Asteroseismology of vibration powered neutron stars

Abstract

There is a general recognition today that basic features of asteroseismology of non-convective finalstage (FS) stars, such as white dwarfs, neutron stars and and strange quark star, can be properlyunderstood working from the model of vibrating solid star, rather than the liquid star model lyingat the base of asteroseismology of convective main-sequence (MS) stars. In accord with this, mostof current investigations of an FS-star vibrations rests on principles of solid-mechanical theory ofcontinuous media, contrary to the study of the MS-star vibrations which are treated in terms of fluid-mechanical theory. This means that super dense matter of FS-stars (whose gravitational pressureis counterbalanced by degeneracy pressure of constituting Fermi-matter), possesses elasticity andviscosity generic to solid state of condensed matter, whereas a fairly dilute matter of the MS (whoseinternal pressure of self-gravity is opposed by radiative pressure) possesses property of fluidity whichis generic to the liquid state of a highly conducting condensed matter. This feature of the MS-starmatter plays crucial role in generation of their magnetic fields in the dynamo processes involvingmacroscopic flows which are supported by energy of nuclear reactions in the central, reactive zone,of these stars. In the meantime, in the FS stars, like white dwarfs and neutron stars, there are nonuclear energy sources to support convection. The prevailing today view, therefore, is that a highlystable to spontaneous decay dipolar magnetic fields of neutron stars are fossil. The extremely largeintensity of magnetic fields of degenerate solid stars is attributed to amplification of fossil magneticfield in the magnetic-flux-conserving process of core-collapse supernova.Even still before discovery of neutron stars, it has been realized that, for absence of nuclearsources of energy, the radiative activity of these of compact objects should be powered by energy ofstored in either rotation or vibrations and that the key role in maintaining the neutron star radiationshould play an ultra strong magnetic field. As is commonly know today, the neutron star capabilityof accommodating such a field is central to understanding pulsating character of magneto-dipoleradiation of radio pulsars whose radiative power is provided, as is commonly believed, by the energyof rigid-body rotation. The discovery of soft gamma-ray repeaters and their identification withmagnetars [1] – quaking neutron stars endowed with ultra strong magnetic fields experiencing decay– has stimulated enhanced interest in the study of models of quake-induced magneto-mechanicalseismic vibrations of neutron star and resultant electromagnetic radiation. Of particular interestin this study are torsional magneto-mechanical vibrations about axis of magnetic dipole momentof the star driven by forces of magnetic-field-dependent stresses in a perfectly conducting matterand in a permanently magnetized non-conducting matter [2]. Most, if not all, reported up to nowcomputations of frequency spectra of poloidal and toroidal Alfv´en vibration modes in pulsars andmagnetars rest on tacitly adopted assumption about constant-in-time magnetic field in which aperfectly conducting neutron star matter undergoes Lorentz-force-driven oscillations [3, 4, 5, 6, 7,8, 9, 10, 11, 12] (see, also, references therein). A special place in the study of the above Alfv´enmodes of pure shear magneto-mechanical seismic vibrations (a-modes) occupies a homogeneousmodel of a solid star with the uniform density ρ and frozen-in poloidal static magnetic field of bothhomogeneous and inhomogeneous internal and dipolar external configuration, and we start section2 with a brief outline of this model. In section 3, a mathematical background of quaking neutronstar model is briefly outlined with emphasis on the loss of vibration energy caused by depletion ofinternal magnetic field pressure and resulting vibration-energy powered magneto-dipole radiation.The decreasing of magnetic field pressure in the star is presumed to be caused by coupling betweenvibrating star and outgoing material which is expelled by quake, but mechanisms of star-envelopeinteraction resulting in the decay of magnetic field, during the time of vibrational relaxation, are2