An Isolated Stellar-mass Black Hole Detected through Astrometric Microlensing^*

Stars with initial masses greater than ∼20 M_⊙ are expected to end their lives as black holes (BHs; e.g., Fryer & Kalogera 2001; Woosley et al. 2002; Heger et al. 2003; Spera et al. 2015; Sukhbold et al. 2016). Objects of these masses constitute roughly 0.1% of all stars, leading to the expectation that the Galaxy should now contain of the order of ≈10⁸ BHs (Shapiro & Teukolsky 1983; van den Heuvel 1992; Brown & Bethe 1994; Samland 1998).

However, the actual detection of stellar-mass BHs is observationally challenging, and determining their masses is even more so. BHs have been identified in the Galaxy and Local Group through X-ray emission due to accretion in short-period binary systems, most of them soft X-ray transients. In such cases, dynamical masses of the BHs can be measured or estimated through radial velocity measurements and light-curve modeling for the optical companion stars (the techniques are reviewed by Remillard & McClintock 2006; Casares & Jonker 2014). Masses of nearly two dozen BHs in X-ray binary systems have been determined using these methods, with varying degrees of precision. These "electromagnetically measured" BH masses show a distribution peaking near 7–8 M_⊙, with few if any below ∼5 M_⊙ (e.g., Özel et al. 2010; Farr et al. 2011; Kreidberg et al. 2012; Corral-Santana et al. 2016). This suggests that a "mass gap" exists between the lowest-mass BHs and the highest measured masses of neutron stars (NSs) in binary radio pulsars of ∼2.1–2.3 M_⊙ (Linares et al. 2018; Cromartie et al. 2020). Recently, however, a few nonaccreting or weakly accreting BHs have been discovered in longer-period spectroscopic binaries in the field (e.g., Thompson et al. 2019; Jayasinghe et al. 2021) and in globular clusters (Giesers et al. 2019), lying in the NS–BH gap with dynamical masses of ∼3–4.5 M_⊙. Precision astrometry of nearby stars by Gaia shows the promise of detecting additional wide-binary systems containing quiescent BHs (e.g., Chawla et al. 2021; Janssens et al. 2022, and references therein) and measuring their masses. At the high-mass end, the BH mass distribution falls off above ∼10 M_⊙, and very few electromagnetic BH masses are known above ∼15 M_⊙, the only exception in the Milky Way being an updated mass determination of 21.1 ± 2.2 M_⊙ for the BH in Cygnus X-1 (Miller-Jones et al. 2021). Among extragalactic X-ray binaries, BH masses as high as 15.65 ± 1.45 M_⊙ and 17 ± 4 M_⊙ have been reported for M33 X-7 (Orosz et al. 2007) and NGC 300 X-1 (Binder et al. 2021), respectively. An even higher mass of at least 23.1 M_⊙ was reported for the compact object in IC 10 X-1 (Silverman & Filippenko 2008), but this has been questioned (Laycock et al. 2015).

The first detections of gravitational waves (GWs) by the Laser Interferometry Gravitational Wave Observatory (LIGO) and Virgo Collaboration (Abbott et al. 2016) revealed a population of massive merging binary BHs, BH–NS pairs, and binary NSs at extragalactic distances. In the source catalogs from the third observing run of the Advanced LIGO and Advanced Virgo Collaboration (Abbott et al. 2021a, 2021b), the inferred masses of the BHs among the premerger systems range from ≈6 to 95 M_⊙, with two low-mass outliers among the secondary components at ∼2.6 and 2.8 M_⊙, which could be either BHs or NSs.

The electromagnetic and GW mass measurements described above are all for BHs in binary systems, including those undergoing mass accretion or mergers. However, there are reasons to believe that a substantial fraction of stellar-mass BHs are single, rather than belonging to binaries. First, about 30% of massive stars are born single (see Sana et al. 2012; de Mink et al. 2014). Second, in a close-binary system, the pair may enter into a common envelope and merge before the supernova (SN) explosion (e.g., Fryer et al. 1999; Zhang & Fryer 2001; Tutukov et al. 2011; Dominik et al. 2012). Lastly, in a wide binary, the "natal kick" imparted to the companion by the SN event may be large enough to detach the two components, producing an isolated BH (e.g., Tauris & van den Heuvel 2006; Belczynski et al. 2016). Being less altered by interactions with companions, single BHs potentially provide a more direct probe of BH formation than those in binaries.

Isolated BHs are extremely difficult to detect directly. They emit no light of their own, and the accretion rate from the interstellar medium is generally likely to be too low to produce detectable X-ray or radio emission (see, however, Agol & Kamionkowski 2002; Fender et al. 2013; Tsuna & Kawanaka2019; Scarcella et al. 2021, for the case of isolated BHs in dense environments). In fact, until now, no isolated stellar-mass BH has ever been unambiguously found within our Galaxy or elsewhere.

Microlensing is the only available method for measuring the masses of isolated BHs. Astrometric microlensing—the relativistic deflection of the apparent position of a background star when a compact object passes in front of it—provides a direct method for measuring the masses of BH lenses. High spatial resolution interferometric observations of microlensing events (Dong et al. 2019; Zang et al. 2020) and observations of rare events where the lens passes over the surface of the source (Yoo et al. 2004) can also yield the masses of BH lenses. In this paper, we describe how the technique of astrometric microlensing is used to determine masses. We discuss our ongoing program of astrometric measurements of microlensing events with the Hubble Space Telescope (HST). Then, we report the first detection of an isolated BH and our measurement of its mass.

A microlensing event occurs when a star or compact object (the lens) passes almost exactly in front of a background star (the source). As predicted by general relativity (Einstein 1936), the lens magnifies the image of the source, producing an apparent amplification of its brightness. The lens also slightly shifts the apparent position of the source (Miyamoto & Yoshii 1995; Høg et al. 1995; Walker 1995)—an analog of the deflection of stellar images during the 1919 solar eclipse (Dyson et al. 1920), which provided support for the general theory of relativity.

Microlensing survey programs, including OGLE (Udalski et al. 2015), MOA (Bond et al. 2001), and KMTNet (Kim et al. 2016), carry out photometric monitoring of rich stellar fields in the Galactic bulge. These surveys typically detect >2000 events toward the Galactic bulge annually. To date, more than 30,000 microlensing events have been discovered and monitored by these survey programs.

The characteristic scale of gravitational microlensing is provided by the angular Einstein radius θ_E, given by

where M_lens is the mass of the lens and

is the relative lens–source parallax, with D_L and D_S being the distances from the observer to the lens and to the source, respectively.

The Einstein radius θ_E, however, cannot be obtained directly from the magnification light curve, whose only characteristic that carries a physical dimension is the timescale. Given the relative proper motion μ_LS between lens and source, we can straightforwardly define , the time for the source to traverse an angular distance of θ_E in the barycentric reference frame, as (more details in Section 2.4). The distribution of the timescale of the observed events peaks around 25 days, with ranging from a fraction of a day to several hundred days (e.g., Wyrzykowski et al. 2015).

If BHs constitute a small but nonnegligible fraction of the total stellar mass of the Galaxy, as described above, then a few of the observed microlensing events are expected to be due to BHs. Equation (1) shows that the angular Einstein radius θ_E is proportional to the square root of the lens mass. Hence, all else being equal, events due to massive compact objects would preferentially tend to be characterized by longer event durations (t_E ≳ 150 days), combined with an apparent lack of light contribution from the lens. However, a degeneracy between lens mass and proper motion remains. Thus, a long-duration event with no light contribution from the lens could arise from a high-mass, nonluminous BH lens with a large Einstein radius—but it could alternatively be due simply to an unusually slow-moving, faint, low-mass ordinary star. If the lens were a luminous massive star, this would generally be recognizable through the contribution of its light, particularly when observations are available in two different bandpasses.

Indeed, several long-duration OGLE and MOA microlensing events have been suggested as being due to BHs (e.g., Bennett et al. 2002; Mao et al. 2002; Minniti et al. 2015; Wyrzykowski & Mandel 2020). However, these claims remain statistical in nature, being derived from assumptions about the transverse velocity distributions.

