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Astron.Nachr./ANxxx(xxxx)x,xxx–xxx Flip-flop phenomenon: observations and theory D. ELSTNER1 and H. KORHONEN1 AstrophysikalischesInstitutPotsdam,AnderSternwarte16,D-14482Potsdam,Germany 5 0 0 Received<date>;accepted<date>;publishedonline<date> 2 n a J 7 Abstract. Inmanyactivestarsthespotsconcentrateontwopermanentactivelongitudeswhichare180◦ apart.Insomeof 1 thesestarsthedominantpartofthespotactivitychangesthelongitudeeveryfewyears.Thisso-calledflip-flopphenomenon hasuptonowbeenreportedin11stars,bothsingleandbinaryalike,andincludingalsotheSun.Toexplainthisphenomenon, 1 anon-axisymmetricdynamomode,givingrisetotwopermanentactivelongitudesatoppositestellarhemispheres,isneeded v togetherwithanoscillatingaxisymmetricmagneticfield.Herewediscusstheobservedcharacteristicsoftheflip-flopphe- 3 nomenonandpresentadynamosolutiontoexplainthem. 4 3 1 Keywords:stars:activity–stars:magneticfields–stars:spots–methods:numerical 0 5 (cid:13)c0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim 0 / h 1. Introduction Com.Jetsuetal.(1991,1993)noticedfromphotometricob- p - servation, that the spot activity on FK Com for 1966–1990 o The global structure and behaviour of the stellar magnetic concentratedontwolongitudes,180◦ apart.Theyalsonoted r t fields are determined by different dynamo modes that have thatduringthe individualobservingseasonsonlyoneofthe s a differentsymmetriesandstabilities(seee.g.Brandenburget active longitudes had spots. This behaviour is very well il- : al. 1989). In slowly rotatingstars, like the Sun, axisymmet- lustrated in Fig. 1 (from Jetsu et a; 1993), which shows the v i ric modes are excited. These modesdo not show any struc- normalizedmagnitudesofFKComfor1966–1990.Thepho- X tureinthelongitudinaldistributionofthespotsandtheyos- tometric minimum is always either around the phase 0.0 or r cillate in time. In more rapidly rotating stars the higher or- 0.5.ThedataJetsuetal.(1993)usedcanbeanalysedtogether a der non-axisymmetric modes get excited (see e.g. Moss et with more recent observations (1991–2003) to estimate the al. 1995; Tuominen, Berdyugina & Korpi 2002). The mag- frequencyatwhichflip-flopeventsoccuronFKCom.There netic configuration in the non-axisymmetricmodes consists is onaverageoneflip-flop eventevery2.6 years,givingfull of two starspots that are 180◦ apart, explaining the perma- cyclelengthof5.2years(Korhonenetal.2004). nent active longitudes seen in many rapidly rotating stars. After the discovery of the flip-flop phenomenon on Thesenon-axisymmetricmodesdonotoscillate.Forexplain- FK Com it has been reported also on other active stars. ingtheflip-flopphenomenon,whereweseebothactivelon- Berdyugina&Tuominen(1998)studiedthephotometricob- gitudes and oscillations, axisymmetric dynamo modes need servationsoffourRSCVn binariesanddiscoveredthatalso toco-existwiththenon-axisymmetricmodes. thesestarshavepermanentactivelongitudesthatarealterna- In thispaperwe describe the observedcharacteristicsof tivelyactive.InthecaseofIIPegRodono` etal.