Slow Rotation Approximation Research Articles

W-mode instability of ultracompact relativistic stars

In this article we show that the mainly spacetime w modes may become unstable in a rotating ultracompact $(R<3M)$ relativistic star. We provide results for the axial modes of a rotating star in the slow rotation approximation, and demonstrate that an instability may become active in stars with an ergosphere. This suggests a close association with the so-called ``ergosphere instability'' discussed by other authors. The unstable modes are found to grow on a time scale of seconds to minutes. We discuss briefly whether the instability is likely to have any astrophysical relevance.

Product angular distributions in dissociative recombination.

The dependence of the dissociative recombination cross section upon the angle between the incoming electron beam and the ion internuclear axis is determined for diatomic molecules. Product angular distributions are derived for the component partial waves of the Coulomb wave function. In agreement with earlier results for dissociative attachment, it is shown that in the slow rotation approximation, if electron capture is dominated by a single partial wave, the product angular distribution is given by the square of the absolute value of the partial wave spherical harmonic describing the incoming electron.

Inertial modes of non-stratified superfluid neutron stars

We present results concerning adiabatic inertial-mode oscillations of non-stratified superfluid neutron stars in Newtonian gravity, using the anelastic and slow-rotation approximations. We consider a simple two-fluid model of a superfluid neutron star, where one fluid consists of the superfluid neutrons and the second fluid contains all the comoving constituents (protons, electrons). The two fluids are assumed to be ``free'' in the sense that vortex-mediated forces like mutual friction or pinning are absent, but they can be coupled by the equation of state, in particular by entrainment. The stationary background consists of the two fluids rotating uniformly around the same axis with potentially different rotation rates. We study the special cases of co-rotating backgrounds, vanishing entrainment, and the purely toroidal r-modes, analytically. We calculate numerically the eigenfunctions and frequencies of inertial modes in the general case of non co-rotating backgrounds, and study their dependence on the relative rotation rate and entrainment. In these non-stratified models we find avoided crossings only between associated mode-pairs, eg. an ``ordinary'' mode and its ``superfluid'' counterpart, while other mode-frequencies generally cross as the background parameters are varied. We confirm (for the first time in a mode calculation) the onset of a ``two-stream instability'' at a critical relative background rotation rate, and study some of the properties of this instability for the inertial modes.

Two-parameter nonlinear spacetime perturbations: gauge transformations and gauge invariance

An implicit fundamental assumption in relativistic perturbation theory is that there exists a parametric family of spacetimes that can be Taylor expanded around a background. The choice of the latter is crucial to obtain a manageable theory, so that it is sometime convenient to construct a perturbative formalism based on two (or more) parameters. The study of perturbations of rotating stars is a good example: in this case one can treat the stationary axisymmetric star using a slow rotation approximation (expansion in the angular velocity Ω), so that the background is spherical. Generic perturbations of the rotating star (say parametrized by λ) are then built on top of the axisymmetric perturbations in Ω. Clearly, any interesting physics requires nonlinear perturbations, as at least terms λΩ need to be considered. In this paper, we analyse the gauge dependence of nonlinear perturbations depending on two parameters, derive explicit higher-order gauge transformation rules and define gauge invariance. The formalism is completely general and can be used in different applications of general relativity or any other spacetime theory.

Slowly rotating charged fluid balls and their matching to an exterior domain

The slow-rotation approximation of Hartle is developed to a setting where a charged rotating fluid is present. The linearized Einstein-Maxwell equations are solved on the background of the Reissner-Nordstrom space-time in the exterior electrovacuum region. The theory is put to action for the charged generalization of the Wahlquist solution found by Garcia. The Garcia solution is transformed to coordinates suitable for the matching and expanded in powers of the angular velocity. The two domains are then matched along the zero pressure surface using the Darmois-Israel procedure. We prove a theorem to the effect that the exterior region is asymptotically flat if and only if the parameter C_{2}, characterizing the magnitude of an external magnetic field, vanishes. We obtain the form of the constant C_{2} for the Garcia solution. We conjecture that the Garcia metric cannot be matched to an asymptotically flat exterior electrovacuum region even to first order in the angular velocity. This conjecture is supported by a high precision numerical analysis.

