View source for Barycenter (astronomy)

{{short description|Center of mass of multiple bodies orbiting each other}}
{{Use American English|date=December 2020}}
{{Use dmy dates|date=December 2020}}
{{redirect|Barycenter|the general concept|Barycenter (physics)}}
{{anchor|Animations}}{{Multiple image
| image1            = orbit1.gif
| image2            = orbit2.gif
| image3            = orbit3.gif
| image4            = orbit4.gif
| image5            = orbit5.gif
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| perrow            = 2/2/1
| header            = Animation of barycenters
| caption1          = Two bodies with similar mass, like the [[90 Antiope]] asteroid system
| caption2          = Two bodies with slightly different masses, like [[Pluto]] and [[Charon (moon)|Charon]]
| caption3          = Two bodies with significant difference in masses, like [[Earth]] and the [[Moon]]
| caption4          = Two bodies with an extreme difference in mass, like the [[Sun]] and [[Earth]]
| caption5          = Two bodies with the same mass with [[Orbital eccentricity|eccentric]] elliptic orbits, common for [[binary star]]s}}
{{Astrodynamics}}

In [[astronomy]], the '''barycenter''' (or '''barycentre'''; {{etymology|grc|''{{wikt-lang|grc|βαρύς}}'' ({{grc-transl|βαρύς}})|heavy||''{{wikt-lang|grc|κέντρον}}'' ({{grc-transl|κέντρον}})|center}})<ref>{{cite OED2|barycentre}}</ref> is the [[center of mass]] of two or more bodies that [[orbit]] one another and is the point about which the bodies orbit. A barycenter is a [[Dynamical_system_(definition)|dynamical]] point, not a physical object. 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 [[two-body problem]].

If one of the two orbiting bodies is much more massive than the other and the bodies are relatively close to one another, the barycenter will typically be located within the more massive object. In this case, rather than the two bodies appearing to orbit a point between them, the less massive body will appear to orbit about the more massive body, while the more massive body might be observed to wobble slightly. This is the case for the [[Lunar theory|Earth–Moon system]], whose barycenter is located on average {{convert|4671|km|mi|abbr=on}} from Earth's center, which is 74% of Earth's radius of {{convert|6378|km|mi|abbr=on}}. When the two bodies are of similar masses, the barycenter will generally be located between them and both bodies will orbit around it. This is the case for [[Pluto]] and [[Charon (moon)|Charon]], one of Pluto's [[natural satellite]]s, as well as for many [[binary asteroid]]s and [[binary star]]s. When the less massive object is far away, the barycenter can be located outside the more massive object. This is the case for [[Jupiter]] and the [[Sun]]; despite the Sun being a thousandfold more massive than Jupiter, their barycenter is slightly outside the Sun due to the relatively large distance between them.<ref name="MacDougal" />

In astronomy, '''barycentric coordinates''' are non-rotating coordinates with the origin at the barycenter of two or more bodies. The [[International Celestial Reference System]] (ICRS) is a barycentric coordinate system centered on the [[Solar System]]'s barycenter.
<!-- 
#transferred lead-content#
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 [[natural satellite|moon]] orbits a [[planet]], or a planet orbits a [[star]], both bodies are actually orbiting around a point that is not at 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 {{convert|1710|km|mi|abbr=on}} below the surface of the Earth ({{convert|4661|km|mi|abbr=on}} from Earth's center, about 74% of Earth's radius), where their respective masses balance. This is the point about which the Earth and Moon orbit as they travel around the [[Sun]].
-->

== Two-body problem ==
{{main|Two-body problem}}

The barycenter is one of the [[focus (geometry)|foci]] of the [[elliptic orbit|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, ''r''<sub>1</sub>, is given by:

:<math>r_1 = a \cdot \frac{m_2}{m_1 + m_2} = \frac{a}{1 + \frac{m_1}{m_2}}</math>

where :
*''r''<sub>1</sub> is the [[distance]] from body 1's center to the barycenter
*''a'' is the distance between the centers of the two bodies
*''m''<sub>1</sub> and ''m''<sub>2</sub> are the [[mass]]es of the two bodies.

