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\section{Theory}  Appleton's equation is given by \begin{equation}  \eta^2=\left(\frac{kc}{\omega}\right)^2=1-\frac{\omega_{pe}^2}{\omega^2\left((1+\frac{i\nu}{\omega})-\frac{(\omega_{ce}^2\sin^2\theta})^2{2\omega^2\left(1-\frac{\omega_{pe}^2}{\omega^2}\right)}\pm\sqrt{\frac{\omega_{ce}^2\sin^2\theta}{4(1-\frac{\omega_{pe}^2}{\omega^2}}+\frac{\omega_{ce}^2cos^2\theta}{\omega^2}}\right)} \eta^2=\left(\frac{kc}{\omega}\right)^2=1-\frac{\omega_{pe}^2}{\omega^2\left((1+\frac{i\nu}{\omega}-\frac{(\omega_{ce}^2\sin^2\theta})^2{2\omega^2\left(1-\frac{\omega_{pe}^2}{\omega^2}\right)}\pm\sqrt{\frac{\omega_{ce}^2\sin^2\theta}{4(1-\frac{\omega_{pe}^2}{\omega^2}}+\frac{\omega_{ce}^2cos^2\theta}{\omega^2}}\right)}  \end{equation}