deletions | additions       diff --git a/Mixed Modes.tex b/Mixed Modes.tex  index 3b0e53e..368485d 100644  --- a/Mixed Modes.tex  +++ b/Mixed Modes.tex 
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\section{Mixed Modes}  Acoustic waves (p-modes) excited by turbulent convection in red giant envelopes can couple to internal gravity waves . This is because, as the star increase its radius, the frequency $\omega$ of p-modes decreases and becomes comparable to the frequency of waves in the g-mode cavity. In this situation some of the energy in the p-modes can leak into the g-modes cavity, where waves with frequency $\omega < N$ can be excited ($N$ is the Brunt-Vaisala frequency). These pulsation modes, called mixed-modes, are characterized by significant displacements in both the envelope and the core of a red giant. Their particular nature has enabled important asteroseismic discoveries, in particular the ability to distinguish He-burning clump stars from red giant stars burning hydrogen in a shell, as well as the measurement of core-envelope differential rotation in RGB stars \cite{Bedding_2011}.  \begin{itemize}  \item Excitation of stochastically excited p-modes 
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\item Tunneling and evolution of tunneling integral for l=1,2  \end{itemize}  \subsection{Tunneling Integral}  Acoustic waves (p-modes) excited by turbulent convection in red giant envelopes can couple to internal gravity waves . This is because, as the star increase its radius, the frequency $\omega$ of p-modes decreases and becomes comparable to the frequency of waves in the g-mode cavity. In this situation some of the energy in the p-modes can leak into the g-modes cavity, where waves with frequency $\omega < N$ can be excited ($N$ is the Brunt-Vaisala frequency). These pulsation modes, called mixed-modes, are characterized by significant displacements in both the envelope and the core of a red giant. Their particular nature has enabled important asteroseismic discoveries, in particular the ability to distinguish He-burning clump stars from red giant stars burning hydrogen in a shell, as well as the measurement of core-envelope differential rotation in RGB stars .