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On possible explanations of pulsations in Maia stars JadwigaDaszyn´ska-Daszkiewicz1,⋆,PrzemysławWalczak1,andAlexeyPamyatnykh2 1InytutAstronomiczny,UniwersytetWrocławski,ul.Kopernika11,Wrocław,Poland 7 2NicolausCopernicusAstronomicalCenter,Bartycka18,00-716,Warsaw,Poland 1 0 2 Abstract.Thelong-timephotometricsurveysinafewyoungopenclustersallowedtoidentifythelightvari- ability in stars located on the HR diagram between the well defined δ Scuti variables and Slowly Pulsating n B-typestars. SeveralobjectsofthistypeweresuggestedalsofromtheanalysisoftheKeplerdata. Assuming a J the pulsational origin of this variability, we try to explain the observed frequencies with pulsational models involvingrotationand/ormodificationofthemeanopacityprofile. 4 ] R S 1 Introduction M ≈ 3.0 M⊙. Thelasthypothesisdemandsthemodifica- tions of the opacity profile and higher degree modes (up . h PulsatingvariablesbetweentheinstabilitystripsofδScuti to 6-8) and worksfor both, low and fast rotating stars of p and Slowly Pulsating B-type (SPB) stars have been sug- B8-A2 spectral type. Sect.2 presents the results of our - gested many years ago by Struve [1]. Although later a o modellingandconclusionsaregiveinSummary. r memberofthePleiadesMaiaappearedtobeaconstantstar t s (Struve[2]),theidearemainedandthesearchhavecontin- a uedformanyyears(McNamara[3], Scholz[4]). Finally, 2 The three explanations [ theanalysisofa7-yearobservationcampaignoftheopen 1 cluster NGC3766led tothe discoveryof alargenumber Thefirsthypothesis: v of variablestars(Mowlaviet al. [5]). Amongstthemthe fastrotatingstarswithunderestimatedmasses 7 authors found 36 new variables, which according to the 3 In this scenario, Maia stars are linked to fast rotating position on the color-magnitudediagram, have the spec- 9 SlowlyPulsatingB-typestars.Thefastrotationcausesthat 0 tral types between late type B and early type A. The pe- masses of the stars seen close equator-on are underesti- 0 riodsoftheirlightvariabilityarefromtherangeofabout mated(Salmonetal.[7]).Weperformpulsationalcompu- 01. 0-.110-[0d.7−1[]d,]a,ncdortrheesppohnodtionmgetotritcheamfrepqliutuednecsieasroefoafbothuet1o.r4- tationsformodelswithmassesM =3−4M⊙intheframe- work of the traditional approximation (e.g. Lee & Saio 7 der of a few millimagnitudes. The very recent work by [12],Townsend[13],Savonije[14]).Asmentionedabove, 1 Mowlavi et al. [6] showed that most of them rotate fast v: (withVrotsini>150kms−1)whichsupportsthehypothe- NonGlyCZ37A6M6Sismaovdeeryls.yoTuhneginoiptieanlchlyudsrtoerg,etnhuasbuwnedaanncaelyasnedd i sisthatMaiapulsatorsarefastrotatingSPBstars(Salmon X metallicityweresetat0.7and0.015,respectively,andthe etal. [7].) r chemical mixture was adopted from Asplund at al. [15]. a This new type of variables was also suggested from Here, we show results obtained with the OPAL opacities the analysis of space data. From the Kepler photometry, (Iglesias & Rogers [16]) but with the OPLIB (Colgan et Balonaetal. [8–10]foundB-typestarswithhighfrequen- al. [17, 18]) and OP data (Seaton [19]), the results are ciesandlowamplitudes.Thesestarsarelocatedbelowthe qualitativelysimilar. rededgeoftheβCepinstabilitystripandsomeofthemcan High order gravity modes as well as mixed gravity- beconsideredasMaiacandidates. Severalcandidatesare