OGLE-2018-BLG-0677Lb: A Super-Earth Near the Galactic Bulge

The event was first detected by the Optical Gravitational Lensing Experiment (OGLE) Early Warning System (Udalski et al. 1994; Udalski 2003), with designations OGLE-2018-BLG-0677 and OGLE-2018-BLG-0680, since it lies in the overlap region of two survey fields. It is located at (R.A., decl.) = (17^h55^m0027, −32°00'5951), which corresponds to (l, b) = (−161, −331) in Galactic coordinates. In combination, the observations from the two fields have a frequency of 1–3 data points per day. The OGLE observations were reduced using the difference image analysis from Wozniak (2000).

KMTNet is a wide-field imaging system, with three telescopes and cameras sharing the same specifications, installed at Cerro-Tololo Inter-American Observatory in Chile (KMTC), the South African Astronomical Observatory in South Africa (KMTS), and the Siding Spring Observatory in Australia (KMTA). The telescopes each have a 1.6 m primary mirror, and a wide-field camera (a mosaic of four 9k × 9k CCDs) that image approximately a 2.0 × 2.0 square degree field of view.

Weather permitting, the network of telescopes and cameras allow a 24 hr per day monitoring of the Galactic bulge. This allows one to trace the light curves of stars continuously and is ideal for the detection of extra-solar planets by microlensing and transit, variable objects, and asteroids and comets.

The event OGLE-2018-BLG-0677 was independently detected by KMTNet and given the designation KMT-2018-BLG-0816 (Kim et al. 2018). The event is located in the BLG01 and BLG41 KMTNet fields, giving an effective observation cadence of 15 minutes.

Photometry was extracted from the KMTNet observations using the software package PYDIA (Albrow 2017), which employs a difference-imaging algorithm based on the modified delta-basis-function approach of Bramich et al. (2013). The light curve of the event, with a single lens single-source (1L1S) model, is shown in Figure 1.

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This is the simplest of models, which considers a point mass lens with a point mass source. The magnification is modeled by a Paczyński (1986) curve,

$$\begin{eqnarray}&&A(u)=\displaystyle \frac{{u}^{2}+2}{u\sqrt{{u}^{2}+4}},\end{eqnarray} \tag{ 1 }$$

where u is the angular separation between source and lens, normalized by the Einstein angle θ_E. Given the relative motion between them, this separation will be a function of time and is assumed rectilinear as

$$\begin{eqnarray}&&u(t)={\left({\tau }^{2}+{u}_{0}^{2}\right)}^{1/2},\end{eqnarray} \tag{ 2 }$$

with $\tau \equiv (t-{t}_{0})/{t}_{{\rm{E}}};$ where u₀ is the impact parameter of the event, t₀ is the time at u = u₀, and t_E is the Einstein radius crossing time. These three parameters characterize completely the light-curve magnification model, A(t). We find these parameters by a Markov Chain Monte Carlo (MCMC) search while reducing the χ² of a linear fit to the observed flux, $F(t)=A(t){F}_{{\rm{S}}}+{F}_{{\rm{B}}}$, where the source and blend flux, F_S and F_B, are determined for each data set.

The best-fit values (after renormalization of data uncertainties as will be discussed in Section 3.4) are presented in Table 1. We note that in Tables 1–3, F_S and F_B are given in a system with 18 as the magnitude zero-point.

Table 1.  Best-fit 1L1S Model Parameters

${\chi }_{\min }^{2}/{N}_{\mathrm{data}}$	1602.51/1557
u0	0.1029 ± 0.0012
t0	8229.5417 ± 0.0007
tE (days)	4.95 ± 0.04
${F}_{{\rm{S}},\mathrm{OGLE}0677}$	0.293 ± 0.009
${F}_{{\rm{B}},\mathrm{OGLE}0677}$	−0.065 ± 0.011

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As can be seen in Figure 1, there is a small anomalous feature in the light curve relative to the 1L1S model that occurs over ∼5 hr during the interval 8229.70–8229.90. The anomaly primarily takes the form of a dip of ∼0.05 mag followed by a smaller and shorter bump. See the residuals in Figure 2. These features are well traced by the KMTC01 and KMTC41 data sets, and they are confirmed by two points from the OGLE-2018-BLG-0680 data, one each on the dip and the subsequent small bump. The OGLE-2018-BLG-0677 data set has a single point just before the start of the anomaly. We have examined the direct and difference images from KMTC and are satisfied that there are no systematic effects that can be attributed to seeing, background, or image cosmetics that could cause these features. Additionally, we have found no evidence that the dip is a repeating phenomenon, such as might be due to a starspot.

