Classical IMFs (Salpeter, Kroupa, Chabrier, Maschberger)

Classical IMFs¶

The single-star initial mass function $\xi(m) \equiv \mathrm{d}N/\mathrm{d}m$ captures the birth-mass distribution of individual stars formed in a single star-formation event. It is the standard input for population synthesis, chemical evolution, and the IMF-inference problems progenax is built to solve. Four functional forms dominate the literature, each with a regime where it is the natural choice:

This chapter derives each form, lists the parameter values that turn it into “the” canonical IMF, and explains why progenax picks Maschberger (2013) as its default in production.

Salpeter (1955): the original power-law¶

The first published IMF, fit to Galactic-disk solar-neighbourhood stars Salpeter, 1955:

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valid for $m \gtrsim 1\,\Msun$. Below $\sim 1\,\Msun$ the actual stellar distribution turns over (the IMF flattens and eventually decreases toward sub-stellar masses), which Salpeter does not capture. Modern usage typically restricts (1) to $m \in [1, m_{\max}]$ and pairs it with a separate low-mass model — which is essentially the Kroupa construction below.

The cumulative is closed-form:

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inverted analytically for inverse-CDF sampling. progenax’s PowerLawIMF.salpeter(m_min=1.0, m_max=150.0) is the Salpeter configuration (internally exponents=[2.35], breakpoints=[]).

Kroupa (2001): multi-segment broken power-law¶

Kroupa (2001) extends Salpeter to a piecewise broken power-law that captures the observed low-mass turnover:

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with continuity coefficients enforced at each break. The four-segment form includes the brown-dwarf regime ($m < 0.08\,\Msun$); a common three-segment usage drops $\alpha_0$ and starts at $0.08\,\Msun$. The two high-mass slopes $\alpha_2 = \alpha_3 = 2.3$ match the Salpeter value (different normalisation conventions account for the small difference between Salpeter’s 2.35 and Kroupa’s 2.3).

PowerLawIMF accepts arbitrary segment slopes and break-points (breakpoints lists the interior breaks only, excluding m_min and m_max):

from progenax.imf import PowerLawIMF

# canonical Kroupa: the two equal high-mass slopes (2.3) merge to one
# segment, so the classmethod uses 3 segments / 2 breaks (exact).
kroupa = PowerLawIMF.kroupa(m_min=0.01, m_max=150.0)

# or build the explicit 4-segment form directly:
kroupa4 = PowerLawIMF(
    exponents=[0.3, 1.3, 2.3, 2.3],
    breakpoints=[0.08, 0.5, 1.0],
    m_min=0.01,
    m_max=150.0,
)

The class internally computes the continuity coefficients $a_i$ (so that $\xi$ is continuous at each break) and the per-segment integrals needed for normalisation and sampling.

Chabrier (2003): lognormal + power-law¶

Chabrier (2003) replaces the broken-power-law low-mass piece with a lognormal:

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with the Chabrier (2003) Table 1 single-star (disk) values $m_c \approx 0.08\,\Msun$ (0.079), $\sigma_{\log m} = 0.69$, and $\alpha_3 = 2.3$, joined continuously at $m = 1\,\Msun$. (The often-quoted $m_c = 0.22$, $\sigma = 0.57$ are the system IMF; progenax’s ChabrierIMF defaults to the single-star disk form — its defaults are m_c=0.08, sigma=0.69, alpha=2.3, m_trans=1.0.) The lognormal provides a smoother low-mass description than the broken power-law, which matters for unresolved-population integrated luminosities and colours. ChabrierIMF is the standard implementation; it shares its high-mass tail slope with PowerLawIMF.

The Chabrier IMF does not admit a closed-form inverse CDF. Sampling uses a Newton solver on $\mathrm{CDF}(m) = u$; the iteration count is fixed at 30 steps which is enough for double-precision convergence across the entire mass range.

Maschberger (2013): smooth and analytically invertible¶

The Maschberger (2013) “L3” form is progenax’s production default because it captures the same physics as Chabrier+Kroupa with a single smooth functional form and a closed-form inverse CDF:

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with default parameters $\alpha = 2.3$ (Salpeter slope), $\beta = 1.4$, $\mu = 0.2\,\Msun$ (peak mass). The two factors interact:

• For $m \gg \mu$: the second factor approaches 1, leaving the high-mass power-law tail $\xi \propto m^{-\alpha}$ (= Salpeter).

• For $m \ll \mu$: the second factor suppresses the distribution, producing the observed turnover below $\sim 0.3\,\Msun$.

The closed-form CDF and its inverse are derived in Maschberger (2013) §3. For $u \sim \mathcal{U}(0, 1)$,

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with $C = \mu / [(1-\beta)(1-\alpha)]$ and $P_{\min}, P_{\max}$ the primitive evaluated at the integration endpoints. progenax’s Maschberger class uses (6) directly, with no Newton solver, no rejection sampling, fully vmap-compatible.

Check yourself¶

Truncated power-law¶

TruncatedIMF wraps any of the above with explicit hard cutoffs at $m_{\min}$ and $m_{\max}$. This is useful for:

• Testing edge cases (e.g. forcing $m_{\max} = 5\,\Msun$ to study a specific stellar regime).

• Cluster-mass-dependent upper limits via the Marks et al. (2012) Weidner-Kroupa relation $m_{\max}(M_{\mathrm{cl}})$.

• Survey selection effects where stars below some completeness threshold are not observed.

The truncation is implemented via inverse-CDF resampling within the truncated range — not by rejection — so all sampled masses are guaranteed inside $[m_{\min}, m_{\max}]$ in a single pass.

Comparison and choice guide¶

For new users: use Maschberger unless you have a specific reason to pick another. For comparison with prior work using a different IMF, match the convention of the comparison; progenax’s flexible PowerLawIMF + ChabrierIMF cover the standard alternatives.

Connection to binary-aware and environment IMFs¶

The IMFs above describe single-star distributions. They feed into the binary-aware framework (Binary-aware IMF recovery) where the primary mass is drawn from $\xi(m \mid \alpha)$ and the secondary is drawn conditionally from the Moe & Di Stefano (2017) mass-ratio distribution. They also feed into the environment-dependent framework (Environment-dependent IMFs) where the slopes $\alpha_{1,2,3}$ become functions of cloud-core density and metallicity per Marks et al. (2012).

Both extensions wrap the classical IMF rather than replacing it: the single-star $\xi$ remains the underlying object, with binary multiplicity and environment dependence layered on top.

Implementation, validation & references¶

• In code: src/progenax/imf/power_law.py (PowerLawIMF, the Salpeter/Kroupa classmethods), chabrier.py (ChabrierIMF), smooth.py (Maschberger), and truncated.py (TruncatedIMF) — see the IMF API.

• Validated in: IMF statistics — the regression suite that checks each sampled distribution against its analytic form.

• Primary sources: Salpeter (1955), Kroupa (2001), Chabrier (2003), and Maschberger (2013) — full notes in the bibliography.