On the Measurements of Numerical Viscosity and Resistivity in Eulerian MHD Codes

We propose a simple ansatz for estimating the value of the numerical resistivity and the numerical viscosity of any Eulerian MHD code. We test this ansatz with the help of simulations of the propagation of (magneto)sonic waves, Alfven waves, and the tearing mode instability using the MHD code Aenus. By comparing the simu- lation results with analytical solutions of the resistive-viscous MHD equations and an empirical ansatz for the growth rate of tearing modes we measure the numerical viscosity and resistivity of Aenus. The comparison shows that the fast-magnetosonic speed and wavelength are the characteristic velocity and length, respectively, of the aforementioned (relatively simple) syst…

Modulating magnetar emission by magneto-elastic oscillations

We present a new numerical tool to calculate the emission of highly magnetized neutron stars (magnetars) and apply it to describe the quasi-periodic oscillations (QPOs) observed in magnetar giant flares. In previous work we have developed a model of magneto-elastic oscillations of magnetars that allows to reproduce the observed frequencies. These QPOs can couple to the star's exterior through the magnetic field and induce currents in the magnetosphere that provide scattering targets for resonant cyclotron scattering of the photons. The scattering is calculated with a Monte-Carlo approach and it is coupled to a code that calculates the momentum distribution of the charge carriers as an one-d…

Imprints of superfluidity on magnetoelastic quasiperiodic oscillations of soft gamma-ray repeaters.

Our numerical simulations show that axisymmetric, torsional, magnetoelastic oscillations of magnetars with a superfluid core can explain the whole range of observed quasiperiodic oscillations (QPOs) in the giant flares of soft gamma-ray repeaters. There exist constant phase QPOs at $f\ensuremath{\lesssim}150\text{ }\text{ }\mathrm{Hz}$ and resonantly excited high-frequency QPOs ($fg500\text{ }\text{ }\mathrm{Hz}$), in good agreement with observations. The range of magnetic field strengths required to match the observed QPO frequencies agrees with that from spin-down estimates. These results suggest that there is at least one superfluid species in magnetar cores.

Termination of the magnetorotational instability via parasitic instabilities in core-collapse supernovae

The magnetorotational instability (MRI) can be a powerful mechanism amplifying the magnetic field in core-collapse supernovae. Whether initially weak magnetic fields can be amplified by this instability to dynamically relevant strengths is still a matter of debate. One of the main uncertainties concerns the process that terminates the growth of the instability. Parasitic instabilities of both Kelvin-Helmholtz and tearing-mode type have been suggested to play a crucial role in this process, disrupting MRI channel flows and quenching magnetic field amplification. We perform two-dimensional and three-dimensional sheering-disc simulations of a differentially rotating protoneutron star layer in …

Numerical Hydrodynamics in Special Relativity

This review is concerned with a discussion of numerical methods for the solution of the equations of special relativistic hydrodynamics (SRHD). Particular emphasis is put on a comprehensive review of the application of high-resolution shock-capturing methods in SRHD. Results obtained with different numerical SRHD methods are compared, and two astrophysical applications of SRHD flows are discussed. An evaluation of the various numerical methods is given and future developments are analyzed.

The exact solution of the Riemann problem with non-zero tangential velocities in relativistic hydrodynamics

We have generalised the exact solution of the Riemann problem in special relativistic hydrodynamics for arbitrary tangential flow velocities. The solution is obtained by solving the jump conditions across shocks plus an ordinary differential equation arising from the self-similarity condition along rarefaction waves, in a similar way as in purely normal flow. The dependence of the solution on the tangential velocities is analysed, and the impact of this result on the development of multidimensional relativistic hydrodynamic codes (of Godunov type) is discussed.

A Numerical Study of Relativistic Jets

Numerical simulations of supersonic jets are able to explain the structures observed in many VLA images of radio sources. The improvements achieved in classical simulations (see Hardee, these proceedings) are in contrast with the almost complete lack of relativistic simulations the reason being that numerical difficulties arise from the highly relativistic flows typical of extragalactic jets. For our study, we have developed a two-dimensional code which is based on (i) an explicit conservative differencing of the special relativistic hydrodynamics (SRH) equations and (ii) the use of an approximate Riemann solver (see Marti et al. 1995a,b and references therein).

