Detectability of axisymmetric magnetic fields from the core to the surface of oscillating post-main sequence stars

Magnetic fields in the stellar interiors are key candidates to explain observed core rotation rates inside solar-like stars along their evolution. Recently, asteroseismic estimates of radial magnetic field amplitudes near the hydrogen-burning shell (H-shell) inside about 24 red-giants (RGs) have been obtained by measuring frequency splittings from their power spectra. Using general Lorentz-stress (magnetic) kernels, we investigated the potential for detectability of near-surface magnetism in a 1.3 $M_{\odot}$ star of super-solar metallicity as it evolves from a mid sub-giant to a late sub-giant into an RG. Based on these sensitivity kernels, we decompose an RG into three zones - deep core, H-shell, and near-surface. The sub-giants instead required decomposition into an inner core, an outer core, and a near-surface layer. Additionally, we find that for a low-frequency g-dominated dipolar mode in the presence of a typical stable magnetic field, ~25% of the frequency shift comes from the H-shell and the remaining from deeper layers. The ratio of the subsurface tangential field to the radial field in H-burning shell decides if subsurface fields may be potentially detectable. For p-dominated dipole modes close to $\nu_\rm{max}$, this ratio is around two orders of magnitude smaller in subgiant phases than the corresponding RG. Further, with the availability of magnetic kernels, we propose lower limits of field strengths in crucial layers in our stellar model during its evolutionary phases. The theoretical prescription outlined here provides the first formal way to devise inverse problems for stellar magnetism and can be seamlessly employed for slow rotators.