Radio emissions from the outer heliosphere

For nearly fifteen years the Voyager 1 and 2 spacecraft have been detecting an unusual radio emission in the outer heliosphere in the frequency range from about 2 to 3 kHz, Two major events have been observed, the first in 1983–84 and the second in 1992–93. In both cases the onset of the radio emission occurred about 400 days after a period of intense solar activity, the first in mid-July 1982, and the second in May–June 1991. These two periods of solar activity produced the two deepest cosmic ray Forbush decreases ever observed. Forbush decreases are indicative of a system of strong shocks and associated disturbances propagating outward through the heliosphere. The radio emission is believed to have been produced when this system of shocks and disturbances interacted with one of the outer boundaries of the heliosphere, most likely in the vicinity of the the heliopause. The emission is believed to be generated by the shock-driven Langmuir-wave mode conversion mechanism, which produces radiation at the plasma frequency (f _p ) and at twice the plasma frequency (2f _p ). From the 400-day travel time and the known speed of the shocks, the distance to the interaction region can be computed, and is estimated to be in the range from about 110 to 160 AU.Abbreviations PWS Plasma Wave Subsystem - AU Astronomical Unit - DSN Deep Space Network - NASA National Aeronautics and Space Administration - GMIR Global Merged Interaction Region - MHD Magnetohydrodynamic - CME coronal mass ejection - f _p plasma frequency - R radial distance - AGC automatic gain control     

Small scale structure of magnetospheric electron density through on-line tracking of plasma resonances

The plasma resonance phenomena observed at f _pe, nf _ce, and f _qn by the GEOS-1 S-301 relaxation sounder are identified through a pattern recognition software process implemented in a mini-computer which receives on-line the compressed data. First, this processing system distributes in real time f _pe and f _ce measurements to the ground media. Second, it drives and controls automatically the S-301 on-board experiment by sending appropriate telecommands: the tracking of resonances is performed by shortening the frequency sweeps to a narrow range centered on the resonance location. Examples of such tracking sequences are presented, exhibiting sampling rates of the electron density measurements from once every 22 s (slowest rate) to once every 86 ms (highest rate available). The results give evidence of the existence of very small scale structures in the magnetospheric density, having characteristic sizes of the order of a few 10² m or/and a few 10^-1 s. The relative amplitude of these density fluctuations is typically 1%. Because of satellite spinning, fixed frequency sounding sequences allow to measure in a few seconds the directivity features of the plasma resonance signals. Examples of directional patterns in the plane perpendicular to the geomagnetic field are presented: the electrostatic nature of the waves received at f _pe, nf _ce, and f _qe being consistent with these patterns, the corresponding k vector orientations become available. The Bernstein modes properties are used to interpret the cf _ce and f _qe results.     

Collective radio-emission from plasmas

Collective radiation processes operating in laboratory and space plasmas are reviewed with an emphasis towards astrophysical applications. Particular stress is placed on the physics involved in the various processes rather than in the detailed derivation of the formulas. Radiation processes from stable non-thermal, weakly turbulent and strongly turbulent magnetized and unmagnetized plasmas are discussed. The general theoretical ideas involved in amplification processes such as stimulated scattering are presented along with their application to free electron and plasma lasers. Direct radio-emission of electromagnetic waves by linear instabilities driven by beams or velocity anisotropies are shown to be of relevance in space applications. Finally, as an example of the computational state of the art pertaining to plasma radiation, a study of the type III solar radio bursts is presented.

Frequently used Symbols

Latin Symbols teB ₀ ambient magnetic field - B ₁ perturbed magnetic field - c speed of light - E ₁ perturbed electric field - H Heaviside function - I unit dyadic - k wavevector of radiation fields - K _D inverse Debye length - m, M electron and ion mass - T _e , T _i electron and ion temperature - u relativistic velocity - V _e , V _i electron and ion thermal speeds - V _P , V _g wave phase and group velocities - W wave spectral energy density Greek Symbols relativistic factor - plasma dielectric tensor - _L , _T longitudinal and transverse components of in isotropic media (i.e., =kk _L /k ²+(l–kk/k ²) _T ) - index of refraction - angle between k and B ₀ - plasma dispersion tensor (i.e. =(c ²/ ²)(kk–k ² l)+) - determinant of - _D Debye length - _e electron cyclotron frequency - _u upper hybrid frequency - wave frequency - _e electron plasma frequency Proceedings of the NASA/JPL Workshop on the Physics of Planetary and Astrophysical Magnetospheres.National Research Council/Naval Research Laboratory Research Associate.     

