A proton with equatorial pitch angle close to mirroring on the shell of the Earth's…

PHYSICS · Subjective · Fermi University Physics · Plasma Physics · The plasma state and single-particle motion · Laboratory, industrial and space plasmas

Plasma Physics, Ch Slow changes and what survives them: adiabatic invariants, mirrors and trapped orbits

Problem

A proton with equatorial pitch angle close to

90^\circ

mirroring on the

L=4

shell of the Earth's dipole with kinetic energy

1\,\mathrm{MeV}

has gyration period

T_g = 0.14\,\mathrm{s}

, bounce period

T_b = 5.5\,\mathrm{s}

and drift period

T_d \approx 11\,\mathrm{min}

(the last from

T_d = 2\pi eB_0R_E^2/3WL

). Because the equatorial dipole field goes as

B_{\rm eq}\propto L^{-3}

and the gyration period depends only on this field,

T_g \propto L^3

. For the same pitch-angle geometry, the bounce path length scales as

L

and the particle's speed as

W^{1/2}

, so

T_b \propto L/W^{1/2}

. The drift-period formula above gives

T_d \propto 1/(WL)

directly. A second proton with the same pitch-angle geometry has kinetic energy

4\,\mathrm{MeV}

and mirrors on the

L=6

shell. Find its gyration, bounce and drift periods.

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