Detectability of axisymmetric magnetic fields from the core to the surface of oscillating post-main sequence stars

For a spherically symmetric, non-rotating, non-magnetic, adiabatic, isotropic reference star, the wave equation over a hydrostatic equilibrium is given by the following eigenvalue problem for a specific mode of stellar oscillation denoted by the subscript $k$ (Christensen–Dalsgaard, 2003; Lavely & Ritzwoller, 1992; Aerts et al., 2010):

where

In these expressions, $\rho_{0}$ is the equilibrium density of the reference star, $c_{s}$ is the sound speed, $\omega_{k}$ and $\boldsymbol{\xi}_{k}$ are the eigenfrequency and eigenfunction of the mode $k$, the unit vector pointed radially outward is denoted by $\hat{r}$, and $g$ is gravity directed radially inward. Upon solving Eqn. (1), the quantization of the three dimensional oscillations are represented by the label $k\equiv(n,\ell,m)$ where the eigenfunctions may be represented in the form (Chandrasekhar & Kendall, 1957):

$(r,\theta,\varphi)$ are the common spherical coordinates, $n$ is radial order, $Y_{\ell m}(\theta,\varphi)$ are spherical harmonics of degree $\ell$ and azimuthal order $m\>\in[-\ell,\,-\ell+1,...,\,0,...,\ell-1,\ell]$, $\boldsymbol{\nabla}_{h}$ is the horizontal gradient, and $U_{n\ell}(r),\>V_{n\ell}(r)$ are profiles of the radial and horizontal components of the eigenfunctions as a function of stellar radius $r$. By virtue of the operator $\mathcal{L}_{0}$ being hermitian, the eigenfunctions are orthogonal by construction. The modes of oscillations in solar-like stars are stochastically excited and intrinsically damped which requires one to solve for the structure and energy equations using the quasi-adiabatic approximation. The total displacement vector is a superposition of all possible $\boldsymbol{\xi}_{nlm}(r,\theta,\varphi,t)\equiv\boldsymbol{\xi}_{n\ell m}(r,\theta,\varphi)\exp\left[-i\omega_{k}t\right]$, with $\omega_{k}=\omega_{r}+i\eta$ (where $\omega_{r},\eta\in\mathbb{R}$) and amplitudes related to the excitation and damping. The contribution due to this intrinsic damping is accounted for in $\eta$. For a reference star, the oscillation power spectrum is $2\ell+1$ fold degenerate for a given multiplet ($n,\ell$). All these $2\ell+1$ values of $m$ for the multiplet have the same angular frequency $\omega_{r}=\omega_{n\ell}=2\pi\nu_{n\ell}$, which for solar-like oscillators are given by the asymptotic theories (eg. Christensen-Dalsgaard & Berthomieu, 1991, Section 3.4.3 of Aerts et al. 2010).

Now, we introduce a perturbation, $\epsilon\>\delta\mathcal{L}(\boldsymbol{\xi})$ to the system which modifies the resonant angular frequency of a mode to $\omega_{n\ell}+\epsilon\>\delta\omega$ and the displacement vector to $\boldsymbol{\xi}_{k}+\epsilon\>\delta\boldsymbol{\xi}_{k}$ (with $\epsilon\ll 1$, indicating that the order of magnitude of the perturbation $\delta\mathcal{L}$ and angular frequency splitting $\delta\omega$ are $\mathcal{O}(\epsilon)$ smaller than the wave generator $\mathcal{L}_{0}$ and the background angular frequency $\omega_{n\ell}$ respectively), yielding

Using the orthogonality of $\boldsymbol{\xi}_{k}$ and retaining terms in Eqn. (4) up to first order in $\epsilon$, an axisymmetric perturbation $\delta\mathcal{L}$ induces the angular frequency splittings given by

where $I_{n\ell}=2\omega_{n\ell}\;\int\limits_{0}^{R_{\rm{star}}}\rm{d}r\;4\pi\>r^{2}\rho_{0}\>[U_{n\ell}^{2}+\ell(\ell+1)V_{n\ell}^{2}]$.

