arxiv: 2604.25816 · v1 · submitted 2026-04-28 · 🌌 astro-ph.EP

Recognition: unknown

Exploring the conditions for forming planetesimals by the streaming instability and planetary systems by pebble accretion

Anders Johansen (University of Copenhagen / Lund University) , Wladimir Lyra (New Mexico State University)

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:17 UTC · model grok-4.3

classification 🌌 astro-ph.EP

keywords streaming instabilitypebble accretionplanetesimal formationprotoplanetary disksStokes numberplanet formationexoplanetsdisk turbulence

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The pith

A narrow range of pebble sizes lets the streaming instability form planetesimals while pebble accretion builds cold giants, super-Earths and rocky embryos together.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper maps the turbulence strength, pebble sizes and initial disk sizes needed for planetesimals to form by streaming instability and for planets to grow by pebble accretion. It identifies an optimum Stokes number range of 0.01 to 0.03 where the streaming instability activates with only slight pebble enrichment and where cold gas giants, super-Earths and rocky embryos can all form. This range connects the two mechanisms under shared disk conditions. The authors also show analytically that the largest planetesimals produced match the mass at which pebble accretion turns efficient, provided formation happens early. Turbulence and disk size then determine which planet classes succeed.

Core claim

By exploring protoplanetary disc conditions in terms of turbulence strength, Stokes number and initial disc size, the paper identifies an optimum Stokes number range between St=0.01 and St=0.03 where all three planetary classes form and where the streaming instability is triggered for a slightly elevated pebble metallicity. Cold gas giants require a turbulence strength of at most δ=10^{-4} and large initial disc sizes to benefit from a prolonged pebble flux; super-Earths and rocky planet embryos tolerate higher turbulence strengths. A higher Stokes number of St=0.1 is detrimental to cold gas giants due to the short-lived pebble flux, while Stokes numbers below 0.003 demand extremely low turb

What carries the argument

The Stokes number of pebbles, which sets how tightly they couple to the gas and thereby controls both their concentration into planetesimals by streaming instability and the duration of the pebble supply available for accretion.

If this is right

All major planet classes form together when pebbles have Stokes numbers between 0.01 and 0.03 and pebble metallicity is only slightly elevated.
Cold gas giants appear only in large, low-turbulence disks that sustain a long pebble flux.
Super-Earths and rocky embryos form at higher turbulence levels consistent with the vertical shear instability.
Planetesimal masses produced by streaming instability reach the threshold where pebble accretion becomes efficient if formation occurs early.
Gas loss to disk winds or outer pressure bumps can widen the conditions that permit cold gas giants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The diversity of observed exoplanet systems can arise from modest variations in a few shared disk parameters rather than entirely separate formation channels.
The mass match between planetesimals and the pebble-accretion threshold suggests a direct transition to rapid growth without needing a separate embryo stage.
Turbulence measurements in disks that do or do not host cold giants could test whether low turbulence is strictly required.
Adding thermal or dynamical torques that slow migration would further relax the conditions needed for outer planets.

Load-bearing premise

Pebble metallicity must reach locally elevated levels to trigger the streaming instability, and the disk must evolve without strong pressure bumps or rapid gas loss that would alter the timing of the pebble flux.

What would settle it

Direct measurement of pebble Stokes numbers in a young disk that contains cold gas giants but shows turbulence above 10^{-4} or pebble metallicities below the modest elevation threshold would falsify the claimed optimum range.

Figures

Figures reproduced from arXiv: 2604.25816 by Anders Johansen (University of Copenhagen / Lund University), Wladimir Lyra (New Mexico State University).

**Figure 1.** Figure 1: The pebble flux M˙ p through the inner disc as a function of time for three values of St (St = 0.003, St = 0.01, St = 0.03) and three different initial protoplanetary disc sizes (R1 = 112 AU, R1 = 55 AU, R1 = 203 AU) for the analytical pebble drift model of Gurrutxaga et al. (2024) with constant Stχ and α = 0.01. The thin black line indicates 1% of the gas flux for the 112 AU disc; the St = 0.003 case foll… view at source ↗

**Figure 2.** Figure 2: The panels show, for two different combinations of the viscosity coefficient α and the turbulent diffusion coefficient δ, the temporal evolution of the pebble metallicity in a large protoplanetary disc (R1 = 112 AU) at 5 AU (exterior of the water ice line, blue lines) and at 0.5 AU (interior of the water ice line, red lines). Water vapor released at the passage of the water ice line is included in the me… view at source ↗

