Synodic Period Calculator (time between alignments of two orbits)

Understanding the synodic period calculator — concept and principle

The synodic period is the time between two successive alignments of two bodies that orbit the same centre, such as Earth and Mars lining up with the Sun. This calculator finds it from the two sidereal orbital periods and also shows how fast the angle between them changes and how many orbits each body completes per cycle. Astronomy students and observers planning oppositions or launch windows use it.

Oppositions repeat when Earth has gained one full lap: about 780 days.

How it is calculated

1/S = |1/P₁ − 1/P₂|

P₁ and P₂ are the sidereal orbital periods in days (measured relative to the stars).
S is the synodic period in days; S ÷ 365.25 gives years.
Relative angular speed = 360° ÷ S per day.
Orbits per cycle = S ÷ P for each body.

The principle

Each body sweeps 360° per orbit, so its angular speed is 360°/P. When both orbit in the same direction, the faster one gradually pulls ahead. The angle between them grows at the difference of the two speeds, 360° × (1/P₁ − 1/P₂) per day. They line up again when the faster body has gained exactly one full lap, which takes S days. It is the same idea as the minute and hour hands of a clock meeting about every 65.5 minutes. For a planet seen from Earth, S is the interval between successive oppositions (outer planets) or similar conjunctions (inner planets).

Worked example

Default inputs: P₁ = 365.256 days (Earth), P₂ = 686.980 days (Mars).

1/365.256 = 0.0027378 per day; 1/686.980 = 0.0014557 per day.
Difference = 0.0012822 per day.
S = 1 ÷ 0.0012822 = 779.9 days = 2.135 years.
Relative angular speed = 360 ÷ 779.9 = 0.4616° per day.
Earth completes 779.9 ÷ 365.256 = 2.135 orbits and Mars 779.9 ÷ 686.98 = 1.135 orbits per cycle.

Each point is the calculator re-run with only “Orbital period of the first body (days)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Orbital period of the first body (days)	Synodic period (days)
91.3	105.3
183	248.8
274	455.6
365	779.9
457	1,361.3
548	2,705.9
731	11,528.2

How to read the result

The closer two periods are, the longer the synodic period: bodies with nearly equal periods drift apart very slowly. Mars oppositions come roughly every 26 months, which is why Mars missions launch on that rhythm. Outer planets far from the Sun have synodic periods only slightly longer than a year, because Earth laps them almost once a year. The number of orbits per cycle always differs by exactly one between the two bodies.

Common mistakes

Entering synodic periods instead of sidereal ones.
Using periods of bodies that orbit different centres, such as a moon and a planet.

Keep in mind: The result is an average based on circular, coplanar motion. Actual alignment dates shift because orbits are elliptical and inclined, so use an ephemeris for exact dates.

Frequently asked questions

What is the difference between sidereal and synodic periods?

The sidereal period is one orbit relative to the stars; the synodic period is the time between repeated alignments as seen from another moving body.

Why is the Moon's synodic month longer than its sidereal month?

Earth moves around the Sun during the month, so the Moon needs extra time to return to the same Sun–Earth–Moon alignment.

Can the formula be used for satellites?

Yes, for two satellites around the same planet in the same direction, such as a spacecraft and a space station.

Sources: MATC astronomy course notes, "Sidereal and Synodic Periods" · NASA NSSDCA Planetary Fact Sheets (D. R. Williams, NASA Goddard)

Checked 9 Nov 2026. Synodic–sidereal relation 1/P = 1 ± 1/S (periods in years), MATC astronomy course notes “Sidereal and Synodic Periods”; sidereal orbit periods from NASA NSSDCA Planetary Fact Sheets (D. R. Williams, NASA Goddard).

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