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{{about||the music album|Orbital Period (album)}}
{{short description|time taken to make one complete orbit}}
{{More citations needed|date=May 2018}}
{{Astrodynamics}}
The '''orbital period''' is the time a given [[astronomical object]] takes to complete one [[orbit]] around another object, and applies in [[astronomy]] usually to [[planet]]s or [[asteroid]]s orbiting the [[Sun]], [[moons]] orbiting planets, [[exoplanets]] orbiting other [[star]]s, or [[binary star]]s.

For objects in the [[Solar System]], this is often referred to as the '''sidereal period''', determined by a 360° revolution of one celestial body around another, e.g. the Earth orbiting the Sun. The term ''sidereal'' denotes that the object returns to the same position relative to the [[fixed stars]] [[celestial sphere|projected in the sky]]. When describing orbits of binary stars, the orbital period is usually referred to as just the '''period'''. For example, [[Jupiter]] has a sidereal period of 11.86 years while the main binary star [[Alpha Centauri]] AB has a period of about 79.91 years.

Another important orbital period definition can refer to the repeated cycles for celestial bodies as observed from the Earth's surface. An example is the so-called '''synodic period''', applying to the elapsed time where planets return to the same kind of phenomena or location, such as when any planet returns between its consecutive observed [[conjunction (astronomy)|conjunctions]] with or [[opposition (astronomy)|oppositions]] to the Sun. For example, [[Jupiter]] has a synodic period of 398.8 days from Earth; thus, Jupiter's opposition occurs once roughly every 13 months.

Periods in astronomy are conveniently expressed in various units of time, often in hours, days, or years. They can be also defined under different specific astronomical definitions that are mostly caused by small complex eternal gravitational influences by other celestial objects. Such variations also include the true placement of the centre of gravity between two astronomical bodies ([[barycenter]]), [[perturbation (astronomy)|perturbation]]s by other planets or bodies, [[orbital resonance]], [[general relativity]], etc. Most are investigated by detailed complex astronomical theories using [[celestial mechanics]] using precise positional observations of celestial objects via [[astrometry]].

==Related periods==
{{see also|Lunar month#Types}}

There are many [[periodic function|periods]] related to the orbits of objects, each of which are often used in the various fields of [[astronomy]] and [[astrophysics]]. Examples of some of the common ones include the following:

* {{anchor|Sidereal period}}The '''sidereal period''' is the amount of time that it takes an object to make a full orbit, relative to the [[star]]s. This is the orbital period in an inertial (non-rotating) [[frame of reference]].
* {{anchor|Synodic period}}The '''synodic period''' is the amount of time that it takes for an object to reappear at the same point in relation to two or more other objects (e.g. the [[Moon]]'s phase and its position relative to the [[Sun]] and [[Earth]] repeats every 29.5 day synodic period, longer than its 27.3 day orbit around the Earth, due to the motion of the Earth about the Sun). The time between two successive [[opposition (planets)|oppositions]] or [[conjunction (astronomy)|conjunctions]] is also an example of the synodic period. For the planets in the solar system, the synodic period (with respect to Earth) differs from the sidereal period due to the Earth's orbiting around the Sun.
* {{anchor|Draconitic period|Draconic period}}The '''draconitic period''' (also '''draconic period''' or '''[[nodal period]]'''), is the time that elapses between two passages of the object through its [[orbital node|ascending node]], the point of its orbit where it crosses the [[ecliptic]] from the southern to the northern hemisphere. This period differs from the sidereal period because both the orbital plane of the object and the plane of the ecliptic precess with respect to the fixed stars, so their intersection, the [[line of nodes]], also precesses with respect to the fixed stars. Although the plane of the ecliptic is often held fixed at the position it occupied at a specific [[epoch (astronomy)|epoch]], the orbital plane of the object still precesses causing the draconitic period to differ from the sidereal period.<ref>{{cite book|title=Satellite Orbits: Models, Methods, and Applications|year=2000|publisher=Springer Science & Business Media|isbn=978-3-540-67280-7|url=https://books.google.com/books?id=hABRnDlDkyQC&pg=PA50|author=Oliver Montenbruck, Eberhard Gill |page=50}}</ref>
* {{anchor|Anomalistic period}}The '''anomalistic period''' is the time that elapses between two passages of an object at its [[periapsis]] (in the case of the planets in the [[Solar System]], called the [[perihelion]]), the point of its closest approach to the attracting body. It differs from the sidereal period because the object's [[semi-major axis]] typically advances slowly.
* {{anchor|Tropical period}}Also, the '''tropical period''' of Earth (a [[tropical year]]) is the interval between two alignments of its rotational axis with the Sun, also viewed as two passages of the object at a [[right ascension]] of [[equinox (celestial coordinates)|0 hr]]. One Earth [[year]] is slightly shorter than the period for the Sun to complete one circuit along the [[ecliptic]] (a [[sidereal year]]) because the [[axial tilt|inclined axis]] and [[celestial equator|equatorial plane]] slowly [[axial precession|precess]] (rotate with respect to [[fixed stars|reference stars]]), realigning with the Sun before the orbit completes. This cycle of axial precession for Earth, known as ''precession of the equinoxes'', recurs roughly every 25,770 years.{{citation needed|date=November 2016}}

