MOA-2019-BLG-008Lb: A New Microlensing Detection of an Object at the Planet/Brown Dwarf Boundary

The microlensing event MOA-2019-BLG-008 was first announced on 2019 February 4 by the MOA Collaboration (Sumi et al. 2003), which operates the 1.8 m MOA survey telescope at Mount John observatory in New Zealand, at equatorial coordinates α = 17^h51^m5589, δ = −29°59'2303 (J2000) (l, b = 3598049, −17203). The event was also independently identified by the Early Warning System (EWS) ³⁵ of the Optical Gravitational Lensing Experiment (OGLE) survey (Udalski 2003; Udalski et al. 2015) as OGLE-2019-BLG-0011. OGLE observations were carried out with the 1.3 m Warsaw telescope at Las Campanas Observatory in Chile, with the 32-chip mosaic CCD camera. The event occurred in OGLE bulge field BLG501, which was imaged about once per hour when not interrupted by weather or the full Moon, providing good coverage of the light curve when the bulge was visible from Chile.

Additional observations were obtained by the ROME/REA survey (Tsapras et al. 2019) using 6 × 1 m telescopes from the southern ring of the global robotic telescope network of the Las Cumbres Observatory (LCO; Brown et al. 2013). The LCO telescopes are located at the Cerro Tololo International Observatory (CTIO) in Chile, South African Astronomical Observatory (SAAO) in South Africa, and Siding Spring Observatory (SSO) in Australia, and they provided good coverage of the light curve, although the event occurred early in the 2019 ROME/REA microlensing season (i.e., ∼March to September of each year, when the Galactic bulge is observable). Observations were acquired in the survey mode.

The event lies in fields BLG02 and BLG42 of the Korea Microlensing Telescopes Network (KMTNet; Kim et al. 2016) and so was intensely monitored by that survey, although KMTNet did not independently discover the event. Observations were also obtained from the Spitzer satellite as part of an effort to constrain the parallax (Yee et al. 2015a). Spitzer observations will be presented in a companion paper (C. Han et al. 2022, in preparation).

This analysis uses all available ground-based observations of MOA-2019-BLG-008. The list of contributing telescopes is given in Table 1. Most data were obtained in the I band (or SDSS-i), but we note that MOA observations were performed with the MOA wide-band red filter, which is specific to that survey (Sako et al. 2008). ROME/REA obtained observations in three different bands (SDSS-i', SDSS-r', and SDSS-g'). The KMTNet survey observations were carried out in the I band, with a complementary V-band observation every 10 I exposures.

The photometric analysis of crowded-field observations is a challenging task. Images of the Galactic bulge contain hundreds of thousands of stars whose point-spread functions (PSFs) often overlap; therefore, aperture and PSF-fitting photometry offer very limited sensitivity to photometric deviations generated by the presence of low-mass planetary companions. For this reason, observers of microlensing events routinely perform difference image analysis (DIA; Tomaney & Crotts 1996; Alard & Lupton 1998; Bramich 2008; Bramich et al. 2013), which offers superior photometric precision under such crowded conditions. Most microlensing teams have developed custom DIA pipelines to reduce their observations. OGLE, MOA, and KMT images were reduced using the photometric pipelines described in Udalski (2003), Bond et al. (2001), and Albrow et al. (2009), respectively. The LCO observations were processed using the pyDANDIA pipeline (ROME/REA, in preparation), a customized reimplementation of the DanDIA pipeline (Bramich 2008; Bramich et al. 2013) in Python. The data sets presented in this paper have been carefully reprocessed to achieve greater photometric accuracy, and it is these data that we used as input when modeling the microlensing event. They are available for download from the online version of the paper.

We note the presence of a very long term baseline trend spanning several observing seasons in the OGLE and MOA photometry that can be seen in Figure 1. As described later in this work, we determined that the source star is a red giant. Many red giants exhibit variability at the ∼10% level (Wray et al. 2004; Wyrzykowski et al. 2006; Percy et al. 2008; Soszyński et al. 2013; Arnold et al. 2020), and it is possible that this is also the case for this source, despite the apparently very long period P ≥ 1000 days. Because this trend manifests over very long timescales (several years), much longer than the duration of the actual microlensing event (weeks), it does not have any noticeable effect on the determination of the parameters of this event, which are primarily derived from the detailed morphology of the microlensing light curve. The baseline over the duration of the microlensing event is effectively flat. Therefore, to increase the speed of the modeling process, we only used observations with JD ≥ 2,457,800 and included data sets with more than 10 measurements in total during the course of the microlensing event. The latter constraint applies only to the LCO data and is limited to the observations acquired by the reactive REA mode on a different target in the same field (Tsapras et al. 2019). We verified that our data selection does not impact the overall results, by exploring models with the full baseline.

