arXiv subjects

Martin Donachie

Publications and source records attributed to Martin Donachie.

At least 19 recordsLinked to original sources

arXiv2023

Systematic Reanalysis of KMTNet microlensing events, Paper I: Updates of the Photometry Pipeline and a New Planet Candidate

In this work, we update and develop algorithms for KMTNet tender-love care (TLC) photometry in order to create an new, mostly automated, TLC pipeline. We then start a project to systematically apply the new TLC pipeline to the historic KMTNet microlensing events, and search for buried planetary signals. We report the discovery of such a planet candidate in the microlensing event MOA-2019-BLG-421/KMT-2019-BLG-2991. The anomalous signal can be explained by either a planet around the lens star or the orbital motion of the source star. For the planetary interpretation, despite many degenerate solutions, the planet is most likely to be a Jovian planet orbiting an M or K dwarf, which is a typical microlensing planet. The discovery proves that the project can indeed increase the sensitivity of historic events and find previously undiscovered signals.

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arXiv2022

OGLE-2018-BLG-0799Lb: a $q \sim 2.7 \times 10^{-3}$ Planet with Spitzer Parallax

We report the discovery and analysis of a planet in the microlensing event OGLE-2018-BLG-0799. The planetary signal was observed by several ground-based telescopes, and the planet-host mass ratio is $q = (2.65 \pm 0.16) \times 10^{-3}$. The ground-based observations yield a constraint on the angular Einstein radius $θ_{\rm E}$, and the microlensing parallax vector $\vecπ_{\rm E}$, is strongly constrained by the Spitzer data. However, the 2019 Spitzer baseline data reveal systematics in the Spitzer photometry, so there is ambiguity in the magnitude of the parallax. In our preferred interpretation, a full Bayesian analysis using a Galactic model indicates that the planetary system is composed of an $M_{\rm planet} = 0.26_{-0.11}^{+0.22}~M_{J}$ planet orbiting an $M_{\rm host} = 0.093_{-0.038}^{+0.082}~M_{\odot}$, at a distance of $D_{\rm L} = 3.71_{-1.70}^{+3.24}$ kpc. An alternate interpretation of the data shifts the localization of the minima along the arc-shaped microlens parallax constraints. This, in turn, yields a more massive host with median mass of $0.13 {M_{\odot}}$ at a distance of 6.3 kpc. This analysis demonstrates the robustness of the osculating circles formalism, but shows that further investigation is needed to assess how systematics affect the specific localization of the microlens parallax vector and, consequently, the inferred physical parameters.

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arXiv2022

Systematic KMTNet Planetary Anomaly Search, Paper I: OGLE-2019-BLG-1053Lb, A Buried Terrestrial Planet

In order to exhume the buried signatures of "missing planetary caustics" in the KMTNet data, we conducted a systematic anomaly search to the residuals from point-source point-lens fits, based on a modified version of the KMTNet EventFinder algorithm. This search reveals the lowest mass-ratio planetary caustic to date in the microlensing event OGLE-2019-BLG-1053, for which the planetary signal had not been noticed before. The planetary system has a planet-host mass ratio of $q = (1.25 \pm 0.13) \times 10^{-5}$. A Bayesian analysis yields estimates of the mass of the host star, $M_{\rm host} = 0.61_{-0.24}^{+0.29}~M_\odot$, the mass of its planet, $M_{\rm planet} = 2.48_{-0.98}^{+1.19}~M_{\oplus}$, the projected planet-host separation, $a_\perp = 3.4_{-0.5}^{+0.5}$ au, and the lens distance of $D_{\rm L} = 6.8_{-0.9}^{+0.6}$ kpc. The discovery of this very low mass-ratio planet illustrates the utility of our method and opens a new window for a large and homogeneous sample to study the microlensing planet-host mass-ratio function down to $q \sim 10^{-5}$.

