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Amplification of neutron star magnetic fields by thermoelectric effects. I - General formalism
Abstract
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.
... In these studies, two main families of numerical methods have been used, according to the discretization of the induction equation in the angular direction. The most common approach relies on the spectral decomposition of potential functions in spherical harmonics; however, it requires an analytical manipulation of the equations [54]. The second family stems from [26] and applies finite-volume methods to evolve the magnetic field components, allowing them to resolve the magnetic discontinuities. ...
... Note that our crustal induction equation neglects terms such as the thermo-electric effect [54], relevant possibly only at high temperatures and in the outermost layers of the star (envelope). ...
... The initial magnetic field in our code is prescribed by using the scalar functions Φ and Ψ for the poloidal and toroidal components as in [54,22,25], which easily allow for the definition of multipoles. The first model, Core, includes the ambipolar diffusion and uses the same initial twisted-torus model as in [50], where the toroidal field is contained within the closed field lines and, automatically, the azimuthal component of the Lorentz force is initially zero everywhere. ...
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.
... Using the notation of Geppert and Wiebicke (1991), the basic idea is to expand the poloidal (Φ) and toroidal (Ψ ) scalar functions in a series of spherical harmonics ...
... where we use D nm and C nm as a shorthand for the nonlinear Hall terms (the full expressions can also be found in Geppert and Wiebicke 1991). These include sums over running indices and coupling constants related to Clebsch-Gordan coefficients (the sum rules to combine angular momentum operators are used to determine which multipoles are coupled to each other). ...
... In this appendix we go through some of the ideas of the mathematical formalism and compare the most common notations. Adopting the notation of Geppert and Wiebicke (1991), the magnetic field can be written in terms of two scalar functions Φ(r, t) and Ψ (r, t) (analogous to the stream functions in hydrodynamics) as follows: ...
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.
... Using the notation of Geppert and Wiebicke (1991), the basic idea is to expand the poloidal (Φ) and toroidal (Ψ ) scalar functions in a series of spherical harmonics ...
... where we use D nm and C nm as a shorthand for the nonlinear Hall terms (the full expressions can also be found in Geppert and Wiebicke 1991). These include sums over running indices and coupling constants related to Clebsch-Gordan coefficients (the sum rules to combine angular momentum operators are used to determine which multipoles are coupled to each other). ...
... In this appendix we go through some of the ideas of the mathematical formalism and compare the most common notations. Adopting the notation of Geppert and Wiebicke (1991), the magnetic field can be written in terms of two scalar functions Φ(r,t) and Ψ (r,t) (analogous to the stream functions in hydrodynamics) as follows: ...
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 application of these mechanisms to compact object astrophysics dates back to Dolginov and Urpin [160] (for white dwarfs) and Blandford et al. [7] (who extended the analysis to NSs). Thermoelectric effects were studied with the Hall effect in great mathematical detail in a series of papers by Geppert and Wiebicke [161][162][163][164][165][166], which laid the foundations for many magnetothermal evolution models (e.g., [36], see also Pons and Viganò [47] for a review), but they ended up being quite overlooked in the last decades. However, the necessity of revisiting them more systematically has recently become apparent for at least two reasons: first, computational advancements [153] showed that the thermal structure of a NS crust can be quite variegated; second, transient bursting phenomena are often linked to some sort of heat deposition in the crust other than the ohmic dissipation that locally enhance thermal gradients [167]. ...
Neutron stars host the strongest magnetic fields that we know of in the Universe. Their magnetic fields are the main means of generating their radiation, either magnetospheric or through the crust. Moreover, the evolution of the magnetic field has been intimately related to explosive events of magnetars, which host strong magnetic fields, and their persistent thermal emission. The evolution of the magnetic field in the crusts of neutron stars has been described within the framework of the Hall effect and Ohmic dissipation. Yet, this description is limited by the fact that the Maxwell stresses exerted on the crusts of strongly magnetised neutron stars may lead to failure and temperature variations. In the former case, a failed crust does not completely fulfil the necessary conditions for the Hall effect. In the latter, the variations of temperature are strongly related to the magnetic field evolution. Finally, sharp gradients of the star's temperature may activate battery terms and alter the magnetic field structure, especially in weakly magnetised neutron stars. In this review, we discuss the recent progress made on these effects. We argue that these phenomena are likely to provide novel insight into our understanding of neutron stars and their observable properties.
