Power-law Momentum Research Articles

Observation of Subdiffusive Dynamic Scaling in a Driven and Disordered Bose Gas.

We explore the dynamics of a tuneable box-trapped Bose gas under strong periodic forcing in the presence of weak disorder. In absence of interparticle interactions, the interplay of the drive and disorder results in an isotropic nonthermal momentum distribution that shows subdiffusive dynamic scaling, with sublinear energy growth and the universal scaling function captured well by a compressed exponential. We explain that this subdiffusion in momentum space can naturally be understood as a random walk in energy space. We also experimentally show that for increasing interaction strength, the gas behavior smoothly crosses over to wave turbulence characterized by a power-law momentum distribution, which opens new possibilities for systematic studies of the interplay of disorder and interactions in driven quantum systems.

Energetic Neutral Atoms from Solar Energetic Particles due to Shocks: Inclusion of Upstream Particles

Many aspects of solar energetic particles are not well understood, including their acceleration mechanism. There has been recent interest in the potential of energetic neutral atoms (ENAs) as remote probes of solar energetic particles (SEPs) and their acceleration. The single accidental observation (in physical units) has been modeled as accelerated by a coronal mass ejection (CME)-driven shock by several authors, all of whom have assumed that the upstream component of the shock can be ignored. In this article, we relax this assumption and model the flux of ENAs at 1 au due to a CME-driven shock with an upstream component. We show the effect of varying parameters of the shock acceleration model, specifically α, the exponent of the power law in momentum of the mean free path, and η, a measure of the relative turbulence level. The main result is that including the upstream component significantly increases the flux at 1 au for typically assumed parameters in the energy range of the STEREO observation. We also derive the form of the ENA transport equation that we used in this study. These results enable a better understanding of potential observations of ENAs due to SEPs.

Universal equation of state for wave turbulence in a quantum gas.

Boyle's 1662 observation that the volume of a gas is, at constant temperature, inversely proportional to pressure, offered a prototypical example of how an equation of state (EoS) can succinctly capture key properties of a many-particle system. Such relationships are now cornerstones of equilibrium thermodynamics1. Extending thermodynamic concepts to far-from-equilibrium systems is of great interest in various contexts, including glasses2,3, active matter4-7 and turbulence8-11, but is in general an open problem. Here, using a homogeneous ultracold atomic Bose gas12, we experimentally construct an EoS for a turbulent cascade of matter waves13,14. Under continuous forcing at a large length scale and dissipation at a small one, the gas exhibits a non-thermal, but stationary, state, which is characterized by a power-law momentum distribution15 sustained by a scale-invariant momentum-space energy flux16. We establish the amplitude of the momentum distribution and the underlying energy flux as equilibrium-like state variables, related by an EoS that does not depend on the details of the energy injection or dissipation, or on the history of the system. Moreover, we show that the equations of state for a wide range of interaction strengths and gas densities can be empirically scaled onto each other. This results in a universal dimensionless EoS that sets benchmarks for the theory and should also be relevant for other turbulent systems.

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

The current paper considers the thermalization of an ensemble of electrons under the influence of an external electric field and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric field strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature T b has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form where λ n are the eigenvalues of the Fokker-Planck operator. Alternatively, a finite difference algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, σ(x) = σ 0/x p , where is the reduced speed. This model with p = 2 is also employed to demonstrate a steady-state Kappa distribution which features prominently in space physics and other fields. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary differential equation.

An investigation of induced free-surface wave oscillations in prismatic open-channel

ABSTRACT This paper reported on the free-surface wave behavior in a rectangular open-channel, induced by sluice-gate maneuvers. Numerical computations used the McCormack scheme to solve the one-dimensional extended St-Venant model embedding the Prandtl power-law momentum correction coefficient. The fundamental pulsation of the hydraulic system was inspected using the analogy with the water-hammer theory, known in pressurized-pipe flows. Different flow scenarios were reported; including the superposition between (i) downstream water-hammer maneuver and sine excitation of upstream flow depth and (ii) sinusoidal fluctuations of upstream lateral inflow and downstream flow-depth. Results highlighted that such sluice-gates maneuvers involved severe scenarios leading to significant amplifications of the depth peak and crest values above the initial value.

