Ultrarelativistic Shock Waves Research Articles

Revisiting first type self-similar solutions of explosions containing ultrarelativistic shocks

We revisit the first type self-similar solutions for ultrarelativistic shock waves produced by explosions propagating into cold external medium whose density profile decreases with radius as ρ ∝ r−k. The first-type solutions proposed by Blandford and McKee (hereafter BM solution) conforms to the global conservation of energy and applies when k < 4. They have been found to be invalid when k > 17/4 because of the divergence of total energy contained in the shocked fluids. So far no attention has been paid to the particle number. We use the BM solution to calculate the total particle number traversed by the shock and find that it diverges when k > 3. This is inconsistent with the finite particles in the surrounding medium. We propose a possible solution when k > 3 based on the conservation of particle number and discuss its implication for the second-type solutions.

Shock acceleration in gamma-ray bursts

Abstract Gamma-ray bursts offer a rather unique window on the fundamental astrophysics of particle acceleration. Sources of high-energy gamma rays, they are also likely sources of cosmic rays, possibly of the so-called ultra-high energy cosmic rays, and they may well turn out to be the strongest sources of high energy neutrinos. Through the interaction of their outflow with the circum-burst medium, these explosions generate ultra-relativistic shock waves that convert part of the bulk kinetic energy into particle energy, ultimately giving rise to the impressive photon power law spectra of the afterglow. The prompt emission may well occur through the interactions of disturbances moving with mildly relativistic relative velocity within the flow itself. However, the detailed acceleration mechanism is not yet understood. This chapter discusses the progress made in the past decade in our understanding of relativistic shock acceleration and its relation to gamma-ray burst phenomenology. It notably discusses the intimate relationship between the electromagnetic micro-instabilities upstream of the collisionless shock and the accelerated particles. It also briefly discusses the possibility of accelerating particles to ultra-high energies and the production of secondary neutrino signals. It concludes with a list of open questions and some perspectives.

On electromagnetic instabilities at ultra-relativistic shock waves

(Abridged) This paper addresses the issue of magnetic field generation in a relativistic shock precursor through micro-instabilities. The level of magnetization of the upstream plasma turns out to be a crucial parameter, notably because the length scale of the shock precursor is limited by the Larmor rotation of the accelerated particles in the background magnetic field and by the speed of the shock wave. We discuss in detail and calculate the growth rates of the following beam plasma instabilities seeded by the accelerated and reflected particle populations: for an unmagnetized shock, the Weibel and filamentation instabilities, as well as the Cerenkov resonant longitudinal and oblique modes; for a magnetized shock, the Weibel instability and the resonant Cerenkov instabilities with the longitudinal electrostatic modes, as well as the Alfven, Whisler and extraordinary modes. All these instabilities are generated upstream, then they are transmitted downstream. The modes excited by Cerenkov resonant instabilities take on particular importance with respect to the magnetisation of the downstream medium since, being plasma eigenmodes, they have a longer lifetime than the Weibel modes. We discuss the main limitation of the wave growth associated with the length of the precursor and the magnetisation of the upstream medium for both oblique and parallel shock waves. We also characterize the proper conditions to obtain Fermi acceleration. We recover some results of most recent particle-in-cell simulations and conclude with some applications to astrophysical cases of interest, pulsar winds and gamma-ray burst external shock waves in particular. (Abridged)

Horizon Structure of the AdS 5 Line Element After the Collision of Two Shock Waves

In this paper we consider collision of two ultrarelativistic shock-waves in AdS5 space and discuss about horizon structure of the metric after the collision. We show that all singularities of such metric are hidden behind the horizon.

Observational evidences of multiple shock waves in X-ray selected BL Lacerate objects

The results of optical R-band photometry of three X-ray selected BL Lacertae objects 1ES 0502+675, 1ES 0806+524 and 1ES 1959+650 from the Einstein Slew Survey are presented. The observations are performed during 1997–2006 with the 70-cm meniscus type telescope of Abastumani Astrophysical Observatory. The objects show clear long-term variability with time scales of 1–3 years without evident periodicity. The main results consist in the discovery of multiple-peak structures on historical light curves as predicted theoretically on the basis of the assumption that long-term variability of blazars are triggered by ultrarelativistic shock waves propagating through their jets. They are direct evidence of the existence of reverse shocks besides the main one arising by a single disturbance. Two-peak maxima are found for 1ES 0502+675 and 1ES 0806+524. The more complicated structure shows 1ES 199+650 - a maximum with four consequent peaks. The relative strengths of the main and reverse shocks are mainly equal according to the shape of the respective peaks. The brightness dip between them is on average 60% less than in the case of consequent main shocks. Optical maximum epochs, covered well by the observations, show that the main shocks are either not always accompanied by reverse ones or the later are not always strong enough to be discovered by the observations.

