------------------------------------------------------------------------
From: "Julio C. Chaname"  jchaname@astronomy.ohio-state.edu 
To: Galactic Center Newsletter <gcnews@aoc.nrao.edu>
Subject:  astro-ph/0102481
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\shortauthors{Chanam\'e et al.\ }
\shorttitle{Microlensing by the Sgr A* Cluster of Black Holes}
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%electronic submission format
%\title{Microlensing by Sgr A* and by the Cluster of Black Holes Around It}
\title{Microlensing by the Cluster of Black Holes Around Sgr A*}
\author{Julio Chanam\'e\,$^1$, Andrew Gould\,$^1$$^,$$^2$, \& Jordi 
Miralda-Escud\'e\,$^1$$^,$$^3$} 
\affil{{}$^1$ Department of Astronomy, The Ohio State University,
Columbus, OH 43210, USA}
\affil{{}$^2$ Laboratoire de Physique Corpusculaire et Cosmologie, 
Coll\`ege de France, F-75231, Paris, France}
\affil{{}$^3$ Alfred P. Sloan Fellow}
\email{jchaname@astronomy.ohio-state.edu, gould@astronomy.ohio-state.edu,
jordi@astronomy.ohio-state.edu}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}

We show that at any given time, the Galactocentric black hole Sgr A*
is expected to be microlensing about 0.7 bulge stars if the threshold
of detectability of the {\it fainter} image is $K_\thr=21$, and
about 4.6 sources if $K_\thr=23$.  The lensed images then provide a unique
probe of the 20,000 stellar-mass black holes that are predicted to cluster 
within 0.7 pc of Sgr A*: if one of these black holes passes close to
a microlensed image, it will give rise to a short (weeks long) microlensing
event.  We show that the mass and projected velocity of the black hole
can both be measured unambiguously from such an event, if the lightcurve
and the mean astrometric displacement can be measured.
For $K_\thr=23$ and moderate magnifications by Sgr A*, these events are
expected only at a rate of 0.05$\,\rm yr^{-1}$, so deeper observations
would be required to see an event within the next decade. However, if
highly magnified images of a star were found, the rate of events by the
stellar black holes would be much higher.

\end{abstract}

\keywords{black hole physics -- 
Galaxy: center -- Galaxy: kinematics and dynamics -- gravitational lensing}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\section{Introduction}

Any black holes formed from the final core collapse of massive stars
within the central 5 pc of the Milky Way during the last
10 Gyr should have migrated to the Galactic center (GC) owing to dynamical
friction off ordinary stars (Morris 1993). 
Miralda-Escud\'e \& Gould (2000, hereafter MG) showed that the majority
of these black holes survive capture by the massive black hole at the
position of Sgr A* and so should still be present in the 
GC today, forming a cluster of black holes in orbit around Sgr A* that
extends to a radius $\sim 0.7\,$pc. Low-mass stars older than
$\sim 1\,$Gyr should have been expelled from this region by the
same process of dynamical friction. If stars with an initial mass
greater than $\sim 30\,M_\odot$ generally produce black holes,
then this black-hole cluster should contain about 20,000 members today.

  MG discussed several methods to test for 
the existence of the black-hole cluster. 
Among these, the only one that detects the black
holes directly is microlensing of a background source, probably
a bulge star.  This method requires first the discovery of the two 
images of a bulge star lensed by Sgr A*, and then the monitoring of 
these images to detect the short-duration (weeks long) microlensing events  
caused by the passage of one of the cluster black holes near an image.

The observational challenge here is to identify the two images of a
faint $(K\sim 21)$ lensed star in a field crowded with $\sim 400\,\rm
arcsec^{-2}$ stars of similar magnitude and brighter (see \S\ 2.1).
Hence, a resolution $\ll 0.\hskip-2pt'' 1$ is required.  Such deep,
high-resolution observations are beyond current capabilities, but
should be achievable within a few years.  It will then be relatively
easy to distinguish the pair of resolved images from the more numerous
field stars by means of their positions, flux ratio, colors and proper
motions, which are all related.  We show in \S\ 2, that there are
likely to be a few such detectable image pairs at any given time.
However, as we show in \S\ 3, the rate of events caused by cluster
black holes is only about 1 per 100 years per image pair, so that
still deeper observations will probably be required to detect the
cluster black holes during this decade.  In \S\ 4, we show that the
masses of these black holes could be measured from their astrometric
effects.  Finally, in \S\ 5, we discuss implications of our results.

