Observations of Microlensed Images with Dual-field Interferometry: On-sky Demonstration and Prospects

In his seminal paper on gravitational microlensing, A. Einstein (1936) realized that the angular separation between the two images of the source star created by a Galactic gravitational lens would be well below the resolution limits of contemporary telescopes. That led him to believe that observing microlensed images would be, in practice, impossible. This notion persisted for many decades until the first optical interferometry facilities were developed.

In point-source point-lens microlensing events, the gravitational lens creates two images of the source star (called the major and minor image). They are separated by approximately 2θ_E, where is the angular Einstein radius. Here, M is the lens mass, π_rel is the lens–source relative parallax, and is a constant. For typical configurations of stellar-mass microlensing events in the Milky Way, θ_E ~ 1 mas, and so the expected separation between the minor and major images is usually smaller than a few milliarcseconds. Interferometric observations of microlensing events, therefore, provide a direct way to precisely measure angular Einstein radii by resolving the images (and, as a consequence, measure the masses of lensing objects).

Although a few tens of thousands of single-lens microlensing events have been discovered so far, only a few have had the angular Einstein radius measured. That was possible for only a small number of events exhibiting finite-source effects (A. Gould 1994; R. J. Nemiroff & W. A. D. T. Wickramasinghe 1994; H. J. Witt & S. Mao 1994), and the probability of such measurements is biased toward low-mass (planetary-mass) lenses. Another route for θ_E measurements has involved astrometric observations (E. Hog et al. 1995; M. Miyamoto & Y. Yoshii 1995; M. A. Walker 1995), leading to the first detection of an isolated black hole (C. Lam et al. 2022; P. Mróz et al. 2022; K. C. Sahu et al. 2022; C. Y. Lam & J. R. Lu 2023). These observations, however, required long-term astrometric monitoring, which is not feasible for a large sample of events. Conversely, interferometric observations are much more efficient, as only one exposure is sufficient to resolve the microlensed images and measure the angular Einstein radius. When combined with the microlensing parallax (π_E ≡ π_rel/θ_E) measurements from the light curve, they allow us to precisely determine the masses, distances, and transverse velocities of isolated objects, including neutron stars and black holes.

F. Delplancke et al. (2001) were the first to discuss the prospects of using the European Southern Observatory Very Large Telescope Interferometer (VLTI) to study gravitational microlensing events. They estimated that the first-generation VLTI instruments would be able to observe dozens of events every year in the mid 2000s. Yet the first successful resolution of microlensed images did not materialize until late 2017, when S. Dong et al. (2019) observed the microlensing event TCP J05074264+2447555 (aka Kojima-1) with the second-generation VLTI instrument GRAVITY (GRAVITY Collaboration et al. 2017).

Why did it take almost two decades to achieve this milestone? The early predictions of VLTI performance underestimated the role and impact of many practical problems encountered during system operations, such as beam stability, vibrations, and air dispersion (e.g., F. Delplancke 2008). Moreover, even when these technical problems were solved or mitigated, the performance of a ground-based interferometer is ultimately limited by atmospheric turbulence, which breaks the coherence of the wave fronts arriving at each telescope. Therefore, the observations must be carried out with exposure times ( ~1 ms) shorter than the typical wave-front-coherence timescale (on the order of 20–30 ms) and shorter than the mechanical vibrations timescale, which dominate in the 10–1000 Hz regime. Even for the largest modern 10 m class telescopes, this requirement sets a limiting magnitude of K ≈ 10 for interferometric observations. Gravitational microlensing events brighter than this limit are exceedingly rare (Figure 1). They occur at most a few times a year, and they exceed the GRAVITY limiting magnitude for a short period of time. The three events with published interferometric observations were unusually bright: TCP J05074264+2447555 (S. Dong et al. 2019; W. Zang et al. 2020), ASASSN-22av (Z. Wu et al. 2024a), and Gaia19bld (E. Bachelet et al. 2022; A. Cassan et al. 2022; K. A. Rybicki et al. 2022) reached V ≈ 11.5, , and I ≈ 9.0, respectively. The former two were observed using the standard on-axis mode of GRAVITY, whose fringe tracker allows minute-long science exposures (S. Lacour et al. 2019) that significantly enhance the VLTI sensitivity (F. Eisenhauer et al. 2023).

Zoom In Zoom Out Reset image size

The dual-field interferometric observations were designed to alleviate this problem and reach fainter sources (e.g., M. Shao & M. M. Colavita 1992; M. M. Colavita et al. 1999; F. Delplancke 2008; J. Woillez et al. 2014). In this observing mode, the target and a nearby bright (K ≲ 10) reference (fringe-tracking) star are observed at the same time. The fringe-tracking star is needed to measure the atmospheric turbulence and stabilize the optical path difference between the telescopes, providing a correction to the science target. That enables observation of the target with exposure times longer than the turbulence and vibration timescales, allowing one to reach fainter objects. The angular separation between the science object and the fringe-tracking star must be smaller than 20″–30″, which is set by the atmospheric conditions. In 2019–2022, the GRAVITY instrument was upgraded to enable such dual-field interferometric observations (GRAVITY+ Collaboration et al. 2022), hence the name of this observing mode, GRAVITY Wide.

