# Magnetic fields of 30 to 100 kG in the cores of red giant stars

Gang Li, Sébastien Deheuvels, Jérôme Ballot, F. Lignieres

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Gang Li, Sébastien Deheuvels, Jérôme Ballot, F. Lignieres. Magnetic fields of 30 to 100 kG in the cores of red giant stars. Nature, 2022, 610 (7930), pp.43-46. 10.1038/s41586-022-05176-0. hal-03821920

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# Magnetic fields of 30 to 100 kG in the cores of red giant stars

https://doi.org/10.1038/s41586-022-05176-0

Received: 15 April 2022

Accepted: 2 August 2022

Published online: 5 October 2022

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Gang Li<fnref n="1" />, Sébastien Deheuvels<fnref n="1" />✉, Jérôme Ballot<fnref n="1" /> & François Lignières<fnref n="1" />

A red giant star is an evolved low- or intermediate-mass star that has exhausted its central hydrogen content, leaving a helium core and a hydrogen-burning shell. Oscillations of stars can be observed as periodic dimmings and brightenings in the optical light curves. In red giant stars, non-radial acoustic waves couple to gravity waves and give rise to mixed modes, which behave as pressure modes in the envelope and gravity modes in the core. These modes have previously been used to measure the internal rotation of red giants<fnref n="1" />,², leading to the conclusion that purely hydrodynamical processes of angular momentum transport from the core are too inefficient³. Magnetic fields could produce the additional required transport⁴⁻⁶. However, owing to the lack of direct measurements of magnetic fields in stellar interiors, little is currently known about their properties. Asteroseismology can provide direct detection of magnetic fields because, like rotation, the fields induce shifts in the oscillation mode frequencies⁷⁻¹². Here we report the measurement of magnetic fields in the cores of three red giant stars observed with the Kepler¹³ satellite. The fields induce shifts that break the symmetry of dipole mode multiplets. We thus measure field strengths ranging from about 30 kilogauss to about 100 kilogauss in the vicinity of the hydrogen-burning shell and place constraints on the field topology.

Rotation lifts the degeneracy between the angular frequencies $\omega$ of oscillation modes with same degree $l$ and radial order $n$ but different azimuthal order $m$. This produces multiplets with $(2l+1)$ components, which can be used to probe the internal rotation of stars. At first order, rotational multiplets are symmetric with respect to the central ($m=0$) component, as is the case for all red giants studied so far². Magnetic fields are known to break this symmetry⁹⁻¹²,¹⁴.

We detected clear asymmetries in the multiplets of three hydrogen-shell burning giants observed with Kepler, namely KIC 8684542, KIC 7518143 and KIC 11515377 (Fig. 1). We notice several common characteristics for the three stars. First, the asymmetries of dipole multiplets, defined as $\delta_{asym} = \omega_{m=-1} + \omega_{m=+1} - 2\omega_{m=0}$ (ref. ¹⁵), consistently have the same sign for each star: they are all positive (or consistent with zero) for KIC 8684542 (Fig. 2) and KIC 7518143, but negative for KIC 11515377. Second, the absolute values of the asymmetries are systematically lower for pressure (p)-dominated modes (dark shaded regions in Fig. 2) than for gravity (g)-dominated modes (light shaded regions). This indicates that the cause of the asymmetries is located in the core. Finally, the detected asymmetries sharply decrease with frequency.

In the presence of magnetic fields, we showed that, under very general assumptions, the average frequency shift of the components in a dipole multiplet is given by

$$ \delta\omega_B = \frac{\zeta I}{\mu_0 \omega^3} \int_{r_i}^{r_o} K(r) \overline{B_r^2} dr, \quad (1) $$

where $\overline{B_r^2}$ is a horizontal average of the squared radial field $B_r$, $r_i$ and $r_o$ are the turning points of the g-mode cavity, $I$ is a term depending on

the core structure (Supplementary Information), and $\mu_0$ is the permeability of the vacuum. The weight function $K(r)$ sharply peaks near the hydrogen-burning shell, so that $\delta\omega_B$ essentially measures $\overline{B_r^2}$ near this shell (Extended Data Fig. 1). The dependence of $\delta\omega_B$ in $\omega^{-3}$ shows that magnetic perturbations are expected to sharply decrease with frequency. The factor $\zeta$ corresponds to the fraction of the mode kinetic energy that is trapped in the g-mode cavity ($\zeta=1$ for pure gravity modes). Equation (1) shows that magnetic shifts are expected to be larger for g-dominated modes than for p-dominated modes. The characteristics of magnetic shifts are thus very similar to those of the detected asymmetries.

