Synodic Period Calculator

Background.

Missions to Mars leave Earth in bursts about 26 months apart, and the spacing is set by geometry rather than engineering. Earth circles the Sun in a year and Mars in 687 days, so Earth keeps lapping Mars, and the two return to the same arrangement, with Mars opposite the Sun in our sky, only every 780 days or so. That interval between repeated alignments is the synodic period. The 687 days, one orbit measured against the fixed stars, is the sidereal period.

Converting between them needs nothing but the two sidereal periods. Each body moves through 360° divided by its period every day, the faster one gains on the slower at the difference of those rates, and an alignment comes round again each time the gain totals a full 360°: 1/S = |1/T₁ − 1/T₂|. Masses and distances play no part. The relation also runs backwards, which is how Copernicus turned dated oppositions and conjunctions into the planets' true orbital periods, and both inverse forms are offered here.

The answer is an average. The orbits are treated as circles in one plane, travelled at constant speed, whereas real ellipses make individual Mars oppositions arrive anywhere from about 764 to 811 days apart.

What is synodic period calculator?

The synodic period of two bodies circling the same primary is the average time between successive identical alignments as seen from the inner one: opposition to opposition, conjunction to conjunction, or new moon to new moon. A sidereal period, by contrast, is a single orbit measured against the distant stars. The synodic period is always longer than the faster body's sidereal period, and it grows without limit as the two sidereal periods approach each other, because the faster body then gains only slowly. The word comes from the Greek synodos, a meeting. The inputs are sidereal periods and the output synodic, or the other way round in the inverse modes.

How to use this calculator.

1. 01Choose the mode: the synodic period from two sidereal periods, or one body's sidereal period from the other's plus a known synodic period.
2. 02Enter periods in days. JPL tabulates planetary periods in Julian years, so multiply by 365.25: Mars's 1.8808476 years becomes 686.9796 days.
3. 03In the synodic mode the order of the two periods does not matter; the faster and slower bodies are sorted for you.
4. 04In 'inner from synodic', the known period must belong to the outer, slower body; in 'outer from synodic', to the inner, faster one. Swapping them gives a plausible but wrong orbit, or an error if the synodic period is shorter than the inner body's period.
5. 05Read the relative angular rate for the degrees per day the faster body gains, and alignments per year for how often the configuration recurs.

The formula.

A body with sidereal period T moves through 360/T degrees of its orbit per day. Two bodies sharing a primary therefore separate in longitude at 360/T_fast − 360/T_slow degrees per day, and the alignment recurs when that separation reaches 360°. Dividing through by 360 gives 1/S = 1/T_fast − 1/T_slow, or in product form S = T_fast × T_slow ÷ (T_slow − T_fast).

The inverse forms come from the same line: 1/T_inner = 1/T_outer + 1/S and 1/T_outer = 1/T_inner − 1/S. The second needs S to be longer than T_inner, since otherwise the result would be zero or negative, so such inputs are refused.

With two equal sidereal periods the faster body never gains, 1/S is zero and the synodic period is infinite, so equal periods are refused with an explanation. Nearly equal periods are allowed but produce an enormous interval, and once it exceeds a hundred times the slower orbit a note warns that the bodies are effectively co-orbital, where their mutual gravity, absent from this kinematic formula, will govern what they actually do.

Days are 86,400 seconds and years are Julian years of 365.25 days. Because the orbits are idealised as circular, coplanar and uniform, the result is the long-run average interval rather than the date of any particular alignment.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NASA JPL Solar System Dynamics, 'Planetary Physical Parameters', column 'Sidereal Orbital Period (y)': Mercury 0.2408467, Venus 0.61519726, Earth 1.0000174, Mars 1.8808476, Jupiter 11.862615, Saturn 29.447498, Uranus 84.016846, Neptune 164.79132. Source of every default and of the cross-checked classical synodic periods. Independent, open access. Retrieved 2026-07-29.
2. [2]NASA JPL Solar System Dynamics, 'Approximate Positions of the Planets' (Standish & Williams), Table 1: mean longitude rates in degrees per century, referred to the mean ecliptic and equinox of J2000 and valid 1800–2050 — Earth/Moon barycentre 35 999.37244981, Mars 19 140.30268499. Used as an INDEPENDENT route to the Earth–Mars synodic period. Independent, open access. Retrieved 2026-07-29.
3. [3]NASA Goddard Space Flight Center, eclipse pages, 'The Moon's Orbit and Rotation' (F. Espenak): sidereal month 27.32166 days (27d 07h 43m 12s), synodic month 29.53059 days (29d 12h 44m 03s), anomalistic month 27.55455 days, draconic month 27.21222 days. Used as an independent check on the lunar case. Independent, open access. Retrieved 2026-07-29.
4. [4]Copernicus, N. (1543). De revolutionibus orbium coelestium, Book V — the derivation of sidereal periods from observed synodic periods, which is the inverse this calculator implements. Print/bibliographic reference.