We consider a general magnetic field $\boldsymbol{B}$ which is represented in terms of the generalized spherical harmonics $Y_{pq}^{\mu}(\theta,\varphi)$ (Dahlen & Tromp, 1999; Phinney & Burridge, 1973) as

Here $\mu\in\{-1,0,1\}$ and $\hat{e}_{-}=(\hat{\theta}-i\hat{\varphi})/\sqrt{2}$, $\hat{e}_{0}=\hat{r}$, $\hat{e}_{+}=-(\hat{\theta}+i\hat{\varphi})/\sqrt{2}$. The configuration of the perturbing field is specified by angular degree $p$ and azimuthal order $q\in\{-p,-p+1,...,p-1,p\}$. When the magnetic field $\boldsymbol{B}$ is the perturbation to the stellar model ($\epsilon\approx|\boldsymbol{B}|^{2}\left(4\pi GM^{2}/R^{4}\right)^{-1}$, a ratio of magnetic pressure to the gas pressure Gough & Thompson, 1990), then $\delta\mathcal{L}(\boldsymbol{\xi})\equiv\delta\mathcal{L}_{B}(\boldsymbol{\xi})$, as derived in Goedbloed & Poedts (2004) and subsequently used in Das et al. (2020), is

For solar-like oscillators, the eigenfunctions become evanescent beyond the stellar surface. This renders the oscillation frequencies of such modes insensitive to the perturbations above the surface. Using Eqns. (3) and (2.2), neglecting contributions due to surface boundary terms, and considering self-coupling in an isolated multiplet $(n,\ell)$, Eqn. (5) (refer Appendix C of Das et al., 2020) takes the following form

where $\mu,\sigma\in\{-1,0,1\}$, $s$ and $t$ are the angular degree and azimuthal order of the Lorentz stress perturbation, $R_{\rm{star}}$ is the radius of the star, and $\mathcal{B}_{st}^{\mu\sigma}(r;n,\ell,m)$ are the Lorentz-stress sensitivity kernels which are dependent on $U_{n\ell}(r)$ and $V_{n\ell}(r)$. $h_{st}^{\mu\sigma}(r)$ denote components of the Lorentz-stress tensor (Das et al., 2020) in the generalized spherical harmonics basis, which captures the amplitude and topology of the magnetic field

Theoretical studies on the calculation of splittings, as mentioned in the introduction, have been carried out for red-giant stars. They have been employed often to compute the impact due to some dominant terms in the magnetic perturbation operator (eg. Bugnet et al., 2021; Loi, 2021; Mathis et al., 2021; Bugnet, 2022; Li et al., 2022). However, most studies have considered the effect of only the radial field strength $B_{r}$ in the H-shell burning region to find frequency splittings. The formalism in Das et al. (2020) can be put to use for any magnetic field topology, provided the oscillation eigenfunctions retain their form as in Eq. (3) in the presence of other perturbations (which breaks down when traditional approximation of rotation is considered for fast rotators) and the magnetic field amplitude remains small enough for first order perturbations to remain valid. We use this formalism to calculate the effects of axisymmetric magnetic fields on frequency splittings in a non-rotating red-giant and its corresponding sub-giants. For both these cases, using a thorough calculation of kernels, we estimate threshold strengths for the potential for detection of different field components at different layers which could let us constrain their configurations.

Duez et al. (2010) and Duez & Mathis (2010) discuss in detail the stable magnetic field configurations that may be present inside a radiative interior. Earlier works (eg. Tayler, 1973; Braithwaite, 2006; Markey & Tayler, 1973; Braithwaite, 2007) show that pure toroidal or poloidal magnetic field models are physically unstable on dynamical timescales inside a star, and instead, a mix of the two is stable (eg. Tayler, 1980; Braithwaite, 2009). The first solution to the stable field configuration obtained semi-analytically in Appendix B of Bugnet et al. (2021) is seen to have a strong poloidal magnetic field component and a weak toroidal component, with the maximum amplitude of the toroidal field being much smaller than that of the maximum radial field amplitude. Hence, we consider an axisymmetric solenoidal magnetic field that may be decomposed into toroidal and poloidal components on the spherical coordinates

where $B_{0}$ is the magnetic-field amplitude scaling factor, and $\beta(r)$ and $\alpha(r)$ are functions parameterizing the radial variations of the poloidal and toroidal counterparts, respectively. Following the same decomposition method as employed in Appendix D.1 of Das et al. (2020), this magnetic field $B_{p_{0}q_{0}}(r)$ is represented in the general spherical harmonics basis as

