CHARACTERIZING LENSES AND LENSED STARS OF HIGH-MAGNIFICATION SINGLE-LENS GRAVITATIONAL MICROLENSING EVENTS WITH LENSES PASSING OVER SOURCE STARS

When an astronomical object (lens) is closely aligned with a background star (source), the light from the source is deflected by the gravity of the lens, resulting in the brightening of the source star. The magnification of the source flux is related to the projected lens-source separation by

$$\begin{equation} A={u^2+2 \over u\sqrt{u^2+4}}, \end{equation} \tag{ 1 }$$

where the separation u is expressed in units of the angular Einstein radius θ_E. The Einstein radius is related to the physical parameters of the lens system by

$$\begin{equation} \theta _{\rm E}=\left(\kappa M \pi _{\rm rel}\right)^{1/2};\qquad \pi _{\rm rel}={\rm AU} \left({1\over D_{\rm L}}-{1\over D_{\rm S}}\right), \end{equation} \tag{ 2 }$$

where κ = 4G/(c²AU) = 8.14 mas M_☉⁻¹, M is the mass of the lens, π_rel is the relative source-lens parallax, and D_L and D_S are the distances to the lens and source star, respectively. The relative motion between the source, lens, and observer leads to light variation in the source star (lensing event). The first microlensing events were detected by Alcock et al. (1993) and Udalski et al. (1993) from experiments based on a proposal by Paczyński (1986). With the development of an observational strategy combined with upgraded instruments, the detection rate of lensing events has dramatically increased from several dozen events per year during the early phase of lensing experiments to more than a thousand events per year in current experiments.

The magnification of the source star flux increases as the lens approaches the source star. For a small fraction of events, the lens-source separation is even smaller than the source radius and the lens passes over the surface of the source star. These events are of scientific importance for various reasons.

First, a high-magnification event with a lens passing over a source star provides a rare chance to measure the brightness profile of a remote star. For such an event, in which the lens-source separation is comparable to the source size near the peak of the event, different parts of the source star are magnified by different amounts. The resulting lensing light curve deviates from the standard form of a point-source event (Witt & Mao 1994; Gould 1994; Nemiroff & Wickramasinghe 1994; Alcock et al. 1997) and the analysis of the deviation enables us to measure the limb-darkening profile of the lensed star (Witt 1995; Loeb & Sasselov 1995; Valls-Gabaud 1998; Bryce et al. 2002; Heyrovský 2003). With the same principle, it is also possible to study irregular surface structures such as spots (Heyrovský & Sasselov 2000; Han et al. 2000; Hendry et al. 2002; Rattenbury et al. 2002).

Second, it is possible to measure the Einstein radius of the lens and the relative lens-source proper motion. The light curve at the moment of the entrance (exit) of the lens into (from) the source surface exhibits inflection of the curvature. The duration of the passage over the source as measured by the interval between the entrance and exit of the lens over the surface of the source star is

$$\begin{equation} \Delta t_{\rm T} = 2\sqrt{{\rho _\star }^2-{u_0}^2}\ t_{\rm E}, \end{equation} \tag{ 3 }$$

where ρ_⋆ is the source radius in units of θ_E (normalized source radius), u₀ is the lens-source separation normalized by θ_E at the moment of the closest approach (impact parameter), and t_E is the timescale for the lens to transit θ_E (Einstein timescale). The impact parameter and the Einstein timescale are measured from the overall shape of the light curve and the duration of the event. With the known u₀ and t_E combined with the measured duration of the passage over the source, the normalized source radius is measured from relation (3). With the addition of information about the angular source size, θ_⋆, the Einstein radius and the lens-source proper motion are measured as θ_E = θ_⋆/ρ_⋆ and μ = θ_E/t_E, respectively. For general lensing events, the Einstein timescale is the only measurable quantity related to the physical parameters of the lens. However, the timescale results from the combination of three physical parameters: the mass of the lens, M, the distance to the lens, D_L, and the lens-source transverse speed, v, and thus the information about the lens is highly degenerate. The Einstein radius, on the other hand, does not depend on v and thus the physical parameters of the lens can be better constrained. For a fraction of events with long timescales, it is possible to additionally measure the lens parallax, π_E = π_rel/θ_E, from the deviation of the light curve induced by the orbital motion of the Earth around the Sun. With the Einstein radius and the lens parallax measured, the physical parameters of the lens are uniquely determined (Gould 1997).

