Astronomy&Astrophysicsmanuscriptno.8800 (cid:13)c ESO2008 February2,2008 LE Invisible sunspots and rate of solar magnetic flux emergence S.Dalla1,L.Fletcher2,andN.A.Walton3 8 0 1 CentreforAstrophysics,UniversityofCentralLancashire,PrestonPR12HE,UK. 0 2 DepartmentofPhysicsandAstronomy,UniversityofGlasgow,GlasgowG128QQ,UK. 2 3 InstituteofAstronomy,UniversityofCambridge,CambridgeCB3OHA,UK. n ReceivedOctober5,2007 a J ABSTRACT 4 Aims.Westudythevisibilityofsunspotsanditsinfluenceonobservedvaluesofsunspotregionparameters. ] Methods.Weuse VirtualObservatory tools provided byAstroGrid toanalyse asample of 6862 sunspot regions. By studying the h distributionsoflocationswheresunspotswerefirstandlastobservedonthesolardisk,wederivethevisibilityfunctionofsunspots, p therateofmagneticfluxemergenceandtheratiobetweenthedurationsofgrowthanddecayphasesofsolaractiveregions. - Results.Wedemonstratethatthevisibilityofsmallsunspotshasastrongcentre-to-limbvariation,farlargerthanwouldbeexpected o fromgeometrical(projection)effects.Thisresultsinalargenumberofyoungspotsbeinginvisible:44%ofnewregionsemergingin r t thewestoftheSungoundetected.Forsunspotregionsthataredetected,largedifferencesexistbetweenactuallocationsandtimesof s fluxemergence,andtheapparentonesderivedfromsunspotdata.Thedurationofthegrowthphaseofsolarregionshasbeen,upto a now,underestimated. [ Keywords.Sunspots–Sun:photosphere–Sun:magneticfields–Sun:activity. 1 v 3 0 1. Introduction seamless access to a variety of astronomical data, and at pro- 7 vidingefficientsoftwaretoolsfordataanalysis.Ourworkflows 0 Thebirthofanewspotonthesolardiskindicatestheemergence processedallentriesinthecatalogueandextractedpropertiesof . of magnetic flux through the photosphere, a process which is 1 individualregions,identifiedbytheirNOAAregionnumber,in- keytothesolarcycle(Solanki2003;Fisheretal.2000)andthe 0 cludingthetimeandlocationoftheirfirstandlastobservation. study of stellar magnetic dynamos. Sunspots also cause varia- 8 The longitude of each region at 12:00UT on the days when it 0 tionsinthetotalsolarirradiance,animportantparameterinde- wasfirstandlastobservedwascalculated. : termining the Sun’s influence on climate (Foukaletal.2006). v Figure1showshistogramsofthelocationsatwhichsunspot The presence of a sunspot is key to a solar region being as- i regions were first and last observed, with 0◦ longitude corre- X signed an active region number by the NOAA Space Weather spondingtothe Earth-Sunline.As the Sunrotates,manyspots PredictionCenter(http://www.swpc.noaa.gov)sothatitsevolu- r firstcomeintoviewneartheeast(or‘rising’)limb,causingthe a tionandactivitycanbetracked(Gallagheretal.2007).Thefor- peak to the left in Fig. 1-a. Similarly, the peak in Fig. 1-b cor- mationandevolutionofactiveregionsarefundamentaltosolar responds to regions that rotated out of view. Regions first ob- dynamicphenomenasuchasflaresandCoronalMassEjections servedsufficientlyfarfromtheeastlimb,aregenerallyassumed andtheireffectontheEarthenvironment. tobe‘new’,indicatingtheemergenceofmagneticfluxthrough The visibility of sunspotsis currentlythoughtto be limited the photosphere.As spots movein longitudeby approximately onlybygeometricaleffectsarising fromprojectionof the solar 14.3◦ each day, regions first observed to the west of −60◦ are sphere onto a 2D image, effects referred to as foreshortening. typically described as new emergences (see inset in Fig. 1-a). In this paper we present results obtained serendipitously while Similarlyregionslastseentotheeastof+60◦ areinterpretedas analysing sunspot data by means of Virtual Observatory tools, havingdecayedtothepointthataspotisnolongervisible(see showingthatthevisibilityofsmallsunspotsismuchpoorerthan insetinFig.1-b). predictedbytheforeshorteningmodel. The insets in Fig. 1 display strong east-west asymmetry. A total of 825 new regions are seen to emerge in the bin [−60◦,−40◦], while only 177 in [+40◦,+60◦], a ratio of 4.7:1. 