The telegraph equation in charged particle transport

T. I. Gombosi, J. Randy Jokipii, J. Kota, K. Lorencz, L. L. Williams

Research output: Contribution to journalArticle

39 Citations (Scopus)

Abstract

We present a new derivation of the telegraph equation which modifies its coefficients. First, an infinite-order partial differential equation is obtained for the velocity-space solid angle-averaged phase-space distribution of particles which underwent at least a few collisions. It is shown that in the lowest order asymptotic expansion this equation simplifies to the well-known diffusion equation. The second-order asymptotic expansion for isotropic small-angle scattering results in a modified telegraph equation with a signal propagation speed of v(5/11)1/2 instead of the usual v/31/2. Our derivation of a modified telegraph equation follows from an expansion of the Boltzmann equation in the relevant smallness parameters and not from a truncation of an eigen-function expansion. This equation is consistent with causality. It is shown that under steady state conditions in a convecting plasma the telegraph equation may be regarded as a diffusion equation with a modified transport coefficient, which describes a combination of diffusion and cosmic-ray inertia. This modified transport coefficient becomes negative for particles with random velocities less than the critical velocity, vc. This negative value is a consequence of the second time derivative term in the telegraph equation and it is closely related to causality.

Original languageEnglish (US)
Pages (from-to)377-384
Number of pages8
JournalAstrophysical Journal
Volume403
Issue number1
StatePublished - 1993

Fingerprint

charged particles
expansion
derivation
transport properties
particle
critical velocity
inertia
partial differential equations
cosmic rays
cosmic ray
phase shift
collision
scattering
collisions
plasma
propagation
coefficients
approximation

Keywords

  • Acceleration of particles
  • Cosmic rays
  • Diffusion

ASJC Scopus subject areas

  • Space and Planetary Science

Cite this

Gombosi, T. I., Jokipii, J. R., Kota, J., Lorencz, K., & Williams, L. L. (1993). The telegraph equation in charged particle transport. Astrophysical Journal, 403(1), 377-384.

The telegraph equation in charged particle transport. / Gombosi, T. I.; Jokipii, J. Randy; Kota, J.; Lorencz, K.; Williams, L. L.

In: Astrophysical Journal, Vol. 403, No. 1, 1993, p. 377-384.

Research output: Contribution to journalArticle

Gombosi, TI, Jokipii, JR, Kota, J, Lorencz, K & Williams, LL 1993, 'The telegraph equation in charged particle transport', Astrophysical Journal, vol. 403, no. 1, pp. 377-384.
Gombosi TI, Jokipii JR, Kota J, Lorencz K, Williams LL. The telegraph equation in charged particle transport. Astrophysical Journal. 1993;403(1):377-384.
Gombosi, T. I. ; Jokipii, J. Randy ; Kota, J. ; Lorencz, K. ; Williams, L. L. / The telegraph equation in charged particle transport. In: Astrophysical Journal. 1993 ; Vol. 403, No. 1. pp. 377-384.
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