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Astronomy&Astrophysicsmanuscriptno.Gstars (cid:13)c ESO2011 January28,2011 The differential rotation of G dwarfs M.Ku¨ker1,G.Ru¨diger1,andL.L.Kitchatinov2,3 1 AstrophysikalischesInstitutPotsdam,AnderSternwarte16,D-14482Potsdam e-mail:[email protected], [email protected] 2 InstituteofSolar-TerrestrialPhysics,POBox291,Irkutsk664033,Russia e-mail:[email protected] 3 PulkovoAstronomicalObservatory,St.Petersburg196140,Russia January28,2011 1 1 ABSTRACT 0 2 AseriesofstellarmodelsofspectraltypeGiscomputedtostudytherotationlawsresultingfrommean-fieldequations.Therotation lawsoftheslowlyrotatingSun,thefastrotatingMOSTstarsǫ Eriandκ1 CetandtherapidrotatorsR58andLQLupcaneasilybe n reproduced.Wealsofindthatdifferencesinthedepthoftheconvectionzonecauselargedifferencesinthesurfacerotationlawand a thattheextremesurfaceshearofHD171488canonlybeexplainedwithaartificiallyshallowconvectionlayer. J Wealsocheckthethermalwindequilibriuminfast-rotatingGdwarfsandfindthatthepolarsubrotation(dΩ/dz < 0)isduetothe 7 baroclineeffectandthattheequatorialsuperrotation(dΩ/dr > 0)isduetotheΛeffectaspartoftheReynoldsstresses.Inthebulk 2 of the convection zones where the meridional flow is slow and smooth the thermal wind equilibrium actually holds between the centrifugalandthepressureforces.Itdoesnothold,however,intheboundingshearlayersincludingtheequatorialregionwherethe ] Reynoldsstressesdominate. R S Keywords.stars:individual:HD171488–stars:interiors–stars:solar-type–stars:rotation . h p 1. Introduction whereδΩisthedifferencebetweentheangularvelocitiesatthe - o equatorandthepolarcaps1.Thepowerlaw(1)wasconfirmed r Many stars show signs of differential rotation. The solar equa- bythefindingsusingspectroscopicmethods(Reiners2006).The t s tor rotates with a shorter period than the polar caps. The dif- lightcurvesofthestarsǫ Eriandκ1 CetrecordedbytheMOST a ference of 132 nHz between the rotation periods found by satellite(Crolletal.2006;Walkeretal.2007)bothindicatewith [ Ulrichetal.(1988)correspondsto a difference of 0.07 rad/day δΩ ≃ 0.062andδΩ ≃ 0.064averysimilarequator-polediffer- 1 between the respective angular velocities or a lapping time of ence of the rotation rate as the Sun. Also the value of 0.11 for v 88 days. Helioseismology has found that this pattern persists the young G star CoRoT-2a (Fro¨hlichetal.2009) well fits the 7 throughout the whole convection zone but not in the radia- common picture of a rotation-independentsurface shear for G 9 tive zone below (Thompsonetal.2003). Stellar differential ro- stars. 2 tation can be inferred from the light curves of rotating spotted It is challenged, however, by recent observations of the 5 stars (see, e.g. Henry et al. 1995; Messina & Guinan 2003), young G dwarfs LQ Lup, R58 and HD171488. Marsden et . 1 frommonitoringmagneticactivity(Donahueetal.1996),spec- al.(2005)reportasurfacedifferentialrotationof0.025±0.015 0 troscopically(Reiners&Schmitt2003),orbyDopplerimaging rad/day for R58 while Jeffers&Donati(2009b) find the much 1 (Barnesetal.2000;Donatietal.2000). largervalue of0.138±0.011rad/day.Donatiet al. (2000)find 1 Whileseveralstudieshavefoundasystematicdependenceof asurfacedifferentialrotationδΩ = 0.12±0.02rad/dayforLQ : v thesurfacedifferentialrotationontherotationrate,nosuchde- Lup.Jeffers&Donati(2009a)determinedthesurfacerotationof i pendenceisfoundwhenthesamplesare combined(Hall1991; HD 171488 using the Zeeman-Dopplerimaging technique and X Barnesetal.2005). Moreover, measurements of surface differ- found a very strong surface differential rotation of 0.5 rad/day. r ential rotation of the rapidly rotating K dwarfs PZ Tel and AB Marsden et al. (2006) report a smaller but still large value of a DorwiththeDopplerimagingtechniqueshowthatstarsrotating 0.402±0.044rad/day.Huberetal.(2009)foundnoevidencefor muchfasterthantheSunshowverysimilarsurfaceshearvalues differentialrotationatallbutcouldnotruleitouteither.Jeffers –asfirstpredictedbyKitchatinov&Ru¨diger(1999).PZ Telis etal.(2010)confirmedthefindingsofMarsdenetal.(2006). ayoungKdwarfwitharotationperiodof0.95days.Itssurface The values found for LQ Lup and R58 are in line with the differentialrotationδΩ = 0.075rad/dayis remarkablyclose to Barneset al. picturebutthe largevaluesfoundfor HD 171488 thatoftheSun(Barnesetal.2000).ABDorisarapidlyrotating are not. The three G dwarfs are similar in their ages, effective K0dwarfwitharotationperiodof0.51days.Itssurfacediffer- temperatures and radii yet HD 171488 shows a much stronger ential rotation was found to vary with time between 0.09 and differential rotation than the other two stars. The studies men- 0.05rad/day(CollierCameron&Donati2002). tionedhavefocusedonrotationrateandeffectivetemperatureas Barnesetal.(2005)proposedadependenceontheeffective the properties determining the surface differential rotation. As temperature(albeitwithlargescatter)andfoundthepowerlaw thestarsareverysimilarbytheirstellarparameterssuchasage 1 thestrengthofthedifferentialrotationisalsoexpressedintermsof δΩ=Ωeq−Ωpole ∝Te8ff.92±0.31 (1) thelappingtimebetweentheequatorandthepoles,Plap =2π/δΩ 2 M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars andeffectivetemperature,weask if therecouldbe adifference where tistheangularmomentumfluxvector in their internal structure that would cause the observeddiffer- ence in their surface rotation. All three stars have just reached ti =rsinθ(ρrsinθΩumi +ρQiφ), (5) thezeroagemainsequenceorareapproachingit.Giventherapid withthetwotransportersofangularmomentumi)themeridional retreatoftheconvectionzoneinthefinalpartofthepre-mainse- flow(um)andii)thezonalfluxoftheturbulentangularmomen- quenceevolutionandtheuncertaintyofthestellarageweaskif tum(Q ). thestrongdifferentialrotationofHD1714888canbeexplained iφ The equation for the meridional flow is derived from the byadifferentdepthofitsouterconvectionzone. Reynolds equation by taking the azimuthal component of its Inthefollowingwecomputemodelconvectionzonesofdif- curl: ferentdepth and their large-scale gasmotions, i.e. rotationand meridionalflow.Themodelsarebasedonthemeanfieldformu- 1 ∂Ω2 1 ∇× ∇·R +rsinθ + (∇ρ×∇p) +...=0 (6) lationoffluiddynamicswhichhasbeenverysuccessfulforthe " ρ # ∂z ρ2 φ Sun,wherethemodelsreproducethesurfacerotation,theinter- φ nalintheconvectionzone,andthesurfacemeridionalflowvery whereR =−ρQ istheReynoldsstressand∂/∂z=cosθ∂/∂r− ij ij well(Kitchatinov&Ru¨diger(1999,KR99),Ku¨ker&Stix(2001, sinθ∂/∂θisthederivativealongtheaxisofrotation.Forveryfast KS01)).Modelshavebeenconstructedforavarietyofstarswith rotationthesecondtermontheLHSofEq.(6)dominatesandthe spectraltypesfromMtoF(Ku¨ker&Ru¨diger2008). rotationrate is constantalongcylindricalsurfaces, as stated by A new scheme allows the computation of stellar rotation theTaylor-Proudmantheorem. lawsandmeridionalflowpatternsbasedonamean-fieldmodel ofthe large-scaleflowsin stellar convectionzonesalsoforfast rotation rates when narrow boundary layers exist. It assumes 2.2.Transportcoefficients strict spherical symmetry for the basic stratification, ignoring Ifsphericalpolarcoordinatesarechosen,theΛeffectappearsin anyflattening thatmightoccurfor veryfastrotation.However, twocomponentsoftheReynoldsstress: theimpactofrotationonthethermalstructurecanbetakeninto accountbyincludingagravitydarkeningtermintheheattrans- Q = Qvisc+Vν sinθΩ (7) rφ rφ t portequationsothatthemodelremainsapplicableevenformod- Q = Qvisc+Hν cosθΩ (8) eratelyflattenedstars. θφ θφ t Here, Qvisc and Qvisc contain only first order derivatives of Ω rφ θφ 2. Theoryofstellarrotationlaws with respect to r and θ and therefore vanish for uniform rota- tion.ThecoefficientsVandHrefertotheverticalandhorizontal While thermal convection is driven by the stratification of the partoftheΛeffectwhileν representstheeddyviscosityinthe star,theconvectivegasmotionscarryangularmomentumaswell t convectionzone.Asthecorrespondingtermscontaintheangu- asheat.Momentumtransportbyturbulentmotionsisknownas larvelocityΩitself,theydonotvanishforrigidrotation.Thus, Reynoldsstressandcanbedescribedasaturbulentviscosityin rigidrotationisnotstress-free. caseofthesimplestshearflows.Inarotating,stratifiedconvec- Forslowrotation,theconvectiveheatfluxisproportionalto tion zone the Reynolds stress is anisotropic and its azimuthal thegradientofthespecificentropy: componentshavea contributionproportionaltotheangularve- locity, Ω, itself rather than its gradient. This form of Reynolds ∂S stress (with ‘Λ effect’) is not compatible with solid body rota- Fconv =−ρTχ , (9) i t∂x tion. i whereχ theturbulentheatconductivity.Theturbulentheatcon- t ductivity is determined by the convection velocity u and the 2.1.Basicequations c convectiveturnovertimeτ : c Our modelconsists of a 1D backgroundmodelwhich assumes 1 hydrostatic equilibrium and spherical symmetry and a system χ = τ u2 (10) ofpartialdifferentialequationsthatdescribetheconvectiveheat t 3 c c flux, the transport of angular momentum, and the meridional Usingstandardmixing-lengththeorywefind flow, assuming axisymmetry. The equation for the convective heattransportthenreads l2g ∂S u2 =− m (11) ∇·(Fconv+Frad)−ρTu¯ ·∇S =0, (2) c 4Cp ∂r where Fconv and Frad aretheconvectiveandradiativeheatflux, and respectively,ρthemassdensity,u¯ themeangasvelocity,T the l gastemperature.S isthespecificentropy, τ = m (12) c u c p S =C log , (3) wherel is the mixinglength.Equation(9) describesa strictly v ργ! m radial,diffusiveheatflux.Inarotatingconvectionzonetheheat where pisthegaspressureandC thespecificheatatconstant fluxisnotstrictlyalignedwiththeentropygradient,i.e. v volumeandγtheadiabateindex.Insphericalpolarcoordinates ∂S theReynoldsequationcanberewrittenasasystemoftwopartial Fconv =−ρTχΦ (13) differentialequations.TheazimuthalcomponentoftheReynolds i t ij∂xj equationexpressestheconservationofangularmomentum, (Ru¨diger 1989, Kitchatinovetal.1994), where the dimension- ∇·t =0, (4) less coefficientsΦ are functionsof the Coriolis numberΩ∗ = ij M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars 3 2τ Ω fulfilling |Φ | ≤ 1. The flux vector is then tilted towards where L isthe stellar luminosityandr theradiusatwhichthe c ij b therotationaxis. boundaryislocated.Astheradiativeheatfluxisprescribedand Theradiativeheatfluxisgivenby wedefinethebottomoftheconvectionzoneasthepointwhere theradiativeheatfluxequalsthetotalheatfluxthisconditionis 16σT3∇T Frad =− i , (14) equivalenttoimposingzeroconvectiveheatflux. i 3κρ At the upper boundary, we assume that the convective heat flux equals the radiative flux is converted into radiation whereσistheStefan-Boltzmannconstantandκtheopacity. (Gilman&Glatzmaier1981), which is expressed through the linearizedcondition 2.3.Backgroundmodel L S F = (1+4 ), (23) Weassumethatthestarisessentiallyinhydrostaticequilibrium r 4πrt2 Cp and that all gas motions constitute small perturbations of that wherer istheradiusofthatboundary.Theboundaryconditions equilibrium which do not change the structure of the star. We t ontherotationaxisareimpliedbytherequirementsforaxisym- particularlyassumethatnonetgastransportoccurs: metry. ∇·(ρu¯)=0 (15) Tosolvetheabovesystemanexpansionintermsofspherical harmonicsisusedforthedependenceonlatitude.Forthespecific Our model convection zone is a spherical layer with adiabatic entropy, the angular velocity and the stream function ψ of the stratificationthatisheatedfrombelowandcooledatthesurface. meridionalflow, Weassumeanidealgaswiththeequationsofstate, 1 ∂ψ 1 ∂ψ u = , u =− , (24) R γ−1 r ρr2sinθ ∂θ θ ρrsinθ ∂r p= ρT =C ρT (16) µ p γ wehave (p is the gaspressure,R the gasconstant,Cp the heatcapacity N forconstantpressure)andhydrostaticequilibrium, S(r,θ) = S (r)P (cosθ), n 2(n−1) dp Xn=1 dr =−gρ. (17) Ω(r,θ) = N Ω (r)P12n−1(cosθ), (25) n sinθ Fromthelawofgravity, Xn=1 N GM(r) 1 g=− , (18) ψ(r,θ) = ψ (r)P (cosθ)sinθ. r2 n 2n Xn=1 (G gravity constant, M(r) the mass contained in the sphere of 1 radiusr),wederive Inthisequation,Pl and Pl arethenormalizedstandardandad- jointLegendrepolynomials.Theexpansionsareconvenientbe- dg(r) g(r) causethefunctionsofcolatitudeθontherighthandsidearethe =−2 +4πGρ. (19) dr r eigenfunctionsof the angular parts of the correspondingdiffu- sionoperators.InusingonlyLegendrepolynomialsofonlyeven For adiabatic stratification, pρ−γ = const., the density can be orodddegreeinacertainexpansionweassumesymmetrywith expressedintermsofthetemperature, respecttotheequator.Theexpansion(25)usuallyconvergesfast T 1/(γ−1) such that N = 5 is sufficient for cases like the solar rotation. ρ=ρ , (20) Fasterrotatingstarsrequirevaluesupto20. e T ! e Applyingtheexpansions(25)tothesystemofpartialdiffer- where ρ and T are the valuesof density and temperatureat a entialequations(2),(4),and(6)transformstheequationsintoa e e referenceradiusr .Theconditionofhydrostaticequilibriumcan system of ordinarynonlineardifferentialequationswith the in- e thenberewrittenintermsofthetemperature,i.e. dependentvariabler.Wefurtherreducetheequationstothesys- tem of first-orderdifferentialequationsby introducingnew de- dT g(r) pendentvariables.Inparticular,theReynoldsstresscomponents =− . (21) dr C R andR areconvenientnewdependentvariables.Theresult- p rθ rφ ing system of first-order differential equations is solved by the Equations (19), (20), and (21) form a system that can be inte- standardrelaxationmethodasdescribedbyPressetal.(1992). grated from the reference radius, where the values of density, The rapidly rotating stars have very thin boundary layers gravity, and temperatureare known from the full stellar evolu- nearthetopandbottomoftheirconvectionzones.Toresolvethe tionmodel. layers,nonuniformnumericalgrid(zerosofChebyshevpolyno- mials)isapplied 2.4.Boundaryconditionsandnumericalmethod 1 i−3/2 r = r +r −(r −r )cos π , (26) Asboundaryconditionsweassumestress-freeandimpenetrable i 2 t b t b n−2 !! boundariesfor the gasmotionand an imposedradialheatflux. 2≤i≤n−1, r =r , r =r , 1 b n t At the lower boundarythe heat flux is constant:corresponding tothestellarluminosityfortheheattransport, where n is the number of radial grid points. The grid has fine spacingneartheboundaries. Tosolveforthesolarrotationlaw, L avalueofnassmallas20issufficient,butweusuallyusehigher F = (22) r 4πr2 resolutions(e.g.n=200),especiallyforfastrotation. b 4 M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars 3. TheSun We first simulate the solar rotation law and compare the re- sults to those from earlier models by Kitchatinov & Ru¨diger (1999) and Ku¨ker & Stix (2001) solving the same set of equations with different numerical methods. KR99 used a simple Kramers’ law for the opacity while KS01 used the opacity from Ahrens et al. (1992). Now an analyti- cal opacity law after Stellingwerf(1975) as distributed with Hansen&Kawaler(1994) is used. A more elaborate treatment is possible but not likely to improve the model as we do not treattheuppermostlayersoftheconvectionzoneandtheatmo- Fig.2. Temperaturedeviationas functionof the latitude at the sphere.Asthesolarconvectionzoneisnotfullyionizeduptothe bottom (left) and top (right) of the solar convection zone. The photosphere, the adiabatic gradient, ∇ = ∂logT/∂logP , ad S polar axis is always warmer than the equator. The differences, and hence the specific heat capacity are not constant. While (cid:0) (cid:1) however,aresmall. ∇ =0.4holdsthroughoutmostoftheconvectionzone,itdrops ad tovaluesaslowas0.1intheupper35,000km.Asweworkwith constantvaluesweeitherhavetoexcludetheuppermostlayeror betweentherotationratesoftheequatorandthepolarcaps,cor- adjust ∇ to reproduce the depth of the convection zone from respondingtoarelativeshear,δΩ/Ω ,of29percent.Thevari- ad eq thestellarevolutionmodel. ation with radius at midlatitudes is weak. The angular velocity decreases with radius at high latitudes while at the equator it slightlyincreaseswithradius(‘superrotation’)in thedeepcon- vectionzone.Inthesurfacelayerthereisanegativegradientof Ω(‘subrotation’)atalllatitudes.Liketherotationprofilescom- putedwiththeKR99andKS01schemes,thisrotationprofileis inexcellentagreementwiththefindingsofhelioseismology. ThebottompanelsofFig.1showthemeridionalflow.There isonecellperhemispherewiththesurfaceflowdirectedtowards the poles and the return flow at the bottom of the convection zone.The amplitudeof this‘counter-clockwise’flow is 14 m/s at the (model) surface and 6 m/s at the bottom. The difference between the flow speeds at the top and bottom respectively is smaller than might be expected from the density stratification andtherequirementofmassconservationbuttheconcentration of the return flow to a thin layer allows a relatively fast flow despitethelargermassdensity. Figure2showsthequantity T δT =(S −S ) , (28) 0 C p at the top and the bottom of the solar convectionzone with S 0 thespecificentropyatthebottomoftheconvectionzoneatthe equator.ThischoiceofS0impliesnegativevaluesofδT atthe topoftheconvectionzone.Ingeneral,thepolaraxisiswarmer thanthemoreequatorwardpartsofthefluid.Unfortunately,the Fig.1. The solar differential rotation and meridional flow as smallnessofthetemperaturedifferencedoesnotallowempirical computed with the new scheme. Top left: Surface rotation rate confirmations.Thoughthe deviationfromthe adiabaticstratifi- vs.co-latitude.Topright:rotationrateasfunctionofradiusat0, cationisonlysmall,theresultingbaroclinetermhasaprofound 15,30,45,60,75,and90◦latitude,respectively,fromtoptobot- impactonthelarge-scalemeridionalflowandthedifferentialro- tom. Bottom left:Streamlinesof the meridionalflow. The flow tation(seebelow). iscounter-clockwise,i.e.directedtowardsthepoleatthesurface andtowardstheequatoratthebottom.Bottomright:Flowspeed 3.1.Fast-rotatingSun asfunctionoflatitudeatthetop(solidblue)andbottom(dashed red)oftheconvectionzone.Positivevaluesindicateaflowaway To study the impactof the basic rotationwe computethe rota- fromthenorthpole. tion law of a hypotheticalfast-rotating Sun with the P = 1.33 dayrotationperiodofHD171488.Figure3showstheresulting rotation pattern. The isocontoursare cylinder-shapedin accord ThetoppanelsinFig.1showtherotationpatternintheso- to the Taylor-Proudmantheorem.Consequently,the radialpro- larconvectionzone.Itagreeswellwiththosecomputedwiththe files in the right diagram show an increase of the rotation rate KR99 codes and the KS01 scheme with the modifications de- with increasing radius at low latitudes. Unlike the case of the scribedinBonannoetal. (2007).Thesurfacerotationisfastest real Sun, there is a pronounced increase in the upper part of attheequatorwithadifference the convection zone at the equator. At both radial boundaries there are pronounced boundary layers with huge gradients of δΩ=0.07rad/day (27) the rotation rate which are caused by the stress-free boundary M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars 5 conditions.Thesurfacerotationisfastestattheequatorandde- 0.69R . Forthis modelstar we find a surface differentialrota- ⋆ creases monotonously towards the polar caps but the slope of tion δΩ = 0.51rad/dayork = 0.10.Thisis in agreementwith the decline has a minimum at mid-latitudes. With a value of theobservedvalue. 0.08rad/dayinsteadof0.07rad/daythehorizontalshearisonly For κ1 Cet we use solar mass and metalicity and an age of slightlystrongerthanfortheSundespiteafactorof20between 1 Gyr. The modelstar has an effective temperatureof 5677 K, theaveragerotationrates. 0.915 R and 0.78 L . The resulting rotation law shows a sur- ⊙ ⊙ The meridionalflow only hasone flow cell per hemisphere face shear δΩ = 0.077 rad/day, or k = 0.12. This is slightly withthesurfaceflowdirectedtowardsthepoles.Theamplitudes largerthantheobservedvalue.Anearliermodelfromthesame are 34 m/s at the surface and 17 m/s at the bottom of the con- evolutionary track, with a stellar age of 167 Myr, an effective vectionzone.AsfortherealSunwithitsmuchslowerrotation, temperatureof5649K,aradiusof0.9R ,andanluminosityof ⊙ thereturnflowis halfasfastasthesurfaceflowbutevenmore 0.74 L yields k = 0.11, or δΩ = 0.072 rad/day. This is still ⊙ concentratedat the bottom of the convectionzone. The flow is notinperfectagreementwiththeobservedvaluebutreasonably fastestatmid-latitudes. close. Possiblereasonsfortheremainingdiscrepancyarethestellar model,whichmightnotbeasufficientlyaccuraterepresentation oftherealstarandanunderestimationofthetotalsurfaceshear bythesin2θderivedfromspotrotationastheobservedspotsdo notcoverthewholerangeoflatitudesfrompoletoequator. 4. Numericalexperiments Wecarryoutseveralnumericalexperimentstocomparetheroles playedbythetwodriversofdifferentialrotation,i.e.theΛeffect andthebaroclineflow. Fig.3. Rotation of a hypothetical fast-rotating Sun with P = 4.1.SunwithoutΛeffect 1.33d. Ourmodelincludestwoeffectsthatarecapabletomaintaindif- ferentialrotation,theΛeffectandthebaroclineflowasduetoa horizontaltemperaturegradient.Thiscanbeseenwhenthebaro- clineterminEq.(6)isrewrittenintermsofthetemperature, 3.2.MOSTstars:ǫEriandκ1Cet 1 g ∂δT ǫEriandκ1Cetareactivedwarfstarsforwhichsurfacedifferen- (∇ρ×∇p) ≈− . (31) ρ2 φ rT ∂θ tialrotationhasbeenderivedfromlightcurvesrecordedbythe MOSTsatellite Crolletal. (2006)deriveda rotationlawofthe Thebaroclinetermcanbeapowerfuldriverofmeridionalflows, form which in turn can drive differential rotation. In our theory the P latitudinal temperature profile is due to the anisotropic heat- eq P(B)= (29) conductivitytensorinthepresenceofrotation.Alwaysthepolar 1−ksin2B axisisslightlywarmerthantheequatorbutwithanunobservable fortheK2Vstarǫ Eri,where P(B)istherotationperiodatthe temperatureexcess.A positivepole-equatortemperaturediffer- latitudeB,andPeqtherotationperiodattheequator(B=0).The encewilldriveaclockwiseflow.Itsangularmomentumtransport parameterkgivestheshearoftherotationlaw.Thegeneralrule leadstoanacceleratedrotationatlowlatitudesandtoslowerro- derivedfromthetheoryofthestellarrotationlawsofKitchatinov tation atthe polarcaps. A baroclineflow (also ‘thermalwind’) &Ru¨diger(1999) can therefore maintain differential rotation with solar-type sur- face rotation.For an illustration we repeatour computationfor P k≃ eq (30) theSunwiththeΛtermscanceledwithintheReynoldsstress. 