OGLE-2018-BLG-0022: First Prediction of an Astrometric Microlensing Signal from a Photometric Microlensing Event

Considering the characteristic caustic-crossing features, we conduct binary-lens modeling of the observed light curve. In the first-round modeling, we assume that the observer and the lens components do not experience any acceleration, and thus the relative lens-source motion is rectilinear. We refer to this model as the standard model.

Standard modeling requires seven lensing parameters, including the time of the closest lens-source approach, t₀, the separation at that time, u₀, the event timescale, t_E, the projected separation, s, and the mass ratio between the lens components, q, the source trajectory angle, α, and the normalized source radius, ρ. The lengths of s, u₀, and ρ are normalized to ${\theta }_{{\rm{E}}}$. We compute finite-source magnifications considering the limb-darkening variation of the source star's surface brightness. The profile of the surface brightness is modeled by S ∝ 1 − Γ_λ(1–1.5 cos ψ), where Γ_λ denotes the linear limb-darkening coefficient and ψ represents the angle between the normal to the source surface and the line of sight toward the source center. We adopt the limb-darkening coefficients from Claret (2000) considering the source type. The determination of the source type is discussed in Section 3.3. The adopted limb-darkening coefficients are Γ_V = 0.726, Γ_R = 0.639, and Γ_I = 0.527. We set the center of mass of the binary lens as the reference position. In the preliminary modeling, we conduct a grid search for s and q while the other parameters are searched for using a downhill approach based on the Markov chain Monte Carlo (MCMC) method. We then refine the solutions found from the preliminary search by allowing all parameters to vary.

The standard modeling yields a unique solution with binary-lens parameters of (s, q) ∼ (0.51, 0.35). This indicates that the lens is a binary composed of masses of a same order with a projected separation smaller than the angular Einstein radius, i.e., close binary (s < 1.0). The event timescale is t_E ∼ 67.6 days, which is substantially longer than typical galactic lensing events. We check for the possible existence of a binary-lens solution with s > 1.0, wide binary, caused by the close/wide binary degeneracy. We find that the fit of the best-fit wide-binary solution is worse than the fit of the close-binary solution by Δχ² > 20,000, indicating that the close/wide degeneracy is clearly resolved.

Although the standard model provides a fit that describes the overall light curve, we find that the model leaves systematic residuals. This can be seen in the bottom panel of Figure 2, which shows a relatively small (≲0.05 mag) but easily noticed deviation from the standard model in the region around the main anomaly features. We also find that the deviation persists throughout the light curve. This suggests the need to consider higher-order effects.

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Noticing the residual from the standard model, we conduct additional modeling considering higher-order effects. It is known that two higher-order effects cause long-term deviations in lensing light curves. The first is the microlens-parallax effect, which is caused by the acceleration of the observer's motion induced by the orbital motion of Earth around the Sun (Gould 1992). The other is the lens-orbital effect, which is caused by the acceleration of the lens motion induced by the orbital motion of the lens (Dominik 1998; Ioka et al. 1999). We test these effects by conducting three sets of additional modeling. In the parallax and orbit models, we separately consider the microlens-parallax and lens-orbital effects, respectively. In the orbit+parallax model, we simultaneously consider both effects. For events affected by the microlens-parallax effect, there usually exist a pair of degenerate solutions with u₀ > 0 and u₀ < 0. This so-called ecliptic degeneracy is caused by the mirror symmetry of the source trajectory with respect to the binary axis (Smith et al. 2003; Skowron et al. 2011). We check this degeneracy when the microlens-parallax effect is considered in modeling.

Considering the higher-order effects requires one to include additional parameters in modeling. Parallax effects are described by two parameters of π_E,N and π_E,E, which denote the two components of the microlens-parallax vector, π_E, projected onto the sky along the north and east directions in the equatorial coordinates, respectively. Under the approximation that the change of the lens position caused by the orbital motion is small, lens-orbital effects are described by two parameters of ds/dt and dα/dt, which represent the change rates of the binary separation and the orientation angle, respectively.

In Table 1, we summarize the results of the additional modeling runs in terms of χ² values. From the comparison of the model fits, we find the following results. First, the fit greatly improves with the consideration of the higher-order effects. We find that the fit improves by Δχ² = 6680.6 and 7455.3 with respect to the standard model by considering the lens-orbital and microlens-parallax effects, respectively, indicating that the higher-order effects are clearly detected. When both effects are simultaneously considered, the fit further improves by Δχ² ∼ 1509.5 and 734.8 with respect to the orbit and parallax models, respectively. To visualize this improvement, we present the residuals of the tested models in the lower panels of Figure 2. In Figure 3, we also present the cumulative distributions of Δχ² as a function of time to show the region of the fit improvement. It is found that the greatest fit improvement occurs in the region around the main features of the light curve, i.e., the hump and caustic spikes, although the fit improves throughout the event. Second, the ecliptic degeneracy between the solutions with u₀ > 0 and u₀ < 0 is also resolved. It is known that this degeneracy is usually very severe even for binary-lens events with well covered caustic features, e.g., Δχ² ∼ 3 for OGLE-2017-BLG-053 (Jung et al. 2018) and Δχ² ∼ 8 for OGLE-2017-BLG-0039 (Han et al. 2018b). For OGLE-2018-BLG-0022, we find that the solution with u₀ > 0 is preferred over the solution with u₀ < 0 by Δχ² ∼ 226.3, which is big enough to clearly resolve the degeneracy.

