The effect of strong pitch angle scattering on the use of artificial auroral streaks for echo detection—Echo 5

During the Echo 5 experiment launched 13 November 1979 from the Poker Flat Research Range (Fairbanks, Alaska), a 0.75 A, 37 keV electron beam was injected both up and down the field line. The objective of the experiment was to test the use of optical and X-ray methods to detect the beam as it interacted with the atmosphere below the rocket for both the downward injections (markers) and the upward injected electrons which mirrored at the Southern Hemisphere and returned echoes. A ground-based TV system and rocket borne photometers and X-ray detectors viewed the interaction region. The artificial auroral streaks created by the markers were easily visible on the ground TV system but the large intensity of photons produced around the rocket masked any response to the markers by the on-board photometers and X-ray detectors. No echoes were detected with any of the detection systems although the power in some of the upward injections was 7.6 times the power in a detected downward injection thus setting an upper limit on the loss-cone echo flux. The magnitude of the bounce averaged pitch angle diffusion coefficient necessary to explain the lack of observable echoes was found to be 4 × 10⁻⁴ s⁻¹. Comparing with calculations done by Lyons (1974) for the pitch angle diffusion of electrons by electrostatic waves, it was found that an equatorial wave electric field of 11 mV m⁻¹ would account for the lack of echoes. Such fields should cause strong pitch angle scattering of up to 10 keV natural electrons and thus be consistent with the presence of diffuse aurora on the Echo 5 trajectory. Direct measurements have also revealed such fields in equatorial regions.     

Pitch-angle distributions in and near the loss cone of (100–350) keV protons

The pitch-angle distributions in and near the loss cone, of ～ (100–200) and ～ (200–350) keV protons observed by the ESRO IB satellite during the period 7–15 October 1969 are presented. The data include periods of relative quiet as well as more disturbed geomagnetic conditions. Spatial characteristics and dynamics of the protons, both on the night-and dayside of the Earth are described. The actual pitch-angle distribution is interpreted as produced by wave-particle interactions, and the diffusion coefficient and lifetime against pitch angle scattering have been estimated from existing theories. During slightly disturbed conditions, the observations suggest an average random walk in pitch angle made by a particle during a crossing of the diffusion region of about one half of the loss cone half angle for 4 ? L ? 6. The lifetime against pitch angle scattering into the loss cone is found to be somewhat less than the charge exchange lifetime for these (100–350) keV protons. The spectral density of interacting waves is tentatively estimated to about 0·1 $\text{γ}{}^2\text{Hz}$, and compares with estimates arrived at from completely different approaches.     

Motion of charged particles in the magnetosphere

The adiabatic motion of charged particles in the magnetosphere has been investigated using Mead-Fairfield magnetospheric field model (Mead and Fairfield, 1975). Since the motion of charged particles in a dipolar field geometry is well understood, we bring out in this paper some important features in characteristic motion due to non-dipolar distortions in the field geometry. We look at the tilt averaged picture of the field configuration and estimate theoretically the parameters like bounce period, longitudinal invariant and the bounce averaged drift velocities of the charged particle in the Mead-Fairfield field geometry. These parameters are evaluated as a function of pitch angle and azimuthal position in the region of ring current (5 to 7 Earth radii from the centre of the Earth) for four ranges of magnetic activity. At different longitudes the non-dipolar contribution as a percentage of dipole value in bounce period and longitudinal invariant show maximum variation for particles close to 90° pitch angles. For any low pitch angle, these effects maximize at the midnight meridian. The radial component of the bounce averaged drift velocity is found to be greatest at the dawn-dusk meridians and the contribution vanishes at the day and midnight meridians for all pitch angles. In the absence of tilt-dependent terms in the model, the latitudinal component of the drift velocity vanishes. On the other hand, the relative non-dipolar contribution to bounce averaged azimuthal drift velocity is very high as compared to similar contribution in other characteristic parameters of particle motion. It is also shown that non-dipolar contribution in bounce period, longitudinal invariant and bounce averaged drift velocities increases in magnitude with increase in distance and magnetic activity.     

Nonlinear generation of ion loss-cone waves by electrostatic turbulence in the magnetosphere

Effects of plasma turbulence on the stability of electrostatic ion loss-cone waves are examined. The turbulence is assumed to be electrostatic with frequencies near 1.5 times the electron gyrofrequency and the frequencies of the generated waves are below the ion plasma frequency ω_pi>. A nonlinear growth rate of the order of 10^?2ω_pi may be obtained, when the amplitude of the turbulence is 20 mV/m. This is comparable to previously found growth rates of the linear ion loss-cone instability, in a plasma with large pitch angle anisotropy. Bounce averaged pitch angle diffusion coefficients are also presented for different models of the ion loss-cone wave spectrum.     

Proton observations supporting the ion cyclotron wave heating theory of SAR arc formation

Low altitude satellite observations of precipitated and locally mirroring protons during periods of ground-based SAR arc observations are presented. The SAR arcs are found to be located in a region with significantly enhanced proton pitch angle scattering and enhanced electron temperature, but inside the plasmapause where the proton pitch angle distribution is anisotropic. The increase in the pitch angle scattering takes place in a localized region having a width of a few tenths of a L-value. The observations can favourably be accounted for by the Cornwall et al. (1971) theory for the SAR arc formation. Using observed proton fluxes and typical energy spectra, the expected Hβ intensity in the SAR arc region is estimated to be a few Rayleighs, and the energy flux from precipitated protons above a few keV to be 10^?2?10^?1erg/cm²s. These estimates are in reasonable agreement with previously published theoretical and experimental values. Simultaneous groundbased observations of Hα emissions were found in the region of intense, isotropic proton precipitation located outside the plasmapause.     

Bounce solutions in viscous fluid cosmology

We investigate the bounce cosmology induced by inhomogeneous viscous fluids in FRW space-time (non necessarily flat), taking into account the early-time acceleration after the bounce. Different forms for the scale factor and several examples of fluids will be considered. We also analyze the relation between bounce and finite-time singularities and between the corresponding fluids realizing this scenarios. In the last part of the work, the study is extended to the framework of f(R)-modified gravity, where the modification of gravity may also be considered as an effective (viscous) fluid producing the bounce.     

Energetic solar-proton precipitation in the auroral zone associated with storm sudden commencements

Measurements by balloon-borne instruments, data from the satellites Explorer 41 and 43 and riometer recordings were used to investigate the influence of magnetospheric processes on the precipitation of energetic solar protons related to the occurrence of two ssc's on 8–9 August 1972. The high-energy protons (E_p ? 30 MeV) had direct access to auroral-zone latitudes. The flux variations of low-energy (some MeV) protons in interplanetary space and the magnetosphere were different from those of the protons precipitated in the auroral zone. These low-energy protons were precipitated mainly during and after the ssc's. The importance of direct proton access, radial diffusion, pitch angle scattering and proton acceleration for the explanation of the low-energy proton behaviour is discussed.     

Interaction of particles with ion cyclotron waves and magnetosonic waves. Observations from GEOS 1 and GEOS 2

Coordinated observations involving ion composition, thermal plasma, energetic particle, and ULF magnetic field data from GEOS 1 and 2 often reveal the presence of electromagnetic ion cyclotron and magnetosonic waves, which are distinguished by their respective polarization characteristics and frequency spectra. The ion cyclotron waves are identified by a magnetic field perturbation that lies in a plane perpendicular to the Earth's magnetic field B₀ and propagate along B₀. They are associated with the abundance of cold He⁺ in the presence of anisotropic pitch angle distributions of ions having energies E > 20 keV, and were observed at frequencies near the He⁺ gyrofrequency. The magnetosonic waves are characterized by a magnetic field perturbation parallel to B₀ and thus seem to be propagating perpendicular to the Earth's magnetic field. They often occur at harmonics (not always including the fundamental) at the proton gyrofrequency and are associated with phase-space-density distributions that peak at energies E ～ 5–30 keV and at a pitch angle of 90°. Such a ring-like distribution is shown to excite instability in the magnetosonic mode near harmonics of the proton gyrofrequency. Magnetosonic waves are associated in other cases with sharp spatial gradients in energetic ion intensity. Such gradients are encountered in the early afternoon sector (as a consequence of the drift shell distortion caused by the convection electric field) and could likewise constitute a source of free energy for plasma instabilities.     