The topmost component belongs to $Y_{10}^{-1}\hat{e}_{-}$, the middle one to $Y_{10}^{0}\hat{e}_{0}$, the bottom-most component belongs to $Y_{10}^{+1}\hat{e}_{+}$. Using Eqn. (9), the Lorentz-stress tensor for this model magnetic field can be expressed as

where $\gamma_{j}=\sqrt{\frac{2j+1}{4\pi}}$, $\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\mathcal{W}_{m_{1}\,m_{2}\,m_{3}}^{\ell_{1}\,\ell_{2}\,\ell_{3}}$ is a Wigner 3j symbol (Wigner, 1993) which takes a value of 0 (Appendix C of Dahlen & Tromp, 1999) when it fails to satisfy one or more of the following selection rules:

1. 1.

   $m_{i}\in\{-\ell_{i},-\ell_{i}+1,...,\ell_{i}-1,\ell_{i}\}$,

2. 2.

   $|\ell_{1}-\ell_{2}|\leq\ell_{3}\leq\ell_{1}+\ell_{2}$,

3. 3.

   $m_{1}+m_{2}+m_{3}=0$,

4. 4.

   $\ell_{1}+\ell_{2}+\ell_{3}$ is an even integer if $m_{1}=m_{2}=m_{3}=0$.

According to these selection rules, $h_{st}^{\mu\sigma}=0$ when $s\notin\{0,1,2\}$ and $t\neq 0$. Also, we have $h_{10}^{\mu\sigma}~{}=~{}0\;\forall\mu,\sigma$ since it is proportional to $\mathcal{W}_{0\,0\,0}^{1\,1\,1}$ which yields zero. For a dipolar magnetic field, due to the selection rule imposed by Wigner 3-$j$ in Eqn. (2.3), we are only required to compute the sensitivity kernels for $s\in\{0,2\}$ and $t=0$ for this particular axisymmetric model of stellar magnetic field (since $h_{st}^{\mu\sigma}=0$ when $t\neq 0$, or $s\neq 0\>\rm{or}\>2$).

The kernels $\mathcal{B}_{st}^{\mu\sigma}(r;n,\ell,m)\equiv\mathcal{B}_{st}^{\mu\sigma}(k,m)$ depend on $U_{n\ell}(r)\equiv U_{k}$ and $V_{n\ell}(r)\equiv V_{k}$ and their first and second order radial derivatives. They describe how different oscillation modes respond to the Lorentz stresses due to stellar internal magnetic fields of different strength and geometric configurations. For axisymmetric magnetic fields as defined in Section 2.3, we are only required to calculate $\mathcal{B}_{st}^{\mu\sigma}$ for $t=0$ (since $h_{st}^{\mu\sigma}=0$ when $t\neq 0$). The sensitivity kernels for self-coupled modes in the presence of an axisymmetric magnetic field in the star, as derived in Appendix C of Das et al. (2020), can be expressed as

where $\tilde{\mathcal{G}}_{s}^{\mu\sigma}(k)=r^{2}\mathcal{G}_{s}^{\mu\sigma}(k)$ ($\mathcal{G}_{s}^{\mu\sigma}(k)$ has been derived in Appendix C.1 of Das et al., 2020) and the kernel components were found to satisfy the following identities:

As a consequence of these symmetries and relations, we have only four independent components $\tilde{\mathcal{G}}_{s}^{\mu\sigma}$ to compute. Using the expressions for $\mathcal{G}_{s}^{\mu\sigma}(k)$ derived in Appendix C.1 of Das et al. (2020), the expressions for these independent $\tilde{\mathcal{G}}_{s}^{\mu\sigma}$ are:

where expressions for the ten $\chi_{i}^{\mu\sigma}(k)$ are shown in the Appendix A.

