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Astron. Astrophys. 336, 411-424 (1998)

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3. The system SMC-47 Tuc: a paradigm for cluster lensing

In this section we thoroughly discuss the basics of microlensing by a globular cluster. We use the system SMC-47 Tuc as an example, but the results, by appropriately scaling them, are valid for other globular clusters as well.

First, we discuss simple models of 47 Tuc which will then be used for the computation of the optical depth and the microlensing event rate for different geometries of lens and source.

The globular cluster 47 Tuc (NGC 104) lies at galactic coordinates [FORMULA], [FORMULA]. For the distance we assume [FORMULA]kpc, although in the literature values up to [FORMULA]kpc are quoted (Harris 1996). Our choice will rather underestimate the optical depth. The position is such that it overlaps with a part of the outer region of the SMC, which makes it an interesting object. Globular clusters are small objects compared to the scale of their distance, hence they are well suited for gravitational lensing, since one may assume the distance of their stars to be the same for all practical purposes.

3.1. Spatial density and velocity dispersion for 47 Tuc

For the calculation of the lensing rate we need to know the spatial distribution of the dark matter in the globular cluster. Since this is not known, we will instead discuss models for the total mass of the cluster. To get the mass density [FORMULA] of the dark objects, we then make the simplifying assumption that [FORMULA] is proportional to the total mass density [FORMULA], i.e. [FORMULA] is given by

[EQUATION]

where [FORMULA] is the total mass of the MACHOs in the cluster. We are aware, that this assumption is oversimplified, however, as long as the content of dark matter in globular clusters is not known, it is one way to parametrize our ignorance. Moreover, since the expected event rate for 47 Tuc is about one event per year or even less, we do not consider multi-mass models for 47 Tuc. We postpone the discussion of them to Sect. 4, when we discuss lensing by globular clusters towards the bulge in which situation the model can be tested due to the higher number of microlensing events.

The simplest model of a globular cluster is a self-gravitating isothermal sphere of identical "particles" (stars). The equilibrium distribution function in phase-space coordinates is

[EQUATION]

where [FORMULA] is the gravitational potential of the cluster, T the temperature, [FORMULA] Boltzmann's constant and m the mass of a "particle". Taking into account the spherical symmetry of the globular cluster and defining the one-dimensional velocity dispersion [FORMULA] as

[EQUATION]

Eq. (9) reads

[EQUATION]

Here r is the radial distance relative to the cluster center. Integrating over all velocities, we get

[EQUATION]

Inserting Eq. (12) into the Poisson equation for the gravitational potential, one obtains

[EQUATION]

A solution of this equation is

[EQUATION]

The mass density given in Eq. (14), together with the integrated velocity distribution Eq. (11), defines the singular isothermal sphere.

To avoid the singularity at the origin we rescale the variables ([FORMULA]) and find a non-singular solution which can be approximated for [FORMULA] by (for details see Binney & Tremaine 1987)

[EQUATION]

and for [FORMULA] with the singular function given in Eq. (14).

An observable quantity, connected with the surface density of stars in a cluster, is the surface brightness. The luminosity function [FORMULA] gives the relative number of stars with absolute magnitude M in the range [[FORMULA], [FORMULA]]. The number surface density of the stars can be computed from the observed surface brightness. Assuming that all stars in the cluster have the same mass, we can compare the above densities with surface brightness measurements. The mass surface density [FORMULA] is derived from the mass density by the integration

[EQUATION]

where b is the projected radial distance. For the density of the singular isothermal sphere, given by Eq. (14), the surface density is

[EQUATION]

Similarly, the density [FORMULA]in Eq. (15) leads to the surface density

[EQUATION]

where [FORMULA].

Since Eq. (15) fits the regular solution of Eq. (13) in the range [FORMULA], we find that the corresponding surface density falls to roughly half of its central value at the core radius [FORMULA]. Knowing [FORMULA] and [FORMULA] from observations, the central density [FORMULA] is determined in the isothermal model by

[EQUATION]

With [FORMULA]pc as given in Lang (1992) and a velocity dispersion of [FORMULA] (Binney & Tremaine 1987), Eq. (19) for 47 Tuc yields [FORMULA]

King (1962) found that the above surface density functions (17), (18) fit well star counts up to a limiting radius [FORMULA] (called tidal radius), where the measured surface density drops sharply to zero. Thus a better fit to the surface density of the cluster is given by

[EQUATION]

With a spherical symmetric density

[EQUATION]

the corresponding mass density becomes

[EQUATION]

where

[EQUATION]

and k is chosen such that [FORMULA] (King 1962). Integrating this density with a tidal radius [FORMULA] (Lang 1992), one finds a total mass of [FORMULA] for 47 Tuc.

