aa r X i v : . [ a s t r o - ph ] S e p Comm. in AsteroseismologyVol. number, publication date (will be inserted in the production process)
Identifying pulsation modes from two-passbandphotometry
Jadwiga Daszy´nska-Daszkiewicz
Instytut Astronomiczny, Uniwersytet Wroc lawski, ul. Kopernika 11, Poland
Abstract
I discuss a prospect for mode identification from two-passband photometryof forthcoming BRITE space mission. Examples of photometric diagnostic dia-grams are shown for three types of main sequence pulsating variables: β Cephei,Slowly Pulsating B-type and δ Scuti stars. I consider also taking into accountthe radial velocity data from simultaneous spectroscopy, which can be carriedout from the ground. With such observations, much better discrimination ofthe spherical harmonic degree, ℓ , can be accomplished and more constraints onstellar parameters and input physics can be derived. Introduction
Nowadays space based observations allow detecting oscillation modes with lowerand lower photometric amplitudes. We are already at the detecton thresholdof the order of − − − mag, resulting in a growing number of frequencypeaks. There are many examples of excellent work both observational andtheoretical based on WIRE and MOST data (e.g. Bruntt et al. 2007, Walker etal. 2005, Barban et al, 2007, Saio et al. 2007). Now we expect similar resultsfrom the just initiated COROT mission.However, for using these rich frequency data for asteroseismic modelling,mode identification is a prerequisite. In the case of main sequence pulsators,we are still far from obtaining very regular patterns in the oscillation spectra,which could help in solving the problem. We are also far from nonlinear theory,which would answer a question about mode selection mechanism. The mainunresolved problem is why most of the theoretically unstable modes are notobserved. Nowakowski (2005) suggested that the dominant effect of limiting Identifying pulsation modes from two-passband photometry the mode amplitude is a collective saturation of the opacity driving mechanism,instead of a resonant mode coupling.In order to identify modes in β Cephei, δ Scuti or Slowly Pulsating B-type stars, we need additional observables from multicolour photometry or/andspectroscopy. The photometric method of mode identification consists in usingamplitude ratios and phase differences in different passbands; it is based on thesemi-analytical formula for the light variation due to linear pulsation derived byDziembowski (1977). Balona & Stobie (1979) showed that modes with differ-ent values of ℓ are located in different regions on the amplitude ratios vs phasedifferences diagram. Since then, the method has been applied to various typesof pulsating variables by many authors (Watson 1988, Garrido et al. 1990,Heynderickx et al. 1994). The next important improvement was includingnonadiabatic calculations by Cugier, Dziembowski & Pamyatnykh (1994), whoapplied the method to β Cep stars. Then, Balona & Evers (1999) emphasizedthe problem of very high sensitivity of photometric amplitudes and phases tothe treatment of convection in the case of δ Sct stars. A photometric identi-fication of ℓ for SPB stars, based on nonadiabatic calculations, was performedby Townsend (2002). All these works were done by assuming that the rotationdoes not influence pulsation. However, main sequence pulsators are very oftenrapid rotators. The next step in developing the method was its extension toclose frequency modes coupled by a fast rotation by Daszy´nska-Daszkiewicz etal. (2002). The photometric method in the version for