Kroupa (2001)

Abstract (paraphrased)¶

Defines an average Galactic-field IMF as a multi-segment power law, with changes in the power-law index at only two masses ($\sim 0.5\,M_\odot$ and $\sim 0.08\,M_\odot$). Quantifies how Poisson noise, unresolved binaries, and dynamical evolution introduce apparent scatter in measured power-law indices, and argues that no convincing evidence for a variable IMF exists once these are accounted for. The resulting canonical broken power law is the standard IMF for star-cluster modelling.

The canonical IMF (verified against the paper, §2.2)¶

Kroupa writes the IMF as a piecewise power law (Eqs. 1–2, p. 234)

with, for single stars, $\xi(m)\,dm$ the number in $[m, m+dm]$ (Eq. 2):

The breaks are at the hydrogen-burning limit ($0.08\,M_\odot$) and at $0.5\,M_\odot$. Kroupa quotes the original Salpeter value as $\alpha = 2.35$ and adopts the rounded $\alpha = 2.3 \pm 0.3$ for the high-mass slope (p. 234). The mean stellar mass of this IMF is $\langle m\rangle = 0.36\,M_\odot$ over 0.01–$50\,M_\odot$ (p. 235).

Because $\alpha_2 = \alpha_3 = 2.3$, the two highest segments can be merged into a single $\ge 0.5\,M_\odot$ segment with no change to the IMF.

Why the IMF appears to vary — the corrections (the paper’s core argument)¶

Kroupa’s central methodological point is that most reported IMF “variations” are artefacts of three observational/dynamical effects, and vanish once corrected — hence a single canonical IMF suffices:

• Unresolved multiplicity. Surveys measure systems, not stars. Because companions are preferentially low-mass, the inferred system MF is flatter at low mass than the single-star IMF. Kroupa derives the single-star IMF by populating systems with a realistic binary fraction and “observing” them — the low-mass slope $\alpha_1$ steepens on correction. (progenax keeps the two distinct: PowerLawIMF.kroupa() is the single-star IMF.)

• Poisson/sampling noise. A finite cluster scatters the measured high-mass slope by $\pm0.5$–0.7 (the quoted per-segment uncertainties); apparent cluster-to-cluster scatter is consistent with one universal IMF.

• Dynamical evolution. Two-body relaxation and tidal stripping preferentially remove low-mass stars, flattening the present-day MF relative to the IMF — an age/dynamical effect, not an IMF difference.

The conclusion — no convincing evidence for a primordial IMF variation in resolved Local Group populations — is precisely the null hypothesis that the environment-dependent extensions (Marks et al. (2012), Jeřábková et al. (2018)) later challenge in the extreme density/metallicity regimes (dense GCs, UCDs, starbursts) that the Galactic field does not probe.

Inputs¶

The canonical IMF has no environmental inputs — it is the universal baseline ($\alpha_3=2.3$). The environment-dependent IMF reduces to exactly this canonical form in the low-density / solar-metallicity limit ($\rho_{\rm cl}\lesssim10^5\,M_\odot\,{\rm pc^{-3}}$, ${\rm [Fe/H]}\gtrsim0$); only outside that regime do $(\rho_{\rm cl}, {\rm [Fe/H]})$ drive $\alpha_3$ away from 2.3. No turbulence quantity enters at any point.

Use in progenax¶

• Classical IMFs (Salpeter, Kroupa, Chabrier, Maschberger) — the canonical multi-segment broken power law.

• Environment-dependent IMFs — the 4-segment basis for the environment-dependent (IGIMF/Marks) IMF.

• progenax.imf.PowerLawIMF.kroupa() — three-segment form ($\alpha=[0.3,1.3,2.3]$, breaks $[0.08,0.5]$), the exact merge of Kroupa’s $\alpha_2=\alpha_3=2.3$.

• progenax.imf.IMFParams.kroupa() — explicit four-segment form ($\alpha=[0.3,1.3,2.3,2.3]$, breaks $[0.08,0.5,1.0]$), kept separate for gradient-based inference of the high-mass slope $\alpha_3$.

Notes¶

The canonical IMF in cluster modelling. Maschberger (2013)'s smooth $L_3$ form approximates it with a single analytic expression. The quoted per-segment uncertainties ($\approx 99\%$ confidence for $m \ge 0.5\,M_\odot$) set the scale below which a measured IMF variation is consistent with a universal IMF.