Period-luminosity relations of fast-rotating B-type stars in the young open cluster NGC3766
H. Saio, S. Ekström, N. Mowlavi, C. Georgy, S. Saesen, P. Eggenberger, T. Semaan, S. J. A. J. Salmon
MMNRAS , 1–11 (2016) Preprint 11 October 2018 Compiled using MNRAS L A TEX style file v3.0
Period-luminosity relations of fast-rotating B-type stars inthe young open cluster NGC 3766
H. Saio, , (cid:63) S. Ekstr¨om, N. Mowlavi, C. Georgy, S. Saesen, P. Eggenberger, T. Semaan and S. J. A. J. Salmon Astronomical Institute, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan Institute of Astronomy, University of Geneva, 51 chemin des Maillettes, 1290 Versoix, Switzerland Institut d’Astrophysique et de G´eophysique, Universit´e de Li`ege, 17 all´ee du 6 Aoˆut, 4000, Li`ege, Belgium
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We study the pulsational properties of rapidly rotating main-sequence B-type starsusing linear non-adiabatic analysis of non-radial low-frequency modes taking into ac-count the effect of rotation. We compare the properties of prograde sectoral g andretrograde r modes excited by the κ mechanism at the Fe opacity peak with the newlydiscovered period-luminosity relation that is obeyed by a group of fast-rotating B-typestars in the young open cluster NGC 3766. The observed relation consists of two se-quences in the period versus magnitude diagram, at periods shorter than 0.5 days.We find that this property is consistent with similar period-luminosity relations pre-dicted for excited sectoral prograde g -modes of azimuthal orders m = − and m = − in fast-rotating stars along an isochrone. We further show that some of the rapidlyrotating stars that have photometric variability with periods longer than a day maybe caused by r -mode pulsation predicted to be excited in these stars. One fast-rotatingstar, in particular, shows both short and long periods that can be explained by thesimultaneous excitation of g - and r -mode pulsations in models of fast-rotating stars. Key words: stars:early-type – stars:massive – stars:oscillations – stars:rotation–openclusters and associations:individual:NGC3766
A large fraction of the main-sequence band on theHertzsprung-Russell (HR) diagram is covered by the pres-ence of pulsating variable stars (see e.g., Jeffery & Saio2016), with a noticeable gap between the blue edge of δ Sctvariables at log T e ff (cid:39) . and the red-edge of Slowly Pulsat-ing B (SPB) stars at log T e ff (cid:39) . . The gap is understood,based on classical (i.e. non-rotating) stellar models, as re-sulting from the fact that the partial ionization zone of He(responsible for the κ mechanism in δ Sct stars) is too close tothe surface while the partial ionization zone of the iron peakelements (responsible for the κ mechanism in SPB stars) istoo deep inside the star.Recently, Mowlavi et al. (2013) discovered in the youngopen cluster NGC 3766 thirty six periodic variable stars thatlie precisely in the gap region of the main sequence, at lu-minosities brighter than δ Sct stars and fainter than SPBstars. The presence of periodic variable stars on the faintside of the SPB instability strip had already been sporad- (cid:63)
E-mail: [email protected] ically reported in the literature, including Maia stars (e.g.Scholz et al. 1998; Percy & Wilson 2000), individual lateB-type stars (Kallinger et al. 2004), and some CoRoT tar-gets (Degroote et al. 2009), but the existence of such apopulation of periodic variables was most clearly revealedby Mowlavi et al. (2013) from the study of an open clus-ter, in which the position of stars can be compared in thecolor-magnitude diagram to the positions of known groupsof pulsating stars. Similar data were also available for an-other young cluster, NGC 884 (Saesen et al. 2013), thoughwith a distribution of periods and stellar parameters thatmade conclusions more difficult to draw. The results pre-sented in Mowlavi et al. (2013) triggered both theoreticalinvestigations (Salmon et al. 2014, who suggested the newvariables to be fast-rotating SPB stars) and observationalstudies (e.g. Lata et al. 2014, 2016; Balona et al. 2016).New key observational results have just been publishedby Mowlavi et al. (2016) on the new periodic variables inNGC 3766, based on spectra obtained with the Very LargeTelescope. The authors first confirm the fast-rotating na-ture of all the new variables with periods less than 0.5 d.Most importantly, their analysis reveals that the photomet- c (cid:13) a r X i v : . [ a s t r o - ph . S R ] F e b H. Saio et al. ric periods of the majority of the fast-rotating stars fall ontwo ridges in the period-luminosity (PL) plane, revealing theexistence of a new PL relation obeyed by those stars. Therelation, they argue, is linked to stellar rotation, with pul-sation playing a key role in the formation of the sequences. In this paper, we investigate the origin of the two PLsequences discovered for the Fast-Rotating Pulsating B-type(FaRPB) stars by Mowlavi et al. (2016), based on pulsationmodels of fast-rotating stars. We consider stellar models ly-ing on isochrones (Sect. 2), and analyze their pulsation prop-erties for both g and r modes using a linear non-adiabaticprescription (Sect. 3). Our pulsation predictions are thencompared to the data of FaRPB stars in NGC 3766 (Sect. 4).Conclusions on the nature of the PL relation of FaRPB starsand on the occurence of r mode pulsation in some periodicstars observed in the cluster are finally drawn in Sect. 5.In Appendixes A and B we discuss the general properties,mainly in the co-rotating frame, of low-frequency pulsationsin a rapidly rotating star. Finally, Appendix C comparesSPB instability regions based on the OPAL and OP opaci-ties with slightly different metallicity. We compute models with rotation using the Geneva stellarevolution code (see Ekstr¨om et al. 2012; Anderson et al.2016). All models start at the zero-age main sequence(ZAMS) with an initial homogeneous chemical compositionof X=0.72 and Z=0.014 needed to reproduce the composi-tion of the Sun at its present age when we evolve a 1 M (cid:12) model up to the present age of the Sun (see Ekstr¨om et al.2012). We also compute a few M (cid:12) models with an initiallarger-than-solar metallicity of Z=0.02 and with X=0.706,that are used for comparison purposes. The opacities aretaken from the OPAL opacity tables (Iglesias & Rogers1996), adapted to the chemical compositions of our mod-els. Several sets of evolutionary tracks are computed, eachstarting with different masses and (uniform) rotation rates;the latter ones being expressed in terms of the Ω init / Ω c ra-tio of the angular velocity to critical angular velocity. Withthe critical angular frequency, the centrifugal force is equalto the gravity at the equator (see Maeder & Meynet 2000).The stellar masses used in this study range from 2.5 to 8 M (cid:12) ,and the initial Ω init / Ω c ratios from 0 and 0.8. To be consis-tent with the pulsation analysis, which assumes solid-bodyrotation, we artificially added a large diffusion coefficient( D arti . = cm s − ) in the angular momentum transportequation (see Chaboyer & Zahn 1992). For models reach-ing the critical velocity in the course of their evolution, amechanical mass-loss is assumed to keep the star at criticalvelocity (Georgy et al. 2011). Models with single or double surface spots for the light vari-ations are inconsistent with the two ridges in the PL plane, be-cause some stars have two periods closely spaced (both belongingto one ridge), or have two separate periods corresponding to thetwo ridges, but the period ratio always deviates slightly from two.
