Outer Core Of Neutron Star Research Articles

On the peculiar rotational evolution of PSR B0950+08

ABSTRACT The long-term rotational evolution of the old, isolated pulsar, PSR B0950+08, is intriguing in that its spin-down rate displays sinusoidal-like oscillations due to alternating variations, both in magnitude and sign, of the second time derivative of the pulse frequency. We show that the large internal temperature to pinning energy ratio towards the base of the crust implied by the recent high surface temperature measurement of PSR B0950+08 leads to linear creep interaction between vortex lines and pinning sites to operate in this pulsar. Vortex lines assume a parabolic shape due to pinning to nuclear clusters and finite tension of vortices acts as a restoring force that tends to bring a vortex back to its straight shape. The resulting low-frequency oscillations of vortex lines combined with the time variable coupling between the internal superfluid components and the external pulsar braking torque give rise to an oscillatory spin-down rate. We apply this model to PSR B0950+08 observations for several external torque models. Our model has potential to constrain the radial extension of the closed magnetic field region in the outer core of neutron stars from the oscillation period of the spin-down rate.

1S0 Pairing Gaps, Chemical Potentials and Entrainment Matrix in Superfluid Neutron-Star Cores for the Brussels–Montreal Functionals

Temperature and velocity-dependent 1S0 pairing gaps, chemical potentials and entrainment matrix in dense homogeneous neutron–proton superfluid mixtures constituting the outer core of neutron stars, are determined fully self-consistently by solving numerically the time-dependent Hartree–Fock–Bogoliubov equations over the whole range of temperatures and flow velocities for which superfluidity can exist. Calculations have been made for npeμ in beta-equilibrium using the Brussels–Montreal functional BSk24. The accuracy of various approximations is assessed and the physical meaning of the different velocities and momentum densities appearing in the theory is clarified. Together with the unified equation of state published earlier, the present results provide consistent microscopic inputs for modeling superfluid neutron-star cores.

Crust-core interface and bulk neutron star properties

The nuclear symmetry energy plays an important role in the description of the properties of finite nuclei as well as neutron stars. Especially, for low values of baryon density, the accurate description of the crust-core interface strongly depends on the symmetry energy. Usually, the well known parabolic approximation is employed for the definition of the symmetry energy without avoiding some drawbacks. In the present paper, a class of nuclear models, suitable for the description of the inner and outer core of neutron stars, is applied in studying the effect of higher orders of the expansion of the energy on the location of the crust-core transition. The thermodynamical and dynamical methods are used for the determination of the transition density ${n}_{\mathrm{t}}$ and pressure ${P}_{\mathrm{t}}$. The corresponding energy density functional is applied for the study of some relevant properties of both nonrotating and slowly rotating neutron stars. We found that the larger the value of the slope parameter $L$, the slower the convergence of the expansion. In addition, a universal relation is presented between ${n}_{\mathrm{t}}$ and $L$, by employing the full expression and dynamical approach. The crustal moment of inertia is very sensitive to the location of the transition while the effects are moderated concerning the critical angular velocity of the $r$-mode instability and minimum mass configuration. The effect on the tidal deformability is less but not negligible. In any case, the use of the parabolic approximation leads to the overestimation of ${n}_{\mathrm{t}}$ and ${P}_{\mathrm{t}}$ and, consequently, on inaccurate predictions. Moreover, in some cases, even the matching process at the interface may affect considerably the predictions, introducing errors of the same order with the one due to the employed method.

Roton instabilities in the superfluid outer core of neutron stars

We study the superfluid dynamics of the outer core of neutron stars by means of a generalized hydrodynamic model made of a neutronic superfluid and a protonic superconductor, coupled by both the dynamic entrainment and the Skyrme SLy4 nucleon-nucleon interactions. The resulting nonlinear equations of motion are probed in the search for dynamical instabilities triggered by the relative motion of the superfluids that could be of potential relevance to observational features of neutron stars. Through linear analysis, the origin and expected growth of the instabilities is explored for varying nuclear-matter density. Differently from previous findings, the dispersion of linear excitations in our model shows rotonic structures below the pair-breaking energy threshold, which lies at the origin of the dynamical instabilities and could eventually lead to emergent vorticity along with modulations of the superfluid density.

