take into account effects of stellar rotation. The numerical results were obtained in Wojtek Dziembowski's group in Warsaw and in Mike Breger's group in Vienna.
THEORETICAL CLUES FOR MODE IDENTIFICATION INSTABILITY RANGES AND ROTATIONAL SPLITTING PATTERNS A.A. PAMYATNYKH Copernicus 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 Scuti variables are presented. Comparison of theoretical and observed instability ranges allows to constrain possible parameters of the stellar models. As an example, models of θ 2 Tau are considered. The main uncertainties in the determination of the theoretical frequency ranges for δ Sct variables are due to unsatisfactory treatment of convection. The structure of rotationally split multiplets for the low-order modes excited in δ Sct stars is discussed. The rotational coupling between close modes of spherical harmonic 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 spectra of different main-sequence variables like β Cep, SPB and δ Sct stars. The main goal is to show typical regularities and non-regularities in these spectra, if we take into account effects of stellar rotation. The numerical results were obtained in 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 the last ten years by Wojtek Dziembowski, Philip Goode, Marie-Jo Goupil, Hideyuki Saio and their collaborators who created corresponding codes for linear analysis of radial and nonradial oscillations (Dziembowski and Goode, 1992; Soufi et al., 1998; Goupil et al., 2000; Saio, 2002). Basic theoretical aspects of stellar pulsation, including effects of rotation on the oscillation frequencies, were summarized recently by Christensen-Dalsgaard and Dziembowski (2000). This paper is a continuation of reviews on pulsational instability domains in the upper 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 models in the HR diagram and studied the effects of variations of different stellar parameters (chemical composition, opacity, convection, overshooting from the convective core, 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 correspond to the ZAMS and TAMS models, respectively. The large dots mark unstable modes. According to Pamyatnykh (2000).
In Section 2 we briefly discuss the theoretical frequency spectra of unstable modes for β Cep, SPB and δ Sct variables. In Section 3 we give an example of model constraints from the comparison of observed and theoretical frequency ranges. In Section 4 we show and discuss the structure of rotationally split multiplets in typical models of a δ Sct-type variable, and in the last section we outline some problems.
2. Structure of the Frequency Spectra of Low-Degree Modes Some properties of the pulsations within the β Cep and δ Sct instability domains are 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 in both cases low-order acoustic and gravity modes are excited by the same classical κ–mechanism. The main difference is that the oscillations of the β Cep and δ Sct variables 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 changes in the stellar interiors, in the region surrounding the convective core. Due to these changes, the p- and g-modes are not separated in frequency already in mid- or early-MS evolution, and the phenomenon of ‘avoided crossing’ between p- and gmodes takes place (Aizenman et al., 1977). This results in a mixed character of the low-order nonradial modes: they are similar to pure acoustic modes in the outer stellar layers and to pure gravity modes in the interior. Both in the β Cep and δ Sct star models we find unstable low-order, low-degree p-, g- and mixed modes. The frequency range of the unstable modes in the 1.8 M models is more extended than that in the 12 M models. This is in agreement with the fact that the observed frequency range of the unstable modes is wider in δ Sct than in β Cep stars. For a more detailed discussion of these frequency spectra see Paper II. In Figure 2 the periods of unstable low-degree modes of an evolutionary sequence of typical SPB models are plotted. This figure is similar to Figure 8 in Dziembowski et al. (1993) and to Figure 4 in Dziembowski (1995), but the newest data 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 accurately described by the asymptotic theory. In each model of a given effective temperature a large number of modes is excited simultaneously. These modes are nearly equidistant in period. For a more detailed discussion of theoretical spectra of the SPB models see Dziembowski et al. (1993) and Paper I.
3. Model Constraints from Comparison of Theoretical and Observed Frequency Ranges Detailed quantitative fitting of observed frequencies of a multiperiodic variable with the theoretical frequency spectrum of an appropriate stellar model seems to be 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 asteroseismology of δ Sct stars are discussed). Also, models usually predict much larger number of unstable modes than it is observed in an individual star. For example, a model of the evolved δ Sct-type variable 4 CVn predicts approximately 500 unstable modes of low degree, < 3, which is 25 times larger than the number of observed modes (at least 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 the ZAMS model, whereas rightmost points correspond to the TAMS model. Numbers close to some modes give corresponding radial orders. A few solid lines connect modes of the same radial order in the sequence of the models.
However, we can try to compare the frequency range as a whole with the corresponding theoretical ranges of unstable modes and to obtain some constraints on model parameters. Figure 3 shows the results of such a comparison for the primary component of θ 2 Tau, a δ Sct-type variable, in which 11 frequencies in the 10.8 to 14.6 c/d range are detected (see Breger et al., 2002; for more details). The normalized growth rates of radial and nonradial modes are plotted against frequency for nine models of different mass and effective temperature. Only axisymmetric modes (m = 0) are shown. The independence of the growth rate on the spherical harmonic degree, , is a typical feature of modes excited by the κ-mechanism. The best fit between the theoretical and observed frequency ranges is achieved for models with Teff ≈ 7800 K (or slightly higher), in agreement with photometric calibrations. The instability range spans two or three radial orders in the range p4 to p6 for radial modes.
