Orbital Period

Information about Orbital Period

The orbital period is the time taken for a planet (or another object) to make one complete orbit.

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:

The sidereal period is the time that it takes the 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 time that it takes for the object to reappear at the same point in the sky, relative to the Sun, as observed from Earth; i.e. returns to the same elongation (and planetary phase). This is the time that elapses between two successive conjunctions with the Sun and is the object's Earth-apparent orbital period. The synodic period differs from the sidereal period since Earth itself revolves around the Sun.
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 object's line of nodes typically precesses or recesses slowly.
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 precesses or recesses slowly.
The tropical period, finally, is the time that elapses between two passages of the object at right ascension zero. It is slightly shorter than the sidereal period because the vernal point precesses.

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.

$\text{[math]}$

and using algebra we obtain

$\text{[math]}$

For a superior planet one derives likewise:

$\text{[math]}$

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

$\text{[math]}$ ,

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:

	Sid. P. (a)	Syn. P. (a)	Syn. P. (d)
Mercury	0.241	0.317	115.9
Venus	0.615	1.599	583.9
Earth	1	—	—
Moon	0.0748	0.0809	29.5306
Mars	1.881	2.135	780.0
4 Vesta	3.629	1.380	504.0
1 Ceres	4.600	1.278	466.7
10 Hygiea	5.557	1.219	445.4
Jupiter	11.87	1.092	398.9
Saturn	29.45	1.035	378.1
Uranus	84.07	1.012	369.7
Neptune	164.9	1.006	367.5
134340 Pluto	248.1	1.004	366.7
136199 Eris	557	1.002	365.9
90377 Sedna	12050	1.00001	365.1

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 run its phases, coming back to the same solar aspect angle for an observer on the planet's surface —the Earth's motion does not affect this value, because an Earth observer is not involved. 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 $\text{[math]}$ (in seconds) of a small body orbiting a central body in a circular or elliptical orbit is:

$\text{[math]}$

and

$\text{[math]}$ (standard gravitational parameter)

where:

$\text{[math]}$ is length of orbit's semi-major axis (m),
$\text{[math]}$ is the standard gravitational parameter,
$\text{[math]}$ is the gravitational constant,
$\text{[math]}$ the mass of the central body (kg).

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

$\text{[math]}$

and for a body of water

$\text{[math]}$

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

$\text{[math]}$

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 $\text{[math]}$ can be calculated as follows:

$\text{[math]}$

where:

$\text{[math]}$ 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),
$\text{[math]}$ and $\text{[math]}$ are the masses of the bodies,
$\text{[math]}$ 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

orbit	center-to-center distance	altitude above the Earth's surface	speed	period/time in space	specific orbital energy
minimum sub-orbital spaceflight (vertical)	6500 km	100 km	0.0 km/s	just reaching space	1.0 MJ/kg
ICBM	up to 7600 km	up to 1200 km	6 to 7 km/s	time in space: 25 min	27 MJ/kg
LEO	6,600 to 8,400 km	200 to 2000 km	circular orbit: 6.9 to 7.8 km/s elliptic orbit: 6.5 to 8.2 km/s	89 to 128 min	32.1 to 38.6 MJ/kg
Molniya orbit	6,900 to 46,300 km	500 to 39,900 km	1.5 to 10.0 km/s	11 h 58 min	54.8 MJ/kg
GEO	42,000 km	35,600 km	3.1 km/s	23 h 56 min	57.5 MJ/kg
Orbit of the Moon	363,000 to 406,000 km	357,000 to 399,000 km	0.97 to 1.08 km/s	27.3 days	61.8 MJ/kg

• • [ e] Orbits
Orbit types	Box orbit • Circular orbit • Ecliptic orbit • Elliptic orbit • Highly Elliptical Orbit • Graveyard orbit • Hyperbolic trajectory • Inclined orbit • Osculating orbit • Parabolic trajectory • Capture orbit • Escape orbit • Semi-synchronous orbit • Subsynchronous orbit • Synchronous orbit • Parking orbit
Earth-centered orbits	Geosynchronous orbit • Geostationary orbit • Sun-synchronous orbit • Low Earth orbit • Medium Earth Orbit • Molniya orbit • Near equatorial orbit • Orbit of the Moon • Polar orbit • Tundra orbit
Other orbits	Areosynchronous orbit • Areostationary orbit • Halo orbit • Lissajous orbit • Lunar orbit • Heliocentric orbit • Heliosynchronous orbit
Orbital elements	Inclination ( $\text{[math]}$ ) • Longitude of the ascending node ( $\text{[math]}$ ) • Eccentricity ( $\text{[math]}$ ) • Argument of periapsis ( $\text{[math]}$ ) • Semi-major axis ( $\text{[math]}$ ) • Mean anomaly at epoch ( $\text{[math]}$ )
Other orbital parameters	True anomaly ( $\text{[math]}$ ) • Semi-minor axis ( $\text{[math]}$ ) • Linear eccentricity ( $\text{[math]}$ ) • Eccentric anomaly ( $\text{[math]}$ ) • Mean longitude ( $\text{[math]}$ ) • True longitude ( $\text{[math]}$ ) • Orbital period ( $\text{[math]}$ )
Orbital maneuvers	Bi-elliptic transfer • Geostationary transfer orbit • Gravitational slingshot • Hohmann transfer orbit • Orbital inclination change • Orbit phasing • Space rendezvous
Other orbital mechanics topics	Apsis • Celestial coordinate system • Delta-v budget • Epoch • Ephemeris • Equatorial coordinate system • Gravity turn • Ground track • Interplanetary Transport Network • Kepler's laws of planetary motion • Lagrangian point • List of orbits • n-body problem • Orbit equation • Orbital state vectors • Perturbation • Retrograde and direct motion • Specific orbital energy • Specific relative angular momentum • The Oberth effect

