deletions | additions       diff --git a/Magnetic Trapping.tex b/Magnetic Trapping.tex  index 3cfb4df..97fd4dd 100644  --- a/Magnetic Trapping.tex  +++ b/Magnetic Trapping.tex 
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For strong enough magnetic fields, there exists a critical magneto-gravity radius, $r_{\rm MG}$, defined as the radius where $\omega=\omega_{\rm MG}$. At this location, magneto-gravity waves become evanescent and can no longer propagate inward. An incoming wave must either reflect or propagate inward as a pure Alfven wave.  The reflection or transition at $r_{\rm MG}$ is analogous to reflection of light between materials of differing refractive indices. In this case it It  is well known that light at small incidence angles, with ${\bf {\hat k}} \cdot {\bf {\hat n}} = \cos \theta $ (where ${\bf {\hat k}}$ is the direction of the wave vector and ${\bf {\hat n}}$ is the surface normal) is transmitted. Light at large incidence angles is reflected. In our case, the location of $r_{\rm MG}$ is similar to such an interface. Just above $r_{\rm MG}$, incoming magneto-gravity waves have $k = \omega/(\sqrt{2} v_A \mu$, whereas just below the  interface because an Alfven wave has $k = \omega/(v_A \mu$. Across the location of $r_{\rm MG}$ a transmitted wave has a sudden jump in wave number and group velocity by a factor of $\sqrt{2}$. Thus,  the wavenumber interface has an effective index of refraction  of $n = \sqrt{2}