Arbitrary Pitch Angle Research Articles

Analytical forms of the first 14 moments of the cosmic ray Fokker–Planck equation

The Fokker–Planck transport equation describing the motion of energetic particles through a plasma is explored analytically. The latter equation provides a pitch-angle and position-dependent distribution function of the charged particles. In the current paper the first 14 moments of this equation are computed exactly for an arbitrary initial pitch angle. Such analytical forms are required in nonlinear treatments of perpendicular transport and other scenarios in plasma physics and astrophysics.

Simple analytical expressions for electron pitch angle diffusion coefficients

In the present paper, we have calculated electron pitch angle diffusion coefficients due to resonant interactions with whistler mode lower band chorus (LBC), upper band chorus (UBC), and electrostatic electron cyclotron harmonic (ECH) waves. Calculations have been performed at two values of the ratio of electron plasma frequency to gyro-frequency and thirteen representative values of electron energies for the plasma sheet electrons. The numerical data of diffusion coefficients have been fitted to simple analytical expressions for each wave mode. These analytical expressions allow for simple evaluation of pitch angle diffusion coefficients for the arbitrary pitch angle, energy, the ambient magnetic field, the wave amplitude, and the ratio of plasma frequency to gyro-frequency. In the case of LBC waves, the analytical coefficients are generally within a factor of two to the numerical coefficients, except at higher pitch angles where the numerical coefficients drop to show negligible values. Likewise, also for UBC waves, the analytical coefficients are generally within a factor of two to the numerical coefficients. The analytical coefficients for ECH waves are generally in agreement with numerical coefficients. However, the analytical expressions developed for ECH waves do not reproduce sharp fluctuations, dips, and gaps, the so-called banded structure observed in the data of numerical coefficients [A. K. Tripathi et al., Phys. Plasmas 23, 042101 (2016)] The effect of the shape variation on the profile of the wave spectral intensity and the effect of cold temperature on the numerical coefficients are also investigated. Applications of the analytical expressions of diffusion coefficients are discussed.

SYNCHROTRON POLARIZATION AND SYNCHROTRON SELF-ABSORPTION SPECTRA FOR A POWER-LAW PARTICLE DISTRIBUTION WITH FINITE ENERGY RANGE

We have derived asymptotic forms for the degree of polarization of the optically thin synchrotron and for synchrotron self-absorption (SSA) spectra assuming a power-law particle distribution of the form N(γ) ~ γ–p with γ1 < γ < γ2, especially for a finite high-energy limit, γ2, in the case of an arbitrary pitch angle. The new results inferred concern more especially the high-frequency range x η2 with parameter η = γ2/γ1. The calculated SSA spectra concern instantaneous photon emission where cooling effects are not considered. They have been obtained by also ignoring likely effects such as Comptonization, pair creation and annihilation, as well as magnetic photon splitting. To that aim, in addition to the two usual absorption frequencies, a third possible one has been derived and expressed in terms of the Lambert W function based on the analytical asymptotic form of the absorption coefficient, αν, for the high-frequency range ν ν2 (with ν2 the synchrotron frequency corresponding to γ2). We have shown that the latter frequency may not have realistic applications in astrophysics, except in the case of an adequate set of parameters allowing one to neglect Comptonization effects. More detailed calculations and discussions are presented.

Numerical estimates of drift loss and Dst effect for outer radiation belt relativistic electrons with arbitrary pitch angle

