Issue 
A&A
Volume 676, August 2023



Article Number  L9  
Number of page(s)  10  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/202346832  
Published online  10 August 2023 
Letter to the Editor
Asymmetries of frequency splittings of dipolar mixed modes: A window on the topology of deep magnetic fields
^{1}
Université ParisSaclay, Université Paris Cité, CEA, CNRS, AIM, 91191 GifsurYvette, France
email: stephane.mathis@cea.fr
^{2}
Institute of Science and Technology Austria (IST Austria), Am Campus 1, Klosterneuburg, Austria
Received:
7
May
2023
Accepted:
11
June
2023
Context. Space asteroseismology is revolutionizing our knowledge of the internal structure and dynamics of stars. A breakthrough is ongoing with the recent discoveries of signatures of strong magnetic fields in the core of red giant stars. The key signature for such a detection is the asymmetry these fields induce in the frequency splittings of observed dipolar mixed gravitoacoustic modes.
Aims. We investigate the ability of the observed asymmetries of the frequency splittings of dipolar mixed modes to constrain the geometrical properties of deep magnetic fields.
Methods. We used the powerful analytical RacahWigner algebra used in quantum mechanics to characterize the geometrical couplings of dipolar mixed oscillation modes with various realistically plausible topologies of fossil magnetic fields. We also computed the induced perturbation of their frequencies.
Results. First, in the case of an oblique magnetic dipole, we provide the exact analytical expression of the asymmetry as a function of the angle between the rotation and magnetic axes. Its value provides a direct measure of this angle. Second, considering a combination of axisymmetric dipolar and quadrupolar fields, we show how the asymmetry is blind to the unraveling of the relative strength and sign of each component. Finally, in the case of a given multipole, we show that a negative asymmetry is a signature of nonaxisymmetric topologies.
Conclusions. Asymmetries of dipolar mixed modes provide a key bit of information on the geometrical topology of deep fossil magnetic fields, but this is insufficient on its own. Asteroseismic constraints should therefore be combined with spectropolarimetric observations and numerical simulations, which aim to predict the more probable stable largescale geometries.
Key words: asteroseismology / stars: magnetic field / stars: oscillations / methods: analytical
© The Authors 2023
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.
1. Introduction
Over the past two decades, space asteroseismology has made Sir Eddington’s dream of “seeing” the interiors of stars a reality. Beginning with helioseismology for our Sun and moving onto asteroseismology for the stars of our galaxy, the frequencies of Solar and stellar oscillation modes have progressively allowed us to constrain the internal solar and stellar structure and dynamics with a precision that has never before been achieved. First, stellar seismology has allowed us to constrain the hydrostatic and thermodynamic equilibrium structures of stars at different evolutionary stages (e.g., Hekker & ChristensenDalsgaard 2017; ChristensenDalsgaard 2021; Aerts 2021). Next, it has demonstrated that stellar interiors are the seats of an extremely intense transport of angular momentum for all stellar masses at all ages (García et al. 2007; Aerts et al. 2019, and references therein). Finally, it has opened a unique window to plausibly constrain deep internal magnetic fields (e.g., Goode & Thompson 1992; Hasan et al. 2005; Bugnet et al. 2021; Mathis et al. 2021) and magnetic activity (e.g., Santos et al. 2021).
In this context, the recent discovery of deep magnetic fields in the core of three red giant stars by Li et al. (2022), followed by ten more magnetized red giants reported in Deheuvels et al. (2023), constitutes an extraordinary breakthrough. Indeed, this is the first time we have gained access to constraints on possible fossil fields in the stably stratified core of low and intermediatemass evolved stars. These fields result from the relaxation towards an equilibrium configuration of past dynamo magnetic fields (e.g., Braithwaite & Spruit 2004; Duez & Mathis 2010) – for instance, those generated in the convective core of mainsequence stars with masses above ∼1.1 M_{⊙} (see Fig. 1 in Bugnet et al. 2021, for details of this formation mechanism). This discovery is of major importance since these fields are one of the possible mechanisms to explain the observed strong transport of angular momentum in stellar interiors (e.g., Spruit 1999, 2002; Mathis & Zahn 2005; Fuller et al. 2019; Eggenberger et al. 2022; Petitdemange et al. 2023). Moreover, before this discovery, such fossil fields have only been observed at the surface of ∼10% early O,B,Atype stars thanks to highprecision spectropolarimetry (Wade et al. 2016), but never in the deep interiors of stars. This discovery strengthens the role of red giant stars as one of the cornerstones of modern stellar astrophysics. Indeed, these stars have also allowed during the last decade to improve our knowledge of stellar evolutionary stages (Bedding et al. 2011) and of the core to the surface rotation contrast (e.g., Mosser et al. 2012; Deheuvels et al. 2014; Gehan et al. 2018), and thus of the internal transport of angular momentum.
