4. Microlensing towards the bulge using dark objects in globular clusters as lenses
One encounters several problems by using globular cluster stars as sources. In fact, globular clusters have at most some stars that can be monitored. The angular size of a globular cluster is small and the stars are highly concentrated towards the center. Furthermore, most globular clusters are located towards the galactic center which restricts the number of objects being well suited for an exploration of the halo. Therefore, microlensing by globular clusters only leads to valuable results, if one is able to simultanously observe dozens of clusters with a very high resolution for several years. At present this seems to be a somewhat to demanding task, however this could be feasible in the near future.
In this section we discuss the possibility to use dark objects in globular clusters towards the bulge as lenses (see also Taillet, Longaretti & Salati 1995, 1996), in front of rich regions of the galactic center or the spiral arms. There is a large sample of possible clusters which could be used for this purpose (see Table 1). The core radius of the dark component of the cluster can be larger than the core radius obtained by surface luminosity measurements. Since the size of the cluster is much smaller than its distance, it can be assumed to be equal for all its members, moreover, the velocity dispersion inside the cluster is also small, hence globular clusters are well suited for determining the lens mass.
Table 1. Optical depth and event rate due to MACHOs in some globular clusters for sources located towards the galactic center. The symbols are defined within the text. Cluster data is adopted from Harris (1996).
4.1. Estimate of optical depth and event rate
In the first part of this section, we give some (conservative) estimates for the optical depth and the event rate due to some globular clusters towards the bulge by scaling the results obtained for 47 Tuc under the assumption that the clusters contain only little dark matter. As was derived in Eq. (39), the optical depth due to a globular cluster is proportional to the surface density of microlensing events and the pure geometrical quantity . Assuming to be proportional to the total visual magnitude , we can calculate the contribution to the optical depth due to the clusters (see Table 1). Since the galactic bar is not too extended, we fix the distance of the sources to be kpc.
Compared with the measured average optical depth towards the galactic center due to the disk and the bar itself (Alcock, Allsman, Alves et al. 1997a), we see from Table 1 that some of the globular clusters can give a very significant contribution to the total optical depth or even dominate it along their line of sight, as in the case of NGC 6553, which lies close to the ideal distance of .
A rough estimate of the microlensing event rate per year due to globular clusters towards Baades Window is obtained by properly rescaling Eq. (40). We compute the event rate in units of as in Eq. (40), in order to be able to easily compare between the different clusters. To that purpose one has to use the scaling factor given by:
where the unprimed quantities belong to 47 Tuc and the primed to the cluster under consideration. A is the surface which corresponds to at the distance of the cluster. For the calculation of the numbers, as given in Table 1, we assumed which is indeed a very conservative value and, therefore, we will get a lower limit for the eventrate.
4.2. Spatial distribution of microlensing events around NGC 6522, NGC 6528 and NGC 6540
In the following we present a rough analysis of the microlensing events around the three globular clusters which lie within the observation fields of MACHO and OGLE.
Within a radius of pc around NGC 6522 and 6528 we found 7 events for each of them (see Table 2). Since the projected areas of these two clusters overlap, 4 events lie within the pc circles around both clusters. NGC 6540, which is nearer to us, hosts a total of 15 events within the pc circle. At first sight, since the covered area is about four times larger, this value seems to be lower than expected. However, NGC 6540 is about 8 times less bright than NGC 6522. Therefore, if the total luminosity roughly scales with the total mass content, we expect NGC 6540 to contain less dark matter than NGC 6522.
Table 2. Microlensing events within a radial distance of pc around NGC 6522, NGC 6528 and NGC 6540. Values marked with an asterisk lie within pc of both NGC 6522 and NGC 6528. For NGC 6540 we also give the finer binning as used for Fig. 8. Data is taken from the alert list of the MACHO collaboration and the event list of the OGLE team. The event duration follows the OGLE convention i.e. , which is half the value as reported in the MACHO alert list.
Within pc we found 3 events for NGC 6522, 2 events for NGC 6528 and 4 events for NGC 6540 (see Table 2). The event rate ratio in units of events per square degree for the pc circle and the pc ring (excluding the innermost pc) are 99/25 for NGC 6522, 74/35 for NGC 6528 and 33/17 for NGC 6540, respectively. For all three clusters the central region shows an increase in microlensing events. Due to the fact that the pc circles around NGC 6522 and NGC 6528 intersect, we can also calculate the event rate in the overlapping region, which we find to be 70 events per square degree. Within our poor statistics, this is what one expects for a line of sight crossing twice the region of influence of a globular cluster. It would also give a first hint, that globular clusters can indeed have a population of dark objects that reaches out as far as pc.
