Spitzer Parallax of OGLE-2018-BLG-0596: A Low-mass-ratio Planet around an M Dwarf

OGLE-2018-BLG-0596 is at (R.A., decl.)_J2000 = (17:56:13.33, −29:11:56.7), corresponding to (l, b) = (0.96, −2.13) in Galactic coordinates. It was discovered as a probable microlensing event by the Optical Gravitational Lensing Experiment (OGLE: Udalski et al. 2015a) survey, and announced on 2018 April 15 through its Early Warning System (Udalski 2003). The event is in the OGLE-IV field BLG505, for which OGLE observations were conducted with a one hour cadence using the 1.3 m Warsaw telescope located at Las Campanas in Chile.

The Microlensing Observations in Astrophysics (MOA: Sumi et al. 2003) survey independently discovered this event on May 15 and named it as MOA-2018-BLG-145. The MOA observations were taken using the 1.8 m MOA-II telescope located at Mt. John Observatory in New Zealand. The MOA observation cadence for the event is 15 minutes.

The event was also independently discovered by the Korea Microlensing Telescope Network (KMTNet: Kim et al. 2016) by employing their post-season event finder algorithm (Kim et al. 2018), and it was cataloged as KMT-2018-BLG-0945. The KMTNet survey used three 1.6 m telescopes positioned at the Cerro Tololo Interamerican Observatory, Chile (KMTC), the South African Astronomical Observatory, South Africa (KMTS), and the Siding Spring Observatory, Australia (KMTA). The KMTNet observations were conducted with a 30 minute cadence.

The great majority of images were obtained in the I band for OGLE and KMTNet and a customized R band for MOA,⁴¹ with some V-band images for the source color measurement. All of the survey data were reduced using the image subtraction methodology (Alard & Lupton 1998), specifically Woźniak (2000) for OGLE, Bond et al. (2001) for MOA, and Albrow et al. (2009) for KMTNet.

In addition to the observations from these high-cadence surveys, the event was observed by two lower-cadence surveys. These surveys used, respectively, the 3.8 m United Kingdom Infrared Telescope (UKIRT: Shvartzvald et al. 2017a) and the 3.6 m Canada–France–Hawaii Telescope (CFHT: Zang et al. 2018) that are both located at the Maunakea Observatory in Hawaii. The UKIRT and CFHT observations for the event were carried out in the H and i band, respectively.

On May 10, the KMTNet group noticed in KMTNet data reduced on the basis of the OGLE alert that the event had shown an anomaly at HJD'(=HJD − 2,450,000) ∼ 8243.5. Because this anomaly occurred when the event was just ∼0.3 mag brighter than its baseline, it was impossible to precisely determine the lensing parameters at that time. Nevertheless, they found from real-time modeling that the anomaly was likely to have been produced by a very low-mass companion to the primary lens, i.e., a planetary companion. In response to this potential importance, the Spitzer team announced OGLE-2018-BLG-0596 as a Spitzer target on May 24. The Spitzer observations for the event were initiated on July 4 (when it first became observable due to Sun-angle restrictions) with a cadence of 1 day. In total, 36 images were taken during 8304 < HJD' < 8341. The data were reduced based on the methods presented by Calchi Novati et al. (2015b).

2.2.1. Is the Event Part of the Spitzer Parallax Sample?

We first model the light curve based on the data acquired from the ground-based observations. For our standard binary-lens model, we introduce seven nonlinear parameters (see the appendix of Jung et al. 2015 for a graphical presentation of the parameters.) This includes three single-lens parameters (t₀, u₀, t_E), three parameters for the binary companion (s, q, α), and one parameter for the source radius ρ. Here, u₀ is the impact parameter (in units of θ_E), s is the companion-host projected separation (in units of θ_E), q = M₂/M₁ is their mass ratio, and α is their orientation angle with respect to ${{\boldsymbol{\mu }}}_{\mathrm{rel}}$. In addition, we introduce two flux parameters (f_S, f_B)_i for each data set in order to model the observed flux f_i(t) as

$$\begin{eqnarray}&&{f}_{i}(t)={f}_{{\rm{S}},i}A(t)+{f}_{{\rm{B}},i},\end{eqnarray} \tag{ 5 }$$

where A(t) is the magnification given by the model and the subscripts "S" and "B" denote the source and any blended light, respectively.

