Mode identification in rapidly rotating stars from BRITE data
MMode identification in rapidly rotatingstars from BRITE data
Daniel R. Reese , Marc-Antoine Dupret and Michel Rieutord ·
1. LESIA, Observatoire de Paris, PSL Research University, CNRS, SorbonneUniversit´es, UPMC Univ. Paris 06, Univ. Paris Diderot, Sorbonne Paris Cit´e, 5place Jules Janssen, 92195 Meudon, France2. Institut d’Astrophysique et G´eophysique de l’Universit´e de Li`ege, All´ee du 6 Aoˆut 17,4000 Li`ege, Belgium3. Universit´e de Toulouse, UPS-OMP, IRAP, Toulouse, France4. CNRS, IRAP, 14 avenue Edouard Belin, 31400 Toulouse, FranceApart from recent progress in Gamma Dor stars, identifying modes in rapidly ro-tating stars is a formidable challenge due to the lack of simple, easily identifiablefrequency patterns. As a result, it is necessary to look to observational methodsfor identifying modes. Two popular techniques are spectroscopic mode identifica-tion based on line profile variations (LPVs) and photometric mode identificationbased on amplitude ratios and phase differences between multiple photometricbands. In this respect, the BRITE constellation is particularly interesting asit provides space-based multi-colour photometry. The present contribution de-scribes the latest developments in obtaining theoretical predictions for amplituderatios and phase differences for pulsation modes in rapidly rotating stars. Thesedevelopments are based on full 2D non-adiabatic pulsation calculations, usingmodels from the ESTER code, the only code to treat in a self-consistent waythe thermal equilibrium of rapidly rotating stars. These predictions are thenspecifically applied to the BRITE photometric bands to explore the prospects ofidentifying modes based on BRITE observations.
Rapidly rotating stars intervene in many areas of astrophysics. For instance, themajority of massive and intermediate mass stars are rapid rotators ( e.g.
Zinnecker& Yorke, 2007; Royer, 2009). Primordial stars are also expected to be rapid rotatorsdue to their low metallicity and hence, opacity ( e.g.
Ekstr¨om et al., 2008). Rapidrotation is also thought to play a key role in the precursors to gamma-ray bursts ( e.g.
Woosley & Heger, 2006). Understanding these stars would yield valuable informationto these areas of astrophysics.Rapid rotation introduces many new phenomena in stars. These include centrifu-gal deformation, gravity darkening, baroclinic flows, and various forms of turbulenceand transport phenomena ( e.g.
Maeder, 2009; Rieutord et al., 2016). As a result,there are many uncertainties in the models and a need for further observational con-straints. Currently, one of the best ways of constraining stellar structure is throughasteroseismology. However, identifying pulsation modes in rapidly rotating stars, i.e. finding the correspondence between observed and theoretical pulsations, is a pta.edu.pl/proc/2018mar21/123 PTA Proceedings (cid:63)
March 21, 2018 (cid:63) vol. 123 (cid:63) a r X i v : . [ a s t r o - ph . S R ] M a r aniel R. Reese, Marc-Antoine Dupret, Michel Rieutordformidable challenge ( e.g. Goupil et al., 2005). Indeed, the pulsation spectra lacksimple frequency patterns, and reliable predictions for mode amplitudes are notavailable given that modes in these stars tend to be excited by the κ mechanism. Asa result, it is necessary to apply mode identification techniques.There are two main types of mode identification techniques. The first is basedon multi-colour photometry and involves looking at ratios between pulsation am-plitudes, and phase differences, in different photometric bands. These signaturesdepend on the geometry of the pulsation mode but are independent of the intrinsicmode amplitudes. With its two colours, the BRITE mission is an ideal source ofspace-based multi-colour photometric observations of pulsating stars. The secondapproach is based on spectroscopy and consists in looking at how the profile of agiven absorption line changes with time. These variations are known as line profilevariations (LPVs) and provide a very rich information which can be complementaryto that provided by the photometric approach. Both of these techniques need to beadapted to rapid rotation.In the following, we will focus on photometric mode identification techniques.We will specifically look at results in the BRITE photometric bands and see up towhat extent mode identification may be constrained. The next section describesthe prerequisites for coming up with reliable predictions as well as the calculationscarried out. This is followed by various results which focus on amplitude ratios, phasedifferences, and complex asteroseismology. A discussion concludes these proceedings. In order to calculate reliable mode visibilities, it is necessary to carry out 2D pulsa-tion calculations which fully take into account the effects of rapid rotation, in orderto correctly calculate the geometry of the modes. For instance, low-degree acousticmodes become island modes at rapid rotation rates (Ligni`eres & Georgeot, 2008,2009). These modes take on an elongated structure which circumvents the equatorand is characterised by a new