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.

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 name sidereal is added as it implies 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. 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.

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 eternal 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.

Other periods related to the orbital period

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 (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 oppositions or 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 (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

Ellipse semi-major and minor axes
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]


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:


  • 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 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 =  = 4/3πa3ρ)

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] we get:

T = 1.41 hours

and for a body made of water (ρ ≈ 1,000 kg/m3)[4]

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]


  • 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

There are observable characteristics of two bodies which orbit a third body in different orbits, and thus have different orbital periods. This is known as their synodic period; it is the time between conjunctions, and since it is observable from either the first or the second body, the two synodic periods will be different, depending from which celestial body one is observing.

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 P1 and P2, so that P1 < P2, 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) Synodic period (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)
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
2060 Chiron 50.42 1.020 372.6
Uranus 84.01 1.012 369.7
Neptune 164.8 1.006 367.5
134340 Pluto 248.1 1.004 366.7
50000 Quaoar 287.5 1.003 366.5
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.

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 Jupiter Saturn Chiron Uranus Neptune Pluto Quaoar Eris
Sol 11.86 29.46 50.42 84.01 164.8 248.1 287.5 557.0
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 star Orbital period
AM Canum Venaticorum 17.146 minutes
Beta Lyrae AB 12.9075 days
Alpha Centauri AB 79.91 years
Proxima CentauriAlpha Centauri AB 500,000 years or more

See also


  1. ^ Oliver Montenbruck, Eberhard Gill. Satellite Orbits: Models, Methods, and Applications. p. 50. Retrieved 1 June 2018.
  2. ^ Bate, Mueller and 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.


  • Bate, Roger B.; Mueller, Donald D.; White, Jerry E. (1971), Fundamentals of Astrodynamics, Dover

13P/Olbers is a periodic comet with an orbital period of 69 years. It fits the classical definition of a Halley-type comet with (20 years < period < 200 years).Heinrich Wilhelm Matthias Olbers (Bremen) discovered the comet on March 6, 1815. Its orbit was first computed by Carl Friedrich Gauss on March 31, Friedrich Bessel calculated an orbital period as 73 years, later as 73.9 years, calculations by other astronomers during that era resulted anywhere between 72 and 77 years.The comet was last detected in 1956. It will next come to perihelion on June 30, 2024. the comet will be closest to the Earth on January 10, 2094 when it passes at a distance of 0.756 AU.There is some speculation that 13P/Olbers has an associated meteor shower on Mars coming from the direction of Beta Canis Major.


144P/Kushida is a periodic comet discovered in January, 1994, by Yoshio Kushida at the Yatsugatake South Base Observatory in Japan. This was the first comet discovery of 1994 and his second discovery within a month.

Based on data gathered during the period of January 9–11, 1994 Syuichi Nakano calculated the date of perihelion to be 1993 December 5.33 and the distance of perihelion as 1.36 AU. The low inclination to the ecliptic suggested to Nakano that the comet could be a short period type. On January 14, 1994 Daniel W. E. Green confirmed Nakano's suggestion and published a short-period orbit on IAU Circular 5922. Based on 29 positions obtained during the period of January 9–13, Green determined a perihelion date of 1993 December 12.99, a perihelion distance of 1.37 AU, and an orbital period of 7.20 years.

Using over 300 positions obtained between January 7 and July 9, 1994 Patrick Rocher refined the calculations and determined the perihelion distance as 1.367 AU, the perihelion date as 1993 December 12.862, and the orbital period as 7.366 years.


14P/Wolf is a periodic comet in the Solar System.

Max Wolf (Heidelberg, Germany) discovered the comet on September 17, 1884. It was later discovered, but not credited to, Ralph Copeland (Dun Echt Observatory, Aberdeen, Scotland) on September 23.Previously, the comet had a perihelion of 2.74 AU and an orbital period of 8.84 a; this changed to a perihelion of 2.43 AU and an orbital period of 8.28 a due to passing 0.125 AU from Jupiter on September 27, 1922. The current values have been from when the comet passed Jupiter again on August 13, 2005. Another close approach to Jupiter on March 10, 2041 will return the comet to parameters similar to the period 1925–2000.The comet nucleus is estimated to be 4.7 kilometers in diameter.

