19 92MNRAS.257. .340S 12 2 1 Accepted 1992January30.Received29;inoriginalform1991July7 Robert ConnonSmithandGeorgeW.CollinsII Differential rotationandpolarhollows WarnerandSwaseyObservatory,CaseWesternReserveUniversity,Cleveland,Ohio44106-7215,USA Whelan 1971).Suchconfigurationshavebeeninvestigatedin hollows intheirpolarregions(e.g.Bodenheimer1971; More recentlyithasbeenshownthatsomeaxisymmetric non-axisymmetric figuresthatarenoteverywhereconvex. liquids, ithasbeenknownthatextremerotationcanproduce Since Darwin(1910)investigatedbinaryfissionofrotating AstronomyCentre,PhysicsandDivision,UniversityofSussex,Palmer,BrightonBN19QH have developedaverypowerfulandnowfastnumerical leagues (e.g.Eriguchi&Sugimoto1981;Müller detail inaseriesofpapersbyEriguchi,Hachisuandcol- Mon. Not.R.astr.Soc.(1992)257,340-352 which hasbeenknownforsometimetofailconverge code thatcanbeusedtostudythree-dimensionalequilibrium stellar modelswithextremedifferentialrotationdevelop (1986b) hasillustratedtheversatüityofnewmethodby objects withhighdensitycontrast(Clement1978).Hachisu code isanewversionoftheSelf-ConsistentFieldmethod, configurations withalmostanydegreeofdistortion.The to differentiallyrotatingincompressible fluidsandHachisu burger’ sequencebifurcatingfromtheclassicalMaclaurin moto (1981)showedthatthereexisteda‘concaveham- constructing variousnon-axisymmetricconfigurations,such as contactbinariesandclosemultiplestars.Eriguchi&Sugi- (Wong 1974).Eriguchi&Hachisu (1985)extendedthiswork known Dyson-Wongsequence oftoroidalconfigurations 1 INTRODUCTION mentioned papers containstrikingdiagrams oftheextreme exist formoregeneralpolytropes. Manyoftheabove- (1986a) showedthatsimilar structures,withpolarholes, sequence andgoingovercontinuously intothealready 1985a,b; Müller&Eriguchi1985;Hachisu 1986a,b; Hachisu,Tohline&Eriguchi1988).Theseauthors 1985; Hachisu1986a,b;Hachisu,Eriguchi&Nomoto © Royal Astronomical Society • Provided by the NASA Astrophysics Data System bizarre shapesofstarsoveralargerangedifferentialrotationandspeed. Differentially rotatingstarsdevelophollowsintheirpolarregionsifthedifferential SUMMARY some resemblancetothickdiscs. We findrangesinwhichtherearenophysicalsolutionsbutarguethatsomestable rotation issufficientlyextreme.Wequantifythisstatement,andexplorethesometimes single starswithhollowsmayexist.Someofthemostextremeconfigurationsbear Key words:stars:rotation. found, noneofClement’smodelswasveryrapidlyrotating, viewing intheequatorialplane(equator-on).Itistherefore very strongvariationinapparentbrightnessastheobserver beyond thesecularstabilitylimit(seeSection4);also, because thenumericalmethodfailedtoconvergeformodels tially rotatingstellarmodelswithrealisticequationsofstate culations thathadbeenmadeuntilveryrecentlyfordifferen- develop andwhethertheseconditionsapplytoreal(single) important toaskunderwhatconditionspolarhollows moved awayfromviewingalongtherotationaxis(pole-on)to then theywóuldhavebizarreobservationalproperties,witha of Clement(1978,1979).Althoughsomehollowswere main-sequence stars(Mlessthanabout10M)werethose displayed bythepolytropesthatwehavejustdescribed, method makesuseexplicitlyofthebarotropicequation rotating stars,sotheycoveronlyasmallrangeofdifferential calculations werenotdesignedtoexaminetheshapesof and withdensitycontrastscharacteristicofthemajority stars. Asfarasweareaware,theonlytwo-dimensionalcal- dwarfs andotherdegenerateconfigurations(Hachisuetal. those foundintheirearlierpolytropic models,buttheircal- rotation. Veryrecently,Eriguchi &Müller(1991)con- state, andhasbeenextendedfrompolytropesonlytowhite shapes thathavebeenconstructed.However,thenew differential rotation,andfound extremeshapessimilarto structed Cowling-typemodels (power-lawopacity,point- & Hachisu1989). rotation speed (slow, inthesensethatcentrifugal forceis culations wererestrictedto a verysmallrangeinsurface source energygeneration)with amuchmoreextreme 1986a,b) ortorelativisticpolytropes(e.g.Komatsu,Eriguchi 0 If realstarspossessedthegeometricalstructuresthatare 19 92MNRAS.257. .340S 2 but awiderangeofrotationspeedanddegreedifferential we explorethequestionofhollowsinaquasi-analytical justified inPaper1,andwediscussitfurtherbelow.]Wetook lem. IntheAppendixwemakeamorerestrictedstudyofan results areindependentoftheprecisechoiceangular rotation, totryobtainsomephysicalinsightintotheprob- manner, usingaparticularangularmomentumdistribution much lessthangravityatthesurfaceequator).Inthispaper the differentialrotationlaw alternative rotationlaw,todemonstratethatourqualitative ANALYTICAL RESULTS where Qistheangularvelocityanditsvalueatpole,p momentum distribution. In ordertodiscusstheshapesofsurfacesrotating tial exists(e.g.Tassoul1978).Itsparticularformwasbriefly is aconstantandythedimensionlessdistancefrom constructed detailedmodelsofrapidlyrotatingstars,using were encounteredonanyofourmodels. tion ofrotationspeedandviewingangle.Despiteourcon- the valuesp=0(uniformrotation),0.2and0.5.Detailed example ofaconservativelaw,forwhichcentrifugalpoten- rotation axis(y=1attheequator).