Stability of planets in triple star systems

A thesis submitted to the Faculty of Science, University of the Witwatersrand South Africa in fulfilment of the requirements for the degree of Doctor of Philosophy. April 2018.This study investigates the space around triple systems to find the regions of secular stability of planetary orbits. Numerical N-body simulations are used to determine empirically the bounds of these regions as a function of the system’s configuration. There have been numerous theoretical studies of the stellar dynamics of triple systems, some with limited numerical checks, but the few purely empirical studies have been confined largely to binaries. Very little has been done on planetary orbits within either of these systems. There has been almost no work on generalised systems, little on retrograde planetary orbits and none on retrograde stellar orbits, with nearly all being on coplanar orbits and for a limited number of orbital parameters. This work expands into, and investigates new areas through 1. Providing a generalised mapping of the regions of planetary stability in triples, by: 2. examining all four types of orbits – P1, P2, S1 and S3; 3. investigating these orbit types for both prograde and retrograde motion of the planets; 4. investigating them for both prograde and retrograde motion of the outer body of the triple; 5. investigating highly-inclined orbits of the outer star, stellar Kozai resonance and its effect on the region of stability for P1 and P2 orbits; 6. extending the number of parameters used to all relevant orbital elements of the triple’s stars, and 7. expanding these elements and mass ratios to wider ranges that will accommodate recent and possible future observational discoveries. This resulted in semi-analytical models describing the stability bounds of each type of orbital configuration found in triples. These relationships can be used to guide searches for planets in triple systems and to determine quickly the feasibility of initial observational estimates of planetary orbital parameters, and to select suitable candidates for a survey of such systems. The geometry of the stable zone indicates not only where to look for planets but the most suitable search method. To highlight how the stability of planets in triple systems differs from that for binaries, an analysis of these systems over the same parameter space was required, resulting in a contribution to the body of empirical work on binaries as well. Key words: methods: numerical – methods: N-body simulations – planet-star interactions – celestial mechanics – stars: hierarchical triples – planetary systems: dynamical evolution and stabilityLG201

Metaheuristic approaches to realistic portfolio optimisation

In this thesis we investigate the application of two heuristic methods, genetic algorithms and tabu/scatter search, to the optimisation of realistic portfolios. The model is based on the classical mean-variance approach, but enhanced with floor and ceiling constraints, cardinality constraints and nonlinear transaction costs which include a substantial illiquidity premium, and is then applied to a large I 00-stock portfolio. It is shown that genetic algorithms can optimise such portfolios effectively and within reasonable times, without extensive tailoring or fine-tuning of the algorithm. This approach is also flexible in not relying on any assumed or restrictive properties of the model and can easily cope with extensive modifications such as the addition of complex new constraints, discontinuous variables and changes in the objective function. The results indicate that that both floor and ceiling constraints have a substantial negative impact on portfolio performance and their necessity should be examined critically relative to their associated administration and monitoring costs. Another insight is that nonlinear transaction costs which are comparable in magnitude to forecast returns will tend to diversify portfolios; the effect of these costs on portfolio risk is, however, ambiguous, depending on the degree of diversification required for cost reduction. Generally, the number of assets in a portfolio invariably increases as a result of constraints, costs and their combination. The implementation of cardinality constraints is essential for finding the bestperforming portfolio. The ability of the heuristic method to deal with cardinality constraints is one of its most powerful features.Decision SciencesM. Sc. (Operations Research