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In principle, the normalized angular source radius ρ = θ_*/θ_E has to be considered (Witt & Mao 1994), but preliminary models indicated that this parameter is loosely constrained. This is because the source trajectory does not pass close to caustics, as can be inferred from Figure 2. However, there exists an upper limit ρ_m where the finite-source effects would start to be significantly visible in the models. Because θ_E ≥ θ_*/ρ_m , this limit introduces constraints on the mass and distance of the lens that can be used for the analysis presented in Section 5. Indeed, it is straightforward to derive (Gould 2000):

$$\begin{eqnarray}&&{\pi }_{l}\geqslant \frac{{\pi }_{{\rm{E}}}{\theta }_{* }}{{\rho }_{m}}+{\pi }_{s},\end{eqnarray} \tag{ 4 }$$

where π_l and π_s are the parallax of the lens and source, respectively. The constraint on the mass can be written as

$$\begin{eqnarray}&&{M}_{L}\geqslant \frac{{\theta }_{* }}{\kappa {\pi }_{{\rm{E}}}{\rho }_{m}}.\end{eqnarray} \tag{ 5 }$$

Therefore, we sample the distribution of ρ around the best model and found that ρ_m ≤ 0.01 (with a conservative 10σ limit), and we consider the source as a point for the rest of the modeling presented in this analysis.

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A binary lens model involves seven parameters. Parameter t₀ is the time when the angular distance u₀ (scaled to θ_E) between the source and the center of mass of the binary lens is minimal. The event duration is characterized by the angular Einstein ring radius crossing time t_E = θ_E/μ_rel, where μ_rel is the lens/source relative proper motion (in the geocentric frame, because pyLIMA follows the geocentric formalism of Gould 2004). The binary separation projected on the plane of the lens is defined as s and the mass ratio between two components as q. The angle between the trajectory and the binary axis (fixed along the x-axis) is defined as α. We also consider a source flux f_s and a blend flux f_b for each of the data sets, adding 2n parameters, where n is the number of data sets (i.e., 29 in this study). As discussed previously, we neglect the last parameter ρ and fit the simple point-source binary lens model.

Following Gould (2004), we define the parallax vector π_E by its north (π_E,N) and east (π_E,E) components. We set the parallax reference time as t_0,par = 2,458,570 HJD (Skowron et al. 2011) for all models considered in this analysis. This coincides with the time of the anomaly peak and corresponds to the calendar date 2019 March 28. At this time, Earth's acceleration vector was nearly parallel to the east direction. Therefore, we expect the π_E,E component to be the better constrained of the two. We found that the light-curve morphology can only be explained by a single geometry, in agreement with previous results from real-time modeling conducted by V. Bozza ³⁶ and C. Han. ³⁷ However, a second solution exists, a consequence of the binary ecliptic degeneracy (Skowron et al. 2011) with (u₀, α, π_E,N) $\iff$ − (u₀, α, π_E,N). Because the magnification pattern is symmetric relative to the binary axis, it exists two symmetric source trajectories can produce identical light curves for static binaries, i.e., (u₀, α) $\iff$ − (u₀, α). Moreover, the projected Earth acceleration can be considered as almost constant during the duration of the event, leading to the π_E,N degeneracy for events located toward the Galactic bulge (Smith et al. 2003; Jiang et al. 2004; Gould 2004; Poindexter et al. 2005; Skowron et al. 2011). This degeneracy is especially severe for events occurring near the equinoxes because the projected Earth acceleration varies slowly (Skowron et al. 2011). We therefore explore both solutions in the following analysis. Note that pyLIMA uses the formalism of Gould (2004), and therefore u₀ ≥ 0 if the lens passes the source on its right. Given the relatively long timescale of the event, we also explored the possibility of orbital motion of the lens (Albrow et al. 2000; Bachelet et al. 2012b) and considered the simplest linear model, parameterized with d s/d t and d α/d t, the linear separation and angular variation over time at time t₀. For completeness, we explored the parameter space for a static model (i.e., without the annual parallax) and found that the best model converges to a similar geometry. However, the residuals display systematic trends around the event peak that are the clear signature of annual parallax, which is reflected in the high χ² value presented in Table 2.