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Figure 3. Normalized growth rates, η, plotted against frequency, f , in test models of the primary component of θ 2 Tau. Positive values of η correspond to unstable modes. The values of M (in solar units) and Teff are given in each panel. The symbol p5 is plotted near the corresponding radial overtone. The thick horizontal line shows the range of frequencies (10.865 to 14.615 cd−1 ) observed in the primary component of θ 2 Tau. From Breger et al. (2002).
The main uncertainty of such a study is the unsatisfactory description of convection in the stellar envelope and its interaction with pulsations. We used the standard mixing-length theory and the assumption of frozen-in convection. This simple assumption is probably incorrect in the hydrogen convection zone. Therefore, the reality of the additional excitation in the hydrogen zone must be examined by using a nonlocal time-dependent treatment of convection. The first promising results in this direction (see Michel et al., 1999; Houdek, 2000) exist. Note that the total 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 total damping. Therefore even small contributions to the driving are important. We note also the nonlocal results by Kupka and Montgomery (2002), who point out the necessity to use very different values of the mixing-length parameter in the hydrogen 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 small value of this parameter in the hydrogen zone and a significantly higher value in the deeper helium zone. We performed some tests and did find a possibility to fit observations with slightly hotter models. However, we probably can exclude modes involving the second radial overtone (mode p3 ) and lower-order modes, as well as all hot models of the primary with Teff > 8000 K. The reason is that all these 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 follow Christensen-Dalsgaard and Dziembowski (2000) and Goupil et al. (2000). The third order expression for a rotationally split frequency may be written in the form: νm = ν0 + m(1 − Cn )
2 3 + (D0 + m2 D1 ) + m 2 T 2π 2π ν0 ν0
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 been omitted. The temporal dependence of the oscillations is assumed to be exp(−iωt), so that prograde modes correspond to m > 0. The frequency ν0 includes effects of the horizontally averaged centrifugal force in the equilibrium model. (In the computations of stellar evolution we assumed solid-body rotation and conservation of global angular momentum during evolution.) The Ledoux constant, C, determines the usual equidistant splitting valid in the limit of slow rotation. The term m stands for transformation of the co-rotating coordinate system to the inertial coordinate system of the observer. The second and third order coefficients D0 , D1 and T are determined by a perturbation method and take into account nonspherically symmetric distortion due to the centrifugal force and second and third order Coriolis effects. The quadratic terms destroy the symmetry of the multiplet and also predict a frequency shift for the radial modes and nonradial axisymmetric modes. The cubic term affects the value of (νm − ν−m )/m, which can be used to determine the rotational velocity, and may result in fictitious variation of the rotation velocity with depth, as shown by Goupil et al. (2000). Moreover, an important additional correction to the frequency, which is not taken into account in Eq. (1), arises when rotation couples close modes of spherical harmonic degree, , differing by 2 and of the same azimuthal order, m. We will illustrate 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 of rotational mode coupling is given by Daszy´nska-Daszkiewicz et al. (2002) and we will follow this description. In Figure 4 we show the behaviour of frequencies of unstable low-degree modes in a typical δ Sct-type model of 1.8 M which evolves from the ZAMS to the TAMS. This plot is similar to Figure 1 in Daszy´nska-Daszkiewicz et al. (these Proceedings) for a β Cep-type model of 12 M . However, the range of unstable frequencies is now much larger; it extends up to 7 radial modes. Both avoided crossing phenomenon and presence of close frequencies of modes with spherical harmonic degree, , differing by 2, are seen very clearly. The modes are designated according 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 mode can significantly change its properties. As mentioned by Daszy´nska-Daszkiewicz et al. (2002), the physical nature of the modes – that is, the relative proportion of the contribution 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 correspond to dominant acoustic and gravity mode characters, respectively. The rotational mode coupling is important when the frequency distance between the 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/s on the ZAMS, and we have also assumed conservation of the global angular momentum during evolution. With these assumptions the rotational velocity on the TAMS is equal to 81 km/s. For these √ velocities we find that the dimensionless rotational frequency is σrot ≡ / 4π G < 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 understood from Figure 4. The two upper panels in Figure 5 show the effects of rotation on low-order and higher-order modes in an extended scale. As noted by Daszy´nska-Daszkiewicz et al. (2002), the coupling strength depends on mode properties. Coupling between acoustic modes is stronger than that involving one or more gravity modes. This is so because the effect of the centrifugal distortion is only 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 is much more pronounced for higher frequencies. Also, due to centrifugal distortion, the asymmetry of the rotational splitting is larger for higher frequencies, as can be seen in the lower panel for dipole modes. The Ledoux constant is small for acoustic modes, therefore linear splitting for these modes is due to the transformation from the co-rotating coordinate system to the inertial coordinate system of the observer. In contrast, the Ledoux constant is about 0.5 for dipole gravity modes, therefore linear splitting can be approximately two times smaller for gravity modes than for acoustic modes. Such a case takes place 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 typically higher rotational velocities and due to a smaller frequency range of unstable modes (see Figure 1). In Figure 6 we show the effect of the rotational coupling on the period ratio of 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 quadrupole modes, very large and nonregular perturbations to the period ratio occur. If rotation is fast enough, this effect must be taken into account. The effect of rotational coupling 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-Dalsgaard and Dziembowski (2000).
