MOST detects SPBe pulsations in HD 127756 & HD 217543: Asteroseismic rotation rates independent of vsini
C. Cameron, H. Saio, R. Kuschnig, G.A.H. Walker, J.M. Matthews, D.B. Guenther, A.F.J. Moffat, S.M. Rucinski, D. Sasselov, W.W. Weiss
aa r X i v : . [ a s t r o - ph ] M a y Draft: October 30, 2018
MOST ⋆ detects SPBe pulsations in HD 127756 & HD 217543:Asteroseismic rotation rates independent of vsini C. Cameron , H. Saio , R. Kuschnig , G.A.H. Walker , J.M. Matthews , D.B. Guenther ,A.F.J. Moffat , S.M. Rucinski , D. Sasselov , W.W. Weiss ABSTRACT
The
MOST (Microvariability and Oscillations of Stars) satellite has discov-ered SPBe (Slowly Pulsating Be) oscillations in the stars HD 127756 (B1/B2Vne) and HD 217543 (B3 Vpe). For HD 127756, 30 significant frequencies areidentified from 31 days of nearly continuous photometry; for HD 217543, up to40 significant frequencies from 26 days of data. In both cases, the oscillationsfall into three distinct frequency ranges, consistent with models of the stars. Thevariations are caused by nonradial g-modes (and possibly r-modes) distorted by ⋆ Based on data from the
MOST satellite, a Canadian Space Agency mission, jointly operated by DynaconInc., the University of Toronto Institute of Aerospace Studies and the University of British Columbia withthe assistance of the University of Vienna. Dept. of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver,BC V6T 1Z1, Canada; [email protected], [email protected] Astronomical Institute, Graduate School of Science, Tohoku University, Sendai, 980-8578, Japan;[email protected] Institut f¨ur Astronomie, Universit¨at Wien T¨urkenschanzstrasse 17, A–1180 Wien, Austria;[email protected], [email protected]; [email protected] Department of Astronomy and Physics, St. Mary’s University Halifax, NS B3H 3C3, Canada; [email protected] D´ept. de physique, Univ. de Montr´eal C.P. 6128, Succ. Centre-Ville, Montr´eal, QC H3C 3J7, Canada;and Obs. du mont M´egantic; moff[email protected] Dept. of Astronomy & Astrophysics, David Dunlap Obs., Univ. Toronto P.O. Box 360, Richmond Hill,ON L4C 4Y6, Canada; [email protected] Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA; [email protected]
MOST , HD163868 and β CMi, are all close to their critical values.
Subject headings: stars: early-type — stars: emission-line, Be — stars: individ-ual(HD 127756, HD 217543, HD 163868, β CMi); techniques: photometric
1. Introduction
Be stars are rapidly rotating B-type stars close to the main sequence that show or haveshown emission lines in their photospheric spectra (see Porter & Rivinius 2003, for a recentreview). Some Be stars (especially those of early type) also exhibit line-profile variationsindicating the presence of nonradial pulsations (e.g. Rivinius et al. 2003). The
MOST (Microvariability and Oscillations of STars) satellite (Matthews et al. 2004) photometricallydetected multiple periods in three Be stars: ζ Oph (O9.5 V; Walker et al. 2005a), HD 163868[B1.5-5 Ve; see section 4.1 for details.](Walker et al. 2005b), and β CMi (B8 Ve; Saio et al.2007). Walker et al. (2005a) suggest the pulsations of ζ Oph ( .
20 c d − ) are well modelledas a combination of low-order, radial and nonradial, p and g-modes; modified by rotation,and consistent with the β Cephei-type variables. The oscillations in the latter two starsare attributed to high-order, nonradial g-mode pulsations excited by the κ - mechanismnear the Fe opacity bump (log T ≈ . ≈ | m | Ωaccording to the azimuthal order m . (Those frequency groups occur around 1 . − and 3 .
3c d − for HD 163868, and around 3 . − for β CMi.) This type of grouping characterizesthe amplitude spectra of SPBe stars, and makes the periods of their light variations close totheir rotation period, or half of it, just as in the λ Eri variables (Balona 1995).Since the discovery of the SPBe variability in the aforementioned stars by the authorsthere have been observations of Be stars by Uytterhoeven et al. (2007) and Guti´errez-Soto et al.(2007) that show seemingly similar characteristics. In particular, Uytterhoeven et al. de-tected three periods (2.234 c d − , 4.713 c d − , 4.671 c d − ) from their ground-based pho-tometry in the Be star V2104 Cyg (B5-7; as described in Uytterhoeven et al. 2007) andGuti´errez-Soto et al. detected multiperiodic photometric variations in the two early type Bestars NW Ser (B2.5 IIIe) and V1446 Aql (B2 IVe). Both space and ground-based campaigns 3 –of Be stars are yielding data that can be used in conjunction with models to determinethe rotation periods of rapidly rotating stars asteroseismically from the observed frequencygroupings alone.In this paper, we report the MOST detections and modelling of SPBe pulsations inanother two Be stars: HD 127756 and HD 217543. HD 127756 is a southern, early-typeBe star (B1/B2 Vne; V=7.56 mag; δ = − ◦ ) for which no v sin i value is available. HD217543 (= V378 and = HR 8758) is an intermediate-type Be star with shell characteristics(B3 Vpe; V = 6.555 mag; δ = +38 ◦ v sin i of 305 kms − for this star and Bernacca & Perinotto (1970) suggest a larger value (370 km s − ). HD217543 also shows marked variations in emission-line strength (Copeland & Heard 1963). Inaddition, we present an alternate model of the SPBe star HD 163868 (Walker et al. 2005b)and discuss the implications of the models and observations of all published data on theSPBe stars observed by MOST to date.
