KMT-2021-BLG-1122L: The first microlensing triple stellar system

We systematically inspect the microlensing data acquired by the KMTNet survey during the previous seasons in order to find anomalous lensing events for which the anomalies in the lensing light curves cannot be explained by the usual binary-lens or binary-source interpretations. From the inspection, we find that interpreting the three lensing events OGLE-2018-BLG-0584, KMT-2018-BLG-2119, and KMT-2021-BLG-1122 requires four-body (lens+source) models, in which either both the lens and source are binaries (2L2S event) or the lens is a triple system (3L1S event). Following the analyses of the 2L2S events presented in \citet{Han2023}, here we present the 3L1S analysis of the KMT-2021-BLG-1122. It is found that the lens of the event KMT-2021-BLG-1122 is composed of three masses, in which the projected separations (normalized to the angular Einstein radius) and mass ratios between the lens companions and the primary are $(s_2, q_2)\sim (1.4, 0.53)$ and $(s_3, q_3) \sim (1.6, 0.24)$. By conducting a Bayesian analysis, we estimate that the masses of the individual lens components are $(M_1, M_2, M_3)\sim (0.47\,M_\odot, 0.24\,M_\odot, 0.11\,M_\odot)$. The companions are separated in projection from the primary by $(a_{\perp,2}, a_{\perp,3})\sim (3.5, 4.0)$~AU. The lens of KMT-2018-BLG-2119 is the first triple stellar system detected via microlensing.


Introduction
Light curves of gravitational microlensing events often exhibit deviations from the smooth and symmetric form of the event involved with a single lens mass and a single source star (Paczyński 1986).The most common cause of such deviations is the binarity of the lens, 2L1S event (Mao & Paczyński 1991), or the source, 1L2S event (Griest & Hu 1993;Han & Gould 1997).Anomalous lensing events comprise about 10% of the more than 3000 lensing events that are being annually detected from the three, currently working, lensing surveys conducted by the Optical Gravitational Lensing Experiment (OGLE; Udalski et al. 2015), Microlensing Observations in Astrophysics (MOA; Bond et al. 2001), and Korea Microlensing Telescope Network (KMTNet; Kim et al. 2016) groups.The prime goal of these survey experiments is to find extrasolar planets, which are detected via anomalous signals in lensing light curves.Planetary microlensing signals appear in various shapes and are often confused with anomalies produced by other causes.Therefore, in order to single out planetary signals from those of other origins, it is essential to analyze all anomalous events.
In the current microlensing experiments, anomalous events are being analyzed almost in real time with the progress of events.This became possible with the implementation of early warning systems: the OGLE early warning system (Udalski et al. 1994), the MOA real-time analysis system (Bond et al. 2001), and the KMTNet EventFinder AlertFinder (Kim et al. 2018a) system 1 .These systems enable one to detect lensing events in their early phases and to promptly identify anomalies appearing in lensing light curves.Analyses of the events with identified anomalies are conducted by multiple modelers of the individual survey groups as the anomalies proceed, and models of the events found from these analyses are circulated or posted on web pages2 to inform researchers in the microlensing community about the nature of the anomalies.In most cases, anomalous events are interpreted with a binary-lens (2L1S) or a binary-source (1L2S) model.
For a minor fraction of anomalous events, it is known that the anomalies cannot be explained by the usual 2L1S or 1L2S interpretation.Han et al. (2023), hereafter Paper I, conducted a systematic investigation of events found from the KMTNet survey during the previous seasons in search for anomalous events for which no plausible models had been presented.From this investigation, they found that the anomalies appearing in the light curves of the events OGLE-2018-BLG-0584 and KMT-2018-BLG-2119 required a four-body (lens+source) model, in which both the lens and source are binaries (2L2S event).In this paper, we present the analysis of another four-body lensing event KMT-2021-BLG-1122.Unlike the two 2L2S events presented in Paper I, the lens system of KMT-2021-BLG-1122 is composed of three masses (3L1S).
We present the analysis of the event KMT-2021-BLG-1122 as follows.In Sect.2, we depict the observations conducted to obtain the photometric data of the event, and describe the characteristic features of the anomaly appearing in the light curve of the event.In Sect.3, we begin by describing the lensing parameters used in the modeling conducted under various configurations of the lens and source system.We then describe, in detail, the analyses conducted under the individual configurations in the following subsections: 2L1S model in Sect. 3.1,2L2S model in Sect. 3.2,and 3L1S model in Sect. 3.3. In Sect. 4, we specify the source star of the event and estimate the angular Einstein radius of the lens system.In Sect.5, we estimate the physical parameters including the masses of the individual lens components, the distance to the lens system, and the projected separations among the lens components.We summarize the results from the analysis and conclude in Sect.6.

