Neutron stars harbor extremely strong magnetic fields within their solid outer crust. The topology of this field strongly influences the surface temperature distribution and, hence, the star's observational properties. In this work, we present the first realistic simulations of the coupled crustal magnetothermal evolution of isolated neutron stars in three dimensions accounting for neutrino emission, obtained with the pseudo-spectral code PARODY. We investigate both the secular evolution, especially in connection with the onset of instabilities during the Hall phase, and the short-term evolution following episodes of localized energy injection. Simulations show that a resistive tearing instability develops in about a Hall time if the initial toroidal field exceeds »10 15 G. This leads to crustal failures because of the huge magnetic stresses coupled with the local temperature enhancement produced by dissipation. Localized heat deposition in the crust results in the appearance of hot spots on the star surface, which can exhibit a variety of patterns. Because the transport properties are strongly influenced by the magnetic field, the hot regions tend to drift away and get deformed following the magnetic field lines while cooling. The shapes obtained with our simulations are reminiscent of those recently derived from NICER X-ray observations of the millisecond pulsar PSRJ0030+0451.

X-ray emission from the surface of isolated neutron stars (NSs) has been now observed in a variety of sources. The ubiquitous presence of pulsations clearly indicates that thermal photons either come from a limited area, possibly heated by some external mechanism, or from the entire (cooling) surface but with an inhomogeneous temperature distribution. In a NS the thermal map is shaped by the magnetic field topology, since heat flows in the crust mostly along the magnetic field lines. Self-consistent surface thermal maps can hence be produced by simulating the coupled magnetic and thermal evolution of the star. We compute the evolution of the neutron star crust in three dimensions for different initial configurations of the magnetic field and use the ensuing thermal surface maps to derive the spectrum and the pulse profile as seen by an observer at infinity, accounting for general-relativistic effects. In particular, we compare cases with a high degree of symmetry with inherently 3D ones, obtained by adding a quadrupole to the initial dipolar field. Axially symmetric fields result in rather small pulsed fractions ( 5%), while more complex configurations produce higher pulsed fractions, up to ∼ 25%. We find that the spectral properties of our axisymmetric model are close to those of the bright isolated NS RX J1856.5-3754 at an evolutionary time comparable with the inferred dynamical age of the source.

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The radio-quiet pulsar PSR J2021+4026 is mostly known because it is the only rotation-powered pulsar that shows variability in its γ-ray emission. Using XMM-Newton archival data, we first confirmed that its flux is steady in the X-ray band, and then we showed that both the spectral and timing X-ray properties, that is to say the narrow pulse profile, the high pulsed fraction of 80–90%, and its dependence on the energy, can be better reproduced using a magnetized atmosphere model instead of simply a blackbody model. With a maximum likelihood analysis in the energy-phase space, we inferred that the pulsar has, in correspondence of one magnetic pole, a hot spot with a temperature of T ∼ 1 MK and colatitude extension of θ ∼ 20°. For the pulsar distance of 1.5 kpc, this corresponds to a cap of R ∼ 5 − 6 km, which is greater than the standard dimension of the dipolar polar caps. The large pulsed fraction further argues against emission from the entire star surface, as it would be expected in the case of secular cooling. An unpulsed (≲40% pulsed fraction), nonthermal component, probably originating in a wind nebula, is also detected. The pulsar geometry derived with our spectral fits in the X-ray is relatively well constrained (χ = 90° and ξ = 20°–25°) and consistent with what is deduced from γ-ray observations, provided that only one of the two hemispheres is active. The evidence for an extended hot spot in PSR J2021+4026, which was also found in other pulsars of a similar age but not in older objects, suggests a possible age dependence of the emitting size of thermal X-rays.