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Magneto-thermal evolution of neutron stars with coupled Ohmic, Hall and ambipolar effects via accurate finite-volume simulations
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Abstract
Simulating the long-term evolution of temperature and magnetic fields in neutron stars is a major effort in astrophysics, having significant impact in several topics. A detailed evolutionary model requires, at the same time, the numerical solution of the heat diffusion equation, the use of appropriate numerical methods to control non-linear terms in the induction equation, and the local calculation of realistic microphysics coefficients. Here we present the latest extension of the magneto-thermal 2D code in which we have coupled the crustal evolution to the core evolution, including ambipolar diffusion. It has also gained in modularity, accuracy, and efficiency. We revise the most suitable numerical methods to accurately simulate magnetar-like magnetic fields, reproducing the Hall-driven magnetic discontinuities. From the point of view of computational performance, most of the load falls on the calculation of microphysics coefficients. To a lesser extent, the thermal evolution part is also computationally expensive because it requires large matrix inversions due to the use of an implicit method. We show two representative case studies: (i) a non-trivial multipolar configuration confined to the crust, displaying long-lived small-scale structures and discontinuities; and (ii) a preliminary study of ambipolar diffusion in normal matter. The latter acts on timescales that are too long to have relevant effects on the timescales of interest but sets the stage for future works where superfluid and superconductivity need to be included.
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In a previous paper, we reported simulations of the evolution of the magnetic field in neutron star (NS) cores through ambipolar diffusion, taking the neutrons as a motionless uniform background. However, in real NSs, neutrons are free to move, and a strong composition gradient leads to stable stratification (stability against convective motions) both of which might impact on the time-scales of evolution. Here, we address these issues by providing the first long-term two-fluid simulations of the evolution of an axially symmetric magnetic field in a neutron star core composed of neutrons, protons, and electrons with density and composition gradients. Again, we find that the magnetic field evolves towards barotropic ‘Grad–Shafranov equillibria’, in which the magnetic force is balanced by the degeneracy pressure gradient and gravitational force of the charged particles. However, the evolution is found to be faster than in the case of motionless neutrons, as the movement of charged particles (which are coupled to the magnetic field, but are also limited by the collisional drag forces exerted by neutrons) is less constrained, since neutrons are now allowed to move. The possible impact of non-axisymmetric instabilities on these equilibria, as well as beta decays, proton superconductivity, and neutron superfluidity, are left for future work.
Magnetars are neutron stars (NSs) with extreme magnetic fields¹ of strength 5 × 10¹³−10¹⁵ G. These fields are generated by dynamo action during the proto-NS phase, and are expected to have both poloidal and toroidal components2–6, although the energy of the toroidal component could be ten times larger⁷. Only the poloidal dipolar field can be measured directly, via NS spin-down⁸. The magnetic field provides heating and governs how this heat flows through the crust⁹. Magnetar thermal X-ray emission in quiescence is modulated with the rotational period of the NS, with a typical pulsed fraction 10–58%, implying that the surface temperature is substantially non-uniform despite the high thermal conductivity of the star’s crust. Poloidal dipolar fields cannot explain this large pulsed fraction10,11. Previous two-dimensional simulations12,13 have shown that a strong, large-scale toroidal magnetic field pushes a hot region into one hemisphere and increases the pulsed fraction. Here, we report three-dimensional magneto-thermal simulations of magnetars with strong, large-scale toroidal magnetic fields. These models, combined with ray propagation in curved spacetime, accurately describe the observed light curves of 10 out of 19 magnetars in quiescence and allow us to further constrain their rotational orientation. We find that the presence of a strong toroidal magnetic field is enough to explain the strong modulation of thermal X-ray emission in quiescence.
The strong magnetic field of neutron stars is intimately coupled to the observed temperature and spectral properties, as well as to the observed timing properties (distribution of spin periods and period derivatives). Thus, a proper theoretical and numerical study of the magnetic field evolution equations, supplemented with detailed calculations of microphysical properties (heat and electrical conductivity, neutrino emission rates) is crucial to understand how the strength and topology of the magnetic field vary as a function of age, which in turn is the key to decipher the physical processes behind the varied neutron star phenomenology. In this review, we go through the basic theory describing the magneto-thermal evolution models of neutron stars, focusing on numerical techniques, and providing a battery of benchmark tests to be used as a reference for present and future code developments. We summarize well-known results from axisymmetric cases, give a new look at the latest 3D advances, and present an overview of the expectations for the field in the coming years.