Imprints of superfluidity on magneto-elastic QPOs of SGRs

Our numerical simulations show that axisymmetric, torsional, magneto-elastic oscillations of magnetars with a superfluid core can explain the whole range of observed quasi-periodic oscillations (QPOs) in the giant flares of soft gamma-ray repeaters. There exist constant phase, magneto-elastic QPOs at both low (f<150 Hz) and high frequencies (f>500 Hz), in full agreement with observations. The range of magnetic field strengths required to match the observed QPO frequencies agrees with that from spin-down estimates. These results strongly suggest that neutrons in magnetar cores are superfluid.

Magneto-elastic oscillations of neutron stars: exploring different magnetic field configurations

We study magneto-elastic oscillations of highly magnetized neutron stars (magnetars) which have been proposed as an explanation for the quasi-periodic oscillations (QPOs) appearing in the decaying tail of the giant flares of soft gamma-ray repeaters (SGRs). We extend previous studies by investigating various magnetic field configurations, computing the Alfv��n spectrum in each case and performing magneto-elastic simulations for a selected number of models. By identifying the observed frequencies of 28 Hz (SGR 1900+14) and 30 Hz (SGR 1806-20) with the fundamental Alfv��n QPOs, we estimate the required surface magnetic field strength. For the magnetic field configurations investigated (dipole…

Magneto‐elastic oscillations modulating the emission of magnetars

Magneto-elastic oscillations of neutron stars are believed to explain observed quasi-periodic oscillations (QPOs) in the decaying tail of the giant flares of highly magnetized neutron stars (magnetars). Strong efforts of the theoretical modelling from different groups have increased our understanding of this phenomenon significantly. Here, we discuss some constraints on the matter in neutron stars that arise if the interpretation of the observations in terms of superfluid, magneto-elastic oscillations is correct. To explain the observed modulation of the light curve of the giant flare, we describe a model that allows the QPOs to couple to the stellar exterior through the magnetic field. In …

Relativistic MHD simulations of stellar core collapse and magnetars

We present results from simulations of magneto-rotational stellar core collapse along with Alfven oscillations in magnetars. These simulations are performed with the CoCoA/CoCoNuT code, which is able to handle ideal MHD flows in dynamical spacetimes in general relativity. Our core collapse simulations highlight the importance of genuine magnetic effects, like the magneto-rotational instability, for the dynamics of the flow. For the modelling of magnetars we use the anelastic approximation to general relativistic MHD, which allows for an effective suppression of fluid modes and an accurate description of Alfven waves. We further compute Alfven oscillation frequencies along individual magneti…

Titanium hidden in dust

Cassiopeia A, one of the most intriguing galactic supernova remnants, has been a target of many observational efforts including most recent observations by ALMA, Hubble, Herschel, Spitzer, NuSTAR, Integral, and other observatories. We use recent gamma-ray lines observations of the radioactive products of Cas A supernova explosive nucleosynthesis as well as spectral energy densities derived for Cas A at infrared wavelengths to speculate about the possibility of radioactive 44Ti being locked into large dust grains. This suggestion is also supported by the possible observation of a pre-supernova outburst about 80 years before the actual Cas A supernova explosion in 1671 AD by Italian astronome…

Magneto-elastic oscillations and the damping of crustal shear modes in magnetars

In a realistic model of magneto-elastic oscillations in magnetars, we find that crustal shear oscillations, often invoked as an explanation of quasi-periodic oscillations (QPOs) seen after giant flares in soft gamma-ray repeaters (SGRs), are damped by resonant absorption on timescales of at most 0.2s, for a lower limit on the dipole magnetic field strength of 5 10 13 G. At higher magnetic field strengths (typical in magnetars) the damping timescale is even shorter, as anticipated by earlier toy-models. We have investigated a range of equations of state and masses and if magnetars are dominated by a dipole magnetic field, our findings exclude torsional shear oscillations of the crust from ex…