(2000)later the flip-flop phenomenonand presenta modelthat can pro- confirmedtheirresults.Berdyugina,Pelt&Tuominen(2002) ducethem. discoveredflip-flopsonyoungsolartypestar,LQHya.And arecentanalysisof120yearsofsunspotdata(Berdyugina& Usoskin2003) suggeststhattheSunalso haspermanentac- 2. Observationsofflip-flop phenomenon tive longitudeswith associated flip-flops. On the Sun a flip- flopeventoccursonaverageevery3.8yearsonthenorthern Theflip-flopphenomenon,inwhichthemainpartofthespot andevery3.65yearsonthesouthernhemisphere. activitychanges180◦ onthestellarsurface,wasfirstdiscov- When the flip-flop phenomenon was discovered, it was ered in the early 1990s on a single, very active, giant, FK notsurewhetherthephenomenonwascausedbyspotmove- Correspondenceto:[email protected] mentacrossthestellardiskoremergenceoffluxonthenew 4 RESULTS z) H n e ( at n r o ati ot R Fig.1. The active longitude structure on FK Com. Normal- ized magnitudes(ordinate)of FK Com for 1966–1990plot- Fig.2.Thesolarrotationlaw. tedagainstthephase(abscissa).Thephaseshavebeendeter- minedusingtheephemerisHJD2439252.895+2.4002466E. 3. Modelling flip-flops ThisfigurehasbeentakenfromJetsuetal.(1993). Themodelconsistsofaturbulentfluidinasphericalshellof innerradiusr andouterradiusr . active longitude. Korhonen et al. (2001) have shown, with in out Wesolvetheinductionequation Doppler images just before and after a flip-flop event on FK Com, that flip-flops are caused by changing the relative ∂hBi =curl(α ◦hBi−η curlhBi), (1) strengthsofthespotgroupsatthetwoactivelongitudeswith- ∂t q T outactualspotmovementonthestellarsurface. insphericalcoordinates(r,θ,ϕ)foranα2Ω-dynamo.Asolar typerotationlaw(seeFig.2)inthecorotatingframewiththe Allthestarsforwhichtheflip-flopphenomenonhasbeen core reported are listed in Table 1. Spectral type, rotation period and the relativedifferentialrotationcoefficientare givento- Ω(r,θ)= 1Ω 1+erf r−rin (Ω −Ω ) (2) 0 s c getherwiththeflip-flopperiod(thelengthofthefullcycle). 2 (cid:20) (cid:18) d1 (cid:19)(cid:21) The flip-flop phenomenonhas so far been detected in many whereΩ =Ω −acos2θisused. s eq differentkindsofstars:bothbinariesandsinglestars;young, Onlythesymmetricpart mainsequenceandevolvedalike.Usually,theflip-flopperiod α =α cosθ(1.−2cos2θ) is between 5 and 10 years, median being 7 years. The stars rr 0 themselves usually have rotation periods< 3 days (median αθθ =α0cosθ(1.−2sin2θ) 2.4days).Anyhow,noclearcorrelationbetweentherotation α =α cosθ ϕϕ 0 periodandtheflip-flopperiodcanbeseen. α =α =2α cos2θsinθ (3) rθ θr 0 Apart from the stars mentioned in Table 1, there are of the α-tensor is included. In orderto saturate the dynamo also two other stars forwhich flip-flopshavebeen reported. wechoosealocalquenchingof ThesestarsaresinglegiantHD199178(Hackman2004)and α RS CVn binary RT Lac (Lanza et al. 2002). In these stars αq = 1+B2/B2 (4) onlyveryfeweventshavebeenobserved,sonoinformation eq ontheflip-flopcyclelengthcanbeobtained. Forα0 we choosea value slightlyabovethe criticalone for thedynamothreshold. The inner boundaryis a perfectconductorand the outer boundary resembles a vacuum condition, by including an Table 1. Starsthatshow flip-flopphenomenon.In the Table outerregionupto1.2stellarradiiintothecomputationalgrid the name of the star, spectral type and age, rotation period, with 10 times higher diffusivity. At the very outer part the flip-flop period and the relative differential rotation coeffi- pseudo vacuum condition is used. In order to see the influ- cientaregiven. enceofthethicknessoftheconvectionzonewehavechosen Name Type Prot Pff ∆Ω/Ω r = 0.7fora thin (resultsshownin Fig. 3) andr = 0.4 Sun single,G2V 27d 71yr 0.19 in in LQHya single,K2V,ZAMS 1.6d 5.22yr 0.0223 forathick(Fig.4)convectionzone. ABDor single,K0V,ZAMS 0.5d 5.54yr 0.055 EKDra single,G1.5V,ZAMS 2.6d 44yr - FKCom single,G7III 2.4d 5.26yr 0.0186 4. Results IIPeg RSCVn,K2IV 6.7d 9.37yr 0.048 sigmaGem RSCVn,K1III 19.6d 14.97yr <0.0049 EIEri RSCVn,G5IV 1.95d 9.07yr -0.15–-0.2010 WiththeparameterΩ0wemodelthestrengthofthedifferen- HR7275 RSCVn,K1III-IV 2.3d 17.57yr - tialrotation.ForΩ =1wehaveasolarrotationlawwithan 1)Berdyugina & Usoskin 2003 2)Berdyugina et al. 2002 3)Ko˝va´ri et al. 2004 0 4)Berdyugina & Ja¨rvinen 2005 5)Collier Cameron & Donati 2002 6)Korhonen oscillatingaxisymmetricdynamosolution.ThecaseΩ0 = 0 et al. 2004 7)Berdyugina & Tuominen 1998 8)Weber 2004 9)Ko˝va´ri et al. 2001 isanα2-dynamowhichgivesamigratingnon-axisymmetric 10)Washuettl2004 dynamobecauseoftheanisotropicα(cf.Ru¨diger,Elstner& Ossendrijver2003). 