2. The magnetic boundary layer approach
In the presence of a magnetic field the full set of equations describing stellar oscillations is very complicated to solve because of the Lorentz force. Hence, we adopted the concept of a magnetic boundary layer used in previous papers dealing with this problem, (Roberts & Soward 1983; Campbell & Papaloizou 1986; Dziembowski & Goode 1996). Although the magnetic field is strong everywhere inside the star, the Lorentz force has an influence on the dynamics of the oscillations only in the outer parts, where the density becomes so small that the magnetic pressure is of the same order as (or larger than) the gas pressure.
is the adiabatic exponent, p the unperturbed gas pressure. and are respectively the Alfvénic and sound speeds, and the photospheric magnetic field (0.5-1.5 kG).
We can divide the star into two regions. The magnetic pressure is neglected compared with the gas pressure in the whole star (where ), except in the very thin layer near the photosphere where . This thin layer, where is non-negligible compared with the unit, is called the boundary magnetic layer. Inside this layer, there are no pure p-modes but rather the so-called magneto-sonic modes whose properties are a mixture of acoustic and Alfvénic modes. The bottom of this layer, , is chosen where the quantity is sufficiently small to neglect the magnetic pressure and to provide a decoupling between p-modes and the magnetic field (see Table 1 in Appendix A). Then, to find the eigen-modes of the star, we must solve two systems of equations. Inside the star (), we solve the system for adiabatic p-modes oscillations without a magnetic field, described for example in Unno et al. (1989). In the magnetic layer, we integrate the complete system of ideal MHD equations still in the case of adiabatic oscillations (see appendix A). The advantage of this method is that, since this layer is very thin and situated at the top of the star, we can adopt a plane-parallel approximation, i.e. we neglect, locally, the curvature of the star. Assuming that the horizontal wave number remains small compared with the radial wave number, i.e. for small values of . We neglect the derivatives with respect to the colatitude for the perturbed quantities. Matching the solutions of the magnetic layer with the solutions of the deep interior, i.e. corresponding to non-magnetic oscillations, one obtains the eigen frequency spectrum of the star.
2.1. The magnetic layer
We divided the magnetic layer into a mesh in the colatitude , and we solved the MHD system of equations for each colatitude point with this plane-parallel approach.
The magnetic field is assumed to be dipolar which is a good approximation for Ap stars (Borra et al. 1982)
x being the dimensionless radial position ().
The Eulerian perturbation of the Lorentz force in the case of MHD approximation is
In the magnetic layer we neglected all the derivatives by compared to the derivatives by x and we assumed large radial wave numbers, since we study high frequency oscillations. Then one gets the following form for the Lorentz force,
In the case of non-axisymmetric oscillations, is not equal to zero, therefore the full system to solve contains the equation of motion projected on . Then, this system for adiabatic motions which was of the fourth order in the axisymmetric case (see Dziembowski & Goode 1996) becomes of the sixth order (see in Appendix A).
We assume small perturbations which leads us to linearize these equations. More details about this system are given in Appendix A. The result of this integration in the magnetic layer, with adequate boundary conditions at the top of the star, leads to the ratio between and the Lagrangian perturbation of pressure and the radial displacement for p-modes at
This relation is then used as a boundary condition to solve numerically the classical system of equations, say without a magnetic field, for the inner part of the star, . The function only depends on the absolute value of m, because of the azimuthal symmetry of the unperturbed magnetic field. Therefore, the perturbations of the frequencies, due to the magnetic field, will depend also on the absolute value of m.
In Fig. 1, we can see two plots of the function which represent its real and imaginary parts, for a given radial order . We see that this function depends on m, for both the real and the imaginary parts. This implies that the corresponding eigen frequencies will also depend on m. The angular dependence of in is influenced by the angular dependence of the "acoustic cut-off frequency", i.e. the maximal frequency for trapped p-modes. It follows from the expression Eq. A.22 in Appendix A that this frequency is
the acoustic waves, at the top of the boundary layer, become partially propagating in the atmosphere. The equality in Eq. 12 corresponds to a null wave number and for . Then, for a given frequency we have two regions in colatitude: from the pole to the acoustic waves are reflected whereas for to the equator they are partially propagating. This change of the mode's nature has a signature in the behaviour of the function . In the case of Fig. 1 .
Note that is equal to the non-magnetic frequency at the pole, and is smaller than everywhere else. For the limiting case of the equator, it is equal to zero: therefore, there are no trapped waves at the equator.
2.2. The interior
Following Roberts & Soward (1983), there is a full decoupling between Alfvénic and acoustic modes for . This decoupling is recovered from Eq. A.10 and A.12 in Appendix A because of the small value of . The Alfvénic modes are assumed to be dissipated well before the center of the star . A WKBJ treatment has been adopted to find them.
The p-modes are described by the system of equations without a magnetic field, as presented in Unno et al. (1989), whose solutions are the radial displacement and the Lagrangian perturbation of the pressure. is the degree of the spherical harmonic . However, as the boundary condition depends on , the solutions are no longer represented by a single , but by linear combinations of associated Legendre functions,
represents the coefficient of angular mixing, it corresponds to the "weight" of each in the sum. nnl is the maximal degree of the Legendre functions in the series.
The effect of magnetic field on p-modes, in the interior, is described through the coefficients .
We use here the following normalization for ,
with the Kronecker symbol.
The relation in Eq. 16 (with taking the same values as k) is a homogeneous system whose solutions are the coefficients . Its discriminant has to be equal to zero for eigen frequencies (for non-trivial solutions ). This last condition is used to find the frequency spectrum of the star.
Because is even in (see the system in appendix A; it only depends on ), the function does not vanish if k and have the same parity. In order to have a non-zero , one must choose for the series Eq. 14 and Eq. 15 either only odd-Legendre functions or only even-Legendre functions, according to the parity of the degree , i.e. the degree of the mode without a magnetic field. As we put , we have then, 18 terms in the series Eq. 14-15.
© European Southern Observatory (ESO) 2000
Online publication: March 28, 2000