Adv. SpaceRes.Vol. 8, No. 7, pp. (7)15—(7)20, 1988 0273—1177/88$0.00 + .50Printed in Great Britain. All rights reserved. Copyright© 1989 COSPAR

MODELLING OFTOTAL SOLARIRRADIANCE VARIABILITY: AN OVERVIEW

Hugh S.HudsonCenterfor AstrophysicsandSpaceSciences,Universityof California, SanDiego,

LaJolla, California92093, U.S.A.

ABSTRACT

Severalcomponentscontributeto the observedvariationsof the total solarirradiance. There hasbeenconsiderableeffort expendedon buildingempiricalmodelsfor specificcomponents,especiallyfor sunspotsandfaculae.Thesemodelstypically usetimeseriesof ground—baseddata asa meansof representingthetotal—irradiancevariability. There are severalreasonsto do this modeling. The modelsmay help toidentify thephysical causeof a variation; the parametersof a model (e.g. the effective temperatureofa sunspot)may be determinablevia correlationwith the total—irradianceobservations;the models maybe usedasproxy representationsfor total—irradiancevariability during periods of no data;finally, themodelsin principle canbeusedasa basisfor “correcting“ thetotal—irradiancedata,asameansfor betteridentificationof additionalcomponentsof variability.

INTRODUCTION: WHY MAKE MODELS?

Sincethebeginningof quality space-borneobservationsof total solar irradiance(late 1978 for Nimbus-7, early 1980 for the Solar Maximum Mission), it hasbeenclear that severaldifferentphenonsenacontribute to the observedvariability. Smith and Gottlieb (/1/) madeinterestingearly assessmentsofthe contributionsto the solar variability from then—known causessuch as sunspots. When the dataactuallyarrived,therewereof coursesomesurprises.Therearenow six separatemechanismsknownnowto contributeto the observedvariability (/2/); of thesesix only threehadreceivedany significantpriordiscussionor successfulprediction.

Models of the solar total irradiancebeganwith attemptsto find solar variability in theAbbott ground—baseddata,andin thedatafrom theMariner—6and—7 probes/3/. Theterm“model“ in this contextdoesnot refer to anumericalmodel baseduponphysical law, suchasthat of a stellarinterior, but insteadtoanempiricalmatchingof the observationswith a combinationof relatedbut independentdata.Perhaps“interpolationformula“ would be a better termfor this kind of treatmentof thedata in theabsenceofactual physics;howeversomeof the componentsof total—irradiancevariationdo havephysicalmodels tohelp explainthem. In thefuture theempiricalandphysicalmodelingmaybecombined,but this hasnothappenedyet.

We can identify severalreasonsfor empiricalmodel—building:

• Help identify the causeof aparticularvariation.

• Leadto the determinationof aphysicalparameter.

• Providea proxy for estimationof variationsin the absenceof data.

• Reducethevarianceof thedata to make subtlereffectsmorevisible.

This paperreviewsthe existing work on empirical modelsof total irradiance,discussesthe degreetowhich themodels haveaccomplishedthepurposeslisted above,and then suggestssousenewproblemsinthis area.

MODELS OF TOTAL SOLAR LRRADIANCE

Simplesunspotblocking

Theoriginaltotal—irradiancemodels(/3/, /4/) dealtfirst with the “sunspotblocking“ contributionthatisthemostobvious perturbationof the solartotal irradiance.Essentially,suchamodeltakestheprojectedareaof thesunspotspresenton thevisiblehemisphere,asfoundin thesynopticdatarecords,andestimatesa total—irradiancedeficit from the fractionof the solar disk areaoccupiedby the spots. Complicationsarisefromphotosphericlimb darkeningandradiationfromthespotsthemselves— penumbrais differentfrom umbra,the umbraeof differentspotsmayhave different effectivetemperatures(e.g. /5/), and theratio of penumbraland umbral areasmay vary. Figure 1 sketcheshow a singledark spot on the solarsurfaceproducesa “dip“ in total irradiancefrom sucha model.

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(7)16 H. S. Hudson

Figure 1. Simple models of solar total—irradiance“dips“ producedby dark spotsand thereradiationof blockedenergyon thesolarsurface(adaptedfrom/21/), shownassketchesof thetimevariationof to-tal irradianceover approximatelyonemonth. Thedashedline in eachcaseshows the level of the un-perturbedbrightness;from top to bottom we havedirect extraction of energy, i.e. no re—radiation;immediatelocal re—radiotion;immediateglobal re— Lradiation;andlong—termstorageof theblockedflux.

