Mon. Not. R. Astron. Soc. 304, 845±850 (1999) The applicability of the astrometric method for determining the physical parameters of gravitational microlenses Cheongho Han1 * and Kyongae Chang21 1 2 Department of Astronomy and Space Science, Chungbuk National University, Chongju, Korea 361-763 Department of Physics, Chongju University, Chongju, Korea 360-764 Accepted 1998 December 7. Received 1998 November 16; in original form 1998 September 2 In this paper, we investigate the applicability of the astrometric method to the determination of the lens parameters for gravitational microlensing events towards both the LMC and the Galactic bulge. For this analysis, we investigate the dependence of the uncertainty of astrometrically determined angular Einstein ring radius, D vE =vE;0 †, on the lens parameters by testing various types of events. In addition, by computing D vE =vE;0 † for events with lensing parameters that are the most probable for a given lens mass under the standard models of Galactic matter density and velocity distributions, we determine the expected distribution of the uncertainties as a function of lens mass. From this study, we ®nd that the values of the angular Einstein ring radius are expected to be measured with uncertainties D vE =vE;0 † , . 10 per cent up to a lens mass of M , 0:1 M( for both Galactic disc±bulge and halo±LMC events with a moderate observational strategy. The uncertainties are relatively large for Galactic bulge±bulge self-lensing events, D vE =vE;0 † , 25 per cent for M , 0:1 M(, but they can be substantially reduced by adopting more aggressive observational strategies. We also ®nd that although astrometric observations can be performed for most photometrically detected Galactic bulge events, a signi®cant fraction (,45 per cent) of LMC events cannot be astrometrically observed owing to the faintness of their source stars. Key words: astrometry ± dark matter ± gravitational lensing. 1 INTRODUCTION Current microlensing experiments (Alcock et al. 1997a, 1997b; Ansari et al. 1996; Udalski et al. 1997; Alard & Guibert 1997) are detecting massive astrophysical compact objects (MACHOs) by monitoring the light variations of stars undergoing gravitational microlensing. The light curve of a microlensing event is related to the lensing parameters by    1=2 u2 ‡ 2 t ÿ t0 2 2 Aˆ ; 1† ; u ˆ b ‡ tE u u2 ‡ 4†1=2 where A is the ampli®cation, u is the lens±source separation, b is the impact parameter, t0 is the time of maximum ampli®cation, and tE is the Einstein ring radius crossing time (Einstein time-scale). Of these lensing parameters, only the time-scale provides information about the lens because it is the only one directly related to the physical parameters of the lens (lens parameters) by   rE 4GM Dol Dls 1=2 tE ˆ ; ; 2† rE ˆ v c2 Dos *E-mail: cheongho@astronomy.chungbuk.ac.kr (CH); kchang@alpha94. chongju.ac.kr (KC) q 1999 RAS where v is the lens±source transverse speed, rE is the physical size of the Einstein ring radius, M is the mass of the lens, and Dol , Dls and Dos are the separations between the observer, lens and source star. However, owing to the degeneracy in the lens parameters, the nature of the lens population is still very uncertain. The uncertainties in the lens parameters can be signi®cantly reduced if one can measure the lens proper motion, m ˆ vE =tE , where vE ˆ rE =Dol is the angular Einstein ring radius. The lens proper motions can be measured both photometrically (Gould 1994; Nemiroff & Wickramasinghe 1994; Witt & Mao 1994) and spectroscopically (Maoz & Gould 1994; Loeb & Sasselov 1995). However, measuring the proper motion with either of these techniques is only possible for several special classes of lensing events. The lens proper motions can be measured in a general way if the separation between the two source-star images can be measured. As this image separation is generally of the order of 1 mas, it cannot be measured with current instrumentation. However, the proposed Space Interferometry Mission (Allen, Shao, & Peterson 1998, hereafter SIM) will have high enough positional accuracy to measure the astrometric displacement in the light centroid caused by gravitational microlensing (Walker 1995; Hùg, Novikov & Polnarev 1995; PaczynÂski 1998; Boden, Shao & Van Buren 1998). The astrometric shift of the source-star image centroid is Downloaded from https://academic.oup.com/mnras/article/304/4/845/1048410 by guest on 15 April 2022 A B S T R AC T 846 Cheongho Han and Kyongae Chang related to the lensing parameters by  ˆ