Table 2.  Best Model Parameters

Parameters (unit)	2L1S−	2L1S+	2L1S-P	2L1S+P	2L1S-POM	2L1S+POM	1L2S
t0 (HJD −2,450,000)	${8546.47}_{-0.06}^{+0.06}$	${8546.47}_{-0.06}^{+0.06}$	${8550.6}_{-0.3}^{+0.2}$	${8547.8}_{-0.3}^{+0.2}$	${8547.5}_{-0.4}^{+0.4}$	${8546.7}_{-0.2}^{+0.2}$	${8541.8}_{-0.1}^{+0.1}$
u0	$-{0.287}_{-0.002}^{+0.002}$	${0.287}_{-0.002}^{+0.002}$	$-{0.293}_{-0.003}^{+0.003}$	${0.294}_{-0.004}^{+0.003}$	$-{0.360}_{-0.005}^{+0.005}$	${0.354}_{-0.004}^{+0.005}$	${0.195}_{-0.004}^{+0.005}$
tE (days)	${93.4}_{-0.5}^{+0.4}$	${93.4}_{-0.5}^{+0.5}$	${83.7}_{-0.5}^{+0.5}$	${87.9}_{-0.5}^{+0.7}$	${79.2}_{-0.7}^{+0.7}$	${80.9}_{-0.7}^{+0.6}$	${114}_{-2}^{+2}$
πE,N	*	*	$-{0.23}_{-0.02}^{+0.02}$	${0.07}_{-0.01}^{+0.01}$	$-{0.16}_{-0.03}^{+0.03}$	${0.11}_{-0.02}^{+0.02}$	${0.109}_{-0.007}^{+0.007}$
πE,E	*	*	${0.107}_{-0.002}^{+0.002}$	${0.110}_{-0.003}^{+0.003}$	${0.136}_{-0.005}^{+0.005}$	${0.142}_{-0.004}^{+0.005}$	${0.1232}_{-0.006}^{+0.006}$
s	${1.5827}_{-0.0009}^{+0.0009}$	${1.5828}_{-0.0009}^{+0.0009}$	${1.664}_{-0.006}^{+0.005}$	${1.620}_{-0.005}^{+0.004}$	${1.558}_{-0.002}^{+0.002}$	${1.545}_{-0.002}^{+0.002}$	*
q	${0.0267}_{-0.0001}^{+0.0001}$	${0.0267}_{-0.0001}^{+0.0001}$	${0.0395}_{-0.0011}^{+0.0009}$	${0.0320}_{-0.0008}^{+0.0007}$	${0.0201}_{-0.0004}^{+0.0004}$	${0.0187}_{-0.0003}^{+0.0003}$	*
α (rad)	$-{2.297}_{-0.001}^{+0.001}$	${2.297}_{-0.001}^{+0.001}$	$-{2.247}_{-0.003}^{+0.003}$	${2.285}_{-0.003}^{+0.003}$	$-{2.243}_{-0.006}^{+0.005}$	${2.257}_{-0.003}^{+0.003}$	*
ds/dt (yr−1)	*	*	*	*	${0.43}_{-0.04}^{+0.04}$	${0.47}_{-0.03}^{+0.03}$	*
d α/dt (yr−1)	*	*	*	*	$-{1.46}_{-0.09}^{+0.09}$	${1.51}_{-0.07}^{+0.06}$	*
Δt0 (days)	*	*	*	*	*	*	${28.2}_{-0.1}^{+0.1}$
Δu0	*	*	*	*	*	*	$-{0.210}_{-0.005}^{+0.004}$
qV	*	*	*	*	*	*	${0.0910}_{-0.006}^{+0.006}$
${q}_{{\mathrm{MOA}}_{R}}$	*	*	*	*	*	*	${0.0916}_{-0.001}^{+0.001}$
qI	*	*	*	*	*	*	${0.0876}_{-0.001}^{+0.001}$
qgp	*	*	*	*	*	*	${0.4}_{-0.3}^{+0.4}$
qrp	*	*	*	*	*	*	${0.11}_{-0.02}^{+0.03}$
qip	*	*	*	*	*	*	${0.10}_{-0.01}^{+0.02}$
χ2	25,336	25,336	22,556	22,735	22,351	22,333	27,203
θ* (μas)	10.4 ± 0.3	10.4 ± 0.3	10.5 ± 0.3	10.5 ± 0.3	11.5 ± 0.3	11.5 ± 0.3	*

Note. Models parameters are defined as 16th, 50th, and 84th percentile MCMCs, except for the χ2, which is reported as the minimum value (i.e., the best model in each case). The angular source radius θ_* for each model is also presented; see Section 4. The two static models are denoted 2L1S, the two models with parallax are denoted 2L1S-P, and models with orbital motion of the lens are 2L1S-POM; + and − indicate positive or negative u₀.

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We explored the possibility that the observed light curve was due to a binary source (Gaudi 1998). Following the approach described in Hwang et al. (2013), we added to the single lens model the extra parameters Δt₀ and Δu₀, the shifts in the time of peak and separation of the second source relative to the first, respectively. Finally, we also added the flux ratio of the two sources in each observed band q_λ . We report our results in Table 2. We also explore models with two different source angular sizes but did not find any significant improvements in the model likelihood.