The sunspot—blockingmodelswerequite successfulwith theACRIM data /6/. Hudsonei al. /7/ defineda “PhotometricSunspotIndex“ (PSI) as

PSI = 3.15 x i07 x ~‚ < (3~i,+ ) (1)

wheres~and~z,representtheareaandverticalangleof theith visible sunspotgroup. The factor(3~s~+2)/2representsthe photosphericlimb—darkeningfunction in thesimplestapproximation,and the numericalcoefficient of thesumrepresentsthebestvaluesfor T~,

1of umbraandpenumbra,and theratio of theirareas(as obtainedfrom /8/).

Hudsonand Willson /9/ found that this simple PSI model could “explain“ approximatelyhalf of thevariancein theACRIM‘ data,in thesensethat with no adjustableparametersthesimplePSItime serieshada linear correlationcoefficientof 0.733againsttheACRIM data. Figure2 (from /9/) showsthetimeseriesinvolved in this correlation.

l370~— • ACRIM

u )~‘s~‘e~.is..4‘ ~ :\~“~,°~t~1) V - ‚ ~ Figure 2. Daily meansof data from the Active1364 ~‚ „ Cavity RadiometerIrracliance Monitor (ACRIM)

~ ~ on board SMM (top line); PSI index (secondline),

.~ ~ ~:~ ‚~ ~ and theresidualsin comparingthetwo (third line).~ The bottom two lines give for referencethe 10—cm

— -4 • ACRIM • PSI radio flux indexand the sunspotnumber. The PSIi~i‘~~“ ~ index“predicts“ thedipsin total irradiance,butone

~ shouldnote thepronouncedoscillationsin theresid-~i0T ~. ualsnear thecenterof theyear: thesecorrespond

~ 50~-A~ ~~Aj\_p\15J~v‘ to poorercorrelationbetweenPSI andACRIM, but1201 themselvesshowa good positive correlationwith

SUNSPOT NUMMER the 10—cmflux (from /20/).

047 7? lOT 37 167 iS? 227 23? 267217DAY OP YEAR

960

Incorporatingthe faculae

Various authorsexploredthis correlationbetweenthe blocking of solarluminosity by the sunspotsandthe total irradiance. The obvious next step was to incorporateinto the models the faculae, whichobviously (from white—light picturesof the Sun)were potential contributorsof excessluminosity. Thefirst observationalquestion— since sunspotblocking implied energystorage— was whetheror not thefaculaesiniply re—radiatedthe storedenergy.

Chapman/10/,/ll/ givesfurther detailon this aspectof the modelingof total irradiance.Theessentialproblemwith handling the faculae is their low contrast,especiallyat disk center,togetherwith thelackof good synopticdataon their occurrence.Accordingly, theuseof a chromosphericemissionline suchasthe CaK line seemsattractive.It must,however,becalibratedasaproxy for a direct measureoffacular

‘ACRIM is the Active Cavity Radiometer Irradiance Monitor ofthe Solar Maximum Mission, R.C. Wilson, Principal

Investigator.

Modelling of Total IrradianceVariability: An Overview (7)17

brightness.This meanseitheraseparatesetof observations/12/ or afreeparameterin themodelfitting.

Sofiaei al./13/ give a particularly elaborateexpressionwith manyfree parameters,incorporatingbothspotsandfaculae:

I,,.~,,— I~,= I, ~ A~[A~+ b~,j.i+ cppm][(as + b

3p + C5~L

2— 1) + Q(af + bfg~+ c

1i.~

2— 1)] (2)

wheretheindicesp, s, and f refer to quiet photosphere,sunspot,andfacularespectively,and Q is anotherparameter.

The limb—darkeningdependencesareworth specialnote. Although thecontrastof faculaegrowstowardsthelimb, their actualintensityneverthelessdeclinesmonotonically.Thus theactualradiationpatternofthefacular excessis not a hollow cone, but a filled one.

