Isothermal neutron star cores

By considering the relativistic expression for isothermal NS cores,T·e ^/2 = constant, we have shown that some of the standard equations of state, when applied to NS cores, correspond to constancy of some adiabatic exponents. It has been shown that the equation of state,P=KE, corresponds to ₁ = to ₂ = ₃ 1 +K and the equation of state, dP/dE=K, corresponds to ₃ 1 +K. The conditions under which different equations of state represent isothermal cores have been obtained: For isothermal NS, the local temperatureT, can be expressed in terms of pressureP, energy densityE, and rest mass density . For example: (a)P =KE :T = constant × (E/); (b)P=KE :T = constant × (P/); (c) dP/dE =K :T ^K ; (d) = ₂ :T = constant × (P/E); and (e) = ₃ :T = constant × (P/)^1/2. Equation of state corresponding to = ₂ is obtained as:P=E/ln(K/E) and the equation corresponding to = ₃ comes out as:E=P ln(K/P). Core-envelope models can be developed for these two cases. When core equation corresponding to = ₂ or = ₃ is used in the core, we can ensure the continuity of dP/dE at the core-envelope boundary, along with the continuity ofP, E, , and . The parameters of isothermal NS cores corresponding to the cases = ₂ and = ₃, have been obtained. The maximum mass of these NS cores comes out to be 2.7 .     

Structure of neutron star cores

After reviewing the outer and central regions of a neutron star, we discuss the central region and the possibility that the core has a solid structure. We present the work of different groups on the solidification problem, suggesting that the neutron star-cores are indeed solid.     

Two-fluid models of superfluid neutron star cores

Both relativistic and non-relativistic two-fluid models of neutron star cores are constructed, using the constrained variational formalism developed by Brandon Carter and co-workers. We consider a mixture of superfluid neutrons and superconducting protons at zero temperature, taking into account mutual entrainment effects. Leptons, which affect the interior composition of the neutron star and contribute to the pressure, are also included. We provide the analytic expression of the Lagrangian density of the system, the so-called master function, from which the dynamical equations can be obtained. All the microscopic parameters of the models are calculated consistently using the non-relativistic nuclear energy density functional theory. For comparison, we have also considered relativistic mean field models. The correspondence between relativistic and non-relativistic hydrodynamical models is discussed in the framework of the recently developed 4D covariant formalism of Newtonian multifluid hydrodynamics. We have shown that entrainment effects can be interpreted in terms of dynamical effective masses that are larger in the relativistic case than in the Newtonian case. With the nuclear models considered in this work, we have found that the neutron relativistic effective mass is even greater than the bare neutron mass in the liquid core of neutron stars.     

On the dynamics of superfluid neutron star cores

We discuss the nature of the various modes of pulsation of superfluid neutron stars using comparatively simple Newtonian models and the Cowling approximation. The matter in these stars is described in terms of a two-fluid model, where one fluid is the neutron superfluid, which is believed to exist in the core and inner crust of mature neutron stars, and the other fluid represents a conglomerate of all other constituents (crust nuclei, protons, electrons, etc.). In our model, we incorporate the non-dissipative interaction known as the entrainment effect, whereby the momentum of one constituent (e.g. the neutrons) carries along part of the mass of the other constituent. We show that there is no independent set of pulsating g-modes in a non-rotating superfluid neutron star core, even though the linearized superfluid equations contain a well-defined (and real-valued) analogue to the so-called Brunt–Väisälä frequency. Instead, what we find are two sets of spheroidal perturbations whose nature is predominately acoustic. In addition, an analysis of the zero-frequency subspace (i.e. the space of time-independent perturbations) reveals two sets of degenerate spheroidal perturbations, which we interpret to be the missing g-modes, and two sets of toroidal perturbations. We anticipate that the degeneracy of all these zero-frequency modes will be broken by the Coriolis force in the case of rotating stars. To illustrate this we consider the toroidal pulsation modes of a slowly rotating superfluid star. This analysis shows that the superfluid equations support a new class of r-modes, in addition to those familiar from, for example, geophysical fluid dynamics. Finally, the role of the entrainment effect on the superfluid mode frequencies is shown explicitly via solutions to dispersion relations that follow from a 'local' analysis of the linearized superfluid equations.     

