Energy losses of solar cosmic rays in interplanetary space

A simple model of solar cosmic ray propagation which includes diffusion, convection, and energy loss by adiabatic deceleration is studied. A Monte Carlo technique is employed to investigate the variation of mean particle energy in the interplanetary medium after the impulsive release of mono-energetic particles at the Sun. At 1 AU typical energy losses are 43% at 20 h and 64% at 60 h after particle release for a diffusion coefficient (r)= ₀r with =+1/2 and ₀=1.33 × 10²¹ cm² s^–1. When ₀ in this model is reduced by a factor of 4, the energy loss is greater by a factor of 2 at 60 h after particle release. When is increased, the energy losses are greater. Using the model parameters above, an increase in solar wind speed from 300 to 600 km s^–-1 gives rise to energy losses that are greater again by factor of 2 at a time of 60 h. Results are compared with an observation by Murray et al. (1971) of a knee in the energy spectrum of solar protons. It is not considered likely that the change in the energy of the knee with time requires, in addition to adiabatic deceleration, another energy change process which acts to increase the energy of particles.Part of this work was performed while the author was at CSIRO, Division of Radiophysics, Epping, NSW, Australia; also supported in part by the U.S. Atomic Energy Commission.     

Energy losses of galactic cosmic rays in the interplanetary medium

Using realistic models of cosmic-ray propagation in interplanetary space we present, for electrons, protons and helium nuclei of a given energy near Earth, calculations of their distribution in energy before entering the solar cavity and their mean energy loss. Interplanetary conditions appropriate for the epochs 1965 and 1969 have been used. Cosmic-ray energies in the range of 20 MeV/nucleon to 1000 MeV/nucleon have been considered.     

Convective transport of low energy cosmic rays in the interplanetary region

A complete solution has been obtained of the steady-state transport equations, including energy losses, for cosmic-rays in the interplanetary region for conditions in which diffusive transport is negligible and convective effects dominate. The region of validity of the solution will in general be a shell between heliocentric radiiR ₁ andR ₂ (R ₂ may be infinite). The precise range of kinetic energyT and heliocentric radiusr in which the solution is valid is not known but it appears to be applicable in the vicinity of Earth to protons withT≤1 MeV. ForT～0.5 MeV near Earth,R ₁ may be ～0.5 AU andR ₁ will decrease asT, observed near Earth, decreases. The solution is simple in form but quite general; it predicts the differential number densityU (r, T) in terms of that observed at radius a (near Earth, say). Thus it may be quite useful in interpreting and co-ordinating steady-state cosmicray observations atT～1 MeV. The differential and integral intensities, differential anisotropy and differential radial-gradient at (r, T) also are determined. A simple interpretation of the solution is given in terms of energy losses due to adiabatic deceleration of the particles as they are being convected outward from the Sun. This leads to the useful notion of following a particle in (r, T) as it increasesr and decreasesT. Particles convected from the outer corona to Earth decrease their kinetic energy by factor ～500.Following a particle the Compton-Getting factor remains constant. Particles observed at (a, T) in convective transport have come from nearer the Sun; they may be of solar origin but may also be of galactic origin having penetrated tor<R ₁   

Radial gradients and anisotropies of cosmic rays in the interplanetary medium

Approximate equations which describe the behavior of cosmic rays in the interplanetary medium under suitable conditions are used to make comparisons between observations and theoretical predictions of radial gradients and radial anisotropies. In the high energy region there appear to be no inconsistencies between theory and observations. In the low energy region it is shown that theoretical predictions of the radial anisotropy expected from large radial gradients of the intensity are not inconsistent with observed radial anisotropies. However, in the latter case there are other inconsistencies, which suggest that some aspects of the observations or of the theory (or both) are unsatisfactory.NAS NASA Resident Postdoctoral Research Associate.     

