bksw_2010_eqn_1

\begin{matrix}

k T \approx m c^2 \beta_s^2 \Rightarrow T \approx \frac{m c^2 \beta_s^2}{k}

\\

a T^4 \gg n k T \Rightarrow a T^3 \gg n k

\\

a \left( \frac{m c^2 \beta_s^2}{k} \right )^3 \gg n k

\\

m c^2 \beta_s^2 \gg \left(\frac{n k^4}{a} \right )^{1/3}

\\

\beta_s \gg \left(\frac{n k^4}{a} \right )^{1/6} \left( m c^2 \right )^{-1/2}

\end{matrix}