Issue 
A&A
Volume 645, January 2021



Article Number  L9  
Number of page(s)  6  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/202040031  
Published online  18 January 2021 
Letter to the Editor
A “nodrift” runaway pileup of pebbles in protoplanetary disks in which midplane turbulence increases with radius
^{1}
ISAS/JAXA, Sagamihara, Kanagawa, Japan
email: hyodo@elsi.jp
^{2}
EarthLife Science Institute, Tokyo Institute of Technology, Meguroku, Tokyo 1528550, Japan
^{3}
Laboratoire J.L. Lagrange, Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, 06304 Nice, France
Received:
1
December
2020
Accepted:
22
December
2020
Context. A notable challenge of planet formation is to find a path to directly form planetesimals from small particles.
Aims. We aim to understand how drifting pebbles pile up in a protoplanetary disk with a nonuniform turbulence structure.
Methods. We consider a disk structure in which the midplane turbulence viscosity increases with the radius in protoplanetary disks, such as in the outer region of a dead zone. We perform 1D diffusionadvection simulations of pebbles that include backreaction (the inertia) to the radial drift and the vertical and radial diffusions of pebbles for a given pebbletogas mass flux.
Results. We report a new mechanism, the “nodrift” runaway pileup, that leads to a runaway accumulation of pebbles in disks, thus favoring the formation of planetesimals by streaming and/or gravitational instabilities. This occurs when pebbles drifting in from the outer disk and entering a dead zone experience a decrease in vertical turbulence. The scale height of the pebble subdisk then decreases, and, for small enough values of the turbulence in the dead zone and high values of the pebbletogas flux ratio, the backreaction of pebbles on gas leads to a significant decrease in their drift velocity and thus their progressive accumulation. This occurs when the ratio of the flux of pebbles to that of the gas is large enough that the effect dominates over any KelvinHelmholtz shear instability. This process is independent of the existence of a pressure bump.
Key words: accretion, accretion disks / planets and satellites: formation / planetdisk interactions
© ESO 2021
1. Introduction
Forming planetesimals directly from small particles in an evolving protoplanetary disk is a major challenge in planet formation due to the “growth barrier” (Blum & Wurm 2000; Zsom et al. 2010) and the “drift barrier” (Whipple et al. 1972; Weidenschilling 1977). Streaming instability (SI) may be a promising mechanism for forming planetesimals directly from pebbles, although it requires special conditions: the Stokes number τ_{s} ≳ 0.01 and a local elevated solidtogas ratio (Z ≡ ρ_{p}/ρ_{g} ≳ 1, where ρ_{p} and ρ_{g} are the spatial densities of pebbles and gas, respectively; Youdin et al. 2004; Youdin & Goodman 2005; Carrera et al. 2015). How, when, and whether such conditions are met in an evolving protoplanetary disk are still a matter of intense debate.
Different mechanisms have been proposed for planetesimal formation via the local enhancement of materials. They include the diffusive redistribution and recondensation of water vapor outside the snow line (e.g., Stevenson & Lunine 1988; Ciesla & Cuzzi 2006; Ros & Johansen 2013), and detailed numerical simulations have been performed (e.g., Drżkowska & Alibert 2017; Schoonenberg & Ormel 2017; Hyodo et al. 2019, 2021; Gárate et al. 2020). Other mechanisms include pileups at pressure maxima. These pressure maxima can be created by the planet’s gravity (e.g., Dipierro & Laibe 2017; Kanagawa et al. 2018), by a sharp change in the local magnetorotational instability (MRI) driven turbulence structure at an evaporation front (e.g., Kretke & Lin 2007; Brauer et al. 2008), by a change in the gas accretion velocity at the inner (e.g., Chatterjee & Tan 2014; Ueda et al. 2019; Charnoz et al. 2019) and outer (e.g., Pinilla et al. 2016) edges of a dead zone, and/or by the change in the gas profile due to gaspebble friction (backreaction) combined with pebble growth (Gonzalez et al. 2017). However, a positive pressure gradient may not be likely because the nonmagnetohydrodynamic (nonMHD) effects erase a clear transition between dead and active zones (e.g., Mori et al. 2017). The backreaction of solids onto the gas would smooth out the bump (e.g., Taki et al. 2016; Kanagawa et al. 2018). Further studies are required to assess these likelihoods in an evolving disk.
