A Wide-orbit Exoplanet OGLE-2012-BLG-0838Lb

The exoplanets known today show a large degree of diversity. For example, we now know a planetary system orbiting a pulsar (PSR1257+12; Wolszczan & Frail 1992), extremely short-period planets (55 Cnc e; Winn et al. 2011), planets with extremely high surface temperatures (KELT-9b; Gaudi et al. 2017), rocky planets in the habitable zone (Kepler-186f; Quintana et al. 2014), a gas giant planet orbiting a brown dwarf (2M1207b; Chauvin et al. 2004), and an Earth-mass planet around an ultra-cool dwarf (OGLE-2016-BLG-1195; Shvartzvald et al. 2017), to name a few. These planets have been discovered using a few different detection techniques, and each technique has distinct capabilities and limitations. By far the largest number of planets have been discovered using the transit technique, and in particular the yield of planets from Kepler, the first mission to statistically explore the population of exoplanets over a broad region of parameter space, was notably high (Coughlin et al. 2016). Kepler exoplanets are on orbits similar to the inner planets in the solar system and in many cases are more compact than that of Mercury. The longest-period confirmed transiting exoplanets are Kepler-1647b (1108 days; Kostov et al. 2016), Kepler-167e (1071 days; Kipping et al. 2016), and Kepler-1654b (1048 days; Beichman et al. 2018). The orbital periods of these planets are shorter than the orbital periods of all solar system gas and ice giants. The lack of a large number of the long-period planets hampers our understanding of the formation of planetary systems as a whole and our ability to place the solar system in the context of known exoplanetary systems in particular.

The main reason for this unsatisfactory situation is that different planet detection techniques have different sensitivities to the wide-orbit planets. For giant planets, the radial velocity (RV) technique is intrinsically limited by the length of the time baseline of the RV surveys themselves (Kane 2011; Sahlmann et al. 2016; Wittenmyer et al. 2017). For example, only recently did Blunt et al. (2019) report the detection of a 3M_Jup minimum mass object on a ${74}_{-22}^{+43}\,\mathrm{yr}$ long highly eccentric orbit via RVs, and in this case the detection required a fairly fortuitous alignment of the orbit of the planet. In particular, the RV data taken during periastron passage of the planet exhibited a signal that is highly unlikely to be produced by other astronomical phenomenon. The limit set by the long-term stability of the spectrographs makes detection of long-period Neptune-mass planets much more difficult than long-period Jupiter-mass planets: the RV signals are $0.5\,{\rm{m}}\,{{\rm{s}}}^{-1}$ and $9\,{\rm{m}}\,{{\rm{s}}}^{-1}$, respectively, for a Neptune-mass and a Jupiter-mass planet on a 10 au edge-on orbit around 1 M_⊙ star. Astrometric detection of planets on relatively wide orbits (semimajor axis up to 5–6 au) can be done using Gaia data or by combining Gaia and Hipparcos data (Perryman et al. 2014; Snellen & Brown 2018), but the astrometric technique is also only sensitive down to Jupiter-mass objects for most stars. The Gaia mission can be extended from nominal 5 yr up to 10 yr and this will increase the number of detected planets by a factor of 3–4 (Perryman et al. 2014). The extension of the Gaia mission improves sensitivity to the wider-orbit and lower-mass planets, but still predicted detections have orbits smaller than the orbit of Saturn. It is possible to improve astrometric constraints on the orbital period by incorporating the RV data (Eisner & Kulkarni 2002; Feng et al. 2019), but the fundamental limitations given earlier still apply.

There are two planet detection techniques that find wide-orbit planets: direct imaging and microlensing. Current direct imaging surveys (Baron et al. 2019; Nielsen et al. 2019) can detect planets with separations from ≈5 au to thousands of astronomical units and more massive than ≈2 Jupiter masses. The direct imaging and microlensing can detect planets at similar separations, but there are significant differences between these techniques. Direct imaging discovers self-luminous planets around nearby young stars, and allows follow-up studies of these directly detected objects, including spectroscopy (Bowler 2016). Contrary, the microlensing planets orbit old stars at a range of distances and there is no possibility for spectroscopic follow-up. The comparison of statistical properties of direct imaging and microlensing planets should give constraints on the planet migration.

The microlensing technique is sensitive to planetary systems that are a few kiloparsec away, and mostly probe the planetary population orbiting the most numerous, low-mass (and hence mostly old) stars. The ongoing microlensing surveys are sensitive to planet/star mass ratios smaller than 10⁻³ even for wide-orbit planets. In fact, the widest-orbit microlensing planet has a mass ratio of 2.4 × 10⁻⁴ (OGLE-2008-BLG-092LAb; Poleski et al. 2014). Microlensing can probe Neptune-mass planets, even for planets that have no detectable stellar host and thus may be unbound (Mróz et al. 2018).