long-period g-modesin rotating stars, for which perturbation approach is no longer adequate, wasformulated by Townsend (2003a) and Daszy´nska-Daszkiewicz et al. (2007).A few years ago, Daszy´nska-Daszkiewicz et al. (2003,2005a) proposed anew method of the identification of ℓ , which uses the amplitudes and phasesthemselves and combines photometry, and radial velocity data. The methodallows also to extract simultaneously a new asteroseismic probe, which yieldsconstraints on stellar parameters and input physics, e.g., convection, opacities.BRITE (BRIght Target Explorer) is the first space-based mission which willperform two-colour photometry of bright stars. It will give an opportunity of notonly detecting low-level oscillations but also of identifying their degrees ℓ . How-ever, much more could be achieved, if ground-based spectroscopic observationsare organized simultaneously. Then another type of information, contained inthe radial velocity and line profile variations, would be supplied.The aim of this paper is to show what can be done for mode identifica-tion from observations with the two BRITE passbands. I will recall the basicformulae and show examples of photometric diagnostic diagrams for models of β Cephei, SPB and δ Scuti stars. Then I will present diagrams which includeamplitudes and phases from photometric and radial velocity variations. Finally,I will discuss uncertainties arising from effects of rotation, convection and at- adwiga Daszy´nska-Daszkiewicz
Mode identification from photometry
In order to calculate photometric amplitudes and phases, two inputs are needed.They come from • nonadiabtic theory of stellar pulsation, • models of stellar atmospheres.We assume linear pulsation theory, which is adequate because of small modeamplitudes in main sequence pulsators, and we use temporally static, plane par-allel atmosphere, which is justified because the eigenfunctions of the consideredmodes are nearly constant in the atmosphere.Let us consider a pulsation mode in the zero-rotation approximation, thegeometry of which can be described by a single spherical harmonic, Y mℓ , withthe degree ℓ and the azimuthal order m . The shape of the radial eigenfunc-tions of the mode are determined by its radial order n . Then the local radialdisplacement is given by δrR = εY mℓ ( θ, φ )e − i ωt , (1) where ε is the intrinsic mode amplitude, ω is the angular pulsation frequency,which of course depends on ( n, ℓ, m ); other symbols have their usual meanings.The corresponding changes of the bolometric flux, F bol , and the local gravity, g , are given by δ F bol F bol = εf Y mℓ ( θ, φ )e − i ωt , (2) and δg eff g eff = − ε (cid:18) ω R GM (cid:19) Y mℓ ( θ, φ )e − i ωt . (3) The complex parameter, f , describes the ratio of the local flux perturbation tothe radial displacement at the level of the photosphere and it is obtained fromnonadiabatic calculations.The complex photometric amplitudes of the light variation in the passband λ due to a pulsation mode with frequency ω can be written in the followingform A λ ( i ) = − . εY mℓ ( i, b λℓ ( D λ ,ℓ + D ,ℓ + D λ ,ℓ ) (4) where D λ ,ℓ = 14 f ∂ log( F λ | b λℓ | ) ∂ log T eff , (5 a ) Identifying pulsation modes from two-passband photometry D ,ℓ = (2 + ℓ )(1 − ℓ ) , (5 b ) D λ ,ℓ = − (cid:18) ω R GM (cid:19) ∂ log( F λ | b λℓ | ) ∂ log g (5 c ) and i is the inclination angle. The partial derivatives of log( F λ | b λℓ | ) over effec-tive temperature and gravity are derived from atmospheric models and b λℓ = Z h λ ( µ ) µP ℓ ( µ ) dµ, (6) is the disc-averaging factor, containing the information about the visibility ofthe mode with a given degree ℓ . The integrals b λℓ are weighted by the limb-darkening law, h λ ( µ ) . The term D λ ,ℓ describes the temperature effects, theterm D ,ℓ stands for the geometrical effects, and the influence of gravity changesis contained in the term D λ ,ℓ . The terms D λ ,ℓ and D λ ,ℓ include the perturbationof the limb-darkening, and their ℓ -dependence arises from the nonlinearity ofthe limb-darkening law. B passband R passband |b |
