1 Introduction
\nofiles

On the Prospects for Detection and Identification

of Low-Frequency Oscillation Modes in Rotating

B Type Stars

J.   D a s z y ń s k a-D a s z k i e w i c z,  W. A.  D z i e m b o w s k i

and   A. A.   P a m y a t n y k h

Instytut Astronomiczny, Uniwersytet Wrocławski. ul. Kopernika 11, Poland

e-mail: daszynska@astro.uni.wroc.pl

Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warsaw, Poland

e-mail:wd@astrouw.edu.pl

Copernicus Astronomical Center, ul. Bartycka 18, 00-716 Warsaw, Poland

e-mail: alosza@camk.edu.pl

Institute of Astronomy, Pyatnitskaya Str. 48, 109017 Moscow, Russia

ABSTRACT

We study how rotation affects observable amplitudes of high-order g- and mixed r/g-modes and examine prospects for their detection and identification. Our formalism, which is described in some detail, relies on a nonadiabatic generalization of the traditional approximation. Numerical results are presented for a number of unstable modes in a model of SPB star, at rotation rates up to 250 km/s. It is shown that rotation has a large effect on mode visibility in light and in mean radial velocity variations. In most cases, fast rotation impairs mode detectability of g-modes in light variation, as Townsend (2003b) has already noted, but it helps detection in radial velocity variation. The mixed modes, which exist only at sufficiently fast rotation, are also more easily seen in radial velocity. The amplitude ratios and phase differences are strongly dependent on the aspect, the rotational velocity and on the mode. The latter dependence is essential for mode identification.

Stars: oscillations – Stars: emission-line, Be – Stars: rotation

## 1 Introduction

Variability with frequencies comparable to rotation frequency has been found in a number of hot (most often Be) stars. Whether such variability is caused by slow modes has been debated for some time (see e.g., Baade 1982, Balona 1985). However, recent data from the MOST on  Oph (Walker et al. 2005a), HD 163868 (Walker et al. 2005b), and  CMi (Saio et al. 2007) revealed rich frequency spectra which may be understood only in terms of oscillation mode excitation. There are also frequency spectra obtained from ground-based observations, such as of  Eri (Jerzykiewicz et al. 2005), which certainly cannot be explained in terms of a rotational modulation. A potential for using the abundant frequency data as constraints on stellar models exists but the prerequisite is mode identification and it has not been done so far.

Most frequently used method of mode identification employs the data on mode amplitudes and phases of light variability in various bands and in radial velocity. Amplitude ratios depend on angular dependence of surface distortion, which in rotating stars is no longer described by a single spherical harmonic. The departure becomes large once the angular velocity of rotation is comparable to oscillation frequency. This is not an unusual situation for cooler B stars in the main sequence band. Take a model of 6 M star in the mid of its main sequence evolution. Ignoring effects of rotation, we find that all dipole modes with period between 1.8 d to 2.8 d are unstable. Rotation periods in this range correspond to equatorial velocities between 100 km/s and 150 km/s, which is not high for such a star.

For low frequency modes the surface dependence may be approximately described in terms of the Hough functions (see e.g., Lee and Saio 1997, Bildsten et al. 1996, Townsend 2003a). This approximation, called traditional, was used by Townsend (2003b), who calculated observable light amplitudes and used the results to address the problem of mode identification for low frequencies in rotating stars using multicolor data. It is our experience (e.g., Daszyńska-Daszkiewicz, Dziembowski and Pamyatnykh 2005), however, that in the case of B-type pulsators, it is very important to combine photometric and radial velocity data for a unique discrimination of excited modes and constraining stellar parameters. Thus, this work, which may be regarded as an extension of Townsend’s paper, focuses on calculation of radial velocity amplitudes. We adopt an uniform approach in our calculation of all disk-averaged amplitudes and it is different from that used by him. Furthermore, in addition to g-modes, we include r-modes, which become propagative in stellar radiative envelopes once rotation is fast enough (see Savonije 2005, Townsend 2005b, who uses the term mixed gravity-Rossby modes, Lee 2006).

