The observational properties of isolated NSs are shaped by their magnetic field and surface temperature. They evolve in a strongly coupled fashion, and modelling them is key in understanding the emission properties of NSs. Much effort was put in tackling this problem in the past but only recently a suitable 3D numerical framework was developed. We present a set of 3D simulations addressing both the long-Term evolution (≈ 104-106 yrs) and short-lived outbursts (â 1 yr). Not only a 3D approach allows one to test complex field geometries, but it is absolutely key to model magnetar outbursts, which observations associate to the appearance of small, inherently asymmetric hot regions. Even though the mechanism that triggers these phenomena is not completely understood, following the evolution of a localised heat injection in the crust serves as a model to study the unfolding of the event

De Grandis, D

Taverna, R

Turolla, R

Wood, TS

Zane, S

English

UCL Discovery

Neutron Star Astrophysics at the Crossroads: Magnetars and the Multimessenger RevolutionProceedings IAU Symposium No. 363, 2023E. Troja & M. G. Baring, eds.doi:10.1017/S174392132200031XModelling Magnetar Behaviour with 3DMagnetothermal SimulationsDavide De Grandis∗,1,2 , Roberto Turolla1,2, Roberto Taverna1 ,Toby S. Wood3 and Silvia Zane2∗email: davide.degrandis@phd.unipd.it1Department of Physics and Astronomy, University of Padova, via Marzolo 8,I-35131 Padova, Italy2Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Surrey,RH5 6NT, United Kingdom3School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU,United KingdomAbstract. The observational properties of isolated NSs are shaped by their magnetic field andsurface temperature. They evolve in a strongly coupled fashion, and modelling them is key inunderstanding the emission properties of NSs. Much effort was put in tackling this problemin the past but only recently a suitable 3D numerical framework was developed. We present aset of 3D simulations addressing both the long-term evolution (≈ 104–106 yrs) and short-livedoutbursts (<∼ 1 yr). Not only a 3D approach allows one to test complex field geometries, but itis absolutely key to model magnetar outbursts, which observations associate to the appearanceof small, inherently asymmetric hot regions. Even though the mechanism that triggers thesephenomena is not completely understood, following the evolution of a localised heat injectionin the crust serves as a model to study the unfolding of the event.Keywords. neutron stars, magnetic fields, numerical methods1. IntroductionWe present a set of simulations of the magnetic and thermal evolution in a NS crustperformed with a suitably adapted version of the 3D code PARODY (Wood & Hollerbach2014). We restrict the integration domain to the NS crust only, relying on the assumptionof complete magnetic flux expulsion from the superconducting core. The core of the staris treated as a boundary condition in prescribing the temperature at the crust bottom.The evolution of the crust is computed by solving the Hall induction equation in theelectron MHD regime (i.e. one in which only electrons can move), and the heat balanceequation∂B∂t=−c∇×[σ−1J+1ecneJ×B+G ·∇T − ∇μe](1.1)Cv∂T∂t=−∇ ·(TG · J− k ·∇T − μeJ)+E · J−Nν (1.2)where B and E are the magnetic and electric field, T is the temperature, ne is theelectron density, μ= c(3π2ne)1/3 is the electron chemical potential, J = c∇×B/4π isthe current, k is the thermal conductivity tensor, G is the thermopower tensor and Cv isthe heat capacity (per unit volume). The electrical and thermal conductivity σ and k are© The Author(s), 2023. Published by Cambridge University Press on behalf of International AstronomicalUnion. This is an Open Access article, distributed under the terms of the Creative Commons Attributionlicence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, andreproduction in any medium, provided the original work is properly cited.https://doi.org/10.1017/S174392132200031X Published online by Cambridge University Press3D simulations of magnetars 281taken as those of a completely degenerate gas of electrons (see De Grandis et al. (2020) fora complete description of the code and microphysical input). Nν is the neutrino emissivitydue to weak processes in the crust (Yakovlev et al. 2001; we considered the contributionsfrom weak plasmon decay, neutrino bremsstrahlung, weak pair annihilation and