Efficient Computation of the Characteristic Polynomial

This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third algorithm could improve both complexity and computational time. We use these results as a basis for the computation of the characteristic polynomial of integer matrices. We first use early termination and Chinese remaindering for dense matrices. Then a probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices

Local simulations of the magnetized Kelvin-Helmholtz instability in neutron-star mergers

Context. Global MHD simulations show Kelvin-Helmholtz (KH) instabilities at the contact surface of two merging neutron stars. That region has been identified as the site of efficient amplification of magnetic fields. However, these global simulations, due to numerical limitations, were unable to determine the saturation level of the field strength, and thus the possible back-reaction of the magnetic field onto the flow. Aims. We investigate the amplification of initially weak fields in KH unstable shear flows, and the back-reaction of the field onto the flow. Methods. We use a high-resolution ideal MHD code to perform 2D and 3D local simulations of shear flows. Results. In 2D, the magnetic field is amplified in less than 0.01ms until it reaches locally equipartition with the kinetic energy. Subsequently, it saturates due to resistive instabilities that disrupt the KH vortex and decelerate the shear flow on a secular time scale. We determine scaling laws of the field amplification with the initial field strength and the grid resolution. In 3D, this hydromagnetic mechanism may be dominated by purely hydrodynamic instabilities limiting the amplification. We find maximum magnetic fields of 10^16 G locally, and r.m.s. maxima within the box of 10^15 G. However, such strong fields exist only for a short period. In the saturated state, the magnetic field is mainly oriented parallel to the shear flow for strong initial fields, while weaker initial fields tend to lead to a more balanced distribution of the field energy. In all models the flow shows small-scale features. The magnetic field is at most in equipartition with the decaying shear flow. (abridged)Comment: 26 pages, 22 figures (figure quality reduced); accepted for publication in Astronomy & Astrophysic

The theory of pulsar winds and nebulae

We review current theoretical ideas on pulsar winds and their surrounding nebulae. Relativistic MHD models of the wind of the aligned rotator, and of the striped wind, together with models of magnetic dissipation are discussed. It is shown that the observational signature of this dissipation is likely to be point-like, rather than extended, and that pulsed emission may be produced. The possible pulse shapes and polarisation properties are described. Particle acceleration at the termination shock of the wind is discussed, and it is argued that two distinct mechanisms must be operating, with the first-order Fermi mechanism producing the high-energy electrons (above 1 TeV) and either magnetic annihilation or resonant absorption of ion cyclotron waves responsible for the 100 MeV to 1 TeV electrons. Finally, MHD models of the morphology of the nebula are discussed and compared with observation.Comment: 33 pages, to appear in Springer Lecture Notes on "Neutron stars and pulsars, 40 years after the discovery", ed W.Becke

Relativistic Shocks: Particle Acceleration and Magnetization

We review the physics of relativistic shocks, which are often invoked as the sources of non-thermal particles in pulsar wind nebulae (PWNe), gamma-ray bursts (GRBs), and active galactic nuclei (AGN) jets, and as possible sources of ultra-high energy cosmic-rays. We focus on particle acceleration and magnetic field generation, and describe the recent progress in the field driven by theory advances and by the rapid development of particle-in-cell (PIC) simulations. In weakly magnetized or quasi parallel-shocks (where the magnetic field is nearly aligned with the flow), particle acceleration is efficient. The accelerated particles stream ahead of the shock, where they generate strong magnetic waves which in turn scatter the particles back and forth across the shock, mediating their acceleration. In contrast, in strongly magnetized quasi-perpendicular shocks, the efficiencies of both particle acceleration and magnetic field generation are suppressed. Particle acceleration, when efficient, modifies the turbulence around the shock on a long time scale, and the accelerated particles have a characteristic energy spectral index of ~ 2.2 in the ultra-relativistic limit. We discuss how this novel understanding of particle acceleration and magnetic field generation in relativistic shocks can be applied to high-energy astrophysical phenomena, with an emphasis on PWNe and GRB afterglows.Comment: 32 pages; 9 figures; invited topical review, comments welcome; submitted for publication in "The Strongest Magnetic Fields in the Universe" (Space Sciences Series of ISSI, Springer), Space Science Review

