Effective Lorentz Factor in Cold Plasma, Razin Effect

I was studying about the synchrotron radiation in plasma medium and got stuck at a point. Usually the Lorentz factor is defined in the following way $$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ where $$\gamma$$ is the Lorentz factor, $$v$$ is the particle's velocity and $$c$$ is the speed of light in the vacuum. But when the particle moves in a medium (for example, plasma medium) with some refractive index it's $$\gamma$$ gets changed. For cold plasma the refractive index is written as, $$n(\omega)^2\approx 1-\frac{\omega_{pe}^2}{\omega^2}$$ where, $$n(\omega)$$ is the refractive index that depends on the frequency $$\omega$$ and $$\omega_p$$ is the plasma frequency. Now in a medium with some refractive index the speed of light gets modified in the following way $$c\rightarrow\frac{c}{n}$$ following this the Lorentz factor also changes as $$\gamma_{*}=\frac{1}{\sqrt{1-\frac{n^2v^2}{c^2}}}\\ \implies \gamma_{*}=\frac{1}{\sqrt{1-\frac{v^2}{c^2}(1-\frac{\omega_{pe}^2}{\omega^2})}}=\frac{1}{\sqrt{1-\frac{v^2}{c^2} + \frac{v^2\omega_{pe}^2}{c^2\omega^2}}}=\frac{1}{\sqrt{\frac{1}{\gamma^2}+ \frac{v^2\omega_{pe}^2}{c^2\omega^2}}}$$ But in this paper just below equation (2) the Lorentz factor in cold plasma is defined as $$\gamma^{-2}_{*}=(\gamma^{-2}+ {\omega_{pe}^2}/{\omega^2})$$.

I do not understand how the $$v^2/c^2$$ term is getting vanished from the denominator in the definition of $$\gamma_{*}$$ ?

Let me start by defining $$\beta = \tfrac{ v }{ c }$$, then we can show that: $$\beta^{2} = \frac{ \gamma^{2} - 1 }{ \gamma^{2} } \tag{0}$$ where $$\gamma$$ is the Lorentz factor.
We can then show that: \begin{align} \gamma^{-2} + \beta^{2} \tilde{\omega}^{-2} & = \gamma^{-2} + \left( \frac{ \gamma^{2} - 1 }{ \gamma^{2} } \right) \tilde{\omega}^{-2} \tag{1a} \\ & = \gamma^{-2} + \tilde{\omega}^{-2} - \gamma^{-2} \tilde{\omega}^{-2} \tag{1b} \end{align} where $$\tilde{\omega} = \frac{ \omega }{ \omega_{pe} }$$ and $$\omega_{pe}$$ is the plasma frequency.
In the limit of large $$\gamma$$, the last term can be dropped leaving one with the expression in that paper.
• It depends on the situation but in general, that last term would be considered a second order term and most approximations try to stop at first order. It may also be the case that this is for a pulsar or magnetar where the emissions span from radio to x-ray frequencies, in which case the $\tilde{\omega}^{-2}$ term can be small as well. The paper cites Landau and Lifshitz as its source, which is not exactly the most transparent of resources either... Sep 27 at 14:13