arXiv subjects

Woong-Tae Kim

Publications and source records attributed to Woong-Tae Kim.

At least 37 records · Page 2Linked to original sources

arXiv2021

Effects of Varying Mass Inflows on Star Formation in Nuclear Rings of Barred Galaxies

Observations indicate that the star formation rate (SFR) of nuclear rings varies considerably with time and is sometimes asymmetric rather than being uniform across a ring. To understand what controls temporal and spatial distributions of ring star formation, we run semi-global, hydrodynamic simulations of nuclear rings subject to time-varying and/or asymmetric mass inflow rates. These controlled variations in the inflow lead to variations in the star formation, while the ring orbital period ($18\,{\rm Myr}$) and radius ($600\,{\rm pc}$) remain approximately constant. We find that both the mass inflow rate and supernova feedback affect the ring SFR. An oscillating inflow rate with period $Δτ_\text{in}$ and amplitude 20 causes large-amplitude (a factor of $\gtrsim 5$), quasi-periodic variations of the SFR, when $Δτ_\text{in} \gtrsim 50\,{\rm Myr}$. We find that the time-varying ISM weight and midplane pressure track each other closely, establishing an instantaneous vertical equilibrium. The measured time-varying depletion time is consistent with the prediction from self-regulation theory provided the time delay between star formation and supernova feedback is taken into account. The supernova feedback is responsible only for small-amplitude (a factor of $\sim 2$) fluctuations of the SFR with a timescale $\lesssim 40\,{\rm Myr}$. Asymmetry in the inflow rate does not necessarily lead to asymmetric star formation in nuclear rings. Only when the inflow rate from one dust lane is suddenly increased by a large factor, the rings undergo a transient period of lopsided star formation.

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arXiv2021

OGLE-2019-BLG-0468Lb,c: two microlensing giant planets around a G-type star

With the aim of interpreting anomalous lensing events with no suggested models, we conducted a project of reinvestigating microlensing data in and before the 2019 season. In this work, we report a multi-planet system OGLE-2019-BLG-0468L found from the project. The light curve of the lensing event OGLE-2019-BLG-0468, which consists of three distinctive anomaly features, could not be explained by the usual binary-lens or binary-source interpretation. We find a solution explaining all anomaly features with a triple-lens interpretation, in which the lens is composed of two planets and their host, making the lens the fourth multi-planet system securely found by microlensing. The two planets have masses $\sim 3.4~M_{\rm J}$ and $\sim 10.2~M_{\rm J}$, and they are orbiting around a G-type star with a mass $\sim 0.9~M_\odot$ and a distance $\sim 4.4$ kpc. The host of the planets is most likely responsible for the light of the baseline object, although the possibility for the host to be a companion to the baseline object cannot be ruled out.

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arXiv2021

Star Formation in Nuclear Rings with the TIGRESS Framework

Nuclear rings are sites of intense star formation at the centers of barred galaxies. To understand what determines the structure and star formation rate (SFR; $\dot{M}_{\rm SF}$) of nuclear rings, we run semi-global, hydrodynamic simulations of nuclear rings subject to constant mass inflow rates $\dot{M}_{\rm in}$. We adopt the TIGRESS framework of Kim \& Ostriker to handle radiative heating and cooling, star formation, and related supernova (SN) feedback. We find that the SN feedback is never strong enough to destroy the ring or quench star formation everywhere in the ring. Under the constant $\dot{M}_{\rm in}$, the ring star formation is very steady and persistent, with the SFR exhibiting only mild temporal fluctuations. The ring SFR is tightly correlated with the inflow rate as $\dot{M}_{\rm SF}\approx 0.8\dot{M}_{\rm in}$, for a range of $\dot{M}_{\rm in}=0.125-8\,M_\odot\,{\rm yr}^{-1}$. Within the ring, vertical dynamical equilibrium is maintained, with the midplane pressure (powered by SN feedback) balancing the weight of the overlying gas. The SFR surface density is correlated nearly linearly with the midplane pressure, as predicted by the pressure-regulated, feedback-modulated star formation theory. Based on our results, we argue that the ring SFR is causally controlled by $\dot{M}_\text{in}$, while the ring gas mass adapts to the SFR to maintain the vertical dynamical equilibrium under the gravitational field arising from both gas and stars.

