Yoshiharu Eriguchi

Dynamical instability of differentially rotating stars

We study the dynamical instability against bar-mode deformation of differentially rotating stars. We performed numerical simulation and linear perturbation analysis adopting polytropic equations of state with the polytropic index n= 1. It is found that rotating stars of a high degree of differential rotation are dynamically unstable even for the ratio of the kinetic energy to the gravitational potential energy of O(0.01). Gravitational waves from the final non-axisymmetric quasi-stationary states are calculated in the quadrupole formula. For rotating stars of mass 1.4 M⊙ and radius several 10 km, gravitational waves have frequency several 100 Hz and effective amplitude ∼5 × 10−22 at a distance of ∼100 Mpc.

Dynamical bar-mode instability of differentially rotating stars: effects of equations of state and velocity profiles

As an extension of our previous work, we investigate the dynamical instability against nonaxisymmetric bar-mode deformations of differentially rotating stars in Newtonian gravity by varying the equations of state and velocity profiles. We performed the numerical simulation and the follow-up linear stability analysis by adopting polytropic equations of state with polytropic indices n = 1, 3/2 and 5/2, and with two types of angular velocity profiles (the so-called j-constant-like and Kepler-like laws). It is confirmed that rotating stars with a high degree of differential rotation are dynamically unstable against bar-mode deformation, even when the ratio of the kinetic energy to the gravitational potential energy β is of order 0.01. The criterion for the onset of bar-mode dynamical instability depends weakly on the polytropic index n and the angular velocity profile, as long as the degree of differential rotation is high. Gravitational waves from the final non-axisymmetric quasi-stationary states are calculated using the quadrupole formula. For proto-neutron stars of mass 1.4 M� , radius ∼30 km and β 0.1, such gravitational waves have a frequency of ∼600‐1400 Hz, and the effective amplitude is larger than 10 −22 at a distance of about 100 Mpc, irrespective of n and the angular

New numerical method for constructing quasiequilibrium sequences of irrotational binary neutron stars in general relativity

We propose a new numerical method to compute quasi-equilibrium sequences of general relativistic irrotational binary neutron star systems. It is a good approximation to assume that (1) the binary star system is irrotational, i.e. the vorticity of the flow field inside component stars vanishes everywhere (irrotational flow), and (2) the binary star system is in quasi-equilibrium, for an inspiraling binary neutron star system just before the coalescence as a result of gravitational wave emission. We can introduce the velocity potential for such an irrotational flow field, which satisfies an elliptic partial differential equation (PDE) with a Neumann type boundary condition at the stellar surface. For a treatment of general relativistic gravity, we use the Wilson--Mathews formulation, which assumes conformal flatness for spatial components of metric. In this formulation, the basic equations are expressed by a system of elliptic PDEs. We have developed a method to solve these PDEs with appropriate boundary conditions. The method is based on the established prescription for computing equilibrium states of rapidly rotating axisymmetric neutron stars or Newtonian binary systems. We have checked the reliability of our new code by comparing our results with those of other computations available. We have also performed several convergence tests. By using this code, we have obtained quasi-equilibrium sequences of irrotational binary star systems with strong gravity as models for final states of real evolution of binary neutron star systems just before coalescence. Analysis of our quasi-equilibrium sequences of binary star systems shows that the systems may not suffer from dynamical instability of the orbital motion and that the maximum density does not increase as the binary separation decreases.

Equilibrium Configurations of Magnetized Rotating Polytropes: Effects of Strong Toroidal Magnetic Fields in Addition to Poloidal Magnetic Fields

We have constructed many equilibrium sequences of magnetized polytropic stars with infinite conductivity for polytropes of indices N = 0.5, 1, and 3. Not only poloidal magnetic fields but also toroidal magnetic fields have been included in addition to rotation. By choosing simplified forms for arbitrary functions appearing in the formulation of magnetized barotropic equilibrium stars, we have obtained strongly magnetized polytropes whose toroidal magnetic fields are of comparable strength to those of poloidal magnetic fields. The exterior magnetic fields of the obtained stars consist of dipole-like poloidal fields, which decrease as r-3 when r → ∞. On the other hand, the interior magnetic fields are mixed poloidal-toroidal, which are composed of tori of twisted field lines around the symmetry axis and of untwisted poloidal fields that continue to the exterior fields penetrating the surfaces of the stars.

Properties of general relativistic, irrotational binary neutron stars in close quasiequilibrium orbits: Polytropic equations of state

We investigate close binary neutron stars in quasiequilibrium states in a general relativistic framework. We assume conformal flatness for the spatial metric and irrotational velocity field for the neutron stars. We adopt the polytropic equation of state. The computation is performed for the polytropic index n(=0.5, 0.66667, 0.8, 1, 1.25), and compactness of neutron stars M/R(=0.03 - 0.3). Results of this paper are as follows. (i) The sequences of the irrotational binary are always terminated at an innermost orbit where a cusp (inner Lagrange point)appears at the inner edges of the stellar surface. The binaries with cusps are found to be dynamically unstable for n=0.5 and stable for n > 0.8 irrespective of M/R 0.13 irrespective of n, which indicates that the realistic binary neutron stars satisfy a necessary condition (q<1) for formation of a black hole before the merger. (vi) The specific angular momentum of any mass element in irrotational binary neutron stars at the innermost orbit appears to be too small to form a disk around black holes formed after the merger.