Figure 1: The disc-averaging factor, b λℓ , as a function of the spherical harmonic degree, ℓ , for the BR Johnson filters. The adopted stellar parameters are log T eff = 3 . and log g = 4 . . The amplitudes and phases of the light variation are given by |A λ | and arg( A λ ) , respectively. Having these numbers, we can construct photometricdiagnostic diagrams in the form A x /A y vs. ϕ x − ϕ y , where x and y denotepassbands. These observables are independent of the intrinsic amplitude, ε ,inclination angle, i , and azimuthal order, m , because the product εY mℓ dropsout. These is an advantage for the identification of ℓ but also a disadvantagebecause the order m is beyond of the reach of the photometric method. adwiga Daszy´nska-Daszkiewicz BR Johnson filters which are not very different from the BRITE passbands: BT1(390-460 nm) and BT2 (550-700 nm). The well known property of the factor b λℓ is that it decreases very rapidly with growing degree ℓ ; this can be seenfrom Fig. 1. In this paper I will consider modes with degrees up to ℓ = 6 .All calculations were done with the Warsaw-New Jersey evolutionary code andnonadiabatic pulsation code of Dziembowski (1977). I used OPAL opacity ta-bles of Iglesias & Rogers (1996) and the solar chemical composition of Grevesse& Noels (1993), assuming the metallicity Z = 0 . . I adopted Kurucz atmo-spheric models in the NOVER-ODFNEW version (Castelli & Kurucz 2004),which have more smooth flux derivatives than the standard Kurucz models,and the Claret nonlinear formula for the limb-darkening law. The standardatmospheric metallicity, [m/H]=0.0, and the microturbulence velocity, ξ t = 2 km/s, were assumed. β Cephei pulsators -0.2 -0.1 0.0 0.1 0.20.81.01.21.41.61.82.02.2 0.00 0.05 0.100.91.01.11.21.31.41.5 =6=4=2=1=5=3=0, p =0, p A B / A R B - R [rad] =6=4=2=3=5=1 =0, p =0, p B - R [rad] Figure 2: The locations of unstable modes with degrees ℓ up to 6 for β Cephei starmodels of 12 M ⊙ on the diagnostic diagrams involving Johnson B and R filters. Theleft panel contains all models in the main sequence evolutionary phase, and the rightpanel, the model with log T eff = 4 . and log g = 3 . . I considered stellar models with a mass of M = 12 M ⊙ during main sequencephase of evolution, corresponding to the temperature range of log T eff = 4 . − . . In Fig. 2, the photometric diagram in the BR passband is presented.The left panel shows all modes which become unstable between ZAMS and Identifying pulsation modes from two-passband photometry
TAMS and the right panel shows unstable modes for only one stellar modelwith log T eff = 4 . and log g = 3 . . Modes with different degrees ℓ arelocated in separated regions. The radial modes are spread over a wide rangeof the amplitude ratios and phase differences, whereas the nonradial modes areconcentrated in small areas. This behaviour results from different contributionsof the temperature ( D λ ,ℓ f ) and the geometrical effects ( D ,ℓ ) to the lightvariation (Daszy´nska-Daszkiewicz et al. 2002)Slowly Pulsating B-type starsIn order to calculate SPB oscillation, I chose the models with M = 5 M ⊙ onthe main sequence, which include log T eff from 4.235 to 4.134. Positions ofmodes with different values of ℓ on the photometric BR diagram are shown inFig. 