Burkhart & Mocz (2019)

The big idea¶

In a turbulent, self-gravitating molecular cloud the volume density $\rho$ is not single-valued — it has a probability distribution function (PDF). Where gravity is weak the PDF is a lognormal set by supersonic turbulence; where gravity wins it grows a power-law tail at high density (collapsing regions). BM19 model the PDF as a piecewise lognormal + power law and show that the density where the two pieces join, $s_t$, is not a free parameter but a mathematically motivated critical density for star formation — the post-shock density where the Jeans length equals the sonic length. The mass fraction in the power-law tail is the self-gravitating (dense) gas fraction $f_\mathrm{dense}$, which sets the star-formation efficiency.

Throughout, work in the log-density $s \equiv \ln(\rho/\rho_0)$ ($\rho_0$ = mean density).

Core equations¶

Lognormal width (Eq. 1). Supersonic isothermal turbulence gives a lognormal of variance

with $\mathcal{M}$ the sonic Mach number and $b\in[1/3,1]$ the driving parameter ($1/3$ solenoidal → 1 compressive). Mass conservation fixes the lognormal mean at $s_0 = -\sigma_s^2/2$, so $\int e^{s} p_\mathrm{LN}\,ds = 1$.

Transition density (Eq. 2). The lognormal joins the power-law tail at

where $\alpha$ is the slope of the tail $p_\mathrm{PL}(s)\propto e^{-\alpha s}$. This is derived, not fitted. For the canonical collapsing value $\alpha = 3/2$ it reduces to $s_t = \sigma_s^2$ (Eq. 16).

PDF slope ↔ radial slope. A spherical region $\rho\propto r^{-\kappa}$ produces a density-PDF power law $p(s)\propto e^{-\alpha s}$ with

So $\alpha=3/2 \Leftrightarrow \kappa=2$ (isothermal collapse; Shu 1977) and $\alpha=2 \Leftrightarrow \kappa=3/2$.

Self-gravitating gas fraction (Eqs. 17–20). Defining dense gas as all mass above $s_t$,

Demanding the PDF be continuous at $s_t$ fixes the tail amplitude $C = p_\mathrm{LN}(s_t)\,e^{\alpha s_t}$, and the mass-weighted integrals evaluate to

(BM19 Eq. 19/20 write the identical result multiplied through by $\alpha-1$.) The $1/(\alpha-1)$ makes $f_\mathrm{dense}\to1$ as $\alpha\to1$ — the limit where the whole cloud is self-gravitating — and is the term a numerical implementation must guard.

Pure-lognormal limit. Removing the power law gives $f_\mathrm{dense}^{LN} = \tfrac12\,\mathrm{erfc}\!\big((s_t-\sigma_s^2/2)/(\sqrt2\sigma_s)\big)$; the true $f_\mathrm{dense}$ exceeds it because the shallower tail adds high-density mass.

Behaviour (their Fig. 5). $f_\mathrm{dense}$ decreases with $\mathcal{M}$ (the PDF widens, $s_t$ moves up) and decreases with $\alpha$ (steeper tail → less dense gas). The instantaneous SFE is $\epsilon_\mathrm{inst}=\epsilon_0\,f_\mathrm{dense}$ (Eq. 24).

The central result: only two parameters, no free critical density. Because $s_t$ is derived from $(\sigma_s,\alpha)$ via Eq. 2, the self-gravitating fraction in Eqs. 18–20 is controlled by just those two numbers — BM19 stress there is “no need to invoke a critical density of collapse.” This is what lets the SFE be predicted from observables (cloud Mach number / PDF width and the measured tail slope) rather than a tuned threshold.

$s_t$ is a critical density (Eqs. 9–15). Equating the Jeans length to the post-shock sonic length gives a critical overdensity $\rho_\mathrm{crit}/\rho_0 = \exp(s_\mathrm{crit}) = \tfrac{\pi^2}{15}\,\alpha_\mathrm{vir}\,\mathcal{M}^2$ (their Eq. 11, with $\alpha_\mathrm{vir}=5v_L^2 R/GM$). For virialised clouds ($\alpha_\mathrm{vir}\approx1$) this matches the post-shock density $\rho_\mathrm{ps}/\rho_0=\mathcal{M}^2$ to within a factor of a few, and BM19 show $s_t\approx s_\mathrm{crit}\approx s_\mathrm{ps}$ in the $\alpha\simeq1.5$–2 limit. They validate $s_t$ and $f_\mathrm{dense}$ against AREPO moving-mesh gravo-turbulent simulations ($b=1/3$, $\mathcal{M}=5,10,16$; their Figs. 4–5), finding the dense fraction (and hence SFE) is weakly anti-correlated with Mach number.

Use in progenax¶

• BM19 dense-gas SFR framework — BM19 1-D PDF theory and the $\alpha$ window.

• The magnification factor ζ — three ways to compute it — the $\alpha\leftrightarrow p$ ($\kappa=3/\alpha$) mapping.

• experimental/gravoturb/theory/density_pdf.py — sigma_s_squared, transition_density, dense_mass_fraction, dense_mass_fraction_lognormal, pdf_slope_to_radial.

• experimental/gravoturb/theory/density_cdf.py — the BM19 volume PDF + inverse-CDF that imprints the BM19 marginal on the 3-D FDF field via the rank copula.

Validation: closed-form $f_\mathrm{dense}$ matches direct quadrature of Eq. 18 to $\mathrm{rel}\,10^{-4}$ (AC1); mass conservation $\int e^{s}p_\mathrm{LN}\,ds=1$ to 10^-3 (AC2).

Notes¶

• The companion Paper I (Burkhart 2018) defines the renormalisation constants $N,C$ and the density shift $s_s$ (Eq. 3) for mass-conserving periodic-box simulations.

• $\alpha$ is the PDF slope, not the radial slope ($\kappa=3/\alpha$). The canonical collapsing window is $\alpha\in[1.5,2]$ (saturating toward $\alpha\simeq1.5$).