Initial results from the ISEE-1 and -2 plasma wave investigation

In this paper we present an initial survey of results from the plasma wave experiments on the ISEE-1 and -2 spacecraft which are in nearly identical orbits passing through the Earth's magnetosphere at radial distances out to about 22.5R _e . Essentially every crossing of the Earth's bow shock can be associated with an intense burst of electrostatic and whistler-mode turbulence at the shock, with substantial wave intensities in both the upstream and downstream regions. Usually the electric and magnetic field spectrum at the shock are quite similar for both spacecraft, although small differences in the detailed structure are sometimes apparent upstream and downstream of the shock, probably due to changes in the motion of the shock or propagation effects. Upstream of the shock emissions are often observed at both the fundamental, f _- ^p , and second harmonic, 2f _p ^- , of the electron plasma frequency. In the magnetosphere high resolution spectrograms of the electric field show an extremely complex distribution of plasma and radio emissions, with numerous resonance and cutoff effects. Electron density profiles can be obtained from emissions near the local electron plasma frequency. Comparisons of high resolution spectrograms of whistler-mode emissions such as chorus detected by the two spacecraft usually show a good overall similarity but marked differences in detailed structure on time scales less than one minute. Other types of locally generated waves, such as the (n+1/2)f _- ^g electron cyclotron waves, show a better correspondence between the two spacecraft. High resolution spectrograms of kilometric radio emissions are also presented which show an extremely complex frequency-time structure with many closely spaced narrow-band emissions.     

Dissipation of magnetic fields in active galactic nuclei

A model for the emission processes causing rapid variability (less than one day) in active galactic nuclei is developed. Relativistic electron beams escape from reconnection sheets in coronae of accretion disks and excite plasma turbulence with a typical frequency , which depends on the electron number densityn (see also the contribution by R. van Oss). The finite lengths of different beams emerging from different reconnection sheets allows that the waves arecoherently scattered to frequencies ²_pe. For Lorentz factors 10³ and densities typical for disk coronaen10⁶ cm ^–3 (derived from iron line observations) one easily reaches the optical, frequency range. The time scale of the variability is then caused by the relaxation of the electron beams. Likewise, this model explains the very rapid variability in the X-ray (less than 10 minutes) by changing the parameters slightly. According to this scenario the higher the variable frequency is, the closer to the central black hole it should originate.     

Banded electron cyclotron harmonic instability-a first comparison of theory and experiment

A study, which is the first of its kind, uses information derived from simultaneously measured wave spectra and particle distributions as the input to a theoretical linear instability model of an electrostatic cyclotron harmonic wave event recorded on GEOS-1. The presence of a hot loss cone component of the particle distribution is established experimentally, and the model accounts reasonably for the observed frequencies and relative strengths of the (n+1/2)f _c and upper hybrid emission features.     

Radio plasma imager simulations and measurements

The Radio Plasma Imager (RPI) will be the first-of-its kind instrument designed to use radio wave sounding techniques to perform repetitive remote sensing measurements of electron number density (N _e) structures and the dynamics of the magnetosphere and plasmasphere. RPI will fly on the Imager for Magnetopause-to-Aurora Global Exploration (IMAGE) mission to be launched early in the year 2000. The design of the RPI is based on recent advances in radio transmitter and receiver design and modern digital processing techniques perfected for ground-based ionospheric sounding over the last two decades. Free-space electromagnetic waves transmitted by the RPI located in the low-density magnetospheric cavity will be reflected at distant plasma cutoffs. The location and characteristics of the plasma at those remote reflection points can then be derived from measurements of the echo amplitude, phase, delay time, frequency, polarization, Doppler shift, and echo direction. The 500 m tip-to-tip X and Y (spin plane) antennas and 20 m Z axis antenna on RPI will be used to measures echoes coming from distances of several R _E. RPI will operate at frequencies between 3 kHz to 3 MHz and will provide quantitative N _e values from 10^–1 to 10⁵ cm^–3. Ray tracing calculations, combined with specific radio imager instrument characteristics, enables simulations of RPI measurements. These simulations have been performed throughout an IMAGE orbit and under different model magnetospheric conditions. They dramatically show that radio sounding can be used quite successfully to measure a wealth of magnetospheric phenomena such as magnetopause boundary motions and plasmapause dynamics. The radio imaging technique will provide a truly exciting opportunity to study global magnetospheric dynamics in a way that was never before possible.     

