= Solution
The stated conversion gives $\Theta_c\simeq0.10$ eV. Hence the ideal longitudinal bath <temperature> is
$$
\boxed{\Theta_\parallel\simeq\frac{(0.10\,\mathrm{eV})^2}{4(5.0\times10^4\,\mathrm{eV})}
=5.0\times10^{-8}\,\mathrm{eV},\qquad
T_{p,\rm eq}\sim5.8\times10^{-4}\,\mathrm K.}
$$
This estimate equates the cooled <proton> thermal-energy scale with the effective <electron> bath scale, not the two particles' random <speeds>. The <proton> root-mean-square <speed> for a one-component convention $m_p\sigma_v^2=k_BT$ would be about $2.2\,\mathrm{m\,s^{-1}}$.
The result is an ideal floor under the hinted cooling model. It is not a promise of a sub-millikelvin stored beam in a real ring, where noise, beam self-fields, intrabeam scattering and other <heat> sources enter the balance.
Back to article page