[Thiswaschosenasan balances gravityatthesurface.Weshowbelowthathollows what happenstothemodelswhentheyrotateatratesup than wereinprinciplepossible,andwehavenowconsidered models andphotometricindiceswerecalculatedasafunc- atmospheres werefittedtothe(onedimensional)interior uniformly rotatingstars,butismoredoubtfulforstrongdiffer- described bytheRoche(pointsource)approximation.As 2 THESHAPESOFROTATINGSTARS: not alwaysatthemostextremevaluesofrotationspeed.We the mostextremerotationspeeds,wherecentrifugalforce our mostrapidlyrotatingmodelshadsmallerrotationspeeds sidering rapiddifferentialrotation,nonoticeablehollows Q =(l-/ry),(1) discussed inPaper1,thisisagoodapproximationfor stars, weshallassumethatthegravitationalpotentialcanbe of thickdiscs. models thatarestableandrelateourresultsbrieflyto reason forthislimitation,estimatetheregionofequilibrium discuss qualitativelywhattheirpropertiesmightbe. photometric indicesforstarswithpolarhollows,althoughwe not exist.Wehavethereforeattemptedtocalculate give reasonsforbelievingthatmostsuchstarsprobablydo develop forlargeenoughvaluesofp,althoughinterestingly cal results,andweexpect the correspondingnumerical ential rotation.However,itenablesustoobtainsomeanalyti- rotating starswehavealsouncoveredaregimeinwhichthere the rangeofmodels thatwouldbeworthconstructing withan with theextraeffortinvolved andisnotappropriateatthis would notreturnimprovedphysical insightcommensurate results tobecorrectbetter than 10percentandconsider- seem tobenoequilibriummodels.Wediscussthephysical exploratory stage,whereweare tryingtoputsomelimitson ably betterinsomecases.Amore exactnumericaltreatment p p In apreviouspaper(Collins&Smith1985,Paper1),we We havesubsequentlyrealized(Smith1987,p.145)that In thecourseofourexplorationpossibleshapes © Royal Astronomical Society • Provided by the NASA Astrophysics Data System x has aninfinitedensitycontrast,ourmodelsrepresentthe tion lawaswehadusedinPaper1,becausewantedto extremes. should havepropertiesintermediatebetweenthesetwo opposite extremetothelow-npolytropicmodels.Realstars accurate code.BecausetheRocheapproximationformally that paperhadcausedustomissasignificantrangeofrealiz- discover whetherourrestrictedrangeofrotationspeedin values ofpmoreextremethanthoseweconsideredinPaper tion. Appendix todemonstratethatourresultsarenotaconse- instability), sowehaveconsideredanotherrotationlawinthe able models.Unfortunately,thelawisformallyunstablefor be written by equation(1),thesurfaceofstarisanequipotential.If quence ofchoosinganunstableangularmomentumdistribu- face atanangle6totherotationaxis,and07=Tsinis Here risthedistancefromcentreofstartosur- This isformallythesamesituationasinabarotropicconfi- total forceisderivablefromapotential,thepressureand conservative rotationlawofthekindweareusing,where (minimum) distanceofthesurfacefromaxisatthatpoint. R isthepolarradius,equationdescribingsurfacecan pressure canbeexpressedasafunctionofthedensityalone. density arebothfunctionsofthepotentialaloneandso 1 (seeSection4,wherewediscusstheconsequencesof barotropic relationbetweenpressureanddensityisonlythat there isamolecularweightgradient.Inthatcase,ifthestar with temperatureandpressuresurfacesnotcoinciding,are guration, andstrictlyspeakingweshouldnotusethis M isthemassofstarandGgravitationalconstant. vary withpressureifpdoes.)However,werecognizethata has aperfectgasequationofstate,theconsequence one situationinwhichgenuinelybaroclinieconfigurations, approximation forarealstar,whichisbaroclinie.(There to representthedynamicalstructureo‘fabaroclinieconfi- conservative forcelawisanapproximation,wecanstilluse it burgh 1966,appendix)thatastarwithconservative guration. Inparticular,thereisthewell-knownresult(Rox- strictly possibleevenwithaconservativerotationlaw:if have large-scalecirculationcurrents(cfEriguchi&Müller GMGM that sense,thereisnosuchthingasapurebarotrope,unless rotation lawcannotbeinhydrostaticequlibriumbutmust thermal effectsaresomehowcompletelysuppressed,which hand sideof equation (2)tobeGM/R we havetacitly has everbeenconstructed(cf Section4),sothereisatpres- these currentsandtheirback-reactionontherotationlaw is impossibleinarealstar.Nofullyself-consistentmodelfor rotating star. ent no‘exact’waytotreatabarocline. Thequasi-barotropic approximation isthesimplest approach,andwetherefore p adopt equation(2)asourdescription ofthesurfacea Thu beconstantonapotentialsurface,whichallowsTto 1991), whichthemselvesintroducebaroclinieeffects-in = p R r n For thepresentpaperwenaturallychosesamerota- With theRocheapproximation,androtationlawgiven There aretwopointsweshouldmakehere.First,witha The secondpointisjusttonote thatbychoosingtheleft- Differential rotationandpolarhollows341 Jo ' or (2) 19 92MNRAS.257. .340S 2 Dy1 2 face (seenoteabove). face isopen,therenoplacewherex=y.]Thendivideequa- This isanexplicitexpression fortheshapeofaclosedsur- Ky,p) = where tion (5)throughbyx,usex=y+zandsolveforztofind meets they-axis(z=0)atareal,finitevalueofy.Ifsur- First weevaluateRjRbytakingx=y=linequation(5): to showthatthereisanexplicitsolutionofequation(5)forz. in thedirectionparalleltorotationaxis.Further,itiseasy To findtheshapeofthissurface,itismostconvenientto [Note thatthisassumesthesurfaceisclosed,i.e. it distance ofthepotentialsurfaceaboveequatorialplane, consider zasafunctionofy,sinceisthedimensionless [,p,a) =^-^m,p)- [y,p)\. (8) dimensionless formas we canwritetheequationforshapeofsurfacein 342 R.C.SmithandG.W.CollinsII unity, butforaratherdifferentreasonfromtheonegivenin value ofaforextremedifferentialrotationislessthan principle. (Inpractice,asweshallseebelow,themaximum velocity andaistheratioofcentrifugalforcetogravityat Here Ristheequatorialradius,Qangular variables: models ofothertypessystem[cf.captiontoFig.2and principally interestedinthispaper,butdoesrestrictpossible is anaturalandappropriateoneforstars,inwhichweare Paper 1.Toreachextremerotation,themodelsof1 x =r/Æ,yrsin0/ÆZ=rcos different valuesonpartsofthesystem.Ourchoice ring, doubleringetc.),theconstantofintegrationcantake more complicated,multiplyconnectedsystems(starplus a maximumvalueofunityforallrotationlaws,atleastin surface equator.ContrarytothestatementinPaper1,ahas some remarksinSection6). Q/ =(1-p),anddefining should allhavehada=1.)Notingfromequation(1)that axis: asingle,rotatingstarwithspheroidaltopology.For assumed asimplyconnectedbodypenetratedbyitsrotation w =Q/Q, p e e ep max p What weareinterested inisknowingwhether ornotthis It isconvenienttodefinethefollowingdimensionless 2 (1+D) 4 (I-P) Jo 4 V 2 (i -p) © Royal Astronomical Society • Provided by the NASA Astrophysics Data System oj(y, p)dÿ, a a a =~ /(1,P). 