Moment of inertia of a neutron star. II. Relativistic corrections

In part I we suggested an approximate equation to determine the contribution of relativistic effects to the moment of inertia of a superdense star. In the present paper it is tested on model neutron stars with nine different variants of the equation of state of superdense matter. It is established that the approximation error does not exceed 5% for stable configurations. A more accurate version of the Ravenhall—Pethick equation [D. G. Ravenhall and C. J. Pethick, Astrophys. J., 424, 846 (1994)] for the moment of inertia as a function of the mass and radius of a neutron star is derived. Translated from Astrofizika, Vol. 40, No. 4, pp. 507–516, October–December, 1997.     

Moment of inertia of a neutron star. I. Relativistic equation for the accumulated moment of inertia

A relativistic, first-order differential equation is derived for the accumulated moment of inertia of a spherically symmetric celestial body. An approximate equation is proposed to describe the contribution of relativistic effects to the moment of inertia of a superdense star. For configurations of an incompressible fluid, this approximation describes the results of the numerical calculations of Chandrasekhar and Miller to within 5% in the entire range of central pressures from 0 to ∞. Translated from Astrofizika, Vol. 40, No. 1, pp. 87–96, January–March, 1997.     

Magnetic deformation of magnetars for the giant flares of the soft gamma-ray repeaters

We present one possible mechanism for the giant flares of the soft gamma-ray repeaters (SGRs) within the framework of the magnetar (superstrongly magnetized neutron star) model, motivated by the positive period increase associated with the August 27 event from SGR 1900+14. From second-order perturbation analysis of the equilibrium of the magnetic polytrope, we find that there exist different equilibrium states separated by the energy of the giant flares and the shift in the moment of inertia to cause the period increase. This suggests that, if we assume that global reconfiguration of the internal magnetic field of     suddenly occurs, the positive period increase     as well as the energy ≳10⁴⁴ erg of the giant flares may be explained. The moment of inertia can increase with a release of energy, because the star shape deformed by the magnetic field can be prolate rather than oblate. In this mechanism, since oscillation of the neutron star will be excited, a ∼ ms-period pulsation of the burst profile and an emission of gravitational waves are expected. The gravitational waves could be detected by planned interferometers such as LIGO, VIRGO and LCGT.     

Dense matter in strong gravitational field of neutron star

Mass, radius and moment of inertia are direct probes of compositions and Equation of State (EoS) of dense matter in neutron star interior. These are computed for novel phases of dense matter involving hyperons and antikaon condensate and their observable consequences are discussed in this article. Furthermore, the relationship between moment of inertia and quadrupole moment is also explored.     

Freely precessing neutron stars: model and observations

We present a model of a freely precessing neutron star, which is then compared against pulsar observations. The aim is to draw conclusions regarding the structure of the star, and to test theoretical ideas of crust–core coupling and superfluidity. We argue that, on theoretical grounds, it is likely that the core neutron superfluid does not participate in the free precession of the crust. We apply our model to the handful of proposed observations of free precession that have appeared in the literature. Assuming crust-only precession, we find that all but one of the observations are consistent with there being no pinned crustal superfluid at all; the maximum amount of pinned superfluid consistent with the observations is about 10⁻¹⁰ of the total stellar moment of inertia. However, the observations do not rule out the possibility that the crust and neutron superfluid core precess as a single unit. In this case the maximum amount of pinned superfluid consistent with the observations is about 10⁻⁸ of the total stellar moment of inertia. Both of these values are many orders of magnitude less than the 10⁻² value predicted by many theories of pulsar glitches. We conclude that superfluid pinning, at least as it affects free precession, needs to be reconsidered.     