2. The
MOST photometry and frequency analysis
The
MOST satellite (launched on 30 June 2003) houses a 15/17.3 cm Rumak-Maksutovtelescope feeding a CCD photometer through a single custom broadband optical filter (350– 700 nm); see Walker et al. (2003) for details.
MOST observed HD 127756 and HD 217543as guide stars for other Primary Science Targets in different observing runs. The guide starsare sampled by subrasters on the CCD and the photometry is primarily processed on-boardbefore downlinking to Earth, by subtracting a mean sky value within the subraster afterapplying a specified threshold. Individual exposure times are set by the Attitude ControlSystem (ACS) star-tracking requirements (about 0.5 and 1.5 sec for the two stars). Thoseexposures are “stacked” on-board to build up signal-to-noise (S/N) and stacked samples areobtained roughly every 20 seconds. Table 1 summarises the observations. The target fieldsare outside the
MOST ’s Continuous Viewing Zone so there is a gap during part of each 101.4min satellite orbit. The duty cycle during each orbit was 30.7% for HD 127756 and 26.1%for HD 217543, but as can be seen in the light curves of Figures 1 and 5 below, the effectiveduty cycle for sampling the timescales of variability in these stars is close to 100%.The frequency analysis of the light curves was done using the CAPER code (Cameron et al.2006; also see Walker et al. 2005b and Saio et al. 2006). CAPER calculates a discrete Fouriertransform of a time series and uses the position of the largest amplitude in the spectrumas an initial guess for the frequency, amplitude and phase parameters in a nonlinear leastsquares fit to the variability. A sinusoid is fitted to the data using all identified parametersand then subtracted from the original light curve. This process is repeated until a predefined 4 –S/N is reached in the amplitude spectrum.A peak with a S/N of ∼ . & σ (Kuschnig et al. 1997)and is adopted as our lower limit to the significance of the extracted periodicities. The S/N ofeach identified periodicity is estimated (before prewhitening that component) by taking themean amplitude in a box around the identified peak and sigma clipping points until the meanconverges. This is done to ensure that high amplitude peaks near the identified frequencydo not skew the local mean amplitude. The S/N calculation method contains two potentialsources of uncertainty: 1. The width of the box used to average the amplitude spectrum (thenoise) and 2. Uncertainties in the fitted amplitude. We estimate the uncertainty in the noisecalculation by varying the width of the averaging box from ± − to ± − in steps of0.1 c d − and then calculate an average noise and the standard error on that average noise.Once amplitude uncertainties are assessed from a bootstrap analysis (described below), wecombine both to arrive at the final uncertainty in S/N. This uncertainty is dominated by theprecision of the amplitude parameter so to limit the size of our data tables we only reportthe amplitude uncertainties, but show the full errorbars in all S/N plots presented in thepaper.Special care is taken to assess the precision of our fitted parameters and to identifyfrequencies that are possibly unresolved. Recently, Breger (2007) discussed the differencebetween frequency resolution and the precision of fit parameters to time series data. Tradi-tionally one estimates resolution in Fourier space as T − (Rayleigh criterion), where T is thelength of the observing run. Loumos & Deeming (1978) suggest an upper limit of 1.5 T − (roughly corresponding to the spacing between the main peak of the window function and thepeak of its first sidelobe) be used when identifying periodicities directly from an amplitudespectrum. However, lower, and arguably more realistic, estimates are used by Kurtz (1980),who estimates frequency resolution as 0 . T − (approximately the half-width of a peak in theamplitude spectrum), and by Kallinger et al. (2008), who suggest ∼ . T − can be usedbased on a large number of simulated data sets. Ultimately, the resolution of frequencies inFourier space is a function of S/N (or significance) of each individual peak; see, for example,Kallinger et al. (2008), and the above criteria are only estimates used when determining thefrequency resolution over the entire frequency range of interest. The precision of fitted pa-rameters, on the other hand, can be estimated by refitting identified parameters to a largenumber of data sets created by sampling the fitted function in the same way as the dataand adding random, normally distributed noise. This Monte Carlo procedure is used, forexample, in Period04 (Lenz & Breger 2005).We assess the precision of our fit parameters and estimate our resolution using a typeof bootstrap analysis (Cameron et al. 2006). By randomly sampling the light curves of 5 –HD 127756 and HD 217543 (with the possibility of replacement) 100000 times and thenrefitting our parameters to those resampled data sets, we build distributions for each ofthe fit parameters, e.g. as in Fig. 4 discussed in the next section. We estimate the 1 and3 σ uncertainties for each parameter as the width of the region, centred on the parameterin question, that contains 68 and 99% of the bootstrap realisations, respectively. Noticethat this differs from a Monte Carlo procedure (as described above) in that there are noassumtions made about the noise of our resampled data sets (we only use the data) andthat the frequency resolution of our data sets can be estimated because the windowing(determined from the temporal sampling of the data) of each of the resampled data sets israndomly changed. Thus, we test the robustness of our fit against both the inherent noiseof the data and the sampling of the data as well. HD 127756 was observed for a total of 30.7 days by
MOST . The light curve is shownin Fig. 1, which shows clear variations with periods near 1 day and 0 . MOST does not suffer from cycle/day aliasing due to daily gaps as experienced in single-site ground-based observations; these periodicities are intrinsic to the star.) Table 2 liststhe frequencies, amplitudes, phases (referenced to the time of the first observation), the 1and 3 σ uncertainties from the bootstrap analysis, and the S/N of the 30 most significantperiodicities. The fit is shown superimposed over two zoomed sections of the light curvelabeled A and B in the lower panels of Fig. 1.The amplitude spectrum of the data along with the fitted points and the residuals fromthe fit are shown in the top panel of Fig. 2. Most of the frequencies gather into three groups; ∼ − , ∼ − , and ∼ − . This property is similar to the frequency groupingsof the SPBe star HD 163868 (Walker et al. 2005b). The lower panel of Fig. 2 plots theS/N for each of the identified periodicities and the window function of the data. Amongthe frequencies listed in Table 2, ν = 0 . − and ν = 0 . − have the fewestobserved cycles (close to the length of the run at 1 / . . − ) and are included toreduce the scatter in the residuals from our fit. They may not be genuine stellar oscillationfrequencies but it should be noted that we have not observed artifacts associated with thebaselines of other MOST observations, especially with such a relatively large amplitude of 7mmag as in the case of ν here.A comparison of the closely spaced frequencies near ∼ − to the window functionand to our fit is given in Fig. 3. Notice that the peak with the largest amplitude has anasymmetric component that is wider than the window function. When that frequency ( ν ) 6 –is prewhitened, significant power remains near that asymmetry and is fitted as ν (shownas the data point with the smallest amplitude in Fig. 3). These frequencies are spaced by ∼ .
