BM19 dense-gas SFR framework

The BM19 framework¶

The Burkhart (2018) and Burkhart & Mocz (2019) papers introduce a forward model — in progenax we call it BM19 — that turns four cloud-scale parameters into a predicted dense-gas SFR:

with $\mathcal{M}$ the sonic Mach number, $b$ the turbulence forcing parameter, $\alpha$ the density-PDF tail slope, and $\eta_{\mathrm{survive}}$ the post-feedback survival fraction. The output is a tuple of cloud-state variables progenax stores in the BM19Result PyTree.

This chapter is the forward chain end-to-end: each step’s physics, the parameter mapping, and the connection to observed SFR.

The forward chain¶

The BM19 forward model has six numbered steps, each implementing a single physical idea:

Each step uses one piece of the framework:

• Step 1 uses Density PDFs and the freefall-density factor, Equation.

• Step 2 uses the closed-form transition density Equation.

• Step 3 uses the lognormal+power-law PDF and the FDF kernel (Density PDFs and the freefall-density factor).

• Step 4 applies a feedback survival factor $\eta_{\mathrm{survive}} \in [0, 1]$ — the fraction of dense gas that is not disrupted by early stellar feedback before it forms stars.

• Step 5 applies the Kritsuk et al. (2011) α↔p mapping ($p = 3/\alpha$), derived once in The α↔p mapping (stated once, used everywhere).

• Step 6 evaluates the PP20 magnification factor (The magnification factor ζ — three ways to compute it).

The four input parameters¶

The Burkhart & Mocz (2019) α window has a clear physical interpretation:

• $\alpha = \alpha_0 = 3.0$ — turbulence-dominated, no significant collapse. $p = 1.0$, marginally collapsing radial profile.

• $\alpha = \alpha_{\mathrm{sat}} = 1.5$ — gravity-saturated, fully collapsing. $p = 2.0$, singular isothermal radial profile.

Real molecular clouds occupy the middle of this range, $\alpha \sim 2.0$ corresponding to $p \sim 1.5$ and $\zeta \sim 1.4$.

Connecting to observed SFR¶

The cloud-integrated SFR predicted by BM19 is

Three quantities at the right combine to produce the canonical “$\mathrm{SFR}/M_{\mathrm{dense}}$” diagnostic that observers measure:

with $M_{\mathrm{dense}} = f_{\mathrm{sub}}\,M_{\mathrm{cloud}}$ and $t_{\mathrm{ff,dense}}$ the free-fall time of the dense gas (not the mean cloud density). For typical Galactic cloud parameters ($\zeta \sim 1.4$, $\varepsilon_{\mathrm{ff,int}} \sim 0.01$, $t_{\mathrm{ff,dense}} \sim 0.3$ Myr), $\mathrm{SFR}/M_{\mathrm{dense}} \sim 5 \times 10^{-2}$ Myr$^{-1}$ — consistent with observed dense-gas SFEs in nearby molecular clouds.

Implementation in gravoturb¶

The clean-room package exposes the forward chain as small, individually testable functions rather than one monolithic bm19_pipeline orchestrator (that pattern, and its BM19Result PyTree, were intentionally dropped). Compose them directly — each is @jax.jit-compatible and differentiable:

import jax.numpy as jnp
from gravoturb.theory.density_pdf import (
    sigma_s_squared, transition_density, dense_mass_fraction, pdf_slope_to_radial,
)
from gravoturb.theory.dense_gas_sfr import magnification_factor
from progenax.cluster.turbulence import spectral_slope_from_mach

mach, b, alpha = 10.0, 0.4, 2.0
sigma_s2 = sigma_s_squared(mach, b)             # ln(1+b²ℳ²)        ≈ 2.833
s_t      = transition_density(alpha, sigma_s2)  # (α−½)σ_s² (Eq. 2)  ≈ 4.250
f_dense  = dense_mass_fraction(mach, b, alpha)    # mass-wtd dense frac ≈ 0.0568
p        = pdf_slope_to_radial(alpha)           # 3/α                = 1.5
zeta     = magnification_factor(p)              # PP20 ζ(p)          = √2 ≈ 1.414
beta     = spectral_slope_from_mach(mach)       # Kim&Ryu density slope ≈ 2.585

print(f"σ_s={jnp.sqrt(sigma_s2):.3f}  s_t={s_t:.3f}  f_dense={f_dense:.4f}  "
      f"p={p:.1f}  ζ={zeta:.3f}  β={beta:.3f}")

Note β ≈ 2.585 here, not the ~4 (Burgers) value an earlier version of this page printed: spectral_slope_from_mach returns the density power-spectrum slope, which flattens with Mach per Kim & Ryu (2005) (see Density PDFs and the freefall-density factor and the per-paper note). To turn these scalars into a 3D realization and read back the realized dense fraction, use gravoturb.realization.pipeline.build_turbulent_field (the AC6 cornerstone).

