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Journal of Geophysical Research: Space Physics

Limiting energy spectrum of a saturated radiation belt


  • Michael Schulz,

  • Gerald T. Davidson


Through a further application of the condition for magnetospheric wave growth in the presence of anisotropic charged-particle distributions, the Kennel-Petschek theory that traditionally imposes an upper bound on the integral flux of charged particles at energies above a certain threshold is extended to provide a limit on the differential flux at any energy above this threshold. Thus, a modest reformulation of the nonrelativistic Kennel-Petschek problem for electrons and protons enables a limiting energy spectrum to be derived, such that (for specified pitch-angle anisotropy s of the energetic particle population) electromagnetic-cyclotron waves at each frequency less than a fraction s/(s + 1) of the equatorial gyrofrequency are marginally stable against spontaneous generation. The limiting spectrum is given in closed form for integer values of s (>0) and computed numerically or by analytical interpolation for non-integer values of s. Asymptotic expansions for energies E barely above and much greater than the minimum resonant energy E* provide estimates of the limiting energy spectrum J*(E) in these extremes, regardless of whether s is an integer. A reconsideration of the original Kennel-Petschek problem, in which the differential energy spectrum is not calculated but specified as a certain power law (JE1-l), enables both the wave frequency ω*/2π corresponding to maximum spatial growth rate and the limiting integral flux I* above the minimum resonant energy E* to be calculated in closed form as functions of l and s.

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