vE ÿ T xà ‡ bÃy ; u2 ‡ 2 3† 2 D E P E N D E N C Y O F L E N S PA R A M E T E R S To show the applicability of the astrometric method to the determination of vE , we begin our analysis by investigating how the uncertainties in an astrometrically determined vE changes with respect to varying lens parameters. We determine the uncertainties in vE by conducting model ®ts to the astrometric shifts of simulated events with various lensing parameters. The result of the ®t is obtained by computing x2 , i.e. N x2 ˆ obs  2 1 X dvc;x ti † ÿ dvc0;x ti † 2 jdvc iˆ1  2 ‡ dvc;y ti † ÿ dvc0;y ti † †; 4† where Nobs is the number of astrometric observations and dvc0;x ; dvc0;y † and dvc;x ; dvc;y † are the centroid shifts of the simulated and model events, respectively. Following the mission speci®cations of SIM, we adopt the positional accuracy of the astrometric measurements to be jdvc ˆ 0:01 mas (http://huey.jpl.nasa.gov/ sim). As our goal is to see the variation of D vE =vE;0 † with respect to individual lensing parameters, the uncertainties are determined under ®xed, but realistic, observational conditions. We will discuss the expected uncertainties under other observational conditions in Section 4. To determine the uncertainties, we assume that the astrometric centroid shifts of events are measured with a frequency f ˆ 1 dÿ1 during tmin # tobs # tmax , where tmin and tmax are the times of the ®rst and last astrometric measurements. As the astrometric observations will be performed for events that are photometrically detected, we set tmin ˆ ÿ0:4tE while tmax ˆ 3:0tE . The uncertainties are determined by the 3j level, which is equivalent to Dx2 ˆ x2 ÿ x2min ˆ 9, where x2min is the best®tting x2 value. To illustrate the sensitivity of this uncertainty on the lens parameters, we have tested a sample event with lensing parameters b; tE ; vE † ˆ 0:5; 11:3 d; 0:22 mas†. Then the dependency of the uncertainty on each lensing parameter is obtained by varying the parameter of interest while holding the other lensing parameters constant. In Fig. 1, we present the uncertainties of the astrometrically determined angular Einstein ring radius with respect to various lensing parameters. From the ®gure, one ®nds that the uncertainties increase signi®cantly with decreasing angular Einstein ring radius and decreasing Einstein time-scale. On the other hand, the dependency of D vE =vE;0 † on the impact parameter is not important. 3 DEPENDENCY ON LENS MASSES In the previous section, we showed that the uncertainty D vE =vE;0 † depends strongly on the size of the angular Einstein ring radius and the duration of the Einstein time-scale. As both vE and tE are related to the mass of the lens, the strong dependency of D vE =vE;0 † on vE and tE implies that the uncertainties for events caused by different populations of lenses will be greatly different. Then the naturally arising questions are: `Is the astrometric observation of lensing events a guaranteed method of determining vE ?' and if not, `To what extent can one determine vE with this method?' To answer these q 1999 RAS, MNRAS 304, 845±850 Downloaded from https://academic.oup.com/mnras/article/304/4/845/1048410 by guest on 15 April 2022 where T ˆ t ÿ t0 †=tE , and the x- and y-axes represent the directions that are parallel and normal to the lens±source transverse motion, respectively. The trajectory of the apparent source-star image traces out an ellipse during the event (astrometric ellipse; Jeong, Han & Park 1999). With the measured astrometric ellipse, one can determine the impact parameter of the event because the shape (i.e. the axis ratio) of the ellipse is related to the impact parameter b. The importance of measuring the astrometric centroid shifts is that one can then determine the angular Einstein ring radius as the size of the astrometric ellipse, i.e. the semimajor axis, is directly proportional