The upgraded instrument opened up an entirely new pathway for characterizing and studying microlensing events because, in the dense fields of the Galactic bulge, the probability of finding a suitable (K ≲ 10, within 30″) fringe-tracking star is as large as 50%–70%. According to the GRAVITY manual (Issue 110),²⁹ the GRAVITY Wide observations are possible for science targets as faint as K_SC ≈ 16–17. However, the limiting magnitude strongly depends on the atmospheric conditions and the separation between the science object and the fringe-tracking star. In addition, closure-phase observations lose the signal-to-noise ratio faster than normal visibility data as the targets become fainter. Thus, in practice, we rarely considered observing targets fainter than K ≈ 13.5–14, which still left us with plenty of candidate events to scrutinize (Figure 1).

As soon as ESO offered regular GRAVITY Wide observations from October 2022 (“Period 110”), we initiated a program (PI: A. Mérand) of interferometric observations of microlensing events. Our primary science goal was to detect and measure precise masses, distances, and transverse velocities of isolated stellar remnants—neutron stars and black holes. Our observations also served as a test bed for verifying the capabilities of the new instrument and planning, executing, and analyzing interferometric follow-up observations of a large number of microlensing events.

This paper presents the first resolution of microlensed images with GRAVITY Wide in the microlensing event OGLE-2023-BLG-0061/KMT-2023-BLG-0496.

The VLTI observations can be carried out with either four 8.2 m Unit Telescopes (UTs) or four 1.8 m Auxiliary Telescopes (ATs). The VLTI UT observing runs are organized every month (around the full Moon) and typically last one week. During the rest of the month, observations are possible with ATs, which may be relocated to more than 10 observing stations.³⁰ GRAVITY Wide observations may be conducted with ATs, provided that the fringe-tracking star is brighter than K_FT ≈ 9–9.5 and the science target is brighter than K_SC ≈ 13–14. For UTs, the limiting magnitudes are larger—K_FT ≈ 10–10.5 and K_SC ≈ 16–17, respectively. As discussed above, the limiting magnitudes of science targets strongly depend on the atmospheric conditions (isoplanatic angle) and the separation of the fringe-tracking star.

We began the selection of microlensing targets about 7–10 days before the planned start of each VLTI UT run. The candidates were chosen from publicly available lists of microlensing alerts published by the Optical Gravitational Lensing Experiment (OGLE) Early Warning System³¹ (EWS; A. Udalski 2003; A. Udalski et al. 2015), the Korea Microlensing Telescope Network (KMTNet) Alert System³² (H.-W. Kim et al. 2018), and the Microlensing Observations in Astrophysics Transient Alerts.³³ In addition, we also checked the transient alerts published by all-sky surveys, such as Gaia (S. T. Hodgkin et al. 2013, 2021) and the All Sky Automated Survey for SuperNovae (ASAS-SN; B. J. Shappee et al. 2014).

The selection of targets was based on several scientific and technical criteria. First, we selected events near or past their maximum brightness, so that the parameters describing their light curves were reasonably well measured. However, the observations had to be secured before the event faded, before the minor image became too faint, and so the contrast ratio between the microlensed images became too large. We required the contrast ratio to be smaller than 10:1, which was conservatively adopted based on our experience with the GRAVITY data. That contrast ratio corresponds to the maximum lens–source separation (in Einstein radius units) of or, equivalently, the minimum amplitude of .

Because the primary scientific motivation of our project was searching for stellar remnants, which are expected to give rise to long-duration events, we selected events with Einstein timescales longer than t_E = 50 days. However, the nature of the lens cannot be known at the time of selecting the targets. The mass of the lens can be determined only after the interferometric data are combined with the full light curve. That usually means waiting several weeks after the interferometric observations are taken, because they are collected close to the maximum magnification.

In the next step, we checked if suitable fringe-tracking and adaptive optics reference stars were located within 30″ of the event. For possible fringe-tracking stars, we queried the Two Micron All Sky Survey (2MASS) Point Source Catalog (M. F. Skrutskie et al. 2006), while for adaptive optics guide stars, we queried Gaia Data Release 3 (Gaia Collaboration et al. 2016, 2023). The guide star had to be brighter than G ≈ 14 for the Multi Application Curvature Adaptive Optics (R. Arsenault et al. 2003) system or brighter than K = 8 for the Coudé Infrared Adaptive Optics (S. Kendrew et al. 2012) system. Finally, we estimated the expected K-band brightness of the event during the planned observations, by assuming that the blending parameter in the K band was identical to that in the I band. The baseline K-band brightness of the event was taken from the VISTA Variables in the Via Lactea (VVV) survey (D. Minniti et al. 2010), the United Kingdom Infra-Red Telescope Galactic Plane Survey (A. Lawrence et al. 2007; P. W. Lucas et al. 2008), or 2MASS (M. F. Skrutskie et al. 2006). We also examined the estimated K-band brightness of the source by using the best-fit I-band source fluxes as cross-checks. For such estimates, we assumed dereddened color (I − K)_0,giant = 1.4 and (I − K)_0,dwarf = 1.0 for sources roughly classified using the extinction-corrected source magnitudes as giants (I₀ < 16.5) and dwarfs (I₀ > 16.5), respectively.

