Relativistic Maxwellian Distribution Research Articles

Relativistic Maxwellian mixture model

For analyzing distribution functions of relativistic plasma, we propose a mixture model composed of relativistic Maxwellian distributions. We first summarize the basic properties of the relativistic Maxwellian distribution, including the derivation of the normalization constant when there is a bulk velocity. We also examine the maximum likelihood estimation of the relativistic Maxwellian distribution. We then introduce a relativistic Maxwellian mixture model (R-MMM), which is a weighted sum of relativistic Maxwellian distributions. We develop an expectation–maximization algorithm for estimating the parameters of R-MMM, namely, the mixing proportion, the bulk velocity, and the temperature of each component. We apply a two-component R-MMM to a distribution function by a particle-in-cell (PIC) simulation of relativistic pair plasma and separate the simulated distribution function into two components. We find that one component has a large bulk velocity while the other is almost stagnant, and that the two components have almost the same temperatures, which is also consistent with the initial temperature of PIC simulation.

Relativistic electron kinetic effects on laser diagnostics in burning plasmas

Toroidal interferometry/polarimetry (TIP), poloidal polarimetry (PoPola), and Thomson scattering systems (TS) are major optical diagnostics being designed and developed for ITER. Each of them relies upon a sophisticated quantitative understanding of the electron response to laser light propagating through a burning plasma. Review of the theoretical results for two different applications is presented: interferometry/polarimetry (I/P) and polarization of Thomson scattered light, unified by the importance of relativistic (quadratic in vTe/c) electron kinetic effects. For I/P applications, rigorous analytical results are obtained perturbatively by expansion in powers of the small parameter τ = Te/me c2, where Te is electron temperature and me is electron rest mass. Experimental validation of the analytical models has been made by analyzing data of more than 1200 pulses collected from high-Te JET discharges. Based on this validation the relativistic analytical expressions are included in the error analysis and design projects of the ITER TIP and PoPola systems. The polarization properties of incoherent Thomson scattered light are being examined as a method of Te measurement relevant to ITER operational regimes. The theory is based on Stokes vector transformation and Mueller matrices formalism. The general approach is subdivided into frequency-integrated and frequency-resolved cases. For each of them, the exact analytical relativistic solutions are presented in the form of Mueller matrix elements averaged over the relativistic Maxwellian distribution function. New results related to the detailed verification of the frequency-resolved solutions are reported. The precise analytic expressions provide output much more rapidly than relativistic kinetic numerical codes allowing for direct real-time feedback control of ITER device operation.

Monte Carlo method for computing relativistic photon-Maxwellian electron scattering cross sections

Compton and inverse Compton scattering from relativistic Maxwellian electrons both have an important feature, i.e. calculating the radiation transport in high-temperature and full-ionized plasma. Description and evaluation of relativistic photon-Maxwellian electron scattering are numerically complex and computationally time consuming. A Monte Carlo method is proposed to simulate photon scattering with relativistic Maxwellian electron and compute the scattering cross-sections. To compute the total cross-section of a photon of energy hν interacting with electrons at temperature Te in the laboratory coordinate, the calculation steps of Monte Carlo scheme are described as follows. The first step is to sample the velocity of an electron, the directions are isotropically sampled, and the speed is sampled from the relativistic Maxwellian distribution at temperature Te. The second step is to transform the photon energy hν into the photon energy in the coordinate in which the electron is at rest. The third step is to use the exact Klein-Nishina formula to compute the cross-sections. The fourth step is go back to the first step, and cycle this many times. The last step is to summarize all computed cross-sections and averaged them, and the average value is what we need. The operation and corresponding formula for each step are described in this paper. A better method of sampling the speed of a relativistic electron is expected to be found.A Monte Carlo processor is developed to compute the scattering cross-section of a photon of any energy, interacting with electrons at any temperature. To check this method, scatterings of the photons of various energies with electrons with various temperatures are simulated, and the results are compared with those from the numerical integration method. The comparison indicates that the simulated cross-sections are in pretty good agreement with those from the multiple integration method for the cases of electron temperature less than 25 keV. But unfortunately, the difference is obvious for the case of temperature more than 25 keV, and the error increases with temperature increasing. Why so? When the temperature is more than 25 keV, the sampling of electron speed is inaccurate when using the present method, which maybe results in this difference. So, we need to find a more accurate method of sampling relativistic electron speed to solve this problem in the future.

