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 J4π*(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 (J4π ∝ E1-l), enables both the wave frequency ω*/2π corresponding to maximum spatial growth rate and the limiting integral flux I4π* above the minimum resonant energy E* to be calculated in closed form as functions of l and s.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.