The presence of superconducting and superfluid components in the core of mature neutron stars calls for the rethinking of
a number of key magnetohydrodynamical notions like resistivity, the induction equation, magnetic energy and flux-freezing.
Using a multifluid magnetohydrodynamics formalism, we investigate how the magnetic field evolution is modified when neutron
star matter is composed of superfluid neutrons, type-II superconducting protons and relativistic electrons. As an application
of this framework, we derive an induction equation where the resistive coupling originates from the mutual friction between
the electrons and the vortex/fluxtube arrays of the neutron and proton condensates. The resulting induction equation allows
the identification of two time-scales that are significantly different from those of standard magnetohydrodynamics. The astrophysical
implications of these results are briefly discussed.
Evolution of the magnetic field in neutron star cores depends on the
properties of matter above nuclear density. The present paper considers
drift and dissipative processes which determine evolution of the
magnetic field in the core region. The generalized Ohm's law is derived
assuming that all species of particles are in the normal
(non-superfluid) state. We discuss in detail a possible influence of the
considered processes on the field decay. Probably the most important of
these processes is dissipation due to the enhancement of the resistivity
in strong magnetic fields. Drift phenomena associated with the partial
pressure gradients, however, can also strongly influence evolution of
the field. These phenomena are of particular importance if there is no
chemical equilibrium in npe-plasma.
Electrical conductivity and resistivity tensors are calculated for a three-component degenerate plasma with a strong magnetic field. The electron and proton magnetization parameters are determined for the npe-plasma of neutron star cores. An expression is obtained for the resistivity tranverse to the magnetic field due to collisions of charged and neutral particles.
We calculate the quiescent X-ray, neutrino, and Alfvén wave emission from a neutron star with a very strong magnetic field, Bdipole ~ 1014 − 1015 G and Binterior ~ (5–10) × 1015 G. These results are compared with observations of quiescent emission from the soft gamma repeaters and from a small class of anomalous X-ray pulsars that we have previously identified with such objects. The magnetic field, rather than rotation, provides the main source of free energy, and the decaying field is capable of powering the quiescent X-ray emission and particle emission observed from these sources. New features that are not present in the decay of the weaker fields associated with ordinary radio pulsars include fracturing of the neutron star crust, strong heating of its core, and effective suppression of thermal conduction perpendicular to the magnetic field. As the magnetic field is forced through the crust by diffusive motions in the core, multiple small-scale fractures are excited, as well as a few large fractures that can power soft gamma repeater bursts. The decay rate of the core field is a very strong function of temperature and therefore of the magnetic flux density. The strongest prediction of the model is that these sources will show no optical emissions associated with X-ray heating of an accretion disk.
We present a conservative second order accurate finite volume discretization of the magnetohydrodynamics equations including the Hall term. The scheme is generalized to three-dimensional block-adaptive grids with Cartesian or generalized coordinates. The second order accurate discretization of the Hall term at grid resolution changes is described in detail. Both explicit and implicit time integration schemes are developed. The stability of the explicit time integration is ensured by including the whistler wave speed for the shortest discrete wave length into the numerical dissipation, but then second order accuracy requires the use of symmetric limiters in the total variation diminishing scheme. The implicit scheme employs a Newton–Krylov–Schwarz type approach, and can achieve significantly better efficiency than the explicit scheme with an appropriate preconditioner. The second order accuracy of the scheme is verified by numerical tests. The parallel scaling and robustness are demonstrated by three-dimensional simulations of planetary magnetospheres.
Three mechanisms that promote the loss of magnetic flux from an isolated neutron star - Ohmic decay, ambipolar diffusion, and Hall drift - are investigated. Equations of motions are solved for charged particles in the presence of a magnetic field and a fixed background of neutrons, while allowing for the creation and destruction of particles by weak interactions. Although these equations apply to normal neutrons and protons, the present interpretations of their solutions are extended to cover cases of neutron superfluidity and proton superconductivity. The equations are manipulated to prove that, in the presence of a magnetic force, the charged particles cannot be simultaneously in magnetostatic equilibrium and chemical equilibrium with the neutrons. The application of the results to real neutron stars is discussed.