With these fitting parameters, we carry out a systematic analysis by following the procedure of Jung et al. (2015). First, we estimate initial values of (t₀, u₀, t_E) = (8277.17, 0.28, 28.83 days) by fitting a 1L1S curve to the data set with the anomaly excluded. We also adopt an initial value of ρ = 0.01 based on t_E and the source brightness estimated from the fit. We next perform a grid search over (s, q), in which (s, q) are held fixed, while (t₀, u₀, t_E, α, ρ) are sought based on a downhill approach. For this approach, as well as for determining the uncertainties of the parameters, we employ a Markov Chain Monte Carlo (MCMC) algorithm. For each set of fitting parameters, the lensing magnification is evaluated by inverse ray-shooting (Kayser et al. 1986; Schneider & Weiss 1987) in the anomaly region and by semianalytic approximations (Gould 2008; Pejcha & Heyrovský 2009) elsewhere. This model magnification is then used to fit the flux parameters (f_S, f_B)_i that minimize the χ² of the observed flux f_i(t).

Figure 2 displays the Δχ² distribution on the (log s, log q) space acquired from the grid search. We identify two local minima, one with s < 1 ("Close") and the other with s > 1 ("Wide"). We find that in both solutions, the lens system responsible for the weak "bump" is composed of two masses with $q\lesssim {10}^{-4}$, implying that the lower-mass component is a planet. We then seed the local solutions into MCMCs and allow all fitting parameters to vary. The two standard solutions derived from this refinement process are given in Tables 1 and 2.

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Table 1.  Lensing Parameters for Close Solutions

Parameters	Standard	Parallax		Orbit+Parallax
		(+, +)	(−, −)	(+, +)	(−, −)
${\chi }_{\mathrm{tot}}^{2}$/dof	25959.6/26562	25901.6/26596	25906.6/26596	25895.4/26594	25897.3/26594
t0 (HJD')	8277.16 ± 0.010	8277.14 ± 0.010	8277.14 ± 0.010	8277.14 ± 0.015	8277.14 ± 0.015
u0	0.285 ± 0.002	0.285 ± 0.004	−0.284 ± 0.004	0.284 ± 0.006	−0.282 ± 0.006
tE (days)	28.881 ± 0.109	28.924 ± 0.110	29.038 ± 0.109	29.003 ± 0.110	29.170 ± 0.116
s	0.566 ± 0.003	0.564 ± 0.005	0.565 ± 0.005	0.512 ± 0.017	0.499 ± 0.018
q (10−4)	1.203 ± 0.089	1.327 ± 0.105	1.313 ± 0.106	1.827 ± 0.132	1.879 ± 0.133
α (rad)	6.072 ± 0.009	6.076 ± 0.011	−6.071 ± 0.011	6.022 ± 0.011	−6.025 ± 0.014
ρ* (10−2)	1.120 ± 0.042	1.139 ± 0.043	1.137 ± 0.042	1.347 ± 0.049	1.362 ± 0.050
πE,N	⋯	−0.041 ± 0.023	0.043 ± 0.022	−0.023 ± 0.022	0.033 ± 0.026
πE,E	⋯	0.177 ± 0.010	0.179 ± 0.010	0.178 ± 0.010	0.177 ± 0.010
ds/dt (yr−1)	⋯	⋯	⋯	−0.580 ± 0.045	−0.734 ± 0.031
dα/dt (yr−1)	⋯	⋯	⋯	−0.637 ± 0.101	0.566 ± 0.135
fS	1.956 ± 0.002	1.955 ± 0.002	1.945 ± 0.002	1.945 ± 0.002	1.930 ± 0.002
fB	−0.044 ± 0.003	−0.043 ± 0.003	−0.032 ± 0.003	−0.032 ± 0.003	−0.017 ± 0.003
${\chi }_{\mathrm{ground}}^{2}$	25959.6	25867.4	25871.6	25860.9	25861.8
${\chi }_{{Spitzer}}^{2}$	⋯	34.1	34.9	34.5	35.1
${\chi }_{\mathrm{penalty}}^{2}$	⋯	0.11	0.14	0.0079	0.43

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Table 2.  Lensing Parameters for Wide Solutions