The key to these major advances is the coupling between acoustic and gravity modes. Mixed gravitoacoustic oscillation modes behave as acoustic modes in the deep convective envelope of red giant stars and as gravity modes in their radiative core (e.g., Shibahashi 1979). Therefore, they allow us to probe these stars near the surface and in the core. In the case of deep magnetic fields, the key parameter is the splitting of the frequencies of mixed modes due to the combined action of rotation and magnetism. In the case of magnetism, the Lorentz force causes an asymmetric frequency splitting as this has been shown by Bugnet et al. (2021) and Bugnet (2022) in the case of axisymmetric dipolar fields and by Li et al. (2022) in the case of a general 3D magnetic topology. If the asymmetry in the power spectrum density provides constraints on the amplitude of the internal magnetic field, the relation between the asymmetry and the 3D topology of the magnetic field, as presented in Li et al. (2022), might be degenerate.
The objective of our study is therefore to explore in more detail the relation between the asymmetry parameter a as defined in Li et al. (2022) and the magnetic field topology. In Sect. 2.2, we first analytically derive an expression of a for a general magnetic topology by using the Racahwigner algebra used in quantum mechanics. This allows us to explore its behavior for a set of chosen magnetic topologies based on spectropolarimetric observations of fossil fields and on the stateoftheart of our theoretical knowledge on the formation and the stability of these fields, presented in Sect. 3. The results we obtain allow us to quantify the strength and weaknesses of a as a window on the topology of deep magnetic fields and to conclude on the crucial importance of continuing studies to predict fossil fields’ magnetic topologies and novel inversion techniques to overrule its potential degeneracy.
2. Mixed mode perturbations by the combined action of magnetism and rotation
2.1. General perturbative formalism
We start by considering an evolved lowmass or intermediatemass star where mixed gravitoacoustic modes propagate and we assume that its central stably stratified radiative core is the seat of a deep steady^{1} magnetic field. Its radial component is expanded on spherical harmonics defined in Eq. (B.1):
where r, θ, and φ are the radius, colatitude, and azimuth, respectively, and L_{B} is the formal limit of the expansion. Following Goupil et al. (2013), we introduce ⟨Ω⟩_{g} (resp. ⟨Ω⟩_{p}) as the averaged rotation of the cavity where mixed modes behave as gravity (resp. acoustic) modes.
In the case of nonrotating nonmagnetic stars, an oscillation eigenmode is purely spheroidal and can be expanded on vectorial spherical harmonics (Rieutord 1987):
where and with ∇_{H} = e_{θ}∂_{θ} + 1/sinθ e_{φ}∂_{φ}, {e_{j}}_{j ≡ r, θ, φ} being the classical spherical unitvector basis. The unperturbed mixed mode frequency ω_{0; n, l} is the eigenvalue of the linearized oscillations equations, which can be written formally as
where ℒ_{0} is a linear operator operating on ξ_{0; n, l, m}, which accounts for the compressibility, buoyancy, and selfgravity of the mode; we refer to Unno et al. (1989) for its detailed expression. Because of the degeneracy with m of Eq. (3), any linear combination of eigenmodes,
is an eigenfunction of ℒ_{0} associated with ω_{0; n, l}.
A deep magnetic field as those detected in Li et al. (2022) and Deheuvels et al. (2023) and moderate rotation trigger firstorder perturbations of the eigenfrequencies (eigenfunctions) such that ω_{n, l, m} = ω_{0; n, l} + ω_{1; n, l, m} with ω_{1; n, l, m} ≪ ω_{0; n, l} (respectively ξ_{n, l, m} = ξ_{0; n, l} + ξ_{1; n, l, m} with ξ_{1; n, l, m} ≪ ξ_{0; n, l}). In the frame rotating with ⟨Ω⟩_{g}, they can be computed by solving the linear system following Li et al. (2022) formalism:
The diagonal R matrix is associated to the Coriolis acceleration and the residual coreenvelope differential rotation in the frame rotating with the gdominated modes propagation cavity. The matrix, M, is associated to the Lorentz force and its elements at a fixed l are given by:
where
with the background magnetic field, B_{0}, the hydrostatic density profile of the star, ρ, the Eulerian perturbation of the magnetic field induced by the oscillation, B′=∇ × (ξ × B_{0}), and the magnetic permeability of vacuum, μ_{0}. We neglect here the indirect terms that are induced by the deformation of the hydrostatic structure of the star by the Lorentz force (Gough & Thompson 1990). This is justified in Appendix A for the case of mixed modes propagating in the core of red giant stars. We have introduced the inner product ⟨f, g⟩=∫f^{*} g ρ r^{2}dr sinθ dθ dφ. Finally, I is the mode inertia defined as:
where R is the radius of the star.