... The application of these mechanisms to compact object astrophysics dates back to Dolginov and Urpin [160] (for white dwarfs) and Blandford et al. [7] (who extended the analysis to NSs). Thermoelectric effects were studied with the Hall effect in great mathematical detail in a series of papers by Geppert and Wiebicke [161][162][163][164][165][166], which laid the foundations for many magnetothermal evolution models (e.g., [36], see also Pons and Viganò [47] for a review), but they ended up being quite overlooked in the last decades. However, the necessity of revisiting them more systematically has recently become apparent for at least two reasons: first, computational advancements [153] showed that the thermal structure of a NS crust can be quite variegated; second, transient bursting phenomena are often linked to some sort of heat deposition in the crust other than the ohmic dissipation that locally enhance thermal gradients [167]. ...
Neutron stars host the strongest magnetic fields that we know of in the Universe. Their magnetic fields are the main means of generating their radiation, either magnetospheric or through the crust. Moreover, the evolution of the magnetic field has been intimately related to explosive events of magnetars, which host strong magnetic fields, and their persistent thermal emission. The evolution of the magnetic field in the crusts of neutron stars has been described within the framework of the Hall effect and Ohmic dissipation. Yet, this description is limited by the fact that the Maxwell stresses exerted on the crusts of strongly magnetised neutron stars may lead to failure and temperature variations. In the former case, a failed crust does not completely fulfil the necessary conditions for the Hall effect. In the latter, the variations of temperature are strongly related to the magnetic field evolution. Finally, sharp gradients of the star’s temperature may activate battery terms and alter the magnetic field structure, especially in weakly magnetised neutron stars. In this review, we discuss the recent progress made on these effects. We argue that these phenomena are likely to provide novel insight into our understanding of neutron stars and their observable properties.
... Ces modèles contiennent une physique complexe avec une équation d'état de la matière ultra-dense encore inconnue et des découplages entre les différentes particules composant l'étoile à neutrons comme les électrons et les neutrons pour l'Effet Hall ou les protons et les neutrons pour la diffusion ambipolaire. L'effet magnéto-thermique, c'est-à-dire une amplification du champ magnétique par des gradients de température, est potentiellement important dans la croûte de l'étoile à neutron (Blandford et al., 1983 ;Geppert et Wiebicke, 1991). Une autre explication possible pour le chauffage de la croûte serait un chauffage lors des phénomènes transitoires observables à plus haute énergie (Li et Beloborodov, 2015). ...
