Expert: Philip A. Stahl - 1/2/2012

Question
Hello. I am very interested in the aurora but want to get beyond the pretty pictures to the physics. I have started self-study of plasma physics but there's a lot to learn and I'd like to know the key parts to learn for application to where the aurora occurs. From what I've read, studied so far plasma orbit theory is maybe the first stuff to master. What do I need to know including the key equations that describe the plasmas?

Answer
Hello,

It is nice to see someone doing self study in basic plasma physics, especially as it can help to better understand phenomena like the aurora.

Probably what you'll most be focusing on is the Earth’s "near magnetosphere".  By “near magnetosphere” is meant the region extending from the ionosphere to a distance of 2 R_E, where 1 R_E = one Earth radius. It includes the auroral oval, the magnetopause (to 2 R_E ); the plasmasphere, plasma sheet, polar cusp and ionosphere. Within this domain we enter plasma “orbit theory” and consider a charged particle (say of charge q) in a uniform and constant magnetic field (B).

The governing equation of motion for charged particles is:

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

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

Thus:

(2) dv/dt = q/ m [v X B] is a centripetal acceleration.

e.g. a = (-r) (v⊥)^2/ r


Note that the velocity has two components:

(3) v = v⊥ + v ‖

where the second term denotes the velocity along B (magnetic induction) which stays constant so that

(4) dv ‖ / dt = 0

Meanwhile:

(5) (v⊥ )^2/ r = q/ m [v⊥ B]


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


(6) r = m/ q [v⊥ / B] = v⊥ / (qB/m)

for which one can have either the electron:

(7a) W(e) = qB/m_e

or the ion, gyrofrequency:

(7a) W(i) = qB/m_i

As shown in Fig. 1, both electrons and ions (e.g. protons) undergo a corkscrew motion around the centers of their paths (guiding centers) and have drift velocities in the same direction in this field orientation.

This drift motion occurs in crossed (E X B) fields and hence this is called “E cross B” drift. In the magnetosphere, special focus is on the motion of charged (e, i) particles undergoing periodic motion between two mirror points.

In space physics, one uses the sine of *the loss cone angle* to obtain the mirror ratio (where
B(min) , B (max) refer to minimum and maximum magnetic induction respectively):

sin(Θ_L) = ± [ (B(min) / B (max)]^½

If one finds that there are particles within the “mirrors” (see Fig. 2) for which the “pitch angle” (φ) has:

(9) sin (φ) > [ (B(min) / B (max )]^½


then these will be reflected within the tube, On the other hand, those particles for which the “less than” condition applies will be lost, on transmission out of the mirror configuration.

Since the adiabatic invariant for particle motion is a constant of the motion:

(10) u= ½ [mv ^2 /B]

we have:

(11) [v⊥^2 /B(max)] = [v ‖ ^ 2 /B_z] = const. or [v⊥^2 /v ‖ ^ 2 ] = B_z /B(max)


where we take B_z = B(min)

that is, the minimum of the magnetic field intensity.


For kinetic energy of particles we must have:

(12) E = ½ m (v⊥^2 + v ‖ ^ 2 ) = const.

So that E is a constant of the motion, as well as u. Both of these are called “adiabatic invariants”.


A third adiabiatic invariant is deduced based on the diagram (Fig.2), showing a charged particle trapped between two “mirror” walls, M1 and M2.

In effect, we are analyzing a system with a trapped charge within a magnetic bottle of length L (or the distance between the mirrors). If the particle is trapped as shown then the particle behavior is periodic.

If we let L change slowly then: v ‖ L = const.

Now, let M1 be stationary and M2 move toward M1 at velocity v_m, then the incident velocity relative to the wall is:

-[(v ‖ + v_m) - v_m]

and:

delta v ‖ = - [-(v ‖ + v_m) - v_m] + v ‖ = 2 v_m

Now, in each reflection the velocity changes by 2 v_m. The number of reflections per second can then be expressed as:

#R = v ‖/ 2L

We can then write:

dv ‖/ dt = 2 v_m (v ‖/ 2L) = (v ‖/ L) (-dL/dt) = -(v ‖/ L) (dL/dt)

whence:

L (dv ‖/ dt ) + dv ‖ (dL/ dt) = 0

so:

d/dt( v ‖/ L) = 0 = const.


It should be noted that all the adiabatic invariants are *approximations* to what are called Poincare invariants. These assume the form:

P = INT_c p*dq

where INT_c denotes an integral for which all points on the closed curve, C, in phase space, conform to the equations of motion.

In respect of the magnetosphere, the time required for a charged particle to move from the equatorial plane to one mirror point (say, M1) and back is given by the bounce period:

t(b) = 4 INT (0 to Θ) [ds/ v‖ ]


where ds is an element of arc length along the field line (B) and an integration is performed between 0 and Θ, for which we find:

v ‖ = v [1 – B/B_max]^½


You need to note that 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 approximation” – assuming magnetic field changes are small over a gyroperiod, gyroradius. Based on this simplification, the electron or ion moves along B-field referenced to a guiding center, such that (Chen, Intro. to Plasma Physics, 1977):

(13 a ) x – xo = - i v⊥ exp (iWt)/ W = r _L sin (Wt)

and

(13b) y – yo = ± v⊥ exp (iWt)/ W = r_ L cos (Wt)

The key point is that the guiding center (xo, yo) is fixed. while r_L 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 referenced in the Introduction.

With “magnetic mirrors” (pinched B-field gradients) present we have the possible loss cone effect. In the auroral context, the loss cone concept has validity in connection with field –aligned potential drops and enhancing parallel current densities (e.g. Birkeland currents). To be specific, in the magnetospheric environment only electrons of small pitch angles contribute to J . Any parallel electric field increases the flux of electrons inside the loss cone and increases J such that (Cravens, p. 430):

J / Jmax = R[ 1 – (1 – 1/R) exp (e φ‖ / k_B T_e ) ]

Lastly, it's useful to be aware of the conditions for which particles are *trapped*, or quasi-trapped:

If particles-orbits are “trapped” one has the condition:

(v / v⊥ ) < (B (max) / B(min) - 1)^½

If “quasi-trapped” then:

(v / v⊥ ) > (B (max) / B(min) - 1)^½

If transient:

(v / v⊥ ) < (B (max) / B(min) - 1)^½


I do hope this makes sense, but if you need more assistance here, don't be afraid to ask. Commendations to you for undertaking a fascinating area of physics, but one definitely not for the faint -hearted!

Btw, an excellent introductory book to put all these quantitative aspects into context is 'Auroras, Magnetic Storms, Solar Flares and Cosmic Rays', 1998, by the American Geophysical Union, Eds. Steven T. Suess and Bruce T. Tsurutani. Special attention ought to be given to Chapter 1 ('Aurora') written by Syun-Ichi Akasofu, and Chapter 2, ('The Earth's Magnetosphere') by S.W.H. Cowley.