On the variation of the Initial Mass Function

TL;DR

The paper investigates whether the Galactic-field IMF is universal and how apparent IMF variations arise from Poisson noise, cluster dynamics, and unresolved binaries. It introduces the alpha-plot to compare IMF slopes across mass ranges, adopts a four-part IMF with breaks near $0.5\,M_\odot$ and $0.08\,M_\odot$, and uses N-body star-cluster simulations to quantify biases in inferred slopes. The results show that most of the observed scatter in IMF determinations can be explained by sampling and dynamical effects, with unresolved binaries biasing slopes by about $Δα\approx0.5$ in the $0.08$–$1\,M_\odot$ range, prompting a revised present-day IMF that is steeper below $1\,M_\odot$ than the traditional Galactic-field IMF. While there is tentative evidence for IMF variations linked to environment (e.g., globular clusters, halo white-dwarf progenitors, and very young clusters), the current data remain inconclusive, and the study provides corrected IMF parameters and a framework for interpreting future observations.

Abstract

(shortened) In this contribution an average or Galactic-field IMF is defined, stressing that there is evidence for a change in the power-law index at only two masses: near 0.5 Msun and 0.08 Msun. Using this supposed universal IMF, the uncertainty inherent to any observational estimate of the IMF is investigated, by studying the scatter introduced by Poisson noise and the dynamical evolution of star clusters. It is found that this apparent scatter reproduces quite well the observed scatter in power-law index determinations, thus defining the fundamental limit within which any true variation becomes undetectable. Determinations of the power-law indices alpha are subject to systematic errors arising mostly from unresolved binaries. The systematic bias is quantified here, with the result that the single-star IMFs for young star-clusters are systematically steeper by d_alpha=0.5 between 0.1 and 1 Msun than the Galactic-field IMF, which is populated by, on average, about 5 Gyr old stars. The MFs in globular clusters appear to be, on average, systematically flatter than the Galactic-field IMF, and the recent detection of ancient white-dwarf candidates in the Galactic halo and absence of associated low-mass stars suggests a radically different IMF for this ancient population. Star-formation in higher-metallicity environments thus appears to produce relatively more low-mass stars.

Figures (11)

• Figure 1: The alpha-plot. The data show the compilation by Scalo (1998) of determinations of $\alpha$ over different mass ranges for Milky-Way (MW) and Large-Magellanic-Cloud (LMC) clusters and OB associations. Unresolved multiple systems are not corrected for. The large open triangles (Muench, Lada & Lada 2000 from Orion Nebula Cluster observations, binary corrections not applied) serve to illustrate the present knowledge for $m<0.1\,M_\odot$. The horizontal long-dashed lines in the BD regime are the Galactic-field IMF (eq. \ref{['eq:imf']}) with associated approximate uncertainties. For $0.08\le m \le 1.0\,M_\odot$ the thick short-dashed lines represent the KTG93 single-star IMF (Kroupa, Tout & Gilmore 1993), which has $\alpha_3=2.7$ for $m>1\,M_\odot$ from Scalo's (1986) determination. The long-dashed lines for $m>1\,M_\odot$ show the approximate average $\alpha=2.3$, which is adopted in the Galactic-field IMF (eq. \ref{['eq:imf']}). The Miller & Scalo (1979) log-normal IMF for a constant star-formation rate and a Galactic disk age of 12 Gyr is plotted as the diagonal long-dash-dotted line. The long-dash-dotted horizontal lines labelled "SN" are those $\alpha_3=0.70 (1.4)$ for which 50 % of the stellar (including BD) mass is in stars with $8 - 50 (8 - 120)\,M_\odot$. The vertical dotted lines delineate the four mass ranges (eq. \ref{['eq:imf']}), and the shaded areas highlight those stellar mass regions where the derivation of the IMF is additionally complicated due to unknown ages, especially for Galactic field stars: for $0.08<m< 0.15\,M_\odot$ long-contraction times make the conversion from an empirical LF to an IMF dependent on the precise knowledge of the age, and for $0.8< m<2.5\,M_\odot$ post-main sequence evolution makes derived masses uncertain in the absence of precise age knowledge. A few of the MW data are labelled by their star-clusters, and Table \ref{['tab:a_m']} lists the $m_{\rm av}<1\,M_\odot$ data.
• Figure 2: The adopted logarithmic IMF (eqs. \ref{['eq:imf']} and \ref{['eq:xiL']}), $\xi_{\rm L}/10^3$, for $10^6$ stars (solid histogram). Two random renditions of this IMF with $10^3$ stars are shown as the heavy and thin dotted histograms. The mass-ranges over which power-law functions are fitted are indicated by the arrowed six regions (eq. \ref{['eq:bin']}), while thin vertical dotted lines indicate the masses at which $\alpha_i$ changes.
• Figure 3: Purely statistical variation of $\alpha$ in the six mass ranges (eq. \ref{['eq:bin']}) for different $N$ as indicated in the key. Large outer squares indicate those $\alpha$ fits obtained with $nb=2$ and 3 mass bins. The open circles, open triangles, vertical and horizontal lines are as in Fig. \ref{['fig:a_m']}.
• Figure 4: Mass functions for single-stars (solid histograms) and systems (dash-dotted histograms) at $t=0$ in models B1E4 (thick histograms) and B1E4d (thin histograms). Note the smaller number of massive stars in model B1E4d, which has a steeper IMF for $m>1\,M_\odot$ with $\alpha_3=2.7$ (Table \ref{['tab:clmods']}). The solid dots are the IMF for $N=10^6$ stars (Fig. \ref{['fig:mf_sv']}) scaled to $N=10^4$, and the vertical dotted lines and arrowed regions are as in Fig. \ref{['fig:mf_sv']}.
• Figure 5: Examples of the evolution of individual clusters. Top panel: The number of systems (thick curves) and all individual stars and BDs (thin curves) within the innermost 3.2 pc. The short-dashed lines are for $N=800$, the solid lines are for $N=3000$, and the dash-dotted lines are for $N=10^4$. Bottom panel: The binary proportion for $R\le3.2$ pc (thick curves), and all $R$ (thin curves) for the same cases as in top panel. In both panels, the horizontal dotted lines indicate the times (3 and 70 Myr) at which the mass functions are observed.