= Solution
Here $G$ is the <gravitational constant>, $r$ is the separation of the orbiting object from the central body's centre, and $J$ is the magnitude of the object's conserved <orbital angular momentum>, not angular momentum per unit <mass>. Approximate the much heavier central body as fixed, neglect other forces, and use the Newtonian gravitational <potential energy> $U=-GMm_0/r$, with zero at infinity.
A <central force> confines the motion to a fixed <orbital plane>. In plane polar coordinates its <kinetic energy> is $T=\tfrac12m_0(\dot r^2+r^2\dot\phi^2)$. Conservation of <angular momentum> gives $J=m_0r^2\dot\phi$; eliminating $\dot\phi$ yields
$$
\boxed{E=\frac12m_0\dot r^2+V_{\rm eff}(r),\qquad V_{\rm eff}(r)=\frac{J^2}{2m_0r^2}-\frac{GMm_0}{r}.}
$$
The centrifugal term belongs to the radial <effective potential>; it is the eliminated transverse <kinetic energy>, not an additional physical interaction. If recoil of the central body mattered, the relative-coordinate reduction would use the <reduced mass>.
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