==Small body orbiting a central body==
[[File:Ellipse semi-major and minor axes.svg|thumb|upright=1.2|The semi-major axis (''a'') and semi-minor axis (''b'') of an ellipse]]
According to [[Kepler's laws of planetary motion|Kepler's Third Law]], the '''orbital period''' ''T'' (in seconds) of two point masses orbiting each other in a circular or [[elliptic orbit]] is:{{sfnp | Bate, Mueller and White | 1971 | p=33 }}

:<math>T = 2\pi\sqrt{\frac{a^3}{\mu}}</math>

where:
* ''a'' is the orbit's [[semi-major axis]]
* ''μ'' = ''GM'' is the [[standard gravitational parameter]]
** ''G'' is the [[gravitational constant]],
** ''M'' is the mass of the more massive body.

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

Inversely, for calculating the distance where a body has to orbit in order to have a given orbital period:

:<math>a = \sqrt[3]{\frac{GMT^2}{4\pi^2}}</math>

where:
* ''a'' is the orbit's semi-major axis in meters,
* ''G'' is the gravitational constant,
* ''M'' is the mass of the more massive body,
* ''T'' is the orbital period in seconds.

For instance, for completing an orbit every 24&nbsp;[[hour]]s around a mass of 100&nbsp;[[kg]], a small body has to orbit at a distance of 1.08&nbsp;[[metre|meters]] from its [[center of mass]].

==Orbital period as a function of central body's density==
When a very small body is in a circular orbit barely above the surface of a sphere of any radius and mean density ''ρ'' (in kg/m<sup>3</sup>), the above equation simplifies to (since {{nowrap|''M''&nbsp;{{=}} ''Vρ''&nbsp;{{=}} {{sfrac|4|3}}{{pi}}''a''<sup>3</sup>''ρ''}})

:<math>T = \sqrt{ \frac {3\pi}{G \rho} }</math>

Thus the orbital period in low orbit depends only on the density of the central body, regardless of its size.

So, for the Earth as the central body (or any other spherically symmetric body with the same mean density, about 5,515&nbsp;kg/m<sup>3</sup>,<ref>{{citation |url=http://www.wolframalpha.com/input/?i=density+of+the+earth |title=Density of the Earth |publisher=wolframalpha.com}}</ref> e.g. [[Mercury (planet)|Mercury]] with 5,427&nbsp;kg/m<sup>3</sup> and [[Venus]] with 5,243&nbsp;kg/m<sup>3</sup>) we get:
:''T'' = 1.41 hours

and for a body made of water (''ρ''&nbsp;≈&nbsp;1,000&nbsp;kg/m<sup>3</sup>)<ref>{{citation |url=http://www.wolframalpha.com/input/?i=density+of+water |title=Density of water |publisher=wolframalpha.com}}</ref>, respectively bodies with a similar density, e.g. Saturn's moons [[Iapetus]] with 1,088&nbsp;kg/m<sup>3</sup> and [[Tethys (moon)|Tethys]] with 984&nbsp;kg/m<sup>3</sup> we get:

:''T'' = 3.30 hours

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" [[time standard|unit of time]] if we have a unit of mass, a unit of length and a unit of density.