to vE . Once the angular Einstein ring radius is determined from the astrometric shifts, one can determine the lens proper motion from vE combined with the event time-scale, which is determined from photometric monitoring of the event. As astrometric centroid shifts of microlensing events exhibit various shapes depending on the combination of the lensing parameters, i.e. b, tE and vE , the uncertainties in the astrometrically determined values of the angular Einstein ring radius, D vE =vE;0 †, will depend on the lensing parameters. The values of the lensing parameters tE and vE are related to the physical parameters of lenses, Dol , v and M, and thus the uncertainties will also depend on the lens parameters. In addition, owing to the differences in the lens parameters for different types of lensing events, the expected values of D vE =vE;0 † for Large Magellanic Cloud (LMC) events will be different from those of Galactic bulge events. As a result, the expected uncertainties of vE will have a wide range. Boden et al. (1998) performed a detailed study of the performance of the astrometric observations of gravitational microlensing events in determining various observables, including vE . From this analysis, they demonstrated the usefulness of astrometric observations of lensing events in determining the lens parameters. In their analysis, however, Boden et al. (1998) considered only LMC events, while the majority of events are detected towards the Galactic bulge. In addition, their estimate of the uncertainties is based on the very limited number of test cases they considered. For example, all the events they considered had a ®xed lens location of Dol ˆ 8 kpc, and their only values for the Einstein time-scale and impact parameter were tE ˆ 0:1 and 0.2 yr and b ˆ 0:4 and 0.8, respectively. They also did not take into account the restriction that the source-star brightness can impose on astrometric observations. Therefore, one cannot make general conclusions about the usefulness of astrometrically monitoring lensing events based on their analysis alone. In this paper, we investigate the applicability of the astrometric method to the determination of the lens parameters for events towards both the LMC and the Galactic bulge. For this analysis, we investigate the dependency of D vE =vE;0 † on the lens parameters by testing various types of events. In addition, by computing D vE =vE;0 † for events with the most probable lensing parameters for a given lens mass under the standard models of the Galactic matter density and velocity distributions, we determine the expected distribution of the uncertainties in vE as a function of the lens mass. From this study, we ®nd that the values of the angular Einstein ring radius are expected to be measured with uncertainties D vE =vE;0 † , . 10 per cent up to the lens mass of M , 0:1 M( for both Galactic disc±bulge and halo±LMC events with a moderate observational strategy. The uncertainties are relatively large for Galactic bulge±bulge self-lensing events, D vE =vE;0 † , 25 per cent for M , 0:1 M( , but they can be substantially reduced by adopting more aggressive observational strategies. We also ®nd that although astrometric observations can be performed for most photometrically detected Galactic bulge events, a signi®cant fraction (,45 per cent) of LMC events cannot be astrometrically observed owing to the faintness of their source stars. Astrometric method 847 obtained by Dos ¥ f tE † ˆ 0 ´ dDos r Dos † ¥ ¥ 0 0 0 dDol r Dol †prE2 dvy dvz vf vy ; vz †     4GM Dol Dls 1=2 ´ d tE ÿ ; c2 v2 Dos 5† for the Einstein time-scale, and Dos ¥ f vE † ˆ 0 ¥ ¥ 0 0 0 dDol r Dol †prE2 dvy dvz vf vy ; vz †   1=2  4GM Dls ; ´ d vE ÿ c2 Dol Dos Figure 1. The dependency of the uncertainties of the astrometrically determined vE on various lensing parameters. The astrometric centroid shifts of the event are assumed to be measured with a frequency f ˆ 1 dÿ1 during ÿ0:4tE # tobs # 3:0tE and with a positional accuracy jdvc ˆ 0:01 mas. To investigate the dependencies, we test an example event with a set of lensing parameters of b; tE ; vE † ˆ 0:5; 11:3 d; 0:22 mas†. The dependency of the uncertainties on each lensing parameter is obtained