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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 $\gamma_{*}$ ?

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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.

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  • $\begingroup$ Thank you for your answer. I have a small confusion in the large gamma limit why does only the last term drop. The first term is also gets small in that limit why can't we drop that as well ? $\endgroup$
    – Sayan
    Sep 27 at 13:55
  • $\begingroup$ 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... $\endgroup$ Sep 27 at 14:13

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