Federrath & Klessen (2012)

The big idea¶

FK12 is the unifying comparison of analytic star-formation-rate (SFR) theories. It derives and compares six models for the dimensionless SFR per free-fall time $\mathrm{SFR}_\mathrm{ff}$ — the Krumholz & McKee (KM), Padoan & Nordlund (PN), and Hennebelle & Chabrier (HC) theories, plus multi-freefall versions of each — all as a single integral over the lognormal density PDF. It then tests all six against MHD simulations ($\mathcal{M}=3$–50, $\mathcal{M}_A=1$–$\infty$, solenoidal/mixed/compressive forcing). The headline: the SFR depends on four parameters and the multi-freefall KM and PN models fit best (to within a factor of 2).

The four controlling parameters (their §1)¶

1. Virial parameter $\alpha_\mathrm{vir} = 2E_\mathrm{kin}/|E_\mathrm{grav}|$.

2. Sonic Mach number $\mathcal{M} = \sigma_v/c_s$.

3. Turbulent forcing parameter $b$ — fraction of energy in compressive modes: $b\approx1/3$ solenoidal (divergence-free), $b\approx0.4$ natural mixture, $b\approx1$ compressive (curl-free).

4. Plasma $\beta = 2\mathcal{M}_A^2/\mathcal{M}^2 = P_\mathrm{th}/P_\mathrm{mag}$ (thermal-to-magnetic pressure; $\mathcal{M}_A$ the Alfvén Mach number).

Comparing forcings, the SFR is >10× higher for compressive than solenoidal forcing at fixed $\mathcal{M}$; magnetic fields reduce the SFR by a factor of ~2.

The density PDF and σ_s² (their §2.1–2.2)¶

The log-density $s=\ln(\rho/\rho_0)$ has a lognormal PDF (Eq. 1) with mean fixed by mass conservation (Eq. 3), $s_0 = -\tfrac12\sigma_s^2$. The width depends on forcing, Mach and magnetization. For the intermediate field scaling $B\propto\rho^{1/2}$ (Molina+2012),

which in the hydrodynamic limit ($\beta\to\infty$, no field) reduces to the relation used throughout gravoturb:

(equivalently Eq. 5, $\sigma_s^2=\ln[1+b^2\mathcal{M}^2\,2\mathcal{M}_A^2/(\mathcal{M}^2+2\mathcal{M}_A^2)]$). This is the same HD relation as Federrath+2010 Eq. 19 and BM19 Eq. 1.

The SFR-per-freefall framework (their §2.3) and the six models¶

The SFR per free-fall time is the mass above a critical density, weighted by the local free-fall rate (the multi-freefall insight — gas at different $\rho$ collapses at different rates):

The six models (Table 1) differ only in the critical density $s_\mathrm{crit}$ (the lower integration limit) and whether the $t_\mathrm{ff}$ factor is kept inside the integral (multi-ff) or set to 1 (single-ff):

Two facts that matter downstream:

• The PN critical density carries the prefactor $0.067\,\theta^{-2}$ — the same form gravoturb.theory.collapse_threshold implements (HD: 0.547 at $\theta=0.35$); see Padoan & Nordlund (2011).

• The KM/HC critical density carries the $(\pi^2/5)\phi_x^2$ form — distinct from PN’s, which is why the PN11 note flags them as different prefactor conventions.

Use in progenax¶

• Density PDFs and the freefall-density factor — the lognormal+power-law PDF (FK12 §2.1) and the $\rho^{3/2}$ free-fall kernel (the $t_\mathrm{ff}(\rho)^{-1}\propto\rho^{1/2}$ weight in Eq. 7).

• Burkhart & Mocz (2019) — BM19 simplifies this framework by tying $s_\mathrm{crit}$ to the PDF transition density $s_t$, removing a free critical density.

• $b\in[1/3,1]$ grounds cluster.turbulence.b_from_environment and the FK10 forcing parameter.

Placement-PMF corollary (gravoturb Phase 1, verified vs the PDF 2026-07-16)¶

The multi-freefall integrand of Eq. 7 — [t_ff(ρ₀)/t_ff(ρ)]·(ρ/ρ₀) = (ρ/ρ₀)^{3/2} via Eq. 8 — is the relative star-formation weight per cell. Normalizing it into a placement PMF cancels the ε/φ_t efficiency prefactors exactly, so where stars form needs no efficiency knob (only how many does, and the IC generator takes N⋆ as input). gravoturb’s placement='multi_freefall' uses p_⋆ ∝ w(s_turb)·e^{(3/2)s_total} with the eligibility gate w on the BM19 transition s_t — the s_t-for-s_crit substitution described under “Use in progenax” (BM19’s derived transition replaces FK12’s assumed critical density; FK12 itself does not license s_t, BM19 does). The derived tail_star_fraction (Σ_{s>s_t} p_⋆ under the gated PMF) then replaces the former free f_sub knob, with the smooth collapse_eligible_fraction (the eligible share of the ungated ρ^{3/2} measure — a different, smaller number) as the analytic hook.

Notes¶

• The HD gravoturb path drops magnetic fields: it uses $\sigma_s^2=\ln(1+b^2\mathcal{M}^2)$, i.e. FK12 Eq. 4 with $\beta\to\infty$. Magnetization ($\beta<\infty$) narrows the PDF and lowers the SFR by ~2× — not modelled here.

• Best-fit efficiencies from the simulations: SFE $=1$–$10\%$, local $\epsilon\approx0.3$–0.7 (best $\sim0.5$); the multi-ff KM and PN models match to a factor of 2 over two orders of magnitude in SFR.

• FK12 is the conceptual parent of BM19: BM19 keeps the lognormal+power-law PDF and the $\rho^{3/2}$ kernel but replaces the assumed critical density with the derived transition density $s_t$.