**Figure 3.** Figure 3: The two panels show the growth time-scale of planetesimals as a function of their distance to the star and their mass (colors and contours), overplotted with the characteristic mass of planetesimals formed by the streaming instability (green line) and 10 times the characteristic mass to represent the upper end of the initial mass function (green dash-dotted line). The left panel shows growth time-scales fo… view at source ↗

**Figure 4.** Figure 4: Top left: Population synthesis for a low turbulence level of δ = 10−5 and an initial planetesimal mass that is 20 times the characteristic mass of the streaming instability (SI CM). Colors mark the water mass fraction in the non-gas material, while open circles indicate two gas fraction intervals (see legend on top). The two black lines show the pebble isolation mass at early (t = 0.3 Myr) and late (t = 5.… view at source ↗

**Figure 5.** Figure 5: The plots illustrate how increasing the turbulence level further than in view at source ↗

**Figure 6.** Figure 6: Comparison of two models with angular momentum transport coefficient α = 10−2 exterior of 5 AU and a lower value (α = 10−3 , top panel) or a higher value (α = 10−1 , top panel) interior of 5 AU. We fix the initial planetesimal mass to M0 = 10−3 ME and starting times between 0.5 and 5 Myr, to probe the effect of early collisional evolution. In the low-α case, we use δ = α = 10−3 to additionally probe the e… view at source ↗

**Figure 8.** Figure 8: Two models with high α = 0.02 and high Z = 0.02, giving the same dust mass as in the nominal model but with half the gas mass. This may represent either an initially higher metallicity or a gradual gas loss to disc winds. The lower migration rates make it possible to form Jupiter analogues both at δ = 3 × 10−5 (upper panel) and at δ = 10−4 (lower panel). migration rate of protoplanets leads to a better agr… view at source ↗

**Figure 9.** Figure 9: Map of conditions for triggering planetesimal formation by the streaming instability as well as the outcome of planet formation simulations, both as a function of Stokes number St and turbulent diffusion coefficient δ. We use here a viscosity parameter for angular momentum transport of α = 10−2 and an initial disc of either 112 AU (top) or 55 AU (bottom). We refer to the main text for a full discussion of … view at source ↗

read the original abstract

The streaming instability and pebble accretion are two physical mechanisms with demonstrated potentials to drive, respectively, the formation of planetesimals and the growth of planetary systems containing a diverse range of planetary types. Here we explore the protoplanetary disc conditions in terms of turbulence strength, Stokes number and initial disc size that are needed to (i) form planetesimals by the streaming instability, (ii) form gas giant planets in cold orbits, (iii) form super-Earths and sub-Neptunes close to the star and (iv) form rocky planet embryos in temperate orbits. We identify an optimum Stokes number range between St= 0.01 and St= 0.03 where all three planetary classes form and where the streaming instability is triggered for a slightly elevated pebble metallicity. Cold gas giants require a turbulence strength of at most $\delta=10^{-4}$ and furthermore need large initial disc sizes to benefit from a prolonged pebble flux; super-Earths and rocky planet embryos tolerate higher turbulence strengths similar to those measured for the vertical shear instability. A higher Stokes number of St=0.1 is detrimental to the formation of cold gas giants due to the short-lived pebble flux. For Stokes numbers below St= 0.003, extremely low values of turbulence ($\delta<10^{-5}$) are required to form cold gas giants. We highlight how loss of gas to disc winds, reduction in the migration speed by thermal or dynamical torques or the presence of pressure bumps in the outer disc could increase the parameter space for the formation of cold gas giants. We derive analytically that the mass of the largest planetesimals formed by the streaming instability is of similar magnitude to the threshold mass beyond which pebble accretion becomes efficient, if planetesimals form in the earliest phases of protoplanetary disc evolution.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper maps out Stokes number and turbulence ranges where streaming instability plus pebble accretion can produce cold giants, super-Earths, and rocky embryos together, plus an analytical claim that largest SI planetesimals match the pebble accretion threshold mass under early formation.

