Flip-flop phenomenon: observations and theory

0.26 MB·

by  D. Elstner

Astron.Nachr./ANxxx(xxxx)x,xxx–xxx
Flip-flop phenomenon: observations and theory
D. ELSTNER1 and H. KORHONEN1
AstrophysikalischesInstitutPotsdam,AnderSternwarte16,D-14482Potsdam,Germany
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7 Abstract. Inmanyactivestarsthespotsconcentrateontwopermanentactivelongitudeswhichare180◦ apart.Insomeof
1 thesestarsthedominantpartofthespotactivitychangesthelongitudeeveryfewyears.Thisso-calledflip-flopphenomenon

hasuptonowbeenreportedin11stars,bothsingleandbinaryalike,andincludingalsotheSun.Toexplainthisphenomenon,
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anon-axisymmetricdynamomode,givingrisetotwopermanentactivelongitudesatoppositestellarhemispheres,isneeded
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togetherwithanoscillatingaxisymmetricmagneticfield.Herewediscusstheobservedcharacteristicsoftheflip-flopphe-
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nomenonandpresentadynamosolutiontoexplainthem.
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1 Keywords:stars:activity–stars:magneticfields–stars:spots–methods:numerical
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5 (cid:13)c0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
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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
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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
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Hackman,T.:2004,A&Asubmitted
Jetsu,L.,Pelt,J.,Tuominen,I.,Nations,H.L.:1991,in:TheSunand
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berg,p.381
Jetsu,L.,Pelt,J.,Tuominen,I.:1993,A&A278,449
Korhonen et al. (2004) found that for FK Com the relative
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surfacedifferentialrotation(∆Ω/Ω)isabout10%oftheso-
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Lanza,A.F.,Catalano,S.,Rodono`,M.,˙Ibanog˘lu,C.,Evren,S.,Tas¸,
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