In the region where Li et al. (2022) and Deheuvels et al. (2023) have detected and probed deep magnetic fields, mixed modes behave as asymptotic gmodes. This allows us to simplify the previous equations. First, in this regime, the acoustic behavior of the mode is weak and can be filtered out. That allows us to neglect the third term in Eq. (7) because of the corresponding anelastic behavior of the modes, where ∇ ⋅ (ρ ξ) ≈ 0. In this regime, modes are rapidly oscillating along the radial direction (i.e., k_{r; n, l} ≫ k_{h; l}, where k_{r; n, l} and k_{h; l} are the local vertical and horizontal wave numbers, as per Eq. (10), respectively) and mostly horizontal (i.e., ξ_{r; n, l} ≪ ξ_{h; n, l}) because of their weak compressibility. The inertia can thus be simplified as follows:
where r_{i} and r_{o} are the inner and outer turning point, respectively (Shibahashi 1979). We define ζ as the ratio of the mode inertia within the cavity where it propagates as a gravity mode with I. Following Hekker & ChristensenDalsgaard (2017) and Mathis et al. (2021), the horizontal component of the displacement is expressed in this regime by its JWKB (Jeffreys, Wentzel, Krammer, Brillouin; Fröman & Fröman 2005) expression:
with the phase where k_{r; n, l} = N/ω_{0; n, l} × k_{h; l} with . In the case of magnetic topologies that vary over scales of L_{B} larger than the radial wavelength of the mode, we demonstrate in Appendix A that ℒ_{L} is simplified as follows:
Following the rapid phase approximation for radial rapidlyoscillating integrals (Mathis et al. 2021), we obtain the general expression for M_{m, m′}:
where dΩ = sinθdθdφ, with Ω being the solid angle, and
The function K peaks in a narrow range at the hydrogen burning shell because of the cubic dependency of N/r.
Finally, we use the RacahWigner algebra detailed in Appendix B, which is intensively used in quantum mechanics, to analytically reduce the horizontal integral . Using Eqs. (B.4), (B.25), (B.23), and (B.8), we finally obtain:
with
where the 𝒦 and 𝒥 coupling coefficients, which involve the 3j and 6j coefficients, have been defined in Eqs. (B.23) and (B.8), respectively; δ is the usual Kronecker symbol. This allows us to provide a precise physical diagnosis on the geometrical couplings between the (squared radial component of the) magnetic field and dipolar mixed modes in the next section.
Solving Eq. (5) leads to find (2l+1) frequencies in the rotating frame ω_{m} = ω_{0; n, l} + ω_{1; m} (m ∈ {−l,…,l}). Next, we take the Dopplershift into account to get the frequencies ω_{m, m′} = ω_{m} − m′⟨Ω⟩_{g} in the inertial frame and finally define the frequency shifts:
2.2. Case of dipolar mixed modes
We focus on dipolar (l = 1) mixed modes, which are the ones that are primarily observed in asteroseismology. In the more general case, its multiplet has nine components. Following Li et al. (2022), we define the average of the frequency shifts of this multiplet as
In this case, the diagonal R matrix is given by
which corresponds to the classical, symmetric (when computed at the first order) rotational splitting. Using the property that the sum of the eigenvalues of a matrix is its trace and that Tr R = 0, we find (using the results of Sect. 2.1 and Appendix C) that:
with
where we compute analytically the horizontal average over the sphere of the squared radial component of the magnetic field using Eq. (B.8):
with
where (⋅⋅⋅) is a 3j coefficient.
