Field-line (Euler-potential) model of the ring current

The equation of a magnetic field line (labeled L) in Dungey's model magnetosphere (dipole field plus uniform southward ΔB) is r=La[1+(r3/2b3)]sin2 θ, where r denotes geocentric distance, θ denotes magnetic colatitude, a is the Earth's radius, and b is the radius of the field model's equatorial neutral line. This model can be generalized (e.g., to accommodate a ring current) by treating b as a function of L and ϕ (magnetic local time) rather than as a constant, so as to yield measured or calculated values of the equatorial magnetic field B0. (In this generalization the equatorial neutral line has a radius b*(ϕ)=(3a/2)L*(ϕ) for some particular ϕ-dependent value of L called L*.) This approach yields an estimate for how a specified distortion of equatorial B0 might map to higher latitudes. It also allows for analytical calculation of the current density J=(c/4π) ∇×B at arbitrary latitude. Since charged particles (of scalar momentum p) scattered strongly in pitch angle satisfy an adiabatic invariant Λ=p3Ψ, where Ψ is the flux-tube volume (per unit magnetic flux), it is of interest to approximate (as well as possible) the flux-tube volume Ψ as a function of L and ϕ. By generalizing the calculation of Schulz [1998a. Particle drift and loss rates under strong pitch angle diffusion in Dungey's model magnetosphere. Journal of Geophysical Research 103, 61–67], we have found such an analytical approximation of Ψ for arbitrarily non-constant b and are using it in bounce-averaged transport simulations of diffuse auroral electrons described by a Hamiltonian function in which the kinetic energy is given by [(Λ/Ψ)2/3c2+(m0c2)2]1/2−m0c2, where m0 is the rest mass and c is the speed of light.