The Equation of State for Cool Relativistic Two-Constituent Superfluid Dynamics

The natural relativistic generalisation of Landau's two constituent superfluid theory can be formulated in terms of a Lagrangian $L$ that is given as a function of the entropy current 4-vector $s^\rho$ and the gradient $\nabla\varphi$ of the superfluid phase scalar. It is shown that in the ``cool" regime, for which the entropy is attributable just to phonons (not rotons), the Lagrangian function $L(\vec s, \nabla\varphi)$ is given by an expression of the form $L=P-3\psi$ where $P$ represents the pressure as a function just of $\nabla\varphi$ in the (isotropic) cold limit. The entropy current dependent contribution $\psi$ represents the generalised pressure of the (non-isotropic) phonon gas, which is obtained as the negative of the corresponding grand potential energy per unit volume, whose explicit form has a simple algebraic dependence on the sound or ``phonon" speed $c_P$ that is determined by the cold pressure function $P$.Comment: 26 pages, RevTeX, no figures, published in Phys. Rev. D. 15 May 199

A Relativistic Mean Field Model for Entrainment in General Relativistic Superfluid Neutron Stars

General relativistic superfluid neutron stars have a significantly more intricate dynamics than their ordinary fluid counterparts. Superfluidity allows different superfluid (and superconducting) species of particles to have independent fluid flows, a consequence of which is that the fluid equations of motion contain as many fluid element velocities as superfluid species. Whenever the particles of one superfluid interact with those of another, the momentum of each superfluid will be a linear combination of both superfluid velocities. This leads to the so-called entrainment effect whereby the motion of one superfluid will induce a momentum in the other superfluid. We have constructed a fully relativistic model for entrainment between superfluid neutrons and superconducting protons using a relativistic $\sigma - \omega$ mean field model for the nucleons and their interactions. In this context there are two notions of ``relativistic'': relativistic motion of the individual nucleons with respect to a local region of the star (i.e. a fluid element containing, say, an Avogadro's number of particles), and the motion of fluid elements with respect to the rest of the star. While it is the case that the fluid elements will typically maintain average speeds at a fraction of that of light, the supranuclear densities in the core of a neutron star can make the nucleons themselves have quite high average speeds within each fluid element. The formalism is applied to the problem of slowly-rotating superfluid neutron star configurations, a distinguishing characteristic being that the neutrons can rotate at a rate different from that of the protons.Comment: 16 pages, 5 figures, submitted to PR

Slowly Rotating General Relativistic Superfluid Neutron Stars with Relativistic Entrainment

Neutron stars that are cold enough should have two or more superfluids/supercondutors in their inner crusts and cores. The implication of superfluidity/superconductivity for equilibrium and dynamical neutron star states is that each individual particle species that forms a condensate must have its own, independent number density current and equation of motion that determines that current. An important consequence of the quasiparticle nature of each condensate is the so-called entrainment effect, i.e. the momentum of a condensate is a linear combination of its own current and those of the other condensates. We present here the first fully relativistic modelling of slowly rotating superfluid neutron stars with entrainment that is accurate to the second-order in the rotation rates. The stars consist of superfluid neutrons, superconducting protons, and a highly degenerate, relativistic gas of electrons. We use a relativistic $\sigma$ - $\omega$ mean field model for the equation of state of the matter and the entrainment. We determine the effect of a relative rotation between the neutrons and protons on a star's total mass, shape, and Kepler, mass-shedding limit.Comment: 30 pages, 10 figures, uses ReVTeX

Relativistic Kinetics of Phonon Gas in Superfluids

The relativistic kinetic theory of the phonon gas in superfluids is developed. The technique of the derivation of macroscopic balance equations from microscopic equations of motion for individual particles is applied to an ensemble of quasi-particles. The necessary expressions are constructed in terms of a Hamilton function of a (quasi-)particle. A phonon contribution into superfluid dynamic parameters is obtained from energy-momentum balance equations for the phonon gas together with the conservation law for superfluids as a whole. Relations between dynamic flows being in agreement with results of relativistic hydrodynamic consideration are found. Based on the kinetic approach a problem of relativistic variation of the speed of sound under phonon influence at low temperature is solved.Comment: 23 pages, Revtex fil

Relativistic Two-stream Instability

We study the (local) propagation of plane waves in a relativistic, non-dissipative, two-fluid system, allowing for a relative velocity in the "background" configuration. The main aim is to analyze relativistic two-stream instability. This instability requires a relative flow -- either across an interface or when two or more fluids interpenetrate -- and can be triggered, for example, when one-dimensional plane-waves appear to be left-moving with respect to one fluid, but right-moving with respect to another. The dispersion relation of the two-fluid system is studied for different two-fluid equations of state: (i) the "free" (where there is no direct coupling between the fluid densities), (ii) coupled, and (iii) entrained (where the fluid momenta are linear combinations of the velocities) cases are considered in a frame-independent fashion (eg. no restriction to the rest-frame of either fluid). As a by-product of our analysis we determine the necessary conditions for a two-fluid system to be causal and absolutely stable and establish a new constraint on the entrainment.Comment: 15 pages, 2 eps-figure

