deletions | additions       diff --git a/section_Magnetic_Trapping_subsection_Magneto__.tex b/section_Magnetic_Trapping_subsection_Magneto__.tex  index a7e573b..dc0fa1a 100644  --- a/section_Magnetic_Trapping_subsection_Magneto__.tex  +++ b/section_Magnetic_Trapping_subsection_Magneto__.tex 
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\label{eqn:k}  k^2 = \frac{\omega^2}{2 v_A^2 \mu^2} \bigg[ 1 \pm \sqrt{1 - \frac{4 \mu^2 v_A^2 N^2 k_\perp^2}{\omega^4}} \bigg] \, ,  \end{equation}  where $k$ is the wavenumber and $\mu = \cos \theta$, where $\theta$ is the angle between ${\bf B}$ and ${\bf k}$. For dipole waves, the horizontal wave number is $k_\perp = \sqrt{\ell(\ell+1)}/r = \sqrt{2}/r$.  The positive and negative roots of equation \ref{eqn:k} correspond to the ``slow" and ``fast" magneto-gravity waves, respectively, which correspond to Alfven waves and gravity waves in the limit of vanishing magnetic field. Pure Alfven waves can also exist. The key feature of equation \ref{eqn:k} is that dipolar magneto-gravity waves become evanescent in regions where the radial magnetic field strength exceeds the critical value  \begin{equation}  \label{eqn:Bc}  B_c= \sqrt{\frac{\pi \rho}{2}} \, \frac{\omega^2 r}{N} \, .  \end{equation}  Low frequency gravity waves cannot exist in these regions because magnetic tension overwhelms the buoyancy force, i.e., the stiff field lines cannot be bent by the placid gravity wave motion. In other words, dipolar waves become evanescent where their frequency $\omega$ is less than the magneto-gravity frequency $\omega_{\rm MG}$ defined as  \begin{equation}  \label{eqn:maggrav}  \omega_{MG} = \sqrt{2 v_{A,r} N k_\perp} \, .  \end{equation}  Figure \ref{fig:Prop} shows a propagation diagram outlining the behavior of solar-like oscillations within a red giant with a magnetized core.  In red giant cores, $B_c$ is typically minimized at the peak in $N$ corresponding to the sharp density gradient within the H-burning shell. Therefore, gravity waves are most susceptible to magnetic alteration in the H-burning shell. The observation of a star with depressed dipole modes thus provides a {\it lower limit} to the internal magnetic field strength, given by equation \ref{eqn:Bc}, evaluated in the H-burning shell. We refer to this field strength as $B_{c,{\rm min}}$, which corresponds to a minimum {\it radial} field strength for magnetic suppression. Magnetic suppression via horizontal fields can also occur, but in general requires much larger field strengths (see Supplementary Material).