• Images
• Videos
• Documents
• Books
• Articles

Two-body problem

Main article: Two-body problem

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. In a simple two-body case, the distance from the center of the primary to the barycenter, r1, is given by:

r 1 = a ⋅ m 2 m 1 + m 2 = a 1 + m 1 m 2 {\displaystyle r_{1}=a\cdot {\frac {m_{2}}{m_{1}+m_{2}}}={\frac {a}{1+{\frac {m_{1}}{m_{2}}}}}}

where :

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

The semi-major axis of the secondary's orbit, r2, is given by r2 = a − r1.

When the barycenter is located within the more massive body, that body will appear to "wobble" rather than to follow a discernible orbit.

Primary–secondary examples

The following table sets out some examples from the Solar System. Figures are given rounded to three significant figures. The terms "primary" and "secondary" are used to distinguish between involved participants, with the larger being the primary and the smaller being the secondary.

• m1 is the mass of the primary in Earth masses (M🜨)
• m2 is the mass of the secondary in Earth masses (M🜨)
• a (km) is the average orbital distance between the centers of the two bodies
• r1 (km) is the distance from the center of the primary to the barycenter
• R1 (km) is the radius of the primary
• ⁠r1/R1⁠ a value less than one means the barycenter lies inside the primary

Primary–secondary examples

Primary	m1(M🜨)	Secondary	m2(M🜨)	a(km)	r1(km)	R1(km)	⁠r1/R1⁠
Earth	1	Moon	0.0123	384,400	4,6713	6,371	0.7334
Pluto	0.0021	Charon	0.000254(0.121 M♇)	19,600	2,110	1,188.3	1.785
Sun	333,000	Earth	1	150,000,000(1 AU)	449	695,700	0.0006456
Sun	333,000	Jupiter	318(0.000955 M☉)	778,000,000(5.20 AU)	742,370	695,700	1.0778
Sun	333,000	Saturn	95.2	1,433,530,000(9.58 AU)	409,700	695,700	0.59

Example with the Sun

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

a R 1 ⋅ m 2 m 1 . {\displaystyle {\frac {a}{R_{1}}}\cdot {\frac {m_{2}}{m_{1}}}.}

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

a R ⊙ ⋅ m p l a n e t m ⊙ > 1 ⇒ a ⋅ m p l a n e t > R ⊙ ⋅ m ⊙ ≈ 2.3 × 10 11 m ⊕ km ≈ 1530 m ⊕ AU {\displaystyle {a \over R_{\odot }}\cdot {m_{\mathrm {planet} } \over m_{\odot }}>1\;\Rightarrow \;{a\cdot m_{\mathrm {planet} }}>{R_{\odot }\cdot m_{\odot }}\approx 2.3\times 10^{11}\;m_{\oplus }\;{\mbox{km}}\approx 1530\;m_{\oplus }\;{\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's orbit (1.02×1010 km, 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, only the motions of the four giant planets (Jupiter, Saturn, Uranus, Neptune) need to be considered. The contributions of all other planets, dwarf planets, etc. are negligible. If the four giant planets were on a straight line on the same side of the Sun, the combined center of mass would lie at about 1.17 solar radii, or just over 810,000 km, above the Sun's surface.⁹

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:

1 1 − e > r 1 R 1 > 1 1 + e {\displaystyle {\frac {1}{1-e}}>{\frac {r_{1}}{R_{1}}}>{\frac {1}{1+e}}}

The Sun–Jupiter system, with eJupiter = 0.0484, just fails to qualify: 1.05 < 1.07 > 0.954.

Relativistic corrections

In classical mechanics (Newtonian gravitation), this definition simplifies calculations and introduces no known problems. In general relativity (Einsteinian gravitation), complications arise because, while it is possible, within reasonable approximations, to define the barycenter, we find that 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.¹⁰

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 synchronized with 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 are as follows:¹¹

Object	Semi-major axis(in AU)	Apoapsis(in AU)	Orbital period(in years)
C/2006 P1 (McNaught)	2,050	4,100	92,600
C/1996 B2 (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, barycentric coordinates are more stable than heliocentric coordinates for a given epoch because the barycentric osculating orbit is not as greatly affected by where Jupiter is on its 11.8 year orbit.¹²

See also

• Barycentric Dynamical Time
• Centers of gravity in non-uniform fields
• Center of mass
• Lagrange point
• Mass point geometry
• Roll center
• Weight distribution

References

1. "barycentre". Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. /wiki/Oxford_English_Dictionary ↩

2. MacDougal, Douglas W. (December 2012). Newton's Gravity: An Introductory Guide to the Mechanics of the Universe. Berlin: Springer Science & Business Media. p. 199. ISBN 978-1-4614-5444-1. 978-1-4614-5444-1 ↩

3. Moore, P. (2005). "SOLAR SYSTEM | Moon". Encyclopedia of Geology. pp. 264–272. doi:10.1016/B0-12-369396-9/00077-0. ISBN 978-0-12-369396-9. barycentre lies 1700 km below the Earth's surface(6370km–1700km) 978-0-12-369396-9 ↩

4. The Earth has a perceptible "wobble". Also see tides. /wiki/Tide ↩

5. Pluto and Charon are sometimes considered a binary system because their barycenter does not lie within either body.[4] /wiki/Pluto ↩

6. The Sun's wobble is barely perceptible. ↩

7. "If You Think Jupiter Orbits the Sun, You're Mistaken". HowStuffWorks. 9 August 2016. The Sol-Jupiter barycenter sits 1.07 times the radius of the sun https://science.howstuffworks.com/jupiter-orbit-sun-barycenter.htm ↩

8. The Sun orbits a barycenter just above its surface.[6] ↩

9. Meeus, Jean (1997), Mathematical Astronomy Morsels, Richmond, Virginia: Willmann-Bell, pp. 165–168, ISBN 0-943396-51-4 0-943396-51-4 ↩

10. Brumberg, Victor A. (1991). Essential Relativistic Celestial Mechanics. London: Adam Hilger. ISBN 0-7503-0062-0. 0-7503-0062-0 ↩

11. Horizons output (30 January 2011). "Barycentric Osculating Orbital Elements for 2007 TG422". Archived from the original on 28 March 2014. Retrieved 31 January 2011. (Select Ephemeris Type:Elements and Center:@0) /wiki/JPL_Horizons_On-Line_Ephemeris_System ↩

12. Kaib, Nathan A.; Becker, Andrew C.; Jones, R. Lynne; Puckett, Andrew W.; Bizyaev, Dmitry; Dilday, Benjamin; Frieman, Joshua A.; Oravetz, Daniel J.; Pan, Kaike; Quinn, Thomas; Schneider, Donald P.; Watters, Shannon (2009). "2006 SQ372: A Likely Long-Period Comet from the Inner Oort Cloud". The Astrophysical Journal. 695 (1): 268–275. arXiv:0901.1690. Bibcode:2009ApJ...695..268K. doi:10.1088/0004-637X/695/1/268. S2CID 16987581. /wiki/The_Astrophysical_Journal ↩