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Orbital period
For the music album, see Orbital Period (album).
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 celestial objects in general the sidereal orbital period (sidereal year) is referred to by the orbital period, determined by a 360° revolution of one celestial body around another, e.g. the Earth orbiting the Sun, relative to the fixed starsprojected in the sky. Orbital periods can be defined in several ways. The tropical period is more particular about the position of the parent star. It is the basis for the solar year, and respectively the calendar year.
The synodic period incorporates not only the orbital relation to the parent star, but also to other celestial objects, making it not a mere different approach to the orbit of an object around its parent, but a period of orbital relations with other objects, normally Earth and their orbits around the Sun. It applies 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 the small complex external gravitational influences of 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.
Related periods
See also: Lunar month § Types
There are many periods related to the orbits of objects, each of which are often used in the various fields of astronomy and astrophysics, particularly they must not be confused with other revolving periods like rotational periods. Examples of some of the common orbital ones include the following:
Small body orbiting a central body
The semi-major axis (a) and semi-minor axis (b) of an ellipse
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 1√2 times (≈ 0.707 times) the escape velocity.
Effect of central body's density
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 metre 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 ρ (in kg/m3), the above equation simplifies to (since
M =  =
4
3
πa3ρ
)
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/m3,[3] e.g. Mercury with 5,427 kg/m3 and Venus with 5,243 kg/m3) we get:
T = 1.41 hours
and for a body made of water (ρ ≈ 1,000 kg/m3),[4] or bodies with a similar density, e.g. Saturn's moons Iapetus with 1,088 kg/m3 and Tethys with 984 kg/m3 we get:
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, such as 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:[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).[citation needed]
In a parabolic or hyperbolic trajectory, the motion is not periodic, and the duration of the full trajectory is infinite.
Synodic period
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 synodic period, applying to the elapsed time where planets return to the same kind of phenomenon 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 T1 and T2, so that T1 < T2, their synodic period is given by:[6]
Examples of sidereal and synodic periods
Table of synodic periods in the Solar System, relative to Earth:[citation needed]
ObjectSidereal period
(yr)
Synodic period
(yr)(d)[7]
Mercury0.240846 (87.9691 days)0.317115.88
Venus0.615 (225 days)1.599583.9
Earth1 (365.25636 solar days)
Mars1.8812.135779.9
Jupiter11.861.092398.9
Saturn29.461.035378.1
Uranus84.011.012369.7
Neptune164.81.006367.5
134340 Pluto248.11.004366.7
Moon0.0748 (27.32 days)0.080929.5306
99942 Apophis (near-Earth asteroid)0.8867.7692,837.6
4 Vesta3.6291.380504.0
1 Ceres4.6001.278466.7
10 Hygiea5.5571.219445.4
2060 Chiron50.421.020372.6
50000 Quaoar287.51.003366.5
136199 Eris5571.002365.9
90377 Sedna120501.0001365.3[citation needed]
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.[citation needed]
Synodic periods relative to other planets
The concept of synodic period applies not just 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:
Orbital period (years)
Relative toMarsJupiterSaturnChironUranusNeptunePlutoQuaoarEris
Sun1.88111.8629.4650.4284.01164.8248.1287.5557.0
Mars2.2362.0091.9541.9241.9031.8951.8931.887
Jupiter19.8515.5113.8112.7812.4612.3712.12
Saturn70.8745.3735.8733.4332.8231.11
2060 Chiron126.172.6563.2861.1455.44
Uranus171.4127.0118.798.93
Neptune490.8386.1234.0
Pluto1810.4447.4
50000 Quaoar594.2
Binary stars
Binary starOrbital period
AM Canum Venaticorum17.146 minutes
Beta Lyrae AB12.9075 days
Alpha Centauri AB79.91 years
Proxima CentauriAlpha Centauri AB500,000 years or more
See also
Notes
  1. ^ Oliver Montenbruck, Eberhard Gill (2000). Satellite Orbits: Models, Methods, and Applications. Springer Science & Business Media. p. 50. ISBN 978-3-540-67280-7.
  2. ^ Bate, Mueller & White (1971), p. 33.
  3. ^ Density of the Earth, wolframalpha.com
  4. ^ Density of water, wolframalpha.com
  5. ^ Bradley W. Carroll, Dale A. Ostlie. An introduction to modern astrophysics. 2nd edition. Pearson 2007.
  6. ^ Hannu Karttunen; et al. (2016). Fundamental Astronomy (6th ed.). Springer. p. 145. ISBN 9783662530450. Retrieved December 7, 2018.
  7. ^ "Questions and Answers - Sten's Space Blog". www.astronomycafe.net.
Bibliography
Bate, Roger B.; Mueller, Donald D.; White, Jerry E. (1971), Fundamentals of Astrodynamics, Dover
External links
Look up synodic in Wiktionary, the free dictionary.
Last edited on 3 June 2021, at 10:47
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