04 c d − which is greater than the Rayleigh criterion for our data ( ∼ .
03 c d − ). Thepoints are clearly separated in frequency within their respective 3 σ errorbars. The peaklabelled as Ax ( ν in Table 2) is spaced from ν at nearly the resolution limit suggested byLoumos & Deeming (1978) ( ∼ .
06 c d − ). This peak is clearly resolved from ν and hasan amplitude that is ∼ ν and ν (shown circled in Fig. 3) havethe smallest frequency separation and are barely resolved within their 3 σ errorbars with aseparation of ∼ .
004 c d − . The bootstrap distributions for these frequencies are plotted inFig 4 and shows the parameters are normally distributed and the frequency distributions of ν and ν nearly overlap.We suggest, based on our bootstrap distributions, that frequencies spaced by less than0 . T − ∼ . − (within their 3 σ errorbars) are at the resolution limit of our data set.Using this resolution criterion, frequency pairs ν and ν ; ν and ν ; and ν and ν shouldbe modelled with caution. We show in section 3.1 that the determination of the rotationfrequency of HD 127756 and the general interpretation of the observed variability dependson the frequency ranges and groupings and does not rely on the individual frequencies beingfully resolved. In section 4.3, we discuss the limits of detailed modelling of the stars presentedin this work. MOST observed HD 217543 as a guide star for a total of 26.1 days. Figure 5 shows thelight curve with clear periods of ∼ . ∼ .
25 days with modulations characteristic ofmore complex multi-periodicity. The fit to the 40 most significant frequencies (see Table 3)is shown in zoomed regions labelled A and B in the lower panels of the plot. Note that inTable 3 there are 6 frequencies with S/N ranging from 3.09 to 3.38. These are below theS/N ∼ . & σ detections (Kuschnig et al. 1997). Theyare included to illustrate that within the S/N errors plotted in the lower panel of Figure 6all identified frequencies reach the S/N ∼ . MOST light curve of HD 127756. The top panel shows the entire light curvespanning a total of 30.7 days. The middle and the bottom panels are expanded light curvesfor the portions A and B, respectively, indicated in the top panel. Solid lines indicate the fitof the 30 significant frequencies (Table 2) from the frequency analysis of the full light curve.The short-term variability seen in the middle panel is a consequence of stray Earthshinemodulated with the
MOST satellite orbital period of ∼ . σ error bars are the fitted parameters (see Table 2). The inverteddash - dot line is the residual amplitude spectrum obtained after the fit was subtracted fromthe light curve. (Lower Panel) The S/N of the identified periodicities with 3 σ uncertaintiesestimated from both the fitted amplitudes and frequencies and the mean of the amplitudespectrum (see section 2 for details). The light grey line represents the window function ofthe data centred on the frequency with the largest amplitude and scaled to the maximumS/N for clarity. 9 –Fig. 3.— A zoomed region around the largest peak in the amplitude spectrum of HD 127756.The window function is shown as the inverted, dotted line and the fit is shown as points with3 σ errorbars. The asymmetry of the largest peak in the amplitude spectrum (width ∼ . − [ dashed line]) is compared to the width of the window function ( ∼ .
