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):
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Copernicus devised a mathematical formula to calculate a planet's sidereal period from its synodic period.
Using the abbreviations
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:
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  779.9 
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 Sunsynodic 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.
In astrodynamics the orbital period (in seconds) of a small body orbiting a central body in a circular or elliptic orbit is:
where:
Note that for all ellipses with a given semimajor axis the orbital period is the same, regardless of eccentricity.
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
In celestial mechanics when both orbiting bodies' masses have to be taken into account the orbital period can be calculated as follows:
where:
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.
orbit  centertocenter distance 
altitude above the Earth's surface 
speed  period/time in space  specific orbital energy 

minimum suborbital spaceflight (vertical)  6,500 km  100 km  0.0 km/s  just reaching space  1.0 MJ/kg 
ICBM  up to 7,600 km  up to 1,200 km  6 to 7 km/s  time in space: 25 min  27 MJ/kg 
Low Earth orbit  6,600 to 8,400 km  200 to 2,000 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  35,786 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 
Binary star  Orbital period 

Beta Lyrae AB  12.9075 days 
Alpha Centauri AB  79.91 yr 
Proxima Centauri  Alpha Centauri AB  500,000 years or more 