Parameters	Standard	Parallax		Orbit+Parallax
		(+, +)	(−, −)	(+, +)	(−, −)
${\chi }_{\mathrm{tot}}^{2}$/dof	26009.5/26562	25915.7/26596	25926.1/26596	25912.2/26594	25924.0/26594
t0 (HJD')	8277.16 ± 0.010	8277.13 ± 0.011	8277.14 ± 0.011	8277.13 ± 0.015	8277.13 ± 0.014
u0	0.286 ± 0.002	0.284 ± 0.004	−0.286 ± 0.004	0.285 ± 0.006	−0.285 ± 0.005
tE (days)	28.833 ± 0.107	28.974 ± 0.110	28.870 ± 0.105	28.889 ± 0.107	28.894 ± 0.105
s	1.769 ± 0.004	1.773 ± 0.006	1.775 ± 0.006	1.811 ± 0.015	1.788 ± 0.016
q (10−4)	0.160 ± 0.015	0.187 ± 0.015	0.181 ± 0.014	0.244 ± 0.027	0.188 ± 0.019
α (rad)	2.897 ± 0.011	2.899 ± 0.018	−2.901 ± 0.015	2.929 ± 0.018	−2.921 ± 0.016
ρ* (10−2)	1.495 ± 0.066	1.587 ± 0.074	1.591 ± 0.058	1.772 ± 0.078	1.572 ± 0.059
πE,N	⋯	−0.061 ± 0.025	−0.071 ± 0.024	−0.075 ± 0.026	−0.078 ± 0.032
πE,E	⋯	0.191 ± 0.010	0.157 ± 0.010	0.193 ± 0.011	0.159 ± 0.011
ds/dt (yr−1)	⋯	⋯	⋯	0.371 ± 0.048	0.151 ± 0.036
dα/dt (yr−1)	⋯	⋯	⋯	0.355 ± 0.089	−0.225 ± 0.091
fS	1.961 ± 0.002	1.949 ± 0.002	1.961 ± 0.002	1.959 ± 0.002	1.958 ± 0.002
fB	−0.048 ± 0.003	−0.035 ± 0.003	−0.047 ± 0.003	−0.045 ± 0.003	−0.045 ± 0.003
${\chi }_{\mathrm{ground}}^{2}$	26009.5	25881.5	25890.3	25878.3	25887.5
${\chi }_{{Spitzer}}^{2}$	⋯	34.1	35.6	33.9	35.6
${\chi }_{\mathrm{penalty}}^{2}$	⋯	0.12	0.16	0.0019	0.94

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We now take into account the microlens parallax in order to simultaneously model the data obtained from the ground and Spitzer. This introduces two additional parameters ${{\boldsymbol{\pi }}}_{{\rm{E}}}=({\pi }_{{\rm{E}},N},{\pi }_{{\rm{E}},E})$, which represent the vector microlens parallax (Gould 1992), i.e.,

$$\begin{eqnarray}&&{{\boldsymbol{\pi }}}_{{\rm{E}}}=\displaystyle \frac{{\pi }_{\mathrm{rel}}}{{\theta }_{{\rm{E}}}}\displaystyle \frac{{{\boldsymbol{\mu }}}_{\mathrm{rel}}}{{\mu }_{\mathrm{rel}}}.\end{eqnarray} \tag{ 6 }$$

Then, the parallax parameters are approximately related to the offset ${\rm{\Delta }}{\boldsymbol{u}}=({\rm{\Delta }}\beta ,{\rm{\Delta }}\tau )$ between the two light curves observed from the ground and Spitzer, i.e.,

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{\pi }}}_{{\rm{E}}}=\displaystyle \frac{\mathrm{au}}{{D}_{\perp }}({\rm{\Delta }}\beta ,{\rm{\Delta }}\tau );\,\,\,\,\,{\rm{\Delta }}\beta ={u}_{0,\oplus }-{u}_{0,\mathrm{sat}};\\ {\rm{\Delta }}\tau =\displaystyle \frac{{t}_{0,\oplus }-{t}_{0,\mathrm{sat}}}{{t}_{{\rm{E}}}},\end{array}\end{eqnarray} \tag{ 7 }$$

where D_⊥ is the projected Earth-Spitzer separation and (Δβ, Δτ) represent the components of the lens-source separation vector that are perpendicular to and parallel with the source trajectory, respectively. For single-lens events, the perpendicular offset Δβ  generally suffers from a fourfold degeneracy

$$\begin{eqnarray}&&{\rm{\Delta }}\beta =\pm {u}_{0,\oplus }-\pm {u}_{0,\mathrm{sat}},\end{eqnarray} \tag{ 8 }$$

due to the rotational symmetry of the lensing magnification about the lens (Refsdal 1966; Gould 1994). These four possible solutions are usually denoted by (+, +), (−, −), (+, −), and (−, +) depending on the signs of u_0,⊕ and u_0,sat. For binary lenses, however, the fourfold degeneracy persists only if the source trajectory is almost parallel to the binary axis, i.e., α ∼ 0 (Zhu et al. 2015), and otherwise is reduced to a twofold degeneracy: (+, +) and (−, −). We therefore expect that OGLE-2018-BLG-0596 may only suffer from a twofold degeneracy, but we need a detailed analysis to draw a definitive conclusion.