In the case where the rotation terms dominate the magnetic ones, the ninecomponent multiplet reduces to a triplet as in the case of the observed stars KIC 8684542, KIC 11515377 and KIC 7518143 studied in Li et al. (2022). As underscored in the research conducted by Li et al. (2022), and further supported by Bugnet et al. (2021) in the axisymmetric case, the magnetic field’s presence induces an asymmetry of the triplet:
where we introduce the asymmetry parameter a
which we compute analytically using the RacahWigner algebra; P_{2}(cosθ) is the Legendre polynomial of order l = 2. It allows us in Sect. 3 to identify the cases where the measurement of δ_{asym} allows us to unravel and characterise the properties of the topology of the field and those where it is blind or where it only provides partial information. Since −1/2 ≤ P_{2}(cosθ) ≤ 1, we also have −1/2 < a < 1 where the minimum and maximum a values correspond to the cases where is totally concentrated around the pole or along the equator, respectively.
3. Exploration of possible magnetic topologies
3.1. Classical oblique dipolar fossil field
The classical model for a fossil field in a stellar radiation zone is the oblique rotator model (e.g., Moss 1990; Mestel et al. 1981) in which a dipolar field is inclined with respect to the rotation axis with an angle β between the direction of the magnetic dipolar moment and the rotation axis. Following Prat et al. (2020), its radial component is given by:
where B_{0} is the field amplitude and b_{r}(r) its radial distribution. Applying the general result given in Eq. (24), we obtain:
where is the rotation matrix used in quantum mechanics (Varshalovich et al. 1988) to project a spherical harmonics on spherical harmonics when the vertical axis of the reference frame, in which the spherical coordinates (θ,φ) are defined, is rotated by an angle β. This leads to a new reference frame with its spherical coordinates (θ′,φ′). If we consider in the reference frame of the magnetic dipole, it projects onto axisymmetric spherical harmonics and , where θ_{B} is the colatitude in this frame, because ∝ cos^{2}θ_{B}. When rotating back to the reference frame with the vertical axis aligned along the rotation axis, we are thus carrying out a rotation with the angle β. This explains why since we are projecting onto the spherical harmonics . When we plot a as a function of β we recover the results reported in Li et al. (2022), where a = 2/5 when β = 0° and a = −1/5 when β = 90°. For a given measured asymmetry a, we can invert the previous equation that leads to
We thus find the critical inclination β = 54.736° for which the sign of a changes.
We can apply this result to the three stars observed by Li et al. (2022), namely: KIC 8684542, KIC 11515377, and KIC 7518143. For KIC 8684542, we get 0° < β < 16.78° since a = 0.47 ± 0.12. For KIC 11515377, we obtain 75.04° < β < 90° since a = −0.24 + 0.08/−0.234. Finally, for KIC 7518143, we get that β < 31° since a > 0.24. These results are represented on Fig. 1. These applications demonstrate how helpful and powerful is the RacahWigner algebra used here to compute analytically a as a function of the considered inclined dipolar magnetic topology and to derive its obliquity. However, one should remember that this is derived assuming we know the type of magnetic topology we are looking for.
Fig. 1. Inclination angle of the dipole field as function of the measured asymmetry. Possible inclination angles for KIC 8684542, KIC 11515377, and KIC 7518143 are represented. 
3.2. Combining dipolar and quadrupolar fields
To investigate the possible degeneracy of the asymmetry of mixed modes as a diagnosis of the magnetic topology, we consider an axisymmetric topology composed by a combination of a dipolar and a quadrupolar components (e.g., Moss 1974, 1985). We chose a topology defined as
where ℛ is the ratio between the dipolar and the quadrupolar components. We get
which we then plotted (as shown in Fig. 2). When comparing the obtained values with those in Fig. 1, we see from Fig. 2 that it is not objectively possible to disentangle between the combination proposed here with an oblique dipole with an angle β < ≈25.88° computed using Eq. (27) with the minimum asymptotic value obtained in Eq. (29), that is, a = 2/7.
Fig. 2. Asymmetry parameter as function of the ratio between the dipolar and the quadrupolar components in the case of a dipolar + quadrupolar field. 