The latitudinal and azimuthal dependence of the amplitudeof linear nonradial pulsations in a non-rotating star (or ro-tating with a period much longer than the pulsation period)is separated by spherical harmonic Y m (cid:96) ( θ, φ ) from the radialdistribution, where θ and φ are the colatitude and azimuthalangles, respectively. The pulsation motion ξ ( r , t ) and pertur-bations δ f ( r , t ) of scalar variables associated with the nonra-dial mode identified with a latitudinal degree (cid:96) , an azimuthalorder m and a radial order n , can be represented as ξ ( r , t ) = e i ω t (cid:0) ξ r , n Y m (cid:96) e r + ξ h , n ∇ h Y m (cid:96) (cid:1) ,δ f ( r , t ) = e i ω t δ f n Y m (cid:96) , (1)where ω is the angular frequency of pulsation, and ∇ h is thehorizontal gradient defined as ∇ h Y m (cid:96) = ∂ Y m (cid:96) ∂θ e θ + θ ∂ Y m (cid:96) ∂φ e φ . (2)In addition to the displacement form given in equation (1)(we call it a spheroidal motion), toroidal motions expressedas η ∝ ( ∇ h Y m (cid:96) (cid:48) ) × e r (3)are possible in general. In a non-rotating star they generateno oscillation and do not couple with spheroidal motions.As a result, toroidal motions have no effect on nonradialpulsations in non-rotating stars.In the presence of rotation, however, the Coriolis forceconnects the spheroidal motion associated with Y m (cid:96) with thetoroidal motions associated with Y m (cid:96) ± ; i.e., (cid:96) (cid:48) = (cid:96) ± , whichbrings coupling among spheroidal motions of (cid:96), (cid:96) ± , . . . .Therefore, in the presence of rotation, the angular depen-dence of a nonradial pulsation cannot be represented by asingle spherical harmonic; i.e., a single (cid:96) cannot be specifiedto a pulsation mode. We note, however, that the even orodd property of a mode with respect to the equator (whichis determined by even or odd (cid:96) − | m | ) is preserved becausethe coupling occurs between (cid:96) and (cid:96) ± .Since the Coriolis force is proportional to | Ω ω | and pul-sational acceleration is proportional to ω , the Coriolis forceeffects are stronger for larger Ω /ω , where ω is the pulsationfrequency in the co-rotating frame and Ω is the rotation fre-quency. Therefore, low frequency g modes are affected bythe Coriolis force more strongly than p modes.Furthermore, stellar rotation makes possible the pres-ence of pulsation modes in which toroidal motions are domi-nant. These modes are called r modes (Papaloizou & Pringle1978). They are normal modes of global Rossby waves, whoserestoring force is associated with the latitudinal gradient ofthe Coriolis force. They propagate in the opposite directionof rotation; i.e., they are retrograde modes in the co-rotatingframe. Since the Coriolis force on the toroidal motion gen-erates spheroidal perturbations, and hence density pertur-bations, normal modes ( r modes) are formed (Papaloizou &Pringle 1978; Provost et al. 1981) and can be excited by the κ mechanism (Saio 1982; Berthomieu & Provost 1983) whichworks on the spheroidal perturbations. In the co-rotatingframe, the frequency range of the r modes lie in the fre-quency range of high-order g modes. In a numerical analysis,a group of r modes is recognized as a series of frequencies MNRAS , 1–11 (2016) eriods of Fast-Rotating Pulsating B-type stars with small radial orders ( n ) leading to a maximum frequency( n = ) in a middle of the g -mode frequency range.In this paper, we perform a linear nonadiabatic analysisof the g and r modes by expressing the pulsational displace-ments ξ and the perturbations δ f of the scalar variables of amode as truncated series (Lee & Baraffe 1995) in the formof ξ = e i ω t J (cid:88) j = (cid:20) ξ jr Y ml j e r + ξ j h ∇ h Y ml j + η j ( ∇ h Y ml (cid:48) j ) × e r (cid:21) , (4) δ f = e i ω t J (cid:88) j = δ f j Y ml j , (5)where l j = | m | + j − + I and l (cid:48) j = l j + − I with I = for even modes and I = for odd modes, and J is the trun-cated length of the series. We adopt J = in most cases.Although no mode in a rotating star can be described by asingle latitudinal degree, we sometimes use, for convenience, (cid:96) e or (cid:96) (cid:48) e to identify a mode, representing the main spheroidaldegree or main toroidal degree, respectively. We note that amode has an azimuthal order m , whose sign indicates the az-imuthal propagation direction of the mode in the co-rotatingframe. In this paper, we adopt the convention that m < corresponds to a prograde mode, which is consistent withthe above equations. The amplitude is expected to grow ifthe imaginary part of ω is negative with a growth rate of −I ( ω ) / R ( ω ) , where I and R mean the imaginary and thereal parts. If a mode has a positive growth rate, we call itan excited mode. Because of the linear analysis, we cannotpredict the amplitude of the excited mode.To avoid the complexity related to the inseparabilityof the angular dependence of the equations, the traditionalapproximation of rotation (TAR) is sometimes used in theliterature, in which the horizontal component of the angularrotation frequency Ω sin θ is neglected. This corresponds toneglecting the Coriolis force associated with the radial dis-placement, and the radial component of the Coriolis forceassociated with the horizontal displacement in the momen-tum equation. In this approximation the angular dependenceof a pulsation mode is separated from the radial dependence,and the set of equations becomes similar to the one for thenon-rotating case. The TAR gives reasonable frequencies for g and r modes, and is useful in understanding the prop-erties of low-frequency modes in a rotating stars (see Ap-pendix A). However, the stability of retrograde g modes aswell as tesseral g modes is significantly affected by the ap-proximation, because the effect of mode interactions are notincluded in the TAR (Lee 2008). We do not use the TARin this paper, because evaluating accurately the stability ofeach mode is important for the present study.In rapidly rotating stars, retrograde g modes andtesseral g modes tend to be damped (Lee 2008; Aprilia et al.2011), while sectoral prograde g modes are less affected byrotation. In our models for rapidly rotating B-type main-sequence stars of intermediate mass stars, all excited ret-rograde modes are found to be r modes. Although sometesseral odd prograde and axisymmetric modes are excited,they have large latitudinal wavenumber, which correspondsto the fact that the terms of large l j ( > | m | + ) in eqs. 4, 5are important (or, in words of the TAR, λ > (cid:96) ( (cid:96) + ; Ap-pendix A). The amplitude distribution on the surface is con-fined to an equatorial region (Townsend 2003, ; Appendix B) Figure 1.