Asymmetries originate from the dependence of magnetic shifts on $|m|$. The asymmetry of dipole multiplets can be directly related to the average magnetic shift by the expression

$$ \delta_{asym} = 3a\delta\omega_B. \quad (2) $$

The coefficient $a$ involves an average of $\overline{B_r^2}$ weighted by the second degree Legendre polynomial $P_2(\cos\theta) = (3\cos^2\theta - 1)/2$, where $\theta$ is the colatitude, and we have shown that $-1/2 \le a \le 1$ (Supplementary Information). For instance, a dipole magnetic field ($B_r - \cos\theta$) yields a positive asymmetry ($a = 2/5$). A field that is entirely concentrated on the poles produces maximal asymmetry ($a = 1$). Conversely, a field concentrated near the equator gives minimal asymmetry ($a = -1/2$).

We then compared the measured asymmetries with those that would be produced by internal magnetic fields. We fit an expression of $\delta_{asym}$ based on equations (1) and (2) to the observed asymmetries. The results are shown in Fig. 2. The agreement with the observed

<footnote n="1">¹IRAP, Université de Toulouse, CNRS, CNES, UPS, Toulouse, France, ✉e-mail: sebastien.deheuvels@irap.omp.eu</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 1 | Asymmetric splittings of two mixed modes in KIC 8684542.**</figcaption></figure>

a, Gravity-dominated mixed mode. b, Pressure-dominated mixed mode. The grey lines show the observed power spectrum in terms of signal-to-noise ratio (S/N) and the black lines show the best-fitting model spectrum. The vertical dotted lines mark the locations of the frequencies. It is noted that the x axis is re-scaled to align the m = ±1 modes.

asymmetries is quite good, with the overall decrease of the asymmetries with frequency being well reproduced, as well as their modulation with the trapping of the modes in the g and p cavities. This confirms that the detected asymmetries are indeed produced by magnetic fields in the cores of these red giants. For completeness, we have also shown that other mechanisms that can produce multiplet asymmetries, such as higher-order rotational effects¹⁶ or near-degeneracy effects¹⁵, cannot account for the observed asymmetries (Supplementary Information).

Equation (1) can then be used to estimate the squared radial magnetic field $\langle B_r^2 \rangle$ averaged in the horizontal direction, and weighted by the function $K(r)$ in the radial direction, yielding

$$ \langle B_r^2 \rangle = \frac{\mu_0 \delta_g \omega_{\text{max}}^3}{3aI}, \qquad (3) $$

where $\omega_{\text{max}} \equiv 2\pi v_{\text{max}}$ corresponds to the angular frequency of maximum power of the oscillations and $\delta_g$ is the asymmetry of pure g modes at $v_{\text{max}}$. The minimal magnetic field that can produce the observed asymmetries is obtained by assuming $a=1$ for stars with positive asymmetries, and $a=-1/2$ for stars with negative asymmetries. To estimate $I$, we computed models that reproduce the seismic properties of each of the three stars, using the stellar evolution code MESA¹⁷ (Supplementary Information). We thus obtained minimal magnetic-field intensities of 65 kG for KIC 8684542, 26 kG for KIC 7518143 and 70 kG for KIC 11515377.