	Alexander \& Sternberg (1999) calculated lensing rates for Sgr A*
considered as an isolated body, and obtained results that are broadly
consistent with those presented here.  Alexander \& Loeb (2001) investigated
microlensing by stars orbiting near Sgr A*, similar to the problem we
treat here.
They obtained rates that are several orders of magnitude higher than those
we find. We trace the sources of the discrepancy in \S\ 5.

\section{Expected number of stars lensed by Sgr A*}

The expected number of sources being lensed by Sgr A* at any given
time can be estimated empirically given three observational inputs: 1)
the $K$-band luminosity function (LF) of the sources, 2) the volume
density of $K$-band light as a function of distance behind the GC, and
3) the extinction as a function of distance behind the GC.  The first
two of these are well determined from observations, as we summarize
below.  The last is not measured.  For purposes of this paper we will
assume that there is no significant additional extinction in the $\sim
1\,$kpc lying behind the GC, above and beyond the $A_K=3$ magnitudes
of extinction known to lie in front of the GC (e.g., Blum, Sellgren,
\& DePoy 1996).

\subsection{Distribution of Stars behind the Galactic Center}

We derive the (dereddened) bulge $K_0$ LF from the $J$-band LF
measured by Zoccali et al.\ (2000, Table 1) using {\it Hubble Space
Telescope} NICMOS $(\Omega_{\rm Zocc}=22.\hskip-2pt''5 \times
22.\hskip-2pt''5)$ $J$ and $H$ images of a field lying at
$(l,b)=(0,-6)$.  We convert this LF to $H_0$ using their reported
extinction and observed $(J,J-H)$ color-magnitude diagram.  We then
convert to $K_0$ using the $(K,H-K)$ relation observed for nearby
main-sequence stars (e.g.\ Henry \& McCarthy 1993).  The resulting
{\it surface density} $K_0$ LF, $(d N_{\rm Zocc}/d K_0)$ normalized to
$\Omega_{\rm Zocc}$ is shown in Figure 1, where the vertical axis
gives the number of stars in the Zoccali et al (2000) field in each
bin of 0.5 mag width in J.  It reaches $K\sim 23.5$, and includes
bulge main sequence stars with masses $M\ga 0.15\,M_\odot$.  At the
bright end we include all stars with $K<$ 16 in a single bin at
$K=16$.  We note that these bright stars are very rare in any case,
and our results depend mostly on the much more numerous and fainter
($K\sim$ 21) main sequence stars.  It is also known that the bright
end of the LF at the GC and the bulge are different (Blum et al.\
1996), while for main-sequence stars this LF is almost the same
throughout the entire bulge (Narayanan, Gould, \& DePoy 1996).




	To convert this to a local {\it volume density} $K$ LF,
$(d N/d K d V)(D_{s})$, at a distance $D_{s}$ along the line of sight of 
Sgr A*, we first write
\begin{equation}
{d N\over d K d V}(D_{s}) = 
{\rho(D_{s})\over\Sigma_{\rm Zocc}}\,
{(d N_{\rm Zocc}/d K_0)|_{K_0 = K - \Delta K}
\over R_0^2\Omega_{\rm Zocc}},
\label{eqn:conv1}
\end{equation}
where $\rho(D_{s})$ is the local bulge mass density on the GC line of sight
at a distance $D_s$ from us,
$\Sigma_{\rm Zocc}$
is the bulge column density toward the Zoccali et al.\ (2000) field,
$R_0=8\,$kpc is the Galactocentric distance, 
$\Delta K=A_K(D_{s}) + 2.5\log(D_{s}/R_0)$, and where we assume
$A_K(D_{s})=3$.  In practice, we set $\Delta K = A_K = 3$, thus
neglecting the dimming due to larger distances, since this contribution
is smaller than the uncertainty introduced by assuming a constant
$A_K(D_{s})$.

