Table of Contents
ISRN High Energy Physics
Volume 2012, Article ID 673250, 5 pages
Research Article

The Characteristics of Cosmic Rays in a Fractal Medium

Faculty of Physics, Shahid Bahonar University of Kerman, Kerman 76169-14111, Iran

Received 18 December 2011; Accepted 12 January 2012

Academic Editors: G. Di Sciascio and A. S. Tonachini

Copyright © 2012 S. Doostmohammadi and S. J. Fatemi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the galactic propagation of cosmic rays with energies below the knee of the cosmic ray spectrum (at 3×106 GeV). Lagutin and his colleagues suggested that propagation in a fractal medium would be an appropriate model for such particles and pointed out that a different diffusion equation to that usually assumed is then required. We present results of calculations using a diffusion equation appropriate to an inhomogeneous medium. These results are encouraging and can fit properties of the energy density of cosmic rays using the supernova model of Erlykin and Wolfendale. The containment time of cosmic rays, in such a modeling of propagation in a fractal medium, is discussed.

1. Introduction

It has long been considered that the supernova remnants (SNRs) are responsible for galactic cosmic rays [1, 2]. They totally release about 1051 erg as their explosion energy which makes them as very likely candidates in the galaxy, capable of delivering the requisite cosmic ray (CR) power (1041erg/s) to the interstellar medium [3]. The most likely particle acceleration mechanism would be diffusive shock acceleration which can also satisfy the requirement of producing a power law energy spectrum such as what is observed. Although there is evidence in the favor of SNR models [4, 5], they also face with some serious problems. One problem is the energy of the knee (~106 GeV) which is above the maximum energy of accelerated protons in all the models (e.g., Axford and Berezhko model [3]). For instance, none of the existing supernova models can easily provide the necessary acceleration for galactic CRs to reach the highest observed energies, higher than 106 GeV.

Also, the total particle energy required to be extracted from the shock energy can be at least a few percent, based on conventional galactic propagation models. It is necessary to ensure that any alternative propagation models do not require a substantial increase in this fraction. The interstellar medium (ISM) is known as an inhomogeneous environment of matter and magnetic fields in different scales with a fractal character. Galactic structures and physical parameters such as shells, clouds, filaments, temperature, density, and degree of ionization are distributed over ranges of spatial scales [6].

However, during the recent decades, from both theoretical and observational points of view, more evidence (e.g., [7]) has been about the existence of multiscale structures in the galaxy. Hence, we like to see the effect of CR propagation in such a fractal medium. In this search, we will examine the issue of the flux, or energy density, of CRs at the Earth in the energy range below the spectral knee. By assuming that the diffusion of CRs in ISM has a fractal structure, a question arises here about whether the calculated particle density as a function of galactic radius is consistent with the observations of the radial gradient and energy density of CRs. We will also examine more directly how results of such propagation differ from more conventional propagation modelling of galactic CR containment times. It is commonly assumed that CR propagation through the galaxy has a form of normal diffusion and causes a spatial distribution of particles around the source.

In this paper, we will examine the suggestion by Lagutin et al. [8] to consider an interstellar medium with scales described by a fractal structure, resulting in an anomalous diffusion equation. Lagutin et al. introduced a parameter, 𝛼, as a key parameter for CR propagation in such interstellar medium, where its value is related to the ISM spectrum of magnetic irregularities. The normal diffusion in a homogeneous medium, which yields a Gaussian distribution of particle densities, is achieved when 𝛼=2. The range of 𝛼<2 corresponds to the so-called superdiffusive regime of anomalous diffusion. In this paper, we will show the usefulness of assuming the inhomogeneous ISM [9, 10] in the range 𝛼<2 [8]. We will attempt to identify a likely value of 𝛼, using CR flux data plus the assumption that those particles originate in supernovae. This supposition which is in agreement with the physical pattern of galactic magnetic fields provides us with a parameter to be included in comparing supernovae source models with observation.

2. Calculating the Radial Gradient of Galactic CRs and the Local Energy Density

We will assume that all CRs in the energy range of interest are of supernova origin. We will also assume that the spatial distribution function of supernovae within our galaxy is as 𝜌𝑠𝑛=𝐴𝐹(𝑅,𝑍) [11, 12] that𝑅𝐹(𝑅,𝑍)=𝑅0𝑎𝑅exp𝑏𝑅0𝑍1𝐻𝑧.(2.1) The assumed values of the parameters in (2.1) are 𝑎=1.69,𝑏=3.33,𝑅0=8.5Kpc (the galactic radius of the solar system), and 𝐻𝑧=0.2Kpc (the vertical scale parameter of galactic supernovae). 𝑍 is distance perpendicular to the galactic disk.

