Excellent reviews of the GC luminosity function (GCLF), including its use as a distance indicator, are given in Harris (2001) and Richtler (2003). The NED D database (2006 release) gives a comprehensive compilation of GCLF distances. Here we summarize some basic facts and discuss the evolution of the GCLF.
In many massive galaxies studied to date, the GCLF can be well-fit by a
Gaussian or t5 distribution; typical parameters for
the normal distribution in Es are peak MV ~ -7.4, with
~ 1.4. In spirals, the
peak is similar, but ~
1.2 may be a more accurate value for the dispersion. The peak luminosity
is a convolution of the peaks for the two GC subpopulations;
line-blanketing effects (especially in bluer bands) cause the metal-poor
GC peak to be slightly brighter than the metal-rich peak, so the exact
location of the peak depends on the color distribution of the GC system
(Ashman, Conti, & Zepf
1995).
The standard peak or turnover of this
distribution corresponds to a mass of ~ 2 × 105
M
. In
absolute luminosity or mass space, this distribution is a broken power
law with a bright-end slope of ~ -1.8 and a relatively flat faint-end
slope of ~ -0.2
(McLaughlin 1994).
In some massive galaxies there appears to be a departure below a power
law for the most massive GCs (e.g.,
Burkert & Smith 2000).
The expectation of significant
dynamical evolution at the faint end of the GCLF has led to suggestions
that only the bright half of the GC mass function (GCMF) represents the
initial mass spectrum of the GC system. The faint half would then
represent the end product of a Hubble time's worth of dynamical
destruction. Recent observational and theoretical work on this problem
offers some supporting evidence, but also raises some interesting
questions.
In the Galaxy, the total sample of GCs is relatively small, especially when divided into subpopulations. This limits our ability to draw general conclusions from this best-studied galaxy. Harris (2001) provided a good overview of the current situation. Within the 8 kpc radius that contains both metal-poor and metal-rich GCs, the subpopulations have quite similar LFs. Considering only the metal-poor GCs, the turnover of the GCLF becomes brighter by nearly ~ 0.5 mag out to 8-9 kpc, then decreases to its initial value. The inner radial trend is the opposite of naive expectations for dynamical evolution, which should preferentially disrupt low-mass GCs as shocks accelerate mass loss through two-body relaxation. There are too few metal-rich GCs to study the radial behavior of the GCLF in detail. The GCs in the far outer halo (beyond ~ 50 kpc or so) are all quite faint (MV < -6), except for the anomalous GC NGC 2419. This cluster has MV = -9.6 and may be the stripped nucleus of a dwarf galaxy (van den Bergh & Mackey 2004). Of course, this level of detail is not expected to be visible in external galaxies, whose GC systems are generally seen in projection about an unknown axis. This is an important limitation in interpreting the results of the theoretical simulations discussed below.
Fall & Zhang (2001) presented a semi-analytic study of GC system evolution in a Milky Way-like galaxy. The mass-loss rate by two-body relation is taken to depend only on the details of the cluster's orbit, and not on the mass or concentration of the GC. This appears to be consistent with results from more detailed N-body simulations (e.g., Baumgardt & Makino 2003). Fall & Zhang found that a wide range of initial GCMFs, including single and broken power laws, eventually evolved to turnover masses similar to those observed. They found that the evolution of the turnover mass was rapid in the first Gyr or two, but that subsequent evolution was slow. The turnover is expected to change substantially with galactocentric radius, due to the preferential destruction of low-mass GCs towards the galaxy center. This can only be avoided if the outer GCs display strong radial anisotropy. However, the assumptions made by Fall & Zhang need to be considered when applying their results to real data. For instance, they used a static spherical potential, while a live and/or nonradial potential could increase phase mixing and erase some radial signatures of evolution.
Vesperini (2000;
2001)
modeled the evolution of GCLFs of two initial
forms, log-normal and power-law functions. The evolution of individual
GCs was determined using analytic formulae derived from the simulations
of
Vesperini & Heggie
(1997).
To summarize their results: GCLFs that
are initially log-normal provide a much better fit to the observed
data. Dynamical evolution can indeed carve away the low-mass end of a
power-law GCLF to produce a log-normal function, but the resulting
turnover masses are generally small
(
105
M), and
vary significantly with galactocentric radius and from galaxy to
galaxy. All three predictions for the evolution of an initial power-law
function are inconsistent with observations. However, a GCLF that is
initially log-normal suffers little evolution in shape or turnover, and
that only in the first few Gyr. If constant initial SN
is assumed, GC destruction leads to a SN-galaxy
luminosity relation of the form SN
L0.67.