==Two bodies orbiting each other==<!-- This section is linked from [[Binary star]] -->
In [[celestial mechanics]], when both orbiting bodies' masses have to be taken into account, the '''orbital period''' ''T'' can be calculated as follows:<ref>Bradley W. Carroll, Dale A. Ostlie. An introduction to modern astrophysics. 2nd edition. Pearson 2007.</ref>

:<math>T= 2\pi\sqrt{\frac{a^3}{G \left(M_1 + M_2\right)}}</math>

where:
* ''a'' is the sum of the [[semi-major axis|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),
* ''M''<sub>1</sub> + ''M''<sub>2</sub> is the sum of the masses of the two bodies,
* ''G'' 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 {{section link|Orbit|Scaling in gravity}}).{{citation needed|date=November 2011}}

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

==Synodic period==

One of the observable characteristics of two bodies which orbit a third body in different orbits, and thus have different orbital periods, is their '''synodic period''', which is the time between [[conjunction (astronomy)|conjunctions]].

An example of this related period description is the repeated cycles for celestial bodies as observed from the Earth's surface, the so-called '''synodic period''', applying to the elapsed time where planets return to the same kind of phenomena or location. For example, when any planet returns between its consecutive observed [[conjunction (astronomy)|conjunctions]] with or [[opposition (astronomy)|oppositions]] to the Sun. For example, [[Jupiter]] has a synodic period of 398.8 days from Earth; thus, Jupiter's opposition occurs once roughly every 13 months.

If the orbital periods of the two bodies around the third are called ''T''<sub>1</sub> and ''T''<sub>2</sub>, so that ''T''<sub>1</sub>&nbsp;<&nbsp;''T''<sub>2</sub>, their synodic period is given by:<ref>{{cite book|title=Fundamental Astronomy|author=Hannu Karttunen|url=https://books.google.ca/books?id=ndd2DQAAQBAJ|accessdate=December 7, 2018|edition=6th|year=2016|publisher=Springer|isbn=9783662530450|display-authors=etal|pages=145}}</ref>

:<math>\frac{1}{T_\mathrm{syn}} = \frac{1}{T_1} - \frac{1}{T_2}</math>

==Examples of sidereal and synodic periods==
Table of synodic periods in the Solar System, relative to Earth:{{citation needed|date=January 2011}}

{| class="wikitable sortable"
! rowspan=2 | Object
! rowspan=2 | Sidereal period <br/>([[Julian year (astronomy)|yr]])
! colspan=2 | Synodic period
|-
! ({{abbr|yr|year}})
! ({{abbr|d|day}})<ref>{{cite web|url=http://www.astronomycafe.net/qadir/q1166.html|title=Questions and Answers - Sten's Space Blog|website=www.astronomycafe.net}}</ref>
|-
| [[Mercury (planet)|Mercury]]
| 0.240846 (87.9691 days)
| 0.317
| 115.88 
|-
| [[Venus]]
| 0.615 (225 days)
| 1.599
| 583.9
|-
| [[Earth]]
| '''1''' (365.25636 [[solar day]]s)
| colspan=2 | —
|-
| [[Moon]]
| 0.0748 (27.32 days)
| 0.0809
| 29.5306
|-
| [[99942 Apophis]] ([[near-Earth asteroid]])
| 0.886
| 7.769
| 2,837.6
|-
| [[Mars]]
| 1.881
| 2.135
| 779.9
|-
| [[4 Vesta]]
| 3.629
| 1.380
| 504.0
|-
| [[Ceres (dwarf planet)|1 Ceres]]
| 4.600
| 1.278
| 466.7
|-
| [[10 Hygiea]]
| 5.557
| 1.219
| 445.4
|-
| [[Jupiter]]
| 11.86
| 1.092
| 398.9
|-
| [[Saturn]]
| 29.46
| 1.035
| 378.1
|-
|[[2060 Chiron]]
|50.42
|1.020
|372.6
|-
| [[Uranus]]
| 84.01
| 1.012
| 369.7
|-
| [[Neptune]]
| 164.8
| 1.006
| 367.5
|-
| [[Pluto|134340 Pluto]]
| 248.1
| 1.004
| 366.7
|-
|[[50000 Quaoar]]
|287.5
|1.003
|366.5
|-
| [[Eris (dwarf planet)|136199 Eris]]
| 557
| 1.002
| 365.9
|-
| [[90377 Sedna]]
| 12050
| 1.0001
| 365.3 {{citation needed|date=December 2013}}
|-
|}

