Efficiency of an analytical propagator with collision detection for Keplerian systems

Efficiency of an analytical propagator with collision detection for Keplerian systems

Aliberti, Dylan (TU Delft Applied Sciences; TU Delft Electrical Engineering, Mathematics and Computer Science)

Visser, P.M. (mentor)
Thijssen, J.M. (graduation committee)
Bouwman, W.G. (graduation committee)
van Gijzen, M.B. (graduation committee)

Delft University of Technology

Applied Mathematics | Applied Physics

2022-09-02

The aim of this thesis was to test the efficiency in practice of an analytical propagator with collision detection for N-body Keplerian systems. This can be used to simulate the evolution of a protoplanetary disk, which gives insight into how planetary systems form. The analytic propagator calculates collisions one by one, while a numerical propagator would compute each time step. The idea of using the analytic propagator is that collisions are rare in astronomical scales, such that jumping from collision to collision and calculating it, is more efficient than calculating all the time steps that are between collisions. Simplifying the orbits of the planetesimals into perfect Keplerian orbits, analytical solutions exist which are used by the analytic propagator.

In this thesis, the runtimes of simulations were measured as well as other properties directly related to the runtime. The overall efficiency of the algorithm with respect to N seemed to be O(N³), which is one power less than previously predicted. The prediction was that the runtime of the full simulation is O(N²ε + N⁴s³/Ia³). Here ε is the maximum eccentricity, s/a is the ratio of a planetesimal's radius to the semi-major axis of its orbit, and I is the maximum inclination. This was calculated by estimating the total number of collisions to be O(N²s²/Ia²) and the runtime for each collision to be O(N²s/a). But the number of collisions turns from quadratic to linear in N, implying that above a certain N almost all planetesimals collide, which reduces the power of N by one. For comparison, the octree code has an algorithmic efficiency of O(N logN) per time step, and the number of steps for a fixed integration time grows as O(N^4/3 log N).

numerical efficiency
simulation
gravitation
analytical propagator
collision detection
celestial mechanics
protoplanetary disks

http://resolver.tudelft.nl/uuid:a5764b08-9dca-4398-9309-fb99fe3a271d

Student theses

bachelor thesis

© 2022 Dylan Aliberti