Lensing light curve and observations
The lensing event KMT-2021-BLG-1122 was detected solely by the KMTNet survey.The event occurred on a faint source star, with a baseline magnitude of I base = 20.36,located toward the Galactic bulge field with equatorial coordinates (RA, Dec) J2000 = (17:35:51.10, −28:26:47.22),which correspond to the Galactic coordinates (l, b) = (−0 • .720, 2 • .076).The source lies in the KMTNet BLG14 field, toward which observations were conducted with a 1 hr cadence.The event was identified by the KMTNet AlertFinder system and an alert was issued on 2021 June 3, which corresponds to the abridged Heliocentric Julian Date of HJD ≡ HJD − 2450000 ∼ 9368.5.
Observations of the event were carried out with the use of the three 1.6 m telescopes operated by the KMTNet group.The telescopes are located in the three continents of the Southern Hemisphere so continuous coverage of lensing events are possible, and the sites of the individual telescopes are the Siding Spring Observatory in Australia (KMTA), the Cerro Tololo Interamerican Observatory in Chile (KMTC), and the South African Astronomical Observatory in South Africa (KMTS).The camera mounted on each telescope is composed of four 9k × 9k CCD chips, which in combination yield a 4 deg 2 field of view.
Images containing the source were mainly taken in the I band, and about one-tenth of images were obtained in the V band.Reductions of the images and photometry of the event were done using the KMTNet pipeline constructed based on the pySIS code of Albrow et al. (2009), and additional photometry was carried out for a subset of the KMTC data set using the pyDIA code of Albrow (2017).The analysis of the event was done based on the I-band light curve constructed from the pySIS reduction, and the pyDIA photometry data were used for the source color measurement.We discuss the detailed procedure of the source color measurement in Sect. 4. According to the recipe described in Yee et al. (2012), we readjusted the error bars of the data determined by the automated pipeline in order for the errors to be consistent with the scatter of data and χ 2 per degree of freedom (d.o.f.) for each data set to reach unity.
Figure 1 shows the lensing light curve of KMT-2021-BLG-1122 constructed with the combined I-band pySIS photometry data obtained from the three KMTNet telescopes.It shows a strong anomaly relative to a 1L1S form, and the anomaly displays a complex pattern that is characterized by the three strong features appearing at HJD ∼ 9369.7 (t 1 ), ∼9371.7 (t 2 ), and ∼9373.3(t 3 ).From the characteristic profiles described in Schneider & Weiss (1986), the anomalies at around t 1 and t 3 appear to be a pair of caustic-crossing features that are produced when a source enters and exits a caustic, respectively.On the other hand, the profile at around t 2 , which is approximately symmetric with respect to the peak, appears to be produced when a source approaches close to or crosses over the cusp of a caustic (Schneider & Weiss 1987).
In general, the light-curve profile of a 2L1S event in the region between a pair of caustic-crossing spikes exhibits a "U"-shape pattern.Occasionally, deviations from this U-shape pattern can arise if a source sweeps one fold of a caustic, for example, OGLE-2016-BLG-0890 (Han et al. 2022c); however, in this case, the resulting deviation is smooth and thus differs from the sharp pattern of the A8, page 2 of 8 observed anomaly at around t 2 .This suggests that a different explanation is needed to explain the origin of the anomaly.