048 c d − [ dashdot line]). The amplitude of the first sidelobe of the window function (labelled Ay) is ∼ ν and ν (both circled) is discussed in section 2.1. 10 –Fig. 4.— A comparison of bootstrap distributions for parameter sets ( ν , A , φ ) and ( ν ,A , φ ) [see Table 2] for 100000 realisations of the HD 127756 light curve. The top panelsare distributions for the fitted phase ( φ ) while the middle and lower panels show distributionsfor the amplitude (A) and frequency ( ν ) parameters, respectively. In each panel symbols areshown (from top to bottom) for the 1 σ ( ⋆ ) and 2 σ ( (cid:4) ) error intervals containing 68 and 95% of the realisations (note that Table 2 lists the 3 σ , or 99%, error interval) centred on thefitted parameter. Below those symbols in each panel are the 1 σ ( (cid:7) ) errorbars obtained fromthe formula definition of standard deviation and the mean ( • ) of the distribution with thestandard error on the mean. These distributions are shown because the frequencies are theclosest to each other. 11 –with HD 127756, most frequencies are grouped around three ranges; ∼ − , ∼ − , ∼ − . The second and the third frequency range is higher by a factor of ∼ ν = 0 . − and ν = 0 . − are close to the length of run (1 / . . − ) but were included to reduce theresiduals in the light curve. They may not be intrinsic stellar pulsations.Frequencies ν and ν of Table 3 overlap within their 3 σ uncertainties. The bootstrapdistributions are given in Fig. 7 and show all parameters are normally distributed like thosein Fig. 4 for HD 127756. However, the long tails on the frequency distributions suggest thatthese frequencies are not fully resolved. If we adopt the same resolution criterion as for HD127756, frequencies spaced less than 0 . T − ∼ . − (within their 3 σ errorbars) areat (or below) the resolution limit of our data set. This means frequency sets ν and ν ; ν and ν ; and ν and ν should be modelled with care. Once again, the resolution ofindividual frequencies is not a requirement for the determination of the rotation frequencyof this star (see section 3.2).
3. Theoretical models
The groupings of frequencies seen in the two stars is consistent with high radial-order g-mode oscillations of which frequencies in the co-rotating frame are smaller than the rotationfrequency meaning that HD 127756 and HD 217543 are two new SPBe stars. The meanfrequencies of the second and third frequency groups for HD 217543 are about twice thecorresponding ones for HD 127756, indicating that the rotation frequency of HD 217543 isabout double that of HD 127756 (see below).The modelling here is the same as that in Walker et al. (2005b) and Saio et al. (2007).The same chemical composition (
X, Z ) = (0 . , .
02) is assumed for all models. We haveconsidered models computed with OP (Opacity Project; Badnell et al. 2005) opacity tablesas well as models with OPAL(95) opacity tables (Iglesias & Rogers 1996), taking into accountthe recent theoretical results (Jeffery & Saio 2006, 2007; Miglio et al. 2007) that OP opacitiestend to excite pulsations in hotter models than those with OPAL opacities. The equationof state in the envelope was obtained by solving Saha’s equation for Hydrogen, Helium, andCarbon. The structure of a convection zone in the envelope was calculated with a localmixing length theory using a mixing length of 1.5 times the pressure scale height. Theperturbation of convective flux was neglected in the stability analysis (described below) andno overshooting from the convective core was assumed. 12 –Fig. 5.—
MOST light curve of HD 217543. (Top panel) The full light curve for a total of26.1 days.The middle and the bottom panels show expanded light curves for the portions Aand B (respectively) indicated in the top panel. Solid lines indicate the fit of the 40 mostsignificant frequencies from the frequency analysis of the full light curve (see Table 3). 13 –Fig. 6.— Fourier amplitude spectrum of the light curve of HD 217543 and the identifiedfrequency parameters from Table 3. The panels and the meaning of the symbols are describedin Fig. 2. 14 –Fig. 7.— A comparison of bootstrap distributions for parameter sets ( ν , A , φ ) and ( ν ,A , φ ) [see Table 3] for 100000 realisations of the HD 217543 light curve. Symbols are thesame as those in Fig. 4. These distributions are shown because the fitted frequencies are theclosest to each other. In this case, the long tails on the frequency distributions suggest thatthese frequencies are not fully resolved. 15 –The stability of nonradial pulsations in rapidly-rotating stars was examined using themethod of Lee & Baraffe (1995), in which the deformation [proportional to P (cos θ )] of theequilibrium structure due to the centrifugal force is included. The angular dependencies ofpulsational perturbations are expanded into terms proportional to spherical harmonics Y ml j for a given azimuthal order m ( Y ml ′ j for toroidal velocity field) with l j = | m | + 2 j ( l ′ j = l j + 1)for even modes and l j = | m | + 2 j + 1 ( l ′ j = l j −
1) for odd modes with j = 0 , , . . . N . Theseries is truncated at N = 9 so we can obtain accurate eigenfunctions for low-degree modeswithin a reasonable computing time. We adopt the convention that a negative m representsa prograde mode (in the co-rotating frame) with respect to the stellar rotation. Even (odd)modes are symmetric (anti-symmetric) with respect to the equatorial plane. We designatethe angular-dependence type of a mode by a set of ( m , ℓ ) in which ℓ is defined as the l j valueof the largest-amplitude component. Taking into account that high surface degrees reducethe visibility of the modes, we consider in this paper (as in our previous analyses) modeswith ℓ ≤ Kozok (1985) gives values of V = 6 .
56, ( B − V ) = − .
22, and ( U − B ) = − .
01 for HD127756. The ( B − V ) value corresponds to log T eff = 4 .
322 according to Code et al’s (1976)calibration. (Note that using Flower (1977)’s table gives log T eff = 4 . ± .
01 in ( B − V ) , which corresponds to ± .
02 in log T eff , we estimate the effectivetemperature of HD 127756 lies in the range of log T eff = 4 . ± . . This range is shownby vertical lines on the HR diagram in Fig. 8 along with some evolutionary tracks. From arelation between ( U − B ) and M v for Be stars, Kozok (1985) estimated M v ≈ − . ± . − . L/L ⊙ ≈ . ± . , ,
10, and 11 M ⊙ because they cover the range of effective temperatures derived above, during the main-sequence evolution phase (Fig.8). Only the 10 and 11 M ⊙ models cross into the estimatedluminosity range during the late stages of main-sequence evolution. A rotation frequency of0 .