2. Dataanalysis Whatisthecauseofthestrongasymmetryinthesecurves?Why We analysed sunspot group data from the USAF/Mount shouldthenumberofnewregionsemerginginanygivenlongi- Wilson catalogue, for times between 1 December 1981 to 31 tudebinnotbeconstant?Whatisthetruerateofmagneticflux December2005,coveringtwo and a half solar cycles. We pro- emergenceontheSun? cessed the data by means of AstroGrid workflows. AstroGrid An asymmetry in the location of emergence of new (http://www.astrogrid.org)(Waltonetal.2006)istheUK’scon- sunspots as viewed from Earth was discovered 100 years ago tributiontoaglobalVirtualObservatory(VO),aimedatallowing (Maunder1907)andattractedtheattentionoffamousphysicists, 2 Dallaetal.:Invisiblesunspots Fig.2.Valuesofthesumsofcountsinpositiveandnegativelongitude binsateithersideoftheEarth-Sunline,asafunctionofabsolutevalue of the longitude, for region emergences (a) and disappearances (b). Dottedlinesindicatethemean.Thesizesoferrorbarsarealsoshown. Thesizeofeachbinis6◦. served emergingin a unitbin atlongitudeλis (Schuster1911) [SeealsoAppendixA]: Ω N(λ)= N 1− s′(λ) (1) Fig.1.Histogramsofnumberofsunspotregionsversustheirlongitude 1 " k # at12:00UTConthedaywhentheywerefirst(a)andlast(b)observed. Eachlongitudebinis6◦.DataarefromtheUSAF/MtWilsoncatalogue whereΩisthesolarrotationrateandkisthelineargrowthcon- ofsunspotgroups,forthetimerange1December1981to31December stant of a region’s area with time. Similarly, we obtain for the 2005. Thetotalnumber of regionsis6862. Longitude 0◦ corresponds numbern(λ)ofregionsseentodisappearatλ: totheEarth-Sunline,negativevaluestoeasternlongitudesandpositive valuestowesternlongitudes.Insetsshowthehistogramsfortherange n(λ)=n 1+ Ω s′(λ) (2) [−60◦,+60◦]. 1 " l # withn thenumberofregionsthatreachpeakareainaunitlon- 1 gitudeintervalandlthedecaytimeconstant,l>0. Considering two bins centred at +λ and −λ and indicating thenumberofregionsseenemerginginthemas N and N re- + − spectively, one can write Eq. (1) for each of the two bins, and who demonstrated that a visibility function that favours obser- obtainexpressionsforN andN .Byaddingthesetogetherand vations in the centre of the disk, and the curve of evolution + − assuming that the visibility function is symmetric with respect of a spot’s size, can produce such asymmetry (Schuster1911; to λ=0, so that s′(−λ)=−s′(+λ), one finds that N +N =2N Minnaert1939). The graphical representation introduced by + − 1 (Schuster1911)[amisprintappearsinSchuster’sexpressionon Minnaert (1939) makes the cause of the asymmetry immedi- hisp.319].Forregionsdisappearinginthesametwobins,from ately clear. However,Minnaerthimself went on to assume that Eq.(2): n + n =2n =2N . Here we assume that the average theonlyfactorlimitingthevisibilityofsunspotsisgeometrical + − 1 1 number n of regions that reach peak area in a unit longitude effects associated with foreshortening.This results in a visibil- 1 binisequaltotheaveragenumberN ofregionsthatemergein ityfunctionproportionaltosecλ,withλthelongitude,andpoor 1 aunitlongitudebin. visibilityonlyverynearthelimb.Thelatterassumptionhasre- Figure2-ashowsvaluesofN +N versus|λ|obtainedfrom mainedundisputeduntilthepresentday.Moreover,appreciation + − the data of Fig. 1-a by adding counts in positive and negative of the cause of the east-west asymmetry appears to have been λ-bins at either side of λ=0. Figure 2-b shows n +n obtained lost, and asymmetries in sunspot parameters have since been + − fromthedisappearencesdataofFig.1-b.N +N andn +n are ascribed, e.g., to observer bias or systematic inclination of the + − + − approximately constant, as predicted by Schuster’s theory, and magneticfluxtubes(Howard1991). we