Interpreting the anomaly
In order to explain the anomalies in the lensing light curve, we conducted a modeling series under various interpretations of the lens-system configuration.We tested three models under the 2L1S, 2L2S, and 3L1S configurations.We decided to exclude 1L2S and 1L3S models because the observed light curve exhibits obvious caustic-crossing features, that is, those around t 1 and t 3 , while 1L2S and 1L3S events do not involve caustics 3 .
In the modeling under each tested interpretation, we searched for a lensing solution, representing a set of lensing parameters that characterize the observed lensing light curve.The basic lensing parameters of a 1L1S event include (t 0 , u 0 , t E ), which denote the time of the closest source approach to the lens, the source-lens separation (impact parameter) at t 0 , and the Einstein timescale, respectively.The Einstein timescale represents the time required for the source to transit the angular Einstein radius of the lens, θ E , and the length of u 0 is scaled to θ E .
A 2L1S system corresponds to the case in which the lens contains an extra component compared to the 1L1S system.Considering the extra lens component requires one to include additional parameters when modeling.These additional parameters are (s, q, α, ρ), and they denote the projected separation (normalized to θ E ); the mass ratio between the lens components (M 1 and M 2 ); the angle (source trajectory angle) between the relative lenssource proper motion, µ, and the axis connecting M 1 and M 2 ; and the ratio between the angular radius of the source, θ * , to the Einstein radius, that is, ρ = θ * /θ E (normalized source radius); respectively.The normalized source radius is included in modeling because a 2L1S event usually involves caustic crossings, during which lensing magnifications are affected by finite-source effects (Bennett & Rhie 1996).For the computation of finite-source magnifications, we utilized the map-making method of Dong et al. (2006).
For the modeling of a 2L2S system, it is required to include extra parameters in addition to those of the 2L1S model in order to describe an extra source.These extra parameters include (t 0,2 , u 0,2 , ρ 2 , q F ), where the first three denote the closest approach time, impact parameter, and normalized radius of the source companion, and the last parameter indicates the flux ratio between the companion (S 2 ) and primary (S 1 ) of the source.In the 2L2S modeling, we use the notations (t 0,1 , u 0,1 , ρ 1 ) to denote the parameters related to S 1 .
A 3L1S system has an extra lens component (M 3 ) compared to the 2L1S system, and this requires one to include additional model parameters.These parameters include (s 3 , q 3 , ψ), which denote the separation and mass ratio between M 1 and M 3 , as well as the orientation angle of M 3 as measured from the M 1 -M 2 axis with a center at the position of M 1 , respectively.In order to distinguish the parameters describing M 2 from those describing M 3 , we use the notations (s 2 , q 2 ) to denote the separation and mass ratio of M 2 .The lensing parameters for the 1L1S, 2L1S, 2L2S, and 3L1S models are summarized in Table 1 of Paper I.

2L1S interpretation
We started by modeling the light curve under the 2L1S interpretation of the anomalies.In this modeling, we searched for the binary parameters s and q via a grid approach, while we sought 3 There has been only one 1L3S microlensing event published to date: OGLE-2015-BLG-1459 (Hwang et al. 2018).
to find the other parameters via a downhill approach using a Markov chain Monte Carlo (MCMC) algorithm.We first constructed a ∆χ 2 map on the plane of the grid parameters, identified local minima, and then refined the individual minima by allowing all lensing parameters to vary.From the 2L1S modeling, we found that no lensing solution could describe all the anomaly features, and this explains why the event had been left without a plausible model.
Although the light curve could not be explained by a 2L1S model, we found a solution that could approximately describe the caustic-crossing features at around t 1 and t 3 .The model curve of this solution, found from modeling the light curve with the exclusion of the data around t 2 , is drawn over the data points in Figs. 1  and 2, and the corresponding configuration of the lens system is presented in the upper inset of the top panel of Fig. 2. The binary lensing parameters of the solution are (s, q) ∼ (1.69, 0.43), which results in a single resonant caustic elongated along the binary axis.According to this solution, the source entered the caustic by crossing the caustic fold lying just above the left side on-axis cusp, passed through the inner region of the caustic, and then exited the caustic by crossing the upper fold of the caustic.The caustic entrance and exit resulted in the anomaly features matching the observed ones well around t 1 and t 3 , respectively.