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We examine the possibility that the anomaly is due to a binary source. In the case that the source consists of two stars, the total flux is given by the linear combination of the individual source fluxes, ${F}_{\mathrm{tot}}={A}_{1}{F}_{1}+{A}_{2}{F}_{2}+{F}_{{\rm{B}}}$, where A₁ and A₂ are the individual magnifications of each source (Gaudi 1998) as parameterized in Section 3.1, but sharing the same value of t_E.

The total magnification of the combined source flux is given by

$$\begin{eqnarray}&&{A}_{\mathrm{tot}}=\displaystyle \frac{{A}_{1}+{A}_{2}{q}_{F}}{1+{q}_{F}},\end{eqnarray} \tag{ 3 }$$

where ${q}_{F}={F}_{2}/{F}_{1}$ is the luminosity ratio (Griest & Hu 1992). In total, there are six parameters: u₀₁ and t₀₁ for A₁; u₀₂ and t₀₂ for A₂; t_E is shared by A₁ and A₂; and the luminosity ratio q_F. We fit the data as described in the previous subsection but using this model instead. The best-fit values found for the data are presented in Table 2, and the light-curve model shown in Figure 2. It is apparent that the best binary source model does not reproduce the anomalous feature in the light curve, but produces a small "bump" deviating almost imperceptibly from the 1L1S model.

Table 2.  Best-fit 1L2S Model Parameters

${\chi }_{\min }^{2}/{N}_{\mathrm{data}}$	1593.08/1557
u01	0.103 ± 0.004
t01	8229.542 ± 0.004
u02	${0.05}_{-0.05}^{+0.31}$
t02	${8229.873}_{-0.061}^{+0.546}$
qF	${0.0031}_{-0.0028}^{+0.0317}$
tE (days)	4.96 ± 0.14
${F}_{{\rm{S}},\mathrm{OGLE}0677}$	0.287 ± 0.009
${F}_{{\rm{B}},\mathrm{OGLE}0677}$	−0.063 ± 0.011

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We adopt the standard parameterization of a binary lens light curve by describing it with seven parameters (Gaudi 2012): s, q, ρ_*, α, u₀, t₀, and t_E. These represent the normalized separation between binary lens components, the mass ratio of the binary lens components, the normalized source radius, the source trajectory angle with respect to the binary axis, the normalized closest approach between the lens center of mass and the source (which occurs at time t₀, the time of closest approach), and the timescale to cross the Einstein radius, respectively. The factor used for the normalized parameters is the angular Einstein radius,

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}=\sqrt{\displaystyle \frac{4{{GM}}_{L}}{{c}^{2}}\left(\displaystyle \frac{1}{{D}_{L}}-\displaystyle \frac{1}{{D}_{S}}\right)},\end{eqnarray} \tag{ 4 }$$

where M_L is the total mass of the lens system, and D_L, D_S are the distances from Earth to the lens and source. A visualization of this combination of parameters can be found in Jung et al. (2015).

The fitting is done by a maximum likelihood estimation, which is equivalent to minimization of the χ². The process is done in two parts. The first is through a fixed value grid search of the (s, q) parameters to find regions where the minimum χ² may be located. For each (s, q), a grid of (r, α) (where r is a reparameterization of u₀, centered-on and normalized to caustics, see McDougall & Albrow 2016) is used to seed a minimization over (${\rho }_{* },{t}_{0},{t}_{{\rm{E}}})$ by a simple Nelder–Mead optimization (Nelder & Mead 1965). This approach for the fixed (s, q, r, α) is due to the relation of these parameters to the geometry.