weaksynchrotron emission). Contrary to neutrino emission from the core, that regulates thethermal evolution throughout the NS life, crustal processes become active above ∼ 3×109 K; therefore, they are important only in the early cooling phases or during transientactivity due to large heat deposition in the crust. We computed the luminosity of thestar (in the local rest frame) assuming patch-wise blackbody emission at the local surfacetemperature. Note that the latter is different from the temperature at the top of the crustdue to the presence of a (magnetised) blanketing envelope (e.g. Bezgonov et al. 2021),which also sets the boundary condition for Eq. 1.2.We studied the long-term evolution adopting the equation of state fromGourgouliatos & Cumming 2014 that describes the inner crust only, i.e. assuming athick blanketing envelope; to describe short term phenomena we added the outer layersof the crust up to ≈ 107 g cm−3, using the profile proposed by Yakovlev et al. (2021) (seetheir Eq. 2) for standard NS parameters, M = 1.4M, R= 12 km.2. Secular Evolution ModelsWe first consider the evolution of a purely dipolar initial field (Fig. 1) with Bd  1013 G.In the first ≈ 105 yr, the evolution is dominated by the Hall term, which displaces energyto the smaller scales. In particular, it tends to favour odd multipoles, pointing to astructure known as the Hall attractor (Gourgouliatos & Cumming 2014). We found thatthis configuration produces stronger field structures around the magnetic equator, whereclosed lines get formed. In turn, this translates into the appearance of a hot equatorialbelt: in fact, not only the field ohmic decay heats the crust, but due to the anisotropicconductivity, which is favoured along the field lines, the magnetic loops act as a thermalinsulator. Note, however, that the presence of a magnetised envelope makes so that thecrustal thermal structure does not directly map to the surface: in fact, at the equator thefield is parallel to the surface, thwarting heat transport in the blanketing envelope. Theresult is a cold equatorial belt, whereas the polar regions are initially warmer. After theequatorial heat trapping has been active for a long time, however, hot structures appeararound the cold equatorial belt, with intermediate temperature polar caps. If the fieldis not symmetric, this mechanism is not as efficient. We ran a simulation with a fieldcomposed initially of a dipolar field and a quadrupolar one tilted by an angle Θq = 45◦. Inthis case, a hot structure appears around the equator of the dipole, but it is asymmetric,having an almost spot-like appearence. The effect of the envelope still produces a coldbelt, which is skewed by the asymmetric field. The thermal map can then be used tosimulate the emission observables using a GR ray-tracing code (Taverna et al. 2015).The pulsed fraction can reach <∼ 20%, whereas in the dipolar case it is <∼ 5%.3. Short-term evolution & Magnetar OutburstsMagnetars are the most strongly magnetised NSs and are characterised by violentactivity in the X/γ-rays. In particular, many of them exhibit outbursts, sudden (≈ hours)increases of the flux (by a factor 10-1000 with respect to the quiescent level), accompaniedby a modification of the X-ray spectrum and of the pulse profile. These are interpretedas the appearance of hotter region(s) on the star surface, that then decay over severalmonths (see Coti Zelati et al. (2018) for a comprehensive overview of outburst properties).In order to model this behaviour, we used the version of the code adapted to short-termevents, injecting heat in a localised patch in the outer crust (ρ 3× 107 g cm−3). Thehttps://doi.org/10.1017/S174392132200031X Published online by Cambridge University Press282 D. De Grandis et al.Figure 1. Two snapshots at t 2 kyr and t 400 kyr of the evolution of the temperatureand poloidal field in an initially dipolar case. The left half of each plot shows the temperaturestructure inside the crust (enlarged for visualisation), with the field lines superimposed; theright half the corresponding surface thermal map, considering the blanketing envelope (fromDe Grandis et al. 2021).Figure 2. Thermal maps and pulsed fractions a field with Bquad/Bdip = 1.25, Θq = 45◦, shownat three times (increasing from top to bottom, left to right). The PF evolution is shown fordifferent viewing geometries (from De Grandis et al. 2021).injection was performed over a short