01 mHz (0.86 c d − ) is adopted to approximately fit the observed two groups of frequencieswith m = − m = − T eff ≈ . , .
32, and 4 .
30 for each mass and for both OP and OPAL opacities. The growth ratesand m values of excited low-degree ( ℓ ≤
2) modes are shown in Fig.9 for the OP opacityand in Fig.10 for the OPAL opacity models. Red (full) lines are for modes symmetric with 16 –Fig. 8.— Evolutionary tracks of models computed for HD 127756. A constant and rigidrotation at a rate of 0.01 mHz (0.86 c d − ) is assumed throughout the evolution. Theevolutionary track of 10 M ⊙ non-rotating models computed using OPAL opacities is shownwith a dashed line. Vertical dash-dotted lines indicate the estimated range in the effectivetemperature for HD 127756 based on photometric indices. 17 –respect to the equatorial plane and blue (broken) lines are for anti-symmetric modes.Generally, more modes are excited in cooler and more luminous models. By comparingthe models with OP and OPAL opacities (Figs 9, 10), a similar number of modes are excitedin cooler and more luminous models using the OPAL opacities compared to the cases thatuse OP opacities. This suggests that the stability boundary seems to shift to redder andmore luminous values for the OPAL opacities, which is consistent with the result of thestability analysis by Miglio et al. (2007) for non-rotating B-stars.Since both high-order g-modes (and r-modes) and some low-order g-modes are excitedin these models, the frequency versus growth-rate diagrams are more complex than those forless massive models of HD 217543 and HD 163868 (see below). We note that some of theexcited low-order g-modes have considerable contributions from high l components whichtend to reduce their visibility in integrated light.In order to be consistent with the observed frequencies of HD 127756 (Fig. 6), at leasttwo groups of frequencies around 0.011 mHz (1 c d − ) and 0.023 mHz (2 c d − ) should beexcited. Among the models with the OP tables shown in Fig. 9, this requirement is metby models of mass 11 M ⊙ , the cooler two models of mass 10 M ⊙ , and the coolest model of9 M ⊙ . On the other hand, among the models with the OPAL opacity tables (Fig.10), thecooler 11 M ⊙ model and the coolest 10 M ⊙ model are more or less consistent with HD 127756.These models (except for the 9 M ⊙ model) are luminous enough to be consistent with therange of the luminosity estimated above.Each panel of Figs. 9, and 10 gives the value of the normalized rotation frequency,Ω ≡ Ω / p GM/R corresponding to Ω = 0 .
01 mHz. Since the radius R refers to a meanradius (which is smaller than the equatorial radius), the critical rotation on the equatorcorresponds to Ω ≈ .
75. The values of Ω in these models indicate that the equatorialrotation speeds on the surface are not far from the critical speed, which seems common inBe stars. Although the angular rotation speed of HD 127756 is much smaller than thatof HD 163868 (0.016 mHz Walker et al. 2005b), the larger radius of HD 127756 makes thesurface rotation velocity near critical.Figure 11 provides a comparison between the observed frequencies with a closely matchedmodel of 10 M ⊙ using OP opacities. Most of the excited frequencies in the lowest frequencygroup ( < .
004 mHz) are odd r-modes of m = 1. Some of the frequencies in this group areretrograde, even g-modes of m = 1, in which high l j components contribute significantly tothe eigenfunction. The observed frequency group at 0.011 mHz ( ∼ − ) is mainly iden-tified with prograde, high-order (n = 23 −
41) g-modes of m = − OP Fig. 9.— Growth rates η and azimuthal order m versus frequencies (in the observers’ frame)of excited low-degree ( ℓ ≤
2) modes are shown for selected models for HD 127756 computedwith OP opacities. Solid (red) lines are for even (symmetric with respect to the equator)modes, while broken (blue) lines for odd modes. 19 –
OPAL
Fig. 10.— The same as Fig. 9 but for models computed with OPAL opacities. 20 –group. The frequency range actually observed for this group is still larger than the predictedrange. The frequency group at 0.023 mHz ( ∼ − ) is mainly covered by prograde eveng-modes of m = − m = 0 mode at 0.032mHz ( ≈ .
8c d − ) is the 5th radial order g-mode with a dominant l j = 2 component. The m = − l j = 2 component. The visibility of relatively high frequency ( > .
026 mHz, 2.25 c d − ) m = − l j components are large in these modes. Fig. 11.— Observed frequencies of HD 127756 are compared with a closely matched 10 M ⊙ model with OP opacities. The bottom and the middle panels show azimuthal orders andgrowth rates of excited modes with ℓ ≤
2. The top panel shows observed frequencies andcorresponding amplitudes. A slightly smaller rotational frequency of 0.0094 mHz (Ω = 0 . Zorec et al. (2005) estimated the parameters of HD 217543 as log T eff = 4 . g =3 .