These fields are expected to produce magnetic shifts in the mode frequencies. We found that magnetic shifts can be detected by their impact on the pattern of pure g modes. In the non-magnetic case, the periods of pure g modes with high radial orders $n_g$ are given by

<figure><img src="image_5.png" /><figcaption>**Fig. 2 | Multiplet asymmetries in KIC 8684542 as a function of mode frequency.** The blue circles indicate multiplets with three detected components, and the black squares indicate multiplets with only two detected components. The vertical error bars indicate 1σ standard deviations. The background shows the $\zeta$ value. Light areas correspond to the g-dominated modes and dark ridges show the locations of the p-dominated modes. The red dashed curve corresponds to a fit of the theoretical expression of magnetic asymmetries (equations (1) and (2)) to the observed asymmetries. See also Extended Data Figs. 2 and 3 for KIC 11515377 and KIC 7518143, respectively.</figcaption></figure>

$P(n_g) = \Delta\Pi_1(n_g + \varepsilon_g)$, where $\Delta\Pi_1$ is the asymptotic period spacing. The offset term $\varepsilon_g$ is strongly constrained for hydrogen-shell burning giants. From Kepler observations, it was found that $\varepsilon_g = 0.28 \pm 0.08$ (ref. ¹⁸), in agreement with theoretical predictions¹⁹. We have shown (Supplementary Information) that magnetic shifts are expected to produce an increase in the measured value of $\varepsilon_g$.

The two red giants KIC 8684542 and KIC 11515377 have measured values of $\varepsilon_g$ that significantly deviate from the expected range ($\varepsilon_g = 0.50 \pm 0.02$ and $\varepsilon_g = 0.50 \pm 0.03$, respectively). This can be used to directly measure the intensity of magnetic shifts in these stars. For KIC 7518143, the measured $\varepsilon_g = 0.30 \pm 0.03$ is consistent with regular red giants, which can be used to derive an upper limit for the magnetic-field intensity.

To retrieve this information, we computed asymptotic expressions of mixed modes including magnetic and rotational perturbations. We optimized the values of the magnetic shift of pure g modes at $v_{\text{max}}$ ($\delta\omega_g$) and the asymmetry coefficient $a$ to reproduce at best the observed mode frequencies. We found an excellent agreement with the observations for the three stars (Fig. 3). We thus obtained field intensities of 102 ± 12 kG for KIC 8684542, 98 ± 24 kG for KIC 11515377 and an upper limit of 41 kG for KIC 7518143. These measurements are fully compatible with the lower limits that were obtained using multiplet asymmetries, which gives further evidence that the oscillation mode perturbations are indeed caused by magnetic fields.

We also obtained measurements of the coefficient $a$ for the three stars, which can be used to place constraints on the geometry of the internal magnetic field. We found $a = 0.47 \pm 0.12$ for KIC 8684542 and $a = -0.24_{-0.23}^{+0.08}$ for KIC 11515377. These measurements would be consistent with a dipolar field aligned with the rotation axis for the first star ($a = 2/5$) and a dipolar field aligned with the equator for the second ($a = -1/5$), although other configurations are naturally possible.

This study provides keys to explore internal magnetic fields along the red giant branch. This type of measurement requires the radial magnetic field to be large enough to produce detectable magnetic splittings, but small enough to remain below a critical value $B_c$ above which magnetic tension hampers the propagation of gravity waves²⁰ (the magnetic fields measured here are indeed below $B_c$; Supplementary
<figure><img src="image_6.png" /><figcaption>**Fig. 3** | Stretched échelle diagram for KIC 8684542. The periods of dipole mixed modes have been stretched²⁹ to make them regularly spaced in period by the asymptotic period spacing of g modes, $ΔΠ_1$. Thus, modes with the same azimuthal order *m* appear nearly vertically aligned in an échelle diagram. The blue circles show the observed frequencies with horizontal error bars indicating 1σ standard deviations. The red crosses correspond to the best-fit asymptotic mixed-mode frequencies obtained by including a magnetic perturbation. See also Extended Data Figs. 4 and 5 for KIC 11515377 and KIC 7518143, respectively.</figcaption></figure>

Information). The range of field intensities that satisfy both conditions represents about one order of magnitude (Supplementary Information). Field strengths above $B_c$ have been invoked to account for red giants with suppressed dipole modes²⁰, implying that about 20% of red giants are strongly magnetized²¹. A systematic search for magnetic perturbations in red giants with detected oscillations will give the prevalence of red giants with magnetic fields below $B_c$.

