The Fokker-Planck (FP) equation ∂tf+μ∂xf=∂μ(1−μ2)∂μf is solved analytically. Foremost among its applications, this equation describes the propagation of energetic particles through a scattering ...medium (in x- direction, with μ being the x- projection of particle velocity). The solution is found in terms of an infinite series of mixed moments of particle distribution, ⟨μjxk⟩. The second moment ⟨x2⟩ (j=0, k=2) was obtained by G. I. Taylor (1920) in his classical study of random walk: ⟨x2⟩=⟨x2⟩0+t/3+exp(−2t)−1/6 (where t is given in units of an average time between collisions). It characterizes a spatial dispersion of a particle cloud released at t=0, with ⟨x2⟩0 being its initial width. This formula distills a transition from ballistic (rectilinear) propagation phase, ⟨x2⟩−⟨x2⟩0≈t2/3 to a time-asymptotic, diffusive phase, ⟨x2⟩−⟨x2⟩0≈t/3. The present paper provides all the higher moments by a recurrence formula. The full set of moments is equivalent to the full solution of the FP equation, expressed in form of an infinite series in moments ⟨μjxk⟩. An explicit, easy-to-use approximation for a point source spreading of a pitch-angle averaged distribution f0(x,t) (starting from f0(x,0)=δ(x), i.e., Green’s function), is also presented and verified by a numerical integration of the FP equation.
The direct measurements of cosmic rays (CRs), after correction for the propagation effects in the interstellar medium, indicate that their source spectra are likely to be significantly steeper than ...the canonical E−2 spectrum predicted by the standard diffusive shock acceleration (DSA) mechanism. The DSA has long been held responsible for the production of galactic CRs in supernova remnant (SNR) shocks. The γ-ray "probes" of the acceleration spectra of CRs on the spot, inside the SNRs, lead to the same conclusion. We show that the steep acceleration spectrum can be attributed to the combination of (i) spherical expansion, (ii) tilting of the magnetic field along the shock surface, and (iii) shock deceleration. Because of (i) and (ii), the DSA is efficient only on two "polar caps" of a spherical shock where the local magnetic field is within 45° to its normal. The shock-produced spectrum observed edge-on steepens with the particle energy because the number of freshly accelerated particles with lower energies continually adds up to a growing acceleration region. We demonstrate the steepening effect by obtaining an exact self-similar solution for the particle acceleration at an expanding shock surface with an arbitrary energy dependence of particle diffusivity κ. We show that its increase toward higher energy steepens the spectrum, which deeply contrasts with the standard DSA spectrum where κ cancels out.
Energetic nonthermal particles (cosmic rays, CRs) are accelerated in supernova remnants, relativistic jets and other astrophysical objects. The CR energy density is typically comparable with that of ...the thermal components and magnetic fields. In this review we discuss mechanisms of magnetic field amplification due to instabilities induced by CRs. We derive CR kinetic and magnetohydrodynamic equations that govern cosmic plasma systems comprising the thermal background plasma, comic rays and fluctuating magnetic fields to study CR-driven instabilities. Both resonant and non-resonant instabilities are reviewed, including the Bell short-wavelength instability, and the firehose instability. Special attention is paid to the longwavelength instabilities driven by the CR current and pressure gradient. The helicity production by the CR current-driven instabilities is discussed in connection with the dynamo mechanisms of cosmic magnetic field amplification.
Recent observations of supernova remnant W44 by the Fermi spacecraft observatory support the idea that the bulk of galactic cosmic rays is accelerated in such remnants by a Fermi mechanism, also ...known as diffusive shock acceleration. However, the W44 expands into weakly ionized dense gas, and so a significant revision of the mechanism is required. Here, we provide the necessary modifications and demonstrate that strong ion-neutral collisions in the remnant surrounding lead to the steepening of the energy spectrum of accelerated particles by exactly one power. The spectral break is caused by Alfven wave evanescence leading to the fractional particle losses. The gamma-ray spectrum generated in collisions of the accelerated protons with the ambient gas is calculated and successfully fitted to the Fermi Observatory data. The parent proton spectrum is best represented by a classical test particle power law ∝E(-2), steepening to E(-3) at E(br)≈7 GeV due to deteriorated particle confinement.
The unexpected energy spectrum of the positron/electron ratio is interpreted astrophysically, with a possible exception of the 100–300 GeV range. The data indicate that this ratio, after a decline ...between 0.5 and 8 GeV, rises steadily with a trend towards saturation at 200–400 GeV. These observations (except for the trend) appear to be in conflict with the diffusive shock acceleration (DSA) mechanism, operating in a single supernova remnant (SNR) shock. We argue that e+/e− ratio can still be explained by the diffusive shock acceleration if positrons are accelerated in a subset of SNR shocks which (i) propagate in clumpy gas media and (ii) are modified by accelerated cosmic ray protons. The protons penetrate into the dense gas clumps upstream to produce positrons and charge the clumps positively. The induced electric field expels positrons into the upstream plasma where they are shock accelerated. Since the shock is modified, these positrons develop a harder spectrum than that of the cosmic ray electrons accelerated in other SNRs. Mixing these populations explains the increase in the e+/e− ratio at E>8 GeV. It decreases at E<8 GeV because of a subshock weakening which also results from the shock modification. Contrary to the expelled positrons, most of the antiprotons, electrons, and heavier nuclei, are left unaccelerated inside the clumps. Scenarios for the 100–300 GeV AMS-02 fraction exceeding the model prediction, including, but not limited to, possible dark matter contribution, are also discussed.
The much-anticipated proof of cosmic ray (CR) acceleration in supernova remnants must hinge on the full consistency of acceleration theory with the observations; direct proof is impossible because of ...CR-orbit scrambling. Recent observations indicate deviations between helium and proton CR rigidity spectra inconsistent with the theory. By considering an initial (injection) phase of the diffusive shock acceleration, where elemental similarity does not apply, we demonstrate that the spectral difference is, in fact, a unique signature of the acceleration mechanism. Collisionless shocks inject more He(2+) when they are stronger and so produce harder He(2+) spectra. The injection bias is due to Alfvén waves driven by the more abundant protons, so the He(2+) ions are harder to trap by these waves. By fitting the p/He ratio to the PAMELA data, we bolster the diffusive shock acceleration case for resolving the century-old mystery of CR origin.
The rate of inelastic nuclear interactions in a short bent silicon crystal was precisely measured for the first time using a 180 GeV/c positive hadron beam produced in the North Experimental Area of ...the CERN SPS. An angular asymmetry dependence on the crystal orientation in the vicinity of the planar channeling minimum has been observed. For the inspected crystal, this probability is about
∼
20
%
larger than in the amorphous case because of the atomic density increase along the particle trajectories in the angular range of volume reflection, whose dimension is determined by the crystal bending angle. Instead, for the opposite angular orientation with respect to the planar channeling, there is a smaller probability excess of
∼
4
%
.
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