The formalism of Das et al. (2020) is by construction independent of the inherent symmetries of the internal magnetic field topology. As discussed in Section 2.4.1 of Das et al. (2020), the axisymmetric nature of the magnetic field topologies simplifies our calculations by setting $t=0$ for all cases. For cases where non-zero values of $t$ are possible, the selection rules for the Wigner 3-j symbol render it mandatory for us to take into consideration coupling among the different degenerate $m$ (say, $m,m^{\prime}\in\{-\ell,-\ell+1,..,0,..,\ell-1,\ell\}$) belonging to the same multiplet. This will result in a more complicated forward model involving eigenvalue problems to calculate the splittings. For slowly rotating stars, such non-axisymmetric magnetic fields could arise in cases where there is a finite obliquity between rotational and magnetic axes.

For the major part of this work, we focus on a model of a typical 4.056 Gyr red-giant branch star with mass $M=1.3\,M_{\odot}$, metallicity $Z=0.025$, radius $R_{\rm{star}}=5.383\,R_{\odot}$. This RGB structure model is generated using MESA (Paxton et al., 2011, 2013, 2015, 2018, 2019) We also generate two models of the same star in its sub-giant phase using MESA at ages of 3.624 Gyr and 3.702 Gyr¹¹1The data is available on Zenodo under an open-source Creative Commons Attribution license: https://doi.org/10.5281/zenodo.10913776 (catalog doi:10.5281/zenodo.10913776) . Figure 1 shows schematic diagrams of the stellar structure in the RG and SG phases. The star in the former SG stage has reached a peak luminosity in the SG evolution, beyond which luminosity starts dropping due to the narrowing H-burning shell around the core, which includes the latter (Iben, 1967). These phases will henceforth be referred to as the mid sub-giant phase (MSG) and the late sub-giant phase (LSG) respectively (see Figure 2). Their frequencies of maximum power ($\nu_{\rm{max}}$) are $\nu_{\rm{max,MSG}}=581.73\>\mu\rm{Hz}$ and $\nu_{\rm{max,LSG}}=542.99\>\mu\rm{Hz}$ respectively, which are much larger than that of the red giant ($\nu_{\rm{max,RG}}=160.928\>\mu\rm{Hz}$). The eigenfunctions and resonant frequencies of $\ell=0,1,2$ are computed with GYRE (Townsend & Teitler, 2013), generated in the range of [$120\mu\rm{Hz},200\mu\rm{Hz}$] for the RG and $[\nu_{\rm{max}}-3\Delta\nu,\nu_{\rm{max}}+3\Delta\nu]$ for the SGs, $\Delta\nu$ being their large frequency separations. The three stages used in this study have been indicated in Figure 2.

We then use unperturbed frequencies, displacement eigenvectors, and relevant structure parameters of the model stars as input to compute the field kernels and the splittings induced by a given magnetic field having a topology mentioned in Section 2.3.

The Brunt-Väisälä frequency $N$, commonly known as buoyancy frequency, is defined as

with $P_{0}$ being the equilibrium pressure profile, and $\Gamma_{1}$ being the adiabatic exponent of the material in the stellar interior. $N^{2}$ is the square of the frequency of oscillation of a blob of plasma in the statically stable stellar interior when it is displaced radially. Hence, it takes negative values in convective regions and positive values in radiative zones. From Figure 2, we note that $N^{2}$ goes to 0 at $R_{\rm{rad}}=0.1364\>R_{\rm{star}}$ for the RG, indicating the boundary between the radiative core and convective envelope.

As we have discussed in the introduction of this work, red-giant stars show the presence of mixed modes, which are formed due to considerable coupling between the $p$-modes in the convective envelope and the $g$-modes within the radiative interior. These mixed modes thus have part of their inertia in the core and part of it in the envelope. For such modes, (Deheuvels et al., 2012; Goupil et al., 2013) defined :

The $g$-mode cavity of the RG extends from the centre to its $R_{\rm{rad}}$.