Over the last few decades, MHD simulations have shown that a region where ionization is too low for the MRI to operate (i.e., a “dead zone”; Gammie 1996) might ubiquitously exist in the inner part of the disk midplane, and that only surface layers are magnetically active, supporting accretion (Fig. 1; Gressel et al. 2015; Simon et al. 2015; Bai & Stone 2013; Bai et al. 2016; Mori et al. 2017). The transition between active and dead zones might be smooth (i.e., the αparameter in the disk midplane α_{mid} increases as a function of the distance to the star r), and α_{dead} = 10^{−5} − 10^{−3} is reported for the αparameter within a dead zone (e.g., Gressel et al. 2015; Simon et al. 2015; Mori et al. 2019). We note that the sound waves propagated from MRIactive surface layers (e.g., Okuzumi & Hirose 2011; Yang et al. 2018) and the vertical shear instability (VSI; e.g., Stoll & Kley 2016; Flock et al. 2020) could induce vertical mixing with an equivalent α for vertical diffusivity up to ∼10^{−3}, even within a dead zone. These results imply nonuniform disk turbulence structures.
Fig. 1.
Schematic illustration of pebble drift and its pileup within a protoplanetary disk with a dead zone. The disk gas accretion is characterized by αparameter a_{acc}, while the midplane diffusivity, being a dead zone (α_{dead} ≪ α_{acc}) in the inner region, is characterized by α_{mid}. During the inward drift of pebbles, the pebble scale height H_{p} decreases as until a KH instability prevents it from becoming smaller. A smaller H_{p} leads to an elevated local midplane concentration of pebbles within a thinner midplane layer. The elevated midplane pebbletogas ratio causes the backreaction to be more effective in reducing the radial drift velocity of pebbles. Such a physical interplay with a sufficiently large pebbletogas mass flux results in a progressive accumulation of pebbles in a runaway fashion (i.e., the “nodrift” runaway pileup). 

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In this Letter, we consider a toy model of a protoplanetary disk with a dead zone located at the inner region of the disk. We consider different αparameters for the gas accretion and for the pebble motion. Including the effects of the backreaction of pebbles onto gas, which slows the drift velocity of pebbles, we study how drifting pebbles within a dead zone pile up. We demonstrate that a runaway pileup of pebbles occurs for a sufficiently large pebbletogas mass flux F_{p/g} when pebbles reach a critical lowlevel of turbulence; we call this the “nodrift” (ND) runaway pileup (the summary of this mechanism is shown by a schematic illustration in Fig. 1). It is worth mentioning that the newly reported ND mechanism does not require the snow line or the pressure maxima.
In Sect. 2, we describe the disk models and settings of our 1D simulations that solve the diffusionadvection of drifting pebbles. In Sect. 3, we show the results of our 1D simulations as well as analytical arguments. In Sect. 4, we summarize this Letter.
2. Models and settings
2.1. Gas structure
Here, we adopt a toy model where the gas accretion toward the central star is characterized by a nondimensional αparameter α_{acc} (Fig. 1). Using the prescription of the classical αaccretion disk model (Shakura & Sunyaev 1973; LyndenBell & Pringle 1974), the surface density of the gas is given as
where Ṁ_{g} and are the gas mass accretion rate and the effective viscosity, respectively (c_{s} is the gas sound velocity and Ω_{K} is the Keplerian orbital frequency).
The gas rotates at subKeplerian speed. The degree of the deviation of the gas rotation frequency from that of Keplerian η is given by
where Ω, P_{g}, and H_{g} are the gas orbital frequency, the gas pressure, and the gas scale height, respectively, and C_{η} is defined as
which depends on the temperature profile (e.g., C_{η} = 11/8 for T ∝ r^{−1/2}).
Using the αdisk prescription, the gas accretion velocity v_{g} is written as
where a negative sign indicates accretion toward the central star.
2.2. Pebbles in the disk midplane
The radial drift velocity of pebbles, including the effects of gassolid friction − drift backreaction (hereafter DriftBKR) − is given as (Ida & Guillot 2016; Schoonenberg & Ormel 2017; Hyodo et al. 2019)
where τ_{s} is the Stokes number of pebbles. Λ ≡ ρ_{g}/(ρ_{g} + ρ_{p}) = 1/(1 + Z) characterizes the strength of the backreaction due to the pileup of pebbles, where Z ≡ ρ_{p}/ρ_{g} is the midplane pebbletogas density ratio ( and , where Σ_{p} and H_{p} are the surface density and scale height of pebbles, respectively).
