2. Limits on zero-metallicity VLM stars
Saumon et al. (1994 ) have computed a grid of fully non-gray atmospheric models for zero-metallicity () VLM stars and brown dwarfs in the mass range . Their calculations take account of all the major sources of opacity for a pure H+He mixture including H2, H and H , as well as other sources tabulated by Lenzuni et al. (1991 ). The resulting spectral energy distributions are found to deviate significantly from blackbody below an effective temperature of 4 000 K. Saumon et al. also present predictions for the photometry of VLM stars () from M through to V bands which show that these stars would appear significantly bluer than stars with metallicity comparable to that measured for the Galactic spheroid () or disc () populations.
In particular, Saumon et al. find that the expected colour for zero-metallicity VLM stars ranges between 1.27 for 0.2- stars to 1.57 at the hydrogen-burning limit (). This is somewhat below the threshold values assumed by BFGK, GF and Flynn et al. (1996 ) in their analyses, whose inferences are based on the number counts of stars with . Whilst one can not say for sure whether any dark halo population has a metallicity substantially less than that of the spheroid, there is clearly a need to re-examine the HST data using the zero-metallicity predictions of Saumon et al. in order to close up this last remaining 'loophole'.
The positions of the 22 HST fields are tabulated in Gould et al. (1996 ), along with the corresponding minimum and maximum I -band magnitudes for each field. (All selections were performed in the I band.) Note that two locations (, and , ) each have 2 fields in very close proximity.
The limiting maximum I -band magnitude, which determines the maximum distance out to which a VLM star will be seen, ranges from 22.56 to 24.40 and is determined for each field according to the ability to discriminate clearly between stellar and extended sources. The minimum I -band magnitude ranges from 17.05 to 19.45 and sets the minimum distance at which the stars can be satisfactorily imaged. Objects with magnitudes below this limit produce saturated images. It should be noted that BFGK calculate the I -band magnitude limits from the HST band (F814W filter) assuming the stellar spectral energy distributions listed by Gunn & Stryker (1983 ). They therefore do not strictly apply to VLM stars, though any differences in calibration will be small and are therefore neglected here.
In computing the minimum and maximum observable distances for each VLM star mass, and for each field, I follow Gould et al. (1996 ) in converting the B -band extinction values determined by Burstein & Heiles (1982 ) to I -band reddenings. The extinction is assumed to be confined to a disc of thickness pc. Within this disc the extinction per unit distance is taken to be uniform.
The halo mass within the volume defined by the minimum and maximum distances , , and the solid angle per field sr, is calculated assuming a spherically-symmetric softened isothermal halo density distribution of the form
where x is the distance measured along the observer's line of sight, l and b are Galactic coordinates, kpc is the Sun's Galactocentric distance, is taken to be the local DM density normalisation and kpc is the assumed halo core radius. For the small solid angles considered here, this gives an integrated halo mass between and for field i of
The integral can be performed analytically though the resulting expression is long.
for n independent fields. The direction of the inequality reflects the fact that the dependency of on m is steeper than the first power of m. Fields 1 and 19, using the order in which they are listed in Tab. 1 of Gould et al. (1996 ), are discarded in this analysis because of their close proximity to fields 2 and 20, respectively. Whilst fields 1 and 2 do not actually overlap, field 1 is nonetheless excluded here to provide consistency with the cluster analysis of Sect. 3.2, where statistical independency of neighbouring fields is an important criteria. is therefore summed over rather than 22 fields.
The median value for is found to range from 330 pc for 0.092- stars up to 1 kpc for 0.2- stars. For the median values are 3.3 kpc and 10.1 kpc for 0.092- and 0.2- stars, respectively.
Applying Eq. 3 to VLM stars at the hydrogen-burning limit mass of , one finds an expectation number of stars in the HST fields of 6 310, and for 0.2- objects the expectation is nearly an order of magnitude larger at 60 100 stars. These numbers take account of the fact that data from one-third of field 4 (i.e. data from one of the three detector chips) had to be discarded by Gould et al. (1996 ) due to problems with receiving the data from HST, and that as much as 2% of each of the fields was discarded due to emission from background galaxies. The number of stars detected in the 20 fields with values in the range 1.2 to 1.7, spanning the range predicted by Saumon et al. (1994 ), is only 75. The 95% CL (Confidence level) upper limit on the average, for a realisation of 75 stars, is 91. Therefore, even if one assumes that all of the stars detected by HST are halo VLM stars right on the hydrogen-burning limit, their contribution to the halo DM can be no more than 1.4% at the 95% CL, and the limit is correspondingly stronger than this for more massive objects. In fact it is likely that a significant fraction of these objects may belong to the disc or spheroid.
The limit of 1.4% is stronger than that inferred by BFGK for solar-metallicity VLM stars from their analysis of one of the HST fields. Their results translate to a 95% CL upper limit of less than 4% for stars with , adopting . The 95% upper limit inferred by GF for low-metallicity stars with corresponds to a halo fraction of less than 0.9% for the same field. However, both of these studies exclude from their analyses stars with , so the limits presented here, which are derived from 20 fields, constitute a completely independent check on previous results. The conclusion is that, regardless of their metallicity, VLM stars do not contribute significantly to the halo DM, at least under the assumption that they are smoothly distributed in the halo.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998