## 3. VLM star clusters## 3.1. RationaleThe rationale for invoking clustered DM comes from a number of
considerations. Firstly, visible globular clusters are observed out to
great distances from the Galactic centre suggesting they may have an
important role in halo formation, though they are identified with the
spheroid population. Secondly, some of the most promising theories for
the formation of large amounts of compact halo baryonic DM (Ashman
1990 ; De Paolis et al. 1995 ) predict that the DM should be
clustered into dark globular clusters with a typical mass of around
(Ashman 1990 ). These theories predict that the
clusters should comprise either brown dwarfs or VLM stars, due to high
gas pressures suppressing the minimum fragmentation mass. They also
explain the typical mass and spatial distribution of the visible
cluster population since the theories are extensions of the Fall-Rees
theory of globular-cluster formation (Fall & Rees 1985 ). The
baryon Jeans mass at the cosmological epoch of recombination is
, which is also close to
for a cosmological density of order unity and a
baryon density satisfying cosmological
nucleosynthetic constraints: , where The effect that clustering can have on number-count statistics is
well illustrated by considering the expectation distances both to the
nearest unclustered VLM star and to the nearest cluster. Let us assume
that VLM stars contribute some fraction to the
halo DM and that their density is everywhere ,
where is given by Eq. 1. Let us also take
the fraction of VLM stars residing in clusters to be everywhere
, so that the fraction of unclustered VLM stars
is correspondingly . Then, for VLM stars with
mass whilst the local number density of VLM star clusters of mass
The expectation value for the distance where is the gamma function. So, using Eq. 4 and Eq. 5, the expectation distances to the nearest unclustered VLM star and star cluster are, respectively, where is normalised to a solid angle 20 times that of a single HST field. The large value for indicates the need to
consider the variation of along the line of
sight in order to obtain a more accurate answer, though the estimate
of Eq. 8 is sufficient to show that one might not see Another important consideration is whether one expects HST to be able to resolve individual stars within clusters. Since BFGK filter their observations to search for only stellar-like sources it is conceivable that a significant fraction of the total number of possible VLM star sources have been discarded because they reside in the cores of unresolvable clusters. Since the number of field galaxies far exceeds the number of stars at the typical limiting magnitudes of the HST fields, it may prove extremely difficult to analyse these fields for the presence of unresolved cluster cores. Using Eq. 8 as a lower limit on the cluster distance one finds
a typical angular separation between neighbouring stars (as projected
along the line of sight) for a cluster with radius compared to a HST pixel resolution of 0.1 arcsec. Since one expects clusters to be centrally concentrated one anticipates Eq. 9 overestimating the angular separation of stars near the core and underestimating the separation near the cluster edge. Therefore it seems that a significant fraction of cluster sources may appear unresolved. This fraction will be assessed more rigorously in Sect. 3.3. For the cluster scenario to be viable it must be consistent with existing dynamical constraints on such objects. These constraints, which are reviewed in Sect. 3.4, provide upper and lower bounds to both the permitted cluster mass and radius. The maximum contribution of such clusters to the halo is therefore also potentially bounded by these constraints. Finally, in Sect. 3.5, I show that this scenario is compatible with MACHO microlensing observations, and I use the MACHO limit on the contribution of VLM stars to compute the allowed ratio of clustered to unclustered stars. Potentially, this ratio provides the strongest constraint on the cluster scenario. ## 3.2. Space density of VLM clustersIn Sect. 2 limits were placed on the space density of unclustered VLM stars from HST number counts, assuming the stellar distribution obeys Poisson statistics. However, since clusters are extended objects, one can not directly apply the same assumption to them, since one may have a cluster which is only partially within the field of view (in fact this will be the case generally for the HST fields due to their small solid angle). However one can use Poisson statistics to assess the likely number
of cluster The description above applies to a cylindrical survey volume (i.e. a circular field of view). The size of the larger volume and number-count limits derived from it are sensitive to the geometry of the field volume on which it is centred (i.e. the shape of the field of view), which for the HST is 'L'-shaped rather than circular. However, for most of the cluster radii considered in this paper their angular sizes are always much larger than the angular size of the field of view itself, so the sensitivity to the shape of the field of view is generally very small, and so for simplicity I assume a cylindrical field volume geometry throughout. Another important consideration when placing limits on cluster
number densities is the statistical dependency of neighbouring fields.
