deletions | additions       diff --git a/Magnetic Constraints.tex b/Magnetic Constraints.tex  index f3c3dd0..7d019a7 100644  --- a/Magnetic Constraints.tex  +++ b/Magnetic Constraints.tex 
...
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 (see Supplementary Material).  Figure \ref{fig:Bc} 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 $\omega = \omega_{\rm max} = 2 \pi \nu_{\rm max}$, and $\nu_{\rm max}$ is the frequency of maximum oscillation power evaluated from scaling relations (\cite{Huber_2011}). At On  the lower subgiant branch, where $\nu_{\rm max} \sim 400\,\mu$Hz, field strengths of order $B_c \sim 10^7 \, {\rm G}$ 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 $r$ at the H-burning shell decreases, and  the value of $N$ increases. By the luminosity bump (near $\nu_{\rm max} \sim 400\,\mu$Hz), field strengths of under $10^4 \, {\rm G}$ are sufficient for magnetic suppression. At the clump, field strengths of only $\sim \! 10^{4} \, {\rm G}$ are sufficient. As discussed in the supplementary material, these field strengths are easily obtained in the descendants of magnetic Ap stars. Magnetic suppression on the lower sub-giant branch (higher $\nu_{\rm max}$) and in higher mass stars ($M \gtrsim 2 M_\odot$) may be less common due to the larger 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}