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Fermi Mechanism

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Fermi Acceleration

Fermi studied how it is posible to transfer macroscopic kinetic energy of moving magnetized plasma to individual particles. He considered an iterative process in which for each encounter a particle gains energy which is proportional to the energy.

Let's write the increase in energy as $\Delta E = \xi E$ after $n$ encounters then:$$E_n = E_0(1 + \xi)^n$$

where $E_0$ is the injection energy in the acceleration region. If the probability of escaping this acceleration region is $P_{esc}$ per encounter, after $n$ the remaining probability is $(1 - P_{esc})^n$. To reach a given energy $E$ we need:$$n = \log\left(\frac{E_n}{E_0}\right)/\log(1 + \xi)$$

after each interaction there is a fraction $(1-P_{esc})$ that remain and the rest escapes the accelerator. If $N_0$ particles entered the "encounter" in the first place, after $n$ interaction those remaining are:$$N(\ge E_n) = N_0(1- P_{esc})^n $$

These particles will always eventually escape since $P_{esc}$ is not 0, but for any given number of cycles, $n$, we can be sure that those remaining particles (whenever they escape) will have more energy than those that escaped at the cycle $n$. We can rewrite:$$\log\left(\frac{N}{N_0}\right) = n(1 - P_{esc}) $$

equalling $n$ with the equation above we have:$$\frac{\log (N/N_0)}{\log (E_n/E_0)} = \frac{\log(1-P_{esc})}{\log(1+ \xi)}$$

For any given energy then we have:$$N(\ge E) = N_0 \left(\frac{E}{E_0}\right)^{-\gamma}$$

where we defined$$\gamma = \log\left(\frac{1}{1-P_{esc}}\right)\frac{1}{\log (1+\xi)} \approx \frac{P_{esc}}{\xi} = \frac{1}{\xi}\cdot\frac{T_{cycle}}{T_{esc}}$$

where $T_{cycle}$ is the characteristic time of acceleration cycle, and $T_{esc}$ is the characteristic time to escape the acceleration region.

Note that $N(\ge E)$ is the integral spectrum, the differntial spectrum is given by:$$ n(E) \propto E^{-1 - \gamma} $$

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Fermi Mechanism

A mechanism working for a time $t$ will produce a maximum energy:$$E\leq E_0 (1+\xi)^{t/T_{cycle}}$$

Two characteristic features are apparent from this equation:

  • High energy particles take longer to accelerate

  • If a given kind of Fermi accelerator has a limited lifetime this will be characterize by a maximum energy per particle that can produce.

General mechanism

In the general mechanism we can imagine a particle encountering something moving at a speed $\beta$. This "something" can be for example a magnetic cloud.

In this general approach, the particle might enter at different angles and exit at difference angles. Let's assume $O^\prime$ to be the reference system where the magnetic cloud is in the rest frame. A particle with energy $E_1$ in the lab reference system will have total energy in this reference system given by the boost transformation with $\beta$ being the speed of the plasma flow (cloud:$$E_1^\prime = \gamma E_1 (1 -\beta \cos\theta_1)$$

Before leaving the gas cloud the particle has an anegy $E_2^\prime$. If we transform this back to the lab reference system we get:$$E_2 = \gamma E_2^\prime (1 +\beta \cos\theta_2^\prime)$$

As the particle suffers from colissioness scatterings inside the cloud the energy in the moving frame just before it escapes is $E_2^\prime = E_1^\prime$ and so we can calculate the increment in energy $\Delta E = E_2 - E_1$ as:$$\xi = \frac{\Delta E}{E_1} = \frac{1 - \beta \cos \theta_1 + \beta\cos\theta_2^\prime - \beta^2\cos\theta_1\cos\theta_2^\prime}{1 - \beta^2} -1$$

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Fermi 2nd order acceleration.

In the second order (first chronologically) Fermi considered encounters with moving clouds of plasma.

  • The scattered angle is uniform so the average value is $\langle \cos\theta_2^\prime\rangle = 0$.

  • The probability of collision at angle $\theta$ with the cloud of speed V is proportional to the relative velocity between the cloud and the particle $c$ if we assume it relativistic (factor $1/2$ is there to have a proper normalization):

$$\frac{dn}{d\cos \theta_1} = \frac{1}{2}\frac{c - V\cos\theta_1}{c}=\frac{1 -\beta \cos\theta_1}{2}, {\;\; \rm for -1 \leq \cos\theta_1 \leq 1}$$

and so:$$\langle \cos\theta_1\rangle = \frac{\int \cos\theta_1 \cdot \frac{dn}{d\cos\theta_1} d\cos\theta_1}{\int \frac{dn}{d\cos\theta_1}d\cos\theta_1} = - \frac{\beta}{3}$$

Particles can gain or lose energy depending on the angles, but on average the gain is$$\xi = \frac{1 + \frac{1}{3}\beta^2}{1 - \beta^2} \sim \frac{4}{3}\beta^2$$

Problems with the 2nd order acceleration

  • The energy increase is second order of $\beta$ and..

  • ... the random velocities of clouds are relatively small: $\beta \sim 10^{-4}$ !!!

  • Some collisions result in energy losses! Only with the average one finds a net increase.

  • Very little chance of a particle gaining significant energy!

  • The theory does not predict the power law exponent

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