It is important to combine the constraints of both the wide-orbit and the free-floating planets (Mróz et al. 2017) in order to fully understand the formation and evolution of planetary systems. The bound-planet parameters that are readily measured for microlensing are the mass ratio (q) and the projected separation (s) in units of the Einstein ring radius (θ_E ). The microlensing planets with the widest orbits are OGLE-2008-BLG-092LAb (s = 5.3; Poleski et al. 2014), OGLE-2011-BLG-0173Lb (s = 4.6; Poleski et al. 2018), and KMT-2016-BLG-1107Lb (s = 3.0; Hwang et al. 2019)—see discussion in Poleski et al. (2018). There are only a few more planets with s > 2. For a typical configuration, θ_E corresponds to around 2.5 au. Hence, the three widest-orbit planets are at projected separations from 7 to 15 au. The distribution of microlensing planets as a whole has already been studied statistically (e.g., Gould et al. 2010; Cassan et al. 2012; Suzuki et al. 2016; Udalski et al. 2018), but the statistical properties of the wide-orbit planets have not yet been comprehensively analyzed, partly due to the small number of known such systems.

The large sample of wide-orbit planets is important for understanding formation of planetary systems. We have detailed knowledge about Uranus and Neptune, but explaining their formation is challenging. The standard core-accretion model (Pollack et al. 1996) cannot reproduce properties of Uranus and Neptune if formed in situ. Three major theoretical approaches to formation of solar system ice giants are migration from closer orbits (Thommes et al. 1999; Tsiganis et al. 2005), pebble accretion (Lambrechts et al. 2014; Venturini & Helled 2017), and collisions of planetary embryos (Izidoro et al. 2015); at this point, none of these theories are favored.

Here we present the discovery of a wide-orbit exoplanet OGLE-2012-BLG-0838Lb. A short-duration anomaly is observed well before the main peak of the event and points to an event with s = 2.1. The wide-orbit planet interpretation is confirmed by detailed modeling. The planetary anomaly was found in pure survey observations by the Optical Gravitational Lensing Experiment (OGLE; Udalski et al. 2015), i.e., the planet detection did not depend on targeted follow-up photometry. This means that the planet can be included in future statistical studies of the wide-orbit planets. For OGLE-2012-BLG-0838, high-resolution imaging and satellite imaging were collected, which helps to constrain the planet properties.

In the next section, we present the data collected for OGLE-2012-BLG-0838. We describe the model fitting in Section 3. In Section 4, we analyze current constraints on the physical properties of the system. We summarize the paper in Section 5.

2.1. OGLE photometry

OGLE is a large-scale photometric survey. It is currently in its fourth phase (OGLE-IV) and operates a 1.3 m telescope at Las Campanas Observatory (Chile) that is equipped with a 32 CCD chip camera (256 M pixels in total). The camera field of view is 1.4 deg², and the pixel scale is 026. OGLE bulge observations are performed in the I band, and we use only these to fit the microlensing model. When the anomaly occurred, the field of OGLE-2012-BLG-0838 was observed once per one or two nights as a part of bulge survey observations aiming at finding the ongoing microlensing events. This cadence of observations is not enough for characterizing planetary anomalies in most cases, but gives targets for follow-up photometric observations. For OGLE-2012-BLG-0838 anomaly, there is a single epoch that deviates by more than 1 mag and four epochs that deviate by less than 0.25 mag. There are 20 OGLE fields that are observed with higher cadence. For the OGLE-2012-BLG-0838 field and CCD camera chip (#32), the median seeing is 146, which is slightly higher than for the same chip in higher-cadence bulge fields (≈135). Additional lower cadence V-band data taken in survey mode on the target exist, but do not cover the anomaly, and we use them only to characterize the source star. Photometry of the OGLE data is performed using a difference image analysis (DIA; Alard 2000; Woźniak 2000). We corrected the native photometric uncertainties following Skowron et al. (2016). We use data from 2012 as well as 2011, which constrain the baseline brightness. For a more detailed description of the OGLE survey, see Udalski et al. (2008) and Udalski et al. (2015).