  The first factor on the right-hand side of equation (\ref{eqn:conv1})
depends only on the spatial distribution of the mass in the bulge, and
not its normalization. To evaluate it, we use a combination of the Kent
(1992) model, which is based on K band images, and the DIRBE data from
Dwek et al.\ (1995). The reason this combination is required is that
the Kent model is axisymmetric, and so does not include the effects of
the bar which are important at large radius, while the DIRBE map cannot
be applied at small radius because of its limited resolution. We rewrite this
term as
\begin{equation}
{\rho(D_{s})\over\Sigma_{\rm Zocc}} = 
{\rho(D_{s})\over\Sigma_{(0,-2)}}\,
{\Sigma_{(0,-2)}\over\Sigma_{\rm Zocc}},
\label{eqn:conv2}
\end{equation}
where $\Sigma_{(0,-2)}$ is the column density in the direction
$(l,b)=(0,-2)$. We determine the first ratio from equation (3) in Kent
(1992), except that we impose a constant density core within 0.7 pc
due to relaxation effects around Sgr A* (MG).  We determine the second
ratio from the DIRBE map (Fig.\ 1a in Dwek et al.\ 1995), obtaining a
value $\Sigma_{(0,-2)}/\Sigma_{\rm Zocc}\simeq 13$.  Note that this
procedure is relatively insensitive to possible disk contamination of
the Zoccali et al.\ (2000) field since to leading order this
contamination affects both the star counts and the DIRBE map equally.
We have also used this same procedure to predict the surface density
of stars near Sgr A*, which is found to be 440 arcsec$^{-2}$ stars
with $K<21$ mag.

\subsection{Number of Lensed Sources}

	The two images of a microlensed source will be magnified by
\begin{equation}
A_\pm(u) = {u_\pm^2\over u_+^2-u_-^2},\quad
u_\pm = {u\pm\sqrt{u^2+4}\over 2},\quad
u\equiv {\theta_\rel\over\theta_\e},
\label{eqn:apm}
\end{equation}
where $\theta_\rel$ is the angular separation of the lens and source,
$\theta_{I\pm}=u_\pm\theta_\e$ are the positions of the two images,
$\theta_\e$ is the Einstein radius,
\begin{equation}
 \theta_\e(D_s) = \sqrt {\frac{4GM_\ast}{c^{2}} \frac{D_{ls}}{D_{l}D_{s}}} =
0.\hskip-2pt'' 55 \,
\biggl({10 D_{ls} \over D_s }\biggr)^{1/2},
\end{equation}
$M_\ast=3\times 10^6\,M_\odot$ 
is the mass of Sgr A*, $D_{l}=R_0$, $D_{s}$ is the distance
to the source, and $D_{ls}=D_s-D_l$.
The angular separation of the two lensed images of a background star at
$D_{ls}\sim $ 1 kpc is $\sim 2\theta_\e\sim 1''$, clearly large enough to 
resolve them. On the other hand, the duration of the ``microlensing event'' 
is very long, given the typical proper motion of bulge stars of a few mas 
per 
year.  This implies
that the usual method of microlensing detection from 
the magnification lightcurve is impractical.  Thus, to detect a lensed
source, it is essential to identify {\it a pair of images}, which can be
done from their relative magnifications 
and proper motions (both of which are unambiguously predicted from their
positions relative to Sgr A*), as well as their common color.


	The expected number of observable lenses $N_{\rm lens}$ is simply the 
number for which the fainter image is brighter than the threshold of 
detectability, $K_\thr$,
\begin{equation}
%\begin{eqnarray*}
N_{\rm lens} = \int_{R_0+0.7\,\rm pc}^{D_{s,\rm max}} d D_s\,D_s^2
\int_0^\infty d\theta\,2\pi\theta
{d N\over d V}\biggl[D_s,K_\thr 
           + 2.5\log A_-\biggl({\theta\over\theta_\e(D_s)}\biggr)\biggr],
%\end{eqnarray*}
\label{eqn:nlens}
\end{equation}
where $dN/dV(D_s,K')= \int^{K'} dK\,(dN/dVdK)(D_s,K)$ is
the cumulative LF. We use $D_{s,\rm max}=9$ kpc, effectively assuming that
additional obscuration becomes important farther than 1 kpc behind the GC.
We find for $K_{\rm thr}=(21,22,23)$, that
$N_{\rm lens}=(0.7,1.9,4.6)$, which can be summarized,
\begin{equation}
N_{\rm lens} \simeq 4.6\times10^{0.4(K_{\rm thr} - 23)},
\qquad (21<K_{\rm thr}<23).
\label{eqn:nlenseval}
\end{equation}
Thus, in order to have a
reasonable probability of finding one lensed star, the inner $1''$
of the GC must be imaged down to at least $K\sim 21$.
Once a lensed image pair is detected, it is mainly the brighter
member that will be useful to search for black-hole cluster members.
Figure 1 shows the distribution of brighter images for the three cases
$K_\thr=(21,22,23)$.  Note that for $K_\thr=21$, the brighter images
will typically be in the range $18<K<20$.  From 0.7 pc and up to 1 kpc 
behind the GC
the contributions of equal logarithmic intervals of $D_{ls}$ are 25\%,
32\%, and 43\%.  Therefore, although most lenses will be from the
outer bulge, with $\theta_\e\ga 0.\hskip-2pt'' 1$, a significant
fraction will be close to Sgr A*, with $\theta_\e\sim 30\,$mas.  
Of course, if there is significant extinction behind the black hole, then
only the closer sources might be left.

