Magneto-Elastic Dynamics Simulations

• Magneto-elastic dynamics simulations are numerical models that couple magnetic fields with elastic deformations, crucial for studying neutron star interiors and advanced materials.
• They extend GRMHD codes to incorporate elasticity and realistic equations of state, accurately capturing oscillatory modes and quasi-periodic oscillation families.
• The simulations reveal three distinct QPO families that correlate with observable magnetar flare signatures, offering insights into internal magnetic fields and crust-core transitions.

Magneto-elastic dynamics simulations refer to the numerical modeling and analysis of coupled phenomena involving magnetic fields and elastic deformations in materials. This class of simulations is pivotal in understanding the internal oscillatory dynamics of highly magnetized astrophysical objects, notably neutron stars ("magnetars"), where strong magnetic fields and elastic crustal properties interact, and also in terrestrial materials science, where magneto-elastic coupling dictates advanced functional behavior. The focus here is on the magneto-elastic oscillations of magnetars, as treated in general-relativistic frameworks (Gabler et al., 2010).

1. Numerical Formulation and Physical Ingredients

The simulation framework in (Gabler et al., 2010) extends a two-dimensional, general-relativistic, ideal magneto-hydrodynamical (GRMHD) code to incorporate elasticity in the neutron star crust. The foundational dynamical variable is the total stress–energy tensor,

$$T^{\mu\nu} = T^{\mu\nu}_{\text{fluid}} + T^{\mu\nu}_{\text{mag}} + T^{\mu\nu}_{\text{elas}}$$

where

$$T^{\mu\nu}_{\text{fluid}+\text{mag}} = \rho h u^\mu u^\nu + P g^{\mu\nu} + b^2 u^\mu u^\nu + \frac{1}{2}b^2 g^{\mu\nu} - b^\mu b^\nu, \ T^{\mu\nu}_{\text{elas}} = -2 \mu_S \Sigma^{\mu\nu}$$

with $\rho$ the rest-mass density, $h$ specific enthalpy, $P$ pressure, $u^\mu$ four-velocity, $b^\mu$ magnetic field in the fluid frame, $\mu_S$ the shear modulus, and $\Sigma^{\mu\nu}$ the elastic shear tensor.

The conservation law is cast as

$$\frac{\partial(\sqrt{-g} U)}{\partial t} + \frac{\partial(\sqrt{-g} F^i)}{\partial x^i} = 0$$

solved numerically via a Riemann-solver-based finite-volume method, with state vector $U = [S_\phi, B^\phi]$. Fluxes in the $r$ direction, for example, are given by: $$F^r = [- (b_\phi B^r)/W - 2\mu_S \Sigma^r_\phi,\; -v^\phi B^r]$$ Boundary conditions impose $\xi^\phi_{,r} = 0$ at the stellar surface and, at the crust–core interface, continuity of traction and parallel electric field, such that

$$\xi^\phi_{\text{core}, r} = (1 + \delta) \xi^\phi_{\text{crust}, r},\;\; \delta = \mu_S/(b_r b^r)$$

The background neutron star is modeled using the LORENE library with microphysical equations of state, assuming axisymmetry, the Cowling approximation, and a conformally flat metric.

2. Magneto-Elastic Eigenmode Structure

In the regime without a magnetic field, the simulations accurately capture pure crustal shear oscillation frequencies to within a few percent of eigenmode calculations, confirming the elasticity implementation. Introduction of a magnetic field fundamentally alters the spectrum: the Alfvén waves in the neutron star core couple to the elastic crustal modes, yielding hybrid "magneto-elastic" oscillations. Each magnetic field line supports independent oscillations, and, in the short-wavelength limit (with standing waves along field lines), the mode frequencies obey

$f = \kappa / (2\pi t_{\text{tot}})$

where $t_{\text{tot}}$ is twice the Alfvén wave travel time along the field line, and $\kappa$ denotes the mode's harmonic. The propagation speed in the crust includes both Alfvén and shear velocities: $\frac{dx}{dt} = \sqrt{v_a^2(x) + v_s^2(x)}$ This encapsulates the intertwined character of the magnetic and elastic responses in the resulting spectrum.

3. Influence of Magnetic Field Strength

The dynamical regime is strongly controlled by the dipole magnetic field strength:

• Weak-field regime ($B < 5 \times 10^{13}$ G): Oscillations are dominated by pure crustal shear modes.
• Intermediate field ($5 \times 10^{13} \lesssim B \lesssim 10^{15}$ G): Magneto-elastic interaction is significant. Oscillations are largely core-confined, and crustal amplitudes are suppressed by efficient magnetic damping.
• Strong-field regime ($B \gtrsim 10^{15}$ G): Magnetic pressure dominates; oscillations resemble those absent an elastic crust.

Changes in $B$ alter Alfvén velocities and thus the standing-wave travel times, modifying both modal frequencies and spatial structures. In the intermediate regime, the coupling damps pure shear oscillations and replaces them with magneto-elastic waves that are distinct in character.

4. Quasi-Periodic Oscillation (QPO) Families and Spectral Localization

Three families of quasi-periodic oscillations (QPOs) emerge in the presence of a strong crust–core coupling:

• Lower QPOs ($L_n^{(\pm)}$): Localized near the equator, these correspond to closed field lines within the core. Identified as turning-point QPOs due to their association with extrema in the Alfvén continuous spectrum (i.e., where the local frequency has a stationary point along the sequence of field lines). The crustal elasticity minimally affects these modes.
• Upper QPOs ($U_n^{(\pm)}$): In models with a crust, fundamental upper QPOs shift from the polar axis (as seen in crustless models) to off-polar latitudes and acquire even parity, with vanishing amplitude in the crust.
• Edge QPOs ($E_n^{(+)}$): Represent a new spectral family attached to field lines at the boundary between those closed within the core and those extending to the exterior. These are "edge" modes linked with discontinuities in the continuum spectrum.

This tri-family structure is directly traceable to spectral features (turning points, continuum edges) in the semi-analytic model of the magneto-elastic continuum, and is a robust prediction for strongly magnetized neutron star interiors.

5. Astrophysical Implications and Observational Connections

Strong magneto-elastic coupling in the $5 \times 10^{13}$–$10^{15}$ G field range efficiently damps pure crust shear modes, constructing a spectrum dominated by hybrid core–crust modes. In this regime:

• The identified QPO families (Lower, Upper, Edge) align with observed quasi-periodic modulation signatures in the aftermath of SGR giant flares, notably for frequency ratios near low-integer values (e.g., 30:92:150 Hz for SGR 1806–20).
• The spectrum's dependence on boundary physics and elastic properties underscores the potential of QPOs as probes of neutron star magnetic topology and dense matter microphysics.
• Strong sensitivity of oscillatory phenomenology to $B$ suggests that QPO measurements may constrain both the internal $B$-field strength and crustal equation of state.

A plausible implication is that the combined presence of these QPO families and their distinctive spectral localization offers a diagnostic of internal magnetic field strength and the nature of the crust–core transition, informing the equation of state for neutron star matter.

6. Summary

Magneto-elastic dynamics simulations, as realized in general-relativistic codes extended for elasticity (Gabler et al., 2010), exhibit a rich interplay between crustal shear and core Alfvén modes. The emergence of three robust QPO families—distinguished by spatial localization and spectral features—enables a direct mapping between measurable oscillation frequencies and the internal structure of strongly magnetized neutron stars. These results motivate both refined models including additional microphysics and continued observational campaigns aimed at QPO detection in SGR activities.