tokens: ( 2011 ) do show the formation of cores within 1 Myr. From Equations ( 31 ) and ( 32 ), is required for km at 5 AU in 10 MMSN. Figure 4 shows (or α for ), which can prepare the conditions for the formation of Jupiter given by –300 km at 5–6 AU in disks with . Saturn’s core forms in the disk after the formation of Jupiter. The gap opening by Jupiter assists the rapid growth of Saturn’s core at the outer edge of the gap, where the radial drift due to gas drag is negligible. The collisional fragments produced during core formation accelerate the growth of the core, and the supply of fragments produced in the outer disk induces further rapid growth. These effects allow the rapid formation of Saturn’s core within years after gap opening by Jupiter, and the rapid formation may explain the massive core of Saturn (Kobayashi et al. 2012 ). For the rapid growth of Saturn’s core in 10 MMSN, 100 km radius or smaller planetesimals are required, which is satisfied for ( for ) in the 10 MMSN disk (see Figure 6 ). This upper limit is similar to the condition for Jupiter. If Jupiter can be formed via other mechanisms in the MMSN disk, then kilometer sized or smaller planetesimals are required for the rapid formation of Saturn’s core, which corresponds to for . Figure 6. Turbulence strength or α , required for the formation of Mars, Jupiter, and Saturn for the MMSN (top panel) and 10 MMSN (bottom panel). The formation of the core required for Jupiter via core accretion is difficult to accomplish with the MMSN. The value of α is estimated under the assumption of . On the other hand, the formation age of Mars estimated from the amount of W isotope in the mantle is 2–4 Myr (Dauphas Pourmand 2011 ). For 10 MMSN (MMSN), the early formation of Mars may be explained by planetesimals with radii of km (5–30 km) (Kobayashi Dauphas 2013 ). The disk inside the snow line ( AU) has (Hayashi 1981 ). From Equations ( 31 ) and ( 32 ), the formation of Mars requires at 1 AU ( at 1.5 AU) in the 10 MMSN disk, while 10 −4 –10 −3 is suitable at 1–1.5 AU in the MMSN (see Figure 6 ). In addition, main-belt asteroids with radii smaller than 50–100 km have a slope of , which is similar to the outcome of collisional cascade (Bottke et al. 2005 ), while larger main-belt asteroids except for the largest bodies (Ceres, Vesta, and Pallas) have shallower slopes. The main-belt asteroids, except for the largest asteroids, have a peak of in the radius range of 50–100 km. The onset of runaway growth causes a similar shaped peak around the runaway radius (see Figures 1 and 5 ). Assuming that the radius of 50–100 km in the main belt corresponds to the planetesimal radius, at 2 AU ( at 3 AU) in the 10 MMSN and at 2 AU ( at 3 AU) in the MMSN. In Figure 6 , the constraints on the turbulence strength ( or α ) are summarized based on these considerations. Jupiter cannot be formed via planetesimal accretion in the MMSN; therefore, there is no hatched region for Jupiter in the MMSN. If the solar system
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