There are several kinds of orbital periods for objects around the Sun :
- The is the time that it takes the object to make one full orbit around the Sun, relative to the Star s. This is considered to be an object's true orbital period.
- The 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 . 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 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 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 , 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.
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:
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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.
