OGLE-2009-BLG-092/MOA-2009-BLG-137: A DRAMATIC REPEATING EVENT WITH THE SECOND PERTURBATION PREDICTED BY REAL-TIME ANALYSIS

When a foreground astronomical object is closely aligned to a background star, the light from the background star (source) is amplified by the gravity of the foreground object (lens). The magnification of this gravitational lensing phenomenon depends on the projected separation between the lens and source star. With the change of the separation, the lensing magnification varies in time. For an event caused by a single-mass lens, the brightness variation is characterized by its non-repeating symmetric light curve (Paczyński 1986).

When a star is gravitationally magnified by a lens composed of two masses (binary lens), the resulting light curves become complicated due to the nonlinear nature of binary-lensing magnifications (Schneider & Weiss 1986). The most important feature of binary lensing is caustics, which represent the set of source positions at which the lensing magnification of a point source becomes infinite. Caustics form a single or multiple sets of closed curves each of which is composed of concave curves that meet at cusps. The number, shape, size, and locations of caustic curves vary depending on the separation and mass ratio between the binary components. As a result, light curves of binary-lensing events exhibit great diversity (Erdl & Schneider 1993; Mao & Paczyński 1991).

In current microlensing experiments, events are observed from the combination of survey and follow-up observations. Survey observations, e.g., the Optical Gravitational Lensing Experiment (OGLE; Udalski et al. 2005) and the Microlensing Observations in Astrophysics (MOA; Sumi et al. 2010), are operated in order to maximize the event rate by monitoring a large area of sky toward the Galactic bulge on a roughly nightly basis using large-format cameras. On the other hand, follow-up observations, e.g., the Microlensing Follow-Up Network (μFUN; Gould et al. 2006), the Probing Lensing Anomalies Network (PLANET; Beaulieu et al. 2006), and RoboNet (Tsapras et al. 2009), are focused on events alerted by survey observations to densely cover various anomalies including planet-induced perturbations. However, the limited number of telescopes available for follow-up observations restricts the number of events that can be monitored at any given time. For the efficient use of telescopes for follow-up observations, then, it is important to judge which events should be focused upon among the events alerted by survey observations. This judgment can be done based on real-time modeling of events. Real-time modeling is also important to judge the time and duration of follow-up observations. Extended coverage of events is often needed to determine the physical parameters of lenses by measuring subtle deviations caused by long-lasting effects. In addition, planets and binaries often induce multiple perturbations, and resolving additional perturbations is very critical for accurate and precise characterization of lenses. Real-time modeling helps to judge how long and when intensive follow-up observations should be carried out. Despite its importance, routine real-time modeling of binary and planetary lensing events has been difficult due to the large number of parameters to be included in modeling combined with the complexity of the χ² surface of the parameter space. With the development of efficient codes from the efforts of theoretical studies on binary-lensing phenomenology, however, it is now possible to routinely release models of light curves just after or even during the progress of perturbations.⁶⁰

In this paper, we present results of the analysis of a dramatic repeating event for which the light curve is characterized by two distinct peaks with perturbations near both peaks. The event is highlighted because the second perturbation, which occurred ∼100 days after the first peak, was predicted by the real-time analysis conducted after the first perturbation. With the dense coverage of the second peak by follow-up observations based on the real-time modeling, we are able to measure the physical parameters of the lens.

The event OGLE-2009-BLG-092/MOA-2009-BLG-137 occurred on a Galactic bulge star located at (α, δ)₂₀₀₀ = (17^h51^m37^s95, −29°32'4323), which corresponds to the Galactic coordinates (l, b) = (015, − 144). The event was independently detected by the OGLE and MOA groups using the 1.3 m Warsaw telescope of Las Campanas Observatory in Chile and 1.8 m telescope of Mt. John Observatory in New Zealand, respectively.