The degeneracy between mass and relative velocity can be lifted if precise astrometry is added to the photometry of the microlensing event. The size of the expected astrometric shift is small—of the order of milliarcseconds—but it is proportional to the angular Einstein radius θ_E, and therefore if this small shift can be measured, then the mass of the lens can be determined unambiguously, as described in detail below.

Photometric microlensing is the apparent transient brightening that results as a background source passes almost directly behind a foreground lens (see reviews by Paczyński 1996; Gaudi 2012, and Tsapras 2018).

With θ _S and θ _L denoting the angular positions of the source and lens as seen by the observer, we can define a dimensionless source–lens separation

As the lens intervenes near the line of sight from the observer to the source, the gravitational bending of light leads to a time-varying magnification

which depends solely on u = ∣ u ∣. This expression holds as long as the finite angular size of the source star can be neglected, which we will assume in the following discussion. We will demonstrate in Section 9.3 that this is a valid approximation for the case analyzed in this paper.

If F_S is the intrinsic source flux and F_B the background flux contributed by any other objects not resolved from the observed source star, the observed flux of the target for a specific telescope and filter is given by

where F_base ≡ F_S + F_B is the baseline flux and g ≡ F_B/F_S is the specific blend ratio.

Microlensing also produces an astrometric shift of the apparent position of the source. If we assume that we can observe the centroid of light formed by the images of the source without any contribution from other bodies such as the lens or other neighboring stars, its shift is described by the vector

(Gould 1992; Paczyński 1998). In contrast to the photometric microlensing signature, the astrometric signature is explicitly proportional to the angular Einstein radius θ_E.

Moreover, while the magnification diverges (for a pointlike source) as u → 0, the astrometric shift becomes maximal for . The light magnification falls rapidly with increasing u: for large separations, u ≫ 1, the brightness enhancement, A(u) − 1, falls as 1/u⁴. On the other hand, the centroid shift, given by Equation (6), decreases more slowly with u, and for large separations it falls only as 1/u. The astrometric perturbation thus has a considerably longer duration than the photometric signal. For more details, see Dominik & Sahu (2000), Sahu et al. (2014), and Bramich (2018).

The photometric and astrometric signatures of a microlensing event are connected because they arise from the same source–lens trajectory, u (t). Specifically, if a fit to the light curve of a microlensing event already yields u as a function of time, the astrometric data then provide a direct measurement of the angular Einstein radius θ_E, as well as the orientation angle of the trajectory.

For constructing the source–lens trajectory u (t), we have to consider the proper motions μ _S and μ _L of the source and lens objects, as well as their parallaxes, π_S and π_L.

Let γ (t) (1 au) denote the projection of Earth's orbit onto a plane perpendicular to the direction toward the source star. The apparent geocentric positions of source and lens star are then given by (see An et al. 2002; Gould 2004)

so that for θ (t) ≡ θ _S(t) − θ _L(t) one finds

where μ _LS ≡ μ _L − μ _S and π_LS ≡ π_L − π_S are the relative proper motion and relative parallax between lens and source, while .

Consequently, with the microlensing parallax parameter π_E = π_LS/θ_E, u (t) ≡ θ (t)/θ_E takes the form (see Dominik et al. 2019)

where

as well as

By construction, δ γ (t₀) = 0 and .

By choosing t₀ so that , we can write u (t) in its components toward northern and eastern directions as

where u₀ ≡ ∣ u ₀∣, , and ψ denotes the direction angle of measured from north toward east. Alternatively, the trajectory can be parameterized as

where , , and ψ^⋆ denotes the direction angle of − μ _LS = μ _S − μ _L measured from north toward east. Note that the starred quantities, ψ^⋆, , , and refer to parameters in the barycentric reference frame, whereas the corresponding unstarred quantities refer to parameters as seen by an observer on Earth.

This applies to any orthonormal reference frame; the only difference is in the specific angle, e.g., ψ_eq, ψ_ecl, and ψ_gal (and , , and ) for equatorial, ecliptic, and galactic coordinates, respectively, which are related by a rotation of the coordinate axes at the target position.

While most discussions of photometric microlensing events choose an ecliptic coordinate frame, the observed astrometric data are more easily described in an equatorial coordinate frame, and therefore we adopt the latter in the following analysis. In this frame,

gives the position angle (PA) of the proper motion of the lens with respect to the source μ _LS, measured from equatorial north toward east. We illustrate the geometry of the source–lens trajectory u (t) in Figure 1.

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As stated above, the only useful physical parameter in a typical microlensing event is the timescale , which is the time it takes the source to traverse the radius of the Einstein ring, which itself depends on the lens mass, M_lens, and the lens–source parallax, π_LS.

However, for long-duration events, the annual parallax tends to lead to prominent departures in the photometric signature (Gould 1992; Alcock et al. 1995), so that a microlensing parallax parameter π_E ≡ π_LS/θ_E can be inferred. Coincidentally, the BH mass lenses tend to imply such long-duration events. On the other hand, the astrometric signature is proportional to the angular Einstein radius θ_E, so that by combining photometric and astrometric observations, M_lens, π_LS, and μ _LS become fully decoupled. Specifically, with π_E from the photometry and θ_E from the astrometry, the definition of θ_E, Equation (1), immediately gives us

where .

In the case of microlensing toward the Galactic bulge, the source often lies at the distance of the bulge itself, which can be verified from its baseline position in a color–magnitude diagram (CMD). If spectroscopic observations are available in addition to baseline photometry—as is the case for the event discussed in this paper—a more accurate source distance can be determined. The lens–source relative parallax, π_LS = π_E θ_E, can then be used to estimate the distance to the lens, using Equation (2). As a bonus, the event timescale gives a direct measure of the relative transverse velocity of lens and source (which, for stellar remnants, might include "kicks" received in SN explosions). This method thus provides independent measurements of three separate physical parameters of the lens: its mass, distance, and transverse velocity.

Some features of astrometric deflections under various scenarios are described by Dominik & Sahu (2000). To illustrate a typical case, we show in Figure 2 the calculated astrometric shifts and light magnification for a nominal event of a BH lens of mass 5 M_⊙, at a distance of 2 kpc from the Sun, passing in front of a background source situated in the Galactic bulge at a distance of 8 kpc. The closest angular approach is assumed to be at a separation of 0.05θ_E. In this case, the size of the angular Einstein ring is θ_E ≃ 4 mas, so that the maximum astrometric shift is ∼1.4 mas (occurring at a separation of ) and the maximum light magnification is a factor of ∼20 (at closest angular approach).

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As Figure 2 illustrates, and as discussed in Section 2.3, the duration of the astrometric deflection is considerably longer than that of the photometric magnification. This makes it necessary to carry out the astrometric measurements over a longer time interval than the photometry. Although the deflection measured at any given epoch provides in principle an estimate of θ_E, it is necessary to observe at multiple epochs in order to separate the shifts caused by microlensing from those caused by the proper motion of the source; observations at a late epoch are particularly useful for this purpose. The figure also shows that the astrometric shift is close to zero at the time of highest magnification; therefore, observations near the photometric peak are also very useful to constrain the source proper motion.

In Figure 3, we plot the maximum astrometric shifts for a source in the Galactic bulge at 8 kpc, as functions of lens mass. The lenses are assumed to be located at distances of 2 and 4 kpc ("disk" lenses), and 6 kpc ("bulge" lenses). The dotted line at the bottom shows the nominal astrometric precision of 0.2 mas achievable with high signal-to-noise ratio (S/N) HST imaging, as discussed below. Therefore, the deflection is detectable at 1σ per epoch for lens masses down to ∼0.5 M_⊙, except at lens distances larger than 6 kpc. The most favorable situation for a precise mass measurement, of course, would be for a nearby, high-mass lens.