100days The resulting rotation and flow patterns are shown in Fig. 4. Therotationisindeedsolar-typebutwithδΩ = 0.04weaker isperfectlyfulfilledbythesestars. thanwithΛeffectandtheisocontoursaredistinctlydisc-shaped An equatorial rotation period P = 11.2 days and a value eq atthepoleseveninthemidlatitudes.Theequatorregionshows k = 0.11+0.03 for the differentialrotation has been reported for −0.02 basicallynostructure. ǫ Eri corresponding with δΩ = 0.062 rad/day. For the G5 V The typical superrotation of the deep convection zone be- starκ1 CetWalkeretal.(2007)foundP = 8.77daysandk = eq neaththeequatorknownasaresultofthehelioseismologydoes 0.09+0.006correspondingwithδΩ=0.064rad/day. −0.005 notoccur. The reason is simple: if only a (fast) meridionalcir- Kitchatinov&Olemskoy(2010)computedrotationlawsfor culation transports the angular momentumwith a pattern sym- modelstarswith 0.8 and1.0solar massesfor ǫ Eriand κ1 Cet. metric with respect to the equator then the angular momen- Theresultswerek =0.127forǫ Eriandk=0.13forκ1 Cet. tum becomes uniform in the equatorial part of the convection Here the calculations for these stars are based on im- zone (except, of course, the boundary layers). Hence, there is proved stellar models computed with the MESA/STARS code r2Ω ≃ constindependentoftheflowdirectionbutincontradic- (Paxtonetal.2010). A star with M = 0.85M , Z = 0.02 and ⋆ ⊙ tiontotheobservation2. anageof0.44Gyrservesasmodelforǫ Eri.Ithasaneffective temperatureof5076K,aradiusof0.76R⊙,andaluminosityof 2 without Λ effect a superrotation beneath the equator can only be 0.34 L⊙. The bottomof the outer convectionzone is located at duetoananticlockwiseflowwhichissufficientlyslow 6 M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars With amplitudes of 4.7 m/s at the top and 2.1 m/s at the bottomoftheconvectionzonethebarocline-inducedclockwise meridional flow is substantially weaker than in the complete model–anditgoesinthe‘wrong’direction. Fig.5. Solarrotationlawwithoutbaroclinicterms.Thestructure disappearsnowatthepolaraxisbutitappearsattheequatorial region. The equatorial plane shows superrotation but the polar axisrotatesalmostrigid. 5. YoungGdwarfs 5.1.CoRoT-2a ThisG7VstarisayoungSun,i.e.ithassolarmassbutismuch younger with an age of 0.5 Gyrs. The star has been observed by the CoRoT satellite and found to have a planet with an or- bital period of 1.743 days(Alonso et al. 2008).Besides plane- tarytransits,thelightcurveshowsperiodicvariationthatismost easilyexplainedbyarotating,spottedsurfacewithbasicrotation periodof4.5 days.Spotmodelingusing circularspotsfindsan Fig.4. SolarrotationlawandmeridionalflowwithoutΛeffect. excellent fit assuming three spots and a solar-type surface dif- Top: The rotation law. Note the steep negative gradient of the ferentialrotation with a difference of 0.11 rad/day (Fro¨hlichet angular velocity at the poles and the almost rigid rotation be- al.2009). neath the equator.Bottom:The meridionalflow is anti-solar:it ispositive(equatorwards)atthetopoftheconvectionzoneand negative(polwards)atthebottomoftheconvectionzone. 4.2.Sunwithoutbaroclineflow Nextthe barocline term is canceled while keeping the Λ effect whichmaintainsthedifferentialrotationtogetherwiththemerid- ionalflowcausedbytheformerthroughthesecondtermonthe LHSofEq.(6)(‘Biermann-Kippenhahnflow’,seeKippenhahn Fig.6. RotationpatternofCoRoT-2a,ayoungSunrotatingwith 1963;Ko¨hler1970). Figure5showstheresultingrotationpat- P=4.5days. tern. The differential rotation is solar-type, i.e. the rotation pe- riodisshortestattheequatorandlongestatthepolarcaps.With δΩ=0.014rad/daythesurfacerotationismuchmorerigidthan Our model star is based on a model from an evolutionary observed.Theisocontourplotshowsadistinctlycylinder-shaped trackfortheSun.Theageis0.5Gyrs,theradius0.9solarradii, pattern in the bulk of the convectionzone while at the top and andtheluminosity75percentofthesolarvalue.Thebottomof bottomboundariesthepatterndeviatesfromthecylindergeom- theconvectionzoneisatafractionalradiusx=0.73. etry and the rotation rate falls off with increasing radius at all latitudes.Thecounterclockwisemeridionalflowhasaverysim- Figure6showstheresultingrotationpattern.Thesurfacedif- ilargeometryasinthefullmodelbutitisfasterwithamplitudes ferentialrotationof0.09rad/dayreproducesthevalueobserved of17and8m/satthetopandbottom. byFro¨hlichetal.(2009).Therotationpatternismorecylindrical thanthatof theSun, butlessthan thatofourfast-rotatingSun. Thesubrotationalongthepolaraxiswhichistypicalforthe Similarly,theboundarylayersaremorepronouncedthaninthe solar rotation law and which is produced by baroclinic flows Sun but less than for the fast-rotating Sun. The flow pattern is (cf. Fig. 4) does not exist here. Ko¨hler (1970) and Ru¨diger et similar to that in the solar convection zone. The amplitude is al. (1998)havegivenseveralnumericalexamplesof thisdirect slightlylargerwithamaximumvalueof18.6m/satthetopand consequenceoftheTaylor-Proudmantheorem. 10.6m/satthebottomoftheconvectionzone. M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars 7 5.2. R58,LQLupandHD171488 Wenowaddresstherecentdifferentialrotationmeasurementsof the fast-rotating G dwarfs R58, LQ Lup and HD 171488. R58 (HD 307938)is a youngactive G dwarf in the open cluster IC 2602. It has a photospherictemperature of 5800 K and rotates with a period of 0.56 days(Marsdenetal.2005). LQ Lup (RX J1508.6-4423) is a post-T Tauri star with an effective temper- ature of 5750±50 K and a rotation periodof 0.31days, anda radiusof1.22±0.12solarradii(Donatietal.2000),correspond- ingtoamassof1.16±0.04solarmasses,andanageof25±10 Myr.Theequator-poleΩ-differenceis0.12rad/day.HD171488 (V889Her)isayoungactiveGdwarf.Strassmeieretal.(2003) found a photospherictemperatureof 5830 K, a rotation period of 1.337 days, a mass of 1.06±0.05 solar masses, a radius of 1.09± 0.05 solar radii, and an age of 30–50 Myr. Marsden et al. (2006) find a rotation period of 1.313 days, a photospheric temperatureof5800K,andaradiusof1.15±0.08solarradii. With 0.4–0.5 rad/day the equator-pole Ω-difference of HD171488isexceptionallylarge.Thek-valueof0.10missesthe k =0.013predictedbyEq.(30)byafactorof7.5.Asthecorre- spondingfactorsforR58andLQLuparemuchsmaller(factor 2–3)the followingcalculationstake the case of HD 171488as Fig.7. InternalrotationofyoungGdwarfswithrotationperiod the main example. As the stars have similar radii and effective of1.33days.Top:deepconvectionzone,δΩ = 0.11rad/dayat temperaturesweinvestigatewhatimpactdifferencesintheinter- thesurface.Bottom:shallowconvectionzone.δΩ=0.5rad/day nalstructurehaveonthesurfacerotationpatterns.Westudytwo atthesurface. modelsthatmainlydifferinthemetalicityand,asaconsequence, the depth of the convection zone. The models were computed with the MESA/STARS code.We use the metalicity as control shearofδΩ=0.5rad/day.Thecontourplotshowsadisc-shaped parameter for the depth of the convection zone and vary mass pattern with flat profiles at low latitudes and a decrease of the andmixinglengthparametertoadjustradiusandeffectivetem- rotationratewithincreasingradiusatthepolarcaps.Theradial peraturetovaluesclosetothoseobservedforthethreeyoungG profiles show pronouncedboundarylayersat the radialbound- dwarfs.Therotationperiodisthe1.33dayperiodofHD171488. aries. The flow pattern is mostly solar-type. There is one large Bothmodelstarsare30Myrold. flowcellwithpolewardflowatthetopandthereturnflowatthe bottomoftheconvectionzone.Inaddition,thereisasmallcell of clockwise flow at low latitudes. The flow speeds at the top 5.2.1. Deepconvectionzone reach 18 m/s and 2.2 m/s, respectively. The return flow at the ThefirstmodelstarhasthehighmetalicityofZ =0.03.Amass bottomoftheconvectionzonereaches3.5m/s. of 1.11 solar masses and a value of 1.6 for the mixing length parameterlead to an effective temperatureof 5685K and a ra- 5.2.3. Truncatedconvectionzone dius of 1.14 solar radii. The bottom of the convection zone is located at x = 0.765 and has a temperature of 1.46 × 106 K. Themodelsabovehave(roughly)thesameeffectivetemperature The rotation law is shown in the top part of Fig. 7. The iso- butdifferinmassandstructure.Tofurtherisolatetheeffectofthe contourplotshowscylinder-shapedcontourssimilartothosefor depth of the convectionzone, we compute a modelof a young thefast-rotatingSun.Thesurfacerotationlawissolar-typewith solarmassstarandartificiallyreducethedepthoftheconvection a fast-rotatingequator. The surface shear is much stronger. We zonebyimposingthelowerboundaryatafractionalradius x = findavalueof0.11rad/day,about1.5timesthesolarvalue.The 0.88insteadof x = 0.78andcomparetherotationpatternwith radialprofilesshowsuperrotationbeneaththe equatorandsub- thatof thefullmodel.Inbothcases the topboundaryis at x = rotation along the rotation axis. The boundarylayers are much 0.98. The truncated convectionzone thus has half the depth of less pronounced and the surface layer resembles the real Sun, the full model. In both cases the rotation law is solar-type but i.e. the rotation rate decreases with increasing radius at all lat- whilethefullconvectionzoneproducesasurfaceshearof0.12 itudes. Themeridionalflow isof solar-type(counterclockwise) rad/day,thetruncatedconvectionzonehas0.32rad/day.Thefull with the surface flow towards the poles and the return flow lo- modelhasatemperaturedifferenceof24Katthetopand124K catedatthebottomofthe convectionzone.Theamplitudesare atthebottom.Incaseofthetruncatedmodelthepolarcapis47 23and14m/s,respectively. Khotterthantheequatoratthetopand254Katthebottom. 5.2.2. Shallowconvectionzone 5.2.4. Veryfastrotation Oursecondmodelstarhasametalicityof0.01.Amassof1.08 Toillustrate theimpactofthe rotationrate,we repeatthe com- solarmassesandamixinglengthparameterof1.0leadtoara- putationfortheshallowconvectionzonemodelwithrotationpe- diusof1.14solarradiiandaneffectivetemperatureof5750K. riodof0.33days.Figure8showstheresultingrotationpattern. Thebottomoftheconvectionzoneliesatafractionalradiusof Theleftplotshowssolar-typerotationbothinthesurfacediffer- 0.89andatemperatureof6×105 K.ThebottompartofFig.7 entialrotationandinthepredominantlyradialisocontours.This showstheresultingrotationlaw.Thesurfacerotationhasatotal is still not cylinder-shaped but closer to the Taylor-Proudman 8 M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars thebottomoftheconvectivezonethedifferencesare750Kand 100Kforaconvectionzoneofsmallorlargedepth,respectively. Thecorrespondingvaluesatthetopare55Kand19K. 6.2.Thermalwindequilibrium We always find a higher temperature at high latitudes than at the equator. This is a consequence of the tilt of the convective heattransportvectortowardstheaxisof rotationcausedbythe Coriolisforce.Insphericalpolarcoordinatesthecomponentsof theheatfluxread Fig.8. DifferentialrotationoftheGdwarfwiththeshallowcon- ∂S ∂S F = −ρTχ −ρTχ , (32) vectionzoneandarotationperiodof0.33days.Left:isocontours r rr ∂r rθ∂θ oftheangularvelocity.Right:therotationprofilesatvariouslat- ∂S ∂S itudes. Fθ = −ρTχθr ∂r −ρTχθθ∂θ. (33) The first term in the horizontal component precludes a purely than the disc-shaped pattern we found for a rotation period of radial stratification. Any variation of the specific entropy with 1.33days.Thesurfaceshearis0.38rad/day,thetemperaturedif- the radiuswill cause a horizontalheat flux and thus build up a ferencebetweenpolarcapsandequatoris1785Katthebottom horizontalgradient. This case is profoundlydifferentfrom that and 184K at the top. The maximumflow speedsare 14 m/sat ofalatitude-dependentbutstillpurelyradialheatflux.Thelatter thetopand8m/satthebottomoftheconvectionzone. wouldhavea muchsmallerimpactandwouldresultina much weakerdifferentialrotation(Ru¨digeretal.2005). The impact on the maintenance of differential rotation can 6. Discussion be seen from the equation for the meridionalflow. For fast ro- 6.1.Shallowvs.deepconvectionzone tation (i.e. large Taylor number)Reynoldsstress and nonlinear termscanbeneglectedandEq.