Structure of the ovaries of the Nimba otter shrew, Micropotamogale lamottei, and the Madagascar hedgehog tenrec, Echinops telfairi

The otter shrews are members of the subfamily Potamogalinae within the family Tenrecidae. No description of the ovaries of any member of this subfamily has been published previously. The lesser hedgehog tenrec, Echinops telfairi, is a member of the subfamily Tenrecinae of the same family and, although its ovaries have not been described, other members of this subfamily have been shown to have ovaries with non-antral follicles. Examination of these two species illustrated that non-antral follicles were characteristic of the ovaries of both species, as was clefting and lobulation of the ovaries. Juvenile otter shrews range from those with only small follicles in the cortex to those with 300- to 400-mu m follicles similar to those seen in non-pregnant and pregnant adults. As in other species, most of the growth of the oocyte occurred when follicles had one to two layers of granulosa cells. When larger follicles became atretic in the Nimba otter shrew, hypertrophy of the theca interna produced nodules of glandular interstitial tissue. In the tenrec, the hypertrophying theca interna cells in most large follicles appeared to undergo degeneration. Both species had some follicular fluid in the intercellular spaces between the more peripheral granulosa cells. It is suggested that this fluid could aid in separation of the cumulus from the remaining granulosa at ovulation. The protruding follicles in lobules and absence of a tunica albuginea might also facilitate ovulation of non-antral follicles. Ovaries with a thin-absent tunica albuginea and follicles with small-absent antra are widespread within both the Eulipotyphla and in the Afrosoricida, suggesting that such features may represent a primitive condition in ovarian development. Lobulated and deeply crypted ovaries are found in both groups but are not as common in the Eulipotyphla making inclusion of this feature as primitive more speculative. Copyright (C) 2005 S. Karger AG, Basel

Variational description of multi-fluid hydrodynamics: Uncharged fluids

We present a formalism for Newtonian multi-fluid hydrodynamics derived from an unconstrained variational principle. This approach provides a natural way of obtaining the general equations of motion for a wide range of hydrodynamic systems containing an arbitrary number of interacting fluids and superfluids. In addition to spatial variations we use ``time shifts'' in the variational principle, which allows us to describe dissipative processes with entropy creation, such as chemical reactions, friction or the effects of external non-conservative forces. The resulting framework incorporates the generalization of the entrainment effect originally discussed in the case of the mixture of two superfluids by Andreev and Bashkin. In addition to the conservation of energy and momentum, we derive the generalized conservation laws of vorticity and helicity, and the special case of Ertel's theorem for the single perfect fluid. We explicitly discuss the application of this framework to thermally conducting fluids, superfluids, and superfluid neutron star matter. The equations governing thermally conducting fluids are found to be more general than the standard description, as the effect of entrainment usually seems to be overlooked in this context. In the case of superfluid He4 we recover the Landau--Khalatnikov equations of the two-fluid model via a translation to the ``orthodox'' framework of superfluidity, which is based on a rather awkward choice of variables. Our two-fluid model for superfluid neutron star matter allows for dissipation via mutual friction and also ``transfusion'' via beta-reactions between the neutron fluid and the proton-electron fluid.Comment: uses RevTeX 4; 20 pages. To appear in PRD. v2: removed discussion of charged fluids and coupling to electromagnetic fields, which are submitted as a separate paper for a clearer presentation v3: fixed typo in Eq.(9), updated some reference

Cosmological Two-stream Instability

Two-stream instability requires, essentially, two things to operate: a relative flow between two fluids and some type of interaction between them. In this letter we provide the first demonstration that this mechanism may be active in a cosmological context. Building on a recently developed formalism for cosmological models with two, interpenetrating fluids with a relative flow between them, we show that two-stream instability may be triggered during the transition from one fluid domination to the other. We also demonstrate that the cosmological expansion eventually shuts down the instability by driving to zero the relativeflow and the coupling between the two fluids.Comment: 9 pages, 3 figures. Final version that is publishe

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

Relativistic superfluid models for rotating neutron stars

This article starts by providing an introductory overview of the theoretical mechanics of rotating neutron stars as developped to account for the frequency variations, and particularly the discontinuous glitches, observed in pulsars. The theory suggests, and the observations seem to confirm, that an essential role is played by the interaction between the solid crust and inner layers whose superfluid nature allows them to rotate independently. However many significant details remain to be clarified, even in much studied cases such as the Crab and Vela. The second part of this article is more technical, concentrating on just one of the many physical aspects that needs further development, namely the provision of a satisfactorily relativistic (local but not microscopic) treatment of the effects of the neutron superfluidity that is involved.Comment: 42 pages LateX. Contribution to Physics of Neutron Star Interiors, ed. D. Blasche, N.K. Glendenning, A. Sedrakian (ECT workshop, Trento, June 2000