The effect of gas drag on the growth of protoplanets -- Analytical expressions for the accretion of small bodies in laminar disks

TL;DR

This paper develops analytical expressions to understand how gas drag influences the accretion of small bodies onto protoplanets in laminar disks, revealing a new settling mechanism that accelerates growth for larger protoplanets.

Contribution

It introduces a novel analytical framework for calculating impact radii considering gas drag effects, including a new settling accretion mode relevant for protoplanets around 1000 km.

Findings

Settling encounters significantly increase impact radii for larger protoplanets.

Accretion of small debris is slow for protoplanets under 50 km.

The settling mechanism provides a faster growth channel for protoplanets of about 1000 km.

Abstract

Planetary bodies form by accretion of smaller bodies. It has been suggested that a very efficient way to grow protoplanets is by accreting particles of size <<km (e.g., chondrules, boulders, or fragments of larger bodies) as they can be kept dynamically cold. We investigate the effects of gas drag on the impact radii and the accretion rates of these particles. As simplifying assumptions we restrict our analysis to 2D settings, a gas drag law linear in velocity, and a laminar disk characterized by a smooth (global) pressure gradient that causes particles to drift in radially. These approximations, however, enable us to cover an arbitrary large parameter space. The framework of the circularly restricted three body problem is used to numerically integrate particle trajectories and to derive their impact parameters. Three accretion modes can be distinguished: hyperbolic encounters, where the 2-body gravitational focusing enhances the impact parameter; three-body encounters, where gas drag enhances the capture probability; and settling encounters, where particles settle towards the protoplanet. An analysis of the observed behavior is presented; and we provide a recipe to analytically calculate the impact radius, which confirms the numerical findings. We apply our results to the sweepup of fragments by a protoplanet at a distance of 5 AU. Accretion of debris on small protoplanets (<50 km) is found to be slow, because the fragments are distributed over a rather thick layer. However, the newly found settling mechanism, which is characterized by much larger impact radii, becomes relevant for protoplanets of ~10^3 km in size and provides a much faster channel for growth.