To conduct a systematic analysis, we first consider additional information extracted from the ground- and space-based observations. As shown in Figure 1, the Spitzer data only cover the falling side of the event and do not cover the anomaly. In such a case, it is difficult to precisely constrain ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ from the data alone. We therefore apply a color constraint on the Spitzer source flux to improve the parallax measurement (Yee et al. 2015b). For this, we derive an IHL color–color relation using the OGLE, UKIRT, and Spitzer data based on the method described by Calchi Novati et al. (2015b).⁴² From a model-independent regression of I- and H-band data (Gould et al. 2010), we first measure the instrumental (I − H) color of the source as (I − H)_S = 2.81 ± 0.01. We next construct (I − H, I) and (I − L, I) instrumental color–magnitude diagrams (CMDs) by cross-matching field stars within 120'' of the source. We then conduct the color–color regression on red giant stars (15 < I < 18; 2.4 < (I − H) < 3.0) to confine the sample to the bulge population. From this, we find (I − L) =1.43(I − H) − 8.72. We thereby derive (I − L)_S = −4.70 ±0.02, where the instrumental Spitzer magnitude is given by $L=25-2.5\mathrm{log}\,{f}_{{\rm{S}},{Spitzer}}$. We impose this color constraint on the model by adding a χ² penalty, i.e.,

$$\begin{eqnarray}&&{\chi }_{\mathrm{penalty}}^{2}=\displaystyle \frac{{\left[2.5\mathrm{log}({R}_{\mathrm{model}}/{R}_{\mathrm{constraint}})\right]}^{2}}{{\sigma }_{\mathrm{constraint}}^{2}},\end{eqnarray} \tag{ 9 }$$

where R is the flux ratio between I and L bands and σ_constraint is the uncertainty of (I − L)_S.

Space-based observations can provide an opportunity not only to measure the microlens parallax but also to constrain the orbital motion of the binary lens (Han et al. 2016). As discussed by Batista et al. (2011), the annual microlens parallax (due to Earth's orbital motion) can be partially degenerate with the lens orbital motion, and so the microlens parallax measured from a single accelerating frame can absorb the lens orbital motion. By contrast, the space-based microlens parallax does not suffer from this degeneracy because it is determined from the feature of the light curves from two different observatories. This enables one to break such degeneracy and detect the lens orbital motion in the ground-based light curve. Therefore, we also take into account the lens orbital effect. To account for this effect, we introduce two linearized parameters (ds/dt, dα/dt), which represent the relative velocity of the lens components projected onto the plane of the sky (Dominik 1998; Jung et al. 2013).

We now model the light curve with a set of parameters (t₀, u₀, t_E, s, q, α, ρ, π_E,N, π_E,E, ds/dt, dα/dt) and the color constraint described above. For each of the Close and Wide configurations obtained from the ground data sets, we first conduct a grid search for the parameters (π_E,N, π_E,E) in order to check the four possibilities of the ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ measurement. We then rerun the MCMC process with various starting points identified in the (π_E,N, π_E,E) space. From this, we find that in both configurations, all MCMC chains converged to two points (see the second and third columns in Tables 1 and 2). That is, they do not suffer from the degeneracy between the pair of [(+, +), (+, −)] or [(−, −), (−, +)], and only suffer from the degeneracy between the pair of [(+, +), (−, −)]. The latter degeneracy is induced by the mirror symmetry of source trajectories relative to the binary axis, i.e., the "ecliptic degeneracy" (Jiang et al. 2004; Skowron et al. 2011). Finally, we further investigate the solutions by including the lens orbital effects.

In Tables 1 and 2, we present the four solutions, i.e., [(+, +), (−, −)] × (Close, Wide), solutions. The corresponding caustic geometries are shown in Figure 3. We find that in each solution, the source star fully envelops a planetary caustic that is located far from the host. In the Close configuration, the anomaly is generated by the envelopment of one of the triangular caustics, while in the Wide configuration it is generated by the envelopment of the quadrilateral caustic (e.g., Hwang et al. 2019). The most important difference between the two sets of solutions is in the mass ratio q, which is almost 10 times smaller in the Wide solutions.

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We find that the ${\chi }^{2}$ difference of the ground data sets between the standard and best-fit (+, +)_Close solution is Δχ² ∼ 99 (see Table 1). This suggests that the microlens parallax is partially constrained by the annual microlens parallax effect. To better understand this, we additionally model the ground-based light curve with the lens orbital effect and the annual parallax effect ("orbit+AP").

However, we find the possibility that the χ² improvement is caused by systematics of the data. From the cumulative distribution of ${\rm{\Delta }}{\chi }^{2}={\chi }_{\mathrm{standard}}^{2}-{\chi }_{\mathrm{orbit}+\mathrm{AP}}^{2}$ as a function of time, we find that there is a long-term inconsistent trend between KMTNet+MOA and the other data sets (see Figure 4). That is, most of the improvement comes from the KMTNet+MOA data, while the improvement from the other data is very minor. This discrepancy implies that these two data sets may not be stable enough to precisely explore the parameter space (e.g., Han et al. 2018). From this, one might further conjecture that our   π_E measurements are affected by false-positive effects caused by the systematics. We therefore step back and carry out a series of tests to verify our solutions.

