Inside-Out Planet Formation. VIII. Onset of Planet Formation and the Transition Disk Phase

Inside-Out Planet Formation (IOPF) is a theory of {\it in situ} formation via pebble accretion of close-in Earth to Super-Earth mass planets at the pressure maximum associated with the dead zone inner boundary (DZIB), whose location is set initially by thermal ionization of alkali metals at $\sim1,200:$K. With midplane disk temperatures determined by viscous accretional heating, the radial location of the DZIB depends on the accretion rate of the disk. Here, we investigate the ability of pebbles to be trapped at the DZIB as a function of the accretion rate and pebble size. We discuss the conditions that are needed for pebble trapping to become efficient when the accretion rate drops to $\sim10^{-9}:M_\odot:{\rm yr}^{-1}$ and the resulting DZIB is at $\sim 0.1:$au, which is the expected evolutionary phase of the disk at the onset of IOPF. This provides an important boundary condition for IOPF theory, i.e., the properties of pebbles when planet formation begins. We find for our fiducial model that typical pebble sizes of $\sim0.5:$mm are needed for pebble trapping to first become efficient at DZIBs near 0.1~au. This model may also provide an explanation for the first emergence of the transition disk phase in protoplanetary disks with accretion rates of $\sim10^{-9}:M_\odot:{\rm yr}^{-1}$.

💡 Research Summary

This paper investigates a critical step in the Inside‑Out Planet Formation (IOPF) scenario: the ability of pebbles to become trapped at the dead‑zone inner boundary (DZIB) when the protoplanetary disk’s accretion rate has declined to roughly 10⁻⁹ M⊙ yr⁻¹, placing the DZIB at about 0.1 au. The authors construct a self‑consistent α‑disk model in which the viscosity α jumps from a low value in the dead zone (α≈10⁻⁴) to a higher value in the MRI‑active inner region. By solving for the radial profiles of surface density, temperature, opacity, and aspect ratio for three accretion rates (10⁻⁸, 10⁻⁹, 10⁻¹⁰ M⊙ yr⁻¹), they locate the DZIB and quantify the width of the region with a positive pressure gradient (δr_DZIB). As the accretion rate drops, the DZIB moves inward (0.73 au → 0.23 au → 0.066 au) and the positive‑gradient zone narrows, while the magnitude of the pressure gradient remains roughly constant (d ln P/d ln r ≈ 1).

Pebble radial drift is expressed as the sum of a pressure‑gradient‑driven term (v_drift) and a gas‑drag term (v_drag). Near a pressure maximum, v_drift vanishes, leaving only the inward gas flow to pull pebbles toward the star. For a pebble with Stokes number τ_fric = 1, the authors estimate v_drift ≈ 2 × 10⁴ cm s⁻¹ and v_drag ≈ 1 cm s⁻¹. Efficient trapping therefore requires pebbles large enough that their outward drift induced by the positive pressure gradient exceeds the inward gas drag.

Using the full disk structure, the authors compute pebble drift velocities over a grid of radii and sizes. They find that the minimum pebble size capable of outward drift (and thus trapping) decreases with lower accretion rates: ≈0.10 cm for ṁ = 10⁻⁸ M⊙ yr⁻¹, ≈0.063 cm for ṁ = 10⁻⁹ M⊙ yr⁻¹, and ≈0.035 cm for ṁ = 10⁻¹⁰ M⊙ yr⁻¹. However, realistic disks feature a distribution of pebble sizes, variable opacities, and τ_fric ≠ 1 for most particles. Accounting for these complexities, the authors adopt a more conservative “typical” pebble size of ~0.5 mm as the threshold for the first efficient trapping at the DZIB when it resides near 0.1 au (Ṁ≈10⁻⁹ M⊙ yr⁻¹).

This result provides a concrete boundary condition for IOPF: the onset of planet formation occurs when the disk has thinned enough that 0.5‑mm pebbles can accumulate at the DZIB, allowing the first Earth‑mass core to grow rapidly via pebble accretion. Once the core reaches the gap‑opening mass, the MRI front retreats, the DZIB moves outward, and a new pebble trap forms, leading to sequential, inside‑out planet formation.

The authors also link this trapping condition to the emergence of the transition‑disk phase. As the accretion rate declines and the inner disk loses mass, the optical depth drops, producing the characteristic inner cavity observed in transition disks. Thus, the same physical evolution that enables pebble trapping and the birth of the first close‑in planet also naturally explains the appearance of transition‑disk signatures.

In summary, the paper demonstrates that (1) the DZIB location is strongly accretion‑rate dependent, (2) the minimum pebble size for trapping scales inversely with the accretion rate, and (3) a realistic pebble size of ~0.5 mm is sufficient for the first trapping event at Ṁ≈10⁻⁹ M⊙ yr⁻¹. These findings solidify the initial conditions of the IOPF model and provide a unified explanation for the simultaneous onset of close‑in planet formation and the transition‑disk phenomenon.