period is the time it takes for a celestial body
in the solar
to complete a return to an appearance it started from, as it is seen from an observer
such as the Earth
and is affected by a third body such as the Sun
For example, the Moon's synodic period is seen from Earth as the time period between the appearances of two full moons. The sidereal period of the Moon is 27.3 days, but it takes 29.5 days for the moon to complete a synodic period as a result of the Earth's move around the sun. If the Earth stood still, the synodic period would have matched the sidereal period.
It is possible to approximately calculate the synodic period from the sideral
period. Let's take variable sy as the synodic period we would like to calculate. The Earth travels 360/366.25 = ~0.983 degrees/day around the Sun, so during the synodic period, it will pass sy*0.985 degrees. The Moon's sidereal period is 360/27.321 = ~13.176 degrees/day. If we divide the two we get the difference in days between the synodic and the sidereal, because it will be the time needed for the Moon to compensate for its appeareance as a result of Earth's orbit around the Sun. So sy = 27.321 + sy*0.983/13.176. If we isolate sy we get sy = ~29.523. After isolation and assigning variables instead of numbers, we get the formula:
synodic = sidereal/(1 - sidereal/observer_sidereal)
And checking with our data:
synodic = 27.321/(1 - 27.321/366.25) = ~29.523
Note that these calculations do not take into account the elliptical orbits
of the objects. Plus, calculating the synodic period of another planet like Mars as seen from Earth, requires a different formula because Mars doesn't orbit the Earth like the Moon does.