3.3. Case of a general field
In the case of the oblique dipolar fossil field, we have been able to demonstrate explicitly in Sect. 3.1 how a is providing a very precise constrain on the inclination angle β of the field. In the case where we have combined axisymmetric dipolar and quadripolar fields, we have shown how a becomes partially blind and a degenerated diagnosis parameter when we do not know a priori the type of configurations we are looking for. It is now interesting to consider a modal magnetic field such as to unravel the potential information provided by a and its sign. In Fig. 3, we represent the variation of a as a function of m, since a(−m) = a(m), for a fixed l. We verify that −1/2 ≤ a ≤ 1. We see that for each l, there is a critical m above which the sign of a changes and becomes negative. This is an interesting result, as it shows that in the case of multipolar magnetic field where a {l,m} mode dominates the magnetic energy spectrum, the sign of a provides us an information on how close the field is to a sectoral (i.e., l = m) configuration and, thus, its degree of nonaxisymmetry. Combined with results obtained thanks to MHD numerical simulations (e.g., Braithwaite & Spruit 2004; Braithwaite & Nordlund 2006; Brun 2007; Braithwaite 2008; EmeriauViard & Brun 2017; Becerra et al. 2022) on fossil field formation and instabilities, the sign of the asymmetry parameter could therefore provide us with valuable information on the initial magnetic conditions and the stability of the observed fields.
Fig. 3. Asymmetry parameter for a given magnetic field with a latitudinal degree 1 ≤ l ≤ 10 and an azimuthal order −l ≤ m ≤ l. 
4. Conclusions
In this Letter, we put to use the powerful RacahWigner algebra used in Quantum mechanics to explore the behavior of the asymmetry of the frequency splittings of dipolar mixed gravitoacoustic modes when they propagate in the magnetised radiative core of an evolved low or intermediatemass star. First, following Li et al. (2022), we studied the classic case of a dipolar fossil field, which can have any inclination angle β with respect to the rotation axis. The RacahWigner algebra allows us to provide an exact analytical expression of the asymmetry as a function of this angle (Eq. (26)). This allows us, thanks to Eq. (27), to constrain its value using the observed asymmetry for the observed stars KIC 8684542, KIC 11515377, and KIC 7518143 (in Fig. 1). Next, we explore the behavior of a more complex, but potentially less stable fossil field configuration^{2}, which has an axisymmetric quadrupolar component in addition to an axisymmetric dipole. We show that the signature of such a field leads to similar values of the asymmetry that the one obtained with a sole dipolar field with an angle β < ≈25.88°. Finally, we explore the more general case of a multipolar field for which a {l,m} mode dominates the magnetic energy spectrum. We show that the asymmetry parameter becomes negative when m → l. In other words, a negative asymmetry is the sign of a strong nonaxisymmetric field. This study therefore demonstrates the difficulty to constrain the possible geometrical configurations of deep fossil fields and the importance of simultaneously developing asteroseismic diagnoses and MHD simulations of their formation and instabilities to unravel their main properties.
Duez & Mathis (2010) have demonstrated that the most stable lowestenergy mixed (poloidal + toroidal) equilibrium configuration is the dipolar one, while higher degree equilibrium configurations have higher energy and are thus less stable.
Acknowledgments
The authors are grateful to the referee for her/his detailed and constructive report, which has allowed us to improve our article. S. M. acknowledges support from the CNES GOLFSOHO and PLATO grants at CEA/DAp and PNPS (CNRS/INSU). We thank R. A. Garcia for fruitful discussions and suggestions.
References
 Aerts, C. 2021, Rev. Mod. Phys., 93, 015001 [Google Scholar]
 Aerts, C., Mathis, S., & Rogers, T. M. 2019, ARA&A, 57, 35 [Google Scholar]
 Becerra, L., Reisenegger, A., Valdivia, J. A., & Gusakov, M. E. 2022, MNRAS, 511, 732 [NASA ADS] [CrossRef] [Google Scholar]
 Bedding, T. R., Mosser, B., Huber, D., et al. 2011, Nature, 471, 608 [Google Scholar]
 Braithwaite, J. 2008, MNRAS, 386, 1947 [Google Scholar]