Neutron star convection is a transient phenomenon and has an extremely high magnetic Reynolds number. In this sense, a neutron star dynamo is the quintessential fast dynamo. The convective motions are only mildly turbulent on scales larger than the approximately 100 cm neutrino mean free path, but the turbulence is well developed on smaller scales. Several fundamental issues in the theory of fast dynamos are raised in the study of a neutron star dynamo, in particular the possibility of dynamo action in mirror-symmetric turbulence. It is argued that in any high magnetic Reynolds number dynamo, most of the magnetic energy becomes concentrated in thin flux ropes when the field pressure exceeds the turbulent pressure at the smallest scale of turbulence. In addition, the possibilities for dynamo action during the various (pre-collapse) stages of convective motion that occur in the evolution of a massive star are examined, and the properties of white dwarf and neutron star progenitors are contrasted.
In this review we study the nuclear pastas as they are expected to be formed in neutron star crusts. We start with a study of the pastas formed in nuclear matter (composed of protons and neutrons), we follow with the role of the electron gas on the formation of pastas, and we then investigate the pastas in neutron star matter (nuclear matter embedded in an electron gas).Nuclear matter (NM) at intermediate temperatures (1 MeV ≲ T ≲ 15 MeV), at saturation and sub-saturation densities, and with proton content ranging from 30% to 50% was found to have liquid, gaseous and liquid-gas mixed phases. The isospin-dependent phase diagram was obtained along with the critical points, and the symmetry energy was calculated and compared to experimental data and other theories. At low temperatures (T ≲ 1 MeV) NM produces crystal-like structures around saturation densities, and pasta-like structures at sub-saturation densities. Properties of the pasta structures were studied with cluster-recognition algorithms, caloric curve, the radial distribution function, the Lindemann coefficient, Kolmogorov statistics, Minkowski functionals; the symmetry energy of the pasta showed a connection with its morphology.Neutron star matter (NSM) is nuclear matter embedded in an electron gas. The electron gas is included in the calculation by the inclusion of an screened Coulomb potential. To connect the NM pastas with those in neutron star matter (NSM), the role the strength and screening length of the Coulomb interaction have on the formation of the pastas in NM was investigated. Pasta was found to exist even without the presence of the electron gas, but the effect of the Coulomb interaction is to form more defined pasta structures, among other effects. Likewise, it was determined that there is a minimal screening length for the developed structures to be independent of the cell size.Neutron star matter was found to have similar phases as NM, phase transitions, symmetry energy, structure function and thermal conductivity. Like in NM, pasta forms at around T ≈ 1.5 MeV, and liquid-to-solid phase changes were detected at T ≈ 0.5 MeV. The structure function and the symmetry energy were also found to depend on the pasta structures.
The activity of magnetars is powered by their intense and dynamic magnetic fields and has been proposed as the trigger to extragalactic fast radio bursts. Here we estimate the frequency of crustal failures in young magnetars, by computing the magnetic stresses in detailed magnetothermal simulations including Hall drift and ohmic dissipation. The initial internal topology at birth is poorly known but is likely to be much more complex than a dipole. Thus, we explore a wide range of initial configurations, finding that the expected rate of crustal failures varies by orders of magnitude depending on the initial magnetic configuration. Our results show that this rate scales with the crustal magnetic energy, rather than with the often used surface value of the dipolar component related to the spin-down torque. The estimated frequency of crustal failures for a given dipolar component can vary by orders of magnitude for different initial conditions, depending on how much magnetic energy is distributed in the crustal nondipolar components, likely dominant in newborn magnetars. The quantitative reliability of the expected event rate could be improved by a better treatment of the magnetic evolution in the core and the elastic/plastic crustal response, not included here. Regardless of that, our results are useful inputs in modeling the outburst rate of young Galactic magnetars, and their relation with the fast radio bursts in our and other galaxies. © 2020. The American Astronomical Society. All rights reserved..
The self-consistent approach to the magnetic field evolution in neutron star (NS) cores, developed recently, is generalized to the case of superfluid and superconducting NSs. Applying this approach to the cold matter of NS cores composed of neutrons, protons, electrons, and muons, we find that, similarly to the case of normal matter, an arbitrary configuration of the magnetic field may result in generation of macroscopic particle velocities, strongly exceeding their diffusive (relative) velocities. This effect substantially accelerates evolution of the magnetic field in the stellar core. An hierarchy of time-scales of such evolution at different stages of NS life is proposed and discussed. It is argued that the magnetic field in the core cannot be considered as frozen or vanishing and that its temporal evolution should affect the observational properties of NSs.