Third, high-magnification events are sensitive to planetary companions of lenses. This is because a planet induces a small caustic near the primary lens and a high-magnification event, resulting from the source trajectory passing close to the primary, has a high chance of producing signals that indicate the existence of the planet (Griest & Safizadeh 1998). For an event with a lens passing over a source star, the planetary signal is weakened by the finite-source effect (Bennett & Rhie 1996). Nevertheless, two of the microlensing planets were discovered through this channel: MOA-2007-BLG-400 (Dong et al. 2009) and MOA-2008-BLG-310 (Janczak et al. 2010).

Fourth, high-magnification events provide a chance to spectroscopically study remote Galactic bulge stars. Most stars in the Galactic bulge are too faint for spectroscopic observations even with large telescopes. However, the enhanced brightness of lensed stars during high-magnification events allows spectroscopic observation to be possible, enabling the population study of Galactic bulge stars (Johnson et al. 2008; Bensby et al. 2009, 2011; Cohen et al. 2009; Epstein et al. 2010).

In this work, we present integrated analysis results for 14 high-magnification events with lenses passing over source stars that have been detected since 2004. Among them, eight events were newly analyzed and one event was reanalyzed with additional data.

The sample of events in our analysis is selected under the definition of a single-lens event where the lens-source separation at the time of the peak magnification is less than the radius of the source star, i.e., u₀ < ρ_*, and thus the lens passes over the surface of the source star. To obtain a sample of events, we begin by searching for high-magnification events that have been detected since 2004. Events with lenses passing over source stars can usually be distinguished by the characteristic features of their light curves near the peak. These features are the inflection of the curvature at the moment when the finite source first touches and completely leaves the lens and the round shape of the light curve during the passage of the lens over the source. To be more objective than visual inspection, we conduct modeling of all high-magnification events with peak magnifications of A_P ⩾ 10 to judge the qualification of events. From these searches, we find that 18 such events exist. Among them, analysis results of 12 events have not been published before. We learn that four unpublished events, MOA-2006-BLG-130/OGLE-2006-BLG-437 (J. Baudry et al. 2012, in preparation), MOA-2009-BLG-411 (P. Fouque et al. 2012, in preparation), MOA-2010-BLG-523 (A. Gould et al. 2012, in preparation), and MOA-2010-BLG-311 (L.-W. Hung et al. 2012, in preparation), are under analysis by other researchers and thus are excluded in our analysis. We note that four known source-crossing events detected before 2004 exist, including MACHO Alert 95-30 (Alcock et al. 1997), OGLE sc26_2218 (Smith 2003), OGLE-2003-BLG-238 (Jiang et al. 2004), and OGLE-2003-BLG-262 (Yoo et al. 2004). We also note that MOA-2007-BLG-400 (Dong et al. 2009) and MOA-2008-BLG-310 (Janczak et al. 2010) exhibit characteristic features of source-crossing single-lens events, but we exclude them from the sample because the lenses of the events turned out to have planetary companions.

In this work, we conduct analyses of nine events. Among them, eight events are newly analyzed in this work. These events include MOA-2007-BLG-176, MOA-2007-BLG-233/OGLE-2007-BLG-302, MOA-2009-BLG-174, MOA-2010-BLG-436, MOA-2011-BLG-093, MOA-2011-BLG-274, OGLE-2011-BLG-0990/MOA-2011-BLG-300, and OGLE-2011-BLG-1101/MOA-BLG-2011-325. For OGLE-2004-BLG-254, which was analyzed before by Cassan et al. (2006), we conduct additional analysis by adding more data sets taken from CTIO and FCO.⁷⁵ In Table 1, we summarize the status of the analysis for all 18 events that have been detected since 2004.