 Braithwaite, J., & Nordlund, Å. 2006, A&A, 450, 1077 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Braithwaite, J., & Spruit, H. C. 2004, Nature, 431, 819 [Google Scholar]
 Brun, A. S. 2007, Astron. Nachr., 328, 1137 [NASA ADS] [CrossRef] [Google Scholar]
 Bugnet, L. 2022, A&A, 667, A68 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Bugnet, L., Prat, V., Mathis, S., et al. 2021, A&A, 650, A53 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 ChristensenDalsgaard, J. 2021, Liv. Rev. Sol. Phys., 18, 2 [NASA ADS] [CrossRef] [Google Scholar]
 Deheuvels, S., Doğan, G., Goupil, M. J., et al. 2014, A&A, 564, A27 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Deheuvels, S., Li, G., Ballot, J., & Lignières, F. 2023, A&A, 670, L16 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Duez, V., & Mathis, S. 2010, A&A, 517, A58 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Duez, V., Mathis, S., & TurckChièze, S. 2010, MNRAS, 402, 271 [Google Scholar]
 Eggenberger, P., Moyano, F. D., & den Hartogh, J. W. 2022, A&A, 664, L16 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 EmeriauViard, C., & Brun, A. S. 2017, ApJ, 846, 8 [Google Scholar]
 Fröman, N., & Fröman, P. O. 2005, Physical Problems Solved by the PhaseIntegral Method (Cambridge: Cambridge University Press) [Google Scholar]
 Fuller, J., & Mathis, S. 2023, MNRAS, 520, 5573 [CrossRef] [Google Scholar]
 Fuller, J., Piro, A. L., & Jermyn, A. S. 2019, MNRAS, 485, 3661 [NASA ADS] [Google Scholar]
 García, R. A., TurckChièze, S., JiménezReyes, S. J., et al. 2007, Science, 316, 1591 [Google Scholar]
 Gehan, C., Mosser, B., Michel, E., Samadi, R., & Kallinger, T. 2018, A&A, 616, A24 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Goode, P. R., & Thompson, M. J. 1992, ApJ, 395, 307 [Google Scholar]
 Gough, D. O., & Thompson, M. J. 1990, MNRAS, 242, 25 [Google Scholar]
 Goupil, M. J., Mosser, B., Marques, J. P., et al. 2013, A&A, 549, A75 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Hasan, S. S., Zahn, J. P., & ChristensenDalsgaard, J. 2005, A&A, 444, L29 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Hekker, S., & ChristensenDalsgaard, J. 2017, A&ARv, 25, 1 [Google Scholar]
 Li, G., Deheuvels, S., Ballot, J., & Lignières, F. 2022, Nature, 610, 43 [NASA ADS] [CrossRef] [Google Scholar]
 Mathis, S., & Zahn, J. P. 2005, A&A, 440, 653 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mathis, S., Bugnet, L., Prat, V., et al. 2021, A&A, 647, A122 [EDP Sciences] [Google Scholar]
 Mestel, L., Nittmann, J., Wood, W. P., & Wright, G. A. E. 1981, MNRAS, 195, 979 [NASA ADS] [CrossRef] [Google Scholar]
 Moss, D. L. 1974, MNRAS, 168, 61 [NASA ADS] [CrossRef] [Google Scholar]
 Moss, D. 1985, MNRAS, 213, 575 [NASA ADS] [Google Scholar]
 Moss, D. 1990, MNRAS, 244, 272 [NASA ADS] [Google Scholar]
 Mosser, B., Goupil, M. J., Belkacem, K., et al. 2012, A&A, 548, A10 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3 [Google Scholar]
 Petitdemange, L., Marcotte, F., & Gissinger, C. 2023, Science, 379, 300 [NASA ADS] [CrossRef] [Google Scholar]
 Prat, V., Mathis, S., Neiner, C., et al. 2020, A&A, 636, A100 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Reisenegger, A. 2009, A&A, 499, 557 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Rieutord, M. 1987, Geophys. Astrophys. Fluid Dyn., 39, 163 [NASA ADS] [CrossRef] [Google Scholar]
 Santos, A. R. G., Breton, S. N., Mathur, S., & García, R. A. 2021, ApJS, 255, 17 [NASA ADS] [CrossRef] [Google Scholar]
 Shibahashi, H. 1979, PASJ, 31, 87 [NASA ADS] [Google Scholar]
 Spruit, H. C. 1999, A&A, 349, 189 [NASA ADS] [Google Scholar]
 Spruit, H. C. 2002, A&A, 381, 923 [CrossRef] [EDP Sciences] [Google Scholar]
 Strugarek, A., Brun, A. S., Mathis, S., & Sarazin, Y. 2013, ApJ, 764, 189 [NASA ADS] [CrossRef] [Google Scholar]
 Unno, W., Osaki, Y., Ando, H., Saio, H., & Shibahashi, H. 1989, Nonradial Oscillations of Stars (Tokyo: University of Tokyo Press) [Google Scholar]
 Varshalovich, D. A., Moskalev, A. N., & Khersonskii, V. K. 1988, Quantum Theory of Angular Momentum (World Scientific) [CrossRef] [Google Scholar]
 Wade, G. A., Neiner, C., Alecian, E., et al. 2016, MNRAS, 456, 2 [Google Scholar]
Appendix A: Asymptotic expression of the Lorentz force
To compute the impact of deep magnetic fields on mixed modes, we can focus on the regime of asymptotic g modes, which are rapidly oscillating along the radial direction (we refer the reader to Sec. 2.1).