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arXiv2021

The Circumnuclear Disk Revealed by ALMA. I. Dense Clouds and Tides in the Galactic Center

Utilizing the Atacama Large Millimeter/submillimeter Array (ALMA), we present CS line maps in five rotational lines ($J_{\rm u}=7, 5, 4, 3, 2$) toward the circumnuclear disk (CND) and streamers of the Galactic Center. Our primary goal is to resolve the compact structures within the CND and the streamers, in order to understand the stability conditions of molecular cores in the vicinity of the supermassive black hole (SMBH) Sgr A*. Our data provide the first homogeneous high-resolution ($1.3'' = 0.05$ pc) observations aiming at resolving density and temperature structures. The CS clouds have sizes of $0.05-0.2$ pc with a broad range of velocity dispersion ($σ_{\rm FWHM}=5-40$ km s$^{-1}$). The CS clouds are a mixture of warm ($T_{\rm k}\ge 50-500$ K, n$_{\rm H_2}$=$10^{3-5}$ cm$^{-3}$) and cold gas ($T_{\rm k}\le 50$ K, n$_{\rm H_2}$=$10^{6-8}$ cm$^{-3}$). A stability analysis based on the unmagnetized virial theorem including tidal force shows that $84^{+16}_{-37}$ % of the total gas mass is tidally stable, which accounts for the majority of gas mass. Turbulence dominates the internal energy and thereby sets the threshold densities $10-100$ times higher than the tidal limit at distance $\ge 1.5$ pc to Sgr A*, and therefore, inhibits the clouds from collapsing to form stars near the SMBH. However, within the central $1.5$ pc, the tidal force overrides turbulence and the threshold densities for a gravitational collapse quickly grow to $\ge 10^{8}$ cm$^{-3}$.

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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

KMT-2018-BLG-1025Lb: microlensing super-Earth planet orbiting a low-mass star

We aim to find missing microlensing planets hidden in the unanalyzed lensing events of previous survey data. For this purpose, we conduct a systematic inspection of high-magnification microlensing events, with peak magnifications $A_{\rm peak}\gtrsim 30$, in the data collected from high-cadence surveys in and before the 2018 season. From this investigation, we identify an anomaly in the lensing light curve of the event KMT-2018-BLG-1025. The analysis of the light curve indicates that the anomaly is caused by a very low mass-ratio companion to the lens. We identify three degenerate solutions, in which the ambiguity between a pair of solutions (solutions B) is caused by the previously known close--wide degeneracy, and the degeneracy between these and the other solution (solution A) is a new type that has not been reported before. The estimated mass ratio between the planet and host is $q\sim 0.8\times 10^{-4}$ for the solution A and $q\sim 1.6\times 10^{-4}$ for the solutions B. From the Bayesian analysis conducted with measured observables, we estimate that the masses of the planet and host and the distance to the lens are $(M_{\rm p}, M_{\rm h}, D_{\rm L})\sim (6.1~M_\oplus, 0.22~M_\odot, 6.7~{\rm kpc})$ for the solution A and $\sim (4.4~M_\oplus, 0.08~M_\odot, 7.5~{\rm kpc})$ for the solutions B. The planet mass is in the category of a super-Earth regardless of the solutions, making the planet the eleventh super-Earth planet, with masses lying between those of Earth and the Solar system's ice giants, discovered by microlensing.