An anomaly alert was issued on 2009 May 4. Based on the alert, the event was intensively observed by follow-up groups of the μFUN, PLANET, and RoboNet collaborations. The telescopes used for these follow-up observations include the μFUN 1.3 m SMARTS telescope of the Cerro Tololo Inter-American Observatory (CTIO) in Chile, μFUN 0.4 m of Auckland Observatory, μFUN 0.4 m of Farm Cove Observatory (FCO) in New Zealand, μFUN 1.0 m Mt. Lemmon Observatory (LOAO) in Arizona, USA, μFUN 0.4 m of Bronberg Observatory in South Africa, RoboNet 2.0 m Liverpool Telescope (LT) in La Palma, Canary Islands, RoboNet 2.0 m Faulkes North (FTN) in Hawaii, RoboNet 2.0 m Faulkes South (FTS) in Australia, and PLANET 1.0 m of Mt. Canopus Observatory in Australia. Dense coverage by the survey and follow-up observations revealed an anomaly near the peak of the light curve.

Just after the perturbation, real-time analysis of the event was conducted using data available at that moment. Modeling showed that the perturbations were produced by a binary lens. The projected separation of the binary system was degenerate between two values, one larger than the Einstein radius and the other smaller. This degeneracy is known as the close/wide binary degeneracy (Dominik 1999b). It was also found that if the perturbation was produced by a wide binary, the source trajectory would pass close to the caustic associated with the other component of the binary, and thus there would be an additional perturbation approximately ∼100 days after first perturbation.

Another modeling conducted at around HJD = 2, 454, 975 with additional acquired data resolved the close/wide binary degeneracy and finally predicted the second anomaly. The region between the two peaks was expected to vary smoothly without any major perturbation. In addition, the time gap between the two peaks is too long to be continuously monitored by follow-up observations. Therefore, observations during this period were conducted mostly by survey groups.

A third modeling was conducted at HJD ∼ 2, 455, 035 to precisely predict the time of the second perturbation. From this, the second perturbation was predicted to occur at HJD ∼2,455,045. With this prediction, another follow-up campaign was prepared to cover the second perturbation. In addition to the telescopes used for the follow-up observations of the first perturbation, additional telescopes participated in the observations of the second peak. These include the PLANET 1.54 m Danish Telescope of La Silla Observatory of the Microlensing Network for the Detections of Small Terrestrial Exoplanets (MiNDSTEp) group, μFUN 1.0 m of Wise Observatory in Israel, μFUN 0.36 m of Kumeu Observatory, μFUN 0.3 m of Molehill Astronomy Observatory (MAO) in New Zealand, and PLANET 1.0 m of SAAO in South Africa. From this campaign, the second peak was also densely resolved.

Figure 1 shows the light curve of OGLE-2009-BLG-092/MOA-2009-BLG-137. In the light curve, the MOA data except during the perturbation regions are binned by 1 day for clarity but the modeling is based on unbinned data. The insets show enlargements of the perturbation regions around the individual peaks. We mark the times when the real-time analyses were conducted.

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Due to their great diversity, describing light curves of binary-lensing events requires to include many parameters. The basic structure of light curves of binary-lens events is characterized by six lensing parameters. The first set of three parameters is needed to describe light curves of standard single-lens events: the time required for the source to transit the Einstein radius, t_E (Einstein timescale), the time of the closest lens–source approach, t₀, and the lens–source separation in units of θ_E at that time, u₀ (impact parameter). Describing the deviation caused by the lens binarity requires an additional set of three binary-lensing parameters: the mass ratio between the lens components, q; the projected binary separation in units of the Einstein radius, s; and the angle of the source trajectory with respect to the binary axis, α.

In addition to these basic parameters, additional parameters are needed to describe detailed structures of lensing light curves. The event OGLE-2009-BLG-092/MOA-2009-BLG-137 exhibits caustic-induced perturbations at both peaks and thus it is required to consider the finite-source effect to describe deviations occurring when the source approaches and crosses over caustics (Nemiroff & Wickramasinghe 1994; Witt & Mao 1994; Gould 1994). The finite-source effect is parameterized by the ratio of the source radius θ_⋆ to the Einstein radius θ_E, i.e., ρ_⋆ = θ_⋆/θ_E (normalized source radius).

