How to form a millisecond magnetar? Magnetic field amplification in protoneutron stars

TL;DR

This paper investigates how magnetorotational instability (MRI) can amplify magnetic fields in protoneutron stars, potentially explaining the origin of magnetars' extreme magnetic fields and their role in energetic supernovae.

Contribution

It provides a detailed analysis of MRI behavior in protoneutron stars considering neutrino effects, buoyancy, and magnetic Prandtl number, advancing understanding of magnetic field amplification mechanisms.

Findings

MRI can exponentially amplify magnetic fields in protoneutron stars.

Neutrino radiation and buoyancy significantly influence MRI growth.

High magnetic Prandtl number affects the efficiency of magnetic field amplification.

Abstract

Extremely strong magnetic fields of the order of $10^{15}\,{\rm G}$ are required to explain the properties of magnetars, the most magnetic neutron stars. Such a strong magnetic field is expected to play an important role for the dynamics of core-collapse supernovae, and in the presence of rapid rotation may power superluminous supernovae and hypernovae associated to long gamma-ray bursts. The origin of these strong magnetic fields remains, however, obscure and most likely requires an amplification over many orders of magnitude in the protoneutron star. One of the most promising agents is the magnetorotational instability (MRI), which can in principle amplify exponentially fast a weak initial magnetic field to a dynamically relevant strength. We describe our current understanding of the MRI in protoneutron stars and show recent results on its dependence on physical conditions specific to protoneutron stars such as neutrino radiation, strong buoyancy effects and large magnetic Prandtl number.

Full text

††volume: 331††journal: SN 1987A 30 Years Later††editors: M. Renaud, A. Markowith, G. Dubner, A. Ray & A. Bykov eds.

1 Introduction

The delayed injection of energy due to the spin down of a fast rotating, highly magnetized neutron star is the most popular model to explain a class of superluminous supernovae (e.g. Kasen & Bildsten 2010; Inserra et al. 2013). The birth of such millisecond magnetars is furthermore a potential central engine for long gamma-ray bursts and hypernovae (e.g. Duncan & Thompson 1992; Metzger et al. 2011; Obergaulinger & Aloy 2017). One of the most fundamental and open question in these models is the origin of the strong magnetic field that is invoked. The most studied mechanism capable of generating such a strong magnetic field is the growth of the magnetorotational instability (MRI) in the protoneutron star (Akiyama et al. 2003; Masada et al. 2007; Obergaulinger et al. 2009; Sawai & Yamada 2014; Guilet et al. 2015; Guilet & Müller 2015; Rembiasz et al. 2016a,b; Mösta et al. 2016). In the following we review several aspects of the physics of the MRI in the physical conditions relevant to protoneutron stars, by describing its linear growth (Section 2) and its non-linear dynamics (Section 3).

2 Linear growth of the MRI

Although it is too often neglected in numerical simulations, neutrino radiation can have a dramatic impact on the linear growth of the MRI. This was shown by Guilet et al. (2015) by applying analytical results for the linear growth of the MRI to a numerical model of a protoneutron star (PNS). In Guilet et al. (2015), we have shown that, depending on the physical conditions, the MRI growth can occur in three main different regimes (Fig. 1):

• •

Viscous regime (dark blue color in Fig. 1) : on length-scales longer than the neutrino mean free path, neutrino viscosity significantly affects the growth of the MRI if $E_{\nu}\equiv\frac{v_{A}^{2}}{\nu\Omega}<1$, where $v_{A}$ is the Alfvén velocity, $\nu$ the viscosity induced by neutrinos and $\Omega$ the rotation angular frequency. The growth of the MRI is then slower and takes place at longer wavelengths compared to the ideal regime. In the viscous regime, the wavelength of the most unstable mode is independent of the magnetic field strength, while the growth rate is proportional to the magnetic field strength (Fig. 2). As a result, a minimum magnetic field strength of $\sim 10^{12}\,{\rm G}$ is required for the MRI to grow on sufficiently short time-scales.

• •

Drag regime (light blue color in Fig. 1) : on length-scales shorter than the neutrino mean free path, neutrino radiation exerts a drag force $-\Gamma\bf{v}$, where $\bf{v}$ is the fluid velocity perturbation and $\Gamma\simeq 6\times 10^{3}\,(T/10\,{\rm MeV})^{6}\,{\rm s^{-1}}$ is a damping rate. This drag has a significant impact on the MRI if the damping rate is larger than the rotation angular frequency ($\Gamma>\Omega$). In this regime, the growth rate of the most unstable mode is independent of the magnetic field strength, but is reduced by a factor of $\Gamma/\Omega$ compared to the ideal regime (Fig. 2). The wavelength of the most unstable mode is not much affected by the neutrino drag.

• •

Ideal regime (orange color in Fig. 1) : this is the classical MRI regime which occurs when neutrino viscosity or drag are negligible. The growth rate of the MRI is then a fraction of the angular frequency, independent of the magnetic field strength, while the most unstable wavelength is proportional to the magnetic field strength.

Fig. 1 shows where these three regimes apply, as a function of magnetic field strength and radius in the PNS (note that qualitatively similar results were obtained by Guilet et al. (2017) in the context of neutron star mergers). Three regions in the PNS can be distinguished:

• •

Deep inside the PNS, the relevant MRI regime is the viscous MRI. The MRI can grow on sufficiently short time-scales if the initial magnetic field is above a critical strength.