Emission mechanism for low-frequency radiation in the outer heliosphere

The question of how low-frequency radio emissions in the outer heliosphere might be generated is considered. It is argued that the free energy contained in an electron beam distribution is first transformed into electrostatic Langmuir waves. The nonlinear interactions of these waves which can produce electromagnetic waves are then treated in the semi-classical formalism. Comparison of the results of the discussed model with electromagnetic radiation coming from upstream of the Earth's bow shock shows that the model adequately explains the generation of plasma waves at planetary shocks. By analogy, this model can provide a quantitative explanation of intensity of radio emissions at 2 to 3 kHz detected by the Voyager plasma wave instrument in the outer heliosphere provided that the electron beams generating Langmuir waves exist also in the postshock plasma due to secondary shocks in the compressed solar wind beyond the termination shock. The field strength of Langmuir waves required to generate the second harmonic emissions are approximately of 100–200 V m^–1 for the primary and 50–100 V m^–1 for the secondary foreshocks. However, only in the secondary foreshock the expected density is consistent with the observed frequency.     

Reception of the NKL (Jim Creek) transmitter onboard GEOS-1

At the beginning of the GEOS lifetime, some attempts have been made for taking advantage of the passes over Alaska. GEOS was then commanded in a fixed mode and the corresponding telemetry data were recorded at the NASA stations. For two passes over Jim Creek (48°2N–121°9W) where a powerful VLF transmitter (f ₀ = 18.6 kHz) is located, GEOS was put in a specific mode in order to study the magnetospheric electromagnetic field in the vicinity of f ₀. The results of one pass (June 11, from 0755 UT) are presented here.During this pass, a strong enhancement of all the e.m. components at f ₀ has been observed for a specific period of time, when GEOS was very near to the exact conjugacy with NKL. The distance, as measured on the ground, over which the signal was above -6 dB from the maximum is of the order of 800 km. During the corresponding period of time (0740–0750 UT), the satellite altitude varied between 8000 and 6000 km. The magnetospheric region where the signal is strong appears to be structured, as if there were many ducts.Preliminary results concerning the polarization characteristics of the signal are presented. In the absence of precise measurements of these characteristics, the comparison between the electric and magnetic components of the received signal is not easy to interpret. An examination of the onboard computed correlograms (in the frequency range from f ₀ -0.6 kHz to f ₀ +3.3 kHz) shows that, for this pass, no VLF emissions were triggered by NKL, at the altitude of the satellite.     

Plasma waves at the dayside magnetopause

Experimental investigations of plasma waves at the magnetopause, including recent results from the AMPTE/IRM satellite, show that both E and B fluctuations typically have a featureless spectrum which monotonically decreases with frequency; integrated rms amplitudes are typically a few mV m^-1 for E and 10 nT for B, though in particular E can be as much as an order of magnitude larger in exceptional cases. Surveys show a lack of correlation between wave parameters and the magnetopause parameters. Under the assumption that crossing the diffusion region would give a pronounced signature in the waves, the survey data allow an upper limit to be placed on the latitudinal extent of the diffusion region, which is about 1000 km — implying that it is not surprising that the wave data surveys have so far failed to detect it. The observed wave turbulence levels have been used to estimate diffusion coefficients under different assumptions for the wave mode, but the resulting diffusion coefficient is always too small to explain either reconnection or boundary layer formation. Recent work of Galeev et al. (1986) indicates that the dominant diffusion process may be magnetic field migration, which is a macroscopic process involving the interaction of tearing mode islands. Assuming this mode to be present at the observed level of B, a particle diffusion coefficient of nearly 10⁹ m² s^-1 is obtained. Another macroscopic diffusive process which could occur at the magnetopause is stochastic E × B scattering, which also implies a diffusion coefficient the order of 10⁹ m² s^-1 if the observed E spectrum is assumed to be a turbulent cascade consisting of convective cells.     