Due to the large time gap between the two peaks of the light curve, the relative lens–source motion may deviate from a rectilinear one due to the acceleration of the observer's motion induced by the Earth's orbital motion around the Sun (Refsdal 1966; Gould 1992; Smith et al. 2003). We consider this so-called parallax effect in the modeling by including the two parallax parameters π_E,N and π_E,E, which are the two components of the microlensing parallax vector $\mbox{\boldmath $\pi $}_{\rm E}$ projected on the sky in the direction of north and east celestial coordinates. The direction of this vector is that of the lens–source relative motion in the frame of the Earth at t₀.⁶¹

We also check the possibility of the change of lens positions caused by its orbital motion. The orbital motion has two effects on lensing magnifications. One causes the binary axis to rotate or, equivalently, makes the source trajectory angle, α, change in time. The other effect causes the separation between the binary components to vary in time (Dominik 1998; Ioka et al. 1999; Albrow et al. 2000). The change in the binary separation alters the shape of the caustic in the course of the event. To the first order, the orbital effect is parameterized by

$$\begin{equation} \alpha (t)=\alpha (t_0)+\omega \left({t-t_0\over t_{\rm E}}\right) \end{equation} \tag{ 1 }$$

and

$$\begin{equation} s(t)=s(t_0)+\dot{s}\left({t-t_0\over t_{\rm E}}\right), \end{equation} \tag{ 2 }$$

where the orbital lensing parameters ω and $\dot{s}$ represent the rates of change in the source trajectory angle and the projected binary separation, respectively.

Due to the sheer size of the parameter space, it is difficult to find binary-lensing solutions from brute-force searches. Searches for solutions become further hampered by the complexity of the χ² surface. This complexity implies that even if a solution that seemingly describes an observed light curve is found, it is difficult to be sure that all possible minima have been investigated (Dominik 1999a, 1999b) and thus a simple downhill approach cannot be used. To avoid these difficulties, we use a hybrid approach in which grid searches are conducted over the space of a subset of parameters and the remaining parameters are allowed to vary so that the model light curve results in minimum χ² at each grid point. See also Dong et al. (2006) and Bennett (2010). We choose s, q, and α as grid parameters because they are related to the features of light curves in a complicated pattern such that a small change in these parameters can result in dramatic changes in the resulting light curves. On the other hand, the other parameters are more directly related to the observed light curve features. We use a Markov Chain Monte Carlo method for χ² minimization. Once the solutions of the individual grid points are determined, the best-fit model is obtained by comparing the χ² minima of the individual grid points.

In addition to the difficulties mentioned above, binary-lens modeling suffers from an additional difficulty that arises due to large computations required for modeling. Most binary-lensing events exhibit perturbations induced by caustic crossings or approaches during which the finite-source effect is important. Calculating finite-source magnifications requires a numerical method for which heavy computations are needed. Considering that modeling requires the production of many light curves of trial models, it is important to apply an efficient method for magnification calculations. In our modeling, we use a customized version of the inverse ray-shooting method to calculate finite-source magnifications. In the usual ray-shooting method, a large number of rays are uniformly shot from the image plane, bent according to the lens equation, and land on the source plane. Then, the lensing magnification corresponding to the location of a finite source is computed by comparing the number density of rays on the source surface with the density on the image plane. The main shortcoming of this method is that only a small fraction of rays land on the source surface and most of the rest of the rays are not used for magnification computations. We reduce computation time by minimizing wasted rays. For this, we first make grids on the image plane. We then find the image positions corresponding to the individual positions of the envelope of the source star and then register the grids corresponding to the image positions. We minimize the number of rays by restricting the region of ray shooting only to the registered grids on the image plane. This scheme is similar to that of Rattenbury et al. (2002). We set the width of the grids to be slightly bigger than the source star. If the width is too small, the region inside the image may not be registered. If the width is too big, the fraction of rays not arriving on the source surface will increase. We find that the optimal grid width corresponds to the diameter of the source. This is because lensing-induced distortions always result in images slimmer than the source and thus grids with a width of the source diameter fill the images, thereby minimizing wasted rays. To further speed up the computation, we use the finite-source magnification calculations based on the numerical ray-shooting method only in the region near the caustic, and a simple semi-analytic hexadecapole approximation (Pejcha & Heyrovský 2009; Gould 2008) is used in other part of light curves.

