3.1. The model
The results presented in this paper have been calculated with the following stellar model
L is the luminosity of the star and the hydrogen fraction at the center. This model has been obtained using usual assumptions for stellar structure and also neglecting all the processes of chemical diffusion.
We found for the non-magnetic acoustic cut-off frequency (assuming an isothermal atmosphere)
denotes the cyclic frequency. Hereafter, we will consider only high frequencies, i.e. from about 1200 to about 2300 , which represent a typical range of observable frequencies (i.e. radial numbers from about 12 to 24).
3.2. The magnetic shift of frequencies
The calculations show that the magnetic field leads to a shift of the real part of the frequency, denoted by , and also creates a positive complex component . The imaginary part comes from the coupling between p-modes and the magnetic field inside the boundary layer. A part of the p-mode's energy is then converted into Alfvénic waves inside this layer, which are dissipated beneath this layer, in the interior (see for instance Roberts & Soward 1983). Therefore, we can write the frequency in the following form
with n the radial order and a constant of the stellar model.
The large separation is
The results are presented in Figs. 2-5. We note that the effect of the magnetic field, i.e. the frequency shift, increases with the radial order n, for both the real and imaginary parts. This is explained by the fact that the effect of the perturbation, far from the outer turning point, is small. Only modes which have frequencies close to the critical cut-off frequency are significantly affected by the magnetic field.
However, the results we have obtained point out that this shift of frequencies strongly depends on the geometry of the mode, say the value of and m which corresponds to the geometric dependence of the Lorentz force in and m, see Eq. 6-8.
The m-dependence of the frequencies is a direct effect of the magnetic field since it breaks the spherical symmetry of the problem (if we neglect the rotation). It leads then to a raising of the degeneracy of the frequencies (see, for exam ple, Eq. 21). This effect is analogous to the Zeemann effect in Quantum Mechanics. However, it remains a partial degeneracy 1 since the frequencies only depend on the absolute value of m, because the unperturbed magnetic field is dipolar and, therefore, does not depend on the longitude (see Eq. 2).
First, we have investigated the influence of the degree in the case of axisymmetric oscillations , in Fig. 2. In that case, the typical range of is about 5-18 for high overtones, say and 2-10 for . We find here the same order of magnitude as Dziembowski & Goode (1996) for axisymmetric oscillations but for a younger star's model.
For both the real and imaginary parts, we see a systematic difference between radial () and non-radial modes (); for large frequencies the shift in the real part is lower than in the radial case, whereas the shift of the imaginary part is larger, for any value of (and for ).
We have investigated how the magnetic effects on oscillations depend on the value of m. The results are presented in Figs. 3-5 for different values of and in each case for its corresponding values of m (). Let us first examine the shift of the real part . As a general result, non-axisymmetric oscillations have greater real shifts of frequency than the case . More, one sees that for any value of , always increases with m. The modification of the frequency from the axisymmetric case is relatively important. For example, in Fig. 5, we note that which was about 10 for increases up to 18 in the case , for kG.
We see, in Figs. 3-5, that the imaginary part, also depends on the value of m. One notes that sectoral modes , have smaller imaginary parts than axisymmetric modes, whereas modes with have larger shifts. The amplitude of the modifications introduced for non-axisymmetric oscillations, as for the real part, is also important. As we can see, for kG in Fig. 5, decreases from 4 (for 2000 ) in the case to about 2.5 for and increases up to in the case of , say more than twice the value of the axisymmetric case.
These results show that the geometry and the stability of non-axisymmetric oscillations are greatly influenced by the magnetic field. As a matter of fact, the imaginary part introduced by magnetic processes is of the same order as the one due to non-adiabatic effects (see for instance Dziembowski & Goode 1996).
We see again that the behaviour of the frequencies with the magnetic field strongly depends on the geometrical nature of modes. For these graphs we extend the calculations up to 1.5 kG. We see that the damping of modes, i.e. , passes through a maximum. This maximum is obtained for a particular whose value depends on the parameters , m (e.g. for ). Such results have already been found by Dziembowski & Goode (1996) but for axisymmetric modes. Note also the behaviour of the mode such as , whose frequencies form a loop, due to the fact that for some magnetic fields (0.9-1.1 kG) both and decrease.
The observed oscillations are interpreted in terms of the oblique pulsator model (see Kurtz 1982, and later improvements by Dziembowski & Goode 1985,1986; Kurtz & Shibahashi 1986; Shibahashi & Takata 1993) which states that the pulsation axis is nearly aligned with the magnetic axis and oblique with the rotation axis. According to oblique pulsator model, the observed modes have, generally, a dipole geometry ( ). Our calculations do not explain this preference since their geometries do not minimize energy losses due to Alfvénic waves. As a matter of fact, we see in Figs. 3 and 6 that in most of the cases the imaginary part is generally smaller for than for the sectoral mode . The same is largely true for other degrees.
3.3. Small separations
As we have seen in the previous sections, the real part of the frequency is shifted up to . This value is of the same order as the small separations in the non-magnetic limit, say
These separations, in the non-magnetic case, are widely influenced by the deep stellar interior. Consequently, they give useful information about these regions and therefore an estimation of the stellar age.
We see that the changes due to deep stellar structure (in the non-magnetic case) are of the same order as the magnetic shifts of frequencies ( ). The non-magnetic separations are degenerate in m, as the corresponding frequencies. However, the magnetic field raises partially the degeneracy for and, therefore, the degeneracy for S. This implies that the small separations in the case of non-axisymmetric oscillations with magnetic field depend on the azimuthal degree m. Let us define this separation by
An example of this separation is given in Fig. 8, for , and , .
