© The Authors 2025

1. Introduction

Despite decades of research, we still lack a coherent picture of planet formation. When stars are young, they are surrounded by a circumstellar disk of gas and dust. The dust component must give rise to planetary building blocks, such as planetesimals and embryos, that can later become rocky planets or the cores of giant planets. The current open question regarding the origin of these building blocks is fundamental: What is the mechanism that converts dust grains into kilometer-size bodies?

Once the disk is established, collisions between micron-sized grains leads to rapid growth until the grains reach millimeter to centimeter sizes, at which point they encounter two main barriers:

• fragmentation barrier: As grains grow in size, their collision speed increases (Ormel & Cuzzi 2007) until it overcomes the adhesion of the grains (e.g., Güttler et al. 2010);

• radial drift barrier: As grains grow, aerodynamic drag makes them drift toward the star with increasing speed (Weidenschilling 1977) until the drift timescale is shorter than the grain growth timescale (Birnstiel et al. 2012).

Current efforts to overcome these barriers generally focus on aerodynamic processes that concentrate dust. If the dust density is sufficiently high, the collective gravity of dust grains can lead to gravitational collapse, giving rise to full size planetesimals, leapfrogging intermediate sizes. Two prominent mechanisms include the SI, which collects dust into filaments (Youdin & Goodman 2005; Johansen et al. 2007), and vortices which trap dust (Barge & Sommeria 1995; Adams & Watkins 1995; Tanga et al. 1996). Both have been shown to produce self-gravitating dust clumps (e.g., Johansen et al. 2007; Lyra et al. 2008, respectively), but both have important limitations.

For a solar-like dust-to-gas ratio of Z = 0.01, dust growth models predict dust sizes of St ≤ 0.1 with a typical maximum value of St ≲ 0.01 (Drazkowska et al. 2021), where St = t_stopΩ_K is the stopping time t_stop normalized by the Keplerian frequency Ω_K. However, the SI requires high (Z, St) to work (e.g., Carrera et al. 2015; Lim et al. 2024), and it has not been shown to form planetesimals for Z = 0.01, St = 0.01. Conversely, vortex trapping struggles for St = 0.03, Z = 0.01 and α = 3 × 10⁻⁴ (Lyra et al. 2024), where α is the turbulence parameter (Shakura & Sunyaev 1973). It is not been shown to work for realistic α ∼ 10⁻³ − 10⁻², (Lesur & Papaloizou 2010; Lyra & Klahr 2011), Z = 0.01, and St = 0.01.

Furthermore, there are many known exoplanets around subsolar metallicity stars (e.g., GJ 9827c, and Kepler 37d & 408b are 0.2 − 2 M_⊕ planets around stars with 0.003 ≤ Z_⋆ ≤ 0.006), so any planetesimal formation model must work for Z <  0.01. Here we find a mechanism that bypasses these problems by simultaneously increasing dust concentration and St, while decreasing turbulence. Recently we proposed a mechanism where the SI and dust growth work in tandem, forming a feedback loop where each process enhances the other (Carrera et al. 2025, “Paper I”): The SI concentrates dust, which dampens turbulence (Johansen et al. 2009) and slows radial drift, promoting grain growth. In turn, grain growth makes the SI more effective. This feedback proved extremely effective in the fragmentation-limited regime, taking the system straight toward the region where the SI is thought to produce planetesimals. Instead, in the drift-limited regime the gains were modest.

Here we present a follow-up investigation with a different mix of mechanisms. We propose that vortex trapping also exhibits a feedback loop with dust growth. Since vortices also collect particles, they also dampen turbulence, which promotes grain growth. At the same time, larger grains (up to St ≲ 1) concentrate more strongly. Because vortices are true dust traps, this mechanism completely eliminates the radial drift barrier, making it especially important in the drift-limited outer disk.

This Letter is organized as follows. We present our model in Section 2. We describe how mass loading affects the fragmentation barrier, and how we combine grain growth and vortices into a feedback loop. Section 3 shows our final results. We discuss in Section 4 and draw conclusions in Section 5.

2. Model

We used the same model for turbulence dampening as in Paper I. We include a summary in Appendices A and B.

