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\section{Theory}  Appleton's equation is given by \begin{equation}  \eta^2=(\frac{kc}{\omega})^2=1-\frac{\omega_{pe}^2}{\omega^2} \eta^2=\left(\frac{kc}{\omega}\right)^2=1-\frac{\omega_{pe}^2}{\omega^2}  \end{equation}