The photon sphere result assumes a non-rotating (Schwartzschild) black-hole. There is additional information about photon spheres around rotating black holes in the help file, although you should research what these are to get a more thorough understanding.
Synodic periods are important when using low-energy transfers from one orbit to another (which might equate to a transfer from one planet to another), where the timing of the arrival of both the target object and the spacecraft is important. The target object could be another planet, or a specific location in the geostationary orbit, and synodic periods dictate when such transfers can take place.
You can read further details on what the software can doand download it from the .