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Astron. Astrophys. 328, 5-11 (1997)

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4. Constraints on cluster membership

Fig. 1 assumes that all stars reside in clusters at the present day, an unrealistic assumption since one expects some fraction of the clusters to have evaporated away over time. As in Paper I one can place limits on the fraction of stars [FORMULA] which must remain in clusters by using the strong limits [FORMULA] on the unclustered scenario (listed in Table 2). Assuming the lower limit on the cluster halo fraction to be given by the lower limit inferred by MACHO, [FORMULA], the present-day halo fraction in stars which have evaporated away from clusters is [FORMULA]. Since HST observations demand [FORMULA] one has

[EQUATION]

The resulting values for [FORMULA] for 0.2- [FORMULA] and 0.092- [FORMULA] stars are given in Table 3.

From Table 3 it is clear that all models require a very high fraction of all stars to reside in clusters at present. Even for hydrogen-burning limit stars the required clustering fraction must be at least 95% at present. Capriotti & Hawley (1996) have undertaken a detailed analysis of cluster mass loss within an isothermal halo potential for a range of cluster masses, density profiles and Galactocentric distances. Their analysis takes account of evaporation, disruption and tidal processes. They find that clusters with masses between [FORMULA] generally survive largely intact to the present day but that less massive clusters survive only if they have high central density concentrations, nearly circular orbits and reside at large distances from the Galactic centre. At the Solar position Capriotti & Hawley find that clusters with a half-mass to tidal radii ratio of 0.3 (comparable to the value for the clusters analysed here and in Paper I) survive more than 95% intact only if they have masses exceeding [FORMULA].

However, there are a number of reasons why these limits may be stronger than applicable to the low-mass star cluster scenario. Firstly, Capriotti & Hawley assume that the clusters comprise [FORMULA] stars (i.e. between 4 and 9 times more massive than the stars considered in the present study). The evaporation timescale scales approximately as [FORMULA] for a fixed cluster mass, so for clusters comprising lower mass stars the evaporation timescale is correspondingly longer. Secondly, the local halo density assumed by Capriotti & Hawley of [FORMULA]  pc-3 at a Galactocentric distance of 8.5 kpc is on the higher end of the values for the halo models analysed in this paper, and is considerably larger than the allowed MACHO lower limit, [FORMULA], on the local density in lenses (by a factor of between 5 and 9. Hence disruption due to close encounters with other clusters is substantially less in the halo models investigated here than for the model analysed by Capriotti & Hawley.

Lastly, a study by Oh & Lin (1992) has shown that the cluster escape rates may be substantially smaller than commonly assumed due to angular momentum transfer arising from the action of the Galactic tidal torque on cluster stars with highly eccentric orbits (which in the absence of the torque would constitute the bulk of the escapees). The rates calculated by Oh & Lin for isotropic cluster models are broadly consistent with the values used by Capriotti & Hawley (1996) and other authors, but for the case of anisotropic stellar orbits the escape rates can be 1-2 orders of magnitude smaller, again implying correspondingly longer evaporation timescales. It therefore appears that, under certain conditions, one may be able to reconcile the high cluster fraction requirements derived in the present study with the findings of cluster dynamical studies, at least for clusters comprising stars close to the hydrogen-burning limit.

In any case, the validity of the figures in Table 3 depend upon just how smoothly distributed are the stars which have evaporated from clusters. If they still have not completely homogenised today, instead maintaining a somewhat lumpy distribution (reflecting their cluster origin), then the limits on [FORMULA] are too strong.

For example, a cluster with a mass [FORMULA] and radius 3 pc represents an over-density of about [FORMULA] over the background average at the Solar neighbourhood [i.e [FORMULA] ]. However, an under-density in the unclustered (or more precisely `post-clustered') stellar population of just a factor 10 ([FORMULA]) over volumes larger than [FORMULA]  pc3, which is roughly the volume probed by 50 HST fields for hydrogen-burning limit stars (and is of order 10 times smaller than the halo volume per cluster), is all that is required to weaken the constraints on [FORMULA] by a factor 10. This would result in a much more comfortable lower limit on [FORMULA] of just 0.5 for 0.092- [FORMULA] stars, rather than 0.95. If the under-density is a factor 5 lower than the background ([FORMULA]) one requires [FORMULA] for the lowest mass stars and for an under-density factor of 2 ([FORMULA]) [FORMULA] must exceed 0.9.

In order to rule out the cluster scenario definitively (say with 95% confidence) one needs a survey that is both sufficiently wide and deep that it might be expected to contain at least 3 clusters on average, regardless of their mass and radius (though their mass and radius must be dynamically permitted). From Fig. 1 it appears that the most difficult dynamically-allowed clusters for HST to exclude are those with a mass of around [FORMULA]. If the halo fraction in low-mass stars is around 40%, typical of the preferred value for the MACHO results, then the local number density of such clusters is around 130 kpc-3 (adopting a local halo density of [FORMULA]  pc-3; in reality of course the average density within the fields is dependent upon the halo model and the field locations). If the clusters comprise hydrogen-burning limit zero-metallicity stars ([FORMULA]) then a HST-type survey will be sensitive to them out to about 3.6 kpc in the I band [using the colour-magnitude relation of Eq. (3), and assuming a limiting I -band sensitivity of 24 mag], so in order to expect to detect at least 3 such clusters, the survey must cover a solid angle of at least 4.5 deg2, or the equivalent of 3 700 HST fields! Therefore, only if HST fails to detect any clusters from 3 700 fields would all dynamically-allowed clusters be ruled out with 95% confidence from explaining the observed microlensing events. For comparison, an all-sky K -band survey over Galactic latitudes [FORMULA] requires a limiting magnitude of about 17.5 in order to produce similar constraints. This should be compared to the expected K -band limit of about 14 for the ground-based DENIS and 2MASS surveys.

An easier alternative is to instead obtain several fields as close to the Galactic centre as is feasible, where the dynamical constraints are much stronger than for the Solar neighbourhood position. For low-mass stars, this necessitates a telescope such as HST with the capability for obtaining very deep fields, since shallow surveys with wide angular coverage essentially only probe the local Solar neighbourhood.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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