by varying the parameter of interest while holding the other lens parameters constant. One ®nds that the uncertainties D vE =vE;0 † increase signi®cantly with decreasing angular Einstein ring radius and with decreasing Einstein time-scale. On the other hand, the dependency of D vE =vE;0 † on the impact parameter is not important. questions, one must determine the expected value of D vE =vE;0 † as a function of lens mass. We determine the relation between D vE =vE;0 † and the lens mass by computing the uncertainties for events with lensing parameters that are the most probable for a given lens mass. Because the values of vE and tE also depend on the lens parameters v and Dol , there is no one-to-one correspondence between the lens mass and the lensing parameters. However, with models of the Galactic matter density and velocity distributions, one can statistically determine the most probable values of the lensing parameters corresponding to individual lens masses. With models for the Galactic matter density and velocity distributions, the distributions of tE and vE for a given lens mass M are for the angular Einstein ring radius. Here r Dol † and r Dos † are the density distributions of lens and source stars, d is the delta function, vy and vz are the components of the transverse velocity v, and f vy ; vz † is their distribution. In the above equations, the factors prE2 and v are included because events with larger cross-sections and higher transverse speeds are more likely to occur. For Galactic bulge events (disk±bulge plus bulge±bulge self-lensing events), vy and vz represent the velocity components that are normal and azimuthal to the Galactic plane, respectively. On the other hand, as we adopt a non-rotating isotropic velocity model for LMC events (halo±LMC events, see Table 1), these variables represent two velocity components of arbitrary direction in the plane normal to the line of sight towards the LMC. We assume that the individual components of the transverse velocities have Gaussian distributions of the form " # v ÿ vÅ †2 i [ fy; zg: 12† f vi † ~ exp ÿ i 2 i ; 2jvi The values of the mean, vÅ i , and the dispersion, j2vi , of the velocity distributions, which are listed in Table 1, are adopted from the models of Han & Gould (1995) for the Galactic bulge events and from Han & Gould (1996) for the LMC events. For the matter density distributions of the individual Galactic components, we adopt an axisymmetric model (Kent 1992) for the Galactic bulge, a double-exponential disc model (Bahcall 1986) for the Galactic disc, and an isothermal sphere model with a core radius (Bahcall, Schmidt & Soneira 1983) for the Galactic halo. The adopted models of the matter density distributions are listed in Table 2. The adopted distances to the Galactic centre and to the LMC are 8.0 and 55 kpc, respectively. In Fig. 2, we present the expected distributions of f vE † and f tE † for various types of events and different values of lens mass. From Table 1. The transverse velocity distribution models for different types of microlensing events. The distributions are assumed to be Gaussian in form,   f vi † ˆ exp ÿ vi ÿ vÅ i †2 =2j2vi ; i [ fy; zg, and the values of the mean and standard deviation of the distributions are listed. Here D ˆ Dol =Dos . Event type bulge±bulge disc±bulge halo±LMC q 1999 RAS, MNRAS 304, 845±850 6† vÅ y km sÿ1 † jvy km sÿ1 † vÅ z km sÿ1 † jvz km sÿ1 † ÿ220 1 ÿ D† 220D 0 ‰1002 1 ‡ D†2 Š1=2 2 2 1=2 ‰302 ‡ 100 pD  Š 250= 2 0 0 0 ‰1002 1 ‡ D†2 Š1=2 2 2 1=2 ‰202 ‡ 100 pD  Š 250= 2 Downloaded from https://academic.oup.com/mnras/article/304/4/845/1048410 by guest on 15 April 2022 ´ dDos r Dos † 848 Cheongho Han and Kyongae Chang Table 2. The model matter density distributions. In the Galactic bulge model, s4 ˆ R4 ‡ z=0:61†4 , R ˆ x2 ‡ y2 †1=2 , where x and z represent the axes directed along the line of sight and towards the Galactic pole. The notation K0 represents a modi®ed Bessel function. The adopted values of the solar Galactocentric distance and the core radius of the halo are R0 ˆ 8 kpc and rc ˆ 2 kpc, respectively. Galactic components disk bulge halo Distribution (M( pcÿ3 )    r R; z† ˆ 0:06 exp ÿ R ÿ R0 †=3500 ‡ z=325 r s† ˆ 1:04 ´ 