The $\zeta_{n\ell}$ (hereafter simplified as $\zeta$ for notational convenience) is the fraction of the total mode inertia existing inside the $g$-mode cavity, which usually lies within the star’s radiative interior. For $g$-dominant modes, $\zeta_{n\ell}\approx 1$ and for a $p$-dominated mode, $\zeta_{n\ell}\ll 1$. Hence, $\zeta$ quantifies the $p$/$g$-mode nature of a mixed mode (e.g. Mosser et al., 2015; Vrard et al., 2016). $\zeta_{n\ell}$ for the RG is plotted in Figure 3.

By definition, the kernel $r^{2}\mathcal{B}_{s0}^{\mu\sigma}(k,m)$ corresponding to different modes of oscillations $(k,m)$ teach us how frequency splittings depend on an axisymmetric Lorentz-stress component $h_{s0}^{\mu\sigma}(r)$. This identifies which $B_{i}B_{j}$ ($i,j\in\{r,\theta,\varphi\}$) components in different regions of the star dominantly contribute to the magnetic splittings.

Table 1 summarizes which component of Lorentz-stress $\mathbf{B}\mathbf{B}$ is sensed by which kernel component. By virtue of Eqns. (14)-(16), we have only used the four independent kernel components. We have also demarcated the parity in $s$ (even or odd) for these kernel sensitivities based on self-coupling assumption. The interested reader is referred to the discussion in Section. 3.2 of Das et al. (2020) where these parity-based selection rules are extensively laid out. Restricting the analysis to $\ell=1$ dipolar modes, the following section investigates the sensitivity of the 6 magnetic kernels (two for $s=0$ and four for $s=2$) to various locations in the modeled stars.

Having dealt with a red-giant star in the last subsection, we proceed to investigate the potential for detection of magnetic fields in the earlier phases of the star - the sub-giant phase. This phase is identified as the transition period between the main sequence and red-giant characterized by the deepening of the convective envelope in low and intermediate-mass stars. In this stage, the $g$-mode trapped in the core starts coupling to the $p$-mode in the envelope and presents us with mixed modes for the first time in the evolution. We now explore which zones of these stars affect splittings the most and whether near-surface fields and core fields could be distinctly measured using different modes in different parts of the power spectrum.

Axisymmetric magnetic field sensitivity kernels were constructed (eg., as in Figures 21 and 22) using the aforementioned theory, and three significant zones were identified for each of them. The significant zones for the MSG stage are the inner core with $r/R_{\rm{star}}\in(0,0.2]$ containing the H-burning shell, the outer core with $r/R_{\rm{star}}\in(0.2,0.7873)$, and the near-surface layers with $r/R_{\rm{star}}\in(0.9,1)$, whereas those for the LSG stage are $r/R_{\rm{star}}\in(0,0.1]$ containing the H-burning shell, the outer core $r/R_{\rm{star}}\in(0.1,0.6637)$, and the near-surface layers with $r/R_{\rm{star}}\in(0.9,1)$. As done in Section 3.2.2, we investigate Lorentz stress component contribution diagrams similar to Figure 5 (normalization was done with respect to the inner radiative zones of both the sub-giants which contain the H-burning shell) for all used modes. Lorentz stress component contribution diagrams for a dipole mode closest to $\nu_{\rm{max}}$ at each sub-giant stage are shown in Figure 7.

Perturbations to the stellar reference model induce frequency splittings. These splittings are detectable in the data obtained from a telescope only when certain criteria are satisfied. For the different cases discussed in this work (also see Appendix D), we calculate $y_{n\ell m}=\delta\nu_{n\ell m}\times\left(B_{0,\rm{ref}}/B_{0}\right)^{2}$, $B_{0,\rm{ref}}$ being a reference value for field strength $B_{0}$ that induces a splitting of $y_{n\ell m}$. A necessary criterion to be satisfied is $\delta\nu_{n\ell m}>\nu_{\rm{min}}$, where $\nu_{\rm{min}}$ is some measure of minimum separation in frequency. This implies that for detectability, the inequality $B_{0}>B_{0,\rm{min}}$, needs to be satisfied. Here $B_{0,\rm{min}}=B_{0,\rm{ref}}\times\sqrt{\nu_{\rm{min}}/y_{n\ell m}}$. The bare minimum and simplest of all such criteria is that the distance between the peaks on the power spectrum should be separated by at least 1 unit of the frequency resolution $\nu_{\rm{res}}$, i.e. $\nu_{\rm{min}}=\nu_{\rm{res}}$, which is typically the reciprocal of the total observation time for the star. While calculating the value of $B_{0,\rm{min}}$, one has to verify that $|\boldsymbol{B}|^{2}(4\pi GM^{2}/R^{4})^{-1}<<1$ and that it is below its corresponding critical field as mentioned in Fuller et al. (2015). Since these splitting measurements are key to inferring the perturbations, in this subsection, we investigate the detectability of splittings in dipolar mixed modes closest to $\nu_{\rm{max}}$ during the 3 evolutionary stages of the star in consideration.