Multiple—parametermodels

The introduction of multiple parametershas in generalnot been particularly fruitful, except in thesense that slightly better empirical fits can be obtained. The problem is that errors (both randomand systematic)in the correlateddata (spot areasand Ca plage areas)confusethe estimationof theparameters.Figure3 /9/ shows the useof two adjustableparametersin a particularmodelof the total—irradiancevariations:the conclusionin this casewas that the assumptionof instantaneousre—radiationof the sunspotdeficit from eachactive regionmadeabadfit to theactualvariationsobservedby ACRIM.Thestrengthofthis conclusion,however,is weakenedby the fact that the synopticdataon sunspotareasare ill—understoodfrom the point of view of errors. The contoursshownin Figure 3 cannotthereforeeasilybe interpretedin termsof x2 confidencelevels.Further,the additionof two freeparametersto theelementaryPSImodel (no free parameters)did not result in a largereductionof theresidualscatter.

l•c_•_.

0.425Figure 3. Contourplot of the standarddeviation —~~0.40 /of residualsin the ACRIM datafor 1980 aftercor-rection for a two-parameter:~:‚;« ~:action ~0.45

of local, simultaneousre—emissionfrom eachactive a .5 .

regioninto s facular limb—darkeningfunction. The 0.5small value of ß favored by the data suggeststheneedfor energy storage(from /9/). With correctestimatesof errorsin thedata,contoursof this sort 0.6

could he usedto derive the allowablerangesof pa-rametersin simplemodels.

.5 LOs

The inferencefrom Figure 3 (and from our classicalknowledgeof the developmentpatternsof spotsand plage)is that a delaymust occurbetweenthe sunspotluminosity deficit andthe facular lmmnosityexcess. Lawrence(/14/) now estimatesthe meandelay at about0.7 solar rotations (.~19 days) fromdirect facularphotometry.

The other sideof the Sun

We measurethe solar flux variationsfrom a single favoredpoint in space(at leastprior to the launchof the Phobosspacecraftin summer1988), so that a conversionis necessaryto infer actual changesin the solar luminosity. Modelling the relativelylargecontributionsfrom active regionsmight be of aidin assessingthe global variationof luminosity /15/, /7/). To approximatethe active—regionpopulationon theinvisible hemisphereof the Sunrequirescertainassumptionsto be madeabout their growth anddecay, so that any global modelling of this type will have uncertaintiesin it due to theunpredictableelementof active—regionformation. Figure4 /7/ shows a global luminosity index compiled for part of1980, illustrating themore gradualvariationof thetotal luminosity resultingfrom theeliminationof therotationalmodulation(Figure 1).

(7)18 H. S. Hudson

DAY 0F YEAR, 980

50 70 90 110 ISO 50 10 90

1370 (a)369 ~ ...... .... ...~

368 :..~: ~ •....

13671366 Figure 4. Inference of a global luminosity index

(b) •_•~ /7/: (a), daily averagevaluesfor total solarirradi--l •:~ ~. ..: ~ ancefrom ACRIM observations;(b) thePSI model

-2 .~ of sunspotdeficit; (c) the global deficit, obtainedby estimatingthe active—regionpopulationon the

(cl invisible hemisphere;and (d) a resulting model ofthe total—irracliancevariations as seenagain from

-I •.~...— ~ --.- ~- the Earth,incorporatingboth (b) and (e) as if the— -2 global re—radiationof sunspotdeficits were instan-

3 (d) taneous. The global index in (e) effectively sup-o ~ pressestherotationalmodulationof apparentsolar-i -.‚ .. I luminosity.

MAR I APR I MAY I JUN I980

Extendingthemodels: a solar—cycledependenceof total irradiance?

The solar total irradianceappearsto showa solar—cyclecomponentof variation (/167/17/). From themodelling point of view, this is interesting for two reasons.First, it illustrates the incompletenessof

any given model — thenew componentcould not he properly modelledin termsof the active—regionphenomenapreviouslyused. Second,the adjustmentof total—irradiancedatafor sunspoteffects [e.g.,from Equation (1)[ and for facular effects hasmadeit possibleto recognizethe new componentmoreeasily. Figure 5 illustrates the long—term variation as it appearsin the ACRIM data. Thesecorrelationssuggestthe relationships

S + PSI = 1365.65 + 1.774 x 10_2 x F10 (3)

and

S + PSI = 1366.73 + 1.526 X 10_2 X R~ (4)

as long—tarinrelationshipsbetweenthe total irradianceS and the 10—cmflux F10 and sunspotnmnberRz, respectively;notethe useof PSI to makeuseof thebestof thecorrelationsshownin Figure5. Notealso the differencein S valuesfor PSI = 0, an indication of theprecisionof theserelationships.

WHAT HAS BEEN LEARNED FROM MODELS?

