Massive planetary cores ( \sim 10 Earth masses ) trigger rapid gas accretion to form gas giant planets such as Jupiter and Saturn . We investigate the core growth and the possibilities for cores to reach such a critical core mass . At the late stage , planetary cores grow through collisions with small planetesimals . Collisional fragmentation of planetesimals , which is induced by gravitational interaction with planetary cores , reduces the amount of planetesimals surrounding them , and thus the final core masses . Starting from small planetesimals that the fragmentation rapidly removes , less massive cores are formed . However , planetary cores acquire atmospheres that enlarge their collisional cross section before rapid gas accretion . Once planetary cores exceed about Mars mass , atmospheres significantly accelerate the growth of cores . We show that , taking into account the effects of fragmentation and atmosphere , initially large planetesimals enable formation of sufficiently massive cores . On the other hand , because the growth of cores is slow for large planetesimals , a massive disk is necessary for cores to grow enough within a disk lifetime . If the disk with 100 km-sized initial planetesimals is 10 times as massive as the minimum mass solar nebula , planetary cores can exceed 10 Earth masses in the Jovian planet region ( > 5 AU ) .