= Solution
Write $s=(r^2+z^2)^{1/2}$. The <Paczyński-Wiita potential> is a spherical <Newtonian gravitational potential> with $\Phi'(s)=GM/(s-a)^2$ and $\Phi''(s)=-2GM/(s-a)^3$. At $z=0$, $s=r$, so
$$
\Phi_{,r}=\frac{GM}{(r-a)^2},\qquad
\Phi_{,rr}=-\frac{2GM}{(r-a)^3},\qquad
\Phi_{,zz}=\frac{GM}{r(r-a)^2}.
$$
Substituting into the orbital formulas gives
$$
\boxed{\Omega^2=\Omega_z^2=\frac{GM}{r(r-a)^2},\qquad
\kappa^2=\frac{GM(r-3a)}{r(r-a)^3}}.
$$
Choosing positive orbital and vertical frequencies, $\Omega=\Omega_z=\sqrt{GM}/[\sqrt r(r-a)]$. For $r>3a$, $\kappa=\Omega\sqrt{(r-3a)/(r-a)}$. For $a<r<3a$, $\kappa^2<0$ and radial disturbances grow at rate $\sqrt{-\kappa^2}$, even though the vertical motion remains stable. Thus \b[circular orbits are radially unstable for $a<r<3a$], and $r=3a$ is the marginal orbit. This recovers the <innermost stable circular orbit> of the <Paczyński-Wiita circular orbit> model.
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