THEORETICAL CLUES FOR MODE IDENTIFICATION -INSTABILITY RANGES AND ROTATIONAL SPLITTING PATTERNS

A.A. PAMYATNYKHCopernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland

Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya Str. 48,109017 Moscow, Russia

Institute of Astronomy, University of Vienna, Türkenschanzstr. 17,A-1180, Vienna, Austria

Abstract. Frequency spectra of unstable modes for typical models of β Cephei, SPB and δ Scutivariables are presented. Comparison of theoretical and observed instability ranges allows to constrainpossible parameters of the stellar models. As an example, models of θ2 Tau are considered. The mainuncertainties in the determination of the theoretical frequency ranges for δ Sct variables are due tounsatisfactory treatment of convection. The structure of rotationally split multiplets for the low-ordermodes excited in δ Sct stars is discussed. The rotational coupling between close modes of sphericalharmonic degree, �, differing by 2, can significantly disturb the frequency spectrum.

Keywords: stellar oscillations, instability ranges, rotational splitting

1. Introduction

In this review I will discuss some properties of the theoretical frequency spectraof different main-sequence variables like β Cep, SPB and δ Sct stars. The maingoal is to show typical regularities and non-regularities in these spectra, if wetake into account effects of stellar rotation. The numerical results were obtainedin Wojtek Dziembowski’s group in Warsaw and in Mike Breger’s group in Vienna.The most important effects of rotation on the frequencies were studied during thelast ten years by Wojtek Dziembowski, Philip Goode, Marie-Jo Goupil, HideyukiSaio and their collaborators who created corresponding codes for linear analysisof radial and nonradial oscillations (Dziembowski and Goode, 1992; Soufi et al.,1998; Goupil et al., 2000; Saio, 2002). Basic theoretical aspects of stellar pulsa-tion, including effects of rotation on the oscillation frequencies, were summarizedrecently by Christensen-Dalsgaard and Dziembowski (2000).

This paper is a continuation of reviews on pulsational instability domains in theupper part of the main sequence – namely, on β Cep and SPB instability domains(Pamyatnykh, 1999; Paper I), and on the δ Sct instability domain (Pamyatnykh,2000; Paper II). In previous papers we considered the position of unstable modelsin the HR diagram and studied the effects of variations of different stellar paramet-ers (chemical composition, opacity, convection, overshooting from the convectivecore, rotation) on the position of the instability domains in the HR diagram.

Astrophysics and Space Science 284: 97–107, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1. Frequencies of low-order p- and g-modes with low degree, �, for models of 12 and 1.8 M�in the main sequence evolutionary phase. In each panel, the leftmost and rightmost points correspondto the ZAMS and TAMS models, respectively. The large dots mark unstable modes. According toPamyatnykh (2000).

In Section 2 we briefly discuss the theoretical frequency spectra of unstablemodes for β Cep, SPB and δ Sct variables. In Section 3 we give an exampleof model constraints from the comparison of observed and theoretical frequencyranges. In Section 4 we show and discuss the structure of rotationally split mul-tiplets in typical models of a δ Sct-type variable, and in the last section we outlinesome problems.

2. Structure of the Frequency Spectra of Low-Degree Modes

Some properties of the pulsations within the β Cep and δ Sct instability domainsare given in Figure 1, where the frequency oscillation spectra for stellar models of

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12 M� and 1.8 M� during their evolution from the ZAMS to the TAMS are plotted.The oscillations of both types of variables are similar in many aspects because inboth cases low-order acoustic and gravity modes are excited by the same classicalκ–mechanism. The main difference is that the oscillations of the β Cep and δ Sctvariables rely on different opacity bumps (see Paper II). For radial modes (� = 0)we see an almost equidistant frequency separation between consecutive modes.The complicated patterns of nonradial modes are caused by evolutionary changesin the stellar interiors, in the region surrounding the convective core. Due to thesechanges, the p- and g-modes are not separated in frequency already in mid- orearly-MS evolution, and the phenomenon of ‘avoided crossing’ between p- and g-modes takes place (Aizenman et al., 1977). This results in a mixed character of thelow-order nonradial modes: they are similar to pure acoustic modes in the outerstellar layers and to pure gravity modes in the interior. Both in the β Cep and δ Sctstar models we find unstable low-order, low-degree p-, g- and mixed modes. Thefrequency range of the unstable modes in the 1.8 M� models is more extendedthan that in the 12 M� models. This is in agreement with the fact that the observedfrequency range of the unstable modes is wider in δ Sct than in β Cep stars. For amore detailed discussion of these frequency spectra see Paper II.