Candidate targets were selected independently by two teams (P.M. and S.D.) and subsequently investigated in more detail. In particular, we paid special attention to the microlensing parallax measurements, which are necessary for the lens mass determination. Because the value of the microlensing parallax is inversely proportional to the square root of the lens mass, we required it to be consistent with zero (or close to zero) during the trigger. We also ran light-curve simulations to ensure that the parallax would be precisely measured (or constrained) by the end of the observing season. We did not consider events with variable source stars, because the variability may affect the microlensing parallax measurements.

The blue solid line in Figure 1 shows the cumulative distribution of the expected K-band peak magnitudes of the events detected by OGLE EWS in 2023. Only events with relatively long timescales (t_E ≥ 50 days) and suitable adaptive optics guide stars (G ≤ 14) within 30″ are presented. Only a few events were bright enough (K < 10.5) for standard GRAVITY on-axis observations. In contrast, the solid red line in Figure 1 shows the distribution of the K-band magnitudes of the brightest star within 30″ of the event (which may serve as a fringe-tracking star). Nearly 100 events could have been considered for GRAVITY Wide observations.

The detection of the microlensing event OGLE-2023-BLG-0061 was announced by the OGLE EWS (A. Udalski 2003; A. Udalski et al. 2015) on 2023 March 13.60 UT (). It was independently identified by the KMTNet Alert System (S.-L. Kim et al. 2016; H.-W. Kim et al. 2018) on 2023 April 20, and it was designated KMT-2023-BLG-0496. The event occurred on a bright red clump star (I = 16.389 ± 0.001, V −I =2.26 ± 0.02) with equatorial coordinates (R.A., decl.)_J2000 = (, ). According to data from the VVV survey, the source star had K = 13.515 ± 0.011 (D. Minniti et al. 2010).

3.1. Photometric Data

3.2. VLTI Data

OGLE-2023-BLG-0061/KMT-2023-BLG-0496 was considered a promising candidate for VLTI observations on 2023 July 20. Although the event was still before the peak, the available data predicted a relatively long timescale (t_E ≈ 100 days). At the same time, the optical brightness indicated that the microlens parallax should be robustly constrained. A nearby (132) bright (G = 13.7, K = 9.8) star, 2MASS 17430426-3515195, could serve as a fringe-tracking and adaptive optics guide star. The first GRAVITY Wide observations were secured with UTs on 2023 July 29 (“epoch 1”). We obtained four sets of medium-resolution observations (R = λ/Δλ ≈ 500) and one set of low-resolution data (R ≈ 22), a new mode never tried before with GRAVITY Wide. The first three medium-resolution exposures were of low quality, and the resulting closure phases were very noisy. We therefore decided not to use them in the subsequent analysis. The preliminary modeling carried out at that time indicated that the medium- and low-resolution data were inconsistent with each other.

We thus attempted to collect additional VLTI data to study the magnitude of the possible systematics affecting the low-resolution data. We triggered on-axis GRAVITY observations with ATs on 2023 August 23, when the event was approaching the peak (at magnification ~90), but the data could not be collected. The next (successful) attempt to gather additional VLTI UT observations took place on 2023 September 29 (“epoch 2”), when we secured six low-resolution exposures with GRAVITY Wide. Table 1 presents the log of VLTI observations and Figure 2 shows the coverage of the UV plane during both VLTI epochs. Each low-resolution observation consisted of 12 30 s exposures; the medium-resolution observation consisted of four 100 s exposures. For each night, a bright-star pair was observed to center the science fringe with the GRAVITY differential delay line before the microlens observation. We did not adjust the science fringe when the telescope was moved to the science target.

Table 1. Log of VLTI Observations

Epoch	Time	HJD	Resolution	Exposure Time
1a	2023-07-29 02:59:45 UT	154.625	Medium	4 × 100 s
1b	2023-07-29 03:21:42 UT	154.640	Low	12 × 30 s
2a	2023-09-28 23:30:05 UT	216.479	Low	12 × 30 s
2b	2023-09-28 23:36:41 UT	216.484	Low	12 × 30 s
2c	2023-09-28 23:43:20 UT	216.488	Low	12 × 30 s
2d	2023-09-28 23:57:23 UT	216.498	Low	12 × 30 s
2e	2023-09-29 00:03:56 UT	216.503	Low	12 × 30 s
2f	2023-09-29 00:10:29 UT	216.507	Low	12 × 30 s

