Ivan R. King
Astronomy Department, University of California,
Berkeley, CA 94720-3411
Giampaolo Piotto
Dipartimento di Astronomia, Università di Padova, Vicolo
dell'Osservatorio 5, I-35122 Padova, Italy
Adrienne M. Cool, Jay Anderson, Craig Sosin
Astronomy Department, University of California,
Berkeley, CA 94720-3411
Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS5-26555.
Keywords: globular clusters,mass functions,luminosity functions,photometry,Hubble Space Telescope
The repair of HST has benefited the study of globular clusters to an unusual extent. For the first time, faint stars are being resolved in the cluster centers, while away from the centers the new HST observations are reaching unprecedentedly faint magnitudes.
Getting accurate magnitudes from HST WFPC2 images is far from trivial. Our first attempts, using DAOPHOT in the conventional way, gave a disappointing accuracy; we decided to devote a long experimental period to developing what we call our method of weighted, neighbor-subtracted aperture photometry. (We have since found, however, that there exists an unconventional choice of DAOPHOT parameters that will produce results that are at least as good. For details of this, and of our method, see Cool & King 1995.)
In addition, our group is developing a quite different approach to star-finding and photometry in globular clusters that we believe shows excellent promise, especially in the crowded fields that are often encountered. These methods will be described in forthcoming papers.
The CMD of NGC 6397 in Figure 1 is an example of what good photometric
methods can achieve (Cool, Piotto, & King 1996). Down to V=24, the
full width of the main sequence is 0. This CMD
comes from WFPC2
images taken during 5 orbits in V and 5 in I. A vital step in the
planning of these observations was to include short exposures, in order
to get a large dynamic range in the PSFs with which the photometry was
carried out. In other clusters, where we had to use material that
lacked such exposures, our photometric accuracy suffered.
Figure: Our color--magnitude diagrams of 3 globular clusters. That of
NGC 6397 illustrates in particular the
precision and faintness that can be achieved with WFPC2 images.
The bends in the main sequence of NGC 6397 are striking, and they offer an
excellent test to theories of the structure of stars of low metallicity.
At least one theoretical group has demonstrated impressive agreement with our
data (Brocato et al. 1996,
see also the paper by Brocato et al., this volume, p. 
).
Below and to the left of the main sequence of NGC 6397 in Fig. 1 is a
group of stars that form a narrow sequence roughly parallel to the lower
main sequence, with magnitudes in the range 
22.6--25.2.
The narrowness of this sequence, and its close match in shape and
position to theoretical white-dwarf cooling tracks, leave little doubt
that these stars are cluster white dwarfs. The brightest 
15 WDs
form a well-defined sequence that is well matched in position and shape
to theoretical cooling sequences for WDs with masses of 0.55
. From
comparison of our observations with the theoretical tracks, we deduce
that the WD mass dispersion is certainly less than 0.05
, and is
probably much less than that value. This is the first detection of a WD
sequence in NGC 6397. (Paresce, De Marchi, & Romaniello [1995b],
with a lower photometric accuracy, were able to demonstrate the presence
of WDs but did not delineate a sequence.)
Figure: Our luminosity function of NGC 6397, compared with LFs by
Paresce et al. (1995b) with HST and by Richer et al. (1989) from the ground.
In determining the luminosity function of a globular cluster it is
important to work from a CMD, so as to distinguish cluster members from
field stars; and it is also important to make careful studies of the
completeness of the measurements, which can seriously affect the faint
end of the LF. We have taken care of both these problems. First,
field-star contamination has been limited by determining the LFs from
the stars within 
of the fiducial main sequences; then a
correction for this contamination was made on the basis of the CMD of
the stars outside these 
boundaries. The completeness
correction has been evaluated via the usual artificial-star
experiments. The final LFs, as corrected for field contamination and
completeness, have been limited to the magnitude range in which the
completeness was >50%. In Figure 2 we illustrate the kind of
problems which arise when these corrections are not properly taken into
account: we compare our LF of NGC 6397 with other LFs of the same
cluster. From our studies of field-star corrections, we believe that
the faintest point of Paresce et al. (1995b) is unreliable. (It is
equivalent to 
on our CMD in Fig. 1, a level at which the MS is
indistinguishable from the field.) The ground-based study of Fahlman
et al. (1989), which did not have a CMD available, has clearly gone
wrong at the faint end, presumably because of field-star problems and/or
overcorrection for incompleteness.
Figure: Luminosity functions in I (
) and V (
)
of M15, M30, and NGC 6397.
In Figure 3 we compare LFs of M15 (NGC 7078), M30 (NGC 7099), and NGC 6397 (Piotto, Cool, & King 1996). All three have similar metallicities, so that we can in effect compare their MFs without actually going through the M--L transformation. (In the V-band plot M15 is missing, because its V images were taken with a different filter.)