Theoretical HR diagram showing evolutionary tracksof non-rotating (black lines) and rotating (red lines) models as-suming an initial rotation of 70% the critical angular velocity.A chemical composition of ( X , Z ) = (0 . , . is adopted. Thenumbers close to the starting or end point of each evolutionarytrack indicate the initial masses in solar unit. The black and reddot-dashed lines are isochrones at × yr for non-rotating androtating models, respectively. The solid and dashed lines in blueare the instability boundaries for (cid:96) = and (cid:96) = g modes, respec-tively, for non-rotating models. The thick parts of the evolution-ary tracks of rotating models indicate where prograde sectoraldipole modes ( m = − ) are excited. and anti-symmetric to the equator, so that they should behardly visible because of the cancellation for a star with alarge v sin i . On the other hand, the latitudinal distributionof a prograde sectroral g mode is affected only mildly byrotation (Lee & Saio 1997; Townsend 2003, Appendix B)so that they should be most visible in a star having a large v sin i . The amplitude distribution of an r mode tends to havea broad peak in the mid-latitude (Lee & Saio 1997; Savonije2005, Appendix B), so that r modes can be detected pho-tometrically if they produce enough temperature variations.The theoretical prediction is consistent with the period spac-ings observed by Van Reeth et al. (2016) in the Kepler dataof γ Dor variables. These authors found that most of theprograde modes detected in rapidly rotating γ Dor stars areprograde sectoral g modes, while all retrograde modes are r modes. Taking into account these facts, we analyse in thispaper the behaviour of both sectoral prograde g modes and r modes in intermediate-mass main-sequence stars. G AND R MODES INROTATING MAIN-SEQUENCE B-TYPESTARS
High order g modes are excited in intermediate-mass main-sequence B-type stars by the κ -mechanism of the opacity MNRAS000
High order g modes are excited in intermediate-mass main-sequence B-type stars by the κ -mechanism of the opacity MNRAS000 , 1–11 (2016)
H. Saio et al. peak at a temperature of ∼ × K (Gautschy & Saio 1993;Dziembowski et al. 1993). The blue lines in Fig. 1 indicatethe region in the HR diagram where high-order g modes areexcited in non-rotating models (solid lines for (cid:96) = , dashedlines for (cid:96) = modes). Stars located in this region of the HRdiagram show multi-periodic light variations with periodsfrom one to a few days, characteristic of g modes. They areSPB stars.The cool-luminous boundary of the SPB instabilityrange arises due to strong radiative damping in the radia-tive core at the termination of the main-sequence evolution;i.e., the boundary indicates the disappearance of the convec-tive core. Since rapidly rotating stars have a more extendedmain sequence in the HR diagram than non-rotating starsdue to rotational mixing in the stellar interior (e.g., Maeder2009), the instability region of g modes is accordingly widerfor models of rotating stars than for models of non-rotatingstars. This is shown, in Fig. 1, by thick-line parts of evo-lutionary tracks for rotating stars starting with an initialangular velocity equal to 70% the critical angular velocity.In addition, the lower luminosity limit of the SPB in-stability range is predicted to be lower (i.e., occurs in lessmassive stars) in models of rapidly rotating stars than inmodels of non-rotating stars, as first found by Townsend(2005a). Salmon et al. (2014) nicely explained it by com-bining the optimal condition of the κ -mechanism with anincrease of the periods (in the co-rotating frame) of sectoralprograde modes due to rotation. It is known that the periods of high radial order g modesof non-rotating stars are equally spaced because the periodsare approximately proportional to n / √ (cid:96) ( (cid:96) + (e.g., Unnoet al. 1989; Aerts et al. 2010). This property of equal pe-riod spacing is preserved for high-order prograde sectoral g modes in rapidly rotating stars if the periods P corot in theco-rotating frame are longer than the rotation period P rot .This is shown by the triangles and squares in the bottom-left panel of Fig. 2. We note that the period spacing ∆ P corot islarger for rotating stars than for non-rotating stars becauseof a decrease in the effective (cid:96) ( (cid:96) + ( = eigenvalue λ in theTAR; see Appendix A).The period spacing properties of g modes in Fig. 2 lookvery different from that of sectoral prograde dipole g modespresented in fig. 4 of Bouabid et al. (2013), in which ∆ P corot gradually increases with many cyclic variations as P corot in-creases. The cyclic variations are caused by the gradient ofthe mean-molecular weight ( µ ) exterior to the convectivecore boundary (Miglio et al. 2008). No cyclic variations oc-cur for our rotating models at an age of × yr, becauseno strong µ gradients are formed due to their young ageand to the action of rotational mixing. The gradual increaseof ∆ P corot is caused by the gradual decrease of λ as a func-tion of the ratio P corot / P rot . The decrease of λ saturates for P corot > P rot (Fig. A1 in Appendix A), so does the increaseof ∆ P corot . The saturation can be seen in fig. 4 of Bouabidet al. (2013) to occur for P corot > ∼ . d ( ≈ P rot ) in the mostrapidly rotating case. Therefore, the period spacings we findfor prograde sectoral g modes in Fig. 2 are consistent withthose of Bouabid et al. (2013).The period