deletions | additions       diff --git a/Magnetic Constraints.tex b/Magnetic Constraints.tex  index 2c7e280..c191c11 100644  --- a/Magnetic Constraints.tex  +++ b/Magnetic Constraints.tex 
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Above, we showed that incoming magneto-gravity waves will become trapped in the core if the magnetic field strength exceeds $B_c$ (equation \ref{eqn:Bc}) at some point within the core. Constraints can also be placed on the internal magnetic fields of stars without suppressed dipole modes. These stars cannot have radial field strengths in excess of $B_{c,{\rm min}}$ within their H-burning shells. However, they may contain larger fields away from the H-burning shell, or they may contain fields that are primarily horizontal (e.g., strong toroidal fields).  Figure \ref{fig:Bc} shows the value of $B_{c,{\rm min}}$ as stars evolve up the RGB. We have evaluated $B_{c,{\rm min}}$ 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}). On the lower sub-giant branch, where $\nu_{\rm max} \gtrsim 250\,\mu$Hz, field strengths of order $B_{c,{\rm min}} \gtrsim 10^6 \, {\rm G}$ are required for magnetic suppression. As stars evolve up the red giant branch, the value of $B_{c,{\rm min}}$ decreases sharply as the value of $r$ at the H-burning shell decreases, and the value of $N$ increases. By the luminosity bump (near $\nu_{\rm max} \sim 40\,\mu$Hz), field strengths of only $B_{c,{\rm min}} \sim \!10^4 \, {\rm G}$ are sufficient for magnetic suppression. Stars therefore become more susceptible to magnetic suppression as they evolve up the RGB. Magnetic suppression on the lower subgiant 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.  %As low-mass stars evolve up the RGB, their cores contract. If magnetic flux is conserved, the strength of their internal magnetic fields will increase.  Stars therefore become more susceptible to magnetic suppression as they evolve up the RGB. Magnetic suppression on the lower subgiant 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.  %We find (see Supplementary material) that the field strengths quoted above are easily obtained in the descendants of magnetic Ap stars, or stars with core magnetic dynamos during the main sequence.   We expect magnetic suppression on the RGB to be easily detectable in the RGB descendants of Sun-like stars, if their cores contain radial fields in excess of $\sim 10^5 \, {\rm G}$. This corresponds to a main sequence field strength of $\sim \! 10^3 \, {\rm G}$ if magnetic flux is conserved within the core as it contracts by a factor of $\sim 10$ between the main sequence and lower RGB. Hence, an observation (or lack thereof) of suppressed dipole modes in solar-mass stars will place tight constraints on the internal field strengths of Sun-like stars.