The disk midplane could be too weakly ionized for the MRI to operate (the dead zone with extremely weak turbulence; Gammie 1996). Thus, the disk midplane could have very different turbulence from that of gas accretion, which controls the disk surface density (i.e., α_{acc}). In this Letter, we use an effective viscous parameter α_{mid} to characterize the radial and vertical diffusion processes of pebbles near the disk midplane (Fig. 1).
The scale height of pebbles characterizes the degree of pebble concentration in the disk midplane (). In the steady state, the scale height of pebbles is regulated by the vertical turbulent stirring (Dubrulle et al. 1995; Youdin & Lithwick 2007; Okuzumi et al. 2012; Hyodo et al. 2019) as
where a coefficient K characterizes the strength of the backreaction onto the diffusivity (Hyodo et al. 2019; Ida et al. 2021) and K = 1 is used (the choice of the K value does not alter the conclusion – see Appendix A for the K = 0 case).
For a small α_{mid} (i.e., for a small H_{p, tur}), a vertical shear KelvinHelmholtz (KH) instability prevents a further decrease in the pebble scale height. This minimum scale height, H_{p, KH}, is (Hyodo et al. 2021)
which is valid for Z ≲ 1, and Z ≫ 1 indicates a gravitational collapse. Thus, the scale height of pebbles H_{p} is given as
2.3. Numerical settings
We performed 1D diffusionadvection simulations that included the backreaction (the inertia) to radial drift of pebbles that slows the pebble drift velocity as pileup proceeds (see details in Hyodo et al. 2019, 2021). The drift velocity of pebbles is given by Eq. (6), while the diffusivity is given as . We included backreaction onto the diffusivity of pebbles (with K = 1). We assumed that pebbles drift outside the snow line, and the sublimation of pebbles was neglected (i.e., the dead zone exists beyond the snow line). The Stokes number of pebbles was set to be constant: τ_{s} = 0.1 (Okuzumi et al. 2016; Ida & Guillot 2016). The disk midplane temperature was set to T(r) = 150 K × (r/3 au)^{−β} (β = 1/2). The surface density of the gas (molecular weight of μ_{g} = 2.34) is described by Σ_{g} = Ṁ_{g}/3πν_{g}, where . We used α_{acc} = 10^{−2} and Ṁ_{g} = 10^{−8} M_{⊙} yr^{−1}. These led to C_{η} = 11/8 for Σ_{g} ∝ r^{−1} and T ∝ r^{−1/2}.
We considered a dead zone in the inner region of the disk midplane, and we used a nondimensional turbulence parameter in the midplane α_{mid}, which differs from the one that characterizes gas accretion, α_{acc} (Eq. (1)). The midplane α_{mid} is modeled as
where: α_{acc} and α_{dead} are those outside and inside a dead zone; r^{*} is the innermost radial distance, where α_{mid} = α_{acc}; and Δr_{tra} is the radial width of the transition from α_{mid} = α_{acc} to α_{mid} = α_{dead}. As shown below, the ND mode occurs for an arbitrary choice of the dead zone structure, that is, irrespective of a sharp or a smooth change between the active and dead zones (i.e., an arbitrary choice of α_{dead}, α_{acc}, r^{*}, and Δr_{tra}), as long as it satisfies that α_{mid} is smaller than a threshold value (Sect. 3.2.2).
At the beginning of the 1D simulations, we set the pebbletogas mass flux F_{p/g} at the outer boundary (r_{out} = 15 au), and F_{p/g} at r_{out} was fixed throughout the simulations. We note that Elbakyan et al. (2020) performed a timedependent simulation of disk formation to find a high F_{p/g} variation between 𝒪(10^{−4}) to 𝒪(1) in a single disk evolution (see also Ida et al. 2021).
