Density PDFs and the freefall-density factor

A turbulent molecular cloud has a distribution of densities, not a single density. The shape of that distribution — the volume-density PDF $p_V(\rho)$ — is the foundational object that every subsequent chapter in the gravoturbulence section consumes. This chapter develops the two halves of the framework and then joins them:

1. The density PDF — the Federrath & Klessen (2012) lognormal core (set by turbulence) joined to a high-density power-law tail (set by self-gravity-driven collapse).

2. The freefall-density factor (FDF) — the kernel $\rho/t_{\mathrm{ff}}(\rho) \propto \rho^{3/2}$ that weights each density by its star-forming efficiency.

3. The cloud-integrated SFR — the convolution of the two, which yields the dense-mass fraction and the magnification factor that BM19 dense-gas SFR framework and The magnification factor ζ — three ways to compute it consume.

The chapter ends with the single, canonical α↔p mapping ($p = 3/\alpha$) that connects the PDF-tail slope to the radial-profile slope — every downstream page refers back to this statement.

Part 1 — The density PDF¶

The lognormal core¶

In a purely turbulent (non-self-gravitating) supersonic medium, the density PDF is well-described by a lognormal:

where $s \equiv \ln(\rho/\langle\rho\rangle)$ is the log-density, $\langle s \rangle = -\sigma_s^2/2$ (so that $\langle\rho\rangle$ is the mean), and $\sigma_s^2$ is the log-density variance. The lognormal form follows from the central limit theorem applied to the cumulative product of compressions and rarefactions a fluid element experiences in a turbulent cascade.

The variance scales with Mach number as

with $b$ the turbulence-driving parameter ($b \in [1/3, 1]$) and $\mathcal{M} = v_{\mathrm{turb}} / c_s$ the sonic Mach number. The forcing parameter has well-defined physical limits:

For a typical Mach $\mathcal{M} = 10$ molecular cloud and $b = 0.4$, $\sigma_s^2 = \ln[1 + 16] \approx 2.83$, so $\sigma_s \approx 1.7$. The lognormal therefore spans about 4 e-folds in $s$ at the $\pm 2\sigma$ level — orders of magnitude in $\rho$.

The power-law tail¶

When self-gravity becomes important — at densities high enough that the local freefall time is shorter than the turbulent crossing time — the density PDF develops a power-law tail at high $s$:

where $\alpha$ is the tail slope and $s_t$ is the transition density where the PDF crosses from lognormal to power-law. Physically, the power-law tail represents self-gravitating gas in approximate isothermal collapse — the Kritsuk et al. (2011) derivation maps $\rho \propto r^{-p}$ to a power-law in $s$ with slope $\alpha = 3/p$ (the α↔p mapping developed at the end of this chapter).

The transition density $s_t$¶

The matching point $s_t$ between lognormal and power-law is constrained by continuity of $p_V(s)$ Burkhart, 2018:

This is a closed-form expression: given $\sigma_s^2$ (set by Mach + forcing) and $\alpha$ (a free parameter), $s_t$ is determined. gravoturb’s transition_density(alpha, sigma_s_sq) returns this value directly; it is differentiable in both arguments. (For $\alpha = 3/2$ it reduces to $s_t = \sigma_s^2$, BM19 Eq. 16.)

For typical molecular-cloud parameters ($\sigma_s^2 \approx 2.83$, $\alpha \approx 2$), $s_t = (\alpha - \tfrac{1}{2})\sigma_s^2 \approx 4.25$, meaning the lognormal-to-power-law transition occurs around $\rho/\langle\rho\rangle \approx e^{4.25} \approx 70$. This matches the Kainulainen et al. (2014) observational dense-gas threshold $s_{\mathrm{th}} \approx 4.2$ (as adopted by Parmentier & Pasquali (2020)).

Evaluating the PDF in gravoturb¶

import jax.numpy as jnp
from gravoturb.theory.density_pdf import sigma_s_squared, transition_density
from gravoturb.theory.density_cdf import log_density_pdf

mach = 10.0          # Sonic Mach number
b = 0.4              # Forcing parameter
alpha = 2.0          # Power-law tail slope

sigma_s_sq = sigma_s_squared(mach, b)            # ≈ 2.833
sigma_s = jnp.sqrt(sigma_s_sq)
s_t = transition_density(alpha, sigma_s_sq)      # ≈ 4.250   (args: alpha, σ_s²)

# Evaluate the full PDF (it takes mach, b, alpha directly — σ_s² and s_t are internal)
s_grid = jnp.linspace(-5, 10, 200)
pdf_values = log_density_pdf(s_grid, mach, b, alpha)

log_density_pdf evaluates the lognormal piece for $s < s_t$ and the power-law piece for $s \ge s_t$, with continuous matching at $s_t$ enforced by the closed-form Equation. The function is JIT-compatible and differentiable in $\mathcal{M}$, $b$, and $\alpha$ — useful for inferring all three from observed cloud properties.