106 s=0:482†ÿ1:85 (for s < 938 pc) r s† ˆ 3:53K0 s=667† (for s $ 938 pc) r r† ˆ 7:9 ´ 10ÿ3 rc2 ‡ R20 †= rc2 ‡ r2 † Figure 2. The distributions of the Einstein ring radii and the Einstein timescales for various types of events and different values of lens mass. From the ®gure, one ®nds that the expected distribution for a given type of event covers a wide range, considering the variety of lens populations and corresponding lens masses. In addition, even with a ®xed lens mass, the distributions for different types of events are substantially different from each other. Therefore, the analysis of D vR =vE;p † based on limited sets of lensing parameters for only a single population of lensing events will lead to erroneous conclusions about the general applicability of the astrometric method to the determination of vE . the ®gure, one can see that the expected distribution for a given type of event covers a wide range, considering the variety of lens populations and corresponding lens masses. In addition, even with a ®xed lens mass, the distributions for different types of events are substantially different from each other. Therefore, the analysis of D vE =vE;0 † based on limited sets of lensing parameters for only a single type of lensing event will lead to erroneous conclusions about the general applicability of the astrometric method to the determination of vE . Once the most probable values of vE and tE for a given lens mass are obtained from their distributions, the expected values of D vE =vE † are determined as before, by carrying out a x2 ®t to the astrometric centroid shifts of the event with the most probable lensing parameters. As the dependency of the uncertainties on the impact parameter is not important, we assume b ˆ 0:5 for all simulated events. The astrometric centroid shifts of the events are assumed to be measured under moderate observational conditions with a frequency f ˆ 0:5 dÿ1 during ÿ0:4tE # tobs # 3:0tE and with a positional accuracy of jdvc ˆ 0:01 mas. Then, the relation between D vE =vE † and lens mass is obtained by repeating the uncertainty determinations for different lens masses. In Fig. 3, we present the resulting relation between the lens mass and the expected uncertainty in the astrometrically determined values of vE . We ®nd that the angular Einstein ring radii are expected to be measured with uncertainties D vE =vE;0 † , . 10 per cent up to the lens mass of M , 0:1 M( for both Galactic disc± bulge and halo±LMC events. For Galactic bulge-bulge self-lensing events, the uncertainty is relatively large: D vE =vE;0 † , 25 per cent for M , 0:1 M(. 4 I M P ROV E D A S T RO M E T R I C O B S E RVAT I O N A L S T R AT E G I E S Up to now, we have investigated the uncertainties D vE =vE;0 † for a ®xed observational strategy. In this section, we investigate the dependency of the uncertainties on observational strategies to ®nd the optimal astrometric observational strategies for better q 1999 RAS, MNRAS 304, 845±850 Downloaded from https://academic.oup.com/mnras/article/304/4/845/1048410 by guest on 15 April 2022 Figure 3. The relation between the lens mass and the expected uncertainties in the astrometrically determined values of angular Einstein ring radii for various types of events. The uncertainties are determined by carrying out x2 ®ts to the astrometric centroid shifts of the events with lensing parameters vE and tE , which are the most probable for a given lens mass under the models of Galactic matter density and velocity distributions. As the dependency of the uncertainty on the impact parameters is not important, we assume b ˆ 0:5 for all events. For the moderate observational conditions, the astrometric centroid shifts of the events are assumed to be measured with f ˆ 0:5 dÿ1 during ÿ0:4tE # tobs # 3:0tE and a positional accuracy of jdvc ˆ 0:01 mas. The D vR =vE;p †±M relation for the Galactic bulge events is re-determined under a new observational strategy with f ˆ 2 dÿ1 and it is represented by a thick solid line. Astrometric method 849 f ˆ 2 dÿ1 . As we had already tested the uncertainties for a long enough observational duration of tmax ˆ 3tE , we only increased the observational frequency for this test. In Fig. 2, we present the newly determined D vE =vE;0 †±M relation (represented by a thick solid line) for the Galactic bulge±bulge self-lensing events, which have the largest uncertainties among the various types of events. One ®nds that the accuracy of the vE determination signi®cantly improves ± the uncertainty decreases by approximately ,30 per cent over the entire range of lens masses. 