One-dimensional kinematics of particle stream flow with application to solar wind simulation

A simple kinematic method for determining the particle velocity distribution of a model solar wind for which the spatial distribution of particles is given as a function of particle travel time has been developed by Hakamada and Akasofu (1982). Here we formalize their method mathematically and derive an inverse procedure for determining the particle distribution from a given velocity distribution. This inverse procedure is then applied to a simulated velocity distribution obtained from an MHD finite difference code.     

Variations of cosmic-ray energy in interplanetary space

A consistent theory of energy exchange between high-energy charged cosmic-ray particles and the random inhomogeneities of a magnetic field frozen in the moving solar wind plasma is developed. It is shown that the mode of the particle energy variations at a given law of plasma velocity variation in space is determined by the specific form of the particle distribution function. The equation for the density of cosmic-ray energy is obtained. Consideration is given to the generation of a charged particle energy spectrum in the course of multiple scatterings by the random inhomogeneities of the magnetic field.     

Impact and explosion crater ejecta,fragment size,and velocity

A model was developed for the mass distribution of fragments that are ejected at a given velocity for impact and explosion craters. The model is semiempirical in nature and is derived from (1) numerical calculations of cratering and the resultant mass versus ejection velocity, (2) observed ejecta blanket particle size distributions, (3) an empirical relationships between maximum ejecta fragment size and crater diameter, (4) measurements of maximum ejecta size versus ejecta velocity, and (5) an assumption on the functional form for the distribution of fragments ejected at a given velocity. This model implies that for planetary impacts into competent rock, the distribution of fragments ejected at a given velocity is broad; e.g., 68% of the mass of the ejecta at a given velocity contains fragments having a mass less than 0.1 times a mass of the largest fragment moving at that velocity. Using this model, we have calculated the largest fragment that can be ejected from asteroids, the Moon, Mars, and Earth as a function of crater diameter. The model is unfortunately dependent on the size-dependent ejection velocity limit for which only limited data are presently available from photography of high explosive-induced rock ejecta. Upon formation of a 50-km-diameter crater on an atmosphereless planet having the planetary gravity and radius of the Moon, Mars, and Earth, fragments having a maximum mean diameter of ≈30, 22, and 17 m could be launched to escape velocity in the ejecta cloud. In addition, we have calculated the internal energy of ejecta versus ejecta velocity. The internal energy of fragments having velocities exceeding the escape velocity of the moon (～2.4 km/sec) will exceed the energy required for incipient melting for solid silicates and thus, the fragments ejected from Mars and the Earth would be melted.     

Observation of Hamiltonian chaos in wave–particle interaction

The motion of charged particle in longitudinal waves is a paradigm for the transition to large scale chaos in Hamiltonian systems. Recently a test cold electron beam has been used to observe its non-self-consistent interaction with externally excited wave(s) in a specially designed Traveling Wave Tube (TWT). The velocity distribution function of the electron beam is recorded with a trochoidal energy analyzer at the output of the TWT. An arbitrary waveform generator is used to launch a prescribed spectrum of waves along the slow wave structure (a 4 m long helix) of the TWT. The resonant velocity domain associated to a single wave is observed, as well as the transition to large scale chaos when the resonant domains of two waves and their secondary resonances overlap. This transition exhibits a “devil’s staircase” behavior when increasing the excitation amplitude in agreement with numerical simulation. A new strategy for control of chaos by building barriers of transport which prevent electrons to escape from a given velocity region as well as its robustness are also successfully tested. Thus generic features of Hamiltonian chaos have been experimentally observed.     

日珥谱线不对称性的研究 Ⅰ.视向速度对谱线轮廓的影响

本文主要从理论上用解析方法讨论日珥视向速度随深度变化对谱线轮廓对称性的影响,得到的结论具有普遍性。第二节的分析表明,日珥的谱线轮廓可表示为二项叠加,其中第一项与源函数无关,只依赖于速度场模型,源函数分布仅通过第二项对谱线轮廓产生影响。然后在源函数不随深度变化的假定下,讨论各种速度场模型的谱线轮廓是否对称。得到的结论为:(1)常源函数与常速度场结合的谱线轮廓为对称轮廓;(2)常源函数与线性对称速度场结合的谱线也是对称轮廓;(3)常源函数与线性非对称速度场结合的谱线轮廓为不对称轮廓。最后,用数值计算对理论分析结果进行了检验。     

Velocity dispersion in particulate disks with size distribution

Quasi-equilibrium solutions for the pre-planetary disk are studied in terms of Hämeen-Anttila's theory (1984) of collisional, self-gravitating systems. The distribution of particle sizes is assumed to follow simple power-law distributions, with a power index in the range of 1.5–5.0. The treatment includes mutual impacts with a velocity dependent coefficient of restitution, as well as gravitational encounters with dynamical friction. The mean gravitational field of the disk is also taken into account. The results indicate that the energy(equi)-partition depends mainly on the index of size distribution, but is also affected by the optical thickness of the system, as well as on the vertical thickness as compared to the particle size. The vertical component of the gravitational field is found to be important, especially when the mass of the system is concentrated on the large particles.     