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In the middle column of Table 2, we present the determined lensing parameters of the best-fit solution, i.e., the orbit+parallax model with u₀ > 0. In Figure 4, we also present the lens-system configuration, which shows the source trajectory (the black curve with an arrow) with respect to the lens components (marked by M₁ and M₂) and caustic (closed curve composed of concave curves). It is found that the source passed almost parallel to the binary axis. The caustic, which is composed of four folds, is located between the lens components. The hump centered at HJD' ∼ 8227 was produced when the source passed the excess magnification region extending from the right on-binary-axis caustic cusp. The source crossed the lower right fold caustic, producing the first caustic spike. Then, the source passed the upper left fold caustic, producing the second spike. To be mentioned is that the source enveloped the left on-axis caustic cusp during the caustic exit. See the left inset of Figure 4, which shows the enlargement of the caustic exit region. As a result, the caustic-crossing pattern differs from that produced when the source passes a regular fold caustic. See the inset in the upper panel of Figure 2.

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We note that OGLE-2018-BLG-0022 is a very rare case in which all the lensing parameters including those describing the higher-order effects are accurately determined without any ambiguity. As mentioned, the event does not suffer from the close/wide degeneracy, and thus there is no ambiguity in the binary separation. Furthermore, the ecliptic degeneracy is resolved with a significant confidence level, and thus the microlens parallax and the resulting lens mass are uniquely determined.

Prompted by the accuracy of modeling, we further check whether the determinations of the complete orbital parameters are possible for this event. This requires two additional parameters s_∥ and ds_∥/dt, which represent the line-of-sight binary separation normalized to ${\theta }_{{\rm{E}}}$ and the change rate of s_∥, respectively (Skowron et al. 2011; Shin et al. 2012). One also needs the information of the angular Einstein radius, and we describe the procedure for the ${\theta }_{{\rm{E}}}$ estimation in Section 3.3. We find that it is difficult to determine the full orbital parameters. In Figure 5, we present the Δχ² distributions of MCMC points in the planes of the higher-order lensing-parameter combinations. It is found that the additional two parameters (s_∥ and ds_∥/dt) are poorly constrained, while the other higher-order parameters (π_E,N, π_E,E, ds/dt, dα/dt) are well constrained. We judge that the difficulty of full characterization of the orbital lens motion is caused by the short duration of the major anomaly features.

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In addition to the ground-based data, there exist data obtained from space-based Spitzer observations. Spitzer observations of the event were conducted in the 3.6 μm channel (L band) with 1 day cadence during the period 8305.7 ≤ HJD' ≤ 8341.0 and, in total, 34 data points were acquired. Data reduction was conducted using the procedure described in Calchi Novati et al. (2015). In Figure 6, we plot the Spitzer data over the data points from the ground-based observations.

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Spitzer data may help to improve the accuracy of ${\pi }_{{\rm{E}}}$ measurement. This is because the Spitzer telescope is in a heliocentric orbit and thus the Earth-satellite separation and the physical Einstein radius, r_E = D_L${\theta }_{{\rm{E}}}$, are of the same order of astronomical units. In this case, the light curve seen from space would be substantially different from the light curve observed from the ground (Refsdal 1966; Gould 1994), and thus space-based data can give an important constraint on the microlens parallax (Han et al. 2018a). We, therefore, check the effect of the Spitzer data on the ${\pi }_{{\rm{E}}}$ measurement. In the analysis with the additional Spitzer data, we impose a constraint of the source color with the measured instrumental value of I − L =−5.21 ± 0.02 following the procedure described in Shin et al. (2017).

In the right column of Table 2, we present the lensing parameters estimated with the additional Spitzer data. From the comparison of the parameters with those estimated from the ground-based data, it is found that the parameters are similar to each other except for the slight differences in the higher-order parameters. In Figure 7, we present the Δχ² distributions of MCMC points in the π_E,E–π_E,N plane obtained from the modelings with (right panel) and without (left panel) the Spitzer data. It is found that the additional Spitzer data make the north component of the microlens-parallax vector slightly smaller than the value estimated from the ground-based data.