Analytical study of whistler mode waves in presence of parallel DC electric field for relativistic plasma in the magnetosphere of Uranus

In present paper, field aligned whistler mode waves are analyzed, in the presence of DC field in background plasma having relativistic distribution function in the magnetosphere of Uranus. The work has been examined for relativistic Maxwellian and loss-cone distribution function. In both the cases, we have studied the effect of various plasma parameters on the growth rate of waves by using the method of characteristics and discussed using data provided by Voyager 2. Growth rate has increased by increasing the magnitude of electric field, temperature anisotropy, energy density and number density of particles for Maxwellian and loss-cone background. However, when relativistic factor λ=1-v2/c2 increases, growth rate decreases. The significant increase in real frequency of whistler waves can be observed. The results can be used for comparative study of planetary magnetospheres. The derivation can also be adapted to study various other instabilities in magnetosphere of Uranus.

Exact relativistic expressions for polarization of incoherent Thomson scattering

We present a derivation of the degree of polarization for incoherent Thomson scattering (TS) using Mueller matrix formalism. An exact analytic solution is obtained for spectrum-integrated matrix elements. The solution is valid for the full range of incident polarizations, scattering angles, and electron thermal motion from non-relativistic to ultra-relativistic. It is based on a newly developed theoretical model, a finite transit time (FTT) correction to previous theoretical work on TS polarization. The Mueller matrix elements are substantially different from previous calculations without the FTT correction, even to the lowest linear order in Te/mec2≪1. Mathematically, the derivation is a unique example of fully analytical integration of the 3D scattering operator over a relativistic Maxwellian distribution function; experimentally, the results have application to the use of the polarization properties of Thomson scattered light as a method of electron temperature measurement. The results can also be used as a reliable tool for benchmarking and verification of numerical codes for frequency resolved properties of TS polarization.

Formulation of the relativistic heat equation and the relativistic kinetic Fokker–Planck equations using GENERIC

In this paper, we formulate the relativistic heat equation and the relativistic kinetic Fokker–Planck equations into the GENERIC (General Equation for Non-Equilibrium Reversible–Irreversible Coupling) framework. We also show that the relativistic Maxwellian distribution is the stationary solution of the latter. The GENERIC formulation provides an alternative justification that the two equations are meaningful relativistic generalizations of their non-relativistic counterparts.

Anomalous skin effects in a weakly magnetized degenerate electron plasma

Fully relativistic analysis of anomalous skin effects for parallel propagating waves in a weakly magnetized degenerate electron plasma is presented and a graphical comparison is made with the results obtained using relativistic Maxwellian distribution function [G. Abbas, M. F. Bashir, and G. Murtaza, Phys. Plasmas 18, 102115 (2011)]. It is found that the penetration depth for R- and L-waves for degenerate case is qualitatively small in comparison with the Maxwellian plasma case. The quantitative reduction due to weak magnetic field in the skin depth in R-wave for degenerate plasma is large as compared to the non-degenerate one. By ignoring the ambient magnetic field, previous results for degenerate field free case are salvaged [A. F. Alexandrov, A. S. Bogdankevich, and A. A. Rukhadze, Principles of Plasma Electrodynamics (Springer-Verlag, Berlin/Heidelberg, 1984), p. 90].

Spreading of ultrarelativistically expanding shell: An application to GRBs

Optically thick energy dominated plasma created in the source of Gamma-Ray Bursts (GRBs) expands radially with acceleration and forms a shell with constant width measured in the laboratory frame. When strong Lorentz factor gradients are present within the shell it is supposed to spread at sufficiently large radii. There are two possible mechanisms of spreading: hydrodynamical and thermal ones. We consider both mechanisms evaluating the amount of spreading that occurs during expansion up to the moment when the expanding shell becomes transparent for photons. We compute the hydrodynamical spreading of an ultrarelativistically expanding shell. In the case of thermal spreading we compute the velocity spread as a function of two parameters: comoving temperature and bulk Lorentz factor of relativistic Maxwellian distribution. Based on this result we determine the value of thermal spreading of relativistically expanding shell. We found that thermal spreading is negligible for typical GRB parameters. Instead hydrodynamical spreading appears to be significant, with the shell width reaching ∼1010cm for total energy E=1054erg and baryonic loading B=10-2. Within the fireshell model such spreading will result in the duration of Proper Gamma-Ray Bursts up to several seconds.

The generation of random variates from a relativistic Maxwellian distribution

A procedure for generating random variates from a relativistic Maxwellian distribution with arbitrary temperature and drift velocity is presented. The algorithm is based on the rejection method and can be used to initialize particle velocities in kinetic simulations of plasmas and gases.