\section{Rate of Microlensing Events by the Cluster Black Holes}

After two images of a background bulge star lensed by Sgr A* have
been identified, monitoring them could reveal short microlensing events
due to one of the cluster black holes passing near one of the images.
If the typical cluster black hole has
a mass comparable to the masses inferred from X-ray binary systems,
$m_{\rm bh}\simeq 7$ \msun (Bailyn et al.\ 1998), the mass ratio of the
binary lens of Sgr A* and the cluster black hole is
$q= m_{\rm bh}/M_{\ast}  \sim 2\times 10^{-6}$.
The lightcurves would be similar to those caused by planets
around ordinary stars (Mao \& Paczy\'nski 1991; Gould \& Loeb 1992;
Albrow et al.\ 1998), although with longer timescales. In this
section, we calculate the rate at which these planet-like events should
take place.

  The rate of microlensing due to cluster black holes depends on their
surface density, their velocity distribution, 
and a {\it linear cross section} giving the size of
the region around an unperturbed image where a cluster black hole will
significantly change the magnification.
We obtain the surface density of black holes, $\Sb(b)$ (where $b$ is the
projected radius), using the model in
MG, where the cluster density varies as
$\rho(r)\propto r^{-7/4}$, the total stellar mass enclosed within
$r_0=1.8$ pc is equal to $M_{\ast}=3\times 10^6$ \msun, and the density
normalization of the black hole cluster is reduced by a factor
$(m_{\rm bh}/\overline{m})^{-1/2} = 0.18$
below that of the cluster of stars, owing to the change in the mean mass
of the objects between the black hole cluster $(m_{\rm bh}\sim 7\,M_\odot)$
at $r \lesssim 0.7$ pc
and the stellar cluster at larger distances
$(\overline{m}\sim 0.23\,M_\odot)$.
At projected radii 
$b \ll 0.7$ pc, the result is 
\begin{equation}
\Sb(b) = 0.18 \,\frac{5}{16 \pi} \,
{M_{\ast}/m_{\rm bh}\over {r_{0}^{5/4}}}
\int_{-\infty}^{\infty} (z^2 + b^2)^{-7/8}\, dz = 
1.6\times 10^{5} \,
\biggl({\theta\over 1''}\biggr)^{-3/4} {\rm pc^{-2}} \, ,
\end{equation}
where $\theta\equiv b/R_0$ is the angular distance from the center. 

We assume an isotropic velocity
distribution. For $\rho(r) \propto r^{-7/4}$, the Jeans equation
(see Binney \& Tremaine 1987) yields a one-dimensional velocity dispersion 
$\sigma(r) = \sqrt{4/11} \; v_{c}(r) = 68.5 (r/{\rm pc})^{-1/2} \kms$, where
$v_c(r)$ is the circular velocity. The projected
two-dimensional velocity dispersion is $\sigma_p(b)=0.99 \sigma(r) =
68 (b/{\rm pc})^{-1/2} \kms$.