Propagation of solar-flare cosmic rays along corotating interplanetary flux-tubes

The propagation of solar-flare cosmic rays along a corotating spiral mean interplanetary magnetic flux-tube is investigated under a variety of initial, inner and outer boundary conditions, by solving the transport equations numerically. The inner boundary condition is shown to have negligible effects on the redistribution of the cosmic rays in interplanetary space, apart from a proportionality constant. The diffusion coefficients have been classified into type I and type II: for type I the peakalong-spiral stays at a finite heliocentric distance, while for type II it travels to infinity. The time dependence of the anisotropy vector is characteristically different in each type and in the case of type I is a function of the observation point. The general propagation characteristics with a free-escape outer boundary are like those of type I. It is shown that closely exponential decay can be produced without a free-escape boundary. The concept of an ‘equilibrium shape’ at late time is introduced. With a source that decays slowly, a maximum and minimum pair appears in the spatial distribution; the maximum subsequently propagates outward and the minimum inward. The impulsive models given do not reproduce satisfactorily all the observed characteristics of solar burst events and a solar source extended in time appears to be indicated.     

Measurements of the interplanetary magnetic field in relation to the modulation of cosmic rays

Field strength distributions and low frequency power spectra are derived from interplanetary field measurements made by the HEOS-1 and HEOS-2 satellites during the years 1969–1973. The spectral analysis involved the use of a technique which is shown to allow correctly for missing data. Comparison spectra, derived by the same technique, are presented for the years 1963–1968. The use of mear-field-aligned co-ordinates enabled the easy separation of the transverse and longitudinal fluctuation spectra. A power law function involving a ‘break point’-frequency was fitted to each spectrum by a least squares technique. The total power level, the power spectral density at zero frequency and the correlation length are found to vary significantly and in a similar way over the solar cycle. The magnitude and phase of these variations are compared with measurements of the cosmic ray neutron monitor rate and the coronal green line intensity and the influence of mid-latitude solar phenomena on the character of the interplanetary field in the ecliptic is demonstrated. The correlation length and zero frequency power density are found to be considerably larger than previously estimated and, contrary to the usual assumption in modulation theory, the rms amplitude of the perturbation field is comparable to the mean field experienced by the high rigidity particles. Although the mean interplanetary field strength is found to be independent of the level of solar activity, during higher activity the most probable vector average decreases by approximately 0.5 γ due to the enhanced directional fluctuation in the field. Power anisotropy measurements suggest that Alfvénic disturbances in the solar wind have fluctuation spectra confined mainly to frequencies larger than 10^?3 Hz. The data are interpreted as indicating that the cosmic ray intensity in the Galaxy is some 75% larger than the intensity recorded by neutron monitors on Earth. Previous failure to find a correlation between neutron monitor intensity and interplanetary field parameters is attributed to a lack of statistical accuracy in the field data. The measured power spectra are used to estimate the magnitude of the parallel diffusion coefficient using the relationships derived by Klimas and Sandri, Jokipii, and Quenby et al.     

Evidence for confinement of low-energy cosmic rays ahead of interplanetary shock waves

Short-lived ( 15 min), low-energy proton increases associated with the passage of interplanetary shock waves have been previously reported. In the present paper, we have examined in a fine time scale ( 1 min) the concurrent particle and magnetic field data, taken by detectors on Explorer 34, for four of these events which occurred on 30 May 1967, 5 June 1967, 29 November 1967, and 11 January 1968. Our results further support the view that these impulsive events are due to confinement of the solar cosmic-ray particles in the region just ahead ( 10⁶ km) of the advancing shock front. Data from the Pioneer 7 spacecraft for a similar event on 30 August 1966, when this spacecraft was 1.9 × 10⁶ km from the Earth, are shown to be consistent with this interpretation.     

