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.

The Model

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}}

They assumed that in the hydro case \epsilon \approx 1

MHD?

For MHD a similar analysis can be carried out although the post shock \beta must be measured. The main modification is that the post shock magnetic pressure must also be taken into account.

The setups

The results

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