deletions | additions       diff --git a/Magnetic Constraints.tex b/Magnetic Constraints.tex  index 639d724..2c60546 100644  --- a/Magnetic Constraints.tex  +++ b/Magnetic Constraints.tex 
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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 suppressed depressed  dipole modes therefore thus  provides a {\it  lower limit limit}  to the internal magnetic  field strength at the location of the H-burning shell,   \begin{equation}  \label{eqn:BHburn}  B_c(r_H) \geq \sqrt{\frac{\pi}{2}} \frac{\sqrt{\rho_H} r_H \omega^2}{N_H} \, ,  \end{equation}  with the $H$ subscript indicating the right hand side of strength, given by  equation \ref{eqn:BHburn} should be \ref{eqn:Bc},  evaluated near in  the H-burning shell where $B_c$ is minimized. 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.  Figure \ref{fig:DipoleBEvol} shows the value of $B_c(r_H)$ $B_c$, evaluated at the H-burning shell,  for dipole modes as a function of stellar radius for an evolving star with $M=1.5M_\odot$. stars evolve up the RGB.  We have evaluated $B_c$ for angular frequencies $\omega = \omega_{\rm max} = 2 \pi \nu_{\rm max}$, where $\nu_{\rm max}$ is the frequency of maximum power, evaluated via the scaling relation   \begin{equation}  \label{eqn:numax}  \nu_{\rm max} = \nu_{{\rm max}, \odot} \bigg(\frac{M}{M_\odot}\bigg) \bigg(\frac{R}{R_\odot}\bigg)^{-2} \bigg(\frac{T}{T_\odot}\bigg)^{-1/2}   \end{equation}  from \cite{Huber_2011}, with $\nu_{{\rm max}, \odot} = 3090 \mu{\rm Hz}$. max}$.  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}