The Lorentz force can be expressed in the following two ways:
where B is the magnetic field. We recognise in the second expansion, the magnetic tension force and the gradient of the magnetic pressure.
We introduce the linear expansion of B:
where B_{0} and B′ are the background magnetic field and the waveinduced magnetic field perturbation, respectively. The latter is computed using the linearised induction equation in its adiabatic limit:
where u = ∂_{t}ξ is the velocity field of the oscillation mode with ξ its Lagrangian displacement; we omit in this appendix any _{n, l} subscripts to lighten notations. Since mixed modes we are considering are behaving as asymptotic gmodes in radiative layers, they are nearly incompressible (i.e.,∇ · u ≈ 0), predominantly horizontal (i.e., ξ_{r} < < ξ_{h}), and rapidly oscillating in the radial direction (i.e., k_{r} > > k_{h}). If we focus on background magnetic topologies (B_{0}), which have characteristic length of variation larger than the mode wavelength, Eq. (A.3) is simplified as:
Thanks to the JWKB approximation, which applies for asymptotic gmodes, this leads to:
where u_{h} = ∂_{t}ξ_{h} = iω_{0}ξ_{h}.
The linearized Lorentz force,
is also simplified into:
where we have assumed that the fluctuations of the magnetic pressure are negligible when compared to those of the gaseous pressure. Using Eq. (A.5) and the JWKB approximation, we finally obtain:
where we have introduced the radial Alfvén velocity and the corresponding pulsation ω_{A} = V_{r; A}k_{r}. The ℒ_{L} operator corresponds to the socalled direct modification of modes’ eigenfrequencies and eigenfunctions by the linearized Lorentz force (Gough & Thompson 1990). It scales as
when compared to the mode acceleration in the nonmagnetic case.
However, indirect effects can also affect the modes. In the case of a non forcefree field, as this is expected for fossil fields (Reisenegger 2009; Duez & Mathis 2010), the hydrostatic structure of the star is distorted by the Lorentz force (Duez et al. 2010; Fuller & Mathis 2023). This modifies the cavity propagation of the modes and thus also contributes to the perturbation of their eigenfrequencies and eigenfunctions. We refer to the seminal article by Gough & Thompson (1990) for the complete mathematical modeling of this effect. Here, we focus on its order of magnitude. It scales as:
where we introduce the Alfvén velocity, , the Universal gravity constant, G, the mass contained in a sphere of radius r, M_{r}, and the radius of the radiative core, r_{c}, which we chose as the characteristic length of variation of the fossil field.
We evaluate the ratio between the direct and the indirect effects induced by the magnetic field in the case of dipolar mixed modes as:
where is the dynamical frequency and we assume that V_{r; A} and V_{A} have the same order of magnitude.
Computing models of typical red giants using the MESA stellar structure and evolution code (Paxton et al. 2011), we can evaluate this ratio. For a 1.5M_{⊙} star with a metallicity of Z = 0.02, we have ω_{0} ≈ 2πν_{max} with ν_{max} ≈ 150μHz, N_{max} ≈ 10^{4}μHz, r_{c} ≈ 0.75R_{⊙}, and M_{r}(r_{c}) ≈ 0.3M_{⊙}, where R_{⊙} and M_{⊙} are the solar radius and mass, respectively, leading to R_{direct}/R_{indirect} ≈ 2803 > > 1. This justifies why the indirect effect has been neglected in studies devoted to red giant stars.
In addition, when considering the full mathematical expression of the perturbation of the wave operator induced by indirect effects (we refer the reader to Eq. 3.24 in Gough & Thompson 1990), we can identify that its dominant terms (i.e., those with the highest derivatives of the eigenfunction because they are rapidly oscillating along the radial direction in the asymptotic regime) scale with the compressibility of the mode, which is weak for highorder gmodes. The second term induced by indirect effects scales as , where ρ_{B} is the perturbation of the hydrostatic background density induced by the Lorentz force (see Eq. 2.27 in Gough & Thompson 1990). As this can be seen, for instance in Duez et al. (2010), the ratio ρ_{B}/ρ is very small in the deep central layers of stars.