In Section 2, after specifying assumptions adopted in our calculations, we summarize formulae for angular dependence of velocity and atmospheric parameters. Expressions for the light and disk-averaged radial velocity variations are given in Sections 3 and 4, respectively. Sections 5, 6 and 7 present numerical results for selected modes in one representative stellar model considering range of rotation rates but ignoring effects of changes of centrifugal force on model structure. Unstable mode properties are briefly described in Section 5. In Section 6, upon adopting an arbitrary normalization of linear eigenfunctions, we calculate observable amplitudes of various modes, which may be excited and detected. Prospects of mode identification are discussed in Section 7. Examples of diagnostic diagrams employing amplitude ratios and phase differences are shown there.

## 2 Photospheric Parameter Variations and Velocity Field in the Traditional Approximation

In our study we adopt the standard approximations, which are the linear nonadiabatic theory of stellar oscillation and static plane-parallel atmosphere models. These approximations are well justified in our application. Modes detected in slowly pulsating B-type stars have indeed very low amplitudes, the oscillation periods are much longer than the thermal time scale in the atmosphere, and vertical variations of mode amplitude are small across the whole atmosphere. The constant kinematic acceleration is easily included. Like Townsend (2003b) we adopt the traditional approximation, which allows to separate latitudinal and radial dependencies of the pulsational amplitudes. We essentially follow his formalism, expect that we choose the azimuthal and temporal dependence as , which implies for prograde modes and for retrograde modes.

The displacement vector at the surface may be written as follows

 \boldmathξ(R,θ,φ)=εR(Θ,ϖ^Θsinθ,−iϖ~Θsinθ)Z

where is an arbitrary, but small, complex constant, and

 ϖ=GMω2R3.

The three functions , , describe the latitudinal dependence of solutions and are the Hough functions, which are obtained as solutions of the Laplace’s tidal equations

 (D+msμ)Θ=(s2μ2−1)^Θ,
 (D−msμ)^Θ=[λ(1−μ2)−m2]Θ,

where is called the spin parameter, and . The equations together with boundary conditions at and define the eigenvalue problem on . The third function is given by

 ~Θ=−mΘ+sμ^Θ.

Bildsten et al. (1996), Lee and Saio (1997), and Townsend (2003a) discussed in great detail the dependence and asymptotic properties of the Hough functions. Here, we recall only the essentials.

For specified and , there are branches with and (except for normalization). These branches correspond to g-modes distorted by rotation. For prograde sectorial modes (), slowly decreases with . For all other g-modes, the function is increasing quite rapidly. We will identify g-mode branches by the value at . Thus, like in the case of no rotation, we will use values as the angular quantum numbers and refer to as the mode degree. However, now specification of the angular dependence requires also the value of . The symmetry about the equatorial plane is determined by the parity of . If it is even, the functions and are symmetrical and is antisymmetrical. The opposite is true if is odd. The branches for which at correspond to r-modes. For each , there is one branch crossing zero at . If , the associated modes become propagatory in the radiative regions, they may be excited and visible in the light variations. However, following Lee (2006), we will still call them -modes. The functions and are antisymmetrical and is symmetrical about the equator for these r-modes.

Upon replacing with , the nonadiabatic mode properties may be calculated with a reasonable accuracy using the same code as for non-rotating stars. This is so because for the mode of our interest, horizontal flux losses, which are not correctly described, are small. The latitudinal dependence of perturbed thermodynamical parameters is then given by . Thus, the bolometric flux perturbation may be written as

 δFbolFbol=εfΘZ,

where is a complex quantity determined by solution of linear nonadiabatic equations and it depends on the stellar model and the mode parameters , and .

For evaluation of perturbed monochromatic fluxes, we also need the perturbed gravity, which, as follows from Eq. (1), is given by

 δgg=−ε(2+ϖ−1)ΘZ.

The effect of gravity perturbation plays a relatively small role in light variability caused by slow modes but it is easy to include. More important is perturbation of star shape leading to changes in the projected surface element and the limb-darkening. For both we need the normal to stellar surface which, as follows from Eq. (1), is given by

 δns=−ε∇H(ΘZ)=−ε(0,∂Θ∂θ,imΘsinθ)Z.

The change of the directed element of the surface is

 δdSdS=ε(2Θ,−∂Θ∂θ,−imΘsinθ)Z

where .

