Non-rotating Neutron Stars Research Articles

The magnetosphere of oscillating neutron stars in general relativity

Just as a rotating magnetized neutron star has material pulled away from its surface to populate a magnetosphere, a similar process can occur as a result of neutron-star pulsations rather than rotation. This is of interest in connection with the overall study of neutron star oscillation modes but with a particular focus on the situation for magnetars. Following a previous Newtonian analysis of the production of a force-free magnetosphere in this way Timokhin et al., we present here a corresponding general-relativistic analysis. We give a derivation of the general relativistic Maxwell equations for small-amplitude arbitrary oscillations of a non-rotating neutron star with a generic magnetic field and show that these can be solved analytically under the assumption of low current density in the magnetosphere. We apply our formalism to toroidal oscillations of a neutron star with a dipole magnetic field and find that the low current density approximation is valid for at least half of the oscillation modes, similarly to the Newtonian case. Using an improved formula for the determination of the last closed field line, we calculate the energy losses resulting from toroidal stellar oscillations for all of the modes for which the size of the polar cap is small. We find that general relativistic effects lead to shrinking of the size of the polar cap and an increase in the energy density of the outflowing plasma. These effects act in opposite directions but the net result is that the energy loss from the neutron star is significantly smaller than suggested by the Newtonian treatment.

Frame Dragging Effect on Properties of Rotating Neutron Stars with Strong Magnetic Field

The general relativistic frame dragging effect on the properties, such as the moments of inertia and the radii of gyration of fast rotating neutron stars with a uniform strong magnetic field, is calculated accurate to the first order in the uniform angular velocity. The results show that compared with the corresponding non-rotating static spherical symmetric neutron star with a weaker magnetic field, a fast rotating neutron star (millisecond pulsar) with a stronger magnetic field has a relative smaller moment of inertia and radius of gyration.

Nonrotating black hole in a post-Newtonian tidal environment

In the first part of the paper we construct the metric of a tidally deformed, nonrotating black hole. The metric is presented as an expansion in powers of $r/b\ensuremath{\ll}1$, in which $r$ is the distance to the black hole and $b$ the characteristic length scale of the tidal field---the typical distance to the remote bodies responsible for the tidal environment. The metric is expanded through order $(r/b{)}^{4}$ and written in terms of a number of tidal multipole moments, the gravitoelectric moments ${\mathcal{E}}_{ab}$, ${\mathcal{E}}_{abc}$, ${\mathcal{E}}_{abcd}$, and the gravitomagnetic moments ${\mathcal{B}}_{ab}$, ${\mathcal{B}}_{abc}$, ${\mathcal{B}}_{abcd}$. It differs from the similar construction of Poisson and Vlasov in that the tidal perturbation is presented in Regge-Wheeler gauge instead of the light-cone gauge employed previously. In the second part of the paper we determine the tidal moments by matching the black-hole metric to a post-Newtonian metric that describes a system of bodies with weak mutual gravity. This extends the previous work of Taylor and Poisson (Paper I in this sequence), which computed only the leading-order tidal moments, ${\mathcal{E}}_{ab}$ and ${\mathcal{B}}_{ab}$. The matching is greatly facilitated by the Regge-Wheeler form of the black-hole metric, and this motivates the work carried out in the first part of the paper. The tidal moments are calculated accurately through the first post-Newtonian approximation, and at this order they are independent of the precise nature of the compact body. The moments therefore apply equally well to a rotating black hole, or to a (rotating or nonrotating) neutron star. As an application of this formalism, we examine the intrinsic geometry of a tidally deformed event horizon and describe it in terms of a deformation function that represents a quadrupolar and octupolar tidal bulge.

Evolution of neutron stars with toroidal magnetic fields: Axisymmetric simulation in full general relativity

