Definition
The sidereal period is the time a body needs to complete one revolution around its central body, measured relative to the distant stars. It is the true orbital period: for a planet, the time to return to the same direction as seen from the Sun; for the Moon, the time to return to the same place among the stars as seen from the Earth.
It is distinct from the synodic period, the time between two identical configurations with the Sun as seen from the Earth, which mixes the planet's motion with the Earth's.
Kepler's third law
Kepler found in 1619 that the square of the period is proportional to the cube of the semi-major axis. Expressed in years and astronomical units for bodies orbiting the Sun:
P² = a³ (P in years, a in AU)P = 2π √(a³ / GM)Newton's form shows that the period depends only on the size of the orbit and the mass of the central body, not on the eccentricity. A comet on a very elongated orbit and a planet on a nearly circular one with the same semi-major axis have the same period. Example: Halley's comet, a = 17.8 AU, P = √(17.8³) = 75 years.
Sidereal periods of the planets
| Planet | a (AU) | Sidereal period | Mean speed |
|---|---|---|---|
| Mercury | 0.387 | 87.97 days | 47.4 km/s |
| Venus | 0.723 | 224.70 days | 35.0 km/s |
| Earth | 1.000 | 365.256 days | 29.8 km/s |
| Mars | 1.524 | 686.98 days | 24.1 km/s |
| Jupiter | 5.203 | 11.862 years | 13.1 km/s |
| Saturn | 9.537 | 29.457 years | 9.7 km/s |
| Uranus | 19.19 | 84.02 years | 6.8 km/s |
| Neptune | 30.07 | 164.8 years | 5.4 km/s |
Neptune, discovered in September 1846, completed its first full orbit since discovery in July 2011. Pluto, discovered in 1930, will need until 2178.
Three kinds of year
Measured against different reference points, the Earth's orbit gives years of different lengths:
| Year | Length | Reference |
|---|---|---|
| Sidereal year | 365.25636 days | The fixed stars |
| Tropical year | 365.24219 days | The vernal equinox, which drifts backwards with precession |
| Anomalistic year | 365.25964 days | Perihelion, which advances slowly |
The tropical year is about 20 minutes shorter than the sidereal year, because the equinox moves to meet the Sun by 50.3″ a year. Calendars follow the tropical year, so that the seasons stay on the same dates; the Gregorian calendar averages 365.2425 days.
Five kinds of month
| Month | Length | Returns to |
|---|---|---|
| Sidereal | 27.32166 days | The same star |
| Tropical | 27.32158 days | The same ecliptic longitude (equinox) |
| Anomalistic | 27.55455 days | Perigee |
| Draconic | 27.21222 days | The ascending node |
| Synodic | 29.53059 days | The same phase |
Each month governs a different phenomenon: the synodic month the phases, the anomalistic month the Moon's distance and apparent size, the draconic month the eclipses. Their near-commensurabilities produce the saros cycle used to predict eclipses.
How periods are measured
No observer can see a planet complete an orbit from outside. Sidereal periods were first derived from the synodic periods, which are directly observable, through 1/P = 1/E ∓ 1/S. Modern ephemerides such as VSOP87, used by Meeus and by this site, do not use a single period at all: they express the mean longitude of each planet as a polynomial in time, whose linear term is the mean motion, plus thousands of periodic terms from the planets' mutual attractions.
Explore it on AstroAlgos
Frequently asked questions
What is the sidereal period of the Earth?
365.25636 days, the sidereal year. The calendar year follows the slightly shorter tropical year of 365.24219 days, measured from equinox to equinox.
How do you calculate a sidereal period?
From the semi-major axis with Kepler's third law, P² = a³ (years and AU), or from an observed synodic period S with 1/P = 1/E − 1/S for an outer planet and 1/P = 1/E + 1/S for an inner one.
Why are there several lengths of month?
Because the Moon's return can be measured against different references: the stars (27.32 days), the Sun and its phases (29.53 days), its perigee (27.55 days) or its node (27.21 days). Each governs a different phenomenon.
