  Ncorpi$\mathcal{O}$N is a $N$-body software developed for the time-efficient
integration of collisional and fragmenting systems of planetesimals or moonlets
orbiting a central mass. It features a fragmentation model, based on crater
scaling and ejecta models, able to realistically simulate a violent impact. The
user of Ncorpi$\mathcal{O}$N can choose between four different built-in modules
to compute self-gravity and detect collisions. One of these makes use of a
mesh-based algorithm to treat mutual interactions in $\mathcal{O}(N)$ time.
Another module, much more efficient than the standard Barnes-Hut tree code, is
a $\mathcal{O}(N)$ tree-based algorithm called FalcON. It relies on fast
multipole expansion for gravity computation and we adapted it to collision
detection as well. Computation time is reduced by building the tree structure
using a three-dimensional Hilbert curve. For the same precision in mutual
gravity computation, Ncorpi$\mathcal{O}$N is found to be up to 25 times faster
than the famous software REBOUND. Ncorpi$\mathcal{O}$N is written entirely in
the C language and only needs a C compiler to run. A python add-on, that
requires only basic python libraries, produces animations of the simulations
from the output files. The name Ncorpi$\mathcal{O}$N, reminding of a scorpion,
comes from the French $N$-corps, meaning $N$-body, and from the mathematical
notation $\mathcal{O}(N)$, due to the running time of the software being almost
linear in the total number $N$ of moonlets. Ncorpi$\mathcal{O}$N is designed
for the study of accreting or fragmenting disks of planetesimal or moonlets. It
detects collisions and computes mutual gravity faster than REBOUND, and unlike
other $N$-body integrators, it can resolve a collision by fragmentation. The
fast multipole expansions are implemented up to order six to allow for a high
precision in mutual gravity computation.