The search for microlensing events in the OGLE data is performed daily (Udalski 2003). The event OGLE-2012-BLG-0838 was discovered on ${{\rm{HJD}}}^{{\rm{{\prime} }}}\equiv {\rm{HJD}}-2,450,000=6082$, i.e., after the anomaly was over (see Figure 1). The planetary nature of the anomaly was first suggested on HJD' = 6126.403 (by A. U.), and subsequently the planetary models were fitted (by C. H.). Event coordinates are ${\rm{R}}.{\rm{A}}.\,=\,{18}^{{\rm{h}}}{12}^{{\rm{m}}}00\buildrel{\rm{s}}\over{.} 74$ and $\mathrm{decl}.\,=\,-25^\circ 42^{\prime} 41\buildrel{\prime\prime}\over{.} 8$, which translate to l = 5720 and b = − 3472. The baseline brightness in the standard photometric system is I = 17.610 mag and (V–I) = 1.851 mag (Szymański et al. 2011).

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2.2. MOA Photometry

2.3. EPOXI Imaging

2.4. VVV Photometry

2.5. SMARTS Photometry

2.6. MagAO Imaging

The H band high-resolution images of the OGLE-2012-BLG-0838 field were taken on HJD' = 6766, with the Magellan Adaptive Optics system (MagAO; Close et al. 2012; Males et al. 2014; Morzinski et al. 2014) on the 6.5 m Clay Telescope at Las Camapanas Observatory (Chile). We used the Clio Wide camera, which has a plate scale of 27.49 mas and a field of view of 14'' × 28''. The integration time for an individual science exposure was 30 s, and we took 10 sets of images with four dithers for each set. Individual dithered frames were astrometrically aligned using the positions of the 10 bright isolated stars, and then the aligned and resampled images were median-combined. We performed the coordinate transformation from the OGLE frame to the MagAO frame using the positions of the six common isolated stars. The position of the source that we identify on MagAO image lies (22, −14) ± (19, 17) mas in the east and north relative to the transformed position of the target centroid on the subtracted OGLE image. The closest star on the MagAO images is about 390 mas away, so the identification of the target is secure (see Figure 2). The MagAO source is isolated with an FWHM of 160 mas. We use SExtractor (Bertin & Arnouts 1996) to perform aperture photometry on the MagAO images. MagAO data are typically calibrated to the Two Micron All-Sky Survey (2MASS) photometric catalog (Skrutskie et al. 2006). Due to the lack of overlapping stars between the MagAO image of OGLE-2012-BLG-0838 and 2MASS catalog, we used the VVV data as a bridge between 2MASS and MagAO to do the photometric calibration. We performed PSF photometry on the extracted VVV image with DoPhot (Schechter et al. 1993) and then we used common isolated stars within 3' of the target to calibrate it to the 2MASS magnitude system. Only stars with H > 12.8 mag are used to avoid detector nonlinearity for VVV. Then we calibrated the MagAO magnitudes using four common isolated stars (marked in Figure 2) between MagAO and VVV.

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The light curve of OGLE-2012-BLG-0838 (Figure 1) presents the main event, which, except short anomaly, is well approximated by the Paczyński (1986) point-source point-lens model. The anomaly is short and high-amplitude, but its detailed shape is not well determined. Such events can be produced by two types of events: (1) a binary source and a single lens (Gaudi 1998), or (2) a single source and a binary lens. Furthermore, the binary-lens case presents two possibilities (e.g., Bhattacharya et al. 2016; Poleski et al. 2018): separation can be larger or smaller than one (called wide and close model, respectively). We discuss all three possibilities below, starting from the binary lens s > 1 (or wide solution), which turns out to be the correct model. The model fitting was performed using data sets that cover the anomaly, i.e., OGLE I band and both MOA data sets.

3.1. Wide Binary-lens Model

To represent a binary-lens model, we use following parameters: t₀—the epoch of the minimum approach, u₀—the minimum separation (normalized to ${\theta }_{{\rm{E}}}$), ${t}_{{\rm{E}}}$—the Einstein timescale, ρ—the source radius (normalized to ${\theta }_{{\rm{E}}}$), α—the angle between the source trajectory and the lens axis, s, and q. For parameter conventions we follow Skowron et al. (2011) and define t₀ and u₀ relative to the primary lens. The first three parameters (t₀, u₀, and ${t}_{{\rm{E}}}$) are constrained by the main subevent, i.e., their values can be obtained by fitting a point-source point-lens model to the data with the short-duration anomaly epochs removed (HJD' from 6077 to 6083). The other parameters are constrained by the time and length of the short-duration anomaly except ρ, and can be reasonably well estimated by visual inspection of the light curve. There are two additional flux parameters for each data set: the source flux and the blending flux. We estimate them separately for each model using linear regression. The linear limb-darkening coefficients are assumed to be Γ_I = 0.46, Γ_MOA − R = 0.51, and Γ_MOA,V = 0.66, which were estimated based on a preliminary fitted model and the color–surface brightness relations by Claret & Bloemen (2011).