Characterizing the energy distribution of laser-generated relativistic electrons in cone-wire targets

Transport of relativistic electrons in a solid Cu wire target has been modeled with the implicit hybrid particle-in-cell code LSP to investigate the electron energy distribution and energy coupling from the high-intensity, short-pulse laser to electrons entering to the wire. Experiments were performed on the TITAN laser using a 1.5 mm long Cu wire attached to a Au cone tip at the laser intensity of 1 × 1020 W/cm2 which was irradiated into the cone. The simulated Cu Kα wire profile and yields matched the measurements using a two-temperature energy distribution. These modeling results show that the cold component of the energy spectrum can be determined with ±100 keV accuracy from the fit to the initial experimental fall-off of the Kα emission while the simulated profiles were relatively insensitive to the hotter component of the electron distribution (>4 MeV). The slope of measured escaped electrons was used to determine the hotter temperature. Using exponential energy distributions, the laser-to-electron-in-wire coupling efficiencies inferred from the fits decreased from 3.4% to 1.5% as the prepulse energy increases up to 1 J. The comparison of the energy couplings using the exponential and Relativistic Maxwellian distribution functions showed that the energy inferred in the cold component is independent of the type of the distribution function.

Landau damping of longitudinal oscillation in ultra-relativistic plasmas with nonextensive distribution

The generalized dispersion equation for longitudinal oscillation in an unmagnetized, collisionless, isotropic and relativistic plasma is derived in the context of nonextensive q-distribution. An analytical expression for the Landau damping is obtained in an ultra-relativistic regime, which is related to q-parameter. In the limit q → 1, the result based on the relativistic Maxwellian distribution is recovered. It is shown that the interactions between the wave and particles are stronger and the waves are more strongly damped for lower values of q-parameter. The results are explained by the increased number of superthermal particles or low velocity particles contained in the plasma with the nonextensive distribution.

Entropy mapping of the outer electron radiation belt between the magnetotail and geosynchronous orbit

[1] The specific entropy (entropy density) S is examined for the outer electron radiation belt at geosynchronous orbit and for the energetic electron population in the Earth's magnetotail. The outer electron radiation belt is measured with the SOPA detectors on board six geosynchronous satellites and the energetic electrons of the magnetotail are measured with instrumentation on board 12 Global Positioning Satellites (GPS) with a magnetic field model used to map the GPS orbit to the magnetotail. Density n and temperature T values are determined from relativistic Maxwellian fits to the electron measurements, enabling the specific entropy S to be calculated. For low temperatures the nonrelativstic specific entropy is S = T/n2/3; for a relativistic Maxwellian distribution a relativistically correct expression for S = S(T,n) is derived and used. The outer electron radiation belt at geosynchronous orbit local midnight (n ∼ 3 × 10−4 cm−3 and T ∼ 140 keV) and the energetic-electron population in the magnetotail (n ∼ 1 × 10−4 cm−3 and T ∼ 50 keV) statistically have the same specific entropy. Hence the two populations are probably the same. This implies adiabatic transport (1) from the magnetotail to the dipole (where the magnetotail electrons are the source of the outer electron radiation belt) or (2) from the dipole to the magnetotail (where the magnetotail electrons are leakage from the radiation belt).

A numerical code for the calculation of relativistic electron cyclotron damping with an arbitrary distribution function at an arbitrary harmonic