3. Results
3.1. Numerical results
Figure 2 shows the results of our 1D simulations for different combinations of F_{p/g} and the midplane turbulence structures (i.e., different choices of α_{dead}, r^{*}, and Δr_{tra} in Eq. (10)) for the case of α_{acc} = 10^{−2}. The top panels show given turbulence structures (i.e., α_{mid}(r)/α_{acc}) and the bottom panels show the resultant midplane pebbletogas ratio in their spatial densities (i.e., ρ_{p}/ρ_{g}). The gray lines in the top panels are the analytically derived critical α_{mid}/α_{acc} below which a runaway pileup of pebbles is expected to occur for a given F_{p/g} (Eq. (23) with τ_{s} = 0.1).
Fig. 2.
α_{mid}/α_{acc} (top panels) and Z = ρ_{p}/ρ_{g} (bottom panels) as a function of the distance to a star in the 1D numerical simulations. Included are the cases where α_{acc} = 10^{−2} and τ_{p} = 0.1. From left to right: cases where (F_{p/g}, α_{dead}, r^{*}, Δr_{tra}) = (0.11, 10^{−4}, 9 au, 5 au), (0.17, 10^{−4}, 9 au, 5 au), (0.47, 10^{−4}, 9 au, 5 au), and (0.17, 10^{−4}, 5 au, 0.5 au). The gray lines in the top panels represent the analytical critical α_{mid}/α_{acc} below which the ND runaway pileup is expected to occur for a given F_{p/g} (Eq. (23) with F_{p/g} > F_{p/g, crit2}). The gray lines in the bottom panels show the critical Z (i.e., Z_{crit} = 1) above which the ND runaway pileup occurs (Sect. 3.2). The blue, green, and red lines in the bottom panels are those at t = 3 × 10^{4} yrs, t = 1 × 10^{5} yrs, and t = 3 × 10^{5} yrs from the beginning of the calculations, respectively. Panel a: system that reaches a steadystate, panels b–d: ND runaway pileups. 

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For a small F_{p/g} (e.g., F_{p/g} = 0.11; panel a in Fig. 2), the system reaches a steady state. For this given F_{p/g}, analytical arguments (the gray line) predict the requirement of a smaller α_{mid}/α_{acc} for a runaway pileup of pebbles to occur (i.e., the black line in Fig. 2 needs to be below the gray line).
For a larger F_{p/g} (e.g., F_{p/g} = 0.17; panel b in Fig. 2) that has the same α_{mid}/α_{acc} structure as that of panel a, pebbles progressively pile up over time. Continuous pileup occurs as the drift velocity of pebbles progressively decreases due to strong DriftBKR and as the pileup efficiently proceeds. For the F_{p/g} given here, the gray line overlaps with the black line (i.e., the analytical requirement for the runaway pileup is met) and the analytical arguments (Sect. 3.2) show a very good accordance with 1D numerical simulations. For an even larger F_{p/g} with the same α_{mid}(r)/α_{acc} as those in panels a and b, a runaway pileup of pebbles occurs more quickly and significantly (panel c in Fig. 2).
Comparing panels b and d in Fig. 2 (the same F_{p/g}, α_{dead}, and α_{acc} but different shapes and locations of the transition between active and dead zones), 1D simulations demonstrate that a runaway pileup of pebbles occurs irrespective of the shape and location of the dead zone as long as F_{p/g} is sufficiently large. This shows the robustness of this new mechanism of runaway pebble pileup.
Numerical results show that when the change in the midplane turbulence on r is smooth (i.e., α_{mid}), the pileup of pebbles efficiently propagates toward the outer region in the disk (compare panels b and d in Fig. 2). This indicates that, when the ND runaway pileup occurs at a given radial distance, a progressive pileup of pebbles could occur even at a greater radial distance where the ND conditions are not met, leading to a global pileup of pebbles for the SI to operate.
When α_{acc} = 10^{−3}, 1D simulations only show a runaway pileup of pebbles for a very large value of F_{p/g} (F_{p/g} ≳ 0.8). This is because the surface density of the gas increases (Eq. (1)) for a smaller α_{acc}, and ρ_{p}/ρ_{g} correspondingly decreases.
3.2. Analytical arguments
3.2.1. Midplane concentration of pebbles
Below, we discuss the concentration of pebbles in the disk midplane considering DriftBKR. Our arguments are made using the analytical equation presented below (Eq. (11)). We solve this equation using two distinct approaches: a direct numerical approach and an analytical approach with some approximation. In both ways, we identify a parameter space (F_{p/g} − α_{acc} − α_{mid} space) where the ND runaway pileup of drifting pebbles could take place.
