Using a recently developed theory, we calculate the wave magnetic field amplitude threshold B&tgr; necessary to allow the nonlinear trapping of energetic gyroresonant and Landau resonant electrons by VLF whistler mode waves in the magnetosphere, propagating at an arbitrary angle, &psgr;, with respect to the earth's magnetic field, B0. This theory predicts that the amplitude threshold is governed near the magnetic equatorial plane by the gradient of &psgr; with respect to distance, z, along B0. Using commonly accepted models of the magnetosphere and computer raytracing techniques, we determine the function &psgr;(z) for magnetic shells in the range 2≤L≤5 and for frequencies 5 kHz≤f≤17.8 kHz.

We then use the functions &psgr;(z) to calculate B&tgr;. It is found that the minimum values of B&tgr; along each L shell generally occur at points of ''second-order'' resonance where both &ugr;z=&ugr;R and &ugr;˙z=&ugr;˙R, where &ugr;z is the particle velocity along the z axis and &ugr;R is the resonance velocity. In the case of gyroresonance, for a given frequency it is found that for most L shells accessible to the input rays, there is a single point of second-order resonance near the magnetic equator, and R&tgr; there is a single point of second-order resonance near the magnetic equator, and B&tgr; these is larger than that associated with ducted waves of the same frequency on the same L shell. However, over a narrow range of L shell, as many as three points of second, order resonance can exist on each magnetic shell for the case of nonducted waves, suggesting that as many as three regions of VLF emission generation can exist on the same magnetic field line. The results of the calculations agree qualitatively with experimental data concerning triggered VLF emissions, which indicates that B&tgr; is generally larger for nonducted waves than for ducted waves and that nonducted waves tend to trigger multiple VLF emissions.

Calculations also indicate that as many as three points of second order resonance can exist for energetic electrons experiencing a nonlinear Landau resonance with nonducted waves on certain L shells. Minimum threshold values of B&tgr; for the nonlinear Landau resonance can be lower than that for the nonlinear gyroresonance interaction, suggesting the possibility that nonducted waves may sometimes trigger VLF emissions via the Landau resonance mechanism.