The fast and slow rotation approximations for the description of the kinetics of triplet radical quenching and electron spin polarization

The specific features of the mechanisms and kinetics of generation of net chemically induced dynamic electron polarization (CIDEP) in triplet radical quenching (TRQ) in liquids is analysed in detail. The problem reduces to the analysis of fairly strong non-adiabatic transitions between states of the triplet radical spin Hamiltonian which are known to determine CIDEP generation in TRQ. The analysis is performed in two limits of fast and slow rotation of the triplet molecule using the previously developed method of treatment for non-adiabatic transitions. The method made it possible to derive analytical formulas for the CIDEP generation probability P e and rate K e, and for the TRQ probability P q and rate K q in the case of relatively strong quenching. It is shown that the dependence of K e on the relative diffusion coefficient D r is of bell shape and cannot be described correctly by the usually applied relation K e = K q P e.

Slowly rotating superfluid Newtonian neutron star model with entrainment

We develop a formalism that can be used to model slowly rotating superfluid Newtonian neutron stars. A simple two-fluid model is used to describe the matter, where one fluid consists of the superfluid neutrons that are believed to exist in the inner crust and core of mature neutron stars, while the other fluid is a charge neutral conglomerate of the remaining constituents (crust nuclei, core superconducting protons, electrons, etc). We include the entrainment effect, which is a non-dissipative interaction between the two fluids whereby a momentum induced in one of the fluids will cause part of the mass of the other fluid to be carried along. The equations that describe rotational equilibria (i.e. axisymmetric and stationary configurations) are approximated using the slow-rotation approximation; an expansion in terms of the rotation rates of the two fluids where only terms up to second-order are kept. Our formalism allows the neutrons to rotate at a rate different from that of the charged constituents. For a particular equation of state that is quadratic in the two mass-densities and relative velocities of the fluids, we find an analytic solution to the slow-rotation equations. This result provides an elegant generalisation to the two-fluid problem of the standard expressions for the one-fluid polytrope E / ρ 2 . The model equation of state includes entrainment and is general enough to allow for realistic values for, say, mass and radius of the star. It also includes a mixed term in the mass densities that can be related to symmetry energy terms that appear in more realistic equations of state. We use the analytic solution to explore how relative rotation between the two fluids, the symmetry energy term, and entrainment affect the neutron star's local distribution of particles, as well as global quantities as the Kepler limit, ellipticity, and moments of inertia.

SLOWLY ROTATING BOSON-FERMION STAR

Relativistic prescription is used to study the slow rotation of stars composed by self-gravitating bosons and fermions (fermions may be considered as neutrons). Previous results demand that purely boson stars are unable to display slow rotation, if one uses relativistic prescription with classical scalar fields. In contrast to this, the present work shows that a combined boson-fermion star in its ground-state can rotate. Its structure and stability are analysed under slow rotation approximations.

R-mode oscillations of rapidly rotating Newtonian stars: A new numerical scheme and its application to the spin evolution of neutron stars

We have developed a new numerical scheme to solve r-mode oscillations of rapidly rotating polytropic stars in Newtonian gravity. In this scheme, Euler perturbations of the density, three components of the velocity are treated as four unknown quantities together with the oscillation frequency. For the basic equations of oscillations, the compatibility equations are used instead of the linearized equations of motion. By using this scheme, we have solved the classical r-mode oscillations of rotational equilibrium sequences of polytropes with the polytropic indices $N=0.5,$ 1.0, and 1.5 for $m=2,3,$ and 4 modes. Here m is the rank of the spherical harmonics ${Y}_{l}^{m}.$ These results have been applied to investigate the evolution of uniformly rotating hot young neutron stars by considering the effect of gravitational radiation and viscosity. We have found that the maximum angular velocities of neutron stars are around 10--20 % of the Keplerian angular velocity irrespective of the softness of matter. This confirms the results obtained from the analysis of r-modes with the slow rotation approximation employed by many authors.

A numerical study of the r-mode instability of rapidly rotating nascent neutron stars

The first results of numerical analysis of classical r-modes of {\it rapidly} rotating compressible stellar models are reported. The full set of linear perturbation equations of rotating stars in Newtonian gravity are numerically solved without the slow rotation approximation. A critical curve of gravitational wave emission induced instability which restricts the rotational frequencies of hot young neutron stars is obtained. Taking the standard cooling mechanisms of neutron stars into account, we also show the `evolutionary curves' along which neutron stars are supposed to evolve as cooling and spinning-down proceed. Rotational frequencies of $1.4M_{\odot}$ stars suffering from this instability decrease to around 100Hz when the standard cooling mechanism of neutron stars is employed. This result confirms the results of other authors who adopted the slow rotation approximation.