Our fixed position results are then used as seeds for a more refined search using an MCMC algorithm implemented using the MORSE code (McDougall & Albrow 2016). A similar two-step process can also be seen in Shin et al. (2019). During this second part of the process, the seed solutions from the grid search are used as starting points, and the search is now continuous in the parameter space.

Once a minimum value for χ² was found, the original magnitude uncertainties for each data set were renormalized via

$$\begin{eqnarray}&&{\sigma }_{i}^{{\prime} }=k\sqrt{{\sigma }_{i}^{2}+{\epsilon }^{2}}.\end{eqnarray} \tag{ 5 }$$

The coefficients k and for each data set are determined in such a way that the reduced χ² for both the higher- and lower-magnification data points for each site are approximately unity (Yee et al. 2012). The renormalization factors are listed in Table 6. The MCMC search was then re-run with the renormalized data uncertainties to corroborate the solution and obtain a new minimized χ². The final model parameter values and their uncertainties are directly obtained from the 68% confidence interval around the medians of the marginalized posterior parameter distributions.

Given the short total time of the event, it was not possible to obtain parallax information (Gould 1992, 2000; Alcock et al. 1995; Buchalter & Kamionkowski 2002). For the rest of our analysis, no higher-order light-curve effects were considered.

3.4.1. Model of OGLE-2018-BLG-0677

From the fitting process described in the previous section, we found two degenerate solutions with similar q. Often this situation corresponds to a well-known wide/close case degeneracy where s < 1 and s > 1 for the two solutions (see, e.g., Griest & Safizadeh 1998; Dominik 1999; Batista et al. 2011). However, in the present case, the degeneracy is between two solutions with s < 1, and is due to a source trajectory passage past two cusps of connected or disconnected caustics, as shown in Figure 3. Each correspond to a minor-image perturbation that demagnifies the source relative to the magnification due solely to the host star. This type of degeneracy has been detected previously for events OGLE-2012-BLG-0950 (Koshimoto et al. 2017) and OGLE-2016-BLG-1067 (Calchi Novati et al. 2019). A thorough explanation of the phenomenon is given in Han et al. (2018) with reference to the event MOA-2016-BLG-319. Following the nomenclature of that paper, we refer to the case where the source passes between the caustics as the "inner" solution, and that where it passes the single connected caustic as the "outer" solution.

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For the current event, OGLE-2018-BLG-0677, we could not break this degeneracy given that both cases produce almost identical light curves, see Figure 3. Table 3 lists the best-fit parameters for both solutions. Apart from the clear difference in geometry, they share several parameters, q, u₀, t₀, and t_E, well within their respective uncertainties. Also, the values of the minimum χ² are close enough that there is no clear statistical difference. Despite this degeneracy, the similarities of q, ρ, and t_E will allow us to infer similar physical properties for the system, which will be discussed in the following section.

Table 3.  Best-fit Binary Model Parameters

	Inner	Outer
${\chi }_{\min }^{2}/{N}_{\mathrm{data}}$	1556.03/1557	1555.96/1557
s	${0.912}_{-0.054}^{+0.002}$	${0.985}_{-0.002}^{+0.059}$
${\mathrm{log}}_{10}q$	$-{4.105}_{-0.0822}^{+0.305}$	$-{4.054}_{-0.109}^{+0.268}$
${\rho }_{* }$	${0.01209}_{-0.00545}^{+0.00013}$	${0.01238}_{-0.00590}^{+0.00016}$
u0	0.102 ± 0.003	0.102 ± 0.003
α	−1.98 ± 0.01	4.307 ± 0.008
t0	8229.544 ± 0.002	8229.544 ± 0.002
tE (days)	4.94 ± 0.11	4.94 ± 0.11
${F}_{{\rm{S}},\mathrm{OGLE}0677}$	0.294 ± 0.009	0.294 ± 0.009
${F}_{{\rm{B}},\mathrm{OGLE}0677}$	−0.069 ± 0.011	−0.070 ± 0.011

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As a note, ρ_* is detected rather weakly in the light-curve analysis. It may be of interest to ask whether the omission of ρ_* affects our subsequent results in any significant way.