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arXiv2022

KMT-2021-BLG-0171Lb and KMT-2021-BLG-1689Lb: Two Microlensing Planets in the KMTNet High-cadence Fields with Followup Observations

Follow-up observations of high-magnification gravitational microlensing events can fully exploit their intrinsic sensitivity to detect extrasolar planets, especially those with small mass ratios. To make followup more uniform and efficient, we develop a system, HighMagFinder, based on the real-time data from the Korean Microlensing Telescope Network (KMTNet) to automatically alert possible ongoing high-magnification events. We started a new phase of follow-up observations with the help of HighMagFinder in 2021. Here we report the discovery of two planets in high-magnification microlensing events, KMT-2021-BLG-0171 and KMT-2021-BLG-1689, which were identified by the HighMagFinder. We find that both events suffer the ``central-resonant'' caustic degeneracy. The planet-host mass-ratio is $q\sim4.7\times10^{-5}$ or $q\sim 2.2\times10^{-5}$ for KMT-2021-BLG-0171, and $q\sim2.5\times10^{-4}$ or $q\sim 1.8\times10^{-4}$ for KMT-2021-BLG-1689. Together with two events reported by Ryu et al. (2022), four cases that suffer such degeneracy have been discovered in the 2021 season alone, indicating that the degenerate solutions may have been missed in some previous studies. We also propose a new factor for weighting the probability of each solution from the phase-space. The resonant interpretations for the two events are disfavored under this consideration. This factor can be included in future statistical studies to weight degenerate solutions.

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arXiv2021

KMT-2021-BLG-0322: Severe degeneracy between triple-lens and higher-order binary-lens interpretations

We investigate the microlensing event KMT-2021-BLG-0322, for which the light curve exhibits three distinctive sets of caustic-crossing features. It is found that the overall features of the light curve are approximately described by a binary-lens (2L1S) model, but the model leaves substantial residuals. We test various interpretations with the aim of explaining the residuals. We find that the residuals can be explained either by considering a nonrectilinear lens-source motion caused by the microlens-parallax and lens-orbital effects or by adding a low-mass companion to the binary lens (3L1S model). The degeneracy between the higher-order 2L1S model and the 3L1S model is very severe, making it difficult to single out a correct solution based on the photometric data. This degeneracy was known before for two previous events (MACHO-97-BLG-41 and OGLE-2013-BLG-0723), which led to the false detections of planets in binary systems, and thus the identification of the degeneracy for KMT-2021-BLG-0322 illustrates that the degeneracy can be not only common but also very severe, emphasizing the need to check both interpretations of deviations from 2L1S models. From the Bayesian analysis conducted with the measured lensing observables of the event timescale, angular Einstein radius, and microlens parallax, it was estimated that the binary lens components have masses $(M_1, M_2) =(0.62^{+0.25}_{-0.26}~M_\odot,0.07^{+0.03}_{-0.03}~M_\odot)$, for both 2L1S and 3L1S solutions, and the mass of the tertiary lens component according to the 3L1S solution is $M_3=6.40^{+2.64}_{-2.78}~M_{\rm J}$.

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arXiv2021

New Giant Planet beyond the Snow Line for an Extended MOA Exoplanet Microlens Sample

Characterizing a planet detected by microlensing is hard if the planetary signal is weak or the lens-source relative trajectory is far from caustics. However, statistical analyses of planet demography must include those planets to accurately determine occurrence rates. As part of a systematic modeling effort in the context of a $>10$-year retrospective analysis of MOA's survey observations to build an extended MOA statistical sample, we analyze the light curve of the planetary microlensing event MOA-2014-BLG-472. This event provides weak constraints on the physical parameters of the lens, as a result of a planetary anomaly occurring at low magnification in the light curve. We use a Bayesian analysis to estimate the properties of the planet, based on a refined Galactic model and the assumption that all Milky Way's stars have an equal planet-hosting probability. We find that a lens consisting of a $1.9^{+2.2}_{-1.2}\,\mathrm{M}_\mathrm{J}$ giant planet orbiting a $0.31^{+0.36}_{-0.19}\,\mathrm{M}_\odot$ host at a projected separation of $0.75\pm0.24\,\mathrm{au}$ is consistent with the observations and is most likely, based on the Galactic priors. The lens most probably lies in the Galactic bulge, at $7.2^{+0.6}_{-1.7}\mathrm{kpc}$ from Earth. The accurate measurement of the measured planet-to-host star mass ratio will be included in the next statistical analysis of cold planet demography detected by microlensing.