Particle Acceleration in Relativistic Shearing Flows: Energy Spectrum

We consider the acceleration of charged particles in relativistic shearing flows, with Lorentz factor up to Γ0 ∼ 20. We present numerical solutions to the particle transport equation and compare these with results from analytical calculations. We show that in the highly relativistic limit the particle energy spectrum that results from acceleration approaches a power law, , with a universal value for the slope of this power law, where α parameterizes the power-law momentum dependence of the particle mean free path. At mildly relativistic flow speeds, the energy spectrum becomes softer and sensitive to the underlying flow profile. We explore different flow examples, including Gaussian and power-law-type velocity profiles, showing that the latter yield comparatively harder spectra, producing for Γ0 ≃ 3 and Kolmogorov turbulence. We provide a comparison with a simplified leaky-box approach and derive an approximate relation for estimating the spectral index as a function of the maximum shear flow speed. These results are of relevance for jetted, high-energy astrophysical sources such as active galactic nuclei, since shear acceleration is a promising mechanism for the acceleration of charged particles to relativistic energies and is likely to contribute to the high-energy radiation observed.

Modeling of Cosmic-Ray Production and Transport and Estimation of Gamma-Ray and Neutrino Emissions in Starburst Galaxies

Abstract Starburst galaxies (SBGs) with copious massive stars and supernova (SN) explosions are the sites of active cosmic-ray production. Based on the predictions of nonlinear diffusive shock acceleration theory, we model the cosmic-ray proton (CRP) production by both pre-SN stellar winds (SWs) and supernova remnants (SNRs) from core-collapse SNe inside the starburst nucleus. Adopting different models for the transport of CRPs, we estimate the γ-ray and neutrino emissions due to pp collisions from nearby SBGs such as M82, NGC 253, and Arp 220. We find that with the current γ-ray observations by Fermi-LAT, Veritas, and H.E.S.S., it would be difficult to constrain CRP production and transport models. Yet, the observations are better reproduced with (1) the combination of a single power-law (PL) momentum distribution for SNR-produced CRPs and a diffusion model in which the CRP diffusion is mediated by the strong Kolmogorov-type turbulence of δ B / B ∼ 1 , and (2) the combination of a double-PL model for SNR-produced CRPs and a diffusion model in which the scattering of CRPs is controlled mostly by self-excited waves rather than the preexisting turbulence. The contribution of SW-produced CRPs could be substantial in Arp 220, where the star formation rate is higher and the slope of the initial mass function would be flatter. We suggest that M82 and NGC 253 might be detectable as point sources of high-energy neutrinos in the upcoming KM3NET and IceCube-Gen2, when optimistic models are applied. Future observations of neutrinos, as well as γ-rays, would provide constraints for the production and diffusion of CRPs in SBGs.

DHybridR: A Hybrid Particle-in-cell Code Including Relativistic Ion Dynamics

Abstract We present the first plasma simulations obtained with the code dHybridR, a hybrid particle-in-cell code with fluid electrons and both thermal and energetic ions that retain relativistic dynamics. dHybridR is constructed to study astrophysical and space-physics problems where a few energetic nonthermal particles (i.e., cosmic rays, CRs) affect the overall dynamics of a nonrelativistic plasma, such as CR-driven instabilities, collisionless shocks, magnetic reconnection, turbulence, etc. In this method paper we provide some applications to linear (resonant/nonresonant CR streaming instability) and strongly nonlinear (parallel shocks) problems that show the capabilities of the code. In particular, we provide the first self-consistent hybrid runs that show the acceleration of relativistic ions at nonrelativistic shocks; CRs develop a power law in momentum, which translates into a broken power law in energy that exhibits a steepening around the ion rest mass, as predicted by the theory of diffusive shock acceleration. We present examples of 2D dHybridR runs relevant for fast shocks in radio supernovae, whose evolution can be followed in real time, and 3D runs of low-Mach-number heliospheric shocks, which can be compared with in situ spacecraft observations.

Growth Rates of the Upper-Hybrid Waves for Power-Law and Kappa Distributions with a Loss-Cone Anisotropy

