Diffusion and heat flow equations for the mid-latitude topside ionosphere

For application to the mid-latitude topside ionosphere, we have derived diffusion and heat flow equations for a gas mixture composed of two major ions, electrons and a number of minor ions. These equations were derived by expanding the velocity distribution of each constituent about its 13 lower order velocity moments. As a consequence, each constituent was allowed to have its own temperature and drift velocity. The restriction to mid-latitudes results because we have assumed that the species temperature and drift velocity differences were small. In deriving the diffusion and thermal conduction equations, we have discovered some new transport effects. For the major ions, we have found that: (1) a temperature gradient in either gas causes thermal diffusion in both gases; (2) a temperature gradient in either gas causes heat to flow in both gases; and (3) a relative drift between the major ion gases induces a heat flow in both gases. Similar transport effects have also been found for the minor ions.     

Diffusion coefficients for three major ions in the topside ionosphere

Published experimental data on ion composition in the topside ionosphere are examined. For certain features (the light ion trough, the high-latitude trough, the high-latitude hole and the mid-latitude total ion concentration trough) it is pointed out that the number of major ions present may be 3 or more. Transport equations derived by Schunk and co-workers are extended to include the case of three major ions in the topside ionosphere. Specific calculations are made for the O⁺, H⁺ and He⁺ ions and the behaviour of the diffusion coefficients is discussed. From a model of the high-latitude ionization hole, described by Heelis et al., representative concentration and temperature profiles are obtained. These profiles are used to demonstrate further the behaviour of the ion diffusion coefficients.     

Interaction of interstellar pick-up ions with the solar wind

The interaction of interstellar pick-up ions with the solar wind is studied by comparing a model for the velocity distribution function of pick-up ions with actual measurements of He⁺ ions in the solar wind. The model includes the effects of pitch-ang'e diffusion due to interplanetary Alfvén waves, adiabatic deceleration in the expanding solar wind and the radial variation of the source function. It is demonstrated that the scattering mean free path is in the range 0.1 AU and that energy diffusion can be neglected as compared with adiabatic deceleration. The effects of adiabatic focusing, of the radial variation of the neutral density and of a variation of the solar wind velocity with distance from the Sun are investigated. With the correct choice of these parameters we can model the measured energy spectra of the pick-up ions reasonably well. It is shown that the measured differential energy density of the pick-up ions does not vary with the solar wind velocity and the direction of the interplanetary magnetic field for a given local neutral gas density and ionization rate. Therefore, the comparison of the model distributions with the measurements leads to a quantitative determination of the local interstellar gas density.Paper dedicated to Professor Hannes Alfvén on the occasion of his 80th birthday, 30 May 1988.     

Physical characteristics of the plasma tail of comet C/2006 M4 (SWAN)

Wide-angle images with narrow-band filters isolating the radiation of CO⁺ ions and continuum of comet C/2006 M4 (SWAN) were acquired. The brightness distribution in the plasma tail shortly after the outburst of the comet on October 26, 2006, has been derived. The model brightness distribution was calculated, and the parameters of the diffusion model of the plasma tail have been obtained. For the date of observations, the magnetic-field induction in the plasma tail, as well as the lifetime of luminous particles and the lengthwise and transverse diffusion coefficients of ions have been estimated. Possible association between the change in the tail structure and that in the physical parameters along the plasma tail of the comet is discussed.     

A non-linear study of the possible effects of electron-ion collisions on the diffusion of a cylindrical column of ionisation in the Earth's ionosphere

The possible effects of electron—ion collisions on the diffusion of a cylindrical plasma irregularity orientated at right angles to the geomagnetic field lines, at altitude 300 km, are discussed. The results obtained indicate that the diffusion may proceed at a rate rather greater than for a case for which the effects of collisions of electrons with ions are assumed to be negligible. Furthermore, it is suggested that if the plasma density is great enough, the production of a 'fin' of ionisation along the magnetic field direction may be inhibited by the action of electron—ion collisions.     