Rossbymodesareconsidered.Inmodelswithmassescor- alsofromtheCoRoTobservations(Degrooteetal. [11]). respondingto B8-A2 type, these modesare stable. Their In this paper, assuming the pulsational origin of this instabilitybeginsfromthemassofaboutM =3M⊙.More- variability,we testa few hypotheses. Because NGC3766 over, including the effects of rotation on g modes, both, is a veryyoungopencluster, we consideronlyZeroAge enhancesthepulsationalinstabilityandcausestheshiftof Main Sequence (ZAMS) models. The first and already modes to the higher frequency range. In Fig.1, we plot studied hypothesis involves the very high rotation. In the instability parameter, η, as a function of mode fre- the second, new scenario, we modify the opacity profile quencyforaZAMSmodelwithamassM =3.2M⊙,effec- in order to excite pulsational modes in the required fre- tivetemperature,logT = 4.119. androtationalvelocity eff quency range in the rotating models with masses below V = 250 kms−1. The parameter η is the normalized rot ⋆e-mail:[email protected] workintegralandforunstablemodesittakespositiveval- 0.2 g-modes M=3.2 M , logTeff=4.1194, ZAMS, OPAL, Vrot=250 km/s mixed Rossby modes (r,-2) (r,-3) (r,-4) (r,-1) (1,+1) (2,+2) 0.1 (1, 0) (3,+3) (2,+1) (3,+2) (4,+4) 0.0 -0.1 0 1 2 3 4 5 6 7 -1 8 9 10 11 12 13 [d ] obs Figure1. Theinstabilityparameter,η,asafunctionofthemodefrequencyforaZAMSmodelwithamassM=3.2M⊙andeffective temperature,logT =4.1194,rotatingwiththevelocityofV =250 kms−1.Only(ℓ, m)and(r, −m)modeswhichreachinstability eff rot (η>0)areshown. ues. For clarity, we depicted only modes which become ble to get the instability in the observed frequencyrange unstable(η>0)insomerangeoffrequencies. for modes with ℓ ≥ 2. Adding opacity at the Z-bump Thesecondhypothesis: (logT = 5.3)makesalsothedipolemodesunstable. This fastrotatingstarswithnon-standardopacities scenario works for, both, slow and fast rotating stars of B8-A2type.Below,weshowtheresultsobtainedwiththe Inthesecond,newscenario,wecontinuedpulsational modified OPAL data. We added 150% of the opacity at computationswiththetraditionalapproximationbutaddi- logT = 5.1 and 50% of the opacity at logT = 5.3. As tionally we modified the mean opacity profile in models one can see, in the mass range corresponding to late B- with masses below M = 3.0 M⊙. The standardopacities type and early A-type stars, M ∈ (2.5, 4), high degree wereincreasedatthedepthcorrespondingtotemperature modes (ℓ > 6) can reach frequencies as high as about 9 logT = 5.1. This modification mimics the new opacity d−1. Considering modes with such high degrees is jus- bumpidentifiedintheKuruczatmospheremodels(Cugier tified by the fact that the amplitudes of the new variable [20]). This new bump was suggested, for example, as stars in NGC3766 are very low and only very long cam- a possible cause of excitation of low frequency g-modes paign(about7years)allowedtodetectthem. in δ Scuti stars as detected in the Kepler data (Balona et al. [21]). TheeffectofincreasingtheOPAL opacitiesby 100%atlogT = 5.1ispresentedinFig.2. Amodelwith 3 Summary the mass M = 2.5 M⊙ on ZAMS was considered. The Here,we discussedthreepossibleexplanationsforpulsa- rotationalrate was V = 200 kms−1. For modelswith rot tions in Maia stars. Each of them has its strengths and such masses, pulsational