moment of inertia factor

For a planetary body with principal moments of inertia

:$\backslash frac\{C\}\{MR^2\}$,

where C is the first principal moment of inertia of the body, M is the mass of the body, and R is the mean radius of the body. For a sphere with uniform density, $C/MR^2=2/5$.{{refn |For a sphere with uniform density we can calculate the moment of inertia and the mass by integrating over disks from the "south pole" to the "north pole". Using a density of 1, a disk of radius {{mvar|r}} has a moment of inertia of

:$\backslash int\_0^r2\backslash pi\; r^3\backslash \; dr=\backslash frac\{\backslash pi\; r^4\}2$

whereas the mass is

:$\backslash int\_0^r2\backslash pi\; r\backslash \; dr=\backslash pi\; r^2$

Letting $r=R\backslash cos\backslash theta$ and integrating over $R\backslash sin\backslash theta$ we get:

:$\backslash begin\{align\}$

\frac C{R^5}&=\frac\pi 2\int_{-1}^1\cos^4\theta\ d\sin\theta\\

& = \frac\pi 2\int_{-1}^1(1-\sin^2\theta)^2\ d\sin\theta\\

& = \frac\pi 2\int_{-1}^1(1-2\sin^2\theta+\sin^4\theta)d\sin\theta\\

& = \frac\pi 2\int_{-1}^1(d\sin\theta-\frac 23d\sin^3\theta+\frac 15d\sin^5\theta)\\

& = \frac 8{15}\pi

\end{align}

:$\backslash begin\{align\}$

\frac M{R^3}&=\pi\int_{-1}^1\cos^2\theta\ d\sin\theta\\

& = \pi\int_{-1}^1(1-\sin^2\theta)d\sin\theta\\

& = \pi\int_{-1}^1(d\sin\theta-\frac 13d\sin^3\theta)\\

& = \frac 43\pi

\end{align}

This gives $C/MR^2=2/5$.|group = note}}{{refn |For several other examples (in which the rotation axis is the axis of symmetry if not otherwise specified), a solid cone has a factor of 3/10; a uniform thin rod (rotating about its center perpendicularly to its axis, so R is length/2) has a factor of 1/3; a hollow cone or a solid cylinder has a factor of 1/2; a hollow sphere has factor of 2/3; a hollow open-ended cylinder has a factor of 1.| group = note}} For a differentiated planet or satellite, where there is an increase of density with depth, . The quantity is a useful indicator of the presence and extent of a planetary core, because a greater departure from the uniform-density value of 2/5 conveys a greater degree of concentration of dense materials towards the center.