For almost all events analyzed in this work, the source-crossing part of the light curve was densely covered. This was possible due to the coordinated work of the survey and follow-up observations. Survey groups issued alerts of events. For a fraction of the events with high magnifications, additional alerts were issued. In other cases, follow-up teams issued high-magnification alerts independently. The peak time of a high-magnification event was predicted by real-time modeling based on the rising part of the light curve. Finally, the peak was densely covered by many telescopes that were prepared for follow-up observations at the predicted time of the peak. For MOA-2010-BLG-436, the rising part of the light curve was not covered by survey observations due to the short timescale of the event and thus no alert was issued. Nevertheless, the event was positioned in a high frequency field of the MOA survey and thus the peak was covered densely enough to be confirmed as an event with the lens passing over the source.

Table 2 shows the observatories of the telescopes that were used for observations of the individual events along with the observed passbands (marked as subscripts after the observatory names) and the numbers of data points (values in parentheses). Also marked are the coordinates (R.A., decl.) of the events. Survey observations were conducted by MOA and OGLE groups using the 1.8 m telescope at Mt. John Observatory in New Zealand and the 1.3 m Warsaw University telescope at Las Campanas Observatory in Chile, respectively. Follow-up observations were carried out by the μFUN, PLANET, RoboNet, and MiNDSTEp groups using 22 telescopes located in eight different countries. These telescopes include 1.3 m SMARTS CTIO, 0.4 m CAO in Chile, 0.4 m Auckland, 0.4 m FCO, 0.4 m Possum, 0.4 m Kumeu, 0.4 m VLO in New Zealand, 1.0 m Lemmon in Arizona, USA, 0.4 m Bronberg in South Africa, 0.6 m Pico dos Dias in Brazil, 0.25 m Craigie, 0.3 m PEST in Australia, 0.28 m SSO in French Polynesia, 1.0 m SAAO, 1.5 m Boyden in South Africa, 1.0 m Canopus, 0.6 m Perth in Australia, 1.5 m Steward in Arizona, USA, 2.0 m FTN in Hawaii, USA, 2.0 m FTS in Australia, 2.0 m LT in La Palma, Spain, and 1.54 m Danish in La Silla, Chile.

Table 2. Observatories

Event	MOA	OGLE	μFUN	PLANET	RoboNet
(R.A., Decl.)J2000					/MiNDSTEp
OGLE-2004-BLG-254		LCOI	CTIOI, V (39/5)	BoydenI (74)
($17^{\rm h}56^{\rm m}36^{\rm s}\hspace{-2.0pt}.20$,$-32^{\circ }33^{\prime }01^{\prime \prime }\hspace{-2.0pt}.80$)		(377)	FCON (129)	CanopusI (59)
				SAAOI (112)
MOA-2007-BLG-176	Mt.JohnR		AucklandR (68)	CanopusI (26)	DanishI (2)
($18^{\rm h}05^{\rm m}00^{\rm s}\hspace{-2.0pt}.41$,$-25^{\circ }47^{\prime }03^{\prime \prime }\hspace{-2.0pt}.69$)	(1388)		CTIOI, V (41/4)	StewardI (4)
			FCON (33)
			LemmonI (66)
			VLON (129)
MOA-2007-BLG-233	Mt.JohnR	LCOI	CTIOI, V (80/5)	CanopusI, V (60/5)	DanishI (125)
/OGLE-2007-BLG-302	(645)	(628)	FCON (23)	PerthI (23)
($17^{\rm h}54^{\rm m}14^{\rm s}\hspace{-2.0pt}.86$,$-31^{\circ }11^{\prime }02^{\prime \prime }\hspace{-2.0pt}.65$)			LemmonI (19)	SAAOI (80)
			SSON (80)
MOA-2009-BLG-174	Mt.JohnR		BronbergN (147)	CanopusI (40)	LTR (7)
($18^{\rm h}02^{\rm m}07^{\rm s}\hspace{-2.0pt}.60$,$-31^{\circ }25^{\prime }24^{\prime \prime }\hspace{-2.0pt}.20$)	(2189)		CAON (111)
			CraigieN (130)
			CTIOI, V (286/7)
			KumeuN (90)
			PossumN (60)
MOA-2010-BLG-436	Mt.JohnR			SAAOI, V (14/3)	FTSI (3)
($18^{\rm h}03^{\rm m}21^{\rm s}\hspace{-2.0pt}.68$,$-27^{\circ }38^{\prime }10^{\prime \prime }\hspace{-2.0pt}.74$)	(2581)
MOA-2011-BLG-093	Mt.JohnR	LCOI	CTIOI, V (76/21)	CanopusI (254)	FTNI (3)
($17^{\rm h}46^{\rm m}17^{\rm s}\hspace{-2.0pt}.83$,$-34^{\circ }20^{\prime }24^{\prime \prime }\hspace{-2.0pt}.76$)	(2247)	(292)	PESTN (124)		FTSI (19)
MOA-2011-BLG-274	Mt.JohnR	LCOI	AucklandR (53)
($17^{\rm h}54^{\rm m}42^{\rm s}\hspace{-2.0pt}.34$,$-28^{\circ }54^{\prime }59^{\prime \prime }\hspace{-2.0pt}.26$)	(3447)	(76)	CTIOI (4)
			FCON (16)
			KumeuR (49)
			PESTN (15)
OGLE-2011-BLG-0990	Mt.JohnR	LCOI	OPDI (275)	CanopusI (10)
/MOA-2011-BLG-300	(1708)	(3434)	PossumR (23)	SAAOI, V (95/6)
($17^{\rm h}51^{\rm m}30^{\rm s}\hspace{-2.0pt}.29$,$-30^{\circ }17^{\prime }47^{\prime \prime }\hspace{-2.0pt}.60$)
OGLE-2011-BLG-1101	Mt.JohnR	LCOI	AucklandR (60)	CanopusI (98)	FTNI (65)
/MOA-2011-BLG-325	(609)	(192)	CTIOI, V (126/12)		FTSI (145)
($18^{\rm h}03^{\rm m}31^{\rm s}\hspace{-2.0pt}.62$,$-26^{\circ }20^{\prime }39^{\prime \prime }\hspace{-2.0pt}.50$)			PossumR (24)		LTI (27)
			SSON (107)
			VLON (113)

