Shigeyuki Karino

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

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.

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.

Linear Stability Analysis of Differentially Rotating Polytropes: New Results for the m = 2 f-Mode Dynamical Instability

We have studied the f-mode oscillations of differentially rotating polytropes by making use of the linear stability analysis. We found that the critical values of T/|W| where the dynamical instability against the m = 2 f-mode oscillations sets in decrease down to T/|W| ~ 0.20 as the degree of differential rotation becomes higher. Here m is an azimuthal mode number and T and W are the rotational energy and gravitational potential energy, respectively. This tendency is almost independent of the compressibility of the polytropes. These are the first exact results of the linear stability analysis for the occurrence of the dynamical instability against the m = 2 f-modes.

Accretion mode of the ultraluminous X-ray source M82 X-2

Periodic pulsations have been found in emission from the ultra-luminous X-ray source (ULX) M82 X-2, strongly suggesting that the emitter is a rotating neutron star rather than a black hole. However, the radiation mechanisms and accretion mode involved have not yet been clearly established. In this paper, we examine the applicability to this object of standard accretion modes for high mass X-ray binaries (HMXBs). We find that spherical wind accretion, which drives OB-type HMXBs, cannot apply here but that there is a natural explanation in terms of an extension of the picture for standard Be-type HMXBs. We show that a neutron star with a moderately strong magnetic field, accreting from a disc-shaped wind emitted by a Be-companion, could be compatible with the observed relation between spin and orbital period. A Roche lobe overflow picture is also possible under certain conditions.

Gravitational radiation reaction driven secular instability of f-mode oscillations in differentially rotating Newtonian stars

We have carried out the linear stability analysis of equilibrium sequences of differentially rotating Newtonian polytropic stars. In particular, we have found critical points along the equilibrium sequences for the gravitational radiation reaction driven secular instability of f-mode oscillations with azimuthal wavenumbers m = 2, 3, and 4. We show that the critical values of T/|W| where the instability sets in strongly depend on the degree of differential rotation of the models. Here, T and W are the rotational kinetic energy and the gravitational energy of the star, respectively. We also find that as the degree of differential rotation is increased, the dependence of the critical value of T/|W| on the equations of state becomes weaker.

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 {\it 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

Radiative Column and Light Curve of X-Ray Binary Pulsars

N = 0.5, 1.0

Orbital parameters of supergiant fast X-ray transients

and 1.5 for