set of quantum numbers (˜ n, ˜ (cid:96), m ) ( e.g. Reese, 2008).Calculating such modes as well as other modes present in rapidly rotating stars re-quires the use of stellar models which fully take into account stellar deformation.Furthermore, non-adiabatic pulsation calculations are required in order to correctlycalculate δT eff /T eff , the variations in effective temperature, as these intervene in theintensity variations used to calculate mode visibilities. In order to carry out suchcalculations, it is necessary for the stellar model to respect the energy conservationequation. This means that the stellar model will be baroclinic, i.e. isobars, isotherms,and isochores will not coincide, and the rotation profile will be non-conservative, i.e. it will depend on both s , the distance to the rotation axis, and on z , the verticalcoordinate.In the work presented here, we will use ESTER models (Espinosa Lara & Rieu-tord, 2013; Rieutord et al., 2016) as these are currently the only rapidly rotatingmodels which satisfy the energy equation locally. Non-adiabatic calculations willbe carried out using the TOP pulsation code ( e.g. Reese et al., 2009, 2017a). Forstars in the mass range of δ Scuti stars, our implementation of non-adiabatic calcu- Evolution STEllaire en Rotation. Two-dimensional Oscillation Program. (cid:63) PTA Proceedings (cid:63)
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March 21, 2018 (cid:63)(cid:63) vol. 123 pta.edu.pl/proc/2018mar21/123 ode identification in rapidly rotating stars from BRITE datalations are not fully reliable. Accordingly, we will also use models from the SCF code (Jackson et al., 2005; MacGregor et al., 2007) along with adiabatic pulsationcalculations. Non-adiabatic effects will be approximated in the same way as is donein Reese et al. (2017b), i.e. using the pseudo non-adiabatic (PNA) approach (seeTable 1). As was shown in Daszy´nska-Daszkiewicz et al. (2002) and Townsend (2003) am-plitude ratios depend on the azimuthal order, m , and the inclination, i , in rotatingstars, thereby complicating the task of mode identification. Nonetheless, Reese et al.(2013) found similar amplitude ratios for modes with the same ( (cid:96), m ) values but dif-ferent radial orders, n , as expected from ray theory (Pasek et al., 2012). Accordingly,Reese et al. (2017b) proposed an alternate mode identification strategy. This strat-egy involves choosing a reference mode, then choosing the N (typically 9) othermodes with the most similar amplitude ratios. When the reference mode happensto be an island mode, the other modes also tend to be island modes with similarquantum numbers. The corresponding frequencies would then follow patterns asexpected from the asymptotic frequency formula ( e.g. Ligni`eres & Georgeot, 2009;Reese et al., 2009). By repeating this procedure, one can hope to group similarmodes together into families and identify recurrent frequency spacings as expectedfrom the asymptotic formula. An open question is whether this strategy still con-tinues to work when using the 2 photometric bands from BRITE rather than the 7bands from the Geneva photometric system.In Table 1, we give the average success rates at finding other island modes usingthe Geneva and BRITE photometric systems for different stellar masses. The thirdcolumn gives the success rate at identifying other island modes if the reference modeis an island mode, whereas the fourth and fifth columns give the success rates atfinding island modes with the same ( (cid:96), | m | ) and (˜ (cid:96), | m | ) values, respectively. Werecall that two modes with the same (˜ (cid:96), | m | ) values will not necessarily have thesame ( (cid:96), | m | ) values as one could be symmetric with respect to the equator and theother anti-symmetric. The last column gives the proportion of island modes in theentire set of modes. As can be seen, the success rates for the BRITE photometricsystem are much lower. Typically, one can expect to identify 1 or 2 other islandmodes out of a set of n = 9 modes, which is insufficient for the purposes of modeidentification.One may then wonder what happens if a supplementary photometric band isincluded. For reasons of normalisation, we prefer not to mix visibilities from theGeneva and BRITE systems. Hence, we use the B and G Geneva bands as represen-tative of the BRITE bands although we do note the latter are much wider. Then weinclude either the U or V1 bands as these are centred around the smallest and high-est wavelengths, respectively, besides the B and G bands. Table 2 gives the successrates for the BRITE photometric system as well as reduced versions of the Genevasystem. Columns 2 to 4 have the same meaning as columns 3 to 5 of Table 1. Ascan be seen, adding one band, especially at small wavelengths, increases the success Self-Consistent Field. pta.edu.pl/proc/2018mar21/123 PTA Proceedings (cid:63)
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Table 1: Success rates for the mode identification strategy using the Geneva and BRITEphotometric systems, for SCF models at 0 . K (where Ω K is the Keplerian break-up rota-tion rate). Photo. . . . . .
