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Most globular clusters have half-mass radii of a few pc with no apparent correlation with their masses. This is different from elliptical galaxies, for which the Faber–Jackson relation suggests a strong positive correlation between mass and radius. Objects that are somewhat in between globular clusters and low-mass galaxies, such as ultracompact dwarf galaxies, have a mass–radius relation consistent with the extension of the relation for bright ellipticals. Here we show that at an age of 10 Gyr a break in the mass–radius relation at∼106M is established because objects below this mass, i.e. globular clusters, have undergone expansion driven by stellar evolution and hard binaries. From numerical simulations we find that the combined energy production of these two effects in the core comes into balance with the flux of energy that is conducted across the half-mass radius by relaxation. An important property of this ‘balanced’ evolution is that the cluster half-mass radius is independent of its initial value and is a function of the number of bound stars and the age only. It is therefore not possible to infer the initial mass–radius relation of globular clusters, and we can only conclude that the present day properties are consistent with the hypothesis that all hot stellar systems formed with the same mass–radius relation and that globular clusters have moved away from this relation because of a Hubble time of stellar and dynamical evolution.

Ever since Isaac Newton in 1687 posed the N-body problem, astronomers have been looking for its solutions in order to understand the evolution of dynamical systems, such as our own solar system, star clusters and galaxies. The main difficulty is that small errors grow exponentially, so that numerical solutions diverge easily from the mathematical solution. This thesis presents two new state of the art N-body algorithms, one of which is designed for high precision (Brutus) and the other for speed (Sakura). The assumption that N-body results are accurate in a statistical sense, is put to the test for three-body configurations. Finally, a new mathematical model is constructed that describes the origin of chaos in a dynamical systems, and explains the short Liapounov time of Comet Halley's orbit.