Under normal conditions in a neutron-star (NS) crust, ions are locked in place in the crustal lattice and only electrons are mobile, and magnetic-field evolution is thus directly related to the electron velocity. The evolution, however, builds magnetic stresses that can become sufficiently large for the crust to exceed its elastic limit, and to flow plastically. We consider the nature of this plastic flow and the back-reaction on the crustal magnetic-field evolution. We formulate a plane-parallel model for the local failure, showing that surface motions are inevitable once the crust yields, in the absence of extra dissipative mechanisms. We perform numerical evolutions of the crustal magnetic field under the joint effect of Hall drift and Ohmic decay, tracking the build-up of magnetic stresses, and diagnosing crustal failure with the von Mises criterion. Beyond this point we solve for the coupled evolution of the plastic velocity and magnetic field. Our results suggest that to have a coexistence of a magnetar corona with small-scale magnetic features, the viscosity of the plastic flow must be roughly 10³⁶–10³⁷ g cm⁻¹s⁻¹. We find significant motion at the surface at a rate of 10–100 cm yr⁻¹, and that the localized magnetic field is weaker than in evolutions without plastic flow. We discuss astrophysical implications, and how our local simulations could be used to build a global model of field evolution in the NS crust.
Central Compact Objects (CCOs) are X-ray sources with luminosity ranging between 1032 and 1034 erg s−1, located at the centres of supernova remnants. Some of them have been confirmed to be neutron stars. Timing observations have allowed the estimation of their dipole magnetic field, placing them in the range ∼1010–1011 G. The decay of their weak dipole fields, mediated by the Hall effect and Ohmic dissipation, cannot provide sufficient thermal energy to power their X-ray luminosity, as opposed to magnetars whose X-ray luminosities are comparable. Motivated by the question of producing high X-ray power through magnetic field decay while maintaining a weak dipole field, we explore the evolution of a crustal magnetic field that does not consist of an ordered axisymmetric structure, but rather comprises a tangled configuration. This can be the outcome of a non-self-excited dynamo, buried inside the crust by fallback material following the supernova explosion. We find that such initial conditions lead to the emergence of the magnetic field from the surface of the star and the formation of a dipolar magnetic field component. An internal tangled magnetic field of the order of 1014 G can provide sufficient Ohmic heating to the crust and power CCOs, while the dipole field it forms is approximately 1010 G, as observed in CCOs.
We examine the effects of plastic flow that appear in a neutron-star crust when a magnetic stress exceeds the threshold. The dynamics involved are described using the Navier–Stokes equation comprising the viscous-flow term, and the velocity fields for the global circulation are determined using quasi-stationary approximation. We simulate the magnetic-field evolution by taking into consideration the Hall drift, Ohmic dissipation, and fluid motion induced by the Lorentz force. The decrease in the magnetic energy is enhanced, as the energy converts to the bulk motion energy and heat. It is found that the bulk velocity induced by the Lorentz force has a significant influence in the low-viscosity and strong-magnetic-field regimes. This effect is crucial near magnetar surfaces.
We report on Bayesian parameter estimation of the mass and equatorial radius of the millisecond pulsar PSR J0030+0451, conditional on pulse-profile modeling of Neutron Star Interior Composition Explorer X-ray spectral-timing event data. We perform relativistic ray-tracing of thermal emission from hot regions of the pulsar's surface. We assume two distinct hot regions based on two clear pulsed components in the phase-folded pulse-profile data; we explore a number of forms (morphologies and topologies) for each hot region, inferring their parameters in addition to the stellar mass and radius. For the family of models considered, the evidence (prior predictive probability of the data) strongly favors a model that permits both hot regions to be located in the same rotational hemisphere. Models wherein both hot regions are assumed to be simply connected circular single-temperature spots, in particular those where the spots are assumed to be reflection-symmetric with respect to the stellar origin, are strongly disfavored. For the inferred configuration, one hot region subtends an angular extent of only a few degrees (in spherical coordinates with origin at the stellar center) and we are insensitive to other structural details; the second hot region is far more azimuthally extended in the form of a narrow arc, thus requiring a larger number of parameters to describe. The inferred mass M and equatorial radius R eq are, respectively, ${1.34}_{-0.16}^{+0.15}\,{M}_{\odot }$ and ${12.71}_{-1.19}^{+1.14}\,\mathrm{km}$, while the compactness ${GM}/{R}_{\mathrm{eq}}{c}^{2}={0.156}_{-0.010}^{+0.008}$ is more tightly constrained; the credible interval bounds reported here are approximately the 16% and 84% quantiles in marginal posterior mass.
