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# Particle Motion

This module allows you to follow the three-dimensional trajectories of charged particles through different magnetic field configurations. The equation of motion for each particle is solved numerically, using the Runge-Kutta integration method.

The mass of the electron used in these exercises is the geometric mean of the proton and electron masses. If the true electron mass were used, we would not be able to display the electron and proton gyro motion on the same plot.

The three plots in this module show different orthogonal views of the field configuration. Each plot is 100 km per side (-50 km to +50 km).

Text boxes allow you to specify the initial positions (km) and velocities (km/s) of the selected particle. The magnetic field is along the positive z-direction (i.e., out of the screen in the leftmost plot and upward on the two plots to the right).

The buttons near the center of the window allow you to start and stop drawing the particle's trajectory. The time step allows you to slow down the motion for vizualization purposes on fast computers. Particle trajectories may be overlaid, which each trajectory being drawn in a different color.

The middle panel on the left shows the particle mass in proton masses, its charge in electron charges, its energy in electron volts, and the time in minutes and seconds since the trajectory began in simulation time (not observers time). The panel on the right shows the first and last pitch angels (i.e., the angle between the velocity vector and the magnetic field) and the minimum and maximum positions in all these coordinate directions.

Any position may be measured with the mouse. The particle coordinates will appear in the boxes below the panels. The writing of the measured postion toggles between box pairs.

## Magnetic field type

You have the option of plotting particle trajectories through the following types of magnetic fields. Note that changing the magnetic field type causes plotting to stop and the initial conditions to be reset.

Changing the magnetic field or the electric field will erase the plots.

• Uniform
• The uniform magnetic field points in the same direction and is the same strength everywhere. You can adjust the strength by entering a value in the Field strength box. The default field strength is 100 nT, and lies along the positive z-axis.

• E⨯B drift
• This option applies an electric field with components across and along the magnetic field. An electric field causes a particle to accelerate parallel to the electric field if it is positively charged and opposite the electric field if it is negatively charged. If the charged particle is moving in a magnetic field, the component of the electric field along the magnetic field causes the particle to accelerate.

The component of the electric field across the magnetic field leads to drift motions perpendicular to the magnetic field because the particle gyro radius alternatively increases and decreases as it is accelerated and decelerated by the electric field. Use the text boxes to specify the magnetic and electric field strengths and the angle of the electric field to the magnetic field.

In the three plots, the electric field is directed along the positive x-axis for an angle of 90°. For 0° it is aligned along the positive z-axis, i.e., parallel to the magnetic field.

• ∇B drift
• Gradients in magnetic fields cause particles to drift perpendicular to the magnetic field and the gradient if the particles have energy perpendicular to the magnetic field. In this option, the magnetic field lines are still straight; i.e., they point in the same direction everywhere, but the field strength is not uniform. We assume the field lines lie along the positive z-direction and field strength increases linearly in the positive x-direction. The field equations are:

Bx = 0
By = 0
Bz = B0 + dBz/dx * (x + 50)

where B0 is the field strength at the left border of the xy-plot (x = -50 km; the right border is at +50 km), and dBz/dx is the gradient of the field strength.

By default, B0 = 10 nT, d(Bz)/dx = 10 nT/10 km, and the resulting field strength on the right hand side of the plots is 110 nT.

• Curvature drift
• Curvature of magnetic field lines causes charged particles to drift if they have energy parallel to the magnetic field. The magnetic field lines in the model are circles of equal field strength in the xy-plane. You can adjust the field strenght using the Field strength text box, and can move the center of the circles along the x-axis using the Horizontal center text box. The field equations are:

Bx = B0 * cos θ
By = B0 * sin θ
Bz = 0
where
B0 is the field strength
θ = arctan(y / (x - xc))
xc is the position of the center of the circle

By default, B0 = 100 nT and xc = 0 km.

• Magnetic mirror
• The magnetic mirror configuration, also called the magnetic bottle, is composed of field lines that approach one another near the ends, and diverge in the middle. The stronger fields at the ends can reflect charged particles so that they are trapped in the magnetic field. The mirror ratio is the ratio of the stronger field strength at the end of the bottle to the field at the middle. The field equations are:

Bx = Br * cos θ
By = Br * sin θ
Bz = B0 * ((1 - 1 / s) * (1 - f) + 1 / s)
where
s = mirror ratio
L = bottle scale length
f = sech(z / L)
r = √(x2 + y2)
Br = -B0 * r *(1 - s) * f * tanh(z / L) / (2 * L)
θ = arctan(y / x)

By default, L = 200 km and s = 100. Changing the value of B0 or the mirror ratio, s, will erase the plots.

• Harris current sheet
• The Harris current sheet produces a magnetic field that is everywhere along the z-direction, but gradually decreases in strength on the approach to x = 0 with a scale length H.

The field changes direction at x = 0. This model is often used to describe the plasma sheet in the magnetotail of the Earth. The equations for the field components are:

Bx = 0
By = 0
Bz = B0 * tanh(x / H)

The default values for B0 and H are 100 nT and 10 km, respectively.

• Dipole
• The dipole field configuration is a very important one in naturally occuring plasma. The converging magnetic field lines of a dipole field reflect charged particles and causes them to return to the equator. Thus the particles can become trapped in this configuration.

In addition, the dipole field has curvature and gradients, causing drifts in the particle motion. Near the Earth, this model is a good approximation of the magnetic field. The magnetic field strength used here is 1000 nT at 1000 km from the center, in the equatorial plane.

## Particle type

You can select to display the trajectories of the following types of particles: H+, He+, He++, O+, e, and H-.