Space Plasma Physics Revisited (2)

2)Magnetic mirrors and solar loops

The basic plasma physics for solar active regions usually starts by examining solar coronal loops to see where they conform to the typical magnetic mirror profile used in standard plasma physics. In space physics, one uses the sine of the loss cone angle to obtain the mirror ratio relating the magnetic inductions at the loop ends:

sin (q _L ) = ± Ö (B_min / B _max )

If one finds that there are particles within the “mirrors” for which the “pitch angle” (a) has:

sin (a )  >  Ö (B_min / B _max )

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

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

 m=  ½  [mv ² /B]  

 we have:

 [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

A simple model for a mirror system is shown below with M1 and M2 the magnetic mirrors separated by a distance  2 L.

If we are careful to change L slowly, then one finds:

 v _|| L = const.

 Now, assume M1 is stationary and M2 moves toward M1 at a velocity,    v _m  , then the incident velocity relative to the wall is:

[(v _||   +  v _m)   -   v _m ]

And:

 

D v _||   =  - [- (v _||   +  v _m)   -   v _m ]  + v _|    =   2 v _m

Thus, with each reflection, the velocity changes by 2 v _m and the number of reflections per second will be:   v _||  / 2L

 Further:

dv _||  /dt  =   2 v _m  (v _||  / 2L)  =  v _||  /L (-dL/dt)

=    - v _||  /L (dL/dt)

So that:   d/dt (v _||  L)  = 0

For kinetic energy of particles we must have:

E   =  ½ m (v_⊥ ²  + v _|| ² ) = const.

Note also that:  v_⊥/ Ω  =   v_⊥/(qB/m) = m v_⊥/qB

Let the guiding center lie on the z-axis (for simplicity) then the average force experienced will be written:

F _z = + ½ q  v_⊥ r _L   (¶B _z/ ¶z) = + ½ q  v_⊥ / Ω _c  ¶B _z/ ¶z

Since:   v_⊥   = r _L  Ω _c     so:  r _L   =  v_⊥/ Ω _c

Where r _L   is the guiding center Larmor  radius.

With appropriate substitutions we get:

F _z = + ½ q  v_⊥ r _L   (m v_⊥/qB) (¶B _z/ ¶z) =

+ ½ m v²_⊥ /B  (¶B _z/ ¶z)

But by previous definition for the adiabatic invariant:

m _m  = m(v⊥)²/ 2B  = const . 

So:   F _z =   m  ¶B _z/ ¶z

Where I have dropped the subscript ‘m’.

In relation to solar flare prognostication we are interested in the criterion for the hydrodynamic loss cone instability  which requires that the particular condition for the ratio of untrapped to trapped particles (Pearlstein et al, 1966)[1] :

 n/ n_o  >   2  W _e /  p w _e  =  0.1

where W _e  is the electron cyclotron frequency; and  w _e  is the electron plasma frequency:

w _e     =  [n_e e²/ m_e  ε_o] ^½

Where n_e is the electron number density, e  is the electron charge,  m_e  is the electron mass and ε_o  is the permittivity of free space.

Problems:

1)A proton moves in a uniform electric and magnetic field, with fields given by: E = 10 V/m (x^) and B = 0.0001 T (z^)where '^' denotes vector direction. (Take m _m  = 8.5  x 10 ^-22  J/T)

a) Find the gyrofrequency and the gyro-radius

b) Find the proton's E X B drift speed

c)Find the gyration speed v  and compare  it to the drift speed

d)Find the gyro-period, gyration energy and magnetic moment of the proton.

2) Consider a plasma mirror machine of length 2L with a mirror ratio of 10 so that B(L) = B(-L) = 10 B(0). A group of N (N > 1) electrons with an isotropic velocity distribution is released at the center of the machine. Ignoring collisions and the effect of space charge, how many electrons escape?

3) Consider a similar mirror configuration for a solar coronal loop for which  B_min =   0.2 B_max . Find the loss cone angle for this loop and also determine whether particles will remain within it. Find the velocity  ratio :    

v _^ / v _||   if    B_max = 0.95 B _z

4) The basic plasma frequency equation relating the ion to the electron plasma frequency can be written:

w ³      =  -  ½ ( w _i ) ² w _e

Find the roots of this equation in terms of the ratio of masses, m _e /m _i.  Show that the real part (or frequency seen in the ion rest frame)  will be:

w _r  =  (½) ^5/3  (m _e /m _i) ^1/3  w _e

Hint:  You may make use of the fact that:

0 =  1 -  w _i  ² / w ^{2   -}-  w _e ² /( w -  w _e ) ²

Where  w _i is the ion plasma frequency and  w _e is the electron plasma frequency

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[1] L.D. Pearlstein, M.N. Rosenbluth, and  D.B. Chang, 1966, Phys. Fluids., Vol. 9, p. 53.