1/2 xl(y, p). QlRl GM (7) (6) (5) (4) (3) == ya z = to lowestorderine,and the star.Wefind,aftersomealgebra, physical solutionstothepartsofequipotentialoutside tive valuesofewillcorrespondtothecontinuation positive forpointsonthephysicalsurfaceofstar.Nega- y =l-£,|£|«1.(12) face. Wewrite p =7dividesclosedsurfacesfromopenones.Althoughthe Since yisnormalizedbytheequatorialradius,emustbe p =1arenon-rotatingattheequator!) about theequator(y=1)andfindlocalshapeofsur- shape equation(7)iscomplicatedingeneral,wecanexpand result forcriticalrotation(a=1),whichshowsthatthevalue given byequation(1),intheparameterspace0izisimaginary:i.e.thereno equator, throughacuspatf=0,tothepartofequipoten- the transitionbetweenequipotentialsurfacesthatareclosed real solutionneary=1forwhichzissmall.Thusp?marks force togravityreachesunity.Forp<%thisoccursatthe there isacorrespondingtransitionbetweenclosedandopen imaginary neary=1);fora1,therearenophysicalstellar equipotentials, whichoccurswhentheratioofcentrifugal models forp>7. at theequator(z-0asy—1)andsurfacesthatareopen{z equator, whereacuspformsfor=1[thehashori- the nextorderin£expansionforz: zontal tangent,dz!dy=0,forp7,ascanbeseenbygoingto z =±©£foralandpv].(17) the outwardnormaltosurfacepointsawayfromrota- at apointawayfromtheequator,asweshallnowdemon- tion axisandwemusthaveanegativepressuregradientifthe strate. Atapointonthesurfaceofstarneartoequator, requires thatthecentrifugalforcebelessthany-com- normal totherotationaxis(thepositiveydirection),this star istobeinequihbriumwithnooutflow.Inthedirection ponent ofgravity,i.e.that Using equation(1),andthedimensionlessvariablesdefined Q rsin0<- above, wecanshowfromequation(18)that /(y, a,p)= centrifugal force_Qr where with theboundarybeinggiven bytheequality.Themodel violated, sowerequire will beunphysicalifthereisany yforwhichtheconditionis and sotheconditionforphysicalmodelsisthata/(y,a,p)^1, which willbesatisfiedifa/ x-l> where/isthemaxi- mum valueof / onthesurface,i.e.itisgiven byevaluating a/(y, a,p)^lVye[0,1], (21) ma max For a<1,wecannotderivesuchsimpleresult.However, For p>7,theratioofcentrifugalforcetogravityvanishes = 3 gravity GM 1/2 3-7p| © Royal Astronomical Society • Provided by the NASA Astrophysics Data System GM 3 or i-py l-pj (l+D)’ sin G. af{y, a,p), (15) (16) (18) (19) (20) 4 382n 23 23 23 The transitionbetweenphysicalandunphysicalmodelsis a maximum,thenyissolutionof between aandpwhichisgivenexplicitlyby then givenbya/(y)=l,whichdefinesarelationship 3ay(l -pyJ^Apil-p)ji+^[/(i,)-I(y,p)] dfjdy =0atfixedpanda.Ifyisthevalueofforwhich/ the boundarybynumericalexploration,usingmethod must besolvediteratively,anditisinfactsimplerjusttofind described inthenextsection.However,wecanfindanesti- 64p(l-p)=27ayyJ. (23) mate fortheboundarybyconsideringconditionQar/ to theconditionthatwouldapplyifmaximumofratio condition ismathematicallymorerestrictive(itcorresponds force togravityvariesoverthestar’ssurface. proximation totheexactconditionabove,anditallowsan Numerically, itturnsouttobeareasonablefirstap- of centrifugalforcetogravityoccurredattheequator). become m analytical argumentthatdisplayshowtheratioofcentrifugal centrifugal force_Qor m mp f(y, p)= m GM =1insteadofQr/GM-since07=sin01,outsidetherangeofinterest, independent ofa.liisnoweasytoshowthat,fory^0,f(y,p) and isgreaterthan1forallotherp.Considerfirstthecase m Thus themaximumratioofcentrifugalforcetogravity occurs fory=3/7p>1,outsidetherangeofinterest,and / decreases asydecreases,becominglessthan1foryI,however, from theequatortowards pole(ydecreasing)andwe gravity increasesabove1aswe movealongthestar’ssurface equator, thismeansthatthe ratioofcentrifugalforceto maximum of/occursaty = 3/7p \ withoutcentrifugalforce gravity GM Differential rotationandpolarhollows343 i-p. i -pyV l-pl 4/7 Iff p / = af{y,p), = m(p),say. (22) (24) (25) (26) 19 92MNRAS.257. .340S =i27 =orr anew for /?>y,y<1,soa/increasesas^decreasesawayfrom Thus the(approximate)boundarybetweenphysicaland fugal forcetogravitywillbegreaterthan1forsomey1,ratioofcentri- models nowbeingthataf<\.However,westillhavethat, fact atsomewhatlargerp,butonlybyafewpercent.[The used orinsteadofrinourcriterion,theexactboundaryis with unphysicalmodelshavingp>pBecausewehave unphysical modelsisgivenby 344 KC.SmithandG.W.CollinsII demonstration ofthisresult,byshowingthatforagivenpthe too tedioustoreproducehere.] critical aisalwayslargerthanl/m(p),straightforward,but hollows, thatC>1(equation11),andhavenotbeenableto which wecanfindequilibriummodels.However,have we haveestimatedtheboundaryin{p,a)planewithin range ofpandaforwhichtherearenophysicalmodels, pointing normaltothesurfacenearequator,andtacitly hollows occur,andwhatshapetheytake,usingthevalueof C is satisfied.Wenowturntoanumericalexplorationofwhere estimate ingeneralforwhatvaluesofpandathiscondition only beenabletowritedownaconditionfortheexistenceof in thedirectionnormaltorotationaxiscanactuallybe gradient inthatdirectioniszero.Thatwillbetrue,for assumes thatthenormalstillpointsawayfromrotation