### 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/3^{1/2}. 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, v_{c}. This negative value is a consequence of the second time derivative term in the telegraph equation and it is closely related to causality.

Original language | English (US) |
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Pages (from-to) | 377-384 |

Number of pages | 8 |

Journal | Astrophysical Journal |

Volume | 403 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1993 |

### Keywords

- Acceleration of particles
- Cosmic rays
- Diffusion

### ASJC Scopus subject areas

- Astronomy and Astrophysics
- Space and Planetary Science

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## Cite this

*Astrophysical Journal*,

*403*(1), 377-384. https://doi.org/10.1086/172209