A study of the magnetic helicity during eight HELIOS-observed solar proton events

We have studied the influence of the magnetic helicity on solar particle propagation using the IMF data observed by the HELIOS spacecraft in the range 0.31–0.95 AU, during eight solar proton events. For this, we have derived power and helicity spectra of the turbulence of the magnetic field during the time of the events. These are used to compute the particle pitch-angle scattering coefficients according to the quasi-linear theory (QLT) treatment of particle propagation in turbulent magnetic fields. The results show that in all the cases the helicity effects are negligible and the particle's mean free paths deduced from the pitch-angle diffusion coefficients are the same regardless of whether or not helicity effects are included in the calculations. The computed mean free paths are quite different in each case.Deceased 10 April, 1995.     

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.     

Weak lensing in the second post-Newtonian approximation: gravitomagnetic potentials and the integrated Sachs–Wolfe effect

Dark matter currents in the large-scale structure give rise to gravitomagnetic terms in the metric, which affect the light propagation. Corrections to the weak-lensing power spectrum due to these gravitomagnetic potentials are evaluated by perturbation theory. A connection between gravitomagnetic lensing and the integrated Sachs–Wolfe (iSW) effect is drawn, which can be described by a line-of-sight integration over the divergence of the gravitomagnetic vector potential. This allows the power spectrum of the iSW-effect to be derived within the framework of the same formalism as derived for gravitomagnetic lensing and reduces the iSW-effect to a second-order lensing phenomenon. The three-dimensional power spectra are projected by means of a generalized Limber-equation to yield the angular power spectra. Gravitomagnetic corrections to the weak-lensing spectrum are negligible at currently accessible scales, and cosmic-variance considerations suggest that the detection of the iSW-effect's contribution to the cosmic microwave background angular power spectrum is too small to be detectable at multipoles probed by the Planck satellite.     

Motion of energetic particles in a magnetospheric model including a convection electric field

A magnetospheric model including convection and corotation electric fields is developed. Drift shells and pitch-angle evolution are obtained through perturbation methods. The asymmetry of the shell is maximum in a plane different from the noon-midnight meridian. As a consequence the dip in fluxes is not maximum in this meridian and the amount of shift is shown to be directly proportional to the convection electric field intensity. It is inversely proportional to the energy or the pitch-angle of the particles.     

Differential rotation and sector structure of solar magnetic fields

The differential rotation and sector structure of solar magnetic fields has been studied using digitized data on photospheric magnetic fields recorded at the Mount Wilson Observatory during the period August 1959–May 1970. The power spectra show considerable power in high-frequency peaks, corresponding to harmonic components with wavelengths less than 1/10 solar rotation. Calculations for a series of shorter time intervals show how the distribution of power over the various harmonic components in the sector pattern varies strongly with the solar cycle. The equatorial rotation rate of solar magnetic fields is about 0.1 km s^-1 faster than that of the photospheric plasma determined from Doppler shifts. It is shown that the Doppler measurements mainly refer to the non-network regions. The differential flow of 0.1 km s^-1 forms streamlines around the magnetic fine structures. The different rotation rates of various solar features can be explained in terms of the rotation rates of magnetic and non-magnetic regions. The rotation rates of the magnetic fields in active and quiet regions agree at the equator. At higher latitudes, however, the background fields deviate less from solid-body rotation, indicating that their source is below the deepest layers to which the sunspot magnetic fields penetrate. This suggests that turbulent diffusion of the field in old active regions may not be the main source for the background magnetic field, but that the source is located close to a rigidly rotating solar core with a synodic rotation period of 26.87 days.     

Time-dependent transport of energetic particles in magnetic turbulence: computer simulations versus analytical theory

We explore numerically the transport of energetic particles in a turbulent magnetic field configuration. A test-particle code is employed to compute running diffusion coefficients as well as particle distribution functions in the different directions of space. Our numerical findings are compared with models commonly used in diffusion theory such as Gaussian distribution functions and solutions of the cosmic ray Fokker–Planck equation. Furthermore, we compare the running diffusion coefficients across the mean magnetic field with solutions obtained from the time-dependent version of the unified non-linear transport theory. In most cases we find that particle distribution functions are indeed of Gaussian form as long as a two-component turbulence model is employed. For turbulence setups with reduced dimensionality, however, the Gaussian distribution can no longer be obtained. It is also shown that the unified non-linear transport theory agrees with simulated perpendicular diffusion coefficients as long as the pure two-dimensional model is excluded.     