in the inertial (observer’s) frame, P inert , is related to the period in the co-rotating frame as P inert = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P corot − m Ω π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − . (6)Because of the rotational ‘advection’ effect, P inert of a pro-grade g mode is shorter than P corot . As seen in the bottompanels of Fig. 2, the period spacings of prograde g modesin the inertial frame are much smaller than those in theco-rotating frame, and decrease with period. This propertyis useful in asteroseismic analyses. In fact, such a period-spacing/period relation was recently observed in the Keplerdata of a SPB star (P´apics et al. 2015) and used to infer theinterior rotation rate and overshooting efficiency (Moravvejiet al. 2016).In addition to g modes, some r modes (odd modes,though) are excited in rotating stars in the SPB range, asshown by Townsend (2005b) and Savonije (2005) using theTAR, and by Lee (2006) without using the TAR. Our calcu-lations confirm that odd r modes of m = ( (cid:96) (cid:48) e = , (cid:96) e = ) areexcited in models of rotating stars, as shown in the upperpanels of Fig. 2 (filled inverted triangles; see also Fig. B1 inAppendix B), while all even r modes of m = ( (cid:96) (cid:48) e = , (cid:96) e = ;lower panels) seem to be damped.In the co-rotating frame, each r -mode series has a lim-iting (maximum) angular frequency of ω = m Ω (cid:96) (cid:48) e ( (cid:96) (cid:48) e + (7)(see Appendix A). The lowest radial order mode has a fre-quency close to this frequency, while modes of higher radialorders have smaller frequencies in the series (e.g., Provostet al. 1981; Saio 1982). In other words, the frequencies ofthe r modes in the co-rotating frame are confined between and ω . Period spacings of sufficiently high order modesare nearly constant in the co-rotating frame (left panels inFig. 2).For the m = odd r modes ( (cid:96) (cid:48) e = ), ω = Ω (phasespeed equals rotation speed, but in the opposite direction).The limiting frequency in the inertial frame is therefore zero;i.e., P rot < P inert < ∞ . As a result, for this series, the periodspacing and the period in the inertial frame increase withdecreasing radial order, as seen in the upper right panel ofFig. 2. We note that r modes with periods of a few days inthe inertial frame are excited, allowing the co-existence ofshort-periods prograde g modes and long-period r modes ina rapidly rotating star.For even r modes with ( m , (cid:96) e , (cid:96) (cid:48) e ) = (1 , , , ω = Ω .This corresponds to a limiting frequency of Ω in the iner-tial frame; i.e, the period range of this r mode series in theinertial frame satisfies P rot < P inert < . P rot .Recently, Van Reeth et al. (2016) identified retrogrademodes in γ Dor stars as r modes based on the period/period-spacing diagram. Those r modes should be even modes of m = judging from Fig. 9 of Van Reeth et al. (2016). In ourintermediate-mass main-sequence stars, however, only oddmodes are excited among the r modes by the κ -mechanismon the Fe opacity peak. This is in agreement with the resultsobtained by Lee (2006). MNRAS , 1–11 (2016) eriods of Fast-Rotating Pulsating B-type stars Figure 2.
Period-spacings versus periods for g and r modes of a rotating M (cid:12) main-sequence model at an age of × yr in the co-rotatingframe (left panels) and in the inertial frame (right panels). Filled and open symbols are for excited and damped modes, respectively.Initial (i.e. at ZAMS) angular rotation velocities are color-coded as indicated in the top panels. Triangles and squares are progradesectoral g modes of m = − and m = − , respectively, while inverted triangles are r modes of m = with (cid:96) (cid:48) e = , (cid:96) e = (upper panels) andwith (cid:96) (cid:48) e = , (cid:96) e = (lower panels). Lower and upper panels are, respectively, for modes having even and odd temperature distributionswith respect to the equator. The color-corded short vertical lines drawn on the bottom horizontal axis indicate the rotation periods. Figure 3.
Periods in the co-rotating frame (left panel) and in the inertial frame (right panel) of the most strongly excited modes (withhighest growth rates) in the models at an age of 100 million (in red) and of 30 million (in black) years versus the luminosity of the models.Triangles, squares, and pentagons are prograde sectoral modes of m = − , m = − , and m = − , respectively (with m = − (cid:96) e ). Models withinitial rotation frequencies of . Ω c and . Ω c are connected by solid and dashed lines, respectively. Dotted lines in the left panel showthe rotation period as a function of luminosity along the isochrones for the two cases of the initial angular velocity of rotation.MNRAS000
Periods in the co-rotating frame (left panel) and in the inertial frame (right panel) of the most strongly excited modes (withhighest growth rates) in the models at an age of 100 million (in red) and of 30 million (in black) years versus the luminosity of the models.Triangles, squares, and pentagons are prograde sectoral modes of m = − , m = − , and m = − , respectively (with m = − (cid:96) e ). Models withinitial rotation frequencies of . Ω c and . Ω c are connected by solid and dashed lines, respectively. Dotted lines in the left panel showthe rotation period as a function of luminosity along the isochrones for the two cases of the initial angular velocity of rotation.MNRAS000 , 1–11 (2016) H. Saio et al.