Therefore in Section 5.2 we give results that propagate the ρ_* distribution, and also show the case where the ρ_* information is omitted from the Bayesian estimates.

3.4.2. Other Local Minima

Five additional seed solutions were identified in our initial grid search close, located close in (s, q) to the solutions reported above. All of these are very shallow in χ² space, and with subsequent MCMC runs, all converged to one of either of the reported solutions. Light curves and caustic geometries for the two most prominent of these local minima are shown in Figure 4.

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To analyze how these local minima are related to the best-fit solutions, we followed the procedure described in Hwang et al. (2017) and ran various "hot" and "cold" MCMC realizations to explore the regions around all the solutions. In Figure 5 we present a scatter plot of the combined samples from the various runs, color coded by their ${\rm{\Delta }}{\chi }^{2}$ relative to the best solutions. We adopt the parameter

$$\begin{eqnarray}&&{\rm{\Delta }}\xi ={u}_{0}\csc \,\alpha -(s-1)/s,\end{eqnarray} \tag{ 6 }$$

introduced by Hwang et al. (2017), which traces the offset between the centers of the source and the major-image caustic as the source crosses the planet–star axis. Three of the identified extra solutions are located within the purple region below the best-fit solutions, with 25 < Δχ² < 36, and are indistinguishable from their surroundings. The two remaining local minima, i.e., those displayed in Figure 4 and labeled as outer and inner local minima in Figure 5, have 16 < Δχ² < 25. They are degenerate, and correspond to lens mass ratios ∼ twice that of the best solutions. They each represent a source trajectory over an on-axis cusp that provides a single bump in magnification, and correspond to a major-image perturbation with degeneracies as discussed in Gaudi & Gould (1997).

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Their light-curve fit parameters are listed in Table 4, without uncertainties as they are too shallow in χ² space to retain an MCMC chain. They have a similar t_E to the best solutions, but a larger ρ_*.

Table 4.  Local Minima Binary Model Parameters

	Inner	Outer
${\chi }_{\min }^{2}/{N}_{\mathrm{data}}$	1580.19/1557	1579.82/1557
s	1.030	1.082
${\mathrm{log}}_{10}q$	−3.732	−3.725
ρ*	0.031	0.031
u0	0.107	0.107
α	1.498	1.498
t0	8229.543	8229.543
tE (days)	4.837	4.852
${F}_{{\rm{S}},\mathrm{OGLE}0677}$	0.279 ± 0.001	0.276 ± 0.001
${F}_{{\rm{B}},\mathrm{OGLE}0677}$	−0.020 ± 0.002	−0.018 ± 0.002

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From Figure 1, it can be seen that there is a subtle anomaly following the peak of the light curve compared to the 1L1S model. Although a binary lens model could in principle represent this small feature found in the data, the justification for the increase in model complexity needs to be strengthened; therefore, to avoid overfitting by assuming a binary lens (Gaudi 1997), we compared with two simpler models. The three models compared are the previously described single lens single-source (1L1S), single lens binary source (1L2S), and binary lens single-source (2L1S) models. Often the χ² value is used as an indicator of the goodness of fit, but this time instead of simply relying on this statistic to compare between different models, we obtain the log evidence given the data, $\mathrm{ln}\,{ \mathcal Z }$, for a more robust model comparison. This was done by applying the nested sampling method (Skilling 2006) for cases (1L1S) and (1L2S). For the case of (2L1S), nested sampling was too inefficient, so the approximation algorithm presented in van Haasteren (2009) for posterior distributions was used instead to estimate $\mathrm{ln}\,{ \mathcal Z }$. For completeness, we also applied the approximate algorithm to the other two cases, which led to the same results as nested sampling. The log odds ratio is obtained by $\mathrm{ln}\,O=\mathrm{ln}\,{{ \mathcal Z }}_{M1}-\mathrm{ln}\,{{ \mathcal Z }}_{M2}$, and a model is preferred as long as ln O > 0, but a strong preference is given when ln O > 5 (Jaynes et al. 2003; Sivia & Skilling 2006).