Success rates . . . . .
IslandModel System Island ( (cid:96), | m | ) (˜ (cid:96), | m | ) prop. Adia. (2 M (cid:12) ) Geneva 0.564 0.359 0.416 0.0115Adia. (2 M (cid:12) ) BRITE 0.145 0.058 0.071 0.0115Adia. (1.8 M (cid:12) ) Geneva 0.554 0.401 0.452 0.0330Adia. (1.8 M (cid:12) ) BRITE 0.258 0.133 0.159 0.0330PNA (1.8 M (cid:12) ) Geneva 0.469 0.303 0.349 0.0330PNA (1.8 M (cid:12) ) BRITE 0.201 0.079 0.102 0.0330
PNA = pseudo non-adiabaticTable 2: Success rates for the mode identification strategy for the BRITE and reducedversions of the Geneva photometric systems. These values are obtained for the 1 . (cid:12) stellar model using pseudo non-adiabatic calculations. Photo. . . . . .
Success rates . . . . .
Bands Island ( (cid:96), | m | ) (˜ (cid:96), | m | )BRITE 0.201 0.079 0.102B, G 0.182 0.060 0.087U, B, G 0.327 0.182 0.211B, V1, G 0.285 0.146 0.187rates appreciably.In summary, this approach is expected to start working for at least 3 photomet-ric bands, and would also require a large number of acoustic modes, preferably inthe asymptotic regime. Hence, additional observations besides those of BRITE areneeded, and δ Scuti stars would be the most suitable targets.
In some cases, nonetheless, observations may only be available in 2 rather than 3bands. Furthermore, the number of observed modes may be too small for the abovestrategy. This raises the question as to how much information can be obtainedfrom both amplitude ratios and phase differences. In what follows, we use full non-adiabatic calculations as these are needed for obtaining reliable phase differences.Accordingly, we will work with 9 M (cid:12)
ZAMS models produced by the ESTER code,with X = 0 . Z = 0 . . . K (whereΩ K = (cid:113) GM/R is the Keplerian break-up rotation rate, R eq being the equatorialradius).As a first step, we compared our amplitude ratios vs. phase differences with thosefrom Handler et al. (2017, hereafter H17) for similar non-rotating models in order54 (cid:63) PTA Proceedings (cid:63)
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March 21, 2018 (cid:63)(cid:63) vol. 123 pta.edu.pl/proc/2018mar21/123 ode identification in rapidly rotating stars from BRITE data φ B − φ R [ ◦ ] A B / A R ‘ =0 ‘ =3 ‘ =1 ‘ =2 n =0 n =1 n =1 n =2 n =0 n =1 n =1 n =2 Ω = 0 . K , | m | = 0 ‘ = 0 ‘ = 1 ‘ = 2 ‘ = 3 φ B − φ R [ ◦ ] A B / A R Ω = 0 . K , | m | = 0 ‘ = 0 ‘ = 1 ‘ = 2 ‘ = 3
80 60 40 20 0 20 40 φ B − φ R [ ◦ ] A B / A R Ω = 0 . K , | m | = 0 ‘ = 0 ‘ = 1 ‘ = 2 ‘ = 3 φ B − φ R [ ◦ ] A B / A R Ω = 0 . K , | m | = 0 ‘ = 0 ‘ = 1 ‘ = 2 ‘ = 3 Fig. 1: Amplitude ratios vs. phase differences for axisymmetric modes in 9 M (cid:12)
ESTERmodels at different rotation rates using the BRITE photometric bands. The lower rightpanel is a zoom in of the lower left panel. Lines connect results for the same ( n, (cid:96) ) valuesbut for stellar inclinations ranging from 2 ◦ to 89 ◦ in increments of 1 ◦ . to validate our calculations. The mass of our model is 9 M (cid:12) whereas those of H17range from 9 . (cid:12) . The amplitude ratios vs. phase differences are plotted inthe upper left panel of Fig. 1, the grey regions corresponding to H17. As can beseen, a qualitative agreement is obtained.We then look at how rotation affects amplitude ratios and phases differencesin the remaining panels of Fig. 1. As expected, these now depend on the stellarinclination, in contrast to the non-rotating case. Furthermore, there are large excur-sions in these diagrams. This is typically caused by mode amplitudes going to zeroat slightly different stellar inclinations in the different photometric bands. Accord-ingly, it seems unlikely that such excursions will be seen in observed stars, since atleast one of the components will likely be below the detection threshold.An important question is then whether amplitude ratios and phase differenceswill be similar for modes with the same ( (cid:96), | m | ) values but different radial orders, andwhether this can help with mode identification. Figure 2 shows amplitude ratios andphase differences for ( (cid:96), | m | ) = (3 ,
2) in the left panel and (2 ,