95, and M = 6 . M ⊙ . These values yield log L/L ⊙ = 3 . M ⊙ rotating with a frequency of 0.02 mHz (1.73c d − ). The rotation frequency was chosen so that high-order g-modes have frequencies inthe observers’ frame consistent with the observed oscillation frequencies for HD 217543 (seebelow). Figure 12 shows the calculated evolutionary tracks with HD 217543 (a big circle)put beside the 7 M ⊙ tracks on the HR diagram in accordance with Zorec et al.’s parameters.Fig. 12.— The position of HD 217543 is shown by a circle along with evolutionary tracks for6, 7, and 8 M ⊙ models rotating at a rate of 0.02 mHz. The dashed line shows the evolutionarytrack of 7 M ⊙ non-rotating models calculated with OPAL opacities.Since the parameters of HD 217543 are relatively well determined, we present the resultsfrom a pulsation analysis for models with a mass 7 M ⊙ and appropriate effective temperatures.Figure 13 shows growth rates and azimuthal order m (prograde modes correspond to m < M ⊙ models rotating at a rate of 0.02mHz (1.7 c d − ). The top and the bottom panels are for models with OP and OPALopacities, respectively. The cooler models have log T eff = 4 . T eff ≈ . − for the cooler models and about 360 km s − for thehotter models, indicating the inclination angle of the rotation axis is very high (70 ◦ − ◦ ).We expect only symmetric (even) modes to be detected when observing the star at such ahigh inclination angle.Prograde, high-order g-modes of m = − − . − . . − .
05 mHz in the inertial frame. The predicted frequency groups agreewell with those detected by
MOST in HD 217543. As is the case for the HD 127756 models,the OP opacities excite a larger number of g-modes compared with OPAL opacities, andamong the models shown, those with OP opacities agree better with observed frequencies.However, these models cannot explain the very low frequencies ( < .
01 mHz) we observedsince very few r-modes are excited and these odd modes should be invisible in the nearlyequator-on orientation.
4. Discussion
HD 127756 and HD 217543 join HD 163868 (Walker et al. 2005b), and β CMi (Saio et al.2007) as members of the SPBe class. We can start to investigate this class in more detailswith four stars.
Walker et al. (2005b) considered HD 163868 as a B5 Ve star (Thackeray et al. 1973),and compared the observed frequencies with a 6 M ⊙ model having log T eff ≈ .
23 and rotatingat a frequency of 0 .
016 mHz (Ω / p GM/R = 0 . B − V ) = − .
19 and ( U − B ) = − .
82 with E ( B − V ) = 0 . B − V ) value corresponds to B3.5 and log T eff = 4 .
251 (Flower 1977), while the ( U − B ) value corresponds to B2 and log T eff = 4 .
331 (Lang 1992). Furthermore, Sterken et al. (1998)obtained b − y = 0 . c = 0 . β index β c = 2 .
630 (meanvalue) for HD 163868. Adopting the relations E ( b − y ) = 0 . E ( B − V ) (Davis & Shobbrook1977) and c = c − . E ( b − y ) (Crawford 1975) yields c = 0 . β c and c into the interpolation formula for log T eff derived by Balona (1984) we 23 – Fig. 13.— Growth rates and azimuthal order m versus pulsation frequencies of excitedmodes with ℓ ≤ M ⊙ models computed with the OP (middle panels)and OPAL (bottom panels) opacities. A rotation frequency of 0.02 mHz (1.728 c d − ) isassumed. Red lines are for even modes symmetric with respect to the equatorial plane, whileblue (broken) lines for the anti-symmetric modes (odd modes). For comparison, an observedamplitude-frequency diagram of HD 217543 is shown in both of the top panels. 24 –obtain log T eff = 4 .
01 HD 163868MOST HD 163868MOST
Fig. 14.— The growth rates and azimuthal orders versus frequencies of excited nonradialpulsations ( ℓ ≤
2) for 8 M ⊙ main-sequence models rotating at a rate of 0 .
016 mHz. The toppanels show observed amplitude versus frequency of HD 163868 as observed by
MOST .The parameters referenced above indicate that the effective temperature of HD 163868should be log T eff ≈ .
30 which corresponds to B2 V rather than B5 V. Therefore, we havere-modelled HD 163868 by adopting a mass of 8 M ⊙ . We have assumed a rotation frequencyof 0 .
016 mHz as before. Figure 14 shows the growth rates and azimuthal orders versusfrequencies of excited nonradial pulsations for two models having slightly different log T eff calculated with OP opacities and two models with OPAL opacities. The cooler model withOP opacities reproduces well three observed frequency groups. The agreement of this modelto the observed data is comparable with that of the old 6 M ⊙ model shown in Walker et al.(2005b). The new model, however, rotates nearly critically (Ω / p GM/R = 0 . m = 1 are excited in addition to odd r-modes in the very low frequencyrange. For these retrograde g-modes, the eigenfunctions are significantly affected by high l j components, which indicates that the visibility of these modes should be low (Thisproperty has already been pointed out for the model of HD 127756 shown in Fig. 11).Dziembowski et al. (2007) and Savonije (2007) argued that these retrograde g-modes of m = 1 are responsible for the very low frequencies observed in HD 163868. In contrastto those results, only a few such modes are excited in our analysis (no such modes werefound excited in our previous models in Walker et al. 2005b).The difference might be explained by the fact that in our method including the effectsof centrifugal deformation yields stronger damping for retrograde g-modes (Lee 2008; privatecommunication), and that our analysis is restricted to low surface-degree modes of ℓ ≤ l j components tend to be significant for retrograde g-modes. Further theoreticaland observational investigations are needed to clarify the nature of the very low frequencies. As we demonstrated in section 3, comparing the observed oscillation frequencies of aSPBe star with theoretical models yields the rotation frequency of the star without referringto v sin i . Using our rotational frequencies, we can derive the equatorial velocity of each starand see how close it is to the critical velocity if an accurate estimate for the equatorial radiusis available.The MOST satellite has detected the SPBe-type variations in four stars so far: HD127756 and HD 217543 (this paper), HD 163868 (Walker et al. 2005b), and β CMi (Saio et al.2007). Figure 15 shows the rotation frequencies derived for these stars as a function ofeffective temperature (bottom panel). This figure shows that the rotation frequency decreasessystematically as the effective temperature increases. This is due to the fact that the hotter(more massive) Be stars have larger radii. The top panel of Fig. 15 shows the rotationfrequency normalized as Ω = Ω / p GM/R , where R is the mean radius taken from thebest model. The normalized rotation frequencies lie between 0.7 and 0.8 for the four casesindicating that these stars rotate nearly critically at the equator. Since the equatorial radiusis larger than the mean radius R , the critical value of Ω is ≈ .
