Orbital period

The orbital period is the time taken for a given object to make one complete orbit about another object.

When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.

There are several kinds of orbital periods for objects around the Sun (or other celestial objects):

• The sidereal period is the temporal cycle that it takes an object to make one full orbit around the Sun, relative to the stars. This is considered to be an object's true orbital period.
• The synodic period is the temporal interval that it takes for an object to reappear at the same point in relation to two other objects (linear nodes), i.e. the Moon relative to the Sun as observed from Earth returns to the same illumination phase. The synodic period is the time that elapses between two successive conjunctions with the Sun-Earth line in the same linear order. The synodic period differs from the sidereal period due to the Earth orbit around the Sun, with one less synodic period than lunar orbit per solar orbit.
• The draconitic period is the time that elapses between two passages of the object at its ascending node, the point of its orbit where it crosses the ecliptic from the southern to the northern hemisphere. It differs from the sidereal period because the node is a planar coincidence rather than a linear coincidence, and the object's line of nodes typically precesses or recesses slowly in relation to orbital cycle.
• The anomalistic period is the time that elapses between two passages of the object at its perihelion, the point of its closest approach to the Sun. It differs from the sidereal period because the object's semimajor axis typically advances slowly.
• The Earth tropical period or year, finally, is the time that elapses between two alignments of the axis of rotation with the Sun, also viewed as two passages of the object at right ascension zero. One Earth year has a shorter interval than solar orbit (sidereal period) because the inclined axis and equatorial plane slowly precesses (rotates in sidereal terms), realigning before orbit completes with an interval equal to the inverse of the precession cycle (about 25,770 years).

Relation between sidereal and synodic period

Copernicus devised a mathematical formula to calculate a planet's sidereal period from its synodic period.

Using the abbreviations

E = the sidereal period of Earth (a sidereal year, not the same as a tropical year)

P = the sidereal period of the other planet

S = the synodic period of the other planet (as seen from Earth)

During the time S, the Earth moves over an angle of (360°/E)S (assuming a circular orbit) and the planet moves (360°/P)S.

Let us consider the case of an inferior planet, i.e. a planet that will complete one orbit more than Earth before the two return to the same position relative to the Sun.

and using algebra we obtain

For a superior planet one derives likewise:

Generally, knowing the sidereal period of the other planet and the Earth, P and E, the synodic period can easily be derived:

which stands for both an inferior planet or superior planet.

The above formulae are easily understood by considering the angular velocities of the Earth and the object: the object's apparent angular velocity is its true (sidereal) angular velocity minus the Earth's, and the synodic period is then simply a full circle divided by that apparent angular velocity.

Table of synodic periods in the Solar System, relative to Earth:

In the case of a planet's moon, the synodic period usually means the Sun-synodic period. That is to say, the time it takes the moon to complete its illumination phases, competing the solar phases for an observer on the planet's surface —the Earth's motion does not determine this value for other planets, because an Earth observer is not orbited by the moons in question. For example, Deimos' synodic period is 1.2648 days, 0.18% longer than Deimos' sidereal period of 1.2624 d.

Calculation

Small body orbiting a central body

In astrodynamics the orbital period (in seconds) of a small body orbiting a central body in a circular or elliptic orbit is:

where:

• is length of orbit's semi-major axis,
• is the standard gravitational parameter,
• is the gravitational constant,
• the mass of the central body.

Note that for all ellipses with a given semi-major axis the orbital period is the same, regardless of eccentricity.

Orbital period as a function of central body's density

For the Earth (and any other spherically symmetric body with the same average density) as central body we get

and for a body of water

T in hours, with R the radius of the body.

Thus, as an alternative for using a very small number like G, the strength of universal gravity can be described using some reference material, like water: the orbital period for an orbit just above the surface of a spherical body of water is 3 hours and 18 minutes. Conversely, this can be used as a kind of "universal" unit of time.

For the Sun as central body we simply get

T in years, with a in astronomical units. This is the same as Kepler's Third Law

Two bodies orbiting each other

In celestial mechanics when both orbiting bodies' masses have to be taken into account the orbital period can be calculated as follows:

where:

• is the sum of the semi-major axes of the ellipses in which the centers of the bodies move, or equivalently, the semi-major axis of the ellipse in which one body moves, in the frame of reference with the other body at the origin (which is equal to their constant separation for circular orbits),
• and are the masses of the bodies,
• is the gravitational constant.

Note that the orbital period is independent of size: for a scale model it would be the same, when densities are the same (see also Orbit#Scaling in gravity).

In a parabolic or hyperbolic trajectory the motion is not periodic, and the duration of the full trajectory is infinite.

Earth orbits

Binary stars

See also

• Geosynchronous orbit derivation
• Sidereal time
• Sidereal year
• Opposition (astronomy)

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