[{"month":"03","title":"Near-degeneracy effects in quadrupolar mixed modes: From an asymptotic description to data fitting","ddc":["520"],"PlanS_conform":"1","article_number":"A321","article_type":"original","author":[{"last_name":"Liagre","first_name":"Bastien Raymond Bernard","full_name":"Liagre, Bastien Raymond Bernard","id":"662f1873-cab4-11f0-a719-8087d302868d"},{"full_name":"Desai, Aayush A","id":"502cfd30-32c1-11ee-a9a4-d8dad5c6739e","last_name":"Desai","first_name":"Aayush A"},{"last_name":"Einramhof","first_name":"Lukas","full_name":"Einramhof, Lukas","id":"f1497a1a-72ef-11ef-b75a-fd877bbf6e8c"},{"id":"d9edb345-f866-11ec-9b37-d119b5234501","full_name":"Bugnet, Lisa Annabelle","orcid":"0000-0003-0142-4000","first_name":"Lisa Annabelle","last_name":"Bugnet"}],"doi":"10.1051/0004-6361/202558023","year":"2026","DOAJ_listed":"1","day":"01","scopus_import":"1","arxiv":1,"quality_controlled":"1","status":"public","corr_author":"1","oa":1,"file":[{"date_updated":"2026-04-07T09:00:50Z","creator":"dernst","access_level":"open_access","relation":"main_file","file_id":"21664","file_size":12287607,"checksum":"560cac19dc70184626b85e71a26ee22e","success":1,"content_type":"application/pdf","date_created":"2026-04-07T09:00:50Z","file_name":"2026_AstronomyAstrophysics_Liagre.pdf"}],"type":"journal_article","abstract":[{"text":"Dipolar (ℓ = 1) mixed modes have revealed a surprisingly weak differential rotation between the core and the envelope of evolved solar-like stars. Quadrupolar (ℓ = 2) mixed modes also contain information regarding internal dynamics but are very rarely characterised due to their low amplitude and the challenging identification of adjacent or overlapping rotationally split multiplets affected by near-degeneracy effects. We aim to extend the broadly used asymptotic seismic diagnostics beyond ℓ = 1 mixed modes by developing an analogue asymptotic description of ℓ = 2 mixed modes while explicitly accounting for near-degeneracy effects that distort their rotational multiplets. We have derived a new asymptotic formulation of near-degenerate mixed ℓ = 2 modes that describes off-diagonal terms representing the interaction between modes of adjacent radial orders. This formalism, expressed directly in the mixed-mode basis, provides analytical expressions for the near-degeneracy effects. We implemented the formalism within a global Bayesian mode-fitting framework for a direct fit of all ℓ = 0, 1, 2 modes in the power spectrum density. We were able to asymptotically model the asymmetric rotational splitting present in various radial orders of ℓ = 2 modes observed in young red giant stars without the need for any numerical stellar modelling. We applied our formalism to the Kepler target KIC 7341231, and it yielded core and envelope rotation rates consistent with previous numerical modelling while providing improved constraints from the global and model-independent approach. We also characterised the new target, KIC 8179973, measuring its rotation rate and mixed-mode parameters for the first time. As our framework relies on a direct global fit, it allows for much better precision on the asteroseismic parameters and rotation rate estimates than standard methods, yielding better constraints for rotation inversions. We have placed the first observational constraints on the asymptotic ℓ = 2 mixed-mode parameters (ΔΠ2, q2, and εg, 2), thus paving the way towards the use of asymptotic seismology beyond ℓ = 1 mixed modes.","lang":"eng"}],"date_created":"2026-04-05T22:01:32Z","date_updated":"2026-04-07T09:01:44Z","article_processing_charge":"No","_id":"21658","OA_type":"diamond","OA_place":"publisher","language":[{"iso":"eng"}],"has_accepted_license":"1","external_id":{"arxiv":["2511.05314 "]},"citation":{"ieee":"B. R. B. Liagre, A. A. Desai, L. Einramhof, and L. A. Bugnet, “Near-degeneracy effects in quadrupolar mixed modes: From an asymptotic description to data fitting,” <i>Astronomy and Astrophysics</i>, vol. 707. EDP Sciences, 2026.","chicago":"Liagre, Bastien Raymond Bernard, Aayush A Desai, Lukas Einramhof, and Lisa Annabelle Bugnet. “Near-Degeneracy Effects in Quadrupolar Mixed Modes: From an Asymptotic Description to Data Fitting.” <i>Astronomy and