Satellite observations often show that relativistic electron fluxes that decrease during a geomagnetic storm main phase do not recover their prestorm level even when the storm has substantially recovered. A possible explanation for such sustained flux dropout is that the electrons that move to larger shells (L shells) aided by the disturbance storm time (Dst) effect associated with the main phase geomagnetic field depression may be suffering drift loss to the magnetopause, resulting in irreversible (nonadiabatic) flux decreases during a geomagnetic storm. In this study, we have numerically evaluated the drift loss effect by combining it with the Dst effect and including off‐equatorially mirroring electrons for three different storm conditions obtained by averaging 95 geomagnetic storms that occurred from 1997 to 2002. Using the Tsyganenko T02 model and our own simplified method, we estimated the storm time flux changes based on the guiding center orbit dynamics. Assuming that there is no competing source mechanism taking place at the same time, our calculations of the electron fluxes at equatorial midnight suggest that the drift loss when combined with the Dst effect can be responsible for flux dropouts, which can be seen even inside the geosynchronous orbit during storm periods. Specifically, by evaluating omnidirectional flux values at three specific times that correspond to the storm onset time, the time of minimum Dst value, and the end of the Dst recovery, we have obtained the following numerical results. First, for the strong storm with −150 nT &lt; Dstmin ≤ −100 nT, the combined drift loss and Dst effect can cause a complete dropout of the electron flux for r ≥ ∼5 RE at the end of the storm recovery. A nearly full recovery of the particle flux by the adiabatic Dst effect is seen only for r &lt; ∼5 RE. For the moderate storm with −100 nT &lt; Dstmin ≤ −50 nT, the overall flux decrease level at the end of the storm recovery is less significant compared to that of the strong storm. However, the combined loss effect can still penetrate into r ∼ 5 RE, leading to some partial dropout of the flux. For the severe storm with Dstmin ≤ −150 nT, the flux dropout is far more significant than for the other two storms, indicating that the combined drift loss and Dst effect can even reduce the flux level at an inner region of r ∼ 4 RE. But in this case, the solar wind dynamic pressure is so high that the dayside magnetopause can cross the geosynchronous orbit. Consequently, the flux dropouts seen in actual observations can be primarily attributed to a fast and direct loss to the magnetopause at times when the magnetopause crosses the geosynchronous orbit. It is possible that our numerical results may quantitatively change to some extent with different magnetospheric models and assumptions and may also change depending on the validity of the fully adiabatic invariants assumption.

On Particle Acceleration around Shocks. I.

We derive a relativistically covariant (although not manifestly so) equation for the distribution function of particles accelerated at shocks, which applies also to extremely relativistic shocks, and arbitrarily anisotropic particle distributions. The theory is formulated for arbitrary pitch angle scattering, and reduces to the well--known case for small angle scatterings via a Fokker--Planck approximation. The boundary conditions for the problem are completely reformulated introducing a physically motivated Green's function; the new formulation allows derivation of the particle spectrum both close and far away from the injection energy in an exact way, while it can be shown to reduce to a power--law at large particle energies. The particle spectral index is also recovered in a novel way. Contact is made with the Newtonian treatment.

Comprehensive computational model of Earth's ring current

A comprehensive ring current model (CRCM) has been developed that couples the Rice Convection Model (RCM) and the kinetic model of Fok and coworkers. The coupled model is able to simulate, for the first time using a self‐consistently calculated electric field, the evolution of an inner magnetosphere plasma distribution that conserves the first two adiabatic invariants. The traditional RCM calculates the ionospheric electric fields and currents consistent with a magnetospheric ion distribution that is assumed to be isotropic in pitch angle. The Fok model calculates the plasma distribution by solving the Boltzmann equation with specified electric and magnetic fields. To combine the RCM and the Fok model, the RCM Birkeland current algorithm has been generalized to arbitrary pitch angle distributions. Given a specification of height‐integrated ionospheric conductance, the RCM component of the CRCM computes the ionospheric electric field and currents. The Fok model then advances the ring current plasma distribution using the electric field computed by the RCM and at the same time calculates losses along particle drift paths. We present the logic of CRCM and the first validation results following the H+ distribution during the previously studied magnetic storm of May 2, 1986. The H+ fluxes calculated by the coupled model agree very well with observations by AMPTE/CCE. In particular, the coupled model is able to reproduce the high H+ flux seen on the dayside at L ∼ 2.3 that the previous simulation, which employed a Stern‐Volland convection model with shielding factor 2, failed to produce. Though the Stern‐Volland and CRCM electric fields differ in several respects, the most notable difference is that the CRCM predicts strong electric fields near Earth in the storm main phase, particularly in the dusk‐midnight quadrant. Thus the CRCM injects particles more deeply and more quickly.

Optical configuration and color-representation range of a variable-pitch dot matrix holographic printer

The optical system configuration and design of a dot matrix holographic printer that can create image grating pixels of variable size, arbitrary pitch, and discretionary angle on a photoresist plate are presented. With the capability to vary spot size, grating orientation, and grating pitch on the fly, this newly developed holographic printer can apply a prespecified color at each specific viewing angle. Diffractive images with various visual effects and the wide color range that are possible by use of this system are examined in detail.

Synchro-curvature radiation—a new and more general radiation in curved magnetic field

We obtain a universal radiation formula for a relativistic charged particle moving in curved magnetic field lines with arbitrary pitch angle. It reduces either to the formula for synchrotron radiation or to that for curvature radiation in appropriate limits. It presents a new radiation mechanism and describes more realistically the radiation in magnetic systems, especially in pulsars.

