## Stability and numerical errors in the N-body problem

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2008##### Item status

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Urminsky, David

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##### Abstract

Despite the wide acceptance that errors incurred in numerical solutions to N-body systems
grow exponentially, most research assumes that the statistical results of these systems are reliable.
However, if one is to accept that the statistical results of N-body solutions are reliable,
it is important to determine if there are any systematic statistical errors resulting from the incurred
growth of errors in individual solutions.
In this thesis we consider numerical solutions to the 3-body problem in which one of
the bodies escapes the system. It is shown for a particular 3-body con guration, known as
the Sitnikov problem, that the mean lifetime of the system is dependent on the accuracy of the
numerical integration. To provide a theoretical explanation of the phenomenon, an approximate
Poincar´e map is developed whose dynamics on a particular surface of section is shown to be
similar to the dynamics of the Sitnikov Problem. In fact there is a set on which the approximate
Poincar´e map is topologically equivalent, like the Sitnikov Problem, to the shift map on the set
of bi-in nite sequences. The structure of the escape regions on the surface of section form a
cantor set-like structure whose boundary can more easily be delineated using the approximate
Poincar´e map than for the Sitnikov problem. Further it is shown that numerical errors destroy
escape regions and can cause orbits to migrate to a region in which escape is faster.
Finally, a relationship between the Lyapunov time, tl, and the lifetime, td, of the 3-body
problem is discussed. Firstly, the Sitnikov problem and the approximate Poincar´e map of the
Sitnikov problem both exhibit a two-part power law relationship beween tl and td like that
for a particular case of the general 3-body problem. Further, it is demonstrated that large
perturbations to the energy of the escaping body in uences the relationship between tl and
td for small tl. Finally, it is shown that the approximate Poincar´e map yields a theoretical
explanation of the phenomenon based on the structure of the escape regions the orbits traverse
before escape.