Neutral Points of Oscillation Modes along Equilibrium Sequences of Rapidly Rotating Polytropes in General Relativity: Application of the Cowling Approximation

The relativistic Cowling approximation in which all metric perturbations are omitted is applied to nonaxisymmetric infinitesimal oscillations of uniformly rotating general relativistic polytropes. Frequencies of lower order f-modes, which are important in analysis of secular instability driven by gravitational radiation, are investigated, and neutral points of the mode along equilibrium sequences of rotating polytropes are determined. Since this approximation becomes more accurate as stars are more relativistic and/or as they rotate more rapidly, we will be able to analyze how a rotation period of a neutron star may be limited by this instability. Possible errors in determining neutral points caused by omitting metric perturbations are also estimated.

Quasi‐radial modes of rotating stars in general relativity

By using the Cowling approximation, quasi-radial modes of rotating general relativistic stars are computed along equilibrium sequences from non-rotating to maximally rotating models. The eigenfrequencies of these modes are decreasing functions of the rotational frequency. The eigenfrequency curve of each mode as a function of the rotational frequency has discontinuities, which arise from the avoided crossing with other curves of axisymmetric modes.

R -mode oscillations of differentially and rapidly rotating Newtonian polytropic stars

For analysis of the r-mode oscillation of hot young neutron stars, it is necessary to consider the effect of differential rotation, because viscosity is not strong enough for differentially rotating young neutron stars to be led to uniformly rotating configurations on a very short time scale after their birth. In this paper, we have developed a numerical scheme to solve the r-mode oscillations of differentially rotating polytropic inviscid stars. This is the extended version of the method which was applied to compute the r-mode oscillations of uniformly rotating Newtonian polytropic stars. By using this new method, we have succeeded in obtaining eigenvalues and eigenfunctions of the r-mode oscillations of differentially rotating polytropic stars. Our numerical results show that as the degree of differential rotation is increased, it becomes more difficult to solve the r-mode oscillations for slightly deformed configurations from a sphere compared to solving the r-mode oscillations of considerably deformed stars. One reason for this seems that for slightly deformed stars a corotation cylinder appears near the stellar surface region if the degree of differential rotation is large enough. This is similar to the situation that the perturbational approach of low-frequency r-mode oscillations for slowly rotating stars in general relativity results in a singular eigenvalue problem.

Newtonian models for black hole-gaseous star close binary systems

Circularly orbiting black hole--gaseous star close binary systems are examined by using numerically exact stationary configurations in the framework of Newtonian gravity. We have chosen a polytropic star for the fluid component of the binary system and considered two ideal situations: (i) a synchronously rotating star and (ii) an irrotationally rotating star. They correspond to a rotating star under the influence of viscosity and to that in the inviscid limit, respectively. By analysing the stationary sequences of binary systems with small separations, we can discuss the final stages of black hole--gaseous star close binary systems. Our computational results show that the binary systems reach the Roche(--Riemann) limit states or the Roche lobe filling states without suffering from hydrodynamical instability caused by the tidal force for a certain realistic parameter range of the mass ratio and the polytropic index. Moreover, some of these stable Roche(--Riemann) limits or Roche lobe filling states survive even under the general relativistic effect. Therefore, at the final stage of the evolution, which is caused by the emission of gravitational waves, Roche lobe overflow is another possibility in addition to the merging of a black hole and a star. For a sufficiently stiff equation of state (the polytropic index N0.3--0.7, depending on the mass ratio), the turning point, which corresponds to the secular instability limit for the synchronous binary case and the dynamical instability limit for the irrotational binary case, disappears on the solution sequence. Therefore, even for a realistic parameter range, our results are different from the semi-analytic results computed by the ellipsoidal approximation in which the turning point always appears.

Frequencies of f-Modes in Differentially Rotating Relativistic Stars and Secular Stability Limits

We have computed the eigenfrequencies of f-modes for constant rest mass sequences of rapidly rotating relativistic inviscid stars in differential rotation. The frequencies have been calculated neglecting the metric perturbations (the relativistic Cowling approximation) and expressed as a function of the ratio between the rotational kinetic energy and the absolute value of the gravitational energy of the stellar model, β ≡ T/|W|. The zeros and the endpoints of these sequences mark, respectively, the onset of the secular instability driven by gravitational radiation reaction and the maximum value of β at which an equilibrium model exists. In differentially rotating stars, the secular stability limits appear at a β larger than those found for uniformly rotating stars. Differential rotation, on the other hand, also allows for the existence of equilibrium models at values of β larger than those for uniformly rotating stars, moving the endpoint of the sequences to larger β. As a result, for some degrees of differential rotation, the onset of the secular instability for f-modes is generally favored by the presence of differential rotation.