Guido De Marchi and Francesco Paresce
European Southern Observatory, Karl-Schwarzschild Strasse 2,
D-85748 Garching, Germany
, we derive deep main sequence
luminosity functions that show a power-law increase with decreasing
luminosity up to 
. Beyond this point they all drop
sharply down to the measurement limit even after photometric
incompleteness is accounted for and in spite of their, presumably, very
different dynamical histories. We conclude that, in these regions of
the clusters, the local stellar population has not been modified by
dynamical evolution and still reflects the intrinsic properties of the
IMF, revealing a deficiency of faint objects. Using the only available
empirical mass-luminosity relation for the clusters' metallicity, we
obtain mass functions that increase with decreasing mass but then
flatten out below 
M
. Using theoretical mass-luminosity
relations would give conflicting results that reveal the still
approximate understanding of the physics of very low mass stars. Until
a reliable empirical mass-luminosity relation is available, the
exploration of the lower end of the IMF as well as of the brown dwarf
region will be subject to considerable uncertainties.
Keywords: low mass stars, luminosity functions, globular clusters
The luminosity function (LF) of stars at the faint end of the H burning main sequence carries fundamental information concerning stellar structure, galactic formation and dynamics. Specifically, its transformation into a mass function (MF) via a theoretical mass-luminosity relation and bolometric correction allows interesting comparisons with physical models of very low mass (VLM) stars and brown dwarfs (BDs; D'Antona 1987, Burrows & Liebert 1993, Burrows et al. 1993, Saumon et al. 1994), with models of star formation (Silk 1989, Larson 1992) and, because VLM stars seem to be the most numerous objects in most stellar populations, with models of the dark matter in galaxies (Zinnecker 1987, Silk 1992). This last issue has come to the fore recently thanks to the results on microlensing of LMC stars reported by Alcock et al. (1993) and Aubourg et al. (1993), which could be explained by the existence in the galactic halo of a dynamically significant population of very low luminosity degenerate dwarfs. For such objects to provide all the dark matter content of the halo, the stellar mass function must rise steeply all the way through the VLM and most of the BD regions (Richstone et al. 1992, Richer & Fahlman 1992).
Recent determinations of the VLM LF in various environments, however, have been plagued by a number of uncertainties especially at the faintest and most important end (Jarrett, Dickman & Herbst 1994, Kroupa, Tout & Gilmore 1990, Hesser 1993, Richer & Fahlman 1992). The same microlensing experiments mentioned above allow for a different explanation (see Sahu 1994), and other studies of the low-mass stellar content of the Galactic halo suggest that low mass and VLM stars do not contribute significantly to the mass budget of the Galaxy (Hu et al. 1994, Bahcall et al. 1994). The main observational problem here is that of constructing a physically coherent and statistically complete sample of these very faint stars. The ambiguity of the ensuing results has been distressing with conflicting claims of flat, rising, and decreasing LFs close to the H burning mass limit. Although the VLM LF should be easier to obtain for stars in the solar neighborhood, it has the disadvantage of representing an average over a complex sample of widely different objects. This may explain a considerable part of, if not all, the observational uncertainties.
A simpler and, perhaps, more reliable approach is that of using the VLM stars in globular clusters (GCs), since they are a much more homogeneous sample. The problem here is that their faintest stars, as a group, are several magnitudes fainter than those in the solar neighborhood, thus requiring deep, accurate observations. At present, only the refurbished Space Telescope has both the spatial resolution and the sensitivity needed to reach the faint end of the main sequence (MS) in nearby GCs without the crowding problems, and the ensuing incompleteness, that are unavoidable in ground-based observations.
Figure: Luminosity functions of four of the clusters observed with the
WFPC2. Solid line: NGC6397; dashed line: M15; dot-dashed line:
M10; dotted line: M55. All LFs have already been corrected for
photometric incompleteness.
