Some natural zero one laws for ordinals below ε0

We are going to prove that every ordinal α with ε_0 > α ≥ ω^ω satisfies a natural zero one law in the following sense. For α < ε_0 let Nα be the number of occurences of ω in the Cantor normal form of α. (Nα is then the number of edges in the unordered tree which can canonically be associated with α.) We prove that for any α with ω ω  ≤ α < ε_0 and any sentence ϕ in the language of linear orders the asymptotic density of ϕ along α is an element of  {0,1}. We further show that for any such sentence ϕ the asymptotic density along ε_0 exists although this limit is in general in between 0 and 1. We also investigate corresponding asymptotic densities for ordinals below ω^ω

Identification of Shocks in the Spectra from Black Holes

We study the spectral properties of a low angular momentum flow as a function of the shock strength, compression ratio, accretion rate and flow geometry. In the absence of a satisfactory description of magnetic fields inside the advective disk, we consider the presence of only stochastic fields and use the ratio of the field energy to the gravitational energy density as a parameter. We not only include `conventional' synchrotron emission and Comptonization by Maxwell-Bolzmann electrons in the gas, but we also compute these effects due to power-law electrons. For strong shocks, a bump is produced due to the post-shock flow. A power-law spectral components due to the thermal and non-thermal electrons appear after this bump.Comment: 8 pages, 5 figures, Astronomy and Space Science (in press), Proceedings of the Hong Kong Conference (2004) Edited by Cheng and Romer

Diffusive propagation of cosmic rays from supernova remnants in the Galaxy. II: anisotropy

We investigate the effects of stochasticity in the spatial and temporal distribution of supernova remnants on the anisotropy of cosmic rays observed at Earth. The calculations are carried out for different choices of the diffusion coefficient D(E) for propagation in the Galaxy. The propagation and spallation of nuclei are taken into account. At high energies we assume that $D(E)\sim(E/Z)^{\delta}$, with $\delta=1/3$ and $\delta=0.6$ being the reference scenarios. The large scale distribution of supernova remnants in the Galaxy is modeled following the distribution of pulsars with and without accounting for the spiral structure of the Galaxy. Our calculations allow us to determine the contribution to anisotropy resulting from both the large scale distribution of SNRs in the Galaxy and the random distribution of the nearest remnants. The naive expectation that the anisotropy amplitude scales as D(E) is shown to be an oversimplification which does not reflect in the predicted anisotropy for any realistic distribution of the sources. The fluctuations in the anisotropy pattern are dominated by nearby sources, so that predicting or explaining the observed anisotropy amplitude and phase becomes close to impossible. We find however that the very weak energy dependence of the anisotropy amplitude below $10^{5}$ GeV and the rise at higher energies, can best be explained if the diffusion coefficient is $D(E)\sim E^{1/3}$. Faster diffusion, for instance with $\delta=0.6$, leads in general to an exceedingly large anisotropy amplitude. The spiral structure introduces interesting trends in the energy dependence of the anisotropy pattern, which qualitatively reflect the trend seen in the data. For large values of the halo size we find that the anisotropy becomes dominated by the large scale regular structure of the source distribution, leading indeed to a monotonic increase of $\delta_A$ with energy.Comment: 21 Pages, to appear in JCA

Three-dimensional fast electron transport for ignition-scale inertial fusion capsules

Three-dimensional hybrid PIC simulations are presented to study electron energy transport and deposition in a full-scale fast ignition configuration. Multi-prong core heating close to ignition is found when a few GA, few PW beam is injected. Resistive beam filamentation in the corona seeds the 3D current pattern that penetrates the core. Ohmic heating is important in the low-density corona, while classical Coulomb deposition heats the core. Here highest energy densities (few Tbar at 10 keV) are observed at densities above 200 g/cc. Energy coupling to the core ranges from 20 to 30%; it is enhanced by beam collimation and decreases when raising the beam particle energy from 1.5 to 5.5 MeV.Comment: 5 pages, 5 figure

Particle Acceleration at Relativistic Shocks

I review the current status of Fermi acceleration theory at relativistic shocks. I first discuss the relativistic shock jump conditions, then describe the non-relativistic Fermi mechanism and the differences introduced by relativistic flows. I present numerical calculations of the accelerated particle spectrum, and examine the maximum energy attainable by this process. I briefly consider the minimum energy for Fermi acceleration, and a possible electron pre-acceleration mechanism.Comment: 17 pages, 4 figures. To appear in "Relativistic Flows in Astrophysics", A.W. Guthmann, M. Georganopoulos, A. Marcowith and K. Manolokou, eds., Lecture Notes in Pysics, Springer Verla

On the multiplicity of the O-star Cyg OB2 #8A and its contribution to the gamma-ray source 3EG J2033+4118

We present the results of an intensive spectroscopic campaign in the optical waveband revealing that Cyg OB2 #8A is an O6 + O5.5 binary system with a period of about 21.9 d. Cyg OB2 #8A is a bright X-ray source, as well as a non-thermal radio emitter. We discuss the binarity of this star in the framework of a campaign devoted to the study of non-thermal emitters, from the radio waveband to gamma-rays. In this context, we attribute the non-thermal radio emission from this star to a population of relativistic electrons, accelerated by the shock of the wind-wind collision. These relativistic electrons could also be responsible for a putative gamma-ray emission through inverse Compton scattering of photospheric UV photons, thus contributing to the yet unidentified EGRET source 3EG J2033+4118.Comment: 8 pages, 4 figures, conference on "The Multiwavelength Approach to Gamma-Ray Sources", to appear in Ap&S

Diffusive propagation of cosmic rays from supernova remnants in the Galaxy. I: spectrum and chemical composition