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First, we fit for the geometric parameters using only the Spitzer data and the color constraint. That is, we exclude the ground observations in order to identify all possible combinations of (π_E,N, π_E,E) that are consistent with the space observations alone. In this modeling, the lensing geometry seen from the ground is set by imposing Gaussian constraints on the fitting parameters (t₀, u₀, t_E, f_S,OGLE, f_B,OGLE) based on the ground-based solution derived in Section 3.1. Next, we include all data sets except for KMTNet and MOA, for which we use only partial data sets. While the parallax parameters are measured from the overall shape of the light curve, the binary parameters are measured from the anomaly. For this event, the overall shape is well characterized by the other data sets, but their coverage near the anomaly is very poor. To account for the anomaly as well as to remove any spurious parallax effects originating from possible systematics, we therefore use only the data near the anomaly region (specifically $8235\lt \mathrm{HJD}^{\prime} \,\lt 8250$) for KMTNet and MOA. With these modified data sets, we then run full MCMC chains incorporating all models obtained from the first step.

From this test, we find that all MCMC chains converged to two points for both the Close and Wide configurations. In addition, the locations of these two points in each configuration are nearly identical to those derived from the full data sets, indicating that the measured parallaxes are consistent with each other. This consistency can be seen in Figure 5, where we show the Δχ² maps in the (π_E,N, π_E,E) plane obtained from the test. We note that the cross mark in each panel represents the location of ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ listed in Tables 1 and 2, i.e., the four solutions. From this figure, one also finds that only the (+, +) and (−, −) models are permitted by the modified ground-based data sets. Therefore, we conclude that our four solutions are not significantly affected by systematics.

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The χ² difference between the (+, +)_Close and (+, +)_Wide solution is Δχ² ∼ 17.⁴³ Mathematically, this implies that the probability of (+, +)_Wide solution relative to the (+, +)_Close solution is lowered by ${P}_{\mathrm{lc}}={e}^{-{\rm{\Delta }}{\chi }^{2}/2}\sim 2\times {10}^{-4}$. However, this relative fit probability depends on the assumption that all data have uncorrelated errors. Unfortunately, such conditions are generally not satisfied for crowded field photometry. Hence, it is difficult to entirely reject the (+, +)_Wide solution from the measured Δχ² alone.

Nevertheless, we can better understand the χ² difference by inspecting the cumulative distribution of ${\rm{\Delta }}{\chi }^{2}={\chi }_{{(+,+)}_{\mathrm{Wide}}}^{2}\,-{\chi }_{{(+,+)}_{\mathrm{Close}}}^{2}$ (see Figure 6). From this, we find that most of the difference comes from the rising part of the anomaly (HJD' ∼ 8241), where the (+, +)_Wide solution provides a relatively poor fit to the data. In this region, the (+, +)_Wide model curve is on average located 0.01 mag above that of the (+, +)_Close solution due to the difference in the lensing magnification field. For the Close configuration, the source passes over the negative planet-host axis during the few days immediately prior to the planetary-caustic anomaly (see Figure 3). Generically, this axis is characterized by a trough (e.g., Gaudi 2012). By contrast, the Wide configuration has no such trough. Moreover, the short-term deviation that favors the Close solution cannot be ascribed to the type of long-term systematics discussed above. Therefore, we consider that the Wide solutions are disfavored. We will further discuss this preference of the data in Section 4.2.

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As discussed by Gaudi (1998), short-term anomalies can also be produced by a binary source, i.e., 1L2S event. In particular, if the binary source (denoted as "S1" and "S2") has a large flux ratio q_F = f_S2/f_S1 and the second source passes very close to the lens, the resulting light curve can take a similar form to that of a 2L1S planetary event. We therefore search for 1L2S solutions based on the method of Jung et al. (2017). In this search, we simultaneously consider the parallax effect, finite-source effect, and orbital motion of the binary source (the xallarap effect). We find that the best-fit 1L2S solution is disfavored by Δχ² ∼ 87. To check the result, we also draw the cumulative Δχ² distribution of the 1L2S solution relative to the best-fit 2L1S solution. As shown in Figure 6, we find that most of the χ² difference comes from the short-lived anomaly region (8240 < HJD' < 8245), in which the 1L2S solution continuously fails to fit the observed light curve. Hence, we exclude the 1L2S solution.

