## 3. HST observations and halo fraction constraintsGould et al. (1997) have calculated the disc luminosity function for M-dwarf stars using data from several HST WFC2 fields. These include 22 fields originally analysed by Gould et al. (1996), along with the Hubble Deep Field, 28 overlapping fields comprising the Groth Strip, and 2 other new fields: a total of 53 WFC2 fields. In Paper I, 20 of the original 22 fields are analysed, the other 2 fields being omitted due to statistical problems introduced by their close proximity to some of the other fields (namely that clusters appearing in these fields could also appear in the other fields and thus be double counted). In this study these 20 fields are combined with the new fields analysed by Gould et al. (1997), making the total number of fields 51. The nearest-neighbour separation between these fields is sufficiently large that double counting is not expected to be a problem for clusters of interest. (The overlapping Groth Strip fields are treated as a single large field for the purpose of this study.) The limiting and saturation for (corresponding to ). The analyses for the unclustered and clustered scenarios proceed as in Paper I, except that the models listed in Table 1 of this paper now replace the model used there. The calculations for the cluster scenario, which are described in detail in Paper I, assume that the surface-brightness profiles of the clusters follow the King (1962) surface-brightness law and take into account cluster resolvability, as well as line-of-sight overlap. Table 2 lists the results for the unclustered scenario. Within the 51 HST WFC2 fields analysed a total of 145 candidate stars with are found, implying a 95% confidence level (CL) upper limit on the average number of 166 stars. This colour range spans the colour predictions of Saumon et al. (1994) for stars with masses in the interval , where the lower value corresponds to the hydrogen-burning limit. Comparison with the expected number tabulated in Table 2 clearly shows that, for all models, even the lowest mass unclustered stars fall well short of providing the halo dark matter density inferred by MACHO. The upper limit on their fractional contribution is shown for 0.2- and 0.092- stars. For the lowest mass stars ranges from 0.5% for models B and D to 1.1% for the lighter halo model C.
One interesting feature of Table 2 is that for the flattened halo model D the expected number counts are enhanced for 0.092- stars relative to the predictions for the spherically symmetric models, producing the highest predicted number-count for these stars. This contrasts with the results for the brighter 0.2- stars, with the heavy halo model B producing the highest number-count prediction. The enhancement for 0.092- stars in model D arises because the flattening preferentially increases the stellar surface density near the Galactic plane, and this is reflected in the counts of 0.092- stars which can be at most only a few kpc from the plane if they are to be detected.
The constraints on the halo fraction for the
clustered scenario as a function of cluster mass The lower plateau shows the 95% CL upper limit halo fraction
for the The upper plateau to the right represents the 95% CL
The dashed lines in the plots of Fig. 1 represent the dynamical constraints derived for the local Solar neighbourhood. In fact some of the HST fields are somewhat closer in to the Galactic centre, where the dynamical constraints are stronger, but most are further away so the limits shown are stronger than applicable for most of the HST fields. The functional form for the constraints are detailed in Paper I and are dependent upon Galactic as well as cluster parameters [consult Lacey & Ostriker (1985); Carr & Lacey (1987); Moore (1993); Moore & Silk (1995); Carr & Sakellariadou (1997) for derivations, and see Carr (1994) for a detailed review of dynamical constraints]. Their variation from plot to plot is due to model variations in the local density and rotation speed (see Table 1). The dynamical constraints are projected onto the plane for direct comparison with the MACHO lower limits. The intersection of the MACHO lower-limit plateau with the dynamical limits therefore represents cluster parameters compatible with MACHO, dynamical limits, and the constraints from the 51 HST fields. For each model it is evident that the region compatible with all limits spans a significant range of masses and radii. For models C and D the maximum permitted cluster mass is around , whilst for models A, B and S one can have cluster masses in excess of . Interestingly, whilst in the unclustered scenario the heavy halo model B is the most strongly constrained in terms of allowed halo fraction , it nonetheless allows a relatively wide range of viable cluster masses in the clustered scenario. Conversely, the permitted cluster mass range for the light halo model C is more restricted. This apparent paradox is due to the fact that the HST, dynamical and microlensing observations limit the halo density normalisation at different positions in the halo, so their intersection is sensitive to the halo density profile. In particular, the HST and dynamical limits essentially apply to the local Solar neighbourhood position ( kpc) for clusters comprising relatively dim hydrogen-burning limit stars, where as the microlensing observations towards the LMC constrain the density of lenses at somewhat larger distances (primarily between 10 and 30 kpc from the Galactic centre, where the product of lens number density and lensing cross-section is largest). Hence, for a given microlensing constraint on the mass density of lenses at 10 to 30 kpc, the local dynamical and number-count constraints are weaker for haloes with rising rotation curves (such as model B) than for models with falling rotation curves (such as model C). The relatively large range in allowed cluster masses and radii for
model S is in apparent contrast to the results of Paper I,
in which the surviving parameter space is shown to be much smaller for
the very similar model adopted there. There are two reasons for this
apparent discrepancy: (1) in Fig. 1 of this paper it is assumed
that the clusters comprise hydrogen-burning limit stars, where as in
Fig. 3 of Paper I the constraints are shown for the brighter
0.2- stars; (2) in this study consistency is
being demanded only with the © European Southern Observatory (ESO) 1997 Online publication: March 24, 1998 |