The effect of heavy ions on the formation and structure of cometary bow shocks

We investigate the interaction of heavy cometary ions with the solar wind and the formation of a bow shock in front of a comet by means of a hybrid (particle ion, fluid electron) simulation code that solves self-consistently for the electromagnetic fields and the motion of the charged particles. This kinetic treatment of the solar wind protons and the heavy cometary ions allows us to examine two important issues. One is the effect of the velocity distribution function of the heavy ions on the shock formation and structure, and the other is the degree of coupling between the two ion species. The result of this study indicate that at high Mach numbers the shock structure is highly dependent upon the velocity distribution of the heavy ions. For example, when the newly created ions comprise a ring distribution in the solar wind frame, most of them turn around downstream of the shock surface and reenter the upstream region to form a large foot that extends about a heavy ions gyroradius upstream of the shock. On the other hand, heavy ions which have been picked up by the solar wind and possesses a Maxwellian distribution can mostly penetrate the shock without returning upstream and affecting the shock structure as much. In either case, however, at high Mach numbers the shock strength is the same. At low Mach numbers, where the shock is weak, the velocity distribution of heavy ions has a smaller effect on the formation of shock and its structure. In this regime, the degree of coupling between the cometary ions and the solar wind protons and the corresponding critical Mach number (at which a shock should begin to form) are determined from a set of Rankine-Hugoniot relations. The results of the simulations suggest that some coupling does occur (evidently, through the electromagnetic fields, since there are no particle collisions in the calculations), but less than that expected from magnetohydrodynamics. For low Mach numbers, it is also shown that shocks have a transitory nature, where they are continuously formed by the protons and subsequently destroyed by the heavy ions.     

Beam propagation and Langmuir wave generation in a plasma with κ distribution function

Using a kappa velocity distribution function for the electrons of the background plasma, the dynamics of a beam of hot electrons streaming through the plasma and the generation of Langmuir waves are investigated in the frame work of quasilinear theory. It is shown that the Langmuir waves are strongly damped by high energy tail of the Kappa distribution function. The spatial expansion of the beam is reduced and the spectral density of Langmuir waves becomes narrower. The height of the plateau in the beam distribution function increases at small velocities and the average velocity of beam is larger than that of a Maxwellian distribution. The influence of Kappa velocity distribution function on the gasdynamical parameters is investigated. It is found that, the height of plateau in the beam distribution function, and its lower velocity boundary are enhanced while, the local beam width in velocity space decreases.     

Jump relations for magnetohydrodynamic shock waves in non-ideal gas flow

The generalized jump relations across the magnetohydrodynamic (MHD) shock front in non-ideal gas are derived considering the equation of state for non-ideal gas as given by Landau and Lifshitz. The jump relations for pressure, density, and particle velocity have been derived, respectively in terms of a compression ratio. Further, the simplified forms of the MHD shock jump relations have been obtained in terms of non-idealness parameter, simultaneously for the two cases viz., (i) when the shock is weak and, (ii) when it is strong. Finally, the cases of strong and weak shocks are explored under two distinct conditions viz., (i) when the applied magnetic field is strong and, (ii) when the field is weak. The aim of this paper is to contribute to the understanding of how shock waves behave in magnetized environment of non-ideal gases.     

A note on adiabatic solutions of the one-dimensional current sheet problem

It has recently been shown that adiabatic solutions of the one-dimensional current sheet problem exist provided that magnetically trapped particles are included in the model together with the current-carrying untrapped “beam” particles. We show here that a formulation of the problem in terms of particle velocity and pitch angle is advantageous, and we derive some general properties of the solutions. In particular it is shown that there is, in general, no discontinuity in the value of the particle distribution function ? across the boundary in velocity space between “beam” and trapped particles, but that there will be a discontinuity in the gradients of ?. An example is given in which the beam population is of bi-Maxwellian form at the outer boundary of the current sheet.     

Steady state charge neutral models of the magnetopause

Plane models of the magnetopause are investigated under the assumption that ionospheric electrons are able to short-circuit electric fields (exact charge neutrality). Using the Vlasov theory a general method is presented for constructing distribution functions that lead to given magnetic field and tangential bulk velocity profiles. As an example we describe the magnetic field transition in terms of error functions and obtain particle distributions in explicit form, including bulk velocities.It is thus shown that bulk velocities in the direction of the magnetic field do not necessarily lead to a non-equilibrium magnetopause which investigations by Parker and Lerche seem to suggest.Of the European Space Research Organisation (ESRO).     