32P/Comas Solà

32P/Comas Solà is a periodic comet with a current orbital period of 8.8 years.

The comet nucleus is estimated to be 8.4 kilometers in diameter.


60P/Tsuchinshan, also known as Tsuchinshan 2, is a periodic comet in the solar system with an orbital period of 6.79 years.

Tsuchinshan is the Wade-Giles transliteration corresponding to the pinyin Zĭjīn Shān, which is Mandarin Chinese for "Purple Mountain".

It was discovered at the Purple Mountain Observatory, Nanking, China on 11 January 1965 with a magnitude estimated as a very faint 15. The elliptical orbit was computed to give a perihelion date of 9 February 1965 with an orbital period of 6.69 years. Revised calculations predicted the next perihelion would be on 28 November 1971 and Elizabeth Roemer of the University of Arizona successfully relocated the comet with the 154-cm reflector at Catalina. It was also observed in 1978, 1985, 1991-1992, and 1998-1999.

The comet peaked at about apparent magnitude 16.3 in 2012. On 29 December 2077 the comet will pass 0.068 AU (10,200,000 km; 6,300,000 mi) from Mars.


63P/Wild is a periodic comet in the Solar System with a current orbital period of 13.21 years.

It was first detected by Paul Wild at the Zimmerwald Observatory of the Astronomical Institute of Bern, Switzerland on a photographic plate exposed on 26 March 1960, who estimated its brightness at a magnitude of 14.3. Its elliptical orbit was then calculated to have an orbital period of 13.17 years.Its predicted reappearance in 1973 was observed by Elizabeth Roemer of the U.S. Naval Observatory, Flagstaff, Arizona at a magnitude of 17.5. Although not found in 1986 it was rediscovered in 1999 with a magnitude of around 12.

The 2013 return was moderately favourable with magnitude again around 12.


70P/Kojima is a periodic comet in the Solar System with a current orbital period of 7.05 years.It was discovered at Ishiki, Aichi, Japan by Nobuhisa Kojima, who estimated its brightness at magnitude 14. Its parabolic orbit was calculated by Kiichirō Furukawa to have a perihelion date of 1 November 1970. This was revised on the basis of further observations to an elliptical orbit with a perihelion of 7 October and an orbital period of 6.16.

Hiroki Kosai and Furukawa relocated the comet on 9 December 1977 at its next predicted apparition with the 105 cm Schmidt telescope at the Kiso Station of the Tokyo Astronomical Observatory, estimating its brightness at magnitude 16. It was subsequently observed in 1985/1986 and 1992/1994 by Spacewatch with magnitudes of 20 and 22.1. The comet then passed close to Jupiter, which reduced the perihelion distance from 2.4 AU (Astronomical Unit) to 1.97 AU, increased the eccentricity from 0.39 to 0.46 and reduced the orbital period from 7.85 to 6.99.


71P/Clark is a periodic comet in the Solar System with an orbital period of 5.5 years.

It was discovered by Michael Clark at Mount John University Observatory, New Zealand on 9 June 1973 with a brightness of apparent magnitude 13. Subsequently it has been observed in 1978, 1984, 1989, 1995, 2000, 2006, 2011 and 2017.


78P/Gehrels, also known as Gehrels 2, is a Jupiter-family periodic comet in the Solar System with a current orbital period of 7.22 years.