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In Figure 4, we present the source trajectory seen from the satellite (blue curve with an arrow marked by Spitzer). In Figure 6, we also present the model light curve for the Spitzer data (blue curve).

We determine the angular Einstein radius, which is the other ingredient needed for the lens mass measurement besides ${\pi }_{{\rm{E}}}$, from the combination of the normalized source radius and the angular source radius, i.e., ${\theta }_{{\rm{E}}}$ = θ_*/ρ. The normalized source radius ρ is determined by analyzing the caustic-crossing parts of the light curve. The angular source radius θ_* is estimated based on the de-reddened color, (V − I)₀, and brightness, I₀, of the source. We determine (V − I)₀ and I₀ using the method of Yoo et al. (2004), which utilizes the centroid of the red giant clump (RGC) in the color–magnitude diagram (CMD) as a reference.

In Figure 8, we mark the position of the source with respect to the RGC centroid in the OGLE-III CMD. We note that the de-reddened source color and brightness are estimated using the I- and V-band pyDIA photometry of the KMTC data set. However, the magnitude of the KMTC data is not calibrated, while the OGLE-III data are calibrated (Szymański et al. 2011). We, therefore, place the source position on the calibrated OGLE-III CMD using the offsets in color and brightness between the RGC centroids of the KMTC and OGLE-III CMDs. With the apparent color and brightness of the source of (V − I, I) = (2.21, 15.86) and the RGC centroid of (V − I, I)_RGC = (2.06, 15.61) combined with the known de-reddened values of the RGC centroid (V − I, I)_RGC,0 = (1.06, 14.35) (Bensby et al. 2011; Nataf et al. 2013), we estimate that the de-reddened color and brightness of the source are (V − I, I)₀ = (1.21, 14.58), indicating that the source is a K-type giant. The measured V − I color is converted into V − K color using the color–color relation of Bessell & Brett (1988). We then estimate the angular source radius using the (V − K)/θ_* relation of Kervella et al. (2004). We estimate that the source has an angular radius of θ_* = 6.54 ± 0.46 μas.

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With the source radius, we estimate the angular Einstein radius of

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}\ =1.31\pm 0.09\,\mathrm{mas}.\end{eqnarray} \tag{ 2 }$$

Combined with the measured event timescale t_E, the relative lens-source proper motion as measured in the geocentric frame is estimated by

$$\begin{eqnarray}&&{\mu }_{\mathrm{geo}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{{t}_{{\rm{E}}}}=6.82\pm 0.48\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 3 }$$

In the heliocentric frame, the proper motion is

$$\begin{eqnarray}&&{\mu }_{\mathrm{helio}}=\left|{\mu }_{\mathrm{geo}}\,\displaystyle \frac{{{\boldsymbol{\pi }}}_{{\rm{E}}}}{{\pi }_{{\rm{E}}}}+{{\boldsymbol{v}}}_{\oplus ,\perp }\displaystyle \frac{{\pi }_{\mathrm{rel}}}{\mathrm{au}}\right|=7.33\pm 0.52\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 4 }$$

Here ${\boldsymbol{v}}$_⊕,⊥ (N, E) = (2.1, 18.5) km s⁻¹ denotes the projected velocity of Earth at t₀, π_rel = au (${D}_{{\rm{L}}}^{-1}-{D}_{{\rm{S}}}^{-1}$), and D_S denotes the distance to the source (Gould 2004; Dong et al. 2009). In Table 3, we summarize the angular Einstein radius and the proper motion. Also listed is the angle of the heliocentric lens-source relative proper motion as measured from north toward east, i.e., ϕ_helio = tan⁻¹ (μ_helio,E/μ_helio,N) ∼ 22°.

Table 3.  Einstein Radius and Proper Motion

Quantity	Value
Angular Einstein radius	1.31 ± 0.09 mas
Proper motion (geocentric)	6.82 ± 0.48 mas yr−1
Proper motion (heliocentric)	7.32 ± 0.52 mas yr−1
ϕhelio	217 ± 15

Note. The angle ϕ_helio represents the angle of the heliocentric lens-source relative proper motion as measured from north toward east.

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Being able to determine ${\pi }_{{\rm{E}}}$ and ${\theta }_{{\rm{E}}}$ without any ambiguity, the mass of the lens is uniquely determined. It is found that the lens is a binary composed of an early M-dwarf primary with a mass of

$$\begin{eqnarray}&&{M}_{1}=0.50\pm 0.05\,{M}_{\odot }\end{eqnarray} \tag{ 5 }$$

and a late M-dwarf companion with a mass of

$$\begin{eqnarray}&&{M}_{2}=0.15\pm 0.01\,{M}_{\odot }.\end{eqnarray} \tag{ 6 }$$

We note that the masses are estimated based on the solution obtained using both the ground-based and Spitzer data.

