Synodic period calculator
Pick two planets, or the Moon and Earth, or enter your own sidereal periods. The calculator gives the synodic period, the time between two identical alignments as seen from one of them, such as one opposition of Mars to the next or one new moon to the next.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Choose the two bodies. Both must orbit the same centre: two planets around the Sun, or the Moon paired with Earth (the Moon’s month against Earth’s year gives the lunar phase cycle).
- For anything else choose Custom period and enter the sidereal period in days, years or hours.
- Read the synodic period in days, Julian years and hours.
1/S = |1/T₁ − 1/T₂|
Order does not matter. The closer the two periods, the longer the synodic period; equal periods never realign.
Worked example
Earth (T₁ = 365.256 days) and Mars (T₂ = 686.98 days):
S = 365.256 × 686.98 ÷ (686.98 − 365.256) = 250,923.6 ÷ 321.724 = 779.93 days, that is 2.1353 years. This is the time from one opposition of Mars to the next, when Mars is best placed for viewing.
Venus and Earth: 224.701 × 365.256 ÷ 140.555 = 583.92 days between inferior conjunctions.
Tips
- Outer planets come round about once a year: Jupiter’s synodic period with Earth is 398.88 days, so its oppositions move about a month later each year.
- Custom periods work for satellites too, for example two spacecraft in different orbits around Earth, or a satellite against Earth’s rotation.
- The periods are averages; real oppositions and conjunctions vary by days or weeks because orbits are elliptical.
Frequently asked questions
What is a synodic period?
The time it takes two bodies orbiting the same centre to come back to the same arrangement, for example Sun, Earth and Mars in a line. It differs from the sidereal period, which is one full orbit measured against the stars.
What is the formula?
1/S = |1/T₁ − 1/T₂|, so S = T₁ × T₂ ÷ |T₂ − T₁|. Each body moves 360° ÷ T per day; the synodic period is how long the faster one takes to gain a full turn on the slower.
Why is the synodic month longer than the sidereal month?
While the Moon goes round Earth in 27.3217 days, Earth moves along its orbit, so the Moon needs about two more days to line up with the Sun again: 27.3217 × 365.256 ÷ (365.256 − 27.3217) = 29.5306 days from one new moon to the next.
Which periods does it use for the planets?
Rounded sidereal periods in days: Mercury 87.969, Venus 224.701, Earth 365.256, Mars 686.98, Jupiter 4332.589, Saturn 10,759.22, Uranus 30,685.4, Neptune 60,189, Pluto 90,560, and the Moon 27.3217.