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Fermi acceleration [1], sometimes referred to diffusive shock acceleration (a subclass of Fermi acceleration), is the acceleration that charged particles undergo when reflected by a magnetic mirror. This is thought to be the primary mechanism by which particles gain energy beyond the thermal energy in astrophysical shock waves. It plays a very important role in many astrophysical models, mainly of shocks including solar flares and supernova remnants. It is studied by using the Fermi-Ulam model.

There are two types of Fermi acceleration: First order Fermi acceleration (in shocks) and Second Order Fermi acceleration (in the environment of moving magnetized gas clouds). In both cases the environment has to be collisionless in order for the mechanism to be effective. This is because Fermi acceleration only applies to particles with energies exceeding the thermal energies, and frequent collisions with surrounding particles will cause severe energy loss and as a result no acceleration will occur.


First order Fermi acceleration

Shock waves typically have moving magnetic inhomogeneities both preceding and following them. Consider the case of a charged particle traveling through the shock wave (from upstream to downstream). If it encounters a moving change in the magnetic field, this can reflect it back through the shock (downstream to upstream) at increased velocity. If a similar process occurs upstream, the particle will again gain energy. These multiple reflections greatly increase its energy. The resulting energy spectrum of many particles undergoing this process (assuming that they do not influence the structure of the shock) turn out to be a power law:
\frac{dN(\epsilon)}{d\epsilon}\propto \epsilon ^{-p}
where p\gtrsim2.
The term "First order" comes from the fact that the energy gain per shock crossing is proportional to βs, the velocity of the shock divided by the speed of light.


The injection problem

A mystery of first order Fermi processes is the injection problem. In the environment of a shock, only particles with energies that exceed the thermal energy by much (a factor of a few at least) can cross the shock and 'enter the game' of acceleration. It is not known at the time what mechanism causes the particles to initially have high enough energy to do so.

Second order Fermi acceleration

Second order Fermi Acceleration relates to the amount of energy gained during the motion of a charged particle in the presence of randomly moving "magnetic mirrors". So, if the magnetic mirror is moving towards the particle, the particle will end up with increased energy upon reflection. The opposite holds if the mirror is receding. This notion was used by Fermi (1949)[2] to explain the mode of formation of cosmic rays. In this case the magnetic mirror is a moving interstellar magnetized cloud. In a random motion environment, Fermi argued, the probability of a head-on collision is greater than a head-tail collision, so particles would, on average, be accelerated. This random process is now called second-order Fermi acceleration, because the mean energy gain per bounce depends on the mirror velocity squared, \beta_m^2. Surprisingly, the resulting energy spectrum anticipated from this physical setup is very similar to the one found for first order Fermi acceleration.


  1. ^ M. A. Lieberman and A. J. Lichtenberg, Phys. Rev. A 5, 1852 (1972)
  2. ^ On the Origin of the Cosmic Radiation, E. Fermi, Physics Review 75, pp. 1169-1174, 1949

Longair, Malcolm S. (1994). High Energy Astrophysics, Volume 2. Cambridge University Press. ISBN 9780521435840.  

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