Charged Particle Trajectories in Earth’s Magnetic Field

Sarah Arveson

MagnetismA charged particle will experience a force when placed in a magnetic field and given an initial velocity

This is given by the Lorenz force

where q is the charge of the particle in Coulombs, m is the mass of the particle in kilograms, v is velocity in meters per second, and B is the magnetic field in Tesla

Consider a uniform magnetic field in the z direction. Assume the only force is due to the magnetic field, and there is no electric field present. The trajectory of a charged particle should follow a helical path with radius of curvature given by

The trajectory will look like so…

Geomagnetism: Earth’s Magnetic Field• Produced by the convecting, rotating iron outer core• Approximately a magnetic dipole field tilted at 11.5 degrees from geographic

North• The magnetic field is given by

• And we arrive at equations for the magnetic field

Where m is the best fit magnetic dipole moment in amp meters squared and Is the permeability of free space

r is the radial direction

is colatitude is longitude

TrajectoriesWe can then solve for the trajectory of charged particles in Earth’s magnetic field using 4th order Runge Kutta:

Simulating Electron TrajectoriesI shoot beams of electrons from 1-2 Earth radii away at the Earth from random locations in the Northern Hemisphere. Each particle is given an initial velocity such that the radius of curvature is around the same magnitude as the Earth’s radius (for the purpose of interesting trajectories.

Particles with high enough energies (velocities) are deflected and fly away, but those that have enough energy to make it to the Earth but a low enough energy to be trapped by the magnetic field will become “trapped” in the magnetic field lines.

Aurora Borealis & Aurora Australis• Cosmic Rays are hitting us on a daily basis- they consist mainly of

protons (hydrogen nuclei) and travel at very high speeds with very high energies.

• At high latitudes, oxygen and nitrogen atoms carried from the solar wind that combine with photons. Light is emitted as the atoms return from their excited states to their ground states.

• Simulation will require lots of extra parameters to account for colors and relativistic effects

Run my code with…

CPT_compressed.mmfield.mudot.m