A Semi-Analytical Model  for Gravitational  Microlensing

This thesis describes the theory and implementation of a semi-analytical model for gravitational microlensing. Gravitational microlensing is observed when a distant background `source' star comes into close alignment with an intermediate `lens' star. The gravitational eld of the lens de ects the paths of light emitted from the source, which causes an increase in its observed brightness. As the alignment of the two stars changes with time, the apparent magni cation of the source follows a well de ned `lightcurve'. A companion body (such as a planet) orbiting the lens star can introduce large deviations from the standard lightcurve, which can be modelled to determine a mass ratio and separation for the companion(s). This provides a means to detect extrasolar planets orbiting the lens star. We show, from basic principles, the development of the standard model of a mi- crolensing event, including the e ect of multiple lens masses and orbital motion. We discuss the two, distinctly di erent, numerical approaches that are used to calculate theoretical lightcurves using this model. The `ray shooting' approaches are discussed with reference to the previously developed modelling code (MLENS), which implemented them. This is followed by a comprehensive description of the `semi-analytical' approaches used in the new software (mlens2) developed during this thesis programme; a key feature of these techniques is the determination of the source magni cation from the roots of a high order polynomial. We also discuss the process of nding the best- t model for an observed microlensing event, with respect to the mlens2 software package. Finally, we demonstrate the capabilities of our semi-analytical model by generating theoretical lightcurves for the microlensing events OGLE-2005-BLG-390 and OGLE-2006-BLG-109 and comparing them to the observational data and published models.