5 R E S T R I C T I O N B Y S O U R C E - S TA R BRIGHTNESS determinations of vE . The accuracy of the astrometric determination of vE can be broadly improved in three ways. First, the uncertainties will be decreased if the astrometric positional accuracy can be improved. However, as the positional accuracy is restricted by the instrumentation and not by the observational strategy, we do not consider this scenario further. Secondly, the accuracy can be improved by increasing the duration of the astrometric observations. Finally, increasing the frequency of astrometric observation will also help to improve the accuracy in determining vE . In Fig. 4, we present the uncertainties as functions of the frequency (upper panel) and the duration (lower panel) of the astrometric measurements of dvc . The uncertainties are obtained for an example event with a set of lensing parameters b; tE ; vE † ˆ 0:5; 11:3 d; 0:22 mas† under various observational conditions. From the ®gure, one ®nds that the accuracy improves signi®cantly with increasing observational frequencies. Longer observations of the event also contribute to decrease the uncertainty of vE determination. However, observations longer than tmax , 3tE do not help for a signi®cant reduction of the uncertainty. SIM can achieve astrometric accuracy of ,4 mas for a star as faint as V , 20 with 14 h of observation (http://huey.nasa.gov/sim), and the observational time decreases with increasing source-star brightness. Therefore, the astrometric observation by SIM can be performed in average up to a frequency of 2 dÿ1 . As we know that measuring dvc with increased frequency will help to improve the accuracy in determining vE , we re-determined the relation between D vE =vE;0 † and the lens mass for a new observational strategy with q 1999 RAS, MNRAS 304, 845±850 6 CONCLUSION In this paper, we have considered the effectiveness of astrometric monitoring of gravitational microlensing events in determining the lens parameters. The main results from this analysis can be summarized as follows: (i) The uncertainties of the astrometrically determined angular Einstein ring radii are strongly dependent on the values of vE and tE for the events. However, the dependency of D vE =vE;0 † on the impact parameter is not important. (ii) The distributions of vE and tE for events caused by various populations of lenses cover wide ranges. Even with a ®xed lens mass, the distributions for different types of lensing events are substantially different from each other. Therefore, the analysis of D vE =vE;0 † based on limited sets of lensing parameters for only a single type of lensing event will lead to erroneous conclusions about Downloaded from https://academic.oup.com/mnras/article/304/4/845/1048410 by guest on 15 April 2022 Figure 4. The dependency of the astrometrically determined vE on the observational frequencies (upper panel) and the duration of measurements (lower panel). The uncertainties are determined for an example event with a set of lensing parameter of b; tE ; vE † ˆ 0:5; 11:3 d; 0:22 mas† under different frequencies and durations of astrometric measurements of the centroid shifts. From the ®gure, one ®nds that the accuracy of vE determination improves signi®cantly with increasing observational frequencies. Longer observations of the event also contribute to decrease the uncertainty of vE determination. However, measurements of dvc longer than tmax , 3tE do not help to a signi®cant reduction of the uncertainty. The applicability of the astrometric method for determining lens parameters is additionally restricted by source-star brightness because the astrometric observations are only possible for events with bright source stars. For both the Galactic bulge and LMC ®elds, the photometric monitoring of source stars is restricted by