2.1. Vortex trapping

Inside a vortex the dust density follows a Gaussian profile with constant density along ellipses of equal aspect ratio (Lyra & Lin 2013). The density peaks at the center of the vortex, and reaches a maximum column dust-to-gas ratio of (see Appendix A)

$$\begin{array}{c}\hfill Z_{\max }=Z_{\mathrm{disk}}(1+\frac{\mathrm{St}}{\alpha })·\end{array}$$(1)

Figure 1 shows the vorticity, pebble column density, and vertical diffusion inside a vortex using the recent simulation data from Lyra et al. (2024). We note that the vortex has a distinct α, independent from the rest of the disk. Vortices produce their own turbulence via the elliptic instability (Lesur & Papaloizou 2010; Lyra & Klahr 2011). We also note the dust concentration inside the vortex with a peak near the center. Lyra et al. (2024) showed that their runs have maximum dust density consistent with Equation (1), at least up to the point where gravitational instability leads to collapse. To estimate the vortex trapping timescale, we start with the drift velocity of solid grains relative to the gas inside a vortex (Lyra & Lin 2013)

$$\begin{array}{c}\hfill {\mathit{v}}_{\mathrm{drift}}=t_{\mathrm{stop}}\mathbf{\nabla }\mathit{h},\end{array}$$(2)

Fig. 1. Gas vorticity (top), pebble column density (middle), and vertical diffusion (bottom) for the 5123 simulation of Lyra et al. (2024).

where h is enthalpy (defined as dh = dP/ρ_g, where P is pressure). We refer to Lyra & Lin (2013) for the full expression of ∇h inside the vortex, but the key result is that, for a typical vortex aspect ratio of 4, HΩ_K² ≤ |∇h|≤4HΩ_K² at the boundary, decreasing toward zero at the center. We further note that, in general, ∇h does not point exclusively toward the center of the vortex, so that the “radial” speed is v_rad ≤ v_drift ¹. Nonetheless, these constraints provide a useful ballpark estimate of the radial drift rate of solid grains

$$\begin{array}{c}\hfill v_{\mathrm{rad}}\lesssim t_{\mathrm{stop}}{\mathrm{\Omega }}_{\mathrm{K}}^{2}H=\mathrm{St}{c}_{\mathrm{s}}.\end{array}$$(3)

We let t_Z be the vortex trapping timescale. Again, for a vortex with an aspect ratio of 4, the radial distance traversed by the grains is between H and 4H, so that

$$\begin{array}{c}\hfill t_Z\gtrsim \frac{H}{v_{\mathrm{rad}}}\gtrsim \frac{H}{\mathrm{St}{c}_{\mathrm{s}}}=\frac{1}{\mathrm{St}{\mathrm{\Omega }}_{\mathrm{K}}}·\end{array}$$(4)

Comparing this expression against the simulations of Lyra et al. (2024), we find support for the t_Z ∝ 1/St scaling, and indeed we find that t_Z appears to be in the vicinity of t_Z ∼ 4(St Ω_K)⁻¹. To cover the range of uncertainty in t_Z, we ran our semi-analytic model twice, spanning an order of magnitude in t_Z

$$\begin{array}{c}\hfill t_Z\in [\frac{1}{\mathrm{St}{\mathrm{\Omega }}_{\mathrm{K}}},\frac{10}{\mathrm{St}{\mathrm{\Omega }}_{\mathrm{K}}}].\end{array}$$(5)

It is worth noting that vortex trapping cannot continue indefinitely. When St ≫ 1, solid grains can no longer be trapped and escape the vortex (e.g., Raettig et al. 2015). In practice, we only explore the parameter space for St ≤ 0.1 because that is the most relevant region for determining whether the local conditions are consistent with planetesimal formation via the SI.

2.2. Vertical sedimentation

Next, we added an expression for dust sedimentation that accounts for mass loading. A common approach is to model ρ_p(z) as a Gaussian profile with scale height $H_{\mathrm{p}}=H\sqrt{\alpha /(\mathrm{St}+\alpha )}$ so that the midplane dust-to-gas ratio is given by ϵ = Z(H/H_p) (Youdin & Lithwick 2007). However, this expression does not account for mass loading. Using the colloid approximation, Yang & Zhu (2020) defined the effective scale height of the dust-gas mixture $\stackrel{\sim }{H}\equiv {\stackrel{\sim }{c}}_{\mathrm{s}}/\mathrm{\Omega }$, and Lim et al. (2024) showed that

$$\begin{array}{c}\hfill H_{\mathrm{p}}=\stackrel{\sim }{H}\sqrt{{\left(\frac{\mathrm{\Pi }}{5}\right)}^2+\frac{\alpha }{\alpha +\mathrm{St}}}\end{array}$$(6)