It was discovered by Tom Gehrels at the Lunar and Planetary Laboratory, Arizona, USA on photographic plates exposed between 29 September and 5 October 1973 at the Palomar Observatory. It had a brightness of apparent magnitude of 15. Brian G. Marsden computed the parabolic and elliptical orbits which suggested an orbital period of 8.76 years, later revising the data to give a perihelion date of 30 November 1963 and orbital period of 7.93 years.The comet's predicted next appearance in 1981 was observed by W. and A. Cochran at the McDonald Observatory, Texas on 8 June 1981. It was observed again in 1989 and in 1997, when favourable conditions meant that brightness increased to magnitude 12. It has subsequently been observed in 2004 when it reached magnitude 10, 2012, and 2018.


89P/Russell is a periodic comet in the Solar System with a current orbital period of 7.28 years.It was discovered on a photographic plate by Kenneth Russell of Siding Spring Observatory in New South Wales, Australia on 28 September 1980. Brightness was estimated at a magnitude of 17. The elliptical orbit calculated by Brian G. Marsden gave a perihelion date of 19 May 1980 and an orbital period of 7.12 years.

It has been observed on each subsequent apparition, most recently in 2009. The next perihelion is computed as 14 December 2016.

C/2013 V5

C/2013 V5 (Oukaimeden) is a retrograde Oort cloud comet discovered on 12 November 2013 by Oukaimeden Observatory at an apparent magnitude of 19.4 using a 0.5-meter (20 in) reflecting telescope.From 5 May 2014 until 18 July 2014 it had an elongation less than 30 degrees from the Sun. By late August 2014 it had brighten to apparent magnitude 8 making it a small telescope and high-end binoculars target for experienced observers. It crossed the celestial equator on 30 August 2014 becoming a southern hemisphere object. On 16 September 2014 the comet passed 0.480 AU (71,800,000 km; 44,600,000 mi) from Earth. The comet peaked around magnitude 6.2 in mid-September 2014 but only had an elongation of about 35 degrees from the Sun. On 20 September 2014 the comet was visible in STEREO HI-1B. The comet came to perihelion (closest approach to the Sun) on 28 September 2014 at a distance of 0.625 AU (93,500,000 km; 58,100,000 mi) from the Sun.C/2013 V5 is dynamically new. It came from the Oort cloud with a loosely bound chaotic orbit that was easily perturbed by galactic tides and passing stars. Before entering the planetary region (epoch 1950), C/2013 V5 had an orbital period of several million years. After leaving the planetary region (epoch 2050), it will have an orbital period of about 6000 years.

Elliptic orbit

In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 (thus excluding the circular orbit). In a wider sense, it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

In a gravitational two-body problem with negative energy, both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit.

Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit, and tundra orbit.

Geosynchronous orbit

A geosynchronous orbit (sometimes abbreviated GSO) is an orbit around Earth of a satellite with an orbital period that matches Earth's rotation on its axis, which takes one sidereal day (about 23 hours, 56 minutes, and 4 seconds). The synchronization of rotation and orbital period means that, for an observer on Earth's surface, an object in geosynchronous orbit returns to exactly the same position in the sky after a period of one sidereal day. Over the course of a day, the object's position in the sky traces out a path, typically in a figure-8 form, whose precise characteristics depend on the orbit's inclination and eccentricity. Satellites are typically launched in an eastward direction. A geosynchronous orbit is 35,786 km (22,236 mi) above the Earth's surface. Those closer to Earth orbit faster than Earth rotates, so from Earth, they appear to move eastward while those that orbit beyond geosynchronous distances appear to move westward.

A special case of geosynchronous orbit is the geostationary orbit, which is a circular geosynchronous orbit in Earth's equatorial plane (that is, directly above the Equator). A satellite in a geostationary orbit appears stationary, always at the same point in the sky, to observers on the surface. Popularly or loosely, the term geosynchronous may be used to mean geostationary. Specifically, geosynchronous Earth orbit (GEO) may be a synonym for geosynchronous equatorial orbit, or geostationary Earth orbit. Communications satellites are often given geostationary or close to geostationary orbits so that the satellite antennas that communicate with them do not have to move, but can be pointed permanently at the fixed location in the sky where the satellite appears.

