Non-axisymmetric low frequency oscillations of rotating and magnetized neutron stars

We investigate non-axisymmetric low frequency modes of a rotating and magnetized neutron star, assuming that the star is threaded by a dipole magnetic field whose strength at the stellar surface, $B_0$, is less than $\sim 10^{12}$G, and whose magnetic axis is aligned with the rotation axis. For modal analysis, we use a neutron star model composed of a fluid ocean, a solid crust, and a fluid core, where we treat the core as being non-magnetic assuming that the magnetic pressure is much smaller than the gas pressure in the core. Here, we are interested in low frequency modes of a rotating and magnetized neutron star whose oscillation frequencies are similar to those of toroidal crust modes of low spherical harmonic degree and low radial order. For a magnetic field of $B_0\sim 10^7$G, we find Alfv'en waves in the ocean have similar frequencies to the toroidal crust modes, and we find no $r$-modes confined in the ocean for this strength of the field. We calculate the toroidal crustal modes, the interfacial modes peaking at the crust/core interface, and the core inertial modes and $r$-modes, and all these modes are found to be insensitive to the magnetic field of strength $B_0\ltsim10^{12}$G. We find the displacement vector of the core $l^\prime=|m|$ $r$-modes have large amplitudes around the rotation axis at the stellar surface even in the presence of a surface magnetic field $B_0\sim10^{10}$G, where $l^\prime$ and $m$ are the spherical harmonic degree and the azimuthal wave number of the $r$-modes, respectively. We suggest that millisecond X-ray variations of accretion powered X-ray millisecond pulsars can be used as a probe into the core $r$-modes destabilized by gravitational wave radiation.

💡 Research Summary

The paper investigates non‑axisymmetric low‑frequency oscillations in rotating, magnetized neutron stars whose surface dipole magnetic field strength lies between roughly 10⁷ G and 10¹² G. The authors adopt a three‑layer stellar model consisting of a fluid ocean, a solid crust, and a fluid core. The core is treated as non‑magnetic because the magnetic pressure is assumed to be far smaller than the gas pressure throughout the core. The magnetic axis is aligned with the rotation axis, simplifying the modal analysis for azimuthal numbers m ≠ 0.

The study focuses on modes whose frequencies are comparable to those of low‑degree, low‑radial‑order toroidal crustal modes. For a surface field of B₀ ≈ 10⁷ G, the authors find that Alfvén waves can propagate in the ocean with frequencies that match the toroidal crustal modes. In this weak‑field regime, no r‑modes are confined to the ocean, indicating that the magnetic tension suppresses oceanic r‑mode propagation.

The authors then compute toroidal crustal modes, interfacial modes localized at the crust‑core boundary, and core inertial modes including the classic r‑modes. All of these modes remain essentially insensitive to magnetic fields up to B₀ ≈ 10¹² G. A particularly striking result concerns the core r‑modes with spherical‑harmonic degree l′ = |m|. Even when a surface field as strong as B₀ ≈ 10¹⁰ G is present, the displacement eigenfunctions of these r‑modes exhibit large amplitudes near the rotation axis and extend to the stellar surface. This behavior reflects the dominance of the Coriolis restoring force over magnetic forces in the deep interior.

The paper argues that such core r‑modes can become unstable through the emission of gravitational radiation (the Chandrasekhar‑Friedman‑Schutz instability). When destabilized, they would modulate the X‑ray flux of accretion‑powered millisecond pulsars at frequencies close to the stellar spin frequency, producing millisecond‑scale variability. Consequently, high‑time‑resolution X‑ray observations of these pulsars could serve as a diagnostic tool for probing the existence and properties of core r‑modes, offering indirect evidence of gravitational‑wave‑driven instabilities in neutron stars. This proposal opens a new observational window onto the internal dynamics of rapidly rotating, weakly magnetized neutron stars.