The Sun has by far the lowest moment of inertia factor value among Solar System bodies; it has by far the highest central density ({{val|162|u=g/cm3}},{{refn |A star's central density tends to increase over the course of its lifetime, aside from during brief core nuclear fusion ignition events like the helium flash.| group = note}} compared to ~13 for Earth{{cite web |last1=Robertson |first1=Eugene C. |date=26 July 2001 |url=http://pubs.usgs.gov/gip/interior/ |title=The Interior of the Earth |publisher=USGS |accessdate=24 March 2007}}{{cite book|author1=Hazlett, James S. |author2=Monroe, Reed |author3=Wicander, Richard |title=Physical geology : exploring the earth|year= 2006|publisher= Thomson|location= Belmont|isbn= 9780495011484|page= 346|edition= 6.}}) and a relatively low average density (1.41 g/cm³ versus 5.5 for Earth). Saturn has the lowest value among the gas giants in part because it has the lowest bulk density ({{val|0.687|u=g/cm3}}).{{cite web |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html |title=Saturn Fact Sheet |publisher=NASA |last=Williams |first=David R. |accessdate=31 July 2007 |date=7 September 2006 |archiveurl=https://web.archive.org/web/20140414074059/http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html |archivedate=14 April 2014 |url-status=live}} Ganymede has the lowest moment of inertia factor among solid bodies in the Solar System because of its fully differentiated interior,{{cite journal |last1=Showman |first1=Adam P. |last2=Malhotra |first2=Renu |title=The Galilean Satellites |date=1999-10-01 |journal=Science |volume=286 |pages=77–84 |doi=10.1126/science.286.5437.77 |url=http://www.lpl.arizona.edu/~showman/publications/showman-malhotra-1999.pdf |pmid=10506564 |issue=5437 }}{{cite journal |last1=Sohl |first1=F. |last2=Spohn |first2=T |last3=Breuer |first3=D. |last4=Nagel |first4=K. |title=Implications from Galileo Observations on the Interior Structure and Chemistry of the Galilean Satellites |journal=Icarus |date=2002 |volume=157 |issue=1 |pages=104–119 |doi=10.1006/icar.2002.6828 |bibcode=2002Icar..157..104S }} a result in part of tidal heating due to the Laplace resonance,{{cite journal |last1=Showman |first1=Adam P. |last2=Stevenson |first2=David J. |last3=Malhotra |first3=Renu |title=Coupled Orbital and Thermal Evolution of Ganymede |journal=Icarus |date=1997 |volume=129 |issue=2 |pages=367–383 |doi=10.1006/icar.1997.5778 |url=http://www.lpl.arizona.edu/~showman/publications/showman-etal-1997.pdf |bibcode=1997Icar..129..367S }} as well as its substantial component of low density water ice. Callisto is similar in size and bulk composition to Ganymede, but is not part of the orbital resonance and is less differentiated. The Moon is thought to have a small core, but its interior is otherwise relatively homogenous.{{cite web |last1= Brown| first1= D.| last2= Anderson|first2= J.|website= NASA|url= http://www.nasa.gov/topics/moonmars/features/lunar_core.html |title= NASA Research Team Reveals Moon Has Earth-Like Core |publisher=NASA |date= 6 January 2011}}{{cite journal|last1= Weber|first1= R. C.|last2= Lin|first2= P.-Y.|last3= Garnero|first3= E. J.|last4= Williams|first4= Q.|last5= Lognonne|first5= P.|title= Seismic Detection of the Lunar Core|journal= Science|volume= 331|issue= 6015|date= 2011-01-21|pages= 309–312|url= http://www.earth.northwestern.edu/people/seth/351/lunarcore.2011.pdf|doi= 10.1126/science.1199375|pmid= 21212323|bibcode= 2011Sci...331..309W|s2cid= 206530647|access-date= 2017-04-10|archive-url= https://web.archive.org/web/20151015035756/http://www.earth.northwestern.edu/people/seth/351/lunarcore.2011.pdf|archive-date= 2015-10-15|url-status= dead}}

The polar moment of inertia is traditionally determined by combining measurements of spin quantities (spin precession rate and/or obliquity) with gravity quantities (coefficients of a spherical harmonic representation of the gravity field). These geodetic data usually require an orbiting spacecraft to collect.

For bodies in hydrostatic equilibrium, the Darwin–Radau relation can provide estimates of the moment of inertia factor on the basis of shape, spin, and gravity quantities.

The moment of inertia factor provides an important constraint for models representing the interior structure of a planet or satellite. At a minimum, acceptable models of the density profile must match the volumetric mass density and moment of inertia factor of the body.

File:Sun poster.svg|The Sun (C/MR² = 0.070)

File:Saturn diagram.svg|Saturn (C/MR² = 0.22)

File:Ganymede diagram.svg|Ganymede (C/MR² = 0.3115)

File:Earth Internal Structure.svg|Earth (C/MR² = 0.3307)

File:Callisto diagram.svg|Callisto (C/MR² = 0.3549)

File:Moon diagram.svg|The Moon (C/MR² = 0.3929)

{{reflist

| group = note

}}

{{reflist

|refs=

{{cite journal |last1= Anderson|first1=J. D.|last2= Schubert|first2= G.|title= Saturn's satellite Rhea is a homogeneous mix of rock and ice|journal= Geophysical Research Letters |volume= 34|issue= 2|pages=L02202|year= 2007|doi= 10.1029/2006GL028100|bibcode= 2007GeoRL..34.2202A|s2cid=128410558 |doi-access= }}