Relativistic MHD simulations of extragalactic jets

We have performed a comprehensive parameter study of the morphology and dynamics of axisymmetric, magnetized, relativistic jets by means of numerical simulations. The simulations have been performed with an upgraded version of the GENESIS code which is based on a second-order accurate finite volume method involving an approximate Riemann solver suitable for relativistic ideal magnetohydrodynamic flows, and a method of lines. Starting from pure hydrodynamic models we consider the effect of a magnetic field of increasing strength (up to β ≡ |b|2/2p ≈ 3.3 times the equipartition value) and different topology (purely toroidal or poloidal). We computed several series of models investigating the …

Coherent magneto-elastic oscillations in superfluid magnetars

We study the effect of superfluidity on torsional oscillations of highly magnetised neutron stars (magnetars) with a microphysical equation of state by means of two-dimensional, magnetohydrodynamical- elastic simulations. The superfluid properties of the neutrons in the neutron star core are treated in a parametric way in which we effectively decouple part of the core matter from the oscillations. Our simulations confirm the existence of two groups of oscillations, namely continuum oscillations that are confined to the neutron star core and are of Alfv\'enic character, and global oscillations with constant phase and that are of mixed magneto-elastic type. The latter might explain the quasi-…

Magnetoelastic oscillations of neutron stars with dipolar magnetic fields

A new general relativistic magnetohydrodynamics code for dynamical spacetimes

We present a new numerical code which solves the general relativistic magneto-hydrodynamics (GRMHD) equations coupled to the Einstein equations for the evolution of a dynamical spacetime within the conformally-flat approximation. This code has been developed with the main objective of studying astrophysical scenarios in which both, high magnetic fields and strong gravitational fields appear, such as the magneto-rotational collapse of stellar cores, the collapsar model of GRBs, and the evolution of neutron stars. The code is based on an existing and thoroughly tested purely hydrodynamics code and on its extension to accommodate weakly magnetized fluids (passive magnetic field approximation).…

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

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 initia…

Hydromagnetic instabilities and magnetic field amplification in core collapse supernovae

Some of the most violent events in the universe, the gamma ray burst, could be related to the gravitational collapse of massive stellar cores. The recent association of long GRBs to some class of type Ic supernova seems to support this view. In such scenario fast rotation, strong magnetic fields and general relativistic effects are key ingredients. It is thus important to understand the mechanism that amplifies the magnetic field under that conditions. I present global simulations of the magneto-rotational collapse of stellar cores in general relativity and semi-global simulations of hydromagnetic instabilities under core collapse conditions. I discuss effect of the magneto-rotational insta…

Semi-global simulations of the magneto-rotational instability in core collapse supernovae

Possible effects of magnetic fields in core collapse supernovae rely on an efficient amplification of the weak pre-collapse fields. It has been suggested that the magneto-rotational instability (MRI) leads to rapid field growth. Although MRI studies exist for accretion discs, the application of their results to core collapse supernovae is inhibited as the physics of supernova cores is substantially different from that of accretion discs. We address the problem of growth and saturation of the MRI by means of semi-global simulations, which combine elements of global and local simulations by taking the presence of global background gradients into account and using a local computational grid. W…

Comparison between the fCCZ4 and BSSN formulations of Einstein equations in spherical polar coordinates

Recently, we generalized a covariant and conformal version of the Z4 system of the Einstein equations using a reference metric approach, that we denote as fCCZ4. We successfully implemented and tested this approach in a 1D code that uses spherical coordinates and assumes spherical symmetry, obtaining from one to three orders of magnitude reduction of the Hamiltonian constraint violations with respect to the BSSN formulation in tests involving neutron star spacetimes. In this work, we show preliminary results obtained with the 3D implementation of the fCCZ4 formulation in a fully 3D code using spherical polar coordinates.

