The many lives of magnetized neutron stars

The magnetic field strength at birth is arguably one of the most important properties to determine the evolutionary path of a neutron star. Objects with very high fields, collectively known as magnetars, are characterized by high X-ray quiescent luminosities, occurrence of outbursts, and, for some of them, sporadic giant flares. While the magnetic field strength is believed to drive their collective behaviour, however, the diversity of their properties, and, especially, the observation of magnetar-like bursts from “low-field” pulsars, has been a theoretical puzzle. In this review, we discuss results of long-term simulations following the coupled evolution of the X-ray luminosity and the timing properties for a large, homogeneous sample of X-ray emitting isolated neutron stars, accounting for a range of initial magnetic field strengths, envelope compositions, and neutron star masses. In addition, by following the evolution of magnetic stresses within the neutron star crust, we can also relate the observed magnetar phenomenology to the physical properties of neutron stars, and in particular to their age and magnetic field strength and topology. The dichotomy of “high-B” field pulsars versus magnetars is naturally explained, and occasional outbursts from old, low B-field neutron stars are predicted. We conclude by speculating on the fate of old magnetars, and by presenting observational diagnostics of the neutron star crustal field topology.

Neutron Stars—Thermal Emitters

Confronting theoretical models with observations of thermal radiation emitted by neutron stars is one of the most important ways to understand the properties of both, superdense matter in the interiors of the neutron stars and dense magnetized plasmas in their outer layers. Here we review the theory of thermal emission from the surface layers of strongly magnetized neutron stars, and the main properties of the observational data. In particular, we focus on the nearby sources for which a clear thermal component has been detected, without being contaminated by other emission processes (magnetosphere, accretion, nebulae). We also discuss the applications of the modern theoretical models of the formation of spectra of strongly magnetized neutron stars to the observed thermally emitting objects.

Neutron Stars—Cooling and Transport

Observations of thermal radiation from neutron stars can potentially provide information about the states of supranuclear matter in the interiors of these stars with the aid of the theory of neutron-star thermal evolution. We review the basics of this theory for isolated neutron stars with strong magnetic fields, including most relevant thermodynamic and kinetic properties in the stellar core, crust, and blanketing envelopes.

The imprint of the crustal magnetic field on the thermal spectra and pulse profiles of isolated neutron stars

Isolated neutron stars (NSs) show a bewildering variety of astrophysical manifestations, presumably shaped by the magnetic field strength and topology at birth. Here, using state-of-the-art calculations of the coupled magnetic and thermal evolution of NSs, we compute the thermal spectra and pulse profiles expected for a variety of initial magnetic field configurations. In particular, we contrast models with purely poloidal magnetic fields to models dominated by a strong internal toroidal component. We find that, while the former displays double-peaked profiles and very low pulsed fractions, in the latter, the anisotropy in the surface temperature produced by the toroidal field often results in a single pulse profile, with pulsed fractions that can exceed the 50–60 per cent level even for perfectly isotropic local emission. We further use our theoretical results to generate simulated ‘observed’ spectra, and show that blackbody (BB) fits result in inferred radii that can be significantly smaller than the actual NS radius, even as low as ∼1–2 km for old NSs with strong internal toroidal fields and a high absorption column density along their line of sight. We compute the size of the inferred BB radius for a few representative magnetic field configurations, NS ages and magnitudes of the column density. Our theoretical results are of direct relevance to the interpretation of X-ray observations of isolated NSs, as well as to the constraints on the equation of state of dense matter through radius measurements.

Unifying the observational diversity of isolated neutron stars via magneto-thermal evolution models

Observations of magnetars and some of the high magnetic field pulsars have shown that their thermal luminosity is systematically higher than that of classical radio-pulsars, thus confirming the idea that magnetic fields are involved in their X-ray emission. Here we present the results of 2D simulations of the fully coupled evolution of temperature and magnetic field in neutron stars, including the state-of-the-art kinetic coefficients and, for the first time, the important effect of the Hall term. After gathering and thoroughly re-analysing in a consistent way all the best available data on isolated, thermally emitting neutron stars, we compare our theoretical models to a data sample of 40 sources. We find that our evolutionary models can explain the phenomenological diversity of magnetars, high-B radio-pulsars, and isolated nearby neutron stars by only varying their initial magnetic field, mass and envelope composition. Nearly all sources appear to follow the expectations of the standard theoretical models. Finally, we discuss the expected outburst rates and the evolutionary links between different classes. Our results constitute a major step towards the grand unification of the isolated neutron star zoo.