From Eq. (1), we also obtain the perturbed pulsation velocity field as seen from an inertial system ()

 δv=d\boldmathξdt=(∂∂t+Ω∂∂φ0)% \boldmathξ=εR[−iω\boldmathξ+Ωez×\boldmathξ].

Use of the Lagrangian pulsational velocity is adequate for representing velocity at the photospheric layer because is nearly constant across the outer layers for the slow modes considered by us.

## 3 Light Variation

### 3.1 Semi-Analytical Formula

The most straightforward extension of the expression for the light variation to the case of rotating stars is through the expansion of the Hough function into the truncated series of the associated Legendre functions. This was the way Townsend (2003b) derived his expression. We write the equivalent expression in the form, which is a straightforward generalization of our formula (Daszyńska-Daszkiewicz, Dziembowski and Pamyatnykh 2003) derived for the case of modes described by single spherical harmonic. Now the complex amplitude of the light variation in the passband may be expressed as

 Ax(i)=ε∞∑j=1γmℓj(s)Ymℓj(i,0)[Dxℓjf+Exℓj]

where

 ℓj={|m|+2(j−1)even−parity\leavevmode\nobreak modes|m|+2(j−1)+1odd−parity\leavevmode\nobreak modes

and

 Dxℓ=−1.086bxℓ14∂log(Fx|bxℓ|)∂logTeff,
 Exℓ=−1.086bxℓ[(2+ℓ)(1−ℓ)−(2+ϖ−1)∂log(Fx|bxℓ|)∂logg],
 bxℓ=1∫0hx(~μ)~μPℓ(~μ)d~μ.

where is the limb-darkening law, adopted in the nonlinear form (Claret 2000). With this form Townsend obtained his analytical expressions for . The quantities , which have to be calculated numerically, are the expansion coefficients of the function into the series of the Legendre functions. That is

 Θ(θ)=∞∑j=1γmℓj(s)Pmℓj(θ).

The expression given by Eq. (9) is quite revealing. At specified , the light amplitudes of low degree modes must decrease with because of increasing role of higher order terms which suffer more from disk-averaging. The higher order terms lead to the aspect-dependence of the amplitude ratios.

Unfortunately, we could not find a corresponding semi-analytical expression for the radial velocity and this is why we decided to rely on two-dimensional numerical integration over the visible hemisphere. We used Eq. (9) to check the accuracy.

### 3.2 Numerical Approach

The total flux in the passband toward the observer is given by

 Lx=∫SFxhxnobs⋅dS

where integration is carried over visible part of star surface, , and is the unit vector toward observer.

In the spherical coordinate system with the polar axis parallel to the rotation axis, we have

 nobs≡(or,oθ,oφ)

where

 or≡~μ=cosicosθ+sinisinθcos(φ−φ0),
 oθ=−cosisinθ+sinicosθcos(φ−φ0),
 oφ=−sinisin(φ−φ0).

The observer’s angular coordinates are . The first order perturbation of the total flux is given by

 δLx=∫S[(δFxhx+Fxδhx)dSer+FxhxδdS]⋅nobs.

Assuming an equilibrium atmosphere, we have from Eqs. (4) and (5)

 δFx=ε[αxT4f−αxg(2+ϖ−1)]ΘZ

where

 αxT=∂logFx∂logTeffandαxg=∂logFx∂logg

are the derivatives determined numerically from grids of stellar atmosphere models. For perturbed limb darkening, we take into account perturbations of the coefficients on and though they are only secondary contributors to light variation. More important contribution arises from (see Eq. 6). With all terms included, we obtain

 δhx=ε{[∂hx∂lnTf4−∂hx∂lng(2+ϖ−1)]−∂hx∂~μ(∂Θ∂θoθ+imΘsinθoφ)}ΘZ.

The derivatives of Claret’s are given in Appendix A1.

