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 pile-up 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 pile-up occurs (Sect. 3.2). The blue, green, and red lines in the bottom panels are those at t = 3 × 10⁴ yrs, t = 1 × 10⁵ yrs, and t = 3 × 10⁵ yrs from the beginning of the calculations, respectively. Panel a: system that reaches a steady-state, panels b–d: ND runaway pile-ups.