Download table as:  ASCIITypeset image

Zoom In Zoom Out Reset image size

The data were reduced with the standard GRAVITY pipeline (version 1.4.2). We first used the Python script run_gravi_reduced.py to reduce the raw data and apply the pixel-to-visibility matrix (P2VM). The default options were used, except that we adopted –gravity_vis.output-phase-sc=SELF_VISPHI to calculate the internal differential phase between each spectral channel and –gravity_vis.opd-pupil-stddev-max-sc=9999 to ignore the poor pupil measurements in the acquisition camera, which do not affect our closure-phase measurements. The pipeline performed the bias and sky subtraction, flat-fielding, wavelength calibration, and spectral extraction. The application of the P2VM converts the pixel detector counts into complex visibilities, taking into account all instrumental effects, including relative throughput, coherence, phase shift, and crosstalk. The dark, bad-pixel, flat-field, wavelength calibration, and P2VM matrix data were reduced from the daily calibration data obtained close in time to our observations. We then used run_gravi_trend.py to calibrate the closure-phase data. For epoch 1, we used the star pair to center the science fringe and calibrate the medium-resolution data. Unfortunately, we did not have calibrator data observed in the low-resolution data on the same night. For epoch 2, we used a bright-star pair observed after the microlens observation for the calibration. The following analyses are based on the closure-phase data from the calibrated medium-resolution and uncalibrated low-resolution data from epoch 1 and the calibrated low-resolution data from epoch 2. In this way, we use all the data with good quality. Meanwhile, when modeling the low-resolution data from epoch 2, we found that the calibration makes little difference to the closure phase.

The light curve of the event can be well fitted by a standard point-source point-lens model with the annual parallax effect (which is caused by the orbital motion of the Earth). This model has five free parameters: the time of closest approach between the lens and the source t₀, their minimum separation (in Einstein radius units) u₀, the Einstein radius crossing timescale t_E, and the northern and eastern components of the microlensing parallax vector, π_E = (π_E,N, π_E,E). The latter is a vector quantity whose direction is parallel to the direction of the relative lens–source proper motion μ_rel. The magnification is calculated using the formula

where . The latter quantity is evaluated in the geocentric frame that is moving with a velocity equal to the Earth's velocity at t_0,par = 2460179 (A. Gould 2004) in which

and

where Δs is the projected position of the Sun. Two models with different signs of u₀ are possible; both models are almost perfectly degenerate, with the u₀ < 0 model being preferred by only Δχ² = 3.1. The best-fit model (with u₀ > 0) is presented in Figure 3. The best-fit parameters and their uncertainties are reported in Table 2. In Table 2, we also report the best-fit source magnitude I_s, baseline magnitude I₀, and dimensionless blending parameter f_s.

Zoom In Zoom Out Reset image size

Figure 4 shows the constraints on the microlensing parallax vector derived using data from different observatories. The blue, red, green, and orange contours mark the constraints from OGLE, KMTC, KMTA, and KMTS data, respectively, whereas the solid black contours mark the best-fit model to all data. The eastern component of π_E, which is parallel to the projected acceleration of the Sun, is relatively well measured in all data sets. However, the northern component π_E,N (perpendicular to the projected acceleration of the Sun) has a considerably larger uncertainty. The northern component is also more susceptible to noise in the data (A. Gould et al. 1994; M. C. Smith et al. 2003; A. Gould 2004); hence, slightly different values of π_E,N are determined using different data sets. The best-fit parameters for individual data sets are reported in Tables 3 and 4.

Zoom In Zoom Out Reset image size

The light-curve model of a microlensing event makes it possible to predict the brightness ratio η between the minor and major image at a given epoch of interferometric observations. Moreover, the detection of the microlensing parallax in the light curve allows us to predict the orientation of microlensed images in the sky. The position angle (PA; north through east) of the microlensing parallax vector can be calculated using the formula

The angle ϕ between the source–lens relative proper motion and source–lens relative position at a given time can also be calculated directly from the parameters of the light-curve model:

Note that in the limit of no parallax, this equation simplifies to , which is equivalent to Equation (8) derived by S. Dong et al. (2019). Then, following S. Dong et al. (2019), the PA of the minor image relative to the major image (north through east) is simply PA = Φ_π + ϕ if u₀ > 0 or PA = Φ_π − ϕ if u₀ < 0. These two cases depend on the trajectory of the lens relative to the source: u₀ > 0 if the lens passes the source on its right and u₀ < 0 on its left (J. Skowron et al. 2011). Note that S. Dong et al. (2019) introduced a different definition of the PA of the major image relative to the minor image: ψ = PA + π.

Table 2 presents the predicted lens–source separation u (in Einstein radius units), the flux ratio between the minor and major image η, and the PAs of the images for both epochs of VLTI observations. Note that the expected lens–source separation and flux ratio of the images are virtually identical for the u₀ > 0 and u₀ < 0 models. However, the expected PAs are different. For the positive u₀ solution, we can predict PA = −164.83 ± 0.95° and PA = 31.52 ± 0.58° for both VLTI epochs. For the negative u₀ solution, we predict PA = −156.60 ± 0.68° and PA = 14.29 ± 1.06°, respectively.