Using the solution of Lagutin et al. [8] for the steady-state superdiffusion model of CR particles, we first calculated the radial gradient of the galactic CR intensity using the supernova distribution described by (2.1) with the assumption of cylindrical symmetry. Assuming the production rate of type II SN explosions is 102year1 in the galaxy, we examined how well the model would predict the observed value of the energy density of CRs. To have the concentration of CRs with energy 𝐸 at a distance 𝑟 from a point source (supernova) which has a power law energy spectrum, we apply the steady-state diffusion equation suggested by Lagutin et al. as follows:𝐷(Δ)𝛼/2𝑁(𝑟,𝐸)=𝑆(𝑟,𝐸),(2.2) where 𝑆(𝑟,𝐸) describes the density distribution of sources, 𝐷 is the anomalous diffusivity, 𝑁(𝑟,𝐸) is the number of particles at distance 𝑟 from SN, and (Δ)𝛼/2 is the fractional Laplacian called the “Riss” operator [13, 14]. Following Lagutin et al. through the Greens function for the above equation, we can find the solution for the steady-state case with CRs propagating diffusively from sources having the assumed spatial distribution of supernovae. We assume that 𝜏0, the galactic lifetime of CRs with 1 GeV energy, equals 4×107 years [15] and that 𝐻𝐺, the vertical scale of the galactic magnetic fields, is 1 Kpc.

The observed intensity of CRs, 𝐼, at an energy 𝐸 is the sum over all sources of the product of the intensity from a source at a particular location and the density of sources there, that is, the sum of all 𝑁(𝑟,𝐸)𝐹(𝑅,𝑍). The intensity, in arbitrary units, in terms of 𝑅 for different 𝛼, (an intrinsic parameter of interstellar medium), is shown in Figure 1. We calculated the radial gradient of CRs, for each 𝛼 value in Figure 1, and obtained an average galactic radial gradient of CRs in each case. Our results for the range 1.6<𝛼<2 are consistent with experimental determinations of the fractional radial gradient, ranging from 3×102 to 6×102kpc1 [16].

Figure 1: The intensity (arb. Units) of CRs as a function of galactic radial distance assuming a supernova origin with CRs propagating in various fractal media (for different values of the parameter 𝛼, see text).

We wished to determine how well our anomalous diffusion model would fit the CR energy density as measured at the Earth. This was based on supernovae occurring relatively locally, in an annulus between 4.5 Kpc and 12.5 Kpc from the galactic centre, each radiating for a total effective time of 𝑇 years, and assuming that a supernova explosion occurs once per 100 years anywhere in the galaxy. We used the flux 𝐹(𝑅,𝑍) from a single supernova derived above (following Lagutin et al.) from the superdiffusion Greens function and integrated this with a Monte Carlo technique over the appropriate part of our galaxy model. The energy density, 𝐼, is now a function of 𝑍,𝑟,𝛼,and𝑇 plus the fraction of supernova energy transferred to CRs. Using our model for the energy density contributed by a single supernova, we determined the propagation parameters which result in the best fit to the observed CR energy density. Testing different values of parameters, such as 𝛼 in a range of 0.5 to 2, 𝑇 from 104 to 107 years and fraction of supernova energy transferred to particles from 0.01 to 0.1, we obtained a series of resulting values of CR energy density in the vicinity of the sun.

To compare our simulated energy density of CRs with observational results, we used a model of Erlykin and Wolfendale which gives an energy spectrum for CRs measured in the vicinity of the sun [16]. This allowed us to calculate a total CR energy density above 104.4 GeV as 1.8×104eVcm3. A comparison of our derived energy densities, as a function of propagation parameters, with the energy density of the Erlykin and Wolfendale model, indicates a best fit for 𝛼=1.8 (with a total possible range 1.6 to 1.9 given uncertainty in the supernova emission time and the fraction of supernova energy being transferred to particles). This value, being below 2, implies a fractal structure for ISM. In this case, we found a best fit of 10% for the fraction of energy going into energetic particles. This fraction is comparable with the value of 16% for the kinetic energy of the supernova ejecta found by A.M. Hillas [2]. The effective emitting time of the supernova was then best fitted at 7×104 years. Our calculations thus support a model of the interstellar medium, within which galactic CRs produced by supernova propagate, being most consistent with CR observations if a fractal and nonhomogeneous structure is incorporated. We expect the resulting flux to be reduced for the superdiffusion case when compared to earlier modeling, but the required energy transfer to CRs is still not excessive, even for our conservative assumption of one supernova per hundred years.

In order to examine this result from another point of view, we have followed the propagation of CRs explicitly through a model of the galactic magnetic field which assumes a fractal structure through a power law distribution of characteristic magnetic field scale sizes. At any location, we assume that one such scale size applies to the magnetic field, which has itself a turbulent structure with a Kolmogorov spectrum up to that scale size. This is an extension of the type of propagation often considered [17] which assumes a Kolmogorov spatial spectrum up to a fixed scale size. Figure 2 shows the result of such modeling. The superdiffusion, fractal model results in shorter galactic containment times. Those times are strongly influenced by the possibility of encountering a region associated with a particularly large turbulence scale which makes particle loss more likely. In this way, the final CR energy density is less than for conventional diffusion for a given distribution and frequency of supernova sources. In the case presented, which corresponds closely to superdiffusion 𝛼 of 1.7, there is an overall reduction in containment by about a factor of five. This is the approximate value of 𝛼 which we found to be required to fit the galactic CR energy density.

