Cosmic‐Ray Diffusion Approximation with Weak Adiabatic Focusing

Schlickeiser, R.; Shalchi, A.

doi:10.1086/591237

Cited by 49 publications

(65 citation statements)

References 30 publications

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“…A constant focusing length L is adopted, which is not a significant limitation as long as L does not change much within one scattering length λ. The condition λ L is in fact the weak focusing limit of Schlickeiser & Shalchi (2008), which yields λ ≈ λ 0 where λ 0 is the scattering length in the absence of focusing. Note, however, that the focusing length in the Parker spiral of magnetic field is not constant, and λ/L is not small sufficiently close to the sun.…”

Section: The Fokker-planck Equation and The Diffusion Approximationmentioning

confidence: 99%

“…The purpose of this paper is to generalise the calculations of Earl (1974Earl ( , 1976 and Beeck & Wibberenz (1986) and derive the telegraph equation for the particle density in the weak focusing limit of Schlickeiser & Shalchi (2008). The resulting equation is valid for an anisotropic scattering rate, complementing A&A 554, A59 (2013) the derivation given by Litvinenko & Noble (2013).…”

Section: Introductionmentioning

confidence: 97%

“…Earl (1981) and Beeck & Wibberenz (1986) obtained the diffusion equation taking adiabatic focusing into account (for a recent discussion, see Litvinenko 2012). Schlickeiser & Shalchi (2008) gave a systematic derivation of the diffusion approximation in a weak focusing limit, when the magnetic focusing length is much greater Appendix A is available in electronic form at http://www.aanda.org than the scattering length of cosmic-ray particles. In this limit, the main effect of a non-uniform mean magnetic field is the convection-like coherent streaming, whereas the parallel diffusion coefficient is independent of the focusing length.…”

Section: Introductionmentioning

confidence: 99%

“…We also use the new transport equation to re-examine the propagation of energetic particles produced in solar flares. As an illustration, we compare the temporal profiles, predicted by the fundamental solution to the telegraph equation, with those calculated by Schlickeiser & Shalchi (2008) in the diffusion approximation. Comparison of the predicted profiles allows us to quantify the accuracy of the diffusion approximation in a concrete application.…”

Section: Introductionmentioning

confidence: 99%

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The telegraph equation for cosmic-ray transport with weak adiabatic focusing

Litvinenko

Schlickeiser

2013

A&A

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Time-dependent solutions of a spatial diffusion equation are often used to describe the transport of solar energetic particles, accelerated in large solar flares. Approximate analytical solutions of the diffusion approximation can complement and guide detailed numerical solutions of the Fokker-Planck equation for the particle distribution function. The accuracy of the diffusion approximation is limited, however, because the signal propagation speed is infinite in the diffusion limit. An improved description of cosmic-ray transport is provided by the telegraph equation, characterised by a finite signal propagation speed. We derive the telegraph equation for the particle density, taking into account adiabatic focusing in a large-scale interplanetary magnetic field in a weak focusing limit. As an illustration, we calculate a propagating pulse solution of the telegraph equation, determine the rise time when the maximum particle intensity is reached at a given distance from the Sun, and compare the results with those obtained in the diffusion approximation. In comparison with the diffusion equation, the telegraph equation predicts an asymmetrical shape of the pulse and a shorter rise time. These potentially significant differences suggest that the more accurate telegraph equation should be used in analysis of the solar energetic particle data, at least to quantify the accuracy of the focused diffusion model.

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Section: The Fokker-planck Equation and The Diffusion Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The telegraph equation for cosmic-ray transport with weak adiabatic focusing

Litvinenko

Schlickeiser

2013

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“…According to the diffusion approximation presented in Schlickeiser et al (2007) and Schlickeiser & Shalchi (2008), the anisotropic part, g(z, p, μ, T ), can be expressed in the weak focusing limit and for negligible perpendicular diffusion as…”

Section: Anisotropy-time Profilesmentioning

confidence: 99%

A diffusive description of the focused transport of solar energetic particles

Artmann

Schlickeiser

Agueda

et al. 2011

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The transport of solar energetic charged particles along the interplanetary magnetic field in the ecliptic plane of the sun can be described roughly by a one-dimensional diffusion equation. Large-scale spatial variations of the guide magnetic field can be taken into account by adding an additional term to the diffusion equation that includes the effect of adiabatic focusing. We solve this equation analytically by assuming a point-like particle injection in time and space and a spatial power-law dependence for the focusing length and the spatial diffusion coefficient. We infer the intensity-and anisotropy-time profiles of solar energetic particles from this solution. Through these the influence of different assumptions for the diffusion parameters can be seen in a mathematically closed form. The comparison of calculated and measured intensity-and anisotropy-time profiles, which are a powerful diagnostic tool for interplanetary particle transport, gives information about the large-scale spatial dependence of the focusing length and the diffusion coefficient. For an exceptionally large solar energetic particle event, which did occur on 2001 April 15, we fit the 27−512 keV electron intensities and anisotropies observed by the Wind spacecraft using the theoretically derived profiles. We find a linear spatial dependence of the mean free path along the guiding magnetic field. We also find the mean free path to be energy independent, which supports the theory of "velocity-dependent diffusion". This means that the intensity profiles for the discussed energies exhibit the same shape if they are plotted against the traveled distance and not against the time. In this case the profiles differ only in their maximum values and we can determine the energy spectra of the solar flare electrons out of the scaling factor we need to fit the data. The derived spectra exhibits a power-law dependence ∝E −3 kin in an energy range from ∼50 keV to ∼500 keV.

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