In principle, sufficiently large clusters could appear in more than
one field and so any limits would need to take such correlations into
account. Even if one is placing limits based on the A lower limit on the expectation number of clusters with mass
Note that for , in Eq. 2, as required. If one assumes that the HST fields contain no cluster stars at all
then the 95% CL upper limit on the allowed halo fraction is
; 3 being the 95% CL upper limit on the
true average number of clusters when none are seen, assuming
statistical independency of neighbouring fields and Poisson
statistics. Note from Eq. 10 that the upper limit
is independent of since
it represents a limit on the The limits on for a range of cluster masses and radii are shown in Fig. 1, assuming . The solid grid depicts the limits assuming the clusters comprise VLM stars of mass , where as the dotted line shows the limits for clusters comprising hydrogen-burning limit stars (). The choice of cluster parameters is guided by the dynamical considerations discussed in Sect. 3.4. The plane at is shown to guide the eye in seeing which part of the parameter space is constrained and which is not. The most strongly constrained region occurs for low cluster masses and large radii, though combinations of cluster parameters for which (where is the average halo density within the survey volumes) are unphysical since they denote clusters whose average density is less than that of the local halo background. (Such 'clusters' might instead best be regarded as voids.) For interesting (i.e. significant) values of and this unphysical region includes the case where the average angular separation between neighbouring stars, given by Eq. 9, exceeds the HST field of view. Whilst HST provides strong limits on clusters of low mass and large radius, it is clear from the figure that a significant fraction of the parameter space is not constrained by HST observations.
It is apparent from Fig. 1 that the relationship between
and where for 0.2- stars
pc and , and for 0.092-
stars pc and
. If instead of providing all of the halo DM one
merely requires that clusters provide at least 40% of the DM, the halo
fraction indicated by the MACHO experiment for a local halo
normalisation (Alcock et al. 1997), then one
requires pc and
for 0.2- stars, or
pc and for hydrogen-burning limit stars.
These inequalities should be compared to the requirement noted above
that clusters must represent local density The limit should also be compared to the dynamical constraints to be discussed in Sect. 3.4. It is important to note that these limits are fairly insensitive to the actual number of VLM stars observed in the HST fields, provided the observed number is relatively small (e.g. ). This is because if even just a portion of one cluster enters one of the fields of view one would generally expect HST to detect hundreds or even thousands of stars in that field due to the high cluster surface density. For the same reason these limits are also relatively insensitive to the surface-density profile of the cluster, at least within the range of reasonable surface-density profiles (i.e. similar to those inferred from the surface-brightness profiles of visible clusters). ## 3.3. The question of resolvabilityThe previous subsection dealt with limits imposed from the absence
of clusters within the HST fields. However, compact clusters can
escape detection even if they do appear in the fields, provided their
surface densities are sufficiently high as not to allow them to be
resolved, or provided the total surface density through all clusters
along the line of sight is sufficiently large. The rigorous
point-source selection criteria of BFGK means that any unresolved
portions of clusters are discarded, in which case only some fraction
is detectable. Thus the The effect on resolvability due to several clusters aligned along the line of sight can be estimated from the ratio of the average cluster surface density to that between and : Assuming the distribution of clusters on the sky is Poissonian, their sky-covering factor is . Therefore the surface density measured along a line of sight through a cluster will be enhanced over that expected for the single cluster alone by a factor which is on average The median line-of-sight surface density between and for the 20 HST fields is for 0.092- stars and for 0.2- stars, corresponding to values for of 1.45 and 2.34, respectively, assuming - clusters with a 10-pc radius and . The resolvable fraction of an individual cluster is sensitive to
its assumed surface-density profile, as well as its mass and radius
and the distance at which it is expected to be observed. I assume here
that the cluster surface density follows the surface-brightness
profile of many observed globular clusters, which are well described
by the King (1962 ) surface-brightness law. Thus the surface number
density as a function of cluster-centric
radius where denotes the central surface density
and is the projected cluster core radius.