Astrophysics</i>. EDP Sciences, 2026. <a href=\"https://doi.org/10.1051/0004-6361/202558023\">https://doi.org/10.1051/0004-6361/202558023</a>.","apa":"Liagre, B. R. B., Desai, A. A., Einramhof, L., &#38; Bugnet, L. A. (2026). Near-degeneracy effects in quadrupolar mixed modes: From an asymptotic description to data fitting. <i>Astronomy and Astrophysics</i>. EDP Sciences. <a href=\"https://doi.org/10.1051/0004-6361/202558023\">https://doi.org/10.1051/0004-6361/202558023</a>","short":"B.R.B. Liagre, A.A. Desai, L. Einramhof, L.A. Bugnet, Astronomy and Astrophysics 707 (2026).","ista":"Liagre BRB, Desai AA, Einramhof L, Bugnet LA. 2026. Near-degeneracy effects in quadrupolar mixed modes: From an asymptotic description to data fitting. Astronomy and Astrophysics. 707, A321.","mla":"Liagre, Bastien Raymond Bernard, et al. “Near-Degeneracy Effects in Quadrupolar Mixed Modes: From an Asymptotic Description to Data Fitting.” <i>Astronomy and Astrophysics</i>, vol. 707, A321, EDP Sciences, 2026, doi:<a href=\"https://doi.org/10.1051/0004-6361/202558023\">10.1051/0004-6361/202558023</a>.","ama":"Liagre BRB, Desai AA, Einramhof L, Bugnet LA. Near-degeneracy effects in quadrupolar mixed modes: From an asymptotic description to data fitting. <i>Astronomy and Astrophysics</i>. 2026;707. doi:<a href=\"https://doi.org/10.1051/0004-6361/202558023\">10.1051/0004-6361/202558023</a>"},"publication_identifier":{"eissn":["1432-0746"],"issn":["0004-6361"]},"department":[{"_id":"LiBu"},{"_id":"IlCa"},{"_id":"GradSch"}],"oa_version":"Published Version","volume":707,"date_published":"2026-03-01T00:00:00Z","publication_status":"published","publisher":"EDP Sciences","acknowledgement":"We thank the referee for their careful and constructive report, which has substantially enhanced both the quality and clarity of the manuscript. L. Bugnet and L. Einramhof gratefully acknowledge support from the European Research Council (ERC) under the Horizon Europe programme (Calcifer; Starting Grant agreement N°101165631). While partially funded by the European Union, views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. The authors acknowledge the great support and feedback provided during the redaction of this article by Pr. Rafael García and Pr. Savita Mathur. We would also like to thank Dr. Emily Hatt for her insights on uncertainty estimates. The authors also thank the members of the Asteroseismology and Stellar Dynamics group of the Institute of Science and Technology Austria (ISTA) for very useful discussions: L. Barrault, S.B. Das, K. Smith. This paper includes data collected by the Kepler mission and obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the Kepler mission is provided by the NASA Science Mission Directorate. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. Software: AstroPy (Astropy Collaboration 2013, 2018), Matplotlib (Hunter 2007), NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), emcee (Foreman-Mackey et al. 2013), celerite (Foreman-Mackey et al. 2017), slepc4py (Dalcin et al. 2011; Hernandez et al. 2005), KADACS (García et al. 2011), sloscillations (Kuszlewicz et al. 2019, 2023).","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"publication":"Astronomy and Astrophysics","file_date_updated":"2026-04-07T09:00:50Z","intvolume":"       707","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"ddc":["520"],"title":"Unveiling complex magnetic field configurations in red giant stars","month":"10","scopus_import":"1","day":"01","year":"2024","doi":"10.1051/0004-6361/202450918","author":[{"full_name":"Das, Srijan B","id":"9ce7c423-dacf-11ed-8942-e09c6cb27149","orcid":"0000-0003-0896-7972","last_name":"Das","first_name":"Srijan B"},{"id":"f1497a1a-72ef-11ef-b75a-fd877bbf6e8c","full_name":"Einramhof, Lukas","first_name":"Lukas","last_name":"Einramhof"},{"id":"d9edb345-f866-11ec-9b37-d119b5234501","full_name":"Bugnet, Lisa Annabelle","orcid":"0000-0003-0142-4000","last_name":"Bugnet","first_name":"Lisa Annabelle"}],"project":[{"name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413","call_identifier":"H2020"}],"article_type":"original","article_number":"A217","corr_author":"1","status":"public","quality_controlled":"1","arxiv":1,"date_created":"2024-11-10T23:02:00Z","abstract":[{"text":"The