La fin de vie des étoiles massives donne lieu à une explosion, appelée supernova. Ces explosions sont provoquées par l'effondrement de leur cœur de fer et la formation d'une étoile à neutrons. Les observations des supernovae montrent que certaines d'entre elles ont des caractéristiques extrêmes comme leur énergie cinétique pour les hypernovae ou leur luminosité pour les supernovae superlumineuses. Un moteur central de ces explosions différent du mécanisme des neutrinos pour les supernovae standards est souvent invoqué pour expliquer ces caractéristiques extrêmes : une explosion magnétorotationnelle. Ce mécanisme suppose la formation d'une proto-étoile à neutrons (PNS) en rotation rapide et avec un fort champ magnétique qui permet d'extraire l'énergie de rotation et obtenir une explosion plus énergétique ou lumineuse. Cette PNS, une fois refroidie en étoile à neutrons, fait partie de la classe des magnétars, qui se distingue par toute une diversité d'émissions à haute énergie dues à la dissipation de leur intense champ magnétique interne. Les observations de ces objets permettent d'inférer que la composante dipolaire de leur champ magnétique est de l'ordre de 10¹⁴-10¹⁵ G.L'origine des magnétars et de leur fort champ magnétique à grande échelle, particulièrement en présence de rotation rapide, reste une question ouverte. Deux mécanismes ont été invoqués pour amplifier le champ magnétique dans les PNS : la dynamo convective ou l'instabilité magnétorotationnelle (MRI). Cette thèse se propose d'étudier en détail le scénario de formation par la MRI. Celle-ci a déjà été étudiée de manière analytique ou dans des simulations numériques locales dans une boite représentant une partie de la PNS. Pour la première fois, cette thèse présente des modèles 3D sphériques simplifiés, ce qui permet d'étudier l'origine du dipôle. Une première étude a été menée pour étudier la génération d'un champ magnétique à grande échelle dans l'approximation incompressible, ce qui permet une plus vaste exploration des paramètres et des simulations plus longues. Nos simulations montrent la présence d'une dynamo auto-entretenue, dont l'état saturé ne dépend pas des conditions initiales du champ magnétique. Bien que cet état soit dominé par le champ magnétique turbulent (≥ 10¹⁵ G), un dipôle représentant 5% du champ magnétique moyen est généré dans toutes les simulations. De manière inédite, ce dipôle est orienté vers le plan équatorial plutôt que vers l'axe de rotation. De plus, la comparaison de ces modèles sphériques avec les modèles locaux montre que l'état turbulent de la MRI a des propriétés similaires, bien que le champ magnétique soit légèrement plus faible dans les modèles globaux. Un modèle basé sur l'approximation anélastique a ensuite été développé afin de prendre en compte les profils de densité et d'entropie d'une structure réaliste de PNS. Les simulations montrent également une dynamo auto-entrenue avec un champ magnétique moyen ≥ 10¹⁴ G et un dipôle équatorial de l'ordre de 4.3% du champ magnétique. De plus, un nouveau comportement à grande échelle apparaît avec ce modèle réaliste : une dynamo de champ moyen qui peut être décrite comme une dynamo αΩ. La comparaison de ce modèle avec des modèles idéalisés montre que la stratification en densité favorise l'apparition d'une dynamo de champ moyen. La force de flottaison limite la turbulence dans le plan équatorial mais a une influence assez faible dans l'ensemble du fait de la forte diffusion thermique due aux neutrinos. Dans l'ensemble, les résultats présentés dans cette thèse confirment la capacité de la MRI de former des magnétars dans le cas d'une rotation rapide.
... In general, all these coefficients depend on the physical conditions (density, temperature, chemical composition), and one or another term may dominate in different regimes. Other possible terms contributing to the electric field are due to the thermoelectric effect [41] and the chemical potential imbalance [16], wherever the gradient of temperature and of chemical potential, respectively, induce a movement of the charges. Here we do not consider such terms, since they require the coupled evolution with the temperature and the chemical composition. ...
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 {\em Simflowny} platform. The code uses high-order numerical schemes for the time and spatial discretization, and relies on the highly-scalable {\em 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.
... Given the existence of strong temperature gradients in certain layers and during certain periods of neutron star life, the transfer of thermal energy into magnetic is a genuine process of the magneto -thermal evolution scenarios. The transfer of thermal into magnetic energy may proceed either via an instability (Blandford et al. 1983;Urpin et al. 1986;Geppert & Wiebicke 1991), or the temperature gradient is sustained by "external" processes as e.g. the bombardment of the polar cap of radio pulsars. Then the thermoelectrically generated electric field acts as a battery. ...
The magnetic and thermal evolution of neutron stars is a very complex process with many non-linear interactions. For a decent understanding of neutron star physics, these evolutions cannot be considered isolated. A brief overview is presented, which describes the main magneto–thermal interactions that determine the fate of both isolated neutron stars and accreting ones. Special attention is devoted to the interplay of thermal and magnetic evolution at the polar cap of radio pulsars. There, a strong meridional temperature gradient is maintained over the lifetime of radio pulsars. It may be strong enough to drive thermoelectric magnetic field creation which perpetuate a toroidal magnetic field around the polar cap rim. Such a local field component may amplify and curve the poloidal surface field at the cap, forming a strong and small scale magnetic field as required for the radio emission of pulsars.
... However, due to the huge meridional temperature gradient a creation and perpetuation of small scale field structures is conceivable. The battery term of the induction equation including thermoelectric effects, ∇Q × ∇T , where Q is the predominantly radius dependent thermopower (see for details Blandford et al. 1983;Urpin et al. 1986;Geppert & Wiebicke 1991), causes the creation of a toroidal field structure. It surrounds the 'hot spot' as a torus, its inner radius corresponds roughly to the scale of the meridional temperature gradient. ...
Models of pulsar radio emission that are based on an inner accelerating region require the existence of very strong and small-scale
surface magnetic field structures at or near the canonical polar cap. The aim of this paper is to identify a mechanism that
creates such field structures and maintains them over a pulsar's lifetime. The likely physical process that can create the
required ‘magnetic spots’ is the Hall drift occurring in the crust of a neutron star. It is demonstrated that the Hall drift
can produce small-scale strong surface magnetic field anomalies (spots) on time-scales of 104 yr by means of non-linear interaction between poloidal and toroidal components of the subsurface magnetic field. These anomalies
are characterized by strengths of about 1014 G and curvature radii of field lines of about 106 cm, both of which are fundamental for generation of observable radio emission.
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.
Well before the radio discovery of pulsars offered the first observational confirmation for their existence (Hewish et al., Nature 217:709–713, 1968), it had been suggested that neutron stars might be endowed with very strong magnetic fields of 10¹⁰–10¹⁴ G (Hoyle et al., Nature 203:914–916, 1964; Pacini, Nature 216:567–568, 1967). It is because of their magnetic fields that these otherwise small ed inert, cooling dead stars emit radio pulses and shine in various part of the electromagnetic spectrum. But the presence of a strong magnetic field has more subtle and sometimes dramatic consequences: In the last decades of observations indeed, evidence mounted that it is likely the magnetic field that makes of an isolated neutron star what it is among the different observational manifestations in which they come. The contribution of the magnetic field to the energy budget of the neutron star can be comparable or even exceed the available kinetic energy. The most magnetised neutron stars in particular, the magnetars, exhibit an amazing assortment of explosive events, underlining the importance of their magnetic field in their lives. In this chapter we review the recent observational and theoretical achievements, which not only confirmed the importance of the magnetic field in the evolution of neutron stars, but also provide a promising unification scheme for the different observational manifestations in which they appear. We focus on the role of their magnetic field as an energy source behind their persistent emission, but also its critical role in explosive events.
A linear stage of the magnetic field generation in outer (rho of less
than about 10 to the 11th g/cu cm) layers of rather hot rotating neutron
stars by thermomagnetic phenomena which accompany the heat transport
from the stellar core is considered. For a neutron star with surface
gravity of 10 to the 14th cm/second-squared whose envelope consists of
degenerate electrons and Fe-56 ions in the fluid phase, the
thermomagnetic instability is developed at surface temperatures, T(e),
of more than about 3 million K and rotation periods of less than about 1
s. Axially-symmetric toroidal modes of magnetic field are unstable. In
particular, at T(e) of 5 million K, the maximal instability increment
equals 1/70 d and corresponds to the magnetic configuration with
wavelength of about 80 m in the meridional direction and maximal field
at a depth of about 40 m. At this T(e) the magnetic field modes with
wavelengths of between about 30 m to 1 km are generated. If T(e) falls
down to 3 million K due to the neutron star cooling, the instability
increment decreases to zero, while the spectrum of wavelengths tightens
to about 120 m. It is possible that at the nonlinear stage the proposed
mechanism leads to the generation of strong magnetic fields of about 10
to the 10th to 10 to the 13th G present in the surface layers of neutron
stars.
The stability of degenerate envelopes of neutron stars which possess a strong magnetic field (≡1011 - 1012G) is considered. It is shown that at early stages of evolution thermomagnetic instability may develop in the envelopes. The characteristic timescale and the conditions necessary for such an instability to develop are determined. Estimates of the small-scale magnetic fields and of the velocity of hydrodynamical motions generated by the instability are performed. Small-scale motions are able to transport the heat efficiently and to change significantly the cooling rate of neutron stars. The coefficient of turbulent "thermal conductivity" is evaluated.
The envelopes of nonmagnetic neutron stars are studied using the best available opacities and equation of state. The general relativistic equations of the structure and evolution of nonmagnetic neutron stars are discussed, and it is shown that they can be reduced to a single equation for calculating the thermal structure of neutron star envelopes. The physical input needed to solve the thermal structure equation is investigated and the numerical results of envelope model calculations are presented. It is shown that the thermal structure of neutron star envelopes is determined by the single parameter T(s) to the 4th/g(s), where T(s) is the effective surface temperature and g(s) the surface gravity of the star. This result is used to derive a number of other scaling relations, and the effects of general relativity on the envelope thermal structure are examined in detail. The results of a sensitivity analysis of the regional opacity needed to obtain a reliable relationship between the temperatures of the inner and outer boundaries of the envelope is presented.
Diese Arbeit befaßt sich mit der mathematischen Grundlage der vorangehenden über die Behandlung sphärischer Dynamomodelle. Es wird eine Darstellung von beliebigen Vektorfeldern im dreidimensionalen euklidischen Raum als Summe aus einem poloidalen und einem toroidalen Anteil begründet. Damit werden bekannte Darstellungen für rotationssymmetrische und für nicht notwendig rotationssymmetrische, jedoch quellenfreie Felder verallgemeinert. Es ergeben sich einige bemerkenswerte Aussagen, die bei vielen Aufgaben weitreichende Folgerungen gestatten. Als Beispiel werden Gleichungen für eine Klasse von kraftfreien Magnetfeldern gelöst. Ferner werden Möglichkeiten der weiteren Verallgemeinerung untersucht.
This paper deals with mathematical fundamentals of the foregoing one on the treatment of spherical dynamo models. A representation of arbitrary vector fields in the three-dimensional euclidian space as a sum of a poloidal and a toroidal part is founded. In this way well-known representations of axisymmetric or not necessarily axisymmetric but solenoidal fields are generalized. The mentioned parts of a field show some remarkable properties allowing far-reaching consequences in special problems. As an example equations governing a class of force-free magnetic fields are solved. Then possibilities of further generalization are investigated.
The authors summarize and extend the analysis of thermoelectric
phenomena in degenerate stars due to Urpin and Yakovlev (1980) and
calculate necessary conditions for field generation. A linearized
calculation of the growth of the field in the solid crust is given,
which demonstrates that small seed fields can grow exponentially for
sufficiently large heat fluxes. Non-linear growth of the field in the
liquid is described. If the field strength can grow to
≡1011 G, the electron gyrofrequency will exceed the
collision frequency and the field growth in the solid will enter the
non-linear phase. The Hall effect will lead to rapid convection of
magnetic flux and the creation of progressively larger scale structure,
perhaps resulting in the establishment of an axisymmetric field
geometry. In the absence of external heat sources, the interior of the
star will cool and the field will decay. The authors outline some of the
observational consequences of this theory for pulsars, binary X-ray
sources, X- and γ-ray bursters and white dwarfs.
The thermal structure of neutron star envelopes is investigated using
approximate analytical formulae. In the degenerate layers the thermal
structure equation is solved exactly for electron-dominated heat
transport. In the nondegenerate layers it is shown that if the opacity
is a power-law function of density and temperature, then the T(rho)
profiles lie along curves of constant thermal conductivity. The two
solutions are matched at intermediate densities to give an approximate
relation between the heat flux and the core temperature of the neutron
star. The dependence on various uncertain factors is found explicitly,
allowing a detailed understanding of the physical processes that control
the heat flux. The possible significance of the results to cooling
calculations that take into account strong magnetic fields is discussed.