Anatoly Spitkovsky

Particle acceleration in relativistic collisionless shocks: Fermi process at last?

We present evidence that relativistic shocks propagating in unmagnetized plasmas can self-consistently accelerate particles. We use long-term two-dimensional particle-in-cell simulations to study the well-developed shock structure in unmagnetized pair plasma. The particle spectrum downstream of such a shock consists of two components: a relativistic Maxwellian, with a characteristic temperature set by the upstream kinetic energy of the flow, and a high-energy tail, extending to energies >100 times that of the thermal peak. This high-energy tail is best fitted as a power law in energy with index –2.4 ± 0.1, modified by an exponential cutoff. The cutoff moves to higher energies with time of the simulation, leaving a larger power-law range. The number of particles in the tail is ~1% of the downstream population, and they carry ~10% of the kinetic energy in the downstream region. Investigating the trajectories of particles in the tail, we find that the energy gains occur as particles bounce between the upstream and downstream regions in the magnetic fields generated by the Weibel instability. We compare this mechanism to the first-order Fermi acceleration and set a lower limit on the efficiency of the shock acceleration process.

Particle Acceleration in Relativistic Magnetized Collisionless Pair Shocks: Dependence of Shock Acceleration on Magnetic Obliquity

We investigate shock structure and particle acceleration in relativistic magnetized collisionless pair shocks by means of 2.5D and 3D particle-in-cell simulations. We explore a range of inclination angles between the pre-shock magnetic field and the shock normal. We find that only magnetic inclinations corresponding to subluminal shocks, where relativistic particles following the magnetic field can escape ahead of the shock, lead to particle acceleration. The downstream spectrum in such shocks consists of a relativistic Maxwellian and a high-energy power-law tail with exponential cutoff. For increasing magnetic inclination in the subluminal range, the high-energy tail accounts for an increasing fraction of particles (from ~1% to ~2%) and energy (from ~4% to ~12%). The spectral index of the power law increases with angle from –2.8 ± 0.1 to –2.3 ± 0.1. For nearly parallel shocks, particle energization mostly proceeds via the diffusive shock acceleration process; the upstream scattering is provided by oblique waves which are generated by the high-energy particles that escape upstream. For larger subluminal inclinations, shock-drift acceleration is the main acceleration mechanism, and the upstream oblique waves regulate injection into the acceleration process. For superluminal shocks, self-generated shock turbulence is not strong enough to overcome the kinematic constraints, and the downstream particle spectrum does not show any significant suprathermal tail. As seen from the upstream frame, efficient acceleration in relativistic (Lorentz factor γ0 5) magnetized (σ 0.03) flows exists only for a very small range of magnetic inclination angles (34°/γ0), so relativistic astrophysical pair shocks have to be either nearly parallel or weakly magnetized to generate nonthermal particles. These findings place constraints on the models of pulsar wind nebulae, gamma-ray bursts, and jets from active galactic nuclei that invoke particle acceleration in relativistic magnetized shocks.

On the Structure of Relativistic Collisionless Shocks in Electron-Ion Plasmas

Relativistic collisionless shocks in electron-ion plasmas are thought to occur in the afterglow phase of gamma-ray bursts (GRBs) and in other environments where relativistic flows interact with the interstellar medium. A particular regime of shocks in an unmagnetized plasma has generated much interest for GRB applications. In this Letter, we present ab initio particle-in-cell simulations of unmagnetized relativistic electron-ion shocks. Using long-term 2.5-dimensional simulations with ion-electron mass ratios from 16 to 1000, we resolve the shock formation and reach a steady state shock structure beyond the initial transient. We find that even at high ion-electron mass ratios initially unmagnetized shocks can be effectively mediated by the ion Weibel instability with a typical shock thickness of ~20 ion skin depths. Upstream of the shock, the interaction with merging ion current filaments heats the electron component, so that the postshock flow achieves near-equipartition between the ions and electrons, with the electron temperature reaching 50% of the ion temperature. This energy exchange helps to explain the large electron energy fraction inferred from GRB afterglow observations.

RELATIVISTIC RECONNECTION: AN EFFICIENT SOURCE OF NON-THERMAL PARTICLES

In magnetized astrophysical outflows, the dissipation of field energy into particle energy via magnetic reconnection is often invoked to explain the observed non-thermal signatures. By means of two- and three-dimensional particle-in-cell simulations, we investigate anti-parallel reconnection in magnetically dominated electron-positron plasmas. Our simulations extend to unprecedentedly long temporal and spatial scales, so we can capture the asymptotic state of the system beyond the initial transients, and without any artificial limitation by the boundary conditions. At late times, the reconnection layer is organized into a chain of large magnetic islands connected by thin X-lines. The plasmoid instability further fragments each X-line into a series of smaller islands, separated by X-points. At the X-points, the particles become unmagnetized and they get accelerated along the reconnection electric field. We provide definitive evidence that the late-time particle spectrum integrated over the whole reconnection region is a power law whose slope is harder than –2 for magnetizations σ 10. Efficient particle acceleration to non-thermal energies is a generic by-product of the long-term evolution of relativistic reconnection in both two and three dimensions. In three dimensions, the drift-kink mode corrugates the reconnection layer at early times, but the long-term evolution is controlled by the plasmoid instability which facilitates efficient particle acceleration, analogous to the two-dimensional physics. Our findings have important implications for the generation of hard photon spectra in pulsar winds and relativistic astrophysical jets.

