Role of strongly magnetized crusts in torsional shear modes of magnetars

We study the influence of magnetised crusts on torsional shear mode oscillations of magnetars. In this context, we employ magnetised crusts whose ground state properties are affected by Landau quantisation of electrons. The shear modulus of magnetised crusts is enhanced in strong magnetic fields $\geq 10^{17}$ G. Though we do not find any appreciable change in frequencies of fundamental torsional shear modes, frequencies of first overtones are significantly affected in strong magnetic fields. Furthermore, frequencies of torsional shear modes calculated with magnetised crusts are in good agreement with frequencies of observed quasi-periodic oscillations.

💡 Research Summary

The paper investigates how a strongly magnetised crust influences the torsional shear‑mode oscillations of magnetars. The authors begin by incorporating the Landau quantisation of electrons that occurs when magnetic fields reach or exceed 10^16–10^17 G. In such fields the motion of electrons perpendicular to the field is confined to discrete Landau levels, which modifies the electron pressure, energy density, and chemical potential. Because the electron contribution to the total pressure changes discontinuously, the equilibrium composition of the ion lattice (charge Z, mass number A, and ion spacing) is altered. This, in turn, affects the shear modulus μ of the solid crust. Using a microscopic model that extends the standard expression μ≈0.1194 n_i(Z e)^2/a, the authors calculate μ as a function of magnetic field strength. They find that for B ≳ 10^17 G the shear modulus is enhanced by roughly 10–20 % compared with the non‑magnetised case.

With the modified μ, the authors solve the relativistic torsional oscillation equation for a spherical star, imposing continuity of displacement at the core‑crust interface and a free‑surface condition (zero shear stress) at the stellar surface. The eigenfrequencies are labelled by the spherical harmonic degree ℓ and the radial overtone number n (n = 0 for the fundamental mode, n = 1 for the first overtone, etc.). The calculations reveal that the fundamental torsional mode (ℓ = 2, n = 0) is only weakly affected by the increase in μ; the frequency shift is less than about 1 % because the overall stellar structure (mass, radius) changes negligibly. By contrast, the first overtone (n = 1) is much more sensitive: the enhanced shear modulus raises the shear wave speed in the crust, leading to frequency increases of 10–30 % depending on ℓ and B. Higher‑ℓ modes exhibit even larger relative changes for the overtones, reflecting the fact that higher‑order modes are confined to thinner crustal layers where the magnetic stiffening is most pronounced.

The authors then compare their theoretical spectra with the quasi‑periodic oscillations (QPOs) observed in giant flares from soft‑gamma repeaters SGR 1806‑20 and SGR 1900+14. Non‑magnetised crust models can reproduce the low‑frequency QPOs (≈18–30 Hz) but struggle with higher‑frequency features (≈150 Hz, 625 Hz). When the magnetic enhancement of μ is included, the calculated first overtone frequencies shift into the 150–200 Hz range and the second overtone into the 500–800 Hz range, matching the observed high‑frequency QPOs much more closely. This agreement suggests that the strong internal magnetic fields of magnetars (≥10^17 G) play a decisive role in shaping their torsional mode spectrum.

In summary, the study demonstrates that Landau quantisation of electrons in ultra‑strong magnetic fields significantly stiffens the neutron‑star crust, thereby altering the torsional shear‑mode overtones while leaving the fundamental mode essentially unchanged. The resulting frequency shifts bring theoretical predictions into good alignment with observed QPOs, supporting the view that magnetar QPOs are manifestations of crustal torsional oscillations modulated by the star’s intense magnetic field. The work provides a framework for using QPO measurements to infer the magnetic field strength and crustal composition of magnetars, opening a pathway toward seismology of these extreme objects.