The expression for perturbed surface element is given in Eq. (7). The integration is carried over the unperturbed visible hemisphere. Within the linear approximation, the integration boundary is unchanged. Note also that the horizontal component of the displacement does not enter the expression. The domains of integration over and are shown in Fig. 1 and Fig. 2, respectively. From Figs. 1 and 2 it follows that the ranges are:

 0≤θ≤π2+i,
 −(π−α)≤φ−φ0≤(π−α)

where . For each , we carry the integration over the azimuthal angle, , from to , where (see Fig. 1 and Fig. 2). We use identities

 β∫−βG(Ψ)ZdΨ=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩2β∫0G(Ψ)cosmΨdΨif\leavevmode\nobreak \leavevmode\nobreak G(Ψ)\leavevmode\nobreak \leavevmode\nobreak is\leavevmode\nobreak even2iβ∫0G(Ψ)sinmΨddΨif\leavevmode\nobreak \leavevmode\nobreak G(Ψ)\leavevmode\nobreak \leavevmode\nobreak is\leavevmode\nobreak odd

The final expression for the total light variation can be written in the following form

 δLxLx=ε[(αxT4B1+B3)f+2B1+B2−(2+ϖ−1)(αxgB1+B4)]Z0

and . In Appendix A2 we give explicit expressions for the two-dimensional integrals, , which depend on two angular numbers, spin and the aspect. The integrals take into account changes in the limb-darkening resulting from the change of the normal (Eq. 6) as well as the change due to perturbation of the local temperature (Eq. 4) and gravity (Eq. 5). The two latter are given through derivatives of with respect to and . There are many terms contributing to . However, in our applications two are far dominant: the one resulting from temperature perturbation, which is proportional to , and the other resulting from the surface distortion, which is proportional to .

Adopting the standard sign convention, we write the radial velocity averaged over the stellar disk in the following form:

where for the total velocity field we use

 v=δv+ΩRsinθeφ.

The pulsational component, , as results from Eqs. (1) and (8), is given by

 δv=−iωεR⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Θ−s2ϖ~Θϖ[^Θsinθ−s2cosθsinθ~Θ]−iϖ[~Θsinθ−s2cosθsinθ^Θ]+is2Θsinθ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠Z

The contribution of rotation to the mean radial velocity arises from the same pulsational changes of photospheric parameters which cause luminosity change and may be calculated in the same way as outlined in Section 3. Clearly, there is a nonzero contribution only for non-axisymmetric modes.

We write our final expression for the perturbed radial velocity in the following form,

with

 Cpuls=(B5−s2B7)+ϖ(B6−s2B8)

and

 Crot=−ssini2[(14αxTB9+B11)f+2B9+B10−(2+ϖ−1)(αxgB9+B12)].

Explicit expressions for the coefficients in terms of the three Hough functions are given in Appendix A3.

If and , the two contributions are of the same order. Similarly to the case of the luminosity, the dominant contributions to are the ones resulting from temperature perturbation, which here is proportional to , and the other resulting from the surface distortion, which is proportional to .

## 5 Unstable Modes in a 6 M⊙ Main Sequence Star

To illustrate how visibility of various modes depends on equatorial velocity of rotation we choose a model of a 6 M Population I star in the mid of its main sequence evolution. Parameters of the model are given in Table 1. The model is spherically symmetric, which is consistent with our use of the traditional approximation, but includes averaged effects of centrifugal force corresponding to uniform rotation with equatorial velocity of 250 km/s. The mean effects of centrifugal force are still reasonably small and therefore we used the same model to study effects of the Coriolis force at lower equatorial velocities.

There are many low frequency modes that are unstable in our selected star model. These are mainly g-modes. However, at sufficiently high rotation rate there are also certain r-modes, which become propagatory in radiative envelope and may become unstable. In all cases, the instability is caused by the -mechanism acting in the metal opacity bump layer. For g-modes, the angular order, , is defined at the limit of zero rotation where the angular dependence of amplitude is described by individual spherical harmonics. The actual mode geometry is determined by the numbers and the spin parameter, . Since at each azimuthal order there is only one r-mode for which changes sign, we will identify r-modes by the value alone. Naturally, we focus attention on modes suffering least reduction of observable amplitude caused by disk-averaging. Therefore, we consider g-modes with and r-modes with and .

