995; Butler et al. 2006) and transit methods (Batalha et al. 2013; Borucki et al. 2011) are mostly sensitive to planets that are relatively more massive or orbit closer to their host stars.

Searching for companions to BDs is very crucial to understanding the formation mechanism of BDs. Though still a matter of debate, most theories suppose that BDs were formed by the direct collapse of molecular clouds on a much smaller scale than stars, and promoted by turbulent fragmentation (Luhman 2012). Some young BDs are observed with excess emission from the surrounding raw material disks (Apai et al. 2005; Luhman et al. 2005; Ricci et al. 2012). The recent direct imaging survey to search for BDs and very low-mass stars (VLMS) found a strong preference to equal-mass binary systems if they have companions (Burgasser et al. 2006; Liu et al. 2010). Burgasser et al. (2006) found a very steep index γ = 4.2 ± 1.0 for the power-law distribution ∝q^γ, which is derived by fitting the observed mass-ratio distribution. Though there are still large uncertainties because the masses of the objects detected by direct imaging were estimated based on assumed ages, lower mass-ratio (i.e., q ≤ 0.3) binaries would be very uncommon among field VLMS and BDs. However, there have been several discoveries of binary BDs (Han et al. 2017) as well as a BD orbiting an M dwarf (Han et al. 2016) with small mass ratios (q < 0.3) by microlensing. In addition, Han et al. (2013) found a system that consists of a several Jupiter-mass planet orbiting a BD and suggested that the planetary-mass companion might have been formed in a protoplanetary disk surrounding the BD. Microlensing can play a meaningful role in understanding the formation of VLMS and BDs by detecting companions of low-mass hosts.

Gravitational microlensing has been used to discover several dozen exoplanets around M-dwarf stars, revealing that massive Jovian planets beyond the snow line are much more uncommon compared to low-mass ones (Gould et al. 2010; Sumi et al. 2010; Cassan et al. 2012; Shvartzvald et al. 2016). Particularly, Suzuki et al. (2016) and Udalski et al. (2018) found a possible break and peak around q ∼ 2 × 10⁻⁴ in the companion-host mass ratio function. They suggested that ice giant mass planets are the most abundant ones outside the snow line. This is qualitatively consistent with predictions from core accretion theory. However, in their results, the estimated abundance of Jovian- and Neptune-mass planets is a power of 10 more than that predicted by the formation models around M-dwarf stars (D. Suzuki et al. 2018, in preparation). However, as the hydrogen and helium in their protoplanetary disks would be lost within several million years, massive Jovian planets are unlikely to be formed around VLMS and BDs because there would, as a result, be insufficient materials to form gas giants (Ida & Lin 2005).

Without an observed microlensing parallax, we cannot determine the absolute masses of the lens system from a microlensing light curve alone. In such cases, a Bayesian analysis is usually applied by using Galactic model priors (comprising a mass function and number density, and velocity distributions) to obtain the posterior probability distributions of the physical parameters of the lens system. The results of the Bayesian analysis generally strongly depend on the priors. In addition, we should pay special attention to low-mass hosts because the mass function for the low-mass region ranging from BDs down to planetary-mass objects is still very uncertain. Gravitational microlensing plays an important role for probing the occurrence of planets in distant orbits and planetary-mass objects unbound to any host stars. By investigating the timescale distribution of short microlensing events in the MOA data set in 2006–2007, Sumi et al. (2011) suggested a possible population excess of unbound or distant Jupiter-mass objects. Unbound planets are thought to derive from various physical processes. There are, for instance, star–planet scattering (Holman & Wiegert 1999; Musielak et al. 2005; Doolin & Blundell 2011; Malmberg et al. 2011; Veras & Raymond 2012; Kaib et al. 2013), planet–planet scattering (Levison et al. 1998; Ford & Rasio 2008; Guillochon et al. 2011), and stellar mass loss (Veras et al. 2011; Veras & Tout 2012; Voyatzis et al. 2013). Mróz et al. (2017) recently updated the study with a larger sample and reported no significant excess of short-timescale (1–2 days) microlensing events and placed 95% upper limits on the frequency of unbound or distant Jupiter-mass objects of 0.25 planet per main-sequence star using the OGLE 2010-2015 observations. They also reported a possible abundance of unbound or distant super-Earth-mass objects (Dai & Guerras 2018). These studies have played an important role for probing the mass function for low-mass objects, while it is still uncertain.

