Steady Wind-Blown Cavities within Infalling Rotating Envelopes: Application to the Broad Velocity Component in Young Protostars

Wind-driven outflows are observed around a broad range of accreting objects throughout the Universe, ranging from forming low-mass stars to super-massive black holes. We study the interaction between a central isotropic wind and an infalling, rotating, envelope, determining the steady-state cavity shape formed at their interface under the assumption of weak mixing. The shape of the resulting wind-blown cavity is elongated and self-similar, with a physical size determined by the ratio between wind ram pressure and envelope thermal pressure. We compute the growth of a warm turbulent mixing-layer between the shocked wind and the deflected envelope, and calculate the resultant broad line profile, under the assumption of a linear (Couette-type) velocity profile across the layer. We then test our model against the warm broad velocity component observed in CO $J$=16--15 by Herschel/HIFI in the protostar Serpens-Main SMM1. Given independent observational constraints on the temperature and density of the dust envelope around SMM1, we find an excellent match to all its observed properties (line profile, momentum, temperature) and to the SMM1 outflow cavity width for a physically reasonable set of parameters: a ratio of wind to infall mass-flux $\simeq 4\%$, a wind speed $v_{\rm w} \simeq 30$ km/s, an interstellar abundance of CO and H$_2$, and a turbulent entrainment efficiency consistent with laboratory experiments. The inferred ratio of ejection to disk accretion rate, $\simeq 6-20\%$, is in agreement with current disk wind theories. Thus, the model provides a new framework to reconcile the modest outflow cavity widths in protostars with the large observed flow velocities. Being self-similar, it is applicable over a broader range of astrophysical contexts as well.