(6)isreducedtothethermalwind Our“shallow”and“deep”convectionzonemodelshavesimilar equation radiiand effectivetemperaturesbutthestars differin massand metalicity.Wehavealsochosendifferentvaluesforthemixing- ∂Ω g ∂δT length parameter.As a result, the depthof the convectionzone 2rsinθΩ − =0. (34) ∂z rT ∂θ differsbetweenthemodels.Wefindverydifferentresultsforthe differentialrotation of these convectionzones. The modelwith It followsthata gradientof the angularvelocity alongthe axis ashallowconvectionzoneshowsmuchstrongerdifferentialro- ofrotationisneededtobalancethehorizontaltemperaturegra- tationthantheoneswithdeeperconvectionzones.Our“shallow dient. Hence,a disc-shapedrotationpatternresults in the polar convection zone” model does indeed reproduce the very large area(only)duetotheactionofthebaroclinicflow.Forwarmer valueof0.5rad/dayfoundbyJeffers&Donati(2009)whilethe poles(δT sinksequatorwards)thez-gradientofΩisnegativeas “deepconvectionzone”modelwithits smaller valueof 0.11is observed.Themeridionalflowwithoutthebarocliniccomponent in rough agreement with the values observed for LQ Lup and onlyyieldsdΩ/dz≃0(Ko¨hler1970,seealsoFig.5).Hence,the R58. empirical finding of negativedΩ/dz at the polar axis by helio- The result from the shallow convection zone model is in seismologyprovestheexistenceofbaroclinicflowsinthesolar line with previous findings on F stars. Extreme values of sur- convectionzone. faceshearhavebeenobservedforFstarsbyReiners(2006)and AswehavedemonstratedbymeansofFig.4thesuperrota- foundintheoreticalmodelsforstarswithveryshallowconvec- tionbeneaththesolarequatorisadirectindicationfortheaction tion zones by Ku¨ker & Ru¨diger (2007). In the latter study, the oftheReynoldsstressintheconvectionzone.Allmodelswith- largest values of surface shear are found for the most massive outΛ-effectbutwithanclockwise(equatorwardatthesurface) starsstudied.Thesestars arenotonlythe hottest,theirconvec- circulationleadtodΩ/dr<0inthebulkoftheconvectionzone ∼ tionzonesarealsoveryshallow. beneath close to the equator. Such clockwise flows are able to To achieve the thin convection zone we had to lower the acceleratethe equatorcomparedwith thepolesbuttheycannot mixing-lengthparametertoa valueof1.0.Thisisnotonlydif- producethesuperrotationbeneaththeequator. ferentfromthevalueusedforthe“deep”convectionzonemodel Figure 9 shows results from a detailed computation of the but much lower than what is usually used in stellar evolution terms on the LHS of Eq. (34) and their sum vs. the fractional models.Hence,themodelisprobablynotrealisticanddoesnot radiusforalatitudeof45◦.Theleftpanelshowstheplotforthe directlyapplytoLQ Lup.Notbeingspecialistsin thefield, we Sun, the rightpanelthat for the deep convectionzone G dwarf leavethequestionwhetherashallowconvectionzoneispossible model.ThesamequantitieswerecomputedforthefastSunand open but we wonder if strong magnetic fields could inhibit the theshallowconvectionzoneGdwarfmodels. convective heat transport in a way that might be mimicked by FortherealSunthetwotermscancelinthebulkofthesolar thischoiceofparameter. convectionzonewherethemeridionalflow isslowandsmooth The“truncatedconvectionzone”modelavoidstheuncertain- but not close to the boundaries. For the deep convection zone tiesaboutthe”shallowconvectionzone”modelandshowseven the boundarylayersare muchthinnerbutalso much morepro- moreclearlytheimpactthedepthoftheconvectionzonehason nounced.Thesefindingsindicatethatforfastrotationthemerid- thesurfacerotation.Boththeverticalandhorizontaltemperature ionalflowismainlydrivenbytheboundarylayerswhilethebulk gradientsaresteepestforthemodelwiththeshallowconvection of the convection zone is in thermal wind equilibrium. In the zoneandflattestforthemodelwiththedeepconvectionzone.At boundary layers the meridional circulation is strongly sheared M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars 9 so that the RHS of Eq. (34) (which consists on 3rd derivatives depthoftheconvectionzonehasabigimpactonstellarrotation. ofthemeridionalvelocitycomponentsmultipliedwiththeeddy Shallowconvectionzonesproducestrongersurfaceshear. viscosity)becomeslarge. The presented mean-field theory for G stars naturally ex- plainstherotationlawsoftheSun,oftheMOSTstarsǫ Eriand κ1Cet,andofsuchfast-rotatingstarslikeR58andLQLup.The Drivers of meridional flow Drivers of meridional flow discrepancy between these stars and the much stronger differ- 0 ential rotation observed for HD 171488 – if real – is hard to 0.0 explain for stars of similar age and spectral type. If, however, -50 -0.2 HD 171488 had a shallower convectionzone for some reason, itsstrongsurfacedifferentialrotationfollowsimmediately. -100 -0.4 During the pre-MS evolution the convection zone retreats -150 from the central region and the originally fully convective star -0.6 formsaradiativezonearoundthecore.Thischangeinthedepth 0.75 0.80 0.85 0.90 0.95 0.80 0.85 0.90 0.95 oftheconvectionzoneshouldbereflectedbythesurfacediffer- Fractional radius Fractional radius entialrotation. Acknowledgements. This work has been funded by the Deutsche Fig.9. Thedriversofangularmomentumvs.fractionalradiusat Forschungsgemeinschaft (RU 488/21). LLK is thankful to the support 45◦latitudefortheSun(left,slowrotation)andthedeepconvec- by the Russian Foundation for Basic Research (projects 10-02-00148, tionzoneG dwarf(right,fastrotation).Dashedline:baroclinic 10-02-00391). term. Dash-dotted line: centrifugalterm. Solid line: both. Note that the balance (34) well holds in the bulk of the convection zones of fast rotation stars where the meridional circulation is References slow and smooth.Equation(34) doesnothold in the boundary Ahrens,B.Stix,M.,Thorn,M.1992,A&A,264,673 areaswheretheflowisfastandshearing. Alonso,R.,Auvergne,M.,Baglin,A.,etal.2008,A&A,482,L21 Barnes,J.R.,CollierCameron,A.,James,D.J.,&Donati,J.-F.2000,MNRAS, 314,162 Thenumericalexperimentsshowthatthebaroclinictermis Barnes,J.R.,CollierCameron,A.,Donati,J.-F.,etal.2005,MNRAS,357,L1 an importantbut notthe sole driverof the solar differentialro- Bonanno,A.,Ku¨ker,M.,&Paterno`,L.2007,A&A,462,1031 CollierCameron,A.,&Donati,J.-F.2002,MNRAS,329,L23 tation. Cancellation of the Λ effect leaves a surface shear of Croll,B.,Walker,G.A.H.,Kuschnig,R.,etal.2006,ApJ,648,607 0.04 rad/daywhile switching off the baroclinic term reduces it Donahue,R.A.,Saar,S.H.,&Baliunas,S.L.1996,ApJ,466,384 to0.014rad/day.ThisisnotbecausetheΛeffectisaninefficient Donati,J.-F.,Mengel,M.,Carter,B.D.,etal. 2000,MNRAS,316,699 transporterofangularmomentumbutbecauseitisbalancedby Fro¨hlich,H.-E.,Ku¨ker,M.,Hatzes,A.P.,&Strassmeier,K.G.2009,A&A,506, 263 themeridionalflow.Cancelingthemeridionalflowaltogetherin- Gilman,P.A.,&Glatzmaier,G.A.1981,ApJS,45,335 creases the surface shear to 0.18rad/day.This is a well-known Hall, D.S. 1991, The Sun and cool stars, Activity, magnetism, dynamos, effect:meridionalflowsdrivenbydifferentialrotationacttore- Springer,p.95 duceit(Ru¨digeretal.1998).Thebaroclinicflowhastheoppo- Hansen,C.J.,&Kawaler,S.D.1994,StellarInteriors,SpringerVerlag sitedirectionandbuildsupthedifferentialrotationalongthepo- Henry,G.W.,Eaton,J.A.,Hamer,J.,&Hall,D.S.1995,ApJ,97,513 Huber,K.F.,Wolter,U.,Czesla,S.etal. 2009,A&A,501,715 lar axis. The observed superrotation beneath the equator, how- Jeffers,S.V.,&Donati,J.-F.2009a,MNRAS,390,635 ever, cannot be produced by meridional circulation but should Jeffers,S.V.,&Donati,J.-F.2009b,ASPConferenceSeries,405,523 beadirectconsequenceoftheexistenceoftheΛeffect. Jeffers,S.V.,Donati,J.-F.,Alecian,E.,&Marsden,S.C.2010,MNRAS,inpress Kippenhahn,R.1963,ApJ,137,664 Kitchatinov,L.L.,Pipin,V.V.,&Ru¨digerG.1994,AN,315,157 7. Conclusions Kitchatinov,L.L.,&Ru¨diger,G.1999,A&A,344,911 Kitchatinov,L.L.,&Olemskoy,S.V.2010,subm. Stellar differential rotation is driven by Reynolds stress, by Ko¨hler,H.1970,SolarPhys.13,3 Ku¨ker,M.,&Stix,M.2001,A&A,366,668 thecentrifugal-induced(‘Biermann-Kippenhahn’)flowand(for Ku¨ker,M.,&Ru¨diger,G.2007,AN,328,1050 stratified density) by the baroclinic flow. In our model both Ku¨ker,M.,&Ru¨diger,G.2008,J.Physics,Conf.Series,Vol.118,012029 meridionalcirculationshaveopposite directions.Thebarocline Marsden,S.C.,Waite,I.A.,Carter,B.D.,Donati,J.-F.2005,MNRAS359,2005 flow becomes important for faster rotation as the convective Marsden,S.C.,Donati,J.-F.,&Semel,M.,etal.2006,MNRAS,370,468 Messina,S.,&Guinan,E.2003,A&A,409,1017 heat-flux deviates from the radial direction. This deviation can Paxton,B.,Bildsten,L.,Dotter,A.,etal.2010,arXiv:1009.1622v1 beinterpretedasatiltoftheheat-fluxvectortowardstherotation Press,W.H.,Teukolski,S.A.,Vetterling,W.T.,&Flannery,B.P.1992,Numerical axis and causes the poles to be slightly warmer than the lower RecipesinFORTRAN,2nded.,CambridgeUniversityPress latitudes.The baroclinicflow is responsibleforthe strongneg- Reiners,A.,&Schmitt,J.H.M.M.2003,A&A,412,813 ative gradient dΩ/dz along the rotation axis, and the Reynolds Reiners,A.2006,A&A,446,267 Ru¨diger1989,Differential rotationandstellarconvection, GordonandBreach stress producesthe typical positive dΩ/dr along the equatorial SciencePublishers midplanebothknownfromhelioseismology.Asaresultofboth Ru¨diger,G.,v.Rekowski,B.,Donahue,R.A.,&Baliunas,S.L.1998,ApJ,494, impactstheisolinesofΩknownfortheSunarealmostradialin 691 mid-latitudes. Ru¨diger,G.,Egorov,P.,Kitchatinov,L.L.,&Ku¨ker,M.,2005,A&A,431,345 Stellingwerf,R.F.1975,ApJ,195,441 Studyingrealstellar modelsimpliesvaryingabunchofpa- Strassmeier,K.G.,Picghler,Weber,Granzer,T.2003,A&A,411,595 rameters as mass, radius, effective temperature and the depth Thompson,M.J.,Christensen-Dalsgaard, J.,Miesch,M.S.,&Toomre,J.2003, of the convection zone are interdependent. Kitchatinov & ARA&A,2003,599 Olemskoy (2010) studied differential rotation along the lower Ulrich,R.K.,Boyden,J.E.,Webster,L.,etal.1988.Sol.Phys.,117,291 ZAMSforfixedrotationrateandfoundthatthesurfaceshearis Walker,G.A.H.,Croll,B.,Kuschnig,R.,etal.2007,ApJ,659,565 afunctionoftheeffectivetemperaturealone.Usingsomeartifi- cialmodels,wefindthatforthesameeffectivetemperaturethe