In order to find as many microlensing events as possible, a large number of stars must be monitored, because the probability of any given star being microlensed is very low. The second phase of the Microlensing Observations in Astrophysics (MOA; Bond et al. 2001; Sumi et al. 2003) collaboration, MOA-II, has continued microlensing survey observations using the 1.8 m MOA-II telescope with a 2.2 deg² wide field of view (FOV) CCD camera (MOA-cam3; Sako et al. 2008) at the Mount John Observatory (MJO) in New Zealand. The MOA observational fields are mainly toward Baade's window in the Galactic bulge and they observe the fields with cadences of ∼10–50 times per night. Six MOA-II fields (∼13 deg²) are observed with a 15 minute cadence, seeking particularly short-time microlensing events. About 600 microlensing event alerts are issued by MOA-II in real time each year.³⁴ Other microlensing survey groups are the Optical Gravitational Lensing Experiment (OGLE; Udalski et al. 2015) collaboration and the Korea Microlensing Telescope Network (KMTNet; Kim et al. 2016). About 2000 microlensing event alerts are issued by OGLE in real time each year,³⁵ and about 2200 alerts are issued by KMTNet each year.³⁶

In this paper, we present the analysis of the short-timescale microlensing event, MOA-2015-BLG-337. We find two competing models that explain the observed data. One comprises a planetary-mass-ratio lens system and the other, a binary mass ratio lens system. Because of the short timescale, ∼6 days, of this event, we conducted our Bayesian analysis with an assumed mass function extending down to BD and also down to planetary-mass regimes. We describe the observations of microlensing and the data sets for this event in Section 2. Our light curve modeling is explained in Section 3. In Section 4, we present the calibration of the source star and the estimation of the angular Einstein radius. In Section 5, we perform a Bayesian analysis in order to determine the probability distribution of the physical parameters of the lens system. In Section 6, we discuss the results of this work. Finally, we present the summary and our conclusion in Section 7.

Event MOA-2015-BLG-337 was first discovered by MOA on 2015 July 10 UT (HJD' ≡ HJD − 2450000 = 7214) and positioned at (R.A., decl.)_J2000 = (18^h07^m4769, −28°10'1300), which corresponds to Galactic coordinates (l, b) = (311, −383). The event is located in MOA field "gb14", which is observed with a 15 minute cadence. OGLE independently found and alerted the event as OGLE-2015-BLG-1598 on 2015 July 11 UT (HJD' = 7215) with their 1.3 m Warsaw telescope with a 1.4 deg² FOV at LCO. KMTNet also observed this event as KMT-2015-BLG-0511 (Kim et al. 2018) in their regular observation survey with their three 1.6 m KMTNet telescopes each having a 4.0 deg² FOV at CTIO, SAAO and SSO.

The MOA observers noticed that the light curve of the MOA-2015-BLG-337 deviated from a single-lens model around HJD' ∼ 7214.9 and the MOA collaboration issued an anomaly alert encouraging follow-up observations. Some modelers in microlensing survey and follow-up groups immediately modeled this event and found two indistinguishable solutions with planetary and binary-lens mass ratios. KMTNet-CTIO and OGLE light curves also cover the part of the anomaly. This event was also observed by the 0.61 m Boller & Chivens telescope at MJUO. The number of data points of each telescope and passband are shown in Table 1. The MOA and B&C data were reduced with MOA's photometry pipeline (Bond et al. 2001) which uses the Difference Image Analysis (DIA) technique (Tomaney & Crotts 1996; Alard & Lupton 1998; Alard 2000). The OGLE data were reduced with the OGLE DIA (Wozniak 2000) photometry pipeline (Udalski et al. 2015). The KMTNet data also were reduced with their PySIS photometry pipeline (Albrow et al. 2009; Kim et al. 2016).

The photometric errors estimated from pipelines are generally underestimated owing to the high stellar densities in the Galactic bulge. We, therefore, renormalized the error bars of each data set following the method described in Yee et al. (2012), i.e., ${\sigma }_{i}^{{\prime} }=k\sqrt{{\sigma }_{i}^{2}+{e}_{\min }^{2}}$, where ${\sigma }_{i}^{{\prime} }$ and σ_i are the renormalized and original error bar of the ith data points in magnitude, k and e_min are parameters for renormalizing. The value of e_min represents systematic errors that dominate at high magnification and can be affected by flat-fielding errors. First, we fit the light curve to find a tentative best-fit model. Then, we fitted the k and e_min values so that the cumulative χ² distribution from the tentative best model sorted by model-magnification is χ²/dof ∼ 1 and close to linear with a slope of 1, respectively. However, we could not constrain e_min in most of data sets because our data sets are not highly magnified, and thus not sensitive to e_min. So, we set e_min = 0.003, which is empirically chosen, and selected the k value in order to be χ²/dof ∼ 1 as shown Table 1. The renormalizing parameters applied for the OGLE-I data are consistent with those taken from Skowron et al. (2016). After the error renormalization, we fitted the light curve again to find the final best model. In general, error renormalization processes do not significantly affect the final results. We confirmed the final best model is consistent with the previous tentative best-fit model before error renormalization.

Figure 1 represents the light curve of this event. It shows a clear asymmetric feature, which cannot be explained by a finite-source point-lens (FSPL) model by Δχ² > 1600. We describe the procedure of binary-lens modeling in the following section.