In each model, prograde g modes are excited only in a lim-ited range of periods, which are, in the co-rotating frame,considerably longer than the rotation period divided by | m | (see the left panel of Fig. 3). For this reason, the periods ofthe excited sectoral prograde g modes in the inertial frameare close to the rotation period divided by | m | (see Eq. 6) . Therefore, we expect a PL relation for each m along anisochrone if the rotation frequency varies in a systematic wayas a function of luminosity. The right panel of Fig. 3 showssuch PL relations for ages of × and × yr, in whichthe periods of the most strongly excited mode (with thehighest growth rate) are connected for (cid:96) e ( = − m ) = (trian-gles), (cid:96) e = (squares), and (cid:96) e = (pentagons). The g modesof each m form a distinctive PL relation for a given initialangular velocity to critical angular velocity ratio. As seen inthis figure, the gradients of the PL relations at a given ageare not sensitive to the initial rotation velocity, while theperiod of the most excited mode at a luminosity shifts up-wards as rotation speed decreases. For this reason, we expecta tight PL relation to be observed in a young cluster, wherewe expect that stars were born with large rotation speedsnot far from the critical speeds because of the matter accre-tion in the formation process. It is such a relation that wasfound in NGC 3766 by Mowlavi et al. (2016), with which wewill compare our model predictions in the next section. Wealso note that the PL relations might be blurred in olderclusters if rotational velocity decreases with time in somestars due to various braking effects.The period luminosity relation becomes slightly steeperand covers a wider range of luminosities due to the effectof evolution. This is shown in Fig. 3, which compares modelpredictions at ages of × yr and × yr. We note thatthe M (cid:12) model enters the SPB instability region during itsevolution, as a result of its luminosity increase with age. Our PL relation predictions for fast-rotating stars are com-pared in Fig. 4 to the PL relations of FaRPB stars discov-ered by Mowlavi et al. (2016) in NGC 3766 (shown as dashedlines in Fig. 4). We take the models at an age of × yr(black lines in Fig. 3), in agreement with the age estimate of . ± . Myr derived by Aidelman et al. (2012). A distanceof 2.1 kpc has moreover been used to convert the luminosi-ties of our models to apparent V magnitudes.The sequences of the prograde dipole ( − m = (cid:96) e = )and quadrupole ( − m = (cid:96) e = ) modes of rapidly rotat-ing models ( Ω ini / Ω c (cid:38) . ) roughly agree with the observedtwo sequences of the PL relation. The predictions of all thefast-rotating models we studied nicely cover the magnituderange of FaRPB stars observed on the second (shortest peri-ods) sequence. The match in magnitude is less complete forthe first (longest periods) sequence, though, the predictionsof the fastest-rotating models (with Ω ini / Ω c = . ) coveringonly 3/4 of the magnitude range. To reproduce the bright-est FaRPB stars on the first sequence, at V (cid:46) . mag, Such groups of g modes are observed in rapidly rotating Bestars (e.g., Walker et al. 2005; Cameron et al. 2008). the dipole g -modes must be excited in models up to ∼ M (cid:12) .Those modes are, however, damped in our models for M > M (cid:12) .Two options can help to solve this discrepancy be-tween observed luminosities of the brightest FaRPB starsand model predictions. The first option is to assume a larger-than-solar metallicity for NGC 3766. This is probable be-cause the metallicity of NGC 3766 has been estimated onlyby a photometric method from the mean ultraviolet excessof bright dwarfs (Tadross 2003), which has a large uncer-tainty. We expect that a larger metallicity yields a higherFe-opacity bump and hence stronger excitation. To quantifythe impact of a higher-than-solar metallicity on the dipole g -mode excitation, M (cid:12) models with a metallicity of Z = . have been computed, and their pulsation properties ana-lyzed. The periods of the excited g and r modes in these Z = . models are shown in Fig. 4 by smaller symbols atlocations slightly shifted leftward from the standard M (cid:12) models. The number of r modes ( m = ) and g modes of (cid:96) e = that are excited is larger in the Z = . models thanin the Z=0.014 models;i.e., a larger metallicity helps excitemore modes. However, dipole g modes in the M (cid:12) models,which are needed to explain the most luminous FaRPB stars,are still not excited even in the most rapidly rotating model,which indicates that the enhancement of metallicity with theOPAL opacity is insufficient to increase the mass limit forthe excitation of dipole g modes to M (cid:12) .The second option that can help to excite dipole g modes in stars more massive than M (cid:12) is to use OP (Badnellet al. 2005) rather than OPAL (Iglesias & Rogers 1996) opac-ities. Test studies using OP opacities on non-rotating mod-els show that the SPB instability range is shifted to slightlyhigher effective temperatures than the ones obtained usingOPAL opacities (see Appendix C and Fig. C1). A similarconclusion is reached by Miglio et al. (2007). The opacity ef-fect would be stronger (Salmon et al. 2012; Moravveji 2016;Daszy´nska-Daszkiewicz et al. 2017) if the Ni and possiblyFe opacities are revised upward compared to OP opacitiesas discussed by Turck-Chi`eze et al. (2016). The combinedeffect of the two factors, higher cluster metallicity and useof OP opacities, may then act to increase the stellar massup to which dipole g modes can be excited (as shown bythe blue solid line in Fig. C1), thereby improving the matchbetween model predictions and observations of FaRBP starsthat obey the newly discovered PL relation in NGC 3766.We note that the gradient of the observational PL rela-tions are slightly steeper than that along a constant Ω init / Ω c ,making the predicted periods slightly longer than the ob-served ones in the less luminous part. This may indicate lessmassive stars to be rotating more rapidly, or a higher metal-licity/opacity as discussed above may reduce the discrep-ancy. Furthermore, although beyond the scope of this paper,a careful statistical comparison between observed PL rela-tions and theoretical ones based on models with improvedmetallicity/opacity would yield a mean M − Ω relation alongthe main sequence of the cluster.In addition to the FaRPB stars satisfying the PL re-lation, periodic variable stars were observed in NGC 3766with periods larger than 1 day (Mowlavi et al. 2013) andin the same magnitude range as the FaRPB stars. Mowlaviet al. (2016) showed that several of them could be explainedby the fact that they are in binary systems, with the pho- MNRAS , 1–11 (2016) eriods of Fast-Rotating Pulsating B-type stars Figure 4.
Period-magnitude diagram. Red markers represent the B-type stars observed in NGC 3766 (circles for stars with v sin i > km/s and diamonds for stars with v sin i < km/s). Green, black and blue markers represent the pulsation periods, in the inertialframe, predicted to be excited in × yr old models having Z=0.014 and initial angular rotation velocities equal to 0, 0.7 and 0.8times the critical angular velocity, respectively. Triangles and squares represent excited prograde sectoral g-modes of m = − and m = − ,respectively (with the convention that negative m values correspond to prograde modes), while inverted triangles represent excited r modesof m = . Model luminosities are converted to apparent V magnitudes using a distance modulus of . mag, a mean color excess of E ( B − V ) = . mag (McSwain et al. 2008; Aidelman et al. 2012), and bolometric corrections from Flower (1996). The continuous linesconnect the most strongly excited prograde sectoral g modes. The dashed lines indicate the two sequences of the PL relation obtainedby Mowlavi et al. (2016) for the FaRPB stars in NGC 3766. The numbers 6, 5, 4, and 3 aligned horizontally at the bottom of the panelnear the x-axis indicate model masses in solar units. The periods of excited modes predicted in M (cid:12) models with Z = . are also shownfor comparison. For clarity, they are reported in the figure using smaller symbols at locations slightly shifted leftward from the standard M (cid:12) models. tometric period linked to their orbital period. But this isnot the case for all of them. Interestingly, one of the sin-gle stars (having star ID 51 in Mowlavi et al. 2013) con-tains both a short (0.23111 d, falling on the second PLrelation) and long (3.4692 d, 1.9777 d) periods. The twolonger periods are much too long to be explained by g -mode pulsations, but they are comparable to the periodspredicted for excited r modes (see Fig. 4, where the r modesare drawn with inverted triangles). The star is rotating at v sin i = ± km/s (Mowlavi et al. 2016), which is com-patible with the requirement of fast rotation to generate g and r mode pulsations. The fact that both short and long periods are simultaneously excited in the same star furthersupports an r -mode pulsation for the long periods in thisstar.The rotation velocities of the × yr models that leadto excited g and r modes are shown in Fig. 5 versus thepredicted pulsation periods. The rotation velocities are cal-culated based on the rotation frequencies and mean radii ofthe stellar models. The equatorial velocities should actuallybe slightly larger than those calculated using mean stellarradii due to the oblateness of fast-rotating stars, but by lessthan a few percents. The observed v sin i versus period distri-bution of FaRPB stars in NGC 3766, plotted with red cir- MNRAS000