We study the stability of neutron stars with toroidal magnetic fields by magnetohydrodynamic simulation in full general relativity under the assumption of axial symmetry. Nonrotating and rigidly rotating neutron stars are prepared for a variety of magnetic field configuration. For modeling the neutron stars, the polytropic equation of state with the adiabatic index $\ensuremath{\Gamma}=2$ is used for simplicity. It is found that nonrotating neutron stars are dynamically unstable for the case where toroidal magnetic field strength varies $\ensuremath{\propto}{\ensuremath{\varpi}}^{2k\ensuremath{-}1}$ with $k\ensuremath{\ge}2$ (here $\ensuremath{\varpi}$ is the cylindrical radius), whereas for $k=1$ the neutron stars are stable. After the onset of the instability, unstable modes grow approximately in the Alfv\'en time scale and, as a result, a convective motion is excited to change the magnetic field profile until a new state, which is stable against axisymmetric perturbation, is reached. We also find that rotation plays a role in stabilization, although the instability still occurs in the Alfv\'en time scale when the ratio of magnetic energy to rotational kinetic energy is larger than a critical value $\ensuremath{\sim}0.2$. Implication for the evolution of magnetized protoneutron stars is discussed.

Gravitational waves from pulsations of neutron stars described by realistic equations of state

In this work we discuss the time evolution of nonspherical perturbations of a nonrotating neutron star described by a realistic equation of state (EOS). We analyze 10 different EOS for a large sample of neutron star models. Various kinds of generic initial data are evolved and the gravitational signals are computed. We focus on the dynamical excitation of fluid and spacetime modes and extract the corresponding frequencies. We employ a constrained numerical algorithm based on standard finite-differencing schemes which permits stable and long-term evolutions. Our code provides accurate waveforms and allows one to capture, via Fourier analysis, the frequencies of the fluid modes with an accuracy comparable to that of frequency-domain calculations. The results we present here are useful for providing comparisons with simulations of nonlinear oscillations of (rotating) neutron star models as well as test beds for 3D nonlinear codes.

NUMERICAL METHODS FOR SOLVING THE RELATIVISTIC OPPENHEIMER–VOLKOFF EQUATIONS

In this paper, three methods for computing non-rotating neutron star models are discussed. The relativistic Oppenheimer–Volkoff (OV) equations of hydrostatic equilibrium are solved using (a) the well-known variable step method, (b) a method based on a modified form of the OV equations, where the thermodynamic enthalpy function is the independent variable, and (c) a new straightforward method, based on the ODEPACK Fortran package for solving systems of differential equations. We present results concerning several equations of state and conclusions concerning the applicability of each method.

Nonrotating and rotating neutron stars in the extended field theoretical model

We study the properties of nonrotating and rotating neutron stars for a new set of equations of state (EOSs) with different high-density behavior obtained using the extended field theoretical model. The high-density behavior for these EOSs are varied by varying the {omega}-meson self-coupling and hyperon-meson couplings in such a way that the quality of fit to the bulk nuclear observables, nuclear matter incompressibility coefficient, and hyperon-nucleon potential depths remain practically unaffected. We find that the largest value for maximum mass for the nonrotating neutron star is 2.1M{sub {center_dot}}. The radius for a neutron star with canonical mass is 12.8-14.1 km, provided only those EOSs are considered for which the maximum mass is larger than 1.6M{sub {center_dot}}, the lower bound on the maximum mass measured so far. Our results for the very recently discovered fastest rotating neutron star indicate that this star is supramassive with mass 1.7M{sub {center_dot}}-2.7M{sub {center_dot}} and circumferential equatorial radius 12-19 km.

Neutron stars in Einstein-aether theory

As current and future experiments probe strong gravitational regimes around neutron stars and black holes, it is desirable to have theoretically sound alternatives to general relativity against which to test observations. Here we study the consequences of one such generalization, Einstein-aether theory, for the properties of nonrotating neutron stars. This theory has a parameter range that satisfies all current weak-field tests. We find that within this range it leads to lower maximum neutron star masses, as well as larger surface redshifts at a particular mass, for a given nuclear equation of state. For nonrotating black holes and neutron stars, the innermost stable circular orbit is only slightly modified in this theory.

Low Mach Number Modeling of Type I X‐Ray Burst Deflagrations

The low Mach number approximation (LMNA) is applied to two-dimensional hydrodynamic modeling of type I X-ray bursts on a rectangular patch on the surface of a nonrotating neutron star. Because such phenomena involve decidedly subsonic flows, the time step increase offered by the LMNA makes routine simulations of these deflagrations feasible in an environment where strong gravity produces significant stratification, while allowing for potentially significant lateral differences in temperature and density. The model is employed to simulate the heating, peak, and initial cooling stages in the deep envelope layers of a burst. During the deflagration, Benard-like cells naturally fill up a vertically expanding convective layer. The Mach number is always less than 0.15 throughout the simulation, thus justifying the low Mach number approximation. While the convective layer is superadiabatic on average, significant fluctuations in adiabaticity occur within it on subconvective timescales. Due to convective layer expansion, significant compositional mixing naturally occurs, but tracer particle penetration through the convective layer boundaries on convective timescales is temporary and spatially limited. Thus, mixing occurs on the relatively slow burst timescale through thermal expansion of the convective layer rather than from mass penetration of the convective layer boundary through particle convection. At the convective layer boundaries where mixing is less efficient, the actual temperature gradient more closely follows the Ledoux criterion.