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3.1. Model Description

The magnification of a standard binary-lens model has seven parameters; the time of the source closest to the center of mass, t₀; the Einstein radius crossing time, t_E; the impact parameter normalized by the Einstein radius, R_E, u₀; the mass ratio of a lens companion relative to a host, q; the projected separation normalized by R_E between binary components, s; the angle between the binary-lens axis and the source trajectory direction, α; and finally the source angular radius relative to the angular Einstein radius, θ_E, ρ. If ρ is measured from the light curve modeling, we can estimate the angular Einstein radius, θ_E = θ_*/ρ, because the source angular radius, θ_*, can be estimated from its apparent magnitude and color corrected for extinction and reddening (Boyajian et al. 2014). Given parameters x = (t_E, t₀, u₀, q, s, α, ρ), the magnification A(t, x) can be calculated at any given time, t. The model light curve can be written as

$$\begin{eqnarray}&&F(t)=A(t,{\boldsymbol{x}}){f}_{{\rm{s}}}+{f}_{{\rm{b}}},\end{eqnarray} \tag{ 1 }$$

where f_s is the un-magnified source flux, and f_b is the blend flux. Each telescope and passband has a corresponding f_s, f_b pair. We applied a linear limb-darkening model for the source, I(θ) = I(0)[1 − u_λ(1 − cos θ)]. Here, θ and u_λ represent the angle between the line of sight and the normal to the stellar surface, and the linear limb-darkening coefficient, respectively. We estimated the effective temperature of the source, T_eff ∼ 6000 K, and assumed a metallicity, log[M/H] = 0.0 and surface gravity log(g/cm s⁻¹) = 4.50. According to the ATLAS model of Claret & Bloemen (2011), the limb-darkening coefficients are u_R = 0.7021, u_I = 0.6055 and u_V = 0.7801. The u_R+I value, corresponding to limb darkening in the MOA-Red passband, is estimated as the mean of u_R and u_I.

3.2. Modeling

We used a Markov Chain Monte Carlo (MCMC) approach (Verde et al. 2003) in order to search for the best-fit model parameters combined with our implementation of the image ray-shooting method (Bennett & Rhie 1996; Bennett 2010). At first, we conducted a broad grid search by using the standard static binary-lens model with 9680 fixed grid points covering wide ranges of the three parameters, q, s and α, with all other parameters allowed to vary. Then, the best 100 models with the smallest χ² were refined with all free parameters. With these procedures, we can find the best-fit model and avoid missing any local minima in the wide range of the parameters space. After that, we found a pair of optimal solutions, one with a planetary-mass ratio of q ∼ 1.1 × 10⁻² and the other with a stellar binary mass ratio of q ∼ 1.7 × 10⁻¹. These parameters are labeled as "Planetary" and "Binary" in Table 2, respectively. We could not find any model with a significant microlensing parallax effect as expected because of the short event timescale. In addition, we examined the possibility of a binary-source point-lens (BSPL) model for this event, and confirmed it could be ruled out by Δχ² > 50.

Table 2.  The Best-fit Model Parameters

Parameters	Units	Planetary	Planetary	Binary	Binary	BSPL
		(close)	(wide)	(close)	(wide)
tE	days	5.382(94)	5.432(90)	5.496(95)	6.104(99)	6.052
t0	HJD-2450000	7214.9183(20)	7214.9243(23)	7214.9120(21)	7215.7390(21)	7215.0536
u0		0.0470(11)	0.0567(18)	0.0458(12)	0.0435(13)	0.0331
q	⋯	0.0108(14)	0.0109(12)	0.178(23)	0.235(54)	⋯
s	⋯	0.606(33)	1.548(64)	0.263(11)	4.71(38)	⋯
α	radian	1.4445(57)	1.4449(63)	2.5359(78)	0.5877(75)	⋯
ρ	⋯	0.0223(30)	0.0263(23)	0.0166(<0.0183)a	0.0152(<0.0176)a	0.0041
t0,2b	HJD-2450000	⋯	⋯	⋯	⋯	7214.7049
u0,2b	⋯	⋯	⋯	⋯	⋯	0.0379
ρ2b	⋯	⋯	⋯	⋯	⋯	0.0048
fit χ2	⋯	21800.18	21800.34	21806.26	21808.21	21852.51

Notes. The numbers in parentheses indicate the 1σ uncertainties derived from the 16th/84th percentile values of the stationary distributions given by MCMC.

^aThis value indicates a 1σ upper limit on ρ. The binary model including the parameter ρ is favored over one without ρ by only Δχ² < 4. This indicates that the finite source effects for the binary models are not significant for fitting, and it is questionable to accept this ρ value directly. Thus, we do not adopt the best-fit ρ value but put an upper limit on ρ. ^bThese are secondary source parameters for binary-source point-lens (BSPL) model.

