deletions | additions       diff --git a/subsection_Ray_Tracing_label_ray__.tex b/subsection_Ray_Tracing_label_ray__.tex  index 7a00616..53a6fa2 100644  --- a/subsection_Ray_Tracing_label_ray__.tex  +++ b/subsection_Ray_Tracing_label_ray__.tex 
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\label{eqn:hamiltonian}  H = \omega = \sqrt{ \frac{{\bf k_\perp}^2 N^2}{{\bf k}^2} + ({\bf k} \cdot {\bf v_A})^2 } \, .  \end{equation}  Here, ${\bf k}$ is the wave vector, ${\bf k_\perp}$ is its component perpendicular to the gravitational field, and ${\bf v_A} = {\bf B}/\sqrt{4 \pi \rho}$. In reality, the Hamiltonian contains additional terms that allow for the existence of pure Alfven Alfv\'en  waves, although we ignore this subtlety here. The equations of motion corresponding to the Hamiltonian of equation \ref{eqn:hamiltonian} are  \begin{equation} 
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Equation \ref{eqn:dxdt} describes the group velocity of the wave, while equation \ref{eqn:dkdt} describes the evolution of its wave vector, which is related to the momentum of the wave. Note that in the absence of a magnetic field in a spherical star, only the radial component of the wave vector changes, and the horizontal component is conserved. This is not surprising because the Hamiltonian is spherically symmetric and thus angular momentum (and hence angular wave vector) is conserved.   However, in the presence of a magnetic field, the last two terms of equation \ref{eqn:dkdt} break the spherical symmetry. Except in the unphysical case of a purely radial field or a constant field, these terms are non-zero, and therefore the angular component of the wave vector must change. At the radius $r_{\rm MG}$ where $v_A \sim \omega^2/(N k_\perp)$, each term in equation \ref{eqn:dkdt} is the same order of magnitude, assuming $|\nabla B|/B \sim 1/r$. Therefore, the rate of change in horizontal wavenumber is comparable to the rate of change in radial wavenumber at field strengths near $B_c$. Upon wave reflection or conversion into Alfven Alfv\'en  waves, the radial wavenumber will generally change by order unity, i.e., the change in radial wavenumber is $|\Delta k_r| \sim |k_r|$. We therefore expect a correspondingly large change in $k_\perp$, such that $|\Delta k_\perp| \sim |k_r|$. Hence, dipole waves will generally obtain high multipole moments when they propagate through strongly magnetized regions of the star.
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