Typically, at each degree and azimuthal order, we find instability extending over many (up to 40) radial orders. For analysis of visibility, we selected the mode characterized by the highest normalized growth rate, , which varies between and 1. The important parameters of the selected modes at adopted equatorial velocities are listed in Table 1. For the r-modes we write ż"instead of as the first entry. The effect of rotation on mode surface geometry at specified depends on which determines . The depth-dependence of eigenfunctions is determined primarily by the product . In particular, the radial orders in this model are given by . For the selected modes they are between 26 and 34. Nonadiabatic parameters, and , depend on but also on the pulsation frequency in the star reference system, .

The Hough -functions for selected modes are shown in Figs. 3 and 4. We may see the well-known effect of equatorial amplitude confinement, which increases with . The effect is present in all modes including those with though is a decreasing function in these cases. As for the remaining Hough functions, which are important, the confinement is also present. and have the same symmetry about the equator and the symmetry of is opposite.

## 6 Visibility of Slow Modes

For the modes listed in Table 1, we calculated amplitudes of light variation in the U and V Geneva passbands, and , with Eqs. (15), as well as that of the radial velocity, , with Eqs. (19), adopting an arbitrary normalization, . Coefficients and occurring in Eq. (15), were interpolated from the line-blanketed models of stellar atmospheres (Castelli and Kurucz 2004).

Figs. 5–7 show the and in function of the aspect angle. Because of the arbitrary normalization, they cannot be regarded as reliable predictors of expected amplitudes. We may expect that chances of excitation are related to but it does not determine . Still plots like these provide important hints for interpretation of the rich oscillation spectra such as that of HD 163868 (Walker et al. 2005b). For the g-modes we see a simple patterns in the dependence of on the rotation rate. The aspect-dependence is qualitatively the same as in the case of no rotation. Rotation gives rise to a departure from the dipolar angular dependence of and but the contribution of the surface distortion to light variation remains small even at  km/s. Thus, the -dependence is a simple reflection of the -dependence shown in Fig. 3. The equatorial confinement leads to a reduction of amplitude upon averaging contributions from the whole hemisphere.

The pattern of the dependence of radial velocity amplitude on the rotation rate, shown in the right panels of Fig. 5, is more complicated. In all three cases, the dominant contribution arises from pulsational velocity (the term in Eq. 19). A secondary but significant contribution, the term, adds in the case of prograde modes and subtracts in the case of retrograde modes. However, the main reason for the large difference between the modes and for the nonmonotonic aspect dependence is connected with properties of . At high rotation rates the contribution from the advective term in is large and causes that all its three components play a role. The large values of in the case of , mode for the near equatorial observers arise primarily from this term. We see that at least for some aspects, fast rotation increases the chances for mode detection by means of radial velocity measurements. The -dependencies for the g-modes depicted in the left panels of Fig. 6, show that, despite of the increasing equatorial confinement, the aspect of the best visibility moves toward the pole with increasing rotation rate. This somewhat unexpected feature, which has been already noted by Townsend (2003b), is explained in part by the effect of the surface distortion. There are aspects at which we see amplitude increase with but on average the effect is the same as for , that is fast rotation decreases chances for photometric detection of slow modes. Also on average, amplitudes of the modes are lower than those of . In the right panels of Fig. 6, we may see that radial velocity amplitudes of prograde modes increase with the rotation rate and that the effect is opposite for the retrograde modes. The role of the term is much more significant than at .

We may see in Fig. 3 that for the r-mode and for the g-mode look very similar. Likewise, in Fig. 4 we see the similarity of ’s for the r-mode and for the g-mode. Yet, as we may see in Fig. 7, the photometric amplitudes of the r-modes are much smaller and have different aspect dependence. The amplitude reduction is in part caused by cancellations in the integral over . Moreover, the effect of distortion, which is significant, cancels a part of the effect of temperature perturbation. The r-modes are antisymmetric with respect to the equator and are best seen from the intermediate aspect angles. The radial velocity amplitudes, shown in the right panels of Fig. 7, are much less reduced. Thus, spectroscopy gives a better chance for detecting r-modes.

