Analytic Expressions for the Inner-rim Structure of Passively Heated Protoplanetary Disks

We analytically derive the expressions for the structure of the inner region of protoplanetary disks based on the results from the recent hydrodynamical simulations. The inner part of a disk can be divided into four regions: a dust- free region with a gas temperature in the optically thin limit, an optically thin dust halo, an optically thick condensation front, and the classical, optically thick region, in order from the innermost to the outermost. We derive the dust-to-gas mass ratio profile in the dust halo using the fact that partial dust condensation regulates the temperature relative to the dust evaporation temperature. Beyond the dust halo, there is an optically thick condensation front where all the available silicate gas condenses out. The curvature of the condensation surface is determined by the condition that the surface temperature must be nearly equal to the characteristic temperature similar to 1200 K. We derive the midplane temperature in the outer two regions using the two-layer approximation, with the additional heating by the condensation front for the outermost region. As a result, the overall temperature profile is step-like, with steep gradients at the borders between the outer three regions. The borders might act as planet traps where the inward migration of planets due to gravitational interaction with the gas disk stops. The temperature at the border between the two outermost regions coincides with the temperature needed to activate magnetorotational instability, suggesting that the inner edge of the dead zone must lie at this border. The radius of the dead zone inner edge predicted from our solution is similar to 2-3 times larger than that expected from the classical optically thick temperature.