To study the dependence of the optical depth and the lensing rate on different mass distributions, we will use the following four models (see Fig. 1).

[FIGURE] Fig. 1. The four different mass density models used in the calculations, plotted as a function of the radial distance r. For clarity the plot ends at [FORMULA] rather than at [FORMULA].

1. Fitted King model:
the density is given by Eq. (22) with central density [FORMULA], core radius [FORMULA] and [FORMULA] as already mentioned above.

2. Inner isothermal model:
the density is given by Eq. (15) with [FORMULA] and [FORMULA] as above, but with [FORMULA] chosen such that the total mass within the tidal radius is the same as in the fitted King model, rather than determined via Eq. (19).

3. Singular model:
the mass density is described by the isothermal sphere, i.e. by Eq. (14) with the same [FORMULA] as mentioned above. The total mass of this model within the tidal radius [FORMULA] is [FORMULA].

4. [FORMULA] - model:
the mass density is given by

[EQUATION]

with [FORMULA] as above and [FORMULA] chosen such that the total mass within the tidal radius is the same as for the fitted King model.

The last three mass distributions are cutted discontinuously at the tidal radius. Thus the integration range for the optical depth and the lensing rate is only a sphere with a radius equal to the tidal radius. For all models the velocity distribution follows a Maxwell distribution as in Eq. (11) with [FORMULA].

The [FORMULA]-model is considered mainly because the integrations can be performed analytically. Hence, we can easily compare the analytic result with the ones obtained numerically for the other models.

Of course there exist more sophisticated models for globular clusters, e.g. the (single or multimass) King-Michie models (King 1966, Gunn & Griffin 1979), that take into account the finite escape velocity from the cluster, which naturally leads to a finite extension of the cluster. However, for the study of the influence of different mass distributions on the measured quantities, we consider the above mentioned models to be sufficient and, therefore, in this section, restrict our discussion to them. In addition, we remind on the well known scaling properties of the King model, which allow to derive the corresponding quantities for a different choice of parameters.

3.2. Optical depth, lensing rate and mean event duration for SMC-47 Tuc

We now discuss the different possibilities for microlensing in the system SMC-47 Tuc, i.e. events where the source and the lens are both located in 47 Tuc; the source is in the SMC and the lens in 47 Tuc; the source is in 47 Tuc and the lens in the halo of the Milky Way, or finally the source resides in the SMC and the lens in the Milky Way. For SMC self-lensing we refer e.g. to the paper by Palanque-Delabrouille, Afonso, Albert et al. (1997).

At the end we also discuss the dependence on the mass function, which itself is independent of the lensing geometry. Since we hope to disentangle the different cases, we also calculate the differential rate [FORMULA]. Throughout this part we will assume all lenses to have the same mass. The velocity distribution of the halo objects, as well as those of the cluster is taken to be a Maxwell function. As already mentioned, we define the amplification threshold to be [FORMULA].

3.2.1. Optical depth for source and deflecting mass in the cluster

For completeness we discuss also this case, although, as we will see, its contribution can be neglected for all practical purposes.

For a pointlike source [FORMULA] is, according to Eq. (3), given by

[EQUATION]

where

[EQUATION]

[FORMULA] is the distance from the observer to the cluster center and D the one to the source. Hence, the integration is cut at the boundary of the cluster. The optical depth for a source located in the center of the cluster ([FORMULA]) is [FORMULA] for the fitted King model. The results for the 4 different models are shown in Fig. 2. Of course the isothermal-sphere model differs most, because its total mass [FORMULA] varies substantially from the others.

[FIGURE] Fig. 2. The optical depth for the four different models as a function of the source position. All sources are located on the line of sight from the observer to the cluster center, but at different radii relative to the center. The optical depth of the singular model is only shown for sources with [FORMULA], since its mass density has a singularity at [FORMULA]. For the plot we assume [FORMULA], and the total mass of the models as given in Sect. 3.