2L2S interpretation
The fact that the two caustic-crossing features can be described by a 2L1S model suggests that the other anomaly feature around t 2 may be explained with the introduction of an extra source (2L2S) or an extra lens component (3L1S) to the 2L1S configuration.In this subsection, we present the analysis conducted under the 2L2S interpretation.
The 2L2S modeling was carried out based on the 2L1S solution.With the initial parameters obtained from the 2L1S modeling, we searched for a 2L2S solution by testing various trajectories of the second source considering the location and magnitude of the anomaly around t 2 .We first checked the case in which the position of the source companion with respect to the primary source does not vary: "static 2L2S model."The model curve of the best-fit static 2L2S solution is presented in Fig. 2, and the lensing parameters are listed in Table 1.The lens-system configuration of the solution is shown in the middle inset of the top panel.The caustic configuration is similar to that of the 2L1S solution, except that there are two source trajectories of S 1 and S 2 , which are marked by black and blue arrowed lines, respectively.According to the solution, the source companion approached the lens slightly earlier than the primary source with an impact parameter greater than that of the primary source.The second source passed the tip of the upper left caustic cusp, and this produced the anomaly feature around t 2 .Although the static 2L2S model can give rise to sharp anomaly features appearing in the inner region between the caustic spikes, it was found that the residual from the 2L2S model was substantial, as shown in the second lower panel of Fig. 2.
We further checked whether the anomalies can be explained by considering the orbital motion of the source: "orbital 2L2S model."We considered the source orbital motion by introducing five extra parameters of (s s , q s , φ, ds s /dt, dφ/dt), where s s and q s represent the normalized separations and mass ratio between the source components, respectively; φ is the orientation angle of S 2 with respect to S 1 ; and (ds s /dt, dφ/dt) represent the change rates of the s s and φ induced by the source orbital motion, respectively.It is found that the consideration of the source orbital motion substantially improves the fit, by ∆χ 2 = 275.5, with respect to A8, page 3 of 8 Fig. 2. Enlarged view around the central anomaly region of the light curve.Three model curves obtained under the 3L1S, 2L2S, and 2L1S lenssystem configurations are drawn over the data points, and the three lower panels show the residuals from the individual models.The two insets in the top panel show the lens-system configurations of the 2L1S and 2L2S models.In each inset, the red figure is the caustic and the line with an arrow represents the source trajectory.For the 2L2S configuration, there are two source trajectories, in which the black and blue ones represent the trajectories of the primary and secondary source stars, respectively.the static model.The model curve of the best-fit orbital 2L2S solution is drawn in the top panel of Fig. 2, the residual is shown in the panel labeled "2L2S (orbit)," and the lensing parameters are listed in Table 1.According to the model, the anomaly features around t 1 and t 3 were produced by the primary source as in the static case, and the feature around t 2 was produced by the secondary source.We note that the secondary source crossed the caustic two more times at HJD ∼ 9368.7 (t 0 ) and ∼9372.7,and the former crossing is not covered by the data and the latter crossing feature is difficult to identify in the data.We note that the source radius ratio ρ 2 /ρ 1 = 1/46 and the source flux ratio q F = 0.15 would lead to inconsistent estimates of θ E , which would argue against the physical plausibility of this solution.However, we have decided not to pursue this issue because this solution would prove to be heavily disfavored by χ 2 .Readers can refer to Sect.3.3 for more details.