The ROME strategy consists of regular monitoring of 20 fields in the Galactic bulge in SDSS-g', SDSS-r', and SDSS-i', as described in Tsapras et al. (2019), and is designed to improve our understanding of the source and blend properties. The photometry is obtained using the pyDANDIA algorithm (R. A. Street et al. 2022, in preparation ³⁸ ) and calibrated to the VPHAS+ catalog (Drew et al. 2016). For this event, we investigated all combinations of filters and telescope sites and selected LSC_A (i.e., LCO dome A in Chile) for the ROME CMD analysis, as it provided the deepest catalog. Figure 4 presents the CMD for stars in a 2' × 2' square centered on the target, while Figure 3 presents a composite image of the ROME observations from LSC_A. The latter presents the variable extinction in the field of view, as well as the two clusters NGC 6451 and Basel 5.

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The first step is to estimate the centroid of the red giant clump (RGC). In Figure 4, stars located within 2' from the event location from the ROME/REA and VPHAS catalogs are displayed in (r − i, i) and (g − i, i) CMDs. We note that the location of the RGC is quite uncertain in the g band for the ROME/REA data. This is due to the high extinction along this line of sight and leads to an accurate g-band calibration of the ROME data. We use the VPHAS magnitudes of these stars to estimate the centroid positions of the RGC in the three bands. We found the magnitudes of the RGC to be ((g), (r), (i))_RGC = (21.989 ± 0.007, 18.836 ±0.005, 17.139 ± 0.005) mag.

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Following the method of Street et al. (2019) and using the intrinsic magnitude of the RGC ${({M}_{g},{M}_{r},{M}_{i})}_{o,c}\,=(1.331\pm 0.056,0.552\pm 0.026,0.262\pm 0.032)$ (Ruiz-Dern et al. 2018), one can estimate the reddening E(g − i) =3.74 ± 0.1 mag, E(r − i) = 1.41 ± 0.1 mag, and extinction A_i = 2.3 ± 0.1 mag. The best model returns a source magnitude of (g, r, i)_s = (24.2 ± 0.8, 19.47 ± 0.05, 17.31 ± 0.03) mag. Because the event occurred at the beginning of the season, the event was poorly covered by the ROME data in the g band and the source brightness is not well constrained. However, we can use the (r − i) color and i magnitude to estimate θ_*. As described in Appendix A, we used the same catalog as Boyajian et al. (2014) to construct a new color–radius relation:

$$\begin{eqnarray}&&\begin{array}{l}{\mathrm{log}}_{10}\left(2{\theta }_{* }\right)=\left(-0.298\pm 0.044\right){\left(r-i\right)}_{o}^{2}\\ \quad +\,\left(0.919\pm 0.058\right){\left(r-i\right)}_{o}-0.2{i}_{o}+\left(0.767\pm 0.010\right).\end{array}\end{eqnarray} \tag{ 6 }$$

This relation returns θ_* = 9.8 ± 1.2 μas, while the second relation using the g band returns θ_* = 22.9 ± 8.4 μas. The latter value is inaccurate owing to large uncertainties in the source brightness in the g band.

The MOA magnitude system can be transformed to the OGLE-III magnitude system (i.e., the Johnson−Cousins system) by using the relation presented in Appendix B. Using the intrinsic color and magnitude of the RGC ((V − I)₀, I₀) = (1.06, 14.32) mag (Bensby et al. 2013; Nataf et al.2013) and and the measured position of the RGC centroid in Figure 5, we could estimate (E(V − I), A_I ) = (2.3 ± 0.1, 2.4 ± 0.1) mag, in good agreement with the previous estimation. Knowing that the transformed magnitudes of the source are (V, I)_s = (20.8 ±0.1, 16.94 ± 0.08) mag, we found (V, I)_0,s = (16.1 ± 0.1, 14.54 ± 0.08) mag, and we ultimately estimate θ_* = 8.9 ±1.3 μas (Adams et al. 2018), in relative agreement with the previous estimation. In Figure 5, we also display the source and blend position using the measurements from OGLE-IV in the I band and the transformed MOA V band. We estimate the source to be (V, I)_s = (20.8 ± 0.1, 16.88 ± 0.01) mag and derive θ_* = 10.4 ± 1.6 μas. This estimate is likely more accurate than the previous one because it relies on a single color transformation (with the highest color term in Equation (B1)).