In Figure 2 the periods of unstable low-degree modes of an evolutionary se-quence of typical SPB models are plotted. This figure is similar to Figure 8 inDziembowski et al. (1993) and to Figure 4 in Dziembowski (1995), but the newestdata on stellar opacity were used (OPAL data of 1996, see Paper I for details).The excited oscillations here are high-order gravity modes which can be accuratelydescribed by the asymptotic theory. In each model of a given effective temperat-ure a large number of modes is excited simultaneously. These modes are nearlyequidistant in period. For a more detailed discussion of theoretical spectra of theSPB models see Dziembowski et al. (1993) and Paper I.

3. Model Constraints from Comparison of Theoretical and ObservedFrequency Ranges

Detailed quantitative fitting of observed frequencies of a multiperiodic variablewith the theoretical frequency spectrum of an appropriate stellar model seems tobe still an open issue. We don’t know any successful example of such a fitting(see, for example, Goupil and Talon, 2002; where problems of asteroseismologyof δ Sct stars are discussed). Also, models usually predict much larger number ofunstable modes than it is observed in an individual star. For example, a model of theevolved δ Sct-type variable 4 CVn predicts approximately 500 unstable modes oflow degree, � < 3, which is 25 times larger than the number of observed modes (atleast 17 frequencies in the range 4.7–9.7 c/d, see the short discussion in Paper II).The mechanism of the modal selection is still unknown.

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Figure 2. Periods of unstable high-order gravity modes of � = 1 and 2 in the sequence of 4 M�models during evolution on the main sequence. Leftmost points in each panel correspond to theZAMS model, whereas rightmost points correspond to the TAMS model. Numbers close to somemodes give corresponding radial orders. A few solid lines connect modes of the same radial order inthe sequence of the models.

However, we can try to compare the frequency range as a whole with the cor-responding theoretical ranges of unstable modes and to obtain some constraints onmodel parameters. Figure 3 shows the results of such a comparison for the primarycomponent of θ2 Tau, a δ Sct-type variable, in which 11 frequencies in the 10.8 to14.6 c/d range are detected (see Breger et al., 2002; for more details). The normal-ized growth rates of radial and nonradial modes are plotted against frequency fornine models of different mass and effective temperature. Only axisymmetric modes(m = 0) are shown. The independence of the growth rate on the spherical harmonicdegree, �, is a typical feature of modes excited by the κ-mechanism. The best fitbetween the theoretical and observed frequency ranges is achieved for models withTeff ≈ 7800 K (or slightly higher), in agreement with photometric calibrations. Theinstability range spans two or three radial orders in the range p4 to p6 for radialmodes.

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Figure 3. Normalized growth rates, η, plotted against frequency, f , in test models of the primarycomponent of θ2 Tau. Positive values of η correspond to unstable modes. The values of M (in solarunits) and Teff are given in each panel. The symbol p5 is plotted near the corresponding radialovertone. The thick horizontal line shows the range of frequencies (10.865 to 14.615 cd−1) observedin the primary component of θ2 Tau. From Breger et al. (2002).

The main uncertainty of such a study is the unsatisfactory description of con-vection in the stellar envelope and its interaction with pulsations. We used thestandard mixing-length theory and the assumption of frozen-in convection. Thissimple assumption is probably incorrect in the hydrogen convection zone. There-fore, the reality of the additional excitation in the hydrogen zone must be examinedby using a nonlocal time-dependent treatment of convection. The first promisingresults in this direction (see Michel et al., 1999; Houdek, 2000) exist. Note that thetotal driving in a δ Sct star (with the main contribution due to the κ-mechanism

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operating in the second helium ionization zone) only slightly exceeds the totaldamping. Therefore even small contributions to the driving are important.

We note also the nonlocal results by Kupka and Montgomery (2002), who pointout the necessity to use very different values of the mixing-length parameter in thehydrogen and helium convection zones if we still use local mixing-length theory.According to nonlocal studies of A-star envelopes, it is necessary to use a smallvalue of this parameter in the hydrogen zone and a significantly higher value inthe deeper helium zone. We performed some tests and did find a possibility tofit observations with slightly hotter models. However, we probably can excludemodes involving the second radial overtone (mode p3) and lower-order modes, aswell as all hot models of the primary with Teff > 8000 K. The reason is that allthese models are stable in the observed frequency range.