3L1S interpretation
We additionally searched for a model under the 3L1S interpretation.Similar to the case of the 2L2S modeling, the 3L1S modeling was done based on the 2L1S solution, from which we adopted the initial parameters of (t 0 , u 0 , t E , s 2 , q 2 , α, ρ).We then conducted preliminary searches for the parameters related to M 3 , that is, (s 3 , q 3 , ψ), via a grid approach, and finally refined the solution by letting all parameters vary.Figure 3 shows the ∆χ 2 map on the log s 3 -log q 3 parameter plane constructed from the preliminary grid search.The map shows that there exists a unique χ 2 minimum at (log s 3 , log q 3 ) ∼ (0.2, −0.6).
It was found that the model identified under the 3L1S interpretation described all the observed anomaly features well in the lensing light curve.The lensing parameters of the best-fit 3L1S solution are listed in Table 2 together with the value of χ 2 /d.o.f.In Fig. 2, we have drawn the model curve (solid curve drawn over the data points) and residual (presented in the panel labeled "3L1S") from the model.The parameters related to M 2 and M 3 are (s 2 , q 2 ) ∼ (1.4,0.53) and (s 3 , q 3 ) ∼ (1.60, 0.24), respectively, indicating that the primary of the lens is accompanied by two lower-mass companions lying in the vicinity of the Einstein ring.If the separations among the lens components were intrinsic (three-dimensional), the three-body lens system would be dynamically unstable, and this indicates that one or both companions are likely to be aligned by chance with the primary.We found that it was difficult to detect microlens-parallax effects A8, page 4 of 8 1.601 ± 0.043 q 3 0.241 ± 0.027 ψ (rad) 1.383 ± 0.010 ρ (10 −3 ) 2.50 ± 0.21 Notes.HJD = HJD − 2450000.(Gould 1992) due to the relatively short timescale, t E ∼ 15 days, of the event compared to the orbital period of Earth.The lens-system configuration of the 3L1S model is shown in Fig. 4, in which the main panel shows the central anomaly region and the inset shows the whole view of the lens system.The coordinates of the configuration is centered at the effective position of the M 1 -M 2 pair defined by Di Stefano & Mao (1996) and An & Han (2002), and the gray curves encompassing the caustic represent the equi-magnification contours.One finds that the caustic induced by M 2 is similar to that of the 2L1S solution, but the 3L1S caustic and the resulting magnification pattern around the caustic are distorted by M 3 , which makes the central part of the caustic nested and self-intersected (Gaudi et al. 1998;Daněk & Heyrovský 2015a,b, 2019).According to the 3L1S solution, the anomaly appearing around t 2 was produced by the source approach close to the tip of the swallow-tail caustic part induced by M 3 .
From the comparison of the 3L1S and 2L2S (orbital) models, we find that the anomaly features in the lensing light curve are better explained by the 3L1S interpretation than by the 2L2S interpretation.In Fig. 5, we present the cumulative distribution of the χ 2 difference, ∆χ 2 = χ 2 2L2S − χ 2 3L1S , between the two mod-Fig.3. Map of ∆χ 2 on the log s 3 -log q 3 parameter plane obtained from the initial grid searches for (s 3 , q 3 ) parameters.Color coding is used to differentiate between points with ≤1nσ (red), ≤2nσ (yellow), ≤3nσ (green), ≤4nσ (cyan), ≤5nσ (blue), and ≤6nσ (purple), where n = 9.The upper panel shows the map around the minima constructed with denser grids.
els in the region of the anomaly for the detailed comparison of the two models.The major differences occur around t 0 and t 1 , at around which point S 2 and S 1 first crossed the caustic according to the 2L2S model, respectively.Besides these regions, the 3L1S model explains the KMTA data point well at the epoch of HJD = 9373.272,while the residual of this data point from the 2L2S model, ∆I = 0.656 mag, is very big.As a whole, the 3L1S model yields a better fit than the 2L2S model by ∆χ 2 = 53.8,despite the fact that 10 lensing parameters are used in the 3L1S modeling compared to the 14 parameters of the 2L2S modeling.

Source star and Einstein radius
We have specified the source of the event not only for the full characterization of the event but also for the estimation of the angular Einstein radius.The Einstein radius was determined from the angular radius of the source, θ * , by the relation where θ * was deduced from the source type, and the normalized source radius ρ was measured from the modeling by analyzing the caustic-crossing parts of the light curve.
We determined the source type by measuring the extinctionand reddening-corrected color and brightness, (V − I, I) 0,S .For this, we utilized the Yoo et al. (2004) method, in which the source is located in the instrumental color-magnitude diagram (CMD), and then its color and magnitude are calibrated using the centroid of the red giant clump (RGC) as a reference.The RGC centroid can be used as a reference for calibration because its de-reddened color and magnitude are known.
A8, page 5 of 8 where the I-and V-band magnitudes were estimated by regressing the KMTC light-curve data of the individual passbands processed using the same pyDIA code with respect to the model.By measuring the offsets in the color and magnitude, ∆(V − I, I), of the source from the RGC centroid, with (V − I, I) RGC = (3.669,18.025), together with the known de-reddened color and magnitude of the RGC centroid, (V − I, I) 0,RGC = (1.060,14.377) (Bensby et al. 2013;Nataf et al. 2013), the de-reddened source color and magnitude were measured as The estimated color and magnitude indicate that the source is a subgiant star of an early K spectral type.The angular source radius was deduced from the measured source color and magnitude.For this, we first converted the V − I color into a V − K color using the color-color relation of Bessell & Brett (1988), and then estimated the angular source radius using the relation between (V − K, V) and θ * of Kervella et al. (2004).This procedure yielded the angular radii of the source and Einstein ring of and  respectively.Together with the measured event timescale, the relative lens-source proper motion was estimated as µ = θ E t E = (12.49± 1.53) mas yr −1 .(6)

