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
A
;
1
;
u
b
tE
u u2 41=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 ;
;
2
rE
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 vE =tE ,
where vE 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
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A B S T R AC T
846
Cheongho Han and Kyongae Chang
related to the lensing parameters by
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
obs
2
1 X
dvc;x ti ÿ dvc0;x ti
2
jdvc i1
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 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 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 ÿ0:4tE while
tmax 3:0tE . The uncertainties are determined by the 3j level,
which is equivalent to Dx2 x2 ÿ x2min 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 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
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where T 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 8 kpc, and their only values for the Einstein time-scale and
impact parameter were tE 0:1 and 0.2 yr and b 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
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
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 1 dÿ1
during ÿ0:4tE # tobs # 3:0tE and with a positional accuracy
jdvc 0:01 mas. To investigate the dependencies, we test an example
event with a set of lensing parameters of
b; tE ; vE
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 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 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 D2 1=2
2 2 1=2
302 100
pD
250= 2
0
0
0
1002 1 D2 1=2
2 2 1=2
202 100
pD
250= 2
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´
dDos r Dos
848
Cheongho Han and Kyongae Chang
Table 2. The model matter density distributions. In the Galactic bulge
model, s4 R4 z=0:614 , R 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 8 kpc
and rc 2 kpc, respectively.
Galactic
components
disk
bulge
halo
Distribution
(M( pcÿ3 )
r R; z 0:06 exp ÿ R ÿ R0 =3500 z=325
r s 1:04 ´ 106 s=0:482ÿ1:85 (for s < 938 pc)
r s 3:53K0 s=667 (for s $ 938 pc)
r r 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 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 0:5 dÿ1 during ÿ0:4tE # tobs # 3:0tE and
with a positional accuracy of jdvc 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
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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 0:5
for all events. For the moderate observational conditions, the astrometric
centroid shifts of the events are assumed to be measured with f 0:5 dÿ1
during ÿ0:4tE # tobs # 3:0tE and a positional accuracy of jdvc 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 2 dÿ1 and it is represented by a
thick solid line.
Astrometric method
849
f 2 dÿ1 . As we had already tested the uncertainties for a long
enough observational duration of tmax 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 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
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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 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
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We would like to thank P. Martini for careful reading of the
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This paper has been typeset from a TEX/LATEX ®le prepared by the author.
q 1999 RAS, MNRAS 304, 845±850
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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 .