Notes. Mt. John: Mt. John Observatory, New Zealand; LCO: Las Campanas Observatory, Chile; Auckland: Auckland Observatory, New Zealand; Bronberg: Bronberg Observatory, South Africa; CAO: CAO San Pedro Observatory, Chile; Cragie: Craigie Observatory, Australia; CTIO: Cerro Tololo Inter-American Observatory, Chile; FCO: Farm Cove Observatory, New Zealand; Kumeu: Kumeu Observatory, New Zealand; Lemmon: Mt Lemmon Observatory, Arizona, USA; OPD: Observatorio do Pico dos Dias, Brazil; PEST: Perth Exoplanet Survey Telescope, Australia; Possum: Possum Observatory, New Zealand; SSO: Southern Stars Observatory, French Polynesia; VLO: Vintage Lane Observatory, New Zealand; Boyden: Boyden Observatory, South Africa; Canopus: Canopus Hill Observatory, Tasmania, Australia; Perth: Perth Observatory, Australia; SAAO: South African Astronomical Observatory, South Africa; Steward: Steward Observatory, Arizona, USA; FTN: Faulkes North, Hawaii; FTS: Faulkes South, Australia; LT: Liverpool Telescope, La Palma, Spain; Danish: Danish Telescope, European Southern Observatory, La Silla, Chile. The subscription after each observatory represents the filter used for observation and the value in parenthesis is the number of data points. The filter "N" denotes that no filter is used.

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The reduction of the data was conducted by using photometry codes that were developed by individual groups. The MOA and OGLE data were reduced by photometry codes developed by Bond et al. (2001) and Udalski (2003), respectively, which are based on the Difference Image Analysis method (Alard & Lupton 1998). The μFUN data were processed using a DoPHOT pipeline (Schechter et al. 1993). For PLANET and MiNDSTEp data, a pySIS pipeline (Albrow et al. 2009) is used. For RoboNet data, a DanDIA pipeline (Bramich 2008) is used.

