View source for Astronomy:Orbital period

{{short description|Time an astronomical object takes to complete one orbit around another object}}
{{Astrodynamics|Equations}}

The '''orbital period''' (also '''revolution period''') is the amount of time a given [[Astronomy:Astronomical object|astronomical object]] takes to complete one [[Physics:Orbit|orbit]] around another object. In [[Astronomy:Astronomy|astronomy]], it usually applies to [[Astronomy:Planet|planet]]s or [[Astronomy:Asteroid|asteroid]]s orbiting the [[Astronomy:Sun|Sun]], moons orbiting planets, [[Astronomy:Exoplanet|exoplanet]]s orbiting other [[Astronomy:Star|star]]s, or [[Astronomy:Binary star|binary star]]s. It may also refer to the time it takes a satellite orbiting a planet or moon to complete one orbit.

For celestial objects in general, the orbital period is determined by a 360° revolution of [[Astronomy:Orbiting body|one body]] around its [[Astronomy:Primary body|primary]], ''e.g.'' Earth around the Sun. 

Periods in astronomy are expressed in units of time, usually hours, days, or years.

==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'' of two point masses orbiting each other in a circular or [[Astronomy:Elliptic orbit|elliptic orbit]] is:{{sfnp|Bate|Mueller|White|1971|p=33}}

:<math qid=Q120630173>T = 2\pi\sqrt{\frac{a^3}{GM}}</math>

where:
* ''a'' is the orbit's semi-major axis
* ''G'' is the [[Physics:Gravitational constant|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 T:

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

For instance, for completing an orbit every 24&nbsp;[[Hour|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 the central body's [[Physics:Center of mass|center of mass]].

In the special case of perfectly circular orbits, the semimajor axis a is equal to the radius of the orbit, and the orbital velocity is constant and equal to

: <math qid=Q120630325>v_\text{o} = \sqrt{\frac{G M}{r}}</math>

where:
* ''r'' is the circular orbit's radius in meters,

This corresponds to {{frac|1|√2}} times (≈ 0.707 times) the [[Astronomy:Escape velocity|escape velocity]].

===Effect of central body's density===
For a perfect sphere of uniform [[Physics:Density|density]], it is possible to rewrite the first equation without measuring the mass as:

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

where:
* ''r'' is the sphere's radius
* ''a'' is the orbit's semi-major axis in metres,
* ''G'' is the gravitational constant,
* ''ρ'' is the density of the sphere in kilograms per cubic metre.

For instance, a small body in circular orbit 10.5 [[Centimetre|cm]] above the surface of a sphere of [[Chemistry:Tungsten|tungsten]] half a metre in radius would travel at slightly more than 1 [[Millimetre|mm]]/[[Second|s]], completing an orbit every hour. If the same sphere were made of [[Chemistry:Lead|lead]] the small body would need to orbit just 6.7 [[Millimetre|mm]] above the surface for sustaining the same orbital period.

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. [[Astronomy:Mercury (planet)|Mercury]] with 5,427&nbsp;kg/m<sup>3</sup> and [[Astronomy:Venus|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> or bodies with a similar density, e.g. Saturn's moons [[Religion:Iapetus|Iapetus]] with 1,088&nbsp;kg/m<sup>3</sup> and 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, such as 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" [[Physics:Time standard|unit of time]] if we have a unit of density.

==Two bodies orbiting each other==<!-- This section is linked from [[Astronomy:Binary star|Binary star]] -->
In [[Physics:Celestial mechanics|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, p. 49 (equation 2.37 simplified).</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 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 [[Physics:Gravitational constant|gravitational constant]].

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

==Related periods==

For celestial objects in general, the '''orbital period''' typically refers to the '''sidereal period''', determined by a 360° revolution of [[Astronomy:Orbiting body|one body]] around its [[Astronomy:Primary body|primary]] relative to the [[Astronomy:Fixed stars|fixed stars]] [[Celestial sphere|projected in the sky]]. For the case of the [[Astronomy:Earth|Earth]] orbiting around the [[Astronomy:Sun|Sun]], this period is referred to as the [[Astronomy:Sidereal year|sidereal year]]. This is the orbital period in an inertial (non-rotating) [[Astronomy:Frame of reference|frame of reference]].

'''Orbital periods''' can be defined in several ways. The '''tropical period''' is more particularly about the position of the parent star. It is the basis for the solar year, and respectively the [[Physics:Calendar year|calendar year]].

The '''synodic period''' refers not to the orbital relation to the parent star, but to other [[Astronomy:Astronomical object|celestial objects]], making it not a merely different approach to the orbit of an object around its parent, but a period of orbital relations with other objects, normally Earth, and their orbits around the Sun. It applies to the elapsed time where planets return to the same kind of phenomenon or location, such as when any planet returns between its consecutive observed [[Astronomy:Conjunction|conjunctions]] with or [[Astronomy:Opposition|oppositions]] to the Sun.  For example, [[Astronomy:Jupiter|Jupiter]] has a synodic period of 398.8 days from Earth; thus, Jupiter's opposition occurs once roughly every 13 months.

