Through use of the condition for magnetospheric whistler-mode wave growth over a broad band of wave frequencies in the presence of anisotropic energetic-electron pitch-angle distributions, Kennel-Petschek theory (which traditionally imposes an upper bound on the integral flux of electrons at energies above a certain threshold) can be generalized to yield a limit on the differential electron flux at any energy above this threshold. Thus, a modest reformulation of the nonrelativistic Kennel-Petschek problem enables a limiting energy spectrum to be derived such that (for specified pitch-angle anisotropy s of the energetic electron 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 that results can be calculated in closed form for integer values of s (> 0) and computed numerically or by analytic 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 for the limiting energy spectrum in these extremes, regardless of whether the anisotropy parameter s is an integer.
↧