Are neutron stars crushed? Gravitomagnetic tidal fields as a mechanism for binary-induced collapse

Numerical simulations of binary neutron stars by Wilson, Mathews, and Marronetti indicated that neutron stars that are stable in isolation can be made to collapse to black holes when placed in a binary. This claim was surprising as it ran counter to the Newtonian expectation that a neutron star in a binary should be more stable, not less. After correcting an error found by Flanagan, Wilson and Mathews found that the compression of the neutron stars was significantly reduced but not eliminated. This has motivated us to ask the following general question: Under what circumstances can general-relativistic tidal interactions cause an otherwise stable neutron star to be compressed? We have found that if a nonrotating neutron star possesses a current-quadrupole moment, interactions with a gravitomagnetic tidal field can lead to a compressive force on the star. If this current quadrupole is induced by the gravitomagnetic tidal field, it is related to the tidal field by an equation-of-state-dependent constant called the gravitomagnetic Love number. This is analogous to the Newtonian Love number that relates the strength of a Newtonian tidal field to the induced mass quadrupole moment of a star. The compressive force is almost never larger than the Newtonian tidalmore » interaction that stabilizes the neutron star against collapse. In the case in which a current quadrupole is already present in the star (perhaps as an artifact of a numerical simulation), the compressive force can exceed the stabilizing one, leading to a net increase in the central density of the star. This increase is small (< or approx. 1%) but could, in principle, cause gravitational collapse in a star that is close to its maximum mass. This paper also reviews the history of the Wilson-Mathews-Marronetti controversy and, in an appendix, extends the discussion of tidally induced changes in the central density to rotating stars.« less

Perturbative Analysis of Universality and Individuality in Gravitational Waves from Neutron Stars

The universality observed in gravitational wave spectra of non-rotating neutron stars is analyzed here. We show that the universality in the axial oscillation mode can be reproduced with a simple stellar model, namely the centrifugal barrier approximation (CBA), which captures the essence of the Tolman VII model of compact stars. Through the establishment of scaled co-ordinate logarithmic perturbation theory (SCLPT), we are able to explain and quantitatively predict such universal behavior. In addition, quasi-normal modes of individual neutron stars characterized by different equations of state can be obtained from those of CBA with SCLPT.

Universality in quasi-normal modes of neutron stars

We study universality in gravitational waves emitted from non-rotating neutron stars characterized by different equations of state (EOSs). We find that the quasi-normal mode frequencies of such waves, including the w-modes and the f-mode, display similar universal scaling behaviours that hold for most EOSs. Such behaviours are shown to stem from the mathematical structure of the axial and the polar gravitational wave equations, and the fact that the mass distribution function can be approximated by a cubic-quintic polynomial in the radius. As a benchmark for other realistic neutron stars, a simple model of neutron stars is adopted here to reproduce the pulsation frequencies and the generic scaling behaviours mentioned above with good accuracy.

Equation of state of dense matter and maximum mass of neutron stars

Theoretical models of the equation of state (EOS) of neutron-star matter (starting with the crust and ending at the densest region of the stellar core) are reviewed. Apart from a broad set of baryonic EOSs, strange quark matter, and even more exotic (abnormal and Q-matter) EOSs are considered. Results of calculations of M_max for non-rotating neutron stars and exotic compact stars are reviewed, with particular emphasis on the dependence on the dense-matter EOS. Rapid rotation increases M_max, and this effect is studied for both neutron stars and exotic stars. Theoretical results are then confronted with measurements of masses of neutron stars in binaries, and the consequences of such a confrontation and their possible impact on the theory of dense matter are discussed.