a™(Pcri<)> ) The shapesoftheconfigurationsdiscussedinlastsection NUMERICAL RESULTS pressure gradient,beingnormaltothesurface,isparallel axis aty.Theconditionthatcentrifugalforceequalsgravity m the surface.Theonlyminordisadvantageofusingthisequa- can bedeterminedbysolvingequation(7)fortheshapeof as ourguide. fugal forcemustexceedgravityforequilibrium.Thisclose into thequadrantcontainingrotationaxis,andcentri- gradient isactuallypositive,thenormaltosurfacepoints example, atthepeakofahollow{cf.Fig.5),where satisfied foranequilibriummodelsolongasthepressure m tion isthatitimplicitlyassumesthesurfaceclosed,and 3 THESHAPESOFROTATINGSTARS: relationship betweenhollowsandunphysicalmodels the rotationaxis.Insideahollow,outwardpressure the boundarybetweenphysical andunphysicalmodels potentials whichcorrespondtounphysicalmodels.However, it cannotbeusedtodeterminetheshapesofopenequi- boundary inFig.4displayhollows.[Itcanfacteasilybe criv going smoothlytozeroasy1. the expression(7)forzbreaks down,givinganegativesquare which definestheboundarybetweenphysicalandunphysical y ~fotypeofhollowjusttoappearat=y: shown thattheconditionC(y)=1foranextremuminzat explains whyallthemodelsonequilibriumsideof root forsomevalueofysignificantly lessthan1insteadof can stillbefoundbylookingfor valuesofpandaforwhich solutions.] see thenextsection-isexactlyequivalenttoequation(23) diagrams, wehave combinedequations(5) and(6)inthe m m m Note thatthisargumenthasmadeuseoftheoutward- Our analyticalconsiderationshaveshownthatthereisa To findtheopenequipotentials thatappearinsomeofour © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 4 4 form polynomial inxforagiven0.Atmostoneofthezerosthis which is,fortherotationlawweareusing,aseventh-order nique thatselectsthezerowewant.Inpractice,wereable polynomial correspondstoaphysicallyrealisticstellar to findseveralzeros,anddiscardtheunphysicalones. model, sowemustnowbecarefultodeviseanumericaltech- F(jc) =(1~p)(l-x)+ax[/(xsin0,/?)-/(1,/?)]==()(28) ferential rotationandspeed.Theobviousapproach However, astudyoftheotherzerosenabledustogainphysi- to solvingequation(28)isuseaniterativetechniquesuch cal insightintohowtheshapeofsurfacedependsondif- for findingthezerosofpolynomialscanmeetwithsomediffi- as Newton-Raphson.However,theuseofNewton-Raphson culties, especiallyif,asisthecasehere,zerosoccurator near anextremumofthefunction,whenthereisdanger up byasimplebisectiontechniqueiftheiterationfailedto that theiterationwillshootofftoanotherzero.Forthisreason, Raphson iterationtofindtherootsofF(x)=0,itwasbacked although ournumericalcodeusedastandardNewton- With suitableinitialguesses,itwaspossibletofindallthe converge andwetookgreatcarewiththeinitialguesses. pj- theequipotentialsarenolongerclosedatequator. detached equatorialringsassociatedwithstarsoflessextreme outer solutionswerestabletheymightbeviablemodelsfor could notmatchdirectlytoastablestar.Nonetheless,ifthetoroidal for valuesof0justawayfromtheequatortherearenoroots with thedoublezeroforp=7;F{x)thenhasapointofinflec- the exteriorregiondemarcatedbyouterequipotential differential rotation. for sufficientlylargep.However,wehavefoundthebound- our isillustratedinFig.3,andconfirmsanalytical ous equipotentialsurfacepassingthroughx=1.Thisbehavi- of F(x)=0intheneighbourhoodx1,andsonocontinu- tion atx=1.Forvaluesofplargerthanthiscriticalvalue7, increases, thenextlargestpositivezerodecreasesandmerges For smallp,thiszerooccursataminimumofF(x).Asp At theequator,functionF{x)alwayshasazeroatjc=1. the characterofshapeequationchangesaspincreases. at x=1isnowamaximumofF{x).Thishastheeffectthat one ofthezerosmovesofftosmallervaluesxandzero surface. value ofylessthan1.ThisboundaryisshowninFig.4;the argument thattherearenostellarmodelswitha=1and y =0andCV 1 ory=0andC=l.Itisclear thatifthe there maybeanextremumat the polefortworeasons:either ary betweenphysicalandunphysicalmodelssimplybyevalu- have foundintheprevioussection ananalyticalexpression hollows inthe(p,a)plane,and theshapesofhollows.We analytical approximation(27)liesslightlybelowitinthe and findingthevalueofaforwhichzbecomesnegativeat a ating z(y)fromequation(7)foragivenpandrangeof a (p, a)plane. and minimaonthesurfacez{y). Notefromequation(9)that C =1(equations10and11) for theappearanceofmaxima The multiplerootpersistsforalldifferentialrotation,but For a<1,weagainexpecttheequipotentialstobeopen We nowturntothemainpoint ofthepaper:location © Royal Astronomical Society • Provided by the NASA Astrophysics Data System /? =0.5.Thereisnozerointheneighbourhoodofx=lforany pj for ahollow)wehaveC>1there.Thisdemonstratesthat pole, whileifthepoleisataminimum(asufficientcondition cient, conditionfornohollows)wemusthaveC<1atthe extremum istobeamaximum(anecessary,thoughnotsuffi- unity attheequator.Thefunctionalwayshasazerox=1,butfor Figure 3.