Figure 2: The containment time ratio for conventional diffusion compared to superdiffusion in a turbulent galactic magnetic field.

3. Conclusions

We have examined CR propagation of 1012 to 1019eV in a galactic model incorporating fractal structure which is shown in Figure 2. Our calculated result with the observed values of radial gradient of galactic CRs given in the range of 3×102 to 6×102Kpc1 and also comparison of our simulated values of energy density of CRs from the expected supernova model (1.8×104eVcm3) confirm a diffusion of cosmic particles of supernova origin in the fractal medium and reject their diffusion in homogenous medium with Gaussian distribution (𝛼=2). We find that a superdiffusion model results in lower galactic containment times for nearly the whole CR energy spectrum, and as a result lower energy density below that expected from conventional propagation in turbulent magnetic field.


  1. V. L. Ginzborg and S. I. Syrovatskiĭ, The Origin of Cosmic Rays, Pergamon Press, New York, NY, USA, 1964.
  2. A. M. Hillas, “Can diffusive shock acceleration in supernova remnants account for high-energy galactic cosmic rays?” Journal of Physics G, vol. 31, no. 5, pp. R95–R131, 2005. View at Publisher · View at Google Scholar
  3. E. G. Berezhko, V. K. Elshin, and L. T. Ksenofontov, “Acceleration of cosmic rays in supernova remnants,” Zhurnal Experimentalnoy i Theoreticheskoy Phisiki, vol. 109, p. 3, 1996 (Russian). View at Google Scholar
  4. C. L. Bhat, M. R. Issa, B. P. Houston, C. J. Mayer, and A. W. Wolfendale, “Cosmic γ rays and the mass of gas in the Galaxy,” Nature, vol. 314, no. 6011, pp. 511–515, 1985. View at Publisher · View at Google Scholar · View at Scopus
  5. J. L. Osborne, A. W. Wolfendale, and L. Zhang, “A search for cosmic-ray acceleration by supernova remnant shocks,” Journal of Physics G, vol. 21, no. 3, article 017, pp. 429–437, 1995. View at Publisher · View at Google Scholar · View at Scopus
  6. B. G. Elmegreen, S. Kim, and L. Staveley-Smith, “A fractal analysis of the H I emission from the large Magellanic cloud,” Astrophysical Journal, vol. 548, no. 2, pp. 749–769, 2001. View at Publisher · View at Google Scholar · View at Scopus
  7. A. C. Cadavid, J. K. Lawrence, and A. A. Ruzmaikin, “Anomalous diffusion of solar magnetic elements,” Astrophysical Journal, vol. 521, no. 2, pp. 844–850, 1999. View at Publisher · View at Google Scholar · View at Scopus
  8. A. A. Lagutin, V. V. Makarov, D. V. Strelnikov, and A. G. Tyumentsev, “Anomalous diffusion of the cosmic ray: steady-state solution,” in Proceedings of the 27th International Cosmic Ray Conference (ICRC '01), p. 1889, Hamburg, Germany, August 2001.
  9. A.A. Lagutin and V. Uchaikin, “Fractional diffusion of cosmic rays,” in Proceedings of the 27th International Cosmic Ray Conference (ICRC '01), p. 1900, Hamburg, Germany, 2001. View at Publisher · View at Google Scholar
  10. J. Feder, Fractals, Plenum Press, New York, NY, USA, 1988.
  11. A. D. Erlykin, A. A. Lagutin, and A. W. Wolfendale, “Properties of the interstellar medium and the propagation of cosmic rays in the Galaxy,” Astroparticle Physics, vol. 19, no. 3, pp. 351–362, 2003. View at Publisher · View at Google Scholar · View at Scopus
  12. M. Pohl and J. A. Esposito, “Electron acceleration in supernova remnants and diffuse gamma rays above 1 GeV,” Astrophysical Journal, vol. 507, no. 1, pp. 327–338, 1998. View at Publisher · View at Google Scholar · View at Scopus
  13. S. G. Samko, A. A. Kilbas, and O. Maritchcv, Fractional Integrals and Derivations and some Application, Nauka, Minsk, Belarus, 1987.
  14. R. Plag, “On the origin of cosmic rays,” 2001, View at Google Scholar
  15. A. D. Erlykin and A. W. Wolfendale, “Supernova remnants and the origin of the cosmic radiation: I. SNR acceleration models and their predictions,” Journal of Physics G, vol. 27, no. 5, pp. 941–958, 2001. View at Publisher · View at Google Scholar · View at Scopus
  16. A. D. Erlykin and A. W. Wolfendale, “The origin of cosmic rays,” Europhysics News, vol. 32, no. 6, pp. 246–249, 2001. View at Publisher · View at Google Scholar
  17. R. W. Clay, “The source energy spectrum of cosmic rays,” Publications of the Astronomical Society of Australia, vol. 19, no. 2, pp. 228–232, 2002. View at Publisher · View at Google Scholar