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arXiv2020

Four Microlensing Planets with Faint-source Stars Identified in the 2016 and 2017 Season Data

Microlensing planets occurring on faint source stars can escape detection due to their weak signals. Occasionally, detections of such planets are not reported due to the difficulty of extracting high-profile scientific issues on the detected planets. For the solid demographic census of microlensing planetary systems based on a complete sample, we investigate the microlensing data obtained in the 2016 and 2017 seasons to search for planetary signals in faint-source lensing events. From this investigation, we find four unpublished microlensing planets including KMT-2016-BLG-2364Lb, KMT-2016-BLG-2397Lb, OGLE-2017-BLG-0604Lb, and OGLE-2017-BLG-1375Lb. We analyze the observed lensing light curves and determine their lensing parameters. From Bayesian analyses conducted with the constraints from the measured parameters, it is found that the masses of the hosts and planets are in the ranges $0.50\lesssim M_{\rm host}/M_\odot\lesssim 0.85$ and $0.5 \lesssim M_{\rm p}/M_{\rm J}\lesssim 13.2$, respectively, indicating that all planets are giant planets around host stars with subsolar masses. The lenses are located in the distance range of $3.8 \lesssim \dl/{\rm kpc}\lesssim 6.4$. It is found that the lenses of OGLE-2017-BLG-0604 and OGLE-2017-BLG-1375 are likely to be in the Galactic disk.

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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

Local Simulations of Spiral Galaxies with the TIGRESS Framework: I. Star Formation and Arm Spurs/Feathers

Spiral arms greatly affect gas flows and star formation in disk galaxies. We use local three-dimensional simulations of the vertically-stratified, self-gravitating, differentially-rotating, interstellar medium (ISM) subject to a stellar spiral potential to study the effects of spiral arms on star formation and formation of arm spurs/feathers. We adopt the TIGRESS framework of Kim & Ostriker (2017) to handle radiative heating and cooling, star formation, and ensuing supernova (SN) feedback. We find that more than 90% of star formation takes place in spiral arms, but the global star formation rate (SFR) in models with spiral arms is enhanced by less than a factor of 2 compared to the no-arm counterpart. This results from a quasi-linear relationship between the SFR surface density Sigma_SFR and the gas surface density Sigma, and supports the picture that spiral arms do not trigger star formation but rather concentrate star-forming regions. Correlated SN feedback produces gaseous spurs/feathers downstream from arms in both magnetized and unmagnetized models. These spurs/feathers are short-lived and have magnetic fields parallel to their length, in contrast to the longer-lived features with perpendicular magnetic fields induced by gravitational instability. SN feedback drives the turbulent component of magnetic fields, with the total magnetic field strength sublinearly proportional to Sigma. The total midplane pressure varies by a factor of ~10 between arm and interarm regions but agrees locally with the total vertical ISM weight, while Sigma_SFR is locally consistent with the prediction of pressure-regulated, feedback-modulated theory.

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arXiv2019

OGLE-2016-BLG-1227L: A Wide-separation Planet from a Very Short-timescale Microlensing Event

We present the analysis of the microlensing event OGLE-2016-BLG-1227. The light curve of this short-duration event appears to be a single-lens event affected by severe finite-source effects. Analysis of the light curve based on single-lens single-source (1L1S) modeling yields very small values of the event timescale, $t_{\rm E}\sim 3.5$ days, and the angular Einstein radius, $θ_{\rm E}\sim 0.009$ mas, making the lens a candidate of a free-floating planet. Close inspection reveals that the 1L1S solution leaves small residuals with amplitude $ΔI\lesssim 0.03$ mag. We find that the residuals are explained by the existence of an additional widely-separated heavier lens component, indicating that the lens is a wide-separation planetary system rather than a free-floating planet. From Bayesian analysis, it is estimated that the planet has a mass of $M_{\rm p} = 0.79^{+1.30}_{-0.39} M_{\rm J}$ and it is orbiting a low-mass host star with a mass of $M_{\rm host}=0.10^{+0.17}_{-0.05} M_\odot$ located with a projected separation of $a_\perp=3.4^{+2.1}_{-1.0}$ au. The planetary system is located in the Galactic bulge with a line-of-sight separation from the source star of $D_{\rm LS}=1.21^{+0.96}_{-0.63}$ kpc. The event shows that there are a range of deviations in the signatures of host stars for apparently isolated planetary lensing events and that it is possible to identify a host even when a deviation is subtle.