Fine structures of radio bursts play an important role in diagnostics of the solar flare plasma. Among them the zebras, which are prevalently assumed to be generated by the double plasma resonance instability, belong to the most important. In this paper we compute the growth rate of this instability for two types of the electron distribution: a) for the power-law distribution and b) for the kappa distribution, in the both cases with the loss-cone type anisotropy. We found that the growth rate of the upper-hybrid waves for the power-law momentum distribution strongly depends on the pitch-angle boundary. The maximum growth rate was found for the pitch-angle $\theta_\mathrm{c} \approx$ 50$^\circ$. For small angles the growth rate profile is very flat and for high pitch-angles the wave absorption occurs. Furthermore, analyzing the growth rate of the upper hybrid waves for the kappa momentum distribution we found that a decrease of the characteristic momentum $p_\kappa$ shifts the maximum of the growth rate to lower values of the ratio of the electron-plasma and electron-cyclotron frequencies, and the frequency widths of the growth rate peaks are very broad. But, if we consider the kappa distribution which is isotropic up to some large momentum $p_m$ and anisotropic with loss-cone above this momentum then distinct peaks of the growth rate appear and thus distinct zebra stripes can be generated. It means that the restriction for small momenta for the anisotropic part of distributions is of principal importance for the zebra stripes generation. Finally, for the 1 August 2010 zebra stripes, the growth rates in dependence on radio frequency were computed. It was shown that in this case the growth rate peaks are more distinct than in usually presented dependencies of growth rates on the ratio of the plasma and cyclotron frequencies.

Implications of a Loop-top Origin for Microwave, Hard X-Ray, and Low-energy Gamma-Ray Emission from Behind-the-limb Flares

The Fermi gamma-ray Space Telescope (Fermi) has detected hard X-ray (HXR) and gamma-ray photons from three flares, which according to \stereo occurred in active regions behind the limb of the Sun as delineated by near Earth instruments. For two of these flares \r has provided HXR images with sources located just above the limb, presumably from the loop top (LT) region of a relatively large loop. Fermi-Gamma-ray Burst Monitor has detected HXRs and gamma-rays, and RSTN has detected microwaves emissions with similar light curves. This paper presents a quantitative analysis of these multi-wavelength observations assuming that HXRs and microwaves are produced by electrons accelerated at the LT source, with emphasize on the importance of the proper treatment of escape of the particles from the acceleration-source region and the trans-relativistic nature of the analysis. The observed spectra are used to determine the magnetic field and relativistic electron spectra. It is found that a simple power-law in momentum (with cut off above a few 100 MeV) agrees with all observations, but in energy space a broken power law spectrum (steepening at rest mass energy) may be required. It is also shown that the production of the $>100$ MeV photons detected by The Fermi-Large Area Telescope at the LT source would require more energy compared to photospheric emission. These energies are smaller than that required for electrons, so that the possibility that all the emissions originate in the LT cannot be ruled out on energetic grounds. However, the differences in the light curves and emission centroids of HXRs and $>100$ MeV gamma-rays favour a different source for the latter.

Particle Acceleration Due to Cosmic-ray Viscosity and Fluid Shear in Astrophysical Jets

Abstract An analytical model describing the acceleration of energetic charged particles due to cosmic-ray viscosity and fluid shear in radio jets based on the diffusive transport equation for cosmic rays is developed. The model indicates that laminar shear is not a significant mechanism for particle acceleration in nonrelativistic jets but may be important in relativistic jets. The model produces a power-law momentum spectrum of particles accelerated by the shear (i.e., as , where f 0 is the isotropic phase space distribution in the scattering frame). For ultra-relativistic jets with no back-flowing cocoon, for the energetic particle diffusion coefficient case κ ∝ p α . Increasing the shear in the background flow gives harder accelerated particle momentum spectra. Monoenergetic source solutions (Green’s functions) and Green’s formula are derived and used to obtain solutions for monoenergetic particle momentum spectra specified on the edge of the jet at radius r = r 2.

Diffusive Cosmic-Ray Acceleration at Shock Waves of Arbitrary Speed with Magnetostatic Turbulence. I. General Theory and Correct Nonrelativistic Speed Limit

Abstract The analytical theory of diffusive acceleration of cosmic rays at parallel stationary shock waves of arbitrary speed with magnetostatic turbulence is developed from first principles. The theory is based on the diffusion approximation to the gyrotropic cosmic-ray particle phase-space distribution functions in the respective rest frames of the up- and downstream medium. We derive the correct cosmic-ray jump conditions for the cosmic-ray current and density, and match the up- and downstream distribution functions at the position of the shock. It is essential to account for the different particle momentum coordinates in the up- and downstream media. Analytical expressions for the momentum spectra of shock-accelerated cosmic rays are calculated. These are valid for arbitrary shock speeds including relativistic shocks. The correctly taken limit for nonrelativistic shock speeds leads to a universal broken power-law momentum spectrum of accelerated particles with velocities well above the injection velocity threshold, where the universal power-law spectral index is independent of the flow compression ratio r. For nonrelativistic shock speeds, we calculate for the first time the injection velocity threshold, settling the long-standing injection problem for nonrelativistic shock acceleration.