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arXiv2021

KMT-2019-BLG-1715: planetary microlensing event with three lens masses and two source stars

We investigate the gravitational microlensing event KMT-2019-BLG-1715, of which light curve shows two short-term anomalies from a caustic-crossing binary-lensing light curve: one with a large deviation and the other with a small deviation. We identify five pairs of solutions, in which the anomalies are explained by adding an extra lens or source component in addition to the base binary-lens model. We resolve the degeneracies by applying a method, in which the measured flux ratio between the first and second source stars is compared with the flux ratio deduced from the ratio of the source radii. Applying this method leaves a single pair of viable solutions, in both of which the major anomaly is generated by a planetary-mass third body of the lens, and the minor anomaly is generated by a faint second source. A Bayesian analysis indicates that the lens comprises three masses: a planet-mass object with $\sim 2.6~M_{\rm J}$ and binary stars of K and M dwarfs lying in the galactic disk. We point out the possibility that the lens is the blend, and this can be verified by conducting high-resolution followup imaging for the resolution of the lens from the source.

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arXiv2021

An Earth-mass Planet in a Time of Covid-19: KMT-2020-BLG-0414Lb

We report the discovery of KMT-2020-BLG-0414Lb, with a planet-to-host mass ratio $q_2 = 0.9$--$1.2 \times 10^{-5} = 3$--$4~q_{\oplus}$ at $1σ$, which is the lowest mass-ratio microlensing planet to date. Together with two other recent discoveries ($4 \lesssim q/q_\oplus \lesssim 6$), it fills out the previous empty sector at the bottom of the triangular $(\log s, \log q)$ diagram, where $s$ is the planet-host separation in units of the angular Einstein radius $θ_{\rm E}$. Hence, these discoveries call into question the existence, or at least the strength, of the break in the mass-ratio function that was previously suggested to account for the paucity of very low-$q$ planets. Due to the extreme magnification of the event, $A_{\rm max}\sim 1450$ for the underlying single-lens event, its light curve revealed a second companion with $q_3 \sim 0.05$ and $|\log s_3| \sim 1$, i.e., a factor $\sim 10$ closer to or farther from the host in projection. The measurements of the microlens parallax $π_{\rm E}$ and the angular Einstein radius $θ_{\rm E}$ allow estimates of the host, planet, and second companion masses, $(M_1, M_2, M_3) \sim (0.3M_{\odot}, 1.0M_{\oplus}, 17M_{J})$, the planet and second companion projected separations, $(a_{\perp,2}, a_{\perp,3}) \sim (1.5, 0.15~{\rm or}~15)$~au, and system distance $D_{\rm L} \sim 1$ kpc. The lens could account for most or all of the blended light ($I \sim 19.3$) and so can be studied immediately with high-resolution photometric and spectroscopic observations that can further clarify the nature of the system. The planet was found as part of a new program of high-cadence follow-up observations of high-magnification events. The detection of this planet, despite the considerable difficulties imposed by Covid-19 (two KMT sites and OGLE were shut down), illustrates the potential utility of this program.

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arXiv2021

OGLE-2017-BLG-1049: Another giant planet microlensing event

We report a giant exoplanet discovery in the microlensing event OGLE-2017-BLG-1049, which is a planet-host star mass ratio of $q=9.53\pm0.39\times10^{-3}$ and has a caustic crossing feature in the Korea Microlensing Telescope Network (KMTNet) observations. The caustic crossing feature yields an angular Einstein radius of $θ_{\rm E}=0.52 \pm 0.11\ {\rm mas}$. However, the microlens parallax is not measured because of the time scale of the event $t_{\rm E}\simeq 29\ {\rm days}$, which is not long enough in this case to determine the microlens parallax. Thus, we perform a Bayesian analysis to estimate physical quantities of the lens system. From this, we find that the lens system has a star with mass $M_{\rm h}=0.55^{+0.36}_{-0.29} \ M_{\odot}$ hosting a giant planet with $M_{\rm p}=5.53^{+3.62}_{-2.87} \ M_{\rm Jup}$, at a distance of $D_{\rm L}=5.67^{+1.11}_{-1.52}\ {\rm kpc}$. The projected star-planet separation in units of the Einstein radius $(θ_{\rm E})$ corresponding to the total mass of the lens system is $a_{\perp}=3.92^{+1.10}_{-1.32}\ \rm{au}$. This means that the planet is located beyond the snow line of the host. The relative lens-source proper motion is $μ_{\rm rel}\sim 7 \ \rm{mas \ yr^{-1}}$, thus the lens and source will be separated from each other within 10 years. Then the flux of the host star can be measured by a 30m class telescope with high-resolution imaging in the future, and thus its mass can be determined.