4 RESULTS Fig.3. The magnetic pressure fromthe dominantradialcomponentof the magneticfield on the stellar surface is shown at threedifferenttimestepsforthethinlayermodel.Wesubtractedrotationandmigration. Fig.4. TheupperpanelisasinFig.3butnowforthethickconvectionzonemodel.ThelowerpanelshowsDopplerimages of FK Com forJune1997andJanuary1998(fromKorhonenetal. 2001).Forthe Dopplerimagesthe greyscale givesthe temperaturescaleof3600K–5700K. For10%ofthesolardifferentialrotationwefoundsimilar mainlydeterminedbythemagneticdiffusivityandvaryonly excitation conditions for a drifting non-axisymmetric mode weakly with the value for α. The field strength saturates and an oscillating axisymmetric mode. Because of the cho- abouttheequipartitionvalue.InTable2wepresentachoice sen positive α in the northern hemisphere, we get a pole- of models in order to illustrate the parameter dependence wardmigrationoftheoscillatingmode.Thedriftofthenon- of different solutions for an axisymmetric (first row), a axisymmetricmodeisoppositetotherotation. non-axisymmetric(secondrow)andtwo flip-flop(thickand thin model; third and fourth row, respectively) solutions. Usingasimpleα-quenching,givenbyEq.4,in3Dsimu- Notice, that the axisymmetric solution appears already for lationswefoundcoexistingsolutionsforbothmodes,show- Ω = 0.1 but with a higher α than is used for the thick ingamagneticflip-flopphenomenon.Wefollowedthesolu- 0 convectionzoneflip-flopmodel(thirdrow).Thisisprobably tioninoursimulationupto100diffusiontimes.Therewere duetothelocalquenchingofthem=1mode. nosignforitbeingonlyatemporaryphenomenon.Thetem- poralbehaviourofthemagneticenergyisshowninFig.5. For the thick convection zone we found models where For an assumed turbulent diffusivity of about themagneticspotsappearalreadyat50◦latitudeandaremi- 1012cm2s−1 we get a period of about 6 years for the gratingpolewardsduringthecycle.Inthiscasetheopposite thin and 9 years for the thick model. These values are spotdoesnotstarttoappearexactly180◦ awayfromtheold References 5. Discussion Forthe first time we foundstable mixedmodesolutionsfor weaklydifferentialrotatingstars. We couldfollowthecycle over100diffusiontimes. Moss(2004)presentedasimilarmodelwithisotropicα. Because there is no preferred m=1 mode in that case, it is probably not possible to have stable mixed mode solutions foralongertime.Alsohedidnotfindsimilarvaluesforthe magnetic energy in both modes. In contrary to our solution hegotanasymmetricdistributionofm=1andm = 0modes withrespecttotheequator.Weobservedasimilarexoticbe- haviourforhighlyover-criticalα. Fig.5. Energydensity in m = 0 and m = 1 modesfor the The assumptionsfor the modelare somewhatuncertain. thickmodelnormalizedtoequipartition.Thetimeunitisthe First,changingthediffusivitywouldchangetheperiodofthe diffusiontimeof30years.Left:ThewholesimulationRight: oscillatingmodeand thereforealso the flip-flop period.The Finaltimewithhigherresolution.Noticetheweakoscillation simple scaling of the solar rotation law may not be the best oftheenergyinthem = 1modesynchronizedwithm = 0 approximationforthestarsshowingflip-flops,alsothesimple due to α-quenching. The