75 according to the Rochemodel. Although the rotation frequency itself is well determined, the ratio to the criticalrotation frequency is affected by uncertainty in the stellar radius . In our analysis the deformation from the centrifugal force was included up to the order Ω . It is worthnoting that this approximation is not accurate for the equatorial region of a nearly critically rotating star,
26 –Fig. 15.— The estimated rotation frequencies versus effective temperatures for the fourSPBe stars observed by the
MOST satellite (bottom panel). The top panel shows normalizedrotation frequencies Ω ≡ Ω / p GM/R , where R is the mean radius. The critical rotation atthe equator occurs when Ω ≈ .
75. Probable errors in Ω, which come from uncertainties inthe stellar radius, are estimated as follows: for β CMi and HD 217543 ∆ log R ≈ |
2∆ log T eff | ;for HD 163868 the difference from the previous model (Walker et al. 2005b) is adopted asa probable error; and for HD 127756 an error of ± . T eff & , . − . β CMi rotates nearly critically.The values of Ω for early type Be stars in Fig. 15 look slightly smaller than the value for β CMi, but not significantly as low as claimed by Cranmer (2005). We need more observationsof SPBe-type oscillations for other Be stars as well as accurate stellar parameters in orderto better understand the connection between rotation speed and the Be phenomena.
In the previous sections we compared theoretical models with observations mainly withrespect to the observed frequency ranges of excited modes rather than the frequencies ofindividual modes. If the models become good enough, it will be possible to compare eachfrequency or frequency spectrum of g-modes to observed periodicities to obtain useful in-formation on stellar structure. As a first step for SPBe stars, we present in this subsectionexploratory comparisons of g-mode frequencies in the corotating frame.The top panel of Fig. 16 compares frequencies of HD 127756 in the co-rotating framewith prograde g-modes in the 10 M ⊙ model shown in Fig. 11, where the second and thirdobserved frequency groups are assumed to be prograde g-modes with m = − m = −
2, respectively. Theoretical frequencies of excited (damped) modes are indicated by solid(dotted) vertical lines, while observed frequencies are indicated by large dots. In this model,excited g-mode groups have radial orders of n = 23 −
41 for m = − ℓ = 1) and n = 26 − m = − ℓ = 2). Generally, frequency spacings of excited g-modes are smaller thanobserved frequency spacings. The former tend to be even smaller than the observationallimit for this data set, which is of order ∼ / (30 . ∼ . ∼ . − breaks the usual g-mode frequency spectrum (wherethe spacing should increase with frequency) because of significant contributions from high l j components. and might affect the stability of g-modes. However, a shift of frequency ranges of excited g-modes in theco-rotating frame would not change our conclusions.
28 –
Fig. 16.— Exploratory comparisons of g-mode frequencies in the co-rotating frame. Thetop panel shows frequencies for HD 127756 compared to the 10 M ⊙ model shown in Fig. 11.The lower panel plots frequencies for HD 217543 compared to the cooler 7 M ⊙ model withthe OP opacity shown in Fig. 13. Observed frequencies are converted to co-rotating framefrequencies assuming that the second and the third frequency groups of each star belong toprograde modes with m = − m = −
2, respectively. The rotation frequency assumed is0 . − ) for HD 127756, while a slightly larger value of 0.021 mHz is adoptedfor HD 217543 (1.814 c d − ) to improve the match. The horizontal axis is the frequency inthe co-rotating frame and the vertical axis indicates azimuthal order m . The frequencies ofexcited and damped g-modes are indicated by solid and dotted bars, respectively. 29 –The bottom panel of Fig. 16 compares HD 217543 to the 7 M ⊙ model with OP opacityshown in the right panel of Fig. 13. Observed frequencies are converted to the co-rotatingframe by assuming the rotation frequency is 0.021 mHz (to have a better agreement we haveemployed a slightly higher rotation frequency than before). Excited g-modes in this modelhave radial orders of n = 13 −
29 for m = − ℓ = 1), and n = 17 −
37 for m = − ℓ = 2).The radial orders tend to be lower and the frequency spacings larger than in the HD 127756model. In the frequency range of 0 .
75 c d − . ν crot . . − for HD 217543, five observedfrequencies agree well with model frequencies of m = − ∼ / . ≈ .
038 c d − ).It is obvious from Fig. 16 that agreement between the models and observations is un-satisfactory. To resolve g-mode frequency spacings, observations with much longer baselinewould be necessary. In addition, models including possible differential rotations might benecessary to fit with observed frequencies. We are confident that in the near future detailedg-mode asteroseismology should be a possibility for Be stars.