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arXiv2019

Origin of Non-axisymmetric Features of Virgo Cluster Early-type Dwarf Galaxies. II. Tidal Effects on Disk Features and Stability

A fraction of dwarf galaxies in the Virgo cluster contain disk features like bars and spiral arms. Using $N$-body simulations, we investigate the effects of tidal forces on the formation of such disk features in disk dwarf galaxies resembling VCC856. We consider 8 Cluster-Galaxy models in which disk dwarf galaxies with differing pericenter distance and spin orientation experience the tidal gravitational force of a Virgo-like NFW halo, and additional 8 Galaxy-Galaxy models in which two dwarf galaxies undergo tidal interactions with different strength. We find that the cluster tidal effect is moderate due to the small galaxy size, making the bars form earlier by $\sim1$--$1.5\Gyr$ compared to the cases in isolation. While the galactic halos significantly lose their mass within the virial radius due to the cluster tidal force, the mass of the stellar disks is nearly unchanged, suggesting that the inner regions of a disk-halo system is secured from the tidal force. The tidal forcing from either the cluster potential or a companion galaxy triggers the formation of two-armed spirals at early time before a bar develops. The tidally-driven arms decay and wind with time, suggesting that they are kinematic density waves. In terms of the strength and pitch angle, the faint arms in VCC856 are best matched with the arms in a marginally unstable galaxy produced by a distant tidal encounter with its neighbor $\sim0.85\Gyr$ ago.

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arXiv2019

The Nuclear Filaments inside the Circumnuclear Disk in the Central 0.5 pc of the Galactic center

We present CS(7-6) line maps toward the central parsec of the Galactic Center (GC), conducted with the Atacama Large Millimeter/submillimeter Array (ALMA). The primary goal is to find and characterize the gas structure in the inner cavity of the circumnuclear disk (CND) in high resolution (1.3"=0.05 pc). Our large field-of-view mosaic maps -- combining interferometric and single-dish data that recover extended emission - provide a first homogeneous look to resolve and link the molecular streamers in the CND with the neutral nuclear filaments newly detected within the central cavity of the CND. We find that the nuclear filaments are rotating with Keplerian velocities in a nearly face-on orbit with an inclination angle of ~10-20 degree (radius <= 0.5 pc). This is in contrast to the CND which is highly inclined at ~65-80 degree (radius ~2-5 pc). Our analysis suggests a highly warped structure from the CND to the nuclear filaments. This result may hint that the nuclear filaments and the CND were created by different external clouds passing by Sgr A*.

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arXiv2019

Properties of Density and Velocity Gaps Induced by a Planet in a Protoplanetary Disk

Gravitational interactions between a protoplanetary disk and its embedded planet is one of the formation mechanisms of gaps and rings found in recent ALMA observations. To quantify the gap properties measured in not only surface density but also rotational velocity profiles, we run two-dimensional hydrodynamic simulations of protoplanetary disks by varying three parameters: the mass ratio $q$ of a planet to a central star, the ratio of the disk scale height $h_p$ to the orbital radius $r_p$ of the planet, and the viscosity parameter $α$. We find the gap depth $δ_Σ$ in the gas surface density depends on a single dimensionless parameter $K\equiv q^2(h_p/r_p)^{-5}α^{-1}$ as $δ_Σ= (1 + 0.046K)^{-1}$, consistent with the previous results of Kanagawa et al. (2015a). The gap depth $δ_V$ in the rotational velocity is given by $δ_V= 0.007 (h_p/r_p) K^{1.38}/(1 +0.06K^{1.03})$. The gap width, in both surface density and rotational velocity, has a minimum of about $4.7 h_p$ when the planet mass $M_p$ is around the disk thermal mass $M_{\text{th}}$, while it increases in a power-law fashion as $M_p/M_{\text{th}}$ increases or decrease from unity. Such a minimum in the gap width arises because spirals from sub-thermal planets have to propagate before they shock the disk gas and open a gap. We compare our relations for the gap depth and width with the previous results, and discuss their applicability to observations.