Part 2 — The freefall-density factor¶

The freefall-density factor (FDF) is the kernel that converts the density-PDF picture into a star formation rate. Its form follows from elementary considerations: the local star formation rate per unit volume scales as

with $t_{\mathrm{ff}}$ the local free-fall time

Combining Equation and Equation:

This is the FDF kernel. High-density gas contributes disproportionately to the cloud-integrated SFR — a $\rho \to 10\rho$ region produces $\sim 32\times$ more SFR per unit volume than the mean-density gas. It is the proximate reason that the magnification factor exists at all: a density-PDF-weighted cloud has higher SFR than a uniform-density cloud of the same mass.

Why $\rho^{3/2}$ and not something else¶

Three alternative scalings are sometimes proposed:

The $\rho^{3/2}$ kernel is appropriate when gravitational free-fall is the rate-limiting step for local star formation. This applies to the dense cores ($\rho \gtrsim 10^4\,\mathrm{cm}^{-3}$) where most stars actually form. The $\rho / t_{\mathrm{cross}}$ kernel would be appropriate if turbulent feedback regulated SFR on the crossing timescale rather than the free-fall — which is the case in some massive-star-feedback-regulated regimes but not in the dense-core regime relevant to dense-gas SFR observations.

The Federrath & Klessen (2012) and Burkhart (2018) frameworks both adopt the $\rho^{3/2}$ kernel; gravoturb inherits this convention.

Part 3 — Combining the PDF and the FDF: the cloud-integrated SFR¶

The density PDF $p_V(\rho)$ describes what density structure a cloud has. The FDF describes how each density contributes to star formation. The cloud-integrated SFR is the convolution of the two.

The full SFR formula¶

For a cloud with volume-density PDF $p_V(\rho)$, mean density $\langle\rho\rangle$, and total mass $M$, the cloud-integrated SFR is

with $\langle t_{\mathrm{ff}}\rangle = \sqrt{3\pi/(32\,G\,\langle\rho\rangle)}$ the mean-density free-fall time and $\varepsilon_{\mathrm{ff,int}} \sim 0.01$ the intrinsic star-formation efficiency per free-fall time — the fraction of a free-fall time’s mass that is actually converted to stars. Observations constrain $\varepsilon_{\mathrm{ff,int}} \sim 0.01$ for typical Galactic dense gas Burkhart, 2018. The integrand is the dimensionless FDF kernel $(\rho/\langle\rho\rangle)^{3/2}$ weighted by the volume PDF.

The lower limit $\rho_t$ — the transition density at which the PDF crosses from lognormal to power-law — encodes the physical assumption that only self-gravitating gas forms stars. Below $\rho_t$, turbulent fluctuations compress and re-expand the gas without forming stars; above $\rho_t$, gravitational collapse dominates and the local SFR follows the FDF kernel. The lognormal core represents turbulent fluctuations that compress and re-expand without forming stars.

The self-gravitating fraction¶

A useful auxiliary quantity is the self-gravitating (dense) fraction $f_{\mathrm{dense}}$, the mass fraction of the cloud above the transition density:

This is not the SFR-weighted fraction; it is the simple mass fraction (kernel $\rho/\langle\rho\rangle = e^s$, exponent 1). The SFR uses the FDF-weighted version with $(\rho/\langle\rho\rangle)^{3/2}$ (exponent $3/2$) in the integrand. The two differ only in the exponent of the dimensionless density kernel — both integrate the volume PDF, both are dimensionless. gravoturb computes both:

from gravoturb.theory.density_pdf import sigma_s_squared, transition_density, dense_mass_fraction
from gravoturb.theory.dense_gas_sfr import magnification_factor

mach, b, alpha = 10.0, 0.4, 2.0
sigma_s_sq = sigma_s_squared(mach, b)        # lognormal variance        ≈ 2.833
s_t = transition_density(alpha, sigma_s_sq)  # transition log-density    ≈ 4.250  (args: alpha, σ_s²)

# f_dense — mass fraction in the dense power-law tail
f_dense = dense_mass_fraction(mach, b, alpha)  # ≈ 0.057

# geometric magnification ζ for the implied radial slope p = 3/α
zeta = magnification_factor(3.0 / alpha)     # ζ(1.5) = √2 ≈ 1.414
print(f"f_dense = {f_dense:.3f}, ζ = {zeta:.3f}")

dense_mass_fraction evaluates Equation for the Burkhart (2018) framework by splitting the integral into a lognormal body (an erf term) and a power-law tail ($M_{\mathrm{PL}} = C\,e^{(1-\alpha)s_t}/(\alpha-1)$, valid for $\alpha > 1$) and returning the ratio $M_{\mathrm{PL}}/(M_{\mathrm{LN}} + M_{\mathrm{PL}})$. For typical Galactic-cloud parameters ($\mathcal{M} = 10$, $b = 0.4$, $\alpha = 2$), $f_{\mathrm{dense}} \sim 0.05$–0.15 — a few percent of cloud mass is in the dense star-forming tail at any instant.

Connecting to the magnification factor ζ¶

For a spatially uniform density (no PDF spread), the FDF integral in Equation reduces trivially to the “top-hat” reference and the cloud SFR is that of a uniform cloud. For a non-uniform cloud, the integral is larger than the top-hat reference — by exactly the magnification factor

This is the same dimensionless ratio that The magnification factor ζ — three ways to compute it evaluates geometrically for a power-law radial profile $\rho(r) \propto r^{-p}$; this chapter’s PDF-based formulation is its density-space dual. They produce the same number for the same physical cloud.

