812 research outputs found
Variational multi-fluid dynamics and causal heat conductivity
No full textWe discuss heat conductivity from the point of view of a variational multi-fluid model, treating entropy as a dynamical entity. We demonstrate that a two-fluid model with a massive fluid component and a massless entropy can reproduce a number of key results from extended irreversible thermodynamics. In particular, we show that the entropy entrainment is intimately linked to the thermal-relaxation time that is required to make heat propagation in solids causal. We also discuss non-local terms that arise naturally in a dissipative multi-fluid model, and relate these terms to those of phonon hydrodynamics. Finally, we formulate a complete heat-conducting two-component model and discuss briefly the new dissipative terms that arise. <br/
A flux-conservative formalism for convective and dissipative multi-fluid systems, with application to Newtonian superfluid neutron stars
No full textWe develop a flux-conservative formalism for a Newtonian multi-fluid system, including dissipation and entrainment (i.e. allowing the momentum of one fluid to be a linear combination of the velocities of all fluids). Maximum use is made of mass, energy, and linear and angular momentum conservation to specify the equations of motion. Also used extensively are insights gleaned from a convective variational action principle, key being the distinction between each velocity and its canonically conjugate momentum. Dissipation is incorporated to second order in the "thermodynamic forces" via the approach pioneered by Onsager. An immediate goal of the investigation is to understand better the number, and form, of independent dissipation terms required for a consistent set of equations of motion in the multi-fluid context. A significant, but seemingly innocuous detail, is that one must be careful to isolate "forces" that can be written as total gradients, otherwise errors can be made in relating the net internal force to the net externally applied force. Our long-range aim is to provide a formalism that can be used to model dynamical multi-fluid systems both perturbatively and via fully nonlinear 3D numerical evolutions. To elucidate the formalism we consider the standard model for a heat-conducting, superfluid neutron star, which is believed to be dominated by superfluid neutrons, superconducting protons, and a highly degenerate, ultra-relativistic gas of normal fluid electrons. We determine that in this case there are, in principle, 19 dissipation coefficients in the final set of equations
A covariant action principle for dissipative fluid dynamics: from formalism to fundamental physics
No full textWe present a new variational framework for dissipative general relativistic fluid dynamics. The model extends the convective variational principle for multi-fluid systems to account for a range of dissipation channels. The key ingredients in the construction are (i) the use of a lower dimensional matter space for each fluid component, and (ii) an extended functional dependence for the associated volume forms. In an effort to make the concepts clear, the formalism is developed step-by-step with model examples considered at each level. Thus we consider a model for heat flow, derive the relativistic Navier-Stokes equations and discuss why the individual dissipative stress tensors need not be spacetime symmetric. We argue that the new formalism, which notably does not involve an expansion away from an assumed equilibrium state, provides a conceptual breakthrough in this area of research. We also provide an ambitious list of directions in which one may want to extend it in the future. This involves an exciting set of problems, relating to both applications and foundational issues
Probing neutron-star superfluidity with gravitational-wave data
No full textWe discuss the possibility that future gravitational-wave detectors may be able to detect various modes of oscillation of old, cold neutron stars. We argue that such detections would provide unique insights into the superfluid nature of neutron-star cores, and could also lead to a much improved understanding of pulsar glitches. Our estimates are based on a detector configuration with several narrow-band (cryogenic) interferometers operating as a "xylophone" which could lead to high sensitivity at high frequencies. We also draw on recent advances in our understanding of the dynamics of pulsating superfluid neutron-star cores
Relativistic fluid dynamics: physics for many different scales
No full textThe relativistic fluid is a highly successful model used to describe the dynamics of many-particle systems moving at high velocities and/or in strong gravity. It takes as input physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process—e.g., drawing on astrophysical observations—an understanding of relativistic features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model systems as “small” as colliding heavy ions in laboratory experiments, and as large as the Universe itself, with “intermediate” sized objects like neutron stars being considered along the way. The purpose of this review is to discuss the mathematical and theoretical physics underpinnings of the relativistic (multi-) fluid model. We focus on the variational principle approach championed by Brandon Carter and collaborators, in which a crucial element is to distinguish the momenta that are conjugate to the particle number density currents. This approach differs from the “standard” text-book derivation of the equations of motion from the divergence of the stress-energy tensor in that one explicitly obtains the relativistic Euler equation as an “integrability” condition on the relativistic vorticity. We discuss the conservation laws and the equations of motion in detail, and provide a number of (in our opinion) interesting and relevant applications of the general theory. The formalism provides a foundation for complex models, e.g., including electromagnetism, superfluidity and elasticity—all of which are relevant for state of the art neutron-star modelling
On the dynamics of superfluid neutron star cores
