The Streaming Instability

Why the streaming instability matters

For decades, planet formation had a gaping hole in the middle. Dust grains a micron across stick together electrostatically and by van der Waals forces, growing to pebbles millimetres to centimetres wide. Gravity takes over once you have a ~100 km planetesimal that can pull in more material. But getting from pebbles to planetesimals — bridging roughly six orders of magnitude in size — resisted every mechanism tried. Boulders bounce or shatter instead of sticking, and worse, they drift into the star before they can grow. The streaming instability is the mechanism that finally closes that gap, and it does so without ever building a meter-scale boulder.

• It bypasses the meter-size barrier. No need for boulders to stick; solids collapse gravitationally as a cloud.
• It is fast. Filaments collapse in years to a few thousand years — matching the ~1 Myr age of the oldest meteorite parent bodies.
• It sets the initial size distribution. Simulations reproduce the observed ~100 km turnover in the Kuiper Belt and asteroid belt.
• It makes binaries. Excess angular momentum in collapsing clouds naturally yields equal-mass, same-colour binaries like those in the cold classicals.
• It feeds later stages. Its planetesimals are the seeds for pebble accretion and core accretion of giant-planet cores.
• It is testable. ALMA rings/gaps, meteorite ages, and New Horizons' Arrokoth all provide observational hooks.

How it works, step by step

The instability lives on a single asymmetry: gas feels pressure, solids do not.

1. Gas orbits sub-Keplerian. The disk's gas pressure normally falls off with radius, adding an outward force that partly offsets stellar gravity. So the gas orbits slightly slower than the local circular (Keplerian) speed — typically about 50 m/s slower, a fraction η ≈ 0.001–0.005 of the orbital velocity.
2. Pebbles feel a headwind. Solids have no pressure support, so they "want" to orbit at the full Keplerian speed and continually plough through the slower gas. The headwind saps their angular momentum, so they spiral inward — this is radial drift.
3. But pebbles push back on the gas. Drag is mutual (Newton's third law). Where solids happen to pile up, they drag the local gas up toward Keplerian speed. The headwind there weakens, and the pile's inward drift slows.
4. A traffic jam forms. Because the clump drifts more slowly, faster-drifting pebbles from farther out catch up and join it — like cars piling up behind a slow truck. The overdensity grows. This is the positive feedback at the heart of the instability.
5. Filaments and rings emerge. The runaway organizes solids into azimuthal filaments, concentrating them far above the smooth-disk background — factors of tens to hundreds in local solid density.
6. Self-gravity takes over. Once a filament's solid density exceeds the Roche (tidal) density, the clump is gravitationally bound against the star's tide and against shear, and it collapses in near free-fall.
7. Planetesimals (and binaries) are born. The cloud collapses to one or more ~100 km bodies. Excess spin angular momentum frequently splits it into a bound pair.

The key numbers

Two dimensionless quantities decide whether the instability turns on: the particle Stokes number (how tightly a grain is coupled to the gas) and the local metallicity (solid-to-gas mass ratio). Below are representative thresholds and the objects they produce.

Quantity	Symbol	Value / range	Meaning
Stokes number	τs = Ω·tstop	~0.001 to a few	Best clumping for mm–dm pebbles; ~1 is the drift-barrier size
Critical metallicity	Z = Σsolid/Σgas	≳ 0.01–0.02	Solids must be pre-concentrated above solar (~0.01)
Sub-Keplerian offset	Δv = η·vK	~30–100 m/s	The headwind; η ≈ 0.001–0.005
Planetesimal diameter	D	~100 km (30–400 km)	Peak of the initial mass function
Mass function slope	dN/dM	∝ M−1.6	Steep power law above the peak
Collapse time	tff	~few–tens of orbits	Years to millennia at 3 AU

The problem it defeats: the meter-size barrier

Radial drift is fastest when a particle's stopping time equals its orbital time — that is, when the Stokes number τ_s = Ω·t_stop ≈ 1. At about 1 AU in the minimum-mass solar nebula, that resonance corresponds to roughly a meter-scale boulder. The peak drift speed is set by the sub-Keplerian offset:

v_{drift, max} ≈ η · v_K ≈ 50 m/s

where η ≈ 0.0015 is the fractional pressure support and v_K ≈ 30 km/s at 1 AU. Fifty metres per second doesn't sound catastrophic, but integrated over an orbit it drains a boulder from 1 AU into the star in only ~100–1000 years — far shorter than the ~10⁵–10⁶ yr it would take to grow by chance collisions. Compounding the trouble, laboratory and numerical collision experiments show that at those sizes and speeds grains tend to bounce or fragment rather than stick. Growth stalls and inventory is lost. The streaming instability sidesteps both traps: it never asks meter boulders to glue together, and it never lets a single boulder wander alone long enough to drift away — it gathers billions of pebbles into a self-gravitating swarm first.