Adiabatic oscillations of non-rotating superfluid neutron stars

We present results concerning the linear (radial and non-radial) oscillations of non-rotating superfluid neutron stars in Newtonian physics. We use a simple two-fluid model to describe the superfluid neutron star, where one fluid consists of the superfluid neutrons, while the second fluid contains all the remaining constituents (protons, electrons). The two fluids are assumed to be “free” in the sense of absence of vortex-mediated forces like mutual friction or pinning, but they can be coupled by the equation of state, in particular by entrainment. We calculate numerically the eigen-frequencies and -modes of adiabatic oscillations, neglecting beta-reactions that would lead to dissipation. We find a doubling of all acoustic-type modes (f-modes, p-modes), and confirm the absence of g-modes in these superfluid models. We show analytically and numerically that only in the case of non-stratified background models (i.e. with no composition gradient) can these acoustic modes be separated into two distinct families, which are characterized by either co- or counter-moving fluids respectively, and which are sometimes referred to as “ordinary” and “superfluid” modes. In the general, stratified case, however, this separation is not possible, and these acoustic modes can not be classified as being either purely “ordinary” or “superfluid”. We show how the properties of the two-fluid modes change as functions of the coupling by entrainment. We find avoided mode-crossings for the stratified models, while the crossings are not avoided in the non-stratified, separable case. The oscillations of normal-fluid neutron stars are recovered as a special case simply by locking the two fluids together. In this effective one-fluid case we find the usual singlet f- and p-modes, and we also find the expected g-modes of stratified neutron star models.

New approach to the evolution of neutron star oscillations

We present a new derivation of the perturbation equations governing the oscillations of relativistic non-rotating neutron star models using the ADM-formalism. This formulation has the advantage that it immediately yields the evolution equations in a hyperbolic form, which is not the case for the Einstein field equations in their original form. We show that the perturbation equations can always be written in terms of spacetime variables only, regardless of any particular gauge. We demonstrate how to obtain the Regge-Wheeler gauge, by choosing appropriate lapse and shift. In addition, not only the 3-metric but also the extrinsic curvature of the initial slice have to satisfy certain conditions in order to preserve the Regge-Wheeler gauge throughout the evolution. We discuss various forms of the equations and show their relation to the formulation of Allen et al. New results are presented for polytropic equations of state. An interesting phenomenon occurs in very compact stars, where the first ring-down phase in the wave signal corresponds to the first quasinormal mode of an equal mass black hole, rather than to one of the proper quasinormal modes of the stellar model. A somewhat heuristic explanation to account for this phenomenon is given. For realistic equations of state, the numerical evolutions exhibit an instability, which does not occur for polytropic equations of state. We show that this instability is related to the behavior of the sound speed at the neutron drip point. As a remedy, we devise a transformation of the radial coordinate $r$ inside the star, which removes this instability and yields stable evolutions for any chosen numerical resolution.

SU(2) Chiral Sigma Model and Properties of Neutron Stars

We discuss the {\it SU}(2) chiral sigma model in the context of nuclear matter using a mean field approach at high density. In this model we include a dynamically generated isoscalar vector field and higher-order terms in the scalar field. With the inclusion of these, we reproduce the empirical values of the nuclear matter saturation density, binding energy, and nuclear incompressibility. The value of the incompressibility is chosen according to recently obtained heavy-ion collision data. We then apply the same dynamical model to neutron-rich matter in beta equilibrium, related to neutron star structure. The maximum mass and corresponding radius of stable non-rotating neutron stars are found to be in the observational limit.

A microscopic study of neutron stars’ structure

AbstractStarting from the nuclear interactions which are used to calculate the properties of atomic nuclei, we derive a new equation of state (EOS) for neutron star matter. With this realistic EOS, we calculate the structure of non-rotating neutron stars.

