Hidden magnetic fields in stellar interiorsprobed by asteroseismology

Hidden magnetic fields in stellar interiors probed by asteroseismology Antoine Fort, Rhita-Maria Ouazzani, Marie-Jo Goupil an[email protected], [email protected], [email protected] -July 4, 2025 Introduction Understanding the distribution and evolution of angular momentum is essential for understanding stellar formation and evolution. Recent missions like CoRoT and Kepler have highlighted flaws in current angular momentum transport models.1, 2 Magnetic field is a promising candidate to explain the missing transport, and hopefully solve the issue. Asteroseismology via perturbative methods has recently enabled the first detection and measurements of magnetic fields in stellar interiors.3Here, we present an upgrade of the oscillation code ACOR4, 5 that will fully incorporate stellar rotation and toroidal magnetic fields in a two-dimensional framework. This code allows us to test the existing perturbative methods and tackle the area of strong and complex internal magnetic fields, thereby allowing realistic seismic diagnostics of magnetic fields. ACOR(Mag) ACOR(Mag) is an extension of the ACOR code (Adiabatic Code of Oscillation including Rotation), designed to solve 2D adiabatic non-radial pulsations equations for rapidly rotating stars in a direct manner. It accounts for both centrifugal deformation and the full Coriolis acceleration. We have developed a modular architecture that takes as input a system of equations described in the most compact and readable form. This system is automatically processed by ACOR through a code generation pipeline built with SymPy. This approach allows for faster analytical derivations, streamlined implementation, and significantly enhanced error control. Stellar models / magnetic field Stellar models : We address the case of gmodes in intermediate mass stars such as those found in γDor or red giants stars. Those are computed using the stellar evolution codes CESAM6or CLES.7 Magnetic field : In this first implementation, we consider only pure toroidal magnetic fields.8, 9 Two configurations have been used : •A gaussian configuration B B B0=B0f(r) sin θe e eφ with f a gaussian, which has been compared to the perturbative approach, and its effects over ∆Pn, resonance dip and magnetic mode-trapping. •The configuration used in the plots is defined as the toroidal part of the realistic mixed magnetic configuration derived by Duez &Mathis, 2011 (see figure). This configuration is tuned such as B B B0= 0 0 0at the surface. In future works, we aim to incorporate other magnetic field configurations, as well as to extend the code to mixed configurations. MHD oscillation equations ∂ρ ∂t +∇ ∇ ∇·(ρv v v) = 0 ∂ ∂t +v v v· ∇ ∇ ∇v v v+∇p ∇p ∇p ρ+∇Φ ∇Φ ∇Φ−1 4πρ (∇ × B B B ∇ × B B B ∇ × B B B)×B B B= 0 ∂ ∂t +v v v· ∇ ∇ ∇δρ ρ−1 Γ1 δp p= 0 ∆Φ −4πGρ = 0 ∂B B B ∂t − ∇ ∇ ∇×(v v v×B B B)=0 For : X=X0(r, θ)+X′(r, θ, ϕ, t) X′and δX, Eulerian and Lagrangian perturbation, and v0 v0 v0= Ω Ω Ω×r r r Numerical method •Spectral decomposition on M spherical harmonics (8M differentiated over 10M variables) •Radial discretization : 4th-order finite differences scheme11 •Solving eigenvalue problem :Newtonlike method, solving the system for the Eigenfunction with an initial guess σ0, producing a deviation δσ and iterating until convergence (A′ 11+δσA′ 12)dy1 dζ + (A′ 11 +δσA′ 12)dy2 dζ = (A11 +δσA12)y1+ (A21 +δσA22)y2 0=(B11 +δσB12)y1+ (B21 +δσB22)y2 Comparison with perturbative method Comparisons with first-order perturbative predictions for a pure toroidal magnetic field. The frequency shift δω due to the magnetic field is given by3: δω =1 2ω ⟨ξ(0)|LL(ξ(0))⟩ ⟨ξ(0)|ξ(0)⟩ where ξ(0) is the unperturbed displacement, and LLis the linear magnetic perturbation operator from the Lorentz force.→ACOR results match perturbative predictions at low field strength and show discrepancies for higher field strengths ∆Pnevolution with increasing B B B0 We investigate the influence of the magnetic field through its signatures on the ∆Pnof g-modes. →ACOR results match ∆Pnkm computed with the magnetic extension of the TAR for toroidal magnetic field Resonance with pure inertial modes We used a model computed by CLES7with rapid rotation, which shows a dip in the ∆Pn pattern due to a resonance between gravito-inertial and pure inertial modes12 trapped in the convective core. It allows us to probe the core of the stars. Conclusion and outlook We presented the implementation of a 2D oscillation code that fully accounts for the effects of both stellar rotation and internal magnetic fields. Recent developments have been showcased, in particular on toroidal magnetic fields. Our results highlight clear effects on the period spacing patterns (∆Pn), which are potentially detectable in observations. These effects are consistent with predictions from the magnetic TAR framework for toroidal fields.8Additionally, we investigated the impact of magnetic fields on the dips in ∆Pnassociated with resonances involving pure inertial modes. Future work will concentrate on a more detailed study of the influence of toroidal fields in γDor and red giants and aim to extend this framework to include poloidal magnetic fields. References [1] Mosser et al. 2012, A&A [2] Ouazzani et al. 2019, A&A [3] Li et al. 2022, Nature [4] Ouazzani et al. 2012, A&A [5] Ouazzani et al. 2015, A&A [6] Morel Lebreton 2008, Ap&SS [7] Scuflaire et al. 2008, Ap&SS [8] Dhouib et al. 2022, A&A [9] Petitdemange et al. 2023, Science [10] Duez & Mathis 2010, A&A [11] Scuflaire et al. 2008, Ap&SS [12] Ouazzani et al. 2020, A&A