Jupiter’s radio spectrum from 74 MHz up to 8 GHz

We carried out a brief campaign in September 1998 to determine Jupiter’s radio spectrum at frequencies spanning a range from 74 MHz up to 8 GHz. Eleven different telescopes were used in this effort, each uniquely suited to observe at a particular frequency. We find that Jupiter’s spectrum is basically flat shortwards of 1-2 GHz, and drops off steeply at frequencies greater than 2 GHz. We compared the 1998 spectrum with a spectrum (330 MHz-8 GHz) obtained in June 1994, and report a large difference in spectral shape, being most pronounced at the lowest frequencies. The difference seems to be linear with log(ν), with the largest deviations at the lowest frequencies (ν).We have compared our spectra with calculations of Jupiter’s synchrotron radiation using several published models. The spectral shape is determined by the energy-dependent spatial distribution of the electrons in Jupiter’s magnetic field, which in turn is determined by the detailed diffusion process across L-shells and in pitch angle, as well as energy-dependent particle losses. The spectral shape observed in September 1998 can be matched well if the electron energy spectrum at L = 6 is modeled by a double power law E^−a (1+(E/E₀))^−b, with a = 0.4, b = 3, E₀ = 100 MeV, and a lifetime against local losses τ₀ = 6 × 10⁷ s. In June 1994 the observations can be matched equally well with two different sets of parameters: (1) a = 0.6, b = 3, E₀ = 100 MeV, τ₀ = 6 × 10⁷ s, or (2) a = 0.4, b = 3, E₀ = 100 MeV, τ₀ = 8.6 × 10⁶ s. We attribute the large variation in spectral shape between 1994 and 1998 to pitch angle scattering, coulomb scattering and/or energy degradation by dust in Jupiter’s inner radiation belts.     

Observations of energetic ions in inverted V events

Simultaneous measurements of keV ions and electrons with the ESRO 1A satellite have shown the following ion characteristics among others. Ions of about 6 keV energy are strongly field-aligned on the flanks of the inverted V events (mainly through the disappearance of the ion flux near 90° pitch angle). Field-aligned electron fluxes are often found in the same regions of the inverted V events where the ions are field-aligned. At the centre of inverted V events isotropization occurs (except in some small events). The 1 keV ion flux at large pitch angles (80°) is generally not reduced very much when the 6 keV, 80° ion flux shows strongly decreased values. The ratio of the 1 to 6 keV ion flux has a maximum near the centre of an inverted V event where the electron spectrum is hardest and the 6 keV ions are isotropic (or nearly isotropic).The observations are interpreted in terms of a model with two oppositely directed field-aligned electrostatic potential drops: one upper accelerating electrons downward and one lower, produced by the electron influx, which accelerates ions downward. Ion scattering in turbulent wave fields is proposed to be responsible for the observation that the 1 keV ion flux at large pitch angles does not decrease strongly where the 6 keV ion flux does and as an explanation of the isotropization at the centre of the event. The source problem for the ions is eliminated by the precipitating electrons ionizing continuously the thin neutral atmosphere even at altitudes of a few thousand kilometers.     

Tidal evolution of Mimas, Enceladus, and Dione

The tidal evolution through several resonances involving Mimas, Enceladus, and/or Dione is studied numerically with an averaged resonance model. We find that, in the Enceladus-Dione 2:1 e-Enceladus type resonance, Enceladus evolves chaotically in the future for some values of k₂/Q. Past evolution of the system is marked by temporary capture into the Enceladus-Dione 4:2 ee^′-mixed resonance. We find that the free libration of the Enceladus-Dione 2:1 e-Enceladus resonance angle of 1.5° can be explained by a recent passage of the system through a secondary resonance. In simulations with passage through the secondary resonance, the system enters the current Enceladus-Dione resonance close to tidal equilibrium and thus the equilibrium value of tidal heating of 1.1(18,000/QS) GW applies. We find that the current anomalously large eccentricity of Mimas can be explained by passage through several past resonances. In all cases, escape from the resonance occurs by unstable growth of the libration angle, sometimes with the help of a secondary resonance. Explanation of the current eccentricity of Mimas by evolution through these resonances implies that the Q of Saturn is below 100,000. Though the eccentricity of Enceladus can be excited to moderate values by capture in the Mimas-Enceladus 3:2 e-Enceladus resonance, the libration amplitude damps and the system does not escape. Thus past occupancy of this resonance and consequent tidal heating of Enceladus is excluded. The construction of a coherent history places constraints on the allowed values of k₂/Q for the satellites.     