can use their mean values (indicated by the dotted lines in The large asymmetries in the insets of Fig. 1, however, Fig. 2) to calculate N , the actual number of sunspot regions 1 demonstrate that the visibility of new sunspot formation in emerginginaunitbin.WefindN =160.55±11.41fromFig.2-a [−60◦,+60◦]ispoor,andfluxappearstoemergeanddisperseat 1 and 158.95±12.19 from Fig. 2-b. Considering that all regions locations(andtimes)thatarefarfromthelocations(andtimes) in the USAF/Mt Wilson catalogue are included in our analy- whereactualemergenceanddecaytookplace. sis, andthatthe catalogueis compiledfromobservationsmade We use the dataof Fig. 1 to derivethe rateof regionemer- at discrete intervals of time, the data fit Schuster’s simple the- gence,theaverageratiobetweentheslopesofgrowthanddecay oryremarkablywell.Thereisexcellentagreementinthevalues phasesofregionevolutionandto determinethevisibilityfunc- of N derivedseparately from emergencesand disappearances. 1 tion.Lets(λ)bethevisibilityfunction,givingtheminimumac- The data shown in Fig. 2 are consistent with a constant value tual(asopposedtoapparent)areathatasunspotregionneedsto of N +N and n +n as would be expected from a visibility + − + − reachto bevisibleatlongitudeλ. sisexpectedtohaveamini- functionthatissymmetricwithrespecttoλ=0.Figure2-adoes mumatλ=0,wherevisibilityisbest.Assumingthat,overagiven displayasmallslope,whilethisisnotseeninFig.2-b.Several time period,the numberof magnetic flux emergencesin a unit authors discussed the issue of whether the magnetic flux tubes longitudebin is a constant N , the number N(λ) of regionsob- ofactiveregionspresentasystematicinclinationwithrespectto 1 Dallaetal.:Invisiblesunspots 3 the direction perpendicularto the solar surface. An analysis of magnetogramsshowedasystematicinclinationofmagneticflux tubesofgrowingactiveregionsof24◦ intheW-Edirection,i.e. trailingthe rotation(Howard1991). Thiswouldresultin better visibilityofyoungregionsinthewest,theoppositeoftheeffect we find. We conclude that our data do not show evidence for a strong inclination although this issue may need to be further investigatedwithotherdata. FromEqs.(1)and(2),bysolvingforthefirstderivatives′(λ), and by dividing one equation by the other, we find an expres- sion for the ratio k/l between the growth and decay constants characterising sunspot region evolution. We find a mean value k/l=1.37±0.26(s.d.),from16longitudebins.(Here,weexclude 4longitudebinsnearλ=0inwhich N/N andn/n arecloseto 1 1 1,givingtwotermsclosetozerotobedividedbyeachotherto find k/l). Our value for k/l implies that, on average, the dura- tion of the decay phase of an active region is only 1.37 times largerthanthatoftherisephase.Thisisverydifferentfrompre- viousestimates, accordingto which active regionsreach maxi- mumdevelopmentveryquickly,e.g.within5daysofemergence (Harvey1993),anddecayslowly.Theratiobeingdifferentfrom 1isthecauseofthelackofcompletesymmetryinthedistribu- tions of emergences and disappearances (see insets of Fig. 1). While in generalthecurveofevolutionof sunspotregionswill becomplex,itslinearenvelope,describingatrianglewithslopes k andl,isausefulfirstapproximation.Bymeansofmodelling, wefoundthattheasymmetryin[−60◦,+60◦]islargelyindepen- dentoftheactualtotallifetime ofregions,while lifetimeisthe keyfactorinfluencingtheheightofthepeakinFig.1-a. Having obtained N and the ratio k/l, values of the deriva- 1 tive of the visibility function, s′, given by Eqs.(1) and (2), de- pend on a single parameter, the growth constant k. In Fig. 3-a we plot s′ data points, calculated using Eqs.(1) (squares) and (2)(triangles)foravalueofthegrowthconstantk=40msh/day Fig.3.