Three‐Dimensional Ring Current Decay Model

This work is an extension of a previous ring current decay model. In the previous work, a two‐dimensional kinetic model was constructed to study the temporal variations of the equatorially mirroring ring current ions, considering charge exchange and Coulomb drag losses along drift paths in a magnetic dipole field. In this work, particles with arbitrary pitch angle are considered. By bounce averaging the kinetic equation of the phase space density, information along magnetic field lines can be inferred from the equator. The three‐dimensional model is used to simulate the recovery phase of a model great magnetic storm, similar to that which occurred in early February 1986. The initial distribution of ring current ions (at the minimum Dst) is extrapolated to all local times from AMPTE/CCE spacecraft observations on the dawnside and duskside of the inner magnetosphere spanning the L value range L = 2.25 to 6.75. Observations by AMPTE/CCE of ring current distributions over subsequent orbits during the storm recovery phase are compared to model outputs. In general, the calculated ion fluxes are consistent with observations, except for H+ fluxes at tens of keV, which are always overestimated. A newly invented visualization idea, designated as a chromogram, is used to display the spatial and energy dependence of the ring current ion differential flux. Important features of storm time ring current, such as day‐night asymmetry during injection and drift hole on the dayside at low energies (&lt;10 keV), are manifested in the chromogram representation. The pitch angle distribution is well fit by the function, jo(1 + Ayn), where y is sine of the equatorial pitch angle. The evolution of the index n is a combined effect of charge exchange loss and particle drift. At low energies (&lt;30 keV), both drift dispersion and charge exchange are important in determining n.

Nonsteady state ionosphere‐plasmasphere coupling of superthermal electrons

Numerical solutions to the nonsteady state kinetic equation which describes the transport of superthermal electrons in the ionosphere and plasmasphere between the magnetically conjugate regions of the ionosphere are presented. The distribution function in time, distance along arbitrary geomagnetic field lines, energy, and pitch angle are among the parameters calculated by the model. This model represents a unified approach by self‐consistently coupling the interaction of the two hemispheres and the trapping of superthermal electrons in the plasmasphere. Our calculations take into account the various ionization and excitation processes and the effect of an inhomogeneous magnetic field (i.e., magnetic mirroring of precipitating electrons and focusing of escaping electrons along magnetic field lines). Omnidirectional flux spectra and pitch angle distributions are shown for various L shells and situations, and the features are described in detail. Nonsteady state calculations predict that a depleted flux tube can take several hours to reach steady state levels again. Plasmaspheric transparencies are calculated for different conditions of illumination, scattering processes in the conjugate ionospheres, and field‐aligned gradients of the thermal plasma density. Plasmaspheric transparency is found to be a complicated function of not only the plasmaspheric thermal plasma but also the ionospheric sources and scattering processes. A phenomenological model is used to describe the energy transmission, reflection, and deposition in the plasmasphere. By studying the ionosphere and plasmasphere as one system rather than two separate ones, substantial corrections are introduced in the values of key parameters describing photoelectron fluxes.

A bounce‐averaged kinetic model of the ring current ion population

A bounce‐averaged ring current kinetic model for arbitrary pitch angle, including losses due to charge exchange and Coulomb collisions along ion drift paths, is developed and solved numerically. Results from simplified model runs, intended to illustrate the effects of adiabatic drifts and collisional losses on the proton population, are presented. The processes of i) particle acceleration under the conditions of time‐independent magnetospheric electric fields; ii) a predominant loss of particles with small pitch angles due to charge exchange; and iii) a buildup of a low‐energy population caused by the Coulomb drag energy degradation, are discussed.

Particle tracing in the magnetosphere: New algorithms and results

We use a fast, efficient method to trace charged particles through realistic magnetospheric electric and magnetic fields, greatly reducing computer simulation times. The method works for particles having arbitrary charge, energy, or pitch angle but which conserve the first two adiabatic invariants. We also apply an efficient method of classifying drift orbits, which greatly simplifies the task of identifying the last closed drift path or other drift boundaries. Finally, we calculate the time‐independent evolution of the bounce‐averaged phase space density along convective drift orbits. With these three tools, convective evolution of the particle distribution from the tail can now be described quantitatively, an essential step in understanding the production of unstable distributions in the magnetosphere. One can also categorize topologically different drift orbits, which is necessary to understand the unique particle signatures of the convecting plasma such as Alfvén layers and the plasmapause. These signatures can then be used to extract the electric and magnetic fields or to test the validity of the model fields. The method is particularly appropriate for particles in the energy range 0.01&lt;E&lt;100 keV, which are influenced by both electric and magnetic fields, and for time periods without invariant destroying waves.

