General Considerations

We start with motions of an individual, charged plasma particle subject to imposed, external electric, magnetic and other force fields. This is obviously simpler than studying motions of finite volumes of plasma since the electromagnetic and collisional interactions between charged plasma particles are ignored. Collective (wave) effects are also ignored here. Nevertheless, since many plasmas are collisionless and since a plasma's internal electromagnetic interactions are often unimportant compared with macroscopic fields (e.g., Earth's magnetic field), orbit theory often describes the motion of the plasma as a whole. An example of this is the flow of the solar wind plasma across its magnetic field.

Orbit theory is also important in understanding the motion of energetic particles, which often act as test particles, and in understanding the acceleration of particles. Examples of the former are the motion of energetic particles in the ring current and Van Allen radiation belts in Earth's inner magnetosphere, while the latter is exemplified in drift acceleration at shock waves. Orbit theory is also important in understanding the creation of particle distributions with free energy for wave growth, for instance in Earth's foreshock.

The basic equation for orbit theory is the (non-relativistic) equation of motion

$$m \frac{d {\bf v}}{d t} = q ( {\bf E} + {\bf v} \times {\bf B} )$$ (2.1)

for a particle with mass m and charge q moving in an electric field ${\bf E}$ and magnetic field ${\bf B}$. This can be generalized by including additional forces such as gravity. The total derivative in equation (2.1) can be separated into

$$\frac{d}{d t} = \frac{\partial}{\partial t} + {\bf v} . {\bf\nabla} \ .$$ (2.2)

One basic technique in orbit theory is to write the particle velocity as the sum

$${\bf v} = {\bf v}_{\parallel} + {\bf v}_{D} + {\bf v}_{g}$$ (2.3)

of three terms. The first, ${\bf v}_{\parallel}$, is the particle velocity parallel to the magnetic field, otherwise known as the particle's parallel velocity. The second, ${\bf v}_{D}$, is the drift velocity of the particle's gyrocenter perpendicular to the magnetic field: this drift velocity is associated with electric or other forces directed perpendicular to the magnetic field or else temporal or spatial variations in electric or magnetic fields. The sum ${\bf v}_{\parallel} + {\bf v}_{D}$ describes the velocity of the particle's gyrocenter. The final component, ${\bf v}_{g}$ is the particle's intrinsic gyromotion or cyclotron motion about its gyrocenter (and the magnetic field).

From Eq. (2.1) the time rate of change of a particle's kinetic energy is

$$\frac{d}{d t} (1/2 m v^{2}) = {\bf v} . m\frac{d {\bf v}}{d t} = q {\bf v} . {\bf E}\ .$$ (2.4)

Energization of a particle therefore requires, as expected, the existence of a non-zero electric field such that at least one component of the particle velocity produces a non-zero value of ${\bf v} . {\bf E}$. Parallel electric fields can therefore energize particles. However, from Eq. (2.4) it can be seen that drifts ${\bf v}_{D}$ can also lead to particle energization.

The analyses below assume time-stationary macroscopic fields ${\bf E}$ and ${\bf B}$ unless otherwise stated. Time-varying electric and magnetic fields are sometimes important, however; for instance in ``betatron'' acceleration. Moreover, the time-varying fields of plasma waves can accelerate charged particles.