In this paper we investigate the effect of stochasticity in the spatial and temporal distribution of supernova remnants on the spectrum and chemical composition of cosmic rays observed at Earth. The calculations are carried out for different choices of the diffusion coefficient D(E) experienced by cosmic rays during propagation in the Galaxy. In particular, at high energies we assume that D(E)\sim E^{\delta}, with $\delta=1/3$ and $\delta=0.6$ being the reference scenarios. The large scale distribution of supernova remnants in the Galaxy is modeled following the distribution of pulsars, with and without accounting for the spiral structure of the Galaxy. We find that the stochastic fluctuations induced by the spatial and temporal distribution of supernovae, together with the effect of spallation of nuclei, lead to mild but sensible violations of the simple, leaky-box-inspired rule that the spectrum observed at Earth is $N(E)\propto E^{-\alpha}$ with $\alpha=\gamma+\delta$, where $\gamma$ is the slope of the cosmic ray injection spectrum at the sources. Spallation of nuclei, even with the small rates appropriate for He, may account for slight differences in spectral slopes between different nuclei, providing a possible explanation for the recent CREAM observations. For $\delta=1/3$ we find that the slope of the proton and helium spectra are $\sim 2.67$ and $\sim 2.6$ respectively at energies above 1 TeV (to be compared with the measured values of $2.66\pm 0.02$ and $2.58\pm 0.02$). For $\delta=0.6$ the hardening of the He spectra is not observed. We also comment on the effect of time dependence of the escape of cosmic rays from supernova remnants, and of a possible clustering of the sources in superbubbles. In a second paper we will discuss the implications of these different scenarios for the anisotropy of cosmic rays.Comment: 28 pages, To appear in JCA

Gamma-ray emission expected from Kepler's SNR

Nonlinear kinetic theory of cosmic ray (CR) acceleration in supernova remnants (SNRs) is used to investigate the properties of Kepler's SNR and, in particular, to predict the gamma-ray spectrum expected from this SNR. Observations of the nonthermal radio and X-ray emission spectra as well as theoretical constraints for the total supernova (SN) explosion energy E_sn are used to constrain the astronomical and particle acceleration parameters of the system. Under the assumption that Kepler's SN is a type Ia SN we determine for any given explosion energy E_sn and source distance d the mass density of the ambient interstellar medium (ISM) from a fit to the observed SNR size and expansion speed. This makes it possible to make predictions for the expected gamma-ray flux. Exploring the expected distance range we find that for a typical explosion energy E_sn=10^51 erg the expected energy flux of TeV gamma-rays varies from 2x10^{-11} to 10^{-13} erg/(cm^2 s) when the distance changes from d=3.4 kpc to 7 kpc. In all cases the gamma-ray emission is dominated by \pi^0-decay gamma-rays due to nuclear CRs. Therefore Kepler's SNR represents a very promising target for instruments like H.E.S.S., CANGAROO and GLAST. A non-detection of gamma-rays would mean that the actual source distance is larger than 7 kpc.Comment: 6 pages, 4 figures. Accepted for publication in Astronomy and Astrophysics, minor typos correcte

New evidence for strong nonthermal effects in Tycho's supernova remnant

For the case of Tycho's supernova remnant (SNR) we present the relation between the blast wave and contact discontinuity radii calculated within the nonlinear kinetic theory of cosmic ray (CR) acceleration in SNRs. It is demonstrated that these radii are confirmed by recently published Chandra measurements which show that the observed contact discontinuity radius is so close to the shock radius that it can only be explained by efficient CR acceleration which in turn makes the medium more compressible. Together with the recently determined new value $E_{sn}=1.2\times 10^{51}$ erg of the SN explosion energy this also confirms our previous conclusion that a TeV gamma-ray flux of $(2-5)\times 10^{-13}$ erg/(cm$^2$s) is to be expected from Tycho's SNR. Chandra measurements and the HEGRA upper limit of the TeV gamma-ray flux together limit the source distance $d$ to $3.3\leq d\leq 4$ kpc.Comment: 5 pages, 4 figures. Accepted for publication in Astrophysics and Space Science, Proc. of "The Multi-Messenger Approach to High-Energy Gamma-ray Sources (Third Workshop on the Nature of Unidentified High-Energy Sources)", Barcelona, July 4-7, 200

Cosmic-ray acceleration in supernova remnants: non-linear theory revised

A rapidly growing amount of evidences, mostly coming from the recent gamma-ray observations of Galactic supernova remnants (SNRs), is seriously challenging our understanding of how particles are accelerated at fast shocks. The cosmic-ray (CR) spectra required to account for the observed phenomenology are in fact as steep as $E^{-2.2}--E^{-2.4}$, i.e., steeper than the test-particle prediction of first-order Fermi acceleration, and significantly steeper than what expected in a more refined non-linear theory of diffusive shock acceleration. By accounting for the dynamical back-reaction of the non-thermal particles, such a theory in fact predicts that the more efficient the particle acceleration, the flatter the CR spectrum. In this work we put forward a self-consistent scenario in which the account for the magnetic field amplification induced by CR streaming produces the conditions for reversing such a trend, allowing --- at the same time --- for rather steep spectra and CR acceleration efficiencies (about 20%) consistent with the hypothesis that SNRs are the sources of Galactic CRs. In particular, we quantitatively work out the details of instantaneous and cumulative CR spectra during the evolution of a typical SNR, also stressing the implications of the observed levels of magnetization on both the expected maximum energy and the predicted CR acceleration efficiency. The latter naturally turns out to saturate around 10-30%, almost independently of the fraction of particles injected into the acceleration process as long as this fraction is larger than about $10^{-4}$.Comment: 24 pages, 5 figures, accepted for publication in JCA