(a)ThebehaviourofthefunctionF(x)forselectedvalues there willbeatransitionbetweensurfaceswithnohollows zero, (b)ThebehaviourofF(x)neartheequator,fora=\and occurs atamaximum.Forp=t,F(x)haspointofinflectionthe of paroundthecriticalvalue7,fora=1.xisnormalizedtobe region wherehollowsaretobefound. hollow when(C-l)changessign.Moreprecisely,ifwe is clearthatincasethereachangetheshapeof and surfaceswithhollowswhenC=1atthepole,i.e. evaluate [C{y)-1]forallpointsonthesurfaceatgivenvalues regions withsimilarlyshapedhollows). Theseboundariesare plane thelinesalongwhichC =1,bothfory0(thebound- of panda,thenslowlyincrease aatfixedp,wefindthat shown inFig.4. ary ofthehollowsregion)and fory^O(theboundariesof at whichC=1.Wehavetherefore mappedoutinthe(p,a) a newextremumoccurswhenever anewvalueofyappears 07^ jt/2. C(0,7?, a)=1definestheboundaryin(/?,planeof What abouttheextremawhichoccuratvaluesofy#0?It 0-7 0-80-9101-11-21-3 Differential rotationandpolarhollows345 19 92MNRAS.257. .340S 346 R.C.SmithandG.W.CollinsII reaches aminimum (withz=z),risestoa maximum again boundary ofthehollowsregionforsmalla,butlarge we hadatfirstexpected.ThecurveC(0,/?,a)=1definesthe p =0.2,0.5themaximumvaluesofawere,respectively,0.713and0.263.Thedasheddash-dottedUneslabelledsd markthe turning pointsareconcaveatthepole.Thefollowingscheme points inthefunctionz{y).Surfaceswithanevennumberof different shapesofthesurfacesbynumberturning Figure 4.The(/?,a)plane.planedividesintothreemainregions:modelsthatareeverywhereconvex,showpolarhollows and is illustratedinFig.5,andsomerealexamplesareplotted small maximum(withz<=Zp)atfinitey.Weclassifythe and largeenoughp,thereareshapeswithhollowsevenwhen these regionshavebeendeterminednumerically,asexplainedinSection3,exceptforoneortwocriticalpointsthat found the numberofturningpointsonsurface;abbreviation‘M&R’standsfor‘moatedandringed’(seeFig.5).Theboundariesbetween all unphysical modelswithequipotentialsthatareopenattheequator.Thehollowsfurthersubdivideintoregionssimple hollows Fig. 6. approximate onsetofsecularanddynamicalinstabilitiesrespectively,usingtheestimatediscussedintext(Section4). analytically. WealsomarktherangeofparameterscoveredbymodelsPaper1;for/?=0thosefull ofa,for and smallerregionswithmorecomplexshapes.TheseshapesareillustratedinFig.5.Thenumberbracketsafterthelabelsfor regionsis no hollows. Fig. 6. zero. Wecallthisasimplehollow. Twoexamplesaregivenin single maximum(withz=Zi> z)andthenfallssmoothlyto C(0, p,a)<1:theseshapesareconvexatthepolebuthave a 2 p0le p Three turningpoints-convex atpole,surfacefallsuntilit The shapesthatoccuraremuchmorecomplicatedthan Two turningpoints-concave atpole,surfacerisestoa One turningpoint-convexatpole,andeverywhereelse: © Royal Astronomical Society • Provided by the NASA Astrophysics Data System (z =Z!),thenfallstozero.Notethatz<£always,butZi can nowbelessthanorgreaterz.Wecallthisshape a are illustratedinFig.6. maximum withz=>,fallstoaminimumand zero. Therearenowfivepossiblecases: rises againtoamaximumwithz=Zibeforefinallyfalling moat ifZi>zandaditch<.Twomoats plane, buttheamplitudeof moats,ringsandditchesis is interesting,andcomplicates theappearanceof(/?,a) 2p they lieinanunstable regionofparameterspace (seebelow). small (typicallyafewpercent ofz)andthesemorecomplex shapes areprobablyoflittle real significance,especiallyas p 3p2 p Four turningpoints-concaveatpole,surfacerisestoa (2) z7, themostrapidlyrotating starssimplydonotexist. lar materialcannotbeinequilibrium. region haveopenequipotentialsattheequator,andstel- ably ingeneral(seeAppendix):modelstheexcluded for relativelylargevaluesofp,butthesimplehollowsonly with thereasonforlimit onphysicalmodels,thatthe Although wecannotsolvethisexplicitlyforageneral reason forthis isrelatedtothewayinwhich theratioof possess polarhollowsofany kind.Wesuspect,byanalogy Nonetheless, thereisasubstantial numberofrapidlyrotat- we intuitivelyexpected.Part of theexplanationisthat,for the valueofaforwhichdp¡da=0.Somealgebrayields value ofp,wecandifferentiateitwithrespecttoaandfind appear forsmallormoderatevaluesofa.Thatisnotwhat a =(l-p) find ananalyticalexpressionforit.Fromequations(8)and it does.However,thereisinfactnoreasontosupposethat between thedifferentregionsareshowninFig.4. We havenotplottedanyrealexamplesofshapeswithfour hollows -fromFig.5itisclearthattheyformanaturaltran- have notfoundanyringedhollows,andbelievethatthey the shallowmoatedhollows;fordeephollows, The shapeswithfourturningpointshaveparticularlylow where C(0,p,a)=1,wehave (10) wecanseethatontheboundaryofhollowsregion, parallel tothea-axisforalargerangeofa,butwewereable cally, becausetheboundaryofhollowsregionrunsnearly very hardtodeterminethecorrespondingvalueofanumeri- hollows stilloccurredforpl/2p;thisregionissmall values consideredinPaper1.(Theinstabilityisconfinedto to invokesomeconstraintmaintaintheangular rents meansthat,evenforstablerotationlaws,itisnecessary Section 2),andtheinteractionofcirculationcurrents also presentinallstarswithconservativerotationlaws(cf. larger p,reachingnearly45percentoftheradiusfor teract theangularmomentumtransportbutweakenoughto netic field,whichissupposedtobestrongenoughcoun- transport bythecirculation.Asuitableconstraintisamag- momentum distributionagainsttheangular stars isacomplexandill-understoodproblem(seeSmith unstable angularmomentumdistributionofthekindweare magnetic fieldmightverywellalsobeenoughtomaintainan considering, andwefeelthatourchoiceofrotationlawisno Appendix someresultsfor a stableconservativelaw,to turbances -cf.Smith1987). Nonetheless,wegiveinthe less realisticthananyotherconservativelaw(especiallyas demonstrate thatourresultsare notcriticallydependenton any suchlawisliabletodisruption bynon-axisymmetricdis- the choiceofangularmomentum distribution. 