Detecting stars, galaxies, and asteroids with Gaia

(Abridged) Gaia aims to make a 3-dimensional map of 1,000 million stars in our Milky Way to unravel its kinematical, dynamical, and chemical structure and evolution. Gaia's on-board detection software discriminates stars from spurious objects like cosmic rays and Solar protons. For this, parametrised point-spread-function-shape criteria are used. This study aims to provide an optimum set of parameters for these filters. We developed an emulation of the on-board detection software, which has 20 free, so-called rejection parameters which govern the boundaries between stars on the one hand and sharp or extended events on the other hand. We evaluate the detection and rejection performance of the algorithm using catalogues of simulated single stars, double stars, cosmic rays, Solar protons, unresolved galaxies, and asteroids. We optimised the rejection parameters, improving - with respect to the functional baseline - the detection performance of single and double stars, while, at the same time, improving the rejection performance of cosmic rays and of Solar protons. We find that the minimum separation to resolve a close, equal-brightness double star is 0.23 arcsec in the along-scan and 0.70 arcsec in the across-scan direction, independent of the brightness of the primary. We find that, whereas the optimised rejection parameters have no significant impact on the detectability of de Vaucouleurs profiles, they do significantly improve the detection of exponential-disk profiles. We also find that the optimised rejection parameters provide detection gains for asteroids fainter than 20 mag and for fast-moving near-Earth objects fainter than 18 mag, albeit this gain comes at the expense of a modest detection-probability loss for bright, fast-moving near-Earth objects. The major side effect of the optimised parameters is that spurious ghosts in the wings of bright stars essentially pass unfiltered.