"Ashfall" Induced by Molecular Outflow in Protostar Evolution. II. Analytical Study on the Maximum Size of Dust Grains Lifted by Outflows

In this paper, we study the dust dynamics in the molecular outflow using an analytical magnetohydrodynamical outflow model. Specifically, we investigate the maximum size of dust grains $a_{\rm d,max}$ that can be lifted by the outflow and whether they can escape into interstellar space. We also investigate the dependence of the maximum size of the dust grains on various outflow parameters, such as the mass ejection rate of the outflow $\dot{M}$, the disk size $r_{\rm disk}$, the mass of the central protostar $M_{*}$ and the internal dust density $\rho_{\rm mat}$. We find the empirical formula for the maximum dust size as a function of the outflow parameters. $a_{\rm d,max}$ depends on the parameters as $a_{\rm d,max}$ $\propto$ $\dot{M}^{1.0}$ $r_{\rm disk}^{-0.44}$ $M_{*}^{-0.82}$ $\rho_{\rm mat}^{-1.0}$. We also find that the dust grains with size of 100 $\rm \mu$m to 1 mm can be lifted by the outflow and can distribute in the well outside of the disk when the mass ejection rate of the outflow has the value of $10^{-7}\,M_{\odot}\rm \,yr^{-1}$ to $10^{-6}\,M_{\odot}\rm \,yr^{-1}$. This result is consistent with recent observations that show a correlation between the mass ejection rate of the outflow and the spectral index $\beta$ of the dust opacity at the envelope scale.