read the letter

The one thing to know is that this work finds an optimum Stokes number window from 0.01 to 0.03 where the streaming instability triggers with only mildly elevated pebble metallicity and where the disc conditions allow all three main planet classes to form. They also derive that the mass of the largest planetesimals matches the pebble accretion efficiency threshold when formation occurs early. The paper does a solid job of varying turbulence, particle Stokes number, and initial disc size to show what is required for cold giants versus super-Earths versus rocky embryos. Cold giants specifically need turbulence below 10 to the minus 4 and large discs to maintain a long pebble supply, while the other types tolerate more turbulence. The analytical mass relation is derived independently of the numerical runs, which keeps it clean. Where it gets soft is the consistency of the assumptions. The mass equivalence is tied to the earliest phases of disc evolution. But the cold giant formation path relies on large initial disc sizes and low turbulence to prolong the pebble flux, which means the process extends to later times when gas surface density has decreased. Since both the planetesimal mass from streaming instability and the accretion threshold depend on local density and radius, the similarity may not hold once you move away from the earliest times. The models also require that pebble metallicity can be locally boosted enough to trigger the instability, and they explore cases without strong pressure bumps or fast gas loss. This kind of parameter exploration is useful for anyone building population synthesis models or interpreting disc observations in terms of formation pathways. A reader looking for quantitative ranges to plug into their own calculations would find it helpful. The paper deserves a serious referee. The analytical part provides a clear link that can be tested, and the numerical survey covers the key parameters even if additional convergence checks would help. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript explores protoplanetary disc conditions in turbulence strength δ, Stokes number St, and initial disc size needed to form planetesimals via the streaming instability (SI) and to produce cold gas giants, super-Earths/sub-Neptunes, and rocky planet embryos via pebble accretion. It identifies an optimum St range of 0.01–0.03 where SI triggers at mildly elevated pebble metallicity and all three planet classes form. Cold gas giants require δ ≤ 10^{-4} and large initial discs for prolonged pebble flux; higher St=0.1 or St<0.003 impose stricter limits. The paper analytically derives that the largest SI planetesimal mass is comparable in magnitude to the pebble-accretion efficiency threshold when formation occurs in the earliest disc phases, and discusses how disc winds, torques, or pressure bumps could expand viable regions for cold giants.

Significance. If the results hold, the work supplies a unified parameter-space framework linking SI planetesimal formation directly to pebble-accretion outcomes across planet classes, yielding testable ranges for St and δ that could inform both simulations and exoplanet population studies. The analytical mass-equivalence result, if shown to be robust beyond the earliest-phase assumption, would provide a concrete physical connection between the two mechanisms. Explicit exploration of mitigating processes such as pressure bumps adds practical value for interpreting observed disc structures.

major comments (2)

[analytical derivation and cold-giant parameter space] The analytical derivation (abstract and corresponding section) states that the largest SI planetesimal mass matches the pebble-accretion threshold mass only if planetesimals form in the earliest phases of disc evolution. However, the reported conditions for cold gas giants (large initial disc sizes combined with δ ≤ 10^{-4}) necessarily extend the pebble flux over later evolutionary stages where Σ_gas has declined. Because both the SI clump mass and the pebble-accretion threshold scale with local gas density and orbital radius, the magnitude equivalence may not hold under the same parameter combinations used for the numerical results; a time-dependent verification or explicit calculation at the reported late times is required to support the central claim.
[numerical results and abstract] The numerical exploration of the St–δ–disc-size space (abstract and results sections) identifies the optimum St=0.01–0.03 window and the boundaries for each planetary class, yet provides no error bars, convergence tests, or details on how post-hoc metallicity thresholds for SI triggering were chosen. These omissions make it difficult to assess the robustness of the reported optimum range and the separation between cold-giant and other-planet regimes.

minor comments (2)

A summary table listing the viable δ and St ranges for each planetary class (cold giants, super-Earths, rocky embryos) would improve readability and allow direct comparison with the analytical mass result.
The manuscript would benefit from a brief statement on the assumed disc-evolution model (e.g., how Σ_gas(t) is parameterized) to clarify the timing assumptions underlying both the numerical runs and the analytical derivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important points regarding the scope of our analytical derivation and the transparency of our numerical parameter-space exploration. We address each major comment below and will incorporate revisions to improve clarity and support for the claims.

read point-by-point responses

Referee: [analytical derivation and cold-giant parameter space] The analytical derivation (abstract and corresponding section) states that the largest SI planetesimal mass matches the pebble-accretion threshold mass only if planetesimals form in the earliest phases of disc evolution. However, the reported conditions for cold gas giants (large initial disc sizes combined with δ ≤ 10^{-4}) necessarily extend the pebble flux over later evolutionary stages where Σ_gas has declined. Because both the SI clump mass and the pebble-accretion threshold scale with local gas density and orbital radius, the magnitude equivalence may not hold under the same parameter combinations used for the numerical results; a time-dependent verification or explicit calculation at the reported late times is required to support the central claim.

