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Ncorpi$\mathcal{O}$N is a $N$-body software developed for the time-efficientintegration of collisional and fragmenting systems of planetesimals or moonletsorbiting a central mass. It features a fragmentation model, based on craterscaling and ejecta models, able to realistically simulate a violent impact. Theuser of Ncorpi$\mathcal{O}$N can choose between four different built-in modulesto compute self-gravity and detect collisions. One of these makes use of amesh-based algorithm to treat mutual interactions in $\mathcal{O}(N)$ time.Another module, much more efficient than the standard Barnes-Hut tree code, isa $\mathcal{O}(N)$ tree-based algorithm called FalcON. It relies on fastmultipole expansion for gravity computation and we adapted it to collisiondetection as well. Computation time is reduced by building the tree structureusing a three-dimensional Hilbert curve. For the same precision in mutualgravity computation, Ncorpi$\mathcal{O}$N is found to be up to 25 times fasterthan the famous software REBOUND. Ncorpi$\mathcal{O}$N is written entirely inthe C language and only needs a C compiler to run. A python add-on, thatrequires only basic python libraries, produces animations of the simulationsfrom 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 mathematicalnotation $\mathcal{O}(N)$, due to the running time of the software being almostlinear in the total number $N$ of moonlets. Ncorpi$\mathcal{O}$N is designedfor the study of accreting or fragmenting disks of planetesimal or moonlets. Itdetects collisions and computes mutual gravity faster than REBOUND, and unlikeother $N$-body integrators, it can resolve a collision by fragmentation. Thefast multipole expansions are implemented up to order six to allow for a highprecision in mutual gravity computation.

Ncorpi$\mathcal{O}$N : A $\mathcal{O}(N)$ software for N-body integration in collisional and fragmenting systems