We explored the parameter space using the Multimodal Ellipsoidal Nested Sampling algorithm (MultiNest; Feroz & Hobson 2008; Feroz et al. 2009). At each step MultiNest approximates the probed distribution by a union of multidimensional ellipsoids. MultiNest can sample the multimodal posterior and search for multiple separated modes, which is an important advantage. The search for multiple modes can be run on all parameters or a selected subset of parameters. Additionally, MultiNest properly calculates Bayesian evidence for each mode. In our practice, MultiNest requires more model evaluations than the Monte Carlo Markov Chain (MCMC) methods, but execution time is not a limiting factor for OGLE-2012-BLG-0838. MultiNest was able to model OGLE data alone, while MCMC methods had poor convergence.

MultiNest found three separate modes whose main difference was the best-fit value of ρ. In particular, MultiNest found that the three modes had values of ρ = 0.0011 ± 0.0007, ρ = 0.0037 ± 0.0002, and ρ = 0.00595 ± 0.00034. The first two modes require fine tuning of u₀, ${t}_{{\rm{E}}}$, and s. MultiNest not only searches for multiple modes and calculates posterior distributions of parameters, but also calculates the posterior probability of each mode. The posterior probabilities for the first and the second modes are smaller than the third one by a factor of 50 and 180, respectively. Additionally, the first mode predicts a large relative lens-source proper motion of $\approx 16\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$, which is a priori unlikely. Thus, the third mode is a priori preferred and we consider only this mode as viable in the rest of the paper (details of the other two models are presented in the Appendix). We present the results of the model fitting in Table 1 and Figure 3. The best model is plotted in Figure 1.

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Table 1.  Wide Binary-lens Model for OGLE-2012-BLG-0838

Parameter	Static Model
t0	6145.909 ± 0.034
u0	0.373 ± 0.011
tE (d)	40.44 ± 0.84
ρ	0.00595 ± 0.00034
α (deg)	12.66 ± 0.11
s	2.153 ± 0.029
q	0.000395 ± 0.000033
Fs/Fbasea	0.875 ± 0.035
χ2/dof	830.72/678

Note.

^aF_s is the source flux and F_base is the baseline flux (i.e., source plus blending). Both are for the OGLE I band.

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It turns out that the combination of the MOA and OGLE data restricts the number of separate modes significantly better than the OGLE data alone. When we fitted only the OGLE data and searched for multiple modes on all parameters, then MultiNest reported only a single mode. When the search for multiple modes was run only on ρ, then 10–30 modes were found, depending on the exact settings. The 1σ ranges of the posterior parameters of these modes were overlapping, which showed that the OGLE data alone do not allow a unique identification of multiple modes. We inspected many modes and in Figure 4 present a few modes, which were selected to show the whole range of the diversity of the light curves. The problems we faced with fitting the OGLE data alone are a less-severe case of the discrete and continuous degeneracies seen in the case of OGLE-2002-BLG-055 (Gaudi & Han 2004), which also had only a single epoch that is much brighter than predicted by the point-source point-lens model.

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We derive the source brightness using posterior distributions. We obtain V_s = 19.596 ± 0.044 mag, I_s = 17.754 ± 0.043 mag, H_s = 15.484 ± 0.043 mag, and ${K}_{s,s}=15.326\,\pm 0.044\,\mathrm{mag}$. We also use calibrations of the MOA photometry to the OGLE-III magnitude system (Szymański et al. 2011) to derive the source brightness from the MOA data. The transformations have a scatter of 0.048 mag in the V band and 0.045 mag in the I band. We obtain ${V}_{s}^{\mathrm{MOA}}\,=19.587\pm 0.043\,\mathrm{mag}$ and ${I}_{s}^{\mathrm{MOA}}=17.662\,\pm 0.043\,\mathrm{mag}$, which are consistent with V_s and I_s derived from the OGLE data within uncertainties.