Saturated magnetic field amplification at supernova shocks

Cosmic ray streaming instabilities at supernova shocks are discussed in the quasi-linear diffusion formalism which takes into account the feedback effect of wave growth on the cosmic ray streaming motion. In particular, the non-resonant instability that leads to magnetic field amplification in the short wavelength regime is considered. The linear growth rate is calculated using kinetic theory for a streaming distribution. We show that the non-resonant instability is actually driven by a compensating current in the background plasma. The non-resonant instability can develop into a non-linear regime generating turbulence. The saturation of the amplified magnetic fields due to particle diffusion in the turbulence is derived analytically. It is shown that the evolution of parallel and perpendicular cosmic ray pressures is predominantly determined by non-resonant diffusion. However, the saturation is determined by resonant diffusion which tends to reduce the streaming motion through pitch angle scattering. The saturated level can exceed the mean background magnetic field.     

Cosmic-Ray Momentum Diffusion in Magnetosonic versus Alfvénic Turbulent Field

Energetic particle transport in a finite amplitude magnetosonic and Alfvénic turbulence is considered using the Monte Carlo particle simulations, which involve integration of particle equations of motion. We show that in the low- plasma the cosmic-ray acceleration can be the most important damping process for magnetosonic waves. Assuming such conditions we derive the momentum diffusion coefficient Dp, for relativistic particles in the presence of anisotropic finite-amplitude turbulent wave fields, for flat and Kolmogorov-type turbulence spectra, respectively. We confirm the possibility of larger values of Dp occurring due to transit-time damping resonance interaction in the presence of isotropic fast-mode waves in comparison to the Alfvén waves of the same amplitude (cf. Schlickeiser and Miller, 1997). The importance of quasi-perpendicular fast-mode waves is stressed for the acceleration of high velocity particles.     

Solar magnetic fields and convection

Recent developments in solar dynamo and other theories of magnetic fields and convection are discussed and extended. A basic requirement of these theories, that surplus fields are eliminated by turbulent or eddy diffusion, is shown to be invalid. A second basic requirement, that strong surface fields are created by granule or supergranule motions, is shown to be improbable. Parker's new thin-filament dynamo, based on the Petschek mechanism, is shown to provide the alternative possibilities: either the magnetic fields halt all convection or a steady state is reached in which the fields are a tangle of long, thin filaments. From the above and other considerations it is concluded that the dynamo and related diffuse-field theories are unacceptable, that solar magnetic fields are not dominated by convection, and that all the fields emerge as strong, concentrated fields (flux ropes) which were wound and twisted from a permanent, primordial field. The discussion may, incidentally, provide the physical elements of a deductive theory of hydromagnetic convection.     

Recovery of the cosmological peculiar velocity from the density field in the weakly non-linear regime

Using third-order perturbation theory, we derive a relation between the divergence of the peculiar velocity and the density. Specifically, we compute the expectation value of the divergence given density. Our calculations assume Gaussian initial conditions and are valid for Gaussian filtering of the evolved density and velocity fields. The mean velocity divergence turns out to be a third-order polynomial in the density contrast. We test the power-spectrum dependence of the coefficients of the polynomial for scale-free and standard CDM spectra and find it rather weak. Over scales larger than about 5  h ⁻¹ Mpc, the scatter in the relation is small compared with that introduced by random errors in the observed density and velocity fields. The relation can be useful for recovering the peculiar velocity from the associated density field, and also for non-linear analyses of the anisotropies of structure in redshift surveys.     

Cosmological Perturbation Theory and the Spherical Collapse model — I. Gaussian initial conditions

We present a simple and intuitive approximation for solving the perturbation theory (PT) of small cosmic fluctuations. We consider only the spherically symmetric or monopole contribution to the PT integrals, which yields the exact result for tree-graphs (i.e. at leading order). We find that the non-linear evolution in Lagrangian space is then given by a simple local transformation over the initial conditions, although it is not local in Euler space. This transformation is found to be described by the spherical collapse (SC) dynamics, as it is the exact solution in the shearless (and therefore local) approximation in Lagrangian space. Taking advantage of this property, it is straightforward to derive the one-point cumulants, ξ_J, for both the unsmoothed and smoothed density fields to arbitrary order in the perturbative regime. To leading-order this reproduces, and provides us with a simple explanation for, the exact results obtained by Bernardeau. We then show that the SC model leads to accurate estimates for the next corrective terms when compared with the results derived in the exact perturbation theory making use of the loop calculations. The agreement is within a few per cent for the hierarchical ratios S _J  = ξ _J /ξ ^{J −1}₂. We compare our analytic results with N -body simulations, which turn out to be in very good agreement up to scales where σ ≈ 1. A similar treatment is presented to estimate higher order corrections in the Zel'dovich approximation. These results represent a powerful and readily usable tool to produce analytical predictions that describe the gravitational clustering of large-scale structure in the weakly non-linear regime.     