The 2L1S best fit has two degenerate solutions, inner and outer (see Griest & Safizadeh 1998), but the difference in χ² between these (Δχ² = 0.07) is small, with a slight preference to the outer model. On the other hand, the log evidence gives a strong preference for the inner model. We need to adopt a model for renormalization, so we decided to choose the inner solution for this purpose.

Table 5 shows the minimum χ² and the log evidence based on the renormalized data error bars, and it is clear that the 2L1S solution is preferred with Δχ² > 46 compared with the 1L1S, and is also preferred from the evidence. The renormalization parameters are found in Table 6.

Table 5.  Minimum χ² and Log Evidence for 1557 Data Points for the Different Lensing Models

Model	1L1S	1L2S	2L1S(inner)	2L1S(outer)
${\chi }_{\mathrm{renorm}}^{2}$	1602.51	1593.08	1556.03	1555.96
$\mathrm{ln}\,{{ \mathcal Z }}_{\mathrm{renorm}}$	−811.33	−815.11	−806.15	−807.23
${\rm{\Delta }}{\chi }_{\mathrm{renorm}}^{2}$	0.00	9.43	46.48	46.55
${\rm{\Delta }}\mathrm{ln}\,{{ \mathcal Z }}_{\mathrm{renorm}}$	0.0	−4.29	5.18	4.1

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One last consideration worth mentioning is the case for 1L2S. From the χ² point of view it seems to improve compared to 1L1S, however the evidence gives an indication that the 1L1S is slightly better. Here the meaning of the evidence becomes clear. It says that the 1L2S model is not better than the 1L1S (the extra model complexity does not outweigh the reduction in χ²), and in fact the difference with the 1L1S model is only between 8229.85 and 8229.88. This can be clearly seen in Figure 2 with the 1L1S and 1L2S best-fit light curves.

We performed a Bayesian analysis to characterize the physical properties of the system. As remarked above, our source of information from the light curve comes only from the distributions of t_E and ρ_*, which restricts how much information we can gather. In other words, we need to limit the number of parameters used in the analysis to the most basic. The main parameters are the total mass of the lens, M_L, the distance to the lens, D_L, the proper motion, μ_rel, and the source distance, D_S. Nevertheless, during the formulation of the analysis, the effective transverse velocity v = D_Lμ_rel will be used as an intermediate step during the parameter sampling.

Our analysis resembles a hierarchical Bayes (see e.g., Gelman et al. 2003),

$$\begin{eqnarray}&&p({\rm{\Theta }}| y)\propto p({\rm{\Theta }})\int p({t}_{{\rm{E}}},{\rho }_{* }| {\rm{\Theta }})p(y| {t}_{{\rm{E}}},{\rho }_{* }){{dt}}_{{\rm{E}}}d{\rho }_{* },\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}&&{\rm{\Theta }}\equiv ({M}_{L},{D}_{L},{D}_{S},v)\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&y\equiv \mathrm{data}.\end{eqnarray} \tag{ 9 }$$

This describes the probability distribution function for ${t}_{{\rm{E}}}$ and the weights based on the distribution of ρ_*, which is shown in Figure 7. The prior sampling distribution for the physical quantities is p(Θ), which includes the information of the Galactic model. For the analysis, t_E and ρ_* act as constraints, and they are hidden in $p({t}_{{\rm{E}}},{\rho }_{* }| {\rm{\Theta }})$, which is the probability of a value of t_E and ρ_* given by the physical parameters.

For this purpose we took a combined approach in which we perform an MCMC simulation to obtain the posterior distribution from the Galactic modeling. Ideally, we would describe the process in observable variables similar to Batista et al. (2011), Yee et al. (2012), or Jung et al. (2018), but in this case we are only able to relate two observables to the physical parameters, t_E and ρ_*. We therefore take advantage of the Monte Carlo simulation to take care of the marginalization. We used two different sampling algorithms in order to verify that our results contain no algorithmic bias. Independently, we ran the Emcee sampler (Foreman-Mackey et al. 2013), which uses the standard Metropolis–Hasting sampling method, and Dynesty (Higson et al. 2018), which uses nested sampling. After convergence, both algorithms arrived at the same posterior distributions.

