Synodic Period Calculator

Background.

A planet's year and the interval at which you can actually see it are two different numbers, and the gap between them catches out almost everyone the first time. Mars takes 687 days to go once round the Sun, but Mars comes to opposition — closest and brightest, opposite the Sun in Earth's sky — only every 780 days. The difference exists because the observer is moving too. By the time Mars has completed a lap, Earth has completed nearly two and has to chase it down before the two line up again. The first figure is the sidereal period, measured against the fixed stars; the second is the synodic period, measured against a moving observer, and this calculator converts between them.

The relation is subtraction of angular rates. The inner body sweeps through 360°/T_fast degrees a day and the outer through 360°/T_slow, so the inner one gains on the outer at the difference of those rates. It needs to gain a full 360° for the alignment to repeat, which gives 1/S = |1/T_fast − 1/T_slow|, or equivalently S = T_fast·T_slow ⁄ |T_slow − T_fast|. No masses, no gravitational constant and no distances appear anywhere: this is pure kinematics.

The calculator also runs the relation backwards, because that is how the heliocentric solar system was actually measured. Copernicus could observe synodic periods directly — an opposition is an event you can date — but sidereal periods are not directly observable from a moving Earth. Rearranging this same equation let him derive them, and combining those with Kepler's third law eventually gave the scale of the solar system. Give this page a synodic period and one sidereal period and it returns the other.

The singularity is worth understanding rather than avoiding. If two bodies share the same sidereal period they never gain on one another at all, 1/S is zero, and the synodic period is infinite: they stay in fixed relative position forever. That is not a numerical failure but a real configuration — Jupiter's Trojan asteroids sit permanently 60° ahead of and behind the planet for exactly this reason. The calculator refuses equal periods with an explanation rather than returning infinity, and it flags the near-equal case too, where the answer becomes astronomically large and mutual gravitational perturbation — which this kinematic formula ignores completely — starts to matter far more than the drift it computes.

One accuracy limit belongs beside the answer. Both orbits are treated as circular, coplanar and uniform, so the result is the MEAN interval. Real orbits are elliptical and inclined, and individual alignments run early or late by weeks: Mars's mean synodic period is 780 days, but actual opposition-to-opposition intervals range from about 764 to 811 days depending on where in its eccentric orbit each one falls. Use this number to plan a decade, not to date a specific event — for that you need an ephemeris.

Periods here are in days of exactly 86 400 seconds and years are Julian years of exactly 365.25 days, the IAU convention. The defaults are Earth and Mars, taken from JPL's tabulated sidereal orbital periods.

What is synodic period calculator?

The synodic period of two bodies orbiting the same primary is the mean interval between successive identical alignments as seen from one of them. For planets seen from Earth it is the time from one opposition to the next, or one conjunction to the next. For the Moon it is the time from one new moon to the next — the synodic month of 29.53 days, which is why the calendar month is roughly that length.

It is always longer than the shorter of the two sidereal periods, and for an outer planet it is longer than Earth's year. The reason is that alignment requires the inner body to gain a complete extra lap on the outer one. The closer the two sidereal periods are, the longer that takes: Mars, whose year is not quite twice Earth's, has a synodic period of 780 days, while Jupiter, whose year is nearly twelve times Earth's, comes to opposition every 399 days — barely more than an Earth year, because Jupiter has hardly moved by the time Earth comes round again.

The word comes from the Greek sunodos, a meeting or conjunction. The distinction between synodic and sidereal quantities runs through positional astronomy: there is a synodic and a sidereal month, a synodic and a sidereal day, and a synodic and a sidereal period for every planet.

How to use this calculator.

1. 01Choose what you are solving for: the synodic period, or one of the sidereal periods if you already know the synodic one.
2. 02Enter the known sidereal period in days. Earth's is pre-filled.
3. 03For the first mode, enter the second body's sidereal period. For the inverse modes, enter the known synodic period instead.
4. 04In 'inner from synodic' the period you supply must be the OUTER body's; in 'outer from synodic' it must be the INNER body's.
5. 05Read the synodic period in days and years, and the relative angular rate if you want to know how fast the alignment drifts.
6. 06Remember this is the mean interval. Individual alignments can be weeks early or late because real orbits are elliptical.

The formula.

Think in angular rates. A body with sidereal period T sweeps through 360/T degrees per day. Two bodies orbiting the same primary therefore separate in longitude at 360/T_fast − 360/T_slow degrees per day. An alignment repeats when that accumulated difference reaches a full 360°, so the time required is 360 ⁄ (360/T_fast − 360/T_slow), which simplifies to 1/S = 1/T_fast − 1/T_slow. The absolute value in the formula just removes the need to know in advance which body is which.

Inverting is straightforward algebra and is the historically important direction. Given the outer body's sidereal period and the synodic period, 1/T_inner = 1/T_outer + 1/S. Given the inner body's sidereal period, 1/T_outer = 1/T_inner − 1/S. The second of these carries a physical constraint the calculator enforces: the synodic period must exceed the inner body's sidereal period, because otherwise 1/T_outer comes out zero or negative, which is not an orbit.

ROUNDING STAGE. Nothing is rounded part-way through. All arithmetic runs at 40 significant digits in a dedicated high-precision decimal context, and rounding happens once, at the return boundary, to 12 significant digits. That same rounded value drives the near-co-orbital note, so the note and the number on screen can never disagree.

SIGNIFICANT FIGURES. Two limits apply, and the second dominates. The tabulated sidereal periods carry seven or eight significant figures, so the arithmetic supports about seven. But the circular-orbit assumption is worth only about two: real Mars oppositions land anywhere in a 764–811 day window around the 780-day mean. Quote the mean to three figures at most, and never treat it as a prediction of a specific date.

INVALID DOMAIN AND THE SINGULARITY. Every period must be strictly positive. Two identical sidereal periods make 1/S = 0 and S infinite; that is the genuine singularity of this relation, and the calculator raises a labelled error explaining it rather than returning infinity. Near-identical periods are allowed but produce an enormous answer, which is flagged: once the synodic period exceeds a hundred times the slower orbit, the two bodies are effectively co-orbital and mutual gravitational perturbation — entirely absent from this formula — will dominate their real behaviour.

APPROXIMATION REGIME. Circular, coplanar, uniform orbits with constant periods. No eccentricity, no inclination, no perturbations, no orbital decay.

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.