deletions | additions       diff --git a/Mode Visibility.tex b/Mode Visibility.tex  index 0780d29..9410ae2 100644  --- a/Mode Visibility.tex  +++ b/Mode Visibility.tex 
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In the absence of magnetic suppression, each mode receives a stochastic energy input $\dot{E}_{\rm in}$ \cite{Dupret_2009}. At its time-averaged equilibrium amplitude, the energy input and damping rates of the mode are equal: $\dot{E}_{\rm in} = \dot{E}_{\rm out} = E_{\alpha} \gamma_\alpha$, where $E_{\alpha}$ is the energy contained in the mode and $\gamma_\alpha$ is its damping rate.  Modes suppressed via the magnetic greenhouse effect have an extra source of damping determined by the rate at which energy leaks through the evanescent region separating the acoustic cavity from the g-mode cavity. For suppressed modes, we assume that any mode energy which leaks into the g-mode cavity is completely lost via the magnetic greenhouse effect.   Given some mode energy contained within the acoustic cavity, $E_{\rm ac}$, the rate at which mode energy leaks into the core is  \begin{equation}  \dot{E}_{\rm leak} = E_{\rm ac} \frac{T^2}{2 t_{\rm cross}} ,  \end{equation}  where $T$ is depends on  the transmission coefficient through the evanescent region (Eq.\ref{eqn:integral}), ($T$)  and$t_{\rm cross}$  is  the wave crossing time $t_{\rm cross}$  across the acoustic cavity. The suppressed mode is also damped by the same mechanisms as a normal mode. In the case of envelope modes for stars low on the RGB, this damping is created by convective motions near the surface of the star \citep{Dupret_2009}. The equilibrium energy of the suppressed mode is  \begin{equation}  \label{eqn:esup}  \dot{E}_{\rm in} = \dot{E}_{\rm out} = E_{\rm ac} \bigg[\gamma_{\rm ac} + \frac{T^2}{2 t_{\rm cross}} \bigg],  \end{equation}  where $\gamma_{\rm ac}$ is the damping rate due to non-adiabatic effects in the acoustic cavity.  Now, we assume that the suppression mechanism is localized to the core and that the energy input $\dot{E}_{\rm in}$ is unaltered. Then we can set equations \ref{eqn:enorm} and \ref{eqn:esup} equal to each other to find  \begin{equation}  E_{\alpha} \gamma_\alpha = E_{\rm ac} \bigg[\gamma_{\rm ac} + \frac{T^2}{2 t_{\rm cross}} \bigg] \, .  \end{equation}  Since the damping mode, which  is produced almost entirely mostly occurring  in the acoustic cavity, the normal mode damping cavity at a  rateis  \begin{equation}  \gamma_\alpha \simeq \frac{E_{\alpha,{\rm ac}}}{E_\alpha}  \gamma_{\rm ac} \, ,  \end{equation}  where $E_{\alpha,{\rm ac}}$ is the mode energy contained in the acoustic cavity. Then we have  \begin{equation}  \label{eqn:ebalance}  E_{\alpha,{\rm ac}} \gamma_{\rm ac} = E_{\rm ac} \bigg[\gamma_{\rm ac} + \frac{T^2}{2 t_{\rm cross}} \bigg] \, .  \end{equation}  The energy of a mode within the envelope is proportional to its surface amplitude squared, hence, ac}. One can show that  the visibility of a mode scales as $V_{\alpha} \propto E_{\alpha,{\rm ac}}$. Then expression for  the ratio ofthe  visibility of the suppressed mode to between modes  that of are suppressed by  the magnetic greenhouse effects and  normal mode is  \begin{equation}  \frac{V_{\rm sup}^2}{V_\alpha^2} = \frac{E_{\rm ac}}{E_{\alpha,{\rm ac}}} \, .  \end{equation}  Then equation \ref{eqn:ebalance} leads to  \begin{equation}  \frac{V_{\rm sup}^2}{V_\alpha^2} = \frac{\gamma_{\rm ac}}{\gamma_{\rm ac} + T^2/(2 t_{\rm cross})} \, .  \end{equation}  Using the fact that the large frequency spacing modes  is$\Delta \nu \simeq (2 t_{\rm cross})^{-1}$ \citep{Chaplin_2013} and defining $\tau_{\rm ac} = \gamma_{\rm ac}^{-1}$, we have our final result:  \begin{equation}  \label{eqn:vis}  \frac{V_{\rm sup}^2}{V_\alpha^2} = \bigg[1 + \Delta \nu \tau_{\rm ac} T^2 \bigg]^{-1} \, .  \end{equation}  Where we used the fact that the large frequency spacing is $\Delta \nu \simeq (2 t_{\rm cross})^{-1}$ \citep{Chaplin_2013} and have defined $\tau_{\rm ac} = \gamma_{\rm ac}^{-1}$.  The value of $T^2$ can be easily calculated from a stellar model, whereas the envelope life-time $\tau_{\rm ac}$ is approximately equal to the life-time of radial modes (becuase they have all their energy in the acoustic cavity). For stars ascending the RGB below the bump, $\tau_{\rm ac} \sim 10 \, {\rm days}$ \citep{Dupret_2009}.