Torsional oscillations of neutron stars*

Motivated by the possibility that torsional oscillations of neutron stars may be observable in the timing of pulsar subpulses and/or in future gravitational-wave detectors, this paper develops the detailed mathematical theory of such torsional oscillations and of the gravitational waves they emit. The oscillations are analysed using the formulation of first-order perturbations of a fully general relativistic spherical stellar model. All sources of damping are ignored except gravitational radiation reaction. The perturbations are resolved into spherical harmonics, which decouple from each other. For each harmonic this paper presents equations of motion, an action principle, an energy conservation law and a Liapunov-type proof that the oscillations are always stable. Each harmonic is then resolved into normal modes with outgoing gravitational waves (time dependence |$e^{i\omega t}$| with ω complex) and an eigenvalue problem is posed for the eigenfunctions and the eigenfrequencies ω. Five methods of solving the eigenvalue problem are presented; three methods are valid in general (the method of resonances, the variational method and the method of energy conservation); one is valid in the slow-motion approximation (wavelength of waves large compared to star) and one is valid in the weak-gravity approximation. For stellar models with weak gravity and with radially constant density and shear modulus the eigenvalue problem is solved analytically. An appendix develops a general theory of action principles for systems with radiative boundary conditions – a theory which is then used to derive the action principles in the body of the paper and which could be useful for a variety of other problems involving physical systems coupled to radiation.

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