tokens: of solids, α tur . The previous work assumed α acc = α tur . We will show that Drift-BKR and Diff-BKR of silicate grains are the most essential factor for the occurrence of the runaway pile-up of silicate grains and that of icy pebbles is also resulted in by Diff-BKR. The ratio between α tur and α acc controls relative importance between the runaway pile-up of silicate grains inside the snow line and that of icy pebbles outside the snow line. We map the parameter regions of F p/g and α tur ∕ α acc for GI (runaway pile-up) of silicate grains, GI of icy pebbles and SI of the icy pebbles. In Sect. 2 , we describe our basic set-ups of numerical appro- aches. In Sect. 3 , we present our numerical results. In Sect. 4 , we discuss the results. In Sect. 5 , we summarize our paper. 2 Numerical methods In this section, we discuss our numerical methods and models. 2.1 Gas disk model In this paper, we use classical α -accretion disk model ( Shakura Sunyaev 1973 ; Lynden-Bell Pringle 1974 ) where the surface density of the gas is expressed as a function of the gas accretion rate Ṁ gas and the effective viscosity ν acc as and ν acc is written using the sound speed c s , the Keplerian orbital frequency Ω and the dimensionless effective viscous parameter α acc given as The above equation assumes a steady state of the background gas. When the gas accretion is dominated by the viscous diffusion, α acc equals the dimensionless turbulence parameter α tur . In contrast, when the gas accretion is regulated by the disk wind-driven accretion ( Bai et al. 2016 ; Suzuki et al. 2016 ), generally α acc is larger than α tur ( Armitage et al. 2013 ; Hasegawa et al. 2017 ). The isothermal sound speed of the gas is written as where k B is the Boltzmann constant, μ gas is the molecular weight of the gas and T is the temperature of the gas. In this paper, the temperature profile is fixed as We note that our disk is somewhat a hot disk but we chose this profile so that we can directly compare with the previous work ( Schoonenberg Ormel 2017 ). Under the above conditions, the gas scale height is written as The midplane gas density is . In this paper, we assume that the above gas profile is unchanged through the simulation and thus μ gas = 2.34 m proton , where m proton is the proton mass. The accretion velocity of the gas v gas is written as where a negative value indicates radial drift to the central star. 2.2 Radial motions of solids and vapor In this work, we describe a pebble as the mixture of many micron-sized silicate grains covered by water ice that is also used in the previous paper (see “many-seeds pebble model” in Schoonenberg Ormel 2017 and the models used by Saito Sirono 2011 and Ida Guillot 2016 ). In our one-dimensional calculation, we adopt the single-size approximation ( Ormel 2014 ; Sato et al. 2016 ) where the pebble size is represented by a single size depending on the radial distance r , and solve the system integrated over the vertical direction of the disk
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