is a better predictor of the dust scale height. The (Π/5)² term estimates the amount of turbulence caused by the SI, ignoring the effect of St. For this work we made two further modifications. First, we assume that inside the vortex the SI is not the dominant source of turbulence so that the (Π/5)² term can be neglected. Second, we use Equation (B.3) for ${\stackrel{\sim }{c}}_{\mathrm{s}}$ instead of the colloid approximation. The two expressions agree for St ≪ 1. This gives us an expression for the midplane dust-to-gas ratio that is valid even when neither St nor ϵ are negligible:

$$\begin{array}{c}\hfill ϵ=Z\frac{H}{H_{\mathrm{p}}}=Z\sqrt{\frac{1+\mathrm{St}+ϵ}{1+\mathrm{St}}}\sqrt{1+\frac{\mathrm{St}}{\alpha }}·\end{array}$$(7)

Equation (7) assumes that the dust has a Gaussian profile. The true profile is more centrally peaked (Lim et al. 2024), meaning that Equation (7) is a conservative estimate. This is a quadratic on ϵ. We let ζ ≡ Z²(1 + St/α)/(1 + St) and solve

$$\begin{array}{c}\hfill ϵ=\frac{\zeta +\sqrt{{\zeta }^2+4\zeta (1+\mathrm{St})}}{2}·\end{array}$$(8)

It should be noted that we did not include Equation (1) in this expression. That would be a good option if we were to treat St as a static quantity, but we are interested (St, Z) as dynamic quantities that evolve together and respond to one another. As a result, the expressions for St_frag (Equation B.4) and Z_max (Equation 1) are dynamic targets that the system is steadily moving toward. This is described in more detail in the next section.

2.3. Feedback loop

The feedback loop arises from the co-evolution of dust growth, vortex trapping, and sedimentation, as each process changes the environment for the others. The model parameters are the column dust-to-gas ratio Z_disk = 0.01, the turbulence parameter α ∈ [10⁻⁴, 3 × 10⁻³], and the classic fragmentation barrier St_x ∈ [0.01, 0.04]. Using St_x as an input allows us to explore the problem without assuming a particular disk model. However, as a point of reference, for the passive disk of Chiang & Youdin (2010) at 40 AU the associated fragmentation velocity is

$$\begin{array}{c}\hfill v_{\mathrm{frag}}=0.52{\mathrm{ms}}^{-1}\sqrt{\frac{\alpha }{{10}^{-4}}·\frac{\mathrm{St}}{0.01}}·\end{array}$$(9)

We recall that our objective is the formation of planetesimals in the drift-dominated outer disk. Therefore, dust grains enter the vortex with a small drift-limited grain size St₀ <  St_x, and then grow inside the vortex. For the sake of simplicity, we set St₀ ≡ 0.1St_x, initialize ϵ with Equation (8), and apply the following algorithm:

1. Update St_frag, t_grow, Z_max, t_Z (Equations B.4, B.6, 1, and 4);

2. Let Δt = 0.1min(t_Z, t_grow) be the iteration timestep;

3. Update (St, Z) at the same time:

   • St = min[St_max,St⋅exp(Δt/t_grow)]

   • Z = min[Z_max,Z⋅exp(Δt/t_Z)]

4. Update ϵ (Equation 8).

3. Results

Figure 2 shows the evolution of (St, ϵ) at the center of the vortex for 24 simulations. We explored a range of turbulence values 10⁻⁴ ≤ α ≤ 3 × 10⁻³, and vortex trapping timescales 1 ≤ t_Z(StΩ_K)≤10. Every simulation has Z = 0.01. We were mainly interested in vortices in the outer disk, where dust grains are limited by radial drift instead of fragmentation. That means that dust grains might enter the vortex well below the fragmentation limit. To capture this, we set the initial grain size to St₀ = 0.1St_x, where St_x is the value of the fragmentation limit most often encountered in the literature (Equation B.5). We treat St_x as an input parameter so that our analysis remains agnostic to the disk model.

Fig. 2. Growth tracks for a range of grain sizes St, levels of turbulence α, and vortex trapping timescales tZ. Every run has Z = 0.01. The vertical dashed lines are Stx, but the growth tracks start at St0 = 0.1Stx as a proxy for the small drift-limited grains entering the vortex. Solid circles mark the start of the growth tracks and open circles mark every 100 orbits. The green region is the SI planetesimal formation (Lim et al. 2024). Most scenarios result in growth tracks that reach this region. However, for large α and small Stx, the growth tracks converge to a steady state outside the planetesimal formation region (the thick circles are multiple iterations plotted on top of each other). As a point of reference, we added the associated fragmentation velocity vfrag for some Stx values for the passive disk of Chiang & Youdin (2010) at 40 AU.

