Evolution of Hubble wedges in episodic protostellar outflows

Young low-mass protostars undergo short phases of high accretion and outburst activity leading to lumpy outflows. Recent observations have shown that the position-velocity and mass-velocity diagrams of such outflows exhibit individual bullet-like features; some of these bullets subscribe to a `Hubble Law' velocity relation, and others are manifest as `Hubble wedges'. In order to explore the origin of these features, we have developed a new episodic outflow model for the SPH code {\sc gandalf}, which mimics the accretion and ejection behaviour of FU Ori type stars. We apply this model to simulations of star formation, invoking two types of initial conditions: spherically symmetric cores in solid-body rotation with $\rho\propto r^{-2}$, and spherically symmetric turbulent cores with density proportional to the density of a Bonnor-Ebert sphere. For a wide range of model parameters, we find that episodic outflows lead to self-regulation of the ejected mass and momentum, and we achieve acceptable results, even with relatively low resolution. Using this model, we find that recently ejected outflow bullets produce a `Hubble wedge' in the position-velocity relation. However, once such a bullet hits the leading shock front, it decelerates and aligns with older bullets to form a `Hubble-law'. Bullets can be identified as bumps in the mass-velocity relation, which can be fit with a power-law, $dM/d\upsilon_{_{\rm RAD}}\propto\upsilon_{_{\rm RAD}}^{-1.5}$.