After considering the static binary-lens model, we attempted to include the microlensing parallax effect. Microlensing parallax is described by a 2D vector ${{\boldsymbol{\pi }}}_{{\bf{E}}}$, whose amplitude is equal to the relative lens-source parallax divided by ${\theta }_{{\rm{E}}}$. If both ${\theta }_{{\rm{E}}}$ and ${{\boldsymbol{\pi }}}_{{\bf{E}}}$ are measured, then both the lens mass (M) and distance (D_l) are measured directly (Gould 2000):

$$\begin{eqnarray}&&M=\displaystyle \frac{{\theta }_{{\rm{E}}}}{\kappa {\pi }_{{\rm{E}}}},\qquad \displaystyle \frac{1}{{D}_{l}}=\displaystyle \frac{{\pi }_{{\rm{E}}}{\theta }_{{\rm{E}}}}{\mathrm{AU}}+\displaystyle \frac{1}{{D}_{s}},\end{eqnarray} \tag{ 1 }$$

where $\kappa =4{\rm{G}}/(\mathrm{AU}\,{{\rm{c}}}^{2})=8.14\,\mathrm{mas}\,{M}_{\odot }^{-1}$ is a constant, and D_s is the source distance. The annual microlensing parallax breaks the assumption that the apparent lens-source relative motion is rectilinear. The effect is undetectable for most events, because during their (typically short) duration, Earth's motion around the Sun can be well approximated by a straight line. OGLE-2012-BLG-0838 has relatively long ${t}_{{\rm{E}}}$ of 40 days. The anomaly additionally increases the chances of measuring ${\pi }_{{\rm{E}}}$, because it provides a well-timed event (An & Gould 2001).

We consider two degenerate scenarios: u₀ < 0 and u₀ > 0. The best-fitting parallax model improves χ² by 23.4 and the uncertainty of ${\pi }_{{\rm{E}},N}$ is large (${\sigma }_{{\pi }_{{\rm{E}},N}}=0.33$). We checked a plot of χ² difference between the best parallax model and the best static model. It revealed that there may be a problem with the quality of the MOA data on a timescale of dozens of days. It is known that low-level systematics may mimic the microlensing parallax signal. Some trends in residuals of static binary-lens model can be seen in Figure 1, e.g., around HJD' = 6100. Thus, we decided to report only a well-established static model and do not present potentially spurious parallax models.

3.2. Close Binary-lens Model

We additionally searched for close (i.e., s < 1) binary-lens models and found two such solutions. The parameters of these models are presented in the Appendix. The first model (marked "close A" in Figure 5) has χ² = 857.84, i.e., it is worse fit to the data by Δχ² = 27.12. The wide model is favored over the close model a priori. First, the wide model predicts the relative lens-source proper motion of $\approx 3\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$ (see below), which is the typical value, while the close model predicts much less likely $\approx 15\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$. Second, a recent statistical analysis of microlensing events (Suzuki et al. 2016) shows that the microlensing planet occurrence rate is increasing with increasing s and decreasing q, and this result is confirmed by a joint analysis of microlensing, RV, and direct imaging results (Clanton & Gaudi 2016). We reject the close model based on Δχ², as well as the two arguments given above.

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The close model with α = 177 deg has a source trajectory that crosses the binary axis outside the caustics (in other words, the source passes all caustics on the same side). There is a second model in which the source trajectory crosses the binary axis between the planetary and central caustics (see the lower part of Figure 3 in Poleski et al. 2018). For OGLE-2012-BLG-0838, the latter model has α = 207.20 ± 0.98 deg (marked "close B" in Figure 5) and a corresponding χ² is 876.34, i.e., which is sufficiently larger (${\rm{\Delta }}{\chi }^{2}\simeq 45.62$) than the best fit (wide model) χ² to be rejected. We compare all three binary-lens models for the anomaly part of the light curve in Figure 5.

3.3. Binary-source Model

Here we discuss a few different pieces of information about the lens and source. We are not able to directly measure the lens mass and distance, but we discuss the prospects for doing so. Thus, we derive the lens properties using Bayesian priors derived using a Galactic simulation. We also discuss the origin of the excess flux.

In Figure 2, we show the MagAO image of OGLE-2016-BLG-0838. The final calibrated H-band brightness of the target is ${H}_{\mathrm{target}}=15.29\pm 0.05\,\mathrm{mag}$, where the uncertainty estimate combines the statistical and systematic components. H_target is brighter than the H-band source flux measured using SMARTS photometry and the difference corresponds to ${H}_{\mathrm{excess}}\,={17.26}_{-0.33}^{+0.46}\,\mathrm{mag}$. Later in this section we discuss where this excess flux comes from.

The relative lens-source proper motion is ${\mu }_{\mathrm{rel}}={\theta }_{{\rm{E}}}/{t}_{{\rm{E}}}\,=3\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$ (see below). We may expect that the lens and source could be resolved in about 10 years from now allowing the lens flux to be measured and leading to an estimate of the lens mass and distance, when combined with the stellar isochrones (Yee 2015) and the constraint on the angular Einstein ring radius from the detection of finite source effects. In some cases, an identification of the lens in the follow-up high-resolution imaging is problematic (Bhattacharya et al. 2017). The future lens flux measurement can definitely use the MagAO image presented here for calibration. We also list nearby stars in Table 2. As one can see in Figure 2, the event is by far the brightest object within the ground-based seeing limit.