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arXiv2021

KMT-2019-BLG-0371 and the Limits of Bayesian Analysis

We show that the perturbation at the peak of the light curve of microlensing event KMT-2019-BLG-0371 is explained by a model with a mass ratio between the host star and planet of $q \sim 0.08$. Due to the short event duration ($t_{\rm E} \sim 6.5\ $ days), the secondary object in this system could potentially be a massive giant planet. A Bayesian analysis shows that the system most likely consists of a host star with a mass $M_{\rm h} = 0.09^{+0.14}_{-0.05}M_{\odot}$ and a massive giant planet with a mass $M_{\rm p} = 7.70^{+11.34}_{-3.90}M_{\rm Jup}$. However, the interpretation of the secondary as a planet (i.e., as having $M_{\rm p} < 13 M_{\rm Jup}$) rests entirely on the Bayesian analysis. Motivated by this event, we conduct an investigation to determine which constraints meaningfully affect Bayesian analyses for microlensing events. We find that the masses inferred from such a Bayesian analysis are determined almost entirely by the measured value of $θ_{\rm E}$ and are relatively insensitive to other factors such as the direction of the event $(\ell, b)$, the lens-source relative proper motion $μ_{\rm rel}$, or the specific Galactic model prior.

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arXiv2021

OGLE-2019-BLG-0960Lb: The Smallest Microlensing Planet

We report the analysis of OGLE-2019-BLG-0960, which contains the smallest mass-ratio microlensing planet found to date (q = 1.2--1.6 x 10^{-5} at 1-sigma). Although there is substantial uncertainty in the satellite parallax measured by Spitzer, the measurement of the annual parallax effect combined with the finite source effect allows us to determine the mass of the host star (M_L = 0.3--0.6 M_Sun), the mass of its planet (m_p = 1.4--3.1 M_Earth), the projected separation between the host and planet (a_perp = 1.2--2.3 au), and the distance to the lens system (D_L = 0.6--1.2 kpc). The lens is plausibly the blend, which could be checked with adaptive optics observations. As the smallest planet clearly below the break in the mass-ratio function (Suzuki et al. 2016; Jung et al. 2019), it demonstrates that current experiments are powerful enough to robustly measure the slope of the mass-ratio function below that break. We find that the cross-section for detecting small planets is maximized for planets with separations just outside of the boundary for resonant caustics and that sensitivity to such planets can be maximized by intensively monitoring events whenever they are magnified by a factor A > 5. Finally, an empirical investigation demonstrates that most planets showing a degeneracy between (s > 1) and (s < 1) solutions are not in the regime (|log s| >> 0) for which the "close"/"wide" degeneracy was derived. This investigation suggests a link between the "close"/"wide" and "inner/outer" degeneracies and also that the symmetry in the lens equation goes much deeper than symmetries uncovered for the limiting cases.

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arXiv2020

KMT-2019-BLG-0842Lb: A Cold Planet Below the Uranus/Sun Mass Ratio

We report the discovery of a cold planet with a very low planet/host mass ratio of $q=(4.09\pm0.27) \times 10^{-5}$, which is similar to the ratio of Uranus/Sun ($q=4.37 \times 10^{-5}$) in the Solar system. The Bayesian estimates for the host mass, planet mass, system distance, and planet-host projected separation are $M_{\rm host}=0.76\pm 0.40 M_\odot$, $M_{\rm planet}=10.3\pm 5.5 M_\oplus$, $D_{\rm L} = 3.3\pm1.3\,{\rm kpc}$, and $a_\perp = 3.3\pm 1.4\,{\rm au}$, respectively. The consistency of the color and brightness expected from the estimated lens mass and distance with those of the blend suggests the possibility that the most blended light comes from the planet host, and this hypothesis can be established if high resolution images are taken during the next (2020) bulge season. We discuss the importance of conducting optimized photometry and aggressive follow-up observations for moderately or very high magnification events to maximize the detection rate of planets with very low mass ratios.