1) butcanbenegativeifp>i.Thusthisrotationlawis 1987 forarecentreview).Theexistenceofcirculationcur- 1977; Moss&Smith1981;MestelWeiss1987).Such a stability ofthe models, usingthewell-known criterionbased How seriousisthisinstability?Thedevelopmentofthe We havealsolookedinanapproximate wayattheglobal 19 92MNRAS.257. .340S =2 virial techniquetoshowthatpolytropicstellarmodelswith index behaveinaverysimilarwaytotheclassical on theratioofkinetictogravitationalenergy,Tj\W\. unstable forT/|W|*0.14.For7/1«0.26,apointofover- that theirpropertiesdependverylittleonn.Inparticular,the incompressible bodiessuchastheMaclaurinspheroidsand Ostriker andcolleagues(Ostriker1970)haveusedthetensor models reachapointofbifurcationandbecomesecularly the totalmass.Itiseasytoshowthat where «isascalefactorwhichdeterminedbyspecifying unstable. ÿw\ After somealgebra,wecanshowthatforourrotationlaw, where Ristheradiusofstar(assumedsphericallysym- P Pcexp(—rla),(32) in themassrange1.5-30Mhaveapointofbifurcationat function ofpanda,bymakingtheusefulapproximationthat stability isreachedandthemodelsbecomedynamically assuming sphericalsymmetry: metric), pisthemeandensityandcentraldensity. the densitydistributionisexponential{cfMestel1968,p. internal angularmomentumdistribution. stars, andfound(Clement1979)thatmodelsofrotatingstars somewhat lowerkineticenergies,correspondingto7/|W| We haveevaluatedthisexpressionwiththefollowing 383): approximations: values ofT/\W\equalto0.14and0.26,evaluatedusingthe above approximations.Theselinesindicatewherethe 0 of course,onlyveryroughindicatorsthedomain respectively, shouldtheybepolytropesofindex3.Theyare, models wouldbecomesecularlyanddynamicallyunstable ing section. Indeed, mostofthemodels not dealtwithinPaperIare c range ofmodelswhichhavesimple hollowsandareglobally likely tobeunstable.However, theredoesseemtobeasmall are secularlyunstableandperhaps dynamicallyunstable. stability, becauseoftheapproximationsused.However, it stable. Weturntotheimplications ofthisresultinthefollow- seems likelythatallmodels having complexpolarhollows R *0.10. Therewasonlyaweakdependenceonmassand a 4 T We haveestimatedthevalueofTj\W\forourmodels,asa Clement hasconsideredhowtheseresultsextendtoreal (ii) n—3polytrope,withp/p*54- (i) R=R,sothatthefirstfactorisa/(1-p); Fig. 4showsUnesdenotingwheretheconfigurationshave c e 1/356 GM \pj32 © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 1+6^ +12/-^ (34) (33) tions onsuchastar.Themannerbywhichthecentrally flux isproportionaltothelocalgravitysomelowpower. been traditionallyusedtorelatethelocalradiativeflux temperature, whilethedynamicsofrotationwilldeter- determine thelocalvalueofeffectiveatmospheric produced stellarenergymakesitswaytothesurfacewill for atmosphereswherethefluxcanberepresentedby The vonZeipelresultholdsstrictlyonlyforconservative local surfacegravity,andLucy(1967)hasgivenasimilar mine thelocalgravity.AtheorembyvonZeipel(1924)has result forfullyconvectivestars.Inbothinstancesthelocal We haveseenthatifarotatinggaseousconfigurationis (Smith 1970;Smith&Worley1974).AndersonShu diffusion equation.Thelatterapproximationisnotvalidin rotation laws(suchastheoneweareconsideringhere)and that theangularmomentumisaxiallyconcentrated,itwill endowed withsufficientdifferentialrotation,distributedso HOLLOWS but thissuggestionisnotborneoutbyobservationalevi- the fluxinconvectivestarsshouldbeindependentofgravity, (1977) suggested,onthebasisofmixing-lengththeory,that stellar atmospheres,butmaynotbeaseriousrestriction dramatic aspectstotheobserver. develop aconcavityinthepolarregions.Shouldrealstars be coolerthaninanon-rotatingstarofthesamemass,butwe we expectthattheequatorialregionsofarotatingstarare dence (e.g.Eaton,Wu&Ruciñski1980).Thus,ingeneral, exist withwell-developedhollowstheywouldpresentsome figurations willbeconsiderablyelevatedintemperature luminosity maybelargeenoughforthepolarregionsalsoto For strongdifferentialrotation,thedecreaseincentral non-rotating stellarcounterpart,atleastforuniformrotation. emerge somewhere,thepolarregionswillbehotterthanina relatively cool.Sincethecentrallyproducedenergymust 5 IMPLICATIONSOFSTELLARPOLAR found inPaper1foreithertherigidlyordifferentiallyrotat- the hollow,substantialregionsofitwillonlybevisibleatlow the majorityofstar.However,becauseconcavity inclination. Thesevariationswillsignificantlyexceedthose compared tothelargehighlydistortedlow-gravityregionsof still expectthemtobehotterthantheequator. would belargelydeterminedbythelow-gravitybulgeof ing models.Forthevastmajorityofsuchstars,spectra angles ofinclination.Thereforelargevariationsincolourand specific luminositywillbeassociatedwithdifferentanglesof broadened butpeculiarlineprofiles.Whilesuchprofiles luminous, cool,low-gravitystarexhibitingrotationally rotationally broadenedprofiles,thecurvaturewouldbe star. Thespectrumwouldbethatofarelativelyunder- profile. Indeed,astheequatorialangularvelocityof considerably lessthanthetypicalrotationallybroadened should showtheupwardcurvingshapecharacteristicof include inanymodel.Thelarge changeintheangularvelo- probably becalled‘sharp-line stars’. models exhibitinghollowsis solow,theseobjectswould leading tocirculation currents,turbulentflow, orboth.Such city frompoletoequatorshould generateconsiderableshear some significantphysicaleffects thatwouldbedifficultto Consider theconstraintsthatdeterminesurfacecondi- Thus wecanexpectthatthepolarhollowsofthesecon- The presenceofthehollow would beaccompaniedby Differential rotationandpolarhollows349 19 92MNRAS.257. .340S be sufficienttogeneratelarge-scaleturbulence,theentire flow couldtransportsufficientenergylaterallytoupsetthe the useofLucy’slawevenforearly-typestars.However, validity ofvonZeipel’slaw.Indeed,shouldthevelocityshear turbulent viscositytoredistributethestellarangular outer envelopemightbeforcedintoconvection,requiring removed. Anumberofauthors(e.g.Tassoul&1982; momentum sothattheconditionsleadingtoholloware 350 R.C.SmithandG.W.CollinsII zontal energytransportbymeridionalcirculationcurrentsor driving allrotatingstarstowardnearlyrigidrotation.Hori- Smith 1987)havesuggestedthatthisindeedtakesplace, should turbulencebewidelypresent,onecouldexpectthe