The effect of a geomagnetic storm on energetic trapped protons at low altitudes

The irreversible changes of the intensity of trapped protons with energy above 1 MeV in the Earth's magnetosphere near the outer boundary of trapping are observed after moderate geomagnetic storms on the low-altitude polar-orbiting satellite Intercosmos-17. These changes are interpreted in terms of nonadiabatical effects of proton motion in the disturbed geomagnetic field (assuming D_st variation) which affects the conditions for stable trapping of protons during the storm. The decrease of proton intensity is due to an adiabatic decrease of energy, an increase of mirror-point altitude and nonadiabatic scattering and losses. The interaction of two types of particle motion—gyrorotation and the ‘bounce’ motion, which leads to the instability of motion, is assumed. The importance of nonadiabatical losses of trapped protons with low equatorial pitch angles for changes near the proton boundary is pointed out.     

The regularizing function in resonance problems

The regularizing function ψ(x) in the theory of resonance removes the singularities discovered by Poincaré (1893), of the form 1/x, at theturning points x ₁ andx ₂ of thecritical argument x, librating in the rangex ₁≤x≤x ₂. This function has been explicitly introduced into the HamiltonianF ₀ of the Ideal Resonance Problem in the author's recent paper (1977) in order to remove the singularities in the second-order perturbations. It is shown here that this procedure can be extended toall orders. ThenF ₀ can be put into the form $$F_0 = B(y) + \Psi (x)$$ where ψ is thecomplete regularizing function, removing theclassical singularity of thesmall divisor, in addition to the singularities of Poincaré.     

Test Particle Simulations of Interaction Between Monochromatic Chorus Waves and Radiation Belt Relativistic Electrons

Chorus waves have been suggested to be effective in acceleration of radiation belt electrons. Here we perform gyro-averaged test-particle simulations to calculate the bounce-averaged pitch angle and energy diffusion coefficients for parallel-propagating monochromatic chorus waves, and perform a comparison of test-particle (TP) model with quasi-linear (QL) theory to evaluate the influence of nonlinear processes. For small amplitude chorus waves, the diffusion coefficients of TP and QL models are in good agreement. As the wave amplitude reaches a threshold value, two nonlinear processes (phase trapping and phase bunching) start to occur, especially at large equatorial pitch angles. Phase trapping yields rapid increases in pitch angle and kinetic energy. In contrast, phase bunching causes overall decreases in pitch angle and kinetic energy. For the waves with amplitudes slightly above the threshold value, the average behavior is dominated by the phase trapping, and TP diffusion coefficients are larger than QL ones. As wave amplitude increases, TP diffusion coefficients become smaller than QL ones, indicating that phase trapping gradually reduces the dominance over phase bunching.     

Instability of an area-preserving polynomial mapping

Fixed points and eigencurves have been studied for the Hénon-Heiles mapping:x′=x+a (y?y ³),y′=y(x′?x′ ³). Eigencurves of order 21 proceed rapidly to infinity fora=1.78, but as ‘a’ decreases, they spiral around the origin repeatedly before escaping to infinity. Fixed pointsx _f on thex-axis have been located for the range 1≤a≤2.4, for ordersn up to 100. Their locations vary continuously witha, as do the eigencurves, and hyperbolic points remain hyperbolic. Forn=3 and 2.4≥a≥2.37, a very detailed study has been made of how escape occurs, with segments of an eigencurve mapping to infinity through various escape channels. Further calculations with ‘a’ decreasing to 2.275 show that this instability is preserved and that the eigencurve will spiral many times around the origin before reaching an escape channel, there being more than 34 turns fora=2.28. The rapid increase of this number is associated with the rapid decrease of the intersection angle between forward and backward eigencurves (at the middle homoclinic point), with decreasing ‘a’, this angle governing the outward motion. By a semi-topological argument, it is shown that escape must occur if the above intersection angle is nonzero. In the absence of a theoretical expression for this angle, one is forced to rely on the numerical evidence. If the angle should attain zero for a valuea=a _c>a_m,wherea _m .is the minimum value for which the fixed points exist, then no escape would be possible fora _c However, on the basis of calculations by Jenkins and Bartlett (1972) forn=6, and the results of the present article forn=3, it appears highly probable thata _c=a_m,and that escape from the neighborhood of a hyperbolic point is always possible. If there is escape from the hyperbolic fixed point forn=4,a=1.6, located atx _f=0.268, then the eigencurve must cross the apparently closed invariant curve of Hénon-Heiles which intersects thex-axis atx?±0.4, so that this curve cannot in fact be closed.     

