Electron injection and acceleration at nonlinear shocks: Results of numerical simulations

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We present results of numerical simulations of electron injection and acceleration at nonlinear high Mach number shocks. The electrons are assumed to be heated at the thermal subshock to an energy Einj, which is treated as a free parameter, above which they are injected by self-generated whistlers to momentum mpvA. This injection mechanism requires Mach numbers greater than 43β-1- (kTe/Einj)1/2, where Te and β- are the upstream electron temperature and plasma beta parameter. Above mpvA electrons are trapped in the shock by Alfvén waves. In the proton precursor region the Alfvén waves are assumed to be generated by protons accelerated at the shock, and have nonlinear intensities. Below GeV, however, electrons of a given rigidity propagate faster than protons with a similar rigidity and therefore diffuse to regions ahead of the proton precursor. In those regions the Alfvén waves are generated by the electrons themselves. The diffusion coefficient appears to increase with decreasing acceleration efficiency. As a result, the number of electrons accelerated to energies GeV and above and, hence, the electron to proton ratio, depend only weakly on the extent of electron heating at the subshock. The negative feedback also renders the electron spectra insensitive to shock compression ratio and smoothing length scale. The estimated e/p ratio at GeV is between ∼1%-10%.

Original languageEnglish
Pages (from-to)327-333
Number of pages7
JournalAstrophysical Journal
Issue number1
StatePublished - 1 May 1994
Externally publishedYes


  • Acceleration of particles
  • Shock waves
  • Supernovae remnants
  • Waves


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