The magnification factor ζ — three ways to compute it

The magnification factor ζ¶

The magnification factor ζ quantifies how much the cloud-integrated star formation rate (SFR) of a centrally-concentrated cloud exceeds that of a uniform-density (“top-hat”) cloud of the same total mass and outer radius. Parmentier & Pasquali (2020) introduced ζ as a compact way to fold the geometry of a molecular cloud into the dense-gas SFR-mass relation, extending earlier “single-mean-density” frameworks Tan et al., 2006Federrath & Klessen, 2012Burkhart, 2018.

This chapter gives the single integral definition of ζ, then derives and implements the three computation modes that gravoturb exposes — each captures a different physical situation:

• PP20 analytic ζ(p) — exact, closed-form, for a pure power-law profile.

• Cored ζ — numerical, for a profile with a flat inner core (realistic molecular clouds).

• Direct 3D ζ — parameter-free, measured directly from an arbitrary 3D density field.

Why ζ exists, and how it is defined¶

The local star formation rate per unit volume scales as $\dot\rho_\star \propto \rho / t_{\mathrm{ff}}(\rho) \propto \rho^{3/2}$, because the local free-fall time $t_{\mathrm{ff}} \propto \rho^{-1/2}$ (the freefall-density factor). A cloud whose density field is not uniform therefore fragments faster in its high-density regions than its mean density would suggest — the cloud-integrated SFR is biased above the SFR a uniform cloud of the same mass would produce.

ζ is the ratio of these two SFRs, at fixed total mass $M$ and outer radius $R$:

where $V_R$ is the cloud volume, $\langle\rho\rangle = M/V_R$ is the volume-averaged density, and the top-hat reference $\mathrm{SFR}_{\mathrm{TH}} \propto M / t_{\mathrm{ff}}(\langle\rho\rangle) \propto M \sqrt{\langle\rho\rangle}$ is the SFR the same gas mass would produce if redistributed uniformly. By construction $\zeta(\rho = \mathrm{const}) = 1$ — a uniform cloud is its own top-hat reference. For ζ > 1 the cloud is “magnified”; centrally concentrated profiles always give ζ > 1, and the steeper the concentration, the larger ζ.

Mode 1 — PP20 analytic ζ(p) for a power-law profile¶

Consider a sphere of outer radius $R$ filled with a pure power-law profile

The upper bound $p < 2$ is set by the convergence threshold of the SFR integrand $\int r^{2-3p/2}\,\mathrm{d}r$ — physically, $p = 2$ is the singular isothermal sphere where the central density runs away. Five steps evaluate Equation analytically.

Step 1 — total mass.

Step 2 — mean density.

Step 3 — SFR integral.

The denominator $(2-p)$ in this step is the source of the $\zeta \to \infty$ divergence as $p \to 2$.

Step 4 — top-hat reference.

Step 5 — combine. Dividing Step 3 by Step 4 collapses every $\rho_R$ and $R$ factor:

This is the canonical analytic form implemented in gravoturb.theory.dense_gas_sfr.magnification_factor. Parmentier & Pasquali (2020) quote the same result in Eq. 6 as

where “2.6” is a numerical approximation to $3^{3/2}/2 = 2.598\ldots$ Equations Equation and Equation therefore agree to 0.08% across the entire physical $0 \le p < 2$ domain. gravoturb uses the unrounded Equation form so that anchor values are exact, and the exact constant is fixed by the physical top-hat lower limit $\zeta(0) = 1$.

Spot values¶

These anchors are exact analytic values; tests/experimental/unit/test_pp20.py and AC3 lock them (the AC suite prints them — see gravoturb/VALIDATION_SUMMARY.md). The ζ column is computed directly with magnification_factor.

The $p = 1.67$ value is observational gold: Kainulainen et al. (2014) derived $p \approx 5/3$ as the median radial slope inferred from the column-density PDFs of 16 nearby molecular clouds (Pipe, Lupus, Perseus, Aquila, Polaris, …). The corresponding $\zeta \approx 1.79$ is a useful sanity check for any PP20 implementation. ζ rises steeply toward $p = 2$ (from $\approx 4.44$ at $p = 1.9$ to $\approx 8.28$ at $p = 1.95$) — the divergence the cored and direct-3D modes below regularise.

