Constrained Evolution of a Radially Magnetized Protoplanetary Disk: Implications for Planetary Migration

Abstract

We consider the inner $\sim$ AU of a protoplanetary disk (PPD), at a stage where angular momentum transport is driven by the mixing of a radial magnetic field into the disk from a T-Tauri wind. Because the radial profile of the imposed magnetic field is well constrained, a deterministic calculation of the disk mass flow becomes possible. The vertical disk profiles obtained in Paper I imply a stronger magnetization in the inner disk, faster accretion, and a secular depletion of the disk material. Inward transport of solids allows the disk to maintain a broad optical absorption layer even when the grain abundance becomes too small to suppress its ionization. Thus a PPD may show a strong middle-to-near infrared spectral excess even while its mass profile departs radically from the minimum-mass solar nebula. The disk surface density is buffered at $\sim 30$ g cm$^{-2}$: below this, X-rays trigger strong enough magnetorotational turbulence at the midplane to loft mm-cm sized particles high in the disk, followed by catastrophic fragmentation. A sharp density gradient bounds the inner depleted disk, and propagates outward to $\sim 1$-2 AU over a few Myr. Earth-mass planets migrate through the inner disk over a similar timescale, whereas the migration of Jupiters is limited by the supply of gas. Gas-mediated migration must stall outside 0.04 AU, where silicates are sublimated and the disk shifts to a much lower column. A transition disk emerges when the dust/gas ratio in the MRI-active layer falls below $X_d \sim 10^{-6}(a_d/\mu{\rm m})$, where $a_d$ is the grain size.