Our approach is similar to that of Yoo et al. (2004), but we describe our Galactic model and the fact that we can define the probability of a physical property given an observed parameter, e.g., $p({\rm{\Theta }}| {t}_{{\rm{E}}}^{(0)})$, as presented in Dominik (1998) and Albrow et al. (2000).

For the Galactic model, the lens mass function, Φ_{log m}, is a power-law or Gaussian distribution depending on the mass ranges according to Chabrier (2003). The mass density distribution, Φ_x(x), considers the disk, bulge, or both depending on the case, where the disk is modeled by a double exponential with 0.3 and 1.0 kpc as the scale heights of a thin and thick disk perpendicular to the Galactic plane, and the corresponding column mass densities are ${{\rm{\Sigma }}}_{\mathrm{thin}}=25\,{M}_{\odot }\,{\mathrm{pc}}^{-2}$ and ${{\rm{\Sigma }}}_{\mathrm{thick}}\,=35\,{M}_{\odot }\,{\mathrm{pc}}^{-2}$. For the bulge we adopt a model of a barred bulge that is tilted by an angle of 20° (see Han & Gould 1995; Grenacher et al. 1999). The probability density of the absolute effective velocity, Φ_ν(ν, x), assumes Gaussian distributions, for which isotropic velocity dispersions are assumed for the Galactic disk and bulge, and the values of ${\sigma }^{\mathrm{disk}}=30\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${\sigma }^{\mathrm{bulge}}=100\,\mathrm{km}\,{{\rm{s}}}^{-1}$are adopted. While the velocity mean for bulge objects is assumed purely random, the disk lenses rotation velocity can described by a Navarro–Frenk–White model (see Navarro et al. 1997). The precise equations for each component of the Galactic model can be found in the appendices of Dominik (2006).

Therefore, the probability of our assumed Galactic model (galactic prior) for the lens is in the form

$$\begin{eqnarray}&&{p}_{i}(m,\zeta ,x)\propto {{\rm{\Phi }}}_{\mathrm{log}m}(\mathrm{log}m){{\rm{\Phi }}}_{\zeta }(\zeta ,x){{\rm{\Phi }}}_{x}(x),\end{eqnarray} \tag{ 10 }$$

selecting from bulge or disk populations accordingly. The parameters in this prescription are the mass, $m={M}_{L}/{M}_{\odot }$, the fractional lens-source distance, x = D_L/D_S, and the effective transverse velocity, $\zeta =v/{v}_{c}$, where ${v}_{c}=100\,\mathrm{km}\,{{\rm{s}}}^{-1}$ is a scaling constant to keep the velocity dimensionless. These distributions and prescriptions can also be found in Dominik (2006), and, as mentioned above, the assumed properties are described in its appendix.

The source distance probability distribution is defined as

$$\begin{eqnarray}&&{p}_{\mathrm{source}}({D}_{S})=\displaystyle \frac{{D}_{S}^{\gamma }\rho ({D}_{S})}{{\int }_{0}^{{D}_{s,\max }}{D}_{S}^{\gamma }\rho ({D}_{S}){{dD}}_{S}},\end{eqnarray} \tag{ 11 }$$

where $\rho ({D}_{S})$ is the density of objects at the source distance as defined in Dominik (2006). We adopted this distribution as we do not have any information on the source location, and we based its definition on the Zhu et al. (2017) argument, but we use a value of γ = 1. For our calculations, we assumed that the source was part of the bulge population.

