The time between successive identical Sun-relative configurations of two orbiting bodies: oppositions of Mars, inferior conjunctions of Venus, full Moons.
S = 1 / |1/P − 1/E|Example: P = 686.98 days (Mars), E = 365.25 days (Earth) gives S = 779.91 days, the mean interval between Mars oppositions.
Sidereal periods are NASA/JPL rounded values (days). Synodic period is computed as 1/|1/P − 1/365.25|.
| Planet | Sidereal (days) | Synodic (days) | Synodic (years) |
|---|
Illumination = (1 − cos(2π × age/29.530589))/2; elongation = 360° × age/29.530589. For a full phase calendar, moonrise times and Moon wallpapers, see the moon-phase-calculator.
Given the synodic period S and one known sidereal period E, solve 1/P = 1/E ± 1/S. The inferior branch (P < E) always exists; the superior branch (P > E) exists only when S > E.
Example: S = 779.91 days with E = 365.25 days gives P = 686.98 days on the superior branch — Mars's sidereal period.
The synodic period is S = 1 / |1/P - 1/E|, where P and E are the sidereal periods of the two bodies in the same unit. Example: Mars (P = 686.98 days) and Earth (E = 365.25 days) give S = 1/|1/686.98 - 1/365.25| = 779.91 days (about 2.14 years) — the mean interval between oppositions of Mars.
A sidereal period is one full orbit measured against the fixed stars; a synodic period is the time between successive identical configurations as seen from a moving observer — typically the Sun. Because the observer moves too, the synodic period is usually longer. The Moon is the classic example: its sidereal month (one orbit vs. the stars) is 27.321661 days, but its synodic month (new Moon to new Moon, the phase cycle we see) is 29.530589 days.
Venus orbits faster than Earth (P = 224.7 days vs. E = 365.25 days), so it laps us regularly. S = 1/|1/224.7 - 1/365.25| = 583.93 days (about 1.60 years) — the mean time between inferior conjunctions, when Venus passes between Earth and the Sun.
The mean synodic month — new Moon to new Moon — is 29.530589 days. It sets the whole lunar phase cycle: at age 29.530589/4 days the Moon is at first quarter, 90 degrees from the Sun with 50.0% of its face illuminated; at half a synodic month it is full.
Invert the formula: 1/P = 1/E ± 1/S, so P = 1/(1/E + 1/S) on the inferior branch (the unknown body orbits faster than the reference) or P = 1/(1/E - 1/S) on the superior branch (valid only when S > E). Example: Venus's synodic period S = 583.93 days vs. Earth gives P = 224.70 days on the inferior branch — Venus's sidereal period.
Mercury has the shortest synodic period, 115.88 days — it laps Earth most frequently. Mars has the longest, 779.91 days, because its orbit is the closest to Earth's speed among the planets, so Earth takes longest to lap it. Jupiter's is only 398.88 days by comparison.
Then 1/P - 1/E = 0 and the synodic period is unbounded — the bodies keep the same relative configuration forever and never drift into a new alignment. The calculator reports this as an infinite synodic period rather than a number.
Use this tool (synodic-period) for S = 1/|1/P - 1/E| — the time between repeated Sun-relative alignments. Use orbital-period for Kepler's sidereal orbital period T = 2π√(a³/μ) of a single body. period-calculator is unrelated: it predicts menstrual cycles ("when will my period start?"). For Moon phase calendars, moonrise times and wallpapers, use moon-phase-calculator.
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