The [[semi-major axis]] of the secondary's orbit, ''r''<sub>2</sub>, is given by {{nowrap|1=''r''<sub>2</sub> = ''a'' − ''r''<sub>1</sub>}}.

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

=== {{anchor|Sun-Jupiter barycenter}} 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.

<div style="text-align: left; direction: ltr; margin-left: 1em;">
{{plainlist|
* ''m''<sub>1</sub> is the mass of the primary in Earth masses ({{earth mass|link=yes}})
* ''m''<sub>2</sub> is the mass of the secondary in Earth masses ({{earth mass|link=yes}})
* ''a'' (km) is the average orbital distance between the centers of the two bodies
* ''r''<sub>1</sub> (km) is the distance from the center of the primary to the barycenter
* ''R''<sub>1</sub> (km) is the radius of the primary
* {{sfrac|''r''<sub>1</sub>|''R''<sub>1</sub>}} a value less than one means the barycenter lies inside the primary
}}

{| class="wikitable" style="text-align: center;"
|+ '''Primary–secondary examples'''<br />
! [[primary and secondary (relationship)|Primary]]
! ''m''<sub>1</sub><br />({{earth mass|link=yes}})
! [[primary and secondary (relationship)|Secondary]]
! ''m''<sub>2</sub><br />({{earth mass}})
! ''a''<br />([[kilometre|km]])
! ''r''<sub>1</sub><br />(km)
! ''R''<sub>1</sub><br />(km)
! {{sfrac|''r''<sub>1</sub>|''R''<sub>1</sub>}}
|- 
| [[Earth]]
| 1
| [[Moon]]
| 0.0123
| 384,400
| 4,670<ref>{{Cite web |title=Center of Gravity - an overview |url=https://www.sciencedirect.com/topics/earth-and-planetary-sciences/center-of-gravity |website=ScienceDirect Topics |quote=barycentre lies 1700 km below the Earth's surface<br/>(6370km–1700km)}}</ref>
| 6,380
| 0.732{{efn|group=PSEx|name=PSEx1|The Earth has a perceptible "wobble". Also see [[tide]]s.}}
|-
| [[Pluto]]
| 0.0021
| [[Charon (moon)|Charon]]
| {{longitem|0.000254<br />{{small|(0.121&nbsp;[[Pluto#Mass and size|''M''<sub>♇</sub>]])}}|style=line-height: 120%; padding: 8px 0 2px;}}
| &nbsp; 19,600
| 2,110
| 1,150
| 1.83{{efn|group=PSEx|name=PSEx2|[[Pluto]] and Charon are sometimes considered a [[Binary system (astronomy)|binary system]] because their barycenter does not lie within either body.<ref name="Olkin-2015" />}}
|-
| [[Sun]]
| 333,000
| Earth
| 1
| {{longitem|150,000,000<br />{{small|(1 [[astronomical unit|AU]])}}|style=line-height: 120%; padding: 8px 0 2px;}}
| 449
| 696,000
| 0.000646{{efn|group=PSEx|name=PSEx3|The Sun's wobble is barely perceptible.}}
|-
| Sun
| 333,000
| [[Jupiter]]
| {{longitem|318<br />{{small|(0.000955&nbsp;{{solar mass|link=yes}})}}|style=line-height: 120%; padding: 8px 0 2px;}}
| {{longitem|778,000,000<br />{{small|(5.20 AU)}}|style=line-height: 120%; padding: 8px 0 2px;}}
| 742,000
| 696,000
| 1.07<ref>{{Cite web |date=2016-08-09 |title=If You Think Jupiter Orbits the Sun, You're Mistaken |url=https://science.howstuffworks.com/jupiter-orbit-sun-barycenter.htm |website=HowStuffWorks |language=en-us |quote=The Sol-Jupiter barycenter sits 1.07 times the radius of the sun}}</ref>{{efn|group=PSEx|name=PSEx4|The Sun orbits a barycenter just above its surface.<ref name="NASA2005" />}}
|-
| Sun
| 333,000
| [[Saturn]]
| 95.2
| {{longitem|1,430,000,000<br />{{small|(9.58 AU)}}|style=line-height: 120%; padding: 8px 0 2px;}}
| 409,000
| 696,000
| 0.588
|}