computations with the standard weaknesses. opacitiesmodeldonotpredictanyinstability. Asonecan Fastrotatingstarsseenequator-oncanaccountforpul- see,itismorethanenoughtoconsidermodeswiththede- sationsoftheMaiatype.Thishypothesishasbeenalready gree ℓ ≤ 3 to cover the frequency range observed in the explored by Salmon et al. [7]. In this case, we demand NGC3766stars. theclose equator-onorientationforallstars. Asa conse- Thethirdhypothesis: quence, all modeswhich have an equatorialnodeare not slowrotatingstarswithnon-standardopacities goodcandidates. Itisworthtomentionthatprogradesec- The third explanation involves modifications of the toralmodesreachthehighestvisibilityontheequator(e.g. opacityprofileandincludingmodeswiththeharmonicde- Salmonetal. [7],Daszyn´ska-Daszkiewiczetal. [22]). On greesupto8. Here,weperformedallpulsationalcompu- theotherhand,giventhefactthattherearenotmanyiden- tationsassumingthezero-rotationapproximation,i.e., all tifiedMaiavariablesthisscenarioisplausible. Moreover, effectsofrotationonpulsationswereneglected. itisstrengthenedbytherecentresultofMowlavietal. [6] We found that by increasing the mean opacity at the whoshowedthatallnew-classvariablesinNGC3766are depth logT = 5.1 in a M = 2.5 M⊙ model, it is possi- fastrotators. 0.2 M=2.5 M , logTeff=4.0451, ZAMS, Vrot=200 km/s g-modes OPAL: =2* at logT=5.1 mixed Rossby modes std all modes with =1,2,3 and r,m=-1,-2 0.1 0.0 -0.1 0 1 2 3 4 5 6 -1 7 8 9 10 11 [d ] obs Figure2.Theinstabilityparameter,η,asafunctionofthemodefrequencyfortheZAMSmodelwithamassM=2.5M⊙andeffective temperature, logT = 4.0451, rotatingwiththevelocityofV = 200 kms−1. ThemodelwascomputedwiththemodifiedOPAL eff rot opacities whichwereartificiallyincreased by100% at logT = 5.1. Modes withthedegreeup to3andmixed-Rossby modeswith m=−1, −2areshown. 10 0.16 0.14 8 0.12 0.1 6 ] 0.08 d / η c [ ν 0.06 4 8 8 8 0.04 2 123 1234567 12345678 12345678 1234567 1234567 1234567 12345678 2345678 2345678 2345678 345678 345678 45678 00.02 0 -0.02 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 M [M ] ⊙ Figure3.ThefrequencyrangeofunstablemodesasafunctionofmassforZAMSmodelscomputedwiththemodifiedOPALopacity profile. Modeswiththeharmonicdegreesuptoℓ =8wereconsidered. Theopacitieswereincreasesby150%atlogT = 5.1andby 50%atlogT =5.3.Thevaluesoftheinstabilityparameterarecodedbycolours. Twonewscenariosinvolvemodificationsofthemean [2] O.Struve,J.Sahade,C.R.Lynds,S.S.Huang,Astro- opacityprofile.Weshowedthatanincreaseoftheopacity phys.J.125,115(1957) atlogT = 5.1isindispensabletogettheinstabilityinthe [3] B.J.McNamara,Astrophys.J.289,213(1985) frequency range 1.4 – 10 d−1 in the models with masses [4] G. Scholz, H. Lehmann, G. Hildebrandt, K. Panov, M <3.0M⊙.Theaboveconclusionsarevalidiftheeffects L.Iliev,Astron.&Astrophys.337,447(1998) of rotation on pulsations are taken in to account. In the [5] N.Mowlavi,F.Barblan,S.Saesen,L.Eyer,Astron. case of g-mode pulsations, we do this via the traditional &Astrophys.554,A108(2013) approximation. If the effects of rotation on pulsation are [6] N. Mowlavi, S. Saesen, T. Semaan, P. Eggenberger, neglected then, additionally to the opacity modifications, F. Barblan, L. Eyer, S. Ekström, C. Georgy, ArXiv thehigherℓmodeshavetobeconsidered. e-prints(2016),1610.01077 Here, we need quite significant modifications of the [7] S.J.A.J. Salmon, J. Montalbán, D.R. Reese, M.A. opacitieswhichhaveto be justified atsome point. There Dupret, P. Eggenberger, Astron. & Astrophys. 