We evaluate θ_* using the method of Yoo et al. (2004). Based on the KMTC star catalog constructed by the pyDIA reduction, we first estimate the instrumental source color (V − I)_S = 2.65 ± 0.01 and magnitude I_S = 17.19 ± 0.01 from regression and the model, respectively. We next measure the centroid of the giant clump (GC) as (V − I, I)_GC = (2.57 ± 0.02, 16.29 ± 0.02). Figure 7 displays the locations of the source and GC in the KMTC CMD. We then compare this centroid to the calibrated centroid of (V − I, I)_0,GC = (1.06 ± 0.07, 14.40 ± 0.09) obtained from Bensby et al. (2013) and Nataf et al. (2013), respectively. This yields an offset Δ(V − I, I) =(1.51 ± 0.07, 1.89 ± 0.09). Using this offset, we estimate the dereddened source position as

$$\begin{eqnarray}\begin{array}{rcl}{\left(V-I,I\right)}_{0,{\rm{S}}} & = & {\left(V-I,I\right)}_{{\rm{S}}}-{\rm{\Delta }}(V-I,I)\\ & = & (1.14\pm 0.07,15.30\pm 0.09).\end{array}\end{eqnarray} \tag{ 10 }$$

We then apply ${(V-I)}_{0,{\rm{S}}}$ to the VIK relation of Bessell & Brett (1988) and derive ${(V-K)}_{0,{\rm{S}}}=2.63\pm 0.07$. Finally, we estimate θ_* from the (V − K)_0,S − θ_* relation of Kervella et al. (2004), i.e.,

$$\begin{eqnarray}&&{\theta }_{* }=4.46\pm 0.38\,\mu \mathrm{as},\end{eqnarray} \tag{ 11 }$$

where the error is primarily due to the uncertainty of the GC position and color/surface-brightness conversion. The derived source star properties are listed in Table 3.

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The source star of OGLE-2018-BLG-0596 is bright (as derived above), and the blend flux associated with the lensing phenomenon is negligible (see Tables 1 and 2). In this case, the proper motion measured by Gaia can be attributed to that of the source. Then, we can use this measurement to estimate the relative probability of our four solutions by comparing the lens projected velocity (from the model) with that expected from the known Galactic velocity distribution.

For this comparison, we first estimate the lens projected velocity using the four MCMC chains summarized in Tables 1 and 2. For each chain, we first derive the angular Einstein radius θ_E based on the measured θ_*. We next estimate the relative lens-source proper motion in the geocentric frame (Equation (3)), and transform it to the heliocentric frame, i.e.,

$$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{\mathrm{rel},\mathrm{hel}}={{\boldsymbol{\mu }}}_{\mathrm{rel}}+{{\boldsymbol{\nu }}}_{\oplus ,\perp }\displaystyle \frac{{\pi }_{\mathrm{rel}}}{\mathrm{au}},\end{eqnarray} \tag{ 12 }$$

where ${{\boldsymbol{\nu }}}_{\oplus ,\perp }=({\nu }_{\oplus ,N},{\nu }_{\oplus ,E})=(0.47,28.61)\,\mathrm{km}\,{{\rm{s}}}^{-1}$ is the projected velocity of Earth at t₀ (Gould 2004). The lens proper motion in the heliocentric frame is then given by

$$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{{\rm{L}},\mathrm{hel}}={{\boldsymbol{\mu }}}_{\mathrm{rel},\mathrm{hel}}+{{\boldsymbol{\mu }}}_{{\rm{G}}},\end{eqnarray} \tag{ 13 }$$

where ${{\boldsymbol{\mu }}}_{{\rm{G}}}(N,E)=(-5.26\pm 0.55,-4.92\pm 0.66)\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$ is the source proper motion measured from Gaia (Luri et al. 2018). We then estimate the lens proper motion ${{\boldsymbol{\mu }}}_{{\rm{L}},\mathrm{gal}}$ in Galactic coordinates with the coordinate transform of Bachelet et al. (2018). We finally derive the lens projected velocity relative to the local standard of rest (LSR) by

$$\begin{eqnarray}&&{{\boldsymbol{\nu }}}_{{\rm{L}}}={{\boldsymbol{\mu }}}_{{\rm{L}},\mathrm{gal}}{D}_{{\rm{L}}}+{{\boldsymbol{\nu }}}_{\odot ,\mathrm{pec}},\end{eqnarray} \tag{ 14 }$$

where ${{\boldsymbol{\nu }}}_{\odot ,\mathrm{pec}}=({\nu }_{\odot ,W},{\nu }_{\odot ,V})=(7,12)\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (Schönrich et al. 2010) is the peculiar motion of the Sun relative to the LSR.⁴⁴ In Figure 8, we show the distributions of ${{\boldsymbol{\nu }}}_{{\rm{L}}}$ obtained from the four MCMC chains. We note that the W and V axis are defined in Cartesian coordinates so that the components point in the direction of the north Galactic pole and the Galactic rotation, respectively.