tical difficultiestobeencounteredinmodellingtheappear- turbulent flowcouldcauseotherproblems.Onemayno used inmodellingrotatingstellaratmospheresmakeuseof locally, makingtheatmospheremodellingfarmorecompli- longer beabletoassumethathydrostaticequilibriumapplies the symmetryofstar(seeCollins1965;&Har- highly efficientquadratureschemesandatheoremregarding ance ofsuchobjects.Thestandardquadraturetechniques further, makingthemmoreisothermalandtherebyweaken- models cannotbereadilyrepresentedbysimplepolynomial type appropriateforthesurfaceofasphere.Functionsthat rington 1966).ThequadratureschemesareoftheGaussian hollow. Thiswouldservetoheatthelocalatmospheres cated. Shouldthehollowbedeep,someofemergentradi- a rotatingstarissymmetricabouttheaxisofrotationand ture accuracywillbedecreasedforthesefigures.Whilethis the figuresexhibitingpolarhollows.Thusoverallquadra- distortions ofasphericalsurface.Thisisparticularlytrue distortions representedbytherapidlydifferentiallyrotating can berepresentedbypolynomialsonthesurfaceof weak spectrallines. ment, butwouldshowanomalouslybroadenedandperhaps in arelativelyhigh-temperatureandhigh-gravityenviron- ing thespectrallines.Theresultantspectrawouldbeformed ation wouldbetrappedandilluminateotherpartsofthe the integrandoffluxintegralforspecificluminosity degree ofthequadrature,problemsassociatedwith problem canreadilybesolvedsimplybyincreasingthe sphere willbeexactlyintegrated.Unfortunately,thelarge grating overanyhemisurfacedefinedbyaplanepassing equatorial plane,thentheintegralcanbeobtainedbyinte- the constructionofmodelatmosphereswhichlocally the integrationtobecarriedoutfrompolewith containing theobserverandaxisofrotation.Thisallows through thecentreofstarandperpendiculartoplane symmetry theoremcannot.Collins(1965)observedthatif on linesofconstantlatitudeandlongitude,greatlyfacilitating inclination functionallyincludedintheexpressionfor will bevisibletotheobserver. Thatisnotthecaseforstars theorem willonlyapplytoconvexsurfaces,asitwastacitly determine thestructureofstar.Unfortunately,this numerical evaluationofthelocal atmospheres,considerable with polarhollows.Inaddition tovastlycomplicatingthe specific intensity.Thusthequadraturepointsmaybelocated effort mustbemadetodefine the stellarsurfacevisibleto assumed thatallpointsonthehemisurfacecutbyplane deterrent totheconstruction of suchmodels,theymayform observer. Whilenoneofthese concernsisafundamental a practicalbarrier byincreasingtherequired computation Finally, wemustobservethattherearesomepurelyprac- © Royal Astronomical Society • Provided by the NASA Astrophysics Data System time beyondacceptablelimits.However,shouldallthese We haveseenhowdifferentiallyrotatingstarsmaydevelop concerns beaddressed,itwouldpossibletomodelthe hollows causeconsiderabledifficultiesinthemodellingof atmospheres oftheseconfigurations. depressions orhollowsinthepolarregions,andthatthese 6 CONCLUSIONS inclination. Thisisjustwhatonewouldexpectasthelogical exhibit awiderangeofspectradependingontheangle such objects.Wehavealsosuggestedthatobjectswould extension ofPaper1.However,in1andPeacock& rotation, thepredictedphotometriceffectswerenotpresent ferentially rotatingmodelsdiscussedhere.Shouldsuchstars in theobservationalliterature,suggestingthatstrongdiffer- Smith (1987)weshowedthat,forfarlessextremedifferential that thisargumentisevenmorerelevantfortheextremedif- ential rotationisnotcommonamongstars.Itwouldseem horizontal shearandvariousinstabilitiesaccompanyingthe Although healsopointsoutthat‘noneoftheseresultshas that centrallycondensedpolytropesseemtomirrorthe In addition,thereisanexcellentchancethattheseextreme differential rotationwillservetoreducetheaspect existence. Indeed,thepotentialanomaliesinspectraof exist, theycannotbenumerous,fortheirphotometricand become dynamicallyunstable.Whilethesecularanddynami- general polytropes.Atevenfasterrotationstheseobjectswill been properlydemonstrated’,theworkofHachisuand behaviour ofthewell-studiedMaclaurinspheroidsandare of therotation,slowlyforcingstartowardrigidrotation. already mentionedthatanumberofworkersbelievethe their existencewouldrequirethattheybeconfusedwith spectroscopic anomalieswouldhavedrawnattentiontotheir unlikely. Certainly,boththeoryandobservationseemto unstable. Theapparentabsenceoftheseobjectsamongreal lity suggestthatthemostextrememodelsareindeed not beenproperlyinvestigated,ourroughestimatesofstabi- cal stabilityofthemodelswehavediscussedinthispaperhas and appearstoconfirmtheexistenceofsecularinstabilityfor others referredtoinSection1hasimprovedthesituation expected tosharetheirsecularinstabilityforfastrotation. stars willbesecularlyunstable.Tassoul(1978)pointsout such starswouldbesomonumentalthatitseemslikely mitigate againsttheexistenceofsuchobjects. early historyofsuchstarsinstabilitiesdevelop,makingthe some otherunusualstellarobject-perhapssymbioticstars. for whichmostoftheproblemsmentionedaboveare existence ofdifferentiallyrotatingmain-sequencestars stars alsosuggeststhattheyareunstable.Perhapsduringthe modelling themimplythatfurtherstudyofshouldbe objects andtheformidableproblemstobeencounteredin matter accretingontoacentral objectcanproduceasur- relevant andtheunderstanding ofwhichwouldgreatly abandoned? No-forthereappearstoexistaclassofobjects tures possesslong-termstability, theirformsmayexistfor described inSection3.While itisunlikelythatthesestruc- rounding envelopenotdissimilar totheconfigurations enrich ourknowledgeofstellar astrophysics.Theflowof some time,obscuring thecentralattracting object. Itseems Should