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Astronomy&Astrophysicsmanuscriptno.Gstars (cid:13)c ESO2011 January28,2011 The differential rotation of G dwarfs M.Ku¨ker1,G.Ru¨diger1,andL.L.Kitchatinov2,3 1 AstrophysikalischesInstitutPotsdam,AnderSternwarte16,D-14482Potsdam e-mail:[email protected], [email protected] 2 InstituteofSolar-TerrestrialPhysics,POBox291,Irkutsk664033,Russia e-mail:[email protected] 3 PulkovoAstronomicalObservatory,St.Petersburg196140,Russia January28,2011 1 1 ABSTRACT 0 2 AseriesofstellarmodelsofspectraltypeGiscomputedtostudytherotationlawsresultingfrommean-fieldequations.Therotation lawsoftheslowlyrotatingSun,thefastrotatingMOSTstarsǫ Eriandκ1 CetandtherapidrotatorsR58andLQLupcaneasilybe n reproduced.Wealsofindthatdifferencesinthedepthoftheconvectionzonecauselargedifferencesinthesurfacerotationlawand a thattheextremesurfaceshearofHD171488canonlybeexplainedwithaartificiallyshallowconvectionlayer. J Wealsocheckthethermalwindequilibriuminfast-rotatingGdwarfsandfindthatthepolarsubrotation(dΩ/dz < 0)isduetothe 7 baroclineeffectandthattheequatorialsuperrotation(dΩ/dr > 0)isduetotheΛeffectaspartoftheReynoldsstresses.Inthebulk 2 of the convection zones where the meridional flow is slow and smooth the thermal wind equilibrium actually holds between the centrifugalandthepressureforces.Itdoesnothold,however,intheboundingshearlayersincludingtheequatorialregionwherethe ] Reynoldsstressesdominate. R S Keywords.stars:individual:HD171488–stars:interiors–stars:solar-type–stars:rotation . h p 1. Introduction whereδΩisthedifferencebetweentheangularvelocitiesatthe - o equatorandthepolarcaps1.Thepowerlaw(1)wasconfirmed r Many stars show signs of differential rotation. The solar equa- bythefindingsusingspectroscopicmethods(Reiners2006).The t s tor rotates with a shorter period than the polar caps. The dif- lightcurvesofthestarsǫ Eriandκ1 CetrecordedbytheMOST a ference of 132 nHz between the rotation periods found by satellite(Crolletal.2006;Walkeretal.2007)bothindicatewith [ Ulrichetal.(1988)correspondsto a difference of 0.07 rad/day δΩ ≃ 0.062andδΩ ≃ 0.064averysimilarequator-polediffer- 1 between the respective angular velocities or a lapping time of ence of the rotation rate as the Sun. Also the value of 0.11 for v 88 days. Helioseismology has found that this pattern persists the young G star CoRoT-2a (Fro¨hlichetal.2009) well fits the 7 throughout the whole convection zone but not in the radia- common picture of a rotation-independentsurface shear for G 9 tive zone below (Thompsonetal.2003). Stellar differential ro- stars. 2 tation can be inferred from the light curves of rotating spotted It is challenged, however, by recent observations of the 5 stars (see, e.g. Henry et al. 1995; Messina & Guinan 2003), young G dwarfs LQ Lup, R58 and HD171488. Marsden et . 1 frommonitoringmagneticactivity(Donahueetal.1996),spec- al.(2005)reportasurfacedifferentialrotationof0.025±0.015 0 troscopically(Reiners&Schmitt2003),orbyDopplerimaging rad/day for R58 while Jeffers&Donati(2009b) find the much 1 (Barnesetal.2000;Donatietal.2000). largervalue of0.138±0.011rad/day.Donatiet al. (2000)find 1 Whileseveralstudieshavefoundasystematicdependenceof asurfacedifferentialrotationδΩ = 0.12±0.02rad/dayforLQ : v thesurfacedifferentialrotationontherotationrate,nosuchde- Lup.Jeffers&Donati(2009a)determinedthesurfacerotationof i pendenceisfoundwhenthesamplesare combined(Hall1991; HD 171488 using the Zeeman-Dopplerimaging technique and X Barnesetal.2005). Moreover, measurements of surface differ- found a very strong surface differential rotation of 0.5 rad/day. r ential rotation of the rapidly rotating K dwarfs PZ Tel and AB Marsden et al. (2006) report a smaller but still large value of a DorwiththeDopplerimagingtechniqueshowthatstarsrotating 0.402±0.044rad/day.Huberetal.(2009)foundnoevidencefor muchfasterthantheSunshowverysimilarsurfaceshearvalues differentialrotationatallbutcouldnotruleitouteither.Jeffers –asfirstpredictedbyKitchatinov&Ru¨diger(1999).PZ Telis etal.(2010)confirmedthefindingsofMarsdenetal.(2006). ayoungKdwarfwitharotationperiodof0.95days.Itssurface The values found for LQ Lup and R58 are in line with the differentialrotationδΩ = 0.075rad/dayis remarkablyclose to Barneset al. picturebutthe largevaluesfoundfor HD 171488 thatoftheSun(Barnesetal.2000).ABDorisarapidlyrotating are not. The three G dwarfs are similar in their ages, effective K0dwarfwitharotationperiodof0.51days.Itssurfacediffer- temperatures and radii yet HD 171488 shows a much stronger ential rotation was found to vary with time between 0.09 and differential rotation than the other two stars. The studies men- 0.05rad/day(CollierCameron&Donati2002). tionedhavefocusedonrotationrateandeffectivetemperatureas Barnesetal.(2005)proposedadependenceontheeffective the properties determining the surface differential rotation. As temperature(albeitwithlargescatter)andfoundthepowerlaw thestarsareverysimilarbytheirstellarparameterssuchasage 1 thestrengthofthedifferentialrotationisalsoexpressedintermsof δΩ=Ωeq−Ωpole ∝Te8ff.92±0.31 (1) thelappingtimebetweentheequatorandthepoles,Plap =2π/δΩ 2 M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars andeffectivetemperature,weask if therecouldbe adifference where tistheangularmomentumfluxvector in their internal structure that would cause the observeddiffer- ence in their surface rotation. All three stars have just reached ti =rsinθ(ρrsinθΩumi +ρQiφ), (5) thezeroagemainsequenceorareapproachingit.Giventherapid withthetwotransportersofangularmomentumi)themeridional retreatoftheconvectionzoneinthefinalpartofthepre-mainse- flow(um)andii)thezonalfluxoftheturbulentangularmomen- quenceevolutionandtheuncertaintyofthestellarageweaskif tum(Q ). thestrongdifferentialrotationofHD1714888canbeexplained iφ The equation for the meridional flow is derived from the byadifferentdepthofitsouterconvectionzone. Reynolds equation by taking the azimuthal component of its Inthefollowingwecomputemodelconvectionzonesofdif- curl: ferentdepth and their large-scale gasmotions, i.e. rotationand meridionalflow.Themodelsarebasedonthemeanfieldformu- 1 ∂Ω2 1 ∇× ∇·R +rsinθ + (∇ρ×∇p) +...=0 (6) lationoffluiddynamicswhichhasbeenverysuccessfulforthe " ρ # ∂z ρ2 φ Sun,wherethemodelsreproducethesurfacerotation,theinter- φ nalintheconvectionzone,andthesurfacemeridionalflowvery whereR =−ρQ istheReynoldsstressand∂/∂z=cosθ∂/∂r− ij ij well(Kitchatinov&Ru¨diger(1999,KR99),Ku¨ker&Stix(2001, sinθ∂/∂θisthederivativealongtheaxisofrotation.Forveryfast KS01)).Modelshavebeenconstructedforavarietyofstarswith rotationthesecondtermontheLHSofEq.(6)dominatesandthe spectraltypesfromMtoF(Ku¨ker&Ru¨diger2008). rotationrate is constantalongcylindricalsurfaces, as stated by A new scheme allows the computation of stellar rotation theTaylor-Proudmantheorem. lawsandmeridionalflowpatternsbasedonamean-fieldmodel ofthe large-scaleflowsin stellar convectionzonesalsoforfast rotation rates when narrow boundary layers exist. It assumes 2.2.Transportcoefficients strict spherical symmetry for the basic stratification, ignoring Ifsphericalpolarcoordinatesarechosen,theΛeffectappearsin anyflattening thatmightoccurfor veryfastrotation.However, twocomponentsoftheReynoldsstress: theimpactofrotationonthethermalstructurecanbetakeninto accountbyincludingagravitydarkeningtermintheheattrans- Q = Qvisc+Vν sinθΩ (7) rφ rφ t portequationsothatthemodelremainsapplicableevenformod- Q = Qvisc+Hν cosθΩ (8) eratelyflattenedstars. θφ θφ t Here, Qvisc and Qvisc contain only first order derivatives of Ω rφ θφ 2. Theoryofstellarrotationlaws with respect to r and θ and therefore vanish for uniform rota- tion.ThecoefficientsVandHrefertotheverticalandhorizontal While thermal convection is driven by the stratification of the partoftheΛeffectwhileν representstheeddyviscosityinthe star,theconvectivegasmotionscarryangularmomentumaswell t convectionzone.Asthecorrespondingtermscontaintheangu- asheat.Momentumtransportbyturbulentmotionsisknownas larvelocityΩitself,theydonotvanishforrigidrotation.Thus, Reynoldsstressandcanbedescribedasaturbulentviscosityin rigidrotationisnotstress-free. caseofthesimplestshearflows.Inarotating,stratifiedconvec- Forslowrotation,theconvectiveheatfluxisproportionalto tion zone the Reynolds stress is anisotropic and its azimuthal thegradientofthespecificentropy: componentshavea contributionproportionaltotheangularve- locity, Ω, itself rather than its gradient. This form of Reynolds ∂S stress (with ‘Λ effect’) is not compatible with solid body rota- Fconv =−ρTχ , (9) i t∂x tion. i whereχ theturbulentheatconductivity.Theturbulentheatcon- t ductivity is determined by the convection velocity u and the 2.1.Basicequations c convectiveturnovertimeτ : c Our modelconsists of a 1D backgroundmodelwhich assumes 1 hydrostatic equilibrium and spherical symmetry and a system χ = τ u2 (10) ofpartialdifferentialequationsthatdescribetheconvectiveheat t 3 c c flux, the transport of angular momentum, and the meridional Usingstandardmixing-lengththeorywefind flow, assuming axisymmetry. The equation for the convective heattransportthenreads l2g ∂S u2 =− m (11) ∇·(Fconv+Frad)−ρTu¯ ·∇S =0, (2) c 4Cp ∂r where Fconv and Frad aretheconvectiveandradiativeheatflux, and respectively,ρthemassdensity,u¯ themeangasvelocity,T the l gastemperature.S isthespecificentropy, τ = m (12) c u c p S =C log , (3) wherel is the mixinglength.Equation(9) describesa strictly v ργ! m radial,diffusiveheatflux.Inarotatingconvectionzonetheheat where pisthegaspressureandC thespecificheatatconstant fluxisnotstrictlyalignedwiththeentropygradient,i.e. v volumeandγtheadiabateindex.Insphericalpolarcoordinates ∂S theReynoldsequationcanberewrittenasasystemoftwopartial Fconv =−ρTχΦ (13) differentialequations.TheazimuthalcomponentoftheReynolds i t ij∂xj equationexpressestheconservationofangularmomentum, (Ru¨diger 1989, Kitchatinovetal.1994), where the dimension- ∇·t =0, (4) less coefficientsΦ are functionsof the Coriolis numberΩ∗ = ij M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars 3 2τ Ω fulfilling |Φ | ≤ 1. The flux vector is then tilted towards where L isthe stellar luminosityandr theradiusatwhichthe c ij b therotationaxis. boundaryislocated.Astheradiativeheatfluxisprescribedand Theradiativeheatfluxisgivenby wedefinethebottomoftheconvectionzoneasthepointwhere theradiativeheatfluxequalsthetotalheatfluxthisconditionis 16σT3∇T Frad =− i , (14) equivalenttoimposingzeroconvectiveheatflux. i 3κρ At the upper boundary, we assume that the convective heat flux equals the radiative flux is converted into radiation whereσistheStefan-Boltzmannconstantandκtheopacity. (Gilman&Glatzmaier1981), which is expressed through the linearizedcondition 2.3.Backgroundmodel L S F = (1+4 ), (23) Weassumethatthestarisessentiallyinhydrostaticequilibrium r 4πrt2 Cp and that all gas motions constitute small perturbations of that wherer istheradiusofthatboundary.Theboundaryconditions equilibrium which do not change the structure of the star. We t ontherotationaxisareimpliedbytherequirementsforaxisym- particularlyassumethatnonetgastransportoccurs: metry. ∇·(ρu¯)=0 (15) Tosolvetheabovesystemanexpansionintermsofspherical harmonicsisusedforthedependenceonlatitude.Forthespecific Our model convection zone is a spherical layer with adiabatic entropy, the angular velocity and the stream function ψ of the stratificationthatisheatedfrombelowandcooledatthesurface. meridionalflow, Weassumeanidealgaswiththeequationsofstate, 1 ∂ψ 1 ∂ψ u = , u =− , (24) R γ−1 r ρr2sinθ ∂θ θ ρrsinθ ∂r p= ρT =C ρT (16) µ p γ wehave (p is the gaspressure,R the gasconstant,Cp the heatcapacity N forconstantpressure)andhydrostaticequilibrium, S(r,θ) = S (r)P (cosθ), n 2(n−1) dp Xn=1 dr =−gρ. (17) Ω(r,θ) = N Ω (r)P12n−1(cosθ), (25) n sinθ Fromthelawofgravity, Xn=1 N GM(r) 1 g=− , (18) ψ(r,θ) = ψ (r)P (cosθ)sinθ. r2 n 2n Xn=1 (G gravity constant, M(r) the mass contained in the sphere of 1 radiusr),wederive Inthisequation,Pl and Pl arethenormalizedstandardandad- jointLegendrepolynomials.Theexpansionsareconvenientbe- dg(r) g(r) causethefunctionsofcolatitudeθontherighthandsidearethe =−2 +4πGρ. (19) dr r eigenfunctionsof the angular parts of the correspondingdiffu- sionoperators.InusingonlyLegendrepolynomialsofonlyeven For adiabatic stratification, pρ−γ = const., the density can be orodddegreeinacertainexpansionweassumesymmetrywith expressedintermsofthetemperature, respecttotheequator.Theexpansion(25)usuallyconvergesfast T 1/(γ−1) such that N = 5 is sufficient for cases like the solar rotation. ρ=ρ , (20) Fasterrotatingstarsrequirevaluesupto20. e T ! e Applyingtheexpansions(25)tothesystemofpartialdiffer- where ρ and T are the valuesof density and temperatureat a entialequations(2),(4),and(6)transformstheequationsintoa e e referenceradiusr .Theconditionofhydrostaticequilibriumcan system of ordinarynonlineardifferentialequationswith the in- e thenberewrittenintermsofthetemperature,i.e. dependentvariabler.Wefurtherreducetheequationstothesys- tem of first-orderdifferentialequationsby introducingnew de- dT g(r) pendentvariables.Inparticular,theReynoldsstresscomponents =− . (21) dr C R andR areconvenientnewdependentvariables.Theresult- p rθ rφ ing system of first-order differential equations is solved by the Equations (19), (20), and (21) form a system that can be inte- standardrelaxationmethodasdescribedbyPressetal.(1992). grated from the reference radius, where the values of density, The rapidly rotating stars have very thin boundary layers gravity, and temperatureare known from the full stellar evolu- nearthetopandbottomoftheirconvectionzones.Toresolvethe tionmodel. layers,nonuniformnumericalgrid(zerosofChebyshevpolyno- mials)isapplied 2.4.Boundaryconditionsandnumericalmethod 1 i−3/2 r = r +r −(r −r )cos π , (26) Asboundaryconditionsweassumestress-freeandimpenetrable i 2 t b t b n−2 !! boundariesfor the gasmotionand an imposedradialheatflux. 2≤i≤n−1, r =r , r =r , 1 b n t At the lower boundarythe heat flux is constant:corresponding tothestellarluminosityfortheheattransport, where n is the number of radial grid points. The grid has fine spacingneartheboundaries. Tosolveforthesolarrotationlaw, L avalueofnassmallas20issufficient,butweusuallyusehigher F = (22) r 4πr2 resolutions(e.g.n=200),especiallyforfastrotation. b 4 M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars 3. TheSun We first simulate the solar rotation law and compare the re- sults to those from earlier models by Kitchatinov & Ru¨diger (1999) and Ku¨ker & Stix (2001) solving the same set of equations with different numerical methods. KR99 used a simple Kramers’ law for the opacity while KS01 used the opacity from Ahrens et al. (1992). Now an analyti- cal opacity law after Stellingwerf(1975) as distributed with Hansen&Kawaler(1994) is used. A more elaborate treatment is possible but not likely to improve the model as we do not treattheuppermostlayersoftheconvectionzoneandtheatmo- Fig.2. Temperaturedeviationas functionof the latitude at the sphere.Asthesolarconvectionzoneisnotfullyionizeduptothe bottom (left) and top (right) of the solar convection zone. The photosphere, the adiabatic gradient, ∇ = ∂logT/∂logP , ad S polar axis is always warmer than the equator. The differences, and hence the specific heat capacity are not constant. While (cid:0) (cid:1) however,aresmall. ∇ =0.4holdsthroughoutmostoftheconvectionzone,itdrops ad tovaluesaslowas0.1intheupper35,000km.Asweworkwith constantvaluesweeitherhavetoexcludetheuppermostlayeror betweentherotationratesoftheequatorandthepolarcaps,cor- adjust ∇ to reproduce the depth of the convection zone from respondingtoarelativeshear,δΩ/Ω ,of29percent.Thevari- ad eq thestellarevolutionmodel. ation with radius at midlatitudes is weak. The angular velocity decreases with radius at high latitudes while at the equator it slightlyincreaseswithradius(‘superrotation’)in thedeepcon- vectionzone.Inthesurfacelayerthereisanegativegradientof Ω(‘subrotation’)atalllatitudes.Liketherotationprofilescom- putedwiththeKR99andKS01schemes,thisrotationprofileis inexcellentagreementwiththefindingsofhelioseismology. ThebottompanelsofFig.1showthemeridionalflow.There isonecellperhemispherewiththesurfaceflowdirectedtowards the poles and the return flow at the bottom of the convection zone.The amplitudeof this‘counter-clockwise’flow is 14 m/s at the (model) surface and 6 m/s at the bottom. The difference between the flow speeds at the top and bottom respectively is smaller than might be expected from the density stratification andtherequirementofmassconservationbuttheconcentration of the return flow to a thin layer allows a relatively fast flow despitethelargermassdensity. Figure2showsthequantity T δT =(S −S ) , (28) 0 C p at the top and the bottom of the solar convectionzone with S 0 thespecificentropyatthebottomoftheconvectionzoneatthe equator.ThischoiceofS0impliesnegativevaluesofδT atthe topoftheconvectionzone.Ingeneral,thepolaraxisiswarmer thanthemoreequatorwardpartsofthefluid.Unfortunately,the Fig.1. The solar differential rotation and meridional flow as smallnessofthetemperaturedifferencedoesnotallowempirical computed with the new scheme. Top left: Surface rotation rate confirmations.Thoughthe deviationfromthe adiabaticstratifi- vs.co-latitude.Topright:rotationrateasfunctionofradiusat0, cationisonlysmall,theresultingbaroclinetermhasaprofound 15,30,45,60,75,and90◦latitude,respectively,fromtoptobot- impactonthelarge-scalemeridionalflowandthedifferentialro- tom. Bottom left:Streamlinesof the meridionalflow. The flow tation(seebelow). iscounter-clockwise,i.e.directedtowardsthepoleatthesurface andtowardstheequatoratthebottom.Bottomright:Flowspeed 3.1.Fast-rotatingSun asfunctionoflatitudeatthetop(solidblue)andbottom(dashed red)oftheconvectionzone.Positivevaluesindicateaflowaway To study the impactof the basic rotationwe computethe rota- fromthenorthpole. tion law of a hypotheticalfast-rotating Sun with the P = 1.33 dayrotationperiodofHD171488.Figure3showstheresulting rotation pattern. The isocontoursare cylinder-shapedin accord ThetoppanelsinFig.1showtherotationpatternintheso- to the Taylor-Proudmantheorem.Consequently,the radialpro- larconvectionzone.Itagreeswellwiththosecomputedwiththe files in the right diagram show an increase of the rotation rate KR99 codes and the KS01 scheme with the modifications de- with increasing radius at low latitudes. Unlike the case of the scribedinBonannoetal. (2007).Thesurfacerotationisfastest real Sun, there is a pronounced increase in the upper part of attheequatorwithadifference the convection zone at the equator. At both radial boundaries there are pronounced boundary layers with huge gradients of δΩ=0.07rad/day (27) the rotation rate which are caused by the stress-free boundary M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars 5 conditions.Thesurfacerotationisfastestattheequatorandde- 0.69R . Forthis modelstar we find a surface differentialrota- ⋆ creases monotonously towards the polar caps but the slope of tion δΩ = 0.51rad/dayork = 0.10.Thisis in agreementwith the decline has a minimum at mid-latitudes. With a value of theobservedvalue. 0.08rad/dayinsteadof0.07rad/daythehorizontalshearisonly For κ1 Cet we use solar mass and metalicity and an age of slightlystrongerthanfortheSundespiteafactorof20between 1 Gyr. The modelstar has an effective temperatureof 5677 K, theaveragerotationrates. 0.915 R and 0.78 L . The resulting rotation law shows a sur- ⊙ ⊙ The meridionalflow only hasone flow cell per hemisphere face shear δΩ = 0.077 rad/day, or k = 0.12. This is slightly withthesurfaceflowdirectedtowardsthepoles.Theamplitudes largerthantheobservedvalue.Anearliermodelfromthesame are 34 m/s at the surface and 17 m/s at the bottom of the con- evolutionary track, with a stellar age of 167 Myr, an effective vectionzone.AsfortherealSunwithitsmuchslowerrotation, temperatureof5649K,aradiusof0.9R ,andanluminosityof ⊙ thereturnflowis halfasfastasthesurfaceflowbutevenmore 0.74 L yields k = 0.11, or δΩ = 0.072 rad/day. This is still ⊙ concentratedat the bottom of the convectionzone. The flow is notinperfectagreementwiththeobservedvaluebutreasonably fastestatmid-latitudes. close. Possiblereasonsfortheremainingdiscrepancyarethestellar model,whichmightnotbeasufficientlyaccuraterepresentation oftherealstarandanunderestimationofthetotalsurfaceshear bythesin2θderivedfromspotrotationastheobservedspotsdo notcoverthewholerangeoflatitudesfrompoletoequator. 4. Numericalexperiments Wecarryoutseveralnumericalexperimentstocomparetheroles playedbythetwodriversofdifferentialrotation,i.e.theΛeffect andthebaroclineflow. Fig.3. Rotation of a hypothetical fast-rotating Sun with P = 4.1.SunwithoutΛeffect 1.33d. Ourmodelincludestwoeffectsthatarecapabletomaintaindif- ferentialrotation,theΛeffectandthebaroclineflowasduetoa horizontaltemperaturegradient.Thiscanbeseenwhenthebaro- clineterminEq.(6)isrewrittenintermsofthetemperature, 3.2.MOSTstars:ǫEriandκ1Cet 1 g ∂δT ǫEriandκ1Cetareactivedwarfstarsforwhichsurfacedifferen- (∇ρ×∇p) ≈− . (31) ρ2 φ rT ∂θ tialrotationhasbeenderivedfromlightcurvesrecordedbythe MOSTsatellite Crolletal. (2006)deriveda rotationlawofthe Thebaroclinetermcanbeapowerfuldriverofmeridionalflows, form which in turn can drive differential rotation. In our theory the P latitudinal temperature profile is due to the anisotropic heat- eq P(B)= (29) conductivitytensorinthepresenceofrotation.Alwaysthepolar 1−ksin2B axisisslightlywarmerthantheequatorbutwithanunobservable fortheK2Vstarǫ Eri,where P(B)istherotationperiodatthe temperatureexcess.A positivepole-equatortemperaturediffer- latitudeB,andPeqtherotationperiodattheequator(B=0).The encewilldriveaclockwiseflow.Itsangularmomentumtransport parameterkgivestheshearoftherotationlaw.Thegeneralrule leadstoanacceleratedrotationatlowlatitudesandtoslowerro- derivedfromthetheoryofthestellarrotationlawsofKitchatinov tation atthe polarcaps. A baroclineflow (also ‘thermalwind’) &Ru¨diger(1999) can therefore maintain differential rotation with solar-type sur- face rotation.For an illustration we repeatour computationfor P k≃ eq (30) theSunwiththeΛtermscanceledwithintheReynoldsstress. 