The Internal Rotation of the Sun

51 Pages·1.049 MB·English

by  Thompson M.J. et al.

1Aug2003 14:6 AR AR194-AA41-15.tex AR194-AA41-15.sgm LaTeX2e(2002/01/18) P1:GCE
10.1146/annurev.astro.41.011802.094848
Annu.Rev.Astron.Astrophys.2003.41:599–643
doi:10.1146/annurev.astro.41.011802.094848
Copyright(cid:176)c 2003byAnnualReviews.Allrightsreserved
T I R S
HE NTERNAL OTATION OF THE UN
Michael J. Thompson
Space&AtmosphericPhysicsGroup,ImperialCollegeLondon,SW72AZ,
UnitedKingdom;email:[email protected]
g
or
s. Jørgen Christensen-Dalsgaard
w
e
vi TeoretiskAstrofysikCenterandInstitutforFysikogAstronomi,AarhusUniversitet,
ualrenly. DK-8000A˚rhusC,Denmark;email:[email protected]
no
s.anuse Mark S. Miesch
urnalonal HighAltitudeObservatory,NationalCenterforAtmosphericResearch,Boulder,
arjopers Colorado80301;email:[email protected]
om For Juri Toomre
ded fr4/09. JILAandDepartmentofAstrophysical&PlanetarySciences,UniversityofColorado,
a2 Boulder,Colorado80309-0440; email:[email protected]
nlo10/
43. Dowarch on KeyWords solarinteriorrotation,solarinteriorconvection,starrotation,
99-6Rese helioseismology,turbulentconvection
strophys. 2003.41:5nstitute for Nuclear EtoTwnufahhrrseeiAltciehrehbolarltssahsatrtabtr-uideeaodvencibnoetewlsbueriuntteHiihovlteitnhenluaaimotplthesotodeehdfitcesehtomhlersneoo,Svtitleuanhoctntegite’oiysrronanshndaurailzarsafotrttaeonircvtaewaeentirosiisoonfuoatntlaelrdstomrioifodoenoevrdp,ueiobresrynuunfdtrodenkuaipfornnfreweodismrwtteaosalndetrtraedioflrtgy.rtaaoeUitlomoeenndflrpitotokhrhubieeeogstrShehdreuleiyvxsnaptp’uatsreinneorcicdontfeiaotacatrlftitmiorpiooonilncmnyss-.,.
Astro. Ay Joint I tchoenvroectatitoionnzaoxnies.aLnadyienrsthoef sroutbaptihoontaolspshheerairchlaavyeerbs.eeSntuddiisecsovoefrtehdeattimtheevbaarsiaetioofnthoef
v. b rotationhaveuncoveredzonal-flowbands,extendingthroughasubstantialfractionof
e
R theconvectionzone,whichmigrateoverthecourseofthesolarcycle,andthereare
nu. hintsofothertemporalvariationsandofajet-likestructure.Atthesametime,build-
n
A ingonearlierworkwithmean-fieldmodels,researchershavemadegreatprogressin
supercomputersimulationsoftheintricateinterplaybetweenturbulentconvectionand
rotationintheSun’sinterior.Suchstudiesarebeginningtotransformourunderstanding
ofhowrotationorganizesitselfinastellarinterior.
1.INTRODUCTION
Rotation is ubiquitous in astrophysical systems. During a system’s formation as
aresultofgravitationalcontraction,itsmomentofinertiaisreduced,sometimes
by many orders of magnitude; typically, therefore, any initial rotation is greatly
0066-4146/03/0922-0599$14.00 599
1Aug2003 14:6 AR AR194-AA41-15.tex AR194-AA41-15.sgm LaTeX2e(2002/01/18) P1:GCE
600 THOMPSONETAL.
enhanced.Thisprincipleappearstoapplyoveravastrangeofscales,fromgalaxies
toneutronstars.Oftentheevolutionofrotationseemscloselylinkedtotheevolution
of the magnetic field, which, being frozen to the partly conducting plasma, is
alsoamplifiedbycontraction.Furthermore,acombinationofrotationalshearand
turbulencemayleadtodynamoeffects,instarsaswellasgalaxies,whichamplify
thefield.Anunderstandingoftheseprocessesisessentialforadescriptionofmost
astrophysicalphenomena.Yet,thedetailsoftheevolutionofrotationandmagnetic
fieldsarepoorlyunderstoodfromatheoreticalpointofview,leadingtoobvious
difficulties in the modeling of the systems, and observations rarely provide the
g
or necessarydetailedinsight.
s.