Aims . In this work, we study the structure of neutron stars under the effect of a poloidal magnetic field and determine the limiting largest magnetic field strength that induces a deformation such that the ratio between the polar and equatorial radii does not exceed 2%. We consider that, under these conditions, the description of magnetic neutron stars in the spherical symmetry regime is still satisfactory.
Methods . We described different compositions of stars (nucleonic, hyperonic, and hybrid) using three state-of-the-art relativistic mean field models (NL3 ωρ , MBF, and CMF, respectively) for the microscopic description of matter, all in agreement with standard experimental and observational data. The structure of stars was described by the general relativistic solution of both Einstein’s field equations assuming spherical symmetry and Einstein-Maxwell’s field equations assuming an axi-symmetric deformation.
Results . We find a limiting magnetic moment on the order of 2 × 10 ³¹ Am ² , which corresponds to magnetic fields on the order of 10 ¹⁶ G at the surface and 10 ¹⁷ G at the center of the star, above which the deformation due to the magnetic field is above 2%, and therefore not negligible. We show that the intensity of the magnetic field developed in the star depends on the equation of state (EoS), and, for a given baryonic mass and fixed magnetic moment, larger fields are attained with softer EoS. We also show that the appearance of exotic degrees of freedom, such as hyperons or a quark core, is disfavored in the presence of a very strong magnetic field. As a consequence, a highly magnetized nucleonic star may suffer an internal conversion due to the decay of the magnetic field, which could be accompanied by a sudden cooling of the star or a gamma ray burst.
We study the coupling of the force-free magnetosphere to the long-term internal evolution of a magnetar. We allow the relation between the poloidal and toroidal stream functions - that characterizes themagnetosphere - to evolve freely without constraining its particular form.We find that, on time-scales of the order of kyr, the energy stored in the magnetosphere gradually increases, as the toroidal region grows and the field lines expand outwards. This continues until a critical point is reached beyond which force-free solutions for the magnetosphere can no longer be constructed, likely leading to some large-scale magnetospheric reorganization. The energy budget available for such events can be as high as several 10⁴⁵ erg for fields of 10¹⁴ G. Subsequently, starting from the new initial conditions, the evolution proceeds in a similar manner. The time-scale to reach the critical point scales inversely with the magnetic field amplitude. Allowing currents to pass through the last few metres below the surface, where the magnetic diffusivity is orders of magnitude larger than in the crust, should give rise to a considerable amount of energy deposition through Joule heating.We estimate that the effective surface temperature could increase locally from ~0.1 keVtõ0.3-0.6 keV, in good agreement with observations. Similarly, the power input from the interior into the magnetosphere could be as high as 10³⁵-10³⁶ erg s⁻¹, which is consistent with peak luminosities observed during magnetar outbursts. Therefore, a detailed treatment of currents flowing through the envelope may be needed to explain the thermal properties of magnetars.
In the interior of neutron stars, the induction equation regulates the long-term evolution of the magnetic fields by means of resistivity, Hall dynamics and ambipolar diffusion. Despite the apparent simplicity and compactness of the equation, the dynamics it describes is not trivial and its understanding relies on accurate numerical simulations. While a few works in 2D have reached a mature stage and a consensus on the general dynamics at least for some simple initial data, only few attempts have been performed in 3D, due to the computational costs and the need for a proper numerical treatment of the intrinsic non-linearity of the equation. Here, we carefully analyze the general induction equation, studying its characteristic structure, and we present a new Cartesian 3D code, generated by the user-friendly, publicly available Simflowny platform. The code uses high-order numerical schemes for the time and spatial discretization, and relies on the highly-scalable SAMRAI architecture for the adaptive mesh refinement. We present the application of the code to several benchmark tests, showing the high order of convergence and accuracy achieved and the capabilities in terms of magnetic shock resolution and three-dimensionality. This paper paves the way for the applications to a realistic, 3D long-term evolution of neutron stars interior and, possibly, of other astrophysical sources.
