Revisiting Single Particle Dynamics In Space Plasma

"Single particle dynamics"  is perhaps the most important introductory aspect of plasma theory, especially for space plasmas (e.g. Earth's magnetosphere, heliosphere etc.). Specifically, it is the domain wherein we enter plasma “orbit theory” and consider a charged particle (say of charge q) in a uniform and constant magnetic field (B). The behavior of such charges, in both magnetic and electric fields, is crucial for understanding plasma behavior as a whole.

The governing equation of motion with F the Lorentz force,  is:

m (dv/dt) = q(v X B) = F

The motion here is such that v will always be perpendicular to the force acting on the particle so v ⊥ F, implying circular motion. Thus:

dv/dt = q/ m [v X B]

Meanwhile, (v⊥ ) ² / r  = q/ m [v⊥ B]

The quantity r  above is none other than the gyro-radius. Solving for it one finds:

r  = m/ q [v⊥ / B] = v_⊥ / (qB/m) =  v_⊥ /   Ω

for which one can have either the electron, or ion gyro-frequency (Ω). The ion gyrofrequency will be:

Ω _i  = qB/ m _i

 And the electron gyrofrequency is:

Ω _e  = qB/ m _e

 These equations help to explain the physical basis for the origin of a preponderance of radio waves (i.e. gyro-magnetic emission) such as from the Sun, and other cosmic objects (e.g. quasars, pulsars). 

 Note that the velocity v has two components, parallel and perpendicular:   

v =     v_||   + v_⊥

where the first term denotes the velocity along B which stays constant so that d(v_|| )/ dt = 0.

 Meanwhile:

v_⊥²/ r  = q/ m [v_⊥ · B]

(setting the centripetal force = to the magnetic force producing it)

From the preceding we obtain the gyro frequencies.

While on this topic it is useful to also consider the "gyration velocity" which  is simply the magnitude:

v⊥ = 2 m _m  B/ m

where m _m is the magnetic moment.

Thus, the proton gyrates at this rate provided m _m is a constant of the motion: viz.   m _m  = m(v⊥)²/ 2B  = const .

The gyro-period is: T = 2 p  / Ω   

Bear in mind the gyration energy:

E  = m _m  B = m/2 (E/B) ²,

Note that the position of a particle at any time t can always be specified for a given coordinate system, if one knows the Larmor radius r  and the gyrofrequency, Ω:

Thus: r(t)= r (cos Ω t x^ - sin Ω t y^)

Which equation can be referenced to the diagram below:

The basics presented above ignore the fact that no general solution exists to the equations of motion for a charged particle moving under the influence of the Lorentz force in a dipole B-field. What happens is that an approximation is needed, called “the guiding center (GC) approximation”. This assumes the  magnetic field changes are small over a gyroperiod, and gyroradius.

Based on this simplification, the electron or ion moves along B-field referenced to a guiding center, such that:

(a ) x – x_o = - i v⊥ exp (i Ω t)/ W = r sin (Ω t)

and

(b) y – y_o = ± v⊥ exp (i Ω t)/ W = r  cos (Ω t)

The key point is that the guiding center (x_o, y_o) is fixed. while r is the "Larmor radius" (or gyroradius).

In the GC approximation, particle motion displays three components: 

1) gyration about a field line (given by the gyrofrequency, or cyclotron frequency); 

2) reflection between two mirror points (embodied by the “bounce period”) and

 3) a gradual longitudinal drift, denoted by the (E x B) drift. 

Suggested Problems:

1)Given the situation shown below for a particle in plasma:

And: v _o = v _xo x^ + v _yo y^ (for initial velocity)

Solve the equation of motion: F = q(E + v X B)

2) (a)If the perpendicular velocity component ( v⊥) is 10⁵  m/s for an electron in a plasma, find its Larmor radius, gyration energy and its gyro-period.

(b) Find the guiding center positions for the electron referenced in (a) if t = T/2.