Axisymmetric p‐mode pulsations of stars with dipole magnetic fields

ABSTRACT The effect of a dipole magnetic field on adiabatic axisymmetric non‐radial p‐mode pulsations is studied numerically. The angular dependence of pulsation, which cannot be represented by a single spherical harmonic in the presence of a magnetic field, is expanded into a series of spherical harmonics with different degrees ℓ. The presence of a magnetic field not only shifts the pulsation frequency, the pulsations are also damped due to the generation of magnetic slow waves. In agreement with the results of Cunha & Gough, who used a different approach from ours, we find that the effect of a magnetic field on the intermediate‐to‐high‐order p‐modes is not monotonic but cyclic with respect to the pulsation frequency and the magnetic field strength. The damping rate of a high‐order p‐mode becomes very small at about 3 kG and 8 kG; the corresponding field strengths are higher for lower overtones. The diminished magnetic damping is favourable for the corresponding modes, if they are excited by the classical κ‐mechanism, to survive even in the presence of a strong magnetic field. This picture could explain the mode selection as observed in the rapidly oscillating Ap stars. For a low‐order p‐mode, the damping rate increases as the strength of the magnetic field increases. We find that in the presence of a magnetic field of a few kG, magnetic damping seems to exceed the driving owing to the κ‐mechanism of oscillations representative of δ Scuti variability. This may explain why δ Scuti‐type oscillations are unlikely to be seen in magnetic Ap stars. The amplitude of a mainly dipole (or quadrupole) mode is strongly confined to the magnetic axis in the outer layers. Furthermore, horizontal motion can be comparable to radial motion even for high‐order p‐modes. We discuss the influence of the magnetic distortion of the eigenfunction on the pulsation amplitude modulation with respect to the rotation phase.