Chabrier (2003) - progenax — differentiable initial conditions for N-body simulations

Abstract (paraphrased)¶

Reviews present-day and initial mass functions across Galactic components and into the substellar regime. The disk IMF is well described by a lognormal form below $\sim 1\,M_\odot$ joined continuously to a power-law tail above. Chabrier reports two distinct parameterisations — one for single objects (individual stars, binaries resolved) and one for stellar systems (unresolved) — which differ in characteristic mass and width. The single-object form extends smoothly into the brown-dwarf regime.

Definitions and the two disk parameterisations (verified, §1.2 + Table 1)¶

Chabrier uses both the logarithmic and linear mass functions (Eqs. 1–2):

\xi(\log m) = \frac{dn}{d\log m}, \qquad \xi(m) = \frac{dn}{dm} = \frac{1}{m\ln 10}\,\xi(\log m).

(1)

The original Salpeter slope is stated as $x = 1.35,\ \alpha = 2.35$ (p. 765).

Single-object disk IMF — Table 1 (p. 769). For $m \le 1\,M_\odot$ a lognormal, for $m > 1\,M_\odot$ a power law $\xi(\log m) = A\,m^{-x}$ :

\xi(\log m)_{m\le 1} = 0.158 \, \exp\!\left[-\frac{(\log m - \log 0.079)^2}{2\,(0.69)^2}\right], \qquad x = 1.3 \pm 0.3 \ \Rightarrow\ \alpha = x+1 = 2.3 \ (dN/dm).

(2)

System IMF — Eq. 18 (p. 770). The unresolved-system MF has a different characteristic mass and width:

\xi(\log m)_{m\le 1} = 0.086 \, \exp\!\left[-\frac{(\log m - \log 0.22)^2}{2\,(0.57)^2}\right].

(3)

Which form does progenax implement?

progenax.imf.ChabrierIMF implements the single-object disk IMF ((2)): $m_c = 0.079$ (rounded to 0.08), $\sigma = 0.69$ , $A_\mathrm{ln} = 0.158$ , and a high-mass slope $\alpha = 2.3$ (Chabrier’s Table 1 $x = 1.3$ , not the system values $m_c \approx 0.22$ , $\sigma \approx 0.57$ ). The lognormal and power-law are joined continuously at $m_\mathrm{trans} = 1\,M_\odot$ by setting $A_\mathrm{pl} = \xi_\mathrm{ln}(m_\mathrm{trans})\,m_\mathrm{trans}^{\alpha}$ (value-continuous; only the slope has a kink). The original Salpeter $\alpha = 2.35$ is available separately via PowerLawIMF.salpeter().

Use in progenax¶

Classical IMFs (Salpeter, Kroupa, Chabrier, Maschberger) — the lognormal + power-law disk IMF.
progenax.imf.ChabrierIMF — single-object disk IMF, sampled via a Newton inverse-CDF.

Notes¶

The single-object vs system distinction is the key subtlety: the system IMF ((3)) is broader-peaked because unresolved binaries shift the apparent characteristic mass upward. For sampling individual stars in an N-body IC (progenax’s use case), the single-object form is the correct choice. For HMC inference where analytic invertibility matters, prefer Maschberger (2013).

References¶

Chabrier, G. (2003). Galactic stellar and substellar initial mass function. Publications of the Astronomical Society of the Pacific, 115, 763–795. 10.1086/376392