Implementation¶

The shipping function is exactly Equation — there is no clip and no P_MAX. The analytic form is well-behaved over the entire physical $0 \le p < 2$ window; the only singularity is at $p = 2$ (not at $p = 1.3$, despite earlier buggy docstrings — see the Historical note). Where gradients near $p \to 2$ would destabilise an HMC chain, use the cored mode below, which stays finite at all $p$.

import jax.numpy as jnp
from jaxtyping import Array, Float

# Exact PP20 constant: 3^{3/2}/2 = 2.5980762... (PP20 Eq. 6 prints it rounded as 2.6).
_PP20_CONST = 3.0**1.5 / 2.0

def magnification_factor(p: Float[Array, ""]) -> Float[Array, ""]:
    # zeta(p) = (3-p)^{3/2} / [ (3^{3/2}/2) (2-p) ];  valid 0 <= p < 2, diverges as p -> 2.
    return (3.0 - p) ** 1.5 / (_PP20_CONST * (2.0 - p))

The function is JIT-compatible, vectorisable via jax.vmap, and differentiable across the physical domain. See the gravoturb.theory.dense_gas_sfr source for the full signature and Cross-cutting physics tests for the regression suite that anchors every spot value above.

Mode 2 — cored profiles: magnification_factor_with_core¶

The PP20 analytic ζ(p) assumes a pure power-law profile, which predicts infinite density at $r = 0$. Real molecular clouds have flat inner cores — set by thermal pressure or magnetic support — that transition to a power-law envelope outside some core radius $r_c$. gravoturb’s magnification_factor_with_core numerically integrates ζ for the cored profile

This profile is flat at $r \ll r_c$ (with $\rho \to \rho_c$) and power-law at $r \gg r_c$ (with $\rho \propto r^{-p}$), producing a finite central density without sacrificing the power-law outer slope that drives the gravitating-tail SFR. This matters for ζ in two ways:

1. At $p \to 2$ the pure power-law $\zeta$ diverges as $1/(2-p)$. The cored ζ stays finite because the flat inner core regularises the $\int \rho^{3/2}\,\mathrm{d}V$ integral.

2. For inference, the cored form gives stable gradients at all $p$ — including $p \to 2$ where the pure form diverges — and exposes $r_c/R$ as a free parameter.

Substituting Equation into Equation, the result has no closed form for general $(p, r_c/R)$. gravoturb evaluates the integral numerically on a fixed (grad-safe) radial grid and delegates the actual ratio to the direct estimator below — the constant $4\pi R^3$ volume factor cancels in the ratio:

import jax.numpy as jnp
from jaxtyping import Array, Float

def magnification_factor_with_core(
    p: Float[Array, ""],
    r_c_over_R: Float[Array, ""],
    n_nodes: int = 2048,
) -> Float[Array, ""]:
    # Cored rho(r) = rho_c [1 + (r/r_c)^2]^{-p/2}, trapezoid over r/R in (0, 1].
    x = jnp.linspace(1.0 / n_nodes, 1.0, n_nodes)        # r/R in (0,1]
    rho = (1.0 + (x / r_c_over_R) ** 2) ** (-p / 2.0)    # rho / rho_c
    w = x**2                                             # dV ~ r^2 dr (4*pi*R^3 cancels)
    return zeta_from_field(rho, w)

The output is differentiable in both $p$ and $r_c/R$. As $r_c/R \to 0$ the profile approaches a pure power law and ζ → the analytic magnification_factor; as $r_c/R \to \infty$ (a core larger than the cloud) it approaches a top-hat and ζ → 1.

Limiting behaviours¶

tests/experimental/unit/test_pp20.py and the AC4 direct-field check verify ζ convergence to the analytic value at $p \in \{0.5, 1.0, 1.5\}$ as $r_c/R \to 0$.