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In the planetary model, there are two statistically indistinguishable solutions, one with a close (s < 1) binary separation and the other with a wide (s > 1) separation. The close model is favored by only Δχ² ≃ 0.2. This severe degeneracy between s and s⁻¹, which is known as close/wide degeneracy, is common for central-caustic crossing events because the shapes of the central-caustic with s and s⁻¹ are similar for smaller q (Griest & Safizadeh 1998; Dominik 1999). In these planetary models, the finite source effect with ρ ∼ 2 × 10⁻² is detected with Δχ² > 35 compared to the point source models, and the signal mostly came from MOA-Red data. This ρ value is relatively large compared to the typical values for other microlensing planetary events and it indicates a small angular Einstein radius, θ_E, and prefers a smaller lens mass in the Bayesian analysis of Section 5.

In the binary model, the model deviates from the planetary model particularly between 7214.5 < HJD − 2450000 <7214.8 (see Figure 1). The difference between these models is only Δχ² ∼ 6 due to the large photometric uncertainties in this period. There is also a close/wide degeneracy in the binary model: the close model is favored over the wide one by Δχ² ∼ 2. The finite source effect was not significantly detected for these binary models, showing only an improvement of χ² < 4.0 compared to the point source model. Thus, we do not adopt the best-fit ρ value but put only an upper limit on ρ in the following analysis.

Our inability to distinguish between the planetary and binary models is owing to the lack of data during the anomaly where there are only subtle differences in the magnification pattern between these models. Figure 2 represents the caustic geometries and the magnification patterns around them for each model. The magnification pattern this event can be explained by the two different central-caustic shapes, which makes it difficult to distinguish the two planetary and stellar binary solutions. Similar to this event, there are a number of microlensing events which can be explained with both stellar binary and planetary-mass ratio lens systems (Gaudi & Han 2004; Choi et al. 2012; Park et al. 2014).

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By using the measurement of ρ (=θ_*/θ_E) from the models, the source angular radius θ_* allows us to estimate θ_E. We can estimate θ_* from the source intrinsic color and magnitude (Boyajian et al. 2014).

We converted the instrumental source magnitude in MOA-Red and MOA-V bands into the standard Kron-Cousin I-band and Johnson V-band scales using the model light curve. With the procedure described in Bond et al. (2017), we cross-referenced stars within 2' of the source between the MOA star catalog reduced by DoPHOT (Schechter et al. 1993) and the OGLE-III star catalog (Szymański et al. 2011). Then, we found the following relations,

$$\begin{eqnarray*}{I}_{\mathrm{OGLE} \mbox{-} \mathrm{III}}-{R}_{\mathrm{MOA}} & = & (28.119\pm 0.005)\\ & & -\,(0.206\pm 0.002){(V-R)}_{\mathrm{MOA}}\\ {V}_{\mathrm{OGLE} \mbox{-} \mathrm{III}}-{V}_{\mathrm{MOA}} & = & (27.901\pm 0.005)\\ & & -\,(0.148\pm 0.002){(V-R)}_{\mathrm{MOA}}.\end{eqnarray*}$$

We tried to correct interstellar extinction and reddening by following the standard procedure in Yoo et al. (2004), which treats the Red Clump Giants (RCG) in the color–magnitude diagrams (CMD) as standard candles. Figure 3 shows the CMD of the OGLE-III star catalog within 2'' of the source, plotted over the CMD of the Baade's window from Holtzman et al. (1998) whose extinction and reddening are matched by using RCG position. The magnitude and color of source and the center of the RCG, (V − I, I)_s = (1.611, 19.682) ± (0.019, 0.012) and (V − I, I)_RCG = (2.050, 15.553) ± (0.013, 0.048) are shown as filled blue and red circles, respectively. We obtained the intrinsic RCG centroid in this field (V − I, I)_RCG,0 = (1.060, 14.348) ± (0.070, 0.040) (Bensby et al. 2013; Nataf et al. 2013), and found the extinction and reddening of RCG centroid to be A_I,RCG = 1.205± 0.062 and E(V − I)_RCG = 0.990 ± 0.071, respectively. The intrinsic source magnitude and color values are (V − I, I)_s,0 = (0.621, 18.478) ± (0.074, 0.064) on the assumption that the source star suffers from the same extinction and reddening as that of the bulge RCGs in the field. From the CMD, we can see that the source star is slightly bluer than other typical bulge dwarf stars in this field. This is possibly because the source does not have the same extinction and reddening as the median values of the RCG stars. However, even if we assume 10% less extinction and reddening from those for the median RCG, the changes in the θ_E and μ_rel are less than 3%, which is much less than the typical uncertainty of these values. We summarize the source color, magnitude, and angular radius in Table 3. We also independently obtained the intrinsic source color using a linear regression from KMTNet-CTIO I and V, ${(V-I)}_{{\rm{s}},0,\mathrm{KMTNet}}=0.688\pm 0.066$, which is consistent with the color from MOA-Red and V above but with a smaller uncertainty. We applied this ${(V-I)}_{{\rm{s}},0,\mathrm{KMTNet}}$ in the following analysis.