0) in the right panel.As can be seen, the left panel corresponds to a case where the amplitude ratiosand phases differences are similar, whereas the right panel shows a case with largerdifferences, especially for i (cid:39) ◦ , due to large excursions at different inclinations.Hence, the answer to the question depends both on the choice of ( (cid:96), | m | ) and on theinclination. A more exhaustive search for a large set of radial orders will be needed pta.edu.pl/proc/2018mar21/123 PTA Proceedings (cid:63) March 21, 2018 (cid:63) vol. 123 (cid:63) A B / A R Ω = 0 . K , ‘ = 3 , | m | = 2 Inclination, i [ ◦ ] φ B − φ R [ ◦ ] UnstableStableProgradeRetrograde A B / A R Ω = 0 . K , ‘ = 2 , | m | = 0 Inclination, i [ ◦ ] φ B − φ R [ ◦ ] UnstableStable
Fig. 2: Amplitude ratios and phase differences as a function of inclination for modes withthe same ( (cid:96), | m | ) values. We note that the zigzags at low inclinations in the left panel arenumerical artifacts due to low mode visibilities in both bands. for δ Scuti type stars once reliable non-adiabatic pulsation calculations are availablefor these.Even with a limited number of modes, one can always apply a χ minimisationto find the modes which best reproduce observed amplitude ratios and phases dif-ferences. In this regard, we recall the work done by Daszy´nska-Daszkiewicz et al.(2015) in modelling the SPB star µ Eridani, where such a minimisation was carriedout in the framework of the traditional approximation, only including excited modes,and taking into account the observed v sin i value. This allowed them to constrainthe mode identification as well as the values of v and i . In 2009, Daszy´nska-Daszkiewicz & Walczak applied what they called “complex as-teroseismology” to the β Cephei star θ Ophiuchi. Complex asteroseismology involvesobservationally determining the f parameter in addition to the mode identification,using both multi-colour photometry and radial velocity measurements from spec-troscopy, where f is the ratio of the bolometric flux perturbations to the radialdisplacement: δ F bol F bol = 4 δT eff T eff = f ξ r R . (1)The f parameter is complex due to non-adiabatic effects which introduce a phaseshift between the flux perturbations and radial displacements. This parameter is in-dependent of latitude in the non-rotating case because δ F bol and ξ r are proportionalto the same spherical harmonic. An open question is what happens to f when thestar rotates rapidly.In Fig. 3, we plot the real and imaginary parts of f as a function of colatitude fortwo different modes. In our definition of f , we used the displacement perpendicularto the stellar surface, ξ v , normalised by the equatorial radius. As can be seen f now depends on the colatitude. Furthermore, sharp spikes occur in the right panelas a result of δT eff and ξ v going to zero at slightly different colatitudes. Hence,the f parameter should be described as an f profile. This raises the question asto whether it would be possible to define some sort of disk-integrated, possibly56 (cid:63) PTA Proceedings (cid:63)
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Colatitude [ ◦ ] f p a r a m e t e r Ω = 0 . K , ( n, ‘, m ) = (2 , , < ( f ) = ( f ) Colatitude [ ◦ ] f p a r a m e t e r Ω = 0 . K , ( n, ‘, m ) = (0 , , < ( f ) = ( f ) Fig. 3: The f parameter as a function of colatitude for two pulsation modes in a rapidlyrotating ESTER model. inclination-dependent, effective f parameter which may still be used to constrainstellar structure, as is done in the non-rotating case. In summary, rotation leads to a much more complicated picture for amplitude ratioand phase differences. It also leads to a latitude-dependant f parameter, f beingthe ratio between the bolometric flux perturbations and radial displacement. Ac-cordingly, this makes mode identification more difficult and complicates complexasteroseismology. Conversely, this may provide tighter constraints on the azimuthalorders and stellar inclination ( e.g. Daszy´nska-Daszkiewicz et al., 2015).Different prospects include extending full non-adiabatic calculations to δ Scutistars, developing a database of mode visibilities and LPVs for the purposes of iden-tification, adapting and/or developing mode identification tools, and fully exploitingpulsation data of rapidly rotating stars from the BRITE mission as well as fromPLATO 2.0 and the ground-based spectroscopic SONG network.
Acknowledgements.
DRR and MR acknowledge the support of the French Agence Nationalede la Recherche (ANR) to the ESRR project under grant ANR-16-CE31-0007. DRR also ac-knowledges financial support from the “Programme National de Physique Stellaire” (PNPS)of CNRS/INSU, France.
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