Since the lower limit of integration is very close to 1, [FORMULA] can be approximated as follows

[EQUATION]

Here

[EQUATION]

and [FORMULA] is the distance from the lens (located at [FORMULA]) to the source. Hence, [FORMULA] is proportional to [FORMULA] weighted with the mass density in the globular cluster along the line of sight from the observer to the source.

3.2.2. Lensing rate for source and deflecting mass in the cluster

We assume the mass function to be independent of the position and all lenses are taken to have the same mass [FORMULA] (in units of [FORMULA]). From Eq. (9) we find that the transverse velocity distribution is given by

[EQUATION]

with [FORMULA]. If the source as well as the deflecting mass are in the cluster, the event rate becomes

[EQUATION]

with [FORMULA] and [FORMULA] defined to be

[EQUATION]

We use the same geometry as in the previous subsection. The result for the fitted King model for a source located at the center of the cluster is [FORMULA] 1/s.

In order to more easily compare between the microlensing rates for different locations of the source and the lens, we introduce the quantity [FORMULA]:

[EQUATION]

where [FORMULA] is the number of microlensing events per unit time in an area of [FORMULA] located at an angular distance [FORMULA] from the cluster center. For simplicity, we will give the rate [FORMULA] in units of pc. In the plane perpendicular to the line of sight through the cluster center, [FORMULA] corresponds to 1.2 pc. Hence, we can define a new quantity

[EQUATION]

which is in units of [FORMULA] [FORMULA]. We call [FORMULA] the surface density of microlensing events. To calculate [FORMULA] we have to add up the lensing rates for all stars located on the line of sight with impact parameter b from the cluster center. From Eq. (27), we see that the lensing rate [FORMULA] depends on D and on b, through [FORMULA] and [FORMULA]. Taking this into account we get

[EQUATION]

where in the ideal situation, which will lead to an upper bound for [FORMULA] we have [FORMULA]. [FORMULA] is the number density of stars (in units of [FORMULA]) in the cluster at distance r from the center and D, b, [FORMULA] are all in units of pc. To describe the distribution of stars in the cluster, we assume that the number density is proportional to the mass density [FORMULA] of the fitted King model, since the mass surface density of this model is proportional to the number surface density of stars in a cluster. Thus [FORMULA] is given by

[EQUATION]

with [FORMULA] (Lang 1992). With Eqs. (27) and (29) this leads to

[EQUATION]

The results for the four models are shown in Fig. 3.

[FIGURE] Fig. 3. The surface density of microlensing events is plotted as a function of the projected radius b for the four different models calculated in the text, assuming [FORMULA]. For the plot we assume [FORMULA], and the total mass of the models as given in Sect. 3.

The number of lensing events per year in the whole cluster is

[EQUATION]

for the fitted King model. On the other hand, the number of lensing events depends also on the telescope resolution. Assuming a distribution according to Eq. (30), we find that stars with a projected radial distance smaller than [FORMULA]pc from the center cannot be resolved if the limiting resolution of the telescope is [FORMULA], whereas if it is [FORMULA] even stars with projected radius b smaller than [FORMULA]pc from the center cannot be resolved. Hence, the number of expected lensing events per year using a telescope with a resolution of [FORMULA] is

[EQUATION]

and the one for a resolution of [FORMULA] is

[EQUATION]

The first line is for the fitted King model, the second for the [FORMULA]-model.

The distribution of the microlensing events as a function of their duration T, using the transverse velocity distribution as defined in Eq. (26) and the definition of [FORMULA] in Eq. (28), is given by

[EQUATION]

The result for the fitted King model assuming a lens mass [FORMULA] is shown in Fig. 4.

[FIGURE] Fig. 4. The distribution of the lensing events with duration [FORMULA] for different geometries of the lens system. For the calculations we used the fitted King model and made the assumption that all lenses have the same mass [FORMULA]. The notation is to be understood as follows: the first entry denotes the position of the source, the second the one of the lens e.g. SMC[FORMULA]Cl means a star of the SMC is lensed by an object in the cluster. MW abbreviates Milky Way; [FORMULA] and D are defined within the text. For the plot we assume [FORMULA], and the total mass of the model as given in Sect. 3.

