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

3. HST observations and halo fraction constraints

Gould 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 I -band magnitudes for the fields are listed in Table 1 of Gould et al. (1997). The Groth Strip is treated as a single field with an angular coverage of 25.98 WFC2 fields (this accounts for overlaps) and magnitude limits corresponding to the modal values listed in Gould et al. (1997). As in Paper I, these limits are translated into star-mass dependent limiting distances by converting the line-of-sight extinction values listed in Burstein & Heiles (1984) to I -band reddenings and using the photometric predictions of Saumon et al. (1994) for zero-metallicity low-mass stars. The predictions for the V and I bands are well fit by the colour-magnitude relation

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.

Table 2. Constraints on unclustered zero-metallicity low-mass stars in the Galactic halo arising from the detection of 145 candidate stars within 51 HST WFC2 fields. The second and third columns give the expected number of detectable stars for a full halo () for stars with masses of and (the hydrogen-burning limit mass), respectively. The last two columns give the 95% confidence upper limit on the maximum halo fraction .

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.

Fig. 1a-e. Comparison of constraints on the halo fraction from HST limits, MACHO observations and dynamical constraints for the 5 halo reference models (A-D, S), assuming halo stars have a mass of and all reside in clusters with mass M and radius R. The lower plateau to the left of each plot corresponds to the 95% CL upper limit for the unclustered scenario inferred from HST counts (see Table 2). The upper plateau on the right corresponds to the 95% CL lower limit halo fraction inferred by MACHO 1st- and 2nd-year observations (Alcock et al. 1997), with the central value for the MACHO halo fraction indicated by the skirting surrounding the plots (see also Table 3). The curved surface joining the lower and upper flat regions corresponds to the 95% CL upper limit on the halo fraction in clusters from HST counts. Also projected onto the plane are the cluster dynamical constraints (dashed lines) for the local Solar neighbourhood. The intersection between these constraints and the MACHO lower-limit plateau indicates cluster parameters compatible with HST, MACHO and dynamical constraints.

The constraints on the halo fraction for the clustered scenario as a function of cluster mass M and radius R are shown in Fig. 1 for the 5 models (A-D, S) assuming all stars reside in clusters and have the hydrogen-burning limit mass of . Each plot is characterised by a lower plateau to the left, an upper plateau to the right and a curved rising surface joining the two. This curved surface between the two flat regions represents the 95% CL upper limit halo fraction in clusters inferred from the presence of only 145 candidate stars within the 51 HST WFC2 fields. The constraints are actually calculated on the basis of no stars being present within these fields, since for clusters there is little difference in the constraints assuming no stars are found or assuming a few hundred stars are found. The reason for this, as discussed in Paper I, is that the clusters considered here contain between 1000 and members each, so the presence of just one cluster within any of the HST fields would typically result in thousands if not millions of candidates being detected.

The lower plateau shows the 95% CL upper limit halo fraction for the unclustered scenario (corresponding to the values listed in Table 2). Clusters with masses and radii within this region have internal densities which are lower than that of the halo background average and are thus unphysical, since they represent local under-densities rather than over-densities. Clearly constraints on clusters cannot be stronger than constraints on a smooth stellar distribution. The intersection of the lower plateau with the curved rising surface therefore denotes the boundary between unphysical and physical cluster parameters.

The upper plateau to the right represents the 95% CL lower limit on the halo fraction inferred from MACHO 1st- and 2nd-year microlensing results (Alcock et al. 1997). It is calculated by taking the 95% CL lower limit on the measured microlensing optical depth for all 8 MACHO events (), subtracting the optical depth contribution expected from non-halo components [corresponding to (Alcock et al. 1996)], and normalising to the optical depth prediction for a full halo () for each model. The top of the skirting surrounding each plot is normalised to the central MACHO value for the halo fraction for comparison, and is calculated in a similar manner to the lower limit ( and , together with , are tabulated in Table 3 for each model). Since this plateau lies below the extrapolation of the HST cluster-fraction constraint [which rises asymptotically over this region - see Fig. 2 of Paper I], it is consistent with both MACHO and HST limits.

Table 3. Microlensing halo fractions and minimum clustering fractions for the reference halo models. Column 2 gives the expected optical depth for a full halo as calculated by Alcock et al. (1996). Column 3 gives the central value for the halo fraction using the 1st+2nd year optical depth estimate of measured by Alcock et al. (1997), and subtracting from it an optical depth of expected from non-halo populations. The 4th column gives the 95% CL lower limit on the halo fraction using the lower limit for the measured optical depth of . The last two columns give the lower limit on the present-day clustering fraction using column 4, Eq. 4 and Table 2.

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 lower limit MACHO halo fraction , rather than with the central value as in Paper I. This latter difference is particularly important because it enlarges both the sizes of the dynamically-permitted region and the MACHO plateau, and hence enlarges their intersection. Since these differences serve to maximise the size of the surviving region, the constraints shown in this paper should be taken as firm limits on allowed cluster parameters.

© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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