Magnetic Shielding Lab (2D equatorial plane)

Charged particle in a dipole field

In the equatorial plane of a magnetic dipole the field is perpendicular to the plane and falls as B = B_s R_E^3 / r^3, with B_s = 3.1e-5 T at the surface. A particle of charge q and momentum p obeys dp/dt = q v x B: the force does no work, so the speed is constant while the direction turns with local gyro-radius r_L = p/(qB). Measured in the Stormer length C_st = sqrt(mu0 M q / (4 pi p)) (that is, (C_st/R_E)^2 = q B_s R_E / p) the equation of motion has no free parameter, so one picture describes every energy.

Forbidden zone and cutoff

Conservation of canonical angular momentum shows that a particle arriving from infinity can never come closer than (sqrt 2 - 1) C_st = 0.414 C_st in the equatorial plane. To reach a radius r the particle needs C_st <= r / 0.414, i.e. momentum p >= q B_s R_E (0.414 R_E / r)^2: about 10 GV at the surface, which is the right order of the observed equatorial geomagnetic cutoff (about 15 GV). Solar-wind particles (keV) are far below this, so they are turned away; in reality aurora electrons enter by following field lines to the poles, which an equatorial slice cannot show.

Integrator

The Boris rotation scheme turns the velocity by exactly 2 atan(B dt/2) each step, so |v| is conserved to rounding error; the step is limited to 0.005 r^3 so tight gyrations near the zone are resolved.

Numerical checks on this page

(1) |v| stays at 1 along a 14-unit trajectory. (2) In a uniform field the Boris orbit is a circle of radius m v/(qB) over ten gyrations. (3) 801 beams with different impact parameters never come closer than the analytic Stormer radius.

Honest simplifications

Pure dipole with no solar-wind compression, magnetopause, ring current or electric fields; equatorial plane only (no mirroring along field lines, no polar precipitation); non-relativistic equations of motion with the relativistic momentum used for the cutoff; the Earth is a disc that simply marks the surface. Real cutoffs depend on latitude (roughly as cos^4) and direction of arrival.

• Pick 10 GeV proton: the red tracks hit the Earth disc, the blue ones are deflected away.
• 1 GeV proton is entirely shielded: the whole beam stays outside the yellow forbidden circle and the Earth is tiny in the picture.
• Switch to an electron: the tracks bend the opposite way around the dipole.
• Slide the target radius to 6.6 R_E (geostationary orbit) and watch the cutoff energy fall.
• Drag the tracer impact parameter to find the path that grazes the dashed Stormer radius.