On Fermi acceleration and magnetohydrodynamic instabilities at ultra-relativistic magnetized shock waves

Fermi acceleration can take place at ultra-relativistic shock waves if the upstream or downstream magnetic field has been remodelled so that most of the magnetic power lies on short spatial scales. The relevant conditions under which Fermi acceleration becomes efficient in the presence of both coherent and short-scale turbulent magnetic fields are addressed. Within the magnetohydrodynamic (MHD) approximation, this paper then studies the amplification of a pre-existing magnetic field through the streaming of cosmic rays upstream of a relativistic shock wave. The magnetic field is assumed to be perpendicular in the shock-front frame, as generally expected in the limit of large shock Lorentz factor. In the MHD regime, compressive instabilities seeded by the net cosmic-ray charge in the shock precursor (as seen in the shock-front frame) develop on the shortest spatial scales but saturate at a moderate level δB/B ∼ 1, which is not sufficient for Fermi acceleration. As we argue, it is possible that other instabilities outside the MHD range provide enough amplification to allow successful Fermi acceleration.

On the collision of two shock waves inAdS5

We consider two ultrarelativistic shock waves propagating and colliding in five-dimensional Anti-de-Sitter spacetime. By transforming to Rosen coordinates, we are able to find the form of the metric shortly after the collision. Using holographic renormalization, we calculate the energy-momentum tensor on the boundary of AdS space for early times after the collision. Via the gauge-gravity duality, this gives some insights on bulk dynamics of systems created by high energy scattering in strongly coupled gauge theories. We find that Bjorken boost-invariance is explicitely violated at early times and we obtain an estimate for the thermalization time in this simple system.

Modeling heavy ion collisions in AdS/CFT

We construct a model of high energy heavy ion collisions as two ultrarelativistic shock waves colliding in AdS5. We point out that shock waves corresponding to physical energy-momentum tensors of the nuclei completely stop almost immediately after the collision in AdS5, which, on the field theory side, corresponds to complete nuclear stopping due to strong coupling effects, likely leading to Landau hydrodynamics. Since in real-life heavy ion collisions the large Bjorken x part of nuclear wave functions continues to move along the light cone trajectories of the incoming nuclei leaving the small-x partons behind, we conclude that a pure large coupling approach is not likely to adequately model nuclear collisions. We show that to account for small-coupling effects one can model the colliding nuclei by two (unphysical) ultrarelativistic shock waves with zero net energy each (but with non-zero energy density). We use this model to study the energy density of the strongly-coupled matter created immediately after the collision. We argue that expansion of the energy density in the powers of proper time τ squared corresponds on the gravity side to a perturbative expansion of the metric in graviton exchanges. Using such expansion we reproduce our earlier result [1] that the energy density of produced matter at mid-rapidity starts out as a constant (of time) in heavy ion collisions at large coupling.

Cosmic‐Ray Acceleration at Ultrarelativistic Shock Waves: Effects of Downstream Short‐Wave Turbulence

The present paper is the last of a series studying the first-order Fermi acceleration processes at relativistic shock waves with the method of Monte Carlo simulations applied to shocks propagating in realistically modeled turbulent magnetic fields. The model of the background magnetic field structure of Niemiec & Ostrowski has been augmented here by a large-amplitude short-wave downstream component, imitating that generated by plasma instabilities at the shock front. Following the recent work of Niemiec & Ostrowski, we have considered ultrarelativistic shocks with the mean magnetic field oriented both oblique and parallel to the shock normal. For both cases, simulations have been performed for different choices of magnetic field perturbations, represented by various wave power spectra within a wide wavevector range. The results show that the introduction of the short-wave component downstream of the shock is not sufficient to produce power-law particle spectra with the universal spectral index 4.2. On the contrary, concave spectra with cutoffs are preferentially formed, the curvature and cutoff energy being dependent on the properties of turbulence. Our results suggest that the electromagnetic emission observed from astrophysical sites with relativistic jets, e.g., active galactic nuclei and gamma-ray bursts, is likely generated by particles accelerated in processes other than the widely invoked first-order Fermi mechanism.

