Particle Orbital Dynamics in the Magnetosphere (3)

We conclude now our treatment of particle orbital dynamics in the magnetosphere.

Assume two magnetic mirrors, one near Earth (e.g. equator) and a second at some distance ℓ = fL where L defines the Earth radii distance, f denotes a multiple of L, and the mirrors approach each other slowly. Then the intermediate plasma particles gain energy all the while according to:

(1) dE/dt = p (dv/dt) = mv(dv/dt) = mv^2/ ℓ [dℓ/dt]

where ℓ is the path length defined by the L-parameter. Then the acceleration (dv/dt) is the Fermi acceleration.

In the numerical model I use to analyze particle trapping and orbits, L is taken at an initial value of 5 (e.g. L = 5 RE) and then reduced by increments to a minimum of ℓ = 1.5 L.

By application of differentials, and since dE/dℓ = mv(dv/dℓ) = 2E/ ℓ, one can show the energy varies according to the change in ℓ by (cf. Hargreaves, 1992):

dE/ E = 2dℓ/ ℓ

Thus, we are looking at a total change in ℓ of 85%. for which: +dE = 1.7E (170% increase in energy corresponding to 85% increase in length)

On this basis, one expects chaotic dynamics to radically emerge as ℓ (less than) L(5).
Analysis proceeds under two regimes.

I) Normal time steps: Computing the orbit position over n t(b) bounce periods, for 1 MeV electrons and protons trapped between the mirrors, initialized at position, M1-M2 = L(2). At the energy E k = 1.0 MeV, the electron bounce period t(e) b = 0.11s, and for protons, t(p) b = 4.7 s. Using these, the total orbital duration will be taken as 50 bounce periods. Thus, the bounce period selects the time increment for each species. It will be shorter for electrons (0.1s), and longer for protons (0.5s). Keeping E k constant, the orbits are then mapped according to:

(2) r(t) = r _L [cos (Wt) - sin (Wt)] and v(t) = r_ L [- sin (Wt)] - cos (Wt) ]

where t _i = it(e) b or t = it(p) b depending on whether electron or proton orbits are analyzed at the particular L. In each case, the B-field is assumed to remain constant, as well as the Larmor radius (r_L)

II) Chaotic time process via Poincare Section time step. As we know, chaotic dynamics arises from extreme sensitivity to initial conditions. In this regime, the time step in (A) is adjusted to reflect an applied force (Fermi force or mv^2/ ℓ [dℓ/dt]) acting once per cycle (bounce period) such that:

t _n = φ_o + 2πn/ W

where φ_o is the phase of the Poincare section, and: φ_o = W to mod 2π. Thus, φ_o is reduced by integrable multiples of 2π so that: 0 Less than) φ_o (less than) 2π.

For purposes of the analysis, at point of initialization, φ_o= 0 (W to = 0) and thence:

t(e) n = nt(e) b + 2πn/ W_e

Or: t(p) n = nt(p) b + 2πn/ W_p

The above time sequence process, as in the case for (A), will be done as the L-parameter is diminished from L = 2RE to L = 0.5RE in increments of 0.1 RE. Once more, the Larmor radii are kept constant, and the value of E _k = 1.0 MeV, as well as the pitch angle, taken to be φ ~ 3o deg. All numerical trials are done using Mathcad following the regimen described above. Two separate models were computed, one according to Method (A) and the other according to Method (B). '

If results warrant it, chaotic emergence is assessed by virtue of the exponential divergence rate for a given orbit (e.g. Chen, 1993) whence:

(3) lambda= lim n->L [(1/ ndt) [ SIGMAi=1 to n {ln (w_i/w_o)}]

where w_o is the norm of the initial tangent vector, w_i is the norm of the tangent vector after the ith step, and D_t is the time step (t = nD_t = nt(e) b + 2pn/ W_e
or nt(p) b + 2πn/ W_p ).

Note that that value L (lambda ->L, as n -> L) at oo, corresponds to the standard Lyapunov exponent.

In the context of periods defined via Poincare section, the Lyapunov exponents are:

(4) L1((xo , yo ) = lim k->oo (1/ T_k) {( lim d_o=1 [ln ½(d_max k/d_o)^½])}


The results for L-parameters from L = 5 down to L = 1.5 are presented in terms of four “polar” (actually r(t) by t) plots per diagram, with the top frames (designated A and B) referenced to electrons and protons, respectively, in the Method (I) regime, and the bottom frames (designated C and D) referenced to electrons and protons, respectively, in the Method (II) regime.

In the attached graphic (generated from Mathcad), particle dynamics are depicted at the L= 5 boundary, and we see quasi-trapped orbits appear in the top frames, for both electrons and protons of ~ 1.0 MeV. Computations show for the electrons(B), ln (Dr(ti)/ Dr (ti-1)] ~ (- 0.001) and for protons(A), ~ (- 0.02). Thus, in either case, lambda (less than) 0, so there is no evidence of dissipation.