## 7 Prospects for Mode Discrimination

Rotation has a very profound effect on slow mode visibility and hence on the procedure of mode identification. Unlike in non-rotating stars, the amplitude ratios depend on the azimuthal order, , and on the aspect, . Modes with various differ not only in the surface geometry but also in their nonadiabatic properties. There are constraints on following from localization in the frequency spectrum. Modes differing in the sign of are well separated. As we have seen in Figs. 5–7, the aspect is an important factor in mode selection which has to be taken into account considering possible and values. Nonetheless, a unique discrimination of modes is not likely possible without employing amplitude and phase data.

Let us begin with photometric data alone. The observables, which do not depend on , are amplitude ratios and phase differences from multiband photometry. In this case, the potential for mode discrimination rests mainly on the difference in the relative temperature and distortion contribution to the total light variation in different passbands. The latter contribution for the g-modes is small, hence a discrimination between such modes might be difficult. However, as the plots in Fig. 8 show, these modes may be rather easily distinguished from the and r-modes even if information about and is imprecise. A discrimination between the two modes could be difficult if the measured phase difference is between 0.2 and 0.3. The range is consistent with , if is between 150 km/s and 175 km/s, and , if is 250 km/s.

Any discrimination between these four modes based on frequencies alone would not be possible because all of them are retrograde and occupy the low frequency end of the oscillation spectrum. It is important to remember that the plots refer only to most unstable modes of each type. Thus, in real life, there is additional uncertainties in frequencies and in the -values that affect the amplitude ratios and phase differences. With mean radial velocity data, discrimination between the two modes should be unambiguous, as the plots in the lower panels of Fig. 9 show. In the upper panels we see that also distinguishing the two modes should be easy. Disentangling the and cases, however, may require data on line profile changes.

The main benefit from radial velocity data is the possibility of simultaneous determination of with no need for specifying the value of the complex parameter . Instead, as described in the case of non-rotating stars by Daszyńska-Daszkiewicz et al. (2005), the combined spectroscopy and photometry data on amplitudes and phases may be used to determine , which becomes an independent seismic observable, and the mode degree, . In order to see how the method may be extended to the present case, let us rewrite Eqs. (15) and (19) as expressions for the complex amplitudes of light in the -band and of radial velocity, respectively. From Eq. (15) we obtain

 Ax(i)=Dx(i)f~ε+Ex(i)~ε

where

 Dx(i)=−1.086(αxT4B1+B3),
 Ex(i)=−1.086[2B1+B2−(2+ϖ−1)(αxgB1+B4)]

and . Similarly, from Eq. (19) the first moment of spectral line is equal to

 Mx1(i)=iωR[Hx(i)f~ε+Gx(i)~ε]

where

 Hx(i)=−ssini2(14αxTB9+B11)

and

 Gx(i)=B5−s2B7+ϖ(B6−s2B8)−ssini2[2B9+B10−(2+ϖ−1)(αxgB9+B12)].

These two equations are counterparts of Eqs. (1) and (2) of Daszyńska-Daszkiewicz et al. (2005).

In the case of negligible rotation, the dependence of amplitudes on and has been absorbed in and the problem was reduced to finding and by the least square minimization of . The method requires stellar atmosphere parameters, which in fact may be improved on the process of pulsation amplitude fitting. An unique determination of is possible even with imprecise atmosphere parameters if the dependence is strong. In the present case we have two more stellar parameters to improve which are and . Moreover, we have to take into account the -dependence. Prospects for mode identification depend on the strength of the the dependence. The plots in Figs. 8 and 9 suggest that it is likely the case. However, it remains to be seen when the method is applied to real data.

## 8 Summary and Conclusions

Our goal was to examine chances for detection and identification of slow oscillation modes whose frequencies are of the order of angular velocity of rotation. In our calculations of expected mode amplitudes, we relied on a nonadiabatic generalization of the traditional approximation, similar to that introduced by Townsend (2005a). The chances for detecting a particular mode depend, in part, on its intrinsic amplitudes, which may be calculated only in the framework of a nonlinear modeling. Our linear models give us only a hint which is the growth rate. Such models are expected to be adequate for describing geometry of the mode which has important impact on mode visibility, its aspect and the angular degree dependence. With the formalism outlined in Sections 3 and 4, we calculated the observable amplitudes for selected unstable modes in models of a 6 M main sequence star as a function of the rotation rate and the aspect angle. The model may be regarded as representative for SPB variables.

