It is thought that magnetic fields must be present in the interiors of stars to resolve certain discrepancies between theory and observation (e.g. angular momentum transport), but such fields are difficult to detect and characterise. Asteroseismology is a powerful technique for inferring the internal structures of stars by measuring their oscillation frequencies, and succeeds particularly with evolved stars, owing to their mixed modes, which are sensitive to the deep interior. The goal of this work is to present a phenomenological study of the combined effects of rotation and magnetism in evolved stars, where both are assumed weak enough that first-order perturbation theory applies, and we focus on the regime where Coriolis and Lorentz forces are comparable. Axisymmetric twisted-torus field configurations are used, which are confined to the core and allowed to be misaligned with respect to the rotation axis. Factors such as the field radius, topology and obliquity are examined. We observe that fields with finer-scale radial structure and/or smaller radial extent produce smaller contributions to the frequency shift. The interplay of rotation and magnetism is shown to be complex: we demonstrate that it is possible for nearly symmetric multiplets of apparently low multiplicity to arise even under a substantial field, which might falsely appear to rule out its presence. Our results suggest that proper modelling of rotation and magnetism, in a simultaneous fashion, may be required to draw robust conclusions about the existence/non-existence of a core magnetic field in any given object.
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