Magnetar formation through a convective dynamo in protoneutron stars Raphaël Raynaud, Jérôme Guilet, Hans-Thomas Janka, Thomas Gastine To cite this version: Raphaël Raynaud, Jérôme Guilet, Hans-Thomas Janka, Thomas Gastine. Magnetar formation through a convective dynamo in protoneutron stars. Science Advances , American Association for the Advancement of Science (AAAS), 2020, 6 (eaay2732), 10.1126/sciadv.aay2732. hal-02428428 HAL Id: hal-02428428 https://hal.archives-ouvertes.fr/hal-02428428 Submitted on 14 Mar 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. SCIENCE ADVANCES | RESEARCH ARTICLE ASTRONOMY Copyright © 2020 The Authors, some rights reserved; Magnetar formation through a convective dynamo exclusive licensee American Association in protoneutron stars for the Advancement Raphaël Raynaud1*, Jérôme Guilet1, Hans-Thomas Janka2, Thomas Gastine3 of Science. No claim to original U.S. Government Works. Distributed The release of spin-down energy by a magnetar is a promising scenario to power several classes of extreme explosive under a Creative transients. However, it lacks a firm basis because magnetar formation still represents a theoretical challenge. Using Commons Attribution the first three-dimensional simulations of a convective dynamo based on a protoneutron star interior model, we NonCommercial demonstrate that the required dipolar magnetic field can be consistently generated for sufficiently fast rotation License 4.0 (CC BY-NC). rates. The dynamo instability saturates in the magnetostrophic regime with the magnetic energy exceeding the kinetic energy by a factor of up to 10. Our results are compatible with the observational constraints on galactic magnetar field strength and provide strong theoretical support for millisecond protomagnetar models of gamma- ray burst and superluminous supernova central engines. Downloaded from INTRODUCTION ri = 12.5 km to ro = 25 km, 15 km below the PNS surface (fig. S1). Magnetars are isolated, slowly rotating neutron stars characterized We take into account the physical diffusion processes to compute by a variable x-ray activity, which is thought to be powered by the the fluid viscosity n, thermal conductivity k, and magnetic diffusivity dissipation of strong magnetic fields (1). Their measured spin-down h consistent with the PNS structure (fig. S2). Convection is driven is related to a dipolar surface magnetic field ranging from 1014 to by a fixed heat flux at the boundaries, which we estimate using the 15 52 http://advances.sciencemag.org/ 10 G. 1D model to Fo ∼ 2 × 10 erg/s. We apply stress-free boundary Although magnetic flux conservation during stellar collapse is conditions for the velocity field for which the total angular momentum commonly invoked to explain pulsar magnetism, it tends to fall of the system is conserved. Because of the extremely large value of short for these stronger field strengths (2). Furthermore, a strong the conductivity, the magnetic boundary condition is set by assuming magnetic field brakes stellar rotation (3) such that a fossil field is that the material outside the convective zone is a perfect electrical probably not compatible with the fast rotation needed for millisecond conductor. For the sake of comparison, we also carried out simulations magnetar central engines. Alternative scenarios preferentially rely with an outer pseudo-vacuum boundary condition. Nevertheless, on the magnetic field amplification by a turbulent dynamo, which the need for local boundary conditions necessarily introduces some could be triggered by the magnetorotational instability (4, 5) or by undesired discontinuities, and a stronger validation of the magnetar convection (6). The former may grow in the differentially rotating central engine scenario would require a direct and simultaneous outer stable layers, while strong convective motions develop deeper modeling of the outer 10 km below the PNS surface. On the other on March 13, 2020 inside the protoneutron star (PNS) during the first 10 s following hand, the restriction of the computational domain to the convective the core collapse of a massive star. However, the generation of a zone allowed us to perform a parameter study consisting of 31 sim- dipolar component compatible with observational constraints has ulations in which we vary the PNS rotation rate W and the magnetic 2 never been demonstrated through direct numerical simulations. diffusivity by changing, respectively, the Ekman number E = no/(Wd ), To that end, we build a self-consistent, three-dimensional (3D) where d = ro − ri is the shell gap, and the magnetic Prandtl number PNS model solving the nonlinear, magneto-hydrodynamic anelastic Pm = no/ho (table S1). We focus on the regime of fast rotation cor- equations governing the flow of an electrically