The synodic period is the time it takes for the earth, sun and planet to achieve the same relative positions to each other. The sidereal period is the time it takes a planet to go around the sun once. Put another way, the synodic period is the period of a planet relative to the sun as seen from the earth. The sidereal period is the period of a planet around the sun, but viewed against the backdrop of the "fixed" stars. (When we talk about the length of a year on different planets, we are talking about the sidereal period.) It is very easy to measure the synodic period directly from earth; it is impossible to measure the sidereal. However, it is very easy to calculate the sidereal period from the synodic.
A planet is in opposition when it is exactly opposite the sun; one can draw a straight line from the sun through the earth and then through the planet. The time it takes for a planet to go from opposition to opposition is the synodic period of that planet. Because an inner planet always goes faster than an outer planet, the synodic period is simply the time it takes the inner planet to lap the outer planet. The following diagram shows two successive oppositions.
Let's first imagine that the earth is the inner planet in the above diagram. Let's also call the time between two successive oppositions S. If we measure S in years, that also happens to be exactly the number of times that the earth has gone around the sun. Since the earth is lapping the outer planet, that means the outer planet has gone around the sun exactly 1 less time than the earth. Calling the time it takes the outer planet to go around the sun T, T is the sidereal period, which is simply time/orbit, or:
If we imagine that the earth is the outer planet, then everything is the same, except that the inner planet is lapping the earth, so that the inner planet has made exactly 1 more trip around the sun than the earth. So we can say:
It is important to note that the sidereal period is rather meaningless in the Ptolemeic system, as the earth is the center of all the orbits. In the Ptolemeic system, the synodic periods are measured and are important, but one would not care about how long it takes a planet to go around the sun. One would care about how long it takes a planet to go around the earth, and this could be measured directly. It wasn't until Copernicus that calculating the sidereal period became important, as Copernicus put the sun at the center, and had the earth just another planet.
The table below shows the sidereal and synodic periods for the planets in our solar system:
|Synodic Period (days)||115.9||583.8||779.9||398.9||378.1||369.7||367.5||366.7|
|Sidereal Period (days)||88.0||224.7||687.0||4332