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Session 2B: MODES Mode Identification Session 2B:MODES Mode identification Comm. in Asteroseismology Vol. 157, 2008, WrocMl aw HELASWorkshop 2008 M. Breger,W.Dziembowski,&M. Thompson,eds. Pulsationalmodeidentificationfrommulti-colourphotometry − an observer’spoint of view G. Handler Institut f¨ur Astronomie,T¨u rkenschanzstraße 17, A-1180 Vienna,Austria Abstract We review the methods available to performmodeidentificationfromtime-series photometric measurements.Following the developments in time,wediscussthe rootsofthe method, itS basicmethodology, extensions andimprovements.Suggestionsonthe optimal choice of filterstoapplythe methodare quoted andseveral practicalapplications arecited, with a focus on howadditional astrophysicalinformationwas obtained at the same time.Finally, some practicalconsiderations concerning observationalmodediscriminationfromphotometry aregiven, some pitfallsare discussedand abrief glimpseatfuturespace data is made. IndividualObjects: 12 Lac,G226-29, PG 1351+489, 44 Tau, ν Eri, θ Oph Whydoweneed observational mode identifications? Thegoalofany proper asteroseismicstudyisthe gain of knowledgeofstellarstructure andevolution.Thisisfacilitated by the comparison of observed pulsational mode spectra with those predicted by theoreticalmodels. Thedecisive requirement forany stellarmodel thatdeservestobecalled seismicmodel is itsuniqueness: each individualoscillationmode observed must be identifiedwithits pulsationalquantum numbers k, ℓ and m beyond any doubt, and(withinreasonablelimits) only onestellarmodel exists thatcan explain themall. Unfortunately, the pulsationmodes in real starsdonot comelabelledwiththeirk,ℓand m. In some cases, we areluckytoobserve afairly complete setofpulsational signals over some rangeoffrequency.Becausethe frequencies of the stellareigenmodes arenot randomly distributed, frequency groups thathavecertain structure(e.g.,frequency ratios of radial modes,equidistantlyspaced frequencies/periods of p/gmodes,rotationally split frequency multiplets)can aid the identificationofthe pulsation modesthatcause them. This technique is called pattern recognition andhas been appliedsuccessfully to whitedwarf stars(e.g. Winget et al. 1991, 1994). For most pulsators, the observed mode spectraare toosparse or toocomplicated (e.g. if rotationallysplit patterns of modes with different kand/or ℓ overlap in frequency)to facilitatepatternrecognition. In such cases, we must resort to observationalmethods that give additional clues towardS ℓ andm.Bothphotometric andspectroscopictechniques have been developed forthispurpose.Whereasthe latter arereviewedbyTelting (2008), we will discussthe photometric methods in what follows. Numerous excellent reviewsonphotometric mode identificationare present in the litera- ture, most recentlybyDaszy´n ska-Daszkiewicz (2008). Becausethe latestoverviewarticleS emphasizedthe theoreticalaspects of mode identification, we hereattempt to streSS the issues relevant to observation. G. Handler 107 Acautionarytale Even in the presence of sparse observationalmodespectra, it is tempting to applypattern recognition methods (alsonamed”mode identificationbymagic numbers”,Breger 1989). One(in)famous example is the β Cephei star 12 Lacertae. Itsfive dominant pulsation fre- quencies have been knownfor alongtime(Jerzykiewicz 1978). Among them,anexactlyequally spaced frequency tripletispresent, andthe obviouS assumption is thatitisdue to rotational splitting.However,all interpretations in thisdirection failedbecausethe splitting was”tooequidistant”.Therefore, hypothesesconcerningresonant mode couplingwereinvoked to explain thistriplet, andseveral papers were written on the subject. Besides the equally spaced triplet, the present authornoticed thatthe frequency ratioof twoofthe fivethen knownpulsation modes wasconsistent with thatofradial fundamental andfirst overtone mode pulsations-to the fourth digit! This wouldalsosolve the dilemma of the equally spaced tripletasthe hypothesizedfirst overtone mode corresponded to oneof the tripletcomponents. Consequently, aphotometric multisitecampaignon12Lacertaewas organized(Han- dler et al. 2006).Fromits multicolourdataitbecame clearthatthe apparent rotationally split mode tripletconsisted of an ℓ =1,anℓ=0,and an ℓ =2mode. Thesuspected radial modes both turnedout to be ℓ =2.Aspectroscopicmultisite campaign (Desmetetal. 