conducting fluid under- responding to periods of a few milliseconds, which translates in low going thermal convection in a rotating spherical shell. This sound- to moderate Rossby numbers Ro = U/(Wd) ∈ [10−2,1], where U is proof approximation, routinely used to model planetary and stellar the root mean square fluid velocity. dynamos, is justified in the convective zone of a PNS because the Figure 1 is representative of the time evolution of dynamos ob- Mach number is low (Ma ≃ 0.02 to 0.05). The equations are integrated tained with short rotation periods. The kinematic phase is character- in time with the benchmarked pseudo-spectral code MagIC (7, 8). ized by an oscillatory mode, which saturates below equipartition, which is typical of an aW dynamo driven by a large-scale differential rotation. More unexpected is the secondary growth following this RESULTS first plateau, which results in a much stronger field dominated by its The anelastic background state is assumed to be steady and isentro- axisymmetric toroidal component. In our set of simulations, the pic and matches the structure of the PNS convective zone 0.2 s after transition takes from 1 to 5 s, which is longer than the kinematic bounce given by a 1D simulation of a core-collapse supernova of a growth but still shorter than the duration of the convective phase 27 MSun progenitor. At this time, the convective zone extends from (9); in any case, the dynamo saturation is independent of the initial magnetic field, even for initial fields as low as 106 G, which suggests that the bifurcation is supercritical. The changes in the field and 1AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, velocity structures are illustrated by the insets in Fig. 1 and the F-91191 Gif-sur-Yvette, France. 2Max-Planck-Institut für Astrophysik, Karl-Schwarzschild- 3 3D renderings of Fig. 2. In the saturated state, the strong toroidal Str. 1, D-85748 Garching, Germany. Institut de Physique du Globe de Paris, Sorbonne Paris Cité, Université Paris-Diderot, UMR7154 CNRS, 1 rue Jussieu, 75005 Paris, France. magnetic field breaks the characteristic columnar structure of rotating *Corresponding author. Email: [email protected] convection (Fig. 2A), leading to the concentration of the most vigorous Raynaud et al., Sci. Adv. 2020; 6 : eaay2732 13 March 2020 1 of 6 SCIENCE ADVANCES | RESEARCH ARTICLE ()η A Kinematic First Saturated state ) ( (G) (s) Fig. 1. Time series of the densities of kinetic energy in the rotating frame Downloaded from (blue) and magnetic energy (red) for a fast-rotating model with rotation period P = 2.1 ms and Pm = 2. The orange line represents the energy of the dipole com- ponent (l = 1 mode of the poloidal potential). The dashed gray horizontal lines show the fiducial range from 1014 to 1015 G for the intensity of the dipole field con- strained by magnetar timing parameters. The insets show slices of the azimuthal Bf t magnetic field at times ∼ 2 s and 10 s indicated by vertical dotted lines. The http://advances.sciencemag.org/ upper x axis is labeled in units of magnetic diffusion time. B radial flows inside the tangent cylinder (Fig. 2B). We also note that in most simulations, the entropy field displays an equatorial symmetry breaking similar to what has been reported for fixed flux Boussinesq convection (10). This hemispheric asymmetry may be related to the emergence of the “Lepton-number emission self-sustained asymmetry” (LESA) observed in nonmagnetized core-collapse supernova simu- lations (11), suggesting that LESA may also take place in fast rotating, strongly magnetized PNS. on March 13, 2020 Our study demonstrates that this super-equipartition state is a new instance of a strong field dynamo characterized by the magneto- strophic balance between the Coriolis force ~ ϱ W e z × u and the Lorentz m−1 force 0 (∇ × B ) × B (12–15). Figure 3 shows that our strong field solutions follow the expected scaling for the ratio of the magnetic −1 and kinetic energies in the rotating frame, EB/EK ∝ Ro , which holds for rotation-dominated convective flows with Ro < 0.2. By contrast, weak field solutions are stable for slower rotation or when we use a pseudo-vacuum magnetic boundary condition, which arti- ficially prevents currents outside of the convective zone, thereby forcing the toroidal magnetic field to vanish at the boundary. Since Fig. 2. Three-dimensional rendering of the weak and strong field solutions. this unphysical constraint seems to impede a further growth of the Snapshots (A) and (B) correspond to the left and right insets in Fig. 1, respectively. toroidal field, we believe that the strong field branch will be achieved Left-right snapshots correspond to the left-right insets in Fig. 1. Magnetic field lines in future models including both the stable and unstable regions.