I have access to "rise" and "set" times for astronomical objects, and want to determine the corresponding times of culmination. Is there a reliable and accurate way to do this? It isn't clear to me from the sources of the data I have whether the times I have are for the geometric centers of the objects, or take into account refraction, or the angular extents of the objects, so I'm hoping there is an approach that is insensitive to these factors. In particular, I'm interested in transit times for the Sun.
To a very good approximation, the transit time can be estimated as the arithmetic mean of the rise and set times. However, this will not work accurately for rapidly moving objects (e.g. Moon). I followed the link in your question not realizing it was in the Mathematica forum and I'm not familiar with how Mathematica computes these times. However, I am willing to bet that atmospheric refraction is accounted for in the traditional way, that is by adding 34 arc minutes to the object's altitude as rise/set. Refraction acts to increase an object's altitude, but the effect is zero at the zenith. Anyway, if refraction is accounted for, the rise/set times should be for either the center of the object's disk or the upper limb of the object's disk. There are two conventions used here. In the U.S. all sunrise/sunset/moonrise/moonset times are for upper limb of Sun's/Moon's disk. However, most European almanacs tabulate these times for the center of Sun's/Moon's disk. All other celestial objects are too small to show a naked eye disk, so the rise/set times refer to the disk's center.