time (≈minutes); we verified that the exact durationof the injection is irrelevant, as long as it is shorter than ≈ 1 hour and the total injectedenergy is the same. We studied the effect of neutrino emission by increasing the amountof heat in the same patch, comprising ∼ 3% of the total volume and ∼ 20% of the surface.As crustal neutrino emission gets triggered above  3× 109 K, more and more heat getsdissipated via this channel rather than carried away by surface photon emission. Hence,the peak luminosity saturates, as shown in Fig. 3. Moreover, the effect of the magneticfield is particularly evident considering different geometries of the field itself. The samepatch of  5 km radius heated with H  4× 1040 erg, yields very different lightcurveswhen located at the magnetic pole or across the equator, again due to the thermallyinsulating effect of the field. The shape of the emerging hotspot is also different in thetwo cases, see Fig. 4. In the models considered above, we took a field B  1013 G asa representative value. This choice is not restrictive, since we found that models withfields in a range ≈ 1012 G–1014 G yield lightcurves that only differ quantitatively betweeneach other. In particular, their timing properties (rise-onset and peak times) showed novariations in all the cases we considered.Our work has shown the key role of the magnetic field in shaping the observed prop-erties of NSs both in the long and the short term and highlighted the importanceof 3D simulations in providing a through assessment of the different aspects of NSmagnetothermal evolution.https://doi.org/10.1017/S174392132200031X Published online by Cambridge University Press3D simulations of magnetars 28310 2 10 1 100 101 102t (days)10331034L(erg/s)Luminosity in local f.o.r., @ B=1 × 1013 G0 100 200 300 400 500 600 700 800H (1042 erg)0102030405060L peak(1033erg/s)Peak luminosity in local f.o.r., @ B=1 × 1013 Glog fitFigure 3. (Left panel) Luminosity curves after the injection of increasing amounts of heat;higher curves correspond to higher H for a NS with T (0) = 108 K, B  1013 G. (Right panel)Peak luminosity as a function of the heat injection. A logarithmic fit to the points has beenadded as a reference.10 2 10 1 100 101 102t (days)10331034L(erg/s)Luminosity in local f.o.r., @ B=1 × 1013 GAccretedIronPoleEquatorPoleEquator(a) Evolution of the luminosity(f) Thermal map at the top of the crust(left) and above the envelope (right).Figure 4. Outburst models for a NS with B = 1013 G and an injection of H  4× 1040 erg intwo different positions w.r.t. the dipolar field, around the pole (continuous lines) and acrossthe equator (dot-dashed lines). Two different envelope compositions are considered, Fe-Ni (redcurves) and a light-element (blue curves, Bezgonov et al. 2021). The thermal maps are shownafter t= 9 h (rise phase) for the Fe-Ni envelope. The top row shows the case of polar injection(pole towards the observer), the bottom one equatorial injection (pole upwards).ReferencesBeznogov M. V., Potekhin A. Y., Yakovlev, D. G. 2001, Phys. Rep., 919:168. doi:10.1016/j.physrep.2021.03.004.Coti Zelati, F., Rea N., Pons, J. A., Campana S., Esposito, P. 2018, MNRAS, 474(1):961–1017.doi: 10.1093/mnras/stx2679.De Grandis, D., Turolla, R., Wood, T. S., Zane, S., Taverna, R., Gourgouliatos, K. N. 2020, ApJ903(1):40. doi: 10.3847/1538-4357/abb6f9.De Grandis, D., Taverna, R., Turolla, R., Gnarini, A., Popov, S. B., Zane, S., Wood, T. S. 2021,ApJ 914(2):118. doi: 10.3847/1538-4357/abfdac.Gourgouliatos, K, N., Cumming, A. 2014, Phys. Rev. Lett., 112(17):171101. doi: 10.1103/PhysRevLett.112.171101.Taverna, R., Turolla, R., Gonzalez Caniulef, D., Zane, S., Muleri, F., Soffitta, P. 2015, MNRAS,454(3):32543266. doi: 10.1093/mnras/stv2168.Wood, T. S., Hollerbach, R. 2015, Phys. Rev. Lett., 114(19):191101. doi: 10.1103/PhysRevLett.114.191101.Yakovlev, D. G., Kaminker, A. D., Gnedin, O. Y., Haensel, P.2001, Phys. Rep. 354(1-2):1–155.doi: 10.1016/S0370-1573(00)00131-9.Yakovlev, D. G., Kaminker, A. D., Potekhin, A. Y., Haensel, P. 2021 MNRAS 500(4):4491–4505.doi: 10.1093/mnras/staa3547.https://doi.org/10.1017/S174392132200031X Published online by Cambridge University Press

Modelling Magnetar Behaviour with 3D Magnetothermal Simulations

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Modelling Magnetar Behaviour with 3D Magnetothermal Simulations

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