Space Plasma Physics Revisited (1)

Space plasma associated with emission nebula with E, v, B components shown.

1.Plasma Orbit Theory – Particle dynamics:

Here we consider a charged particle (say of charge q) in a uniform and constant magnetic field (B).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  (F) 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]

Where v⊥  is a positive, constant velocity denoting the speed in the plane perpendicular to the magnetic induction, 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) . These equations 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 has two components:   

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)

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

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

where the denominator denotes the gyro-frequency.Ω   = qB/ m 

So that the ion gyrofrequency will be:

Ω _i  = qB/ m _i

And the electron gyrofrequency is:

Ω _e  = qB/ m _e

While on this topic it is useful to consider this as well: The "gyration velocity" is simply the magnitude:

v⊥ = Ö (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,

These equations explain the physical basis for the origin of a preponderance of radio waves (i.e. via gyro-magnetic emission). 

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^)



Geometry for gyrofrequency in terms of position r, y

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 (F =q( v X B) )in a dipole B-field. What happens is that an approximation is needed, called “the guiding center 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 preceding equation pair (a, b) describes a circular orbit around the guiding center (x_o, y_o), with the direction of gyration always such that the magnetic field generated by the charged particle is opposite to the externally imposed field.  (Plasma particle then tend to reduce the magnetic field and we say plasmas are "diamagnetic")

In a "crossed" E-B field (e.g. E X B), for example, we would have:






















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

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. In this instance we allow for  a slight inhomogeneity in E  -  start by expanding E about the guiding center (Taylor series expansion). Without getting into the details - which the mathematically inclined reader can check for himself, viz. 

E = 

E( r₀ ) + [(r -   r₀ ) · Ñ] E +  ½  [(r -   r₀ ) · Ñ]² E  +    ……

We arrive at:

d[v]/dt  =  q([E] +  [v] X B)

Or: q[E] +   v _d  X B  = 0

So that:   v _d     =    [E] X B/  B ²

=   (1 +    _¼ r ²  v⊥² ) E _o  X B/  B ²

Where  E _o  is the E-field at the guiding center.

With such an electric field present we will find the motion to be the sum of two motions: 1) the circular Larmor gyration and 2) a drift of the guiding center.  The equation of motion will then be found to be:

m (dv/dt) = q(E +  v X B)

Selected Problems:

1) Find the ion and electron gyrofrequencies for an ion and electron in solar plasma with a magnetic induction (field strength) of B = 0.0001 T.

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

3) Thence or otherwise obtain the gyration energy in eV. (1.6 x 10 ^-19  J = 1 eV)

4) Find the guiding center positions for the electron referenced above (previous problems) if t = T/4.