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Next, we construct the Galactic velocity distribution in the LSR frame based on the model of Robin et al. (2003). Because the lens distance is D_L ∼ 6 kpc (as derived below), it is expected that the lens is located in the Galactic disk or outer bulge. Therefore, we separately consider the bulge, thin disk, and thick disk distributions. We use ${\bar{\nu }}_{\mathrm{gal},W}\,=$ $({\bar{\nu }}_{\mathrm{bulge},W},{\bar{\nu }}_{\mathrm{thin},W},{\bar{\nu }}_{\mathrm{thick},W})\,=$ $(0,0,0)\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${\bar{\sigma }}_{\mathrm{gal},W}\,=$ $({\bar{\sigma }}_{\mathrm{bulge},W},{\bar{\sigma }}_{\mathrm{thin},W},{\bar{\sigma }}_{\mathrm{thick},W})\,=$ $(100,20,42)\,\mathrm{km}\,{{\rm{s}}}^{-1}$ for the W-direction and ${\bar{\nu }}_{\mathrm{gal},V}\,=$ $({\bar{\nu }}_{\mathrm{bulge},V},{\bar{\nu }}_{\mathrm{thin},V},{\bar{\nu }}_{\mathrm{thick},V})\,=$ $(-220,0,0)\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${\bar{\sigma }}_{\mathrm{gal},V}\,=$ $({\bar{\sigma }}_{\mathrm{bulge},V},{\bar{\sigma }}_{\mathrm{thin},V},{\bar{\sigma }}_{\mathrm{thick},V})\,=$ $(115,30,51)\,\mathrm{km}\,{{\rm{s}}}^{-1}$ for the V-direction with the asymmetric drift of (ν_ad,bulge, ν_ad,thin, ν_ad,thick) =(0, 0, −53) km s⁻¹.

In each solution, we separately apply three model distributions to the kth chain link and derive a probability that the lens has the projected velocity expected from the model distribution, i.e.,

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{gal},k} & = & \left[\displaystyle \frac{{e}^{-{\left({\nu }_{{\rm{L}},W}-{\nu }_{\mathrm{gal},{\rm{W}}}\right)}^{2}/2{\bar{\sigma }}_{\mathrm{gal},W}^{2}}{e}^{-{\left({\nu }_{{\rm{L}},V}-{\nu }_{\mathrm{gal},V}\right)}^{2}/2{\bar{\sigma }}_{\mathrm{gal},V}^{2}}}{{\bar{\sigma }}_{\mathrm{gal},W}{\bar{\sigma }}_{\mathrm{gal},V}}\right]\\ & \, & \times \,{\rho }_{{\rm{d}},\mathrm{gal}}({D}_{{\rm{L}}}){\rho }_{{\rm{d}},\mathrm{bulge}}({D}_{{\rm{S}}}),\end{array}\end{eqnarray} \tag{ 15 }$$

where ${{\boldsymbol{\nu }}}_{\mathrm{gal}}={\bar{{\boldsymbol{\nu }}}}_{\mathrm{gal}}+{{\boldsymbol{\nu }}}_{\mathrm{ad}}$ and ρ_d,gal = (ρ_d,bulge, ρ_d,thin, ρ_d,thick) is the Galactic density profile presented in Jung et al. (2018b). We then estimate the three probabilities (P_bulge, P_thin, P_thick) by P_gal = ΣP_gal,k, and find the total probability P_tot,gal by combining these three probabilities, i.e., P_tot,gal = P_bulge + P_thin + P_thick. Finally, we derive the net relative probability P_net by multiplying the fit probability ${P}_{\mathrm{lc}}={e}^{-({\chi }^{2}-{\chi }_{\mathrm{best}}^{2})/2}$ by P_tot,gal.

The results are listed in Table 4. We find that in both the Close and Wide configurations, the lens system favors the bulge populations. This is mainly because the direction of lens projected velocity ${{\boldsymbol{\nu }}}_{{\rm{L}}}$ is opposite with respect to the LSR and its magnitude is relatively high compared to the rotational velocity v_rot =220 km s⁻¹ (see Figure 8). From Table 4, we also find that the Galactic-model probabilities P_tot,gal are comparable to each other and do not lend significant weight to either solution, implying that the preference for the Close solutions discussed in Section 3.4 is not significantly affected by the Galactic prior. Therefore, we can consider that from the balance of evidence P_net, the Close configuration is strongly favored although the Wide configuration cannot be completely ruled out.

