Seismic signature of a magnetic field in the $\gamma$ Doradus star KIC 2309579

Technical Summary: Seismic signature of a magnetic field in the γ Doradus star KIC 2309579

Problem and Context
While asteroseismology has successfully revealed internal rotation, ages, and masses of stars, stellar models consistently overestimate core rotation rates, suggesting a missing physical mechanism for angular momentum transport. Magnetic fields are a primary candidate for this mechanism. Although internal magnetic fields have recently been detected in the radiative cores of red giants and the radiative envelope of a slowly rotating $\delta$ Scuti–$\gamma$ Doradus star, no seismic evidence of an internal magnetic field has been established in typical, rapidly rotating $\gamma$ Doradus stars. These stars, which possess convective cores and radiative envelopes, exhibit high radial order gravito-inertial (g-i) modes that are theoretically sensitive to magnetic fields. However, their rapid rotation (periods $\sim$1 day) complicates analysis, as rotation cannot be treated as a simple perturbation; instead, the Traditional Approximation of Rotation (TAR) is required.

Methodology
The authors employed a multi-step approach to identify and model magnetic signatures in $\gamma$ Doradus stars:

1. Candidate Identification via $\delta K_a$: The study utilized a seismic variable, $\delta K_a$, defined as a specific combination of $\ell=1$ and $\ell=2$ Kelvin mode frequencies. According to asymptotic theory under TAR, $\delta K_a$ should follow a universal curve dependent on the spin parameter ($s_{n,1}$) and vanish at high spin values in the absence of magnetic fields. Conversely, a radial magnetic field component induces an increase in $\delta K_a$ proportional to $s_{n,1}^3$. The authors analyzed a sample of 155 $\gamma$ Dor stars (from Li et al. 2020) to identify deviations from the theoretical asymptotic trend.
2. Spectral Analysis of KIC 2309579: The star KIC 2309579 was selected as the best candidate showing a deviation consistent with a magnetic profile. The authors reprocessed Kepler light curves using the FELIX code to extract oscillation frequencies (SNR > 4.7). They identified $\ell=1$ and $\ell=2$ Kelvin modes in the 13–17 $\mu$Hz and 28–32 $\mu$Hz ranges, respectively.
3. Bayesian Modeling: A magnetic asymptotic model was fitted to the observed frequencies using a Bayesian approach (ABIM code). The model incorporated the TAR and the WKB approximation, including frequency shifts induced by a magnetic field ($\nu_B$) proportional to $s^3$. The fit included free parameters for the buoyancy radius ($\Pi_0$), rotation frequency ($\nu_{rot}$), phase offsets ($\epsilon_{g\ell}$), and the magnetic parameter $\nu_B$.
4. Exclusion of Alternatives: The authors rigorously tested alternative explanations for the observed spectral features, including glitches, radial/latitudinal differential rotation, and mode coupling with pure inertial modes (dips).
5. Field Strength Inference: To convert the seismic parameter $\nu_B$ into a physical magnetic field strength ($B_r$), the authors computed stellar structure models using CESAM2k20, calibrated on the star's buoyancy radius, luminosity, effective temperature, and metallicity. They evaluated the magnetic weight function $K_r(r)$ to determine where the field is likely located.

Key Results

• Magnetic Detection: The analysis of KIC 2309579 revealed a clear departure in $\delta K_a$ from the median trend of non-magnetic stars and the asymptotic prediction. The deviation follows the $s_{n,1}^3$ scaling expected for a magnetic field.
• Model Fit: The magnetic model successfully reproduced the observed frequencies for both $\ell=1$ and $\ell=2$ modes, as visualized in stretched period échelle diagrams. In contrast, models attempting to explain the spectrum via "dips" (coupling with inertial modes) failed to reproduce the long-period modes for both degrees simultaneously.
• Field Strength and Location: The derived seismic parameter corresponds to an integrated magnetic quantity $I B_{eq}^2 = 2.23 \pm 0.16 \times 10^{-20} \text{ cm s}^{-2} \text{g}^{-1} \text{G}^2$.
  • If the field were located below the convective envelope, the required strength ($\sim$140–700 G) would exceed the critical field limit for that region, likely suppressing g-modes entirely, which contradicts observations.
  • If the field is located in the layers just above the convective core (the region that was previously convective), the required field strength is $B_r \approx 4 \pm 0.3$ kG. This value is below the critical field limit for all tested stellar models.
• Conclusion on Origin: The results favor the scenario where the magnetic field is a remnant of a dynamo-generated field from the convective core, left behind as the core retreated during the star's evolution.

Significance
The paper claims the first seismic detection of an internal magnetic field in a typical, rapidly rotating $\gamma$ Doradus star. While previous detections in red giants and a slow rotator provided constraints on angular momentum transport, this detection extends the evidence to a different evolutionary stage and rotation regime. The inferred field strength ($\sim$4 kG) is consistent with extrapolations from red giant cores and supports the hypothesis that magnetic fields generated by core dynamos during the main sequence persist and influence the radiative envelope. The methodology, specifically the use of $\delta K_a$ and the magnetic asymptotic model, is presented as a tool that can be extended to other $\gamma$ Dor stars observed by Kepler, TESS, and the upcoming PLATO mission.