4. Rotational Splitting and Rotational Mode Coupling

To illustrate the effects of rotation on stellar oscillation frequencies, I will followChristensen-Dalsgaard and Dziembowski (2000) and Goupil et al. (2000). Thethird order expression for a rotationally split frequency may be written in the form:

νm = ν0 + m(1 − Cn�)�

2π+ �2

2πν0(D0 + m2D1) + m

�3

ν20

T

where subscript m denotes the azimuthal order of a mode. The subscripts (n, �)

which denote, respectively, the radial order and the degree of the mode, have beenomitted. The temporal dependence of the oscillations is assumed to be exp(−iωt),so that prograde modes correspond to m > 0. The frequency ν0 includes effectsof the horizontally averaged centrifugal force in the equilibrium model. (In thecomputations of stellar evolution we assumed solid-body rotation and conservationof global angular momentum during evolution.) The Ledoux constant, C, determ-ines the usual equidistant splitting valid in the limit of slow rotation. The termm� stands for transformation of the co-rotating coordinate system to the inertialcoordinate system of the observer. The second and third order coefficients D0,D1 and T are determined by a perturbation method and take into account non-spherically symmetric distortion due to the centrifugal force and second and thirdorder Coriolis effects. The quadratic terms destroy the symmetry of the multipletand also predict a frequency shift for the radial modes and nonradial axisymmetricmodes. The cubic term affects the value of (νm − ν−m)/m, which can be usedto determine the rotational velocity, and may result in fictitious variation of therotation velocity with depth, as shown by Goupil et al. (2000).

Moreover, an important additional correction to the frequency, which is nottaken into account in Eq. (1), arises when rotation couples close modes of sphericalharmonic degree, �, differing by 2 and of the same azimuthal order, m. We willillustrate how this effect can be significant at typical velocities of rotation and for

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Figure 4. Evolution of frequency spectra of unstable modes with � = 0, 2 (left panel) and with � = 1,3 (right panel) in a sequence of δ Sct models with mass of 1.8 M�. For simplicity, only modes of� = 1 are identified in the right panel. Dotted vertical line corresponds to the model of Teff = 7515 K.Effects of rotation on frequency spectrum of this model are presented in Figure 5.

typical models of main sequence stars. A detailed discussion of the occurrence ofrotational mode coupling is given by Daszynska-Daszkiewicz et al. (2002) and wewill follow this description.

In Figure 4 we show the behaviour of frequencies of unstable low-degree modesin a typical δ Sct-type model of 1.8 M� which evolves from the ZAMS to theTAMS. This plot is similar to Figure 1 in Daszynska-Daszkiewicz et al. (theseProceedings) for a β Cep-type model of 12 M�. However, the range of unstablefrequencies is now much larger; it extends up to 7 radial modes. Both avoidedcrossing phenomenon and presence of close frequencies of modes with sphericalharmonic degree, �, differing by 2, are seen very clearly. The modes are designatedaccording to the avoided-crossing principle, i.e. each mode preserves its initial des-

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ignation on the ZAMS besides the fact that in the course of the evolution this modecan significantly change its properties. As mentioned by Daszynska-Daszkiewiczet al. (2002), the physical nature of the modes – that is, the relative proportion of thecontribution from acoustic and gravity propagation zones to the mode’s energy –is reflected in the slope of the σ (log Teff) function. Slow and rapid rises correspondto dominant acoustic and gravity mode characters, respectively.

The rotational mode coupling is important when the frequency distance betweenthe modes becomes comparable to the rotational frequency. For our model of 1.8 M�we assumed uniform (solid-body) rotation with equatorial velocity of 100 km/son the ZAMS, and we have also assumed conservation of the global angular mo-mentum during evolution. With these assumptions the rotational velocity on theTAMS is equal to 81 km/s. For these velocities we find that the dimensionlessrotational frequency is σrot ≡ �/