The existing EPOXI data have not yet been reduced. We use representative parallax wide binary-lens models (all are within 2σ) fitted to the OGLE data to predict the magnification as seen by EPOXI—see Figure 6. We also show a histogram of the amplitude (i.e., difference between maximum and minimum) of magnification predicted for EPOXI in Figure 7. The lens mass and distance can be measured directly if the microlensing parallax is measured. Some of the magnification curves are almost flat. If the true magnification curve is almost flat, then the parallax measurement is unlikely. If the highest magnification is ≲4, then the magnification curve can be reasonably well approximated as a linear function of time. In this case, it will be necessary to remove potential systematic linear trends in the EPOXI photometry in a manner that is independent of the photometry of the source in the EPOXI data in order to measure ${{\boldsymbol{\pi }}}_{{\bf{E}}}$.

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The event parameters could be constrained better if the proper motion of the source was known. The baseline object is included in the Gaia DR2 (Gaia Collaboration et al. 2018). The astrometric χ²/dof (keyword astrometric_gof_al) is 3.9 while values  >3 indicate that the fit to the data is unreliable. The Gaia proper-motion measurement interpretation is additionally hindered by the fact that the baseline object is a blend of sources with detectable contribution from other stars (see Table 1). It is unlikely that Gaia DR2 proper motion can put useful constraints on system properties.

To measure ${\theta }_{{\rm{E}}}$, we use the method developed by Yoo et al. (2004). We present the color–magnitude diagram of stars lying within 2' from the target in Figure 8. The red clump has an observed color of (V–I)_RC = 2.003 ± 0.008 mag and a brightness of I_RC = 15.489 ± 0.030 mag. We compare these values with the extinction-corrected values from Bensby et al. (2011) and Nataf et al. (2013), respectively, to obtain E(V–I) = 0.943 mag and A_I = 1.205 mag. The extinction-corrected source properties are ${I}_{s,0}=16.550\pm 0.043\,\mathrm{mag}$ and ${V}_{s,0}=17.450\pm 0.066\,\mathrm{mag}$ and using the Bessell & Brett (1988) color–color relations we obtain ${(V-K)}_{s,0}=2.022\,\pm 0.147\,\mathrm{mag}$. The estimated ${(V-K)}_{s,0}$ and ${V}_{s,0}$ correspond to ${\theta }_{\star }=1.93\pm 0.14\,\mu {\rm{as}}$ (Kervella et al. 2004). When combined with ρ for the wide model we obtain ${\theta }_{{\rm{E}}}=0.325\pm 0.029\,\mathrm{mas}$.

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We do not have an interesting constraint on the microlensing parallax. Hence, we must use the Bayesian simulations of the Galaxy to derive the lens mass and distance. For this purpose, we use an approach similar to that presented by Clanton & Gaudi (2014). The lenses are drawn from the density profiles of a double-exponential disk (Zheng et al. 2001) and boxy Gaussian bulge (model G2 by Dwek et al. 1995), which are normalized according to Gould et al. (1996) and Han & Gould (2003), respectively. The lens mass function is taken from the model 1 in Sumi et al. (2011). For the source distance we use the boxy Gaussian bulge distribution, i.e., model G2 by Dwek et al. (1995) and weight it by D_s². The kinematics of the disk and bulge follow Clanton & Gaudi (2014). The event rate Γ is weighted according to (Clanton & Gaudi 2014)

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{4}{\rm{\Gamma }}}{{{dD}}_{l}{{dM}}_{l}{d}^{2}{{\boldsymbol{\mu }}}_{{\bf{rel}}}}=2{r}_{{\rm{E}}}{\mu }_{\mathrm{rel}}{D}_{l}\nu \displaystyle \frac{{d}^{2}{\rm{\Gamma }}}{{d}^{2}{{\boldsymbol{\mu }}}_{{\bf{rel}}}}\displaystyle \frac{d{\rm{\Gamma }}}{{{dM}}_{l}},\end{eqnarray} \tag{ 2 }$$

where ${r}_{{\rm{E}}}={D}_{l}{\theta }_{{\rm{E}}}$ is the Einstein ring radius projected on the lens plane and ν is the position-dependent density of lenses. In this simulation, we drawn a total of 1.5 · 10⁹ events. The results of the simulations are presented in Table 3 and in Figure 9.

