R-Modes in Superfluid Neutron Stars

The analogs of r-modes in superfluid neutron stars are studied here. These modes, which are governed primarily by the Coriolis force, are identical to their ordinary-fluid counterparts at the lowest order in the small angular-velocity expansion used here. The equations that determine the next order terms are derived and solved numerically for fairly realistic superfluid neutron-star models. The damping of these modes by superfluid ``mutual friction'' (which vanishes at the lowest order in this expansion) is found to have a characteristic time-scale of about 10^4 s for the m=2 r-mode in a ``typical'' superfluid neutron-star model. This time-scale is far too long to allow mutual friction to suppress the recently discovered gravitational radiation driven instability in the r-modes. However, the strength of the mutual friction damping depends very sensitively on the details of the neutron-star core superfluid. A small fraction of the presently acceptable range of superfluid models have characteristic mutual friction damping times that are short enough (i.e. shorter than about 5 s) to suppress the gravitational radiation driven instability completely.Comment: 15 pages, 8 figure

r-modes in Relativistic Superfluid Stars

We discuss the modal properties of the $r$-modes of relativistic superfluid neutron stars, taking account of the entrainment effects between superfluids. In this paper, the neutron stars are assumed to be filled with neutron and proton superfluids and the strength of the entrainment effects between the superfluids are represented by a single parameter $\eta$. We find that the basic properties of the $r$-modes in a relativistic superfluid star are very similar to those found for a Newtonian superfluid star. The $r$-modes of a relativistic superfluid star are split into two families, ordinary fluid-like $r$-modes ($r^o$-mode) and superfluid-like $r$-modes ($r^s$-mode). The two superfluids counter-move for the $r^s$-modes, while they co-move for the $r^o$-modes. For the $r^o$-modes, the quantity $\kappa\equiv\sigma/\Omega+m$ is almost independent of the entrainment parameter $\eta$, where $m$ and $\sigma$ are the azimuthal wave number and the oscillation frequency observed by an inertial observer at spatial infinity, respectively. For the $r^s$-modes, on the other hand, $\kappa$ almost linearly increases with increasing $\eta$. It is also found that the radiation driven instability due to the $r^s$-modes is much weaker than that of the $r^o$-modes because the matter current associated with the axial parity perturbations almost completely vanishes.Comment: 14 pages, 4 figures. To appear in Physical Review

Reducing orbital eccentricity in binary black hole simulations

Binary black hole simulations starting from quasi-circular (i.e., zero radial velocity) initial data have orbits with small but non-zero orbital eccentricities. In this paper the quasi-equilibrium initial-data method is extended to allow non-zero radial velocities to be specified in binary black hole initial data. New low-eccentricity initial data are obtained by adjusting the orbital frequency and radial velocities to minimize the orbital eccentricity, and the resulting ($\sim 5$ orbit) evolutions are compared with those of quasi-circular initial data. Evolutions of the quasi-circular data clearly show eccentric orbits, with eccentricity that decays over time. The precise decay rate depends on the definition of eccentricity; if defined in terms of variations in the orbital frequency, the decay rate agrees well with the prediction of Peters (1964). The gravitational waveforms, which contain $\sim 8$ cycles in the dominant l=m=2 mode, are largely unaffected by the eccentricity of the quasi-circular initial data. The overlap between the dominant mode in the quasi-circular evolution and the same mode in the low-eccentricity evolution is about 0.99.Comment: 27 pages, 9 figures; various minor clarifications; accepted to the "New Frontiers" special issue of CQ

R-modes of neutron stars with the superfluid core

We investigate the modal properties of the $r$-modes of rotating neutron stars with the core filled with neutron and proton superfluids, taking account of entrainment effects between the superfluids. The stability of the $r$-modes against gravitational radiation reaction is also examined considering viscous dissipation due to shear and a damping mechanism called mutual friction between the superfluids in the core. We find the $r$-modes in the superfluid core are split into ordinary $r$-modes and superfluid $r$-modes, which we call, respectively, $r^o$- and $r^s$-modes. The two superfluids in the core flow together for the $r^o$-modes, while they counter-move for the $r^s$-modes. For the $r^o$-modes, the coefficient $\kappa_0\equiv\lim_{\Omega\to 0}\omega/\Omega$ is equal to $2m/[l^\prime(l^\prime+1)]$, almost independent of the parameter $\eta$ that parameterizes the entrainment effects between the superfluids, where $\Omega$ is the angular frequency of rotation, $\omega$ the oscillation frequency observed in the corotating frame of the star, and $l^\prime$ and $m$ are the indices of the spherical harmonic function representing the angular dependence of the $r$-modes. For the $r^s$-modes, on the other hand, $\kappa_0$ is equal to $2m/[l^\prime(l^\prime+1)]$ at $\eta=0$ (no entrainment), and it almost linearly increases as $\eta$ is increased from $\eta=0$. The mutual friction in the superfluid core is found ineffective to stabilize the $r$-mode instability caused by the $r^o$-mode except in a few narrow regions of $\eta$. The $r$-mode instability caused by the $r^s$-modes, on the other hand, is extremely weak and easily damped by dissipative processes in the star.Comment: 22 pages, 22 figures, accepted for publication in the Astrophysical Journa