Going throughthe list of purposesfrom the Introductionabove,let us see how well modellinghashelpedin understandingsolar total—irradiancevariations.

• Identify thecause. The largestindividual effectson the total solarirradiance,namely the sunspotsand the faculae,wererathersharplyidentifiedby meansof the variationspredictedby niodels. Thisdoesnot meanthat the causeswould not havebeen correctlyfound without the isiodels,but thequantitativeagreementhasbeenreassuring.

• Determinea parameter. Although thereare many parametersthat are interestingand need to hedetermined,themodel—buildingeffort hasnot yet reachedthe level of sophisticationat which anyrealresultscould be obtained.

• Proxy data. The inclusion of the secular(solar—cycle?) term in total—irradiancemodeling (/17/,/16/) maynow make it possibleto go backwardsin time more successfully(e.g. /18/) in ordertoassesstotal—irradianceeffects on climate.

• Reducethe variance. The eliminationof the PSI dependencemadethe facular contribution andthesolar—cycle(?) termeasierto recognize. Unfortunately,the scatterof residualsremainsquitelargeevenaftermodel correctionsfor sunspotsand faculae. This situationmaypersistuntil bettersupportingdatabecomeavailable.

Modellingof TotalIrradianceVariability: An Overview (7)19

1371. 1371 IlIllIllIllilIlIl 1371 IiIIjiIiIIIiiI

1370 1370 1370

E~j1369 _EEE ~ 1369 •~ • I 1369

+ 3.

— 1368 ~ 1368 1368E Li

;

1367 E E s 1367 1367 • E

E •

3366 kIL~IIIIlI;,Il, 1366 hTIhlhI~Ihl~~ 1366 NI III 111111,O ~i0O 200 300 0 100 200 300 0 100 200 300

rio rio rio

Figure5. Correlationof total irradiance,asmeasuredby theACRIM instrumenton board the SolarMaximum Mission (daily means)with the solar 10—cm flux index /17/. Left panel shows the totalirradiance,middle theresidualsaftercorrectionfor sunspots[the PSI indexof Equation(1)[, and rightthe residuals after an additional and similar correctionfor faculae. The two cloudsof points comefrom 1980 and from 1984—1986,respectivelyperiodsof high andlow solar activity, and their separationindicatesthe magnitudeof thelong—termvariationtentativelyidentifiedwith the solarmagneticcycle.

FUTURE PROSPECTS

After theinitial excitementin the early1980‘swith theACRIM data,little progressin modelinghasbeenmade.This is unfortunatebut may change,becausenow we havea muchlongertimne seriesof excellenttotal—irradiancedataavailablefor use. Formanypurposes,thenewerdataarein factbetterthanthe olderones,becauseof the reducedfrequencyof occurrenceof active regionsduring solarminimum conditions.This mayhelp to limit theconfusionamongdifferentkindsof variation. The following list suggestssomeapplicationsof thesedata.

1. Albregtsonet al./19/ haveobserveda variation of effective temperaturesin sunspotumbraewithphasein the solar cycle. If presentsystematically,this shouldshowup as a variation of the multi-plicativeconstantof thePSI index.

2. Limb—darkeningcoefficientsfor sunspotsand faculaemaybe derivablefrom adjustmentof param-eters in modelsof thePSI type.

3. With theadventof the Phobossolarirradiancedata, therewill be achanceto study thedirectivityof solar variability.

4. Hypotheticallow—frequencyvariability due to othercauses(e.g. large—scaleconvectivecells) mayhe discoveredby meansof improvementsin thetreatmentof sunspotsandfaculae.

5. Different solar propertieshavedifferent solar—cyclevariations. Therelationshipbetweenthe solarluminosity and theseproperties(e.g. the 10—cm flux) will be an interestingapplicationof total—irradiancemodels in theincreasingphaseof thecycle 22.

Many other specific researchproblemsprobably can be approachedby thetechniquesof modelling thetotal irradiancein termsof independentdatasets, suchassunspots.I would like to point out, however,that the weak link in thesestudieswill be in the independentdata, which are normally compiledas apart of the varioussynopticprograms,and in manycasesarenot well—understoodas quantitativedatasets. On the other hand,theobservationsof total irradiancefrom spacehave superbquality; it is likelythat suchdatawill continueto beobtainedfromvariousvehicles,and that spectralirradiancedatawillbe addedto the database.

Acknowledgement.This researchwas supportedby NASA underthe SMM GuestInvestigatorProgram.

(7)20 H. S. Hudson

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