Exploring the probing power of gamma-Dor's inertial dip for core magnetism: case of a toroidal field

TL;DR

This study extends the analytical framework for inertial dips in γ-Doradus stars by incorporating a toroidal magnetic field with a bi-layer Alfvén frequency, enabling a controlled examination of core and envelope magnetism on dip formation. Using the Traditional Approximation of Rotation and Magnetism (TARM) and a bi-layer rotational/Alfvén structure, the authors derive a coupling between core magneto-inertial modes and envelope magneto-gravito-inertial modes, yielding Lorentzian-like dip profiles. They find that stronger core fields push the inertial dip to lower spin parameters and narrow the dip, while envelope fields add curvature to the PSP and can suppress envelope modes if sufficiently strong, revealing a degeneracy with differential rotation. The work identifies regimes where core magnetism could be probed through inertial dips and outlines observational strategies, while acknowledging simplifications (toroidal bi-layer fields, uniform zones) and calling for future numerical MHD extensions to explore more realistic field geometries and transition-region physics.

Abstract

Gamma-Dor stars are ideal targets for studies of stellar innermost dynamical properties due to their rich asteroseismic spectrum of gravity modes. Integrating internal magnetism to the picture appears as the next milestone of detailed asteroseismic studies, for its prime importance on stellar evolution. The inertial dip in prograde dipole modes period-spacing pattern of gamma-Dors stands out as a unique window on the convective core structure and dynamics. Recent studies have highlighted the dependence of the dip structure on core density stratification, contrast of the near-core Brunt-Väisälä frequency and rotation rate, as well as the core-to-near-core differential rotation. In the meantime, the effect of magnetism has been derived on envelope modes. We aim to revisit the inertial dip formation including core and envelope magnetism, and explore the probing power of this feature on dynamo-generated core fields. We consider a toroidal magnetic field with a bi-layer (core and envelope) Alfvén frequency. This configuration allows us to revisit the coupling problem using our knowledge on both core magneto-inertial modes and envelope magneto-gravito-inertial modes. We stay in an analytical framework to exhibit the magnetic effects on the inertial dip shape and location, setting up a laboratory towards the comprehension of magnetic effects on the dip structure. We show a shift of the inertial dip towards lower spin parameter values and a thinner dip with increasing core magnetic field, quite similar to the signature of differential rotation. The magnetic effects become sizeable when the ratio between the magnetic and the Coriolis effects is large enough. We explore the degeneracy of the magnetic effects with differential rotation. We study the detectability of core magnetism, considering observational constraints on the modes periods and potential gravito-inertial mode suppression.

Figures (11)

• Figure 1: Magnetic star with a bi-layer Alfvén frequency, $\omega_{\rm A, core}$ in the core, $\omega_{\rm A, env}$ in the envelope, and a bi-layer rotation rate, $\Omega_{\rm core}$ in the core, $\Omega_{\rm env}$ in the envelope. The cavity for $m$-$g$-$i$ modes lies between $r_{\rm a}$ and $r_{\rm b}$ in the radiative zone. They get an evanescent character in the region $[R_{\rm core};r_{\rm a}]$ when applying the TARM. Magneto-inertial modes propagate in the convective core below the location $R_{\rm core}$.
• Figure 2: Magnetic field profiles of uniform Alfvén frequency, taking the 1.5 $\rm M_{\odot}$$fz$ model, at the equator. As an example, a magnetic field profile with a bi-layer Alfvén frequency corresponding to $\rm Le_{core} = 1.0\times 10^{-2}$ and $\rm Le_{env} = 2.0\times 10^{-2}$ would follow the blue profile in the core (turquoise zone) and a red profile in the envelope (pink zone).
• Figure 3: Hierarchy of the characteristic angular frequencies hypothesized in this work. Quantities related to the envelope are specified in the top panel, while quantities related to the core are noted in the bottom panel. Values for the linear frequencies for the $fz$ model are denoted in parentheses. An Alfvén frequency of $\omega_{\rm A,env}\approx5\times10^{-2}\mu\rm Hz$ corresponds to an equatorial field strength of $\rm B_{0}^{\varphi}\approx 9\times 10^{4}\rm G$ at the bottom of the radiative zone for the three considered models.
• Figure 4: PSPs containing inertial dips obtained by solving Eq. \ref{['eq:coupling_cont']}, from a situation with no magnetic fields (black) to a $\rm Le=2\times 10^{-2}$. The coupling parameter is fixed at $\epsilon =1.5\times 10^{-2}$, the buoyancy travel time $\Pi_{0,\rm M} = 4670\rm s$, the rotation rate $\Omega=2\pi\times 1.14 \, \rm c.d^{-1}$, parameters adapted to the $im$ model.
• Figure 5: Inertial dips overplotted for different core Lehnert number for a fixed envelope Lehnert number $\rm Le_{env} = 10^{-3}$ and coupling parameter $\epsilon = 1.5 \times 10^{-2}$, in a uniformly rotating star, in the framework of a continuous $N$ at $r_{a}$. Dots are obtained by solving Eq. \ref{['eq:coupling_cont']}, and continuous line by applying the dip profile given by Eq. \ref{['eq:coupl_disc']}.