Inference: what BM19 enables¶

The differentiability of Equation end-to-end means BM19 can sit inside an HMC posterior over cloud parameters. Given observational constraints on:

• $\sigma_s$ from column-density PDFs (e.g. Kainulainen et al. (2014) 16-cloud sample).

• SFR from infrared luminosities or YSO counts.

• Mean cloud density from extinction maps or molecular-line surveys.

Because the forward functions above are differentiable end-to-end, NUTS can in principle recover posteriors on $(\mathcal{M}, b, \alpha)$ — the cloud’s turbulent state — from observed substructure. The clean-room package ships a differentiable physics-direct inference layer (gravoturb.inference) that does exactly this, predicting summary statistics analytically and differentiating those rather than the stochastic simulator. That layer — the 3-D recovery of $\alpha$ via a peaks-over-threshold tail block and the projected $\beta$ estimator — is developed in full in Differentiable inference — natal cloud parameters from cluster substructure.

The 2026-04-28 PP20 fix and BM19¶

The transcription bug fix documented in the magnification-factor historical note affects step 6 of the BM19 chain — the magnification_factor(p) call. Pre-fix, the buggy form zeta = (3 - p) / (2.6 - 2 * p) ** 1.5 produced silently wrong ζ values for typical $p \in [1, 2]$:

The buggy denominator $(2.6 - 2p)$ goes negative for $p > 1.3$, so raising it to the $3/2$ power returned complex/NaN values there — the spurious “$p = 1.3$ domain limit” earlier docstrings rationalised. The shipping magnification_factor has no such pole below $p = 2$ and applies no clip (the magnification-factor chapter shows the exact source); the only true singularity is the physical one at $p = 2$, regularised by the cored ζ. All committed BM19 outputs from before 2026-04-28 carry these errors; see PP20 ζ(p) transcription fix for the full history and which validation artefacts need regeneration.

Domain of validity¶

1. Burkhart & Mocz (2019) α window assumed. Outside $\alpha \in [1.5, 3.0]$, the framework’s mapping to PP20 ζ becomes uncertain. progenax allows external $\alpha$ values but warns at construction.

2. Single-cloud assumption. BM19 describes one cloud at one moment in time. Galactic-disc SFR predictions need an integration over the cloud-mass function — not currently included.

3. No metallicity dependence. Moe et al. (2019) finds binary fractions are metallicity-dependent; cloud-PDF parameters likely are too. progenax does not currently include this.

4. Static feedback. $\eta_{\mathrm{survive}}$ is a single scalar, not a time-resolved feedback model. For evolved-cluster studies where feedback acts on a timescale comparable to $t_{\mathrm{ff}}$, a time-resolved framework is needed.

Implementation, validation & references¶

• In code: the forward chain is composed from src/experimental/gravoturb/theory/density_pdf.py (sigma_s_squared, transition_density, dense_mass_fraction, pdf_slope_to_radial) and src/experimental/gravoturb/theory/dense_gas_sfr.py (magnification_factor), with spectral_slope_from_mach in src/progenax/cluster/turbulence.py. This experimental subsystem is repo-only with no generated website API page; the module reference is the package source and its VALIDATION_SUMMARY.md.

• Validated in: gravoturbulent PP20; the PP20 transcription fix is recorded in PP20 ζ(p) transcription fix.

• Primary sources: the framework is Burkhart (2018) (Part I, the “BM19” name) extended by Burkhart & Mocz (2019) (Part II); the α↔p correspondence is Kritsuk et al. (2011); the ζ BM19 consumes is Parmentier & Pasquali (2020); the Federrath & Klessen (2012) density-PDF parameterisation underpins the whole framework; the Kainulainen et al. (2014) anchor at $p \approx 5/3$ is the strongest local-cloud constraint. Full notes in the bibliography.