√4πG <ρ > ≈ 0.12 − 0.13. It is clear from

Figure 4 that the coupling must be significant in many cases.Figure 5 illustrates all the main effects of rotation on the frequency spectrum

of unstable modes in the 1.8 M� model with Teff = 7515 K and Vrot = 92 km/s.The gravitational or acoustic character of the modes involved can be easily un-derstood from Figure 4. The two upper panels in Figure 5 show the effects ofrotation on low-order and higher-order modes in an extended scale. As noted byDaszynska-Daszkiewicz et al. (2002), the coupling strength depends on mode prop-erties. Coupling between acoustic modes is stronger than that involving one ormore gravity modes. This is so because the effect of the centrifugal distortion isonly important in the acoustic cavity and it increases with the mode frequency.Such a tendency is seen very clearly in the two upper panels, where the coupling ismuch more pronounced for higher frequencies. Also, due to centrifugal distortion,the asymmetry of the rotational splitting is larger for higher frequencies, as can beseen in the lower panel for dipole modes.

The Ledoux constant is small for acoustic modes, therefore linear splitting forthese modes is due to the transformation from the co-rotating coordinate system tothe inertial coordinate system of the observer. In contrast, the Ledoux constant isabout 0.5 for dipole gravity modes, therefore linear splitting can be approximatelytwo times smaller for gravity modes than for acoustic modes. Such a case takesplace in our model for low-order dipole modes, � = 1, see lower panel.

The rotational splitting results in the widening of the frequency instability range.This effect is relatively more important for β Cep than for δ Sct stars due to typ-ically higher rotational velocities and due to a smaller frequency range of unstablemodes (see Figure 1).

In Figure 6 we show the effect of the rotational coupling on the period ratioof the two lowest radial modes in the same evolutionary sequence of the 1.8 M�models. Due to rotational coupling between radial modes and closest quadrupolemodes, very large and nonregular perturbations to the period ratio occur. If rotationis fast enough, this effect must be taken into account. The effect of rotationalcoupling on the period ratio is, probably, not important for well-studied high-

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Figure 5. Effects of rotation on the frequencies of close pairs of � = 0 and 2 (upper and middle panel)and � = 1 and 3 modes (lower panel) for the 1.8 M� model with Teff = 7515 K and Vrot = 92 km/s.At each step upward a new effect is added. In an analogy with Figure 11 from Christensen-Dalsgaardand Dziembowski (2000).

amplitude δ Sct-type variables like AI Vel which rotate slowly (see Petersen andChristensen-Dalsgaard, 1996). This may not be true for one of the best studied δ Sctvariables, FG Vir, which has v sin i of about 20 km/s, but there are indications fromspectroscopy in favour of a low inclination angle of the rotation axis, therefore truerotation velocity may be around 80 km/s (Mantegazza and Poretti, 2002).

5. Some Conclusions

We have seen that simple regular patterns in the theoretical frequency spectra aresignificantly disturbed – both in the spacing between modes of consecutive order

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Figure 6. Variation of the period ratio of the radial first overtone to the fundamental mode duringevolution of the 1.8 M� model from the ZAMS to the TAMS. According to Goupil et al. (inpreparation).

and in the rotational splitting. For the radial order of modes, the equidistant patternis disturbed by avoided crossing phenomenon, whereas the rotational splitting isasymmetric due to second order and higher order effects.

However, if we have a rich observed frequency spectrum it is still expedientto search for statistically significant equidistant spacings to infer information onstellar mean density (as was suggested, for example, by Handler et al., 1997; forthe XX Pyx) and/or on stellar rotation.

The important effect of rotation is the coupling between close frequency modesof spherical harmonic degree, �, differing by 2, and of the same azimuthal order,m. Such a coupling may occur at typical rotation velocities of stars in the upperpart of the main sequence. It must be a rather often phenomenon because close fre-

THEORETICAL CLUES FOR MODE IDENTIFICATION 107

quencies of relevant modes occur over wide ranges of the frequencies and effectivetemperatures of the models in the instability domains.

The rotational coupling has also strong influence on photometric diagnosticdiagrams (amplitude ratio versus phase difference in two passbands of multicolourphotometry), as discussed by Daszynska-Daszkiewicz in these Proceedings (themore detailed results of this study are given by Daszynska-Daszkiewicz et al.,2002).

Acknowledgements

All numerical results presented here were obtained in the Wojtek Dziembowski’sgroup in Warsaw and in the Mike Breger’s group in Vienna. I am indebted to M.-J.Goupil for useful discussions during the Conference. It is my pleasure to thank theorganizers of the Conference for the very nice scientific atmosphere and for thehospitality. The work was supported by Polish KBN grant No. 5 P03D 012 20.

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