crowding. Current microlensing experiments towards the Galactic bulge reach the photometric detection limit when the stellar density of the ®elds arrives at ,104 star degÿ2 (C. Alcock 1997, private communication). Based on the model luminosity function of Han (1997), this corresponds to V , 20:5, which approximately coincides with the astrometric detection limit of SIM (V , 20, http:// huey.jpl.nasa.gov/sim). Therefore, the astrometric observations of microlensing events can be achieved for most of the photometrically detected Galactic bulge events. By using the R-band luminosity function provided by Gould (1998), we also determine the fraction of LMC events with source stars bright enough for astrometric observation. With the photometric determination limit of V , 21 (Alcock et al. 1997b), which corresponds to R , 21:1, we ®nd that only about half (,55 per cent) of LMC source stars are brighter than the astrometric detection limit (R , 20:25). Considering that some faint source stars can become brighter than the astrometric detection limit when they are highly ampli®ed, the number of events for which astrometric shifts can be measured will be higher than this estimate. However, as an event can be astrometrically observed only for a short time owing to the very fast motion of the centroid shift near the peak ampli®cation, this increase in the fraction of events for vE determination will be small. Therefore, for LMC events the major restriction in the general applicability of the astrometric vE determination comes from the faintness of the source stars and not from the large uncertainties in the determined values of vE . 850 Cheongho Han and Kyongae Chang REFE REN CE S AC K N O W L E D G M E N T S Alard C., Guibert, J., 1997, A&A, 326, 1 Alcock C. et al., 1997a, ApJ, 479, 119 Alcock C. et al., 1997b, ApJ, 486, 697 Allen R. J., Peterson D. M., Shao M., 1997, Proc. SPIE, 2871, 504 Ansari R. et al., 1996, A&A, 314, 94 Bahcall J. N., 1986, ARA&A, 24, 577 Bahcall J. N., Schmidt M., Soneira R. M., 1983, ApJ, 265, 730 Boden A. F., Shao M., Van Buren D., 1998, ApJ, 502, 538 Gould A., 1994, ApJ, 421, L71 Gould A., 1998, ApJ, submitted (astro-ph/9807350) Han C., 1997, ApJ, 484, 555 Han C., Gould A., 1995, ApJ, 447, 53 Han C., Gould A., 1996, ApJ, 473, 233 Hùg E., Novikov I. D., Polnarev A. G., 1995, A&A, 294, 287 Jeong Y., Han C., Park S.-H., 1999, ApJ, 511, in press Kent S. M., 1992, ApJ, 387, 181 Loeb A., Sasselov D., 1995, ApJ, 449, L33 Maoz D., Gould A., 1994, ApJ, 425, L67 Nemiroff R. J., Wickramasinghe W. A. D. T., 1994, ApJ, 424, L21 PaczynÂski B., 1998, ApJ, 404, L23 Udalski A. et al., 1997, Acta Astron., 47, 169 Walker M. A., 1995, ApJ, 453, 37 Witt H. J., Mao S., 1994, ApJ, 430, 505 We would like to thank P. Martini for careful reading of the manuscript. This paper has been typeset from a TEX/LATEX ®le prepared by the author. q 1999 RAS, MNRAS 304, 845±850 Downloaded from https://academic.oup.com/mnras/article/304/4/845/1048410 by guest on 15 April 2022 the general applicability of the astrometric method in determining lens parameters. (iii) Measurements of dvc with increased frequencies during longer times of observations help to improve the accuracy of vE determination. However, measurements longer than tmax , 3tE do not contribute to a signi®cant reduction in D vE =vE;0 †. (iv) With a moderate observational strategy, the value of vE can be determined with an uncertainty D vE =vE;0 † , . 10 per cent up to a lens mass of M , 0:1 M( for the Galactic disc±bulge and halo± LMC events. The uncertainties for the Galactic bulge±bulge selflensing events is relatively large with D vE =vE;0 † , 25 per cent for M , 0:1 M( . However, the uncertainty can be reduced substantially by adopting a more aggressive observational strategy. (v) Astrometric observations are possible for most photometrically detected Galactic bulge events. However, nearly half of the LMC events are not astrometrically observable owing to the faintness of their source stars. Therefore, for LMC events the major restriction in the general applicability of the astrometric determination of vE comes from the source-star brightness and not from the large uncertainties in the determined value of vE .