Dynamic modelling of nitrous oxide emissions from three Swedish sludge liquor treatment systems

The objective of this paper is to model the dynamics and validate the results of nitrous oxide (N2O) emissions from three Swedish nitrifying/denitrifying, nitritation and anammox systems treating real anaerobic digester sludge liquor. The Activated Sludge Model No. 1 is extended to describe N2O production by both heterotrophic and autotrophic denitrification. In addition, mass transfer equations are implemented to characterize the dynamics of N2O in the water and the gas phases. The biochemical model is simulated and validated for two hydraulic patterns: (1) a sequencing batch reactor; and (2) a moving-bed biofilm reactor. Results show that the calibrated model is partly capable of reproducing the behaviour of N2O as well as the nitritation/nitrification/denitrification dynamics. However, the results emphasize that additional work is required before N2O emissions from sludge liquor treatment plants can be generally predicted with high certainty by simulations. Continued efforts should focus on determining the switching conditions for different N2O formation pathways and, if full-scale data are used, more detailed modelling of the measurement devices might improve the conclusions that can be drawn.</jats:p

R-mode oscillations and rocket effect in rotating superfluid neutron stars. I. Formalism

We derive the hydrodynamical equations of r-mode oscillations in neutron stars in presence of a novel damping mechanism related to particle number changing processes. The change in the number densities of the various species leads to new dissipative terms in the equations which are responsible of the {\it rocket effect}. We employ a two-fluid model, with one fluid consisting of the charged components, while the second fluid consists of superfluid neutrons. We consider two different kind of r-mode oscillations, one associated with comoving displacements, and the second one associated with countermoving, out of phase, displacements.Comment: 10 page

Time-Independent Gravitational Fields

This article reviews, from a global point of view, rigorous results on time independent spacetimes. Throughout attention is confined to isolated bodies at rest or in uniform rotation in an otherwise empty universe. The discussion starts from first principles and is, as much as possible, self-contained.Comment: 47 pages, LaTeX, uses Springer cl2emult styl

Nonradial Oscillations of Neutron Stars with a Solid Crust -- Analysis in the Relativistic Cowling Approximation--

Nonradial oscillations of relativistic neutron stars with a solid crust are computed in the relativistic Cowling approximation, in which all metric perturbations are ignored. For the modal analysis, we employ three-component relativistic neutron star models with a solid crust, a fluid core, and a fluid ocean. As a measure for the relativistic effects on the oscillation modes, we calculate the relative frequency difference defined as $\delta\sigma/\sigma\equiv(\sigma_{GR}-\sigma_N)/\sigma_{GR}$, where $\sigma_{GR}$ and $\sigma_R$ are, respectively, the relativistic and the Newtonian oscillation frequencies. The relative difference $\delta\sigma/\sigma$ takes various values for different oscillation modes of the neutron star model, and the value of $\delta\sigma/\sigma$ for a given mode depends on the physical properties of the models. We find that $|\delta\sigma/\sigma|$ is less than $\sim0.1$ for most of the oscillation modes we calculate, although there are a few exceptions such as the fundamental (nodeless) toroidal torsional modes in the crust, the surface gravity modes confined in the surface ocean, and the core gravity modes trapped in the fluid core. We also find that the modal properties, represented by the eigenfunctions, are not strongly affected by introducing general relativity. It is however shown that the mode characters of the two interfacial modes, associated with the core/crust and crust/ocean interfaces, have been interchanged between the two through an avoided crossing when we move from Newtonian dynamics to general relativistic dynamics.Comment: 14 pages, 9 figures, To appear in Astronomy and Astrophysic

Compact Zernike phase contrast X-ray microscopy using a single-element optic

We demonstrate Zernike phase contrast in a compact soft x-ray microscope using a single-element optic. The optic is a combined imaging zone plate and a Zernike phase plate and does not require any additional alignment or components. Contrast is increased and inversed in an image of a test object using the Zernike zone plate. This type of optic may be implemented into any existing x-ray microscope where phase contrast is of interest

Compact Zernike phase contrast X-ray microscopy using a single-element optic

We demonstrate Zernike phase contrast in a compact soft x-ray microscope using a single-element optic. The optic is a combined imaging zone plate and a Zernike phase plate and does not require any additional alignment or components. Contrast is increased and inversed in an image of a test object using the Zernike zone plate. This type of optic may be implemented into any existing x-ray microscope where phase contrast is of interest