The initial mass functions (IMFs) of SSCs have important consequences for cluster masses, their long-term survival, and potentially, for the eventual remnants left by their dissolution. Are the IMFs of clusters power law? Is there evidence for top-heavy IMFs? How do the IMFs of SSCs compare to Galactic IMFs? Is there evidence for initial mass segregation in young clusters? The most massive star in the universe is likely to be in an SSC; is there a fundamental limit to the mass of stars? There is a review of the outpouring of recent IMF work on massive young clusters by Elmegreen (2008).
Mass functions have been determined for the large star clusters of the Local Group. Many appear to be Salpeter. The Salpeter mass functions are defined as (M) ~ M-, where = 2.35, or, expressed in logarithmic mass intervals as L d logM ~ M d logM, with = -1.35 (Scalo 1986, 1998). The Kroupa IMF is Salpeter at higher masses, and flattens to = 1.3 for stars below 0.5 M (Kroupa 2001). Kroupa IMFs often cannot be distinguished from Salpeter in extragalactic SSCs. We will adopt the convention here and note that while in many cases these are referred to as IMFs, what is observed is actually a present day mass function (PDMF), from which an IMF may be inferred.
The PDMF of R136 in the LMC has been measured down to 0.6 M (Massey and Hunter 1998), where it is consistent with Salpeter, ~ -1.3, and then flattens below 2 M to ~ -0.3 (Sirianni et al. 2000). For a Salpeter or Kroupa power law IMF, flattening or turnover of the power law corresponds to an effective "characteristic mass" for the cluster (Lada and Lada 2003). R136 thus appears to have a characteristic mass of 1 - 2 M, as compared to ~ 0.5 - 1 M for the much smaller nearby Galactic embedded clusters (Lada and Lada 2003). The southern cluster NGC 3603 has a somewhat flatter-than-Salpeter power law slope of = -0.7 to -0.9 from 0.4 to 20 M (Sung and Bessell 2004, Stolte et al. 2006, Harayama et al. 2008). Arches has a very similar power law PDMF with = -0.8 down to 1.3 M (Stolte et al. 2002), which may correspond to an IMF that is close to Salpeter = -1.0 - 1.1 (Kim et al. 2007). The Arches cluster is mass segregrated, with a flatter slope in the center, ~ 0, than in the outer parts of the cluster, and the mass function may turn over at 6 - 7 M in the core (Stolte et al. 2005); however, the MF at larger radii does not show this effect (Kim et al. 2007). Trends appear to be toward flatter MF power laws and higher characteristic masses for the largest clusters in the Local Group, but the numbers of clusters are very small, and dynamical effects are poorly understood as yet (Stolte et al. 2002).
Beyond the Local Group, it is more difficult to determine IMFs. Observational quantities for more distant SSCs are integral properties such as total luminosity and dynamical masses from cluster velocity dispersions. Constraints on IMFs from these integral properties require assumptions about mass cutoffs, cluster ages, and cluster structure. There is accumulating evidence, however, that IMFs in starbursts and the IMFs in SSCs, if Salpeter, may have higher characteristic cutoffs than typical Galactic clusters. Rieke et al. (1980) modeled the starburst in M82 from its IR emission properties, and concluded that the IMF of the starburst must have a low mass cutoff of 3 - 8 M. Sternberg (1998) used visible mass-to-light ratios to argue for a cutoff of 1 M for NGC1705-1, although he does not find a cutoff for NGC 1569A. If the IMF in the M82 SSCs is Salpeter, then McCrady et al. (2003) find that the individual M82 clusters must have truncated mass functions, although some clusters show strong evidence for mass segregation and possible dynamical evolution, which complicates this interpretation (McMillan and Portegies Zwart 2003, Boily et al. 2005, McCrady et al. 2005, McCrady and Graham 2007).
What the low mass cutoffs are for SSCs, whether this varies with environment and how, and what is the likely cause of the low mass cutoffs, effective characteristic masses, and the effects of mass segregation on cluster masses and evolution will be fertile ground for SSC research in the next decade.
At the other end of the IMF, there is the question of what is the limiting mass of a star. If the cluster IMFs bear any resemblance to the Galactic power law Salpeter function, then newly-formed clusters consisting of hundreds of thousands to millions of stars are the place to find the most massive and rare O stars. In addition, in dense star clusters there is the possibility of the formation of very massive stars through stellar collisions, which becomes a viable mechanism in dense environments (Bonnell et al. 1998, Clarke and Bonnell 2008). What is the limiting mass of stars in the local universe? Where are the most massive stars found?
