  In a standard theory of the formation of the planets in our Solar System,
terrestrial planets and cores of gas giants are formed through accretion of
kilometer-sized objects (planetesimals) in a protoplanetary disk. Gravitational
$N$-body simulations of a disk system made up of numerous planetesimals are the
most direct way to study the accretion process. However, the use of $N$-body
simulations has been limited to idealized models (e.g. perfect accretion)
and/or narrow spatial ranges in the radial direction, due to the limited number
of simulation runs and particles available. We have developed new $N$-body
simulation code equipped with a particle-particle particle-tree (${\rm P^3T}$)
scheme for studying the planetary system formation process: GPLUM. For each
particle, GPLUM uses the fourth-order Hermite scheme to calculate gravitational
interactions with particles within cut-off radii and the Barnes-Hut tree scheme
for particles outside the cut-off radii. In existing implementations, ${\rm
P^3T}$ schemes use the same cut-off radius for all particles, making a
simulation become slower when the mass range of the planetesimal population
becomes wider. We have solved this problem by allowing each particle to have an
appropriate cut-off radius depending on its mass, its distance from the central
star, and the local velocity dispersion of planetesimals. In addition to
achieving a significant speed-up, we have also improved the scalability of the
code to reach a good strong-scaling performance up to 1024 cores in the case of
$N=10^6$. GPLUM is freely available from https://github.com/YotaIshigaki/GPLUM
with MIT license.