Since the integrated mass of the cluster must be where . For the purpose of simplification I adopt for all clusters, a value which is typical for visible clusters, though in reality there is a large dispersion about this value. Clusters appear resolved in the HST fields provided is less than where arcsec for the HST. If
then one expects that at least some of the
cluster is unresolvable. From Eq. 16 to 18, the mass fraction of
a cluster at distance if or unity otherwise. Note the factor from Eq. 15, which enters in the upper integral limit. This is because, as argued above, one expects the observed line-of-sight density to be , rather than just . Thus, for HST field where is given by Eq. 1. Hence, the
Fig. 2 shows the upper limits on the effective halo fraction
, assuming . Unlike the
plot in Fig. 1, the surface does not strictly represent a
95% CL constraint, since for that one would need to evaluate the
joint 95% CL variation in the cluster number
and in the resolvable fraction
. Such an evaluation would require detailed
Monte-Carlo simulations of the cluster distribution for each parameter
set , and for each field
Fig. 2 shows that the problem of cluster resolvability only becomes important for high-mass, compact clusters, but that it can have a very strong effect on detection in this regime. For the highest mass and most compact clusters considered the resolvable fraction rapidly approaches zero, so that no point-source observation of a VLM star is expected, even if clusters appear within the field of view. The density enhancement due to the superposition of clusters is found to have little effect on , since only becomes significantly larger than unity for clusters with large radius (in which case in Eq. 15), and these clusters anyway have such low surface densities that the enhancement has little consequence. As for Fig. 1, one can crudely parameterise the portion of the plot which permits to be unity by using Eq. 12 with pc and for 0.2- stars, or pc and for 0.092- stars. If one instead simply demands consistency with MACHO () then the allowed range is slightly larger with pc and , or pc and for 0.2- or 0.092- stars, respectively. ## 3.4. Dynamical considerationsOne can provide strong constraints on the permissible mass and radius of clusters which contribute significantly to halo DM by considering their dynamical effects on the visible stellar population (e.g. Lacey & Ostriker 1985 ; Carr & Lacey 1987 ; Moore 1993 ; Moore & Silk 1995 ). Such considerations lead to the conclusion that viable cluster parameters are bounded above and below, forming an 'island' in parameter space (Kerins & Carr 1994 ; Moore & Silk 1995 ). This is important because it means that one cannot invoke arbitrarily massive and compact clusters in order to force agreement between various observations. Furthermore, in principle cluster parameters which are compatible with dynamical limits may be incompatible with source-count and MACHO limits, thereby ruling out the entire scenario. The issue of dynamical constraints is somewhat complicated by the
fact that the limits depend (sensitively in some instances) on
Galactic as well as cluster parameters. Whilst I do not explicitly
present the dependencies of the constraints on these other parameters
(these can be found in the references cited above), I assume the
following values for the relevant parameters: a typical Galactocentric
distance for clusters in the vicinity of the HST fields of 8 kpc
(in fact most of the fields are further from the Galactic centre,
where their dynamical effects would be weaker than assumed here); a
cluster halo fraction (required to provide
consistency with MACHO observations); an average halo mass density
, giving a cluster number density
; a halo core radius
kpc; a one-dimensional halo velocity dispersion of
156 km s An upper limit on the radius of clusters comes from the requirement that they do not disrupt one another due to collisions occurring within the lifetime of the Galaxy. This implies pc for the parameters adopted above. Clusters also need to avoid tidal disruption due to the differential force from the Galactic potential acting across the cluster diameter; a constraint which requires pc. Since at any time there is always a finite fraction of cluster members whose velocities exceed the cluster escape velocity, clusters slowly evaporate over time. That they do not evaporate by the present day requires . The final constraint comes from considering the effect of close or direct collisions between dark clusters and observed diffuse globular clusters (Carr 1978 ; Moore 1993 ; Moore & Silk 1995 ). Moore (1993 ) showed that the requirement that observed diffuse clusters show no sign of disruption today leads to the limit for dark clusters with small radius, though this value is derived from a rather unusually diffuse cluster that may have atypical properties which result in too strong a constraint. A more robust limit from other less diffuse and more representative clusters demands for (B. Moore, private communication). For clusters with larger radius Moore & Silk (1995 ) derive a limit corresponding to pc. These limits, for an assumed star mass , are
