Barycentric coordinates (astronomy)
Encyclopedia
In astronomy
, barycentric coordinates are non-rotating coordinates with origin at the center of mass
of two or more bodies.
The barycenter (or barycentre; from the Greek
βαρύκεντρον) is the point between two objects where they balance each other. For example, it is the center of mass where two or more celestial bodies orbit
each other. When a moon
orbits a planet
, or a planet orbits a star
, both bodies are actually orbiting around a point that lies outside the center of the primary (the larger body). For example, the moon does not orbit the exact center of the Earth
, but a point on a line between the center of the Earth and the Moon, approximately 1,710 km below the surface of the Earth, where their respective masses balance. This is the point about which the Earth and Moon orbit as they travel around the Sun
.
of the elliptical orbit
of each body. This is an important concept in the fields of astronomy
, astrophysics
, and the like (see two-body problem
). In a simple two-body case, r1, the distance from the center of the primary to the barycenter is given by:
where:
r1 is essentially the semi-major axis
of the primary's orbit around the barycenter—and r2 = a − r1 the semi-major axis of the secondary's orbit. Where the barycenter is located within the more massive body, that body will appear to "wobble" rather than following a discernible orbit.
. Figures are given rounded to three significant figures
. The last two columns show R1, the radius of the first (more massive) body, and r1/R1, the ratio of the distance to the barycenter and that radius: a value less than one shows that the barycenter lies inside the first body.
Hence, the barycenter of the Sun-planet system will lie outside the Sun only if:
That is, where the planet is heavy and far from the Sun.
If Jupiter had Mercury
's orbit (57,900,000 km, 0.387 AU), the Sun-Jupiter barycenter would be only 5,500 km from the center of the Sun (r1/R1 ~ 0.08). But even if the Earth had Eris'
orbit (68 AU), the Sun-Earth barycenter would still be within the Sun (just over 30,000 km from the center).
To calculate the actual motion of the Sun, you would need to sum all the influences from all the planet
s, comet
s, asteroid
s, etc. of the Solar System
(see n-body problem
). If all the planets were aligned on the same side of the Sun, the combined center of mass would lie about 500,000 km above the Sun's surface.
The calculations above are based on the mean distance between the bodies and yield the mean value r1. But all celestial orbits are elliptical, and the distance between the bodies varies between the apses
, depending on the eccentricity, e. Hence, the position of the barycenter varies too, and it is possible in some systems for the barycenter to be sometimes inside and sometimes outside the more massive body. This occurs where:
Note that the Sun-Jupiter system, with eJupiter = 0.0484, just fails to qualify: 1.05 ≯ 1.07 > 0.954.
, this definition simplifies calculations and introduces no known problems. In the General Theory of Relativity
, problems arise because, while it is possible, within reasonable approximations, to define the barycentre, the associated coordinate system does not fully reflect the inequality of clock rates at different locations. Brumberg explains how to set up barycentric coordinates in General Theory of Relativity.
The coordinate systems involve a world-time, i.e., a global time coordinate that could be set up by telemetry
. Individual clocks of similar construction will not agree with this standard, because they are subject to differing gravitational potentials or move at various velocities, so the world-time must be slaved to some ideal clock; that one is assumed to be very far from the whole self-gravitating system. This time standard is called Barycentric Coordinate Time
, abbreviated "TCB."
For objects at such high eccentricity, the Sun's barycentric coordinates are more stable than heliocentric coordinates.
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, barycentric coordinates are non-rotating coordinates with origin at the center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
of two or more bodies.
The barycenter (or barycentre; from the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
βαρύκεντρον) is the point between two objects where they balance each other. For example, it is the center of mass where two or more celestial bodies orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
each other. When a moon
Natural satellite
A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....
orbits a planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
, or a planet orbits a star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
, both bodies are actually orbiting around a point that lies outside the center of the primary (the larger body). For example, the moon does not orbit the exact center of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
, but a point on a line between the center of the Earth and the Moon, approximately 1,710 km below the surface of the Earth, where their respective masses balance. This is the point about which the Earth and Moon orbit as they travel around the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
.
Two-body problem
The barycenter is one of the fociFocus (geometry)
In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...
of the elliptical orbit
Elliptic orbit
In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity less than 1; this includes the special case of a circular orbit, with eccentricity equal to zero. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 . In a...
of each body. This is an important concept in the fields of astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, astrophysics
Astrophysics
Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...
, and the like (see two-body problem
Two-body problem
In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus In...
