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The thermal evolution of isolated neutron stars is a key element in unravelling their internal structure and composition and establishing evolutionary connections among different observational subclasses. Previous studies have predominantly focused on one-dimensional or axisymmetric two-dimensional models. In this study, we present the thermal evolution component of the novel three-dimensional magnetothermal code MATINS (MAgneto-Thermal evolution of Isolated Neutron Star). MATINS employs a finite volume scheme and integrates a realistic background structure, along with state-of-the-art microphysical calculations for the conductivities, neutrino emissivities, heat capacity, and superfluid gap models. This paper outlines the methodology employed to solve the thermal evolution equations in MATINS, along with the microphysical implementation that is essential for the thermal component. We test the accuracy of the code and present simulations with non-evolving magnetic fields of different configurations (all with electrical currents confined to the crust and a magnetic field that does not thread the core), to produce temperature maps of the neutron star surface. Additionally, for a specific magnetic field configuration, we show one fully coupled evolution of magnetic field and temperature. Subsequently, we use a ray-tracing code to link the neutron star surface temperature maps obtained by MATINS with the phase-resolved spectra and pulsed profiles that would be detected by distant observers. This study, together with our previous article focused on the magnetic formalism, presents in detail the most advanced evolutionary code for isolated neutron stars, with the aim of comparison with their timing properties, thermal luminosities and the associated X-ray light curves., QC 20240906

We present the first 3D fully coupled magneto-thermal simulations of neutron stars (including the most realistic background structure and microphysical ingredients so far) applied to a very complex initial magnetic field topology in the crust, similar to what recently obtained by proto-neutron star dynamo simulations. In such configurations, most of the energy is stored in the toroidal field, while the dipolar component is a few percent of the mean magnetic field. This initial feature is maintained during the long-term evolution (∼106 yr), since the Hall term favours a direct cascade (compensating for Ohmic dissipation) rather than a strong inverse cascade, for such an initial field topology. The surface dipolar component, responsible for the dominant electromagnetic spin-down torque, does not show any increase in time, when starting from this complex initial topology. This is at contrast with the timing properties of young pulsars and magnetars which point to higher values of the surface dipolar fields. A possibility is that the deep-seated magnetic field (currents in the core) is able to self-organize in large scales (during the collapse or in the early life of a neutron star). Alternatively, the dipolar field might be lower than is usually thought, with magnetosphere substantially contributing to the observed high spin-down, via e.g., strong winds or strong coronal magnetic loops, which can also provide a natural explanation to the tiny surface hotspots inferred from X-ray data.

Using Physics-Informed Neural Networks (PINNs) to solve a specific boundary value problem is becoming more popular as an alternative to traditional methods. However, depending on the specific problem, they could be computationally expensive and potentially less accurate. The functionality of PINNs for real-world physical problems can significantly improve if they become more flexible and adaptable. To address this, our work explores the idea of training a PINN for general boundary conditions and source terms expressed through a limited number of coefficients, introduced as additional inputs in the network. Although this process increases the dimensionality and is computationally costly, using the trained network to evaluate new general solutions is much faster. Our results indicate that PINN solutions are relatively accurate, reliable, and well-behaved. We applied this idea to the astrophysical scenario of the magnetic field evolution in the interior of a neutron star connected to a force-free magnetosphere. Solving this problem through a global simulation in the entire domain is expensive due to the elliptic solver’s needs for the exterior solution. The computational cost with a PINN was more than an order of magnitude lower than the similar case solved with a finite difference scheme, arguably at the cost of accuracy. These results pave the way for the future extension to 3D of this (or a similar) problem, where generalised boundary conditions are very costly to implement.

Using physics-informed neural networks (PINNs) to solve a specific boundary value problem is becoming more popular as an alternative to traditional methods. However, depending on the specific problem, they could be computationally expensive and potentially less accurate. The functionality of PINNs for real-world physical problems can significantly improve if they become more flexible and adaptable. To address this, our work explores the idea of training a PINN for general boundary conditions and source terms expressed through a limited number of coefficients, introduced as additional inputs in the network. Although this process increases the dimensionality and is computationally costly, using the trained network to evaluate new general solutions is much faster. Our results indicate that PINN solutions are relatively accurate, reliable, and well behaved. We applied this idea to the astrophysical scenario of the magnetic field evolution in the interior of a neutron star connected to a force-free magnetosphere. Solving this problem through a global simulation in the entire domain is expensive due to the elliptic solver’s needs for the exterior solution. The computational cost with a PINN was more than an order of magnitude lower than the similar case solved with a finite difference scheme, arguably at the cost of accuracy. These results pave the way for the future extension to three-dimensional of this (or a similar) problem, where generalized boundary conditions are very costly to implement.

We perform the first 3D fully coupled magneto-thermal simulations of neutron stars (including the most realistic background structure and microphysical ingredients so far) applied to a very complex initial magnetic field topology in the crust, similar to what was recently obtained by proto-neutron stars dynamo simulations. In such configurations, most of the energy is stored in the toroidal field, while the dipolar component is a few per cent of the mean magnetic field. This initial feature is maintained during the long-term evolution (∼106 yr), since the Hall term favours a direct cascade (compensating for Ohmic dissipation) rather than a strong inverse cascade, for such an initial field topology. The surface dipolar component, responsible for the dominant electromagnetic spin-down torque, does not show any increase in time, when starting from this complex initial topology. This is in contrast to the timing properties of young pulsars and magnetars which point to higher values of the surface dipolar fields. A possibility is that the deep-seated magnetic field (currents in the core) is able to self-organize in large scales (during the collapse or in the early life of a neutron star). Alternatively, the dipolar field might be lower than is usually thought, with magnetosphere substantially contributing to the observed high spin-down, via e.g. strong winds or strong coronal magnetic loops, which can also provide a natural explanation to the tiny surface hotspots inferred from X-ray data.

Neutron stars cool down during their lifetime through the combination of neutrino emission from the interior and photon cooling from the surface. Strongly magnetized neutron stars, called magnetars, are no exception, but the effect of their strong fields adds further complexities to the cooling theory. Besides other factors, modelling the outermost hundred meters (the envelope) plays a crucial role in predicting their surface temperatures. In this letter, we revisit the influence of envelopes on the cooling properties of neutron stars, with special focus on the critical effects of the magnetic field. We explore how our understanding of the relation between the internal and surface temperatures has evolved over the past two decades, and how different assumptions about the neutron star envelope and field topology lead to radically different conclusions on the surface temperature and its cooling with age. In particular, we find that relatively old magnetars with core-threading magnetic fields are actually much cooler than a rotation-powered pulsar of the same age. This is at variance with what is typically observed in crustal-confined models. Our results have important implications for the estimates of the X-ray luminosities of aged magnetars, and the subsequent population study of the different neutron star classes.

The long-term evolution of the internal, strong magnetic fields of neutron stars needs a specific numerical modelling. The diversity of the observed phenomenology of neutron stars indicates that their magnetic topology is rather complex and 3D simulations are required, for example, to explain the observed bursting mechanisms and the creation of surface hotspots. We present MATINS, a new 3D numerical code for magnetothermal evolution in neutron stars, based on a finite-volume scheme that employs the cubed-sphere system of coordinates. In this first work, we focus on the crustal magnetic evolution, with the inclusion of realistic calculations for the neutron star structure, composition, and electrical conductivity assuming a simple temperature evolution profile. MATINS follows the evolution of strong fields (1014 − 1015 Gauss) with complex non-axisymmetric topologies and dominant Hall-drift terms, and it is suitable for handling sharp current sheets. After introducing the technical description of our approach and some tests, we present long-term simulations of the non-linear field evolution in realistic neutron star crusts. The results show how the non-axisymmetric Hall cascade redistributes the energy over different spatial scales. Following the exploration of different initial topologies, we conclude that during a few tens of kyr, an equipartition of energy between the poloidal and toroidal components happens at small-scales. However, the magnetic field keeps a strong memory of the initial large scales, which are much harder to be restructured or created. This indicates that large-scale configuration attained during the neutron star formation is crucial to determine the field topology at any evolution stage.

Neutron stars cool down during their lifetime through the combination of neutrino emission from the interior and photon cooling from the surface. Strongly magnetised neutron stars, called magnetars, are no exception, but the effect of their strong fields adds further complexities to the cooling theory. Besides other factors, modelling the outermost hundred meters (the envelope) plays a crucial role in predicting their surface temperatures. In this letter, we revisit the influence of envelopes on the cooling properties of neutron stars, with special focus on the critical effects of the magnetic field. We explore how our understanding of the relation between the internal and surface temperatures has evolved over the past two decades, and how different assumptions about the neutron star envelope and field topology lead to radically different conclusions on the surface temperature and its cooling with age. In particular, we find that relatively old magnetars with core-threading magnetic fields are actually much cooler than arotation-powered pulsar of the same age. This is at variance with what is typically observed in crustal-confined models. Our results have important implications for the estimates of the X-ray luminosities of aged magnetars, and the subsequent population study of the different neutron star classes.