projected onto the plane in Fig. 3 (bold
lines), together with the corresponding HST limits from Fig. 2,
which assume a clustering fraction . Regions of
parameter space for which HST counts permit
are projected onto the plane , corresponding to
the halo fraction preferred by MACHO microlensing observations towards
the LMC (Alcock et al. 1997). Thus the flat region of the
plane at small cluster radii and large masses
simultaneously satisfies MACHO observations and HST counts. It is
clear that this region and the region bounded by the dynamical limits
are almost mutually exclusive, with only a small area of the
plane satisfying all three requirements. This
is due to the strong limits inferred from the absence of dynamical
effects on presently-observed diffuse globular clusters, and from
evaporation considerations. The surviving parameter region is
characterised by a cluster mass and radius
pc and is relatively insensitive to the
VLM star mass within the range considered in this paper. Clearly, for
the cluster scenario to be viable
## 3.5. Microlensing signaturesThe effect of clustering halo objects can be to increase or decrease the observed microlensing rate, depending on whether or not the observer's line of sight intersects with more or less than the expected average number of clusters. One can therefore derive firm upper and lower limits on the halo lens fraction by assuming it to be clustered. Maoz (1994 ) has shown that there are two key features that should
help to discriminate between microlensing events in the clustered and
unclustered cases. The first is that the microlensing optical depth
through a cluster is , where
is the Einstein radius and Whether clustering could significantly affect the statistics for
the halo DM fraction depends upon how many
clusters one expects to see within the solid angle of observation. The
MACHO LMC search covers a solid angle of 11 deg for a halo with a density profile given by Eq. 1. Thus, for clusters which are simultaneously compatible with dynamical limits and HST source counts (implying from Fig. 3) one expects to be within the MACHO field of view if and . If the clusters all have the same mass and comprise objects with
the same MF, then the observed rate will depend only on
. Assuming the cluster distribution on the sky
obeys Poisson statistics, the probability of there being The MACHO data yields a likely halo fraction if the lenses are assumed to be unclustered. The inferred value for would be the same for the cluster scenario provided , otherwise one should take . The 1- variation on is less than 3% for , so the inferred halo fractions for the clustered and unclustered regimes are virtually identical in this case. Only for do Poisson fluctuations in the number of clusters become important, though in this case the microlensing signal is anyway dominated by the smoothly distributed VLM stars. The conclusion therefore is that for cluster mass scales compatible with dynamical limits their microlensing signatures are indistinguishable from the unclustered case. It is interesting to consider what the inferred halo fraction from microlensing, together with the HST source-count limits, imply for the likely value for the clustering efficiency . The halo considered in this paper has a microlensing optical depth towards the LMC, assuming it is completely comprised of lenses. The 95% CL lower limit on the measured optical depth is (Alcock et al. 1997), compared to the optical depth contribution expected from all non-halo components of . Subtracting the contribution from these components gives a lower limit on the halo fraction of . From Sect. 2 the maximum contribution from unclustered zero-metallicity VLM stars to the halo is 1.4% at the 95% CL. Therefore, consistency between MACHO and HST observations requires that . That is, one requires a present-day clustering efficiency of 92% or better. This is certainly a very strong demand for the cluster scenario to meet. If one adopts the central value for the measured optical depth () then the required clustering efficiency must be at least 97%. © European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |