Download Charged Particle Trajectories in Earth*s Magnetic Field

Charged Particle Trajectories in
Earth’s Magnetic Field
Sarah Arveson
Magnetism
A charged particle will experience a force when placed in a magnetic field and
given an initial velocity
This is given by the Lorenz force
F = q × (E + v ´ B)
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
v2
q B
å F = ma Þ qvB = m( r ) Þ r = m × v
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
¶V 1 ¶V
1 ¶V
The magnetic field is given by
•
And we arrive at equations for the magnetic field
B(r) = -m 0 ÑV (r) = (
, × ,
)
¶r r ¶q r sin(q ) ¶f
m × r mr cos(q ) m cos(q )
V (r) =
=
=
3
3
4p r
4p r
4p r 2
-m 0 m cos(q )
Where m is the best fit magnetic dipole moment
Br (r, q , f ) =
in amp meters squared and 0 Is the
2p r 3
permeability of free space
-m 0 msin(q )
Bq (r, q , f ) =
REarth = 6.3712 ×10 6 m
3
4p r
r is the radial direction
kg × m
Bf (r, q , f ) = 0
m 0 = 4p ×10-7 2 2
m
q
is colatitude
f
is longitude
As
m = 7.94 ×10 22
B0 = 3×10 -5 T
Trajectories
We can then solve for the trajectory of charged particles in Earth’s magnetic
field using 4th order Runge Kutta:
å F = Fmagnetic =q × (v ´ B) = ma Þ a =
x
y
vx
vy
z
vz
u = v du =
ax
x
vy
ay
vz
az
q
× (v ´ B)
m
Simulating Electron Trajectories
I 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.m
mfield.m
udot.m