\def\bmhr{{\hat{\bm\rho}}}
\def\bmhp{{\hat{\bm\phi}}}
\def\bmbetahat{{\hat{\bm\beta}}}
\def\bfs{{\bf s}}
\def\bfhz{{\hat{\bf z}}}
As you say, the velocity 
$\bmbetahat = \cos\psi(t)\bmhp+\sin\psi(t)\bfz$.

\noindent
But $\psi(t) = \psi_0\cos\omega_p t$ and $\psi_0\sim 2$ mrad.
\begin{figure}[h]
\centering
\includegraphics{betaz.pdf}
\includegraphics{betap.pdf}
\end{figure}
 