We also note that the flux ratio of the images is tightly constrained by the light-curve model, with a precision better than 0.3%. Conversely, the uncertainty of the PA is dominated by the uncertainty in the determination of the angle Φ_π, which itself is dominated by the uncertainty of π_E,N. As we observed small systematic differences between π_E,N determined using different data sets (Figure 4), these differences propagate to systematic variations in Φ_π and therefore PA. In particular, the angle Φ_π can vary from 18.56° to 25.90° (Tables 3 and 4).

The VLTI/GRAVITY observations of OGLE-2023-BLG-0061/KMT-2023-BLG-0496 provide a complementary view of the microlensing event. In particular, the spatial resolution of the interferometer allows us to resolve the microlensed images, providing a precise measurement of the angular Einstein radius. This information is not included in the light curve of the event. We first separately analyze the VLTI data collected during epochs 1a, 1b, and 2, because they were taken at different times and using different instrument configurations. That will allow us to study the consistency between the model parameters derived using different data sets and the consistency with the light-curve model (Sections 5.1, 5.2, and 5.3). In Section 5.4, we combine all VLTI data sets to derive the final parameters of the system. The results presented in this section were independently checked using the PMOIRED software³⁴ (A. Mérand 2022) and we found virtually identical results.

5.1. No-lens-light Model

In the simplest case, if the lens is dark and there are no other blended stars in the GRAVITY field of view, the two images of the source created by microlensing look like a mundane “binary star.” We start by fitting such a simple binary star model to the closure-phase data. We assume that both images of the source star can be considered as point-like. We place the major image in the origin of the coordinate system, whereas the position of the minor image is parameterized by a vector (Δα, Δδ). If we denote the flux ratio between the minor and major image as η, then the complex visibility is

where λ is the wavelength of observations and (U, V) is the separation of the telescopes in the UV plane. The visibility can be calculated for all possible pairs of telescopes (, , ,…, where the indices denote different telescopes). For a triangle of telescopes, one can define the bispectrum as the product , and then the closure phase (denoted in Equation (7) as T3) is the argument of this bispectrum (see, N. Dalal & B. F. Lane 2003). Closure phases can be calculated for all possible triangles formed by the telescopes of the interferometer. For VLTI, there are four possible closure phases. In theory, one of them is not independent. However, because the measured closure phases may be affected by noise in the data, we employ all four closure-phase sets in the fits.

We first conduct the grid search to find the initial parameters of the model. We keep the flux ratio fixed to η = 0.591 (epoch 1) and η = 0.426 (epoch 2)—that is, the best-fit values from the light-curve model (Table 2). We then evaluate the goodness-of-the-fit statistic χ² on a grid of 201 × 201 points uniformly distributed over the range  −10 mas≤(Δα, Δδ)≤10 mas. For each case, we find there is only one significant minimum of the χ² surface.

We then refine the results of the grid search by allowing all parameters to vary. This is achieved by maximizing the log-likelihood function defined as

where is the number of exposures in the epoch and Λ is the number of spectral channels. Because the error bars calculated from the pipeline may be underestimated, we add a constant error term σ₀ in quadrature. The best-fit parameters and their uncertainties are calculated using the Markov Chain Monte Carlo (MCMC) algorithm coded by D. Foreman-Mackey et al. (2013). We assume flat (noninformative) priors on all parameters of the model.

The results of the fits—separately for VLTI epochs 1a, 1b, and 2—are reported in Table 5. The best-fitting closure-phase models are shown in Figures 5 and 6. In addition to four model parameters, we report four derived quantities: the separation between the images , their PA (north through east) , the source–lens separation u (in Einstein radius units), and the angular Einstein radius θ_E. The latter quantity is calculated using the formula

derived under the assumption that the flux ratio between the microlensed images determines their separation in Einstein radius units. Similarly, the source–lens separation is

The results of the fits to the individual epoch 2 exposures are presented in Table 6.

Table 5. Best-fit Parameters of the Closure-phase Model Without the Lens Light

Parameter	Epoch 1a	Epoch 1b	Epoch 2
Δα (mas)	−0.760 ± 0.012	−0.652 ± 0.030	1.163  ±  0.007
Δδ (mas)	−2.467 ± 0.018	−2.545 ± 0.029	2.309  ±  0.022
η	0.6168  ±  0.0096	0.6358  ±  0.0127	0.4414  ±  0.0098
σ0 (deg)	7.04  ±  0.33		2.56  ±  0.26
s (mas)	2.581  ±  0.015	2.627  ±  0.022	2.585  ±  0.020
PA (deg)	−162.87 ± 0.36	−165.63 ± 0.77	26.74  ±  0.27
u	0.2424  ±  0.0075	0.2266  ±  0.0107	0.4120  ±  0.0119
θE (mas)	1.2814  ±  0.0066	1.3050  ±  0.0107	1.2660  ±  0.0083
χ2/dof	824.5/924	49.6/48	282.6/308