Neutron star cores

Current theories, and the astrophysical implications, of the nature of high density neutron star matter are reviewed. Suggestions are made for a compromise between the alternatives of neutron crystallization and pion condensation.     

Electrical conductivity of neutron star cores in the presence of a magnetic field

General theory of electrical conductivity of a multicomponent mixture of degenerate fermions in a magnetic fieldB, developed in the preceding article (this volume), is applied to a matter in neutron star interiors at densities ₀, where ₀ = 2.8×10¹⁴ g cm^–3 is the standard nuclear matter density. A model of free-particle mixture ofn, p, e is used, with account for appearance of ^–-hyperons at > _c , where _c 4₀. The electric resistivities along and acrossB, and , and the Hall resistivity _H are calculated and fitted by simple analytical formulae at _c and > _c for the cases of normal or superfluid neutrons provided other particles are normal. Charge transport alongB is produced by electrons, due to their Coulombic collisions with other charged particles; is independent ofB and almost independent of the neutron superfluidity. Charge transport acrossB at largeB may be essentially determined by other charged particles. If _c , one has = [1 + (B/B ₀)²] for the normal neutrons, and for the superfluid neutrons, while _H = B/B _e for both cases. HereB _e 10⁹ T ₈ ² G,B ₀10¹¹ T ₈ ² G, andT ₈ is temperature in units of 10⁸ K. Accordingly for the normal neutrons atBB ₀, the transverse resistivity suffers an enhancement, _{1/4 1}. When 5₀ andB varies from 0 toBB _p 10¹³ T ₈ ² G, increases by a factor of about 10³–10⁴ and _H changes sign. WhenBB _p , remains constant for the superfluid neutrons, and _H B ² for the normal neutrons, while _H B for any neutron state. Strong dependence of resistivity onB, T, and may affect evolution of magnetic fields in neutron star cores. In particular, the enhancement of at highB may noticeably speed up the Ohmic decay of those electric currents which are perpendicular toB.     

Equation of state of neutron star cores and spin down of isolated pulsars

We study possible impact of a softening of the equation of state by a phase transition, or appearance of hyperons, on the spin evolution of isolated pulsars. Numerical simulations are performed using exact 2-D simulations in general relativity. The equation of state of dense matter at supranuclear densities is poorly known. Therefore, the accent is put on the general correlations between evolution and equation of state, and mathematical strictness. General conjectures referring to the structure of the one-parameter families of stationary configurations are formulated. The interplay of the back bending phenomenon and stability with respect to axisymmetric perturbations is described. Changes of pulsar parameters in a corequake following instability are discussed, for a broad choice of phase transitions predicted by different theories of dense matter. The energy release in a corequake, at a given initial pressure, is shown to be independent of the angular momentum of collapsing configuration. This result holds for various types of phases transition, with and without metastability. We critically review observations of pulsars that could be relevant for the detection of the signatures of the phase transition in neutron star cores. This work was partially supported by the Polish MNiI Grant no. 1P03D-008-27.     

The stochastic background of gravitational waves from neutron star formation at cosmological distances

We investigate the stochastic gravitational wave background that results from neutron star birth throughout the Universe. The neutron star birth rate, as a function of redshift, is calculated using an observation-based model for the evolving star formation rate, together with an estimate of the rate of core-collapse supernovae in the nearby Universe and an estimate of the neutron star/black hole branching ratio. Using three sample waveforms, based on numerical models of stellar core collapse by Zwerger & Müller, the spectral flux density, spectral strain, spectral energy density and duty cycle of the background have been computed. Our results show, contrary to recent claims, that the spectrum of the stochastic background clearly reflects the different physics in the core-collapse models. For a star formation model that is corrected for dust extinction, the neutron star formation rate throughout the Universe is high enough to result in a nearly continuous background of gravitational waves, with spectral features that can be related to emission mechanisms.     

