Johannes Ruoff

Radial oscillations of relativistic stars

We present a new survey of the radial oscillation modes of neutron stars. This study complements and corrects earlier studies of radial oscillations. We present an extensive list of frequencies for the most common equations of state and some more recent ones. In order to check the accuracy, we use two different numerical schemes which yield the same results. The stimulation for this work comes from the need of the groups that evolve the full nonlinear Einstein equations to have reliable results from perturbation theory for comparison.

On the r-mode spectrum of relativistic stars: the inclusion of the radiation reaction

We consider both mode calculations and time-evolutions of axial r modes for relativistic uniformly rotating non-barotropic neutron stars, using the slow-rotation formalism, in which rotational corrections are considered up to linear order in the angular velocity Ω. We study various stellar models, such as uniform density models, polytropic models with different polytropic indices n, and some models based on realistic equations of state. For weakly relativistic uniform density models and polytropes with small values of n, we can recover the growth times predicted from Newtonian theory when standard multipole formulae for the gravitational radiation are used. However, for more compact models, we find that relativistic linear perturbation theory predicts a weakening of the instability compared to the Newtonian results. When turning to polytropic equations of state, we find that for certain ranges of the polytropic index n, the r mode disappears, and instead of a growth, the time-evolutions show a rapid decay of the amplitude. This is clearly at variance with the Newtonian predictions. It is, however, fully consistent with our previous results obtained in the low-frequency approximation.

On the r-mode spectrum of relativistic stars in the low-frequency approximation

ABSTRA C T The axial modes for non-barotropic relativistic rotating neutron stars with uniform angular velocity are studied, using the slow-rotation formalism together with the low-frequency approximation, first investigated by Kojima. The time-independent form of the equations leads to a singular eigenvalue problem, which admits a continuous spectrum. We show that for la 2, it is nevertheless also possible to find discrete mode solutions (the r modes). However, under certain conditions related to the equation of state and the compactness of the stellar model, the eigenfrequency lies inside the continuous band and the associated velocity perturbation is divergent; hence these solutions have to be discarded as being unphysical. We corroborate our results by explicitly integrating the time-dependent equations. For stellar models admitting a physical r-mode solution, it can indeed be excited by arbitrary initial data. For models admitting only an unphysical mode solution, the evolutions do not show any tendency to oscillate with the respective frequency. For higher values of l it seems that in certain cases there are no mode solutions at all.

New approach to the evolution of neutron star oscillations

We present a new derivation of the perturbation equations governing the oscillations of relativistic nonrotating 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 three-metric but also the extrinsic curvature of the initial slice has 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 ringdown 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.

Inertial modes of slowly rotating relativistic stars in the Cowling approximation

We study oscillations of slowly rotating relativistic barotropic as well as non-barotropic polytropic stars in the Cowling approximation, including first-order rotational corrections. By taking into account the coupling between the polar and axial equations, we find that, in contrast with previous results, the m = 2 r modes are essentially unaffected by the continuous spectrum and exist even for very relativistic stellar models. In order to solve the infinite system of coupled equations numerically we have to truncate it at some value l max. Although the time-dependent equations are regular and can be evolved numerically without any problems, the eigenvalue equations possess a singular structure, which is related to the existence of a continuous spectrum. This prevents the numerical computation of an eigenmode if its eigenfrequency falls inside the continuous spectrum. The properties of the latter depend strongly on the cut-off value l max and it can consist of several either disconnected or overlapping patches, which are broader the more relativistic the stellar model is. By discussing the dependence of the continuous spectrum as a function of both the cut-off value l max and the compactness M/R, we demonstrate how it affects the inertial modes. By evolving the time-dependent equations we are able to show that some of the inertial modes can actually exist inside the continuous spectrum, while some cannot. For more compact and therefore more relativistic stellar models the width of the continuous spectrum increases strongly and consequently some of the inertial modes, which exist in less relativistic stars, disappear.

Close-limit approximation to neutron star collisions

We develop a close-limit approximation to the head-on collision of two neutron stars similar to that used to treat the merger of black hole binaries. This approximation can serve as a useful benchmark test for future fully non-linear studies. For neutron star binaries, the close-limit approximation involves assuming that the merged object can be approximated as a perturbed, stable neutron star during the ring-down phase of the coalescence. We introduce a prescription for the construction of initial data sets, discuss the physical plausibility of the various assumptions involved, and briefly investigate the character of the gravitational radiation produced during the merger. The numerical results show that several of the merged objects fluid pulsation modes are excited to a significant level.

Excitation of neutron star oscillations by an orbiting particle

The excitation of neutron stars is expected to be an important source of gravitational radiation. Therefore it is of fundamental importance to investigate mechanisms that trigger oscillations in neutron stars in order to characterize the emitted radiation. We present results from a numerical study of the even-parity gravitational radiation generated from a particle orbiting a neutron star. In contrast to previous calculations performed in the Fourier domain, we use the direct time evolution of the perturbation equations. We focus our investigation on those conditions on the orbital parameters that favor the excitation of w modes. We find that for astrophysically realistic conditions, there is practically no w-mode contribution to the emitted radiation. Only for particles with ultrarelativistic orbital speeds

Evolution equations for the perturbations of slowly rotating relativistic stars

g~0.9c

w-mode instability of ultracompact relativistic stars

does the w-mode significantly contribute to the total emitted gravitational energy. We also stress the importance of setting consistent initial configurations.

PERTURBATIONS OF SLOWLY ROTATING RELATIVISTIC STARS

We present a new derivation of the equations governing the oscillations of slowly rotating relativistic stars. Previous investigations have been mostly carried out in the Regge–Wheeler gauge. However, in this gauge the process of linearizing the Einstein field equations leads to perturbation equations which as such cannot be used to perform numerical time evolutions. It is only through the tedious process of combining and rearranging the perturbation variables in a clever way that the system can be cast into a set of hyperbolic first order equations, which is then well suited for the numerical integration. The equations remain quite lengthy, and we therefore rederive the perturbation equations in a different gauge, which has been first proposed by Battiston et al. (1970). Using the ADM formalism, one is immediately lead to a first order hyperbolic evolution system, which is remarkably simple and can be numerically integrated without many further manipulations. Moreover, the symmetry between the polar and the axial equations becomes directly apparent.