weexpectthepresenceofsuchobjects?Wehave Does theunlikelypossibilityofexistencethese 19 92MNRAS.257. .340S blanket thatsurroundsit.Manyofthemodellingproblems central object,weshallhavetounderstandtheaccreting that inmanycases,ifwearetounderstandthenatureof described abovewillhavetobedealtwithifthisis unicorn’ butmayserveasthefoundationformodellingof ing starsisnotsimplyaninvestigationofsome‘stellar accomplished. Thusthestudyofrapidlydifferentiallyrotat- by theworkonpolytropesdiscussedearlier(e.g.Hachisuet thick accretiondiscs.Astartonthisproblemhasbeenmade geometrical problemsdescribedinthelastsectionwillneed to bemademodeltheappearanceofsuchdiscsthen to betackled.Experienceinmodellingtheatmospheresof rapidly rotatingstarsmayproveveryusefulinthemorechal- ACKNOWLEDGMENTS discs. lenging problemofmodellingthespectrathickaccretion al 1986a,b;Hachisuet1988),butifaseriousattemptis During partofthisworkGWCIIwassupportedbytheSERC to ProfessorsLeonMestelandPeterSweetforusefulcon- hospitality extendedduringhisstay.Theauthorsaregrateful as aSeniorVisitingFellow.HewishestothanktheAstro- versations andtothereferee,DrY.Eriguchi,forhishelpful nomy Centre,andespeciallyProfessorRogerTayler,forthe suggestions. Anderson, L.&Shu,F.H.,1977.Astrophys.J.,214,798. REFERENCES Bodenheimer, R,1971.Astrophys.J.,167,153. Collins, G.W,II,1965.Astrophys.J.,142,265. Clement, M.J.,1979.Astrophys.230,230. Clement, M.J.,1978.Astrophys.222,967. Collins, G.W,II&Harrington,J.R,1966.Astrophys.J.,146,152. Eaton, J.A.,Wu,C.-C.&Ruciñski,S.M.,1980.Astrophys.J.,239, Darwin, SirG.H.,1910.In:ScientificPapers,Vol.HI,Figuresof Collins, G.W,II&Smith,R.C,1985.Mon.Not.astr.Soc.,213, Eriguchi, Y.&Sugimoto,D.,1981.Prog,theor.Phys.,65,1870. Eriguchi, Y.&Hachisu,I.,1985.Astr.Astrophys.,148,289. Eriguchi, Y.&Müller,E.,1985a.Astr.Astrophys.,146,260. Eriguchi, Y.&Müller,E.,1991.Astr.Astrophys.,248,435. Eriguchi, Y.&Müller,E.,1985b.Astr.Astrophys.,147,161. Fricke, K.,1967.DoctoralDissertation,GöttingerSternwarte. Fricke, K,,1968.Z.Astrophys.,68,317. Goldreich, P.&Schubert,G.,1967.Astrophys.J.,150,571. Hachisu, I.,1986a.Astrophys.J.Suppl,61,479. Hachisu, I.,Eriguchi,Y.&Nomoto,K.,1986a.Astrophys.J.,308, Hachisu, I.,1986b.Astrophys.J.Suppl,62,461. Hachisu, I.,Eriguchi,Y.&Nomoto,K.,1986b.Astrophys.J.,311, Komatsu, H.,Eriguchi,Y.&Hachisu, I.,1989.Mon.Not.R.astr. Hachisu, I.,Tohline,J.E.&Eriguchi, Y,1988.Astrophys.J.Suppl, Mestel, L.,1968.Mon.Not.R.astr. Soc.,138,359. Lucy, L.B.,1967.Z.Astrophys.,65, 89. Mestel, L.&Weiss, N.O.,1987.Mon.Not.R.astr. Soc.,226,123. Mestel, L.&Moss,D.L.,1977.Mon. Not.R.astr.Soc.,178,27. University Press,Cambridge. 519 (Paper1). 919. Equilibrium ofRotatingLiquid,pp.161-162,Cambridge 214. 66,315. Soc., 239,153. 161 © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 2 2 2 i(y,p)=l Moss, D.&Smith,R.C,\9%\.Rep.Prog.Phys.,44,831. Müller, E.&Eriguchi,Y.,1985.Astr.Astrophys.,152,325. Roxburgh, I.W.,1966.Mon.Not.R.astr.Soc.,132,201. Peacock, T.&Smith,R.C.,1987.Observatory,107,12. Ostriker, J.R,1970.In:StellarRotation,IAUColloq.No.4,p.147, Tassoul, J.-L.&M.,1982.Astrophys.J.Suppl,49,317. Tassoul, J.-L.,1978.TheoryofRotatingStars,p.268,Princeton Smith, R.C,1987.In:PhysicsofBeStars,p.123,edsSlettebak,A. Smith, R.C,1970.Mon.Not.astr.Soc.,148,275. Smith, R.C.&Worley,R.,1974.Mon.Not.astr.Soc.,167,199. This showsthatthereisagainacriticalvalueofp,this Wong, C.-Y,1974.Astrophys.J.,190,675. Whelan, J.A.J.,1971.D.Phil.thesis,UniversityofSussex. von Zeipel,H.,1924.Mon.Not.R.astr.Soc.,84,665. p =3wehave APPENDIX: ANALTERNATIVEROTATION we simplyconsider directlythecondition that z=0:for physical solutions(closedsurfaces) andtheunphysicalones derive ananalyticalexpression fortheboundarybetween Again thereisacuspwithhorizontaltangent:for= 1, Z= ±£ (open surfaces).Inthiscase,the algebraissomewhateasierif time, abovewhichtheequipotentialsurfacesarenotclosed. LAW ernative stablelawwhosedimensionlessformis Since therotationlaw(1)isunstableto Goldreich-Schubert-Fricke instability,weconsiderthealt- the limitofsmallpisuniformrotation,asbefore,but Note thatpcannowcoverthewholerangefrom0toinfinity; rotation law(41)usedbyEriguchi&Müller(1991)isofthe limit oflargepisnowuniformangularmomentum.[The of Section2,wefindeasilythat numerical calculationsforp=20and16.7.]Inthenotation where theshapeofsurfaceisagaingivenbyequations(7) above form,withtheirBequaltoour1/p.Theymake equator, usingequation(12),andwefind(fora=1) and (9)forzdz/dy. D=-a(l+p)— -L For a<1,thecriticalvalueof pislarger,andwecanagain As before,wecanexpandtheshapeequation(7)about ed. Slettebak,A.,Reidel,Dordrecht. & Snow,T.R,CambridgeUniversityPress,Cambridge. University Press,Princeton,NJ. 2 1+pV ' 2 (i+py) 1 ,-V Differential rotationandpolarhollows351 2(1+//)’ 1/2 c-- 32 (1 +Z))(1+py «(!+/ (37) (38) (35) (36) 19 92MNRAS.257. .340S 2 2 2 2 2 -+2 352 R.CSmithandG.W.CollinsII Figure Al.The(p,a)planeforthestablerotationlawdiscussedin pattern tothatinFig.4.Theasymptoticformsoftheboundaries, hollows. Weexpectthemorecomplexshapestofollowasimilar unphysical solutionsandtheboundaryofregionsimple the Appendix.Weshowonlyboundarybetweenphysicaland and thesignificantturningpoints,aremarked. y arenegative(thatthisconditionisalsonecessarywas where itcanbeshownthatthedenominatornevervanishes less than1. we canfindthevaluesofpandaforwhichthereisroot negative atsomey<1,soifweexaminetherootsofz()=0 equator, y=\.Foropensurfaces,weexpectthatatthe closed surfacesweexpectthatzwillbezeroonlyatthe confirmed numerically).Thisconditioncanbeshownto avoid zvanishingatsomey<1,itissufficientthattheroots in therange0