3. Again, the left panel contains all unstable modes, and the right panel,one model with log T eff = 4 . and log g = 4 . . The well-known propertyof this type of diagrams is the zero phase difference and the same amplituderatio for all modes with ℓ = 1 . This is because the light variation of the ℓ = 1 mode in SPB models is totaly dominated by the temperature effects. We cansee an overlapping of domains with different values of ℓ . This can be partlyremoved by considering only one model (the right panel). Another instructive A B / A R B - R [rad] =1=2=3=4=5=6 B - R [rad] Figure 3: The same as in Fig. 2 but for Slowly Pulsating B-type star models of 5 M ⊙ . The left panel contains all models in the main sequence evolutionary phase andthe right panel, the model with log T eff = 4 . and log g = 4 . . information can be drawn from instability conditions. Fig.4 shows the amplituderatio and the phase difference as a function of frequency for the model with adwiga Daszy´nska-Daszkiewicz =1=2=3=4=5=6 A B / A R frequency [c/d] B - R [ r ad ] frequency [c/d] Figure 4: Amplitude ratios (on the left) and phase differences (on the right) in the BR bands as a function of oscillation frequency for the SPB model with log T eff = 4 . and log g = 4 . . log T eff = 4 . and log g = 4 . . As one can see, modes with ℓ = 1 areunstable only for the lowest frequencies ( ν < . c/d). The instability is shiftedto the higher frequencies for higher ℓ -modes, e.g., the ℓ = 5 modes becomeunstable for ν > . c/d. δ Scuti pulsatorsAs representatives of δ Sct pulsator, I took models with M = 2 M ⊙ in themain sequence phase. The corresponding log T eff range is (3.963, 3.854). Allcalculations were made under assumptions of the mixing length theory andthe convective flux freezing approximation. The mixing length parameter was α conv = 0 . , i.e., an assumption of inefficient convective transport. The photo-metric BR diagram for this type of pulsators is presented in Fig. 5. All unstablemodes of M = 2 M ⊙ models are shown on the left hand side, and modes inthe ( log T eff = 3 . , log g = 4 . ) model on the right hand side. As we cansee, there is some overlap, especially for the ℓ ≤ modes. Fixing the modelparameters helps in removing this ambiguity.The common opinion about δ Sct pulsation modes is that the main infor-mation about the ℓ values is contained in the phase differences. In fact, this istrue only for low degree modes with ℓ ≤ , but for higher ℓ ’s we can learn muchalso from the amplitude ratios. In Fig. 6, the amplitude ratio (the left panel)and phase difference (the right panel) as a function of oscillation frequency aredepicted. As we can see, not much can be we achieved for low degree modes Identifying pulsation modes from two-passband photometry -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.050.81.01.21.41.61.82.02.22.4 -0.15 -0.10 -0.05 0.001.01.21.41.61.82.02.2 =3 =6=4=5 =2 =1 =0 A B / A R B - R [rad] =2 =1=0=6=4 =3=5 B - R [rad] Figure 5: The same as in Fig. 2 but for δ Scuti star models of 2 M ⊙ . The left panelcontaines all models in the main sequence evolutionary phase, and the right panel,the model with log T eff = 3 . and log g = 4 . .