In the case of a planet's [[natural satellite|moon]], the synodic period usually means the Sun-synodic period, namely, the time it takes the moon to complete its illumination phases, completing the solar phases for an astronomer 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 (moon)|Deimos]]'s synodic period is 1.2648&nbsp;days, 0.18% longer than Deimos's sidereal period of 1.2624&nbsp;d.{{citation needed|date=November 2011}}

=== Synodic periods relative to other planets ===
The concept of synodic period does not just apply to the Earth, but also to other planets as well, and the formula for computation is the same as the one given above. Here is a table which lists the synodic periods of some planets relative to each other:
{| class="wikitable sortable"
|+ Orbital period (years)
|-
! Relative to
! Jupiter
! Saturn
! Chiron
! Uranus
! Neptune
! Pluto
! Quaoar
! Eris
|-
! [[Sun|Sol]]
| 11.86
| 29.46
| 50.42
| 84.01
| 164.8
| 248.1
| 287.5
| 557.0
|-
! [[Jupiter]]
| {{n/a|}}
| 19.85
| 15.51
| 13.81
| 12.78
| 12.46
| 12.37
| 12.12
|-
! [[Saturn]]
| {{n/a|}}
| {{n/a|}}
| 70.87
| 45.37
| 35.87
| 33.43
| 32.82
| 31.11
|-
! [[2060 Chiron]]
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| 126.1
| 72.65
| 63.28
| 61.14
| 55.44
|-
! [[Uranus]]
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| 171.4
| 127.0
| 118.7
| 98.93
|-
! [[Neptune]]
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| 490.8
| 386.1
| 234.0
|-
! [[Pluto|134340 Pluto]]
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| 1810.4
| 447.4
|-
! [[50000 Quaoar]]
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| 594.2
|}<!--- The Earth-synodic period can also be calculated, but it varies with the planet's position with respect to the Earth; in the Deimos example, at Mars opposition the Earth-synodic period would be about 1.2604&nbsp;d because Earth's motion overtakes Mars. At Mars conjunction, Deimos's Earth-synodic period would be 1.2692 d because Earth's motion now accentuates Mars' apparent motion. Since the Earth--extra-planetary moon synodic period reflects Earth-planet-moon alignments, it is fairly meaningless. --->

==Binary stars==
{| class="wikitable"
![[Binary star]]!!Orbital period
|-
| [[AM Canum Venaticorum]]
| style="text-align: right" | 17.146 minutes
|-
| [[Beta Lyrae]] AB
| style="text-align: right" | 12.9075 days
|-
| [[Alpha Centauri]] AB
| style="text-align: right" | 79.91 years
|-
| [[Proxima Centauri]] – [[Alpha Centauri]] AB
|  style="text-align: right" | 500,000 years or more
|}

==See also==
* [[Geosynchronous orbit derivation]]
* [[Rotation period]] – time that it takes to complete one revolution around its axis of rotation
* [[Satellite revisit period]]
* [[Sidereal time]]
* [[Sidereal year]]
* [[Opposition (astronomy)]]
* [[List of periodic comets]]

==Notes==
{{Reflist}}

==Bibliography==
*{{Citation |last1=Bate |first1=Roger B.|last2= Mueller|first2=Donald D.| last3=White|first3=Jerry E.| title=Fundamentals of Astrodynamics|publisher= Dover| date=1971| ref=harv}}

== External links ==
{{Wiktionary|synodic}}

{{Orbits}}

{{DEFAULTSORT:Orbital Period}}
[[Category:Time in astronomy]]
[[Category:Orbits|Period]]