The error bars estimated from the different observatories are rescaled so that χ² per degree of freedom becomes unity for the data set of each observatory where χ² is computed based on the best-fit model. According to this simple scheme, however, we find a systematic tendency for some data sets with error bars near the peak of a light curve to be overestimated. We find that this is caused by the inclusion of redundant data at the baseline in the error normalization. In this case, the data at the baseline greatly outnumber accurate data points near the peak and thus error-bar normalization is mostly dominated by the baseline data. To minimize this systematic tendency, we restrict the range of data for error normalization not to be too wide so that error estimation is not dominated by data at the baseline, but not to be too narrow so that lensing parameters can be measured accurately. For the final data set used for modeling, we eliminate data points lying beyond 3σ from the best-fit model.

In Figures 1–9, we present the light curves of the individual events. In each figure, the lower two panels show the overall shape of the light curve and residual, and the upper two panels show the enlargement of the peak region of the light curve and residual. For each figure, we mark the moments of the lens' entrance and exit of the source by two dotted vertical lines. Also marked is the peak source magnification. We note that the same color of data points is used for each observatory throughout the light curves, and colors of data points are chosen to match those of the labels of the observatories. We note that the magnitude scale corresponds to one of the observatories in the list, while data from the other observatories have adjusted blends and are vertically shifted to match the first light curve. The choice of reference is based on data from the survey observation, i.e., OGLE and MOA data. If both OGLE and MOA data are available, the OGLE data are used for reference.

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Modeling the light curve of each event is conducted by searching for a set of lensing parameters that best describes the observed light curve. For all events, the light curves appear to have a standard form except the peak region, and thus we start with a simple single-lens modeling. The light curve of a standard single-lensing event is characterized by three parameters: the time of the closest lens-source approach, t₀, the lens-source separation at that moment, u₀, and the Einstein timescale, t_E. Based on the initial solution, we refine this solution by considering additional second-order effects.

In order to precisely describe the peak region of the light curve of an event with a lens passing over a source, additional parameters are needed to describe the deviation caused by the finite-source effect. To the first-order approximation, the finite-source effect is described by the normalized source radius, ρ_⋆. For a more refined description of the deviation, additional parameters of the limb-darkening coefficients, u_λ, are needed to account for the variation of the deviation caused by the brightness profile of the source star surface. With the coefficients, the limb-darkening profile is modeled by the standard linear law:

$$\begin{equation} I=I_0{\left[1-u_\lambda \left(1-{\rm cos}\phi \right)\right]}, \end{equation} \tag{ 4 }$$

where I₀ is the intensity of light at the center of the stellar disk and ϕ is the angle between the normal and stellar surface and the line of sight toward the observer.

For an event with a timescale comparable to the orbital period of the Earth, the position of the observer changes by Earth's orbital motion during the event and the resulting light curve deviates from a symmetric standard form. This parallax effect is described by two parameters of π_{E, N} and π_{E, E} that represent the two components of the lens parallax vector $\mbox{\boldmath $\pi $}_{\rm E}$ projected on the sky in the north and east equatorial coordinates, respectively. The direction of the parallax vector corresponds to the lens-source relative motion in Earth's frame. The size of the parallax vector corresponds to the ratio of Earth's orbit, i.e., 1 AU, to the Einstein radius projected on the observer's plane.

For a high-magnification event, the peak of the light curve can exhibit additional deviations if the lens has a companion. For a planetary companion located near the Einstein ring of the primary lens or a binary companion with a separation from the primary substantially smaller or larger than the Einstein radius, a small caustic is induced near the location of the primary lens. Then, the source trajectory of a high-magnification event passing close to the primary lens has a high chance of approaching the caustic, resulting in a perturbation near the peak of the light curve. A description of the perturbation induced by a lens companion requires three additional parameters of the mass ratio between the binary lens components, q, the projected separation in units of the Einstein radius, s, and the angle between the source trajectory and the binary axis, α.