{{Notelist|group=PSEx}}
</div>

=== Example with the Sun ===
[[File:Solar system barycenter.svg|thumb|Motion of the [[Solar System]]'s barycenter relative to the Sun]]

If {{nowrap|''m''<sub>1</sub> ≫ ''m''<sub>2</sub>}}—which is true for the Sun and any planet—then the ratio {{sfrac|''r''<sub>1</sub>|''R''<sub>1</sub>}} approximates to:

:<math>\frac{a}{R_1} \cdot \frac{m_2}{m_1}.</math>

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

:<math>{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}</math>

—that is, where the planet is massive ''and'' far from the Sun.

If Jupiter had [[Mercury (planet)|Mercury]]'s orbit ({{convert|57900000|km|AU|abbr=on|disp=comma}}), the Sun–Jupiter barycenter would be approximately 55,000&nbsp;km from the center of the Sun ({{nowrap|{{sfrac|''r''<sub>1</sub>|''R''<sub>1</sub>}} ≈ 0.08}}). But even if the Earth had [[Eris (dwarf planet)|Eris]]'s orbit ({{convert|68|AU|km|order=flip|abbr=on|disp=comma}}), the Sun–Earth barycenter would still be within the Sun (just over 30,000&nbsp;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&nbsp;km, above the Sun's surface.<ref name=Meeus1997/>

The calculations above are based on the mean distance between the bodies and yield the mean value ''r''<sub>1</sub>. But all celestial orbits are elliptical, and the distance between the bodies varies between the [[apsis|apses]], depending on the [[eccentricity (orbit)|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:

:<math>\frac{1}{1-e} > \frac{r_1}{R_1} > \frac{1}{1+e}</math>

The Sun–Jupiter system, with ''e''<sub>Jupiter</sub>&nbsp;=&nbsp;0.0484, just fails to qualify: {{nowrap|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. [[Victor Brumberg|Brumberg]] explains how to set up barycentric coordinates in general relativity.<ref name="Brumberg" />

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 potential]]s 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:<ref name="barycenter" />

{| class="wikitable" style="margin-left: 12px; text-align: right;"
|-
! [[Small Solar System body|Object]]
! [[Semi-major axis]]<br />(in [[Astronomical unit|AU]])
! [[Apoapsis]]<br />(in AU)
! [[Orbital period]]<br />(in years)
|-
| align=left | [[C/2006 P1|C/2006 P1 (McNaught)]]
| 2,050
| 4,100
| 92,600
|-
| align=left | [[Comet Hyakutake|C/1996 B2 (Hyakutake)]]
| 1,700
| 3,410
| 70,000
|-
| align=left | [[C/2006 M4 (SWAN)]]
| 1,300
| 2,600
| 47,000
|-
| align=left |{{mpl|(308933) 2006 SQ|372}}
| 799
| 1,570
| 22,600
|-
| align=left | {{mpl|(87269) 2000 OO|67}}
| 549
| 1,078
| 12,800
|-
| align=left | [[90377 Sedna]]
| 506
| 937
| 11,400
|-
| align=left | {{mpl|2007 TG|422}}
| 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.<ref name="Kaib2009" />

== See also ==
{{div col|colwidth=22em}}
* [[Barycentric Dynamical Time]]
* [[Centers of gravity in non-uniform fields]]
* [[Center of mass]]
* [[Lagrange point]]
* [[Mass point geometry]]
* [[Roll center]]
* [[Weight distribution]]
{{div col end}}