569, are some results thatsupportssuchapproach,e.g., byin- A18(2014) creasingtheopacityatthedepthlogT = 5.1itispossible [8] L.A. Balona, A. Pigulski, P.D. Cat, G. Handler, toexplainlowfrequencyg-modesinδScutistarsdetected J. Gutiérrez-Soto, C.A. Engelbrecht, F. Frescura, intheKeplerdata(Balonaetal.[21]).Giventhatthereare M. Briquet, J. Cuypers, J. Daszyn´ska-Daszkiewicz stillmanyuncertaintiesincomputationsofstellaropacities etal.,MNRAS413,2403(2011) and, for example, in case of hybridB-type pulsatorssig- [9] L.A.Balona,A.S.Baran,J.Daszyn´ska-Daszkiewicz, nificant modifications have been suggested (Daszyn´ska- P.DeCat,MNRAS451,1445(2015) Daszkiewicz et al. [23]), such hypothesis for Maia vari- [10] L.A.Balona,C.A.Engelbrecht,Y.C.Joshi,S.Joshi, ablesisreasonable. Additionally,thesetwohypothesisdo K.Sharma,E. Semenko,G. Pandey,N.K.Chakrad- not have any requirements for the inclination angle and hari,D.Mkrtichian,B.P.Hemaetal.,MNRAS460, allunstablemodescanbeconsidered. Thethirdexplana- 1318(2016) tion needs the higher degreemodes, up to 8, which have [11] P.Degroote,C.Aerts,M.Ollivier,A.Miglio,J.De- poorervisibility.However,giventhelowvaluesoftheob- bosscher, J. Cuypers, M. Briquet, J. Montalbán, servedamplitudesandthelackofknowledgeoftheselec- A.Thoul,A.Noelsetal.,Astron.&Astrophys.506, tionmechanism,weshouldallowsuchapossibility.More- 471(2009) over, the often forgottenfact is that while the mode visi- bilityisgettingworsewithhigherℓ accordingtothedisc [12] U.Lee,H.Saio,Astrophys.J.491,839(1997) averagingeffects, it increasesas ℓ2 accordingto the geo- [13] R.H.D.Townsend,MNRAS340,1020(2003) metricalfactor(ℓ−1)(ℓ+2)(e.g.Daszyn´ska-Daszkiewicz [14] G.J.Savonije,Astron.&Astrophys.443,557(2005) et al. [24]). The explanation with the opacity modifica- [15] M.Asplund,N.Grevesse,A.J.Sauval,P.Scott,Ann. tionsandhigherℓmodesworksforbothfastandslowro- Rev.Astron.Astrophys.47,481(2009),0909.0948 tators. [16] C.A. Iglesias, F.J. Rogers, Astrophys. J. 464, 943 Todecidewhichscenarioispreferred,firstly,morede- (1996) terminations of the rotational velocity of these stars are [17] J. Colgan, D.P. Kilcrease, N.H. Magee, J. Abdal- needed. Then,pulsationalanalysisofmoreMaiastarshas lah,M.E.Sherrill,C.J.Fontes,P.Hakel,H.L.Zhang, tobeperformedtakingintoaccountvariouseffectsonpul- HighEnergyDensityPhysics14,33(2015) sationalinstability.Itcannotberuledoutthatrotationand [18] J. Colgan, D.P. Kilcrease, N.H. Magee, M.E. Sher- opacity modification can work together and hold the key rill,J.Abdallah,J.,P.Hakel,C.J.Fontes,J.A.Guzik, tosolvingtheMaiastarproblem. K.A.Mussack,Astrophys.J.817,116(2016) [19] M.J.Seaton,MNRAS362,L1(2005) Acknowledgements [20] H.Cugier,Astron.&Astrophys.565,A76(2014) [21] L.A. Balona, J. Daszyn´ska-Daszkiewicz, A.A. This work was financially supported by the Polish Pamyatnykh,MNRAS452,3073(2015) National Science Centre grant 2015/17/B/ST9/02082. [22] J. Daszyn´ska-Daszkiewicz, W.A. Dziembowski, Calculations have been carried out using resources A.A.Pamyatnykh,AcA57,11(2007) provided by Wrocław Centre for Networking and Super- computing(http://wcss.pl),grantno.265. 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