Probing magnetic field Geometries

The recent measurement of magnetic field strength inside the radiative interior of red giant stars opens the way towards the full 3D characterization of the geometry of stable large-scale magnetic fields. However, current measurements, which are limited to dipolar (l = 1) mixed modes, do not properly constrain the topology of magnetic fields due to degeneracies on the observed magnetic field signature on such l = 1 mode frequencies. Efforts focused towards unambiguous detections of magnetic field configurations are now key to better understand angular momentum transport in stars. We investigate the detectability of complex magnetic field topologies (as the ones observed at the surface of stars with a radiative envelope with spectropolarimetry) inside the radiative interior of red giants. We focus on a field composed of a combination of a dipole and a quadrupole (quadrudipole), and on an offset field. We explore the potential of probing such magnetic field topologies from a combined measurement of magnetic signatures on l = 1 and quadrupolar (l = 2) mixed mode oscillation frequencies. We first derive the asymptotic theoretical formalism for computing the asymmetric signature in frequency pattern for l = 2 modes due to a quadrudipole magnetic field. To access asymmetry parameters for more complex magnetic field topologies, we numerically perform a grid search over the parameter space to map the degeneracy of the signatures of given topologies. We demonstrate the crucial role played by l = 2 mixed modes in accessing internal magnetic fields with a quadrupolar component. The degeneracy of the quadrudipole compared to pure dipolar fields is lifted when considering magnetic asymmetries in both l = 1 and l = 2 mode frequencies. In addition to the analytical derivation for the quadrudipole, we present the prospect for complex magnetic field inversions using magnetic sensitivity kernels from standard perturbation analysis for forward modeling. Using this method, we explore the detectability of offset magnetic fields from l = 1 and l = 2 frequencies and demonstrate that offset fields may be mistaken for weak and centered magnetic fields, resulting in underestimating magnetic field strength in stellar cores. We emphasize the need to characterize l = 2 mixed-mode frequencies, (along with the currently characterized l = 1 mixed modes), to unveil the higher-order components of the geometry of buried magnetic fields, and better constrain angular momentum transport inside stars.