*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})_{min}/ B

_{max})

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

m=
½ [mv

^{2}/B]
[v

_{^}^{2}/B_{max}] = [v_{||}_{ }^{2}/B_{z}] = const. or
[v

_{^}^{2}/ v_{||}_{ }^{2}] = B_{z }/B_{max}_{z}= B

_{min}

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

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

_{||}

_{ }L = const.

_{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
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

Note also that:

_{⊥}^{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**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

**v**

^{2}

_{⊥ }/B (¶B

_{z}/ ¶z)

But
by previous definition for the adiabatic invariant:

m

_{m }= m(**v**⊥)^{2}/ 2B = const ._{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}e^{2}/ 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.

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

v

_{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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