Summary of talk given on 29 November 2006
The initial mass function (IMF) describes the frequency distribution by mass of a given set of stars. All stars in the set must have formed very recently, approximately contemporaneously, and in an environment where the stars can be assumed to have developed under similar conditions and with similar composition (e.g., a single cluster).
The IMF usually takes the form of a multipart power-law function. Each part specifies the frequency distribution for a particular mass range and is characterized by a different exponent; generally, the lower the masses encompassed by the mass range, the smaller in absolute magnitude the exponent, and the shallower the slope of the IMF.
A standard IMF is given by Kroupa (2001, 2002):
with exponents
where dN = ξ(m) dm is the number of stars in the mass range m to m + dm.
k is a normalization constant specified by the total stellar mass Mecl in the entire set of stars being described:
α3 = 2.35 is known as the Salpeter exponent; E. Salpeter described an IMF that has served to the present day for stellar masses greater than about one to five solar masses.
The IMF is often assumed to be universal within the Local Group of galaxies, and particularly within the range of stellar metallicities found in the Milky Way.
Given the form of the standard IMF and sets of previous observations, stars
more massive than about 80 M
should be statistically so rare that very
massive stars are unlikely to be observed in any one cluster. No stars more
massive than about 150 M have been observed. Whether or not this
suggests a fundamental stellar upper mass limit has been a subject of some
dispute.
Spherical accretion models of star formation predict an upper mass limit of
ten to forty M, due to radiation pressure feedback on the accreting
material; however, stars have been observed still accreting in this range
(Chini, et al., 2004). Bonnell, Bate, and Zinnecker (1998) propose that
protostars found in the dense cores of young clusters, of about ten solar
masses, may merge to form very massive stars. Additionally, massive stars
may form through disc accretion; in such a scenario, radiation could exit
through the polar regions of the star (Nakano, 1989). There is little
theoretical consensus on the formation of extremely massive stars.
Weidner and Kroupa (2004) therefore undertake an analysis to show that self-consistently incorporating a fundamental upper mass limit into a solution of the IMF produces a distribution in which the highest stellar mass predicted to be found within a given cluster falls in concurrence with the highest masses observed in that cluster.
The number of stars counted above a mass m is:
Weidner and Kroupa distinguish between a fundamental upper mass limit, mmax*, and a local highest stellar mass in a cluster, mmax.
Weidner and Kroupa assume therefore for purposes of calculation that the most massive star expected in a cluster follows:
Weidner and Kroupa set two cases:
1. mmax* = 150 M
2. mmax* → ∞
In each case, Weidner and Kroupa solve for both the normalization constant k and the local highest stellar mass mmax, additionally using
with mlow set to 0.01 M throughout their analysis. Weidner and Kroupa also assume
mmax > m1, αi ≠ 1, and α3 > 1.
Weidner and Kroupa use R136 as a sample cluster; R136 has about 105 stars, Mecl = 5 ⋅
104 M to 2.5 ⋅ 105 M.
Massey & Hunter (1998) confirm a Salpeter power-law IMF for m > one to five
M in R136. Weidner and Kroupa calculate from their analysis that without a
fundamental upper mass limit, mmax for R136 should be greater than 750
M, based on the estimate of Mecl. (mmax begins to rise above 200 M
for Mecl > 104.5 M.) The number of stars expected to have masses
greater than 150 M ranges between ten and forty for our Mecl range for
R136. See figure 2.
Figure 2. Number of stars (logarithmic) above mass m for R136 for different mass estimates (dotted line: MR136 = 2.5 ⋅ 105 M; dashed line: MR136 = 5 ⋅ 104 M, Selman et al. 1999). The vertical solid line marks m = 150 M.1
The observed mmax of R136, however, is about 150 M. Inserting a
fundamental upper mass limit of 150 M, as in case 1 of their analysis,
does result in an expected local mmax of about 150 M for R136. See figure 4.
Figure 4. Double logarithmic plot of the maximal stellar mass versus cluster mass. Shown are three cases: finite total upper mass limit of mmax* = 150 M (dotted line); mmax* = 1000 M (short-dashed line); and no limit, mmax* = ∞ (long-dashed line). The vertical lines mark the empirical mass interval for R136 in the LMC.1
Weidner and Kroupa next suggest that modifying α3 may appropriately influence the
structure of the IMF. They show that setting α3 > 2.8 produces the same
effects in R136 as specifying an mmax* of 150 M; local mmax falls to
about 150 M. See figure 6.
