Since the discovery in 1958 of the Earth's radiation belts by James Van Allen and colleagues using Explorer 1 [Van Allen et al., 1958], radiation belt physics has been an active area of theoretical and experimental research. Extensive progress has been made in understanding the basic physical processes occurring in the radiation belts of Earth and the other magnetized planets [e.g., Roederer, 1970; Schulz and Lanzerotti, 1974; Lyons and Williams, 1984; Schulz, 1991; Lemaire et al., 1996; Santos-Costa et al., 2003; Bolton et al., 2004], though many unsolved problems remain. For Earth, current problems include quantifying the variability of the energetic electron flux in the outer radiation belt and understanding the physical processes that control the transport, acceleration and loss of radiation belt electrons [Summers et al., 2007a, 2007b; Li et al., 2007; Baker and Kanekal, 2008; Hudson et al., 2008].
 It has been recognized since the very early studies [Dungey, 1963; Cornwall, 1964; Kennel and Petschek, 1966; Kennel, 1969; Roberts, 1969] that electromagnetic waves can scatter magnetospheric electrons into the loss cone by pitch angle diffusion and hence cause electron precipitation into the atmosphere. Dungey  and Cornwall  considered precipitation of radiation belt electrons by externally generated whistler mode waves, for instance by lightning-generated whistlers. Kennel and Petschek  showed that a trapped anisotropic electron population, if sufficiently anisotropic, will itself generate whistler mode waves which will scatter electrons and lead to precipitation. This process in which the pitch angle diffusion is self-excited is inherently nonlinear since the trapped particles act back on themselves by means of the waves. Nevertheless, Kennel and Petschek  assumed that the whistler mode turbulence is weak and were able to use a linear perturbation technique, and, in a seminal paper, obtained the widely quoted result that there exists an upper limit on the integral omnidirectional flux of particles that can be stably trapped in a dipole field; if the limiting flux is exceeded then increased particle precipitation results. Kennel and Petschek  argued that the limiting flux is attained in the steady state situation of marginal stability when the wave energy lost due to internal reflection is balanced by the increase in wave energy due to convective wave growth. We provide a brief technical derivation of the Kennel-Petschek flux limit at the end of our paper in Appendix B. The Kennel-Petschek limit is not an actual physical limit on the flux of particles that can be stably trapped in a planetary magnetosphere, and, further, the limit applies only to the case of weak diffusion wherein the loss cone is partially empty. While a sufficiently intense particle source will lead to an increase in trapped flux beyond the Kennel-Petschek limit and to an increased rate of precipitation, the precipitation rate itself cannot exceed the strong diffusion rate [Kennel, 1969]. Therefore, in principle, if the particle source population continues to increase beyond that required to produce strong diffusion, then the trapped particle flux could increase indefinitely. In general, a trapped particle distribution, in addition to being subject to the self-induced or nonparasitic pitch angle scattering postulated in the Kennel and Petschek  scenario, could also be subject to parasitic scattering by externally generated waves. For instance, energetic electrons in the slot region can be scattered by plasmaspheric hiss [Lyons et al., 1972] while those in the Earth's outer zone can be subject to intense precipitation due to scattering by electromagnetic ion cyclotron waves, under appropriate conditions [Thorne and Kennel, 1971; Lyons and Thorne, 1972; Summers and Thorne, 2003]. The integral flux of stably trapped particles is expected to be much less than the Kennel-Petschek limit if significant precipitation losses occur due to parasitic pitch angle scattering. Other particle loss processes not accounted for in the Kennel and Petschek  formulation include loss due to sweeping by planetary satellites or to absorption by dust clouds each of which occurs, for example, at Saturn [e.g., Santos-Costa et al., 2003]. Jovian radiation belt particles likewise suffer losses due to satellites and rings [e.g., Santos-Costa and Bourdarie, 2001; Sicard and Bourdarie, 2004].
 The self-limiting flux concept has been addressed in various studies. Schulz  incorporated the Kennel-Petschek limit on the trapped flux in a nonlinear phenomenological model of particle saturation in the Earth's outer radiation belt. Schulz and Davidson  extended the Kennel-Petschek theory to derive a limiting form of the particle differential flux, which at large energy E varies asymptotically as 1/E. Comparisons of the Kennel-Petschek integral flux limit with observed particle fluxes in the Earth's outer zone have been made by Baker et al. , Davidson et al. , and other authors, as well as by Kennel and Petschek . Observed values of the electron flux in the region 5 ≤ L ≤ 10, in particular at geosynchronous orbit L = 6.6, were found to be generally below, and often near, the Kennel-Petschek limit. Typically, observed values were within a factor of 2 or 3 of the Kennel-Petschek limit, though some of the data suggest that electron fluxes exhibit greater temporal variability than can be accounted for by the steady state Kennel-Petschek theory. Baker et al.  present data at Earth's geosynchronous orbit that provide evidence for an experimental upper limit to the electron integral flux that well exceeds the Kennel-Petschek limit. Barbosa and Coroniti , assuming an ultrarelativistic electron population, calculated the Kennel-Petschek flux limit for Jupiter's inner magnetosphere (4 ≤ L ≤ 12), and concluded that within theoretical and experimental uncertainties observed fluxes were near the stably trapped limit. Using data from the LECP instrument on the Voyager 2 spacecraft, Mauk et al.  found that measured electron integral fluxes in the inner magnetosphere of Uranus near L = 4.73 were in excess of the Kennel-Petschek limit by an order of magnitude. At the same time, intense whistler mode wave activity was observed by the Voyager 2 plasma wave experiment [Gurnett et al., 1986].
 The Kennel-Petschek concept of self-limitation of radiation belt particle fluxes remains important in magnetospheric physics, though the conditions relating to its validity are imperfectly understood and warrant further investigation. Almost all previous investigations of flux limiting, including the original work by Kennel and Petschek , have used the nonrelativistic approximation. Here for the first time, we present an exact, fully relativistic analysis of the Kennel-Petschek trapping limit. When relativistic effects are included, wave growth properties can be significantly different from those derived from a nonrelativistic treatment, for electron thermal energies above 100 keV, especially in regions of low plasma density [e.g., Xiao et al., 1998]. We present in section 2 our derivation of the limiting electron integral flux in a fully relativistic regime. To achieve this, we derive the temporal growth rate for field-aligned whistler mode waves in a relativistic plasma modeled by our adopted distribution function (equation (1)), and we apply the condition that waves generated at the magnetic equator acquire a specified convective gain over a given growth region. It is important to note that we apply the latter condition to replace the condition used by Kennel and Petschek  involving wave reflection. The idea of Kennel and Petschek  that wave packets generated near the equator are subsequently reflected back from higher latitudes has not been borne out by observations. In section 3 we briefly examine in the context of the present study the aforementioned limiting energy spectrum derived by Schulz and Davidson . In section 4, we apply our trapping limits to Earth, Jupiter and Uranus, and compare our results with observed energetic electron fluxes. Finally, in section 5 we summarize the results and state our conclusions.