  We calculate the linear cross section assuming a Chang-Refsdal lens
(Chang \& Refsdal 1979; Schneider, Ehlers \& Falco 1992), of a point mass in 
the external shear
$\gamma=(\theta_\e/\theta_I)^2$ caused by Sgr A*, where $\theta_I$ is the
angular distance from the image to Sgr A*. We consider events on the
brighter of the unperturbed images, with unperturbed magnification
$A_+$ given by equation (2). Our criterion for the detectability of the
short microlensing event is that the magnification of this image is
increased to at least $(1+\delta) A_+$. If $\xi$ is the
axis joining Sgr A* and the cluster black hole, and $\eta$ the
perpendicular axis, we find that the contours of constant $\delta$ 
have lengths along the $(\xi, \eta)$ axes of
\begin{equation}
L_{\xi}=\frac{2\,(2\gamma -\lambda +1)}{\sqrt{\lambda 
-\gamma}}\sqrt{q}R_0\theta_\e,
\;\;\;\;\;\;
L_{\eta}=\frac{2\,(2\gamma +\lambda -1)}{\sqrt{\lambda 
+\gamma}}\sqrt{q}R_0\theta_\e
\,,
\end{equation}
\noindent where 
\begin{equation}
\lambda=\left(1-\frac{1}{(1+\delta)A_{+}} \right)^{1/2}.
\end{equation} 
The case $\lambda=1$ corresponds to requiring a caustic crossing.

  To compute the rate of events, we notice first that the surface
density of caustics in the source plane is $\Sb/(1-\gamma^2)$, and the
velocity dispersion of these caustics is anisotropic. To render the
calculation more transparent, it is convenient to make a linear
transformation of the source plane to $\xi'=\xi/(1+\gamma)$, and
$\eta'=\eta/(1-\gamma)$ 
(with constant $\gamma$;
this transformation would be locally equivalent to the lens plane in
the absence of the perturbing black hole). In this transformed source
plane, the surface density and velocity dispersion of the caustics are
the same as the surface density and velocity dispersion of the black
holes in the lens plane, and the dimensions of contours of constant
$\delta$ are modified by factors $L_{\xi'} = L_{\xi}/(1+\gamma)$,
$L_{\eta'} = L_{\eta}/(1-\gamma)$.

  The rate of events is just the product $\Gamma = \Sb\, \sigma_p\,
\overline{L}'$, where $\overline{L}'$ 
is the average length of the $\delta$ contour
over all possible directions of motion of the black hole, and where
$\Sb$ and $\sigma_p$ are evaluated at the projected radius of the
brighter
unperturbed image, $b=\gamma^{-1/2} R_0\theta_\e$. 
To be definite,
we choose $\lambda=1$ and we approximate $\overline{L}'$ as the geometric
mean of 
$L_{\xi'}$ and $L_{\eta'}$, which is
$\overline{L}' = 4\gamma/(1-\gamma^2)^{3/4}q^{1/2}R_0\theta_\e$. 
This yields an event rate
\begin{equation}
\Gamma \simeq 1.3 \times 10^{-2}\, \frac{\gamma^{13/8}}{(1-\gamma^2)^{3/4}}\,
\biggl({\theta_\e\over 1''}\biggr)^{-1/4}\, {\rm yr}^{-1}\, .
\end{equation}
For a bulge star at $D_{ls}$=500 pc
(i.e., $\theta_\e \simeq 0.\hskip-2pt ''4$), 
a typical
shear of $\gamma$=0.6 (i.e., impact parameter $u$=0.52,
$A_{+}$=1.556), this implies approximately 0.01 microlensing events
per year.

  If a very highly magnified pair of images is found, the rate of
events is enhanced. For example, for $\gamma=0.99$, when each image has
an unperturbed magnification of 50 (which would last for 1 year if the
total Sgr A* event has an Einstein radius crossing time of 100
years), the rate is enhanced to $\sim$ one event per year (we have
assumed 
$\overline{L}'\simeq L_{\eta'}/\sqrt{2}$, 
more appropriate for this
case).

  The timescale for these events is \,$\Delta t \simeq \overline{L}'/\sigma_p$, 
or 
\begin{equation}
\Delta t \,\simeq \, 240 (\gamma^{-1}-\gamma)^{-3/4}
\biggl({\theta_\e\over 1''}\biggr)^{3/2}
\biggl({m_{\rm bh}\over 7\,M_\odot}\biggr)^{1/2}
\,\, {\rm days}\, ,
\end{equation}
For the same parameters used
previously, with $\gamma=0.6$ this gives an event duration of $\sim$
60 days. For $\gamma=0.99$, the event duration would be increased to
$\sim$ one year (using now $\overline{L}'\simeq \sqrt{2} L_{\xi'}$). 
Since this is of order of the inverse of the event rate, we would be in
the ``optically thick'' lensing regime where
several cluster black holes are significantly affecting
the magnification at a typical time.