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Table 3.  Posterior Physical Parameters Statistics

Parameter	Unit	Value
μrel	$\mathrm{mas}\,{\mathrm{yr}}^{-1}$	2.96 ± 0.27
Ds	kpc	8.17 ± 0.91
Dl	kpc	${6.32}_{-1.04}^{+0.83}$
Ml	${M}_{\odot }$	${0.40}_{-0.20}^{+0.29}$
Mp	${M}_{\mathrm{Jup}}$	${0.167}_{-0.83}^{+0.121}$
${r}_{{\rm{E}}}$	AU	${2.06}_{-0.38}^{+0.33}$
r⊥a	AU	${4.43}_{-0.81}^{+0.71}$

Note.

^aInstantaneous projected star–planet separation: ${r}_{\perp }={{sD}}_{l}{\theta }_{{\rm{E}}}$.

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There are four possible sources contributing to the measured excess flux: the lens, unrelated ambient star(s), a companion to the source star, and a companion to the lens star. Following the method developed by Koshimoto et al. (2017) and fully described by Koshimoto et al. (2020), we calculate the probabilities of all possible combinations of the four sources that explain the observed excess flux, assuming that none of them is a stellar remnant. We use posterior distributions of D_s, D_l, and M_l from the above Bayesian simulations as a prior to analyze the origin of the excess flux. We use the luminosity function from Zoccali et al. (2003) which is normalized to the stellar density in the OGLE-2012-BLG-0838 field to calculate the ambient stars flux distribution, where the normalization is done by comparing the number of the red clump stars in this field to that in the Zoccali et al. (2003) field using the OGLE-III catalog (Szymański et al. 2011). For the flux distribution of a companion to the source or the lens star, we use the binary distribution, which is based on Ward-Duong et al. (2015) and on the summary in a review paper of stellar multiplicity by Duchêne & Kraus (2013). We consider only companions to the source or lens whose mass ratio and semimajor axis are consistent with both the light curve and the MagAO image where no signal of a detectable stellar companion is shown. The prior distributions of parameters (M_l, D_l, and H-band magnitudes) are shown in Figure 10.

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After deriving the prior distributions, we apply an H_excess constraint as detailed by Koshimoto et al. (2020). We present the resulting posterior distributions in Figure 11. The measured H_excess is brighter than in prior (see center panel in Figure 10), hence, adding an H_excess constraint increases the estimate of the lens mass (from ${0.40}_{-0.20}^{+0.29}\,{M}_{\odot }$ to ${0.54}_{-0.29}^{+0.33}\,{M}_{\odot }$). Also the posterior probabilities that the lens companion, the source companion, or the ambient star contribute to H_excess are higher than the corresponding prior probabilities. In particular, the probability that the source companion contributes to H_excess increased from 0.37 to 0.65. The main origin of H_excess is therefore more likely to be the lens or the source companion.

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We can estimate the expected RV signal from OGLE-2012-BLG-0838Lb. We assume the median values for stellar host scenario from Table 3, i.e., M_l = 0.40 M_⊙ and ${r}_{\perp }=4.43\,\mathrm{au}$. We estimate the semimajor axis assuming a random position of the planet and a circular orbit, $a={(3/2)}^{1/2}{r}_{\perp }=5.4\,\mathrm{au}$. The orbital period is $P={({a}^{3}/M)}^{1/2}=19.9\,\mathrm{yr}$. For the edge-on configuration, the RV signal would be $K=3.2\,{\rm{m}}\,{{\rm{s}}}^{-1}$. Detecting planets with similar properties around nearby stars would be challenging for the RV surveys. The longest-period RV planets with well-measured RV curves are HR 5183b (P ≈ 74 yr and $K\,=38.3\,{\rm{m}}\,{{\rm{s}}}^{-1}$; Blunt et al. 2019), HD 30177c (P = 20.8 yr and $K=35.8\,{\rm{m}}\,{{\rm{s}}}^{-1}$ or P = 31.8 yr and K = 59.4 m s⁻¹; Wittenmyer et al. 2017), and GJ 676Ac ($P=20.4\,\mathrm{yr}$, $K=90.0\,{\rm{m}}\,{{\rm{s}}}^{-1}$; Sahlmann et al. 2016), though for neither of them the RV data cover the full orbital period. The amplitudes of the RV signals for these three planets are more than an order of magnitude larger than predicted for OGLE-2012-BLG-0838Lb.

We have presented the microlensing discovery of a wide-orbit planet OGLE-2012-BLG-0838Lb. Alternative models of observed light curve were considered and found inadequate. Finding planets on similar orbits around local low-mass stars presents a challenge. The lens physical properties are constrained but not directly measured. We have discussed additional existing and future data that can measure the physical parameters of the lens system directly.

