Orbital Period

In Astronomy, the orbital period indicates the duration put by a Astre (star, Planet, Astéroïde) to carry out a complete Orbite. For example, the Ground has one 365,25 days orbital period.

If this rotation compared to the Sun such as is observed on Ground, one speaks about synodical Period; it is the apparent orbital period. If it is relating to the star S, one speaks about sidereal Period; the latter is regarded as the period of real rotation of the object.

There exist other types of orbital periods:

the anomalistic period is the duration between two passages of the object to its Périastre; according to whether this last precess or recess, this period is shorter or long that the sidereal period;
the draconitic period is the duration between two passages of the object to its ascending Nœud or going down, it will thus depend on the Précession S of the two implied plans (the orbit of the object and the datum-line, generally the ecliptic );
the period tropic is the duration between two passages of the object to the Right ascension zero; because of the Precession of the equinoxes, this period is slightly and systematically shorter than the sidereal péride.

Calculations

Orbiting body of negligible mass

The orbital period

P \,

of a negligible body of mass orbiting around a central body can be calculated in the following way:

P = 2 \ pi \ sqrt {\ frac {a^3} {GM}}

where:

$a \,$ is the length of the Equatorial radius of the orbit,
$G \,$ is the Constante of gravitation,
$M \,$ is the mass of the central object.

Two bodies

When one takes account of the mass of the two bodies, the orbital period

P \,

can be calculated in the following way:

P = 2 \ pi \ sqrt {\ frac {a^3} {G \ left (M_1 + M_2 \ right)}}

where:

$a \,$ is the sum of the Equatorial radius S of the ellipses in which the center of the bodies move where, in an equivalent way, the equatorial radius of the ellipse in which one of the bodies moves in the reference mark having as origin the other body (which is equal to their distance for circular orbits),
$M_1 \,$ and $M_2 \,$ is the masses of the bodies,
$G \,$ is the Constante of gravitation.

One can note that the orbital period is independent of the size.

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