Parmentier & Pasquali (2020)

The big idea¶

Two clouds with the same dense-gas mass $M_\mathrm{dg}$ can form stars at different rates if their density gradients differ: a centrally-concentrated clump has more of its mass at high density, where the local free-fall time $\tau_\mathrm{ff}\propto\rho^{-1/2}$ is short, so it forms stars faster than a uniform (“top-hat”) clump of the same mass. PP20 capture this geometric boost with a single magnification factor $\zeta$, so that

$\zeta=1$ is the top-hat lower limit; $\zeta>1$ for any centrally-concentrated profile.

Physical & observational context (Sections 1–3)¶

The dense-gas star-formation law — $N_\mathrm{YSO}\propto M_\mathrm{dg}$, equivalently $\mathrm{SFR}\propto M_\mathrm{dg}$ — is observationally near-linear (Lada et al. 2010 find $N_\mathrm{YSO}=0.18\,M_\mathrm{dg}$) but carries real scatter. PP20’s thesis is that the scatter is not purely random: it partly tracks the cloud’s internal density gradient. Steeper gradients (larger $p$) push more mass into the short-free-fall-time inner region, raising $\mathrm{SFR}/M_\mathrm{dg}$. The two key inputs PP20 separate (their Eq. 2) are the intrinsic efficiency per free-fall time $\epsilon_\mathrm{ff,int}$ (a true SF-physics number) and the purely geometric boost $\zeta$ — so that measuring a cloud’s gradient unlocks its $\zeta$ and lets one read $\epsilon_\mathrm{ff,int}$ cleanly from the data.

The radial slope $p$ is inferred observationally from the cloud $\rho$-pdf or projected $\Sigma$-pdf: for a power-law tail of index $n$ in the $\Sigma$-pdf, $p = 1 + 2/n$ (Kritsuk et al. 2011); $p\equiv\kappa$ in the Kainulainen et al. (2014) notation. The Kainulainen sample has a mean $p \approx 1.67$ (range $1<p<2.2$), which is the canonical observational anchor: $\zeta(1.67)\approx1.79$. PP20 also apply the framework to the Central Molecular Zone (CMZ), whose clouds sit ~10× below the nearby-cloud locus in $(p,\,\mathrm{SFR}/M_\mathrm{dg})$.

Core equations¶

Why $\zeta$ has this form. The dense-gas SFR is the mass-weighted free-fall rate, $\propto\langle\rho^{1/2}\rangle_\mathrm{mass}$, while the “naive” rate uses the mean density, $\propto\langle\rho\rangle^{1/2}$. Hence

Closed form for a power-law sphere $\rho\propto r^{-p}$ (Eq. 6, in Eq. 9). Performing the integrals over the sphere gives

PP20 print the constant as the rounded 2.6; the exact value is $3^{3/2}/2 = 2.598$, fixed by the physical top-hat limit $\zeta(0)=1$. The factor $(2-p)$ in the denominator means $\zeta$ diverges only as $p\to2$ — there is no pole at $p=1.3$ (a previously caught transcription fabrication). Spot values:

Forbidden regions (their Fig. 1). Equation (6) is the upper limit on $\zeta$ (pure power law), and $\zeta=1$ is the lower limit (top-hat); real cored profiles lie between. For $p\ge2$ a pure power law would drive the central density — and the SFR — to infinity, so PP20 add a flat inner core: a cored profile $\rho(r)=\rho_c[1+(r/r_c)^2]^{-p/2}$ has a finite $\zeta$ even for $p\ge2$, obtained by numerically integrating the ratio above over $r/R\in[0,1]$. The core damps the boost: a relative core size $r_c/r_\mathrm{clump}>0.1$ already reduces $\zeta$ significantly (their Fig. 1), because it shrinks the centre-to-edge density contrast.

Link to BM19. The PDF slope $\alpha$ (Burkhart & Mocz) and the radial slope $p$ are the same quantity: $p = \kappa = 3/\alpha$. So $\alpha\in[1.5,3]\Leftrightarrow p\in[1,2]$.

Use in progenax¶

• The magnification factor ζ — three ways to compute it — full $\zeta(p)$ derivation, equivalence proof, spot values, and the generalisation to cored and direct-3D profiles.

• experimental/gravoturb/theory/dense_gas_sfr.py — magnification_factor (analytic), magnification_factor_with_core (trapezoid), zeta_from_field (field estimator).

Validation: anchors $\zeta(0)=1,\ \zeta(1)=1.089,\ \zeta(1.5)=\sqrt2,\ \zeta(1.67)=1.79$ to $<0.1\%$ (AC3); the direct field estimator matches the analytic $\zeta(p)$ to $\sim3\%$ on a sampled power law (AC4).

Notes¶

• $\zeta$ is a Part-III / SFR-interpretation quantity: it is not needed to generate the gravoturbulent ICs, only to interpret the dense-gas SFR afterwards.

• The 2026-04-28 transcription fix at PP20 ζ(p) transcription fix removed the spurious $p=1.3$ pole; the clean-room pp20.py re-derives the exact $3^{3/2}/2$ constant from first principles.