Mode 3 — direct 3D ζ: zeta_from_field¶

The PP20 and cored modes both parameterise the cloud’s density profile. For inferences using simulation snapshots or detailed observational density maps, that parameterisation throws away most of the information in the data. zeta_from_field instead measures ζ directly from a sampled density field and its volume weights:

This is exactly $\langle\rho^{1/2}\rangle_{\mathrm{mass}}/\langle\rho\rangle^{1/2}$ for cells of volume $w_i$ — the discrete form of Equation. The implementation is the shipping source verbatim:

import jax.numpy as jnp
from jaxtyping import Array, Float

def zeta_from_field(
    rho: Float[Array, " n"], weights: Float[Array, " n"]
) -> Float[Array, ""]:
    # zeta = sum(rho^{3/2} w) * sqrt(sum w) / (sum(rho w))^{3/2}.  rho need not be normalized.
    num = jnp.sum(rho**1.5 * weights) * jnp.sqrt(jnp.sum(weights))
    den = jnp.sum(rho * weights) ** 1.5
    return num / den

rho is the flattened density field (any shape; need not be normalized) and weights the matching per-cell volume element. For a uniform voxel grid the volume element cancels exactly between numerator and denominator, so passing weights = ones_like(rho) gives the correct unitless ζ. This is the estimator magnification_factor_with_core (Mode 2) delegates to, and the right tool for measuring ζ from a 3D field with no parametric assumption.

Soft-mask weights for differentiable tail inference¶

A common use is to weight the field by a soft identification of the gravitating tail rather than its full volume. The natural “tail” definition — “everything above the transition density $\rho_t$” — is a hard step function, whose gradient is zero almost everywhere; inferring $\rho_t$ or any density-PDF parameter through it is impossible with autodiff. Replacing the step with a sigmoid in log-density,

gives a continuous, well-behaved gradient $\partial w/\partial s$ (the sigmoid derivative). Large $\kappa$ approaches the hard threshold; small $\kappa$ gives a smooth transition (a typical $\kappa \sim 10$ gives a transition width $\sim 0.1$ in $s$). This is the technical step that lets the inference layer infer $s_t$ and the PDF parameters from observed dense-gas structure:

import jax
import jax.numpy as jnp
from gravoturb.theory.density_pdf import sigma_s_squared, transition_density
from gravoturb.theory.dense_gas_sfr import zeta_from_field

# BM19 transition density is closed-form in (sigma_s, alpha)
sigma_s_sq = sigma_s_squared(10.0, 0.4)
s_t = transition_density(2.0, sigma_s_sq)            # args: alpha, σ_s²

# soft tail weights via a sigmoid in log-density space
s = jnp.log(rho_grid.ravel() / jnp.mean(rho_grid))
collapse_weights = jax.nn.sigmoid(10.0 * (s - s_t))

# direct ζ over the soft-masked field
zeta_direct = zeta_from_field(rho_grid.ravel(), collapse_weights)

In the full subsystem the density field and soft mask come from gravoturb.realization.pipeline.build_turbulent_field (the AC6 cornerstone); the realized dense fraction is gravoturb.realization.placement.f_tail_actual. The soft-mask reparameterisation of a discrete threshold is the standard differentiable-programming technique; here it makes $\partial\zeta/\partial s_t$ analytic and HMC-compatible (see Differentiable inference — natal cloud parameters from cluster substructure).

Which ζ-mode when¶

For HMC-based inference of cloud parameters, all three are differentiable. The choice depends on the level of cloud parameterisation in the inference target: power-law fit → PP20; cored fit → cored; gridded field → direct 3D.

Historical note — the 2026-04-28 transcription fix¶

Implementation, validation & references¶

• In code: src/experimental/gravoturb/theory/dense_gas_sfr.py (magnification_factor, magnification_factor_with_core, zeta_from_field). 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: physics tests (the ζ spot-value regression suite) and gravoturbulent PP20; the AC3/AC4 acceptance checks lock the anchors.

• Primary sources: ζ follows Parmentier & Pasquali (2020) directly; the integral definition Equation and the α↔p correspondence come from Tan et al. (2006), Federrath & Klessen (2012), and Kritsuk et al. (2011); the Kainulainen et al. (2014) observational anchor provides the $p = 1.67$ check value; the cored profile Equation is a standard King (1966)-style generalisation, and the direct-3D soft-mask measurement is original to gravoturb. Full notes in the bibliography. The Burkhart (2018)Burkhart & Mocz (2019) framework that consumes ζ is BM19 dense-gas SFR framework.