). In a simple two-body case, r1, the distance from the center of the primary to the barycenter is given by:
where:
- a is the distance between the centers of the two bodies;
- m1 and m2 are the massMassMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
es of the two bodies.
r1 is essentially the semi-major axis
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
of the primary's orbit around the barycenter—and r2 = a − r1 the semi-major axis of the secondary's orbit. Where the barycenter is located within the more massive body, that body will appear to "wobble" rather than following a discernible orbit.
Examples
The following table sets out some examples from the Solar SystemSolar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
. Figures are given rounded to three significant figures
Significant figures
The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...
. The last two columns show R1, the radius of the first (more massive) body, and r1/R1, the ratio of the distance to the barycenter and that radius: a value less than one shows that the barycenter lies inside the first body.
| Larger body |
m1 (mE=1) |
Smaller body |
m2 (mE=1) |
a (km Kilometre The kilometre is a unit of length in the metric system, equal to one thousand metres and is therefore exactly equal to the distance travelled by light in free space in of a second... ) |
r1 (km) |
R1 (km) |
r1/R1 |
|---|---|---|---|---|---|---|---|
| Remarks | |||||||
| Earth Earth Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets... |
1 | Moon Moon The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more... |
0.0123 | 384,000 | 4,670 | 6,380 | 0.732 |
| The Earth has a perceptible "wobble"; see tide Tide Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the moon and the sun and the rotation of the Earth.... s. |
|||||||
| Pluto Pluto Pluto, formal designation 134340 Pluto, is the second-most-massive known dwarf planet in the Solar System and the tenth-most-massive body observed directly orbiting the Sun... |
0.0021 | Charon Charon (moon) Charon is the largest satellite of the dwarf planet Pluto. It was discovered in 1978 at the United States Naval Observatory Flagstaff Station. Following the 2005 discovery of two other natural satellites of Pluto , Charon may also be referred to as Pluto I... |
0.000254 (0.121 mPluto) |
19,600 | 2,110 | 1,150 | 1.83 |
| Both bodies have distinct orbits around the barycenter, and as such Pluto and Charon were considered as a double planet Double planet In astronomy, double planet and binary planet are informal terms used to describe a binary system of two astronomical objects that each satisfy the definition of planet and that are near enough to each other to have a significant gravitational effect on each other compared with the effect of the... by many before the redefinition of planet Planet A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,... in August 2006. |
|||||||
| Sun Sun The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields... |
333,000 | Earth | 1 | 150,000,000 (1 AU Astronomical unit An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance.... ) |
449 | 696,000 | 0.000646 |
| The Sun's wobble is barely perceptible. | |||||||
| Sun | 333,000 | Jupiter Jupiter Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,... |
318 (0.000955 mSun) |
778,000,000 (5.20 AU) |
742,000 | 696,000 | 1.07 |
| The Sun orbits a barycenter just above its surface. | |||||||
Inside or outside the Sun?
If m1 ≫ m2—which is true for the Sun and any planet—then the ratio r1/R1 approximates to:Hence, the barycenter of the Sun-planet system will lie outside the Sun only if:
That is, where the planet is heavy and far from the Sun.
If Jupiter had Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
's orbit (57,900,000 km, 0.387 AU), the Sun-Jupiter barycenter would be only 5,500 km from the center of the Sun (r1/R1 ~ 0.08). But even if the Earth had Eris'
Eris (dwarf planet)
Eris, formal designation 136199 Eris, is the most massive known dwarf planet in the Solar System and the ninth most massive body known to orbit the Sun directly...
orbit (68 AU), the Sun-Earth barycenter would still be within the Sun (just over 30,000 km from the center).
To calculate the actual motion of the Sun, you would need to sum all the influences from all the planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s, comet
Comet
A comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...
s, asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
s, etc. of the Solar System
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
(see n-body problem
N-body problem
The n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...
). If all the planets were aligned on the same side of the Sun, the combined center of mass would lie about 500,000 km above the Sun's surface.
The calculations above are based on the mean distance between the bodies and yield the mean value r1. But all celestial orbits are elliptical, and the distance between the bodies varies between the apses
Apsis
An apsis , plural apsides , is the point of greatest or least distance of a body from one of the foci of its elliptical orbit. In modern celestial mechanics this focus is also the center of attraction, which is usually the center of mass of the system...
, depending on the eccentricity, e. Hence, the position of the barycenter varies too, and it is possible in some systems for the barycenter to be sometimes inside and sometimes outside the more massive body. This occurs where:
Note that the Sun-Jupiter system, with eJupiter = 0.0484, just fails to qualify: 1.05 ≯ 1.07 > 0.954.
