Neutron star models

Neutron star models are calculated using an equation of state discussed in an earlier paper. A maximum mass for a neutron star of 1.74 solar masses is found. The central density of thie star is 3.3×10¹⁵ g/cm³. The lightest stars have masses of 0.02 (resp. 0.03) solar masses with central densities 2.2×10¹⁴ g/cm³ (resp. 1.9×10¹⁴).     

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.     

Shock waves on neutron star cores

A numerical model of the transient behavior of a radiation-dominated shock was calculated in order to demonstrate the relatively large initial escape of internal energy that takes place when the opacity law is a positive power of temperature, as for neutrinos,K∼T ². Attention was given to the consequences of diffusive transport of radiation versus local production of internal energy by duplicating the calculation with and without artificial viscosity. It is concluded that a shock formed on the neutron star core of an imploding supernova may radiate its internal energy in electron neutrinos more effectively than had hithertofore been considered.     

Neutron star electrodynamics in curved space

We calculate the fields surrounding and the power radiated by a slowly rotating neutron star with a frozen-in magnetic dipole field, tilted with respect to the rotation axis, including the effects of spacetime curvature. The general relativistic effects suppress the radiated power relative to flat space by factors up to 1/7 for magnetic dipole radiation and 1/50 for the associated electric quadrupole radiation. This suppression exceeds that which might be expected from a surface red shift alone.Numerical results are found using power series which describe the behavior of electromagnetic fields exterior to a black hole or slowly rotating neutron star. These new solutions, appropriate near the stellar surface, converge for all radii exterior to the neutron star (or black hole) making analytic continuation of the power series unnecessary as well as allowing matching to a linear combination of asymptotic expansions, appropriate for large radius. Typical numerical values for these functions are presented as well as techniques for accelerating the convergence of their respective power series which make them attractive alternatives to numerical integration.Supported in part by NSF Grant PHY 77 28356.     

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.     

Millimetre and submillimetre observations of high-mass star forming cores

I summarize recent millimetre and submillimetre observations of cloud cores where massive star formation is currently taking place. The first systematic continuum surveys in this wavelength range obtained with single dish telescopes and high-resolution data of NGC 2024 and W51A obtained with the Plateau de Bure interferometer are presented in more detail. Also given is a discussion of observing methods and some of the difficulties involved with the observations.     

Neutron star surface emission: Beyond the dipole model

Recent Chandra and XMM-Newton observations of a number of X-ray “dim” pulsating neutron stars revealed quite unexpected features in the emission from these sources. Their soft thermal spectrum, believed to originate directly from the star surface, shows evidence for a phase-varying absorption line at some hundred eVs. The pulse modulation is relatively large (pulsed fractions in the range ～8–35% in amplitude), the pulse shape is often non-sinusoidal, and the hard X-ray color appears to be anti-correlated in phase with the total emission. Moreover, the prototype of this class, RX J0720.4-3125, has been found to undergo rather sensible changes both in its spectral and timing properties over a timescale of a few years. By modeling the light curves of two sources, RBS 1223 and RX J0720.4-3125, it has been found evidence for two hot regions located at a slightly non antipodal direction. All these new findings are difficult to reconcile with the standard picture of a cooling neutron star endowed with a purely dipolar magnetic field. Here we present more realistic models of surface emission, where the effects of different neutron star thermal and magnetic surface distributions are accounted for. We show how a star-centered field made of a dipolar and a quadrupolar component can influence the properties of the observed light curves and we present results that account self-consistently for toroidal and poloidal crustal field configurations.     