No full textWe discuss the nature of the various modes of pulsation of superfluid neutron stars using comparatively simple Newtonian models and the Cowling approximation. The matter in these stars is described in terms of a two-fluid model, where one fluid is the neutron superfluid, which is believed to exist in the core and inner crust of mature neutron stars, and the other fluid represents a conglomerate of all other constituents (crust nuclei, protons, electrons, etc.). In our model, we incorporate the non-dissipative interaction known as the entrainment effect, whereby the momentum of one constituent (e.g. the neutrons) carries along part of the mass of the other constituent. We show that there is no independent set of pulsating g-modes in a non-rotating superfluid neutron star core, even though the linearized superfluid equations contain a well-defined (and real-valued) analogue to the so-called Brunt–Väisälä frequency. Instead, what we find are two sets of spheroidal perturbations whose nature is predominately acoustic. In addition, an analysis of the zero-frequency subspace (i.e. the space of time-independent perturbations) reveals two sets of degenerate spheroidal perturbations, which we interpret to be the missing g-modes, and two sets of toroidal perturbations. We anticipate that the degeneracy of all these zero-frequency modes will be broken by the Coriolis force in the case of rotating stars. To illustrate this we consider the toroidal pulsation modes of a slowly rotating superfluid star. This analysis shows that the superfluid equations support a new class of r-modes, in addition to those familiar from, for example, geophysical fluid dynamics. Finally, the role of the entrainment effect on the superfluid mode frequencies is shown explicitly via solutions to dispersion relations that follow from a 'local' analysis of the linearized superfluid equations
Slowly rotating relativistic superfluid stars
No full textWe present a general formalism to treat slowly rotating general relativistic superfluid neutron stars. As a first approximation, their matter content can be described in terms of a two-fluid model, where one fluid is the neutron superfluid, which is believed to exist in the core and inner crust of mature neutron stars, and the other fluid represents a conglomerate of all other constituents (crust nuclei, protons, electrons, etc). We obtain a system of equations, good to second order in the rotational velocities, that determines the metric and the matter variables, irrespective of the equation of state for the two fluids. In particular, allowance is made for the so-called entrainment effect, whereby the momentum of one constituent (e.g. the neutrons) carries along part of the mass of the other constituent. As an illustration of the developed framework, we consider a simplified equation of state for which the two fluids are described by different polytropes. We determine numerically the effects of the two fluids on the rotational frame-dragging, the induced changes in the neutron and proton densities and the inertial mass, as well as the change in shape of the star. We further discuss issues regarding conservation of the two baryon numbers, the mass-shedding (Kepler) limit and chemical equilibrium
Entropy entrainment and dissipation in finite temperature superfluids
No full text: Neutron stars are expected to contain several distinct superfluid components, ranging from the neutron superfluid which coexists with the elastic crust to the mixed neutron superfluid/proton superconductor in the outer core and more exotic phases like superfluid hyperons and colour-flavour-locked superconducting quarks in the deep core. These different phases may have significant effect on the dynamics of the system. Building on a general variational framework for multifluid dynamics, we consider the behaviour of superfluid systems at finite temperatures (as required to understand various dissipation channels). As a demonstration of the validity of the underlying principles, such as treating the excitations in the system as a massless "entropy" fluid, we show that the model is formally equivalent to the traditional two-fluid approach for superfluid helium. In particular, we demonstrate how the entropy entrainment is related to the "normal fluid density". We also show how the superfluid constraint of irrotationality reduces the number of dissipation coefficients in the system. The analysis provides insight into the more general problem where vortices are present in the superfluid, and we discuss how the so-called mutual friction force can be accounted for. The end product is a formalism for finite temperature effects in a single condensate that can be applied to both low temperature laboratory systems and the various superfluid phases in a neutron star. This provides a key step towards the modelling of more realistic neutron star dynamics, and the understanding of a range of phenomena from pulsar glitches to magnetar seismology and the gravitational-wave-driven r-mode instability
Inertial modes of non-stratified superfluid neutron stars
No full textWe present results concerning adiabatic inertial-mode oscillations of non-stratified superfluid neutron stars in Newtonian gravity, using the anelastic and slow-rotation approximations. We consider a simple two-fluid model of a superfluid neutron star, where one fluid consists of the superfluid neutrons and the second fluid contains all the comoving constituents (protons, electrons). The two fluids are assumed to be 'free' in the sense that vortex-mediated forces such as mutual friction or pinning are absent, but they can be coupled by the equation of state, in particular by entrainment. The stationary background consists of the two fluids rotating uniformly around the same axis with potentially different rotation rates. We study the special cases of corotating backgrounds, vanishing entrainment, and the purely toroidal r modes analytically. We calculate numerically the eigenfunctions and frequencies of inertial modes in the general case of non-corotating backgrounds, and study their dependence on the relative rotation rate and entrainment. In these non-stratified models, we find avoided crossings only between associated mode pairs, e.g. an 'ordinary' mode and its 'superfluid' counterpart, while other mode frequencies generally cross as the background parameters are varied. We confirm (for the first time in a mode calculation) the onset of a 'two-stream instability' at a critical relative background rotation rate, and we study some of the properties of this instability for the inertial modes
Lagrangian perturbation theory of nonrelativistic superfluid stars
No full textWe develop a Lagrangian perturbation framework for rotating non-relativistic superfluid neutron stars. This leads to the first generalization of classic work on the stability properties of rotating stars to models that account for the presence of potentially weakly coupled superfluid components. Our analysis is based on the standard two-fluid model expected to be relevant for the conditions that prevail in the outer core of mature neutron stars. We discuss the implications of our results for dynamic and secular instabilities of a simple neutron star model in which the two fluids are allowed to assume different (uniform) rotation rates
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