Padoan & Nordlund (2011)

The big idea¶

PN11 derive the critical density for gravitational collapse in supersonic turbulence from first principles, rather than from the turbulent-pressure / sonic-scale argument of Krumholz & McKee (2005). The physical picture: turbulence drives shocks; the post-shock gas is dense and, if a post-shock region’s Bonnor–Ebert mass falls below its actual mass, it collapses. The minimum density at which this happens is the critical density $\rho_\mathrm{crit}$. The star-formation rate then follows from the mass fraction of the turbulent density PDF lying above $\rho_\mathrm{crit}$.

In gravoturb PN11 is kept as a clearly-labelled classical alternative to the BM19 transition density $s_t$ — not the default path.

Core equations¶

Shock jump → characteristic post-shock density. Balancing thermal and ram pressure for a shock of velocity $v_0/2$ (Eq. 3) gives the post-shock density (Eq. 4)

where $\mathcal{M}_{S,0}$ is the rms sonic Mach number.

Critical density for collapse (hydrodynamic case, Eq. 8). Setting the Bonnor–Ebert mass equal to the mass of a post-shock region, $M_\mathrm{HD}(\rho_\mathrm{cr})=M_\mathrm{BE}(\rho_\mathrm{cr})$, yields

where $\theta\le1$ is the fraction of the cloud size set by the turbulence integral scale. PN11 adopt $\theta = 0.35$ (their Section 2, after Wang & George 2002), giving the numerical critical density (Eq. 11)

In log-density this is the PN11 critical threshold

Virial parameter (Eqs. 1, 9). Equivalently $\alpha_\mathrm{vir} = 5\sigma_{v,\mathrm{1D}}^2 R/(GM)$ (Bertoldi & McKee 1992; Eq. 1) or $\alpha_\mathrm{vir} = 5 v_0^2 / (\pi G \rho_0 L_0^2)$ (Eq. 9, a uniform sphere of radius $L_0/2$, mean density $\rho_0$, 3-D rms velocity $v_0$). gravoturb’s virial_parameter helper uses the Eq. 1 form. $\alpha_\mathrm{vir}\approx1$ is virial equilibrium; the SFR decreases with increasing $\alpha_\mathrm{vir}$ (more turbulent support relative to gravity) and increases with $\mathcal{M}_{S,0}$.

MHD generalisation (Eq. 18). Including magnetic pressure ($\beta_0 = 2c_s^2/v_{A,0}^2$ the ratio of gas to magnetic pressure),

which reduces to the HD case as $\beta_0\to\infty$. gravoturb implements only the HD form (Eq. 8/11); the MHD form is noted but not coded.

Use in progenax¶

• experimental/gravoturb/theory/collapse_threshold.py — critical_overdensity (Eq. 8), critical_log_density (classical-alternative threshold), THETA_PN11 = 0.35.

Validation: prefactor $0.067\,\theta^{-2}=0.547$ at $\theta=0.35$; $\partial s_\mathrm{crit}/\partial\mathcal{M}=2/\mathcal{M}$.

Notes¶

• Prefactor provenance. The correct prefactor is 0.547 (PN11 Eq. 11). A prior implementation used 0.242 (~2.3× too small) — corrected in the clean-room rewrite.

• PN11’s $\rho_\mathrm{crit}$ is distinct from the Krumholz–McKee (2005) turbulent- pressure form and from the Federrath–Klessen (2012) $(\pi^2\phi_x^2/5)$ form; BM19 also recast their own $s_t$ in terms of an $\alpha_\mathrm{vir}\mathcal{M}^2$ critical density (BM19 Eq. 11/15), so the forms are related but not identical.