22 24 26 28 30 32 34 36 38 401.01.21.41.61.82.02.2 22 24 26 28 30 32 34 36 38 40-0.16-0.14-0.12-0.10-0.08-0.06-0.04-0.02 =0=1=2=3=4=5=6 A B / A R frequency [c/d] =0 =6=2=4=5=3=1 B - R [ r ad ] frequency [c/d] Figure 6: Amplitude ratios (on the left) and phase differences (on the right) in the BR bands as a function of oscillation frequency for the δ Sct model with log T eff = 3 . and log g = 4 . . also from the A B /A B vs. frequency plot because the ℓ ≤ modes are excitedwith very close frequencies. Another interesting property to note is that forvery high frequency mode ( ν > c/d), the phase differences are almost thesame for all degrees ℓ . adwiga Daszy´nska-Daszkiewicz Adding radial velocity measurements
The radial velocity variation averaged over stellar disc is expressed by the well-known Dziembowski’s (1977) formula V rad ( i ) = i ωRεY mℓ ( i, (cid:18) u λℓ + GMR ω v λℓ (cid:19) , (7) where u λℓ = Z h λ ( µ ) µ P ℓ ( µ ) dµ, (8 a ) and v λℓ = ℓ Z h λ ( µ ) µ ( P ℓ − ( µ ) − µP ℓ ( µ )) dµ. (8 b ) From observations, the radial velocity variations are determined by calculatingthe first moment, M λ , of a well isolated spectral line.In this section, I show examples of diagnostic diagrams constructed fromthe radial velocity variation and the light variation in the Johnson R filter.Unstable oscillation modes for β Cep, SPB and δ Sct star models considered inthe previous section are presented in Fig 7, 8 and 9, respectively. As we can -3 -2 -1 0 1 20.00.20.40.60.81.01.21.4 -3 -2 -1 0 1 20.00.20.40.60.81.01.21.4 =6=2 =4 =5=3=1=0, p =0, p A V r ad / A R [ k m / s / mm ag ] Vrad - R [rad] =6=2 =4=5=3=1=0, p =0, p Vrad - R [rad] Figure 7: The location of unstable modes with degrees ℓ up to 6 for β Cep modelsof 12 M ⊙ on the diagnostic diagrams constructed with amplitudes and phases forthe R passband and the radial velocity variations. As in Fig. 2, the left panel showsall main sequence models, and the right panel, one model with log T eff = 4 . and log g = 3 . . Identifying pulsation modes from two-passband photometry -1 0 1 2 3 4 50.00.51.01.52.02.53.03.54.0 -1 0 1 2 3 4 50.00.51.01.52.02.53.03.54.0 =6=2=4 =5 =3=1 A V r ad / A R [ k m / s / mm ag ] Vrad - R [rad] =6=2=4 =5 =3=1 Vrad - R [rad] Figure 8: The same as in Fig. 7 but for SPB models of 5 M ⊙ . In the right panelunstable modes for the model with log T eff = 4 . and log g = 4 . are plotted. -2 -1 0 1 20.00.10.20.30.4 -2 -1 0 1 20.00.10.20.30.4 =0 =6=2 =4=5=3 =1 A V r ad / A R [ k m / s / mm ag ] Vrad - R [rad] =0 =6=2 =4=5=3 =1 Vrad - R [rad] Figure 9: The same as in Fig. 7 but for δ Sct models of 2 M ⊙ . In the right panelunstable modes for the model with log T eff = 3 . and log g = 4 . are plotted. see, the configuration of ℓ modes domains differs from that in the photometricdiagrams. In particular, there are larger phase differences between light andradial velocity variation than between two passnands.Moreover, with two photometric passbands and the radial velocity data,one can apply the method of simultaneous extracting from observations the adwiga Daszy´nska-Daszkiewicz ℓ and the nonadiabatic parameter f . Determination of the empiricalvalue of f allows to avoid, in the process of the ℓ -identification, the inputfrom pulsation models, which still needs many improvements, for example thepulsation-convection interaction, opacity tables, effects of diffusion, mixing etc.On the other hand, the parameter f constitutes a new asteroseismic probe,giving information on subphotospheric layers and is complementary to oscillationfrequencies determined by stellar interior. Comparing empirical and theoreticalvalues of f , one can draw conclusions about the efficiency of convection in δ Sct stars (Daszy´nska-Daszkiewicz et al 2003,2005b) or about opacities in β Cepstars (Daszy´nska-Daszkiewicz et al. 2005a).
