Erratum for: A&A, 614, A62 (2018), https://doi.org/10.1051/0004-6361/201732221

1 Introduction

In the numerical models presented in the original paper, a programing error led to an overestimation of water vapor and dust diffusion, resulting in a nonphysically long dust lifetime in the modeled protoplanetary disks. Correcting the error led to some changes in the planetesimal formation outcomes. Despite these changes, the conclusions of the original paper stay the same. Here, we present the updated results¹.

2 Model

Correcting the dust diffusion prescription led to numerical oscillations occurring inside of the water snow line location in the models with low α_t. In order to mitigate these oscillations, we slightly changed the prescription for the fragmentation threshold velocity υ_f from the previous step function to the following sigmoid function: $${\upsilon }_{\text{f}}\left(r\right)=10\text{\hspace{0.17em}m\hspace{0.17em}}{\text{s}}^{-1}/\left(1+9\cdot \exp \left(-400\cdot \left(\frac{{\text{Σ}}_{\text{ice}}}{{\text{Σ}}_{\text{d}}}-0.01\right)\right)\right).$$(1)

This function introduces a smooth transition between υ_f,out = 10 m s⁻¹ outside of the water snow line and υ_f,in = 1 m s⁻¹ inside of the snow line used in the original paper.

Furthermore, we changed the prescription for evaporation and condensation of water. In the original paper, following Drążkowska & Alibert (2017), we assumed that condensation of water vapor on dust grains is instantaneous. Here, we applied a more consistent prescription presented in Schoonenberg & Ormel (2017, their Eqs (12)–(15)), where evaporation and condensation are both time-dependent, leading to a slightly smoother transition in the density of the solids.

The planetesimal formation outcomes do not change after introducing these smoothing routines. Since in the new models the solids are lost from the disk faster, we only ran the models for 5 Myrs in contrast to 10 Myrs in the original paper.

Fig. 3 Summary of models showing the influence of the viscosity parameter αv and the midplane turbulence strength αt. The triangles denote models where planetesimals are formed both during the disk buildup and in the protoplanetary disk stage. The circles mark models where planetesimals only form after the disk is fully formed, and squares mean no planetesimal formation at all.

3 Results

The corrected diffusion prescription leads to a shorter lifetime of the solids in the disk, which overall makes it harder to form planetesimals. However, at the same time, lowering the previously nonphysically high diffusion stimulates the formation of a significant dust density enhancement outside of the snow line, which promotes planetesimal formation.

In Fig. 3, we present the results of the parameter study exploring models with different values for the viscosity parameter α_v and the midplane turbulence strength α_t corresponding to the same parameter space as covered in the original work. The main difference from the original results is that planetesimals do not form at all in the models with α_t ≈ α_v. This is because the disk is too small to allow for a strong concentration of solids in the long term. In this sense, the error present in the original version of the code was equivalent to having a nonphysically large disk able to supply pebble flux over a very long time. The fact that a large initial disk size promotes planetesimal formation was already discussed in Drążkowska & Alibert (2017, their Sect. 3.3.2).

Overall, we observe similar trends as reported in the original work: planetesimals either only form during the disk phase or both during the infall and the disk phases. The high α_v values promote outward diffusion of water vapor and thus support planetesimal formation during the disk formation stage when it is triggered by the cold finger effect. At the same time, however, the low midplane turbulence represented by the α_t value is necessary to allow for the formation of large enough pebbles and a dense midplane layer needed to trigger the planetesimal formation via streaming instability.

Figure 4 presents the mass budget of the run corresponding to α_v = 10⁻³ and α_t = 10⁻⁵ highlighted in the original work. As presented in the original work, about 1.5 M_⊕ of planetesimals are formed during the infall phase, but only 225 M_⊕ instead of the previously reported 900 M_⊕ of planetesimals are formed during the disk phase. This reduction is a direct consequence of the shorter retention of solids also visible in Fig. 4.

Figure 5 presents the distribution of planetesimals that formed in the model with α_v = 10⁻³ and α_t = 10⁻⁵ as a function of radial distance and time. The distribution resembles the one presented in the original work; however, it is more radially condensed as a consequence of the lower diffusion. Because of the corrected (shortened with respect to the original work) supply of pebbles, planetesimal formation ceases at about 3 Myrs instead of continuing throughout the whole disk lifetime.

Fig. 4 Time evolution of star mass, gas disk, dust and water, and planetesimal reservoir for the model with αv = 10−3 and αt = 10−5.

Fig. 5 Planetesimal formation obtained in the model with αv = 10−3 and αt = 10−5. Upper panel: Surface density of planetesimals at the end of the disk buildup stage (at 7 × 105 yr, gray dashed line) and at the end of the disk lifetime (at 5 × 106 yr, red solid line). The black dotted line corresponds to the minimum mass solar nebula (MMSN). Lower panel: Radial and time distribution of planetesimal formation. The light blue solid line shows the location of the snow line.

Acknowledgements

We thank Raphael Marschall and Alessandro Morbidelli for their help with identifying the bug in the original code.

References

© The Authors 2023

All Figures

	Fig. 3 Summary of models showing the influence of the viscosity parameter αv and the midplane turbulence strength αt. The triangles denote models where planetesimals are formed both during the disk buildup and in the protoplanetary disk stage. The circles mark models where planetesimals only form after the disk is fully formed, and squares mean no planetesimal formation at all.
In the text

	Fig. 4 Time evolution of star mass, gas disk, dust and water, and planetesimal reservoir for the model with αv = 10−3 and αt = 10−5.
In the text

Fig. 5 Planetesimal formation obtained in the model with αv = 10−3 and αt = 10−5. Upper panel: Surface density of planetesimals at the end of the disk buildup stage (at 7 × 105 yr, gray dashed line) and at the end of the disk lifetime (at 5 × 106 yr, red solid line). The black dotted line corresponds to the minimum mass solar nebula (MMSN). Lower panel: Radial and time distribution of planetesimal formation. The light blue solid line shows the location of the snow line.

In the text