Combining spectral and shock-capturing methods: A new numerical approach for 3D relativistic core collapse simulations

We present a new three-dimensional general relativistic hydrodynamics code which is intended for simulations of stellar core collapse to a neutron star, as well as pulsations and instabilities of rotating relativistic stars. Contrary to the common approach followed in most existing three-dimensional numerical relativity codes which are based in Cartesian coordinates, in this code both the metric and the hydrodynamics equations are formulated and solved numerically using spherical polar coordinates. A distinctive feature of this new code is the combination of two types of accurate numerical schemes specifically designed to solve each system of equations. More precisely, the code uses spectra…

Grid-based Methods in Relativistic Hydrodynamics and Magnetohydrodynamics

An overview of grid-based numerical methods used in relativistic hydrodynamics (RHD) and magnetohydrodynamics (RMHD) is presented. Special emphasis is put on a comprehensive review of the application of high-resolution shock-capturing methods. Results of a set of demanding test bench simulations obtained with different numerical methods are compared in an attempt to assess the present capabilities and limits of the various numerical strategies. Applications to three astrophysical phenomena are briefly discussed to motivate the need for and to demonstrate the success of RHD and RMHD simulations in their understanding. The review further provides FORTRAN programs to compute the exact solution…

Magneto-elastic torsional oscillations of magnetars

We extend a general-relativistic ideal magneto-hydrodynamical code to include the effects of elasticity. Using this numerical tool we analyse the magneto-elastic oscillations of highly magnetised neutron stars (magnetars). In simulations without magnetic field we are able to recover the purely crustal shear oscillations within an accuracy of about a few per cent. For dipole magnetic fields between 5 x 10^13 and 10^15 G the Alfv\'en oscillations become modified substantially by the presence of the crust. Those quasi-periodic oscillations (QPOs) split into three families: Lower QPOs near the equator, Edge QPOs related to the last open field line and Upper QPOs at larger distance from the equa…

Fully Covariant and Conformal Formulation of the Z4 System Compared to the BSSN Formulation in Spherical Symmetry

We have generalized a covariant and conformal version of the Z4 system of the Einstein equations by adopting a reference metric approach, that we denote as fCCZ4, well suited for curvilinear as well as Cartesian coordinates. We implement this formalism in spherical polar coordinates under the assumption of spherical symmetry using a partially-implicit Runge-Kutta (PIRK) method, without using any regularization scheme, and show that our code can evolve both vacuum and non-vacuum spacetimes without encountering instabilities. We have performed several tests and compared the Hamiltonian constraint violations of the fCCZ4 system, for different choices of certain free parameters, with these of B…

Constraining properties of high-density matter in neutron stars with magneto-elastic oscillations

We discuss torsional oscillations of highly magnetised neutron stars (magnetars) using two-dimensional, magneto-elastic-hydrodynamical simulations. Our model is able to explain both the low- and high-frequency quasi-periodic oscillations (QPOs) observed in magnetars. The analysis of these oscillations provides constraints on the breakout magnetic-field strength, on the fundamental QPO frequency, and on the frequency of a particularly excited overtone. More importantly, we show how to use this information to generically constraint properties of high-density matter in neutron stars, employing Bayesian analysis. In spite of current uncertainties and computational approximations, our model-depe…

Hydrodynamic simulations of the interaction of supernova shock waves with a clumpy environment: the case of the RX J0852.0-4622 (Vela Jr.) supernova remnant

Observations in all electromagnetic bands show that many supernova remnants (SNRs) have a very aspherical shape. This can be the result of asymmetries in the supernova explosion or a clumpy circumstellar medium. We study the generation of inhomogeneities and the mixing of elements arising from these two sources in multidimensional hydrodynamic simulations of the propagation of a supernova blast wave into a cloudy environment. We model a specific SNR, Vela Jr (RX J0852.0-4622). By comparing our results with recent observations, we can constrain the properties of the explosion. We find that a very energetic explosion of several 10^{51} erg occurring roughly about 800 years ago is consistent w…

