deletions | additions
diff --git a/subsection_Mode_Visibility_Here_we__.tex b/subsection_Mode_Visibility_Here_we__.tex
index 1e1d96a..3b077ba 100644
--- a/subsection_Mode_Visibility_Here_we__.tex
+++ b/subsection_Mode_Visibility_Here_we__.tex
...
\subsection{Mode Visibility}
Here we estimate the visibility of modes suppressed via the magnetic greenhouse effect.
In To do this, we consider the
absence energy balance between driving and damping of
magnetic suppression, each a mode. 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, such that
\begin{equation}
\label{eqn:enorm}
\dot{E}_{\rm in} = \dot{E}_{\rm out} = E_{\alpha} \gamma_\alpha \, ,
...
t_{\rm cross} = \int^R_{r_2} \frac{dr}{v_s} \, .
\end{equation}
A 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}. \cite{Houdek_1999,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 convective motions 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}
\label{edamp}
E_{\alpha} \gamma_\alpha = E_{\rm ac} \bigg[\gamma_{\rm ac} + \frac{T^2}{2 t_{\rm cross}} \bigg] \, .
\end{equation}
Since the The damping
of a non-suppressed mode is
produced almost entirely in localized to the acoustic cavity,
we have so its energy loss rate can be written
\begin{equation}
{E_\alpha} \gamma_\alpha \simeq E_{\alpha,{\rm ac}} \gamma_{\rm ac} \, ,
\end{equation}
where $E_{\alpha,{\rm ac}}$ is the mode energy contained in the acoustic cavity.
Then Inserting this into equation \ref{edamp}, 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] \, .
...
\begin{equation}
\frac{V_{\rm sup}^2}{V_{\rm norm}^2} = \frac{\gamma_{\rm ac}}{\gamma_{\rm ac} + T^2/(2 t_{\rm cross})} \, .
\end{equation}
Using the fact that the large frequency separation is $\Delta \nu \simeq (2 t_{\rm cross})^{-1}$
\citep{Chaplin_2013} \cite{Chaplin_2013} and defining $\tau = \gamma_{\rm ac}^{-1}$, we have our final result:
\begin{equation}
\frac{V_{\rm sup}^2}{V_{\rm norm}^2} = \bigg[1 + \Delta \nu \tau T^2 \bigg]^{-1} \, .
\end{equation}
The damping time $\tau$ is the lifetime of wave energy located in the acoustic cavity. It is not equal to the lifetime of a non-suppressed dipole mode, because much of
its the dipole mode energy resides within the core. Instead,
it $\tau$ is approximately equal to the lifetime of a radial mode, because all of its energy is in the acoustic cavity. Thus, $\tau$ can be equated with observed/theoretical lifetimes of radial modes.