There are many [[Periodic function|periods]] related to the orbits of objects, each of which are often used in the various fields of [[Astronomy:Astronomy|astronomy]] and [[Physics:Astrophysics|astrophysics]], particularly they must not be confused with other revolving periods like rotational periods. Examples of some of the common orbital ones include the following:

* {{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. In common usage, these two objects are typically Earth and the Sun. The time between two successive [[Astronomy:Opposition (planets)|oppositions]] or two successive [[Astronomy:Conjunction|conjunctions]] is also equal to the synodic period. For celestial bodies in the solar system, the synodic period (with respect to Earth and the Sun) differs from the tropical period owing to Earth's motion around the Sun. For example, the synodic period of the [[Astronomy:Moon|Moon]]'s orbit as seen from [[Astronomy:Earth|Earth]], relative to the [[Astronomy:Sun|Sun]], is 29.5 mean solar days, since the Moon's phase and position relative to the Sun and Earth repeats after this period. This is longer than the sidereal period of its orbit around Earth, which is 27.3 mean solar days, owing to the motion of Earth around the Sun.
* {{anchor|Draconitic period|Draconic period}}The '''draconitic period''' (also '''draconic period''' or '''[[Astronomy:Nodal period|nodal period]]'''), is the time that elapses between two passages of the object through its [[Astronomy:Orbital node|ascending node]], the point of its orbit where it crosses the [[Astronomy:Ecliptic|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 [[Astronomy:Line of nodes|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 [[Astronomy:Epoch|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 [[Astronomy:Solar System|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 [[Astronomy:Tropical year|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|right ascension]] of [[Astronomy:Equinox (celestial coordinates)|0 hr]]. One Earth [[Year|year]] is slightly shorter than the period for the Sun to complete one circuit along the [[Astronomy:Ecliptic|ecliptic]] (a [[Astronomy:Sidereal year|sidereal year]]) because the [[Astronomy:Axial tilt|inclined axis]] and [[Celestial equator|equatorial plane]] slowly [[Astronomy:Axial precession|precess]] (rotate with respect to [[Astronomy: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 25,772 years.<ref>{{Cite web|url=http://demonstrations.wolfram.com/PrecessionOfTheEarthsAxis/|title=Precession of the Earth's Axis - Wolfram Demonstrations Project|website=demonstrations.wolfram.com|language=en|access-date=2019-02-10}}</ref>

Periods can be also defined under different specific astronomical definitions that are mostly caused by the small complex external gravitational influences of other celestial objects. Such variations also include the true placement of the [[Physics:Center of mass|centre of gravity]] between two [[Astronomy:Astronomical object|astronomical bodies]] ([[Barycenter|barycenter]]), [[Astronomy:Perturbation|perturbation]]s by other planets or bodies, orbital resonance, [[Physics:General relativity|general relativity]], etc. Most are investigated by detailed complex astronomical theories using [[Physics:Celestial mechanics|celestial mechanics]] using precise positional observations of celestial objects via [[Astronomy:Astrometry|astrometry]].

===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 [[Astronomy:Conjunction|conjunctions]].

An example of this related period description is the repeated cycles for celestial bodies as observed from the Earth's surface, the '''synodic period''', applying to the elapsed time where planets return to the same kind of phenomenon or {{nowrap|location{{tsp}}{{mdash}}{{tsp}}}}for example, when any planet returns between its consecutive observed [[Astronomy:Conjunction|conjunctions]] with or [[Astronomy:Opposition|oppositions]] to the Sun. For example, [[Astronomy:Jupiter|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.com/books?id=ndd2DQAAQBAJ|access-date=December 7, 2018|edition=6th|year=2016|publisher=Springer|isbn=9783662530450|display-authors=0|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}}


In the case of a planet's [[Astronomy: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, [[Astronomy: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 applies not just 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
! Mars
! Jupiter
! Saturn
! Chiron
! Uranus
! Neptune
! Pluto
! Quaoar
! Eris
|-
! [[Astronomy:Sun|Sun]]
| 1.881
| 11.86
| 29.46
| 50.42
| 84.01
| 164.8
| 248.1
| 287.5
| 557.0
|-
! [[Astronomy:Mars|Mars]]
| {{n/a|}}
| 2.236
| 2.009
| 1.954
| 1.924
| 1.903
| 1.895
| 1.893
| 1.887
|-
! [[Astronomy:Jupiter|Jupiter]]
| {{n/a|}}
| {{n/a|}}
| 19.85
| 15.51
| 13.81
| 12.78
| 12.46
| 12.37
| 12.12
|-
! [[Astronomy:Saturn|Saturn]]
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| 70.87
| 45.37
| 35.87
| 33.43
| 32.82
| 31.11
|-
! [[Astronomy:2060 Chiron|2060 Chiron]]
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| 126.1
| 72.65
| 63.28
| 61.14
| 55.44
|-
! [[Astronomy:Uranus|Uranus]]
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| 171.4
| 127.0
| 118.7
| 98.93
|-
! [[Astronomy:Neptune|Neptune]]
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| 490.8
| 386.1
| 234.0
|-
! [[Astronomy:Pluto|Pluto]]
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| {{n/a|}}
| 1810.4
| 447.4
|-
! [[Astronomy:50000 Quaoar|50000 Quaoar]]
| {{n/a|}}
| {{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. --->

===Example of orbital periods: binary stars===
{| class="wikitable"
![[Astronomy:Binary star|Binary star]]!!Orbital period.
|-
| [[Astronomy:AM Canum Venaticorum|AM Canum Venaticorum]]
| style="text-align: right" | 17.146 minutes
|-
| [[Astronomy:Beta Lyrae|Beta Lyrae]] AB
| style="text-align: right" | 12.9075 days
|-
| [[Astronomy:Alpha Centauri|Alpha Centauri]] AB
| style="text-align: right" | 79.91 years
|-
| [[Astronomy:Proxima Centauri|Proxima Centauri]] – [[Astronomy:Alpha 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
* [[Astronomy:Satellite revisit period|Satellite revisit period]]
* [[Astronomy:Sidereal time|Sidereal time]]
* [[Astronomy:Sidereal year|Sidereal year]]
* [[Astronomy:Opposition|Opposition]]
* 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}}

== External links ==

{{Orbits}}

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

{{Sourceattribution|Orbital period}}