Simulating a stochastic background of gravitational waves from neutron star formation at cosmological distances

We develop a temporal simulation of the potentially detectable gravitational wave background from neutron star formation at cosmological distances. By using a recent model for the evolving star formation rate, we investigate the statistical distribution of gravitational wave amplitudes due to supernovae that result in neutron star formation in the Einstein–de Sitter cosmology. We find that the gravitational wave amplitude distribution in our frame is highly skewed, with skewness related to the distribution of sources, and that the potentially detectable gravitational wave strain is dominated by sources at a redshift of     . Time traces of the simulation, using selected waveforms, are presented graphically and are also made available as web-based audio files. The method developed can readily be extended to different cosmologies, as well as to incorporate other waveforms and source types. This type of simulation will be useful in testing and optimizing detection strategies for gravitational wave backgrounds due to various types of individually undetectable astrophysical sources.     

Self-similar pressure oscillations in neutron star envelopes as probes of neutron star structure

We study eigenmodes of acoustic oscillations of high multipolarity l ∼ 100–1000 and high frequency (∼100 kHz), localized in neutron star envelopes. We show that the oscillation problem is self-similar. Once the oscillation spectrum is calculated for a given equation of state (EOS) in the envelope and given stellar mass M and radius R , it can be rescaled to a star with any M and R (but the same EOS in the envelope). For l ≳ 300, the modes can be subdivided into the outer and inner ones. The outer modes are mainly localized in the outer envelope. The inner modes are mostly localized near the neutron drip point, being associated with the softening of the EOS after the neutron drip. We calculate oscillation spectra for the EOSs of cold-catalyzed and accreted matter and show that the spectra of the inner modes are essentially different. A detection and identification of high-frequency pressure modes would allow one to infer M and R and determine also the EOS in the envelope (accreted or ground state) providing a new, potentially powerful method to explore the main parameters and internal structure of neutron stars.     

Nuclear reactions on accreting neutron star surface

The production of X-rays and gamma-rays in bursts is believed to be due to the rapid burning of matter accreted onto a neutron star surface from its companion, most probably a giant star. The accreted matter consists mainly of hydrogen and helium and a very small amount of heavy elements. Due to the infall of matter, the temperature at the bottom layers is raised to a value of the order of 10⁸ K. The neutron star surface density is>10⁷ g cm^–3. As hydrogen burning is a slow process under any temperature and density conditions, we consider the helium-burning reactions as the source of gamma-rays in the neutron star surface. Under high-density conditions the ordinary laboratory reaction rates should become modified. At high-density conditions, the strong screening effect due to the polarising cloud of electrons around the ions become important and enhances the reaction rates considerably. The helium-burning reactions are calculated under such conditions. The abundances of helium-burning products such as¹²C, 1¹⁶O, and²⁰Ne, etc., are computed. Under high-density and temperature conditions carbon is found to be more abundant than oxygen. Neon is completely absent in almost all the relevant physical conditions in which a strong screening effect is operative. It is suggested that explosive burning of accreted helium of 10^–13 M will account for the observed energy of gamm-ray burst.     

Nucleosynthesis in neutron star atmospheres

The composition of neutron star atmospheres is calculated as a function of time including effects of diffusion, cooling and thermonuclear reactions. A seven-component nuclear reaction network with includes He⁴, C¹², O¹⁶, Ne²⁰, Mg²⁴, Si²⁸ and Fe⁵⁶ is utilized. Neutron star models with different initial nuclear abundances are compared as to subsequent nucleosynthesis. It is found that the final abundances are independent of original composition assuming He⁴ as the major initial constituent. The final composition of the atmosphere is predominantly Fe⁵⁶. Mass loss from an evolving neutron star is examined as a possible source of cosmic rays. It is found that a neutron star contributes only Fe⁵⁶ significantly to the cosmic-ray spectrum.     

