Radio Astronomical Investigations With the Aid of Artificial Satellites

Abstract

We give a recursive formula to count maximal small copies of the Mandelbrot set and its higher degree analogues. This formula is used to compute the asymptotic growth of the number of maximal small copies of period n.

Particle acceleration at shocks in the presence of a braided magnetic field

“…There are two distinct physical effects which come into play here: 'microscopic' diffusion i.e., that which would pertain in a uniform magnetic field as a result of fluctuations with a length scale approximately equal to the gyro radius of the particle concerned, and 'macroscopic' diffusion of the field lines themselves, which results from fluctuations of much longer length scale. The importance of braiding, or 'field line wandering' as it is called in the astrophysical literature (Getmantsev 1963, Jokipii & Parker 1969, Jokipii 1973 can be quantified in terms of a single parameter Λ related to the correlation lengths of the macroscopic magnetic field fluctuations and to the relative strength of the macroscopic and microscopic fluctuations. In the case of plasmas in which braiding is important, a further distinction must be made concerning time scales.…”

Interstellar Turbulence II: Implications and Effects

“…Magnetic fields with irregularities smaller than the gyroradius cause particles to skip over field lines, giving anomalous diffusion (Parker 1964;Chuvilgin & Ptuskin 1993;Casse, Lemoine & Pelletier 2002;Otsuka & Hada 2003;Erlykin et al 2003). Braided field lines can lead to sub-diffusion (Getmantsev 1963), when the mean squared particle position in the cross field direction increases as the square root of time, instead of directly with time as in normal diffusion. Compound diffusion has both cross-field diffusion and diffusion along the field lines (Kirk et al 1996).…”

Diffusion and drift of cosmic rays in highly turbulent magnetic fields

“…While a plateau is attained at large t for ρ = 0.1 and σ 2 = 1, hence corresponding to the usual diffusion relation ∆x 2 ⊥ ∝ t, for lower values of rigidity and turbulence level the phenomenon of subdiffusion shows up. As an approach to understanding the subdiffusive regime, it has been proposed and investigated the so-called compound diffusion, in which particles are assumed to be strictly tied to the field lines, while they scatter back and forth along the lines [29,22,30,31]. For the limiting case of compound diffusion, it turns out that m = 1/2, while in the case of three-dimensional particle transport m is expected to have a smooth dependence with rigidity and turbulence, such that 1/2 ≤ m(ρ, σ 2 ) < 1.…”