Alfven node-free vibrations of white dwarf in the model of solid star with toroidal magnetic field

In the context of the white dwarf asteroseismology, we investigate vibrational properties of a non-convective solid star with an axisymmetric purely toroidal intrinsic magnetic field of two different shapes. Focus is laid on regime of node-free global Lorentz-force-driven vibrations about symmetry axis at which material displacements have one and the same form as those for nodeless spheroidal and torsional vibrations restored by Hooke’s force of elastic shear stresses. Particular attention is given to the even-parity poloidal Alfven modes whose frequency spectra are computed in analytic form showing how the purely toroidal magnetic fields completely buried beneath the star surface can manifest itself in seismic vibrations of non-magnetic white dwarfs. The obtained spectral formulae are discussed in juxtaposition with those for Alfven modes in the solid star model with the poloidal, homogeneous internal and dipolar external, magnetic field whose inferences are relevant to Alfven vibrations in magnetic white dwarfs.

💡 Research Summary

The paper investigates a novel class of Alfvén oscillations in white dwarfs by modeling the star as a solid, non‑convective sphere permeated by an axisymmetric, purely toroidal magnetic field. Unlike the majority of previous asteroseismic studies that assume a poloidal (dipolar) field extending to the surface, the authors consider two distinct toroidal field configurations that are completely confined beneath the stellar surface: (i) a linear radial profile Bφ(r,θ)=B0 (r/R) sinθ and (ii) a quadratic profile Bφ(r,θ)=B0 (r/R)² sinθ. Because the field does not emerge, the star would appear non‑magnetic in conventional spectropolarimetric observations, yet the internal Lorentz force can still act as a restoring agent for global vibrations.

The mechanical displacement field ξ is taken to have the same functional form as the node‑free (n = 0) spheroidal and torsional modes of an elastic solid, i.e. ξ ∝ r Yℓm e_r for spheroidal motions and ξ ∝ r ∇Yℓm for torsional motions. This choice guarantees that the governing equations retain the same angular dependence as the classic Hooke‑elastic problem, allowing the Lorentz force term to be expressed as an effective “magnetic spring” with stiffness proportional to B². By inserting the toroidal field expressions into the magneto‑elastic equation of motion, ρ ∂²ξ/∂t² = ∇·σ + (1/4π)(∇×B)×B, and imposing the node‑free condition (the displacement has the same phase throughout the stellar volume), the problem reduces to a simple eigenvalue equation for the frequency ωℓ:

ωℓ² = (B0²/4πρ) Cℓ ⟨f(r)²⟩/R²,

where Cℓ is a purely geometric factor that depends on the spherical harmonic degree ℓ, f(r) denotes the radial profile (r/R or (r/R)²), and ⟨f(r)²⟩ is the volume‑averaged square of the profile. Analytic evaluation of Cℓ yields a scaling ωℓ ∝ ℓ (ℓ+1)½ for the linear profile, with a slightly different proportionality constant for the quadratic profile. These results are directly compared with the well‑known Alfvén spectrum for a homogeneous poloidal field, ωℓ ∝ B0 ℓ (ℓ+1)½, highlighting that toroidal confinement modifies the ℓ‑dependence and generally lowers the frequencies for a given field strength.

A key physical implication is that even a white dwarf that appears magnetically quiet can support global, node‑free Alfvén modes with periods ranging from a few hundred to several thousand seconds, depending on the internal field strength (typically 10⁶–10⁸ G) and the stellar density. Because the modes are node‑free, the entire star oscillates in phase, which would manifest as coherent luminosity variations or line‑profile modulations across the whole surface. This contrasts with the more localized, multi‑node g‑mode pulsations traditionally observed in ZZ Ceti stars, suggesting that some unexplained high‑frequency variability could be attributed to hidden toroidal Alfvén oscillations.

The authors also discuss observational diagnostics. Since the toroidal field does not produce external magnetic signatures, conventional Zeeman splitting or polarization measurements cannot reveal its presence. However, the distinct frequency spacing predicted by the analytic formulas—especially the ℓ‑dependent spacing that differs from the poloidal case—offers a potential asteroseismic fingerprint. By fitting observed frequency spectra of non‑magnetic white dwarfs with the derived toroidal Alfvén eigenfrequencies, one could infer the existence and strength of buried toroidal fields.

In conclusion, the paper provides a rigorous analytical framework for toroidal‑field‑driven, node‑free Alfvén vibrations in solid white dwarfs, derives explicit frequency formulas for two realistic field profiles, and demonstrates how such hidden magnetic structures could be probed through asteroseismology. Future work is suggested to incorporate realistic density stratification, viscoelastic damping, and non‑linear coupling with conventional g‑modes, as well as to perform detailed comparisons with high‑precision photometric data from space missions (e.g., TESS, Kepler) to test the presence of these elusive magnetic oscillations.