The orbital period is the time a given astronomical object takes to complete one orbit around another object, and applies in astronomy usually to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars.
For objects in the Solar System, this is often referred to as the sidereal period, determined by a 360° revolution of one celestial body around another, e.g. the Earth orbiting the Sun. The term sidereal denotes that the object returns to the same position relative to the fixed stars projected in the sky. When describing orbits of binary stars, the orbital period is usually referred to as just the period. For example, Jupiter has a sidereal period of 11.86 years while the main binary star Alpha Centauri AB has a period of about 79.91 years.
Another important orbital period definition can refer to the repeated cycles for celestial bodies as observed from the Earth's surface. An example is the so-called synodic period, applying to the elapsed time where planets return to the same kind of phenomena or location, such as when any planet returns between its consecutive observed conjunctions with or oppositions to the Sun. For example, Jupiter has a synodic period of 398.8 days from Earth; thus, Jupiter's opposition occurs once roughly every 13 months.
Periods in astronomy are conveniently expressed in various units of time, often in hours, days, or years. They can be also defined under different specific astronomical definitions that are mostly caused by small complex external gravitational influences by other celestial objects. Such variations also include the true placement of the centre of gravity between two astronomical bodies (barycenter), perturbations by other planets or bodies, orbital resonance, general relativity, etc. Most are investigated by detailed complex astronomical theories using celestial mechanics using precise positional observations of celestial objects via astrometry.
There are many periods related to the orbits of objects, each of which are often used in the various fields of astronomy and astrophysics. Examples of some of the common ones include the following:
According to Kepler's Third Law, the orbital period T (in seconds) of two point masses orbiting each other in a circular or elliptic orbit is:^{[2]}
where:
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:
where:
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 the central body's center of mass.
In the special case of perfectly circular orbits, the orbital velocity is constant and equal (in m/s) to
where:
This corresponds to times (? 0.707 times) the escape velocity.
For a perfect sphere of uniform density, it is possible to rewrite the first equation without measuring the mass as:
where:
For instance, a small body in circular orbit 10.5 cm above the surface of a sphere of tungsten half a meter in radius would travel at slightly more than 1 mm/s, completing an orbit every hour. If the same sphere were made of lead the small body would need to orbit just 6.7 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 ? (in kg/m^{3}), the above equation simplifies to (since )
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 kg/m^{3},^{[3]} e.g. Mercury with 5,427 kg/m^{3} and Venus with 5,243 kg/m^{3}) we get:
and for a body made of water (? ? 1,000 kg/m^{3})^{[4]}, respectively bodies with a similar density, e.g. Saturn's moons Iapetus with 1,088 kg/m^{3} and Tethys with 984 kg/m^{3} we get:
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.
In celestial mechanics, when both orbiting bodies' masses have to be taken into account, the orbital period T can be calculated as follows:^{[5]}
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.
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 conjunctions.
An example of this related period description is the repeated cycles for celestial bodies as observed from the Earth's surface, the so-called synodic period, applying to the elapsed time where planets return to the same kind of phenomena or location. For example, when any planet returns between its consecutive observed conjunctions with or oppositions to the Sun. For example, 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_{1} and T_{2}, so that T_{1} < T_{2}, their synodic period is given by:^{[6]}
Table of synodic periods in the Solar System, relative to Earth:^{[]}
Object | Sidereal period (yr) |
Synodic period | |
---|---|---|---|
(yr) | (d)^{[7]} | ||
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) | -- | |
Mars | 1.881 | 2.135 | 779.9 |
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 |
Moon | 0.0748 (27.32 days) | 0.0809 | 29.5306 |
99942 Apophis (near-Earth asteroid) | 0.886 | 7.769 | 2,837.6 |
4 Vesta | 3.629 | 1.380 | 504.0 |
1 Ceres | 4.600 | 1.278 | 466.7 |
10 Hygiea | 5.557 | 1.219 | 445.4 |
2060 Chiron | 50.42 | 1.020 | 372.6 |
50000 Quaoar | 287.5 | 1.003 | 366.5 |
136199 Eris | 557 | 1.002 | 365.9 |
90377 Sedna | 12050 | 1.0001 | 365.3^{[]} |
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.^{[]}
The concept of synodic period does not just apply 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:
Relative to | Mars | Jupiter | Saturn | Chiron | Uranus | Neptune | Pluto | Quaoar | Eris |
---|---|---|---|---|---|---|---|---|---|
Sun | 1.881 | 11.86 | 29.46 | 50.42 | 84.01 | 164.8 | 248.1 | 287.5 | 557.0 |
Mars | 2.236 | 2.009 | 1.954 | 1.924 | 1.903 | 1.895 | 1.893 | 1.887 | |
Jupiter | 19.85 | 15.51 | 13.81 | 12.78 | 12.46 | 12.37 | 12.12 | ||
Saturn | 70.87 | 45.37 | 35.87 | 33.43 | 32.82 | 31.11 | |||
2060 Chiron | 126.1 | 72.65 | 63.28 | 61.14 | 55.44 | ||||
Uranus | 171.4 | 127.0 | 118.7 | 98.93 | |||||
Neptune | 490.8 | 386.1 | 234.0 | ||||||
134340 Pluto | 1810.4 | 447.4 | |||||||
50000 Quaoar | 594.2 |
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 |