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For estimation of the source angular radius θ_*, we used the following empirical relation,

$$\begin{eqnarray}&&\mathrm{log}{\theta }_{\mathrm{LD}}=0.5014+0.4197(V-I)-0.2I,\end{eqnarray} \tag{ 2 }$$

where θ_LD ≡ 2θ_* (Fukui et al. 2015). This empirical relation is derived by using a subsample of FGK stars between 3900 K < T_eff < 7000 K from Boyajian et al. (2014) and the accuracy of this relation is better than 2%. According to this relation, we found that θ_* ≡ θ_LD/2 = 0.621 ± 0.045 μas, and angular Einstein radius θ_E and the lens-source relative proper motion μ_rel are given as,

$$\begin{eqnarray*}&&{\theta }_{{\rm{E}}}=\displaystyle \frac{{\theta }_{* }}{\rho }\\ &&\,\,=\,\left\{\begin{array}{l}0.028\pm 0.004\,\mathrm{mas}\,(\mathrm{for}\,\mathrm{the}\,\mathrm{planetary}\,\mathrm{close}\,\mathrm{model})\\ 0.024\pm 0.003\,\mathrm{mas}\,(\mathrm{for}\,\mathrm{the}\,\mathrm{planetary}\,\mathrm{wide}\,\mathrm{model})\\ \gt 0.034\,\mathrm{mas}\,(\mathrm{for}\,\mathrm{the}\,\mathrm{binary}\,\mathrm{close}\,\mathrm{model})\\ \gt 0.035\,\mathrm{mas}\,(\mathrm{for}\,\mathrm{the}\,\mathrm{binary}\,\mathrm{wide}\,\mathrm{model})\end{array}\right.\end{eqnarray*}$$

$$\begin{eqnarray*}&&{\mu }_{\mathrm{rel}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{{t}_{{\rm{E}}}}\\ &&\,\,=\,\left\{\begin{array}{l}1.90\pm 0.29\,\mathrm{mas}\,{\mathrm{yr}}^{-1}\,(\mathrm{for}\,\mathrm{the}\,\mathrm{planetary}\,\mathrm{close}\,\mathrm{model})\\ 1.59\pm 0.18\,\mathrm{mas}\,{\mathrm{yr}}^{-1}\,(\mathrm{for}\,\mathrm{the}\,\mathrm{planetary}\,\mathrm{wide}\,\mathrm{model})\\ \gt 2.26\,\mathrm{mas}\,{\mathrm{yr}}^{-1}\,(\mathrm{for}\,\mathrm{the}\,\mathrm{binary}\,\mathrm{close}\,\mathrm{model})\\ \gt 2.11\,\mathrm{mas}\,{\mathrm{yr}}^{-1}\,(\mathrm{for}\,\mathrm{the}\,\mathrm{binary}\,\mathrm{wide}\,\mathrm{model}),\end{array}\right.\end{eqnarray*}$$

where only the lower limits are given for the binary models.

Since no significant parallax signal is detected in this event, we cannot directly measure the lens properties, the host mass, M_host, the distance, D_L, or lens-source relative transverse velocity, v_⊥. We, therefore, performed a Bayesian approach to explore the probability distribution of the lens characteristics (Beaulieu et al. 2006; Gould et al. 2006; Bennett et al. 2008). We assumed the Galactic model (Han & Gould 1995) as the priors for Galactic mass density and velocity. The observed t_E and θ_E can constrain the lens physical parameters in the Bayesian analysis. Both the observed t_E and θ_E value in this event are smaller than average. This suggests that the host mass is likely to be very low because both t_E and θ_E are proportional to $\sqrt{M}$. Therefore, we need a mass function extending to a very low-mass regime in the Bayesian analysis. We applied the broken power-law mass function used in Sumi et al. (2011) and Mróz et al. (2017) as follows,

$$\begin{eqnarray}{dN}/{dM}=\left\{\begin{array}{ll}{a}_{0}{M}^{-{\alpha }_{\mathrm{pl}}} & (0.001\leqslant M/{M}_{\odot }\leqslant 0.01)\\ {a}_{1}{M}^{-{\alpha }_{\mathrm{bd}}} & (0.01\leqslant M/{M}_{\odot }\leqslant 0.08)\\ {a}_{2}{M}^{-{\alpha }_{\mathrm{ms}2}} & (0.08\leqslant M/{M}_{\odot }\leqslant {M}_{\mathrm{break}})\\ {a}_{3}{M}^{-{\alpha }_{\mathrm{ms}1}} & ({M}_{\mathrm{break}}\leqslant M/{M}_{\odot }\leqslant 1.0),\end{array}\right.\end{eqnarray} \tag{ 3 }$$