amplitude δ Sct-type variables like AI Vel which rotate slowly (see Petersen and Christensen-Dalsgaard, 1996). This may not be true for one of the best studied δ Sct variables, FG Vir, which has v sin i of about 20 km/s, but there are indications from spectroscopy in favour of a low inclination angle of the rotation axis, therefore true rotation 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 are significantly 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 during evolution of the 1.8 M model from the ZAMS to the TAMS. According to Goupil et al. (in preparation).
and in the rotational splitting. For the radial order of modes, the equidistant pattern is disturbed by avoided crossing phenomenon, whereas the rotational splitting is asymmetric due to second order and higher order effects. However, if we have a rich observed frequency spectrum it is still expedient to search for statistically significant equidistant spacings to infer information on stellar mean density (as was suggested, for example, by Handler et al., 1997; for the XX Pyx) and/or on stellar rotation. The important effect of rotation is the coupling between close frequency modes of 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 upper part of the main sequence. It must be a rather often phenomenon because close fre-
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quencies of relevant modes occur over wide ranges of the frequencies and effective temperatures of the models in the instability domains. The rotational coupling has also strong influence on photometric diagnostic diagrams (amplitude ratio versus phase difference in two passbands of multicolour photometry), as discussed by Daszy´nska-Daszkiewicz in these Proceedings (the more detailed results of this study are given by Daszy´nska-Daszkiewicz et al., 2002). Acknowledgements All numerical results presented here were obtained in the Wojtek Dziembowski’s group 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 the organizers of the Conference for the very nice scientific atmosphere and for the hospitality. The work was supported by Polish KBN grant No. 5 P03D 012 20. References Aizenman, M.L., Smeyers, P. and Weigert, A.: 1977, A&A 58, 41. Breger, M., Pamyatnykh, A.A., Zima, W. et al.: 2002, MNRAS 336, 249. Christensen-Dalsgaard, J. and Dziembowski, W.A.: 2000, in: C. Ibanoglu (ed.), Variable Stars as Essential Astrophysical Tools, Kluwer Academic Publishers, NATO Science Series, Series C, Mathematical and Physical Sciences, vol. 544, 1. Daszy´nska-Daszkiewicz, J., Dziembowski, W.A., Pamyatnykh, A.A. and Goupil, M.-J.: 2002, A&A 392, 151. Dziembowski, W.A.: 1995, in: R.K. Ulrich, E.J. Rhodes Jr. and W. Däppen (eds.), GONG’94: Helioand Astero-Seismology, A.S.P. Conf. Ser. 76, 586. Dziembowski, W.A. and Goode, P.R.: 1992, ApJ 394, 670. Dziembowski, W.A., Moskalik, P. and Pamyatnykh, A.A.: 1993, MNRAS 265, 588. Goupil, M.-J., Dziembowski, W.A., Pamyatnykh, A.A. and Talon, S.: 2000, in: M. Breger and M.H. Montgomery (eds.), Delta Scuti and Related Stars, A.S.P. Conf. Ser. 210, 267. Goupil, M.-J. and Talon, S.: 2002, in: C. Aerts, T.R. Bedding and J. Christensen-Dalsgaard (eds.), Radial and Nonradial Pulsations as Probes of Stellar Physics, A.S.P. Conf. Ser. 259, 306. Handler, G., Pikall, H., O’Donoghue, D. et al.: 1997, MNRAS 286, 303. Houdek, G.A.: 2000, in: M. Breger and M.H. Montgomery (eds.), Delta Scuti and Related Stars, A.S.P. Conf. Ser. 210, 454. Kupka, F. and Montgomery, M.H.: 2002, MNRAS 330, L6. Mantegazza, L. and Poretti, E.: 2002, A&A (in press). Michel, E., Hernández, M.M., Houdek, G. et al.: 1999, A&A 342, 153. Pamyatnykh, A.A.: 1999, Acta Astr. 49, 119 (Paper I). Pamyatnykh, A.A.: 2000, in: M. Breger and M.H. Montgomery (eds.), Delta Scuti and Related Stars, A.S.P. Conf. Ser. 210, 215 (Paper II). Petersen, J.O. and Christensen-Dalsgaard, J.: 1996, A&A 312, 463. Saio, H.: 2002, in: C. Aerts, T.R. Bedding and J. Christensen-Dalsgaard (eds.), Radial and Nonradial Pulsations as Probes of Stellar Physics, A.S.P. Conf. Ser. 259, 177. Soufi, F., Goupil, M.-J. and Dziembowski, W.A.: 1998, A&A 334, 911.