Transport,charge exchange and loss of energetic heavy ions in the earth's radiation belts: Applicability and limitations of theory

Ions in the trapping region of the earth's magnetosphere are subject to physical and chemical interactions which control their absolute and relative abundances. Charge exchange reactions act to establish a distribution of ionic states that is largely determined by the chemical properties of the individual species. Convection (“drift”) mechanisms and cross-L diffusion cause ions to be distributed over the entire trapping region with flux intensities determined by the nature and strength of the ion source, transport and loss mechanisms which in general are dependent on energy, mass and charge. Current theories describe ion transport through path tracing for individual particles or by radial diffusion for a population as a whole based on stochastic analysis; a comprehensive treatment of the combined convection and diffusion for trapped and non-trapped ions is yet to be developed. Even in studies where diffusion is the sole transport mechanism considered, only equatorially mirroring particles ($\text{α}{}_0\text{=}\text{π}\text{2}$) have been theoretically treated. There are clearly both upper and lower bounds on the ion energy beyond which diffusion theory ceases to be valid: at high energies where the ion gyroradius becomes too large for the adiabatic approximations to be valid and at low energies where convective drift is a dominant process. In spite of the known shortcomings of the diffusion theory and associated modeling, intriguing theoretical predictions of the relative ionic composition of the radiation belts have been made and some of them are now confirmed by direct observation. Among them is the predicted importance of ions heavier than protons at ring current energies of tens of keV which follows from the charge exchange chemistry.     

Diffusion of a multi-component plasma

Diffusion in a weakly ionized plasma composed of negative ions as well as positive ions is examined using appropriate linearized fluid equations. For initial, electrically neutral, density perturbations of the form exp(ikx) the diffusion process is characterized by electrical non-neutrality and by three stages or time scales. For equal positive and negative ion diffusion coefficients these stages are in general (1) equilibration of the electron gas so that pressure gradient and electric forces are balanced (2) ambipolar-like diffusion of all three species, and concluding with (3) free ion diffusion. The details of the process are governed by the product kλ_e (wave number times electron Debye length) and the ambient ratio of negative ion to electron number density. Numerical and analytic results for separate positive and negative diffusion coefficients show added complexity which is briefly described. These results or the more complete numerical solutions find application to the lower D region of the ionosphere.     

Interplanetary Pick-Up Ion Acceleration A Study of Anisotropic Phase-Space Diffusion

Interplanetary pick-up ions originate from ionizations of neutral interstellar atoms in the heliosphere. Over the past periods it was generally expected that after pick-up by the frozen-in solar wind magnetic fields these ions quickly isotropize in velocity space by strong pitch- angle scattering, they do, however, not assimilate to the ambient solar wind ions. Meanwhile careful investigations of pick-up ion data obtained with the plasma analyzers on AMPTE and ULYSSES could clearly reveal that, especially at periods of flow-aligned fields, noticeably anisotropic distributions must prevail. To better understand the evolutionary tracks of pick-up ions in interplanetary phase-space we carried out an injection study which takes into account all relevant convection and diffusion processes, i.e. describing pitch angle scattering, adiabatic cooling, drifts and energy diffusion. As demonstrated here particles injected at 1 AU establish a distribution function with substantial anisotropies up to distances beyond 6 AU. Only under the action of fairly strong isotropic turbulence levels a trend towards isotropy can be recognized. The bulk velocity of the injected pick-up ions turns out to be remarkably smaller than the solar wind velocity. It also is obvious that pick-ups are strongly spread out from that solar wind plasma parcel into which they were originally implanted. As one consequence it must be concluded that the derivation of interstellar He gas parameters, using He pick-up ion flux data, require appreciable caution. Due to anisotropic spatial diffusion the location of the LISM helium cone axis, i.e. the LISM wind vector, and the LISM helium temperature are hidden in the associated He⁺pick-up ion flux patterns. This revised version was published online in July 2006 with corrections to the Cover Date.     

Resonant interaction of hydromagnetic waves with charged particles

The effect of the resonant interaction of a distribution of hydromagnetic waves on a distribution of particles is described by a diffusion equation in momentum space. The diffusion coefficients and other coefficients describing systematic acceleration and diffusion in energy space are derived in general and for a number of particular cases. It is shown that the resonant acceleration of slow ions by hydromagnetic waves is ineffective. The time evolution of the energy spectrum for ultrarelativistic particles due to interaction with hydromagnetic waves is found and applied to the case of the Crab Nebula in an accompanying article.     