migration period in Table 2 is di- α-quenchingnon-linearitymaynotbeadequate.Anyhow,the rectlytakenfromthem=1field. mainpropertiesremainfordifferentquenchingforms.Also, thesimplediagonalformoftheαtensorisnotjustified.Nev- Table2.Parametersforasampleofrunswiththeenergyden- ertheless,theresultsfromourmodelareencouraging. sityinequipartitionunitsE0forthemodem =0andE1for The flip-flop phenomenonappears in a limited range of m = 1,respectively.P0denotestheperiodoftheoscillation thestrengthofdifferentialrotation.Thethicknessofthecon- indiffusiontimes(30years)andP1themigrationperiod.The vection zone is not very important. To what extent a large dynamo-numberCα =α0rstarη−1isused. scalemeridionalflowchangesthisbehaviourhastobeinves- rin Ω0 Cα E0 E1 P0 P1 tigated. 0.4 0.1 20 1.4 10−9 0.27 Acknowledgements. This project has been supported by the 0.4 0.11 10 10−6 0.1 0.23 DeutscheForschungsgemeinschaft grantKO2320/1.Thisresearch 0.4 0.12 10 0.06 0.06 0.3 0.2 hasmadeuseoftheSimbaddatabase, operatedattheCDS,Stras- 0.7 0.11 21 0.1 0.4 0.23 0.46 bourg,France. References spot. The distance can shrink down to 90◦. It depends also onα.Forweaklyovercriticalαwefindnearly180◦distance Berdyugina,S.V.,Tuominen,I.:1998,A&A336,L25 betweenoldandnewspot.Inallmodelswefoundacounter- Berdyugina,S.V.,Pelt,J.,Tuominen,I.:2002,A&A394,505 rotating migration of the magnetic pattern. The period was Berdyugina,S.V.,Usoskin,I.:2003,A&A405,1121 twice the flip-flop oscillation period for the thin layer and Berdyugina,S.V.,Ja¨rvinen,S.P.:2005,thisvolumeofAN nearlyequaltotheflip-flopperiodforourthicklayermodel. Brandenburg, A., Krause, F., Meinel, R., Moss, D., Tuominen, I.: This migration should be carefully considered in the future 1989,A&A213,411 CollierCameron,A.,Donati,J.-F.:2002,MNRAS329,L23 dataanalysis. Hackman,T.:2004,A&Asubmitted Jetsu,L.,Pelt,J.,Tuominen,I.,Nations,H.L.:1991,in:TheSunand Cool Stars: activity, magnetism, dynamos, Tuominen I., Moss 4.1. ComparisontoFKCom D., Ru¨diger G. (eds.), Proc. IAU Coll. 130, Springer, Heidel- berg,p.381 Jetsu,L.,Pelt,J.,Tuominen,I.:1993,A&A278,449 Korhonen et al. (2004) found that for FK Com the relative Korhonen, H., Berdyugina, S.V., Strassmeier, K.G., Tuominen, I.: surfacedifferentialrotation(∆Ω/Ω)isabout10%oftheso- 2001,A&A,379,L30 lar value. This implies that the 10% scaling of the solar ro- Korhonen,H.,Berdyugina,S.V.,Tuominen,I.:2004,AN325,402 tationlaw,thathasbeenusedinthemodel,isanappropriate Ko˝va´ri,Zs.,Strassmeier,K.G.,Bartus,J.,Washuettl,A.,Weber,M., firstapproximationforthisstar.InFig.4themodelingresults Rice,J.B.:2001,A&A373,199 forthethickconvectionzonemodelarecomparedtotheob- Ko˝va´ri, Zs.,Strassmeier, K.G., Granzer, T., Weber, M., Ola´h, K., servedsurfacetemperaturedistributionsonFKComforJune Rice,J.B:2004,A&A417,1047 Lanza,A.F.,Catalano,S.,Rodono`,M.,˙Ibanog˘lu,C.,Evren,S.,Tas¸, 1997andJanuary1998(Korhonenetal.2001).Inthecalcula- G.,Çakırlı,O¨.,Devlen,A.:2002,A&A386,583 tionsthespotsappearathighlatitudes.Thisisalsoconsistent Moss,D.,Barker,D.M.,Brandenburg,A.,Tuominen,I.:1995,A&A with the Doppler imaging results for FK Com, which show 294,155 spotsmainlyathighlatitudes,andbasicallyneverlowerthan Moss,D.:2004,MNRAS latitude 45◦. Also, the spot size looks reasonable and is in Rodono`, M., Messina, S., Lanza, A.F., Cutispoto, G., Teriaca, L.: linewiththeobservations. 2000,A&A358,624 References References Ru¨diger,G.,Elstner,D.,Ossendrijver,M.:2003,A&A406,15 Tuominen,I.,Berdyugina,S.V.,Korpi,M.J.:2002,AN323,367 Washuettl,A.:2004,PhDThesis,UniversityofPotsdam Weber,M.:2004,PhDThesis,UniversityofPotsdam