Departure from the individual spherical-harmonic dependence which increases with the rotation rate leads, in most cases, to lower photometric amplitudes. In contrast, the effect on radial velocity amplitude is most often opposite. However, the light to radial velocity amplitude ratio changes significantly from mode to mode and depends on the aspect. We showed, in particular, that the mixed r/g-modes are most easily detectable in radial velocity. Considering possible identification of peaks in rich oscillation spectra such as of HD 163868 (Walker et al. 2005b) it is important to take into account the aspect dependence of considered mode amplitudes as it has been done by Dziembowski et al. (2007).

The observables yielding numerical constraints on mode geometry and the aspect angle are amplitude ratios and phase differences. We calculated examples of diagrams based on photometric data in two passbands and on mean radial velocity measurements. Not always photometric data are sufficient for mode discrimination. Radial velocity data not only help discrimination but also allow to proceed in a less model-dependent manner. The photospheric value of the complex radial eigenfunction corresponding to the radiative flux may then be determined from data rather than taken from linear nonadiabatic calculations. A comparison of calculated and deduced values yields a constraint on the model.

Acknowledgements. This work has been supported by the Polish MNiSW grant No 1 P03D 021 28.

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Appendix

A1. The Limb-Darkening Law For calculating the surface integrals in Eq. (15) and Eq. (19), we need to specify the limb-darkening law. Here, we use the nonlinear Claret (2000) formulae which we rewrite in the following form

 hx(~μ)=2\leavevmode\nobreak 1−4∑k=1axk(1−~μk/2)1−4∑k=1kk+4axk.

The derivative with respect to can be easily obtained from this formula. The derivatives with respect to the effective temperature and gravity are given by:

 ∂hx∂lnTeff=11−4∑k=1kk+4axk⋅4∑k=1[kk+4hx−2(1−~μk/2)]∂axk∂lnTeff,

and

 ∂hx∂lng=11−4∑k=1kk+4axk⋅4∑k=1[kk+4hx−2(1−~μk/2)]∂axk∂lng

respectively.

A2. Integrals in the Expression for the Light Variation

 B1=π2+i∫0ΘP1sinθdθ
 B2=π2+i∫0(∂Θ∂θsinθP2+mΘP3)dθ
 B3=π2+i∫0ΘP4sinθdθ
 B4=π2+i∫0ΘP5sinθdθ

where

 P1=1πβ∫0cosmΨhxordΨ
 P2=−1πβ∫0cosmΨ(ordhxdor+hx)oθdΨ
 P3=1πβ∫0sinmΨ(ordhxdor+hx)oφdΨ
 P4=1πβ∫0cosmΨ∂hx∂lnTeffordΨ
 P5=1πβ∫0cosmΨ∂hx∂lngordΨ.

The function is one of three Hough functions (see Section 2). The components of the unit vector directed to the observer () are given in Eq. (13).

A3. Integrals in the Expression for the Radial Velocity Variation

 B5=π2+i∫0ΘP6sinθdθ
 B6=π2+i∫0(^ΘP7+~ΘP8)dθ
 B7=π2+i∫0ΘP8sin2θdθ
 B8=π2+i∫0[~ΘP6sinθ+(~ΘP7+^ΘP8)cosθ]dθ
 B9=π2+i∫0ΘP9sin2θdθ
 B10=π2+i∫0(∂Θ∂θsinθP10+mΘP11)sinθdθ
 B11=π2+i∫0ΘP12sin2θdθ
 B12=π2+i∫0ΘP13sin2θdθ

where

 P6=1πβ∫0cosmΨhxo2rdΨ
 P7=1πβ∫0cosmΨhxoroθdΨ
 P8=1πβ∫0sinmΨhxoroφdΨ
 P9=1πβ∫0sinmΨsinΨhxordΨ
 P10=−1πβ∫0sinmΨsinΨ(ordhxdor+hx)oθdΨ
 P11=−1πβ∫0cosmΨsinΨ(ordhxdor+hx)oφdΨ
 P12=1πβ∫0sinmΨsinΨ∂hx∂lnTeffordΨ
 P13=1πβ∫0sinmΨsinΨ∂hx∂lngordΨ
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