Burkhart (2018)

The big idea¶

Earlier analytic star-formation-rate (SFR) models — Krumholz & McKee (2005), Padoan & Nordlund (2011), Hennebelle & Chabrier (2011), Federrath & Klessen (2012) — integrate the freefall-weighted density over a purely lognormal (LN) density PDF set by supersonic turbulence. Burkhart (2018) extends this to the form actually seen in simulations and column-density observations of giant molecular clouds: a piecewise LN + power-law (PL) PDF, where self-gravity carves a high-density power-law tail onto the turbulent lognormal body. The central physical narrative: gas becomes gravitationally unstable past a critical density $\rho_{\rm crit}$ and forms the PL tail; as the cloud collapses, the transition density $\rho_t$ between LN and PL moves to lower density while the PL slope $\alpha$ becomes increasingly shallow, and the SFR accelerates beyond the LN-only prediction. This explains why star-formation efficiency per free-fall time increases with shallower PL slopes, and why depletion times vary across local and extragalactic clouds — without invoking extreme variations in turbulence.

This is the SFR Part I of the framework progenax adopts; the companion Burkhart & Mocz (2019) (Part II, BM19) makes the construction self-consistent by deriving the transition density $s_t$ from the condition that the Jeans length equals the sonic length, and gives the closed-form self-gravitating mass fraction $f_{\rm dense}$.

Core results¶

The turbulent lognormal width (Eq. 5). The LN body width is set by the sonic Mach number and the forcing parameter,

with $s\equiv\ln(\rho/\rho_0)$ (Eq. 3) and the mass-conserving mean $s_0=-\tfrac12\sigma_s^2$ (Eq. 4).

The piecewise LN + PL PDF (Eq. 18). The density PDF is a lognormal body joined at $s_t$ to a power-law tail,

normalised by $N$ (Eq. 19, a closed form in $C$, $\alpha$, $s_t$, $\sigma_s$). Requiring $p_{\rm LN+PL}$ to be continuous and differentiable at $s_t$ fixes the amplitude $C$ and the transition $s_t$ analytically. This is exactly the log_density_pdf implemented in gravoturb.

The α sign convention (Eq. 6). Burkhart writes the tail as $p_{\rm PL}(s)=C\,e^{-\alpha s}$ for $s>s_t$ and notes explicitly that “in our definition of the PL slope $\alpha$ is positive since the minus sign appears in the exponent separately.” progenax uses this same convention throughout (log_density_pdf, the peaks-over-threshold tail block), so $\alpha$ is positive and a steeper tail means a larger $\alpha$.

The SFR integral (Eqs. 7–8). The SFR per free-fall time is the freefall-weighted integral over the PDF above the critical density,

split into an LN integral from $\rho_{\rm crit}$ to $\rho_t$ plus a PL integral from $\rho_t$ to the maximum density. The paper reviews how the critical density $\rho_{\rm crit}$ differs between the KM05, PN11, and Hennebelle–Chabrier models (Eqs. 9–17).

Use in progenax¶

• The piecewise PDF (2) is log_density_pdf; (1) is sigma_s_squared. These 1-point scalars are the inputs the differentiable-inference layer (Differentiable inference — natal cloud parameters from cluster substructure) recovers from observed substructure.

• The SFR integral (3) is the forward chain of Density PDFs and the freefall-density factor and BM19 dense-gas SFR framework; the geometric, radial-profile dual is the Parmentier & Pasquali ζ (Parmentier & Pasquali (2020)).

• The positive-α convention (Eq. 6) is the one used by the BM19 tail and the peaks-over-threshold estimator.

Notes¶

• Burkhart (2018) establishes the LN+PL PDF and its SFR; the self-consistent transition density $s_t=(\alpha-\tfrac12)\sigma_s^2$ (Jeans = sonic) and the closed-form $f_{\rm dense}$ used in the code are the Part II results of Burkhart & Mocz (2019) (Burkhart & Mocz (2019)).

• The LN+PL form is motivated by observations: the dense star-forming gas PDF takes a piecewise LN+PL shape in both 3-D density and column density (Kainulainen et al. 2009; Collins et al. 2012; Burkhart et al. 2017; Kainulainen, Federrath & Henning (2014)); some GMCs may be fully power-law.