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The Gaia mission (Gaia Collaboration et al. 2016) recently released their "Early Data Release 3" data set (EDR3; Gaia Collaboration et al. 2020), which significantly increases the volume and precision of the Gaia catalog. We queried the Gaia catalog requesting all stars within a 3' radius ³⁹ around the coordinates of the event to generate a Gaia CMD, which we present in Figure 6. We limit our study to stars with a Renormalized Unit Weight Error (RUWE; a statistical criterion of the data quality) better than 1.4. ⁴⁰ MOA-2019-BLG-008 is in the catalog (at 63 mas; Gaia EDR3 4056394717636682112) and appears in a sparse location of the CMD. The reported parallax is p = 0.39 ± 0.12 mas, which corresponds to a distance of D = 2.56 ± 0.79 kpc. This object is also significantly redder and brighter than the blend discussed in the previous section. Indeed, by using the magnitude transformation from Gaia to the Johnson−Cousins system ⁴¹ (Bachelet et al. 2019), we found this object to be ((V − I), I) =(2.25 ± 0.07, 16.17 ± 0.05) mag, and this is very likely the sum of the source and the blend previously discussed ((V − I), I)_tot = (2.30, 16.19) mag. This is confirmed by several useful metrics available in the catalog. First, we compute the corrected BP and RP excess flux factor (Evans et al. 2018; Riello et al. 2020) and find 0.28, ⁴² which corresponds to a blend probability of ∼0.3 (see Figure 19 of Riello et al. 2020). Second, we note that the fraction R of visits that indicated a significant blend (defined by "phot_bp_n_blended_transits" and "phot_rp_n_blended_transits" for the BP and RP bands, respectively) divided by the number of visits used for the astrometric solution ("astrometric_matched_transits") is very high for both bands, R = 36/39 ∼ 90%.

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Following Mróz et al. (2020), we plot in Figure 6 the distribution of galactic proper motions of the disk (gray) and bulge (red) populations. Note that Gaia EDR3 provides proper motion in equatorial coordinates, which we transformed using the same method as in Bachelet et al. (2019). The disk population, approximated by the main-sequence population, is estimated from the Gaia CMD by using all stars with G_BP − G_RP ≤ 2.5. The bulge population is estimated from the RGC population of the CMD (i.e., 2.8 < G_BP − G_RP ≤ 3.5 and 17.8 ≤ G ≤ 19). Because this object is blended, it is difficult to extract meaningful constraints from the proper-motion distribution. However, it will be possible to do so when the source and lens will be sufficiently separated, as discussed later.

As can be seen in the different CMDs, there is a significant blend flux in the data sets, which likely belongs to the population of foreground stars of the Galactic disk. The measurements from MOA/OGLE are (V, I)_b = (18.5 ± 0.1, 16.88 ± 0.01) mag and are consistent with a late F dwarf located at ∼2.5 kpc (Bessell & Brett 1988; Pecaut & Mamajek 2013; assuming half extinction). Measurements from the ROME survey indicate that the blend brightnesses are (g, r, i)_b = (19.43 ± 0.01, 17.91 ± 0.01, 16.97 ± 0.02) mag, consistent with a G dwarf at ∼2.0 kpc (Finlator et al. 2000; Schlafly et al. 2010). These results are in agreement with the lens properties derived in the next section.

The source and blend have similar brightness in Cousins I band, but the former is much redder. Therefore, the object identified at this location by Gaia is dominated by light from the blend. At the epoch J2016, Gaia measures a total offset of 60 ± 15 mas with respect to the event location measured in 2019 (i.e., during peak magnification). The reported error on the distance has been computed from the north and east components' error from ground surveys (of the order of ∼15 mas) and neglecting Gaia errors (of the order of ∼0.1 mas). Similarly, the magnified source and the baseline object in the KMTNet images are separated by ∼0.068 pixels, which is equivalent to 27 mas. Because the blend and source have similar brightness, this indicates a separation between the blend and the source of ∼60 mas. Therefore, the hypothesis that the blend, or a potential companion, is the lens is probable. Assuming that the blend is the lens, π_E ∼ 0.2, a source distance D_S ∼ 8 kpc, and that the light detected by Gaia is solely due to the blend, we can estimate θ_E ∼ (0.39 − 0.125)/0.2 ∼ 1.3 mas and M_L ∼ 0.8 M_⊙. So the astrometric solution is also compatible with a G dwarf at ∼2.5 kpc, with the notable exception of the relative proper motion. Indeed, we can estimate the geocentric relative proper motion to be μ_geo =θ_E/t_E ∼ 1.3/80 ∼ 6 mas yr⁻¹. Because this event peaked in early March, the heliocentric correction v_⊕ π_rel/au ∼ (0.2, −0.4) is small, and we can therefore assume μ_geo = μ_hel (Dong et al. 2009). The separation between the lens and the source at the Gaia epoch (J2016) is therefore expected to be ∼18 mas, significantly smaller than the previously measured ∼60 mas. However, this argument cannot, by itself, rule out the hypothesis that the blend is the lens owing to the relatively large errors.

Therefore, both photometric and astrometric arguments provide sufficient evidence that the lens represents a significant fraction of the blended light, but only high-resolution imaging in the near future will provide a conclusive answer to this puzzle.