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arXiv2019

Modeling UV Radiation Feedback from Massive Stars: III. Escape of Radiation from Star-forming Giant Molecular Clouds

Using a suite of radiation hydrodynamic simulations of star cluster formation in turbulent clouds, we study the escape fraction of ionizing (Lyman continuum) and non-ionizing (FUV) radiation for a wide range of cloud masses and sizes. The escape fraction increases as H II regions evolve and reaches unity within a few dynamical times. The cumulative escape fraction before the onset of the first supernova explosion is in the range 0.05-0.58; this is lower for higher initial cloud surface density, and higher for less massive and more compact clouds due to rapid destruction. Once H II regions break out of their local environment, both ionizing and non-ionizing photons escape from clouds through fully ionized, low-density sightlines. Consequently, dust becomes the dominant absorber of ionizing radiation at late times and the escape fraction of non-ionizing radiation is only slightly larger than that of ionizing radiation. The escape fraction is determined primarily by the mean $\langle τ\rangle$ and width $σ$ of the optical-depth distribution in the large-scale cloud, increasing for smaller $\langle τ\rangle$ and/or larger $σ$. The escape fraction exceeds (sometimes by three orders of magnitude) the naive estimate $e^{-\langle τ\rangle}$ due to non-zero $σ$ induced by turbulence. We present two simple methods to estimate, within $\sim20\%$, the escape fraction of non-ionizing radiation using the observed dust optical depth in clouds projected on the plane of sky. We discuss implications of our results for observations, including inference of star formation rates in individual molecular clouds, and accounting for diffuse ionized gas on galactic scales.

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arXiv2019

Spectroscopic Mass and Host-star Metallicity Measurements for Newly Discovered Microlensing Planet OGLE-2018-BLG-0740Lb

We report the discovery of the microlensing planet OGLE-2018-BLG-0740Lb. The planet is detected with a very strong signal of $Δχ^2\sim 4630$, but the interpretation of the signal suffers from two types of degeneracies. One type is caused by the previously known close/wide degeneracy, and the other is caused by an ambiguity between two solutions, in which one solution requires to incorporate finite-source effects, while the other solution is consistent with a point-source interpretation. Although difficult to be firmly resolved based on only the photometric data, the degeneracy is resolved in strong favor of the point-source solution with the additional external information obtained from astrometric and spectroscopic observations. The small astrometric offset between the source and baseline object supports that the blend is the lens and this interpretation is further secured by the consistency of the spectroscopic distance estimate of the blend with the lensing parameters of the point-source solution. The estimated mass of the host is $1.0\pm 0.1~M_\odot$ and the mass of the planet is $4.5\pm 0.6~M_{\rm J}$ (close solution) or $4.8\pm 0.6~M_{\rm J}$ (wide solution) and the lens is located at a distance of $3.2\pm 0.5$~kpc. The bright nature of the lens, with $I\sim 17.1$ ($V\sim 18.2$), combined with its dominance of the observed flux suggest that radial-velocity (RV) follow-up observations of the lens can be done using high-resolution spectrometers mounted on large telescopes, e.g., VLT/ESPRESSO, and this can potentially not only measure the period and eccentricity of the planet but also probe for close-in planets. We estimate that the expected RV amplitude would be $\sim 60\sin i ~{\rm m~s}^{-1}$.

