Abstract:
We formulate deformation of relativistic stars due to the magnetic stress, considering the magnetic fields to be perturbations from spherical stars. The ellipticity for the dipole magnetic field is calculated for some stellar models. We have found that the ellipticity becomes large with increase of a relativistic factor for the models with the same energy ratio of the magnetic energy to the gravitational energy.

Abstract:
The purpose of these notes is to give a brief review of superfluidity in neutron stars. After a short presentation explaining why and how superfluidity is expected in the crust and core of neutron stars, consequences on thermal evolution and rotational dynamics are discussed. The second part summarizes a formalism that has been recently developped to describe the hydrodynamics of superfluids or superconductors in the framework of general relativity. As an application, one can compute the oscillations of a two-component relativistic neutron star.

Abstract:
Now the hypothesis of existence of scalar fields of a various nature and energy density in the modern Universe is intensively explored. It can explain a nature of the dark (non- baryon) matter in the Universe and an existence of positive $\Lambda $-term (see e.g. gr-qc/9904398). One of component of such field has a cluster nature and organizes in the closed gravitational configurations from galactic scales up to relativistic microscopic stars. In the authors paper astro-ph/0008334 the hypothesis of detonation of such fields was considered. As a result of phase transition behind the wavefront a relativistic plasma of high energy density can appear. This process is similar to a relativistic detonation and it can create macroscopic fireballs sufficient for an explanation of the phenomenon of gamma-ray bursts (see e.g. the review astro-ph/9907392). In astro-ph/0008334 it was supposed that the front of such ''detonation'' wave is entered by the flow of scalar fields with constant energy density. If the size of the formed plasma configuration is commensurable with the size of scalaron cluster, this hypothesis is not correct. It is necessary to take into account a modification of the energy density of the scalar field from centre to a periphery. It is changes the dynamics of the fireball on principle. The indicated problem in framework of special relativity is considered in this paper.

Abstract:
Models for the internal composition of Dense Compact Stars are reviewed as well as macroscopic properties derived by observations of relativistic processes. Modeling of pure neutron matter Neutron Stars is presented and crust properties are studied by means of a two fluid model.

Abstract:
We analyze damping of oscillations of general relativistic superfluid neutron stars. To this aim we extend the method of decoupling of superfluid and normal oscillation modes first suggested in [Gusakov & Kantor PRD 83, 081304(R) (2011)]. All calculations are made self-consistently within the finite temperature superfluid hydrodynamics. The general analytic formulas are derived for damping times due to the shear and bulk viscosities. These formulas describe both normal and superfluid neutron stars and are valid for oscillation modes of arbitrary multipolarity. We show that: (i) use of the ordinary one-fluid hydrodynamics is a good approximation, for most of the stellar temperatures, if one is interested in calculation of the damping times of normal f-modes; (ii) for radial and p-modes such an approximation is poor; (iii) the temperature dependence of damping times undergoes a set of rapid changes associated with resonance coupling of neighboring oscillation modes. The latter effect can substantially accelerate viscous damping of normal modes in certain stages of neutron-star thermal evolution.

Abstract:
Neutrino asymmetry in general relativistic radiative spacetime exterior to spinning stars is investigating by making use of Newmann-Penrose (NP) spin coefficient formalism. It is shown that neutrino current depends on the direction of rotation of the star. The solution is obtained in test field approximation where the neutrinos do not generate gravitational fields.

Abstract:
We have calculated the non-radial oscillation in slowly rotating relativistic stars with the Cowling approximation. The frequencies are compared with those based on the complete linearized equations of general relativity. It is found that the results with the approximation differ by less than about $20 %$ for typical relativistic stellar models. The approximation is more accurate for higher-order modes as in the Newtonian case.

Abstract:
We study structures of general relativistic compact stars with exotic matter. Our study is based on axisymetric and stationary formalism including purely toroidal magnetic field. We also study the finite size effects of quark-hadron mixed phase on structures of magnetars. For hybrid stars, we find a characteristic distribution of magnetic field, which has a discontinuity originated in the quark-hadron mixed phase. These distributions of magnetic field will change astrophysical phenomena, such as cooling processes.

Abstract:
We analyze the stability of relativistic, quasi-equilibrium binary neutron stars in synchronous circular orbit. We explore stability against radial collapse to black holes prior to merger, and against orbital plunge. We apply theorems based on turning points along uniformly rotating sequences of constant angular momentum and rest mass to locate the onset of secular instabilities. We find that inspiraling binary neutron stars are stable against radial collapse to black holes all the way down to the innermost stable circular orbit.

Abstract:
We study the various linear responses of neutron stars to external relativistic tidal fields. We focus on three different tidal responses, associated to three different tidal coefficients: (i) a gravito-electric-type coefficient G\mu_\ell=[length]^{2\ell+1} measuring the \ell^{th}-order mass multipolar moment GM_{a_1... a_\ell} induced in a star by an external \ell^{th}-order gravito-electric tidal field G_{a_1... a_\ell}; (ii) a gravito-magnetic-type coefficient G\sigma_\ell=[length]^{2\ell+1} measuring the \ell^{th} spin multipole moment G S_{a_1... a_\ell} induced in a star by an external \ell^{th}-order gravito-magnetic tidal field H_{a_1... a_\ell}; and (iii) a dimensionless ``shape'' Love number h_\ell measuring the distortion of the shape of the surface of a star by an external \ell^{th}-order gravito-electric tidal field. All the dimensionless tidal coefficients G\mu_\ell/R^{2\ell+1}, G\sigma_\l/R^{2\ell+1} and h_\ell (where R is the radius of the star) are found to have a strong sensitivity to the value of the star's ``compactness'' c\equiv GM/(c_0^2 R) (where we indicate by c_0 the speed of light). In particular, G\mu_\l/R^{2\l+1}\sim k_\ell is found to strongly decrease, as c increases, down to a zero value as c is formally extended to the ``black-hole (BH) limit'' c^{BH}=1/2. The shape Love number h_\ell is also found to significantly decrease as c increases, though it does not vanish in the formal limit c\to c^{BH}. The formal vanishing of \mu_\ell and \sigma_\ell as c\to c^{BH} is a consequence of the no-hair properties of black holes; this suggests, but in no way proves, that the effective action describing the gravitational interactions of black holes may not need to be augmented by nonminimal worldline couplings.