Clusters of Galaxies: magnetic fields and nonthermal emission

The nonthermal particle content of galaxy clusters should in part have a cosmological component generated during the early starburst phase of the member galaxies. This is reviewed in the framework of a simple cluster formation model suggested previously. It implies a nonthermal energy fraction of about 10 percent for the Intracluster gas. We also propose a mechanism for the early generation of Intracluster magnetic fields in terms of Galactic Winds. It results in typical field strengths of about 0.1 microGauss. Such comparatively weak fields are consistent with an inverse Compton origin of the excess EUV and hard X-ray emission of the Coma cluster, given the radio synchrotron emission. The required relativistic electrons must have been accelerated rather recently, less than a few billion years ago, presumably in cluster accretion shocks. This is in contrast to the hadronic nonthermal component which accumulates on cosmological time scales, and whose pion-decay TeV gamma-ray emission is expected to be larger, or of the same order as the inverse Compton TeV emission. This gamma-radiation characterizes the energetic history of cluster formation and should be observable with future arrays of imaging atmospheric Cherenkov telescopes.Comment: 16 pages, 8 figures; invited talk presented at the VERITAS Workshop on TeV Astrophysics of Extragalactic Sources, submitted to Astroparticle Physic

Shape and evolution of wind-blown bubbles of massive stars: on the effect of the interstellar magnetic field

The winds of massive stars create large (>10 pc) bubbles around their progenitors. As these bubbles expand they encounter the interstellar coherent magnetic field which, depending on its strength, can influence the shape of the bubble. We wish to investigate if, and how much, the interstellar magnetic field can contribute to the shape of an expanding circumstellar bubble around a massive star. We use the MPI-AMRVAC code to make magneto-hydrodynamical simulations of bubbles, using a single star model, combined with several different field strengths: B = 5, 10, and 20 muG for the interstellar magnetic field. This covers the typical field strengths of the interstellar magnetic fields found in the galactic disk and bulge. Furthermore, we present two simulations that include both a 5 muG interstellar magnetic field and a 10,000 K interstellar medium and two different ISM densities to demonstrate how the magnetic field can combine with other external factors to influence the morphology of the circumstellar bubbles. Our results show that low magnetic fields, as found in the galactic disk, inhibit the growth of the circumstellar bubbles in the direction perpendicular to the field. As a result, the bubbles become ovoid, rather than spherical. Strong interstellar fields, such as observed for the galactic bulge, can completely stop the expansion of the bubble in the direction perpendicular to the field, leading to the formation of a tube-like bubble. When combined with a warm, high-density ISM the bubble is greatly reduced in size, causing a dramatic change in the evolution of temporary features inside the bubble. The magnetic field of the interstellar medium can affect the shape of circumstellar bubbles. This effect may have consequences for the shape and evolution of circumstellar nebulae and supernova remnants, which are formed within the main wind-blown bubble.Comment: Proposed for acceptance for publication in Astronomy & Astrophysics. The published version will contain animations of each simulatio

Generic design of Chinese remaindering schemes

We propose a generic design for Chinese remainder algorithms. A Chinese remainder computation consists in reconstructing an integer value from its residues modulo non coprime integers. We also propose an efficient linear data structure, a radix ladder, for the intermediate storage and computations. Our design is structured into three main modules: a black box residue computation in charge of computing each residue; a Chinese remaindering controller in charge of launching the computation and of the termination decision; an integer builder in charge of the reconstruction computation. We then show that this design enables many different forms of Chinese remaindering (e.g. deterministic, early terminated, distributed, etc.), easy comparisons between these forms and e.g. user-transparent parallelism at different parallel grains

Magnetic fields in supernova remnants and pulsar-wind nebulae

We review the observations of supernova remnants (SNRs) and pulsar-wind nebulae (PWNe) that give information on the strength and orientation of magnetic fields. Radio polarimetry gives the degree of order of magnetic fields, and the orientation of the ordered component. Many young shell supernova remnants show evidence for synchrotron X-ray emission. The spatial analysis of this emission suggests that magnetic fields are amplified by one to two orders of magnitude in strong shocks. Detection of several remnants in TeV gamma rays implies a lower limit on the magnetic-field strength (or a measurement, if the emission process is inverse-Compton upscattering of cosmic microwave background photons). Upper limits to GeV emission similarly provide lower limits on magnetic-field strengths. In the historical shell remnants, lower limits on B range from 25 to 1000 microGauss. Two remnants show variability of synchrotron X-ray emission with a timescale of years. If this timescale is the electron-acceleration or radiative loss timescale, magnetic fields of order 1 mG are also implied. In pulsar-wind nebulae, equipartition arguments and dynamical modeling can be used to infer magnetic-field strengths anywhere from about 5 microGauss to 1 mG. Polarized fractions are considerably higher than in SNRs, ranging to 50 or 60% in some cases; magnetic-field geometries often suggest a toroidal structure around the pulsar, but this is not universal. Viewing-angle effects undoubtedly play a role. MHD models of radio emission in shell SNRs show that different orientations of upstream magnetic field, and different assumptions about electron acceleration, predict different radio morphology. In the remnant of SN 1006, such comparisons imply a magnetic-field orientation connecting the bright limbs, with a non-negligible gradient of its strength across the remnant.Comment: 20 pages, 24 figures; to be published in SpSciRev. Minor wording change in Abstrac