Impact of plasma sheath on rocket-based E-region ion measurements

We model the particle velocity distribution functions around the entrance window of the Suprathermal Ion Imager (SII). The SII sensor was mounted on a 1 m boom carried by the scientific payload of NASA rocket 36.234 as part of Joule II mission to investigate Joule heating in the E-region ionosphere. The rocket flew above Northern Alaska on 19 January 2007. The payload was spin-stabilized with a period of 1.6 s, giving an apparent rotation of the ion flow velocity in the frame of reference of the payload. The SII sensor is an electrostatic analyzer that measures two dimensional slices of the distribution of the kinetic energies and arrival-angles of low energy ions. The study is concerned with the interpretation of data obtained from the SII sensor. For this purpose, we numerically investigate ram velocity effects on ions velocity distributions in the vicinity of the SII sensor aperture at an altitudes of approximately 150 km. The electrostatic sheath profiles surrounding the SII sensor, boom and payload are calculated numerically with the PIC code PTetra. It is observed that the direction of the ion flow velocity modifies the plasma sheath potential profile. This in turn impacts the velocity distributions of NO⁺ and \(\mathrm{O}_{2}^{+}\) ions at the aperture of the particle sensor. The velocity distribution functions at the sensor aperture are calculated by using test-particle modeling. These particle distribution functions are then used to inject particles in the sensor, and calculate the fluxes on the sensor microchannel plate (MCP), from which comparisons with the measurements can be made.     

A new distribution function for relativistic counterstreaming plasmas

The particle distribution function that describes two interpenetrating plasma streams is re-investigated. It is shown how, based on the Maxwell–Boltzmann–Jüttner distribution function, which has been derived almost a century ago, a counterstreaming distribution function can be derived that uses velocity space. Such is necessary for various analytical calculations and numerical simulations that are reliant on velocity coordinates rather than momentum space. The application to the electrostatic two-stream instability illustrates the differences caused by the use of the relativistic distribution function.     

Effect of general loss-cone distribution function on electrostatic ion-cyclotron instability in an inhomogeneous plasma: particle aspect analysis

The electrostatic ion-cyclotron instability (EICI) in low β (ratio of plasma to magnetic pressure), anisotropic, inhomogeneous plasma is studied by investigating the trajectories of the particles using the general loss-cone distribution function (Dory-Guest-Harris type) for the plasma ions. In particular, the role of the loss-cone feature as determined by the loss-cone indices, in driving the drift-cyclotron loss-cone (DCLC) instability is analysed. It is found that for both long and short wavelength DCLC mode the loss-cone indices and the perpendicular thermal velocity affect the dispersion equation and the growth rate of the wave by virtue of their occurrence in the temperature anisotropy. The dispersion relation for the DCLC mode derived here using the particle aspect analysis approach and the general loss-cone distribution function considers the ion diamagnetic drift and also includes the effects of the parallel propagation and the ion temperature anisotropy. It is also found that the diamagnetic drift velocity due to the density gradient of the plasma ions in the presence of the general loss-cone distribution acts as a source of free energy for the wave and leads to the generation of the DCLC instability with enhanced growth rate. The particle aspect analysis approach used to study the EICI in inhomogeneous plasma gives a fairly good explanation for the particle energisation, wave emission by the wave–particle interaction and the results obtained using this particle aspect analysis approach are in agreement with the previous theoretical findings using the kinetic approach.     

Cosmic energy equation and clustering of non-point mass system of galaxies

Cosmic energy equation is an important equation for studying the gravitational galaxy clustering in the expanding universe. We derive the distribution function for fluctuations in particle number by using the cosmic energy equation for extended structures (galaxies with halos). From spatial distribution function, containing particle fluctuations, we derive the velocity distribution function to understand the influence of particle fluctuations on the velocities of galaxies.With the help of cosmic energy equation we try to find out the physical constraints for the application of quasi-equilibrium approximation.     

Auroral ion velocity distributions for a polarization collision model

We have calculated the effect that convection electric fields have on the velocity distribution of auroral ions at the altitudes where the plasma is weakly-ionized and where the various ion-neutral collision frequencies are much smaller than the ion cyclotron frequencies, i.e. between about 130 and 300 km. The appropriate Boltzmann equation has been solved by expanding the ion velocity distribution function in a generalized orthogonal polynomial series about a bi-Maxwellian weight factor. We have retained enough terms in the series expansion to enable us to obtain reliable quantitative results for electric field strengths as large as 90 mV m^?1. Although we have considered a range of ion-neutral scattering mechanisms, our main emphasis has been devoted to the long-range polarization interaction. In general, we have found that to lowest order the ion velocity distribution is better represented by a two-temperature or bi-Maxwellian distribution than by a one-temperature Maxwellian, with there being different ion temperatures parallel and perpendicular to the geomagnetic field. However, the departures from this zeroth-order bi-Maxwellian distribution become significant when the ion drift velocity approaches (or exceeds) the neutral thermal speed.     

