Environment-dependent IMFs

The canonical IMF is treated as universal in most population-synthesis work — a single $\xi(m \mid \alpha)$ applied to every star-forming environment. This is observationally inadequate at the extremes: ultra-compact dwarfs and massive globular cluster progenitors require top-heavy IMFs ($\alpha_3 < 2.3$) to reproduce their stellar populations and integrated colours, while metal-poor populations show hints of low-mass-suppressed IMFs Marks et al., 2012Jeřábková et al., 2018. progenax implements the Marks et al. (2012) cluster-scale variation and the Jeřábková et al. (2018) density relation for the high-mass slope (the cluster-IMF input to the IGIMF framework) as fully differentiable models. The galaxy-wide IGIMF integral itself is background theory here, not an implemented sampler — see the scope note below.

Two physical drivers of IMF variation¶

Both papers attribute IMF variation to two environmental parameters:

The two effects are correlated: starbursts that produce dense cores are typically metal-enriched, so cancelling rather than reinforcing the effects. Marks et al. (2012) resolves the degeneracy via the Fundamental Plane parameterisation in §Paragraph.

Notation: 3-segment vs 4-segment IMF¶

The progenax 4-segment IMF:

Continuity coefficients $a_i$ are computed from $a_0 = 1$ via $a_i = a_{i-1}\,m_i^{\alpha_i - \alpha_{i-1}}$. progenax’s environment code currently exposes BirthEnvironment, env_to_imf_params, and the alpha3_* helper functions; it does not export an EnvironmentIMF class. The brown-dwarf segment $\alpha_0$ is held fixed at 0.3 regardless of environment, reflecting the lack of strong observational constraints on sub-stellar IMF variation.

The upper-mass cutoff is $m_{\max,\star} \approx 150\,\Msun$, with an optional cluster-mass-dependent $m_{\max}$ via the Weidner-Kroupa relation when the cluster IMF (rather than individual stars) is being sampled.

Marks+2012: high-mass slope variation¶

Marks et al. (2012) derives empirical relations between the high-mass slope $\alpha_3$ and four candidate environmental parameters $\lambda$:

```{list-table} Marks et al. (2012) Table 3 — linear fit coefficients (page 2251). :header-rows: 1

• • Parameter $\lambda$

  • $p_\lambda$

  • $q_\lambda$

  • $\lambda_{\mathrm{lim}}$

  • Top-heavy when

• • $\log_{10}(M_{\mathrm{cl}}/10^6\,\Msun)$

  • -0.94

  • +2.14

  • 0.68

  • $\lambda > \lambda_{\mathrm{lim}}$

• • $\log_{10}(M_{\mathrm{ecl}}/10^6\,\Msun)$

  • -0.77

  • +1.59

  • 0.27

  • $\lambda > \lambda_{\mathrm{lim}}$

• • $\log_{10}(\rho_{\mathrm{cl}}/10^6\,\Msun\,\mathrm{pc}^{-3})$

  • -0.43

  • +1.86

  • 0.095

  • $\lambda > \lambda_{\mathrm{lim}}$

• • [Fe/H]

  • +0.66

  • +2.63

  • -0.5

  • $\lambda < \lambda_{\mathrm{lim}}$

Mass and density become top-heavy *above* their thresholds (intuition:
denser, more massive clouds form more massive stars). Metallicity is
top-heavy *below* its threshold (intuition: metal-poor clouds fragment
into more massive stars).

### Verification on Marks+12 Table 4

Setting [Fe/H] $= -2.0$ and applying the [Fe/H] coefficients:

```{math}
\alpha_3 \;=\; 0.66 \cdot (-2.0) + 2.63 \;=\; 1.31

Marks+12 Table 4 reads $\alpha_3 = 1.31$ at [Fe/H] $= -2.0$ — exact agreement. progenax’s regression suite locks every Marks+12 Table 4 entry to within 10<sup>-3</sup>.

Low-mass slope variation (tentative)¶

Marks et al. (2012) extends the metallicity dependence to the low-mass slopes following Kroupa (2001):

with $\Delta\alpha \approx 0.5$ and canonical values $\alpha_1^{\mathrm{c}} = 1.3$, $\alpha_2^{\mathrm{c}} = 2.3$.

The Fundamental Plane: $\alpha_3(\rho_{\mathrm{cl}}, [\mathrm{Fe/H}])$¶

The single-parameter fits in (2) capture density and metallicity effects separately. Marks et al. (2012) combine them into a 2D “Fundamental Plane” via a coordinate rotation:

The optimal rotation angle $\vartheta = 98°$ gives $\cos\vartheta \approx -0.139$ and $\sin\vartheta \approx +0.990$, i.e. the rotated coordinate is dominated by $\log\rho_{\mathrm{cl}}$ with a weak negative metallicity contribution. The fit reads

↗ model card

Substituting the rotation back yields the expanded form (Marks et al. (2012) Eq. 15):

↗ model card

valid when $\hat x \ge -0.87$, with $\alpha_3 = 2.3$ otherwise.