The concentration of pebbles at the midplane is written as
where h_{p/g} ≡ H_{p}/H_{g} and v_{p} are functions of Λ(Z). We directly find a solution (i.e., solve for Z) for Eq. (11) for a given F_{p/g} and α_{mid}. The color contours in Fig. 3 show directly obtained ρ_{p}/ρ_{g} and the red region is where no steadystate solution is found (i.e., the radial drift of pebbles is halted due to DriftBKR). The dashed lines in Fig. 3 are analytically derived critical F_{p/g, crit}, above which no steadystate solution is found for ρ_{p}/ρ_{g} (i.e., the ND runaway pileup). The analytical estimations reproduce the direct solutions well.
Fig. 3.
Midplane pebbletogas ratio Z ≡ ρ_{p}/ρ_{g} in the α_{mid}/α_{acc} − F_{p/g} space. Included are the cases where r = 5 au, C_{η} = 11/8 for T ∝ r^{−1/2}, τ_{s} = 0.1, and Ri = 0.5. Left and right panels: cases where α_{acc} = 10^{−2} and α_{acc} = 10^{−3}, respectively. The color contours are obtained by directly solving Eq. (11). The red regions indicate the nodrift runaway pileup. The diagonal dashed black line (Eq. (16)) and the horizontal dashed black line (Eq. (20)) are the analytically derived critical F_{p/g, crit} above which the ND runaway pileup takes place. The critical Z above which the ND runaway pileup occurs (i.e., Z_{crit} = 1) is shown by the white lines. The parameter map is very weakly dependent on the radial distance to the star (see Eqs. (16) and (20)), and the ND mode can occur irrespective of the shape and position of a dead zone. 

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The F_{p/g, crit} can be divided into two different regimes. In the first regime, H_{p, tur} > H_{p, KH, max}, where H_{p, KH, max} is the maximum scale height of pebbles regulated by a KH instability. In this case, using Eqs. (5) and (6), the vertically averaged metallicity of pebbles is given as
where an approximation is made for α_{acc} ≪ τ_{s} ≪ 1, Z ≪ 1, Λ ≃ 1. Solving this equation gives
where and . The Z_{Σ} has a real solution when 1−4ab > 0 and a critical F_{p/g, crit1} for the first regime is given as
where h_{p/g} ≃ (τ_{s}/(α_{mid}(1+Z)^{−K}))^{−1/2} ≃ (τ_{s}/α_{mid})^{−1/2} for Z ≪ 1 and α_{mid} ≪ 1 + τ_{s} (Eq. (9)). The above criteria for F_{p/g, crit1} are independent of r, and thus the ND mechanism occurs irrespective of the position and shape of a dead zone as long as it satisfies its criteria (i.e., Eq. (16)). The diagonal dashed black line in Fig. 3 shows Eq. (16) and is in accordance with the direct solutions of Eq. (11) (color contours in Fig. 3). The critical Z above which the ND occurs, Z_{crit}, is obtained by inserting Eq. (16) into Eq. (11), and Z_{crit} ∼ 1 for the first regime. Analytical Z_{crit} = 1 agrees with the direct solution (color contours in Fig. 3). We note that because F_{p/g, crit1} is proportional to C_{η}, it increases with a steeper Σ_{g} gradient at an outer boundary of a dead zone. However, the ND mode does not need a sharp boundary. If the boundary is very gradual, the change in C_{η} is negligible.