Before we can interpret these results, we must determine what corrections, if any, are required to convert these observed local LFs to global LFs. Our multi-mass model of NGC 6397 (cf. Section 8) shows that the LF in its envelope differs little from the global LF (as is clearly shown in Fig. 5); the mass-segregation effects, while strong, are largely confined to the small central regions. To first order, these results are applicable to M15 and M30 as well, since all three clusters have collapsed cores and similar surface brightness profiles, and since the fields analyzed here are out in the envelope in all three cases.
The LFs of M15 and M30 are strikingly similar, suggesting that they were formed with identical mass functions, which have changed similarly or not at all. (The alternative scenario, that they were born with different mass functions but have changed so as to become similar, seems excessively contrived.) Relative to the other two clusters, however, NGC 6397 is markedly deficient in faint stars. As indicated above, the deficiency is very unlikely to be due to differences between the local and global LFs; instead it might be due to the short relaxation time in NGC 6397, or, even more likely, to the strong tidal shocks that the cluster gets in its frequent passages through a dense part of the Galactic plane (Dauphole et al. 1996).
The difference between the NGC 6397 LF and the near-twin LFs of M30
and M15 is apparent only when they are compared over a large range of
magnitudes. De Marchi & Paresce (1996) have asserted---incorrectly,
as it turns out---that NGC 6397 and M15 have very similar MFs. The
problem is the shortness of the interval over which they compared
their LFs---only 
--10.1. Our HST LFs for NGC 6397
and M15 span a larger range of magnitudes, allowing a comparison from
--10.5. Our LF measurements are in reasonable
agreement with those of De Marchi & Paresce for M15, and with Paresce
et al. (1995b) for NGC 6397, for the ranges in which we overlap with
them. The larger magnitude range that we measure, however, shows that
the two LFs have different overall shapes; and while they match each
other at the bright end, there is an unmistakable deficiency of faint
stars in NGC 6397 relative to M15 (as well as M30).
Unlike the inhomogeneous population of the halo field, each globular cluster very probably represents a single pure act of star formation, and the stars that we find in the cluster today bear witness to that event in the distant past. One of the important lines of evidence is in the mass functions of the clusters, which HST can follow to significantly lower masses than ever before.
Figure: Mass functions of 5 globular clusters. Four of them are from
our own data, while the MF of 
Centauri is taken from Elson
et al. (1995).
The most uncertain step in deriving the mass function of a globular cluster is the transformation from luminosity function to mass function. This transformation is extremely sensitive to the mass--luminosity relation, whose slope enters the transformation directly. Unfortunately, M--L relations are poorly known for stars of low metal abundance, and are surely abundance-dependent besides.
We have converted our LFs into MFs, trying several M--L relations available in the literature (Alexander et al. 1996, Baraffe et al. 1995, Bergbusch & VandenBerg 1992, D'Antona & Mazzitelli 1995). Caution must still be exercised in interpreting the resulting MFs, however, given the underlying problem of the paucity of observational constraints on any of these relations. In Figure 4 we show the mass functions of five clusters, derived in this case from the mass--luminosity relations of D'Antona & Mazzitelli (1995). Of all the M--L relations available to us, this set leads to the steepest slope of the resulting mass functions. It is interesting to note that even with these M--L transformations the slopes of the MFs barely approach the critical slope of 2.0, beyond which the lowest-mass stars would dominate the total. With other M--L relations, therefore, the low-mass stars are not a dominant contributor---contrary to the assertions of Richer et al. (1991), whose steep MFs we do not confirm.
Figure: Mass functions in NGC 6397 at radii 7
and 4, as
observed, in stars per arcmin
. The numbers in the 7
field are
higher because of the higher density at the cluster center, but the mass
function is quite different. The solid lines are from a dynamical
model fitted to the cluster; the numbers have been fitted to the
observations at 4 but not at 7
. Also shown (dashed line) is
the global mass function of the model; the right-hand ordinate scale
applies to it.
Differences in the radial distributions of stars of different mass have been seen a number of times from the ground, significantly but weakly (Sandage 1954, Oort & van Herk 1959, Richer & Fahlman 1989, Drukier et al. 1993, and many other places). In HST images, however, we can see faint stars all the way in to the cluster center, and the effects are tremendous. They have already been seen and commented on (Shara et al. 1995, Paresce, De Marchi & Jedrzejewski 1995a). What we have done is to quantify the effect by fitting a cluster model (King, Sosin, & Cool 1995).
Our first impression of the images of the center of NGC 6397 was astonishment; the proportion of faint stars at the center was very small. But when we fitted a dynamical model to the cluster, it became clear that this was just what was to be expected. Figure 5 shows the mass functions observed at the center and in an outlying field, and the fit of our theoretical model to them.
The model is a multi-mass embodiment of the algorithm that has given rise to the name ``King models'' (King 1966). It gives a good fit to the radial density profile of the cluster and to the mass function that we have measured in it. It is noteworthy that we are able to use a King model to fit a cluster with a post-collapse core; the stabilization of the core by binaries is probably what makes this possible.
This work was supported by NASA Grant NAG5-1607.
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