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Star Formation in Self-Gravitating Turbulent Fluids

We present a model of star formation in self-gravitating turbulent gas. We treat the turbulent velocity $v_T$ as a dynamical variable, and assume that it is adiabatically heated by the collapse. The theory predicts the run of density, infall velocity, and turbulent velocity, and the rate of star formation in compact massive gas clouds. The turbulent pressure is dynamically important at all radii, a result of the adiabatic heating. The system evolves toward a coherent spatial structure with a fixed run of density, $ρ(r,t)\toρ(r)$; mass flows through this structure onto the central star or star cluster. We define the sphere of influence of the accreted matter by $m_*=M_g(r_*)$, where $m_*$ is the stellar plus disk mass in the nascent star cluster and $M_g(r)$ is the gas mass inside radius $r$. The density is given by a broken power law with a slope $-1.5$ inside $r_*$ and $\sim -1.6$ to $-1.8$ outside $r_*$. Both $v_T$ and the infall velocity $|u_r|$ decrease with decreasing $r$ for $r>r_*$; $v_T(r)\sim r^p$, the size-linewidth relation, with $p\approx0.2-0.3$, 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 $r>r_*$. For $r<r_*$, 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 super-linearly with time, $\dot M_*=ϕM_{\rm cl}(t/τ_{ff})^2$, with $ϕ$ a dimensionless number somewhat less than unity, $M_{\rm cl}$ the clump mass and $τ_{ff}$ the free-fall time of the clump. We suggest that small values of p can be used as a tracer of convergent collapsing flows.