Vesperini et al. (2003)
explored M87 in detail, and (reminiscent of
Fall & Zhang 2001)
found that the observed
log-normal GCLF, which has a constant turnover with radius (see also
Harris et al. 1999),
could only evolve from a power-law initial GCLF if
there were (unobserved) strong radial anisotropy in the GC
kinematics. Of course, the details of the simulations are quite
important. For example,
Vesperini & Zepf
(2003)
showed that a
power-law GCLF and a concentration-mass relation for individual GCs can
result in a log-normal GCLF with little radial variation in turnover
mass. This is because low-concentration GCs are preferentially destroyed
at all radii. This is consistent with the study of
Smith & Burkert
(2002),
who found that the slope of the low-mass part of the GCLF in the
Galaxy depends upon GC concentration. Another relevant piece of evidence
is the similarity in the GCLF turnover among spirals, Es, and even dEs
(see Section 10.1). This suggests
either initial GCLFs close to
log-normal (such that little dynamical evolution occurs), or that the
dominant destruction processes are not specific to particular galaxy
types-for example, disk shocking in spirals.
The sole galaxy with a significant subpopulation of intermediate-age GCs
and evidence for dynamical evolution is the merger remnant
NGC 1316. HST/ACS observations by
Goudfrooij et al. (2004)
have clearly demonstrated dynamical evolution of its GC system. When they
divided the red (1.03 
V - I 1.40)
GCs into two equal radial bins, they found that the outer (beyond ~ 9.4
kpc) GCs have the power law LF seen for many systems of YMCs in merging
and starbursting galaxies. By contrast, the inner GCs show a LF turnover
characteristic of old GC systems. As Goudfrooij et al. argued,
these observations would appear to provide the conclusive link between
YMCs in mergers and old GCs in present-day Es.
However, the difficulty with this interpretation is that the observed location of the metal-rich peak of the inner GCs in NGC 1316 is at MV ~ -6. If the metal-rich GCs were formed in the merger, are ~ 3 Gyr old, and have solar metallicity (consistent with spectroscopic results; Goudfrooij et al. 2001), Maraston (2005) models with either a Salpeter or Kroupa IMF predict that age-fading to 12 Gyr will result in a peak at MV ~ -4.6. Old metal-rich GCs are observed to have a GCLF turnover at MV ~ -7.2 (e.g., Larsen et al. 2001). Thus ~ 2.6 mag of additional evolution of the GCLF turnover (~ 1.2 mag of dynamical evolution and ~ 1.4 mag of age fading) would be required to turn the new metal-rich GC subpopulation of NGC 1316 into that of a normal E. The 1.2 mag of dynamical evolution needed is far beyond that predicted by even the most "optimistic" models for GC destruction. For example, Fall & Zhang (2001) models for GC system evolution, assuming a power-law initial GCMF, predict < 0.2 mag of evolution in the GCLF peak from 3 to 12 Gyr using the same Maraston (2005) models. This is consistent with the finding of Whitmore et al. (2002) that little evolution in the GCLF peak is expected after the first 1.5-2 Gyr, as age-fading balances GC destruction. The implication is that in a Hubble time, the metal-rich GC system of NGC 1316 may not look like that of a normal E galaxy. Fall & Zhang (2001) models, designed to study GC destruction in a Milky Way-like galaxy, may even have limited applicability to NGC 1316. Other models (e.g., those of Vesperini discussed above) predict less evolution, so the expected difference between the suitably evolved young GCs in NGC 1316 and the old metal-rich GCs in local Es could be even greater. A caveat here is that the observed GCLF peak could be the convolution of a brighter, more evolved GC population in the innermost regions with a less evolved population in the outer parts of the bin. Such a convolution would tend to lessen the difference between the observed and expected turnover luminosity.
Thus, while there is some evidence for the paradigm of the evolution of a power-law LF to a log-normal LF, there are still unresolved issues. The installation of WFC3 on a potential future HST servicing mission would allow much more efficient and accurate age-dating of YMCs through its wide-field U-band imaging capability. This would also make it possible to investigate the mass function as a function of age in more detail, and hence directly address questions of dynamical evolution.