The second regime appears when a KH instability plays a role, which corresponds to H_{p, tur} < H_{p, KH, max}. Thus, in this case, the scale height of pebbles is described by H_{p, KH}, which is independent of α_{mid}. As ρ_{p}/ρ_{g} increases from zero, H_{p, KH} initially increases and reaches its maximum H_{p, KH, max} at Z = ρ_{p}/ρ_{g} = 1/2. As ρ_{p}/ρ_{g} further increases, H_{p, KH} decreases^{1}. By smoothly connecting with the critical value of Z_{crit} = 1 in the first regime (i.e., Z_{crit} = 1 as well in the second regime, which agrees with the direct solution; see the color contours in Fig. 3), the critical F_{p/g} in the second regime F_{p/g, crit2} adopts (H_{p, KH} with Z = 1), and Z_{crit} is given as
where and an approximation is made for α_{acc} ≪ τ_{s} ≪ 1 and Λ ≃ 1. The is Λ using . Here, Z = 1 and . Thus, F_{p/g, crit2} with C_{η} = 11/8 is given as
which is independent of α_{mid}. We note that the above criterion for F_{p/g, crit2} has a very weak dependence on r through H_{g}/r. Equation (20) is plotted by a horizontal dashed black line in Fig. 3, and it shows a good consistency with the direct solution of Eq. (11). Including a KH instability (i.e., the second regime and Eq. (20)) prevents the scale height of pebbles from becoming increasingly smaller as α_{mid} decreases, reducing the parameter range of the ND runaway pileup^{2}.
3.2.2. Criteria for the ND runaway pileup
Summarizing the analytical arguments above, the scale height of pebbles decreases with decreasing α_{mid} until a KH instability plays a role. As pebbles drift inward from the active to the dead zones in the disk midplane, α_{mid} becomes smaller with decreasing distance to the star r, while keeping a constant F_{p/g}. This implies that the evolutionary path in the α_{mid}/α_{acc} − F_{p/g} space (Fig. 3) is a horizontal shift from right to left for a given fixed F_{p/g} as pebbles drift inward. Therefore, for the ND mode to take place, the following two conditions need to be met.
First, F_{p/g} needs to be sufficiently large to satisfy F_{p/g} ≥ F_{p/g, crit2}. As F_{p/g, crit2} ∝ 1/α_{acc} (Eq. (20)), the ND mode is limited to relatively large F_{p/g} values for the α_{acc} = 10^{−3} case (F_{p/g} ≳ 0.8 at 5 au), while only moderate F_{p/g} ≳ 0.08 is required in the case of α_{acc} = 10^{−2}.
Second, a sufficiently small α_{mid}/α_{acc} is required in order to have a sufficiently small scale height of pebbles. Rewriting Eq. (16) gives a critical α_{mid}/α_{acc} below which the ND occurs for a given F_{p/g} as
which is independent of r. These two criteria are in very good accordance with the 1D simulations (Sect. 3.1).
4. Conclusions
In this Letter, we studied how drifting pebbles pile up in a protoplanetary disk with a nonuniform turbulence structure, including the backreaction of pebbles onto the gas that slows the drift velocity of pebbles as pileup proceeds. We considered that gas accretion is regulated by an αparameter that is distinct from that of midplane turbulence. In the disk midplane, the turbulence strength can decrease with decreasing radial distance (e.g., the case where the inner region of the disk is an MRIinactive deadzone and the outer region is MRIactive). Thus, drifting pebbles are further concentrated in the midplane as the scale height decreases.
We demonstrated a new mechanism for a runaway pileup of pebbles to occur for a moderate F_{p/g} when pebbles reach a critical level of low turbulence in the dead zone via a continuous slowing of their drift velocity due to the backreaction (the “nodrift” (ND) runaway pileup). The ND runaway pileup occurs irrespective of the shape of the dead zone. Our results imply that SI could also occur as a natural consequence of pebble drift in a protoplanetary disk with a dead zone.
The minimum scale height of pebbles in a dead zone is a critical parameter that characterizes the ND mode. Further studies that include detailed physical processes – such as a KH instability, VSI, vertical shear streaming instability, and the velocity fluctuation by the sound waves propagated from MRIactive surface layers – are required.
The vertical shear streaming instability (VSSI; Lin 2020) might affect the pebble scale height when the midplane diffusivity is very small. If the VSSI dominated over a KH instability for the scale height of pebbles, the boundary of the ND regime would be regulated by the VSSI.
Acknowledgments
We thank Satoshi Okuzumi and Shoji Mori for helpful comments and fruitful discussion. R.H. acknowledges the financial support of JSPS GrantsinAid (JP17J01269, 18K13600). R.H. also acknowledges JAXA’s International Top Young program. S.I. acknowledges the financial support (JSPS Kakenhi 15H02065, MEXT Kakenhi 18H05438). T.G. was partially supported by a JSPS Long Term Fellowship at the University of Tokyo.
