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tokens: proportional to the disk pebble surface density, but weakly dependent on time. In the default model we adopt a constant pebble flux of Ṁ peb = 100 M ⊕ Myr −1 (consistent with Eq. ( 14 ) of Lambrechts Johansen 2014 ) and neglect the time dependence for simplicity. A lower Ṁ peb means an intrinsic metal-deficient/less massive disk and the planetesimals and planets take longer to grow by pebble accretion. On the other hand, the pebble flux cannot be too high as the density ratio between pebbles and gas would otherwise exceed unity ( ρ peb ∕ ρ gas > 1) and the streaming instability will be triggered over the entire disk. The above inequality can be written as Σ peb ∕ H peb > Σ gas ∕ H gas where the scale height of the pebble is given by ( Youdin Lithwick 2007 ) The pebble flux therefore needs to be smaller than 2 πrv r Σ gas H peb ∕ H gas . Our default value is well below this limit. We also explore two additional pebble fluxes, Ṁ peb = 200 M ⊕ Myr −1 and 50 M ⊕ Myr −1 , in Sect. 4 . The pebble mass can be measured from (sub)millimeter dust continuum emission from the young protoplanetary disks. The inferred values are from a few Earth mass to a few hundreds of Earth mass ( Ricci et al. 2010 ; Andrews et al. 2013 ; Ansdell et al. 2017 ), which is correlated with the gas disk accretion rates ( Manara et al. 2016 ). For a typical T Tauri star with a disk accretion rate of 10 −8 M ⊙ yr −1 , the dust mass is ~ 100 M ⊕ (Fig 1 of Manara et al. 2016 ), consistent with our adopted fiducial value. In Eq. ( 8 ), α t is the coefficient of turbulent gas diffusivity, which is different from the concept of turbulent viscosity α ν . We nevertheless note that the above two quantities are approximated to be the same when the turbulence is driven by magnetorotational instability ( Johansen Klahr 2005 ; Zhu et al. 2015 ; Yang et al. 2018 ). The value of α t can be constrained from the moleculeline broadening measurements ( Flaherty et al. 2015 , 2017 ) and the level of dust settling ( Pinte et al. 2016 ). We adopt a fiducial value of α t = 10 −3 and test the influence of a lower turbulent disk ( α t = 10 −4 ) in Sect. 4 . For pebbles we adopt a fiducial aerodynamical size of τ s = 0.1. This chosen value is consistent with the sophisticated dust coagulation model ( Birnstiel et al. 2012 ) and disk observations ( Tazzari et al. 2016 ). A lower τ s = 0.03 is also explored in Sect. 4 . A fraction of pebbles will be accreted onto a planetesimal when pebbles drift through its orbit. The pebble accretion efficiency ( ε PA = Ṁ PA ∕ Ṁ peb ) is taken from Liu Ormel (2018) and Ormel Liu (2018) . This quantity depends on the disk properties ( τ s , η , α t ) and the planet properties ( m p , a , e , i ). In the limit of 2D and 3D pebble accretion these are given by and respectively, where Δ v is the relative velocity between the planet and the pebble, h peb = H peb ∕ r , is the aspect ratio of the pebble disk. In Eq. ( 10 ) we have already assume that i h peb . The above