Therefore, the Galactic prior, which has the Galactic model information, is sampled from the probability distribution

$$\begin{eqnarray}&&p({\rm{\Theta }})\propto {\rm{\Omega }}(m,\zeta ,x){p}_{i}(m,\zeta ,x){p}_{\mathrm{source}}({D}_{S}).\end{eqnarray} \tag{ 12 }$$

The lens for bulge or disk populations are selected accordingly. Meanwhile, the information of the prior is weighted by

$$\begin{eqnarray}&&{\rm{\Omega }}(m,\zeta ,x)\propto {\left(M/{M}_{\odot }\right)}^{1/2}\zeta \sqrt{x(1-x)}\end{eqnarray} \tag{ 13 }$$

(Dominik 2006). The parameter t_E then becomes intrinsic and is defined as

$$\begin{eqnarray}&&{t}_{{\rm{E}}}(m,\zeta ,x,{D}_{S})=\displaystyle \frac{2{r}_{{\rm{E}},\odot }\sqrt{{mx}(1-x)}}{\zeta {v}_{c}},\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}&&{r}_{{\rm{E}},\odot }=\sqrt{\displaystyle \frac{{{GM}}_{\odot }}{{c}^{2}}{D}_{S}}\end{eqnarray} \tag{ 15 }$$

is a scale length defined as the Einstein radius of a solar mass lens located halfway between the observer and source.

The angular source size parameter,

$$\begin{eqnarray}&&{\rho }_{* }(m,x,{D}_{S})={\theta }_{* }/{\theta }_{{\rm{E}}}(m,x,{D}_{S}),\end{eqnarray} \tag{ 16 }$$

where θ_* is the source angular radius given in the previous section and ${\theta }_{{\rm{E}}}(m,x,{D}_{S})$ is the Einstein angle for an MCMC realization.

Given that ρ_* and t_E are derived from the prior parameters, their joint probability is

$$\begin{eqnarray}&&p({t}_{{\rm{E}}},{\rho }_{* }| {\rm{\Theta }})\propto \delta ({t}_{{\rm{E}}}-{t}_{{\rm{E}}}({\rm{\Theta }}))\delta ({\rho }_{* }-{\rho }_{* }({\rm{\Theta }})).\end{eqnarray} \tag{ 17 }$$

Additionally, there is a weight given by the information contained in the probability distribution $p(y| {t}_{{\rm{E}}},{\rho }_{* })$, which corresponds to the posterior distributions of our solution from our light-curve fitting.

The simplest to represent is the value of ${t}_{{\rm{E}}}$, as it is well constrained and reduces simply to

$$\begin{eqnarray}&&p(y| {t}_{{\rm{E}}})\propto \exp \left(-\displaystyle \frac{{\left({t}_{{\rm{E}}}-{t}_{{\rm{E}},\mathrm{best}}\right)}^{2}}{2{\sigma }_{{t}_{{\rm{E}}}}^{2}}\right).\end{eqnarray} \tag{ 18 }$$

Here ${t}_{E,\mathrm{best}}$ and ${\sigma }_{{t}_{{\rm{E}}}}$ correspond to the values from the best fit of the distribution from the light-curve fitting.

For the case of ρ_*, it is not as simple because the constraint is not well approximated as a Gaussian. We introduce the information by the use of ${\rm{\Delta }}{\chi }^{2}={\chi }^{2}-{\chi }_{\min }^{2}$ of each of the light-curve fitting samples. From the lower envelope of these samples, binned in ρ_*, we obtain a numerical function Δχ²(ρ_*), see Figure 8. The probability of ρ_* is then given by

$$\begin{eqnarray}&&p(y| {\rho }_{* })\propto \exp \left(-\displaystyle \frac{{\rm{\Delta }}{\chi }^{2}({\rho }_{* })}{2}\right).\end{eqnarray} \tag{ 19 }$$

The previous weights are combined as

$$\begin{eqnarray}&&p(y| {t}_{{\rm{E}}},{\rho }_{* })\propto p(y| {\rho }_{* })p(y| {t}_{{\rm{E}}}).\end{eqnarray} \tag{ 20 }$$

The information of the Galactic model is introduced into Equation (20) by the restriction imposed by the Dirac δ function in Equation (17).

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The combination of the prior and these weights gives the desired posterior probability for the physical parameters, where μ_rel is appropriately obtained from ζ. We note that both outer and inner models lead to the same final results. This is expected given the similarity of the t_E and ρ_* distributions.

