Nearby supernova remnants and the cosmic-ray spectral hardening at high energies

Recent measurements of cosmic-ray spectra of several individual nuclear species by the CREAM, TRACER, and ATIC experiments indicate a change in the spectral index of the power laws at TeV energies. Possible explanations among others include non linear diffusive shock acceleration of cosmic-rays, different cosmic-ray propagation properties at higher and lower energies in the Galaxy and the presence of nearby sources. In this paper, we show that if supernova remnants are the main sources of cosmic rays in our Galaxy, the effect of the nearby remnants can be responsible for the observed spectral changes. Using a rigidity dependent escape of cosmic-rays from the supernova remnants, we explain the apparent observed property that the hardening of the helium spectrum occurs at relatively lower energies as compared to the protons and also that the spectral hardening does not persist beyond $\sim (20-30)$ TeV energies.Comment: 6 pages, MNRAS accepted, minor text correction

Cosmic-Ray proton spectrum below 100 TeV in the local region

The propagation of cosmic-ray protons in the Galaxy is discussed under the framework of a three dimensional convection-diffusion model. Starting with the assumption of a uniform and continuous distribution of cosmic-ray sources injecting CRs continuously in the Galaxy and by invoking a supernova explosion at various distances from the Earth, it is found that only those sources located within a distance of ~ 1.5 kpc can produce appreciable temporal fluctuations in the CR proton flux observed at the Earth. So, the construction of the local CR proton spectrum is discussed by seperating the contributions of the distant sources from that of the nearby sources. The contribution from the distant sources is treated in the framework of a continuous source distribution model both in space as well as time, but that of the nearby sources in a discrete space-time source model. The study predicts the presence of at least one old nearby source with a characteristic age of ~ 10^5 yrs located at a distance of ~ 0.1 kpc to explain the observed proton flux below ~ 100 GeV.Comment: 8 pages, 6 figures, 1 table, uses mn2e.cls, minor text corrections, accepted for publication in MNRA

Revisiting the effect of nearby supernova remnants on local cosmic rays

In an earlier paper, the effect of the nearby known supernova remnants (SNRs) on the local cosmic-rays (CRs) was studied, considering different possible forms of the particle injection time. The present work is a continuation of the previous work, but assumes a more realistic model of CR propagation in the Galaxy. The previous work assumed an unbounded three-dimensional diffusion region, whereas the present one considers a flat cylindrical disc bounded in both the radial and vertical directions. The study has found that the effect of the vertical halo boundary $H$ on the local SNR contribution to the observed CR anisotropy is negligible as long as $H\gtrsim 2kpc$. Considering the values of the halo height $H\gtrsim 2kpc$ obtained by different authors, the present work suggests that the study of the effect of local sources on the CR anisotropy can be carried out without having much information on $H$ and hence, using the much simpler three-dimentional unbounded solution. Finally, the present work discusses about the possibility of explaining the observed anisotropy below the knee by a single dominant source with properly chosen source parameters, and claims that the source may be an \textit{undetected} old SNR with a characteristic age of $\sim 1.5\times 10^5 yr$ located at a distance of $\sim 0.57 kpc$ from the Sun.Comment: 5 pages, 2 figures, accepted in MNRAS Letters, minor text correction

Musical Actions of Dihedral Groups

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.Comment: 27 pages, 11 figures. To appear in the American Mathematical Monthly