Initial mass functions

The stellar initial mass function $\xi(m) \equiv \mathrm{d}N/\mathrm{d}m$ is the birth-mass distribution of stars formed in a single star-formation event. It governs chemical enrichment, supernova rates, and the integrated colours of stellar populations Salpeter, 1955Kroupa, 2001Chabrier, 2003. progenax implements the IMF as one of its three orthogonal IC ingredients (What is an initial condition?) and provides multiple parameterisations ranging from textbook (Salpeter, Kroupa, Chabrier) through smooth analytically-invertible (Maschberger) to physically-detailed (binary-aware Moe & Di Stefano, and an environment-dependent IMF that maps cluster birth conditions to the high-mass slope $\alpha_3$ via the Marks+2012 / Jeřábková+2018 relations).

Map of the IMF chapters¶

The chapters split a large topic into focussed pieces. New readers should start with Classical IMFs (Salpeter, Kroupa, Chabrier, Maschberger), then jump to Binary-aware IMF recovery for the binary-aware framework end to end before drilling into the sub-chapters as needed.

Common API contract¶

Every IMF in progenax satisfies the IMFProtocol:

class IMFProtocol(Protocol):
    m_min: float
    m_max: float

    def logpdf(self, m: Float[Array, "..."]) -> Float[Array, "..."]:
        """Log-PDF for likelihood evaluation."""
        ...

    def cdf(self, m: Float[Array, "..."]) -> Float[Array, "..."]:
        """Cumulative number fraction below mass m."""
        ...

    def ppf(self, u: Float[Array, "..."]) -> Float[Array, "..."]:
        """Inverse CDF."""
        ...

    def sample(
        self,
        key: PRNGKey,
        n: int,
    ) -> Float[Array, "N"]:
        """Draw N stellar masses."""
        ...

    def mean_mass(self) -> float:
        """Expected mass."""
        ...

sample uses inverse-CDF for the analytically-invertible IMFs (Maschberger; truncated power-law) and a fixed-iteration Newton solver for the rest. logpdf is the workhorse for the scalar IMF families. Binary-aware inference is described conceptually in this section, but the current BinaryIMF API exposes sampling helpers (sample_primaries, sample_mass_ratios, sample_systems, and sample_all_masses) rather than an exact logpdf. Environment-dependent IMF support currently maps a BirthEnvironment to IMF parameters rather than exposing an EnvironmentIMF class.

Notation conventions¶

The convention $\alpha = 2.35$ for the Salpeter slope is universal in this section. Marks et al. (2012) use a different segmentation (3-segment vs progenax’s 4-segment); the Environment-dependent IMFs chapter documents the conversion in detail.

Composability with profiles, velocities, and modifiers¶

IMFs compose orthogonally with Spatial density profiles and Velocity distribution functions: any IMF can pair with any spatial profile and velocity DF. The IMF determines the masses in (masses, positions, velocities); the profile determines the positions; the DF determines the velocities. The mass-segregation modifier (Mass segregation) couples IMF and profile by re-pairing high-mass particles to low-energy orbits, but the underlying $\xi(m)$ is preserved.

Implementation, validation & references¶

• In code: the IMF family lives under src/progenax/imf/ (power_law.py, chabrier.py, smooth.py, truncated.py, the binary/ package, and the environment/ package). See the IMF API; each chapter below carries its exact module path.

• Validated in: IMF statistics (distributions + Marks+12 numerical anchors), binary-aware recovery, and environment IMF.

• Primary sources: canonical IMFs Salpeter (1955), Kroupa (2001), Chabrier (2003), Maschberger (2013); binary statistics Sana et al. (2012), Moe & Di Stefano (2017), Moe et al. (2019); environment dependence Marks et al. (2012), Jeřábková et al. (2018). Full notes in the bibliography; each chapter below points at the specific result(s) used.