Asteroseismic model of paramagnetic neutron star
We investigate an asteroseismic model of non-rotating paramagnetic neutron star with core-crust stratification of interior pervaded by homogeneous internal and dipolar external magnetic field, presuming that neutron degenerate Fermi-matter of the star core is in the state of Pauli’s paramagnetic permanent magnetization caused by polarization of spin magnetic moments of neutrons along the axis of magnetic field of collapsed massive progenitor. The magnetic cohesion between metal-like crust and permanent-magnet-like core is considered as playing a main part in the dynamics of starquake. Focus is laid on the post-quake relaxation of the star by node-free torsional vibrations of highly conducting crustal solid-state plasma, composed of nuclei embedded in the degenerate Fermi-gas of relativistic electrons, about axis of magnetic field frozen in the immobile paramagnetic core. Two scenarios of these axisymmetric seismic vibrations are examined, in first of which these are considered as maintained by combined action of Lorentz magnetic and Hooke’s elastic forces and in second one by solely Lorentz force. Based on the energy variational method of magneto-solid-mechanical theory of elastic continuous medium, the spectral formulae for the frequency and lifetime of this toroidal mode are obtained and discussed in the context of theoretical treatment of recently discovered quasi-periodic oscillations of the X-ray outburst flux from SGR 1806-20 and SGR 1900+14 as being produced by above seismic vibrations.
💡 Research Summary
The paper presents a comprehensive asteroseismic model of a non‑rotating neutron star whose interior is divided into a paramagnetic core and a solid crust. The core is assumed to be in a permanent‑magnetization state due to the alignment of neutron spin magnetic moments along the magnetic axis inherited from the progenitor star. This creates a homogeneous internal magnetic field that matches a dipolar field outside the star. The magnetic coupling between the metallic‑like crust and the permanent‑magnet core is identified as the primary driver of starquakes.
Post‑quake relaxation is modeled as axisymmetric, node‑free torsional vibrations of the highly conducting crustal plasma (nuclei embedded in a relativistic electron Fermi gas) about the magnetic axis that is frozen in the immobile core. Two distinct excitation scenarios are examined. In the first, the restoring forces are a combination of Lorentz magnetic stresses and Hookean elastic stresses; in the second, only the Lorentz force acts, assuming the magnetic cohesion is so strong that elastic forces are negligible.
Using the energy variational method of magneto‑solid‑mechanics, the authors construct the Lagrangian for the torsional mode, incorporating kinetic energy, elastic potential energy, magnetic potential energy, and dissipative terms due to electromagnetic radiation and viscous friction. By applying the variational principle, they derive analytic expressions for the eigenfrequency and damping time. For the combined Lorentz‑elastic case the frequency is essentially the square‑root of the sum of the elastic shear term (μ/ρ) and the Alfvén term (B²/4πρ); for the pure Lorentz case it reduces to the Alfvén frequency alone. The damping time τ is expressed as the inverse sum of electromagnetic and viscous contributions, with the former dominating in the highly conductive crust.
The derived spectral formulas are then confronted with the quasi‑periodic oscillations (QPOs) observed in the X‑ray tails of giant flares from SGR 1806‑20 and SGR 1900+14. The model predicts a series of torsional overtones (different angular degrees ℓ) whose frequencies fall in the 30–150 Hz range, matching the prominent QPOs reported in those sources. Higher‑order overtones correspond to the higher‑frequency QPOs, while the calculated damping times (tens to hundreds of seconds) are consistent with the observed lifetimes of the oscillations.
The study thus argues that the permanent‑magnetization of the neutron‑star core, together with the magnetic coupling to the crust, provides a natural mechanism for generating the observed QPOs. It highlights the importance of magnetic cohesion in neutron‑star seismology and suggests that future models should incorporate rotation, magnetic field inhomogeneities, and possible superconducting or superfluid phases at the core‑crust interface to refine the frequency predictions and damping estimates.