deletions | additions       diff --git a/Magnetic Constraints.tex b/Magnetic Constraints.tex  index 9d15bcd..0f0044f 100644  --- a/Magnetic Constraints.tex  +++ b/Magnetic Constraints.tex 
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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).  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 $B_c$ 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 $B_c$, 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}). 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 $\nu_{\rm max} \sim 40\,\mu$Hz), field strengths of only $\sim \!10^4 \, {\rm G}$ are sufficient for magnetic suppression.   As low mass low-mass  stars evolve up the RGB, their cores are radiative and are contracting. contract.  If magnetic flux is conserved, the strength of their internal magnetic fields will increase, while the value of $B_c$ decreases. Stars therefore become more susceptible to magnetic suppression as they evolve up the RGB. We find (see supplementary material) that the field strengths quoted above are easily obtained in the descendants of magnetic Ap stars. However, 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 expect magnetic suppression on the RGB to be easily detectable in the 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.