Diffusion tensor for Galactic cosmic rays in the interplanetary medium with a magnetic field

The Boltzmann kinetic equation is analyzed in the MHD approximation. This analysis requires an explicit expression for the collision integral F _c. In the classical theory, F _c=?vf _μ ⁽¹⁾ Ω_μ, where f _μ ⁽¹⁾ is the first spherical harmonic in the Galactic-cosmic-ray (GCR) distribution, Ω_μ are the components of a unit particle velocity vector, and the frequency ν of collisions between GCRs and interplanetary magnetic-field nonuniformities is assumed to be a scalar. The assumption that ν_ij is a tensor (which is the result of anisotropy in the interplanetary medium) distinguishes this study from others. Since the anisotropic GCR effects in the heliomagnetosphere are marginal, the nondiagonal elements of tensor ν_ij were set equal to zero. Our analysis has yielded the diffusion-tensor components D _∥, and D _A, which are expressed in terms of interplanetary parameters. The energy dependencies of D _∥, and D _A are in good agreement with the experimental data and calculations by other authors.     

Energy losses and modulation of galactic cosmic rays

Numerical solutions of the cosmic-ray transport equation for the interplanetary region and including the effects of diffusion, convection, and energy loss, have been obtained for a galactic differential number-density spectrum of gaussian form, central kinetic energyT ₀, and a width at half height maximum of 0.1T ₀. These solutions have been used to show the redistribution in energy, and the reduction in number density, within the solar cavity. The mean energy loss is shown to be usefully approximated by the force-field potential when /T ₀1/2. The principal finding is that galactic cosmic-ray protons and helium nuclei, with kinetic energies less than about 80 MeV/nucleon, virtually, are completely excluded from near Earth by convective effects. As a result of this exclusion, it is found that it is not possible to demodulate the proton and helium-nuclei spectra, observed in 1964–65, to obtain information about low-energy galactic cosmic rays.     

Spatial dependence of the pitch-angle and associated spatial diffusion coefficients for cosmic rays in interplanetary space

The spatial dependence of the pitch-angle and associated spatial diffusion coefficients for cosmic ray particles in interplanetary space is calculated in the WKB approximation. The model considers only Alfven waves of solar origin to be responsible for scattering of moderate energy particles. After developing the general theory results are presented for the asymptotic case corresponding to radial distancesr greater than about 1 to 2 AU. The radial diffusion coefficient _r increases with energyE like _r E , wherev2/3. The radial mean free path turns out to increase proportional tor ³ at medium and low heliographic latitudes. This behaviour is consistent with a very small radial cosmic ray gradient and the existence of a free boundary for particle diffusion. At equal radial distances the high latitude mean free path is not only much smaller than the one calculated at the lower latitudes but in addition increases only weakly with distance. Some conceivable dynamical implications for the outer solar system are indicated.     

The effect of a bounded interplanetary diffusion medium on the propagation of solar flare cosmic rays

An analysis of solar flare data indicates that the graph of log(nt ^3/(2–)) deviates late in the solar event from the straight line predicted for the infinite, unbounded interplanetary medium. It is shown by mathematical analysis, utilizing a model based on the radial diffusion coefficient D = Mr , with 1, that the deviation can be ascribed to the loss of flare particles through an external boundary at about 5–6 AU from the Sun. An inner region terminating at 5–6 AU, followed by an extensive region of increasingly less resistance to the diffusion of flare particles is also feasible and it is shown that measurements taken at the Earth cannot predict the extent of this outer region. The results are applicable to either the isotropic or highly anisotropic models. The constant diffusion model is shown to be inadequate since it requires a boundary 1.5 AU from the Sun. In view of the present and previous studies of solar flare data, it is asserted that the fundamental principle governing the diffusion of solar flare particles through interplanetary space is the radial diffusion coefficient mode of propagation.     

Sector structure of the interplanetary magnetic field and anisotropy of 50–1000 GV cosmic rays

It is demonstrated that, at high rigidities (50 GV and beyond), all the main features of cosmic-ray anisotropy of solar origin can be explained in terms of regular particle motion —without diffusion being involved — in the large-scale interplanetary magnetic field (IMF). A simple model of the IMF is adopted with a corotating warped neutral sheet separating the regions of alternative polarities; the warped shape is indispensable for obtaining any form of anisotropy. Energy losses occurring along various computed trajectories are calculated to give the sidereal, solar and antisidereal intensity waves. The reliability of the variations obtained are checked by changing the parameters of the IMF model. Both the sense and amplitude of the polarity-dependent sidereal vector are compatible with those established experimentally. Also reproduced are the predictions of corotation in addition to the 3-hour phase of the semi-diurnal wave. The corotation is found to be near perfect at 50 GV, while it reduces at 100 GV. The model presented accounts for the change of solar daily vector that was observed in 1969.     