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Although an unambiguous determination of lens mass is possible from a combination of photometry and astrometry, as we have just discussed, the expected astrometric shifts are extremely small, of the order of milliarcseconds or less. HST has demonstrated its capability to carry out submilliarcsecond astrometry through a variety of techniques. For example, high-S/N HST observations of isolated sources were used to achieve submilliarcsecond accuracy, leading to the measurement of proper motions for several distant hypervelocity stars (Brown et al. 2015). A collection of sources was used as probes to achieve an astrometric accuracy of ∼12 μas, in order to measure the transverse velocity of M31 (Sohn et al. 2012). Spatial-scan techniques have been used to achieve an astrometric accuracy of ∼30 μas (Casertano et al. 2016; Riess et al. 2018) in the trigonometric parallax of Cepheids used for accurate determination of H₀ and to measure the distance to the globular cluster NGC 6397 (Brown et al. 2018). Recently, our group used the astrometric microlensing technique to measure the mass of the nearby white dwarf Stein 2051 B, achieving an astrometric precision of ∼0.2 mas per epoch (Sahu et al. 2017). Kains et al. (2017) looked for astrometric deflections in HST observations of 10 microlensing events with timescales of <50 days. They achieved an astrometric precision of 0.2 mas per epoch (but did not detect any deflections). From the ground, Zurlo et al. (2018) used the Very Large Telescope (VLT) to measure the mass of Proxima Centauri through astrometric microlensing. Lu et al. (2016) employed the Keck telescope to look specifically for isolated BHs by monitoring three microlensing events, where they achieved a final positional error of 0.26–0.68 mas. The timescales of those events were 60–160 days, and there were no detections of astrometric deflections.

In 2009, we began a multicycle HST program of astrometry of long-duration microlensing events in the direction of the Galactic bulge in order to detect isolated BHs and measure their masses. Our aim is to select events having timescales ≳200 days, light curves showing no evidence for a light contribution by a luminous lens, and preferably a high magnification factor. We then obtain high-resolution HST imaging as the events proceed, in order to measure the astrometric deflections of the background sources. To date we have monitored eight long-duration events. For some of them, there is no clear detection of an astrometric signal, but our data analysis is still in progress, and the results will be discussed in separate publications. In the present paper we analyze and discuss our findings for an event that clearly shows a large astrometric deflection, consistent with a high-mass lens.

MOA-2011-BLG-191/OGLE-2011-BLG-0462 (hereafter designated MOA-11-191/OGLE-11-462) was a long-duration and high-magnification microlensing event in the direction of the Galactic bulge. It was discovered independently by both MOA and OGLE ground-based microlensing survey programs and announced by both teams nearly simultaneously on 2011 June 2, through their public-alert websites. ⁷⁵ The target was also covered by the Wise Microlensing Survey. Table 1 gives details of this remarkable event.

Table 1. Basic Data for MOA-11-191/OGLE-11-462 Microlensing Event

Parameter	Value	Sources and Notes a
Event designation (MOA)	MOA-2011-BLG-191	(1)
Event designation (OGLE)	OGLE-2011-BLG-0462	(1)
J2000 R.A., α	17:51:40.2082	(2)
J2000 decl., δ	−29:53:26.502	(2)
Galactic coordinates, (l, b)	35986, −162	(2)
Baseline F606W magnitude	21.943 ± 0.014	(3)
Baseline F814W magnitude	19.578 ± 0.012	(3)
Baseline (F606W − F814W) color	2.365 ± 0.026	(3)
Peak magnification,	369	(4)
Date of peak magnification, t0	2011 July 20.825	(4)
Timescale, tE	270.7 ± 11.2 days	(4)

Note.

^a Sources and notes: (1) MOA and OGLE websites; the event was first alerted by MOA. (2) This paper, from astrometric analysis in Section 5.2 in Gaia EDR3 frame at average epoch 2013.5. (3) This paper, Vegamag scale, from photometric analysis in Section 5.3. (4) This paper, from Table 6.

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MOA-11-191/OGLE-11-462 occurred in an extremely crowded Galactic bulge field, less than 2° from the Galactic center. The observed peak magnification factor of this event was only about 20 in the ground-based data, but this was strongly diluted by blending with neighboring stars. It soon became apparent, based on findings disseminated through internal communications in the microlensing groups, that the undiluted event actually had an extremely high magnification factor, approaching 400. Blending also made the apparent timescale of the event appear shorter than the actual value, which was inferred to be longer than 200 days. It was clear from the ground-based observations that there was blending, for two reasons. First, the light curve for a typical event has a characteristic shape that is completely determined by the timescale and the maximum magnification, except for distortions due, e.g., to the lens–source relative parallax. The shape of the observed light curve was inconsistent with the expected shape unless the light at baseline was highly diluted by a blend, thus implying that the real magnification was much larger than the observed value. Second, as the source brightened, its centroid position in the ground-based images was seen to change, again consistent with blending with a neighboring star. Note that this shift is due simply to blending and scales with the separation of the two stars; it is unrelated to the much smaller relativistic deflection of the source itself, which is discussed below.

Figure 4 shows an region centered on the source, as imaged by us in the F814W (I-band) filter by HST with its Wide Field Camera 3 (WFC3). The source star is encircled in green. A conspicuous neighbor, nearly 20 times brighter than the unmagnified source, lies at a separation of only 04. The cyan circles in the figure have diameters of and , corresponding to the generally best seeing in the ground-based survey observations and more typical seeing, respectively. Thus, in ground-based images, the source is indeed blended with the bright neighbor and a number of fainter stars, depending on the seeing.

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High-magnification events are generally very sensitive to perturbations due to planets around the lensing objects (Mao & Paczyński 1991; Griest & Safizadeh 1998). Thus, considerable interest was aroused by MOA-11-191/OGLE-11-462 among groups engaged in searches for such planets. As a result, intensive photometric monitoring of this event was carried out by multiple groups, providing valuable data for our analysis.

We used the flat-fielded and CTE-corrected (_flc) images produced by the Space Telescope Science Institute pipeline reductions (Sahu et al. 2021; Dressel 2021) for the analysis. As noted above, WFC3 suffered from increasingly poor CTE during this period, so it was essential to take it into account. The _flc products were produced using the v2.0 pixel-based CTE model described by Anderson (2021).

To measure stellar positions in individual frames, we used an updated version of the star-measuring algorithm described in Anderson & King (2006). The routine goes through each exposure pixel by pixel and identifies as a potential star any local maximum that is sufficiently bright and isolated. The routine uses the spatially variable effective point-spread functions (PSFs) provided at the WFC3/UVIS website ⁷⁶ to fit the PSF to the star images in the individual _flc exposures, in order to determine a position and flux for each star in the raw pixel frame of that exposure. Finally, the positions are corrected for geometric distortion using the distortion solutions provided by Bellini et al. (2011).

As the positions of individual stars are expected to change during the ∼6 yr course of our observations owing to their proper motions, we needed to determine their proper motions to properly specify the reference frame. For this, we began with the Gaia Early Data Release 3 (EDR3; Gaia Collaboration et al.2021) positions and motions for the bright but unsaturated HST stars in the field. The reference frame was constructed to place the bright star close to MOA-11-191/OGLE-11-462 at the center of the reference frame at (x, y) = (1000, 1000) at the 2016.0 epoch, with a plate scale of 40 mas pixel⁻¹ and north up. (Note that the Gaia catalog could be incomplete in this region because of the high source density.) Using the Gaia positions and motions, we determined the position for each Gaia star in this frame at each epoch in order to properly transform the distortion-corrected observations at that epoch into the reference frame. This ensures that the proper motions that we derive represent absolute proper motions. After this initial setup of the reference frame based on the brighter stars, we incorporated high-precision HST stars that were too faint to be found with high precision in the Gaia catalog and solved for their accurate positions and motions. We then used their time-dependent positions to improve the reference frame. Even after allowing individual solutions to improve on the basis of HST observations, there remains very good agreement between our proper motions and those of Gaia. Figure 6 plots the proper motions of our reference stars derived from our HST observations against the Gaia proper motions, where the red points are for brighter stars with G < 18 and blue points are for fainter stars with G ≥ 18. The agreement is imperfect, of course, since the HST observations have a 6 yr baseline and have higher S/N in individual measurements, resulting in higher accuracy in proper-motion measurements, particularly at fainter magnitudes. The agreement is better for the brighter sample, for which Gaia proper-motion errors are typically smaller (<0.2 mas yr⁻¹). Note that the MOA-11-191/OGLE-11-462 source itself is too faint at baseline for inclusion in the Gaia catalog.

