Maschberger (2013)

Abstract (paraphrased)¶

Proposes the $L_3$ IMF, a three-parameter functional form that is a heavy-tailed approximation to the lognormal: a low-mass power law and a high-mass power law joined smoothly. The standard Kroupa (2001) and Chabrier (2003) IMFs are essentially indistinguishable from it. Its decisive practical advantage is that the cumulative distribution function and its inverse (the quantile / mass-generating function) are closed-form, so sampling needs no special functions and no Newton iteration.

The $L_3$ functional form (verified against the paper, Table 1)¶

With the auxiliary function $G(m)$ (Table 1, Eq. 1),

the pdf, CDF, and quantile are (Table 1, Eqs. 2–4)

The canonical single-star parameters (Table 1; system/binary values in parentheses):

The effective low-mass exponent is $\gamma = \alpha + \beta(1-\alpha) = 0.48$ (single-star; Table 1, Eq. 5), i.e. $p(m)\propto m^{-0.48}$ as $m\to 0$. The limits $m_l, m_u$ “are only needed for the normalization” (Table 1 caption). Maschberger states the Salpeter exponent as $\alpha = +2.35$ in the linear convention (§2.1).

Use in progenax¶

• Classical IMFs (Salpeter, Kroupa, Chabrier, Maschberger) — progenax production default; closed-form inverse CDF.

• Binary-aware IMF recovery — single-star backbone for the binary-aware IMF.

• progenax.imf.Maschberger — uses the analytic primitive $P(m) = \tfrac{\mu}{(1-\alpha)(1-\beta)}\,G(m)$ ((1)) and the closed-form quantile (3) (no Newton, no error function). progenax defaults to $m_\mathrm{max} = 300\,M_\odot$ (vs the paper’s $m_u = 150$); since the limit only sets the normalization, this is a convention choice.

Notes¶

progenax default IMF. The closed-form inverse CDF makes it the only canonical IMF that samples in $\mathcal{O}(1)$ per particle without Newton iterations — important for fast, differentiable IC generation. The infamous “peak” of the $L_3$ pdf is at the scale parameter $\mu$, which is not where the two power laws cross (Table 1, Eq. 11).