Abstract: Star formation in molecular cloud cores is traditionally described by the expansion wave collapse solution (EWCS) of isothermal spheres. While the cloud cores are actually not isothermal, we adopt the polytropic equation of state (EoS) in the modeling of star formation process. A new type of envelope expansion with core collapse (EECC) solution is found; and through comparisons with observations, it is demonstrated that infall and outflow motions co-exist in realistic star forming cores. It is also realized that the introduction of polytropic EoS naturally causes the variation of protostellar mass accretion rate, leading to a simple and physical solution to the “luminosity problem”.

 We will also briefly present our recent progress on the decay of turbulent motions in cloud cores.