The configuration of the fields used for the calculation of the magnetic splittings is of the form given in Eqn. (2.3), where for a poloidal field in the core (H-burning shell region in RG and inner core in SG) we have $\alpha(r)=0$ and

and for a toroidal field in the envelope we have $\beta(r)=0$ and

for a toroidal field in the envelope (refer Appendix D.1 for details on the choice of such toy topologies and $B_{0}$), where the coefficients $\beta_{1},\beta_{2},\alpha_{1},\alpha_{2}$ and $B_{0}$ (check Table 2) were chosen keeping the structure of the star in mind. The poloidal magnetic fields in the core have magnitudes $\sim B_{0}$ close to the centre and $\sim 0$ in the envelope, whereas the toroidal fields in the envelope have magnitudes $\sim B_{0}/2$ close to the surface and $\sim 0$ in the core.

Figure 8 demonstrates the detectability of magnetic splittings for the most visible dipolar mixed modes (closest to $\nu_{\rm{max}}$) in our 1.3 $M_{\odot}$ star during its MSG, LSG and RGB phases. In Figure 5 we already noted that specific field components at the H-burning shell and the near-surface envelope dominantly affect the splittings. Similarly, Figure 7 shows that in the MSG and LSG phases, magnetic splittings are primarily influenced by field components in the inner core H-burning shell, the outer core and near-surface envelope. To have a consistent model of magnetic field across the different evolutionary phases of the star, we use inner poloidal and outer toroidal fields as per Eqns. (26) & (27) with values of $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ specified in Table 2. We already showed that for MSG and LSG phases, there are three zones of potential magnetic detectability. Therefore, the two-zone model of the magnetic field does not provide explicit control over the outer-core field. Nevertheless, it is easy to see that we could estimate the limit of detectability of field strength of the outer core by using a three-zone model instead of a two-zone model.

For the RG, where we have clearly distinguishable $g$ and $p$-dominated modes, we have two vertical lines showing the detectability for the inner poloidal field (to which the $g$-dominated modes are sensitive) and the outer toroidal field (to which the $p$-dominated modes are sensitive). This is not the case for MSG and LSG and hence all the cases of splitting are plotted along one vertical line.

A minimum splitting of 7.9 $\rm{n}Hz$ and 10.6 $\rm{n}Hz$ is needed to be detectable by 4-years of Kepler and 3-years of PLATO. Treating these as the $\nu_{\rm{res}}$ and using $B_{0}^{2}$ proportionality to splitting strength, we calculate the threshold values for detectability of field strengths of different topologies at different layers of the star. For Kepler observations for our choice of RG and the model magnetic field, we find a threshold of $B_{0}=$32 kG poloidal field in the H-burning shell and 190 G toroidal field in the near-surface envelope to be detectable. Similarly, H-burning shell threshold of $B_{0}=$2.1 MG and 1.7 MG for MSG and LSG, respectively, and near-surface threshold of toroidal $B_{0}$ around 114 G and 119 G for MSG and LSG, respectively are required for these splittings to be detectable.

We want to emphasize that these limits are specific to our choice of stellar model and magnetic field model. But the procedure of constructing Lorentz-stress diagrams as in Figures 5 & 7 is general and can be followed in a similar fashion for the choice of star at hand to obtain estimates of magnetic field detectability thresholds for a given satellite measurement.