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Thorn {\tt IDBrillData} provides three choices for the $q$ function:
an exponential form, ({\tt IDBrillData::q\_function = "exp"})
\begin{equation}
-q = a \; \frac{\rho^{2+b}}{r^2} \left( \frac{z}{\sigma_z} \right)^2
-e^{-(\rho - \rho_0^2)} \left[ 1 + d \frac{\rho^m}{1 + e \rho^m}
+q = a \; \frac{\rho^{2+b}}{r^2} e^{-\left( \frac{z}{\sigma_z} \right)^2}
+e^{-(\rho - \rho_0)^2} \left[ 1 + d \frac{\rho^m}{1 + e \rho^m}
\cos^2 \left( n \phi + \phi_0 \right) \right]
\end{equation}
a generalized form of the $q$ function first written down by Eppley