{{cite journal|last1= Durante|first1= D.|last2= Hemingway|first2= D.J.|last3= Racioppa|first3= P.|last4= Iess|first4= L.|last5= Stevenson|first5= D.J.|title= Titan's gravity field and interior structure after Cassini|journal= Icarus|volume= 326|pages= 123–132|year= 2019|doi= 10.1016/j.icarus.2019.03.003|bibcode= 2019Icar..326..123D|url= https://authors.library.caltech.edu/93651/1/1-s2.0-S0019103518307267-main.pdf|hdl= 11573/1281269|s2cid= 127984873|hdl-access= free}}

{{cite book|last1= Fortney|first1= J.J.|last2= Helled|first2= R.|last3= Nettlemann|first3= N.|last4= Stevenson|first4= D.J.|last5= Marley|first5= M.S.|last6= Hubbard|first6= W.B.|last7= Iess|first7= L.|editor1= Baines, K.H.|editor2= Flasar, F.M.|editor3= Krupp, N.|editor4= Stallard, T.|title= Saturn in the 21st Century|chapter= The Interior of Saturn|chapter-url= https://books.google.com/books?id=p358DwAAQBAJ&pg=PA51 |pages= 44–68|date=6 December 2018|publisher= Cambridge University Press|isbn= 978-1-108-68393-7}}

{{cite book|last=Hubbard|first=William B.|title=Planetary interiors|date=1984|publisher=Van Nostrand Reinhold|url=https://archive.org/details/planetaryinterio0000hubb|url-access=registration|location=New York, N.Y.|isbn= 978-0442237042|oclc= 10147326}}

{{cite book|last2=Lissauer|first2=Jack J.|last1=de Pater|first1=Imke|title=Planetary sciences|date=2015|url= https://books.google.com/books?id=stFpBgAAQBAJ&pg=PA250|publisher=Cambridge University Press|location=New York|isbn= 978-0521853712|edition=2nd updated|oclc= 903194732}}

{{cite journal|last1=Margot|first1=Jean-Luc|last2= Peale|first2= Stanton J.|last3=Solomon|first3=Sean C.|last4= Hauck|first4= Steven A.|last5=Ghigo|first5=Frank D.|last6=Jurgens|first6=Raymond F.|last7= Yseboodt|first7= Marie|last8= Giorgini|first8=Jon D.|last9= Padovan|first9= Sebastiano|last10=Campbell|first10=Donald B.|title=Mercury's moment of inertia from spin and gravity data|journal=Journal of Geophysical Research: Planets|volume= 117|issue=E12|date=2012|pages=E00L09–|issn=0148-0227|doi=10.1029/2012JE004161|bibcode = 2012JGRE..117.0L09M |doi-access=free}}

{{cite journal|last1= McKinnon|first1=W. B.|title= Effect of Enceladus's rapid synchronous spin on interpretation of Cassini gravity|journal= Geophysical Research Letters|volume= 42|issue= 7|year= 2015|pages= 2137–2143|doi= 10.1002/2015GL063384|bibcode=2015GeoRL..42.2137M|doi-access= free}}

{{cite journal|last1= Mao|first1= X.|last2= McKinnon|first2=W. B.|title= Faster paleospin and deep-seated uncompensated mass as possible explanations for Ceres' present-day shape and gravity|journal= Icarus|volume= 299|year= 2018|pages= 430–442|doi= 10.1016/j.icarus.2017.08.033|bibcode= 2018Icar..299..430M}}

{{cite book | last1= Murray | first1= Carl D. | last2= Dermott | first2= Stanley F. |title= Solar System Dynamics | url= https://books.google.com/books?id=I_8LBAAAQBAJ | date=13 February 2000 | publisher= Cambridge University Press | location= Cambridge | isbn= 978-1139936156 | oclc= 40857034}}

{{cite journal|last1= Ni|first1= D.|title= Empirical models of Jupiter's interior from Juno data|journal= Astronomy & Astrophysics|volume= 613|year= 2018|pages= A32|doi= 10.1051/0004-6361/201732183|doi-access= free}}