3.2.3. Optical depth and lensing rate for the source in the SMC and the deflecting mass in the cluster

What changes in the calculation of [FORMULA] and [FORMULA] is itemized below:

!!!!beginitemize

LI the distance D of the source is now about [FORMULA]kpc, corresponding to the distance of the SMC from the Sun;

LI the x-integration goes from [FORMULA],
to [FORMULA];

LI the velocity distribution gets shifted, because of the motion of 47 Tuc relative to the SMC with a velocity of about [FORMULA] [FORMULA] [FORMULA]km/s. Hence, a velocity drift has to be introduced in the Maxwell distribution, such that

[EQUATION]

where we assume the x-axis to be parallel to the velocity drift. The resulting transverse velocity distribution is then given similarly to Eq. (26). With this, the integral over [FORMULA] in Eq. (4) yields

[EQUATION]

!!!enditemize

Thus, the lensing rate and the optical depth become

[EQUATION]

For the extreme case of a star in the SMC lying on the line of sight going through the center of the cluster, the optical depth is [FORMULA] and the lensing rate [FORMULA] 1/s.

Since the tidal radius is smaller than about [FORMULA]pc, the quantity [FORMULA] as well as [FORMULA] does not vary more than 10% over the integration range, and the variation of [FORMULA] as well as [FORMULA] is directly proportional to the surface density of the cluster. Assuming a tidal radius of [FORMULA]pc we find (with [FORMULA] an average value for x)

[EQUATION]

[EQUATION]

The mean event duration is

[EQUATION]

which is nearly independent of the angle between source and cluster center, since both [FORMULA] and [FORMULA] are proportional to [FORMULA] as given by Eqs. (37) and (38).

Of course the chance to find a lensing event in this case depends not only on the optical depth, but also on the number of SMC stars in the background of 47 Tuc. Hesser, Harris, Vandenberg et al. (1987) finds an average of [FORMULA] with a magnitude in the intervall of [FORMULA]. In the following we assume a constant number of [FORMULA], although there is a gradient in the density of the SMC stars across the whole 47 Tuc region decreasing from the southeast to the northwest side.

According to Eq. (29) we get with the above mentioned proportionality between [FORMULA] and [FORMULA] a surface density of microlensing events of

[EQUATION]

where [FORMULA] is the number of SMC stars per [FORMULA]. [FORMULA] is shown in Fig. 5.

[FIGURE] Fig. 5. The surface density of microlensing events is shown for the different geometries of the lens system. For the calculations we used the fitted King model and made the assumption, that all lenses have the same mass [FORMULA]. For the abbreviations we refer to Fig. 4. For the plot we assume [FORMULA], and the total mass of the models as given in Sect. 3.

The number of events per year in the whole area of 47 Tuc is then given by

[EQUATION]

for the fitted King model. The expected number of events per year for an observation done with a resolution of [FORMULA] is

[EQUATION]

and for an observation with a resolution of [FORMULA]

[EQUATION]

The first value is for the fitted King-model and the second for the [FORMULA]-model.

For comparison, if we insert a velocity [FORMULA] arcsec/year for the cluster and a density of 50 SMC stars per [FORMULA] in the formula for the event rate per year, as given in the paper of Paczynski (1994) one obtains [FORMULA] for the rate in the whole area of the cluster, [FORMULA] for the rate with a resolution of 0.1" and [FORMULA] for the rate with a resolution of 1". The factor f parametrizes the fraction of dark matter [FORMULA]. Paczynski used in his calculations the total mass as given by the singular isothermal sphere model, which gives [FORMULA]. The difference to our values is mainly due to the different adopted mass surface densities. Since the total mass of the fitted King model is [FORMULA], this explains, with the help of Eq. (40), just the factor 12 between Paczynski's results for the number of events and ours for the fitted King model for the rate in the whole area of the cluster. A multi-mass model fitted to radial velocity and surface brightness observations of 47 Tuc leads to a total mass of [FORMULA] (Meylan 1989), which is about 3 times bigger than the value for the fitted King model. In addition, the fitted King model concentrates much more mass in the center, whereas in the singular isothermal sphere model, there is still a lot of mass in the outer regions of the cluster. From these considerations, we see that the uncertainties are quite important and they are mainly due to the poor knowledge of the dark matter content in globular clusters.

