Protostellar disks subject to infall: a one-dimensional inviscid model and comparison with ALMA observations

Abstract

A new one-dimensional, inviscid, and vertically integrated disk model with prescribed infall is presented. The flow is computed using a second-order shock-capturing scheme. Included are vertical infall, radial infall at the outer radial boundary, radiative cooling, stellar irradiation, and heat addition at the disk-surface shock. Simulation parameters are chosen to target the L1527 IRS disk which has been observed using ALMA (Atacama Large Millimeter Array). The results give an outer envelope of radial infall and $u_\phi \propto 1/r$ which encounters a radial shock at $r_\mathrm{shock} \sim 1.5\ \times$ the centrifugal radius ($r_\mathrm{c}$) across which the radial velocity is greatly reduced and the gas temperature rises from a pre-shock value of $\approx 25$ K to $\approx 180$ K over a spatially thin region calculated using a separate shock structure code. At $r_\mathrm{c}$, the azimuthal velocity $u_\phi$ transitions from being $\propto 1/r$ to being nearly Keplerian. These results qualitatively agree with recent ALMA observations which indicate a radial shock where SO is sublimated as well as a transition from a $u_\phi \sim 1/r$ region to a Keplerian inner disk. However, in one set of observations, the position-velocity map of cyclic-C$_3$H$_2$, together with a certain ballistic maximum velocity relation, has suggested that the radial shock coincides with a ballistic centrifugal barrier, which places the shock at $r_\mathrm{shock} = 0.5 r_\mathrm{c}$, i.e, inward of $r_\mathrm{c}$, rather than outward as given by our simulations. It is argued that radial velocity plots from previous magnetic rotating-collapse simulations also indicate that the radial shock is located outward of $r_\mathrm{c}$. The discrepancy with observations is analyzed and discussed, but remains unresolved.