For each event, we search for a solution of the best-fit lensing parameters by minimizing χ² in the parameter space. For the χ² minimization, we use a Markov Chain Monte Carlo (MCMC) method. We compute finite magnifications by using the ray-shooting technique (Schneider & Weiss 1986; Kayser et al. 1986; Wambsganss 1997). In this method, rays are uniformly shot from the image plane, bent according to the lens equation, and land on the source plane. Then, a finite magnification is computed by comparing the number densities of rays on the image and source planes. Precise computation of finite magnifications by using this numerical technique requires a large number of rays and thus demands heavy computation. To minimize computation, we limit finite-magnification computation by using the ray-shooting method only when the lens is close to the source. Once a solution of the parameters is found, we estimate the uncertainties of the individual parameters based on the chain of solutions obtained from MCMC runs.

In Table 3, we present the lensing parameters of the best-fit solutions of the individual events determined from modeling. To provide integrated results of events with lenses passing over source stars, we also provide solutions of events that were previously analyzed. For OGLE-2004-BLG-254, we provide solutions to both the previous analysis and this work for comparison.

For all events analyzed in this work, we are able to measure the limb-darkening coefficients of source stars. In Table 4, we present the measured limb-darkening coefficients. We measure the coefficients corresponding to the individual data sets covering the peak of each light curve instead of the individual passbands. This is because the characteristics of filters used for different telescopes are different from one another even though they are denoted by a single representative band. Thus the joint fitting of data measured in different filter systems may result in erroneous measurements of limb-darkening coefficients (Fouque et al. 2010). To compare with the theoretical values, we also provide values of coefficients predicted by Claret (2000) for the Bessell V, R, and I filters. Also provided are the source types and the adopted values of log g and T_eff where the typical uncertainties of the surface gravity and the effective temperature are Δ(log g) = 0.5 and ΔT_eff = 250 K, respectively. We adopt a solar metallicity. We note that the measured coefficients are generally in good agreement with theoretical values, u_th. From the table, it is found that for 23 out of the total 29 measurements, the measured coefficients are within the 20% range of the fractional difference as measured by f_u = (u − u_th)/u_th. The cases with large differences where f_u > 20% include u_I(Canopus) for OGLE-2004-BLG-254, u_R(MOA), u_R(Auckland), and u_I(Lemmon) for MOA-2007-BLG-176, u_I(CTIO) for MOA-2009-BLG-174, and u_I(CTIO) for OGLE-2011-BLG-1101/MOA-2011-BLG-325. By inspecting individual data points on the light curves, we find that the major reasons for the large differences between the measured and theoretical values are due to poor coverage [u_I(Canopus) for OGLE-2004-BLG-254, u_R(MOA, Auckland) and u_I(Lemmon) for MOA-2007-BLG-176, u_I(CTIO) for MOA-2009-BLG-174] or poor data quality [u_I(CTIO) for OGLE-2011-BLG-1101/MOA-2011-BLG-325]. Other possible reasons for differences from the predicted values include differences of individual filters from the standard Bessel filters, as well as differences in the method to compute the theoretical values (Heyrovský 2007).

The source type of each event is determined based on the location of the source in the color–magnitude diagram (CMD) of stars in the same field. CMDs are obtained from CTIO images taken in V and I bands. To locate the lensed star in the CMD, it is required to measure the fraction of blended light in the observed light curve. This is done by including a blending parameter in the process of light curve modeling. For MOA-2011-BLG-274, a CMD taken from CTIO is available but images were taken after the event and thus we could not determine the source color and magnitude by the usual method. Instead we employed the method of Gould et al. (2010). In this method, we first measure the source instrumental magnitudes by fitting the OGLE (I_OGLE) and PEST (unfiltered, N_PEST) data to the light curve model. We then align each of these data sets to CTIO (V/I) using comparison stars, which effectively transforms N_PEST/I_OGLE to (V/I)_CTIO. In Figure 10, we present the CMDs of stars in the fields of the individual events and the locations of source stars.⁷⁶ For MOA-2010-BLG-436 and OGLE-2011-BLG-0990/MOA-2011-BLG-300, there exist SAAO data taken in I and V bands, but the number and quality of V-band data are not numerous or good enough to specify the source type.