recent measurement of magnetic field strength inside the radiative interior of red giant stars has opened the way toward 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 toward unambiguous detections of magnetic field configurations are now key to better understand angular momentum transport in stars. We investigated the detectability of complex magnetic field topologies (such as the ones observed at the surface of stars with a radiative envelope with spectropolarimetry) inside the radiative interior of red giants. We focused on a field composed of a combination of a dipole and a quadrupole (quadrudipole) and on an offset field. We explored 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 derived the asymptotic theoretical formalism for computing the asymmetric signature in the frequency pattern for ℓ = 2 modes due to a quadrudipole magnetic field. To access asymmetry parameters for more complex magnetic field topologies, we numerically performed 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 explored 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 the 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 to better constrain angular momentum transport inside stars.","lang":"eng"}],"type":"journal_article","oa":1,"file":[{"file_id":"18534","file_size":5306256,"checksum":"d43bbe6ed8ce4512e65e2d0d87070cf6","date_updated":"2024-11-11T09:01:11Z","creator":"dernst","access_level":"open_access","relation":"main_file","success":1,"content_type":"application/pdf","date_created":"2024-11-11T09:01:11Z","file_name":"2024_AstronomyAstrophysics_Das.pdf"}],"_id":"18528","article_processing_charge":"No","date_updated":"2025-09-08T14:36:39Z","department":[{"_id":"LiBu"}],"publication_identifier":{"issn":["0004-6361"],"eissn":["1432-0746"]},"citation":{"apa":"Das, S. B., Einramhof, L., &#38; Bugnet, L. A. (2024). Unveiling complex magnetic field configurations in red giant stars. <i>Astronomy and Astrophysics</i>. EDP Sciences. <a href=\"https://doi.org/10.1051/0004-6361/202450918\">https://doi.org/10.1051/0004-6361/202450918</a>","short":"S.B. Das, L. Einramhof, L.A. Bugnet, Astronomy and Astrophysics 690 (2024).","ista":"Das SB, Einramhof L, Bugnet LA. 2024. Unveiling complex magnetic field configurations in red giant stars. Astronomy and Astrophysics. 690, A217.","ama":"Das SB, Einramhof L, Bugnet LA. Unveiling complex magnetic field configurations in red giant stars. <i>Astronomy and Astrophysics</i>. 2024;690. doi:<a href=\"https://doi.org/10.1051/0004-6361/202450918\">10.1051/0004-6361/202450918</a>","mla":"Das, Srijan B., et al. “Unveiling Complex Magnetic Field Configurations in Red Giant Stars.” <i>Astronomy and Astrophysics</i>, vol. 690, A217, EDP Sciences, 2024, doi:<a href=\"https://doi.org/10.1051/0004-6361/202450918\">10.1051/0004-6361/202450918</a>.","ieee":"S. B. Das, L. Einramhof, and L. A. Bugnet, “Unveiling complex magnetic field configurations in red giant stars,” <i>Astronomy and Astrophysics</i>, vol. 690. EDP Sciences, 2024.","chicago":"Das, Srijan B, Lukas Einramhof, and Lisa Annabelle Bugnet. “Unveiling Complex Magnetic Field Configurations in Red Giant Stars.” <i>Astronomy and Astrophysics</i>. EDP Sciences, 2024. <a href=\"https://doi.org/10.1051/0004-6361/202450918\">https://doi.org/10.1051/0004-6361/202450918</a>."},"external_id":{"isi":["001336485200015"],"arxiv":["2405.20133"]},"has_accepted_license":"1","language":[{"iso":"eng"}],"OA_type":"hybrid","OA_place":"publisher","publisher":"EDP Sciences","publication_status":"published","date_published":"2024-10-01T00:00:00Z","volume":690,"oa_version":"Published Version","ec_funded":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","intvolume":"       690","file_date_updated":"2024-11-11T09:01:11Z","publication":"Astronomy and Astrophysics","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"isi":1,"acknowledgement":"The authors thank S. Mathis, L. Barrault, S. Torres, A. Cristea, and K. M. Smith for very useful discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curíe grant agreement No 101034413. The authors thank the anonymous referee for valuable comments and suggestions to improve the manuscript."}]