In Table 1, we summarize the results of modeling. We test five different models. The first model is based on a static binary lens (no orbital effect) without the parallax effect (standard model). The second model includes the parallax effect. Finally, the orbital motion of the lens is additionally considered in the third model. When the parallax or orbital motion is considered, a pair of solutions resulting from the mirror-image source trajectories with impact parameters u₀ > 0 and u₀ < 0 result in slightly different light curves due to the asymmetry of the source trajectories with respect to the binary axis. We, therefore, check both models with u₀ > 0 and u₀ < 0 whenever the parallax or orbital effect is considered.

Table 1. Fit Parameters

Parameter	Standard	Model
		Parallax		Parallax+Orbital motion
		u0 > 0	u0 < 0	u0 > 0	u0 < 0
χ2/dof	6639.03/6569	6616.06/6567	6598.27/6567	6510.61/6565	6503.02/6565
t0 (HJD)	4990.441 ± 0.053	4990.501 ± 0.047	4990.106 ± 0.054	4988.473 ± 0.058	4987.744 ± 0.047
u0	−0.062 ± 0.001	0.060 ± 0.001	−0.063 ± 0.001	0.053 ± 0.001	−0.051 ± 0.001
tE (days)	35.53 ± 0.01	35.44 ± 0.04	35.48 ± 0.03	35.23 ± 0.05	35.25 ± 0.03
s	2.921 ± 0.001	2.923 ± 0.001	2.927 ± 0.001	2.909 ± 0.002	2.8946 ± 0.001
q	0.562 ± 0.001	0.565 ± 0.001	0.557 ± 0.002	0.530 ± 0.001	0.518 ± 0.001
α (rad)	6.2466 ± 0.0001	0.0343 ± 0.0003	6.2410 ± 0.0002	0.0411 ± 0.0003	6.2420 ± 0.0002
ρ⋆	0.0101 ± 0.0001	0.0100 ± 0.0001	0.0094 ± 0.0001	0.0091 ± 0.0001	0.0090 ± 0.0001
πE,N	...	−0.055 ± 0.009	−0.125 ± 0.005	−0.018 ± 0.016	0.059 ± 0.008
πE,E	...	−0.019 ± 0.013	0.057 ± 0.009	−0.039 ± 0.013	−0.054 ± 0.012
$\dot{s}$ (t−1E)	...	...	...	0.0315 ± 0.0016	0.0464 ± 0.0013
ω (t−1E)	...	...	...	−0.0068 ± 0.0007	0.0099 ± 0.0006

Note. The parameters of the best-fit solution are marked in bold fonts.

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From the table, it is found that the model including both the parallax and orbital effects resulting from a source trajectory with u₀ < 0 provides the best fit to the observed light curve. It is also found that the parallax effect improves the fit by Δχ²/dof = 40.8/6567, while the orbital effect further improves the fit by Δχ²/dof = 136.0/6565. The differences in the goodness of fit between the solutions can be seen in Figure 2, where we present the residuals of data from the individual models. One finds that the "parallax + orbit" solution removes the systematic residuals that are present in the other solutions. We note that the amount of parallax π_E = (π²_E,E + π²_E,N)^1/2 = 0.080 is substantially smaller than the typical values of events for which parallaxes are measured. The orbit-induced changes of the binary separation during the time gap Δt between the two peaks of $\Delta s= \dot{s}(\Delta t/t_{\rm E})\sim 0.13$ and the source trajectory angle of Δα = ω(Δt/t_E) ∼ 16 are very small. Despite their small amplitudes, both parallax and orbital effects are measurable, thanks to the dense coverage of the second perturbation from follow-up observations combined with the fact that the event had a long effective timescale and there were two disconnected deviations. Figure 3 presents the contours of χ² in the space of the combinations of the parallax and orbital lensing parameters. We note that the tangential and radial velocities of the companion relative to the primary are v_t = r_⊥ω and $v_{\rm r} =r_\perp (\dot{s}/s)$, respectively, where r_⊥ represents the projected binary separation. We, therefore, set the ordinate of the lower panels as $\dot{s}/s$ so that it is in the same order of ω.