Jeřábková+2018: the IGIMF framework¶

The cluster-scale IMF variation in Marks et al. (2012) propagates to galaxy-wide IMFs via the Integrated Galaxy-wide IMF (IGIMF) framework of Jeřábková et al. (2018). Stars in a galaxy do not form in a single cluster — they form in a population of embedded clusters (ECMF) over some star-formation timescale $\delta t \sim 10$ Myr. The galaxy-wide IMF is

where $\xi_{\mathrm{cl}}(m \mid M_{\mathrm{ecl}}, [\mathrm{Fe/H}])$ is the Marks et al. (2012) cluster IMF and $\xi_{\mathrm{ECMF}}$ is the embedded-cluster mass function. The upper integration limit $M_{\mathrm{ecl,max}}(\mathrm{SFR})$ is the mass of the most massive cluster a galaxy of star-formation rate SFR can produce — a “Weidner-Kroupa” cap that scales with SFR.

The qualitative IGIMF predictions:

• Low SFR ($\sim 10^{-3}\,\Msun/\mathrm{yr}$): galaxy populated by small clusters, each with bottom-heavy IMF. gwIMF is bottom-heavy (steeper than Salpeter at high masses).

• High SFR ($\gtrsim 10\,\Msun/\mathrm{yr}$): galaxy populated by many massive clusters with top-heavy IMFs. gwIMF is top-heavy.

• Metal-poor: shifts the entire predicted gwIMF toward top-heavy at fixed SFR.

This SFR-and-metallicity-dependent gwIMF has consequences for chemical evolution, supernova rates, and the ionising-photon budget at high redshift — see Jeřábková et al. (2018) Fig. 5–7 for the predicted trends.

Domain of validity¶

The fits in this chapter are derived from samples with limited parameter coverage. Extrapolation outside these ranges yields predictions of unknown reliability:

The current implementation exposes differentiable helper functions for these relations. There is no EnvironmentIMF class; the public API is the functional env_to_imf_params() (which takes a clamp_domain flag, default True, to clip inputs to the calibrated ranges above) plus the alpha3_* / x_* helpers.

Implementation in progenax¶

from progenax.imf import BirthEnvironment, env_to_imf_params, PowerLawIMF

env = BirthEnvironment.from_cluster_mass(
    M_ecl=1e6,
    FeH=-1.5,
)
params = env_to_imf_params(env, model="marks_plane")

imf = PowerLawIMF(
    exponents=[params.alpha0, params.alpha1, params.alpha2, params.alpha3],
    breakpoints=[params.m_break0, params.m_break1, params.m_break2],
    m_min=params.m_min,
    m_max=params.m_max,
)
masses = imf.sample(key, 10000)

The env_to_imf_params mapping is differentiable in $\rho_{\mathrm{cl}}$, [Fe/H], and $M_{\mathrm{ecl}}$ (and the SFE $\varepsilon$); gradients flow from IMFParams back to BirthEnvironment. The Fundamental Plane threshold (5) uses jax.nn.sigmoid rather than jnp.where to keep gradients flowing through the threshold region — this matters when fitting $\alpha_3$ directly to data near the boundary $\hat x = -0.87$.

See `progenax.imf` for the full API and IMF statistics validation for the regression suite that locks Marks et al. (2012) Table 4 + Eq. 15 numerical values.

Connection to other chapters¶

The environment IMF feeds into:

• Binary-aware IMF recovery — binary-aware likelihood, where the IMF $\alpha$ is inferred from observed system masses. Misspecifying the environment (using a universal IMF for a metal-poor cluster) adds bias on top of the binary contamination bias. They compose: a naive single-star, universal-IMF fit to a metal-poor binary-rich cluster returns an $\alpha$ that is biased by both effects.

• BM19 dense-gas SFR framework — the BM19 dense-gas SFR framework predicts cloud/ECMF-scale properties that the IGIMF folds into a galaxy-wide IMF. progenax implements the cluster-scale link (cloud properties $\to \alpha_3$); the galaxy-wide ECMF integration is background theory, not an implemented end-to-end chain.

Implementation, validation & references¶

• In code: the environment package is src/progenax/imf/environment/ — birth_environment.py (BirthEnvironment), mapping.py (env_to_imf_params, the alpha3_* / x_* helpers), coefficients.py, and density.py. There is no EnvironmentIMF class; the public API is functional. See the IMF API.

• Validated in: environment IMF — the regression suite that locks Marks+12 Table 4 + Eq. 15 to within 10<sup>-3</sup> and pins the corrected -0.87 Fundamental-Plane threshold.

• Primary sources: the cluster-scale variation follows Marks et al. (2012) (with the Marks et al. (2014) erratum); the IGIMF aggregation follows Jeřábková et al. (2018); the Kroupa (2001) broken-power-law baseline anchors the canonical $(\alpha_1, \alpha_2, \alpha_3) = (1.3, 2.3, 2.3)$ reference. Full notes in the Marks et al. (2012) per-paper page.