\section{Measuring the Black Hole Masses}

	The black holes generate Chang-Refsdal magnification patterns
and so are formally identical to planetary lenses.  Gould \& Loeb
(1992) showed that for the planetary case one could determine the
mass ratio $q$ from the photometric light curve alone: the
effective range of the planet is a factor $q^{1/2}$ smaller
than that of the parent star, so $q$ is proportional to the square
of the ratio of the durations of the planetary perturbation to
the event as a whole.  Moreover, the structure of the planetary 
perturbation, and so the constant of proportionality,
can be determined from easily measured features of the light curve.

	In presenting their argument, Gould \& Loeb (1992) explicitly
assumed that the orbital speed of the planet was small compared
to the transverse speed of the system.  This assumption, which
is well justified in the planetary case, implies that the transverse
velocities are essentially the same for both objects and therefore
allows one to infer a ratio of effective ranges from a ratio of
timescales.  Unfortunately, the assumption breaks down in the
present case: unlike planets, the orbiting black holes have speeds 
that are the same order or larger than the sources being lensed.

	It is nevertheless possible to recover the black-hole mass
by incorporating astrometric information which, while generally
unavailable for planets, can be obtained for black holes.  First,
the normalized lens-source separation, $u$, can be determined either
from the flux ratio $\cal R$ of the two images, 
$u={\cal R}^{1/4}-{\cal R}^{-1/4}$,
or from the measured fractional offset 
$\Delta\equiv(\theta_{I+}+\theta_{I-})/(\theta_{I+}-\theta_{I-})$ 
of Sgr A* from the midpoint of the two images, 
$u=2(1/\Delta -1)^{-1/2}$.  One can then determine the
Einstein radius, 
$\theta_\e = (\theta_{I+}-\theta_{I-})/({\cal R}^{1/4}+{\cal R}^{-1/4})$, or
$\theta_\e = \sqrt{\Delta^{-1}-\Delta}(\theta_{I+}-\theta_{I-})/2$.

	Next, the shear $\gamma = {\cal R}^{-1/2}$ or 
$\gamma = (1-\Delta)/(1+\Delta)$
determines the geometry of the 
Chang-Refsdal lens up to an overall scale factor $q$.  For fixed
source trajectory through this geometry, the astrometric deviation is
proportional to $q\theta_\e$. Hence, if this trajectory were known, then 
measurement of the astrometric deviation would yield $q$.

	If the source passes through the caustic, then the trajectory
can be easily determined from the photometric light curve alone.  However,
if the source passes outside a cusp, the photometric light curve is
potentially subject to two interpretations: the approached cusp could be
along the Sgr-A*/black-hole axis, or perpendicular to it.  These two cases
are nevertheless easily resolved astrometrically since the two 
astrometric deviations are at right angles to one another.  Once it is 
resolved, one knows from the height of the photometric peak of the
deviation, how close the source was to the cusp when it crossed the cusp
axis.  The astrometric deviation at this point then gives $q$.

	The astrometric deviation is of order 
$q\theta_\e\sim 1.5\,{\rm mas}(\theta_\e/1'')$.  Recall that just to
resolve the imaged stars requires {\it resolution} $\ll 0.\hskip-2pt ''1$, so
{\it centroiding} the images to sub-mas levels should be feasible.  Once
the $\gamma$, $q\theta_\e$, and the trajectory are determined, the proper 
motion and hence the transverse velocity can be found from the light
curve.

\section{Discussion}

  We have shown that if stars down to $K=21$ are resolved
within an arc second of Sgr A*, a pair of lensed images from a
background bulge star is likely to be found, and as many as 5 are
expected if $K=23$ is reached. These images will need to be
distinguished from the much more numerous stars in the vicinity of
Sgr A* by finding a pair of images with the
correct relations between magnifications, positions and proper motions
expected for a point mass lens.

  The detection of these lenses would show that Sgr A* is
concentrated within a radius $\la 1''$, but this has already
been demonstrated at much smaller radii by the direct measurement of
stellar accelerations (Ghez et al.\ 2000).  A much more interesting application
would be to test the predicted presence of a cluster of stellar black
holes around Sgr A*. Unfortunately, we find that these events should
be quite rare: only about one per century for every lensed image that is
monitored. However, this rate would be enhanced if a pair of highly
magnified images were found.  (Our actual calculation was a bit 
conservative in that it was restricted to caustic-crossing events,
which are exceptionally easy to recognize.  But even if we assumed
that events with photometric deviations of $\delta\geq 0.3$ were 
detectable, this would only have increased the rate by a factor $\la 1.5$.)

