Understanding the Synodic Period
The synodic period is the time it takes for two orbiting bodies to return to the same relative alignment as seen from one of them — for example, the time between two successive oppositions of Mars as seen from Earth, or between two successive new moons as seen from Earth. It differs from the sidereal period, which is the time a single body takes to complete one full orbit relative to the fixed stars. Because both bodies are moving, the synodic period is always longer than the shorter of the two sidereal periods (for bodies orbiting the same direction around a shared center).
Deriving the formula
Each body sweeps out 360° (one full orbit) in its sidereal period T, so its average angular speed is 360°/T, or in rate terms 1/T orbits per unit time. The two bodies realign whenever the faster one has gained exactly one full lap (360°) on the slower one. The relative angular speed is the difference of the two rates, 1/T1 - 1/T2, so the time to gain one full lap — the synodic period S — satisfies 1/S = |1/T1 - 1/T2|. Solving for S gives the equivalent form S = (T1 × T2) / |T1 - T2|. Enter both sidereal periods in the same unit; the calculator handles the subtraction, inversion, and unit conversion for you.
Working with units and periods
- Both T1 and T2 must be sidereal periods (relative to the fixed stars), not synodic periods of some other pair — mixing the two types gives a meaningless result.
- Keep both periods in the same time unit (days or years) before comparing; this calculator lets you enter both in whichever unit you pick and converts internally.
- For Earth-based astronomy, one input is usually Earth's sidereal year (365.256 days) and the other is the target body's sidereal period (Moon: 27.322 days; Mars: 686.98 days; Jupiter: 4332.59 days).
- The result S is symmetric in form but its meaning depends on which body is "faster": the faster body laps the slower one once every S, in the direction of the faster body's motion.
Knowing the limits
This formula assumes both bodies orbit in the same direction (prograde) around a common center with roughly constant angular speed — true to a good approximation for planets and moons in near-circular, coplanar orbits. It does not directly account for orbital eccentricity (which makes the actual time between real-world events like oppositions vary slightly from one cycle to the next), inclination, or retrograde motion (a body orbiting the opposite direction would use 1/S = 1/T1 + 1/T2 instead of the difference). If the two periods are equal, the bodies stay in fixed relative alignment and the synodic period is infinite.