DIFFUSIVE COSMIC RAY ACCELERATION AT RELATIVISTIC SHOCK WAVES WITH MAGNETOSTATIC TURBULENCE. II. INFLUENCE OF A FINITE DOWNSTREAM MEDIUM

ABSTRACT The diffusive acceleration of relativistic cosmic rays at parallel shock waves with magnetostatic turbulence and a finite size of the downstream medium is investigated. For ultrarelativistic shock speeds with Lorentz factor , both the differential momentum spectrum at the shock and the volume-integrated momentum spectrum are power-law distribution functions with different spectral indices as compared to the case of an infinitely extended downstream medium. However, the spectral differences are only modest as compared to the case of nonrelativistic shocks. The behavior of the momentum spectrum of shock-accelerated particles depends sensitively on the relativistic shock wave Peclet number , i.e., the ratio between the diffusion and convection timescales of cosmic rays to propagate from the shock position to the downstream boundary z 0. For large values of the free-escape boundary has no influence on the effectiveness of particle acceleration, still providing a flat momentum power-law spectrum of the accelerated particles. In the opposite case of small Peclet numbers at all momenta, the momentum spectrum at the shock steepens to the greater spectral index , whereas the volume-integrated momentum spectrum flattens by the same factor for its power-law spectral index, where s denotes the spectral index of the downstream power spectrum of magnetostatic turbulence. This effectiveness of relativistic shocks in generating flat power-law momentum spectra irrespective of the Peclet number differs completely from the behavior of nonrelativistic shocks.

RE-ACCELERATION MODEL FOR THE 'SAUSAGE' RADIO RELIC

The Sausage radio relic is the arc-like radio structure in the cluster CIZA J2242.8+5301, whose observed properties can be best understood by synchrotron emission from relativistic electrons accelerated at a merger-driven shock. However, there remain a few puzzles that cannot be explained by the shock acceleration model with only in-situ injection. In particular, the Mach number inferred from the observed radio spectral index, $M_{\rm radio}\approx 4.6$, while the Mach number estimated from X-ray observations, $M_{\rm X-ray}\approx 2.7$. In an attempt to resolve such a discrepancy, here we consider the re-acceleration model in which a shock of $M_s\approx 3$ sweeps through the intracluster gas with a pre-existing population of relativistic electrons. We find that observed brightness profiles at multi frequencies provide strong constraints on the spectral shape of pre-existing electrons. The models with a power-law momentum spectrum with the slope, $s\approx 4.1$, and the cutoff Lorentz factor, $\gamma_{e,c}\approx 3-5\times 10^4$ can reproduce reasonably well the observed spatial profiles of radio fluxes and integrated radio spectrum of the Sausage relic. The possible origins of such relativistic electrons in the intracluster medium remain to be investigated further.

RE-ACCELERATION MODEL FOR THE 'TOOTHBRUSH' RADIO RELIC

The Toothbrush radio relic associated with the merging cluster 1RXS J060303.3 is presumed to be produced by relativistic electrons accelerated at merger-driven shocks. Since the shock Mach number inferred from the observed radio spectral index, M radio ≈ 2.8, is larger than that estimated from X-ray observations, M X ≲ 1.5, we consider the re-acceleration model in which a weak shock of M s ≈ 1.2 - 1.5 sweeps through the intracluster plasma with a preshock population of relativistic electrons. We nd the models with a power-law momentum spectrum with the slope, s ≈ 4.6, and the cutoff Lorentz factor,γ e,c ≈ 7-8 x 10 4 can reproduce reasonably well the observed proles of radio uxes and integrated radio spectrum of the head portion of the Toothbrush relic. This study conrms the strong connection between the ubiquitous presence of fossil relativistic plasma originated from AGNs and the shock-acceleration model of radio relics in the intracluster medium.

TIME-DEPENDENT DIFFUSIVE SHOCK ACCELERATION IN SLOW SUPERNOVA REMNANT SHOCKS

Recent gamma ray observations show that middle aged supernova remnants interacting with molecular clouds can be sources of both GeV and TeV emission. Models involving re-acceleration of pre-existing cosmic rays in the ambient medium and direct interaction between supernova remnant and molecular clouds have been proposed to explain the observed gamma ray emission. For the re-acceleration process, standard DSA theory in the test particle limit produces a steady state particle spectrum that is too flat compared to observations, which suggests that the high energy part of the observed spectrum has not yet reached a steady state. We derive a time dependent DSA solution in the test particle limit for situations involving re-acceleration of pre-existing cosmic rays in the preshock medium. Simple estimates with our time dependent DSA solution plus a molecular cloud interaction model can reproduce the overall shape of the spectra of IC 443 and W44 from GeV to TeV energies through pure $\pi^0$-decay emission. We allow for a power law momentum dependence of the diffusion coefficient, finding that a power law index of 0.5 is favored.

