Synodic Period

Definition

The synodic period of a planet or of the Moon is the time it takes to come back to the same position relative to the Sun, as seen from the Earth: from one opposition to the next, from one inferior conjunction to the next, or from one full Moon to the next. The word comes from the Greek synodos, meeting.

It differs from the sidereal period, the time for one turn around the Sun relative to the stars, because the Earth is moving too. The synodic period is what an observer actually experiences: how often a planet returns to the evening sky, how often the Moon goes through its phases.

The formula

Think of two runners on a circular track. The faster gains one full lap on the slower once every synodic period. With E the Earth's sidereal period and P the planet's:

outer planet (P > E):   1/S = 1/E − 1/Pinner planet (P < E):   1/S = 1/P − 1/Ethe Moon:               1/S = 1/P_moon − 1/E

Example for Jupiter: 1/S = 1/365.256 − 1/4332.6 = 0.0025070, so S = 398.9 days. Every year Jupiter comes to opposition about 34 days later than the previous year.

Synodic periods of the planets

Two cases stand out. Mars and Venus, whose periods are close to the Earth's, have the longest synodic periods, because the Earth gains on them, or they gain on the Earth, very slowly. Distant planets barely move during a year, so their synodic periods approach one year from above.

The synodic month

For the Moon, the synodic period is the month of phases, or lunation: 29.530589 days on average. It is about 2.2 days longer than the sidereal month of 27.321662 days, because while the Moon circles the Earth, the Earth moves about 27° along its orbit and the Moon has to catch up to be opposite the Sun again.

Individual lunations vary between about 29.27 and 29.83 days, because the Moon's and the Earth's orbits are eccentric. Meeus computes the instants of the phases from a mean lunation formula in an integer k plus periodic corrections that reach about 14 hours. Lunar calendars, such as the Islamic calendar, count synodic months; twelve of them make 354.37 days, about 11 days less than a year.

Cycles built on synodic periods

• Venus: five synodic periods (2,919.6 days) are almost exactly eight years (2,922 days). Venus therefore repeats its evening and morning appearances on nearly the same dates every eight years, and its five inferior conjunctions trace a pentagram around the zodiac.
• Mars: missions to Mars can only leave in the window that comes once per synodic period, every 26 months, when the Earth and Mars are placed for an efficient transfer ellipse.
• Retrograde loops: every superior planet goes through one retrograde loop per synodic period, centred on opposition.
• The Metonic cycle: 235 synodic months are almost exactly 19 years, which is why the phases of the Moon fall on nearly the same dates every 19 years.

Measuring the sidereal period from the synodic one

Historically the formula was used backwards. Ancient astronomers could time oppositions or conjunctions, hence synodic periods, directly from the sky. Copernicus derived from them the sidereal periods of the planets, and with them their relative distances from the Sun, one of the strongest arguments for the heliocentric system.

Explore it on AstroAlgos

• Earth to Mars — launch windows every synodic period
• Heliocentric view — watch the Earth lap the outer planets
• Opposition — the start of each synodic cycle for outer planets
• Sidereal period — one turn relative to the stars