Cosmic ray fluctuations in compressible turbulent medium

The influence of compressibility of the medium on cosmic ray (CR) fluctuations has been investigated. The CR transport equation has been used to obtain an equation for the second moment of CR particle density (correlation function of the particle density). It is shown that the effects due to the compressibility of the medium has an essential influence on CR fluctuations. The relations between CR power spectra and random velocity field have been determined. For the turbulence which is created by an ensemble of weak sound waves we have obtained the connection between the spectral indices of CR power spectra and the velocity field. It is shown that the spectral indices of CR power spectra and the velocity field of random sound waves coincide.     

Dynamics of charged particles and perpendicular diffusion in turbulent magnetic field

A test particle code is employed to explore the dynamics of charged particles and perpendicular diffusion in turbulent magnetic field, where a three-dimensional (3D) isotropic turbulence model is used in this paper. The obtained perpendicular diffusion at different particle energies is compared with that of the nonlinear guiding center (NLGC) theory. It is found that the NLGC theory is consistent with test particle simulations when the particle energies are small. However, the difference between the NLGC theory and test particle simulations tends to increase when the particle energy is sufficiently large, and the threshold is related to the turbulence bend-over length. In the NLGC theory, the gyrocenter of a charged particle is assumed to follow the magnetic field line. Therefore, when the particle has sufficiently large energy, its gyroradius will be larger than the turbulence bend-over length. Then the particle can cross the magnetic field lines, and the difference between the test particle simulations and NLGC theory occurs.     

Non-linear momentum diffusion of heliospheric cosmic rays

Besides parallel and perpendicular spatial diffusion, momentum diffusion can be seen as the third important process of cosmic ray transport. In this paper, the recently derived weakly non-linear theory is applied for a simple quasi-magnetostatic composite model to determine the momentum diffusion coefficient. It is demonstrated that non-linear effects are essential and cannot be neglected. Therefore, the weakly non-linear transport theory has to be preferred over the traditional quasi-linear approach. Within this improved theory, we find for the rigidity dependence of the momentum diffusion coefficient   A ∼ R ^1.4  for relativistic and   A ∼ R ^0.4  for non-relativistic cosmic rays.     

Diffusion of cosmic rays in a multiphase interstellar medium swept-up by a supernova remnant blast wave

Supernova remnants (SNRs) are one of the most energetic astrophysical events and are thought to be the dominant source of Galactic cosmic rays (CRs). A recent report on observations from the Fermi satellite has shown a signature of pion decay in the gamma-ray spectra of SNRs. This provides strong evidence that high-energy protons are accelerated in SNRs. The actual gamma-ray emission from pion decay should depend on the diffusion of CRs in the interstellar medium. In order to quantitatively analyse the diffusion of high-energy CRs from acceleration sites, we have performed test particle numerical simulations of CR protons using a three-dimensional magnetohydrodynamics (MHD) simulation of an interstellar medium swept-up by a blast wave. We analyse the diffusion of CRs at a length scale of order a few pc in our simulated SNR, and find the diffusion of CRs is precisely described by a Bohm diffusion, which is required for efficient acceleration at least for particles with energies above 30 TeV for a realistic interstellar medium. Although we find the possibility of a superdiffusive process (travel distance ∝ t^0.75) in our simulations, its effect on CR diffusion at the length scale of the turbulence in the SNR is limited.     

The relation between far-UV and visible extinctions

For directions of sufficient reddening (E(B−V)>∼0.25), there is a simple relation between the slope of the extinction curve in the far-UV and E(B−V). Regardless of direction, the far-UV extinction curve is proportional to 1/λⁿ e^{−2E(B−V)/λ} (λ in μm, n=4), in accordance with the idea that reddened stars spectra are contaminated by scattered light (Zagury, 2001b).This relation is not compatible with the standard theory of extinction which states that far-UV and visible extinctions are due to different classes of particle. In that model the two (far-UV and visible) extinctions vary thus independently according to the proportion of each type of particle.In preceding papers I have shown that the standard theory cannot explain UV observations of nebulae, and is contradicted by the UV spectra of stars with very low reddening: for how long shall the standard theory be considered as the interpretation of the extinction curve?     

Galactic cosmic ray modulation from 1965–1970

Numerical solutions of the cosmic-ray equation of transport within the solar cavity and including the effects of diffusion, convection, and energy losses due to adiabatic deceleration, have been used to reproduce the modulation of galactic electrons, protons and helium nuclei observed during the period 1965–1970. Kinetic energies between 10 and 10⁴ MeV/nucleon are considered. Computed and observed spectra (where data is available) are given for the years 1965, 1968, 1969 and 1970 together with the diffusion coefficients. These diffusion coefficients are assumed to be of separable form in rigidity and radial dependence, and are consistent with the available magneticfield power spectra. The force-field solutions are given for these diffusion coefficients and galactic spectra and compared with the numerical solutions. For each of the above years we have (i) determined the radial density gradients near Earth; (ii) found the mean energy losses suffered by galactic particles as they diffuse to the vicinity of the Earth's orbit; (iii) shown quantitatively the exclusion of low-energy galactic protons and helium nuclei from near Earth by convective effects; and (iv), for nuclei of a given energy near Earth, obtained their distribution in energy before entering the solar cavity. It is shown that the energy losses and convection lead to near-Earth nuclei spectra at kinetic energies ≤100 MeV/nucleon in which the differential intensity is proportional to the kinetic energy with little dependence on the form of the galactic spectrum. This dependence is in agreement with the observed spectra of all species of atomic nuclei and we argue that this provides strong observational evidence for the presence of energy losses in the propagation process; and for the exclusion of low energy galactic nuclei from near Earth.     

The behaviour of outer radiation belt electrons (> 1·5 MeV) during a geomagnetically disturbed time on 8 March 1970 and its relationship to magnetospheric processes

Omnidirectional intensities of electrons with energies E_e > 1·5 MeV detected by a low orbiting polar satellite (GRS-A/AZUR) in the outer radiation belt are examined during disturbed times including the main phase of a very strong geomagnetic storm on 8 March 1970. The particle intensity features are discussed in relationship with proposed magnetospheric processes. It is found that a superposition of the two following effects can explain the particle behavior in the trapping region:(A) Radial diffusion. After the southward turning of the interplanetary field an inward motion of both the energetic electron belt and the plasmapause took place. This effect was observed at L > 3 R_E and we attribute it to enhanced magnetospheric electric field fluctuations. Later, a strong interplanetary shock impinged upon the magnetosphere which was related to the triggering of intense magnetospheric substorms; a further inward diffusion occurred at L ? 3 R_E, accompanied by an inward movement of the electron slot. A rough estimation of the diffusion coefficient leads to a power spectrum of the electric field fluctuations which seems to be consistent with experimentally determined power spectra (Mozer, 1971).(B) Adiabatic response to ring current changes. Large energetic electron intensity decreases within the outer radiation belt are shown to be adiabatic changes due to ring current variations. The influence of the inflation of the magnetosphere due to the developing ring current is simultaneously observed by the decrease of the solar proton outoff (1·7-2·5 MeV).     

Perturbative signature of substructures in strong gravitational lenses

In the perturbative approach, substructures in the lens can be reduced to their effect on the two perturbative fields f ₁ and  d f ₀/dθ  . A simple generic model of an elliptical lens with a substructure situated near the critical radius is investigated in detail. Analytical expressions are derived for each perturbative field, and basic properties are analysed. The power spectrum of the fields is well approximated by a power law, resulting in significant tails at high frequencies. Another feature of the perturbation by a substructure is that the ratio of the power spectrum at order n of the two fields   R _n   is almost one. The ratio   R _n ≃ 1  is specific to substructures, for instance a higher order distortion  ( n > 2)  but with autosimilar isophotes will result in   R _n ∝ 1/ n ²  . Finally, the problem of reconstructing the perturbative field is investigated. Local field models are implemented and fitted to maximize image similarity in the source plane. The non-linear optimization is greatly facilitated, since in the perturbative approach the circular source solution is always known. Examples of image distortions in the subcritical regime due to substructures are presented, and analysed for different source shapes. Provided enough images and signal are available, the substructure field can be identified confidently. These results suggest that the perturbative method is an efficient tool to estimate the contribution of substructures to the mass distribution of lenses.