We note, in these graphs, that the small separations do not have the same behaviour with the magnetic field, depending on the mode's geometry. In the case of even degrees (, ) the small separations with are smaller than the axisymmetric case and decrease with . In the case of odd degrees (, ) we see that with . More, one can note that for a given value of m (i.e. 0 or 1) the small separations decrease with the value of .
These results show that it is very complicated to get information on stellar structure in terms of these small separations, because of the perturbations introduced by the magnetic field.
3.4. Angular geometry of modes
In this subsection we discuss the influence of the angular dependence of the Lorentz force on the geometry of p-modes oscillations. As we have seen in the previous subsection, the boundary condition for p-modes depends on which is a direct consequence of the -dependence of . We cannot represent a mode in the interior by a single spherical harmonic, as for non-magnetic oscillations, but we need to expand the solutions , as a series of associated Legendre functions, Eq. 14 and Eq. 15. We saw that this angular expansion involved only functions which have the same parity for and the same value of m.
The calculations show that, when increases, the number of terms with a significant amplitude in the series increases and the "weight" of these terms, say the value of , changes.
The contribution of these terms is found in the energy of the mode. The kinetic energy of a mode can be written as
with the displacement, and in spherical coordinates
where represent the Eulerian perturbation of the gravitational potential.
Since form an orthogonal basis, we have
is the mean density of the star. In order to represent the weight of each component of the total energy of the mode, we plot the following ratio
Several plots of are given in Figs. 9-13, as function of the magnetic field. We plot only the components k which have non negligible amplitudes. We see clearly that the energy of the main component of the mode decreases when increases, whereas the energy of new components increases.
Let us examine, for example, the case of in Fig. 9. We see that when is relatively small, kG, about 100% of the energy is contained in the component , . However, the components get non negligible amplitudes when kG. And for kG the energies of the three components are equal, each of them has about 30% of the total energy of the mode . Nonetheless, the component also increases with the magnetic field and becomes dominant for kG. We see that the presence of another component depends on the value of m; see for example the case of and in Fig. 13. The values of the degrees, involved in this mixing, strongly depend on m; the case has larger degrees for the Legendre functions than the case . One realizes that the identification of modes is very difficult in the presence of a strong magnetic field because the dominant component, and then its geometry, of a given mode depends on the magnetic field. In the present case, for kG the dominant component is and for it is . The magnetic field totally changes the geometrical nature of the mode. One can note that for magnetic fields above 1.1 kG the mode , or becomes undetectable for observers because the spatial averaging of the component vanishes.
To the contrary, we can expect to see modes which are invisible without a magnetic field, say with , but for which in the case of strong the dominant component is obtained for . This is the case of , as we can see on Fig. 14. However, we should note that one cannot arbitrarily increase the magnetic field because of the main assumption we made for the magnetic layer, i.e. the plane parallel approximation. To be valid the magnetic layer must be situated close to the top of the star. When increases, the base of the boundary layer has to be deeper (say for greater densities) in the star in order to the ratio (see Eq. 1) remains small to insure the decoupling between acoustic and Alfvénic modes. We give in Table 1 in Appendix A several positions of for different values of .
We also assumed small horizontal wave numbers in the magnetic layer compared with the radial ones. This assumption still remains valid if the degree of Legendre function is not too large. As the series Eq. 14 and Eq. 15 require higher orders when the magnetic field increases, we have a limitation on the value of . In our case, the plots of show that even in the case of magnetic fields above 1 kG the maximal amplitude is for small degrees of Legendre functions, i.e. , except for and and for which some polynomials of large degrees have strong components in the expansions Eq. 14,15. For the cases of and we stop the calculations for kG and kG, respectively, because they involve degrees larger than 20 for higher magnetic fields, and then are incompatible with the assumptions we made.
3.5. Contribution to pulsational damping
From the system of linearized equations Eq. A.1-A.4 in appendix A, one can derive an equation for energy conservation (by multiplying with the complex conjugate of and integrating over a given volume )
This formula is useful for two reasons. First, it provides a way to estimate the numerical precision of the frequencies (see in Appendix C). Second, it can be used to find the angular location of the damping zone. To do this, we take the imaginary part of Eq. 33 and neglect the contribution of the magnetic field since, at we have . Then, we get
Neglecting the effect of the gravitational potential which is small for high frequencies, one can write the imaginary part of the frequency as
Then, one sees that the imaginary part of frequency comes from the imaginary part of the acoustic flux through the fitting surface. corresponds to this flux normalized by the inertia of the mode. One can verify in Fig. 15 that , first, depends on , but its contribution to comes from a region located essentially between and the magnetic pole . The location of the maximum of this function depends on the value of : one sees in Fig. 15 (a) that the maximum is for for kG, i.e. for defined in Sect. 2.1, whereas for kG the maximum is for which corresponds to an extremum of the Legendre polynomial . We find again that the presence of the mixing of spherical harmonics described in Sect. 3.4. One can check in Fig. 9 that the Legendre polynomial of degree 5 starts to dominate the angular geometry of the mode from kG. Note that even in the case of (full line) one sees the contribution of with the second extremum of close to .
One sees that the integral of the function g decreases from kG which leads to a decay of , which corresponds to the Fig. 6. For the case of the non-axisymmetric mode , we find the same kind of results, except that for kG the maximum of the function g is for .
© European Southern Observatory (ESO) 2000
Online publication: March 28, 2000