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arXiv2020

One Planet or Two Planets? The Ultra-sensitive Extreme-magnification Microlensing Event KMT-2019-BLG-1953

We present the analysis of a very high-magnification ($A\sim 900$) microlensing event KMT-2019-BLG-1953. A single-lens single-source (1L1S) model appears to approximately delineate the observed light curve, but the residuals from the model exhibit small but obvious deviations in the peak region. A binary lens (2L1S) model with a mass ratio $q\sim 2\times 10^{-3}$ improves the fits by $Δχ^2=181.8$, indicating that the lens possesses a planetary companion. From additional modeling by introducing an extra planetary lens component (3L1S model) and an extra source companion (2L2S model), it is found that the residuals from the 2L1S model further diminish, but claiming these interpretations is difficult due to the weak signals with $Δχ^2=16.0$ and $13.5$ for the 3L1S and 2L2L models, respectively. From a Bayesian analysis, we estimate that the host of the planets has a mass of $M_{\rm host}=0.31^{+0.37}_{-0.17}~M_\odot$ and that the planetary system is located at a distance of $D_{\rm L}=7.04^{+1.10}_{-1.33}~{\rm kpc}$ toward the Galactic center. The mass of the securely detected planet is $M_{\rm p}=0.64^{+0.76}_{-0.35}~M_{\rm J}$. The signal of the potential second planet could have been confirmed if the peak of the light curve had been more densely observed by followup observations, and thus the event illustrates the need for intensive followup observations for very high-magnification events even in the current generation of high-cadence surveys.

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arXiv2020

OGLE-2018-BLG-1269Lb: A Jovian Planet With A Bright, $I=16$ Host

We report the discovery of a planet in the microlensing event OGLE-2018-BLG-1269, with planet-host mass ratio $q \sim 6\times10^{-4}$, i.e., $0.6$ times smaller than the Jupiter/Sun mass ratio. Combined with the $Gaia$ parallax and proper motion, a strong one-dimensional constraint on the microlens parallax vector allows us to significantly reduce the uncertainties of lens physical parameters. A Bayesian analysis that ignores any information about light from the host yields that the planet is a cold giant $(M_{2} = 0.69_{-0.22}^{+0.44}\,M_{\rm J})$ orbiting a Sun-like star $(M_{1} = 1.13_{-0.35}^{+0.72}\,M_{\odot})$ at a distance of $D_{\rm L} = 2.56_{-0.62}^{+0.92}\,{\rm kpc}$. The projected planet-host separation is $a_{\perp} = 4.61_{-1.17}^{+1.70}\,{\rm au}$. Using {\it Gaia} astrometry, we show that the blended light lies $\lesssim 12\,$mas from the host and therefore must be either the host star or a stellar companion to the host. An isochrone analysis favors the former possibility at $>99.6\%$. The host is therefore a subgiant. For host metallicities in the range of $0.0 \leq {\rm [Fe/H]} \leq +0.3$, the host and planet masses are then in the range of $1.16 \leq M_{1}/M_{\odot} \leq 1.38$ and $0.74 \leq M_{2}/M_{\rm J} \leq 0.89$, respectively. Low host metallicities are excluded. The brightness and proximity of the lens make the event a strong candidate for spectroscopic followup both to test the microlensing solution and to further characterize the system.

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arXiv2020

OGLE-2017-BLG-0406: ${\it Spitzer}$ Microlens Parallax Reveals Saturn-mass Planet orbiting M-dwarf Host in the Inner Galactic Disk

We report the discovery and analysis of the planetary microlensing event OGLE-2017-BLG-0406, which was observed both from the ground and by the ${\it Spitzer}$ satellite in a solar orbit. At high magnification, the anomaly in the light curve was densely observed by ground-based-survey and follow-up groups, and it was found to be explained by a planetary lens with a planet/host mass ratio of $q=7.0 \times 10^{-4}$ from the light-curve modeling. The ground-only and ${\it Spitzer}$-"only" data each provide very strong one-dimensional (1-D) constraints on the 2-D microlens parallax vector $\bf{π_{\rm E}}$. When combined, these yield a precise measurement of $\bf{π_{\rm E}}$, and so of the masses of the host $M_{\rm host}=0.56\pm0.07\,M_\odot$ and planet $M_{\rm planet} = 0.41 \pm 0.05\,M_{\rm Jup}$. The system lies at a distance $D_{\rm L}=5.2 \pm 0.5 \ {\rm kpc}$ from the Sun toward the Galactic bulge, and the host is more likely to be a disk population star according to the kinematics of the lens. The projected separation of the planet from the host is $a_{\perp} = 3.5 \pm 0.3 \ {\rm au}$, i.e., just over twice the snow line. The Galactic-disk kinematics are established in part from a precise measurement of the source proper motion based on OGLE-IV data. By contrast, the ${\it Gaia}$ proper-motion measurement of the source suffers from a catastrophic $10\,σ$ error.