First and foremost, we find that this mechanism is extremely effective. Nearly every scenario leads to (St, ϵ) evolution tracks that enter the planetesimal formation region for the SI (green region; Lim et al. 2024). We find that vortex trapping + dust growth is a far more powerful mechanism than the SI + dust growth that we explored in Paper I. The mechanism in Paper I was ineffective in the drift-limited regime and required Z >  0.01 in the fragmentation-limited regime. Replacing the SI with vortices allows us to run all models with Z = 0.01. A companion work (Eriksson et al. 2025) shows that this mechanism can form planetesimals in the ultra-low metallicity disks of the early universe (Z ≥ 0.0004).

Second, we find that this mechanism can stall for strong turbulence relative to the grain size (α = 3 × 10⁻³, St_x ≤ 0.02 in Figure 2). It is worth noting that the vortex in Figure 1 already exhibits some turbulence dampening. In Lyra et al. (2024) the low-dust vortex had α ∼ 3 × 10⁻³ and the high-dust vortex had α ∼ 3 × 10⁻⁴, showing that dust dampens turbulence. However, their runs did not include coagulation, so they do not model the full effect of the mechanism presented in this Letter.

4. Discussion

4.1. Bouncing barrier versus dust traps

Perhaps the most important caveat is that grain sizes may be limited by bouncing rather than fragmentation. Meaning that growth stalls because grain collisions result in bouncing instead of sticking (Zsom et al. 2010). Importantly, if present, the bouncing barrier can lead to grains that are much smaller than those of the fragmentation limit (Zsom et al. 2010). However, it is not clear that this is the case. Whether collisions lead to sticking or bouncing depends on the detailed properties of the grains, such as their shape, surface tension, and porosity. Furthermore, Jungmann & Wurm (2021) have made a strong case that the bouncing barrier may be overcome by electrostatic forces.

Suppose that the bouncing barrier is present. The bouncing barrier resembles fragmentation in that they are both a limit on the collision speed between grains $v_{\mathrm{coll}}=c_{\mathrm{s}}\sqrt{3\alpha \mathrm{St}}$. The critical difference is that bounce-limited grains are around an order of magnitude smaller (Dominik & Dullemond 2024). That means that the (St, ϵ) evolution tracks should have a similar shape to those in Figure 2, but will require either higher Z or lower α to reach the planetesimal formation region.

This leads us to an important process that we omitted: pebble flux. We kept the total dust mass inside the vortex constant. In a real disk there is a steady influx of dust grains drifting from the outer disk, which are captured by the vortex. Therefore, Z grows over the lifetime of the vortex. This might be the key to overcoming the bouncing barrier if it is present.

5. Conclusions

We presented a novel mechanism where dust growth and vortex trapping work in tandem, as each one changes the environment in a way that enhances the other. Vortices eliminate the radial drift barrier and concentrate grains, which dampens turbulence. Lower turbulence allows fragmentation-limited grains to grow, and larger grains concentrate more strongly. This new interaction is potentially more powerful than that involving the SI that we reported in Paper I. Our conclusions are as follows:

1. Unlike the mechanism described in Paper I, the one presented here is effective for Z = 0.01 and St = 0.01 (Figure 2), making it fully compatible with dust evolution models.

2. The mechanism presented here is active wherever vortices form. Crucially, it is active in the drift-dominated outer disk, where the mechanism of Paper I is not effective.

We did find that, for sufficiently high turbulence (α ≥ 3 × 10⁻³; higher than suggested by vortex simulations) the system can stall below the planetesimal formation threshold. However, in a real disk this would be mitigated by the fact that vortices continuously trap grains as they drift from the outer disk. The total dust mass in the vortex increases for as long as the vortex lives.

Altogether, the combination of vortices and dust growth in a feedback loop appears to bridge the gap between the dust growth barriers and planetesimal formation mechanisms. This novel mechanism works entirely within the (St, Z) constraints predicted by dust evolution models, and it is effective anywhere that vortices form.

Acknowledgments

DC, WL, and JBS acknowledge support from NASA under Emerging Worlds through grant 80NSSC25K7414. J.L. acknowledges support from NASA under the Future Investigators in NASA Earth and Space Science and Technology grant 80NSSC22K1322. LE acknowledges the support from NASA via the Emerging Worlds program (80NSSC25K7117), as well as the Institute for Advanced Computational Science Postdoctoral Fellowship.