Paczynski (1994) proposed to measure the microlensing of background stars by MACHOs in globular clusters, because in this way the mass of the lens can be determined more precisely than in the lensing experiments under way. Since the lens is in the cluster, it has the transverse velocity and the position of the cluster. To estimate the inherent error of the method we adopt the values below, where the transverse velocity [FORMULA] is assumed to be precisely known up to the cluster dispersion velocity and the distance up to the cluster size: [FORMULA], [FORMULA] and [FORMULA]. It follows then an uncertainty in the mass determination of [FORMULA] The uncertainty in the lens mass with the currently reported lensing events is more like a factor 3 or even bigger (Jetzer & Massó 1994, Jetzer 1994).

The distribution of the lensing events as a function of the duration T is:

[EQUATION]

where y is the angle between [FORMULA] and the projected MACHO velocity in the plane perpendicular to the line of sight. The numerical results, with the above mentioned values for [FORMULA] and [FORMULA] are shown in Fig. 4.

3.2.4. Optical depth and lensing rate for the source in the cluster and the deflecting mass in the halo of the Milky Way

To calculate the optical depth and the lensing rate we set the position of the source equal to the position of the cluster. In the reference frame, where the origin is at the galactic center and the [FORMULA] plane is the symmetry plane of the galaxy, the coordinates of a lens, located on the line of sight from the observer to the cluster center, are [FORMULA], [FORMULA], and [FORMULA], where [FORMULA] and [FORMULA] are the galactic latitude and longitude of 47 Tuc. With the number density distribution Eq. (5), we find the optical depth (with [FORMULA]) to be [FORMULA].

Let's now look at the lensing rate. Since the velocity distribution in the halo is assumed to be Maxwellian, the integration over [FORMULA] in Eq. (9) leads to

[EQUATION]

with [FORMULA]. Thus, in a model where all lenses have the same mass, the lensing rate becomes with Eq. (4).

[EQUATION]

According to Eq. (29) the surface density of microlensing events in this case is

[EQUATION]

[FORMULA] is the number surface density of the stars in the cluster in units of [FORMULA].

The time scale calculated with [FORMULA] and [FORMULA] as obtained above is

[EQUATION]

For the distribution of the lensing events we obtain

[EQUATION]

with the MACHO mass density [FORMULA] as in Eqs. (5) and (6).

[EQUATION]

The numerical results with [FORMULA] and [FORMULA] for [FORMULA] are shown in Fig. 4 where the motion of the cluster relative to the Milky Way halo is neglected.

For a list of globular clusters which might be used as targets for a systematic microlensing search, we refer to the papers of Gyuk & Holder (1997) and Rhoads & Malhotra (1997).

3.3. Optical depth and lensing rate for a source in the SMC and the deflecting mass in the halo of the Milky Way

Since this is a standard case we restrict to the presentation of the results. With a distance [FORMULA]kpc and the galactic coordinates of the SMC [FORMULA] [FORMULA], we get for the optical depth and the lensing rate

[EQUATION]

Therefore, the time scale is [FORMULA] days. The result for [FORMULA] is shown in Fig. 4.

3.3.1. Dependence of the lensing rate [FORMULA] on the mass function

Up to now, we assumed all lenses in the cluster to have the same mass, which is, of course, rather unphysical. Therefore, we discuss now the variation of [FORMULA] with the mass function [FORMULA]. However, we will still make the assumption that the mass function does not depend on the position. To estimate the variation of [FORMULA] we choose a mass function of the form

[EQUATION]

with [FORMULA] in the interval [-1, 3] and the mass m in the range [FORMULA]. With Eq. (47) the lensing rate [FORMULA] becomes (with [FORMULA]=1)

[EQUATION]

[FORMULA] is the lensing rate computed under the assumption that all lenses have mass [FORMULA].

For the limiting cases [FORMULA] and [FORMULA] this leads to a lensing rate

[EQUATION]

In Fig. 6 [FORMULA] is shown for different upper and lower limits of the MACHO mass.

[FIGURE] Fig. 6. [FORMULA] is plotted as a funtion of the slope [FORMULA]. This corresponds to the dependence of the lensing rate on the slope of the mass function. [FORMULA] is the lensing rate for a model where all MACHOs have the same mass [FORMULA]

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Online publication: July 20, 1998
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