Inertial modes of neutron stars with the superfluid core

We investigate the modal properties of inertial modes of rotating neutron stars with the core filled with neutron and proton superfluids, taking account of entrainment effects between the superfluids. In this paper, the entrainment effects are modeled by introducing a parameter $\eta$ so that no entrainment state is realized at $\eta=0$. We find that inertial modes of rotating neutron stars with the superfluid core are split into two families, which we call ordinary fluid inertial modes ($i^o$-mode) and superfluid inertial modes ($i^s$-mode). The two superfluids in the core counter-move for the $i^s$-modes. For the $i^o$-modes, $\kappa_0=\lim_{\Omega\to 0}\omega/\Omega$ is only weakly dependent on the entrainment parameter $\eta$, where $\Omega$ and $\omega$ are the angular frequency of rotation and the oscillation frequency observed in the corotating frame of the star, respectively. For the $i^s$-modes, on the other hand, $|\kappa_0|$ almost linearly increases as $\eta$ increases. Avoided crossings as functions of $\eta$ are therefore quite common between $i^o$- and $i^s$-modes. We find that some of the $i^s$-modes that are unstable against the gravitational radiation reaction at $\eta=0$ become stable when $\eta$ is larger than $\eta_{crit}$, the value of which depends on the mode. Since the radiation driven instability associated with the current multipole radiation is quite weak for the inertial modes and the mutual friction damping in the superfluid core is strong, the instability caused by the inertial modes will be easily suppressed unless the entrainment parameter $\eta$ is extremely small and the mutual friction damping is sufficiently weak.Comment: 19 pages, 20 figures. To appear in MNRA

Light curves from rapidly rotating neutron stars

We calculate light curves produced by a hot spot of a rapidly rotating neutron star, assuming that the spot is perturbed by a core $r$-mode, which is destabilized by emitting gravitational waves. To calculate light curves, we take account of relativistic effects such as the Doppler boost due to the rapid rotation and light bending assuming the Schwarzschild metric around the neutron star. We assume that the core $r$-modes penetrate to the surface fluid ocean to have sufficiently large amplitudes to disturb the spot. For a $l'=m$ core $r$-mode, the oscillation frequency $\omega\approx2m\Omega/[l'(l'+1)]$ defined in the co-rotating frame of the star will be detected by a distant observer, where $l'$ and $m$ are respectively the spherical harmonic degree and the azimuthal wave number of the mode, and $\Omega$ is the spin frequency of the star. In a linear theory of oscillation, using a parameter $A$ we parametrize the mode amplitudes such that ${\rm max}\left(|\xi_\theta|,|\xi_\phi|\right)/R=A$ at the surface, where $\xi_\theta$ and $\xi_\phi$ are the $\theta$ and $\phi$ components of the displacement vector of the mode and $R$ is the radius of the star. For the $l'=m=2$ $r$-mode with $\omega=2\Omega/3$, we find that the fractional Fourier amplitudes at $\omega=2\Omega/3$ in light curves depend on the angular distance $\theta_s$ of the spot centre measured from the rotation axis and become comparable to or even larger than $A\sim0.001$ for small values of $\theta_s$.Comment: 10 pages, 9 figures, submitted to M

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.Comment: 4 pages, 2 figures; MNRAS,316,L1(2000

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

A rare case of isolated duodenal metastases from hepatocellular carcinoma associated with p53 and ki-67 expression: a case report

Hepatocellular carcinoma (HCC) is the most common primary tumor of the liver worldwide. The incidence of HCC is increasing in North America secondary to rises in chronic liver disease from alcohol abuse and viral hepatitis. HCC most commonly metastasizes hematogenously or through lymphatics to the lungs and regional lymph nodes. Involvement of small bowel is rare and typically results from direct invasion and extension. We examined the molecular features related to this extremely rare case of isolated duodenal metastasis of HCC and noted p53 and Ki-67 positive staining. Here, we review the possible molecular and immunohistochemical studies that may aid definitive diagnosis and the evidence for the management of metastatic hepatocellular carcinoma

The rotational modes of relativistic stars: Numerical results

We study the inertial modes of slowly rotating, fully relativistic compact stars. The equations that govern perturbations of both barotropic and non-barotropic models are discussed, but we present numerical results only for the barotropic case. For barotropic stars all inertial modes are a hybrid mixture of axial and polar perturbations. We use a spectral method to solve for such modes of various polytropic models. Our main attention is on modes that can be driven unstable by the emission of gravitational waves. Hence, we calculate the gravitational-wave growth timescale for these unstable modes and compare the results to previous estimates obtained in Newtonian gravity (i.e. using post-Newtonian radiation formulas). We find that the inertial modes are slightly stabilized by relativistic effects, but that previous conclusions concerning eg. the unstable r-modes remain essentially unaltered when the problem is studied in full general relativity.Comment: RevTeX, 29 pages, 31 eps figure

Secular Instability of g-Modes in Rotating Neutron Stars

Gravitational radiation tends to drive gravity modes in rotating neutron stars unstable. For an inviscid star, the instability sets in when the rotation frequency is about 0.7 times the corresponding mode frequency of the nonrotating star. Neutron stars with spin frequencies \go 100 Hz are susceptible to this instability, with growth time of order years. However, it is likely that viscous dissipation suppresses the instability except for a narrow range of temperatures around $10^9$ K. We also show that the viscosity driven instability of g-modes is absent.Comment: 8 pages, 5 figures; MNRAS, in press. 2 figures added; More details are given for the mode propertie

R-modes of a neutron star with a magnetic dipole field

We study $r$-modes of a rotating magnetized neutron star, assuming a magnetic dipole field whose axis is aligned with the axis of rotation. We approach the problem by applying a singular perturbation theory to the oscillations of rotating stars. In this treatment, we divide the star into a thin surface magnetic layer and a non-magnetic core. We integrate linearized ideal MHD equations in the surface magnetic layer and non-magnetic oscillation equations in the core, and match the two integrations at the interface to obtain a complete solution. For a polytropic neutron star model of mass $M=1.4M_\odot$ and radius $R=10^6$cm, the magnetic dipole field becomes effective on the modal properties of the $r$-modes only when the field strength $B_S$ is much greater than $10^{14}$G. We also find that the damping effects caused by very short magnetic perturbations in the surface layer are not important for the $r$-mode instability of rapidly rotating neutron stars if the field strength $B_S$ is smaller than $10^{12}$G.Comment: Accepted for publication in M

Josephson effect test for triplet pairing symmetry

The critical current modulation and the spontaneous flux of the vortex states in corner Josephson junctions between Sr$_2$RuO$_4$ and a conventional s-wave superconductor are calculated as a function of the crystal orientation, and the magnetic field. For Sr$_2$RuO$_4$ we assume two nodeless p-wave pairing states. Also we use the nodal $f$-wave states $B_{1g}\times E_u$ and $B_{2g} \times E_u$, and one special p-wave state having line nodes. It is seen that the critical current depends solely on the topology of the gap.Comment: 22 pages, 12 figure