Oscillations of highly magnetized non-rotating neutron stars

Neutron stars (NSs), remnants of core-collapse supernovae, often possess strong magnetic fields, reaching 10^14-16 G in magnetars. These highly magnetized NSs are potential sources for phenomena like fast radio bursts, gamma-ray bursts, and superluminous supernovae, and their pulsations are speculated to produce detectable gravitational waves. Studying these pulsations is crucial for understanding NS structure and equations of state. However, consistently solving the nonlinear Einstein and Maxwell equations for highly magnetized NSs presents a significant computational challenge. Recent advances in numerical solvers now allow for more detailed investigations. This work focuses on the pulsation modes of non-rotating NSs with strong purely toroidal magnetic fields, utilizing two-dimensional axisymmetric general-relativistic magnetohydrodynamics (GRMHD) simulations to explore the influence of extreme magnetic fields on stellar oscillations with minimal assumptions. Previous studies using Newtonian approaches or the Cowling approximation (neglecting spacetime evolution) have provided insights, but the limitations of these approaches necessitate full GRMHD simulations for accurate eigenfrequency determination. The study uses the XNS code for constructing equilibrium models and the Gmunu code for dynamic spacetime evolution, enabling a systematic investigation of oscillation modes as a function of magnetic field strength.

The literature extensively explores the oscillations of non-magnetized neutron stars using perturbative calculations and dynamical simulations, with and without spacetime evolution. Studies incorporating magnetic fields have employed Newtonian or general-relativistic approaches with the Cowling approximation. However, the Cowling approximation has been shown to overestimate oscillation frequencies. Previous work has explored equilibrium models with simple field configurations (purely toroidal or poloidal), but these are often unstable. Simulations suggest that NS magnetic fields rapidly rearrange into a mixed, roughly axisymmetric configuration known as a "twisted torus." While some studies using GRMHD simulations have begun to address the dynamics of magnetized NSs, accurate eigenfrequency determination for highly magnetized NSs remains a challenge. This research leverages recent advancements in GRMHD simulation codes (XNS and Gmunu) to address this gap.

The study employs a two-pronged approach using two distinct codes: XNS and Gmunu. First, twelve equilibrium models of non-rotating, axisymmetric NSs were constructed using the XNS code, encompassing a range of magnetic-to-binding energy ratios (H/W). This included one non-magnetized reference model and eleven magnetized models with increasing H/W ratios, ranging from near zero to approximately 0.34. All models shared a similar mass and polytropic equation of state. Next, the Gmunu code was used to dynamically evolve these equilibrium models in a full GRMHD simulation, introducing three different initial fluid perturbations (ℓ = 0, 2, and 4) to excite stellar oscillations. The simulations were conducted for a time span of 10 ms, maintaining consistency with the polytropic equation of state used in the equilibrium model construction. A key aspect of the methodology involves careful selection of simulation parameters such as atmospheric density and grid refinement. Adaptive mesh refinement (AMR) was employed to enhance resolution near the stellar surface. Subsequently, a Fourier analysis was performed on the simulation data. Fast Fourier transforms (FFTs) were used to extract the eigenfrequencies of the excited oscillation modes. Parabolic interpolation of prominent FFT peaks provided a measurement of the eigenfrequencies and their uncertainties. The spatial distribution of FFT amplitudes at the eigenfrequencies were used to construct eigenfunctions. This process allows for identification of oscillation modes across simulations and analysis of the eigenfrequency dependence on the magnetic-to-binding energy ratio. The use of parabolic interpolation simplifies the process in this ideal GRMHD simulation context where physical damping of oscillations is absent.

The simulations revealed six dominant oscillation modes: the fundamental quasi-radial (ℓ = 0) mode F and its first overtone H₁; the fundamental quadrupole (ℓ = 2) mode 2f and its first overtone 2p₁; and the fundamental hexadecapole (ℓ = 4) mode 4f and its first overtone 4p₁. The study identified a threshold in the magnetic-to-binding energy ratio (H/W) for magnetization effects on stellar oscillations. For H/W ≤ 10^-2, oscillations remained largely insensitive to magnetization, even with maximum magnetic field strengths reaching ~10¹⁷ G. However, for H/W ≥ 10^-1, a significant suppression of oscillation frequencies was observed. This suppression was particularly noticeable in the higher-order ℓ = 4 modes, which were sometimes masked by the ℓ = 2 modes in the most magnetized models. Further analysis linked this frequency suppression to a decrease in stellar compactness. The compactness (M/R_circ, where M is the gravitational mass and R_circ is the circumferential radius) remained nearly constant for H/W ≤ 10^-2 but decreased significantly for H/W > 10^-1. A strong quasilinear correlation was found between the eigenfrequencies of the oscillation modes and the stellar compactness. This correlation is consistent with previous findings for non-magnetized NSs. The results indicate that a strong toroidal field causes significant NS deformation, impacting the propagation of seismic waves within the star and consequently altering the oscillation mode frequencies.

The findings demonstrate that the impact of strong magnetic fields on neutron star oscillations is not simply a minor perturbation. While Newtonian studies and those using the Cowling approximation predicted minor frequency shifts approximately proportional to B², the full GRMHD simulations reveal a much more substantial effect for H/W ≥ 10^-1. The strong toroidal magnetic field causes significant deformation and a consequent decrease in compactness, leading to the suppression of oscillation modes. This effect becomes dominant when the magnetic field's influence on the stellar structure is no longer a small correction. The observed quasilinear relationship between eigenfrequency and compactness provides a powerful tool for understanding the influence of magnetic fields on NS oscillations. The results have implications for gravitational wave detection, as the excited oscillation modes are potential sources of gravitational waves detectable by next-generation detectors like KAGRA, ET, and NEMO. The use of purely toroidal fields in this study, while simplifying the calculations, is a limitation because such configurations are generally unstable in reality. Future studies should investigate more realistic field configurations, such as purely poloidal fields or the mixed "twisted torus" configuration, and should also incorporate rotation.

This study provides the first systematic investigation of the effects of strong purely toroidal magnetic fields on the oscillations of non-rotating neutron stars using full GRMHD simulations. The key finding is the identification of a threshold in the magnetic-to-binding energy ratio beyond which oscillation mode frequencies are significantly suppressed, a phenomenon linked to a reduction in stellar compactness. This work highlights the limitations of previous analytical and approximate approaches and paves the way for future studies investigating more realistic magnetic field configurations and incorporating rotation to enhance the model's realism.

The simulations were limited to two dimensions and employed purely toroidal magnetic fields, which are known to be unstable in three dimensions. The absence of rotation also limits the model's realism. While the use of a polytropic equation of state simplifies the computations, it is not a perfect representation of the true nuclear equation of state in neutron stars. The initial perturbation amplitudes were relatively small, potentially missing some less prominent oscillation modes. Future research would benefit from extending the model to three dimensions, considering more realistic magnetic field configurations (including poloidal and mixed fields), incorporating rotation, and using more accurate equations of state.