Synodic Period Calculator: Planetary Conjunctions and Oppositions
Enter the sidereal orbital period of any two bodies orbiting the same star (or the same planet) and this calculator finds their synodic period: the time between successive conjunctions or oppositions as seen from one of the bodies. Choose from built-in solar system presets or enter any custom orbital period. The show-your-work panel walks through every step of the calculation with your actual numbers.
Formula
Worked example
Earth (P1 = 365.25 days) observing Mars (P2 = 686.97 days): 1/365.25 = 0.002738, 1/686.97 = 0.001456, difference = 0.001282, so S = 1/0.001282 = 780 days (about 2.14 years). Mars oppositions therefore occur roughly every 26 months.
What is the synodic period?
The synodic period is the time it takes for two orbiting bodies to return to the same relative configuration as seen from one of them. For Earth and Mars, it is the gap between successive oppositions (when Mars rises exactly as the Sun sets). For the Moon, it is the gap between successive full moons, which is the familiar 29.53-day lunar month. The key point is that the synodic period depends on the observer's location: an astronomer on Mars would measure a different synodic period for Earth than an astronomer on Earth measures for Mars.
The synodic period differs from the sidereal period. The sidereal period is the time a body takes to complete one orbit relative to the distant stars, measured from an inertial vantage point. Because Earth is itself moving around the Sun, it takes Mars longer to catch up with Earth again than it would if Earth were stationary, so the synodic period is always longer than the sidereal period for outer planets. For inner planets such as Venus the relationship is reversed: Venus orbits faster than Earth, so it laps Earth periodically and the synodic period (584 days) is much longer than its sidereal period (225 days).
The synodic period formula
The relationship between sidereal periods P1 and P2 and the synodic period S follows directly from angular velocities. Each body sweeps 360 degrees in its sidereal period, so its angular velocity is 360/P degrees per day. The faster body gains on the slower at a rate equal to the difference of those velocities. One synodic period elapses when the faster body has gained exactly 360 degrees on the slower:
1/S = |1/P1 - 1/P2|
This form handles both cases automatically. If P1 is Earth (365.25 days) and P2 is Mars (686.97 days), then 1/S = |1/365.25 - 1/686.97| = 0.001282, giving S = 780 days. If instead P2 is Venus (224.70 days), the same formula gives 1/S = |1/365.25 - 1/224.70| = 0.001717, so S = 582 days, matching the known 584-day Venus synodic period to within rounding.
The older textbook form splits into two cases: 1/S = 1/P_inner - 1/P_outer for inferior planets, and 1/S = 1/P_outer - 1/P_inner for superior planets. Both are equivalent to taking the absolute difference; the absolute-value form is simpler and works universally.
How to use this calculator
Select a preset from each dropdown to auto-fill the sidereal periods for any two solar system bodies, or choose "Custom" and type in any period you like. The calculator works for moons, asteroids, and exoplanets as long as you know the sidereal orbital periods in the same unit (this tool uses Earth days). Switch the output unit between days and years depending on which is more readable for your case: nearby planets such as Mercury produce short synodic periods where days are natural, while outer planets produce multi-year synodic periods where "years" is easier to grasp.
The results panel shows the synodic period in both units, the number of conjunctions per Earth year, and the relative angular speed. The "show your work" panel beneath the results walks through every arithmetic step with your exact input numbers.
Inferior and superior planets
Astronomers classify planets by their position relative to Earth and the Sun. Inferior planets (Mercury and Venus) orbit closer to the Sun than Earth and display a full cycle of phases, similar to the Moon. They reach inferior conjunction when they pass between Earth and the Sun, and superior conjunction when they are on the far side of the Sun. Superior planets (Mars through Neptune) orbit farther from the Sun than Earth; they reach opposition (directly opposite the Sun, rising at sunset) and conjunction (behind the Sun). Opposition is the best time to observe a superior planet because it is closest to Earth and fully illuminated.