Inertial Modes of Slowly Rotating Isentropic Stars

We investigate inertial mode oscillations of slowly and uniformly rotating, isentropic, Newtonian stars. Inertial mode oscillations are induced by the Coriolis force due to the star's rotation, and their characteristic frequencies are comparable with the rotation frequency $\Omega$ of the star. So called r-mode oscillations form a sub-class of the inertial modes. In this paper, we use the term ``r-modes'' to denote the inertial modes for which the toroidal motion dominates the spheroidal motion, and the term ``inertial modes'' to denote the inertial modes for which the toroidal and spheroidal motions have comparable amplitude to each other. Using the slow rotation approximation consistent up to the order of $\Omega^3$, we study the properties of the inertial modes and r-modes, by taking account of the effect of the rotational deformation of the equilibrium on the eigenfrequencies and eigenfunctions. The eigenfrequencies of the r-modes and inertial modes calculated in this paper are in excellent agreement with those obtained by Lindblom et al (1999) and Lockitch & Friedman (1998). We also estimate the dissipation timescales due to the gravitational radiation and several viscous processes for polytropic neutron star models. We find that for the inertial modes, the mass multipole gravitational radiation dominates the current multipole radiation, which is dominating in the case of the r-modes. It is also found that the inertial mode instability is more unstable than previously reported by Lockitch & Friedman (1998), and survives the viscous damping processes relevant in neutron stars.

Gravitational Waves from Relativistic Stars

Stellar pulsations in rotating relativistic stars are reviewed. Slow rotation approximation is applied to solving the Einstein equations. The rotational effects on the non-axisymmetric oscillations are explicitly shown in the polar and axial modes.

Mass inflation in a rotating charged black hole.

The structure of the Cauchy horizon of a charged rotating black hole is analyzed under the combined effect of an ingoing and outgoing flux of gravitational waves. In particular, by means of an axisymmetric realization of the Ori model, the growth of the mass parameter near the Cauchy horizon is studied in the slow rotation approximation. It is shown that the mass parameter inflates, while the angular momentum per unit mass deflates, but initial deviations from spherical symmetry survive. \textcopyright{} 1996 The American Physical Society.

Fast Rotation of Neutron Stars

We show that for realistic equations of state of dense matter, the universal proportionality factor relating the maximum rotation rate of neutron stars due to mass-shedding limit to the mass and radius of maximum allowable mass configuration of non-rotating models results from a universal proportionality between masses and radii of static maximum-mass neutron stars and those of maximally rotating configurations. These empirical relations cannot be obtained in the slow rotation approximation.

Interior solutions for rotating fluid spheres.

Equations depicting the interior of rotating fluid spheres are investigated. The analysis leads to a class of solutions which contains all previously known interiors incorporating slow uniform rotation, and several new metrics are presented. One is the slow-rotation analog of the Schwarzschild interior, which until now was known only through numerical integration of the field equations. Models are developed for a solution by Tolman, a recent interior presented by the author, and the slow-rotation approximation of the Wahlquist interior. Expressions for inertial-frame dragging and the moments of inertia for these metrics are presented.

Gravi-inertial reference frames in Schwarzschild and Kerr spaces

Gravi-inertial reference frames — the analogue of inertial frames in two-dimensional space — are constructed in Schwarzschild and Kerr spaces. With their help the energy, momentum, and angular momentum of these systems are determined. A form is found for writing Kerr's solution in Bondi-Sachs coordinates in the slow-rotation approximation. A new solution of Einstein's equations is found in this approximation which describes the gravitational field of a rotating and radiating source.

Rotating massive stars in general relativity

Equilibrium models of uniformly rotating massive stars are investigated, using a weak field, slow rotation approximation, which is shown to be adequate for all cases of interest. The fate of radial perturbations about these equilibrium configurations is investigated using a linear­ized stability analysis to determine the oscillation frequency σ in a peturbation ∝ e iσ t . An eigenvalue equation for σ 2 is obtained which can be made self adjoint with respect to the spatial metric, and a variational principle to determine σ 2 is derived. Numerical determinations of σ 2 have been carried out for a variety of masses, radii and rotational velocities, and these results are incorporated in a simple formula that gives the dependence of σ 2 on these quantities. The condition for instability, σ 2 negative, is determined, and it is found that for large masses and maximum rotation velocity, so that when centrifugal force balances gravity at the surface, a massive star becomes unstable when its radius is 208 times the Schwarzschild radius 2 GM/c 2 .