Expert: Philip A. Stahl - 4/22/2008

Question
Hello. My questions seems to have gone to James Gort by mistake. I was really looking for your help here.
Can you explain a bit about how Fermi acceleration works and how one can best quantify it?

Answer
Hello, Jani

First, let's revisit the plasma mirror machine problem you posed last year. As you recall we laid out the "machine" thus:

----------------!--------------
     
----------------0-------------- B = (L)
B = (-L)

<--------- L -------------- -->


This can be a solar loop (say coronal) loop but make the adjustment that the end points (at B = ± L) are “pinched” and thus of narrower bore than at the midpoint. (This is hard to show in a sketch so I leave it to you to do it mentally!)

Then this yields higher magnetic induction, B, at those points which will be “B_max”.

Given the pinched condtion the "mirror points" will be located at (+/- L). We consider particles trapped in this mirror domain (and again if you consult that problem you will see that I gave the special condition for particles being "trapped".

We define what is called the “loss cone angle”:

sin (THETA)_L  =   ± [B_min/  B_max]^1/2

Particles (e.g. electrons) will be TRAPPED provided:

THETA (O) >  (THETA)_L

Thus, THETA (O) =  (THETA)_L

which is said to be the "loss cone" of the system or machine.

Now, how will Fermi acceleration work?

For this we consider only particles moving ALPNG the B-field, e.g. from +L to -L.

Let the particle "bounce" between the mirrors at +L and -L and imagine these mirrors are ever moved closer and closer together, such that at any given time:

L'(t) <  L

where L is the original length and L'(t) is that at any time t.

It can be seen that if the path shortens by (dL/dt) then the particle speed must increase by (dL/dt) at each reflection (of the particle against the moving mirror point)

One can write the rate of change of velocity, v (bearing in mind there is a mirror reflection every t = L/v):

dv/dt =  (dL/dt)(v/L)

or dv/v  =  dL/L

Let the particle now have mass m, then its rate of increase in energy, E is:

dE/dt =  mv (dv/dt) = (mv)^2/ L  * (dL/dt)

You should further be able to show from this that the relative increment of energy increase (dE/E) is related to the relative path shortening by:

dE/ E  =  2 (dL)/ L

Say then that the mirror distance shortens from L to 9L/10 or a factor of 10%, then the energy will be found to increase by:

dE =   [2 (0.1)] E/L  = 0.2 [E/L]

or by 20% from the original value

In more practical terms for the case of coronal loops that demonstrate mirror points, we can quantify the Fermi acceleration using:

Delta E = 1/ 2m[p_perp ^2  + (L_o/ L)^2  p_par^2]  


where L is the instantaneous separation and L_o  is the original parallel path length. As L decreases, the particle energy increases by Delta E as it bounces back and forth between the mirrors.

Note that 'p_perp' denotes the particle momentum perpendicular to the B-field and 'p_par' denotes the component of momentum parallel to the B-field which is most directly affected by the change in the ratio (L_o/L).

Hope this helps.