# A Keplerian method for Collision Detection in Protoplanetary Disk Simulations

### Calculating the computational efficiency and the consecutive collision times in N-body systems as a function of Keplerian parameters

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## Abstract

This report has been made to gain more insight and knowledge of the Keplerian method for collision detection simulations by simplifying N-body systems to a list or particles with orbital parameters. Collision detection is an essential part in modeling the evolution of protoplanetary disk becoming planetary systems by the merging of planetesimal objects. Presumably Kep- lerian systems allow for a computationally efficient algorithm, having its computational time influenced by orbital parameters, unlike the efficient octree method which only scales with the number of particles.

The aim of this research was to create a collision detection model for a simplified proto- planetary disk system using Kepler systems and analyse the results of the model to check the reliability of this simulation method, i.e. if the quality of the results are trustworthy, perform- ing consistently well, and are accurate. In addition, the method was tested on its computational efficiency by investigating the run time of separate parts of the algorithm.

In the initialisation of the algorithm a particle list is created and sorted by increasing apoap- sis. Then by a sweep and prune filter the list is checked for possible collision pairs. For pairs that have a small enough minimal orbital intersection distance that they can collide, the colli- sion time is calculated and added to a time-ranked collision list. The algorithm keeps merging the pairs on top op the list, creating a new particle with new possible collision pairs and new collision times, and updates the time-ranked collision list after each merge. An empty collision list implies the end of the simulation. Various parameters as well as length of the lists are saved within the algorithm for subsequent examination.

The results of the Keplerian model found showed similarities to other simulation studies, and include the perceived critical eccentricity of 0.02, preservation of shape of body radius distribution, and the formation of large bodies. Simulations showed three stages of collisions, characterised by collisions occurring homogeneously throughout the disk, large body colli- sions, a final stage of small body collisions. The performance of the algorithm could be sum- marised by the number of checks by the sweep and prune filter #checks ∝ N 2 and #checks ∝ ε for eccentricity ε < 0.1, the number of pairs in the collision pair list was found to be propor- tional to the number of starting particles N, the body radius s, the inverse of the inclination 1/I, and the eccentricity ε, so #pairs ∝ N2, s, 1/I, ε. For N < 1000 the computation time of the algorithm appeared to be proportional to N2, but for N < 10.000 proportional to to a higher power. For ε < 10−9 and N < 10.000 the computation time was found to be runtime ∝ s.

The conclusion of this research is that the Keplerian model could be used to model N-body systems for collision detection because of its behavioural resemblance to observable astrophys- ical systems. The computation time of the Keplerian system scales with a higher order per N than for instance the three code algorithm, but has showed to be influenced by orbital parame- ters in which this method differentiates itself from other simulation methods, with a remarkable influence on the total computation time by the body radius s of the system. More research could be done for higher orders of N to better model reality as well as providing more insight on the proportionality of the computation time. Recommendations to the algorithm consist of adding planetary spin as a particle parameter, and methods to incorporate apsidal precession in a com- petent manner.