General relativistic electromagnetic fields of a slowly rotating magnetized neutron star – II. Solution of the induction equations

We have solved numerically the general relativistic induction equations in the interior background space–time of a slowly rotating magnetized neutron star. The analytic form of these equations was discussed recently (Paper I), where corrections due to both the space–time curvature and the dragging of reference frames were shown to be present. Through a number of calculations we have investigated the evolution of the magnetic field with different rates of stellar rotation, different inclination angles between the magnetic moment and the rotation axis, as well as different values of the electrical conductivity. All of these calculations have been performed for a constant-temperature relativistic polytropic star and make use of a consistent solution of the initial-value problem which avoids the use of artificial analytic functions. Our results show that there exist general relativistic effects introduced by the rotation of the space–time which tend to decrease the decay rate of the magnetic field. The rotation-induced corrections are however generally hidden by the high electrical conductivity of the neutron star matter, and when realistic values for the electrical conductivity are considered, these corrections become negligible even for the fastest known pulsar.     

Relativistic Correction of Neutrino Emission in Neutron Stars

By the relativistic mean field theory and relevant weak-interactional cooling theory, the relativistic cooling properties in the conventional and hyperonic neutron star matter are studied. Also a comparison between the relativistic and non-relativistic results after taking consideration of the gravity correction is performed. The results show that the relativistic effect of neutrino emission reduces the neutrino emissivity, luminosity, and the cooling rate of stellar objects, in comparison with the non-relativistic case. In the neutron star matter without hyperon, the amplitude of the cooling rate reduction caused by the relativistic effect is maximal after taking the gravity correction into consideration, it attains 56% for a 2 M_⊙ neutron star composed of conventional neutron star matter, and in the hyperonic matter the amplitude of reduction is minimal, about 38%.     

Magnetized relativistic jets and long-duration GRBs from magnetar spin-down during core-collapse supernovae

We use ideal axisymmetric relativistic magnetohydrodynamic simulations to calculate the spin-down of a newly formed millisecond,   B ∼ 10¹⁵ G  , magnetar and its interaction with the surrounding stellar envelope during a core-collapse supernova (SN) explosion. The mass, angular momentum and rotational energy lost by the neutron star are determined self-consistently given the thermal properties of the cooling neutron star's atmosphere and the wind's interaction with the surrounding star. The magnetar drives a relativistic magnetized wind into a cavity created by the outgoing SN shock. For high spin-down powers  (∼10⁵¹–10⁵² erg s⁻¹)  , the magnetar wind is superfast at almost all latitudes, while for lower spin-down powers  (∼10⁵⁰ erg s⁻¹)  , the wind is subfast but still super-Alfvénic. In all cases, the rates at which the neutron star loses mass, angular momentum and energy are very similar to the corresponding free wind values (≲30 per cent differences), in spite of the causal contact between the neutron star and the stellar envelope. In addition, in all cases that we consider, the magnetar drives a collimated  (∼5–10°)  relativistic jet out along the rotation axis of the star. Nearly all of the spin-down power of the neutron star escapes via this polar jet, rather than being transferred to the more spherical SN explosion. The properties of this relativistic jet and its expected late-time evolution in the magnetar model are broadly consistent with observations of long duration gamma-ray bursts (GRBs) and their associated broad-lined Type Ic SN.     

Statistical studies of pulsar glitches

Shemar & Lyne have previously presented observations and an analysis of 32 glitches and their subsequent relaxations observed in a total of 15 pulsars. These data are brought together in this paper with those published by other authors. We show quantitatively how glitch activity decreases linearly with decreasing rate of slow-down. As indicated previously from studies of the Vela pulsar, the analysis suggests that 1.7 per cent of the moment of inertia of a typical neutron star is normally contained in pinned superfluid which releases its excess angular momentum at the time of a glitch. There is a broad range of glitch amplitude and there is a strong indication that pulsars with large magnetic fields suffer many small glitches while others show a smaller number of large glitches. Transient effects following glitches are very marked in young pulsars and decrease linearly with decreasing rate of slow-down, suggesting that the amount of loosely pinned superfluid decreases with age. We suggest that the low braking index of the Vela and Crab pulsars cannot be caused by a decreasing moment of inertia and should be attributed to step increases in the effective magnetic moment of the neutron star at the glitches.     

Neutron star models including the effects of hyperon formation

A new series of neutron star models has been computed. The equation of state used included the effects of nuclear dissolution, nuclear forces, and the presence of hyperons. The nuclear forces were introduced through use of a generalized form of the Levinger-SimmonsV andV potentials. The maximum stable masses obtained were 2.28 and 2.37M , respectively. Details are given of the structure of the outer layers which are expected to be crystalline. Expressions are given for the angular momentum and rotational energy of a neutron star and the relevant moment of inertia is tabulated for the models.     