Electric currents and voltage drops along auroral field lines

The current state of knowledge concerning Birkeland currents (j _∥) and parallel electric field (E _∥) is briefly reviewed. Four types of j _∥ are discussed-the primary ‘region 1’ sheets, the ‘region 2’ sheets which parallel them and which seem to close in the partial ring current, the cusp currents which appear to correlate with interplanetary B _y, and the ‘Harang filament’. The energy required by E _∥ and by the associated particle acceleration processes seems to be derived from j _∥. Much of the evidence for e _∥ comes from particles, from ‘inverted V’ spectra, rising ion beams and expanded loss cones, while ‘conies’ may signify acceleration by Electrostatic Ion Cyclotron (EIC) waves, associated with beams accelerated by E _∥. Different theoretical studies predict for E _∥ a smooth, disordered or abrupt structure, and evidence for all 3 types can be deduced from S3-3 electric field probe observations.

     

Mechanisms for the discrete aurora — A review

The V-shock is identified as the primary mechanism for the acceleration of electrons responsible for the discrete aurora. A brief review of the evidence supporting the V-shock model is given, including the dynamics of auroral striations, anomalous motion of barium plasma at high altitudes and in-situ observations of large electric fields. The V-shock is a nonlinear, n = 0 ion cyclotron mode soliton, Doppler shifted to zero frequency. The V-shock is also shown to be a generalization of the one-dimensional double layer model, which is an ion acoustic soliton Doppler shifted to zero frequency. The essential difference between the double layer theory and the theory for the oblique, current-driven, laminar electrostatic shock is that the plasma dielectric constant in directions perpendicular to the magnetic field is c ²/V _a ^/2 , where V _a is the Alfvén velocity; but the plasma dielectric constant parallel to the magnetic field is unity. Otherwise, in the limit that the shock thickness perpendicular to the magnetic field is much larger than an ion gyroradius, the equations describing the double layer and the oblique shock are the same. The V-shock, while accounting for the acceleration of auroral electrons, requires an energy source and mechanism for generating large potential differences perpendicular to the magnetic field. An energy source is the earthward streaming protons coming from the distant magnetospheric tail. It is shown how these protons can be energized by the cross-tail electric field, which is the tailward extension of the polar cap dawn-to-dusk electric field. The local, large cross-field potential differences associated with the V-shock are seen to be the result of a non-linear, E × B drift turbulent cascade which transfers energy from small- to large-scale sizes. Energy at the smallest scale sizes comes from the kinetic energy in the ion cyclotron motion of the earthward streaming protons, which are unstable against the zero-frequency flute-mode instability. The review points out the gaps in our understanding of the mechanism of the diffuse aurora and the mechanism of the auroral substorm.     

Substorm Trigger Processes in the Magnetotail: Recent Observations and Outstanding Issues

The present article reviews recent studies about near-Earth substorm processes. A focus is placed on the relationship between two fundamental processes, that is, tail current disruption (TCD) and the formation of a near-Earth neutral line (NENL). The former is inferred to cause dipolarization, and the latter is often associated with the fast plasma flow in the plasma sheet. Whereas it is inferred from the directions of fast plasma flows that the NENL is formed at 20–30 R _E from the Earth, dipolarization is most manifest in the near-Earth (6.6–12 R _E) region. The observation of the fast plasma flow prior to substorm (Pi2) onsets favors the idea that the NENL is formed first and dipolarization is the effect of the pile-up of magnetic flux convected earthward from the NENL, which is called the pile-up model. The present paper addresses several outstanding issues regarding this model, including (1) the interpretation of plasma flow deceleration in terms of the flux pile up, (2) highly irregular magnetic fluctuations observed in the near- Earth region, (3) the spatial coherency of the fast plasma flow, (4) the spatial structure and expansion of dipolarization region, and (5) the explosive growth phase. The paper also proposes the possibility that TCD is an independent process, but the formation of the NENL sets a favorable condition for it.     

Interaction of non-perpendicular/parallel solar wind shock waves with the earth's magnetosphere