Download table as:  ASCIITypeset image

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 6. Best-fit Parameters of the Closure-phase Model for Individual Epoch 2 Data Without the Lens Light

Parameter	Epoch 2a	Epoch 2b	Epoch 2c	Epoch 2d	Epoch 2e	Epoch 2f
Δα (mas)	1.114 ± 0.016	1.165 ± 0.017	1.200 ± 0.015	1.200 ± 0.012	1.115 ± 0.022	1.179 ± 0.018
Δδ (mas)	2.429 ± 0.085	2.374 ± 0.082	2.252 ± 0.054	2.283 ± 0.054	2.248 ± 0.045	2.335 ± 0.054
η	0.4010 ± 0.0238	0.4300 ± 0.0253	0.4698 ± 0.0253	0.4440 ± 0.0248	0.3649 ± 0.0306	0.4289 ± 0.0244
σ0 (deg)
s (mas)	2.672 ± 0.079	2.644 ± 0.075	2.552 ± 0.047	2.579 ± 0.048	2.509 ± 0.035	2.615 ± 0.047
u	0.4608 ± 0.0295	0.4252 ± 0.0292	0.3800 ± 0.0283	0.4088 ± 0.0269	0.3649 ± 0.0306	0.4289 ± 0.0244
PA (deg)	24.64 ± 0.77	26.13 ± 0.80	28.04 ± 0.65	27.71 ± 0.60	26.36 ± 0.82	26.77 ± 0.70
θE (mas)	1.3021 ± 0.0343	1.2933 ± 0.0328	1.2535 ± 0.0201	1.2633 ± 0.0199	1.2342 ± 0.0142	1.2790 ± 0.0193
χ2/dof	51.6/48	50.6/48	47.5/48	40.2/48	28.6/48	46.5/48

Download table as:  ASCIITypeset image

There is some tension between the medium- and low-resolution interferograms obtained during epoch 1: the measured Δα position differs by 3.3σ, the Δδ position by 2.3σ, and the PA by 3.2σ. On the other hand, the inferred angular Einstein radii are formally consistent between the three different data sets, although the differences between individual measurements can amount up to 1.5σ–2.9σ. These tensions indicate that at least one of the analyzed data sets may suffer from unaccounted-for, low-level systematic errors. Another explanation involves possible correlated noise in the closure-phase data that was not taken into account in evaluating the likelihood function (see J. Kammerer et al. 2020). Thus, the error bars reported in Tables 5 and 6 may be underestimated.

Further checks are possible because the flux ratio η and position angle PA of the images can be independently measured using the light-curve data (Section 4). In particular, there are two possible light-curve models differing by the sign of u₀, which predict the PAs of the microlensed images as (−1648, 315) (positive u₀) or (−1566, 143) (negative u₀), during the VLTI epochs 1 and 2, respectively. The measured angles (−1629, 267) (Table 5) seem to favor the u₀ > 0 model.

However, while the light-curve and closure-phase models agree well for epoch 1, the expected PAs differ by almost 48 (that is, 7.5σ) during epoch 2. That is illustrated in Figure 7, which presents 1σ, 2σ, and 3σ confidence ellipses in the (PA, η) plane. The gray contours mark the constraints from three different VLTI data sets, whereas the black contours are calculated based on the light-curve model (u₀ > 0). The blue, red, green, and orange contours mark the constraints on (PA, η) from OGLE, KMTC, KMTA, and KMTS data, respectively.

Zoom In Zoom Out Reset image size

The PAs of the images measured from interferometric observations can be projected onto the (π_E,N, π_E,E) plane (Figure 4). The dotted and dashed lines in this figure correspond to the best-fitting values of the PAs during epochs 1a and 2, respectively. The dashed line does not intersect color contours from the light-curve model, which further exemplifies the tension discussed above.

The light-curve model also predicts the flux ratios between the microlensed images: η = 0.5907 ± 0009 for epoch 1 and η = 0.4264 ± 0.0013 for epoch 2. The former value is in 2.7σ–3.5σ tension with the closure-phase fits to epoch 1a and 1b data, respectively.

5.2. Luminous Lens Model

5.3. Luminous Blend Model

We also consider the model with a luminous blend, which has two additional parameters compared to the luminous lens model, the offset of the blend (Δα_b, Δδ_b) relative to the lens in the sky. We searched for possible blends on a grid of 201 × 201 positions uniformly spread over the range  −20 ≤Δα_b, Δδ_b ≤20 mas. We kept the position of the blend fixed but allowed the other parameters (u, PA, θ_E, and η_b) to vary. The best-fit parameters were found using a downhill approach using the Nelder–Nead algorithm. We adopted a prior on the lens–source separation u from the light-curve model, and we required η_b ≥0.