The relativistic expressions for the anti-Hermitian parts of the relativistic dielectric tensor elements can be expressed as a single integral over the parallel momentum variable, allowing an arbitrary electron distribution function. A computer program has been written for the calculation of this single integral. The numerical results are tested for a relativistic Maxwellian distribution function and agree with analytical expressions for this case. The numerical code is therefore an essential element in a more general validation, evaluation and demonstration of powerful analytical results presented. The computer program is then applied to the calculation of relativistic electron cyclotron harmonic damping at any arbitrary harmonic, for any distorted electron distribution function, distorted for example by an electric field or by RF power sources, as for instance by both electron cyclotron and lower-hybrid waves, as calculated from a relativistic Fokker–Planck code. For generality, we also include the case of relativistic Landau damping, also used to check the code. Program summaryProgram title: DAMPINGCatalogue identifier: AEIS_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIS_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 84 108No. of bytes in distributed program, including test data, etc.: 2 581 423Distribution format: tar.gzProgramming language: FORTRANComputer: Any computer or work stationOperating system: Any with a Fortran compilerRAM: 8 Mbytes, using double precision arithmetic, with 400×200 grid points, on an ALPHA serverClassification: 19.8External routines: Requires a link to the NAG libraries for the Bessel functions subroutines.Nature of problem: The computer program calculates generalized expressions for the anti-Hermitian part of the relativistic dielectric tensor in a plasma for an arbitrary distribution function, with comparison to the analytical expressions derived in [1], associated with a relativistic Maxwellian distribution, for any n-th cyclotron harmonic and for the relativistic Landau damping case when n=0. The anti-Hermitian parts of the dielectric tensor are important to know, since they determine the damping or emission of an electromagnetic wave within the plasma medium. New analytical expressions are derived for the relativistic Landau damping and for the electron cyclotron damping including the case n||=1, for the case of an arbitrary distribution function. This is important since studies of the RF heating of tokamak plasmas in a fully self-consistent way involve the simultaneous description of the temporal evolution of the heated species velocity distribution function using a Fokker–Planck code, coupled to the damping of the wave on the evolving distribution function. We can evaluate numerically the degree by which a non-Maxwellian nature of the relativistic distribution function changes the Landau damping and the electron cyclotron damping, with respect to the Maxwellian case. In addition this is also important in the study of electron cyclotron emission, which can be a sensitive indicator of non-thermal electron distributions.Solution method: The relativistic expressions for the anti-Hermitian parts of the dielectric tensor elements can be expressed as a single integral over the parallel momentum variable [1]. This single integral is calculated along the resonance curve, for an arbitrary distribution function calculated from a relativistic Fokker–Planck code. Since Fokker–Planck codes usually evolve the distribution functions using a spherical coordinate system in momentum space, the present code has to first transform the distribution function calculated by the Fokker–Planck code from a spherical coordinate system to a cylindrical coordinate system in momentum space, where the single integral over the parallel momentum variable can be calculated along the resonance curves. In the evaluation of the term in Eq. (8), cubic spline are used for the calculation of the derivatives.Running time: An execution on an ALPHA server 4000 from Digital requires less than 10 s CPU time on a single processor for 400×200 grid points.

Monte Carlo approach to Thomson scattering in relativistic fusion plasmas with allowance for ultraintense laser radiation

Thomson scattering (TS) of electromagnetic radiation by relativistic electrons is of great interest as an electron distribution diagnostics in high temperature plasma devices. A Monte Carlo approach to TS simulation has been implemented. It is based on first principles, i.e. on the integration of electron trajectories and on the computation and averaging of radiated Liénard–Wiechert fields. The treatment is relativistic. Nonlinear effects, arising from ultraintense lasers (both linear and circular polarizations having been considered) and/or from highly energetic electrons, are included from the outset. TS spectra from isotropic monoenergetic electron distribution functions and from the relativistic Maxwellian distribution have been extensively studied, both for low-intensity and for ultraintense incoming laser. A particularly simple quadratic dependence of S2(ω) on ω has been found in the case of monoenergetic distributions. Numerical results are checked against known analytical results for low-intensity lasers, and found to be in agreement with them. For high-intensity lasers, new effects on scattered spectra arise, like an overall red-shift or the appearance of higher harmonics. The numerical methods developed in this work are general enough to deal with anisotropic and/or non-Maxwellian distribution functions.

Dispersion relation of transverse oscillation in relativistic plasmas with non-extensive distribution

AbstractThe generalized dispersion equation for superluminal transverse oscillation in an unmagnetized, collisionless, isotropic and relativistic plasma with non-extensiveq-distribution is derived. The analytical dispersion relation is obtained in an ultra-relativistic regime, which is related toq-parameter and temperature. In the limitq→ 1, the result based on the relativistic Maxwellian distribution is recovered. Using the numerical method, we obtain the full dispersion curve that cannot be given by an analytic method. It is shown that the numerical solution is in good agreement with the analytical result in the long-wavelength and short-wavelength region for ultra-relativistic plasmas.

Dispersion relation of longitudinal oscillation in relativistic plasmas with nonextensive distribution

The generalized dispersion equation for longitudinal oscillation in an unmagnetized, collisionless, isotropic and relativistic plasma is derived in the context of nonextensive q-distribution. The analytical expression for the dispersion relation is obtained in an ultra-relativistic regime, which is related to the q-parameter and temperature. In the limit q→1, the result based on the relativistic Maxwellian distribution is recovered. Using the numerical simulation method, we obtain the full dispersion curve which cannot be given by an analytic method. It is shown that the numerical solution is in good agreement with the analytical result in the long-wavelength, short-wavelength and near light wave region for ultra-relativistic plasmas.

