Orbital period

The orbital period is the time taken for a given object to make one complete orbit around 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.

Periods related to the orbital period

There are several periods related to the orbits of objects, each of which is used in the fields of astronomy and astrophysics:

• The sidereal period is the temporal cycle that it takes an object to make a full orbit, relative to the stars. This is the orbital period in an inertial (non-rotating) frame of reference.
• 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 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.
• The draconitic period, or draconic period, is the time that elapses between two passages of the object through its 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, the orbital plane of the object still precesses causing the draconitic period to differ from the sidereal 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 semimajor axis typically advances slowly.
• Also, the Earth's tropical period (or simply its "year") is the time that elapses between two alignments of its axis of rotation with the Sun, also viewed as two passages of the object at right ascension zero. One Earth year has a slightly shorter interval than the solar orbit (sidereal period) because the inclined axis and equatorial plane slowly precess (rotate with respect to the stars), realigning with the Sun before the orbit completes. The Earth's precession cycle completes in about 25,770 years.

Small body orbiting a central body

According to Kepler's Third Law, the orbital period T (in seconds) of two bodies orbiting each other in a circular or elliptic orbit is:

${\displaystyle T=2\pi {\sqrt {\frac {r^{3}}{\mu }}}}$

where:

• r is the orbit's semi-major axis in meters
• μ = GM is the standard gravitational parameter in m³/s²
  • 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 pulse a given orbital period:

${\displaystyle r={\sqrt[{3}]{\frac {GMT^{2}}{4\pi ^{2}}}}}$

where:

• r 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 hours around a mass of 100 kg, a small body has to orbit at a distance of 1.08 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/m³), the above equation simplifies to (since M = Vρ = 4/3πa³ρ):

$T = \sqrt{ \frac {3\pi}{G \rho} }$

So, for the Earth as the central body (or any other spherically symmetric body with the same mean density, about 5,515 kg/m³)^[1] we get:

T = 1.41 hours

and for a body made of water (ρ ≈ 1,000 kg/m³)^[2]

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" unit of time if we have a unit of mass, a unit of length and a unit of density.

Two bodies orbiting each other

In celestial mechanics, when both orbiting bodies' masses have to be taken into account, the orbital period T can be calculated as follows:^[3]

${\displaystyle T=2\pi {\sqrt {\frac {r^{3}}{G\left(M_{1}+M_{2}\right)}}}}$

where:

• r 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),
• M₁ + M₂ 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 Orbit#Scaling in gravity).

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

Synodic period

When two bodies orbit a third body in different orbits, and thus different orbital periods, their respective, synodic period can be found. If the orbital periods of the two bodies around the third are called P₁ and P₂, so that P₁ < P₂, their synodic period is given by:

${\displaystyle {\frac {1}{P_{\mathrm {syn} }}}={\frac {1}{P_{1}}}-{\frac {1}{P_{2}}}}$

Examples of sidereal and synodic periods

Table of synodic periods in the Solar System, relative to Earth:

Object	Sidereal period (yr)	Synodic period (yr)	Synodic period (d) [4]
Mercury	0.240846 (87.9691 days)	0.317	115.88
Venus	0.615 (225 days)	1.599	583.9
Earth	1 (365.25636 solar days)	—	—
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
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
Uranus	84.01	1.012	369.7
Neptune	164.8	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 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's synodic period is 1.2648 days, 0.18% longer than Deimos's sidereal period of 1.2624 d.

Binary stars

Binary star	Orbital period
AM Canum Venaticorum	17.146 minutes
Beta Lyrae AB	12.9075 days
Alpha Centauri AB	79.91 years
Proxima Centauri – Alpha Centauri AB	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
• Sidereal time
• Sidereal year
• Opposition (astronomy)
• List of periodic comets

Notes

1. Density of the Earth, wolframalpha.com
2. Density of water, wolframalpha.com
3. Bradley W. Carroll, Dale A. Ostlie. An introduction to modern astrophysics. 2nd edition. Pearson 2007.
4. http://www.astronomycafe.net/qadir/q1166.html

External links