Star formation in self-gravitating turbulent fluids

We present a model of star formation in self-gravitating turbulent gas in which the turbulent velocity, v_T, is a dynamical variable and is adiabatically heated by the collapse. The theory predicts the run of density, infall, and turbulent velocity and the rate of star formation in compact massive clouds. The adiabatic heating ensures that the turbulent pressure is dynamically important at all radii. The system evolves toward a coherent spatial structure with a fixed run of density, $\rho (r,t)\to \rho \left(r\right)$; mass flows through this structure onto the central star or star cluster. We define the sphere of influence of the accreted matter by $m_{\ast }=M_g\left(r_{\ast }\right)$, where m_* is the stellar plus disk mass in the nascent star cluster and M_g(r) is the gas mass inside radius r. Both v_T and the infall velocity, $|u_r|,$ decrease with decreasing r for $r>r_{\ast }$; $v_T\left(r\right)\sim r^p$, the size–line-width relation, with $p\approx 0.2-0.3$, explaining the observation that Larson's Law is altered in massive star-forming regions. The infall velocity is generally smaller than the turbulent velocity at $r>r_{\ast }$. For $r<r_{\ast }$, the infall and turbulent velocities are again similar, and both increase with decreasing r as $r^{-1/2}$, with a magnitude about half of the free-fall velocity. The accreted (stellar) mass grows superlinearly with time, ${\dot{M}}_{\ast }=\varphi M_{cl}{\left(t/{\tau }_{ff}\right)}^2$, with ϕ a dimensionless number somewhat less than unity, $M_{cl}$ the clump mass, and ${\tau }_{ff}$ the free-fall time of the clump. We suggest that small values of p can be used as a tracer of convergent collapsing flows.