Acceleration of Pick-up Ions at the Solar Wind Termination Shock

It is generally accepted that pick-up ions act as a seed population for anomalous cosmic rays originating at the solar wind termination shock. We believe that the ion pre-acceleration process operating in the heliosphere up to the termination shock can be very important to inject the ions into the shock acceleration process. The pick-up ions pre-accelerated by solar wind turbulences have already a pronounced high energy tail when they reach the shock. Some fraction of these ions can experience further acceleration up to energies of anomalous cosmic rays by means of shock drift and diffusive acceleration. In the present paper the shock drift acceleration of pick-up ions suffering multiple reflection due to abrupt changes in both the strength and direction of the magnetic field through the shock is considered. The reflection process operates for high velocity particles different from the reflection by the electric cross-shock potential. During the first reflection the mean kinetic energy of pick-up ions increases by approximately a factor of 10. Reflected particles have highly anisotropic velocity distribution. Subsequent excursion of the particles in the turbulent upstream flow leads to diffusion in pitch-angle space and, as a result, the particles can return to the shock again suffering, thus, multiple encounters. In order to describe the motion of particles in the upstream and down streamparts of the flow we solve the Fokker-Plank transport equation for anisotropic velocity distribution function. This revised version was published online in July 2006 with corrections to the Cover Date.     

Non-extensive statistics and slarr neutrinos

In this paper we will show that, assuming the existence of a long-range microscopic memory of the random force, acting in the solar core, mainly on the electrons and the protons rather than on the light and heavy ions (or, equally, assuming the existence of an anomalous diffusion of solar core constituents of light mass and of normal diffusion of heavy ions), the equilibrium statistical distribution that these particles must obey, is that of the Tsallis non-extensive statistics, the distribution differing very slightly from the usual Maxwellian distribution. Due to the high-energy depleted tail of the distribution, the nuclear rates are reduced and, using earlier results on the standard solar model neutrino fluxes, calculated by Clayton and collaborators, we can evaluate fluxes in good agreement with the experimental data. While the proton distribution is only very slightly different from the Maxwell one there is a slightly larger difference with the electron distribution. This revised version was published online in July 2006 with corrections to the Cover Date.     

Sources of sodium in the lunar exosphere: Modeling using ground-based observations of sodium emission and spacecraft data of the plasma

Observations of the equatorial lunar sodium emission are examined to quantify the effect of precipitating ions on source rates for the Moon’s exospheric volatile species. Using a model of exospheric sodium transport under lunar gravity forces, the measured emission intensity is normalized to a constant lunar phase angle to minimize the effect of different viewing geometries. Daily averages of the solar Lyman α flux and ion flux are used as the input variables for photon-stimulated desorption (PSD) and ion sputtering, respectively, while impact vaporization due to the micrometeoritic influx is assumed constant. Additionally, a proxy term proportional to both the Lyman α and to the ion flux is introduced to assess the importance of ion-enhanced diffusion and/or chemical sputtering. The combination of particle transport and constrained regression models demonstrates that, assuming sputtering yields that are typical of protons incident on lunar soils, the primary effect of ion impact on the surface of the Moon is not direct sputtering but rather an enhancement of the PSD efficiency. It is inferred that the ion-induced effects must double the PSD efficiency for flux typical of the solar wind at 1 AU. The enhancement in relative efficiency of PSD due to the bombardment of the lunar surface by the plasma sheet ions during passages through the Earth’s magnetotail is shown to be approximately two times higher than when it is due to solar wind ions. This leads to the conclusion that the priming of the surface is more efficiently carried out by the energetic plasma sheet ions.     

Temperature anisotropy of the Jovian sulfur nebula

We discuss the apparent paradox between the reported observation of a 3-eV gyration energy of Jupiter's ionized sulfur nebula and its observed thickness. We present an observation of the thickness of the cloud taken nearly edge on and show that this implies a large bounce-averaged anisotropy of the sulfur in temperature, T₆ ? T_⊥. From these observations, we construct a self-consistent model of the sulfur nebula in which the sulfur ions are injected by Io as ions and remain sufficiently collisionless in the magnetosphere to maintain the anisotropy for a time longer than a characteristic diffusion time. We also show that the proton-electron plasma is collisionally thermalized and provides an adequate means of tapping the rotational energy of the planet to provide the power radiated in the sulfur lines.     