The grid search results for the epoch 2 data are presented in Figure 8. There is only one local minimum around (Δα_b, Δδ_b) = (−6.0, 15.8) mas that may be statistically significant (the χ² statistics is improved by Δχ² = 26.6). We explored this local minimum using the MCMC approach and found u = 0.4295 ± 0.0016, PA = 26.00 ± 0.24°, θ_E = 1.2672 ± 0.0054 mas, Δα_b = −5.91 ± 0.18 mas, Δδ_b = 15.73 ± 0.32 mas, and η_b = 0.0426 ± 0.0094. Thus, even if we consider this solution statistically significant, the blend does not explain the tension between the PA measured using the light-curve and interferometric data.

Zoom In Zoom Out Reset image size

5.4. Final Closure-phase Models

The mass and distance to the lens can be obtained from

where D_s ≈ 8 kpc is the source distance. As the interferometric data do not provide strong evidence for the light from the lens, we adopt θ_E = 1.2802 ± 0.0085 mas (Table 8) as our final measurement. Using π_E = 0.3330 ± 0.0084, we find M = 0.472 ± 0.012 M_⊙, π_rel = 0.426 ± 0.011 mas, and D_l = 1.81 ± 0.05 kpc. The lens is, therefore, most likely a main-sequence star located in the nearby Galactic disk. Alternatively, it may be a white dwarf. The mass function of field white dwarfs peaks at approximately 0.6 M_⊙ and falls steeply toward lower masses (e.g., P. E. Tremblay et al. 2016; S. Torres et al. 2021; T. Cunningham et al. 2024). Because white dwarfs less massive than 0.5 M_⊙ are much less frequent than main-sequence stars of that mass, we find this possibility unlikely.

If the lens is a main-sequence star, its absolute magnitude is M_K = 5.76 ± 0.04 (M. J. Pecaut & E. E. Mamajek 2013), and its apparent K-band magnitude is 17.20 ± 0.07, assuming K-band extinction of 0.15 mag toward the lens (D. M. Nataf et al. 2013). That corresponds to η_b ≈ 0.03 in the K band, consistent with the limits on the lens flux from the VLTI data.

The trajectory of the source in the sky is presented in Figure 9 by a gray solid line. The open symbols (red square and blue circle) mark the positions of the source during the two epochs of VLTI observations, whereas the filled symbols mark the positions of the images of the source.

Zoom In Zoom Out Reset image size

The start of the scientific operations of GRAVITY Wide opens up completely new possibilities for the follow-up of microlensing events. Only very few of the brightest events could have been observed with the standard on-axis GRAVITY mode. In contrast, GRAVITY Wide observations provide the opportunity for studies of a large sample of events and measurements of masses, distances, and transverse velocities of isolated objects, including neutron stars and black holes. Interferometric observations are very efficient at the same time—only a 1 hr exposure is sufficient for determining the angular Einstein radius. That is in stark contrast to astrometric microlensing measurements, which require time-consuming monitoring lasting several years (C. Lam et al. 2022; K. C. Sahu et al. 2022). Astrometric measurements are also more susceptible to systematic errors due to blending than interferometric observations (P. Mróz et al. 2022). Thus, as argued by A. Gould (2023), interferometric observations with GRAVITY Wide (and its successor, GRAVITY+) hold promise for routine detections of isolated neutron stars and stellar remnants.

In this paper, we report the first successful observation of a microlensing event with GRAVITY Wide and the resolution of microlensed images in the event OGLE-2023-BLG-0061/KMT-2023-BLG-0496. These observations serve as a test bed for verifying the capabilities of the new instrument. In particular, some observations were carried out using the low-resolution mode, which had never been tried before with GRAVITY Wide. The comparison of medium- and low-resolution interferograms (epochs 1a and 1b) reveals low-level systematic differences between the inferred parameters of the lensing system (Table 5). Similar systematic differences can be seen in models fitted to individual exposures taken during epoch 2 (Table 6). Judging from Figures 5 and 6, the systematic and correlated errors in the closure-phase data can reach up to 5°–10°. We defer a detailed investigation of the systematic errors in the closure-phase data to a separate study.

We use the bootstrapping method to deal with these systematic and correlated errors. Thanks to the large number of observations, we are still able to measure the angular Einstein radius with a subpercent precision, θ_E = 1.2802 ± 0.0085 mas. By combining the information from the light-curve and the closure-phase data, we measure the mass of the lens with a precision of 2.6%, M = 0.472 ± 0.012 M_⊙.

Independent tests of the accuracy of the closure-phase model are possible with the light-curve model. The model of the microlensing event light curve predicts the brightness ratio of the microlensed images and their PA in the sky. While the predicted flux ratio matches that inferred from the closure-phase data reasonably well (Figure 7), there is a tension in the PAs measured using the epoch 2 data. This tension cannot be explained by additional blended light in the closure-phase data (Section 5.3).

One possible explanation for this systematic difference is the unaccounted-for systematic errors in the closure-phase data. Alternatively, the problem may lie in the light-curve model. However, we find this unlikely, because the residuals from the best-fit light-curve model are smaller than  ≈0.01 mag, and there is no strong evidence that the light-curve model is inadequate. A binary lens model can be ruled out, because the binary lens would produce additional images of the source, which should have been detected in our grid search. Still, the unaccounted-for orbital motion of the source (the “xallarap effect”) could, in principle, partly explain the discrepancy. The xallarap effect would not create additional images but would deflect the path of the source on the sky. Our modeling, however, does not provide strong evidence for xallarap in the light-curve data (Appendix).