(a)Derivativeofthevisibilityfunctionfromemergence(squares) and using k/l=1.37 (msh indicatesmillionthsof the visible so- anddisappearance(triangles)data,fork=40msh/dayandk/l=1.37,with larhemisphere).Wefitthefunctions′(λ)=c tan−1(c λ)tothe msh=millionths of the visible solar hemisphere; the solid line shows 1 2 data and obtain a best fit when c =117.0and c =4.7 (the solid the best fit to the data points. (b) Visibility function s(λ), giving the 1 2 lineinFig.3-a).Byintegrationweobtains(λ)=c /c [xtan−1x− minimumareaasunspot needstoreachtobedetectedatlongitudeλ, 1/2ln(1+x2)]+A ,wherex=c λ(withλinra1dia2ns)andA forseveralvaluesofk.ThedottedlineshowsthefunctionAmin secλ. min 2 min is the minimum of the visibility function, set equal to 8 msh, theapproximateminimumarearequiredatthediskcentrefora grouptobeincludedinthecatalogue.Figure3-bshowss(λ)for withforeshortening.Itshowsthattheminimumarearequiredfor thefitofFig.3-a(solidline)andforthoseobtainedfortwoother a spot to be detected at λ=±30◦ is more than twice the thresh- valuesofk.ThedottedlineshowsAmin secλ,thevisibilityfunc- old area at λ=0◦ if k=14 msh/day and almost 4 times if k=30 tion expectedfrom geometricalconsiderationsdescribing fore- msh/day.Thecentre-to-limbvariationinvisibilityisremarkably shortening. large. Sunspotregionsareinrealitycharacterisedbyadistribution We investigated whether the asymmetries shown in Fig. 1 ofgrowthanddecayconstants.ThecurvesofFig.3-btherefore display any solar cycle dependence, and found no evidence of need to be interpreted as the visibility functions arising when it.TheUSAF/MountWilsonlocationsandtimesoffirstappear- the most probable value of k in the distribution is equal to the ancesagreewiththosefromSOHO/MDIcontinuumdata(asver- numerical value given. Recurrent sunspots were found to have ifiedmanuallyfora sampleofregions).Itisknownthatseeing adecayrateconstantl≈10msh/day(MartinezPilletetal.1993), associatedwithground-baseddatadoesnotcausesignificantre- giving k=14 msh/day using our k/l ratio. Due to the very slow ductionofvisibility(Gyorietal.2004). decayofrecurrentspotsthelatterisalikelygoodlowerlimitforl How many spots are affected by the visibility effects here andconsequentlyk.Evenwhenk=14msh/day,sunspotvisibility described? The distribution of sunspot areas measured at a is much worse than expected from projection effects only. To single longitudinal location on the solar disk is lognormal recoverthe geometricalvisibilitycurvefromthe data, it would (Bogdanetal.1988) and the number of spots with area at benecessarytoassumek=3msh/day,avaluenotconsistentwith CentralMeridianaround10mshismorethan2ordersofmagni- observations. tudelargerthanthenumberofspotswithareaofabout100msh. Thereforethemajorityofsunspotswillcrossthevisibilitycurve showninFig.3-b. 3. Discussion Our results have a number of important implications. The Figure 3-b demonstrates that the visibility of small sunspots is first is that the radiative processes that make a small region muchpoorerthanexpectedfromgeometricaleffectsassociated of strong magnetic field appear as a dark spot, have a strong 4 Dallaetal.:Invisiblesunspots centre-to-limbvariation.Thismayproveimportantforthestudy Council and through the EU’s Framework 6 programme. We of sunspots’ 3D structure and will require further investiga- thank Dr Frank Hill for pointing out that an east-west asym- tion. Whether larger spots are also affected by the same pro- metry had beenreportedin the literature and Dr HughHudson cess will also need further study. Reports of centre-to-limb forcomments.L.F.acknowledgesthesupportofPPARCRolling variations of corrected sunspot areas have appeared in the Grant PP/C000234/1 and financial support by the European literature (Gyorietal.2004; Hoytetal.1983). Faculae have a CommissionthroughtheSOLAIRENetwork(MTRN-CT-2006- large centre-to-limb variation, and their contrast changes sign 035484). as one moves towards the disk centre, resulting