Proton fraction in neutron star matter: dynamical mean-field approach

Dynamical mean field theory (DMFT) is used to study neutron matter, both with and without admixture of the proton fraction. The system is approximated by the lattice Hubbard model. The corresponding equation of state as a function of temperature/density/asymmetry is investigated. The results are compared with the standard mean field (MF) approach where the effect of local correlations is neglected. Whereas the influence of the correlations on the properties of a pure neutron matter is found to be moderate, it becomes strong when the proton admixture is taken into account. In particular, we calculate the proton fraction, energy density and pressure in outer core of neutron stars, taking into account the beta equilibrium condition. The DMFT predicts that the proton fraction is several times the MF based calculations, whereas the DMFT results for energy density and pressure are 30%–40% lower than the corresponding MF estimates. Physical implications of our findings for a neutron star dynamics are discussed.

Superfluid Phase Transitions and Effects of Thermal Pairing Fluctuations in Asymmetric Nuclear Matter

We investigate superfluid phase transitions of asymmetric nuclear matter at finite temperature (T) and density (ρ) with a low proton fraction (Yp ≤ 0.2), which is relevant to the inner crust and outer core of neutron stars. A strong-coupling theory developed for two-component atomic Fermi gases is generalized to the four-component case, and is applied to the system of spin-1/2 neutrons and protons. The phase shifts of neutron-neutron (nn), proton-proton (pp) and neutron-proton (np) interactions up to k = 2 fm−1 are described by multi-rank separable potentials. We show that the critical temperature {{boldsymbol{T}}}_{{bf{c}}}^{{bf{n}}{bf{n}}} of the neutron superfluidity at Yp = 0 agrees well with Monte Carlo data at low densities and takes a maximum value {{boldsymbol{T}}}_{{bf{c}}}^{{bf{n}}{bf{n}}}= 1.68 MeV at {boldsymbol{rho }}{boldsymbol{/}}{{boldsymbol{rho }}}_{{bf{0}}}{boldsymbol{=}}{bf{0.14}} with ρ0 = 0.17 fm−3. Also, the critical temperature {{boldsymbol{T}}}_{{bf{c}}}^{{bf{n}}{bf{n}}} of the proton superconductivity for Yp ≤ 0.2 is substantially suppressed at low densities due to np-pairing fluctuations, and starts to dominate over {{boldsymbol{T}}}_{{bf{c}}}^{{bf{n}}{bf{n}}} only above {boldsymbol{rho }}{boldsymbol{/}}{{boldsymbol{rho }}}_{{bf{0}}}{boldsymbol{=}}{bf{0.70}}(0.77) for Yp = 0.1(0.2), and (iii) the deuteron condensation temperature {{boldsymbol{T}}}_{{bf{c}}}^{{bf{d}}} is suppressed at Yp ≤ 0.2 due to a large mismatch of the two Fermi surfaces.

Dispersion and decay of collective modes in neutron star cores

We calculate the frequencies of collective modes of neutrons, protons and electrons in the outer core of neutron stars. The neutrons and protons are treated in a hydrodynamic approximation and the electrons are regarded as collisionless. The coupling of the nucleons to the electrons leads to Landau damping of the collective modes and to significant dispersion of the low-lying modes. We investigate the sensitivity of the mode frequencies to the strength of entrainment between neutrons and protons, which is not well characterized. The contribution of collective modes to the thermal conductivity is evaluated.

Microscopic vortex velocity in the inner crust and outer core of neutron stars

Treatment of the vortex motion in the superfluids of the inner crust and the outer core of neutron stars is a key ingredient in modeling a number of pulsar phenomena, including glitches and magnetic field evolution. After recalculating the microscopic vortex velocity in the inner crust, we evaluate the velocity for the vortices in the outer core for the first time. The vortex motion between pinning sites is found to be substantially faster in the inner crust than in the outer core, $v_0^{\rm crust} \sim 10^{7}\mbox{\cms} \gg v_0^{\rm core} \sim 1\mbox{\cms}$. One immediate result is that vortex creep is always in the nonlinear regime in the outer core in contrast to the inner crust, where both nonlinear and linear regimes of vortex creep are possible. Other implications for pulsar glitches and magnetic field evolution are also presented.