w
e
vi
ualrenly. 1.1.RotationandDynamicsofStellarInteriors
no
s.anuse Rotationplaysanimportantroleintheunderstandingofstellarformationandevo-
urnalonal lEuetnioenn(sf2o0r0a3re).cSentatrcsoamrepbiloartinofnroonmthceongternaecrtiaolntoopficinotfesrstetelllalarrrogtaasticolno,usdese,Mimapeadretirn&g
arjopers rapidrotationtotheprotostar.Theinfallingmaterialtendstobeorganizedinan
om For accretiondisk,which,probablythroughtheactionofmagneticfields,oftenleads
ded fr4/09. totheformationofcollimatedjets.Fromsuchdisks,evidenceforwhichhasbeen
a2 foundinmanyyoungstars,planetarysystemsmaysubsequentlyform.Therota-
43. Downloarch on 10/ tmcilouangstnreaerttsei,cwoofhrtoihgseiennm,ebewemtlwybeeferosnrmctahenedbpsertoaatrsosisustmaerseatdanbtdolitshhhaeevddeistthhker.oOsuabgmsheetrahvgeaetii,nostnhesoraowcfttiyhooanut,ntlhgiekserteleysluloalrft
99-6Rese isawiderangeofrotationrates.Thesubsequentevolutionoftherotationofastar
003.41:5Nuclear isstedlelaterrwmiinndedanbdytthheeriendteisrtprliabyutbioetnwoefeanntghuelparomssoibmleelnotsusmofinanthgeulsaterlmlaorminetenrtiuomr.iFnoar
phys. 2ute for sthtarrosuogfhraelmataivgenleytilcoawllmydaossm,iinncaltueddisntgeltlhaerwSuinnd,alningkueldart-omthoemoeuntteurmcolonsvsetcatkioenszpolancee.
stronstit Thestrengthofthewindisrelatedtothemagneticactivityofthestar,whichinturn
v. Astro. Aby Joint I icissognaevcnelecortsiaeollncyozauospnsleui,nmpgeredbsetutowmbeaeebnclyarudoseteapdteionbndyiadnnygdnoatnmhtehoaepnrrgooutcaletasiros-enmsroawmteitehoniftnuthomersjluotassrts..bTTehnhueissa,tlthehatehdrees
e totheSkumanichlaw,accordingtowhichthesurfaceangularvelocitydecreases
R
u. graduallywithagetast¡1=2 (Skumanich1972;forarecentupdateontheobser-
n
n
A vationalevidence,seeBarnes2001).Theredistributionofangularmomentumby
convectionisarapidprocess;inthesolarcase,atleast,thishasledtodifferential
rotation, with the equator spinning more rapidly than the poles. In contrast, the
lossofangularmomentumthroughastellarwindtospindownthestarproceeds
typicallymuchmoreslowly.Anunderstandingofangular-momentumtransportin
radiativeregions,andhencepredictionsoftheresultingrotationprofilethereand
itscouplingtoconvectiveregions,issofarratheruncertain.
Rotation has a dynamical effect on stellar structure and evolution through
the modification by the centrifugal acceleration of hydrostatic balance. In most
cases,however,thiseffectismodest.Muchmoreimportanteffectsarisefromthe
1Aug2003 14:6 AR AR194-AA41-15.tex AR194-AA41-15.sgm LaTeX2e(2002/01/18) P1:GCE
THEINTERNALROTATIONOFTHESUN 601
redistributionofchemicalelementsthatmaybecausedbycirculationandinsta-
bilities induced by rotation. In particular, the nonuniform heating along equipo-
tentials, induced by rotation, drives the large-scale Eddington-Sweet circulation
(e.g.,Sweet1950).Indeed,withoutsucheffects,thesurfacecompositionofmany
stars, possessing thin outer convection zones, would be drastically modified by
settlingandselectiveradiativelevitation.Modificationsofthisnatureareseenin
afewso-called‘peculiar’AandBstars,whichindicatesthatlittlemixingistak-
ing place. However, in the ‘normal’ majority of stars, mixing—likely caused by
rotation—must overwhelm the effects of settling and levitation. Rotationally in-
g
or ducedmixingprocessesinstellarinteriorsalsoappeartoaccountforseveralcases
s.
w ofsurface-abundancechangesduringstellarevolution,asobserved,forexample,
e
vi inmassivestarsorinredgiants(forareview,seeMaeder&Meynet2000).Mixing
e
ualrnly. mayalsoaffectastar’sevolutionbychangingitscompositionstructure.Thus,an
no
s.anuse understanding of stellar rotation is crucial for the modeling of stellar structure
urnalonal amnadyebveoliumtipoonr.taInntdeinedt,heasSduins.cuAssgeedneinraSlerecvtiioenw7o.3f,mroixtaintigoninalslytarinsdwuacsedgimveinxibnyg
arjopers Pinsonneault(1997).