Resonant tidal excitations of rotating neutron stars in coalescing binaries

In a coalescing neutron star–neutron star or neutron star–black hole binary, oscillation modes of the neutron star can be resonantly excited by the companion during the final minutes of the binary inspiral, when the orbital frequency sweeps up from a few Hz to a few thousand Hz. The resulting resonant energy transfer between the orbit and the neutron star speeds up or slows down the inspiral, depending on whether the resonant mode has positive or negative energy, and induces a phase change in the emitted gravitational waves from the binary. While only g-modes can be excited for a non-rotating neutron star, f-modes and r-modes can also be excited when the neutron star is spinning. A tidal resonance, designated by the index ( jk, m) ({ jk} specifies the angular order of the mode, as in the spherical harmonic Yjk), occurs when the mode frequency equals m times the orbital frequency. For the f-mode resonance to occur before coalescence, the neutron star must have rapid rotation, with spin frequency νs≳710 Hz for the (22,2)-resonance and νs≳570 Hz for the (33,3)-resonance (assuming canonical neutron star mass, M=1.4 M⊙, and radius, R=10 km; however, for R=15 km, these critical spin frequencies become 330 and 260 Hz, respectively). Although current observations suggest that such high rotation rates may be unlikely for coalescing binary neutron stars, these rates are physically allowed. Because of their strong tidal coupling, the f-mode resonances induce a large change in the number of orbital cycles of coalescence, ΔNorb, with the maximum ΔNorb∼10–1000 for the (22,2)-resonance and ΔNorb∼1 for the (33,3)-resonance. Such f-mode resonant effects, if present, must be included in constructing the templates of waveforms used in searching for gravitational wave signals. Higher order f-mode resonances can occur at slower rotation rates, but the induced orbital change is much smaller (ΔNorb≲0.1). For the dominant g-mode (22,2)-resonance, even modest rotation (νs≲100 Hz) can enhance the resonant effect on the orbit by shifting the resonance to a smaller orbital frequency. However, because of the weak coupling between the g-mode and the tidal potential, ΔNorb lies in the range 10−3–10−2 (depending strongly on the neutron star equation of state) and is probably negligible for the purpose of detecting gravitational waves. Resonant excitations of r-modes require misaligned spin–orbit inclinations, and the dominant resonances correspond to octopolar excitations of the jk=2 mode, with (jk, m)=(22,3) and (22,1). Since the tidal coupling of the r-mode depends strongly on rotation rate, ΔNorb≲10−2(R/10 km)10(M/1.4 M⊙)−20/3 is negligible for canonical neutron star parameters, but can be appreciable if the neutron star radius is larger.

The imprint of the equation of state on the axial w-modes of oscillating neutron stars

We discuss the dependence of the pulsation frequencies of the axial quasi-normal modes of a nonrotating neutron star upon the equation of state describing the star interior. The continued fraction method has been used to compute the complex frequencies for a set of equations of state based on different physical assumptions and spanning a wide range of stiffness. The numerical results show that the detection of axial gravitational waves would allow to discriminate between the models underlying the different equation of states, thus providing relevant information on both the structure of neutron star matter and the nature of the hadronic interactions.

Quasi‐normal Modes of Rotating Relativistic Stars: Neutral Modes for Realistic Equations of State

We compute zero-frequency (neutral) quasi-normal f-modes of fully relativistic and rapidly rotating neutron stars, using several realistic equations of state (EOSs) for neutron star matter. The zero-frequency modes signal the onset of the gravitational radiation-driven instability. We find that the l=m=2 (bar) f-mode is unstable for stars with gravitational mass as low as 1.0 - 1.2 M_\odot, depending on the EOS. For 1.4 M_\odot neutron stars, the bar mode becomes unstable at 83 % - 93 % of the maximum allowed rotation rate. For a wide range of EOSs, the bar mode becomes unstable at a ratio of rotational to gravitational energies T/W \sim 0.07-0.09 for 1.4 M_\odot stars and T/W \sim 0.06 for maximum mass stars. This is to be contrasted with the Newtonian value of T/W \sim 0.14. We construct the following empirical formula for the critical value of T/W for the bar mode, (T/W)_2 = 0.115 - 0.048 M / M_{max}^{sph}, which is insensitive to the EOS to within 4 - 6 %. This formula yields an estimate for the neutral mode sequence of the bar mode as a function only of the star's mass, M, given the maximum allowed mass, M_{max}^{sph}, of a nonrotating neutron star. The recent discovery of the fast millisecond pulsar in the supernova remnant N157B, supports the suggestion that a fraction of proto-neutron stars are born in a supernova collapse with very large initial angular momentum. Thus, in a fraction of newly born neutron stars the instability is a promising source of continuous gravitational waves. It could also play a major role in the rotational evolution (through the emission of angular momentum) of merged binary neutron stars, if their post-merger angular momentum exceeds the maximum allowed to form a Kerr black hole.