  If an event is detected toward the black-hole cluster, it need not
  necessarily be due to a black hole: there will also be events in
  this direction caused by ordinary stars surrounding the black-hole
  cluster.  The old stellar population should have a core radius of
  $r_c\sim 0.7\,$pc, caused by scatterings from the cluster black
  holes. For events seen at projected radius $\theta$, the surface
  mass density of these stars is a factor
  $(R_0\theta_\e/r_c)^{3/4}/0.18\simeq 0.45(\theta/0.\hskip-2pt
  ''5)^{3/4}$ smaller than that of the black holes.  The duration of
  these events due to old stars would be similar to the ones from the
  cluster black holes, since the lower masses are roughly compensated
  by the lower velocity dispersion.  However, as discussed in \S\ 4,
  they could be distinguished astrometrically.

  Alexander \& Loeb (2001) predicted the rate for such stellar
microlensing near Sgr A*, although ignoring the cluster of black
holes and the resulting core in the stellar distribution. 
Their results differ strongly from ours: they predict that, with
a limiting magnitude $K=21$, almost ten microlensed images magnified
by a factor greater than 5 will be observed at any given time. Our
prediction is that about 5 stars lensed by Sgr A* would be found whose
faintest
image is brighter than $K=23$, and whose brightest image would typically
be brighter than $K=21$; but the probability that any of these images
is significantly magnified by a stellar lens at any random time is only
a few times $10^{-3}$. We think this difference can be explained by
the following three factors.  First, they normalized their density
of lenses three times higher than the stellar density computed by
Genzel et al.\ (2000).  Second, they assumed a density of sources that are
brighter than the turnoff and are at a distance $R_0$ behind the GC that is 
about 300 times higher than the density observed in the solar neighborhood 
(Bahcall 1984).
Third, they claim that the probability to have a high magnification
is strongly enhanced when the projected stellar lenses account for
only a few percent of the mass of Sgr A*, which we think is
incorrect.

  Finally, we mention that more microlensing events by cluster black
holes might be found by monitoring to the same depth all the stars
over the entire area of the black hole cluster, within $\sim 20''$
of Sgr A*. In this case,
we would not know a priori which of the source stars are at a large
distance behind the GC, and the magnification by Sgr A*
would be negligible. The events from the cluster black holes
would tend to have Einstein timescales $t_\e$ that were 
$(m_{\rm bh}/\overline{m})^{1/2}\sim 6$  times longer than those of 
ordinary bulge events (since, far away from Sgr A*, the velocities of the 
black holes are similar to those of other bulge stars).  They
could not be unambiguously distinguished on the basis of timescales
alone because the full-width at half-maximum of the mass estimate
is almost a factor 100 (Gould 2000).  However, since most of this
width is due to the large dispersion in proper motions $\mu$, it could
largely be removed by measuring the astrometric deviation, which yields
the Einstein radius $\theta_\e$, and so $\mu=\theta_\e/t_\e$.
Since the surface density of black holes is
$\Sb \propto \theta^{-3/4}$, and their velocity dispersion
$\sigma\propto \theta^{-1/2}$, the number of microlensing events they
produce within $\theta$ is proportional to $\theta^{3/4}$, so there
should be about 10 times more events from the entire cluster, within
$20''$, than events that are affected by the magnification of
Sgr A*.

\begin{acknowledgements}

JC thanks Scott Gaudi for useful discussions
and insights.  Work by AG was supported in part by grant AST 97-27520
from the NSF, and in part by a grant from Le Minist\`ere de
L'\'Education Nationale de la Recherche et de la Technologie.

\end{acknowledgements}


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\clearpage

\begin{figure}
\plotone{f1.eps}
\caption[junk]{\label{fig:one}
Dereddened surface density $K_{0}$ luminosity function
{\it derived} from the Zoccali et al.\ (2000) data, shown as filled squares.
These are all the stars contained inside the 0.76 pc$^{2}$ of the Zoccali
et al.\ (2000) field.  The first data point (indicated by
a star) represents all stars brighter than $K_{0}\sim 16$.  Open symbols
represent the distribution of brighter images of bulge stars (number
per magnitude interval) lensed by Sgr A$^{\ast}$ for three detection
thresholds.
}
\end{figure}

\end{document}