ACCELERATION OF PARTICLES AT THE TERMINATION SHOCK OF A RELATIVISTIC STRIPED WIND

The relativistic wind of obliquely rotating pulsars consists of toroidal stripes of opposite magnetic field polarity, separated by current sheets of hot plasma. By means of two- and three-dimensional particle-in-cell simulations, we investigate particle acceleration and magnetic field dissipation at the termination shock of a relativistic striped wind. At the shock, the flow compresses and the alternating fields annihilate by driven magnetic reconnection. Irrespective of the stripe wavelength λ or the wind magnetization σ (in the regime σ 1 of magnetically dominated flows), shock-driven reconnection transfers all the magnetic energy of alternating fields to the particles, whose average Lorentz factor increases by a factor of σ with respect to the pre-shock value. The shape of the post-shock spectrum depends primarily on the ratio λ/(rL σ), where rL is the relativistic Larmor radius in the wind. The spectrum becomes broader as the value of λ/(rL σ) increases, passing from a relativistic Maxwellian to a flat power-law tail with slope around –1.5, populated by particles accelerated by the reconnection electric field. Close to the equatorial plane of the wind, where the stripes are symmetric, the highest energy particles resulting from magnetic reconnection can escape ahead of the shock, and be injected into a Fermi-like acceleration process. In the post-shock spectrum, they populate a power-law tail with slope around –2.5, which extends beyond the flat component produced by reconnection. Our study suggests that the spectral break between the radio and the optical band in Pulsar Wind Nebulae can be a natural consequence of particle acceleration at the termination shock of striped pulsar winds.

RESISTIVE SOLUTIONS FOR PULSAR MAGNETOSPHERES

The current state of the art in the modeling of pulsar magnetospheres invokes either the vacuum or force-free limits for the magnetospheric plasma. Neither of these limits can simultaneously account for both the plasma currents and the accelerating electric fields that are needed to explain the morphology and spectra of high-energy emission from pulsars. To better understand the structure of such magnetospheres, we combine accelerating fields and force-free solutions by considering models of magnetospheres filled with resistive plasma. We formulate Ohms law in the minimal velocity fluid frame and construct a family of resistive solutions that smoothly bridges the gap between the vacuum and the force-free magnetosphere solutions. The spin-down luminosity, open field line potential drop, and the fraction of open field lines all transition between the vacuum and force-free values as the plasma conductivity varies from zero to infinity. For fixed inclination angle, we find that the spin-down luminosity depends linearly on the open field line potential drop. We consider the implications of our resistive solutions for the spin-down of intermittent pulsars and sub-pulse drift phenomena in radio pulsars.

The Maximum Energy of Accelerated Particles in Relativistic Collisionless Shocks

The afterglow emission from gamma-ray bursts (GRBs) is usually interpreted as synchrotron radiation from electrons accelerated at the GRB external shock that propagates with relativistic velocities into the magnetized interstellar medium. By means of multi-dimensional particle-in-cell simulations, we investigate the acceleration performance of weakly magnetized relativistic shocks, in the magnetization range 0 ? 10?1. The pre-shock magnetic field is orthogonal to the flow, as generically expected for relativistic shocks. We find that relativistic perpendicular shocks propagating in electron-positron plasmas are efficient particle accelerators if the magnetization is ? 10?3. For electron-ion plasmas, the transition to efficient acceleration occurs for ? 3 ? 10?5. Here, the acceleration process proceeds similarly for the two species, since the electrons enter the shock nearly in equipartition with the ions, as a result of strong pre-heating in the self-generated upstream turbulence. In both electron-positron and electron-ion shocks, we find that the maximum energy of the accelerated particles scales in time as ?maxt 1/2. This scaling is shallower than the so-called (and commonly assumed) Bohm limit ?maxt, and it naturally results from the small-scale nature of the Weibel turbulence generated in the shock layer. In magnetized plasmas, the energy of the accelerated particles increases until it reaches a saturation value ?sat/?0 mic 2 ~ ??1/4, where ?0 mic 2 is the mean energy per particle in the upstream bulk flow. Further energization is prevented by the fact that the self-generated turbulence is confined within a finite region of thickness ??1/2 around the shock. Our results can provide physically grounded inputs for models of non-thermal emission from a variety of astrophysical sources, with particular relevance to GRB afterglows.