The first galactic model we use to generate a stellar population is the Besançon model ⁴³ (Robin et al. 2003), version M1612. This version consists of an ellipsoidal bulge titled by ∼10° from the Sun−Galactic center direction and populated with stellar masses drawn from a broken power-law initial mass function (IMF) ${dN}/{dM}\propto {M}^{\alpha }$, with α = −1 and α = −2.35 for 0.15 ≤ 0.7 M_⊙ and M > 0.7 M_⊙, respectively (Robin et al. 2012; Penny et al. 2019). The disk is modeled by a thin disk component with a two-slope power-law IMF, with α = 1.6 and α = 3.0 for M ≤ 1 M_⊙ and M > 1 M_⊙ (Robin et al. 2012), while the density distribution is derived from Einasto (1979). The outer part of the disk model has recently been updated and is described in Amôres et al. (2017). The thick-disk and halo population is fully described in Robin et al. (2014), while the kinematics of the population are described in Bienaymé et al. (2015). We select the Marshall et al. (2006) 3D map to estimate the extinction for the simulation. Finally, we note that the Besançon model has been used in several studies for microlensing predictions. Based on the original work of Kerins et al. (2009), Awiphan et al. (2016) and Specht et al. (2020) developed the MaBμls − 2 software that computes theoretical maps of the distribution of optical depth, event rate, and timescales of microlensing events, which are in good agreement with observations. In particular, MaBμls − 2 predictions of event rate and optical depth are in excellent agreement with the 8 yr of observations from the OGLE survey (Mróz et al. 2019). The Besançon model has also been used by Penny et al. (2013) to simulate the potential yields of a microlensing exoplanet survey with the Euclid space telescope. More recently, Penny et al. (2019) and Johnson et al. (2020) used an updated version of the Besançon galactic model to estimate the expected number of detections of bound and unbound planets from the Roman (formerly known as WFIRST) microlensing survey (Spergel et al. 2015).

The second simulation was made using the "Galaxy model" (GalMod, version 18.21), which is a theoretical stellar population synthesis model (Pasetto et al. 2018) ⁴⁴ simulating a mock catalog for a given field of view and photometric system. Similarly as for the Besançon model, the parameter range in magnitude and color permits the simulation of faint lens stars down to the dwarf and brown dwarf regime. Briefly, GalMod consists of the sum of several stellar populations, including a thin and a thick disk, a stellar halo, and a bulge immersed in a halo of dark matter. Stars are generated using the multiple-star population consistency theorem described in Pasetto et al. (2019) with a kinematics model from Pasetto et al. (2016). For our simulation, we used the Rosin−Rammler star formation rate (SFR; Chiosi 1980) for the bulge and the tilted bar. The thin disk is a combination of five different stellar populations with various ages and kinematics, with a constant SFR, while the thick disk is drawn from a single population. We used the same IMF for all the different components of the model (Kroupa 2001).

We first requested samples from the two models within a 2' × 2' cone along the line of sight to the event, and we set the maximum distance to 10 kpc. We then draw samples of lens and source star combinations and apply a sequence of rules. First, the source has to be more distant than the lens. Then, we consider an event only if the angular separation between the source and the lens is below 10''. Following the approach described in Shin et al. (2019), we proceed to compute the associated event parameters (i.e., t_E, π_E, θ_*, and I_s in this case) and compare them with our measured observables derived from modeling. Each such combination contributes to the final derived distribution with a weight ${w}_{i}=\tfrac{1}{\sqrt{2\pi | {C}^{-1}| }}\exp (-{\delta }_{i}^{2}/2)$, with ${\delta }_{i}^{2}$ being the Mahalanobis distance:

$$\begin{eqnarray}&&{\delta }_{i}^{2}={{\boldsymbol{r}}}_{{\boldsymbol{i}}}^{\top }{{\boldsymbol{C}}}^{-1}{{\boldsymbol{r}}}_{{\boldsymbol{i}}},\end{eqnarray} \tag{ 7 }$$

where r _i = (t_E − t_E,i , π_E − π_E,i , θ_* − θ_*,i , I_s − I_s,i ) are the differences between the best-fit model parameters and the simulated parameters and C is the covariance matrix. Note that we also reject models with ρ ≥ 0.01, following the discussion presented in Section 3.1.

Because the galactic models return a finite number of stars (168,134 for the Besançon model and 64,679 for the GalMod model) and the event parameters are slightly unusual (with t_E ∼ 80 days and θ_* ∼ 10 μas), a large fraction of lens and source combinations have a null weight. For instance, the Besançon model contains only ∼0.4% of stars with ∣θ_* − 11.5∣ < 3σ, and about 1% of events are expected to have t_E ∼ 80 days (Mróz et al. 2019). Therefore, the vast majority of trials (≥99.9%) have null weights w_i , and it would therefore require several thousands of billions of trials to obtain meaningful parameter distributions. In light of this, we adjusted our strategy and adopted a Markov Chain Monte Carlo (MCMC) approach. Using the Mahalanobis as the log-likelihood, we adapt the galactic models to define priors on the modeling parameters: the source and lens distances D_s and D_l , the proper motion of the source and lens, the mass of the lens M_l , the angular radius of the source θ_*, and the magnitude of the source I_s . We use a kernel density estimation (KDE) algorithm to derive continuous distributions from the galactic model samples. This allows a prior estimation across the entire parameter space, but at the cost of a somewhat smoother distribution and the use of extrapolation.