The additional testing of the model presented in this paper will be possible thanks to the data from the Gaia satellite (Gaia Collaboration et al. 2016). OGLE-2023-BLG-0061/KMT-2023-BLG-0496 was observed and alerted by the Gaia Photometric Science Alerts system (S. T. Hodgkin et al. 2021) as Gaia23ckg. Time-series astrometric data collected by Gaia, when made public later this decade, may reveal deflection due to astrometric microlensing effects and enable independent measurements of the angular Einstein radius.

We thank David Bennett and Łukasz Wyrzykowski for their comments on the manuscript.

Based on observations collected at the European Southern Observatory under ESO program 108.220D. This research was funded in part by the National Science Centre, Poland, grants OPUS 2021/41/B/ST9/00252 and SONATA 2023/51/D/ST9/00187 awarded to P.M. This research is partly supported by the National Natural Science Foundation of China (grant No. 12133005) and the science research grants from the China Manned Space Project with No. CMS-CSST-2021-B12. S.D. acknowledges the New Cornerstone Science Foundation through the XPLORER PRIZE.

GRAVITY+ is developed by the Max Planck Institute for Extraterrestrial Physics, the Institute National des Sciences de l'Univers du CNRS (INSU), with its institutes LESIA/Paris Observatory-PSL, IPAG/Grenoble Observatory, Lagrange/Côte d'Azur Observatory, and CRAL/Lyon Observatory, the Max Planck Institute for Astronomy, the University of Cologne, the CENTRA—Centro de Astrofisica e Gravitação, the University of Southampton, the Katholieke Universiteit Leuven, and the European Southern Observatory. D.D. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement CoG—866070).

This research has made use of the KMTNet system operated by the Korea Astronomy and Space Science Institute (KASI) at the three host sites of CTIO in Chile, SAAO in South Africa, and SSO in Australia. Data transfer from the host site to KASI was supported by the Korea Research Environment Open NETwork (KREONET). This research was supported by KASI under the R&D program (project No. 2024-1-832-01) supervised by the Ministry of Science and ICT. W.Zang, H.Y., S.M., R.K., J.Z., and W.Zhu acknowledge support by the National Natural Science Foundation of China (grant No. 12133005). W.Zang acknowledges the support from the Harvard–Smithsonian Center for Astrophysics through the CfA Fellowship. J.C.Y. and I.-G.S. acknowledge support from U.S. NSF grant No. AST-2108414. Y.S. acknowledges support from BSF grant No. 2020740. The work by C.H. was supported by grants from the National Research Foundation of Korea (2019R1A2C2085965 and 2020R1A4A2002885). J.C.Y. acknowledges support from a Scholarly Studies grant from the Smithsonian Institution.

In this section, we explore whether the orbital motion of the source (xallarap effect) can explain the discrepancy between the PAs of the microlensed images measured using VLTI data and calculated from the light-curve model. Compared to the standard point-source point-lens model, models including the xallarap effect have seven additional parameters describing the shape of the source's orbit. These are P_ξ—the orbital period; a_ξ—the semimajor axis (expressed in Einstein radius units); i_ξ—the inclination of the orbit; Ω_ξ—the longitude of the ascending node of the orbit; u_ξ—the argument of latitude at the reference epoch (which is taken to be t_0,ξ = t_0,par); e_ξ—the eccentricity of the orbit; and ω_ξ—the argument of periapsis of the orbit.

For simplicity, we consider only circular orbits, keeping e_ξ = 0 and ω_ξ = 0. We also assume that the companion to the source is dark, given there is no strong evidence for the additional light in the VLTI data (Section 5.3). We employ the MULENSMODEL package by R. Poleski & J. C. Yee (2019) to calculate the source trajectory and magnification.

We search for best-fit xallarap models on a grid of orbital periods P_ξ spanning from 10 to 251 days. We keep the orbital period fixed, but the remaining parameters (a_ξ, i_ξ, Ω_ξ, u_ξ, and the standard microlensing parameters) are allowed to vary. The best-fit parameters are found using the MCMC approach (D. Foreman-Mackey et al. 2013). Simultaneously, we calculate the PA of the minor image relative to the major image during the two epochs of VLTI observations (Section 4). We assume uniform priors on all parameters.

The results of our calculations for the OGLE data are presented in Table 9, where we report the χ² statistic of the best-fit model and the predicted PAs. There is no strong evidence for xallarap in the data. The best-fit model, for the orbital period of P_ξ = 79.4 days, is favored over the model without xallarap by only Δχ² = 11.6, which is not statistically significant given the increased number of free parameters. The predicted PAs are all similar to those calculated using the standard point-source point-lens model. However, their uncertainties are typically larger, which quantifies the additional degrees of freedom on the position of the source due to orbital motion.