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In the next step of our analysis, we used only the HST observations because the Gaia measurements have much higher uncertainties for the fainter stars and the HST observations have a longer baseline of 6 yr compared to the 3 yr of Gaia. In this step of the transformation, we used stars (1) with brightness similar to the average brightness of the target, (2) with color similar to the source's color (see Figure 7), and (3) lying within 350 pixels of the source. The first criterion minimizes any residual shift caused by CTE effects. We note that we already used the most recent CTE correction software for our analysis. Since the CTE effects on the position measurements are differential, using linear transformations based on stars of similar brightness should remove any residual CTE effects. (It is worth noting here that the images with the highest astrometric deflection were taken when WFC3/UIVS was young and when CTE losses were small.) The second criterion ensures that the stars used in the transformation belong to the bulge, which helps in minimizing errors due to parallax, as described in more detail below. The third criterion minimizes residuals in the distortion solution.

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We employed an iterative procedure to measure the positions and proper motions of the stars, starting from the revised values in each iteration. We rejected the highest-sigma point after each complete iteration. We repeated this procedure until the highest-sigma point was no more than a preset tolerance, for which we adopted 6σ. Only a small number of points were rejected by this procedure, mostly affected by cosmic-ray hits on the detector. Then, at each epoch the reference-star positions were corrected for proper motion, and the positions of the source were determined relative to this adjusted frame. The estimated uncertainty in the position of the source star relative to the adjusted frame is ∼0.4 mas in each individual exposure.

As an illustration, Figure 8 shows the proper motions as measured for nine representative stars. Figure 9 shows the errors in the proper-motion measurements of the reference stars. We note that all the reference stars are within about 1.5 mag of each other (See Figure 7). Figure 10 shows the the histograms of the residuals of each measurement from that star's proper-motion solution along the R.A. and decl. directions. Both distributions are consistent with a Gaussian distribution (the red curve). As shown by previous similar studies, the final reference-frame positions are expected to be internally accurate to better than 0.01 pixels (Anderson et al. 2008; Bellini et al. 2015).

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We specifically solved for the proper motions of the reference stars, but we ignored their parallaxes. The reason for adopting this approach is the following.

Our choice of reference stars ensures that a large fraction of them belong to the Galactic bulge and hence have similar parallactic motion to the source star. We note that the source parallax is small to begin with (≲0.2 mas). Then, the source position is referenced to stars chosen to be at comparable distance, so any remaining impact of the source parallax on the astrometry or photometry of the event is expected to be negligible.

As described earlier, there is a bright star ∼10 pixels away from our source. The target star is close to the brightness of this neighbor in the first two epochs and slowly fades to its nominal brightness, at which point it is about 3 mag fainter than the bright neighbor. For accurate astrometry of the source, we wanted to make sure that the position measurements of the source are not affected by the presence of this bright neighbor. So we wanted to subtract the PSF of the bright star before measuring the positions of the source at every epoch. However, subtracting the bright star is not just a matter of subtracting a standard PSF. The separation is ∼10 pixels, and the available library PSFs go out to 12 pixels and are tapered and not very accurate in the wings. Thus, we needed to make a more extended PSF model.

To make an extended PSF, we carefully selected stars that (1) are within ∼350 pixels of the bright neighbor, (2) have brightness and color similar to the neighbor, and (3) are fairly isolated. We found 18 such stars (excluding the bright star itself), which provided a good sample to make the required extended PSFs. We used the images of these 18 stars to produce a separate well-sampled, extended PSF for each individual exposure. For illustration, Figure 11 shows the stacked PSFs in F606W and F814W. We have taken particular care to make sure that the PSF is well characterized in the wings since the source lies in the wings of the bright star, and subtracting the wings correctly is crucial for accurate astrometry.

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We then took this PSF model for each exposure and subtracted it from the neighbor star in each exposure. Figure 12 shows the original images (first and third rows) and the subtracted images (second and fourth rows) in the F606W and F814W filters. The residuals are very small, particularly in the wings of the PSF. We found that the astrometric position of the source changes by ∼0.03 pixels (1.2 mas) after this subtraction, which could have a significant effect on the mass determination of the lens, so this extra step of neighbor subtraction was crucial in improving the analysis/results. The resulting astrometric positions of the source were used for further analysis as described in the next section.

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In addition to the measured positions, the analysis algorithm provides PSF-based photometry of all the stars in the field. To set a calibrated zero-point, we used standard aperture photometry to determine the fluxes of a few isolated stars in the field within an aperture with a 10-pixel radius. These fluxes were then corrected to an infinite aperture, using encircled-energy measurements from Calamida et al. (2021), and the photometric zero-point in the image headers (PHOTFLAM) was used to convert these fluxes to the Vegamag scale. The mean difference between these values and the values obtained by the PSF fitting was then applied to all of the PSF magnitudes to convert them to Vegamag.

As described above, the source star lies on the wings of the PSF of the neighboring bright star. Hence, for accurate photometry of the source, it was critical to correctly subtract the light contribution from the bright neighbor. The photometry for the source was carried out after subtracting the superposed flux from the neighbor star using a high-fidelity PSF as described above. The resultant time-series HST photometry of the source is shown in Figure 13, along with the model light curve described below in Section 8. There is no detectable color change as the event progresses and the star fades: the color of the source has remained constant to within 0.01 mag during the 6 yr of observations with HST. There is also no detectable blending as described in more detail in Section 8.2. The source at baseline brightness has apparent magnitudes of m_F606W = 21.943 ± 0.014 and m_F814W = 19.578 ± 0.012.

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We made a stack of all the images for each filter for every epoch. The HST images allow us to detect and measure magnitudes of stars as faint as V ≃ 25. Figure 7 shows the CMD based on this photometry, where we also show the position of the source and the bright neighbor 04 away.

Ground-based photometry of MOA-11-191/OGLE-11-462 suffers from significant amounts of blending with neighboring stars, as shown in our HST images (Figure 4). Moreover, the amount of blending changes markedly depending on image quality. There is a bright neighbor star only 04 away, along with two fainter stars within 05 of the source. At larger separations, there are three more stars within , whose combined brightness is greater than that of the baseline source, and there are several more stars within 15 that are also brighter than the source. Since in the available ground-based imaging data the measured image quality is seldom better than , the ground-based photometry will always include the light from at least the three closest stars within 05. In a fraction of the data, taken under poorer seeing conditions, the source is blended with an increasing number of neighbors.

Our HST images allow us to place constraints on the expected blending parameter, g, defined as the ratio of the flux from neighbors included in the photometry to the flux from the unmagnified source itself (Equation (5)), in the ground-based observations. The bright neighbor is 18.88 times brighter than the source at baseline in F814W, and the contribution from the two fainter stars is an additional 0.19 times that of the source. Thus, the expected blending factor due to these three stars is g = 19.07 in the F814W filter. Since the bright neighbor is similar in color to the source (see Figure 7), we adopt this value of g for both OGLE (I band) and MOA (R band) data as an initial estimate, but we keep it as a variable in our analysis. The final values (Section 5.2) differ significantly between OGLE and MOA, possibly due in part to differences in processing between the two data sets.

The effect of blending can be reduced substantially by basing the photometry on difference images. It can be further reduced by restricting the analysis to images taken under good seeing. It is obvious, however, that variable blending will affect the noise characteristics of ground-based photometry; it is difficult to include such effects deterministically because of the imperfect knowledge of the blending (at the subpercent level) for individual images.

7.2.1. Photometric Constraints on Parallax and Lens Trajectory

As discussed in Section 2.3, the light curve of a long-duration microlensing event such as MOA-11-191/OGLE-11-462 can show distortion by the relative parallactic motions of the source and lens (e.g., Gould 1992; Alcock et al. 1995). Specifically, the light curve is sensitive to π_E and φ_LS because their combination modifies the relative path of source and lens and thus the shape of the light curve. We note that, in our formalism, φ_LS corresponds to the PA of the path of the lens relative to the source without parallax in equatorial coordinates (not to be confused with the instantaneous path of the lens at the time of closest angular approach in ecliptic coordinates; see Section 2.2).

