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

## TL;DR

This paper provides analytical models for the structure and temperature profile of the inner regions of passively heated protoplanetary disks, identifying key zones and potential planet trap locations.

## Contribution

It introduces new analytical expressions for the disk's inner structure based on recent hydrodynamical simulation results, including detailed temperature and dust-to-gas ratio profiles.

## Key findings

- The temperature profile is step-like with steep gradients at region borders.
- The borders may act as planet traps, halting inward planetary migration.
- The predicted dead-zone inner edge is 2-3 times larger than classical estimates.

## Abstract

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: dust-free region with gas temperature in the optically thin limit, optically thin dust halo, optically thick condensation front and the classical optically thick region in order from the inside. We derive the dust-to-gas mass ratio profile in the dust halo using the fact that partial dust condensation regulates the temperature 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 $\sim 1200{\,\rm K}$. We derive the mid-plane 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 $\sim$ 2-3 times larger than that expected from the classical optically thick temperature.