Uncertainties
There are many sources of uncertainties in all results presented in this paper.Firstly, all calculation were done assuming the zero rotation approximation. Ef-fects of rotation can spoil the nice mode separation in the diagnostic diagrams,making them dependent on the inclination, rotation rate and the azimuthalorder m . One such effect is rotational mode coupling. It takes place if thefrequency distance between modes, with the degrees ℓ differing by 2 and thesame m , is of the order of the rotation frequency. In such a situation, the pho-tometric amplitude for the coupled mode has to be calculated as superpositionof all mode amplitudes satisfying the conditions ℓ k = ℓ j + 2 and m k = m j .The effect of rotational mode coupling on the photometric diagrams for β Cepstar was studied by Daszy´nska-Daszkiewicz et al. (2002). Examples for δ Sctstars can be found in Daszy´nska-Daszkiewicz (2007). Another case when therotation has to be included is when the pulsational frequency, ω , is of the or-der of the rotational frequency, Ω , so that the perturbation approach fails. Ithappens often in the case of the rapidly rotating SPB stars, in which high or-der gravity modes are excited. Such slow modes are no longer described bythe spherical harmonics, and more complicated formalisms are needed. Onepossibility is the use of the traditional approximation which allows expressingthe angular dependence of eigenfunctions by the Hough functions (e.g. Lee &Saio 1997, Townsend 2003b). The formula for the light variation due to lowfrequency oscillation was given by Townsend (2003a). Daszy´nska-Daszkiewiczet al. (2007) discussed a prospect for mode identification from diagnostic dia-grams and derived the expression for the radial velocity variation.Another uncertainty comes from the input physics and atmospheric models.The effect of metallicity parameter, Z , and opacities on the diagnostic diagramsfor β Cep star models was considered by Cugier et al. (1994). For a highervalue of Z , the separation of modes with different ℓ ’s is much better. Similarly,computations with the OP tables, instead of OPAL, lead to a little better ℓ dis-2 Identifying pulsation modes from two-passband photometry crimination, especially the radial modes are much more spread. Then, Cugier& Daszy´nska (2001) checked the effect of the atmospheric metallicity param-eter, [m/H], and the microturbulence velocity, ξ t , on the diagnostic propertiesof photometric diagrams. For β Cep stars these parameters have negligibleimpact.The main problem in applying the method of diagnostic diagrams to δ Sctmodes is that photometric amplitudes and phases exhibit a strong dependenceon the treatment of convection (Balona & Evers 1999). To circumvent thisproblem, Daszy´nska-Daszkiewicz et al. (2003, 2005b) invented the method ofsimultaneous determination of degree ℓ and the nonadiabatic parameter f fromobservations. Then we can identify the ℓ -value independently of the pulsationmodels and, by comparing empirical and theoretical f values, constraints onconvection can be inferred. The result was that the convective transport in δ Sctstars studied by us is rather inefficient. The progress in modelling δ Sct pulsationwith time-dependent convection treatment was achieved e.g. Grigah`cene et al.(2005), Dupret et al. (2005a) and Dupret et al. (2005b).In the case of δ Sct models, the uncertainties in atmospheric models canplay much more important role than in the B-type pulsators. In Kurucz stan-dard models, the flux derivatives over effective temperature and gravity arenot smooth at the temperature where convective transport becomes important.This is because of using an overshooting approximation that moves the fluxhigher in the atmosphere, above the top of the nominal convection zone. Inthe NOVER-ODFNEW models computed by F. Castelli the problem of non-smooth derivatives does not exist. There are also NEMO models (N¯ ew Mo delGrid of Stellar Atmospheres) which include different treatment of convection.The models were computed by the Vienna group with modified versions of theKurucz ATLAS9 code. The grids have smaller steps in T eff and log g than in Ku-rucz’s computations, and the flux derivatives are perfectly smooth. The effectof using various atmospheric models in the calculation of δ Sct observables wasdiscussed by Daszy´nska-Daszkiewicz et al. (2004) and Daszy´nska-Daszkiewicz(2007).
Conclusions
There is a potential for mode identification from the BRITE photometry. As wecould see, two-colour information can yield some constraints on the sphericalharmonic degree, ℓ , of the oscillation modes excited in main sequence pulsators.However, space observations should be followed up by simultaneous ground-based spectroscopy. The advantages of adding the spectroscopic variations isobvious. By combing photometry and the radial velocity data, we will improvesignificantly the discrimination of ℓ and infer better seismic constraints on stel- adwiga Daszy´nska-Daszkiewicz m , becomes possible. With such unprece-dented data, we can hope for a great step in asteroseismology of main sequencepulsators with the BRITE-Constellation. Acknowledgments.
The author thanks Werner Weiss for inviting herto participate in the BRITE Workshop and Miko laj Jerzykiewicz for carefullyreading the manuscript. This work was supported by the Polish MNiSW grantNo. 1 P03D 021 28.
References