In principle, a sufficiently accurate light curve can provide good constraints on both π_E and φ_LS. However, as discussed in the previous subsection, the photometry is significantly affected by blending. We attempted to model the light curve alone but found that it can be fitted with a range of parameter combinations, in which the values of π_E and φ_LS are strongly correlated. In addition, the derived value of π_E varies with the specific subset of photometric data chosen for analysis, as well as with the assumed blending factor for those data. The derived value of π_E ranges from 0.07 to 0.12, with larger values corresponding to larger values of φ_LS, ranging from 330° to 358°. The reason is that increasing φ_LS makes the lens move in a more northerly direction as seen in the bottom panel of Figure 15. Since parallax is predominantly in the east–west direction, in order to produce a fixed change in u, the value of π_E has to increase with φ_LS, so that the change in position due to parallax can compensate for a more northerly motion of the lens. Several different combinations of these quantities can reproduce the observed light curve, with differences between solutions of the order of 1 mmag at early and late times and ∼5 mmag near the peak. Systematic differences in the data at this level could be caused by small variations in blending associated with changes in the ground-based seeing, or other minor secular variations in the photometry. Therefore, we conclude that, when photometry alone is used to constrain the parameters of the event, only a reliable joint constraint on π_E and φ_LS can be derived. Fortunately, astrometry provides a robust independent estimate of φ_LS, allowing us to break this degeneracy and determine the two quantities separately.

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7.2.2. Astrometric Deflection and Orientation of the Relative Motion

7.2.3. Constraining the Lens Trajectory Orientation

Since the parallactic effect is unimportant for constraining φ_LS, we first fitted the photometry using a light-curve model that neglects parallax. The resultant model (with t₀ −2,450,000 = 5763.33, t_E = 231.56 days, , and g = 19.3) predicts the total deflection in units of θ_E as a function of time, through Equation (6).

As described in Section 5.2, we have accurate measurements of the (x, y) positions of the source at the eight epochs of HST observations. These positions are affected by both the proper motion of the source and its deflections, which are a function of θ_E and φ_LS. We fitted a model to the positions, whose parameters are the x (R.A.) and y (decl.) components of the proper motion, θ_E, and φ_LS. This fit resulted in values of θ_E = 5.2 ± 0.5 mas and φ_LS = 3379 ± 50.

Components of the resultant deflections as a function of time, after subtracting the best-fitting proper motion, are shown in the top two panels of Figure 16. These panels plot the deflections in the R.A. and decl. directions. The bottom left panel shows the total amount of deflection, again as a function of time. Note that the total deflection reaches a maximum of ∼2 mas in late 2012.

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The bottom right panel of Figure 16 plots the R.A. versus decl. deflections. These deflections are always along the line joining the lens to the source, and thus at large separations their direction is opposite to the direction of the relative motion of the lens. The solid black line passes through the origin, at the φ_LS angle derived above. The dotted blue lines indicate the allowed range based on the uncertainties. Although the best-fit value from photometry alone is φ_LS = 3548 (shown by the solid magenta line), the allowed range of φ_LS (shown by the dashed magenta lines) has a flat probability distribution. The combined constraint from photometry and astrometry is used in our subsequent analysis.

Note that for clarity in Figure 16 we have shown the mean deflections at each epoch, and not the individual measurements. In particular, this avoids a confusing overlap of points in the bottom right panel, where the deflections are not a monotonic function of time. The individual measurements are shown in the next section.

In addition to OGLE and MOA, photometric time-series data were obtained at several more observatories (see Table 2). These data typically cover a relatively narrow range of ∼20 days around the peak, beíng primarily aimed at searching for planet-related distortions of the light curve. These photometric series, unlike those obtained by the survey programs, do not provide significant constraints on the lens–source model—especially since each set of observations can have a different baseline magnitude and blending parameter. Nevertheless, in our initial approach, we included all the data in our analysis but also carried out analyses separately for "OGLE-only," "MOA-only," and "OGLE+MOA-only" data sets.

Ground-based time-series photometry is susceptible to systematic noise, and this must not be mistaken for real features of the light curve. To improve the robustness of our solution, we model the photometric uncertainties, and moreover force the model to follow the bulk of the data by explicitly down-weighting outliers. As implemented for the SIGNALMEN microlensing anomaly detector (Dominik et al. 2007, 2019), we specifically adopt a bi-square weight function with regard to the median residual and a Gaussian distribution for the uncertainties with revised standard deviation in magnitude of

where σ denotes the reported error bar, κ is a scaling factor, and σ₀ corresponds to a systematic error added in quadrature. For the plot of the various data sets as shown in Figure 14, we give the respective estimated values of κ and σ₀ in Table 4. If the size of the error bars does not vary substantially, there is a degeneracy between κ and σ₀, and either of the parameters provides modified constant error bars (while it does not matter which).

We find two viable models, significantly only distinguished by the sign of u₀, and will refer in the following to the model with u_0,−. The microlensing parallax parameter π_E is constrained by the wing of the light curve and much less sensitive to the peak region, and therefore it is mostly constrained by the microlensing survey data. From various combinations of data sets, we consistently find π_E = 0.10 ± 0.02 but some variation in the trajectory angle ψ, correlated with t_E and the blend fraction, yielding visually indistinguishable model light curves.

However, the angle of lens–source proper motion, φ_LS, follows robustly from the astrometric data (see Section 7.2.2), given that the centroid shift to first order (i.e., neglecting the small distortion caused by parallax) traces an ellipse (a highly flattened ellipse resembling a line in our case) whose semimajor axis is parallel to μ _LS. If we restrict this angle to the range 333° ≤ φ_LS ≤ 343°, as suggested by the astrometric data, the photometric light curve does not change substantially, and we find π_E ≃ 0.086.

We emphasize here that it is incorrect to say that there is a discrepancy between the paths determined from photometry and astrometry, since there is a correlation between φ_LS and π_E in the photometric solution. However, restricting the trajectory angle φ_LS as robustly derived in the last section from the orientation of the centroid shifts to the range 333° ≤ φ_LS ≤ 343° suggests π_E ≃ 0.086.

We now turn to a full analysis in which we fit the astrometric and photometric data simultaneously in order to obtain all of the parameters. Such a solution is important, since the crucial parameters of θ_E and π_E are derived from two different types of data. A simultaneous solution is also essential for a correct estimate of the uncertainties in the model parameters.

We follow the same plane-of-sky approach described in Section 7.2, which makes it easier to work with, and also show the actual paths of the lens and the source, as well as the deflections. We follow a different parameterization procedure where the model parameters we optimize contain all terms needed to characterize the positions of the lens and the source on the sky as a function of time; these include the reference positions and proper motions of both lens and source, their relative parallax, and the angular Einstein radius of the lens. In principle, the source parallax is also needed; however, its parallax in the reference system we use is close to zero, and it is not meaningfully constrained by the observations. As discussed below in Section 9.2, the best constraints on the source distance come instead from photometry and high-resolution spectroscopy.

From these parameters, the undeflected paths of the source and lens can be determined. The deflection of the lensed image of the source is then computed, and the resulting deflected source positions are matched to the observed positions. The same calculation also yields the source magnification; in order to match the observed photometry, the model must include a baseline magnitude of the source and a blending parameter for each photometric data set. Consistent with the previous approach, we found that most of the photometric data sets cover too short a time interval to yield meaningful constraints on the event parameters in the presence of significant blending; therefore, we limit the model optimization to the MOA and OGLE photometric data sets and validate the resulting model for the other data sets separately (see Section 8.1). In addition, in order to avoid undue impact from any secular variations in photometric responses, we only include OGLE and MOA photometric measurements within ±2 yr from the peak of the event. We adopt the approximate values of φ_LS, t_E, t₀, and u₀ from the analysis of the previous section as our initial estimates, but we leave all parameters free in the optimization; the baseline magnitude and blending parameters for MOA and OGLE are also separately optimized.

As discussed in the previous subsection, the results of the optimization depend to some extent on the relative weighting of astrometry and photometry. Because the number of photometric measurements greatly exceeds that of astrometric measurements and the nominal photometric uncertainties are very small, an optimization using nominal errors disproportionately weights photometry, resulting in a poor match to the astrometry. In order to obtain a more balanced weighting of astrometry and photometry, we scaled the photometric errors by different amounts in different temporal bins making sure that the scaled errors are compatible with the statistical dispersions in the measurements, and validated the solution based on a reasonable match to the astrometric data. In our final model, most photometric points are still at an uncertainty below 10 mmag. This solution has a total astrometric χ² of 136 with 106 points. (The solution with nominal weights has a higher astrometric χ² of 149.) The parameters of the final model are given in Table 5.