In [Gusakov et al. Phys. Rev. D 96, 103012 (2017)], we proposed a self-consistent method to study the quasistationary evolution of the magnetic field in neutron-star cores. Here, we apply it to calculate the instantaneous particle velocities and other parameters of interest, which are fixed by specifying the magnetic field configuration. Interestingly, we found that the magnetic field can lead to generation of a macroscopic fluid motion with the velocity significantly exceeding the diffusion particle velocities. This result calls into question the standard view on the magnetic field evolution in neutron stars and suggests a new, shorter time scale for such evolution.
It has been suggested that, when the pressure within stellar matter becomes high enough, a new phase consisting of neutrons will be formed. In this paper we study the gravitational equilibrium of masses of neutrons, using the equation of state for a cold Fermi gas, and general relativity. For masses under $\frac{1}{3}\ensuremath{\bigodot}$ only one equilibrium solution exists, which is approximately described by the nonrelativistic Fermi equation of state and Newtonian gravitational theory. For masses $\frac{1}{3}\ensuremath{\bigodot}<m<\frac{3}{4}\ensuremath{\bigodot}$ two solutions exist, one stable and quasi-Newtonian, one more condensed, and unstable. For masses greater than $\frac{3}{4}\ensuremath{\bigodot}$ there are no static equilibrium solutions. These results are qualitatively confirmed by comparison with suitably chosen special cases of the analytic solutions recently discovered by Tolman. A discussion of the probable effect of deviations from the Fermi equation of state suggests that actual stellar matter after the exhaustion of thermonuclear sources of energy will, if massive enough, contract indefinitely, although more and more slowly, never reaching true equilibrium.
A formalism is presented for investigating the influence of
thermoelectric effects on the magnetic field in the envelope of neutron
stars. It is based on the simultaneous solution of the time-dependent
heat transport equation and induction equation. The inclusion of the
thermoelectric effects and of the anisotropy of the transport
coefficients (the electric and heat conductivity and the thermopower)
leads to nonlinear terms in these equations. The magnetic field is
decomposed into toroidal and poloidal components, in order to transform
the vector induction equation in a set of scalar equations. Subsequent
expansion of all searched functions in a series of spherical harmonics
makes it possible to study the coupling properties between modes of
different multipolarities. The special cases of linearized equations and
of axial symmetry are discussed.
Over the past decade, the numerical modeling of the magnetic field evolution in astrophysical scenarios has become an increasingly important field. In the crystallized crust of neutron stars the evolution of the magnetic field is governed by the Hall induction equation. In this equation the relative contribution of the two terms (Hall term and Ohmic dissipation) varies depending on the local conditions of temperature and magnetic field strength. This results in the transition from the purely parabolic character of the equations to the hyperbolic regime as the magnetic Reynolds number increases, which presents severe numerical problems. Up to now, most attempts to study this problem were based on spectral methods, but they failed in representing the transition to large magnetic Reynolds numbers. We present a new code based on upwind finite differences techniques that can handle situations with arbitrary low magnetic diffusivity and it is suitable for studying the formation of sharp current sheets during the evolution. The code is thoroughly tested in different limits and used to illustrate the evolution of the crustal magnetic field in a neutron star in some representative cases. Our code, coupled to cooling codes, can be used to perform long-term simulations of the magneto-thermal evolution of neutron stars.
Nuclear shapes in the inner crust of a neutron star are studied with due consideration for nuclear surface diffuseness and lattice types. The nuclear shapes considered are cylinder, slab and cylindrical hole in addition to sphere and spherical hole. The Thomas-Fermi calculations are performed in the zero-temperature approximation with four energy density functionals. These functionals are constructed so as to reproduce gross properties of laboratory nuclei and to be consistent with the equation of state of pure neutron matter by Friedman and Pandharipande. The nucleon distributions are parametrized in a way that allows a neutron gas outside the nuclei as well as different nuclear surfaces for neutrons and protons. This study confirms the liquid-drop result that the nuclear shape changes from sphere to cylinder, slab, cylindrical hole and spherical hole successively as the matter density increases. These shape changes occur at densities between 1.0 × 1014 and 1.5 × 1014 g/cm3, and are accompanied by first-order phase transitions. The density range for each nuclear shape to exist stably becomes narrower successively in the above order. At a fixed matter density, the energy difference between the phases of two successive nuclear shapes is of the order of 0.1–1 keV per nucleon. The existence of non-spherical nuclear shapes reduces the energy by an amount up to 3–5 keV per nucleon compared with the case in which only spherical and spherical hole nuclei are considered. This energy reduction corresponds to 4–6 MeV per spherical nucleus.