5. Conclusion
Precise photometry by the
MOST satellite has revealed high-order g-mode pulsationsin two more rapidly rotating Be stars; HD 127756 and HD 217543. High radial order g-modes with pulsation frequencies in the co-rotating frame that are much smaller than therotation frequency appear in groups depending on the azimuthal order m in an observationalamplitude-frequency diagram. Theoretical models indicate that, in rapidly rotating stars,high-order g-modes are excited near the Fe opacity bump at T ∼ × K as in SPBstars. One difference from slowly rotating SPB stars is the fact that among the high-order g-modes, prograde modes are predominantly excited. These modes have frequencies of ∼ | m | Ωin the observers’ frame with Ω being the rotation frequency. For m = − −
2, expectedfrequencies are Ω and 2Ω consistent with observed frequencies of HD 127756 and HD 217543as well as those of previously discovered in HD 163868 (Walker et al. 2005b).An SPBe star provides an opportunity to determine the rotation frequency withoutreferring to v sin i . We have determined rotation frequencies of ≈ .
01 mHz ( ∼ − ) and ≈ .
02 mHz ( ∼ − ) for HD 127756 and HD 217543, respectively. Combining these resultswith previously determined rotation frequencies for HD 163868 and β CMi, we have foundthat the rotation frequencies of Be stars systematically decrease with increasing effectivetemperature (or increasing the stellar radius). This indicates that the rotation velocity of 30 –Be stars stays close to the critical value independently of the effective temperature.Further observations of SPBe stars are needed to provide details of the properties ofrotation velocities among the Be stars. In order to do a detailed comparison of g-mode spectra(g-mode asteroseismology) between models and observations it is necessary to observe theSPBe stars at different epochs to both increase the frequency resolution and to confirmthe observed periodicities. We expect that such detailed analysis will become possible andprovide information about the interior structure of the Be stars in the near future.This research has made use of the SIMBAD database, operated at CDS, Strasbourg,France. The Natural Sciences and Engineering Research Council of Canada supports theresearch of C.C., D.B.G., J.M.M., A.F.J.M., J.F.R., and S.M.R.. H.S. is supported by the21st Century COE programme of MEXT, Japan. R.K. is supported by the Canadian SpaceAgency. W.W.W. is supported by the Austrian Space Agency and the Austrian ScienceFund (P17580-N2). A.F.J.M. is also supported by FQRNT (Qu´ebec) and C.C is partiallysupported by a Walter C. Sumner Memorial Fellowship.
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This preprint was prepared with the AAS L A TEX macros v5.2.
33 –Table 1. Summary of the
MOST observations of HD 127756 and HD 217543HD 127756 HD 217543Spectral type B1/B2 Vne B3 VpeV [mag] 7.59 6.56Dates (2006) May 5 – Jun 5 Sep 19 – Oct 15Duration [days] 30.7 26.1Duty cycle [%] ∼ ∼ MOST photometry ν [c d − ] A [mmag] φ [rad] ∼ S/N σ ν σ ν σ A σ A σ φ σ φ ± ± ± ± ± ± ± ± ± ± ± ± ± ± . . . . .
06 0 . − . . . . .
06 0 . ± ± ± ± ± ± . . . . .
10 0 . − . . . . .
10 0 . ± ± . . . . .
08 0 . − . . . . .
08 0 . ± . . . . .
08 0 . − . . . . .
08 0 . ± ± . . . . .
34 1 . − . . . . .
34 1 . a Phases are referenced to the first observation in the data set. 35 –Table 3. HD 217543 periodicities from
MOST photometry ν [c d − ] A [mmag] φ [rad] ∼ S/N σ ν σ ν σ A σ A σ φ σ φ + 0 . . . . .
07 0 .
581 0.0269 12.9 1.01 10.80 − . . . . .
07 0 .
30+ 0 . . . . .
14 0 .
432 0.0806 7.1 0.09 5.48 − . . . . .
14 0 .
433 0.1201 13.3 1.76 8.28 ± ± ± ± ± ± ± ± ± ± ± ± ± . . . . .
17 0 . − . . . . .
17 0 .
46+ 0 . . . . .
10 0 . − . . . . .
10 0 .
44+ 0 . . . . .
05 0 . − . . . . .
05 0 .
18+ 0 . . . . .
08 0 . − . . . . .
08 1 .
28+ 0 . . . . .
05 0 . − . . . . .
05 0 .
41+ 0 . . . . .
02 0 . − . . . . .
02 0 .
02+ 0 . . . . .
07 1 . − . . . . .
07 0 .
28+ 0 . . . . .
05 0 . − . . . . .
05 0 .
52+ 0 . . . . .
06 0 . − . . . . .
06 0 .
53 36 –Table 3—Continued ν [c d − ] A [mmag] φ [rad] ∼ S/N σ ν σ ν σ A σ A σ φ σ φ + 0 . . . . .
08 0 . − . . . . .
08 0 .
39+ 0 . . . . .
18 0 . − . . . . .
18 0 . ± ± . . . . .
18 0 . − . . . . .
18 0 . ± . . . . .
15 0 . − . . . . .
15 0 . ± . . . . .
12 0 . − . . . . .
12 0 .
24+ 0 . . . . .
16 1 . − . . . . .
16 0 .
35+ 0 . . . . .
27 1 . − . . . . .
27 0 .
73+ 0 . . . . .
14 0 . − . . . . .
14 0 .
36+ 0 . . . . .
20 0 . − . . . . .
20 0 .
41+ 0 . . . . .
18 0 . − . . . . .
18 0 . ± . . . . .
17 0 . − . . . . .
17 0 ..