Ring current protons in the upper atmosphere within the plasmasphere

keV protons observed by the ESRO 1A satellite in the upper atmosphere equatorward of the main precipitation zone are described and discussed. The protons are highly anisotropic (empty loss cone) between the low-latitude boundary of the main precipitation zone and approximately L=4 during quiet and moderately disturbed conditions (K_p=0?4). Between L=4 and L=2.7 the proton flux is generally enhanced compared to that at L values somewhat above 4 and only moderately anisotropic. Substorms push the outer main precipitation zone equatorwards, but the boundaries of the innermost, moderately anisotropic region (at L=2.7 and L=4) move only when strong magnetic storms compress the plasmasphere to within L=4. It is suggested that the moderately anisotropic zone is caused by the ion-cyclotron instability for which the growth rate may have a broad maximum between about L=2.7 and L=4. For proton energies in the keV range the instability is excited only in regions with cold plasma densities above several hundred ions per cubic centimeter. It is finally concluded that the observations of low-latitude proton precipitation lend further support to the mechanism of ion-cyclotron instability as the cause of proton pitch angle diffusion, as proposed by Cornwall et al. (1970).     

Theory of solar radio pulsation

We show that the observed modulation of some coronal microwave, X-ray and Type III emission into pulses of 10 sec intervals is a consequence of the stimulation of electron cyclotron waves propagated in the whistler mode in dipole-like bipolar regions of dimension 0.2 R . Assuming that a power law spectrum of 10 keV electrons with a slope similar to solar flare protons can be trapped in a bipolar region, we show that whistlers can be generated by pitch angle instability. The resultant 10 sec bounce motion of whistler wave trains leads to enhanced, modulated emission in microwave and X-ray frequencies by pitch angle scattering of MeV electrons, and to modulated Type III emission by scattering with coherent plasma waves. A direct prediction of the theory is the existence of sympathetic pulsations at two sources a fraction of a solar radius apart. A second test of the theory is that modulated Type III emission should show strong polarization.This work was conducted under U.S. Air Force Space and Missile Systems Organization (SAMSO) Contract No. F04701-69-C-0066.     

The estimate of the Venus magnetotail length

We consider the process of flux tubes straightening in the Venus magnetotail on the basis of MHD model. We estimate the distance x _t, where flux tubes are fully straightened due to the magnetic tension and the magnetotail with the characteristic geometry of field lines (“slingshot” geometry) ends. We investigate the influence of the transversal current sheet scale on the process of flux tubes straightening. The assumption of a thin current sheet allows to obtain a lower estimate of the magnetotail length, x _t > 31R _V (R _V is the Venus radius), while the assumption of a broad current sheet allows to obtain an upper estimate, x _t < 44R _V. We show that kinetic effects associated with the losses of particles with small pitch angles from the flux tube and the influx of magnetosheath plasma into the flux tube do not significantly affect the estimate of the magnetotail length. The model predicts the existence of energetic fluxes of protons H⁺ (2–5 keV) and oxygen ions O⁺ (35–80 keV) in the distant tail. We discuss the magnetotail structure at x > x _t.     

On the Temporal Variability of the “Strahl” and Its Relationship with Solar Wind Characteristics: STEREO SWEA Observations

The “strahl” is a specific population of the solar wind, constituted by strongly field aligned electrons flowing away from the Sun, with energies >60 eV. Using the Solar Wind Electron Analyzer (SWEA) onboard STEREO, we investigate the short time scale fluctuations of this population. It is shown that its phase space density (PSD) at times presents fluctuations larger than 50% at scales of the order of minutes and less. The fluctuations are particularly strong for periods of a few tens of hours in high-speed streams, following the crossing of the corotating interaction region, when the strahl is also the most collimated in pitch angle. The amplitude of the fluctuations tends to decrease in conjunction with a broadening in pitch angle. Generally, the strongly fluctuating strahl is observed when the magnetic field is also highly perturbed. That SWEA is able to perform a very rapid 3D analysis at a given energy is essential since it can be demonstrated that the observed magnetic turbulence can only marginally perturb the PSD measurements.