We have investigated the LF of VLM stars in six GCs (NGC 6397,
M15, 47Tucanae, M10, M55, and NGC6656) at a distance of
approximately one half-mass radius from their centers with very deep
and photometrically accurate WFPC2 observations in the F606W and F814W
bands (Paresce, De Marchi, & Romaniello 1995, De Marchi & Paresce
1995a,1995b, De Marchi et al. 1996). In all cases, the LFs that we
derive (see Figure1) increase slowly following a power-law up to 
, in accordance with ground-based observations, but then
drop sharply from there down to the measurement limit. Photometric
completeness is not the cause of the observed trend as even in the last
bin it is always 
, and because the limited amount of crowding
in our WFPC2 images allows us to assign strong reliability to the
completeness correction. Independent confirmation of the drop of
GC LFs at the faint MS end is given by Cool, King & Piotto (1996) for
NGC6397 and NGC7099, and by H. Richer (priv. comm.) for M4.
The consistency of all these results suggests that low-luminosity stars (red dwarfs) are not abundant in GCs, in very good agreement with the behavior found for halo red dwarfs by Bahcall et al. (1994). Yet, does this imply that low-mass stars are not produced in any significant amount in GCs? Answering this question, however, requires addressing two more issues, namely the shape of the mass-luminosity (M-L) relation at low masses, and the effects of cluster dynamics on the local stellar population. We will first concentrate on the latter.
According to Richer et al. (1991), near the half-mass radius
(where our observations were taken) the present-day mass function
(PDMF) of a GC should be as close as possible to the initial MF (IMF):
here both internal dynamical mechanisms, such as mass segregation
(Pryor, Smith, & McClure 1986) and evaporation (Spitzer 1987), as well
as tidal interactions with the Galaxy (Stiavelli et al. 1992) should
have negligible effects on the local stellar population. The objects
used in our investigation (with the exception of 47Tuc) had been
selected to have very similar metallicity (on average [Fe/H]
) in order to allow their LFs to be directly compared, yet with
completely different dynamical histories (both internal and due to the
interaction with the Galaxy) to investigate the effects of dynamics on
the PDMF at the VLM end. Then, the marked similarity of the LFs shown
in Figure1 immediately implies that all the clusters in our sample
(excluding 47Tuc for the moment) have the same PDMF, as the MFs of
clusters with the same metal content must be similar in order for their
LFs to agree. But because these objects evolved under possibly widely
different dynamical conditions (see Aguilar, Hut & Ostryker 1986), in
order for them to all have the same PDMF at the half-mass radius, they
must have been born with the same IMF. Any alternative scenario in
which GCs that were formed with different IMFs now have the same PDMF
at the half-mass radius would look highly contrived.
This conclusion also strengthens Richer et al.'s (1991) proposal that the stellar population near the half-mass radius is relatively insensitive to either internal or external dynamical modifications. Of course, this does not mean that dynamics cannot modify stellar populations in GCs. In fact, this has been clearly demonstrated by the existence of objects in cluster cores whose origin must be dynamical (De Marchi & Paresce 1994a,1994b), as well as by the first direct confirmation that mass segregation is at work in GCs through two body relaxation and energy equipartition (Paresce, De Marchi & Jedrzejewski 1995, King, Sosin & Cool 1995, De Marchi & Paresce 1996).
The discussion presented above implies that, once converted into a MF, the LFs of the clusters we studied will provide us directly with the IMF. Although this is an extremely powerful way to investigate the star formation process in the early phases of our Galaxy, the attempt to secure the shape of the IMF at the VLM end is somewhat hampered by our rather approximate knowledge of the M-L relation, whose slope at the faint end is unfortunately still uncertain at present. The only available semi-empirical determination of this relation for a GC is that obtained by Fahlman et al. (1989) for NGC6397, which has the advantage of being applicable to all the objects in our sample (except for 47Tuc), as they all share the same low metallicity.
Figure: Mass functions derived from the LFs of Figure1 by using the
M-L relation of Fahlman et al. (1989). Solid line: NGC6397; crosses
and dashed line: M15; triangles: M10; squares: M55.