OGLE Team acknowledges Marcin Kubiak and Grzegorz Pietrzyński, former members of the team, for their contribution to the collection of the OGLE photometric data over the past years. Work by R.P. was partly supported by Polish National Agency for Academic Exchange grant "Polish Returns 2019." We thank Michael Albrow for information about the EPOXI observations. We thank Ping Chen and Ji Wang for their help. This work was supported by NASA contract NNG16PJ32C. The OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to A.U. The MOA project is supported by JSPS KAKENHI grants No. JSPS24253004, JSPS26247023, JSPS23340064, JSPS15H00781, JP16H06287, and JP17H02871. The work of N.K. was supported by JSPS KAKENHI grant No. JP18J00897. The work of J.-P.B. and J.-B.M. was supported by the ANR COLD-WORLDS project, grant ANR-18-CE31-0002-01 of the French Agence Nationale de la Recherche. Work by C.H. was supported by the grants of National Research Foundation of Korea (2017R1A4A1015178 and 2019R1A2C2085965). This paper includes data gathered with the 6.5 m Magellan Clay Telescope at Las Camapanas Observatory, Chile. X.X. and S.D. acknowledge Project 11573003 supported by NSFC. This research uses data obtained through the Telescope Access Program (TAP), which has been funded by "the Strategic Priority Research Program-The Emergence of Cosmological Structures" of the Chinese Academy of Sciences (grant No.11 XDB09000000) and the Special Fund for Astronomy from the Ministry of Finance. Work by A.G. was supported by AST-1516842 from the US NSF and by JPL grant 1500811. A.G. received support from the European Research Council under the European Union's Seventh Framework Programme (FP 7) ERC grant Agreement No. [321035]. Based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 179.B-2002.

Software: Astropy (Astropy Collaboration et al. 2013, 2018), SExtractor (Bertin & Arnouts 1996), DoPhot (Schechter et al. 1993), MulensModel (Poleski & Yee 2018, 2019), Poleski et al. (2014; https://arxiv.org/src/1408.6223v3/anc), MultiNest (Feroz & Hobson 2008; Feroz et al. 2009), Numerical Recipes (Press et al. 1992), SM https://www.astro.princeton.edu/~rhl/sm/, AstroML (Ivezić et al. 2014), corner.py (Foreman-Mackey 2016).

Tables A1, A2, and A3 present parameters of rejected wide binary-lens models, close binary-lens models, and a binary-source model, respectively. Indexes 1 and 2 indicate parameters for the first and second source, respectively.

Table A1.  Two Rejected Wide Binary-lens Models for OGLE-2012-BLG-0838

Parameter
t0	6145.909 ± 0.029	6145.907 ± 0.025
u0	0.3585 ± 0.0017	0.34289 ± 0.0028
tE (d)	41.53 ± 0.14	42.88 ± 0.26
ρ	0.00111 ± 0.00073	0.00372 ± 0.00020
α (deg)	12.611 ± 0.046	12.438 ± 0.044
s	2.1199 ± 0.0044	2.0764 ± 0.0076
q	0.00389 ± 0.00019	0.00371 ± 0.00013
Fs/Fbase	0.829 ± 0.013	0.781 ± 0.050
${\chi }^{2}/\mathrm{dof}$	835.16/678	836.23/678

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Table A2.  Close Binary-lens Models for OGLE-2012-BLG-0838

Parameter	Close A Model	Close B Model
t0	6146.197 ± 0.041	6145.866 ± 0.043
u0	0.3869 ± 0.0092	0.341 ± 0.017
tE (d)	39.02 ± 0.69	44.8 ± 1.5
ρ	0.00123 ± 0.00012	0.000541 ± 0.000087
α (deg)	176.68 ± 0.92	207.20 ± 0.98
s	0.4554 ± 0.0053	0.4964 ± 0.0097
q	0.0159 ± 0.0021	0.0120 ± 0.0016
Fs/Fbase	0.920 ± 0.030	0.768 ± 0.049
${\chi }^{2}/\mathrm{dof}$	857.84/678	876.34/678

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Table A3.  Binary-source Model for OGLE-2012-BLG-0838

Parameter	Value
t0,1	6145.943 ± 0.035
u0,1	0.417 ± 0.016
tE (d)	37.37 ± 0.95
t0,2	6079.3968 ± 0.0071
u0,2	0.00037 ± 0.00028
ρ2	0.00781 ± 0.00029
${F}_{s,1}/{F}_{\mathrm{base}}$	1.027 ± 0.052
${F}_{s,2}/{F}_{\mathrm{base}}$	0.00875 ± 0.00026
${\chi }^{2}/\mathrm{dof}$	988.98/678

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