Radiative-condensation instability in gravitating strongly coupled dusty plasma with polarization force

The radiative-condensation instability (RCI) in self-gravitating strongly coupled dusty plasma (SCDP) is investigated considering the effects of dust thermal velocity and polarization force on the massive dust particulates. In particular, the outer core of the dense neutron star which is supposed to be strongly coupled in nature with temperature T∼107 K and number density n∼1.3×1030 cm−3 is analyzed. The modified generalized hydrodynamic (GH) equations and electron temperature perturbation equation with radiative effects are solved using the linear perturbation method. In the classical hydrodynamic limit, the modified condition of Jeans instability owing to radiative condensation, polarization force and dust thermal velocity is obtained. In the kinetic limit, velocity of compressional mode also modifies the condition of Jeans instability. The dust thermal velocity and viscoelastic effects have stabilizing whereas polarization force and radiative cooling have destabilizing influence on the growth rate of the Jeans instability. The radiative effects stabilize the growth rate of unstable radiative modes. In isobaric mode (short wavelength), the basic condition of radiative instability is obtained which is unaffected due to the presence of polarization force and viscoelastic effects. The radiative cooling time in the outer core of neutron star is estimated and compared with the gravitational free fall time, and it is found that the cooling takes place too fast for self-gravity to be important.

How neutron stars constrain the nuclear equation of state

Recent neutron star observations set new constraints for the equation of state of baryonic matter. A chiral effective field theory approach is used for the description of neutron-dominated nuclear matter present in the outer core of neutron stars. Possible hybrid stars with quark matter in the inner core are discussed using a three-flavor Nambu–Jona-Lasinio model.

Elementary excitations in homogeneous superfluid neutron star matter: Role of the proton component

The thermal evolution of neutron stars depends on the elementary excitations affecting the stellar matter. In particular, the low-energy excitations, whose energy is proportional to the transferred momentum, can play a major role in the emission and propagation of neutrinos. In this paper, we focus on the density modes associated with the proton component in the homogeneous matter of the outer core of neutron stars (at density between one and three times the nuclear saturation density, where the baryonic constituents are expected to be neutrons and protons). In this region, it is predicted that the protons are superconducting. We study the respective roles of the proton pairing and Coulomb interaction in determining the properties of the modes associated with the proton component. This study is performed in the framework of the random phase approximation, generalized in order to describe the response of a superfluid system. The formalism we use ensures that the generalized Ward's identities are satisfied. An important conclusion of this work is the presence of a pseudo-Goldstone mode associated with the superconducting protons in neutron-star matter. Indeed, the Goldstone mode, which characterizes a pure superfluid, is suppressed in usual superconductors because of the long-range Coulomb interaction, which allows a plasmon mode. However, for the proton component of stellar matter, the Coulomb field is screened by the electrons and a pseudo-Goldstone mode occurs, with a velocity increased by the Coulomb interaction.

QUASI-STATIONARY ELECTROMAGNETIC EFFECTS IN CONDUCTORS AND SUPERCONDUCTORS IN SCHWARZSCHILD SPACE–TIME

The general principles needed to compute the effect of a stationary gravitational field on the quasistationary electromagnetic phenomena in normal conductors and superconductors are formulated from general relativistic point of view. Generalization of the skin effect, that is the general relativistic modification of the penetration depth (of the time-dependent magnetic field in the conductor) due to its relativistic coupling to the gravitational field is obtained. The effect of the gravitational field on the penetration and coherence depths in superconductors is also studied. As an illustration of the foregoing general results, we discuss their application to superconducting systems in the outer core of neutron stars. The relevance of these effects to electrodynamics of magnetized neutron stars has been shown.

Covariant vortex in superconducting-superfluid-normal fluid mixtures with a stiff equation of state

Using a covariant description, we obtain the integrals of motion for a cylindrically symmetric, stationary vortex configuration in a mixture of interacting superconductors, superfluids and normal fluids. We then integrate the stress-energy density and find a very simple, closed expression for the energy per unit length and the relevant stress coefficients of the vortex with respect to a vortex-free reference state. This result is found assuming a ``stiff'' equation of state for the fluid mixture, which is the least compressible but still causal equation of state (contrary to the incompressible case). As an illustration for these general results, we discuss some applications to ``real'' superfluid-superconducting systems that are contained as special cases. These include the two-fluid model for He-II, uncharged binary superfluid mixtures, conventional superconductors and the superfluid neutron-proton-electron plasma in the outer core of neutron stars.