Physical lens parameters
In this section, we estimate the physical lens parameters of the mass M and distance D L to the lens.These parameters can A8, page 6 of 8 sometimes be constrained by measuring lensing observables of (t E , θ E , π E ), where π E denotes the microlens parallax.The first two observables are related to the physical parameters by where κ = 4G/(c 2 AU) 8.14 mas M −1 and π rel = π L − π S = AU(1/D L − 1/D S ) represents the relative lens-source parallax.With the additional measurement of the observable π E , the physical parameters can be uniquely constrained by the relations (Gould 1992) For KMT-2021-BLG-1122, the observables t E and θ E were securely measured, but the other observable π E could not be measured, and thus we estimated the physical lens parameters by conducting a Bayesian analysis using a Galactic model together with the constraints provided by the measured observables.
In the first step of the Bayesian analysis, a large number of artificial lensing events were produced from a Monte Carlo simulation.For each simulated event, we derived the locations of the lens and source and their transverse velocity from a Galactic model, and assigned the lens mass from a mass-function model.In the Monte Carlo simulation, we adopted the Galactic model of Jung et al. (2021), whose detailed description is given therein.With the simulated events, we then constructed the posteriors of the lens mass and distance by giving a weight to each simulated event of where (t E , θ E ) and [σ(t E ), σ(θ E )] represent the measured values of the observables and their uncertainties, respectively, and (t E,i , θ E,i ) indicate the observables of each simulated event.
Figure 7 shows the posteriors of the primary lens mass (top panel), and distances to the lens (middle panel) and source (bottom panel).We additionally present the D S posterior to show the relative location of the lens and source.For each posterior, we have marked the contributions by the disk and bulge lens populations with curves drawn in blue and red, respectively.The masses of the individual lens components estimated from the Bayesian analysis are M 2 = 0.24 +0.13 −0.17 M , and The estimated masses of all the lens components are in the mass regime of M dwarfs, and thus the lens is a triple system composed of three low-mass stars.The estimated distance to the lens system is and the probabilities for the lens to be in the disk and bulge are 33% and 67%, respectively.For each parameter, we present the median as a representative value and the lower and upper limits were estimated as the 16% and 84% ranges of the posterior distribution.The projected separations of the secondary and tertiary lens components from the primary are respectively.As noted, the fact that the projected separations a ⊥,2 and a ⊥,3 are similar to each other suggests that one or both lens companions are aligned with the primary by chance.We note that KMT-2021-BLG-1122L is the first triple stellar system detected via microlensing.In total, there exist 12 confirmed 3L1S events: OGLE-2006-BLG-109 (Gaudi et al. 2008;Bennett et al. 2010), OGLE-2012-BLG-0026 (Han et al. 2013), OGLE-2018-BLG-1011(Han et al. 2019), OGLE-2019-BLG-0468 (Han et al. 2022d), OGLE-2021-BLG-1077 (Han et al. 2022a), OGLE-2006-BLG-284 (Bennett et al. 2020), OGLE-2007-BLG-349 (Bennett et al. 2016), OGLE-2008-BLG-092 (Poleski et al. 2014), OGLE-2013-BLG-0341 (Gould et al. 2014), OGLE-2018-BLG-1700 (Han et al. 2020b), KMT-2019-BLG-1715(Han et al. 2021b), and KMT-2020-BLG-0414 (Zang et al. 2021) 4 .Among them, the lenses of the first five events are two-planet systems, and the lenses of the other seven events are binary-star systems possessing planets.Therefore, the common trait of the previously discovered microlensing triple systems is that at least one of the lens components is a planet.Two planets with similar separations from a host 4 Besides these confirmed cases, there are six candidate triple-lensing events, including OGLE-2014-BLG-1722 (Suzuki et al. 2018), OGLE-2018-BLG-0532 (Ryu et al. 2020), KMT-2019-BLG-1953(Han et al. 2020a), KMT-2019-BLG-0304 (Han et al. 2021a), OGLE-2019-BLG-1470 (Kuang et al. 2022), and KMT-2021-BLG-0240 (Han et al. 2022b).For these events, 3L1S interpretations yielded best-fit models, but the signals of the tertiary lens components were not firmly confirmed either due to the subtlety of the signals or the degeneracy with other interpretations.Under the 3L1S interpretations of these events, the lenses belong to either the category of two-planet systems or the category of planets in binaries similar to the cases of the confirmed 3L1S events.
A8, page 7 of 8 can be dynamically stable if the planets are in mean-motion resonance (Madsen & Zhu 2019).A planet in a binary system can be dynamically stable if the planet orbits one star of a wide binary stellar system or the barycenter of a closely spaced binary system.Microlensing detection of a triple stellar system is difficult because a system with similar masses and intrinsic separations among the components would be dynamically unstable.One way for such a system to be detected via microlensing is for the companions of the triple system to be closely aligned with the primary so that they lie around the Einstein ring of the primary.This alignment requires a projection effect with a low probability, and this explains the relative rareness of triple stellar systems detected by microlensing.