100days The resulting rotation and flow patterns are shown in Fig. 4. Therotationisindeedsolar-typebutwithδΩ = 0.04weaker isperfectlyfulfilledbythesestars. thanwithΛeffectandtheisocontoursaredistinctlydisc-shaped An equatorial rotation period P = 11.2 days and a value eq atthepoleseveninthemidlatitudes.Theequatorregionshows k = 0.11+0.03 for the differentialrotation has been reported for −0.02 basicallynostructure. ǫ Eri corresponding with δΩ = 0.062 rad/day. For the G5 V The typical superrotation of the deep convection zone be- starκ1 CetWalkeretal.(2007)foundP = 8.77daysandk = eq neaththeequatorknownasaresultofthehelioseismologydoes 0.09+0.006correspondingwithδΩ=0.064rad/day. −0.005 notoccur. The reason is simple: if only a (fast) meridionalcir- Kitchatinov&Olemskoy(2010)computedrotationlawsfor culation transports the angular momentumwith a pattern sym- modelstarswith 0.8 and1.0solar massesfor ǫ Eriand κ1 Cet. metric with respect to the equator then the angular momen- Theresultswerek =0.127forǫ Eriandk=0.13forκ1 Cet. tum becomes uniform in the equatorial part of the convection Here the calculations for these stars are based on im- zone (except, of course, the boundary layers). Hence, there is proved stellar models computed with the MESA/STARS code r2Ω ≃ constindependentoftheflowdirectionbutincontradic- (Paxtonetal.2010). A star with M = 0.85M , Z = 0.02 and ⋆ ⊙ tiontotheobservation2. anageof0.44Gyrservesasmodelforǫ Eri.Ithasaneffective temperatureof5076K,aradiusof0.76R⊙,andaluminosityof 2 without Λ effect a superrotation beneath the equator can only be 0.34 L⊙. The bottomof the outer convectionzone is located at duetoananticlockwiseflowwhichissufficientlyslow 6 M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars With amplitudes of 4.7 m/s at the top and 2.1 m/s at the bottomoftheconvectionzonethebarocline-inducedclockwise meridional flow is substantially weaker than in the complete model–anditgoesinthe‘wrong’direction. Fig.5. Solarrotationlawwithoutbaroclinicterms.Thestructure disappearsnowatthepolaraxisbutitappearsattheequatorial region. The equatorial plane shows superrotation but the polar axisrotatesalmostrigid. 5. YoungGdwarfs 5.1.CoRoT-2a ThisG7VstarisayoungSun,i.e.ithassolarmassbutismuch younger with an age of 0.5 Gyrs. The star has been observed by the CoRoT satellite and found to have a planet with an or- bital period of 1.743 days(Alonso et al. 2008).Besides plane- tarytransits,thelightcurveshowsperiodicvariationthatismost easilyexplainedbyarotating,spottedsurfacewithbasicrotation periodof4.5 days.Spotmodelingusing circularspotsfindsan Fig.4. SolarrotationlawandmeridionalflowwithoutΛeffect. excellent fit assuming three spots and a solar-type surface dif- Top: The rotation law. Note the steep negative gradient of the ferentialrotation with a difference of 0.11 rad/day (Fro¨hlichet angular velocity at the poles and the almost rigid rotation be- al.2009). neath the equator.Bottom:The meridionalflow is anti-solar:it ispositive(equatorwards)atthetopoftheconvectionzoneand negative(polwards)atthebottomoftheconvectionzone. 4.2.Sunwithoutbaroclineflow Nextthe barocline term is canceled while keeping the Λ effect whichmaintainsthedifferentialrotationtogetherwiththemerid- ionalflowcausedbytheformerthroughthesecondtermonthe LHSofEq.(6)(‘Biermann-Kippenhahnflow’,seeKippenhahn Fig.6. RotationpatternofCoRoT-2a,ayoungSunrotatingwith 1963;Ko¨hler1970). Figure5showstheresultingrotationpat- P=4.5days. tern. The differential rotation is solar-type, i.e. the rotation pe- riodisshortestattheequatorandlongestatthepolarcaps.With δΩ=0.014rad/daythesurfacerotationismuchmorerigidthan Our model star is based on a model from an evolutionary observed.Theisocontourplotshowsadistinctlycylinder-shaped trackfortheSun.Theageis0.5Gyrs,theradius0.9solarradii, pattern in the bulk of the convectionzone while at the top and andtheluminosity75percentofthesolarvalue.Thebottomof bottomboundariesthepatterndeviatesfromthecylindergeom- theconvectionzoneisatafractionalradiusx=0.73. etry and the rotation rate falls off with increasing radius at all latitudes.Thecounterclockwisemeridionalflowhasaverysim- Figure6showstheresultingrotationpattern.Thesurfacedif- ilargeometryasinthefullmodelbutitisfasterwithamplitudes ferentialrotationof0.09rad/dayreproducesthevalueobserved of17and8m/satthetopandbottom. byFro¨hlichetal.(2009).Therotationpatternismorecylindrical thanthatof theSun, butlessthan thatofourfast-rotatingSun. Thesubrotationalongthepolaraxiswhichistypicalforthe Similarly,theboundarylayersaremorepronouncedthaninthe solar rotation law and which is produced by baroclinic flows Sun but less than for the fast-rotating Sun. The flow pattern is (cf. Fig. 4) does not exist here. Ko¨hler (1970) and Ru¨diger et similar to that in the solar convection zone. The amplitude is al. (1998)havegivenseveralnumericalexamplesof thisdirect slightlylargerwithamaximumvalueof18.6m/satthetopand consequenceoftheTaylor-Proudmantheorem. 10.6m/satthebottomoftheconvectionzone. M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars 7 5.2. R58,LQLupandHD171488 Wenowaddresstherecentdifferentialrotationmeasurementsof the fast-rotating G dwarfs R58, LQ Lup and HD 171488. R58 (HD 307938)is a youngactive G dwarf in the open cluster IC 2602. It has a photospherictemperature of 5800 K and rotates with a period of 0.56 days(Marsdenetal.2005). LQ Lup (RX J1508.6-4423) is a post-T Tauri star with an effective temper- ature of 5750±50 K and a rotation periodof 0.31days, anda radiusof1.22±0.12solarradii(Donatietal.2000),correspond- ingtoamassof1.16±0.04solarmasses,andanageof25±10 Myr.Theequator-poleΩ-differenceis0.12rad/day.HD171488 (V889Her)isayoungactiveGdwarf.Strassmeieretal.(2003) found a photospherictemperatureof 5830 K, a rotation period of 1.337 days, a mass of 1.06±0.05 solar masses, a radius of 1.09± 0.05 solar radii, and an age of 30–50 Myr. Marsden et al. (2006) find a rotation period of 1.313 days, a photospheric temperatureof5800K,andaradiusof1.15±0.08solarradii. With 0.4–0.5 rad/day the equator-pole Ω-difference of HD171488isexceptionallylarge.Thek-valueof0.10missesthe k =0.013predictedbyEq.(30)byafactorof7.5.Asthecorre- spondingfactorsforR58andLQLuparemuchsmaller(factor 2–3)the followingcalculationstake the case of HD 171488as Fig.7. InternalrotationofyoungGdwarfswithrotationperiod the main example. As the stars have similar radii and effective of1.33days.Top:deepconvectionzone,δΩ = 0.11rad/dayat temperaturesweinvestigatewhatimpactdifferencesintheinter- thesurface.Bottom:shallowconvectionzone.δΩ=0.5rad/day nalstructurehaveonthesurfacerotationpatterns.Westudytwo atthesurface. modelsthatmainlydifferinthemetalicityand,asaconsequence, the depth of the convection zone. The models were computed with the MESA/STARS code.We use the metalicity as control shearofδΩ=0.5rad/day.Thecontourplotshowsadisc-shaped parameter for the depth of the convection zone and vary mass pattern with flat profiles at low latitudes and a decrease of the andmixinglengthparametertoadjustradiusandeffectivetem- rotationratewithincreasingradiusatthepolarcaps.Theradial peraturetovaluesclosetothoseobservedforthethreeyoungG profiles show pronouncedboundarylayersat the radialbound- dwarfs.Therotationperiodisthe1.33dayperiodofHD171488. aries. The flow pattern is mostly solar-type. There is one large Bothmodelstarsare30Myrold. flowcellwithpolewardflowatthetopandthereturnflowatthe bottomoftheconvectionzone.Inaddition,thereisasmallcell of clockwise flow at low latitudes. The flow speeds at the top 5.2.1. Deepconvectionzone reach 18 m/s and 2.2 m/s, respectively. The return flow at the ThefirstmodelstarhasthehighmetalicityofZ =0.03.Amass bottomoftheconvectionzonereaches3.5m/s. of 1.11 solar masses and a value of 1.6 for the mixing length parameterlead to an effective temperatureof 5685K and a ra- 5.2.3. Truncatedconvectionzone dius of 1.14 solar radii. The bottom of the convection zone is located at x = 0.765 and has a temperature of 1.46 × 106 K. Themodelsabovehave(roughly)thesameeffectivetemperature The rotation law is shown in the top part of Fig. 7. The iso- butdifferinmassandstructure.Tofurtherisolatetheeffectofthe contourplotshowscylinder-shapedcontourssimilartothosefor depth of the convectionzone, we compute a modelof a young thefast-rotatingSun.Thesurfacerotationlawissolar-typewith solarmassstarandartificiallyreducethedepthoftheconvection a fast-rotatingequator. The surface shear is much stronger. We zonebyimposingthelowerboundaryatafractionalradius x = findavalueof0.11rad/day,about1.5timesthesolarvalue.The 0.88insteadof x = 0.78andcomparetherotationpatternwith radialprofilesshowsuperrotationbeneaththe equatorandsub- thatof thefullmodel.Inbothcases the topboundaryis at x = rotation along the rotation axis. The boundarylayers are much 0.98. The truncated convectionzone thus has half the depth of less pronounced and the surface layer resembles the real Sun, the full model. In both cases the rotation law is solar-type but i.e. the rotation rate decreases with increasing radius at all lat- whilethefullconvectionzoneproducesasurfaceshearof0.12 itudes. Themeridionalflow isof solar-type(counterclockwise) rad/day,thetruncatedconvectionzonehas0.32rad/day.Thefull with the surface flow towards the poles and the return flow lo- modelhasatemperaturedifferenceof24Katthetopand124K catedatthebottomofthe convectionzone.Theamplitudesare atthebottom.Incaseofthetruncatedmodelthepolarcapis47 23and14m/s,respectively. Khotterthantheequatoratthetopand254Katthebottom. 5.2.2. Shallowconvectionzone 5.2.4. Veryfastrotation Oursecondmodelstarhasametalicityof0.01.Amassof1.08 Toillustrate theimpactofthe rotationrate,we repeatthe com- solarmassesandamixinglengthparameterof1.0leadtoara- putationfortheshallowconvectionzonemodelwithrotationpe- diusof1.14solarradiiandaneffectivetemperatureof5750K. riodof0.33days.Figure8showstheresultingrotationpattern. Thebottomoftheconvectionzoneliesatafractionalradiusof Theleftplotshowssolar-typerotationbothinthesurfacediffer- 0.89andatemperatureof6×105 K.ThebottompartofFig.7 entialrotationandinthepredominantlyradialisocontours.This showstheresultingrotationlaw.Thesurfacerotationhasatotal is still not cylinder-shaped but closer to the Taylor-Proudman 8 M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars thebottomoftheconvectivezonethedifferencesare750Kand 100Kforaconvectionzoneofsmallorlargedepth,respectively. Thecorrespondingvaluesatthetopare55Kand19K. 6.2.Thermalwindequilibrium We always find a higher temperature at high latitudes than at the equator. This is a consequence of the tilt of the convective heattransportvectortowardstheaxisof rotationcausedbythe Coriolisforce.Insphericalpolarcoordinatesthecomponentsof theheatfluxread Fig.8. DifferentialrotationoftheGdwarfwiththeshallowcon- ∂S ∂S F = −ρTχ −ρTχ , (32) vectionzoneandarotationperiodof0.33days.Left:isocontours r rr ∂r rθ∂θ oftheangularvelocity.Right:therotationprofilesatvariouslat- ∂S ∂S itudes. Fθ = −ρTχθr ∂r −ρTχθθ∂θ. (33) The first term in the horizontal component precludes a purely than the disc-shaped pattern we found for a rotation period of radial stratification. Any variation of the specific entropy with 1.33days.Thesurfaceshearis0.38rad/day,thetemperaturedif- the radiuswill cause a horizontalheat flux and thus build up a ferencebetweenpolarcapsandequatoris1785Katthebottom horizontalgradient. This case is profoundlydifferentfrom that and 184K at the top. The maximumflow speedsare 14 m/sat ofalatitude-dependentbutstillpurelyradialheatflux.Thelatter thetopand8m/satthebottomoftheconvectionzone. wouldhavea muchsmallerimpactandwouldresultina much weakerdifferentialrotation(Ru¨digeretal.2005). The impact on the maintenance of differential rotation can 6. Discussion be seen from the equation for the meridionalflow. For fast ro- 6.1.Shallowvs.deepconvectionzone tation (i.e. large Taylor number)Reynoldsstress and nonlinear termscanbeneglectedandEq.