Figure 17 shows the reconstructed motion of the lens (magenta) and of the source (black) in the plane of the sky, based on our adopted model. The predicted apparent source trajectory is shown by the green solid line. The astrometric measurements are shown individually by the small filled circles, and their epoch averages are shown as red triangles. Cyan squares show the model position at each epoch; gray lines connect the model lens position to the undeflected and deflected source positions at each epoch.

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Figure 18 presents the measured and predicted source positions separately for the x and y coordinates. To improve the legibility of the plot, the fitted proper motion of the source has been subtracted from both model and data; therefore, the points shown represent the deflection of the source. The black line is our final adopted model, which takes photometry and astrometry into account.

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We can now check the consistency of the final model with our measured HST photometry in the F606W and F814W filters and also constrain the amount of blending. We used the final model to calculate the magnifications at the HST observation epochs and varied the baseline magnitudes and the blending factor to fit the HST photometry. The resulting fit is excellent (see Figure 13) and yields baseline source magnitudes of m_F606W =21.943 ± 0.012 and m_F814W = 19.578 ± 0.012, with corresponding blending factors for the HST photometry of g =−0.012 ± 0.015 and −0.006 ± 0.012, respectively. The bottom two panels of Figure 13 show the residuals of the observations relative to the model in F606W and F814W at the epochs of HST observations, where we have assumed the minimum physically allowed value of g = 0. (It makes little difference if we instead use the slightly negative values of g from the formal fit.) The stringent constraints on blending and lack of color variation make it unlikely that our deflection measurements could be affected by blending with a binary companion or a field star lying within the HST PSF.

The path of the lens in the sky plane, as derived from the above analysis, is shown in Figure 15. We note that there are two solutions corresponding to u_0,+ and u_0,− (Gould 2004). However, at t₀, the angular separation between the source and the lens is dominated by the parallactic motion of the lens (the separation caused by the parallactic motion is ∼0.04, compared to the actual value of u₀ ≃ 0.00271). Thus, the path without parallax in both solutions lies on the u_0,+ side.

The paths of the lens for the u_0,+ and u_0,− solutions are shown in Figure 15, where the dashed lines show the path of the lens without parallax, and the solid lines show the path of the lens after the parallactic motion is taken into account. The respective paths are separated only by 0.02 mas at the time of maximum magnification, and they quickly merge. The u_0,− solution is the preferred solution and is the one used in our analysis. However, we verified that both solutions provide practically identical deflections (since the deflection measurements are at much higher u, where the two paths nearly merge), and the results are the same for all practical purposes in both solutions.

Table 5. Parameters of the Full Fit to Astrometry and Photometry

Parameter	Units	Value	Uncertainty (1σ)	Notes a
μS (R.A.)	mas yr−1	−2.263	0.029	(1)
μS (decl.)	mas yr−1	−3.597	0.030	(1)
θE	mas	5.18	0.51	(2)
	days	270.7	11.2	(3)
φLS	deg	342.5	4.9	(4)
(HJD −2,450,000.0)	days	5765.00	0.87	(5)
πE		0.0894	0.0135	(6)
		0.0422	0.0072	(7)
MOA baseline R magnitude	mag	16.5147	0.0016
MOA blending parameter		16.07	0.66
OGLE baseline I magnitude	mag	16.4063	0.0015
OGLE blending parameter		18.80	0.79
Derived parameters:
u0		0.00271		(8)
t0 (HJD −2,450,000.0)	days	5763.32		(9)

Note.

^a Notes: (1) Undeflected proper motion of the source in Gaia EDR3 absolute frame. (2) Angular Einstein radius. (3) Angular Einstein radius θ_E divided by absolute value of lens–source proper motion μ_LS. (4) Orientation angle of the lens proper motion relative to the source (N through E). (5) Time of closest angular approach without parallax motion. (6) Relative parallax of lens and source in units of θ_E. (7) Impact parameter in units of θ_E without parallax motion. (8) Impact parameter derived using the model-fit parameters above and after including parallax motion, in units of θ_E. (9) Time of closest angular approach derived as above and after including parallax motion.

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The first two rows of Table 5 give the proper motions of the source. We use the values of θ_E, , and φ_LS given in the next three rows to determine the proper motion of the lens with respect to the source as −2.10 ± 0.22 mas yr⁻¹ and 6.66 ± 0.67 mas yr⁻¹ in R.A. and decl., respectively. The resulting absolute proper motions of the lens are given below in Table 6.

In a target-of-opportunity program focusing on high-dispersion spectroscopy of high-amplitude microlensing events in the Galactic bulge, Bensby et al. (2013) obtained four spectra of MOA-11-191/OGLE-11-462 on 2011 July 20 and 21. These observations were made almost precisely around the dates of maximum magnification. The authors employed spectrographs on three different large telescopes, VLT, Magellan, and Keck I, at spectral resolutions ranging from 46,000 to 90,000. The aim of their program was to determine chemical compositions of dwarfs and subgiant stars in the Galactic bulge.

Bensby et al. (2013) carried out a non-LTE model atmosphere analysis of the spectra, obtaining an effective temperature and surface gravity of T_eff = 5382 ± 92 K and , consistent with a late G-type subgiant. The metallicity was found to be slightly above solar, at [Fe/H] = +0.26 ± 0.14, and the radial velocity is + 134.0 km s⁻¹.

As discussed above (e.g., Figures 4 and 5), the HST images show that MOA-11-191/OGLE-11-462 is accompanied by a neighboring star only 04 away, which would have been included in the spectrograph apertures. However, at the dates of the observations, MOA-11-191/OGLE-11-462 was ∼3.2 mag brighter than the neighbor in the F606W bandpass; thus, the contamination was about 5%, which should have a minimal effect on the spectroscopic investigation.

As shown in Figure 7, the neighbor has a color similar to that of the baseline source but is more luminous. We verified its small effect on the spectroscopy by artificially contaminating the source spectrum with a 5% flux from a giant star with the same effective temperature and metallicity, but with . This produced a smaller change in the line strengths of the combined light than would a change of for the source star by its 0.13 dex uncertainty.

We estimate the distance to the source using two independent sets of theoretical stellar isochrones: "Padova" isochrones, which we obtained using the PARSEC web tool ⁷⁸ (Bressan et al. 2012; Marigo et al. 2017), and "BaSTI" isochrones ⁷⁹ (Hidalgo et al. 2018; Pietrinferni et al. 2021). In Figure 19 we plot the position of the source in the distance-independent versus plane (red filled circle with error bars). Superposed are PARSEC isochrones, calculated for the measured metallicity of [M/H] = +0.26 (blue curves), and BaSTI isochrones, for a metallicity of [M/H] = +0.3 (orange curves). Both sets of isochrones agree well around the location of the source star, yielding an age of about 4 Gyr, but with considerable uncertainty.

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At the position of the source in the diagram, a star on the PARSEC 4 Gyr isochrone has absolute magnitudes (Vegamag scale) in the native HST/WFC3 bandpasses of M_F606W =+3.04 and M_F814W = +2.40 and an intrinsic color of . The BaSTI models—which have been updated recently, using new filter throughput curves released in 2020 October by the WFC3 team (Calamida et al. 2021)—yield similar absolute magnitudes of +3.01 and +2.38 and an intrinsic color of 0.63. Taking the averages and using the observed baseline color of m_F606W −m_F814W = 2.365 (Table 1), we find a color excess of E(m_F606W − m_F814W) = 1.73. Based on the values of A_λ /A_V given for the WFC3 filters at the PARSEC website (derived from the interstellar extinction curves of Cardelli et al. 1989 and O'Donnell 1994, with R_V = 3.1), this color excess corresponds to an extinction of A_F814W = 3.33 mag. Thus, using an apparent magnitude at baseline of m_F814W = 19.581 from Table 1, we find a true distance modulus of , giving a linear distance of D_S = 5.9 ± 1.3 kpc. The quoted error is the formal uncertainty (dominated by the uncertainty in the spectroscopic ), but there likely are additional systematic errors, given the large amount of extinction, the assumption of a standard value of R_V , the high sensitivity to , and other sources. For example, if a low value of R_V ≃ 2.5, as deduced by Nataf et al. (2013) for the Galactic bulge region, were used, the distance would increase by ∼2.8 kpc.