The distributions obtained from the Galactic modeling are presented in Figure 9, also contour levels for the distributions of the lens mass and the ratio of the lens and source distributions can be found in Figure 10, and the corresponding estimators for the median and error values are in Table 7. As noted above, the outer and inner solutions produce indistinguishable distributions.

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Table 7.  Estimators from the Bayesian Results

${\mathrm{log}}_{10}\,{M}_{L}/{M}_{\odot }$	${D}_{L}/{D}_{S}$	DS (kpc)	μrel (mas yr−1)
$-{0.94}_{-0.47}^{+0.34}$	${0.92}_{-0.12}^{+0.05}$	${8.37}_{-1.07}^{+1.06}$	${7.02}_{-1.68}^{+2.48}$

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From the resulting distributions of the basic four parameters, the lens mass, lens distance, source distance, and proper motion, we can propagate the information to derive the physical mass of the binary lens components and their physical separation. Additionally, the host-mass distribution is essentially identical to the total lens-mass distribution in Figure 9. In this case, the binary-lens mass ratio is small, and the host consists of a low-mass star or brown dwarf with a super-Earth companion. As mentioned previously, given the overlapping of the distributions of ${t}_{{\rm{E}}}$ and q for the inner and outer solutions, the Bayesian analysis returns indistinguishable distributions, and for these reasons it is justifiable to say that both give the same derived planet mass distribution. This is not the case for the projected separation between host and planet, which has a small but not negligible difference. The derived distributions are shown in Figure 11 and their estimators in Table 8.

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Table 8.  Estimators for the Planet Mass and Projected Separation

	${\mathrm{log}}_{10}{M}_{p}/{M}_{\oplus }$	${a}_{\perp }$(Inner)	${a}_{\perp }$(Outer)
${t}_{{\rm{E}}}+{\rho }_{* }$	${0.60}_{-0.48}^{+0.40}$	${0.63}_{-0.17}^{+0.20}$ au	${0.72}_{-0.19}^{+0.23}$ au
${t}_{{\rm{E}}}$ only	${0.73}_{-0.45}^{+0.37}$	${0.64}_{-0.21}^{+0.19}$ au	${0.80}_{-0.26}^{+0.23}$ au

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Furthermore, as mentioned at the end of Section 3.4.1, the value for ρ_* is weakly constrained. The relatively broad and highly non-Gaussian form of this constraint is shown in Figure 8. For comparison purposes, in Figure 11, we show alongside the complete Galactic modeling, which includes the information of t_E and ρ_*, the derived distributions produced by removing the information about ρ_* and using only t_E. In such a case, the planet mass distribution is shifted a little higher, and the projected separation distributions are a little broader. Their estimators are also included in Table 8.

The inner and outer solutions for the planet in OGLE-2018-BLG-0667 both imply that the secondary lens is a super-Earth/Sub-Neptune planet with a mass of ${M}_{\mathrm{planet}}={3.96}_{-2.66}^{+5.88}{M}_{\oplus }$, while the host is a dwarf star or brown dwarf with a mass of ${M}_{\mathrm{host}}={0.12}_{-0.08}^{+0.14}\,{M}_{\odot }$.

The projected separation between the star and planet is ${0.63}_{-0.17}^{+0.20}$ au and ${0.72}_{-0.19}^{+0.23}$ au for the inner and outer solutions, respectively. The lens system is estimated to be at a distance of ${7.58}_{-1.35}^{+1.15}$ kpc, with a 66.9% probability of lying in the bulge (Figure 9).

For completeness, we note the physical implications of the solutions corresponding to the two local minima discussed in Section 3.4.2. Since these are degenerate with each other in q, t_E, and ρ_*, they imply the same primary and secondary lens mass and distance. The local minima would imply a total lens mass of 0.1 M_⊙ and a planet mass of 6.43 M_⊕, values very similar to the favored solutions, but a lens located at a distance of 0.99 D_S. The projected separation of the planet from its host would be smaller, either 0.20 or 0.22 au, for the inner and outer local minima respectively.