Eq. \ref{eq:1719451} is an initial equation. $$v = \sqrt{ \frac{K + (4/3) G}{\rho} } \label{eq:1719451}$$ Eq. \ref{eq:2178289} is an initial equation. $$K = f \frac{E}{a^3} \label{eq:2178289}$$ Based on the assumption $$K >> G, drop non-dominant term in Eq. \ref{1719451}; yeilds Eq. \ref{9155336} $$v = \sqrt{ \frac{K}{\rho} } \label{eq:9155336}$$ Substitute LHS of Eq. \ref{eq:2178289} into Eq. \ref{eq:9155336}; yields Eq. \ref{eq:5077893}. $$v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} } \label{eq:5077893}$$ Eq. \ref{eq:2438445} is an initial equation. $$\rho = \frac{m}{a^3} \label{eq:2438445}$$ Multiply both sides of Eq. \ref{eq:2438445} by \(a^3$$; yields Eq. \ref{eq:7834577}. $$a^3 \rho = m \label{eq:7834577}$$ Substitute RHS of Eq. \ref{eq:7834577} into Eq. \ref{eq:5077893}; yields Eq. \ref{eq:5020923}. $$v = \sqrt{f} \sqrt{\frac{E}{m}} \label{eq:5020923}$$ Based on the assumption $$\sqrt{f} \approx 2, drop non-dominant term in Eq. \ref{5020923}; yeilds Eq. \ref{4534919} $$v = \sqrt{\frac{E}{m}} \label{eq:4534919}$$ Eq. \ref{eq:5961293} is an initial equation. $$E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2} \label{eq:5961293}$$ Eq. \ref{eq:6901924} is an assumption. $$E_{\rm Rydberg} = E \label{eq:6901924}$$ Eq. \ref{eq:9431422} is an initial equation. $$\alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c} \label{eq:9431422}$$ Multiply both sides of Eq. \ref{eq:9431422} by \(c$$; yields Eq. \ref{eq:6181437}. $$\alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar} \label{eq:6181437}$$ Substitute LHS of Eq. \ref{eq:6901924} into Eq. \ref{eq:5961293}; yields Eq. \ref{eq:3642765}. $$E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2} \label{eq:3642765}$$ Substitute LHS of Eq. \ref{eq:3642765} into Eq. \ref{eq:4534919}; yields Eq. \ref{eq:2063484}. $$v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} } \label{eq:2063484}$$ Simplify Eq. \ref{eq:2063484}; yields Eq. \ref{eq:4586348}. $$v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}} \label{eq:4586348}$$ Substitute LHS of Eq. \ref{eq:6181437} into Eq. \ref{eq:4586348}; yields Eq. \ref{eq:5883117}. $$v = \alpha c \sqrt{ \frac{m_e}{2 m} } \label{eq:5883117}$$ Eq. \ref{eq:6979804} is an initial equation. $$m = A m_p \label{eq:6979804}$$ Substitute LHS of Eq. \ref{eq:6979804} into Eq. \ref{eq:5883117}; yields Eq. \ref{eq:8323044}. $$v = \alpha c \sqrt{ \frac{m_e}{A m_p} } \label{eq:8323044}$$ The maximum of Eq. \ref{eq:8323044} with respect to $$A$$ is Eq. \ref{eq:9568206} $$v_u = \alpha c \sqrt{ \frac{m_e}{m_p} } \label{eq:9568206}$$ Eq. \ref{eq:9568206} is one of the final equations.