The interaction of travelling interplanetary shock waves with the bow shock-magnetosphere system is considered. We consider the general case when the interplanetary magnetic field is oblique to the Sun-planetary axis, thus, the interplanetary shock is neither parallel nor perpendicular. We find that an ensemble of shocks are produced after the interaction for a representative range of shock Mach numbers. First, we find that the system S ₊ R _– CS _– S ₊ appears after the collision of travelling fast shock waves S ₊ (Mach number M = 2 to 7) with the bow shock. Here, S _– and R _– represent the slow shock wave and slow rarefaction wave, and C represents the contact surface. It is shown that in the presence of an interplanetary field that is inclined by 45° to the radial solar wind velocity vector, the waves R _– and S _– are weak waves and, to the first degree of approximation, the situation is similar to the previously studied normal perpendicular case. The configuration, R ₊ C _m S _– S ₊ or R ₊ C _m R _– S ₊ where C _m is the magnetopause, appears as the result of the fast shock wave's collision with the magnetopause. In this case the waves S _– and R _– are weak. The fast rarefaction wave reflected from the magnetosphere is developed similar to the case for the collision of a perpendicular shock. The shock wave intensity is varied for Mach numbers from 2 to 10. Thus, in the limits of the first approximation, the validity of the one-dimensional consideration of the nonstationary interaction of travelling interplanetary shock waves with the bow shock-magnetosphere system is proved. The appearance of the fast rarefaction wave, R ₄, decreasing the pressure on the magnetosphere of the Earth after the abrupt shock-like contraction, is proved. A possible geomagnetic effect during the global perturbation of the SSC or SI⁺ type is discussed.An invited paper presented at STIP Workshop on Shock Waves in the Solar Corona and Interplanetary Space, 15–19 June, 1980, Smolenice, Czechoslovakia.     

The Electric Antennas for the STEREO/WAVES Experiment

The STEREO/WAVES experiment is designed to measure the electric component of radio emission from interplanetary radio bursts and in situ plasma waves and fluctuations in the solar wind. Interplanetary radio bursts are generated from electron beams at interplanetary shocks and solar flares and are observed from near the Sun to 1 AU, corresponding to frequencies of approximately 16 MHz to 10 kHz. In situ plasma waves occur in a range of wavelengths larger than the Debye length in the solar wind plasma λ _D ≈10 m and appear Doppler-shifted into the frequency regime down to a fraction of a Hertz. These phenomena are measured by STEREO/WAVES with a set of three orthogonal electric monopole antennas. This paper describes the electrical and mechanical design of the antenna system and discusses efforts to model the antenna pattern and response and methods for in-flight calibration.     

Physics of Substorm Growth Phase,Onset, and Dipolarization

A new scenario of substorm growth phase, onset and dipolarization during expansion phase and the corresponding physical processes are presented. During the growth phase, as a result of enhanced plasma convection, the plasma pressure and its gradient continue to be enhanced over the quiet-time values in the plasma sheet. Toward the late growth phase, a strong cross-tail current sheet is formed in the near-Earth plasma sheet region, where a local magnetic well is formed. The equatorial plasma beta (β _eq ) can reach a local maximum with value larger than 50 and the cross-tail current density can be enhanced to over 10nA/m² as obtained from 3D quasi-static magnetospheric equilibrium solutions for the growth phase. The most unstable kinetic ballooning instabilities (KBI) are expected to be located in the tailward side of the strong cross-tail current sheet region. The field lines in the most unstable KBI region map to the transition region between the region-l and region-2 currents in the ionosphere, which is consistent with the observed initial brightening location of the breakup arc in the intense proton precipitation region. The KBI explains the AMPTE/CCE observations that a low frequency instability with a wave period of 50–75 seconds is excited about 2–3 min prior to substorm onset and grows exponentially to a large amplitude at the onset of current disruption (or current reduction). At the onset of current disruption higher frequency instabilities are excited so that the plasma and electromagnetic field fluctuations form a strong turbulent state. Plasma transport takes place due to the strong turbulence to relax the ambient plasma pressure profile so that the plasma pressure and current density are reduced and the ambient magnetic field intensity increases by more than a factor of 2–3 in the high-β _eq region and the field line geometry recovers from tail-like to dipole-like – dipolarization.     

Study of magnetospheric dynamics using energetic solar particles

Information can be obtained from energetic particle measurements through the chemical composition, energy spectrum, directional anisotropy, temporal and spatial intensity variations. This is equivalent to saying that there is a distribution functionf _k(p,r,t) wherek corresponds to thekth particle species of momentump at positionr and timet.Particle transport is described by the Boltzmann equation, and because the densities are generally low in the case of cosmic rays or energetic solar flare particles, collective transport effects can be neglected. In the absence of magnetospheric motion it is relatively easy to treat the problems of particle transport as simple propagation of charged particles in a stationary magnetic field configuration using, for instance, trajectory calculations in model fields. The method here is to use correlated measurements of the particle distribution at two points along a dynamic trajectory, and in this way to learn something about the geomagnetic field. This approach provides a good basis from which to study magnetospheric dynamics. If the magnetosphere moves, large scale electric fields, turbulent electromagnetic fields and sources and sinks affect the propagation of energetic particles considerably. These effects change the distribution functionf _k(p,r,t) and can thus be detected.In this paper, we shall show the importance of the single particle approximation (trajectories in a reference field) in forming the basis of our understanding of the quiet-time penetration of cosmic rays into the magnetosphere, we shall consider the steady dynamics such as wave-particle inter-action and field line reconnection, which is believed to exist nearly all the time, and finally we shall review the work which has been done in the much more complex and less well-understood field of impulsive dynamics such as geomagnetic storms and substorms. This last topic is only just beginning to be investigated in detail, and it is hoped that the study of impulsive dynamics, using energetic particles, may be as successful as the study of the quiet magnetosphere and the steady dynamics.     

