Axisymmetric magnetic modes of neutron stars having mixed poloidal and toroidal magnetic fields

We calculate axisymmetric magnetic modes of a neutron star possessing a mixed poloidal and toroidal magnetic field, where the toroidal field is assumed to be proportional to a dimensionless parameter ζ ₀. Here, we assume an isentropic structure for the neutron star and consider no effects of rotation. Ignoring the equilibrium deformation due to the magnetic field, we employ a polytrope of the index n = 1 as the background model for our modal analyses. For the mixed poloidal and toroidal magnetic field with ζ₀ ≠ 0 axisymmetric spheroidal and toroidal modes are coupled.We compute axisymmetric spheroidal and toroidal magnetic modes as a function of the parameter ζ ₀ froζ ₀ to ~1 for the surface field strengths B_S = 10¹⁴ G and 10¹⁵ G. We find that the frequency ω of the magnetic modes decreases with increasing ζ ₀. We also find that the frequency of the spheroidal magnetic modes is almost exactly proportional to BS for ζ ₀ ≲ 1 but that this proportionality holds only when ζ ₀ ≪ 1 for the toroidal magnetic modes. The wave patterns of the spheroidal magnetic modes and toroidal magnetic modes are not strongly affected by the coupling so long as ζ ₀ ≲ 1.We find no unstable modes having ω² ζ ₀.