om For TheSunexhibitsmagneticactivitythatvariescyclically,withaperiodofap-
ded fr4/09. proximately22years.(Thenumberofsunspotschangeswithaperiodof11years,
a2 whereasthemagneticpolarityhasaperiodof22years.)Thisismostevidentin
43. Downloarch on 10/ toohffetthhpeerescsooelrnaocrneaato.mnSoitmshpeihlsaeorrlema,rsasugucnrhefataiccsetaohcfetisdvuiinstytsrpihboautsstibboeunetnoisfinarecfefltirevrceetderedigniinoontmhseaarnnysdtaotrhtsheethsrtrarousupcgteuhcrteas
99-6Rese varietyofobservations;inmanycases,cyclicvariationssimilartothatobservedin
003.41:5Nuclear tShcehSriujvnearre&seZewn.aa(Fno2r0a0g0e.n)eIrtailsogveenrveireawllyofasssoulamreadndthsatetlulanrdmeralyginnegticthaectmivaigtyn,esteice
phys. 2ute for atacttiiovnityanidsscoomnveefcotirvmeomfodtiyonna.mSoupapcotirotnf,oirntvhoilsvcinogmienstefrraocmtioonbsbeertvweedecnosrtreelllaatriornos-
stronstit betweenvariousmeasuresofactivityandpropertiesofstellarrotationandconvec-
v. Astro. Aby Joint I ttmihoeanrtieo(feno.rgae.b,,otHhuaattrtthamenaiunnntnedre&irosrNtaroonytdaeitnsiog1no9af8l7spt,reoSllpcaehrrrmtiijevasgeornfe&tsitcaZrawsc.atDiavnyitny2a0ims00do)ea.pcIettniowdneoniustldsoinmfoiinlllafoorwlry-,
e believed to be responsible for magnetic fields of various objects ranging from
R
u. galaxies(e.g.,Becketal.1996)toaccretiondisks(e.g.,Vishniac&Brandenburg
n
n
A 1997)andplanets(Glatzmaier&Roberts1997,Proctor&Gilbert1994).
Rotationplaysimportantrolesinotherareasofsolarandstellarastrophysics.
Arapidlyrotatingsolarcore,notobviouslyinconsistentwiththegeneralunder-
standingoftheevolutionofsolarrotation,wasinvokedinearlyattemptstoexplain
the observed solar neutrino deficit (e.g., Bartenwerfer 1973, Demarque, Mengel
&Sweigart1973).Furthermore,rapidinteriorrotationcoulddistorttheSun’sex-
ternalgravitationalfieldandhenceaffecttestsofgravitationaltheoriesbasedon
observations of planetary motion (e.g., Dicke 1964); the resulting oblateness of
theSun’ssurfacemightbedifficulttodiscernamongotherlatitudinalvariations
intheSun’ssurfaceproperties.
1Aug2003 14:6 AR AR194-AA41-15.tex AR194-AA41-15.sgm LaTeX2e(2002/01/18) P1:GCE
602 THOMPSONETAL.
Observationaldataonstellarrotationwere,untilfairlyrecently,limitedtode-
terminationsofthesolar-surfacerotationasafunctionoflatitudeandtomeasure-
mentsofgeneralrotationrates,orprojectedrotationalvelocities,ofdistantstars.
However,overthepasttwodecades,ourknowledgeofthesolarinteriorhasbeen
revolutionizedthroughhelioseismology,i.e.,theanalysisofobservedoscillations
ofthesolarsurface.Inparticular,rotationcausesasplittingofthefrequenciesof
oscillations,fromwhichpropertiesoftheinternalrotationcanbeinferred.Here,
wereviewinformationontheSun’sinternalrotationandotheraspectsofitsinter-
nal dynamics obtained from helioseismology and other sources, and we discuss
g
or thecurrentunderstandingoftheseresultsfromtheoreticalconsiderationsandnu-
s.
w mericalmodeling.Previousreviewsofsolarrotationweregiveninthesevolumes
e
vi byGilman(1974)andHoward(1984),thelatterimmediatelybeforethefirstex-
e
ualrnly. tensivehelioseismicresultsontherotationofthesolarinteriorwerepresentedby
no
s.anuse Duvall et al. (1984). Among the many general reviews on helioseismology, we
urnalonal mwhenerteioans TthhoosmepbsyonGeotuaglh. (&19T9o6o)manred(S1c9h9o1u)eatnadl.C(1h9ri9s8te)ndsiesnc-uDssaelsdghaaerlidos(e2i0s0m2i)c,
arjopers inferenceofsolarrotation.Beck(2000)providedadetailedreviewofsurfaceand
om For helioseismicobservationsofsolarrotation.Miesch(2000)reviewedthenumerical
ded fr4/09. modelingofsolarconvectionandrotation.