{{cite journal|last1=Park|first1=R. S.|last2=Konopliv|first2=A. S.|last3=Bills|first3=B. G.|last4=Rambaux|first4= N.|last5=Castillo-Rogez|first5=J. C.|last6=Raymond|first6=C. A.|last7=Vaughan|first7=A. T.|last8=Ermakov|first8=A. I.|last9=Zuber|first9=M. T.|last10= Fu|first10=R. R.|last11=Toplis|first11=M. J.|last12=Russell|first12=C. T.|last13=Nathues|first13= A.|last14=Preusker|first14=F.|title=A partially differentiated interior for (1) Ceres deduced from its gravity field and shape|journal= Nature|date=2016-08-03|doi= 10.1038/nature18955|volume=537|issue=7621|pages=515–517|pmid=27487219|bibcode = 2016Natur.537..515P |s2cid=4459985|url=https://resolver.sub.uni-goettingen.de/purl?gro-2/86708 }}

{{cite book|last1=Schubert|first1=G.|last2=Anderson|first2=J. D.|last3= Spohn|first3= T.|last4= McKinnon|first4=W. B.|editor1-last= Bagenal| editor1-first= F.| editor2-last= Dowling| editor2-first= T. E.| editor3-last= McKinnon| editor3-first= W. B.|title=Jupiter : the planet, satellites, and magnetosphere|chapter= Interior composition, structure and dynamics of the Galilean satellites|date= 2004|publisher= Cambridge University Press|location=New York|url= https://books.google.com/books?id=aMERHqj9ivcC|chapter-url = https://books.google.com/books?id=aMERHqj9ivcC&pg=PA281|isbn= 978-0521035453|pages= 281–306|oclc= 54081598}}

{{cite web | url = http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html | title = Sun Fact Sheet | last = Williams | first = D. R. | date = | website = Planetary Fact Sheets | publisher = NASA | access-date = 2017-01-26}}

{{cite journal|last1=Williams|first1=James G.|title= Contributions to the Earth's obliquity rate, precession, and nutation|journal= The Astronomical Journal|volume= 108|date= 1994|pages= 711|issn= 0004-6256|doi= 10.1086/117108|bibcode=1994AJ....108..711W|s2cid=122370108 |doi-access= free}}

{{cite journal|last1=Williams|first1=James G.|last2=Newhall|first2=XX|last3=Dickey|first3=Jean O.|title=Lunar moments, tides, orientation, and coordinate frames|journal=Planetary and Space Science|volume= 44|issue=10|date= 1996|pages= 1077–1080|issn= 0032-0633|doi= 10.1016/0032-0633(95)00154-9|bibcode = 1996P&SS...44.1077W }}

{{cite book|last1=Yoder|first1=C.|editor1-last=Ahrens|editor1-first=T.|title=Astrometric and Geodetic Properties of Earth and the Solar System|date=1995|publisher=AGU|location=Washington, DC|isbn=978-0-87590-851-9|url=http://trs-new.jpl.nasa.gov/dspace/handle/2014/32032|oclc=703657999|access-date=2016-08-19|archive-url=https://web.archive.org/web/20160304063858/http://trs-new.jpl.nasa.gov/dspace/handle/2014/32032|archive-date=2016-03-04|url-status=dead}}

{{cite journal |last1=Konopliv |first1=Alex S. |last2=Asmar |first2=Sami W. |last3=Folkner |first3=William M. |last4=Karatekin |first4=Özgür |last5=Nunes |first5=Daniel C. |last6=Smrekar |first6=Suzanne E. |last7=Yoder |first7=Charles F. |last8=Zuber |first8=Maria T. |title=Mars high resolution gravity fields from MRO, Mars seasonal gravity, and other dynamical parameters |journal=Icarus |date=January 2011 |volume=211 |issue=1 |pages=401–428 |doi=10.1016/j.icarus.2010.10.004|bibcode=2011Icar..211..401K }}

}}

Category:Astrophysics

Category:Planetary science

Category:Moment (physics)