Probing shock geometry via the charge to mass ratio dependence of heavy ion spectra from multiple spacecraft observations of the 2013 November 4 event

In large Solar Energetic Particle(SEP) events, ions can be accelerated at coronal mass ejection(CME)-driven shocks to very high energies. The spectra of heavy ions in many large SEP events show features such as roll-overs or spectral breaks. In some events when the spectra are plotted in terms of energy/nucleon, they can be shifted relative to each other to make the spectral breaks align. The amount of shift is charge to mass ratio(Q/A) dependent and varies from event to event. This can be understood if the spectra of heavy ions are organized by the diffusion coefficients(Cohen et al. 2005). In the work of Li et al.(2009), the Q/A dependence of the scaling is related to shock geometry when the CME-driven shock is close to the Sun. For events where multiple in-situ spacecraft observations exist, one may expect that different spacecraft are connected to different portions of the CME-driven shock that have different shock geometries, therefore yielding different Q/A dependence. In this work, we examine one SEP event which occurred on 2013 November 4. We study the Q/A dependence of the energy scaling for heavy ion spectra using helium, oxygen and iron ions. Observations from STEREO-A, STEREO-B and ACE are examined. We find that the scalings are different for different spacecraft. We suggest that this is because ACE, STEREOA and STEREO-B are connected to different parts of the shock that have different shock geometries. Our analysis indicates that studying the Q/A scaling of in-situ particle spectra can serve as a powerful tool to remotely examine the shock geometry for large SEP events.     

The evolution of charged particles in a model of contracting cloud

The evolution of the charged particles are followed during contraction of a model of an interstellar cloud, with initial density number of n = 10 cm^–3. The contraction is followed up to density increase by five orders of magnitude. Special care is given to the details of the negative ions. In addition, we have tested the ambipolar diffusion according to the results of the ion density.The results predict the importance of atomic ions in the diffuse regions. H⁺ and C⁺ are distinctly enhanced in the beginning of contraction but decrease as contraction proceeds. Molecular ions enhance as contraction proceeds and becomes important in dense regions. The most enhanced molecular ions are HCO⁺, O₂ ⁺, C₂H₃ ⁺, H₃O⁺ and SO⁺, H₃ ⁺ is less abundant. The atomic ions (except metalic ions) decrease noticeably as density increases. In general the negative ions are of negligible fractional abundances. It has also been found that the time of ambipolar diffusion is shorter than the dynamical time, hence the magnetic field should be weakened in the central core as the central density increases to n = 10⁴ cm^–3.     

On the origin of time delays in hard X-ray and gamma-ray emission of solar flares

The possibility of a strong pitch-angle diffusion regime as well as of turbulent propagation of energetic ions and electrons in flaring loops has been shown. The strong diffusion regime suggests that two regions with a high level of small-scale turbulence are formed in the magnetic trap. Such additional turbulent mirrors scatter energetic particles and, therefore, the flux of precipitating particles decreases and the mean lifetime of electrons and protons in a flaring loop grows. We cannot rule out that the turbulent propagation of the particles can be responsible for the energy dependence of hard X-ray delays as well as the time lag of the gamma-ray line peaks with respect to the hard X-ray peaks as the electrons and ions are accelerated simultaneously. The trap plus turbulent propagation model may also explain the lack of abundant population of 10–100 keV electrons in interplanetary space in proton-rich events and offers new possibilities for flare plasma diagnostics.     

The emission and absorption of waves by charged particles in magnetized plasmas

A semiclassical theory describing the emission and absorption of waves is applied to the interaction of charged particles with waves in magnetized plasmas. Spontaneous emission of all cold plasma wave modes is calculated in detail. The method gives the absorption coefficient for the waves and a diffusion equation in momentum space for the particles describing the effects of the induced processes.Coefficients describing the systematic change of particle parameters follow from the diffusion equation. Applications of astrophysical interest are outlined.     

Crustal impurities and the internal temperature of a neutron star crust

When the upper part of a neutron star crystallizes to form the crust, the constituting ions are trapped in the lattice as a result of the low diffusion rates in the solid phase. As a consequence, the local composition of the crust corresponds to the equilibrium condition at the melting point and not at the present internal temperature. The inclusion of the small entropic contribution to the free energy does not lead to marked changes in our view of the microscopic structure of a neutron star crust, because the melting temperature is much smaller than the typical energies at play in the crystal cell. However, mixing between layers characterized by different nuclear species is found to be an important effect for the production of impurities. We show that one should expect an increase of the thermal diffusion time through the crust at small temperatures, because of the decrease of thermal conductivity in relatively thin impurity-rich layers acting as isolating shields.