Analytical estimate of f_water at 1 au as a function of the initial disk radius (r_d,0) and the initial disk accretion rate (Ṁ_g,0). Panels a: dependence on the planet mass (M_pl,0). Panels b: Stokes number (St) of pebbles. The other parameters are Ṁ_pe = 10⁻⁹M_⊙ yr⁻¹ and t_diff = 3 × 10⁶ yr. In the panelsa, we use St = 0.1. In the panelsb, M_pl,0 = 1M_⊕. The middle panels in a and b are identical. The color scales are log₁₀f_water.