Heating and Nonthermal Particle Acceleration in Relativistic, Transverse Magnetosonic Shock Waves in Proton‐Electron‐Positron Plasmas

We report the results of 1D particle-in-cell simulations of ultrarelativistic shock waves in proton-electron-positron plasmas. We consider magnetized shock waves, in which the upstream medium carries a large scale magnetic field, directed transverse to the flow. Relativistic cyclotron instability of each species as the incoming particles encounter the increasing magnetic field within the shock front provides the basic plasma heating mechanism. The most significant new results come from simulations with mass ratio $m_p/m_\pm = 100$. We show that if the protons provide a sufficiently large fraction of the upstream flow energy density (including particle kinetic energy and Poynting flux), a substantial fraction of the shock heating goes into the formation of suprathermal power-law spectra of pairs. Cyclotron absorption by the pairs of the high harmonic ion cyclotron waves, emitted by the protons, provides the non-thermal acceleration mechanism. As the proton fraction increases, the non-thermal efficiency increases and the pairs' power-law spectra harden. We suggest that the varying power law spectra observed in synchrotron sources powered by magnetized winds and jets might reflect the correlation of the proton to pair content enforced by the underlying electrodynamics of these sources' outflows, and that the observed correlation between the X-ray spectra of rotation powered pulsars with the X-ray spectra of their nebulae might reflect the same correlation.

On the Efficiency of Fermi Acceleration at Relativistic Shocks

It is shown that Fermi acceleration at an ultra-relativistic shock wave cannot operate on a particle for more than 1 1/2 Fermi cycle (i.e., u -> d -> u -> d) if the particle Larmor radius is much smaller than the coherence length of the magnetic field on both sides of the shock, as is usually assumed. This conclusion is shown to be in excellent agreement with recent numerical simulations. We thus argue that efficient Fermi acceleration at ultra-relativistic shock waves requires significant non-linear processing of the far upstream magnetic field with strong amplification of the small scale magnetic power. The streaming or transverse Weibel instabilities are likely to play a key role in this respect.

Cosmic Ray Acceleration at Ultrarelativistic Shock Waves: Effects of a “Realistic” Magnetic Field Structure

First-order Fermi acceleration processes at ultrarelativistic (γ ~ 5-30) shock waves are studied with Monte Carlo simulations. The accelerated particle spectra are derived by integrating the exact particle trajectories in a turbulent magnetic field near the shock. The magnetic field model upstream of the shock assumes finite-amplitude perturbations within a wide wavevector range and with a predefined wave power spectrum, imposed on the mean field component inclined at some angle to the shock normal. The downstream field structure is obtained as the compressed upstream field. We show that the main acceleration process at superluminal shocks is the particle compression at the shock. Formation of energetic spectral tails is possible in a limited energy range only for highly perturbed magnetic fields. Cutoffs in the spectra occur at low energies within the resonance energy range considered. These spectral features result from the anisotropic character of particle transport in the magnetic field downstream of the shock, where field compression produces effectively two-dimensional perturbations. We also present results for parallel shocks. Because of the turbulent field compression at the shock, the acceleration process becomes inefficient for larger turbulence amplitudes, and features observed in oblique shocks are recovered in this case. For small-amplitude perturbations, particle spectra are formed in the wide energy range, and modifications of the acceleration process due to the existence of long-wave perturbations are observed, as reported previously for mildly relativistic shocks. The critical turbulence amplitude required for efficient acceleration at parallel shocks decreases with increasing shock Lorentz factor γ. In both subluminal and superluminal shocks, an increase of γ leads to steeper spectra with lower cutoff energies. The spectra obtained for the "realistic" background conditions assumed in our simulations do not converge to the "universal" spectral index claimed in the literature. Thus, the role of the first-order Fermi acceleration in astrophysical sources hosting relativistic shocks requires serious reanalysis.