== References ==
{{Reflist
|30em
|refs=

<ref name="barycenter">{{cite web
 |date=2011-01-30 
 |author=[[JPL Horizons On-Line Ephemeris System|Horizons]] output
 |url=http://home.surewest.net/kheider/astro/2007TG422Barycenter.txt
 |title=Barycentric Osculating Orbital Elements for 2007 TG422
 |access-date=2011-01-31 
 |url-status=dead
 |archive-url=https://web.archive.org/web/20140328023855/http://home.surewest.net/kheider/astro/2007TG422Barycenter.txt
 |archive-date=2014-03-28
 }} (Select Ephemeris Type:Elements and Center:@0)</ref>

<ref name="Brumberg">{{cite book
 |title=Essential Relativistic Celestial Mechanics
 |author-link=Victor A. Brumberg
 |last=Brumberg
 |first=Victor A.
 |publisher=Adam Hilger
 |location=London
 |date=1991
 |isbn=0-7503-0062-0}}</ref>

<ref name="Kaib2009">{{cite journal
 |last1=Kaib |first1=Nathan A.
 |last2=Becker |first2=Andrew C.
 |last3=Jones |first3=R. Lynne
 |last4=Puckett |first4=Andrew W.
 |last5=Bizyaev |first5=Dmitry
 |last6=Dilday |first6=Benjamin
 |last7=Frieman |first7=Joshua A.
 |last8=Oravetz |first8=Daniel J.
 |last9=Pan |first9=Kaike
 |last10=Quinn |first10=Thomas
 |last11=Schneider |first11=Donald P.
 |last12=Watters |first12=Shannon
 |title=2006 SQ<sub>372</sub>: A Likely Long-Period Comet from the Inner Oort Cloud
 |journal=[[The Astrophysical Journal]]
 |volume=695
 |issue=1
 |pages=268–275
 |date=2009
 |doi=10.1088/0004-637X/695/1/268
 |arxiv=0901.1690
 |bibcode=2009ApJ...695..268K|s2cid=16987581
 }}</ref>

<ref name="MacDougal">{{cite book
 |title=Newton's Gravity: An Introductory Guide to the Mechanics of the Universe
 |url=https://archive.org/details/newtonsgravityin00macd
 |url-access=limited
 |last=MacDougal
 |first=Douglas W.
 |page=[https://archive.org/details/newtonsgravityin00macd/page/n214 199]
 |publisher=[[Springer Science & Business Media]]
 |location=Berlin
 |date=December 2012
 |isbn=978-1-4614-5444-1}}</ref>

<ref name=Meeus1997>{{citation
 |title=Mathematical Astronomy Morsels
 |last=Meeus
 |first=Jean
 |pages=165–168
 |publisher=Willmann-Bell
 |location=Richmond, Virginia
 |year=1997
 |isbn=0-943396-51-4}}</ref>

<ref name="NASA2005">{{cite web
 |date=2005-09-08
 |title=What's a Barycenter?
 |publisher=Space Place @ NASA
 |url=http://spaceplace.nasa.gov/en/kids/barycntr.shtml
 |access-date=2011-01-20
 |archive-url=https://web.archive.org/web/20101223075230/http://spaceplace.nasa.gov/en/kids/barycntr.shtml
 |archive-date=23 December 2010 
 |url-status=live}}</ref>

<ref name="Olkin-2015">{{cite journal
 |title  =Evidence that Pluto's atmosphere does not collapse from occultations including the 2013 May 04 event
 |author1=Olkin, C. B.
 |author2=Young, L. A.
 |author3=Borncamp, D.
 |date   =January 2015
 |journal=Icarus
 |volume =246
 |pages  =220–225
 |bibcode=2015Icar..246..220O
 |doi    =10.1016/j.icarus.2014.03.026
 |url    =https://www.researchgate.net/publication/262937542
 |display-authors=etal|doi-access=free
 |hdl=10261/167246
 |hdl-access=free
 }}</ref>

}} <!-- end of reflist -->
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[[Category:Astronomical coordinate systems]]
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[[Category:Mass]]
[[Category:Geometric centers]]