High frequency electromagnetic modes in a weakly magnetized relativistic electron plasma

Using the linearized Vlasov-Maxwell model, the polarization tensor for a weakly magnetized electron plasma is derived. For isotropic relativistic Maxwellian velocity distribution function, dispersion relations are obtained for both parallel and perpendicular propagations. The integrals (called Meijer G functions) that arise due to relativistic effects are examined in various limits and dispersion relations are derived for the nonrelativistic, weakly, strongly, and ultrarelativistic Maxwellian velocity distributions. It is generally observed that the propagation domains of the modes are enlarged as one proceeds from the nonrelativistic to the highly relativistic regime. Resultantly, due to the relativistic effects, the Whistler mode is suppressed in the R-wave, the nonpropagation band of X-mode is reduced, and the X-mode itself approaches the O-mode. Further, the results derived in the ultra- and nonrelativistic limits found to be in agreement with the earlier calculations [G. Abbas et al. Phys. Scr. 76, 649 (2007); F. F. Chen, Introduction to Plasma Physics and Controlled Fusion (Plenum, New York, 1984), Vol. 1].

SIMULTANEOUS MULTI-WAVELENGTH OBSERVATIONS OF Sgr A* DURING 2007 APRIL 1-11

We report the detection of variable emission from Sgr A* in almost all wavelength bands (i.e., centimeter, millimeter, submillimeter, near-IR, and X-rays) during a multi-wavelength observing campaign. Three new moderate flares are detected simultaneously in both near-IR and X-ray bands. The ratio of X-ray to near-IR flux in the flares is consistent with inverse Compton scattering of near-IR photons by submillimeter emitting relativistic particles which follow scaling relations obtained from size measurements of Sgr A*. We also find that the flare statistics in near-IR wavelengths is consistent with the probability of flare emission being inversely proportional to the flux. At millimeter wavelengths, the presence of flare emission at 43 GHz (7 mm) using the Very Long Baseline Array with milliarcsecond spatial resolution indicates the first direct evidence that hourly timescale flares are localized within the inner 30 × 70 Schwarzschild radii of Sgr A*. We also show several cross-correlation plots between near-IR, millimeter, and submillimeter light curves that collectively demonstrate the presence of time delays between the peaks of emission up to 5 hr. The evidence for time delays at millimeter and submillimeter wavelengths are consistent with the source of emission initially being optically thick followed by a transition to an optically thin regime. In particular, there is an intriguing correlation between the optically thin near-IR and X-ray flare and optically thick radio flare at 43 GHz that occurred on 2007 April 4. This would be the first evidence of a radio flare emission at 43 GHz delayed with respect to the near-IR and X-ray flare emission. The time delay measurements support the expansion of hot self-absorbed synchrotron plasma blob and weaken the hot spot model of flare emission. In addition, a simultaneous fit to 43 and 84 GHz light curves, using an adiabatic expansion model of hot plasma, appears to support a power law rather than a relativistic Maxwellian distribution of particles.

Perpendicularly propagating electromagnetic modes in a strongly magnetized hot relativistic Maxwellian plasma

Using kinetic theory, new dispersion relations based on relativistic Maxwellian velocity distribution are obtained for ordinary and extraordinary modes in a strongly magnetized hot plasma. It is observed that, unlike the non-relativistic Maxwellian distribution case where only real dispersion relation appears for the relativistic case both the real dispersion relations and the damping decrements occur. To illustrate the results graphically, we have plotted the phase velocity squared and the growth rate versus frequency for different values of n. For O-mode, the non-propagation region disappears as we increase the values of the harmonic number n while for the X-mode, the propagation region occurs only for the large harmonics. As regards damping, for O-mode the damping rate is small for small n and increases with n. For X-mode, on the contrary, the damping rate decreases as n increases.

Growth rates of the Weibel and tearing mode instabilities in a relativistic pair plasma

We present an algorithm for solving the linear dispersion relation in an inhomogeneous, magnetised, relativistic plasma. The method is a generalisation of a previously reported algorithm that was limited to the homogeneous case. The extension involves projecting the spatial dependence of the perturbations onto a set of basis functions that satisfy the boundary conditions (spectral Galerkin method). To test this algorithm in the homogeneous case, we derive an analytical expression for the growth rate of the Weibel instability for a relativistic Maxwellian distribution and compare it with the numerical results. In the inhomogeneous case, we present solutions of the dispersion relation for the relativistic tearing mode, making no assumption about the thickness of the current sheet, and check the numerical method against the analytical expression.