The Star Formation Rate of Supersonic MHD Turbulence (Padoan and Nordlund 2011)

Attempt to determine SFR(\alpha_{vir}, M_S, \beta) for isothermal self-gravitating MHD turbulence.

Post shock density

If we balance ram pressure of flow at any scale with thermal pressure of shocked material then we get post shock densities (\rho_0) that are

\rho_l c_s^2=\rho_0 v_{shock}^2 = \rho_0 \left(v_l/2\right)^2 \rightarrow \rho_l = \rho_0 \mathcal{M}^2_l/4 where \mathcal{M}_l=\frac{v_l}{c_s}

and if v_l \propto l^{1/2} then

\rho_l \propto l

Shock thickness

Flows on a given scale will be coherent for a time

l/v_{l}

and will built up a sheet with a surface density of

\rho_0 l

and will therefore have a thickness of

\rho_0 l / \rho_l = (\rho_0 \mathcal{M}_l^2/4) or \lambda_l = 4 c_s^2 l/v_l^2.

Since v_l \propto l^{1/2}, this thickness is independent of scale and

\lambda_{HD}=4 L_0 / \mathcal{M}_0^2

So flows on all scales will produce shocks of the same thickness, but the post shock densities will increase with scale.

Shock instability

If \lambda_{HD} > \lambda_J(\rho_{cr,HD}) then the shocked layer will be unstable to collapse so we can define a critical density based on the shock layer thickness

\lambda_{HD} > \lambda_J(\rho_{cr,HD}) \rightarrow 4 L_0 / \mathcal{M}_0^2 > c_s \sqrt{\frac{\pi}{G\rho_{cr,HD}}} \rightarrow \rho_{cr,HD} > \frac{c_s^2 M_0^4}{16 \pi G L_0^2} \approx .067 \alpha_{vir} M_0^2 \rho_0

So for a given cloud, we can estimate the densities we need to reach in the shocked layers to trigger collapse. Then using the lognormal density distribution seen for supersonic turbulence

p(x)=\frac{x^{-1}}{2\pi \sigma^2} \exp \left [ -\frac{(\ln(x)+\sigma^2/2)^2}{2 \sigma^2} \right ]

where x = \rho/ \rho_0 and \sigma^2 \approx \ln \left [ 1 + \frac{\mathcal{M}}{2} \right ] ^2

We can calculate the amount of gas above the critical density M_{>\rho_{cr}} = \displaystyle \int_{x_{cr}}^\infty {x p(x) dx} and the characteristic time for that mass to form stars as \tau_{ff,cr} with some efficiency \epsilon giving a star formation rate of \epsilon \frac{M_{\rho_{>cr}}}{\tau_{ff,cr}} or the standard "star formation rate per free fall time" SFR_{ff}=\epsilon \frac{\tau_{ff,0}}{\tau_{ff,cr}} M_{\rho_{>cr}}