Stimulation of VLF amplification in the magnetosphere

Among the various plasma instabilities that exert influence on the dynamic equilibrium state of the magnetosphere, the cyclotron-resonance interaction appears to be the most accessible to artificial stimulation. The strength of the interaction is sensitive to both the background magnetoplasma parameters and the hot energetic particle distribution. Thus, proper modification of one or more conditions can induce significant wave amplification at the expense of hot plasma energy density. Several methods of hot and cold plasma injection have been investigated with the linear theory to assess their effectiveness as a means of stimulating amplification.Only the interaction of VLF waves (3–30 kHz) with hot electrons (0.1–100 keV) is treated here. The injection of a dense jet of barium that travels upward along the geomagnetic field causes appreciable amplification when the jet is within 30° of the geomagnetic equator. Injection of a geosynchronous lithium cloud stimulates amplification of both VLF and ULF waves, but the magnitude depends critically on the state of geomagnetic activity. Conventional hot electron beams may also amplify narrow frequency bands, but the net wave energy is severely limited by the beam energy.Although the cyclotron-resonance is recognized as a dominant interaction in magnetospheric dynamics, its properties have never been confirmed quantitatively by appropriate spacecraft experiments. Controlled injections would provide important insight into this fundamental process because the induced amplification has a well-defined signature.     

Some problems of gamma-astronomy

The modern state of gamma-ray astronomy is reviewed, the paper being mainly devoted to the theoretical models that describe generation of gamma-ray emission under astrophysical conditions. Basic information on the processes of generation and absorption of gamma-rays, as well as the results of observations for various gamma-ray photon energies are reported.In the region of soft gamma-ray emission (i.e., for energies less than tens of MeV), where emission in gamma-ray lines dominates, we also discuss the nature of gamma-ray bursts, the origin of gamma-ray emission from the galactic centre, etc.Discrete sources and, in particular, the mysterious source Cyg X-3 are discussed in the region of very high (E > 10¹² eV) and ultra-high (E > 10¹⁵ eV) energy gamma-ray emission.A larger portion of the review is devoted to the analysis of cosmic-ray origin on the basis of the available gamma-ray data in the region from several tens of MeV to several GeV. The peculiarity of this energy range is, in particular, in the fact that the diffuse galactic emission was observed mainly there. We also discuss the problem of determination of the cosmic-ray density gradient from the gamma-ray data.The origin of high-latitude gamma-ray emission, the problem of galactic gamma-ray halo, etc., are discussed.The theoretical models explaining the nature of unidentified gamma-ray sources, as well as the results of measurements and theoretical estimations of a gamma-ray flux from SN1987A are analysed.List of Notations m electron mass, m = 9.108 × 10^–28 g, - M proton mass, M = 1.672 × 10^–24 g, - e electron charge, e = 4.803 × 10^–10 CGS - c velocity of light, c = 2.9979 × 10¹⁰ cm s^–1, - k Boltzmann constant, k = 1.380 × 10^–16 erg grad^–1, - e electron - p proton - gamma-ray photon - p antiproton - ^{0 0}-meson - -lepton - e ⁺ positron - r, , x radio-frequency, gamma-ray, and X-ray emission bands - E total energy of a particle - E _k kinetic energy - p particle momentum - spectral index for particles - spectral index for emission - n particle density (concentration) - H magnetic field strength - T temperature - _ph energy of low-energy photons - emission frequency - r _H Larmor radius of relativistic particles - k wave number - , z cylindric coordinates, in this case the coordinate (radius) along the galactic disk, z perpendicular to the galactic disk - M solar mass, M = 1.99 × 10³³ g.