The long-term evolution of the internal, strong magnetic fields of neutron stars needs a specific numerical modeling. The diversity of the observed phenomenology of neutron stars indicates that their magnetic topology is rather complex and three-dimensional simulations are required, for example, to explain the observed bursting mechanisms and the creation of surface hotspots. We present MATINS, a new three dimensions numerical code for magneto-thermal evolution in neutron stars, based on a finite-volume scheme that employs the cubed-sphere system of coordinates. In this first work, we focus on the crustal magnetic evolution, with the inclusion of realistic calculations for the neutron star structure, composition and electrical conductivity assuming a simple temperature evolution profile. MATINS follows the evolution of strong fields (1014 − 1015 Gauss) with complex non-axisymmetric topologies and dominant Hall-drift terms, and it is suitable for handling sharp current sheets. After introducing the technical description of our approach and some tests, we present long-term simulations of the non-linear field evolution in realistic neutron star crusts. The results show how the non-axisymmetric Hall cascade redistributes the energy over different spatial scales. Following the exploration of different initial topologies, we conclude that during a few tens of kyr, an equipartition of energy between the poloidal and toroidal components happens at small-scales. However, the magnetic field keeps a strong memory of the initial large-scales, which are much harder to be restructured or created. This indicates that large-scale configuration attained during the neutron star formation is crucial to determine the field topology at any evolution stage.

Neutron star models with maxim mass close to 2 M reach high central densities, which may activate nucleonic and hyperon direct Urca neutrino emission. To alleviate the tension between fast theoretical cooling rates and theal luminosity observations of moderately magnetized, isolated theally emitting stars (with L1031 erg s-1 at t 105.3 yr), some internal heating source is required. The power supplied by the internal heater is estimated for both a phenomenological source in the inner crust and Joule heating due to magnetic field decay, assuming different superfluidity models and compositions of the outer stellar envelope. It is found that a theal power of W(t) ≈ 1034 erg s-1 allows neutron star models to match observations of moderately magnetized, isolated stars with ages t 105.3 yr. The requisite W(t) can be supplied by Joule heating due to crust-confined initial magnetic configurations with (i) mixed poloidal-toroidal fields, with surface strength Bdip = 1013 G at the pole of the dipolar poloidal component and 90 per cent of the magnetic energy stored in the toroidal component; and (ii) poloidal-only configurations with Bdip = 1014 G.

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.

The detection of a short hard X-ray burst and an associated bright soft X-ray source by the Swift satellite in 2020 October heralded a new magnetar in outburst, SGR J1830-0645. Pulsations at a period of ∼10.4 s were detected in prompt follow-up X-ray observations. We present here the analysis of the Swift/Burst Alert Telescope burst, of XMM-Newton and the Nuclear Spectroscopic Telescope Array observations performed at the outburst peak, and of a Swift/X-ray Telescope monitoring campaign over the subsequent month. The burst was single-peaked, lasted ∼6 ms, and released a fluence of ≈ 5 × 10-9 erg cm-2 (15-50 keV). The spectrum of the X-ray source at the outburst peak was well described by an absorbed double-blackbody model plus a power-law component detectable up to ∼25 keV. The unabsorbed X-ray flux decreased from ∼5 × 10-11 to ∼2.5 × 10-11 erg cm-2 s-1 one month later (0.3-10 keV). Based on our timing analysis, we estimate a dipolar magnetic field ≈5.5 × 1014 G at pole, a spin-down luminosity ≈2.4 × 1032 erg s-1, and a characteristic age ≈24 kyr. The spin modulation pattern appears highly pulsed in the soft X-ray band, and becomes smoother at higher energies. Several short X-ray bursts were detected during our campaign. No evidence for periodic or single-pulse emission was found at radio frequencies in observations performed with the Sardinia Radio Telescope and Parkes. According to magneto-thermal evolutionary models, the real age of SGR J1830-0645 is close to the characteristic age, and the dipolar magnetic field at birth was slightly larger, ∼1015 G.

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.

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.

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.