Of course, there is now the problem how to distinguish between the events due to MACHOs located in the globular cluster and those due to MACHOs in the disk or bulge, which will define our "background". Since this cannot be decided for a single event, we assume that the events which lie in the ring from 12 to pc are due to MACHOs in the disk or bulge. This way we certainly overestimate the "background". Moreover, the common events within pc around both NGC 6522 and NGC 6528 were counted as "background" events for both clusters. This way leading also to a higher background rate. The so estimated event rate per area is then subtracted from the value in the inner pc region. The leftover events should be due to MACHOs located in the globular cluster. We find the following values (in parenthesis the "background"): 2 (1) hence a total of 3 events for NGC 6522, 1 (1) event for NGC 6528 and 2 (2) events for NGC 6540. We see that the number of observed events in the inner pc is roughly twice as high as one would get due to MACHOs in the disk or bulge alone. By assuming that all the events in the ring from 12 to pc are due to MACHOs in the disk or bulge, we have certainly underestimated the contribution from the globular cluster, since some of these events might also be associated with the cluster.
Of course, we must discuss the shortcomings of our analysis. We tacitly assumed that the product of total observation time and background star density is the same for the 12 and pc regions for a given cluster. This should at least be well fullfilled for the relatively small regions around NGC 6522 and NGC 6528. For NGC 6522 and NGC 6528 we added MACHO and OGLE data, hence we use a different normalisation for them than for NGC 6540. In addition, we did not take into account the different efficiencies. Moreover, our evaluation is based upon a very poor statistics, and thus one must take the above results with all the necessary caution. However, since all our estimates were performed very conservatively and for all three clusters we get the same behaviour and since also the overlapping region of NGC 6522 and NGC 6528 shows an increase in microlensing events, we think it's fair to state, that globular clusters can -within some pc- at least double the optical depth and also the event rate due to MACHOs in the disk and the bulge; the latter quantities being (Alcock, Allsman, Alves et al. 1997a) and for a typical mass of .
Stars in globular clusters can also act as sources for microlensing, however, as discussed in Sect. 3.2.4 their contribution, unless for the very central region of pc, is very small as compared with lensing due to sources in the bulge. For NGC 6522 and NGC 6528 there is a small probability that the source lies in NGC 6528 and the lens in NGC 6522. Knowing the distances and the relative motion of the two clusters this system might yield the most accurate mass determination for a lensing object.
It is interesting to note, that the above mentioned event rate ratios scale with the total luminosity i.e. the brightest cluster NGC 6522 shows also the largest increase of events towards the center.
For the mean event duration as given in Table 2, we calculate the typical mass of a MACHO for km/s and km/s. Obviously, one should consider only the events due to MACHOs in the cluster. However, since it is not possible to distinguish them, we have just taken the average value as a first approximation. This should not be far from the true value (as can be seen by inspection of Table 2). The results are given in Table 3. We see that depending on the MACHOs can be either Jupiter type objects, brown dwarfs, M-stars or even white dwarfs.
Table 3. Typical masses for the microlensing events around NGC 6522, NGC 6528 and NGC 6540. The upper value corresponds to km/s, the lower to km/s. d denotes the distance from the cluster center.
Below we compute the optical depth and the event rate for four different King models as a function of the distance from the cluster center in the lens plane. The parameters are tuned such that the average of the values in the interval from 1 to pc roughly corresponds to the above mentioned values for the optical depth and the event rate . The tidal radius was assumed to be pc, the distance to the lens is set to be kpc and the one to the source kpc. For the calculation of the event rate we again take the rather low value km/s. Since we are not able to reproduce the mass function, we rather study a bi-mass model, hence the cluster consists of a heavy component (component 1) and a light one described by one of the other components as defined below.
We find that a population of low mass objects as described by a King model with a total mass of can lead to the desired enhancement of the optical depth and the lensing event rate up to distances of pc from the cluster center (see Figs. 9 and 10). Of course, the models and mass values given above have to be taken as an illustration, nevertheless it is clear that the rather high observed microlensing rate of the clusters imply a substantial dark matter component. Although there is a large inherent uncertainty in the event rate due to the poor knowledge of , it is important to note, that the required optical depth and event rate cannot be due to the heavy component alone.
© European Southern Observatory (ESO) 1998
Online publication: July 20, 1998