Spectroscopy of individual stars is the most reliable way to identify the most massive stars, but is only possible within the Local Group. Ultraviolet spectroscopy using the STIS instrument on HST has allowed the classification of 45 known stars of spectral class O2 and O3; of these, 35 are found in the LMC cluster R136 (Walborn et al. 2002). R136 alone contains more than 65 O stars (Massey and Hunter 1998). In addition to these visible O stars, there is also a significant population of infrared O stars and Wolf-Rayet stars in the Galactic Center, including the three large star clusters, Arches, Quintuplet, and Galactic Center. These infrared clusters, each about an order of magnitude less massive than R136, contain an estimated 360 O stars among them, nearly 60 Wolf-Rayet stars, and 2-3 LBVs (Figer 2008).
The highest inferred stellar mass in the Local Group is ~ 170 M in the LMC, and ~ 200 M in the Galaxy; however the latter could eventually turn out to be a binary system; the largest dynamical mass measured is 90 M (Walborn et al. 2002). Weidner & Kroupa (2004) estimate that based on its IMF, observed to be Salpeter, there should be one 750 M star in the R136 cluster, and instead the upper mass limit appears to be 200 M (Koen 2006). Oey & Clarke (2005) extend this to a larger sample of OB associations, obtaining a cutoff of 120 - 200 M, although they caution that this cutoff is sensitive to the IMF slope for stars > 10 M. Figer (2005) finds that there should be one star of M ~ 500 M in the Galactic center, where the current upper mass limit instead seems to be 130 M. These observations argue for a stellar upper limit close to the observed 150 - 200 M. However, given the small statistics, it may be that the absence of extremely massive stars is simply an evolutionary effect (Elmegreen 2008). For example, modeling suggests that the Pistol Star in the Galactic Center may have had an initial mass of 200-250 M (Najarro and Figer 1998, Figer et al. 1998). Stars with masses of several hundred times solar would evolve rapidly (Yungelson et al. 2008), will lose significant fractions of their initial masses, and in addition, might be dust-enshrouded for most of their lives. Moreover small number statistics at the upper mass end mean that for most clusters, the power law slopes of IMFs are uncertain to a few tenths for clusters of the size of NGC 3603 (Elmegreen 2008). Supermassive stars are elusive by nature.
If extremely high mass stars do exist and we have not seen them simply because of their rapid evolution, the best place to look for them is in the youngest and largest clusters. These will probably be found in starbursts. There is a good chance that the youngest regions containing the most massive stars will be deeply embedded, perhaps extinguished even in the near-infrared. In these cases, nebular diagnostics provide another way of gauging the upper end of the mass function. The mid-infrared has a number of fine structure lines that can be observed in large H II regions in external galaxies (Helou et al. 2000, Thornley et al. 2000, Dale et al. 2006), including fine structure lines of Ar, Ne, S, and O, that can provide valuable nebular diagnostics of the input stellar radiation field even in embedded sources.
Line ratios of the mid-IR nebular fine structure lines in starburst galaxies measured with the SWS spectrometer on ISO revealed unexpectedly low excitation H II regions. Nebular models of the ISO lines indicate upper mass cutoffs of Mupper ~ 30 M for the IMFs in these starbursts (Thornley et al. 2000). Given the luminosities and inferred stellar masses, the low upper mass cutoffs are surprising. However, there are a number of possible explanations for low excitation that would allow for the presence of more massive stars (Rigby and Rieke 2004). One of these is lack of spatial resolution. Starburst regions often have extended regions of ionized gas which, when combined with the comparatively hard spectra of the compact SSC H II regions, will tend to wash out the high excitation lines. That the lines of [O IV]25.9 µm, [Ne III]15.55 µm, and [S IV]10.51 µm have been detected in dwarf starbursts suggests that hot stars are indeed present in some SSCs (Beck et al. 1996, Lutz et al. 1998, Crowther et al. 1999, 2006, Beck et al. 2007). Strong dependences on metallicity and differences in the input radiation fields from existing stellar models that must be considered in these interpretations (Crowther et al. 1999, Martín-Hernández et al. 2002, Rigby and Rieke 2004).
High spatial resolution and spectroscopy from space will allow great improvements in our knowledge of the most massive stars and the upper mass cutoffs of SSC IMFs in external galaxies in the next decade. Optical and ultraviolet emission lines including Wolf-Rayet features will continue to be important diagnostics of the high mass stellar content and ages of SSCs in an enlarged sample of sources. Access to mid-IR fine structure lines from space, via JWST, will give valuable information on the most massive stars in embedded young SSC nebulae, allowing observation of homonuclear line ratios of Ar and Ne that are not possible from the ground. The improvement in spatial resolution is also extremely important for isolating the spectral signatures of compact SSC nebulae. The MIRI IFU on JWST, with order-of-magnitude improvements in sensitivity and spatial resolution over previous instruments, will be an extremely powerful tool for isolating compact SSC nebulae from more diffuse and potentially lower excitation ionized gas within the galaxies. The form of the cluster IMFs is key input to the question of the long term survivability of clusters.