a2
nlo10/
43. Dowarch on 1.2.StellarRotation
99-6Rese Swehveenraslutfeficchineinqtulyesfaasrte,acvaauislaebslaetmoeinavsuersatibglaetebrthoeadroentaitnigonooffthdeistsapnetcsttraarlsl.iRneost.atTiohnis,
phys. 2003.41:5ute for Nuclear iplm.irenio.eg,vhovitdfrboestesisgpianhomtsi.,seWiwabslihetuhertreoaecosiofuriftsrfiheectchitleeifnnoiternn-tcohulfiem-nsiabnitgecihorlitnoncfaaotsnimtoganplresoannibngeenltaewtnioenoeftanhthsetterhawetrioissrttoeaicttuiaaotnlinsiofaenolnrvsameex,lisoasascmasitnuypdmlveit,rnhoigtet,
stronstit random orientation of the rotation axes. In practice, modern observations allow
Astro. Ay Joint I meAasudrieremcetndteotefrvmrointsaitnioinaosflothwearost5atkiomnsp¡e1r.iodispossiblewhenthestardisplays
v. b aninhomogeneoussurface,resultinginobservableperiodicvariationsasthestar
e
R rotates.Thisistypicallythecaseforlowermain-sequencestarsthatshowsolar-
u.
n likeactivity.Ifsufficientlylargestarspotsarepresent,broadbandphotometrycan
n
A
show the rotational modulation as demonstrated, for example, by observations
of stars in the Pleiades cluster (Krishnamurthi et al. 1998). Alternatively, other
activity measures may be observed. In an extensive program to study activity
cyclesinotherstars(e.g.,Wilson1978),fluxesinthecalciumHandKlineshave
beenobserved;thesefluxesaremeasuresofchromosphericactivitythatalsoshow
rotationalmodulation(e.g.,Donahue,Saar&Baliunas1996).
Finally, observations of rotational splittings of stellar oscillation frequencies
would evidently provide measures of stellar rotation. So far, this technique has
only been successfully applied in the case of white dwarfs (e.g., Winget et al.
1994). However, it holds great promise for a broad range of stars, including
1Aug2003 14:6 AR AR194-AA41-15.tex AR194-AA41-15.sgm LaTeX2e(2002/01/18) P1:GCE
THEINTERNALROTATIONOFTHESUN 603
Figure1 Observedrotationvelocities,intermsofv sini,fortwostellarclusters
g rot
or of different age: (a) Pleiades (100 Myrs old) and (b) Hyades (600 Myrs old). The
s.
w
e uppersolidlineindicatestheupperenvelopeofrotationinthePleiades,whereasthe
vi
e lowersolidlineshowstheaveragevelocityintheHyades.Theabscissaisthecolor
ualrnly. index,correctedforinterstellarabsorption,whichprovidesameasureoftheeffective
no
s.anuse temperature,increasingtowardtheleft.[FromSoderblom,Jones&Fischer(2001).]
urnalonal
arjopers thosewithsolar-likeoscillations,withtheadventofplannedasteroseismicspace
m or projects.
oF
aded fr24/09. rotaTthioenstteelnldarsdtoatsalogwenderoawllnywinidthicaintecrtehaasti,nagtlaegaes,tifnoraclocworedramncaeinw-siethquthenecgeesntearrasl,
nlo10/ picture of angular-momentum loss through a magnetically coupled stellar wind.
43. Dowarch on Tstheellaervcolluusttieornsooffdroiftfaetrieonntwagitehs,satesldlaermaognestcraantedbeininFfiegrurered1f.rom observations of
99-6Rese InthePleiades,arelativelyyoungcluster,thelargescatterinrotationalvelocity
Astro. Astrophys. 2003.41:5y Joint Institute for Nuclear ptoDtadhhsrnooeeemEnatsachuaigvaenrmheileadcusfloatieteeubefen,dmltclclShytaetieaHrhoraHfeeneroflafyrs&fneoaeicddfgdctBintesKfviafsavade,llver.iririfaueoaTtfrnnetetihaimtaaroitiestaisnpinolo(seccn1nrreoo9ihassnut9tasauli6shdnttri)isoeometb,fwhntoeheteeuleiedtnnforhdoedfcesrdhmtmttocaahhawrenaaessatgtniesloeot,authstnotrhfeiweoatehudrrnriiodstbeehtttehaoxsyststaraiietynmmoohnnfoetepthaif,lctaleehwtol,hpleSlyheeoiufiraurcsincntothihatnidhiqretvdshsauee.seeiexnHymrb.dfeoieeeegnwerrpitgrnoeeeeenrnvdfnpdsoecrferueert,nonthbecomdadyeft.
ev. b activeregionsshowedalatitudevariationsimilartowhatisobservedinthesolar
R
u. butterfly diagram. In fact, by applying the same techinque to solar data as that
n
n usedforthedistantstars,Donahue,Saar&Baliunas(1996)wereabletorecover
A
the solar differential rotation over the latitude band spanned by the solar active
regions.Also,Reiners&Schmitt(2002a,b)showedthatdifferentialrotationcanbe
inferredfromFouriertransformanalysisofrotationallybroadenedspectrallines,
providedthatdataofsufficientqualityandwithsufficientlyhighspectralresolution
areavailable.