2008) confirms these results. We conclude: apparent recognition of frequency patterns wherethereare none canlead to alot of unnecessary work. Thebasic idea Dziembowski(1977) expressedthe fluxvariations of anonradiallyoscillatingstarmathemat- ically. He thenpointed outthatthe Baade-Wesselinkmethod(developed to determine the radiiofCepheids by combining radial velocity andtwo-colourphotometryoverthe pulsation cycle) canbeusedtoinfer the stellarradiusifthe spherical harmonicdegree ℓ of the oscillation is knownor, vice versa,toinfer ℓ if the star’s radius is known. Buta &Smith (1979) andBalona &Stobie(1979a,b)reformulated Dziembowski’s(1977) expressionsfor easier usewithobservational data andpresented the first applications of the method. In particular,Balona &Stobie(1979b)showedthatpulsation modes of different ℓ separate in aphotometric amplitude ratiovs. phase shiftdiagram. Robinson et al. (1982) adapted the calculations to the light variations of pulsating white dwarfstars andrefined the methodbyincluding stellarmodel atmospheres,replacingthe previously used blackbodyspectral distributionsorempirical colour-brightneSS relations,and at the same time providingamore realisticlimb-darkening law. This work hadtwo important ”side” results: first,Robinsonetal. (1982) showed thatthe oscillations of whitedwarf starS arecausedbyg-mode(as opposedtor-mode) pulsation andsecond, thattheseare due temperaturevariations only. Generalconsiderations and optimaluse of themethod Watson (1988) explored the behaviourofphotometric amplitudes andphasesfor alarge varietyofpulsating star models. He started with the expression forthe fluxchangefor nonradial pulsation in the linearregime: Δm(λ, t)=−1.086ǫPℓ,|m|(µ0)((T1 + T2)cos(ωt + ψT )+(T3+T4+T5)cos(ωt)), whereΔm(λ,t)isthe time andwavelength dependent magnitude variationofanoscillation, −1.086ǫ is an amplitude parameter transformedfromfluxes to magnitudes, Pℓ,|m| is the 108 Pulsationalmodeidentification from multi-colourphotometry associated Lagrange polynomial, µ0 is the cosine of the inclinationofthe stellarpulsation axis with respect to the observer, ω is the angularpulsation frequency, t is time and ψT is the phaselag between the changes in temperatureand localgeometry. Theterm T1 is the localtemperaturechangeonthe stellarsurface, T2 is the temperature-dependent limb darkening variation, T3 is the localgeometrychangeonthe stellarsurface, T4 is the local surfacepressure change and T5 is the gravity-dependent limbdarkening variation. Therelativeimportanceofthe Ti terms wasevaluated formodelsofdifferent typesof pulsators, rangingfromwhite dwarftoRCrB stars, Mirasand Cepheids, δ Scuti and β Cephei stars, as well as solar-likeoscillatorsand roAp stars. For instance, the geometrytermwas found to be of virtually no importance forthe high radial overtone pulsators(Watson 1988). Theresults of thisworkallowanoptimal choice of photometric filtersfor efficient mode discrimination:one wants to observeatone wavelength wherethe geometry variationiS significant, while the temperaturetermiswellconstrained at the same time. Finally,Watson(1988) computed ”areas of interest” in amplitude ratiovs. phase shift diagrams, i.e. the loci in whichmodes of agiven ℓ wouldbecontained (alsocalled ”diagnostic diagrams” in the literature). Such diagrams arestill in usefor mode identificationtoday. Adaptations, extensions, improvements and additional astrophysics After thisverybroad work,specialistsonvarious classesofpulsators took the stage. Garrido et al. (1990) performedcomputationsofdiagnostic diagrams for δ Scuti starsinthe Str¨omgren photometric system,tryingtofind the most efficient filter combinations formode identification(usingtwo filters, v and y seem to be best). Later, Heynderickx et al. (1994) tested β Cephei modelsinthe Genevaand Walraven systems, finding thatthe mode discrim- inationisbestdonevia the evaluationofamplitude ratios,and thatknowledge of ultraviolet amplitudes is crucial. Cugier et al. (1994),alsoexamining modelsofβCephei stars, performednonadiabatic calculations of photometric andradial velocity amplitudevariations andphase shifts forthe first time.Theirresults impliedthattheseparametersmay be dependent on stellarmassand thatdiscriminationofthe overtone of radial modes mayalsobepossible. An apparent setback formodeidentificationofδScuti starswas discussedby Balona &Evers (1999):the shallow convection zones of these starsaffect the theoretical predictions of the photometric observables, depending on theassumedconvective efficiency. However, Daszy´nska-Daszkiewicz et al. (2003) realizedthat thisdependence maybeturned into an asset: when introducing asuitablethird filter into theobservations, onecan constrain the convective efficiency simultaneously with obtainingpulsational mode identifications.In other words, onecan learnadditional physics when applying the photometric method. Thephotometric methodhas been extended further andfurtherinthe recent past.For instance, Garrido (2000) modernizedWatson’s(1988) work withafocus on δ Scuti stars, andpresented diagnostic diagrams for γ Doradus starsfor the first time.Balona et al. (2001) pointed outthatthe incorporationofJohnson Ifilter data greatlyimprovesthe relative accuracyofthe measuredphase shifts andthereforeprovided clearermodediscriminationfor
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