where α_ms1 = 2.0 and α_ms2 = 1.3 are the power-law indexes for main-sequence stars. We adopted the power-law index for the BDs of α_bd = 0.49 with M_break = 0.7 (Sumi et al. 2011) and α_bd = 0.8 with M_break = 0.5 (Mróz et al. 2017). Mróz et al. (2017) assumed the low-mass end of the mass function of M/M_⊙ = 0.01 and found that the upper limit of the abundance of Jupiter-mass objects is 0.25 per star with 95% confidence. We applied the case for extending the BD's slope to the planetary-mass regime with α_bd = α_pl, and the case with sharp decline below 0.01 M_⊙ with α_pl = −4.0. The mass functions with α_pl = −4.0 are similar to those in Mróz et al. (2017) but with a slightly gentle cutoff. Figure 4 shows these mass functions. In these mass functions, the relative fractions of number densities between main-sequence stars, BDs, and planetary-mass objects for (α_bd, α_pl) = (0.49, 0.49), (0.49, −4.0), (0.8, 0.8), and (0.8, −4.0) are 1:0.72:0.27, 1:0.72:0.04, 1:0.99:0.70, and 1:0.99:0.07, respectively. Here, the model with (α_bd, α_pl) = (0.49, 0.49) has 0.27 planetary-mass objects per main-sequence stars, which is just slightly higher than the 95% upper limit of Jupiter-mass planetary-mass objects by Mróz et al. (2017). Thus, the abundance of planetary-mass objects in this model can be considered as an upper limit. The model with (α_bd, α_pl) = (0.8, 0.8) has even more planetary-mass objects; thus, it can be considered as an extreme case. On the other hand, the models with (α_bd, α_pl) = (0.49, −4.0) and (0.8, −4.0) can be considered as reference cases if we assume much lower abundance of planetary-mass objects than the upper limit of 0.25 per star. In this analysis, we omitted the mass functions of stellar remnants because the host mass is too low to be affected. Here, the number of main-sequence stars decreases linearly to zero from 1 M_⊙ to 1.2 M_⊙ to avoid the unphysically sharp cutoff at 1 M_⊙. We assumed main-sequence stars with masses above 1.2 M_⊙ have developed into stellar remnants due to their short lifetime (Sumi et al. 2011). We also assumed that the probability of the lens star hosting a companion with the measured physical parameters is independent of the lens host mass because we do not have any information about that for very low-mass hosts.

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The lens parameters and the probability distributions of the lens properties derived from our Bayesian analysis are given in Tables 4 and 5 and Figures 5 and 6. We combined the probability distribution of the close and wide models by weighting the probabilities of the wide models parameters by ${e}^{-{\rm{\Delta }}{\chi }^{2}/2}$, in which ${\rm{\Delta }}{\chi }^{2}={\chi }_{\mathrm{wide}}^{2}-{\chi }_{\mathrm{close}}^{2}$. However, the probability distribution of the projected separation between the host and the companion r_⊥ for the binary model is not combined because the difference of the s values between the close and wide models is so large, s_wide/s_close ≃ 18, compared to that value of the planetary models of ≃2.5.

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Table 4.  Lens Properties Derived from the Bayesian Analysis with α_bd = 0.49

Lens Parameters	αpl = −4.0			αpl = 0.49
	Planetary	Binary		Planetary	Binary
		(s < 1)	(s > 1)		(s < 1)	(s > 1)
Mhost	${25.4}_{-14.2}^{+57.5}{M}_{\mathrm{Jup}}$	${75.5}_{-45.3}^{+126.6}{M}_{\mathrm{Jup}}$		${9.8}_{-6.8}^{+37.6}{M}_{\mathrm{Jup}}$	${76.4}_{-45.1}^{+127.9}{M}_{\mathrm{Jup}}$
Mcomp	${87.2}_{-48.6}^{197.4}{M}_{\oplus }$	${17.8}_{-10.7}^{+29.8}{M}_{\mathrm{Jup}}$		${33.7}_{-23.2}^{+129.0}{M}_{\oplus }$	${18.0}_{-10.6}^{+30.1}{M}_{\mathrm{Jup}}$
r⊥	${0.28}_{-0.10}^{+0.06}\,\mathrm{au}$	${0.19}_{-0.06}^{+0.07}\,\mathrm{au}$	${3.2}_{-1.0}^{+1.2}\,\mathrm{au}$	${0.24}_{-0.09}^{+0.06}\,\mathrm{au}$	${0.19}_{-0.06}^{+0.07}\,\mathrm{au}$	${3.3}_{-1.0}^{+1.2}\,\mathrm{au}$
DL	${7.4}_{-1.0}^{+1.1}\,\mathrm{kpc}$	${6.3}_{-1.3}^{+1.2}\,\mathrm{kpc}$		${7.1}_{-1.0}^{+1.1}\,\mathrm{kpc}$	${6.3}_{-1.3}^{+1.2}\,\mathrm{kpc}$

Note. These results are obtained by Equation (3) with α_bd = 0.49 and M_break = 0.7 as a prior initial mass function (IMF).

