[Paper Review] Torsional Oscillations of Relativistic Stars with Dipole Magnetic Fields II. Global Alfvén Modes

Hajime Sotani, Kostas D. Kokkotas+2 more|arXiv (Cornell University)|Nov 21, 2006

Pulsars and Gravitational Waves Research3 citations

TL;DR

This paper investigates global torsional Alfvén modes in relativistic stars with dipole magnetic fields, proposing they explain quasi-periodic oscillations (QPOs) in soft gamma repeaters (SGRs) better than crustal modes. Using general relativistic ideal MHD simulations, it finds excellent agreement between computed Alfvén mode frequencies and observed SGR QPOs for magnetic fields of (0.8–1.2)×10¹⁶ G, with avoided crossings preventing pure crustal mode identification at high B-fields.

ABSTRACT

We investigate torsional Alfvén modes of relativistic stars with a global dipole magnetic field. It has been noted recently (Glampedakis et al. 2006) that such oscillation modes could serve as as an alternative explanation (in contrast to torsional crustal modes) for the SGR phenomenon, if the magnetic field is not confined to the crust. We compute global Alfvén modes for a representative sample of equations of state and magnetar masses, in the ideal MHD approximation and ignoring $\ell \pm 2$ terms in the eigenfunction. We find that the presence of a realistic crust has a negligible effect on Alfvén modes for $B > 4 imes 10^{15}$ G. Furthermore, we find strong avoided crossings between torsional Alfvén modes and torsional crust modes. For magnetar-like magnetic field strengths, the spacing between consecutive Alfvén modes is of the same order as the gap of avoided crossings. As a result, it is not possible to identify modes of predominantly crustal character and all oscillations are predominantly Alfvén-like. Interestingly, we find excellent agreement between our computed frequencies and observed frequencies in two SGRs, for a maximum magnetic field strenght in the range of (0.8--1.2)$ imes 10^{16}$ G.

Motivation & Objective

• To investigate whether global torsional Alfvén modes in relativistic stars with dipole magnetic fields can explain the observed quasi-periodic oscillations (QPOs) in soft gamma repeaters (SGRs).
• To determine if Alfvén modes can resolve inconsistencies in the crustal torsional mode model, particularly the small frequency spacing between observed QPOs like 26 Hz and 29 Hz in SGR 1806-20.
• To assess the impact of the stellar crust on Alfvén mode frequencies, especially at high magnetic fields.
• To compare computed Alfvén mode frequencies with observed QPOs in SGR 1806-20 and SGR 1900+14 to infer magnetic field strengths and stellar properties.
• To derive an empirical formula for the fundamental Alfvén mode frequency as a function of magnetic field and compactness for future inference applications.

Proposed method

• Solves the ideal magnetohydrodynamics (MHD) equations in a general relativistic framework using the Tolman-Oppenheimer-Volkoff (TOV) equilibrium model for nonrotating stars.
• Models the star with a global dipole magnetic field using the vector potential formalism, assuming the magnetic field does not perturb the equilibrium configuration.
• Ignores ℓ±2 coupling terms in the eigenfunction to simplify the eigenvalue problem, focusing on pure torsional Alfvén modes.
• Performs numerical computation of global Alfvén mode frequencies for multiple equations of state and magnetar masses, including models with realistic crusts.
• Analyzes avoided level crossings between Alfvén and crustal torsional modes to assess mode hybridization at high magnetic fields.
• Compares computed mode frequencies with observed QPOs in SGR 1806-20 (18–1837 Hz) and SGR 1900+14 (28–155 Hz) to determine best-fit magnetic field strengths.

Experimental results

Research questions

• RQ1 Can global torsional Alfvén modes in magnetized relativistic stars explain the observed QPOs in SGRs more consistently than crustal torsional modes?
• RQ2 What is the effect of the stellar crust on Alfvén mode frequencies, particularly at magnetic fields above 4×10¹⁵ G?
• RQ3 How do avoided crossings between Alfvén and crustal modes influence the identification of oscillation modes in high-B stars?
• RQ4 What magnetic field strength is required to reproduce the observed QPO frequencies in SGR 1806-20 and SGR 1900+14 using Alfvén mode models?
• RQ5 Can an empirical formula be derived to relate the fundamental Alfvén mode frequency to stellar compactness and magnetic field strength?

Key findings

• The presence of a realistic crust has a negligible effect on Alfvén mode frequencies for magnetic fields above 4×10¹⁵ G.
• Strong avoided crossings between torsional Alfvén modes and crustal modes occur at magnetar-like field strengths, preventing the existence of purely crustal modes.
• The spacing between consecutive Alfvén modes is comparable to the avoided crossing gap, leading to hybridized, Alfvén-like modes that dominate the spectrum.
• Excellent agreement is found between computed Alfvén mode frequencies and observed QPOs in SGR 1806-20 and SGR 1900+14, with best-fit magnetic field strengths of (0.8–1.2)×10¹⁶ G.
• The fundamental Alfvén mode frequency for ℓ=2 is well described by an empirical formula depending on magnetic field strength and stellar compactness, enabling future inference of stellar parameters.
• For SGR 1806-20, the model reproduces all observed QPOs up to 150 Hz with a magnetic field of B/Bₘᵤ ≈ 1.25; for SGR 1900+14, the fit is excellent at B/Bₘᵤ ≈ 1.94.