Animations
Images are representative (made by hand), not simulated. Two bodies of similar mass orbiting a common barycenter. (similar to the 90 Antiope 90 Antiope The most remarkable feature of Antiope is that it consists of two components of almost equal size , making it a truly "double" asteroid. Its binary nature was discovered on 10 August 2000 by a group of astronomers using adaptive optics at the Keck Telescope on Mauna Kea. The "secondary" is... system). |
 Two bodies with a difference in mass orbiting a common barycenter external to both bodies, as in the Pluto Pluto Pluto, formal designation 134340 Pluto, is the second-most-massive known dwarf planet in the Solar System and the tenth-most-massive body observed directly orbiting the Sun... –Charon Charon (moon) Charon is the largest satellite of the dwarf planet Pluto. It was discovered in 1978 at the United States Naval Observatory Flagstaff Station. Following the 2005 discovery of two other natural satellites of Pluto , Charon may also be referred to as Pluto I... system. |
 Two bodies with a major difference in mass orbiting a common barycenter internal to one body (similar to the Earth Earth Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets... –Moon Moon The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more... system). |
 Two bodies with an extreme difference in mass orbiting a common barycenter internal to one body (similar to the Sun Sun The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields... –Earth Earth Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets... system). |
 Two bodies with similar mass orbiting a common barycenter, external to both bodies, with elliptic orbit Elliptic orbit In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity less than 1; this includes the special case of a circular orbit, with eccentricity equal to zero. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 . In a... s (a common situation for binary star Binary star A binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary... s) |
Relativistic corrections
Within classical mechanicsClassical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, this definition simplifies calculations and introduces no known problems. In the General Theory of Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, problems arise because, while it is possible, within reasonable approximations, to define the barycentre, the associated coordinate system does not fully reflect the inequality of clock rates at different locations. Brumberg explains how to set up barycentric coordinates in General Theory of Relativity.
The coordinate systems involve a world-time, i.e., a global time coordinate that could be set up by telemetry
Telemetry
Telemetry is a technology that allows measurements to be made at a distance, usually via radio wave transmission and reception of the information. The word is derived from Greek roots: tele = remote, and metron = measure...
. Individual clocks of similar construction will not agree with this standard, because they are subject to differing gravitational potentials or move at various velocities, so the world-time must be slaved to some ideal clock; that one is assumed to be very far from the whole self-gravitating system. This time standard is called Barycentric Coordinate Time
Barycentric Coordinate Time
Barycentric Coordinate Time is a coordinate time standard intended to be used as the independent variable of time for all calculations pertaining to orbits of planets, asteroids, comets, and interplanetary spacecraft in the Solar system...
, abbreviated "TCB."
Selected barycentric orbital elements
Barycentric osculating orbital elements for some objects in the Solar System:| Object Small Solar System body A small Solar System body is an object in the Solar System that is neither a planet nor a dwarf planet, nor a satellite of a planet or dwarf planet:... |
Semi-major axis Semi-major axis The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape... (in AU Astronomical unit An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance.... ) |
Apoapsis (in AU) |
Orbital period Orbital period The orbital period is the time taken for a given object to make one complete orbit about 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... (in years) |
|---|---|---|---|
| C/2006 P1 (McNaught) | 2050 | 4100 | 92600 |
| Comet Hyakutake Comet Hyakutake Comet Hyakutake is a comet, discovered on January 31, 1996, which passed very close to Earth in March of that year. It was dubbed The Great Comet of 1996; its passage near the Earth was one of the closest cometary approaches of the previous 200 years. Hyakutake appeared very bright in the night... |
1700 | 3410 | 70000 |
| C/2006 M4 (SWAN) | 1300 | 2600 | 47000 |
| 2006 SQ372 2006 SQ372 ' is a small trans-Neptunian object discovered through the Sloan Digital Sky Survey by Andrew Becker, Andrew W. Puckett, and Jeremy Martin Kubica on images first taken on September 27, 2006 .... |
799 | 1570 | 22600 |
| 2000 OO67 | 549 | 1078 | 12800 |
| 90377 Sedna 90377 Sedna 90377 Sedna is a trans-Neptunian object discovered in 2003, which was about three times as far from the Sun as Neptune. For most of its orbit it is even further from the Sun, with its aphelion estimated at 960 astronomical units , making it one of the most distant known objects in the Solar System... |
506 | 937 | 11400 |
| 2007 TG422 2007 TG422 ', also written as 2007 TG422, is a scattered-disc object with a perihelion distance of 35.5 AU, which is just inside the influence of Neptune, and a aphelion distance similar to Sedna's.-Orbit:... |
501 | 967 | 11200 |
For objects at such high eccentricity, the Sun's barycentric coordinates are more stable than heliocentric coordinates.