When we apply Fahlman et al.'s M-L relation to the LFs in
Figure1, the corresponding MFs (Figure2) all show a power-law
increase in number with decreasing mass down to 
M
,
followed by a flattening below this limit and, at least certainly for
NGC6397, by a drop towards the expected mass limit of the normal
hydrogen burning sequence at 
M
. Since we expect
these MFs to correspond to the IMF, as explained above, we conclude
that the observed MF flattening below 
M
has to be an
intrinsic feature of the IMF itself and could be the direct consequence
of some limitations in the star formation process (Larson 1992), with
strong implications for the mass budget of the Galaxy and the nature of
dark matter.
This conclusion relies on the validity of Fahlman et al.'s
(1989) M-L relation. Theoretical M-L relations appropriate for the
metallicity of our clusters have been recently computed by D'Antona &
Mazzitelli (1996) and by Alexander et al. (1996). Adopting the former,
our LFs translate into MFs that keep increasing, although mildly, to
the detection limit, while, with Alexander et al.'s relation, a
flattening results below 
M
. This discrepancy reveals
the uncertainty still accompanying our understanding of the physics of
VLM stars and justifies our decision of using the semiempirical
relation of Fahlman et al., which we have no reason at present to
believe to be in any significant error.
An intriguing possibility is that the shape of the IMF may be
sensitive to metallicity. In Figure3 we compare the LFs of NGC6397
and 47Tuc, as well as their MFs. For 47Tuc we have used the M-L
relation of Bergbusch & VandenBerg (1992) for [Fe/H]
Z, which
is consistent with that of D'Antona & Mazzitelli (1996). (At such a
relatively high metallicity the M-L relation is better defined than at
lower Z.) The two LFs are different, in that 47Tuc's reaches a
maximum located 
mag fainter, and after that it drops more
slowly. As it is quite unlikely that this difference is due to dynamics
(see above), a viable explanation that remains is metallicity. This may
be understood, at least intuitively, by noticing that since 47Tuc is
times more metal rich than NGC6397 ([Fe/H]
vs.
[Fe/H]
; Djorgovski 1993), one would expect a star of a given
mass to appear fainter in 47Tuc than in NGC6397. Then, ideally, if
the mass distribution were similar, the LF peak should move to fainter
magnitudes, as observed.
With the adopted M-L relations, the MFs that we obtain are
in fact similar, in that they both increase with decreasing mass down
to 
M
and then flatten out. The plateau level that they
reach, however, is not the same, and 47Tuc's MF slope is shallower.
McClure et al. (1986), comparing deep LFs of 7 clusters, came to the
conclusion that the slope of the MF (in the 
--
M
range) is correlated with the metal content, with steeper slopes at
lower metallicities. On the face of it, our result would then confirm
this trend, subject, however, to the uncertainty in the theoretical M-L
relation which is, unfortunately, still poorly defined at present in
the low-mass range.
With the HST and the new generation 10-m class telescopes,
obtaining a deep and accurate census of very faint stars in clusters is
becoming more and more possible today. The biggest problem at present,
however, is the lack of a reliable empirical M-L relation at very-low
masses, which is mandatory to secure the shape of the IMF at the lowest
and most interesting end. A large interferometric program aimed at
measuring masses and luminosities of as many astrometric binary and
multiple systems as possible out to 
pc is being defined at
ESO as one of the main goals of the VLT Interferometer. The knowledge of
the M-L relation that this project could deliver would prove
fundamental and would permit, for the first time, to extend the
investigation of the IMF into the BD regime.
Figure: Left: because of its lower metallicity, the LF of NGC6397
(dashed line) reaches a maximum at a brighter magnitude than that of
47Tuc (solid line), and has a sharper drop. Right: MFs of NGC6397
(dashed line) and 47Tuc (solid line). The two clusters show the same
shape (increase followed by a flattening below 
M
), but
the plateau level is different, suggesting that the IMF could be
sensitive to metallicity.
This paper is 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.
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