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In Table 5, we present the measured Einstein radii. The Einstein radius of each event is determined from the angular source radius, θ_⋆, and the normalized source radius, ρ_⋆, as θ_E = θ_⋆/ρ_⋆. The normalized source radius is measured from modeling. To measure the angular source radius, we use the method of Yoo et al. (2004), where the de-reddened V − I color is measured from the location of the source in the CMD, V − I is converted into V − K using the relation of Bessel & Brett (1988), and then the angular source radius is inferred from the V − K color and the surface brightness relation given by Kervella et al. (2004). In this process, we use the centroid of bulge clump giants as a reference for the calibration of the color and brightness of a source with the assumption that the source and clump giants experience the same amount of extinction and reddening. We note that no CMD is available for MOA-2010-BLG-436 and OGLE-2011-BLG-0990/MOA-2011-BLG-300 and thus the Einstein radius is not provided. Also provided in Table 5 are the relative lens-source proper motions as measured by μ = θ_E/t_E.

We note that the measured Einstein radii of some events are substantially smaller than a typical value. These events include OGLE-2004-BLG-254 (θ_E ∼ 0.14 mas), MOA-2007-BLG-176 (∼0.14 mas), MOA-2007-BLG-233/OGLE-2007-BLG-302 (∼0.17 mas), MOA-2011-BLG-093 (∼0.07 mas), and MOA-2011-BLG-274 (∼0.08 mas). The lens mass and distance are related to the Einstein radius by

$$\begin{equation} M = 0.019 \,M_\odot \biggl ({D_{\rm S}\over 8 {\rm \,kpc}}\biggr) \biggl ({D_{\rm L}\over {D_{\rm S} - D_{\rm L}}}\biggr) \biggl ({\theta _{\rm E}\over 0.14 \,{\rm mas}}\biggr)^2. \end{equation} \tag{ 5 }$$

Hence, the small θ_E of these events implies that the lenses are either very close to the source or very low mass objects. Most of these events have proper motions that are typical of bulge lenses (2–7 mas yr⁻¹) and so may be quite close to the source (see Table 5). But MOA-2011-BLG-274 has a substantially higher proper motion, μ ∼ 11 mas yr⁻¹. It is therefore a good candidate to be classified as a substellar object or even a free-floating planet (Sumi et al. 2011). Because of its high proper motion, it should be possible to detect the lens within a few years using high-resolution infrared imaging, provided it is luminous. In this case a null result would confirm its substellar nature.

For MOA-2009-BLG-174, the lens parallax is measured as Δχ² ∼ 16.2. The measured parallax parameters are

$$\begin{equation} \pi _{{\rm E},\parallel }=-0.049\pm 0.006 ;\qquad \pi _{{\rm E},\perp }=0.038\pm 0.065, \end{equation} \tag{ 6 }$$

where π_{E, ∥} and π_{E, ⊥} are the components of the lens parallax vector that are parallel with and perpendicular to the projected position of the Sun. These values correspond to the standard parallax components of (π_{E, N}, π_{E, E}) = (0.025 ± 0.052, −0.057 ± 0.028). In Figure 11, we present contours of χ² in the space of the parallax parameters. Combined with the measured Einstein radius, the physical parameters of the lens are uniquely determined as

$$\begin{equation} M={{\theta _{\rm E}}\over {\kappa \pi _{\rm E}}}=0.84\pm 0.37\ M_{\odot }, \end{equation} \tag{ 7 }$$

and

$$\begin{equation} D_{\rm L}={{\rm AU}\over {\pi _{\rm E}\theta _{\rm E}+\pi _{\rm S}}} =6.39\pm 1.11\ {\rm kpc}, \end{equation} \tag{ 8 }$$

respectively. We find that the measured lens mass is consistent with the de-reddened color of blended light (V − I)_{0, b} ∼ 1.4, which approximately corresponds to the color of an early K-type main-sequence star with a mass equivalent to the estimated lens mass, suggesting that the blend is very likely to be the lens. We mark the position of the blend in the corresponding CMD in Figure 10.