With the determined ${\pi }_{{\rm{E}}}$ and ${\theta }_{{\rm{E}}}$, the distance to the lens is determined by

$$\begin{eqnarray}&&{D}_{{\rm{L}}}=\displaystyle \frac{\mathrm{au}}{{\pi }_{{\rm{E}}}{\theta }_{{\rm{E}}}\ +{\pi }_{{\rm{S}}}}=2.21\pm 0.18\,\mathrm{kpc},\end{eqnarray} \tag{ 7 }$$

indicating that the lens is in the disk. Here π_S = au/D_S. The source distance is estimated using the relation ${D}_{{\rm{S}}}\,={d}_{\mathrm{GC}}/(\cos l+\sin l\cos {\theta }_{\mathrm{bar}}/\sin {\theta }_{\mathrm{bar}})\sim 7.87\mathrm{kpc}$, where d_GC ∼ 8160 pc is the distance to the Galactic center and θ_bar = 40° is the orientation angle of the bulge bar (Nataf et al. 2013). Once the distance is estimated, the projected separation between the lens components is estimated by

$$\begin{eqnarray}&&{a}_{\perp }={{sD}}_{{\rm{L}}}{\theta }_{{\rm{E}}}\ =1.54\pm 0.13\,\mathrm{au}.\end{eqnarray} \tag{ 8 }$$

For the source of the event, the parallax is not measured but the proper motion (in the heliocentric frame) is listed in the Gaia archive³⁸ with values

$$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{{\rm{S}}}(N,E)=(-6.12,-2.76)\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 9 }$$

The proper motion indicates that the source is a typical bulge star. Since the relative lens-source proper motion (in the heliocentric frame), ${\boldsymbol{\mu }}$, is related to the proper motions of the source, ${\boldsymbol{\mu }}$_S, and the lens, ${\boldsymbol{\mu }}$_L, by ${\boldsymbol{\mu }}$ = ${\boldsymbol{\mu }}$_L − ${\boldsymbol{\mu }}$_S, the Gaia measurement of the source proper motion allows us to estimate the lens proper motion by the relation

$$\begin{eqnarray}&&{{\boldsymbol{\mu }}}_{{\rm{L}}}(N,E)={{\boldsymbol{\mu }}}_{{\rm{S}}}+{\boldsymbol{\mu }}\ =(0.69,-0.05)\,\mathrm{mas}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 10 }$$

Then, the projected lens velocity in the heliocentric frame is

$$\begin{eqnarray}&&{{\boldsymbol{v}}}_{{\rm{L}},\perp }(N,E)={{\boldsymbol{\mu }}}_{{\rm{L}}}{D}_{{\rm{L}}}=(7.2,-0.5)\,\mathrm{km}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 11 }$$

which is very typical for disk stars. Therefore, the estimated lens distance is consistent with the additional constraint from the Gaia observation.

We also check the validity of the lensing solution by computing the projected kinetic-to-potential energy ratio of the lens system. From the determined physical parameters M and a_⊥combined with the lensing parameters s, ds/dt, and dα/dt, the ratio is computed by

$$\begin{eqnarray}&&{\left(\displaystyle \frac{\mathrm{KE}}{\mathrm{PE}}\right)}_{\perp }=\displaystyle \frac{{\left({a}_{\perp }/\mathrm{au}\right)}^{3}}{8{\pi }^{2}(M/{M}_{\odot })}\left[{\left(\displaystyle \frac{1}{s}\displaystyle \frac{{ds}/{dt}}{{\mathrm{yr}}^{-1}}\right)}^{2}+{\left(\displaystyle \frac{d\alpha /{dt}}{{\mathrm{yr}}^{-1}}\right)}^{2}\right]\sim 0.09.\end{eqnarray} \tag{ 12 }$$

In order for the lens to be a gravitationally bound system, the ratio should meet the condition of (KE/PE)_⊥ ≤ KE/PE ≤ 1.0. The estimated kinetic-to-potential energy ratio satisfies this condition. In Table 4, we summarize the determined physical lens parameters.

Given that the distance to the lens is small, the lens might comprise an important fraction of the blended light, e.g., OGLE-2017-BLG-0039 (Han et al. 2018b). We check this possibility by inspecting the agreement between the positions of the lens and blend in the CMD. In Figure 8, we place the locations of the blend and lens. The lens location is estimated based on the mass and distance. Considering the close distance to the lens, we assume that the lens experiences ∼1/3 of the total extinction and reddening toward the bulge field of A_I ∼ 1.26 and E(V − I) ∼ 1.02, respectively (Nataf et al. 2013). The estimated color and brightness of the lens are (V − I, I)_L ∼ (2.3, 19.8), while those of the blend are (V − I, I)_b ∼ (1.7, 18.9). The lens is substantially fainter and redder than the blend and this indicates that the lens is not the main source of the blended light.