Particle drift and loss rates under strong pitch angle diffusion in Dungey's model magnetosphere

Abstract

Strong pitch angle diffusion so thoroughly randomizes the first two adiabatic invariants of charged‐particle motion that it conserves (as a first approximation) the surrounded phase space volume Λ ≡ p3Ψ0, where Ψ0 = ∮(ds/B) is the flux tube volume per unit magnetic flux and p is the common scalar momentum of the particles involved. (This conservation law is widely used in plasma physics. Its nonrelativistic form is usually derived from the adiabatic gas law but is not really a “fluid” result. Since the conservation law does not couple particles with different energies in the same flux tube, such particles can have different values of Λ.) Here the flux tube volume Ψ is computed for selected field lines in Dungey's model magnetosphere (dipole moment μE ≈ −30.2 μT−RE3 plus uniform southward ΔBz ≈ −9.0 COS6Λ* μT, Λ* being the invariant latitude of the quiet time auroral oval, which maps outward to a circular neutral line at radial distance b = (μE/ΔBz)1/3 in the equatorial plane). The integral ∮(ds/B) is fitted within 0.2% at all L values by the expression Ψ0 ≈ (L4a4/μE){(32/35) − [2.045 + 1.045(r0/b)3 + 0.095(r0/b)6 + 0.075(r0/b)9] ln [1−(r0/b)3]}, where r0 (equatorial radius of the field line) is obtained by solving the cubic equation (r0/b) = (La/b)[1 + (1/2)(r0/b)3]. The particle lifetime τ against strong pitch angle diffusion is given in turn by τ ≈ [4ΨBnBs/(Bn + Bs)(1 – η)](m/p), where Bn and Bs (which can differ in an offset‐dipole field model) are the field intensities at the northern and southern foot points of the field line, m is the relativistic mass, and η is the backscatter coefficient. The same expression for Ψ thus enters both the quasi‐adiabatic Hamiltonian (to whose derivative with respect to 1/L the ensemble‐averaged gradient‐curvature drift rate is directly proportional) and the particle loss rate 1/τ (by which the phase space density ƒ = J4π/4πp2 is exponentially attenuated along the ensemble averaged drift trajectory). This construction provides a means of numerically simulating kinematical aspects of electron motion in the outer magnetosphere and precipitation in the diffuse aurora, as well as ion motion in part of the plasma sheet (regarded as a source of the ring current). Its use in magnetospheric plasma simulations should save computer time and reveal analytical structure relevant to magnetospheric dynamics.