Central compact objects and the hidden magnetic field scenario

Central compact objects (CCOs) are X-ray sources lying close to the centre of supernova remnants, with inferred values of the surface magnetic fields significantly lower (≲1011 G) than those of standard pulsars. In this paper, we revise the hidden magnetic field scenario, presenting the first 2D simulations of the submergence and re-emergence of the magnetic field in the crust of a neutron star. A post-supernova accretion stage of about 10−4–10−3 M⊙ over a vast region of the surface is required to bury the magnetic field into the inner crust. When accretion stops, the field re-emerges on a typical time-scale of 1–100 kyr, depending on the submergence conditions. After this stage, the surface magnetic field is restored close to its birth values. A possible observable consequence of the hidden magnetic field is the anisotropy of the surface temperature distribution, in agreement with observations of several of these sources. We conclude that the hidden magnetic field model is viable as an alternative to the antimagnetar scenario, and it could provide the missing link between CCOs and the other classes of isolated neutron stars.

Modelling of the surface emission of the low magnetic field magnetar SGR 0418+5729

We perform a detailed modelling of the post-outburst surface emission of the low magnetic field magnetar SGR 0418+5729. The dipolar magnetic field of this source, B=6×1012G estimated from its spin-down rate, is in the observed range of magnetic fields for normal pulsars. The source is further characterized by a high pulse fraction and a single-peak profile. Using synthetic temperature distribution profiles, and fully accounting for the general-relativistic effects of light deflection and gravitational redshift, we generate synthetic X-ray spectra and pulse profiles that we fit to the observations. We find that asymmetric and symmetric surface temperature distributions can reproduce equally well the observed pulse profiles and spectra of SGR 0418. None the less, the modelling allows us to place constraints on the system geometry (i.e. the angles ψ and ξ that the rotation axis makes with the line of sight and the dipolar axis, respectively), as well as on the spot size and temperature contrast on the neutron star surface. After performing an analysis iterating between the pulse profile and spectra, as done in similar previous works, we further employed, for the first time in this context, a Markov-Chain Monte Carlo approach to extract constraints on the model parameters from the pulse profiles and spectra, simultaneously. We find that, to reproduce the observed spectrum and flux modulation: (a) the angles must be restricted to 65° ≲ ψ + ξ ≲ 125° or 235° ≲ ψ + ξ ≲ 295°; (b) the temperature contrast between the poles and the equator must be at least a factor of ∼6, and (c) the size of the hottest region ranges between 0.2 and 0.7 km (including uncertainties on the source distance). Lastly, we interpret our findings within the context of internal and external heating models.

Population synthesis of isolated neutron stars with magneto-rotational evolution – II. From radio-pulsars to magnetars

Population synthesis studies constitute a powerful method to reconstruct the birth distribution of periods and magnetic fields of the pulsar population. When this method is applied to populations in different wavelengths, it can break the degeneracy in the inferred properties of initial distributions that arises from single-band studies. In this context, we extend previous works to include X-ray thermal emitting pulsars within the same evolutionary model as radio-pulsars. We find that the cumulative distribution of the number of X-ray pulsars can be well reproduced by several models that, simultaneously, reproduce the characteristics of the radio-pulsar distribution. However, even considering the most favourable magneto-thermal evolution models with fast field decay, lognormal distributions of the initial magnetic field overpredict the number of visible sources with periods longer than 12 s. We then show that the problem can be solved with different distributions of magnetic field, such as a truncated lognormal distribution, or a binormal distribution with two distinct populations. We use the observational lack of isolated neutron stars (NSs) with spin periods P > 12 s to establish an upper limit to the fraction of magnetars born with B > 1015 G (less than 1 per cent). As future detections keep increasing the magnetar and high-B pulsar statistics, our approach can be used to establish a severe constraint on the maximum magnetic field at birth of NSs.