Therefore, we can focus in our work on the direct effects and neglect the indirect terms.
Appendix B: Spherical harmonics and RacahWigner algebra
B.1. Scalar quantities
B.1.1. Definition and properties
The spherical harmonics of degree l and order m are defined by:
where is the associated Legendre polynomial and the normalization coefficient is
They obey the orthogonality relation:
where dΩ = sinθ dθ dφ and the complex conjugate spherical harmonics is given by:
Using these properties, every function f(θ, φ) can be expanded as:
where
B.1.2. Products of spherical harmonics
Using the normalization and the orthogonality of spherical harmonics (Eqs. B.3 & B.5) and their complex conjugate (Eq. B.4), we can write:
where we define the integral following Varshalovich et al. (1988):
with the 3jWigner coefficients, which are related to the classical ClebschGordan coefficients by:
B.2. Vector fields
B.2.1. Spheroidal and toroidal basis
Following Rieutord (1987), we expand any vector field X(r,θ,φ) in vectorial spherical harmonics as
where we introduce:
with the horizontal gradient . These vector functions obey the following orthogonality relations:
We note that and represent the poloidal basis and is the toroidal basis of X.
B.2.2. Spin vector harmonics
Definition and properties:
Unfortunately, if the (R,S,T) basis is very well adapted to compute linear spectral projections, for instance of linear differential operators (Rieutord 1987), it fails for computing the spectral projections of scalar products and of vectorial products. Since we have here to compute the projection of on spherical harmonics in Eq. (12), we thus follow Varshalovich et al. (1988), Strugarek et al. (2013) and we introduce the spin1 vector harmonics (SVH). They are defined by:
with q = m − μ and the directional unit vectors:
They form an orthonormal basis over the space of all vector functions that satisfy the vector Laplace equation in spherical coordinates. Their orthonormality condition follows from the properties of the scalar spherical harmonics:
As in the (R,S,T) basis, any vector field X(r,θ,φ) can be expanded on the SVH basis as:
with the radialdependent coefficients
Scalar product:
Following Varshalovich et al. (1988), we can compute the projection of a scalar product of two SVHs on spherical harmonics as follows:
with
The {...} denote the 6j symbol of the RacahWigner algebra. It can be computed using 3j symbols as
where the sum is over all possible values of α, β, γ, δ, ϵ and ϕ for which the 3j symbols are defined.
B.3. Relations between the two basis
Using the definitions of , , and the results derived by Varshalovich et al. (1988), we obtain:
and
Therefore, the relations between the coefficients of the projection of a vector X on the (R,S,T) and on the basis are given by:
and
This allows us to obtain the expansion on spherical harmonics of the scalar product of two general vectors initially projected on the (R,S,T) basis using the SVH basis where this product can be analytically computed using the RacahWigner algebra.
Appendix C: Matrix elements in the dipolar case
In the case of dipolar highorder g modes, the matrix elements, M_{m, m′}, defined in Eq. (6) can be simplified and expressed analytically as ω_{B} and a using the RacahWigner algebra. We obtain
where dΩ = sinθdθdφ and ω_{0; n, l} and ξ_{h; n, l} have been simplified onto ω_{0} and ξ_{h}, respectively. By expanding 1 + cos^{2}θ on spherical harmonics, using the definition of B_{r} given in Eq. (1) and the coupling integrals (𝒥) provided in Eq. (B.8), and assuming the JWKB approximation, we derive its final expression:
where K(r) and ℐ have been defined in Eq. (13).
Applying the same method to the other matrix elements, we get:
and
while .
If we define b = ζω_{B}/ω_{R}, where ω_{B} and ω_{R} are given in Eqs. (20) and (18), respectively, which evaluates the relative strength of magnetic and rotation terms, and c = M_{0, 1}/ω_{B} and d = M_{−1, 1}/ω_{B}, which are related to the nonaxisymmetry of , the matrix ζM + R is simplified as:
All Figures
Fig. 1. Inclination angle of the dipole field as function of the measured asymmetry. Possible inclination angles for KIC 8684542, KIC 11515377, and KIC 7518143 are represented. 

In the text 
Fig. 2. Asymmetry parameter as function of the ratio between the dipolar and the quadrupolar components in the case of a dipolar + quadrupolar field. 

In the text 
Fig. 3. Asymmetry parameter for a given magnetic field with a latitudinal degree 1 ≤ l ≤ 10 and an azimuthal order −l ≤ m ≤ l. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.