First and second type self-similar solutions of implosions and explosions containing ultrarelativistic shocks

We derive self-similar solutions for ultrarelativistic shock waves propagating into cold material of power law density profile in radius ρ∝r−k. We treat both implosions and explosions in three geometries: planar, cylindrical, and spherical. For spherical explosions these are the first type solutions of Blandford and McKee for k<4; they are the second type solutions found by Best and Sari for k>5−3∕4. In addition we find new, hollow (with evacuated interior), first type solutions that may be applicable for 4<k<17∕4. This “sequence” with increasing k of first type solutions, hollow first type solutions, and then second type solutions is reminiscent of the nonrelativistic sequence. However, although in the nonrelativistic case there is a range of k which corresponds to a “gap”—a range in k with neither first nor second type solution which separates the hollow first type solutions and the second type solutions, here there is an “overlap”: a range of k for which current considerations allow for both hollow first and second type solutions. Further understanding is needed to determine which of the two solutions apply in this overlap regime. We provide similar exploration for the other geometries and for imploding configurations. Interestingly, we find a gap for imploding spherical shocks and exploding planar shocks and an overlap for imploding planar solutions. Cylindrical configurations have no hollow solutions and exhibit a direct transition from first type to second type solutions, without a gap or an overlap region.

Self‐similar Evolution of Relativistic Shock Waves Emerging from Plane‐parallel Atmospheres

We study the evolution of the ultra-relativistic shock wave in a plane-parallel atmosphere adjacent to a vacuum and the subsequent breakout phenomenon. When the density distribution has a power law with the distance from the surface, there is a self-similar motion of the fluid before and after the shock emergence. The time evolution of the Lorentz factor of the shock front is assumed to follow a power law when the time is measured from the moment at which the shock front reaches the surface. The power index is found to be determined by the condition for the flow to extend through a critical point. The energy spectrum of the ejected matter as a result of the shock breakout is derived and its dependence on the strength of the explosion is also deduced. The results are compared with the self-similar solution for the same problem with non-relativistic treatment.

Conversion of bulk kinetic energy into radiation in AGNs and GRBs: Particle transport effects

We investigate the spatial structure of collisionless collision fronts in relativistic outflows interacting with ambient material. As a result of the interaction, ambient particles are picked up by the outflow and generate transverse plasma waves via streaming instabilities. Pick-up particle transport under the influence of self-generated turbulence inside such interaction regions is studied. We extend our previous momentum space modeling to include also a spatial dimension. We find that the following possibilities are consistent with quasi-linear equations of particle transport and wave generation: (i) if background waves have small intensities inside the outflow region, leading to inefficient scattering across the pitch-angle, θ , of $90^\circ$, particles are isotropized in the backward hemisphere (relative to the outflow velocity vector) and self-generated waves have a steep, ${\propto} k^{-3}$ wavenumber spectrum; (ii) if background waves have large intensities, enabling particles to cross $\theta=90^\circ$, particles can be fully isotropized. In case (i), however, the calculated self-generated wave amplitudes are close to the magnitude of the ordered field for reasonable choices of model parameters, giving the particles a chance to be scattered across the resonance gap by non-resonant processes. If the resonance gap is filled, a large fraction of the pick-up particles is expected to return to the upstream region, and an ultra-relativistic shock wave is predicted to form in front of the outflow, where the two relativistic particle populations (ambient and reflected) mix and form a relativistic plasma. Reflection of pick-up protons decreases the $\pi^0$-decay luminosity of relativistic outflows, leading to a need to update parameters of previous modeling. An example of outflow parameters reproducing typical TeV-blazar observations is presented.

Particle acceleration in ultra-relativistic oblique shock waves

We perform Monte Carlo simulations of diffusive shock acceleration at highly relativistic oblique shock waves. High upstream flow Lorentz gamma factors ( Γ) are used, which are relevant to models of ultra-relativistic particle shock acceleration in active galactic nuclei (AGN) central engines and relativistic jets and gamma ray burst (GRB) fireballs. We investigate numerically the acceleration properties in the relativistic and ultra-relativistic flow regime ( Γ∼10–10 3), such as angular distribution, acceleration time constant, particle energy gain versus number of crossings and spectral shapes. We perform calculations for sub-luminal and super-luminal shocks. For the first case, the dependence on whether or not the scattering is pitch angle diffusion or large angle scattering is studied. The large angle model exhibits a distinctive structure in the basic power-law spectrum which is not nearly so obvious for small angle scattering. However, both models yield significant ‘speed-up’ or faster acceleration rates when compared with the conventional, non-relativistic expression for the time constant, or alternatively with the time scale r g / c where r g is Larmor radius. The Γ 2 energization for the first crossing cycle and the significantly large energy gain for subsequent crossings as well as the high ‘speed-up’ factors found, are important in supporting the Vietri and Waxman work on GRB ultra-high energy cosmic ray, neutrino and gamma-ray output. Secondly, for super-luminal shocks, we calculate the energy gain for a number of different inclinations and the spectral shapes of the accelerated particles are given. In this investigation we consider only large angle scattering, partly because of computational time limitations and partly because this model provides the most favourable situation for acceleration. We use high gamma flows with Lorentz factors in the range 10–40, which are relevant to AGN accretion disks and jet ultra-relativistic shock configurations. We closely follow the particle’s trajectory along the magnetic field lines during shock crossings where the equivalent of a guiding centre approximation is inappropriate, constantly measuring its phase space co-ordinates in the fluid frames where E =0 . We find that a super-luminal ‘shock drift’ mechanism is less efficient in accelerating particles to the highest energies observed, compared to the first order Fermi acceleration applying in the sub-luminal case, suggesting that the former cannot stand as a sole acceleration mechanism for the ultra-high energy cosmic rays observed.