Which formulation when¶

The two formulations apply to different observational situations:

For inference: the radial-profile formulation is what most observational papers report (clouds are characterised by $r_{\mathrm{eff}}$ and $\rho(r)$). The PDF-based formulation is what cloud simulations output. The 3D formulation is what cosmological simulations output when probed at the cloud scale.

progenax’s BM19 dense-gas SFR framework forward chain uses the PDF-based formulation because the Burkhart (2018)Burkhart & Mocz (2019) framework is parameterised in PDF space.

The α↔p mapping (stated once, used everywhere)¶

The PDF tail-slope $\alpha$ and the radial-profile slope $p$ are not independent. Under spherical symmetry and a power-law correspondence between volume and density Kritsuk et al., 2011Federrath & Klessen, 2012, the canonical relation is

This is the single statement the rest of the gravoturbulence section refers back to: BM19 dense-gas SFR framework applies it in step 5 of the forward chain, and The magnification factor ζ — three ways to compute it consumes the resulting $p$ to compute $\zeta(p)$. For the Burkhart & Mocz (2019) canonical α window $[\alpha_{\mathrm{sat}}, \alpha_0] = [1.5, 3.0]$, the corresponding $p$ window is $[1.0, 2.0]$ — from “marginally collapsing” to “singular isothermal” radial profiles. tests/experimental/unit/test_bm19.py verifies the $p = 3/\alpha$ mapping (pdf_slope_to_radial).

The mapping is what lets the BM19 forward chain hand off to the magnification-factor calculation seamlessly: BM19 infers $\alpha$ from cloud observations, Equation converts it to a radial-profile slope, and $\zeta(p)$ gives the geometric SFR boost.

Domain of validity¶

1. Supersonic turbulence required — at low Mach the lognormal variance shrinks below $\sim 1$ and the lognormal approximation breaks down. gravoturb’s parameterisation works for $\mathcal{M} \gtrsim 3$.

2. Self-gravity required for the power-law tail — at very low density (diffuse warm neutral medium) there is no gravitating tail and the PDF is purely lognormal. The Burkhart (2018) framework assumes a power-law tail exists.

3. Free-fall regime for the kernel — the $\rho^{3/2}$ kernel assumes that local gravitational collapse on a free-fall timescale dominates over turbulent or thermal stabilisation. At very low density other timescales matter; at very high density (proto-stellar cores) hydrostatic- equilibrium effects matter. The isothermal $t_{\mathrm{ff}}$ of Equation ignores thermal-pressure support; the standard treatment lumps this into $\varepsilon_{\mathrm{ff,int}}$.

4. No magnetic-field or feedback support — magnetic fields slow collapse via ambipolar diffusion (Federrath & Klessen (2012) give a magnetic correction; gravoturb does not include it), and the kernel assumes star formation does not back-react on the gas. For clouds older than a few $t_{\mathrm{ff}}$, feedback becomes important and the kernel under-predicts SFR.

5. Spherical/cylindrical symmetry assumed in the α↔p mapping; for highly filamentary or sheet-like clouds the mapping is approximate.

6. Single-cloud, static description. The PDF describes one cloud at a time-averaged instant; multi-cloud line-of-sight superposition requires a separate treatment, and the PDF parameters themselves evolve on the cloud’s free-fall timescale.

7. 3-D field realisation. When a sampled 3-D field is built from the PDF (gravoturb.realization.pipeline.build_turbulent_field), the one-point statistics are imposed by a rank / empirical-CDF copula (Gaussian anamorphosis), which reproduces the dense-tail mass fraction $f_{\mathrm{dense}}$ at any power-spectrum slope $\beta$. For a very extreme threshold $s_t$, where the tail count-probability $1 - F_V(s_t) \lesssim 1/N_{\mathrm{grid}}^3$, the tail is genuinely unresolved at that grid and a warning is emitted (see the α wall in the inference chapter).

Implementation, validation & references¶

• In code: src/experimental/gravoturb/theory/density_pdf.py (sigma_s_squared, transition_density, dense_mass_fraction, pdf_slope_to_radial) and src/experimental/gravoturb/theory/density_cdf.py (log_density_pdf); 3-D field realisation is src/experimental/gravoturb/realization/pipeline.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 $p = 3/\alpha$ mapping is pinned by tests/experimental/unit/test_bm19.py.

• Primary sources: the lognormal core is Federrath & Klessen (2012) (the $\sigma_s^2 = \ln(1 + b^2 \mathcal{M}^2)$ derivation); the power-law tail and the analytic α↔p mapping are Kritsuk et al. (2011); the transition-density matching and PDF+FDF combination are Burkhart (2018); Tan et al. (2006) is the earlier single-mean-density framework; observational verification of the form and dense-gas threshold is Kainulainen et al. (2014). Full notes in the bibliography. The full forward chain is BM19 dense-gas SFR framework.