Past Blog Tie-in:
• Orbital Resonances And Synodic Period
As the orbital (sidereal) period of body B increases with distance from the central mass, it approaches that of the Body A, and the synodic period between them becomes progressively longer, approaching infinity. (Of course, the two bodies cannot have precisely the same orbital period unless they share an orbit, in which case the concept of a synodic period between them becomes meaningless.)
As the orbital period of Body C continues to increase with distance from the central mass, its becomes progressively longer than that of Body A, but the synodic period between them becomes progressively shorter, approaching zero.
At numerous points as the orbit of either Body B or Body C changes with distance from the central mass, the two bodies will achieve various states of mean-motion resonance with Body A, of varying orders.
Assuming the orbital period of Body A is 1.0 unit, with the orbital period of Body B increasing in steps of 0.1 units over the range [0.1, 0.9] units, and then in steps of 0.5 units over the range [1.5, 10.0] units; then, the value of their synodic period forms the following graph:
So, if a worldbuilder wishes to start out by specifying particular synodic periods between the bodies and then calculate their actual sidereal periods, it is important to realize that the greater the magnitude of the synodic period, the closer the two objects will be orbiting to one another.