Effect of rotational and tidal distortions on the periods of small adiabatic oscillations of stellar models

Kopal's method of representing the inner structure of a rotationally and tidally-distorted star by a rotationally-and tidally-distorted Roche model has been used in conjunction with an averaging concept introduced by Kippenhahn and Thomas of representing the equipotential surfaces of a rotationally and tidally distorted stellar model by the equipotential surfaces of an equivalent spherical model to determine the combined effects of rotation and tidal distortion on the periods of small adiabatic oscillations of a stellar model.     

Ohm’s law for plasma in general relativity and Cowling’s theorem

The general-relativistic Ohm’s law for a two-component plasma which includes the gravitomagnetic force terms even in the case of quasi-neutrality has been derived. The equations that describe the electromagnetic processes in a plasma surrounding a neutron star are obtained by using the general relativistic form of Maxwell equations in a geometry of slow rotating gravitational object. In addition to the general-relativistic effect first discussed by Khanna and Camenzind (Astron. Astrophys. 307:665, 1996) we predict a mechanism of the generation of azimuthal current under the general relativistic effect of dragging of inertial frames on radial current in a plasma around neutron star. The azimuthal current being proportional to the angular velocity ω of the dragging of inertial frames can give valuable contribution on the evolution of the stellar magnetic field if ω exceeds 2.7×10¹⁷(n/σ) s⁻¹ (n is the number density of the charged particles, σ is the conductivity of plasma). Thus in general relativity a rotating neutron star, embedded in plasma, can in principle generate axial-symmetric magnetic fields even in axisymmetry. However, classical Cowling’s antidynamo theorem, according to which a stationary axial-symmetric magnetic field can not be sustained against ohmic diffusion, has to be hold in the general-relativistic case for the typical plasma being responsible for the rotating neutron star.     

Mechanisms of optical,X-ray andγ-radiation from Crab pulsar

The synchrotron mechanism of radiation from the Crab pulsar has been investigated on the assumption that the mechanism acts in a source moving with relativistic velocity round a neutron star. A detailed matching has been made of the theoretical spectra of synchrotron radiation from relativistic electrons with the results of measurements of the radiation flux from the Crab pulsar in the infrared, optical and X-ray ranges. The parameters of the radiating region (intensity of the magnetic field, source dimensions, density and lifetime of radiating electrons) have been found. They are expressed through the ratio of the energy density of the magnetic field in the source to that of radiating electrons. The level of Compton-radiation in this region is estimated. Possible values of at which the level will correspond to the available results of measurements of the-radiation flux from the Crab pulsar are given. An estimate is presented for the surface magnetic field of the neutron star which does not contradict those obtained from considerations of the magnetic flux conservation when compressing the object up to the neutron star dimensions.     

Equation of state constraints from neutron stars

Recent measurements of thermal radiation from neutron stars have suggested a rather broad range of radiation radii ( ). Sources in M13 and Omega Cen imply R _∞∼12–14 km, but X7 in 47 Tuc implies R _∞∼16–20 km and RX J1856-3754 R _∞>17 km. If these measurements are all correct, only a limited selection of EOS’s could be consistent with them, but a broad range of neutron star masses (up to 2 M_⊙) would also be necessary. The surviving equations of state are incompatible with significant softening above nuclear saturation densities, such as would occur with Boson condensates, a low-density quark-hadron transition, or hyperons. Other potential constraints, such as from QPO’s, radio pulsar mass and moment of inertia measurements, and neutron star cooling, are compared. US DOE Grant DE-FG02-87ER-40317.     

Gravitational collapse of magnetic neutron stars

Pulsars are presently believed to be rotating neutron stars with frozen-in magnetic fields. Because of the high density of neutron stars, general relativistic effects are important since they effect both the structure and stability of such stars. Besides this, the magnetic field outside the star is also affected. Instead of falling of asr ^(2+l) as in flat space, it is shown that each magnetic multipole varies as a hypergeometric function of radius. A closed form of these hypergeometric functions is given in terms of Legendre functions of the second kind. If the mass of a neutron star exceeds about 2.4m , the star becomes unstable and coliapses. For a quasistatically collapsing body, it is shown that the magnetic field seen by a distant observer vanishes as the radius approaches the gravitational radius.This work was supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research under AFOSR Grant 70-1866.     