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A high-magnification event is an important target in a planet search due to its high efficiency of planetary perturbations. Unfortunately, we find no statistically significant deviations from the single-lens fit for any of the events analyzed in this work. However, it is still possible to place limits on the range of the planetary separation and mass ratio. For this purpose, we construct the so-called exclusion diagrams, which show the confidence levels of excluding the existence of a planet as a function of the normalized star–planet separation and the planet/star mass ratio. We construct diagrams by adopting the Gaudi & Sackett (2000) method. In this method, binary models are fitted to the observed data with the three binary parameters (s, q, α) that are held fixed as grid parameters, while other lensing parameters are allowed to vary to minimize χ². Then, the confidence level of the exclusion for planets with s and q is estimated as the fraction of binary models not consistent with the best-fit single-lens model among all tested models with various values of α. For fitting binary models, it is required to produce many light curves with finite magnifications. We produce light curves by using the "map-making method" (Dong et al. 2006), where a magnification map for a given s and q is constructed and light curves with various source trajectories are produced based on the map. In Figure 12, we present the obtained exclusion diagrams for all analyzed events. Here we adopt a threshold for planet detection at Δχ²_th = χ²_s − χ²_p = 200, where χ²_p and χ²_s represent the χ² values for the best-fit planetary and single-lens models, respectively. For most events, the constraint on the excluded parameter space is not strong mainly due to the severe finite-source effect. However, the constraint is strong for MOA-2009-BLG-174 because of the small source size (ρ_⋆ ∼ 0.002) and dense coverage of the peak.

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We provide integrated analysis results for 14 high-magnification lensing events with lenses passing over the surface of source stars that have been detected since 2004. Among them, eight events are newly analyzed in this work. The newly analyzed events are MOA-2007-BLG-176, MOA-2007-BLG-233/OGLE-2007-BLG-302, MOA-2009-BLG-174, MOA-2010-BLG-436, MOA-2011-BLG-093, MOA-2011-BLG-274, OGLE-2011-BLG-0990/MOA-2011-BLG-300, and OGLE-2011-BLG-1101/MOA-2011-BLG-325. Information about the lenses and lensed stars obtained from the analysis is summarized as follows.

• 1.  
  For all newly analyzed events, we measure the linear limb-darkening coefficients of the surface brightness profile of the source stars.
• 2.  
  For all events with available CMDs of field stars, we measure the Einstein radii and the lens-source proper motions. Among them, five events (OGLE-2004-BLG-254, MOA-2007-BLG-176, MOA-2007-BLG-233/OGLE-2007-BLG-302, MOA-2011-BLG-093, and MOA-2011-BLG-274) are found to have Einstein radii of less than 0.2 mas, making the lenses of the events candidates for very low mass stars or brown dwarfs.
• 3.  
  The measured timescale t_E ∼ 2.7 days combined with the small Einstein radius of ∼0.08 mas for the event MOA-2011-BLG-274 suggests the possibility that the lens is a free-floating planet.
• 4.  
  For MOA-2009-BLG-174, we additionally measure the lens parallax and thus uniquely determine the physical parameters of the lens. The measured lens mass of ∼0.8 M_☉ is consistent with that of a star blended with the source, suggesting the possibility that the blend comes from the lens.
• 5.  
  We find no statistically significant planetary signals for any of the events analyzed in this work. However, it is still possible to place constraints on the range of the planetary separation and mass ratio. For this purpose, we provide exclusion diagrams showing the confidence levels of excluding the existence of a planet as a function of the separation and mass ratio.

Work by C.H. was supported by the Creative Research Initiative Program (2009-0081561) of the National Research Foundation of Korea. The MOA experiment was supported by JSPS17340074, JSPS18253002, JSPS20340052, JSPS22403003, and JSPS23340064. The OGLE project has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. 246678. Work by B.S.G. and A.G. was supported in part by NSF grant AST-1103471. Work by B.S.G., A.G., R.W.P., and J.C.Y. was supported in part by NASA grant NNX08AF40G. Work by J.C.Y. was supported by a National Science Foundation Graduate Research Fellowship under grant No. 2009068160. C.B.H. acknowledges the support of NSF Graduate Research Fellowship 2011082275. T.S. was supported by the grants JSPS18749004, MEXT19015005, and JSPS20740104. F.F., D.R., and J.S. were supported by the Communauté française de Belgique–Actions de recherche concertées–Académie universitaire Wallonie-Europe.