We present the posterior distribution for the model 2L1S+POM in Figure 7 and the derived results for all models in Table 4. Results from the galactic model of Dominik (2006) are also presented for comparison. As a supplementary control, we also derived posterior distributions from the recent galactic model of Koshimoto et al. (2021), especially designed for microlensing studies, and found consistent results.

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Table 4. Derived Parameters from the Galactic Models and the Event Modeling, Defined as the 16th, 50th, and 84th Percentile MCMC Chains

Models	DS (kpc) a	μl,s (mas yr−1)	μb,s (mas yr−1)	DL (kpc)	μl,l (mas yr−1)	μb,l (mas yr−1)	μrel (mas yr−1)	θE (mas)	ML (M⊙)	Mp (MJup)
	${6.7}_{-1.1}^{+1.8}$	$-{4.3}_{-2.0}^{+1.2}$	$-{1.1}_{-2.4}^{+1.5}$	${2.2}_{-0.3}^{+0.2}$	${0.6}_{-2.1}^{+1.4}$	$-{0.5}_{-2.5}^{+1.1}$	${5.4}_{-0.6}^{+1.1}$	${1.2}_{-0.1}^{+0.2}$	${0.6}_{-0.1}^{+0.1}$	${25}_{-3}^{+5}$
2L1S-P	${8.0}_{-0.6}^{+0.6}$	$-{5.2}_{-0.8}^{+0.7}$	$-{0.4}_{-0.4}^{+0.4}$	${2.3}_{-0.2}^{+0.2}$	$-{0.2}_{-0.7}^{+0.6}$	$-{0.6}_{-0.5}^{+0.5}$	${5.2}_{-0.4}^{+0.8}$	${1.2}_{-0.1}^{+0.2}$	${0.6}_{-0.1}^{+0.1}$	${24}_{-2}^{+4}$
	8.0 ± 0.5	*	*	${2.6}_{-0.9}^{+1.2}$	*	*	*	*	${0.5}_{-0.2}^{+0.4}$	${21}_{-6}^{+15}$
	${6.6}_{-0.9}^{+1.3}$	$-{3.9}_{-2.2}^{+1.4}$	$-{0.8}_{-1.9}^{+1.1}$	${3.2}_{-0.4}^{+0.3}$	${0.4}_{-1.8}^{+1.4}$	$-{1.2}_{-1.5}^{+2.1}$	${4.8}_{-0.4}^{+0.8}$	${1.2}_{-0.1}^{+0.2}$	${1.1}_{-0.1}^{+0.2}$	${36}_{-4}^{+6}$
2L1S+P	${8.0}_{-0.6}^{+0.6}$	$-{5.2}_{-0.7}^{+0.7}$	$-{0.4}_{-0.4}^{+0.4}$	${3.5}_{-0.4}^{+0.3}$	$-{0.3}_{-0.6}^{+0.8}$	$-{0.3}_{-0.6}^{+0.8}$	${5.0}_{-0.4}^{+1.1}$	${1.2}_{-0.1}^{+0.3}$	${1.1}_{-0.1}^{+0.3}$	${37}_{-4}^{+10}$
	8.0 ± 0.5	*	*	${2.6}_{-0.9}^{+1.2}$	*	*	*	*	${0.8}_{-0.4}^{+0.4}$	${27}_{-13}^{+12}$
	${8.3}_{-0.9}^{+1.0}$	$-{7.5}_{-1.4}^{+1.7}$	${0.0}_{-1.9}^{+1.6}$	${2.4}_{-0.3}^{+0.3}$	$-{0.3}_{-0.6}^{+0.8}$	$-{1.4}_{-1.7}^{+1.5}$	${6.3}_{-0.7}^{+1.4}$	${1.4}_{-0.2}^{+0.3}$	${0.8}_{-0.1}^{+0.2}$	${17}_{-2}^{+4}$
2L1S-POM	${8.0}_{-0.7}^{+0.5}$	$-{5.6}_{-0.7}^{+0.6}$	$-{0.4}_{-0.4}^{+0.3}$	${2.6}_{-0.2}^{+0.2}$	${0.0}_{-0.6}^{+0.5}$	$-{0.1}_{-0.6}^{+0.6}$	${5.7}_{-0.3}^{+0.6}$	${1.2}_{-0.1}^{+0.1}$	${0.7}_{-0.1}^{+0.1}$	${15}_{-1}^{+2}$
	8.0 ± 0.5	*	*	${2.6}_{-0.9}^{+1.2}$	*	*	*	*	${0.6}_{-0.3}^{+0.3}$	${12}_{-6}^{+6}$
	${8.5}_{-0.7}^{+1.0}$	$-{7.6}_{-1.3}^{+1.5}$	${0.0}_{-1.5}^{+1.6}$	${2.7}_{-0.3}^{+0.3}$	$-{1.8}_{-1.4}^{+1.7}$	$-{0.2}_{-1.6}^{+1.6}$	${6.3}_{-0.7}^{+1.1}$	${1.4}_{-0.2}^{+0.3}$	${0.9}_{-0.1}^{+0.2}$	${18}_{-2}^{+3}$
2L1S+POM	${8.0}_{-0.6}^{+0.5}$	$-{5.6}_{-0.7}^{+0.6}$	$-{0.4}_{-0.4}^{+0.3}$	${2.8}_{-0.2}^{+0.2}$	$-{0.1}_{-0.6}^{+0.7}$	$-{0.6}_{-0.4}^{+0.3}$	${5.6}_{-0.4}^{+0.7}$	${1.3}_{-0.1}^{+0.2}$	${0.9}_{-0.1}^{+0.1}$	${17}_{-1}^{+2}$
	8.0 ± 0.5	*	*	${2.6}_{-0.9}^{+1.2}$	*	*	*	*	${0.6}_{-0.3}^{+0.4}$	${12}_{-5}^{+9}$