Modulating the magnetosphere of magnetars by internal magneto-elastic oscillations

We couple internal torsional, magneto-elastic oscillations of highly magnetized neutron stars (magnetars) to their magnetospheres. The corresponding axisymmetric perturbations of the external magnetic field configuration evolve as a sequence of linear, force-free equilibria that are completely determined by the background magnetic field configuration and by the perturbations of the magnetic field at the surface. The perturbations are obtained from simulations of magneto-elastic oscillations in the interior of the magnetar. While such oscillations can excite travelling Alfv\'en waves in the exterior of the star only in a very limited region close to the poles, they still modulate the near ma…

Morphology and Dynamics of Relativistic Jets

We present a comprehensive analysis of the morphology and dynamics of relativistic pressure-matched axisymmetric jets. The numerical simulations have been carried out with a high-resolution shock-capturing hydrocode based on an approximate relativistic Riemann solver derived from the spectral decomposition of the Jacobian matrices of relativistic hydrodynamics. We discuss the dependence of the jet morphology on several parameters, paying special attention to the relativistic effects caused by high Lorentz factors and large internal energies of the beam flow. The parameter space of our analysis is spanned by the ratio of the beam and ambient medium rest mass density (η), the beam Mach number…

Simulations of Precessing Jets

We report on the results of a three-dimensional, relativistic, hydrodynamical simulation of a precessing jet through which a compact blob of matter is set to propagate. We conclude that the morphology of superluminal sources is the result of a complex combination of phase motions, viewing angle selection effects, and non-linear interactions between perturbations and the underlying jet and/or external medium.

High-order methods for the simulation of hydromagnetic instabilities in core-collapse supernovae

AbstractWe present an assessment of the accuracy of a recently developed MHD code used to study hydromagnetic flows in supernovae and related events. The code, based on the constrained transport formulation, incorporates unprecedented ultra-high-order methods (up to 9th order) for the reconstruction and the most accurate approximate Riemann solvers. We estimate the numerical resistivity of these schemes in tearing instability simulations.

Fully covariant and conformal formulation of the Z4 system in a reference-metric approach: Comparison with the BSSN formulation in spherical symmetry

We adopt a reference-metric approach to generalize a covariant and conformal version of the Z4 system of the Einstein equations. We refer to the resulting system as ``fully covariant and conformal", or fCCZ4 for short, since it is well suited for curvilinear as well as Cartesian coordinates. We implement this fCCZ4 formalism in spherical polar coordinates under the assumption of spherical symmetry using a partially-implicit Runge-Kutta (PIRK) method and show that our code can evolve both vacuum and non-vacuum spacetimes without encountering instabilities. Our method does not require regularization of the equations to handle coordinate singularities, nor does it depend on constraint-preservi…

On the maximum magnetic field amplification by the magnetorotational instability in core-collapse supernovae

Whether the magnetorotational instability (MRI) can amplify initially weak magnetic fields to dynamically relevant strengths in core collapse supernovae is still a matter of active scientific debate. Recent numerical studies have shown that the first phase of MRI growth dominated by channel flows is terminated by parasitic instabilities of the Kelvin-Helmholtz type that disrupt MRI channel flows and quench further magnetic field growth. However, it remains to be prop- erly assessed by what factor the initial magnetic field can be amplified and how it depends on the initial field strength and the amplitude of the perturbations. Different termination criteria leading to different estimates of…

Axisymmetric core collapse simulations using characteristic numerical relativity

We present results from axisymmetric stellar core collapse simulations in general relativity. Our hydrodynamics code has proved robust and accurate enough to allow for a detailed analysis of the global dynamics of the collapse. Contrary to traditional approaches based on the 3+1 formulation of the gravitational field equations, our framework uses a foliation based on a family of outgoing light cones, emanating from a regular center, and terminating at future null infinity. Such a coordinate system is well adapted to the study of interesting dynamical spacetimes in relativistic astrophysics such as stellar core collapse and neutron star formation. Perhaps most importantly this procedure allo…