Drift-shell splitting at arbitrary pitch angle

Analytical expressions are developed to approximate the bounce period and second invariant as functions of particle energy, equatorial pitch angle, and parameters of the guiding field line. The equatorial pitch angle is not restricted to values near π/2, as in the harmonic-oscillator approximation, but terms indicating deviation of the magnetic field from that of a dipole are included to first order only. There follow approximate analytical expressions for tracing drift shells in the magnetosphere. Two important cases are considered separately: (a) shell splitting caused by dawn-dusk asymmetry of the magnetosphere's electrostatic potential, superimposed on a dipolar magnetic field, and (b) shell splitting caused by day-night asymmetry of the earth's magnetic field in the absence of electric fields. In each case, an expression is derived for the ratio DLL/Dxx, i.e., for the relationship between pitch-angle diffusion and radial diffusion in the presence of drift-shell splitting at arbitrary equatorial pitch angle.