On the role of external and internal source in generating energy and pitch angle distributions of inner‐zone protons

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Abstract
We separate the steady‐state distribution function f of energetic inner‐zone protons into two parts: f=f1 + f2, where (1) f1 is a solution of the homogeneous radial diffusion equation, associated with the external (diffusive) source, and (2) f2 is a solution of the inhomogeneous diffusion equation, related to the internal (Crand)source. By making special assumptions about the energy dependence of source and diffusion processes, we are able to evaluate analytically f as a function of energy E, pitch angle α, and dipole shell parameter L. The analysis shows that the external source preferentially supplies protons of energies ≲30 MeV that mirror close to the equator, while the internal source is most important for protons having energies ≳50 MeV or pitch angles ≲50°. Agreement between measured and computed energy spectra and pitch angle distributions for L values from L= 1.2 to L1.7 requires diffusion coefficients DLL(e) ≈ 2 × 10−11 L5 E−1 s−1 or DLL(m) ≈ 3.75 × 10−12 L9 E−1 sin2.7 α s−1 (E in MeV), caused by electrostatic and magnetic fluctuations, respectively.