ORBit is a CORBA compliant Object Request Broker (ORB). The current version is called ORBit2 and is compliant with CORBA version 2.4. It is developed under the GPL license and is used as middleware for the GNOME project.
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Periodicity is the quality of occurring at regular intervals or periods (in time or space) and can occur in different contexts:

A clock marks time at periodic intervals.
A metronome ticks at periodic intervals of time.

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planet, as defined by the International Astronomical Union (IAU), is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, not massive enough to cause thermonuclear fusion in its core, and has cleared its neighbouring region of
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STARS can mean:

Shock Trauma Air Rescue Society
Special Tactics And Rescue Service, a fictional task force that appears in Capcom's Resident Evil video game franchise.

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The Sun

Observation data
Mean distance
from Earth 1.49610¹¹ m
(8.31 min at light speed)
Visual brightness (V) −26.74^m ^[1]
Absolute magnitude 4.
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STAR is an acronym for:

Organizations:

Society for Telescopy, Astronomy, and Radio, a non-profit astronomy club in New Jersey
Special Tasks and Rescue or Special Tactics and Response, synonyms for SWAT

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EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001. Their greatest hit, their debut single "time after time", peaked at #13 in the Oricon singles chart.
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Elongation is an astronomical term that refers to the angle between the Sun and a planet, as viewed from Earth.

When an inferior planet is visible after sunset, it is near its greatest eastern elongation.
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Planetary phase is the term used to describe the appearance of the illuminated section of a planet. Like lunar phases, the planetary phase depends on the relative position of the sun, the planet and the observer.
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Conjunction is a term used in positional astronomy and astrology. It means that, as seen from some place (usually the Earth), two celestial bodies appear near one another in the sky. The event is also sometimes known as an appulse.
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An orbital node is one of the two points where an orbit crosses a plane of reference which it is inclined to.^[1] An orbit which is contained in the plane of reference (called non-inclined) has no nodes.
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ecliptic is the apparent path that the Sun traces out in the sky, as it appears to move in the sky in relation to the stars, this apparent path aligns with the planets throughout the course of the year.
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semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae.

Ellipse

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape.
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Right ascension (abbrev. RA; symbol α) is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system. The other coordinate is the declination.
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equinox can have two meanings:

The moment when the Sun is positioned directly over the Earth's equator and, by extension, the apparent position of the Sun at that moment - see below.

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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities.
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The sidereal year is the time taken for the Sun to return to the same position with respect to the stars of the celestial sphere. It is the orbital period of Earth, equal to 365.25636042 mean solar days (31,558,149.540 seconds), that is 365.25636042 earth rotations or sidereal days.
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A tropical year (also known as a solar year) is the length of time the Sun, as seen from the Earth, takes to return to the same position along the ecliptic (its path among the stars on the celestial sphere) relative to the equinoxes and solstices.
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degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of plane angle, representing ¹⁄₃₆₀ of a full rotation.
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The terms "inferior planet" and "superior planet" were originally used in the Ptolemaic cosmology to differentiate those planets (Mercury and Venus) that were between the stationary Earth and the orbiting Sun from those planets (Mars, Jupiter, and Saturn), which lay beyond
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Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Arabic^[1] mathematician, astronomer, astrologer and geographer,
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Julian year (symbol: a) is a unit of measurement of time defined as exactly 365.25 days of 86,400 SI seconds each, totalling 31,557,600 seconds. That is the average length of the year in the Julian calendar used in Western societies in previous centuries, and for which the
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Orbital Period

Information about Orbital Period

Relation between sidereal and synodic period

Calculation

Small body orbiting a central body

Orbital period as a function of central body's density

Two bodies orbiting each other

Earth orbits

See also

Ellipse