This research report by C. F. Kennel and H. E. Petschek from AVCO Everett Research Laboratory, supported by the National Aeronautics and Space Administration (NASA) and the Department of the Navy, investigates the limits on stably trapped particle fluxes in the magnetosphere. The study focuses on the effects of whistler mode noise on electron pitch angle diffusion and ion cyclotron noise on ion diffusion, leading to particle precipitation into the ionosphere. The authors derive an upper limit on trapped equatorial particle fluxes, which is consistent with observed electron fluxes > 40 keV and proton fluxes > 120 keV from Explorers XIV and XII. They find that beyond L = 4, the fluxes are just below their limit, suggesting the presence of an unspecified acceleration source. The limiting proton and electron fluxes are roughly equal, explaining the higher densities of high-energy protons compared to electrons. The required equatorial whistler mode wideband noise intensity is consistent with observed lifetimes and the limiting trapped particle intensity. The report also discusses the conditions for first adiabatic invariant violation, the linear theory of whistler and ion cyclotron modes, and the nonlinear behavior of weak turbulence, including pitch angle diffusion in finite plasmas. The authors conclude that the maximum stably trapped flux is determined by the balance between wave growth and wave escape rates, and that this limit is consistent with observed data.This research report by C. F. Kennel and H. E. Petschek from AVCO Everett Research Laboratory, supported by the National Aeronautics and Space Administration (NASA) and the Department of the Navy, investigates the limits on stably trapped particle fluxes in the magnetosphere. The study focuses on the effects of whistler mode noise on electron pitch angle diffusion and ion cyclotron noise on ion diffusion, leading to particle precipitation into the ionosphere. The authors derive an upper limit on trapped equatorial particle fluxes, which is consistent with observed electron fluxes > 40 keV and proton fluxes > 120 keV from Explorers XIV and XII. They find that beyond L = 4, the fluxes are just below their limit, suggesting the presence of an unspecified acceleration source. The limiting proton and electron fluxes are roughly equal, explaining the higher densities of high-energy protons compared to electrons. The required equatorial whistler mode wideband noise intensity is consistent with observed lifetimes and the limiting trapped particle intensity. The report also discusses the conditions for first adiabatic invariant violation, the linear theory of whistler and ion cyclotron modes, and the nonlinear behavior of weak turbulence, including pitch angle diffusion in finite plasmas. The authors conclude that the maximum stably trapped flux is determined by the balance between wave growth and wave escape rates, and that this limit is consistent with observed data.