The measurement of magnetic fields in the radiative cores of red giants will allow progress on the origin and evolution of the stellar magnetic field²²,²³. The origin of the detected fields could be a convective core dynamo occurring during the main sequence. Our best-fit models indeed indicate that the hydrogen-burning shell of the three red giants was convective at the very beginning of the main sequence. The dynamo fields generated at that time may have survived when the core became radiative as they can relax into stable configurations²⁴ and ohmic diffusion is negligible²⁵. Assuming magnetic flux conservation, their strengths should range from 3 kG to 5 kG to account for the detected fields (Supplementary Information). This is smaller than the amplitudes predicted by numerical simulations of core convection²⁶ and dynamo scaling laws¹¹ by about one order of magnitude, although this could be due to magnetic energy loss during the relaxation process²⁴ or the post-main-sequence evolution⁶. Another possibility is that the detected fields are the remnants of the internal magnetic field of an Ap star. These main-sequence intermediate-mass stars have a strong surface oblique dipole field²⁷, whose internal part could have survived the post-main-sequence evolution.

Constraints on the field strength and geometry are key to identifying how the redistribution of angular momentum operates inside stars. Our detection of strong magnetic fields in these red giants first suggests that

uniform rotation is enforced in their core up to the hydrogen-burning shell. Therefore, the jump in rotation rate between the core and the envelope, which was revealed by asteroseismology²⁸, must occur at higher radii. Second, we show that the Tayler-Spruit dynamo, which has been proposed as a possible solution for the angular momentum transport in red giants, is not at work in the vicinity of their hydrogen-burning shell. Indeed, the predicted amplitude of the radial magnetic field, of the order of $10^{-2}$ G (ref. ⁵), is far too low to account for the $10^5$-G radial fields measured here.

## Online content

Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-022-05176-0.

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**Data availability**

Kepler data are publicly available from the Mikulski Archive for Space
Telescopes (MAST) portal at https://archive.stsci.edu. Spectra are available
at https://doi.org/10.5281/zenodo.6818371.

**Code availability**

This study makes use of the stellar evolution code MESA, which is avail-
able at https://docs.mesastar.org.

**Acknowledgements** We acknowledge support from the project BEAMING ANR-18-CE31-0001 of the French National Research Agency (ANR) and from the Centre National d'Etudes Spatiales (CNES).

**Author contributions** G.L. discovered the three stars with asymmetric splittings. G.L. and S.D. measured the asymmetries and rotation rates. S.D. measured the absolute magnetic shifts and supervised the whole project. J.B. and F.L. developed the theoretical framework used to interpret the observations. All the authors contributed to writing the manuscript.

**Competing interests** The authors declare no competing interests.

**Additional information**

**Supplementary information** The online version contains supplementary material available at
https://doi.org/10.1038/s41586-022-05176-0.

**Correspondence and requests for materials** should be addressed to Sébastien Deheuvels.
**Peer review information** *Nature* thanks Matteo Cantiello, Margarida Cunha and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

**Reprints and permissions information** is available at http://www.nature.com/reprints.
<figure><img src="image_7.png" /></figure>

**Extended Data Fig. 1 | Shape of the weight function K(m) as a function of the normalized mass.** The function K(m) is shown for the stellar model representative of KIC 11515377. The blue shaded region indicates the

hydrogen-burning shell. The vertical dashed line corresponds to the maximal extent of the initial convective core at the beginning of the main sequence.
<figure><img src="image_8.png" /></figure>

Extended Data Fig. 2 | Multiplet asymmetries in KIC 11515377 as a function of mode frequency. Symbols have the same meaning as in Fig. 2.
<figure><img src="image_9.png" /><figcaption>Extended Data Fig. 3 | Multiplet asymmetries in KIC 7518143 as a function of mode frequency. Symbols have the same meaning as in Fig. 2.</figcaption></figure>
KIC11515377

<figure><img src="image_10.png" /><figcaption>Extended Data Fig. 4 | Stretched échelle diagram for KIC 11515377. Symbols have the same meaning as in Fig. 3.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>KIC7518143</figcaption></figure>

Extended Data Fig. 5 | Stretched échelle diagram for KIC 7518143. Symbols have the same meaning as in Fig. 3.