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In Figure 1, we present the model light curve of the best-fit solution. In Figure 4, we also present the geometry of the lens system based on the best-fit solution. In the figure, the filled circles represent the locations of the lens components, where M₁ is the heavier component, the two closed curves are the caustics, and the curve with an arrow represents the source trajectory. The two upper panels show enlargements of the regions around the individual caustics. The open circles on the source trajectory represent the source star at the time of observations where the size indicates its finite size and the colors correspond to those of the data points of different observatories adopted in Figure 1. All lengths are normalized by the Einstein radius corresponding to the total mass of the binary.

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To determine the physical parameters of the lens system, it is required to measure both the Einstein radius and the lens parallax. The lens parallax is directly measured from modeling. The Einstein radius is measured from the normalized source radius ρ_⋆ combined with the angular size of the source star, θ_⋆. The angular source size is estimated based on the de-reddened magnitude I₀ and color (V − I)₀ of the source star measured from the offset between the source and the centroid of clump giants in the instrumental color–magnitude diagram under the assumption that source and clump giants experience the same amount of extinction (Yoo et al. 2004). In Figure 5, we present the location of the source in the color–magnitude diagram constructed by using the V- and I-band images taken from the CTIO. With the known clump centroid of [(V − I)₀, I₀]_c = (1.04, 14.27) and the measured offsets of Δ(V − I) = (V − I)_S − (V − I)_c = 0.123, and ΔI = I_S − I_c = 0.429, the de-reddened color and magnitude of the source star are measured as [(V − I)₀, I₀]_S = (1.16, 14.69), respectively. Here we adopt the distance to the clump of 8.0 kpc toward the field which is estimated by using the Galactic model of Han & Gould (2003). Then, the angular source size is determined by first transforming from (V − I)₀ to (V − K)₀ using the color–color relation of Bessell & Brett (1988) and then applying the relation between (V − K)₀ and the angular stellar radius of Kervella et al. (2004). The determined angular radius of the source star is θ_⋆ = 6.11 ± 0.53 μas, implying that the source star is a Galactic bulge clump giant star. The uncertainty of θ_⋆ is estimated from the combination of those of the colors and magnitudes of the source and the clump centroid and an additional 7% intrinsic error in the conversion process from the color to the source radius (Yee et al. 2009). Then, the Einstein radius is measured as

$$\begin{equation} \theta _{\rm E}={\theta _\star \over \rho _\star }=0.68\pm 0.06\ {\rm mas}. \end{equation} \tag{ 3 }$$

Combining this with the Einstein timescale yields the relative proper motion between the lens and source of

$$\begin{equation} \mu = {\theta _{\rm E}\over t_{\rm E}} =7.01 \pm 0.61 \ {\rm mas}\ {\rm yr}^{-1}. \end{equation} \tag{ 4 }$$

With the measured Einstein radius and lens parallax, the mass of the lens and the distance to the lens are determined, respectively, as

$$\begin{equation} M={\theta _{\rm E}\over \kappa \pi _{\rm E}},\quad D_{\rm L}={{\rm AU}\over \pi _{\rm E}\theta _{\rm E}+\pi _{\rm S}}, \end{equation} \tag{ 5 }$$

where κ = 4G/(c²AU), π_S = AU/D_S is the parallax of the source, and D_L and D_S represent the distances to the lens and source, respectively. For the best-fit model, the determined values are

$$\begin{equation} M=1.04 \pm 0.16 \, M_\odot \end{equation} \tag{ 6 }$$

and

$$\begin{equation} D_{\rm L}=5.6 \pm 0.7\ {\rm kpc}, \end{equation} \tag{ 7 }$$

respectively. From the normalized separation together with the physical Einstein radius r_E = D_Lθ_E, the projected separation between the binary companions is estimated as

$$\begin{equation} r_\perp =s r_{\rm E}=10.9\pm 1.3\ {\rm AU}. \end{equation} \tag{ 8 }$$

With the known mass ratio, the masses of the individual binary components are estimated, respectively, as

$$\begin{equation} M_1={M \over 1+q}=0.69 \pm 0.11\, M_\odot \end{equation} \tag{ 9 }$$

and

$$\begin{equation} M_2= {q \over 1+q}M=0.36 \pm 0.06\, M_\odot. \end{equation} \tag{ 10 }$$

Therefore, the event OGLE-2009-BLG-092/MOA-2009-BLG-137 occurred on a bulge clump giant and was produced by a binary lens composed of a K- and M-type main-sequence stars.

