Synodic Period Calculator
Find how often two bodies line up as seen from a common center, and for three or more, the longer beat cycle on which they all realign.
Bodies
Pick a body from the list, or choose Custom… to type your own period. The period box stays grayed out and only becomes active when you choose Custom…. Add as many bodies as you like.
Reference tables
Synodic cycle table
Every pair's synodic period, how often the two bodies return to the same alignment, computed from their sidereal periods with the formula 1 / S = | 1 / P₁ − 1 / P₂ |. Read across a row and down a column to find a pair; the diagonal is blank because a body has no synodic period with itself. Choose a unit:
Synodic period in years (1 year = 365.25 days).
These ten are the classic set. For custom periods or extra bodies (Ceres, Chiron, Eris, the lunar nodes), use the calculator above.
Orbital period ratios
How many times longer one body's orbit is than another's: the longer sidereal period divided by the shorter. A ratio close to a small whole number or a simple fraction marks a near-resonance, where the two bodies return to a similar relative position after just a few orbits.
Standard sidereal periods
Mean values used by the picker. The lunar nodes entry is the retrograde nodal cycle, included for cycle work.
| Body | Sidereal period (days) | Years |
|---|---|---|
| Mercury | 87.969 | 0.2408 |
| Venus | 224.701 | 0.6152 |
| Earth | 365.256 | 1.0000 |
| Mars | 686.980 | 1.8808 |
| Jupiter | 4,332.589 | 11.8620 |
| Saturn | 10,759.220 | 29.4571 |
| Uranus | 30,688.500 | 84.0205 |
| Neptune | 60,182.000 | 164.7693 |
| Pluto | 90,560.000 | 247.9398 |
| Moon (sidereal) | 27.322 | 0.0748 |
| Ceres | 1,681.630 | 4.6041 |
| Chiron | 18,413.000 | 50.4120 |
| Eris | 203,830.000 | 558.0561 |
| Lunar Nodes (regression) (retrograde nodal cycle, not an orbit) | 6,798.380 | 18.6130 |
How the synodic period is calculated
The synodic period is the time for two orbiting bodies to return to the same relative alignment (for example, two planets lining up again as seen from the Sun). From their sidereal periods P₁ and P₂ it is:
1 / S = | 1 / P₁ − 1 / P₂ |
The faster body laps the slower one, and the absolute value makes the order of the two bodies irrelevant. The result is shown in both years and days (1 year = 365.25 days). If the two rates are equal there is no synodic period, because they never lap each other.
Synodic period vs sidereal period
These two are easy to mix up. The sidereal period is the true orbital period: the time a body takes to go once around the Sun against the fixed stars. The synodic period is the time between repeats of a configuration, such as one planet catching up to another as seen from a shared center. The calculator takes sidereal periods as input and returns a synodic period.
| Property | Sidereal period | Synodic period |
|---|---|---|
| Measured against | The fixed stars | Another moving body |
| Describes | One full orbit | Return to the same alignment |
| Mercury example | About 87.969 days | About 115.88 days from Earth |
| Used here as | Input (P) | Output (S) |
The synodic value can be longer or shorter than either orbit. Two bodies with nearly equal periods crawl past each other and take a long time to realign, while a fast inner body laps a slow outer body briskly. That is why Mercury's synodic period with Earth, about 115.88 days, is longer than its 87.969 day orbit.
Retrograde bodies (the lunar nodes)
Most bodies travel prograde. The lunar nodes travel retrograde, so when paired with a planet their relative speed is the sum of the two rates: 1/S = 1/P₁ + 1/P₂. For example, Jupiter and the lunar nodes realign about every 7.24 years, not the 32.7 years a plain difference would suggest.
Three or more bodies: the realignment cycle
For three or more bodies we first compute every pairwise synodic period, then take their least common multiple (LCM). That LCM is the beat cycle on which all the pairwise alignments come back into phase together, the full realignment of the group.
Because orbital periods are irrational numbers, an exact LCM does not strictly exist. We approximate it by rounding each pairwise period to two decimals of a day before taking the integer LCM. Treat the result as an approximate long-term beat rather than an exact prediction.
Worked example: Earth and Mars
Earth is about 365.256 days, Mars about 686.980 days.
1/S = |1/365.256 − 1/686.980| = 0.001282, so S is about 779.9 days, roughly 2.135 years. That matches the familiar 26 month spacing of Mars oppositions.
More worked examples
Each uses sidereal orbital periods and the formula above, with results rounded for readability.
Venus and Earth
Venus is about 224.701 days, Earth about 365.256 days.
1/S = |1/224.701 − 1/365.256|, so S is about 583.9 days, roughly 1.6 years. This is the rhythm behind Venus appearing as the morning and evening star.
Mercury and Earth
Mercury is about 87.969 days, Earth about 365.256 days.
1/S = |1/87.969 − 1/365.256|, so S is about 115.88 days, roughly 0.32 years. Because Mercury is fast and close to the Sun, it laps Earth several times a year.
Jupiter and Saturn: the great conjunction
Jupiter is about 4332.59 days, Saturn about 10759.22 days.
1/S = |1/4332.59 − 1/10759.22|, so S is about 7253 days, roughly 19.86 years. This is the great conjunction interval: the two largest planets line up about once every twenty years, most recently in December 2020.
Why synodic periods matter
A synodic period answers a practical question: how often does a sky event come back around? A few of the things it governs:
- Oppositions and conjunctions. An outer planet reaches opposition, its closest and brightest approach to Earth, once per synodic period. Mars does this about every 779.9 days, the familiar roughly 26 month spacing of good Mars viewing.
- Retrograde loops. A planet appears to slow, reverse, and loop backward against the stars once each synodic period, near opposition for outer planets.
- Launch windows. Mission planners aim for the part of the synodic cycle when two planets are best placed for a transfer, which is why Mars missions cluster about every 26 months.
- Long term cycle studies. Pairings such as Jupiter and Saturn realigning about every 19.86 years, or the multi-body beat cycle from the calculator above, are used to study long rhythms in the solar system.
In short, the sidereal period tells you how a body moves on its own, while the synodic period tells you how often you will see something happen.
Frequently asked questions
What is the difference between a synodic and a sidereal period?
The sidereal period is one full orbit against the fixed stars. The synodic period is the time for a body to return to the same alignment relative to another moving body, such as the Sun and Earth. This calculator takes sidereal periods in and gives a synodic period out.
Why can a planet's synodic period be longer than its orbit?
Because both bodies are moving. While the faster body completes an orbit, the slower one moves on too, so the faster body has to keep going to catch up to the same alignment. For bodies with close orbital speeds this takes a long time, which can make the synodic period far longer than either orbit.
What are the synodic periods of Venus, Mars, and Mercury?
As seen from Earth, Venus is about 583.9 days, Mars about 779.9 days, and Mercury about 115.88 days. You can reproduce any of these by entering the two sidereal periods above.
Why are Mars launch windows about 26 months apart?
Earth and Mars return to the same favorable alignment once per synodic period, about 779.9 days, which is roughly 26 months. That is when a transfer between the two planets needs the least energy, so missions cluster around it.
Are synodic periods exact?
No. Orbital periods are not whole numbers and vary slightly over time, so a synodic period is a mean value. Real intervals wobble around it, which is why successive Jupiter and Saturn conjunctions can fall anywhere from about 18 years 10 months to 20 years 8 months apart.