A semi-synchronous orbit has an orbital period of half a sidereal day (i.e., 11 hours and 58 minutes). Relative to Earth's surface, it has twice this period and hence appears to go around Earth once every day. Examples include the Molniya orbit and the orbits of the satellites in the Global Positioning System.

Ground track

A ground track or ground trace is the path on the surface of a planet directly below an aircraft or satellite. In the case of a satellite, it is the projection of the satellite's orbit onto the surface of the Earth (or whatever body the satellite is orbiting).

A satellite ground track may be thought of as a path along the Earth's surface which traces the movement of an imaginary line between the satellite and the center of the Earth. In other words, the ground track is the set of points at which the satellite will pass directly overhead, or cross the zenith, in the frame of reference of a ground observer.

Mean anomaly

In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical two-body problem. It is the angular distance from the pericenter which a fictitious body would have if it moved in a circular orbit, with constant speed, in the same orbital period as the actual body in its elliptical orbit.

Molniya orbit

A Molniya orbit (Russian: Молния, IPA: [ˈmolnʲɪjə] (listen), "Lightning") is a type of satellite orbit, designed to provide coverage over high latitudes. It is a highly elliptical orbit with an inclination of 63.4 degrees, an argument of perigee of 270 degrees and an orbital period of approximately half a sidereal day. The name comes from a series of Soviet/Russian Molniya communications satellites which have been using this type of orbit since the mid-1960s.The high inclination of Molniya orbits provides a high angle of view to communications and monitoring satellites covering high latitudes. It has a long dwell time over the hemisphere of interest, while moving very quickly over the other. Geostationary orbits, which are necessarily inclined over the equator, can only view these regions from a low angle, and are unable to view latitudes above 81 degrees.

P/2010 B2 (WISE)

P/2010 B2 is a periodic comet in the Solar System. It is the first comet discovered by the space observatory WISE and was first observed on January 22, 2010 and has since been followed by ground observatories, among them the Mauna Kea Observatory.The comet has an orbital period of 4.7 years, an aphelion of 4 astronomical units and a perihelion of 1.6 astronomical units.

Semi-major and semi-minor axes

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter.

The semi-major axis (more properly, major semi-axis) is one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (more properly, minor semi-axis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

The length of the semi-major axis of an ellipse is related to the semi-minor axis's length through the eccentricity and the semi-latus rectum , as follows:

The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the center to either vertex of the hyperbola.

A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping fixed. Thus and tend to infinity, faster than .

The major and minor axes are the axes of symmetry for the curve: in an ellipse, the minor axis is the shorter one; in a hyperbola, it is the one that does not intersect the hyperbola.

Sidereal year

A sidereal year (from Latin sidus "asterism, star") is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence, it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic.

It equals 365.256 363 004 SI days for the J2000.0 epoch.The sidereal year differs from the tropical year, "the period of time required for the ecliptic longitude of the sun to increase 360 degrees", due to the precession of the equinoxes.

The sidereal year is 20 min 24.5 s longer than the mean tropical year at J2000.0 (365.242 190 402 SI days).Before the discovery of the precession of the equinoxes by Hipparchus in the Hellenistic period, the difference between sidereal and tropical year was unknown. For naked-eye observation, the shift of the constellations relative to the equinoxes only becomes apparent over centuries or "ages", and pre-modern calendars such as Hesiod's Works and Days would give the times of the year for sowing, harvest, and so on by reference to the first visibility of stars, effectively using the sidereal year. The South and Southeast Asian solar New Year, based on Indic influences, is traditionally reckoned by the sun's entry into Aries and thus the sidereal year, but is also supposed to align with the spring equinox and have relevance to the harvesting and planting season and thus the tropical year. As these have diverged, in some countries and cultures the date has been fixed according to the tropical year while in others the astronomical calculation and sidereal year is still used.

Gravitational orbits
Orbital mechanics

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