Summary and conclusion
We have presented the analysis of the microlensing event KMT-2021-BLG-1122, which was investigated as a part of the project to analyze anomalous lensing events in the previous KMTNet data with no suggested plausible models.We have confirmed that the light curve, characterized by three major anomaly features, of the event could not be explained with the usual 2L1S or 1L2S interpretations, but it was found that a 2L1S solution obtained from the modeling with the exclusion of the data around the second anomaly feature could approximately describe the other two caustic-crossing features.
We found that all the anomaly features could be explained by introducing a tertiary lens component, and thus the lens is a triple system.The masses of the individual lens components estimated from the Bayesian analysis are (M 1 , M 2 , M 3 ) ∼ (0.47 M , 0.24 M , 0.11 M ), and the companions are separated in projection from the primary by (a ⊥,2 , a ⊥,3 ) ∼ (3.5, 4.0) AU.The estimated masses of all the lens components are in the mass regime of M dwarfs, and thus the lens is a triple system composed of three low-mass stars.This is the first triple stellar-lens system detected via microlensing.

Fig. 1 .
Fig. 1.Light curve of KMT-2021-BLG-1122 constructed with the combined data obtained from the three KMTNet telescopes (KMTA, KMTC, and KMTS).The curve drawn over the data points is the bestfit model found under the 2L1S interpretation.The lower panel displays the whole view, and the upper panel shows the enlarged view of the anomaly region.The epochs marked by t 1 , t 2 , and t 3 indicate the three major anomaly features.

Fig. 4 .
Fig. 4. Lens-system configuration of the 3L1S model.The main panel shows the central anomaly region, and the inset displays the whole view of the lens system, where blue dots marked by M 1 , M 2 , and M 3 denote the positions of the individual lens components.Gray curves around the caustic represent the equi-magnification contours.The dotted circle centered at the origin of the coordinates represents the Einstein ring of the lens system.

Figure 6
Figure6shows the position of the source in the instrumental CMD constructed from the pyDIA photometry of stars lying around the source.We note that the CMD shows only bright stars due to the severe extinction, A I = 3.3, toward the field.The measured instrumental color and magnitude of the source are (V − I, I) S = (3.501± 0.056, 21.080 ± 0.037),(2)

Fig. 5 .
Fig. 5. Cumulative distribution of the χ 2 difference, ∆χ 2 = χ 2 2L2S −χ 2 3L1S , between the 2L2S (orbital) and 3L1S models in the region of the major anomaly.The light curve in the upper panel is shown to indicate the epochs of major χ 2 differences.

Fig. 6 .
Fig. 6.Position of the source (blue dot) with respect to the centroid of red giant clump (RGC, red dot) in the instrumental color-magnitude diagram of stars lying in the vicinity of the source.

Fig. 7 .
Fig. 7. Posteriors of the primary lens mass (top panel) and distance to the lens (middle panel) and source (bottom panel).In each panel, the blue and red curves represent the contributions by the disk and bulge lens populations, respectively, and the black curve is sum of the two lens populations.The solid vertical line indicates the median value, and the two dotted lines represent the 1σ range of the distribution.

Table 1 .
Lensing parameters of the 2L2S solutions.

Table 2 .
Lensing parameters of the best-fit 3L1S solution.