Inside-Out Planet Formation. V. Structure of the Inner Disk as Implied by the MRI

The large population of Earth to super-Earth sized planets found very close to their host stars has motivated consideration of $in$ $situ$ formation models. In particular, Inside-Out Planet Formation is a scenario in which planets coalesce sequentially in the disk, at the local gas pressure maximum near the inner boundary of the dead zone. The pressure maximum arises from a decline in viscosity, going from the active innermost disk (where thermal ionization of alkalis yields high viscosities via the magneto-rotational instability (MRI)) to the adjacent dead zone (where the MRI is quenched). Previous studies of the pressure maximum, based on $\alpha$-disk models, have assumed ad hoc values for the viscosity parameter $\alpha$ in the active zone, ignoring the detailed physics of the MRI. Here we explicitly couple the MRI criteria to the $\alpha$-disk equations, to find steady-state (constant accretion rate) solutions for the disk structure. We consider the effects of both Ohmic and ambipolar resistivities, and find solutions for a range of disk accretion rates ($\dot{M}$ = $10^{-10}$ - $10^{-8}$ ${\rm M}_{\odot}$/yr), stellar masses ($M_{\ast}$ = 0.1 - 1 ${\rm M}_{\odot}$), and fiducial values of the $non$-MRI $\alpha$-viscosity in the dead zone ($\alpha_{\rm {DZ}} = 10^{-5}$ - $10^{-3}$). We find that: (1) A midplane pressure maximum forms radially $outside$ the inner boundary of the dead zone; (2) Hall resistivity dominates near the midplane in the inner disk, which may explain why close-in planets do $not$ form in $\sim$50% of systems; (3) X-ray ionization can be competitive with thermal ionization in the inner disk, because of the low surface density there in steady-state; and (4) our inner disk solutions are viscously unstable to surface density perturbations.