References

Appendix A: Vortex trapping

Here we present a concise high-level derivation of Equation 1; we skipped some details that can be found in Lyra & Lin (2013). In addition to vertical sedimentation, a vortex leads to a horizontal concentration of pebbles. The steady-state dust density was also shown to have Gaussian stratification in both midplane directions. Since each Gaussian contributes a $\sqrt{1+\mathrm{St}/\alpha }$ factor, the column density is enhanced by (1 + St/α).

We start by introducing coordinates (a, ν), where a is the semiminor axis of ellipses with equal aspect ratio and ν is the azimuth angle in the vortex reference frame. The conversion to local Cartesian coordinates (x, y) in the vortex frame is

$$\begin{array}{cc}\hfill x& =acos\nu ,\hfill \end{array}$$(A.1)

$$\begin{array}{cc}\hfill y& =a\chi sin\nu ,\hfill \end{array}$$(A.2)

where χ >  1 is the aspect ratio of the vortex. The Jacobian of this coordinate transformation is

$$\begin{array}{c}\hfill J=\left[\begin{array}{cc}\frac{\partial x}{\partial a}& \frac{\partial y}{\partial a}\\ \frac{\partial x}{\partial \nu }& \frac{\partial y}{\partial \nu }\end{array}\right]=\left[\begin{array}{cc}cos\nu & \chi sin\nu \\ -asin\nu & a\chi cos\nu \end{array}\right].\end{array}$$(A.3)

Thus, the total dust mass in the vortex is

$$\begin{array}{c}\hfill m_{\mathrm{p}}={\displaystyle \int {\rho }_{\mathrm{p}}}(a,z)dV={\displaystyle \int {\rho }_{\mathrm{p}}}(a,z)|J|dad\nu dz,\end{array}$$(A.4)

where |J|=aχ is the determinant of the Jacobian. The gas mass has the same expression. In this model ρ_p and ρ_g are constant along ν. For this Appendix, we are not interested in the vertical density profile as we treat it differently from Lyra & Lin (2013) to account for mass loading. Therefore, we let ρ_g, max(z) and ρ_p, max(z) be the gas and dust densities at the center of the vortex, but we do follow Lyra & Lin (2013) in modeling the density as Gaussian on a:

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{p}}(a,z)& ={\rho }_{\mathrm{p},\max }(z)exp\left(\frac{-a^2}{2H_V^2}\right),\hfill \end{array}$$(A.5)

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{g}}(a,z)& ={\rho }_{\mathrm{g},\max }(z)exp\left(\frac{-a^2}{2H_g^2}\right).\hfill \end{array}$$(A.6)

Here H_g = H/f(χ), $H_V=H_{\mathit{g}}/\sqrt{1+\mathrm{St}/\alpha }$, and f(χ) is a factor of order unity (Lyra & Lin 2013). Therefore,

$$\begin{array}{cc}\hfill m_{\mathrm{p}}& ={\displaystyle {\int }_0^{2\pi }}d\nu {\displaystyle {\int }_0^{\infty }}exp\left(\frac{-a^2}{2H_V^2}\right)a\chi da{\displaystyle {\int }_{-\infty }^{\infty }}{\rho }_{\mathrm{p},\max }(z)dz\hfill \end{array}$$(A.7)

$$\begin{array}{cc}& =2\pi \chi H_V^2{\displaystyle {\int }_{-\infty }^{\infty }}{\rho }_{\mathrm{p},\max }(z)dz,\hfill \end{array}$$(A.8)

$$\begin{array}{cc}\hfill m_{\mathrm{g}}& =2\pi \chi H_g^2{\displaystyle {\int }_{-\infty }^{\infty }}{\rho }_{\mathrm{g},\max }(z)dz.\hfill \end{array}$$(A.9)

We let Z_disk ≡ m_p/m_g and Z_max = (∫ρ_p, max(z) dz)  ÷ (∫ρ_g, max(z) dz). We divide Equations A.8 and A.9 and we obtain

$$\begin{array}{c}\hfill Z_{\max }=Z_{\mathrm{disk}}\frac{H_g^2}{H_V^2}=Z_{\mathrm{disk}}(1+\frac{\mathrm{St}}{\alpha }).\end{array}$$(A.10)

Appendix B: Mass loading and the fragmentation barrier

We used the same model of turbulence dampening as in Paper I, and we refer to that paper for details. What follows is a short summary of the model.

