The anomalous precession frequency ($\omega_a)$ depends on the amplitude of vertical betatron oscillations.
The vertical component of the muon momentum contributes to $\omega_a$
through the second term in Eq.~(\ref{eq:au}).
The vertical component of the momentum for a muon with betatron amplitude $y/sqrt{\beta_y}$ is
$$\psi_y=p_y/p = \frac{y}{\beta_y}\cos\phi_y$$
If $\psi=0$, rotation of the polarization is about the vertical axis. For finite $\psi$ the change in the
polarization about the vertical direction $\Delta P_v\propto B\cos\psi\Delta s$, and there is in addition
a rotation about the radial direction, $\Delta P_r\propto B\sin\psi\Delta s$.
As long as we are far from any spin/betatron resonance, the net precession in the radial direciton averages to zero in a betatron period.
The net change in component of polarization perpendicular to the velocity is proportional to
$\int B\cos\psi ds$, where $s$ is along the trajectory of the muon. Since $s=z/\cos\psi$, the precession (and average spin tune)
is independent of the angle $\psi$. There is nevertheless an increase of the path length with pitch angle.
The path length over one betatron period is
$$\Delta P=\int_0^{\lambda}(1+\left(\frac{y}{\beta_y}\cos\phi_y\right)^2)ds \sim 2\pi\beta+\pi\beta\left(\frac{y^2}{\beta_y}\right)$$
and the fractional change in the cylcotron frequency $$\frac{\Delta \omega_c}{\omega_c} \sim  -\frac{1}{2}\left(\frac{y}{\beta_y}\right)^2$$
As $\omega_a =\omega_c \nu$, we find that the fractional change in anomalous precession frequency due to vertical betatron motion is
$$\frac{\Delta \omega_a}{\omega_a} \sim  -\frac{1}{2}\left(\frac{y}{\beta_y}\right)^2$$
(An analysis of the dependence of $\omega_a$ on vertical betatron motion, including resonance effects is given by Farley \cite{farley})
The dependence of $\omega_a$ on vertical betatron motion, along with the measured vertical distribution is shown in Fig. ~\ref{fig:cp_vs_y}

The measured distribution of vertical phase space is used to compute the size of the effect. A fiber harp measurement of the vertical width over the first \SI{30}{\micro\second} of the fill is shown in Fig.~\ref{fig:fibervertical}.

\begin{figure}[]
	\centering
	\begin{tikzpicture}
		\draw (0, 0) node[inner sep=0]{\includegraphics[width=0.95\columnwidth]{fig23.pdf}};
		\draw (0, 0) node[rotate=45,opacity=0.4]{{\fontsize{45}{54}\selectfont\textcolor{red}{\textbf{Preliminary}}}}; % rule of thumb: \fontsize{size}{1.2*size}
	\end{tikzpicture}
	%\includegraphics[width=0.95\columnwidth]{fig23.pdf}
	\caption{\textcolor{orange}{David R.} Evolution of the vertical width.}\label{fig:fibervertical}
\end{figure}