• •

At intermediate radii, MRI growth can take place both in the viscous regime at wavelengths longer than the neutrino mean free path, and in the drag regime at length-scales shorter than the mean free path. Since the growth rate in the viscous regime is proportional to the magnetic field strength, the growth is faster in the viscous regime above a critical magnetic field strength while it is faster in the drag regime for weaker magnetic fields. In addition, in-between these regimes, the MRI growth occurs in a mixed regime (shown in green in Fig. 1) where electron neutrinos are diffusing and thus induce a viscosity, while the other species are free streaming and exert a drag.

• •

Near the PNS surface, the viscous regime is irrelevant because the neutrino mean free path is longer than the wavelength of the MRI. Furthermore, in this region the neutrino drag does not affect much the growth of the MRI because the damping rate is smaller than the angular frequency, i.e. $\Gamma<\Omega$. As a consequence the MRI growth takes place in the ideal regime without much impact of neutrino radiation.

For the sake of simplicity and in order to show the impact of neutrinos in a clear way, this analysis neglected the effect of buoyancy. As shown analytically by Menou et al. (2004) and Masada et al. (2007) and confirmed by numerical simulations in Guilet & Müller (2015), entropy and composition gradients can act against the MRI, but this is alleviated by the diffusion due to neutrinos.

3 Non-linear dynamics of the MRI

In order to know the efficiency of magnetic field amplification, numerical simulations of the non-linear dynamics driven by the MRI are necessary. These simulations have so far mostly been performed in local or semi-global models describing a small portion of the PNS (e.g. Obergaulinger et al. 2009; Masada et al. 2012; Guilet et al. 2015; Rembiasz et al. 2016a,b). The first phase of MRI growth is often dominated by channel flows, which are the fastest growing MRI modes in the presence of a poloidal magnetic field. These modes, which are uniform in the horizontal direction, are approximate non-linear solutions and are therefore potentially able to grow well into the non-linear regime. Their growth is terminated when parasitic instabilities (either the Kelvin-Helmholtz instability or resistive tearing modes) are able to destroy their structures (Goodman & Xu 1994; Rembiasz et al. 2016a,b). By studying in detail this process with two different numerical codes, Rembiasz et al. (2016b) have been able to show that channel modes can only amplify the magnetic field by a factor $\sim 10$ under the physical conditions prevailing in a PNS. A further process such as a turbulent MRI-driven dynamo is therefore necessary to reach very strong magnetic fields.

In Guilet & Müller (2015), we have studied the turbulent phase following the disruption of the channel modes, taking for the first time into account both the viscosity and diffusion due to neutrinos and the buoyancy force due to entropy/composition gradients. This was made possible owing to the use of the Boussinesq approximation, which we showed to be well suited to a local model of a PNS. In addition to drastically reducing the computing time this approximation avoids artifacts at the boundaries caused by global gradients in fully compressible simulations. Fig. 3 shows snapshots of the turbulent phase in a buoyantly unstable case (left panel) and a buoyantly stable case (right panel). While the buoyantly unstable dynamics is turbulent and largely non-axisymmetric, the buoyantly stable dynamics contains more large-scale axisymmetric structures. A channel flow at the largest scale available in the box can be distinguished even during the turbulent phase, which is further illustrated in Fig. 4 by the horizontally averaged space-time diagram (left panel). Fig. 4 (right panel) furthermore shows the kinetic, magnetic and thermal energies and energy injection rates as a function of the dimensionless buoyancy parameter $N^{2}/\Omega^{2}$ (where $N$ is the Brunt-Väisälä frequency). The magnetic energy decreases with $N^{2}/\Omega$ but, interestingly, it becomes almost constant in the stable buoyancy regime ($N^{2}>0$) suggesting an efficient magnetic field amplification even in the presence of a strong, stable stratification.

While these simulations were able to use realistic values for the viscosity due to neutrinos, they have the shortcoming of vastly overestimating the resistivity (like all other simulations, be it by including it explicitly like here or by suffering from a large numerical resistivity (for estimates see Rembiasz et al. 2016c)). Preliminary results show that this is likely to cause a significant underestimate of the efficiency of magnetic field amplification, thus confirming the strong dependence of the MRI on the magnetic Prandtl number (the ratio of viscosity to resistivity) known in the context of accretion disks (e.g. Fromang et al. 2007).

4 Perspectives

The presence of structures on the largest scales described by the local models of Section 3 stresses the necessity of global models encompassing the whole PNS. Such models are extremely challenging computationally because it is necessary to resolve the very small length-scales where MRI grows. The closest to a global model was published by Mösta et al. (2015) who simulated one eighth of a PNS (in an octant symmetry) and claimed to obtain an MRI-driven dynamo. Even this extremely computationally intensive simulation could not, however, answer the fundamental question of the origin of magnetars dipolar magnetic fields because it was actually initialized with a dipolar magnetic field strong enough to lead to magnetar formation by simple flux conservation. More numerical simulations of global models of PNSs are therefore necessary to establish firmly whether the MRI can generate a sufficiently strong dipolar magnetic to explain the birth of millisecond magnetars. We are currently making steps in this direction by developing numerical models of idealized PNSs in quasi-incompressible approximations (Boussinesq or anelastic approximations). By drastically reducing the computational requirements, these approximations should enable insights into the fundamental physical process generating the dipolar magnetic field.

\acknowledgment

J.G. acknowledges support from the Max-Planck-Princeton Center for Plasma Physics and from the European Research Council (grant MagBURST–715368). H.T.J. is grateful for funding by the European Research Council through grant ERC-AdG COCO2CASA-3411 and by the Deutsche Forschung Gemeinschaft through Cluster of Excellence ”Universe” EXC-153. M.A., P.C.D., M.O. and T.R. acknowledge support from the European Research Council (grant CAMAP-259276) and from grants AYA2013-9340979-P, AYA2015-66899-C2-1-P and PROME-TEOII/2014-069.