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Table 5.  Lens Properties Derived from the Bayesian Analysis with α_bd = 0.8

Lens Parameters	αpl = −4.0			αpl = 0.8
	Planetary	Binary		Planetary	Binary
		(s < 1)	(s > 1)		(s < 1)	(s > 1)
Mhost	${19.7}_{-9.3}^{+43.2}{M}_{\mathrm{Jup}}$	${64.3}_{-39.9}^{+114.0}{M}_{\mathrm{Jup}}$		${6.3}_{-3.9}^{+19.6}{M}_{\mathrm{Jup}}$	${62.0}_{-40.4}^{+113.5}{M}_{\mathrm{Jup}}$
Mcomp	${67.7}_{-31.9}^{+148.5}{M}_{\oplus }$	${15.1}_{-9.4}^{+26.8}{M}_{\mathrm{Jup}}$		${21.6}_{-13.4}^{+67.3}{M}_{\oplus }$	${14.6}_{-9.5}^{+26.7}{M}_{\mathrm{Jup}}$
r⊥	${0.28}_{-0.10}^{+0.06}\,\mathrm{au}$	${0.17}_{-0.06}^{+0.07}\,\mathrm{au}$	${3.1}_{-1.0}^{+1.2}\,\mathrm{au}$	${0.23}_{+0.06}^{-0.08}\,\mathrm{au}$	${0.17}_{-0.06}^{+0.07}\,\mathrm{au}$	${3.1}_{-1.1}^{+1.3}\,\mathrm{au}$
DL	${7.4}_{-1.0}^{+1.0}\,\mathrm{kpc}$	${6.2}_{-1.3}^{+1.2}\,\mathrm{kpc}$		${7.0}_{-1.0}^{+1.1}\,\mathrm{kpc}$	${6.2}_{-1.4}^{+1.2}\,\mathrm{kpc}$

Note. These results are obtained by Equation (3) with α_bd = 0.8 and M_break = 0.5 as a prior IMF.

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As one can see in these tables, the results do not depend strongly on the value of α_bd. However, the host mass for the planetary model strongly depends on the assumed α_pl. For the planetary models with α_pl of −4.0, the lens system is likely a BD orbited by a Saturn-mass giant planet. The lens masses for the planetary models with α_pl = 0.49 and 0.8 are lower than that with α_pl = −4.0 as expected, but consistent to within 1σ. This is because of the strong restrictions from the observed values of t_E and θ_E. In the planetary models with α_pl = 0.49, the host, which is on the boundary between a super giant planetary-mass object and a very low-mass BD, is orbited by a super-Neptune-mass planet or a sub-Saturn-mass planet. The assumption of α_pl = −4.0, i.e., the sharp decline in the lens mass function below 0.01 M_⊙ seems unphysical because there are many claims of BDs with masses below 0.01 M_⊙. Thus, we do not consider the models with α_pl = −4.0 as a final result but as a reference for comparison. For the binary model, on the other hand, the dependency of the Bayesian results on α_bd and α_pl is not strong. This is because we only have a lower limit on θ_E, thus a relatively more massive host is allowed in the Bayesian analysis. This model indicates that the lens system is a BD binary with a small mass ratio. In addition, we conducted the independent Bayesian analysis with a different Galactic model by using the same procedure as Bennett et al. (2014). We confirmed that the results of the Bayesian analysis were consistent with those in Tables 4 and 5. We found that the derived lens-source proper motion, μ, for the binary solutions is preferred than that of the planetary solutions by the Galactic model prior. However, this preference is not so large compared to the preference of the planetary model by the Δχ² ∼ 6 from the light curve fitting. So, we concluded that this event has an ambiguity between two solutions while the planetary solution is slightly preferred.

We represent in Figure 7 the distribution of detected bound exoplanets and very low-mass companions in the plane of the host and companion masses. The two solutions for this event are shown by purple circles. The planets detected by microlensing are represented as red circles, in which open and filled circles indicate planets whose masses were estimated by Bayesian analyses and directly measured by higher order microlensing effects, respectively. The green circles indicate the microlensing binary events with large mass ratios of q > 0.1, which are BD binaries or BDs orbiting around VLMS (Choi et al. 2013; Jung et al. 2015; Han et al. 2016, 2017). They are likely formed by a different mechanism from that of planetary systems with q < 0.03. The binary model of this work belongs to this group. The favored planetary models can be one of a new class of planetary system, having an extremely low-mass host with a planetary-mass ratio (q < 0.03). A similar system is MOA-2011-BLG-262L, where one of the models has a sub-Earth-mass moon orbiting a gas giant primary (Bennett et al. 2014). And, Mróz et al. (2017) also found several short-timescale binary events that could have very low-mass hosts. If such low host mass planetary systems exist, this offers a challenge to planet formation theory. However, the priors in our Bayesian analysis are highly uncertain at low masses for two reasons. First, the mass function in the low-mass region is highly uncertain, and second, we have no idea if the low-mass objects are likely to host planetary-mass ratio companions. Thus, we need to measure the lens mass and distance to know the abundances of planets in these low-mass primaries.