From an observer on another planet the roles shift. A hypothetical observer on Mars would see Earth as an inferior planet, passing through phases and reaching inferior conjunction when Earth is between Mars and the Sun. This calculator handles any configuration: just enter the two sidereal periods in any order and the formula returns the correct synodic period regardless of which body is "inner" or "outer".
Practical applications
Knowing the synodic period lets you plan observations. Mars oppositions recur every 780 days (about 26 months), and the orbital geometry is most favorable for observing surface features near opposition. Jupiter oppositions recur every 399 days, just over a year, so Jupiter is well placed in the evening sky for several months each year. Venus reaches inferior conjunction every 584 days and switches from the "evening star" to the "morning star" around that event.
The synodic period also governs spacecraft launch windows. Because travel time from Earth to Mars depends on the relative positions of the two planets, launch windows open roughly every 780 days when the geometry is favorable. Mission planners also use it to time the return leg and to predict when a spacecraft in a phasing orbit will rendezvous with its target. The same principle applies to any two bodies in orbit around a common center, from moons around a giant planet to spacecraft in different Earth orbits.
Sidereal and synodic periods of solar system bodies
| Body | Sidereal period (days) | Synodic period (days) | Type |
|---|---|---|---|
| Mercury | 87.97 | 116 | Inferior |
| Venus | 224.7 | 584 | Inferior |
| Mars | 686.97 | 780 | Superior |
| Jupiter | 4332.59 | 399 | Superior |
| Saturn | 10759.22 | 378 | Superior |
| Uranus | 30688.5 | 370 | Superior |
| Neptune | 60195 | 368 | Superior |
| Moon | 27.32 | 29.53 | Satellite |
Sidereal periods from NASA planetary fact sheets; synodic periods relative to Earth as the observer.
Frequently asked questions
What is the difference between the synodic period and the sidereal period?
The sidereal period is the true orbital period of a body relative to the distant stars, measured from an inertial frame. The synodic period is the time between successive same-configuration events (such as full moon or opposition) as seen from a particular moving observer. Because the observer is itself orbiting, the synodic period is always different from either body's sidereal period.
Why does Mars have a synodic period longer than a year even though its orbit is shorter than two years?
Earth is chasing Mars around the Sun. Mars takes 687 days to complete one orbit, but by the time it has done so, Earth has also moved. Earth must travel a bit further to catch up with Mars again, which adds several months to the next opposition. The synodic period of 780 days reflects that extra catch-up time. For very slow outer planets such as Neptune, their sidereal period is so much longer than Earth's that each opposition arrives just a few days later than the one before: Neptune's synodic period is only 368 days.
Can this calculator work for moons around a planet, not just planets around the Sun?
Yes. The formula 1/S = |1/P1 - 1/P2| applies to any two bodies orbiting the same central body. Enter the sidereal orbital periods of two moons around Jupiter (or two spacecraft in different orbits around Earth) and the calculator returns their synodic period relative to each other.
What happens when both bodies have the same sidereal period?
If P1 = P2, the denominator of the formula is zero and the synodic period is mathematically infinite: the bodies orbit at the same angular speed and never change their relative configuration. In practice, perfect equality means they are in a co-orbital resonance (like a Trojan asteroid at a stable Lagrange point) or the same object.
How do I find the date of the next opposition or conjunction?
Take any past opposition or conjunction date as a reference event. Add the synodic period (in days) to that date to get the next one, and keep adding to get future events. For example, Mars opposition occurred on November 19, 2022. Adding 780 days gives approximately January 16, 2025, which matches the 2025 opposition within a few days. Small deviations occur because planetary orbits are ellipses, not perfect circles.
Why is Venus's synodic period so much longer than its sidereal period?
Venus orbits the Sun in 225 days, but its synodic period as seen from Earth is 584 days. Venus moves faster than Earth but must complete more than one full lap of its own orbit before the geometry repeats. Think of two cars on a circular track: the faster car (Venus) laps the slower one (Earth) after it has gained a full lap, which takes longer than either car's individual lap time when the speed difference is small relative to the lap speeds involved.