Simulated magnetic field expulsion in neutron star cores

The study of long-term evolution of neutron star (NS) magnetic fields is key to understanding the rich diversity of NS observations, and to unifying their nature despite the different emission mechanisms and observed properties. Such studies in principle permit a deeper understanding of the most important parameters driving their apparent variety, e.g. radio pulsars, magnetars, X-ray dim isolated NSs, gamma-ray pulsars. We describe, for the first time, the results from self-consistent magnetothermal simulations considering not only the effects of the Hall-driven field dissipation in the crust, but also adding a complete set of proposed driving forces in a superconducting core. We emphasize how each of these core-field processes drive magnetic evolution and affect observables, and show that when all forces are considered together in vectorial form, the net expulsion of core magnetic flux is negligible, and will have no observable effect in the crust (consequently in the observed surface emission) on megayear time-scales. Our new simulations suggest that strong magnetic fields in NS cores (and the signatures on the NS surface) will persist long after the crustal magnetic field has evolved and decayed, due to the weak combined effects of dissipation and expulsion in the stellar core.

Are pulsars born with a hidden magnetic field?

The observation of several neutron stars in the centre of supernova remnants and with significantly lower values of the dipolar magnetic field than the average radio-pulsar population has motivated a lively debate about their formation and origin, with controversial interpretations. A possible explanation requires the slow rotation of the protoneutron star at birth, which is unable to amplify its magnetic field to typical pulsar levels. An alternative possibility, the hidden magnetic field scenario, considers the accretion of the fallback of the supernova debris on to the neutron star as responsible for the submergence (or screening) of the field and its apparently low value. In this paper, we study under which conditions the magnetic field of a neutron star can be buried into the crust due to an accreting, conducting fluid. For this purpose, we consider a spherically symmetric calculation in general relativity to estimate the balance between the incoming accretion flow and the magnetosphere. Our study analyses several models with different specific entropy, composition, and neutron star masses. The main conclusion of our work is that typical magnetic fields of a few times 1012 G can be buried by accreting only 10−3–10−2 M⊙, a relatively modest amount of mass. In view of this result, the central compact object scenario should not be considered unusual, and we predict that anomalously weak magnetic fields should be common in very young (< few kyr) neutron stars.

Force-free twisted magnetospheres of neutron stars

Context. The X-ray spectra observed in the persistent emission of magnetars are evidence for the existence of a magnetosphere. The high-energy part of the spectra is explained by resonant cyclotron upscattering of soft thermal photons in a twisted magnetosphere, which has motivated an increasing number of efforts to improve and generalize existing magnetosphere models. Aims. We want to build more general configurations of twisted, force-free magnetospheres as a first step to understanding the role played by the magnetic field geometry in the observed spectra. Methods. First we reviewed and extended previous analytical works to assess the viability and limitations of semi-analytical approaches. Second, we built a numerical code able to relax an initial configuration of a nonrotating magnetosphere to a force-free geometry, provided any arbitrary form of the magnetic field at the star surface. The numerical code is based on a finite-difference time-domain, divergence-free, and conservative scheme, based of the magneto-frictional method used in other scenarios. Results. We obtain new numerical configurations of twisted magnetospheres, with distributions of twist and currents that differ from previous analytical solutions. The range of global twist of the new family of solutions is similar to the existing semi-analytical models (up to some radians), but the achieved geometry may be quite different. Conclusions. The geometry of twisted, force-free magnetospheres shows a wider variety of possibilities than previously considered. This has implications for the observed spectra and opens the possibility of implementing alternative models in simulations of radiative transfer aiming at providing spectra to be compared with observations.

Pulsar timing irregularities and the imprint of magnetic field evolution

Context. The rotational evolution of isolated neutron stars is dominated by the magnetic field anchored to the solid crust of the star. Assuming that the core field evolves on much longer timescales, the crustal field evolves mainly though Ohmic dissipation and the Hall drift, and it may be subject to relatively rapid changes with remarkable effects on the observed timing properties. Aims. We investigate whether changes of the magnetic field structure and strength during the star evolution may have observable consequences in the braking index n. This is the most sensitive quantity to reflect small variations of the timing properties that are caused by magnetic field rearrangements. Methods. We performed axisymmetric, long-term simulations of the magneto-thermal evolution of neutron stars with state-of-the-art microphysical inputs to calculate the evolution of the braking index. Relatively rapid magnetic field modifications can be expected only in the crust of neutron stars, where we focus our study. Results. We find that the effect of the magnetic field evolution on the braking index can be divided into three qualitatively different stages depending on the age and the internal temperature: a first stage that may be different for standard pulsars (with n ~ 3) or low field neutron stars that accreted fallback matter during the supernova explosion (systematically n < 3); in a second stage, the evolution is governed by almost pure Ohmic field decay, and a braking index n > 3 is expected; in the third stage, at late times, when the interior temperature has dropped to very low values, Hall oscillatory modes in the neutron star crust result in braking indices of a high absolute value and both positive and negative signs. Conclusions. Current magneto-thermal evolution models predict a large contribution to the timing noise and, in particular, to the braking index, from temporal variations of the magnetic field. Models with strong (≳ 1014 G) multipolar or toroidal components, even with a weak (~1012 G) dipolar field are consistent with the observed trend of the timing properties.

The outburst decay of the low magnetic field magnetar SGR 0418+5729

We report on the long-term X-ray monitoring of the outburst decay of the low magnetic field magnetar SGR 0418+5729 using all the available X-ray data obtained with RXTE, Swift, Chandra, and XMM-Newton observations from the discovery of the source in 2009 June up to 2012 August. The timing analysis allowed us to obtain the first measurement of the period derivative of SGR 0418+5729: ˙ P = 4(1) × 10−15 s s−1, significant at a ∼3.5σ confidence level. This leads to a surface dipolar magnetic field of Bdip 6 × 1012 G. This measurement confirms SGR 0418+5729 as the lowest magnetic field magnetar. Following the flux and spectral evolution from the beginning of the outburst up to ∼1200 days, we observe a gradual cooling of the tiny hot spot responsible for the X-ray emission, from a temperature of ∼0.9 to 0.3 keV. Simultaneously, the X-ray flux decreased by about three orders of magnitude: from about 1.4 × 10−11 to 1.2 × 10−14 erg s−1 cm−2. Deep radio, millimeter, optical, and gamma-ray observations did not detect the source counterpart, implying stringent limits on its multi-band emission, as well as constraints on the presence of a fossil disk. By modeling the magneto-thermal secular evolution of SGR 0418+5729, we infer a realistic age of ∼550 kyr, and a dipolar magnetic field at birth of ∼1014 G. The outburst characteristics suggest the presence of a thin twisted bundle with a small heated spot at its base. The bundle untwisted in the first few months following the outburst, while the hot spot decreases in temperature and size. We estimate the outburst rate of low magnetic field magnetars to be about one per year per galaxy, and we briefly discuss the consequences of such a result in several other astrophysical contexts.

No anticorrelation between cyclotron line energy and X-ray flux in 4U 0115+634

We report on an outburst of the high mass X-ray binary 4U 0115+634 with a pulse period of 3.6 s in 2008 March/April as observed with RXTE and INTEGRAL. During the outburst the neutron star’s luminosity varied by a factor of 10 in the 3–50 keV band. In agreement with earlier work we find evidence of five cyclotron resonance scattering features at ~10.7, 21.8, 35.5, 46.7, and 59.7 keV. Previous work had found an anticorrelation between the fundamental cyclotron line energy and the X-ray flux. We show that this apparent anticorrelation is probably due to the unphysical interplay of parameters of the cyclotron line with the continuum models used previously, e.g., the negative and positive exponent power law (NPEX). For this model, we show that cyclotron line modeling erroneously leads to describing part of the exponential cutoff and the continuum variability, and not the cyclotron lines. When the X-ray continuum is modeled with a simple exponentially cutoff power law modified by a Gaussian emission feature around 10 keV, the correlation between the line energy and the flux vanishes, and the line parameters remain virtually constant over the outburst. We therefore conclude that the previously reported anticorrelation is an artifact of the assumptions adopted in the modeling of the continuum.

Population synthesis of isolated neutron stars with magneto-rotational evolution

We revisit the population synthesis of isolated radio-pulsars incorporating recent advances on the evolution of the magnetic field and the angle between the magnetic and rotational axes from new simulations of the magneto-thermal evolution and magnetosphere models, respectively. An interesting novelty in our approach is that we do not assume the existence of a death line. We discuss regions in parameter space that are more consistent with the observational data. In particular, we find that any broad distribution of birth spin periods with P0 ≲ 0.5 s can fit the data, and that if the alignment angle is allowed to vary consistently with the torque model, realistic magnetospheric models are favoured compared to models with classical magneto-dipolar radiation losses. Assuming that the initial magnetic field is given by a lognormal distribution, our optimal model has mean strength 〈log B0[G]〉 ≈ 13.0–13.2 with width σ(log B0) = 0.6–0.7. However, there are strong correlations between parameters. This degeneracy in the parameter space can be broken by an independent estimate of the pulsar birth rate or by future studies correlating this information with the population in other observational bands (X-rays and γ-rays).