Neutron star masses: dwarfs, giants and neighbors

We discuss three topics related to the neutron star (NS) mass spectrum. At first we discuss the possibility to form low-mass (M≲1M _⊙) objects. In our opinion this and suggest this is possible only due to fragmentation of rapidly rotating proto-NSs. Such low-mass NSs should have very high spatial velocities which could allow identification. A critical assessment of this scenario is given. However, the mechanism has its own problems, and so formation of such objects is not very probable. Secondly, we discuss mass growth due to accretion for NSs in close binary systems. With the help of numerical population synthesis calculations we derive the mass spectrum of massive (M>1.8M _⊙) NSs. Finally, we discuss the role of the mass spectrum in population studies of young cooling NSs. We formulate a kind of mass constraint which can be helpful, in our opinion, in discussing different competitive models of the thermal evolution of NSs. S.B.P. wants to thank the Organizers for support and hospitality. The work of S.B.P. was supported by the RFBR grant 06-02-16025 and by the “Dynasty” Foundation (Russia). The work of M.E.P.—by the RFBR grant 04-02-16720 and that of H.G. by DFG grant 436 ARM 17/4/05.     

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.     

Neutron star retention and millisecond pulsar production in globular clusters

We investigate the conditions by which neutron star retention in globular clusters is favoured. We find that neutron stars formed in massive binaries are far more likely to be retained. Such binaries are likely to then evolve into contact before encountering other stars, possibly producing a single neutron star after a common envelope phase. A large fraction of the single neutron stars in globular clusters are then likely to exchange into binaries containing moderate-mass main-sequence stars, replacing the lower-mass components of the original systems. These binaries will become intermediate-mass X-ray binaries (IMXBs), once the moderate-mass star evolves off the main sequence, as mass is transferred on to the neutron star, possibly spinning it up in the process. Such systems may be responsible for the population of millisecond pulsars (MSPs) that has been observed in globular clusters. Additionally, the period of mass-transfer (and thus X-ray visibility) in the vast majority of such systems will have occurred 5–10 Gyr ago, thus explaining the observed relative paucity of X-ray binaries today, given the MSP population.     

Neutron star models based on an improved equation of state

Using an equation of state for cold degenerate matter which takes nuclear forces and nuclear clustering into account, neutron star models are constructed. Stable models were obtained in the mass range above 0.065M and density range 10^14.08 to 10^15.4 gm/cm³. All of these models were found to be bound. The outer crystalline layer of the star was found to have a thickness of 200 m or more depending on the mass of the model.     

Models of dense cores in translucent regions of low mass star formation

We have constructed models for a region of low mass star formation where stellar winds ablate material from dark dense cores and return it to a translucent intercore medium from which subsequent generations of cores condense. Depletion of gas phase species onto grains plays a major role in the chemistry. For reasonable agreement between model core chemical fractional abundances and measured TMC-1 fractional abundances to obtain, the core collapse, once started, must be relatively uninhibited by turbulence or magnetic fields and the core lifetime must fall in a limited range determined by the assumed depletion rates. In a core with the TMC-1 fractional abundances, CH, OH, C₂H, H₂CO, HCN, HNC, and CN are the only simple species that have been detected in TMC-1 at radio and millimeter wavelengths to have fractional abundances that are roughly constant or increasing with time; this result bears considerably on previous work concerned with searches for spectroscopic evidence for and the diagnosis of collapse during protostellar formation, but depends on the fractions of the OH and CH emissions that are associated with the core centre rather than more extended gas or a core-stellar wind boundary layer. Model results for the abundance ratios of H₂O, CH₄, and NH₃ ices are in good agreement with those inferred for Halley's Comet.     

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.     

Neutron star high‐mass binaries as the origin of SGR/AXP

A close high‐mass binary system consisting of a neutron star (NS) and a massive OB supergiant companion is expected to lead to a Thorne‐Żytkow object (TZO) structure, which consists of a NS core and a stellar envelope. We use the scenario machine program to calculate the formation tracks of TZOs in close high‐mass NS binaries and their subsequent evolution. We propose and demonstrate that the explosion and instant contraction of a TZO structure leave its stellar remnant as a soft gamma‐ray repeater and an anomalous X‐ray pulsar respectively. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)