Table 1: Two mass density/shape models considered in this study. For model 9 1, an exact expression for both $\breve{\kappa}_Z$ and gZ can be derived analytically from Eqs. (8) and (9) provided $H=\rm const.$ (see the Appendix B, Eq. (B.2); 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 $R \gg a$.
Model Properties  
1 mass density 9 $\rho_1=\rho_0 \sqrt{\frac{R}{a}}\frac{k}{E(k)}$
(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 gZ known; see Eq. (B.2)
  potential unknown
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

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