Figure 6. The mass limits (mmax) as a function of the IMF exponent α3 (above 1 M) in the limited case (mmax*= 150 M) and the unlimited case (mmax*= ∞) for the two mass limits of R136.1
The rarity of observed massive stars gives statistical
validity to a Salpeter IMF only for m < 40 M or so (though the Salpeter
IMF is consistent with observations to about 130 M), so Weidner and Kroupa speculate
that allowing α3 to modify itself above about mborder = 40 M in
order to produce an IMF consistent with mmax = 150 M may do the trick.
This results in a too-sharp downturn of the IMF slope around mborder (for
varying mborder up to 100 M) that is not consistent with observations. See figure 7.
Figure 7. The power-law exponent α needed to produce a high-mass limit of 150 M for R136 (solid line: MR136 = 2.5 ⋅ 105 M and dotted line: MR136 = 5 ⋅ 104 M) when the IMF is Salpeter up to a certain mass limit mborder.1
Weidner and Kroupa finally note that the IMF may need to be binary-corrected for the high rate of companions among massive stars, stating that a binary-corrected IMF with α3 > 2.7 or so indicates that available stellar observations cannot constrain mmax* because there are no nearby clusters that are sufficiently massive and young.
Weidner and Kroupa conclude that case 1 of their analysis, with a fundamental upper mass
limit of 150 M, offers a consistent solution to the issue of missing
supermassive stars with a Salpeter IMF up to 130 M while requiring few
assumptions. Weidner and Kroupa offer their analysis in contrast to the proposal of
Elmegreen (2000) that any fundamental upper mass limit is simply a cut-off
placed on analysis done in case 2.
Some objections to Weidner and Kroupa's conclusion are possible. The age of the stars in R136 ranges from one to three million years, with an upper limit of five million years having largely been ruled out. Mecl includes all old and young low- and high-mass stars. If some older supermassive stars have already evolved to supernovae, and thus cannot be observed, the standard IMF may overestimate the number of high-mass stars expected to be observed as extrapolated from observations of intermediate-mass stars, which will not have evolved to supernovae within three million years (or at all, for the lowest-mass stars). The deficit of supermassive stars may thus be an artifact of age.
Using only sufficiently young stars and a lower Mecl young (2 ⋅ 104
M) that encompasses only those stars, Figer (2005) calculates a deficit
of only four supermassive stars, which reduces the possible statistical
significance of R136's lack of stars having masses above 150 M.
Reassuringly, Figer's own study of the Arches cluster (2005), which has a
well-constrained age of about two million years, finds the same lack of
stars with masses above 150 M. Using Poisson statistics and assuming a
Salpeter power-law IMF (the actual IMF observed is somewhat shallower),
Figer calculates that for the case of the Arches cluster, the probability
that the data set is consistent with an infinite fundamental upper mass
limit is no greater than 10-8. See the figure below from Figer (2005).
Figure from Figer (2005). The counts in each bin have been reduced by counts in nearby background fields. Error bars represent the Poisson errors based on the background subtracted counts. Two lines are drawn through the average counts in the four highest populated mass bins, with slopes inferred from the data (d(log N)/d(log m) = Γ = -0.9) and that of Salpeter (Γ = -1.35). For both lines, there is a clear deficit of stars with initial masses greater than approx. 150 M, as seen in the hatched regions. In addition, both slopes predict that at least one star in the cluster should have a mass far beyond that observed if there is no upper mass cut-off. Nmissing is the difference between the number of stars expected with Minitial > 130 M and the number observed. MMAX is the initial mass at which the mass function predicts the existence of one star. tage is the assumed cluster age. Z is the assumed abundance of metals in the cluster stars.1
It is highly likely, then, that there indeed exists a fundamental stellar upper mass limit, although what physical mechanism may be involved is not at all clear.
Interestingly, theories of massive star growth inhibition via radiative feedback rely on metallicity-induced opacity of the accreting material for the accretion to be suppressed by radiation. Massive stars should form more readily given a low-metallicity medium. The metallicities of stars found in Arches and in R136 differ, with those in R136 being of lower metallicity, but the local maximum stellar masses observed in both clusters are nevertheless approximately identical. This suggests that radiative feedback on accretion may not be the physical mechanism inhibiting the growth of highly massive stars.
References
Figer (2005). Nature 434, 192. [astro-ph/0503193]
Elmegreen (2000). ApJ 539, 342. [astro-ph/0005455]
Weidner and Kroupa (2004). MNRAS 348, 187. [astro-ph/0310860]
Kroupa (2005). Nature 434, 148. [ADS]
1 All figure captions taken directly from Weidner and Kroupa (2004) and Figer (2005).