[Paper Review] Planetesimal Formation without Thresholds. II: Gravitational Instability of Solids in Turbulent Protoplanetary Disks

Andrew N. Youdin|arXiv (Cornell University)|Aug 30, 2005

Astro and Planetary Science1 references3 citations

TL;DR

This paper demonstrates that solids in turbulent protoplanetary disks can undergo gravitational instability to form planetesimals without requiring threshold conditions, provided turbulence is weak (αg ≲ 10⁻⁶–10⁻³). Growth occurs rapidly—faster than disk lifetime and radial drift—especially in regions with high particle surface density, low gas density, or short orbital times, such as near the star or in the outer disk, enabling in situ formation of short-period extrasolar planets.

ABSTRACT

We show that small solids in low mass, turbulent protoplanetary disks collect into self-gravitating rings. Growth is faster than disk lifetimes and radial drift times for moderately strong turbulence, characterized by dimensionless diffusivities, $α_g < 10^{-6} -- 10^{-3}$ when particles are mm-sized. This range reflects a strong dependance on disk models. Growth is faster for higher particle surface densities. Lower gas densities and larger solids also give faster growth, as long as aerodynamic coupling is tight. In simple power law models, growth is slowest around ~0.3 AU, where drag coupling is strongest for mm-sized solids. Growth is much faster close to the star where orbital times are short, with implications for in situ formation of short period extrasolar planets. Growth times also decrease toward the outer disk where lower gas densities allow greater particle settling. Beyond roughly Kuiper Belt distances however, solids are sufficiently decoupled from gas that dissipative gravitational instabilities are less effective. Turbulence not only slows growth, but also increases radial wavelengths. The initial solid mass in an unstable ring can be ~0.01 M_Earth or greater, huge compared to km-sized planetesimals. Nonlinear fragmentation, which has not been studied in detail, will lower the final planetesimal mass. We consider applications to the asteroid belt and discuss the alternate hypothesis of collisional agglomeration.

Motivation & Objective

• To investigate whether gravitational instability of solids (GIS) can form planetesimals in turbulent protoplanetary disks without requiring strict stability thresholds.
• To determine the conditions under which GIS growth timescales are shorter than disk lifetimes and radial drift timescales.
• To assess the role of turbulence strength (quantified by αg) and particle properties (size, surface density) in enabling rapid planetesimal formation.
• To explore the implications for in situ formation of short-period extrasolar planets and the origin of compositional gradients in the asteroid belt.
• To evaluate the validity of local linear analysis under realistic disk conditions and the impact of eddy turnover times on growth rates.

Proposed method

• Uses a linear, axisymmetric dispersion relation (eq. [I.13]) to model the dimensionless growth rate γ of gravitational instabilities in solids, with growth time t_grow = 1/(γΩ).
• Applies stability parameters Q_T (velocity dispersion) and Q_R (density contrast) to quantify gravitational instability, derived from turbulent stirring and particle settling balance.
• Models turbulent diffusion via a dimensionless diffusivity αg, with eddy turnover time t0 = 1/Ω (orbital timescale), and explores alternative fast eddy scenarios.
• Derives particle scale height h from vertical balance between settling and turbulent diffusion (eq. [I.29]), and velocity dispersion c from turbulent kicks and epicyclic motion (eqs. [I.25], [I.32]).
• Imposes constraints: t_grow < 0.1 t_disk (~10⁵ yr), t_grow < t_drift, and λ_f < R/2 to ensure local applicability and physical relevance.
• Analyzes power-law disk models with varying αg, particle size, and radial location to map growth times across the disk.

Experimental results

Research questions

• RQ1 Can gravitational instability of solids form planetesimals in turbulent disks without requiring a threshold condition?
• RQ2 What range of turbulence levels (αg) allows GIS growth to occur faster than disk lifetime and radial drift timescales?
• RQ3 How does the growth rate of GIS vary with radial distance in the disk, particularly near the star and in the outer regions?
• RQ4 What role do particle size, surface density, and aerodynamic coupling play in accelerating GIS growth?
• RQ5 How do fast eddy turnover times affect the stability parameters and growth rates compared to orbital turnover times?

Key findings

• Gravitational instability of solids can form planetesimals without thresholds in turbulent disks when αg ≲ 10⁻⁶–10⁻³, particularly for mm-sized solids.
• Growth times are fastest near the star due to short orbital times, enabling in situ formation of hot Neptunes and Jupiters.
• Growth is also fast in the outer disk (up to ~100 AU) where low gas density enhances particle settling and reduces Q_R, but slows beyond ~100 AU due to decoupling.
• The initial mass in an unstable ring can reach ~10⁻² M⊕, far exceeding typical kilometer-sized planetesimals, suggesting potential for direct formation of large bodies.
• Turbulence increases radial wavelengths and slows growth, but growth remains viable for αg ≤ 10⁻³, with fastest growth at τs ≈ 0.3 AU where drag coupling is strongest.
• Fast eddies (t0 ≪ 1/Ω) promote growth for loosely coupled particles (τs > 1) but may hinder it for tightly coupled ones (τs < 1), though collisional damping could enhance growth in the latter case.