Spectra of the spreading layers on the neutron star surface and constraints on the neutron star equation of state

Spectra of the spreading layers on the neutron star surface are calculated on the basis of the Inogamov–Sunyaev model taking into account general relativity correction to the surface gravity and considering various chemical composition of the accreting matter. Local (at a given latitude) spectra are similar to the X-ray burst spectra and are described by a diluted blackbody. Total spreading layer spectra are integrated accounting for the light bending, gravitational redshift and the relativistic Doppler effect and aberration. They depend slightly on the inclination angle and on the luminosity. These spectra also can be fitted by a diluted blackbody with the colour temperature depending mainly on a neutron star compactness. Owing to the fact that the flux from the spreading layer is close to the critical Eddington, we can put constraints on a neutron star radius without the need to know precisely the emitting region area or the distance to the source. The boundary layer spectra observed in the luminous low-mass X-ray binaries, and described by a blackbody of colour temperature   T _c= 2.4 ± 0.1 keV  , restrict the neutron star radii to   R = 14.8 ± 1.5 km  (for a  1.4-M_⊙  star and solar composition of the accreting matter), which corresponds to the hard equation of state.     

Evolution of neutron star magnetic fields

During the evolution of the neutron star its magnetic field first decays exponentially with the time and then may becomes quasi-stationary. The non-decaying magnetic field of the neutron star is generated by a degenerate electron gas which is in the Landau orbital ferromagnetism (LOFER) state. Possibly, due to the neutron star transition into the LOFER state, magnetic fields remained sufficiently strong in the case of such old magnetic neutron stars as powerful X-ray sources (e.g., Her X-1), millisecond pulsars and the binary pulsar PSR 0655+64.     

Statified model of a neutron star

Evolutionary calculations based on realistic equations of state indicate the stratified nature of the distribution of hadron matter in the interiors of neutron stars. In the proposed model, the stratified structure of a neutron star is treated as a rigid inert core surrounded by a dynamical layer. The physical basis for the model is the concept of the stellar matter of the peripheral envelope as an elastic Fermi continuum, the motions of which are described by the equations of nuclear elastodynamics, proposed in the macroscopic theory of collective processes in laboratory nuclear physics. It is shown that the vibrational dynamics of a neutron star is characterized by two branches of gravitational—elastic, spheroidal (s-mode) and torsional (t-mode) nonradial eigenvibrations. Estimates obtained for the periods of global, gravitational nonradial modes suggest that variations in the intensity of micropulses observed in the millisecond range of the spectra of C-pulsars may be ascribed to these vibrations. The proposed two-component model of a neutron star enables one to consider a glitch in a pulsar’s radio emission as a starquake due to the passage of the companion through periastron of the binary system. Translated from Astrofizika, Vol. 42, No. 2, pp. 235–252, April–June, 1999     

Evolution of neutron star magnetic fields

This paper reviews the current status of the theoretical models of the evolution of the magnetic fields of neutron stars other than magnetars. It appears that the magnetic fields of neutron stars decay significantly only if they are in binary systems. Three major physical models for this, namely spindown-induced flux expulsion, ohmic evolution of crustal field and diamagnetic screening of the field by accreted plasma, are reviewed.     

Neutrino oscillations in neutron star matter

We discuss the hypothesis suggested by Mazurek (1979) that neutrino oscillations (v _e v ) could transfer leptonic zero-point energy (e ^-,v _e ; <4/3) to baryons during the gravitational collapse of a massive star (M10M ) and that subsequently the collapse ends in a stellar explosion (>4/3). The estimation of the lengths of neutrino oscillations if occuring in vacuum or dense matter, respectively, shows, however, that vacuum oscillations can be suppressed in dense matter and, therefore, should have no influence on the neutrino emission of neutron stars.