Motion is generated in a rotating spherical shell, by a slight differential rotation of the inner core. We show how the numerical solution tends, with decreasing Ekman number, to the asymptotic limit of Proudman [J. Fluid Mech. 1 (1956) 505–516]. Starting from geophysically large values, we show that the main qualitative features of the asymptotic solution show up only when the Ekman number is decreased below 10−6. Then, we impose a dipolar and force-free magnetic field with internal sources. Both the inner core and the liquid shell are electrically conducting. The first effect of the Lorentz force is to smooth out the change in angular velocity at the tangent cylinder. As the Elsasser number is further increased, the Proudman–Taylor constraint is violated, Ekman layers are changed into Hartmann type layers, shear at the inner sphere boundary vanishes, and the flow tends to a bulk rotation together with the inner sphere. Unexpectedly, for increasing strength of the field, there is a super-rotation (the angular velocity does not reach a maximum at the inner core boundary but in the interior of the fluid) localized in an equatorial torus. At a given field strength, the amplitude of this phenomenon depends on the Ekman number and tends to vanish in the magnetostrophic limit.
We extend our previous numerical model of Hall drift in neutron stars (Hollerbach & Rüdiger 2002) to include variations in density across the depth of the crust. For purely toroidal fields, the results are in perfect agreement with the analytic model of Vainshtein, Chitre & Olinto (2000), who showed that the field would develop very fine structures, which would cause it to decay on the Hall time-scale rather than the much longer ohmic time-scale. However, if we include poloidal fields (which were not considered by Vainshtein et al.), the behaviour is rather different. The field still develops fine structures, but not as fine as before, scaling roughly as R-1/2B rather than the R-1B scaling obtained for purely toroidal fields (where the Hall parameter RB measures the ratio of the ohmic time-scale to the Hall time-scale). As a result, the field still decays considerably faster than if ohmic decay alone were acting, but ultimately still on the ohmic time-scale. This is true even if the initial poloidal field is much weaker than the initial toroidal field (with the sign of the toroidal field also playing an important role). Finally, we consider the possible implications for the magnetic fields of real neutron stars.
The equilibrium of a nonneutral plasma in a toroidal vessel with a toroidal magnetic field is analyzed. In the zero inertia limit it is heuristically shown from force balance considerations that there is an electrostatic hoop force and a force due to diamagnetism along the major radius. The problem of equilibrium is formulated in terms of solutions of a 2D partial difference equation. This equation is solved in the large-aspect-ratio limit and a general expression for the shift of the potential axis is obtained which shows that the shift is approximately epsilon and that it depends solely on the internal capacitance of the cloud. The simulation study is based upon the modified MHD equations and the nonlocal nature of the mode is investigated. Applications to sub-Alfvenic plasma expansions, electromagnetic waves in the earth's magnetosphere, and plasma switches are discussed.
On 1 April 2001, the Polar satellite crossed a subsolar magnetopause associated with antiparallel magnetic fields. Over a width approximately 6 magnetosheath ion skin depths (approximately 3 magnetospheric ion skin depths), perpendicular ion flows different from E x B/B(2) as well as Hall magnetic and electric field signatures were observed. At a smaller scale, the electron flow decoupled from the magnetic field near a deep minimum in the magnetic field strength. Separatrices were identified as boundaries of low frequency electric field turbulence associated with density minima and parallel electric fields. The reconnection rate was less than 2% of the asymptotic Alfvén speed.
With early critical views by Lighthill [1] as the starting point, a revision is performed of the classical magnetohydrodynamic (MHD) theory for the magnetized plasma properties. The importance of retaining the Hall term and using a two-fluid plasma description is stressed. The plasma Hall Magnetohydrodynamics (HMHD) description is derived by accounting for the noncentral character of the internal particle-particle forces sustaining the plasma charge neutrality. Plasma properties with similarities to the Meissner effect are predicted. The HMHD model and its extension to a current-tube plasma description are used to explain experimentally observed characteristics of magnetic confinement plasmas, with application also to cosmic plasmas.
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