A unified theory of gyrotron and peniotron interactions

The thin beam model of an axis-encircling relativistic electron beam in a uniform, constant, external magnetic field in a cylindrical waveguide is used to develop a unified theory of the various interactions of such a beam with the electromagnetic field. The model includes the gyrotron interaction (longitudinal displacement of the beam) and the peniotron interaction (transverse displacement of the beam). Space-charge effects are included. The resulting self-fields influence the boundary conditions of the perturbed electromagnetic field across the electron beam. For a beam with only azimuthal equilibrium flow and for propagation perpendicular to the equilibrium external magnetic field the well-known gyrotron instabilities are recovered. In addition, it is shown that the transverse beam modes (slow peniotron) are unstable for resonant interaction with the transverse electric (TE) waveguide mode. A general dispersion relation for propagation at any angle is derived for a beam with arbitrary energy and arbitrary pitch angle. This dispersion relation includes gyrotron, cyclotron autoresonance maser (CARM), peniotron, and harmonic autoresonant peniotron (HARP) interactions and the coupling between TE and transverse magnetic (TM) waveguide modes. For weak space charge, analytic expressions for the growth rates of the Doppler-shifted cyclotron resonance interaction of TE and TM modes with the gyrotron and slow peniotron modes are obtained.

Stability of astrophysical disks and rings

Spiral perturbations with arbitrary pitch angle in gaseous and stellar disks are examined. In particular, minimum values are found for Toomre's stability margin Q, which assure complete stability of disks, ie., no growth of perturbations for any finite time interval. The corresponding values for a flat rotation curve are Q = vC'3 " in a gas disk and Q = 3 for a stellar disk. The most unstable perturbations for the stellar disk parameters of the solar neighborhood are four-arm spirals: m = 4. Approximate dispersion relations describing perturbations in thin gravitating rings are derived. A close analogy with potential oscillations of an electron plasma is found for the two-ring case so that the general criteria of beam instability derived in plasma physics can be used. Investigation of several of the simplest models of paired rings of differing masses has significantly simplified the calculation of the minimum mass of the thin ring for instability as compared to the case of two identical rings studied previously. It is suggested that the azimuthal structure ("clumping") in several ringlets within the Encke division immediately adjacent to the inner edge of the A ring of Saturn is a result of the development of the analogue of beam instability (in the nonlinear mode).

A uniformly valid asymptotic density-potential relation for spiral galaxies

An asymptotic relation between density and potential perturbations which is uniformly valid for arbitrary pitch angle β for spiral galaxies with number of spiral arms m ⩾ 2, is given. For small pitch angles (¦β¦⪡ 1) 2 2 The pitch angle will be denoted by β throughout so as to leave i for the imaginary unit. , it is reduced to various known asymptotic relations. When ¦β¦is not small, this new result is suitable for discussing the open spiral waves forming barred spiral galaxies. The final result is given by Eq. (5.40), where \\ ̂ gy 1 and \\ ̂ gs 1 are the complex amplitudes of the potential perturbation ψ 1 and the density perturbation σ 1 defined by Eqs. (5.32) and (5.33), respectively. The region in the (− kr, m) plane where the present result is uniformly valid is given by Eq. (6.1) and shown in Fig. 2. In Eq. (6.1), k is the radial wave number and K is the total wave number. When β = 0, Eq. (5.40) is reduced to Eq. (6.4), an asymptotic relation obtained by Shu (1970). When ¦β¦⪡ 1, Eq. (5.40) is reduced to Eq. (6.4), an equivalent form of Bertin and Mark's result (1979). When β = π 2 , Eq. (5.40) is reduced to Eq. (6.5), an asymptotic relation for the “open limit”. For the middle range of the pitch angle, neither the “tightly wound limit” result for ¦β¦⪡ 1 nor the “open limit” result for β ≈ π 2 is valid, but the present result is still valid.