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Appendix A: The case of K = 0
Here, we show the case of K = 0, where the diffusivity of pebbles is a fixed constant: (i.e., the backreaction onto the diffusivity is neglected). Figure A.1 is the same as Fig. 2 but for the case of K = 0.
Fig. A.1.
Same as Fig. 2 but for the case of K = 0. 

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As seen in the case of K = 1 (Fig. 2), a runaway pileup of pebbles within a dead zone occurs as long as F_{p/g} meets the conditions discussed in Sect. 3.2.2 (the analytical arguments were made for K = 0). Compared to the case of K = 1, the radial width and the timescale of the pileup are in general narrower and longer in the case of K = 0 because the diffusivity is a constant for K = 0, while it becomes smaller for K = 1 as pileup proceeds. These additional simulations with K = 0 further show the robustness of the physical mechanism presented in this Letter and demonstrate that DriftBKR, which slows the drift velocity of pebbles, is a fundamental ingredient for the ND mechanism.
All Figures
Fig. 1.
Schematic illustration of pebble drift and its pileup within a protoplanetary disk with a dead zone. The disk gas accretion is characterized by αparameter a_{acc}, while the midplane diffusivity, being a dead zone (α_{dead} ≪ α_{acc}) in the inner region, is characterized by α_{mid}. During the inward drift of pebbles, the pebble scale height H_{p} decreases as until a KH instability prevents it from becoming smaller. A smaller H_{p} leads to an elevated local midplane concentration of pebbles within a thinner midplane layer. The elevated midplane pebbletogas ratio causes the backreaction to be more effective in reducing the radial drift velocity of pebbles. Such a physical interplay with a sufficiently large pebbletogas mass flux results in a progressive accumulation of pebbles in a runaway fashion (i.e., the “nodrift” runaway pileup). 

Open with DEXTER  
In the text 
Fig. 2.
α_{mid}/α_{acc} (top panels) and Z = ρ_{p}/ρ_{g} (bottom panels) as a function of the distance to a star in the 1D numerical simulations. Included are the cases where α_{acc} = 10^{−2} and τ_{p} = 0.1. From left to right: cases where (F_{p/g}, α_{dead}, r^{*}, Δr_{tra}) = (0.11, 10^{−4}, 9 au, 5 au), (0.17, 10^{−4}, 9 au, 5 au), (0.47, 10^{−4}, 9 au, 5 au), and (0.17, 10^{−4}, 5 au, 0.5 au). The gray lines in the top panels represent the analytical critical α_{mid}/α_{acc} below which the ND runaway pileup is expected to occur for a given F_{p/g} (Eq. (23) with F_{p/g} > F_{p/g, crit2}). The gray lines in the bottom panels show the critical Z (i.e., Z_{crit} = 1) above which the ND runaway pileup occurs (Sect. 3.2). The blue, green, and red lines in the bottom panels are those at t = 3 × 10^{4} yrs, t = 1 × 10^{5} yrs, and t = 3 × 10^{5} yrs from the beginning of the calculations, respectively. Panel a: system that reaches a steadystate, panels b–d: ND runaway pileups. 

Open with DEXTER  
In the text 
Fig. 3.
Midplane pebbletogas ratio Z ≡ ρ_{p}/ρ_{g} in the α_{mid}/α_{acc} − F_{p/g} space. Included are the cases where r = 5 au, C_{η} = 11/8 for T ∝ r^{−1/2}, τ_{s} = 0.1, and Ri = 0.5. Left and right panels: cases where α_{acc} = 10^{−2} and α_{acc} = 10^{−3}, respectively. The color contours are obtained by directly solving Eq. (11). The red regions indicate the nodrift runaway pileup. The diagonal dashed black line (Eq. (16)) and the horizontal dashed black line (Eq. (20)) are the analytically derived critical F_{p/g, crit} above which the ND runaway pileup takes place. The critical Z above which the ND runaway pileup occurs (i.e., Z_{crit} = 1) is shown by the white lines. The parameter map is very weakly dependent on the radial distance to the star (see Eqs. (16) and (20)), and the ND mode can occur irrespective of the shape and position of a dead zone. 

Open with DEXTER  
In the text 
Fig. A.1.
Same as Fig. 2 but for the case of K = 0. 

Open with DEXTER  
In the text 
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