The nominal 5.9 kpc distance places the source on the near edge of the Galactic bulge. However, it cannot be ruled out that the source is actually located closer to the center of the Galactic bulge, at a distance of ≈ 8 kpc. The star's high radial velocity is consistent with stars in the bulge but not large enough to rule out disk membership.

Taking the spectroscopy, photometry, and derived distance at face value, these parameters correspond to an angular diameter of the source of ∼0.0038 mas.

We can estimate the effect of the finite size of a small source on the light curve by expanding A(u) around u = u₀, giving

where the prime symbol denotes the derivative. If we average over the face of a spherically symmetric source star, the term with the first derivative cancels out, leaving the quadratic term as the first one that contributes to the difference between finite and pointlike source. Averaging the latter over an extent δ explicitly gives

For u ≪ 1, one finds A(u) ≃ 1/u and thereby . Consequently, we find the largest differences between finite and pointlike sources for the smallest u. For a source of angular radius θ_⋆, we find a source size parameter ρ_⋆ ≡ θ_⋆/θ_E, and if u₀ ≫ ρ_⋆, we can approximate the magnification difference by

With the angular diameter of the source 2 θ_⋆ ≃ 0.0038 mas and the angular Einstein radius θ_E = 5.2 mas, we find ρ_⋆ ≡ θ_⋆/θ_E = 0.00037 while u₀ = 0.00271. Inserting into Equation (20) gives ΔA = 1.5 as compared to A(u₀) = 369 at closest angular approach between lens and source, in close agreement with a numerical evaluation of the average of the exact magnification function A(u). With a blend ratio g ≃ 20, comparing for A and A + ΔA gives a difference of 4 mmag between finite source and pointlike source, which is of the order of the systematic errors of the photometry.

Accounting for the finite source size will thus result in a very small change in u₀, which has very little effect on other parameters, evidenced by their practically identical values for our u_0,+ and u_0,− solutions. Similar to the case of the long-duration microlensing event OGLE-2014-BLG-1186 discussed by Dominik et al. (2019), the measurement of the parallax parameter π_E comes from the wings of the photometric light curve, not from the peak. Additionally, none of the astrometric deflection measurements are close to the peak. With the finite size of the source making little difference, we can safely neglect any limb-darkening effects.

As shown by Equation (16), the mass of the MOA-11-191/OGLE-11-462 lens can be determined from the values of its angular Einstein radius, θ_E, and the relative lens–source parallax, π_E. In Section 8, we derived θ_E = 5.18 ± 0.51 mas and π_E = 0.089 ± 0.014. These yield a mass of M_lens = 7.1 ±1.3 M_⊙.

An object with a mass this large cannot be a single (or double) NS or white dwarf. It can only be a BH—or an ordinary star, or conceivably a binary (or higher multiple) containing stars, BHs, and/or other compact companions. To distinguish between these possibilities, we consider the observational constraints on the luminosity and binarity of the lens.

The mass determination for the lens is independent of assumptions about the distances to the lens and source. However, to constrain the optical luminosity of the lens, we first do need to determine its distance. This can be obtained from Equation (2), using the value of π_LS = 0.463 ± 0.051 mas (from Section 8) and the distance to the source (from Section 9.2) D_S = 5.9 ± 1.3 kpc.

These values yield a lens distance of D_L = 1.58 ± 0.18 kpc. Note that the derived distance of the lens is only a weak function of the adopted source distance. If the source were at the distance of the Galactic center, the lens distance would only increase to ∼1.7 kpc.

10.3.1. From the Baseline Magnitude

10.3.2. Direct Limits from Final-epoch HST Imaging

A much stronger constraint on its optical luminosity comes from the fact that the lens was not detected in our final-epoch HST frames. The parameters θ_E = 5.18 ± 0.51 mas and t_E = 270.7 ± 11.2 days imply that the proper motion of the lens with respect to the source is 6.99 ± 0.75 mas yr⁻¹. At peak magnification in 2011 July, the separation between lens and source was ∼0.01 mas. Therefore, at the epoch of our last HST observation in 2017 (6.1 yr after the peak), the lens–source separation had increased to ∼42.6 mas. At this large a separation, a luminous lens would cause the PSF of the source in our images to show clear signs of elongation. In order to search for such a distortion, we subtracted a "library" F814W PSF ⁸¹ from the final-epoch F814W images (in the individual unresampled _flc frames). We saw no indication of elongation.

In order to place the most stringent limit possible on the lens brightness, we fit the above-referenced PSF to stars of similar brightness and color to that of the source and adjusted the shape of the PSF to optimize the fit (again, in the unresampled frame). We saw a ∼1% variation of the PSF, which is typical of minor, "breathing"-related changes in telescope focus.

Figure 20 shows stacks of the source in our 2017 data at successive stages of analysis and illustrates the process of searching for photometric evidence of the lens. In particular, panel (d) shows the subtraction with the improved PSF. For illustration purposes, we show in panel (f) the residual pattern expected for a lens that is 10% of the brightness of the source. There is no such signature visible in panel (d). We quantify this as follows.

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Using the improved PSF, we modeled the source image as a superposition of two stars separated by 42 mas (about 1 detector pixel) along the implied direction of proper motion. We did this simultaneously in the 4 × 250 s F814W images from 2017. We found a best-fit flux for the lens of −1% of the source flux. We performed the same fits to similar-brightness neighboring stars and found a distribution between −2% and +1%. Finally, we added to each neighbor star a companion of 2.5% at the presumed location and were able to recover the added flux at a level of 2.5% ± 1%. We would clearly detect a lens if it were there.

We therefore conclude that the brightest the lens could be at the presumed location in 2017 is ∼1% of the source flux. We also explored how bright a lens could be at an offset of 42 mas in any direction from the source. We were able to rule out any lens brighter than 2.5% of the source flux.

A luminosity of ∼1% of that of the source corresponds to an apparent F814W or I magnitude of ∼24.6. At the distance and extinction of the lens, the limit on its absolute magnitude is M_I ≳ 10.2 (corresponding to a main-sequence star of about 0.2 M_⊙ or less, or a white dwarf of ≈0.6 M_⊙ with a cooling age older than ∼10⁷ yr).

These considerations leave a single BH—or a BH with a mass greater than ∼3.55 M_⊙ in a binary paired with another BH, or with an NS or faint star—as the only viable possibilities for the lens.

Having established that the MOA-11-191/OGLE-11-462 lens is an extremely faint, or nonluminous, object with a mass of 7.1 ± 1.3 M_⊙, we now consider in more detail whether the lens could be a binary system rather than a single, isolated BH.

MOA-11-191/OGLE-11-462 was a high-magnification event, making its light curve especially sensitive to lens binarity. As described above, the event was intensely monitored photometrically by several microlensing groups in search of planetary-companion signatures, and no manifestation of a binary lens was seen.

In fact, lens binarity will strongly affect both the photometric and the astrometric signature if the angular separation between the components is around the angular Einstein radius, θ_E, and it therefore has to be either much smaller or much larger. In either case, such a binary lens would produce a diamond-shaped caustic.

For a close binary, this caustic would be located around the center of mass of the lens, and both the photometric and astrometric microlensing signatures would be strongly altered if the source–lens trajectory passed over it or came close. Specifically, for a mass ratio q and a separation d θ_E, the caustic extends to (Schneider & Weiß 1986; Erdl & Schneider 1993; Dominik 1999)

Comparing this to u₀ ≃ 0.00271 yields the constraint d < 0.07 for the equal-mass case, with less than 11% variation over the range q ∈ [0.4, 1]. With D_L θ_E = 8.2 au, this implies a separation smaller than 0.6 au. Considering a systematic noise floor of ∼5 mmag, deviations from a point-lens light curve exceed this for the given u₀ ≃ 0.00271 and blend ratio g ≃ 20 if d > 0.022, so that our data imply an upper limit to the separation of ∼0.18 au.

In such a case, since one of the components has to have a mass of at least 3.55 ± 0.55 M_⊙, the merging timescale for such a system is ≲10⁷ yr (Carroll & Ostlie 2006). Given that there is no evidence of active star formation in this location, this seems unlikely.

In the case of widely separated binary components, the observed properties of the lens refer to one of them, leading to the same values as by assuming that the lens is a single isolated body, so that the mass determination of the putative BH remains valid.