Period-magnitude diagram. Red markers represent the B-type stars observed in NGC 3766 (circles for stars with v sin i > km/s and diamonds for stars with v sin i < km/s). Green, black and blue markers represent the pulsation periods, in the inertialframe, predicted to be excited in × yr old models having Z=0.014 and initial angular rotation velocities equal to 0, 0.7 and 0.8times the critical angular velocity, respectively. Triangles and squares represent excited prograde sectoral g-modes of m = − and m = − ,respectively (with the convention that negative m values correspond to prograde modes), while inverted triangles represent excited r modesof m = . Model luminosities are converted to apparent V magnitudes using a distance modulus of . mag, a mean color excess of E ( B − V ) = . mag (McSwain et al. 2008; Aidelman et al. 2012), and bolometric corrections from Flower (1996). The continuous linesconnect the most strongly excited prograde sectoral g modes. The dashed lines indicate the two sequences of the PL relation obtainedby Mowlavi et al. (2016) for the FaRPB stars in NGC 3766. The numbers 6, 5, 4, and 3 aligned horizontally at the bottom of the panelnear the x-axis indicate model masses in solar units. The periods of excited modes predicted in M (cid:12) models with Z = . are also shownfor comparison. For clarity, they are reported in the figure using smaller symbols at locations slightly shifted leftward from the standard M (cid:12) models. tometric period linked to their orbital period. But this isnot the case for all of them. Interestingly, one of the sin-gle stars (having star ID 51 in Mowlavi et al. 2013) con-tains both a short (0.23111 d, falling on the second PLrelation) and long (3.4692 d, 1.9777 d) periods. The twolonger periods are much too long to be explained by g -mode pulsations, but they are comparable to the periodspredicted for excited r modes (see Fig. 4, where the r modesare drawn with inverted triangles). The star is rotating at v sin i = ± km/s (Mowlavi et al. 2016), which is com-patible with the requirement of fast rotation to generate g and r mode pulsations. The fact that both short and long periods are simultaneously excited in the same star furthersupports an r -mode pulsation for the long periods in thisstar.The rotation velocities of the × yr models that leadto excited g and r modes are shown in Fig. 5 versus thepredicted pulsation periods. The rotation velocities are cal-culated based on the rotation frequencies and mean radii ofthe stellar models. The equatorial velocities should actuallybe slightly larger than those calculated using mean stellarradii due to the oblateness of fast-rotating stars, but by lessthan a few percents. The observed v sin i versus period distri-bution of FaRPB stars in NGC 3766, plotted with red cir- MNRAS000 , 1–11 (2016)
H. Saio et al.
Figure 5.
Projected rotation velocity versus photometric periodfor observational quantities, or rotation velocity versus predictedexcited pulsation periods for quantities derived from stellar mod-els (model rotation velocities are calculated using mean stellarradii and rotation frequencies). The meaning of the symbols arethe same as in Fig. 4. Solid red lines connect the photometricperiods of individual multi-periodic stars. cles and diamonds in Fig. 5, shows good agreement with ourmodel predictions: the majority of stars whose periods fallon either or both PL sequences have sufficiently high v sin i values to be consistent with our explanation that these se-quences are formed by prograde sectoral g modes of m = − and m = − in rapidly rotating stars. Such prograde sectoralmodes have pulsation amplitudes that are maximum at theequator and that decrease with decreasing inclination angle.FaRPB stars are thus seen close to equator-on, and theirequatorial rotation velocity should be close to (yet statisti-cally larger than) the measured v sin i . For the (cid:96) e = , m = − mode, for example, the pulsation amplitude is predicted toreach half the maximum amplitude at i (cid:39) o (see Fig. 2of Salmon et al. 2014), and the equatorial rotation velocitywould be 25% larger than the measured v sin i . The fact thatno FaRPB star is observed with a low v sin i further supportsa prograde sectoral g -mode origin for their pulsation periods.The observed distribution of the photometric amplitude A phot of the dominant period of variability versus v sin i alsosupports a dipole g -mode origin of the pulsation. This dis-tribution is shown in Fig. 6 for all FaRPB stars with periodsless than 0.55 d. An upper envelope is visible in the figure,showing a lack of small-amplitude variable stars with large v sin i . This is consistent with prograde sectoral g modes hav-ing their maximum pulsation amplitude at the equator. Suchpulsating stars would have both a small v sin i and a smallvariability amplitude when seen non equator-on.Pulsating stars with periods longer than ∼ . d forma more inhomogeneous group. While all stars with periodsshorter than 0.5 d are fast rotating, long-period pulsators are lll ll llllllll l ll l lll ll ll l llll A phot (mmag)150200250300350 v s i n i ( k m / s ) P ( d ) NGC 3766
Dominant period
Figure 6.
Projected rotational velocity v sin i versus photometricvariability amplitude A phot of all FaRPB stars with photometricperiods smaller than 0.55 d. Note that the horizontal axis is scaledlogarithmically. Only the dominant period (i.e. with the largestvariability amplitude) is plotted for multi-periodic stars. The pe-riod is shown in color according to the color scale drawn on theright of the figure. observed with a variety of v sin i values (see Fig. 5). The originof the long-period variables with small v sin i can be g -modepulsations similar to the classical (i.e. non-rotating) SPBstars. They are also, on the mean, brighter than the fast-rotating stars (see Fig. 4). The long-period variables withlarge v sin i , on the other hand, can be attributed to r -modepulsations. Among them is the fast-rotating hybrid star hav-ing both g - and r - mode pulsation frequencies. The distinc-tion between g - and r -mode pulsations would be apparent ifperiod spacings could be obtained from future observations(cf. Fig. 2). We found that the PL relations of short-period ( (cid:46) . d)FaRPB stars discovered by Mowlavi et al. (2016) inNGC 3766 can be explained by m = − and m = − pro-grade sectoral g modes of rapidly rotating stars born withinitial angular velocities larger than about 70 % of the criti-cal angular velocities.Such PL relations would be blurred in older clusters ifthe rotation of B-type stars is slowed down by such mech-anisms as, for example, magnetic breaking, tidal effects inbinary stars, or possibly angular momentum transport byinternal gravity waves (Rogers 2015). It would be interest-ing to obtain the PL relations in open clusters of variousages. This may provide a way to obtain information on therotation slow-down timescale of B-type stars.Some rapidly rotating B-type stars in NGC 3766 alsoshow long period variations ( > ∼ d). We showed that some ofthem can be explained as stars pulsating in r modes excitedby the same kappa-mechanism as g modes. MNRAS , 1–11 (2016) eriods of Fast-Rotating Pulsating B-type stars ACKNOWLEDGEMENTS
We thank the anonymous referee of this paper for usefulcomments. HS thanks George Meynet for useful discussionsand his hospitality in Geneva Observatory, and Umin Leefor helpful discussions.