1.3.Solar-SurfaceRotation
Detailedobservationshavebeencarriedoutofthesolar-surfacerotationbytrack-
ingthemotionofsurfacefeaturessuchassunspotsand,morerecently,byusing
1Aug2003 14:6 AR AR194-AA41-15.tex AR194-AA41-15.sgm LaTeX2e(2002/01/18) P1:GCE
604 THOMPSONETAL.
Doppler-velocitymeasurements.Itwasfirmlyestablishedbythenineteenthcen-
tury,throughcarefultrackingofsunspotsatdifferentlatitudesontheSun’ssurface,
that the Sun is not rotating as a solid body, but rather shows surface differential
rotationwithlatitude:attheequator,therotationperiodisapproximately25days,
but it increases gradually toward the poles where the period is estimated to be
approximately36days.Thismustsomehowresultfromtheredistributionofan-
gularmomentumbymotionswithintheconvectionzone.Thelatitudevariationis
traditionallyparametrizedasanexpansionin cos2(cid:181),where(cid:181) isco-latitude(e.g.,
Snodgrass1983).TheDopplerrotationrate,forexample,isapproximatedby
g
s.or ˜
view 2… D(451:5¡65:3cos2(cid:181) ¡66:7cos4(cid:181))nHz (1)
e
ualrnly. (Ulrich et al. 1988). Examples of such surface-rotation fits are shown in
no
s.anuse Figure2.ThedifferencebetweentheDopplerandmagnetic-featurerotationrates
m arjournalor personal msoufartyfhaecineDd,iwocpahpteelreetrhrafoettaathttuieorenmsmaagipngpehattricebneftelsayltiugdrhoeteslysanmreootarcneocrrharoperisedpd.oTnadhtestooamnaoenmydaemlpotauhtseblryieanhleimgathohtritaohtnee
ded fro4/09. F ilnartcheellsS,uwn.hiGchizodno,mDinuavtaellth&e DSochpopuler(-2fe0a0t3u)rehsaivgenaalr,gaupepdeathrattothheavseuwpearvger-alnikue-
a2
nlo10/
43. Dowarch on
99-6Rese
003.41:5Nuclear
phys. 2ute for
stronstit
Astro. Ay Joint I
v. b
e
R
u.
n
n
A
Figure2 Near-surfacesolar-rotationratesasdeterminedfromsurfacespectroscopic
Doppler-velocitymeasurements(solidline),trackingmagneticfeatures(dashedline
fromtrackingsunspots,dot-dashedlinefrommagnetogramcorrelationtracking),and
trackingDopplerfeaturesresultingfromlarge-scale(supergranular)convectiveflow
patterns(dottedline);thecurvescorrespondtofitsoftheformgiveninEquation1,
withcoefficientsadaptedfromBeck(2000).Arotationrateof320nHzcorresponds
toaperiodof36days,andarateof460nHzcorrespondstoaperiodof25days.
1Aug2003 14:6 AR AR194-AA41-15.tex AR194-AA41-15.sgm LaTeX2e(2002/01/18) P1:GCE
THEINTERNALROTATIONOFTHESUN 605
properties; thus, the observed rate may reflect the propagation speed of these
wave-likepatterns.
Foratleastonecentury,surfacedifferentialrotationoftheSunhasnotchanged
by more than approximately 5% of the equatorial value (Gilman 1974, Howard
1984,Ru¨diger1989).However,temporalvariationsontheorderofafewpercent
havebeendetected.Doppler-velocitymeasurementsoftheSun’ssurfacerotation
reveal narrow bands of weakly slower and faster rotation, superimposed on the
rather smoothly varying rotation rate as a function of latitude. These variations,
referred to as torsional oscillations, migrate toward the equator on a timescale
g
or comparabletothesolaractivitycycle(Howard&LaBonte1980).Ulrich(2001)
s.
w recentlyprovidedanextensiveanalysisoftheseandotherrelatedflowsthatalso
e
vi emphasizedthecloserelationbetweenthelatitudesoftheflowsandthedominant
e
ualrnly. magnetic activity. In addition, Doppler observations reveal large-scale nonrota-
no
s.anuse tional flows on the solar surface; these include flows from the equator toward
urnalonal tHhaethpaowleawyh1i9c9h6m,Hayatrheapwreasyenettmal.er1i9d9io6n).alcirculationintheconvectionzone(e.g.,
arjopers
om For 1.4.SolarInternalRotation
ded fr4/09. Thepropertiesofthesolarinteriorhavebeenprobedingreatdetailfromobserva-
a2
nlo10/ tionsofoscillationsofthesolarsurface.Mostoscillationmodesarepredominantly
43. Dowarch on stitaalnlydisnugrfaaccoeu-sgtriacvwitayvwesa,vwehs.erAenasaloytsheesrso,fatthreeilratfirveeqlyuesnhcoiretswhaavveeleynigeltdhes,dasrteriensgseenn-t
003.41:599-6Nuclear Rese chaisonavndshesdtorpewarpionnthvtisindooevFndeimrgiunmorfdeoeers3lets.nooTcffehstsrhoooeluafgsrtohhhloeyaudrStriuotnhnstete’asrtciioaocnrn.sgvtAureulcnactriteouxvnraeemlz.ooFpcnuileteryt,hotaehfsremtharoeofrtureaen,tsictouhtnileotsornabotosenfesrrlvhoaaottiatwtituoisodnnaes
phys. 2ute for vcaanritastuiobntlweritdhetlaaitlistu.dImepsiomrtialanrtatomtohnagtothfethseeissutrhfaecneeraor-tasutirofna,cealsthhoeaurglhayweirt,hwshigenreifiin-
stronstit the rotation increases with depth (at least at low latitudes) in the outer 5%. The
Astro. Ay Joint I raatedibaetitvweeienntethrieors,uorfnacthe-eeqouthaetorrhiaalnadn,drohtiagthes-laattitaudneearraltyesu.nTihfoertmranrastieti,oinntbeertmweedein-
v. b thesetwodifferentregimestakesplaceinanarrowregion,thetachocline,where
e
R
u. thereisevidentlyastrongradialrotationalshear.Thisregionisgenerallybelieved
nn toplayanimportantroleinthegenerationofthelarge-scalesolarmagneticfield
A
(e.g.,Weiss1994).
The rotational profile shown in Figure 3 raises a number of theoretical ques-
tions.Simpletheoreticalconsiderationsandearlynumericalmodelssuggestedthat
rotationintheconvectionzoneshouldbe“constantoncylinders,”withtheangular
velocity depending predominantly on the distance from the rotation axis; this is
manifestlynotsatisfiedintheSun.Thus,theremustbeprocessesthatredistribute
angularmomentumintheconvectionzone,resultingintheobservedprofile.Fur-
thermore,ifangularmomentumhasbeenlostfromthesolarsurfaceinthesolar
wind, mechanisms are required to slow down the solar core, and in the process
establishandmaintainthesharpgradientsinthetachocline.
1Aug2003 14:6 AR AR194-AA41-15.tex AR194-AA41-15.sgm LaTeX2e(2002/01/18) P1:GCE
606 THOMPSONETAL.
2.HELIOSEISMICPROBINGOFTHESOLARINTERIOR
The Sun oscillates simultaneously in millions of modes, with periods typically
between approximately 3 and 15 min; the horizontal scales of the modes range
fromradialoscillations,wheretheSunpreservesitssphericalshapeasitoscillates,
tomodeswithalocalhorizontalwavelengthofonlyafewthousandkilometerson
the solar surface. The amplitude of an individual mode, as observed in Doppler
velocity,isatmostapproximately20cms¡1,orafewpartspermillioninintensity.
Modes with amplitudes as low as 3 mm s¡1 have been detected (Garc´ıa et al.
g
or 2001). The detection of such small effects in the Sun’s turbulent atmosphere is
s.
w only possible because the modes are coherent over the Sun’s surface, and over
e
evi timescalesofweeksormonths.
ualrnly. Thedetailedobservationsofsolaroscillationshaverequiredthedevelopmentof
no
s.anuse extremelystablededicatedinstrumentationdeployedinlargecoordinatedprojects.
arjournalpersonal LlIanottineorgni-orNruneontfwintohgreke)fSf(uoCnrth)sai(pnFlcoilnsusdeaettta1hl9e.91g19r)o9un6ne)dtwa-nbodarskeIsRd,IBfSoirS(IOonbNtesre(nBravitaritmoioninanlgohRfaemdseisSakroc-lahavreoOrnasgctheilde-
m or
oF Doppler velocity. Results of major importance for our understanding of the dy-
ded fr4/09. namicsofthesolarconvectionzonehavebeenobtainedfromtheGONG(Global
43. Downloaarch on 10/2 O(aOMnssdcciciiHllhllaaeettlliiisoooonnsnpsNDhaeeottrpwLicpoolOwerkrbFIGsmerreraovqguauepteor)n)r(cy(HSy))acsrhp(vGeaercayreebecrrtrieaeatfllat.e.l1.tS91aO99l.6H9)15O9n)9eiant5lwss)tooraurnckmdaarerVsniewItsRoentGlhltOeahseG(fSVOroOaLmrHiFatObh(iGel(iSlMtoyoblDoaarfIl
99-6Rese SolarIrradianceandGravityOscillations)(Fro¨hlichetal.1995)instruments,which
003.41:5Nuclear emphasizelarge-scalemodesatlowfrequency.
Astro. Astrophys. 2y Joint Institute for 2.1.BTmhaahosremdicoeosbAnsoiescfrptvheeecdtSssuoonla.frSHopseeccliiilfiloacstaieolilnsysm,cthaoenlobbegehydaevsicorriboefdaimnotedremissodfesthcreibgelodbbayl,aorspnhoerrmicaall,
v. b Ym((cid:181);`)Dc Pm(cos(cid:181))exp(im`) (2)
Re l lm l
nu. asfunctionofco-latitude(cid:181) andlongitude`;here, Pm isaLegendrefunction,and
An l
c isanormalizationconstant.Thesphericalharmonicsarecharacterizedbytwo
lm
integernumbers,theirdegreelandtheirazimuthalorderm.Inaddition,amodeis
identifiedbyitsradialorder,n,whichisgivenapproximatelyasthenumberofnodes
intheradialdirection.Thedependenceontimetofanoscillationwithagivenn,l,
andmcanbewrittenas exp(¡i! t),where! ,which,asindicated,depends
nlm nlm
onallthreecharacteristicnumbers,istheangularfrequency.Themeanmultiplet
frequency! ,obtainedasanaverageovermof! ,isapproximatelydetermined
nl nlm
bythesphericallysymmetriccomponentofthestellarstructureascharacterized,
forexample,bythevariationofthesoundspeedcanddensity‰withdistancerto
1Aug2003 14:6 AR AR194-AA41-15.tex AR194-AA41-15.sgm LaTeX2e(2002/01/18) P1:GCE
THEINTERNALROTATIONOFTHESUN 607
thecenter.DifferentmodesaresensitivetodifferentregionsoftheSun,depending
ontheirfrequency,degree,andazimuthalorder.Themodesareconfinedbetween
latitudes§cos¡1(m=L),where L Dl C1=2.Thusmodeswithm Dl resideina
narrowregionneartheequator,whereasmodesoflowmextendtonearthepoles.
Mostoftheobservedmodesareessentiallystandingacousticwaves(pmodes).In
theradialdirection,thesearelargelyconfinedoutsideaninnerturningpointatthe
distancer Dr fromthecenter,wherethehorizontalphasespeed! r=L ofthe
t nlm
modeequalsthelocalsoundspeedc(r).Thus,low-degreemodesextendovermuch
oftheSun,whereashigh-degreemodesareconfinedclosetothesolarsurface.In
g
or addition,atmoderateandhighdegree,oneobservesfmodes,whichcorrespondto
ws. n D0andhavethenatureofsurface-gravitywaves.Forfmodes,thedisplacement
e
vi varies with r as exp(¡kz), where z D R ¡r is depth, k D L=R, and R is the
e
ualrnly. surface radius; these modes also become increasingly confined near the surface
no
s.anuse withincreasingdegree.Byexploitingthedifferentsensitivitiesofthemodes,he-
urnalonal liosBeyisampoplloyginygissuacbhletetochmnaiqkueeinsfteortehnecmesualtbiopulettlforceaqluizeendciceosn!diti,oonnseimnsaidyecothnestSruainn.
arjopers theradialstructureoftheSunbydeterminingcorrectionstontlhesoundspeedand
om For density of solar models. Such analyses show that current solar models are good
ded fr4/09. approximations to the actual solar structure (e.g., Basu et al. 1997, Gough et al.
a2 1996,Turck-Chie`zeetal.2001).
