The past three decades of investigation on nuclear physics and pulsar astrophysics have seen gradual recognition that elastodynamic approach to the continuum mechanics of nuclear matter provides proper account of macroscopic motions of degenerate Fermi-matter constituting interior of the nuclear material objects, the densest of all known today. This paper focuses on one theoretical issue of this development which is concerned with oscillatory behavior of a viscoelastic solid globe in the regime of quasistatic, force-free, non-compressional oscillations less inves- Keywords: Viscoelastic solid globe models, spheroidal and torsional modes of quasistatic shear vibrations; nuclear giant resonances; non-radial pulsations of neutron stars.

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We present several exact solutions of the eigenfrequency problem for torsional shear vibrations in homogeneous and non-homogeneous models of the neutron star crust governed by the canonical equation of solid mechanics with a restoring force of Hookean elasticity. Particular attention is given to the regime of large-lengthscale nodeless axisymmetric differentially rotational oscillations, which are treated in spherical polar coordinates, reflecting the real geometry of the neutron star crust. We highlight the distinction between analytic solutions, computed as a function of the multipole degree of nodeless torsional oscillations and fractional depth of the crust, caused by different boundary conditions imposed on the toroidal field of material displacements. The relevance of the considered models to quasiperiodic oscillations, recently detected during the flare of SGR 1806−20 and SGR 1900+14, is discussed.

Within the framework of Newtonian magneto-solid mechanics, relying on equations appropriate for a perfectly conducting elastic continuous medium threaded by a uniform magnetic field, the asteroseismic model of a neutron star undergoing axisymmetric global torsional nodeless vibrations under the combined action of Hooke's elastic and Lorentz magnetic forces is considered with emphasis on a toroidal Alfvén mode of differentially rotational vibrations about the dipole magnetic-moment axis of the star. The obtained spectral equation for frequency is applied to -pole identification of quasi-periodic oscillations (QPOs) of the X-ray flux during the giant flares of SGR 1806-20 and SGR 1900+14. Our calculations suggest that detected QPOs can consistently be interpreted, within the framework of this model, as being produced by global torsional nodeless vibrations of a quaking magnetar if they are considered to be restored by the joint action of bulk forces of shear elastic and magnetic field stresses.

If quark stars exist, they may be enveloped in thin electron layers (electron seas), which uniformly surround the entire star. These layers will be affected by the magnetic fields of quark stars in such a way that the electron seas would transmit hydromagnetic cyclotron waves, as studied in this paper. Particular attention is devoted to vortex hydrodynamical oscillations of the electron sea. The frequency spectrum of these oscillations is derived in analytic form. If the thermal X-ray spectra of quark stars are modulated by vortex hydrodynamical vibrations, the thermal spectra of compact stars, foremost central compact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence of these vibrational modes observationally. The central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the hydro-cyclotron oscillation model.

A neutron star is the cosmic nuclear object in which the energy of gravitational pull is brought to equilibrium by elastic energy stored in the neutron Fermi-continuum. Evidence for the viscoelastic behaviour of a stellar nuclear matter provides a seismological model of pulsar glitches interpreted as a sudden release of the elastic energy. In laboratory nuclear physics, the signatures of viscoelasticity of nuclear matter are found in the current investigations on the collective nuclear dynamics, in which a heavy nucleus is modelled by a spherical piece of viscoelastic Fermicontinuum compressed to the normal nuclear density. It is plausible to expect, therefore, that the motions of self-gravitating nuclear matter constituting the interior of neutron stars should be governed by the equations of an elastic solid, rather than by hydrodynamic equations describing the behaviour of gaseous plasma inside the main sequence stars. In this paper, we present arguments that elastodynamic equations, originally introduced in the context of nuclear collective dynamics, can provide a proper account of elasticity in the large scale motions of neutron matter under its own gravity. Emphasis is placed on mathematical physics underlying the constructive description of the continuum mechanics and the rheology of macroscopic nuclear matter. The capability of the elastodynamic approach is examined by analysis of oscillatory dynamics of a neutron star in the standard homogenous model, operating with a spherical mass of self-gravitating degenerate neutron matter whose viscoelastic behaviour is described in terms of the spheroidal and torsional gravitational-elastic eigenmodes, inherenly related to viscoelasticy. The energy variational principle is utilized to compute the frequencies of viscoelastic gravitational pulsations and their relaxation time. The method is demonstrated for both the idealized homogeneous model and the neutron star models constructed on realistic equations of state. Finally, we derive analytic conditions for the stability of a neutron star to linear elastic deformations accompanying the nonradial pulsations, and discuss the fingerprints of these pulsations in the electromagnetic activity of radiopulsars.

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