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The second order Fermi mechanism describes the interaction of charged particles with magnetic clouds. This model leads to a collision-less acceleration of cosmic rays up to ultra high energies.

A rough computation (classical/extreme cases) of this phenomenon is described in the book of Claus Grupen, "Astroparticle physics" on page 68:

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My question concerns equation 5.13 & 5.14. In my opinion, the expressions for $\Delta E_1$ & $\Delta E_2$ are wrong.

Indeed, in the frame of the cloud the particle's velocity is equal to $v_{in}=(v+u)$. After the collision it becomes $v_{out}=-(v+u)$

Going back to the lab frame: \begin{equation} v^*_{out}=-(v+u)-u=-v-2u \end{equation}

$\Delta E_1$ then yields: \begin{equation} \Delta E_1=\frac{1}{2}m(v+2u)^2-\frac{1}{2}mv^2 \end{equation}

Similarly, for $\Delta E_2$: \begin{equation} \Delta E_2=\frac{1}{2}m(v-2u)^2-\frac{1}{2}mv^2 \end{equation}

Is there really a mistake or am I doing it wrong?

innisfree
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AxelAE
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    I think you may be correct but I am not sure. Your $\Delta E_{1}$, when expanded, has the form $\approx 2 \ m \left( u^{2} + u \ v \right)$, which looks right (but my memory is not to be blindly trusted). If you normalize by the initial particle energy, then $\Delta E_{1}/E_{o} \approx 4 \left( u^{2}/v^{2} + u / v \right)$, which also looks correct to me. – honeste_vivere Jan 23 '16 at 21:55
  • Thanks for the answer. I found a PhD thesis on the HISPARC experiment and the guy has the same expressions as me for both delta's (http://www.nikhef.nl/pub/services/biblio/theses_pdf/thesis_D_Fokkema.pdf). If we are correct, it's quite surprising that such mistakes appear in published books... – AxelAE Jan 25 '16 at 14:03
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    Unfortunately, it's not as surprising as you might think. It is generally very difficult to catch one's own mistakes and the people editing the books are often so busy they haven't the time to check every single expression (especially in large books). I used to find this very offensive until I started writing review papers and realized how daunting it can be to check every single line in dozens of pages of writing and equations. Books are even worse since they are often much longer and can contain many more topics than a focused review paper... – honeste_vivere Jan 25 '16 at 14:07

1 Answers1

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AFAICT, the extra factor of 2 should be there. A particle of energy $E$ in the cloud's reference frame is (with $c=1$ throughout), $$ E'=\gamma v\left(E+up\right) $$ where $p$ is the particle momentum. Transforming back to the observer frame, this is $$ E''=\gamma v\left(E'+up'\right) $$ where $p'=\gamma v\left(p+uE\right)$. The energy gain is then the difference, $$ E''-E=\Delta E\propto\left(2uv-2u^2\right) $$ which carries the factor of 2 you quote.

Two presentations that I am aware of both contain this factor as well:

I don't (any more) have Kirk's Plasma Astrophysics Saas-Fee lecture on particle acceleration (Springer link to book), but I suspect it is also correctly given there.

Kyle Kanos
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