For each solution, we now estimate the physical parameters a_j by imposing the Galactic prior. In the kth set of MCMC parameters, we evaluate the physical parameters a_j,k with the measured θ_* and weight them by the probability P_gal,k. We then derive the mean and 68% uncertainty range of a_j using all weighted a_j,k.

The results are listed in Table 5, which includes the lens physical properties (M₁, M₂, a_⊥) and the event's phase-space coordinates $({D}_{{\rm{L}}},{D}_{{\rm{S}}},{\mu }_{\mathrm{rel}},{{\boldsymbol{\mu }}}_{\mathrm{rel},\mathrm{hel}},\phi ,{{\boldsymbol{\mu }}}_{{\rm{L}},\mathrm{hel}},{{\boldsymbol{\nu }}}_{{\rm{L}}})$. Here, ϕ is the orientation angle of ${{\boldsymbol{\mu }}}_{\mathrm{rel},\mathrm{hel}}$ as measured north through east. To investigate the physical validity of these measurements, we also show the ratio of the projected kinetic to potential energy (Dong et al. 2009), i.e., (KE/PE)_⊥. For all four solutions, the low values of μ_rel and the large values of D_L favor the bulge lenses as predicted from the Galactic prior. However, the estimated lens masses for the Close and Wide configuration differ from each other due primarily to the difference in mass ratios q between the two configurations, but also, to a much smaller degree, because of the difference in the normalized source radii ρ.

Table 5.  Physical Parameters

Parameters	(+, +)Close	(−, −)Close	(+, +)Wide	(−, −)Wide
θE (mas)	0.336 ± 0.034	0.324 ± 0.033	0.256 ± 0.027	0.276 ± 0.028
M1 (M⊙)	0.231 ± 0.028	0.229 ± 0.031	0.154 ± 0.019	0.196 ± 0.021
M2 (M⊕)	13.93 ± 1.56	14.69 ± 1.72	1.19 ± 0.16	1.30 ± 0.16
a⊥ (au)	0.97 ± 0.13	0.92 ± 0.14	2.77 ± 0.37	3.02 ± 0.39
DL (kpc)	5.65 ± 0.75	5.65 ± 0.80	5.95 ± 0.77	6.11 ± 0.79
DS (kpc)	8.58 ± 1.42	8.38 ± 1.47	8.69 ± 1.35	8.67 ± 1.33
μrel (mas yr−1)	4.22 ± 0.42	4.07 ± 0.41	3.24 ± 0.33	3.49 ± 0.35
μrel,hel,N (mas yr−1)	−0.44 ± 0.50	0.64 ± 0.57	−1.13 ± 0.36	−1.14 ± 0.65
μrel,hel,E (mas yr−1)	4.53 ± 0.46	3.98 ± 0.43	3.33 ± 0.36	3.24 ± 0.33
ϕ (deg) (E of N)	95.5 ± 10.9	81.3 ± 10.9	108.7 ± 11.8	107.4 ± 14.3
μL,hel,N (mas yr−1)	−5.67 ± 0.79	−4.57 ± 0.80	−6.37 ± 0.76	−6.37 ± 0.94
μL,hel,E (mas yr−1)	−0.45 ± 0.41	−0.66 ± 0.44	−1.63 ± 0.44	−1.44 ± 0.41
νL,W (km s−1)	−98.8 ± 18.7	−73.6 ± 22.2	−97.1 ± 16.7	−103.5 ± 20.4
${\nu }_{{\rm{L}},V}$ (km s−1)	−97.4 ± 20.1	−82.5 ± 18.6	−141.3 ± 22.8	−141.1 ± 24.7
(KE/PE)⊥ (10−2)	8.67 ± 2.86	10.06 ± 3.22	50.78 ± 21.92	11.91 ± 8.19

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The most favored (+, +)_Close solution suggests that the host is a mid M-dwarf star with M₁ = 0.23 ± 0.03 M_⊙, and that the companion is a planet with M₂ = 13.93 ± 1.56 M_⊕. The projected planet-host separation is a_⊥ = 0.97 ± 0.13 au. Hence, this interpretation indicates that the planet is a cold Uranus lying projected outside the snow-line distance, i.e., ${a}_{{sl}}\,=2.7({M}_{1}/{M}_{\odot })\sim 0.62\,\mathrm{au}$. On the other hand, the (+, +)_Wide solution corresponds to an Earth-mass planet (M₂ = 1.19 ±0.16 M_⊕) orbiting a late M-dwarf (M₂ = 0.15 ± 0.02 M_⊙). This planet is colder because the projected separation $({a}_{\perp }=2.77\,\pm 0.37\,\mathrm{au})$ is about 7 times larger than the snow line.