Dynamics of oscillating relativistic tori around Kerr black holes

We present a comprehensive numerical study of the dynamics of relativistic axisymmetric accretion tori with a power-law distribution of specific angular momentum orbiting in the background space–time of a Kerr black hole. By combining general relativistic hydrodynamics simulations with a linear perturbative approach we investigate the main dynamical properties of these objects over a large parameter space. The astrophysical implications of our results extend and improve two interesting results that have been recently reported in the literature. First, the induced quasi-periodic variation of the mass quadrupole moment makes relativistic tori of nuclear matter densities, as those formed during the last stages of binary neutron star mergers, promising sources of gravitational radiation, potentially detectable by interferometric instruments. Secondly, p-mode oscillations in relativistic tori of low rest-mass densities could be used to explain high-frequency quasi-periodic oscillations observed in X-ray binaries containing a black hole candidate under conditions more generic than those considered so far.     

The physics of strong magnetic fields in neutron stars

In this paper we present a new result, namely that the primal magnetic field of the collapsed core during a supernova explosion will, as a result of the conservation of magnetic flux, receive a massive boost to more than 90 times its original value by the Pauli paramagnetization of the highly degenerate relativistic electron gas just after the formation of the neutron star. Thus, the observed super-strong magnetic field of neutron stars may originate from the induced Pauli paramagnetization of the highly degenerate relativistic electron gas in the interior of the neutron star. We therefore have an apparently natural explanation for the surface magnetic field of a neutron star.     

Constraints the properties of neutron star matter from the mass of neutron star PSR J1614-2230

The constraints on the properties of neutron star matter from the mass of neutron star PSR J1614-2230 are examined in the framework of the relativistic mean field theory. We find that there are little differences between the σ potentials of large mass neutron star and those of canonnical mass neutron star. For potentials of ω, ρ, neutrons and electrons, the values corresponding to the large mass neutron star are larger than those to the canonnical mass neutron star as the baryon number density is more than a certain value. We also find that for the relative particle number density of electrons, muons, neutrons and protons and the pressure of the neutron star, the values corresponding to the large mass neutron star are far larger than those to the canonnical mass neutron star. For the relative particle number density of hyperons Λ, Σ^?, Σ⁰, Σ⁺ and Ξ^?, the values corresponding to the large mass neutron star are far smaller than those to the canonnical mass neutron star. These mean that the larger mass of neutron star is more advantageous to the production of protons but is not advantageous to the production of hyperons.     

Can a slowly rotating neutron star be a radio pulsar?

It is shown that the radius of curvature of magnetic field lines in the polar region of a rotating magnetized neutron star can be significantly less than the usual radius of curvature of the dipole magnetic field. The magnetic field in the polar cap is distorted by toroidal electric currents flowing in the neutron star crust. These currents close up the magnetospheric currents driven by the electron–positron plasma generation process in the pulsar magnetosphere. Owing to the decrease in the radius of curvature, electron–positron plasma generation becomes possible even for slowly rotating neutron stars, with   PB ^−2/3₁₂ < 10 s  , where P is the period of star rotation and   B ₁₂= B /10¹² G  is the magnitude of the magnetic field on the star surface.     

Describing neutron stars by the relativistic mean field theory

Neutron stars are studied in the framework of the relativistic mean field theory of interacting nucleons, hyperons, and mesons. Within the hadronic freedom, the cores of neutron stars are found to be dominated by hyperons when the density is sufficiently high. The influence of hyperon coupling constants on the transition from a neutron star to a hyperon-dominated strange neutron star is also investigated. It is found that the transition density gets its minimum value when the ratio of hyperon coupling constant to nucleon's takes the value of 0.65, and the calculated maximum mass of the neutron star is 1.4 M which lies within the range of the observational results.