Rapid temporal evolution of radiation from nonthermal electrons in solar flares

Solutions of the time dependent Fokker-Planck equation was found for accelerated electrons undergoing Coulomb collisions in a magnetized, fully ionized plasma. An exact solution was found for arbitrary pitch angle and energy distribution in a uniform background plasma. Then, for an inhomogeneous plasma, a solution was found for particles with small pitch angles. These solutions were used to calculate the temporal evolution of bremsstrahlung x-rays from short bursts of nonthermal electron beams, and these spectra were compared with observed high time resolution spectra of short timescale solar hard x-ray bursts. It is shown that the observed softening in time of the spectra rules out a homogeneous background and therefore the possibility of electrons being confined to the corona either because of converging magnetic field or high densities. The inhomogeneous solution was also applied to a model with constant coronal density and exponentially rising chromospheric density. The spectra are shown to be consistent with that produced by a collimated beam of electrons accelerated in the corona with certain given conditions. These conditions could be violated if large pitch angle electrons are present.

The radial distribution of radiation belt protons: Approximate solution of the steady state transport equation at arbitrary pitch angle

We present quasi‐analytic solutions to the proton diffusion equation, with energy degradation and charge exchange losses included. The transport mode is modeled as D ∝ M−l Lg(K,l), where M, K, and L are the adiabatic invariants, and l refers to the spectral index of the fluctuating electric field. At the outer boundary, say L = 7, a distribution function of the form f ∝ exp −[M/M0(K)]v is maintained; at the inner boundary (L ≈ 1), f is required to vanish. We find that (1) the diffusion flux is directed outward, when energy loss and charge exchange are in approximate balance, (2) the shape of the proton flux is strongly affected by the spectral index of the fluctuating fields, (3) the proton flux exhibits both positive and negative gradients with respect to energy, and (4) the anisotropy of the proton flux increases with decreasing L value, and may pass through a minimum with respect to energy.

Study on contact mechanism and electric resistance characteristics of twisted conductors

AbstractThis paper analyzes the mechanical and electrical characteristics of the hand‐twisted conductor splices in communication cables. the validity of the analysis is verified by experiment. Concerning the mechanical characteristics, it is shown that in the physical contact of the conductors the central axis is twisted with an arbitrary pitch angle on a quasi‐cylinder with the radius equal to that of the conductor. Based on such representation of the contact mechanism, the contact pressure and contact width of the twisted conductors are calculated. Concerning the electrical characteristics, it is assumed that there is a similarity between the current distribution through the twisted conductor contact and the electric lines of force produced by the charge on the surface of contact. Based on such assumption, the constriction resistance of the twisted contact is calculated with consideration of the contact pressure and contact width. Estimation is also made on the growth process of the film resistance and the boundary film thickness due to the chemical substances formed on the conductors in contact. the electrical equivalent circuit for the twisted conductor contact is derived, including the constriction resistance, film resistance and conductor resistance. Using the derived equivalent circuit, an expression is derived for estimation of the contact resistance of the twisted conductors.

Emission, absorption, and polarization of gyrosynchrotron radiation of mildly relativistic particles

In this paper are presented approximate analytic expressions for the emissivity and absorption coefficient of synchrotron radiation of mildly relativistic particles with an arbitrary energy spectrum and pitch angle distribution. From these, an expression for the degree of polarization is derived. The analytic results are compared with numerical results for both thermal and nonthermal (power law) distributions of particles.

Influence of collisions and incoherence on ion-cyclotron transport in mirror plasmas

The effects of classical Coulomb collisions and wave incoherency on drift-cyclotron-loss-cone (drift-cone) mode anomalous transport are studied analytically and by simulation. Pitch-angle collisions have a cumulative effect on the gyrophase angle, and the spatially varying magnetic field in a mirror well contributes a large geometrical amplification of the gyrophase-angle diffusion. Analytical calculations of this effect for particles with arbitrary pitch angle are confirmed by Monte Carlo simulations. However, for the levels of drift-cone turbulence observed in the 2XIIB magnetic mirror experiment and for its relative collisionality, the effect of pitch-angle collisions on anomalous transport is small. In particular, heating of ions to 40 keV energies cannot be explained by this mechanism. Wave incoherency also destroys superadiabaticity, leads to velocity transport, and provides an approximate model of the finite frequency bandwidth of drift-cone turbulence, e.g., due to bursting effects in 2XIIB. Monte Carlo simulations corroborate the analytical transport theory presented. The simulations exhibit significantly enhanced velocity-space transport and heating of ions to nearly 40 keV energies due to wave incoherency for parameters at the limit of applicability to the 2XIIB experiment. These same considerations of resonant wave-particle interactions are also directly relevant to electron and ion-cyclotron resonance heating and to velocity diffusion driven by unstable Alfvén-ion-cyclotron waves in other mirror experiments.