Authors: We agree that the analytical equivalence is presented under the explicit condition of planetesimal formation in the earliest disc phases, as stated in the abstract and derivation section. The cold-gas-giant regime in our numerical exploration does rely on large initial disc sizes and low δ to maintain pebble flux over extended times. However, the largest SI planetesimals are expected to form early when gas densities are highest, after which they can continue to grow via pebble accretion as the disc evolves. The result is intended as an order-of-magnitude physical link rather than a precise equality at all epochs. We will add a clarifying paragraph in the discussion section that (i) reiterates the early-phase assumption, (ii) notes the scaling of both masses with declining Σ_gas, and (iii) provides a brief analytic estimate of the mass ratio at a representative later time (e.g., 1 Myr) for the fiducial cold-giant parameters. This will strengthen the connection without requiring a full time-dependent re-simulation of the entire parameter space. revision: yes
Referee: [numerical results and abstract] The numerical exploration of the St–δ–disc-size space (abstract and results sections) identifies the optimum St=0.01–0.03 window and the boundaries for each planetary class, yet provides no error bars, convergence tests, or details on how post-hoc metallicity thresholds for SI triggering were chosen. These omissions make it difficult to assess the robustness of the reported optimum range and the separation between cold-giant and other-planet regimes.

Authors: The metallicity thresholds for SI triggering are taken directly from the established analytic and simulation results in the literature (Youdin & Johansen 2007; Johansen et al. 2009; Li & Youdin 2021), where the critical pebble-to-gas ratio depends on St and δ. These are not chosen post-hoc but are applied consistently across the St–δ grid as the minimum metallicity needed for the instability to operate. We will insert a short methods subsection that (i) states the exact functional form and references used for the thresholds, (ii) explains that the optimum St window emerges from the overlap of SI-viable metallicities with the pebble-accretion efficiency requirements for each planet class, and (iii) discusses the sensitivity of the reported boundaries to modest variations (±20 %) in the adopted thresholds. Because the study employs semi-analytic prescriptions rather than suites of high-resolution hydrodynamical runs, conventional numerical convergence tests and statistical error bars are not directly applicable; we will nevertheless add a robustness paragraph quantifying how the St=0.01–0.03 optimum shifts under reasonable changes in disc parameters. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical derivation and parameter exploration are independent of fitted outcomes

full rationale

The paper's central results consist of (i) an identified optimum Stokes number window obtained by exploring turbulence, St, and disc size parameters and (ii) an explicit analytical derivation of planetesimal mass equivalence to the pebble-accretion threshold, conditioned on formation in the earliest disc phases. Neither step reduces to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation. The conditional phrasing of the analytical result is stated openly rather than smuggled in, and the numerical boundaries are presented as outcomes of the explored models rather than forced by construction. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The claims rest on standard assumptions of streaming instability and pebble accretion models plus three explored parameters (turbulence strength, Stokes number, initial disc size) and an implicit assumption that pebble metallicity can be locally raised without additional physics.

free parameters (3)

turbulence strength δ
Varied across runs to find thresholds for each planet class; not derived from first principles.
Stokes number St
Explored as input to locate the 0.01-0.03 optimum window.
initial disc size
Used to control pebble flux duration for cold giants.

axioms (2)

domain assumption Pebble metallicity can be locally elevated enough to trigger streaming instability without additional concentration mechanisms.
Invoked to allow SI in the explored parameter space.
domain assumption Disc evolution follows standard viscous or wind-driven models without strong pressure bumps altering pebble drift.
Stated as a baseline that could be relaxed to enlarge the cold-giant parameter space.

pith-pipeline@v0.9.0 · 5646 in / 1681 out tokens · 54942 ms · 2026-05-07T14:17:17.988951+00:00 · methodology

discussion (0)

Reference graph

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