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APPENDIX A: EIGENVALUE OF LAPLACE’STIDAL EQUATION
Although we do not use the traditional approximation of ro-tation (TAR) in our linear nonadiabatic analysis in this pa-per, the TAR is useful in understanding the property of low-frequency oscillations of a rotating star. In this Appendixsection we discuss the properties of g and r modes usingthe TAR. Under the TAR, the angular dependence of a pul-sation mode of a rotating star is separable from the radialdependence. The angular dependence is obtained by solvingLaplace’s tidal equation (e.g., Lee & Saio 1997, and refer-ences therein) with an eigenvalue λ , which represents thelatitudinal degree of the equatorial concentration; a larger λ means the eigenfunction to be more strongly concentratedtoward the equator. The value of λ for a given m depends onthe spin parameter Ω /ω , where Ω is the rotation frequency,and ω is the pulsation frequency in the co-rotating frame.Fig. A1 shows λ as a function of the spin parameter for thecases of azimuthal orders of m = (blue), ± (black), and ± (red). The λ is ordered by an integer k as in Lee & Saio(1997), in which the g modes and the r modes correspondto k ≥ and k ≤ − , respectively.First, we discuss the properties of g modes ( k ≥ ). As Ω → , λ reduces to (cid:96) ( (cid:96) + with (cid:96) = | m | + k , where (cid:96) is the lati-tudinal degree of spherical harmonics Y m (cid:96) ( θ, φ ) . Thus, the sec-toral modes correspond to k = , and even and odd tesseralmodes correspond to even and odd k , respectively. As seenin Fig. A1, except for the prograde sectoral (i.e., m < and k = ) modes, λ becomes very large as Ω /ω increases, in-dicating the eigenfunctions to become concentrated to theequator as the spin parameter increases. (This means that inour numerical calculations with expanding eigenfunctions aseqs. 4, 5, terms associated with larger l j become important.)For this reason, all the tesseral g modes and retrograde sec-toral ( m > , k = ) g modes are expected to be invisible if Ω /ω > , which is consistent with the result of Townsend MNRAS000
Although we do not use the traditional approximation of ro-tation (TAR) in our linear nonadiabatic analysis in this pa-per, the TAR is useful in understanding the property of low-frequency oscillations of a rotating star. In this Appendixsection we discuss the properties of g and r modes usingthe TAR. Under the TAR, the angular dependence of a pul-sation mode of a rotating star is separable from the radialdependence. The angular dependence is obtained by solvingLaplace’s tidal equation (e.g., Lee & Saio 1997, and refer-ences therein) with an eigenvalue λ , which represents thelatitudinal degree of the equatorial concentration; a larger λ means the eigenfunction to be more strongly concentratedtoward the equator. The value of λ for a given m depends onthe spin parameter Ω /ω , where Ω is the rotation frequency,and ω is the pulsation frequency in the co-rotating frame.Fig. A1 shows λ as a function of the spin parameter for thecases of azimuthal orders of m = (blue), ± (black), and ± (red). The λ is ordered by an integer k as in Lee & Saio(1997), in which the g modes and the r modes correspondto k ≥ and k ≤ − , respectively.First, we discuss the properties of g modes ( k ≥ ). As Ω → , λ reduces to (cid:96) ( (cid:96) + with (cid:96) = | m | + k , where (cid:96) is the lati-tudinal degree of spherical harmonics Y m (cid:96) ( θ, φ ) . Thus, the sec-toral modes correspond to k = , and even and odd tesseralmodes correspond to even and odd k , respectively. As seenin Fig. A1, except for the prograde sectoral (i.e., m < and k = ) modes, λ becomes very large as Ω /ω increases, in-dicating the eigenfunctions to become concentrated to theequator as the spin parameter increases. (This means that inour numerical calculations with expanding eigenfunctions aseqs. 4, 5, terms associated with larger l j become important.)For this reason, all the tesseral g modes and retrograde sec-toral ( m > , k = ) g modes are expected to be invisible if Ω /ω > , which is consistent with the result of Townsend MNRAS000 , 1–11 (2016) H. Saio et al.
Figure A1.
The eigenvalue of Laplace’s tidal equation λ as afunction of the spin parameter Ω /ω , where ω is the angular fre-quency of pulsation in the co-rotating frame. The value of az-imuthal order m is color-coded as indicated. Solid and dashedlines are used for even and odd modes (with respect to the equa-tor), respectively. The way of ordering λ by integers k is adoptedfrom Lee & Saio (1997). (2003) (see also Appendix B). On the other hand, for theprograde ( m < ) sectoral ( k = ) g modes, λ decreases andapproaches m as Ω /ω increases; i.e., the prograde sectoral g modes are least concentrated toward the equator and mostvisible among the g modes for a given m in a rapidly rotatingstar.The r modes correspond to λ with a negative k . They arealways retrograde modes ( m > ) in the co-rotating frame.The r modes can exist in the range of spin parameter where λ is positive; i.e., m Ω /ω > ( | m | + | k + | )( | m | + | k | ) . In otherwords, the frequencies of r modes are bounded as ω r mode < m Ω ( m + | k + | )( m + | k | ) ≤ Ω (A1)with m ≥ and k ≤ − . (The last inequality indicates that the r modes are seen as prograde modes in the inertial frame.)At the limiting frequency, λ = , which corresponds to thefact that oscillation velocities are purely toroidal with di-minishing horizontal divergence; i.e., ∇ h · η = (see eqs. 2and 3). (The horizontal divergence of a g mode at Ω = is proportional to (cid:96) ( (cid:96) + .) The r modes are characterizedas retrograde modes (in the co-rotating frame) dominatedby toroidal motions, which corresponds to small λ . How-ever, the modes of k = − become to have g -mode characterwith large λ as Ω /ω increases, while the r -mode character isretained for the modes with k ≤ − . The amplitude of a typi-cal r mode is broadly confined to the mid-latitude zone (e.g.,Lee & Saio 1997; Savonije 2005, ; see also Appendix B), sothat r modes can be photometrically visible (in particular,longer period ones in the inertial frame) if sufficiently largetemperature variations are generated. Figure B1.