in their be- ing darker than the surroundingphotosphere at the disk centre References (Lawrenceetal.1993).Thisdemonstratesthattheappearanceof photosphericmagneticfluxtubesstronglydependsontheview- BogdanT.J.,GilmanP.A.,LercheI.,andHowardR.,Astrophys.J.,327,451 ingangle. (1988) The second implication is that actual distributions of DallaS.,FletcherL.,andWaltonN.A.,Astron.Astrophys.468,1103(2007) sunspot lifetimes and areas may differ from the ap- Fisher G.H.,Fan Y., Longcope D.W.,Linton M.G.,and Pevtsov A.A.,Solar Phys.192,119(2000) parent ones derived from observations. The latter have FoukalP.,FrolichC.,SpruitH.&WigleyT.M.,Nature443,161(2006) been used to constrain mechanisms of sunspot formation GallagherP.T.,McAteerR.T.J.,YoungC.A.,IrelandJ.,HewettR.J.andConlon anddecay(Solanki2003;Petrovay&VanDriel-Gesztelyi1997; P.,SolarActivity Monitoring, inSpaceWeather(edLilensten), Astrophysics MartinezPilletetal.1993).Alargenumberofregionsreported andSpaceSciencelibrary,Vol.344,p.15,Springer(2007) GyoriL.,BaranyiT.,TurmonM.,andPapJ.M.,Adv.SpaceRes.34,269(2004) ofshortdurationmayinfacthavelongerlifetimes,andbecross- HarveyK.L.,MagneticbipolesontheSun,PhDthesis,UniversityofUtrecht inginandoutofthevisibilitycurve.TheGnevyshev-Waldmeier (1993) law, stating that a sunspot’s lifetime increases linearly with its HowardR.F.,SolarPhys.134,233(1991) maximumsize,mayneedtobereassessedinlightofourresults. HoytD.V.,EddyJ.A.,andHudsonH.S.,Astrophys.J.275,878(1983) Poorvisibilityofregionemergencemeansthattheactualtimeof LawrenceJ.K.,K.P.TopkaandH.P.Jones,J.Geophys.Res.98,18911(1993) Martinez Pillet V., Moreno-Insertis F., and Vazquez M., Astron. Astrophys. magneticfluxemergencecanbemuchearlierthantheapparent 274,521(1993) time, e.g. for a region seen to emerge at λ=−50◦, by approxi- MaunderA.S.D.,Mon.Not.R.Astron.Soc.67,451(1907) mately2days.Ontheotherhand,alargefractionofnewemer- MinnaertM.,Astron.Nachr.269,48(1939)[AnEnglishtranslationisavailable gencesinthewesternportionofthesolardiskgoundetected.By at:http://www.star.uclan.ac.uk/∼sdalla/minnaert1939.pdf] PetrovayK.andVanDriel-GesztelyiL.,SolarPhys.176,249(1997) using the data in the inset of Fig. 1-a for λ>0 and the value of SchusterA.,Proc.RoyalSoc.LondonA85,309(1911) actualnumberofemergencesN =160obtainedfromthedataof 1 SolankiS.K.,Astron.Astrophys.Rev.11,153(2003) Fig.2,weobtainthat44%ofnewspotsemergingin[0◦,+60◦] Walton,N.A.,Gonzalez-SolarezE.,DallaS.,RichardsA.&TeddsJ.,Astron.& wereinvisible. Geophys.47,3.22(2006) Thepresenceofasunspotiskeytoasolaractiveregionbe- ing assigned an Active Region Number by the NOAA Space Weather Prediction Center (see e.g. Dalla et al. (2007) for the fulllistofcriteria).AregionthathasbeengivenaNOAAnum- berismonitoredanditsactivitytracked(Gallagheretal.2007). NewregionsemerginginthewestoftheSunwiththeirspotsbe- inginvisiblearemissed andnottracked.We concludethatcur- rentcriteriaforassigningActiveRegionNumbersmayneedto berevisedandthatEUVsolarimagesmayneedtoberoutinely usedtosupplementwhite-lightinformation. Sunspots are well known to cause depletions in the total solar irradiance(TSI) (Foukaletal.2006) and many modelsof TSI variation need as inputinformationon the numberand ar- easofsunspots.WhilethesunspotsthatmostaffectTSIarethe largest ones, of area typically well above the visibility thresh- old shown in Fig. 3-b, our results impact TSI studies because theydemonstratethattheapparentageandstageofdevelopment of a sunspot may not correspond to the actual ones. The latter information is required when studying the time dependence of sunspoteffectsonTSIandwhethertheageofaregionisanim- portantfactorindeterminingthemagnitudeofTSIdecrease. The asymmetry in the distribution of emergences was ob- tained,unexpectedly,duringastudyaimingatcrosscorrelating cataloguesofsunspotregionsandflares,bymeansofAstroGrid workflows(Dallaetal.2007).Thisdemonstratestheusefulness of VO tools in making new science possible, by provision of bettertoolsforanalysisoflargedatasets. Acknowledgements Thisresearchhasmadeuseofdata obtainedusing,orsoftware provided by, the UK’s AstroGrid Virtual Observatory Project, which is funded by the Science and Technology Facilities Dallaetal.:Invisiblesunspots,OnlineMaterialp1 Online Material Dallaetal.:Invisiblesunspots,OnlineMaterialp2 AppendixA: Schuster’sequationandMinnaert’s graphicalrepresentation The fact that a visibility function favouring sunspot observa- tions in the centre of the solar disk, should result in an east- westasymmetryinthe numberofregionsseenemergingatthe Sun, is not immediately intuitive. In this Appendix we sum- marise Schuster’s derivation of Eq.(1) of our paper (Schuster 1911) and describe Minnaert’s graphical representation, from whichthecauseoftheasymmetrybecomesimmediatelyappar- ent(Minnaert1939). Two phenomena combine to produce the effect here de- scribed. The first is the fact that sunspot regions are evolving: theirevolutioncanbecharacterisedintermsoftheirareaandin azero-thorderapproximationcanbedescribedbyacurvesuch asshownin Fig. A.1:a growthphasewith slopek anda decay with slope −l (l>0). The second phenomenonis the solar rota- tion, which carries sunspots which emerged in the east of the Sun towards regions of better visibility (i.e. towards the centre Fig.A.2. Schematic of Minnaert’s graphical representation (adapted ofthedisk)andregionsthatemergeinthewesttowardsregions from Minnnaert 1939). The black curve isthe visibilityfunction s(λ) ofworsevisibility. andthegreyparallelcurvesrepresentthegrowthcurvesofsunspotre- The combination of region evolution and rotation, together gions. with the specific form of the visibility function, determines whether a sunspot region is seen or not, and the location and expressesthefactthattheareaatthelocationwherethespotis timeofitsfirstappearance(andofitsdisappearance)toanEarth firstseenisequaltos(λ). observer.ThisisclearfromMinnaert’sgraphicalrepresentation Eq.A.1canbere-arrangedtogive: (Minnaert 1939), as shown in Fig. A.2. The curve in the dia- gram represents the visibility function s(λ), giving the actual Ω Ω area that a sunspot region needs to reach to be visible at lon- λ− s(λ)=λ − A (A.2) 1 0 k k gitude λ. The parallel lines at a slope represent growth phases ofsunspots.Whenasunspotregiongrowingalongagivenline which,differentiated,gives: crosses s(λ), it becomes visible. From this graphical represen- Ω tation it can be seen that if the visibility function has a strong dλ 1− s′(λ) =dλ (A.3) centre-to-limbvariation,manyspotsforminginthewestarein- " k # 1 visible. An east-west asymmetry in the number of regions ob- If N indicates the number of sunspots emerging in a unit servedemergingthusresults,andtheasymmetrydependsonthe 1 gradients′ofthevisibilityfunction. longitudebinandN(λ)thenumberofregionsobservedemerging ina unitbinatlongitudeλ,then N dλ =N(λ)dλ,whichgives, The process qualitatively represented by Minnaert’s graph 1 1 usingEq.A.3: wasquantitativelydescribedbySchuster(1911),asfollows.Let λ indicate the longitude at which a sunspot forms, and λ the 1 Ω longitudeatwhichitisfirstseenbecauseithascrossedthevisi- N(λ)= N 1− s′(λ) (A.4) 1 bilitycurves(λ).Thetwolongitudesarerelatedbythefollowing " k # equation: (Schuster1911)(Eq(1)ofourpaper). Applying the same derivation to the decay phase, charac- λ−λ A +k 1 = s(λ) (A.1) terisedbyaslope−l,weobtainedEq.(2)ofourpaper. 0 Ω where A is the area of the spot when it first emerges, k is the 0 slopeofthegrowthphaseandΩthesolarrotationrate.Eq.A.1 Fig.A.1. Schematic of time evolution of a sunspot’s area. Here A is 0 theareaofthespotatthetimewhenitfirstemergesthroughthephoto- sphere.