Core magnetic field imprint in the non-radial oscillations of red giant stars

Magnetic fields in red giant stars remain a poorly understood topic, particularly in what concerns their intensity in regions far below the surface. In this work, we propose that gravity-dominated mixed modes of high absolute radial order and low angular degree can be used to probe the magnetic field in their radiative cores. Using two poloidal, axisymmetric configurations for the field in the core and the classical perturbative approach, we derive an analytical expression for the magnetic frequency splitting of these oscillation modes. Considering three distinct red giant models, with masses of 1.3\(M_\odot\), 1.6\(M_\odot\) and 2.0\(M_\odot\), we find that a field strength of $10^5$ G is necessary in the core of these stars to induce a frequency splitting of the order of a $\mu$Hz in dipole and quadrupole oscillation modes. Moreover, taking into account observational limits, we estimate that magnetic fields in the cores of red giants that do not present observable magnetic splittings cannot exceed $10^4$ G. Given the general absence of observable splittings in the oscillation spectra of these stars, and assuming that present mode suppression mechanisms are not biased towards certain azimuthal orders and retain all peaks in each multiplet, our results lead us to conclude that internal fields with the considered configurations and strengths above $10^4$ G are not prevalent in red giants.