Barycenter

The barycenter (or barycentre; from the Greek βαρύ-ς heavy + κέντρ-ον centre[1]) is the center of mass of two or more bodies that are orbiting each other, or the point around which they both orbit. It is an important concept in fields such as astronomy and astrophysics. The distance from a body's center of mass to the barycenter can be calculated as a simple two-body problem.

In cases where one of the two objects is considerably more massive than the other (and relatively close), the barycenter will typically be located within the more massive object. Rather than appearing to orbit a common center of mass with the smaller body, the larger will simply be seen to "wobble" slightly. This is the case for the Earth–Moon system, where the barycenter is located on average 4,671 km from the Earth's center, well within the planet's radius of 6,378 km. When the two bodies are of similar masses, the barycenter will generally be located between them and both bodies will follow an orbit around it. This is the case for Pluto and Charon, as well as for many binary asteroids and binary stars. It is also the case for Jupiter and the Sun, despite the 1,000-fold difference in mass, due to the relatively large distance between them.[citation needed]

In astronomy, barycentric coordinates are non-rotating coordinates with the origin at the center of mass of two or more bodies. The International Celestial Reference System is a barycentric one, based on the barycenter of the Solar System.

In geometry, the term "barycenter" is synonymous with centroid, the geometric center of a two-dimensional shape.

Two-body problem

Main article: Two-body problem
Barycentric view of the PlutoCharon system as seen by New Horizons

The barycenter is one of the foci of the elliptical orbit of each body. This is an important concept in the fields of astronomy and astrophysics. If a is the distance between the centers of the two bodies (the semi-major axis of the system), r1 is the semi-major axis of the primary's orbit around the barycenter, and r2 = a − r1 is the semi-major axis of the secondary's orbit. When the barycenter is located within the more massive body, that body will appear to "wobble" rather than to follow a discernible orbit. In a simple two-body case, r1, the distance from the center of the primary to the barycenter is given by:

${\displaystyle r_{1}=a\cdot {m_{2} \over m_{1}+m_{2}}={a \over 1+m_{1}/m_{2}}}$

where :

r1 is the distance from body 1 to the barycenter
a is the distance between the centers of the two bodies
m1 and m2 are the masses of the two bodies.

Primary–secondary examples

The following table sets out some examples from the Solar System. 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. The term primary–secondary is used to distinguish between the different degrees of relationship of the involved participants.

Primary–secondary examples
Larger
body
m1
(M)
Smaller
body
m2
(M)
a
(km)
r1
(km)
R1
(km)
r1 / R1
Earth 1 Moon 0.0123 384,000 4,670 6,380 0.732
The Earth has a perceptible "wobble". Also see tides.
Pluto 0.0021 Charon
0.000254
(0.121 M)
19,600 2,110 1,150 1.83
Pluto and Charon have distinct orbits around their barycenter, and as such they were considered as a double planet by many before the redefinition of a planet in 2006.
Sun 333,000 Earth 1
150,000,000
(1 AU)
449 696,000 0.000646
The Sun's wobble is barely perceptible.
Sun 333,000 Jupiter
318
(0.000955 M)
778,000,000
(5.20 AU)
742,000 696,000 1.07
The Sun orbits a barycenter just above its surface.[2]

Inside or outside the Sun?

Motion of the Solar System's barycenter relative to the Sun

If m1 ≫ m2 — which is true for the Sun and any planet — then the ratio r1/R1 approximates to:

${\displaystyle {a \over R_{1}}\cdot {m_{2} \over m_{1}}}$

Hence, the barycenter of the Sun–planet system will lie outside the Sun only if:

${\displaystyle {a \over R_{\bigodot }}\cdot {m_{planet} \over m_{\bigodot }}>1\;\Rightarrow \;{a\cdot m_{planet}}>{R_{\bigodot }\cdot m_{\bigodot }}\approx 2.3\times 10^{11}\;m_{Earth}\;{\mbox{km}}\approx 1530\;m_{Earth}\;{\mbox{AU}}}$

That is, where the planet is massive and far from the Sun.

If Jupiter had Mercury's orbit (57,900,000 km, 0.387 AU), the Sun–Jupiter barycenter would be approximately 55,000 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 planets, comets, asteroids, 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:

${\displaystyle {1 \over {1-e}}>{r_{1} \over R_{1}}>{1 \over {1+e}}}$

Note that the Sun–Jupiter system, with eJupiter = 0.0484, just fails to qualify: 1.05  1.07 > 0.954.

Gallery

Images are representative (made by hand), not simulated.

Relativistic corrections

In classical mechanics, this definition simplifies calculations and introduces no known problems. In general relativity, problems arise because, while it is possible, within reasonable approximations, to define the barycenter, 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 relativity.[3]

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 is assumed to be very far from the whole self-gravitating system. This time standard is called Barycentric Coordinate Time, "TCB".

Selected barycentric orbital elements

Barycentric osculating orbital elements for some objects in the Solar System:[4]

Object
Semi-major axis
(in AU)
Apoapsis
(in AU)
Orbital period
(in years)
C/2006 P1 (McNaught) 2,050 4,100 92,600
Comet Hyakutake 1,700 3,410 70,000
C/2006 M4 (SWAN) 1,300 2,600 47,000
(308933) 2006 SQ372 799 1,570 22,600
(87269) 2000 OO67 549 1,078 12,800
90377 Sedna 506 937 11,400
2007 TG422 501 967 11,200

For objects at such high eccentricity, the Sun's barycentric coordinates are more stable than heliocentric coordinates.[5]