Long-Term Evolution of Magnetic Turbulence in Relativistic Collisionless Shocks: Electron-Positron Plasmas

We study the long term evolution of magnetic fields generated by an initially unmagnetized collisionless relativistic e+e− shock. Our two-dimensional particle-in-cell numerical simulations show that downstream of such a Weibel-mediated shock, particle distributions are approximately isotropic, relativistic Maxwellians, and the magnetic turbulence is highly intermittent spatially. The nonpropagating magnetic fields decay in amplitude and do not merge. The fields start with magnetic energy density ~ 0.1-0.2 of equipartition, but rapid downstream decay drives the fields to much smaller values, below ~10−3 of equipartition after ~103 skin depths. To construct a theory to follow field decay to these smaller values, we hypothesize that the observed damping is a variant of Landau damping. The model is based on the small value of the downstream magnetic energy density, which only weakly perturbs particle orbits, for homogeneous turbulence. Using linear kinetic theory, we find a simple analytic form for the damping rates for small-amplitude, subluminous electromagnetic fields. Our theory predicts that overall magnetic energy decays as (ωpt)−q with q ~ 1, which compares with simulations. However, our theory predicts overly rapid damping of short-wavelength modes. Magnetic trapping of particles within the highly spatially intermittent downstream magnetic structures may be the origin of this discrepancy and may allow for some of this initial magnetic energy to persist. Absent additional physical processes that create longer wavelength, more persistent fields, we conclude that initially unmagnetized relativistic shocks in electron-positron plasmas are unable to form persistent downstream magnetic fields. These results put interesting constraints on synchrotron models for the prompt and afterglow emission from GRBs. We also comment on the relevance of these results for relativistic electron-ion shocks.

Simulations of ion acceleration at non-relativistic shocks. I. Acceleration efficiency

We use two-dimensional and three-dimensional hybrid (kinetic ions-fluid electrons) simulations to investigate particle acceleration and magnetic field amplification at non-relativistic astrophysical shocks. We show that diffusive shock acceleration operates for quasi-parallel configurations (i.e., when the background magnetic field is almost aligned with the shock normal) and, for large sonic and Alfvenic Mach numbers, produces universal power-law spectra ∝p –4, where p is the particle momentum. The maximum energy of accelerated ions increases with time, and it is only limited by finite box size and run time. Acceleration is mainly efficient for parallel and quasi-parallel strong shocks, where 10%-20% of the bulk kinetic energy can be converted to energetic particles and becomes ineffective for quasi-perpendicular shocks. Also, the generation of magnetic turbulence correlates with efficient ion acceleration and vanishes for quasi-perpendicular configurations. At very oblique shocks, ions can be accelerated via shock drift acceleration, but they only gain a factor of a few in momentum and their maximum energy does not increase with time. These findings are consistent with the degree of polarization and the morphology of the radio and X-ray synchrotron emission observed, for instance, in the remnant of SN 1006. We also discuss the transition from thermal to non-thermal particles in the ion spectrum (supra-thermal region) and we identify two dynamical signatures peculiar of efficient particle acceleration, namely, the formation of an upstream precursor and the alteration of standard shock jump conditions.

Propagation of Thermonuclear Flames on Rapidly Rotating Neutron Stars: Extreme Weather during Type I X-Ray Bursts

We analyze the global hydrodynamic flow in the ocean of an accreting, rapidly rotating, nonmagnetic neutron star in a low-mass X-ray binary during a type I X-ray burst. We use both analytical arguments and numerical simulations of simplified models for ocean burning. Our analysis extends previous work by taking into account the rapid rotation of the star and the lift-up of the burning ocean during the burst. We find a new regime for the spreading of a nuclear burning front, where the flame is carried along a coherent shear flow across the front. If turbulent viscosity is weak, the speed of flame propagation is v_(flame) ~ (gh)^(1/2)/ft_n ~ 20 km s^(-1), where h is the scale height of the burning ocean, g is the local gravitational acceleration, t_n is the timescale for fast nuclear burning during the burst, and f is the Coriolis parameter, i.e., twice the local vertical component of the spin vector. If turbulent viscosity is dynamically important, the flame speed increases and reaches the maximum value, v^(max)_(flame_ ~ (gh/ft_n)^(1/2) ~ 300 km s^(-1), when the eddy overturn frequency is comparable to the Coriolis parameter f. We show that, as a result of rotationally reduced gravity, the thermonuclear runaway which ignites the ocean is likely to begin on the equator. The equatorial belt is ignited at the beginning of the burst, and the flame then propagates from the equator to the poles. Inhomogeneous cooling (equator first, poles second) of the hot ashes drives strong zonal currents which may be unstable to the formation of Jupiter-type vortices; we conjecture that these vortices are responsible for coherent modulation of X-ray flux in the tails of some bursts. We consider the effect of strong zonal currents on the frequency of modulation of the X-ray flux and show that the large values of the frequency drifts observed in some bursts can be accounted for within our model combined with the model of homogeneous radial expansion. Additionally, if vortices or other inhomogeneities are trapped in the forward zonal flows around the propagating burning front, fast chirps with large frequency ranges (~25-500 Hz) may be detectable during the burst rise. Finally, we argue that an MHD dynamo within the burning front can generate a small-scale magnetic field, which may enforce vertically rigid flow in the fronts wake and can explain the coherence of oscillations in the burst tail.