The effect of gas drag on the growth of protoplanets -- Analytical expressions for the accretion of small bodies in laminar disks (1007.0916v1)

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.

• The paper presents analytical formulations that extend traditional accretion models by incorporating gas drag effects on protoplanetary growth.
• It identifies three distinct encounter modes—hyperbolic, three-body, and settling—that illustrate unique dynamics influenced by gas drag.
• Numerical simulations validate that small planetesimals experience enhanced accretion due to increased effective cross-sections from gas drag.

Analytical Exploration of Gas Drag Effects on Protoplanetary Accretion

The paper titled "The effect of gas drag on the growth of protoplanets" by Ormel and Klahr addresses a critical problem in understanding the early stages of planet formation—specifically, how gas drag influences the accretion rates of small bodies onto forming protoplanets. The authors utilize both numerical simulations and analytical formulations to develop a comprehensive model that encapsulates the effects of gas drag under laminar disk conditions, an area that significantly impacts theoretical models of planetary formation.

Main Contributions

1. Accretion Modes Identification: The research delineates three distinct modes of particle-protoplanet encounters influenced by gas drag: hyperbolic, three-body, and settling encounters. Each mode is characterized by different dynamics—hyperbolic encounters are largely influenced by gravitational focusing, three-body interactions involve complex gravity and drag interactions, and settling encounters are predominantly influenced by drag forces that allow particles to settle relatively gently onto protoplanets.
2. Analytical Framework: By employing the circularly restricted three-body problem and including a linear gas drag law, the authors extend traditional gas-free accretion theories to incorporate a gaseous environment, providing a cleaning framework that retains only three dimensionless parameters: the Stokes number (St), the dimensionless headwind velocity (ζ_w), and the protoplanet size (α_p).
3. Numerical Simulations and Validation: The paper conducts a large set of numerical integrations to determine the impact radii across a diverse range of St and ζ_w values. This empirical dataset validates the analytical expressions developed and offers insights into the critical dependence of accretion rates on particle size—a noteworthy factor being that small planetesimals experience enhanced accretion rates due to an increased effective cross-section from gas drag.
4. Application to Protoplanetary Growth Rates: The model predicts significantly different growth times based on the protoplanet's size and the disk's turbulence, highlighting a rapid growth mechanism for protoplanets (∼10^3 km size) due to the settling regime, particularly in outer disk regions. However, accretion of small particles by planetesimals (∼ km size) is consistently slower—a finding which suggests that these small bodies must accrete higher-mass particles to speed up growth.

Implications and Future Directions

The paper provides a foundational model that can pave the way for understanding accretion dynamics in protoplanetary disks, directly impacting how we model planet formation timescales under different environmental conditions. The findings suggest a nuanced view of accretion, where gas drag can significantly accelerate the growth of already substantial protoplanetary bodies but may present significant barriers for smaller sizes.

Under realistic disk conditions, factors such as disk turbulence, the presence of atmospheres around growing protoplanets, and the complex interplay of resonant interactions may further influence the accretion rates. Future research could expand on this work by considering more turbulent and realistic disk scenarios with non-linear drag components and addressing the potential feedback mechanisms between particles and surrounding gas.

This paper, therefore, forms a critical touchstone for future computational and theoretical investigations that aim to integrate these microscale interactions into the macroscale architecture evolution of planetary systems. As observational data from telescopic arrays and space missions continue to provide empirical measurements of forming planetary systems, this model offers a valuable lens through which such observations can be interpreted and expanded.