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In order to improve the situation to measure the physical parameters of such short-timescale microlensing events, first we have to obtain better data coverage and accuracy in the light curves to distinguish between competing models. Second, we can constrain the lens mass from the lens flux measurements through high-resolution Adaptive Optics (AO) imaging from the ground or the HST directly. Although we will not be able to detect the flux from such a low-mass object, we will be able to find upper limit constraints on the lens mass. Third, the mass can be measured by observing microlensing parallax. Because annual significant parallax cannot be measured in these short-timescale events, we should attempt to measure the terrestrial parallax effect (Gould et al. 2009; Yee et al. 2009), as this would allow us to determine the lens mass directly from only the light curve when also combining finite source effects (Freeman et al. 2015). The small Einstein radius of the low-mass object is suitable for observing this effect given the short baseline of observatories spread across the Earth's surface. This would be implemented more frequently through improvements of microlensing surveys (Park et al. 2012) for rapid event identification and follow-up organization (Brown et al. 2013). Space-based parallax is more powerful in order to measure the lens mass and this has been demonstrated by the simultaneous observations by the Spitzer satellite and ground-based telescopes (Gould & Yee 2014; Street et al. 2016).

In addition, we also advocate for the importance of considering all viable solutions in all ambiguous events which can be interpreted by both a planetary and a stellar binary-lens system (Gaudi & Han 2004; Choi et al. 2012; Park et al. 2014; Hwang et al. 2018) in order to conduct comprehensive microlensing statistical analyses, such as that of Suzuki et al. (2016) and Udalski et al. (2018). Missing the solutions could lead to incorrect statistical results so that we should fully examine all the solutions in all microlensing events including ambiguous events, particularly in short-timescale events.

NASA's WFIRST mission (Green et al. 2012; Spergel et al. 2013) will conduct a space-based microlensing survey toward the Galactic bulge in the near-infrared and is planned to launch in mid 2020s. WFIRST fulfills all of these requirements with its high precision, high cadence, and high spatial resolution survey. It is very sensitive to such short-timescale planetary events and will enable us to determine the lens masses directly using the space parallax. The PRIME (PRime-focus Infrared Microlensing Experiment) project is planning to conduct a microlensing survey toward the central region of the Galactic bulge ($| b| \leqslant 2^\circ$) in the H-band by using a new 1.8 m wide FOV telescope in South Africa. As the PRIME telescope can observe the same fields as WFIRST, simultaneously, the space-based parallax will be observed regularly without the current alert and follow-up strategy. The longer coverage of the PRIME survey can also add time series outside of the WFIRST's observing window of 72 days duration. These future microlensing surveys will reveal the frequency of exoplanets around low-mass BDs and even the frequency of planetary-mass objects orbited by exo-moons.

We analyzed the short-timescale microlensing event MOA-2015-BLG-337 and found there exist two very degenerate solutions with mass ratios of q ∼ 10⁻² and q ∼ 10⁻¹. The former planetary solutions are favored over the latter binary solution by Δχ² ∼ 6. There are degeneracies between the close and the wide solutions with Δχ² ≃ 0.2 and Δχ² ≃ 2.0 in the planetary and the binary models, respectively. We measured the finite source effect for only the planetary models and obtained a large value of ρ ∼ 10⁻², implying a very low-mass lens. We could not detect a significant microlens parallax signal in the light curve. Therefore, we conducted a Bayesian analysis by applying the observed t_E and θ_E values to estimate the probability distribution of the physical properties of the lens system. Since the host is likely to have a lower mass than the main-sequence stars, we used a mass function extending to BD and planetary masses as a prior. The Bayesian results for binary models do not depend greatly on the value of α_bd and α_pl. On the other hand, the results for planetary models strongly depend on α_pl, while the results are consistent with each other within 1σ. So, there are two competing models of the lens system: (1) a BD/planetary-mass boundary object orbited by a super-Neptune (the planetary model with α_pl = 0.49) and (2) a BD binary (the binary model).

This research has made use of the KMTNet system operated by the Korea Astronomy and Space Science Institute (KASI) and the data were obtained at three host sites of CTIO in Chile, SAAO in South Africa, and SSO in Australia. The OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU. Work by C.R. was supported by an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center, administered by USRA through a contract with NASA. Work by N.K. is supported by JSPS KAKENHI grant No. JP15J01676. Work by Y.H. is supported by JSPS KAKENHI grant No. JP1702146. N.J.R. is a Royal Society of New Zealand Rutherford Discovery Fellow. This work was supported by JSPS KAKENHI grant No. JP17H02871. Work by C.H. was supported by the grant (2017R1A4A1015178) of National Research Foundation of Korea. Work by W.Z., Y.K.J., and A.G. was supported by AST1516842 from the US NSF. W.Z., I.G.S., and A.G. were supported by JPL grant 1500811.