(6)isreducedtothethermalwind Our“shallow”and“deep”convectionzonemodelshavesimilar equation radiiand effectivetemperaturesbutthestars differin massand metalicity.Wehavealsochosendifferentvaluesforthemixing- ∂Ω g ∂δT length parameter.As a result, the depthof the convectionzone 2rsinθΩ − =0. (34) ∂z rT ∂θ differsbetweenthemodels.Wefindverydifferentresultsforthe differentialrotation of these convectionzones. The modelwith It followsthata gradientof the angularvelocity alongthe axis ashallowconvectionzoneshowsmuchstrongerdifferentialro- ofrotationisneededtobalancethehorizontaltemperaturegra- tationthantheoneswithdeeperconvectionzones.Our“shallow dient. Hence,a disc-shapedrotationpatternresults in the polar convection zone” model does indeed reproduce the very large area(only)duetotheactionofthebaroclinicflow.Forwarmer valueof0.5rad/dayfoundbyJeffers&Donati(2009)whilethe poles(δT sinksequatorwards)thez-gradientofΩisnegativeas “deepconvectionzone”modelwithits smaller valueof 0.11is observed.Themeridionalflowwithoutthebarocliniccomponent in rough agreement with the values observed for LQ Lup and onlyyieldsdΩ/dz≃0(Ko¨hler1970,seealsoFig.5).Hence,the R58. empirical finding of negativedΩ/dz at the polar axis by helio- The result from the shallow convection zone model is in seismologyprovestheexistenceofbaroclinicflowsinthesolar line with previous findings on F stars. Extreme values of sur- convectionzone. faceshearhavebeenobservedforFstarsbyReiners(2006)and AswehavedemonstratedbymeansofFig.4thesuperrota- foundintheoreticalmodelsforstarswithveryshallowconvec- tionbeneaththesolarequatorisadirectindicationfortheaction tion zones by Ku¨ker & Ru¨diger (2007). In the latter study, the oftheReynoldsstressintheconvectionzone.Allmodelswith- largest values of surface shear are found for the most massive outΛ-effectbutwithanclockwise(equatorwardatthesurface) starsstudied.Thesestars arenotonlythe hottest,theirconvec- circulationleadtodΩ/dr<0inthebulkoftheconvectionzone ∼ tionzonesarealsoveryshallow. beneath close to the equator. Such clockwise flows are able to To achieve the thin convection zone we had to lower the acceleratethe equatorcomparedwith thepolesbuttheycannot mixing-lengthparametertoa valueof1.0.Thisisnotonlydif- producethesuperrotationbeneaththeequator. ferentfromthevalueusedforthe“deep”convectionzonemodel Figure 9 shows results from a detailed computation of the but much lower than what is usually used in stellar evolution terms on the LHS of Eq. (34) and their sum vs. the fractional models.Hence,themodelisprobablynotrealisticanddoesnot radiusforalatitudeof45◦.Theleftpanelshowstheplotforthe directlyapplytoLQ Lup.Notbeingspecialistsin thefield, we Sun, the rightpanelthat for the deep convectionzone G dwarf leavethequestionwhetherashallowconvectionzoneispossible model.ThesamequantitieswerecomputedforthefastSunand open but we wonder if strong magnetic fields could inhibit the theshallowconvectionzoneGdwarfmodels. convective heat transport in a way that might be mimicked by FortherealSunthetwotermscancelinthebulkofthesolar thischoiceofparameter. convectionzonewherethemeridionalflow isslowandsmooth The“truncatedconvectionzone”modelavoidstheuncertain- but not close to the boundaries. For the deep convection zone tiesaboutthe”shallowconvectionzone”modelandshowseven the boundarylayersare muchthinnerbutalso much morepro- moreclearlytheimpactthedepthoftheconvectionzonehason nounced.Thesefindingsindicatethatforfastrotationthemerid- thesurfacerotation.Boththeverticalandhorizontaltemperature ionalflowismainlydrivenbytheboundarylayerswhilethebulk gradientsaresteepestforthemodelwiththeshallowconvection of the convection zone is in thermal wind equilibrium. In the zoneandflattestforthemodelwiththedeepconvectionzone.At boundary layers the meridional circulation is strongly sheared M.Ku¨ker,G.Ru¨diger&L.L.Kitchatinov:DifferentialrotationofGstars 9 so that the RHS of Eq. (34) (which consists on 3rd derivatives depthoftheconvectionzonehasabigimpactonstellarrotation. ofthemeridionalvelocitycomponentsmultipliedwiththeeddy Shallowconvectionzonesproducestrongersurfaceshear. viscosity)becomeslarge. The presented mean-field theory for G stars naturally ex- plainstherotationlawsoftheSun,oftheMOSTstarsǫ Eriand κ1Cet,andofsuchfast-rotatingstarslikeR58andLQLup.The Drivers of meridional flow Drivers of meridional flow discrepancy between these stars and the much stronger differ- 0 ential rotation observed for HD 171488 – if real – is hard to 0.0 explain for stars of similar age and spectral type. If, however, -50 -0.2 HD 171488 had a shallower convectionzone for some reason, itsstrongsurfacedifferentialrotationfollowsimmediately. -100 -0.4 During the pre-MS evolution the convection zone retreats -150 from the central region and the originally fully convective star -0.6 formsaradiativezonearoundthecore.Thischangeinthedepth 0.75 0.80 0.85 0.90 0.95 0.80 0.85 0.90 0.95 oftheconvectionzoneshouldbereflectedbythesurfacediffer- Fractional radius Fractional radius entialrotation. Acknowledgements. This work has been funded by the Deutsche Fig.9. Thedriversofangularmomentumvs.fractionalradiusat Forschungsgemeinschaft (RU 488/21). LLK is thankful to the support 45◦latitudefortheSun(left,slowrotation)andthedeepconvec- by the Russian Foundation for Basic Research (projects 10-02-00148, tionzoneG dwarf(right,fastrotation).Dashedline:baroclinic 10-02-00391). term. Dash-dotted line: centrifugalterm. Solid line: both. Note that the balance (34) well holds in the bulk of the convection zones of fast rotation stars where the meridional circulation is References slow and smooth.Equation(34) doesnothold in the boundary Ahrens,B.Stix,M.,Thorn,M.1992,A&A,264,673 areaswheretheflowisfastandshearing. Alonso,R.,Auvergne,M.,Baglin,A.,etal.2008,A&A,482,L21 Barnes,J.R.,CollierCameron,A.,James,D.J.,&Donati,J.-F.2000,MNRAS, 314,162 Thenumericalexperimentsshowthatthebaroclinictermis Barnes,J.R.,CollierCameron,A.,Donati,J.-F.,etal.2005,MNRAS,357,L1 an importantbut notthe sole driverof the solar differentialro- Bonanno,A.,Ku¨ker,M.,&Paterno`,L.2007,A&A,462,1031 CollierCameron,A.,&Donati,J.-F.2002,MNRAS,329,L23 tation. Cancellation of the Λ effect leaves a surface shear of Croll,B.,Walker,G.A.H.,Kuschnig,R.,etal.2006,ApJ,648,607 0.04 rad/daywhile switching off the baroclinic term reduces it Donahue,R.A.,Saar,S.H.,&Baliunas,S.L.1996,ApJ,466,384 to0.014rad/day.ThisisnotbecausetheΛeffectisaninefficient Donati,J.-F.,Mengel,M.,Carter,B.D.,etal. 2000,MNRAS,316,699 transporterofangularmomentumbutbecauseitisbalancedby Fro¨hlich,H.-E.,Ku¨ker,M.,Hatzes,A.P.,&Strassmeier,K.G.2009,A&A,506, 263 themeridionalflow.Cancelingthemeridionalflowaltogetherin- Gilman,P.A.,&Glatzmaier,G.A.1981,ApJS,45,335 creases the surface shear to 0.18rad/day.This is a well-known Hall, D.S. 1991, The Sun and cool stars, Activity, magnetism, dynamos, effect:meridionalflowsdrivenbydifferentialrotationacttore- Springer,p.95 duceit(Ru¨digeretal.1998).Thebaroclinicflowhastheoppo- Hansen,C.J.,&Kawaler,S.D.1994,StellarInteriors,SpringerVerlag sitedirectionandbuildsupthedifferentialrotationalongthepo- Henry,G.W.,Eaton,J.A.,Hamer,J.,&Hall,D.S.1995,ApJ,97,513 Huber,K.F.,Wolter,U.,Czesla,S.etal. 2009,A&A,501,715 lar axis. The observed superrotation beneath the equator, how- Jeffers,S.V.,&Donati,J.-F.2009a,MNRAS,390,635 ever, cannot be produced by meridional circulation but should Jeffers,S.V.,&Donati,J.-F.2009b,ASPConferenceSeries,405,523 beadirectconsequenceoftheexistenceoftheΛeffect. Jeffers,S.V.,Donati,J.-F.,Alecian,E.,&Marsden,S.C.2010,MNRAS,inpress Kippenhahn,R.1963,ApJ,137,664 Kitchatinov,L.L.,Pipin,V.V.,&Ru¨digerG.1994,AN,315,157 7. Conclusions Kitchatinov,L.L.,&Ru¨diger,G.1999,A&A,344,911 Kitchatinov,L.L.,&Olemskoy,S.V.2010,subm. Stellar differential rotation is driven by Reynolds stress, by Ko¨hler,H.1970,SolarPhys.13,3 Ku¨ker,M.,&Stix,M.2001,A&A,366,668 thecentrifugal-induced(‘Biermann-Kippenhahn’)flowand(for Ku¨ker,M.,&Ru¨diger,G.2007,AN,328,1050 stratified density) by the baroclinic flow. In our model both Ku¨ker,M.,&Ru¨diger,G.2008,J.Physics,Conf.Series,Vol.118,012029 meridionalcirculationshaveopposite directions.Thebarocline Marsden,S.C.,Waite,I.A.,Carter,B.D.,Donati,J.-F.2005,MNRAS359,2005 flow becomes important for faster rotation as the convective Marsden,S.C.,Donati,J.-F.,&Semel,M.,etal.2006,MNRAS,370,468 Messina,S.,&Guinan,E.2003,A&A,409,1017 heat-flux deviates from the radial direction. This deviation can Paxton,B.,Bildsten,L.,Dotter,A.,etal.2010,arXiv:1009.1622v1 beinterpretedasatiltoftheheat-fluxvectortowardstherotation Press,W.H.,Teukolski,S.A.,Vetterling,W.T.,&Flannery,B.P.1992,Numerical axis and causes the poles to be slightly warmer than the lower RecipesinFORTRAN,2nded.,CambridgeUniversityPress latitudes.The baroclinicflow is responsibleforthe strongneg- Reiners,A.,&Schmitt,J.H.M.M.2003,A&A,412,813 ative gradient dΩ/dz along the rotation axis, and the Reynolds Reiners,A.2006,A&A,446,267 Ru¨diger1989,Differential rotationandstellarconvection, GordonandBreach stress producesthe typical positive dΩ/dr along the equatorial SciencePublishers midplanebothknownfromhelioseismology.Asaresultofboth Ru¨diger,G.,v.Rekowski,B.,Donahue,R.A.,&Baliunas,S.L.1998,ApJ,494, impactstheisolinesofΩknownfortheSunarealmostradialin 691 mid-latitudes. Ru¨diger,G.,Egorov,P.,Kitchatinov,L.L.,&Ku¨ker,M.,2005,A&A,431,345 Stellingwerf,R.F.1975,ApJ,195,441 Studyingrealstellar modelsimpliesvaryingabunchofpa- Strassmeier,K.G.,Picghler,Weber,Granzer,T.2003,A&A,411,595 rameters as mass, radius, effective temperature and the depth Thompson,M.J.,Christensen-Dalsgaard, J.,Miesch,M.S.,&Toomre,J.2003, of the convection zone are interdependent. Kitchatinov & ARA&A,2003,599 Olemskoy (2010) studied differential rotation along the lower Ulrich,R.K.,Boyden,J.E.,Webster,L.,etal.1988.Sol.Phys.,117,291 ZAMSforfixedrotationrateandfoundthatthesurfaceshearis Walker,G.A.H.,Croll,B.,Kuschnig,R.,etal.2007,ApJ,659,565 afunctionoftheeffectivetemperaturealone.Usingsomeartifi- cialmodels,wefindthatforthesameeffectivetemperaturethe

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