The most common way to model mass loading is to treat the gas-dust mixture as a colloidal suspension where dust contributes to the inertial of the fluid, but not to its pressure (Chang & Oishi 2010; Shi & Chiang 2013; Laibe & Price 2014; Lin & Youdin 2017; Chen & Lin 2018). This approach is valid in the limit as St → 0, but for our investigation we are interested in the case where St is not negligible, so that the fluid might not be well approximated by a colloid. In Paper I we approach the problem from the point of view of energy conservation, where there is a finite energy source that has to be partitioned between the gas and dust components,

$$\begin{array}{c}\hfill E=\frac{1}{2}{\rho }_{\mathrm{g}}{v}_{\mathrm{g}}^2+\frac{1}{2}{\rho }_{\mathrm{p}}{v}_{\mathrm{p}}^2=\frac{1}{2}{\rho }_{\mathrm{g}}{v}_{\mathrm{g}}^2(1+ϵ\frac{v_{\mathrm{p}}^2}{v_{\mathrm{g}}^2})=\mathrm{Const},\end{array}$$(B.1)

where v_g and v_p are the root-mean-squared velocities of the gas and dust and ϵ is the dust-to-gas ratio. Using the fact that $v_{\mathrm{p}}=v_{\mathrm{g}}/\sqrt{1+\mathrm{St}}$ (Voelk et al. 1980; Cuzzi et al. 1993; Schräpler & Henning 2004), Paper I showed that the gas velocity can be expressed as

$$\begin{array}{cc}\hfill v_{\mathrm{g}}& =\sqrt{\alpha }{\stackrel{\sim }{c}}_{\mathrm{s}},\hfill \end{array}$$(B.2)

$$\begin{array}{cc}\hfill {\stackrel{\sim }{c}}_{\mathrm{s}}& \equiv c_{\mathrm{s}}\sqrt{\frac{1+\mathrm{St}}{1+\mathrm{St}+ϵ}},\hfill \end{array}$$(B.3)

where ${\stackrel{\sim }{c}}_{\mathrm{s}}$ is the effective sound speed of the gas-dust mixture. In the limit as St → 0, Equations B.2 and B.3 reduce to the colloid approximation.

The fragmentation barrier occurs when the collision speed between grains $v_{\mathrm{coll}}=v_{\mathrm{g}}\sqrt{3\mathrm{St}}$ (Ormel & Cuzzi 2007) reaches the fragmentation speed v_frag of the grain material. Combined with Equation B.2 we obtain

$$\begin{array}{cc}\hfill {\mathrm{St}}_{\mathrm{frag}}& ={\mathrm{St}}_{\mathrm{x}}(1+\frac{ϵ}{1+\mathrm{St}}),\hfill \end{array}$$(B.4)

$$\begin{array}{cc}\hfill {\mathrm{St}}_{\mathrm{x}}& \equiv \frac{v_{\mathrm{frag}}^2}{3\alpha c_{\mathrm{s}}^2},\hfill \end{array}$$(B.5)

where St_x is the usual definition of St_frag commonly found in the literature. In other words, mass loading boosts the fragmentation barrier by a factor of 1 + ϵ/(1 + St). Birnstiel et al. (2012) derive the grain growth rate t_grow = 1/(ZΩ_K). For dust growth inside a vortex we replace this with

$$\begin{array}{c}\hfill t_{\mathrm{grow}}=\frac{1}{Z{\mathrm{\Omega }}_{\mathrm{V}}},\end{array}$$(B.6)

where Ω_V is the vortex frequency. In practice, Ω_V ≈ Ω_K, and we adopt the typical value of Ω_V = 0.5Ω_K (Lyra & Lin 2013).

All Figures

	Fig. 1. Gas vorticity (top), pebble column density (middle), and vertical diffusion (bottom) for the 5123 simulation of Lyra et al. (2024).
In the text

Fig. 2. Growth tracks for a range of grain sizes St, levels of turbulence α, and vortex trapping timescales tZ. Every run has Z = 0.01. The vertical dashed lines are Stx, but the growth tracks start at St0 = 0.1Stx as a proxy for the small drift-limited grains entering the vortex. Solid circles mark the start of the growth tracks and open circles mark every 100 orbits. The green region is the SI planetesimal formation (Lim et al. 2024). Most scenarios result in growth tracks that reach this region. However, for large α and small Stx, the growth tracks converge to a steady state outside the planetesimal formation region (the thick circles are multiple iterations plotted on top of each other). As a point of reference, we added the associated fragmentation velocity vfrag for some Stx values for the passive disk of Chiang & Youdin (2010) at 40 AU.

In the text