Notes. Columns display the event physical parameters, while rows correspond to the different models with parallax and orbital motion presented in Section 3. For each microlensing model, the first rows are results from the Besançon model, the second rows are from the GalMod model, and the third rows represent results from Dominik (2006).

^a The source distance and errors have been fixed for Dominik (2006).

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Despite the relatively broad distributions, we find that galactic models are in good agreement for all models. The main differences are seen in the distribution of the source and lens proper motions. The GalMod model has a much narrower distribution of stellar proper motions, as can be seen in the first line of Figure 7, which directly propagates to the source and lens proper motions. However, the relative proper motions of the two galactic models are in 1σ agreement, at ∼5.5 mas yr⁻¹. This is because the relative proper motion is strongly constrained by t_E, which is well determined from the models in this case. For all microlensing models, the derived mass and distance of the host are compatible with the measured blend light, with the exception of the 2L1S-P model, which is much fainter with V_l ∼ 20.5. The companion is an object at the planet/brown dwarf boundary.

While the binary source and static binary models can be safely discarded owing to their very high Δχ² values relative to the best-fit model, the selection of the best overall model with/without orbital motion (Δχ² ∼ 200) and the sign of u₀ (Δχ² ∼ 50) is less trivial. In principle, the Δχ² between the various models is statistically significant. However, despite error bar rescaling, data set residuals can be affected by low-level systematics, leading to errors that are not normally distributed (Bachelet et al. 2017). Because they modify the source trajectory in a similar way, the orbital motion and parallax parameters are often correlated (Batista et al. 2011), which is also the case for this event. The north component π_E,N of the parallax vector is in agreement between all nonstatic models, suggesting that the parallax signal is strong and real, which is expected for an event duration of t_E ∼ 80 days. For models including orbital motion, we can use the results from galactic models to verify whether the system is bound or not (Dong et al. 2009; Udalski et al. 2018). The condition for bound systems is (K_E + P_E ) < 0, where K_E and P_E are the kinetic and potential energies, and this can be rewritten in terms of (projected) escape velocity ratio (Dong et al. 2009):

$$\begin{eqnarray}&&\frac{{v}_{\perp }^{2}}{{v}_{\mathrm{esc},\perp }^{2}}=\frac{{({a}_{\perp })}^{3}}{8{\pi }^{2}{M}_{l}}\left[{\left(\frac{1}{s}\frac{{ds}}{{dt}}\right)}^{2}+{\left(\frac{d\alpha }{{dt}}\right)}^{2}\right]\leqslant 1.\end{eqnarray} \tag{ 8 }$$

We found $\frac{{v}_{\perp }^{2}}{{v}_{\mathrm{esc},\perp }^{2}} \tilde 4$ and $\frac{{v}_{\perp }^{2}}{{v}_{\mathrm{esc},\perp }^{2}} \tilde 6$ for the 2L1S-POM and 2L1S+POM models, respectively. Taken at face value, the ratio of projected velocities indicates that the companion is not bound and that these models are unlikely. However, the relative errors are large, i.e., ≥100%, and therefore the models with orbital motion cannot be completely ruled out. But given the relatively low improvement in the χ² of the orbital motion models, we decided to not explore more sophisticated models, such as the full Keplerian parameterization (Skowron et al. 2011).