deletions | additions       diff --git a/Magnetic Constraints.tex b/Magnetic Constraints.tex  index 2c60546..7571b57 100644  --- a/Magnetic Constraints.tex  +++ b/Magnetic Constraints.tex 
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\subsection{Constraints \section{Constraints  on the Interior Magnetic Field} Above, we showed that incoming magneto-gravity waves will become trapped in the core if the magneto-gravity frequency $\omega_{\rm MG}$ exceeds the wave frequency $\omega$ at some point within the core. Rearranging equation \ref{eqn:vamin}, magnetic suppression requires the field to exceed a critical value:  \begin{equation} 
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\end{equation}  somewhere in the g mode cavity. For incoming dipole waves with $k_\perp = \sqrt{l(l+1)}/r = \sqrt{2}/r$, magnetic suppression requires field strengths of at least $B_c(l=1) = \sqrt{\pi \rho r \omega^2/(2 N)}$.  In red giant cores, the right hand side of equation \ref{eqn:Bc} 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. This lower limit corresponds to a minimum {\it radial} field strength. Magnetic suppression via horizontal fields can also occur, but in general requires much larger field strengths. strengths (see Supplementary Material).  Figure \ref{fig:DipoleBEvol} shows the value of $B_c$, evaluated at the H-burning shell, for dipole modes as stars evolve up the RGB. We have evaluated $B_c$ for angular frequencies $\omega = \omega_{\rm max} = 2 \pi \nu_{\rm max}$. max}$, and $\nu_{\rm max}$ is the frequency of maximum oscillation power evaluated from scaling relations (\cite{Huber_2011}).  At the lower subgiant branch, where the stellar radius is $R\sim 3 R_\odot$, field strengths near $B_c \sim 10^7 \, {\rm G}$ are required for magnetic suppression. As the star evolves up the red giant branch, the value of $B_c$ decreases sharply as the value of $r_H$ decreases and $N_H$ increases. By the bump, field strengths of under $10^4 \, {\rm G}$ are sufficient for magnetic suppression. At the clump, field strengths of $\sim \! 3 \times 10^{4} \, {\rm G}$ are sufficient. As discussed above, these field strengths are easily attainable for the descendants of magnetic Ap stars. Magnetic suppression on the early sub-giant branch is likely to be less common due to the higher field strengths required. Equation \ref{eqn:BHburn} can be rearranged to provide an upper limit to the frequencies of oscillation modes which can be magnetically suppressed:  \begin{equation}