Impact of Fermi-LAT and AMS-02 results on cosmic-ray astrophysics

This article reviews a few topics relevant to Galactic cosmic-ray astrophysics, focusing on the recent AMS-02 data release and Fermi Large Area Telescope data on the diffuse Galactic gamma-ray emissivity. Calculations are made of the diffuse cosmic-ray induced p+p --> pi^0 --> 2 gamma spectra, normalized to the AMS-02 cosmic-ray proton spectrum at ~ 10 - 100 GV, with and without a hardening in the cosmic-ray proton spectrum at rigidities R >~ 300 GV. A single power-law momentum "shock" spectrum for the local interstellar medium cosmic-ray proton spectrum cannot be ruled out from the gamma-ray emissivity data alone without considering the additional contribution of electron bremsstrahlung. Metallicity corrections are discussed, and a maximal range of nuclear enhancement factors from 1.52 to 1.92 is estimated.Origins of the 300 GV cosmic-ray proton and alpha-particle hardening are discussed.

Evidence of hadronic interaction in Tycho Supernova Remnant using Fermi-LAT data

The Fermi Large Area Telescope (LAT) has observed Tycho Supernova Remnant in the MeV-GeV energy range. The spectrum has been studied using the first three years of data and new data are being collected. We present a multiwavelength model of the observed spectrum from radio to TeV energy range, based on the hypothesis of hadronic origin of γ-rays. As described by the Fermi acceleration theory, a single proton population was considered, modeled with a simple power-law in momentum. The photon emissivity is computed following Kamae et al (2006) [T. Kamae, et al., ApJ 647 (2006) 692]. The leptonic component is also taken into account according to Giordano et al. (2012) [F. Giordano, et al., ApJ 744 (2012) L2] prescriptions and it turns out to be negligible with respect to the hadronic one. The model returns a spectral index of 2.23(±0.05) and an acceleration efficiency of 5% of the total kinetic energy expelled in Supernova explosion and it may provide a hint of the acceleration processes in SNRs up to energies close to the knee of cosmic ray spectrum. This work shows that experimental data can be easily explained with a simple model, representing a good test for the acceleration theory.

COSMIC RAY ACCELERATION AT PERPENDICULAR SHOCKS IN SUPERNOVA REMNANTS

Supernova remnants (SNRs) are believed to accelerate particles up to high energies through the mechanism of diffusive shock acceleration (DSA). Except for direct plasma simulations, all modeling efforts must rely on a given form of the diffusion coefficient, a key parameter that embodies the interactions of energetic charged particles with the magnetic turbulence. The so-called Bohm limit is commonly employed. In this paper we revisit the question of acceleration at perpendicular shocks, by employing a realistic model of perpendicular diffusion. Our coefficient reduces to a power-law in momentum for low momenta (of index $\alpha$), but becomes independent of the particle momentum at high momenta (reaching a constant value $\kappa_{\infty}$ above some characteristic momentum $p_{\rm c}$). We first provide simple analytical expressions of the maximum momentum that can be reached at a given time with this coefficient. Then we perform time-dependent numerical simulations to investigate the shape of the particle distribution that can be obtained when the particle pressure back-reacts on the flow. We observe that, for a given index $\alpha$ and injection level, the shock modifications are similar for different possible values of $p_{\rm c}$, whereas the particle spectra differ markedly. Of particular interest, low values of $p_{\rm c}$ tend to remove the concavity once thought to be typical of non-linear DSA, and result in steep spectra, as required by recent high-energy observations of Galactic SNRs.

RELATIVISTIC NON-THERMAL BREMSSTRAHLUNG RADIATION

By applying a method of virtual quanta we derive formulae for relativistic non-thermal bremsstrahlung radiation from relativistic electrons as well as from protons and heavier particles with power-law momentum distribution N(p)dp = k p-qdp. We show that emission which originates from an electron scattering on an ion, represents the most significant component of relativistic non-thermal bremsstrahlung. Radiation from an ion scattering on electron, known as inverse bremsstrahlung, is shown to be negligible in overall non-thermal bremsstrahlung emission. These results arise from theory refinement, where we introduce the dependence of relativistic kinetic energy of an incident particle, upon the energy of scattered photon. In part, it is also a consequence of a different mass of particles and relativistic effects.