Heliospheric propagation of galactic cosmic rays depending on the scattering characteristics of the turbulent interplanetary magnetic field

The effect of the solar wind on the spectrum of cosmic rays accelerated in the Galaxy is studied. The coefficient of cosmic-ray diffusion in the interplanetary turbulent magnetic field is assumed to be independent of the particle energy and a power-law function of the distance from the Sun. The particle spectrum at the heliospheric boundary is specified as a power-law function of the total particle energy.     

Influence of finite injections and of interplanetary propagation on time-intensity and time-anisotropy profiles of solar cosmic rays

For an observer in space the intensities and anisotropies of solar cosmic-ray events are governed by the duration and the functional shape of the injection processes near the Sun and by the propagation along the interplanetary magnetic field from the Sun to the observer. We study the influence of four different types of solar injections (Gaussian, exponential, step-function and coronal diffusion), and of a purely diffusive interplanetary propagation, where the diffusion coefficient has a power law dependence on the radial distance from the Sun, =Mr on both the time-intensity and the time-anisotropy profiles at 1 AU. The main results are as follows: A slow quasi-exponential decay of the intensity can be modelled in some cases; all finite injections produce high anisotropies during the main phase of an event; an effective solar injection length can be determined from simultaneous inspection of the intensities and anisotropies; the intensities and anisotropies do to first order not depend on the analytic shape of the various injection profiles. The model is applied to the November 18, 1968 solar event as observed by Pioneer 9 in the 7.5–21.5 MeV and 21.5–60 MeV energy channels. We obtain local diffusion coefficients in the range M= (2.5–5) × 10²¹ cm² s^–1 and injection periods of the order of 10–20 hr. Closer inspection reveals the change of interplanetary propagation conditions during the event.     

On the acceleration of cosmic rays

The intensive acceleration of energetic charged particles in perpendicular shock waves which has been known to take place in the interplanetary medium has been utilized in this work in order to account for the energization of cosmic rays. It is proposed that cosmic rays can be accelerated up to 10¹⁴–10¹⁵ eV in successive perpendicular shock waves which appear inside supernova shells in our Galaxy.     

Understanding galactic cosmic rays

The case is made for most cosmic rays having come from galactic sources. ‘Structure’, i.e. a lack of smoothness in the energy spectrum, is apparent, strengthening the view that most cosmic rays come from discrete sources, supernova remnants being most likely.     

Relativistic heavy cosmic rays

During three balloon flights of a 1 m² sr ionizationchamber erenkov counter detector system, we have measured the atmospheric attenuation, flux, and charge composition of cosmic-ray nuclei with 16Z30 and rigidity greater than 4.5 GV.The attenuation mean-free-path in air of VH (20Z30) nuclei is found to be 19.7±1.6 g cm^–2, a value somewhat greater than the best previous measurement. The attenuation mean-free-path of iron is found to be 15.6±2.2 g cm^–2, consistent with predictions of geometric cross-section formulae.We measure an absolute flux of VH nuclei 10 to 20% higher than earlier experiments at similar geomagnetic cutoff and level of solar activity. The relative abundances of evencharged nuclei are found to be in good agreement with results of other recent high-resolution counter experiments.We calculate that our observed cosmic ray chemical composition implies relative abundances at the cosmic-ray source of Ca/Fe=0.12±0.04 and S/Fe=0.14±0.05. The results are consistent with all other elements of charge between 16 and 26 being absent at the source and being produced by cosmic-ray fragmentation in interstellar hydrogen. The results show the ratios A/Fe and S/Fe to be significantly lower in the cosmic-ray source than in the solar system.