Jump relations across a shock in non-ideal gas flow

Generalized forms of jump relations are obtained for one dimensional shock waves propagating in a non-ideal gas which reduce to Rankine-Hugoniot conditions for shocks in idea gas when non-idealness parameter becomes zero. The equation of state for non-ideal gas is considered as given by Landau and Lifshitz. The jump relations for pressure, density, temperature, particle velocity, and change in entropy across the shock are derived in terms of upstream Mach number. Finally, the useful forms of the shock jump relations for weak and strong shocks, respectively, are obtained in terms of the non-idealness parameter. It is observed that the shock waves may arise in flow of real fluids where upstream Mach number is less than unity.     

Cosmological perturbation theory and the spherical collapse model — III. The velocity divergence field and the Ω dependence

Cosmological perturbation theory (PT) is a useful tool to study the cumulants of the density and velocity fields in the large-scale structure of the Universe. In Papers I and II of this series we saw that the spherical collapse (SC) model provides the exact solution to PT at tree-level and gives a good approximation to the loop corrections (next-to-leading orders), indicating negligible tidal effects. Here, we derive predictions for the (smoothed) cumulants of the velocity divergence field θ ≡ ▽ ⊙  v for an irrotational fluid in the SC model. By comparing these with the exact analytic results of Scoccimarro &38; Frieman, it is shown that, at least for the unsmoothed case, the loop corrections to the cumulants of θ are dominated by tidal effects. However, most of the tidal contribution seems to cancel out when computing the hierarchical ratios, T _J  = 〈θ ^J 〉 / 〈θ²〉  ^{J −1}. We also extend the work presented in Papers I and II to give predictions for the cumulants of the density and velocity divergence fields in non-flat spaces. In particular, we show the equivalence between the spherically symmetric solution to the equations of motion in the SC model (given in terms of the density) and that of the Lagrangian PT approach (given in terms of the displacement field). It is shown that the Ω dependence is very weak for both cosmic fields even at one loop (a 10 per cent effect at most), except for the overall factor f (Ω) that couples to the velocity divergence.     

Ejection behavior characteristics in experimental cratering in sandstone targets

Abstract– Within the frame of the MEMIN research unit (Multidisciplinary Experimental and Numerical Impact Research Network), impact experiments on sandstone targets were carried out to systematically study the influence of projectile mass, velocity, and target water saturation on the cratering and ejection processes. The projectiles were accelerated with two‐stage light‐gas guns (Ernst‐Mach‐Institute) onto fine‐grained targets (Seeberger sandstone) with about 23% porosity. Collection of the ejecta on custom‐designed catchers allowed determination of particle shape, size distribution, ejection angle, and microstructures. Mapping of the ejecta imprints on the catcher surface enabled linking of the different patterns to ejection stages observed on high‐speed videos. The increase in projectile mass from 0.067 to 7.1 g correlates with an increase in the total ejected mass; ejecta angles, however, are similar in range for all experiments. The increase in projectile velocity from 2.5 to 5.1 km s^?1 correlates with a total ejecta mass increase as well as in an increase in comminution efficiency, and a widening of the ejecta cone. A higher degree of water saturation of the target yields an increase in total ejecta mass up to 400% with respect to dry targets, higher ejecta velocity, and a steeper cone. These data, in turn, suggest that the reduced impedance contrast between the quartz grains of the target and the pores plays a primary role in the ejecta mass increase, while vaporization of water determines the ejecta behavior concerning ejecta velocity and particle distribution.     

Analyzing the Solar Proton Event of 22 October 1989, Using the Method of Spectrographic Global Survey

Using ground-level observations of cosmic-ray (CR) intensities from a worldwide network of stations during the ground-level enhancement (GLE) of 22–23 October 1989, variations of the particle distribution function in all phases of the event were investigated.It is shown that time intensity profiles of 2–4 GV rigidity particles differ greatly from those of higher-energy particles. After the high-energy particle intensity attains a maximum, there is an abrupt decrease in intensity below the background level, followed by the phase of slow recovery to background values. The angular distribution of high-energy particles across the celestial sphere shows, along with an increased intensity, also regions with decreased relative background intensity.To explain the detected phenomenon, it is concluded that it is necessary to consider the solar CR propagation process in the heliosphere not in terms of the movement of particles in the external electromagnetic field but with proper account of the self-consistency of fields with the particle distribution function.