Particle acceleration in ultra-relativistic parallel shock waves

Monte Carlo computations for highly relativistic parallel shock particle acceleration are presented for upstream flow gamma factors, Γ=(1− V 1 2/ c 2) −0.5 with values between 5 and 10 3. The results show that the spectral shape at the shock depends on whether or not the particle scattering is small angle with δθ< Γ −1 or large angle, which is possible if λ>2 r g Γ 2 where λ is the scattering mean free path along the field line and r g the gyroradius, these quantities being measured in the plasma flow frame. The large angle scattering case exhibits distinctive structure superimposed on the basic power-law spectrum, largely absent in the pitch angle case. Also, both cases yield an acceleration rate faster than estimated by the conventional, non-relativistic expression, T=[ c/( V 1− V 2)][ λ 1/ V 1+ λ 2/ V 2] where ‘1’ and ‘2’ refer to upstream and downstream and λ is the mean free path. A Γ 2 energy enhancement factor in the first shock crossing cycle and a significant energy multiplication in the subsequent shock cycles are also observed. The results may be important for our understanding of the production of very high energy cosmic rays and the high energy neutrino and gamma-ray output from gamma-ray bursts and active galactic nuclei.

Comment on the first-order Fermi acceleration at ultra-relativistic shocks

The first-order Fermi acceleration process at an ultra-relativistic shock wave is expected to create a particle spectrum with the unique asymptotic spectral index sigma_{gamma >> 1} approximately 2.2. Below, we discuss this result and differences in its various derivations, which -- explicitly or implicitly -- always require highly turbulent conditions downstream of the shock. In the presence of medium amplitude turbulence the generated particle spectrum can be much steeper than the above asymptotic one. We also note problems with application of the pitch angle diffusion model for particle transport near the ultra-relativistic shocks.

Hidden source of high-energy neutrinos in collapsing galactic nucleus

We propose a model of a short-lived very powerful source of high energy (HE) neutrinos. It is formed as a result of the dynamical evolution of a galactic nucleus prior to its collapse into a massive black hole and formation of high-luminosity AGN. This stage can be referred to as “pre-AGN”. A dense central stellar cluster in the galactic nucleus at the late stage of evolution consists of compact stars (neutron stars and stellar mass black holes). This cluster is sunk deep into the massive gas envelope produced by destructive collisions of a primary stellar population. Frequent collisions of neutron stars in a central stellar cluster are accompanied by the generation of ultrarelativistic fireballs and shock waves. These repeating fireballs result in a formation of the expanding rarefied cavity inside the envelope. The charged particles are effectively accelerated in the cavity and, due to pp collisions in the gas envelope, they produce HE neutrinos. All HE particles, except neutrinos, are absorbed in the thick envelope. The duration of this pre-AGN phase is ∼1 year, the number of the sources can be a few per cosmological horizon. HE neutrino signal can be detected by underground neutrino telescope with effective area S∼1 km 2.

Efficiency of cosmic ray reflections from an ultrarelativistic shock wave

The process of cosmic ray acceleration up to energies in excess of $10^{20}$ eV at relativistic shock waves with large Lorentz factors, $\Gamma \gg 1$ requires $\sim \Gamma^2$ particle energy gains at single reflections from the shock (cf. Gallant & Achterberg 1999). In the present comment, applying numerical simulations we address an efficiency problem arising for such models. The actual efficiency of the acceleration process is expected to be substantially lower than the estimates of previous authors.