With θ_E referring to the mass of the object identified, in the wide-binary case one finds the caustic extending to (e.g., Dominik 1999)

Considering that any companion should be dark, as well as the noise floor of our photometric data, we exclude any companion more massive than 10% of the identified lens within 230 au.

To further quantify the limits on a binary companion to the lens, we performed a Rhie et al. (2000)−type analysis in which we simulate hypothetical binary-lens light curves with the same epochs and properties as the real data. We then fit those light curves with a point-lens model to determine the potential signal from a given lens companion. For this test, we use a subset of the data that provides good coverage of the overall light curve and that is sufficient to provide dense coverage of the peak, where a potential companion would have the greatest effect (Griest & Safizadeh 1998). Specifically, we use the OGLE and MOA survey data sets and the microFUN data from Auckland Observatory, Kumeu Observatory, Vintage Lane Observatory, the Wise Observatory 0.46 m, and the Weizmann Institute. Setting the requirement that the χ² due to the planet is >50, we find that the sensitivity limits can be approximated with a piecewise function

Thus, for q = 0.1, this analysis also excludes companions with separations 0.4 au ≲ a_⊥ ≲ 180 au.

To summarize this section, we have found that the lens that produced the MOA-11-191/OGLE-11-462 microlensing event has a mass of 7.1 ± 1.3 M_⊙. There are stringent limits on its optical luminosity, ruling out the possibility that the lens could be an ordinary star of that mass. Its mass is greater than possible for a white dwarf or NS, or for a binary pair of white dwarfs and/or NSs. Even if the lens is a binary, at least one of its components must still be a BH—but only under unlikely circumstances could it actually be a binary at all. In the rest of our discussion, we will assume that the lens is an isolated, single BH.

The BH lies in the direction of, but is considerably closer than, the Galactic bulge. The fifth row in Table 6 gives the rectangular components of its Galactic position relative to the Sun. The BH is located in the Galactic disk at a distance from us of ∼1580 pc toward the Galactic center and about 45 pc below the Galactic plane. This places it approximately between the Scutum–Centaurus and Sagittarius–Carina spiral arms of the Milky Way, in the terminology of Reid et al. (2019). There are no conspicuous star-forming regions, young stellar objects, or SN remnants at this location.

Most of the stars contained in the small FOV of our HST frames lie at considerably larger distances than the BH itself. In order to study a larger sample of the relatively nearby stellar population surrounding the spatial location of the BH, we selected stars from Gaia EDR3 that satisfy the following criteria: they lie within of the BH on the sky, have trigonometric parallaxes with fractional errors of less than 10% that fall in the range 0.55–0.70 mas, and are brighter than G = 17.

Figure 21 shows the CMD of this population in the Gaia magnitude and color system. The CMD is afflicted by severe differential extinction, which blurs the stellar sequences. Nevertheless, it shows two prominent components: a moderately young and sparse population of main-sequence stars bluer than G_BP − G_RP ≃ 1.0, and a denser older component of redder stars. There is little or no evidence for an extremely young population of massive stars, at the BH's location. However, with a transverse proper motion of ∼ 45 km s⁻¹ (Table 6), it would take only 0.2 Myr for the lens to traverse , so it is possible that the lens originated much farther away.

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The sixth row in Table 6 gives the components of the BH's space velocity relative to the Sun in the V and W directions (in the directions of Galactic rotation and toward the north Galactic pole, respectively). The third component of the space velocity, U, is undetermined since the radial velocity of the BH is unknown; changing the radial velocity to −100 and +100 km s⁻¹ changes V and W only by about ±0.2 and ±2.8 km s⁻¹, respectively. The relatively low value of V indicates that the BH is not a member of an extreme Galactic halo population and may be moving in a roughly circular Galactic orbit (unless the radial velocity is large). However, it does have a moderately high W velocity perpendicular to the Galactic plane, so the orbit has a modest inclination to the Galactic plane. Apart from these considerations, there are few constraints on the age of the BH, except that it has likely had time to move far from its birthplace.

Figure 22 plots the absolute proper motion of the BH (black filled circle with error bars, from the fourth row of Table 6) and compares it with the proper motions of the neighboring Gaia stars from the sample described above. The directions of Galactic longitude and latitude are indicated at the lower left, showing that the dispersions are higher in the longitude direction than perpendicular to it. Members of the "young" population from Figure 21 are plotted with blue points, and members of the older population are plotted with red points. The older population, which has proper-motion dispersions of σ_μ,l = 2.59 mas yr⁻¹ and σ_μ,b = 1.59 mas yr⁻¹, is dynamically hotter than the younger population, which has dispersions of σ_μ,l = 1.08 mas yr⁻¹ and σ_μ,b = 1.05 mas yr⁻¹.

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The BH itself is a prominent outlier, predominantly in b, relative to the older and younger populations (∼4σ and 6σ, respectively). This suggests that it may have received a "natal kick" from the SN explosion associated with its birth—assuming that the BH arose from the surrounding population. Relative to the mean proper motion of the younger population, the BH's motion is offset by about 6.0 mas yr⁻¹, corresponding to a tangential velocity offset of ∼45 km s⁻¹. However, considering that its radial velocity is unknown, we cannot exclude the possibility that the BH might simply be passing through the surrounding population and could conceivably be significantly older. In a recent paper, Andrews & Kalogera (2022) have given an extensive discussion of the possible natal kick of MOA-11-191/OGLE-11-462.

As reviewed in Section 1.1, it is generally believed that only stars above ∼20 M_⊙ collapse to form BHs. To form an isolated ∼7 M_⊙ BH, such as the one discussed in this paper, either its progenitor was a single star that experienced considerable mass loss through stellar winds, it was produced through fallback in a weak SN explosion (e.g., Fryer et al. 2012), or it was born in a close binary that lost mass through a common-envelope mass ejection. In the binary scenario, the binary evidently became unbound, presumably when its lower-mass companion itself became an SN, and its mass loss and SN kick detached the binary.

The MOA-11-191/OGLE-11-462 BH's inferred mass of ∼.1 M_⊙ lies remarkably close to the peak of the Galactic X-ray–binary BH mass distribution (see Section 1.1). This may suggest a similar evolutionary path with mass loss through close-binary interactions, but a larger sample of mass measurements for isolated BHs is needed to distinguish these different formation scenarios.

There is no known X-ray or radio source at the position of MOA-11-191/OGLE-11-462. Considering the high extinction toward the BH, it is possible that it lies in a region of high density of interstellar matter. If so, the accretion rate of material from the interstellar matter by the BH could conceivably be large enough for it to be detected in deep X-ray and/or radio observations, as discussed in Section 1.2, and these would be worth attempting.

Finally, we note the potential of future facilities, such as the Very Large Telescope Interferometer (VLTI), the Nancy Grace Roman Space Telescope, and the Rubin Observatory, in this area. The GRAVITY instrument with VLTI is currently being upgraded to further improve its sensitivity (Eisenhauer 2019), which will make it possible to measure θ_E from spatially resolved microlensed images (Dong et al. 2019) of a large number of microlensed sources in the Milky Way. Roman will detect and characterize several thousand microlensing events (Spergel et al. 2015; Penny et al. 2019); with the same aperture size as HST, its astrometric capabilities are expected to be similar (WFIRST Astrometry Working Group et al. 2019). According to current plans, Roman will observe microlensing events over a ∼2 deg² region in the Galactic bulge with a ∼15-minute cadence for several months each year; thus, it will make thousands of photometric and astrometric observations of every long-duration event, leading to very high accuracy in both kinds of measurements. With such data, Roman should detect numerous BHs, yielding exquisite determinations of their masses, distances, and velocities, and providing important insights to our understanding of the formation and evolution of BHs. Rubin's wide-angle survey will provide deep, long-baseline photometry and consistent data reduction, making it ideal for the discovery and photometric characterization of microlensing events (Sajadian & Poleski 2019), probing many different lines of sight (and hence stellar populations) across the Milky Way and Magellanic Clouds (Street et al. 2018b). In the bulge, it will complement the Roman data by filling in gaps in the Roman survey cadence (Street et al. 2018a; Strader et al. 2018). Rubin discoveries may need high-resolution time-series imaging follow-up for full characterization of the events.