Orbital period

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.

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Astrodynamics

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.

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. Examples of some of the common ones include the following:

  • The sidereal period is the amount of time 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. In common usage, these two objects are typically the Earth and the Sun. The time between two successive oppositions or two successive conjunctions is also equal to the synodic period. For celestial bodies in the solar system, the synodic period (with respect to Earth and the Sun) differs from the sidereal period due to the Earth's motion around the Sun. For example, the synodic period of the Moon's orbit as seen from the Earth, relative to the Sun, is 29.5 mean solar days, since the Moon's phase and position relative to the Sun and Earth repeats after this period. This is longer than the sidereal period of its orbit around the Earth, which is 27.3 mean solar days, due to the motion of the Earth around the Sun.
  • The draconitic period (also draconic period or nodal 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.[1]
  • 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 semi-major axis typically advances slowly.
  • Also, the tropical period of Earth (a tropical year) is the interval between two alignments of its rotational axis with the Sun, also viewed as two passages of the object at a right ascension of 0 hr. One Earth year is slightly shorter than the period for the Sun to complete one circuit along the ecliptic (a sidereal year) because the inclined axis and equatorial plane slowly precess (rotate with respect to reference stars), realigning with the Sun before the orbit completes. This cycle of axial precession for Earth, known as precession of the equinoxes, recurs roughly every 25,770 years.

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:

  • a 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 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:

  • r is the circular orbit's radius in meters,
  • G is the gravitational constant,
  • M is the mass of the central body.

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:

  • r is the sphere's radius
  • a is the orbit's semi-major axis in meters,
  • G is the gravitational constant,
  • ρ is the density of the sphere in kilograms per cubic metre.

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 ρ (in kg/m3), the above equation simplifies to (since M = Vρ = 4/3πa3ρ)

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], respectively 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, 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:[5]

where:

  • a 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

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 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:

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.

Synodic periods relative to other planets

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:

Orbital period (years)
Relative to Mars Jupiter Saturn Chiron Uranus Neptune Pluto Quaoar Eris
Sol 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 stars
Binary starOrbital 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
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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
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