nlo10/
43. Dowarch on i2m.1.p1.orRtOanTAtTaIsOpNeActLoSfPLtIhTeTIsNoGlarOmFoFRdEeQsUisENthCeIiErSsenFsoirtitvhietyprtoesseonltaprurroptoastieo,nth,ewmhiocsht
99-6Rese affects the frequencies through advection and, to a lesser degree, the Coriolis
phys. 2003.41:5ute for Nuclear fbwoeercingeeh.gtSlefouclntaecrdtirotoonta,atoigoronkoeidsrnaseuplfp,firsocuixceihnmttlahytaitosltnohZ.weRTrZhtoh…etaaettifhofeingcahtlecsrap-nolirtbdtieenrqgeu–faf!nentciltfimseiodsfionrbotttaeairtnmioesndoacfsana
Astro. Astroy Joint Instit –!nlm ·!nlm ¡!nl0 Dm 0 0 Knlm(r;(cid:181))˜(r;(cid:181))rdrd(cid:181): (3)
ev. b The kernels Knlm can be calculated from the eigenfunctions for a nonrotating
R model(Schou,Christensen-Dalsgaard&Thompson1994);giventhehelioseismic
u.
n constraints on the structure of solar models, the kernels can, to a very good ap-
n
A
proximation,betakenasknown.Itshouldbenotedthatthekernelsdependonm2,
notm,sothattherotationalsplitting! ¡! isanoddfunctionofm.Also,the
nlm nl0
kernelsaresymmetricalaroundtheequator.Asaresult,therotationalsplittingis
onlysensitivetothecomponentof˜thatissimilarlysymmetrical.Asdescribed
inSection2.2,thedatacanalsobegivenintheformofso-calledacoefficients:
these too can be related to the underlying rotation rate ˜ by kernels, in a form
similartoEquation3.
Thekernelsvaryfrommodetomode,reflectingtheregionsofconfinement.A
fewexamplesareillustratedinFigure4.Theobservedmodesincludesomethat
1Aug2003 14:6 AR AR194-AA41-15.tex AR194-AA41-15.sgm LaTeX2e(2002/01/18) P1:GCE
608 THOMPSONETAL.
Figure4 ContourplotsofrotationalkernelsK inasolarquadrant,withtheequator
nlm
asthehorizontalaxis.Themodesallhavefrequenciesnear2mHz;thefollowingpairs
g of(l;m)areincluded:(a)(5;2);(b)(20;8);(c)(20;17);and(d)(20;20).Thedotted
or
ws. circlemarksthelocationofthelowerradialturningpointrt,andthedottedlineshows
vie thelatitudinalturningpointatco-latitude sin¡1 m=L.
e
ualrnly.
no
urnals.anonal use panendetthreatweehsosleenrtainalgleytooftihnetesromlaerdcieantetepre,ontehtreartsiothnadtaerpethtrsa,pwpietdhvaesriymnielaarrvthaerisautirofancien,
arjopers latitudinalextent.Thus,theobservedfrequencysplittingsprovideasimilarlywide
m or rangeofaveragesoftheinternalrotation.However,theamountofinformationthat
oF
aded fr24/09. iasffaevcatimlaobsletodfetpheenodbssegrrveeadtlymoondetsh,einloccluadtiionngimnotdheesSoufnh.igRhegdieognresenweahretrheethseurlafargcee
nlo10/ numberofmvaluesensuregoodresolutioninthelatitudedirection.Incontrast,
43. Dowarch on opnrolyviadilnimgietsesdennutimalblyernooflmatiotduedseorfestohleultoiowne.sAtdlseog,retheesspoeunnetdrastpeeteodthisehsioglharincotrhee,
99-6Rese solarcore.Thus,thepmodes,regardedasasuperpositionofpropagatingsound
strophys. 2003.41:5nstitute for Nuclear wwetfhofahferevesscerihtesmo,orpsftplidtechinisetdtyaclnwiotcrteeleectotooinmnthstehiedeicneefrtrnhketieqesrurrn.eeeTnglhcsioiefenoes;rfcifsseopcnfhtuseoerrtqnihcuetaehrlnelryteklydse,yurtnmcheeemldsseebitsnyrsiicliilttuirsvosistttrmayatitaioesldlnr,evidn˜oulFucDimegdue˜.rTea(rnh5)de;,
Astro. Ay Joint I !nlm D!nl0CmflnlZRKnl(r)˜(r)dr: (4)
ev. b 0
R R
nnu. The kernels are normalized such that Knl(r)dr D 1. It is particularly striking
A that,accordingtoFigure5b,only10%ofthesplittingarisesintheinner30%of
theradiusforthemodesconsidered.
2.1.2. INVERSIONS TO INFER THE ROTATION PROFILE The analysis of the oscilla-
tiondatatoinferpropertiesofthesolarinteriorisoftencharacterizedasinversion
(Gough & Thompson 1991). The goal of the analysis is to infer the properties
of ˜(r;(cid:181)) from the data, taking into account also the observational errors from
relationssuchasEquation3.Ideally,wewishtodetermineanapproximation˜¯ to
thetrueangularvelocity.Inmostcasesconsideredsofar,theanalysiscorresponds
eitherimplicitlyorexplicitlytomakinglinearcombinationsofthedata.Thus,to