A worked example: the Roche-density trigger

For a clump to collapse rather than shear apart, its internal solid density ρ_d must exceed the Roche density set by the star's tide at orbital radius r:

ρ_Roche ≈ 9 M_⊙ / (4π r³)

Here M_⊙ = 1.989 × 10³⁰ kg is the solar mass and r is measured in metres. At r = 3 AU ≈ 4.5 × 10¹¹ m — near the middle of the asteroid belt — this evaluates to roughly ρ_Roche ≈ 1.6 × 10⁻⁵ kg/m³. Compare that to the smooth-disk pebble density there, of order 10⁻⁹ to 10⁻¹⁰ kg/m³. The streaming instability has to raise the local solid density by a factor of roughly 10⁴–10⁵ to cross the threshold — exactly the concentration factor high-resolution simulations achieve inside filaments. Once ρ_d > ρ_Roche, self-gravity wins, the free-fall time t_ff ≈ (3π / 32 G ρ_d)^1/2 is only a handful of orbits, and a ~100 km planetesimal drops out. Symbols: G = 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻² (gravitational constant), Ω = orbital angular frequency, t_stop = aerodynamic stopping time, v_K = √(GM_⊙/r) = local Keplerian speed.

History and evidence

Andrew Youdin and Jeremy Goodman published the linear analysis in The Astrophysical Journal in 2005, showing that a drag-coupled dust–gas mixture is linearly unstable to axisymmetric perturbations with growth rates of order the orbital frequency. Anders Johansen, Youdin, Hubert Klahr, and collaborators then showed in a series of shearing-box simulations (2007 onward, including a widely-cited 2007 Nature paper) that the nonlinear outcome is gravitationally bound clumps — genuine planetesimals — forming in just a few orbits. Later work by Johansen, Jacob Simon, Rixin Li and others mapped the critical metallicity and derived the characteristic ~100 km size and its power-law tail.

The observational case is strong. New Horizons' 1 January 2019 flyby of Arrokoth (486958, in the cold classical Kuiper Belt) revealed a gently-joined bilobate contact binary of two similar lobes that met at only a few metres per second — a textbook fossil of low-velocity gravitational collapse rather than a violent collision. The abundance of nearly equal-mass, matched-colour binaries among the cold classicals points the same way. And meteorite chronology — the ~1 Myr formation window inferred for iron-meteorite parent bodies and for the calcium–aluminium-rich inclusions in chondrites — demands the near-instantaneous planetesimal assembly the streaming instability provides. ALMA images of ringed disks such as HL Tau and TW Hya, meanwhile, reveal the pressure bumps thought to pre-concentrate solids to the critical metallicity in the first place.

Common misconceptions

• "Pebbles just stick together into planetesimals." No — sticking stalls at the bounce/fragmentation barrier. The streaming instability uses collective gravity, not glue.
• "It's the gravitational instability of the dust layer." Distinct. Goldreich–Ward direct dust-layer collapse is usually stopped by turbulent stirring; the streaming instability is an aerodynamic drag instability that first concentrates solids, then lets gravity finish.
• "Any pebble size works." Only marginally-coupled particles clump well — Stokes numbers roughly 0.001 to a few. Very small dust stays glued to the gas; very large boulders decouple.
• "Solar metallicity is enough." Not by itself. Local solid-to-gas ratio must be enhanced above ~0.01–0.02, so the disk needs pressure bumps, snow lines, or radial pile-ups to seed it.
• "It builds planets directly." It builds planetesimals (~100 km). Growing those to planets still needs pebble accretion, runaway/oligarchic growth, and gas capture.
• "The gas orbits faster than the pebbles." The reverse. Pressure support makes gas orbit slower than Keplerian; the pebbles want to go faster and feel a headwind.