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arXiv2019

A Fast Poisson Solver of Second-Order Accuracy for Isolated Systems in Three-Dimensional Cartesian and Cylindrical Coordinates

We present an accurate and efficient method to calculate the gravitational potential of an isolated system in three-dimensional Cartesian and cylindrical coordinates subject to vacuum (open) boundary conditions. Our method consists of two parts: an interior solver and a boundary solver. The interior solver adopts an eigenfunction expansion method together with a tridiagonal matrix solver to solve the Poisson equation subject to the zero boundary condition. The boundary solver employs James's method to calculate the boundary potential due to the screening charges required to keep the zero boundary condition for the interior solver. A full computation of gravitational potential requires running the interior solver twice and the boundary solver once. We develop a method to compute the discrete Green's function in cylindrical coordinates, which is an integral part of the James algorithm to maintain second-order accuracy. We implement our method in the {\tt Athena++} magnetohydrodynamics code, and perform various tests to check that our solver is second-order accurate and exhibits good parallel performance.

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arXiv2019

Effects of Gas on Formation and Evolution of Stellar Bars and Nuclear Rings in Disk Galaxies

We run self-consistent simulations of Milky Way-sized, isolated disk galaxies to study formation and evolution of a stellar bar as well as a nuclear ring in the presence of gas. We consider two sets of models with cold or warm disks that differ in the radial velocity dispersions, and vary the gas fraction $f_{\rm gas}$ by fixing the total disk mass. A bar forms earlier and more strongly in the cold disks with larger $f_{\rm gas}$, while gas progressively delays the bar formation in the warm disks . The bar formation enhances a central mass concentration which in turn makes the bar decay temporarily, after which it regrows in size and strength, eventually becoming stronger in models with smaller $f_{\rm gas}$. Although all bars rotate fast in the beginning, they rapidly turn to slow rotators. In our models, only the gas-free, warm disk undergoes rapid buckling instability, while other disks thicken more gradually via vertical heating. The gas driven inward by the bar potential readily forms a star-forming nuclear ring. The ring is very small when it first forms and grows in size over time. The ring star formation rate is episodic and bursty due to feedback, and well correlated with the mass inflow rate to the ring. Some expanding shells produced by star formation feedback are sheared out in the bar regions and collide with dust lanes to appear as filamentary interbar spurs. The bars and nuclear rings formed in our simulations have properties similar to those in the Milky Way.

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arXiv2018

KMT-2016-BLG-2052L: Microlensing Binary Composed of M Dwarfs Revealed from a Very Long Time-scale Event

We present the analysis of a binary microlensing event KMT-2016-BLG-2052, for which the lensing-induced brightening of the source star lasted for 2 seasons. We determine the lens mass from the combined measurements of the microlens parallax $\pie$ and angular Einstein radius $\thetae$. The measured mass indicates that the lens is a binary composed of M dwarfs with masses of $M_1\sim 0.34~M_\odot$ and $M_2\sim 0.17~M_\odot$. The measured relative lens-source proper motion of $μ\sim 3.9~{\rm mas}~{\rm yr}^{-1}$ is smaller than $\sim 5~{\rm mas}~{\rm yr}^{-1}$ of typical Galactic lensing events, while the estimated angular Einstein radius of $\thetae\sim 1.2~{\rm mas}$ is substantially greater than the typical value of $\sim 0.5~{\rm mas}$. Therefore, it turns out that the long time scale of the event is caused by the combination of the slow $μ$ and large $\thetae$ rather than the heavy mass of the lens. From the simulation of Galactic lensing events with very long time scales ($t_{\rm E}\gtrsim 100$ days), we find that the probabilities that long time-scale events are produced by lenses with masses $\geq 1.0~M_\odot$ and $\geq 3.0~M_\odot$ are $\sim 19\%$ and 2.6\%, respectively, indicating that events produced by heavy lenses comprise a minor fraction of long time-scale events. The results indicate that it is essential to determine lens masses by measuring both $\pie$ and $\thetae$ in order to firmly identify heavy stellar remnants such as neutron stars and black holes.

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