Growth rates of excited modes versus the periodsin the co-rotating frame in the non-rotating (botom panel) androtating (top panel) models of M (cid:12) at an age of × yr. Filledblack triangles and squares are prograde sectoral m = − and m = − ( (cid:96) = and (cid:96) = in the non-rotating model) modes, respectively.A cross is for a single prograde tesseral mode of ( m , (cid:96) e ) = ( − , excited in the rotating model. Filled blue circles are axisymmetric( m = ) odd tesseral modes of (cid:96) e = . Red inverted triangles areexcited odd r modes of m = . The short vertical bar indicates therotation period. APPENDIX B: GROWTH RATES ANDAMPLITUDE DISTRIBUTION ON THESURFACE
Fig. B1 shows growth rates versus periods in the co-rotatingframe for excited modes of | m | ≤ in a rotating M (cid:12) modelwith Ω init = . Ω c (top panel) and a non-rotating M (cid:12) model(botom panel) at an age of × yr. The number of progradesectoral g modes excited in the rotating star is comparableto the number of excited (cid:96) = and 2 g -modes in the non-rotating model, while only one prograde tesseral, ( m , (cid:96) e ) = ( − , , g mode ( × ) is excited in the rotating model. A fewaxisymmetric odd modes ( m = , (cid:96) e = ) are excited, whileall the even axisymmetric modes of (cid:96) e = are found to bedamped. Since the period in the co-rotating frame of all theexcited tesseral g modes are longer than the rotation period,the corresponding spin parameters are greater than 2.We note that some m = even modes with periods of ∼ . d ( Ω /ω < ) corresponding to high (cid:96) ( ≈ ) are excited.Those modes are not shown here, because they should havelow visibilities. Balona & Dziembowski (1999) first foundsuch g modes of high (cid:96) s to be excited in B-type main-sequence stars.Fig. B2 shows the amplitude distributions of luminos-ity variation on the stellar surface of selected modes fromFig. B1. The amplitude distributions of the odd tesseral g -modes (blue and black solid lines in the upper panel) aresignificantly modified from the corresponding Legendre func- MNRAS , 1–11 (2016) eriods of Fast-Rotating Pulsating B-type stars Figure B2.
Amplitude of local luminosity variation (i.e., lin-ear variation of R T ) versus polar-angle (co-latitude) for selectedmodes shown in Fig. B1. Upper and lower panels are for oddand even modes, respectively. Each distribution is normalizedas unity at the maximum. Corresponding azimuthal order m iscolor-corded as indicated. Dotted lines show amplitude distri-butions expected in a non-rotating case;i.e., Legendre function P | m | (cid:96) (cos θ ) . Amplitude distributions for two r modes are plotted(red lines in the top panel); one at P corot = . d and the otherat P corot = . d. The former having a broad peak around θ ≈ ◦ is the lowest radial order ( n = ) r mode exited, and the latter( n = ) with the maximum growth rate. The amplitude distri-butions in the bottom panel are for the sectoral prograde modeshaving maximum growth rates. tions, and shifted toward the equator ( θ = ◦ ) by the effectof rotation. We do not expect to detect any of these tesseralodd modes in stars having large values of v sin i , because ofcancellation. In contrast to the tesseral g modes, rotationaleffects on the prograde sectoral modes are gentle (solid linesin the bottom panel of Fig. B2); the amplitude distributionis shifted toward the equator without changing the shapesignificantly. We expect these sectoral prograde modes to bemost visible in stars with large v sin i .All excited retrograde modes are found to be r modes(inverted red triangles in Fig. B1 and red solid lines in thetop panel of Fig. B2); i.e. no retrograde g modes are ex-cited. These r modes are odd modes corresponding to thesequence of k = − m = in Fig. A1. The λ of the sequenceincreases rapidly as the spin parameter ( Ω /ω ) increases (sodoes P corot ), and hence the property shifts from a pure r mode (with (cid:96) e = ) to that mixed with the g mode prop-erty. This means that components associated with (cid:96) j > become important in the expansions of eqs. 4, 5 as Ω /ω increases. Correspondingly, the distribution of amplitude,in Fig. B2 (red lines in the top panel), shifts from thatclose to the Legendre function P for the shortest period(excited) r mode ( P corot = . d) to that concentrated to-ward the equator (gets similar to tesseral g modes) for the Figure C1.
Theoretical HR diagram showing the instabilityboundaries of g modes for non-rotating models obtained usingthe OP and OPAL opacities. Corresponding chemical composi-tions and opacity tables are color coded as indicated. Solid anddashed lines are for (cid:96) = and (cid:96) = modes, respectively. Thethin red lines represent evolutionary tracks of models computedwith the MESA code. The numbers along the ZAMS indicate thestellar mass of the track, in solar units. mode at P corot = . d (with a maximum growth rate). Inother words, the visibility of odd r modes would decrease as P corot increases (i.e., as P inert decreases ). APPENDIX C: SPB INSTABILITY RANGESOBTAINED BY OPAL AND OP OPACITIES
In the Geneva evolution code OPAL opacity tables (Iglesias& Rogers 1996) are used. To see the difference in the excita-tion of g modes between OPAL and OP (Badnell et al. 2005)opacities, we have also analysed the excitation of g modesfor non-rotating models obtained by the Modules of Exper-iments in Stellar Astrophysics (MESA Paxton et al. 2013)using the OP opacity. Fig. C1 shows the difference. In accor-dance with Miglio et al. (2007), the SPB instability rangeobtained by the OP opacity is shifted slightly to higher T e ff compared with the case of the OPAL opacity. Furthermore,if the OP opacity is used with a slightly higher metallicityof Z = . (the blue lines in Fig. C1), the instability regionalso extends to lower luminosity, covering the instability re-gions obtained with OPAL and OP opacities for Z = . .Thus, combining a slightly higher-than-solar metallicity andthe OP opacity would solve the lack of excitation in theluminous part of Fig. 4. This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS000