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Fix resonant wavenumber definition #9

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@nj-vs-vh nj-vs-vh commented Jul 26, 2024

Aside from what seems like a simple typo ($\Omega v_\parallel$ instead of $\Omega / v_\parallel$), I think there's an issue with expression of $k_{res}$ in terms of $r_L$. In the text it was simply $r_L^{-1}$, which can't be the case since there's also a pitch angle dependence. But also in the lectures we wrote $\frac{1}{r_L \mu}$ (as per my and Luis's notes), which also seems to miss the $\sin \theta$ from the definition of Larmor radius.

So, here I tried to write down the correct definition, with $\tan$ originating from $v_\perp / v_\parallel$

Fixed typo in k_res definition; expanded the expression in terms of Larmor radius
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nj-vs-vh commented Jul 26, 2024

Also, the tangent makes sense in case of $\theta \to 0$ -- it produces $k_{res} \to 0$, implying $\lambda_{res} \to \infty$, which corresponds to straight line particle trajectory.

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