---
DOAJ_listed: '1'
OA_place: publisher
OA_type: diamond
PlanS_conform: '1'
_id: '21658'
abstract:
- lang: eng
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.
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).'
article_number: A321
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Bastien Raymond Bernard
full_name: Liagre, Bastien Raymond Bernard
id: 662f1873-cab4-11f0-a719-8087d302868d
last_name: Liagre
- first_name: Aayush A
full_name: Desai, Aayush A
id: 502cfd30-32c1-11ee-a9a4-d8dad5c6739e
last_name: Desai
- first_name: Lukas
full_name: Einramhof, Lukas
id: f1497a1a-72ef-11ef-b75a-fd877bbf6e8c
last_name: Einramhof
- first_name: Lisa Annabelle
full_name: Bugnet, Lisa Annabelle
id: d9edb345-f866-11ec-9b37-d119b5234501
last_name: Bugnet
orcid: 0000-0003-0142-4000
citation:
ama: 'Liagre BRB, Desai AA, Einramhof L, Bugnet LA. Near-degeneracy effects in quadrupolar
mixed modes: From an asymptotic description to data fitting. Astronomy and
Astrophysics. 2026;707. doi:10.1051/0004-6361/202558023'
apa: 'Liagre, B. R. B., Desai, A. A., Einramhof, L., & Bugnet, L. A. (2026).
Near-degeneracy effects in quadrupolar mixed modes: From an asymptotic description
to data fitting. Astronomy and Astrophysics. EDP Sciences. https://doi.org/10.1051/0004-6361/202558023'
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.” Astronomy and Astrophysics.
EDP Sciences, 2026. https://doi.org/10.1051/0004-6361/202558023.'
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,”
Astronomy and Astrophysics, vol. 707. EDP Sciences, 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.” Astronomy and
Astrophysics, vol. 707, A321, EDP Sciences, 2026, doi:10.1051/0004-6361/202558023.'
short: B.R.B. Liagre, A.A. Desai, L. Einramhof, L.A. Bugnet, Astronomy and Astrophysics
707 (2026).
corr_author: '1'
date_created: 2026-04-05T22:01:32Z
date_published: 2026-03-01T00:00:00Z
date_updated: 2026-04-07T09:01:44Z
day: '01'
ddc:
- '520'
department:
- _id: LiBu
- _id: IlCa
- _id: GradSch
doi: 10.1051/0004-6361/202558023
external_id:
arxiv:
- '2511.05314 '
file:
- access_level: open_access
checksum: 560cac19dc70184626b85e71a26ee22e
content_type: application/pdf
creator: dernst
date_created: 2026-04-07T09:00:50Z
date_updated: 2026-04-07T09:00:50Z
file_id: '21664'
file_name: 2026_AstronomyAstrophysics_Liagre.pdf
file_size: 12287607
relation: main_file
success: 1
file_date_updated: 2026-04-07T09:00:50Z
has_accepted_license: '1'
intvolume: ' 707'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '03'
oa: 1
oa_version: Published Version
publication: Astronomy and Astrophysics
publication_identifier:
eissn:
- 1432-0746
issn:
- 0004-6361
publication_status: published
publisher: EDP Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Near-degeneracy effects in quadrupolar mixed modes: From an asymptotic description
to data fitting'
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 707
year: '2026'
...
---
OA_place: publisher
OA_type: hybrid
_id: '18528'
abstract:
- lang: eng
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.
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.
article_number: A217
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Srijan B
full_name: Das, Srijan B
id: 9ce7c423-dacf-11ed-8942-e09c6cb27149
last_name: Das
orcid: 0000-0003-0896-7972
- first_name: Lukas
full_name: Einramhof, Lukas
id: f1497a1a-72ef-11ef-b75a-fd877bbf6e8c
last_name: Einramhof
- first_name: Lisa Annabelle
full_name: Bugnet, Lisa Annabelle
id: d9edb345-f866-11ec-9b37-d119b5234501
last_name: Bugnet
orcid: 0000-0003-0142-4000
citation:
ama: Das SB, Einramhof L, Bugnet LA. Unveiling complex magnetic field configurations
in red giant stars. Astronomy and Astrophysics. 2024;690. doi:10.1051/0004-6361/202450918
apa: Das, S. B., Einramhof, L., & Bugnet, L. A. (2024). Unveiling complex magnetic
field configurations in red giant stars. Astronomy and Astrophysics. EDP
Sciences. https://doi.org/10.1051/0004-6361/202450918
chicago: Das, Srijan B, Lukas Einramhof, and Lisa Annabelle Bugnet. “Unveiling Complex
Magnetic Field Configurations in Red Giant Stars.” Astronomy and Astrophysics.
EDP Sciences, 2024. https://doi.org/10.1051/0004-6361/202450918.
ieee: S. B. Das, L. Einramhof, and L. A. Bugnet, “Unveiling complex magnetic field
configurations in red giant stars,” Astronomy and Astrophysics, vol. 690.
EDP Sciences, 2024.
ista: Das SB, Einramhof L, Bugnet LA. 2024. Unveiling complex magnetic field configurations
in red giant stars. Astronomy and Astrophysics. 690, A217.
mla: Das, Srijan B., et al. “Unveiling Complex Magnetic Field Configurations in
Red Giant Stars.” Astronomy and Astrophysics, vol. 690, A217, EDP Sciences,
2024, doi:10.1051/0004-6361/202450918.
short: S.B. Das, L. Einramhof, L.A. Bugnet, Astronomy and Astrophysics 690 (2024).
corr_author: '1'
date_created: 2024-11-10T23:02:00Z
date_published: 2024-10-01T00:00:00Z
date_updated: 2025-09-08T14:36:39Z
day: '01'
ddc:
- '520'
department:
- _id: LiBu
doi: 10.1051/0004-6361/202450918
ec_funded: 1
external_id:
arxiv:
- '2405.20133'
isi:
- '001336485200015'
file:
- access_level: open_access
checksum: d43bbe6ed8ce4512e65e2d0d87070cf6
content_type: application/pdf
creator: dernst
date_created: 2024-11-11T09:01:11Z
date_updated: 2024-11-11T09:01:11Z
file_id: '18534'
file_name: 2024_AstronomyAstrophysics_Das.pdf
file_size: 5306256
relation: main_file
success: 1
file_date_updated: 2024-11-11T09:01:11Z
has_accepted_license: '1'
intvolume: ' 690'
isi: 1
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: Astronomy and Astrophysics
publication_identifier:
eissn:
- 1432-0746
issn:
- 0004-6361
publication_status: published
publisher: EDP Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: Unveiling complex magnetic field configurations in red giant stars
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 690
year: '2024'
...