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arXiv2020

A Gas Giant Planet in the OGLE-2006-BLG-284L Stellar Binary System

We present the analysis of microlensing event OGLE-2006-BLG-284, which has a lens system that consists of two stars and a gas giant planet with a mass ratio of $q_p = (1.26\pm 0.19) \times 10^{-3}$ to the primary. The mass ratio of the two stars is $q_s = 0.289\pm 0.011$, and their projected separation is $s_s = 2.1\pm 0.7\,$AU, while the projected separation of the planet from the primary is $s_p = 2.2\pm 0.8\,$AU. For this lens system to have stable orbits, the three-dimensional separation of either the primary and secondary stars or the planet and primary star must be much larger than that these projected separations. Since we do not know which is the case, the system could include either a circumbinary or a circumstellar planet. Because there is no measurement of the microlensing parallax effect or lens system brightness, we can only make a rough Bayesian estimate of the lens system masses and brightness. We find host star and planet masses of $M_{L1} = 0.35^{+0.30}_{-0.20}\,M_\odot$, $M_{L2} = 0.10^{+0.09}_{-0.06}\,M_\odot$, and $m_p = 144^{+126}_{-82}\,M_\oplus$, and the $K$-band magnitude of the combined brightness of the host stars is $K_L = 19.7^{+0.7}_{-1.0}$. The separation between the lens and source system will be $\sim 90\,$mas in mid-2020, so it should be possible to detect the host system with follow-up adaptive optics or Hubble Space Telescope observations.

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arXiv2019

MOA-bin-29b : A Microlensing Gas Giant Planet Orbiting a Low-mass Host Star

We report the discovery of a gas giant planet orbiting a low-mass host star in the microlensing event MOA-bin-29 that occurred in 2006. We find five degenerate solutions with the planet/host-star mass ratio of $q \sim 10^{-2}$. The Einstein radius crossing time of all models are relatively short ($\sim 4-7$ days), which indicates that the mass of host star is likely low. The measured lens-source proper motion is $5-9$ ${\rm mas}\ {\rm yr}^{-1}$ depending on the models. Since only finite source effects are detected, we conduct a Bayesian analysis in order to obtain the posterior probability distribution of the lens physical properties. As a result, we find the lens system is likely to be a gas giant orbiting a brown dwarf or a very late M-dwarf in the Galactic bulge. The probability distributions of the physical parameters for the five degenerate models are consistent within the range of error. By combining these probability distributions, we conclude that the lens system is a gas giant with a mass of $M_{\rm p} = 0.63^{+1.13}_{-0.39}\ M_{\rm Jup}$ orbiting a brown dwarf with a mass of $M_{\rm h} = 0.06^{+0.11}_{-0.04}\ M_\odot$ at a projected star-planet separation of $r_\perp = 0.53^{+0.89}_{-0.18}\ {\rm au}$. The lens distance is $D_{\rm L} = 6.89^{+1.19}_{-1.19}\ {\rm kpc}$, i.e., likely within the Galactic bulge.

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arXiv2019

Candidate Brown-dwarf Microlensing Events with Very Short Timescales and Small Angular Einstein Radii

Short-timescale microlensing events are likely to be produced by substellar brown dwarfs (BDs), but it is difficult to securely identify BD lenses based on only event timescales $t_{\rm E}$ because short-timescale events can also be produced by stellar lenses with high relative lens-source proper motions. In this paper, we report three strong candidate BD-lens events found from the search for lensing events not only with short timescales ($t_{\rm E} \lesssim 6~{\rm days}$) but also with very small angular Einstein radii ($θ_{\rm E}\lesssim 0.05~{\rm mas}$) among the events that have been found in the 2016--2019 observing seasons. These events include MOA-2017-BLG-147, MOA-2017-BLG-241, and MOA-2019-BLG-256, in which the first two events are produced by single lenses and the last event is produced by a binary lens. From the Bayesian analysis conducted with the combined $t_{\rm E}$ and $θ_{\rm E}$ constraint, it is estimated that the lens masses of the individual events are $0.051^{+0.100}_{-0.027}~M_\odot$, $0.044^{+0.090}_{-0.023}~M_\odot$, and $0.046^{+0.067}_{-0.023}~M_\odot/0.038^{+0.056}_{-0.019}~M_\odot$ and the probability of the lens mass smaller than the lower limit of stars is $\sim 80\%$ for all events. We point out that routine lens mass measurements of short time-scale lensing events require survey-mode space-based observations.

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