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In principle, it is possible to constrain the physical orbital parameters such as the semimajor axis, orbital period, and inclination of the orbital plane from the determined orbital lensing parameters of ω and $\dot{s}$ (Dong et al. 2009; Bennett et al. 2010). However, we find that this is difficult because the magnitudes of the changes of the binary separation and source trajectory angle are too small. Nevertheless, it is still possible to check the consistency of the orbital parameters using the parameter

$$\begin{equation} \eta = {(r_\perp /{\rm AU})^3 \over 8 \pi ^2 (M/M_\odot)} \left[ \omega ^2 +\left({\dot{s}\over s}\right)^2\right] \left({{\rm yr} \over t_{\rm E}} \right)^2, \end{equation} \tag{ 11 }$$

which represents the ratio between kinetic and potential energies. To be a bound system, the parameter should be less than unity. Based on the obtained orbital lensing parameters of the best-fit solution, we find η = 0.60, implying that the result is consistent.

We analyzed the light curve of a dramatic repeating binary-lens event OGLE-2009-BLG-092/MOA-2009-BLG-137 for which the light curve was characterized by two distinct peaks separated by ∼100 days with perturbations near both peaks. By precisely predicting the occurrence of the second perturbation from the analysis of data conducted just after the first perturbation, we demonstrated that real-time modelings can be routinely done for anomalous events. Data covering the second peak obtained from follow-up observations enabled us to uniquely determine the physical parameters of the lens system. From the analysis of the data, we found that the event occurred on a bulge clump giant and it was produced by a binary lens composed of a K- and M-type main-sequence stars. The estimated masses of the individual masses of the binary components were M₁ = 0.69 ± 0.11 M_☉ and M₂ = 0.36 ± 0.06 M_☉, respectively, and they were separated in projection by r_⊥ = 10.9 ± 1.3 AU. The measured distance to the lens was D_L = 5.6 ± 0.7 kpc. Real-time modeling of anomalous lensing events is important not only for the efficient use of observational resources but also for precise characterizations of lenses including planetary systems.

Aside from the lens being able to be well characterized by real-time analysis, the event is also important because of its repeating nature. One of the original methods for distinguishing microlensing from variable stars was "non-repeating events." But di Stefano & Mao (1996) pointed out that this would cause one to miss some binary events and urged that selection should not be done blindly against repeating events. Nevertheless, there have been relatively few repeating binary-lens events reported in the literature, e.g., Jaroszyński & Skowron (2008). Even when they are detected, it is often difficult to distinguish them from those produced by binary-source stars. The event OGLE-2009-BLG-092/MOA-2009-BLG-137 results from the special case where the source trajectory passes the central perturbation regions associated with both lens components and thus the binary nature is unambiguously revealed. In current lensing experiments, the majority of binary lenses are detected through the channel of high-magnification events for which the lens binarity can easily be identified from central perturbations (Han & Hwang 2009). A fraction of these events will result in repeating events, although the second peak will not be as dramatic as that of OGLE-2009-BLG-092/MOA-2009-BLG-137. A careful analysis of these repeating events can provide an independent way to study the statistics of wide binary stars (Skowron et al. 2009).

We acknowledge the support of Creative Research Initiative Program (2009-0081561) of National Research Foundation of Korea (C.H.), Korea Astronomy and Space Science Institute (C.-U.L.), NSF AST-0757888 (A.G.), NASA NNG04GL51G (B.S.G., A.G., and R.W.P.), Royal Society University Research Fellow (M.D.), JSPS20740104 (T.S.), JSPS19340058 (Y.M.), and JSPS20340052 and JSPS18253002 (MOA). The MiNDSTEp monitoring campaign is powered by ARTEMiS (Automated Terrestrial Exoplanet Microlensing Search; Dominik et al. 2008). Astronomical research at Armagh Observatory is funded by the Department of Culture, Arts and Leisure, Northern Ireland, UK (T.C.H.).