\begin{table}%t1
%\centering
\par
\caption{Two mass density/shape models considered in this study. For model%$^
\ref{note:virtual}%$ 
1, an exact expression for both $\breve{\kappa}_Z$ and $g_Z$ can be derived analytically from Eqs.~(\ref{eq:mlesskz}) and (\ref{eq:kmu}) provided $H=\rm const.$ (see the Appendix~\ref{ap:Vertical field density test-pair}, Eq.~(\ref{eq:gzmodel1}); the kernel and field are finite for $k^\pm \ne 1$). Model 2 is a vertically parabolic profile. In this case, there is no solution in a closed form, whatever the shape of the system. Note that $\rho_1 \propto 1/a^2$ for $R \ll a$ whereas  $\rho_1 \sim \rm const.$ for $R \gg a$.}
\label{tab:mdmodels}
\begin{tabular}{cll}
\hline\hline
Model & Properties \\  \hline 
1                 & mass density 
\ref{note:virtual}%$ 
& $\rho_1=\rho_0 \sqrt{\frac{R}{a}}\frac{k}{E(k)}$ \\
  (virtual)       & shape & $H=\rm const.\;$ (flat system) \\
                  & secondary kernels & $\breve{\kappa}^{\rm ref.}_R$ and $\breve{\kappa}^{\rm ref.}_\Psi$ unknown \\
                  &                   & $\breve{\kappa}^{\rm ref.}_Z = G\rho_0\ln \left( \frac{1-{k^+}^2}{1-{k^-}^2} \right)$ \\
                  & radial field    & unknown \\
                  & vertical field    &  $g_Z$ known; see Eq.~(\ref{eq:gzmodel1})\\
                  & potential & unknown \\ \hline 
2                 &  mass density &  $\rho_2=\rho_0(a) \left[ 1-\left(\frac{z}{H} \right)^2 \right]$ \\
 (realistic)      & shape & any \\
                  & secondary kernels & unknown \\
                  & field & unknown\\ 
                  & potential & unknown\\ \hline 
\end{tabular}
\end{table}